LMFDB - Embedded newform 25.5.c.a.18.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,5,Mod(7,25)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("25.7");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	25.c (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.58424907710$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			18.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	25.18
Dual form			25.5.c.a.7.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{2} +(6.00000 + 6.00000i) q^{3} +14.0000i q^{4} +12.0000 q^{6} +(26.0000 - 26.0000i) q^{7} +(30.0000 + 30.0000i) q^{8} -9.00000i q^{9} +O(q^{10})$	\(q+(1.00000 - 1.00000i) q^{2} +(6.00000 + 6.00000i) q^{3} +14.0000i q^{4} +12.0000 q^{6} +(26.0000 - 26.0000i) q^{7} +(30.0000 + 30.0000i) q^{8} -9.00000i q^{9} -8.00000 q^{11} +(-84.0000 + 84.0000i) q^{12} +(-139.000 - 139.000i) q^{13} -52.0000i q^{14} -164.000 q^{16} +(1.00000 - 1.00000i) q^{17} +(-9.00000 - 9.00000i) q^{18} -180.000i q^{19} +312.000 q^{21} +(-8.00000 + 8.00000i) q^{22} +(166.000 + 166.000i) q^{23} +360.000i q^{24} -278.000 q^{26} +(540.000 - 540.000i) q^{27} +(364.000 + 364.000i) q^{28} +480.000i q^{29} +572.000 q^{31} +(-644.000 + 644.000i) q^{32} +(-48.0000 - 48.0000i) q^{33} -2.00000i q^{34} +126.000 q^{36} +(251.000 - 251.000i) q^{37} +(-180.000 - 180.000i) q^{38} -1668.00i q^{39} -1688.00 q^{41} +(312.000 - 312.000i) q^{42} +(-1474.00 - 1474.00i) q^{43} -112.000i q^{44} +332.000 q^{46} +(-2474.00 + 2474.00i) q^{47} +(-984.000 - 984.000i) q^{48} +1049.00i q^{49} +12.0000 q^{51} +(1946.00 - 1946.00i) q^{52} +(3331.00 + 3331.00i) q^{53} -1080.00i q^{54} +1560.00 q^{56} +(1080.00 - 1080.00i) q^{57} +(480.000 + 480.000i) q^{58} +3660.00i q^{59} +1592.00 q^{61} +(572.000 - 572.000i) q^{62} +(-234.000 - 234.000i) q^{63} -1336.00i q^{64} -96.0000 q^{66} +(-874.000 + 874.000i) q^{67} +(14.0000 + 14.0000i) q^{68} +1992.00i q^{69} -6068.00 q^{71} +(270.000 - 270.000i) q^{72} +(791.000 + 791.000i) q^{73} -502.000i q^{74} +2520.00 q^{76} +(-208.000 + 208.000i) q^{77} +(-1668.00 - 1668.00i) q^{78} -9120.00i q^{79} +5751.00 q^{81} +(-1688.00 + 1688.00i) q^{82} +(-5654.00 - 5654.00i) q^{83} +4368.00i q^{84} -2948.00 q^{86} +(-2880.00 + 2880.00i) q^{87} +(-240.000 - 240.000i) q^{88} -2160.00i q^{89} -7228.00 q^{91} +(-2324.00 + 2324.00i) q^{92} +(3432.00 + 3432.00i) q^{93} +4948.00i q^{94} -7728.00 q^{96} +(6551.00 - 6551.00i) q^{97} +(1049.00 + 1049.00i) q^{98} +72.0000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 12 q^{3} + 24 q^{6} + 52 q^{7} + 60 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 12 * q^3 + 24 * q^6 + 52 * q^7 + 60 * q^8	$ 2 q + 2 q^{2} + 12 q^{3} + 24 q^{6} + 52 q^{7} + 60 q^{8} - 16 q^{11} - 168 q^{12} - 278 q^{13} - 328 q^{16} + 2 q^{17} - 18 q^{18} + 624 q^{21} - 16 q^{22} + 332 q^{23} - 556 q^{26} + 1080 q^{27} + 728 q^{28} + 1144 q^{31} - 1288 q^{32} - 96 q^{33} + 252 q^{36} + 502 q^{37} - 360 q^{38} - 3376 q^{41} + 624 q^{42} - 2948 q^{43} + 664 q^{46} - 4948 q^{47} - 1968 q^{48} + 24 q^{51} + 3892 q^{52} + 6662 q^{53} + 3120 q^{56} + 2160 q^{57} + 960 q^{58} + 3184 q^{61} + 1144 q^{62} - 468 q^{63} - 192 q^{66} - 1748 q^{67} + 28 q^{68} - 12136 q^{71} + 540 q^{72} + 1582 q^{73} + 5040 q^{76} - 416 q^{77} - 3336 q^{78} + 11502 q^{81} - 3376 q^{82} - 11308 q^{83} - 5896 q^{86} - 5760 q^{87} - 480 q^{88} - 14456 q^{91} - 4648 q^{92} + 6864 q^{93} - 15456 q^{96} + 13102 q^{97} + 2098 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 12 * q^3 + 24 * q^6 + 52 * q^7 + 60 * q^8 - 16 * q^11 - 168 * q^12 - 278 * q^13 - 328 * q^16 + 2 * q^17 - 18 * q^18 + 624 * q^21 - 16 * q^22 + 332 * q^23 - 556 * q^26 + 1080 * q^27 + 728 * q^28 + 1144 * q^31 - 1288 * q^32 - 96 * q^33 + 252 * q^36 + 502 * q^37 - 360 * q^38 - 3376 * q^41 + 624 * q^42 - 2948 * q^43 + 664 * q^46 - 4948 * q^47 - 1968 * q^48 + 24 * q^51 + 3892 * q^52 + 6662 * q^53 + 3120 * q^56 + 2160 * q^57 + 960 * q^58 + 3184 * q^61 + 1144 * q^62 - 468 * q^63 - 192 * q^66 - 1748 * q^67 + 28 * q^68 - 12136 * q^71 + 540 * q^72 + 1582 * q^73 + 5040 * q^76 - 416 * q^77 - 3336 * q^78 + 11502 * q^81 - 3376 * q^82 - 11308 * q^83 - 5896 * q^86 - 5760 * q^87 - 480 * q^88 - 14456 * q^91 - 4648 * q^92 + 6864 * q^93 - 15456 * q^96 + 13102 * q^97 + 2098 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.00000	−	1.00000i	0.250000	−	0.250000i	−0.570971	−	0.820971i	$-0.693433\pi$
$2$	1.00000	−	1.00000i	0.250000	−	0.250000i	0.820971	+	0.570971i	$0.193433\pi$
$3$	6.00000	+	6.00000i	0.666667	+	0.666667i	0.956943	−	0.290276i	$-0.0937472\pi$
$3$	6.00000	+	6.00000i	0.666667	+	0.666667i	−0.290276	+	0.956943i	$0.593747\pi$
$4$			14.0000i			0.875000i
$5$		0			0
$5$		0			0
$6$	12.0000			0.333333
$7$	26.0000	−	26.0000i	0.530612	−	0.530612i	−0.390142	−	0.920755i	$-0.627574\pi$
$7$	26.0000	−	26.0000i	0.530612	−	0.530612i	0.920755	+	0.390142i	$0.127574\pi$
$8$	30.0000	+	30.0000i	0.468750	+	0.468750i
$9$		−	9.00000i		−	0.111111i
$10$		0			0
$11$	−8.00000			−0.0661157			−0.0330579	−	0.999453i	$-0.510525\pi$
$11$	−8.00000			−0.0661157			−0.0330579	+	0.999453i	$0.510525\pi$
$12$	−84.0000	+	84.0000i	−0.583333	+	0.583333i
$13$	−139.000	−	139.000i	−0.822485	−	0.822485i	0.163979	−	0.986464i	$-0.447567\pi$
$13$	−139.000	−	139.000i	−0.822485	−	0.822485i	−0.986464	+	0.163979i	$0.947567\pi$
$14$		−	52.0000i		−	0.265306i
$15$		0			0
$16$	−164.000			−0.640625
$17$	1.00000	−	1.00000i	0.00346021	−	0.00346021i	−0.705375	−	0.708835i	$-0.749221\pi$
$17$	1.00000	−	1.00000i	0.00346021	−	0.00346021i	0.708835	+	0.705375i	$0.249221\pi$
$18$	−9.00000	−	9.00000i	−0.0277778	−	0.0277778i
$19$		−	180.000i		−	0.498615i	−0.968424	−	0.249307i	$-0.919797\pi$
$19$		−	180.000i		−	0.498615i	0.968424	−	0.249307i	$-0.0802030\pi$
$20$		0			0
$21$	312.000			0.707483
$22$	−8.00000	+	8.00000i	−0.0165289	+	0.0165289i
$23$	166.000	+	166.000i	0.313800	+	0.313800i	0.846380	−	0.532580i	$-0.178777\pi$
$23$	166.000	+	166.000i	0.313800	+	0.313800i	−0.532580	+	0.846380i	$0.678777\pi$
$24$			360.000i			0.625000i
$25$		0			0
$26$	−278.000			−0.411243
$27$	540.000	−	540.000i	0.740741	−	0.740741i
$28$	364.000	+	364.000i	0.464286	+	0.464286i
$29$			480.000i			0.570749i	0.958416	+	0.285375i	$0.0921180\pi$
$29$			480.000i			0.570749i	−0.958416	+	0.285375i	$0.907882\pi$
$30$		0			0
$31$	572.000			0.595213			0.297607	−	0.954689i	$-0.403812\pi$
$31$	572.000			0.595213			0.297607	+	0.954689i	$0.403812\pi$
$32$	−644.000	+	644.000i	−0.628906	+	0.628906i
$33$	−48.0000	−	48.0000i	−0.0440771	−	0.0440771i
$34$		−	2.00000i		−	0.00173010i
$35$		0			0
$36$	126.000			0.0972222
$37$	251.000	−	251.000i	0.183346	−	0.183346i	−0.609466	−	0.792812i	$-0.708616\pi$
$37$	251.000	−	251.000i	0.183346	−	0.183346i	0.792812	+	0.609466i	$0.208616\pi$
$38$	−180.000	−	180.000i	−0.124654	−	0.124654i
$39$		−	1668.00i		−	1.09665i
$40$		0			0
$41$	−1688.00			−1.00416			−0.502082	−	0.864820i	$-0.667432\pi$
$41$	−1688.00			−1.00416			−0.502082	+	0.864820i	$0.667432\pi$
$42$	312.000	−	312.000i	0.176871	−	0.176871i
$43$	−1474.00	−	1474.00i	−0.797188	−	0.797188i	0.185463	−	0.982651i	$-0.440621\pi$
$43$	−1474.00	−	1474.00i	−0.797188	−	0.797188i	−0.982651	+	0.185463i	$0.940621\pi$
$44$		−	112.000i		−	0.0578512i
$45$		0			0
$46$	332.000			0.156900
$47$	−2474.00	+	2474.00i	−1.11996	+	1.11996i	−0.128218	+	0.991746i	$0.540926\pi$
$47$	−2474.00	+	2474.00i	−1.11996	+	1.11996i	−0.991746	+	0.128218i	$0.959074\pi$
$48$	−984.000	−	984.000i	−0.427083	−	0.427083i
$49$			1049.00i			0.436901i
$50$		0			0
$51$	12.0000			0.00461361
$52$	1946.00	−	1946.00i	0.719675	−	0.719675i
$53$	3331.00	+	3331.00i	1.18583	+	1.18583i	0.978209	+	0.207622i	$0.0665724\pi$
$53$	3331.00	+	3331.00i	1.18583	+	1.18583i	0.207622	+	0.978209i	$0.433428\pi$
$54$		−	1080.00i		−	0.370370i
$55$		0			0
$56$	1560.00			0.497449
$57$	1080.00	−	1080.00i	0.332410	−	0.332410i
$58$	480.000	+	480.000i	0.142687	+	0.142687i
$59$			3660.00i			1.05142i	0.850663	+	0.525711i	$0.176201\pi$
$59$			3660.00i			1.05142i	−0.850663	+	0.525711i	$0.823799\pi$
$60$		0			0
$61$	1592.00			0.427842			0.213921	−	0.976851i	$-0.431376\pi$
$61$	1592.00			0.427842			0.213921	+	0.976851i	$0.431376\pi$
$62$	572.000	−	572.000i	0.148803	−	0.148803i
$63$	−234.000	−	234.000i	−0.0589569	−	0.0589569i
$64$		−	1336.00i		−	0.326172i
$65$		0			0
$66$	−96.0000			−0.0220386
$67$	−874.000	+	874.000i	−0.194698	+	0.194698i	−0.797723	−	0.603025i	$-0.793962\pi$
$67$	−874.000	+	874.000i	−0.194698	+	0.194698i	0.603025	+	0.797723i	$0.293962\pi$
$68$	14.0000	+	14.0000i	0.00302768	+	0.00302768i
$69$			1992.00i			0.418399i
$70$		0			0
$71$	−6068.00			−1.20373			−0.601865	−	0.798598i	$-0.705575\pi$
$71$	−6068.00			−1.20373			−0.601865	+	0.798598i	$0.705575\pi$
$72$	270.000	−	270.000i	0.0520833	−	0.0520833i
$73$	791.000	+	791.000i	0.148433	+	0.148433i	0.777418	−	0.628985i	$-0.216529\pi$
$73$	791.000	+	791.000i	0.148433	+	0.148433i	−0.628985	+	0.777418i	$0.716529\pi$
$74$		−	502.000i		−	0.0916728i
$75$		0			0
$76$	2520.00			0.436288
$77$	−208.000	+	208.000i	−0.0350818	+	0.0350818i
$78$	−1668.00	−	1668.00i	−0.274162	−	0.274162i
$79$		−	9120.00i		−	1.46130i	−0.682750	−	0.730652i	$-0.739216\pi$
$79$		−	9120.00i		−	1.46130i	0.682750	−	0.730652i	$-0.260784\pi$
$80$		0			0
$81$	5751.00			0.876543
$82$	−1688.00	+	1688.00i	−0.251041	+	0.251041i
$83$	−5654.00	−	5654.00i	−0.820729	−	0.820729i	0.165484	−	0.986213i	$-0.447081\pi$
$83$	−5654.00	−	5654.00i	−0.820729	−	0.820729i	−0.986213	+	0.165484i	$0.947081\pi$
$84$			4368.00i			0.619048i
$85$		0			0
$86$	−2948.00			−0.398594
$87$	−2880.00	+	2880.00i	−0.380499	+	0.380499i
$88$	−240.000	−	240.000i	−0.0309917	−	0.0309917i
$89$		−	2160.00i		−	0.272693i	−0.990661	−	0.136346i	$-0.956464\pi$
$89$		−	2160.00i		−	0.272693i	0.990661	−	0.136346i	$-0.0435360\pi$
$90$		0			0
$91$	−7228.00			−0.872841
$92$	−2324.00	+	2324.00i	−0.274575	+	0.274575i
$93$	3432.00	+	3432.00i	0.396809	+	0.396809i
$94$			4948.00i			0.559982i
$95$		0			0
$96$	−7728.00			−0.838542
$97$	6551.00	−	6551.00i	0.696248	−	0.696248i	−0.267351	−	0.963599i	$-0.586148\pi$
$97$	6551.00	−	6551.00i	0.696248	−	0.696248i	0.963599	+	0.267351i	$0.0861483\pi$
$98$	1049.00	+	1049.00i	0.109225	+	0.109225i
$99$			72.0000i			0.00734619i
$100$		0			0
$101$	16102.0			1.57847			0.789236	−	0.614090i	$-0.210477\pi$
$101$	16102.0			1.57847			0.789236	+	0.614090i	$0.210477\pi$
$102$	12.0000	−	12.0000i	0.00115340	−	0.00115340i
$103$	−994.000	−	994.000i	−0.0936940	−	0.0936940i	0.658706	−	0.752400i	$-0.271104\pi$
$103$	−994.000	−	994.000i	−0.0936940	−	0.0936940i	−0.752400	+	0.658706i	$0.771104\pi$
$104$		−	8340.00i		−	0.771080i
$105$		0			0
$106$	6662.00			0.592916
$107$	8326.00	−	8326.00i	0.727225	−	0.727225i	−0.242841	−	0.970066i	$-0.578079\pi$
$107$	8326.00	−	8326.00i	0.727225	−	0.727225i	0.970066	+	0.242841i	$0.0780793\pi$
$108$	7560.00	+	7560.00i	0.648148	+	0.648148i
$109$			17010.0i			1.43170i	0.698255	+	0.715849i	$0.253960\pi$
$109$			17010.0i			1.43170i	−0.698255	+	0.715849i	$0.746040\pi$
$110$		0			0
$111$	3012.00			0.244461
$112$	−4264.00	+	4264.00i	−0.339923	+	0.339923i
$113$	5161.00	+	5161.00i	0.404182	+	0.404182i	0.879704	−	0.475522i	$-0.157741\pi$
$113$	5161.00	+	5161.00i	0.404182	+	0.404182i	−0.475522	+	0.879704i	$0.657741\pi$
$114$		−	2160.00i		−	0.166205i
$115$		0			0
$116$	−6720.00			−0.499405
$117$	−1251.00	+	1251.00i	−0.0913872	+	0.0913872i
$118$	3660.00	+	3660.00i	0.262856	+	0.262856i
$119$		−	52.0000i		−	0.00367206i
$120$		0			0
$121$	−14577.0			−0.995629
$122$	1592.00	−	1592.00i	0.106960	−	0.106960i
$123$	−10128.0	−	10128.0i	−0.669443	−	0.669443i
$124$			8008.00i			0.520812i
$125$		0			0
$126$	−468.000			−0.0294785
$127$	−12574.0	+	12574.0i	−0.779590	+	0.779590i	−0.979761	−	0.200171i	$-0.935850\pi$
$127$	−12574.0	+	12574.0i	−0.779590	+	0.779590i	0.200171	+	0.979761i	$0.435850\pi$
$128$	−11640.0	−	11640.0i	−0.710449	−	0.710449i
$129$		−	17688.0i		−	1.06292i
$130$		0			0
$131$	20272.0			1.18128			0.590642	−	0.806934i	$-0.298875\pi$
$131$	20272.0			1.18128			0.590642	+	0.806934i	$0.298875\pi$
$132$	672.000	−	672.000i	0.0385675	−	0.0385675i
$133$	−4680.00	−	4680.00i	−0.264571	−	0.264571i
$134$			1748.00i			0.0973491i
$135$		0			0
$136$	60.0000			0.00324394
$137$	10351.0	−	10351.0i	0.551494	−	0.551494i	−0.375378	−	0.926872i	$-0.622487\pi$
$137$	10351.0	−	10351.0i	0.551494	−	0.551494i	0.926872	+	0.375378i	$0.122487\pi$
$138$	1992.00	+	1992.00i	0.104600	+	0.104600i
$139$		−	27060.0i		−	1.40055i	−0.713874	−	0.700274i	$-0.753061\pi$
$139$		−	27060.0i		−	1.40055i	0.713874	−	0.700274i	$-0.246939\pi$
$140$		0			0
$141$	−29688.0			−1.49329
$142$	−6068.00	+	6068.00i	−0.300932	+	0.300932i
$143$	1112.00	+	1112.00i	0.0543792	+	0.0543792i
$144$			1476.00i			0.0711806i
$145$		0			0
$146$	1582.00			0.0742166
$147$	−6294.00	+	6294.00i	−0.291268	+	0.291268i
$148$	3514.00	+	3514.00i	0.160427	+	0.160427i
$149$			16350.0i			0.736453i	0.929736	+	0.368227i	$0.120035\pi$
$149$			16350.0i			0.736453i	−0.929736	+	0.368227i	$0.879965\pi$
$150$		0			0
$151$	1052.00			0.0461383			0.0230692	−	0.999734i	$-0.492656\pi$
$151$	1052.00			0.0461383			0.0230692	+	0.999734i	$0.492656\pi$
$152$	5400.00	−	5400.00i	0.233726	−	0.233726i
$153$	−9.00000	−	9.00000i	−0.000384468	−	0.000384468i
$154$			416.000i			0.0175409i
$155$		0			0
$156$	23352.0			0.959566
$157$	−6499.00	+	6499.00i	−0.263662	+	0.263662i	−0.826540	−	0.562878i	$-0.809694\pi$
$157$	−6499.00	+	6499.00i	−0.263662	+	0.263662i	0.562878	+	0.826540i	$0.309694\pi$
$158$	−9120.00	−	9120.00i	−0.365326	−	0.365326i
$159$			39972.0i			1.58111i
$160$		0			0
$161$	8632.00			0.333012
$162$	5751.00	−	5751.00i	0.219136	−	0.219136i
$163$	19286.0	+	19286.0i	0.725884	+	0.725884i	0.969797	−	0.243913i	$-0.0784313\pi$
$163$	19286.0	+	19286.0i	0.725884	+	0.725884i	−0.243913	+	0.969797i	$0.578431\pi$
$164$		−	23632.0i		−	0.878644i
$165$		0			0
$166$	−11308.0			−0.410364
$167$	15526.0	−	15526.0i	0.556707	−	0.556707i	−0.371661	−	0.928368i	$-0.621212\pi$
$167$	15526.0	−	15526.0i	0.556707	−	0.556707i	0.928368	+	0.371661i	$0.121212\pi$
$168$	9360.00	+	9360.00i	0.331633	+	0.331633i
$169$			10081.0i			0.352964i
$170$		0			0
$171$	−1620.00			−0.0554017
$172$	20636.0	−	20636.0i	0.697539	−	0.697539i
$173$	16891.0	+	16891.0i	0.564369	+	0.564369i	0.930545	−	0.366176i	$-0.119333\pi$
$173$	16891.0	+	16891.0i	0.564369	+	0.564369i	−0.366176	+	0.930545i	$0.619333\pi$
$174$			5760.00i			0.190250i
$175$		0			0
$176$	1312.00			0.0423554
$177$	−21960.0	+	21960.0i	−0.700948	+	0.700948i
$178$	−2160.00	−	2160.00i	−0.0681732	−	0.0681732i
$179$		−	10620.0i		−	0.331450i	−0.986172	−	0.165725i	$-0.947004\pi$
$179$		−	10620.0i		−	0.331450i	0.986172	−	0.165725i	$-0.0529965\pi$
$180$		0			0
$181$	24122.0			0.736302			0.368151	−	0.929766i	$-0.379991\pi$
$181$	24122.0			0.736302			0.368151	+	0.929766i	$0.379991\pi$
$182$	−7228.00	+	7228.00i	−0.218210	+	0.218210i
$183$	9552.00	+	9552.00i	0.285228	+	0.285228i
$184$			9960.00i			0.294187i
$185$		0			0
$186$	6864.00			0.198404
$187$	−8.00000	+	8.00000i	−0.000228774	+	0.000228774i
$188$	−34636.0	−	34636.0i	−0.979968	−	0.979968i
$189$		−	28080.0i		−	0.786092i
$190$		0			0
$191$	−45188.0			−1.23867			−0.619336	−	0.785126i	$-0.712598\pi$
$191$	−45188.0			−1.23867			−0.619336	+	0.785126i	$0.712598\pi$
$192$	8016.00	−	8016.00i	0.217448	−	0.217448i
$193$	−39199.0	−	39199.0i	−1.05235	−	1.05235i	−0.998552	−	0.0537986i	$-0.982867\pi$
$193$	−39199.0	−	39199.0i	−1.05235	−	1.05235i	−0.0537986	−	0.998552i	$-0.517133\pi$
$194$		−	13102.0i		−	0.348124i
$195$		0			0
$196$	−14686.0			−0.382289
$197$	−28349.0	+	28349.0i	−0.730475	+	0.730475i	−0.970714	−	0.240239i	$-0.922774\pi$
$197$	−28349.0	+	28349.0i	−0.730475	+	0.730475i	0.240239	+	0.970714i	$0.422774\pi$
$198$	72.0000	+	72.0000i	0.00183655	+	0.00183655i
$199$			1800.00i			0.0454534i	0.999742	+	0.0227267i	$0.00723476\pi$
$199$			1800.00i			0.0454534i	−0.999742	+	0.0227267i	$0.992765\pi$
$200$		0			0
$201$	−10488.0			−0.259598
$202$	16102.0	−	16102.0i	0.394618	−	0.394618i
$203$	12480.0	+	12480.0i	0.302846	+	0.302846i
$204$			168.000i			0.00403691i
$205$		0			0
$206$	−1988.00			−0.0468470
$207$	1494.00	−	1494.00i	0.0348666	−	0.0348666i
$208$	22796.0	+	22796.0i	0.526905	+	0.526905i
$209$			1440.00i			0.0329663i
$210$		0			0
$211$	18392.0			0.413108			0.206554	−	0.978435i	$-0.433775\pi$
$211$	18392.0			0.413108			0.206554	+	0.978435i	$0.433775\pi$
$212$	−46634.0	+	46634.0i	−1.03760	+	1.03760i
$213$	−36408.0	−	36408.0i	−0.802486	−	0.802486i
$214$		−	16652.0i		−	0.363613i
$215$		0			0
$216$	32400.0			0.694444
$217$	14872.0	−	14872.0i	0.315827	−	0.315827i
$218$	17010.0	+	17010.0i	0.357924	+	0.357924i
$219$			9492.00i			0.197911i
$220$		0			0
$221$	−278.000			−0.00569194
$222$	3012.00	−	3012.00i	0.0611152	−	0.0611152i
$223$	866.000	+	866.000i	0.0174144	+	0.0174144i	0.715760	−	0.698346i	$-0.246080\pi$
$223$	866.000	+	866.000i	0.0174144	+	0.0174144i	−0.698346	+	0.715760i	$0.746080\pi$
$224$			33488.0i			0.667411i
$225$		0			0
$226$	10322.0			0.202091
$227$	45226.0	−	45226.0i	0.877681	−	0.877681i	−0.115614	−	0.993294i	$-0.536883\pi$
$227$	45226.0	−	45226.0i	0.877681	−	0.877681i	0.993294	+	0.115614i	$0.0368835\pi$
$228$	15120.0	+	15120.0i	0.290859	+	0.290859i
$229$		−	39120.0i		−	0.745981i	−0.927835	−	0.372991i	$-0.878332\pi$
$229$		−	39120.0i		−	0.745981i	0.927835	−	0.372991i	$-0.121668\pi$
$230$		0			0
$231$	−2496.00			−0.0467757
$232$	−14400.0	+	14400.0i	−0.267539	+	0.267539i
$233$	33121.0	+	33121.0i	0.610087	+	0.610087i	0.942969	−	0.332882i	$-0.108021\pi$
$233$	33121.0	+	33121.0i	0.610087	+	0.610087i	−0.332882	+	0.942969i	$0.608021\pi$
$234$			2502.00i			0.0456936i
$235$		0			0
$236$	−51240.0			−0.919994
$237$	54720.0	−	54720.0i	0.974203	−	0.974203i
$238$	−52.0000	−	52.0000i	−0.000918014	−	0.000918014i
$239$			88440.0i			1.54829i	0.633007	+	0.774146i	$0.281820\pi$
$239$			88440.0i			1.54829i	−0.633007	+	0.774146i	$0.718180\pi$
$240$		0			0
$241$	20312.0			0.349718			0.174859	−	0.984593i	$-0.444053\pi$
$241$	20312.0			0.349718			0.174859	+	0.984593i	$0.444053\pi$
$242$	−14577.0	+	14577.0i	−0.248907	+	0.248907i
$243$	−9234.00	−	9234.00i	−0.156379	−	0.156379i
$244$			22288.0i			0.374362i
$245$		0			0
$246$	−20256.0			−0.334721
$247$	−25020.0	+	25020.0i	−0.410103	+	0.410103i
$248$	17160.0	+	17160.0i	0.279006	+	0.279006i
$249$		−	67848.0i		−	1.09430i
$250$		0			0
$251$	74752.0			1.18652			0.593260	−	0.805011i	$-0.297840\pi$
$251$	74752.0			1.18652			0.593260	+	0.805011i	$0.297840\pi$
$252$	3276.00	−	3276.00i	0.0515873	−	0.0515873i
$253$	−1328.00	−	1328.00i	−0.0207471	−	0.0207471i
$254$			25148.0i			0.389795i
$255$		0			0
$256$	−1904.00			−0.0290527
$257$	−37799.0	+	37799.0i	−0.572287	+	0.572287i	−0.932767	−	0.360480i	$-0.882613\pi$
$257$	−37799.0	+	37799.0i	−0.572287	+	0.572287i	0.360480	+	0.932767i	$0.382613\pi$
$258$	−17688.0	−	17688.0i	−0.265729	−	0.265729i
$259$		−	13052.0i		−	0.194571i
$260$		0			0
$261$	4320.00			0.0634166
$262$	20272.0	−	20272.0i	0.295321	−	0.295321i
$263$	33586.0	+	33586.0i	0.485564	+	0.485564i	0.906903	−	0.421339i	$-0.138440\pi$
$263$	33586.0	+	33586.0i	0.485564	+	0.485564i	−0.421339	+	0.906903i	$0.638440\pi$
$264$		−	2880.00i		−	0.0413223i
$265$		0			0
$266$	−9360.00			−0.132286
$267$	12960.0	−	12960.0i	0.181795	−	0.181795i
$268$	−12236.0	−	12236.0i	−0.170361	−	0.170361i
$269$		−	28530.0i		−	0.394273i	−0.980376	−	0.197137i	$-0.936836\pi$
$269$		−	28530.0i		−	0.394273i	0.980376	−	0.197137i	$-0.0631642\pi$
$270$		0			0
$271$	−7468.00			−0.101687			−0.0508435	−	0.998707i	$-0.516191\pi$
$271$	−7468.00			−0.101687			−0.0508435	+	0.998707i	$0.516191\pi$
$272$	−164.000	+	164.000i	−0.00221670	+	0.00221670i
$273$	−43368.0	−	43368.0i	−0.581894	−	0.581894i
$274$		−	20702.0i		−	0.275747i
$275$		0			0
$276$	−27888.0			−0.366100
$277$	−6499.00	+	6499.00i	−0.0847007	+	0.0847007i	−0.748188	−	0.663487i	$-0.769076\pi$
$277$	−6499.00	+	6499.00i	−0.0847007	+	0.0847007i	0.663487	+	0.748188i	$0.269076\pi$
$278$	−27060.0	−	27060.0i	−0.350137	−	0.350137i
$279$		−	5148.00i		−	0.0661348i
$280$		0			0
$281$	−97928.0			−1.24021			−0.620104	−	0.784520i	$-0.712909\pi$
$281$	−97928.0			−1.24021			−0.620104	+	0.784520i	$0.712909\pi$
$282$	−29688.0	+	29688.0i	−0.373321	+	0.373321i
$283$	−59854.0	−	59854.0i	−0.747344	−	0.747344i	0.226636	−	0.973980i	$-0.427227\pi$
$283$	−59854.0	−	59854.0i	−0.747344	−	0.747344i	−0.973980	+	0.226636i	$0.927227\pi$
$284$		−	84952.0i		−	1.05326i
$285$		0			0
$286$	2224.00			0.0271896
$287$	−43888.0	+	43888.0i	−0.532822	+	0.532822i
$288$	5796.00	+	5796.00i	0.0698785	+	0.0698785i
$289$			83519.0i			0.999976i
$290$		0			0
$291$	78612.0			0.928331
$292$	−11074.0	+	11074.0i	−0.129879	+	0.129879i
$293$	−28499.0	−	28499.0i	−0.331967	−	0.331967i	0.521366	−	0.853333i	$-0.325423\pi$
$293$	−28499.0	−	28499.0i	−0.331967	−	0.331967i	−0.853333	+	0.521366i	$0.825423\pi$
$294$			12588.0i			0.145634i
$295$		0			0
$296$	15060.0			0.171886
$297$	−4320.00	+	4320.00i	−0.0489746	+	0.0489746i
$298$	16350.0	+	16350.0i	0.184113	+	0.184113i
$299$		−	46148.0i		−	0.516191i
$300$		0			0
$301$	−76648.0			−0.845995
$302$	1052.00	−	1052.00i	0.0115346	−	0.0115346i
$303$	96612.0	+	96612.0i	1.05232	+	1.05232i
$304$			29520.0i			0.319425i
$305$		0			0
$306$	−18.0000			−0.000192234
$307$	117926.	−	117926.i	1.25122	−	1.25122i	0.296043	−	0.955175i	$-0.404333\pi$
$307$	117926.	−	117926.i	1.25122	−	1.25122i	0.955175	−	0.296043i	$-0.0956671\pi$
$308$	−2912.00	−	2912.00i	−0.0306966	−	0.0306966i
$309$		−	11928.0i		−	0.124925i
$310$		0			0
$311$	3892.00			0.0402395			0.0201197	−	0.999798i	$-0.493595\pi$
$311$	3892.00			0.0402395			0.0201197	+	0.999798i	$0.493595\pi$
$312$	50040.0	−	50040.0i	0.514053	−	0.514053i
$313$	49961.0	+	49961.0i	0.509967	+	0.509967i	0.914516	−	0.404549i	$-0.132571\pi$
$313$	49961.0	+	49961.0i	0.509967	+	0.509967i	−0.404549	+	0.914516i	$0.632571\pi$
$314$			12998.0i			0.131831i
$315$		0			0
$316$	127680.			1.27864
$317$	−17099.0	+	17099.0i	−0.170158	+	0.170158i	−0.787049	−	0.616891i	$-0.788392\pi$
$317$	−17099.0	+	17099.0i	−0.170158	+	0.170158i	0.616891	+	0.787049i	$0.288392\pi$
$318$	39972.0	+	39972.0i	0.395277	+	0.395277i
$319$		−	3840.00i		−	0.0377355i
$320$		0			0
$321$	99912.0			0.969633
$322$	8632.00	−	8632.00i	0.0832530	−	0.0832530i
$323$	−180.000	−	180.000i	−0.00172531	−	0.00172531i
$324$			80514.0i			0.766975i
$325$		0			0
$326$	38572.0			0.362942
$327$	−102060.	+	102060.i	−0.954465	+	0.954465i
$328$	−50640.0	−	50640.0i	−0.470702	−	0.470702i
$329$			128648.i			1.18853i
$330$		0			0
$331$	−143128.			−1.30638			−0.653189	−	0.757195i	$-0.726569\pi$
$331$	−143128.			−1.30638			−0.653189	+	0.757195i	$0.726569\pi$
$332$	79156.0	−	79156.0i	0.718138	−	0.718138i
$333$	−2259.00	−	2259.00i	−0.0203717	−	0.0203717i
$334$		−	31052.0i		−	0.278353i
$335$		0			0
$336$	−51168.0			−0.453231
$337$	−103249.	+	103249.i	−0.909130	+	0.909130i	−0.996202	−	0.0870719i	$-0.972249\pi$
$337$	−103249.	+	103249.i	−0.909130	+	0.909130i	0.0870719	+	0.996202i	$0.472249\pi$
$338$	10081.0	+	10081.0i	0.0882410	+	0.0882410i
$339$			61932.0i			0.538909i
$340$		0			0
$341$	−4576.00			−0.0393529
$342$	−1620.00	+	1620.00i	−0.0138504	+	0.0138504i
$343$	89700.0	+	89700.0i	0.762437	+	0.762437i
$344$		−	88440.0i		−	0.747363i
$345$		0			0
$346$	33782.0			0.282185
$347$	104626.	−	104626.i	0.868922	−	0.868922i	−0.123431	−	0.992353i	$-0.539390\pi$
$347$	104626.	−	104626.i	0.868922	−	0.868922i	0.992353	+	0.123431i	$0.0393899\pi$
$348$	−40320.0	−	40320.0i	−0.332937	−	0.332937i
$349$		−	94800.0i		−	0.778319i	−0.921170	−	0.389159i	$-0.872766\pi$
$349$		−	94800.0i		−	0.778319i	0.921170	−	0.389159i	$-0.127234\pi$
$350$		0			0
$351$	−150120.			−1.21850
$352$	5152.00	−	5152.00i	0.0415806	−	0.0415806i
$353$	−63569.0	−	63569.0i	−0.510148	−	0.510148i	0.404424	−	0.914572i	$-0.367472\pi$
$353$	−63569.0	−	63569.0i	−0.510148	−	0.510148i	−0.914572	+	0.404424i	$0.867472\pi$
$354$			43920.0i			0.350474i
$355$		0			0
$356$	30240.0			0.238606
$357$	312.000	−	312.000i	0.00244804	−	0.00244804i
$358$	−10620.0	−	10620.0i	−0.0828626	−	0.0828626i
$359$		−	141840.i		−	1.10055i	−0.834983	−	0.550275i	$-0.814523\pi$
$359$		−	141840.i		−	1.10055i	0.834983	−	0.550275i	$-0.185477\pi$
$360$		0			0
$361$	97921.0			0.751383
$362$	24122.0	−	24122.0i	0.184076	−	0.184076i
$363$	−87462.0	−	87462.0i	−0.663752	−	0.663752i
$364$		−	101192.i		−	0.763736i
$365$		0			0
$366$	19104.0			0.142614
$367$	−142174.	+	142174.i	−1.05557	+	1.05557i	−0.0572103	+	0.998362i	$0.518221\pi$
$367$	−142174.	+	142174.i	−1.05557	+	1.05557i	−0.998362	+	0.0572103i	$0.981779\pi$
$368$	−27224.0	−	27224.0i	−0.201028	−	0.201028i
$369$			15192.0i			0.111574i
$370$		0			0
$371$	173212.			1.25843
$372$	−48048.0	+	48048.0i	−0.347208	+	0.347208i
$373$	−10309.0	−	10309.0i	−0.0740967	−	0.0740967i	0.669087	−	0.743184i	$-0.266685\pi$
$373$	−10309.0	−	10309.0i	−0.0740967	−	0.0740967i	−0.743184	+	0.669087i	$0.766685\pi$
$374$			16.0000i			0.000114387i
$375$		0			0
$376$	−148440.			−1.04997
$377$	66720.0	−	66720.0i	0.469433	−	0.469433i
$378$	−28080.0	−	28080.0i	−0.196523	−	0.196523i
$379$			115380.i			0.803253i	0.915804	+	0.401626i	$0.131555\pi$
$379$			115380.i			0.803253i	−0.915804	+	0.401626i	$0.868445\pi$
$380$		0			0
$381$	−150888.			−1.03945
$382$	−45188.0	+	45188.0i	−0.309668	+	0.309668i
$383$	−62654.0	−	62654.0i	−0.427121	−	0.427121i	0.460525	−	0.887647i	$-0.347661\pi$
$383$	−62654.0	−	62654.0i	−0.427121	−	0.427121i	−0.887647	+	0.460525i	$0.847661\pi$
$384$		−	139680.i		−	0.947266i
$385$		0			0
$386$	−78398.0			−0.526175
$387$	−13266.0	+	13266.0i	−0.0885764	+	0.0885764i
$388$	91714.0	+	91714.0i	0.609217	+	0.609217i
$389$			132690.i			0.876878i	0.898761	+	0.438439i	$0.144468\pi$
$389$			132690.i			0.876878i	−0.898761	+	0.438439i	$0.855532\pi$
$390$		0			0
$391$	332.000			0.00217162
$392$	−31470.0	+	31470.0i	−0.204797	+	0.204797i
$393$	121632.	+	121632.i	0.787522	+	0.787522i
$394$			56698.0i			0.365237i
$395$		0			0
$396$	−1008.00			−0.00642792
$397$	88451.0	−	88451.0i	0.561205	−	0.561205i	−0.368444	−	0.929650i	$-0.620110\pi$
$397$	88451.0	−	88451.0i	0.561205	−	0.561205i	0.929650	+	0.368444i	$0.120110\pi$
$398$	1800.00	+	1800.00i	0.0113633	+	0.0113633i
$399$		−	56160.0i		−	0.352762i
$400$		0			0
$401$	63202.0			0.393045			0.196522	−	0.980499i	$-0.437035\pi$
$401$	63202.0			0.393045			0.196522	+	0.980499i	$0.437035\pi$
$402$	−10488.0	+	10488.0i	−0.0648994	+	0.0648994i
$403$	−79508.0	−	79508.0i	−0.489554	−	0.489554i
$404$			225428.i			1.38116i
$405$		0			0
$406$	24960.0			0.151423
$407$	−2008.00	+	2008.00i	−0.0121220	+	0.0121220i
$408$	360.000	+	360.000i	0.00216263	+	0.00216263i
$409$		−	52890.0i		−	0.316175i	−0.987425	−	0.158087i	$-0.949467\pi$
$409$		−	52890.0i		−	0.316175i	0.987425	−	0.158087i	$-0.0505327\pi$
$410$		0			0
$411$	124212.			0.735326
$412$	13916.0	−	13916.0i	0.0819823	−	0.0819823i
$413$	95160.0	+	95160.0i	0.557897	+	0.557897i
$414$		−	2988.00i		−	0.0174333i
$415$		0			0
$416$	179032.			1.03453
$417$	162360.	−	162360.i	0.933699	−	0.933699i
$418$	1440.00	+	1440.00i	0.00824157	+	0.00824157i
$419$		−	256980.i		−	1.46376i	−0.681431	−	0.731882i	$-0.738642\pi$
$419$		−	256980.i		−	1.46376i	0.681431	−	0.731882i	$-0.261358\pi$
$420$		0			0
$421$	186632.			1.05298			0.526492	−	0.850180i	$-0.323507\pi$
$421$	186632.			1.05298			0.526492	+	0.850180i	$0.323507\pi$
$422$	18392.0	−	18392.0i	0.103277	−	0.103277i
$423$	22266.0	+	22266.0i	0.124440	+	0.124440i
$424$			199860.i			1.11172i
$425$		0			0
$426$	−72816.0			−0.401243
$427$	41392.0	−	41392.0i	0.227018	−	0.227018i
$428$	116564.	+	116564.i	0.636322	+	0.636322i
$429$			13344.0i			0.0725056i
$430$		0			0
$431$	−208028.			−1.11987			−0.559935	−	0.828537i	$-0.689174\pi$
$431$	−208028.			−1.11987			−0.559935	+	0.828537i	$0.689174\pi$
$432$	−88560.0	+	88560.0i	−0.474537	+	0.474537i
$433$	151271.	+	151271.i	0.806826	+	0.806826i	0.984152	−	0.177326i	$-0.0567447\pi$
$433$	151271.	+	151271.i	0.806826	+	0.806826i	−0.177326	+	0.984152i	$0.556745\pi$
$434$		−	29744.0i		−	0.157914i
$435$		0			0
$436$	−238140.			−1.25274
$437$	29880.0	−	29880.0i	0.156465	−	0.156465i
$438$	9492.00	+	9492.00i	0.0494777	+	0.0494777i
$439$			158640.i			0.823159i	0.911374	+	0.411579i	$0.135023\pi$
$439$			158640.i			0.823159i	−0.911374	+	0.411579i	$0.864977\pi$
$440$		0			0
$441$	9441.00			0.0485446
$442$	−278.000	+	278.000i	−0.00142298	+	0.00142298i
$443$	−252974.	−	252974.i	−1.28905	−	1.28905i	−0.935367	−	0.353679i	$-0.884930\pi$
$443$	−252974.	−	252974.i	−1.28905	−	1.28905i	−0.353679	−	0.935367i	$-0.615070\pi$
$444$			42168.0i			0.213903i
$445$		0			0
$446$	1732.00			0.00870719
$447$	−98100.0	+	98100.0i	−0.490969	+	0.490969i
$448$	−34736.0	−	34736.0i	−0.173071	−	0.173071i
$449$		−	123750.i		−	0.613836i	−0.951736	−	0.306918i	$-0.900702\pi$
$449$		−	123750.i		−	0.613836i	0.951736	−	0.306918i	$-0.0992978\pi$
$450$		0			0
$451$	13504.0			0.0663910
$452$	−72254.0	+	72254.0i	−0.353659	+	0.353659i
$453$	6312.00	+	6312.00i	0.0307589	+	0.0307589i
$454$		−	90452.0i		−	0.438840i
$455$		0			0
$456$	64800.0			0.311634
$457$	32201.0	−	32201.0i	0.154183	−	0.154183i	−0.625800	−	0.779983i	$-0.715227\pi$
$457$	32201.0	−	32201.0i	0.154183	−	0.154183i	0.779983	+	0.625800i	$0.215227\pi$
$458$	−39120.0	−	39120.0i	−0.186495	−	0.186495i
$459$		−	1080.00i		−	0.00512623i
$460$		0			0
$461$	75142.0			0.353574			0.176787	−	0.984249i	$-0.443430\pi$
$461$	75142.0			0.353574			0.176787	+	0.984249i	$0.443430\pi$
$462$	−2496.00	+	2496.00i	−0.0116939	+	0.0116939i
$463$	−235714.	−	235714.i	−1.09957	−	1.09957i	−0.994461	−	0.105111i	$-0.966480\pi$
$463$	−235714.	−	235714.i	−1.09957	−	1.09957i	−0.105111	−	0.994461i	$-0.533520\pi$
$464$		−	78720.0i		−	0.365636i
$465$		0			0
$466$	66242.0			0.305043
$467$	−28574.0	+	28574.0i	−0.131020	+	0.131020i	−0.769576	−	0.638556i	$-0.779532\pi$
$467$	−28574.0	+	28574.0i	−0.131020	+	0.131020i	0.638556	+	0.769576i	$0.279532\pi$
$468$	−17514.0	−	17514.0i	−0.0799638	−	0.0799638i
$469$			45448.0i			0.206618i
$470$		0			0
$471$	−77988.0			−0.351549
$472$	−109800.	+	109800.i	−0.492854	+	0.492854i
$473$	11792.0	+	11792.0i	0.0527066	+	0.0527066i
$474$		−	109440.i		−	0.487101i
$475$		0			0
$476$	728.000			0.00321305
$477$	29979.0	−	29979.0i	0.131759	−	0.131759i
$478$	88440.0	+	88440.0i	0.387073	+	0.387073i
$479$		−	26520.0i		−	0.115585i	−0.998329	−	0.0577926i	$-0.981594\pi$
$479$		−	26520.0i		−	0.115585i	0.998329	−	0.0577926i	$-0.0184062\pi$
$480$		0			0
$481$	−69778.0			−0.301598
$482$	20312.0	−	20312.0i	0.0874296	−	0.0874296i
$483$	51792.0	+	51792.0i	0.222008	+	0.222008i
$484$		−	204078.i		−	0.871175i
$485$		0			0
$486$	−18468.0			−0.0781893
$487$	−41374.0	+	41374.0i	−0.174449	+	0.174449i	−0.788931	−	0.614482i	$-0.789365\pi$
$487$	−41374.0	+	41374.0i	−0.174449	+	0.174449i	0.614482	+	0.788931i	$0.289365\pi$
$488$	47760.0	+	47760.0i	0.200551	+	0.200551i
$489$			231432.i			0.967845i
$490$		0			0
$491$	−149288.			−0.619244			−0.309622	−	0.950860i	$-0.600203\pi$
$491$	−149288.			−0.619244			−0.309622	+	0.950860i	$0.600203\pi$
$492$	141792.	−	141792.i	0.585762	−	0.585762i
$493$	480.000	+	480.000i	0.00197491	+	0.00197491i
$494$			50040.0i			0.205052i
$495$		0			0
$496$	−93808.0			−0.381309
$497$	−157768.	+	157768.i	−0.638714	+	0.638714i
$498$	−67848.0	−	67848.0i	−0.273576	−	0.273576i
$499$		−	284100.i		−	1.14096i	−0.821312	−	0.570480i	$-0.806757\pi$
$499$		−	284100.i		−	1.14096i	0.821312	−	0.570480i	$-0.193243\pi$
$500$		0			0
$501$	186312.			0.742276
$502$	74752.0	−	74752.0i	0.296630	−	0.296630i
$503$	117406.	+	117406.i	0.464039	+	0.464039i	0.899977	−	0.435938i	$-0.143583\pi$
$503$	117406.	+	117406.i	0.464039	+	0.464039i	−0.435938	+	0.899977i	$0.643583\pi$
$504$		−	14040.0i		−	0.0552721i
$505$		0			0
$506$	−2656.00			−0.0103735
$507$	−60486.0	+	60486.0i	−0.235309	+	0.235309i
$508$	−176036.	−	176036.i	−0.682141	−	0.682141i
$509$			234960.i			0.906898i	0.891282	+	0.453449i	$0.149807\pi$
$509$			234960.i			0.906898i	−0.891282	+	0.453449i	$0.850193\pi$
$510$		0			0
$511$	41132.0			0.157521
$512$	184336.	−	184336.i	0.703186	−	0.703186i
$513$	−97200.0	−	97200.0i	−0.369344	−	0.369344i
$514$			75598.0i			0.286144i
$515$		0			0
$516$	247632.			0.930052
$517$	19792.0	−	19792.0i	0.0740472	−	0.0740472i
$518$	−13052.0	−	13052.0i	−0.0486427	−	0.0486427i
$519$			202692.i			0.752492i
$520$		0			0
$521$	−171218.			−0.630774			−0.315387	−	0.948963i	$-0.602134\pi$
$521$	−171218.			−0.630774			−0.315387	+	0.948963i	$0.602134\pi$
$522$	4320.00	−	4320.00i	0.0158541	−	0.0158541i
$523$	332666.	+	332666.i	1.21620	+	1.21620i	0.968952	+	0.247248i	$0.0795262\pi$
$523$	332666.	+	332666.i	1.21620	+	1.21620i	0.247248	+	0.968952i	$0.420474\pi$
$524$			283808.i			1.03362i
$525$		0			0
$526$	67172.0			0.242782
$527$	572.000	−	572.000i	0.00205956	−	0.00205956i
$528$	7872.00	+	7872.00i	0.0282369	+	0.0282369i
$529$		−	224729.i		−	0.803060i
$530$		0			0
$531$	32940.0			0.116825
$532$	65520.0	−	65520.0i	0.231500	−	0.231500i
$533$	234632.	+	234632.i	0.825910	+	0.825910i
$534$		−	25920.0i		−	0.0908976i
$535$		0			0
$536$	−52440.0			−0.182530
$537$	63720.0	−	63720.0i	0.220967	−	0.220967i
$538$	−28530.0	−	28530.0i	−0.0985683	−	0.0985683i
$539$		−	8392.00i		−	0.0288860i
$540$		0			0
$541$	13862.0			0.0473621			0.0236811	−	0.999720i	$-0.492461\pi$
$541$	13862.0			0.0473621			0.0236811	+	0.999720i	$0.492461\pi$
$542$	−7468.00	+	7468.00i	−0.0254218	+	0.0254218i
$543$	144732.	+	144732.i	0.490868	+	0.490868i
$544$			1288.00i			0.00435229i
$545$		0			0
$546$	−86736.0			−0.290947
$547$	180026.	−	180026.i	0.601673	−	0.601673i	−0.339083	−	0.940756i	$-0.610117\pi$
$547$	180026.	−	180026.i	0.601673	−	0.601673i	0.940756	+	0.339083i	$0.110117\pi$
$548$	144914.	+	144914.i	0.482558	+	0.482558i
$549$		−	14328.0i		−	0.0475380i
$550$		0			0
$551$	86400.0			0.284584
$552$	−59760.0	+	59760.0i	−0.196125	+	0.196125i
$553$	−237120.	−	237120.i	−0.775386	−	0.775386i
$554$			12998.0i			0.0423503i
$555$		0			0
$556$	378840.			1.22548
$557$	−422549.	+	422549.i	−1.36197	+	1.36197i	−0.490560	+	0.871408i	$0.663208\pi$
$557$	−422549.	+	422549.i	−1.36197	+	1.36197i	−0.871408	+	0.490560i	$0.836792\pi$
$558$	−5148.00	−	5148.00i	−0.0165337	−	0.0165337i
$559$			409772.i			1.31135i
$560$		0			0
$561$	−96.0000			−0.000305032
$562$	−97928.0	+	97928.0i	−0.310052	+	0.310052i
$563$	−105314.	−	105314.i	−0.332253	−	0.332253i	0.521188	−	0.853442i	$-0.325489\pi$
$563$	−105314.	−	105314.i	−0.332253	−	0.332253i	−0.853442	+	0.521188i	$0.825489\pi$
$564$		−	415632.i		−	1.30662i
$565$		0			0
$566$	−119708.			−0.373672
$567$	149526.	−	149526.i	0.465105	−	0.465105i
$568$	−182040.	−	182040.i	−0.564248	−	0.564248i
$569$		−	85830.0i		−	0.265103i	−0.991176	−	0.132551i	$-0.957683\pi$
$569$		−	85830.0i		−	0.265103i	0.991176	−	0.132551i	$-0.0423170\pi$
$570$		0			0
$571$	−142168.			−0.436043			−0.218022	−	0.975944i	$-0.569960\pi$
$571$	−142168.			−0.436043			−0.218022	+	0.975944i	$0.569960\pi$
$572$	−15568.0	+	15568.0i	−0.0475818	+	0.0475818i
$573$	−271128.	−	271128.i	−0.825781	−	0.825781i
$574$			87776.0i			0.266411i
$575$		0			0
$576$	−12024.0			−0.0362413
$577$	154601.	−	154601.i	0.464366	−	0.464366i	−0.435717	−	0.900084i	$-0.643505\pi$
$577$	154601.	−	154601.i	0.464366	−	0.464366i	0.900084	+	0.435717i	$0.143505\pi$
$578$	83519.0	+	83519.0i	0.249994	+	0.249994i
$579$		−	470388.i		−	1.40313i
$580$		0			0
$581$	−294008.			−0.870977
$582$	78612.0	−	78612.0i	0.232083	−	0.232083i
$583$	−26648.0	−	26648.0i	−0.0784021	−	0.0784021i
$584$			47460.0i			0.139156i
$585$		0			0
$586$	−56998.0			−0.165983
$587$	−222974.	+	222974.i	−0.647110	+	0.647110i	−0.952294	−	0.305184i	$-0.901282\pi$
$587$	−222974.	+	222974.i	−0.647110	+	0.647110i	0.305184	+	0.952294i	$0.401282\pi$
$588$	−88116.0	−	88116.0i	−0.254859	−	0.254859i
$589$		−	102960.i		−	0.296782i
$590$		0			0
$591$	−340188.			−0.973967
$592$	−41164.0	+	41164.0i	−0.117456	+	0.117456i
$593$	−148049.	−	148049.i	−0.421014	−	0.421014i	0.464539	−	0.885553i	$-0.346220\pi$
$593$	−148049.	−	148049.i	−0.421014	−	0.421014i	−0.885553	+	0.464539i	$0.846220\pi$
$594$			8640.00i			0.0244873i
$595$		0			0
$596$	−228900.			−0.644397
$597$	−10800.0	+	10800.0i	−0.0303023	+	0.0303023i
$598$	−46148.0	−	46148.0i	−0.129048	−	0.129048i
$599$		−	31800.0i		−	0.0886285i	−0.999018	−	0.0443143i	$-0.985890\pi$
$599$		−	31800.0i		−	0.0886285i	0.999018	−	0.0443143i	$-0.0141103\pi$
$600$		0			0
$601$	−71848.0			−0.198914			−0.0994571	−	0.995042i	$-0.531711\pi$
$601$	−71848.0			−0.198914			−0.0994571	+	0.995042i	$0.531711\pi$
$602$	−76648.0	+	76648.0i	−0.211499	+	0.211499i
$603$	7866.00	+	7866.00i	0.0216331	+	0.0216331i
$604$			14728.0i			0.0403710i
$605$		0			0
$606$	193224.			0.526158
$607$	13526.0	−	13526.0i	0.0367106	−	0.0367106i	−0.688513	−	0.725224i	$-0.741736\pi$
$607$	13526.0	−	13526.0i	0.0367106	−	0.0367106i	0.725224	+	0.688513i	$0.241736\pi$
$608$	115920.	+	115920.i	0.313582	+	0.313582i
$609$			149760.i			0.403795i
$610$		0			0
$611$	687772.			1.84231
$612$	126.000	−	126.000i	0.000336409	−	0.000336409i
$613$	303461.	+	303461.i	0.807573	+	0.807573i	0.984266	−	0.176693i	$-0.0565399\pi$
$613$	303461.	+	303461.i	0.807573	+	0.807573i	−0.176693	+	0.984266i	$0.556540\pi$
$614$		−	235852.i		−	0.625609i
$615$		0			0
$616$	−12480.0			−0.0328892
$617$	−122399.	+	122399.i	−0.321520	+	0.321520i	−0.849350	−	0.527830i	$-0.823006\pi$
$617$	−122399.	+	122399.i	−0.321520	+	0.321520i	0.527830	+	0.849350i	$0.323006\pi$
$618$	−11928.0	−	11928.0i	−0.0312313	−	0.0312313i
$619$			110220.i			0.287660i	0.989602	+	0.143830i	$0.0459418\pi$
$619$			110220.i			0.287660i	−0.989602	+	0.143830i	$0.954058\pi$
$620$		0			0
$621$	179280.			0.464888
$622$	3892.00	−	3892.00i	0.0100599	−	0.0100599i
$623$	−56160.0	−	56160.0i	−0.144694	−	0.144694i
$624$			273552.i			0.702539i
$625$		0			0
$626$	99922.0			0.254984
$627$	−8640.00	+	8640.00i	−0.0219775	+	0.0219775i
$628$	−90986.0	−	90986.0i	−0.230704	−	0.230704i
$629$		−	502.000i		−	0.00126883i
$630$		0			0
$631$	620372.			1.55809			0.779047	−	0.626966i	$-0.215703\pi$
$631$	620372.			1.55809			0.779047	+	0.626966i	$0.215703\pi$
$632$	273600.	−	273600.i	0.684986	−	0.684986i
$633$	110352.	+	110352.i	0.275406	+	0.275406i
$634$			34198.0i			0.0850790i
$635$		0			0
$636$	−559608.			−1.38347
$637$	145811.	−	145811.i	0.359345	−	0.359345i
$638$	−3840.00	−	3840.00i	−0.00943387	−	0.00943387i
$639$			54612.0i			0.133748i
$640$		0			0
$641$	−722888.			−1.75936			−0.879680	−	0.475565i	$-0.842244\pi$
$641$	−722888.			−1.75936			−0.879680	+	0.475565i	$0.842244\pi$
$642$	99912.0	−	99912.0i	0.242408	−	0.242408i
$643$	393026.	+	393026.i	0.950603	+	0.950603i	0.998836	−	0.0482328i	$-0.0153589\pi$
$643$	393026.	+	393026.i	0.950603	+	0.950603i	−0.0482328	+	0.998836i	$0.515359\pi$
$644$			120848.i			0.291385i
$645$		0			0
$646$	−360.000			−0.000862656
$647$	338626.	−	338626.i	0.808931	−	0.808931i	−0.175541	−	0.984472i	$-0.556167\pi$
$647$	338626.	−	338626.i	0.808931	−	0.808931i	0.984472	+	0.175541i	$0.0561674\pi$
$648$	172530.	+	172530.i	0.410880	+	0.410880i
$649$		−	29280.0i		−	0.0695155i
$650$		0			0
$651$	178464.			0.421103
$652$	−270004.	+	270004.i	−0.635148	+	0.635148i
$653$	−254669.	−	254669.i	−0.597241	−	0.597241i	0.342336	−	0.939577i	$-0.388782\pi$
$653$	−254669.	−	254669.i	−0.597241	−	0.597241i	−0.939577	+	0.342336i	$0.888782\pi$
$654$			204120.i			0.477233i
$655$		0			0
$656$	276832.			0.643293
$657$	7119.00	−	7119.00i	0.0164926	−	0.0164926i
$658$	128648.	+	128648.i	0.297133	+	0.297133i
$659$			603660.i			1.39002i	0.718999	+	0.695011i	$0.244600\pi$
$659$			603660.i			1.39002i	−0.718999	+	0.695011i	$0.755400\pi$
$660$		0			0
$661$	14792.0			0.0338551			0.0169275	−	0.999857i	$-0.494612\pi$
$661$	14792.0			0.0338551			0.0169275	+	0.999857i	$0.494612\pi$
$662$	−143128.	+	143128.i	−0.326594	+	0.326594i
$663$	−1668.00	−	1668.00i	−0.00379463	−	0.00379463i
$664$		−	339240.i		−	0.769433i
$665$		0			0
$666$	−4518.00			−0.0101859
$667$	−79680.0	+	79680.0i	−0.179101	+	0.179101i
$668$	217364.	+	217364.i	0.487119	+	0.487119i
$669$			10392.0i			0.0232192i
$670$		0			0
$671$	−12736.0			−0.0282871
$672$	−200928.	+	200928.i	−0.444940	+	0.444940i
$673$	−300409.	−	300409.i	−0.663258	−	0.663258i	0.292888	−	0.956147i	$-0.405384\pi$
$673$	−300409.	−	300409.i	−0.663258	−	0.663258i	−0.956147	+	0.292888i	$0.905384\pi$
$674$			206498.i			0.454565i
$675$		0			0
$676$	−141134.			−0.308843
$677$	256051.	−	256051.i	0.558662	−	0.558662i	−0.370264	−	0.928926i	$-0.620733\pi$
$677$	256051.	−	256051.i	0.558662	−	0.558662i	0.928926	+	0.370264i	$0.120733\pi$
$678$	61932.0	+	61932.0i	0.134727	+	0.134727i
$679$		−	340652.i		−	0.738876i
$680$		0			0
$681$	542712.			1.17024
$682$	−4576.00	+	4576.00i	−0.00983824	+	0.00983824i
$683$	−341954.	−	341954.i	−0.733038	−	0.733038i	0.238183	−	0.971220i	$-0.423448\pi$
$683$	−341954.	−	341954.i	−0.733038	−	0.733038i	−0.971220	+	0.238183i	$0.923448\pi$
$684$		−	22680.0i		−	0.0484765i
$685$		0			0
$686$	179400.			0.381219
$687$	234720.	−	234720.i	0.497321	−	0.497321i
$688$	241736.	+	241736.i	0.510698	+	0.510698i
$689$		−	926018.i		−	1.95066i
$690$		0			0
$691$	−717688.			−1.50307			−0.751536	−	0.659692i	$-0.770687\pi$
$691$	−717688.			−1.50307			−0.751536	+	0.659692i	$0.770687\pi$
$692$	−236474.	+	236474.i	−0.493823	+	0.493823i
$693$	1872.00	+	1872.00i	0.00389798	+	0.00389798i
$694$		−	209252.i		−	0.434461i
$695$		0			0
$696$	−172800.			−0.356718
$697$	−1688.00	+	1688.00i	−0.00347462	+	0.00347462i
$698$	−94800.0	−	94800.0i	−0.194580	−	0.194580i
$699$			397452.i			0.813449i
$700$		0			0
$701$	267352.			0.544061			0.272030	−	0.962289i	$-0.412305\pi$
$701$	267352.			0.544061			0.272030	+	0.962289i	$0.412305\pi$
$702$	−150120.	+	150120.i	−0.304624	+	0.304624i
$703$	−45180.0	−	45180.0i	−0.0914188	−	0.0914188i
$704$			10688.0i			0.0215651i
$705$		0			0
$706$	−127138.			−0.255074
$707$	418652.	−	418652.i	0.837557	−	0.837557i
$708$	−307440.	−	307440.i	−0.613330	−	0.613330i
$709$			159360.i			0.317020i	0.987357	+	0.158510i	$0.0506690\pi$
$709$			159360.i			0.317020i	−0.987357	+	0.158510i	$0.949331\pi$
$710$		0			0
$711$	−82080.0			−0.162367
$712$	64800.0	−	64800.0i	0.127825	−	0.127825i
$713$	94952.0	+	94952.0i	0.186778	+	0.186778i
$714$		−	624.000i		−	0.00122402i
$715$		0			0
$716$	148680.			0.290019
$717$	−530640.	+	530640.i	−1.03219	+	1.03219i
$718$	−141840.	−	141840.i	−0.275138	−	0.275138i
$719$		−	364680.i		−	0.705430i	−0.935731	−	0.352715i	$-0.885258\pi$
$719$		−	364680.i		−	0.705430i	0.935731	−	0.352715i	$-0.114742\pi$
$720$		0			0
$721$	−51688.0			−0.0994304
$722$	97921.0	−	97921.0i	0.187846	−	0.187846i
$723$	121872.	+	121872.i	0.233146	+	0.233146i
$724$			337708.i			0.644265i
$725$		0			0
$726$	−174924.			−0.331876
$727$	−716374.	+	716374.i	−1.35541	+	1.35541i	−0.475925	+	0.879486i	$0.657887\pi$
$727$	−716374.	+	716374.i	−1.35541	+	1.35541i	−0.879486	+	0.475925i	$0.842113\pi$
$728$	−216840.	−	216840.i	−0.409144	−	0.409144i
$729$		−	576639.i		−	1.08505i
$730$		0			0
$731$	−2948.00			−0.00551687
$732$	−133728.	+	133728.i	−0.249574	+	0.249574i
$733$	−641029.	−	641029.i	−1.19308	−	1.19308i	−0.976198	−	0.216883i	$-0.930411\pi$
$733$	−641029.	−	641029.i	−1.19308	−	1.19308i	−0.216883	−	0.976198i	$-0.569589\pi$
$734$			284348.i			0.527786i
$735$		0			0
$736$	−213808.			−0.394701
$737$	6992.00	−	6992.00i	0.0128726	−	0.0128726i
$738$	15192.0	+	15192.0i	0.0278934	+	0.0278934i
$739$			607140.i			1.11173i	0.831272	+	0.555866i	$0.187613\pi$
$739$			607140.i			1.11173i	−0.831272	+	0.555866i	$0.812387\pi$
$740$		0			0
$741$	−300240.			−0.546805
$742$	173212.	−	173212.i	0.314608	−	0.314608i
$743$	248026.	+	248026.i	0.449283	+	0.449283i	0.895116	−	0.445833i	$-0.147093\pi$
$743$	248026.	+	248026.i	0.449283	+	0.449283i	−0.445833	+	0.895116i	$0.647093\pi$
$744$			205920.i			0.372008i
$745$		0			0
$746$	−20618.0			−0.0370484
$747$	−50886.0	+	50886.0i	−0.0911921	+	0.0911921i
$748$	−112.000	−	112.000i	−0.000200177	−	0.000200177i
$749$		−	432952.i		−	0.771749i
$750$		0			0
$751$	169052.			0.299737			0.149869	−	0.988706i	$-0.452115\pi$
$751$	169052.			0.299737			0.149869	+	0.988706i	$0.452115\pi$
$752$	405736.	−	405736.i	0.717477	−	0.717477i
$753$	448512.	+	448512.i	0.791014	+	0.791014i
$754$		−	133440.i		−	0.234716i
$755$		0			0
$756$	393120.			0.687831
$757$	166301.	−	166301.i	0.290204	−	0.290204i	−0.546957	−	0.837161i	$-0.684214\pi$
$757$	166301.	−	166301.i	0.290204	−	0.290204i	0.837161	+	0.546957i	$0.184214\pi$
$758$	115380.	+	115380.i	0.200813	+	0.200813i
$759$		−	15936.0i		−	0.0276628i
$760$		0			0
$761$	557842.			0.963256			0.481628	−	0.876376i	$-0.340046\pi$
$761$	557842.			0.963256			0.481628	+	0.876376i	$0.340046\pi$
$762$	−150888.	+	150888.i	−0.259863	+	0.259863i
$763$	442260.	+	442260.i	0.759676	+	0.759676i
$764$		−	632632.i		−	1.08384i
$765$		0			0
$766$	−125308.			−0.213561
$767$	508740.	−	508740.i	0.864779	−	0.864779i
$768$	−11424.0	−	11424.0i	−0.0193685	−	0.0193685i
$769$			678720.i			1.14773i	0.818952	+	0.573863i	$0.194556\pi$
$769$			678720.i			1.14773i	−0.818952	+	0.573863i	$0.805444\pi$
$770$		0			0
$771$	−453588.			−0.763050
$772$	548786.	−	548786.i	0.920807	−	0.920807i
$773$	164341.	+	164341.i	0.275034	+	0.275034i	0.831123	−	0.556089i	$-0.187699\pi$
$773$	164341.	+	164341.i	0.275034	+	0.275034i	−0.556089	+	0.831123i	$0.687699\pi$
$774$			26532.0i			0.0442882i
$775$		0			0
$776$	393060.			0.652733
$777$	78312.0	−	78312.0i	0.129714	−	0.129714i
$778$	132690.	+	132690.i	0.219219	+	0.219219i
$779$			303840.i			0.500691i
$780$		0			0
$781$	48544.0			0.0795854
$782$	332.000	−	332.000i	0.000542906	−	0.000542906i
$783$	259200.	+	259200.i	0.422777	+	0.422777i
$784$		−	172036.i		−	0.279890i
$785$		0			0
$786$	243264.			0.393761
$787$	−323074.	+	323074.i	−0.521618	+	0.521618i	−0.918060	−	0.396442i	$-0.870245\pi$
$787$	−323074.	+	323074.i	−0.521618	+	0.521618i	0.396442	+	0.918060i	$0.370245\pi$
$788$	−396886.	−	396886.i	−0.639166	−	0.639166i
$789$			403032.i			0.647419i
$790$		0			0
$791$	268372.			0.428928
$792$	−2160.00	+	2160.00i	−0.00344353	+	0.00344353i
$793$	−221288.	−	221288.i	−0.351894	−	0.351894i
$794$		−	176902.i		−	0.280603i
$795$		0			0
$796$	−25200.0			−0.0397717
$797$	−510299.	+	510299.i	−0.803356	+	0.803356i	−0.983619	−	0.180262i	$-0.942305\pi$
$797$	−510299.	+	510299.i	−0.803356	+	0.803356i	0.180262	+	0.983619i	$0.442305\pi$
$798$	−56160.0	−	56160.0i	−0.0881904	−	0.0881904i
$799$			4948.00i			0.00775061i
$800$		0			0
$801$	−19440.0			−0.0302992
$802$	63202.0	−	63202.0i	0.0982612	−	0.0982612i
$803$	−6328.00	−	6328.00i	−0.00981376	−	0.00981376i
$804$		−	146832.i		−	0.227148i
$805$		0			0
$806$	−159016.			−0.244777
$807$	171180.	−	171180.i	0.262849	−	0.262849i
$808$	483060.	+	483060.i	0.739909	+	0.739909i
$809$			1.18656e6i			1.81298i	0.422229	+	0.906489i	$0.361248\pi$
$809$			1.18656e6i			1.81298i	−0.422229	+	0.906489i	$0.638752\pi$
$810$		0			0
$811$	−431008.			−0.655305			−0.327653	−	0.944798i	$-0.606258\pi$
$811$	−431008.			−0.655305			−0.327653	+	0.944798i	$0.606258\pi$
$812$	−174720.	+	174720.i	−0.264991	+	0.264991i
$813$	−44808.0	−	44808.0i	−0.0677914	−	0.0677914i
$814$			4016.00i			0.00606101i
$815$		0			0
$816$	−1968.00			−0.00295559
$817$	−265320.	+	265320.i	−0.397490	+	0.397490i
$818$	−52890.0	−	52890.0i	−0.0790436	−	0.0790436i
$819$			65052.0i			0.0969824i
$820$		0			0
$821$	−639368.			−0.948560			−0.474280	−	0.880374i	$-0.657291\pi$
$821$	−639368.			−0.948560			−0.474280	+	0.880374i	$0.657291\pi$
$822$	124212.	−	124212.i	0.183831	−	0.183831i
$823$	74966.0	+	74966.0i	0.110679	+	0.110679i	0.760277	−	0.649599i	$-0.225063\pi$
$823$	74966.0	+	74966.0i	0.110679	+	0.110679i	−0.649599	+	0.760277i	$0.725063\pi$
$824$		−	59640.0i		−	0.0878382i
$825$		0			0
$826$	190320.			0.278949
$827$	144226.	−	144226.i	0.210879	−	0.210879i	−0.593762	−	0.804641i	$-0.702358\pi$
$827$	144226.	−	144226.i	0.210879	−	0.210879i	0.804641	+	0.593762i	$0.202358\pi$
$828$	20916.0	+	20916.0i	0.0305083	+	0.0305083i
$829$		−	685170.i		−	0.996987i	−0.866894	−	0.498493i	$-0.833887\pi$
$829$		−	685170.i		−	0.996987i	0.866894	−	0.498493i	$-0.166113\pi$
$830$		0			0
$831$	−77988.0			−0.112934
$832$	−185704.	+	185704.i	−0.268272	+	0.268272i
$833$	1049.00	+	1049.00i	0.00151177	+	0.00151177i
$834$		−	324720.i		−	0.466850i
$835$		0			0
$836$	−20160.0			−0.0288455
$837$	308880.	−	308880.i	0.440899	−	0.440899i
$838$	−256980.	−	256980.i	−0.365941	−	0.365941i
$839$		−	445560.i		−	0.632969i	−0.948598	−	0.316484i	$-0.897498\pi$
$839$		−	445560.i		−	0.632969i	0.948598	−	0.316484i	$-0.102502\pi$
$840$		0			0
$841$	476881.			0.674245
$842$	186632.	−	186632.i	0.263246	−	0.263246i
$843$	−587568.	−	587568.i	−0.826805	−	0.826805i
$844$			257488.i			0.361470i
$845$		0			0
$846$	44532.0			0.0622202
$847$	−379002.	+	379002.i	−0.528293	+	0.528293i
$848$	−546284.	−	546284.i	−0.759673	−	0.759673i
$849$		−	718248.i		−	0.996458i
$850$		0			0
$851$	83332.0			0.115068
$852$	509712.	−	509712.i	0.702175	−	0.702175i
$853$	319781.	+	319781.i	0.439496	+	0.439496i	0.891842	−	0.452347i	$-0.149413\pi$
$853$	319781.	+	319781.i	0.439496	+	0.439496i	−0.452347	+	0.891842i	$0.649413\pi$
$854$		−	82784.0i		−	0.113509i
$855$		0			0
$856$	499560.			0.681774
$857$	−576449.	+	576449.i	−0.784873	+	0.784873i	−0.980649	−	0.195776i	$-0.937277\pi$
$857$	−576449.	+	576449.i	−0.784873	+	0.784873i	0.195776	+	0.980649i	$0.437277\pi$
$858$	13344.0	+	13344.0i	0.0181264	+	0.0181264i
$859$		−	807540.i		−	1.09440i	−0.837001	−	0.547202i	$-0.815693\pi$
$859$		−	807540.i		−	1.09440i	0.837001	−	0.547202i	$-0.184307\pi$
$860$		0			0
$861$	−526656.			−0.710429
$862$	−208028.	+	208028.i	−0.279967	+	0.279967i
$863$	−507914.	−	507914.i	−0.681975	−	0.681975i	0.278470	−	0.960445i	$-0.410173\pi$
$863$	−507914.	−	507914.i	−0.681975	−	0.681975i	−0.960445	+	0.278470i	$0.910173\pi$
$864$			695520.i			0.931713i
$865$		0			0
$866$	302542.			0.403413
$867$	−501114.	+	501114.i	−0.666651	+	0.666651i
$868$	208208.	+	208208.i	0.276349	+	0.276349i
$869$			72960.0i			0.0966152i
$870$		0			0
$871$	242972.			0.320273
$872$	−510300.	+	510300.i	−0.671108	+	0.671108i
$873$	−58959.0	−	58959.0i	−0.0773609	−	0.0773609i
$874$		−	59760.0i		−	0.0782326i
$875$		0			0
$876$	−132888.			−0.173172
$877$	673901.	−	673901.i	0.876187	−	0.876187i	−0.116951	−	0.993138i	$-0.537312\pi$
$877$	673901.	−	673901.i	0.876187	−	0.876187i	0.993138	+	0.116951i	$0.0373119\pi$
$878$	158640.	+	158640.i	0.205790	+	0.205790i
$879$		−	341988.i		−	0.442622i
$880$		0			0
$881$	−1.33753e6			−1.72326			−0.861631	−	0.507536i	$-0.830556\pi$
$881$	−1.33753e6			−1.72326			−0.861631	+	0.507536i	$0.830556\pi$
$882$	9441.00	−	9441.00i	0.0121361	−	0.0121361i
$883$	−131554.	−	131554.i	−0.168726	−	0.168726i	0.617693	−	0.786419i	$-0.288067\pi$
$883$	−131554.	−	131554.i	−0.168726	−	0.168726i	−0.786419	+	0.617693i	$0.788067\pi$
$884$		−	3892.00i		−	0.00498045i
$885$		0			0
$886$	−505948.			−0.644523
$887$	327826.	−	327826.i	0.416674	−	0.416674i	−0.467382	−	0.884056i	$-0.654803\pi$
$887$	327826.	−	327826.i	0.416674	−	0.416674i	0.884056	+	0.467382i	$0.154803\pi$
$888$	90360.0	+	90360.0i	0.114591	+	0.114591i
$889$			653848.i			0.827320i
$890$		0			0
$891$	−46008.0			−0.0579533
$892$	−12124.0	+	12124.0i	−0.0152376	+	0.0152376i
$893$	445320.	+	445320.i	0.558431	+	0.558431i
$894$			196200.i			0.245484i
$895$		0			0
$896$	−605280.			−0.753946
$897$	276888.	−	276888.i	0.344127	−	0.344127i
$898$	−123750.	−	123750.i	−0.153459	−	0.153459i
$899$			274560.i			0.339717i
$900$		0			0
$901$	6662.00			0.00820644
$902$	13504.0	−	13504.0i	0.0165978	−	0.0165978i
$903$	−459888.	−	459888.i	−0.563997	−	0.563997i
$904$			309660.i			0.378921i
$905$		0			0
$906$	12624.0			0.0153794
$907$	−105274.	+	105274.i	−0.127970	+	0.127970i	−0.768191	−	0.640221i	$-0.778843\pi$
$907$	−105274.	+	105274.i	−0.127970	+	0.127970i	0.640221	+	0.768191i	$0.278843\pi$
$908$	633164.	+	633164.i	0.767970	+	0.767970i
$909$		−	144918.i		−	0.175386i
$910$		0			0
$911$	1.59209e6			1.91837			0.959183	−	0.282787i	$-0.0912588\pi$
$911$	1.59209e6			1.91837			0.959183	+	0.282787i	$0.0912588\pi$
$912$	−177120.	+	177120.i	−0.212950	+	0.212950i
$913$	45232.0	+	45232.0i	0.0542631	+	0.0542631i
$914$		−	64402.0i		−	0.0770916i
$915$		0			0
$916$	547680.			0.652734
$917$	527072.	−	527072.i	0.626803	−	0.626803i
$918$	−1080.00	−	1080.00i	−0.00128156	−	0.00128156i
$919$		−	515880.i		−	0.610826i	−0.952220	−	0.305413i	$-0.901205\pi$
$919$		−	515880.i		−	0.610826i	0.952220	−	0.305413i	$-0.0987946\pi$
$920$		0			0
$921$	1.41511e6			1.66829
$922$	75142.0	−	75142.0i	0.0883936	−	0.0883936i
$923$	843452.	+	843452.i	0.990050	+	0.990050i
$924$		−	34944.0i		−	0.0409288i
$925$		0			0
$926$	−471428.			−0.549786
$927$	−8946.00	+	8946.00i	−0.0104104	+	0.0104104i
$928$	−309120.	−	309120.i	−0.358948	−	0.358948i
$929$			1.60023e6i			1.85418i	0.374844	+	0.927088i	$0.377696\pi$
$929$			1.60023e6i			1.85418i	−0.374844	+	0.927088i	$0.622304\pi$
$930$		0			0
$931$	188820.			0.217846
$932$	−463694.	+	463694.i	−0.533826	+	0.533826i
$933$	23352.0	+	23352.0i	0.0268263	+	0.0268263i
$934$			57148.0i			0.0655100i
$935$		0			0
$936$	−75060.0			−0.0856755
$937$	908801.	−	908801.i	1.03512	−	1.03512i	0.0357569	−	0.999361i	$-0.488616\pi$
$937$	908801.	−	908801.i	1.03512	−	1.03512i	0.999361	−	0.0357569i	$-0.0113842\pi$
$938$	45448.0	+	45448.0i	0.0516546	+	0.0516546i
$939$			599532.i			0.679957i
$940$		0			0
$941$	455962.			0.514931			0.257466	−	0.966287i	$-0.417113\pi$
$941$	455962.			0.514931			0.257466	+	0.966287i	$0.417113\pi$
$942$	−77988.0	+	77988.0i	−0.0878873	+	0.0878873i
$943$	−280208.	−	280208.i	−0.315106	−	0.315106i
$944$		−	600240.i		−	0.673567i
$945$		0			0
$946$	23584.0			0.0263533
$947$	−463274.	+	463274.i	−0.516580	+	0.516580i	−0.916535	−	0.399955i	$-0.869026\pi$
$947$	−463274.	+	463274.i	−0.516580	+	0.516580i	0.399955	+	0.916535i	$0.369026\pi$
$948$	766080.	+	766080.i	0.852427	+	0.852427i
$949$		−	219898.i		−	0.244168i
$950$		0			0
$951$	−205188.			−0.226877
$952$	1560.00	−	1560.00i	0.00172128	−	0.00172128i
$953$	1.03428e6	+	1.03428e6i	1.13881	+	1.13881i	0.988663	+	0.150151i	$0.0479759\pi$
$953$	1.03428e6	+	1.03428e6i	1.13881	+	1.13881i	0.150151	+	0.988663i	$0.452024\pi$
$954$		−	59958.0i		−	0.0658795i
$955$		0			0
$956$	−1.23816e6			−1.35476
$957$	23040.0	−	23040.0i	0.0251570	−	0.0251570i
$958$	−26520.0	−	26520.0i	−0.0288963	−	0.0288963i
$959$		−	538252.i		−	0.585259i
$960$		0			0
$961$	−596337.			−0.645721
$962$	−69778.0	+	69778.0i	−0.0753995	+	0.0753995i
$963$	−74934.0	−	74934.0i	−0.0808028	−	0.0808028i
$964$			284368.i			0.306004i
$965$		0			0
$966$	103584.			0.111004
$967$	1.10253e6	−	1.10253e6i	1.17906	−	1.17906i	0.199076	−	0.979984i	$-0.436206\pi$
$967$	1.10253e6	−	1.10253e6i	1.17906	−	1.17906i	0.979984	−	0.199076i	$-0.0637940\pi$
$968$	−437310.	−	437310.i	−0.466701	−	0.466701i
$969$		−	2160.00i		−	0.00230042i
$970$		0			0
$971$	−264368.			−0.280395			−0.140198	−	0.990124i	$-0.544774\pi$
$971$	−264368.			−0.280395			−0.140198	+	0.990124i	$0.544774\pi$
$972$	129276.	−	129276.i	0.136831	−	0.136831i
$973$	−703560.	−	703560.i	−0.743148	−	0.743148i
$974$			82748.0i			0.0872247i
$975$		0			0
$976$	−261088.			−0.274086
$977$	−866249.	+	866249.i	−0.907515	+	0.907515i	−0.996071	−	0.0885566i	$-0.971775\pi$
$977$	−866249.	+	866249.i	−0.907515	+	0.907515i	0.0885566	+	0.996071i	$0.471775\pi$
$978$	231432.	+	231432.i	0.241961	+	0.241961i
$979$			17280.0i			0.0180293i
$980$		0			0
$981$	153090.			0.159078
$982$	−149288.	+	149288.i	−0.154811	+	0.154811i
$983$	−1.08205e6	−	1.08205e6i	−1.11980	−	1.11980i	−0.991770	−	0.128034i	$-0.959133\pi$
$983$	−1.08205e6	−	1.08205e6i	−1.11980	−	1.11980i	−0.128034	−	0.991770i	$-0.540867\pi$
$984$		−	607680.i		−	0.627603i
$985$		0			0
$986$	960.000			0.000987455
$987$	−771888.	+	771888.i	−0.792355	+	0.792355i
$988$	−350280.	−	350280.i	−0.358840	−	0.358840i
$989$		−	489368.i		−	0.500314i
$990$		0			0
$991$	−54988.0			−0.0559913			−0.0279957	−	0.999608i	$-0.508912\pi$
$991$	−54988.0			−0.0559913			−0.0279957	+	0.999608i	$0.508912\pi$
$992$	−368368.	+	368368.i	−0.374333	+	0.374333i
$993$	−858768.	−	858768.i	−0.870918	−	0.870918i
$994$			315536.i			0.319357i
$995$		0			0
$996$	949872.			0.957517
$997$	−73099.0	+	73099.0i	−0.0735396	+	0.0735396i	−0.742920	−	0.669380i	$-0.766560\pi$
$997$	−73099.0	+	73099.0i	−0.0735396	+	0.0735396i	0.669380	+	0.742920i	$0.266560\pi$
$998$	−284100.	−	284100.i	−0.285240	−	0.285240i
$999$		−	271080.i		−	0.271623i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.5.c.a.18.1		2
3.2	odd	2		225.5.g.b.118.1		2
4.3	odd	2		400.5.p.a.193.1		2
5.2	odd	4	inner	25.5.c.a.7.1		2
5.3	odd	4		5.5.c.a.2.1	✓	2
5.4	even	2		5.5.c.a.3.1	yes	2
15.2	even	4		225.5.g.b.82.1		2
15.8	even	4		45.5.g.b.37.1		2
15.14	odd	2		45.5.g.b.28.1		2
20.3	even	4		80.5.p.d.17.1		2
20.7	even	4		400.5.p.a.257.1		2
20.19	odd	2		80.5.p.d.33.1		2
40.3	even	4		320.5.p.c.257.1		2
40.13	odd	4		320.5.p.h.257.1		2
40.19	odd	2		320.5.p.c.193.1		2
40.29	even	2		320.5.p.h.193.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.5.c.a.2.1	✓	2	5.3	odd	4
5.5.c.a.3.1	yes	2	5.4	even	2
25.5.c.a.7.1		2	5.2	odd	4	inner
25.5.c.a.18.1		2	1.1	even	1	trivial
45.5.g.b.28.1		2	15.14	odd	2
45.5.g.b.37.1		2	15.8	even	4
80.5.p.d.17.1		2	20.3	even	4
80.5.p.d.33.1		2	20.19	odd	2
225.5.g.b.82.1		2	15.2	even	4
225.5.g.b.118.1		2	3.2	odd	2
320.5.p.c.193.1		2	40.19	odd	2
320.5.p.c.257.1		2	40.3	even	4
320.5.p.h.193.1		2	40.29	even	2
320.5.p.h.257.1		2	40.13	odd	4
400.5.p.a.193.1		2	4.3	odd	2
400.5.p.a.257.1		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	25.c (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} +(6.00000 + 6.00000i) q^{3} +14.0000i q^{4} +12.0000 q^{6} +(26.0000 - 26.0000i) q^{7} +(30.0000 + 30.0000i) q^{8} -9.00000i q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.00000i) q^{2} +(6.00000 + 6.00000i) q^{3} +14.0000i q^{4} +12.0000 q^{6} +(26.0000 - 26.0000i) q^{7} +(30.0000 + 30.0000i) q^{8} -9.00000i q^{9} -8.00000 q^{11} +(-84.0000 + 84.0000i) q^{12} +(-139.000 - 139.000i) q^{13} -52.0000i q^{14} -164.000 q^{16} +(1.00000 - 1.00000i) q^{17} +(-9.00000 - 9.00000i) q^{18} -180.000i q^{19} +312.000 q^{21} +(-8.00000 + 8.00000i) q^{22} +(166.000 + 166.000i) q^{23} +360.000i q^{24} -278.000 q^{26} +(540.000 - 540.000i) q^{27} +(364.000 + 364.000i) q^{28} +480.000i q^{29} +572.000 q^{31} +(-644.000 + 644.000i) q^{32} +(-48.0000 - 48.0000i) q^{33} -2.00000i q^{34} +126.000 q^{36} +(251.000 - 251.000i) q^{37} +(-180.000 - 180.000i) q^{38} -1668.00i q^{39} -1688.00 q^{41} +(312.000 - 312.000i) q^{42} +(-1474.00 - 1474.00i) q^{43} -112.000i q^{44} +332.000 q^{46} +(-2474.00 + 2474.00i) q^{47} +(-984.000 - 984.000i) q^{48} +1049.00i q^{49} +12.0000 q^{51} +(1946.00 - 1946.00i) q^{52} +(3331.00 + 3331.00i) q^{53} -1080.00i q^{54} +1560.00 q^{56} +(1080.00 - 1080.00i) q^{57} +(480.000 + 480.000i) q^{58} +3660.00i q^{59} +1592.00 q^{61} +(572.000 - 572.000i) q^{62} +(-234.000 - 234.000i) q^{63} -1336.00i q^{64} -96.0000 q^{66} +(-874.000 + 874.000i) q^{67} +(14.0000 + 14.0000i) q^{68} +1992.00i q^{69} -6068.00 q^{71} +(270.000 - 270.000i) q^{72} +(791.000 + 791.000i) q^{73} -502.000i q^{74} +2520.00 q^{76} +(-208.000 + 208.000i) q^{77} +(-1668.00 - 1668.00i) q^{78} -9120.00i q^{79} +5751.00 q^{81} +(-1688.00 + 1688.00i) q^{82} +(-5654.00 - 5654.00i) q^{83} +4368.00i q^{84} -2948.00 q^{86} +(-2880.00 + 2880.00i) q^{87} +(-240.000 - 240.000i) q^{88} -2160.00i q^{89} -7228.00 q^{91} +(-2324.00 + 2324.00i) q^{92} +(3432.00 + 3432.00i) q^{93} +4948.00i q^{94} -7728.00 q^{96} +(6551.00 - 6551.00i) q^{97} +(1049.00 + 1049.00i) q^{98} +72.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 12 q^{3} + 24 q^{6} + 52 q^{7} + 60 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 12 * q^3 + 24 * q^6 + 52 * q^7 + 60 * q^8	\( 2 q + 2 q^{2} + 12 q^{3} + 24 q^{6} + 52 q^{7} + 60 q^{8} - 16 q^{11} - 168 q^{12} - 278 q^{13} - 328 q^{16} + 2 q^{17} - 18 q^{18} + 624 q^{21} - 16 q^{22} + 332 q^{23} - 556 q^{26} + 1080 q^{27} + 728 q^{28} + 1144 q^{31} - 1288 q^{32} - 96 q^{33} + 252 q^{36} + 502 q^{37} - 360 q^{38} - 3376 q^{41} + 624 q^{42} - 2948 q^{43} + 664 q^{46} - 4948 q^{47} - 1968 q^{48} + 24 q^{51} + 3892 q^{52} + 6662 q^{53} + 3120 q^{56} + 2160 q^{57} + 960 q^{58} + 3184 q^{61} + 1144 q^{62} - 468 q^{63} - 192 q^{66} - 1748 q^{67} + 28 q^{68} - 12136 q^{71} + 540 q^{72} + 1582 q^{73} + 5040 q^{76} - 416 q^{77} - 3336 q^{78} + 11502 q^{81} - 3376 q^{82} - 11308 q^{83} - 5896 q^{86} - 5760 q^{87} - 480 q^{88} - 14456 q^{91} - 4648 q^{92} + 6864 q^{93} - 15456 q^{96} + 13102 q^{97} + 2098 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 12 * q^3 + 24 * q^6 + 52 * q^7 + 60 * q^8 - 16 * q^11 - 168 * q^12 - 278 * q^13 - 328 * q^16 + 2 * q^17 - 18 * q^18 + 624 * q^21 - 16 * q^22 + 332 * q^23 - 556 * q^26 + 1080 * q^27 + 728 * q^28 + 1144 * q^31 - 1288 * q^32 - 96 * q^33 + 252 * q^36 + 502 * q^37 - 360 * q^38 - 3376 * q^41 + 624 * q^42 - 2948 * q^43 + 664 * q^46 - 4948 * q^47 - 1968 * q^48 + 24 * q^51 + 3892 * q^52 + 6662 * q^53 + 3120 * q^56 + 2160 * q^57 + 960 * q^58 + 3184 * q^61 + 1144 * q^62 - 468 * q^63 - 192 * q^66 - 1748 * q^67 + 28 * q^68 - 12136 * q^71 + 540 * q^72 + 1582 * q^73 + 5040 * q^76 - 416 * q^77 - 3336 * q^78 + 11502 * q^81 - 3376 * q^82 - 11308 * q^83 - 5896 * q^86 - 5760 * q^87 - 480 * q^88 - 14456 * q^91 - 4648 * q^92 + 6864 * q^93 - 15456 * q^96 + 13102 * q^97 + 2098 * q^98

Introduction

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