LMFDB - Embedded newform 325.4.d.b.324.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,4,Mod(324,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("325.324");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	325.d (of order $2$, degree $1$, not 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:	$19.1756207519$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	325.324
Dual form			325.4.d.b.324.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -3.00000i q^{6} -15.0000 q^{7} -21.0000 q^{8} +26.0000 q^{9} +O(q^{10})$	\(q+3.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -3.00000i q^{6} -15.0000 q^{7} -21.0000 q^{8} +26.0000 q^{9} +48.0000i q^{11} -1.00000i q^{12} +(39.0000 + 26.0000i) q^{13} -45.0000 q^{14} -71.0000 q^{16} +45.0000i q^{17} +78.0000 q^{18} -6.00000i q^{19} +15.0000i q^{21} +144.000i q^{22} +162.000i q^{23} +21.0000i q^{24} +(117.000 + 78.0000i) q^{26} -53.0000i q^{27} -15.0000 q^{28} +144.000 q^{29} +264.000i q^{31} -45.0000 q^{32} +48.0000 q^{33} +135.000i q^{34} +26.0000 q^{36} -303.000 q^{37} -18.0000i q^{38} +(26.0000 - 39.0000i) q^{39} -192.000i q^{41} +45.0000i q^{42} -97.0000i q^{43} +48.0000i q^{44} +486.000i q^{46} -111.000 q^{47} +71.0000i q^{48} -118.000 q^{49} +45.0000 q^{51} +(39.0000 + 26.0000i) q^{52} -414.000i q^{53} -159.000i q^{54} +315.000 q^{56} -6.00000 q^{57} +432.000 q^{58} +522.000i q^{59} +376.000 q^{61} +792.000i q^{62} -390.000 q^{63} +433.000 q^{64} +144.000 q^{66} -36.0000 q^{67} +45.0000i q^{68} +162.000 q^{69} +357.000i q^{71} -546.000 q^{72} -1098.00 q^{73} -909.000 q^{74} -6.00000i q^{76} -720.000i q^{77} +(78.0000 - 117.000i) q^{78} +830.000 q^{79} +649.000 q^{81} -576.000i q^{82} +438.000 q^{83} +15.0000i q^{84} -291.000i q^{86} -144.000i q^{87} -1008.00i q^{88} -438.000i q^{89} +(-585.000 - 390.000i) q^{91} +162.000i q^{92} +264.000 q^{93} -333.000 q^{94} +45.0000i q^{96} -852.000 q^{97} -354.000 q^{98} +1248.00i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{2} + 2 q^{4} - 30 q^{7} - 42 q^{8} + 52 q^{9}+O(q^{10}) $ 2 * q + 6 * q^2 + 2 * q^4 - 30 * q^7 - 42 * q^8 + 52 * q^9	$ 2 q + 6 q^{2} + 2 q^{4} - 30 q^{7} - 42 q^{8} + 52 q^{9} + 78 q^{13} - 90 q^{14} - 142 q^{16} + 156 q^{18} + 234 q^{26} - 30 q^{28} + 288 q^{29} - 90 q^{32} + 96 q^{33} + 52 q^{36} - 606 q^{37} + 52 q^{39} - 222 q^{47} - 236 q^{49} + 90 q^{51} + 78 q^{52} + 630 q^{56} - 12 q^{57} + 864 q^{58} + 752 q^{61} - 780 q^{63} + 866 q^{64} + 288 q^{66} - 72 q^{67} + 324 q^{69} - 1092 q^{72} - 2196 q^{73} - 1818 q^{74} + 156 q^{78} + 1660 q^{79} + 1298 q^{81} + 876 q^{83} - 1170 q^{91} + 528 q^{93} - 666 q^{94} - 1704 q^{97} - 708 q^{98}+O(q^{100}) $ 2 * q + 6 * q^2 + 2 * q^4 - 30 * q^7 - 42 * q^8 + 52 * q^9 + 78 * q^13 - 90 * q^14 - 142 * q^16 + 156 * q^18 + 234 * q^26 - 30 * q^28 + 288 * q^29 - 90 * q^32 + 96 * q^33 + 52 * q^36 - 606 * q^37 + 52 * q^39 - 222 * q^47 - 236 * q^49 + 90 * q^51 + 78 * q^52 + 630 * q^56 - 12 * q^57 + 864 * q^58 + 752 * q^61 - 780 * q^63 + 866 * q^64 + 288 * q^66 - 72 * q^67 + 324 * q^69 - 1092 * q^72 - 2196 * q^73 - 1818 * q^74 + 156 * q^78 + 1660 * q^79 + 1298 * q^81 + 876 * q^83 - 1170 * q^91 + 528 * q^93 - 666 * q^94 - 1704 * q^97 - 708 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$-1$

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 $

$2$	3.00000			1.06066			0.530330	−	0.847791i	$-0.322068\pi$
$2$	3.00000			1.06066			0.530330	+	0.847791i	$0.322068\pi$
$3$		−	1.00000i		−	0.192450i	−0.995360	−	0.0962250i	$-0.969323\pi$
$3$		−	1.00000i		−	0.192450i	0.995360	−	0.0962250i	$-0.0306768\pi$
$4$	1.00000			0.125000
$5$		0			0
$5$		0			0
$6$		−	3.00000i		−	0.204124i
$7$	−15.0000			−0.809924			−0.404962	−	0.914334i	$-0.632715\pi$
$7$	−15.0000			−0.809924			−0.404962	+	0.914334i	$0.632715\pi$
$8$	−21.0000			−0.928078
$9$	26.0000			0.962963
$10$		0			0
$11$			48.0000i			1.31569i	0.753155	+	0.657843i	$0.228531\pi$
$11$			48.0000i			1.31569i	−0.753155	+	0.657843i	$0.771469\pi$
$12$		−	1.00000i		−	0.0240563i
$13$	39.0000	+	26.0000i	0.832050	+	0.554700i
$13$	39.0000	+	26.0000i	0.832050	+	0.554700i
$14$	−45.0000			−0.859054
$15$		0			0
$16$	−71.0000			−1.10938
$17$			45.0000i			0.642006i	0.947078	+	0.321003i	$0.104020\pi$
$17$			45.0000i			0.642006i	−0.947078	+	0.321003i	$0.895980\pi$
$18$	78.0000			1.02138
$19$		−	6.00000i		−	0.0724471i	−0.999344	−	0.0362235i	$-0.988467\pi$
$19$		−	6.00000i		−	0.0724471i	0.999344	−	0.0362235i	$-0.0115328\pi$
$20$		0			0
$21$			15.0000i			0.155870i
$22$			144.000i			1.39550i
$23$			162.000i			1.46867i	0.678789	+	0.734333i	$0.262505\pi$
$23$			162.000i			1.46867i	−0.678789	+	0.734333i	$0.737495\pi$
$24$			21.0000i			0.178609i
$25$		0			0
$26$	117.000	+	78.0000i	0.882523	+	0.588348i
$27$		−	53.0000i		−	0.377772i
$28$	−15.0000			−0.101240
$29$	144.000			0.922073			0.461037	−	0.887381i	$-0.347478\pi$
$29$	144.000			0.922073			0.461037	+	0.887381i	$0.347478\pi$
$30$		0			0
$31$			264.000i			1.52954i	0.644302	+	0.764771i	$0.277148\pi$
$31$			264.000i			1.52954i	−0.644302	+	0.764771i	$0.722852\pi$
$32$	−45.0000			−0.248592
$33$	48.0000			0.253204
$34$			135.000i			0.680950i
$35$		0			0
$36$	26.0000			0.120370
$37$	−303.000			−1.34629			−0.673147	−	0.739509i	$-0.735058\pi$
$37$	−303.000			−1.34629			−0.673147	+	0.739509i	$0.735058\pi$
$38$		−	18.0000i		−	0.0768417i
$39$	26.0000	−	39.0000i	0.106752	−	0.160128i
$40$		0			0
$41$		−	192.000i		−	0.731350i	−0.930743	−	0.365675i	$-0.880838\pi$
$41$		−	192.000i		−	0.731350i	0.930743	−	0.365675i	$-0.119162\pi$
$42$			45.0000i			0.165325i
$43$		−	97.0000i		−	0.344008i	−0.985096	−	0.172004i	$-0.944976\pi$
$43$		−	97.0000i		−	0.344008i	0.985096	−	0.172004i	$-0.0550243\pi$
$44$			48.0000i			0.164461i
$45$		0			0
$46$			486.000i			1.55776i
$47$	−111.000			−0.344490			−0.172245	−	0.985054i	$-0.555102\pi$
$47$	−111.000			−0.344490			−0.172245	+	0.985054i	$0.555102\pi$
$48$			71.0000i			0.213499i
$49$	−118.000			−0.344023
$50$		0			0
$51$	45.0000			0.123554
$52$	39.0000	+	26.0000i	0.104006	+	0.0693375i
$53$		−	414.000i		−	1.07297i	−0.843911	−	0.536484i	$-0.819752\pi$
$53$		−	414.000i		−	1.07297i	0.843911	−	0.536484i	$-0.180248\pi$
$54$		−	159.000i		−	0.400688i
$55$		0			0
$56$	315.000			0.751672
$57$	−6.00000			−0.0139424
$58$	432.000			0.978007
$59$			522.000i			1.15184i	0.817506	+	0.575920i	$0.195356\pi$
$59$			522.000i			1.15184i	−0.817506	+	0.575920i	$0.804644\pi$
$60$		0			0
$61$	376.000			0.789211			0.394605	−	0.918851i	$-0.370881\pi$
$61$	376.000			0.789211			0.394605	+	0.918851i	$0.370881\pi$
$62$			792.000i			1.62232i
$63$	−390.000			−0.779927
$64$	433.000			0.845703
$65$		0			0
$66$	144.000			0.268563
$67$	−36.0000			−0.0656433			−0.0328216	−	0.999461i	$-0.510449\pi$
$67$	−36.0000			−0.0656433			−0.0328216	+	0.999461i	$0.510449\pi$
$68$			45.0000i			0.0802508i
$69$	162.000			0.282645
$70$		0			0
$71$			357.000i			0.596734i	0.954451	+	0.298367i	$0.0964419\pi$
$71$			357.000i			0.596734i	−0.954451	+	0.298367i	$0.903558\pi$
$72$	−546.000			−0.893704
$73$	−1098.00			−1.76043			−0.880214	−	0.474578i	$-0.842601\pi$
$73$	−1098.00			−1.76043			−0.880214	+	0.474578i	$0.842601\pi$
$74$	−909.000			−1.42796
$75$		0			0
$76$		−	6.00000i		−	0.00905588i
$77$		−	720.000i		−	1.06561i
$78$	78.0000	−	117.000i	0.113228	−	0.169842i
$79$	830.000			1.18205			0.591027	−	0.806652i	$-0.298723\pi$
$79$	830.000			1.18205			0.591027	+	0.806652i	$0.298723\pi$
$80$		0			0
$81$	649.000			0.890261
$82$		−	576.000i		−	0.775714i
$83$	438.000			0.579238			0.289619	−	0.957142i	$-0.406471\pi$
$83$	438.000			0.579238			0.289619	+	0.957142i	$0.406471\pi$
$84$			15.0000i			0.0194837i
$85$		0			0
$86$		−	291.000i		−	0.364876i
$87$		−	144.000i		−	0.177453i
$88$		−	1008.00i		−	1.22106i
$89$		−	438.000i		−	0.521662i	−0.965384	−	0.260831i	$-0.916003\pi$
$89$		−	438.000i		−	0.521662i	0.965384	−	0.260831i	$-0.0839965\pi$
$90$		0			0
$91$	−585.000	−	390.000i	−0.673897	−	0.449265i
$92$			162.000i			0.183583i
$93$	264.000			0.294360
$94$	−333.000			−0.365386
$95$		0			0
$96$			45.0000i			0.0478416i
$97$	−852.000			−0.891830			−0.445915	−	0.895075i	$-0.647122\pi$
$97$	−852.000			−0.891830			−0.445915	+	0.895075i	$0.647122\pi$
$98$	−354.000			−0.364892
$99$			1248.00i			1.26696i
$100$		0			0
$101$	396.000			0.390133			0.195067	−	0.980790i	$-0.437508\pi$
$101$	396.000			0.390133			0.195067	+	0.980790i	$0.437508\pi$
$102$	135.000			0.131049
$103$			182.000i			0.174107i	0.996204	+	0.0870534i	$0.0277451\pi$
$103$			182.000i			0.174107i	−0.996204	+	0.0870534i	$0.972255\pi$
$104$	−819.000	−	546.000i	−0.772207	−	0.514805i
$105$		0			0
$106$		−	1242.00i		−	1.13805i
$107$			612.000i			0.552937i	0.961023	+	0.276469i	$0.0891642\pi$
$107$			612.000i			0.552937i	−0.961023	+	0.276469i	$0.910836\pi$
$108$		−	53.0000i		−	0.0472215i
$109$		−	1083.00i		−	0.951675i	−0.879533	−	0.475838i	$-0.842145\pi$
$109$		−	1083.00i		−	0.951675i	0.879533	−	0.475838i	$-0.157855\pi$
$110$		0			0
$111$			303.000i			0.259094i
$112$	1065.00			0.898509
$113$			90.0000i			0.0749247i	0.999298	+	0.0374623i	$0.0119274\pi$
$113$			90.0000i			0.0749247i	−0.999298	+	0.0374623i	$0.988073\pi$
$114$	−18.0000			−0.0147882
$115$		0			0
$116$	144.000			0.115259
$117$	1014.00	+	676.000i	0.801234	+	0.534156i
$118$			1566.00i			1.22171i
$119$		−	675.000i		−	0.519976i
$120$		0			0
$121$	−973.000			−0.731029
$122$	1128.00			0.837085
$123$	−192.000			−0.140748
$124$			264.000i			0.191193i
$125$		0			0
$126$	−1170.00			−0.827237
$127$		−	2086.00i		−	1.45750i	−0.684780	−	0.728750i	$-0.740102\pi$
$127$		−	2086.00i		−	1.45750i	0.684780	−	0.728750i	$-0.259898\pi$
$128$	1659.00			1.14560
$129$	−97.0000			−0.0662044
$130$		0			0
$131$	−1467.00			−0.978415			−0.489208	−	0.872167i	$-0.662714\pi$
$131$	−1467.00			−0.978415			−0.489208	+	0.872167i	$0.662714\pi$
$132$	48.0000			0.0316505
$133$			90.0000i			0.0586766i
$134$	−108.000			−0.0696252
$135$		0			0
$136$		−	945.000i		−	0.595831i
$137$	414.000			0.258178			0.129089	−	0.991633i	$-0.458795\pi$
$137$	414.000			0.258178			0.129089	+	0.991633i	$0.458795\pi$
$138$	486.000			0.299790
$139$	2419.00			1.47609			0.738046	−	0.674750i	$-0.235749\pi$
$139$	2419.00			1.47609			0.738046	+	0.674750i	$0.235749\pi$
$140$		0			0
$141$			111.000i			0.0662971i
$142$			1071.00i			0.632932i
$143$	−1248.00	+	1872.00i	−0.729811	+	1.09472i
$144$	−1846.00			−1.06829
$145$		0			0
$146$	−3294.00			−1.86721
$147$			118.000i			0.0662073i
$148$	−303.000			−0.168287
$149$			930.000i			0.511333i	0.966765	+	0.255666i	$0.0822948\pi$
$149$			930.000i			0.511333i	−0.966765	+	0.255666i	$0.917705\pi$
$150$		0			0
$151$			1683.00i			0.907024i	0.891250	+	0.453512i	$0.149829\pi$
$151$			1683.00i			0.907024i	−0.891250	+	0.453512i	$0.850171\pi$
$152$			126.000i			0.0672365i
$153$			1170.00i			0.618228i
$154$		−	2160.00i		−	1.13025i
$155$		0			0
$156$	26.0000	−	39.0000i	0.0133440	−	0.0200160i
$157$		−	1874.00i		−	0.952621i	−0.879277	−	0.476310i	$-0.841974\pi$
$157$		−	1874.00i		−	0.952621i	0.879277	−	0.476310i	$-0.158026\pi$
$158$	2490.00			1.25376
$159$	−414.000			−0.206493
$160$		0			0
$161$		−	2430.00i		−	1.18951i
$162$	1947.00			0.944264
$163$	−1194.00			−0.573750			−0.286875	−	0.957968i	$-0.592616\pi$
$163$	−1194.00			−0.573750			−0.286875	+	0.957968i	$0.592616\pi$
$164$		−	192.000i		−	0.0914188i
$165$		0			0
$166$	1314.00			0.614375
$167$	2388.00			1.10652			0.553260	−	0.833008i	$-0.313383\pi$
$167$	2388.00			1.10652			0.553260	+	0.833008i	$0.313383\pi$
$168$		−	315.000i		−	0.144659i
$169$	845.000	+	2028.00i	0.384615	+	0.923077i
$170$		0			0
$171$		−	156.000i		−	0.0697638i
$172$		−	97.0000i		−	0.0430011i
$173$		−	1566.00i		−	0.688213i	−0.938931	−	0.344106i	$-0.888182\pi$
$173$		−	1566.00i		−	0.688213i	0.938931	−	0.344106i	$-0.111818\pi$
$174$		−	432.000i		−	0.188217i
$175$		0			0
$176$		−	3408.00i		−	1.45959i
$177$	522.000			0.221672
$178$		−	1314.00i		−	0.553306i
$179$	657.000			0.274338			0.137169	−	0.990548i	$-0.456200\pi$
$179$	657.000			0.274338			0.137169	+	0.990548i	$0.456200\pi$
$180$		0			0
$181$	−1222.00			−0.501826			−0.250913	−	0.968010i	$-0.580731\pi$
$181$	−1222.00			−0.501826			−0.250913	+	0.968010i	$0.580731\pi$
$182$	−1755.00	−	1170.00i	−0.714776	−	0.476517i
$183$		−	376.000i		−	0.151884i
$184$		−	3402.00i		−	1.36304i
$185$		0			0
$186$	792.000			0.312216
$187$	−2160.00			−0.844678
$188$	−111.000			−0.0430612
$189$			795.000i			0.305967i
$190$		0			0
$191$	1260.00			0.477332			0.238666	−	0.971102i	$-0.423290\pi$
$191$	1260.00			0.477332			0.238666	+	0.971102i	$0.423290\pi$
$192$		−	433.000i		−	0.162756i
$193$	342.000			0.127553			0.0637764	−	0.997964i	$-0.479686\pi$
$193$	342.000			0.127553			0.0637764	+	0.997964i	$0.479686\pi$
$194$	−2556.00			−0.945928
$195$		0			0
$196$	−118.000			−0.0430029
$197$	81.0000			0.0292945			0.0146472	−	0.999893i	$-0.495337\pi$
$197$	81.0000			0.0292945			0.0146472	+	0.999893i	$0.495337\pi$
$198$			3744.00i			1.34381i
$199$	−1996.00			−0.711019			−0.355509	−	0.934673i	$-0.615693\pi$
$199$	−1996.00			−0.711019			−0.355509	+	0.934673i	$0.615693\pi$
$200$		0			0
$201$			36.0000i			0.0126331i
$202$	1188.00			0.413799
$203$	−2160.00			−0.746809
$204$	45.0000			0.0154443
$205$		0			0
$206$			546.000i			0.184668i
$207$			4212.00i			1.41427i
$208$	−2769.00	−	1846.00i	−0.923056	−	0.615371i
$209$	288.000			0.0953176
$210$		0			0
$211$	2833.00			0.924321			0.462161	−	0.886796i	$-0.347074\pi$
$211$	2833.00			0.924321			0.462161	+	0.886796i	$0.347074\pi$
$212$		−	414.000i		−	0.134121i
$213$	357.000			0.114841
$214$			1836.00i			0.586478i
$215$		0			0
$216$			1113.00i			0.350602i
$217$		−	3960.00i		−	1.23881i
$218$		−	3249.00i		−	1.00940i
$219$			1098.00i			0.338794i
$220$		0			0
$221$	−1170.00	+	1755.00i	−0.356121	+	0.534181i
$222$			909.000i			0.274811i
$223$	3507.00			1.05312			0.526561	−	0.850138i	$-0.323481\pi$
$223$	3507.00			1.05312			0.526561	+	0.850138i	$0.323481\pi$
$224$	675.000			0.201341
$225$		0			0
$226$			270.000i			0.0794696i
$227$	−228.000			−0.0666647			−0.0333324	−	0.999444i	$-0.510612\pi$
$227$	−228.000			−0.0666647			−0.0333324	+	0.999444i	$0.510612\pi$
$228$	−6.00000			−0.00174281
$229$			5493.00i			1.58510i	0.609808	+	0.792549i	$0.291247\pi$
$229$			5493.00i			1.58510i	−0.609808	+	0.792549i	$0.708753\pi$
$230$		0			0
$231$	−720.000			−0.205076
$232$	−3024.00			−0.855756
$233$			3627.00i			1.01980i	0.860235	+	0.509898i	$0.170317\pi$
$233$			3627.00i			1.01980i	−0.860235	+	0.509898i	$0.829683\pi$
$234$	3042.00	+	2028.00i	0.849837	+	0.566558i
$235$		0			0
$236$			522.000i			0.143980i
$237$		−	830.000i		−	0.227486i
$238$		−	2025.00i		−	0.551518i
$239$		−	6075.00i		−	1.64418i	−0.569357	−	0.822090i	$-0.692808\pi$
$239$		−	6075.00i		−	1.64418i	0.569357	−	0.822090i	$-0.307192\pi$
$240$		0			0
$241$		−	210.000i		−	0.0561298i	−0.999606	−	0.0280649i	$-0.991065\pi$
$241$		−	210.000i		−	0.0561298i	0.999606	−	0.0280649i	$-0.00893451\pi$
$242$	−2919.00			−0.775374
$243$		−	2080.00i		−	0.549103i
$244$	376.000			0.0986514
$245$		0			0
$246$	−576.000			−0.149286
$247$	156.000	−	234.000i	0.0401864	−	0.0602796i
$248$		−	5544.00i		−	1.41953i
$249$		−	438.000i		−	0.111474i
$250$		0			0
$251$	7092.00			1.78344			0.891719	−	0.452589i	$-0.149499\pi$
$251$	7092.00			1.78344			0.891719	+	0.452589i	$0.149499\pi$
$252$	−390.000			−0.0974908
$253$	−7776.00			−1.93230
$254$		−	6258.00i		−	1.54591i
$255$		0			0
$256$	1513.00			0.369385
$257$			5805.00i			1.40897i	0.709718	+	0.704486i	$0.248823\pi$
$257$			5805.00i			1.40897i	−0.709718	+	0.704486i	$0.751177\pi$
$258$	−291.000			−0.0702204
$259$	4545.00			1.09040
$260$		0			0
$261$	3744.00			0.887923
$262$	−4401.00			−1.03777
$263$			792.000i			0.185691i	0.995681	+	0.0928457i	$0.0295963\pi$
$263$			792.000i			0.185691i	−0.995681	+	0.0928457i	$0.970404\pi$
$264$	−1008.00			−0.234993
$265$		0			0
$266$			270.000i			0.0622359i
$267$	−438.000			−0.100394
$268$	−36.0000			−0.00820541
$269$	−5472.00			−1.24027			−0.620137	−	0.784493i	$-0.712923\pi$
$269$	−5472.00			−1.24027			−0.620137	+	0.784493i	$0.712923\pi$
$270$		0			0
$271$		−	2331.00i		−	0.522502i	−0.965271	−	0.261251i	$-0.915865\pi$
$271$		−	2331.00i		−	0.522502i	0.965271	−	0.261251i	$-0.0841351\pi$
$272$		−	3195.00i		−	0.712225i
$273$	−390.000	+	585.000i	−0.0864611	+	0.129692i
$274$	1242.00			0.273839
$275$		0			0
$276$	162.000			0.0353306
$277$			1384.00i			0.300204i	0.988671	+	0.150102i	$0.0479602\pi$
$277$			1384.00i			0.300204i	−0.988671	+	0.150102i	$0.952040\pi$
$278$	7257.00			1.56563
$279$			6864.00i			1.47289i
$280$		0			0
$281$			4062.00i			0.862344i	0.902270	+	0.431172i	$0.141900\pi$
$281$			4062.00i			0.862344i	−0.902270	+	0.431172i	$0.858100\pi$
$282$			333.000i			0.0703187i
$283$		−	3764.00i		−	0.790624i	−0.918547	−	0.395312i	$-0.870636\pi$
$283$		−	3764.00i		−	0.790624i	0.918547	−	0.395312i	$-0.129364\pi$
$284$			357.000i			0.0745917i
$285$		0			0
$286$	−3744.00	+	5616.00i	−0.774082	+	1.16112i
$287$			2880.00i			0.592338i
$288$	−1170.00			−0.239385
$289$	2888.00			0.587828
$290$		0			0
$291$			852.000i			0.171633i
$292$	−1098.00			−0.220053
$293$	4227.00			0.842812			0.421406	−	0.906872i	$-0.361537\pi$
$293$	4227.00			0.842812			0.421406	+	0.906872i	$0.361537\pi$
$294$			354.000i			0.0702235i
$295$		0			0
$296$	6363.00			1.24947
$297$	2544.00			0.497030
$298$			2790.00i			0.542350i
$299$	−4212.00	+	6318.00i	−0.814670	+	1.22200i
$300$		0			0
$301$			1455.00i			0.278621i
$302$			5049.00i			0.962044i
$303$		−	396.000i		−	0.0750812i
$304$			426.000i			0.0803710i
$305$		0			0
$306$			3510.00i			0.655730i
$307$	−306.000			−0.0568871			−0.0284436	−	0.999595i	$-0.509055\pi$
$307$	−306.000			−0.0568871			−0.0284436	+	0.999595i	$0.509055\pi$
$308$		−	720.000i		−	0.133201i
$309$	182.000			0.0335069
$310$		0			0
$311$	−2106.00			−0.383988			−0.191994	−	0.981396i	$-0.561495\pi$
$311$	−2106.00			−0.383988			−0.191994	+	0.981396i	$0.561495\pi$
$312$	−546.000	+	819.000i	−0.0990742	+	0.148611i
$313$			10051.0i			1.81507i	0.419979	+	0.907534i	$0.362037\pi$
$313$			10051.0i			1.81507i	−0.419979	+	0.907534i	$0.637963\pi$
$314$		−	5622.00i		−	1.01041i
$315$		0			0
$316$	830.000			0.147757
$317$	−2154.00			−0.381643			−0.190821	−	0.981625i	$-0.561115\pi$
$317$	−2154.00			−0.381643			−0.190821	+	0.981625i	$0.561115\pi$
$318$	−1242.00			−0.219019
$319$			6912.00i			1.21316i
$320$		0			0
$321$	612.000			0.106413
$322$		−	7290.00i		−	1.26166i
$323$	270.000			0.0465115
$324$	649.000			0.111283
$325$		0			0
$326$	−3582.00			−0.608554
$327$	−1083.00			−0.183150
$328$			4032.00i			0.678750i
$329$	1665.00			0.279010
$330$		0			0
$331$			10770.0i			1.78844i	0.447630	+	0.894219i	$0.352268\pi$
$331$			10770.0i			1.78844i	−0.447630	+	0.894219i	$0.647732\pi$
$332$	438.000			0.0724047
$333$	−7878.00			−1.29643
$334$	7164.00			1.17364
$335$		0			0
$336$		−	1065.00i		−	0.172918i
$337$			2171.00i			0.350926i	0.984486	+	0.175463i	$0.0561422\pi$
$337$			2171.00i			0.350926i	−0.984486	+	0.175463i	$0.943858\pi$
$338$	2535.00	+	6084.00i	0.407946	+	0.979071i
$339$	90.0000			0.0144193
$340$		0			0
$341$	−12672.0			−2.01240
$342$		−	468.000i		−	0.0739957i
$343$	6915.00			1.08856
$344$			2037.00i			0.319267i
$345$		0			0
$346$		−	4698.00i		−	0.729960i
$347$			7047.00i			1.09021i	0.838368	+	0.545105i	$0.183510\pi$
$347$			7047.00i			1.09021i	−0.838368	+	0.545105i	$0.816490\pi$
$348$		−	144.000i		−	0.0221816i
$349$		−	6873.00i		−	1.05416i	−0.849814	−	0.527082i	$-0.823286\pi$
$349$		−	6873.00i		−	1.05416i	0.849814	−	0.527082i	$-0.176714\pi$
$350$		0			0
$351$	1378.00	−	2067.00i	0.209550	−	0.314326i
$352$		−	2160.00i		−	0.327069i
$353$	9318.00			1.40495			0.702475	−	0.711709i	$-0.252078\pi$
$353$	9318.00			1.40495			0.702475	+	0.711709i	$0.252078\pi$
$354$	1566.00			0.235119
$355$		0			0
$356$		−	438.000i		−	0.0652077i
$357$	−675.000			−0.100069
$358$	1971.00			0.290979
$359$			4128.00i			0.606873i	0.952852	+	0.303437i	$0.0981341\pi$
$359$			4128.00i			0.606873i	−0.952852	+	0.303437i	$0.901866\pi$
$360$		0			0
$361$	6823.00			0.994751
$362$	−3666.00			−0.532267
$363$			973.000i			0.140687i
$364$	−585.000	−	390.000i	−0.0842372	−	0.0561581i
$365$		0			0
$366$		−	1128.00i		−	0.161097i
$367$			2536.00i			0.360703i	0.983602	+	0.180352i	$0.0577236\pi$
$367$			2536.00i			0.360703i	−0.983602	+	0.180352i	$0.942276\pi$
$368$		−	11502.0i		−	1.62930i
$369$		−	4992.00i		−	0.704263i
$370$		0			0
$371$			6210.00i			0.869022i
$372$	264.000			0.0367951
$373$		−	92.0000i		−	0.0127710i	−0.999980	−	0.00638550i	$-0.997967\pi$
$373$		−	92.0000i		−	0.0127710i	0.999980	−	0.00638550i	$-0.00203258\pi$
$374$	−6480.00			−0.895917
$375$		0			0
$376$	2331.00			0.319713
$377$	5616.00	+	3744.00i	0.767211	+	0.511474i
$378$			2385.00i			0.324527i
$379$		−	10182.0i		−	1.37998i	−0.723817	−	0.689992i	$-0.757614\pi$
$379$		−	10182.0i		−	1.37998i	0.723817	−	0.689992i	$-0.242386\pi$
$380$		0			0
$381$	−2086.00			−0.280496
$382$	3780.00			0.506287
$383$	579.000			0.0772468			0.0386234	−	0.999254i	$-0.487703\pi$
$383$	579.000			0.0772468			0.0386234	+	0.999254i	$0.487703\pi$
$384$		−	1659.00i		−	0.220470i
$385$		0			0
$386$	1026.00			0.135290
$387$		−	2522.00i		−	0.331267i
$388$	−852.000			−0.111479
$389$	−2106.00			−0.274495			−0.137247	−	0.990537i	$-0.543826\pi$
$389$	−2106.00			−0.274495			−0.137247	+	0.990537i	$0.543826\pi$
$390$		0			0
$391$	−7290.00			−0.942893
$392$	2478.00			0.319280
$393$			1467.00i			0.188296i
$394$	243.000			0.0310715
$395$		0			0
$396$			1248.00i			0.158370i
$397$	1974.00			0.249552			0.124776	−	0.992185i	$-0.460179\pi$
$397$	1974.00			0.249552			0.124776	+	0.992185i	$0.460179\pi$
$398$	−5988.00			−0.754149
$399$	90.0000			0.0112923
$400$		0			0
$401$		−	11886.0i		−	1.48020i	−0.672499	−	0.740098i	$-0.734779\pi$
$401$		−	11886.0i		−	1.48020i	0.672499	−	0.740098i	$-0.265221\pi$
$402$			108.000i			0.0133994i
$403$	−6864.00	+	10296.0i	−0.848437	+	1.27266i
$404$	396.000			0.0487667
$405$		0			0
$406$	−6480.00			−0.792111
$407$		−	14544.0i		−	1.77130i
$408$	−945.000			−0.114668
$409$		−	1254.00i		−	0.151605i	−0.997123	−	0.0758023i	$-0.975848\pi$
$409$		−	1254.00i		−	0.151605i	0.997123	−	0.0758023i	$-0.0241518\pi$
$410$		0			0
$411$		−	414.000i		−	0.0496864i
$412$			182.000i			0.0217633i
$413$		−	7830.00i		−	0.932903i
$414$			12636.0i			1.50006i
$415$		0			0
$416$	−1755.00	−	1170.00i	−0.206841	−	0.137894i
$417$		−	2419.00i		−	0.284074i
$418$	864.000			0.101100
$419$	−5823.00			−0.678931			−0.339466	−	0.940618i	$-0.610246\pi$
$419$	−5823.00			−0.678931			−0.339466	+	0.940618i	$0.610246\pi$
$420$		0			0
$421$		−	7341.00i		−	0.849830i	−0.905233	−	0.424915i	$-0.860304\pi$
$421$		−	7341.00i		−	0.849830i	0.905233	−	0.424915i	$-0.139696\pi$
$422$	8499.00			0.980391
$423$	−2886.00			−0.331731
$424$			8694.00i			0.995797i
$425$		0			0
$426$	1071.00			0.121808
$427$	−5640.00			−0.639201
$428$			612.000i			0.0691171i
$429$	1872.00	+	1248.00i	0.210678	+	0.140452i
$430$		0			0
$431$		−	7485.00i		−	0.836519i	−0.908328	−	0.418260i	$-0.862640\pi$
$431$		−	7485.00i		−	0.836519i	0.908328	−	0.418260i	$-0.137360\pi$
$432$			3763.00i			0.419091i
$433$		−	15203.0i		−	1.68732i	−0.536878	−	0.843660i	$-0.680396\pi$
$433$		−	15203.0i		−	1.68732i	0.536878	−	0.843660i	$-0.319604\pi$
$434$		−	11880.0i		−	1.31396i
$435$		0			0
$436$		−	1083.00i		−	0.118959i
$437$	972.000			0.106401
$438$			3294.00i			0.359346i
$439$	1762.00			0.191562			0.0957809	−	0.995402i	$-0.469465\pi$
$439$	1762.00			0.191562			0.0957809	+	0.995402i	$0.469465\pi$
$440$		0			0
$441$	−3068.00			−0.331282
$442$	−3510.00	+	5265.00i	−0.377723	+	0.566585i
$443$		−	7317.00i		−	0.784743i	−0.919807	−	0.392372i	$-0.871655\pi$
$443$		−	7317.00i		−	0.784743i	0.919807	−	0.392372i	$-0.128345\pi$
$444$			303.000i			0.0323868i
$445$		0			0
$446$	10521.0			1.11700
$447$	930.000			0.0984060
$448$	−6495.00			−0.684955
$449$		−	5016.00i		−	0.527215i	−0.964630	−	0.263608i	$-0.915088\pi$
$449$		−	5016.00i		−	0.527215i	0.964630	−	0.263608i	$-0.0849124\pi$
$450$		0			0
$451$	9216.00			0.962227
$452$			90.0000i			0.00936558i
$453$	1683.00			0.174557
$454$	−684.000			−0.0707086
$455$		0			0
$456$	126.000			0.0129397
$457$	9870.00			1.01028			0.505141	−	0.863037i	$-0.331440\pi$
$457$	9870.00			1.01028			0.505141	+	0.863037i	$0.331440\pi$
$458$			16479.0i			1.68125i
$459$	2385.00			0.242532
$460$		0			0
$461$		−	14541.0i		−	1.46907i	−0.678570	−	0.734536i	$-0.737400\pi$
$461$		−	14541.0i		−	1.46907i	0.678570	−	0.734536i	$-0.262600\pi$
$462$	−2160.00			−0.217516
$463$	−2112.00			−0.211993			−0.105997	−	0.994366i	$-0.533803\pi$
$463$	−2112.00			−0.211993			−0.105997	+	0.994366i	$0.533803\pi$
$464$	−10224.0			−1.02293
$465$		0			0
$466$			10881.0i			1.08166i
$467$		−	3276.00i		−	0.324615i	−0.986740	−	0.162307i	$-0.948106\pi$
$467$		−	3276.00i		−	0.324615i	0.986740	−	0.162307i	$-0.0518936\pi$
$468$	1014.00	+	676.000i	0.100154	+	0.0667695i
$469$	540.000			0.0531661
$470$		0			0
$471$	−1874.00			−0.183332
$472$		−	10962.0i		−	1.06900i
$473$	4656.00			0.452607
$474$		−	2490.00i		−	0.241286i
$475$		0			0
$476$		−	675.000i		−	0.0649970i
$477$		−	10764.0i		−	1.03323i
$478$		−	18225.0i		−	1.74392i
$479$		−	15453.0i		−	1.47404i	−0.675870	−	0.737020i	$-0.736232\pi$
$479$		−	15453.0i		−	1.47404i	0.675870	−	0.737020i	$-0.263768\pi$
$480$		0			0
$481$	−11817.0	−	7878.00i	−1.12018	−	0.746790i
$482$		−	630.000i		−	0.0595347i
$483$	−2430.00			−0.228921
$484$	−973.000			−0.0913787
$485$		0			0
$486$		−	6240.00i		−	0.582412i
$487$	−3660.00			−0.340555			−0.170278	−	0.985396i	$-0.554466\pi$
$487$	−3660.00			−0.340555			−0.170278	+	0.985396i	$0.554466\pi$
$488$	−7896.00			−0.732449
$489$			1194.00i			0.110418i
$490$		0			0
$491$	747.000			0.0686591			0.0343296	−	0.999411i	$-0.489070\pi$
$491$	747.000			0.0686591			0.0343296	+	0.999411i	$0.489070\pi$
$492$	−192.000			−0.0175936
$493$			6480.00i			0.591977i
$494$	468.000	−	702.000i	0.0426241	−	0.0639362i
$495$		0			0
$496$		−	18744.0i		−	1.69684i
$497$		−	5355.00i		−	0.483309i
$498$		−	1314.00i		−	0.118236i
$499$			15804.0i			1.41780i	0.705307	+	0.708902i	$0.250809\pi$
$499$			15804.0i			1.41780i	−0.705307	+	0.708902i	$0.749191\pi$
$500$		0			0
$501$		−	2388.00i		−	0.212950i
$502$	21276.0			1.89162
$503$		−	12078.0i		−	1.07064i	−0.844650	−	0.535319i	$-0.820191\pi$
$503$		−	12078.0i		−	1.07064i	0.844650	−	0.535319i	$-0.179809\pi$
$504$	8190.00			0.723833
$505$		0			0
$506$	−23328.0			−2.04952
$507$	2028.00	−	845.000i	0.177646	−	0.0740193i
$508$		−	2086.00i		−	0.182188i
$509$		−	16110.0i		−	1.40287i	−0.712731	−	0.701437i	$-0.752542\pi$
$509$		−	16110.0i		−	1.40287i	0.712731	−	0.701437i	$-0.247458\pi$
$510$		0			0
$511$	16470.0			1.42581
$512$	−8733.00			−0.753804
$513$	−318.000			−0.0273685
$514$			17415.0i			1.49444i
$515$		0			0
$516$	−97.0000			−0.00827556
$517$		−	5328.00i		−	0.453240i
$518$	13635.0			1.15654
$519$	−1566.00			−0.132447
$520$		0			0
$521$	3915.00			0.329212			0.164606	−	0.986359i	$-0.447365\pi$
$521$	3915.00			0.329212			0.164606	+	0.986359i	$0.447365\pi$
$522$	11232.0			0.941784
$523$			16184.0i			1.35311i	0.736392	+	0.676555i	$0.236528\pi$
$523$			16184.0i			1.35311i	−0.736392	+	0.676555i	$0.763472\pi$
$524$	−1467.00			−0.122302
$525$		0			0
$526$			2376.00i			0.196955i
$527$	−11880.0			−0.981975
$528$	−3408.00			−0.280898
$529$	−14077.0			−1.15698
$530$		0			0
$531$			13572.0i			1.10918i
$532$			90.0000i			0.00733458i
$533$	4992.00	−	7488.00i	0.405680	−	0.608520i
$534$	−1314.00			−0.106484
$535$		0			0
$536$	756.000			0.0609221
$537$		−	657.000i		−	0.0527964i
$538$	−16416.0			−1.31551
$539$		−	5664.00i		−	0.452627i
$540$		0			0
$541$			7923.00i			0.629642i	0.949151	+	0.314821i	$0.101945\pi$
$541$			7923.00i			0.629642i	−0.949151	+	0.314821i	$0.898055\pi$
$542$		−	6993.00i		−	0.554198i
$543$			1222.00i			0.0965765i
$544$		−	2025.00i		−	0.159598i
$545$		0			0
$546$	−1170.00	+	1755.00i	−0.0917058	+	0.137559i
$547$			14389.0i			1.12473i	0.826888	+	0.562367i	$0.190109\pi$
$547$			14389.0i			1.12473i	−0.826888	+	0.562367i	$0.809891\pi$
$548$	414.000			0.0322723
$549$	9776.00			0.759981
$550$		0			0
$551$		−	864.000i		−	0.0668015i
$552$	−3402.00			−0.262317
$553$	−12450.0			−0.957374
$554$			4152.00i			0.318414i
$555$		0			0
$556$	2419.00			0.184512
$557$	10383.0			0.789842			0.394921	−	0.918715i	$-0.370772\pi$
$557$	10383.0			0.789842			0.394921	+	0.918715i	$0.370772\pi$
$558$			20592.0i			1.56224i
$559$	2522.00	−	3783.00i	0.190822	−	0.286232i
$560$		0			0
$561$			2160.00i			0.162558i
$562$			12186.0i			0.914654i
$563$		−	16425.0i		−	1.22954i	−0.788706	−	0.614770i	$-0.789249\pi$
$563$		−	16425.0i		−	1.22954i	0.788706	−	0.614770i	$-0.210751\pi$
$564$			111.000i			0.00828713i
$565$		0			0
$566$		−	11292.0i		−	0.838583i
$567$	−9735.00			−0.721043
$568$		−	7497.00i		−	0.553815i
$569$	−12213.0			−0.899817			−0.449908	−	0.893075i	$-0.648543\pi$
$569$	−12213.0			−0.899817			−0.449908	+	0.893075i	$0.648543\pi$
$570$		0			0
$571$	−6383.00			−0.467811			−0.233906	−	0.972259i	$-0.575151\pi$
$571$	−6383.00			−0.467811			−0.233906	+	0.972259i	$0.575151\pi$
$572$	−1248.00	+	1872.00i	−0.0912264	+	0.136840i
$573$		−	1260.00i		−	0.0918626i
$574$			8640.00i			0.628269i
$575$		0			0
$576$	11258.0			0.814381
$577$	6426.00			0.463636			0.231818	−	0.972759i	$-0.425533\pi$
$577$	6426.00			0.463636			0.231818	+	0.972759i	$0.425533\pi$
$578$	8664.00			0.623486
$579$		−	342.000i		−	0.0245476i
$580$		0			0
$581$	−6570.00			−0.469139
$582$			2556.00i			0.182044i
$583$	19872.0			1.41169
$584$	23058.0			1.63381
$585$		0			0
$586$	12681.0			0.893937
$587$	−21330.0			−1.49980			−0.749901	−	0.661551i	$-0.769899\pi$
$587$	−21330.0			−1.49980			−0.749901	+	0.661551i	$0.769899\pi$
$588$			118.000i			0.00827591i
$589$	1584.00			0.110811
$590$		0			0
$591$		−	81.0000i		−	0.00563772i
$592$	21513.0			1.49355
$593$	12084.0			0.836813			0.418407	−	0.908260i	$-0.362589\pi$
$593$	12084.0			0.836813			0.418407	+	0.908260i	$0.362589\pi$
$594$	7632.00			0.527180
$595$		0			0
$596$			930.000i			0.0639166i
$597$			1996.00i			0.136836i
$598$	−12636.0	+	18954.0i	−0.864088	+	1.29613i
$599$	−2394.00			−0.163299			−0.0816496	−	0.996661i	$-0.526019\pi$
$599$	−2394.00			−0.163299			−0.0816496	+	0.996661i	$0.526019\pi$
$600$		0			0
$601$	−21971.0			−1.49121			−0.745604	−	0.666389i	$-0.767839\pi$
$601$	−21971.0			−1.49121			−0.745604	+	0.666389i	$0.767839\pi$
$602$			4365.00i			0.295522i
$603$	−936.000			−0.0632121
$604$			1683.00i			0.113378i
$605$		0			0
$606$		−	1188.00i		−	0.0796356i
$607$			15406.0i			1.03017i	0.857141	+	0.515083i	$0.172239\pi$
$607$			15406.0i			1.03017i	−0.857141	+	0.515083i	$0.827761\pi$
$608$			270.000i			0.0180098i
$609$			2160.00i			0.143724i
$610$		0			0
$611$	−4329.00	−	2886.00i	−0.286633	−	0.191088i
$612$			1170.00i			0.0772785i
$613$	9630.00			0.634506			0.317253	−	0.948341i	$-0.397240\pi$
$613$	9630.00			0.634506			0.317253	+	0.948341i	$0.397240\pi$
$614$	−918.000			−0.0603379
$615$		0			0
$616$			15120.0i			0.988965i
$617$	14748.0			0.962289			0.481144	−	0.876641i	$-0.340221\pi$
$617$	14748.0			0.962289			0.481144	+	0.876641i	$0.340221\pi$
$618$	546.000			0.0355394
$619$		−	3672.00i		−	0.238433i	−0.992868	−	0.119217i	$-0.961962\pi$
$619$		−	3672.00i		−	0.238433i	0.992868	−	0.119217i	$-0.0380383\pi$
$620$		0			0
$621$	8586.00			0.554822
$622$	−6318.00			−0.407281
$623$			6570.00i			0.422506i
$624$	−1846.00	+	2769.00i	−0.118428	+	0.177642i
$625$		0			0
$626$			30153.0i			1.92517i
$627$		−	288.000i		−	0.0183439i
$628$		−	1874.00i		−	0.119078i
$629$		−	13635.0i		−	0.864329i
$630$		0			0
$631$			19875.0i			1.25390i	0.779059	+	0.626950i	$0.215697\pi$
$631$			19875.0i			1.25390i	−0.779059	+	0.626950i	$0.784303\pi$
$632$	−17430.0			−1.09704
$633$		−	2833.00i		−	0.177886i
$634$	−6462.00			−0.404793
$635$		0			0
$636$	−414.000			−0.0258116
$637$	−4602.00	−	3068.00i	−0.286245	−	0.190830i
$638$			20736.0i			1.28675i
$639$			9282.00i			0.574633i
$640$		0			0
$641$	1710.00			0.105368			0.0526840	−	0.998611i	$-0.483222\pi$
$641$	1710.00			0.105368			0.0526840	+	0.998611i	$0.483222\pi$
$642$	1836.00			0.112868
$643$	16452.0			1.00903			0.504513	−	0.863404i	$-0.331672\pi$
$643$	16452.0			1.00903			0.504513	+	0.863404i	$0.331672\pi$
$644$		−	2430.00i		−	0.148689i
$645$		0			0
$646$	810.000			0.0493329
$647$		−	25902.0i		−	1.57390i	−0.617017	−	0.786950i	$-0.711659\pi$
$647$		−	25902.0i		−	1.57390i	0.617017	−	0.786950i	$-0.288341\pi$
$648$	−13629.0			−0.826231
$649$	−25056.0			−1.51546
$650$		0			0
$651$	−3960.00			−0.238410
$652$	−1194.00			−0.0717188
$653$			18108.0i			1.08518i	0.839999	+	0.542589i	$0.182556\pi$
$653$			18108.0i			1.08518i	−0.839999	+	0.542589i	$0.817444\pi$
$654$	−3249.00			−0.194260
$655$		0			0
$656$			13632.0i			0.811342i
$657$	−28548.0			−1.69523
$658$	4995.00			0.295935
$659$	32904.0			1.94500			0.972502	−	0.232894i	$-0.0748195\pi$
$659$	32904.0			1.94500			0.972502	+	0.232894i	$0.0748195\pi$
$660$		0			0
$661$		−	15318.0i		−	0.901363i	−0.892685	−	0.450682i	$-0.851181\pi$
$661$		−	15318.0i		−	0.901363i	0.892685	−	0.450682i	$-0.148819\pi$
$662$			32310.0i			1.89692i
$663$	1755.00	+	1170.00i	0.102803	+	0.0685355i
$664$	−9198.00			−0.537578
$665$		0			0
$666$	−23634.0			−1.37507
$667$			23328.0i			1.35422i
$668$	2388.00			0.138315
$669$		−	3507.00i		−	0.202673i
$670$		0			0
$671$			18048.0i			1.03835i
$672$		−	675.000i		−	0.0387481i
$673$		−	7729.00i		−	0.442691i	−0.975195	−	0.221346i	$-0.928955\pi$
$673$		−	7729.00i		−	0.442691i	0.975195	−	0.221346i	$-0.0710449\pi$
$674$			6513.00i			0.372213i
$675$		0			0
$676$	845.000	+	2028.00i	0.0480769	+	0.115385i
$677$		−	19242.0i		−	1.09236i	−0.837667	−	0.546182i	$-0.816081\pi$
$677$		−	19242.0i		−	1.09236i	0.837667	−	0.546182i	$-0.183919\pi$
$678$	270.000			0.0152939
$679$	12780.0			0.722314
$680$		0			0
$681$			228.000i			0.0128296i
$682$	−38016.0			−2.13447
$683$	22518.0			1.26153			0.630767	−	0.775973i	$-0.282740\pi$
$683$	22518.0			1.26153			0.630767	+	0.775973i	$0.282740\pi$
$684$		−	156.000i		−	0.00872048i
$685$		0			0
$686$	20745.0			1.15459
$687$	5493.00			0.305052
$688$			6887.00i			0.381634i
$689$	10764.0	−	16146.0i	0.595175	−	0.892763i
$690$		0			0
$691$			9168.00i			0.504728i	0.967632	+	0.252364i	$0.0812081\pi$
$691$			9168.00i			0.504728i	−0.967632	+	0.252364i	$0.918792\pi$
$692$		−	1566.00i		−	0.0860266i
$693$		−	18720.0i		−	1.02614i
$694$			21141.0i			1.15634i
$695$		0			0
$696$			3024.00i			0.164690i
$697$	8640.00			0.469531
$698$		−	20619.0i		−	1.11811i
$699$	3627.00			0.196260
$700$		0			0
$701$	1170.00			0.0630389			0.0315195	−	0.999503i	$-0.489965\pi$
$701$	1170.00			0.0630389			0.0315195	+	0.999503i	$0.489965\pi$
$702$	4134.00	−	6201.00i	0.222262	−	0.333393i
$703$			1818.00i			0.0975351i
$704$			20784.0i			1.11268i
$705$		0			0
$706$	27954.0			1.49017
$707$	−5940.00			−0.315978
$708$	522.000			0.0277090
$709$		−	1662.00i		−	0.0880363i	−0.999031	−	0.0440181i	$-0.985984\pi$
$709$		−	1662.00i		−	0.0880363i	0.999031	−	0.0440181i	$-0.0140159\pi$
$710$		0			0
$711$	21580.0			1.13827
$712$			9198.00i			0.484143i
$713$	−42768.0			−2.24639
$714$	−2025.00			−0.106140
$715$		0			0
$716$	657.000			0.0342922
$717$	−6075.00			−0.316423
$718$			12384.0i			0.643686i
$719$	−30960.0			−1.60586			−0.802930	−	0.596073i	$-0.796727\pi$
$719$	−30960.0			−1.60586			−0.802930	+	0.596073i	$0.796727\pi$
$720$		0			0
$721$		−	2730.00i		−	0.141013i
$722$	20469.0			1.05509
$723$	−210.000			−0.0108022
$724$	−1222.00			−0.0627283
$725$		0			0
$726$			2919.00i			0.149221i
$727$			8372.00i			0.427098i	0.976932	+	0.213549i	$0.0685023\pi$
$727$			8372.00i			0.427098i	−0.976932	+	0.213549i	$0.931498\pi$
$728$	12285.0	+	8190.00i	0.625429	+	0.416953i
$729$	15443.0			0.784586
$730$		0			0
$731$	4365.00			0.220855
$732$		−	376.000i		−	0.0189855i
$733$	2739.00			0.138018			0.0690091	−	0.997616i	$-0.478016\pi$
$733$	2739.00			0.138018			0.0690091	+	0.997616i	$0.478016\pi$
$734$			7608.00i			0.382584i
$735$		0			0
$736$		−	7290.00i		−	0.365099i
$737$		−	1728.00i		−	0.0863659i
$738$		−	14976.0i		−	0.746984i
$739$		−	6756.00i		−	0.336297i	−0.985762	−	0.168148i	$-0.946221\pi$
$739$		−	6756.00i		−	0.336297i	0.985762	−	0.168148i	$-0.0537788\pi$
$740$		0			0
$741$	−234.000	−	156.000i	−0.0116008	−	0.00773388i
$742$			18630.0i			0.921737i
$743$	−29643.0			−1.46366			−0.731828	−	0.681490i	$-0.761332\pi$
$743$	−29643.0			−1.46366			−0.731828	+	0.681490i	$0.761332\pi$
$744$	−5544.00			−0.273189
$745$		0			0
$746$		−	276.000i		−	0.0135457i
$747$	11388.0			0.557785
$748$	−2160.00			−0.105585
$749$		−	9180.00i		−	0.447837i
$750$		0			0
$751$	18128.0			0.880826			0.440413	−	0.897795i	$-0.354832\pi$
$751$	18128.0			0.880826			0.440413	+	0.897795i	$0.354832\pi$
$752$	7881.00			0.382168
$753$		−	7092.00i		−	0.343223i
$754$	16848.0	+	11232.0i	0.813751	+	0.542500i
$755$		0			0
$756$			795.000i			0.0382459i
$757$			6410.00i			0.307761i	0.988089	+	0.153881i	$0.0491772\pi$
$757$			6410.00i			0.307761i	−0.988089	+	0.153881i	$0.950823\pi$
$758$		−	30546.0i		−	1.46369i
$759$			7776.00i			0.371872i
$760$		0			0
$761$		−	28290.0i		−	1.34758i	−0.738921	−	0.673792i	$-0.764664\pi$
$761$		−	28290.0i		−	1.34758i	0.738921	−	0.673792i	$-0.235336\pi$
$762$	−6258.00			−0.297511
$763$			16245.0i			0.770784i
$764$	1260.00			0.0596665
$765$		0			0
$766$	1737.00			0.0819326
$767$	−13572.0	+	20358.0i	−0.638926	+	0.958390i
$768$		−	1513.00i		−	0.0710881i
$769$			27960.0i			1.31114i	0.755136	+	0.655568i	$0.227571\pi$
$769$			27960.0i			1.31114i	−0.755136	+	0.655568i	$0.772429\pi$
$770$		0			0
$771$	5805.00			0.271157
$772$	342.000			0.0159441
$773$	5649.00			0.262847			0.131423	−	0.991326i	$-0.458045\pi$
$773$	5649.00			0.262847			0.131423	+	0.991326i	$0.458045\pi$
$774$		−	7566.00i		−	0.351362i
$775$		0			0
$776$	17892.0			0.827687
$777$		−	4545.00i		−	0.209847i
$778$	−6318.00			−0.291146
$779$	−1152.00			−0.0529842
$780$		0			0
$781$	−17136.0			−0.785114
$782$	−21870.0			−1.00009
$783$		−	7632.00i		−	0.348334i
$784$	8378.00			0.381651
$785$		0			0
$786$			4401.00i			0.199718i
$787$	−756.000			−0.0342420			−0.0171210	−	0.999853i	$-0.505450\pi$
$787$	−756.000			−0.0342420			−0.0171210	+	0.999853i	$0.505450\pi$
$788$	81.0000			0.00366181
$789$	792.000			0.0357363
$790$		0			0
$791$		−	1350.00i		−	0.0606833i
$792$		−	26208.0i		−	1.17583i
$793$	14664.0	+	9776.00i	0.656663	+	0.437775i
$794$	5922.00			0.264690
$795$		0			0
$796$	−1996.00			−0.0888773
$797$		−	31194.0i		−	1.38638i	−0.720753	−	0.693192i	$-0.756204\pi$
$797$		−	31194.0i		−	1.38638i	0.720753	−	0.693192i	$-0.243796\pi$
$798$	270.000			0.0119773
$799$		−	4995.00i		−	0.221164i
$800$		0			0
$801$		−	11388.0i		−	0.502341i
$802$		−	35658.0i		−	1.56998i
$803$		−	52704.0i		−	2.31617i
$804$			36.0000i			0.00157913i
$805$		0			0
$806$	−20592.0	+	30888.0i	−0.899904	+	1.34986i
$807$			5472.00i			0.238691i
$808$	−8316.00			−0.362074
$809$	−17055.0			−0.741189			−0.370594	−	0.928795i	$-0.620846\pi$
$809$	−17055.0			−0.741189			−0.370594	+	0.928795i	$0.620846\pi$
$810$		0			0
$811$			35520.0i			1.53795i	0.639280	+	0.768974i	$0.279232\pi$
$811$			35520.0i			1.53795i	−0.639280	+	0.768974i	$0.720768\pi$
$812$	−2160.00			−0.0933512
$813$	−2331.00			−0.100556
$814$		−	43632.0i		−	1.87875i
$815$		0			0
$816$	−3195.00			−0.137068
$817$	−582.000			−0.0249224
$818$		−	3762.00i		−	0.160801i
$819$	−15210.0	−	10140.0i	−0.648938	−	0.432625i
$820$		0			0
$821$			1095.00i			0.0465478i	0.999729	+	0.0232739i	$0.00740899\pi$
$821$			1095.00i			0.0465478i	−0.999729	+	0.0232739i	$0.992591\pi$
$822$		−	1242.00i		−	0.0527004i
$823$		−	2554.00i		−	0.108174i	−0.998536	−	0.0540868i	$-0.982775\pi$
$823$		−	2554.00i		−	0.108174i	0.998536	−	0.0540868i	$-0.0172248\pi$
$824$		−	3822.00i		−	0.161585i
$825$		0			0
$826$		−	23490.0i		−	0.989494i
$827$	−21522.0			−0.904950			−0.452475	−	0.891777i	$-0.649459\pi$
$827$	−21522.0			−0.904950			−0.452475	+	0.891777i	$0.649459\pi$
$828$			4212.00i			0.176784i
$829$	13124.0			0.549838			0.274919	−	0.961467i	$-0.411349\pi$
$829$	13124.0			0.549838			0.274919	+	0.961467i	$0.411349\pi$
$830$		0			0
$831$	1384.00			0.0577743
$832$	16887.0	+	11258.0i	0.703668	+	0.469112i
$833$		−	5310.00i		−	0.220865i
$834$		−	7257.00i		−	0.301306i
$835$		0			0
$836$	288.000			0.0119147
$837$	13992.0			0.577819
$838$	−17469.0			−0.720115
$839$		−	23424.0i		−	0.963869i	−0.876207	−	0.481935i	$-0.839934\pi$
$839$		−	23424.0i		−	0.963869i	0.876207	−	0.481935i	$-0.160066\pi$
$840$		0			0
$841$	−3653.00			−0.149781
$842$		−	22023.0i		−	0.901381i
$843$	4062.00			0.165958
$844$	2833.00			0.115540
$845$		0			0
$846$	−8658.00			−0.351854
$847$	14595.0			0.592078
$848$			29394.0i			1.19032i
$849$	−3764.00			−0.152156
$850$		0			0
$851$		−	49086.0i		−	1.97726i
$852$	357.000			0.0143552
$853$	31077.0			1.24743			0.623714	−	0.781653i	$-0.285623\pi$
$853$	31077.0			1.24743			0.623714	+	0.781653i	$0.285623\pi$
$854$	−16920.0			−0.677975
$855$		0			0
$856$		−	12852.0i		−	0.513169i
$857$			19422.0i			0.774146i	0.922049	+	0.387073i	$0.126514\pi$
$857$			19422.0i			0.774146i	−0.922049	+	0.387073i	$0.873486\pi$
$858$	5616.00	+	3744.00i	0.223458	+	0.148972i
$859$	−1744.00			−0.0692718			−0.0346359	−	0.999400i	$-0.511027\pi$
$859$	−1744.00			−0.0692718			−0.0346359	+	0.999400i	$0.511027\pi$
$860$		0			0
$861$	2880.00			0.113996
$862$		−	22455.0i		−	0.887263i
$863$	−19179.0			−0.756501			−0.378251	−	0.925703i	$-0.623474\pi$
$863$	−19179.0			−0.756501			−0.378251	+	0.925703i	$0.623474\pi$
$864$			2385.00i			0.0939113i
$865$		0			0
$866$		−	45609.0i		−	1.78967i
$867$		−	2888.00i		−	0.113128i
$868$		−	3960.00i		−	0.154852i
$869$			39840.0i			1.55521i
$870$		0			0
$871$	−1404.00	−	936.000i	−0.0546185	−	0.0364123i
$872$			22743.0i			0.883228i
$873$	−22152.0			−0.858799
$874$	2916.00			0.112855
$875$		0			0
$876$			1098.00i			0.0423493i
$877$	29217.0			1.12496			0.562479	−	0.826812i	$-0.309848\pi$
$877$	29217.0			1.12496			0.562479	+	0.826812i	$0.309848\pi$
$878$	5286.00			0.203182
$879$		−	4227.00i		−	0.162199i
$880$		0			0
$881$	−15633.0			−0.597831			−0.298916	−	0.954280i	$-0.596625\pi$
$881$	−15633.0			−0.597831			−0.298916	+	0.954280i	$0.596625\pi$
$882$	−9204.00			−0.351377
$883$			30589.0i			1.16580i	0.812544	+	0.582900i	$0.198082\pi$
$883$			30589.0i			1.16580i	−0.812544	+	0.582900i	$0.801918\pi$
$884$	−1170.00	+	1755.00i	−0.0445151	+	0.0667727i
$885$		0			0
$886$		−	21951.0i		−	0.832346i
$887$			25884.0i			0.979819i	0.871773	+	0.489910i	$0.162970\pi$
$887$			25884.0i			0.979819i	−0.871773	+	0.489910i	$0.837030\pi$
$888$		−	6363.00i		−	0.240460i
$889$			31290.0i			1.18046i
$890$		0			0
$891$			31152.0i			1.17130i
$892$	3507.00			0.131640
$893$			666.000i			0.0249573i
$894$	2790.00			0.104375
$895$		0			0
$896$	−24885.0			−0.927845
$897$	6318.00	+	4212.00i	0.235175	+	0.156783i
$898$		−	15048.0i		−	0.559196i
$899$			38016.0i			1.41035i
$900$		0			0
$901$	18630.0			0.688852
$902$	27648.0			1.02060
$903$	1455.00			0.0536206
$904$		−	1890.00i		−	0.0695359i
$905$		0			0
$906$	5049.00			0.185145
$907$			12305.0i			0.450475i	0.974304	+	0.225237i	$0.0723158\pi$
$907$			12305.0i			0.450475i	−0.974304	+	0.225237i	$0.927684\pi$
$908$	−228.000			−0.00833309
$909$	10296.0			0.375684
$910$		0			0
$911$	29772.0			1.08276			0.541378	−	0.840779i	$-0.317903\pi$
$911$	29772.0			1.08276			0.541378	+	0.840779i	$0.317903\pi$
$912$	426.000			0.0154674
$913$			21024.0i			0.762095i
$914$	29610.0			1.07157
$915$		0			0
$916$			5493.00i			0.198137i
$917$	22005.0			0.792442
$918$	7155.00			0.257244
$919$	−47644.0			−1.71015			−0.855076	−	0.518502i	$-0.826490\pi$
$919$	−47644.0			−1.71015			−0.855076	+	0.518502i	$0.826490\pi$
$920$		0			0
$921$			306.000i			0.0109479i
$922$		−	43623.0i		−	1.55819i
$923$	−9282.00	+	13923.0i	−0.331008	+	0.496513i
$924$	−720.000			−0.0256345
$925$		0			0
$926$	−6336.00			−0.224853
$927$			4732.00i			0.167658i
$928$	−6480.00			−0.229220
$929$		−	21924.0i		−	0.774277i	−0.922022	−	0.387138i	$-0.873464\pi$
$929$		−	21924.0i		−	0.774277i	0.922022	−	0.387138i	$-0.126536\pi$
$930$		0			0
$931$			708.000i			0.0249235i
$932$			3627.00i			0.127475i
$933$			2106.00i			0.0738985i
$934$		−	9828.00i		−	0.344306i
$935$		0			0
$936$	−21294.0	−	14196.0i	−0.743607	−	0.495738i
$937$		−	32398.0i		−	1.12956i	−0.825242	−	0.564779i	$-0.808961\pi$
$937$		−	32398.0i		−	1.12956i	0.825242	−	0.564779i	$-0.191039\pi$
$938$	1620.00			0.0563911
$939$	10051.0			0.349310
$940$		0			0
$941$			2097.00i			0.0726464i	0.999340	+	0.0363232i	$0.0115646\pi$
$941$			2097.00i			0.0726464i	−0.999340	+	0.0363232i	$0.988435\pi$
$942$	−5622.00			−0.194453
$943$	31104.0			1.07411
$944$		−	37062.0i		−	1.27782i
$945$		0			0
$946$	13968.0			0.480062
$947$	20016.0			0.686835			0.343417	−	0.939183i	$-0.388415\pi$
$947$	20016.0			0.686835			0.343417	+	0.939183i	$0.388415\pi$
$948$		−	830.000i		−	0.0284358i
$949$	−42822.0	−	28548.0i	−1.46476	−	0.976509i
$950$		0			0
$951$			2154.00i			0.0734471i
$952$			14175.0i			0.482578i
$953$			24993.0i			0.849531i	0.905304	+	0.424765i	$0.139643\pi$
$953$			24993.0i			0.849531i	−0.905304	+	0.424765i	$0.860357\pi$
$954$		−	32292.0i		−	1.09590i
$955$		0			0
$956$		−	6075.00i		−	0.205523i
$957$	6912.00			0.233473
$958$		−	46359.0i		−	1.56346i
$959$	−6210.00			−0.209105
$960$		0			0
$961$	−39905.0			−1.33950
$962$	−35451.0	−	23634.0i	−1.18814	−	0.792090i
$963$			15912.0i			0.532458i
$964$		−	210.000i		−	0.00701623i
$965$		0			0
$966$	−7290.00			−0.242807
$967$	−40959.0			−1.36210			−0.681051	−	0.732236i	$-0.738477\pi$
$967$	−40959.0			−1.36210			−0.681051	+	0.732236i	$0.738477\pi$
$968$	20433.0			0.678452
$969$		−	270.000i		−	0.00895113i
$970$		0			0
$971$	−48933.0			−1.61723			−0.808617	−	0.588335i	$-0.799784\pi$
$971$	−48933.0			−1.61723			−0.808617	+	0.588335i	$0.799784\pi$
$972$		−	2080.00i		−	0.0686379i
$973$	−36285.0			−1.19552
$974$	−10980.0			−0.361213
$975$		0			0
$976$	−26696.0			−0.875531
$977$	47388.0			1.55177			0.775884	−	0.630876i	$-0.217304\pi$
$977$	47388.0			1.55177			0.775884	+	0.630876i	$0.217304\pi$
$978$			3582.00i			0.117116i
$979$	21024.0			0.686343
$980$		0			0
$981$		−	28158.0i		−	0.916428i
$982$	2241.00			0.0728240
$983$	16803.0			0.545201			0.272600	−	0.962127i	$-0.412116\pi$
$983$	16803.0			0.545201			0.272600	+	0.962127i	$0.412116\pi$
$984$	4032.00			0.130625
$985$		0			0
$986$			19440.0i			0.627886i
$987$		−	1665.00i		−	0.0536956i
$988$	156.000	−	234.000i	0.00502330	−	0.00753495i
$989$	15714.0			0.505234
$990$		0			0
$991$	−57526.0			−1.84397			−0.921985	−	0.387226i	$-0.873433\pi$
$991$	−57526.0			−1.84397			−0.921985	+	0.387226i	$0.873433\pi$
$992$		−	11880.0i		−	0.380232i
$993$	10770.0			0.344185
$994$		−	16065.0i		−	0.512627i
$995$		0			0
$996$		−	438.000i		−	0.0139343i
$997$			25000.0i			0.794140i	0.917788	+	0.397070i	$0.129973\pi$
$997$			25000.0i			0.794140i	−0.917788	+	0.397070i	$0.870027\pi$
$998$			47412.0i			1.50381i
$999$			16059.0i			0.508593i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.4.d.b.324.1		2
5.2	odd	4		13.4.b.a.12.2	yes	2
5.3	odd	4		325.4.c.b.51.1		2
5.4	even	2		325.4.d.a.324.2		2
13.12	even	2		325.4.d.a.324.1		2
15.2	even	4		117.4.b.a.64.1		2
20.7	even	4		208.4.f.b.129.1		2
40.27	even	4		832.4.f.c.129.2		2
40.37	odd	4		832.4.f.e.129.2		2
65.2	even	12		169.4.c.b.22.1		2
65.7	even	12		169.4.c.c.146.1		2
65.12	odd	4		13.4.b.a.12.1	✓	2
65.17	odd	12		169.4.e.d.23.1		4
65.22	odd	12		169.4.e.d.23.2		4
65.32	even	12		169.4.c.b.146.1		2
65.37	even	12		169.4.c.c.22.1		2
65.38	odd	4		325.4.c.b.51.2		2
65.42	odd	12		169.4.e.d.147.1		4
65.47	even	4		169.4.a.b.1.1		1
65.57	even	4		169.4.a.c.1.1		1
65.62	odd	12		169.4.e.d.147.2		4
65.64	even	2	inner	325.4.d.b.324.2		2
195.47	odd	4		1521.4.a.i.1.1		1
195.77	even	4		117.4.b.a.64.2		2
195.122	odd	4		1521.4.a.d.1.1		1
260.207	even	4		208.4.f.b.129.2		2
520.77	odd	4		832.4.f.e.129.1		2
520.467	even	4		832.4.f.c.129.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.b.a.12.1	✓	2	65.12	odd	4
13.4.b.a.12.2	yes	2	5.2	odd	4
117.4.b.a.64.1		2	15.2	even	4
117.4.b.a.64.2		2	195.77	even	4
169.4.a.b.1.1		1	65.47	even	4
169.4.a.c.1.1		1	65.57	even	4
169.4.c.b.22.1		2	65.2	even	12
169.4.c.b.146.1		2	65.32	even	12
169.4.c.c.22.1		2	65.37	even	12
169.4.c.c.146.1		2	65.7	even	12
169.4.e.d.23.1		4	65.17	odd	12
169.4.e.d.23.2		4	65.22	odd	12
169.4.e.d.147.1		4	65.42	odd	12
169.4.e.d.147.2		4	65.62	odd	12
208.4.f.b.129.1		2	20.7	even	4
208.4.f.b.129.2		2	260.207	even	4
325.4.c.b.51.1		2	5.3	odd	4
325.4.c.b.51.2		2	65.38	odd	4
325.4.d.a.324.1		2	13.12	even	2
325.4.d.a.324.2		2	5.4	even	2
325.4.d.b.324.1		2	1.1	even	1	trivial
325.4.d.b.324.2		2	65.64	even	2	inner
832.4.f.c.129.1		2	520.467	even	4
832.4.f.c.129.2		2	40.27	even	4
832.4.f.e.129.1		2	520.77	odd	4
832.4.f.e.129.2		2	40.37	odd	4
1521.4.a.d.1.1		1	195.122	odd	4
1521.4.a.i.1.1		1	195.47	odd	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	325.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -3.00000i q^{6} -15.0000 q^{7} -21.0000 q^{8} +26.0000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -3.00000i q^{6} -15.0000 q^{7} -21.0000 q^{8} +26.0000 q^{9} +48.0000i q^{11} -1.00000i q^{12} +(39.0000 + 26.0000i) q^{13} -45.0000 q^{14} -71.0000 q^{16} +45.0000i q^{17} +78.0000 q^{18} -6.00000i q^{19} +15.0000i q^{21} +144.000i q^{22} +162.000i q^{23} +21.0000i q^{24} +(117.000 + 78.0000i) q^{26} -53.0000i q^{27} -15.0000 q^{28} +144.000 q^{29} +264.000i q^{31} -45.0000 q^{32} +48.0000 q^{33} +135.000i q^{34} +26.0000 q^{36} -303.000 q^{37} -18.0000i q^{38} +(26.0000 - 39.0000i) q^{39} -192.000i q^{41} +45.0000i q^{42} -97.0000i q^{43} +48.0000i q^{44} +486.000i q^{46} -111.000 q^{47} +71.0000i q^{48} -118.000 q^{49} +45.0000 q^{51} +(39.0000 + 26.0000i) q^{52} -414.000i q^{53} -159.000i q^{54} +315.000 q^{56} -6.00000 q^{57} +432.000 q^{58} +522.000i q^{59} +376.000 q^{61} +792.000i q^{62} -390.000 q^{63} +433.000 q^{64} +144.000 q^{66} -36.0000 q^{67} +45.0000i q^{68} +162.000 q^{69} +357.000i q^{71} -546.000 q^{72} -1098.00 q^{73} -909.000 q^{74} -6.00000i q^{76} -720.000i q^{77} +(78.0000 - 117.000i) q^{78} +830.000 q^{79} +649.000 q^{81} -576.000i q^{82} +438.000 q^{83} +15.0000i q^{84} -291.000i q^{86} -144.000i q^{87} -1008.00i q^{88} -438.000i q^{89} +(-585.000 - 390.000i) q^{91} +162.000i q^{92} +264.000 q^{93} -333.000 q^{94} +45.0000i q^{96} -852.000 q^{97} -354.000 q^{98} +1248.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{2} + 2 q^{4} - 30 q^{7} - 42 q^{8} + 52 q^{9}+O(q^{10}) \) 2 * q + 6 * q^2 + 2 * q^4 - 30 * q^7 - 42 * q^8 + 52 * q^9	\( 2 q + 6 q^{2} + 2 q^{4} - 30 q^{7} - 42 q^{8} + 52 q^{9} + 78 q^{13} - 90 q^{14} - 142 q^{16} + 156 q^{18} + 234 q^{26} - 30 q^{28} + 288 q^{29} - 90 q^{32} + 96 q^{33} + 52 q^{36} - 606 q^{37} + 52 q^{39} - 222 q^{47} - 236 q^{49} + 90 q^{51} + 78 q^{52} + 630 q^{56} - 12 q^{57} + 864 q^{58} + 752 q^{61} - 780 q^{63} + 866 q^{64} + 288 q^{66} - 72 q^{67} + 324 q^{69} - 1092 q^{72} - 2196 q^{73} - 1818 q^{74} + 156 q^{78} + 1660 q^{79} + 1298 q^{81} + 876 q^{83} - 1170 q^{91} + 528 q^{93} - 666 q^{94} - 1704 q^{97} - 708 q^{98}+O(q^{100}) \) 2 * q + 6 * q^2 + 2 * q^4 - 30 * q^7 - 42 * q^8 + 52 * q^9 + 78 * q^13 - 90 * q^14 - 142 * q^16 + 156 * q^18 + 234 * q^26 - 30 * q^28 + 288 * q^29 - 90 * q^32 + 96 * q^33 + 52 * q^36 - 606 * q^37 + 52 * q^39 - 222 * q^47 - 236 * q^49 + 90 * q^51 + 78 * q^52 + 630 * q^56 - 12 * q^57 + 864 * q^58 + 752 * q^61 - 780 * q^63 + 866 * q^64 + 288 * q^66 - 72 * q^67 + 324 * q^69 - 1092 * q^72 - 2196 * q^73 - 1818 * q^74 + 156 * q^78 + 1660 * q^79 + 1298 * q^81 + 876 * q^83 - 1170 * q^91 + 528 * q^93 - 666 * q^94 - 1704 * q^97 - 708 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more