LMFDB - Embedded newform 325.4.c.b.51.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,4,Mod(51,325)]

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([0, 1]))

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

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

chi := DirichletCharacter("325.51");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	325.c (of order $2$, degree $1$, 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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			51.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	325.51
Dual form			325.4.c.b.51.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.00000i q^{6} -15.0000i q^{7} +21.0000i q^{8} -26.0000 q^{9} +O(q^{10})$	\(q+3.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.00000i q^{6} -15.0000i q^{7} +21.0000i q^{8} -26.0000 q^{9} -48.0000i q^{11} -1.00000 q^{12} +(-26.0000 - 39.0000i) q^{13} +45.0000 q^{14} -71.0000 q^{16} +45.0000 q^{17} -78.0000i q^{18} -6.00000i q^{19} -15.0000i q^{21} +144.000 q^{22} -162.000 q^{23} +21.0000i q^{24} +(117.000 - 78.0000i) q^{26} -53.0000 q^{27} +15.0000i q^{28} -144.000 q^{29} -264.000i q^{31} -45.0000i q^{32} -48.0000i q^{33} +135.000i q^{34} +26.0000 q^{36} -303.000i q^{37} +18.0000 q^{38} +(-26.0000 - 39.0000i) q^{39} +192.000i q^{41} +45.0000 q^{42} +97.0000 q^{43} +48.0000i q^{44} -486.000i q^{46} -111.000i q^{47} -71.0000 q^{48} +118.000 q^{49} +45.0000 q^{51} +(26.0000 + 39.0000i) q^{52} +414.000 q^{53} -159.000i q^{54} +315.000 q^{56} -6.00000i q^{57} -432.000i q^{58} +522.000i q^{59} +376.000 q^{61} +792.000 q^{62} +390.000i q^{63} -433.000 q^{64} +144.000 q^{66} -36.0000i q^{67} -45.0000 q^{68} -162.000 q^{69} -357.000i q^{71} -546.000i q^{72} +1098.00i q^{73} +909.000 q^{74} +6.00000i q^{76} -720.000 q^{77} +(117.000 - 78.0000i) q^{78} -830.000 q^{79} +649.000 q^{81} -576.000 q^{82} -438.000i q^{83} +15.0000i q^{84} +291.000i q^{86} -144.000 q^{87} +1008.00 q^{88} -438.000i q^{89} +(-585.000 + 390.000i) q^{91} +162.000 q^{92} -264.000i q^{93} +333.000 q^{94} -45.0000i q^{96} -852.000i q^{97} +354.000i q^{98} +1248.00i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{4} - 52 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^4 - 52 * q^9	$ 2 q + 2 q^{3} - 2 q^{4} - 52 q^{9} - 2 q^{12} - 52 q^{13} + 90 q^{14} - 142 q^{16} + 90 q^{17} + 288 q^{22} - 324 q^{23} + 234 q^{26} - 106 q^{27} - 288 q^{29} + 52 q^{36} + 36 q^{38} - 52 q^{39} + 90 q^{42} + 194 q^{43} - 142 q^{48} + 236 q^{49} + 90 q^{51} + 52 q^{52} + 828 q^{53} + 630 q^{56} + 752 q^{61} + 1584 q^{62} - 866 q^{64} + 288 q^{66} - 90 q^{68} - 324 q^{69} + 1818 q^{74} - 1440 q^{77} + 234 q^{78} - 1660 q^{79} + 1298 q^{81} - 1152 q^{82} - 288 q^{87} + 2016 q^{88} - 1170 q^{91} + 324 q^{92} + 666 q^{94}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^4 - 52 * q^9 - 2 * q^12 - 52 * q^13 + 90 * q^14 - 142 * q^16 + 90 * q^17 + 288 * q^22 - 324 * q^23 + 234 * q^26 - 106 * q^27 - 288 * q^29 + 52 * q^36 + 36 * q^38 - 52 * q^39 + 90 * q^42 + 194 * q^43 - 142 * q^48 + 236 * q^49 + 90 * q^51 + 52 * q^52 + 828 * q^53 + 630 * q^56 + 752 * q^61 + 1584 * q^62 - 866 * q^64 + 288 * q^66 - 90 * q^68 - 324 * q^69 + 1818 * q^74 - 1440 * q^77 + 234 * q^78 - 1660 * q^79 + 1298 * q^81 - 1152 * q^82 - 288 * q^87 + 2016 * q^88 - 1170 * q^91 + 324 * q^92 + 666 * q^94

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 $

$2$			3.00000i			1.06066i	0.847791	+	0.530330i	$0.177932\pi$
$2$			3.00000i			1.06066i	−0.847791	+	0.530330i	$0.822068\pi$
$3$	1.00000			0.192450			0.0962250	−	0.995360i	$-0.469323\pi$
$3$	1.00000			0.192450			0.0962250	+	0.995360i	$0.469323\pi$
$4$	−1.00000			−0.125000
$5$		0			0
$5$		0			0
$6$			3.00000i			0.204124i
$7$		−	15.0000i		−	0.809924i	−0.914334	−	0.404962i	$-0.867285\pi$
$7$		−	15.0000i		−	0.809924i	0.914334	−	0.404962i	$-0.132715\pi$
$8$			21.0000i			0.928078i
$9$	−26.0000			−0.962963
$10$		0			0
$11$		−	48.0000i		−	1.31569i	−0.753155	−	0.657843i	$-0.771469\pi$
$11$		−	48.0000i		−	1.31569i	0.753155	−	0.657843i	$-0.228531\pi$
$12$	−1.00000			−0.0240563
$13$	−26.0000	−	39.0000i	−0.554700	−	0.832050i
$13$	−26.0000	−	39.0000i	−0.554700	−	0.832050i
$14$	45.0000			0.859054
$15$		0			0
$16$	−71.0000			−1.10938
$17$	45.0000			0.642006			0.321003	−	0.947078i	$-0.395980\pi$
$17$	45.0000			0.642006			0.321003	+	0.947078i	$0.395980\pi$
$18$		−	78.0000i		−	1.02138i
$19$		−	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
$22$	144.000			1.39550
$23$	−162.000			−1.46867			−0.734333	−	0.678789i	$-0.762505\pi$
$23$	−162.000			−1.46867			−0.734333	+	0.678789i	$0.762505\pi$
$24$			21.0000i			0.178609i
$25$		0			0
$26$	117.000	−	78.0000i	0.882523	−	0.588348i
$27$	−53.0000			−0.377772
$28$			15.0000i			0.101240i
$29$	−144.000			−0.922073			−0.461037	−	0.887381i	$-0.652522\pi$
$29$	−144.000			−0.922073			−0.461037	+	0.887381i	$0.652522\pi$
$30$		0			0
$31$		−	264.000i		−	1.52954i	−0.644302	−	0.764771i	$-0.722852\pi$
$31$		−	264.000i		−	1.52954i	0.644302	−	0.764771i	$-0.277148\pi$
$32$		−	45.0000i		−	0.248592i
$33$		−	48.0000i		−	0.253204i
$34$			135.000i			0.680950i
$35$		0			0
$36$	26.0000			0.120370
$37$		−	303.000i		−	1.34629i	−0.739509	−	0.673147i	$-0.764942\pi$
$37$		−	303.000i		−	1.34629i	0.739509	−	0.673147i	$-0.235058\pi$
$38$	18.0000			0.0768417
$39$	−26.0000	−	39.0000i	−0.106752	−	0.160128i
$40$		0			0
$41$			192.000i			0.731350i	0.930743	+	0.365675i	$0.119162\pi$
$41$			192.000i			0.731350i	−0.930743	+	0.365675i	$0.880838\pi$
$42$	45.0000			0.165325
$43$	97.0000			0.344008			0.172004	−	0.985096i	$-0.444976\pi$
$43$	97.0000			0.344008			0.172004	+	0.985096i	$0.444976\pi$
$44$			48.0000i			0.164461i
$45$		0			0
$46$		−	486.000i		−	1.55776i
$47$		−	111.000i		−	0.344490i	−0.985054	−	0.172245i	$-0.944898\pi$
$47$		−	111.000i		−	0.344490i	0.985054	−	0.172245i	$-0.0551020\pi$
$48$	−71.0000			−0.213499
$49$	118.000			0.344023
$50$		0			0
$51$	45.0000			0.123554
$52$	26.0000	+	39.0000i	0.0693375	+	0.104006i
$53$	414.000			1.07297			0.536484	−	0.843911i	$-0.319752\pi$
$53$	414.000			1.07297			0.536484	+	0.843911i	$0.319752\pi$
$54$		−	159.000i		−	0.400688i
$55$		0			0
$56$	315.000			0.751672
$57$		−	6.00000i		−	0.0139424i
$58$		−	432.000i		−	0.978007i
$59$			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	$0.370881\pi$
$62$	792.000			1.62232
$63$			390.000i			0.779927i
$64$	−433.000			−0.845703
$65$		0			0
$66$	144.000			0.268563
$67$		−	36.0000i		−	0.0656433i	−0.999461	−	0.0328216i	$-0.989551\pi$
$67$		−	36.0000i		−	0.0656433i	0.999461	−	0.0328216i	$-0.0104493\pi$
$68$	−45.0000			−0.0802508
$69$	−162.000			−0.282645
$70$		0			0
$71$		−	357.000i		−	0.596734i	−0.954451	−	0.298367i	$-0.903558\pi$
$71$		−	357.000i		−	0.596734i	0.954451	−	0.298367i	$-0.0964419\pi$
$72$		−	546.000i		−	0.893704i
$73$			1098.00i			1.76043i	0.474578	+	0.880214i	$0.342601\pi$
$73$			1098.00i			1.76043i	−0.474578	+	0.880214i	$0.657399\pi$
$74$	909.000			1.42796
$75$		0			0
$76$			6.00000i			0.00905588i
$77$	−720.000			−1.06561
$78$	117.000	−	78.0000i	0.169842	−	0.113228i
$79$	−830.000			−1.18205			−0.591027	−	0.806652i	$-0.701277\pi$
$79$	−830.000			−1.18205			−0.591027	+	0.806652i	$0.701277\pi$
$80$		0			0
$81$	649.000			0.890261
$82$	−576.000			−0.775714
$83$		−	438.000i		−	0.579238i	−0.957142	−	0.289619i	$-0.906471\pi$
$83$		−	438.000i		−	0.579238i	0.957142	−	0.289619i	$-0.0935286\pi$
$84$			15.0000i			0.0194837i
$85$		0			0
$86$			291.000i			0.364876i
$87$	−144.000			−0.177453
$88$	1008.00			1.22106
$89$		−	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
$92$	162.000			0.183583
$93$		−	264.000i		−	0.294360i
$94$	333.000			0.365386
$95$		0			0
$96$		−	45.0000i		−	0.0478416i
$97$		−	852.000i		−	0.891830i	−0.895075	−	0.445915i	$-0.852878\pi$
$97$		−	852.000i		−	0.891830i	0.895075	−	0.445915i	$-0.147122\pi$
$98$			354.000i			0.364892i
$99$			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	$0.437508\pi$
$102$			135.000i			0.131049i
$103$	−182.000			−0.174107			−0.0870534	−	0.996204i	$-0.527745\pi$
$103$	−182.000			−0.174107			−0.0870534	+	0.996204i	$0.527745\pi$
$104$	819.000	−	546.000i	0.772207	−	0.514805i
$105$		0			0
$106$			1242.00i			1.13805i
$107$	612.000			0.552937			0.276469	−	0.961023i	$-0.410836\pi$
$107$	612.000			0.552937			0.276469	+	0.961023i	$0.410836\pi$
$108$	53.0000			0.0472215
$109$		−	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
$112$			1065.00i			0.898509i
$113$	−90.0000			−0.0749247			−0.0374623	−	0.999298i	$-0.511927\pi$
$113$	−90.0000			−0.0749247			−0.0374623	+	0.999298i	$0.511927\pi$
$114$	18.0000			0.0147882
$115$		0			0
$116$	144.000			0.115259
$117$	676.000	+	1014.00i	0.534156	+	0.801234i
$118$	−1566.00			−1.22171
$119$		−	675.000i		−	0.519976i
$120$		0			0
$121$	−973.000			−0.731029
$122$			1128.00i			0.837085i
$123$			192.000i			0.140748i
$124$			264.000i			0.191193i
$125$		0			0
$126$	−1170.00			−0.827237
$127$	−2086.00			−1.45750			−0.728750	−	0.684780i	$-0.759898\pi$
$127$	−2086.00			−1.45750			−0.728750	+	0.684780i	$0.759898\pi$
$128$		−	1659.00i		−	1.14560i
$129$	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	$0.662714\pi$
$132$			48.0000i			0.0316505i
$133$	−90.0000			−0.0586766
$134$	108.000			0.0696252
$135$		0			0
$136$			945.000i			0.595831i
$137$			414.000i			0.258178i	0.991633	+	0.129089i	$0.0412053\pi$
$137$			414.000i			0.258178i	−0.991633	+	0.129089i	$0.958795\pi$
$138$		−	486.000i		−	0.299790i
$139$	−2419.00			−1.47609			−0.738046	−	0.674750i	$-0.764251\pi$
$139$	−2419.00			−1.47609			−0.738046	+	0.674750i	$0.764251\pi$
$140$		0			0
$141$		−	111.000i		−	0.0662971i
$142$	1071.00			0.632932
$143$	−1872.00	+	1248.00i	−1.09472	+	0.729811i
$144$	1846.00			1.06829
$145$		0			0
$146$	−3294.00			−1.86721
$147$	118.000			0.0662073
$148$			303.000i			0.168287i
$149$			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	$-0.850171\pi$
$151$		−	1683.00i		−	0.907024i	0.891250	−	0.453512i	$-0.149829\pi$
$152$	126.000			0.0672365
$153$	−1170.00			−0.618228
$154$		−	2160.00i		−	1.13025i
$155$		0			0
$156$	26.0000	+	39.0000i	0.0133440	+	0.0200160i
$157$	−1874.00			−0.952621			−0.476310	−	0.879277i	$-0.658026\pi$
$157$	−1874.00			−0.952621			−0.476310	+	0.879277i	$0.658026\pi$
$158$		−	2490.00i		−	1.25376i
$159$	414.000			0.206493
$160$		0			0
$161$			2430.00i			1.18951i
$162$			1947.00i			0.944264i
$163$			1194.00i			0.573750i	0.957968	+	0.286875i	$0.0926165\pi$
$163$			1194.00i			0.573750i	−0.957968	+	0.286875i	$0.907384\pi$
$164$		−	192.000i		−	0.0914188i
$165$		0			0
$166$	1314.00			0.614375
$167$			2388.00i			1.10652i	0.833008	+	0.553260i	$0.186617\pi$
$167$			2388.00i			1.10652i	−0.833008	+	0.553260i	$0.813383\pi$
$168$	315.000			0.144659
$169$	−845.000	+	2028.00i	−0.384615	+	0.923077i
$170$		0			0
$171$			156.000i			0.0697638i
$172$	−97.0000			−0.0430011
$173$	1566.00			0.688213			0.344106	−	0.938931i	$-0.388182\pi$
$173$	1566.00			0.688213			0.344106	+	0.938931i	$0.388182\pi$
$174$		−	432.000i		−	0.188217i
$175$		0			0
$176$			3408.00i			1.45959i
$177$			522.000i			0.221672i
$178$	1314.00			0.553306
$179$	−657.000			−0.274338			−0.137169	−	0.990548i	$-0.543800\pi$
$179$	−657.000			−0.274338			−0.137169	+	0.990548i	$0.543800\pi$
$180$		0			0
$181$	−1222.00			−0.501826			−0.250913	−	0.968010i	$-0.580731\pi$
$181$	−1222.00			−0.501826			−0.250913	+	0.968010i	$0.580731\pi$
$182$	−1170.00	−	1755.00i	−0.476517	−	0.714776i
$183$	376.000			0.151884
$184$		−	3402.00i		−	1.36304i
$185$		0			0
$186$	792.000			0.312216
$187$		−	2160.00i		−	0.844678i
$188$			111.000i			0.0430612i
$189$			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	$0.423290\pi$
$192$	−433.000			−0.162756
$193$		−	342.000i		−	0.127553i	−0.997964	−	0.0637764i	$-0.979686\pi$
$193$		−	342.000i		−	0.127553i	0.997964	−	0.0637764i	$-0.0203145\pi$
$194$	2556.00			0.945928
$195$		0			0
$196$	−118.000			−0.0430029
$197$			81.0000i			0.0292945i	0.999893	+	0.0146472i	$0.00466253\pi$
$197$			81.0000i			0.0292945i	−0.999893	+	0.0146472i	$0.995337\pi$
$198$	−3744.00			−1.34381
$199$	1996.00			0.711019			0.355509	−	0.934673i	$-0.384307\pi$
$199$	1996.00			0.711019			0.355509	+	0.934673i	$0.384307\pi$
$200$		0			0
$201$		−	36.0000i		−	0.0126331i
$202$			1188.00i			0.413799i
$203$			2160.00i			0.746809i
$204$	−45.0000			−0.0154443
$205$		0			0
$206$		−	546.000i		−	0.184668i
$207$	4212.00			1.41427
$208$	1846.00	+	2769.00i	0.615371	+	0.923056i
$209$	−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	$0.347074\pi$
$212$	−414.000			−0.134121
$213$		−	357.000i		−	0.114841i
$214$			1836.00i			0.586478i
$215$		0			0
$216$		−	1113.00i		−	0.350602i
$217$	−3960.00			−1.23881
$218$	3249.00			1.00940
$219$			1098.00i			0.338794i
$220$		0			0
$221$	−1170.00	−	1755.00i	−0.356121	−	0.534181i
$222$	909.000			0.274811
$223$		−	3507.00i		−	1.05312i	−0.850138	−	0.526561i	$-0.823481\pi$
$223$		−	3507.00i		−	1.05312i	0.850138	−	0.526561i	$-0.176519\pi$
$224$	−675.000			−0.201341
$225$		0			0
$226$		−	270.000i		−	0.0794696i
$227$		−	228.000i		−	0.0666647i	−0.999444	−	0.0333324i	$-0.989388\pi$
$227$		−	228.000i		−	0.0666647i	0.999444	−	0.0333324i	$-0.0106120\pi$
$228$			6.00000i			0.00174281i
$229$			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
$232$		−	3024.00i		−	0.855756i
$233$	−3627.00			−1.01980			−0.509898	−	0.860235i	$-0.670317\pi$
$233$	−3627.00			−1.01980			−0.509898	+	0.860235i	$0.670317\pi$
$234$	−3042.00	+	2028.00i	−0.849837	+	0.566558i
$235$		0			0
$236$		−	522.000i		−	0.143980i
$237$	−830.000			−0.227486
$238$	2025.00			0.551518
$239$		−	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	$0.00893451\pi$
$241$			210.000i			0.0561298i	−0.999606	+	0.0280649i	$0.991065\pi$
$242$		−	2919.00i		−	0.775374i
$243$	2080.00			0.549103
$244$	−376.000			−0.0986514
$245$		0			0
$246$	−576.000			−0.149286
$247$	−234.000	+	156.000i	−0.0602796	+	0.0401864i
$248$	5544.00			1.41953
$249$		−	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	$0.149499\pi$
$252$		−	390.000i		−	0.0974908i
$253$			7776.00i			1.93230i
$254$		−	6258.00i		−	1.54591i
$255$		0			0
$256$	1513.00			0.369385
$257$	5805.00			1.40897			0.704486	−	0.709718i	$-0.251177\pi$
$257$	5805.00			1.40897			0.704486	+	0.709718i	$0.251177\pi$
$258$			291.000i			0.0702204i
$259$	−4545.00			−1.09040
$260$		0			0
$261$	3744.00			0.887923
$262$		−	4401.00i		−	1.03777i
$263$	−792.000			−0.185691			−0.0928457	−	0.995681i	$-0.529596\pi$
$263$	−792.000			−0.185691			−0.0928457	+	0.995681i	$0.529596\pi$
$264$	1008.00			0.234993
$265$		0			0
$266$		−	270.000i		−	0.0622359i
$267$		−	438.000i		−	0.100394i
$268$			36.0000i			0.00820541i
$269$	5472.00			1.24027			0.620137	−	0.784493i	$-0.287077\pi$
$269$	5472.00			1.24027			0.620137	+	0.784493i	$0.287077\pi$
$270$		0			0
$271$			2331.00i			0.522502i	0.965271	+	0.261251i	$0.0841351\pi$
$271$			2331.00i			0.522502i	−0.965271	+	0.261251i	$0.915865\pi$
$272$	−3195.00			−0.712225
$273$	−585.000	+	390.000i	−0.129692	+	0.0864611i
$274$	−1242.00			−0.273839
$275$		0			0
$276$	162.000			0.0353306
$277$	1384.00			0.300204			0.150102	−	0.988671i	$-0.452040\pi$
$277$	1384.00			0.300204			0.150102	+	0.988671i	$0.452040\pi$
$278$		−	7257.00i		−	1.56563i
$279$			6864.00i			1.47289i
$280$		0			0
$281$		−	4062.00i		−	0.862344i	−0.902270	−	0.431172i	$-0.858100\pi$
$281$		−	4062.00i		−	0.862344i	0.902270	−	0.431172i	$-0.141900\pi$
$282$	333.000			0.0703187
$283$	3764.00			0.790624			0.395312	−	0.918547i	$-0.370636\pi$
$283$	3764.00			0.790624			0.395312	+	0.918547i	$0.370636\pi$
$284$			357.000i			0.0745917i
$285$		0			0
$286$	−3744.00	−	5616.00i	−0.774082	−	1.16112i
$287$	2880.00			0.592338
$288$			1170.00i			0.239385i
$289$	−2888.00			−0.587828
$290$		0			0
$291$		−	852.000i		−	0.171633i
$292$		−	1098.00i		−	0.220053i
$293$		−	4227.00i		−	0.842812i	−0.906872	−	0.421406i	$-0.861537\pi$
$293$		−	4227.00i		−	0.842812i	0.906872	−	0.421406i	$-0.138463\pi$
$294$			354.000i			0.0702235i
$295$		0			0
$296$	6363.00			1.24947
$297$			2544.00i			0.497030i
$298$	−2790.00			−0.542350
$299$	4212.00	+	6318.00i	0.814670	+	1.22200i
$300$		0			0
$301$		−	1455.00i		−	0.278621i
$302$	5049.00			0.962044
$303$	396.000			0.0750812
$304$			426.000i			0.0803710i
$305$		0			0
$306$		−	3510.00i		−	0.655730i
$307$		−	306.000i		−	0.0568871i	−0.999595	−	0.0284436i	$-0.990945\pi$
$307$		−	306.000i		−	0.0568871i	0.999595	−	0.0284436i	$-0.00905509\pi$
$308$	720.000			0.133201
$309$	−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	$0.561495\pi$
$312$	819.000	−	546.000i	0.148611	−	0.0990742i
$313$	−10051.0			−1.81507			−0.907534	−	0.419979i	$-0.862037\pi$
$313$	−10051.0			−1.81507			−0.907534	+	0.419979i	$0.862037\pi$
$314$		−	5622.00i		−	1.01041i
$315$		0			0
$316$	830.000			0.147757
$317$		−	2154.00i		−	0.381643i	−0.981625	−	0.190821i	$-0.938885\pi$
$317$		−	2154.00i		−	0.381643i	0.981625	−	0.190821i	$-0.0611151\pi$
$318$			1242.00i			0.219019i
$319$			6912.00i			1.21316i
$320$		0			0
$321$	612.000			0.106413
$322$	−7290.00			−1.26166
$323$		−	270.000i		−	0.0465115i
$324$	−649.000			−0.111283
$325$		0			0
$326$	−3582.00			−0.608554
$327$		−	1083.00i		−	0.183150i
$328$	−4032.00			−0.678750
$329$	−1665.00			−0.279010
$330$		0			0
$331$		−	10770.0i		−	1.78844i	−0.447630	−	0.894219i	$-0.647732\pi$
$331$		−	10770.0i		−	1.78844i	0.447630	−	0.894219i	$-0.352268\pi$
$332$			438.000i			0.0724047i
$333$			7878.00i			1.29643i
$334$	−7164.00			−1.17364
$335$		0			0
$336$			1065.00i			0.172918i
$337$	2171.00			0.350926			0.175463	−	0.984486i	$-0.443858\pi$
$337$	2171.00			0.350926			0.175463	+	0.984486i	$0.443858\pi$
$338$	−6084.00	−	2535.00i	−0.979071	−	0.407946i
$339$	−90.0000			−0.0144193
$340$		0			0
$341$	−12672.0			−2.01240
$342$	−468.000			−0.0739957
$343$		−	6915.00i		−	1.08856i
$344$			2037.00i			0.319267i
$345$		0			0
$346$			4698.00i			0.729960i
$347$	7047.00			1.09021			0.545105	−	0.838368i	$-0.316490\pi$
$347$	7047.00			1.09021			0.545105	+	0.838368i	$0.316490\pi$
$348$	144.000			0.0221816
$349$		−	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
$352$	−2160.00			−0.327069
$353$		−	9318.00i		−	1.40495i	−0.711709	−	0.702475i	$-0.752078\pi$
$353$		−	9318.00i		−	1.40495i	0.711709	−	0.702475i	$-0.247922\pi$
$354$	−1566.00			−0.235119
$355$		0			0
$356$			438.000i			0.0652077i
$357$		−	675.000i		−	0.100069i
$358$		−	1971.00i		−	0.290979i
$359$			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
$362$		−	3666.00i		−	0.532267i
$363$	−973.000			−0.140687
$364$	585.000	−	390.000i	0.0842372	−	0.0561581i
$365$		0			0
$366$			1128.00i			0.161097i
$367$	2536.00			0.360703			0.180352	−	0.983602i	$-0.442276\pi$
$367$	2536.00			0.360703			0.180352	+	0.983602i	$0.442276\pi$
$368$	11502.0			1.62930
$369$		−	4992.00i		−	0.704263i
$370$		0			0
$371$		−	6210.00i		−	0.869022i
$372$			264.000i			0.0367951i
$373$	92.0000			0.0127710			0.00638550	−	0.999980i	$-0.497967\pi$
$373$	92.0000			0.0127710			0.00638550	+	0.999980i	$0.497967\pi$
$374$	6480.00			0.895917
$375$		0			0
$376$	2331.00			0.319713
$377$	3744.00	+	5616.00i	0.511474	+	0.767211i
$378$	−2385.00			−0.324527
$379$		−	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
$382$			3780.00i			0.506287i
$383$		−	579.000i		−	0.0772468i	−0.999254	−	0.0386234i	$-0.987703\pi$
$383$		−	579.000i		−	0.0772468i	0.999254	−	0.0386234i	$-0.0122973\pi$
$384$		−	1659.00i		−	0.220470i
$385$		0			0
$386$	1026.00			0.135290
$387$	−2522.00			−0.331267
$388$			852.000i			0.111479i
$389$	2106.00			0.274495			0.137247	−	0.990537i	$-0.456174\pi$
$389$	2106.00			0.274495			0.137247	+	0.990537i	$0.456174\pi$
$390$		0			0
$391$	−7290.00			−0.942893
$392$			2478.00i			0.319280i
$393$	−1467.00			−0.188296
$394$	−243.000			−0.0310715
$395$		0			0
$396$		−	1248.00i		−	0.158370i
$397$			1974.00i			0.249552i	0.992185	+	0.124776i	$0.0398213\pi$
$397$			1974.00i			0.249552i	−0.992185	+	0.124776i	$0.960179\pi$
$398$			5988.00i			0.754149i
$399$	−90.0000			−0.0112923
$400$		0			0
$401$			11886.0i			1.48020i	0.672499	+	0.740098i	$0.265221\pi$
$401$			11886.0i			1.48020i	−0.672499	+	0.740098i	$0.734779\pi$
$402$	108.000			0.0133994
$403$	−10296.0	+	6864.00i	−1.27266	+	0.848437i
$404$	−396.000			−0.0487667
$405$		0			0
$406$	−6480.00			−0.792111
$407$	−14544.0			−1.77130
$408$			945.000i			0.114668i
$409$		−	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
$412$	182.000			0.0217633
$413$	7830.00			0.932903
$414$			12636.0i			1.50006i
$415$		0			0
$416$	−1755.00	+	1170.00i	−0.206841	+	0.137894i
$417$	−2419.00			−0.284074
$418$		−	864.000i		−	0.101100i
$419$	5823.00			0.678931			0.339466	−	0.940618i	$-0.389754\pi$
$419$	5823.00			0.678931			0.339466	+	0.940618i	$0.389754\pi$
$420$		0			0
$421$			7341.00i			0.849830i	0.905233	+	0.424915i	$0.139696\pi$
$421$			7341.00i			0.849830i	−0.905233	+	0.424915i	$0.860304\pi$
$422$			8499.00i			0.980391i
$423$			2886.00i			0.331731i
$424$			8694.00i			0.995797i
$425$		0			0
$426$	1071.00			0.121808
$427$		−	5640.00i		−	0.639201i
$428$	−612.000			−0.0691171
$429$	−1872.00	+	1248.00i	−0.210678	+	0.140452i
$430$		0			0
$431$			7485.00i			0.836519i	0.908328	+	0.418260i	$0.137360\pi$
$431$			7485.00i			0.836519i	−0.908328	+	0.418260i	$0.862640\pi$
$432$	3763.00			0.419091
$433$	15203.0			1.68732			0.843660	−	0.536878i	$-0.180396\pi$
$433$	15203.0			1.68732			0.843660	+	0.536878i	$0.180396\pi$
$434$		−	11880.0i		−	1.31396i
$435$		0			0
$436$			1083.00i			0.118959i
$437$			972.000i			0.106401i
$438$	−3294.00			−0.359346
$439$	−1762.00			−0.191562			−0.0957809	−	0.995402i	$-0.530535\pi$
$439$	−1762.00			−0.191562			−0.0957809	+	0.995402i	$0.530535\pi$
$440$		0			0
$441$	−3068.00			−0.331282
$442$	5265.00	−	3510.00i	0.566585	−	0.377723i
$443$	7317.00			0.784743			0.392372	−	0.919807i	$-0.371655\pi$
$443$	7317.00			0.784743			0.392372	+	0.919807i	$0.371655\pi$
$444$			303.000i			0.0323868i
$445$		0			0
$446$	10521.0			1.11700
$447$			930.000i			0.0984060i
$448$			6495.00i			0.684955i
$449$		−	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
$452$	90.0000			0.00936558
$453$		−	1683.00i		−	0.174557i
$454$	684.000			0.0707086
$455$		0			0
$456$	126.000			0.0129397
$457$			9870.00i			1.01028i	0.863037	+	0.505141i	$0.168560\pi$
$457$			9870.00i			1.01028i	−0.863037	+	0.505141i	$0.831440\pi$
$458$	−16479.0			−1.68125
$459$	−2385.00			−0.242532
$460$		0			0
$461$			14541.0i			1.46907i	0.678570	+	0.734536i	$0.262600\pi$
$461$			14541.0i			1.46907i	−0.678570	+	0.734536i	$0.737400\pi$
$462$		−	2160.00i		−	0.217516i
$463$			2112.00i			0.211993i	0.994366	+	0.105997i	$0.0338033\pi$
$463$			2112.00i			0.211993i	−0.994366	+	0.105997i	$0.966197\pi$
$464$	10224.0			1.02293
$465$		0			0
$466$		−	10881.0i		−	1.08166i
$467$	−3276.00			−0.324615			−0.162307	−	0.986740i	$-0.551894\pi$
$467$	−3276.00			−0.324615			−0.162307	+	0.986740i	$0.551894\pi$
$468$	−676.000	−	1014.00i	−0.0667695	−	0.100154i
$469$	−540.000			−0.0531661
$470$		0			0
$471$	−1874.00			−0.183332
$472$	−10962.0			−1.06900
$473$		−	4656.00i		−	0.452607i
$474$		−	2490.00i		−	0.241286i
$475$		0			0
$476$			675.000i			0.0649970i
$477$	−10764.0			−1.03323
$478$	18225.0			1.74392
$479$		−	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
$482$	−630.000			−0.0595347
$483$			2430.00i			0.228921i
$484$	973.000			0.0913787
$485$		0			0
$486$			6240.00i			0.582412i
$487$		−	3660.00i		−	0.340555i	−0.985396	−	0.170278i	$-0.945534\pi$
$487$		−	3660.00i		−	0.340555i	0.985396	−	0.170278i	$-0.0544665\pi$
$488$			7896.00i			0.732449i
$489$			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	$0.489070\pi$
$492$		−	192.000i		−	0.0175936i
$493$	−6480.00			−0.591977
$494$	−468.000	−	702.000i	−0.0426241	−	0.0639362i
$495$		0			0
$496$			18744.0i			1.69684i
$497$	−5355.00			−0.483309
$498$	1314.00			0.118236
$499$			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
$502$			21276.0i			1.89162i
$503$	12078.0			1.07064			0.535319	−	0.844650i	$-0.320191\pi$
$503$	12078.0			1.07064			0.535319	+	0.844650i	$0.320191\pi$
$504$	−8190.00			−0.723833
$505$		0			0
$506$	−23328.0			−2.04952
$507$	−845.000	+	2028.00i	−0.0740193	+	0.177646i
$508$	2086.00			0.182188
$509$		−	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
$512$		−	8733.00i		−	0.753804i
$513$			318.000i			0.0273685i
$514$			17415.0i			1.49444i
$515$		0			0
$516$	−97.0000			−0.00827556
$517$	−5328.00			−0.453240
$518$		−	13635.0i		−	1.15654i
$519$	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	$0.447365\pi$
$522$			11232.0i			0.941784i
$523$	−16184.0			−1.35311			−0.676555	−	0.736392i	$-0.736528\pi$
$523$	−16184.0			−1.35311			−0.676555	+	0.736392i	$0.736528\pi$
$524$	1467.00			0.122302
$525$		0			0
$526$		−	2376.00i		−	0.196955i
$527$		−	11880.0i		−	0.981975i
$528$			3408.00i			0.280898i
$529$	14077.0			1.15698
$530$		0			0
$531$		−	13572.0i		−	1.10918i
$532$	90.0000			0.00733458
$533$	7488.00	−	4992.00i	0.608520	−	0.405680i
$534$	1314.00			0.106484
$535$		0			0
$536$	756.000			0.0609221
$537$	−657.000			−0.0527964
$538$			16416.0i			1.31551i
$539$		−	5664.00i		−	0.452627i
$540$		0			0
$541$		−	7923.00i		−	0.629642i	−0.949151	−	0.314821i	$-0.898055\pi$
$541$		−	7923.00i		−	0.629642i	0.949151	−	0.314821i	$-0.101945\pi$
$542$	−6993.00			−0.554198
$543$	−1222.00			−0.0965765
$544$		−	2025.00i		−	0.159598i
$545$		0			0
$546$	−1170.00	−	1755.00i	−0.0917058	−	0.137559i
$547$	14389.0			1.12473			0.562367	−	0.826888i	$-0.309891\pi$
$547$	14389.0			1.12473			0.562367	+	0.826888i	$0.309891\pi$
$548$		−	414.000i		−	0.0322723i
$549$	−9776.00			−0.759981
$550$		0			0
$551$			864.000i			0.0668015i
$552$		−	3402.00i		−	0.262317i
$553$			12450.0i			0.957374i
$554$			4152.00i			0.318414i
$555$		0			0
$556$	2419.00			0.184512
$557$			10383.0i			0.789842i	0.918715	+	0.394921i	$0.129228\pi$
$557$			10383.0i			0.789842i	−0.918715	+	0.394921i	$0.870772\pi$
$558$	−20592.0			−1.56224
$559$	−2522.00	−	3783.00i	−0.190822	−	0.286232i
$560$		0			0
$561$		−	2160.00i		−	0.162558i
$562$	12186.0			0.914654
$563$	16425.0			1.22954			0.614770	−	0.788706i	$-0.289249\pi$
$563$	16425.0			1.22954			0.614770	+	0.788706i	$0.289249\pi$
$564$			111.000i			0.00828713i
$565$		0			0
$566$			11292.0i			0.838583i
$567$		−	9735.00i		−	0.721043i
$568$	7497.00			0.553815
$569$	12213.0			0.899817			0.449908	−	0.893075i	$-0.351457\pi$
$569$	12213.0			0.899817			0.449908	+	0.893075i	$0.351457\pi$
$570$		0			0
$571$	−6383.00			−0.467811			−0.233906	−	0.972259i	$-0.575151\pi$
$571$	−6383.00			−0.467811			−0.233906	+	0.972259i	$0.575151\pi$
$572$	1872.00	−	1248.00i	0.136840	−	0.0912264i
$573$	1260.00			0.0918626
$574$			8640.00i			0.628269i
$575$		0			0
$576$	11258.0			0.814381
$577$			6426.00i			0.463636i	0.972759	+	0.231818i	$0.0744674\pi$
$577$			6426.00i			0.463636i	−0.972759	+	0.231818i	$0.925533\pi$
$578$		−	8664.00i		−	0.623486i
$579$		−	342.000i		−	0.0245476i
$580$		0			0
$581$	−6570.00			−0.469139
$582$	2556.00			0.182044
$583$		−	19872.0i		−	1.41169i
$584$	−23058.0			−1.63381
$585$		0			0
$586$	12681.0			0.893937
$587$		−	21330.0i		−	1.49980i	−0.661551	−	0.749901i	$-0.730101\pi$
$587$		−	21330.0i		−	1.49980i	0.661551	−	0.749901i	$-0.269899\pi$
$588$	−118.000			−0.00827591
$589$	−1584.00			−0.110811
$590$		0			0
$591$			81.0000i			0.00563772i
$592$			21513.0i			1.49355i
$593$		−	12084.0i		−	0.836813i	−0.908260	−	0.418407i	$-0.862589\pi$
$593$		−	12084.0i		−	0.836813i	0.908260	−	0.418407i	$-0.137411\pi$
$594$	−7632.00			−0.527180
$595$		0			0
$596$		−	930.000i		−	0.0639166i
$597$	1996.00			0.136836
$598$	−18954.0	+	12636.0i	−1.29613	+	0.864088i
$599$	2394.00			0.163299			0.0816496	−	0.996661i	$-0.473981\pi$
$599$	2394.00			0.163299			0.0816496	+	0.996661i	$0.473981\pi$
$600$		0			0
$601$	−21971.0			−1.49121			−0.745604	−	0.666389i	$-0.767839\pi$
$601$	−21971.0			−1.49121			−0.745604	+	0.666389i	$0.767839\pi$
$602$	4365.00			0.295522
$603$			936.000i			0.0632121i
$604$			1683.00i			0.113378i
$605$		0			0
$606$			1188.00i			0.0796356i
$607$	15406.0			1.03017			0.515083	−	0.857141i	$-0.327761\pi$
$607$	15406.0			1.03017			0.515083	+	0.857141i	$0.327761\pi$
$608$	−270.000			−0.0180098
$609$			2160.00i			0.143724i
$610$		0			0
$611$	−4329.00	+	2886.00i	−0.286633	+	0.191088i
$612$	1170.00			0.0772785
$613$		−	9630.00i		−	0.634506i	−0.948341	−	0.317253i	$-0.897240\pi$
$613$		−	9630.00i		−	0.634506i	0.948341	−	0.317253i	$-0.102760\pi$
$614$	918.000			0.0603379
$615$		0			0
$616$		−	15120.0i		−	0.988965i
$617$			14748.0i			0.962289i	0.876641	+	0.481144i	$0.159779\pi$
$617$			14748.0i			0.962289i	−0.876641	+	0.481144i	$0.840221\pi$
$618$		−	546.000i		−	0.0355394i
$619$		−	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
$622$		−	6318.00i		−	0.407281i
$623$	−6570.00			−0.422506
$624$	1846.00	+	2769.00i	0.118428	+	0.177642i
$625$		0			0
$626$		−	30153.0i		−	1.92517i
$627$	−288.000			−0.0183439
$628$	1874.00			0.119078
$629$		−	13635.0i		−	0.864329i
$630$		0			0
$631$		−	19875.0i		−	1.25390i	−0.779059	−	0.626950i	$-0.784303\pi$
$631$		−	19875.0i		−	1.25390i	0.779059	−	0.626950i	$-0.215697\pi$
$632$		−	17430.0i		−	1.09704i
$633$	2833.00			0.177886
$634$	6462.00			0.404793
$635$		0			0
$636$	−414.000			−0.0258116
$637$	−3068.00	−	4602.00i	−0.190830	−	0.286245i
$638$	−20736.0			−1.28675
$639$			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	$0.483222\pi$
$642$			1836.00i			0.112868i
$643$		−	16452.0i		−	1.00903i	−0.863404	−	0.504513i	$-0.831672\pi$
$643$		−	16452.0i		−	1.00903i	0.863404	−	0.504513i	$-0.168328\pi$
$644$		−	2430.00i		−	0.148689i
$645$		0			0
$646$	810.000			0.0493329
$647$	−25902.0			−1.57390			−0.786950	−	0.617017i	$-0.788341\pi$
$647$	−25902.0			−1.57390			−0.786950	+	0.617017i	$0.788341\pi$
$648$			13629.0i			0.826231i
$649$	25056.0			1.51546
$650$		0			0
$651$	−3960.00			−0.238410
$652$		−	1194.00i		−	0.0717188i
$653$	−18108.0			−1.08518			−0.542589	−	0.839999i	$-0.682556\pi$
$653$	−18108.0			−1.08518			−0.542589	+	0.839999i	$0.682556\pi$
$654$	3249.00			0.194260
$655$		0			0
$656$		−	13632.0i		−	0.811342i
$657$		−	28548.0i		−	1.69523i
$658$		−	4995.00i		−	0.295935i
$659$	−32904.0			−1.94500			−0.972502	−	0.232894i	$-0.925181\pi$
$659$	−32904.0			−1.94500			−0.972502	+	0.232894i	$0.925181\pi$
$660$		0			0
$661$			15318.0i			0.901363i	0.892685	+	0.450682i	$0.148819\pi$
$661$			15318.0i			0.901363i	−0.892685	+	0.450682i	$0.851181\pi$
$662$	32310.0			1.89692
$663$	−1170.00	−	1755.00i	−0.0685355	−	0.102803i
$664$	9198.00			0.537578
$665$		0			0
$666$	−23634.0			−1.37507
$667$	23328.0			1.35422
$668$		−	2388.00i		−	0.138315i
$669$		−	3507.00i		−	0.202673i
$670$		0			0
$671$		−	18048.0i		−	1.03835i
$672$	−675.000			−0.0387481
$673$	7729.00			0.442691			0.221346	−	0.975195i	$-0.428955\pi$
$673$	7729.00			0.442691			0.221346	+	0.975195i	$0.428955\pi$
$674$			6513.00i			0.372213i
$675$		0			0
$676$	845.000	−	2028.00i	0.0480769	−	0.115385i
$677$	−19242.0			−1.09236			−0.546182	−	0.837667i	$-0.683919\pi$
$677$	−19242.0			−1.09236			−0.546182	+	0.837667i	$0.683919\pi$
$678$		−	270.000i		−	0.0152939i
$679$	−12780.0			−0.722314
$680$		0			0
$681$		−	228.000i		−	0.0128296i
$682$		−	38016.0i		−	2.13447i
$683$		−	22518.0i		−	1.26153i	−0.775973	−	0.630767i	$-0.782740\pi$
$683$		−	22518.0i		−	1.26153i	0.775973	−	0.630767i	$-0.217260\pi$
$684$		−	156.000i		−	0.00872048i
$685$		0			0
$686$	20745.0			1.15459
$687$			5493.00i			0.305052i
$688$	−6887.00			−0.381634
$689$	−10764.0	−	16146.0i	−0.595175	−	0.892763i
$690$		0			0
$691$		−	9168.00i		−	0.504728i	−0.967632	−	0.252364i	$-0.918792\pi$
$691$		−	9168.00i		−	0.504728i	0.967632	−	0.252364i	$-0.0812081\pi$
$692$	−1566.00			−0.0860266
$693$	18720.0			1.02614
$694$			21141.0i			1.15634i
$695$		0			0
$696$		−	3024.00i		−	0.164690i
$697$			8640.00i			0.469531i
$698$	20619.0			1.11811
$699$	−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	$0.489965\pi$
$702$	−6201.00	+	4134.00i	−0.333393	+	0.222262i
$703$	−1818.00			−0.0975351
$704$			20784.0i			1.11268i
$705$		0			0
$706$	27954.0			1.49017
$707$		−	5940.00i		−	0.315978i
$708$		−	522.000i		−	0.0277090i
$709$		−	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
$712$	9198.00			0.484143
$713$			42768.0i			2.24639i
$714$	2025.00			0.106140
$715$		0			0
$716$	657.000			0.0342922
$717$		−	6075.00i		−	0.316423i
$718$	−12384.0			−0.643686
$719$	30960.0			1.60586			0.802930	−	0.596073i	$-0.203273\pi$
$719$	30960.0			1.60586			0.802930	+	0.596073i	$0.203273\pi$
$720$		0			0
$721$			2730.00i			0.141013i
$722$			20469.0i			1.05509i
$723$			210.000i			0.0108022i
$724$	1222.00			0.0627283
$725$		0			0
$726$		−	2919.00i		−	0.149221i
$727$	8372.00			0.427098			0.213549	−	0.976932i	$-0.431498\pi$
$727$	8372.00			0.427098			0.213549	+	0.976932i	$0.431498\pi$
$728$	−8190.00	−	12285.0i	−0.416953	−	0.625429i
$729$	−15443.0			−0.784586
$730$		0			0
$731$	4365.00			0.220855
$732$	−376.000			−0.0189855
$733$		−	2739.00i		−	0.138018i	−0.997616	−	0.0690091i	$-0.978016\pi$
$733$		−	2739.00i		−	0.138018i	0.997616	−	0.0690091i	$-0.0219837\pi$
$734$			7608.00i			0.382584i
$735$		0			0
$736$			7290.00i			0.365099i
$737$	−1728.00			−0.0863659
$738$	14976.0			0.746984
$739$		−	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
$742$	18630.0			0.921737
$743$			29643.0i			1.46366i	0.681490	+	0.731828i	$0.261332\pi$
$743$			29643.0i			1.46366i	−0.681490	+	0.731828i	$0.738668\pi$
$744$	5544.00			0.273189
$745$		0			0
$746$			276.000i			0.0135457i
$747$			11388.0i			0.557785i
$748$			2160.00i			0.105585i
$749$		−	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	$0.354832\pi$
$752$			7881.00i			0.382168i
$753$	7092.00			0.343223
$754$	−16848.0	+	11232.0i	−0.813751	+	0.542500i
$755$		0			0
$756$		−	795.000i		−	0.0382459i
$757$	6410.00			0.307761			0.153881	−	0.988089i	$-0.450823\pi$
$757$	6410.00			0.307761			0.153881	+	0.988089i	$0.450823\pi$
$758$	30546.0			1.46369
$759$			7776.00i			0.371872i
$760$		0			0
$761$			28290.0i			1.34758i	0.738921	+	0.673792i	$0.235336\pi$
$761$			28290.0i			1.34758i	−0.738921	+	0.673792i	$0.764664\pi$
$762$		−	6258.00i		−	0.297511i
$763$	−16245.0			−0.770784
$764$	−1260.00			−0.0596665
$765$		0			0
$766$	1737.00			0.0819326
$767$	20358.0	−	13572.0i	0.958390	−	0.638926i
$768$	1513.00			0.0710881
$769$			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
$772$			342.000i			0.0159441i
$773$		−	5649.00i		−	0.262847i	−0.991326	−	0.131423i	$-0.958045\pi$
$773$		−	5649.00i		−	0.262847i	0.991326	−	0.131423i	$-0.0419547\pi$
$774$		−	7566.00i		−	0.351362i
$775$		0			0
$776$	17892.0			0.827687
$777$	−4545.00			−0.209847
$778$			6318.00i			0.291146i
$779$	1152.00			0.0529842
$780$		0			0
$781$	−17136.0			−0.785114
$782$		−	21870.0i		−	1.00009i
$783$	7632.00			0.348334
$784$	−8378.00			−0.381651
$785$		0			0
$786$		−	4401.00i		−	0.199718i
$787$		−	756.000i		−	0.0342420i	−0.999853	−	0.0171210i	$-0.994550\pi$
$787$		−	756.000i		−	0.0342420i	0.999853	−	0.0171210i	$-0.00545006\pi$
$788$		−	81.0000i		−	0.00366181i
$789$	−792.000			−0.0357363
$790$		0			0
$791$			1350.00i			0.0606833i
$792$	−26208.0			−1.17583
$793$	−9776.00	−	14664.0i	−0.437775	−	0.656663i
$794$	−5922.00			−0.264690
$795$		0			0
$796$	−1996.00			−0.0888773
$797$	−31194.0			−1.38638			−0.693192	−	0.720753i	$-0.743796\pi$
$797$	−31194.0			−1.38638			−0.693192	+	0.720753i	$0.743796\pi$
$798$		−	270.000i		−	0.0119773i
$799$		−	4995.00i		−	0.221164i
$800$		0			0
$801$			11388.0i			0.502341i
$802$	−35658.0			−1.56998
$803$	52704.0			2.31617
$804$			36.0000i			0.00157913i
$805$		0			0
$806$	−20592.0	−	30888.0i	−0.899904	−	1.34986i
$807$	5472.00			0.238691
$808$			8316.00i			0.362074i
$809$	17055.0			0.741189			0.370594	−	0.928795i	$-0.379154\pi$
$809$	17055.0			0.741189			0.370594	+	0.928795i	$0.379154\pi$
$810$		0			0
$811$		−	35520.0i		−	1.53795i	−0.639280	−	0.768974i	$-0.720768\pi$
$811$		−	35520.0i		−	1.53795i	0.639280	−	0.768974i	$-0.279232\pi$
$812$		−	2160.00i		−	0.0933512i
$813$			2331.00i			0.100556i
$814$		−	43632.0i		−	1.87875i
$815$		0			0
$816$	−3195.00			−0.137068
$817$		−	582.000i		−	0.0249224i
$818$	3762.00			0.160801
$819$	15210.0	−	10140.0i	0.648938	−	0.432625i
$820$		0			0
$821$		−	1095.00i		−	0.0465478i	−0.999729	−	0.0232739i	$-0.992591\pi$
$821$		−	1095.00i		−	0.0465478i	0.999729	−	0.0232739i	$-0.00740899\pi$
$822$	−1242.00			−0.0527004
$823$	2554.00			0.108174			0.0540868	−	0.998536i	$-0.482775\pi$
$823$	2554.00			0.108174			0.0540868	+	0.998536i	$0.482775\pi$
$824$		−	3822.00i		−	0.161585i
$825$		0			0
$826$			23490.0i			0.989494i
$827$		−	21522.0i		−	0.904950i	−0.891777	−	0.452475i	$-0.850541\pi$
$827$		−	21522.0i		−	0.904950i	0.891777	−	0.452475i	$-0.149459\pi$
$828$	−4212.00			−0.176784
$829$	−13124.0			−0.549838			−0.274919	−	0.961467i	$-0.588651\pi$
$829$	−13124.0			−0.549838			−0.274919	+	0.961467i	$0.588651\pi$
$830$		0			0
$831$	1384.00			0.0577743
$832$	11258.0	+	16887.0i	0.469112	+	0.703668i
$833$	5310.00			0.220865
$834$		−	7257.00i		−	0.301306i
$835$		0			0
$836$	288.000			0.0119147
$837$			13992.0i			0.577819i
$838$			17469.0i			0.720115i
$839$		−	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
$842$	−22023.0			−0.901381
$843$		−	4062.00i		−	0.165958i
$844$	−2833.00			−0.115540
$845$		0			0
$846$	−8658.00			−0.351854
$847$			14595.0i			0.592078i
$848$	−29394.0			−1.19032
$849$	3764.00			0.152156
$850$		0			0
$851$			49086.0i			1.97726i
$852$			357.000i			0.0143552i
$853$		−	31077.0i		−	1.24743i	−0.781653	−	0.623714i	$-0.785623\pi$
$853$		−	31077.0i		−	1.24743i	0.781653	−	0.623714i	$-0.214377\pi$
$854$	16920.0			0.677975
$855$		0			0
$856$			12852.0i			0.513169i
$857$	19422.0			0.774146			0.387073	−	0.922049i	$-0.373486\pi$
$857$	19422.0			0.774146			0.387073	+	0.922049i	$0.373486\pi$
$858$	−3744.00	−	5616.00i	−0.148972	−	0.223458i
$859$	1744.00			0.0692718			0.0346359	−	0.999400i	$-0.488973\pi$
$859$	1744.00			0.0692718			0.0346359	+	0.999400i	$0.488973\pi$
$860$		0			0
$861$	2880.00			0.113996
$862$	−22455.0			−0.887263
$863$			19179.0i			0.756501i	0.925703	+	0.378251i	$0.123474\pi$
$863$			19179.0i			0.756501i	−0.925703	+	0.378251i	$0.876526\pi$
$864$			2385.00i			0.0939113i
$865$		0			0
$866$			45609.0i			1.78967i
$867$	−2888.00			−0.113128
$868$	3960.00			0.154852
$869$			39840.0i			1.55521i
$870$		0			0
$871$	−1404.00	+	936.000i	−0.0546185	+	0.0364123i
$872$	22743.0			0.883228
$873$			22152.0i			0.858799i
$874$	−2916.00			−0.112855
$875$		0			0
$876$		−	1098.00i		−	0.0423493i
$877$			29217.0i			1.12496i	0.826812	+	0.562479i	$0.190152\pi$
$877$			29217.0i			1.12496i	−0.826812	+	0.562479i	$0.809848\pi$
$878$		−	5286.00i		−	0.203182i
$879$		−	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	$0.596625\pi$
$882$		−	9204.00i		−	0.351377i
$883$	−30589.0			−1.16580			−0.582900	−	0.812544i	$-0.698082\pi$
$883$	−30589.0			−1.16580			−0.582900	+	0.812544i	$0.698082\pi$
$884$	1170.00	+	1755.00i	0.0445151	+	0.0667727i
$885$		0			0
$886$			21951.0i			0.832346i
$887$	25884.0			0.979819			0.489910	−	0.871773i	$-0.337030\pi$
$887$	25884.0			0.979819			0.489910	+	0.871773i	$0.337030\pi$
$888$	6363.00			0.240460
$889$			31290.0i			1.18046i
$890$		0			0
$891$		−	31152.0i		−	1.17130i
$892$			3507.00i			0.131640i
$893$	−666.000			−0.0249573
$894$	−2790.00			−0.104375
$895$		0			0
$896$	−24885.0			−0.927845
$897$	4212.00	+	6318.00i	0.156783	+	0.235175i
$898$	15048.0			0.559196
$899$			38016.0i			1.41035i
$900$		0			0
$901$	18630.0			0.688852
$902$			27648.0i			1.02060i
$903$		−	1455.00i		−	0.0536206i
$904$		−	1890.00i		−	0.0695359i
$905$		0			0
$906$	5049.00			0.185145
$907$	12305.0			0.450475			0.225237	−	0.974304i	$-0.427684\pi$
$907$	12305.0			0.450475			0.225237	+	0.974304i	$0.427684\pi$
$908$			228.000i			0.00833309i
$909$	−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	$0.317903\pi$
$912$			426.000i			0.0154674i
$913$	−21024.0			−0.762095
$914$	−29610.0			−1.07157
$915$		0			0
$916$		−	5493.00i		−	0.198137i
$917$			22005.0i			0.792442i
$918$		−	7155.00i		−	0.257244i
$919$	47644.0			1.71015			0.855076	−	0.518502i	$-0.173510\pi$
$919$	47644.0			1.71015			0.855076	+	0.518502i	$0.173510\pi$
$920$		0			0
$921$		−	306.000i		−	0.0109479i
$922$	−43623.0			−1.55819
$923$	−13923.0	+	9282.00i	−0.496513	+	0.331008i
$924$	720.000			0.0256345
$925$		0			0
$926$	−6336.00			−0.224853
$927$	4732.00			0.167658
$928$			6480.00i			0.229220i
$929$		−	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
$932$	3627.00			0.127475
$933$	−2106.00			−0.0738985
$934$		−	9828.00i		−	0.344306i
$935$		0			0
$936$	−21294.0	+	14196.0i	−0.743607	+	0.495738i
$937$	−32398.0			−1.12956			−0.564779	−	0.825242i	$-0.691039\pi$
$937$	−32398.0			−1.12956			−0.564779	+	0.825242i	$0.691039\pi$
$938$		−	1620.00i		−	0.0563911i
$939$	−10051.0			−0.349310
$940$		0			0
$941$		−	2097.00i		−	0.0726464i	−0.999340	−	0.0363232i	$-0.988435\pi$
$941$		−	2097.00i		−	0.0726464i	0.999340	−	0.0363232i	$-0.0115646\pi$
$942$		−	5622.00i		−	0.194453i
$943$		−	31104.0i		−	1.07411i
$944$		−	37062.0i		−	1.27782i
$945$		0			0
$946$	13968.0			0.480062
$947$			20016.0i			0.686835i	0.939183	+	0.343417i	$0.111585\pi$
$947$			20016.0i			0.686835i	−0.939183	+	0.343417i	$0.888415\pi$
$948$	830.000			0.0284358
$949$	42822.0	−	28548.0i	1.46476	−	0.976509i
$950$		0			0
$951$		−	2154.00i		−	0.0734471i
$952$	14175.0			0.482578
$953$	−24993.0			−0.849531			−0.424765	−	0.905304i	$-0.639643\pi$
$953$	−24993.0			−0.849531			−0.424765	+	0.905304i	$0.639643\pi$
$954$		−	32292.0i		−	1.09590i
$955$		0			0
$956$			6075.00i			0.205523i
$957$			6912.00i			0.233473i
$958$	46359.0			1.56346
$959$	6210.00			0.209105
$960$		0			0
$961$	−39905.0			−1.33950
$962$	−23634.0	−	35451.0i	−0.792090	−	1.18814i
$963$	−15912.0			−0.532458
$964$		−	210.000i		−	0.00701623i
$965$		0			0
$966$	−7290.00			−0.242807
$967$		−	40959.0i		−	1.36210i	−0.732236	−	0.681051i	$-0.761523\pi$
$967$		−	40959.0i		−	1.36210i	0.732236	−	0.681051i	$-0.238477\pi$
$968$		−	20433.0i		−	0.678452i
$969$		−	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	$0.799784\pi$
$972$	−2080.00			−0.0686379
$973$			36285.0i			1.19552i
$974$	10980.0			0.361213
$975$		0			0
$976$	−26696.0			−0.875531
$977$			47388.0i			1.55177i	0.630876	+	0.775884i	$0.282696\pi$
$977$			47388.0i			1.55177i	−0.630876	+	0.775884i	$0.717304\pi$
$978$	−3582.00			−0.117116
$979$	−21024.0			−0.686343
$980$		0			0
$981$			28158.0i			0.916428i
$982$			2241.00i			0.0728240i
$983$		−	16803.0i		−	0.545201i	−0.962127	−	0.272600i	$-0.912116\pi$
$983$		−	16803.0i		−	0.545201i	0.962127	−	0.272600i	$-0.0878837\pi$
$984$	−4032.00			−0.130625
$985$		0			0
$986$		−	19440.0i		−	0.627886i
$987$	−1665.00			−0.0536956
$988$	234.000	−	156.000i	0.00753495	−	0.00502330i
$989$	−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	$0.873433\pi$
$992$	−11880.0			−0.380232
$993$		−	10770.0i		−	0.344185i
$994$		−	16065.0i		−	0.512627i
$995$		0			0
$996$			438.000i			0.0139343i
$997$	25000.0			0.794140			0.397070	−	0.917788i	$-0.370027\pi$
$997$	25000.0			0.794140			0.397070	+	0.917788i	$0.370027\pi$
$998$	−47412.0			−1.50381
$999$			16059.0i			0.508593i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.4.c.b.51.2		2
5.2	odd	4		325.4.d.a.324.1		2
5.3	odd	4		325.4.d.b.324.2		2
5.4	even	2		13.4.b.a.12.1	✓	2
13.12	even	2	inner	325.4.c.b.51.1		2
15.14	odd	2		117.4.b.a.64.2		2
20.19	odd	2		208.4.f.b.129.2		2
40.19	odd	2		832.4.f.c.129.1		2
40.29	even	2		832.4.f.e.129.1		2
65.4	even	6		169.4.e.d.23.2		4
65.9	even	6		169.4.e.d.23.1		4
65.12	odd	4		325.4.d.b.324.1		2
65.19	odd	12		169.4.c.c.146.1		2
65.24	odd	12		169.4.c.b.22.1		2
65.29	even	6		169.4.e.d.147.2		4
65.34	odd	4		169.4.a.c.1.1		1
65.38	odd	4		325.4.d.a.324.2		2
65.44	odd	4		169.4.a.b.1.1		1
65.49	even	6		169.4.e.d.147.1		4
65.54	odd	12		169.4.c.c.22.1		2
65.59	odd	12		169.4.c.b.146.1		2
65.64	even	2		13.4.b.a.12.2	yes	2
195.44	even	4		1521.4.a.i.1.1		1
195.164	even	4		1521.4.a.d.1.1		1
195.194	odd	2		117.4.b.a.64.1		2
260.259	odd	2		208.4.f.b.129.1		2
520.259	odd	2		832.4.f.c.129.2		2
520.389	even	2		832.4.f.e.129.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.b.a.12.1	✓	2	5.4	even	2
13.4.b.a.12.2	yes	2	65.64	even	2
117.4.b.a.64.1		2	195.194	odd	2
117.4.b.a.64.2		2	15.14	odd	2
169.4.a.b.1.1		1	65.44	odd	4
169.4.a.c.1.1		1	65.34	odd	4
169.4.c.b.22.1		2	65.24	odd	12
169.4.c.b.146.1		2	65.59	odd	12
169.4.c.c.22.1		2	65.54	odd	12
169.4.c.c.146.1		2	65.19	odd	12
169.4.e.d.23.1		4	65.9	even	6
169.4.e.d.23.2		4	65.4	even	6
169.4.e.d.147.1		4	65.49	even	6
169.4.e.d.147.2		4	65.29	even	6
208.4.f.b.129.1		2	260.259	odd	2
208.4.f.b.129.2		2	20.19	odd	2
325.4.c.b.51.1		2	13.12	even	2	inner
325.4.c.b.51.2		2	1.1	even	1	trivial
325.4.d.a.324.1		2	5.2	odd	4
325.4.d.a.324.2		2	65.38	odd	4
325.4.d.b.324.1		2	65.12	odd	4
325.4.d.b.324.2		2	5.3	odd	4
832.4.f.c.129.1		2	40.19	odd	2
832.4.f.c.129.2		2	520.259	odd	2
832.4.f.e.129.1		2	40.29	even	2
832.4.f.e.129.2		2	520.389	even	2
1521.4.a.d.1.1		1	195.164	even	4
1521.4.a.i.1.1		1	195.44	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	325.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.00000i q^{6} -15.0000i q^{7} +21.0000i q^{8} -26.0000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.00000i q^{6} -15.0000i q^{7} +21.0000i q^{8} -26.0000 q^{9} -48.0000i q^{11} -1.00000 q^{12} +(-26.0000 - 39.0000i) q^{13} +45.0000 q^{14} -71.0000 q^{16} +45.0000 q^{17} -78.0000i q^{18} -6.00000i q^{19} -15.0000i q^{21} +144.000 q^{22} -162.000 q^{23} +21.0000i q^{24} +(117.000 - 78.0000i) q^{26} -53.0000 q^{27} +15.0000i q^{28} -144.000 q^{29} -264.000i q^{31} -45.0000i q^{32} -48.0000i q^{33} +135.000i q^{34} +26.0000 q^{36} -303.000i q^{37} +18.0000 q^{38} +(-26.0000 - 39.0000i) q^{39} +192.000i q^{41} +45.0000 q^{42} +97.0000 q^{43} +48.0000i q^{44} -486.000i q^{46} -111.000i q^{47} -71.0000 q^{48} +118.000 q^{49} +45.0000 q^{51} +(26.0000 + 39.0000i) q^{52} +414.000 q^{53} -159.000i q^{54} +315.000 q^{56} -6.00000i q^{57} -432.000i q^{58} +522.000i q^{59} +376.000 q^{61} +792.000 q^{62} +390.000i q^{63} -433.000 q^{64} +144.000 q^{66} -36.0000i q^{67} -45.0000 q^{68} -162.000 q^{69} -357.000i q^{71} -546.000i q^{72} +1098.00i q^{73} +909.000 q^{74} +6.00000i q^{76} -720.000 q^{77} +(117.000 - 78.0000i) q^{78} -830.000 q^{79} +649.000 q^{81} -576.000 q^{82} -438.000i q^{83} +15.0000i q^{84} +291.000i q^{86} -144.000 q^{87} +1008.00 q^{88} -438.000i q^{89} +(-585.000 + 390.000i) q^{91} +162.000 q^{92} -264.000i q^{93} +333.000 q^{94} -45.0000i q^{96} -852.000i q^{97} +354.000i q^{98} +1248.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} - 52 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^4 - 52 * q^9	\( 2 q + 2 q^{3} - 2 q^{4} - 52 q^{9} - 2 q^{12} - 52 q^{13} + 90 q^{14} - 142 q^{16} + 90 q^{17} + 288 q^{22} - 324 q^{23} + 234 q^{26} - 106 q^{27} - 288 q^{29} + 52 q^{36} + 36 q^{38} - 52 q^{39} + 90 q^{42} + 194 q^{43} - 142 q^{48} + 236 q^{49} + 90 q^{51} + 52 q^{52} + 828 q^{53} + 630 q^{56} + 752 q^{61} + 1584 q^{62} - 866 q^{64} + 288 q^{66} - 90 q^{68} - 324 q^{69} + 1818 q^{74} - 1440 q^{77} + 234 q^{78} - 1660 q^{79} + 1298 q^{81} - 1152 q^{82} - 288 q^{87} + 2016 q^{88} - 1170 q^{91} + 324 q^{92} + 666 q^{94}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^4 - 52 * q^9 - 2 * q^12 - 52 * q^13 + 90 * q^14 - 142 * q^16 + 90 * q^17 + 288 * q^22 - 324 * q^23 + 234 * q^26 - 106 * q^27 - 288 * q^29 + 52 * q^36 + 36 * q^38 - 52 * q^39 + 90 * q^42 + 194 * q^43 - 142 * q^48 + 236 * q^49 + 90 * q^51 + 52 * q^52 + 828 * q^53 + 630 * q^56 + 752 * q^61 + 1584 * q^62 - 866 * q^64 + 288 * q^66 - 90 * q^68 - 324 * q^69 + 1818 * q^74 - 1440 * q^77 + 234 * q^78 - 1660 * q^79 + 1298 * q^81 - 1152 * q^82 - 288 * q^87 + 2016 * q^88 - 1170 * q^91 + 324 * q^92 + 666 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more