LMFDB - Embedded newform 475.2.b.c.324.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.b (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:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.4
Root			$0.445042i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.2.b.c.324.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.246980i q^{2} -0.801938i q^{3} +1.93900 q^{4} +0.198062 q^{6} -1.69202i q^{7} +0.972853i q^{8} +2.35690 q^{9} +O(q^{10})$	\(q+0.246980i q^{2} -0.801938i q^{3} +1.93900 q^{4} +0.198062 q^{6} -1.69202i q^{7} +0.972853i q^{8} +2.35690 q^{9} -0.911854 q^{11} -1.55496i q^{12} +1.55496i q^{13} +0.417895 q^{14} +3.63773 q^{16} -5.29590i q^{17} +0.582105i q^{18} +1.00000 q^{19} -1.35690 q^{21} -0.225209i q^{22} +4.24698i q^{23} +0.780167 q^{24} -0.384043 q^{26} -4.29590i q^{27} -3.28083i q^{28} -5.00969 q^{29} +1.82908 q^{31} +2.84415i q^{32} +0.731250i q^{33} +1.30798 q^{34} +4.57002 q^{36} -6.29590i q^{37} +0.246980i q^{38} +1.24698 q^{39} +4.18060 q^{41} -0.335126i q^{42} +7.31767i q^{43} -1.76809 q^{44} -1.04892 q^{46} +2.04892i q^{47} -2.91723i q^{48} +4.13706 q^{49} -4.24698 q^{51} +3.01507i q^{52} -2.70171i q^{53} +1.06100 q^{54} +1.64609 q^{56} -0.801938i q^{57} -1.23729i q^{58} -9.87800 q^{59} +0.542877 q^{61} +0.451747i q^{62} -3.98792i q^{63} +6.57301 q^{64} -0.180604 q^{66} +13.9976i q^{67} -10.2687i q^{68} +3.40581 q^{69} -12.8780 q^{71} +2.29291i q^{72} -2.80731i q^{73} +1.55496 q^{74} +1.93900 q^{76} +1.54288i q^{77} +0.307979i q^{78} -1.59419 q^{79} +3.62565 q^{81} +1.03252i q^{82} +12.2349i q^{83} -2.63102 q^{84} -1.80731 q^{86} +4.01746i q^{87} -0.887100i q^{88} -2.91723 q^{89} +2.63102 q^{91} +8.23490i q^{92} -1.46681i q^{93} -0.506041 q^{94} +2.28083 q^{96} -1.55496i q^{97} +1.02177i q^{98} -2.14914 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9}+O(q^{10}) $ 6 * q - 8 * q^4 + 10 * q^6 + 6 * q^9	$ 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9} + 2 q^{11} + 14 q^{14} + 36 q^{16} + 6 q^{19} + 2 q^{24} + 18 q^{26} + 14 q^{29} - 10 q^{31} + 18 q^{34} - 22 q^{36} - 2 q^{39} + 2 q^{41} + 30 q^{44} + 12 q^{46} + 14 q^{49} - 16 q^{51} + 26 q^{54} - 70 q^{56} - 20 q^{59} - 34 q^{61} - 98 q^{64} + 22 q^{66} - 6 q^{69} - 38 q^{71} + 10 q^{74} - 8 q^{76} - 36 q^{79} - 2 q^{81} + 14 q^{84} + 4 q^{86} - 4 q^{89} - 14 q^{91} - 22 q^{94} + 36 q^{96} - 40 q^{99}+O(q^{100}) $ 6 * q - 8 * q^4 + 10 * q^6 + 6 * q^9 + 2 * q^11 + 14 * q^14 + 36 * q^16 + 6 * q^19 + 2 * q^24 + 18 * q^26 + 14 * q^29 - 10 * q^31 + 18 * q^34 - 22 * q^36 - 2 * q^39 + 2 * q^41 + 30 * q^44 + 12 * q^46 + 14 * q^49 - 16 * q^51 + 26 * q^54 - 70 * q^56 - 20 * q^59 - 34 * q^61 - 98 * q^64 + 22 * q^66 - 6 * q^69 - 38 * q^71 + 10 * q^74 - 8 * q^76 - 36 * q^79 - 2 * q^81 + 14 * q^84 + 4 * q^86 - 4 * q^89 - 14 * q^91 - 22 * q^94 + 36 * q^96 - 40 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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 $

$2$	0.246980i	0.174641i	0.996180	+	0.0873205i	$0.0278304\pi$
$2$	0.246980i	0.174641i	−0.996180	+	0.0873205i	$0.972170\pi$
$3$	− 0.801938i	− 0.462999i	−0.972835	−	0.231499i	$-0.925637\pi$
$3$	− 0.801938i	− 0.462999i	0.972835	−	0.231499i	$-0.0743632\pi$
$4$	1.93900	0.969501
$5$	0	0
$5$	0	0
$6$	0.198062	0.0808586
$7$	− 1.69202i	− 0.639524i	−0.947498	−	0.319762i	$-0.896397\pi$
$7$	− 1.69202i	− 0.639524i	0.947498	−	0.319762i	$-0.103603\pi$
$8$	0.972853i	0.343955i
$9$	2.35690	0.785632
$10$	0	0
$11$	−0.911854	−0.274934	−0.137467	−	0.990506i	$-0.543896\pi$
$11$	−0.911854	−0.274934	−0.137467	+	0.990506i	$0.543896\pi$
$12$	− 1.55496i	− 0.448878i
$13$	1.55496i	0.431268i	0.976474	+	0.215634i	$0.0691818\pi$
$13$	1.55496i	0.431268i	−0.976474	+	0.215634i	$0.930818\pi$
$14$	0.417895	0.111687
$15$	0	0
$16$	3.63773	0.909432
$17$	− 5.29590i	− 1.28444i	−0.766519	−	0.642222i	$-0.778013\pi$
$17$	− 5.29590i	− 1.28444i	0.766519	−	0.642222i	$-0.221987\pi$
$18$	0.582105i	0.137204i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.35690	−0.296099
$22$	− 0.225209i	− 0.0480148i
$23$	4.24698i	0.885556i	0.896631	+	0.442778i	$0.146007\pi$
$23$	4.24698i	0.885556i	−0.896631	+	0.442778i	$0.853993\pi$
$24$	0.780167	0.159251
$25$	0	0
$26$	−0.384043	−0.0753170
$27$	− 4.29590i	− 0.826746i
$28$	− 3.28083i	− 0.620019i
$29$	−5.00969	−0.930276	−0.465138	−	0.885238i	$-0.653995\pi$
$29$	−5.00969	−0.930276	−0.465138	+	0.885238i	$0.653995\pi$
$30$	0	0
$31$	1.82908	0.328513	0.164257	−	0.986418i	$-0.447477\pi$
$31$	1.82908	0.328513	0.164257	+	0.986418i	$0.447477\pi$
$32$	2.84415i	0.502779i
$33$	0.731250i	0.127294i
$34$	1.30798	0.224316
$35$	0	0
$36$	4.57002	0.761671
$37$	− 6.29590i	− 1.03504i	−0.855671	−	0.517520i	$-0.826855\pi$
$37$	− 6.29590i	− 1.03504i	0.855671	−	0.517520i	$-0.173145\pi$
$38$	0.246980i	0.0400654i
$39$	1.24698	0.199677
$40$	0	0
$41$	4.18060	0.652901	0.326450	−	0.945214i	$-0.394147\pi$
$41$	4.18060	0.652901	0.326450	+	0.945214i	$0.394147\pi$
$42$	− 0.335126i	− 0.0517110i
$43$	7.31767i	1.11593i	0.829863	+	0.557967i	$0.188418\pi$
$43$	7.31767i	1.11593i	−0.829863	+	0.557967i	$0.811582\pi$
$44$	−1.76809	−0.266549
$45$	0	0
$46$	−1.04892	−0.154654
$47$	2.04892i	0.298865i	0.988772	+	0.149433i	$0.0477447\pi$
$47$	2.04892i	0.298865i	−0.988772	+	0.149433i	$0.952255\pi$
$48$	− 2.91723i	− 0.421066i
$49$	4.13706	0.591009
$50$	0	0
$51$	−4.24698	−0.594696
$52$	3.01507i	0.418114i
$53$	− 2.70171i	− 0.371108i	−0.982634	−	0.185554i	$-0.940592\pi$
$53$	− 2.70171i	− 0.371108i	0.982634	−	0.185554i	$-0.0594080\pi$
$54$	1.06100	0.144384
$55$	0	0
$56$	1.64609	0.219968
$57$	− 0.801938i	− 0.106219i
$58$	− 1.23729i	− 0.162464i
$59$	−9.87800	−1.28601	−0.643003	−	0.765864i	$-0.722312\pi$
$59$	−9.87800	−1.28601	−0.643003	+	0.765864i	$0.722312\pi$
$60$	0	0
$61$	0.542877	0.0695082	0.0347541	−	0.999396i	$-0.488935\pi$
$61$	0.542877	0.0695082	0.0347541	+	0.999396i	$0.488935\pi$
$62$	0.451747i	0.0573719i
$63$	− 3.98792i	− 0.502430i
$64$	6.57301	0.821626
$65$	0	0
$66$	−0.180604	−0.0222308
$67$	13.9976i	1.71008i	0.518562	+	0.855040i	$0.326467\pi$
$67$	13.9976i	1.71008i	−0.518562	+	0.855040i	$0.673533\pi$
$68$	− 10.2687i	− 1.24527i
$69$	3.40581	0.410012
$70$	0	0
$71$	−12.8780	−1.52834	−0.764169	−	0.645016i	$-0.776851\pi$
$71$	−12.8780	−1.52834	−0.764169	+	0.645016i	$0.776851\pi$
$72$	2.29291i	0.270222i
$73$	− 2.80731i	− 0.328571i	−0.986413	−	0.164286i	$-0.947468\pi$
$73$	− 2.80731i	− 0.328571i	0.986413	−	0.164286i	$-0.0525319\pi$
$74$	1.55496	0.180760
$75$	0	0
$76$	1.93900	0.222419
$77$	1.54288i	0.175827i
$78$	0.307979i	0.0348717i
$79$	−1.59419	−0.179360	−0.0896800	−	0.995971i	$-0.528584\pi$
$79$	−1.59419	−0.179360	−0.0896800	+	0.995971i	$0.528584\pi$
$80$	0	0
$81$	3.62565	0.402850
$82$	1.03252i	0.114023i
$83$	12.2349i	1.34295i	0.741025	+	0.671477i	$0.234340\pi$
$83$	12.2349i	1.34295i	−0.741025	+	0.671477i	$0.765660\pi$
$84$	−2.63102	−0.287068
$85$	0	0
$86$	−1.80731	−0.194888
$87$	4.01746i	0.430717i
$88$	− 0.887100i	− 0.0945652i
$89$	−2.91723	−0.309226	−0.154613	−	0.987975i	$-0.549413\pi$
$89$	−2.91723	−0.309226	−0.154613	+	0.987975i	$0.549413\pi$
$90$	0	0
$91$	2.63102	0.275806
$92$	8.23490i	0.858547i
$93$	− 1.46681i	− 0.152101i
$94$	−0.506041	−0.0521941
$95$	0	0
$96$	2.28083	0.232786
$97$	− 1.55496i	− 0.157882i	−0.996879	−	0.0789410i	$-0.974846\pi$
$97$	− 1.55496i	− 0.157882i	0.996879	−	0.0789410i	$-0.0251539\pi$
$98$	1.02177i	0.103214i
$99$	−2.14914	−0.215997
$100$	0	0
$101$	−16.6015	−1.65191	−0.825955	−	0.563737i	$-0.809363\pi$
$101$	−16.6015	−1.65191	−0.825955	+	0.563737i	$0.809363\pi$
$102$	− 1.04892i	− 0.103858i
$103$	− 4.84548i	− 0.477439i	−0.971089	−	0.238720i	$-0.923272\pi$
$103$	− 4.84548i	− 0.477439i	0.971089	−	0.238720i	$-0.0767277\pi$
$104$	−1.51275	−0.148337
$105$	0	0
$106$	0.667267	0.0648107
$107$	4.46681i	0.431823i	0.976413	+	0.215912i	$0.0692723\pi$
$107$	4.46681i	0.431823i	−0.976413	+	0.215912i	$0.930728\pi$
$108$	− 8.32975i	− 0.801530i
$109$	−18.8267	−1.80327	−0.901635	−	0.432498i	$-0.857632\pi$
$109$	−18.8267	−1.80327	−0.901635	+	0.432498i	$0.857632\pi$
$110$	0	0
$111$	−5.04892	−0.479222
$112$	− 6.15511i	− 0.581603i
$113$	20.0368i	1.88491i	0.334337	+	0.942453i	$0.391488\pi$
$113$	20.0368i	1.88491i	−0.334337	+	0.942453i	$0.608512\pi$
$114$	0.198062	0.0185502
$115$	0	0
$116$	−9.71379	−0.901903
$117$	3.66487i	0.338818i
$118$	− 2.43967i	− 0.224589i
$119$	−8.96077	−0.821433
$120$	0	0
$121$	−10.1685	−0.924411
$122$	0.134079i	0.0121390i
$123$	− 3.35258i	− 0.302292i
$124$	3.54660	0.318494
$125$	0	0
$126$	0.984935	0.0877449
$127$	17.8702i	1.58573i	0.609399	+	0.792863i	$0.291411\pi$
$127$	17.8702i	1.58573i	−0.609399	+	0.792863i	$0.708589\pi$
$128$	7.31170i	0.646269i
$129$	5.86831	0.516676
$130$	0	0
$131$	7.44265	0.650267	0.325134	−	0.945668i	$-0.394591\pi$
$131$	7.44265	0.650267	0.325134	+	0.945668i	$0.394591\pi$
$132$	1.41789i	0.123412i
$133$	− 1.69202i	− 0.146717i
$134$	−3.45712	−0.298650
$135$	0	0
$136$	5.15213	0.441791
$137$	− 5.68664i	− 0.485843i	−0.970046	−	0.242921i	$-0.921894\pi$
$137$	− 5.68664i	− 0.485843i	0.970046	−	0.242921i	$-0.0781057\pi$
$138$	0.841166i	0.0716048i
$139$	−3.61596	−0.306701	−0.153351	−	0.988172i	$-0.549006\pi$
$139$	−3.61596	−0.306701	−0.153351	+	0.988172i	$0.549006\pi$
$140$	0	0
$141$	1.64310	0.138374
$142$	− 3.18060i	− 0.266910i
$143$	− 1.41789i	− 0.118570i
$144$	8.57374	0.714479
$145$	0	0
$146$	0.693349	0.0573820
$147$	− 3.31767i	− 0.273637i
$148$	− 12.2078i	− 1.00347i
$149$	−3.29052	−0.269570	−0.134785	−	0.990875i	$-0.543034\pi$
$149$	−3.29052	−0.269570	−0.134785	+	0.990875i	$0.543034\pi$
$150$	0	0
$151$	−10.2131	−0.831133	−0.415566	−	0.909563i	$-0.636417\pi$
$151$	−10.2131	−0.831133	−0.415566	+	0.909563i	$0.636417\pi$
$152$	0.972853i	0.0789088i
$153$	− 12.4819i	− 1.00910i
$154$	−0.381059	−0.0307066
$155$	0	0
$156$	2.41789	0.193587
$157$	− 14.3448i	− 1.14484i	−0.819960	−	0.572420i	$-0.806005\pi$
$157$	− 14.3448i	− 1.14484i	0.819960	−	0.572420i	$-0.193995\pi$
$158$	− 0.393732i	− 0.0313236i
$159$	−2.16660	−0.171823
$160$	0	0
$161$	7.18598	0.566335
$162$	0.895461i	0.0703540i
$163$	− 19.5308i	− 1.52977i	−0.644167	−	0.764885i	$-0.722796\pi$
$163$	− 19.5308i	− 1.52977i	0.644167	−	0.764885i	$-0.277204\pi$
$164$	8.10620	0.632988
$165$	0	0
$166$	−3.02177	−0.234535
$167$	− 11.8823i	− 0.919481i	−0.888053	−	0.459741i	$-0.847942\pi$
$167$	− 11.8823i	− 0.919481i	0.888053	−	0.459741i	$-0.152058\pi$
$168$	− 1.32006i	− 0.101845i
$169$	10.5821	0.814008
$170$	0	0
$171$	2.35690	0.180236
$172$	14.1890i	1.08190i
$173$	15.4722i	1.17633i	0.808741	+	0.588164i	$0.200149\pi$
$173$	15.4722i	1.17633i	−0.808741	+	0.588164i	$0.799851\pi$
$174$	−0.992230	−0.0752208
$175$	0	0
$176$	−3.31708	−0.250034
$177$	7.92154i	0.595420i
$178$	− 0.720497i	− 0.0540035i
$179$	−2.16421	−0.161761	−0.0808803	−	0.996724i	$-0.525773\pi$
$179$	−2.16421	−0.161761	−0.0808803	+	0.996724i	$0.525773\pi$
$180$	0	0
$181$	16.8974	1.25597	0.627986	−	0.778225i	$-0.283879\pi$
$181$	16.8974	1.25597	0.627986	+	0.778225i	$0.283879\pi$
$182$	0.649809i	0.0481670i
$183$	− 0.435353i	− 0.0321822i
$184$	−4.13169	−0.304592
$185$	0	0
$186$	0.362273	0.0265631
$187$	4.82908i	0.353138i
$188$	3.97285i	0.289750i
$189$	−7.26875	−0.528724
$190$	0	0
$191$	5.92394	0.428641	0.214320	−	0.976763i	$-0.431246\pi$
$191$	5.92394	0.428641	0.214320	+	0.976763i	$0.431246\pi$
$192$	− 5.27114i	− 0.380412i
$193$	− 4.43535i	− 0.319264i	−0.987177	−	0.159632i	$-0.948969\pi$
$193$	− 4.43535i	− 0.319264i	0.987177	−	0.159632i	$-0.0510307\pi$
$194$	0.384043	0.0275727
$195$	0	0
$196$	8.02177	0.572984
$197$	− 16.4722i	− 1.17359i	−0.809734	−	0.586797i	$-0.800388\pi$
$197$	− 16.4722i	− 1.17359i	0.809734	−	0.586797i	$-0.199612\pi$
$198$	− 0.530795i	− 0.0377220i
$199$	24.4131	1.73060	0.865300	−	0.501255i	$-0.167128\pi$
$199$	24.4131	1.73060	0.865300	+	0.501255i	$0.167128\pi$
$200$	0	0
$201$	11.2252	0.791765
$202$	− 4.10023i	− 0.288491i
$203$	8.47650i	0.594934i
$204$	−8.23490	−0.576558
$205$	0	0
$206$	1.19673	0.0833804
$207$	10.0097i	0.695721i
$208$	5.65651i	0.392209i
$209$	−0.911854	−0.0630743
$210$	0	0
$211$	−4.34050	−0.298813	−0.149406	−	0.988776i	$-0.547736\pi$
$211$	−4.34050	−0.298813	−0.149406	+	0.988776i	$0.547736\pi$
$212$	− 5.23862i	− 0.359790i
$213$	10.3274i	0.707619i
$214$	−1.10321	−0.0754140
$215$	0	0
$216$	4.17928	0.284364
$217$	− 3.09485i	− 0.210092i
$218$	− 4.64981i	− 0.314925i
$219$	−2.25129	−0.152128
$220$	0	0
$221$	8.23490	0.553939
$222$	− 1.24698i	− 0.0836918i
$223$	26.2379i	1.75702i	0.477725	+	0.878509i	$0.341461\pi$
$223$	26.2379i	1.75702i	−0.477725	+	0.878509i	$0.658539\pi$
$224$	4.81236	0.321540
$225$	0	0
$226$	−4.94869	−0.329182
$227$	− 14.6853i	− 0.974699i	−0.873207	−	0.487349i	$-0.837964\pi$
$227$	− 14.6853i	− 0.974699i	0.873207	−	0.487349i	$-0.162036\pi$
$228$	− 1.55496i	− 0.102980i
$229$	21.6407	1.43006	0.715029	−	0.699095i	$-0.246413\pi$
$229$	21.6407	1.43006	0.715029	+	0.699095i	$0.246413\pi$
$230$	0	0
$231$	1.23729	0.0814078
$232$	− 4.87369i	− 0.319973i
$233$	27.1183i	1.77658i	0.459286	+	0.888289i	$0.348105\pi$
$233$	27.1183i	1.77658i	−0.459286	+	0.888289i	$0.651895\pi$
$234$	−0.905149	−0.0591715
$235$	0	0
$236$	−19.1535	−1.24678
$237$	1.27844i	0.0830435i
$238$	− 2.21313i	− 0.143456i
$239$	11.5308	0.745865	0.372933	−	0.927858i	$-0.378352\pi$
$239$	11.5308	0.745865	0.372933	+	0.927858i	$0.378352\pi$
$240$	0	0
$241$	11.8194	0.761354	0.380677	−	0.924708i	$-0.375691\pi$
$241$	11.8194	0.761354	0.380677	+	0.924708i	$0.375691\pi$
$242$	− 2.51142i	− 0.161440i
$243$	− 15.7952i	− 1.01326i
$244$	1.05264	0.0673883
$245$	0	0
$246$	0.828020	0.0527926
$247$	1.55496i	0.0989396i
$248$	1.77943i	0.112994i
$249$	9.81163	0.621787
$250$	0	0
$251$	−9.66487	−0.610041	−0.305021	−	0.952346i	$-0.598663\pi$
$251$	−9.66487	−0.610041	−0.305021	+	0.952346i	$0.598663\pi$
$252$	− 7.73258i	− 0.487107i
$253$	− 3.87263i	− 0.243470i
$254$	−4.41358	−0.276933
$255$	0	0
$256$	11.3402	0.708761
$257$	− 14.1860i	− 0.884897i	−0.896794	−	0.442449i	$-0.854110\pi$
$257$	− 14.1860i	− 0.884897i	0.896794	−	0.442449i	$-0.145890\pi$
$258$	1.44935i	0.0902328i
$259$	−10.6528	−0.661932
$260$	0	0
$261$	−11.8073	−0.730854
$262$	1.83818i	0.113563i
$263$	− 21.7942i	− 1.34389i	−0.740603	−	0.671943i	$-0.765460\pi$
$263$	− 21.7942i	− 1.34389i	0.740603	−	0.671943i	$-0.234540\pi$
$264$	−0.711399	−0.0437836
$265$	0	0
$266$	0.417895	0.0256228
$267$	2.33944i	0.143171i
$268$	27.1414i	1.65792i
$269$	24.7265	1.50760	0.753800	−	0.657104i	$-0.228219\pi$
$269$	24.7265	1.50760	0.753800	+	0.657104i	$0.228219\pi$
$270$	0	0
$271$	−13.2295	−0.803636	−0.401818	−	0.915720i	$-0.631622\pi$
$271$	−13.2295	−0.803636	−0.401818	+	0.915720i	$0.631622\pi$
$272$	− 19.2650i	− 1.16811i
$273$	− 2.10992i	− 0.127698i
$274$	1.40449	0.0848481
$275$	0	0
$276$	6.60388	0.397507
$277$	0.560335i	0.0336673i	0.999858	+	0.0168336i	$0.00535856\pi$
$277$	0.560335i	0.0336673i	−0.999858	+	0.0168336i	$0.994641\pi$
$278$	− 0.893068i	− 0.0535626i
$279$	4.31096	0.258091
$280$	0	0
$281$	27.6039	1.64671	0.823355	−	0.567527i	$-0.192100\pi$
$281$	27.6039	1.64671	0.823355	+	0.567527i	$0.192100\pi$
$282$	0.405813i	0.0241658i
$283$	− 15.9608i	− 0.948769i	−0.880318	−	0.474385i	$-0.842671\pi$
$283$	− 15.9608i	− 0.948769i	0.880318	−	0.474385i	$-0.157329\pi$
$284$	−24.9705	−1.48172
$285$	0	0
$286$	0.350191	0.0207072
$287$	− 7.07367i	− 0.417546i
$288$	6.70337i	0.395000i
$289$	−11.0465	−0.649796
$290$	0	0
$291$	−1.24698	−0.0730992
$292$	− 5.44339i	− 0.318550i
$293$	25.3327i	1.47995i	0.672632	+	0.739977i	$0.265164\pi$
$293$	25.3327i	1.47995i	−0.672632	+	0.739977i	$0.734836\pi$
$294$	0.819396	0.0477882
$295$	0	0
$296$	6.12498	0.356007
$297$	3.91723i	0.227301i
$298$	− 0.812691i	− 0.0470779i
$299$	−6.60388	−0.381912
$300$	0	0
$301$	12.3817	0.713666
$302$	− 2.52243i	− 0.145150i
$303$	13.3134i	0.764832i
$304$	3.63773	0.208638
$305$	0	0
$306$	3.08277	0.176230
$307$	11.5574i	0.659613i	0.944049	+	0.329806i	$0.106983\pi$
$307$	11.5574i	0.659613i	−0.944049	+	0.329806i	$0.893017\pi$
$308$	2.99164i	0.170464i
$309$	−3.88577	−0.221054
$310$	0	0
$311$	−18.3424	−1.04010	−0.520052	−	0.854135i	$-0.674087\pi$
$311$	−18.3424	−1.04010	−0.520052	+	0.854135i	$0.674087\pi$
$312$	1.21313i	0.0686798i
$313$	− 18.9119i	− 1.06896i	−0.845181	−	0.534481i	$-0.820507\pi$
$313$	− 18.9119i	− 1.06896i	0.845181	−	0.534481i	$-0.179493\pi$
$314$	3.54288	0.199936
$315$	0	0
$316$	−3.09113	−0.173890
$317$	− 20.2784i	− 1.13895i	−0.822008	−	0.569475i	$-0.807146\pi$
$317$	− 20.2784i	− 1.13895i	0.822008	−	0.569475i	$-0.192854\pi$
$318$	− 0.535107i	− 0.0300073i
$319$	4.56810	0.255765
$320$	0	0
$321$	3.58211	0.199934
$322$	1.77479i	0.0989052i
$323$	− 5.29590i	− 0.294672i
$324$	7.03013	0.390563
$325$	0	0
$326$	4.82371	0.267160
$327$	15.0978i	0.834912i
$328$	4.06711i	0.224569i
$329$	3.46681	0.191132
$330$	0	0
$331$	−4.77479	−0.262446	−0.131223	−	0.991353i	$-0.541890\pi$
$331$	−4.77479	−0.262446	−0.131223	+	0.991353i	$0.541890\pi$
$332$	23.7235i	1.30200i
$333$	− 14.8388i	− 0.813160i
$334$	2.93469	0.160579
$335$	0	0
$336$	−4.93602	−0.269282
$337$	− 24.3967i	− 1.32897i	−0.747300	−	0.664487i	$-0.768650\pi$
$337$	− 24.3967i	− 1.32897i	0.747300	−	0.664487i	$-0.231350\pi$
$338$	2.61356i	0.142159i
$339$	16.0683	0.872710
$340$	0	0
$341$	−1.66786	−0.0903196
$342$	0.582105i	0.0314766i
$343$	− 18.8442i	− 1.01749i
$344$	−7.11901	−0.383832
$345$	0	0
$346$	−3.82132	−0.205435
$347$	9.99761i	0.536700i	0.963322	+	0.268350i	$0.0864783\pi$
$347$	9.99761i	0.536700i	−0.963322	+	0.268350i	$0.913522\pi$
$348$	7.78986i	0.417580i
$349$	−21.9584	−1.17541	−0.587703	−	0.809077i	$-0.699967\pi$
$349$	−21.9584	−1.17541	−0.587703	+	0.809077i	$0.699967\pi$
$350$	0	0
$351$	6.67994	0.356549
$352$	− 2.59345i	− 0.138231i
$353$	− 36.8786i	− 1.96285i	−0.191848	−	0.981425i	$-0.561448\pi$
$353$	− 36.8786i	− 1.96285i	0.191848	−	0.981425i	$-0.438552\pi$
$354$	−1.95646	−0.103985
$355$	0	0
$356$	−5.65651	−0.299795
$357$	7.18598i	0.380322i
$358$	− 0.534516i	− 0.0282500i
$359$	−16.5187	−0.871824	−0.435912	−	0.899989i	$-0.643574\pi$
$359$	−16.5187	−0.871824	−0.435912	+	0.899989i	$0.643574\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	4.17331i	0.219344i
$363$	8.15452i	0.428001i
$364$	5.10156	0.267394
$365$	0	0
$366$	0.107523	0.00562034
$367$	6.83148i	0.356600i	0.983976	+	0.178300i	$0.0570598\pi$
$367$	6.83148i	0.356600i	−0.983976	+	0.178300i	$0.942940\pi$
$368$	15.4494i	0.805353i
$369$	9.85325	0.512940
$370$	0	0
$371$	−4.57135	−0.237333
$372$	− 2.84415i	− 0.147462i
$373$	− 4.76271i	− 0.246604i	−0.992369	−	0.123302i	$-0.960652\pi$
$373$	− 4.76271i	− 0.246604i	0.992369	−	0.123302i	$-0.0393483\pi$
$374$	−1.19269	−0.0616723
$375$	0	0
$376$	−1.99330	−0.102796
$377$	− 7.78986i	− 0.401198i
$378$	− 1.79523i	− 0.0923368i
$379$	−14.3773	−0.738514	−0.369257	−	0.929327i	$-0.620388\pi$
$379$	−14.3773	−0.738514	−0.369257	+	0.929327i	$0.620388\pi$
$380$	0	0
$381$	14.3308	0.734190
$382$	1.46309i	0.0748583i
$383$	− 30.6708i	− 1.56721i	−0.621261	−	0.783603i	$-0.713379\pi$
$383$	− 30.6708i	− 1.56721i	0.621261	−	0.783603i	$-0.286621\pi$
$384$	5.86353	0.299222
$385$	0	0
$386$	1.09544	0.0557565
$387$	17.2470i	0.876713i
$388$	− 3.01507i	− 0.153067i
$389$	12.9215	0.655148	0.327574	−	0.944825i	$-0.393769\pi$
$389$	12.9215	0.655148	0.327574	+	0.944825i	$0.393769\pi$
$390$	0	0
$391$	22.4916	1.13745
$392$	4.02475i	0.203281i
$393$	− 5.96854i	− 0.301073i
$394$	4.06829	0.204958
$395$	0	0
$396$	−4.16719	−0.209409
$397$	29.8471i	1.49798i	0.662579	+	0.748992i	$0.269462\pi$
$397$	29.8471i	1.49798i	−0.662579	+	0.748992i	$0.730538\pi$
$398$	6.02954i	0.302234i
$399$	−1.35690	−0.0679298
$400$	0	0
$401$	−28.7101	−1.43371	−0.716856	−	0.697221i	$-0.754420\pi$
$401$	−28.7101	−1.43371	−0.716856	+	0.697221i	$0.754420\pi$
$402$	2.77240i	0.138275i
$403$	2.84415i	0.141677i
$404$	−32.1903	−1.60153
$405$	0	0
$406$	−2.09352	−0.103900
$407$	5.74094i	0.284568i
$408$	− 4.13169i	− 0.204549i
$409$	13.6203	0.673479	0.336739	−	0.941598i	$-0.390676\pi$
$409$	13.6203	0.673479	0.336739	+	0.941598i	$0.390676\pi$
$410$	0	0
$411$	−4.56033	−0.224945
$412$	− 9.39539i	− 0.462878i
$413$	16.7138i	0.822432i
$414$	−2.47219	−0.121501
$415$	0	0
$416$	−4.42253	−0.216833
$417$	2.89977i	0.142002i
$418$	− 0.225209i	− 0.0110153i
$419$	2.13946	0.104519	0.0522596	−	0.998634i	$-0.483358\pi$
$419$	2.13946	0.104519	0.0522596	+	0.998634i	$0.483358\pi$
$420$	0	0
$421$	−15.0562	−0.733795	−0.366897	−	0.930261i	$-0.619580\pi$
$421$	−15.0562	−0.733795	−0.366897	+	0.930261i	$0.619580\pi$
$422$	− 1.07202i	− 0.0521849i
$423$	4.82908i	0.234798i
$424$	2.62837	0.127645
$425$	0	0
$426$	−2.55065	−0.123579
$427$	− 0.918559i	− 0.0444522i
$428$	8.66115i	0.418653i
$429$	−1.13706	−0.0548979
$430$	0	0
$431$	4.37435	0.210705	0.105353	−	0.994435i	$-0.466403\pi$
$431$	4.37435	0.210705	0.105353	+	0.994435i	$0.466403\pi$
$432$	− 15.6273i	− 0.751869i
$433$	− 33.1564i	− 1.59340i	−0.604377	−	0.796698i	$-0.706578\pi$
$433$	− 33.1564i	− 1.59340i	0.604377	−	0.796698i	$-0.293422\pi$
$434$	0.764365	0.0366907
$435$	0	0
$436$	−36.5050	−1.74827
$437$	4.24698i	0.203161i
$438$	− 0.556023i	− 0.0265678i
$439$	−32.2368	−1.53858	−0.769290	−	0.638900i	$-0.779390\pi$
$439$	−32.2368	−1.53858	−0.769290	+	0.638900i	$0.779390\pi$
$440$	0	0
$441$	9.75063	0.464316
$442$	2.03385i	0.0967405i
$443$	21.7362i	1.03272i	0.856373	+	0.516358i	$0.172713\pi$
$443$	21.7362i	1.03272i	−0.856373	+	0.516358i	$0.827287\pi$
$444$	−9.78986	−0.464606
$445$	0	0
$446$	−6.48022	−0.306847
$447$	2.63879i	0.124811i
$448$	− 11.1217i	− 0.525450i
$449$	24.9584	1.17786	0.588929	−	0.808185i	$-0.299550\pi$
$449$	24.9584	1.17786	0.588929	+	0.808185i	$0.299550\pi$
$450$	0	0
$451$	−3.81210	−0.179505
$452$	38.8514i	1.82742i
$453$	8.19029i	0.384814i
$454$	3.62697	0.170222
$455$	0	0
$456$	0.780167	0.0365347
$457$	− 13.1347i	− 0.614414i	−0.951643	−	0.307207i	$-0.900606\pi$
$457$	− 13.1347i	− 0.614414i	0.951643	−	0.307207i	$-0.0993944\pi$
$458$	5.34481i	0.249747i
$459$	−22.7506	−1.06191
$460$	0	0
$461$	−35.3726	−1.64746	−0.823732	−	0.566979i	$-0.808112\pi$
$461$	−35.3726	−1.64746	−0.823732	+	0.566979i	$0.808112\pi$
$462$	0.305586i	0.0142171i
$463$	14.6963i	0.682997i	0.939882	+	0.341498i	$0.110934\pi$
$463$	14.6963i	0.682997i	−0.939882	+	0.341498i	$0.889066\pi$
$464$	−18.2239	−0.846022
$465$	0	0
$466$	−6.69766	−0.310263
$467$	33.2121i	1.53687i	0.639927	+	0.768435i	$0.278964\pi$
$467$	33.2121i	1.53687i	−0.639927	+	0.768435i	$0.721036\pi$
$468$	7.10620i	0.328484i
$469$	23.6843	1.09364
$470$	0	0
$471$	−11.5036	−0.530060
$472$	− 9.60984i	− 0.442329i
$473$	− 6.67264i	− 0.306809i
$474$	−0.315748	−0.0145028
$475$	0	0
$476$	−17.3749	−0.796379
$477$	− 6.36765i	− 0.291555i
$478$	2.84787i	0.130259i
$479$	−24.6219	−1.12500	−0.562502	−	0.826796i	$-0.690161\pi$
$479$	−24.6219	−1.12500	−0.562502	+	0.826796i	$0.690161\pi$
$480$	0	0
$481$	9.78986	0.446379
$482$	2.91915i	0.132964i
$483$	− 5.76271i	− 0.262212i
$484$	−19.7168	−0.896217
$485$	0	0
$486$	3.90110	0.176958
$487$	29.5646i	1.33970i	0.742496	+	0.669851i	$0.233642\pi$
$487$	29.5646i	1.33970i	−0.742496	+	0.669851i	$0.766358\pi$
$488$	0.528139i	0.0239077i
$489$	−15.6625	−0.708282
$490$	0	0
$491$	36.2978	1.63810	0.819049	−	0.573724i	$-0.194502\pi$
$491$	36.2978	1.63810	0.819049	+	0.573724i	$0.194502\pi$
$492$	− 6.50066i	− 0.293073i
$493$	26.5308i	1.19489i
$494$	−0.384043	−0.0172789
$495$	0	0
$496$	6.65371	0.298760
$497$	21.7899i	0.977409i
$498$	2.42327i	0.108589i
$499$	−7.84415	−0.351152	−0.175576	−	0.984466i	$-0.556179\pi$
$499$	−7.84415	−0.351152	−0.175576	+	0.984466i	$0.556179\pi$
$500$	0	0
$501$	−9.52888	−0.425719
$502$	− 2.38703i	− 0.106538i
$503$	− 20.4166i	− 0.910330i	−0.890407	−	0.455165i	$-0.849580\pi$
$503$	− 20.4166i	− 0.910330i	0.890407	−	0.455165i	$-0.150420\pi$
$504$	3.87966	0.172814
$505$	0	0
$506$	0.956459	0.0425198
$507$	− 8.48619i	− 0.376885i
$508$	34.6504i	1.53736i
$509$	14.7530	0.653916	0.326958	−	0.945039i	$-0.393976\pi$
$509$	14.7530	0.653916	0.326958	+	0.945039i	$0.393976\pi$
$510$	0	0
$511$	−4.75004	−0.210129
$512$	17.4242i	0.770048i
$513$	− 4.29590i	− 0.189668i
$514$	3.50365	0.154539
$515$	0	0
$516$	11.3787	0.500918
$517$	− 1.86831i	− 0.0821683i
$518$	− 2.63102i	− 0.115600i
$519$	12.4077	0.544639
$520$	0	0
$521$	37.1487	1.62751	0.813756	−	0.581206i	$-0.197419\pi$
$521$	37.1487	1.62751	0.813756	+	0.581206i	$0.197419\pi$
$522$	− 2.91617i	− 0.127637i
$523$	− 16.3623i	− 0.715472i	−0.933823	−	0.357736i	$-0.883549\pi$
$523$	− 16.3623i	− 0.715472i	0.933823	−	0.357736i	$-0.116451\pi$
$524$	14.4313	0.630434
$525$	0	0
$526$	5.38271	0.234698
$527$	− 9.68664i	− 0.421957i
$528$	2.66009i	0.115765i
$529$	4.96316	0.215790
$530$	0	0
$531$	−23.2814	−1.01033
$532$	− 3.28083i	− 0.142242i
$533$	6.50066i	0.281575i
$534$	−0.577793	−0.0250036
$535$	0	0
$536$	−13.6176	−0.588191
$537$	1.73556i	0.0748950i
$538$	6.10693i	0.263289i
$539$	−3.77240	−0.162489
$540$	0	0
$541$	−2.08947	−0.0898335	−0.0449168	−	0.998991i	$-0.514302\pi$
$541$	−2.08947	−0.0898335	−0.0449168	+	0.998991i	$0.514302\pi$
$542$	− 3.26742i	− 0.140348i
$543$	− 13.5506i	− 0.581514i
$544$	15.0623	0.645792
$545$	0	0
$546$	0.521106	0.0223013
$547$	1.86054i	0.0795511i	0.999209	+	0.0397756i	$0.0126643\pi$
$547$	1.86054i	0.0795511i	−0.999209	+	0.0397756i	$0.987336\pi$
$548$	− 11.0264i	− 0.471025i
$549$	1.27950	0.0546079
$550$	0	0
$551$	−5.00969	−0.213420
$552$	3.31336i	0.141026i
$553$	2.69740i	0.114705i
$554$	−0.138391	−0.00587968
$555$	0	0
$556$	−7.01134	−0.297347
$557$	− 9.80061i	− 0.415265i	−0.978207	−	0.207633i	$-0.933424\pi$
$557$	− 9.80061i	− 0.415265i	0.978207	−	0.207633i	$-0.0665758\pi$
$558$	1.06472i	0.0450732i
$559$	−11.3787	−0.481266
$560$	0	0
$561$	3.87263	0.163502
$562$	6.81759i	0.287583i
$563$	− 38.0170i	− 1.60222i	−0.598514	−	0.801112i	$-0.704242\pi$
$563$	− 38.0170i	− 1.60222i	0.598514	−	0.801112i	$-0.295758\pi$
$564$	3.18598	0.134154
$565$	0	0
$566$	3.94198	0.165694
$567$	− 6.13467i	− 0.257632i
$568$	− 12.5284i	− 0.525680i
$569$	7.32975	0.307279	0.153640	−	0.988127i	$-0.450901\pi$
$569$	7.32975	0.307279	0.153640	+	0.988127i	$0.450901\pi$
$570$	0	0
$571$	37.8775	1.58513	0.792563	−	0.609791i	$-0.208746\pi$
$571$	37.8775	1.58513	0.792563	+	0.609791i	$0.208746\pi$
$572$	− 2.74930i	− 0.114954i
$573$	− 4.75063i	− 0.198460i
$574$	1.74705	0.0729206
$575$	0	0
$576$	15.4919	0.645496
$577$	28.6993i	1.19477i	0.801955	+	0.597384i	$0.203793\pi$
$577$	28.6993i	1.19477i	−0.801955	+	0.597384i	$0.796207\pi$
$578$	− 2.72827i	− 0.113481i
$579$	−3.55688	−0.147819
$580$	0	0
$581$	20.7017	0.858852
$582$	− 0.307979i	− 0.0127661i
$583$	2.46357i	0.102030i
$584$	2.73110	0.113014
$585$	0	0
$586$	−6.25667	−0.258461
$587$	3.72348i	0.153684i	0.997043	+	0.0768422i	$0.0244838\pi$
$587$	3.72348i	0.153684i	−0.997043	+	0.0768422i	$0.975516\pi$
$588$	− 6.43296i	− 0.265291i
$589$	1.82908	0.0753661
$590$	0	0
$591$	−13.2097	−0.543373
$592$	− 22.9028i	− 0.941297i
$593$	− 27.7399i	− 1.13914i	−0.821943	−	0.569570i	$-0.807110\pi$
$593$	− 27.7399i	− 1.13914i	0.821943	−	0.569570i	$-0.192890\pi$
$594$	−0.967476	−0.0396960
$595$	0	0
$596$	−6.38032	−0.261348
$597$	− 19.5778i	− 0.801266i
$598$	− 1.63102i	− 0.0666975i
$599$	23.5579	0.962551	0.481276	−	0.876569i	$-0.340174\pi$
$599$	23.5579	0.962551	0.481276	+	0.876569i	$0.340174\pi$
$600$	0	0
$601$	−7.36898	−0.300587	−0.150293	−	0.988641i	$-0.548022\pi$
$601$	−7.36898	−0.300587	−0.150293	+	0.988641i	$0.548022\pi$
$602$	3.05802i	0.124635i
$603$	32.9909i	1.34349i
$604$	−19.8033	−0.805783
$605$	0	0
$606$	−3.28813	−0.133571
$607$	− 28.4198i	− 1.15352i	−0.816912	−	0.576762i	$-0.804316\pi$
$607$	− 28.4198i	− 1.15352i	0.816912	−	0.576762i	$-0.195684\pi$
$608$	2.84415i	0.115346i
$609$	6.79763	0.275454
$610$	0	0
$611$	−3.18598	−0.128891
$612$	− 24.2024i	− 0.978323i
$613$	6.48129i	0.261777i	0.991397	+	0.130888i	$0.0417829\pi$
$613$	6.48129i	0.261777i	−0.991397	+	0.130888i	$0.958217\pi$
$614$	−2.85443	−0.115195
$615$	0	0
$616$	−1.50099	−0.0604767
$617$	24.4650i	0.984924i	0.870334	+	0.492462i	$0.163903\pi$
$617$	24.4650i	0.984924i	−0.870334	+	0.492462i	$0.836097\pi$
$618$	− 0.959706i	− 0.0386051i
$619$	22.2457	0.894128	0.447064	−	0.894502i	$-0.352470\pi$
$619$	22.2457	0.894128	0.447064	+	0.894502i	$0.352470\pi$
$620$	0	0
$621$	18.2446	0.732130
$622$	− 4.53020i	− 0.181645i
$623$	4.93602i	0.197757i
$624$	4.53617	0.181592
$625$	0	0
$626$	4.67084	0.186684
$627$	0.731250i	0.0292033i
$628$	− 27.8146i	− 1.10992i
$629$	−33.3424	−1.32945
$630$	0	0
$631$	−15.5888	−0.620581	−0.310290	−	0.950642i	$-0.600426\pi$
$631$	−15.5888	−0.620581	−0.310290	+	0.950642i	$0.600426\pi$
$632$	− 1.55091i	− 0.0616919i
$633$	3.48081i	0.138350i
$634$	5.00836	0.198907
$635$	0	0
$636$	−4.20105	−0.166582
$637$	6.43296i	0.254883i
$638$	1.12823i	0.0446670i
$639$	−30.3521	−1.20071
$640$	0	0
$641$	8.17496	0.322892	0.161446	−	0.986882i	$-0.448384\pi$
$641$	8.17496	0.322892	0.161446	+	0.986882i	$0.448384\pi$
$642$	0.884707i	0.0349166i
$643$	11.1836i	0.441038i	0.975383	+	0.220519i	$0.0707750\pi$
$643$	11.1836i	0.441038i	−0.975383	+	0.220519i	$0.929225\pi$
$644$	13.9336	0.549062
$645$	0	0
$646$	1.30798	0.0514617
$647$	− 16.2121i	− 0.637362i	−0.947862	−	0.318681i	$-0.896760\pi$
$647$	− 16.2121i	− 0.637362i	0.947862	−	0.318681i	$-0.103240\pi$
$648$	3.52722i	0.138562i
$649$	9.00730	0.353567
$650$	0	0
$651$	−2.48188	−0.0972725
$652$	− 37.8702i	− 1.48311i
$653$	24.7952i	0.970312i	0.874427	+	0.485156i	$0.161237\pi$
$653$	24.7952i	0.970312i	−0.874427	+	0.485156i	$0.838763\pi$
$654$	−3.72886	−0.145810
$655$	0	0
$656$	15.2079	0.593769
$657$	− 6.61655i	− 0.258136i
$658$	0.856232i	0.0333794i
$659$	20.2868	0.790262	0.395131	−	0.918625i	$-0.370699\pi$
$659$	20.2868	0.790262	0.395131	+	0.918625i	$0.370699\pi$
$660$	0	0
$661$	20.4222	0.794332	0.397166	−	0.917747i	$-0.369994\pi$
$661$	20.4222	0.794332	0.397166	+	0.917747i	$0.369994\pi$
$662$	− 1.17928i	− 0.0458339i
$663$	− 6.60388i	− 0.256473i
$664$	−11.9028	−0.461917
$665$	0	0
$666$	3.66487	0.142011
$667$	− 21.2760i	− 0.823812i
$668$	− 23.0398i	− 0.891437i
$669$	21.0411	0.813498
$670$	0	0
$671$	−0.495024	−0.0191102
$672$	− 3.85922i	− 0.148872i
$673$	28.7254i	1.10728i	0.832755	+	0.553641i	$0.186762\pi$
$673$	28.7254i	1.10728i	−0.832755	+	0.553641i	$0.813238\pi$
$674$	6.02549	0.232093
$675$	0	0
$676$	20.5187	0.789181
$677$	− 44.0062i	− 1.69130i	−0.533740	−	0.845648i	$-0.679214\pi$
$677$	− 44.0062i	− 1.69130i	0.533740	−	0.845648i	$-0.320786\pi$
$678$	3.96854i	0.152411i
$679$	−2.63102	−0.100969
$680$	0	0
$681$	−11.7767	−0.451284
$682$	− 0.411927i	− 0.0157735i
$683$	8.48427i	0.324642i	0.986738	+	0.162321i	$0.0518979\pi$
$683$	8.48427i	0.324642i	−0.986738	+	0.162321i	$0.948102\pi$
$684$	4.57002	0.174739
$685$	0	0
$686$	4.65412	0.177695
$687$	− 17.3545i	− 0.662116i
$688$	26.6197i	1.01487i
$689$	4.20105	0.160047
$690$	0	0
$691$	20.2586	0.770673	0.385336	−	0.922776i	$-0.374085\pi$
$691$	20.2586	0.770673	0.385336	+	0.922776i	$0.374085\pi$
$692$	30.0006i	1.14045i
$693$	3.63640i	0.138135i
$694$	−2.46921	−0.0937297
$695$	0	0
$696$	−3.90840	−0.148147
$697$	− 22.1400i	− 0.838614i
$698$	− 5.42327i	− 0.205274i
$699$	21.7472	0.822553
$700$	0	0
$701$	−23.4101	−0.884188	−0.442094	−	0.896969i	$-0.645764\pi$
$701$	−23.4101	−0.884188	−0.442094	+	0.896969i	$0.645764\pi$
$702$	1.64981i	0.0622680i
$703$	− 6.29590i	− 0.237454i
$704$	−5.99362	−0.225893
$705$	0	0
$706$	9.10826	0.342794
$707$	28.0901i	1.05644i
$708$	15.3599i	0.577260i
$709$	−30.3472	−1.13971	−0.569857	−	0.821744i	$-0.693001\pi$
$709$	−30.3472	−1.13971	−0.569857	+	0.821744i	$0.693001\pi$
$710$	0	0
$711$	−3.75733	−0.140911
$712$	− 2.83804i	− 0.106360i
$713$	7.76809i	0.290917i
$714$	−1.77479	−0.0664199
$715$	0	0
$716$	−4.19641	−0.156827
$717$	− 9.24698i	− 0.345335i
$718$	− 4.07979i	− 0.152256i
$719$	38.3230	1.42921	0.714604	−	0.699529i	$-0.246607\pi$
$719$	38.3230	1.42921	0.714604	+	0.699529i	$0.246607\pi$
$720$	0	0
$721$	−8.19865	−0.305334
$722$	0.246980i	0.00919163i
$723$	− 9.47842i	− 0.352506i
$724$	32.7640	1.21767
$725$	0	0
$726$	−2.01400	−0.0747466
$727$	− 2.69069i	− 0.0997923i	−0.998754	−	0.0498961i	$-0.984111\pi$
$727$	− 2.69069i	− 0.0997923i	0.998754	−	0.0498961i	$-0.0158890\pi$
$728$	2.55960i	0.0948650i
$729$	−1.78986	−0.0662910
$730$	0	0
$731$	38.7536	1.43335
$732$	− 0.844150i	− 0.0312007i
$733$	18.9651i	0.700491i	0.936658	+	0.350246i	$0.113902\pi$
$733$	18.9651i	0.700491i	−0.936658	+	0.350246i	$0.886098\pi$
$734$	−1.68724	−0.0622770
$735$	0	0
$736$	−12.0790	−0.445240
$737$	− 12.7638i	− 0.470160i
$738$	2.43355i	0.0895803i
$739$	−29.8278	−1.09723	−0.548616	−	0.836075i	$-0.684845\pi$
$739$	−29.8278	−1.09723	−0.548616	+	0.836075i	$0.684845\pi$
$740$	0	0
$741$	1.24698	0.0458089
$742$	− 1.12903i	− 0.0414480i
$743$	8.78448i	0.322271i	0.986932	+	0.161136i	$0.0515157\pi$
$743$	8.78448i	0.322271i	−0.986932	+	0.161136i	$0.948484\pi$
$744$	1.42699	0.0523161
$745$	0	0
$746$	1.17629	0.0430671
$747$	28.8364i	1.05507i
$748$	9.36360i	0.342367i
$749$	7.55794	0.276161
$750$	0	0
$751$	−18.1142	−0.660998	−0.330499	−	0.943806i	$-0.607217\pi$
$751$	−18.1142	−0.660998	−0.330499	+	0.943806i	$0.607217\pi$
$752$	7.45340i	0.271798i
$753$	7.75063i	0.282449i
$754$	1.92394	0.0700656
$755$	0	0
$756$	−14.0941	−0.512598
$757$	0.222816i	0.00809840i	0.999992	+	0.00404920i	$0.00128890\pi$
$757$	0.222816i	0.00809840i	−0.999992	+	0.00404920i	$0.998711\pi$
$758$	− 3.55091i	− 0.128975i
$759$	−3.10560	−0.112726
$760$	0	0
$761$	−6.07798	−0.220327	−0.110163	−	0.993913i	$-0.535137\pi$
$761$	−6.07798	−0.220327	−0.110163	+	0.993913i	$0.535137\pi$
$762$	3.53942i	0.128220i
$763$	31.8552i	1.15323i
$764$	11.4865	0.415568
$765$	0	0
$766$	7.57507	0.273698
$767$	− 15.3599i	− 0.554613i
$768$	− 9.09411i	− 0.328156i
$769$	14.6267	0.527453	0.263726	−	0.964598i	$-0.415048\pi$
$769$	14.6267	0.527453	0.263726	+	0.964598i	$0.415048\pi$
$770$	0	0
$771$	−11.3763	−0.409706
$772$	− 8.60015i	− 0.309526i
$773$	− 4.05728i	− 0.145930i	−0.997334	−	0.0729651i	$-0.976754\pi$
$773$	− 4.05728i	− 0.145930i	0.997334	−	0.0729651i	$-0.0232462\pi$
$774$	−4.25965	−0.153110
$775$	0	0
$776$	1.51275	0.0543044
$777$	8.54288i	0.306474i
$778$	3.19136i	0.114416i
$779$	4.18060	0.149786
$780$	0	0
$781$	11.7429	0.420192
$782$	5.55496i	0.198645i
$783$	21.5211i	0.769102i
$784$	15.0495	0.537482
$785$	0	0
$786$	1.47411	0.0525797
$787$	− 19.5657i	− 0.697442i	−0.937227	−	0.348721i	$-0.886616\pi$
$787$	− 19.5657i	− 0.697442i	0.937227	−	0.348721i	$-0.113384\pi$
$788$	− 31.9396i	− 1.13780i
$789$	−17.4776	−0.622218
$790$	0	0
$791$	33.9028	1.20544
$792$	− 2.09080i	− 0.0742934i
$793$	0.844150i	0.0299767i
$794$	−7.37163	−0.261609
$795$	0	0
$796$	47.3370	1.67782
$797$	− 38.9051i	− 1.37809i	−0.724718	−	0.689046i	$-0.758030\pi$
$797$	− 38.9051i	− 1.37809i	0.724718	−	0.689046i	$-0.241970\pi$
$798$	− 0.335126i	− 0.0118633i
$799$	10.8509	0.383876
$800$	0	0
$801$	−6.87561	−0.242938
$802$	− 7.09080i	− 0.250385i
$803$	2.55986i	0.0903355i
$804$	21.7657	0.767617
$805$	0	0
$806$	−0.702447	−0.0247426
$807$	− 19.8291i	− 0.698017i
$808$	− 16.1508i	− 0.568183i
$809$	22.5730	0.793625	0.396812	−	0.917900i	$-0.370116\pi$
$809$	22.5730	0.793625	0.396812	+	0.917900i	$0.370116\pi$
$810$	0	0
$811$	−46.3605	−1.62794	−0.813968	−	0.580909i	$-0.802697\pi$
$811$	−46.3605	−1.62794	−0.813968	+	0.580909i	$0.802697\pi$
$812$	16.4359i	0.576789i
$813$	10.6093i	0.372083i
$814$	−1.41789	−0.0496972
$815$	0	0
$816$	−15.4494	−0.540836
$817$	7.31767i	0.256013i
$818$	3.36393i	0.117617i
$819$	6.20105	0.216682
$820$	0	0
$821$	−17.8194	−0.621901	−0.310951	−	0.950426i	$-0.600647\pi$
$821$	−17.8194	−0.621901	−0.310951	+	0.950426i	$0.600647\pi$
$822$	− 1.12631i	− 0.0392846i
$823$	− 32.5394i	− 1.13425i	−0.823631	−	0.567126i	$-0.808055\pi$
$823$	− 32.5394i	− 1.13425i	0.823631	−	0.567126i	$-0.191945\pi$
$824$	4.71394	0.164218
$825$	0	0
$826$	−4.12797	−0.143630
$827$	− 39.8256i	− 1.38487i	−0.721479	−	0.692436i	$-0.756537\pi$
$827$	− 39.8256i	− 1.38487i	0.721479	−	0.692436i	$-0.243463\pi$
$828$	19.4088i	0.674502i
$829$	−38.7525	−1.34593	−0.672966	−	0.739674i	$-0.734980\pi$
$829$	−38.7525	−1.34593	−0.672966	+	0.739674i	$0.734980\pi$
$830$	0	0
$831$	0.449354	0.0155879
$832$	10.2208i	0.354341i
$833$	− 21.9095i	− 0.759118i
$834$	−0.716185	−0.0247994
$835$	0	0
$836$	−1.76809	−0.0611505
$837$	− 7.85756i	− 0.271597i
$838$	0.528402i	0.0182533i
$839$	−2.76749	−0.0955445	−0.0477723	−	0.998858i	$-0.515212\pi$
$839$	−2.76749	−0.0955445	−0.0477723	+	0.998858i	$0.515212\pi$
$840$	0	0
$841$	−3.90302	−0.134587
$842$	− 3.71858i	− 0.128151i
$843$	− 22.1366i	− 0.762425i
$844$	−8.41624	−0.289699
$845$	0	0
$846$	−1.19269	−0.0410054
$847$	17.2054i	0.591183i
$848$	− 9.82808i	− 0.337498i
$849$	−12.7995	−0.439279
$850$	0	0
$851$	26.7385	0.916586
$852$	20.0248i	0.686037i
$853$	24.1390i	0.826503i	0.910617	+	0.413252i	$0.135607\pi$
$853$	24.1390i	0.826503i	−0.910617	+	0.413252i	$0.864393\pi$
$854$	0.226865	0.00776317
$855$	0	0
$856$	−4.34555	−0.148528
$857$	3.16229i	0.108022i	0.998540	+	0.0540109i	$0.0172006\pi$
$857$	3.16229i	0.108022i	−0.998540	+	0.0540109i	$0.982799\pi$
$858$	− 0.280831i	− 0.00958743i
$859$	−4.86426	−0.165967	−0.0829833	−	0.996551i	$-0.526445\pi$
$859$	−4.86426	−0.165967	−0.0829833	+	0.996551i	$0.526445\pi$
$860$	0	0
$861$	−5.67264	−0.193323
$862$	1.08038i	0.0367978i
$863$	4.54958i	0.154870i	0.996997	+	0.0774348i	$0.0246730\pi$
$863$	4.54958i	0.154870i	−0.996997	+	0.0774348i	$0.975327\pi$
$864$	12.2182	0.415671
$865$	0	0
$866$	8.18896	0.278272
$867$	8.85862i	0.300855i
$868$	− 6.00092i	− 0.203684i
$869$	1.45367	0.0493122
$870$	0	0
$871$	−21.7657	−0.737502
$872$	− 18.3156i	− 0.620245i
$873$	− 3.66487i	− 0.124037i
$874$	−1.04892	−0.0354802
$875$	0	0
$876$	−4.36526	−0.147488
$877$	18.6595i	0.630086i	0.949077	+	0.315043i	$0.102019\pi$
$877$	18.6595i	0.630086i	−0.949077	+	0.315043i	$0.897981\pi$
$878$	− 7.96184i	− 0.268699i
$879$	20.3153	0.685217
$880$	0	0
$881$	19.4306	0.654632	0.327316	−	0.944915i	$-0.393856\pi$
$881$	19.4306	0.654632	0.327316	+	0.944915i	$0.393856\pi$
$882$	2.40821i	0.0810885i
$883$	25.6437i	0.862979i	0.902118	+	0.431490i	$0.142012\pi$
$883$	25.6437i	0.862979i	−0.902118	+	0.431490i	$0.857988\pi$
$884$	15.9675	0.537044
$885$	0	0
$886$	−5.36839	−0.180354
$887$	31.0062i	1.04109i	0.853835	+	0.520544i	$0.174271\pi$
$887$	31.0062i	1.04109i	−0.853835	+	0.520544i	$0.825729\pi$
$888$	− 4.91185i	− 0.164831i
$889$	30.2368	1.01411
$890$	0	0
$891$	−3.30606	−0.110757
$892$	50.8753i	1.70343i
$893$	2.04892i	0.0685644i
$894$	−0.651728	−0.0217970
$895$	0	0
$896$	12.3716	0.413305
$897$	5.29590i	0.176825i
$898$	6.16421i	0.205702i
$899$	−9.16315	−0.305608
$900$	0	0
$901$	−14.3080	−0.476668
$902$	− 0.941511i	− 0.0313489i
$903$	− 9.92931i	− 0.330427i
$904$	−19.4929	−0.648324
$905$	0	0
$906$	−2.02284	−0.0672042
$907$	17.7676i	0.589964i	0.955503	+	0.294982i	$0.0953136\pi$
$907$	17.7676i	0.589964i	−0.955503	+	0.294982i	$0.904686\pi$
$908$	− 28.4748i	− 0.944971i
$909$	−39.1280	−1.29779
$910$	0	0
$911$	41.7313	1.38262	0.691309	−	0.722559i	$-0.257034\pi$
$911$	41.7313	1.38262	0.691309	+	0.722559i	$0.257034\pi$
$912$	− 2.91723i	− 0.0965992i
$913$	− 11.1564i	− 0.369224i
$914$	3.24400	0.107302
$915$	0	0
$916$	41.9614	1.38644
$917$	− 12.5931i	− 0.415862i
$918$	− 5.61894i	− 0.185453i
$919$	−30.4088	−1.00309	−0.501547	−	0.865130i	$-0.667236\pi$
$919$	−30.4088	−1.00309	−0.501547	+	0.865130i	$0.667236\pi$
$920$	0	0
$921$	9.26828	0.305400
$922$	− 8.73630i	− 0.287715i
$923$	− 20.0248i	− 0.659123i
$924$	2.39911	0.0789249
$925$	0	0
$926$	−3.62969	−0.119279
$927$	− 11.4203i	− 0.375091i
$928$	− 14.2483i	− 0.467724i
$929$	28.8219	0.945616	0.472808	−	0.881165i	$-0.343240\pi$
$929$	28.8219	0.945616	0.472808	+	0.881165i	$0.343240\pi$
$930$	0	0
$931$	4.13706	0.135587
$932$	52.5824i	1.72239i
$933$	14.7095i	0.481567i
$934$	−8.20270	−0.268401
$935$	0	0
$936$	−3.56538	−0.116538
$937$	50.4601i	1.64846i	0.566255	+	0.824230i	$0.308392\pi$
$937$	50.4601i	1.64846i	−0.566255	+	0.824230i	$0.691608\pi$
$938$	5.84953i	0.190994i
$939$	−15.1661	−0.494928
$940$	0	0
$941$	1.10082	0.0358857	0.0179428	−	0.999839i	$-0.494288\pi$
$941$	1.10082	0.0358857	0.0179428	+	0.999839i	$0.494288\pi$
$942$	− 2.84117i	− 0.0925702i
$943$	17.7549i	0.578180i
$944$	−35.9335	−1.16954
$945$	0	0
$946$	1.64801	0.0535813
$947$	13.9353i	0.452836i	0.974030	+	0.226418i	$0.0727015\pi$
$947$	13.9353i	0.452836i	−0.974030	+	0.226418i	$0.927299\pi$
$948$	2.47889i	0.0805107i
$949$	4.36526	0.141702
$950$	0	0
$951$	−16.2620	−0.527333
$952$	− 8.71751i	− 0.282536i
$953$	− 41.3400i	− 1.33913i	−0.742751	−	0.669567i	$-0.766480\pi$
$953$	− 41.3400i	− 1.33913i	0.742751	−	0.669567i	$-0.233520\pi$
$954$	1.57268	0.0509174
$955$	0	0
$956$	22.3582	0.723117
$957$	− 3.66334i	− 0.118419i
$958$	− 6.08111i	− 0.196472i
$959$	−9.62192	−0.310708
$960$	0	0
$961$	−27.6544	−0.892079
$962$	2.41789i	0.0779561i
$963$	10.5278i	0.339254i
$964$	22.9178	0.738133
$965$	0	0
$966$	1.42327	0.0457930
$967$	5.26875i	0.169432i	0.996405	+	0.0847158i	$0.0269982\pi$
$967$	5.26875i	0.169432i	−0.996405	+	0.0847158i	$0.973002\pi$
$968$	− 9.89248i	− 0.317956i
$969$	−4.24698	−0.136433
$970$	0	0
$971$	−5.15346	−0.165382	−0.0826911	−	0.996575i	$-0.526351\pi$
$971$	−5.15346	−0.165382	−0.0826911	+	0.996575i	$0.526351\pi$
$972$	− 30.6270i	− 0.982361i
$973$	6.11828i	0.196143i
$974$	−7.30186	−0.233967
$975$	0	0
$976$	1.97484	0.0632130
$977$	− 4.77612i	− 0.152802i	−0.997077	−	0.0764008i	$-0.975657\pi$
$977$	− 4.77612i	− 0.152802i	0.997077	−	0.0764008i	$-0.0243428\pi$
$978$	− 3.86831i	− 0.123695i
$979$	2.66009	0.0850168
$980$	0	0
$981$	−44.3726	−1.41671
$982$	8.96482i	0.286079i
$983$	− 28.9758i	− 0.924186i	−0.886832	−	0.462093i	$-0.847099\pi$
$983$	− 28.9758i	− 0.924186i	0.886832	−	0.462093i	$-0.152901\pi$
$984$	3.26157	0.103975
$985$	0	0
$986$	−6.55257	−0.208676
$987$	− 2.78017i	− 0.0884937i
$988$	3.01507i	0.0959220i
$989$	−31.0780	−0.988222
$990$	0	0
$991$	−38.5042	−1.22313	−0.611564	−	0.791195i	$-0.709459\pi$
$991$	−38.5042	−1.22313	−0.611564	+	0.791195i	$0.709459\pi$
$992$	5.20219i	0.165170i
$993$	3.82908i	0.121512i
$994$	−5.38165	−0.170696
$995$	0	0
$996$	19.0248	0.602822
$997$	− 10.1491i	− 0.321427i	−0.987001	−	0.160713i	$-0.948621\pi$
$997$	− 10.1491i	− 0.321427i	0.987001	−	0.160713i	$-0.0513795\pi$
$998$	− 1.93735i	− 0.0613256i
$999$	−27.0465	−0.855714

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.b.c.324.4		6
5.2	odd	4		475.2.a.h.1.1	yes	3
5.3	odd	4		475.2.a.d.1.3	✓	3
5.4	even	2	inner	475.2.b.c.324.3		6
15.2	even	4		4275.2.a.z.1.3		3
15.8	even	4		4275.2.a.bn.1.1		3
20.3	even	4		7600.2.a.bw.1.1		3
20.7	even	4		7600.2.a.bn.1.3		3
95.18	even	4		9025.2.a.be.1.1		3
95.37	even	4		9025.2.a.w.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.d.1.3	✓	3	5.3	odd	4
475.2.a.h.1.1	yes	3	5.2	odd	4
475.2.b.c.324.3		6	5.4	even	2	inner
475.2.b.c.324.4		6	1.1	even	1	trivial
4275.2.a.z.1.3		3	15.2	even	4
4275.2.a.bn.1.1		3	15.8	even	4
7600.2.a.bn.1.3		3	20.7	even	4
7600.2.a.bw.1.1		3	20.3	even	4
9025.2.a.w.1.3		3	95.37	even	4
9025.2.a.be.1.1		3	95.18	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.246980i q^{2} -0.801938i q^{3} +1.93900 q^{4} +0.198062 q^{6} -1.69202i q^{7} +0.972853i q^{8} +2.35690 q^{9} +O(q^{10})\)	\(q+0.246980i q^{2} -0.801938i q^{3} +1.93900 q^{4} +0.198062 q^{6} -1.69202i q^{7} +0.972853i q^{8} +2.35690 q^{9} -0.911854 q^{11} -1.55496i q^{12} +1.55496i q^{13} +0.417895 q^{14} +3.63773 q^{16} -5.29590i q^{17} +0.582105i q^{18} +1.00000 q^{19} -1.35690 q^{21} -0.225209i q^{22} +4.24698i q^{23} +0.780167 q^{24} -0.384043 q^{26} -4.29590i q^{27} -3.28083i q^{28} -5.00969 q^{29} +1.82908 q^{31} +2.84415i q^{32} +0.731250i q^{33} +1.30798 q^{34} +4.57002 q^{36} -6.29590i q^{37} +0.246980i q^{38} +1.24698 q^{39} +4.18060 q^{41} -0.335126i q^{42} +7.31767i q^{43} -1.76809 q^{44} -1.04892 q^{46} +2.04892i q^{47} -2.91723i q^{48} +4.13706 q^{49} -4.24698 q^{51} +3.01507i q^{52} -2.70171i q^{53} +1.06100 q^{54} +1.64609 q^{56} -0.801938i q^{57} -1.23729i q^{58} -9.87800 q^{59} +0.542877 q^{61} +0.451747i q^{62} -3.98792i q^{63} +6.57301 q^{64} -0.180604 q^{66} +13.9976i q^{67} -10.2687i q^{68} +3.40581 q^{69} -12.8780 q^{71} +2.29291i q^{72} -2.80731i q^{73} +1.55496 q^{74} +1.93900 q^{76} +1.54288i q^{77} +0.307979i q^{78} -1.59419 q^{79} +3.62565 q^{81} +1.03252i q^{82} +12.2349i q^{83} -2.63102 q^{84} -1.80731 q^{86} +4.01746i q^{87} -0.887100i q^{88} -2.91723 q^{89} +2.63102 q^{91} +8.23490i q^{92} -1.46681i q^{93} -0.506041 q^{94} +2.28083 q^{96} -1.55496i q^{97} +1.02177i q^{98} -2.14914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9}+O(q^{10}) \) 6 * q - 8 * q^4 + 10 * q^6 + 6 * q^9	\( 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9} + 2 q^{11} + 14 q^{14} + 36 q^{16} + 6 q^{19} + 2 q^{24} + 18 q^{26} + 14 q^{29} - 10 q^{31} + 18 q^{34} - 22 q^{36} - 2 q^{39} + 2 q^{41} + 30 q^{44} + 12 q^{46} + 14 q^{49} - 16 q^{51} + 26 q^{54} - 70 q^{56} - 20 q^{59} - 34 q^{61} - 98 q^{64} + 22 q^{66} - 6 q^{69} - 38 q^{71} + 10 q^{74} - 8 q^{76} - 36 q^{79} - 2 q^{81} + 14 q^{84} + 4 q^{86} - 4 q^{89} - 14 q^{91} - 22 q^{94} + 36 q^{96} - 40 q^{99}+O(q^{100}) \) 6 * q - 8 * q^4 + 10 * q^6 + 6 * q^9 + 2 * q^11 + 14 * q^14 + 36 * q^16 + 6 * q^19 + 2 * q^24 + 18 * q^26 + 14 * q^29 - 10 * q^31 + 18 * q^34 - 22 * q^36 - 2 * q^39 + 2 * q^41 + 30 * q^44 + 12 * q^46 + 14 * q^49 - 16 * q^51 + 26 * q^54 - 70 * q^56 - 20 * q^59 - 34 * q^61 - 98 * q^64 + 22 * q^66 - 6 * q^69 - 38 * q^71 + 10 * q^74 - 8 * q^76 - 36 * q^79 - 2 * q^81 + 14 * q^84 + 4 * q^86 - 4 * q^89 - 14 * q^91 - 22 * q^94 + 36 * q^96 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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