LMFDB - Embedded newform 525.6.d.m.274.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,6,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	525.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$84.2015054018$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 121x^{6} + 4400x^{4} + 47104x^{2} + 153664 $ x^8 + 121x^6 + 4400x^4 + 47104*x^2 + 153664
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.3
Root			$-7.76014i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.6.d.m.274.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.76014i q^{2} +9.00000i q^{3} +17.8614 q^{4} +33.8412 q^{6} -49.0000i q^{7} -187.486i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q-3.76014i q^{2} +9.00000i q^{3} +17.8614 q^{4} +33.8412 q^{6} -49.0000i q^{7} -187.486i q^{8} -81.0000 q^{9} +277.018 q^{11} +160.752i q^{12} -569.200i q^{13} -184.247 q^{14} -133.408 q^{16} -181.165i q^{17} +304.571i q^{18} -970.770 q^{19} +441.000 q^{21} -1041.63i q^{22} -2007.80i q^{23} +1687.37 q^{24} -2140.27 q^{26} -729.000i q^{27} -875.207i q^{28} -328.100 q^{29} -2639.79 q^{31} -5497.91i q^{32} +2493.16i q^{33} -681.205 q^{34} -1446.77 q^{36} +7107.29i q^{37} +3650.23i q^{38} +5122.80 q^{39} -14747.0 q^{41} -1658.22i q^{42} +2559.60i q^{43} +4947.92 q^{44} -7549.61 q^{46} +5334.50i q^{47} -1200.67i q^{48} -2401.00 q^{49} +1630.49 q^{51} -10166.7i q^{52} -87.4724i q^{53} -2741.14 q^{54} -9186.79 q^{56} -8736.93i q^{57} +1233.70i q^{58} +4803.10 q^{59} -19794.3 q^{61} +9925.97i q^{62} +3969.00i q^{63} -24941.9 q^{64} +9374.63 q^{66} +1824.04i q^{67} -3235.85i q^{68} +18070.2 q^{69} -32627.4 q^{71} +15186.3i q^{72} -77826.0i q^{73} +26724.4 q^{74} -17339.3 q^{76} -13573.9i q^{77} -19262.4i q^{78} +38316.5 q^{79} +6561.00 q^{81} +55450.6i q^{82} +20777.0i q^{83} +7876.86 q^{84} +9624.47 q^{86} -2952.90i q^{87} -51936.9i q^{88} +48797.8 q^{89} -27890.8 q^{91} -35862.1i q^{92} -23758.1i q^{93} +20058.5 q^{94} +49481.2 q^{96} -86326.6i q^{97} +9028.09i q^{98} -22438.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 98 q^{4} - 270 q^{6} - 648 q^{9}+O(q^{10}) $ 8 * q - 98 * q^4 - 270 * q^6 - 648 * q^9	$ 8 q - 98 q^{4} - 270 q^{6} - 648 q^{9} + 372 q^{11} + 1470 q^{14} + 5634 q^{16} + 5484 q^{19} + 3528 q^{21} + 9774 q^{24} + 9618 q^{26} - 1140 q^{29} - 18828 q^{31} - 4290 q^{34} + 7938 q^{36} + 10980 q^{39} - 34008 q^{41} - 69180 q^{44} - 60970 q^{46} - 19208 q^{49} - 2916 q^{51} + 21870 q^{54} - 53214 q^{56} - 19560 q^{59} - 120816 q^{61} - 200290 q^{64} - 42624 q^{66} - 50868 q^{69} + 2028 q^{71} - 283368 q^{74} + 5200 q^{76} - 298320 q^{79} + 52488 q^{81} - 43218 q^{84} + 192570 q^{86} + 15924 q^{89} - 59780 q^{91} - 465184 q^{94} - 211518 q^{96} - 30132 q^{99}+O(q^{100}) $ 8 * q - 98 * q^4 - 270 * q^6 - 648 * q^9 + 372 * q^11 + 1470 * q^14 + 5634 * q^16 + 5484 * q^19 + 3528 * q^21 + 9774 * q^24 + 9618 * q^26 - 1140 * q^29 - 18828 * q^31 - 4290 * q^34 + 7938 * q^36 + 10980 * q^39 - 34008 * q^41 - 69180 * q^44 - 60970 * q^46 - 19208 * q^49 - 2916 * q^51 + 21870 * q^54 - 53214 * q^56 - 19560 * q^59 - 120816 * q^61 - 200290 * q^64 - 42624 * q^66 - 50868 * q^69 + 2028 * q^71 - 283368 * q^74 + 5200 * q^76 - 298320 * q^79 + 52488 * q^81 - 43218 * q^84 + 192570 * q^86 + 15924 * q^89 - 59780 * q^91 - 465184 * q^94 - 211518 * q^96 - 30132 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 3.76014i	− 0.664705i	−0.943155	−	0.332352i	$-0.892158\pi$
$2$	− 3.76014i	− 0.664705i	0.943155	−	0.332352i	$-0.107842\pi$
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	17.8614	0.558168
$5$	0	0
$5$	0	0
$6$	33.8412	0.383767
$7$	− 49.0000i	− 0.377964i
$7$	− 49.0000i	− 0.377964i
$8$	− 187.486i	− 1.03572i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	277.018	0.690282	0.345141	−	0.938551i	$-0.387831\pi$
$11$	277.018	0.690282	0.345141	+	0.938551i	$0.387831\pi$
$12$	160.752i	0.322258i
$13$	− 569.200i	− 0.934129i	−0.884223	−	0.467064i	$-0.845312\pi$
$13$	− 569.200i	− 0.934129i	0.884223	−	0.467064i	$-0.154688\pi$
$14$	−184.247	−0.251235
$15$	0	0
$16$	−133.408	−0.130281
$17$	− 181.165i	− 0.152038i	−0.997106	−	0.0760190i	$-0.975779\pi$
$17$	− 181.165i	− 0.152038i	0.997106	−	0.0760190i	$-0.0242210\pi$
$18$	304.571i	0.221568i
$19$	−970.770	−0.616925	−0.308463	−	0.951236i	$-0.599814\pi$
$19$	−970.770	−0.616925	−0.308463	+	0.951236i	$0.599814\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	− 1041.63i	− 0.458834i
$23$	− 2007.80i	− 0.791409i	−0.918378	−	0.395704i	$-0.870500\pi$
$23$	− 2007.80i	− 0.791409i	0.918378	−	0.395704i	$-0.129500\pi$
$24$	1687.37	0.597974
$25$	0	0
$26$	−2140.27	−0.620920
$27$	− 729.000i	− 0.192450i
$28$	− 875.207i	− 0.210968i
$29$	−328.100	−0.0724454	−0.0362227	−	0.999344i	$-0.511533\pi$
$29$	−328.100	−0.0724454	−0.0362227	+	0.999344i	$0.511533\pi$
$30$	0	0
$31$	−2639.79	−0.493361	−0.246681	−	0.969097i	$-0.579340\pi$
$31$	−2639.79	−0.493361	−0.246681	+	0.969097i	$0.579340\pi$
$32$	− 5497.91i	− 0.949123i
$33$	2493.16i	0.398534i
$34$	−681.205	−0.101060
$35$	0	0
$36$	−1446.77	−0.186056
$37$	7107.29i	0.853492i	0.904371	+	0.426746i	$0.140340\pi$
$37$	7107.29i	0.853492i	−0.904371	+	0.426746i	$0.859660\pi$
$38$	3650.23i	0.410073i
$39$	5122.80	0.539320
$40$	0	0
$41$	−14747.0	−1.37007	−0.685035	−	0.728510i	$-0.740213\pi$
$41$	−14747.0	−1.37007	−0.685035	+	0.728510i	$0.740213\pi$
$42$	− 1658.22i	− 0.145050i
$43$	2559.60i	0.211107i	0.994414	+	0.105553i	$0.0336614\pi$
$43$	2559.60i	0.211107i	−0.994414	+	0.105553i	$0.966339\pi$
$44$	4947.92	0.385293
$45$	0	0
$46$	−7549.61	−0.526053
$47$	5334.50i	0.352249i	0.984368	+	0.176124i	$0.0563561\pi$
$47$	5334.50i	0.352249i	−0.984368	+	0.176124i	$0.943644\pi$
$48$	− 1200.67i	− 0.0752179i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	1630.49	0.0877792
$52$	− 10166.7i	− 0.521401i
$53$	− 87.4724i	− 0.00427742i	−0.999998	−	0.00213871i	$-0.999319\pi$
$53$	− 87.4724i	− 0.00427742i	0.999998	−	0.00213871i	$-0.000680772\pi$
$54$	−2741.14	−0.127922
$55$	0	0
$56$	−9186.79	−0.391466
$57$	− 8736.93i	− 0.356182i
$58$	1233.70i	0.0481548i
$59$	4803.10	0.179635	0.0898177	−	0.995958i	$-0.471372\pi$
$59$	4803.10	0.179635	0.0898177	+	0.995958i	$0.471372\pi$
$60$	0	0
$61$	−19794.3	−0.681106	−0.340553	−	0.940225i	$-0.610614\pi$
$61$	−19794.3	−0.681106	−0.340553	+	0.940225i	$0.610614\pi$
$62$	9925.97i	0.327940i
$63$	3969.00i	0.125988i
$64$	−24941.9	−0.761168
$65$	0	0
$66$	9374.63	0.264908
$67$	1824.04i	0.0496418i	0.999692	+	0.0248209i	$0.00790155\pi$
$67$	1824.04i	0.0496418i	−0.999692	+	0.0248209i	$0.992098\pi$
$68$	− 3235.85i	− 0.0848627i
$69$	18070.2	0.456920
$70$	0	0
$71$	−32627.4	−0.768133	−0.384066	−	0.923306i	$-0.625477\pi$
$71$	−32627.4	−0.768133	−0.384066	+	0.923306i	$0.625477\pi$
$72$	15186.3i	0.345240i
$73$	− 77826.0i	− 1.70930i	−0.519208	−	0.854648i	$-0.673773\pi$
$73$	− 77826.0i	− 1.70930i	0.519208	−	0.854648i	$-0.326227\pi$
$74$	26724.4	0.567320
$75$	0	0
$76$	−17339.3	−0.344348
$77$	− 13573.9i	− 0.260902i
$78$	− 19262.4i	− 0.358488i
$79$	38316.5	0.690745	0.345373	−	0.938466i	$-0.387753\pi$
$79$	38316.5	0.690745	0.345373	+	0.938466i	$0.387753\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	55450.6i	0.910692i
$83$	20777.0i	0.331046i	0.986206	+	0.165523i	$0.0529312\pi$
$83$	20777.0i	0.331046i	−0.986206	+	0.165523i	$0.947069\pi$
$84$	7876.86	0.121802
$85$	0	0
$86$	9624.47	0.140324
$87$	− 2952.90i	− 0.0418264i
$88$	− 51936.9i	− 0.714940i
$89$	48797.8	0.653018	0.326509	−	0.945194i	$-0.394128\pi$
$89$	48797.8	0.653018	0.326509	+	0.945194i	$0.394128\pi$
$90$	0	0
$91$	−27890.8	−0.353068
$92$	− 35862.1i	− 0.441739i
$93$	− 23758.1i	− 0.284842i
$94$	20058.5	0.234141
$95$	0	0
$96$	49481.2	0.547976
$97$	− 86326.6i	− 0.931569i	−0.884898	−	0.465785i	$-0.845772\pi$
$97$	− 86326.6i	− 0.931569i	0.884898	−	0.465785i	$-0.154228\pi$
$98$	9028.09i	0.0949578i
$99$	−22438.5	−0.230094
$100$	0	0
$101$	4054.22	0.0395461	0.0197731	−	0.999804i	$-0.493706\pi$
$101$	4054.22	0.0395461	0.0197731	+	0.999804i	$0.493706\pi$
$102$	− 6130.85i	− 0.0583472i
$103$	− 176509.i	− 1.63936i	−0.572824	−	0.819678i	$-0.694152\pi$
$103$	− 176509.i	− 1.63936i	0.572824	−	0.819678i	$-0.305848\pi$
$104$	−106717.	−0.967497
$105$	0	0
$106$	−328.908	−0.00284322
$107$	60355.4i	0.509632i	0.966990	+	0.254816i	$0.0820149\pi$
$107$	60355.4i	0.509632i	−0.966990	+	0.254816i	$0.917985\pi$
$108$	− 13020.9i	− 0.107419i
$109$	3502.67	0.0282380	0.0141190	−	0.999900i	$-0.495506\pi$
$109$	3502.67	0.0282380	0.0141190	+	0.999900i	$0.495506\pi$
$110$	0	0
$111$	−63965.6	−0.492764
$112$	6536.99i	0.0492416i
$113$	− 216658.i	− 1.59617i	−0.602547	−	0.798083i	$-0.705848\pi$
$113$	− 216658.i	− 1.59617i	0.602547	−	0.798083i	$-0.294152\pi$
$114$	−32852.1	−0.236756
$115$	0	0
$116$	−5860.31	−0.0404367
$117$	46105.2i	0.311376i
$118$	− 18060.3i	− 0.119404i
$119$	−8877.09	−0.0574649
$120$	0	0
$121$	−84312.0	−0.523511
$122$	74429.1i	0.452734i
$123$	− 132723.i	− 0.791011i
$124$	−47150.2	−0.275378
$125$	0	0
$126$	14924.0	0.0837449
$127$	− 161251.i	− 0.887143i	−0.896239	−	0.443571i	$-0.853711\pi$
$127$	− 161251.i	− 0.887143i	0.896239	−	0.443571i	$-0.146289\pi$
$128$	− 82147.9i	− 0.443171i
$129$	−23036.4	−0.121882
$130$	0	0
$131$	−149939.	−0.763371	−0.381686	−	0.924292i	$-0.624656\pi$
$131$	−149939.	−0.763371	−0.381686	+	0.924292i	$0.624656\pi$
$132$	44531.3i	0.222449i
$133$	47567.7i	0.233176i
$134$	6858.64	0.0329971
$135$	0	0
$136$	−33965.8	−0.157469
$137$	366841.i	1.66985i	0.550367	+	0.834923i	$0.314488\pi$
$137$	366841.i	1.66985i	−0.550367	+	0.834923i	$0.685512\pi$
$138$	− 67946.4i	− 0.303717i
$139$	−304758.	−1.33788	−0.668941	−	0.743316i	$-0.733252\pi$
$139$	−304758.	−1.33788	−0.668941	+	0.743316i	$0.733252\pi$
$140$	0	0
$141$	−48010.5	−0.203371
$142$	122683.i	0.510581i
$143$	− 157679.i	− 0.644812i
$144$	10806.0	0.0434270
$145$	0	0
$146$	−292636.	−1.13618
$147$	− 21609.0i	− 0.0824786i
$148$	126946.i	0.476392i
$149$	−275811.	−1.01776	−0.508881	−	0.860837i	$-0.669941\pi$
$149$	−275811.	−1.01776	−0.508881	+	0.860837i	$0.669941\pi$
$150$	0	0
$151$	52129.0	0.186053	0.0930265	−	0.995664i	$-0.470346\pi$
$151$	52129.0	0.186053	0.0930265	+	0.995664i	$0.470346\pi$
$152$	182005.i	0.638963i
$153$	14674.4i	0.0506793i
$154$	−51039.7	−0.173423
$155$	0	0
$156$	91500.3	0.301031
$157$	− 433842.i	− 1.40470i	−0.711833	−	0.702348i	$-0.752135\pi$
$157$	− 433842.i	− 1.40470i	0.711833	−	0.702348i	$-0.247865\pi$
$158$	− 144075.i	− 0.459142i
$159$	787.252	0.00246957
$160$	0	0
$161$	−98382.2	−0.299124
$162$	− 24670.3i	− 0.0738561i
$163$	− 76428.4i	− 0.225313i	−0.993634	−	0.112656i	$-0.964064\pi$
$163$	− 76428.4i	− 0.225313i	0.993634	−	0.112656i	$-0.0359359\pi$
$164$	−263401.	−0.764729
$165$	0	0
$166$	78124.5	0.220048
$167$	− 112603.i	− 0.312433i	−0.987723	−	0.156217i	$-0.950070\pi$
$167$	− 112603.i	− 0.312433i	0.987723	−	0.156217i	$-0.0499298\pi$
$168$	− 82681.1i	− 0.226013i
$169$	47303.9	0.127403
$170$	0	0
$171$	78632.4	0.205642
$172$	45718.0i	0.117833i
$173$	− 151002.i	− 0.383591i	−0.981435	−	0.191796i	$-0.938569\pi$
$173$	− 151002.i	− 0.383591i	0.981435	−	0.191796i	$-0.0614311\pi$
$174$	−11103.3	−0.0278022
$175$	0	0
$176$	−36956.4	−0.0899307
$177$	43227.9i	0.103713i
$178$	− 183486.i	− 0.434064i
$179$	−139241.	−0.324814	−0.162407	−	0.986724i	$-0.551926\pi$
$179$	−139241.	−0.324814	−0.162407	+	0.986724i	$0.551926\pi$
$180$	0	0
$181$	−130806.	−0.296777	−0.148389	−	0.988929i	$-0.547409\pi$
$181$	−130806.	−0.296777	−0.148389	+	0.988929i	$0.547409\pi$
$182$	104873.i	0.234686i
$183$	− 178148.i	− 0.393237i
$184$	−376434.	−0.819679
$185$	0	0
$186$	−89333.7	−0.189336
$187$	− 50186.0i	− 0.104949i
$188$	95281.5i	0.196614i
$189$	−35721.0	−0.0727393
$190$	0	0
$191$	851248.	1.68839	0.844194	−	0.536037i	$-0.180079\pi$
$191$	851248.	1.68839	0.844194	+	0.536037i	$0.180079\pi$
$192$	− 224477.i	− 0.439460i
$193$	− 213023.i	− 0.411656i	−0.978588	−	0.205828i	$-0.934011\pi$
$193$	− 213023.i	− 0.411656i	0.978588	−	0.205828i	$-0.0659887\pi$
$194$	−324600.	−0.619218
$195$	0	0
$196$	−42885.1	−0.0797382
$197$	859147.i	1.57726i	0.614871	+	0.788628i	$0.289208\pi$
$197$	859147.i	1.57726i	−0.614871	+	0.788628i	$0.710792\pi$
$198$	84371.7i	0.152945i
$199$	−733372.	−1.31278	−0.656389	−	0.754422i	$-0.727917\pi$
$199$	−733372.	−1.31278	−0.656389	+	0.754422i	$0.727917\pi$
$200$	0	0
$201$	−16416.4	−0.0286607
$202$	− 15244.4i	− 0.0262865i
$203$	16076.9i	0.0273818i
$204$	29122.7	0.0489955
$205$	0	0
$206$	−663697.	−1.08969
$207$	162632.i	0.263803i
$208$	75935.8i	0.121699i
$209$	−268921.	−0.425852
$210$	0	0
$211$	−655973.	−1.01433	−0.507166	−	0.861848i	$-0.669307\pi$
$211$	−655973.	−1.01433	−0.507166	+	0.861848i	$0.669307\pi$
$212$	− 1562.38i	− 0.00238752i
$213$	− 293646.i	− 0.443482i
$214$	226945.	0.338755
$215$	0	0
$216$	−136677.	−0.199325
$217$	129350.i	0.186473i
$218$	− 13170.5i	− 0.0187699i
$219$	700434.	0.986863
$220$	0	0
$221$	−103119.	−0.142023
$222$	240519.i	0.327543i
$223$	38111.0i	0.0513202i	0.999671	+	0.0256601i	$0.00816876\pi$
$223$	38111.0i	0.0513202i	−0.999671	+	0.0256601i	$0.991831\pi$
$224$	−269397.	−0.358735
$225$	0	0
$226$	−814663.	−1.06098
$227$	− 121434.i	− 0.156414i	−0.996937	−	0.0782068i	$-0.975081\pi$
$227$	− 121434.i	− 0.156414i	0.996937	−	0.0782068i	$-0.0249195\pi$
$228$	− 156054.i	− 0.198809i
$229$	−857933.	−1.08110	−0.540548	−	0.841313i	$-0.681783\pi$
$229$	−857933.	−1.08110	−0.540548	+	0.841313i	$0.681783\pi$
$230$	0	0
$231$	122165.	0.150632
$232$	61514.0i	0.0750333i
$233$	6106.65i	0.00736908i	0.999993	+	0.00368454i	$0.00117283\pi$
$233$	6106.65i	0.00736908i	−0.999993	+	0.00368454i	$0.998827\pi$
$234$	173362.	0.206973
$235$	0	0
$236$	85790.0	0.100267
$237$	344848.i	0.398802i
$238$	33379.1i	0.0381972i
$239$	547252.	0.619716	0.309858	−	0.950783i	$-0.399719\pi$
$239$	547252.	0.619716	0.309858	+	0.950783i	$0.399719\pi$
$240$	0	0
$241$	−1.22391e6	−1.35740	−0.678701	−	0.734415i	$-0.737457\pi$
$241$	−1.22391e6	−1.35740	−0.678701	+	0.734415i	$0.737457\pi$
$242$	317025.i	0.347980i
$243$	59049.0i	0.0641500i
$244$	−353553.	−0.380171
$245$	0	0
$246$	−499055.	−0.525788
$247$	552563.i	0.576288i
$248$	494922.i	0.510985i
$249$	−186993.	−0.191130
$250$	0	0
$251$	936360.	0.938121	0.469060	−	0.883166i	$-0.344593\pi$
$251$	936360.	0.938121	0.469060	+	0.883166i	$0.344593\pi$
$252$	70891.8i	0.0703225i
$253$	− 556197.i	− 0.546295i
$254$	−606326.	−0.589688
$255$	0	0
$256$	−1.10703e6	−1.05575
$257$	1.17669e6i	1.11129i	0.831419	+	0.555647i	$0.187529\pi$
$257$	1.17669e6i	1.11129i	−0.831419	+	0.555647i	$0.812471\pi$
$258$	86620.2i	0.0810158i
$259$	348257.	0.322590
$260$	0	0
$261$	26576.1	0.0241485
$262$	563791.i	0.507417i
$263$	− 204460.i	− 0.182271i	−0.995838	−	0.0911357i	$-0.970950\pi$
$263$	− 204460.i	− 0.182271i	0.995838	−	0.0911357i	$-0.0290497\pi$
$264$	467432.	0.412771
$265$	0	0
$266$	178861.	0.154993
$267$	439180.i	0.377020i
$268$	32579.8i	0.0277084i
$269$	495673.	0.417652	0.208826	−	0.977953i	$-0.433036\pi$
$269$	495673.	0.417652	0.208826	+	0.977953i	$0.433036\pi$
$270$	0	0
$271$	1.47160e6	1.21721	0.608607	−	0.793472i	$-0.291729\pi$
$271$	1.47160e6	1.21721	0.608607	+	0.793472i	$0.291729\pi$
$272$	24168.8i	0.0198077i
$273$	− 251017.i	− 0.203844i
$274$	1.37937e6	1.10995
$275$	0	0
$276$	322759.	0.255038
$277$	− 1.33367e6i	− 1.04436i	−0.852836	−	0.522179i	$-0.825119\pi$
$277$	− 1.33367e6i	− 1.04436i	0.852836	−	0.522179i	$-0.174881\pi$
$278$	1.14593e6i	0.889296i
$279$	213823.	0.164454
$280$	0	0
$281$	652621.	0.493055	0.246527	−	0.969136i	$-0.420710\pi$
$281$	652621.	0.493055	0.246527	+	0.969136i	$0.420710\pi$
$282$	180526.i	0.135182i
$283$	− 1.12665e6i	− 0.836225i	−0.908395	−	0.418113i	$-0.862692\pi$
$283$	− 1.12665e6i	− 0.836225i	0.908395	−	0.418113i	$-0.137308\pi$
$284$	−582769.	−0.428747
$285$	0	0
$286$	−592894.	−0.428610
$287$	722601.i	0.517838i
$288$	445330.i	0.316374i
$289$	1.38704e6	0.976884
$290$	0	0
$291$	776939.	0.537842
$292$	− 1.39008e6i	− 0.954074i
$293$	700326.i	0.476575i	0.971195	+	0.238287i	$0.0765860\pi$
$293$	700326.i	0.476575i	−0.971195	+	0.238287i	$0.923414\pi$
$294$	−81252.8	−0.0548239
$295$	0	0
$296$	1.33251e6	0.883980
$297$	− 201946.i	− 0.132845i
$298$	1.03709e6i	0.676511i
$299$	−1.14284e6	−0.739278
$300$	0	0
$301$	125421.	0.0797908
$302$	− 196012.i	− 0.123670i
$303$	36488.0i	0.0228320i
$304$	129508.	0.0803737
$305$	0	0
$306$	55177.6	0.0336868
$307$	− 1.69500e6i	− 1.02642i	−0.858264	−	0.513208i	$-0.828457\pi$
$307$	− 1.69500e6i	− 1.02642i	0.858264	−	0.513208i	$-0.171543\pi$
$308$	− 242448.i	− 0.145627i
$309$	1.58858e6	0.946483
$310$	0	0
$311$	1.97900e6	1.16023	0.580115	−	0.814534i	$-0.303007\pi$
$311$	1.97900e6	1.16023	0.580115	+	0.814534i	$0.303007\pi$
$312$	− 960452.i	− 0.558585i
$313$	− 2.76052e6i	− 1.59269i	−0.604844	−	0.796344i	$-0.706764\pi$
$313$	− 2.76052e6i	− 1.59269i	0.604844	−	0.796344i	$-0.293236\pi$
$314$	−1.63131e6	−0.933709
$315$	0	0
$316$	684385.	0.385552
$317$	− 1.44228e6i	− 0.806123i	−0.915173	−	0.403061i	$-0.867946\pi$
$317$	− 1.44228e6i	− 0.806123i	0.915173	−	0.403061i	$-0.132054\pi$
$318$	− 2960.18i	− 0.00164153i
$319$	−90889.5	−0.0500077
$320$	0	0
$321$	−543199.	−0.294236
$322$	369931.i	0.198829i
$323$	175870.i	0.0937961i
$324$	117188.	0.0620186
$325$	0	0
$326$	−287381.	−0.149766
$327$	31524.1i	0.0163032i
$328$	2.76484e6i	1.41901i
$329$	261391.	0.133138
$330$	0	0
$331$	1.23419e6	0.619175	0.309588	−	0.950871i	$-0.399809\pi$
$331$	1.23419e6	0.619175	0.309588	+	0.950871i	$0.399809\pi$
$332$	371106.i	0.184779i
$333$	− 575690.i	− 0.284497i
$334$	−423402.	−0.207676
$335$	0	0
$336$	−58832.9	−0.0284297
$337$	− 2.55329e6i	− 1.22469i	−0.790591	−	0.612345i	$-0.790226\pi$
$337$	− 2.55329e6i	− 1.22469i	0.790591	−	0.612345i	$-0.209774\pi$
$338$	− 177869.i	− 0.0846854i
$339$	1.94992e6	0.921547
$340$	0	0
$341$	−731269.	−0.340558
$342$	− 295669.i	− 0.136691i
$343$	117649.i	0.0539949i
$344$	479889.	0.218648
$345$	0	0
$346$	−567790.	−0.254975
$347$	− 2.29710e6i	− 1.02413i	−0.858946	−	0.512067i	$-0.828880\pi$
$347$	− 2.29710e6i	− 1.02413i	0.858946	−	0.512067i	$-0.171120\pi$
$348$	− 52742.8i	− 0.0233461i
$349$	3.02696e6	1.33028	0.665141	−	0.746718i	$-0.268371\pi$
$349$	3.02696e6	1.33028	0.665141	+	0.746718i	$0.268371\pi$
$350$	0	0
$351$	−414947.	−0.179773
$352$	− 1.52302e6i	− 0.655162i
$353$	1.04227e6i	0.445186i	0.974911	+	0.222593i	$0.0714521\pi$
$353$	1.04227e6i	0.445186i	−0.974911	+	0.222593i	$0.928548\pi$
$354$	162543.	0.0689382
$355$	0	0
$356$	871595.	0.364493
$357$	− 79893.8i	− 0.0331774i
$358$	523566.i	0.215906i
$359$	1.36860e6	0.560454	0.280227	−	0.959934i	$-0.409590\pi$
$359$	1.36860e6	0.560454	0.280227	+	0.959934i	$0.409590\pi$
$360$	0	0
$361$	−1.53370e6	−0.619403
$362$	491847.i	0.197269i
$363$	− 758808.i	− 0.302249i
$364$	−498168.	−0.197071
$365$	0	0
$366$	−669862.	−0.261386
$367$	1.73831e6i	0.673694i	0.941559	+	0.336847i	$0.109361\pi$
$367$	1.73831e6i	0.673694i	−0.941559	+	0.336847i	$0.890639\pi$
$368$	267856.i	0.103106i
$369$	1.19450e6	0.456690
$370$	0	0
$371$	−4286.15	−0.00161671
$372$	− 424352.i	− 0.158990i
$373$	5.11889e6i	1.90504i	0.304480	+	0.952519i	$0.401517\pi$
$373$	5.11889e6i	1.90504i	−0.304480	+	0.952519i	$0.598483\pi$
$374$	−188706.	−0.0697601
$375$	0	0
$376$	1.00014e6	0.364832
$377$	186754.i	0.0676733i
$378$	134316.i	0.0483502i
$379$	2.44380e6	0.873913	0.436956	−	0.899483i	$-0.356056\pi$
$379$	2.44380e6	0.873913	0.436956	+	0.899483i	$0.356056\pi$
$380$	0	0
$381$	1.45126e6	0.512192
$382$	− 3.20081e6i	− 1.12228i
$383$	838560.i	0.292104i	0.989277	+	0.146052i	$0.0466566\pi$
$383$	838560.i	0.292104i	−0.989277	+	0.146052i	$0.953343\pi$
$384$	739331.	0.255865
$385$	0	0
$386$	−800997.	−0.273629
$387$	− 207328.i	− 0.0703689i
$388$	− 1.54191e6i	− 0.519972i
$389$	−4.85691e6	−1.62737	−0.813684	−	0.581307i	$-0.802541\pi$
$389$	−4.85691e6	−1.62737	−0.813684	+	0.581307i	$0.802541\pi$
$390$	0	0
$391$	−363743.	−0.120324
$392$	450153.i	0.147960i
$393$	− 1.34945e6i	− 0.440733i
$394$	3.23051e6	1.04841
$395$	0	0
$396$	−400782.	−0.128431
$397$	− 963218.i	− 0.306725i	−0.988170	−	0.153362i	$-0.950990\pi$
$397$	− 963218.i	− 0.306725i	0.988170	−	0.153362i	$-0.0490101\pi$
$398$	2.75758e6i	0.872610i
$399$	−428110.	−0.134624
$400$	0	0
$401$	5.59717e6	1.73823	0.869116	−	0.494608i	$-0.164688\pi$
$401$	5.59717e6	1.73823	0.869116	+	0.494608i	$0.164688\pi$
$402$	61727.8i	0.0190509i
$403$	1.50257e6i	0.460863i
$404$	72414.0	0.0220734
$405$	0	0
$406$	60451.3	0.0182008
$407$	1.96885e6i	0.589150i
$408$	− 305692.i	− 0.0909147i
$409$	3.33795e6	0.986669	0.493335	−	0.869840i	$-0.335778\pi$
$409$	3.33795e6	0.986669	0.493335	+	0.869840i	$0.335778\pi$
$410$	0	0
$411$	−3.30157e6	−0.964086
$412$	− 3.15269e6i	− 0.915036i
$413$	− 235352.i	− 0.0678958i
$414$	611518.	0.175351
$415$	0	0
$416$	−3.12941e6	−0.886603
$417$	− 2.74282e6i	− 0.772426i
$418$	1.01118e6i	0.283066i
$419$	−334485.	−0.0930770	−0.0465385	−	0.998916i	$-0.514819\pi$
$419$	−334485.	−0.0930770	−0.0465385	+	0.998916i	$0.514819\pi$
$420$	0	0
$421$	6.26297e6	1.72217	0.861083	−	0.508464i	$-0.169786\pi$
$421$	6.26297e6	1.72217	0.861083	+	0.508464i	$0.169786\pi$
$422$	2.46655e6i	0.674231i
$423$	− 432095.i	− 0.117416i
$424$	−16399.8	−0.00443021
$425$	0	0
$426$	−1.10415e6	−0.294784
$427$	969919.i	0.257434i
$428$	1.07803e6i	0.284460i
$429$	1.41911e6	0.372282
$430$	0	0
$431$	6.62663e6	1.71830	0.859150	−	0.511723i	$-0.170993\pi$
$431$	6.62663e6	1.71830	0.859150	+	0.511723i	$0.170993\pi$
$432$	97254.4i	0.0250726i
$433$	1.89828e6i	0.486564i	0.969956	+	0.243282i	$0.0782241\pi$
$433$	1.89828e6i	0.486564i	−0.969956	+	0.243282i	$0.921776\pi$
$434$	486373.	0.123949
$435$	0	0
$436$	62562.5	0.0157615
$437$	1.94911e6i	0.488240i
$438$	− 2.63373e6i	− 0.655972i
$439$	163569.	0.0405078	0.0202539	−	0.999795i	$-0.493553\pi$
$439$	163569.	0.0405078	0.0202539	+	0.999795i	$0.493553\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	387742.i	0.0944034i
$443$	7.64692e6i	1.85130i	0.378380	+	0.925650i	$0.376481\pi$
$443$	7.64692e6i	1.85130i	−0.378380	+	0.925650i	$0.623519\pi$
$444$	−1.14251e6	−0.275045
$445$	0	0
$446$	143303.	0.0341128
$447$	− 2.48230e6i	− 0.587605i
$448$	1.22215e6i	0.287694i
$449$	187990.	0.0440068	0.0220034	−	0.999758i	$-0.492996\pi$
$449$	187990.	0.0440068	0.0220034	+	0.999758i	$0.492996\pi$
$450$	0	0
$451$	−4.08518e6	−0.945735
$452$	− 3.86980e6i	− 0.890928i
$453$	469161.i	0.107418i
$454$	−456608.	−0.103969
$455$	0	0
$456$	−1.63805e6	−0.368905
$457$	4.31881e6i	0.967327i	0.875254	+	0.483663i	$0.160694\pi$
$457$	4.31881e6i	0.967327i	−0.875254	+	0.483663i	$0.839306\pi$
$458$	3.22594e6i	0.718610i
$459$	−132069.	−0.0292597
$460$	0	0
$461$	−1.30938e6	−0.286955	−0.143477	−	0.989654i	$-0.545828\pi$
$461$	−1.30938e6	−0.286955	−0.143477	+	0.989654i	$0.545828\pi$
$462$	− 459357.i	− 0.100126i
$463$	− 2.23361e6i	− 0.484233i	−0.970247	−	0.242117i	$-0.922158\pi$
$463$	− 2.23361e6i	− 0.484233i	0.970247	−	0.242117i	$-0.0778416\pi$
$464$	43771.1	0.00943827
$465$	0	0
$466$	22961.8	0.00489826
$467$	6.81739e6i	1.44653i	0.690573	+	0.723263i	$0.257358\pi$
$467$	6.81739e6i	1.44653i	−0.690573	+	0.723263i	$0.742642\pi$
$468$	823502.i	0.173800i
$469$	89378.0	0.0187628
$470$	0	0
$471$	3.90458e6	0.811002
$472$	− 900513.i	− 0.186052i
$473$	709057.i	0.145723i
$474$	1.29668e6	0.265086
$475$	0	0
$476$	−158557.	−0.0320751
$477$	7085.27i	0.00142581i
$478$	− 2.05774e6i	− 0.411928i
$479$	−6.97889e6	−1.38978	−0.694892	−	0.719114i	$-0.744548\pi$
$479$	−6.97889e6	−1.38978	−0.694892	+	0.719114i	$0.744548\pi$
$480$	0	0
$481$	4.04547e6	0.797272
$482$	4.60208e6i	0.902271i
$483$	− 885440.i	− 0.172700i
$484$	−1.50593e6	−0.292207
$485$	0	0
$486$	222032.	0.0426408
$487$	3.11796e6i	0.595728i	0.954608	+	0.297864i	$0.0962742\pi$
$487$	3.11796e6i	0.595728i	−0.954608	+	0.297864i	$0.903726\pi$
$488$	3.71114e6i	0.705436i
$489$	687855.	0.130084
$490$	0	0
$491$	4.79298e6	0.897226	0.448613	−	0.893726i	$-0.351918\pi$
$491$	4.79298e6	0.897226	0.448613	+	0.893726i	$0.351918\pi$
$492$	− 2.37061e6i	− 0.441517i
$493$	59440.2i	0.0110145i
$494$	2.07771e6	0.383061
$495$	0	0
$496$	352169.	0.0642757
$497$	1.59874e6i	0.290327i
$498$	703121.i	0.127045i
$499$	1.95717e6	0.351866	0.175933	−	0.984402i	$-0.443706\pi$
$499$	1.95717e6	0.351866	0.175933	+	0.984402i	$0.443706\pi$
$500$	0	0
$501$	1.01342e6	0.180384
$502$	− 3.52084e6i	− 0.623573i
$503$	6.96696e6i	1.22779i	0.789388	+	0.613894i	$0.210398\pi$
$503$	6.96696e6i	1.22779i	−0.789388	+	0.613894i	$0.789602\pi$
$504$	744130.	0.130489
$505$	0	0
$506$	−2.09138e6	−0.363125
$507$	425735.i	0.0735562i
$508$	− 2.88017e6i	− 0.495174i
$509$	−4.46653e6	−0.764145	−0.382073	−	0.924132i	$-0.624790\pi$
$509$	−4.46653e6	−0.764145	−0.382073	+	0.924132i	$0.624790\pi$
$510$	0	0
$511$	−3.81347e6	−0.646053
$512$	1.53385e6i	0.258588i
$513$	707692.i	0.118727i
$514$	4.42451e6	0.738682
$515$	0	0
$516$	−411462.	−0.0680308
$517$	1.47775e6i	0.243151i
$518$	− 1.30949e6i	− 0.214427i
$519$	1.35902e6	0.221466
$520$	0	0
$521$	−3.58666e6	−0.578890	−0.289445	−	0.957195i	$-0.593471\pi$
$521$	−3.58666e6	−0.578890	−0.289445	+	0.957195i	$0.593471\pi$
$522$	− 99929.7i	− 0.0160516i
$523$	− 1.04975e7i	− 1.67816i	−0.544010	−	0.839079i	$-0.683095\pi$
$523$	− 1.04975e7i	− 1.67816i	0.544010	−	0.839079i	$-0.316905\pi$
$524$	−2.67811e6	−0.426089
$525$	0	0
$526$	−768797.	−0.121157
$527$	478237.i	0.0750096i
$528$	− 332608.i	− 0.0519215i
$529$	2.40508e6	0.373672
$530$	0	0
$531$	−389051.	−0.0598785
$532$	849625.i	0.130151i
$533$	8.39398e6i	1.27982i
$534$	1.65138e6	0.250607
$535$	0	0
$536$	341981.	0.0514151
$537$	− 1.25317e6i	− 0.187532i
$538$	− 1.86380e6i	− 0.277615i
$539$	−665120.	−0.0986117
$540$	0	0
$541$	3.05301e6	0.448472	0.224236	−	0.974535i	$-0.428011\pi$
$541$	3.05301e6	0.448472	0.224236	+	0.974535i	$0.428011\pi$
$542$	− 5.53342e6i	− 0.809088i
$543$	− 1.17725e6i	− 0.171344i
$544$	−996028.	−0.144303
$545$	0	0
$546$	−943860.	−0.135496
$547$	2.62968e6i	0.375781i	0.982190	+	0.187890i	$0.0601650\pi$
$547$	2.62968e6i	0.375781i	−0.982190	+	0.187890i	$0.939835\pi$
$548$	6.55228e6i	0.932054i
$549$	1.60333e6	0.227035
$550$	0	0
$551$	318509.	0.0446934
$552$	− 3.38790e6i	− 0.473242i
$553$	− 1.87751e6i	− 0.261077i
$554$	−5.01479e6	−0.694190
$555$	0	0
$556$	−5.44339e6	−0.746762
$557$	1.29900e7i	1.77407i	0.461699	+	0.887036i	$0.347240\pi$
$557$	1.29900e7i	1.77407i	−0.461699	+	0.887036i	$0.652760\pi$
$558$	− 804004.i	− 0.109313i
$559$	1.45693e6	0.197201
$560$	0	0
$561$	451674.	0.0605924
$562$	− 2.45395e6i	− 0.327736i
$563$	− 430568.i	− 0.0572494i	−0.999590	−	0.0286247i	$-0.990887\pi$
$563$	− 430568.i	− 0.0572494i	0.999590	−	0.0286247i	$-0.00911278\pi$
$564$	−857534.	−0.113515
$565$	0	0
$566$	−4.23636e6	−0.555843
$567$	− 321489.i	− 0.0419961i
$568$	6.11716e6i	0.795571i
$569$	−1.13578e6	−0.147067	−0.0735333	−	0.997293i	$-0.523428\pi$
$569$	−1.13578e6	−0.147067	−0.0735333	+	0.997293i	$0.523428\pi$
$570$	0	0
$571$	8.74188e6	1.12206	0.561028	−	0.827797i	$-0.310406\pi$
$571$	8.74188e6	1.12206	0.561028	+	0.827797i	$0.310406\pi$
$572$	− 2.81636e6i	− 0.359913i
$573$	7.66123e6i	0.974792i
$574$	2.71708e6	0.344209
$575$	0	0
$576$	2.02030e6	0.253723
$577$	6.57366e6i	0.821992i	0.911637	+	0.410996i	$0.134819\pi$
$577$	6.57366e6i	0.821992i	−0.911637	+	0.410996i	$0.865181\pi$
$578$	− 5.21545e6i	− 0.649340i
$579$	1.91721e6	0.237669
$580$	0	0
$581$	1.01808e6	0.125124
$582$	− 2.92140e6i	− 0.357506i
$583$	− 24231.4i	− 0.00295262i
$584$	−1.45912e7	−1.77035
$585$	0	0
$586$	2.63332e6	0.316781
$587$	− 225452.i	− 0.0270060i	−0.999909	−	0.0135030i	$-0.995702\pi$
$587$	− 225452.i	− 0.0270060i	0.999909	−	0.0135030i	$-0.00429826\pi$
$588$	− 385966.i	− 0.0460369i
$589$	2.56263e6	0.304367
$590$	0	0
$591$	−7.73232e6	−0.910629
$592$	− 948169.i	− 0.111194i
$593$	− 1.12379e7i	− 1.31235i	−0.754608	−	0.656175i	$-0.772173\pi$
$593$	− 1.12379e7i	− 1.31235i	0.754608	−	0.656175i	$-0.227827\pi$
$594$	−759345.	−0.0883025
$595$	0	0
$596$	−4.92636e6	−0.568081
$597$	− 6.60035e6i	− 0.757933i
$598$	4.29724e6i	0.491402i
$599$	−1.04392e7	−1.18878	−0.594388	−	0.804178i	$-0.702606\pi$
$599$	−1.04392e7	−1.18878	−0.594388	+	0.804178i	$0.702606\pi$
$600$	0	0
$601$	−5.11417e6	−0.577550	−0.288775	−	0.957397i	$-0.593248\pi$
$601$	−5.11417e6	−0.577550	−0.288775	+	0.957397i	$0.593248\pi$
$602$	− 471599.i	− 0.0530373i
$603$	− 147747.i	− 0.0165473i
$604$	931095.	0.103849
$605$	0	0
$606$	137200.	0.0151765
$607$	1.05947e7i	1.16713i	0.812067	+	0.583564i	$0.198342\pi$
$607$	1.05947e7i	1.16713i	−0.812067	+	0.583564i	$0.801658\pi$
$608$	5.33721e6i	0.585538i
$609$	−144692.	−0.0158089
$610$	0	0
$611$	3.03640e6	0.329046
$612$	262104.i	0.0282876i
$613$	− 1.82007e7i	− 1.95630i	−0.207894	−	0.978151i	$-0.566661\pi$
$613$	− 1.82007e7i	− 1.95630i	0.207894	−	0.978151i	$-0.433339\pi$
$614$	−6.37343e6	−0.682263
$615$	0	0
$616$	−2.54491e6	−0.270222
$617$	1.21685e7i	1.28684i	0.765515	+	0.643418i	$0.222484\pi$
$617$	1.21685e7i	1.28684i	−0.765515	+	0.643418i	$0.777516\pi$
$618$	− 5.97328e6i	− 0.629132i
$619$	−3.72606e6	−0.390862	−0.195431	−	0.980717i	$-0.562611\pi$
$619$	−3.72606e6	−0.390862	−0.195431	+	0.980717i	$0.562611\pi$
$620$	0	0
$621$	−1.46369e6	−0.152307
$622$	− 7.44130e6i	− 0.771211i
$623$	− 2.39109e6i	− 0.246817i
$624$	−683422.	−0.0702632
$625$	0	0
$626$	−1.03800e7	−1.05867
$627$	− 2.42029e6i	− 0.245866i
$628$	− 7.74901e6i	− 0.784056i
$629$	1.28759e6	0.129763
$630$	0	0
$631$	−5.82320e6	−0.582221	−0.291111	−	0.956689i	$-0.594025\pi$
$631$	−5.82320e6	−0.582221	−0.291111	+	0.956689i	$0.594025\pi$
$632$	− 7.18379e6i	− 0.715420i
$633$	− 5.90376e6i	− 0.585625i
$634$	−5.42317e6	−0.535833
$635$	0	0
$636$	14061.4	0.00137843
$637$	1.36665e6i	0.133447i
$638$	341757.i	0.0332404i
$639$	2.64282e6	0.256044
$640$	0	0
$641$	1.32008e7	1.26898	0.634490	−	0.772931i	$-0.281210\pi$
$641$	1.32008e7	1.26898	0.634490	+	0.772931i	$0.281210\pi$
$642$	2.04250e6i	0.195580i
$643$	− 7.67590e6i	− 0.732153i	−0.930585	−	0.366077i	$-0.880701\pi$
$643$	− 7.67590e6i	− 0.732153i	0.930585	−	0.366077i	$-0.119299\pi$
$644$	−1.75724e6	−0.166962
$645$	0	0
$646$	661294.	0.0623467
$647$	1.93427e6i	0.181659i	0.995866	+	0.0908293i	$0.0289518\pi$
$647$	1.93427e6i	0.181659i	−0.995866	+	0.0908293i	$0.971048\pi$
$648$	− 1.23009e6i	− 0.115080i
$649$	1.33055e6	0.123999
$650$	0	0
$651$	−1.16415e6	−0.107660
$652$	− 1.36512e6i	− 0.125762i
$653$	3.51680e6i	0.322748i	0.986893	+	0.161374i	$0.0515926\pi$
$653$	3.51680e6i	0.322748i	−0.986893	+	0.161374i	$0.948407\pi$
$654$	118535.	0.0108368
$655$	0	0
$656$	1.96736e6	0.178494
$657$	6.30390e6i	0.569765i
$658$	− 982865.i	− 0.0884971i
$659$	−6.53542e6	−0.586219	−0.293109	−	0.956079i	$-0.594690\pi$
$659$	−6.53542e6	−0.586219	−0.293109	+	0.956079i	$0.594690\pi$
$660$	0	0
$661$	−5.12993e6	−0.456675	−0.228338	−	0.973582i	$-0.573329\pi$
$661$	−5.12993e6	−0.456675	−0.228338	+	0.973582i	$0.573329\pi$
$662$	− 4.64074e6i	− 0.411569i
$663$	− 928073.i	− 0.0819971i
$664$	3.89540e6	0.342872
$665$	0	0
$666$	−2.16468e6	−0.189107
$667$	658759.i	0.0573339i
$668$	− 2.01124e6i	− 0.174390i
$669$	−342999.	−0.0296297
$670$	0	0
$671$	−5.48337e6	−0.470155
$672$	− 2.42458e6i	− 0.207116i
$673$	− 4.99829e6i	− 0.425386i	−0.977119	−	0.212693i	$-0.931777\pi$
$673$	− 4.99829e6i	− 0.425386i	0.977119	−	0.212693i	$-0.0682235\pi$
$674$	−9.60073e6	−0.814057
$675$	0	0
$676$	844912.	0.0711123
$677$	− 1.47440e7i	− 1.23635i	−0.786039	−	0.618177i	$-0.787872\pi$
$677$	− 1.47440e7i	− 1.23635i	0.786039	−	0.618177i	$-0.212128\pi$
$678$	− 7.33197e6i	− 0.612557i
$679$	−4.23000e6	−0.352100
$680$	0	0
$681$	1.09290e6	0.0903055
$682$	2.74967e6i	0.226371i
$683$	− 2.36725e7i	− 1.94174i	−0.239599	−	0.970872i	$-0.577016\pi$
$683$	− 2.36725e7i	− 1.94174i	0.239599	−	0.970872i	$-0.422984\pi$
$684$	1.40448e6	0.114783
$685$	0	0
$686$	442376.	0.0358907
$687$	− 7.72139e6i	− 0.624171i
$688$	− 341471.i	− 0.0275032i
$689$	−49789.3	−0.00399566
$690$	0	0
$691$	6.61635e6	0.527137	0.263569	−	0.964641i	$-0.415100\pi$
$691$	6.61635e6	0.527137	0.263569	+	0.964641i	$0.415100\pi$
$692$	− 2.69711e6i	− 0.214108i
$693$	1.09948e6i	0.0869673i
$694$	−8.63742e6	−0.680746
$695$	0	0
$696$	−553626.	−0.0433205
$697$	2.67163e6i	0.208303i
$698$	− 1.13818e7i	− 0.884245i
$699$	−54959.8	−0.00425454
$700$	0	0
$701$	−9.31100e6	−0.715651	−0.357825	−	0.933788i	$-0.616482\pi$
$701$	−9.31100e6	−0.715651	−0.357825	+	0.933788i	$0.616482\pi$
$702$	1.56026e6i	0.119496i
$703$	− 6.89955e6i	− 0.526541i
$704$	−6.90937e6	−0.525420
$705$	0	0
$706$	3.91906e6	0.295917
$707$	− 198657.i	− 0.0149470i
$708$	772110.i	0.0578890i
$709$	8.44416e6	0.630871	0.315436	−	0.948947i	$-0.397849\pi$
$709$	8.44416e6	0.630871	0.315436	+	0.948947i	$0.397849\pi$
$710$	0	0
$711$	−3.10364e6	−0.230248
$712$	− 9.14888e6i	− 0.676344i
$713$	5.30017e6i	0.390450i
$714$	−300412.	−0.0220532
$715$	0	0
$716$	−2.48704e6	−0.181301
$717$	4.92527e6i	0.357793i
$718$	− 5.14612e6i	− 0.372536i
$719$	4.13696e6	0.298442	0.149221	−	0.988804i	$-0.452323\pi$
$719$	4.13696e6	0.298442	0.149221	+	0.988804i	$0.452323\pi$
$720$	0	0
$721$	−8.64893e6	−0.619618
$722$	5.76694e6i	0.411720i
$723$	− 1.10152e7i	− 0.783696i
$724$	−2.33637e6	−0.165651
$725$	0	0
$726$	−2.85322e6	−0.200907
$727$	− 1.17731e6i	− 0.0826144i	−0.999146	−	0.0413072i	$-0.986848\pi$
$727$	− 1.17731e6i	− 0.0826144i	0.999146	−	0.0413072i	$-0.0131522\pi$
$728$	5.22913e6i	0.365680i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	463711.	0.0320962
$732$	− 3.18197e6i	− 0.219492i
$733$	− 2.08224e6i	− 0.143143i	−0.997435	−	0.0715717i	$-0.977199\pi$
$733$	− 2.08224e6i	− 0.143143i	0.997435	−	0.0715717i	$-0.0228015\pi$
$734$	6.53629e6	0.447808
$735$	0	0
$736$	−1.10387e7	−0.751144
$737$	505292.i	0.0342668i
$738$	− 4.49150e6i	− 0.303564i
$739$	2.53632e6	0.170841	0.0854205	−	0.996345i	$-0.472777\pi$
$739$	2.53632e6	0.170841	0.0854205	+	0.996345i	$0.472777\pi$
$740$	0	0
$741$	−4.97307e6	−0.332720
$742$	16116.5i	0.00107464i
$743$	557180.i	0.0370274i	0.999829	+	0.0185137i	$0.00589343\pi$
$743$	557180.i	0.0370274i	−0.999829	+	0.0185137i	$0.994107\pi$
$744$	−4.45430e6	−0.295017
$745$	0	0
$746$	1.92477e7	1.26629
$747$	− 1.68294e6i	− 0.110349i
$748$	− 896390.i	− 0.0585792i
$749$	2.95741e6	0.192623
$750$	0	0
$751$	2.49240e7	1.61257	0.806284	−	0.591529i	$-0.201476\pi$
$751$	2.49240e7	1.61257	0.806284	+	0.591529i	$0.201476\pi$
$752$	− 711665.i	− 0.0458914i
$753$	8.42724e6i	0.541624i
$754$	702223.	0.0449828
$755$	0	0
$756$	−638026.	−0.0406007
$757$	1.83178e7i	1.16181i	0.813972	+	0.580905i	$0.197301\pi$
$757$	1.83178e7i	1.16181i	−0.813972	+	0.580905i	$0.802699\pi$
$758$	− 9.18903e6i	− 0.580894i
$759$	5.00577e6	0.315404
$760$	0	0
$761$	−1.64608e6	−0.103036	−0.0515180	−	0.998672i	$-0.516406\pi$
$761$	−1.64608e6	−0.103036	−0.0515180	+	0.998672i	$0.516406\pi$
$762$	− 5.45694e6i	− 0.340456i
$763$	− 171631.i	− 0.0106729i
$764$	1.52044e7	0.942404
$765$	0	0
$766$	3.15310e6	0.194163
$767$	− 2.73393e6i	− 0.167803i
$768$	− 9.96326e6i	− 0.609535i
$769$	2.79809e7	1.70626	0.853131	−	0.521698i	$-0.174701\pi$
$769$	2.79809e7	1.70626	0.853131	+	0.521698i	$0.174701\pi$
$770$	0	0
$771$	−1.05902e7	−0.641606
$772$	− 3.80489e6i	− 0.229773i
$773$	− 1.86200e6i	− 0.112081i	−0.998429	−	0.0560403i	$-0.982152\pi$
$773$	− 1.86200e6i	− 0.112081i	0.998429	−	0.0560403i	$-0.0178475\pi$
$774$	−779582.	−0.0467745
$775$	0	0
$776$	−1.61850e7	−0.964846
$777$	3.13431e6i	0.186247i
$778$	1.82626e7i	1.08172i
$779$	1.43159e7	0.845231
$780$	0	0
$781$	−9.03837e6	−0.530228
$782$	1.36772e6i	0.0799801i
$783$	239185.i	0.0139421i
$784$	320312.	0.0186116
$785$	0	0
$786$	−5.07412e6	−0.292957
$787$	2.51593e7i	1.44798i	0.689812	+	0.723989i	$0.257693\pi$
$787$	2.51593e7i	1.44798i	−0.689812	+	0.723989i	$0.742307\pi$
$788$	1.53455e7i	0.880373i
$789$	1.84014e6	0.105234
$790$	0	0
$791$	−1.06162e7	−0.603294
$792$	4.20689e6i	0.238313i
$793$	1.12669e7i	0.636241i
$794$	−3.62183e6	−0.203881
$795$	0	0
$796$	−1.30990e7	−0.732751
$797$	3.41809e7i	1.90606i	0.302869	+	0.953032i	$0.402056\pi$
$797$	3.41809e7i	1.90606i	−0.302869	+	0.953032i	$0.597944\pi$
$798$	1.60975e6i	0.0894853i
$799$	966426.	0.0535552
$800$	0	0
$801$	−3.95262e6	−0.217673
$802$	− 2.10461e7i	− 1.15541i
$803$	− 2.15592e7i	− 1.17990i
$804$	−293219.	−0.0159975
$805$	0	0
$806$	5.64987e6	0.306338
$807$	4.46106e6i	0.241132i
$808$	− 760108.i	− 0.0409588i
$809$	−1.22270e7	−0.656825	−0.328412	−	0.944534i	$-0.606514\pi$
$809$	−1.22270e7	−0.656825	−0.328412	+	0.944534i	$0.606514\pi$
$810$	0	0
$811$	−2.76565e7	−1.47654	−0.738270	−	0.674505i	$-0.764357\pi$
$811$	−2.76565e7	−1.47654	−0.738270	+	0.674505i	$0.764357\pi$
$812$	287155.i	0.0152836i
$813$	1.32444e7i	0.702759i
$814$	7.40314e6	0.391611
$815$	0	0
$816$	−217520.	−0.0114360
$817$	− 2.48479e6i	− 0.130237i
$818$	− 1.25512e7i	− 0.655844i
$819$	2.25916e6	0.117689
$820$	0	0
$821$	−7.63597e6	−0.395372	−0.197686	−	0.980265i	$-0.563343\pi$
$821$	−7.63597e6	−0.395372	−0.197686	+	0.980265i	$0.563343\pi$
$822$	1.24143e7i	0.640832i
$823$	− 2.27659e7i	− 1.17162i	−0.810450	−	0.585808i	$-0.800777\pi$
$823$	− 2.27659e7i	− 1.17162i	0.810450	−	0.585808i	$-0.199223\pi$
$824$	−3.30929e7	−1.69792
$825$	0	0
$826$	−884956.	−0.0451306
$827$	3.58309e6i	0.182177i	0.995843	+	0.0910886i	$0.0290347\pi$
$827$	3.58309e6i	0.182177i	−0.995843	+	0.0910886i	$0.970965\pi$
$828$	2.90483e6i	0.147246i
$829$	1.25347e7	0.633473	0.316737	−	0.948513i	$-0.397413\pi$
$829$	1.25347e7	0.633473	0.316737	+	0.948513i	$0.397413\pi$
$830$	0	0
$831$	1.20030e7	0.602961
$832$	1.41970e7i	0.711029i
$833$	434977.i	0.0217197i
$834$	−1.03134e7	−0.513435
$835$	0	0
$836$	−4.80330e6	−0.237697
$837$	1.92441e6i	0.0949474i
$838$	1.25771e6i	0.0618687i
$839$	1.13383e7	0.556089	0.278044	−	0.960568i	$-0.410314\pi$
$839$	1.13383e7	0.556089	0.278044	+	0.960568i	$0.410314\pi$
$840$	0	0
$841$	−2.04035e7	−0.994752
$842$	− 2.35496e7i	− 1.14473i
$843$	5.87359e6i	0.284665i
$844$	−1.17166e7	−0.566167
$845$	0	0
$846$	−1.62474e6	−0.0780471
$847$	4.13129e6i	0.197869i
$848$	11669.5i	0 0.000557267i
$849$	1.01399e7	0.482795
$850$	0	0
$851$	1.42700e7	0.675462
$852$	− 5.24492e6i	− 0.247537i
$853$	6.01942e6i	0.283258i	0.989920	+	0.141629i	$0.0452340\pi$
$853$	6.01942e6i	0.283258i	−0.989920	+	0.141629i	$0.954766\pi$
$854$	3.64703e6	0.171117
$855$	0	0
$856$	1.13158e7	0.527837
$857$	2.34123e7i	1.08891i	0.838791	+	0.544454i	$0.183263\pi$
$857$	2.34123e7i	1.08891i	−0.838791	+	0.544454i	$0.816737\pi$
$858$	− 5.33605e6i	− 0.247458i
$859$	−4.09430e6	−0.189320	−0.0946600	−	0.995510i	$-0.530176\pi$
$859$	−4.09430e6	−0.189320	−0.0946600	+	0.995510i	$0.530176\pi$
$860$	0	0
$861$	−6.50341e6	−0.298974
$862$	− 2.49170e7i	− 1.14216i
$863$	− 2.52373e6i	− 0.115349i	−0.998335	−	0.0576747i	$-0.981631\pi$
$863$	− 2.52373e6i	− 0.115349i	0.998335	−	0.0576747i	$-0.0183686\pi$
$864$	−4.00797e6	−0.182659
$865$	0	0
$866$	7.13779e6	0.323421
$867$	1.24833e7i	0.564005i
$868$	2.31036e6i	0.104083i
$869$	1.06144e7	0.476809
$870$	0	0
$871$	1.03824e6	0.0463718
$872$	− 656701.i	− 0.0292467i
$873$	6.99245e6i	0.310523i
$874$	7.32893e6	0.324535
$875$	0	0
$876$	1.25107e7	0.550835
$877$	− 2.00245e7i	− 0.879148i	−0.898206	−	0.439574i	$-0.855129\pi$
$877$	− 2.00245e7i	− 0.879148i	0.898206	−	0.439574i	$-0.144871\pi$
$878$	− 615041.i	− 0.0269258i
$879$	−6.30293e6	−0.275151
$880$	0	0
$881$	2.66240e7	1.15567	0.577834	−	0.816154i	$-0.303898\pi$
$881$	2.66240e7	1.15567	0.577834	+	0.816154i	$0.303898\pi$
$882$	− 731275.i	− 0.0316526i
$883$	− 1.10325e6i	− 0.0476180i	−0.999717	−	0.0238090i	$-0.992421\pi$
$883$	− 1.10325e6i	− 0.0476180i	0.999717	−	0.0238090i	$-0.00757936\pi$
$884$	−1.84185e6	−0.0792727
$885$	0	0
$886$	2.87535e7	1.23057
$887$	− 2.73981e7i	− 1.16926i	−0.811300	−	0.584629i	$-0.801240\pi$
$887$	− 2.73981e7i	− 1.16926i	0.811300	−	0.584629i	$-0.198760\pi$
$888$	1.19926e7i	0.510366i
$889$	−7.90131e6	−0.335308
$890$	0	0
$891$	1.81752e6	0.0766980
$892$	680715.i	0.0286453i
$893$	− 5.17858e6i	− 0.217311i
$894$	−9.33379e6	−0.390584
$895$	0	0
$896$	−4.02525e6	−0.167503
$897$	− 1.02856e7i	− 0.426822i
$898$	− 706870.i	− 0.0292515i
$899$	866114.	0.0357418
$900$	0	0
$901$	−15846.9	−0.000650330 0
$902$	1.53608e7i	0.628634i
$903$	1.12879e6i	0.0460672i
$904$	−4.06202e7	−1.65318
$905$	0	0
$906$	1.76411e6	0.0714011
$907$	− 3.42182e7i	− 1.38115i	−0.723263	−	0.690573i	$-0.757358\pi$
$907$	− 3.42182e7i	− 1.38115i	0.723263	−	0.690573i	$-0.242642\pi$
$908$	− 2.16897e6i	− 0.0873051i
$909$	−328392.	−0.0131820
$910$	0	0
$911$	1.64929e7	0.658419	0.329210	−	0.944257i	$-0.393218\pi$
$911$	1.64929e7	0.658419	0.329210	+	0.944257i	$0.393218\pi$
$912$	1.16558e6i	0.0464038i
$913$	5.75562e6i	0.228515i
$914$	1.62393e7	0.642987
$915$	0	0
$916$	−1.53238e7	−0.603433
$917$	7.34700e6i	0.288527i
$918$	496599.i	0.0194491i
$919$	−3.72827e7	−1.45619	−0.728097	−	0.685474i	$-0.759595\pi$
$919$	−3.72827e7	−1.45619	−0.728097	+	0.685474i	$0.759595\pi$
$920$	0	0
$921$	1.52550e7	0.592601
$922$	4.92345e6i	0.190740i
$923$	1.85715e7i	0.717535i
$924$	2.18203e6	0.0840778
$925$	0	0
$926$	−8.39867e6	−0.321872
$927$	1.42972e7i	0.546452i
$928$	1.80386e6i	0.0687596i
$929$	9.50700e6	0.361413	0.180707	−	0.983537i	$-0.442162\pi$
$929$	9.50700e6	0.361413	0.180707	+	0.983537i	$0.442162\pi$
$930$	0	0
$931$	2.33082e6	0.0881322
$932$	109073.i	0.00411318i
$933$	1.78110e7i	0.669860i
$934$	2.56343e7	0.961512
$935$	0	0
$936$	8.64407e6	0.322499
$937$	3.86595e7i	1.43849i	0.694756	+	0.719246i	$0.255513\pi$
$937$	3.86595e7i	1.43849i	−0.694756	+	0.719246i	$0.744487\pi$
$938$	− 336073.i	− 0.0124717i
$939$	2.48447e7	0.919539
$940$	0	0
$941$	−1.34511e7	−0.495202	−0.247601	−	0.968862i	$-0.579642\pi$
$941$	−1.34511e7	−0.495202	−0.247601	+	0.968862i	$0.579642\pi$
$942$	− 1.46818e7i	− 0.539077i
$943$	2.96090e7i	1.08429i
$944$	−640772.	−0.0234031
$945$	0	0
$946$	2.66615e6	0.0968628
$947$	− 2.80215e7i	− 1.01535i	−0.861549	−	0.507675i	$-0.830505\pi$
$947$	− 2.80215e7i	− 1.01535i	0.861549	−	0.507675i	$-0.169495\pi$
$948$	6.15946e6i	0.222598i
$949$	−4.42986e7	−1.59670
$950$	0	0
$951$	1.29805e7	0.465415
$952$	1.66433e6i	0.0595177i
$953$	1.28981e6i	0.0460039i	0.999735	+	0.0230019i	$0.00732239\pi$
$953$	1.28981e6i	0.0460039i	−0.999735	+	0.0230019i	$0.992678\pi$
$954$	26641.6	0.000947740 0
$955$	0	0
$956$	9.77467e6	0.345905
$957$	− 818006.i	− 0.0288720i
$958$	2.62416e7i	0.923796i
$959$	1.79752e7	0.631142
$960$	0	0
$961$	−2.16607e7	−0.756595
$962$	− 1.52115e7i	− 0.529950i
$963$	− 4.88879e6i	− 0.169877i
$964$	−2.18608e7	−0.757657
$965$	0	0
$966$	−3.32938e6	−0.114794
$967$	2.76643e7i	0.951379i	0.879613	+	0.475690i	$0.157801\pi$
$967$	2.76643e7i	0.951379i	−0.879613	+	0.475690i	$0.842199\pi$
$968$	1.58073e7i	0.542212i
$969$	−1.58283e6	−0.0541532
$970$	0	0
$971$	1.83816e6	0.0625656	0.0312828	−	0.999511i	$-0.490041\pi$
$971$	1.83816e6	0.0625656	0.0312828	+	0.999511i	$0.490041\pi$
$972$	1.05470e6i	0.0358065i
$973$	1.49331e7i	0.505672i
$974$	1.17240e7	0.395983
$975$	0	0
$976$	2.64071e6	0.0887353
$977$	− 3.71707e7i	− 1.24585i	−0.782283	−	0.622923i	$-0.785945\pi$
$977$	− 3.71707e7i	− 1.24585i	0.782283	−	0.622923i	$-0.214055\pi$
$978$	− 2.58643e6i	− 0.0864677i
$979$	1.35179e7	0.450766
$980$	0	0
$981$	−283716.	−0.00941265
$982$	− 1.80223e7i	− 0.596390i
$983$	4.31945e7i	1.42576i	0.701288	+	0.712878i	$0.252609\pi$
$983$	4.31945e7i	1.42576i	−0.701288	+	0.712878i	$0.747391\pi$
$984$	−2.48836e7	−0.819267
$985$	0	0
$986$	223503.	0.00732136
$987$	2.35252e6i	0.0768670i
$988$	9.86953e6i	0.321665i
$989$	5.13918e6	0.167072
$990$	0	0
$991$	3.76939e7	1.21923	0.609616	−	0.792697i	$-0.291324\pi$
$991$	3.76939e7	1.21923	0.609616	+	0.792697i	$0.291324\pi$
$992$	1.45133e7i	0.468260i
$993$	1.11078e7i	0.357481i
$994$	6.01149e6	0.192982
$995$	0	0
$996$	−3.33996e6	−0.106682
$997$	− 1.74026e7i	− 0.554468i	−0.960802	−	0.277234i	$-0.910582\pi$
$997$	− 1.74026e7i	− 0.554468i	0.960802	−	0.277234i	$-0.0894178\pi$
$998$	− 7.35923e6i	− 0.233887i
$999$	5.18121e6	0.164255

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.6.d.m.274.3		8
5.2	odd	4		525.6.a.k.1.4	✓	4
5.3	odd	4		525.6.a.p.1.1	yes	4
5.4	even	2	inner	525.6.d.m.274.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.6.a.k.1.4	✓	4	5.2	odd	4
525.6.a.p.1.1	yes	4	5.3	odd	4
525.6.d.m.274.3		8	1.1	even	1	trivial
525.6.d.m.274.6		8	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.76014i q^{2} +9.00000i q^{3} +17.8614 q^{4} +33.8412 q^{6} -49.0000i q^{7} -187.486i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q-3.76014i q^{2} +9.00000i q^{3} +17.8614 q^{4} +33.8412 q^{6} -49.0000i q^{7} -187.486i q^{8} -81.0000 q^{9} +277.018 q^{11} +160.752i q^{12} -569.200i q^{13} -184.247 q^{14} -133.408 q^{16} -181.165i q^{17} +304.571i q^{18} -970.770 q^{19} +441.000 q^{21} -1041.63i q^{22} -2007.80i q^{23} +1687.37 q^{24} -2140.27 q^{26} -729.000i q^{27} -875.207i q^{28} -328.100 q^{29} -2639.79 q^{31} -5497.91i q^{32} +2493.16i q^{33} -681.205 q^{34} -1446.77 q^{36} +7107.29i q^{37} +3650.23i q^{38} +5122.80 q^{39} -14747.0 q^{41} -1658.22i q^{42} +2559.60i q^{43} +4947.92 q^{44} -7549.61 q^{46} +5334.50i q^{47} -1200.67i q^{48} -2401.00 q^{49} +1630.49 q^{51} -10166.7i q^{52} -87.4724i q^{53} -2741.14 q^{54} -9186.79 q^{56} -8736.93i q^{57} +1233.70i q^{58} +4803.10 q^{59} -19794.3 q^{61} +9925.97i q^{62} +3969.00i q^{63} -24941.9 q^{64} +9374.63 q^{66} +1824.04i q^{67} -3235.85i q^{68} +18070.2 q^{69} -32627.4 q^{71} +15186.3i q^{72} -77826.0i q^{73} +26724.4 q^{74} -17339.3 q^{76} -13573.9i q^{77} -19262.4i q^{78} +38316.5 q^{79} +6561.00 q^{81} +55450.6i q^{82} +20777.0i q^{83} +7876.86 q^{84} +9624.47 q^{86} -2952.90i q^{87} -51936.9i q^{88} +48797.8 q^{89} -27890.8 q^{91} -35862.1i q^{92} -23758.1i q^{93} +20058.5 q^{94} +49481.2 q^{96} -86326.6i q^{97} +9028.09i q^{98} -22438.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 98 q^{4} - 270 q^{6} - 648 q^{9}+O(q^{10}) \) 8 * q - 98 * q^4 - 270 * q^6 - 648 * q^9	\( 8 q - 98 q^{4} - 270 q^{6} - 648 q^{9} + 372 q^{11} + 1470 q^{14} + 5634 q^{16} + 5484 q^{19} + 3528 q^{21} + 9774 q^{24} + 9618 q^{26} - 1140 q^{29} - 18828 q^{31} - 4290 q^{34} + 7938 q^{36} + 10980 q^{39} - 34008 q^{41} - 69180 q^{44} - 60970 q^{46} - 19208 q^{49} - 2916 q^{51} + 21870 q^{54} - 53214 q^{56} - 19560 q^{59} - 120816 q^{61} - 200290 q^{64} - 42624 q^{66} - 50868 q^{69} + 2028 q^{71} - 283368 q^{74} + 5200 q^{76} - 298320 q^{79} + 52488 q^{81} - 43218 q^{84} + 192570 q^{86} + 15924 q^{89} - 59780 q^{91} - 465184 q^{94} - 211518 q^{96} - 30132 q^{99}+O(q^{100}) \) 8 * q - 98 * q^4 - 270 * q^6 - 648 * q^9 + 372 * q^11 + 1470 * q^14 + 5634 * q^16 + 5484 * q^19 + 3528 * q^21 + 9774 * q^24 + 9618 * q^26 - 1140 * q^29 - 18828 * q^31 - 4290 * q^34 + 7938 * q^36 + 10980 * q^39 - 34008 * q^41 - 69180 * q^44 - 60970 * q^46 - 19208 * q^49 - 2916 * q^51 + 21870 * q^54 - 53214 * q^56 - 19560 * q^59 - 120816 * q^61 - 200290 * q^64 - 42624 * q^66 - 50868 * q^69 + 2028 * q^71 - 283368 * q^74 + 5200 * q^76 - 298320 * q^79 + 52488 * q^81 - 43218 * q^84 + 192570 * q^86 + 15924 * q^89 - 59780 * q^91 - 465184 * q^94 - 211518 * q^96 - 30132 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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