LMFDB - Embedded newform 325.6.b.a.274.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.a.274.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.00000i q^{2} -6.00000i q^{3} +7.00000 q^{4} +30.0000 q^{6} -244.000i q^{7} +195.000i q^{8} +207.000 q^{9} +794.000 q^{11} -42.0000i q^{12} +169.000i q^{13} +1220.00 q^{14} -751.000 q^{16} -1534.00i q^{17} +1035.00i q^{18} -2706.00 q^{19} -1464.00 q^{21} +3970.00i q^{22} +702.000i q^{23} +1170.00 q^{24} -845.000 q^{26} -2700.00i q^{27} -1708.00i q^{28} +5038.00 q^{29} -3634.00 q^{31} +2485.00i q^{32} -4764.00i q^{33} +7670.00 q^{34} +1449.00 q^{36} -7058.00i q^{37} -13530.0i q^{38} +1014.00 q^{39} -294.000 q^{41} -7320.00i q^{42} -7618.00i q^{43} +5558.00 q^{44} -3510.00 q^{46} -3020.00i q^{47} +4506.00i q^{48} -42729.0 q^{49} -9204.00 q^{51} +1183.00i q^{52} -626.000i q^{53} +13500.0 q^{54} +47580.0 q^{56} +16236.0i q^{57} +25190.0i q^{58} +30066.0 q^{59} -5806.00 q^{61} -18170.0i q^{62} -50508.0i q^{63} -36457.0 q^{64} +23820.0 q^{66} -12436.0i q^{67} -10738.0i q^{68} +4212.00 q^{69} +4734.00 q^{71} +40365.0i q^{72} +14694.0i q^{73} +35290.0 q^{74} -18942.0 q^{76} -193736. i q^{77} +5070.00i q^{78} +39804.0 q^{79} +34101.0 q^{81} -1470.00i q^{82} +41776.0i q^{83} -10248.0 q^{84} +38090.0 q^{86} -30228.0i q^{87} +154830. i q^{88} -7970.00 q^{89} +41236.0 q^{91} +4914.00i q^{92} +21804.0i q^{93} +15100.0 q^{94} +14910.0 q^{96} -78050.0i q^{97} -213645. i q^{98} +164358. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{4} + 60 q^{6} + 414 q^{9} + 1588 q^{11} + 2440 q^{14} - 1502 q^{16} - 5412 q^{19} - 2928 q^{21} + 2340 q^{24} - 1690 q^{26} + 10076 q^{29} - 7268 q^{31} + 15340 q^{34} + 2898 q^{36} + 2028 q^{39}+ \cdots + 328716 q^{99}+O(q^{100}) $ 2 * q + 14 * q^4 + 60 * q^6 + 414 * q^9 + 1588 * q^11 + 2440 * q^14 - 1502 * q^16 - 5412 * q^19 - 2928 * q^21 + 2340 * q^24 - 1690 * q^26 + 10076 * q^29 - 7268 * q^31 + 15340 * q^34 + 2898 * q^36 + 2028 * q^39 - 588 * q^41 + 11116 * q^44 - 7020 * q^46 - 85458 * q^49 - 18408 * q^51 + 27000 * q^54 + 95160 * q^56 + 60132 * q^59 - 11612 * q^61 - 72914 * q^64 + 47640 * q^66 + 8424 * q^69 + 9468 * q^71 + 70580 * q^74 - 37884 * q^76 + 79608 * q^79 + 68202 * q^81 - 20496 * q^84 + 76180 * q^86 - 15940 * q^89 + 82472 * q^91 + 30200 * q^94 + 29820 * q^96 + 328716 * q^99

Character values

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

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

Coefficient data

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

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

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

$2$	5.00000i	0.883883i	0.897044	+	0.441942i	$0.145710\pi$
$2$	5.00000i	0.883883i	−0.897044	+	0.441942i	$0.854290\pi$
$3$	− 6.00000i	− 0.384900i	−0.981307	−	0.192450i	$-0.938357\pi$
$3$	− 6.00000i	− 0.384900i	0.981307	−	0.192450i	$-0.0616434\pi$
$4$	7.00000	0.218750
$5$	0	0
$5$	0	0
$6$	30.0000	0.340207
$7$	− 244.000i	− 1.88211i	−0.338255	−	0.941054i	$-0.609837\pi$
$7$	− 244.000i	− 1.88211i	0.338255	−	0.941054i	$-0.390163\pi$
$8$	195.000i	1.07723i
$9$	207.000	0.851852
$10$	0	0
$11$	794.000	1.97851	0.989256	−	0.146192i	$-0.0467017\pi$
$11$	794.000	1.97851	0.989256	+	0.146192i	$0.0467017\pi$
$12$	− 42.0000i	− 0.0841969i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	1220.00	1.66356
$15$	0	0
$16$	−751.000	−0.733398
$17$	− 1534.00i	− 1.28737i	−0.765291	−	0.643685i	$-0.777405\pi$
$17$	− 1534.00i	− 1.28737i	0.765291	−	0.643685i	$-0.222595\pi$
$18$	1035.00i	0.752938i
$19$	−2706.00	−1.71966	−0.859832	−	0.510576i	$-0.829432\pi$
$19$	−2706.00	−1.71966	−0.859832	+	0.510576i	$0.829432\pi$
$20$	0	0
$21$	−1464.00	−0.724424
$22$	3970.00i	1.74877i
$23$	702.000i	0.276705i	0.990383	+	0.138353i	$0.0441807\pi$
$23$	702.000i	0.276705i	−0.990383	+	0.138353i	$0.955819\pi$
$24$	1170.00	0.414627
$25$	0	0
$26$	−845.000	−0.245145
$27$	− 2700.00i	− 0.712778i
$28$	− 1708.00i	− 0.411711i
$29$	5038.00	1.11241	0.556203	−	0.831047i	$-0.312258\pi$
$29$	5038.00	1.11241	0.556203	+	0.831047i	$0.312258\pi$
$30$	0	0
$31$	−3634.00	−0.679173	−0.339587	−	0.940575i	$-0.610287\pi$
$31$	−3634.00	−0.679173	−0.339587	+	0.940575i	$0.610287\pi$
$32$	2485.00i	0.428994i
$33$	− 4764.00i	− 0.761530i
$34$	7670.00	1.13788
$35$	0	0
$36$	1449.00	0.186343
$37$	− 7058.00i	− 0.847573i	−0.905762	−	0.423787i	$-0.860701\pi$
$37$	− 7058.00i	− 0.847573i	0.905762	−	0.423787i	$-0.139299\pi$
$38$	− 13530.0i	− 1.51998i
$39$	1014.00	0.106752
$40$	0	0
$41$	−294.000	−0.0273141	−0.0136571	−	0.999907i	$-0.504347\pi$
$41$	−294.000	−0.0273141	−0.0136571	+	0.999907i	$0.504347\pi$
$42$	− 7320.00i	− 0.640306i
$43$	− 7618.00i	− 0.628304i	−0.949373	−	0.314152i	$-0.898280\pi$
$43$	− 7618.00i	− 0.628304i	0.949373	−	0.314152i	$-0.101720\pi$
$44$	5558.00	0.432800
$45$	0	0
$46$	−3510.00	−0.244575
$47$	− 3020.00i	− 0.199417i	−0.995017	−	0.0997085i	$-0.968209\pi$
$47$	− 3020.00i	− 0.199417i	0.995017	−	0.0997085i	$-0.0317910\pi$
$48$	4506.00i	0.282285i
$49$	−42729.0	−2.54233
$50$	0	0
$51$	−9204.00	−0.495509
$52$	1183.00i	0.0606703i
$53$	− 626.000i	− 0.0306115i	−0.999883	−	0.0153058i	$-0.995128\pi$
$53$	− 626.000i	− 0.0306115i	0.999883	−	0.0153058i	$-0.00487216\pi$
$54$	13500.0	0.630013
$55$	0	0
$56$	47580.0	2.02747
$57$	16236.0i	0.661899i
$58$	25190.0i	0.983237i
$59$	30066.0	1.12446	0.562232	−	0.826979i	$-0.309943\pi$
$59$	30066.0	1.12446	0.562232	+	0.826979i	$0.309943\pi$
$60$	0	0
$61$	−5806.00	−0.199780	−0.0998901	−	0.994998i	$-0.531849\pi$
$61$	−5806.00	−0.199780	−0.0998901	+	0.994998i	$0.531849\pi$
$62$	− 18170.0i	− 0.600310i
$63$	− 50508.0i	− 1.60328i
$64$	−36457.0	−1.11258
$65$	0	0
$66$	23820.0	0.673104
$67$	− 12436.0i	− 0.338449i	−0.985577	−	0.169225i	$-0.945874\pi$
$67$	− 12436.0i	− 0.338449i	0.985577	−	0.169225i	$-0.0541264\pi$
$68$	− 10738.0i	− 0.281612i
$69$	4212.00	0.106504
$70$	0	0
$71$	4734.00	0.111451	0.0557253	−	0.998446i	$-0.482253\pi$
$71$	4734.00	0.111451	0.0557253	+	0.998446i	$0.482253\pi$
$72$	40365.0i	0.917643i
$73$	14694.0i	0.322725i	0.986895	+	0.161363i	$0.0515889\pi$
$73$	14694.0i	0.322725i	−0.986895	+	0.161363i	$0.948411\pi$
$74$	35290.0	0.749156
$75$	0	0
$76$	−18942.0	−0.376177
$77$	− 193736.i	− 3.72378i
$78$	5070.00i	0.0943564i
$79$	39804.0	0.717561	0.358781	−	0.933422i	$-0.383193\pi$
$79$	39804.0	0.717561	0.358781	+	0.933422i	$0.383193\pi$
$80$	0	0
$81$	34101.0	0.577503
$82$	− 1470.00i	− 0.0241425i
$83$	41776.0i	0.665628i	0.942992	+	0.332814i	$0.107998\pi$
$83$	41776.0i	0.665628i	−0.942992	+	0.332814i	$0.892002\pi$
$84$	−10248.0	−0.158468
$85$	0	0
$86$	38090.0	0.555348
$87$	− 30228.0i	− 0.428165i
$88$	154830.i	2.13132i
$89$	−7970.00	−0.106656	−0.0533278	−	0.998577i	$-0.516983\pi$
$89$	−7970.00	−0.106656	−0.0533278	+	0.998577i	$0.516983\pi$
$90$	0	0
$91$	41236.0	0.522003
$92$	4914.00i	0.0605293i
$93$	21804.0i	0.261414i
$94$	15100.0	0.176261
$95$	0	0
$96$	14910.0	0.165120
$97$	− 78050.0i	− 0.842255i	−0.907001	−	0.421127i	$-0.861634\pi$
$97$	− 78050.0i	− 0.842255i	0.907001	−	0.421127i	$-0.138366\pi$
$98$	− 213645.i	− 2.24713i
$99$	164358.	1.68540
$100$	0	0
$101$	−23010.0	−0.224447	−0.112223	−	0.993683i	$-0.535797\pi$
$101$	−23010.0	−0.224447	−0.112223	+	0.993683i	$0.535797\pi$
$102$	− 46020.0i	− 0.437972i
$103$	− 121706.i	− 1.13037i	−0.824966	−	0.565183i	$-0.808806\pi$
$103$	− 121706.i	− 1.13037i	0.824966	−	0.565183i	$-0.191194\pi$
$104$	−32955.0	−0.298771
$105$	0	0
$106$	3130.00	0.0270570
$107$	− 70142.0i	− 0.592269i	−0.955146	−	0.296134i	$-0.904302\pi$
$107$	− 70142.0i	− 0.592269i	0.955146	−	0.296134i	$-0.0956976\pi$
$108$	− 18900.0i	− 0.155920i
$109$	195878.	1.57914	0.789568	−	0.613663i	$-0.210305\pi$
$109$	195878.	1.57914	0.789568	+	0.613663i	$0.210305\pi$
$110$	0	0
$111$	−42348.0	−0.326231
$112$	183244.i	1.38034i
$113$	100238.i	0.738476i	0.929335	+	0.369238i	$0.120381\pi$
$113$	100238.i	0.738476i	−0.929335	+	0.369238i	$0.879619\pi$
$114$	−81180.0	−0.585042
$115$	0	0
$116$	35266.0	0.243339
$117$	34983.0i	0.236261i
$118$	150330.i	0.993895i
$119$	−374296.	−2.42297
$120$	0	0
$121$	469385.	2.91451
$122$	− 29030.0i	− 0.176582i
$123$	1764.00i	0.0105132i
$124$	−25438.0	−0.148569
$125$	0	0
$126$	252540.	1.41711
$127$	39286.0i	0.216137i	0.994143	+	0.108068i	$0.0344665\pi$
$127$	39286.0i	0.216137i	−0.994143	+	0.108068i	$0.965533\pi$
$128$	− 102765.i	− 0.554396i
$129$	−45708.0	−0.241834
$130$	0	0
$131$	211460.	1.07659	0.538295	−	0.842757i	$-0.319069\pi$
$131$	211460.	1.07659	0.538295	+	0.842757i	$0.319069\pi$
$132$	− 33348.0i	− 0.166585i
$133$	660264.i	3.23660i
$134$	62180.0	0.299150
$135$	0	0
$136$	299130.	1.38680
$137$	26302.0i	0.119726i	0.998207	+	0.0598628i	$0.0190663\pi$
$137$	26302.0i	0.119726i	−0.998207	+	0.0598628i	$0.980934\pi$
$138$	21060.0i	0.0941371i
$139$	−1344.00	−0.00590014	−0.00295007	−	0.999996i	$-0.500939\pi$
$139$	−1344.00	−0.00590014	−0.00295007	+	0.999996i	$0.500939\pi$
$140$	0	0
$141$	−18120.0	−0.0767557
$142$	23670.0i	0.0985093i
$143$	134186.i	0.548741i
$144$	−155457.	−0.624747
$145$	0	0
$146$	−73470.0	−0.285251
$147$	256374.i	0.978545i
$148$	− 49406.0i	− 0.185407i
$149$	49086.0	0.181131	0.0905653	−	0.995891i	$-0.471133\pi$
$149$	49086.0	0.181131	0.0905653	+	0.995891i	$0.471133\pi$
$150$	0	0
$151$	−357998.	−1.27773	−0.638864	−	0.769320i	$-0.720595\pi$
$151$	−357998.	−1.27773	−0.638864	+	0.769320i	$0.720595\pi$
$152$	− 527670.i	− 1.85248i
$153$	− 317538.i	− 1.09665i
$154$	968680.	3.29138
$155$	0	0
$156$	7098.00	0.0233520
$157$	45450.0i	0.147158i	0.997289	+	0.0735791i	$0.0234422\pi$
$157$	45450.0i	0.147158i	−0.997289	+	0.0735791i	$0.976558\pi$
$158$	199020.i	0.634241i
$159$	−3756.00	−0.0117824
$160$	0	0
$161$	171288.	0.520790
$162$	170505.i	0.510446i
$163$	− 5892.00i	− 0.0173698i	−0.999962	−	0.00868488i	$-0.997235\pi$
$163$	− 5892.00i	− 0.0173698i	0.999962	−	0.00868488i	$-0.00276452\pi$
$164$	−2058.00	−0.00597497
$165$	0	0
$166$	−208880.	−0.588338
$167$	212772.i	0.590369i	0.955440	+	0.295184i	$0.0953811\pi$
$167$	212772.i	0.590369i	−0.955440	+	0.295184i	$0.904619\pi$
$168$	− 285480.i	− 0.780373i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−560142.	−1.46490
$172$	− 53326.0i	− 0.137442i
$173$	− 503178.i	− 1.27822i	−0.769114	−	0.639111i	$-0.779302\pi$
$173$	− 503178.i	− 1.27822i	0.769114	−	0.639111i	$-0.220698\pi$
$174$	151140.	0.378448
$175$	0	0
$176$	−596294.	−1.45104
$177$	− 180396.i	− 0.432806i
$178$	− 39850.0i	− 0.0942710i
$179$	−581724.	−1.35701	−0.678507	−	0.734594i	$-0.737373\pi$
$179$	−581724.	−1.35701	−0.678507	+	0.734594i	$0.737373\pi$
$180$	0	0
$181$	202202.	0.458764	0.229382	−	0.973337i	$-0.426330\pi$
$181$	202202.	0.458764	0.229382	+	0.973337i	$0.426330\pi$
$182$	206180.i	0.461390i
$183$	34836.0i	0.0768954i
$184$	−136890.	−0.298076
$185$	0	0
$186$	−109020.	−0.231059
$187$	− 1.21800e6i	− 2.54708i
$188$	− 21140.0i	− 0.0436225i
$189$	−658800.	−1.34153
$190$	0	0
$191$	−340608.	−0.675572	−0.337786	−	0.941223i	$-0.609678\pi$
$191$	−340608.	−0.675572	−0.337786	+	0.941223i	$0.609678\pi$
$192$	218742.i	0.428232i
$193$	− 275614.i	− 0.532608i	−0.963889	−	0.266304i	$-0.914197\pi$
$193$	− 275614.i	− 0.532608i	0.963889	−	0.266304i	$-0.0858026\pi$
$194$	390250.	0.744455
$195$	0	0
$196$	−299103.	−0.556135
$197$	538218.i	0.988081i	0.869439	+	0.494041i	$0.164481\pi$
$197$	538218.i	0.988081i	−0.869439	+	0.494041i	$0.835519\pi$
$198$	821790.i	1.48970i
$199$	853840.	1.52842	0.764212	−	0.644965i	$-0.223128\pi$
$199$	853840.	1.52842	0.764212	+	0.644965i	$0.223128\pi$
$200$	0	0
$201$	−74616.0	−0.130269
$202$	− 115050.i	− 0.198385i
$203$	− 1.22927e6i	− 2.09367i
$204$	−64428.0	−0.108392
$205$	0	0
$206$	608530.	0.999111
$207$	145314.i	0.235712i
$208$	− 126919.i	− 0.203408i
$209$	−2.14856e6	−3.40238
$210$	0	0
$211$	−1.00112e6	−1.54804	−0.774019	−	0.633162i	$-0.781757\pi$
$211$	−1.00112e6	−1.54804	−0.774019	+	0.633162i	$0.781757\pi$
$212$	− 4382.00i	− 0.00669627i
$213$	− 28404.0i	− 0.0428974i
$214$	350710.	0.523496
$215$	0	0
$216$	526500.	0.767828
$217$	886696.i	1.27828i
$218$	979390.i	1.39577i
$219$	88164.0	0.124217
$220$	0	0
$221$	259246.	0.357052
$222$	− 211740.i	− 0.288350i
$223$	− 21364.0i	− 0.0287687i	−0.999897	−	0.0143844i	$-0.995421\pi$
$223$	− 21364.0i	− 0.0287687i	0.999897	−	0.0143844i	$-0.00457884\pi$
$224$	606340.	0.807414
$225$	0	0
$226$	−501190.	−0.652727
$227$	− 880748.i	− 1.13445i	−0.823561	−	0.567227i	$-0.808016\pi$
$227$	− 880748.i	− 1.13445i	0.823561	−	0.567227i	$-0.191984\pi$
$228$	113652.i	0.144790i
$229$	13030.0	0.0164193	0.00820967	−	0.999966i	$-0.497387\pi$
$229$	13030.0	0.0164193	0.00820967	+	0.999966i	$0.497387\pi$
$230$	0	0
$231$	−1.16242e6	−1.43328
$232$	982410.i	1.19832i
$233$	1.20700e6i	1.45652i	0.685300	+	0.728260i	$0.259671\pi$
$233$	1.20700e6i	1.45652i	−0.685300	+	0.728260i	$0.740329\pi$
$234$	−174915.	−0.208827
$235$	0	0
$236$	210462.	0.245977
$237$	− 238824.i	− 0.276189i
$238$	− 1.87148e6i	− 2.14162i
$239$	187038.	0.211804	0.105902	−	0.994377i	$-0.466227\pi$
$239$	187038.	0.211804	0.105902	+	0.994377i	$0.466227\pi$
$240$	0	0
$241$	271690.	0.301322	0.150661	−	0.988585i	$-0.451860\pi$
$241$	271690.	0.301322	0.150661	+	0.988585i	$0.451860\pi$
$242$	2.34692e6i	2.57609i
$243$	− 860706.i	− 0.935059i
$244$	−40642.0	−0.0437019
$245$	0	0
$246$	−8820.00	−0.00929246
$247$	− 457314.i	− 0.476949i
$248$	− 708630.i	− 0.731628i
$249$	250656.	0.256200
$250$	0	0
$251$	102648.	0.102841	0.0514205	−	0.998677i	$-0.483625\pi$
$251$	102648.	0.102841	0.0514205	+	0.998677i	$0.483625\pi$
$252$	− 353556.i	− 0.350717i
$253$	557388.i	0.547465i
$254$	−196430.	−0.191040
$255$	0	0
$256$	−652799.	−0.622558
$257$	− 221182.i	− 0.208890i	−0.994531	−	0.104445i	$-0.966693\pi$
$257$	− 221182.i	− 0.208890i	0.994531	−	0.104445i	$-0.0333066\pi$
$258$	− 228540.i	− 0.213753i
$259$	−1.72215e6	−1.59523
$260$	0	0
$261$	1.04287e6	0.947605
$262$	1.05730e6i	0.951579i
$263$	− 1.40317e6i	− 1.25090i	−0.780265	−	0.625449i	$-0.784916\pi$
$263$	− 1.40317e6i	− 1.25090i	0.780265	−	0.625449i	$-0.215084\pi$
$264$	928980.	0.820345
$265$	0	0
$266$	−3.30132e6	−2.86077
$267$	47820.0i	0.0410517i
$268$	− 87052.0i	− 0.0740358i
$269$	582954.	0.491195	0.245597	−	0.969372i	$-0.421016\pi$
$269$	582954.	0.491195	0.245597	+	0.969372i	$0.421016\pi$
$270$	0	0
$271$	−1.04690e6	−0.865930	−0.432965	−	0.901411i	$-0.642533\pi$
$271$	−1.04690e6	−0.865930	−0.432965	+	0.901411i	$0.642533\pi$
$272$	1.15203e6i	0.944154i
$273$	− 247416.i	− 0.200919i
$274$	−131510.	−0.105824
$275$	0	0
$276$	29484.0	0.0232977
$277$	1.10461e6i	0.864987i	0.901637	+	0.432493i	$0.142366\pi$
$277$	1.10461e6i	0.864987i	−0.901637	+	0.432493i	$0.857634\pi$
$278$	− 6720.00i	− 0.00521504i
$279$	−752238.	−0.578555
$280$	0	0
$281$	908826.	0.686618	0.343309	−	0.939223i	$-0.388452\pi$
$281$	908826.	0.686618	0.343309	+	0.939223i	$0.388452\pi$
$282$	− 90600.0i	− 0.0678431i
$283$	449254.i	0.333446i	0.986004	+	0.166723i	$0.0533186\pi$
$283$	449254.i	0.333446i	−0.986004	+	0.166723i	$0.946681\pi$
$284$	33138.0	0.0243798
$285$	0	0
$286$	−670930.	−0.485023
$287$	71736.0i	0.0514082i
$288$	514395.i	0.365440i
$289$	−933299.	−0.657319
$290$	0	0
$291$	−468300.	−0.324184
$292$	102858.i	0.0705961i
$293$	1.96083e6i	1.33435i	0.744901	+	0.667175i	$0.232497\pi$
$293$	1.96083e6i	1.33435i	−0.744901	+	0.667175i	$0.767503\pi$
$294$	−1.28187e6	−0.864919
$295$	0	0
$296$	1.37631e6	0.913034
$297$	− 2.14380e6i	− 1.41024i
$298$	245430.i	0.160098i
$299$	−118638.	−0.0767442
$300$	0	0
$301$	−1.85879e6	−1.18254
$302$	− 1.78999e6i	− 1.12936i
$303$	138060.i	0.0863896i
$304$	2.03221e6	1.26120
$305$	0	0
$306$	1.58769e6	0.969309
$307$	− 1.79385e6i	− 1.08627i	−0.839644	−	0.543137i	$-0.817236\pi$
$307$	− 1.79385e6i	− 1.08627i	0.839644	−	0.543137i	$-0.182764\pi$
$308$	− 1.35615e6i	− 0.814576i
$309$	−730236.	−0.435078
$310$	0	0
$311$	2.41233e6	1.41428	0.707141	−	0.707072i	$-0.249985\pi$
$311$	2.41233e6	1.41428	0.707141	+	0.707072i	$0.249985\pi$
$312$	197730.i	0.114997i
$313$	2.15436e6i	1.24296i	0.783430	+	0.621480i	$0.213468\pi$
$313$	2.15436e6i	1.24296i	−0.783430	+	0.621480i	$0.786532\pi$
$314$	−227250.	−0.130071
$315$	0	0
$316$	278628.	0.156967
$317$	2.59616e6i	1.45105i	0.688195	+	0.725526i	$0.258403\pi$
$317$	2.59616e6i	1.45105i	−0.688195	+	0.725526i	$0.741597\pi$
$318$	− 18780.0i	− 0.0104142i
$319$	4.00017e6	2.20091
$320$	0	0
$321$	−420852.	−0.227964
$322$	856440.i	0.460317i
$323$	4.15100e6i	2.21384i
$324$	238707.	0.126329
$325$	0	0
$326$	29460.0	0.0153528
$327$	− 1.17527e6i	− 0.607810i
$328$	− 57330.0i	− 0.0294237i
$329$	−736880.	−0.375325
$330$	0	0
$331$	−917226.	−0.460157	−0.230079	−	0.973172i	$-0.573898\pi$
$331$	−917226.	−0.460157	−0.230079	+	0.973172i	$0.573898\pi$
$332$	292432.i	0.145606i
$333$	− 1.46101e6i	− 0.722007i
$334$	−1.06386e6	−0.521817
$335$	0	0
$336$	1.09946e6	0.531291
$337$	2.23894e6i	1.07391i	0.843611	+	0.536954i	$0.180425\pi$
$337$	2.23894e6i	1.07391i	−0.843611	+	0.536954i	$0.819575\pi$
$338$	− 142805.i	− 0.0679910i
$339$	601428.	0.284239
$340$	0	0
$341$	−2.88540e6	−1.34375
$342$	− 2.80071e6i	− 1.29480i
$343$	6.32497e6i	2.90284i
$344$	1.48551e6	0.676830
$345$	0	0
$346$	2.51589e6	1.12980
$347$	3.41808e6i	1.52391i	0.647631	+	0.761954i	$0.275760\pi$
$347$	3.41808e6i	1.52391i	−0.647631	+	0.761954i	$0.724240\pi$
$348$	− 211596.i	− 0.0936611i
$349$	−2.35691e6	−1.03581	−0.517905	−	0.855438i	$-0.673288\pi$
$349$	−2.35691e6	−1.03581	−0.517905	+	0.855438i	$0.673288\pi$
$350$	0	0
$351$	456300.	0.197689
$352$	1.97309e6i	0.848770i
$353$	3.76395e6i	1.60771i	0.594827	+	0.803854i	$0.297221\pi$
$353$	3.76395e6i	1.60771i	−0.594827	+	0.803854i	$0.702779\pi$
$354$	901980.	0.382550
$355$	0	0
$356$	−55790.0	−0.0233309
$357$	2.24578e6i	0.932601i
$358$	− 2.90862e6i	− 1.19944i
$359$	−3.28216e6	−1.34407	−0.672037	−	0.740517i	$-0.734581\pi$
$359$	−3.28216e6	−1.34407	−0.672037	+	0.740517i	$0.734581\pi$
$360$	0	0
$361$	4.84634e6	1.95725
$362$	1.01101e6i	0.405494i
$363$	− 2.81631e6i	− 1.12180i
$364$	288652.	0.114188
$365$	0	0
$366$	−174180.	−0.0679666
$367$	− 2.42605e6i	− 0.940233i	−0.882604	−	0.470116i	$-0.844212\pi$
$367$	− 2.42605e6i	− 0.940233i	0.882604	−	0.470116i	$-0.155788\pi$
$368$	− 527202.i	− 0.202935i
$369$	−60858.0	−0.0232676
$370$	0	0
$371$	−152744.	−0.0576142
$372$	152628.i	0.0571843i
$373$	− 2.80635e6i	− 1.04441i	−0.852820	−	0.522204i	$-0.825110\pi$
$373$	− 2.80635e6i	− 1.04441i	0.852820	−	0.522204i	$-0.174890\pi$
$374$	6.08998e6	2.25132
$375$	0	0
$376$	588900.	0.214819
$377$	851422.i	0.308526i
$378$	− 3.29400e6i	− 1.18575i
$379$	−3.15392e6	−1.12785	−0.563927	−	0.825825i	$-0.690710\pi$
$379$	−3.15392e6	−1.12785	−0.563927	+	0.825825i	$0.690710\pi$
$380$	0	0
$381$	235716.	0.0831911
$382$	− 1.70304e6i	− 0.597127i
$383$	475044.i	0.165477i	0.996571	+	0.0827384i	$0.0263666\pi$
$383$	475044.i	0.165477i	−0.996571	+	0.0827384i	$0.973633\pi$
$384$	−616590.	−0.213387
$385$	0	0
$386$	1.37807e6	0.470764
$387$	− 1.57693e6i	− 0.535222i
$388$	− 546350.i	− 0.184243i
$389$	−150566.	−0.0504490	−0.0252245	−	0.999682i	$-0.508030\pi$
$389$	−150566.	−0.0504490	−0.0252245	+	0.999682i	$0.508030\pi$
$390$	0	0
$391$	1.07687e6	0.356222
$392$	− 8.33216e6i	− 2.73869i
$393$	− 1.26876e6i	− 0.414379i
$394$	−2.69109e6	−0.873349
$395$	0	0
$396$	1.15051e6	0.368681
$397$	241686.i	0.0769618i	0.999259	+	0.0384809i	$0.0122519\pi$
$397$	241686.i	0.0769618i	−0.999259	+	0.0384809i	$0.987748\pi$
$398$	4.26920e6i	1.35095i
$399$	3.96158e6	1.24577
$400$	0	0
$401$	−3.19679e6	−0.992780	−0.496390	−	0.868100i	$-0.665341\pi$
$401$	−3.19679e6	−0.992780	−0.496390	+	0.868100i	$0.665341\pi$
$402$	− 373080.i	− 0.115143i
$403$	− 614146.i	− 0.188369i
$404$	−161070.	−0.0490977
$405$	0	0
$406$	6.14636e6	1.85056
$407$	− 5.60405e6i	− 1.67693i
$408$	− 1.79478e6i	− 0.533778i
$409$	−423282.	−0.125119	−0.0625593	−	0.998041i	$-0.519926\pi$
$409$	−423282.	−0.125119	−0.0625593	+	0.998041i	$0.519926\pi$
$410$	0	0
$411$	157812.	0.0460824
$412$	− 851942.i	− 0.247267i
$413$	− 7.33610e6i	− 2.11636i
$414$	−726570.	−0.208342
$415$	0	0
$416$	−419965.	−0.118982
$417$	8064.00i	0.00227096i
$418$	− 1.07428e7i	− 3.00731i
$419$	1.13159e6	0.314887	0.157444	−	0.987528i	$-0.449675\pi$
$419$	1.13159e6	0.314887	0.157444	+	0.987528i	$0.449675\pi$
$420$	0	0
$421$	3.47699e6	0.956088	0.478044	−	0.878336i	$-0.341346\pi$
$421$	3.47699e6	0.956088	0.478044	+	0.878336i	$0.341346\pi$
$422$	− 5.00562e6i	− 1.36829i
$423$	− 625140.i	− 0.169874i
$424$	122070.	0.0329757
$425$	0	0
$426$	142020.	0.0379163
$427$	1.41666e6i	0.376008i
$428$	− 490994.i	− 0.129559i
$429$	805116.	0.211210
$430$	0	0
$431$	3.41044e6	0.884335	0.442168	−	0.896932i	$-0.354210\pi$
$431$	3.41044e6	0.884335	0.442168	+	0.896932i	$0.354210\pi$
$432$	2.02770e6i	0.522750i
$433$	3.40722e6i	0.873335i	0.899623	+	0.436667i	$0.143841\pi$
$433$	3.40722e6i	0.873335i	−0.899623	+	0.436667i	$0.856159\pi$
$434$	−4.43348e6	−1.12985
$435$	0	0
$436$	1.37115e6	0.345436
$437$	− 1.89961e6i	− 0.475840i
$438$	440820.i	0.109793i
$439$	7.09114e6	1.75612	0.878061	−	0.478549i	$-0.158837\pi$
$439$	7.09114e6	1.75612	0.878061	+	0.478549i	$0.158837\pi$
$440$	0	0
$441$	−8.84490e6	−2.16569
$442$	1.29623e6i	0.315592i
$443$	8.23508e6i	1.99369i	0.0793445	+	0.996847i	$0.474717\pi$
$443$	8.23508e6i	1.99369i	−0.0793445	+	0.996847i	$0.525283\pi$
$444$	−296436.	−0.0713631
$445$	0	0
$446$	106820.	0.0254282
$447$	− 294516.i	− 0.0697172i
$448$	8.89551e6i	2.09400i
$449$	1.29601e6	0.303383	0.151691	−	0.988428i	$-0.451528\pi$
$449$	1.29601e6	0.303383	0.151691	+	0.988428i	$0.451528\pi$
$450$	0	0
$451$	−233436.	−0.0540414
$452$	701666.i	0.161542i
$453$	2.14799e6i	0.491798i
$454$	4.40374e6	1.00273
$455$	0	0
$456$	−3.16602e6	−0.713020
$457$	1.68196e6i	0.376725i	0.982100	+	0.188363i	$0.0603180\pi$
$457$	1.68196e6i	0.376725i	−0.982100	+	0.188363i	$0.939682\pi$
$458$	65150.0i	0.0145128i
$459$	−4.14180e6	−0.917608
$460$	0	0
$461$	−3.20663e6	−0.702743	−0.351372	−	0.936236i	$-0.614285\pi$
$461$	−3.20663e6	−0.702743	−0.351372	+	0.936236i	$0.614285\pi$
$462$	− 5.81208e6i	− 1.26685i
$463$	5.26370e6i	1.14114i	0.821249	+	0.570570i	$0.193278\pi$
$463$	5.26370e6i	1.14114i	−0.821249	+	0.570570i	$0.806722\pi$
$464$	−3.78354e6	−0.815837
$465$	0	0
$466$	−6.03499e6	−1.28739
$467$	− 8.26813e6i	− 1.75435i	−0.480175	−	0.877173i	$-0.659427\pi$
$467$	− 8.26813e6i	− 1.75435i	0.480175	−	0.877173i	$-0.340573\pi$
$468$	244881.i	0.0516821i
$469$	−3.03438e6	−0.636999
$470$	0	0
$471$	272700.	0.0566413
$472$	5.86287e6i	1.21131i
$473$	− 6.04869e6i	− 1.24311i
$474$	1.19412e6	0.244119
$475$	0	0
$476$	−2.62007e6	−0.530024
$477$	− 129582.i	− 0.0260765i
$478$	935190.i	0.187210i
$479$	−3.65468e6	−0.727797	−0.363899	−	0.931439i	$-0.618555\pi$
$479$	−3.65468e6	−0.727797	−0.363899	+	0.931439i	$0.618555\pi$
$480$	0	0
$481$	1.19280e6	0.235075
$482$	1.35845e6i	0.266334i
$483$	− 1.02773e6i	− 0.200452i
$484$	3.28570e6	0.637549
$485$	0	0
$486$	4.30353e6	0.826483
$487$	7.13084e6i	1.36244i	0.732077	+	0.681221i	$0.238551\pi$
$487$	7.13084e6i	1.36244i	−0.732077	+	0.681221i	$0.761449\pi$
$488$	− 1.13217e6i	− 0.215210i
$489$	−35352.0	−0.00668562
$490$	0	0
$491$	5.72551e6	1.07179	0.535896	−	0.844284i	$-0.319974\pi$
$491$	5.72551e6	1.07179	0.535896	+	0.844284i	$0.319974\pi$
$492$	12348.0i	0.00229977i
$493$	− 7.72829e6i	− 1.43208i
$494$	2.28657e6	0.421568
$495$	0	0
$496$	2.72913e6	0.498105
$497$	− 1.15510e6i	− 0.209762i
$498$	1.25328e6i	0.226451i
$499$	7.17251e6	1.28950	0.644748	−	0.764395i	$-0.276962\pi$
$499$	7.17251e6	1.28950	0.644748	+	0.764395i	$0.276962\pi$
$500$	0	0
$501$	1.27663e6	0.227233
$502$	513240.i	0.0908994i
$503$	2.90611e6i	0.512143i	0.966658	+	0.256072i	$0.0824283\pi$
$503$	2.90611e6i	0.512143i	−0.966658	+	0.256072i	$0.917572\pi$
$504$	9.84906e6	1.72710
$505$	0	0
$506$	−2.78694e6	−0.483895
$507$	171366.i	0.0296077i
$508$	275002.i	0.0472799i
$509$	8.37125e6	1.43217	0.716087	−	0.698011i	$-0.245931\pi$
$509$	8.37125e6	1.43217	0.716087	+	0.698011i	$0.245931\pi$
$510$	0	0
$511$	3.58534e6	0.607404
$512$	− 6.55248e6i	− 1.10466i
$513$	7.30620e6i	1.22574i
$514$	1.10591e6	0.184634
$515$	0	0
$516$	−319956.	−0.0529013
$517$	− 2.39788e6i	− 0.394549i
$518$	− 8.61076e6i	− 1.40999i
$519$	−3.01907e6	−0.491988
$520$	0	0
$521$	5.37332e6	0.867258	0.433629	−	0.901092i	$-0.357233\pi$
$521$	5.37332e6	0.867258	0.433629	+	0.901092i	$0.357233\pi$
$522$	5.21433e6i	0.837572i
$523$	− 5.26875e6i	− 0.842274i	−0.906997	−	0.421137i	$-0.861631\pi$
$523$	− 5.26875e6i	− 0.842274i	0.906997	−	0.421137i	$-0.138369\pi$
$524$	1.48022e6	0.235504
$525$	0	0
$526$	7.01587e6	1.10565
$527$	5.57456e6i	0.874347i
$528$	3.57776e6i	0.558505i
$529$	5.94354e6	0.923434
$530$	0	0
$531$	6.22366e6	0.957877
$532$	4.62185e6i	0.708005i
$533$	− 49686.0i	− 0.00757558i
$534$	−239100.	−0.0362849
$535$	0	0
$536$	2.42502e6	0.364589
$537$	3.49034e6i	0.522315i
$538$	2.91477e6i	0.434159i
$539$	−3.39268e7	−5.03004
$540$	0	0
$541$	−6.07956e6	−0.893056	−0.446528	−	0.894770i	$-0.647340\pi$
$541$	−6.07956e6	−0.893056	−0.446528	+	0.894770i	$0.647340\pi$
$542$	− 5.23451e6i	− 0.765381i
$543$	− 1.21321e6i	− 0.176578i
$544$	3.81199e6	0.552274
$545$	0	0
$546$	1.23708e6	0.177589
$547$	− 7.88715e6i	− 1.12707i	−0.826091	−	0.563536i	$-0.809441\pi$
$547$	− 7.88715e6i	− 1.12707i	0.826091	−	0.563536i	$-0.190559\pi$
$548$	184114.i	0.0261900i
$549$	−1.20184e6	−0.170183
$550$	0	0
$551$	−1.36328e7	−1.91296
$552$	821340.i	0.114730i
$553$	− 9.71218e6i	− 1.35053i
$554$	−5.52305e6	−0.764548
$555$	0	0
$556$	−9408.00	−0.00129066
$557$	− 5.88545e6i	− 0.803788i	−0.915686	−	0.401894i	$-0.868352\pi$
$557$	− 5.88545e6i	− 0.803788i	0.915686	−	0.401894i	$-0.131648\pi$
$558$	− 3.76119e6i	− 0.511375i
$559$	1.28744e6	0.174260
$560$	0	0
$561$	−7.30798e6	−0.980370
$562$	4.54413e6i	0.606890i
$563$	3.91526e6i	0.520583i	0.965530	+	0.260291i	$0.0838186\pi$
$563$	3.91526e6i	0.520583i	−0.965530	+	0.260291i	$0.916181\pi$
$564$	−126840.	−0.0167903
$565$	0	0
$566$	−2.24627e6	−0.294728
$567$	− 8.32064e6i	− 1.08692i
$568$	923130.i	0.120058i
$569$	−9.78180e6	−1.26660	−0.633298	−	0.773908i	$-0.718299\pi$
$569$	−9.78180e6	−1.26660	−0.633298	+	0.773908i	$0.718299\pi$
$570$	0	0
$571$	−1.08198e7	−1.38877	−0.694386	−	0.719603i	$-0.744324\pi$
$571$	−1.08198e7	−1.38877	−0.694386	+	0.719603i	$0.744324\pi$
$572$	939302.i	0.120037i
$573$	2.04365e6i	0.260028i
$574$	−358680.	−0.0454389
$575$	0	0
$576$	−7.54660e6	−0.947753
$577$	1.48792e7i	1.86055i	0.366865	+	0.930274i	$0.380431\pi$
$577$	1.48792e7i	1.86055i	−0.366865	+	0.930274i	$0.619569\pi$
$578$	− 4.66650e6i	− 0.580993i
$579$	−1.65368e6	−0.205001
$580$	0	0
$581$	1.01933e7	1.25278
$582$	− 2.34150e6i	− 0.286541i
$583$	− 497044.i	− 0.0605652i
$584$	−2.86533e6	−0.347650
$585$	0	0
$586$	−9.80413e6	−1.17941
$587$	1.22649e7i	1.46916i	0.678525	+	0.734578i	$0.262620\pi$
$587$	1.22649e7i	1.46916i	−0.678525	+	0.734578i	$0.737380\pi$
$588$	1.79462e6i	0.214057i
$589$	9.83360e6	1.16795
$590$	0	0
$591$	3.22931e6	0.380313
$592$	5.30056e6i	0.621609i
$593$	1.54878e7i	1.80864i	0.426856	+	0.904320i	$0.359621\pi$
$593$	1.54878e7i	1.80864i	−0.426856	+	0.904320i	$0.640379\pi$
$594$	1.07190e7	1.24649
$595$	0	0
$596$	343602.	0.0396223
$597$	− 5.12304e6i	− 0.588291i
$598$	− 593190.i	− 0.0678330i
$599$	−9.75710e6	−1.11110	−0.555551	−	0.831483i	$-0.687493\pi$
$599$	−9.75710e6	−1.11110	−0.555551	+	0.831483i	$0.687493\pi$
$600$	0	0
$601$	−7.57967e6	−0.855981	−0.427990	−	0.903783i	$-0.640778\pi$
$601$	−7.57967e6	−0.855981	−0.427990	+	0.903783i	$0.640778\pi$
$602$	− 9.29396e6i	− 1.04522i
$603$	− 2.57425e6i	− 0.288309i
$604$	−2.50599e6	−0.279503
$605$	0	0
$606$	−690300.	−0.0763583
$607$	1.36231e7i	1.50073i	0.661022	+	0.750367i	$0.270123\pi$
$607$	1.36231e7i	1.50073i	−0.661022	+	0.750367i	$0.729877\pi$
$608$	− 6.72441e6i	− 0.737726i
$609$	−7.37563e6	−0.805853
$610$	0	0
$611$	510380.	0.0553083
$612$	− 2.22277e6i	− 0.239892i
$613$	1.20366e7i	1.29376i	0.762592	+	0.646880i	$0.223927\pi$
$613$	1.20366e7i	1.29376i	−0.762592	+	0.646880i	$0.776073\pi$
$614$	8.96924e6	0.960140
$615$	0	0
$616$	3.77785e7	4.01137
$617$	8.55509e6i	0.904715i	0.891837	+	0.452358i	$0.149417\pi$
$617$	8.55509e6i	0.904715i	−0.891837	+	0.452358i	$0.850583\pi$
$618$	− 3.65118e6i	− 0.384558i
$619$	1.33018e7	1.39535	0.697675	−	0.716414i	$-0.254218\pi$
$619$	1.33018e7	1.39535	0.697675	+	0.716414i	$0.254218\pi$
$620$	0	0
$621$	1.89540e6	0.197230
$622$	1.20617e7i	1.25006i
$623$	1.94468e6i	0.200737i
$624$	−761514.	−0.0782918
$625$	0	0
$626$	−1.07718e7	−1.09863
$627$	1.28914e7i	1.30958i
$628$	318150.i	0.0321909i
$629$	−1.08270e7	−1.09114
$630$	0	0
$631$	9.16681e6	0.916526	0.458263	−	0.888817i	$-0.348472\pi$
$631$	9.16681e6	0.916526	0.458263	+	0.888817i	$0.348472\pi$
$632$	7.76178e6i	0.772981i
$633$	6.00674e6i	0.595840i
$634$	−1.29808e7	−1.28256
$635$	0	0
$636$	−26292.0	−0.00257739
$637$	− 7.22120e6i	− 0.705116i
$638$	2.00009e7i	1.94535i
$639$	979938.	0.0949394
$640$	0	0
$641$	9.96437e6	0.957866	0.478933	−	0.877851i	$-0.341024\pi$
$641$	9.96437e6	0.957866	0.478933	+	0.877851i	$0.341024\pi$
$642$	− 2.10426e6i	− 0.201494i
$643$	− 6.64194e6i	− 0.633530i	−0.948504	−	0.316765i	$-0.897403\pi$
$643$	− 6.64194e6i	− 0.633530i	0.948504	−	0.316765i	$-0.102597\pi$
$644$	1.19902e6	0.113923
$645$	0	0
$646$	−2.07550e7	−1.95678
$647$	844766.i	0.0793370i	0.999213	+	0.0396685i	$0.0126302\pi$
$647$	844766.i	0.0793370i	−0.999213	+	0.0396685i	$0.987370\pi$
$648$	6.64969e6i	0.622106i
$649$	2.38724e7	2.22477
$650$	0	0
$651$	5.32018e6	0.492010
$652$	− 41244.0i	− 0.00379963i
$653$	− 5.79681e6i	− 0.531993i	−0.963974	−	0.265997i	$-0.914299\pi$
$653$	− 5.79681e6i	− 0.531993i	0.963974	−	0.265997i	$-0.0857010\pi$
$654$	5.87634e6	0.537233
$655$	0	0
$656$	220794.	0.0200322
$657$	3.04166e6i	0.274914i
$658$	− 3.68440e6i	− 0.331743i
$659$	1.12406e7	1.00827	0.504136	−	0.863624i	$-0.331811\pi$
$659$	1.12406e7	1.00827	0.504136	+	0.863624i	$0.331811\pi$
$660$	0	0
$661$	−1.54928e7	−1.37920	−0.689599	−	0.724191i	$-0.742213\pi$
$661$	−1.54928e7	−1.37920	−0.689599	+	0.724191i	$0.742213\pi$
$662$	− 4.58613e6i	− 0.406725i
$663$	− 1.55548e6i	− 0.137429i
$664$	−8.14632e6	−0.717037
$665$	0	0
$666$	7.30503e6	0.638170
$667$	3.53668e6i	0.307809i
$668$	1.48940e6i	0.129143i
$669$	−128184.	−0.0110731
$670$	0	0
$671$	−4.60996e6	−0.395268
$672$	− 3.63804e6i	− 0.310774i
$673$	723294.i	0.0615570i	0.999526	+	0.0307785i	$0.00979864\pi$
$673$	723294.i	0.0615570i	−0.999526	+	0.0307785i	$0.990201\pi$
$674$	−1.11947e7	−0.949210
$675$	0	0
$676$	−199927.	−0.0168269
$677$	− 7.57359e6i	− 0.635082i	−0.948244	−	0.317541i	$-0.897143\pi$
$677$	− 7.57359e6i	− 0.635082i	0.948244	−	0.317541i	$-0.102857\pi$
$678$	3.00714e6i	0.251235i
$679$	−1.90442e7	−1.58522
$680$	0	0
$681$	−5.28449e6	−0.436652
$682$	− 1.44270e7i	− 1.18772i
$683$	1.65552e7i	1.35794i	0.734164	+	0.678972i	$0.237574\pi$
$683$	1.65552e7i	1.35794i	−0.734164	+	0.678972i	$0.762426\pi$
$684$	−3.92099e6	−0.320447
$685$	0	0
$686$	−3.16248e7	−2.56577
$687$	− 78180.0i	− 0.00631981i
$688$	5.72112e6i	0.460797i
$689$	105794.	0.00849010
$690$	0	0
$691$	−2.04593e7	−1.63003	−0.815016	−	0.579438i	$-0.803272\pi$
$691$	−2.04593e7	−1.63003	−0.815016	+	0.579438i	$0.803272\pi$
$692$	− 3.52225e6i	− 0.279611i
$693$	− 4.01034e7i	− 3.17211i
$694$	−1.70904e7	−1.34696
$695$	0	0
$696$	5.89446e6	0.461234
$697$	450996.i	0.0351634i
$698$	− 1.17846e7i	− 0.915535i
$699$	7.24199e6	0.560615
$700$	0	0
$701$	1.52050e7	1.16867	0.584334	−	0.811514i	$-0.301356\pi$
$701$	1.52050e7	1.16867	0.584334	+	0.811514i	$0.301356\pi$
$702$	2.28150e6i	0.174734i
$703$	1.90989e7i	1.45754i
$704$	−2.89469e7	−2.20125
$705$	0	0
$706$	−1.88198e7	−1.42103
$707$	5.61444e6i	0.422433i
$708$	− 1.26277e6i	− 0.0946764i
$709$	1.80833e7	1.35102	0.675509	−	0.737351i	$-0.263924\pi$
$709$	1.80833e7	1.35102	0.675509	+	0.737351i	$0.263924\pi$
$710$	0	0
$711$	8.23943e6	0.611256
$712$	− 1.55415e6i	− 0.114893i
$713$	− 2.55107e6i	− 0.187931i
$714$	−1.12289e7	−0.824311
$715$	0	0
$716$	−4.07207e6	−0.296847
$717$	− 1.12223e6i	− 0.0815236i
$718$	− 1.64108e7i	− 1.18801i
$719$	2.08096e7	1.50121	0.750604	−	0.660752i	$-0.229763\pi$
$719$	2.08096e7	1.50121	0.750604	+	0.660752i	$0.229763\pi$
$720$	0	0
$721$	−2.96963e7	−2.12747
$722$	2.42317e7i	1.72998i
$723$	− 1.63014e6i	− 0.115979i
$724$	1.41541e6	0.100355
$725$	0	0
$726$	1.40816e7	0.991537
$727$	− 2.59006e7i	− 1.81750i	−0.417344	−	0.908749i	$-0.637039\pi$
$727$	− 2.59006e7i	− 1.81750i	0.417344	−	0.908749i	$-0.362961\pi$
$728$	8.04102e6i	0.562319i
$729$	3.12231e6	0.217599
$730$	0	0
$731$	−1.16860e7	−0.808859
$732$	243852.i	0.0168209i
$733$	1.96307e7i	1.34951i	0.738043	+	0.674754i	$0.235750\pi$
$733$	1.96307e7i	1.34951i	−0.738043	+	0.674754i	$0.764250\pi$
$734$	1.21303e7	0.831056
$735$	0	0
$736$	−1.74447e6	−0.118705
$737$	− 9.87418e6i	− 0.669626i
$738$	− 304290.i	− 0.0205659i
$739$	1.67436e7	1.12781	0.563906	−	0.825839i	$-0.309298\pi$
$739$	1.67436e7	1.12781	0.563906	+	0.825839i	$0.309298\pi$
$740$	0	0
$741$	−2.74388e6	−0.183578
$742$	− 763720.i	− 0.0509242i
$743$	− 5.57725e6i	− 0.370637i	−0.982679	−	0.185318i	$-0.940668\pi$
$743$	− 5.57725e6i	− 0.370637i	0.982679	−	0.185318i	$-0.0593316\pi$
$744$	−4.25178e6	−0.281604
$745$	0	0
$746$	1.40318e7	0.923135
$747$	8.64763e6i	0.567016i
$748$	− 8.52597e6i	− 0.557173i
$749$	−1.71146e7	−1.11471
$750$	0	0
$751$	1.24035e7	0.802499	0.401250	−	0.915969i	$-0.368576\pi$
$751$	1.24035e7	0.802499	0.401250	+	0.915969i	$0.368576\pi$
$752$	2.26802e6i	0.146252i
$753$	− 615888.i	− 0.0395835i
$754$	−4.25711e6	−0.272701
$755$	0	0
$756$	−4.61160e6	−0.293459
$757$	4.37170e6i	0.277275i	0.990343	+	0.138637i	$0.0442723\pi$
$757$	4.37170e6i	0.277275i	−0.990343	+	0.138637i	$0.955728\pi$
$758$	− 1.57696e7i	− 0.996892i
$759$	3.34433e6	0.210719
$760$	0	0
$761$	−2.10490e7	−1.31756	−0.658780	−	0.752335i	$-0.728927\pi$
$761$	−2.10490e7	−1.31756	−0.658780	+	0.752335i	$0.728927\pi$
$762$	1.17858e6i	0.0735312i
$763$	− 4.77942e7i	− 2.97210i
$764$	−2.38426e6	−0.147781
$765$	0	0
$766$	−2.37522e6	−0.146262
$767$	5.08115e6i	0.311870i
$768$	3.91679e6i	0.239623i
$769$	−2.26551e7	−1.38150	−0.690748	−	0.723096i	$-0.742718\pi$
$769$	−2.26551e7	−1.38150	−0.690748	+	0.723096i	$0.742718\pi$
$770$	0	0
$771$	−1.32709e6	−0.0804017
$772$	− 1.92930e6i	− 0.116508i
$773$	− 1.15053e7i	− 0.692545i	−0.938134	−	0.346272i	$-0.887447\pi$
$773$	− 1.15053e7i	− 0.692545i	0.938134	−	0.346272i	$-0.112553\pi$
$774$	7.88463e6	0.473074
$775$	0	0
$776$	1.52198e7	0.907305
$777$	1.03329e7i	0.614003i
$778$	− 752830.i	− 0.0445911i
$779$	795564.	0.0469712
$780$	0	0
$781$	3.75880e6	0.220506
$782$	5.38434e6i	0.314859i
$783$	− 1.36026e7i	− 0.792898i
$784$	3.20895e7	1.86454
$785$	0	0
$786$	6.34380e6	0.366263
$787$	967112.i	0.0556596i	0.999613	+	0.0278298i	$0.00885964\pi$
$787$	967112.i	0.0556596i	−0.999613	+	0.0278298i	$0.991140\pi$
$788$	3.76753e6i	0.216143i
$789$	−8.41904e6	−0.481471
$790$	0	0
$791$	2.44581e7	1.38989
$792$	3.20498e7i	1.81557i
$793$	− 981214.i	− 0.0554091i
$794$	−1.20843e6	−0.0680253
$795$	0	0
$796$	5.97688e6	0.334343
$797$	− 2.85072e7i	− 1.58968i	−0.606821	−	0.794838i	$-0.707556\pi$
$797$	− 2.85072e7i	− 1.58968i	0.606821	−	0.794838i	$-0.292444\pi$
$798$	1.98079e7i	1.10111i
$799$	−4.63268e6	−0.256723
$800$	0	0
$801$	−1.64979e6	−0.0908547
$802$	− 1.59840e7i	− 0.877502i
$803$	1.16670e7i	0.638516i
$804$	−522312.	−0.0284964
$805$	0	0
$806$	3.07073e6	0.166496
$807$	− 3.49772e6i	− 0.189061i
$808$	− 4.48695e6i	− 0.241781i
$809$	−1.08912e7	−0.585065	−0.292533	−	0.956256i	$-0.594498\pi$
$809$	−1.08912e7	−0.585065	−0.292533	+	0.956256i	$0.594498\pi$
$810$	0	0
$811$	1.28535e7	0.686228	0.343114	−	0.939294i	$-0.388518\pi$
$811$	1.28535e7	0.686228	0.343114	+	0.939294i	$0.388518\pi$
$812$	− 8.60490e6i	− 0.457990i
$813$	6.28141e6i	0.333297i
$814$	2.80203e7	1.48221
$815$	0	0
$816$	6.91220e6	0.363405
$817$	2.06143e7i	1.08047i
$818$	− 2.11641e6i	− 0.110590i
$819$	8.53585e6	0.444669
$820$	0	0
$821$	−9.60605e6	−0.497378	−0.248689	−	0.968583i	$-0.580000\pi$
$821$	−9.60605e6	−0.497378	−0.248689	+	0.968583i	$0.580000\pi$
$822$	789060.i	0.0407315i
$823$	− 1.42909e7i	− 0.735460i	−0.929933	−	0.367730i	$-0.880135\pi$
$823$	− 1.42909e7i	− 0.735460i	0.929933	−	0.367730i	$-0.119865\pi$
$824$	2.37327e7	1.21767
$825$	0	0
$826$	3.66805e7	1.87062
$827$	2.40317e7i	1.22186i	0.791685	+	0.610930i	$0.209204\pi$
$827$	2.40317e7i	1.22186i	−0.791685	+	0.610930i	$0.790796\pi$
$828$	1.01720e6i	0.0515620i
$829$	−1.10830e7	−0.560107	−0.280053	−	0.959984i	$-0.590352\pi$
$829$	−1.10830e7	−0.560107	−0.280053	+	0.959984i	$0.590352\pi$
$830$	0	0
$831$	6.62766e6	0.332934
$832$	− 6.16123e6i	− 0.308574i
$833$	6.55463e7i	3.27292i
$834$	−40320.0	−0.00200727
$835$	0	0
$836$	−1.50399e7	−0.744270
$837$	9.81180e6i	0.484100i
$838$	5.65796e6i	0.278323i
$839$	6.89303e6	0.338069	0.169034	−	0.985610i	$-0.445935\pi$
$839$	6.89303e6	0.338069	0.169034	+	0.985610i	$0.445935\pi$
$840$	0	0
$841$	4.87030e6	0.237446
$842$	1.73849e7i	0.845070i
$843$	− 5.45296e6i	− 0.264279i
$844$	−7.00787e6	−0.338633
$845$	0	0
$846$	3.12570e6	0.150149
$847$	− 1.14530e8i	− 5.48543i
$848$	470126.i	0.0224504i
$849$	2.69552e6	0.128344
$850$	0	0
$851$	4.95472e6	0.234528
$852$	− 198828.i	− 0.00938380i
$853$	683466.i	0.0321621i	0.999871	+	0.0160810i	$0.00511898\pi$
$853$	683466.i	0.0321621i	−0.999871	+	0.0160810i	$0.994881\pi$
$854$	−7.08332e6	−0.332347
$855$	0	0
$856$	1.36777e7	0.638011
$857$	− 7.89742e6i	− 0.367310i	−0.982991	−	0.183655i	$-0.941207\pi$
$857$	− 7.89742e6i	− 0.367310i	0.982991	−	0.183655i	$-0.0587930\pi$
$858$	4.02558e6i	0.186685i
$859$	−3.52556e7	−1.63021	−0.815107	−	0.579310i	$-0.803322\pi$
$859$	−3.52556e7	−1.63021	−0.815107	+	0.579310i	$0.803322\pi$
$860$	0	0
$861$	430416.	0.0197870
$862$	1.70522e7i	0.781649i
$863$	− 1.76565e7i	− 0.807007i	−0.914978	−	0.403503i	$-0.867792\pi$
$863$	− 1.76565e7i	− 0.807007i	0.914978	−	0.403503i	$-0.132208\pi$
$864$	6.70950e6	0.305778
$865$	0	0
$866$	−1.70361e7	−0.771926
$867$	5.59979e6i	0.253002i
$868$	6.20687e6i	0.279623i
$869$	3.16044e7	1.41970
$870$	0	0
$871$	2.10168e6	0.0938690
$872$	3.81962e7i	1.70110i
$873$	− 1.61564e7i	− 0.717476i
$874$	9.49806e6	0.420587
$875$	0	0
$876$	617148.	0.0271725
$877$	− 6.40016e6i	− 0.280991i	−0.990081	−	0.140495i	$-0.955131\pi$
$877$	− 6.40016e6i	− 0.280991i	0.990081	−	0.140495i	$-0.0448695\pi$
$878$	3.54557e7i	1.55221i
$879$	1.17650e7	0.513592
$880$	0	0
$881$	−1.14571e7	−0.497318	−0.248659	−	0.968591i	$-0.579990\pi$
$881$	−1.14571e7	−0.497318	−0.248659	+	0.968591i	$0.579990\pi$
$882$	− 4.42245e7i	− 1.91422i
$883$	− 2.42296e7i	− 1.04579i	−0.852397	−	0.522896i	$-0.824852\pi$
$883$	− 2.42296e7i	− 1.04579i	0.852397	−	0.522896i	$-0.175148\pi$
$884$	1.81472e6	0.0781051
$885$	0	0
$886$	−4.11754e7	−1.76219
$887$	8.66087e6i	0.369617i	0.982775	+	0.184809i	$0.0591665\pi$
$887$	8.66087e6i	0.369617i	−0.982775	+	0.184809i	$0.940833\pi$
$888$	− 8.25786e6i	− 0.351427i
$889$	9.58578e6	0.406793
$890$	0	0
$891$	2.70762e7	1.14260
$892$	− 149548.i	− 0.00629316i
$893$	8.17212e6i	0.342930i
$894$	1.47258e6	0.0616219
$895$	0	0
$896$	−2.50747e7	−1.04343
$897$	711828.i	0.0295389i
$898$	6.48003e6i	0.268155i
$899$	−1.83081e7	−0.755516
$900$	0	0
$901$	−960284.	−0.0394083
$902$	− 1.16718e6i	− 0.0477663i
$903$	1.11528e7i	0.455159i
$904$	−1.95464e7	−0.795511
$905$	0	0
$906$	−1.07399e7	−0.434692
$907$	− 7.84287e6i	− 0.316561i	−0.987394	−	0.158280i	$-0.949405\pi$
$907$	− 7.84287e6i	− 0.316561i	0.987394	−	0.158280i	$-0.0505950\pi$
$908$	− 6.16524e6i	− 0.248162i
$909$	−4.76307e6	−0.191195
$910$	0	0
$911$	−942576.	−0.0376288	−0.0188144	−	0.999823i	$-0.505989\pi$
$911$	−942576.	−0.0376288	−0.0188144	+	0.999823i	$0.505989\pi$
$912$	− 1.21932e7i	− 0.485436i
$913$	3.31701e7i	1.31695i
$914$	−8.40979e6	−0.332981
$915$	0	0
$916$	91210.0	0.00359173
$917$	− 5.15962e7i	− 2.02626i
$918$	− 2.07090e7i	− 0.811059i
$919$	2.00734e7	0.784030	0.392015	−	0.919959i	$-0.371778\pi$
$919$	2.00734e7	0.784030	0.392015	+	0.919959i	$0.371778\pi$
$920$	0	0
$921$	−1.07631e7	−0.418107
$922$	− 1.60332e7i	− 0.621143i
$923$	800046.i	0.0309108i
$924$	−8.13691e6	−0.313530
$925$	0	0
$926$	−2.63185e7	−1.00863
$927$	− 2.51931e7i	− 0.962904i
$928$	1.25194e7i	0.477216i
$929$	−1.10181e7	−0.418858	−0.209429	−	0.977824i	$-0.567160\pi$
$929$	−1.10181e7	−0.418858	−0.209429	+	0.977824i	$0.567160\pi$
$930$	0	0
$931$	1.15625e8	4.37196
$932$	8.44899e6i	0.318614i
$933$	− 1.44740e7i	− 0.544358i
$934$	4.13406e7	1.55064
$935$	0	0
$936$	−6.82168e6	−0.254508
$937$	3.59532e7i	1.33779i	0.743356	+	0.668896i	$0.233233\pi$
$937$	3.59532e7i	1.33779i	−0.743356	+	0.668896i	$0.766767\pi$
$938$	− 1.51719e7i	− 0.563032i
$939$	1.29261e7	0.478415
$940$	0	0
$941$	1.28845e7	0.474345	0.237172	−	0.971468i	$-0.423779\pi$
$941$	1.28845e7	0.474345	0.237172	+	0.971468i	$0.423779\pi$
$942$	1.36350e6i	0.0500643i
$943$	− 206388.i	− 0.00755797i
$944$	−2.25796e7	−0.824680
$945$	0	0
$946$	3.02435e7	1.09876
$947$	− 1.18911e7i	− 0.430871i	−0.976518	−	0.215436i	$-0.930883\pi$
$947$	− 1.18911e7i	− 0.430871i	0.976518	−	0.215436i	$-0.0691171\pi$
$948$	− 1.67177e6i	− 0.0604164i
$949$	−2.48329e6	−0.0895079
$950$	0	0
$951$	1.55769e7	0.558510
$952$	− 7.29877e7i	− 2.61010i
$953$	− 4.40094e7i	− 1.56969i	−0.619694	−	0.784844i	$-0.712743\pi$
$953$	− 4.40094e7i	− 1.56969i	0.619694	−	0.784844i	$-0.287257\pi$
$954$	647910.	0.0230486
$955$	0	0
$956$	1.30927e6	0.0463322
$957$	− 2.40010e7i	− 0.847130i
$958$	− 1.82734e7i	− 0.643288i
$959$	6.41769e6	0.225337
$960$	0	0
$961$	−1.54232e7	−0.538723
$962$	5.96401e6i	0.207779i
$963$	− 1.45194e7i	− 0.504525i
$964$	1.90183e6	0.0659142
$965$	0	0
$966$	5.13864e6	0.177176
$967$	2.11144e7i	0.726128i	0.931764	+	0.363064i	$0.118269\pi$
$967$	2.11144e7i	0.726128i	−0.931764	+	0.363064i	$0.881731\pi$
$968$	9.15301e7i	3.13961i
$969$	2.49060e7	0.852109
$970$	0	0
$971$	2.44293e7	0.831502	0.415751	−	0.909478i	$-0.363519\pi$
$971$	2.44293e7	0.831502	0.415751	+	0.909478i	$0.363519\pi$
$972$	− 6.02494e6i	− 0.204544i
$973$	327936.i	0.0111047i
$974$	−3.56542e7	−1.20424
$975$	0	0
$976$	4.36031e6	0.146518
$977$	5.15549e7i	1.72796i	0.503527	+	0.863980i	$0.332036\pi$
$977$	5.15549e7i	1.72796i	−0.503527	+	0.863980i	$0.667964\pi$
$978$	− 176760.i	− 0.00590931i
$979$	−6.32818e6	−0.211019
$980$	0	0
$981$	4.05467e7	1.34519
$982$	2.86276e7i	0.947339i
$983$	1.38938e7i	0.458604i	0.973355	+	0.229302i	$0.0736444\pi$
$983$	1.38938e7i	0.458604i	−0.973355	+	0.229302i	$0.926356\pi$
$984$	−343980.	−0.0113252
$985$	0	0
$986$	3.86415e7	1.26579
$987$	4.42128e6i	0.144463i
$988$	− 3.20120e6i	− 0.104333i
$989$	5.34784e6	0.173855
$990$	0	0
$991$	3.31496e7	1.07225	0.536123	−	0.844140i	$-0.319888\pi$
$991$	3.31496e7	1.07225	0.536123	+	0.844140i	$0.319888\pi$
$992$	− 9.03049e6i	− 0.291361i
$993$	5.50336e6i	0.177115i
$994$	5.77548e6	0.185405
$995$	0	0
$996$	1.75459e6	0.0560438
$997$	9.45871e6i	0.301366i	0.988582	+	0.150683i	$0.0481473\pi$
$997$	9.45871e6i	0.301366i	−0.988582	+	0.150683i	$0.951853\pi$
$998$	3.58626e7i	1.13976i
$999$	−1.90566e7	−0.604132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.a.274.2		2
5.2	odd	4		325.6.a.a.1.1		1
5.3	odd	4		65.6.a.a.1.1	✓	1
5.4	even	2	inner	325.6.b.a.274.1		2
15.8	even	4		585.6.a.a.1.1		1
20.3	even	4		1040.6.a.a.1.1		1
65.38	odd	4		845.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.a.1.1	✓	1	5.3	odd	4
325.6.a.a.1.1		1	5.2	odd	4
325.6.b.a.274.1		2	5.4	even	2	inner
325.6.b.a.274.2		2	1.1	even	1	trivial
585.6.a.a.1.1		1	15.8	even	4
845.6.a.a.1.1		1	65.38	odd	4
1040.6.a.a.1.1		1	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.00000i q^{2} -6.00000i q^{3} +7.00000 q^{4} +30.0000 q^{6} -244.000i q^{7} +195.000i q^{8} +207.000 q^{9} +794.000 q^{11} -42.0000i q^{12} +169.000i q^{13} +1220.00 q^{14} -751.000 q^{16} -1534.00i q^{17} +1035.00i q^{18} -2706.00 q^{19} -1464.00 q^{21} +3970.00i q^{22} +702.000i q^{23} +1170.00 q^{24} -845.000 q^{26} -2700.00i q^{27} -1708.00i q^{28} +5038.00 q^{29} -3634.00 q^{31} +2485.00i q^{32} -4764.00i q^{33} +7670.00 q^{34} +1449.00 q^{36} -7058.00i q^{37} -13530.0i q^{38} +1014.00 q^{39} -294.000 q^{41} -7320.00i q^{42} -7618.00i q^{43} +5558.00 q^{44} -3510.00 q^{46} -3020.00i q^{47} +4506.00i q^{48} -42729.0 q^{49} -9204.00 q^{51} +1183.00i q^{52} -626.000i q^{53} +13500.0 q^{54} +47580.0 q^{56} +16236.0i q^{57} +25190.0i q^{58} +30066.0 q^{59} -5806.00 q^{61} -18170.0i q^{62} -50508.0i q^{63} -36457.0 q^{64} +23820.0 q^{66} -12436.0i q^{67} -10738.0i q^{68} +4212.00 q^{69} +4734.00 q^{71} +40365.0i q^{72} +14694.0i q^{73} +35290.0 q^{74} -18942.0 q^{76} -193736. i q^{77} +5070.00i q^{78} +39804.0 q^{79} +34101.0 q^{81} -1470.00i q^{82} +41776.0i q^{83} -10248.0 q^{84} +38090.0 q^{86} -30228.0i q^{87} +154830. i q^{88} -7970.00 q^{89} +41236.0 q^{91} +4914.00i q^{92} +21804.0i q^{93} +15100.0 q^{94} +14910.0 q^{96} -78050.0i q^{97} -213645. i q^{98} +164358. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{4} + 60 q^{6} + 414 q^{9} + 1588 q^{11} + 2440 q^{14} - 1502 q^{16} - 5412 q^{19} - 2928 q^{21} + 2340 q^{24} - 1690 q^{26} + 10076 q^{29} - 7268 q^{31} + 15340 q^{34} + 2898 q^{36} + 2028 q^{39}+ \cdots + 328716 q^{99}+O(q^{100}) \) 2 * q + 14 * q^4 + 60 * q^6 + 414 * q^9 + 1588 * q^11 + 2440 * q^14 - 1502 * q^16 - 5412 * q^19 - 2928 * q^21 + 2340 * q^24 - 1690 * q^26 + 10076 * q^29 - 7268 * q^31 + 15340 * q^34 + 2898 * q^36 + 2028 * q^39 - 588 * q^41 + 11116 * q^44 - 7020 * q^46 - 85458 * q^49 - 18408 * q^51 + 27000 * q^54 + 95160 * q^56 + 60132 * q^59 - 11612 * q^61 - 72914 * q^64 + 47640 * q^66 + 8424 * q^69 + 9468 * q^71 + 70580 * q^74 - 37884 * q^76 + 79608 * q^79 + 68202 * q^81 - 20496 * q^84 + 76180 * q^86 - 15940 * q^89 + 82472 * q^91 + 30200 * q^94 + 29820 * q^96 + 328716 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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