LMFDB - Embedded newform 525.4.d.h.274.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,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, 4, names="a")

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
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:	$30.9760027530$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{65})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 33x^{2} + 256 $ x^4 + 33*x^2 + 256
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.1
Root			$-4.53113i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.4.d.h.274.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.53113i q^{2} +3.00000i q^{3} -12.5311 q^{4} +13.5934 q^{6} +7.00000i q^{7} +20.5311i q^{8} -9.00000 q^{9} -19.0623 q^{11} -37.5934i q^{12} -2.93774i q^{13} +31.7179 q^{14} -7.21984 q^{16} +6.49806i q^{17} +40.7802i q^{18} +5.43580 q^{19} -21.0000 q^{21} +86.3735i q^{22} +49.3774i q^{23} -61.5934 q^{24} -13.3113 q^{26} -27.0000i q^{27} -87.7179i q^{28} +291.494 q^{29} +244.307 q^{31} +196.963i q^{32} -57.1868i q^{33} +29.4436 q^{34} +112.780 q^{36} +193.121i q^{37} -24.6303i q^{38} +8.81323 q^{39} +315.113 q^{41} +95.1537i q^{42} -300.996i q^{43} +238.872 q^{44} +223.735 q^{46} -86.5058i q^{47} -21.6595i q^{48} -49.0000 q^{49} -19.4942 q^{51} +36.8132i q^{52} +509.677i q^{53} -122.340 q^{54} -143.718 q^{56} +16.3074i q^{57} -1320.80i q^{58} +83.3852 q^{59} -5.25291 q^{61} -1106.99i q^{62} -63.0000i q^{63} +834.706 q^{64} -259.121 q^{66} -205.992i q^{67} -81.4281i q^{68} -148.132 q^{69} +1004.31 q^{71} -184.780i q^{72} -1007.29i q^{73} +875.055 q^{74} -68.1168 q^{76} -133.436i q^{77} -39.9339i q^{78} +863.237 q^{79} +81.0000 q^{81} -1427.82i q^{82} +1334.72i q^{83} +263.154 q^{84} -1363.85 q^{86} +874.483i q^{87} -391.370i q^{88} -326.249 q^{89} +20.5642 q^{91} -618.755i q^{92} +732.922i q^{93} -391.969 q^{94} -590.889 q^{96} -1526.77i q^{97} +222.025i q^{98} +171.560 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 34 q^{4} + 6 q^{6} - 36 q^{9} - 44 q^{11} + 14 q^{14} - 174 q^{16} - 204 q^{19} - 84 q^{21} - 198 q^{24} + 108 q^{26} + 392 q^{29} + 300 q^{31} + 924 q^{34} + 306 q^{36} + 132 q^{39} - 352 q^{41}+ \cdots + 396 q^{99}+O(q^{100}) $ 4 * q - 34 * q^4 + 6 * q^6 - 36 * q^9 - 44 * q^11 + 14 * q^14 - 174 * q^16 - 204 * q^19 - 84 * q^21 - 198 * q^24 + 108 * q^26 + 392 * q^29 + 300 * q^31 + 924 * q^34 + 306 * q^36 + 132 * q^39 - 352 * q^41 + 504 * q^44 - 1040 * q^46 - 196 * q^49 + 696 * q^51 - 54 * q^54 - 462 * q^56 + 1688 * q^59 - 408 * q^61 + 1678 * q^64 - 456 * q^66 - 1560 * q^69 + 3340 * q^71 + 2436 * q^74 + 824 * q^76 + 1776 * q^79 + 324 * q^81 + 714 * q^84 - 2424 * q^86 - 1176 * q^89 + 308 * q^91 + 2560 * q^94 - 90 * q^96 + 396 * q^99

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 $

$2$	− 4.53113i	− 1.60200i	−0.598667	−	0.800998i	$-0.704303\pi$
$2$	− 4.53113i	− 1.60200i	0.598667	−	0.800998i	$-0.295697\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	−12.5311	−1.56639
$5$	0	0
$5$	0	0
$6$	13.5934	0.924913
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	20.5311i	0.907356i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−19.0623	−0.522499	−0.261249	−	0.965271i	$-0.584135\pi$
$11$	−19.0623	−0.522499	−0.261249	+	0.965271i	$0.584135\pi$
$12$	− 37.5934i	− 0.904356i
$13$	− 2.93774i	− 0.0626756i	−0.999509	−	0.0313378i	$-0.990023\pi$
$13$	− 2.93774i	− 0.0626756i	0.999509	−	0.0313378i	$-0.00997677\pi$
$14$	31.7179	0.605498
$15$	0	0
$16$	−7.21984	−0.112810
$17$	6.49806i	0.0927066i	0.998925	+	0.0463533i	$0.0147600\pi$
$17$	6.49806i	0.0927066i	−0.998925	+	0.0463533i	$0.985240\pi$
$18$	40.7802i	0.533999i
$19$	5.43580	0.0656347	0.0328173	−	0.999461i	$-0.489552\pi$
$19$	5.43580	0.0656347	0.0328173	+	0.999461i	$0.489552\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	86.3735i	0.837041i
$23$	49.3774i	0.447648i	0.974630	+	0.223824i	$0.0718541\pi$
$23$	49.3774i	0.447648i	−0.974630	+	0.223824i	$0.928146\pi$
$24$	−61.5934	−0.523862
$25$	0	0
$26$	−13.3113	−0.100406
$27$	− 27.0000i	− 0.192450i
$28$	− 87.7179i	− 0.592040i
$29$	291.494	1.86652	0.933261	−	0.359200i	$-0.116950\pi$
$29$	291.494	1.86652	0.933261	+	0.359200i	$0.116950\pi$
$30$	0	0
$31$	244.307	1.41545	0.707724	−	0.706489i	$-0.249722\pi$
$31$	244.307	1.41545	0.707724	+	0.706489i	$0.249722\pi$
$32$	196.963i	1.08808i
$33$	− 57.1868i	− 0.301665i
$34$	29.4436	0.148516
$35$	0	0
$36$	112.780	0.522130
$37$	193.121i	0.858077i	0.903286	+	0.429038i	$0.141147\pi$
$37$	193.121i	0.858077i	−0.903286	+	0.429038i	$0.858853\pi$
$38$	− 24.6303i	− 0.105147i
$39$	8.81323	0.0361858
$40$	0	0
$41$	315.113	1.20030	0.600151	−	0.799887i	$-0.295107\pi$
$41$	315.113	1.20030	0.600151	+	0.799887i	$0.295107\pi$
$42$	95.1537i	0.349584i
$43$	− 300.996i	− 1.06748i	−0.845650	−	0.533738i	$-0.820787\pi$
$43$	− 300.996i	− 1.06748i	0.845650	−	0.533738i	$-0.179213\pi$
$44$	238.872	0.818437
$45$	0	0
$46$	223.735	0.717130
$47$	− 86.5058i	− 0.268472i	−0.990949	−	0.134236i	$-0.957142\pi$
$47$	− 86.5058i	− 0.268472i	0.990949	−	0.134236i	$-0.0428580\pi$
$48$	− 21.6595i	− 0.0651309i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−19.4942	−0.0535242
$52$	36.8132i	0.0981745i
$53$	509.677i	1.32093i	0.750855	+	0.660467i	$0.229642\pi$
$53$	509.677i	1.32093i	−0.750855	+	0.660467i	$0.770358\pi$
$54$	−122.340	−0.308304
$55$	0	0
$56$	−143.718	−0.342948
$57$	16.3074i	0.0378942i
$58$	− 1320.80i	− 2.99016i
$59$	83.3852	0.183997	0.0919985	−	0.995759i	$-0.470674\pi$
$59$	83.3852	0.183997	0.0919985	+	0.995759i	$0.470674\pi$
$60$	0	0
$61$	−5.25291	−0.0110257	−0.00551283	−	0.999985i	$-0.501755\pi$
$61$	−5.25291	−0.0110257	−0.00551283	+	0.999985i	$0.501755\pi$
$62$	− 1106.99i	− 2.26754i
$63$	− 63.0000i	− 0.125988i
$64$	834.706	1.63029
$65$	0	0
$66$	−259.121	−0.483266
$67$	− 205.992i	− 0.375611i	−0.982206	−	0.187806i	$-0.939862\pi$
$67$	− 205.992i	− 0.375611i	0.982206	−	0.187806i	$-0.0601375\pi$
$68$	− 81.4281i	− 0.145215i
$69$	−148.132	−0.258450
$70$	0	0
$71$	1004.31	1.67872	0.839362	−	0.543573i	$-0.182929\pi$
$71$	1004.31	1.67872	0.839362	+	0.543573i	$0.182929\pi$
$72$	− 184.780i	− 0.302452i
$73$	− 1007.29i	− 1.61499i	−0.589876	−	0.807494i	$-0.700823\pi$
$73$	− 1007.29i	− 1.61499i	0.589876	−	0.807494i	$-0.299177\pi$
$74$	875.055	1.37464
$75$	0	0
$76$	−68.1168	−0.102810
$77$	− 133.436i	− 0.197486i
$78$	− 39.9339i	− 0.0579695i
$79$	863.237	1.22939	0.614695	−	0.788765i	$-0.289279\pi$
$79$	863.237	1.22939	0.614695	+	0.788765i	$0.289279\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 1427.82i	− 1.92288i
$83$	1334.72i	1.76512i	0.470200	+	0.882560i	$0.344182\pi$
$83$	1334.72i	1.76512i	−0.470200	+	0.882560i	$0.655818\pi$
$84$	263.154	0.341815
$85$	0	0
$86$	−1363.85	−1.71009
$87$	874.483i	1.07764i
$88$	− 391.370i	− 0.474093i
$89$	−326.249	−0.388565	−0.194283	−	0.980946i	$-0.562238\pi$
$89$	−326.249	−0.388565	−0.194283	+	0.980946i	$0.562238\pi$
$90$	0	0
$91$	20.5642	0.0236892
$92$	− 618.755i	− 0.701192i
$93$	732.922i	0.817210i
$94$	−391.969	−0.430091
$95$	0	0
$96$	−590.889	−0.628202
$97$	− 1526.77i	− 1.59815i	−0.601232	−	0.799075i	$-0.705323\pi$
$97$	− 1526.77i	− 1.59815i	0.601232	−	0.799075i	$-0.294677\pi$
$98$	222.025i	0.228857i
$99$	171.560	0.174166
$100$	0	0
$101$	96.8716	0.0954365	0.0477182	−	0.998861i	$-0.484805\pi$
$101$	96.8716	0.0954365	0.0477182	+	0.998861i	$0.484805\pi$
$102$	88.3307i	0.0857455i
$103$	1321.99i	1.26466i	0.774700	+	0.632329i	$0.217901\pi$
$103$	1321.99i	1.26466i	−0.774700	+	0.632329i	$0.782099\pi$
$104$	60.3152	0.0568691
$105$	0	0
$106$	2309.41	2.11613
$107$	− 1745.71i	− 1.57724i	−0.614883	−	0.788619i	$-0.710797\pi$
$107$	− 1745.71i	− 1.57724i	0.614883	−	0.788619i	$-0.289203\pi$
$108$	338.340i	0.301452i
$109$	−476.856	−0.419032	−0.209516	−	0.977805i	$-0.567189\pi$
$109$	−476.856	−0.419032	−0.209516	+	0.977805i	$0.567189\pi$
$110$	0	0
$111$	−579.362	−0.495411
$112$	− 50.5389i	− 0.0426382i
$113$	1641.65i	1.36666i	0.730108	+	0.683332i	$0.239470\pi$
$113$	1641.65i	1.36666i	−0.730108	+	0.683332i	$0.760530\pi$
$114$	73.8910	0.0607064
$115$	0	0
$116$	−3652.75	−2.92370
$117$	26.4397i	0.0208919i
$118$	− 377.829i	− 0.294763i
$119$	−45.4864	−0.0350398
$120$	0	0
$121$	−967.630	−0.726995
$122$	23.8016i	0.0176631i
$123$	945.339i	0.692994i
$124$	−3061.45	−2.21715
$125$	0	0
$126$	−285.461	−0.201833
$127$	844.016i	0.589719i	0.955541	+	0.294859i	$0.0952728\pi$
$127$	844.016i	0.589719i	−0.955541	+	0.294859i	$0.904727\pi$
$128$	− 2206.46i	− 1.52363i
$129$	902.988	0.616308
$130$	0	0
$131$	−2796.20	−1.86493	−0.932463	−	0.361265i	$-0.882345\pi$
$131$	−2796.20	−1.86493	−0.932463	+	0.361265i	$0.882345\pi$
$132$	716.615i	0.472525i
$133$	38.0506i	0.0248076i
$134$	−933.377	−0.601728
$135$	0	0
$136$	−133.413	−0.0841179
$137$	2057.13i	1.28287i	0.767179	+	0.641433i	$0.221660\pi$
$137$	2057.13i	1.28287i	−0.767179	+	0.641433i	$0.778340\pi$
$138$	671.206i	0.414035i
$139$	1745.12	1.06489	0.532444	−	0.846465i	$-0.321274\pi$
$139$	1745.12	1.06489	0.532444	+	0.846465i	$0.321274\pi$
$140$	0	0
$141$	259.517	0.155002
$142$	− 4550.65i	− 2.68931i
$143$	56.0000i	0.0327479i
$144$	64.9786	0.0376033
$145$	0	0
$146$	−4564.15	−2.58720
$147$	− 147.000i	− 0.0824786i
$148$	− 2420.02i	− 1.34408i
$149$	1173.57	0.645254	0.322627	−	0.946526i	$-0.395434\pi$
$149$	1173.57	0.645254	0.322627	+	0.946526i	$0.395434\pi$
$150$	0	0
$151$	1540.07	0.829994	0.414997	−	0.909823i	$-0.363783\pi$
$151$	1540.07	0.829994	0.414997	+	0.909823i	$0.363783\pi$
$152$	111.603i	0.0595540i
$153$	− 58.4826i	− 0.0309022i
$154$	−604.615	−0.316372
$155$	0	0
$156$	−110.440	−0.0566811
$157$	2544.53i	1.29348i	0.762712	+	0.646738i	$0.223867\pi$
$157$	2544.53i	1.29348i	−0.762712	+	0.646738i	$0.776133\pi$
$158$	− 3911.44i	− 1.96948i
$159$	−1529.03	−0.762642
$160$	0	0
$161$	−345.642	−0.169195
$162$	− 367.021i	− 0.178000i
$163$	− 594.708i	− 0.285774i	−0.989739	−	0.142887i	$-0.954361\pi$
$163$	− 594.708i	− 0.285774i	0.989739	−	0.142887i	$-0.0456385\pi$
$164$	−3948.72	−1.88014
$165$	0	0
$166$	6047.81	2.82772
$167$	− 928.498i	− 0.430236i	−0.976588	−	0.215118i	$-0.930986\pi$
$167$	− 928.498i	− 0.430236i	0.976588	−	0.215118i	$-0.0690135\pi$
$168$	− 431.154i	− 0.198001i
$169$	2188.37	0.996072
$170$	0	0
$171$	−48.9222	−0.0218782
$172$	3771.82i	1.67209i
$173$	− 315.642i	− 0.138716i	−0.997592	−	0.0693578i	$-0.977905\pi$
$173$	− 315.642i	− 0.138716i	0.997592	−	0.0693578i	$-0.0220950\pi$
$174$	3962.39	1.72637
$175$	0	0
$176$	137.626	0.0589431
$177$	250.156i	0.106231i
$178$	1478.28i	0.622480i
$179$	1445.49	0.603581	0.301791	−	0.953374i	$-0.402416\pi$
$179$	1445.49	0.603581	0.301791	+	0.953374i	$0.402416\pi$
$180$	0	0
$181$	−1843.81	−0.757180	−0.378590	−	0.925564i	$-0.623591\pi$
$181$	−1843.81	−0.757180	−0.378590	+	0.925564i	$0.623591\pi$
$182$	− 93.1790i	− 0.0379499i
$183$	− 15.7587i	− 0.00636567i
$184$	−1013.77	−0.406176
$185$	0	0
$186$	3320.97	1.30917
$187$	− 123.868i	− 0.0484391i
$188$	1084.02i	0.420532i
$189$	189.000	0.0727393
$190$	0	0
$191$	−244.074	−0.0924637	−0.0462318	−	0.998931i	$-0.514721\pi$
$191$	−244.074	−0.0924637	−0.0462318	+	0.998931i	$0.514721\pi$
$192$	2504.12i	0.941246i
$193$	− 1733.03i	− 0.646355i	−0.946338	−	0.323178i	$-0.895249\pi$
$193$	− 1733.03i	− 0.646355i	0.946338	−	0.323178i	$-0.104751\pi$
$194$	−6918.01	−2.56023
$195$	0	0
$196$	614.025	0.223770
$197$	− 358.230i	− 0.129557i	−0.997900	−	0.0647787i	$-0.979366\pi$
$197$	− 358.230i	− 0.129557i	0.997900	−	0.0647787i	$-0.0206342\pi$
$198$	− 777.362i	− 0.279014i
$199$	3203.63	1.14120	0.570601	−	0.821227i	$-0.306710\pi$
$199$	3203.63	1.14120	0.570601	+	0.821227i	$0.306710\pi$
$200$	0	0
$201$	617.977	0.216859
$202$	− 438.938i	− 0.152889i
$203$	2040.46i	0.705479i
$204$	244.284	0.0838398
$205$	0	0
$206$	5990.12	2.02598
$207$	− 444.397i	− 0.149216i
$208$	21.2100i	0.00707044i
$209$	−103.619	−0.0342940
$210$	0	0
$211$	4943.16	1.61280	0.806401	−	0.591369i	$-0.201412\pi$
$211$	4943.16	1.61280	0.806401	+	0.591369i	$0.201412\pi$
$212$	− 6386.83i	− 2.06910i
$213$	3012.92i	0.969211i
$214$	−7910.05	−2.52673
$215$	0	0
$216$	554.340	0.174621
$217$	1710.15i	0.534989i
$218$	2160.70i	0.671288i
$219$	3021.86	0.932414
$220$	0	0
$221$	19.0896	0.00581044
$222$	2625.16i	0.793646i
$223$	3160.15i	0.948965i	0.880265	+	0.474482i	$0.157365\pi$
$223$	3160.15i	0.948965i	−0.880265	+	0.474482i	$0.842635\pi$
$224$	−1378.74	−0.411255
$225$	0	0
$226$	7438.51	2.18939
$227$	3651.11i	1.06755i	0.845628	+	0.533773i	$0.179226\pi$
$227$	3651.11i	1.06755i	−0.845628	+	0.533773i	$0.820774\pi$
$228$	− 204.350i	− 0.0593571i
$229$	4083.70	1.17842	0.589210	−	0.807980i	$-0.299439\pi$
$229$	4083.70	1.17842	0.589210	+	0.807980i	$0.299439\pi$
$230$	0	0
$231$	400.307	0.114019
$232$	5984.70i	1.69360i
$233$	− 3682.51i	− 1.03540i	−0.855561	−	0.517702i	$-0.826788\pi$
$233$	− 3682.51i	− 1.03540i	0.855561	−	0.517702i	$-0.173212\pi$
$234$	119.802	0.0334687
$235$	0	0
$236$	−1044.91	−0.288211
$237$	2589.71i	0.709789i
$238$	206.105i	0.0561336i
$239$	−2658.78	−0.719591	−0.359796	−	0.933031i	$-0.617154\pi$
$239$	−2658.78	−0.719591	−0.359796	+	0.933031i	$0.617154\pi$
$240$	0	0
$241$	−4820.39	−1.28842	−0.644209	−	0.764850i	$-0.722813\pi$
$241$	−4820.39	−1.28842	−0.644209	+	0.764850i	$0.722813\pi$
$242$	4384.46i	1.16464i
$243$	243.000i	0.0641500i
$244$	65.8249	0.0172705
$245$	0	0
$246$	4283.45	1.11017
$247$	− 15.9690i	− 0.00411369i
$248$	5015.91i	1.28432i
$249$	−4004.17	−1.01909
$250$	0	0
$251$	1672.27	0.420530	0.210265	−	0.977644i	$-0.432567\pi$
$251$	1672.27	0.420530	0.210265	+	0.977644i	$0.432567\pi$
$252$	789.461i	0.197347i
$253$	− 941.245i	− 0.233896i
$254$	3824.34	0.944727
$255$	0	0
$256$	−3320.09	−0.810569
$257$	3697.74i	0.897506i	0.893656	+	0.448753i	$0.148132\pi$
$257$	3697.74i	0.897506i	−0.893656	+	0.448753i	$0.851868\pi$
$258$	− 4091.56i	− 0.987322i
$259$	−1351.84	−0.324323
$260$	0	0
$261$	−2623.45	−0.622174
$262$	12670.0i	2.98760i
$263$	7319.00i	1.71600i	0.513646	+	0.858002i	$0.328294\pi$
$263$	7319.00i	1.71600i	−0.513646	+	0.858002i	$0.671706\pi$
$264$	1174.11	0.273717
$265$	0	0
$266$	172.412	0.0397416
$267$	− 978.747i	− 0.224338i
$268$	2581.32i	0.588354i
$269$	815.097	0.184749	0.0923743	−	0.995724i	$-0.470554\pi$
$269$	815.097	0.184749	0.0923743	+	0.995724i	$0.470554\pi$
$270$	0	0
$271$	−5106.02	−1.14453	−0.572267	−	0.820068i	$-0.693936\pi$
$271$	−5106.02	−1.14453	−0.572267	+	0.820068i	$0.693936\pi$
$272$	− 46.9150i	− 0.0104582i
$273$	61.6926i	0.0136769i
$274$	9321.13	2.05515
$275$	0	0
$276$	1856.26	0.404833
$277$	− 1398.72i	− 0.303398i	−0.988427	−	0.151699i	$-0.951526\pi$
$277$	− 1398.72i	− 0.303398i	0.988427	−	0.151699i	$-0.0484744\pi$
$278$	− 7907.39i	− 1.70595i
$279$	−2198.77	−0.471816
$280$	0	0
$281$	−7102.38	−1.50780	−0.753901	−	0.656988i	$-0.771830\pi$
$281$	−7102.38	−1.50780	−0.753901	+	0.656988i	$0.771830\pi$
$282$	− 1175.91i	− 0.248313i
$283$	4465.18i	0.937907i	0.883223	+	0.468953i	$0.155369\pi$
$283$	4465.18i	0.937907i	−0.883223	+	0.468953i	$0.844631\pi$
$284$	−12585.1	−2.62954
$285$	0	0
$286$	253.743	0.0524621
$287$	2205.79i	0.453671i
$288$	− 1772.67i	− 0.362692i
$289$	4870.78	0.991405
$290$	0	0
$291$	4580.32	0.922692
$292$	12622.5i	2.52970i
$293$	7590.61i	1.51348i	0.653718	+	0.756738i	$0.273208\pi$
$293$	7590.61i	1.51348i	−0.653718	+	0.756738i	$0.726792\pi$
$294$	−666.076	−0.132130
$295$	0	0
$296$	−3964.98	−0.778581
$297$	514.681i	0.100555i
$298$	− 5317.61i	− 1.03369i
$299$	145.058	0.0280566
$300$	0	0
$301$	2106.97	0.403468
$302$	− 6978.26i	− 1.32965i
$303$	290.615i	0.0551003i
$304$	−39.2456	−0.00740425
$305$	0	0
$306$	−264.992	−0.0495052
$307$	− 9480.12i	− 1.76241i	−0.472737	−	0.881203i	$-0.656734\pi$
$307$	− 9480.12i	− 1.76241i	0.472737	−	0.881203i	$-0.343266\pi$
$308$	1672.10i	0.309340i
$309$	−3965.98	−0.730151
$310$	0	0
$311$	7078.01	1.29054	0.645268	−	0.763956i	$-0.276746\pi$
$311$	7078.01	1.29054	0.645268	+	0.763956i	$0.276746\pi$
$312$	180.945i	0.0328334i
$313$	5593.84i	1.01017i	0.863070	+	0.505084i	$0.168539\pi$
$313$	5593.84i	1.01017i	−0.863070	+	0.505084i	$0.831461\pi$
$314$	11529.6	2.07214
$315$	0	0
$316$	−10817.3	−1.92571
$317$	− 3567.81i	− 0.632139i	−0.948736	−	0.316070i	$-0.897637\pi$
$317$	− 3567.81i	− 0.632139i	0.948736	−	0.316070i	$-0.102363\pi$
$318$	6928.24i	1.22175i
$319$	−5556.54	−0.975255
$320$	0	0
$321$	5237.14	0.910618
$322$	1566.15i	0.271050i
$323$	35.3222i	0.00608477i
$324$	−1015.02	−0.174043
$325$	0	0
$326$	−2694.70	−0.457809
$327$	− 1430.57i	− 0.241928i
$328$	6469.62i	1.08910i
$329$	605.541	0.101473
$330$	0	0
$331$	−4389.67	−0.728936	−0.364468	−	0.931216i	$-0.618749\pi$
$331$	−4389.67	−0.728936	−0.364468	+	0.931216i	$0.618749\pi$
$332$	− 16725.6i	− 2.76487i
$333$	− 1738.09i	− 0.286026i
$334$	−4207.14	−0.689236
$335$	0	0
$336$	151.617	0.0246172
$337$	− 2348.83i	− 0.379671i	−0.981816	−	0.189835i	$-0.939205\pi$
$337$	− 2348.83i	− 0.379671i	0.981816	−	0.189835i	$-0.0607955\pi$
$338$	− 9915.79i	− 1.59570i
$339$	−4924.94	−0.789044
$340$	0	0
$341$	−4657.05	−0.739570
$342$	221.673i	0.0350488i
$343$	− 343.000i	− 0.0539949i
$344$	6179.79	0.968581
$345$	0	0
$346$	−1430.21	−0.222222
$347$	− 558.436i	− 0.0863931i	−0.999067	−	0.0431965i	$-0.986246\pi$
$347$	− 558.436i	− 0.0863931i	0.999067	−	0.0431965i	$-0.0137542\pi$
$348$	− 10958.3i	− 1.68800i
$349$	−3233.89	−0.496006	−0.248003	−	0.968759i	$-0.579774\pi$
$349$	−3233.89	−0.496006	−0.248003	+	0.968759i	$0.579774\pi$
$350$	0	0
$351$	−79.3190	−0.0120619
$352$	− 3754.56i	− 0.568519i
$353$	− 7516.35i	− 1.13330i	−0.823959	−	0.566650i	$-0.808239\pi$
$353$	− 7516.35i	− 1.13330i	0.823959	−	0.566650i	$-0.191761\pi$
$354$	1133.49	0.170181
$355$	0	0
$356$	4088.27	0.608646
$357$	− 136.459i	− 0.0202302i
$358$	− 6549.70i	− 0.966935i
$359$	6577.76	0.967021	0.483511	−	0.875338i	$-0.339361\pi$
$359$	6577.76	0.967021	0.483511	+	0.875338i	$0.339361\pi$
$360$	0	0
$361$	−6829.45	−0.995692
$362$	8354.56i	1.21300i
$363$	− 2902.89i	− 0.419731i
$364$	−257.693	−0.0371065
$365$	0	0
$366$	−71.4048	−0.0101978
$367$	− 8307.17i	− 1.18155i	−0.806835	−	0.590777i	$-0.798821\pi$
$367$	− 8307.17i	− 1.18155i	0.806835	−	0.590777i	$-0.201179\pi$
$368$	− 356.497i	− 0.0504992i
$369$	−2836.02	−0.400101
$370$	0	0
$371$	−3567.74	−0.499266
$372$	− 9184.34i	− 1.28007i
$373$	− 4551.09i	− 0.631760i	−0.948799	−	0.315880i	$-0.897700\pi$
$373$	− 4551.09i	− 0.631760i	0.948799	−	0.315880i	$-0.102300\pi$
$374$	−561.261	−0.0775992
$375$	0	0
$376$	1776.06	0.243599
$377$	− 856.335i	− 0.116985i
$378$	− 856.383i	− 0.116528i
$379$	1788.29	0.242370	0.121185	−	0.992630i	$-0.461331\pi$
$379$	1788.29	0.242370	0.121185	+	0.992630i	$0.461331\pi$
$380$	0	0
$381$	−2532.05	−0.340474
$382$	1105.93i	0.148126i
$383$	− 1358.47i	− 0.181240i	−0.995886	−	0.0906199i	$-0.971115\pi$
$383$	− 1358.47i	− 0.181240i	0.995886	−	0.0906199i	$-0.0288848\pi$
$384$	6619.37	0.879670
$385$	0	0
$386$	−7852.60	−1.03546
$387$	2708.97i	0.355825i
$388$	19132.2i	2.50333i
$389$	−9722.54	−1.26723	−0.633615	−	0.773649i	$-0.718430\pi$
$389$	−9722.54	−1.26723	−0.633615	+	0.773649i	$0.718430\pi$
$390$	0	0
$391$	−320.858	−0.0414999
$392$	− 1006.03i	− 0.129622i
$393$	− 8388.61i	− 1.07672i
$394$	−1623.19	−0.207551
$395$	0	0
$396$	−2149.84	−0.272812
$397$	4788.04i	0.605302i	0.953101	+	0.302651i	$0.0978717\pi$
$397$	4788.04i	0.605302i	−0.953101	+	0.302651i	$0.902128\pi$
$398$	− 14516.1i	− 1.82820i
$399$	−114.152	−0.0143227
$400$	0	0
$401$	9681.41	1.20565	0.602826	−	0.797873i	$-0.294041\pi$
$401$	9681.41	1.20565	0.602826	+	0.797873i	$0.294041\pi$
$402$	− 2800.13i	− 0.347408i
$403$	− 717.712i	− 0.0887141i
$404$	−1213.91	−0.149491
$405$	0	0
$406$	9245.58	1.13017
$407$	− 3681.32i	− 0.448344i
$408$	− 400.238i	− 0.0485655i
$409$	−11113.1	−1.34353	−0.671767	−	0.740763i	$-0.734464\pi$
$409$	−11113.1	−1.34353	−0.671767	+	0.740763i	$0.734464\pi$
$410$	0	0
$411$	−6171.40	−0.740663
$412$	− 16566.1i	− 1.98095i
$413$	583.696i	0.0695443i
$414$	−2013.62	−0.239043
$415$	0	0
$416$	578.627	0.0681959
$417$	5235.37i	0.614814i
$418$	469.510i	0.0549389i
$419$	1230.09	0.143421	0.0717107	−	0.997425i	$-0.477154\pi$
$419$	1230.09	0.143421	0.0717107	+	0.997425i	$0.477154\pi$
$420$	0	0
$421$	−12356.5	−1.43044	−0.715222	−	0.698897i	$-0.753674\pi$
$421$	−12356.5	−1.43044	−0.715222	+	0.698897i	$0.753674\pi$
$422$	− 22398.1i	− 2.58370i
$423$	778.552i	0.0894906i
$424$	−10464.2	−1.19856
$425$	0	0
$426$	13651.9	1.55267
$427$	− 36.7703i	− 0.00416731i
$428$	21875.7i	2.47057i
$429$	−168.000	−0.0189070
$430$	0	0
$431$	−7375.27	−0.824256	−0.412128	−	0.911126i	$-0.635214\pi$
$431$	−7375.27	−0.824256	−0.412128	+	0.911126i	$0.635214\pi$
$432$	194.936i	0.0217103i
$433$	− 690.067i	− 0.0765877i	−0.999267	−	0.0382939i	$-0.987808\pi$
$433$	− 690.067i	− 0.0765877i	0.999267	−	0.0382939i	$-0.0121923\pi$
$434$	7748.92	0.857051
$435$	0	0
$436$	5975.55	0.656369
$437$	268.406i	0.0293812i
$438$	− 13692.5i	− 1.49372i
$439$	−8408.79	−0.914191	−0.457095	−	0.889418i	$-0.651110\pi$
$439$	−8408.79	−0.914191	−0.457095	+	0.889418i	$0.651110\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	− 86.4976i	− 0.00930830i
$443$	6568.55i	0.704473i	0.935911	+	0.352236i	$0.114579\pi$
$443$	6568.55i	0.704473i	−0.935911	+	0.352236i	$0.885421\pi$
$444$	7260.06	0.776007
$445$	0	0
$446$	14319.0	1.52024
$447$	3520.72i	0.372537i
$448$	5842.94i	0.616190i
$449$	−2954.55	−0.310543	−0.155271	−	0.987872i	$-0.549625\pi$
$449$	−2954.55	−0.310543	−0.155271	+	0.987872i	$0.549625\pi$
$450$	0	0
$451$	−6006.76	−0.627156
$452$	− 20571.7i	− 2.14073i
$453$	4620.21i	0.479197i
$454$	16543.7	1.71020
$455$	0	0
$456$	−334.810	−0.0343835
$457$	8144.84i	0.833697i	0.908976	+	0.416849i	$0.136865\pi$
$457$	8144.84i	0.833697i	−0.908976	+	0.416849i	$0.863135\pi$
$458$	− 18503.8i	− 1.88782i
$459$	175.448	0.0178414
$460$	0	0
$461$	2495.26	0.252095	0.126048	−	0.992024i	$-0.459771\pi$
$461$	2495.26	0.252095	0.126048	+	0.992024i	$0.459771\pi$
$462$	− 1813.84i	− 0.182657i
$463$	− 5755.66i	− 0.577728i	−0.957370	−	0.288864i	$-0.906722\pi$
$463$	− 5755.66i	− 0.577728i	0.957370	−	0.288864i	$-0.0932776\pi$
$464$	−2104.54	−0.210562
$465$	0	0
$466$	−16685.9	−1.65871
$467$	4143.73i	0.410598i	0.978699	+	0.205299i	$0.0658166\pi$
$467$	4143.73i	0.410598i	−0.978699	+	0.205299i	$0.934183\pi$
$468$	− 331.319i	− 0.0327248i
$469$	1441.95	0.141968
$470$	0	0
$471$	−7633.60	−0.746789
$472$	1711.99i	0.166951i
$473$	5737.67i	0.557755i
$474$	11734.3	1.13708
$475$	0	0
$476$	569.996	0.0548860
$477$	− 4587.09i	− 0.440312i
$478$	12047.3i	1.15278i
$479$	6765.96	0.645396	0.322698	−	0.946502i	$-0.395410\pi$
$479$	6765.96	0.645396	0.322698	+	0.946502i	$0.395410\pi$
$480$	0	0
$481$	567.339	0.0537805
$482$	21841.8i	2.06404i
$483$	− 1036.93i	− 0.0976848i
$484$	12125.5	1.13876
$485$	0	0
$486$	1101.06	0.102768
$487$	− 6360.42i	− 0.591824i	−0.955215	−	0.295912i	$-0.904377\pi$
$487$	− 6360.42i	− 0.591824i	0.955215	−	0.295912i	$-0.0956235\pi$
$488$	− 107.848i	− 0.0100042i
$489$	1784.12	0.164992
$490$	0	0
$491$	−7072.54	−0.650060	−0.325030	−	0.945704i	$-0.605374\pi$
$491$	−7072.54	−0.650060	−0.325030	+	0.945704i	$0.605374\pi$
$492$	− 11846.2i	− 1.08550i
$493$	1894.15i	0.173039i
$494$	−72.3576	−0.00659012
$495$	0	0
$496$	−1763.86	−0.159677
$497$	7030.15i	0.634498i
$498$	18143.4i	1.63258i
$499$	18473.9	1.65732	0.828661	−	0.559751i	$-0.189103\pi$
$499$	18473.9	1.65732	0.828661	+	0.559751i	$0.189103\pi$
$500$	0	0
$501$	2785.49	0.248397
$502$	− 7577.28i	− 0.673687i
$503$	11379.2i	1.00869i	0.863501	+	0.504347i	$0.168267\pi$
$503$	11379.2i	1.00869i	−0.863501	+	0.504347i	$0.831733\pi$
$504$	1293.46	0.114316
$505$	0	0
$506$	−4264.90	−0.374700
$507$	6565.11i	0.575082i
$508$	− 10576.5i	− 0.923730i
$509$	−6064.48	−0.528101	−0.264051	−	0.964509i	$-0.585059\pi$
$509$	−6064.48	−0.528101	−0.264051	+	0.964509i	$0.585059\pi$
$510$	0	0
$511$	7051.02	0.610408
$512$	− 2607.89i	− 0.225105i
$513$	− 146.767i	− 0.0126314i
$514$	16755.0	1.43780
$515$	0	0
$516$	−11315.5	−0.965379
$517$	1649.00i	0.140276i
$518$	6125.38i	0.519563i
$519$	946.926	0.0800875
$520$	0	0
$521$	2682.88	0.225603	0.112801	−	0.993618i	$-0.464018\pi$
$521$	2682.88	0.225603	0.112801	+	0.993618i	$0.464018\pi$
$522$	11887.2i	0.996720i
$523$	− 4309.02i	− 0.360268i	−0.983642	−	0.180134i	$-0.942347\pi$
$523$	− 4309.02i	− 0.360268i	0.983642	−	0.180134i	$-0.0576532\pi$
$524$	35039.6	2.92120
$525$	0	0
$526$	33163.3	2.74903
$527$	1587.52i	0.131221i
$528$	412.879i	0.0340308i
$529$	9728.87	0.799611
$530$	0	0
$531$	−750.467	−0.0613323
$532$	− 476.817i	− 0.0388584i
$533$	− 925.720i	− 0.0752296i
$534$	−4434.83	−0.359389
$535$	0	0
$536$	4229.25	0.340813
$537$	4336.47i	0.348478i
$538$	− 3693.31i	− 0.295966i
$539$	934.051	0.0746427
$540$	0	0
$541$	4081.47	0.324355	0.162178	−	0.986762i	$-0.448148\pi$
$541$	4081.47	0.324355	0.162178	+	0.986762i	$0.448148\pi$
$542$	23136.0i	1.83354i
$543$	− 5531.44i	− 0.437158i
$544$	−1279.88	−0.100872
$545$	0	0
$546$	279.537	0.0219104
$547$	− 8844.82i	− 0.691366i	−0.938351	−	0.345683i	$-0.887647\pi$
$547$	− 8844.82i	− 0.691366i	0.938351	−	0.345683i	$-0.112353\pi$
$548$	− 25778.2i	− 2.00947i
$549$	47.2762	0.00367522
$550$	0	0
$551$	1584.51	0.122509
$552$	− 3041.32i	− 0.234506i
$553$	6042.66i	0.464666i
$554$	−6337.80	−0.486042
$555$	0	0
$556$	−21868.4	−1.66803
$557$	11144.7i	0.847787i	0.905712	+	0.423894i	$0.139337\pi$
$557$	11144.7i	0.847787i	−0.905712	+	0.423894i	$0.860663\pi$
$558$	9962.90i	0.755848i
$559$	−884.249	−0.0669047
$560$	0	0
$561$	371.603	0.0279663
$562$	32181.8i	2.41549i
$563$	− 21857.5i	− 1.63621i	−0.575071	−	0.818104i	$-0.695025\pi$
$563$	− 21857.5i	− 1.63621i	0.575071	−	0.818104i	$-0.304975\pi$
$564$	−3252.05	−0.242794
$565$	0	0
$566$	20232.3	1.50252
$567$	567.000i	0.0419961i
$568$	20619.6i	1.52320i
$569$	−23496.4	−1.73115	−0.865573	−	0.500783i	$-0.833046\pi$
$569$	−23496.4	−1.73115	−0.865573	+	0.500783i	$0.833046\pi$
$570$	0	0
$571$	11067.8	0.811164	0.405582	−	0.914059i	$-0.367069\pi$
$571$	11067.8	0.811164	0.405582	+	0.914059i	$0.367069\pi$
$572$	− 701.743i	− 0.0512961i
$573$	− 732.222i	− 0.0533839i
$574$	9994.72	0.726780
$575$	0	0
$576$	−7512.36	−0.543429
$577$	− 20482.9i	− 1.47784i	−0.673791	−	0.738922i	$-0.735335\pi$
$577$	− 20482.9i	− 1.47784i	0.673791	−	0.738922i	$-0.264665\pi$
$578$	− 22070.1i	− 1.58823i
$579$	5199.10	0.373173
$580$	0	0
$581$	−9343.07	−0.667153
$582$	− 20754.0i	− 1.47815i
$583$	− 9715.60i	− 0.690187i
$584$	20680.8	1.46537
$585$	0	0
$586$	34394.1	2.42458
$587$	23444.3i	1.64847i	0.566248	+	0.824235i	$0.308394\pi$
$587$	23444.3i	1.64847i	−0.566248	+	0.824235i	$0.691606\pi$
$588$	1842.08i	0.129194i
$589$	1328.01	0.0929025
$590$	0	0
$591$	1074.69	0.0748000
$592$	− 1394.30i	− 0.0967996i
$593$	4404.69i	0.305024i	0.988302	+	0.152512i	$0.0487362\pi$
$593$	4404.69i	0.305024i	−0.988302	+	0.152512i	$0.951264\pi$
$594$	2332.09	0.161089
$595$	0	0
$596$	−14706.2	−1.01072
$597$	9610.89i	0.658874i
$598$	− 657.277i	− 0.0449466i
$599$	−3327.05	−0.226945	−0.113472	−	0.993541i	$-0.536197\pi$
$599$	−3327.05	−0.226945	−0.113472	+	0.993541i	$0.536197\pi$
$600$	0	0
$601$	−14244.8	−0.966818	−0.483409	−	0.875395i	$-0.660602\pi$
$601$	−14244.8	−0.966818	−0.483409	+	0.875395i	$0.660602\pi$
$602$	− 9546.97i	− 0.646354i
$603$	1853.93i	0.125204i
$604$	−19298.8	−1.30010
$605$	0	0
$606$	1316.81	0.0882704
$607$	− 11446.5i	− 0.765402i	−0.923872	−	0.382701i	$-0.874994\pi$
$607$	− 11446.5i	− 0.765402i	0.923872	−	0.382701i	$-0.125006\pi$
$608$	1070.65i	0.0714156i
$609$	−6121.38	−0.407308
$610$	0	0
$611$	−254.132	−0.0168266
$612$	732.852i	0.0484049i
$613$	− 19436.4i	− 1.28063i	−0.768111	−	0.640316i	$-0.778803\pi$
$613$	− 19436.4i	− 1.28063i	0.768111	−	0.640316i	$-0.221197\pi$
$614$	−42955.6	−2.82337
$615$	0	0
$616$	2739.59	0.179190
$617$	20530.1i	1.33956i	0.742558	+	0.669781i	$0.233612\pi$
$617$	20530.1i	1.33956i	−0.742558	+	0.669781i	$0.766388\pi$
$618$	17970.4i	1.16970i
$619$	−5833.35	−0.378776	−0.189388	−	0.981902i	$-0.560650\pi$
$619$	−5833.35	−0.378776	−0.189388	+	0.981902i	$0.560650\pi$
$620$	0	0
$621$	1333.19	0.0861499
$622$	− 32071.4i	− 2.06744i
$623$	− 2283.74i	− 0.146864i
$624$	−63.6301	−0.00408212
$625$	0	0
$626$	25346.4	1.61829
$627$	− 310.856i	− 0.0197997i
$628$	− 31885.9i	− 2.02609i
$629$	−1254.91	−0.0795493
$630$	0	0
$631$	24776.6	1.56314	0.781568	−	0.623820i	$-0.214420\pi$
$631$	24776.6	1.56314	0.781568	+	0.623820i	$0.214420\pi$
$632$	17723.2i	1.11549i
$633$	14829.5i	0.931152i
$634$	−16166.2	−1.01268
$635$	0	0
$636$	19160.5	1.19460
$637$	143.949i	0.00895366i
$638$	25177.4i	1.56235i
$639$	−9038.77	−0.559574
$640$	0	0
$641$	27219.4	1.67723	0.838613	−	0.544728i	$-0.183367\pi$
$641$	27219.4	1.67723	0.838613	+	0.544728i	$0.183367\pi$
$642$	− 23730.1i	− 1.45881i
$643$	7091.79i	0.434950i	0.976066	+	0.217475i	$0.0697821\pi$
$643$	7091.79i	0.434950i	−0.976066	+	0.217475i	$0.930218\pi$
$644$	4331.28	0.265026
$645$	0	0
$646$	160.049	0.00974777
$647$	− 27773.0i	− 1.68758i	−0.536670	−	0.843792i	$-0.680318\pi$
$647$	− 27773.0i	− 1.68758i	0.536670	−	0.843792i	$-0.319682\pi$
$648$	1663.02i	0.100817i
$649$	−1589.51	−0.0961382
$650$	0	0
$651$	−5130.46	−0.308876
$652$	7452.37i	0.447634i
$653$	21380.4i	1.28129i	0.767839	+	0.640643i	$0.221332\pi$
$653$	21380.4i	1.28129i	−0.767839	+	0.640643i	$0.778668\pi$
$654$	−6482.09	−0.387568
$655$	0	0
$656$	−2275.06	−0.135406
$657$	9065.59i	0.538329i
$658$	− 2743.78i	− 0.162559i
$659$	17232.3	1.01863	0.509315	−	0.860580i	$-0.329899\pi$
$659$	17232.3	1.01863	0.509315	+	0.860580i	$0.329899\pi$
$660$	0	0
$661$	26577.7	1.56392	0.781962	−	0.623326i	$-0.214219\pi$
$661$	26577.7	1.56392	0.781962	+	0.623326i	$0.214219\pi$
$662$	19890.1i	1.16775i
$663$	57.2689i	0.00335466i
$664$	−27403.4	−1.60159
$665$	0	0
$666$	−7875.49	−0.458212
$667$	14393.2i	0.835544i
$668$	11635.1i	0.673917i
$669$	−9480.44	−0.547885
$670$	0	0
$671$	100.132	0.00576090
$672$	− 4136.22i	− 0.237438i
$673$	− 31695.2i	− 1.81540i	−0.419624	−	0.907698i	$-0.637838\pi$
$673$	− 31695.2i	− 1.81540i	0.419624	−	0.907698i	$-0.362162\pi$
$674$	−10642.9	−0.608231
$675$	0	0
$676$	−27422.7	−1.56024
$677$	− 20440.3i	− 1.16039i	−0.814477	−	0.580195i	$-0.802976\pi$
$677$	− 20440.3i	− 1.16039i	0.814477	−	0.580195i	$-0.197024\pi$
$678$	22315.5i	1.26405i
$679$	10687.4	0.604044
$680$	0	0
$681$	−10953.3	−0.616348
$682$	21101.7i	1.18479i
$683$	− 22896.9i	− 1.28276i	−0.767223	−	0.641381i	$-0.778362\pi$
$683$	− 22896.9i	− 1.28276i	0.767223	−	0.641381i	$-0.221638\pi$
$684$	613.051	0.0342699
$685$	0	0
$686$	−1554.18	−0.0864997
$687$	12251.1i	0.680361i
$688$	2173.14i	0.120422i
$689$	1497.30	0.0827904
$690$	0	0
$691$	−23764.0	−1.30829	−0.654143	−	0.756371i	$-0.726971\pi$
$691$	−23764.0	−1.30829	−0.654143	+	0.756371i	$0.726971\pi$
$692$	3955.35i	0.217283i
$693$	1200.92i	0.0658287i
$694$	−2530.34	−0.138401
$695$	0	0
$696$	−17954.1	−0.977800
$697$	2047.62i	0.111276i
$698$	14653.2i	0.794600i
$699$	11047.5	0.597791
$700$	0	0
$701$	26259.5	1.41485	0.707423	−	0.706791i	$-0.249858\pi$
$701$	26259.5	1.41485	0.707423	+	0.706791i	$0.249858\pi$
$702$	359.405i	0.0193232i
$703$	1049.77i	0.0563196i
$704$	−15911.4	−0.851822
$705$	0	0
$706$	−34057.6	−1.81554
$707$	678.101i	0.0360716i
$708$	− 3134.73i	− 0.166399i
$709$	−12783.0	−0.677116	−0.338558	−	0.940945i	$-0.609939\pi$
$709$	−12783.0	−0.677116	−0.338558	+	0.940945i	$0.609939\pi$
$710$	0	0
$711$	−7769.14	−0.409797
$712$	− 6698.26i	− 0.352567i
$713$	12063.3i	0.633623i
$714$	−618.315	−0.0324087
$715$	0	0
$716$	−18113.6	−0.945444
$717$	− 7976.35i	− 0.415456i
$718$	− 29804.7i	− 1.54916i
$719$	27609.0	1.43205	0.716025	−	0.698075i	$-0.245960\pi$
$719$	27609.0	1.43205	0.716025	+	0.698075i	$0.245960\pi$
$720$	0	0
$721$	−9253.95	−0.477996
$722$	30945.1i	1.59509i
$723$	− 14461.2i	− 0.743868i
$724$	23105.1	1.18604
$725$	0	0
$726$	−13153.4	−0.672407
$727$	− 31306.2i	− 1.59709i	−0.601937	−	0.798544i	$-0.705604\pi$
$727$	− 31306.2i	− 1.59709i	0.601937	−	0.798544i	$-0.294396\pi$
$728$	422.206i	0.0214945i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	1955.89	0.0989621
$732$	197.475i	0.00997113i
$733$	− 15765.8i	− 0.794441i	−0.917723	−	0.397220i	$-0.869975\pi$
$733$	− 15765.8i	− 0.794441i	0.917723	−	0.397220i	$-0.130025\pi$
$734$	−37640.8	−1.89285
$735$	0	0
$736$	−9725.53	−0.487076
$737$	3926.68i	0.196256i
$738$	12850.4i	0.640959i
$739$	−3966.51	−0.197443	−0.0987216	−	0.995115i	$-0.531475\pi$
$739$	−3966.51	−0.197443	−0.0987216	+	0.995115i	$0.531475\pi$
$740$	0	0
$741$	47.9070	0.00237504
$742$	16165.9i	0.799823i
$743$	8224.50i	0.406094i	0.979169	+	0.203047i	$0.0650844\pi$
$743$	8224.50i	0.406094i	−0.979169	+	0.203047i	$0.934916\pi$
$744$	−15047.7	−0.741500
$745$	0	0
$746$	−20621.6	−1.01208
$747$	− 12012.5i	− 0.588373i
$748$	1552.20i	0.0758745i
$749$	12220.0	0.596140
$750$	0	0
$751$	18929.2	0.919754	0.459877	−	0.887983i	$-0.347894\pi$
$751$	18929.2	0.919754	0.459877	+	0.887983i	$0.347894\pi$
$752$	624.558i	0.0302863i
$753$	5016.82i	0.242793i
$754$	−3880.16	−0.187410
$755$	0	0
$756$	−2368.38	−0.113938
$757$	34906.8i	1.67597i	0.545695	+	0.837984i	$0.316266\pi$
$757$	34906.8i	1.67597i	−0.545695	+	0.837984i	$0.683734\pi$
$758$	− 8102.96i	− 0.388276i
$759$	2823.74	0.135040
$760$	0	0
$761$	−13683.4	−0.651803	−0.325902	−	0.945404i	$-0.605668\pi$
$761$	−13683.4	−0.651803	−0.325902	+	0.945404i	$0.605668\pi$
$762$	11473.0i	0.545438i
$763$	− 3337.99i	− 0.158379i
$764$	3058.52	0.144834
$765$	0	0
$766$	−6155.42	−0.290345
$767$	− 244.964i	− 0.0115321i
$768$	− 9960.28i	− 0.467982i
$769$	−41837.3	−1.96189	−0.980943	−	0.194294i	$-0.937758\pi$
$769$	−41837.3	−1.96189	−0.980943	+	0.194294i	$0.937758\pi$
$770$	0	0
$771$	−11093.2	−0.518175
$772$	21716.9i	1.01245i
$773$	19640.0i	0.913843i	0.889507	+	0.456921i	$0.151048\pi$
$773$	19640.0i	0.913843i	−0.889507	+	0.456921i	$0.848952\pi$
$774$	12274.7	0.570031
$775$	0	0
$776$	31346.4	1.45009
$777$	− 4055.53i	− 0.187248i
$778$	44054.1i	2.03010i
$779$	1712.89	0.0787814
$780$	0	0
$781$	−19144.4	−0.877131
$782$	1453.85i	0.0664827i
$783$	− 7870.34i	− 0.359212i
$784$	353.772	0.0161157
$785$	0	0
$786$	−38009.9	−1.72489
$787$	− 24935.3i	− 1.12941i	−0.825293	−	0.564705i	$-0.808990\pi$
$787$	− 24935.3i	− 1.12941i	0.825293	−	0.564705i	$-0.191010\pi$
$788$	4489.02i	0.202938i
$789$	−21957.0	−0.990736
$790$	0	0
$791$	−11491.5	−0.516551
$792$	3522.33i	0.158031i
$793$	15.4317i	0 0.000691041i
$794$	21695.2	0.969692
$795$	0	0
$796$	−40145.1	−1.78757
$797$	1168.33i	0.0519251i	0.999663	+	0.0259625i	$0.00826506\pi$
$797$	1168.33i	0.0519251i	−0.999663	+	0.0259625i	$0.991735\pi$
$798$	517.237i	0.0229448i
$799$	562.120	0.0248891
$800$	0	0
$801$	2936.24	0.129522
$802$	− 43867.7i	− 1.93145i
$803$	19201.2i	0.843829i
$804$	−7743.95	−0.339686
$805$	0	0
$806$	−3252.05	−0.142120
$807$	2445.29i	0.106665i
$808$	1988.88i	0.0865949i
$809$	35175.7	1.52869	0.764345	−	0.644807i	$-0.223062\pi$
$809$	35175.7	1.52869	0.764345	+	0.644807i	$0.223062\pi$
$810$	0	0
$811$	−15256.5	−0.660577	−0.330288	−	0.943880i	$-0.607146\pi$
$811$	−15256.5	−0.660577	−0.330288	+	0.943880i	$0.607146\pi$
$812$	− 25569.3i	− 1.10506i
$813$	− 15318.1i	− 0.660797i
$814$	−16680.5	−0.718245
$815$	0	0
$816$	140.745	0.00603806
$817$	− 1636.16i	− 0.0700635i
$818$	50354.7i	2.15233i
$819$	−185.078	−0.00789639
$820$	0	0
$821$	−15971.9	−0.678956	−0.339478	−	0.940614i	$-0.610250\pi$
$821$	−15971.9	−0.678956	−0.339478	+	0.940614i	$0.610250\pi$
$822$	27963.4i	1.18654i
$823$	2312.41i	0.0979409i	0.998800	+	0.0489705i	$0.0155940\pi$
$823$	2312.41i	0.0979409i	−0.998800	+	0.0489705i	$0.984406\pi$
$824$	−27142.0	−1.14750
$825$	0	0
$826$	2644.80	0.111410
$827$	− 10422.4i	− 0.438238i	−0.975698	−	0.219119i	$-0.929682\pi$
$827$	− 10422.4i	− 0.438238i	0.975698	−	0.219119i	$-0.0703183\pi$
$828$	5568.79i	0.233731i
$829$	13213.4	0.553584	0.276792	−	0.960930i	$-0.410729\pi$
$829$	13213.4	0.553584	0.276792	+	0.960930i	$0.410729\pi$
$830$	0	0
$831$	4196.17	0.175167
$832$	− 2452.15i	− 0.102179i
$833$	− 318.405i	− 0.0132438i
$834$	23722.2	0.984929
$835$	0	0
$836$	1298.46	0.0537179
$837$	− 6596.30i	− 0.272403i
$838$	− 5573.67i	− 0.229761i
$839$	10119.6	0.416409	0.208205	−	0.978085i	$-0.433238\pi$
$839$	10119.6	0.416409	0.208205	+	0.978085i	$0.433238\pi$
$840$	0	0
$841$	60579.9	2.48390
$842$	55988.7i	2.29157i
$843$	− 21307.1i	− 0.870530i
$844$	−61943.4	−2.52628
$845$	0	0
$846$	3527.72	0.143364
$847$	− 6773.41i	− 0.274778i
$848$	− 3679.79i	− 0.149015i
$849$	−13395.5	−0.541501
$850$	0	0
$851$	−9535.80	−0.384116
$852$	− 37755.3i	− 1.51816i
$853$	35378.1i	1.42007i	0.704165	+	0.710037i	$0.251322\pi$
$853$	35378.1i	1.42007i	−0.704165	+	0.710037i	$0.748678\pi$
$854$	−166.611	−0.00667602
$855$	0	0
$856$	35841.4	1.43112
$857$	− 6697.57i	− 0.266960i	−0.991052	−	0.133480i	$-0.957385\pi$
$857$	− 6697.57i	− 0.266960i	0.991052	−	0.133480i	$-0.0426152\pi$
$858$	761.230i	0.0302890i
$859$	24298.4	0.965135	0.482568	−	0.875859i	$-0.339704\pi$
$859$	24298.4	0.965135	0.482568	+	0.875859i	$0.339704\pi$
$860$	0	0
$861$	−6617.37	−0.261927
$862$	33418.3i	1.32045i
$863$	24942.9i	0.983853i	0.870637	+	0.491926i	$0.163707\pi$
$863$	24942.9i	0.983853i	−0.870637	+	0.491926i	$0.836293\pi$
$864$	5318.00	0.209401
$865$	0	0
$866$	−3126.78	−0.122693
$867$	14612.3i	0.572388i
$868$	− 21430.1i	− 0.838002i
$869$	−16455.3	−0.642355
$870$	0	0
$871$	−605.152	−0.0235417
$872$	− 9790.39i	− 0.380212i
$873$	13741.0i	0.532716i
$874$	1216.18	0.0470686
$875$	0	0
$876$	−37867.4	−1.46052
$877$	16276.6i	0.626705i	0.949637	+	0.313353i	$0.101452\pi$
$877$	16276.6i	0.626705i	−0.949637	+	0.313353i	$0.898548\pi$
$878$	38101.3i	1.46453i
$879$	−22771.8	−0.873806
$880$	0	0
$881$	26636.5	1.01862	0.509311	−	0.860582i	$-0.329900\pi$
$881$	26636.5	1.01862	0.509311	+	0.860582i	$0.329900\pi$
$882$	− 1998.23i	− 0.0762855i
$883$	− 21788.3i	− 0.830392i	−0.909732	−	0.415196i	$-0.863713\pi$
$883$	− 21788.3i	− 0.830392i	0.909732	−	0.415196i	$-0.136287\pi$
$884$	−239.215	−0.00910142
$885$	0	0
$886$	29763.0	1.12856
$887$	26813.2i	1.01499i	0.861654	+	0.507496i	$0.169429\pi$
$887$	26813.2i	1.01499i	−0.861654	+	0.507496i	$0.830571\pi$
$888$	− 11895.0i	− 0.449514i
$889$	−5908.11	−0.222893
$890$	0	0
$891$	−1544.04	−0.0580554
$892$	− 39600.2i	− 1.48645i
$893$	− 470.229i	− 0.0176211i
$894$	15952.8	0.596803
$895$	0	0
$896$	15445.2	0.575879
$897$	435.174i	0.0161985i
$898$	13387.4i	0.497488i
$899$	71214.2	2.64196
$900$	0	0
$901$	−3311.91	−0.122459
$902$	27217.4i	1.00470i
$903$	6320.92i	0.232942i
$904$	−33704.8	−1.24005
$905$	0	0
$906$	20934.8	0.767672
$907$	− 15543.0i	− 0.569014i	−0.958674	−	0.284507i	$-0.908170\pi$
$907$	− 15543.0i	− 0.569014i	0.958674	−	0.284507i	$-0.0918300\pi$
$908$	− 45752.6i	− 1.67219i
$909$	−871.844	−0.0318122
$910$	0	0
$911$	48711.1	1.77154	0.885768	−	0.464128i	$-0.153632\pi$
$911$	48711.1	1.77154	0.885768	+	0.464128i	$0.153632\pi$
$912$	− 117.737i	− 0.00427485i
$913$	− 25442.8i	− 0.922273i
$914$	36905.3	1.33558
$915$	0	0
$916$	−51173.3	−1.84587
$917$	− 19573.4i	− 0.704876i
$918$	− 794.976i	− 0.0285818i
$919$	−1030.47	−0.0369883	−0.0184941	−	0.999829i	$-0.505887\pi$
$919$	−1030.47	−0.0369883	−0.0184941	+	0.999829i	$0.505887\pi$
$920$	0	0
$921$	28440.4	1.01753
$922$	− 11306.3i	− 0.403855i
$923$	− 2950.40i	− 0.105215i
$924$	−5016.30	−0.178598
$925$	0	0
$926$	−26079.6	−0.925518
$927$	− 11897.9i	− 0.421553i
$928$	57413.6i	2.03092i
$929$	−879.756	−0.0310698	−0.0155349	−	0.999879i	$-0.504945\pi$
$929$	−879.756	−0.0310698	−0.0155349	+	0.999879i	$0.504945\pi$
$930$	0	0
$931$	−266.354	−0.00937638
$932$	46146.0i	1.62185i
$933$	21234.0i	0.745092i
$934$	18775.8	0.657776
$935$	0	0
$936$	−542.836	−0.0189564
$937$	18668.1i	0.650864i	0.945565	+	0.325432i	$0.105510\pi$
$937$	18668.1i	0.650864i	−0.945565	+	0.325432i	$0.894490\pi$
$938$	− 6533.64i	− 0.227432i
$939$	−16781.5	−0.583221
$940$	0	0
$941$	29613.4	1.02590	0.512948	−	0.858420i	$-0.328553\pi$
$941$	29613.4	1.02590	0.512948	+	0.858420i	$0.328553\pi$
$942$	34588.8i	1.19635i
$943$	15559.5i	0.537313i
$944$	−602.028	−0.0207567
$945$	0	0
$946$	25998.1	0.893521
$947$	− 20738.9i	− 0.711640i	−0.934554	−	0.355820i	$-0.884202\pi$
$947$	− 20738.9i	− 0.711640i	0.934554	−	0.355820i	$-0.115798\pi$
$948$	− 32452.0i	− 1.11181i
$949$	−2959.15	−0.101220
$950$	0	0
$951$	10703.4	0.364966
$952$	− 933.888i	− 0.0317936i
$953$	− 45776.5i	− 1.55598i	−0.628279	−	0.777988i	$-0.716240\pi$
$953$	− 45776.5i	− 1.55598i	0.628279	−	0.777988i	$-0.283760\pi$
$954$	−20784.7	−0.705377
$955$	0	0
$956$	33317.5	1.12716
$957$	− 16669.6i	− 0.563064i
$958$	− 30657.4i	− 1.03392i
$959$	−14399.9	−0.484878
$960$	0	0
$961$	29895.1	1.00349
$962$	− 2570.68i	− 0.0861561i
$963$	15711.4i	0.525746i
$964$	60404.9	2.01817
$965$	0	0
$966$	−4698.44	−0.156491
$967$	− 34461.0i	− 1.14601i	−0.819552	−	0.573005i	$-0.805778\pi$
$967$	− 34461.0i	− 1.14601i	0.819552	−	0.573005i	$-0.194222\pi$
$968$	− 19866.5i	− 0.659643i
$969$	−105.967	−0.00351304
$970$	0	0
$971$	−22762.8	−0.752309	−0.376154	−	0.926557i	$-0.622754\pi$
$971$	−22762.8	−0.752309	−0.376154	+	0.926557i	$0.622754\pi$
$972$	− 3045.06i	− 0.100484i
$973$	12215.9i	0.402490i
$974$	−28819.9	−0.948099
$975$	0	0
$976$	37.9251	0.00124381
$977$	− 4809.57i	− 0.157494i	−0.996895	−	0.0787470i	$-0.974908\pi$
$977$	− 4809.57i	− 0.157494i	0.996895	−	0.0787470i	$-0.0250919\pi$
$978$	− 8084.10i	− 0.264316i
$979$	6219.04	0.203025
$980$	0	0
$981$	4291.70	0.139677
$982$	32046.6i	1.04139i
$983$	− 27591.6i	− 0.895256i	−0.894220	−	0.447628i	$-0.852269\pi$
$983$	− 27591.6i	− 0.895256i	0.894220	−	0.447628i	$-0.147731\pi$
$984$	−19408.9	−0.628793
$985$	0	0
$986$	8582.63	0.277207
$987$	1816.62i	0.0585853i
$988$	200.109i	0.00644365i
$989$	14862.4	0.477854
$990$	0	0
$991$	−22263.4	−0.713643	−0.356822	−	0.934173i	$-0.616140\pi$
$991$	−22263.4	−0.713643	−0.356822	+	0.934173i	$0.616140\pi$
$992$	48119.5i	1.54012i
$993$	− 13169.0i	− 0.420851i
$994$	31854.5	1.01646
$995$	0	0
$996$	50176.8	1.59630
$997$	− 30378.2i	− 0.964983i	−0.875901	−	0.482491i	$-0.839732\pi$
$997$	− 30378.2i	− 0.964983i	0.875901	−	0.482491i	$-0.160268\pi$
$998$	− 83707.4i	− 2.65502i
$999$	5214.26	0.165137

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.d.h.274.1		4
5.2	odd	4		105.4.a.f.1.2	✓	2
5.3	odd	4		525.4.a.k.1.1		2
5.4	even	2	inner	525.4.d.h.274.4		4
15.2	even	4		315.4.a.i.1.1		2
15.8	even	4		1575.4.a.w.1.2		2
20.7	even	4		1680.4.a.bg.1.2		2
35.27	even	4		735.4.a.p.1.2		2
105.62	odd	4		2205.4.a.z.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.f.1.2	✓	2	5.2	odd	4
315.4.a.i.1.1		2	15.2	even	4
525.4.a.k.1.1		2	5.3	odd	4
525.4.d.h.274.1		4	1.1	even	1	trivial
525.4.d.h.274.4		4	5.4	even	2	inner
735.4.a.p.1.2		2	35.27	even	4
1575.4.a.w.1.2		2	15.8	even	4
1680.4.a.bg.1.2		2	20.7	even	4
2205.4.a.z.1.1		2	105.62	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.53113i q^{2} +3.00000i q^{3} -12.5311 q^{4} +13.5934 q^{6} +7.00000i q^{7} +20.5311i q^{8} -9.00000 q^{9} -19.0623 q^{11} -37.5934i q^{12} -2.93774i q^{13} +31.7179 q^{14} -7.21984 q^{16} +6.49806i q^{17} +40.7802i q^{18} +5.43580 q^{19} -21.0000 q^{21} +86.3735i q^{22} +49.3774i q^{23} -61.5934 q^{24} -13.3113 q^{26} -27.0000i q^{27} -87.7179i q^{28} +291.494 q^{29} +244.307 q^{31} +196.963i q^{32} -57.1868i q^{33} +29.4436 q^{34} +112.780 q^{36} +193.121i q^{37} -24.6303i q^{38} +8.81323 q^{39} +315.113 q^{41} +95.1537i q^{42} -300.996i q^{43} +238.872 q^{44} +223.735 q^{46} -86.5058i q^{47} -21.6595i q^{48} -49.0000 q^{49} -19.4942 q^{51} +36.8132i q^{52} +509.677i q^{53} -122.340 q^{54} -143.718 q^{56} +16.3074i q^{57} -1320.80i q^{58} +83.3852 q^{59} -5.25291 q^{61} -1106.99i q^{62} -63.0000i q^{63} +834.706 q^{64} -259.121 q^{66} -205.992i q^{67} -81.4281i q^{68} -148.132 q^{69} +1004.31 q^{71} -184.780i q^{72} -1007.29i q^{73} +875.055 q^{74} -68.1168 q^{76} -133.436i q^{77} -39.9339i q^{78} +863.237 q^{79} +81.0000 q^{81} -1427.82i q^{82} +1334.72i q^{83} +263.154 q^{84} -1363.85 q^{86} +874.483i q^{87} -391.370i q^{88} -326.249 q^{89} +20.5642 q^{91} -618.755i q^{92} +732.922i q^{93} -391.969 q^{94} -590.889 q^{96} -1526.77i q^{97} +222.025i q^{98} +171.560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 34 q^{4} + 6 q^{6} - 36 q^{9} - 44 q^{11} + 14 q^{14} - 174 q^{16} - 204 q^{19} - 84 q^{21} - 198 q^{24} + 108 q^{26} + 392 q^{29} + 300 q^{31} + 924 q^{34} + 306 q^{36} + 132 q^{39} - 352 q^{41}+ \cdots + 396 q^{99}+O(q^{100}) \) 4 * q - 34 * q^4 + 6 * q^6 - 36 * q^9 - 44 * q^11 + 14 * q^14 - 174 * q^16 - 204 * q^19 - 84 * q^21 - 198 * q^24 + 108 * q^26 + 392 * q^29 + 300 * q^31 + 924 * q^34 + 306 * q^36 + 132 * q^39 - 352 * q^41 + 504 * q^44 - 1040 * q^46 - 196 * q^49 + 696 * q^51 - 54 * q^54 - 462 * q^56 + 1688 * q^59 - 408 * q^61 + 1678 * q^64 - 456 * q^66 - 1560 * q^69 + 3340 * q^71 + 2436 * q^74 + 824 * q^76 + 1776 * q^79 + 324 * q^81 + 714 * q^84 - 2424 * q^86 - 1176 * q^89 + 308 * q^91 + 2560 * q^94 - 90 * q^96 + 396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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