LMFDB - Embedded newform 531.5.b.a.296.28

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,5,Mod(296,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("531.296");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	531.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$54.8894503975$
Analytic rank:	$0$
Dimension:	$76$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			296.28
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.49

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.07317i q^{2} +11.7020 q^{4} +48.7224i q^{5} -18.1528 q^{7} -57.4309i q^{8} +O(q^{10})$	\(q-2.07317i q^{2} +11.7020 q^{4} +48.7224i q^{5} -18.1528 q^{7} -57.4309i q^{8} +101.010 q^{10} -148.657i q^{11} +163.320 q^{13} +37.6339i q^{14} +68.1670 q^{16} -244.444i q^{17} +48.5334 q^{19} +570.147i q^{20} -308.191 q^{22} -859.818i q^{23} -1748.87 q^{25} -338.590i q^{26} -212.423 q^{28} +199.947i q^{29} +1348.43 q^{31} -1060.22i q^{32} -506.775 q^{34} -884.449i q^{35} +1444.27 q^{37} -100.618i q^{38} +2798.17 q^{40} -2065.75i q^{41} +1174.58 q^{43} -1739.57i q^{44} -1782.55 q^{46} +3415.30i q^{47} -2071.48 q^{49} +3625.71i q^{50} +1911.16 q^{52} +2949.55i q^{53} +7242.90 q^{55} +1042.53i q^{56} +414.525 q^{58} +453.188i q^{59} +6279.67 q^{61} -2795.54i q^{62} -1107.34 q^{64} +7957.34i q^{65} +4903.39 q^{67} -2860.48i q^{68} -1833.62 q^{70} +924.553i q^{71} -4742.11 q^{73} -2994.23i q^{74} +567.936 q^{76} +2698.54i q^{77} +7235.69 q^{79} +3321.26i q^{80} -4282.65 q^{82} +2645.08i q^{83} +11909.9 q^{85} -2435.10i q^{86} -8537.49 q^{88} +6034.79i q^{89} -2964.72 q^{91} -10061.6i q^{92} +7080.51 q^{94} +2364.66i q^{95} +6822.95 q^{97} +4294.53i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) $ 76 * q - 632 * q^4 - 48 * q^7	$ 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) $ 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.07317i	− 0.518293i	−0.965838	−	0.259147i	$-0.916559\pi$
$2$	− 2.07317i	− 0.518293i	0.965838	−	0.259147i	$-0.0834413\pi$
$3$	0	0
$3$	0	0
$4$	11.7020	0.731372
$5$	48.7224i	1.94890i	0.224614	+	0.974448i	$0.427888\pi$
$5$	48.7224i	1.94890i	−0.224614	+	0.974448i	$0.572112\pi$
$6$	0	0
$7$	−18.1528	−0.370466	−0.185233	−	0.982695i	$-0.559304\pi$
$7$	−18.1528	−0.370466	−0.185233	+	0.982695i	$0.559304\pi$
$8$	− 57.4309i	− 0.897358i
$9$	0	0
$10$	101.010	1.01010
$11$	− 148.657i	− 1.22857i	−0.789085	−	0.614283i	$-0.789445\pi$
$11$	− 148.657i	− 1.22857i	0.789085	−	0.614283i	$-0.210555\pi$
$12$	0	0
$13$	163.320	0.966390	0.483195	−	0.875513i	$-0.339476\pi$
$13$	163.320	0.966390	0.483195	+	0.875513i	$0.339476\pi$
$14$	37.6339i	0.192010i
$15$	0	0
$16$	68.1670	0.266277
$17$	− 244.444i	− 0.845828i	−0.906170	−	0.422914i	$-0.861007\pi$
$17$	− 244.444i	− 0.845828i	0.906170	−	0.422914i	$-0.138993\pi$
$18$	0	0
$19$	48.5334	0.134442	0.0672208	−	0.997738i	$-0.478587\pi$
$19$	48.5334	0.134442	0.0672208	+	0.997738i	$0.478587\pi$
$20$	570.147i	1.42537i
$21$	0	0
$22$	−308.191	−0.636758
$23$	− 859.818i	− 1.62537i	−0.582707	−	0.812683i	$-0.698006\pi$
$23$	− 859.818i	− 1.62537i	0.582707	−	0.812683i	$-0.301994\pi$
$24$	0	0
$25$	−1748.87	−2.79819
$26$	− 338.590i	− 0.500873i
$27$	0	0
$28$	−212.423	−0.270948
$29$	199.947i	0.237749i	0.992909	+	0.118875i	$0.0379286\pi$
$29$	199.947i	0.237749i	−0.992909	+	0.118875i	$0.962071\pi$
$30$	0	0
$31$	1348.43	1.40316	0.701579	−	0.712592i	$-0.252479\pi$
$31$	1348.43	1.40316	0.701579	+	0.712592i	$0.252479\pi$
$32$	− 1060.22i	− 1.03537i
$33$	0	0
$34$	−506.775	−0.438387
$35$	− 884.449i	− 0.721999i
$36$	0	0
$37$	1444.27	1.05498	0.527492	−	0.849560i	$-0.323132\pi$
$37$	1444.27	1.05498	0.527492	+	0.849560i	$0.323132\pi$
$38$	− 100.618i	− 0.0696802i
$39$	0	0
$40$	2798.17	1.74886
$41$	− 2065.75i	− 1.22888i	−0.788964	−	0.614439i	$-0.789382\pi$
$41$	− 2065.75i	− 1.22888i	0.788964	−	0.614439i	$-0.210618\pi$
$42$	0	0
$43$	1174.58	0.635249	0.317625	−	0.948216i	$-0.397115\pi$
$43$	1174.58	0.635249	0.317625	+	0.948216i	$0.397115\pi$
$44$	− 1739.57i	− 0.898540i
$45$	0	0
$46$	−1782.55	−0.842416
$47$	3415.30i	1.54608i	0.634355	+	0.773042i	$0.281266\pi$
$47$	3415.30i	1.54608i	−0.634355	+	0.773042i	$0.718734\pi$
$48$	0	0
$49$	−2071.48	−0.862755
$50$	3625.71i	1.45028i
$51$	0	0
$52$	1911.16	0.706791
$53$	2949.55i	1.05004i	0.851091	+	0.525019i	$0.175942\pi$
$53$	2949.55i	1.05004i	−0.851091	+	0.525019i	$0.824058\pi$
$54$	0	0
$55$	7242.90	2.39435
$56$	1042.53i	0.332441i
$57$	0	0
$58$	414.525	0.123224
$59$	453.188i	0.130189i
$59$	453.188i	0.130189i
$60$	0	0
$61$	6279.67	1.68763	0.843814	−	0.536635i	$-0.180305\pi$
$61$	6279.67	1.68763	0.843814	+	0.536635i	$0.180305\pi$
$62$	− 2795.54i	− 0.727247i
$63$	0	0
$64$	−1107.34	−0.270347
$65$	7957.34i	1.88339i
$66$	0	0
$67$	4903.39	1.09231	0.546156	−	0.837684i	$-0.316091\pi$
$67$	4903.39	1.09231	0.546156	+	0.837684i	$0.316091\pi$
$68$	− 2860.48i	− 0.618615i
$69$	0	0
$70$	−1833.62	−0.374207
$71$	924.553i	0.183407i	0.995786	+	0.0917033i	$0.0292311\pi$
$71$	924.553i	0.183407i	−0.995786	+	0.0917033i	$0.970769\pi$
$72$	0	0
$73$	−4742.11	−0.889869	−0.444935	−	0.895563i	$-0.646773\pi$
$73$	−4742.11	−0.889869	−0.444935	+	0.895563i	$0.646773\pi$
$74$	− 2994.23i	− 0.546791i
$75$	0	0
$76$	567.936	0.0983269
$77$	2698.54i	0.455142i
$78$	0	0
$79$	7235.69	1.15938	0.579690	−	0.814837i	$-0.303174\pi$
$79$	7235.69	1.15938	0.579690	+	0.814837i	$0.303174\pi$
$80$	3321.26i	0.518947i
$81$	0	0
$82$	−4282.65	−0.636919
$83$	2645.08i	0.383957i	0.981399	+	0.191978i	$0.0614903\pi$
$83$	2645.08i	0.383957i	−0.981399	+	0.191978i	$0.938510\pi$
$84$	0	0
$85$	11909.9	1.64843
$86$	− 2435.10i	− 0.329246i
$87$	0	0
$88$	−8537.49	−1.10246
$89$	6034.79i	0.761872i	0.924601	+	0.380936i	$0.124398\pi$
$89$	6034.79i	0.761872i	−0.924601	+	0.380936i	$0.875602\pi$
$90$	0	0
$91$	−2964.72	−0.358014
$92$	− 10061.6i	− 1.18875i
$93$	0	0
$94$	7080.51	0.801325
$95$	2364.66i	0.262013i
$96$	0	0
$97$	6822.95	0.725151	0.362576	−	0.931954i	$-0.381897\pi$
$97$	6822.95	0.725151	0.362576	+	0.931954i	$0.381897\pi$
$98$	4294.53i	0.447160i
$99$	0	0
$100$	−20465.2	−2.04652
$101$	− 16538.9i	− 1.62130i	−0.585529	−	0.810651i	$-0.699113\pi$
$101$	− 16538.9i	− 1.62130i	0.585529	−	0.810651i	$-0.300887\pi$
$102$	0	0
$103$	−8759.60	−0.825676	−0.412838	−	0.910804i	$-0.635462\pi$
$103$	−8759.60	−0.825676	−0.412838	+	0.910804i	$0.635462\pi$
$104$	− 9379.62i	− 0.867198i
$105$	0	0
$106$	6114.94	0.544227
$107$	− 443.036i	− 0.0386964i	−0.999813	−	0.0193482i	$-0.993841\pi$
$107$	− 443.036i	− 0.0386964i	0.999813	−	0.0193482i	$-0.00615912\pi$
$108$	0	0
$109$	−4233.44	−0.356320	−0.178160	−	0.984002i	$-0.557014\pi$
$109$	−4233.44	−0.356320	−0.178160	+	0.984002i	$0.557014\pi$
$110$	− 15015.8i	− 1.24097i
$111$	0	0
$112$	−1237.42	−0.0986466
$113$	− 21957.0i	− 1.71956i	−0.510667	−	0.859779i	$-0.670601\pi$
$113$	− 21957.0i	− 1.71956i	0.510667	−	0.859779i	$-0.329399\pi$
$114$	0	0
$115$	41892.4	3.16767
$116$	2339.77i	0.173883i
$117$	0	0
$118$	939.536	0.0674760
$119$	4437.35i	0.313350i
$120$	0	0
$121$	−7457.78	−0.509377
$122$	− 13018.8i	− 0.874687i
$123$	0	0
$124$	15779.3	1.02623
$125$	− 54757.7i	− 3.50449i
$126$	0	0
$127$	1737.12	0.107701	0.0538507	−	0.998549i	$-0.482850\pi$
$127$	1737.12	0.107701	0.0538507	+	0.998549i	$0.482850\pi$
$128$	− 14667.8i	− 0.895249i
$129$	0	0
$130$	16496.9	0.976150
$131$	29390.6i	1.71264i	0.516448	+	0.856319i	$0.327254\pi$
$131$	29390.6i	1.71264i	−0.516448	+	0.856319i	$0.672746\pi$
$132$	0	0
$133$	−881.019	−0.0498060
$134$	− 10165.6i	− 0.566138i
$135$	0	0
$136$	−14038.7	−0.759011
$137$	25468.0i	1.35692i	0.734639	+	0.678458i	$0.237351\pi$
$137$	25468.0i	1.35692i	−0.734639	+	0.678458i	$0.762649\pi$
$138$	0	0
$139$	−10545.0	−0.545781	−0.272891	−	0.962045i	$-0.587980\pi$
$139$	−10545.0	−0.545781	−0.272891	+	0.962045i	$0.587980\pi$
$140$	− 10349.8i	− 0.528050i
$141$	0	0
$142$	1916.76	0.0950584
$143$	− 24278.6i	− 1.18727i
$144$	0	0
$145$	−9741.90	−0.463348
$146$	9831.22i	0.461213i
$147$	0	0
$148$	16900.8	0.771586
$149$	8582.60i	0.386586i	0.981141	+	0.193293i	$0.0619169\pi$
$149$	8582.60i	0.386586i	−0.981141	+	0.193293i	$0.938083\pi$
$150$	0	0
$151$	2198.87	0.0964374	0.0482187	−	0.998837i	$-0.484646\pi$
$151$	2198.87	0.0964374	0.0482187	+	0.998837i	$0.484646\pi$
$152$	− 2787.32i	− 0.120642i
$153$	0	0
$154$	5594.53	0.235897
$155$	65698.9i	2.73461i
$156$	0	0
$157$	40248.9	1.63288	0.816441	−	0.577430i	$-0.195944\pi$
$157$	40248.9	1.63288	0.816441	+	0.577430i	$0.195944\pi$
$158$	− 15000.8i	− 0.600899i
$159$	0	0
$160$	51656.3	2.01782
$161$	15608.1i	0.602142i
$162$	0	0
$163$	−33968.7	−1.27851	−0.639254	−	0.768995i	$-0.720757\pi$
$163$	−33968.7	−1.27851	−0.639254	+	0.768995i	$0.720757\pi$
$164$	− 24173.3i	− 0.898768i
$165$	0	0
$166$	5483.70	0.199002
$167$	− 54290.6i	− 1.94667i	−0.229398	−	0.973333i	$-0.573676\pi$
$167$	− 54290.6i	− 1.94667i	0.229398	−	0.973333i	$-0.426324\pi$
$168$	0	0
$169$	−1887.61	−0.0660904
$170$	− 24691.3i	− 0.854370i
$171$	0	0
$172$	13744.8	0.464604
$173$	− 51715.6i	− 1.72794i	−0.503540	−	0.863972i	$-0.667969\pi$
$173$	− 51715.6i	− 1.72794i	0.503540	−	0.863972i	$-0.332031\pi$
$174$	0	0
$175$	31746.9	1.03663
$176$	− 10133.5i	− 0.327140i
$177$	0	0
$178$	12511.2	0.394873
$179$	10992.9i	0.343088i	0.985176	+	0.171544i	$0.0548756\pi$
$179$	10992.9i	0.343088i	−0.985176	+	0.171544i	$0.945124\pi$
$180$	0	0
$181$	−19701.0	−0.601355	−0.300678	−	0.953726i	$-0.597213\pi$
$181$	−19701.0	−0.601355	−0.300678	+	0.953726i	$0.597213\pi$
$182$	6146.37i	0.185556i
$183$	0	0
$184$	−49380.2	−1.45854
$185$	70368.5i	2.05605i
$186$	0	0
$187$	−36338.2	−1.03916
$188$	39965.7i	1.13076i
$189$	0	0
$190$	4902.36	0.135799
$191$	31720.9i	0.869520i	0.900546	+	0.434760i	$0.143167\pi$
$191$	31720.9i	0.869520i	−0.900546	+	0.434760i	$0.856833\pi$
$192$	0	0
$193$	−10926.1	−0.293325	−0.146662	−	0.989187i	$-0.546853\pi$
$193$	−10926.1	−0.293325	−0.146662	+	0.989187i	$0.546853\pi$
$194$	− 14145.2i	− 0.375841i
$195$	0	0
$196$	−24240.3	−0.630995
$197$	− 32760.1i	− 0.844138i	−0.906564	−	0.422069i	$-0.861304\pi$
$197$	− 32760.1i	− 0.844138i	0.906564	−	0.422069i	$-0.138696\pi$
$198$	0	0
$199$	26763.6	0.675833	0.337916	−	0.941176i	$-0.390278\pi$
$199$	26763.6	0.675833	0.337916	+	0.941176i	$0.390278\pi$
$200$	100439.i	2.51098i
$201$	0	0
$202$	−34288.0	−0.840310
$203$	− 3629.60i	− 0.0880779i
$204$	0	0
$205$	100648.	2.39496
$206$	18160.2i	0.427942i
$207$	0	0
$208$	11133.0	0.257328
$209$	− 7214.81i	− 0.165171i
$210$	0	0
$211$	1841.68	0.0413666	0.0206833	−	0.999786i	$-0.493416\pi$
$211$	1841.68	0.0413666	0.0206833	+	0.999786i	$0.493416\pi$
$212$	34515.5i	0.767968i
$213$	0	0
$214$	−918.489	−0.0200561
$215$	57228.2i	1.23803i
$216$	0	0
$217$	−24477.9	−0.519822
$218$	8776.65i	0.184678i
$219$	0	0
$220$	84756.1	1.75116
$221$	− 39922.6i	− 0.817399i
$222$	0	0
$223$	−70637.6	−1.42045	−0.710225	−	0.703974i	$-0.751407\pi$
$223$	−70637.6	−1.42045	−0.710225	+	0.703974i	$0.751407\pi$
$224$	19245.9i	0.383568i
$225$	0	0
$226$	−45520.7	−0.891235
$227$	− 15595.1i	− 0.302647i	−0.988484	−	0.151324i	$-0.951646\pi$
$227$	− 15595.1i	− 0.302647i	0.988484	−	0.151324i	$-0.0483536\pi$
$228$	0	0
$229$	28251.7	0.538734	0.269367	−	0.963038i	$-0.413186\pi$
$229$	28251.7	0.538734	0.269367	+	0.963038i	$0.413186\pi$
$230$	− 86850.2i	− 1.64178i
$231$	0	0
$232$	11483.1	0.213346
$233$	− 61486.4i	− 1.13258i	−0.824208	−	0.566288i	$-0.808379\pi$
$233$	− 61486.4i	− 1.13258i	0.824208	−	0.566288i	$-0.191621\pi$
$234$	0	0
$235$	−166402.	−3.01316
$236$	5303.18i	0.0952165i
$237$	0	0
$238$	9199.40	0.162407
$239$	37030.8i	0.648288i	0.946008	+	0.324144i	$0.105076\pi$
$239$	37030.8i	0.648288i	−0.946008	+	0.324144i	$0.894924\pi$
$240$	0	0
$241$	18268.4	0.314533	0.157266	−	0.987556i	$-0.449732\pi$
$241$	18268.4	0.314533	0.157266	+	0.987556i	$0.449732\pi$
$242$	15461.3i	0.264006i
$243$	0	0
$244$	73484.4	1.23428
$245$	− 100927.i	− 1.68142i
$246$	0	0
$247$	7926.48	0.129923
$248$	− 77441.8i	− 1.25913i
$249$	0	0
$250$	−113522.	−1.81635
$251$	− 72994.5i	− 1.15862i	−0.815106	−	0.579312i	$-0.803321\pi$
$251$	− 72994.5i	− 1.15862i	0.815106	−	0.579312i	$-0.196679\pi$
$252$	0	0
$253$	−127818.	−1.99687
$254$	− 3601.34i	− 0.0558209i
$255$	0	0
$256$	−48126.3	−0.734349
$257$	− 11378.0i	− 0.172266i	−0.996284	−	0.0861331i	$-0.972549\pi$
$257$	− 11378.0i	− 0.172266i	0.996284	−	0.0861331i	$-0.0274510\pi$
$258$	0	0
$259$	−26217.6	−0.390836
$260$	93116.4i	1.37746i
$261$	0	0
$262$	60931.7	0.887648
$263$	48957.8i	0.707799i	0.935283	+	0.353900i	$0.115145\pi$
$263$	48957.8i	0.707799i	−0.935283	+	0.353900i	$0.884855\pi$
$264$	0	0
$265$	−143709.	−2.04641
$266$	1826.50i	0.0258141i
$267$	0	0
$268$	57379.2	0.798886
$269$	− 124579.i	− 1.72163i	−0.508914	−	0.860817i	$-0.669953\pi$
$269$	− 124579.i	− 1.72163i	0.508914	−	0.860817i	$-0.330047\pi$
$270$	0	0
$271$	−9306.75	−0.126724	−0.0633621	−	0.997991i	$-0.520182\pi$
$271$	−9306.75	−0.126724	−0.0633621	+	0.997991i	$0.520182\pi$
$272$	− 16663.0i	− 0.225225i
$273$	0	0
$274$	52799.5	0.703280
$275$	259981.i	3.43777i
$276$	0	0
$277$	−21764.8	−0.283658	−0.141829	−	0.989891i	$-0.545298\pi$
$277$	−21764.8	−0.283658	−0.141829	+	0.989891i	$0.545298\pi$
$278$	21861.7i	0.282875i
$279$	0	0
$280$	−50794.7	−0.647892
$281$	− 20634.3i	− 0.261323i	−0.991427	−	0.130661i	$-0.958290\pi$
$281$	− 20634.3i	− 0.261323i	0.991427	−	0.130661i	$-0.0417101\pi$
$282$	0	0
$283$	127917.	1.59719	0.798594	−	0.601871i	$-0.205578\pi$
$283$	127917.	1.59719	0.798594	+	0.601871i	$0.205578\pi$
$284$	10819.1i	0.134138i
$285$	0	0
$286$	−50333.7	−0.615356
$287$	37499.1i	0.455257i
$288$	0	0
$289$	23768.0	0.284575
$290$	20196.6i	0.240150i
$291$	0	0
$292$	−55492.0	−0.650826
$293$	122560.i	1.42763i	0.700335	+	0.713814i	$0.253034\pi$
$293$	122560.i	1.42763i	−0.700335	+	0.713814i	$0.746966\pi$
$294$	0	0
$295$	−22080.4	−0.253725
$296$	− 82946.0i	− 0.946699i
$297$	0	0
$298$	17793.2	0.200365
$299$	− 140425.i	− 1.57074i
$300$	0	0
$301$	−21321.9	−0.235338
$302$	− 4558.64i	− 0.0499829i
$303$	0	0
$304$	3308.38	0.0357988
$305$	305960.i	3.28901i
$306$	0	0
$307$	118945.	1.26202	0.631012	−	0.775773i	$-0.282640\pi$
$307$	118945.	1.26202	0.631012	+	0.775773i	$0.282640\pi$
$308$	31578.1i	0.332878i
$309$	0	0
$310$	136205.	1.41733
$311$	137534.i	1.42197i	0.703208	+	0.710984i	$0.251750\pi$
$311$	137534.i	1.42197i	−0.703208	+	0.710984i	$0.748250\pi$
$312$	0	0
$313$	−84348.8	−0.860974	−0.430487	−	0.902597i	$-0.641658\pi$
$313$	−84348.8	−0.860974	−0.430487	+	0.902597i	$0.641658\pi$
$314$	− 83442.9i	− 0.846311i
$315$	0	0
$316$	84671.8	0.847939
$317$	171375.i	1.70541i	0.522396	+	0.852703i	$0.325038\pi$
$317$	171375.i	1.70541i	−0.522396	+	0.852703i	$0.674962\pi$
$318$	0	0
$319$	29723.4	0.292091
$320$	− 53952.3i	− 0.526878i
$321$	0	0
$322$	32358.3	0.312086
$323$	− 11863.7i	− 0.113714i
$324$	0	0
$325$	−285625.	−2.70415
$326$	70423.0i	0.662643i
$327$	0	0
$328$	−118638.	−1.10274
$329$	− 61997.3i	− 0.572771i
$330$	0	0
$331$	166238.	1.51731	0.758653	−	0.651495i	$-0.225858\pi$
$331$	166238.	1.51731	0.758653	+	0.651495i	$0.225858\pi$
$332$	30952.6i	0.280815i
$333$	0	0
$334$	−112554.	−1.00894
$335$	238905.i	2.12880i
$336$	0	0
$337$	−7776.63	−0.0684749	−0.0342375	−	0.999414i	$-0.510900\pi$
$337$	−7776.63	−0.0684749	−0.0342375	+	0.999414i	$0.510900\pi$
$338$	3913.34i	0.0342542i
$339$	0	0
$340$	139369.	1.20562
$341$	− 200454.i	− 1.72387i
$342$	0	0
$343$	81188.0	0.690087
$344$	− 67457.0i	− 0.570046i
$345$	0	0
$346$	−107215.	−0.895581
$347$	152224.i	1.26422i	0.774878	+	0.632110i	$0.217811\pi$
$347$	152224.i	1.26422i	−0.774878	+	0.632110i	$0.782189\pi$
$348$	0	0
$349$	33115.2	0.271879	0.135940	−	0.990717i	$-0.456595\pi$
$349$	33115.2	0.271879	0.135940	+	0.990717i	$0.456595\pi$
$350$	− 65816.9i	− 0.537281i
$351$	0	0
$352$	−157608.	−1.27202
$353$	− 100888.i	− 0.809633i	−0.914398	−	0.404816i	$-0.867335\pi$
$353$	− 100888.i	− 0.809633i	0.914398	−	0.404816i	$-0.132665\pi$
$354$	0	0
$355$	−45046.4	−0.357440
$356$	70618.8i	0.557212i
$357$	0	0
$358$	22790.2	0.177820
$359$	106137.i	0.823524i	0.911291	+	0.411762i	$0.135086\pi$
$359$	106137.i	0.823524i	−0.911291	+	0.411762i	$0.864914\pi$
$360$	0	0
$361$	−127966.	−0.981925
$362$	40843.6i	0.311678i
$363$	0	0
$364$	−34693.0	−0.261842
$365$	− 231047.i	− 1.73426i
$366$	0	0
$367$	73226.3	0.543669	0.271835	−	0.962344i	$-0.412370\pi$
$367$	73226.3	0.543669	0.271835	+	0.962344i	$0.412370\pi$
$368$	− 58611.2i	− 0.432798i
$369$	0	0
$370$	145886.	1.06564
$371$	− 53542.7i	− 0.389003i
$372$	0	0
$373$	28374.9	0.203947	0.101973	−	0.994787i	$-0.467484\pi$
$373$	28374.9	0.203947	0.101973	+	0.994787i	$0.467484\pi$
$374$	75335.5i	0.538587i
$375$	0	0
$376$	196144.	1.38739
$377$	32655.3i	0.229758i
$378$	0	0
$379$	4193.13	0.0291917	0.0145959	−	0.999893i	$-0.495354\pi$
$379$	4193.13	0.0291917	0.0145959	+	0.999893i	$0.495354\pi$
$380$	27671.2i	0.191629i
$381$	0	0
$382$	65763.0	0.450666
$383$	− 89395.7i	− 0.609423i	−0.952445	−	0.304712i	$-0.901440\pi$
$383$	− 89395.7i	− 0.609423i	0.952445	−	0.304712i	$-0.0985601\pi$
$384$	0	0
$385$	−131479.	−0.887024
$386$	22651.6i	0.152028i
$387$	0	0
$388$	79841.8	0.530355
$389$	159295.i	1.05269i	0.850270	+	0.526347i	$0.176439\pi$
$389$	159295.i	1.05269i	−0.850270	+	0.526347i	$0.823561\pi$
$390$	0	0
$391$	−210178.	−1.37478
$392$	118967.i	0.774201i
$393$	0	0
$394$	−67917.4	−0.437511
$395$	352540.i	2.25951i
$396$	0	0
$397$	108240.	0.686760	0.343380	−	0.939197i	$-0.388428\pi$
$397$	108240.	0.686760	0.343380	+	0.939197i	$0.388428\pi$
$398$	− 55485.7i	− 0.350279i
$399$	0	0
$400$	−119215.	−0.745096
$401$	49561.6i	0.308217i	0.988054	+	0.154109i	$0.0492506\pi$
$401$	49561.6i	0.308217i	−0.988054	+	0.154109i	$0.950749\pi$
$402$	0	0
$403$	220226.	1.35600
$404$	− 193538.i	− 1.18578i
$405$	0	0
$406$	−7524.79	−0.0456502
$407$	− 214701.i	− 1.29612i
$408$	0	0
$409$	−206659.	−1.23540	−0.617700	−	0.786413i	$-0.711936\pi$
$409$	−206659.	−1.23540	−0.617700	+	0.786413i	$0.711936\pi$
$410$	− 208661.i	− 1.24129i
$411$	0	0
$412$	−102504.	−0.603877
$413$	− 8226.63i	− 0.0482305i
$414$	0	0
$415$	−128875.	−0.748292
$416$	− 173155.i	− 1.00057i
$417$	0	0
$418$	−14957.6	−0.0856068
$419$	− 55323.4i	− 0.315123i	−0.987509	−	0.157562i	$-0.949637\pi$
$419$	− 55323.4i	− 0.315123i	0.987509	−	0.157562i	$-0.0503633\pi$
$420$	0	0
$421$	−73603.1	−0.415271	−0.207636	−	0.978206i	$-0.566577\pi$
$421$	−73603.1	−0.415271	−0.207636	+	0.978206i	$0.566577\pi$
$422$	− 3818.12i	− 0.0214400i
$423$	0	0
$424$	169396.	0.942260
$425$	427501.i	2.36679i
$426$	0	0
$427$	−113994.	−0.625209
$428$	− 5184.38i	− 0.0283015i
$429$	0	0
$430$	118644.	0.641665
$431$	93027.5i	0.500791i	0.968144	+	0.250396i	$0.0805607\pi$
$431$	93027.5i	0.500791i	−0.968144	+	0.250396i	$0.919439\pi$
$432$	0	0
$433$	−322218.	−1.71860	−0.859298	−	0.511476i	$-0.829099\pi$
$433$	−322218.	−1.71860	−0.859298	+	0.511476i	$0.829099\pi$
$434$	50746.9i	0.269420i
$435$	0	0
$436$	−49539.5	−0.260603
$437$	− 41729.9i	− 0.218517i
$438$	0	0
$439$	174407.	0.904973	0.452486	−	0.891771i	$-0.350537\pi$
$439$	174407.	0.904973	0.452486	+	0.891771i	$0.350537\pi$
$440$	− 415967.i	− 2.14859i
$441$	0	0
$442$	−82766.5	−0.423653
$443$	68335.4i	0.348208i	0.984727	+	0.174104i	$0.0557028\pi$
$443$	68335.4i	0.348208i	−0.984727	+	0.174104i	$0.944297\pi$
$444$	0	0
$445$	−294029.	−1.48481
$446$	146444.i	0.736210i
$447$	0	0
$448$	20101.4	0.100154
$449$	− 3228.26i	− 0.0160131i	−0.999968	−	0.00800657i	$-0.997451\pi$
$449$	− 3228.26i	− 0.0160131i	0.999968	−	0.00800657i	$-0.00254860\pi$
$450$	0	0
$451$	−307087.	−1.50976
$452$	− 256940.i	− 1.25764i
$453$	0	0
$454$	−32331.4	−0.156860
$455$	− 144448.i	− 0.697733i
$456$	0	0
$457$	−67099.2	−0.321281	−0.160640	−	0.987013i	$-0.551356\pi$
$457$	−67099.2	−0.321281	−0.160640	+	0.987013i	$0.551356\pi$
$458$	− 58570.7i	− 0.279222i
$459$	0	0
$460$	490223.	2.31674
$461$	− 154767.i	− 0.728245i	−0.931351	−	0.364122i	$-0.881369\pi$
$461$	− 154767.i	− 0.728245i	0.931351	−	0.364122i	$-0.118631\pi$
$462$	0	0
$463$	−182419.	−0.850957	−0.425479	−	0.904968i	$-0.639894\pi$
$463$	−182419.	−0.850957	−0.425479	+	0.904968i	$0.639894\pi$
$464$	13629.8i	0.0633072i
$465$	0	0
$466$	−127472.	−0.587006
$467$	− 275387.i	− 1.26273i	−0.775486	−	0.631364i	$-0.782495\pi$
$467$	− 275387.i	− 1.26273i	0.775486	−	0.631364i	$-0.217505\pi$
$468$	0	0
$469$	−89010.3	−0.404664
$470$	344979.i	1.56170i
$471$	0	0
$472$	26027.0	0.116826
$473$	− 174609.i	− 0.780446i
$474$	0	0
$475$	−84878.7	−0.376194
$476$	51925.7i	0.229176i
$477$	0	0
$478$	76771.3	0.336003
$479$	344780.i	1.50270i	0.659906	+	0.751348i	$0.270596\pi$
$479$	344780.i	1.50270i	−0.659906	+	0.751348i	$0.729404\pi$
$480$	0	0
$481$	235879.	1.01953
$482$	− 37873.5i	− 0.163020i
$483$	0	0
$484$	−87270.6	−0.372544
$485$	332430.i	1.41324i
$486$	0	0
$487$	152407.	0.642610	0.321305	−	0.946976i	$-0.395879\pi$
$487$	152407.	0.642610	0.321305	+	0.946976i	$0.395879\pi$
$488$	− 360647.i	− 1.51441i
$489$	0	0
$490$	−209240.	−0.871468
$491$	− 64272.2i	− 0.266600i	−0.991076	−	0.133300i	$-0.957443\pi$
$491$	− 64272.2i	− 0.266600i	0.991076	−	0.133300i	$-0.0425574\pi$
$492$	0	0
$493$	48875.9	0.201095
$494$	− 16433.0i	− 0.0673382i
$495$	0	0
$496$	91918.7	0.373629
$497$	− 16783.2i	− 0.0679459i
$498$	0	0
$499$	−102636.	−0.412193	−0.206096	−	0.978532i	$-0.566076\pi$
$499$	−102636.	−0.412193	−0.206096	+	0.978532i	$0.566076\pi$
$500$	− 640772.i	− 2.56309i
$501$	0	0
$502$	−151330.	−0.600507
$503$	− 152496.i	− 0.602729i	−0.953509	−	0.301364i	$-0.902558\pi$
$503$	− 152496.i	− 0.602729i	0.953509	−	0.301364i	$-0.0974421\pi$
$504$	0	0
$505$	805815.	3.15975
$506$	264988.i	1.03496i
$507$	0	0
$508$	20327.7	0.0787698
$509$	18182.0i	0.0701787i	0.999384	+	0.0350893i	$0.0111716\pi$
$509$	18182.0i	0.0701787i	−0.999384	+	0.0350893i	$0.988828\pi$
$510$	0	0
$511$	86082.7	0.329666
$512$	− 134910.i	− 0.514641i
$513$	0	0
$514$	−23588.6	−0.0892845
$515$	− 426789.i	− 1.60916i
$516$	0	0
$517$	507707.	1.89947
$518$	54353.7i	0.202567i
$519$	0	0
$520$	456997.	1.69008
$521$	34399.8i	0.126730i	0.997990	+	0.0633651i	$0.0201833\pi$
$521$	34399.8i	0.126730i	−0.997990	+	0.0633651i	$0.979817\pi$
$522$	0	0
$523$	120401.	0.440175	0.220087	−	0.975480i	$-0.429366\pi$
$523$	120401.	0.440175	0.220087	+	0.975480i	$0.429366\pi$
$524$	343927.i	1.25258i
$525$	0	0
$526$	101498.	0.366848
$527$	− 329617.i	− 1.18683i
$528$	0	0
$529$	−459446.	−1.64181
$530$	297934.i	1.06064i
$531$	0	0
$532$	−10309.6	−0.0364267
$533$	− 337377.i	− 1.18758i
$534$	0	0
$535$	21585.8	0.0754153
$536$	− 281606.i	− 0.980195i
$537$	0	0
$538$	−258274.	−0.892311
$539$	307938.i	1.05995i
$540$	0	0
$541$	72749.1	0.248561	0.124281	−	0.992247i	$-0.460338\pi$
$541$	72749.1	0.248561	0.124281	+	0.992247i	$0.460338\pi$
$542$	19294.5i	0.0656803i
$543$	0	0
$544$	−259164.	−0.875743
$545$	− 206263.i	− 0.694430i
$546$	0	0
$547$	−403445.	−1.34837	−0.674186	−	0.738561i	$-0.735505\pi$
$547$	−403445.	−1.34837	−0.674186	+	0.738561i	$0.735505\pi$
$548$	298025.i	0.992410i
$549$	0	0
$550$	538986.	1.78177
$551$	9704.11i	0.0319634i
$552$	0	0
$553$	−131348.	−0.429511
$554$	45122.2i	0.147018i
$555$	0	0
$556$	−123398.	−0.399169
$557$	− 495395.i	− 1.59676i	−0.602151	−	0.798382i	$-0.705689\pi$
$557$	− 495395.i	− 1.59676i	0.602151	−	0.798382i	$-0.294311\pi$
$558$	0	0
$559$	191832.	0.613899
$560$	− 60290.2i	− 0.192252i
$561$	0	0
$562$	−42778.5	−0.135442
$563$	− 485634.i	− 1.53212i	−0.642770	−	0.766059i	$-0.722215\pi$
$563$	− 485634.i	− 1.53212i	0.642770	−	0.766059i	$-0.277785\pi$
$564$	0	0
$565$	1.06980e6	3.35124
$566$	− 265194.i	− 0.827811i
$567$	0	0
$568$	53097.9	0.164581
$569$	− 277909.i	− 0.858376i	−0.903215	−	0.429188i	$-0.858800\pi$
$569$	− 277909.i	− 0.858376i	0.903215	−	0.429188i	$-0.141200\pi$
$570$	0	0
$571$	−14581.8	−0.0447238	−0.0223619	−	0.999750i	$-0.507119\pi$
$571$	−14581.8	−0.0447238	−0.0223619	+	0.999750i	$0.507119\pi$
$572$	− 284107.i	− 0.868340i
$573$	0	0
$574$	77742.1	0.235957
$575$	1.50371e6i	4.54809i
$576$	0	0
$577$	214622.	0.644647	0.322323	−	0.946630i	$-0.395536\pi$
$577$	214622.	0.644647	0.322323	+	0.946630i	$0.395536\pi$
$578$	− 49275.2i	− 0.147494i
$579$	0	0
$580$	−113999.	−0.338880
$581$	− 48015.6i	− 0.142243i
$582$	0	0
$583$	438471.	1.29004
$584$	272344.i	0.798532i
$585$	0	0
$586$	254089.	0.739930
$587$	598824.i	1.73789i	0.494906	+	0.868947i	$0.335203\pi$
$587$	598824.i	1.73789i	−0.494906	+	0.868947i	$0.664797\pi$
$588$	0	0
$589$	65444.1	0.188643
$590$	45776.5i	0.131504i
$591$	0	0
$592$	98451.8	0.280918
$593$	162672.i	0.462597i	0.972883	+	0.231299i	$0.0742974\pi$
$593$	162672.i	0.462597i	−0.972883	+	0.231299i	$0.925703\pi$
$594$	0	0
$595$	−216198.	−0.610687
$596$	100433.i	0.282739i
$597$	0	0
$598$	−291126.	−0.814102
$599$	− 205591.i	− 0.572994i	−0.958081	−	0.286497i	$-0.907509\pi$
$599$	− 205591.i	− 0.572994i	0.958081	−	0.286497i	$-0.0924908\pi$
$600$	0	0
$601$	−453485.	−1.25549	−0.627746	−	0.778418i	$-0.716022\pi$
$601$	−453485.	−1.25549	−0.627746	+	0.778418i	$0.716022\pi$
$602$	44203.9i	0.121974i
$603$	0	0
$604$	25731.1	0.0705316
$605$	− 363361.i	− 0.992722i
$606$	0	0
$607$	−11946.3	−0.0324232	−0.0162116	−	0.999869i	$-0.505161\pi$
$607$	−11946.3	−0.0324232	−0.0162116	+	0.999869i	$0.505161\pi$
$608$	− 51456.0i	− 0.139197i
$609$	0	0
$610$	634309.	1.70467
$611$	557787.i	1.49412i
$612$	0	0
$613$	157880.	0.420152	0.210076	−	0.977685i	$-0.432629\pi$
$613$	157880.	0.420152	0.210076	+	0.977685i	$0.432629\pi$
$614$	− 246593.i	− 0.654099i
$615$	0	0
$616$	154979.	0.408425
$617$	347767.i	0.913520i	0.889590	+	0.456760i	$0.150990\pi$
$617$	347767.i	0.913520i	−0.889590	+	0.456760i	$0.849010\pi$
$618$	0	0
$619$	99468.6	0.259600	0.129800	−	0.991540i	$-0.458566\pi$
$619$	99468.6	0.259600	0.129800	+	0.991540i	$0.458566\pi$
$620$	768806.i	2.00002i
$621$	0	0
$622$	285132.	0.736996
$623$	− 109548.i	− 0.282248i
$624$	0	0
$625$	1.57488e6	4.03170
$626$	174870.i	0.446237i
$627$	0	0
$628$	470991.	1.19424
$629$	− 353044.i	− 0.892335i
$630$	0	0
$631$	−679438.	−1.70644	−0.853220	−	0.521551i	$-0.825354\pi$
$631$	−679438.	−1.70644	−0.853220	+	0.521551i	$0.825354\pi$
$632$	− 415553.i	− 1.04038i
$633$	0	0
$634$	355289.	0.883901
$635$	84636.4i	0.209899i
$636$	0	0
$637$	−338313.	−0.833758
$638$	− 61621.8i	− 0.151389i
$639$	0	0
$640$	714648.	1.74475
$641$	242746.i	0.590795i	0.955375	+	0.295397i	$0.0954520\pi$
$641$	242746.i	0.590795i	−0.955375	+	0.295397i	$0.904548\pi$
$642$	0	0
$643$	540528.	1.30736	0.653682	−	0.756769i	$-0.273223\pi$
$643$	540528.	1.30736	0.653682	+	0.756769i	$0.273223\pi$
$644$	182646.i	0.440390i
$645$	0	0
$646$	−24595.5	−0.0589374
$647$	− 267157.i	− 0.638201i	−0.947721	−	0.319101i	$-0.896619\pi$
$647$	− 267157.i	− 0.638201i	0.947721	−	0.319101i	$-0.103381\pi$
$648$	0	0
$649$	67369.3	0.159946
$650$	592151.i	1.40154i
$651$	0	0
$652$	−397500.	−0.935066
$653$	412898.i	0.968315i	0.874981	+	0.484158i	$0.160874\pi$
$653$	412898.i	0.968315i	−0.874981	+	0.484158i	$0.839126\pi$
$654$	0	0
$655$	−1.43198e6	−3.33775
$656$	− 140816.i	− 0.327223i
$657$	0	0
$658$	−128531.	−0.296863
$659$	− 174367.i	− 0.401507i	−0.979642	−	0.200754i	$-0.935661\pi$
$659$	− 174367.i	− 0.401507i	0.979642	−	0.200754i	$-0.0643391\pi$
$660$	0	0
$661$	−634783.	−1.45286	−0.726428	−	0.687243i	$-0.758821\pi$
$661$	−634783.	−1.45286	−0.726428	+	0.687243i	$0.758821\pi$
$662$	− 344639.i	− 0.786409i
$663$	0	0
$664$	151909.	0.344547
$665$	− 42925.3i	− 0.0970667i
$666$	0	0
$667$	171918.	0.386429
$668$	− 635306.i	− 1.42374i
$669$	0	0
$670$	495291.	1.10334
$671$	− 933514.i	− 2.07336i
$672$	0	0
$673$	−3331.58	−0.00735564	−0.00367782	−	0.999993i	$-0.501171\pi$
$673$	−3331.58	−0.00735564	−0.00367782	+	0.999993i	$0.501171\pi$
$674$	16122.3i	0.0354901i
$675$	0	0
$676$	−22088.7	−0.0483366
$677$	394187.i	0.860051i	0.902817	+	0.430026i	$0.141496\pi$
$677$	394187.i	0.860051i	−0.902817	+	0.430026i	$0.858504\pi$
$678$	0	0
$679$	−123856.	−0.268644
$680$	− 683997.i	− 1.47923i
$681$	0	0
$682$	−415575.	−0.893471
$683$	207457.i	0.444721i	0.974965	+	0.222360i	$0.0713761\pi$
$683$	207457.i	0.444721i	−0.974965	+	0.222360i	$0.928624\pi$
$684$	0	0
$685$	−1.24086e6	−2.64449
$686$	− 168317.i	− 0.357667i
$687$	0	0
$688$	80067.3	0.169153
$689$	481721.i	1.01475i
$690$	0	0
$691$	−177681.	−0.372121	−0.186061	−	0.982538i	$-0.559572\pi$
$691$	−177681.	−0.372121	−0.186061	+	0.982538i	$0.559572\pi$
$692$	− 605174.i	− 1.26377i
$693$	0	0
$694$	315586.	0.655237
$695$	− 513779.i	− 1.06367i
$696$	0	0
$697$	−504959.	−1.03942
$698$	− 68653.5i	− 0.140913i
$699$	0	0
$700$	371501.	0.758166
$701$	− 20623.3i	− 0.0419684i	−0.999780	−	0.0209842i	$-0.993320\pi$
$701$	− 20623.3i	− 0.0419684i	0.999780	−	0.0209842i	$-0.00667996\pi$
$702$	0	0
$703$	70095.6	0.141834
$704$	164614.i	0.332139i
$705$	0	0
$706$	−209157.	−0.419627
$707$	300228.i	0.600637i
$708$	0	0
$709$	−228304.	−0.454173	−0.227087	−	0.973875i	$-0.572920\pi$
$709$	−228304.	−0.454173	−0.227087	+	0.973875i	$0.572920\pi$
$710$	93389.0i	0.185259i
$711$	0	0
$712$	346584.	0.683673
$713$	− 1.15941e6i	− 2.28064i
$714$	0	0
$715$	1.18291e6	2.31387
$716$	128638.i	0.250925i
$717$	0	0
$718$	220039.	0.426827
$719$	− 270128.i	− 0.522531i	−0.965267	−	0.261265i	$-0.915860\pi$
$719$	− 270128.i	− 0.522531i	0.965267	−	0.261265i	$-0.0841397\pi$
$720$	0	0
$721$	159011.	0.305885
$722$	265295.i	0.508925i
$723$	0	0
$724$	−230540.	−0.439815
$725$	− 349682.i	− 0.665268i
$726$	0	0
$727$	−628019.	−1.18824	−0.594120	−	0.804376i	$-0.702500\pi$
$727$	−628019.	−1.18824	−0.594120	+	0.804376i	$0.702500\pi$
$728$	170266.i	0.321267i
$729$	0	0
$730$	−479001.	−0.898856
$731$	− 287118.i	− 0.537312i
$732$	0	0
$733$	16824.2	0.0313131	0.0156565	−	0.999877i	$-0.495016\pi$
$733$	16824.2	0.0313131	0.0156565	+	0.999877i	$0.495016\pi$
$734$	− 151811.i	− 0.281780i
$735$	0	0
$736$	−911594.	−1.68285
$737$	− 728921.i	− 1.34198i
$738$	0	0
$739$	859356.	1.57356	0.786781	−	0.617232i	$-0.211746\pi$
$739$	859356.	1.57356	0.786781	+	0.617232i	$0.211746\pi$
$740$	823449.i	1.50374i
$741$	0	0
$742$	−111003.	−0.201617
$743$	− 225378.i	− 0.408258i	−0.978944	−	0.204129i	$-0.934564\pi$
$743$	− 225378.i	− 0.408258i	0.978944	−	0.204129i	$-0.0654362\pi$
$744$	0	0
$745$	−418165.	−0.753417
$746$	− 58826.0i	− 0.105704i
$747$	0	0
$748$	−425228.	−0.760010
$749$	8042.35i	0.0143357i
$750$	0	0
$751$	126773.	0.224775	0.112388	−	0.993664i	$-0.464150\pi$
$751$	126773.	0.224775	0.112388	+	0.993664i	$0.464150\pi$
$752$	232811.i	0.411687i
$753$	0	0
$754$	67700.1	0.119082
$755$	107134.i	0.187946i
$756$	0	0
$757$	−484470.	−0.845425	−0.422712	−	0.906264i	$-0.638922\pi$
$757$	−484470.	−0.845425	−0.422712	+	0.906264i	$0.638922\pi$
$758$	− 8693.09i	− 0.0151299i
$759$	0	0
$760$	135805.	0.235119
$761$	1.03234e6i	1.78259i	0.453421	+	0.891297i	$0.350204\pi$
$761$	1.03234e6i	1.78259i	−0.453421	+	0.891297i	$0.649796\pi$
$762$	0	0
$763$	76848.8	0.132004
$764$	371197.i	0.635942i
$765$	0	0
$766$	−185333.	−0.315860
$767$	74014.6i	0.125813i
$768$	0	0
$769$	−1.04466e6	−1.76654	−0.883269	−	0.468867i	$-0.844662\pi$
$769$	−1.04466e6	−1.76654	−0.883269	+	0.468867i	$0.844662\pi$
$770$	272579.i	0.459739i
$771$	0	0
$772$	−127856.	−0.214530
$773$	− 890.549i	− 0.00149039i	−1.00000		0.000745193i	$-0.999763\pi$
$773$	− 890.549i	− 0.00149039i	1.00000		0.000745193i	$-0.000237202\pi$
$774$	0	0
$775$	−2.35824e6	−3.92631
$776$	− 391848.i	− 0.650721i
$777$	0	0
$778$	330245.	0.545604
$779$	− 100258.i	− 0.165212i
$780$	0	0
$781$	137441.	0.225327
$782$	435734.i	0.712539i
$783$	0	0
$784$	−141206.	−0.229732
$785$	1.96102e6i	3.18231i
$786$	0	0
$787$	−488745.	−0.789102	−0.394551	−	0.918874i	$-0.629100\pi$
$787$	−488745.	−0.789102	−0.394551	+	0.918874i	$0.629100\pi$
$788$	− 383358.i	− 0.617379i
$789$	0	0
$790$	730877.	1.17109
$791$	398582.i	0.637037i
$792$	0	0
$793$	1.02559e6	1.63091
$794$	− 224399.i	− 0.355943i
$795$	0	0
$796$	313187.	0.494285
$797$	− 439572.i	− 0.692012i	−0.938232	−	0.346006i	$-0.887538\pi$
$797$	− 439572.i	− 0.692012i	0.938232	−	0.346006i	$-0.112462\pi$
$798$	0	0
$799$	834850.	1.30772
$800$	1.85418e6i	2.89716i
$801$	0	0
$802$	102750.	0.159747
$803$	704946.i	1.09326i
$804$	0	0
$805$	−760465.	−1.17351
$806$	− 456567.i	− 0.702804i
$807$	0	0
$808$	−949845.	−1.45489
$809$	− 996878.i	− 1.52316i	−0.648072	−	0.761579i	$-0.724425\pi$
$809$	− 996878.i	− 1.52316i	0.648072	−	0.761579i	$-0.275575\pi$
$810$	0	0
$811$	−952365.	−1.44798	−0.723989	−	0.689812i	$-0.757693\pi$
$811$	−952365.	−1.44798	−0.723989	+	0.689812i	$0.757693\pi$
$812$	− 42473.4i	− 0.0644177i
$813$	0	0
$814$	−445112.	−0.671770
$815$	− 1.65504e6i	− 2.49168i
$816$	0	0
$817$	57006.2	0.0854040
$818$	428440.i	0.640300i
$819$	0	0
$820$	1.17778e6	1.75160
$821$	340183.i	0.504691i	0.967637	+	0.252346i	$0.0812020\pi$
$821$	340183.i	0.504691i	−0.967637	+	0.252346i	$0.918798\pi$
$822$	0	0
$823$	−21454.6	−0.0316754	−0.0158377	−	0.999875i	$-0.505041\pi$
$823$	−21454.6	−0.0316754	−0.0158377	+	0.999875i	$0.505041\pi$
$824$	503072.i	0.740928i
$825$	0	0
$826$	−17055.2	−0.0249976
$827$	− 48359.9i	− 0.0707089i	−0.999375	−	0.0353545i	$-0.988744\pi$
$827$	− 48359.9i	− 0.0707089i	0.999375	−	0.0353545i	$-0.0112560\pi$
$828$	0	0
$829$	−764789.	−1.11284	−0.556420	−	0.830901i	$-0.687825\pi$
$829$	−764789.	−1.11284	−0.556420	+	0.830901i	$0.687825\pi$
$830$	267179.i	0.387834i
$831$	0	0
$832$	−180851.	−0.261261
$833$	506360.i	0.729742i
$834$	0	0
$835$	2.64517e6	3.79385
$836$	− 84427.4i	− 0.120801i
$837$	0	0
$838$	−114695.	−0.163326
$839$	977835.i	1.38913i	0.719432	+	0.694563i	$0.244402\pi$
$839$	977835.i	1.38913i	−0.719432	+	0.694563i	$0.755598\pi$
$840$	0	0
$841$	667302.	0.943475
$842$	152592.i	0.215232i
$843$	0	0
$844$	21551.3	0.0302544
$845$	− 91968.7i	− 0.128803i
$846$	0	0
$847$	135380.	0.188707
$848$	201062.i	0.279601i
$849$	0	0
$850$	886284.	1.22669
$851$	− 1.24181e6i	− 1.71474i
$852$	0	0
$853$	180237.	0.247712	0.123856	−	0.992300i	$-0.460474\pi$
$853$	180237.	0.247712	0.123856	+	0.992300i	$0.460474\pi$
$854$	236329.i	0.324041i
$855$	0	0
$856$	−25444.0	−0.0347246
$857$	− 1.15844e6i	− 1.57730i	−0.614845	−	0.788648i	$-0.710782\pi$
$857$	− 1.15844e6i	− 1.57730i	0.614845	−	0.788648i	$-0.289218\pi$
$858$	0	0
$859$	−1.01178e6	−1.37120	−0.685601	−	0.727978i	$-0.740460\pi$
$859$	−1.01178e6	−1.37120	−0.685601	+	0.727978i	$0.740460\pi$
$860$	669681.i	0.905464i
$861$	0	0
$862$	192862.	0.259557
$863$	479943.i	0.644419i	0.946668	+	0.322209i	$0.104426\pi$
$863$	479943.i	0.644419i	−0.946668	+	0.322209i	$0.895574\pi$
$864$	0	0
$865$	2.51971e6	3.36758
$866$	668013.i	0.890736i
$867$	0	0
$868$	−286439.	−0.380183
$869$	− 1.07563e6i	− 1.42438i
$870$	0	0
$871$	800821.	1.05560
$872$	243130.i	0.319747i
$873$	0	0
$874$	−86513.3	−0.113256
$875$	994006.i	1.29829i
$876$	0	0
$877$	505182.	0.656824	0.328412	−	0.944535i	$-0.393487\pi$
$877$	505182.	0.656824	0.328412	+	0.944535i	$0.393487\pi$
$878$	− 361576.i	− 0.469041i
$879$	0	0
$880$	493727.	0.637561
$881$	486038.i	0.626207i	0.949719	+	0.313104i	$0.101369\pi$
$881$	486038.i	0.626207i	−0.949719	+	0.313104i	$0.898631\pi$
$882$	0	0
$883$	−826157.	−1.05960	−0.529799	−	0.848123i	$-0.677733\pi$
$883$	−826157.	−1.05960	−0.529799	+	0.848123i	$0.677733\pi$
$884$	− 467173.i	− 0.597823i
$885$	0	0
$886$	141671.	0.180474
$887$	− 894353.i	− 1.13674i	−0.822773	−	0.568371i	$-0.807574\pi$
$887$	− 894353.i	− 1.13674i	0.822773	−	0.568371i	$-0.192426\pi$
$888$	0	0
$889$	−31533.6	−0.0398997
$890$	609574.i	0.769567i
$891$	0	0
$892$	−826598.	−1.03888
$893$	165756.i	0.207858i
$894$	0	0
$895$	−535600.	−0.668643
$896$	266261.i	0.331659i
$897$	0	0
$898$	−6692.75	−0.00829950
$899$	269615.i	0.333599i
$900$	0	0
$901$	721001.	0.888151
$902$	636644.i	0.782498i
$903$	0	0
$904$	−1.26101e6	−1.54306
$905$	− 959880.i	− 1.17198i
$906$	0	0
$907$	−1.12743e6	−1.37049	−0.685246	−	0.728312i	$-0.740305\pi$
$907$	−1.12743e6	−1.37049	−0.685246	+	0.728312i	$0.740305\pi$
$908$	− 182493.i	− 0.221348i
$909$	0	0
$910$	−299466.	−0.361630
$911$	1.38711e6i	1.67137i	0.549208	+	0.835686i	$0.314929\pi$
$911$	1.38711e6i	1.67137i	−0.549208	+	0.835686i	$0.685071\pi$
$912$	0	0
$913$	393208.	0.471716
$914$	139108.i	0.166518i
$915$	0	0
$916$	330600.	0.394015
$917$	− 533522.i	− 0.634473i
$918$	0	0
$919$	−601069.	−0.711694	−0.355847	−	0.934544i	$-0.615808\pi$
$919$	−601069.	−0.711694	−0.355847	+	0.934544i	$0.615808\pi$
$920$	− 2.40592e6i	− 2.84253i
$921$	0	0
$922$	−320859.	−0.377444
$923$	150998.i	0.177242i
$924$	0	0
$925$	−2.52585e6	−2.95205
$926$	378186.i	0.441045i
$927$	0	0
$928$	211987.	0.246158
$929$	427806.i	0.495696i	0.968799	+	0.247848i	$0.0797233\pi$
$929$	427806.i	0.495696i	−0.968799	+	0.247848i	$0.920277\pi$
$930$	0	0
$931$	−100536.	−0.115990
$932$	− 719511.i	− 0.828334i
$933$	0	0
$934$	−570925.	−0.654464
$935$	− 1.77049e6i	− 2.02521i
$936$	0	0
$937$	−547905.	−0.624059	−0.312030	−	0.950072i	$-0.601009\pi$
$937$	−547905.	−0.624059	−0.312030	+	0.950072i	$0.601009\pi$
$938$	184534.i	0.209735i
$939$	0	0
$940$	−1.94722e6	−2.20374
$941$	974010.i	1.09998i	0.835172	+	0.549989i	$0.185368\pi$
$941$	974010.i	1.09998i	−0.835172	+	0.549989i	$0.814632\pi$
$942$	0	0
$943$	−1.77616e6	−1.99738
$944$	30892.4i	0.0346664i
$945$	0	0
$946$	−361994.	−0.404500
$947$	1.11732e6i	1.24588i	0.782268	+	0.622942i	$0.214063\pi$
$947$	1.11732e6i	1.24588i	−0.782268	+	0.622942i	$0.785937\pi$
$948$	0	0
$949$	−774482.	−0.859961
$950$	175968.i	0.194979i
$951$	0	0
$952$	254841.	0.281187
$953$	1.04491e6i	1.15051i	0.817973	+	0.575256i	$0.195098\pi$
$953$	1.04491e6i	1.15051i	−0.817973	+	0.575256i	$0.804902\pi$
$954$	0	0
$955$	−1.54552e6	−1.69460
$956$	433333.i	0.474140i
$957$	0	0
$958$	714789.	0.778837
$959$	− 462315.i	− 0.502691i
$960$	0	0
$961$	894753.	0.968850
$962$	− 489017.i	− 0.528414i
$963$	0	0
$964$	213776.	0.230041
$965$	− 532343.i	− 0.571659i
$966$	0	0
$967$	882303.	0.943550	0.471775	−	0.881719i	$-0.343613\pi$
$967$	882303.	0.943550	0.471775	+	0.881719i	$0.343613\pi$
$968$	428307.i	0.457093i
$969$	0	0
$970$	689186.	0.732475
$971$	1.63098e6i	1.72986i	0.501895	+	0.864929i	$0.332637\pi$
$971$	1.63098e6i	1.72986i	−0.501895	+	0.864929i	$0.667363\pi$
$972$	0	0
$973$	191422.	0.202193
$974$	− 315966.i	− 0.333060i
$975$	0	0
$976$	428066.	0.449377
$977$	− 209686.i	− 0.219675i	−0.993950	−	0.109838i	$-0.964967\pi$
$977$	− 209686.i	− 0.219675i	0.993950	−	0.109838i	$-0.0350331\pi$
$978$	0	0
$979$	897111.	0.936011
$980$	− 1.18105e6i	− 1.22974i
$981$	0	0
$982$	−133247.	−0.138177
$983$	− 1.18352e6i	− 1.22481i	−0.790545	−	0.612404i	$-0.790203\pi$
$983$	− 1.18352e6i	− 1.22481i	0.790545	−	0.612404i	$-0.209797\pi$
$984$	0	0
$985$	1.59615e6	1.64514
$986$	− 101328.i	− 0.104226i
$987$	0	0
$988$	92755.3	0.0950221
$989$	− 1.00992e6i	− 1.03251i
$990$	0	0
$991$	375098.	0.381942	0.190971	−	0.981596i	$-0.438836\pi$
$991$	375098.	0.381942	0.190971	+	0.981596i	$0.438836\pi$
$992$	− 1.42963e6i	− 1.45278i
$993$	0	0
$994$	−34794.6	−0.0352159
$995$	1.30399e6i	1.31713i
$996$	0	0
$997$	295629.	0.297411	0.148706	−	0.988882i	$-0.452489\pi$
$997$	295629.	0.297411	0.148706	+	0.988882i	$0.452489\pi$
$998$	212783.i	0.213637i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.28	✓	76
3.2	odd	2	inner	531.5.b.a.296.49	yes	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.28	✓	76	1.1	even	1	trivial
531.5.b.a.296.49	yes	76	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.07317i q^{2} +11.7020 q^{4} +48.7224i q^{5} -18.1528 q^{7} -57.4309i q^{8} +O(q^{10})\)	\(q-2.07317i q^{2} +11.7020 q^{4} +48.7224i q^{5} -18.1528 q^{7} -57.4309i q^{8} +101.010 q^{10} -148.657i q^{11} +163.320 q^{13} +37.6339i q^{14} +68.1670 q^{16} -244.444i q^{17} +48.5334 q^{19} +570.147i q^{20} -308.191 q^{22} -859.818i q^{23} -1748.87 q^{25} -338.590i q^{26} -212.423 q^{28} +199.947i q^{29} +1348.43 q^{31} -1060.22i q^{32} -506.775 q^{34} -884.449i q^{35} +1444.27 q^{37} -100.618i q^{38} +2798.17 q^{40} -2065.75i q^{41} +1174.58 q^{43} -1739.57i q^{44} -1782.55 q^{46} +3415.30i q^{47} -2071.48 q^{49} +3625.71i q^{50} +1911.16 q^{52} +2949.55i q^{53} +7242.90 q^{55} +1042.53i q^{56} +414.525 q^{58} +453.188i q^{59} +6279.67 q^{61} -2795.54i q^{62} -1107.34 q^{64} +7957.34i q^{65} +4903.39 q^{67} -2860.48i q^{68} -1833.62 q^{70} +924.553i q^{71} -4742.11 q^{73} -2994.23i q^{74} +567.936 q^{76} +2698.54i q^{77} +7235.69 q^{79} +3321.26i q^{80} -4282.65 q^{82} +2645.08i q^{83} +11909.9 q^{85} -2435.10i q^{86} -8537.49 q^{88} +6034.79i q^{89} -2964.72 q^{91} -10061.6i q^{92} +7080.51 q^{94} +2364.66i q^{95} +6822.95 q^{97} +4294.53i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 76 * q - 632 * q^4 - 48 * q^7	\( 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) \) 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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