LMFDB - Embedded newform 531.5.b.a.296.67

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			296.67
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.40563i q^{2} -25.0320 q^{4} -18.5372i q^{5} +29.7242 q^{7} -57.8559i q^{8} +O(q^{10})$	\(q+6.40563i q^{2} -25.0320 q^{4} -18.5372i q^{5} +29.7242 q^{7} -57.8559i q^{8} +118.743 q^{10} -32.7748i q^{11} -137.661 q^{13} +190.402i q^{14} -29.9096 q^{16} +443.487i q^{17} -630.893 q^{19} +464.025i q^{20} +209.943 q^{22} -445.913i q^{23} +281.371 q^{25} -881.805i q^{26} -744.057 q^{28} -606.766i q^{29} +1621.00 q^{31} -1117.28i q^{32} -2840.81 q^{34} -551.004i q^{35} +873.840 q^{37} -4041.26i q^{38} -1072.49 q^{40} -1861.83i q^{41} +2797.58 q^{43} +820.420i q^{44} +2856.35 q^{46} +1713.95i q^{47} -1517.47 q^{49} +1802.36i q^{50} +3445.94 q^{52} -3467.53i q^{53} -607.554 q^{55} -1719.72i q^{56} +3886.71 q^{58} -453.188i q^{59} +4806.75 q^{61} +10383.5i q^{62} +6678.35 q^{64} +2551.85i q^{65} -3864.25 q^{67} -11101.4i q^{68} +3529.52 q^{70} -9178.02i q^{71} -3348.84 q^{73} +5597.49i q^{74} +15792.5 q^{76} -974.203i q^{77} +7875.77 q^{79} +554.441i q^{80} +11926.2 q^{82} +1055.87i q^{83} +8221.02 q^{85} +17920.2i q^{86} -1896.21 q^{88} -13370.5i q^{89} -4091.86 q^{91} +11162.1i q^{92} -10978.9 q^{94} +11695.0i q^{95} -1572.85 q^{97} -9720.37i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	6.40563i	1.60141i	0.599061	+	0.800703i	$0.295541\pi$
$2$	6.40563i	1.60141i	−0.599061	+	0.800703i	$0.704459\pi$
$3$	0	0
$3$	0	0
$4$	−25.0320	−1.56450
$5$	− 18.5372i	− 0.741489i	−0.928735	−	0.370745i	$-0.879103\pi$
$5$	− 18.5372i	− 0.741489i	0.928735	−	0.370745i	$-0.120897\pi$
$6$	0	0
$7$	29.7242	0.606616	0.303308	−	0.952893i	$-0.401909\pi$
$7$	29.7242	0.606616	0.303308	+	0.952893i	$0.401909\pi$
$8$	− 57.8559i	− 0.903998i
$9$	0	0
$10$	118.743	1.18743
$11$	− 32.7748i	− 0.270866i	−0.990787	−	0.135433i	$-0.956757\pi$
$11$	− 32.7748i	− 0.270866i	0.990787	−	0.135433i	$-0.0432425\pi$
$12$	0	0
$13$	−137.661	−0.814562	−0.407281	−	0.913303i	$-0.633523\pi$
$13$	−137.661	−0.814562	−0.407281	+	0.913303i	$0.633523\pi$
$14$	190.402i	0.971439i
$15$	0	0
$16$	−29.9096	−0.116834
$17$	443.487i	1.53456i	0.641314	+	0.767278i	$0.278390\pi$
$17$	443.487i	1.53456i	−0.641314	+	0.767278i	$0.721610\pi$
$18$	0	0
$19$	−630.893	−1.74763	−0.873813	−	0.486262i	$-0.838360\pi$
$19$	−630.893	−1.74763	−0.873813	+	0.486262i	$0.838360\pi$
$20$	464.025i	1.16006i
$21$	0	0
$22$	209.943	0.433766
$23$	− 445.913i	− 0.842937i	−0.906843	−	0.421468i	$-0.861515\pi$
$23$	− 445.913i	− 0.842937i	0.906843	−	0.421468i	$-0.138485\pi$
$24$	0	0
$25$	281.371	0.450194
$26$	− 881.805i	− 1.30445i
$27$	0	0
$28$	−744.057	−0.949052
$29$	− 606.766i	− 0.721481i	−0.932666	−	0.360741i	$-0.882524\pi$
$29$	− 606.766i	− 0.721481i	0.932666	−	0.360741i	$-0.117476\pi$
$30$	0	0
$31$	1621.00	1.68679	0.843393	−	0.537297i	$-0.180555\pi$
$31$	1621.00	1.68679	0.843393	+	0.537297i	$0.180555\pi$
$32$	− 1117.28i	− 1.09110i
$33$	0	0
$34$	−2840.81	−2.45745
$35$	− 551.004i	− 0.449799i
$36$	0	0
$37$	873.840	0.638305	0.319153	−	0.947703i	$-0.396602\pi$
$37$	873.840	0.638305	0.319153	+	0.947703i	$0.396602\pi$
$38$	− 4041.26i	− 2.79866i
$39$	0	0
$40$	−1072.49	−0.670305
$41$	− 1861.83i	− 1.10758i	−0.832658	−	0.553788i	$-0.813182\pi$
$41$	− 1861.83i	− 1.10758i	0.832658	−	0.553788i	$-0.186818\pi$
$42$	0	0
$43$	2797.58	1.51302	0.756511	−	0.653981i	$-0.226903\pi$
$43$	2797.58	1.51302	0.756511	+	0.653981i	$0.226903\pi$
$44$	820.420i	0.423770i
$45$	0	0
$46$	2856.35	1.34988
$47$	1713.95i	0.775895i	0.921682	+	0.387947i	$0.126816\pi$
$47$	1713.95i	0.775895i	−0.921682	+	0.387947i	$0.873184\pi$
$48$	0	0
$49$	−1517.47	−0.632017
$50$	1802.36i	0.720943i
$51$	0	0
$52$	3445.94	1.27439
$53$	− 3467.53i	− 1.23444i	−0.786792	−	0.617219i	$-0.788259\pi$
$53$	− 3467.53i	− 1.23444i	0.786792	−	0.617219i	$-0.211741\pi$
$54$	0	0
$55$	−607.554	−0.200844
$56$	− 1719.72i	− 0.548380i
$57$	0	0
$58$	3886.71	1.15538
$59$	− 453.188i	− 0.130189i
$59$	− 453.188i	− 0.130189i
$60$	0	0
$61$	4806.75	1.29179	0.645895	−	0.763426i	$-0.276484\pi$
$61$	4806.75	1.29179	0.645895	+	0.763426i	$0.276484\pi$
$62$	10383.5i	2.70123i
$63$	0	0
$64$	6678.35	1.63046
$65$	2551.85i	0.603989i
$66$	0	0
$67$	−3864.25	−0.860827	−0.430414	−	0.902632i	$-0.641632\pi$
$67$	−3864.25	−0.860827	−0.430414	+	0.902632i	$0.641632\pi$
$68$	− 11101.4i	− 2.40082i
$69$	0	0
$70$	3529.52	0.720311
$71$	− 9178.02i	− 1.82067i	−0.413867	−	0.910337i	$-0.635822\pi$
$71$	− 9178.02i	− 1.82067i	0.413867	−	0.910337i	$-0.364178\pi$
$72$	0	0
$73$	−3348.84	−0.628418	−0.314209	−	0.949354i	$-0.601739\pi$
$73$	−3348.84	−0.628418	−0.314209	+	0.949354i	$0.601739\pi$
$74$	5597.49i	1.02219i
$75$	0	0
$76$	15792.5	2.73417
$77$	− 974.203i	− 0.164312i
$78$	0	0
$79$	7875.77	1.26194	0.630971	−	0.775807i	$-0.282657\pi$
$79$	7875.77	1.26194	0.630971	+	0.775807i	$0.282657\pi$
$80$	554.441i	0.0866314i
$81$	0	0
$82$	11926.2	1.77368
$83$	1055.87i	0.153270i	0.997059	+	0.0766348i	$0.0244176\pi$
$83$	1055.87i	0.153270i	−0.997059	+	0.0766348i	$0.975582\pi$
$84$	0	0
$85$	8221.02	1.13786
$86$	17920.2i	2.42296i
$87$	0	0
$88$	−1896.21	−0.244862
$89$	− 13370.5i	− 1.68799i	−0.536353	−	0.843994i	$-0.680198\pi$
$89$	− 13370.5i	− 1.68799i	0.536353	−	0.843994i	$-0.319802\pi$
$90$	0	0
$91$	−4091.86	−0.494126
$92$	11162.1i	1.31878i
$93$	0	0
$94$	−10978.9	−1.24252
$95$	11695.0i	1.29585i
$96$	0	0
$97$	−1572.85	−0.167164	−0.0835822	−	0.996501i	$-0.526636\pi$
$97$	−1572.85	−0.167164	−0.0835822	+	0.996501i	$0.526636\pi$
$98$	− 9720.37i	− 1.01212i
$99$	0	0
$100$	−7043.30	−0.704330
$101$	12809.2i	1.25569i	0.778340	+	0.627843i	$0.216062\pi$
$101$	12809.2i	1.25569i	−0.778340	+	0.627843i	$0.783938\pi$
$102$	0	0
$103$	413.910	0.0390150	0.0195075	−	0.999810i	$-0.493790\pi$
$103$	413.910	0.0390150	0.0195075	+	0.999810i	$0.493790\pi$
$104$	7964.50i	0.736363i
$105$	0	0
$106$	22211.7	1.97684
$107$	− 15603.9i	− 1.36291i	−0.731861	−	0.681454i	$-0.761348\pi$
$107$	− 15603.9i	− 1.36291i	0.731861	−	0.681454i	$-0.238652\pi$
$108$	0	0
$109$	23274.9	1.95900	0.979502	−	0.201436i	$-0.0645608\pi$
$109$	23274.9	1.95900	0.979502	+	0.201436i	$0.0645608\pi$
$110$	− 3891.76i	− 0.321633i
$111$	0	0
$112$	−889.038	−0.0708735
$113$	− 20189.0i	− 1.58109i	−0.612402	−	0.790546i	$-0.709797\pi$
$113$	− 20189.0i	− 1.58109i	0.612402	−	0.790546i	$-0.290203\pi$
$114$	0	0
$115$	−8266.00	−0.625028
$116$	15188.6i	1.12876i
$117$	0	0
$118$	2902.95	0.208485
$119$	13182.3i	0.930887i
$120$	0	0
$121$	13566.8	0.926632
$122$	30790.3i	2.06868i
$123$	0	0
$124$	−40577.0	−2.63898
$125$	− 16801.6i	− 1.07530i
$126$	0	0
$127$	−12135.4	−0.752393	−0.376197	−	0.926540i	$-0.622768\pi$
$127$	−12135.4	−0.752393	−0.376197	+	0.926540i	$0.622768\pi$
$128$	24902.5i	1.51993i
$129$	0	0
$130$	−16346.2	−0.967232
$131$	29285.0i	1.70649i	0.521514	+	0.853243i	$0.325367\pi$
$131$	29285.0i	1.70649i	−0.521514	+	0.853243i	$0.674633\pi$
$132$	0	0
$133$	−18752.8	−1.06014
$134$	− 24753.0i	− 1.37853i
$135$	0	0
$136$	25658.3	1.38724
$137$	− 4399.11i	− 0.234382i	−0.993109	−	0.117191i	$-0.962611\pi$
$137$	− 4399.11i	− 0.234382i	0.993109	−	0.117191i	$-0.0373889\pi$
$138$	0	0
$139$	−16380.4	−0.847802	−0.423901	−	0.905709i	$-0.639340\pi$
$139$	−16380.4	−0.847802	−0.423901	+	0.905709i	$0.639340\pi$
$140$	13792.7i	0.703712i
$141$	0	0
$142$	58791.0	2.91564
$143$	4511.81i	0.220637i
$144$	0	0
$145$	−11247.8	−0.534971
$146$	− 21451.4i	− 1.00635i
$147$	0	0
$148$	−21874.0	−0.998630
$149$	22971.4i	1.03470i	0.855773	+	0.517352i	$0.173082\pi$
$149$	22971.4i	1.03470i	−0.855773	+	0.517352i	$0.826918\pi$
$150$	0	0
$151$	−13853.9	−0.607600	−0.303800	−	0.952736i	$-0.598255\pi$
$151$	−13853.9	−0.607600	−0.303800	+	0.952736i	$0.598255\pi$
$152$	36500.9i	1.57985i
$153$	0	0
$154$	6240.38	0.263130
$155$	− 30048.9i	− 1.25073i
$156$	0	0
$157$	8382.97	0.340094	0.170047	−	0.985436i	$-0.445608\pi$
$157$	8382.97	0.340094	0.170047	+	0.985436i	$0.445608\pi$
$158$	50449.3i	2.02088i
$159$	0	0
$160$	−20711.3	−0.809037
$161$	− 13254.4i	− 0.511339i
$162$	0	0
$163$	4249.18	0.159930	0.0799651	−	0.996798i	$-0.474519\pi$
$163$	4249.18	0.159930	0.0799651	+	0.996798i	$0.474519\pi$
$164$	46605.5i	1.73281i
$165$	0	0
$166$	−6763.54	−0.245447
$167$	− 40728.5i	− 1.46038i	−0.683245	−	0.730189i	$-0.739432\pi$
$167$	− 40728.5i	− 1.46038i	0.683245	−	0.730189i	$-0.260568\pi$
$168$	0	0
$169$	−9610.43	−0.336488
$170$	52660.8i	1.82217i
$171$	0	0
$172$	−70029.0	−2.36713
$173$	− 11562.3i	− 0.386324i	−0.981167	−	0.193162i	$-0.938126\pi$
$173$	− 11562.3i	− 0.386324i	0.981167	−	0.193162i	$-0.0618742\pi$
$174$	0	0
$175$	8363.53	0.273095
$176$	980.280i	0.0316464i
$177$	0	0
$178$	85646.7	2.70315
$179$	− 26376.1i	− 0.823199i	−0.911365	−	0.411599i	$-0.864970\pi$
$179$	− 26376.1i	− 0.823199i	0.911365	−	0.411599i	$-0.135030\pi$
$180$	0	0
$181$	−33691.5	−1.02840	−0.514201	−	0.857670i	$-0.671911\pi$
$181$	−33691.5	−1.02840	−0.514201	+	0.857670i	$0.671911\pi$
$182$	− 26210.9i	− 0.791297i
$183$	0	0
$184$	−25798.7	−0.762013
$185$	− 16198.6i	− 0.473296i
$186$	0	0
$187$	14535.2	0.415659
$188$	− 42903.7i	− 1.21389i
$189$	0	0
$190$	−74913.8	−2.07518
$191$	60835.4i	1.66759i	0.552073	+	0.833796i	$0.313837\pi$
$191$	60835.4i	1.66759i	−0.552073	+	0.833796i	$0.686163\pi$
$192$	0	0
$193$	37876.5	1.01685	0.508423	−	0.861108i	$-0.330229\pi$
$193$	37876.5	1.01685	0.508423	+	0.861108i	$0.330229\pi$
$194$	− 10075.1i	− 0.267698i
$195$	0	0
$196$	37985.5	0.988793
$197$	52067.4i	1.34163i	0.741623	+	0.670816i	$0.234056\pi$
$197$	52067.4i	1.34163i	−0.741623	+	0.670816i	$0.765944\pi$
$198$	0	0
$199$	58282.8	1.47175	0.735875	−	0.677117i	$-0.236771\pi$
$199$	58282.8	1.47175	0.735875	+	0.677117i	$0.236771\pi$
$200$	− 16279.0i	− 0.406974i
$201$	0	0
$202$	−82051.2	−2.01086
$203$	− 18035.6i	− 0.437662i
$204$	0	0
$205$	−34513.3	−0.821255
$206$	2651.35i	0.0624789i
$207$	0	0
$208$	4117.38	0.0951688
$209$	20677.4i	0.473372i
$210$	0	0
$211$	48803.8	1.09620	0.548098	−	0.836414i	$-0.315352\pi$
$211$	48803.8	1.09620	0.548098	+	0.836414i	$0.315352\pi$
$212$	86799.5i	1.93128i
$213$	0	0
$214$	99952.9	2.18257
$215$	− 51859.3i	− 1.12189i
$216$	0	0
$217$	48182.9	1.02323
$218$	149090.i	3.13716i
$219$	0	0
$220$	15208.3	0.314221
$221$	− 61050.9i	− 1.24999i
$222$	0	0
$223$	76803.9	1.54445	0.772225	−	0.635349i	$-0.219144\pi$
$223$	76803.9	1.54445	0.772225	+	0.635349i	$0.219144\pi$
$224$	− 33210.3i	− 0.661877i
$225$	0	0
$226$	129323.	2.53197
$227$	28589.3i	0.554820i	0.960752	+	0.277410i	$0.0894760\pi$
$227$	28589.3i	0.554820i	−0.960752	+	0.277410i	$0.910524\pi$
$228$	0	0
$229$	−76462.1	−1.45806	−0.729030	−	0.684482i	$-0.760028\pi$
$229$	−76462.1	−1.45806	−0.729030	+	0.684482i	$0.760028\pi$
$230$	− 52948.9i	− 1.00092i
$231$	0	0
$232$	−35105.0	−0.652218
$233$	− 40137.7i	− 0.739335i	−0.929164	−	0.369667i	$-0.879472\pi$
$233$	− 40137.7i	− 0.739335i	0.929164	−	0.369667i	$-0.120528\pi$
$234$	0	0
$235$	31771.9	0.575317
$236$	11344.2i	0.203681i
$237$	0	0
$238$	−84440.8	−1.49073
$239$	26767.6i	0.468612i	0.972163	+	0.234306i	$0.0752818\pi$
$239$	26767.6i	0.468612i	−0.972163	+	0.234306i	$0.924718\pi$
$240$	0	0
$241$	1680.86	0.0289399	0.0144699	−	0.999895i	$-0.495394\pi$
$241$	1680.86	0.0289399	0.0144699	+	0.999895i	$0.495394\pi$
$242$	86903.9i	1.48391i
$243$	0	0
$244$	−120323.	−2.02101
$245$	28129.7i	0.468634i
$246$	0	0
$247$	86849.4	1.42355
$248$	− 93784.4i	− 1.52485i
$249$	0	0
$250$	107625.	1.72200
$251$	− 43157.9i	− 0.685036i	−0.939511	−	0.342518i	$-0.888720\pi$
$251$	− 43157.9i	− 0.685036i	0.939511	−	0.342518i	$-0.111280\pi$
$252$	0	0
$253$	−14614.7	−0.228323
$254$	− 77734.5i	− 1.20489i
$255$	0	0
$256$	−52662.3	−0.803562
$257$	− 50727.5i	− 0.768028i	−0.923327	−	0.384014i	$-0.874541\pi$
$257$	− 50727.5i	− 0.768028i	0.923327	−	0.384014i	$-0.125459\pi$
$258$	0	0
$259$	25974.2	0.387206
$260$	− 63878.1i	− 0.944943i
$261$	0	0
$262$	−187589.	−2.73278
$263$	− 45750.0i	− 0.661424i	−0.943732	−	0.330712i	$-0.892711\pi$
$263$	− 45750.0i	− 0.661424i	0.943732	−	0.330712i	$-0.107289\pi$
$264$	0	0
$265$	−64278.5	−0.915322
$266$	− 120123.i	− 1.69771i
$267$	0	0
$268$	96730.1	1.34677
$269$	− 38115.8i	− 0.526746i	−0.964694	−	0.263373i	$-0.915165\pi$
$269$	− 38115.8i	− 0.526746i	0.964694	−	0.263373i	$-0.0848349\pi$
$270$	0	0
$271$	−22465.1	−0.305893	−0.152947	−	0.988234i	$-0.548876\pi$
$271$	−22465.1	−0.305893	−0.152947	+	0.988234i	$0.548876\pi$
$272$	− 13264.5i	− 0.179289i
$273$	0	0
$274$	28179.1	0.375341
$275$	− 9221.88i	− 0.121942i
$276$	0	0
$277$	−107669.	−1.40323	−0.701615	−	0.712556i	$-0.747538\pi$
$277$	−107669.	−1.40323	−0.701615	+	0.712556i	$0.747538\pi$
$278$	− 104927.i	− 1.35768i
$279$	0	0
$280$	−31878.8	−0.406617
$281$	38399.3i	0.486307i	0.969988	+	0.243153i	$0.0781818\pi$
$281$	38399.3i	0.486307i	−0.969988	+	0.243153i	$0.921818\pi$
$282$	0	0
$283$	−31777.5	−0.396777	−0.198389	−	0.980123i	$-0.563571\pi$
$283$	−31777.5	−0.396777	−0.198389	+	0.980123i	$0.563571\pi$
$284$	229745.i	2.84845i
$285$	0	0
$286$	−28901.0	−0.353330
$287$	− 55341.5i	− 0.671873i
$288$	0	0
$289$	−113160.	−1.35486
$290$	− 72048.9i	− 0.856705i
$291$	0	0
$292$	83828.2	0.983161
$293$	5251.78i	0.0611746i	0.999532	+	0.0305873i	$0.00973776\pi$
$293$	5251.78i	0.0611746i	−0.999532	+	0.0305873i	$0.990262\pi$
$294$	0	0
$295$	−8400.84	−0.0965337
$296$	− 50556.8i	− 0.577027i
$297$	0	0
$298$	−147147.	−1.65698
$299$	61384.9i	0.686624i
$300$	0	0
$301$	83155.7	0.917823
$302$	− 88742.8i	− 0.973015i
$303$	0	0
$304$	18869.7	0.204183
$305$	− 89103.9i	− 0.957849i
$306$	0	0
$307$	5997.74	0.0636371	0.0318186	−	0.999494i	$-0.489870\pi$
$307$	5997.74	0.0636371	0.0318186	+	0.999494i	$0.489870\pi$
$308$	24386.3i	0.257066i
$309$	0	0
$310$	192482.	2.00293
$311$	− 53405.2i	− 0.552157i	−0.961135	−	0.276079i	$-0.910965\pi$
$311$	− 53405.2i	− 0.552157i	0.961135	−	0.276079i	$-0.0890350\pi$
$312$	0	0
$313$	57812.3	0.590108	0.295054	−	0.955481i	$-0.404662\pi$
$313$	57812.3	0.590108	0.295054	+	0.955481i	$0.404662\pi$
$314$	53698.2i	0.544628i
$315$	0	0
$316$	−197147.	−1.97431
$317$	− 95275.8i	− 0.948121i	−0.880492	−	0.474061i	$-0.842788\pi$
$317$	− 95275.8i	− 0.948121i	0.880492	−	0.474061i	$-0.157212\pi$
$318$	0	0
$319$	−19886.6	−0.195425
$320$	− 123798.i	− 1.20897i
$321$	0	0
$322$	84902.8	0.818861
$323$	− 279793.i	− 2.68183i
$324$	0	0
$325$	−38733.9	−0.366711
$326$	27218.7i	0.256113i
$327$	0	0
$328$	−107718.	−1.00125
$329$	50945.8i	0.470670i
$330$	0	0
$331$	−132376.	−1.20824	−0.604121	−	0.796892i	$-0.706476\pi$
$331$	−132376.	−1.20824	−0.604121	+	0.796892i	$0.706476\pi$
$332$	− 26430.7i	− 0.239791i
$333$	0	0
$334$	260891.	2.33866
$335$	71632.5i	0.638294i
$336$	0	0
$337$	−18547.6	−0.163316	−0.0816580	−	0.996660i	$-0.526022\pi$
$337$	−18547.6	−0.163316	−0.0816580	+	0.996660i	$0.526022\pi$
$338$	− 61560.8i	− 0.538854i
$339$	0	0
$340$	−205789.	−1.78018
$341$	− 53128.0i	− 0.456893i
$342$	0	0
$343$	−116473.	−0.990008
$344$	− 161856.i	− 1.36777i
$345$	0	0
$346$	74063.6	0.618661
$347$	− 107681.i	− 0.894297i	−0.894460	−	0.447149i	$-0.852439\pi$
$347$	− 107681.i	− 0.894297i	0.894460	−	0.447149i	$-0.147561\pi$
$348$	0	0
$349$	−150396.	−1.23476	−0.617382	−	0.786663i	$-0.711807\pi$
$349$	−150396.	−1.23476	−0.617382	+	0.786663i	$0.711807\pi$
$350$	53573.6i	0.437336i
$351$	0	0
$352$	−36618.7	−0.295541
$353$	− 120299.i	− 0.965413i	−0.875782	−	0.482707i	$-0.839654\pi$
$353$	− 120299.i	− 0.965413i	0.875782	−	0.482707i	$-0.160346\pi$
$354$	0	0
$355$	−170135.	−1.35001
$356$	334692.i	2.64086i
$357$	0	0
$358$	168956.	1.31828
$359$	37128.9i	0.288086i	0.989571	+	0.144043i	$0.0460104\pi$
$359$	37128.9i	0.288086i	−0.989571	+	0.144043i	$0.953990\pi$
$360$	0	0
$361$	267705.	2.05420
$362$	− 215815.i	− 1.64689i
$363$	0	0
$364$	102428.	0.773062
$365$	62078.2i	0.465965i
$366$	0	0
$367$	229188.	1.70161	0.850803	−	0.525484i	$-0.176116\pi$
$367$	229188.	1.70161	0.850803	+	0.525484i	$0.176116\pi$
$368$	13337.1i	0.0984839i
$369$	0	0
$370$	103762.	0.757940
$371$	− 103070.i	− 0.748829i
$372$	0	0
$373$	−67878.4	−0.487881	−0.243941	−	0.969790i	$-0.578440\pi$
$373$	−67878.4	−0.487881	−0.243941	+	0.969790i	$0.578440\pi$
$374$	93107.0i	0.665639i
$375$	0	0
$376$	99162.2	0.701407
$377$	83528.0i	0.587692i
$378$	0	0
$379$	−165620.	−1.15302	−0.576508	−	0.817091i	$-0.695585\pi$
$379$	−165620.	−1.15302	−0.576508	+	0.817091i	$0.695585\pi$
$380$	− 292750.i	− 2.02735i
$381$	0	0
$382$	−389689.	−2.67049
$383$	− 58913.9i	− 0.401624i	−0.979630	−	0.200812i	$-0.935642\pi$
$383$	− 58913.9i	− 0.401624i	0.979630	−	0.200812i	$-0.0643581\pi$
$384$	0	0
$385$	−18059.0	−0.121835
$386$	242623.i	1.62838i
$387$	0	0
$388$	39371.6	0.261529
$389$	− 78199.0i	− 0.516776i	−0.966041	−	0.258388i	$-0.916809\pi$
$389$	− 78199.0i	− 0.516776i	0.966041	−	0.258388i	$-0.0831912\pi$
$390$	0	0
$391$	197757.	1.29353
$392$	87794.7i	0.571342i
$393$	0	0
$394$	−333524.	−2.14850
$395$	− 145995.i	− 0.935716i
$396$	0	0
$397$	−122506.	−0.777277	−0.388638	−	0.921390i	$-0.627054\pi$
$397$	−122506.	−0.777277	−0.388638	+	0.921390i	$0.627054\pi$
$398$	373338.i	2.35687i
$399$	0	0
$400$	−8415.70	−0.0525981
$401$	103707.i	0.644939i	0.946580	+	0.322469i	$0.104513\pi$
$401$	103707.i	0.644939i	−0.946580	+	0.322469i	$0.895487\pi$
$402$	0	0
$403$	−223149.	−1.37399
$404$	− 320642.i	− 1.96452i
$405$	0	0
$406$	115529.	0.700875
$407$	− 28639.9i	− 0.172895i
$408$	0	0
$409$	−55315.2	−0.330672	−0.165336	−	0.986237i	$-0.552871\pi$
$409$	−55315.2	−0.330672	−0.165336	+	0.986237i	$0.552871\pi$
$410$	− 221079.i	− 1.31516i
$411$	0	0
$412$	−10361.0	−0.0610391
$413$	− 13470.6i	− 0.0789747i
$414$	0	0
$415$	19573.0	0.113648
$416$	153806.i	0.888767i
$417$	0	0
$418$	−132452.	−0.758062
$419$	− 142368.i	− 0.810931i	−0.914110	−	0.405466i	$-0.867109\pi$
$419$	− 142368.i	− 0.810931i	0.914110	−	0.405466i	$-0.132891\pi$
$420$	0	0
$421$	109029.	0.615145	0.307573	−	0.951525i	$-0.400483\pi$
$421$	109029.	0.615145	0.307573	+	0.951525i	$0.400483\pi$
$422$	312619.i	1.75546i
$423$	0	0
$424$	−200617.	−1.11593
$425$	124784.i	0.690848i
$426$	0	0
$427$	142877.	0.783621
$428$	390598.i	2.13227i
$429$	0	0
$430$	332191.	1.79660
$431$	− 101136.i	− 0.544440i	−0.962235	−	0.272220i	$-0.912242\pi$
$431$	− 101136.i	− 0.544440i	0.962235	−	0.272220i	$-0.0877579\pi$
$432$	0	0
$433$	−187634.	−1.00078	−0.500388	−	0.865801i	$-0.666809\pi$
$433$	−187634.	−1.00078	−0.500388	+	0.865801i	$0.666809\pi$
$434$	308642.i	1.63861i
$435$	0	0
$436$	−582619.	−3.06487
$437$	281324.i	1.47314i
$438$	0	0
$439$	237768.	1.23374	0.616870	−	0.787065i	$-0.288400\pi$
$439$	237768.	1.23374	0.616870	+	0.787065i	$0.288400\pi$
$440$	35150.5i	0.181563i
$441$	0	0
$442$	391069.	2.00175
$443$	175383.i	0.893675i	0.894615	+	0.446837i	$0.147450\pi$
$443$	175383.i	0.893675i	−0.894615	+	0.446837i	$0.852550\pi$
$444$	0	0
$445$	−247853.	−1.25162
$446$	491977.i	2.47329i
$447$	0	0
$448$	198508.	0.989060
$449$	294798.i	1.46229i	0.682224	+	0.731143i	$0.261013\pi$
$449$	294798.i	1.46229i	−0.682224	+	0.731143i	$0.738987\pi$
$450$	0	0
$451$	−61021.2	−0.300005
$452$	505371.i	2.47362i
$453$	0	0
$454$	−183132.	−0.888492
$455$	75851.8i	0.366389i
$456$	0	0
$457$	372380.	1.78301	0.891505	−	0.453011i	$-0.149650\pi$
$457$	372380.	1.78301	0.891505	+	0.453011i	$0.149650\pi$
$458$	− 489787.i	− 2.33495i
$459$	0	0
$460$	206915.	0.977858
$461$	41703.3i	0.196231i	0.995175	+	0.0981156i	$0.0312815\pi$
$461$	41703.3i	0.196231i	−0.995175	+	0.0981156i	$0.968719\pi$
$462$	0	0
$463$	−44181.8	−0.206102	−0.103051	−	0.994676i	$-0.532860\pi$
$463$	−44181.8	−0.206102	−0.103051	+	0.994676i	$0.532860\pi$
$464$	18148.1i	0.0842938i
$465$	0	0
$466$	257107.	1.18398
$467$	− 216232.i	− 0.991485i	−0.868469	−	0.495743i	$-0.834896\pi$
$467$	− 216232.i	− 0.991485i	0.868469	−	0.495743i	$-0.165104\pi$
$468$	0	0
$469$	−114862.	−0.522191
$470$	203519.i	0.921317i
$471$	0	0
$472$	−26219.6	−0.117691
$473$	− 91690.0i	− 0.409826i
$474$	0	0
$475$	−177515.	−0.786771
$476$	− 329979.i	− 1.45637i
$477$	0	0
$478$	−171463.	−0.750439
$479$	153685.i	0.669825i	0.942249	+	0.334913i	$0.108707\pi$
$479$	153685.i	0.669825i	−0.942249	+	0.334913i	$0.891293\pi$
$480$	0	0
$481$	−120294.	−0.519940
$482$	10766.9i	0.0463445i
$483$	0	0
$484$	−339605.	−1.44972
$485$	29156.3i	0.123951i
$486$	0	0
$487$	−345723.	−1.45771	−0.728853	−	0.684670i	$-0.759946\pi$
$487$	−345723.	−1.45771	−0.728853	+	0.684670i	$0.759946\pi$
$488$	− 278099.i	− 1.16778i
$489$	0	0
$490$	−180189.	−0.750473
$491$	− 20495.1i	− 0.0850133i	−0.999096	−	0.0425067i	$-0.986466\pi$
$491$	− 20495.1i	− 0.0850133i	0.999096	−	0.0425067i	$-0.0135344\pi$
$492$	0	0
$493$	269093.	1.10715
$494$	556325.i	2.27968i
$495$	0	0
$496$	−48483.5	−0.197074
$497$	− 272809.i	− 1.10445i
$498$	0	0
$499$	259780.	1.04329	0.521645	−	0.853162i	$-0.325318\pi$
$499$	259780.	1.04329	0.521645	+	0.853162i	$0.325318\pi$
$500$	420579.i	1.68231i
$501$	0	0
$502$	276454.	1.09702
$503$	204298.i	0.807473i	0.914875	+	0.403736i	$0.132289\pi$
$503$	204298.i	0.807473i	−0.914875	+	0.403736i	$0.867711\pi$
$504$	0	0
$505$	237448.	0.931077
$506$	− 93616.4i	− 0.365638i
$507$	0	0
$508$	303773.	1.17712
$509$	392891.i	1.51648i	0.651976	+	0.758239i	$0.273940\pi$
$509$	392891.i	1.51648i	−0.651976	+	0.758239i	$0.726060\pi$
$510$	0	0
$511$	−99541.4	−0.381208
$512$	61104.6i	0.233096i
$513$	0	0
$514$	324941.	1.22993
$515$	− 7672.74i	− 0.0289292i
$516$	0	0
$517$	56174.4	0.210163
$518$	166381.i	0.620074i
$519$	0	0
$520$	147640.	0.546005
$521$	− 249929.i	− 0.920748i	−0.887725	−	0.460374i	$-0.847715\pi$
$521$	− 249929.i	− 0.920748i	0.887725	−	0.460374i	$-0.152285\pi$
$522$	0	0
$523$	−361636.	−1.32211	−0.661056	−	0.750337i	$-0.729891\pi$
$523$	−361636.	−1.32211	−0.661056	+	0.750337i	$0.729891\pi$
$524$	− 733063.i	− 2.66980i
$525$	0	0
$526$	293058.	1.05921
$527$	718893.i	2.58847i
$528$	0	0
$529$	81002.2	0.289458
$530$	− 411744.i	− 1.46580i
$531$	0	0
$532$	469420.	1.65859
$533$	256302.i	0.902190i
$534$	0	0
$535$	−289254.	−1.01058
$536$	223570.i	0.778186i
$537$	0	0
$538$	244156.	0.843534
$539$	49734.9i	0.171192i
$540$	0	0
$541$	397532.	1.35824	0.679121	−	0.734026i	$-0.262361\pi$
$541$	397532.	1.35824	0.679121	+	0.734026i	$0.262361\pi$
$542$	− 143903.i	− 0.489860i
$543$	0	0
$544$	495501.	1.67435
$545$	− 431452.i	− 1.45258i
$546$	0	0
$547$	−364724.	−1.21896	−0.609480	−	0.792801i	$-0.708622\pi$
$547$	−364724.	−1.21896	−0.609480	+	0.792801i	$0.708622\pi$
$548$	110119.i	0.366691i
$549$	0	0
$550$	59071.9	0.195279
$551$	382804.i	1.26088i
$552$	0	0
$553$	234101.	0.765513
$554$	− 689684.i	− 2.24714i
$555$	0	0
$556$	410034.	1.32639
$557$	− 74648.2i	− 0.240607i	−0.992737	−	0.120304i	$-0.961613\pi$
$557$	− 74648.2i	− 0.240607i	0.992737	−	0.120304i	$-0.0383868\pi$
$558$	0	0
$559$	−385117.	−1.23245
$560$	16480.3i	0.0525520i
$561$	0	0
$562$	−245971.	−0.778775
$563$	− 522150.i	− 1.64732i	−0.567083	−	0.823661i	$-0.691928\pi$
$563$	− 522150.i	− 1.64732i	0.567083	−	0.823661i	$-0.308072\pi$
$564$	0	0
$565$	−374248.	−1.17236
$566$	− 203555.i	− 0.635402i
$567$	0	0
$568$	−531002.	−1.64589
$569$	− 67984.0i	− 0.209982i	−0.994473	−	0.104991i	$-0.966519\pi$
$569$	− 67984.0i	− 0.209982i	0.994473	−	0.104991i	$-0.0334814\pi$
$570$	0	0
$571$	−368588.	−1.13050	−0.565248	−	0.824921i	$-0.691219\pi$
$571$	−368588.	−1.13050	−0.565248	+	0.824921i	$0.691219\pi$
$572$	− 112940.i	− 0.345188i
$573$	0	0
$574$	354497.	1.07594
$575$	− 125467.i	− 0.379485i
$576$	0	0
$577$	−137388.	−0.412665	−0.206332	−	0.978482i	$-0.566153\pi$
$577$	−137388.	−0.412665	−0.206332	+	0.978482i	$0.566153\pi$
$578$	− 724858.i	− 2.16969i
$579$	0	0
$580$	281554.	0.836963
$581$	31385.0i	0.0929758i
$582$	0	0
$583$	−113648.	−0.334367
$584$	193750.i	0.568088i
$585$	0	0
$586$	−33640.9	−0.0979654
$587$	− 95419.4i	− 0.276924i	−0.990368	−	0.138462i	$-0.955784\pi$
$587$	− 95419.4i	− 0.276924i	0.990368	−	0.138462i	$-0.0442159\pi$
$588$	0	0
$589$	−1.02268e6	−2.94787
$590$	− 53812.6i	− 0.154590i
$591$	0	0
$592$	−26136.2	−0.0745760
$593$	577212.i	1.64144i	0.571329	+	0.820721i	$0.306428\pi$
$593$	577212.i	1.64144i	−0.571329	+	0.820721i	$0.693572\pi$
$594$	0	0
$595$	244363.	0.690242
$596$	− 575022.i	− 1.61880i
$597$	0	0
$598$	−393209.	−1.09956
$599$	− 470493.i	− 1.31129i	−0.755068	−	0.655647i	$-0.772396\pi$
$599$	− 470493.i	− 1.31129i	0.755068	−	0.655647i	$-0.227604\pi$
$600$	0	0
$601$	528715.	1.46377	0.731885	−	0.681429i	$-0.238641\pi$
$601$	528715.	1.46377	0.731885	+	0.681429i	$0.238641\pi$
$602$	532664.i	1.46981i
$603$	0	0
$604$	346791.	0.950592
$605$	− 251491.i	− 0.687087i
$606$	0	0
$607$	−432534.	−1.17393	−0.586967	−	0.809611i	$-0.699678\pi$
$607$	−432534.	−1.17393	−0.586967	+	0.809611i	$0.699678\pi$
$608$	704886.i	1.90683i
$609$	0	0
$610$	570766.	1.53391
$611$	− 235944.i	− 0.632015i
$612$	0	0
$613$	−339482.	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$613$	−339482.	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$614$	38419.3i	0.101909i
$615$	0	0
$616$	−56363.4	−0.148537
$617$	− 267332.i	− 0.702232i	−0.936332	−	0.351116i	$-0.885802\pi$
$617$	− 267332.i	− 0.702232i	0.936332	−	0.351116i	$-0.114198\pi$
$618$	0	0
$619$	165165.	0.431060	0.215530	−	0.976497i	$-0.430852\pi$
$619$	165165.	0.431060	0.215530	+	0.976497i	$0.430852\pi$
$620$	752185.i	1.95678i
$621$	0	0
$622$	342094.	0.884228
$623$	− 397429.i	− 1.02396i
$624$	0	0
$625$	−135598.	−0.347131
$626$	370324.i	0.945002i
$627$	0	0
$628$	−209843.	−0.532077
$629$	387537.i	0.979516i
$630$	0	0
$631$	564821.	1.41857	0.709287	−	0.704919i	$-0.249017\pi$
$631$	564821.	1.41857	0.709287	+	0.704919i	$0.249017\pi$
$632$	− 455660.i	− 1.14079i
$633$	0	0
$634$	610301.	1.51833
$635$	224956.i	0.557892i
$636$	0	0
$637$	208897.	0.514817
$638$	− 127386.i	− 0.312954i
$639$	0	0
$640$	461622.	1.12701
$641$	− 425875.i	− 1.03649i	−0.855232	−	0.518246i	$-0.826585\pi$
$641$	− 425875.i	− 1.03649i	0.855232	−	0.518246i	$-0.173415\pi$
$642$	0	0
$643$	−34160.8	−0.0826239	−0.0413119	−	0.999146i	$-0.513154\pi$
$643$	−34160.8	−0.0826239	−0.0413119	+	0.999146i	$0.513154\pi$
$644$	331785.i	0.799991i
$645$	0	0
$646$	1.79225e6	4.29470
$647$	− 341938.i	− 0.816844i	−0.912793	−	0.408422i	$-0.866079\pi$
$647$	− 341938.i	− 0.816844i	0.912793	−	0.408422i	$-0.133921\pi$
$648$	0	0
$649$	−14853.1	−0.0352637
$650$	− 248115.i	− 0.587253i
$651$	0	0
$652$	−106366.	−0.250211
$653$	175603.i	0.411818i	0.978571	+	0.205909i	$0.0660150\pi$
$653$	175603.i	0.411818i	−0.978571	+	0.205909i	$0.933985\pi$
$654$	0	0
$655$	542863.	1.26534
$656$	55686.7i	0.129403i
$657$	0	0
$658$	−326340.	−0.753734
$659$	53477.9i	0.123141i	0.998103	+	0.0615707i	$0.0196110\pi$
$659$	53477.9i	0.123141i	−0.998103	+	0.0615707i	$0.980389\pi$
$660$	0	0
$661$	105581.	0.241647	0.120824	−	0.992674i	$-0.461446\pi$
$661$	105581.	0.241647	0.120824	+	0.992674i	$0.461446\pi$
$662$	− 847953.i	− 1.93489i
$663$	0	0
$664$	61088.5	0.138555
$665$	347624.i	0.786081i
$666$	0	0
$667$	−270565.	−0.608163
$668$	1.01952e6i	2.28476i
$669$	0	0
$670$	−458851.	−1.02217
$671$	− 157540.i	− 0.349902i
$672$	0	0
$673$	−298173.	−0.658322	−0.329161	−	0.944274i	$-0.606766\pi$
$673$	−298173.	−0.658322	−0.329161	+	0.944274i	$0.606766\pi$
$674$	− 118809.i	− 0.261535i
$675$	0	0
$676$	240569.	0.526436
$677$	− 32750.2i	− 0.0714557i	−0.999362	−	0.0357279i	$-0.988625\pi$
$677$	− 32750.2i	− 0.0714557i	0.999362	−	0.0357279i	$-0.0113750\pi$
$678$	0	0
$679$	−46751.7	−0.101405
$680$	− 475634.i	− 1.02862i
$681$	0	0
$682$	340318.	0.731671
$683$	− 202267.i	− 0.433594i	−0.976217	−	0.216797i	$-0.930439\pi$
$683$	− 202267.i	− 0.433594i	0.976217	−	0.216797i	$-0.0695609\pi$
$684$	0	0
$685$	−81547.4	−0.173792
$686$	− 746085.i	− 1.58540i
$687$	0	0
$688$	−83674.3	−0.176773
$689$	477344.i	1.00553i
$690$	0	0
$691$	540120.	1.13119	0.565593	−	0.824684i	$-0.308647\pi$
$691$	540120.	1.13119	0.565593	+	0.824684i	$0.308647\pi$
$692$	289428.i	0.604404i
$693$	0	0
$694$	689767.	1.43213
$695$	303647.i	0.628636i
$696$	0	0
$697$	825699.	1.69964
$698$	− 963378.i	− 1.97736i
$699$	0	0
$700$	−209356.	−0.427257
$701$	265495.i	0.540281i	0.962821	+	0.270141i	$0.0870702\pi$
$701$	265495.i	0.540281i	−0.962821	+	0.270141i	$0.912930\pi$
$702$	0	0
$703$	−551300.	−1.11552
$704$	− 218881.i	− 0.441635i
$705$	0	0
$706$	770591.	1.54602
$707$	380744.i	0.761719i
$708$	0	0
$709$	−483177.	−0.961200	−0.480600	−	0.876940i	$-0.659581\pi$
$709$	−483177.	−0.961200	−0.480600	+	0.876940i	$0.659581\pi$
$710$	− 1.08982e6i	− 2.16191i
$711$	0	0
$712$	−773565.	−1.52594
$713$	− 722826.i	− 1.42185i
$714$	0	0
$715$	83636.5	0.163600
$716$	660248.i	1.28790i
$717$	0	0
$718$	−237834.	−0.461343
$719$	416279.i	0.805242i	0.915367	+	0.402621i	$0.131901\pi$
$719$	416279.i	0.805242i	−0.915367	+	0.402621i	$0.868099\pi$
$720$	0	0
$721$	12303.1	0.0236671
$722$	1.71482e6i	3.28961i
$723$	0	0
$724$	843366.	1.60894
$725$	− 170726.i	− 0.324807i
$726$	0	0
$727$	283110.	0.535656	0.267828	−	0.963467i	$-0.413694\pi$
$727$	283110.	0.535656	0.267828	+	0.963467i	$0.413694\pi$
$728$	236738.i	0.446689i
$729$	0	0
$730$	−397649.	−0.746199
$731$	1.24069e6i	2.32182i
$732$	0	0
$733$	−961667.	−1.78985	−0.894925	−	0.446217i	$-0.852771\pi$
$733$	−961667.	−1.78985	−0.894925	+	0.446217i	$0.852771\pi$
$734$	1.46809e6i	2.72496i
$735$	0	0
$736$	−498212.	−0.919726
$737$	126650.i	0.233169i
$738$	0	0
$739$	360059.	0.659302	0.329651	−	0.944103i	$-0.393069\pi$
$739$	360059.	0.659302	0.329651	+	0.944103i	$0.393069\pi$
$740$	405483.i	0.740474i
$741$	0	0
$742$	660225.	1.19918
$743$	− 912337.i	− 1.65264i	−0.563203	−	0.826319i	$-0.690431\pi$
$743$	− 912337.i	− 1.65264i	0.563203	−	0.826319i	$-0.309569\pi$
$744$	0	0
$745$	425827.	0.767221
$746$	− 434804.i	− 0.781296i
$747$	0	0
$748$	−363845.	−0.650300
$749$	− 463814.i	− 0.826762i
$750$	0	0
$751$	−165236.	−0.292971	−0.146485	−	0.989213i	$-0.546796\pi$
$751$	−165236.	−0.292971	−0.146485	+	0.989213i	$0.546796\pi$
$752$	− 51263.6i	− 0.0906511i
$753$	0	0
$754$	−535049.	−0.941133
$755$	256813.i	0.450529i
$756$	0	0
$757$	930347.	1.62350	0.811752	−	0.584002i	$-0.198514\pi$
$757$	930347.	1.62350	0.811752	+	0.584002i	$0.198514\pi$
$758$	− 1.06090e6i	− 1.84645i
$759$	0	0
$760$	676625.	1.17144
$761$	544322.i	0.939910i	0.882690	+	0.469955i	$0.155730\pi$
$761$	544322.i	0.939910i	−0.882690	+	0.469955i	$0.844270\pi$
$762$	0	0
$763$	691828.	1.18836
$764$	− 1.52283e6i	− 2.60895i
$765$	0	0
$766$	377380.	0.643164
$767$	62386.3i	0.106047i
$768$	0	0
$769$	−809294.	−1.36853	−0.684264	−	0.729234i	$-0.739876\pi$
$769$	−809294.	−1.36853	−0.684264	+	0.729234i	$0.739876\pi$
$770$	− 115679.i	− 0.195108i
$771$	0	0
$772$	−948125.	−1.59086
$773$	1.00316e6i	1.67885i	0.543474	+	0.839426i	$0.317109\pi$
$773$	1.00316e6i	1.67885i	−0.543474	+	0.839426i	$0.682891\pi$
$774$	0	0
$775$	456103.	0.759381
$776$	90998.6i	0.151116i
$777$	0	0
$778$	500914.	0.827568
$779$	1.17462e6i	1.93563i
$780$	0	0
$781$	−300808.	−0.493159
$782$	1.26676e6i	2.07147i
$783$	0	0
$784$	45387.0	0.0738413
$785$	− 155397.i	− 0.252176i
$786$	0	0
$787$	−652885.	−1.05411	−0.527057	−	0.849830i	$-0.676704\pi$
$787$	−652885.	−1.05411	−0.527057	+	0.849830i	$0.676704\pi$
$788$	− 1.30335e6i	− 2.09899i
$789$	0	0
$790$	935189.	1.49846
$791$	− 600100.i	− 0.959116i
$792$	0	0
$793$	−661703.	−1.05224
$794$	− 784726.i	− 1.24474i
$795$	0	0
$796$	−1.45894e6	−2.30256
$797$	− 378935.i	− 0.596552i	−0.954480	−	0.298276i	$-0.903588\pi$
$797$	− 378935.i	− 0.596552i	0.954480	−	0.298276i	$-0.0964115\pi$
$798$	0	0
$799$	−760115.	−1.19065
$800$	− 314371.i	− 0.491205i
$801$	0	0
$802$	−664307.	−1.03281
$803$	109757.i	0.170217i
$804$	0	0
$805$	−245700.	−0.379152
$806$	− 1.42941e6i	− 2.20032i
$807$	0	0
$808$	741090.	1.13514
$809$	− 142668.i	− 0.217987i	−0.994042	−	0.108993i	$-0.965237\pi$
$809$	− 142668.i	− 0.217987i	0.994042	−	0.108993i	$-0.0347628\pi$
$810$	0	0
$811$	−616812.	−0.937802	−0.468901	−	0.883251i	$-0.655350\pi$
$811$	−616812.	−0.937802	−0.468901	+	0.883251i	$0.655350\pi$
$812$	451468.i	0.684723i
$813$	0	0
$814$	183457.	0.276875
$815$	− 78768.1i	− 0.118586i
$816$	0	0
$817$	−1.76497e6	−2.64420
$818$	− 354328.i	− 0.529540i
$819$	0	0
$820$	863937.	1.28486
$821$	− 466894.i	− 0.692679i	−0.938109	−	0.346340i	$-0.887424\pi$
$821$	− 466894.i	− 0.692679i	0.938109	−	0.346340i	$-0.112576\pi$
$822$	0	0
$823$	−337365.	−0.498081	−0.249040	−	0.968493i	$-0.580115\pi$
$823$	−337365.	−0.498081	−0.249040	+	0.968493i	$0.580115\pi$
$824$	− 23947.1i	− 0.0352695i
$825$	0	0
$826$	86287.8	0.126471
$827$	1.23300e6i	1.80282i	0.432966	+	0.901410i	$0.357467\pi$
$827$	1.23300e6i	1.80282i	−0.432966	+	0.901410i	$0.642533\pi$
$828$	0	0
$829$	1.02212e6	1.48728	0.743642	−	0.668578i	$-0.233097\pi$
$829$	1.02212e6	1.48728	0.743642	+	0.668578i	$0.233097\pi$
$830$	125377.i	0.181996i
$831$	0	0
$832$	−919348.	−1.32811
$833$	− 672980.i	− 0.969866i
$834$	0	0
$835$	−754993.	−1.08285
$836$	− 517597.i	− 0.740592i
$837$	0	0
$838$	911956.	1.29863
$839$	444263.i	0.631126i	0.948905	+	0.315563i	$0.102193\pi$
$839$	444263.i	0.631126i	−0.948905	+	0.315563i	$0.897807\pi$
$840$	0	0
$841$	339116.	0.479465
$842$	698398.i	0.985097i
$843$	0	0
$844$	−1.22166e6	−1.71500
$845$	178151.i	0.249502i
$846$	0	0
$847$	403262.	0.562109
$848$	103712.i	0.144225i
$849$	0	0
$850$	−799323.	−1.10633
$851$	− 389657.i	− 0.538051i
$852$	0	0
$853$	−531236.	−0.730112	−0.365056	−	0.930986i	$-0.618950\pi$
$853$	−531236.	−0.730112	−0.365056	+	0.930986i	$0.618950\pi$
$854$	915215.i	1.25490i
$855$	0	0
$856$	−902779.	−1.23207
$857$	994021.i	1.35342i	0.736248	+	0.676712i	$0.236596\pi$
$857$	994021.i	1.35342i	−0.736248	+	0.676712i	$0.763404\pi$
$858$	0	0
$859$	−320156.	−0.433885	−0.216943	−	0.976184i	$-0.569608\pi$
$859$	−320156.	−0.433885	−0.216943	+	0.976184i	$0.569608\pi$
$860$	1.29814e6i	1.75520i
$861$	0	0
$862$	647838.	0.871870
$863$	− 1.28196e6i	− 1.72129i	−0.509209	−	0.860643i	$-0.670062\pi$
$863$	− 1.28196e6i	− 1.72129i	0.509209	−	0.860643i	$-0.329938\pi$
$864$	0	0
$865$	−214333.	−0.286455
$866$	− 1.20192e6i	− 1.60265i
$867$	0	0
$868$	−1.20612e6	−1.60085
$869$	− 258127.i	− 0.341817i
$870$	0	0
$871$	531957.	0.701197
$872$	− 1.34659e6i	− 1.77094i
$873$	0	0
$874$	−1.80205e6	−2.35909
$875$	− 499414.i	− 0.652296i
$876$	0	0
$877$	911572.	1.18520	0.592600	−	0.805497i	$-0.298102\pi$
$877$	911572.	1.18520	0.592600	+	0.805497i	$0.298102\pi$
$878$	1.52305e6i	1.97572i
$879$	0	0
$880$	18171.7	0.0234655
$881$	1.17613e6i	1.51532i	0.652649	+	0.757660i	$0.273658\pi$
$881$	1.17613e6i	1.51532i	−0.652649	+	0.757660i	$0.726342\pi$
$882$	0	0
$883$	−1.15237e6	−1.47799	−0.738994	−	0.673712i	$-0.764699\pi$
$883$	−1.15237e6	−1.47799	−0.738994	+	0.673712i	$0.764699\pi$
$884$	1.52823e6i	1.95562i
$885$	0	0
$886$	−1.12344e6	−1.43114
$887$	− 453852.i	− 0.576855i	−0.957502	−	0.288428i	$-0.906868\pi$
$887$	− 453852.i	− 0.576855i	0.957502	−	0.288428i	$-0.0931325\pi$
$888$	0	0
$889$	−360713.	−0.456414
$890$	− 1.58765e6i	− 2.00436i
$891$	0	0
$892$	−1.92256e6	−2.41630
$893$	− 1.08132e6i	− 1.35597i
$894$	0	0
$895$	−488940.	−0.610393
$896$	740205.i	0.922011i
$897$	0	0
$898$	−1.88837e6	−2.34171
$899$	− 983568.i	− 1.21698i
$900$	0	0
$901$	1.53781e6	1.89431
$902$	− 390879.i	− 0.480429i
$903$	0	0
$904$	−1.16805e6	−1.42930
$905$	624546.i	0.762549i
$906$	0	0
$907$	−645880.	−0.785122	−0.392561	−	0.919726i	$-0.628411\pi$
$907$	−645880.	−0.785122	−0.392561	+	0.919726i	$0.628411\pi$
$908$	− 715649.i	− 0.868017i
$909$	0	0
$910$	−485878.	−0.586738
$911$	− 1.28705e6i	− 1.55082i	−0.631461	−	0.775408i	$-0.717544\pi$
$911$	− 1.28705e6i	− 1.55082i	0.631461	−	0.775408i	$-0.282456\pi$
$912$	0	0
$913$	34606.0	0.0415155
$914$	2.38533e6i	2.85532i
$915$	0	0
$916$	1.91400e6	2.28114
$917$	870473.i	1.03518i
$918$	0	0
$919$	−1.26310e6	−1.49558	−0.747788	−	0.663938i	$-0.768884\pi$
$919$	−1.26310e6	−1.49558	−0.747788	+	0.663938i	$0.768884\pi$
$920$	478237.i	0.565024i
$921$	0	0
$922$	−267135.	−0.314246
$923$	1.26346e6i	1.48305i
$924$	0	0
$925$	245873.	0.287361
$926$	− 283012.i	− 0.330052i
$927$	0	0
$928$	−677929.	−0.787206
$929$	363947.i	0.421703i	0.977518	+	0.210851i	$0.0676236\pi$
$929$	363947.i	0.421703i	−0.977518	+	0.210851i	$0.932376\pi$
$930$	0	0
$931$	957363.	1.10453
$932$	1.00473e6i	1.15669i
$933$	0	0
$934$	1.38510e6	1.58777
$935$	− 269442.i	− 0.308207i
$936$	0	0
$937$	1.01683e6	1.15817	0.579083	−	0.815268i	$-0.303411\pi$
$937$	1.01683e6	1.15817	0.579083	+	0.815268i	$0.303411\pi$
$938$	− 735761.i	− 0.836241i
$939$	0	0
$940$	−795316.	−0.900086
$941$	1.20432e6i	1.36007i	0.733180	+	0.680034i	$0.238035\pi$
$941$	1.20432e6i	1.36007i	−0.733180	+	0.680034i	$0.761965\pi$
$942$	0	0
$943$	−830217.	−0.933616
$944$	13554.7i	0.0152105i
$945$	0	0
$946$	587332.	0.656298
$947$	1.23315e6i	1.37504i	0.726165	+	0.687520i	$0.241301\pi$
$947$	1.23315e6i	1.37504i	−0.726165	+	0.687520i	$0.758699\pi$
$948$	0	0
$949$	461005.	0.511885
$950$	− 1.13710e6i	− 1.25994i
$951$	0	0
$952$	762673.	0.841520
$953$	943059.i	1.03837i	0.854661	+	0.519186i	$0.173765\pi$
$953$	943059.i	1.03837i	−0.854661	+	0.519186i	$0.826235\pi$
$954$	0	0
$955$	1.12772e6	1.23650
$956$	− 670048.i	− 0.733145i
$957$	0	0
$958$	−984451.	−1.07266
$959$	− 130760.i	− 0.142180i
$960$	0	0
$961$	1.70412e6	1.84525
$962$	− 770557.i	− 0.832635i
$963$	0	0
$964$	−42075.3	−0.0452765
$965$	− 702125.i	− 0.753980i
$966$	0	0
$967$	613188.	0.655753	0.327877	−	0.944721i	$-0.393667\pi$
$967$	613188.	0.655753	0.327877	+	0.944721i	$0.393667\pi$
$968$	− 784920.i	− 0.837673i
$969$	0	0
$970$	−186764.	−0.198495
$971$	− 1.73048e6i	− 1.83539i	−0.397288	−	0.917694i	$-0.630049\pi$
$971$	− 1.73048e6i	− 1.83539i	0.397288	−	0.917694i	$-0.369951\pi$
$972$	0	0
$973$	−486893.	−0.514290
$974$	− 2.21457e6i	− 2.33438i
$975$	0	0
$976$	−143768.	−0.150925
$977$	− 871924.i	− 0.913460i	−0.889605	−	0.456730i	$-0.849021\pi$
$977$	− 871924.i	− 0.913460i	0.889605	−	0.456730i	$-0.150979\pi$
$978$	0	0
$979$	−438217.	−0.457218
$980$	− 704145.i	− 0.733179i
$981$	0	0
$982$	131284.	0.136141
$983$	− 398184.i	− 0.412075i	−0.978544	−	0.206038i	$-0.933943\pi$
$983$	− 398184.i	− 0.412075i	0.978544	−	0.206038i	$-0.0660570\pi$
$984$	0	0
$985$	965186.	0.994806
$986$	1.72371e6i	1.77300i
$987$	0	0
$988$	−2.17402e6	−2.22715
$989$	− 1.24748e6i	− 1.27538i
$990$	0	0
$991$	−392895.	−0.400064	−0.200032	−	0.979789i	$-0.564105\pi$
$991$	−392895.	−0.400064	−0.200032	+	0.979789i	$0.564105\pi$
$992$	− 1.81112e6i	− 1.84045i
$993$	0	0
$994$	1.74751e6	1.76867
$995$	− 1.08040e6i	− 1.09129i
$996$	0	0
$997$	983000.	0.988925	0.494462	−	0.869199i	$-0.335365\pi$
$997$	983000.	0.988925	0.494462	+	0.869199i	$0.335365\pi$
$998$	1.66406e6i	1.67073i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.67	yes	76
3.2	odd	2	inner	531.5.b.a.296.10	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.10	✓	76	3.2	odd	2	inner
531.5.b.a.296.67	yes	76	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+6.40563i q^{2} -25.0320 q^{4} -18.5372i q^{5} +29.7242 q^{7} -57.8559i q^{8} +O(q^{10})\)	\(q+6.40563i q^{2} -25.0320 q^{4} -18.5372i q^{5} +29.7242 q^{7} -57.8559i q^{8} +118.743 q^{10} -32.7748i q^{11} -137.661 q^{13} +190.402i q^{14} -29.9096 q^{16} +443.487i q^{17} -630.893 q^{19} +464.025i q^{20} +209.943 q^{22} -445.913i q^{23} +281.371 q^{25} -881.805i q^{26} -744.057 q^{28} -606.766i q^{29} +1621.00 q^{31} -1117.28i q^{32} -2840.81 q^{34} -551.004i q^{35} +873.840 q^{37} -4041.26i q^{38} -1072.49 q^{40} -1861.83i q^{41} +2797.58 q^{43} +820.420i q^{44} +2856.35 q^{46} +1713.95i q^{47} -1517.47 q^{49} +1802.36i q^{50} +3445.94 q^{52} -3467.53i q^{53} -607.554 q^{55} -1719.72i q^{56} +3886.71 q^{58} -453.188i q^{59} +4806.75 q^{61} +10383.5i q^{62} +6678.35 q^{64} +2551.85i q^{65} -3864.25 q^{67} -11101.4i q^{68} +3529.52 q^{70} -9178.02i q^{71} -3348.84 q^{73} +5597.49i q^{74} +15792.5 q^{76} -974.203i q^{77} +7875.77 q^{79} +554.441i q^{80} +11926.2 q^{82} +1055.87i q^{83} +8221.02 q^{85} +17920.2i q^{86} -1896.21 q^{88} -13370.5i q^{89} -4091.86 q^{91} +11162.1i q^{92} -10978.9 q^{94} +11695.0i q^{95} -1572.85 q^{97} -9720.37i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 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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