LMFDB - Embedded newform 531.5.b.a.296.2

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			296.2
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.75

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.80173i q^{2} -44.8670 q^{4} +42.3216i q^{5} -83.7809 q^{7} +225.212i q^{8} +O(q^{10})$	\(q-7.80173i q^{2} -44.8670 q^{4} +42.3216i q^{5} -83.7809 q^{7} +225.212i q^{8} +330.181 q^{10} +79.9124i q^{11} +182.579 q^{13} +653.636i q^{14} +1039.17 q^{16} +201.361i q^{17} -385.359 q^{19} -1898.84i q^{20} +623.455 q^{22} -258.108i q^{23} -1166.12 q^{25} -1424.43i q^{26} +3758.99 q^{28} +1250.23i q^{29} +229.370 q^{31} -4503.95i q^{32} +1570.96 q^{34} -3545.74i q^{35} -504.893 q^{37} +3006.47i q^{38} -9531.33 q^{40} +2916.72i q^{41} -767.722 q^{43} -3585.43i q^{44} -2013.69 q^{46} +2520.35i q^{47} +4618.24 q^{49} +9097.72i q^{50} -8191.76 q^{52} -4215.55i q^{53} -3382.02 q^{55} -18868.5i q^{56} +9753.93 q^{58} -453.188i q^{59} -3836.27 q^{61} -1789.48i q^{62} -18511.8 q^{64} +7727.03i q^{65} -2946.00 q^{67} -9034.46i q^{68} -27662.9 q^{70} -2974.94i q^{71} +679.596 q^{73} +3939.04i q^{74} +17289.9 q^{76} -6695.13i q^{77} -7160.18 q^{79} +43979.4i q^{80} +22755.5 q^{82} -7711.22i q^{83} -8521.92 q^{85} +5989.56i q^{86} -17997.2 q^{88} +13282.8i q^{89} -15296.6 q^{91} +11580.5i q^{92} +19663.1 q^{94} -16309.0i q^{95} -1941.78 q^{97} -36030.2i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 7.80173i	− 1.95043i	−0.221255	−	0.975216i	$-0.571015\pi$
$2$	− 7.80173i	− 1.95043i	0.221255	−	0.975216i	$-0.428985\pi$
$3$	0	0
$3$	0	0
$4$	−44.8670	−2.80418
$5$	42.3216i	1.69286i	0.532497	+	0.846432i	$0.321254\pi$
$5$	42.3216i	1.69286i	−0.532497	+	0.846432i	$0.678746\pi$
$6$	0	0
$7$	−83.7809	−1.70981	−0.854907	−	0.518781i	$-0.826386\pi$
$7$	−83.7809	−1.70981	−0.854907	+	0.518781i	$0.826386\pi$
$8$	225.212i	3.51894i
$9$	0	0
$10$	330.181	3.30181
$11$	79.9124i	0.660433i	0.943905	+	0.330216i	$0.107122\pi$
$11$	79.9124i	0.660433i	−0.943905	+	0.330216i	$0.892878\pi$
$12$	0	0
$13$	182.579	1.08035	0.540174	−	0.841553i	$-0.318358\pi$
$13$	182.579	1.08035	0.540174	+	0.841553i	$0.318358\pi$
$14$	653.636i	3.33488i
$15$	0	0
$16$	1039.17	4.05927
$17$	201.361i	0.696751i	0.937355	+	0.348376i	$0.113267\pi$
$17$	201.361i	0.696751i	−0.937355	+	0.348376i	$0.886733\pi$
$18$	0	0
$19$	−385.359	−1.06748	−0.533738	−	0.845650i	$-0.679213\pi$
$19$	−385.359	−1.06748	−0.533738	+	0.845650i	$0.679213\pi$
$20$	− 1898.84i	− 4.74710i
$21$	0	0
$22$	623.455	1.28813
$23$	− 258.108i	− 0.487917i	−0.969786	−	0.243958i	$-0.921554\pi$
$23$	− 258.108i	− 0.487917i	0.969786	−	0.243958i	$-0.0784460\pi$
$24$	0	0
$25$	−1166.12	−1.86579
$26$	− 1424.43i	− 2.10715i
$27$	0	0
$28$	3758.99	4.79464
$29$	1250.23i	1.48660i	0.668961	+	0.743298i	$0.266739\pi$
$29$	1250.23i	1.48660i	−0.668961	+	0.743298i	$0.733261\pi$
$30$	0	0
$31$	229.370	0.238679	0.119339	−	0.992854i	$-0.461922\pi$
$31$	229.370	0.238679	0.119339	+	0.992854i	$0.461922\pi$
$32$	− 4503.95i	− 4.39839i
$33$	0	0
$34$	1570.96	1.35897
$35$	− 3545.74i	− 2.89448i
$36$	0	0
$37$	−504.893	−0.368804	−0.184402	−	0.982851i	$-0.559035\pi$
$37$	−504.893	−0.368804	−0.184402	+	0.982851i	$0.559035\pi$
$38$	3006.47i	2.08204i
$39$	0	0
$40$	−9531.33	−5.95708
$41$	2916.72i	1.73511i	0.497339	+	0.867556i	$0.334310\pi$
$41$	2916.72i	1.73511i	−0.497339	+	0.867556i	$0.665690\pi$
$42$	0	0
$43$	−767.722	−0.415209	−0.207605	−	0.978213i	$-0.566567\pi$
$43$	−767.722	−0.415209	−0.207605	+	0.978213i	$0.566567\pi$
$44$	− 3585.43i	− 1.85198i
$45$	0	0
$46$	−2013.69	−0.951648
$47$	2520.35i	1.14095i	0.821316	+	0.570473i	$0.193240\pi$
$47$	2520.35i	1.14095i	−0.821316	+	0.570473i	$0.806760\pi$
$48$	0	0
$49$	4618.24	1.92347
$50$	9097.72i	3.63909i
$51$	0	0
$52$	−8191.76	−3.02950
$53$	− 4215.55i	− 1.50073i	−0.661024	−	0.750365i	$-0.729878\pi$
$53$	− 4215.55i	− 1.50073i	0.661024	−	0.750365i	$-0.270122\pi$
$54$	0	0
$55$	−3382.02	−1.11802
$56$	− 18868.5i	− 6.01673i
$57$	0	0
$58$	9753.93	2.89950
$59$	− 453.188i	− 0.130189i
$59$	− 453.188i	− 0.130189i
$60$	0	0
$61$	−3836.27	−1.03098	−0.515489	−	0.856896i	$-0.672390\pi$
$61$	−3836.27	−1.03098	−0.515489	+	0.856896i	$0.672390\pi$
$62$	− 1789.48i	− 0.465526i
$63$	0	0
$64$	−18511.8	−4.51948
$65$	7727.03i	1.82888i
$66$	0	0
$67$	−2946.00	−0.656270	−0.328135	−	0.944631i	$-0.606420\pi$
$67$	−2946.00	−0.656270	−0.328135	+	0.944631i	$0.606420\pi$
$68$	− 9034.46i	− 1.95382i
$69$	0	0
$70$	−27662.9	−5.64549
$71$	− 2974.94i	− 0.590148i	−0.955474	−	0.295074i	$-0.904656\pi$
$71$	− 2974.94i	− 0.590148i	0.955474	−	0.295074i	$-0.0953443\pi$
$72$	0	0
$73$	679.596	0.127528	0.0637639	−	0.997965i	$-0.479690\pi$
$73$	679.596	0.127528	0.0637639	+	0.997965i	$0.479690\pi$
$74$	3939.04i	0.719327i
$75$	0	0
$76$	17289.9	2.99340
$77$	− 6695.13i	− 1.12922i
$78$	0	0
$79$	−7160.18	−1.14728	−0.573641	−	0.819107i	$-0.694469\pi$
$79$	−7160.18	−1.14728	−0.573641	+	0.819107i	$0.694469\pi$
$80$	43979.4i	6.87179i
$81$	0	0
$82$	22755.5	3.38422
$83$	− 7711.22i	− 1.11935i	−0.828711	−	0.559676i	$-0.810925\pi$
$83$	− 7711.22i	− 1.11935i	0.828711	−	0.559676i	$-0.189075\pi$
$84$	0	0
$85$	−8521.92	−1.17950
$86$	5989.56i	0.809838i
$87$	0	0
$88$	−17997.2	−2.32402
$89$	13282.8i	1.67691i	0.544967	+	0.838457i	$0.316542\pi$
$89$	13282.8i	1.67691i	−0.544967	+	0.838457i	$0.683458\pi$
$90$	0	0
$91$	−15296.6	−1.84720
$92$	11580.5i	1.36821i
$93$	0	0
$94$	19663.1	2.22534
$95$	− 16309.0i	− 1.80709i
$96$	0	0
$97$	−1941.78	−0.206375	−0.103188	−	0.994662i	$-0.532904\pi$
$97$	−1941.78	−0.206375	−0.103188	+	0.994662i	$0.532904\pi$
$98$	− 36030.2i	− 3.75159i
$99$	0	0
$100$	52320.1	5.23201
$101$	− 16298.3i	− 1.59771i	−0.601521	−	0.798857i	$-0.705438\pi$
$101$	− 16298.3i	− 1.59771i	0.601521	−	0.798857i	$-0.294562\pi$
$102$	0	0
$103$	1958.02	0.184562	0.0922812	−	0.995733i	$-0.470584\pi$
$103$	1958.02	0.184562	0.0922812	+	0.995733i	$0.470584\pi$
$104$	41119.0i	3.80168i
$105$	0	0
$106$	−32888.6	−2.92707
$107$	− 2114.30i	− 0.184671i	−0.995728	−	0.0923356i	$-0.970567\pi$
$107$	− 2114.30i	− 0.184671i	0.995728	−	0.0923356i	$-0.0294333\pi$
$108$	0	0
$109$	−10576.3	−0.890187	−0.445094	−	0.895484i	$-0.646830\pi$
$109$	−10576.3	−0.890187	−0.445094	+	0.895484i	$0.646830\pi$
$110$	26385.6i	2.18063i
$111$	0	0
$112$	−87062.8	−6.94059
$113$	− 5532.80i	− 0.433299i	−0.976249	−	0.216650i	$-0.930487\pi$
$113$	− 5532.80i	− 0.433299i	0.976249	−	0.216650i	$-0.0695129\pi$
$114$	0	0
$115$	10923.5	0.825976
$116$	− 56093.9i	− 4.16869i
$117$	0	0
$118$	−3535.65	−0.253925
$119$	− 16870.2i	− 1.19132i
$120$	0	0
$121$	8255.01	0.563828
$122$	29929.5i	2.01085i
$123$	0	0
$124$	−10291.1	−0.669299
$125$	− 22900.9i	− 1.46566i
$126$	0	0
$127$	14892.4	0.923329	0.461664	−	0.887055i	$-0.347252\pi$
$127$	14892.4	0.923329	0.461664	+	0.887055i	$0.347252\pi$
$128$	72360.9i	4.41656i
$129$	0	0
$130$	60284.2	3.56711
$131$	− 30423.0i	− 1.77280i	−0.462920	−	0.886400i	$-0.653198\pi$
$131$	− 30423.0i	− 1.77280i	0.462920	−	0.886400i	$-0.346802\pi$
$132$	0	0
$133$	32285.7	1.82519
$134$	22983.9i	1.28001i
$135$	0	0
$136$	−45349.0	−2.45183
$137$	436.991i	0.0232826i	0.999932	+	0.0116413i	$0.00370563\pi$
$137$	436.991i	0.0232826i	−0.999932	+	0.0116413i	$0.996294\pi$
$138$	0	0
$139$	33387.8	1.72806	0.864028	−	0.503443i	$-0.167934\pi$
$139$	33387.8	1.72806	0.864028	+	0.503443i	$0.167934\pi$
$140$	159087.i	8.11666i
$141$	0	0
$142$	−23209.6	−1.15104
$143$	14590.3i	0.713498i
$144$	0	0
$145$	−52911.6	−2.51660
$146$	− 5302.02i	− 0.248734i
$147$	0	0
$148$	22653.0	1.03420
$149$	1393.78i	0.0627800i	0.999507	+	0.0313900i	$0.00999339\pi$
$149$	1393.78i	0.0627800i	−0.999507	+	0.0313900i	$0.990007\pi$
$150$	0	0
$151$	29864.5	1.30979	0.654894	−	0.755721i	$-0.272713\pi$
$151$	29864.5	1.30979	0.654894	+	0.755721i	$0.272713\pi$
$152$	− 86787.5i	− 3.75639i
$153$	0	0
$154$	−52233.6	−2.20246
$155$	9707.30i	0.404050i
$156$	0	0
$157$	−11447.0	−0.464398	−0.232199	−	0.972668i	$-0.574592\pi$
$157$	−11447.0	−0.464398	−0.232199	+	0.972668i	$0.574592\pi$
$158$	55861.8i	2.23769i
$159$	0	0
$160$	190614.	7.44587
$161$	21624.5i	0.834247i
$162$	0	0
$163$	20806.3	0.783106	0.391553	−	0.920156i	$-0.371938\pi$
$163$	20806.3	0.783106	0.391553	+	0.920156i	$0.371938\pi$
$164$	− 130865.i	− 4.86558i
$165$	0	0
$166$	−60160.8	−2.18322
$167$	19061.8i	0.683487i	0.939793	+	0.341743i	$0.111017\pi$
$167$	19061.8i	0.683487i	−0.939793	+	0.341743i	$0.888983\pi$
$168$	0	0
$169$	4774.06	0.167153
$170$	66485.7i	2.30054i
$171$	0	0
$172$	34445.4	1.16432
$173$	− 18508.8i	− 0.618422i	−0.950993	−	0.309211i	$-0.899935\pi$
$173$	− 18508.8i	− 0.618422i	0.950993	−	0.309211i	$-0.100065\pi$
$174$	0	0
$175$	97698.3	3.19015
$176$	83042.8i	2.68087i
$177$	0	0
$178$	103629.	3.27071
$179$	− 57298.4i	− 1.78828i	−0.447785	−	0.894141i	$-0.647787\pi$
$179$	− 57298.4i	− 1.78828i	0.447785	−	0.894141i	$-0.352213\pi$
$180$	0	0
$181$	7164.74	0.218697	0.109349	−	0.994003i	$-0.465124\pi$
$181$	7164.74	0.218697	0.109349	+	0.994003i	$0.465124\pi$
$182$	119340.i	3.60283i
$183$	0	0
$184$	58129.0	1.71695
$185$	− 21367.9i	− 0.624335i
$186$	0	0
$187$	−16091.2	−0.460157
$188$	− 113080.i	− 3.19942i
$189$	0	0
$190$	−127238.	−3.52461
$191$	20059.2i	0.549853i	0.961465	+	0.274926i	$0.0886535\pi$
$191$	20059.2i	0.549853i	−0.961465	+	0.274926i	$0.911347\pi$
$192$	0	0
$193$	4916.30	0.131985	0.0659924	−	0.997820i	$-0.478979\pi$
$193$	4916.30	0.131985	0.0659924	+	0.997820i	$0.478979\pi$
$194$	15149.3i	0.402520i
$195$	0	0
$196$	−207206.	−5.39375
$197$	− 5246.46i	− 0.135187i	−0.997713	−	0.0675933i	$-0.978468\pi$
$197$	− 5246.46i	− 0.135187i	0.997713	−	0.0675933i	$-0.0215320\pi$
$198$	0	0
$199$	28884.6	0.729390	0.364695	−	0.931127i	$-0.381173\pi$
$199$	28884.6	0.729390	0.364695	+	0.931127i	$0.381173\pi$
$200$	− 262624.i	− 6.56559i
$201$	0	0
$202$	−127155.	−3.11623
$203$	− 104745.i	− 2.54180i
$204$	0	0
$205$	−123440.	−2.93731
$206$	− 15276.0i	− 0.359976i
$207$	0	0
$208$	189731.	4.38542
$209$	− 30795.0i	− 0.704997i
$210$	0	0
$211$	20692.8	0.464787	0.232393	−	0.972622i	$-0.425344\pi$
$211$	20692.8	0.464787	0.232393	+	0.972622i	$0.425344\pi$
$212$	189139.i	4.20832i
$213$	0	0
$214$	−16495.2	−0.360189
$215$	− 32491.2i	− 0.702893i
$216$	0	0
$217$	−19216.8	−0.408096
$218$	82513.5i	1.73625i
$219$	0	0
$220$	151741.	3.13514
$221$	36764.3i	0.752734i
$222$	0	0
$223$	−89912.4	−1.80805	−0.904024	−	0.427483i	$-0.859401\pi$
$223$	−89912.4	−1.80805	−0.904024	+	0.427483i	$0.859401\pi$
$224$	377345.i	7.52042i
$225$	0	0
$226$	−43165.4	−0.845120
$227$	− 53703.8i	− 1.04221i	−0.853494	−	0.521103i	$-0.825521\pi$
$227$	− 53703.8i	− 1.04221i	0.853494	−	0.521103i	$-0.174479\pi$
$228$	0	0
$229$	−51847.5	−0.988682	−0.494341	−	0.869268i	$-0.664591\pi$
$229$	−51847.5	−0.988682	−0.494341	+	0.869268i	$0.664591\pi$
$230$	− 85222.4i	− 1.61101i
$231$	0	0
$232$	−281566.	−5.23124
$233$	32321.7i	0.595364i	0.954665	+	0.297682i	$0.0962135\pi$
$233$	32321.7i	0.595364i	−0.954665	+	0.297682i	$0.903787\pi$
$234$	0	0
$235$	−106665.	−1.93147
$236$	20333.1i	0.365074i
$237$	0	0
$238$	−131617.	−2.32358
$239$	42581.4i	0.745459i	0.927940	+	0.372730i	$0.121578\pi$
$239$	42581.4i	0.745459i	−0.927940	+	0.372730i	$0.878422\pi$
$240$	0	0
$241$	−79341.7	−1.36605	−0.683026	−	0.730394i	$-0.739337\pi$
$241$	−79341.7	−1.36605	−0.683026	+	0.730394i	$0.739337\pi$
$242$	− 64403.4i	− 1.09971i
$243$	0	0
$244$	172122.	2.89105
$245$	195451.i	3.25616i
$246$	0	0
$247$	−70358.4	−1.15325
$248$	51656.9i	0.839895i
$249$	0	0
$250$	−178667.	−2.85866
$251$	39819.1i	0.632039i	0.948753	+	0.316020i	$0.102347\pi$
$251$	39819.1i	0.632039i	−0.948753	+	0.316020i	$0.897653\pi$
$252$	0	0
$253$	20626.0	0.322236
$254$	− 116186.i	− 1.80089i
$255$	0	0
$256$	268351.	4.09472
$257$	− 96050.7i	− 1.45423i	−0.686514	−	0.727117i	$-0.740860\pi$
$257$	− 96050.7i	− 1.45423i	0.686514	−	0.727117i	$-0.259140\pi$
$258$	0	0
$259$	42300.4	0.630587
$260$	− 346688.i	− 5.12852i
$261$	0	0
$262$	−237352.	−3.45773
$263$	− 16871.8i	− 0.243922i	−0.992535	−	0.121961i	$-0.961082\pi$
$263$	− 16871.8i	− 0.243922i	0.992535	−	0.121961i	$-0.0389182\pi$
$264$	0	0
$265$	178409.	2.54053
$266$	− 251884.i	− 3.55990i
$267$	0	0
$268$	132178.	1.84030
$269$	80242.5i	1.10892i	0.832210	+	0.554460i	$0.187075\pi$
$269$	80242.5i	1.10892i	−0.832210	+	0.554460i	$0.812925\pi$
$270$	0	0
$271$	18544.0	0.252503	0.126251	−	0.991998i	$-0.459705\pi$
$271$	18544.0	0.252503	0.126251	+	0.991998i	$0.459705\pi$
$272$	209249.i	2.82830i
$273$	0	0
$274$	3409.29	0.0454112
$275$	− 93187.1i	− 1.23223i
$276$	0	0
$277$	37320.5	0.486394	0.243197	−	0.969977i	$-0.421804\pi$
$277$	37320.5	0.486394	0.243197	+	0.969977i	$0.421804\pi$
$278$	− 260482.i	− 3.37046i
$279$	0	0
$280$	798544.	10.1855
$281$	135243.i	1.71278i	0.516329	+	0.856390i	$0.327298\pi$
$281$	135243.i	1.71278i	−0.516329	+	0.856390i	$0.672702\pi$
$282$	0	0
$283$	−45480.0	−0.567868	−0.283934	−	0.958844i	$-0.591640\pi$
$283$	−45480.0	−0.567868	−0.283934	+	0.958844i	$0.591640\pi$
$284$	133476.i	1.65488i
$285$	0	0
$286$	113830.	1.39163
$287$	− 244366.i	− 2.96672i
$288$	0	0
$289$	42974.7	0.514538
$290$	412802.i	4.90846i
$291$	0	0
$292$	−30491.4	−0.357612
$293$	− 72732.8i	− 0.847218i	−0.905845	−	0.423609i	$-0.860763\pi$
$293$	− 72732.8i	− 0.847218i	0.905845	−	0.423609i	$-0.139237\pi$
$294$	0	0
$295$	19179.6	0.220392
$296$	− 113708.i	− 1.29780i
$297$	0	0
$298$	10873.9	0.122448
$299$	− 47125.1i	− 0.527120i
$300$	0	0
$301$	64320.5	0.709931
$302$	− 232994.i	− 2.55465i
$303$	0	0
$304$	−400454.	−4.33317
$305$	− 162357.i	− 1.74530i
$306$	0	0
$307$	−115647.	−1.22703	−0.613517	−	0.789681i	$-0.710246\pi$
$307$	−115647.	−1.22703	−0.613517	+	0.789681i	$0.710246\pi$
$308$	300390.i	3.16653i
$309$	0	0
$310$	75733.7	0.788072
$311$	125412.i	1.29663i	0.761371	+	0.648316i	$0.224526\pi$
$311$	125412.i	1.29663i	−0.761371	+	0.648316i	$0.775474\pi$
$312$	0	0
$313$	36683.0	0.374434	0.187217	−	0.982319i	$-0.440053\pi$
$313$	36683.0	0.374434	0.187217	+	0.982319i	$0.440053\pi$
$314$	89306.1i	0.905778i
$315$	0	0
$316$	321256.	3.21719
$317$	165277.i	1.64473i	0.568963	+	0.822363i	$0.307345\pi$
$317$	165277.i	1.64473i	−0.568963	+	0.822363i	$0.692655\pi$
$318$	0	0
$319$	−99908.6	−0.981797
$320$	− 783449.i	− 7.65087i
$321$	0	0
$322$	168709.	1.62714
$323$	− 77596.3i	− 0.743765i
$324$	0	0
$325$	−212908.	−2.01570
$326$	− 162325.i	− 1.52739i
$327$	0	0
$328$	−656882.	−6.10576
$329$	− 211157.i	− 1.95081i
$330$	0	0
$331$	23137.7	0.211186	0.105593	−	0.994409i	$-0.466326\pi$
$331$	23137.7	0.211186	0.105593	+	0.994409i	$0.466326\pi$
$332$	345979.i	3.13887i
$333$	0	0
$334$	148715.	1.33309
$335$	− 124679.i	− 1.11098i
$336$	0	0
$337$	95901.0	0.844429	0.422215	−	0.906496i	$-0.361253\pi$
$337$	95901.0	0.844429	0.422215	+	0.906496i	$0.361253\pi$
$338$	− 37245.9i	− 0.326021i
$339$	0	0
$340$	382353.	3.30755
$341$	18329.5i	0.157631i
$342$	0	0
$343$	−185762.	−1.57895
$344$	− 172900.i	− 1.46110i
$345$	0	0
$346$	−144400.	−1.20619
$347$	38255.4i	0.317712i	0.987302	+	0.158856i	$0.0507806\pi$
$347$	38255.4i	0.317712i	−0.987302	+	0.158856i	$0.949219\pi$
$348$	0	0
$349$	14688.3	0.120592	0.0602961	−	0.998181i	$-0.480796\pi$
$349$	14688.3	0.120592	0.0602961	+	0.998181i	$0.480796\pi$
$350$	− 762215.i	− 6.22217i
$351$	0	0
$352$	359921.	2.90484
$353$	− 117943.i	− 0.946501i	−0.880928	−	0.473250i	$-0.843081\pi$
$353$	− 117943.i	− 0.946501i	0.880928	−	0.473250i	$-0.156919\pi$
$354$	0	0
$355$	125904.	0.999040
$356$	− 595961.i	− 4.70238i
$357$	0	0
$358$	−447026.	−3.48792
$359$	− 86009.1i	− 0.667353i	−0.942688	−	0.333676i	$-0.891711\pi$
$359$	− 86009.1i	− 0.667353i	0.942688	−	0.333676i	$-0.108289\pi$
$360$	0	0
$361$	18180.5	0.139506
$362$	− 55897.3i	− 0.426554i
$363$	0	0
$364$	686313.	5.17988
$365$	28761.6i	0.215887i
$366$	0	0
$367$	78155.5	0.580266	0.290133	−	0.956986i	$-0.406300\pi$
$367$	78155.5	0.580266	0.290133	+	0.956986i	$0.406300\pi$
$368$	− 268219.i	− 1.98058i
$369$	0	0
$370$	−166706.	−1.21772
$371$	353182.i	2.56597i
$372$	0	0
$373$	167998.	1.20750	0.603748	−	0.797175i	$-0.293673\pi$
$373$	167998.	1.20750	0.603748	+	0.797175i	$0.293673\pi$
$374$	125540.i	0.897506i
$375$	0	0
$376$	−567613.	−4.01492
$377$	228265.i	1.60604i
$378$	0	0
$379$	142529.	0.992255	0.496128	−	0.868250i	$-0.334755\pi$
$379$	142529.	0.992255	0.496128	+	0.868250i	$0.334755\pi$
$380$	731735.i	5.06742i
$381$	0	0
$382$	156496.	1.07245
$383$	− 60981.3i	− 0.415718i	−0.978159	−	0.207859i	$-0.933350\pi$
$383$	− 60981.3i	− 0.415718i	0.978159	−	0.207859i	$-0.0666496\pi$
$384$	0	0
$385$	283349.	1.91161
$386$	− 38355.6i	− 0.257427i
$387$	0	0
$388$	87121.9	0.578714
$389$	175364.i	1.15889i	0.815013	+	0.579443i	$0.196730\pi$
$389$	175364.i	1.15889i	−0.815013	+	0.579443i	$0.803270\pi$
$390$	0	0
$391$	51972.9	0.339956
$392$	1.04008e6i	6.76856i
$393$	0	0
$394$	−40931.4	−0.263672
$395$	− 303030.i	− 1.94219i
$396$	0	0
$397$	235211.	1.49237	0.746185	−	0.665739i	$-0.231883\pi$
$397$	235211.	1.49237	0.746185	+	0.665739i	$0.231883\pi$
$398$	− 225350.i	− 1.42263i
$399$	0	0
$400$	−1.21180e6	−7.57372
$401$	− 141762.i	− 0.881600i	−0.897605	−	0.440800i	$-0.854695\pi$
$401$	− 141762.i	− 0.881600i	0.897605	−	0.440800i	$-0.145305\pi$
$402$	0	0
$403$	41878.1	0.257856
$404$	731254.i	4.48029i
$405$	0	0
$406$	−817193.	−4.95761
$407$	− 40347.2i	− 0.243570i
$408$	0	0
$409$	−186833.	−1.11688	−0.558441	−	0.829544i	$-0.688600\pi$
$409$	−186833.	−1.11688	−0.558441	+	0.829544i	$0.688600\pi$
$410$	963048.i	5.72902i
$411$	0	0
$412$	−87850.5	−0.517547
$413$	37968.5i	0.222599i
$414$	0	0
$415$	326351.	1.89491
$416$	− 822326.i	− 4.75179i
$417$	0	0
$418$	−240254.	−1.37505
$419$	161050.i	0.917342i	0.888606	+	0.458671i	$0.151674\pi$
$419$	161050.i	0.917342i	−0.888606	+	0.458671i	$0.848326\pi$
$420$	0	0
$421$	47543.2	0.268240	0.134120	−	0.990965i	$-0.457179\pi$
$421$	47543.2	0.268240	0.134120	+	0.990965i	$0.457179\pi$
$422$	− 161439.i	− 0.906535i
$423$	0	0
$424$	949393.	5.28098
$425$	− 234810.i	− 1.29999i
$426$	0	0
$427$	321406.	1.76278
$428$	94862.2i	0.517852i
$429$	0	0
$430$	−253488.	−1.37094
$431$	302832.i	1.63022i	0.579304	+	0.815111i	$0.303324\pi$
$431$	302832.i	1.63022i	−0.579304	+	0.815111i	$0.696676\pi$
$432$	0	0
$433$	−108158.	−0.576874	−0.288437	−	0.957499i	$-0.593136\pi$
$433$	−108158.	−0.576874	−0.288437	+	0.957499i	$0.593136\pi$
$434$	149924.i	0.795963i
$435$	0	0
$436$	474527.	2.49625
$437$	99464.2i	0.520839i
$438$	0	0
$439$	−377285.	−1.95767	−0.978837	−	0.204642i	$-0.934397\pi$
$439$	−377285.	−1.95767	−0.978837	+	0.204642i	$0.934397\pi$
$440$	− 761672.i	− 3.93425i
$441$	0	0
$442$	286825.	1.46816
$443$	− 298914.i	− 1.52313i	−0.648086	−	0.761567i	$-0.724430\pi$
$443$	− 298914.i	− 1.52313i	0.648086	−	0.761567i	$-0.275570\pi$
$444$	0	0
$445$	−562151.	−2.83879
$446$	701472.i	3.52647i
$447$	0	0
$448$	1.55094e6	7.72748
$449$	330877.i	1.64125i	0.571470	+	0.820623i	$0.306373\pi$
$449$	330877.i	1.64125i	−0.571470	+	0.820623i	$0.693627\pi$
$450$	0	0
$451$	−233082.	−1.14593
$452$	248240.i	1.21505i
$453$	0	0
$454$	−418983.	−2.03275
$455$	− 647377.i	− 3.12705i
$456$	0	0
$457$	−138028.	−0.660898	−0.330449	−	0.943824i	$-0.607200\pi$
$457$	−138028.	−0.660898	−0.330449	+	0.943824i	$0.607200\pi$
$458$	404500.i	1.92836i
$459$	0	0
$460$	−490106.	−2.31619
$461$	− 224330.i	− 1.05556i	−0.849380	−	0.527782i	$-0.823024\pi$
$461$	− 224330.i	− 1.05556i	0.849380	−	0.527782i	$-0.176976\pi$
$462$	0	0
$463$	−20956.7	−0.0977599	−0.0488800	−	0.998805i	$-0.515565\pi$
$463$	−20956.7	−0.0977599	−0.0488800	+	0.998805i	$0.515565\pi$
$464$	1.29920e6i	6.03449i
$465$	0	0
$466$	252165.	1.16122
$467$	120193.i	0.551119i	0.961284	+	0.275559i	$0.0888631\pi$
$467$	120193.i	0.551119i	−0.961284	+	0.275559i	$0.911137\pi$
$468$	0	0
$469$	246818.	1.12210
$470$	832173.i	3.76719i
$471$	0	0
$472$	102063.	0.458127
$473$	− 61350.5i	− 0.274218i
$474$	0	0
$475$	449373.	1.99168
$476$	756915.i	3.34067i
$477$	0	0
$478$	332208.	1.45397
$479$	− 105143.i	− 0.458256i	−0.973396	−	0.229128i	$-0.926413\pi$
$479$	− 105143.i	− 0.458256i	0.973396	−	0.229128i	$-0.0735875\pi$
$480$	0	0
$481$	−92182.8	−0.398437
$482$	619002.i	2.66439i
$483$	0	0
$484$	−370377.	−1.58108
$485$	− 82179.3i	− 0.349365i
$486$	0	0
$487$	−335236.	−1.41349	−0.706745	−	0.707468i	$-0.749837\pi$
$487$	−335236.	−1.41349	−0.706745	+	0.707468i	$0.749837\pi$
$488$	− 863974.i	− 3.62795i
$489$	0	0
$490$	1.52486e6	6.35092
$491$	− 281597.i	− 1.16806i	−0.811732	−	0.584031i	$-0.801475\pi$
$491$	− 281597.i	− 1.16806i	0.811732	−	0.584031i	$-0.198525\pi$
$492$	0	0
$493$	−251747.	−1.03579
$494$	548917.i	2.24933i
$495$	0	0
$496$	238355.	0.968860
$497$	249243.i	1.00904i
$498$	0	0
$499$	−141232.	−0.567193	−0.283597	−	0.958944i	$-0.591528\pi$
$499$	−141232.	−0.567193	−0.283597	+	0.958944i	$0.591528\pi$
$500$	1.02749e6i	4.10997i
$501$	0	0
$502$	310658.	1.23275
$503$	− 366037.i	− 1.44674i	−0.690463	−	0.723368i	$-0.742593\pi$
$503$	− 366037.i	− 1.44674i	0.690463	−	0.723368i	$-0.257407\pi$
$504$	0	0
$505$	689769.	2.70471
$506$	− 160919.i	− 0.628500i
$507$	0	0
$508$	−668175.	−2.58919
$509$	− 239791.i	− 0.925543i	−0.886478	−	0.462772i	$-0.846855\pi$
$509$	− 239791.i	− 0.925543i	0.886478	−	0.462772i	$-0.153145\pi$
$510$	0	0
$511$	−56937.2	−0.218049
$512$	− 935830.i	− 3.56991i
$513$	0	0
$514$	−749361.	−2.83638
$515$	82866.6i	0.312439i
$516$	0	0
$517$	−201407.	−0.753518
$518$	− 330016.i	− 1.22992i
$519$	0	0
$520$	−1.74022e6	−6.43573
$521$	− 306784.i	− 1.13020i	−0.825021	−	0.565102i	$-0.808837\pi$
$521$	− 306784.i	− 1.13020i	0.825021	−	0.565102i	$-0.191163\pi$
$522$	0	0
$523$	38644.5	0.141281	0.0706406	−	0.997502i	$-0.477496\pi$
$523$	38644.5	0.141281	0.0706406	+	0.997502i	$0.477496\pi$
$524$	1.36499e6i	4.97126i
$525$	0	0
$526$	−131629.	−0.475753
$527$	46186.2i	0.166300i
$528$	0	0
$529$	213221.	0.761937
$530$	− 1.39190e6i	− 4.95513i
$531$	0	0
$532$	−1.44856e6	−5.11816
$533$	532532.i	1.87453i
$534$	0	0
$535$	89480.5	0.312623
$536$	− 663474.i	− 2.30937i
$537$	0	0
$538$	626030.	2.16287
$539$	369055.i	1.27032i
$540$	0	0
$541$	131501.	0.449297	0.224649	−	0.974440i	$-0.427877\pi$
$541$	131501.	0.449297	0.224649	+	0.974440i	$0.427877\pi$
$542$	− 144676.i	− 0.492489i
$543$	0	0
$544$	906920.	3.06458
$545$	− 447606.i	− 1.50697i
$546$	0	0
$547$	−479130.	−1.60132	−0.800661	−	0.599118i	$-0.795518\pi$
$547$	−479130.	−1.60132	−0.800661	+	0.599118i	$0.795518\pi$
$548$	− 19606.5i	− 0.0652888i
$549$	0	0
$550$	−727021.	−2.40337
$551$	− 481786.i	− 1.58691i
$552$	0	0
$553$	599886.	1.96164
$554$	− 291164.i	− 0.948678i
$555$	0	0
$556$	−1.49801e6	−4.84579
$557$	270204.i	0.870925i	0.900207	+	0.435462i	$0.143415\pi$
$557$	270204.i	0.870925i	−0.900207	+	0.435462i	$0.856585\pi$
$558$	0	0
$559$	−140170.	−0.448571
$560$	− 3.68464e6i	− 11.7495i
$561$	0	0
$562$	1.05513e6	3.34066
$563$	− 366903.i	− 1.15754i	−0.815493	−	0.578768i	$-0.803534\pi$
$563$	− 366903.i	− 1.15754i	0.815493	−	0.578768i	$-0.196466\pi$
$564$	0	0
$565$	234157.	0.733516
$566$	354822.i	1.10759i
$567$	0	0
$568$	669992.	2.07670
$569$	434957.i	1.34345i	0.740800	+	0.671725i	$0.234447\pi$
$569$	434957.i	1.34345i	−0.740800	+	0.671725i	$0.765553\pi$
$570$	0	0
$571$	−107372.	−0.329319	−0.164660	−	0.986350i	$-0.552653\pi$
$571$	−107372.	−0.329319	−0.164660	+	0.986350i	$0.552653\pi$
$572$	− 654623.i	− 2.00078i
$573$	0	0
$574$	−1.90647e6	−5.78638
$575$	300984.i	0.910348i
$576$	0	0
$577$	−337882.	−1.01488	−0.507438	−	0.861688i	$-0.669408\pi$
$577$	−337882.	−1.01488	−0.507438	+	0.861688i	$0.669408\pi$
$578$	− 335277.i	− 1.00357i
$579$	0	0
$580$	2.37398e6	7.05702
$581$	646053.i	1.91389i
$582$	0	0
$583$	336875.	0.991131
$584$	153053.i	0.448763i
$585$	0	0
$586$	−567442.	−1.65244
$587$	240650.i	0.698408i	0.937047	+	0.349204i	$0.113548\pi$
$587$	240650.i	0.698408i	−0.937047	+	0.349204i	$0.886452\pi$
$588$	0	0
$589$	−88389.8	−0.254784
$590$	− 149634.i	− 0.429860i
$591$	0	0
$592$	−524671.	−1.49707
$593$	244292.i	0.694703i	0.937735	+	0.347351i	$0.112919\pi$
$593$	244292.i	0.694703i	−0.937735	+	0.347351i	$0.887081\pi$
$594$	0	0
$595$	713974.	2.01673
$596$	− 62534.6i	− 0.176047i
$597$	0	0
$598$	−367657.	−1.02811
$599$	− 174479.i	− 0.486284i	−0.969991	−	0.243142i	$-0.921822\pi$
$599$	− 174479.i	− 0.486284i	0.969991	−	0.243142i	$-0.0781781\pi$
$600$	0	0
$601$	606326.	1.67864	0.839320	−	0.543638i	$-0.182954\pi$
$601$	606326.	1.67864	0.839320	+	0.543638i	$0.182954\pi$
$602$	− 501811.i	− 1.38467i
$603$	0	0
$604$	−1.33993e6	−3.67289
$605$	349365.i	0.954484i
$606$	0	0
$607$	196262.	0.532671	0.266336	−	0.963880i	$-0.414187\pi$
$607$	196262.	0.532671	0.266336	+	0.963880i	$0.414187\pi$
$608$	1.73564e6i	4.69517i
$609$	0	0
$610$	−1.26666e6	−3.40410
$611$	460163.i	1.23262i
$612$	0	0
$613$	446151.	1.18730	0.593651	−	0.804723i	$-0.297686\pi$
$613$	446151.	1.18730	0.593651	+	0.804723i	$0.297686\pi$
$614$	902244.i	2.39325i
$615$	0	0
$616$	1.50782e6	3.97365
$617$	449757.i	1.18143i	0.806880	+	0.590715i	$0.201154\pi$
$617$	449757.i	1.18143i	−0.806880	+	0.590715i	$0.798846\pi$
$618$	0	0
$619$	−69998.7	−0.182688	−0.0913438	−	0.995819i	$-0.529116\pi$
$619$	−69998.7	−0.182688	−0.0913438	+	0.995819i	$0.529116\pi$
$620$	− 435537.i	− 1.13303i
$621$	0	0
$622$	978427.	2.52899
$623$	− 1.11285e6i	− 2.86721i
$624$	0	0
$625$	240379.	0.615371
$626$	− 286190.i	− 0.730309i
$627$	0	0
$628$	513590.	1.30226
$629$	− 101666.i	− 0.256965i
$630$	0	0
$631$	105676.	0.265409	0.132704	−	0.991156i	$-0.457634\pi$
$631$	105676.	0.265409	0.132704	+	0.991156i	$0.457634\pi$
$632$	− 1.61256e6i	− 4.03721i
$633$	0	0
$634$	1.28944e6	3.20793
$635$	630269.i	1.56307i
$636$	0	0
$637$	843193.	2.07801
$638$	779460.i	1.91493i
$639$	0	0
$640$	−3.06243e6	−7.47663
$641$	− 489338.i	− 1.19095i	−0.803374	−	0.595474i	$-0.796964\pi$
$641$	− 489338.i	− 1.19095i	0.803374	−	0.595474i	$-0.203036\pi$
$642$	0	0
$643$	−433833.	−1.04930	−0.524651	−	0.851318i	$-0.675804\pi$
$643$	−433833.	−1.04930	−0.524651	+	0.851318i	$0.675804\pi$
$644$	− 970226.i	− 2.33938i
$645$	0	0
$646$	−605385.	−1.45066
$647$	− 211659.i	− 0.505624i	−0.967515	−	0.252812i	$-0.918645\pi$
$647$	− 211659.i	− 0.505624i	0.967515	−	0.252812i	$-0.0813553\pi$
$648$	0	0
$649$	36215.3	0.0859810
$650$	1.66105e6i	3.93148i
$651$	0	0
$652$	−933517.	−2.19597
$653$	− 8242.36i	− 0.0193297i	−0.999953	−	0.00966485i	$-0.996924\pi$
$653$	− 8242.36i	− 0.0193297i	0.999953	−	0.00966485i	$-0.00307646\pi$
$654$	0	0
$655$	1.28755e6	3.00111
$656$	3.03098e6i	7.04328i
$657$	0	0
$658$	−1.64739e6	−3.80491
$659$	257557.i	0.593065i	0.955023	+	0.296533i	$0.0958303\pi$
$659$	257557.i	0.593065i	−0.955023	+	0.296533i	$0.904170\pi$
$660$	0	0
$661$	114063.	0.261060	0.130530	−	0.991444i	$-0.458332\pi$
$661$	114063.	0.261060	0.130530	+	0.991444i	$0.458332\pi$
$662$	− 180514.i	− 0.411903i
$663$	0	0
$664$	1.73666e6	3.93893
$665$	1.36638e6i	3.08979i
$666$	0	0
$667$	322693.	0.725335
$668$	− 855244.i	− 1.91662i
$669$	0	0
$670$	−972713.	−2.16688
$671$	− 306565.i	− 0.680892i
$672$	0	0
$673$	88490.5	0.195374	0.0976870	−	0.995217i	$-0.468856\pi$
$673$	88490.5	0.195374	0.0976870	+	0.995217i	$0.468856\pi$
$674$	− 748194.i	− 1.64700i
$675$	0	0
$676$	−214197.	−0.468728
$677$	− 115407.i	− 0.251800i	−0.992043	−	0.125900i	$-0.959818\pi$
$677$	− 115407.i	− 0.251800i	0.992043	−	0.125900i	$-0.0401818\pi$
$678$	0	0
$679$	162684.	0.352863
$680$	− 1.91924e6i	− 4.15061i
$681$	0	0
$682$	143002.	0.307449
$683$	− 224241.i	− 0.480699i	−0.970686	−	0.240350i	$-0.922738\pi$
$683$	− 224241.i	− 0.480699i	0.970686	−	0.240350i	$-0.0772621\pi$
$684$	0	0
$685$	−18494.2	−0.0394143
$686$	1.44927e6i	3.07964i
$687$	0	0
$688$	−797796.	−1.68545
$689$	− 769670.i	− 1.62131i
$690$	0	0
$691$	−33452.1	−0.0700595	−0.0350297	−	0.999386i	$-0.511153\pi$
$691$	−33452.1	−0.0700595	−0.0350297	+	0.999386i	$0.511153\pi$
$692$	830431.i	1.73417i
$693$	0	0
$694$	298458.	0.619676
$695$	1.41302e6i	2.92536i
$696$	0	0
$697$	−587315.	−1.20894
$698$	− 114594.i	− 0.235207i
$699$	0	0
$700$	−4.38342e6	−8.94576
$701$	− 211249.i	− 0.429892i	−0.976626	−	0.214946i	$-0.931042\pi$
$701$	− 211249.i	− 0.429892i	0.976626	−	0.214946i	$-0.0689575\pi$
$702$	0	0
$703$	194565.	0.393690
$704$	− 1.47932e6i	− 2.98482i
$705$	0	0
$706$	−920155.	−1.84609
$707$	1.36549e6i	2.73180i
$708$	0	0
$709$	−467290.	−0.929596	−0.464798	−	0.885417i	$-0.653873\pi$
$709$	−467290.	−0.929596	−0.464798	+	0.885417i	$0.653873\pi$
$710$	− 982269.i	− 1.94856i
$711$	0	0
$712$	−2.99146e6	−5.90096
$713$	− 59202.2i	− 0.116455i
$714$	0	0
$715$	−617485.	−1.20785
$716$	2.57080e6i	5.01467i
$717$	0	0
$718$	−671020.	−1.30163
$719$	− 13876.2i	− 0.0268419i	−0.999910	−	0.0134210i	$-0.995728\pi$
$719$	− 13876.2i	− 0.0268419i	0.999910	−	0.0134210i	$-0.00427216\pi$
$720$	0	0
$721$	−164045.	−0.315567
$722$	− 141840.i	− 0.272097i
$723$	0	0
$724$	−321460.	−0.613267
$725$	− 1.45791e6i	− 2.77367i
$726$	0	0
$727$	50908.4	0.0963209	0.0481605	−	0.998840i	$-0.484664\pi$
$727$	50908.4	0.0963209	0.0481605	+	0.998840i	$0.484664\pi$
$728$	− 3.44499e6i	− 6.50017i
$729$	0	0
$730$	224390.	0.421073
$731$	− 154589.i	− 0.289298i
$732$	0	0
$733$	212190.	0.394928	0.197464	−	0.980310i	$-0.436729\pi$
$733$	212190.	0.394928	0.197464	+	0.980310i	$0.436729\pi$
$734$	− 609748.i	− 1.13177i
$735$	0	0
$736$	−1.16250e6	−2.14605
$737$	− 235422.i	− 0.433422i
$738$	0	0
$739$	405142.	0.741853	0.370927	−	0.928662i	$-0.379040\pi$
$739$	405142.	0.741853	0.370927	+	0.928662i	$0.379040\pi$
$740$	958711.i	1.75075i
$741$	0	0
$742$	2.75543e6	5.00475
$743$	541860.i	0.981543i	0.871288	+	0.490771i	$0.163285\pi$
$743$	541860.i	0.981543i	−0.871288	+	0.490771i	$0.836715\pi$
$744$	0	0
$745$	−58986.9	−0.106278
$746$	− 1.31067e6i	− 2.35514i
$747$	0	0
$748$	721965.	1.29037
$749$	177138.i	0.315753i
$750$	0	0
$751$	997939.	1.76939	0.884696	−	0.466169i	$-0.154366\pi$
$751$	997939.	1.76939	0.884696	+	0.466169i	$0.154366\pi$
$752$	2.61908e6i	4.63141i
$753$	0	0
$754$	1.78086e6	3.13247
$755$	1.26391e6i	2.21729i
$756$	0	0
$757$	818723.	1.42871	0.714357	−	0.699781i	$-0.246719\pi$
$757$	818723.	1.42871	0.714357	+	0.699781i	$0.246719\pi$
$758$	− 1.11197e6i	− 1.93533i
$759$	0	0
$760$	3.67299e6	6.35905
$761$	156319.i	0.269925i	0.990851	+	0.134962i	$0.0430913\pi$
$761$	156319.i	0.269925i	−0.990851	+	0.134962i	$0.956909\pi$
$762$	0	0
$763$	886093.	1.52206
$764$	− 899994.i	− 1.54189i
$765$	0	0
$766$	−475760.	−0.810830
$767$	− 82742.5i	− 0.140649i
$768$	0	0
$769$	−1.01798e6	−1.72142	−0.860708	−	0.509099i	$-0.829979\pi$
$769$	−1.01798e6	−1.72142	−0.860708	+	0.509099i	$0.829979\pi$
$770$	− 2.21061e6i	− 3.72847i
$771$	0	0
$772$	−220579.	−0.370110
$773$	− 785241.i	− 1.31415i	−0.753826	−	0.657074i	$-0.771794\pi$
$773$	− 785241.i	− 1.31415i	0.753826	−	0.657074i	$-0.228206\pi$
$774$	0	0
$775$	−267472.	−0.445323
$776$	− 437313.i	− 0.726221i
$777$	0	0
$778$	1.36814e6	2.26033
$779$	− 1.12399e6i	− 1.85219i
$780$	0	0
$781$	237734.	0.389753
$782$	− 405478.i	− 0.663062i
$783$	0	0
$784$	4.79915e6	7.80786
$785$	− 484453.i	− 0.786163i
$786$	0	0
$787$	−246970.	−0.398744	−0.199372	−	0.979924i	$-0.563890\pi$
$787$	−246970.	−0.398744	−0.199372	+	0.979924i	$0.563890\pi$
$788$	235392.i	0.379088i
$789$	0	0
$790$	−2.36416e6	−3.78811
$791$	463543.i	0.740861i
$792$	0	0
$793$	−700422.	−1.11382
$794$	− 1.83505e6i	− 2.91077i
$795$	0	0
$796$	−1.29596e6	−2.04535
$797$	448612.i	0.706244i	0.935577	+	0.353122i	$0.114880\pi$
$797$	448612.i	0.706244i	−0.935577	+	0.353122i	$0.885120\pi$
$798$	0	0
$799$	−507500.	−0.794955
$800$	5.25213e6i	8.20645i
$801$	0	0
$802$	−1.10599e6	−1.71950
$803$	54308.1i	0.0842236i
$804$	0	0
$805$	−915184.	−1.41227
$806$	− 326722.i	− 0.502931i
$807$	0	0
$808$	3.67057e6	5.62226
$809$	121214.i	0.185206i	0.995703	+	0.0926032i	$0.0295188\pi$
$809$	121214.i	0.185206i	−0.995703	+	0.0926032i	$0.970481\pi$
$810$	0	0
$811$	−288619.	−0.438817	−0.219408	−	0.975633i	$-0.570413\pi$
$811$	−288619.	−0.438817	−0.219408	+	0.975633i	$0.570413\pi$
$812$	4.69960e6i	7.12768i
$813$	0	0
$814$	−314778.	−0.475068
$815$	880557.i	1.32569i
$816$	0	0
$817$	295849.	0.443226
$818$	1.45762e6i	2.17840i
$819$	0	0
$820$	5.53839e6	8.23675
$821$	− 556943.i	− 0.826274i	−0.910669	−	0.413137i	$-0.864433\pi$
$821$	− 556943.i	− 0.826274i	0.910669	−	0.413137i	$-0.135567\pi$
$822$	0	0
$823$	−380877.	−0.562322	−0.281161	−	0.959661i	$-0.590720\pi$
$823$	−380877.	−0.562322	−0.281161	+	0.959661i	$0.590720\pi$
$824$	440970.i	0.649464i
$825$	0	0
$826$	296220.	0.434164
$827$	− 345964.i	− 0.505848i	−0.967486	−	0.252924i	$-0.918608\pi$
$827$	− 345964.i	− 0.505848i	0.967486	−	0.252924i	$-0.0813923\pi$
$828$	0	0
$829$	−210809.	−0.306747	−0.153373	−	0.988168i	$-0.549014\pi$
$829$	−210809.	−0.306747	−0.153373	+	0.988168i	$0.549014\pi$
$830$	− 2.54610e6i	− 3.69590i
$831$	0	0
$832$	−3.37987e6	−4.88262
$833$	929934.i	1.34018i
$834$	0	0
$835$	−806724.	−1.15705
$836$	1.38168e6i	1.97694i
$837$	0	0
$838$	1.25646e6	1.78921
$839$	− 253512.i	− 0.360142i	−0.983654	−	0.180071i	$-0.942367\pi$
$839$	− 253512.i	− 0.360142i	0.983654	−	0.180071i	$-0.0576328\pi$
$840$	0	0
$841$	−855787.	−1.20997
$842$	− 370919.i	− 0.523184i
$843$	0	0
$844$	−928422.	−1.30335
$845$	202046.i	0.282967i
$846$	0	0
$847$	−691612.	−0.964042
$848$	− 4.38068e6i	− 6.09186i
$849$	0	0
$850$	−1.83193e6	−2.53554
$851$	130317.i	0.179946i
$852$	0	0
$853$	−444428.	−0.610806	−0.305403	−	0.952223i	$-0.598791\pi$
$853$	−444428.	−0.610806	−0.305403	+	0.952223i	$0.598791\pi$
$854$	− 2.50752e6i	− 3.43818i
$855$	0	0
$856$	476166.	0.649847
$857$	1937.16i	0.00263757i	0.999999	+	0.00131878i	$0.000419782\pi$
$857$	1937.16i	0.00263757i	−0.999999	+	0.00131878i	$0.999580\pi$
$858$	0	0
$859$	207057.	0.280610	0.140305	−	0.990108i	$-0.455192\pi$
$859$	207057.	0.280610	0.140305	+	0.990108i	$0.455192\pi$
$860$	1.45778e6i	1.97104i
$861$	0	0
$862$	2.36261e6	3.17964
$863$	− 3789.72i	− 0.00508845i	−0.999997	−	0.00254422i	$-0.999190\pi$
$863$	− 3789.72i	− 0.00508845i	0.999997	−	0.00254422i	$-0.000809852\pi$
$864$	0	0
$865$	783320.	1.04690
$866$	843816.i	1.12515i
$867$	0	0
$868$	862201.	1.14438
$869$	− 572187.i	− 0.757702i
$870$	0	0
$871$	−537877.	−0.709000
$872$	− 2.38191e6i	− 3.13252i
$873$	0	0
$874$	775993.	1.01586
$875$	1.91866e6i	2.50600i
$876$	0	0
$877$	650612.	0.845907	0.422954	−	0.906151i	$-0.360993\pi$
$877$	650612.	0.845907	0.422954	+	0.906151i	$0.360993\pi$
$878$	2.94347e6i	3.81831i
$879$	0	0
$880$	−3.51450e6	−4.53835
$881$	− 250290.i	− 0.322472i	−0.986916	−	0.161236i	$-0.948452\pi$
$881$	− 250290.i	− 0.322472i	0.986916	−	0.161236i	$-0.0515481\pi$
$882$	0	0
$883$	546029.	0.700317	0.350158	−	0.936691i	$-0.386128\pi$
$883$	546029.	0.700317	0.350158	+	0.936691i	$0.386128\pi$
$884$	− 1.64950e6i	− 2.11081i
$885$	0	0
$886$	−2.33204e6	−2.97077
$887$	94435.2i	0.120029i	0.998198	+	0.0600146i	$0.0191147\pi$
$887$	94435.2i	0.120029i	−0.998198	+	0.0600146i	$0.980885\pi$
$888$	0	0
$889$	−1.24770e6	−1.57872
$890$	4.38575e6i	5.53686i
$891$	0	0
$892$	4.03409e6	5.07010
$893$	− 971239.i	− 1.21793i
$894$	0	0
$895$	2.42496e6	3.02732
$896$	− 6.06246e6i	− 7.55150i
$897$	0	0
$898$	2.58141e6	3.20114
$899$	286765.i	0.354818i
$900$	0	0
$901$	848848.	1.04564
$902$	1.81844e6i	2.23505i
$903$	0	0
$904$	1.24605e6	1.52475
$905$	303223.i	0.370224i
$906$	0	0
$907$	−402864.	−0.489715	−0.244858	−	0.969559i	$-0.578741\pi$
$907$	−402864.	−0.489715	−0.244858	+	0.969559i	$0.578741\pi$
$908$	2.40953e6i	2.92254i
$909$	0	0
$910$	−5.05066e6	−6.09910
$911$	− 1.12201e6i	− 1.35195i	−0.736925	−	0.675974i	$-0.763723\pi$
$911$	− 1.12201e6i	− 1.35195i	0.736925	−	0.675974i	$-0.236277\pi$
$912$	0	0
$913$	616222.	0.739257
$914$	1.07686e6i	1.28904i
$915$	0	0
$916$	2.32624e6	2.77245
$917$	2.54887e6i	3.03116i
$918$	0	0
$919$	−1.55423e6	−1.84028	−0.920142	−	0.391584i	$-0.871927\pi$
$919$	−1.55423e6	−1.84028	−0.920142	+	0.391584i	$0.871927\pi$
$920$	2.46011e6i	2.90656i
$921$	0	0
$922$	−1.75016e6	−2.05881
$923$	− 543161.i	− 0.637566i
$924$	0	0
$925$	588764.	0.688110
$926$	163498.i	0.190674i
$927$	0	0
$928$	5.63096e6	6.53862
$929$	− 1.59966e6i	− 1.85352i	−0.375659	−	0.926758i	$-0.622584\pi$
$929$	− 1.59966e6i	− 1.85352i	0.375659	−	0.926758i	$-0.377416\pi$
$930$	0	0
$931$	−1.77968e6	−2.05325
$932$	− 1.45018e6i	− 1.66951i
$933$	0	0
$934$	937713.	1.07492
$935$	− 681007.i	− 0.778984i
$936$	0	0
$937$	348901.	0.397395	0.198698	−	0.980061i	$-0.436329\pi$
$937$	348901.	0.397395	0.198698	+	0.980061i	$0.436329\pi$
$938$	− 1.92561e6i	− 2.18858i
$939$	0	0
$940$	4.78574e6	5.41619
$941$	− 622331.i	− 0.702817i	−0.936222	−	0.351409i	$-0.885703\pi$
$941$	− 622331.i	− 0.702817i	0.936222	−	0.351409i	$-0.114297\pi$
$942$	0	0
$943$	752829.	0.846590
$944$	− 470940.i	− 0.528472i
$945$	0	0
$946$	−478640.	−0.534844
$947$	− 220680.i	− 0.246073i	−0.992402	−	0.123036i	$-0.960737\pi$
$947$	− 220680.i	− 0.246073i	0.992402	−	0.123036i	$-0.0392632\pi$
$948$	0	0
$949$	124080.	0.137775
$950$	− 3.50589e6i	− 3.88464i
$951$	0	0
$952$	3.79938e6	4.19217
$953$	1.26497e6i	1.39282i	0.717644	+	0.696411i	$0.245221\pi$
$953$	1.26497e6i	1.39282i	−0.717644	+	0.696411i	$0.754779\pi$
$954$	0	0
$955$	−848936.	−0.930826
$956$	− 1.91050e6i	− 2.09041i
$957$	0	0
$958$	−820295.	−0.893797
$959$	− 36611.5i	− 0.0398090i
$960$	0	0
$961$	−870910.	−0.943033
$962$	719185.i	0.777124i
$963$	0	0
$964$	3.55982e6	3.83066
$965$	208066.i	0.223432i
$966$	0	0
$967$	−443472.	−0.474256	−0.237128	−	0.971478i	$-0.576206\pi$
$967$	−443472.	−0.474256	−0.237128	+	0.971478i	$0.576206\pi$
$968$	1.85913e6i	1.98408i
$969$	0	0
$970$	−641141.	−0.681412
$971$	173433.i	0.183947i	0.995761	+	0.0919735i	$0.0293175\pi$
$971$	173433.i	0.183947i	−0.995761	+	0.0919735i	$0.970682\pi$
$972$	0	0
$973$	−2.79726e6	−2.95466
$974$	2.61542e6i	2.75692i
$975$	0	0
$976$	−3.98654e6	−4.18501
$977$	− 1.34964e6i	− 1.41393i	−0.707249	−	0.706964i	$-0.750064\pi$
$977$	− 1.34964e6i	− 1.41393i	0.707249	−	0.706964i	$-0.249936\pi$
$978$	0	0
$979$	−1.06146e6	−1.10749
$980$	− 8.76930e6i	− 9.13088i
$981$	0	0
$982$	−2.19695e6	−2.27822
$983$	98454.2i	0.101889i	0.998701	+	0.0509445i	$0.0162232\pi$
$983$	98454.2i	0.101889i	−0.998701	+	0.0509445i	$0.983777\pi$
$984$	0	0
$985$	222038.	0.228852
$986$	1.96406e6i	2.02023i
$987$	0	0
$988$	3.15677e6	3.23392
$989$	198155.i	0.202588i
$990$	0	0
$991$	1.07801e6	1.09768	0.548842	−	0.835926i	$-0.315069\pi$
$991$	1.07801e6	1.09768	0.548842	+	0.835926i	$0.315069\pi$
$992$	− 1.03307e6i	− 1.04980i
$993$	0	0
$994$	1.94452e6	1.96807
$995$	1.22244e6i	1.23476i
$996$	0	0
$997$	1.03149e6	1.03771	0.518856	−	0.854862i	$-0.326358\pi$
$997$	1.03149e6	1.03771	0.518856	+	0.854862i	$0.326358\pi$
$998$	1.10185e6i	1.10627i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.2	✓	76
3.2	odd	2	inner	531.5.b.a.296.75	yes	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.2	✓	76	1.1	even	1	trivial
531.5.b.a.296.75	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)

\(f(q)\)	\(=\)	\(q-7.80173i q^{2} -44.8670 q^{4} +42.3216i q^{5} -83.7809 q^{7} +225.212i q^{8} +O(q^{10})\)	\(q-7.80173i q^{2} -44.8670 q^{4} +42.3216i q^{5} -83.7809 q^{7} +225.212i q^{8} +330.181 q^{10} +79.9124i q^{11} +182.579 q^{13} +653.636i q^{14} +1039.17 q^{16} +201.361i q^{17} -385.359 q^{19} -1898.84i q^{20} +623.455 q^{22} -258.108i q^{23} -1166.12 q^{25} -1424.43i q^{26} +3758.99 q^{28} +1250.23i q^{29} +229.370 q^{31} -4503.95i q^{32} +1570.96 q^{34} -3545.74i q^{35} -504.893 q^{37} +3006.47i q^{38} -9531.33 q^{40} +2916.72i q^{41} -767.722 q^{43} -3585.43i q^{44} -2013.69 q^{46} +2520.35i q^{47} +4618.24 q^{49} +9097.72i q^{50} -8191.76 q^{52} -4215.55i q^{53} -3382.02 q^{55} -18868.5i q^{56} +9753.93 q^{58} -453.188i q^{59} -3836.27 q^{61} -1789.48i q^{62} -18511.8 q^{64} +7727.03i q^{65} -2946.00 q^{67} -9034.46i q^{68} -27662.9 q^{70} -2974.94i q^{71} +679.596 q^{73} +3939.04i q^{74} +17289.9 q^{76} -6695.13i q^{77} -7160.18 q^{79} +43979.4i q^{80} +22755.5 q^{82} -7711.22i q^{83} -8521.92 q^{85} +5989.56i q^{86} -17997.2 q^{88} +13282.8i q^{89} -15296.6 q^{91} +11580.5i q^{92} +19663.1 q^{94} -16309.0i q^{95} -1941.78 q^{97} -36030.2i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 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

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Representations

Groups

Database

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