LMFDB - Embedded newform 8032.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.a (trivial)

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:	yes
Analytic conductor:	$64.1358429035$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00046 q^{3} +2.74997 q^{5} +3.93356 q^{7} +6.00279 q^{9} +O(q^{10})$	$q-3.00046 q^{3} +2.74997 q^{5} +3.93356 q^{7} +6.00279 q^{9} -2.93372 q^{11} -2.73033 q^{13} -8.25118 q^{15} +0.166965 q^{17} -0.325661 q^{19} -11.8025 q^{21} +0.107540 q^{23} +2.56232 q^{25} -9.00976 q^{27} +7.02207 q^{29} +4.61647 q^{31} +8.80253 q^{33} +10.8171 q^{35} -4.53796 q^{37} +8.19226 q^{39} -5.07016 q^{41} -12.0166 q^{43} +16.5075 q^{45} -5.57042 q^{47} +8.47286 q^{49} -0.500972 q^{51} +7.42213 q^{53} -8.06764 q^{55} +0.977134 q^{57} -13.1811 q^{59} -1.26757 q^{61} +23.6123 q^{63} -7.50832 q^{65} -10.3405 q^{67} -0.322671 q^{69} -10.9702 q^{71} +2.67662 q^{73} -7.68814 q^{75} -11.5400 q^{77} -7.51524 q^{79} +9.02510 q^{81} -15.2176 q^{83} +0.459148 q^{85} -21.0695 q^{87} +6.98606 q^{89} -10.7399 q^{91} -13.8515 q^{93} -0.895557 q^{95} +2.36281 q^{97} -17.6105 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9	$ 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9} - 31 q^{11} + 5 q^{13} - 8 q^{15} - q^{17} - 29 q^{19} + 6 q^{21} - 27 q^{23} + 34 q^{25} - 36 q^{27} + q^{29} + 9 q^{31} - 8 q^{33} - 51 q^{35} + 5 q^{37} + 8 q^{39} - 3 q^{41} - 43 q^{43} - 42 q^{47} + 41 q^{49} - 54 q^{51} - 11 q^{53} + 10 q^{55} + 2 q^{57} - 49 q^{59} + 21 q^{61} - 15 q^{63} - 14 q^{65} - 68 q^{67} - 10 q^{69} - 44 q^{71} + 25 q^{73} - 51 q^{75} - 20 q^{77} + 49 q^{79} + 6 q^{81} - 88 q^{83} - 5 q^{85} - 16 q^{87} - 16 q^{89} - 46 q^{91} - 4 q^{93} - 28 q^{95} - 4 q^{97} - 71 q^{99}+O(q^{100}) $ 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9 - 31 * q^11 + 5 * q^13 - 8 * q^15 - q^17 - 29 * q^19 + 6 * q^21 - 27 * q^23 + 34 * q^25 - 36 * q^27 + q^29 + 9 * q^31 - 8 * q^33 - 51 * q^35 + 5 * q^37 + 8 * q^39 - 3 * q^41 - 43 * q^43 - 42 * q^47 + 41 * q^49 - 54 * q^51 - 11 * q^53 + 10 * q^55 + 2 * q^57 - 49 * q^59 + 21 * q^61 - 15 * q^63 - 14 * q^65 - 68 * q^67 - 10 * q^69 - 44 * q^71 + 25 * q^73 - 51 * q^75 - 20 * q^77 + 49 * q^79 + 6 * q^81 - 88 * q^83 - 5 * q^85 - 16 * q^87 - 16 * q^89 - 46 * q^91 - 4 * q^93 - 28 * q^95 - 4 * q^97 - 71 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−3.00046	−1.73232	−0.866160	−	0.499768i	$-0.833419\pi$
$3$	−3.00046	−1.73232	−0.866160	+	0.499768i	$0.833419\pi$
$4$	0	0
$5$	2.74997	1.22982	0.614911	−	0.788596i	$-0.289192\pi$
$5$	2.74997	1.22982	0.614911	+	0.788596i	$0.289192\pi$
$6$	0	0
$7$	3.93356	1.48674	0.743372	−	0.668878i	$-0.233225\pi$
$7$	3.93356	1.48674	0.743372	+	0.668878i	$0.233225\pi$
$8$	0	0
$9$	6.00279	2.00093
$10$	0	0
$11$	−2.93372	−0.884551	−0.442275	−	0.896879i	$-0.645829\pi$
$11$	−2.93372	−0.884551	−0.442275	+	0.896879i	$0.645829\pi$
$12$	0	0
$13$	−2.73033	−0.757257	−0.378629	−	0.925549i	$-0.623604\pi$
$13$	−2.73033	−0.757257	−0.378629	+	0.925549i	$0.623604\pi$
$14$	0	0
$15$	−8.25118	−2.13045
$16$	0	0
$17$	0.166965	0.0404949	0.0202475	−	0.999795i	$-0.493555\pi$
$17$	0.166965	0.0404949	0.0202475	+	0.999795i	$0.493555\pi$
$18$	0	0
$19$	−0.325661	−0.0747117	−0.0373559	−	0.999302i	$-0.511894\pi$
$19$	−0.325661	−0.0747117	−0.0373559	+	0.999302i	$0.511894\pi$
$20$	0	0
$21$	−11.8025	−2.57552
$22$	0	0
$23$	0.107540	0.0224237	0.0112119	−	0.999937i	$-0.496431\pi$
$23$	0.107540	0.0224237	0.0112119	+	0.999937i	$0.496431\pi$
$24$	0	0
$25$	2.56232	0.512464
$26$	0	0
$27$	−9.00976	−1.73393
$28$	0	0
$29$	7.02207	1.30397	0.651983	−	0.758234i	$-0.273937\pi$
$29$	7.02207	1.30397	0.651983	+	0.758234i	$0.273937\pi$
$30$	0	0
$31$	4.61647	0.829142	0.414571	−	0.910017i	$-0.363932\pi$
$31$	4.61647	0.829142	0.414571	+	0.910017i	$0.363932\pi$
$32$	0	0
$33$	8.80253	1.53232
$34$	0	0
$35$	10.8171	1.82843
$36$	0	0
$37$	−4.53796	−0.746037	−0.373018	−	0.927824i	$-0.621677\pi$
$37$	−4.53796	−0.746037	−0.373018	+	0.927824i	$0.621677\pi$
$38$	0	0
$39$	8.19226	1.31181
$40$	0	0
$41$	−5.07016	−0.791827	−0.395913	−	0.918288i	$-0.629572\pi$
$41$	−5.07016	−0.791827	−0.395913	+	0.918288i	$0.629572\pi$
$42$	0	0
$43$	−12.0166	−1.83251	−0.916256	−	0.400593i	$-0.868804\pi$
$43$	−12.0166	−1.83251	−0.916256	+	0.400593i	$0.868804\pi$
$44$	0	0
$45$	16.5075	2.46079
$46$	0	0
$47$	−5.57042	−0.812530	−0.406265	−	0.913755i	$-0.633169\pi$
$47$	−5.57042	−0.812530	−0.406265	+	0.913755i	$0.633169\pi$
$48$	0	0
$49$	8.47286	1.21041
$50$	0	0
$51$	−0.500972	−0.0701501
$52$	0	0
$53$	7.42213	1.01951	0.509754	−	0.860320i	$-0.329736\pi$
$53$	7.42213	1.01951	0.509754	+	0.860320i	$0.329736\pi$
$54$	0	0
$55$	−8.06764	−1.08784
$56$	0	0
$57$	0.977134	0.129425
$58$	0	0
$59$	−13.1811	−1.71603	−0.858016	−	0.513623i	$-0.828303\pi$
$59$	−13.1811	−1.71603	−0.858016	+	0.513623i	$0.828303\pi$
$60$	0	0
$61$	−1.26757	−0.162295	−0.0811476	−	0.996702i	$-0.525859\pi$
$61$	−1.26757	−0.162295	−0.0811476	+	0.996702i	$0.525859\pi$
$62$	0	0
$63$	23.6123	2.97487
$64$	0	0
$65$	−7.50832	−0.931292
$66$	0	0
$67$	−10.3405	−1.26329	−0.631645	−	0.775258i	$-0.717620\pi$
$67$	−10.3405	−1.26329	−0.631645	+	0.775258i	$0.717620\pi$
$68$	0	0
$69$	−0.322671	−0.0388451
$70$	0	0
$71$	−10.9702	−1.30193	−0.650963	−	0.759110i	$-0.725635\pi$
$71$	−10.9702	−1.30193	−0.650963	+	0.759110i	$0.725635\pi$
$72$	0	0
$73$	2.67662	0.313275	0.156637	−	0.987656i	$-0.449935\pi$
$73$	2.67662	0.313275	0.156637	+	0.987656i	$0.449935\pi$
$74$	0	0
$75$	−7.68814	−0.887750
$76$	0	0
$77$	−11.5400	−1.31510
$78$	0	0
$79$	−7.51524	−0.845531	−0.422765	−	0.906239i	$-0.638941\pi$
$79$	−7.51524	−0.845531	−0.422765	+	0.906239i	$0.638941\pi$
$80$	0	0
$81$	9.02510	1.00279
$82$	0	0
$83$	−15.2176	−1.67035	−0.835175	−	0.549984i	$-0.814634\pi$
$83$	−15.2176	−1.67035	−0.835175	+	0.549984i	$0.814634\pi$
$84$	0	0
$85$	0.459148	0.0498016
$86$	0	0
$87$	−21.0695	−2.25888
$88$	0	0
$89$	6.98606	0.740521	0.370261	−	0.928928i	$-0.379268\pi$
$89$	6.98606	0.740521	0.370261	+	0.928928i	$0.379268\pi$
$90$	0	0
$91$	−10.7399	−1.12585
$92$	0	0
$93$	−13.8515	−1.43634
$94$	0	0
$95$	−0.895557	−0.0918822
$96$	0	0
$97$	2.36281	0.239907	0.119953	−	0.992780i	$-0.461725\pi$
$97$	2.36281	0.239907	0.119953	+	0.992780i	$0.461725\pi$
$98$	0	0
$99$	−17.6105	−1.76992
$100$	0	0
$101$	−3.43213	−0.341509	−0.170755	−	0.985314i	$-0.554621\pi$
$101$	−3.43213	−0.341509	−0.170755	+	0.985314i	$0.554621\pi$
$102$	0	0
$103$	−7.44891	−0.733963	−0.366982	−	0.930228i	$-0.619609\pi$
$103$	−7.44891	−0.733963	−0.366982	+	0.930228i	$0.619609\pi$
$104$	0	0
$105$	−32.4565	−3.16743
$106$	0	0
$107$	11.0107	1.06444	0.532220	−	0.846606i	$-0.321358\pi$
$107$	11.0107	1.06444	0.532220	+	0.846606i	$0.321358\pi$
$108$	0	0
$109$	−8.76192	−0.839239	−0.419620	−	0.907700i	$-0.637837\pi$
$109$	−8.76192	−0.839239	−0.419620	+	0.907700i	$0.637837\pi$
$110$	0	0
$111$	13.6160	1.29237
$112$	0	0
$113$	18.0466	1.69768	0.848839	−	0.528651i	$-0.177302\pi$
$113$	18.0466	1.69768	0.848839	+	0.528651i	$0.177302\pi$
$114$	0	0
$115$	0.295733	0.0275772
$116$	0	0
$117$	−16.3896	−1.51522
$118$	0	0
$119$	0.656765	0.0602056
$120$	0	0
$121$	−2.39327	−0.217570
$122$	0	0
$123$	15.2128	1.37170
$124$	0	0
$125$	−6.70355	−0.599583
$126$	0	0
$127$	−9.32011	−0.827026	−0.413513	−	0.910498i	$-0.635698\pi$
$127$	−9.32011	−0.827026	−0.413513	+	0.910498i	$0.635698\pi$
$128$	0	0
$129$	36.0553	3.17449
$130$	0	0
$131$	16.8072	1.46846	0.734228	−	0.678903i	$-0.237544\pi$
$131$	16.8072	1.46846	0.734228	+	0.678903i	$0.237544\pi$
$132$	0	0
$133$	−1.28101	−0.111077
$134$	0	0
$135$	−24.7765	−2.13242
$136$	0	0
$137$	−5.43218	−0.464102	−0.232051	−	0.972704i	$-0.574544\pi$
$137$	−5.43218	−0.464102	−0.232051	+	0.972704i	$0.574544\pi$
$138$	0	0
$139$	2.34636	0.199016	0.0995080	−	0.995037i	$-0.468273\pi$
$139$	2.34636	0.199016	0.0995080	+	0.995037i	$0.468273\pi$
$140$	0	0
$141$	16.7139	1.40756
$142$	0	0
$143$	8.01003	0.669832
$144$	0	0
$145$	19.3105	1.60365
$146$	0	0
$147$	−25.4225	−2.09681
$148$	0	0
$149$	−9.54919	−0.782300	−0.391150	−	0.920327i	$-0.627923\pi$
$149$	−9.54919	−0.782300	−0.391150	+	0.920327i	$0.627923\pi$
$150$	0	0
$151$	16.4093	1.33537	0.667685	−	0.744444i	$-0.267285\pi$
$151$	16.4093	1.33537	0.667685	+	0.744444i	$0.267285\pi$
$152$	0	0
$153$	1.00225	0.0810274
$154$	0	0
$155$	12.6951	1.01970
$156$	0	0
$157$	2.80788	0.224093	0.112047	−	0.993703i	$-0.464259\pi$
$157$	2.80788	0.224093	0.112047	+	0.993703i	$0.464259\pi$
$158$	0	0
$159$	−22.2698	−1.76611
$160$	0	0
$161$	0.423016	0.0333384
$162$	0	0
$163$	2.54219	0.199120	0.0995599	−	0.995032i	$-0.468257\pi$
$163$	2.54219	0.199120	0.0995599	+	0.995032i	$0.468257\pi$
$164$	0	0
$165$	24.2067	1.88449
$166$	0	0
$167$	−13.9939	−1.08288	−0.541439	−	0.840740i	$-0.682120\pi$
$167$	−13.9939	−1.08288	−0.541439	+	0.840740i	$0.682120\pi$
$168$	0	0
$169$	−5.54530	−0.426562
$170$	0	0
$171$	−1.95487	−0.149493
$172$	0	0
$173$	−14.5985	−1.10990	−0.554950	−	0.831884i	$-0.687263\pi$
$173$	−14.5985	−1.10990	−0.554950	+	0.831884i	$0.687263\pi$
$174$	0	0
$175$	10.0790	0.761902
$176$	0	0
$177$	39.5494	2.97272
$178$	0	0
$179$	14.8908	1.11299	0.556494	−	0.830852i	$-0.312146\pi$
$179$	14.8908	1.11299	0.556494	+	0.830852i	$0.312146\pi$
$180$	0	0
$181$	12.0980	0.899235	0.449618	−	0.893221i	$-0.351560\pi$
$181$	12.0980	0.899235	0.449618	+	0.893221i	$0.351560\pi$
$182$	0	0
$183$	3.80328	0.281147
$184$	0	0
$185$	−12.4792	−0.917493
$186$	0	0
$187$	−0.489828	−0.0358198
$188$	0	0
$189$	−35.4404	−2.57791
$190$	0	0
$191$	2.15986	0.156282	0.0781408	−	0.996942i	$-0.475102\pi$
$191$	2.15986	0.156282	0.0781408	+	0.996942i	$0.475102\pi$
$192$	0	0
$193$	−6.46314	−0.465227	−0.232614	−	0.972569i	$-0.574728\pi$
$193$	−6.46314	−0.465227	−0.232614	+	0.972569i	$0.574728\pi$
$194$	0	0
$195$	22.5284	1.61329
$196$	0	0
$197$	−14.7624	−1.05177	−0.525887	−	0.850554i	$-0.676267\pi$
$197$	−14.7624	−1.05177	−0.525887	+	0.850554i	$0.676267\pi$
$198$	0	0
$199$	9.20868	0.652786	0.326393	−	0.945234i	$-0.394167\pi$
$199$	9.20868	0.652786	0.326393	+	0.945234i	$0.394167\pi$
$200$	0	0
$201$	31.0262	2.18842
$202$	0	0
$203$	27.6217	1.93866
$204$	0	0
$205$	−13.9428	−0.973806
$206$	0	0
$207$	0.645543	0.0448683
$208$	0	0
$209$	0.955399	0.0660863
$210$	0	0
$211$	10.8029	0.743705	0.371853	−	0.928292i	$-0.378723\pi$
$211$	10.8029	0.743705	0.371853	+	0.928292i	$0.378723\pi$
$212$	0	0
$213$	32.9157	2.25535
$214$	0	0
$215$	−33.0452	−2.25366
$216$	0	0
$217$	18.1591	1.23272
$218$	0	0
$219$	−8.03110	−0.542692
$220$	0	0
$221$	−0.455869	−0.0306651
$222$	0	0
$223$	9.48570	0.635209	0.317605	−	0.948223i	$-0.397122\pi$
$223$	9.48570	0.635209	0.317605	+	0.948223i	$0.397122\pi$
$224$	0	0
$225$	15.3810	1.02540
$226$	0	0
$227$	20.8801	1.38586	0.692931	−	0.721004i	$-0.256319\pi$
$227$	20.8801	1.38586	0.692931	+	0.721004i	$0.256319\pi$
$228$	0	0
$229$	−3.00461	−0.198550	−0.0992752	−	0.995060i	$-0.531652\pi$
$229$	−3.00461	−0.198550	−0.0992752	+	0.995060i	$0.531652\pi$
$230$	0	0
$231$	34.6252	2.27817
$232$	0	0
$233$	−29.5205	−1.93395	−0.966977	−	0.254865i	$-0.917969\pi$
$233$	−29.5205	−1.93395	−0.966977	+	0.254865i	$0.917969\pi$
$234$	0	0
$235$	−15.3185	−0.999267
$236$	0	0
$237$	22.5492	1.46473
$238$	0	0
$239$	−12.5422	−0.811289	−0.405644	−	0.914031i	$-0.632953\pi$
$239$	−12.5422	−0.811289	−0.405644	+	0.914031i	$0.632953\pi$
$240$	0	0
$241$	−3.05416	−0.196736	−0.0983679	−	0.995150i	$-0.531362\pi$
$241$	−3.05416	−0.196736	−0.0983679	+	0.995150i	$0.531362\pi$
$242$	0	0
$243$	−0.0502061	−0.00322072
$244$	0	0
$245$	23.3001	1.48859
$246$	0	0
$247$	0.889162	0.0565760
$248$	0	0
$249$	45.6599	2.89358
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−0.315494	−0.0198349
$254$	0	0
$255$	−1.37766	−0.0862722
$256$	0	0
$257$	−18.0435	−1.12553	−0.562763	−	0.826618i	$-0.690262\pi$
$257$	−18.0435	−1.12553	−0.562763	+	0.826618i	$0.690262\pi$
$258$	0	0
$259$	−17.8503	−1.10917
$260$	0	0
$261$	42.1520	2.60914
$262$	0	0
$263$	27.3650	1.68740	0.843700	−	0.536816i	$-0.180373\pi$
$263$	27.3650	1.68740	0.843700	+	0.536816i	$0.180373\pi$
$264$	0	0
$265$	20.4106	1.25381
$266$	0	0
$267$	−20.9614	−1.28282
$268$	0	0
$269$	−12.2396	−0.746259	−0.373129	−	0.927779i	$-0.621715\pi$
$269$	−12.2396	−0.746259	−0.373129	+	0.927779i	$0.621715\pi$
$270$	0	0
$271$	5.51255	0.334864	0.167432	−	0.985884i	$-0.446453\pi$
$271$	5.51255	0.334864	0.167432	+	0.985884i	$0.446453\pi$
$272$	0	0
$273$	32.2247	1.95033
$274$	0	0
$275$	−7.51713	−0.453300
$276$	0	0
$277$	−8.72645	−0.524321	−0.262161	−	0.965024i	$-0.584435\pi$
$277$	−8.72645	−0.524321	−0.262161	+	0.965024i	$0.584435\pi$
$278$	0	0
$279$	27.7117	1.65905
$280$	0	0
$281$	25.0316	1.49326	0.746630	−	0.665240i	$-0.231671\pi$
$281$	25.0316	1.49326	0.746630	+	0.665240i	$0.231671\pi$
$282$	0	0
$283$	−12.0357	−0.715450	−0.357725	−	0.933827i	$-0.616447\pi$
$283$	−12.0357	−0.715450	−0.357725	+	0.933827i	$0.616447\pi$
$284$	0	0
$285$	2.68709	0.159169
$286$	0	0
$287$	−19.9438	−1.17724
$288$	0	0
$289$	−16.9721	−0.998360
$290$	0	0
$291$	−7.08952	−0.415595
$292$	0	0
$293$	5.46596	0.319325	0.159662	−	0.987172i	$-0.448959\pi$
$293$	5.46596	0.319325	0.159662	+	0.987172i	$0.448959\pi$
$294$	0	0
$295$	−36.2476	−2.11042
$296$	0	0
$297$	26.4321	1.53375
$298$	0	0
$299$	−0.293621	−0.0169805
$300$	0	0
$301$	−47.2679	−2.72448
$302$	0	0
$303$	10.2980	0.591603
$304$	0	0
$305$	−3.48576	−0.199594
$306$	0	0
$307$	−10.0339	−0.572664	−0.286332	−	0.958131i	$-0.592436\pi$
$307$	−10.0339	−0.572664	−0.286332	+	0.958131i	$0.592436\pi$
$308$	0	0
$309$	22.3502	1.27146
$310$	0	0
$311$	−17.0235	−0.965314	−0.482657	−	0.875810i	$-0.660328\pi$
$311$	−17.0235	−0.965314	−0.482657	+	0.875810i	$0.660328\pi$
$312$	0	0
$313$	15.9910	0.903863	0.451931	−	0.892053i	$-0.350735\pi$
$313$	15.9910	0.903863	0.451931	+	0.892053i	$0.350735\pi$
$314$	0	0
$315$	64.9330	3.65856
$316$	0	0
$317$	−34.8409	−1.95686	−0.978429	−	0.206581i	$-0.933766\pi$
$317$	−34.8409	−1.95686	−0.978429	+	0.206581i	$0.933766\pi$
$318$	0	0
$319$	−20.6008	−1.15342
$320$	0	0
$321$	−33.0371	−1.84395
$322$	0	0
$323$	−0.0543739	−0.00302544
$324$	0	0
$325$	−6.99597	−0.388067
$326$	0	0
$327$	26.2898	1.45383
$328$	0	0
$329$	−21.9116	−1.20802
$330$	0	0
$331$	12.0139	0.660344	0.330172	−	0.943921i	$-0.392893\pi$
$331$	12.0139	0.660344	0.330172	+	0.943921i	$0.392893\pi$
$332$	0	0
$333$	−27.2404	−1.49277
$334$	0	0
$335$	−28.4360	−1.55362
$336$	0	0
$337$	31.3283	1.70656	0.853281	−	0.521451i	$-0.174609\pi$
$337$	31.3283	1.70656	0.853281	+	0.521451i	$0.174609\pi$
$338$	0	0
$339$	−54.1481	−2.94092
$340$	0	0
$341$	−13.5434	−0.733418
$342$	0	0
$343$	5.79358	0.312824
$344$	0	0
$345$	−0.887335	−0.0477725
$346$	0	0
$347$	9.51824	0.510966	0.255483	−	0.966814i	$-0.417766\pi$
$347$	9.51824	0.510966	0.255483	+	0.966814i	$0.417766\pi$
$348$	0	0
$349$	−0.698937	−0.0374133	−0.0187066	−	0.999825i	$-0.505955\pi$
$349$	−0.698937	−0.0374133	−0.0187066	+	0.999825i	$0.505955\pi$
$350$	0	0
$351$	24.5996	1.31303
$352$	0	0
$353$	−4.58680	−0.244131	−0.122065	−	0.992522i	$-0.538952\pi$
$353$	−4.58680	−0.244131	−0.122065	+	0.992522i	$0.538952\pi$
$354$	0	0
$355$	−30.1677	−1.60114
$356$	0	0
$357$	−1.97060	−0.104295
$358$	0	0
$359$	−1.64998	−0.0870828	−0.0435414	−	0.999052i	$-0.513864\pi$
$359$	−1.64998	−0.0870828	−0.0435414	+	0.999052i	$0.513864\pi$
$360$	0	0
$361$	−18.8939	−0.994418
$362$	0	0
$363$	7.18093	0.376901
$364$	0	0
$365$	7.36061	0.385272
$366$	0	0
$367$	19.6899	1.02780	0.513902	−	0.857849i	$-0.328200\pi$
$367$	19.6899	1.02780	0.513902	+	0.857849i	$0.328200\pi$
$368$	0	0
$369$	−30.4351	−1.58439
$370$	0	0
$371$	29.1954	1.51575
$372$	0	0
$373$	−5.65369	−0.292737	−0.146368	−	0.989230i	$-0.546759\pi$
$373$	−5.65369	−0.292737	−0.146368	+	0.989230i	$0.546759\pi$
$374$	0	0
$375$	20.1138	1.03867
$376$	0	0
$377$	−19.1726	−0.987438
$378$	0	0
$379$	−11.2209	−0.576380	−0.288190	−	0.957573i	$-0.593054\pi$
$379$	−11.2209	−0.576380	−0.288190	+	0.957573i	$0.593054\pi$
$380$	0	0
$381$	27.9647	1.43267
$382$	0	0
$383$	9.28462	0.474422	0.237211	−	0.971458i	$-0.423767\pi$
$383$	9.28462	0.474422	0.237211	+	0.971458i	$0.423767\pi$
$384$	0	0
$385$	−31.7345	−1.61734
$386$	0	0
$387$	−72.1330	−3.66673
$388$	0	0
$389$	17.5617	0.890413	0.445206	−	0.895428i	$-0.353130\pi$
$389$	17.5617	0.890413	0.445206	+	0.895428i	$0.353130\pi$
$390$	0	0
$391$	0.0179555	0.000908047 0
$392$	0	0
$393$	−50.4296	−2.54383
$394$	0	0
$395$	−20.6667	−1.03985
$396$	0	0
$397$	7.74329	0.388625	0.194312	−	0.980940i	$-0.437752\pi$
$397$	7.74329	0.388625	0.194312	+	0.980940i	$0.437752\pi$
$398$	0	0
$399$	3.84361	0.192421
$400$	0	0
$401$	−3.86752	−0.193135	−0.0965674	−	0.995326i	$-0.530786\pi$
$401$	−3.86752	−0.193135	−0.0965674	+	0.995326i	$0.530786\pi$
$402$	0	0
$403$	−12.6045	−0.627874
$404$	0	0
$405$	24.8187	1.23325
$406$	0	0
$407$	13.3131	0.659907
$408$	0	0
$409$	40.1626	1.98591	0.992956	−	0.118481i	$-0.0378025\pi$
$409$	40.1626	1.98591	0.992956	+	0.118481i	$0.0378025\pi$
$410$	0	0
$411$	16.2991	0.803973
$412$	0	0
$413$	−51.8486	−2.55130
$414$	0	0
$415$	−41.8479	−2.05423
$416$	0	0
$417$	−7.04018	−0.344759
$418$	0	0
$419$	−16.8974	−0.825493	−0.412746	−	0.910846i	$-0.635430\pi$
$419$	−16.8974	−0.825493	−0.412746	+	0.910846i	$0.635430\pi$
$420$	0	0
$421$	13.1589	0.641324	0.320662	−	0.947194i	$-0.396095\pi$
$421$	13.1589	0.641324	0.320662	+	0.947194i	$0.396095\pi$
$422$	0	0
$423$	−33.4381	−1.62581
$424$	0	0
$425$	0.427817	0.0207522
$426$	0	0
$427$	−4.98604	−0.241291
$428$	0	0
$429$	−24.0338	−1.16036
$430$	0	0
$431$	−4.88561	−0.235331	−0.117666	−	0.993053i	$-0.537541\pi$
$431$	−4.88561	−0.235331	−0.117666	+	0.993053i	$0.537541\pi$
$432$	0	0
$433$	36.4686	1.75257	0.876285	−	0.481793i	$-0.160014\pi$
$433$	36.4686	1.75257	0.876285	+	0.481793i	$0.160014\pi$
$434$	0	0
$435$	−57.9404	−2.77803
$436$	0	0
$437$	−0.0350217	−0.00167532
$438$	0	0
$439$	−22.2710	−1.06293	−0.531467	−	0.847079i	$-0.678359\pi$
$439$	−22.2710	−1.06293	−0.531467	+	0.847079i	$0.678359\pi$
$440$	0	0
$441$	50.8608	2.42194
$442$	0	0
$443$	35.9423	1.70767	0.853834	−	0.520545i	$-0.174271\pi$
$443$	35.9423	1.70767	0.853834	+	0.520545i	$0.174271\pi$
$444$	0	0
$445$	19.2114	0.910710
$446$	0	0
$447$	28.6520	1.35519
$448$	0	0
$449$	−34.0044	−1.60476	−0.802382	−	0.596810i	$-0.796435\pi$
$449$	−34.0044	−1.60476	−0.802382	+	0.596810i	$0.796435\pi$
$450$	0	0
$451$	14.8745	0.700411
$452$	0	0
$453$	−49.2355	−2.31329
$454$	0	0
$455$	−29.5344	−1.38459
$456$	0	0
$457$	−15.6171	−0.730539	−0.365269	−	0.930902i	$-0.619023\pi$
$457$	−15.6171	−0.730539	−0.365269	+	0.930902i	$0.619023\pi$
$458$	0	0
$459$	−1.50431	−0.0702153
$460$	0	0
$461$	21.5196	1.00227	0.501133	−	0.865370i	$-0.332917\pi$
$461$	21.5196	1.00227	0.501133	+	0.865370i	$0.332917\pi$
$462$	0	0
$463$	−28.1634	−1.30886	−0.654432	−	0.756120i	$-0.727092\pi$
$463$	−28.1634	−1.30886	−0.654432	+	0.756120i	$0.727092\pi$
$464$	0	0
$465$	−38.0913	−1.76644
$466$	0	0
$467$	−26.0081	−1.20351	−0.601756	−	0.798680i	$-0.705532\pi$
$467$	−26.0081	−1.20351	−0.601756	+	0.798680i	$0.705532\pi$
$468$	0	0
$469$	−40.6748	−1.87819
$470$	0	0
$471$	−8.42494	−0.388201
$472$	0	0
$473$	35.2533	1.62095
$474$	0	0
$475$	−0.834447	−0.0382870
$476$	0	0
$477$	44.5535	2.03996
$478$	0	0
$479$	−38.2378	−1.74713	−0.873564	−	0.486709i	$-0.838197\pi$
$479$	−38.2378	−1.74713	−0.873564	+	0.486709i	$0.838197\pi$
$480$	0	0
$481$	12.3901	0.564942
$482$	0	0
$483$	−1.26925	−0.0577527
$484$	0	0
$485$	6.49765	0.295043
$486$	0	0
$487$	1.04285	0.0472560	0.0236280	−	0.999721i	$-0.492478\pi$
$487$	1.04285	0.0472560	0.0236280	+	0.999721i	$0.492478\pi$
$488$	0	0
$489$	−7.62775	−0.344939
$490$	0	0
$491$	−9.19477	−0.414954	−0.207477	−	0.978240i	$-0.566525\pi$
$491$	−9.19477	−0.414954	−0.207477	+	0.978240i	$0.566525\pi$
$492$	0	0
$493$	1.17244	0.0528040
$494$	0	0
$495$	−48.4283	−2.17669
$496$	0	0
$497$	−43.1520	−1.93563
$498$	0	0
$499$	−18.8954	−0.845873	−0.422936	−	0.906159i	$-0.639001\pi$
$499$	−18.8954	−0.845873	−0.422936	+	0.906159i	$0.639001\pi$
$500$	0	0
$501$	41.9881	1.87589
$502$	0	0
$503$	17.1532	0.764825	0.382412	−	0.923992i	$-0.375093\pi$
$503$	17.1532	0.764825	0.382412	+	0.923992i	$0.375093\pi$
$504$	0	0
$505$	−9.43823	−0.419996
$506$	0	0
$507$	16.6385	0.738941
$508$	0	0
$509$	−5.35314	−0.237274	−0.118637	−	0.992938i	$-0.537852\pi$
$509$	−5.35314	−0.237274	−0.118637	+	0.992938i	$0.537852\pi$
$510$	0	0
$511$	10.5286	0.465759
$512$	0	0
$513$	2.93413	0.129545
$514$	0	0
$515$	−20.4843	−0.902644
$516$	0	0
$517$	16.3421	0.718724
$518$	0	0
$519$	43.8022	1.92270
$520$	0	0
$521$	−31.4614	−1.37835	−0.689175	−	0.724595i	$-0.742027\pi$
$521$	−31.4614	−1.37835	−0.689175	+	0.724595i	$0.742027\pi$
$522$	0	0
$523$	27.7693	1.21426	0.607132	−	0.794601i	$-0.292320\pi$
$523$	27.7693	1.21426	0.607132	+	0.794601i	$0.292320\pi$
$524$	0	0
$525$	−30.2417	−1.31986
$526$	0	0
$527$	0.770788	0.0335760
$528$	0	0
$529$	−22.9884	−0.999497
$530$	0	0
$531$	−79.1233	−3.43366
$532$	0	0
$533$	13.8432	0.599616
$534$	0	0
$535$	30.2790	1.30907
$536$	0	0
$537$	−44.6792	−1.92805
$538$	0	0
$539$	−24.8570	−1.07067
$540$	0	0
$541$	4.09558	0.176083	0.0880414	−	0.996117i	$-0.471939\pi$
$541$	4.09558	0.176083	0.0880414	+	0.996117i	$0.471939\pi$
$542$	0	0
$543$	−36.2995	−1.55776
$544$	0	0
$545$	−24.0950	−1.03212
$546$	0	0
$547$	14.9278	0.638265	0.319132	−	0.947710i	$-0.396609\pi$
$547$	14.9278	0.638265	0.319132	+	0.947710i	$0.396609\pi$
$548$	0	0
$549$	−7.60892	−0.324741
$550$	0	0
$551$	−2.28681	−0.0974215
$552$	0	0
$553$	−29.5616	−1.25709
$554$	0	0
$555$	37.4435	1.58939
$556$	0	0
$557$	−27.2107	−1.15296	−0.576478	−	0.817113i	$-0.695573\pi$
$557$	−27.2107	−1.15296	−0.576478	+	0.817113i	$0.695573\pi$
$558$	0	0
$559$	32.8092	1.38768
$560$	0	0
$561$	1.46971	0.0620513
$562$	0	0
$563$	23.8475	1.00505	0.502526	−	0.864562i	$-0.332404\pi$
$563$	23.8475	1.00505	0.502526	+	0.864562i	$0.332404\pi$
$564$	0	0
$565$	49.6275	2.08784
$566$	0	0
$567$	35.5007	1.49089
$568$	0	0
$569$	0.545596	0.0228726	0.0114363	−	0.999935i	$-0.496360\pi$
$569$	0.545596	0.0228726	0.0114363	+	0.999935i	$0.496360\pi$
$570$	0	0
$571$	34.0455	1.42476	0.712379	−	0.701794i	$-0.247618\pi$
$571$	34.0455	1.42476	0.712379	+	0.701794i	$0.247618\pi$
$572$	0	0
$573$	−6.48057	−0.270730
$574$	0	0
$575$	0.275553	0.0114913
$576$	0	0
$577$	−0.166808	−0.00694429	−0.00347214	−	0.999994i	$-0.501105\pi$
$577$	−0.166808	−0.00694429	−0.00347214	+	0.999994i	$0.501105\pi$
$578$	0	0
$579$	19.3924	0.805922
$580$	0	0
$581$	−59.8593	−2.48338
$582$	0	0
$583$	−21.7745	−0.901807
$584$	0	0
$585$	−45.0708	−1.86345
$586$	0	0
$587$	−6.74322	−0.278322	−0.139161	−	0.990270i	$-0.544441\pi$
$587$	−6.74322	−0.278322	−0.139161	+	0.990270i	$0.544441\pi$
$588$	0	0
$589$	−1.50340	−0.0619466
$590$	0	0
$591$	44.2939	1.82201
$592$	0	0
$593$	−34.9486	−1.43517	−0.717583	−	0.696473i	$-0.754752\pi$
$593$	−34.9486	−1.43517	−0.717583	+	0.696473i	$0.754752\pi$
$594$	0	0
$595$	1.80608	0.0740422
$596$	0	0
$597$	−27.6303	−1.13083
$598$	0	0
$599$	46.2196	1.88848	0.944240	−	0.329257i	$-0.106798\pi$
$599$	46.2196	1.88848	0.944240	+	0.329257i	$0.106798\pi$
$600$	0	0
$601$	−9.60816	−0.391925	−0.195963	−	0.980611i	$-0.562783\pi$
$601$	−9.60816	−0.391925	−0.195963	+	0.980611i	$0.562783\pi$
$602$	0	0
$603$	−62.0717	−2.52775
$604$	0	0
$605$	−6.58142	−0.267573
$606$	0	0
$607$	14.5826	0.591890	0.295945	−	0.955205i	$-0.404365\pi$
$607$	14.5826	0.591890	0.295945	+	0.955205i	$0.404365\pi$
$608$	0	0
$609$	−82.8780	−3.35838
$610$	0	0
$611$	15.2091	0.615294
$612$	0	0
$613$	3.37071	0.136142	0.0680708	−	0.997680i	$-0.478316\pi$
$613$	3.37071	0.136142	0.0680708	+	0.997680i	$0.478316\pi$
$614$	0	0
$615$	41.8348	1.68694
$616$	0	0
$617$	38.0121	1.53031	0.765156	−	0.643845i	$-0.222662\pi$
$617$	38.0121	1.53031	0.765156	+	0.643845i	$0.222662\pi$
$618$	0	0
$619$	−20.1794	−0.811077	−0.405539	−	0.914078i	$-0.632916\pi$
$619$	−20.1794	−0.811077	−0.405539	+	0.914078i	$0.632916\pi$
$620$	0	0
$621$	−0.968913	−0.0388812
$622$	0	0
$623$	27.4801	1.10097
$624$	0	0
$625$	−31.2461	−1.24984
$626$	0	0
$627$	−2.86664	−0.114483
$628$	0	0
$629$	−0.757680	−0.0302107
$630$	0	0
$631$	29.5055	1.17459	0.587297	−	0.809372i	$-0.300192\pi$
$631$	29.5055	1.17459	0.587297	+	0.809372i	$0.300192\pi$
$632$	0	0
$633$	−32.4138	−1.28833
$634$	0	0
$635$	−25.6300	−1.01710
$636$	0	0
$637$	−23.1337	−0.916591
$638$	0	0
$639$	−65.8519	−2.60506
$640$	0	0
$641$	−12.6949	−0.501419	−0.250709	−	0.968062i	$-0.580664\pi$
$641$	−12.6949	−0.501419	−0.250709	+	0.968062i	$0.580664\pi$
$642$	0	0
$643$	−6.71927	−0.264982	−0.132491	−	0.991184i	$-0.542298\pi$
$643$	−6.71927	−0.264982	−0.132491	+	0.991184i	$0.542298\pi$
$644$	0	0
$645$	99.1510	3.90407
$646$	0	0
$647$	−15.5410	−0.610978	−0.305489	−	0.952196i	$-0.598820\pi$
$647$	−15.5410	−0.610978	−0.305489	+	0.952196i	$0.598820\pi$
$648$	0	0
$649$	38.6697	1.51792
$650$	0	0
$651$	−54.4858	−2.13547
$652$	0	0
$653$	−25.1774	−0.985269	−0.492635	−	0.870236i	$-0.663966\pi$
$653$	−25.1774	−0.985269	−0.492635	+	0.870236i	$0.663966\pi$
$654$	0	0
$655$	46.2194	1.80594
$656$	0	0
$657$	16.0672	0.626840
$658$	0	0
$659$	−13.9761	−0.544433	−0.272217	−	0.962236i	$-0.587757\pi$
$659$	−13.9761	−0.544433	−0.272217	+	0.962236i	$0.587757\pi$
$660$	0	0
$661$	−11.7986	−0.458914	−0.229457	−	0.973319i	$-0.573695\pi$
$661$	−11.7986	−0.458914	−0.229457	+	0.973319i	$0.573695\pi$
$662$	0	0
$663$	1.36782	0.0531217
$664$	0	0
$665$	−3.52272	−0.136605
$666$	0	0
$667$	0.755157	0.0292398
$668$	0	0
$669$	−28.4615	−1.10039
$670$	0	0
$671$	3.71868	0.143558
$672$	0	0
$673$	−7.90922	−0.304878	−0.152439	−	0.988313i	$-0.548713\pi$
$673$	−7.90922	−0.304878	−0.152439	+	0.988313i	$0.548713\pi$
$674$	0	0
$675$	−23.0859	−0.888575
$676$	0	0
$677$	−30.6221	−1.17690	−0.588452	−	0.808532i	$-0.700262\pi$
$677$	−30.6221	−1.17690	−0.588452	+	0.808532i	$0.700262\pi$
$678$	0	0
$679$	9.29424	0.356680
$680$	0	0
$681$	−62.6500	−2.40075
$682$	0	0
$683$	−36.9536	−1.41399	−0.706994	−	0.707219i	$-0.749949\pi$
$683$	−36.9536	−1.41399	−0.706994	+	0.707219i	$0.749949\pi$
$684$	0	0
$685$	−14.9383	−0.570763
$686$	0	0
$687$	9.01524	0.343953
$688$	0	0
$689$	−20.2649	−0.772030
$690$	0	0
$691$	−6.96790	−0.265071	−0.132536	−	0.991178i	$-0.542312\pi$
$691$	−6.96790	−0.265071	−0.132536	+	0.991178i	$0.542312\pi$
$692$	0	0
$693$	−69.2719	−2.63142
$694$	0	0
$695$	6.45242	0.244754
$696$	0	0
$697$	−0.846539	−0.0320649
$698$	0	0
$699$	88.5753	3.35022
$700$	0	0
$701$	−45.2465	−1.70894	−0.854468	−	0.519504i	$-0.826117\pi$
$701$	−45.2465	−1.70894	−0.854468	+	0.519504i	$0.826117\pi$
$702$	0	0
$703$	1.47784	0.0557377
$704$	0	0
$705$	45.9626	1.73105
$706$	0	0
$707$	−13.5005	−0.507737
$708$	0	0
$709$	38.7551	1.45548	0.727739	−	0.685854i	$-0.240571\pi$
$709$	38.7551	1.45548	0.727739	+	0.685854i	$0.240571\pi$
$710$	0	0
$711$	−45.1124	−1.69185
$712$	0	0
$713$	0.496457	0.0185925
$714$	0	0
$715$	22.0273	0.823775
$716$	0	0
$717$	37.6325	1.40541
$718$	0	0
$719$	19.1344	0.713591	0.356795	−	0.934183i	$-0.383869\pi$
$719$	19.1344	0.713591	0.356795	+	0.934183i	$0.383869\pi$
$720$	0	0
$721$	−29.3007	−1.09122
$722$	0	0
$723$	9.16390	0.340809
$724$	0	0
$725$	17.9928	0.668235
$726$	0	0
$727$	−18.4596	−0.684629	−0.342314	−	0.939585i	$-0.611211\pi$
$727$	−18.4596	−0.684629	−0.342314	+	0.939585i	$0.611211\pi$
$728$	0	0
$729$	−26.9246	−0.997209
$730$	0	0
$731$	−2.00635	−0.0742074
$732$	0	0
$733$	−42.7761	−1.57997	−0.789986	−	0.613125i	$-0.789912\pi$
$733$	−42.7761	−1.57997	−0.789986	+	0.613125i	$0.789912\pi$
$734$	0	0
$735$	−69.9111	−2.57871
$736$	0	0
$737$	30.3361	1.11744
$738$	0	0
$739$	2.76514	0.101717	0.0508587	−	0.998706i	$-0.483804\pi$
$739$	2.76514	0.101717	0.0508587	+	0.998706i	$0.483804\pi$
$740$	0	0
$741$	−2.66790	−0.0980077
$742$	0	0
$743$	−19.3498	−0.709877	−0.354938	−	0.934890i	$-0.615498\pi$
$743$	−19.3498	−0.709877	−0.354938	+	0.934890i	$0.615498\pi$
$744$	0	0
$745$	−26.2599	−0.962090
$746$	0	0
$747$	−91.3481	−3.34225
$748$	0	0
$749$	43.3110	1.58255
$750$	0	0
$751$	−0.947873	−0.0345884	−0.0172942	−	0.999850i	$-0.505505\pi$
$751$	−0.947873	−0.0345884	−0.0172942	+	0.999850i	$0.505505\pi$
$752$	0	0
$753$	−3.00046	−0.109343
$754$	0	0
$755$	45.1250	1.64227
$756$	0	0
$757$	−3.21907	−0.116999	−0.0584995	−	0.998287i	$-0.518632\pi$
$757$	−3.21907	−0.116999	−0.0584995	+	0.998287i	$0.518632\pi$
$758$	0	0
$759$	0.946628	0.0343604
$760$	0	0
$761$	18.6442	0.675853	0.337927	−	0.941172i	$-0.390274\pi$
$761$	18.6442	0.675853	0.337927	+	0.941172i	$0.390274\pi$
$762$	0	0
$763$	−34.4655	−1.24773
$764$	0	0
$765$	2.75617	0.0996494
$766$	0	0
$767$	35.9887	1.29948
$768$	0	0
$769$	21.0506	0.759103	0.379552	−	0.925171i	$-0.376078\pi$
$769$	21.0506	0.759103	0.379552	+	0.925171i	$0.376078\pi$
$770$	0	0
$771$	54.1390	1.94977
$772$	0	0
$773$	−16.7289	−0.601698	−0.300849	−	0.953672i	$-0.597270\pi$
$773$	−16.7289	−0.601698	−0.300849	+	0.953672i	$0.597270\pi$
$774$	0	0
$775$	11.8289	0.424905
$776$	0	0
$777$	53.5593	1.92143
$778$	0	0
$779$	1.65115	0.0591587
$780$	0	0
$781$	32.1836	1.15162
$782$	0	0
$783$	−63.2672	−2.26098
$784$	0	0
$785$	7.72157	0.275595
$786$	0	0
$787$	13.5037	0.481355	0.240678	−	0.970605i	$-0.422630\pi$
$787$	13.5037	0.481355	0.240678	+	0.970605i	$0.422630\pi$
$788$	0	0
$789$	−82.1078	−2.92311
$790$	0	0
$791$	70.9872	2.52401
$792$	0	0
$793$	3.46087	0.122899
$794$	0	0
$795$	−61.2413	−2.17201
$796$	0	0
$797$	2.82916	0.100214	0.0501070	−	0.998744i	$-0.484044\pi$
$797$	2.82916	0.100214	0.0501070	+	0.998744i	$0.484044\pi$
$798$	0	0
$799$	−0.930064	−0.0329033
$800$	0	0
$801$	41.9359	1.48173
$802$	0	0
$803$	−7.85246	−0.277107
$804$	0	0
$805$	1.16328	0.0410003
$806$	0	0
$807$	36.7243	1.29276
$808$	0	0
$809$	19.6030	0.689204	0.344602	−	0.938749i	$-0.388014\pi$
$809$	19.6030	0.689204	0.344602	+	0.938749i	$0.388014\pi$
$810$	0	0
$811$	15.9333	0.559493	0.279747	−	0.960074i	$-0.409750\pi$
$811$	15.9333	0.559493	0.279747	+	0.960074i	$0.409750\pi$
$812$	0	0
$813$	−16.5402	−0.580091
$814$	0	0
$815$	6.99094	0.244882
$816$	0	0
$817$	3.91333	0.136910
$818$	0	0
$819$	−64.4694	−2.25274
$820$	0	0
$821$	50.2700	1.75443	0.877217	−	0.480094i	$-0.159397\pi$
$821$	50.2700	1.75443	0.877217	+	0.480094i	$0.159397\pi$
$822$	0	0
$823$	−11.8847	−0.414276	−0.207138	−	0.978312i	$-0.566415\pi$
$823$	−11.8847	−0.414276	−0.207138	+	0.978312i	$0.566415\pi$
$824$	0	0
$825$	22.5549	0.785260
$826$	0	0
$827$	−29.9550	−1.04164	−0.520819	−	0.853667i	$-0.674373\pi$
$827$	−29.9550	−1.04164	−0.520819	+	0.853667i	$0.674373\pi$
$828$	0	0
$829$	43.2395	1.50177	0.750884	−	0.660434i	$-0.229628\pi$
$829$	43.2395	1.50177	0.750884	+	0.660434i	$0.229628\pi$
$830$	0	0
$831$	26.1834	0.908292
$832$	0	0
$833$	1.41467	0.0490154
$834$	0	0
$835$	−38.4827	−1.33175
$836$	0	0
$837$	−41.5933	−1.43767
$838$	0	0
$839$	−54.1566	−1.86969	−0.934846	−	0.355054i	$-0.884463\pi$
$839$	−54.1566	−1.86969	−0.934846	+	0.355054i	$0.884463\pi$
$840$	0	0
$841$	20.3095	0.700327
$842$	0	0
$843$	−75.1064	−2.58680
$844$	0	0
$845$	−15.2494	−0.524595
$846$	0	0
$847$	−9.41408	−0.323471
$848$	0	0
$849$	36.1128	1.23939
$850$	0	0
$851$	−0.488015	−0.0167289
$852$	0	0
$853$	−6.21577	−0.212824	−0.106412	−	0.994322i	$-0.533936\pi$
$853$	−6.21577	−0.212824	−0.106412	+	0.994322i	$0.533936\pi$
$854$	0	0
$855$	−5.37584	−0.183850
$856$	0	0
$857$	24.1031	0.823347	0.411674	−	0.911331i	$-0.364944\pi$
$857$	24.1031	0.823347	0.411674	+	0.911331i	$0.364944\pi$
$858$	0	0
$859$	−27.5776	−0.940934	−0.470467	−	0.882418i	$-0.655915\pi$
$859$	−27.5776	−0.940934	−0.470467	+	0.882418i	$0.655915\pi$
$860$	0	0
$861$	59.8406	2.03936
$862$	0	0
$863$	−40.2085	−1.36871	−0.684357	−	0.729147i	$-0.739917\pi$
$863$	−40.2085	−1.36871	−0.684357	+	0.729147i	$0.739917\pi$
$864$	0	0
$865$	−40.1453	−1.36498
$866$	0	0
$867$	50.9243	1.72948
$868$	0	0
$869$	22.0476	0.747915
$870$	0	0
$871$	28.2329	0.956635
$872$	0	0
$873$	14.1834	0.480037
$874$	0	0
$875$	−26.3688	−0.891427
$876$	0	0
$877$	−36.2228	−1.22316	−0.611578	−	0.791184i	$-0.709465\pi$
$877$	−36.2228	−1.22316	−0.611578	+	0.791184i	$0.709465\pi$
$878$	0	0
$879$	−16.4004	−0.553172
$880$	0	0
$881$	45.6618	1.53838	0.769192	−	0.639018i	$-0.220659\pi$
$881$	45.6618	1.53838	0.769192	+	0.639018i	$0.220659\pi$
$882$	0	0
$883$	30.9032	1.03997	0.519987	−	0.854174i	$-0.325937\pi$
$883$	30.9032	1.03997	0.519987	+	0.854174i	$0.325937\pi$
$884$	0	0
$885$	108.760	3.65591
$886$	0	0
$887$	−33.6194	−1.12883	−0.564414	−	0.825492i	$-0.690898\pi$
$887$	−33.6194	−1.12883	−0.564414	+	0.825492i	$0.690898\pi$
$888$	0	0
$889$	−36.6612	−1.22958
$890$	0	0
$891$	−26.4771	−0.887017
$892$	0	0
$893$	1.81407	0.0607055
$894$	0	0
$895$	40.9491	1.36878
$896$	0	0
$897$	0.880999	0.0294157
$898$	0	0
$899$	32.4172	1.08117
$900$	0	0
$901$	1.23923	0.0412849
$902$	0	0
$903$	141.826	4.71966
$904$	0	0
$905$	33.2690	1.10590
$906$	0	0
$907$	−43.0256	−1.42864	−0.714321	−	0.699819i	$-0.753264\pi$
$907$	−43.0256	−1.42864	−0.714321	+	0.699819i	$0.753264\pi$
$908$	0	0
$909$	−20.6023	−0.683336
$910$	0	0
$911$	58.5446	1.93967	0.969834	−	0.243765i	$-0.0783825\pi$
$911$	58.5446	1.93967	0.969834	+	0.243765i	$0.0783825\pi$
$912$	0	0
$913$	44.6442	1.47751
$914$	0	0
$915$	10.4589	0.345761
$916$	0	0
$917$	66.1122	2.18322
$918$	0	0
$919$	−3.33134	−0.109891	−0.0549454	−	0.998489i	$-0.517498\pi$
$919$	−3.33134	−0.109891	−0.0549454	+	0.998489i	$0.517498\pi$
$920$	0	0
$921$	30.1063	0.992036
$922$	0	0
$923$	29.9523	0.985892
$924$	0	0
$925$	−11.6277	−0.382317
$926$	0	0
$927$	−44.7142	−1.46861
$928$	0	0
$929$	−0.323806	−0.0106237	−0.00531187	−	0.999986i	$-0.501691\pi$
$929$	−0.323806	−0.0106237	−0.00531187	+	0.999986i	$0.501691\pi$
$930$	0	0
$931$	−2.75928	−0.0904317
$932$	0	0
$933$	51.0784	1.67223
$934$	0	0
$935$	−1.34701	−0.0440520
$936$	0	0
$937$	−37.3092	−1.21884	−0.609419	−	0.792849i	$-0.708597\pi$
$937$	−37.3092	−1.21884	−0.609419	+	0.792849i	$0.708597\pi$
$938$	0	0
$939$	−47.9803	−1.56578
$940$	0	0
$941$	−53.6719	−1.74966	−0.874828	−	0.484434i	$-0.839025\pi$
$941$	−53.6719	−1.74966	−0.874828	+	0.484434i	$0.839025\pi$
$942$	0	0
$943$	−0.545248	−0.0177557
$944$	0	0
$945$	−97.4599	−3.17037
$946$	0	0
$947$	−4.92155	−0.159929	−0.0799645	−	0.996798i	$-0.525481\pi$
$947$	−4.92155	−0.159929	−0.0799645	+	0.996798i	$0.525481\pi$
$948$	0	0
$949$	−7.30805	−0.237229
$950$	0	0
$951$	104.539	3.38990
$952$	0	0
$953$	−2.76000	−0.0894053	−0.0447026	−	0.999000i	$-0.514234\pi$
$953$	−2.76000	−0.0894053	−0.0447026	+	0.999000i	$0.514234\pi$
$954$	0	0
$955$	5.93953	0.192199
$956$	0	0
$957$	61.8120	1.99810
$958$	0	0
$959$	−21.3678	−0.690001
$960$	0	0
$961$	−9.68822	−0.312523
$962$	0	0
$963$	66.0947	2.12987
$964$	0	0
$965$	−17.7734	−0.572147
$966$	0	0
$967$	−7.49804	−0.241121	−0.120560	−	0.992706i	$-0.538469\pi$
$967$	−7.49804	−0.241121	−0.120560	+	0.992706i	$0.538469\pi$
$968$	0	0
$969$	0.163147	0.00524103
$970$	0	0
$971$	38.8875	1.24796	0.623980	−	0.781440i	$-0.285515\pi$
$971$	38.8875	1.24796	0.623980	+	0.781440i	$0.285515\pi$
$972$	0	0
$973$	9.22955	0.295886
$974$	0	0
$975$	20.9912	0.672255
$976$	0	0
$977$	28.5221	0.912502	0.456251	−	0.889851i	$-0.349192\pi$
$977$	28.5221	0.912502	0.456251	+	0.889851i	$0.349192\pi$
$978$	0	0
$979$	−20.4952	−0.655029
$980$	0	0
$981$	−52.5959	−1.67926
$982$	0	0
$983$	−31.3832	−1.00097	−0.500484	−	0.865746i	$-0.666845\pi$
$983$	−31.3832	−1.00097	−0.500484	+	0.865746i	$0.666845\pi$
$984$	0	0
$985$	−40.5960	−1.29350
$986$	0	0
$987$	65.7449	2.09268
$988$	0	0
$989$	−1.29227	−0.0410918
$990$	0	0
$991$	−48.9319	−1.55437	−0.777187	−	0.629270i	$-0.783354\pi$
$991$	−48.9319	−1.55437	−0.777187	+	0.629270i	$0.783354\pi$
$992$	0	0
$993$	−36.0473	−1.14393
$994$	0	0
$995$	25.3236	0.802811
$996$	0	0
$997$	2.62508	0.0831370	0.0415685	−	0.999136i	$-0.486765\pi$
$997$	2.62508	0.0831370	0.0415685	+	0.999136i	$0.486765\pi$
$998$	0	0
$999$	40.8859	1.29357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.g.1.3	✓	30
4.3	odd	2		8032.2.a.j.1.28	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.3	✓	30	1.1	even	1	trivial
8032.2.a.j.1.28	yes	30	4.3	odd	2

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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00046 q^{3} +2.74997 q^{5} +3.93356 q^{7} +6.00279 q^{9} +O(q^{10})\)	\(q-3.00046 q^{3} +2.74997 q^{5} +3.93356 q^{7} +6.00279 q^{9} -2.93372 q^{11} -2.73033 q^{13} -8.25118 q^{15} +0.166965 q^{17} -0.325661 q^{19} -11.8025 q^{21} +0.107540 q^{23} +2.56232 q^{25} -9.00976 q^{27} +7.02207 q^{29} +4.61647 q^{31} +8.80253 q^{33} +10.8171 q^{35} -4.53796 q^{37} +8.19226 q^{39} -5.07016 q^{41} -12.0166 q^{43} +16.5075 q^{45} -5.57042 q^{47} +8.47286 q^{49} -0.500972 q^{51} +7.42213 q^{53} -8.06764 q^{55} +0.977134 q^{57} -13.1811 q^{59} -1.26757 q^{61} +23.6123 q^{63} -7.50832 q^{65} -10.3405 q^{67} -0.322671 q^{69} -10.9702 q^{71} +2.67662 q^{73} -7.68814 q^{75} -11.5400 q^{77} -7.51524 q^{79} +9.02510 q^{81} -15.2176 q^{83} +0.459148 q^{85} -21.0695 q^{87} +6.98606 q^{89} -10.7399 q^{91} -13.8515 q^{93} -0.895557 q^{95} +2.36281 q^{97} -17.6105 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9	\( 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9} - 31 q^{11} + 5 q^{13} - 8 q^{15} - q^{17} - 29 q^{19} + 6 q^{21} - 27 q^{23} + 34 q^{25} - 36 q^{27} + q^{29} + 9 q^{31} - 8 q^{33} - 51 q^{35} + 5 q^{37} + 8 q^{39} - 3 q^{41} - 43 q^{43} - 42 q^{47} + 41 q^{49} - 54 q^{51} - 11 q^{53} + 10 q^{55} + 2 q^{57} - 49 q^{59} + 21 q^{61} - 15 q^{63} - 14 q^{65} - 68 q^{67} - 10 q^{69} - 44 q^{71} + 25 q^{73} - 51 q^{75} - 20 q^{77} + 49 q^{79} + 6 q^{81} - 88 q^{83} - 5 q^{85} - 16 q^{87} - 16 q^{89} - 46 q^{91} - 4 q^{93} - 28 q^{95} - 4 q^{97} - 71 q^{99}+O(q^{100}) \) 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9 - 31 * q^11 + 5 * q^13 - 8 * q^15 - q^17 - 29 * q^19 + 6 * q^21 - 27 * q^23 + 34 * q^25 - 36 * q^27 + q^29 + 9 * q^31 - 8 * q^33 - 51 * q^35 + 5 * q^37 + 8 * q^39 - 3 * q^41 - 43 * q^43 - 42 * q^47 + 41 * q^49 - 54 * q^51 - 11 * q^53 + 10 * q^55 + 2 * q^57 - 49 * q^59 + 21 * q^61 - 15 * q^63 - 14 * q^65 - 68 * q^67 - 10 * q^69 - 44 * q^71 + 25 * q^73 - 51 * q^75 - 20 * q^77 + 49 * q^79 + 6 * q^81 - 88 * q^83 - 5 * q^85 - 16 * q^87 - 16 * q^89 - 46 * q^91 - 4 * q^93 - 28 * q^95 - 4 * q^97 - 71 * q^99

Introduction

L-functions

Modular forms

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