LMFDB - Embedded newform 8032.2.a.j.1.8

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:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-1.62030 q^{3} -2.98882 q^{5} -1.58362 q^{7} -0.374619 q^{9} +O(q^{10})$	$q-1.62030 q^{3} -2.98882 q^{5} -1.58362 q^{7} -0.374619 q^{9} +1.17659 q^{11} -1.64357 q^{13} +4.84280 q^{15} -0.163190 q^{17} -0.993147 q^{19} +2.56594 q^{21} -3.53802 q^{23} +3.93307 q^{25} +5.46790 q^{27} -0.843088 q^{29} -7.13984 q^{31} -1.90643 q^{33} +4.73315 q^{35} +5.41349 q^{37} +2.66308 q^{39} -0.419638 q^{41} -3.12205 q^{43} +1.11967 q^{45} -12.0580 q^{47} -4.49216 q^{49} +0.264417 q^{51} -7.57856 q^{53} -3.51663 q^{55} +1.60920 q^{57} -6.22744 q^{59} -13.9170 q^{61} +0.593252 q^{63} +4.91234 q^{65} -1.54402 q^{67} +5.73266 q^{69} -4.38103 q^{71} -9.05579 q^{73} -6.37276 q^{75} -1.86327 q^{77} +5.44650 q^{79} -7.73580 q^{81} +10.0707 q^{83} +0.487745 q^{85} +1.36606 q^{87} -10.0532 q^{89} +2.60278 q^{91} +11.5687 q^{93} +2.96834 q^{95} -11.9569 q^{97} -0.440773 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.62030	−0.935482	−0.467741	−	0.883865i	$-0.654932\pi$
$3$	−1.62030	−0.935482	−0.467741	+	0.883865i	$0.654932\pi$
$4$	0	0
$5$	−2.98882	−1.33664	−0.668321	−	0.743873i	$-0.732987\pi$
$5$	−2.98882	−1.33664	−0.668321	+	0.743873i	$0.732987\pi$
$6$	0	0
$7$	−1.58362	−0.598550	−0.299275	−	0.954167i	$-0.596745\pi$
$7$	−1.58362	−0.598550	−0.299275	+	0.954167i	$0.596745\pi$
$8$	0	0
$9$	−0.374619	−0.124873
$10$	0	0
$11$	1.17659	0.354756	0.177378	−	0.984143i	$-0.443239\pi$
$11$	1.17659	0.354756	0.177378	+	0.984143i	$0.443239\pi$
$12$	0	0
$13$	−1.64357	−0.455844	−0.227922	−	0.973679i	$-0.573193\pi$
$13$	−1.64357	−0.455844	−0.227922	+	0.973679i	$0.573193\pi$
$14$	0	0
$15$	4.84280	1.25041
$16$	0	0
$17$	−0.163190	−0.0395793	−0.0197896	−	0.999804i	$-0.506300\pi$
$17$	−0.163190	−0.0395793	−0.0197896	+	0.999804i	$0.506300\pi$
$18$	0	0
$19$	−0.993147	−0.227844	−0.113922	−	0.993490i	$-0.536341\pi$
$19$	−0.993147	−0.227844	−0.113922	+	0.993490i	$0.536341\pi$
$20$	0	0
$21$	2.56594	0.559933
$22$	0	0
$23$	−3.53802	−0.737728	−0.368864	−	0.929483i	$-0.620253\pi$
$23$	−3.53802	−0.737728	−0.368864	+	0.929483i	$0.620253\pi$
$24$	0	0
$25$	3.93307	0.786614
$26$	0	0
$27$	5.46790	1.05230
$28$	0	0
$29$	−0.843088	−0.156557	−0.0782787	−	0.996932i	$-0.524942\pi$
$29$	−0.843088	−0.156557	−0.0782787	+	0.996932i	$0.524942\pi$
$30$	0	0
$31$	−7.13984	−1.28235	−0.641176	−	0.767394i	$-0.721553\pi$
$31$	−7.13984	−1.28235	−0.641176	+	0.767394i	$0.721553\pi$
$32$	0	0
$33$	−1.90643	−0.331868
$34$	0	0
$35$	4.73315	0.800048
$36$	0	0
$37$	5.41349	0.889973	0.444986	−	0.895537i	$-0.353209\pi$
$37$	5.41349	0.889973	0.444986	+	0.895537i	$0.353209\pi$
$38$	0	0
$39$	2.66308	0.426434
$40$	0	0
$41$	−0.419638	−0.0655365	−0.0327683	−	0.999463i	$-0.510432\pi$
$41$	−0.419638	−0.0655365	−0.0327683	+	0.999463i	$0.510432\pi$
$42$	0	0
$43$	−3.12205	−0.476109	−0.238054	−	0.971252i	$-0.576510\pi$
$43$	−3.12205	−0.476109	−0.238054	+	0.971252i	$0.576510\pi$
$44$	0	0
$45$	1.11967	0.166910
$46$	0	0
$47$	−12.0580	−1.75885	−0.879423	−	0.476041i	$-0.842071\pi$
$47$	−12.0580	−1.75885	−0.879423	+	0.476041i	$0.842071\pi$
$48$	0	0
$49$	−4.49216	−0.641737
$50$	0	0
$51$	0.264417	0.0370257
$52$	0	0
$53$	−7.57856	−1.04099	−0.520497	−	0.853863i	$-0.674253\pi$
$53$	−7.57856	−1.04099	−0.520497	+	0.853863i	$0.674253\pi$
$54$	0	0
$55$	−3.51663	−0.474182
$56$	0	0
$57$	1.60920	0.213144
$58$	0	0
$59$	−6.22744	−0.810743	−0.405372	−	0.914152i	$-0.632858\pi$
$59$	−6.22744	−0.810743	−0.405372	+	0.914152i	$0.632858\pi$
$60$	0	0
$61$	−13.9170	−1.78189	−0.890943	−	0.454115i	$-0.849955\pi$
$61$	−13.9170	−1.78189	−0.890943	+	0.454115i	$0.849955\pi$
$62$	0	0
$63$	0.593252	0.0747427
$64$	0	0
$65$	4.91234	0.609300
$66$	0	0
$67$	−1.54402	−0.188632	−0.0943162	−	0.995542i	$-0.530066\pi$
$67$	−1.54402	−0.188632	−0.0943162	+	0.995542i	$0.530066\pi$
$68$	0	0
$69$	5.73266	0.690132
$70$	0	0
$71$	−4.38103	−0.519932	−0.259966	−	0.965618i	$-0.583711\pi$
$71$	−4.38103	−0.519932	−0.259966	+	0.965618i	$0.583711\pi$
$72$	0	0
$73$	−9.05579	−1.05990	−0.529950	−	0.848029i	$-0.677789\pi$
$73$	−9.05579	−1.05990	−0.529950	+	0.848029i	$0.677789\pi$
$74$	0	0
$75$	−6.37276	−0.735863
$76$	0	0
$77$	−1.86327	−0.212339
$78$	0	0
$79$	5.44650	0.612779	0.306389	−	0.951906i	$-0.400879\pi$
$79$	5.44650	0.612779	0.306389	+	0.951906i	$0.400879\pi$
$80$	0	0
$81$	−7.73580	−0.859534
$82$	0	0
$83$	10.0707	1.10541	0.552703	−	0.833379i	$-0.313597\pi$
$83$	10.0707	1.10541	0.552703	+	0.833379i	$0.313597\pi$
$84$	0	0
$85$	0.487745	0.0529034
$86$	0	0
$87$	1.36606	0.146457
$88$	0	0
$89$	−10.0532	−1.06564	−0.532818	−	0.846230i	$-0.678867\pi$
$89$	−10.0532	−1.06564	−0.532818	+	0.846230i	$0.678867\pi$
$90$	0	0
$91$	2.60278	0.272845
$92$	0	0
$93$	11.5687	1.19962
$94$	0	0
$95$	2.96834	0.304545
$96$	0	0
$97$	−11.9569	−1.21404	−0.607020	−	0.794687i	$-0.707635\pi$
$97$	−11.9569	−1.21404	−0.607020	+	0.794687i	$0.707635\pi$
$98$	0	0
$99$	−0.440773	−0.0442994
$100$	0	0
$101$	−14.0705	−1.40006	−0.700032	−	0.714111i	$-0.746831\pi$
$101$	−14.0705	−1.40006	−0.700032	+	0.714111i	$0.746831\pi$
$102$	0	0
$103$	−2.08528	−0.205469	−0.102735	−	0.994709i	$-0.532759\pi$
$103$	−2.08528	−0.205469	−0.102735	+	0.994709i	$0.532759\pi$
$104$	0	0
$105$	−7.66913	−0.748431
$106$	0	0
$107$	3.60919	0.348913	0.174457	−	0.984665i	$-0.444183\pi$
$107$	3.60919	0.348913	0.174457	+	0.984665i	$0.444183\pi$
$108$	0	0
$109$	6.51716	0.624230	0.312115	−	0.950044i	$-0.398963\pi$
$109$	6.51716	0.624230	0.312115	+	0.950044i	$0.398963\pi$
$110$	0	0
$111$	−8.77150	−0.832554
$112$	0	0
$113$	−1.23242	−0.115937	−0.0579683	−	0.998318i	$-0.518462\pi$
$113$	−1.23242	−0.115937	−0.0579683	+	0.998318i	$0.518462\pi$
$114$	0	0
$115$	10.5745	0.986079
$116$	0	0
$117$	0.615711	0.0569225
$118$	0	0
$119$	0.258430	0.0236902
$120$	0	0
$121$	−9.61563	−0.874148
$122$	0	0
$123$	0.679941	0.0613083
$124$	0	0
$125$	3.18887	0.285221
$126$	0	0
$127$	−3.39765	−0.301493	−0.150746	−	0.988572i	$-0.548168\pi$
$127$	−3.39765	−0.301493	−0.150746	+	0.988572i	$0.548168\pi$
$128$	0	0
$129$	5.05867	0.445391
$130$	0	0
$131$	16.6286	1.45284	0.726422	−	0.687249i	$-0.241182\pi$
$131$	16.6286	1.45284	0.726422	+	0.687249i	$0.241182\pi$
$132$	0	0
$133$	1.57276	0.136376
$134$	0	0
$135$	−16.3426	−1.40655
$136$	0	0
$137$	−5.00630	−0.427717	−0.213859	−	0.976865i	$-0.568603\pi$
$137$	−5.00630	−0.427717	−0.213859	+	0.976865i	$0.568603\pi$
$138$	0	0
$139$	4.82352	0.409126	0.204563	−	0.978853i	$-0.434423\pi$
$139$	4.82352	0.409126	0.204563	+	0.978853i	$0.434423\pi$
$140$	0	0
$141$	19.5377	1.64537
$142$	0	0
$143$	−1.93381	−0.161713
$144$	0	0
$145$	2.51984	0.209261
$146$	0	0
$147$	7.27866	0.600334
$148$	0	0
$149$	−3.91848	−0.321015	−0.160507	−	0.987035i	$-0.551313\pi$
$149$	−3.91848	−0.321015	−0.160507	+	0.987035i	$0.551313\pi$
$150$	0	0
$151$	1.15716	0.0941687	0.0470844	−	0.998891i	$-0.485007\pi$
$151$	1.15716	0.0941687	0.0470844	+	0.998891i	$0.485007\pi$
$152$	0	0
$153$	0.0611339	0.00494238
$154$	0	0
$155$	21.3397	1.71405
$156$	0	0
$157$	−3.62133	−0.289014	−0.144507	−	0.989504i	$-0.546160\pi$
$157$	−3.62133	−0.289014	−0.144507	+	0.989504i	$0.546160\pi$
$158$	0	0
$159$	12.2796	0.973832
$160$	0	0
$161$	5.60286	0.441567
$162$	0	0
$163$	2.75567	0.215841	0.107920	−	0.994160i	$-0.465581\pi$
$163$	2.75567	0.215841	0.107920	+	0.994160i	$0.465581\pi$
$164$	0	0
$165$	5.69800	0.443589
$166$	0	0
$167$	−2.45454	−0.189938	−0.0949691	−	0.995480i	$-0.530275\pi$
$167$	−2.45454	−0.189938	−0.0949691	+	0.995480i	$0.530275\pi$
$168$	0	0
$169$	−10.2987	−0.792206
$170$	0	0
$171$	0.372051	0.0284515
$172$	0	0
$173$	−13.8852	−1.05567	−0.527837	−	0.849346i	$-0.676997\pi$
$173$	−13.8852	−1.05567	−0.527837	+	0.849346i	$0.676997\pi$
$174$	0	0
$175$	−6.22847	−0.470828
$176$	0	0
$177$	10.0903	0.758436
$178$	0	0
$179$	24.3703	1.82152	0.910762	−	0.412931i	$-0.135495\pi$
$179$	24.3703	1.82152	0.910762	+	0.412931i	$0.135495\pi$
$180$	0	0
$181$	−14.1325	−1.05046	−0.525229	−	0.850961i	$-0.676020\pi$
$181$	−14.1325	−1.05046	−0.525229	+	0.850961i	$0.676020\pi$
$182$	0	0
$183$	22.5497	1.66692
$184$	0	0
$185$	−16.1800	−1.18958
$186$	0	0
$187$	−0.192007	−0.0140410
$188$	0	0
$189$	−8.65906	−0.629854
$190$	0	0
$191$	14.2931	1.03421	0.517107	−	0.855921i	$-0.327009\pi$
$191$	14.2931	1.03421	0.517107	+	0.855921i	$0.327009\pi$
$192$	0	0
$193$	16.6255	1.19673	0.598364	−	0.801224i	$-0.295818\pi$
$193$	16.6255	1.19673	0.598364	+	0.801224i	$0.295818\pi$
$194$	0	0
$195$	−7.95947	−0.569990
$196$	0	0
$197$	−21.8935	−1.55984	−0.779922	−	0.625876i	$-0.784742\pi$
$197$	−21.8935	−1.55984	−0.779922	+	0.625876i	$0.784742\pi$
$198$	0	0
$199$	−10.6646	−0.755993	−0.377997	−	0.925807i	$-0.623387\pi$
$199$	−10.6646	−0.755993	−0.377997	+	0.925807i	$0.623387\pi$
$200$	0	0
$201$	2.50178	0.176462
$202$	0	0
$203$	1.33513	0.0937076
$204$	0	0
$205$	1.25423	0.0875989
$206$	0	0
$207$	1.32541	0.0921222
$208$	0	0
$209$	−1.16853	−0.0808288
$210$	0	0
$211$	−23.2647	−1.60161	−0.800805	−	0.598925i	$-0.795595\pi$
$211$	−23.2647	−1.60161	−0.800805	+	0.598925i	$0.795595\pi$
$212$	0	0
$213$	7.09859	0.486387
$214$	0	0
$215$	9.33127	0.636387
$216$	0	0
$217$	11.3068	0.767553
$218$	0	0
$219$	14.6731	0.991518
$220$	0	0
$221$	0.268213	0.0180420
$222$	0	0
$223$	−15.4103	−1.03195	−0.515974	−	0.856604i	$-0.672570\pi$
$223$	−15.4103	−1.03195	−0.515974	+	0.856604i	$0.672570\pi$
$224$	0	0
$225$	−1.47340	−0.0982267
$226$	0	0
$227$	−6.67168	−0.442815	−0.221408	−	0.975181i	$-0.571065\pi$
$227$	−6.67168	−0.442815	−0.221408	+	0.975181i	$0.571065\pi$
$228$	0	0
$229$	14.1477	0.934909	0.467454	−	0.884017i	$-0.345171\pi$
$229$	14.1477	0.934909	0.467454	+	0.884017i	$0.345171\pi$
$230$	0	0
$231$	3.01906	0.198640
$232$	0	0
$233$	−16.8873	−1.10633	−0.553163	−	0.833073i	$-0.686579\pi$
$233$	−16.8873	−1.10633	−0.553163	+	0.833073i	$0.686579\pi$
$234$	0	0
$235$	36.0394	2.35095
$236$	0	0
$237$	−8.82498	−0.573244
$238$	0	0
$239$	0.0647487	0.00418825	0.00209412	−	0.999998i	$-0.499333\pi$
$239$	0.0647487	0.00418825	0.00209412	+	0.999998i	$0.499333\pi$
$240$	0	0
$241$	18.3270	1.18054	0.590272	−	0.807204i	$-0.299020\pi$
$241$	18.3270	1.18054	0.590272	+	0.807204i	$0.299020\pi$
$242$	0	0
$243$	−3.86937	−0.248220
$244$	0	0
$245$	13.4263	0.857774
$246$	0	0
$247$	1.63230	0.103861
$248$	0	0
$249$	−16.3176	−1.03409
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−4.16280	−0.261713
$254$	0	0
$255$	−0.790294	−0.0494902
$256$	0	0
$257$	1.70658	0.106454	0.0532269	−	0.998582i	$-0.483049\pi$
$257$	1.70658	0.106454	0.0532269	+	0.998582i	$0.483049\pi$
$258$	0	0
$259$	−8.57289	−0.532694
$260$	0	0
$261$	0.315836	0.0195498
$262$	0	0
$263$	13.2357	0.816147	0.408073	−	0.912949i	$-0.366201\pi$
$263$	13.2357	0.816147	0.408073	+	0.912949i	$0.366201\pi$
$264$	0	0
$265$	22.6510	1.39144
$266$	0	0
$267$	16.2892	0.996883
$268$	0	0
$269$	8.49001	0.517645	0.258823	−	0.965925i	$-0.416665\pi$
$269$	8.49001	0.517645	0.258823	+	0.965925i	$0.416665\pi$
$270$	0	0
$271$	−3.16642	−0.192347	−0.0961733	−	0.995365i	$-0.530660\pi$
$271$	−3.16642	−0.192347	−0.0961733	+	0.995365i	$0.530660\pi$
$272$	0	0
$273$	−4.21729	−0.255242
$274$	0	0
$275$	4.62762	0.279056
$276$	0	0
$277$	11.7903	0.708413	0.354207	−	0.935167i	$-0.384751\pi$
$277$	11.7903	0.708413	0.354207	+	0.935167i	$0.384751\pi$
$278$	0	0
$279$	2.67472	0.160131
$280$	0	0
$281$	−6.71797	−0.400760	−0.200380	−	0.979718i	$-0.564218\pi$
$281$	−6.71797	−0.400760	−0.200380	+	0.979718i	$0.564218\pi$
$282$	0	0
$283$	14.4366	0.858168	0.429084	−	0.903265i	$-0.358836\pi$
$283$	14.4366	0.858168	0.429084	+	0.903265i	$0.358836\pi$
$284$	0	0
$285$	−4.80961	−0.284897
$286$	0	0
$287$	0.664546	0.0392269
$288$	0	0
$289$	−16.9734	−0.998433
$290$	0	0
$291$	19.3738	1.13571
$292$	0	0
$293$	24.6385	1.43940	0.719699	−	0.694286i	$-0.244280\pi$
$293$	24.6385	1.43940	0.719699	+	0.694286i	$0.244280\pi$
$294$	0	0
$295$	18.6127	1.08367
$296$	0	0
$297$	6.43349	0.373309
$298$	0	0
$299$	5.81498	0.336289
$300$	0	0
$301$	4.94413	0.284975
$302$	0	0
$303$	22.7984	1.30974
$304$	0	0
$305$	41.5954	2.38174
$306$	0	0
$307$	−12.8647	−0.734229	−0.367114	−	0.930176i	$-0.619654\pi$
$307$	−12.8647	−0.734229	−0.367114	+	0.930176i	$0.619654\pi$
$308$	0	0
$309$	3.37879	0.192213
$310$	0	0
$311$	1.52893	0.0866977	0.0433489	−	0.999060i	$-0.486197\pi$
$311$	1.52893	0.0866977	0.0433489	+	0.999060i	$0.486197\pi$
$312$	0	0
$313$	−5.80997	−0.328399	−0.164199	−	0.986427i	$-0.552504\pi$
$313$	−5.80997	−0.328399	−0.164199	+	0.986427i	$0.552504\pi$
$314$	0	0
$315$	−1.77313	−0.0999043
$316$	0	0
$317$	−23.1078	−1.29786	−0.648930	−	0.760848i	$-0.724783\pi$
$317$	−23.1078	−1.29786	−0.648930	+	0.760848i	$0.724783\pi$
$318$	0	0
$319$	−0.991970	−0.0555397
$320$	0	0
$321$	−5.84798	−0.326402
$322$	0	0
$323$	0.162071	0.00901788
$324$	0	0
$325$	−6.46427	−0.358573
$326$	0	0
$327$	−10.5598	−0.583957
$328$	0	0
$329$	19.0953	1.05276
$330$	0	0
$331$	−2.14779	−0.118053	−0.0590266	−	0.998256i	$-0.518800\pi$
$331$	−2.14779	−0.118053	−0.0590266	+	0.998256i	$0.518800\pi$
$332$	0	0
$333$	−2.02800	−0.111133
$334$	0	0
$335$	4.61481	0.252134
$336$	0	0
$337$	29.7700	1.62168	0.810839	−	0.585270i	$-0.199011\pi$
$337$	29.7700	1.62168	0.810839	+	0.585270i	$0.199011\pi$
$338$	0	0
$339$	1.99690	0.108457
$340$	0	0
$341$	−8.40067	−0.454922
$342$	0	0
$343$	18.1992	0.982663
$344$	0	0
$345$	−17.1339	−0.922459
$346$	0	0
$347$	16.6783	0.895339	0.447669	−	0.894199i	$-0.352254\pi$
$347$	16.6783	0.895339	0.447669	+	0.894199i	$0.352254\pi$
$348$	0	0
$349$	34.0088	1.82045	0.910226	−	0.414112i	$-0.135908\pi$
$349$	34.0088	1.82045	0.910226	+	0.414112i	$0.135908\pi$
$350$	0	0
$351$	−8.98687	−0.479684
$352$	0	0
$353$	1.97341	0.105034	0.0525169	−	0.998620i	$-0.483276\pi$
$353$	1.97341	0.105034	0.0525169	+	0.998620i	$0.483276\pi$
$354$	0	0
$355$	13.0941	0.694964
$356$	0	0
$357$	−0.418734	−0.0221618
$358$	0	0
$359$	−12.8846	−0.680026	−0.340013	−	0.940421i	$-0.610431\pi$
$359$	−12.8846	−0.680026	−0.340013	+	0.940421i	$0.610431\pi$
$360$	0	0
$361$	−18.0137	−0.948087
$362$	0	0
$363$	15.5802	0.817750
$364$	0	0
$365$	27.0662	1.41671
$366$	0	0
$367$	−24.9734	−1.30360	−0.651801	−	0.758390i	$-0.725986\pi$
$367$	−24.9734	−1.30360	−0.651801	+	0.758390i	$0.725986\pi$
$368$	0	0
$369$	0.157204	0.00818373
$370$	0	0
$371$	12.0015	0.623088
$372$	0	0
$373$	13.6494	0.706737	0.353369	−	0.935484i	$-0.385036\pi$
$373$	13.6494	0.706737	0.353369	+	0.935484i	$0.385036\pi$
$374$	0	0
$375$	−5.16693	−0.266819
$376$	0	0
$377$	1.38567	0.0713658
$378$	0	0
$379$	34.0417	1.74860	0.874301	−	0.485384i	$-0.161320\pi$
$379$	34.0417	1.74860	0.874301	+	0.485384i	$0.161320\pi$
$380$	0	0
$381$	5.50522	0.282041
$382$	0	0
$383$	17.7167	0.905280	0.452640	−	0.891693i	$-0.350482\pi$
$383$	17.7167	0.905280	0.452640	+	0.891693i	$0.350482\pi$
$384$	0	0
$385$	5.56898	0.283822
$386$	0	0
$387$	1.16958	0.0594531
$388$	0	0
$389$	2.68417	0.136093	0.0680464	−	0.997682i	$-0.478323\pi$
$389$	2.68417	0.136093	0.0680464	+	0.997682i	$0.478323\pi$
$390$	0	0
$391$	0.577368	0.0291988
$392$	0	0
$393$	−26.9433	−1.35911
$394$	0	0
$395$	−16.2786	−0.819067
$396$	0	0
$397$	−25.6509	−1.28738	−0.643692	−	0.765285i	$-0.722598\pi$
$397$	−25.6509	−1.28738	−0.643692	+	0.765285i	$0.722598\pi$
$398$	0	0
$399$	−2.54835	−0.127577
$400$	0	0
$401$	7.23002	0.361050	0.180525	−	0.983570i	$-0.442220\pi$
$401$	7.23002	0.361050	0.180525	+	0.983570i	$0.442220\pi$
$402$	0	0
$403$	11.7348	0.584552
$404$	0	0
$405$	23.1210	1.14889
$406$	0	0
$407$	6.36947	0.315723
$408$	0	0
$409$	−16.9655	−0.838892	−0.419446	−	0.907780i	$-0.637776\pi$
$409$	−16.9655	−0.838892	−0.419446	+	0.907780i	$0.637776\pi$
$410$	0	0
$411$	8.11172	0.400122
$412$	0	0
$413$	9.86187	0.485271
$414$	0	0
$415$	−30.0996	−1.47753
$416$	0	0
$417$	−7.81557	−0.382730
$418$	0	0
$419$	21.1411	1.03281	0.516405	−	0.856345i	$-0.327270\pi$
$419$	21.1411	1.03281	0.516405	+	0.856345i	$0.327270\pi$
$420$	0	0
$421$	−25.6283	−1.24905	−0.624524	−	0.781006i	$-0.714707\pi$
$421$	−25.6283	−1.24905	−0.624524	+	0.781006i	$0.714707\pi$
$422$	0	0
$423$	4.51717	0.219632
$424$	0	0
$425$	−0.641836	−0.0311336
$426$	0	0
$427$	22.0391	1.06655
$428$	0	0
$429$	3.13336	0.151280
$430$	0	0
$431$	20.6872	0.996465	0.498233	−	0.867043i	$-0.333983\pi$
$431$	20.6872	0.996465	0.498233	+	0.867043i	$0.333983\pi$
$432$	0	0
$433$	29.1703	1.40183	0.700917	−	0.713243i	$-0.252774\pi$
$433$	29.1703	1.40183	0.700917	+	0.713243i	$0.252774\pi$
$434$	0	0
$435$	−4.08291	−0.195760
$436$	0	0
$437$	3.51377	0.168087
$438$	0	0
$439$	−40.0926	−1.91352	−0.956758	−	0.290885i	$-0.906050\pi$
$439$	−40.0926	−1.91352	−0.956758	+	0.290885i	$0.906050\pi$
$440$	0	0
$441$	1.68285	0.0801356
$442$	0	0
$443$	1.26538	0.0601200	0.0300600	−	0.999548i	$-0.490430\pi$
$443$	1.26538	0.0601200	0.0300600	+	0.999548i	$0.490430\pi$
$444$	0	0
$445$	30.0472	1.42437
$446$	0	0
$447$	6.34913	0.300304
$448$	0	0
$449$	−12.6854	−0.598659	−0.299330	−	0.954150i	$-0.596763\pi$
$449$	−12.6854	−0.598659	−0.299330	+	0.954150i	$0.596763\pi$
$450$	0	0
$451$	−0.493743	−0.0232495
$452$	0	0
$453$	−1.87496	−0.0880932
$454$	0	0
$455$	−7.77925	−0.364697
$456$	0	0
$457$	−10.3520	−0.484248	−0.242124	−	0.970245i	$-0.577844\pi$
$457$	−10.3520	−0.484248	−0.242124	+	0.970245i	$0.577844\pi$
$458$	0	0
$459$	−0.892305	−0.0416492
$460$	0	0
$461$	−8.05873	−0.375332	−0.187666	−	0.982233i	$-0.560092\pi$
$461$	−8.05873	−0.375332	−0.187666	+	0.982233i	$0.560092\pi$
$462$	0	0
$463$	15.1752	0.705250	0.352625	−	0.935765i	$-0.385289\pi$
$463$	15.1752	0.705250	0.352625	+	0.935765i	$0.385289\pi$
$464$	0	0
$465$	−34.5768	−1.60346
$466$	0	0
$467$	−5.72283	−0.264821	−0.132411	−	0.991195i	$-0.542272\pi$
$467$	−5.72283	−0.264821	−0.132411	+	0.991195i	$0.542272\pi$
$468$	0	0
$469$	2.44514	0.112906
$470$	0	0
$471$	5.86766	0.270367
$472$	0	0
$473$	−3.67338	−0.168902
$474$	0	0
$475$	−3.90611	−0.179225
$476$	0	0
$477$	2.83907	0.129992
$478$	0	0
$479$	−3.11724	−0.142430	−0.0712152	−	0.997461i	$-0.522688\pi$
$479$	−3.11724	−0.142430	−0.0712152	+	0.997461i	$0.522688\pi$
$480$	0	0
$481$	−8.89744	−0.405689
$482$	0	0
$483$	−9.07834	−0.413079
$484$	0	0
$485$	35.7371	1.62274
$486$	0	0
$487$	16.7067	0.757054	0.378527	−	0.925590i	$-0.376431\pi$
$487$	16.7067	0.757054	0.378527	+	0.925590i	$0.376431\pi$
$488$	0	0
$489$	−4.46502	−0.201915
$490$	0	0
$491$	28.5030	1.28632	0.643161	−	0.765731i	$-0.277623\pi$
$491$	28.5030	1.28632	0.643161	+	0.765731i	$0.277623\pi$
$492$	0	0
$493$	0.137583	0.00619643
$494$	0	0
$495$	1.31739	0.0592124
$496$	0	0
$497$	6.93786	0.311206
$498$	0	0
$499$	−26.4157	−1.18253	−0.591265	−	0.806477i	$-0.701371\pi$
$499$	−26.4157	−1.18253	−0.591265	+	0.806477i	$0.701371\pi$
$500$	0	0
$501$	3.97710	0.177684
$502$	0	0
$503$	9.26459	0.413087	0.206544	−	0.978437i	$-0.433778\pi$
$503$	9.26459	0.413087	0.206544	+	0.978437i	$0.433778\pi$
$504$	0	0
$505$	42.0542	1.87139
$506$	0	0
$507$	16.6870	0.741095
$508$	0	0
$509$	−29.5907	−1.31158	−0.655792	−	0.754941i	$-0.727665\pi$
$509$	−29.5907	−1.31158	−0.655792	+	0.754941i	$0.727665\pi$
$510$	0	0
$511$	14.3409	0.634404
$512$	0	0
$513$	−5.43043	−0.239759
$514$	0	0
$515$	6.23254	0.274639
$516$	0	0
$517$	−14.1874	−0.623961
$518$	0	0
$519$	22.4983	0.987564
$520$	0	0
$521$	−5.39958	−0.236560	−0.118280	−	0.992980i	$-0.537738\pi$
$521$	−5.39958	−0.236560	−0.118280	+	0.992980i	$0.537738\pi$
$522$	0	0
$523$	26.0623	1.13962	0.569812	−	0.821775i	$-0.307016\pi$
$523$	26.0623	1.13962	0.569812	+	0.821775i	$0.307016\pi$
$524$	0	0
$525$	10.0920	0.440451
$526$	0	0
$527$	1.16515	0.0507546
$528$	0	0
$529$	−10.4824	−0.455757
$530$	0	0
$531$	2.33291	0.101240
$532$	0	0
$533$	0.689704	0.0298744
$534$	0	0
$535$	−10.7872	−0.466372
$536$	0	0
$537$	−39.4873	−1.70400
$538$	0	0
$539$	−5.28544	−0.227660
$540$	0	0
$541$	−9.68826	−0.416531	−0.208265	−	0.978072i	$-0.566782\pi$
$541$	−9.68826	−0.416531	−0.208265	+	0.978072i	$0.566782\pi$
$542$	0	0
$543$	22.8989	0.982684
$544$	0	0
$545$	−19.4786	−0.834373
$546$	0	0
$547$	−37.6699	−1.61065	−0.805325	−	0.592834i	$-0.798009\pi$
$547$	−37.6699	−1.61065	−0.805325	+	0.592834i	$0.798009\pi$
$548$	0	0
$549$	5.21356	0.222509
$550$	0	0
$551$	0.837310	0.0356706
$552$	0	0
$553$	−8.62516	−0.366779
$554$	0	0
$555$	26.2165	1.11283
$556$	0	0
$557$	20.8747	0.884489	0.442245	−	0.896895i	$-0.354182\pi$
$557$	20.8747	0.884489	0.442245	+	0.896895i	$0.354182\pi$
$558$	0	0
$559$	5.13131	0.217031
$560$	0	0
$561$	0.311110	0.0131351
$562$	0	0
$563$	6.73312	0.283767	0.141884	−	0.989883i	$-0.454684\pi$
$563$	6.73312	0.283767	0.141884	+	0.989883i	$0.454684\pi$
$564$	0	0
$565$	3.68350	0.154966
$566$	0	0
$567$	12.2505	0.514474
$568$	0	0
$569$	−20.4087	−0.855578	−0.427789	−	0.903879i	$-0.640707\pi$
$569$	−20.4087	−0.855578	−0.427789	+	0.903879i	$0.640707\pi$
$570$	0	0
$571$	2.96961	0.124274	0.0621372	−	0.998068i	$-0.480208\pi$
$571$	2.96961	0.124274	0.0621372	+	0.998068i	$0.480208\pi$
$572$	0	0
$573$	−23.1592	−0.967488
$574$	0	0
$575$	−13.9153	−0.580307
$576$	0	0
$577$	−0.741003	−0.0308484	−0.0154242	−	0.999881i	$-0.504910\pi$
$577$	−0.741003	−0.0308484	−0.0154242	+	0.999881i	$0.504910\pi$
$578$	0	0
$579$	−26.9383	−1.11952
$580$	0	0
$581$	−15.9481	−0.661641
$582$	0	0
$583$	−8.91687	−0.369299
$584$	0	0
$585$	−1.84025	−0.0760851
$586$	0	0
$587$	−6.58905	−0.271959	−0.135980	−	0.990712i	$-0.543418\pi$
$587$	−6.58905	−0.271959	−0.135980	+	0.990712i	$0.543418\pi$
$588$	0	0
$589$	7.09091	0.292176
$590$	0	0
$591$	35.4740	1.45921
$592$	0	0
$593$	20.9066	0.858531	0.429266	−	0.903178i	$-0.358772\pi$
$593$	20.9066	0.858531	0.429266	+	0.903178i	$0.358772\pi$
$594$	0	0
$595$	−0.772400	−0.0316653
$596$	0	0
$597$	17.2799	0.707218
$598$	0	0
$599$	5.05927	0.206716	0.103358	−	0.994644i	$-0.467041\pi$
$599$	5.05927	0.206716	0.103358	+	0.994644i	$0.467041\pi$
$600$	0	0
$601$	9.75337	0.397848	0.198924	−	0.980015i	$-0.436255\pi$
$601$	9.75337	0.397848	0.198924	+	0.980015i	$0.436255\pi$
$602$	0	0
$603$	0.578420	0.0235551
$604$	0	0
$605$	28.7394	1.16842
$606$	0	0
$607$	39.3713	1.59803	0.799017	−	0.601309i	$-0.205354\pi$
$607$	39.3713	1.59803	0.799017	+	0.601309i	$0.205354\pi$
$608$	0	0
$609$	−2.16331	−0.0876618
$610$	0	0
$611$	19.8182	0.801759
$612$	0	0
$613$	34.8197	1.40636	0.703178	−	0.711014i	$-0.251764\pi$
$613$	34.8197	1.40636	0.703178	+	0.711014i	$0.251764\pi$
$614$	0	0
$615$	−2.03223	−0.0819472
$616$	0	0
$617$	−1.27306	−0.0512513	−0.0256256	−	0.999672i	$-0.508158\pi$
$617$	−1.27306	−0.0512513	−0.0256256	+	0.999672i	$0.508158\pi$
$618$	0	0
$619$	−39.5959	−1.59149	−0.795746	−	0.605630i	$-0.792921\pi$
$619$	−39.5959	−1.59149	−0.795746	+	0.605630i	$0.792921\pi$
$620$	0	0
$621$	−19.3456	−0.776310
$622$	0	0
$623$	15.9204	0.637837
$624$	0	0
$625$	−29.1963	−1.16785
$626$	0	0
$627$	1.89337	0.0756139
$628$	0	0
$629$	−0.883426	−0.0352245
$630$	0	0
$631$	20.5498	0.818075	0.409038	−	0.912517i	$-0.365864\pi$
$631$	20.5498	0.818075	0.409038	+	0.912517i	$0.365864\pi$
$632$	0	0
$633$	37.6959	1.49828
$634$	0	0
$635$	10.1550	0.402988
$636$	0	0
$637$	7.38317	0.292532
$638$	0	0
$639$	1.64121	0.0649255
$640$	0	0
$641$	12.8652	0.508146	0.254073	−	0.967185i	$-0.418230\pi$
$641$	12.8652	0.508146	0.254073	+	0.967185i	$0.418230\pi$
$642$	0	0
$643$	30.2703	1.19374	0.596872	−	0.802337i	$-0.296410\pi$
$643$	30.2703	1.19374	0.596872	+	0.802337i	$0.296410\pi$
$644$	0	0
$645$	−15.1195	−0.595329
$646$	0	0
$647$	−32.6811	−1.28483	−0.642414	−	0.766358i	$-0.722067\pi$
$647$	−32.6811	−1.28483	−0.642414	+	0.766358i	$0.722067\pi$
$648$	0	0
$649$	−7.32715	−0.287616
$650$	0	0
$651$	−18.3204	−0.718032
$652$	0	0
$653$	−0.106092	−0.00415170	−0.00207585	−	0.999998i	$-0.500661\pi$
$653$	−0.106092	−0.00415170	−0.00207585	+	0.999998i	$0.500661\pi$
$654$	0	0
$655$	−49.6998	−1.94193
$656$	0	0
$657$	3.39247	0.132353
$658$	0	0
$659$	−37.0595	−1.44363	−0.721817	−	0.692084i	$-0.756693\pi$
$659$	−37.0595	−1.44363	−0.721817	+	0.692084i	$0.756693\pi$
$660$	0	0
$661$	−31.3822	−1.22062	−0.610312	−	0.792161i	$-0.708956\pi$
$661$	−31.3822	−1.22062	−0.610312	+	0.792161i	$0.708956\pi$
$662$	0	0
$663$	−0.434587	−0.0168779
$664$	0	0
$665$	−4.70071	−0.182286
$666$	0	0
$667$	2.98286	0.115497
$668$	0	0
$669$	24.9693	0.965369
$670$	0	0
$671$	−16.3746	−0.632134
$672$	0	0
$673$	13.9800	0.538891	0.269445	−	0.963016i	$-0.413160\pi$
$673$	13.9800	0.538891	0.269445	+	0.963016i	$0.413160\pi$
$674$	0	0
$675$	21.5056	0.827753
$676$	0	0
$677$	−14.0441	−0.539757	−0.269879	−	0.962894i	$-0.586984\pi$
$677$	−14.0441	−0.539757	−0.269879	+	0.962894i	$0.586984\pi$
$678$	0	0
$679$	18.9351	0.726664
$680$	0	0
$681$	10.8101	0.414246
$682$	0	0
$683$	12.1965	0.466685	0.233342	−	0.972395i	$-0.425034\pi$
$683$	12.1965	0.466685	0.233342	+	0.972395i	$0.425034\pi$
$684$	0	0
$685$	14.9630	0.571705
$686$	0	0
$687$	−22.9236	−0.874591
$688$	0	0
$689$	12.4559	0.474531
$690$	0	0
$691$	0.498484	0.0189632	0.00948161	−	0.999955i	$-0.496982\pi$
$691$	0.498484	0.0189632	0.00948161	+	0.999955i	$0.496982\pi$
$692$	0	0
$693$	0.698015	0.0265154
$694$	0	0
$695$	−14.4167	−0.546855
$696$	0	0
$697$	0.0684806	0.00259389
$698$	0	0
$699$	27.3626	1.03495
$700$	0	0
$701$	11.1342	0.420532	0.210266	−	0.977644i	$-0.432567\pi$
$701$	11.1342	0.420532	0.210266	+	0.977644i	$0.432567\pi$
$702$	0	0
$703$	−5.37639	−0.202775
$704$	0	0
$705$	−58.3947	−2.19927
$706$	0	0
$707$	22.2822	0.838009
$708$	0	0
$709$	28.7770	1.08074	0.540372	−	0.841426i	$-0.318284\pi$
$709$	28.7770	1.08074	0.540372	+	0.841426i	$0.318284\pi$
$710$	0	0
$711$	−2.04036	−0.0765195
$712$	0	0
$713$	25.2609	0.946027
$714$	0	0
$715$	5.77981	0.216153
$716$	0	0
$717$	−0.104913	−0.00391803
$718$	0	0
$719$	−20.4623	−0.763115	−0.381557	−	0.924345i	$-0.624612\pi$
$719$	−20.4623	−0.763115	−0.381557	+	0.924345i	$0.624612\pi$
$720$	0	0
$721$	3.30229	0.122984
$722$	0	0
$723$	−29.6953	−1.10438
$724$	0	0
$725$	−3.31592	−0.123150
$726$	0	0
$727$	14.5079	0.538069	0.269034	−	0.963131i	$-0.413295\pi$
$727$	14.5079	0.538069	0.269034	+	0.963131i	$0.413295\pi$
$728$	0	0
$729$	29.4770	1.09174
$730$	0	0
$731$	0.509487	0.0188440
$732$	0	0
$733$	13.5787	0.501542	0.250771	−	0.968046i	$-0.419316\pi$
$733$	13.5787	0.501542	0.250771	+	0.968046i	$0.419316\pi$
$734$	0	0
$735$	−21.7546	−0.802432
$736$	0	0
$737$	−1.81668	−0.0669184
$738$	0	0
$739$	−44.7528	−1.64626	−0.823129	−	0.567855i	$-0.807774\pi$
$739$	−44.7528	−1.64626	−0.823129	+	0.567855i	$0.807774\pi$
$740$	0	0
$741$	−2.64483	−0.0971602
$742$	0	0
$743$	−8.99526	−0.330004	−0.165002	−	0.986293i	$-0.552763\pi$
$743$	−8.99526	−0.330004	−0.165002	+	0.986293i	$0.552763\pi$
$744$	0	0
$745$	11.7117	0.429082
$746$	0	0
$747$	−3.77268	−0.138035
$748$	0	0
$749$	−5.71556	−0.208842
$750$	0	0
$751$	−21.3575	−0.779345	−0.389672	−	0.920954i	$-0.627412\pi$
$751$	−21.3575	−0.779345	−0.389672	+	0.920954i	$0.627412\pi$
$752$	0	0
$753$	1.62030	0.0590471
$754$	0	0
$755$	−3.45856	−0.125870
$756$	0	0
$757$	−13.1946	−0.479566	−0.239783	−	0.970826i	$-0.577076\pi$
$757$	−13.1946	−0.479566	−0.239783	+	0.970826i	$0.577076\pi$
$758$	0	0
$759$	6.74500	0.244828
$760$	0	0
$761$	15.1524	0.549275	0.274637	−	0.961548i	$-0.411442\pi$
$761$	15.1524	0.549275	0.274637	+	0.961548i	$0.411442\pi$
$762$	0	0
$763$	−10.3207	−0.373633
$764$	0	0
$765$	−0.182718	−0.00660620
$766$	0	0
$767$	10.2352	0.369572
$768$	0	0
$769$	27.2035	0.980982	0.490491	−	0.871446i	$-0.336817\pi$
$769$	27.2035	0.980982	0.490491	+	0.871446i	$0.336817\pi$
$770$	0	0
$771$	−2.76518	−0.0995856
$772$	0	0
$773$	−4.38361	−0.157667	−0.0788337	−	0.996888i	$-0.525120\pi$
$773$	−4.38361	−0.157667	−0.0788337	+	0.996888i	$0.525120\pi$
$774$	0	0
$775$	−28.0815	−1.00872
$776$	0	0
$777$	13.8907	0.498325
$778$	0	0
$779$	0.416763	0.0149321
$780$	0	0
$781$	−5.15468	−0.184449
$782$	0	0
$783$	−4.60992	−0.164745
$784$	0	0
$785$	10.8235	0.386308
$786$	0	0
$787$	−25.4509	−0.907228	−0.453614	−	0.891198i	$-0.649865\pi$
$787$	−25.4509	−0.907228	−0.453614	+	0.891198i	$0.649865\pi$
$788$	0	0
$789$	−21.4458	−0.763491
$790$	0	0
$791$	1.95169	0.0693939
$792$	0	0
$793$	22.8735	0.812262
$794$	0	0
$795$	−36.7014	−1.30167
$796$	0	0
$797$	13.0719	0.463030	0.231515	−	0.972831i	$-0.425632\pi$
$797$	13.0719	0.463030	0.231515	+	0.972831i	$0.425632\pi$
$798$	0	0
$799$	1.96775	0.0696139
$800$	0	0
$801$	3.76611	0.133069
$802$	0	0
$803$	−10.6550	−0.376006
$804$	0	0
$805$	−16.7460	−0.590218
$806$	0	0
$807$	−13.7564	−0.484248
$808$	0	0
$809$	−25.6731	−0.902619	−0.451309	−	0.892368i	$-0.649043\pi$
$809$	−25.6731	−0.902619	−0.451309	+	0.892368i	$0.649043\pi$
$810$	0	0
$811$	14.0420	0.493080	0.246540	−	0.969133i	$-0.420706\pi$
$811$	14.0420	0.493080	0.246540	+	0.969133i	$0.420706\pi$
$812$	0	0
$813$	5.13057	0.179937
$814$	0	0
$815$	−8.23621	−0.288502
$816$	0	0
$817$	3.10066	0.108478
$818$	0	0
$819$	−0.975050	−0.0340710
$820$	0	0
$821$	1.05848	0.0369412	0.0184706	−	0.999829i	$-0.494120\pi$
$821$	1.05848	0.0369412	0.0184706	+	0.999829i	$0.494120\pi$
$822$	0	0
$823$	−46.6099	−1.62472	−0.812359	−	0.583157i	$-0.801817\pi$
$823$	−46.6099	−1.62472	−0.812359	+	0.583157i	$0.801817\pi$
$824$	0	0
$825$	−7.49814	−0.261052
$826$	0	0
$827$	51.6717	1.79680	0.898401	−	0.439176i	$-0.144730\pi$
$827$	51.6717	1.79680	0.898401	+	0.439176i	$0.144730\pi$
$828$	0	0
$829$	−49.9301	−1.73415	−0.867073	−	0.498182i	$-0.834001\pi$
$829$	−49.9301	−1.73415	−0.867073	+	0.498182i	$0.834001\pi$
$830$	0	0
$831$	−19.1039	−0.662708
$832$	0	0
$833$	0.733074	0.0253995
$834$	0	0
$835$	7.33620	0.253879
$836$	0	0
$837$	−39.0399	−1.34942
$838$	0	0
$839$	−12.4125	−0.428527	−0.214264	−	0.976776i	$-0.568735\pi$
$839$	−12.4125	−0.428527	−0.214264	+	0.976776i	$0.568735\pi$
$840$	0	0
$841$	−28.2892	−0.975490
$842$	0	0
$843$	10.8851	0.374904
$844$	0	0
$845$	30.7810	1.05890
$846$	0	0
$847$	15.2275	0.523222
$848$	0	0
$849$	−23.3917	−0.802801
$850$	0	0
$851$	−19.1530	−0.656558
$852$	0	0
$853$	−50.3308	−1.72330	−0.861648	−	0.507507i	$-0.830567\pi$
$853$	−50.3308	−1.72330	−0.861648	+	0.507507i	$0.830567\pi$
$854$	0	0
$855$	−1.11200	−0.0380295
$856$	0	0
$857$	24.5848	0.839799	0.419900	−	0.907571i	$-0.362065\pi$
$857$	24.5848	0.839799	0.419900	+	0.907571i	$0.362065\pi$
$858$	0	0
$859$	37.2830	1.27208	0.636041	−	0.771656i	$-0.280571\pi$
$859$	37.2830	1.27208	0.636041	+	0.771656i	$0.280571\pi$
$860$	0	0
$861$	−1.07677	−0.0366961
$862$	0	0
$863$	7.20392	0.245224	0.122612	−	0.992455i	$-0.460873\pi$
$863$	7.20392	0.245224	0.122612	+	0.992455i	$0.460873\pi$
$864$	0	0
$865$	41.5005	1.41106
$866$	0	0
$867$	27.5020	0.934017
$868$	0	0
$869$	6.40831	0.217387
$870$	0	0
$871$	2.53771	0.0859869
$872$	0	0
$873$	4.47928	0.151601
$874$	0	0
$875$	−5.04994	−0.170719
$876$	0	0
$877$	29.5216	0.996872	0.498436	−	0.866926i	$-0.333908\pi$
$877$	29.5216	0.996872	0.498436	+	0.866926i	$0.333908\pi$
$878$	0	0
$879$	−39.9219	−1.34653
$880$	0	0
$881$	10.5797	0.356439	0.178220	−	0.983991i	$-0.442966\pi$
$881$	10.5797	0.356439	0.178220	+	0.983991i	$0.442966\pi$
$882$	0	0
$883$	−10.7707	−0.362464	−0.181232	−	0.983440i	$-0.558009\pi$
$883$	−10.7707	−0.362464	−0.181232	+	0.983440i	$0.558009\pi$
$884$	0	0
$885$	−30.1582	−1.01376
$886$	0	0
$887$	30.9265	1.03841	0.519205	−	0.854650i	$-0.326228\pi$
$887$	30.9265	1.03841	0.519205	+	0.854650i	$0.326228\pi$
$888$	0	0
$889$	5.38057	0.180459
$890$	0	0
$891$	−9.10188	−0.304925
$892$	0	0
$893$	11.9754	0.400742
$894$	0	0
$895$	−72.8386	−2.43473
$896$	0	0
$897$	−9.42202	−0.314592
$898$	0	0
$899$	6.01951	0.200762
$900$	0	0
$901$	1.23674	0.0412018
$902$	0	0
$903$	−8.01099	−0.266589
$904$	0	0
$905$	42.2394	1.40409
$906$	0	0
$907$	36.4117	1.20903	0.604515	−	0.796594i	$-0.293367\pi$
$907$	36.4117	1.20903	0.604515	+	0.796594i	$0.293367\pi$
$908$	0	0
$909$	5.27106	0.174830
$910$	0	0
$911$	15.1473	0.501852	0.250926	−	0.968006i	$-0.419265\pi$
$911$	15.1473	0.501852	0.250926	+	0.968006i	$0.419265\pi$
$912$	0	0
$913$	11.8491	0.392149
$914$	0	0
$915$	−67.3971	−2.22808
$916$	0	0
$917$	−26.3332	−0.869600
$918$	0	0
$919$	−6.46405	−0.213229	−0.106615	−	0.994300i	$-0.534001\pi$
$919$	−6.46405	−0.213229	−0.106615	+	0.994300i	$0.534001\pi$
$920$	0	0
$921$	20.8448	0.686858
$922$	0	0
$923$	7.20052	0.237008
$924$	0	0
$925$	21.2916	0.700065
$926$	0	0
$927$	0.781186	0.0256575
$928$	0	0
$929$	−31.2727	−1.02602	−0.513012	−	0.858381i	$-0.671470\pi$
$929$	−31.2727	−1.02602	−0.513012	+	0.858381i	$0.671470\pi$
$930$	0	0
$931$	4.46138	0.146216
$932$	0	0
$933$	−2.47733	−0.0811042
$934$	0	0
$935$	0.573877	0.0187678
$936$	0	0
$937$	19.8724	0.649203	0.324602	−	0.945851i	$-0.394770\pi$
$937$	19.8724	0.649203	0.324602	+	0.945851i	$0.394770\pi$
$938$	0	0
$939$	9.41391	0.307211
$940$	0	0
$941$	2.85297	0.0930041	0.0465020	−	0.998918i	$-0.485193\pi$
$941$	2.85297	0.0930041	0.0465020	+	0.998918i	$0.485193\pi$
$942$	0	0
$943$	1.48469	0.0483481
$944$	0	0
$945$	25.8804	0.841889
$946$	0	0
$947$	12.1087	0.393480	0.196740	−	0.980456i	$-0.436965\pi$
$947$	12.1087	0.393480	0.196740	+	0.980456i	$0.436965\pi$
$948$	0	0
$949$	14.8838	0.483149
$950$	0	0
$951$	37.4416	1.21413
$952$	0	0
$953$	22.3713	0.724676	0.362338	−	0.932047i	$-0.381979\pi$
$953$	22.3713	0.724676	0.362338	+	0.932047i	$0.381979\pi$
$954$	0	0
$955$	−42.7196	−1.38237
$956$	0	0
$957$	1.60729	0.0519564
$958$	0	0
$959$	7.92806	0.256010
$960$	0	0
$961$	19.9773	0.644428
$962$	0	0
$963$	−1.35207	−0.0435698
$964$	0	0
$965$	−49.6907	−1.59960
$966$	0	0
$967$	−3.73544	−0.120124	−0.0600618	−	0.998195i	$-0.519130\pi$
$967$	−3.73544	−0.120124	−0.0600618	+	0.998195i	$0.519130\pi$
$968$	0	0
$969$	−0.262604	−0.00843607
$970$	0	0
$971$	13.3281	0.427720	0.213860	−	0.976864i	$-0.431396\pi$
$971$	13.3281	0.427720	0.213860	+	0.976864i	$0.431396\pi$
$972$	0	0
$973$	−7.63861	−0.244883
$974$	0	0
$975$	10.4741	0.335439
$976$	0	0
$977$	28.1433	0.900384	0.450192	−	0.892932i	$-0.351356\pi$
$977$	28.1433	0.900384	0.450192	+	0.892932i	$0.351356\pi$
$978$	0	0
$979$	−11.8285	−0.378040
$980$	0	0
$981$	−2.44145	−0.0779495
$982$	0	0
$983$	30.9235	0.986306	0.493153	−	0.869943i	$-0.335844\pi$
$983$	30.9235	0.986306	0.493153	+	0.869943i	$0.335844\pi$
$984$	0	0
$985$	65.4357	2.08496
$986$	0	0
$987$	−30.9402	−0.984837
$988$	0	0
$989$	11.0459	0.351239
$990$	0	0
$991$	−49.4444	−1.57065	−0.785326	−	0.619082i	$-0.787505\pi$
$991$	−49.4444	−1.57065	−0.785326	+	0.619082i	$0.787505\pi$
$992$	0	0
$993$	3.48007	0.110437
$994$	0	0
$995$	31.8746	1.01049
$996$	0	0
$997$	−1.25432	−0.0397247	−0.0198624	−	0.999803i	$-0.506323\pi$
$997$	−1.25432	−0.0397247	−0.0198624	+	0.999803i	$0.506323\pi$
$998$	0	0
$999$	29.6005	0.936517

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.j.1.8	yes	30
4.3	odd	2		8032.2.a.g.1.23	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.23	✓	30	4.3	odd	2
8032.2.a.j.1.8	yes	30	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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-1.62030 q^{3} -2.98882 q^{5} -1.58362 q^{7} -0.374619 q^{9} +O(q^{10})\)	\(q-1.62030 q^{3} -2.98882 q^{5} -1.58362 q^{7} -0.374619 q^{9} +1.17659 q^{11} -1.64357 q^{13} +4.84280 q^{15} -0.163190 q^{17} -0.993147 q^{19} +2.56594 q^{21} -3.53802 q^{23} +3.93307 q^{25} +5.46790 q^{27} -0.843088 q^{29} -7.13984 q^{31} -1.90643 q^{33} +4.73315 q^{35} +5.41349 q^{37} +2.66308 q^{39} -0.419638 q^{41} -3.12205 q^{43} +1.11967 q^{45} -12.0580 q^{47} -4.49216 q^{49} +0.264417 q^{51} -7.57856 q^{53} -3.51663 q^{55} +1.60920 q^{57} -6.22744 q^{59} -13.9170 q^{61} +0.593252 q^{63} +4.91234 q^{65} -1.54402 q^{67} +5.73266 q^{69} -4.38103 q^{71} -9.05579 q^{73} -6.37276 q^{75} -1.86327 q^{77} +5.44650 q^{79} -7.73580 q^{81} +10.0707 q^{83} +0.487745 q^{85} +1.36606 q^{87} -10.0532 q^{89} +2.60278 q^{91} +11.5687 q^{93} +2.96834 q^{95} -11.9569 q^{97} -0.440773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) \) 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

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