LMFDB - Embedded newform 1296.4.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,4,Mod(1,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1296.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1296.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:	$76.4664753674$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{105}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 26 $ x^2 - x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.62348$ of defining polynomial
Character	$\chi$	$=$	1296.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.8704 q^{5} -24.8704 q^{7} +O(q^{10})$	$q-10.8704 q^{5} -24.8704 q^{7} -42.7409 q^{11} +15.1296 q^{13} +13.8704 q^{17} -143.352 q^{19} +19.1296 q^{23} -6.83384 q^{25} -226.093 q^{29} -59.3887 q^{31} +270.352 q^{35} -84.1860 q^{37} -203.259 q^{41} -325.890 q^{43} +10.9436 q^{47} +275.538 q^{49} +140.186 q^{53} +464.611 q^{55} +114.741 q^{59} -755.056 q^{61} -164.465 q^{65} +767.445 q^{67} +335.854 q^{71} +167.279 q^{73} +1062.98 q^{77} +25.3155 q^{79} +287.721 q^{83} -150.777 q^{85} +860.817 q^{89} -376.279 q^{91} +1558.30 q^{95} -402.149 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 9 q^{5} - 19 q^{7}+O(q^{10}) $ 2 * q + 9 * q^5 - 19 * q^7	$ 2 q + 9 q^{5} - 19 q^{7} - 24 q^{11} + 61 q^{13} - 3 q^{17} - 133 q^{19} + 69 q^{23} + 263 q^{25} - 237 q^{29} - 211 q^{31} + 387 q^{35} + 262 q^{37} - 468 q^{41} + 86 q^{43} + 483 q^{47} - 33 q^{49} - 150 q^{53} + 837 q^{55} + 168 q^{59} - 1049 q^{61} + 747 q^{65} + 1166 q^{67} - 312 q^{71} - 311 q^{73} + 1173 q^{77} - 349 q^{79} + 1221 q^{83} - 486 q^{85} + 492 q^{89} - 107 q^{91} + 1764 q^{95} - 128 q^{97}+O(q^{100}) $ 2 * q + 9 * q^5 - 19 * q^7 - 24 * q^11 + 61 * q^13 - 3 * q^17 - 133 * q^19 + 69 * q^23 + 263 * q^25 - 237 * q^29 - 211 * q^31 + 387 * q^35 + 262 * q^37 - 468 * q^41 + 86 * q^43 + 483 * q^47 - 33 * q^49 - 150 * q^53 + 837 * q^55 + 168 * q^59 - 1049 * q^61 + 747 * q^65 + 1166 * q^67 - 312 * q^71 - 311 * q^73 + 1173 * q^77 - 349 * q^79 + 1221 * q^83 - 486 * q^85 + 492 * q^89 - 107 * q^91 + 1764 * q^95 - 128 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−10.8704	−0.972280	−0.486140	−	0.873881i	$-0.661596\pi$
$5$	−10.8704	−0.972280	−0.486140	+	0.873881i	$0.661596\pi$
$6$	0	0
$7$	−24.8704	−1.34288	−0.671438	−	0.741060i	$-0.734323\pi$
$7$	−24.8704	−1.34288	−0.671438	+	0.741060i	$0.734323\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−42.7409	−1.17153	−0.585766	−	0.810480i	$-0.699206\pi$
$11$	−42.7409	−1.17153	−0.585766	+	0.810480i	$0.699206\pi$
$12$	0	0
$13$	15.1296	0.322784	0.161392	−	0.986890i	$-0.448402\pi$
$13$	15.1296	0.322784	0.161392	+	0.986890i	$0.448402\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	13.8704	0.197887	0.0989433	−	0.995093i	$-0.468454\pi$
$17$	13.8704	0.197887	0.0989433	+	0.995093i	$0.468454\pi$
$18$	0	0
$19$	−143.352	−1.73091	−0.865454	−	0.500989i	$-0.832970\pi$
$19$	−143.352	−1.73091	−0.865454	+	0.500989i	$0.832970\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	19.1296	0.173426	0.0867129	−	0.996233i	$-0.472364\pi$
$23$	19.1296	0.173426	0.0867129	+	0.996233i	$0.472364\pi$
$24$	0	0
$25$	−6.83384	−0.0546707
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−226.093	−1.44774	−0.723869	−	0.689937i	$-0.757638\pi$
$29$	−226.093	−1.44774	−0.723869	+	0.689937i	$0.757638\pi$
$30$	0	0
$31$	−59.3887	−0.344082	−0.172041	−	0.985090i	$-0.555036\pi$
$31$	−59.3887	−0.344082	−0.172041	+	0.985090i	$0.555036\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	270.352	1.30565
$36$	0	0
$37$	−84.1860	−0.374056	−0.187028	−	0.982355i	$-0.559886\pi$
$37$	−84.1860	−0.374056	−0.187028	+	0.982355i	$0.559886\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−203.259	−0.774238	−0.387119	−	0.922030i	$-0.626530\pi$
$41$	−203.259	−0.774238	−0.387119	+	0.922030i	$0.626530\pi$
$42$	0	0
$43$	−325.890	−1.15576	−0.577881	−	0.816121i	$-0.696120\pi$
$43$	−325.890	−1.15576	−0.577881	+	0.816121i	$0.696120\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.9436	0.0339636	0.0169818	−	0.999856i	$-0.494594\pi$
$47$	10.9436	0.0339636	0.0169818	+	0.999856i	$0.494594\pi$
$48$	0	0
$49$	275.538	0.803318
$50$	0	0
$51$	0	0
$52$	0	0
$53$	140.186	0.363321	0.181661	−	0.983361i	$-0.441853\pi$
$53$	140.186	0.363321	0.181661	+	0.983361i	$0.441853\pi$
$54$	0	0
$55$	464.611	1.13906
$56$	0	0
$57$	0	0
$58$	0	0
$59$	114.741	0.253186	0.126593	−	0.991955i	$-0.459596\pi$
$59$	114.741	0.253186	0.126593	+	0.991955i	$0.459596\pi$
$60$	0	0
$61$	−755.056	−1.58484	−0.792419	−	0.609978i	$-0.791178\pi$
$61$	−755.056	−1.58484	−0.792419	+	0.609978i	$0.791178\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−164.465	−0.313836
$66$	0	0
$67$	767.445	1.39938	0.699689	−	0.714447i	$-0.253322\pi$
$67$	767.445	1.39938	0.699689	+	0.714447i	$0.253322\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	335.854	0.561387	0.280694	−	0.959797i	$-0.409436\pi$
$71$	335.854	0.561387	0.280694	+	0.959797i	$0.409436\pi$
$72$	0	0
$73$	167.279	0.268199	0.134099	−	0.990968i	$-0.457186\pi$
$73$	167.279	0.268199	0.134099	+	0.990968i	$0.457186\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1062.98	1.57322
$78$	0	0
$79$	25.3155	0.0360534	0.0180267	−	0.999838i	$-0.494262\pi$
$79$	25.3155	0.0360534	0.0180267	+	0.999838i	$0.494262\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	287.721	0.380500	0.190250	−	0.981736i	$-0.439070\pi$
$83$	287.721	0.380500	0.190250	+	0.981736i	$0.439070\pi$
$84$	0	0
$85$	−150.777	−0.192401
$86$	0	0
$87$	0	0
$88$	0	0
$89$	860.817	1.02524	0.512620	−	0.858615i	$-0.328675\pi$
$89$	860.817	1.02524	0.512620	+	0.858615i	$0.328675\pi$
$90$	0	0
$91$	−376.279	−0.433459
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1558.30	1.68293
$96$	0	0
$97$	−402.149	−0.420949	−0.210475	−	0.977599i	$-0.567501\pi$
$97$	−402.149	−0.420949	−0.210475	+	0.977599i	$0.567501\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1331.43	1.31170	0.655852	−	0.754890i	$-0.272310\pi$
$101$	1331.43	1.31170	0.655852	+	0.754890i	$0.272310\pi$
$102$	0	0
$103$	518.505	0.496017	0.248009	−	0.968758i	$-0.420224\pi$
$103$	518.505	0.496017	0.248009	+	0.968758i	$0.420224\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1471.87	1.32982	0.664912	−	0.746922i	$-0.268469\pi$
$107$	1471.87	1.32982	0.664912	+	0.746922i	$0.268469\pi$
$108$	0	0
$109$	−643.668	−0.565616	−0.282808	−	0.959176i	$-0.591266\pi$
$109$	−643.668	−0.565616	−0.282808	+	0.959176i	$0.591266\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1023.73	−0.852249	−0.426125	−	0.904665i	$-0.640121\pi$
$113$	−1023.73	−0.852249	−0.426125	+	0.904665i	$0.640121\pi$
$114$	0	0
$115$	−207.947	−0.168618
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−344.963	−0.265737
$120$	0	0
$121$	495.780	0.372487
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1433.09	1.02544
$126$	0	0
$127$	31.4481	0.0219730	0.0109865	−	0.999940i	$-0.496503\pi$
$127$	31.4481	0.0219730	0.0109865	+	0.999940i	$0.496503\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1937.36	−1.29212	−0.646059	−	0.763287i	$-0.723584\pi$
$131$	−1937.36	−1.29212	−0.646059	+	0.763287i	$0.723584\pi$
$132$	0	0
$133$	3565.23	2.32439
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1158.59	−0.722521	−0.361261	−	0.932465i	$-0.617654\pi$
$137$	−1158.59	−0.722521	−0.361261	+	0.932465i	$0.617654\pi$
$138$	0	0
$139$	−2311.60	−1.41055	−0.705277	−	0.708931i	$-0.749178\pi$
$139$	−2311.60	−1.41055	−0.705277	+	0.708931i	$0.749178\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−646.651	−0.378151
$144$	0	0
$145$	2457.73	1.40761
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2950.21	1.62209	0.811043	−	0.584987i	$-0.198900\pi$
$149$	2950.21	1.62209	0.811043	+	0.584987i	$0.198900\pi$
$150$	0	0
$151$	1726.40	0.930412	0.465206	−	0.885202i	$-0.345980\pi$
$151$	1726.40	0.930412	0.465206	+	0.885202i	$0.345980\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	645.581	0.334544
$156$	0	0
$157$	1283.39	0.652392	0.326196	−	0.945302i	$-0.394233\pi$
$157$	1283.39	0.652392	0.326196	+	0.945302i	$0.394233\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−475.761	−0.232889
$162$	0	0
$163$	−1033.93	−0.496831	−0.248415	−	0.968654i	$-0.579910\pi$
$163$	−1033.93	−0.496831	−0.248415	+	0.968654i	$0.579910\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−282.617	−0.130956	−0.0654778	−	0.997854i	$-0.520857\pi$
$167$	−282.617	−0.130956	−0.0654778	+	0.997854i	$0.520857\pi$
$168$	0	0
$169$	−1968.10	−0.895811
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3532.72	1.55253	0.776266	−	0.630405i	$-0.217111\pi$
$173$	3532.72	1.55253	0.776266	+	0.630405i	$0.217111\pi$
$174$	0	0
$175$	169.960	0.0734160
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4052.74	−1.69227	−0.846135	−	0.532969i	$-0.821076\pi$
$179$	−4052.74	−1.69227	−0.846135	+	0.532969i	$0.821076\pi$
$180$	0	0
$181$	−2830.97	−1.16257	−0.581283	−	0.813702i	$-0.697449\pi$
$181$	−2830.97	−1.16257	−0.581283	+	0.813702i	$0.697449\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	915.137	0.363688
$186$	0	0
$187$	−592.834	−0.231831
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4509.24	1.70826	0.854129	−	0.520061i	$-0.174091\pi$
$191$	4509.24	1.70826	0.854129	+	0.520061i	$0.174091\pi$
$192$	0	0
$193$	−3221.12	−1.20136	−0.600678	−	0.799491i	$-0.705103\pi$
$193$	−3221.12	−1.20136	−0.600678	+	0.799491i	$0.705103\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3784.20	−1.36859	−0.684297	−	0.729204i	$-0.739891\pi$
$197$	−3784.20	−1.36859	−0.684297	+	0.729204i	$0.739891\pi$
$198$	0	0
$199$	2926.27	1.04240	0.521200	−	0.853435i	$-0.325485\pi$
$199$	2926.27	1.04240	0.521200	+	0.853435i	$0.325485\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5623.03	1.94413
$204$	0	0
$205$	2209.51	0.752776
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6126.99	2.02781
$210$	0	0
$211$	−313.474	−0.102277	−0.0511385	−	0.998692i	$-0.516285\pi$
$211$	−313.474	−0.102277	−0.0511385	+	0.998692i	$0.516285\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3542.57	1.12373
$216$	0	0
$217$	1477.02	0.462059
$218$	0	0
$219$	0	0
$220$	0	0
$221$	209.854	0.0638746
$222$	0	0
$223$	710.386	0.213323	0.106661	−	0.994295i	$-0.465984\pi$
$223$	710.386	0.213323	0.106661	+	0.994295i	$0.465984\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	32.6616	0.00954991	0.00477496	−	0.999989i	$-0.498480\pi$
$227$	32.6616	0.00954991	0.00477496	+	0.999989i	$0.498480\pi$
$228$	0	0
$229$	−5503.85	−1.58823	−0.794114	−	0.607768i	$-0.792065\pi$
$229$	−5503.85	−1.58823	−0.794114	+	0.607768i	$0.792065\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6788.81	1.90880	0.954399	−	0.298534i	$-0.0964977\pi$
$233$	6788.81	1.90880	0.954399	+	0.298534i	$0.0964977\pi$
$234$	0	0
$235$	−118.962	−0.0330221
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−429.389	−0.116213	−0.0581064	−	0.998310i	$-0.518506\pi$
$239$	−429.389	−0.116213	−0.0581064	+	0.998310i	$0.518506\pi$
$240$	0	0
$241$	−4843.65	−1.29463	−0.647317	−	0.762221i	$-0.724109\pi$
$241$	−4843.65	−1.29463	−0.647317	+	0.762221i	$0.724109\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2995.22	−0.781050
$246$	0	0
$247$	−2168.86	−0.558709
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2400.87	0.603752	0.301876	−	0.953347i	$-0.402387\pi$
$251$	2400.87	0.603752	0.301876	+	0.953347i	$0.402387\pi$
$252$	0	0
$253$	−817.614	−0.203174
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2980.99	−0.723538	−0.361769	−	0.932268i	$-0.617827\pi$
$257$	−2980.99	−0.723538	−0.361769	+	0.932268i	$0.617827\pi$
$258$	0	0
$259$	2093.74	0.502312
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5560.82	1.30378	0.651891	−	0.758313i	$-0.273976\pi$
$263$	5560.82	1.30378	0.651891	+	0.758313i	$0.273976\pi$
$264$	0	0
$265$	−1523.88	−0.353250
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6288.50	−1.42534	−0.712671	−	0.701499i	$-0.752515\pi$
$269$	−6288.50	−1.42534	−0.712671	+	0.701499i	$0.752515\pi$
$270$	0	0
$271$	−6854.90	−1.53655	−0.768275	−	0.640119i	$-0.778885\pi$
$271$	−6854.90	−1.53655	−0.768275	+	0.640119i	$0.778885\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	292.084	0.0640485
$276$	0	0
$277$	898.172	0.194823	0.0974114	−	0.995244i	$-0.468944\pi$
$277$	898.172	0.194823	0.0974114	+	0.995244i	$0.468944\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−238.770	−0.0506897	−0.0253448	−	0.999679i	$-0.508068\pi$
$281$	−238.770	−0.0506897	−0.0253448	+	0.999679i	$0.508068\pi$
$282$	0	0
$283$	−2071.16	−0.435045	−0.217523	−	0.976055i	$-0.569798\pi$
$283$	−2071.16	−0.435045	−0.217523	+	0.976055i	$0.569798\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5055.14	1.03971
$288$	0	0
$289$	−4720.61	−0.960841
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6577.76	1.31153	0.655763	−	0.754967i	$-0.272347\pi$
$293$	6577.76	1.31153	0.655763	+	0.754967i	$0.272347\pi$
$294$	0	0
$295$	−1247.28	−0.246168
$296$	0	0
$297$	0	0
$298$	0	0
$299$	289.422	0.0559790
$300$	0	0
$301$	8105.03	1.55205
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8207.78	1.54091
$306$	0	0
$307$	5237.30	0.973644	0.486822	−	0.873501i	$-0.338156\pi$
$307$	5237.30	0.973644	0.486822	+	0.873501i	$0.338156\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5704.99	1.04019	0.520097	−	0.854107i	$-0.325896\pi$
$311$	5704.99	1.04019	0.520097	+	0.854107i	$0.325896\pi$
$312$	0	0
$313$	5077.48	0.916920	0.458460	−	0.888715i	$-0.348401\pi$
$313$	5077.48	0.916920	0.458460	+	0.888715i	$0.348401\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2868.91	−0.508309	−0.254155	−	0.967164i	$-0.581797\pi$
$317$	−2868.91	−0.508309	−0.254155	+	0.967164i	$0.581797\pi$
$318$	0	0
$319$	9663.41	1.69607
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1988.36	−0.342523
$324$	0	0
$325$	−103.393	−0.0176468
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−272.172	−0.0456089
$330$	0	0
$331$	−2031.35	−0.337320	−0.168660	−	0.985674i	$-0.553944\pi$
$331$	−2031.35	−0.337320	−0.168660	+	0.985674i	$0.553944\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8342.46	−1.36059
$336$	0	0
$337$	9798.28	1.58382	0.791909	−	0.610640i	$-0.209088\pi$
$337$	9798.28	1.58382	0.791909	+	0.610640i	$0.209088\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2538.32	0.403103
$342$	0	0
$343$	1677.81	0.264120
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4556.56	−0.704925	−0.352463	−	0.935826i	$-0.614656\pi$
$347$	−4556.56	−0.704925	−0.352463	+	0.935826i	$0.614656\pi$
$348$	0	0
$349$	3348.89	0.513644	0.256822	−	0.966459i	$-0.417325\pi$
$349$	3348.89	0.513644	0.256822	+	0.966459i	$0.417325\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1862.27	−0.280789	−0.140395	−	0.990096i	$-0.544837\pi$
$353$	−1862.27	−0.280789	−0.140395	+	0.990096i	$0.544837\pi$
$354$	0	0
$355$	−3650.87	−0.545826
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6179.46	0.908466	0.454233	−	0.890883i	$-0.349913\pi$
$359$	6179.46	0.908466	0.454233	+	0.890883i	$0.349913\pi$
$360$	0	0
$361$	13690.8	1.99604
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1818.39	−0.260765
$366$	0	0
$367$	−6873.95	−0.977704	−0.488852	−	0.872367i	$-0.662584\pi$
$367$	−6873.95	−0.977704	−0.488852	+	0.872367i	$0.662584\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3486.48	−0.487896
$372$	0	0
$373$	1270.03	0.176299	0.0881494	−	0.996107i	$-0.471905\pi$
$373$	1270.03	0.176299	0.0881494	+	0.996107i	$0.471905\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3420.69	−0.467306
$378$	0	0
$379$	−2490.54	−0.337548	−0.168774	−	0.985655i	$-0.553981\pi$
$379$	−2490.54	−0.337548	−0.168774	+	0.985655i	$0.553981\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	312.425	0.0416820	0.0208410	−	0.999783i	$-0.493366\pi$
$383$	312.425	0.0416820	0.0208410	+	0.999783i	$0.493366\pi$
$384$	0	0
$385$	−11555.1	−1.52961
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9643.17	−1.25688	−0.628442	−	0.777856i	$-0.716307\pi$
$389$	−9643.17	−1.25688	−0.628442	+	0.777856i	$0.716307\pi$
$390$	0	0
$391$	265.335	0.0343186
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−275.191	−0.0350540
$396$	0	0
$397$	−2260.32	−0.285749	−0.142874	−	0.989741i	$-0.545634\pi$
$397$	−2260.32	−0.285749	−0.142874	+	0.989741i	$0.545634\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10169.3	−1.26641	−0.633204	−	0.773985i	$-0.718261\pi$
$401$	−10169.3	−1.26641	−0.633204	+	0.773985i	$0.718261\pi$
$402$	0	0
$403$	−898.526	−0.111064
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3598.18	0.438219
$408$	0	0
$409$	−949.833	−0.114832	−0.0574159	−	0.998350i	$-0.518286\pi$
$409$	−949.833	−0.114832	−0.0574159	+	0.998350i	$0.518286\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2853.65	−0.339998
$414$	0	0
$415$	−3127.65	−0.369953
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11799.3	−1.37573	−0.687866	−	0.725837i	$-0.741453\pi$
$419$	−11799.3	−1.37573	−0.687866	+	0.725837i	$0.741453\pi$
$420$	0	0
$421$	−6412.93	−0.742392	−0.371196	−	0.928555i	$-0.621052\pi$
$421$	−6412.93	−0.742392	−0.371196	+	0.928555i	$0.621052\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−94.7882	−0.0108186
$426$	0	0
$427$	18778.6	2.12824
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12042.7	−1.34589	−0.672945	−	0.739693i	$-0.734971\pi$
$431$	−12042.7	−1.34589	−0.672945	+	0.739693i	$0.734971\pi$
$432$	0	0
$433$	7279.83	0.807959	0.403980	−	0.914768i	$-0.367627\pi$
$433$	7279.83	0.807959	0.403980	+	0.914768i	$0.367627\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2742.27	−0.300184
$438$	0	0
$439$	3598.70	0.391245	0.195623	−	0.980679i	$-0.437327\pi$
$439$	3598.70	0.391245	0.195623	+	0.980679i	$0.437327\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14786.4	−1.58583	−0.792913	−	0.609335i	$-0.791437\pi$
$443$	−14786.4	−1.58583	−0.792913	+	0.609335i	$0.791437\pi$
$444$	0	0
$445$	−9357.45	−0.996822
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−114.489	−0.0120336	−0.00601681	−	0.999982i	$-0.501915\pi$
$449$	−114.489	−0.0120336	−0.00601681	+	0.999982i	$0.501915\pi$
$450$	0	0
$451$	8687.47	0.907044
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4090.31	0.421444
$456$	0	0
$457$	6311.14	0.646001	0.323001	−	0.946399i	$-0.395308\pi$
$457$	6311.14	0.646001	0.323001	+	0.946399i	$0.395308\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13744.8	1.38863	0.694317	−	0.719670i	$-0.255707\pi$
$461$	13744.8	1.38863	0.694317	+	0.719670i	$0.255707\pi$
$462$	0	0
$463$	−15648.6	−1.57074	−0.785369	−	0.619028i	$-0.787527\pi$
$463$	−15648.6	−1.57074	−0.785369	+	0.619028i	$0.787527\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7395.79	−0.732840	−0.366420	−	0.930450i	$-0.619417\pi$
$467$	−7395.79	−0.732840	−0.366420	+	0.930450i	$0.619417\pi$
$468$	0	0
$469$	−19086.7	−1.87919
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13928.8	1.35401
$474$	0	0
$475$	979.645	0.0946299
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6261.07	−0.597235	−0.298617	−	0.954373i	$-0.596525\pi$
$479$	−6261.07	−0.597235	−0.298617	+	0.954373i	$0.596525\pi$
$480$	0	0
$481$	−1273.70	−0.120739
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4371.54	0.409281
$486$	0	0
$487$	10314.7	0.959763	0.479881	−	0.877333i	$-0.340680\pi$
$487$	10314.7	0.959763	0.479881	+	0.877333i	$0.340680\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2760.90	−0.253763	−0.126881	−	0.991918i	$-0.540497\pi$
$491$	−2760.90	−0.253763	−0.126881	+	0.991918i	$0.540497\pi$
$492$	0	0
$493$	−3136.01	−0.286488
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8352.82	−0.753874
$498$	0	0
$499$	9585.99	0.859976	0.429988	−	0.902835i	$-0.358518\pi$
$499$	9585.99	0.859976	0.429988	+	0.902835i	$0.358518\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8829.60	0.782689	0.391344	−	0.920244i	$-0.372010\pi$
$503$	8829.60	0.782689	0.391344	+	0.920244i	$0.372010\pi$
$504$	0	0
$505$	−14473.2	−1.27534
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4741.74	−0.412916	−0.206458	−	0.978456i	$-0.566194\pi$
$509$	−4741.74	−0.412916	−0.206458	+	0.978456i	$0.566194\pi$
$510$	0	0
$511$	−4160.30	−0.360158
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5636.37	−0.482268
$516$	0	0
$517$	−467.739	−0.0397894
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2753.22	0.231518	0.115759	−	0.993277i	$-0.463070\pi$
$521$	2753.22	0.231518	0.115759	+	0.993277i	$0.463070\pi$
$522$	0	0
$523$	−17115.3	−1.43098	−0.715489	−	0.698624i	$-0.753796\pi$
$523$	−17115.3	−1.43098	−0.715489	+	0.698624i	$0.753796\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−823.747	−0.0680891
$528$	0	0
$529$	−11801.1	−0.969924
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3075.22	−0.249911
$534$	0	0
$535$	−15999.9	−1.29296
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−11776.7	−0.941113
$540$	0	0
$541$	17880.1	1.42093	0.710467	−	0.703731i	$-0.248484\pi$
$541$	17880.1	1.42093	0.710467	+	0.703731i	$0.248484\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6996.94	0.549938
$546$	0	0
$547$	13069.5	1.02159	0.510795	−	0.859703i	$-0.329351\pi$
$547$	13069.5	1.02159	0.510795	+	0.859703i	$0.329351\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	32410.9	2.50590
$552$	0	0
$553$	−629.608	−0.0484153
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9507.62	−0.723251	−0.361626	−	0.932323i	$-0.617778\pi$
$557$	−9507.62	−0.723251	−0.361626	+	0.932323i	$0.617778\pi$
$558$	0	0
$559$	−4930.58	−0.373061
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20444.6	−1.53044	−0.765221	−	0.643768i	$-0.777370\pi$
$563$	−20444.6	−1.53044	−0.765221	+	0.643768i	$0.777370\pi$
$564$	0	0
$565$	11128.3	0.828625
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2646.06	−0.194954	−0.0974770	−	0.995238i	$-0.531077\pi$
$569$	−2646.06	−0.194954	−0.0974770	+	0.995238i	$0.531077\pi$
$570$	0	0
$571$	−1757.03	−0.128773	−0.0643864	−	0.997925i	$-0.520509\pi$
$571$	−1757.03	−0.128773	−0.0643864	+	0.997925i	$0.520509\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−130.728	−0.00948130
$576$	0	0
$577$	−7515.43	−0.542238	−0.271119	−	0.962546i	$-0.587394\pi$
$577$	−7515.43	−0.542238	−0.271119	+	0.962546i	$0.587394\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7155.75	−0.510964
$582$	0	0
$583$	−5991.67	−0.425642
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4952.49	−0.348230	−0.174115	−	0.984725i	$-0.555706\pi$
$587$	−4952.49	−0.348230	−0.174115	+	0.984725i	$0.555706\pi$
$588$	0	0
$589$	8513.50	0.595573
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−17115.3	−1.18523	−0.592613	−	0.805487i	$-0.701904\pi$
$593$	−17115.3	−1.18523	−0.592613	+	0.805487i	$0.701904\pi$
$594$	0	0
$595$	3749.90	0.258371
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8414.13	0.573943	0.286972	−	0.957939i	$-0.407351\pi$
$599$	8414.13	0.573943	0.286972	+	0.957939i	$0.407351\pi$
$600$	0	0
$601$	28094.2	1.90680	0.953399	−	0.301712i	$-0.0975582\pi$
$601$	28094.2	1.90680	0.953399	+	0.301712i	$0.0975582\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5389.34	−0.362162
$606$	0	0
$607$	−1430.85	−0.0956775	−0.0478388	−	0.998855i	$-0.515233\pi$
$607$	−1430.85	−0.0956775	−0.0478388	+	0.998855i	$0.515233\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	165.572	0.0109629
$612$	0	0
$613$	14438.1	0.951306	0.475653	−	0.879633i	$-0.342212\pi$
$613$	14438.1	0.951306	0.475653	+	0.879633i	$0.342212\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25444.9	−1.66025	−0.830125	−	0.557577i	$-0.811731\pi$
$617$	−25444.9	−1.66025	−0.830125	+	0.557577i	$0.811731\pi$
$618$	0	0
$619$	3479.46	0.225931	0.112966	−	0.993599i	$-0.463965\pi$
$619$	3479.46	0.225931	0.112966	+	0.993599i	$0.463965\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−21408.9	−1.37677
$624$	0	0
$625$	−14724.1	−0.942340
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1167.70	−0.0740208
$630$	0	0
$631$	−11151.7	−0.703552	−0.351776	−	0.936084i	$-0.614422\pi$
$631$	−11151.7	−0.703552	−0.351776	+	0.936084i	$0.614422\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−341.855	−0.0213639
$636$	0	0
$637$	4168.77	0.259298
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1092.78	0.0673356	0.0336678	−	0.999433i	$-0.489281\pi$
$641$	1092.78	0.0673356	0.0336678	+	0.999433i	$0.489281\pi$
$642$	0	0
$643$	−31694.0	−1.94384	−0.971922	−	0.235305i	$-0.924391\pi$
$643$	−31694.0	−1.94384	−0.971922	+	0.235305i	$0.924391\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13719.9	−0.833672	−0.416836	−	0.908982i	$-0.636861\pi$
$647$	−13719.9	−0.833672	−0.416836	+	0.908982i	$0.636861\pi$
$648$	0	0
$649$	−4904.12	−0.296616
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13285.2	−0.796155	−0.398078	−	0.917352i	$-0.630323\pi$
$653$	−13285.2	−0.796155	−0.398078	+	0.917352i	$0.630323\pi$
$654$	0	0
$655$	21059.9	1.25630
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2593.98	−0.153334	−0.0766670	−	0.997057i	$-0.524428\pi$
$659$	−2593.98	−0.153334	−0.0766670	+	0.997057i	$0.524428\pi$
$660$	0	0
$661$	−15875.3	−0.934159	−0.467079	−	0.884215i	$-0.654694\pi$
$661$	−15875.3	−0.934159	−0.467079	+	0.884215i	$0.654694\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−38755.6	−2.25996
$666$	0	0
$667$	−4325.06	−0.251075
$668$	0	0
$669$	0	0
$670$	0	0
$671$	32271.8	1.85669
$672$	0	0
$673$	21434.7	1.22771	0.613853	−	0.789420i	$-0.289619\pi$
$673$	21434.7	1.22771	0.613853	+	0.789420i	$0.289619\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20333.0	1.15430	0.577150	−	0.816638i	$-0.304165\pi$
$677$	20333.0	1.15430	0.577150	+	0.816638i	$0.304165\pi$
$678$	0	0
$679$	10001.6	0.565283
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−27149.0	−1.52097	−0.760487	−	0.649353i	$-0.775040\pi$
$683$	−27149.0	−1.52097	−0.760487	+	0.649353i	$0.775040\pi$
$684$	0	0
$685$	12594.4	0.702493
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2120.95	0.117274
$690$	0	0
$691$	−22710.2	−1.25027	−0.625134	−	0.780517i	$-0.714956\pi$
$691$	−22710.2	−1.25027	−0.625134	+	0.780517i	$0.714956\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	25128.0	1.37146
$696$	0	0
$697$	−2819.29	−0.153211
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20079.5	1.08187	0.540936	−	0.841064i	$-0.318070\pi$
$701$	20079.5	1.08187	0.540936	+	0.841064i	$0.318070\pi$
$702$	0	0
$703$	12068.2	0.647457
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−33113.2	−1.76146
$708$	0	0
$709$	−28983.7	−1.53527	−0.767634	−	0.640889i	$-0.778566\pi$
$709$	−28983.7	−1.53527	−0.767634	+	0.640889i	$0.778566\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1136.08	−0.0596726
$714$	0	0
$715$	7029.37	0.367669
$716$	0	0
$717$	0	0
$718$	0	0
$719$	14496.2	0.751901	0.375951	−	0.926640i	$-0.377316\pi$
$719$	14496.2	0.751901	0.375951	+	0.926640i	$0.377316\pi$
$720$	0	0
$721$	−12895.4	−0.666090
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1545.08	0.0791488
$726$	0	0
$727$	−5000.50	−0.255101	−0.127550	−	0.991832i	$-0.540711\pi$
$727$	−5000.50	−0.255101	−0.127550	+	0.991832i	$0.540711\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4520.24	−0.228710
$732$	0	0
$733$	17515.6	0.882609	0.441305	−	0.897357i	$-0.354516\pi$
$733$	17515.6	0.882609	0.441305	+	0.897357i	$0.354516\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−32801.3	−1.63942
$738$	0	0
$739$	20169.2	1.00397	0.501985	−	0.864876i	$-0.332603\pi$
$739$	20169.2	1.00397	0.501985	+	0.864876i	$0.332603\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36702.6	−1.81223	−0.906116	−	0.423029i	$-0.860967\pi$
$743$	−36702.6	−1.81223	−0.906116	+	0.423029i	$0.860967\pi$
$744$	0	0
$745$	−32070.1	−1.57712
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−36606.0	−1.78579
$750$	0	0
$751$	33320.1	1.61900	0.809499	−	0.587122i	$-0.199739\pi$
$751$	33320.1	1.61900	0.809499	+	0.587122i	$0.199739\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−18766.7	−0.904622
$756$	0	0
$757$	26515.6	1.27309	0.636543	−	0.771241i	$-0.280364\pi$
$757$	26515.6	1.27309	0.636543	+	0.771241i	$0.280364\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5684.35	0.270772	0.135386	−	0.990793i	$-0.456773\pi$
$761$	5684.35	0.270772	0.135386	+	0.990793i	$0.456773\pi$
$762$	0	0
$763$	16008.3	0.759553
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1735.98	0.0817244
$768$	0	0
$769$	−398.377	−0.0186812	−0.00934060	−	0.999956i	$-0.502973\pi$
$769$	−398.377	−0.0186812	−0.00934060	+	0.999956i	$0.502973\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8437.17	0.392579	0.196290	−	0.980546i	$-0.437111\pi$
$773$	8437.17	0.392579	0.196290	+	0.980546i	$0.437111\pi$
$774$	0	0
$775$	405.853	0.0188112
$776$	0	0
$777$	0	0
$778$	0	0
$779$	29137.6	1.34013
$780$	0	0
$781$	−14354.7	−0.657683
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13951.0	−0.634308
$786$	0	0
$787$	−16277.9	−0.737286	−0.368643	−	0.929571i	$-0.620177\pi$
$787$	−16277.9	−0.737286	−0.368643	+	0.929571i	$0.620177\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	25460.5	1.14447
$792$	0	0
$793$	−11423.7	−0.511560
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5112.11	−0.227202	−0.113601	−	0.993526i	$-0.536239\pi$
$797$	−5112.11	−0.227202	−0.113601	+	0.993526i	$0.536239\pi$
$798$	0	0
$799$	151.793	0.00672094
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7149.64	−0.314204
$804$	0	0
$805$	5171.72	0.226434
$806$	0	0
$807$	0	0
$808$	0	0
$809$	13141.2	0.571100	0.285550	−	0.958364i	$-0.407824\pi$
$809$	13141.2	0.571100	0.285550	+	0.958364i	$0.407824\pi$
$810$	0	0
$811$	18614.2	0.805957	0.402979	−	0.915209i	$-0.367975\pi$
$811$	18614.2	0.805957	0.402979	+	0.915209i	$0.367975\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11239.2	0.483059
$816$	0	0
$817$	46717.1	2.00052
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11320.0	0.481208	0.240604	−	0.970623i	$-0.422655\pi$
$821$	11320.0	0.481208	0.240604	+	0.970623i	$0.422655\pi$
$822$	0	0
$823$	−10866.6	−0.460249	−0.230125	−	0.973161i	$-0.573913\pi$
$823$	−10866.6	−0.460249	−0.230125	+	0.973161i	$0.573913\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13059.3	−0.549114	−0.274557	−	0.961571i	$-0.588531\pi$
$827$	−13059.3	−0.549114	−0.274557	+	0.961571i	$0.588531\pi$
$828$	0	0
$829$	−21203.7	−0.888341	−0.444171	−	0.895942i	$-0.646502\pi$
$829$	−21203.7	−0.888341	−0.444171	+	0.895942i	$0.646502\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3821.83	0.158966
$834$	0	0
$835$	3072.17	0.127326
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14480.6	−0.595858	−0.297929	−	0.954588i	$-0.596296\pi$
$839$	−14480.6	−0.595858	−0.297929	+	0.954588i	$0.596296\pi$
$840$	0	0
$841$	26729.0	1.09595
$842$	0	0
$843$	0	0
$844$	0	0
$845$	21394.0	0.870979
$846$	0	0
$847$	−12330.3	−0.500204
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1610.44	−0.0648710
$852$	0	0
$853$	−6467.23	−0.259594	−0.129797	−	0.991541i	$-0.541433\pi$
$853$	−6467.23	−0.259594	−0.129797	+	0.991541i	$0.541433\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3894.64	−0.155237	−0.0776186	−	0.996983i	$-0.524732\pi$
$857$	−3894.64	−0.155237	−0.0776186	+	0.996983i	$0.524732\pi$
$858$	0	0
$859$	37653.8	1.49561	0.747805	−	0.663918i	$-0.231108\pi$
$859$	37653.8	1.49561	0.747805	+	0.663918i	$0.231108\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47067.5	1.85654	0.928271	−	0.371905i	$-0.121295\pi$
$863$	47067.5	1.85654	0.928271	+	0.371905i	$0.121295\pi$
$864$	0	0
$865$	−38402.2	−1.50950
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1082.01	−0.0422377
$870$	0	0
$871$	11611.1	0.451697
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−35641.6	−1.37703
$876$	0	0
$877$	−7443.54	−0.286603	−0.143301	−	0.989679i	$-0.545772\pi$
$877$	−7443.54	−0.286603	−0.143301	+	0.989679i	$0.545772\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	13781.9	0.527040	0.263520	−	0.964654i	$-0.415116\pi$
$881$	13781.9	0.527040	0.263520	+	0.964654i	$0.415116\pi$
$882$	0	0
$883$	−12230.3	−0.466119	−0.233060	−	0.972462i	$-0.574874\pi$
$883$	−12230.3	−0.466119	−0.233060	+	0.972462i	$0.574874\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1674.87	−0.0634011	−0.0317005	−	0.999497i	$-0.510092\pi$
$887$	−1674.87	−0.0634011	−0.0317005	+	0.999497i	$0.510092\pi$
$888$	0	0
$889$	−782.128	−0.0295070
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1568.79	−0.0587878
$894$	0	0
$895$	44055.1	1.64536
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13427.4	0.498140
$900$	0	0
$901$	1944.44	0.0718964
$902$	0	0
$903$	0	0
$904$	0	0
$905$	30773.8	1.13034
$906$	0	0
$907$	−17088.1	−0.625580	−0.312790	−	0.949822i	$-0.601264\pi$
$907$	−17088.1	−0.625580	−0.312790	+	0.949822i	$0.601264\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	21288.4	0.774222	0.387111	−	0.922033i	$-0.373473\pi$
$911$	21288.4	0.774222	0.387111	+	0.922033i	$0.373473\pi$
$912$	0	0
$913$	−12297.4	−0.445768
$914$	0	0
$915$	0	0
$916$	0	0
$917$	48182.8	1.73516
$918$	0	0
$919$	10413.4	0.373784	0.186892	−	0.982380i	$-0.440159\pi$
$919$	10413.4	0.373784	0.186892	+	0.982380i	$0.440159\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5081.32	0.181207
$924$	0	0
$925$	575.313	0.0204499
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6823.43	0.240979	0.120490	−	0.992715i	$-0.461554\pi$
$929$	6823.43	0.240979	0.120490	+	0.992715i	$0.461554\pi$
$930$	0	0
$931$	−39499.0	−1.39047
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6444.36	0.225404
$936$	0	0
$937$	41049.8	1.43120	0.715602	−	0.698508i	$-0.246152\pi$
$937$	41049.8	1.43120	0.715602	+	0.698508i	$0.246152\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2416.12	−0.0837017	−0.0418508	−	0.999124i	$-0.513325\pi$
$941$	−2416.12	−0.0837017	−0.0418508	+	0.999124i	$0.513325\pi$
$942$	0	0
$943$	−3888.26	−0.134273
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2394.07	−0.0821509	−0.0410755	−	0.999156i	$-0.513078\pi$
$947$	−2394.07	−0.0821509	−0.0410755	+	0.999156i	$0.513078\pi$
$948$	0	0
$949$	2530.86	0.0865702
$950$	0	0
$951$	0	0
$952$	0	0
$953$	50651.3	1.72168	0.860838	−	0.508879i	$-0.169940\pi$
$953$	50651.3	1.72168	0.860838	+	0.508879i	$0.169940\pi$
$954$	0	0
$955$	−49017.4	−1.66091
$956$	0	0
$957$	0	0
$958$	0	0
$959$	28814.7	0.970257
$960$	0	0
$961$	−26264.0	−0.881608
$962$	0	0
$963$	0	0
$964$	0	0
$965$	35015.0	1.16805
$966$	0	0
$967$	−25034.9	−0.832542	−0.416271	−	0.909241i	$-0.636663\pi$
$967$	−25034.9	−0.832542	−0.416271	+	0.909241i	$0.636663\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	49553.7	1.63775	0.818875	−	0.573972i	$-0.194598\pi$
$971$	49553.7	1.63775	0.818875	+	0.573972i	$0.194598\pi$
$972$	0	0
$973$	57490.4	1.89420
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35176.2	1.15188	0.575939	−	0.817492i	$-0.304636\pi$
$977$	35176.2	1.15188	0.575939	+	0.817492i	$0.304636\pi$
$978$	0	0
$979$	−36792.1	−1.20110
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33349.3	1.08207	0.541036	−	0.840999i	$-0.318032\pi$
$983$	33349.3	1.08207	0.541036	+	0.840999i	$0.318032\pi$
$984$	0	0
$985$	41135.8	1.33066
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6234.14	−0.200439
$990$	0	0
$991$	−23066.3	−0.739378	−0.369689	−	0.929156i	$-0.620536\pi$
$991$	−23066.3	−0.739378	−0.369689	+	0.929156i	$0.620536\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−31809.7	−1.01350
$996$	0	0
$997$	−55769.9	−1.77156	−0.885782	−	0.464102i	$-0.846377\pi$
$997$	−55769.9	−1.77156	−0.885782	+	0.464102i	$0.846377\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.r.1.1		2
3.2	odd	2		1296.4.a.l.1.2		2
4.3	odd	2		162.4.a.g.1.1		2
9.2	odd	6		144.4.i.b.49.1		4
9.4	even	3		432.4.i.b.289.2		4
9.5	odd	6		144.4.i.b.97.1		4
9.7	even	3		432.4.i.b.145.2		4
12.11	even	2		162.4.a.f.1.2		2
36.7	odd	6		54.4.c.b.37.2		4
36.11	even	6		18.4.c.b.13.2	yes	4
36.23	even	6		18.4.c.b.7.2	✓	4
36.31	odd	6		54.4.c.b.19.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.4.c.b.7.2	✓	4	36.23	even	6
18.4.c.b.13.2	yes	4	36.11	even	6
54.4.c.b.19.2		4	36.31	odd	6
54.4.c.b.37.2		4	36.7	odd	6
144.4.i.b.49.1		4	9.2	odd	6
144.4.i.b.97.1		4	9.5	odd	6
162.4.a.f.1.2		2	12.11	even	2
162.4.a.g.1.1		2	4.3	odd	2
432.4.i.b.145.2		4	9.7	even	3
432.4.i.b.289.2		4	9.4	even	3
1296.4.a.l.1.2		2	3.2	odd	2
1296.4.a.r.1.1		2	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)

\(f(q)\)	\(=\)	\(q-10.8704 q^{5} -24.8704 q^{7} +O(q^{10})\)	\(q-10.8704 q^{5} -24.8704 q^{7} -42.7409 q^{11} +15.1296 q^{13} +13.8704 q^{17} -143.352 q^{19} +19.1296 q^{23} -6.83384 q^{25} -226.093 q^{29} -59.3887 q^{31} +270.352 q^{35} -84.1860 q^{37} -203.259 q^{41} -325.890 q^{43} +10.9436 q^{47} +275.538 q^{49} +140.186 q^{53} +464.611 q^{55} +114.741 q^{59} -755.056 q^{61} -164.465 q^{65} +767.445 q^{67} +335.854 q^{71} +167.279 q^{73} +1062.98 q^{77} +25.3155 q^{79} +287.721 q^{83} -150.777 q^{85} +860.817 q^{89} -376.279 q^{91} +1558.30 q^{95} -402.149 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 9 q^{5} - 19 q^{7}+O(q^{10}) \) 2 * q + 9 * q^5 - 19 * q^7	\( 2 q + 9 q^{5} - 19 q^{7} - 24 q^{11} + 61 q^{13} - 3 q^{17} - 133 q^{19} + 69 q^{23} + 263 q^{25} - 237 q^{29} - 211 q^{31} + 387 q^{35} + 262 q^{37} - 468 q^{41} + 86 q^{43} + 483 q^{47} - 33 q^{49} - 150 q^{53} + 837 q^{55} + 168 q^{59} - 1049 q^{61} + 747 q^{65} + 1166 q^{67} - 312 q^{71} - 311 q^{73} + 1173 q^{77} - 349 q^{79} + 1221 q^{83} - 486 q^{85} + 492 q^{89} - 107 q^{91} + 1764 q^{95} - 128 q^{97}+O(q^{100}) \) 2 * q + 9 * q^5 - 19 * q^7 - 24 * q^11 + 61 * q^13 - 3 * q^17 - 133 * q^19 + 69 * q^23 + 263 * q^25 - 237 * q^29 - 211 * q^31 + 387 * q^35 + 262 * q^37 - 468 * q^41 + 86 * q^43 + 483 * q^47 - 33 * q^49 - 150 * q^53 + 837 * q^55 + 168 * q^59 - 1049 * q^61 + 747 * q^65 + 1166 * q^67 - 312 * q^71 - 311 * q^73 + 1173 * q^77 - 349 * q^79 + 1221 * q^83 - 486 * q^85 + 492 * q^89 - 107 * q^91 + 1764 * q^95 - 128 * q^97

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