LMFDB - Embedded newform 1008.4.h.a.575.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(575,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("1008.575");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 144x^{10} + 12024x^{8} - 766296x^{6} + 11751192x^{4} + 565147728x^{2} + 9666232489 $ x^12 - 144x^10 + 12024x^8 - 766296x^6 + 11751192x^4 + 565147728*x^2 + 9666232489
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{4}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.2
Root			$8.41502 - 0.655192i$ of defining polynomial
Character	$\chi$	$=$	1008.575
Dual form			1008.4.h.a.575.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-14.1054i q^{5} +7.00000i q^{7} +O(q^{10})$	$q-14.1054i q^{5} +7.00000i q^{7} -13.1065 q^{11} +30.2418 q^{13} +78.2552i q^{17} -72.0861i q^{19} -217.553 q^{23} -73.9633 q^{25} +3.49080i q^{29} -305.218i q^{31} +98.7380 q^{35} +191.923 q^{37} -246.011i q^{41} +200.959i q^{43} -338.976 q^{47} -49.0000 q^{49} +423.136i q^{53} +184.873i q^{55} -366.860 q^{59} -730.595 q^{61} -426.573i q^{65} +1024.29i q^{67} -79.3885 q^{71} +127.786 q^{73} -91.7458i q^{77} +413.860i q^{79} -228.695 q^{83} +1103.82 q^{85} +193.891i q^{89} +211.692i q^{91} -1016.81 q^{95} +1004.12 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 192 q^{13} + 84 q^{25} + 72 q^{37} - 588 q^{49} - 1800 q^{61} + 3144 q^{85} - 1152 q^{97}+O(q^{100}) $ 12 * q + 192 * q^13 + 84 * q^25 + 72 * q^37 - 588 * q^49 - 1800 * q^61 + 3144 * q^85 - 1152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 14.1054i	− 1.26163i	−0.775934	−	0.630814i	$-0.782721\pi$
$5$	− 14.1054i	− 1.26163i	0.775934	−	0.630814i	$-0.217279\pi$
$6$	0	0
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−13.1065	−0.359252	−0.179626	−	0.983735i	$-0.557489\pi$
$11$	−13.1065	−0.359252	−0.179626	+	0.983735i	$0.557489\pi$
$12$	0	0
$13$	30.2418	0.645197	0.322598	−	0.946536i	$-0.395444\pi$
$13$	30.2418	0.645197	0.322598	+	0.946536i	$0.395444\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	78.2552i	1.11645i	0.829689	+	0.558226i	$0.188518\pi$
$17$	78.2552i	1.11645i	−0.829689	+	0.558226i	$0.811482\pi$
$18$	0	0
$19$	− 72.0861i	− 0.870404i	−0.900333	−	0.435202i	$-0.856677\pi$
$19$	− 72.0861i	− 0.870404i	0.900333	−	0.435202i	$-0.143323\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−217.553	−1.97231	−0.986153	−	0.165840i	$-0.946967\pi$
$23$	−217.553	−1.97231	−0.986153	+	0.165840i	$0.946967\pi$
$24$	0	0
$25$	−73.9633	−0.591706
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.49080i	0.0223526i	0.999938	+	0.0111763i	$0.00355760\pi$
$29$	3.49080i	0.0223526i	−0.999938	+	0.0111763i	$0.996442\pi$
$30$	0	0
$31$	− 305.218i	− 1.76835i	−0.467160	−	0.884173i	$-0.654723\pi$
$31$	− 305.218i	− 1.76835i	0.467160	−	0.884173i	$-0.345277\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	98.7380	0.476851
$36$	0	0
$37$	191.923	0.852754	0.426377	−	0.904546i	$-0.359790\pi$
$37$	191.923	0.852754	0.426377	+	0.904546i	$0.359790\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 246.011i	− 0.937086i	−0.883441	−	0.468543i	$-0.844779\pi$
$41$	− 246.011i	− 0.937086i	0.883441	−	0.468543i	$-0.155221\pi$
$42$	0	0
$43$	200.959i	0.712698i	0.934353	+	0.356349i	$0.115979\pi$
$43$	200.959i	0.712698i	−0.934353	+	0.356349i	$0.884021\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−338.976	−1.05202	−0.526008	−	0.850480i	$-0.676312\pi$
$47$	−338.976	−1.05202	−0.526008	+	0.850480i	$0.676312\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	423.136i	1.09665i	0.836267	+	0.548323i	$0.184734\pi$
$53$	423.136i	1.09665i	−0.836267	+	0.548323i	$0.815266\pi$
$54$	0	0
$55$	184.873i	0.453242i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−366.860	−0.809509	−0.404755	−	0.914425i	$-0.632643\pi$
$59$	−366.860	−0.809509	−0.404755	+	0.914425i	$0.632643\pi$
$60$	0	0
$61$	−730.595	−1.53349	−0.766747	−	0.641950i	$-0.778126\pi$
$61$	−730.595	−1.53349	−0.766747	+	0.641950i	$0.778126\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 426.573i	− 0.813999i
$66$	0	0
$67$	1024.29i	1.86772i	0.357633	+	0.933862i	$0.383584\pi$
$67$	1024.29i	1.86772i	−0.357633	+	0.933862i	$0.616416\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−79.3885	−0.132700	−0.0663499	−	0.997796i	$-0.521135\pi$
$71$	−79.3885	−0.132700	−0.0663499	+	0.997796i	$0.521135\pi$
$72$	0	0
$73$	127.786	0.204880	0.102440	−	0.994739i	$-0.467335\pi$
$73$	127.786	0.204880	0.102440	+	0.994739i	$0.467335\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 91.7458i	− 0.135784i
$78$	0	0
$79$	413.860i	0.589403i	0.955589	+	0.294701i	$0.0952202\pi$
$79$	413.860i	0.589403i	−0.955589	+	0.294701i	$0.904780\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−228.695	−0.302440	−0.151220	−	0.988500i	$-0.548320\pi$
$83$	−228.695	−0.302440	−0.151220	+	0.988500i	$0.548320\pi$
$84$	0	0
$85$	1103.82	1.40855
$86$	0	0
$87$	0	0
$88$	0	0
$89$	193.891i	0.230926i	0.993312	+	0.115463i	$0.0368352\pi$
$89$	193.891i	0.230926i	−0.993312	+	0.115463i	$0.963165\pi$
$90$	0	0
$91$	211.692i	0.243861i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1016.81	−1.09813
$96$	0	0
$97$	1004.12	1.05107	0.525533	−	0.850773i	$-0.323866\pi$
$97$	1004.12	1.05107	0.525533	+	0.850773i	$0.323866\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 218.638i	− 0.215399i	−0.994183	−	0.107700i	$-0.965652\pi$
$101$	− 218.638i	− 0.215399i	0.994183	−	0.107700i	$-0.0343485\pi$
$102$	0	0
$103$	965.671i	0.923790i	0.886935	+	0.461895i	$0.152830\pi$
$103$	965.671i	0.923790i	−0.886935	+	0.461895i	$0.847170\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	333.264	0.301102	0.150551	−	0.988602i	$-0.451895\pi$
$107$	333.264	0.301102	0.150551	+	0.988602i	$0.451895\pi$
$108$	0	0
$109$	−1891.50	−1.66214	−0.831068	−	0.556170i	$-0.812270\pi$
$109$	−1891.50	−1.66214	−0.831068	+	0.556170i	$0.812270\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	255.337i	0.212567i	0.994336	+	0.106284i	$0.0338951\pi$
$113$	255.337i	0.212567i	−0.994336	+	0.106284i	$0.966105\pi$
$114$	0	0
$115$	3068.69i	2.48832i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−547.787	−0.421979
$120$	0	0
$121$	−1159.22	−0.870938
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 719.895i	− 0.515115i
$126$	0	0
$127$	− 82.2137i	− 0.0574432i	−0.999587	−	0.0287216i	$-0.990856\pi$
$127$	− 82.2137i	− 0.0574432i	0.999587	−	0.0287216i	$-0.00914363\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1925.25	−1.28404	−0.642021	−	0.766687i	$-0.721904\pi$
$131$	−1925.25	−1.28404	−0.642021	+	0.766687i	$0.721904\pi$
$132$	0	0
$133$	504.602	0.328982
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2062.05i	1.28593i	0.765894	+	0.642967i	$0.222297\pi$
$137$	2062.05i	1.28593i	−0.765894	+	0.642967i	$0.777703\pi$
$138$	0	0
$139$	363.818i	0.222005i	0.993820	+	0.111002i	$0.0354061\pi$
$139$	363.818i	0.222005i	−0.993820	+	0.111002i	$0.964594\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−396.365	−0.231788
$144$	0	0
$145$	49.2392	0.0282007
$146$	0	0
$147$	0	0
$148$	0	0
$149$	961.387i	0.528590i	0.964442	+	0.264295i	$0.0851393\pi$
$149$	961.387i	0.528590i	−0.964442	+	0.264295i	$0.914861\pi$
$150$	0	0
$151$	− 2080.34i	− 1.12116i	−0.828099	−	0.560581i	$-0.810578\pi$
$151$	− 2080.34i	− 1.12116i	0.828099	−	0.560581i	$-0.189422\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4305.23	−2.23099
$156$	0	0
$157$	3163.02	1.60788	0.803939	−	0.594712i	$-0.202734\pi$
$157$	3163.02	1.60788	0.803939	+	0.594712i	$0.202734\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1522.87i	− 0.745461i
$162$	0	0
$163$	2325.36i	1.11740i	0.829369	+	0.558701i	$0.188700\pi$
$163$	2325.36i	1.11740i	−0.829369	+	0.558701i	$0.811300\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4001.48	−1.85415	−0.927077	−	0.374870i	$-0.877687\pi$
$167$	−4001.48	−1.85415	−0.927077	+	0.374870i	$0.877687\pi$
$168$	0	0
$169$	−1282.44	−0.583721
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1284.23i	0.564384i	0.959358	+	0.282192i	$0.0910616\pi$
$173$	1284.23i	0.564384i	−0.959358	+	0.282192i	$0.908938\pi$
$174$	0	0
$175$	− 517.743i	− 0.223644i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−644.970	−0.269315	−0.134657	−	0.990892i	$-0.542993\pi$
$179$	−644.970	−0.269315	−0.134657	+	0.990892i	$0.542993\pi$
$180$	0	0
$181$	2374.61	0.975155	0.487578	−	0.873080i	$-0.337881\pi$
$181$	2374.61	0.975155	0.487578	+	0.873080i	$0.337881\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2707.15i	− 1.07586i
$186$	0	0
$187$	− 1025.65i	− 0.401087i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4428.79	−1.67778	−0.838889	−	0.544302i	$-0.816795\pi$
$191$	−4428.79	−1.67778	−0.838889	+	0.544302i	$0.816795\pi$
$192$	0	0
$193$	−3114.28	−1.16151	−0.580753	−	0.814080i	$-0.697242\pi$
$193$	−3114.28	−1.16151	−0.580753	+	0.814080i	$0.697242\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3892.82i	− 1.40788i	−0.710260	−	0.703939i	$-0.751423\pi$
$197$	− 3892.82i	− 1.40788i	0.710260	−	0.703939i	$-0.248577\pi$
$198$	0	0
$199$	− 1837.29i	− 0.654482i	−0.944941	−	0.327241i	$-0.893881\pi$
$199$	− 1837.29i	− 0.654482i	0.944941	−	0.327241i	$-0.106119\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−24.4356	−0.00844848
$204$	0	0
$205$	−3470.10	−1.18225
$206$	0	0
$207$	0	0
$208$	0	0
$209$	944.799i	0.312694i
$210$	0	0
$211$	1753.48i	0.572105i	0.958214	+	0.286053i	$0.0923433\pi$
$211$	1753.48i	0.572105i	−0.958214	+	0.286053i	$0.907657\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2834.62	0.899160
$216$	0	0
$217$	2136.52	0.668372
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2366.58i	0.720331i
$222$	0	0
$223$	2467.93i	0.741098i	0.928813	+	0.370549i	$0.120830\pi$
$223$	2467.93i	0.741098i	−0.928813	+	0.370549i	$0.879170\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3753.03	−1.09735	−0.548673	−	0.836037i	$-0.684867\pi$
$227$	−3753.03	−1.09735	−0.548673	+	0.836037i	$0.684867\pi$
$228$	0	0
$229$	−4115.92	−1.18772	−0.593860	−	0.804568i	$-0.702397\pi$
$229$	−4115.92	−1.18772	−0.593860	+	0.804568i	$0.702397\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2719.40i	0.764608i	0.924037	+	0.382304i	$0.124869\pi$
$233$	2719.40i	0.764608i	−0.924037	+	0.382304i	$0.875131\pi$
$234$	0	0
$235$	4781.41i	1.32725i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3670.35	0.993370	0.496685	−	0.867931i	$-0.334550\pi$
$239$	3670.35	0.993370	0.496685	+	0.867931i	$0.334550\pi$
$240$	0	0
$241$	3043.03	0.813357	0.406678	−	0.913571i	$-0.366687\pi$
$241$	3043.03	0.813357	0.406678	+	0.913571i	$0.366687\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	691.166i	0.180233i
$246$	0	0
$247$	− 2180.01i	− 0.561582i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1036.93	0.260758	0.130379	−	0.991464i	$-0.458381\pi$
$251$	1036.93	0.260758	0.130379	+	0.991464i	$0.458381\pi$
$252$	0	0
$253$	2851.37	0.708554
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 929.028i	− 0.225491i	−0.993624	−	0.112745i	$-0.964036\pi$
$257$	− 929.028i	− 0.225491i	0.993624	−	0.112745i	$-0.0359645\pi$
$258$	0	0
$259$	1343.46i	0.322311i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1676.15	0.392987	0.196494	−	0.980505i	$-0.437045\pi$
$263$	1676.15	0.392987	0.196494	+	0.980505i	$0.437045\pi$
$264$	0	0
$265$	5968.52	1.38356
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 1717.55i	− 0.389297i	−0.980873	−	0.194649i	$-0.937643\pi$
$269$	− 1717.55i	− 0.389297i	0.980873	−	0.194649i	$-0.0623567\pi$
$270$	0	0
$271$	3930.25i	0.880981i	0.897757	+	0.440490i	$0.145195\pi$
$271$	3930.25i	0.880981i	−0.897757	+	0.440490i	$0.854805\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	969.402	0.212571
$276$	0	0
$277$	−4632.58	−1.00486	−0.502428	−	0.864619i	$-0.667560\pi$
$277$	−4632.58	−1.00486	−0.502428	+	0.864619i	$0.667560\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 4487.15i	− 0.952600i	−0.879283	−	0.476300i	$-0.841978\pi$
$281$	− 4487.15i	− 0.952600i	0.879283	−	0.476300i	$-0.158022\pi$
$282$	0	0
$283$	− 3633.83i	− 0.763281i	−0.924311	−	0.381641i	$-0.875359\pi$
$283$	− 3633.83i	− 0.763281i	0.924311	−	0.381641i	$-0.124641\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1722.08	0.354185
$288$	0	0
$289$	−1210.88	−0.246464
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3355.27i	0.669000i	0.942396	+	0.334500i	$0.108567\pi$
$293$	3355.27i	0.669000i	−0.942396	+	0.334500i	$0.891433\pi$
$294$	0	0
$295$	5174.71i	1.02130i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6579.20	−1.27253
$300$	0	0
$301$	−1406.72	−0.269375
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10305.4i	1.93470i
$306$	0	0
$307$	1849.50i	0.343832i	0.985112	+	0.171916i	$0.0549957\pi$
$307$	1849.50i	0.343832i	−0.985112	+	0.171916i	$0.945004\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1872.43	0.341401	0.170700	−	0.985323i	$-0.445397\pi$
$311$	1872.43	0.341401	0.170700	+	0.985323i	$0.445397\pi$
$312$	0	0
$313$	−114.858	−0.0207417	−0.0103709	−	0.999946i	$-0.503301\pi$
$313$	−114.858	−0.0207417	−0.0103709	+	0.999946i	$0.503301\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 7436.55i	− 1.31760i	−0.752320	−	0.658798i	$-0.771065\pi$
$317$	− 7436.55i	− 1.31760i	0.752320	−	0.658798i	$-0.228935\pi$
$318$	0	0
$319$	− 45.7523i	− 0.00803021i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5641.11	0.971764
$324$	0	0
$325$	−2236.78	−0.381767
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 2372.83i	− 0.397625i
$330$	0	0
$331$	− 5161.23i	− 0.857059i	−0.903528	−	0.428530i	$-0.859032\pi$
$331$	− 5161.23i	− 0.857059i	0.903528	−	0.428530i	$-0.140968\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	14448.1	2.35637
$336$	0	0
$337$	−5415.65	−0.875398	−0.437699	−	0.899122i	$-0.644206\pi$
$337$	−5415.65	−0.875398	−0.437699	+	0.899122i	$0.644206\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4000.35i	0.635281i
$342$	0	0
$343$	− 343.000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6577.28	−1.01754	−0.508770	−	0.860902i	$-0.669900\pi$
$347$	−6577.28	−1.01754	−0.508770	+	0.860902i	$0.669900\pi$
$348$	0	0
$349$	8587.72	1.31716	0.658582	−	0.752509i	$-0.271157\pi$
$349$	8587.72	1.31716	0.658582	+	0.752509i	$0.271157\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2518.85i	0.379788i	0.981805	+	0.189894i	$0.0608144\pi$
$353$	2518.85i	0.379788i	−0.981805	+	0.189894i	$0.939186\pi$
$354$	0	0
$355$	1119.81i	0.167418i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10688.2	1.57132	0.785659	−	0.618660i	$-0.212324\pi$
$359$	10688.2	1.57132	0.785659	+	0.618660i	$0.212324\pi$
$360$	0	0
$361$	1662.60	0.242397
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 1802.48i	− 0.258482i
$366$	0	0
$367$	451.872i	0.0642712i	0.999484	+	0.0321356i	$0.0102308\pi$
$367$	451.872i	0.0642712i	−0.999484	+	0.0321356i	$0.989769\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2961.95	−0.414493
$372$	0	0
$373$	8510.92	1.18144	0.590722	−	0.806875i	$-0.298843\pi$
$373$	8510.92	1.18144	0.590722	+	0.806875i	$0.298843\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	105.568i	0.0144218i
$378$	0	0
$379$	− 11356.2i	− 1.53912i	−0.638574	−	0.769561i	$-0.720475\pi$
$379$	− 11356.2i	− 1.53912i	0.638574	−	0.769561i	$-0.279525\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10796.9	1.44046	0.720229	−	0.693736i	$-0.244037\pi$
$383$	10796.9	1.44046	0.720229	+	0.693736i	$0.244037\pi$
$384$	0	0
$385$	−1294.11	−0.171309
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 4634.81i	− 0.604099i	−0.953292	−	0.302049i	$-0.902329\pi$
$389$	− 4634.81i	− 0.604099i	0.953292	−	0.302049i	$-0.0976708\pi$
$390$	0	0
$391$	− 17024.7i	− 2.20198i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5837.67	0.743608
$396$	0	0
$397$	−8193.89	−1.03587	−0.517934	−	0.855421i	$-0.673299\pi$
$397$	−8193.89	−1.03587	−0.517934	+	0.855421i	$0.673299\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 7505.04i	− 0.934623i	−0.884093	−	0.467311i	$-0.845223\pi$
$401$	− 7505.04i	− 0.934623i	0.884093	−	0.467311i	$-0.154777\pi$
$402$	0	0
$403$	− 9230.32i	− 1.14093i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2515.44	−0.306353
$408$	0	0
$409$	708.588	0.0856661	0.0428330	−	0.999082i	$-0.486362\pi$
$409$	708.588	0.0856661	0.0428330	+	0.999082i	$0.486362\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2568.02i	− 0.305966i
$414$	0	0
$415$	3225.84i	0.381566i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11975.7	−1.39631	−0.698153	−	0.715949i	$-0.745994\pi$
$419$	−11975.7	−1.39631	−0.698153	+	0.715949i	$0.745994\pi$
$420$	0	0
$421$	1813.85	0.209980	0.104990	−	0.994473i	$-0.466519\pi$
$421$	1813.85	0.209980	0.104990	+	0.994473i	$0.466519\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 5788.01i	− 0.660611i
$426$	0	0
$427$	− 5114.16i	− 0.579606i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9988.53	1.11631	0.558156	−	0.829736i	$-0.311509\pi$
$431$	9988.53	1.11631	0.558156	+	0.829736i	$0.311509\pi$
$432$	0	0
$433$	−1385.76	−0.153800	−0.0768998	−	0.997039i	$-0.524502\pi$
$433$	−1385.76	−0.153800	−0.0768998	+	0.997039i	$0.524502\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	15682.6i	1.71670i
$438$	0	0
$439$	− 15869.5i	− 1.72531i	−0.505795	−	0.862653i	$-0.668801\pi$
$439$	− 15869.5i	− 1.72531i	0.505795	−	0.862653i	$-0.331199\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10484.7	−1.12448	−0.562240	−	0.826974i	$-0.690060\pi$
$443$	−10484.7	−1.12448	−0.562240	+	0.826974i	$0.690060\pi$
$444$	0	0
$445$	2734.92	0.291343
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2234.73i	0.234885i	0.993080	+	0.117442i	$0.0374695\pi$
$449$	2234.73i	0.234885i	−0.993080	+	0.117442i	$0.962530\pi$
$450$	0	0
$451$	3224.36i	0.336650i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2986.01	0.307663
$456$	0	0
$457$	−17478.3	−1.78906	−0.894529	−	0.447011i	$-0.852489\pi$
$457$	−17478.3	−1.78906	−0.894529	+	0.447011i	$0.852489\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18276.9i	1.84651i	0.384190	+	0.923254i	$0.374481\pi$
$461$	18276.9i	1.84651i	−0.384190	+	0.923254i	$0.625519\pi$
$462$	0	0
$463$	− 15906.7i	− 1.59665i	−0.602230	−	0.798323i	$-0.705721\pi$
$463$	− 15906.7i	− 1.59665i	0.602230	−	0.798323i	$-0.294279\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8595.46	0.851714	0.425857	−	0.904790i	$-0.359973\pi$
$467$	8595.46	0.851714	0.425857	+	0.904790i	$0.359973\pi$
$468$	0	0
$469$	−7170.06	−0.705933
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 2633.88i	− 0.256038i
$474$	0	0
$475$	5331.72i	0.515023i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	19485.8	1.85872	0.929360	−	0.369174i	$-0.120359\pi$
$479$	19485.8	1.85872	0.929360	+	0.369174i	$0.120359\pi$
$480$	0	0
$481$	5804.08	0.550194
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 14163.6i	− 1.32606i
$486$	0	0
$487$	− 10181.7i	− 0.947386i	−0.880690	−	0.473693i	$-0.842921\pi$
$487$	− 10181.7i	− 0.947386i	0.880690	−	0.473693i	$-0.157079\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3098.15	−0.284760	−0.142380	−	0.989812i	$-0.545476\pi$
$491$	−3098.15	−0.284760	−0.142380	+	0.989812i	$0.545476\pi$
$492$	0	0
$493$	−273.173	−0.0249556
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 555.720i	− 0.0501558i
$498$	0	0
$499$	1116.79i	0.100189i	0.998744	+	0.0500944i	$0.0159522\pi$
$499$	1116.79i	0.100189i	−0.998744	+	0.0500944i	$0.984048\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2573.33	−0.228109	−0.114055	−	0.993474i	$-0.536384\pi$
$503$	−2573.33	−0.228109	−0.114055	+	0.993474i	$0.536384\pi$
$504$	0	0
$505$	−3083.99	−0.271754
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 16767.6i	− 1.46014i	−0.683373	−	0.730069i	$-0.739488\pi$
$509$	− 16767.6i	− 1.46014i	0.683373	−	0.730069i	$-0.260512\pi$
$510$	0	0
$511$	894.502i	0.0774373i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13621.2	1.16548
$516$	0	0
$517$	4442.80	0.377939
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 8773.54i	− 0.737766i	−0.929476	−	0.368883i	$-0.879740\pi$
$521$	− 8773.54i	− 0.737766i	0.929476	−	0.368883i	$-0.120260\pi$
$522$	0	0
$523$	8809.57i	0.736550i	0.929717	+	0.368275i	$0.120051\pi$
$523$	8809.57i	0.736550i	−0.929717	+	0.368275i	$0.879949\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	23884.9	1.97427
$528$	0	0
$529$	35162.5	2.88999
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 7439.82i	− 0.604605i
$534$	0	0
$535$	− 4700.84i	− 0.379879i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	642.220	0.0513217
$540$	0	0
$541$	−976.663	−0.0776156	−0.0388078	−	0.999247i	$-0.512356\pi$
$541$	−976.663	−0.0776156	−0.0388078	+	0.999247i	$0.512356\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	26680.4i	2.09700i
$546$	0	0
$547$	19767.5i	1.54515i	0.634922	+	0.772576i	$0.281032\pi$
$547$	19767.5i	1.54515i	−0.634922	+	0.772576i	$0.718968\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	251.638	0.0194558
$552$	0	0
$553$	−2897.02	−0.222773
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17079.5i	1.29925i	0.760254	+	0.649626i	$0.225075\pi$
$557$	17079.5i	1.29925i	−0.760254	+	0.649626i	$0.774925\pi$
$558$	0	0
$559$	6077.37i	0.459831i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7784.65	−0.582742	−0.291371	−	0.956610i	$-0.594111\pi$
$563$	−7784.65	−0.582742	−0.291371	+	0.956610i	$0.594111\pi$
$564$	0	0
$565$	3601.64	0.268181
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8845.46i	0.651706i	0.945420	+	0.325853i	$0.105651\pi$
$569$	8845.46i	0.651706i	−0.945420	+	0.325853i	$0.894349\pi$
$570$	0	0
$571$	− 1.50330i	0 0.000110177i	−1.00000		5.50886e-5i	$-0.999982\pi$
$571$	− 1.50330i	0 0.000110177i	1.00000		5.50886e-5i	$-1.75352e-5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	16091.0	1.16703
$576$	0	0
$577$	2938.69	0.212026	0.106013	−	0.994365i	$-0.466191\pi$
$577$	2938.69	0.212026	0.106013	+	0.994365i	$0.466191\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1600.86i	− 0.114311i
$582$	0	0
$583$	− 5545.85i	− 0.393972i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19938.8	1.40198	0.700989	−	0.713173i	$-0.252742\pi$
$587$	19938.8	1.40198	0.700989	+	0.713173i	$0.252742\pi$
$588$	0	0
$589$	−22001.9	−1.53917
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 19762.6i	− 1.36855i	−0.729222	−	0.684277i	$-0.760118\pi$
$593$	− 19762.6i	− 1.36855i	0.729222	−	0.684277i	$-0.239882\pi$
$594$	0	0
$595$	7726.77i	0.532381i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10471.7	0.714293	0.357147	−	0.934048i	$-0.383750\pi$
$599$	10471.7	0.714293	0.357147	+	0.934048i	$0.383750\pi$
$600$	0	0
$601$	−14290.6	−0.969924	−0.484962	−	0.874535i	$-0.661167\pi$
$601$	−14290.6	−0.969924	−0.484962	+	0.874535i	$0.661167\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	16351.3i	1.09880i
$606$	0	0
$607$	− 26516.3i	− 1.77308i	−0.462648	−	0.886542i	$-0.653101\pi$
$607$	− 26516.3i	− 1.77308i	0.462648	−	0.886542i	$-0.346899\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10251.2	−0.678757
$612$	0	0
$613$	25135.2	1.65612	0.828058	−	0.560642i	$-0.189446\pi$
$613$	25135.2	1.65612	0.828058	+	0.560642i	$0.189446\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4354.81i	0.284146i	0.989856	+	0.142073i	$0.0453768\pi$
$617$	4354.81i	0.284146i	−0.989856	+	0.142073i	$0.954623\pi$
$618$	0	0
$619$	91.2299i	0.00592381i	0.999996	+	0.00296190i	$0.000942805\pi$
$619$	91.2299i	0.00592381i	−0.999996	+	0.00296190i	$0.999057\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1357.24	−0.0872819
$624$	0	0
$625$	−19399.8	−1.24159
$626$	0	0
$627$	0	0
$628$	0	0
$629$	15019.0i	0.952059i
$630$	0	0
$631$	1993.10i	0.125743i	0.998022	+	0.0628716i	$0.0200259\pi$
$631$	1993.10i	0.125743i	−0.998022	+	0.0628716i	$0.979974\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1159.66	−0.0724720
$636$	0	0
$637$	−1481.85	−0.0921710
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 27606.0i	− 1.70105i	−0.525937	−	0.850523i	$-0.676285\pi$
$641$	− 27606.0i	− 1.70105i	0.525937	−	0.850523i	$-0.323715\pi$
$642$	0	0
$643$	3174.43i	0.194693i	0.995251	+	0.0973464i	$0.0310355\pi$
$643$	3174.43i	0.194693i	−0.995251	+	0.0973464i	$0.968965\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23316.4	−1.41679	−0.708395	−	0.705816i	$-0.750580\pi$
$647$	−23316.4	−1.41679	−0.708395	+	0.705816i	$0.750580\pi$
$648$	0	0
$649$	4808.26	0.290818
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 28631.7i	− 1.71584i	−0.513782	−	0.857921i	$-0.671756\pi$
$653$	− 28631.7i	− 1.71584i	0.513782	−	0.857921i	$-0.328244\pi$
$654$	0	0
$655$	27156.4i	1.61998i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−538.740	−0.0318457	−0.0159229	−	0.999873i	$-0.505069\pi$
$659$	−538.740	−0.0318457	−0.0159229	+	0.999873i	$0.505069\pi$
$660$	0	0
$661$	−15272.0	−0.898656	−0.449328	−	0.893367i	$-0.648336\pi$
$661$	−15272.0	−0.898656	−0.449328	+	0.893367i	$0.648336\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 7117.64i	− 0.415053i
$666$	0	0
$667$	− 759.435i	− 0.0440861i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9575.57	0.550910
$672$	0	0
$673$	−23556.7	−1.34925	−0.674625	−	0.738161i	$-0.735695\pi$
$673$	−23556.7	−1.34925	−0.674625	+	0.738161i	$0.735695\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 7988.75i	− 0.453519i	−0.973951	−	0.226760i	$-0.927187\pi$
$677$	− 7988.75i	− 0.453519i	0.973951	−	0.226760i	$-0.0728132\pi$
$678$	0	0
$679$	7028.87i	0.397266i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18508.0	−1.03688	−0.518439	−	0.855115i	$-0.673487\pi$
$683$	−18508.0	−1.03688	−0.518439	+	0.855115i	$0.673487\pi$
$684$	0	0
$685$	29086.1	1.62237
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12796.4i	0.707552i
$690$	0	0
$691$	24965.2i	1.37441i	0.726462	+	0.687207i	$0.241163\pi$
$691$	24965.2i	1.37441i	−0.726462	+	0.687207i	$0.758837\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5131.81	0.280087
$696$	0	0
$697$	19251.7	1.04621
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 19893.7i	− 1.07186i	−0.844262	−	0.535931i	$-0.819961\pi$
$701$	− 19893.7i	− 1.07186i	0.844262	−	0.535931i	$-0.180039\pi$
$702$	0	0
$703$	− 13835.0i	− 0.742241i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1530.47	0.0814132
$708$	0	0
$709$	10872.8	0.575931	0.287966	−	0.957641i	$-0.407021\pi$
$709$	10872.8	0.575931	0.287966	+	0.957641i	$0.407021\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	66401.1i	3.48772i
$714$	0	0
$715$	5590.90i	0.292430i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32431.5	−1.68218	−0.841092	−	0.540892i	$-0.818087\pi$
$719$	−32431.5	−1.68218	−0.841092	+	0.540892i	$0.818087\pi$
$720$	0	0
$721$	−6759.69	−0.349160
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 258.191i	− 0.0132262i
$726$	0	0
$727$	− 20606.8i	− 1.05126i	−0.850714	−	0.525629i	$-0.823830\pi$
$727$	− 20606.8i	− 1.05126i	0.850714	−	0.525629i	$-0.176170\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−15726.1	−0.795693
$732$	0	0
$733$	−2770.73	−0.139617	−0.0698085	−	0.997560i	$-0.522239\pi$
$733$	−2770.73	−0.139617	−0.0698085	+	0.997560i	$0.522239\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 13425.0i	− 0.670983i
$738$	0	0
$739$	19617.9i	0.976532i	0.872695	+	0.488266i	$0.162370\pi$
$739$	19617.9i	0.976532i	−0.872695	+	0.488266i	$0.837630\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2537.18	−0.125276	−0.0626381	−	0.998036i	$-0.519951\pi$
$743$	−2537.18	−0.125276	−0.0626381	+	0.998036i	$0.519951\pi$
$744$	0	0
$745$	13560.8	0.666884
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2332.85i	0.113806i
$750$	0	0
$751$	− 28280.3i	− 1.37412i	−0.726602	−	0.687059i	$-0.758901\pi$
$751$	− 28280.3i	− 1.37412i	0.726602	−	0.687059i	$-0.241099\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−29344.1	−1.41449
$756$	0	0
$757$	−30129.0	−1.44658	−0.723288	−	0.690546i	$-0.757370\pi$
$757$	−30129.0	−1.44658	−0.723288	+	0.690546i	$0.757370\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 11243.4i	− 0.535573i	−0.963478	−	0.267787i	$-0.913708\pi$
$761$	− 11243.4i	− 0.535573i	0.963478	−	0.267787i	$-0.0862922\pi$
$762$	0	0
$763$	− 13240.5i	− 0.628229i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11094.5	−0.522293
$768$	0	0
$769$	11539.8	0.541138	0.270569	−	0.962701i	$-0.412788\pi$
$769$	11539.8	0.541138	0.270569	+	0.962701i	$0.412788\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	23116.8i	1.07562i	0.843066	+	0.537810i	$0.180748\pi$
$773$	23116.8i	1.07562i	−0.843066	+	0.537810i	$0.819252\pi$
$774$	0	0
$775$	22574.9i	1.04634i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−17734.0	−0.815643
$780$	0	0
$781$	1040.51	0.0476726
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 44615.8i	− 2.02854i
$786$	0	0
$787$	− 1401.68i	− 0.0634875i	−0.999496	−	0.0317438i	$-0.989894\pi$
$787$	− 1401.68i	− 0.0634875i	0.999496	−	0.0317438i	$-0.0101060\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1787.36	−0.0803429
$792$	0	0
$793$	−22094.5	−0.989405
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6424.11i	0.285513i	0.989758	+	0.142756i	$0.0455965\pi$
$797$	6424.11i	0.285513i	−0.989758	+	0.142756i	$0.954403\pi$
$798$	0	0
$799$	− 26526.7i	− 1.17453i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1674.83	−0.0736034
$804$	0	0
$805$	−21480.8	−0.940495
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23279.8i	1.01171i	0.862618	+	0.505856i	$0.168823\pi$
$809$	23279.8i	1.01171i	−0.862618	+	0.505856i	$0.831177\pi$
$810$	0	0
$811$	24122.3i	1.04445i	0.852808	+	0.522225i	$0.174898\pi$
$811$	24122.3i	1.04445i	−0.852808	+	0.522225i	$0.825102\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	32800.3	1.40975
$816$	0	0
$817$	14486.4	0.620336
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 25808.0i	− 1.09708i	−0.836123	−	0.548542i	$-0.815183\pi$
$821$	− 25808.0i	− 1.09708i	0.836123	−	0.548542i	$-0.184817\pi$
$822$	0	0
$823$	− 45232.8i	− 1.91582i	−0.287073	−	0.957909i	$-0.592682\pi$
$823$	− 45232.8i	− 1.91582i	0.287073	−	0.957909i	$-0.407318\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−42035.1	−1.76748	−0.883740	−	0.467979i	$-0.844982\pi$
$827$	−42035.1	−1.76748	−0.883740	+	0.467979i	$0.844982\pi$
$828$	0	0
$829$	−24480.4	−1.02562	−0.512810	−	0.858502i	$-0.671395\pi$
$829$	−24480.4	−1.02562	−0.512810	+	0.858502i	$0.671395\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3834.51i	− 0.159493i
$834$	0	0
$835$	56442.6i	2.33925i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26751.5	−1.10079	−0.550396	−	0.834904i	$-0.685523\pi$
$839$	−26751.5	−1.10079	−0.550396	+	0.834904i	$0.685523\pi$
$840$	0	0
$841$	24376.8	0.999500
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18089.3i	0.736439i
$846$	0	0
$847$	− 8114.53i	− 0.329184i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−41753.4	−1.68189
$852$	0	0
$853$	26977.8	1.08289	0.541444	−	0.840737i	$-0.317878\pi$
$853$	26977.8	1.08289	0.541444	+	0.840737i	$0.317878\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 25583.7i	− 1.01975i	−0.860250	−	0.509873i	$-0.829692\pi$
$857$	− 25583.7i	− 1.01975i	0.860250	−	0.509873i	$-0.170308\pi$
$858$	0	0
$859$	− 31259.1i	− 1.24161i	−0.783963	−	0.620807i	$-0.786805\pi$
$859$	− 31259.1i	− 1.24161i	0.783963	−	0.620807i	$-0.213195\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45428.8	−1.79190	−0.895952	−	0.444150i	$-0.853506\pi$
$863$	−45428.8	−1.79190	−0.895952	+	0.444150i	$0.853506\pi$
$864$	0	0
$865$	18114.7	0.712043
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 5424.27i	− 0.211744i
$870$	0	0
$871$	30976.5i	1.20505i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5039.27	0.194695
$876$	0	0
$877$	32177.7	1.23895	0.619477	−	0.785015i	$-0.287345\pi$
$877$	32177.7	1.23895	0.619477	+	0.785015i	$0.287345\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	43816.2i	1.67560i	0.545977	+	0.837800i	$0.316159\pi$
$881$	43816.2i	1.67560i	−0.545977	+	0.837800i	$0.683841\pi$
$882$	0	0
$883$	30769.3i	1.17267i	0.810068	+	0.586336i	$0.199430\pi$
$883$	30769.3i	1.17267i	−0.810068	+	0.586336i	$0.800570\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20431.3	0.773411	0.386705	−	0.922203i	$-0.373613\pi$
$887$	20431.3	0.773411	0.386705	+	0.922203i	$0.373613\pi$
$888$	0	0
$889$	575.496	0.0217115
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24435.5i	0.915679i
$894$	0	0
$895$	9097.59i	0.339775i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1065.45	0.0395271
$900$	0	0
$901$	−33112.6	−1.22435
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 33494.8i	− 1.23028i
$906$	0	0
$907$	18439.6i	0.675057i	0.941315	+	0.337529i	$0.109591\pi$
$907$	18439.6i	0.675057i	−0.941315	+	0.337529i	$0.890409\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	11953.8	0.434737	0.217369	−	0.976090i	$-0.430253\pi$
$911$	11953.8	0.434737	0.217369	+	0.976090i	$0.430253\pi$
$912$	0	0
$913$	2997.39	0.108652
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 13476.7i	− 0.485322i
$918$	0	0
$919$	44910.6i	1.61204i	0.591889	+	0.806019i	$0.298382\pi$
$919$	44910.6i	1.61204i	−0.591889	+	0.806019i	$0.701618\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2400.85	−0.0856174
$924$	0	0
$925$	−14195.2	−0.504580
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 5294.66i	− 0.186988i	−0.995620	−	0.0934942i	$-0.970196\pi$
$929$	− 5294.66i	− 0.186988i	0.995620	−	0.0934942i	$-0.0298037\pi$
$930$	0	0
$931$	3532.22i	0.124343i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−14467.3	−0.506023
$936$	0	0
$937$	52165.8	1.81877	0.909383	−	0.415960i	$-0.136554\pi$
$937$	52165.8	1.81877	0.909383	+	0.415960i	$0.136554\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	29115.8i	1.00866i	0.863511	+	0.504330i	$0.168260\pi$
$941$	29115.8i	1.00866i	−0.863511	+	0.504330i	$0.831740\pi$
$942$	0	0
$943$	53520.6i	1.84822i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14121.4	0.484567	0.242284	−	0.970205i	$-0.422104\pi$
$947$	14121.4	0.484567	0.242284	+	0.970205i	$0.422104\pi$
$948$	0	0
$949$	3864.48	0.132188
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 5587.57i	− 0.189926i	−0.995481	−	0.0949629i	$-0.969727\pi$
$953$	− 5587.57i	− 0.189926i	0.995481	−	0.0949629i	$-0.0302732\pi$
$954$	0	0
$955$	62470.0i	2.11673i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−14434.4	−0.486037
$960$	0	0
$961$	−63366.8	−2.12705
$962$	0	0
$963$	0	0
$964$	0	0
$965$	43928.2i	1.46539i
$966$	0	0
$967$	10370.3i	0.344867i	0.985021	+	0.172434i	$0.0551630\pi$
$967$	10370.3i	0.344867i	−0.985021	+	0.172434i	$0.944837\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12244.2	−0.404671	−0.202335	−	0.979316i	$-0.564853\pi$
$971$	−12244.2	−0.404671	−0.202335	+	0.979316i	$0.564853\pi$
$972$	0	0
$973$	−2546.73	−0.0839098
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34539.2i	1.13102i	0.824742	+	0.565510i	$0.191320\pi$
$977$	34539.2i	1.13102i	−0.824742	+	0.565510i	$0.808680\pi$
$978$	0	0
$979$	− 2541.24i	− 0.0829606i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17119.3	−0.555465	−0.277732	−	0.960659i	$-0.589583\pi$
$983$	−17119.3	−0.555465	−0.277732	+	0.960659i	$0.589583\pi$
$984$	0	0
$985$	−54909.9	−1.77622
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 43719.4i	− 1.40566i
$990$	0	0
$991$	4303.90i	0.137959i	0.997618	+	0.0689797i	$0.0219744\pi$
$991$	4303.90i	0.137959i	−0.997618	+	0.0689797i	$0.978026\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−25915.7	−0.825713
$996$	0	0
$997$	10385.4	0.329899	0.164950	−	0.986302i	$-0.447254\pi$
$997$	10385.4	0.329899	0.164950	+	0.986302i	$0.447254\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.h.a.575.2	yes	12
3.2	odd	2	inner	1008.4.h.a.575.12	yes	12
4.3	odd	2	inner	1008.4.h.a.575.1	✓	12
12.11	even	2	inner	1008.4.h.a.575.11	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.4.h.a.575.1	✓	12	4.3	odd	2	inner
1008.4.h.a.575.2	yes	12	1.1	even	1	trivial
1008.4.h.a.575.11	yes	12	12.11	even	2	inner
1008.4.h.a.575.12	yes	12	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-14.1054i q^{5} +7.00000i q^{7} +O(q^{10})\)	\(q-14.1054i q^{5} +7.00000i q^{7} -13.1065 q^{11} +30.2418 q^{13} +78.2552i q^{17} -72.0861i q^{19} -217.553 q^{23} -73.9633 q^{25} +3.49080i q^{29} -305.218i q^{31} +98.7380 q^{35} +191.923 q^{37} -246.011i q^{41} +200.959i q^{43} -338.976 q^{47} -49.0000 q^{49} +423.136i q^{53} +184.873i q^{55} -366.860 q^{59} -730.595 q^{61} -426.573i q^{65} +1024.29i q^{67} -79.3885 q^{71} +127.786 q^{73} -91.7458i q^{77} +413.860i q^{79} -228.695 q^{83} +1103.82 q^{85} +193.891i q^{89} +211.692i q^{91} -1016.81 q^{95} +1004.12 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 192 q^{13} + 84 q^{25} + 72 q^{37} - 588 q^{49} - 1800 q^{61} + 3144 q^{85} - 1152 q^{97}+O(q^{100}) \) 12 * q + 192 * q^13 + 84 * q^25 + 72 * q^37 - 588 * q^49 - 1800 * q^61 + 3144 * q^85 - 1152 * q^97

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