LMFDB - Embedded newform 1008.4.h.b.575.20

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:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.20
Character	$\chi$	$=$	1008.575
Dual form			1008.4.h.b.575.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.3621i q^{5} -7.00000i q^{7} +O(q^{10})$	$q+10.3621i q^{5} -7.00000i q^{7} -38.1254 q^{11} +73.5614 q^{13} -33.1123i q^{17} +65.0032i q^{19} +3.43204 q^{23} +17.6260 q^{25} -133.962i q^{29} +46.7231i q^{31} +72.5350 q^{35} +69.9543 q^{37} -18.1505i q^{41} +311.903i q^{43} +337.282 q^{47} -49.0000 q^{49} +507.417i q^{53} -395.061i q^{55} -426.674 q^{59} +787.915 q^{61} +762.254i q^{65} +596.439i q^{67} -1128.93 q^{71} -315.280 q^{73} +266.878i q^{77} -514.130i q^{79} +184.445 q^{83} +343.115 q^{85} +884.539i q^{89} -514.930i q^{91} -673.572 q^{95} +945.059 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 192 q^{13} - 984 q^{25} + 720 q^{37} - 1176 q^{49} + 2736 q^{61} + 3408 q^{85} + 1152 q^{97}+O(q^{100}) $ 24 * q - 192 * q^13 - 984 * q^25 + 720 * q^37 - 1176 * q^49 + 2736 * q^61 + 3408 * 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
$5$	10.3621i	0.926818i	0.886144	+	0.463409i	$0.153374\pi$
$5$	10.3621i	0.926818i	−0.886144	+	0.463409i	$0.846626\pi$
$6$	0	0
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−38.1254	−1.04502	−0.522511	−	0.852633i	$-0.675005\pi$
$11$	−38.1254	−1.04502	−0.522511	+	0.852633i	$0.675005\pi$
$12$	0	0
$13$	73.5614	1.56940	0.784702	−	0.619873i	$-0.212816\pi$
$13$	73.5614	1.56940	0.784702	+	0.619873i	$0.212816\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 33.1123i	− 0.472407i	−0.971704	−	0.236203i	$-0.924097\pi$
$17$	− 33.1123i	− 0.472407i	0.971704	−	0.236203i	$-0.0759032\pi$
$18$	0	0
$19$	65.0032i	0.784882i	0.919777	+	0.392441i	$0.128369\pi$
$19$	65.0032i	0.784882i	−0.919777	+	0.392441i	$0.871631\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.43204	0.0311144	0.0155572	−	0.999879i	$-0.495048\pi$
$23$	3.43204	0.0311144	0.0155572	+	0.999879i	$0.495048\pi$
$24$	0	0
$25$	17.6260	0.141008
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 133.962i	− 0.857795i	−0.903353	−	0.428897i	$-0.858902\pi$
$29$	− 133.962i	− 0.857795i	0.903353	−	0.428897i	$-0.141098\pi$
$30$	0	0
$31$	46.7231i	0.270701i	0.990798	+	0.135350i	$0.0432160\pi$
$31$	46.7231i	0.270701i	−0.990798	+	0.135350i	$0.956784\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	72.5350	0.350304
$36$	0	0
$37$	69.9543	0.310822	0.155411	−	0.987850i	$-0.450330\pi$
$37$	69.9543	0.310822	0.155411	+	0.987850i	$0.450330\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 18.1505i	− 0.0691374i	−0.999402	−	0.0345687i	$-0.988994\pi$
$41$	− 18.1505i	− 0.0691374i	0.999402	−	0.0345687i	$-0.0110058\pi$
$42$	0	0
$43$	311.903i	1.10616i	0.833129	+	0.553079i	$0.186547\pi$
$43$	311.903i	1.10616i	−0.833129	+	0.553079i	$0.813453\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	337.282	1.04676	0.523379	−	0.852100i	$-0.324671\pi$
$47$	337.282	1.04676	0.523379	+	0.852100i	$0.324671\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	507.417i	1.31508i	0.753421	+	0.657539i	$0.228402\pi$
$53$	507.417i	1.31508i	−0.753421	+	0.657539i	$0.771598\pi$
$54$	0	0
$55$	− 395.061i	− 0.968545i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−426.674	−0.941496	−0.470748	−	0.882268i	$-0.656016\pi$
$59$	−426.674	−0.941496	−0.470748	+	0.882268i	$0.656016\pi$
$60$	0	0
$61$	787.915	1.65381	0.826903	−	0.562344i	$-0.190100\pi$
$61$	787.915	1.65381	0.826903	+	0.562344i	$0.190100\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	762.254i	1.45455i
$66$	0	0
$67$	596.439i	1.08756i	0.839227	+	0.543781i	$0.183008\pi$
$67$	596.439i	1.08756i	−0.839227	+	0.543781i	$0.816992\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1128.93	−1.88703	−0.943515	−	0.331330i	$-0.892503\pi$
$71$	−1128.93	−1.88703	−0.943515	+	0.331330i	$0.892503\pi$
$72$	0	0
$73$	−315.280	−0.505490	−0.252745	−	0.967533i	$-0.581333\pi$
$73$	−315.280	−0.505490	−0.252745	+	0.967533i	$0.581333\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	266.878i	0.394981i
$78$	0	0
$79$	− 514.130i	− 0.732205i	−0.930575	−	0.366102i	$-0.880692\pi$
$79$	− 514.130i	− 0.732205i	0.930575	−	0.366102i	$-0.119308\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	184.445	0.243922	0.121961	−	0.992535i	$-0.461082\pi$
$83$	184.445	0.243922	0.121961	+	0.992535i	$0.461082\pi$
$84$	0	0
$85$	343.115	0.437835
$86$	0	0
$87$	0	0
$88$	0	0
$89$	884.539i	1.05349i	0.850022	+	0.526747i	$0.176588\pi$
$89$	884.539i	1.05349i	−0.850022	+	0.526747i	$0.823412\pi$
$90$	0	0
$91$	− 514.930i	− 0.593179i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−673.572	−0.727443
$96$	0	0
$97$	945.059	0.989239	0.494620	−	0.869110i	$-0.335307\pi$
$97$	945.059	0.989239	0.494620	+	0.869110i	$0.335307\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	721.171i	0.710488i	0.934774	+	0.355244i	$0.115602\pi$
$101$	721.171i	0.710488i	−0.934774	+	0.355244i	$0.884398\pi$
$102$	0	0
$103$	851.122i	0.814209i	0.913382	+	0.407105i	$0.133462\pi$
$103$	851.122i	0.814209i	−0.913382	+	0.407105i	$0.866538\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−334.313	−0.302049	−0.151025	−	0.988530i	$-0.548257\pi$
$107$	−334.313	−0.302049	−0.151025	+	0.988530i	$0.548257\pi$
$108$	0	0
$109$	−1363.31	−1.19800	−0.598998	−	0.800751i	$-0.704434\pi$
$109$	−1363.31	−1.19800	−0.598998	+	0.800751i	$0.704434\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 430.386i	− 0.358295i	−0.983822	−	0.179147i	$-0.942666\pi$
$113$	− 430.386i	− 0.358295i	0.983822	−	0.179147i	$-0.0573339\pi$
$114$	0	0
$115$	35.5633i	0.0288374i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−231.786	−0.178553
$120$	0	0
$121$	122.547	0.0920711
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1477.91i	1.05751i
$126$	0	0
$127$	748.195i	0.522768i	0.965235	+	0.261384i	$0.0841789\pi$
$127$	748.195i	0.522768i	−0.965235	+	0.261384i	$0.915821\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1134.80	0.756854	0.378427	−	0.925631i	$-0.376465\pi$
$131$	1134.80	0.756854	0.378427	+	0.925631i	$0.376465\pi$
$132$	0	0
$133$	455.022	0.296658
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2270.35i	1.41583i	0.706296	+	0.707916i	$0.250365\pi$
$137$	2270.35i	1.41583i	−0.706296	+	0.707916i	$0.749635\pi$
$138$	0	0
$139$	873.032i	0.532731i	0.963872	+	0.266366i	$0.0858229\pi$
$139$	873.032i	0.532731i	−0.963872	+	0.266366i	$0.914177\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2804.56	−1.64006
$144$	0	0
$145$	1388.13	0.795020
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1603.29i	0.881523i	0.897624	+	0.440762i	$0.145292\pi$
$149$	1603.29i	0.881523i	−0.897624	+	0.440762i	$0.854708\pi$
$150$	0	0
$151$	2265.91i	1.22117i	0.791950	+	0.610586i	$0.209066\pi$
$151$	2265.91i	1.22117i	−0.791950	+	0.610586i	$0.790934\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−484.152	−0.250890
$156$	0	0
$157$	3057.94	1.55446	0.777229	−	0.629218i	$-0.216625\pi$
$157$	3057.94	1.55446	0.777229	+	0.629218i	$0.216625\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 24.0243i	− 0.0117601i
$162$	0	0
$163$	1774.51i	0.852703i	0.904558	+	0.426352i	$0.140201\pi$
$163$	1774.51i	0.852703i	−0.904558	+	0.426352i	$0.859799\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4031.99	−1.86829	−0.934146	−	0.356891i	$-0.883837\pi$
$167$	−4031.99	−1.86829	−0.934146	+	0.356891i	$0.883837\pi$
$168$	0	0
$169$	3214.28	1.46303
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 368.363i	− 0.161885i	−0.996719	−	0.0809426i	$-0.974207\pi$
$173$	− 368.363i	− 0.161885i	0.996719	−	0.0809426i	$-0.0257930\pi$
$174$	0	0
$175$	− 123.382i	− 0.0532961i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2967.94	1.23930	0.619648	−	0.784880i	$-0.287275\pi$
$179$	2967.94	1.23930	0.619648	+	0.784880i	$0.287275\pi$
$180$	0	0
$181$	2690.23	1.10477	0.552385	−	0.833589i	$-0.313718\pi$
$181$	2690.23	1.10477	0.552385	+	0.833589i	$0.313718\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	724.876i	0.288075i
$186$	0	0
$187$	1262.42i	0.493676i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1422.21	0.538783	0.269392	−	0.963031i	$-0.413177\pi$
$191$	1422.21	0.538783	0.269392	+	0.963031i	$0.413177\pi$
$192$	0	0
$193$	1211.88	0.451985	0.225993	−	0.974129i	$-0.427437\pi$
$193$	1211.88	0.451985	0.225993	+	0.974129i	$0.427437\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5286.06i	1.91176i	0.293763	+	0.955878i	$0.405092\pi$
$197$	5286.06i	1.91176i	−0.293763	+	0.955878i	$0.594908\pi$
$198$	0	0
$199$	− 1680.38i	− 0.598587i	−0.954161	−	0.299294i	$-0.903249\pi$
$199$	− 1680.38i	− 0.598587i	0.954161	−	0.299294i	$-0.0967510\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−937.731	−0.324216
$204$	0	0
$205$	188.078	0.0640778
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 2478.27i	− 0.820219i
$210$	0	0
$211$	− 4310.20i	− 1.40629i	−0.711048	−	0.703143i	$-0.751779\pi$
$211$	− 4310.20i	− 1.40629i	0.711048	−	0.703143i	$-0.248221\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3231.98	−1.02521
$216$	0	0
$217$	327.062	0.102315
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2435.79i	− 0.741398i
$222$	0	0
$223$	− 1684.04i	− 0.505704i	−0.967505	−	0.252852i	$-0.918631\pi$
$223$	− 1684.04i	− 0.505704i	0.967505	−	0.252852i	$-0.0813685\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2078.98	0.607872	0.303936	−	0.952692i	$-0.401699\pi$
$227$	2078.98	0.607872	0.303936	+	0.952692i	$0.401699\pi$
$228$	0	0
$229$	−3807.47	−1.09871	−0.549355	−	0.835589i	$-0.685126\pi$
$229$	−3807.47	−1.09871	−0.549355	+	0.835589i	$0.685126\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1124.01i	0.316037i	0.987436	+	0.158018i	$0.0505105\pi$
$233$	1124.01i	0.316037i	−0.987436	+	0.158018i	$0.949489\pi$
$234$	0	0
$235$	3494.96i	0.970154i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4607.08	1.24689	0.623446	−	0.781867i	$-0.285732\pi$
$239$	4607.08	1.24689	0.623446	+	0.781867i	$0.285732\pi$
$240$	0	0
$241$	1037.01	0.277177	0.138589	−	0.990350i	$-0.455743\pi$
$241$	1037.01	0.277177	0.138589	+	0.990350i	$0.455743\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 507.745i	− 0.132403i
$246$	0	0
$247$	4781.73i	1.23180i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5283.50	−1.32865	−0.664326	−	0.747443i	$-0.731282\pi$
$251$	−5283.50	−1.32865	−0.664326	+	0.747443i	$0.731282\pi$
$252$	0	0
$253$	−130.848	−0.0325152
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5839.70i	1.41739i	0.705513	+	0.708697i	$0.250717\pi$
$257$	5839.70i	1.41739i	−0.705513	+	0.708697i	$0.749283\pi$
$258$	0	0
$259$	− 489.680i	− 0.117480i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−641.249	−0.150346	−0.0751732	−	0.997170i	$-0.523951\pi$
$263$	−641.249	−0.150346	−0.0751732	+	0.997170i	$0.523951\pi$
$264$	0	0
$265$	−5257.93	−1.21884
$266$	0	0
$267$	0	0
$268$	0	0
$269$	267.522i	0.0606361i	0.999540	+	0.0303181i	$0.00965202\pi$
$269$	267.522i	0.0606361i	−0.999540	+	0.0303181i	$0.990348\pi$
$270$	0	0
$271$	3465.85i	0.776884i	0.921473	+	0.388442i	$0.126987\pi$
$271$	3465.85i	0.776884i	−0.921473	+	0.388442i	$0.873013\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−671.999	−0.147357
$276$	0	0
$277$	−2470.65	−0.535910	−0.267955	−	0.963431i	$-0.586348\pi$
$277$	−2470.65	−0.535910	−0.267955	+	0.963431i	$0.586348\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	343.043i	0.0728263i	0.999337	+	0.0364132i	$0.0115932\pi$
$281$	343.043i	0.0728263i	−0.999337	+	0.0364132i	$0.988407\pi$
$282$	0	0
$283$	− 4492.77i	− 0.943702i	−0.881678	−	0.471851i	$-0.843586\pi$
$283$	− 4492.77i	− 0.943702i	0.881678	−	0.471851i	$-0.156414\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−127.054	−0.0261315
$288$	0	0
$289$	3816.57	0.776832
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2293.43i	− 0.457282i	−0.973511	−	0.228641i	$-0.926572\pi$
$293$	− 2293.43i	− 0.457282i	0.973511	−	0.228641i	$-0.0734282\pi$
$294$	0	0
$295$	− 4421.26i	− 0.872595i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	252.466	0.0488310
$300$	0	0
$301$	2183.32	0.418088
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8164.49i	1.53278i
$306$	0	0
$307$	4077.28i	0.757989i	0.925399	+	0.378995i	$0.123730\pi$
$307$	4077.28i	0.757989i	−0.925399	+	0.378995i	$0.876270\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2345.29	−0.427619	−0.213809	−	0.976875i	$-0.568587\pi$
$311$	−2345.29	−0.427619	−0.213809	+	0.976875i	$0.568587\pi$
$312$	0	0
$313$	−7298.20	−1.31795	−0.658976	−	0.752164i	$-0.729010\pi$
$313$	−7298.20	−1.31795	−0.658976	+	0.752164i	$0.729010\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2967.24i	− 0.525732i	−0.964832	−	0.262866i	$-0.915332\pi$
$317$	− 2967.24i	− 0.525732i	0.964832	−	0.262866i	$-0.0846677\pi$
$318$	0	0
$319$	5107.34i	0.896414i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2152.41	0.370784
$324$	0	0
$325$	1296.59	0.221299
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 2360.97i	− 0.395637i
$330$	0	0
$331$	− 11705.2i	− 1.94373i	−0.235537	−	0.971865i	$-0.575685\pi$
$331$	− 11705.2i	− 1.94373i	0.235537	−	0.971865i	$-0.424315\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6180.39	−1.00797
$336$	0	0
$337$	−3065.44	−0.495505	−0.247753	−	0.968823i	$-0.579692\pi$
$337$	−3065.44	−0.495505	−0.247753	+	0.968823i	$0.579692\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1781.34i	− 0.282888i
$342$	0	0
$343$	343.000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3546.89	0.548724	0.274362	−	0.961627i	$-0.411533\pi$
$347$	3546.89	0.548724	0.274362	+	0.961627i	$0.411533\pi$
$348$	0	0
$349$	−3078.90	−0.472235	−0.236117	−	0.971725i	$-0.575875\pi$
$349$	−3078.90	−0.472235	−0.236117	+	0.971725i	$0.575875\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 9776.28i	− 1.47405i	−0.675867	−	0.737024i	$-0.736230\pi$
$353$	− 9776.28i	− 1.47405i	0.675867	−	0.737024i	$-0.263770\pi$
$354$	0	0
$355$	− 11698.1i	− 1.74893i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5788.46	0.850984	0.425492	−	0.904962i	$-0.360101\pi$
$359$	5788.46	0.850984	0.425492	+	0.904962i	$0.360101\pi$
$360$	0	0
$361$	2633.58	0.383960
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3266.98i	− 0.468497i
$366$	0	0
$367$	8924.51i	1.26936i	0.772775	+	0.634680i	$0.218868\pi$
$367$	8924.51i	1.26936i	−0.772775	+	0.634680i	$0.781132\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3551.92	0.497052
$372$	0	0
$373$	−5445.11	−0.755863	−0.377932	−	0.925834i	$-0.623365\pi$
$373$	−5445.11	−0.755863	−0.377932	+	0.925834i	$0.623365\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 9854.40i	− 1.34623i
$378$	0	0
$379$	− 8145.34i	− 1.10395i	−0.833860	−	0.551977i	$-0.813874\pi$
$379$	− 8145.34i	− 1.10395i	0.833860	−	0.551977i	$-0.186126\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5779.93	0.771124	0.385562	−	0.922682i	$-0.374008\pi$
$383$	5779.93	0.771124	0.385562	+	0.922682i	$0.374008\pi$
$384$	0	0
$385$	−2765.43	−0.366076
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 15128.6i	− 1.97186i	−0.167165	−	0.985929i	$-0.553461\pi$
$389$	− 15128.6i	− 1.97186i	0.167165	−	0.985929i	$-0.446539\pi$
$390$	0	0
$391$	− 113.643i	− 0.0146986i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5327.49	0.678620
$396$	0	0
$397$	6772.50	0.856177	0.428088	−	0.903737i	$-0.359187\pi$
$397$	6772.50	0.856177	0.428088	+	0.903737i	$0.359187\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4076.84i	− 0.507701i	−0.967244	−	0.253850i	$-0.918303\pi$
$401$	− 4076.84i	− 0.507701i	0.967244	−	0.253850i	$-0.0816970\pi$
$402$	0	0
$403$	3437.02i	0.424839i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2667.04	−0.324816
$408$	0	0
$409$	−11111.3	−1.34333	−0.671663	−	0.740857i	$-0.734420\pi$
$409$	−11111.3	−1.34333	−0.671663	+	0.740857i	$0.734420\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2986.72i	0.355852i
$414$	0	0
$415$	1911.25i	0.226071i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10945.1	1.27614	0.638071	−	0.769977i	$-0.279732\pi$
$419$	10945.1	1.27614	0.638071	+	0.769977i	$0.279732\pi$
$420$	0	0
$421$	1514.48	0.175323	0.0876617	−	0.996150i	$-0.472061\pi$
$421$	1514.48	0.175323	0.0876617	+	0.996150i	$0.472061\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 583.638i	− 0.0666132i
$426$	0	0
$427$	− 5515.40i	− 0.625080i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8625.85	−0.964020	−0.482010	−	0.876166i	$-0.660093\pi$
$431$	−8625.85	−0.964020	−0.482010	+	0.876166i	$0.660093\pi$
$432$	0	0
$433$	−1536.03	−0.170478	−0.0852389	−	0.996361i	$-0.527165\pi$
$433$	−1536.03	−0.170478	−0.0852389	+	0.996361i	$0.527165\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	223.094i	0.0244211i
$438$	0	0
$439$	3087.77i	0.335697i	0.985813	+	0.167849i	$0.0536820\pi$
$439$	3087.77i	0.335697i	−0.985813	+	0.167849i	$0.946318\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13688.6	1.46809	0.734046	−	0.679100i	$-0.237630\pi$
$443$	13688.6	1.46809	0.734046	+	0.679100i	$0.237630\pi$
$444$	0	0
$445$	−9165.72	−0.976397
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 8353.15i	− 0.877973i	−0.898494	−	0.438986i	$-0.855338\pi$
$449$	− 8353.15i	− 0.877973i	0.898494	−	0.438986i	$-0.144662\pi$
$450$	0	0
$451$	691.996i	0.0722501i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5335.78	0.549769
$456$	0	0
$457$	13101.5	1.34105	0.670525	−	0.741887i	$-0.266069\pi$
$457$	13101.5	1.34105	0.670525	+	0.741887i	$0.266069\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 1123.82i	− 0.113539i	−0.998387	−	0.0567695i	$-0.981920\pi$
$461$	− 1123.82i	− 0.113539i	0.998387	−	0.0567695i	$-0.0180800\pi$
$462$	0	0
$463$	17036.2i	1.71002i	0.518615	+	0.855008i	$0.326448\pi$
$463$	17036.2i	1.71002i	−0.518615	+	0.855008i	$0.673552\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18662.0	1.84920	0.924599	−	0.380941i	$-0.124400\pi$
$467$	18662.0	1.84920	0.924599	+	0.380941i	$0.124400\pi$
$468$	0	0
$469$	4175.07	0.411060
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 11891.4i	− 1.15596i
$474$	0	0
$475$	1145.75i	0.110675i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9261.15	−0.883409	−0.441704	−	0.897161i	$-0.645626\pi$
$479$	−9261.15	−0.883409	−0.441704	+	0.897161i	$0.645626\pi$
$480$	0	0
$481$	5145.93	0.487805
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9792.83i	0.916845i
$486$	0	0
$487$	− 10363.0i	− 0.964260i	−0.876100	−	0.482130i	$-0.839863\pi$
$487$	− 10363.0i	− 0.964260i	0.876100	−	0.482130i	$-0.160137\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10480.3	0.963277	0.481638	−	0.876370i	$-0.340042\pi$
$491$	10480.3	0.963277	0.481638	+	0.876370i	$0.340042\pi$
$492$	0	0
$493$	−4435.78	−0.405228
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7902.50i	0.713230i
$498$	0	0
$499$	− 14130.2i	− 1.26765i	−0.773478	−	0.633823i	$-0.781485\pi$
$499$	− 14130.2i	− 1.26765i	0.773478	−	0.633823i	$-0.218515\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19015.4	−1.68559	−0.842796	−	0.538233i	$-0.819092\pi$
$503$	−19015.4	−1.68559	−0.842796	+	0.538233i	$0.819092\pi$
$504$	0	0
$505$	−7472.88	−0.658493
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 9566.03i	− 0.833019i	−0.909131	−	0.416509i	$-0.863253\pi$
$509$	− 9566.03i	− 0.833019i	0.909131	−	0.416509i	$-0.136747\pi$
$510$	0	0
$511$	2206.96i	0.191057i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8819.45	−0.754624
$516$	0	0
$517$	−12859.0	−1.09389
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 10241.7i	− 0.861223i	−0.902537	−	0.430612i	$-0.858298\pi$
$521$	− 10241.7i	− 0.861223i	0.902537	−	0.430612i	$-0.141702\pi$
$522$	0	0
$523$	− 12917.5i	− 1.08001i	−0.841663	−	0.540003i	$-0.818423\pi$
$523$	− 12917.5i	− 1.08001i	0.841663	−	0.540003i	$-0.181577\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1547.11	0.127881
$528$	0	0
$529$	−12155.2	−0.999032
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1335.18i	− 0.108505i
$534$	0	0
$535$	− 3464.20i	− 0.279945i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1868.14	0.149289
$540$	0	0
$541$	−15734.0	−1.25038	−0.625192	−	0.780471i	$-0.714979\pi$
$541$	−15734.0	−1.25038	−0.625192	+	0.780471i	$0.714979\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 14126.8i	− 1.11032i
$546$	0	0
$547$	651.130i	0.0508964i	0.999676	+	0.0254482i	$0.00810128\pi$
$547$	651.130i	0.0508964i	−0.999676	+	0.0254482i	$0.991899\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8707.93	0.673268
$552$	0	0
$553$	−3598.91	−0.276747
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 10465.8i	− 0.796138i	−0.917355	−	0.398069i	$-0.869680\pi$
$557$	− 10465.8i	− 0.796138i	0.917355	−	0.398069i	$-0.130320\pi$
$558$	0	0
$559$	22944.0i	1.73601i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6747.88	−0.505132	−0.252566	−	0.967580i	$-0.581274\pi$
$563$	−6747.88	−0.505132	−0.252566	+	0.967580i	$0.581274\pi$
$564$	0	0
$565$	4459.72	0.332074
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1282.28i	0.0944746i	0.998884	+	0.0472373i	$0.0150417\pi$
$569$	1282.28i	0.0944746i	−0.998884	+	0.0472373i	$0.984958\pi$
$570$	0	0
$571$	− 24900.2i	− 1.82494i	−0.409141	−	0.912471i	$-0.634172\pi$
$571$	− 24900.2i	− 1.82494i	0.409141	−	0.912471i	$-0.365828\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	60.4932	0.00438738
$576$	0	0
$577$	−9656.97	−0.696751	−0.348375	−	0.937355i	$-0.613267\pi$
$577$	−9656.97	−0.696751	−0.348375	+	0.937355i	$0.613267\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1291.12i	− 0.0921937i
$582$	0	0
$583$	− 19345.5i	− 1.37428i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1431.32	0.100642	0.0503209	−	0.998733i	$-0.483976\pi$
$587$	1431.32	0.100642	0.0503209	+	0.998733i	$0.483976\pi$
$588$	0	0
$589$	−3037.15	−0.212468
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 5170.60i	− 0.358063i	−0.983843	−	0.179031i	$-0.942704\pi$
$593$	− 5170.60i	− 0.358063i	0.983843	−	0.179031i	$-0.0572964\pi$
$594$	0	0
$595$	− 2401.80i	− 0.165486i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−23838.2	−1.62604	−0.813022	−	0.582233i	$-0.802179\pi$
$599$	−23838.2	−1.62604	−0.813022	+	0.582233i	$0.802179\pi$
$600$	0	0
$601$	1200.61	0.0814872	0.0407436	−	0.999170i	$-0.487027\pi$
$601$	1200.61	0.0814872	0.0407436	+	0.999170i	$0.487027\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1269.85i	0.0853331i
$606$	0	0
$607$	− 15585.4i	− 1.04216i	−0.853507	−	0.521081i	$-0.825529\pi$
$607$	− 15585.4i	− 1.04216i	0.853507	−	0.521081i	$-0.174471\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24810.9	1.64279
$612$	0	0
$613$	4904.86	0.323174	0.161587	−	0.986858i	$-0.448339\pi$
$613$	4904.86	0.323174	0.161587	+	0.986858i	$0.448339\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9680.19i	0.631621i	0.948822	+	0.315810i	$0.102276\pi$
$617$	9680.19i	0.631621i	−0.948822	+	0.315810i	$0.897724\pi$
$618$	0	0
$619$	20038.4i	1.30115i	0.759442	+	0.650575i	$0.225472\pi$
$619$	20038.4i	1.30115i	−0.759442	+	0.650575i	$0.774528\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6191.78	0.398183
$624$	0	0
$625$	−13111.1	−0.839109
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 2316.35i	− 0.146834i
$630$	0	0
$631$	− 7092.80i	− 0.447480i	−0.974649	−	0.223740i	$-0.928173\pi$
$631$	− 7092.80i	− 0.447480i	0.974649	−	0.223740i	$-0.0718267\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7752.90	−0.484511
$636$	0	0
$637$	−3604.51	−0.224201
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5324.04i	0.328061i	0.986455	+	0.164030i	$0.0524495\pi$
$641$	5324.04i	0.328061i	−0.986455	+	0.164030i	$0.947551\pi$
$642$	0	0
$643$	27990.5i	1.71670i	0.513065	+	0.858350i	$0.328510\pi$
$643$	27990.5i	1.71670i	−0.513065	+	0.858350i	$0.671490\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7653.54	0.465056	0.232528	−	0.972590i	$-0.425300\pi$
$647$	7653.54	0.465056	0.232528	+	0.972590i	$0.425300\pi$
$648$	0	0
$649$	16267.1	0.983884
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20016.4i	1.19955i	0.800171	+	0.599773i	$0.204742\pi$
$653$	20016.4i	1.19955i	−0.800171	+	0.599773i	$0.795258\pi$
$654$	0	0
$655$	11759.0i	0.701466i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26101.0	1.54287	0.771435	−	0.636308i	$-0.219539\pi$
$659$	26101.0	1.54287	0.771435	+	0.636308i	$0.219539\pi$
$660$	0	0
$661$	2639.00	0.155288	0.0776438	−	0.996981i	$-0.475260\pi$
$661$	2639.00	0.155288	0.0776438	+	0.996981i	$0.475260\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4715.01i	0.274948i
$666$	0	0
$667$	− 459.762i	− 0.0266897i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−30039.6	−1.72826
$672$	0	0
$673$	27859.4	1.59569	0.797846	−	0.602862i	$-0.205973\pi$
$673$	27859.4	1.59569	0.797846	+	0.602862i	$0.205973\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	562.192i	0.0319155i	0.999873	+	0.0159578i	$0.00507973\pi$
$677$	562.192i	0.0319155i	−0.999873	+	0.0159578i	$0.994920\pi$
$678$	0	0
$679$	− 6615.41i	− 0.373897i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6345.33	−0.355487	−0.177743	−	0.984077i	$-0.556880\pi$
$683$	−6345.33	−0.355487	−0.177743	+	0.984077i	$0.556880\pi$
$684$	0	0
$685$	−23525.7	−1.31222
$686$	0	0
$687$	0	0
$688$	0	0
$689$	37326.3i	2.06389i
$690$	0	0
$691$	9288.10i	0.511340i	0.966764	+	0.255670i	$0.0822961\pi$
$691$	9288.10i	0.511340i	−0.966764	+	0.255670i	$0.917704\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9046.49	−0.493745
$696$	0	0
$697$	−601.006	−0.0326610
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17248.1i	0.929319i	0.885490	+	0.464659i	$0.153823\pi$
$701$	17248.1i	0.929319i	−0.885490	+	0.464659i	$0.846177\pi$
$702$	0	0
$703$	4547.25i	0.243959i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5048.20	0.268539
$708$	0	0
$709$	28176.8	1.49253	0.746263	−	0.665651i	$-0.231846\pi$
$709$	28176.8	1.49253	0.746263	+	0.665651i	$0.231846\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	160.356i	0.00842268i
$714$	0	0
$715$	− 29061.2i	− 1.52004i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16528.5	−0.857313	−0.428657	−	0.903467i	$-0.641013\pi$
$719$	−16528.5	−0.857313	−0.428657	+	0.903467i	$0.641013\pi$
$720$	0	0
$721$	5957.85	0.307742
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2361.21i	− 0.120956i
$726$	0	0
$727$	27053.5i	1.38013i	0.723746	+	0.690067i	$0.242419\pi$
$727$	27053.5i	1.38013i	−0.723746	+	0.690067i	$0.757581\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	10327.8	0.522557
$732$	0	0
$733$	15109.9	0.761390	0.380695	−	0.924701i	$-0.375685\pi$
$733$	15109.9	0.761390	0.380695	+	0.924701i	$0.375685\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 22739.5i	− 1.13653i
$738$	0	0
$739$	13145.5i	0.654351i	0.944964	+	0.327176i	$0.106097\pi$
$739$	13145.5i	0.654351i	−0.944964	+	0.327176i	$0.893903\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36890.1	1.82149	0.910744	−	0.412972i	$-0.135509\pi$
$743$	36890.1	1.82149	0.910744	+	0.412972i	$0.135509\pi$
$744$	0	0
$745$	−16613.6	−0.817012
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2340.19i	0.114164i
$750$	0	0
$751$	11371.0i	0.552509i	0.961084	+	0.276255i	$0.0890933\pi$
$751$	11371.0i	0.552509i	−0.961084	+	0.276255i	$0.910907\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−23479.7	−1.13180
$756$	0	0
$757$	7923.05	0.380407	0.190204	−	0.981745i	$-0.439085\pi$
$757$	7923.05	0.380407	0.190204	+	0.981745i	$0.439085\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11234.6i	0.535156i	0.963536	+	0.267578i	$0.0862234\pi$
$761$	11234.6i	0.535156i	−0.963536	+	0.267578i	$0.913777\pi$
$762$	0	0
$763$	9543.18i	0.452800i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−31386.8	−1.47759
$768$	0	0
$769$	−21520.7	−1.00917	−0.504587	−	0.863361i	$-0.668355\pi$
$769$	−21520.7	−1.00917	−0.504587	+	0.863361i	$0.668355\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 36498.0i	− 1.69824i	−0.528199	−	0.849121i	$-0.677132\pi$
$773$	− 36498.0i	− 1.69824i	0.528199	−	0.849121i	$-0.322868\pi$
$774$	0	0
$775$	823.542i	0.0381710i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1179.84	0.0542647
$780$	0	0
$781$	43040.8	1.97199
$782$	0	0
$783$	0	0
$784$	0	0
$785$	31686.8i	1.44070i
$786$	0	0
$787$	− 14754.9i	− 0.668306i	−0.942519	−	0.334153i	$-0.891550\pi$
$787$	− 14754.9i	− 0.668306i	0.942519	−	0.334153i	$-0.108450\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3012.70	−0.135423
$792$	0	0
$793$	57960.1	2.59549
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 39753.2i	− 1.76679i	−0.468630	−	0.883395i	$-0.655252\pi$
$797$	− 39753.2i	− 1.76679i	0.468630	−	0.883395i	$-0.344748\pi$
$798$	0	0
$799$	− 11168.2i	− 0.494496i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12020.2	0.528248
$804$	0	0
$805$	248.943	0.0108995
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 8432.49i	− 0.366466i	−0.983070	−	0.183233i	$-0.941344\pi$
$809$	− 8432.49i	− 0.366466i	0.983070	−	0.183233i	$-0.0586562\pi$
$810$	0	0
$811$	31425.2i	1.36065i	0.732910	+	0.680326i	$0.238162\pi$
$811$	31425.2i	1.36065i	−0.732910	+	0.680326i	$0.761838\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18387.8	−0.790301
$816$	0	0
$817$	−20274.7	−0.868203
$818$	0	0
$819$	0	0
$820$	0	0
$821$	40546.9i	1.72363i	0.507226	+	0.861813i	$0.330671\pi$
$821$	40546.9i	1.72363i	−0.507226	+	0.861813i	$0.669329\pi$
$822$	0	0
$823$	− 5380.66i	− 0.227895i	−0.993487	−	0.113948i	$-0.963650\pi$
$823$	− 5380.66i	− 0.227895i	0.993487	−	0.113948i	$-0.0363496\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34962.0	1.47007	0.735034	−	0.678030i	$-0.237166\pi$
$827$	34962.0	1.47007	0.735034	+	0.678030i	$0.237166\pi$
$828$	0	0
$829$	39045.5	1.63583	0.817917	−	0.575337i	$-0.195129\pi$
$829$	39045.5	1.63583	0.817917	+	0.575337i	$0.195129\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1622.50i	0.0674867i
$834$	0	0
$835$	− 41780.1i	− 1.73157i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−36225.1	−1.49062	−0.745309	−	0.666719i	$-0.767698\pi$
$839$	−36225.1	−1.49062	−0.745309	+	0.666719i	$0.767698\pi$
$840$	0	0
$841$	6443.29	0.264188
$842$	0	0
$843$	0	0
$844$	0	0
$845$	33306.8i	1.35596i
$846$	0	0
$847$	− 857.826i	− 0.0347996i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	240.086	0.00967103
$852$	0	0
$853$	−10991.5	−0.441196	−0.220598	−	0.975365i	$-0.570801\pi$
$853$	−10991.5	−0.441196	−0.220598	+	0.975365i	$0.570801\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1133.29i	− 0.0451719i	−0.999745	−	0.0225859i	$-0.992810\pi$
$857$	− 1133.29i	− 0.0451719i	0.999745	−	0.0225859i	$-0.00718994\pi$
$858$	0	0
$859$	− 9768.48i	− 0.388005i	−0.981001	−	0.194003i	$-0.937853\pi$
$859$	− 9768.48i	− 0.388005i	0.981001	−	0.194003i	$-0.0621470\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6597.16	−0.260220	−0.130110	−	0.991500i	$-0.541533\pi$
$863$	−6597.16	−0.260220	−0.130110	+	0.991500i	$0.541533\pi$
$864$	0	0
$865$	3817.03	0.150038
$866$	0	0
$867$	0	0
$868$	0	0
$869$	19601.4i	0.765170i
$870$	0	0
$871$	43874.9i	1.70682i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10345.4	0.399700
$876$	0	0
$877$	−31391.9	−1.20870	−0.604350	−	0.796719i	$-0.706567\pi$
$877$	−31391.9	−1.20870	−0.604350	+	0.796719i	$0.706567\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 11064.9i	− 0.423138i	−0.977363	−	0.211569i	$-0.932143\pi$
$881$	− 11064.9i	− 0.423138i	0.977363	−	0.211569i	$-0.0678573\pi$
$882$	0	0
$883$	4648.99i	0.177181i	0.996068	+	0.0885906i	$0.0282363\pi$
$883$	4648.99i	0.177181i	−0.996068	+	0.0885906i	$0.971764\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22764.3	−0.861726	−0.430863	−	0.902417i	$-0.641791\pi$
$887$	−22764.3	−0.861726	−0.430863	+	0.902417i	$0.641791\pi$
$888$	0	0
$889$	5237.37	0.197588
$890$	0	0
$891$	0	0
$892$	0	0
$893$	21924.4i	0.821582i
$894$	0	0
$895$	30754.2i	1.14860i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6259.10	0.232206
$900$	0	0
$901$	16801.8	0.621252
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27876.5i	1.02392i
$906$	0	0
$907$	28162.2i	1.03099i	0.856891	+	0.515497i	$0.172393\pi$
$907$	28162.2i	1.03099i	−0.856891	+	0.515497i	$0.827607\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−39552.2	−1.43844	−0.719222	−	0.694780i	$-0.755502\pi$
$911$	−39552.2	−1.43844	−0.719222	+	0.694780i	$0.755502\pi$
$912$	0	0
$913$	−7032.05	−0.254903
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7943.60i	− 0.286064i
$918$	0	0
$919$	17361.9i	0.623194i	0.950214	+	0.311597i	$0.100864\pi$
$919$	17361.9i	0.623194i	−0.950214	+	0.311597i	$0.899136\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−83045.5	−2.96151
$924$	0	0
$925$	1233.02	0.0438284
$926$	0	0
$927$	0	0
$928$	0	0
$929$	40213.4i	1.42019i	0.704105	+	0.710096i	$0.251348\pi$
$929$	40213.4i	1.42019i	−0.704105	+	0.710096i	$0.748652\pi$
$930$	0	0
$931$	− 3185.16i	− 0.112126i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−13081.4	−0.457548
$936$	0	0
$937$	43518.4	1.51727	0.758636	−	0.651515i	$-0.225866\pi$
$937$	43518.4	1.51727	0.758636	+	0.651515i	$0.225866\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	39122.6i	1.35532i	0.735373	+	0.677662i	$0.237007\pi$
$941$	39122.6i	1.35532i	−0.735373	+	0.677662i	$0.762993\pi$
$942$	0	0
$943$	− 62.2933i	− 0.00215117i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	49638.3	1.70330	0.851651	−	0.524109i	$-0.175602\pi$
$947$	49638.3	1.70330	0.851651	+	0.524109i	$0.175602\pi$
$948$	0	0
$949$	−23192.5	−0.793318
$950$	0	0
$951$	0	0
$952$	0	0
$953$	53364.1i	1.81389i	0.421254	+	0.906943i	$0.361590\pi$
$953$	53364.1i	1.81389i	−0.421254	+	0.906943i	$0.638410\pi$
$954$	0	0
$955$	14737.1i	0.499354i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15892.4	0.535134
$960$	0	0
$961$	27607.9	0.926721
$962$	0	0
$963$	0	0
$964$	0	0
$965$	12557.7i	0.418908i
$966$	0	0
$967$	− 35948.9i	− 1.19549i	−0.801686	−	0.597746i	$-0.796063\pi$
$967$	− 35948.9i	− 1.19549i	0.801686	−	0.597746i	$-0.203937\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	23629.3	0.780948	0.390474	−	0.920614i	$-0.372311\pi$
$971$	23629.3	0.780948	0.390474	+	0.920614i	$0.372311\pi$
$972$	0	0
$973$	6111.23	0.201354
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 28235.6i	− 0.924603i	−0.886723	−	0.462302i	$-0.847024\pi$
$977$	− 28235.6i	− 0.924603i	0.886723	−	0.462302i	$-0.152976\pi$
$978$	0	0
$979$	− 33723.4i	− 1.10092i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−26052.7	−0.845324	−0.422662	−	0.906287i	$-0.638904\pi$
$983$	−26052.7	−0.845324	−0.422662	+	0.906287i	$0.638904\pi$
$984$	0	0
$985$	−54774.9	−1.77185
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1070.46i	0.0344174i
$990$	0	0
$991$	51723.6i	1.65797i	0.559267	+	0.828987i	$0.311083\pi$
$991$	51723.6i	1.65797i	−0.559267	+	0.828987i	$0.688917\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	17412.3	0.554782
$996$	0	0
$997$	7212.80	0.229119	0.114560	−	0.993416i	$-0.463454\pi$
$997$	7212.80	0.229119	0.114560	+	0.993416i	$0.463454\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.h.b.575.20	yes	24
3.2	odd	2	inner	1008.4.h.b.575.5	✓	24
4.3	odd	2	inner	1008.4.h.b.575.6	yes	24
12.11	even	2	inner	1008.4.h.b.575.19	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.4.h.b.575.5	✓	24	3.2	odd	2	inner
1008.4.h.b.575.6	yes	24	4.3	odd	2	inner
1008.4.h.b.575.19	yes	24	12.11	even	2	inner
1008.4.h.b.575.20	yes	24	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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+10.3621i q^{5} -7.00000i q^{7} +O(q^{10})\)	\(q+10.3621i q^{5} -7.00000i q^{7} -38.1254 q^{11} +73.5614 q^{13} -33.1123i q^{17} +65.0032i q^{19} +3.43204 q^{23} +17.6260 q^{25} -133.962i q^{29} +46.7231i q^{31} +72.5350 q^{35} +69.9543 q^{37} -18.1505i q^{41} +311.903i q^{43} +337.282 q^{47} -49.0000 q^{49} +507.417i q^{53} -395.061i q^{55} -426.674 q^{59} +787.915 q^{61} +762.254i q^{65} +596.439i q^{67} -1128.93 q^{71} -315.280 q^{73} +266.878i q^{77} -514.130i q^{79} +184.445 q^{83} +343.115 q^{85} +884.539i q^{89} -514.930i q^{91} -673.572 q^{95} +945.059 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 192 q^{13} - 984 q^{25} + 720 q^{37} - 1176 q^{49} + 2736 q^{61} + 3408 q^{85} + 1152 q^{97}+O(q^{100}) \) 24 * q - 192 * q^13 - 984 * q^25 + 720 * q^37 - 1176 * q^49 + 2736 * q^61 + 3408 * q^85 + 1152 * q^97

Introduction

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