LMFDB - Embedded newform 1008.3.d.c.449.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,3,Mod(449,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.449"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-2.57794i$ of defining polynomial
Character	$\chi$	$=$	1008.449
Dual form			1008.3.d.c.449.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.15587i q^{5} +2.64575 q^{7} +6.06910i q^{11} -12.5830 q^{13} -17.2941i q^{17} +25.8745 q^{19} -2.41618i q^{23} -1.58301 q^{25} -55.8014i q^{29} -15.2915 q^{31} -13.6412i q^{35} -23.7490 q^{37} -15.4676i q^{41} -27.7490 q^{43} +28.4617i q^{47} +7.00000 q^{49} +46.6690i q^{53} +31.2915 q^{55} -80.6675i q^{59} -62.7085 q^{61} +64.8764i q^{65} -116.081 q^{67} -49.7896i q^{71} +30.7085 q^{73} +16.0573i q^{77} -142.664 q^{79} -28.5763i q^{83} -89.1660 q^{85} -168.203i q^{89} -33.2915 q^{91} -133.406i q^{95} +113.956 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{13} + 40 q^{19} + 36 q^{25} - 40 q^{31} + 32 q^{37} + 16 q^{43} + 28 q^{49} + 104 q^{55} - 272 q^{61} - 168 q^{67} + 144 q^{73} - 232 q^{79} - 272 q^{85} - 112 q^{91} + 96 q^{97}+O(q^{100}) $ 4 * q - 8 * q^13 + 40 * q^19 + 36 * q^25 - 40 * q^31 + 32 * q^37 + 16 * q^43 + 28 * q^49 + 104 * q^55 - 272 * q^61 - 168 * q^67 + 144 * q^73 - 232 * q^79 - 272 * q^85 - 112 * q^91 + 96 * q^97

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
$5$	− 5.15587i	− 1.03117i	−0.856837	−	0.515587i	$-0.827574\pi$
$5$	− 5.15587i	− 1.03117i	0.856837	−	0.515587i	$-0.172426\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.06910i	0.551736i	0.961195	+	0.275868i	$0.0889653\pi$
$11$	6.06910i	0.551736i	−0.961195	+	0.275868i	$0.911035\pi$
$12$	0	0
$13$	−12.5830	−0.967923	−0.483962	−	0.875089i	$-0.660803\pi$
$13$	−12.5830	−0.967923	−0.483962	+	0.875089i	$0.660803\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 17.2941i	− 1.01730i	−0.860974	−	0.508649i	$-0.830145\pi$
$17$	− 17.2941i	− 1.01730i	0.860974	−	0.508649i	$-0.169855\pi$
$18$	0	0
$19$	25.8745	1.36182	0.680908	−	0.732369i	$-0.261585\pi$
$19$	25.8745	1.36182	0.680908	+	0.732369i	$0.261585\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 2.41618i	− 0.105051i	−0.998620	−	0.0525257i	$-0.983273\pi$
$23$	− 2.41618i	− 0.105051i	0.998620	−	0.0525257i	$-0.0167271\pi$
$24$	0	0
$25$	−1.58301	−0.0633202
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 55.8014i	− 1.92418i	−0.272724	−	0.962092i	$-0.587925\pi$
$29$	− 55.8014i	− 1.92418i	0.272724	−	0.962092i	$-0.412075\pi$
$30$	0	0
$31$	−15.2915	−0.493274	−0.246637	−	0.969108i	$-0.579326\pi$
$31$	−15.2915	−0.493274	−0.246637	+	0.969108i	$0.579326\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 13.6412i	− 0.389747i
$36$	0	0
$37$	−23.7490	−0.641865	−0.320933	−	0.947102i	$-0.603996\pi$
$37$	−23.7490	−0.641865	−0.320933	+	0.947102i	$0.603996\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 15.4676i	− 0.377259i	−0.982048	−	0.188629i	$-0.939596\pi$
$41$	− 15.4676i	− 0.377259i	0.982048	−	0.188629i	$-0.0604045\pi$
$42$	0	0
$43$	−27.7490	−0.645326	−0.322663	−	0.946514i	$-0.604578\pi$
$43$	−27.7490	−0.645326	−0.322663	+	0.946514i	$0.604578\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	28.4617i	0.605569i	0.953059	+	0.302785i	$0.0979162\pi$
$47$	28.4617i	0.605569i	−0.953059	+	0.302785i	$0.902084\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	46.6690i	0.880548i	0.897863	+	0.440274i	$0.145119\pi$
$53$	46.6690i	0.880548i	−0.897863	+	0.440274i	$0.854881\pi$
$54$	0	0
$55$	31.2915	0.568936
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 80.6675i	− 1.36725i	−0.729836	−	0.683623i	$-0.760403\pi$
$59$	− 80.6675i	− 1.36725i	0.729836	−	0.683623i	$-0.239597\pi$
$60$	0	0
$61$	−62.7085	−1.02801	−0.514004	−	0.857788i	$-0.671838\pi$
$61$	−62.7085	−1.02801	−0.514004	+	0.857788i	$0.671838\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	64.8764i	0.998098i
$66$	0	0
$67$	−116.081	−1.73255	−0.866276	−	0.499565i	$-0.833493\pi$
$67$	−116.081	−1.73255	−0.866276	+	0.499565i	$0.833493\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 49.7896i	− 0.701261i	−0.936514	−	0.350631i	$-0.885967\pi$
$71$	− 49.7896i	− 0.701261i	0.936514	−	0.350631i	$-0.114033\pi$
$72$	0	0
$73$	30.7085	0.420664	0.210332	−	0.977630i	$-0.432545\pi$
$73$	30.7085	0.420664	0.210332	+	0.977630i	$0.432545\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.0573i	0.208537i
$78$	0	0
$79$	−142.664	−1.80587	−0.902937	−	0.429773i	$-0.858594\pi$
$79$	−142.664	−1.80587	−0.902937	+	0.429773i	$0.858594\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 28.5763i	− 0.344293i	−0.985071	−	0.172147i	$-0.944930\pi$
$83$	− 28.5763i	− 0.344293i	0.985071	−	0.172147i	$-0.0550703\pi$
$84$	0	0
$85$	−89.1660	−1.04901
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 168.203i	− 1.88992i	−0.327189	−	0.944959i	$-0.606101\pi$
$89$	− 168.203i	− 1.88992i	0.327189	−	0.944959i	$-0.393899\pi$
$90$	0	0
$91$	−33.2915	−0.365841
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 133.406i	− 1.40427i
$96$	0	0
$97$	113.956	1.17480	0.587400	−	0.809297i	$-0.300152\pi$
$97$	113.956	1.17480	0.587400	+	0.809297i	$0.300152\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	40.2764i	0.398777i	0.979921	+	0.199388i	$0.0638955\pi$
$101$	40.2764i	0.398777i	−0.979921	+	0.199388i	$0.936104\pi$
$102$	0	0
$103$	−29.1216	−0.282734	−0.141367	−	0.989957i	$-0.545150\pi$
$103$	−29.1216	−0.282734	−0.141367	+	0.989957i	$0.545150\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 181.901i	− 1.70001i	−0.526774	−	0.850005i	$-0.676599\pi$
$107$	− 181.901i	− 1.70001i	0.526774	−	0.850005i	$-0.323401\pi$
$108$	0	0
$109$	−88.1621	−0.808826	−0.404413	−	0.914576i	$-0.632524\pi$
$109$	−88.1621	−0.808826	−0.404413	+	0.914576i	$0.632524\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 64.8190i	− 0.573620i	−0.957988	−	0.286810i	$-0.907405\pi$
$113$	− 64.8190i	− 0.573620i	0.957988	−	0.286810i	$-0.0925948\pi$
$114$	0	0
$115$	−12.4575	−0.108326
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 45.7558i	− 0.384503i
$120$	0	0
$121$	84.1660	0.695587
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 120.735i	− 0.965880i
$126$	0	0
$127$	223.579	1.76047	0.880233	−	0.474543i	$-0.157387\pi$
$127$	223.579	1.76047	0.880233	+	0.474543i	$0.157387\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	207.300i	1.58244i	0.611531	+	0.791220i	$0.290554\pi$
$131$	207.300i	1.58244i	−0.611531	+	0.791220i	$0.709446\pi$
$132$	0	0
$133$	68.4575	0.514718
$134$	0	0
$135$	0	0
$136$	0	0
$137$	175.775i	1.28303i	0.767112	+	0.641514i	$0.221693\pi$
$137$	175.775i	1.28303i	−0.767112	+	0.641514i	$0.778307\pi$
$138$	0	0
$139$	117.166	0.842921	0.421460	−	0.906847i	$-0.361518\pi$
$139$	117.166	0.842921	0.421460	+	0.906847i	$0.361518\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 76.3675i	− 0.534039i
$144$	0	0
$145$	−287.705	−1.98417
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 186.316i	− 1.25044i	−0.780448	−	0.625220i	$-0.785009\pi$
$149$	− 186.316i	− 1.25044i	0.780448	−	0.625220i	$-0.214991\pi$
$150$	0	0
$151$	−182.081	−1.20583	−0.602917	−	0.797804i	$-0.705995\pi$
$151$	−182.081	−1.20583	−0.602917	+	0.797804i	$0.705995\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	78.8410i	0.508652i
$156$	0	0
$157$	68.3765	0.435519	0.217759	−	0.976002i	$-0.430125\pi$
$157$	68.3765	0.435519	0.217759	+	0.976002i	$0.430125\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 6.39261i	− 0.0397057i
$162$	0	0
$163$	36.7451	0.225430	0.112715	−	0.993627i	$-0.464045\pi$
$163$	36.7451	0.225430	0.112715	+	0.993627i	$0.464045\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	240.708i	1.44137i	0.693264	+	0.720684i	$0.256172\pi$
$167$	240.708i	1.44137i	−0.693264	+	0.720684i	$0.743828\pi$
$168$	0	0
$169$	−10.6680	−0.0631241
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 341.029i	− 1.97126i	−0.168905	−	0.985632i	$-0.554023\pi$
$173$	− 341.029i	− 1.97126i	0.168905	−	0.985632i	$-0.445977\pi$
$174$	0	0
$175$	−4.18824	−0.0239328
$176$	0	0
$177$	0	0
$178$	0	0
$179$	138.410i	0.773240i	0.922239	+	0.386620i	$0.126357\pi$
$179$	138.410i	0.773240i	−0.922239	+	0.386620i	$0.873643\pi$
$180$	0	0
$181$	−135.328	−0.747669	−0.373834	−	0.927495i	$-0.621957\pi$
$181$	−135.328	−0.747669	−0.373834	+	0.927495i	$0.621957\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	122.447i	0.661875i
$186$	0	0
$187$	104.959	0.561281
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 168.469i	− 0.882036i	−0.897498	−	0.441018i	$-0.854617\pi$
$191$	− 168.469i	− 0.882036i	0.897498	−	0.441018i	$-0.145383\pi$
$192$	0	0
$193$	143.498	0.743513	0.371757	−	0.928330i	$-0.378756\pi$
$193$	143.498	0.743513	0.371757	+	0.928330i	$0.378756\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 296.510i	− 1.50513i	−0.658521	−	0.752563i	$-0.728817\pi$
$197$	− 296.510i	− 1.50513i	0.658521	−	0.752563i	$-0.271183\pi$
$198$	0	0
$199$	142.243	0.714789	0.357395	−	0.933953i	$-0.383665\pi$
$199$	142.243	0.714789	0.357395	+	0.933953i	$0.383665\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 147.636i	− 0.727273i
$204$	0	0
$205$	−79.7490	−0.389020
$206$	0	0
$207$	0	0
$208$	0	0
$209$	157.035i	0.751364i
$210$	0	0
$211$	−110.996	−0.526048	−0.263024	−	0.964789i	$-0.584720\pi$
$211$	−110.996	−0.526048	−0.263024	+	0.964789i	$0.584720\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	143.070i	0.665443i
$216$	0	0
$217$	−40.4575	−0.186440
$218$	0	0
$219$	0	0
$220$	0	0
$221$	217.611i	0.984667i
$222$	0	0
$223$	104.251	0.467493	0.233747	−	0.972298i	$-0.424901\pi$
$223$	104.251	0.467493	0.233747	+	0.972298i	$0.424901\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	163.161i	0.718773i	0.933189	+	0.359386i	$0.117014\pi$
$227$	163.161i	0.718773i	−0.933189	+	0.359386i	$0.882986\pi$
$228$	0	0
$229$	−278.413	−1.21578	−0.607889	−	0.794022i	$-0.707983\pi$
$229$	−278.413	−1.21578	−0.607889	+	0.794022i	$0.707983\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	404.042i	1.73408i	0.498235	+	0.867042i	$0.333982\pi$
$233$	404.042i	1.73408i	−0.498235	+	0.867042i	$0.666018\pi$
$234$	0	0
$235$	146.745	0.624447
$236$	0	0
$237$	0	0
$238$	0	0
$239$	257.204i	1.07617i	0.842892	+	0.538083i	$0.180851\pi$
$239$	257.204i	1.07617i	−0.842892	+	0.538083i	$0.819149\pi$
$240$	0	0
$241$	249.616	1.03575	0.517875	−	0.855456i	$-0.326723\pi$
$241$	249.616	1.03575	0.517875	+	0.855456i	$0.326723\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 36.0911i	− 0.147311i
$246$	0	0
$247$	−325.579	−1.31813
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 69.2909i	− 0.276059i	−0.990428	−	0.138030i	$-0.955923\pi$
$251$	− 69.2909i	− 0.276059i	0.990428	−	0.138030i	$-0.0440769\pi$
$252$	0	0
$253$	14.6640	0.0579606
$254$	0	0
$255$	0	0
$256$	0	0
$257$	255.873i	0.995614i	0.867288	+	0.497807i	$0.165861\pi$
$257$	255.873i	0.995614i	−0.867288	+	0.497807i	$0.834139\pi$
$258$	0	0
$259$	−62.8340	−0.242602
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 14.6690i	− 0.0557756i	−0.999611	−	0.0278878i	$-0.991122\pi$
$263$	− 14.6690i	− 0.0557756i	0.999611	−	0.0278878i	$-0.00887812\pi$
$264$	0	0
$265$	240.620	0.907998
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 74.9792i	− 0.278733i	−0.990241	−	0.139367i	$-0.955493\pi$
$269$	− 74.9792i	− 0.278733i	0.990241	−	0.139367i	$-0.0445066\pi$
$270$	0	0
$271$	211.203	0.779345	0.389673	−	0.920953i	$-0.372588\pi$
$271$	211.203	0.779345	0.389673	+	0.920953i	$0.372588\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 9.60742i	− 0.0349361i
$276$	0	0
$277$	408.494	1.47471	0.737354	−	0.675507i	$-0.236075\pi$
$277$	408.494	1.47471	0.737354	+	0.675507i	$0.236075\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	200.395i	0.713149i	0.934267	+	0.356575i	$0.116055\pi$
$281$	200.395i	0.713149i	−0.934267	+	0.356575i	$0.883945\pi$
$282$	0	0
$283$	−358.450	−1.26661	−0.633303	−	0.773904i	$-0.718301\pi$
$283$	−358.450	−1.26661	−0.633303	+	0.773904i	$0.718301\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 40.9235i	− 0.142590i
$288$	0	0
$289$	−10.0850	−0.0348961
$290$	0	0
$291$	0	0
$292$	0	0
$293$	325.352i	1.11042i	0.831711	+	0.555209i	$0.187362\pi$
$293$	325.352i	1.11042i	−0.831711	+	0.555209i	$0.812638\pi$
$294$	0	0
$295$	−415.911	−1.40987
$296$	0	0
$297$	0	0
$298$	0	0
$299$	30.4028i	0.101682i
$300$	0	0
$301$	−73.4170	−0.243910
$302$	0	0
$303$	0	0
$304$	0	0
$305$	323.317i	1.06006i
$306$	0	0
$307$	338.037	1.10110	0.550548	−	0.834803i	$-0.314419\pi$
$307$	338.037	1.10110	0.550548	+	0.834803i	$0.314419\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 311.179i	− 1.00057i	−0.865859	−	0.500287i	$-0.833228\pi$
$311$	− 311.179i	− 1.00057i	0.865859	−	0.500287i	$-0.166772\pi$
$312$	0	0
$313$	401.069	1.28137	0.640686	−	0.767803i	$-0.278650\pi$
$313$	401.069	1.28137	0.640686	+	0.767803i	$0.278650\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 84.1485i	− 0.265453i	−0.991153	−	0.132726i	$-0.957627\pi$
$317$	− 84.1485i	− 0.265453i	0.991153	−	0.132726i	$-0.0423731\pi$
$318$	0	0
$319$	338.664	1.06164
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 447.476i	− 1.38537i
$324$	0	0
$325$	19.9190	0.0612891
$326$	0	0
$327$	0	0
$328$	0	0
$329$	75.3027i	0.228884i
$330$	0	0
$331$	223.498	0.675221	0.337610	−	0.941286i	$-0.390381\pi$
$331$	223.498	0.675221	0.337610	+	0.941286i	$0.390381\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	598.499i	1.78656i
$336$	0	0
$337$	−234.583	−0.696092	−0.348046	−	0.937477i	$-0.613155\pi$
$337$	−234.583	−0.696092	−0.348046	+	0.937477i	$0.613155\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 92.8057i	− 0.272157i
$342$	0	0
$343$	18.5203	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	83.6161i	0.240969i	0.992715	+	0.120484i	$0.0384447\pi$
$347$	83.6161i	0.240969i	−0.992715	+	0.120484i	$0.961555\pi$
$348$	0	0
$349$	298.959	0.856617	0.428309	−	0.903632i	$-0.359110\pi$
$349$	298.959	0.856617	0.428309	+	0.903632i	$0.359110\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	428.052i	1.21261i	0.795232	+	0.606306i	$0.207349\pi$
$353$	428.052i	1.21261i	−0.795232	+	0.606306i	$0.792651\pi$
$354$	0	0
$355$	−256.708	−0.723123
$356$	0	0
$357$	0	0
$358$	0	0
$359$	144.536i	0.402608i	0.979529	+	0.201304i	$0.0645179\pi$
$359$	144.536i	0.402608i	−0.979529	+	0.201304i	$0.935482\pi$
$360$	0	0
$361$	308.490	0.854543
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 158.329i	− 0.433778i
$366$	0	0
$367$	610.243	1.66279	0.831394	−	0.555684i	$-0.187543\pi$
$367$	610.243	1.66279	0.831394	+	0.555684i	$0.187543\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	123.475i	0.332816i
$372$	0	0
$373$	0.332021	0.000890137 0	0.000445068	−	1.00000i	$-0.499858\pi$
$373$	0.332021	0.000890137 0	0.000445068		1.00000i	$0.499858\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	702.149i	1.86246i
$378$	0	0
$379$	393.652	1.03866	0.519330	−	0.854574i	$-0.326182\pi$
$379$	393.652	1.03866	0.519330	+	0.854574i	$0.326182\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 540.699i	− 1.41175i	−0.708338	−	0.705874i	$-0.750555\pi$
$383$	− 540.699i	− 1.41175i	0.708338	−	0.705874i	$-0.249445\pi$
$384$	0	0
$385$	82.7895	0.215038
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 474.397i	− 1.21953i	−0.792582	−	0.609765i	$-0.791264\pi$
$389$	− 474.397i	− 1.21953i	0.792582	−	0.609765i	$-0.208736\pi$
$390$	0	0
$391$	−41.7856	−0.106869
$392$	0	0
$393$	0	0
$394$	0	0
$395$	735.557i	1.86217i
$396$	0	0
$397$	400.288	1.00828	0.504141	−	0.863622i	$-0.331809\pi$
$397$	400.288	1.00828	0.504141	+	0.863622i	$0.331809\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 651.065i	− 1.62360i	−0.583933	−	0.811802i	$-0.698487\pi$
$401$	− 651.065i	− 1.62360i	0.583933	−	0.811802i	$-0.301513\pi$
$402$	0	0
$403$	192.413	0.477452
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 144.135i	− 0.354140i
$408$	0	0
$409$	−189.114	−0.462381	−0.231190	−	0.972909i	$-0.574262\pi$
$409$	−189.114	−0.462381	−0.231190	+	0.972909i	$0.574262\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 213.426i	− 0.516770i
$414$	0	0
$415$	−147.336	−0.355026
$416$	0	0
$417$	0	0
$418$	0	0
$419$	829.772i	1.98036i	0.139791	+	0.990181i	$0.455357\pi$
$419$	829.772i	1.98036i	−0.139791	+	0.990181i	$0.544643\pi$
$420$	0	0
$421$	−226.324	−0.537587	−0.268794	−	0.963198i	$-0.586625\pi$
$421$	−226.324	−0.537587	−0.268794	+	0.963198i	$0.586625\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	27.3766i	0.0644156i
$426$	0	0
$427$	−165.911	−0.388551
$428$	0	0
$429$	0	0
$430$	0	0
$431$	436.460i	1.01267i	0.862338	+	0.506334i	$0.169000\pi$
$431$	436.460i	1.01267i	−0.862338	+	0.506334i	$0.831000\pi$
$432$	0	0
$433$	404.243	0.933587	0.466793	−	0.884366i	$-0.345409\pi$
$433$	404.243	0.933587	0.466793	+	0.884366i	$0.345409\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 62.5175i	− 0.143061i
$438$	0	0
$439$	182.818	0.416442	0.208221	−	0.978082i	$-0.433233\pi$
$439$	182.818	0.416442	0.208221	+	0.978082i	$0.433233\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 331.212i	− 0.747658i	−0.927498	−	0.373829i	$-0.878045\pi$
$443$	− 331.212i	− 0.747658i	0.927498	−	0.373829i	$-0.121955\pi$
$444$	0	0
$445$	−867.231	−1.94883
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 12.8425i	− 0.0286025i	−0.999898	−	0.0143013i	$-0.995448\pi$
$449$	− 12.8425i	− 0.0286025i	0.999898	−	0.0143013i	$-0.00455239\pi$
$450$	0	0
$451$	93.8745	0.208147
$452$	0	0
$453$	0	0
$454$	0	0
$455$	171.647i	0.377245i
$456$	0	0
$457$	−310.672	−0.679807	−0.339904	−	0.940460i	$-0.610394\pi$
$457$	−310.672	−0.679807	−0.339904	+	0.940460i	$0.610394\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	389.770i	0.845489i	0.906249	+	0.422744i	$0.138933\pi$
$461$	389.770i	0.845489i	−0.906249	+	0.422744i	$0.861067\pi$
$462$	0	0
$463$	−152.081	−0.328469	−0.164234	−	0.986421i	$-0.552515\pi$
$463$	−152.081	−0.328469	−0.164234	+	0.986421i	$0.552515\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 86.8679i	− 0.186013i	−0.995666	−	0.0930063i	$-0.970352\pi$
$467$	− 86.8679i	− 0.186013i	0.995666	−	0.0930063i	$-0.0296477\pi$
$468$	0	0
$469$	−307.122	−0.654843
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 168.412i	− 0.356050i
$474$	0	0
$475$	−40.9595	−0.0862305
$476$	0	0
$477$	0	0
$478$	0	0
$479$	549.643i	1.14748i	0.819038	+	0.573740i	$0.194508\pi$
$479$	549.643i	1.14748i	−0.819038	+	0.573740i	$0.805492\pi$
$480$	0	0
$481$	298.834	0.621276
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 587.540i	− 1.21142i
$486$	0	0
$487$	690.745	1.41837	0.709184	−	0.705024i	$-0.249064\pi$
$487$	690.745	1.41837	0.709184	+	0.705024i	$0.249064\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 732.380i	− 1.49161i	−0.666165	−	0.745804i	$-0.732066\pi$
$491$	− 732.380i	− 1.49161i	0.666165	−	0.745804i	$-0.267934\pi$
$492$	0	0
$493$	−965.033	−1.95747
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 131.731i	− 0.265052i
$498$	0	0
$499$	454.405	0.910632	0.455316	−	0.890330i	$-0.349526\pi$
$499$	454.405	0.910632	0.455316	+	0.890330i	$0.349526\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	304.594i	0.605554i	0.953061	+	0.302777i	$0.0979138\pi$
$503$	304.594i	0.605554i	−0.953061	+	0.302777i	$0.902086\pi$
$504$	0	0
$505$	207.660	0.411208
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 74.1436i	− 0.145665i	−0.997344	−	0.0728326i	$-0.976796\pi$
$509$	− 74.1436i	− 0.145665i	0.997344	−	0.0728326i	$-0.0232039\pi$
$510$	0	0
$511$	81.2470	0.158996
$512$	0	0
$513$	0	0
$514$	0	0
$515$	150.147i	0.291548i
$516$	0	0
$517$	−172.737	−0.334115
$518$	0	0
$519$	0	0
$520$	0	0
$521$	214.511i	0.411730i	0.978580	+	0.205865i	$0.0660007\pi$
$521$	214.511i	0.411730i	−0.978580	+	0.205865i	$0.933999\pi$
$522$	0	0
$523$	−932.486	−1.78296	−0.891478	−	0.453064i	$-0.850331\pi$
$523$	−932.486	−1.78296	−0.891478	+	0.453064i	$0.850331\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	264.452i	0.501807i
$528$	0	0
$529$	523.162	0.988964
$530$	0	0
$531$	0	0
$532$	0	0
$533$	194.629i	0.365158i
$534$	0	0
$535$	−937.859	−1.75301
$536$	0	0
$537$	0	0
$538$	0	0
$539$	42.4837i	0.0788195i
$540$	0	0
$541$	−186.340	−0.344436	−0.172218	−	0.985059i	$-0.555093\pi$
$541$	−186.340	−0.344436	−0.172218	+	0.985059i	$0.555093\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	454.552i	0.834041i
$546$	0	0
$547$	−792.170	−1.44821	−0.724104	−	0.689691i	$-0.757747\pi$
$547$	−792.170	−1.44821	−0.724104	+	0.689691i	$0.757747\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1443.83i	− 2.62039i
$552$	0	0
$553$	−377.454	−0.682556
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 399.627i	− 0.717463i	−0.933441	−	0.358732i	$-0.883209\pi$
$557$	− 399.627i	− 0.717463i	0.933441	−	0.358732i	$-0.116791\pi$
$558$	0	0
$559$	349.166	0.624626
$560$	0	0
$561$	0	0
$562$	0	0
$563$	349.379i	0.620567i	0.950644	+	0.310283i	$0.100424\pi$
$563$	349.379i	0.620567i	−0.950644	+	0.310283i	$0.899576\pi$
$564$	0	0
$565$	−334.199	−0.591502
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 755.362i	− 1.32753i	−0.747943	−	0.663763i	$-0.768958\pi$
$569$	− 755.362i	− 1.32753i	0.747943	−	0.663763i	$-0.231042\pi$
$570$	0	0
$571$	−95.1503	−0.166638	−0.0833190	−	0.996523i	$-0.526552\pi$
$571$	−95.1503	−0.166638	−0.0833190	+	0.996523i	$0.526552\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.82483i	0.00665187i
$576$	0	0
$577$	756.065	1.31034	0.655169	−	0.755482i	$-0.272597\pi$
$577$	756.065	1.31034	0.655169	+	0.755482i	$0.272597\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 75.6059i	− 0.130131i
$582$	0	0
$583$	−283.239	−0.485830
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 688.642i	− 1.17316i	−0.809893	−	0.586578i	$-0.800475\pi$
$587$	− 688.642i	− 1.17316i	0.809893	−	0.586578i	$-0.199525\pi$
$588$	0	0
$589$	−395.660	−0.671749
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1016.01i	1.71334i	0.515866	+	0.856669i	$0.327470\pi$
$593$	1016.01i	1.71334i	−0.515866	+	0.856669i	$0.672530\pi$
$594$	0	0
$595$	−235.911	−0.396489
$596$	0	0
$597$	0	0
$598$	0	0
$599$	590.259i	0.985408i	0.870197	+	0.492704i	$0.163991\pi$
$599$	590.259i	0.985408i	−0.870197	+	0.492704i	$0.836009\pi$
$600$	0	0
$601$	300.761	0.500434	0.250217	−	0.968190i	$-0.419498\pi$
$601$	300.761	0.500434	0.250217	+	0.968190i	$0.419498\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 433.949i	− 0.717271i
$606$	0	0
$607$	−696.413	−1.14730	−0.573652	−	0.819099i	$-0.694474\pi$
$607$	−696.413	−1.14730	−0.573652	+	0.819099i	$0.694474\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 358.134i	− 0.586144i
$612$	0	0
$613$	557.393	0.909288	0.454644	−	0.890673i	$-0.349767\pi$
$613$	557.393	0.909288	0.454644	+	0.890673i	$0.349767\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 91.9868i	− 0.149087i	−0.997218	−	0.0745436i	$-0.976250\pi$
$617$	− 91.9868i	− 0.149087i	0.997218	−	0.0745436i	$-0.0237500\pi$
$618$	0	0
$619$	1117.42	1.80520	0.902599	−	0.430483i	$-0.141657\pi$
$619$	1117.42	1.80520	0.902599	+	0.430483i	$0.141657\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 445.022i	− 0.714322i
$624$	0	0
$625$	−662.069	−1.05931
$626$	0	0
$627$	0	0
$628$	0	0
$629$	410.717i	0.652969i
$630$	0	0
$631$	−962.219	−1.52491	−0.762456	−	0.647040i	$-0.776007\pi$
$631$	−962.219	−1.52491	−0.762456	+	0.647040i	$0.776007\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1152.74i	− 1.81535i
$636$	0	0
$637$	−88.0810	−0.138275
$638$	0	0
$639$	0	0
$640$	0	0
$641$	224.442i	0.350144i	0.984556	+	0.175072i	$0.0560158\pi$
$641$	224.442i	0.350144i	−0.984556	+	0.175072i	$0.943984\pi$
$642$	0	0
$643$	332.384	0.516927	0.258464	−	0.966021i	$-0.416784\pi$
$643$	332.384	0.516927	0.258464	+	0.966021i	$0.416784\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 910.554i	− 1.40735i	−0.710523	−	0.703674i	$-0.751542\pi$
$647$	− 910.554i	− 1.40735i	0.710523	−	0.703674i	$-0.248458\pi$
$648$	0	0
$649$	489.579	0.754359
$650$	0	0
$651$	0	0
$652$	0	0
$653$	148.304i	0.227112i	0.993532	+	0.113556i	$0.0362241\pi$
$653$	148.304i	0.227112i	−0.993532	+	0.113556i	$0.963776\pi$
$654$	0	0
$655$	1068.81	1.63177
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 45.5636i	− 0.0691405i	−0.999402	−	0.0345703i	$-0.988994\pi$
$659$	− 45.5636i	− 0.0691405i	0.999402	−	0.0345703i	$-0.0110063\pi$
$660$	0	0
$661$	404.214	0.611520	0.305760	−	0.952109i	$-0.401090\pi$
$661$	404.214	0.611520	0.305760	+	0.952109i	$0.401090\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 352.958i	− 0.530764i
$666$	0	0
$667$	−134.826	−0.202138
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 380.584i	− 0.567190i
$672$	0	0
$673$	1016.49	1.51038	0.755190	−	0.655506i	$-0.227544\pi$
$673$	1016.49	1.51038	0.755190	+	0.655506i	$0.227544\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	232.547i	0.343496i	0.985141	+	0.171748i	$0.0549414\pi$
$677$	232.547i	0.343496i	−0.985141	+	0.171748i	$0.945059\pi$
$678$	0	0
$679$	301.498	0.444032
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 118.245i	− 0.173126i	−0.996246	−	0.0865628i	$-0.972412\pi$
$683$	− 118.245i	− 0.173126i	0.996246	−	0.0865628i	$-0.0275883\pi$
$684$	0	0
$685$	906.272	1.32302
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 587.237i	− 0.852303i
$690$	0	0
$691$	−48.2510	−0.0698278	−0.0349139	−	0.999390i	$-0.511116\pi$
$691$	−48.2510	−0.0698278	−0.0349139	+	0.999390i	$0.511116\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 604.093i	− 0.869198i
$696$	0	0
$697$	−267.498	−0.383785
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 236.466i	− 0.337326i	−0.985674	−	0.168663i	$-0.946055\pi$
$701$	− 236.466i	− 0.337326i	0.985674	−	0.168663i	$-0.0539450\pi$
$702$	0	0
$703$	−614.494	−0.874103
$704$	0	0
$705$	0	0
$706$	0	0
$707$	106.561i	0.150723i
$708$	0	0
$709$	−1341.98	−1.89277	−0.946387	−	0.323036i	$-0.895297\pi$
$709$	−1341.98	−1.89277	−0.946387	+	0.323036i	$0.895297\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	36.9470i	0.0518191i
$714$	0	0
$715$	−393.741	−0.550687
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 284.617i	− 0.395852i	−0.980217	−	0.197926i	$-0.936579\pi$
$719$	− 284.617i	− 0.395852i	0.980217	−	0.197926i	$-0.0634205\pi$
$720$	0	0
$721$	−77.0484	−0.106863
$722$	0	0
$723$	0	0
$724$	0	0
$725$	88.3338i	0.121840i
$726$	0	0
$727$	−430.450	−0.592090	−0.296045	−	0.955174i	$-0.595668\pi$
$727$	−430.450	−0.592090	−0.296045	+	0.955174i	$0.595668\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	479.894i	0.656489i
$732$	0	0
$733$	−387.814	−0.529078	−0.264539	−	0.964375i	$-0.585220\pi$
$733$	−387.814	−0.529078	−0.264539	+	0.964375i	$0.585220\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 704.508i	− 0.955913i
$738$	0	0
$739$	−352.907	−0.477547	−0.238773	−	0.971075i	$-0.576745\pi$
$739$	−352.907	−0.477547	−0.238773	+	0.971075i	$0.576745\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1068.60i	− 1.43822i	−0.694897	−	0.719109i	$-0.744550\pi$
$743$	− 1068.60i	− 1.43822i	0.694897	−	0.719109i	$-0.255450\pi$
$744$	0	0
$745$	−960.620	−1.28942
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 481.265i	− 0.642544i
$750$	0	0
$751$	−1224.15	−1.63002	−0.815011	−	0.579446i	$-0.803269\pi$
$751$	−1224.15	−1.63002	−0.815011	+	0.579446i	$0.803269\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	938.786i	1.24343i
$756$	0	0
$757$	−96.3399	−0.127265	−0.0636327	−	0.997973i	$-0.520269\pi$
$757$	−96.3399	−0.127265	−0.0636327	+	0.997973i	$0.520269\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 948.889i	− 1.24690i	−0.781864	−	0.623449i	$-0.785731\pi$
$761$	− 948.889i	− 1.24690i	0.781864	−	0.623449i	$-0.214269\pi$
$762$	0	0
$763$	−233.255	−0.305708
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1015.04i	1.32339i
$768$	0	0
$769$	−998.633	−1.29861	−0.649306	−	0.760527i	$-0.724941\pi$
$769$	−998.633	−1.29861	−0.649306	+	0.760527i	$0.724941\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 86.4704i	− 0.111863i	−0.998435	−	0.0559317i	$-0.982187\pi$
$773$	− 86.4704i	− 0.111863i	0.998435	−	0.0559317i	$-0.0178129\pi$
$774$	0	0
$775$	24.2065	0.0312342
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 400.217i	− 0.513757i
$780$	0	0
$781$	302.178	0.386911
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 352.540i	− 0.449096i
$786$	0	0
$787$	489.814	0.622382	0.311191	−	0.950347i	$-0.399272\pi$
$787$	489.814	0.622382	0.311191	+	0.950347i	$0.399272\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 171.495i	− 0.216808i
$792$	0	0
$793$	789.061	0.995033
$794$	0	0
$795$	0	0
$796$	0	0
$797$	500.193i	0.627595i	0.949490	+	0.313798i	$0.101601\pi$
$797$	500.193i	0.627595i	−0.949490	+	0.313798i	$0.898399\pi$
$798$	0	0
$799$	492.219	0.616044
$800$	0	0
$801$	0	0
$802$	0	0
$803$	186.373i	0.232096i
$804$	0	0
$805$	−32.9595	−0.0409435
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 310.515i	− 0.383826i	−0.981412	−	0.191913i	$-0.938531\pi$
$809$	− 310.515i	− 0.383826i	0.981412	−	0.191913i	$-0.0614691\pi$
$810$	0	0
$811$	45.9190	0.0566202	0.0283101	−	0.999599i	$-0.490987\pi$
$811$	45.9190	0.0566202	0.0283101	+	0.999599i	$0.490987\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 189.453i	− 0.232458i
$816$	0	0
$817$	−717.992	−0.878815
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1208.66i	− 1.47218i	−0.676883	−	0.736091i	$-0.736670\pi$
$821$	− 1208.66i	− 1.47218i	0.676883	−	0.736091i	$-0.263330\pi$
$822$	0	0
$823$	119.182	0.144814	0.0724069	−	0.997375i	$-0.476932\pi$
$823$	119.182	0.144814	0.0724069	+	0.997375i	$0.476932\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 872.026i	− 1.05445i	−0.849727	−	0.527223i	$-0.823233\pi$
$827$	− 872.026i	− 1.05445i	0.849727	−	0.527223i	$-0.176767\pi$
$828$	0	0
$829$	737.822	0.890015	0.445007	−	0.895527i	$-0.353201\pi$
$829$	737.822	0.890015	0.445007	+	0.895527i	$0.353201\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 121.059i	− 0.145328i
$834$	0	0
$835$	1241.06	1.48630
$836$	0	0
$837$	0	0
$838$	0	0
$839$	896.131i	1.06809i	0.845455	+	0.534047i	$0.179329\pi$
$839$	896.131i	1.06809i	−0.845455	+	0.534047i	$0.820671\pi$
$840$	0	0
$841$	−2272.79	−2.70249
$842$	0	0
$843$	0	0
$844$	0	0
$845$	55.0027i	0.0650920i
$846$	0	0
$847$	222.682	0.262907
$848$	0	0
$849$	0	0
$850$	0	0
$851$	57.3819i	0.0674288i
$852$	0	0
$853$	311.875	0.365621	0.182810	−	0.983148i	$-0.441481\pi$
$853$	311.875	0.365621	0.182810	+	0.983148i	$0.441481\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	360.891i	0.421109i	0.977582	+	0.210555i	$0.0675270\pi$
$857$	360.891i	0.421109i	−0.977582	+	0.210555i	$0.932473\pi$
$858$	0	0
$859$	−1211.10	−1.40989	−0.704946	−	0.709261i	$-0.749029\pi$
$859$	−1211.10	−1.40989	−0.704946	+	0.709261i	$0.749029\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	511.500i	0.592699i	0.955079	+	0.296350i	$0.0957694\pi$
$863$	511.500i	0.592699i	−0.955079	+	0.296350i	$0.904231\pi$
$864$	0	0
$865$	−1758.30	−2.03272
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 865.843i	− 0.996367i
$870$	0	0
$871$	1460.65	1.67698
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 319.435i	− 0.365068i
$876$	0	0
$877$	134.591	0.153467	0.0767337	−	0.997052i	$-0.475551\pi$
$877$	134.591	0.153467	0.0767337	+	0.997052i	$0.475551\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	405.447i	0.460212i	0.973166	+	0.230106i	$0.0739073\pi$
$881$	405.447i	0.460212i	−0.973166	+	0.230106i	$0.926093\pi$
$882$	0	0
$883$	552.235	0.625408	0.312704	−	0.949851i	$-0.398765\pi$
$883$	552.235	0.625408	0.312704	+	0.949851i	$0.398765\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1204.00i	1.35738i	0.734423	+	0.678692i	$0.237453\pi$
$887$	1204.00i	1.35738i	−0.734423	+	0.678692i	$0.762547\pi$
$888$	0	0
$889$	591.535	0.665393
$890$	0	0
$891$	0	0
$892$	0	0
$893$	736.434i	0.824674i
$894$	0	0
$895$	713.624	0.797345
$896$	0	0
$897$	0	0
$898$	0	0
$899$	853.286i	0.949151i
$900$	0	0
$901$	807.098	0.895780
$902$	0	0
$903$	0	0
$904$	0	0
$905$	697.734i	0.770977i
$906$	0	0
$907$	−1320.41	−1.45580	−0.727901	−	0.685682i	$-0.759504\pi$
$907$	−1320.41	−1.45580	−0.727901	+	0.685682i	$0.759504\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	965.020i	1.05930i	0.848217	+	0.529649i	$0.177676\pi$
$911$	965.020i	1.05930i	−0.848217	+	0.529649i	$0.822324\pi$
$912$	0	0
$913$	173.433	0.189959
$914$	0	0
$915$	0	0
$916$	0	0
$917$	548.463i	0.598106i
$918$	0	0
$919$	−720.089	−0.783557	−0.391779	−	0.920060i	$-0.628140\pi$
$919$	−720.089	−0.783557	−0.391779	+	0.920060i	$0.628140\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	626.502i	0.678767i
$924$	0	0
$925$	37.5948	0.0406430
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 931.689i	− 1.00289i	−0.865188	−	0.501447i	$-0.832801\pi$
$929$	− 931.689i	− 1.00289i	0.865188	−	0.501447i	$-0.167199\pi$
$930$	0	0
$931$	181.122	0.194545
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 541.158i	− 0.578778i
$936$	0	0
$937$	1370.97	1.46315	0.731575	−	0.681760i	$-0.238785\pi$
$937$	1370.97	1.46315	0.731575	+	0.681760i	$0.238785\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	227.485i	0.241748i	0.992668	+	0.120874i	$0.0385697\pi$
$941$	227.485i	0.241748i	−0.992668	+	0.120874i	$0.961430\pi$
$942$	0	0
$943$	−37.3725	−0.0396315
$944$	0	0
$945$	0	0
$946$	0	0
$947$	658.944i	0.695823i	0.937527	+	0.347911i	$0.113109\pi$
$947$	658.944i	0.695823i	−0.937527	+	0.347911i	$0.886891\pi$
$948$	0	0
$949$	−386.405	−0.407171
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 217.669i	− 0.228404i	−0.993458	−	0.114202i	$-0.963569\pi$
$953$	− 217.669i	− 0.228404i	0.993458	−	0.114202i	$-0.0364311\pi$
$954$	0	0
$955$	−868.604	−0.909533
$956$	0	0
$957$	0	0
$958$	0	0
$959$	465.056i	0.484939i
$960$	0	0
$961$	−727.170	−0.756680
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 739.857i	− 0.766692i
$966$	0	0
$967$	1018.75	1.05352	0.526760	−	0.850014i	$-0.323407\pi$
$967$	1018.75	1.05352	0.526760	+	0.850014i	$0.323407\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	408.547i	0.420749i	0.977621	+	0.210374i	$0.0674683\pi$
$971$	408.547i	0.420749i	−0.977621	+	0.210374i	$0.932532\pi$
$972$	0	0
$973$	309.992	0.318594
$974$	0	0
$975$	0	0
$976$	0	0
$977$	452.365i	0.463015i	0.972833	+	0.231507i	$0.0743657\pi$
$977$	452.365i	0.463015i	−0.972833	+	0.231507i	$0.925634\pi$
$978$	0	0
$979$	1020.84	1.04274
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1732.56i	− 1.76252i	−0.472628	−	0.881262i	$-0.656695\pi$
$983$	− 1732.56i	− 1.76252i	0.472628	−	0.881262i	$-0.343305\pi$
$984$	0	0
$985$	−1528.77	−1.55205
$986$	0	0
$987$	0	0
$988$	0	0
$989$	67.0466i	0.0677923i
$990$	0	0
$991$	−394.745	−0.398330	−0.199165	−	0.979966i	$-0.563823\pi$
$991$	−394.745	−0.398330	−0.199165	+	0.979966i	$0.563823\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 733.387i	− 0.737072i
$996$	0	0
$997$	165.349	0.165846	0.0829232	−	0.996556i	$-0.473574\pi$
$997$	165.349	0.165846	0.0829232	+	0.996556i	$0.473574\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.d.c.449.1		4
3.2	odd	2	inner	1008.3.d.c.449.4		4
4.3	odd	2		252.3.c.a.197.1	✓	4
8.3	odd	2		4032.3.d.h.449.4		4
8.5	even	2		4032.3.d.e.449.4		4
12.11	even	2		252.3.c.a.197.4	yes	4
24.5	odd	2		4032.3.d.e.449.1		4
24.11	even	2		4032.3.d.h.449.1		4
28.3	even	6		1764.3.bk.e.1745.1		8
28.11	odd	6		1764.3.bk.d.1745.4		8
28.19	even	6		1764.3.bk.e.557.4		8
28.23	odd	6		1764.3.bk.d.557.1		8
28.27	even	2		1764.3.c.f.197.4		4
36.7	odd	6		2268.3.bg.c.701.4		8
36.11	even	6		2268.3.bg.c.701.1		8
36.23	even	6		2268.3.bg.c.2213.4		8
36.31	odd	6		2268.3.bg.c.2213.1		8
84.11	even	6		1764.3.bk.d.1745.1		8
84.23	even	6		1764.3.bk.d.557.4		8
84.47	odd	6		1764.3.bk.e.557.1		8
84.59	odd	6		1764.3.bk.e.1745.4		8
84.83	odd	2		1764.3.c.f.197.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.c.a.197.1	✓	4	4.3	odd	2
252.3.c.a.197.4	yes	4	12.11	even	2
1008.3.d.c.449.1		4	1.1	even	1	trivial
1008.3.d.c.449.4		4	3.2	odd	2	inner
1764.3.c.f.197.1		4	84.83	odd	2
1764.3.c.f.197.4		4	28.27	even	2
1764.3.bk.d.557.1		8	28.23	odd	6
1764.3.bk.d.557.4		8	84.23	even	6
1764.3.bk.d.1745.1		8	84.11	even	6
1764.3.bk.d.1745.4		8	28.11	odd	6
1764.3.bk.e.557.1		8	84.47	odd	6
1764.3.bk.e.557.4		8	28.19	even	6
1764.3.bk.e.1745.1		8	28.3	even	6
1764.3.bk.e.1745.4		8	84.59	odd	6
2268.3.bg.c.701.1		8	36.11	even	6
2268.3.bg.c.701.4		8	36.7	odd	6
2268.3.bg.c.2213.1		8	36.31	odd	6
2268.3.bg.c.2213.4		8	36.23	even	6
4032.3.d.e.449.1		4	24.5	odd	2
4032.3.d.e.449.4		4	8.5	even	2
4032.3.d.h.449.1		4	24.11	even	2
4032.3.d.h.449.4		4	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.15587i q^{5} +2.64575 q^{7} +6.06910i q^{11} -12.5830 q^{13} -17.2941i q^{17} +25.8745 q^{19} -2.41618i q^{23} -1.58301 q^{25} -55.8014i q^{29} -15.2915 q^{31} -13.6412i q^{35} -23.7490 q^{37} -15.4676i q^{41} -27.7490 q^{43} +28.4617i q^{47} +7.00000 q^{49} +46.6690i q^{53} +31.2915 q^{55} -80.6675i q^{59} -62.7085 q^{61} +64.8764i q^{65} -116.081 q^{67} -49.7896i q^{71} +30.7085 q^{73} +16.0573i q^{77} -142.664 q^{79} -28.5763i q^{83} -89.1660 q^{85} -168.203i q^{89} -33.2915 q^{91} -133.406i q^{95} +113.956 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{13} + 40 q^{19} + 36 q^{25} - 40 q^{31} + 32 q^{37} + 16 q^{43} + 28 q^{49} + 104 q^{55} - 272 q^{61} - 168 q^{67} + 144 q^{73} - 232 q^{79} - 272 q^{85} - 112 q^{91} + 96 q^{97}+O(q^{100}) \) 4 * q - 8 * q^13 + 40 * q^19 + 36 * q^25 - 40 * q^31 + 32 * q^37 + 16 * q^43 + 28 * q^49 + 104 * q^55 - 272 * q^61 - 168 * q^67 + 144 * q^73 - 232 * q^79 - 272 * q^85 - 112 * q^91 + 96 * q^97

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