LMFDB - Embedded newform 9576.2.a.by.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$76.4647449756$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.585786 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+0.585786 q^{5} -1.00000 q^{7} -2.82843 q^{11} -5.65685 q^{13} +6.24264 q^{17} +1.00000 q^{19} -8.48528 q^{23} -4.65685 q^{25} -1.75736 q^{29} +3.17157 q^{31} -0.585786 q^{35} -6.00000 q^{37} +3.17157 q^{41} -3.17157 q^{43} +13.4142 q^{47} +1.00000 q^{49} -9.07107 q^{53} -1.65685 q^{55} +13.6569 q^{59} +2.00000 q^{61} -3.31371 q^{65} -14.8284 q^{67} +12.7279 q^{71} -3.17157 q^{73} +2.82843 q^{77} +13.6569 q^{79} +3.75736 q^{83} +3.65685 q^{85} +11.1716 q^{89} +5.65685 q^{91} +0.585786 q^{95} -11.6569 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 4 * q^5 - 2 * q^7	$ 2 q + 4 q^{5} - 2 q^{7} + 4 q^{17} + 2 q^{19} + 2 q^{25} - 12 q^{29} + 12 q^{31} - 4 q^{35} - 12 q^{37} + 12 q^{41} - 12 q^{43} + 24 q^{47} + 2 q^{49} - 4 q^{53} + 8 q^{55} + 16 q^{59} + 4 q^{61} + 16 q^{65} - 24 q^{67} - 12 q^{73} + 16 q^{79} + 16 q^{83} - 4 q^{85} + 28 q^{89} + 4 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 - 2 * q^7 + 4 * q^17 + 2 * q^19 + 2 * q^25 - 12 * q^29 + 12 * q^31 - 4 * q^35 - 12 * q^37 + 12 * q^41 - 12 * q^43 + 24 * q^47 + 2 * q^49 - 4 * q^53 + 8 * q^55 + 16 * q^59 + 4 * q^61 + 16 * q^65 - 24 * q^67 - 12 * q^73 + 16 * q^79 + 16 * q^83 - 4 * q^85 + 28 * q^89 + 4 * q^95 - 12 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.585786	0.261972	0.130986	−	0.991384i	$-0.458186\pi$
$5$	0.585786	0.261972	0.130986	+	0.991384i	$0.458186\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	−5.65685	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	−5.65685	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.24264	1.51406	0.757031	−	0.653379i	$-0.226649\pi$
$17$	6.24264	1.51406	0.757031	+	0.653379i	$0.226649\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.48528	−1.76930	−0.884652	−	0.466252i	$-0.845604\pi$
$23$	−8.48528	−1.76930	−0.884652	+	0.466252i	$0.845604\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.75736	−0.326333	−0.163167	−	0.986599i	$-0.552171\pi$
$29$	−1.75736	−0.326333	−0.163167	+	0.986599i	$0.552171\pi$
$30$	0	0
$31$	3.17157	0.569631	0.284816	−	0.958582i	$-0.408068\pi$
$31$	3.17157	0.569631	0.284816	+	0.958582i	$0.408068\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.585786	−0.0990160
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.17157	0.495316	0.247658	−	0.968847i	$-0.420339\pi$
$41$	3.17157	0.495316	0.247658	+	0.968847i	$0.420339\pi$
$42$	0	0
$43$	−3.17157	−0.483660	−0.241830	−	0.970319i	$-0.577748\pi$
$43$	−3.17157	−0.483660	−0.241830	+	0.970319i	$0.577748\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	13.4142	1.95666	0.978332	−	0.207042i	$-0.0663836\pi$
$47$	13.4142	1.95666	0.978332	+	0.207042i	$0.0663836\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.07107	−1.24601	−0.623003	−	0.782219i	$-0.714088\pi$
$53$	−9.07107	−1.24601	−0.623003	+	0.782219i	$0.714088\pi$
$54$	0	0
$55$	−1.65685	−0.223410
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.6569	1.77797	0.888985	−	0.457935i	$-0.151411\pi$
$59$	13.6569	1.77797	0.888985	+	0.457935i	$0.151411\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.31371	−0.411015
$66$	0	0
$67$	−14.8284	−1.81158	−0.905790	−	0.423726i	$-0.860722\pi$
$67$	−14.8284	−1.81158	−0.905790	+	0.423726i	$0.860722\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.7279	1.51053	0.755263	−	0.655422i	$-0.227509\pi$
$71$	12.7279	1.51053	0.755263	+	0.655422i	$0.227509\pi$
$72$	0	0
$73$	−3.17157	−0.371205	−0.185602	−	0.982625i	$-0.559424\pi$
$73$	−3.17157	−0.371205	−0.185602	+	0.982625i	$0.559424\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.82843	0.322329
$78$	0	0
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.75736	0.412424	0.206212	−	0.978507i	$-0.433886\pi$
$83$	3.75736	0.412424	0.206212	+	0.978507i	$0.433886\pi$
$84$	0	0
$85$	3.65685	0.396642
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.1716	1.18418	0.592092	−	0.805870i	$-0.298302\pi$
$89$	11.1716	1.18418	0.592092	+	0.805870i	$0.298302\pi$
$90$	0	0
$91$	5.65685	0.592999
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.585786	0.0601004
$96$	0	0
$97$	−11.6569	−1.18357	−0.591787	−	0.806094i	$-0.701577\pi$
$97$	−11.6569	−1.18357	−0.591787	+	0.806094i	$0.701577\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.07107	0.504590	0.252295	−	0.967650i	$-0.418815\pi$
$101$	5.07107	0.504590	0.252295	+	0.967650i	$0.418815\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.72792	−0.843760	−0.421880	−	0.906652i	$-0.638630\pi$
$107$	−8.72792	−0.843760	−0.421880	+	0.906652i	$0.638630\pi$
$108$	0	0
$109$	12.1421	1.16301	0.581503	−	0.813544i	$-0.302465\pi$
$109$	12.1421	1.16301	0.581503	+	0.813544i	$0.302465\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−11.4142	−1.07376	−0.536879	−	0.843659i	$-0.680397\pi$
$113$	−11.4142	−1.07376	−0.536879	+	0.843659i	$0.680397\pi$
$114$	0	0
$115$	−4.97056	−0.463507
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.24264	−0.572262
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	8.48528	0.752947	0.376473	−	0.926427i	$-0.377137\pi$
$127$	8.48528	0.752947	0.376473	+	0.926427i	$0.377137\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.89949	0.864923	0.432461	−	0.901652i	$-0.357645\pi$
$131$	9.89949	0.864923	0.432461	+	0.901652i	$0.357645\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.343146	0.0293169	0.0146585	−	0.999893i	$-0.495334\pi$
$137$	0.343146	0.0293169	0.0146585	+	0.999893i	$0.495334\pi$
$138$	0	0
$139$	4.48528	0.380437	0.190218	−	0.981742i	$-0.439080\pi$
$139$	4.48528	0.380437	0.190218	+	0.981742i	$0.439080\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	16.0000	1.33799
$144$	0	0
$145$	−1.02944	−0.0854901
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.51472	−0.124091	−0.0620453	−	0.998073i	$-0.519762\pi$
$149$	−1.51472	−0.124091	−0.0620453	+	0.998073i	$0.519762\pi$
$150$	0	0
$151$	−8.48528	−0.690522	−0.345261	−	0.938507i	$-0.612210\pi$
$151$	−8.48528	−0.690522	−0.345261	+	0.938507i	$0.612210\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.85786	0.149227
$156$	0	0
$157$	3.65685	0.291849	0.145924	−	0.989296i	$-0.453384\pi$
$157$	3.65685	0.291849	0.145924	+	0.989296i	$0.453384\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.48528	0.668734
$162$	0	0
$163$	−18.4853	−1.44788	−0.723939	−	0.689863i	$-0.757671\pi$
$163$	−18.4853	−1.44788	−0.723939	+	0.689863i	$0.757671\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.1716	1.32878	0.664388	−	0.747388i	$-0.268692\pi$
$167$	17.1716	1.32878	0.664388	+	0.747388i	$0.268692\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.82843	0.671213	0.335606	−	0.942002i	$-0.391059\pi$
$173$	8.82843	0.671213	0.335606	+	0.942002i	$0.391059\pi$
$174$	0	0
$175$	4.65685	0.352025
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.0711	0.827490	0.413745	−	0.910393i	$-0.364220\pi$
$179$	11.0711	0.827490	0.413745	+	0.910393i	$0.364220\pi$
$180$	0	0
$181$	13.3137	0.989600	0.494800	−	0.869007i	$-0.335241\pi$
$181$	13.3137	0.989600	0.494800	+	0.869007i	$0.335241\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.51472	−0.258407
$186$	0	0
$187$	−17.6569	−1.29120
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.17157	0.663632	0.331816	−	0.943344i	$-0.392339\pi$
$191$	9.17157	0.663632	0.331816	+	0.943344i	$0.392339\pi$
$192$	0	0
$193$	−11.6569	−0.839079	−0.419539	−	0.907737i	$-0.637808\pi$
$193$	−11.6569	−0.839079	−0.419539	+	0.907737i	$0.637808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.9706	1.35160	0.675798	−	0.737087i	$-0.263799\pi$
$197$	18.9706	1.35160	0.675798	+	0.737087i	$0.263799\pi$
$198$	0	0
$199$	14.1421	1.00251	0.501255	−	0.865300i	$-0.332872\pi$
$199$	14.1421	1.00251	0.501255	+	0.865300i	$0.332872\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.75736	0.123342
$204$	0	0
$205$	1.85786	0.129759
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.82843	−0.195646
$210$	0	0
$211$	−4.97056	−0.342188	−0.171094	−	0.985255i	$-0.554730\pi$
$211$	−4.97056	−0.342188	−0.171094	+	0.985255i	$0.554730\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.85786	−0.126705
$216$	0	0
$217$	−3.17157	−0.215300
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−35.3137	−2.37546
$222$	0	0
$223$	4.82843	0.323335	0.161668	−	0.986845i	$-0.448313\pi$
$223$	4.82843	0.323335	0.161668	+	0.986845i	$0.448313\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−28.4853	−1.89063	−0.945317	−	0.326152i	$-0.894248\pi$
$227$	−28.4853	−1.89063	−0.945317	+	0.326152i	$0.894248\pi$
$228$	0	0
$229$	−6.48528	−0.428559	−0.214280	−	0.976772i	$-0.568740\pi$
$229$	−6.48528	−0.428559	−0.214280	+	0.976772i	$0.568740\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.48528	0.424865	0.212432	−	0.977176i	$-0.431861\pi$
$233$	6.48528	0.424865	0.212432	+	0.977176i	$0.431861\pi$
$234$	0	0
$235$	7.85786	0.512591
$236$	0	0
$237$	0	0
$238$	0	0
$239$	13.6569	0.883388	0.441694	−	0.897166i	$-0.354378\pi$
$239$	13.6569	0.883388	0.441694	+	0.897166i	$0.354378\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.585786	0.0374245
$246$	0	0
$247$	−5.65685	−0.359937
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.24264	−0.267793	−0.133897	−	0.990995i	$-0.542749\pi$
$251$	−4.24264	−0.267793	−0.133897	+	0.990995i	$0.542749\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.65685	0.595467	0.297734	−	0.954649i	$-0.403769\pi$
$263$	9.65685	0.595467	0.297734	+	0.954649i	$0.403769\pi$
$264$	0	0
$265$	−5.31371	−0.326419
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0.343146	0.0209220	0.0104610	−	0.999945i	$-0.496670\pi$
$269$	0.343146	0.0209220	0.0104610	+	0.999945i	$0.496670\pi$
$270$	0	0
$271$	16.9706	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$271$	16.9706	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.1716	0.794276
$276$	0	0
$277$	11.3137	0.679775	0.339887	−	0.940466i	$-0.389611\pi$
$277$	11.3137	0.679775	0.339887	+	0.940466i	$0.389611\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.2426	−0.611025	−0.305512	−	0.952188i	$-0.598828\pi$
$281$	−10.2426	−0.611025	−0.305512	+	0.952188i	$0.598828\pi$
$282$	0	0
$283$	−14.1421	−0.840663	−0.420331	−	0.907371i	$-0.638086\pi$
$283$	−14.1421	−0.840663	−0.420331	+	0.907371i	$0.638086\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.17157	−0.187212
$288$	0	0
$289$	21.9706	1.29239
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−18.4853	−1.07992	−0.539961	−	0.841690i	$-0.681561\pi$
$293$	−18.4853	−1.07992	−0.539961	+	0.841690i	$0.681561\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	48.0000	2.77591
$300$	0	0
$301$	3.17157	0.182806
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.17157	0.0670841
$306$	0	0
$307$	12.1421	0.692988	0.346494	−	0.938052i	$-0.387372\pi$
$307$	12.1421	0.692988	0.346494	+	0.938052i	$0.387372\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.0416	−1.13646	−0.568228	−	0.822871i	$-0.692371\pi$
$311$	−20.0416	−1.13646	−0.568228	+	0.822871i	$0.692371\pi$
$312$	0	0
$313$	−30.9706	−1.75056	−0.875280	−	0.483617i	$-0.839323\pi$
$313$	−30.9706	−1.75056	−0.875280	+	0.483617i	$0.839323\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.2132	0.854459	0.427229	−	0.904143i	$-0.359490\pi$
$317$	15.2132	0.854459	0.427229	+	0.904143i	$0.359490\pi$
$318$	0	0
$319$	4.97056	0.278298
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.24264	0.347350
$324$	0	0
$325$	26.3431	1.46125
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−13.4142	−0.739550
$330$	0	0
$331$	14.1421	0.777322	0.388661	−	0.921381i	$-0.372938\pi$
$331$	14.1421	0.777322	0.388661	+	0.921381i	$0.372938\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.68629	−0.474583
$336$	0	0
$337$	−3.17157	−0.172767	−0.0863833	−	0.996262i	$-0.527531\pi$
$337$	−3.17157	−0.172767	−0.0863833	+	0.996262i	$0.527531\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8.97056	−0.485783
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.9706	0.696296	0.348148	−	0.937440i	$-0.386811\pi$
$347$	12.9706	0.696296	0.348148	+	0.937440i	$0.386811\pi$
$348$	0	0
$349$	6.68629	0.357909	0.178954	−	0.983857i	$-0.442729\pi$
$349$	6.68629	0.357909	0.178954	+	0.983857i	$0.442729\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−19.8995	−1.05914	−0.529572	−	0.848265i	$-0.677647\pi$
$353$	−19.8995	−1.05914	−0.529572	+	0.848265i	$0.677647\pi$
$354$	0	0
$355$	7.45584	0.395715
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.4853	0.658948	0.329474	−	0.944165i	$-0.393129\pi$
$359$	12.4853	0.658948	0.329474	+	0.944165i	$0.393129\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.85786	−0.0972451
$366$	0	0
$367$	34.1421	1.78220	0.891102	−	0.453802i	$-0.149933\pi$
$367$	34.1421	1.78220	0.891102	+	0.453802i	$0.149933\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.07107	0.470946
$372$	0	0
$373$	35.6569	1.84624	0.923121	−	0.384510i	$-0.125629\pi$
$373$	35.6569	1.84624	0.923121	+	0.384510i	$0.125629\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.94113	0.511994
$378$	0	0
$379$	9.17157	0.471112	0.235556	−	0.971861i	$-0.424309\pi$
$379$	9.17157	0.471112	0.235556	+	0.971861i	$0.424309\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.51472	0.179594	0.0897969	−	0.995960i	$-0.471378\pi$
$383$	3.51472	0.179594	0.0897969	+	0.995960i	$0.471378\pi$
$384$	0	0
$385$	1.65685	0.0844411
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.4558	0.783642	0.391821	−	0.920041i	$-0.371845\pi$
$389$	15.4558	0.783642	0.391821	+	0.920041i	$0.371845\pi$
$390$	0	0
$391$	−52.9706	−2.67884
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	−21.3137	−1.06970	−0.534852	−	0.844946i	$-0.679633\pi$
$397$	−21.3137	−1.06970	−0.534852	+	0.844946i	$0.679633\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.3848	−1.21772	−0.608859	−	0.793279i	$-0.708372\pi$
$401$	−24.3848	−1.21772	−0.608859	+	0.793279i	$0.708372\pi$
$402$	0	0
$403$	−17.9411	−0.893711
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.9706	0.841200
$408$	0	0
$409$	26.6274	1.31664	0.658321	−	0.752738i	$-0.271267\pi$
$409$	26.6274	1.31664	0.658321	+	0.752738i	$0.271267\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13.6569	−0.672010
$414$	0	0
$415$	2.20101	0.108043
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.7279	−1.01263	−0.506313	−	0.862350i	$-0.668992\pi$
$419$	−20.7279	−1.01263	−0.506313	+	0.862350i	$0.668992\pi$
$420$	0	0
$421$	−18.4853	−0.900917	−0.450459	−	0.892797i	$-0.648740\pi$
$421$	−18.4853	−0.900917	−0.450459	+	0.892797i	$0.648740\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−29.0711	−1.41015
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	30.5858	1.47327	0.736633	−	0.676293i	$-0.236415\pi$
$431$	30.5858	1.47327	0.736633	+	0.676293i	$0.236415\pi$
$432$	0	0
$433$	−40.6274	−1.95243	−0.976215	−	0.216807i	$-0.930436\pi$
$433$	−40.6274	−1.95243	−0.976215	+	0.216807i	$0.930436\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.48528	−0.405906
$438$	0	0
$439$	−2.20101	−0.105048	−0.0525242	−	0.998620i	$-0.516727\pi$
$439$	−2.20101	−0.105048	−0.0525242	+	0.998620i	$0.516727\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−23.3137	−1.10767	−0.553834	−	0.832627i	$-0.686836\pi$
$443$	−23.3137	−1.10767	−0.553834	+	0.832627i	$0.686836\pi$
$444$	0	0
$445$	6.54416	0.310223
$446$	0	0
$447$	0	0
$448$	0	0
$449$	36.8701	1.74001	0.870003	−	0.493047i	$-0.164117\pi$
$449$	36.8701	1.74001	0.870003	+	0.493047i	$0.164117\pi$
$450$	0	0
$451$	−8.97056	−0.422407
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.31371	0.155349
$456$	0	0
$457$	−4.68629	−0.219215	−0.109608	−	0.993975i	$-0.534959\pi$
$457$	−4.68629	−0.219215	−0.109608	+	0.993975i	$0.534959\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−26.5269	−1.23548	−0.617741	−	0.786382i	$-0.711952\pi$
$461$	−26.5269	−1.23548	−0.617741	+	0.786382i	$0.711952\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.89949	0.458094	0.229047	−	0.973415i	$-0.426439\pi$
$467$	9.89949	0.458094	0.229047	+	0.973415i	$0.426439\pi$
$468$	0	0
$469$	14.8284	0.684713
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.97056	0.412467
$474$	0	0
$475$	−4.65685	−0.213671
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−24.7279	−1.12985	−0.564924	−	0.825143i	$-0.691094\pi$
$479$	−24.7279	−1.12985	−0.564924	+	0.825143i	$0.691094\pi$
$480$	0	0
$481$	33.9411	1.54758
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.82843	−0.310063
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.65685	−0.0747728	−0.0373864	−	0.999301i	$-0.511903\pi$
$491$	−1.65685	−0.0747728	−0.0373864	+	0.999301i	$0.511903\pi$
$492$	0	0
$493$	−10.9706	−0.494089
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.7279	−0.570925
$498$	0	0
$499$	−35.5980	−1.59358	−0.796792	−	0.604253i	$-0.793471\pi$
$499$	−35.5980	−1.59358	−0.796792	+	0.604253i	$0.793471\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	34.1838	1.52418	0.762089	−	0.647472i	$-0.224174\pi$
$503$	34.1838	1.52418	0.762089	+	0.647472i	$0.224174\pi$
$504$	0	0
$505$	2.97056	0.132188
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18.9706	−0.840855	−0.420428	−	0.907326i	$-0.638120\pi$
$509$	−18.9706	−0.840855	−0.420428	+	0.907326i	$0.638120\pi$
$510$	0	0
$511$	3.17157	0.140302
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.68629	0.206503
$516$	0	0
$517$	−37.9411	−1.66865
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.02944	0.220344	0.110172	−	0.993913i	$-0.464860\pi$
$521$	5.02944	0.220344	0.110172	+	0.993913i	$0.464860\pi$
$522$	0	0
$523$	28.1421	1.23057	0.615285	−	0.788305i	$-0.289041\pi$
$523$	28.1421	1.23057	0.615285	+	0.788305i	$0.289041\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19.7990	0.862458
$528$	0	0
$529$	49.0000	2.13043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−17.9411	−0.777116
$534$	0	0
$535$	−5.11270	−0.221041
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.82843	−0.121829
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.11270	0.304675
$546$	0	0
$547$	43.5980	1.86412	0.932058	−	0.362310i	$-0.118012\pi$
$547$	43.5980	1.86412	0.932058	+	0.362310i	$0.118012\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.75736	−0.0748660
$552$	0	0
$553$	−13.6569	−0.580749
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.82843	−0.374072	−0.187036	−	0.982353i	$-0.559888\pi$
$557$	−8.82843	−0.374072	−0.187036	+	0.982353i	$0.559888\pi$
$558$	0	0
$559$	17.9411	0.758829
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.51472	0.316708	0.158354	−	0.987382i	$-0.449381\pi$
$563$	7.51472	0.316708	0.158354	+	0.987382i	$0.449381\pi$
$564$	0	0
$565$	−6.68629	−0.281294
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−46.5269	−1.95051	−0.975255	−	0.221084i	$-0.929041\pi$
$569$	−46.5269	−1.95051	−0.975255	+	0.221084i	$0.929041\pi$
$570$	0	0
$571$	20.9706	0.877591	0.438795	−	0.898587i	$-0.355405\pi$
$571$	20.9706	0.877591	0.438795	+	0.898587i	$0.355405\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	39.5147	1.64788
$576$	0	0
$577$	15.6569	0.651803	0.325902	−	0.945404i	$-0.394332\pi$
$577$	15.6569	0.651803	0.325902	+	0.945404i	$0.394332\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.75736	−0.155882
$582$	0	0
$583$	25.6569	1.06260
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.0711	0.622050	0.311025	−	0.950402i	$-0.399328\pi$
$587$	15.0711	0.622050	0.311025	+	0.950402i	$0.399328\pi$
$588$	0	0
$589$	3.17157	0.130682
$590$	0	0
$591$	0	0
$592$	0	0
$593$	11.6152	0.476980	0.238490	−	0.971145i	$-0.423348\pi$
$593$	11.6152	0.476980	0.238490	+	0.971145i	$0.423348\pi$
$594$	0	0
$595$	−3.65685	−0.149916
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−41.8995	−1.71197	−0.855983	−	0.517003i	$-0.827048\pi$
$599$	−41.8995	−1.71197	−0.855983	+	0.517003i	$0.827048\pi$
$600$	0	0
$601$	13.3137	0.543077	0.271539	−	0.962428i	$-0.412467\pi$
$601$	13.3137	0.543077	0.271539	+	0.962428i	$0.412467\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.75736	−0.0714468
$606$	0	0
$607$	−41.9411	−1.70234	−0.851169	−	0.524892i	$-0.824106\pi$
$607$	−41.9411	−1.70234	−0.851169	+	0.524892i	$0.824106\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−75.8823	−3.06987
$612$	0	0
$613$	44.2843	1.78862	0.894312	−	0.447443i	$-0.147665\pi$
$613$	44.2843	1.78862	0.894312	+	0.447443i	$0.147665\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.9411	−1.28590	−0.642951	−	0.765908i	$-0.722290\pi$
$617$	−31.9411	−1.28590	−0.642951	+	0.765908i	$0.722290\pi$
$618$	0	0
$619$	28.4853	1.14492	0.572460	−	0.819933i	$-0.305989\pi$
$619$	28.4853	1.14492	0.572460	+	0.819933i	$0.305989\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11.1716	−0.447580
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−37.4558	−1.49346
$630$	0	0
$631$	−35.4558	−1.41147	−0.705737	−	0.708473i	$-0.749384\pi$
$631$	−35.4558	−1.41147	−0.705737	+	0.708473i	$0.749384\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.97056	0.197251
$636$	0	0
$637$	−5.65685	−0.224133
$638$	0	0
$639$	0	0
$640$	0	0
$641$	11.6985	0.462062	0.231031	−	0.972946i	$-0.425790\pi$
$641$	11.6985	0.462062	0.231031	+	0.972946i	$0.425790\pi$
$642$	0	0
$643$	26.8284	1.05801	0.529005	−	0.848619i	$-0.322565\pi$
$643$	26.8284	1.05801	0.529005	+	0.848619i	$0.322565\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.7279	0.972155	0.486077	−	0.873916i	$-0.338427\pi$
$647$	24.7279	0.972155	0.486077	+	0.873916i	$0.338427\pi$
$648$	0	0
$649$	−38.6274	−1.51626
$650$	0	0
$651$	0	0
$652$	0	0
$653$	41.5980	1.62785	0.813927	−	0.580967i	$-0.197325\pi$
$653$	41.5980	1.62785	0.813927	+	0.580967i	$0.197325\pi$
$654$	0	0
$655$	5.79899	0.226585
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−7.75736	−0.302184	−0.151092	−	0.988520i	$-0.548279\pi$
$659$	−7.75736	−0.302184	−0.151092	+	0.988520i	$0.548279\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.585786	−0.0227158
$666$	0	0
$667$	14.9117	0.577383
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.65685	−0.218380
$672$	0	0
$673$	−1.31371	−0.0506397	−0.0253199	−	0.999679i	$-0.508060\pi$
$673$	−1.31371	−0.0506397	−0.0253199	+	0.999679i	$0.508060\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27.9411	1.07386	0.536932	−	0.843625i	$-0.319583\pi$
$677$	27.9411	1.07386	0.536932	+	0.843625i	$0.319583\pi$
$678$	0	0
$679$	11.6569	0.447349
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.10051	0.0803736	0.0401868	−	0.999192i	$-0.487205\pi$
$683$	2.10051	0.0803736	0.0401868	+	0.999192i	$0.487205\pi$
$684$	0	0
$685$	0.201010	0.00768020
$686$	0	0
$687$	0	0
$688$	0	0
$689$	51.3137	1.95490
$690$	0	0
$691$	−4.97056	−0.189089	−0.0945446	−	0.995521i	$-0.530139\pi$
$691$	−4.97056	−0.189089	−0.0945446	+	0.995521i	$0.530139\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.62742	0.0996636
$696$	0	0
$697$	19.7990	0.749940
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.07107	−0.190717
$708$	0	0
$709$	9.65685	0.362671	0.181335	−	0.983421i	$-0.441958\pi$
$709$	9.65685	0.362671	0.181335	+	0.983421i	$0.441958\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−26.9117	−1.00785
$714$	0	0
$715$	9.37258	0.350515
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30.3848	1.13316	0.566580	−	0.824006i	$-0.308266\pi$
$719$	30.3848	1.13316	0.566580	+	0.824006i	$0.308266\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.18377	0.303937
$726$	0	0
$727$	18.6274	0.690853	0.345426	−	0.938446i	$-0.387734\pi$
$727$	18.6274	0.690853	0.345426	+	0.938446i	$0.387734\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−19.7990	−0.732292
$732$	0	0
$733$	−15.1716	−0.560375	−0.280187	−	0.959945i	$-0.590397\pi$
$733$	−15.1716	−0.560375	−0.280187	+	0.959945i	$0.590397\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	41.9411	1.54492
$738$	0	0
$739$	1.79899	0.0661769	0.0330885	−	0.999452i	$-0.489466\pi$
$739$	1.79899	0.0661769	0.0330885	+	0.999452i	$0.489466\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.9289	0.914554	0.457277	−	0.889324i	$-0.348825\pi$
$743$	24.9289	0.914554	0.457277	+	0.889324i	$0.348825\pi$
$744$	0	0
$745$	−0.887302	−0.0325082
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.72792	0.318911
$750$	0	0
$751$	51.7990	1.89017	0.945086	−	0.326822i	$-0.105978\pi$
$751$	51.7990	1.89017	0.945086	+	0.326822i	$0.105978\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.97056	−0.180897
$756$	0	0
$757$	15.6569	0.569058	0.284529	−	0.958667i	$-0.408163\pi$
$757$	15.6569	0.569058	0.284529	+	0.958667i	$0.408163\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4.78680	−0.173521	−0.0867606	−	0.996229i	$-0.527652\pi$
$761$	−4.78680	−0.173521	−0.0867606	+	0.996229i	$0.527652\pi$
$762$	0	0
$763$	−12.1421	−0.439575
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−77.2548	−2.78951
$768$	0	0
$769$	34.9706	1.26107	0.630535	−	0.776161i	$-0.282835\pi$
$769$	34.9706	1.26107	0.630535	+	0.776161i	$0.282835\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	30.4853	1.09648	0.548240	−	0.836321i	$-0.315298\pi$
$773$	30.4853	1.09648	0.548240	+	0.836321i	$0.315298\pi$
$774$	0	0
$775$	−14.7696	−0.530538
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.17157	0.113633
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.14214	0.0764561
$786$	0	0
$787$	−12.1421	−0.432820	−0.216410	−	0.976303i	$-0.569435\pi$
$787$	−12.1421	−0.432820	−0.216410	+	0.976303i	$0.569435\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11.4142	0.405843
$792$	0	0
$793$	−11.3137	−0.401762
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.4853	0.371408	0.185704	−	0.982606i	$-0.440543\pi$
$797$	10.4853	0.371408	0.185704	+	0.982606i	$0.440543\pi$
$798$	0	0
$799$	83.7401	2.96251
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.97056	0.316564
$804$	0	0
$805$	4.97056	0.175189
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.97056	−0.245072	−0.122536	−	0.992464i	$-0.539103\pi$
$809$	−6.97056	−0.245072	−0.122536	+	0.992464i	$0.539103\pi$
$810$	0	0
$811$	−11.0294	−0.387296	−0.193648	−	0.981071i	$-0.562032\pi$
$811$	−11.0294	−0.387296	−0.193648	+	0.981071i	$0.562032\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10.8284	−0.379303
$816$	0	0
$817$	−3.17157	−0.110959
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.1716	0.389891	0.194945	−	0.980814i	$-0.437547\pi$
$821$	11.1716	0.389891	0.194945	+	0.980814i	$0.437547\pi$
$822$	0	0
$823$	15.8579	0.552770	0.276385	−	0.961047i	$-0.410863\pi$
$823$	15.8579	0.552770	0.276385	+	0.961047i	$0.410863\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.10051	0.0730417	0.0365209	−	0.999333i	$-0.488372\pi$
$827$	2.10051	0.0730417	0.0365209	+	0.999333i	$0.488372\pi$
$828$	0	0
$829$	−6.28427	−0.218262	−0.109131	−	0.994027i	$-0.534807\pi$
$829$	−6.28427	−0.218262	−0.109131	+	0.994027i	$0.534807\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.24264	0.216295
$834$	0	0
$835$	10.0589	0.348102
$836$	0	0
$837$	0	0
$838$	0	0
$839$	49.2548	1.70047	0.850233	−	0.526407i	$-0.176461\pi$
$839$	49.2548	1.70047	0.850233	+	0.526407i	$0.176461\pi$
$840$	0	0
$841$	−25.9117	−0.893506
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.1299	0.382882
$846$	0	0
$847$	3.00000	0.103081
$848$	0	0
$849$	0	0
$850$	0	0
$851$	50.9117	1.74523
$852$	0	0
$853$	34.9706	1.19737	0.598685	−	0.800985i	$-0.295690\pi$
$853$	34.9706	1.19737	0.598685	+	0.800985i	$0.295690\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.51472	0.325017	0.162508	−	0.986707i	$-0.448042\pi$
$857$	9.51472	0.325017	0.162508	+	0.986707i	$0.448042\pi$
$858$	0	0
$859$	42.3431	1.44473	0.722365	−	0.691512i	$-0.243055\pi$
$859$	42.3431	1.44473	0.722365	+	0.691512i	$0.243055\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.2426	−1.23371	−0.616857	−	0.787075i	$-0.711594\pi$
$863$	−36.2426	−1.23371	−0.616857	+	0.787075i	$0.711594\pi$
$864$	0	0
$865$	5.17157	0.175839
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−38.6274	−1.31035
$870$	0	0
$871$	83.8823	2.84224
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.65685	0.191237
$876$	0	0
$877$	41.3137	1.39506	0.697532	−	0.716553i	$-0.254281\pi$
$877$	41.3137	1.39506	0.697532	+	0.716553i	$0.254281\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.3848	0.552017	0.276009	−	0.961155i	$-0.410988\pi$
$881$	16.3848	0.552017	0.276009	+	0.961155i	$0.410988\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−42.4264	−1.42454	−0.712270	−	0.701906i	$-0.752333\pi$
$887$	−42.4264	−1.42454	−0.712270	+	0.701906i	$0.752333\pi$
$888$	0	0
$889$	−8.48528	−0.284587
$890$	0	0
$891$	0	0
$892$	0	0
$893$	13.4142	0.448890
$894$	0	0
$895$	6.48528	0.216779
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5.57359	−0.185890
$900$	0	0
$901$	−56.6274	−1.88653
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.79899	0.259247
$906$	0	0
$907$	47.5980	1.58046	0.790232	−	0.612807i	$-0.209960\pi$
$907$	47.5980	1.58046	0.790232	+	0.612807i	$0.209960\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	47.5563	1.57561	0.787806	−	0.615923i	$-0.211217\pi$
$911$	47.5563	1.57561	0.787806	+	0.615923i	$0.211217\pi$
$912$	0	0
$913$	−10.6274	−0.351716
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.89949	−0.326910
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−72.0000	−2.36991
$924$	0	0
$925$	27.9411	0.918699
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−0.786797	−0.0258140	−0.0129070	−	0.999917i	$-0.504109\pi$
$929$	−0.786797	−0.0258140	−0.0129070	+	0.999917i	$0.504109\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.3431	−0.338257
$936$	0	0
$937$	58.9706	1.92648	0.963242	−	0.268635i	$-0.0865724\pi$
$937$	58.9706	1.92648	0.963242	+	0.268635i	$0.0865724\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26.2843	−0.856843	−0.428421	−	0.903579i	$-0.640930\pi$
$941$	−26.2843	−0.856843	−0.428421	+	0.903579i	$0.640930\pi$
$942$	0	0
$943$	−26.9117	−0.876365
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.4853	0.405717	0.202859	−	0.979208i	$-0.434977\pi$
$947$	12.4853	0.405717	0.202859	+	0.979208i	$0.434977\pi$
$948$	0	0
$949$	17.9411	0.582394
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.95837	0.0634378	0.0317189	−	0.999497i	$-0.489902\pi$
$953$	1.95837	0.0634378	0.0317189	+	0.999497i	$0.489902\pi$
$954$	0	0
$955$	5.37258	0.173853
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.343146	−0.0110808
$960$	0	0
$961$	−20.9411	−0.675520
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.82843	−0.219815
$966$	0	0
$967$	−19.1716	−0.616516	−0.308258	−	0.951303i	$-0.599746\pi$
$967$	−19.1716	−0.616516	−0.308258	+	0.951303i	$0.599746\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	35.3137	1.13327	0.566635	−	0.823969i	$-0.308245\pi$
$971$	35.3137	1.13327	0.566635	+	0.823969i	$0.308245\pi$
$972$	0	0
$973$	−4.48528	−0.143792
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.1543	1.57259	0.786293	−	0.617854i	$-0.211998\pi$
$977$	49.1543	1.57259	0.786293	+	0.617854i	$0.211998\pi$
$978$	0	0
$979$	−31.5980	−1.00988
$980$	0	0
$981$	0	0
$982$	0	0
$983$	28.2843	0.902128	0.451064	−	0.892492i	$-0.351045\pi$
$983$	28.2843	0.902128	0.451064	+	0.892492i	$0.351045\pi$
$984$	0	0
$985$	11.1127	0.354080
$986$	0	0
$987$	0	0
$988$	0	0
$989$	26.9117	0.855742
$990$	0	0
$991$	−32.2843	−1.02554	−0.512772	−	0.858525i	$-0.671381\pi$
$991$	−32.2843	−1.02554	−0.512772	+	0.858525i	$0.671381\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.28427	0.262629
$996$	0	0
$997$	8.34315	0.264230	0.132115	−	0.991234i	$-0.457823\pi$
$997$	8.34315	0.264230	0.132115	+	0.991234i	$0.457823\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.by.1.1		2
3.2	odd	2		3192.2.a.r.1.2	✓	2
12.11	even	2		6384.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.r.1.2	✓	2	3.2	odd	2
6384.2.a.bh.1.2		2	12.11	even	2
9576.2.a.by.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q+0.585786 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+0.585786 q^{5} -1.00000 q^{7} -2.82843 q^{11} -5.65685 q^{13} +6.24264 q^{17} +1.00000 q^{19} -8.48528 q^{23} -4.65685 q^{25} -1.75736 q^{29} +3.17157 q^{31} -0.585786 q^{35} -6.00000 q^{37} +3.17157 q^{41} -3.17157 q^{43} +13.4142 q^{47} +1.00000 q^{49} -9.07107 q^{53} -1.65685 q^{55} +13.6569 q^{59} +2.00000 q^{61} -3.31371 q^{65} -14.8284 q^{67} +12.7279 q^{71} -3.17157 q^{73} +2.82843 q^{77} +13.6569 q^{79} +3.75736 q^{83} +3.65685 q^{85} +11.1716 q^{89} +5.65685 q^{91} +0.585786 q^{95} -11.6569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 4 * q^5 - 2 * q^7	\( 2 q + 4 q^{5} - 2 q^{7} + 4 q^{17} + 2 q^{19} + 2 q^{25} - 12 q^{29} + 12 q^{31} - 4 q^{35} - 12 q^{37} + 12 q^{41} - 12 q^{43} + 24 q^{47} + 2 q^{49} - 4 q^{53} + 8 q^{55} + 16 q^{59} + 4 q^{61} + 16 q^{65} - 24 q^{67} - 12 q^{73} + 16 q^{79} + 16 q^{83} - 4 q^{85} + 28 q^{89} + 4 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 - 2 * q^7 + 4 * q^17 + 2 * q^19 + 2 * q^25 - 12 * q^29 + 12 * q^31 - 4 * q^35 - 12 * q^37 + 12 * q^41 - 12 * q^43 + 24 * q^47 + 2 * q^49 - 4 * q^53 + 8 * q^55 + 16 * q^59 + 4 * q^61 + 16 * q^65 - 24 * q^67 - 12 * q^73 + 16 * q^79 + 16 * q^83 - 4 * q^85 + 28 * q^89 + 4 * q^95 - 12 * q^97

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