LMFDB - Embedded newform 4788.2.a.n.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4788,2,Mod(1,4788)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4788.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4788, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,4,0,-2,0,0,0,3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.2323724878$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 532)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4788.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.23607 q^{5} -1.00000 q^{7} +2.61803 q^{11} +3.47214 q^{13} +5.85410 q^{17} +1.00000 q^{19} +8.23607 q^{23} +12.9443 q^{25} +4.61803 q^{29} -3.85410 q^{31} -4.23607 q^{35} -8.23607 q^{37} -11.5623 q^{41} +4.47214 q^{43} -7.47214 q^{47} +1.00000 q^{49} -6.09017 q^{53} +11.0902 q^{55} -11.0000 q^{59} -4.23607 q^{61} +14.7082 q^{65} -7.56231 q^{67} +3.76393 q^{71} +3.61803 q^{73} -2.61803 q^{77} -4.47214 q^{79} -10.5623 q^{83} +24.7984 q^{85} +15.7082 q^{89} -3.47214 q^{91} +4.23607 q^{95} -3.47214 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 2 q^{7} + 3 q^{11} - 2 q^{13} + 5 q^{17} + 2 q^{19} + 12 q^{23} + 8 q^{25} + 7 q^{29} - q^{31} - 4 q^{35} - 12 q^{37} - 3 q^{41} - 6 q^{47} + 2 q^{49} - q^{53} + 11 q^{55} - 22 q^{59}+ \cdots + 2 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 - 2 * q^7 + 3 * q^11 - 2 * q^13 + 5 * q^17 + 2 * q^19 + 12 * q^23 + 8 * q^25 + 7 * q^29 - q^31 - 4 * q^35 - 12 * q^37 - 3 * q^41 - 6 * q^47 + 2 * q^49 - q^53 + 11 * q^55 - 22 * q^59 - 4 * q^61 + 16 * q^65 + 5 * q^67 + 12 * q^71 + 5 * q^73 - 3 * q^77 - q^83 + 25 * q^85 + 18 * q^89 + 2 * q^91 + 4 * q^95 + 2 * q^97

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
$5$	4.23607	1.89443	0.947214	−	0.320603i	$-0.103886\pi$
$5$	4.23607	1.89443	0.947214	+	0.320603i	$0.103886\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.61803	0.789367	0.394683	−	0.918817i	$-0.370854\pi$
$11$	2.61803	0.789367	0.394683	+	0.918817i	$0.370854\pi$
$12$	0	0
$13$	3.47214	0.962997	0.481499	−	0.876447i	$-0.340093\pi$
$13$	3.47214	0.962997	0.481499	+	0.876447i	$0.340093\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.85410	1.41983	0.709914	−	0.704288i	$-0.248734\pi$
$17$	5.85410	1.41983	0.709914	+	0.704288i	$0.248734\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.23607	1.71734	0.858669	−	0.512530i	$-0.171292\pi$
$23$	8.23607	1.71734	0.858669	+	0.512530i	$0.171292\pi$
$24$	0	0
$25$	12.9443	2.58885
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.61803	0.857547	0.428774	−	0.903412i	$-0.358946\pi$
$29$	4.61803	0.857547	0.428774	+	0.903412i	$0.358946\pi$
$30$	0	0
$31$	−3.85410	−0.692217	−0.346109	−	0.938194i	$-0.612497\pi$
$31$	−3.85410	−0.692217	−0.346109	+	0.938194i	$0.612497\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.23607	−0.716026
$36$	0	0
$37$	−8.23607	−1.35400	−0.677001	−	0.735982i	$-0.736721\pi$
$37$	−8.23607	−1.35400	−0.677001	+	0.735982i	$0.736721\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.5623	−1.80573	−0.902864	−	0.429925i	$-0.858540\pi$
$41$	−11.5623	−1.80573	−0.902864	+	0.429925i	$0.858540\pi$
$42$	0	0
$43$	4.47214	0.681994	0.340997	−	0.940064i	$-0.389235\pi$
$43$	4.47214	0.681994	0.340997	+	0.940064i	$0.389235\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.47214	−1.08992	−0.544962	−	0.838461i	$-0.683456\pi$
$47$	−7.47214	−1.08992	−0.544962	+	0.838461i	$0.683456\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.09017	−0.836549	−0.418275	−	0.908321i	$-0.637365\pi$
$53$	−6.09017	−0.836549	−0.418275	+	0.908321i	$0.637365\pi$
$54$	0	0
$55$	11.0902	1.49540
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	0	0
$61$	−4.23607	−0.542373	−0.271186	−	0.962527i	$-0.587416\pi$
$61$	−4.23607	−0.542373	−0.271186	+	0.962527i	$0.587416\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	14.7082	1.82433
$66$	0	0
$67$	−7.56231	−0.923883	−0.461941	−	0.886910i	$-0.652847\pi$
$67$	−7.56231	−0.923883	−0.461941	+	0.886910i	$0.652847\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.76393	0.446697	0.223348	−	0.974739i	$-0.428301\pi$
$71$	3.76393	0.446697	0.223348	+	0.974739i	$0.428301\pi$
$72$	0	0
$73$	3.61803	0.423459	0.211729	−	0.977328i	$-0.432090\pi$
$73$	3.61803	0.423459	0.211729	+	0.977328i	$0.432090\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.61803	−0.298353
$78$	0	0
$79$	−4.47214	−0.503155	−0.251577	−	0.967837i	$-0.580949\pi$
$79$	−4.47214	−0.503155	−0.251577	+	0.967837i	$0.580949\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.5623	−1.15936	−0.579682	−	0.814843i	$-0.696823\pi$
$83$	−10.5623	−1.15936	−0.579682	+	0.814843i	$0.696823\pi$
$84$	0	0
$85$	24.7984	2.68976
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.7082	1.66507	0.832533	−	0.553975i	$-0.186890\pi$
$89$	15.7082	1.66507	0.832533	+	0.553975i	$0.186890\pi$
$90$	0	0
$91$	−3.47214	−0.363979
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.23607	0.434611
$96$	0	0
$97$	−3.47214	−0.352542	−0.176271	−	0.984342i	$-0.556404\pi$
$97$	−3.47214	−0.352542	−0.176271	+	0.984342i	$0.556404\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.4164	1.43449	0.717243	−	0.696823i	$-0.245404\pi$
$101$	14.4164	1.43449	0.717243	+	0.696823i	$0.245404\pi$
$102$	0	0
$103$	6.52786	0.643210	0.321605	−	0.946874i	$-0.395778\pi$
$103$	6.52786	0.643210	0.321605	+	0.946874i	$0.395778\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.1803	−1.46754	−0.733769	−	0.679399i	$-0.762241\pi$
$107$	−15.1803	−1.46754	−0.733769	+	0.679399i	$0.762241\pi$
$108$	0	0
$109$	0.527864	0.0505602	0.0252801	−	0.999680i	$-0.491952\pi$
$109$	0.527864	0.0505602	0.0252801	+	0.999680i	$0.491952\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.5623	1.55805	0.779025	−	0.626992i	$-0.215714\pi$
$113$	16.5623	1.55805	0.779025	+	0.626992i	$0.215714\pi$
$114$	0	0
$115$	34.8885	3.25337
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.85410	−0.536645
$120$	0	0
$121$	−4.14590	−0.376900
$122$	0	0
$123$	0	0
$124$	0	0
$125$	33.6525	3.00997
$126$	0	0
$127$	−15.7082	−1.39388	−0.696939	−	0.717131i	$-0.745455\pi$
$127$	−15.7082	−1.39388	−0.696939	+	0.717131i	$0.745455\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.90983	−0.428974	−0.214487	−	0.976727i	$-0.568808\pi$
$131$	−4.90983	−0.428974	−0.214487	+	0.976727i	$0.568808\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.41641	−0.462755	−0.231377	−	0.972864i	$-0.574323\pi$
$137$	−5.41641	−0.462755	−0.231377	+	0.972864i	$0.574323\pi$
$138$	0	0
$139$	2.52786	0.214411	0.107205	−	0.994237i	$-0.465810\pi$
$139$	2.52786	0.214411	0.107205	+	0.994237i	$0.465810\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.09017	0.760158
$144$	0	0
$145$	19.5623	1.62456
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.4164	−0.853345	−0.426673	−	0.904406i	$-0.640314\pi$
$149$	−10.4164	−0.853345	−0.426673	+	0.904406i	$0.640314\pi$
$150$	0	0
$151$	1.09017	0.0887168	0.0443584	−	0.999016i	$-0.485876\pi$
$151$	1.09017	0.0887168	0.0443584	+	0.999016i	$0.485876\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−16.3262	−1.31135
$156$	0	0
$157$	−11.5623	−0.922772	−0.461386	−	0.887199i	$-0.652648\pi$
$157$	−11.5623	−0.922772	−0.461386	+	0.887199i	$0.652648\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.23607	−0.649093
$162$	0	0
$163$	−3.14590	−0.246406	−0.123203	−	0.992382i	$-0.539317\pi$
$163$	−3.14590	−0.246406	−0.123203	+	0.992382i	$0.539317\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.23607	0.482561	0.241281	−	0.970455i	$-0.422433\pi$
$167$	6.23607	0.482561	0.241281	+	0.970455i	$0.422433\pi$
$168$	0	0
$169$	−0.944272	−0.0726363
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.70820	−0.586044	−0.293022	−	0.956106i	$-0.594661\pi$
$173$	−7.70820	−0.586044	−0.293022	+	0.956106i	$0.594661\pi$
$174$	0	0
$175$	−12.9443	−0.978495
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.79837	0.508134	0.254067	−	0.967187i	$-0.418232\pi$
$179$	6.79837	0.508134	0.254067	+	0.967187i	$0.418232\pi$
$180$	0	0
$181$	10.7984	0.802637	0.401318	−	0.915939i	$-0.368552\pi$
$181$	10.7984	0.802637	0.401318	+	0.915939i	$0.368552\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−34.8885	−2.56506
$186$	0	0
$187$	15.3262	1.12077
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.0902	1.16424	0.582122	−	0.813102i	$-0.302223\pi$
$191$	16.0902	1.16424	0.582122	+	0.813102i	$0.302223\pi$
$192$	0	0
$193$	−9.27051	−0.667306	−0.333653	−	0.942696i	$-0.608281\pi$
$193$	−9.27051	−0.667306	−0.333653	+	0.942696i	$0.608281\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.90983	0.706046	0.353023	−	0.935615i	$-0.385154\pi$
$197$	9.90983	0.706046	0.353023	+	0.935615i	$0.385154\pi$
$198$	0	0
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.61803	−0.324122
$204$	0	0
$205$	−48.9787	−3.42082
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.61803	0.181093
$210$	0	0
$211$	−0.381966	−0.0262956	−0.0131478	−	0.999914i	$-0.504185\pi$
$211$	−0.381966	−0.0262956	−0.0131478	+	0.999914i	$0.504185\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	18.9443	1.29199
$216$	0	0
$217$	3.85410	0.261633
$218$	0	0
$219$	0	0
$220$	0	0
$221$	20.3262	1.36729
$222$	0	0
$223$	−21.7639	−1.45742	−0.728710	−	0.684822i	$-0.759880\pi$
$223$	−21.7639	−1.45742	−0.728710	+	0.684822i	$0.759880\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.14590	−0.540662	−0.270331	−	0.962767i	$-0.587133\pi$
$227$	−8.14590	−0.540662	−0.270331	+	0.962767i	$0.587133\pi$
$228$	0	0
$229$	24.4721	1.61716	0.808582	−	0.588383i	$-0.200235\pi$
$229$	24.4721	1.61716	0.808582	+	0.588383i	$0.200235\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	27.0902	1.77474	0.887368	−	0.461062i	$-0.152531\pi$
$233$	27.0902	1.77474	0.887368	+	0.461062i	$0.152531\pi$
$234$	0	0
$235$	−31.6525	−2.06478
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.6525	−1.01247	−0.506237	−	0.862394i	$-0.668964\pi$
$239$	−15.6525	−1.01247	−0.506237	+	0.862394i	$0.668964\pi$
$240$	0	0
$241$	−11.2361	−0.723779	−0.361889	−	0.932221i	$-0.617868\pi$
$241$	−11.2361	−0.723779	−0.361889	+	0.932221i	$0.617868\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.23607	0.270632
$246$	0	0
$247$	3.47214	0.220927
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.85410	0.495747	0.247873	−	0.968792i	$-0.420268\pi$
$251$	7.85410	0.495747	0.247873	+	0.968792i	$0.420268\pi$
$252$	0	0
$253$	21.5623	1.35561
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.90983	0.618158	0.309079	−	0.951036i	$-0.399979\pi$
$257$	9.90983	0.618158	0.309079	+	0.951036i	$0.399979\pi$
$258$	0	0
$259$	8.23607	0.511764
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.43769	0.335303	0.167651	−	0.985846i	$-0.446382\pi$
$263$	5.43769	0.335303	0.167651	+	0.985846i	$0.446382\pi$
$264$	0	0
$265$	−25.7984	−1.58478
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.32624	0.446689	0.223344	−	0.974740i	$-0.428303\pi$
$269$	7.32624	0.446689	0.223344	+	0.974740i	$0.428303\pi$
$270$	0	0
$271$	21.2705	1.29209	0.646046	−	0.763299i	$-0.276422\pi$
$271$	21.2705	1.29209	0.646046	+	0.763299i	$0.276422\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	33.8885	2.04356
$276$	0	0
$277$	−8.18034	−0.491509	−0.245754	−	0.969332i	$-0.579036\pi$
$277$	−8.18034	−0.491509	−0.245754	+	0.969332i	$0.579036\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.4164	0.979321	0.489660	−	0.871913i	$-0.337121\pi$
$281$	16.4164	0.979321	0.489660	+	0.871913i	$0.337121\pi$
$282$	0	0
$283$	−8.09017	−0.480911	−0.240455	−	0.970660i	$-0.577297\pi$
$283$	−8.09017	−0.480911	−0.240455	+	0.970660i	$0.577297\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.5623	0.682501
$288$	0	0
$289$	17.2705	1.01591
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−0.236068	−0.0137912	−0.00689562	−	0.999976i	$-0.502195\pi$
$293$	−0.236068	−0.0137912	−0.00689562	+	0.999976i	$0.502195\pi$
$294$	0	0
$295$	−46.5967	−2.71297
$296$	0	0
$297$	0	0
$298$	0	0
$299$	28.5967	1.65379
$300$	0	0
$301$	−4.47214	−0.257770
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−17.9443	−1.02749
$306$	0	0
$307$	19.8541	1.13313	0.566567	−	0.824016i	$-0.308271\pi$
$307$	19.8541	1.13313	0.566567	+	0.824016i	$0.308271\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.7426	−0.779274	−0.389637	−	0.920969i	$-0.627400\pi$
$311$	−13.7426	−0.779274	−0.389637	+	0.920969i	$0.627400\pi$
$312$	0	0
$313$	−7.23607	−0.409007	−0.204503	−	0.978866i	$-0.565558\pi$
$313$	−7.23607	−0.409007	−0.204503	+	0.978866i	$0.565558\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.5279	0.647469	0.323735	−	0.946148i	$-0.395061\pi$
$317$	11.5279	0.647469	0.323735	+	0.946148i	$0.395061\pi$
$318$	0	0
$319$	12.0902	0.676920
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.85410	0.325731
$324$	0	0
$325$	44.9443	2.49306
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.47214	0.411952
$330$	0	0
$331$	−11.0344	−0.606508	−0.303254	−	0.952910i	$-0.598073\pi$
$331$	−11.0344	−0.606508	−0.303254	+	0.952910i	$0.598073\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−32.0344	−1.75023
$336$	0	0
$337$	−1.38197	−0.0752805	−0.0376402	−	0.999291i	$-0.511984\pi$
$337$	−1.38197	−0.0752805	−0.0376402	+	0.999291i	$0.511984\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−10.0902	−0.546413
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.9787	−1.07251	−0.536257	−	0.844055i	$-0.680162\pi$
$347$	−19.9787	−1.07251	−0.536257	+	0.844055i	$0.680162\pi$
$348$	0	0
$349$	−1.43769	−0.0769580	−0.0384790	−	0.999259i	$-0.512251\pi$
$349$	−1.43769	−0.0769580	−0.0384790	+	0.999259i	$0.512251\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.9787	−0.584338	−0.292169	−	0.956367i	$-0.594377\pi$
$353$	−10.9787	−0.584338	−0.292169	+	0.956367i	$0.594377\pi$
$354$	0	0
$355$	15.9443	0.846234
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−27.2705	−1.43928	−0.719641	−	0.694346i	$-0.755694\pi$
$359$	−27.2705	−1.43928	−0.719641	+	0.694346i	$0.755694\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15.3262	0.802212
$366$	0	0
$367$	−22.7082	−1.18536	−0.592679	−	0.805439i	$-0.701930\pi$
$367$	−22.7082	−1.18536	−0.592679	+	0.805439i	$0.701930\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.09017	0.316186
$372$	0	0
$373$	23.7984	1.23223	0.616117	−	0.787655i	$-0.288705\pi$
$373$	23.7984	1.23223	0.616117	+	0.787655i	$0.288705\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.0344	0.825816
$378$	0	0
$379$	−13.0000	−0.667765	−0.333883	−	0.942615i	$-0.608359\pi$
$379$	−13.0000	−0.667765	−0.333883	+	0.942615i	$0.608359\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.4164	−0.736644	−0.368322	−	0.929698i	$-0.620068\pi$
$383$	−14.4164	−0.736644	−0.368322	+	0.929698i	$0.620068\pi$
$384$	0	0
$385$	−11.0902	−0.565207
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.43769	−0.123596	−0.0617980	−	0.998089i	$-0.519683\pi$
$389$	−2.43769	−0.123596	−0.0617980	+	0.998089i	$0.519683\pi$
$390$	0	0
$391$	48.2148	2.43833
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−18.9443	−0.953190
$396$	0	0
$397$	−3.70820	−0.186109	−0.0930547	−	0.995661i	$-0.529663\pi$
$397$	−3.70820	−0.186109	−0.0930547	+	0.995661i	$0.529663\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.72949	−0.136304	−0.0681521	−	0.997675i	$-0.521710\pi$
$401$	−2.72949	−0.136304	−0.0681521	+	0.997675i	$0.521710\pi$
$402$	0	0
$403$	−13.3820	−0.666603
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21.5623	−1.06880
$408$	0	0
$409$	14.8541	0.734488	0.367244	−	0.930125i	$-0.380301\pi$
$409$	14.8541	0.734488	0.367244	+	0.930125i	$0.380301\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.0000	0.541275
$414$	0	0
$415$	−44.7426	−2.19633
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.5279	−1.00285	−0.501426	−	0.865201i	$-0.667191\pi$
$419$	−20.5279	−1.00285	−0.501426	+	0.865201i	$0.667191\pi$
$420$	0	0
$421$	−10.1246	−0.493443	−0.246722	−	0.969086i	$-0.579353\pi$
$421$	−10.1246	−0.493443	−0.246722	+	0.969086i	$0.579353\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	75.7771	3.67573
$426$	0	0
$427$	4.23607	0.204998
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	0	0
$433$	−36.5967	−1.75873	−0.879364	−	0.476151i	$-0.842032\pi$
$433$	−36.5967	−1.75873	−0.879364	+	0.476151i	$0.842032\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.23607	0.393985
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.5623	−0.501830	−0.250915	−	0.968009i	$-0.580732\pi$
$443$	−10.5623	−0.501830	−0.250915	+	0.968009i	$0.580732\pi$
$444$	0	0
$445$	66.5410	3.15435
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.27051	0.295924	0.147962	−	0.988993i	$-0.452729\pi$
$449$	6.27051	0.295924	0.147962	+	0.988993i	$0.452729\pi$
$450$	0	0
$451$	−30.2705	−1.42538
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−14.7082	−0.689531
$456$	0	0
$457$	−27.7984	−1.30035	−0.650177	−	0.759783i	$-0.725305\pi$
$457$	−27.7984	−1.30035	−0.650177	+	0.759783i	$0.725305\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−31.2148	−1.45382	−0.726909	−	0.686734i	$-0.759044\pi$
$461$	−31.2148	−1.45382	−0.726909	+	0.686734i	$0.759044\pi$
$462$	0	0
$463$	−4.05573	−0.188486	−0.0942428	−	0.995549i	$-0.530043\pi$
$463$	−4.05573	−0.188486	−0.0942428	+	0.995549i	$0.530043\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	40.4508	1.87184	0.935921	−	0.352210i	$-0.114570\pi$
$467$	40.4508	1.87184	0.935921	+	0.352210i	$0.114570\pi$
$468$	0	0
$469$	7.56231	0.349195
$470$	0	0
$471$	0	0
$472$	0	0
$473$	11.7082	0.538344
$474$	0	0
$475$	12.9443	0.593924
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.3262	0.563200	0.281600	−	0.959532i	$-0.409135\pi$
$479$	12.3262	0.563200	0.281600	+	0.959532i	$0.409135\pi$
$480$	0	0
$481$	−28.5967	−1.30390
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.7082	−0.667865
$486$	0	0
$487$	33.1803	1.50354	0.751772	−	0.659423i	$-0.229199\pi$
$487$	33.1803	1.50354	0.751772	+	0.659423i	$0.229199\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	23.4164	1.05677	0.528384	−	0.849006i	$-0.322798\pi$
$491$	23.4164	1.05677	0.528384	+	0.849006i	$0.322798\pi$
$492$	0	0
$493$	27.0344	1.21757
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.76393	−0.168835
$498$	0	0
$499$	35.6180	1.59448	0.797241	−	0.603661i	$-0.206292\pi$
$499$	35.6180	1.59448	0.797241	+	0.603661i	$0.206292\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.9443	0.577157	0.288578	−	0.957456i	$-0.406817\pi$
$503$	12.9443	0.577157	0.288578	+	0.957456i	$0.406817\pi$
$504$	0	0
$505$	61.0689	2.71753
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−32.8328	−1.45529	−0.727644	−	0.685954i	$-0.759385\pi$
$509$	−32.8328	−1.45529	−0.727644	+	0.685954i	$0.759385\pi$
$510$	0	0
$511$	−3.61803	−0.160052
$512$	0	0
$513$	0	0
$514$	0	0
$515$	27.6525	1.21851
$516$	0	0
$517$	−19.5623	−0.860349
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−39.1803	−1.71652	−0.858261	−	0.513214i	$-0.828455\pi$
$521$	−39.1803	−1.71652	−0.858261	+	0.513214i	$0.828455\pi$
$522$	0	0
$523$	34.5967	1.51281	0.756405	−	0.654103i	$-0.226954\pi$
$523$	34.5967	1.51281	0.756405	+	0.654103i	$0.226954\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−22.5623	−0.982829
$528$	0	0
$529$	44.8328	1.94925
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−40.1459	−1.73891
$534$	0	0
$535$	−64.3050	−2.78015
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.61803	0.112767
$540$	0	0
$541$	−26.3607	−1.13333	−0.566667	−	0.823947i	$-0.691767\pi$
$541$	−26.3607	−1.13333	−0.566667	+	0.823947i	$0.691767\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.23607	0.0957826
$546$	0	0
$547$	35.3951	1.51339	0.756693	−	0.653770i	$-0.226814\pi$
$547$	35.3951	1.51339	0.756693	+	0.653770i	$0.226814\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.61803	0.196735
$552$	0	0
$553$	4.47214	0.190175
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.3262	0.776508	0.388254	−	0.921552i	$-0.373078\pi$
$557$	18.3262	0.776508	0.388254	+	0.921552i	$0.373078\pi$
$558$	0	0
$559$	15.5279	0.656759
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.8328	1.59446	0.797232	−	0.603674i	$-0.206297\pi$
$563$	37.8328	1.59446	0.797232	+	0.603674i	$0.206297\pi$
$564$	0	0
$565$	70.1591	2.95161
$566$	0	0
$567$	0	0
$568$	0	0
$569$	38.0132	1.59359	0.796797	−	0.604247i	$-0.206526\pi$
$569$	38.0132	1.59359	0.796797	+	0.604247i	$0.206526\pi$
$570$	0	0
$571$	0.527864	0.0220904	0.0110452	−	0.999939i	$-0.496484\pi$
$571$	0.527864	0.0220904	0.0110452	+	0.999939i	$0.496484\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	106.610	4.44594
$576$	0	0
$577$	−23.6869	−0.986099	−0.493050	−	0.870001i	$-0.664118\pi$
$577$	−23.6869	−0.986099	−0.493050	+	0.870001i	$0.664118\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.5623	0.438198
$582$	0	0
$583$	−15.9443	−0.660344
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.5836	−0.436832	−0.218416	−	0.975856i	$-0.570089\pi$
$587$	−10.5836	−0.436832	−0.218416	+	0.975856i	$0.570089\pi$
$588$	0	0
$589$	−3.85410	−0.158806
$590$	0	0
$591$	0	0
$592$	0	0
$593$	26.5279	1.08937	0.544684	−	0.838641i	$-0.316649\pi$
$593$	26.5279	1.08937	0.544684	+	0.838641i	$0.316649\pi$
$594$	0	0
$595$	−24.7984	−1.01663
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12.5623	−0.513282	−0.256641	−	0.966507i	$-0.582616\pi$
$599$	−12.5623	−0.513282	−0.256641	+	0.966507i	$0.582616\pi$
$600$	0	0
$601$	−3.14590	−0.128324	−0.0641619	−	0.997940i	$-0.520437\pi$
$601$	−3.14590	−0.128324	−0.0641619	+	0.997940i	$0.520437\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−17.5623	−0.714009
$606$	0	0
$607$	3.00000	0.121766	0.0608831	−	0.998145i	$-0.480608\pi$
$607$	3.00000	0.121766	0.0608831	+	0.998145i	$0.480608\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−25.9443	−1.04959
$612$	0	0
$613$	−14.0902	−0.569097	−0.284548	−	0.958662i	$-0.591844\pi$
$613$	−14.0902	−0.569097	−0.284548	+	0.958662i	$0.591844\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.4377	−0.661757	−0.330878	−	0.943673i	$-0.607345\pi$
$617$	−16.4377	−0.661757	−0.330878	+	0.943673i	$0.607345\pi$
$618$	0	0
$619$	−36.3820	−1.46231	−0.731157	−	0.682209i	$-0.761019\pi$
$619$	−36.3820	−1.46231	−0.731157	+	0.682209i	$0.761019\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−15.7082	−0.629336
$624$	0	0
$625$	77.8328	3.11331
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−48.2148	−1.92245
$630$	0	0
$631$	12.8885	0.513085	0.256542	−	0.966533i	$-0.417417\pi$
$631$	12.8885	0.513085	0.256542	+	0.966533i	$0.417417\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−66.5410	−2.64060
$636$	0	0
$637$	3.47214	0.137571
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.6738	0.935057	0.467529	−	0.883978i	$-0.345144\pi$
$641$	23.6738	0.935057	0.467529	+	0.883978i	$0.345144\pi$
$642$	0	0
$643$	23.1803	0.914143	0.457072	−	0.889430i	$-0.348898\pi$
$643$	23.1803	0.914143	0.457072	+	0.889430i	$0.348898\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.41641	−0.173627	−0.0868135	−	0.996225i	$-0.527668\pi$
$647$	−4.41641	−0.173627	−0.0868135	+	0.996225i	$0.527668\pi$
$648$	0	0
$649$	−28.7984	−1.13044
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.1246	−0.709271	−0.354635	−	0.935005i	$-0.615395\pi$
$653$	−18.1246	−0.709271	−0.354635	+	0.935005i	$0.615395\pi$
$654$	0	0
$655$	−20.7984	−0.812660
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.6180	−0.725256	−0.362628	−	0.931934i	$-0.618120\pi$
$659$	−18.6180	−0.725256	−0.362628	+	0.931934i	$0.618120\pi$
$660$	0	0
$661$	−25.0557	−0.974555	−0.487277	−	0.873247i	$-0.662010\pi$
$661$	−25.0557	−0.974555	−0.487277	+	0.873247i	$0.662010\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.23607	−0.164268
$666$	0	0
$667$	38.0344	1.47270
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.0902	−0.428131
$672$	0	0
$673$	−12.5623	−0.484241	−0.242121	−	0.970246i	$-0.577843\pi$
$673$	−12.5623	−0.484241	−0.242121	+	0.970246i	$0.577843\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.3820	1.16767	0.583837	−	0.811871i	$-0.301551\pi$
$677$	30.3820	1.16767	0.583837	+	0.811871i	$0.301551\pi$
$678$	0	0
$679$	3.47214	0.133248
$680$	0	0
$681$	0	0
$682$	0	0
$683$	10.2361	0.391672	0.195836	−	0.980637i	$-0.437258\pi$
$683$	10.2361	0.391672	0.195836	+	0.980637i	$0.437258\pi$
$684$	0	0
$685$	−22.9443	−0.876656
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−21.1459	−0.805595
$690$	0	0
$691$	−14.2918	−0.543686	−0.271843	−	0.962342i	$-0.587633\pi$
$691$	−14.2918	−0.543686	−0.271843	+	0.962342i	$0.587633\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.7082	0.406185
$696$	0	0
$697$	−67.6869	−2.56382
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−20.3607	−0.769012	−0.384506	−	0.923122i	$-0.625628\pi$
$701$	−20.3607	−0.769012	−0.384506	+	0.923122i	$0.625628\pi$
$702$	0	0
$703$	−8.23607	−0.310629
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14.4164	−0.542185
$708$	0	0
$709$	35.2918	1.32541	0.662706	−	0.748880i	$-0.269408\pi$
$709$	35.2918	1.32541	0.662706	+	0.748880i	$0.269408\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−31.7426	−1.18877
$714$	0	0
$715$	38.5066	1.44006
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16.9443	0.631915	0.315957	−	0.948773i	$-0.397674\pi$
$719$	16.9443	0.631915	0.315957	+	0.948773i	$0.397674\pi$
$720$	0	0
$721$	−6.52786	−0.243110
$722$	0	0
$723$	0	0
$724$	0	0
$725$	59.7771	2.22007
$726$	0	0
$727$	37.0689	1.37481	0.687404	−	0.726275i	$-0.258750\pi$
$727$	37.0689	1.37481	0.687404	+	0.726275i	$0.258750\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	26.1803	0.968315
$732$	0	0
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.7984	−0.729282
$738$	0	0
$739$	3.29180	0.121091	0.0605453	−	0.998165i	$-0.480716\pi$
$739$	3.29180	0.121091	0.0605453	+	0.998165i	$0.480716\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−41.5410	−1.52399	−0.761996	−	0.647582i	$-0.775780\pi$
$743$	−41.5410	−1.52399	−0.761996	+	0.647582i	$0.775780\pi$
$744$	0	0
$745$	−44.1246	−1.61660
$746$	0	0
$747$	0	0
$748$	0	0
$749$	15.1803	0.554678
$750$	0	0
$751$	10.0902	0.368196	0.184098	−	0.982908i	$-0.441064\pi$
$751$	10.0902	0.368196	0.184098	+	0.982908i	$0.441064\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.61803	0.168067
$756$	0	0
$757$	7.52786	0.273605	0.136802	−	0.990598i	$-0.456317\pi$
$757$	7.52786	0.273605	0.136802	+	0.990598i	$0.456317\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−46.5967	−1.68913	−0.844565	−	0.535452i	$-0.820141\pi$
$761$	−46.5967	−1.68913	−0.844565	+	0.535452i	$0.820141\pi$
$762$	0	0
$763$	−0.527864	−0.0191100
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−38.1935	−1.37909
$768$	0	0
$769$	24.9443	0.899513	0.449757	−	0.893151i	$-0.351511\pi$
$769$	24.9443	0.899513	0.449757	+	0.893151i	$0.351511\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	29.7082	1.06853	0.534265	−	0.845317i	$-0.320588\pi$
$773$	29.7082	1.06853	0.534265	+	0.845317i	$0.320588\pi$
$774$	0	0
$775$	−49.8885	−1.79205
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.5623	−0.414263
$780$	0	0
$781$	9.85410	0.352607
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−48.9787	−1.74813
$786$	0	0
$787$	31.4164	1.11987	0.559937	−	0.828535i	$-0.310825\pi$
$787$	31.4164	1.11987	0.559937	+	0.828535i	$0.310825\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16.5623	−0.588888
$792$	0	0
$793$	−14.7082	−0.522304
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.9787	0.813948	0.406974	−	0.913440i	$-0.366584\pi$
$797$	22.9787	0.813948	0.406974	+	0.913440i	$0.366584\pi$
$798$	0	0
$799$	−43.7426	−1.54750
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.47214	0.334264
$804$	0	0
$805$	−34.8885	−1.22966
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−11.0689	−0.389161	−0.194581	−	0.980887i	$-0.562335\pi$
$809$	−11.0689	−0.389161	−0.194581	+	0.980887i	$0.562335\pi$
$810$	0	0
$811$	−48.8328	−1.71475	−0.857376	−	0.514691i	$-0.827907\pi$
$811$	−48.8328	−1.71475	−0.857376	+	0.514691i	$0.827907\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13.3262	−0.466798
$816$	0	0
$817$	4.47214	0.156460
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−44.3607	−1.54820	−0.774099	−	0.633064i	$-0.781797\pi$
$821$	−44.3607	−1.54820	−0.774099	+	0.633064i	$0.781797\pi$
$822$	0	0
$823$	−8.76393	−0.305491	−0.152746	−	0.988266i	$-0.548812\pi$
$823$	−8.76393	−0.305491	−0.152746	+	0.988266i	$0.548812\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.29180	−0.253561	−0.126780	−	0.991931i	$-0.540464\pi$
$827$	−7.29180	−0.253561	−0.126780	+	0.991931i	$0.540464\pi$
$828$	0	0
$829$	−6.11146	−0.212260	−0.106130	−	0.994352i	$-0.533846\pi$
$829$	−6.11146	−0.212260	−0.106130	+	0.994352i	$0.533846\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.85410	0.202833
$834$	0	0
$835$	26.4164	0.914177
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34.4721	−1.19011	−0.595055	−	0.803685i	$-0.702870\pi$
$839$	−34.4721	−1.19011	−0.595055	+	0.803685i	$0.702870\pi$
$840$	0	0
$841$	−7.67376	−0.264612
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.00000	−0.137604
$846$	0	0
$847$	4.14590	0.142455
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−67.8328	−2.32528
$852$	0	0
$853$	−31.6180	−1.08258	−0.541290	−	0.840836i	$-0.682064\pi$
$853$	−31.6180	−1.08258	−0.541290	+	0.840836i	$0.682064\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49.9230	1.70534	0.852668	−	0.522453i	$-0.174983\pi$
$857$	49.9230	1.70534	0.852668	+	0.522453i	$0.174983\pi$
$858$	0	0
$859$	−19.4508	−0.663654	−0.331827	−	0.943340i	$-0.607665\pi$
$859$	−19.4508	−0.663654	−0.331827	+	0.943340i	$0.607665\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.1591	0.958545	0.479273	−	0.877666i	$-0.340901\pi$
$863$	28.1591	0.958545	0.479273	+	0.877666i	$0.340901\pi$
$864$	0	0
$865$	−32.6525	−1.11022
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−11.7082	−0.397174
$870$	0	0
$871$	−26.2574	−0.889697
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−33.6525	−1.13766
$876$	0	0
$877$	54.5410	1.84172	0.920860	−	0.389894i	$-0.127488\pi$
$877$	54.5410	1.84172	0.920860	+	0.389894i	$0.127488\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.21478	0.175690	0.0878452	−	0.996134i	$-0.472002\pi$
$881$	5.21478	0.175690	0.0878452	+	0.996134i	$0.472002\pi$
$882$	0	0
$883$	−45.0689	−1.51669	−0.758344	−	0.651854i	$-0.773991\pi$
$883$	−45.0689	−1.51669	−0.758344	+	0.651854i	$0.773991\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.1246	0.541411	0.270706	−	0.962662i	$-0.412743\pi$
$887$	16.1246	0.541411	0.270706	+	0.962662i	$0.412743\pi$
$888$	0	0
$889$	15.7082	0.526836
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−7.47214	−0.250045
$894$	0	0
$895$	28.7984	0.962623
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−17.7984	−0.593609
$900$	0	0
$901$	−35.6525	−1.18776
$902$	0	0
$903$	0	0
$904$	0	0
$905$	45.7426	1.52054
$906$	0	0
$907$	46.1935	1.53383	0.766915	−	0.641749i	$-0.221791\pi$
$907$	46.1935	1.53383	0.766915	+	0.641749i	$0.221791\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6.58359	0.218124	0.109062	−	0.994035i	$-0.465215\pi$
$911$	6.58359	0.218124	0.109062	+	0.994035i	$0.465215\pi$
$912$	0	0
$913$	−27.6525	−0.915163
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.90983	0.162137
$918$	0	0
$919$	2.63932	0.0870631	0.0435316	−	0.999052i	$-0.486139\pi$
$919$	2.63932	0.0870631	0.0435316	+	0.999052i	$0.486139\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13.0689	0.430168
$924$	0	0
$925$	−106.610	−3.50531
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−16.4508	−0.539735	−0.269867	−	0.962897i	$-0.586980\pi$
$929$	−16.4508	−0.539735	−0.269867	+	0.962897i	$0.586980\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	64.9230	2.12321
$936$	0	0
$937$	−38.2148	−1.24842	−0.624211	−	0.781256i	$-0.714580\pi$
$937$	−38.2148	−1.24842	−0.624211	+	0.781256i	$0.714580\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	24.4721	0.797769	0.398884	−	0.917001i	$-0.369397\pi$
$941$	24.4721	0.797769	0.398884	+	0.917001i	$0.369397\pi$
$942$	0	0
$943$	−95.2279	−3.10105
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.4508	0.859537	0.429769	−	0.902939i	$-0.358595\pi$
$947$	26.4508	0.859537	0.429769	+	0.902939i	$0.358595\pi$
$948$	0	0
$949$	12.5623	0.407790
$950$	0	0
$951$	0	0
$952$	0	0
$953$	31.4508	1.01879	0.509396	−	0.860532i	$-0.329869\pi$
$953$	31.4508	1.01879	0.509396	+	0.860532i	$0.329869\pi$
$954$	0	0
$955$	68.1591	2.20558
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.41641	0.174905
$960$	0	0
$961$	−16.1459	−0.520835
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−39.2705	−1.26416
$966$	0	0
$967$	−24.6869	−0.793878	−0.396939	−	0.917845i	$-0.629928\pi$
$967$	−24.6869	−0.793878	−0.396939	+	0.917845i	$0.629928\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−50.8885	−1.63309	−0.816546	−	0.577281i	$-0.804114\pi$
$971$	−50.8885	−1.63309	−0.816546	+	0.577281i	$0.804114\pi$
$972$	0	0
$973$	−2.52786	−0.0810396
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.7639	1.30415	0.652077	−	0.758153i	$-0.273898\pi$
$977$	40.7639	1.30415	0.652077	+	0.758153i	$0.273898\pi$
$978$	0	0
$979$	41.1246	1.31435
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−33.5836	−1.07115	−0.535575	−	0.844488i	$-0.679905\pi$
$983$	−33.5836	−1.07115	−0.535575	+	0.844488i	$0.679905\pi$
$984$	0	0
$985$	41.9787	1.33755
$986$	0	0
$987$	0	0
$988$	0	0
$989$	36.8328	1.17122
$990$	0	0
$991$	−0.763932	−0.0242671	−0.0121336	−	0.999926i	$-0.503862\pi$
$991$	−0.763932	−0.0242671	−0.0121336	+	0.999926i	$0.503862\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−29.6525	−0.940047
$996$	0	0
$997$	22.6312	0.716737	0.358368	−	0.933580i	$-0.383333\pi$
$997$	22.6312	0.716737	0.358368	+	0.933580i	$0.383333\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.n.1.2		2
3.2	odd	2		532.2.a.d.1.2	✓	2
12.11	even	2		2128.2.a.e.1.1		2
21.20	even	2		3724.2.a.d.1.1		2
24.5	odd	2		8512.2.a.s.1.1		2
24.11	even	2		8512.2.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.a.d.1.2	✓	2	3.2	odd	2
2128.2.a.e.1.1		2	12.11	even	2
3724.2.a.d.1.1		2	21.20	even	2
4788.2.a.n.1.2		2	1.1	even	1	trivial
8512.2.a.s.1.1		2	24.5	odd	2
8512.2.a.z.1.2		2	24.11	even	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:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.a (trivial)

\(f(q)\)	\(=\)	\(q+4.23607 q^{5} -1.00000 q^{7} +2.61803 q^{11} +3.47214 q^{13} +5.85410 q^{17} +1.00000 q^{19} +8.23607 q^{23} +12.9443 q^{25} +4.61803 q^{29} -3.85410 q^{31} -4.23607 q^{35} -8.23607 q^{37} -11.5623 q^{41} +4.47214 q^{43} -7.47214 q^{47} +1.00000 q^{49} -6.09017 q^{53} +11.0902 q^{55} -11.0000 q^{59} -4.23607 q^{61} +14.7082 q^{65} -7.56231 q^{67} +3.76393 q^{71} +3.61803 q^{73} -2.61803 q^{77} -4.47214 q^{79} -10.5623 q^{83} +24.7984 q^{85} +15.7082 q^{89} -3.47214 q^{91} +4.23607 q^{95} -3.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 2 q^{7} + 3 q^{11} - 2 q^{13} + 5 q^{17} + 2 q^{19} + 12 q^{23} + 8 q^{25} + 7 q^{29} - q^{31} - 4 q^{35} - 12 q^{37} - 3 q^{41} - 6 q^{47} + 2 q^{49} - q^{53} + 11 q^{55} - 22 q^{59}+ \cdots + 2 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 - 2 * q^7 + 3 * q^11 - 2 * q^13 + 5 * q^17 + 2 * q^19 + 12 * q^23 + 8 * q^25 + 7 * q^29 - q^31 - 4 * q^35 - 12 * q^37 - 3 * q^41 - 6 * q^47 + 2 * q^49 - q^53 + 11 * q^55 - 22 * q^59 - 4 * q^61 + 16 * q^65 + 5 * q^67 + 12 * q^71 + 5 * q^73 - 3 * q^77 - q^83 + 25 * q^85 + 18 * q^89 + 2 * q^91 + 4 * q^95 + 2 * q^97

Introduction

L-functions

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