LMFDB - Embedded newform 4284.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4284,2,Mod(1,4284)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4284, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4284.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4284 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4284.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:	$34.2079122259$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 476)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	4284.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30278 q^{5} +1.00000 q^{7} +6.60555 q^{13} -1.00000 q^{17} +6.60555 q^{19} +0.302776 q^{25} +4.30278 q^{31} +2.30278 q^{35} -2.60555 q^{37} -3.90833 q^{41} +7.30278 q^{43} -4.60555 q^{47} +1.00000 q^{49} +3.69722 q^{53} -9.21110 q^{59} -7.90833 q^{61} +15.2111 q^{65} -1.69722 q^{67} +7.81665 q^{71} -7.90833 q^{73} +12.6056 q^{79} -6.00000 q^{83} -2.30278 q^{85} +16.6056 q^{89} +6.60555 q^{91} +15.2111 q^{95} -6.30278 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} + 2 q^{7} + 6 q^{13} - 2 q^{17} + 6 q^{19} - 3 q^{25} + 5 q^{31} + q^{35} + 2 q^{37} + 3 q^{41} + 11 q^{43} - 2 q^{47} + 2 q^{49} + 11 q^{53} - 4 q^{59} - 5 q^{61} + 16 q^{65} - 7 q^{67} - 6 q^{71}+ \cdots - 9 q^{97}+O(q^{100}) $ 2 * q + q^5 + 2 * q^7 + 6 * q^13 - 2 * q^17 + 6 * q^19 - 3 * q^25 + 5 * q^31 + q^35 + 2 * q^37 + 3 * q^41 + 11 * q^43 - 2 * q^47 + 2 * q^49 + 11 * q^53 - 4 * q^59 - 5 * q^61 + 16 * q^65 - 7 * q^67 - 6 * q^71 - 5 * q^73 + 18 * q^79 - 12 * q^83 - q^85 + 26 * q^89 + 6 * q^91 + 16 * q^95 - 9 * 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
$5$	2.30278	1.02983	0.514916	−	0.857240i	$-0.327823\pi$
$5$	2.30278	1.02983	0.514916	+	0.857240i	$0.327823\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	6.60555	1.83205	0.916025	−	0.401121i	$-0.131379\pi$
$13$	6.60555	1.83205	0.916025	+	0.401121i	$0.131379\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	6.60555	1.51542	0.757709	−	0.652593i	$-0.226319\pi$
$19$	6.60555	1.51542	0.757709	+	0.652593i	$0.226319\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0.302776	0.0605551
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	4.30278	0.772801	0.386401	−	0.922331i	$-0.373718\pi$
$31$	4.30278	0.772801	0.386401	+	0.922331i	$0.373718\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.30278	0.389240
$36$	0	0
$37$	−2.60555	−0.428350	−0.214175	−	0.976795i	$-0.568706\pi$
$37$	−2.60555	−0.428350	−0.214175	+	0.976795i	$0.568706\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.90833	−0.610378	−0.305189	−	0.952292i	$-0.598720\pi$
$41$	−3.90833	−0.610378	−0.305189	+	0.952292i	$0.598720\pi$
$42$	0	0
$43$	7.30278	1.11366	0.556831	−	0.830626i	$-0.312017\pi$
$43$	7.30278	1.11366	0.556831	+	0.830626i	$0.312017\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.60555	−0.671789	−0.335894	−	0.941900i	$-0.609039\pi$
$47$	−4.60555	−0.671789	−0.335894	+	0.941900i	$0.609039\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.69722	0.507853	0.253926	−	0.967224i	$-0.418278\pi$
$53$	3.69722	0.507853	0.253926	+	0.967224i	$0.418278\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.21110	−1.19918	−0.599592	−	0.800306i	$-0.704670\pi$
$59$	−9.21110	−1.19918	−0.599592	+	0.800306i	$0.704670\pi$
$60$	0	0
$61$	−7.90833	−1.01256	−0.506279	−	0.862370i	$-0.668979\pi$
$61$	−7.90833	−1.01256	−0.506279	+	0.862370i	$0.668979\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	15.2111	1.88671
$66$	0	0
$67$	−1.69722	−0.207349	−0.103674	−	0.994611i	$-0.533060\pi$
$67$	−1.69722	−0.207349	−0.103674	+	0.994611i	$0.533060\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.81665	0.927666	0.463833	−	0.885923i	$-0.346474\pi$
$71$	7.81665	0.927666	0.463833	+	0.885923i	$0.346474\pi$
$72$	0	0
$73$	−7.90833	−0.925600	−0.462800	−	0.886463i	$-0.653155\pi$
$73$	−7.90833	−0.925600	−0.462800	+	0.886463i	$0.653155\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.6056	1.41824	0.709118	−	0.705090i	$-0.249093\pi$
$79$	12.6056	1.41824	0.709118	+	0.705090i	$0.249093\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−2.30278	−0.249771
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.6056	1.76018	0.880092	−	0.474802i	$-0.157480\pi$
$89$	16.6056	1.76018	0.880092	+	0.474802i	$0.157480\pi$
$90$	0	0
$91$	6.60555	0.692450
$92$	0	0
$93$	0	0
$94$	0	0
$95$	15.2111	1.56063
$96$	0	0
$97$	−6.30278	−0.639950	−0.319975	−	0.947426i	$-0.603675\pi$
$97$	−6.30278	−0.639950	−0.319975	+	0.947426i	$0.603675\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−19.8167	−1.97183	−0.985915	−	0.167245i	$-0.946513\pi$
$101$	−19.8167	−1.97183	−0.985915	+	0.167245i	$0.946513\pi$
$102$	0	0
$103$	−7.21110	−0.710531	−0.355266	−	0.934765i	$-0.615610\pi$
$103$	−7.21110	−0.710531	−0.355266	+	0.934765i	$0.615610\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.39445	0.134806	0.0674032	−	0.997726i	$-0.478529\pi$
$107$	1.39445	0.134806	0.0674032	+	0.997726i	$0.478529\pi$
$108$	0	0
$109$	8.42221	0.806701	0.403350	−	0.915046i	$-0.367846\pi$
$109$	8.42221	0.806701	0.403350	+	0.915046i	$0.367846\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.60555	−0.433254	−0.216627	−	0.976254i	$-0.569506\pi$
$113$	−4.60555	−0.433254	−0.216627	+	0.976254i	$0.569506\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.8167	−0.967471
$126$	0	0
$127$	14.9083	1.32290	0.661450	−	0.749989i	$-0.269941\pi$
$127$	14.9083	1.32290	0.661450	+	0.749989i	$0.269941\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	21.2111	1.85322	0.926611	−	0.376021i	$-0.122708\pi$
$131$	21.2111	1.85322	0.926611	+	0.376021i	$0.122708\pi$
$132$	0	0
$133$	6.60555	0.572774
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.6972	1.08480	0.542399	−	0.840121i	$-0.317516\pi$
$137$	12.6972	1.08480	0.542399	+	0.840121i	$0.317516\pi$
$138$	0	0
$139$	7.09167	0.601508	0.300754	−	0.953702i	$-0.402762\pi$
$139$	7.09167	0.601508	0.300754	+	0.953702i	$0.402762\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.7250	−1.45209	−0.726044	−	0.687649i	$-0.758643\pi$
$149$	−17.7250	−1.45209	−0.726044	+	0.687649i	$0.758643\pi$
$150$	0	0
$151$	−11.1194	−0.904886	−0.452443	−	0.891793i	$-0.649447\pi$
$151$	−11.1194	−0.904886	−0.452443	+	0.891793i	$0.649447\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.90833	0.795856
$156$	0	0
$157$	−7.21110	−0.575509	−0.287754	−	0.957704i	$-0.592909\pi$
$157$	−7.21110	−0.575509	−0.287754	+	0.957704i	$0.592909\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.6056	0.987343	0.493671	−	0.869648i	$-0.335655\pi$
$163$	12.6056	0.987343	0.493671	+	0.869648i	$0.335655\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.7250	1.83589	0.917947	−	0.396703i	$-0.129846\pi$
$167$	23.7250	1.83589	0.917947	+	0.396703i	$0.129846\pi$
$168$	0	0
$169$	30.6333	2.35641
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−17.3028	−1.31551	−0.657753	−	0.753234i	$-0.728493\pi$
$173$	−17.3028	−1.31551	−0.657753	+	0.753234i	$0.728493\pi$
$174$	0	0
$175$	0.302776	0.0228877
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.90833	−0.740583	−0.370292	−	0.928916i	$-0.620742\pi$
$179$	−9.90833	−0.740583	−0.370292	+	0.928916i	$0.620742\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.7250	−1.49961	−0.749803	−	0.661661i	$-0.769852\pi$
$191$	−20.7250	−1.49961	−0.749803	+	0.661661i	$0.769852\pi$
$192$	0	0
$193$	−11.8167	−0.850581	−0.425291	−	0.905057i	$-0.639828\pi$
$193$	−11.8167	−0.850581	−0.425291	+	0.905057i	$0.639828\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.60555	0.328132	0.164066	−	0.986449i	$-0.447539\pi$
$197$	4.60555	0.328132	0.164066	+	0.986449i	$0.447539\pi$
$198$	0	0
$199$	21.1194	1.49712	0.748558	−	0.663069i	$-0.230746\pi$
$199$	21.1194	1.49712	0.748558	+	0.663069i	$0.230746\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−21.0278	−1.44761	−0.723805	−	0.690004i	$-0.757609\pi$
$211$	−21.0278	−1.44761	−0.723805	+	0.690004i	$0.757609\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.8167	1.14689
$216$	0	0
$217$	4.30278	0.292091
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.60555	−0.444337
$222$	0	0
$223$	−9.02776	−0.604543	−0.302272	−	0.953222i	$-0.597745\pi$
$223$	−9.02776	−0.604543	−0.302272	+	0.953222i	$0.597745\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.51388	−0.365969	−0.182984	−	0.983116i	$-0.558576\pi$
$227$	−5.51388	−0.365969	−0.182984	+	0.983116i	$0.558576\pi$
$228$	0	0
$229$	10.7889	0.712950	0.356475	−	0.934305i	$-0.383978\pi$
$229$	10.7889	0.712950	0.356475	+	0.934305i	$0.383978\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.8167	0.905159	0.452580	−	0.891724i	$-0.350504\pi$
$233$	13.8167	0.905159	0.452580	+	0.891724i	$0.350504\pi$
$234$	0	0
$235$	−10.6056	−0.691830
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.3028	−0.731116	−0.365558	−	0.930789i	$-0.619122\pi$
$239$	−11.3028	−0.731116	−0.365558	+	0.930789i	$0.619122\pi$
$240$	0	0
$241$	8.48612	0.546639	0.273320	−	0.961923i	$-0.411878\pi$
$241$	8.48612	0.546639	0.273320	+	0.961923i	$0.411878\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.30278	0.147119
$246$	0	0
$247$	43.6333	2.77632
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.18335	−0.264050	−0.132025	−	0.991246i	$-0.542148\pi$
$251$	−4.18335	−0.264050	−0.132025	+	0.991246i	$0.542148\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.21110	−0.200303	−0.100152	−	0.994972i	$-0.531933\pi$
$257$	−3.21110	−0.200303	−0.100152	+	0.994972i	$0.531933\pi$
$258$	0	0
$259$	−2.60555	−0.161901
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.4222	1.13596	0.567981	−	0.823042i	$-0.307725\pi$
$263$	18.4222	1.13596	0.567981	+	0.823042i	$0.307725\pi$
$264$	0	0
$265$	8.51388	0.523003
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.2111	−0.927437	−0.463719	−	0.885983i	$-0.653485\pi$
$269$	−15.2111	−0.927437	−0.463719	+	0.885983i	$0.653485\pi$
$270$	0	0
$271$	−1.21110	−0.0735692	−0.0367846	−	0.999323i	$-0.511712\pi$
$271$	−1.21110	−0.0735692	−0.0367846	+	0.999323i	$0.511712\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.4222	−1.34722	−0.673610	−	0.739087i	$-0.735257\pi$
$277$	−22.4222	−1.34722	−0.673610	+	0.739087i	$0.735257\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3.69722	−0.220558	−0.110279	−	0.993901i	$-0.535174\pi$
$281$	−3.69722	−0.220558	−0.110279	+	0.993901i	$0.535174\pi$
$282$	0	0
$283$	−9.51388	−0.565541	−0.282771	−	0.959188i	$-0.591254\pi$
$283$	−9.51388	−0.565541	−0.282771	+	0.959188i	$0.591254\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.90833	−0.230701
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.6333	1.26383	0.631916	−	0.775037i	$-0.282269\pi$
$293$	21.6333	1.26383	0.631916	+	0.775037i	$0.282269\pi$
$294$	0	0
$295$	−21.2111	−1.23496
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	7.30278	0.420925
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−18.2111	−1.04276
$306$	0	0
$307$	−1.21110	−0.0691213	−0.0345606	−	0.999403i	$-0.511003\pi$
$307$	−1.21110	−0.0691213	−0.0345606	+	0.999403i	$0.511003\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.88057	0.106637	0.0533187	−	0.998578i	$-0.483020\pi$
$311$	1.88057	0.106637	0.0533187	+	0.998578i	$0.483020\pi$
$312$	0	0
$313$	33.3305	1.88395	0.941977	−	0.335679i	$-0.108966\pi$
$313$	33.3305	1.88395	0.941977	+	0.335679i	$0.108966\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.60555	−0.367543
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.60555	−0.253912
$330$	0	0
$331$	5.69722	0.313148	0.156574	−	0.987666i	$-0.449955\pi$
$331$	5.69722	0.313148	0.156574	+	0.987666i	$0.449955\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.90833	−0.213535
$336$	0	0
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	29.2111	1.56363	0.781817	−	0.623508i	$-0.214293\pi$
$349$	29.2111	1.56363	0.781817	+	0.623508i	$0.214293\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.6333	0.832077	0.416039	−	0.909347i	$-0.363418\pi$
$353$	15.6333	0.832077	0.416039	+	0.909347i	$0.363418\pi$
$354$	0	0
$355$	18.0000	0.955341
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.90833	−0.364608	−0.182304	−	0.983242i	$-0.558355\pi$
$359$	−6.90833	−0.364608	−0.182304	+	0.983242i	$0.558355\pi$
$360$	0	0
$361$	24.6333	1.29649
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−18.2111	−0.953213
$366$	0	0
$367$	6.33053	0.330451	0.165226	−	0.986256i	$-0.447165\pi$
$367$	6.33053	0.330451	0.165226	+	0.986256i	$0.447165\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.69722	0.191950
$372$	0	0
$373$	−30.9361	−1.60181	−0.800905	−	0.598792i	$-0.795648\pi$
$373$	−30.9361	−1.60181	−0.800905	+	0.598792i	$0.795648\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−6.78890	−0.348722	−0.174361	−	0.984682i	$-0.555786\pi$
$379$	−6.78890	−0.348722	−0.174361	+	0.984682i	$0.555786\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−18.4222	−0.941331	−0.470665	−	0.882312i	$-0.655986\pi$
$383$	−18.4222	−0.941331	−0.470665	+	0.882312i	$0.655986\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.1194	0.513075	0.256538	−	0.966534i	$-0.417418\pi$
$389$	10.1194	0.513075	0.256538	+	0.966534i	$0.417418\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	29.0278	1.46054
$396$	0	0
$397$	−21.9361	−1.10094	−0.550470	−	0.834855i	$-0.685552\pi$
$397$	−21.9361	−1.10094	−0.550470	+	0.834855i	$0.685552\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.6056	0.829242	0.414621	−	0.909994i	$-0.363914\pi$
$401$	16.6056	0.829242	0.414621	+	0.909994i	$0.363914\pi$
$402$	0	0
$403$	28.4222	1.41581
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−26.6056	−1.31556	−0.657780	−	0.753210i	$-0.728504\pi$
$409$	−26.6056	−1.31556	−0.657780	+	0.753210i	$0.728504\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9.21110	−0.453249
$414$	0	0
$415$	−13.8167	−0.678233
$416$	0	0
$417$	0	0
$418$	0	0
$419$	17.5139	0.855609	0.427804	−	0.903871i	$-0.359287\pi$
$419$	17.5139	0.855609	0.427804	+	0.903871i	$0.359287\pi$
$420$	0	0
$421$	−30.9361	−1.50773	−0.753866	−	0.657028i	$-0.771813\pi$
$421$	−30.9361	−1.50773	−0.753866	+	0.657028i	$0.771813\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.302776	−0.0146868
$426$	0	0
$427$	−7.90833	−0.382711
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.6333	0.753030	0.376515	−	0.926411i	$-0.377122\pi$
$431$	15.6333	0.753030	0.376515	+	0.926411i	$0.377122\pi$
$432$	0	0
$433$	3.81665	0.183417	0.0917083	−	0.995786i	$-0.470767\pi$
$433$	3.81665	0.183417	0.0917083	+	0.995786i	$0.470767\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−7.90833	−0.377444	−0.188722	−	0.982031i	$-0.560434\pi$
$439$	−7.90833	−0.377444	−0.188722	+	0.982031i	$0.560434\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.2111	1.00777	0.503885	−	0.863771i	$-0.331904\pi$
$443$	21.2111	1.00777	0.503885	+	0.863771i	$0.331904\pi$
$444$	0	0
$445$	38.2389	1.81270
$446$	0	0
$447$	0	0
$448$	0	0
$449$	32.2389	1.52145	0.760723	−	0.649077i	$-0.224845\pi$
$449$	32.2389	1.52145	0.760723	+	0.649077i	$0.224845\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	15.2111	0.713107
$456$	0	0
$457$	−3.09167	−0.144622	−0.0723112	−	0.997382i	$-0.523037\pi$
$457$	−3.09167	−0.144622	−0.0723112	+	0.997382i	$0.523037\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	31.8167	1.48185	0.740925	−	0.671588i	$-0.234388\pi$
$461$	31.8167	1.48185	0.740925	+	0.671588i	$0.234388\pi$
$462$	0	0
$463$	−16.6972	−0.775986	−0.387993	−	0.921662i	$-0.626832\pi$
$463$	−16.6972	−0.775986	−0.387993	+	0.921662i	$0.626832\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	35.4500	1.64043	0.820214	−	0.572056i	$-0.193854\pi$
$467$	35.4500	1.64043	0.820214	+	0.572056i	$0.193854\pi$
$468$	0	0
$469$	−1.69722	−0.0783705
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14.5139	−0.663156	−0.331578	−	0.943428i	$-0.607581\pi$
$479$	−14.5139	−0.663156	−0.331578	+	0.943428i	$0.607581\pi$
$480$	0	0
$481$	−17.2111	−0.784759
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.5139	−0.659041
$486$	0	0
$487$	5.21110	0.236138	0.118069	−	0.993005i	$-0.462330\pi$
$487$	5.21110	0.236138	0.118069	+	0.993005i	$0.462330\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.3305	1.14315	0.571575	−	0.820550i	$-0.306332\pi$
$491$	25.3305	1.14315	0.571575	+	0.820550i	$0.306332\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.81665	0.350625
$498$	0	0
$499$	1.57779	0.0706318	0.0353159	−	0.999376i	$-0.488756\pi$
$499$	1.57779	0.0706318	0.0353159	+	0.999376i	$0.488756\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21.1472	−0.942906	−0.471453	−	0.881891i	$-0.656270\pi$
$503$	−21.1472	−0.942906	−0.471453	+	0.881891i	$0.656270\pi$
$504$	0	0
$505$	−45.6333	−2.03066
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11.0278	0.488797	0.244398	−	0.969675i	$-0.421410\pi$
$509$	11.0278	0.488797	0.244398	+	0.969675i	$0.421410\pi$
$510$	0	0
$511$	−7.90833	−0.349844
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.6056	−0.731728
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−39.3583	−1.72432	−0.862159	−	0.506638i	$-0.830888\pi$
$521$	−39.3583	−1.72432	−0.862159	+	0.506638i	$0.830888\pi$
$522$	0	0
$523$	10.7889	0.471766	0.235883	−	0.971782i	$-0.424202\pi$
$523$	10.7889	0.471766	0.235883	+	0.971782i	$0.424202\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.30278	−0.187432
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−25.8167	−1.11824
$534$	0	0
$535$	3.21110	0.138828
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	42.0555	1.80811	0.904054	−	0.427419i	$-0.140577\pi$
$541$	42.0555	1.80811	0.904054	+	0.427419i	$0.140577\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	19.3944	0.830767
$546$	0	0
$547$	−31.2111	−1.33449	−0.667245	−	0.744838i	$-0.732527\pi$
$547$	−31.2111	−1.33449	−0.667245	+	0.744838i	$0.732527\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	12.6056	0.536043
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−33.6333	−1.42509	−0.712544	−	0.701627i	$-0.752457\pi$
$557$	−33.6333	−1.42509	−0.712544	+	0.701627i	$0.752457\pi$
$558$	0	0
$559$	48.2389	2.04029
$560$	0	0
$561$	0	0
$562$	0	0
$563$	27.6333	1.16461	0.582303	−	0.812972i	$-0.302152\pi$
$563$	27.6333	1.16461	0.582303	+	0.812972i	$0.302152\pi$
$564$	0	0
$565$	−10.6056	−0.446179
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.88057	−0.204604	−0.102302	−	0.994753i	$-0.532621\pi$
$569$	−4.88057	−0.204604	−0.102302	+	0.994753i	$0.532621\pi$
$570$	0	0
$571$	−9.02776	−0.377800	−0.188900	−	0.981996i	$-0.560492\pi$
$571$	−9.02776	−0.377800	−0.188900	+	0.981996i	$0.560492\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	28.2389	1.17560	0.587800	−	0.809007i	$-0.299994\pi$
$577$	28.2389	1.17560	0.587800	+	0.809007i	$0.299994\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.00000	−0.248922
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.6056	0.437738	0.218869	−	0.975754i	$-0.429763\pi$
$587$	10.6056	0.437738	0.218869	+	0.975754i	$0.429763\pi$
$588$	0	0
$589$	28.4222	1.17112
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−43.8167	−1.79933	−0.899667	−	0.436576i	$-0.856191\pi$
$593$	−43.8167	−1.79933	−0.899667	+	0.436576i	$0.856191\pi$
$594$	0	0
$595$	−2.30278	−0.0944046
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−43.9638	−1.79631	−0.898157	−	0.439675i	$-0.855094\pi$
$599$	−43.9638	−1.79631	−0.898157	+	0.439675i	$0.855094\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−25.3305	−1.02983
$606$	0	0
$607$	−21.5139	−0.873221	−0.436611	−	0.899651i	$-0.643821\pi$
$607$	−21.5139	−0.873221	−0.436611	+	0.899651i	$0.643821\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−30.4222	−1.23075
$612$	0	0
$613$	29.1472	1.17724	0.588622	−	0.808408i	$-0.299671\pi$
$613$	29.1472	1.17724	0.588622	+	0.808408i	$0.299671\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.0000	0.966204	0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000	0.966204	0.483102	+	0.875564i	$0.339510\pi$
$618$	0	0
$619$	−43.6333	−1.75377	−0.876885	−	0.480700i	$-0.840383\pi$
$619$	−43.6333	−1.75377	−0.876885	+	0.480700i	$0.840383\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	16.6056	0.665287
$624$	0	0
$625$	−26.4222	−1.05689
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.60555	0.103890
$630$	0	0
$631$	−8.11943	−0.323229	−0.161615	−	0.986854i	$-0.551670\pi$
$631$	−8.11943	−0.323229	−0.161615	+	0.986854i	$0.551670\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	34.3305	1.36237
$636$	0	0
$637$	6.60555	0.261721
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.8444	0.507324	0.253662	−	0.967293i	$-0.418365\pi$
$641$	12.8444	0.507324	0.253662	+	0.967293i	$0.418365\pi$
$642$	0	0
$643$	38.1472	1.50438	0.752189	−	0.658947i	$-0.228998\pi$
$643$	38.1472	1.50438	0.752189	+	0.658947i	$0.228998\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18.4222	−0.724252	−0.362126	−	0.932129i	$-0.617949\pi$
$647$	−18.4222	−0.724252	−0.362126	+	0.932129i	$0.617949\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28.0555	−1.09790	−0.548949	−	0.835856i	$-0.684972\pi$
$653$	−28.0555	−1.09790	−0.548949	+	0.835856i	$0.684972\pi$
$654$	0	0
$655$	48.8444	1.90851
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−38.3028	−1.49206	−0.746032	−	0.665910i	$-0.768043\pi$
$659$	−38.3028	−1.49206	−0.746032	+	0.665910i	$0.768043\pi$
$660$	0	0
$661$	−14.1833	−0.551668	−0.275834	−	0.961205i	$-0.588954\pi$
$661$	−14.1833	−0.551668	−0.275834	+	0.961205i	$0.588954\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	15.2111	0.589861
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−20.1833	−0.778011	−0.389005	−	0.921235i	$-0.627181\pi$
$673$	−20.1833	−0.778011	−0.389005	+	0.921235i	$0.627181\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−48.4222	−1.86102	−0.930508	−	0.366271i	$-0.880634\pi$
$677$	−48.4222	−1.86102	−0.930508	+	0.366271i	$0.880634\pi$
$678$	0	0
$679$	−6.30278	−0.241878
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−41.4500	−1.58604	−0.793019	−	0.609196i	$-0.791492\pi$
$683$	−41.4500	−1.58604	−0.793019	+	0.609196i	$0.791492\pi$
$684$	0	0
$685$	29.2389	1.11716
$686$	0	0
$687$	0	0
$688$	0	0
$689$	24.4222	0.930412
$690$	0	0
$691$	29.4861	1.12170	0.560852	−	0.827916i	$-0.310473\pi$
$691$	29.4861	1.12170	0.560852	+	0.827916i	$0.310473\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	16.3305	0.619452
$696$	0	0
$697$	3.90833	0.148038
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.8444	0.711744	0.355872	−	0.934535i	$-0.384184\pi$
$701$	18.8444	0.711744	0.355872	+	0.934535i	$0.384184\pi$
$702$	0	0
$703$	−17.2111	−0.649129
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−19.8167	−0.745282
$708$	0	0
$709$	12.1833	0.457555	0.228778	−	0.973479i	$-0.426527\pi$
$709$	12.1833	0.457555	0.228778	+	0.973479i	$0.426527\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	29.3028	1.09281	0.546405	−	0.837521i	$-0.315996\pi$
$719$	29.3028	1.09281	0.546405	+	0.837521i	$0.315996\pi$
$720$	0	0
$721$	−7.21110	−0.268555
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−22.4222	−0.831594	−0.415797	−	0.909458i	$-0.636497\pi$
$727$	−22.4222	−0.831594	−0.415797	+	0.909458i	$0.636497\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.30278	−0.270103
$732$	0	0
$733$	−30.2389	−1.11690	−0.558449	−	0.829539i	$-0.688603\pi$
$733$	−30.2389	−1.11690	−0.558449	+	0.829539i	$0.688603\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−25.3583	−0.932820	−0.466410	−	0.884569i	$-0.654453\pi$
$739$	−25.3583	−0.932820	−0.466410	+	0.884569i	$0.654453\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.81665	−0.0666466	−0.0333233	−	0.999445i	$-0.510609\pi$
$743$	−1.81665	−0.0666466	−0.0333233	+	0.999445i	$0.510609\pi$
$744$	0	0
$745$	−40.8167	−1.49541
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.39445	0.0509520
$750$	0	0
$751$	18.6056	0.678926	0.339463	−	0.940619i	$-0.389755\pi$
$751$	18.6056	0.678926	0.339463	+	0.940619i	$0.389755\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−25.6056	−0.931881
$756$	0	0
$757$	−45.7250	−1.66190	−0.830951	−	0.556345i	$-0.812203\pi$
$757$	−45.7250	−1.66190	−0.830951	+	0.556345i	$0.812203\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−37.8167	−1.37085	−0.685426	−	0.728142i	$-0.740384\pi$
$761$	−37.8167	−1.37085	−0.685426	+	0.728142i	$0.740384\pi$
$762$	0	0
$763$	8.42221	0.304904
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−60.8444	−2.19696
$768$	0	0
$769$	−53.2666	−1.92084	−0.960422	−	0.278550i	$-0.910146\pi$
$769$	−53.2666	−1.92084	−0.960422	+	0.278550i	$0.910146\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−25.8167	−0.928560	−0.464280	−	0.885688i	$-0.653687\pi$
$773$	−25.8167	−0.928560	−0.464280	+	0.885688i	$0.653687\pi$
$774$	0	0
$775$	1.30278	0.0467971
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−25.8167	−0.924978
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16.6056	−0.592678
$786$	0	0
$787$	2.42221	0.0863423	0.0431711	−	0.999068i	$-0.486254\pi$
$787$	2.42221	0.0863423	0.0431711	+	0.999068i	$0.486254\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.60555	−0.163755
$792$	0	0
$793$	−52.2389	−1.85506
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.422205	−0.0149553	−0.00747764	−	0.999972i	$-0.502380\pi$
$797$	−0.422205	−0.0149553	−0.00747764	+	0.999972i	$0.502380\pi$
$798$	0	0
$799$	4.60555	0.162933
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−40.6056	−1.42762	−0.713808	−	0.700342i	$-0.753031\pi$
$809$	−40.6056	−1.42762	−0.713808	+	0.700342i	$0.753031\pi$
$810$	0	0
$811$	4.09167	0.143678	0.0718390	−	0.997416i	$-0.477113\pi$
$811$	4.09167	0.143678	0.0718390	+	0.997416i	$0.477113\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	29.0278	1.01680
$816$	0	0
$817$	48.2389	1.68766
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.6056	0.370136	0.185068	−	0.982726i	$-0.440749\pi$
$821$	10.6056	0.370136	0.185068	+	0.982726i	$0.440749\pi$
$822$	0	0
$823$	43.8722	1.52929	0.764644	−	0.644453i	$-0.222915\pi$
$823$	43.8722	1.52929	0.764644	+	0.644453i	$0.222915\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−27.6333	−0.960904	−0.480452	−	0.877021i	$-0.659527\pi$
$827$	−27.6333	−0.960904	−0.480452	+	0.877021i	$0.659527\pi$
$828$	0	0
$829$	−26.6056	−0.924049	−0.462024	−	0.886867i	$-0.652877\pi$
$829$	−26.6056	−0.924049	−0.462024	+	0.886867i	$0.652877\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	54.6333	1.89066
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−18.4222	−0.636005	−0.318003	−	0.948090i	$-0.603012\pi$
$839$	−18.4222	−0.636005	−0.318003	+	0.948090i	$0.603012\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	70.5416	2.42671
$846$	0	0
$847$	−11.0000	−0.377964
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−16.4222	−0.562286	−0.281143	−	0.959666i	$-0.590713\pi$
$853$	−16.4222	−0.562286	−0.281143	+	0.959666i	$0.590713\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.5416	1.17992	0.589960	−	0.807433i	$-0.299144\pi$
$857$	34.5416	1.17992	0.589960	+	0.807433i	$0.299144\pi$
$858$	0	0
$859$	37.4500	1.27778	0.638888	−	0.769300i	$-0.279395\pi$
$859$	37.4500	1.27778	0.638888	+	0.769300i	$0.279395\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.9083	−0.643647	−0.321823	−	0.946800i	$-0.604296\pi$
$863$	−18.9083	−0.643647	−0.321823	+	0.946800i	$0.604296\pi$
$864$	0	0
$865$	−39.8444	−1.35475
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−11.2111	−0.379874
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−10.8167	−0.365670
$876$	0	0
$877$	−32.6056	−1.10101	−0.550506	−	0.834831i	$-0.685565\pi$
$877$	−32.6056	−1.10101	−0.550506	+	0.834831i	$0.685565\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.69722	0.326708	0.163354	−	0.986568i	$-0.447769\pi$
$881$	9.69722	0.326708	0.163354	+	0.986568i	$0.447769\pi$
$882$	0	0
$883$	−14.6695	−0.493667	−0.246833	−	0.969058i	$-0.579390\pi$
$883$	−14.6695	−0.493667	−0.246833	+	0.969058i	$0.579390\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.9083	1.64218	0.821090	−	0.570798i	$-0.193366\pi$
$887$	48.9083	1.64218	0.821090	+	0.570798i	$0.193366\pi$
$888$	0	0
$889$	14.9083	0.500009
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−30.4222	−1.01804
$894$	0	0
$895$	−22.8167	−0.762677
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−3.69722	−0.123172
$902$	0	0
$903$	0	0
$904$	0	0
$905$	32.2389	1.07166
$906$	0	0
$907$	−22.8444	−0.758536	−0.379268	−	0.925287i	$-0.623824\pi$
$907$	−22.8444	−0.758536	−0.379268	+	0.925287i	$0.623824\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−37.8167	−1.25292	−0.626461	−	0.779453i	$-0.715497\pi$
$911$	−37.8167	−1.25292	−0.626461	+	0.779453i	$0.715497\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.2111	0.700452
$918$	0	0
$919$	24.3305	0.802590	0.401295	−	0.915949i	$-0.368560\pi$
$919$	24.3305	0.802590	0.401295	+	0.915949i	$0.368560\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	51.6333	1.69953
$924$	0	0
$925$	−0.788897	−0.0259388
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−53.9361	−1.76959	−0.884793	−	0.465985i	$-0.845700\pi$
$929$	−53.9361	−1.76959	−0.884793	+	0.465985i	$0.845700\pi$
$930$	0	0
$931$	6.60555	0.216488
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−6.78890	−0.221784	−0.110892	−	0.993832i	$-0.535371\pi$
$937$	−6.78890	−0.221784	−0.110892	+	0.993832i	$0.535371\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0.275019	0.00896537	0.00448269	−	0.999990i	$-0.498573\pi$
$941$	0.275019	0.00896537	0.00448269	+	0.999990i	$0.498573\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.8444	−1.58723	−0.793615	−	0.608420i	$-0.791804\pi$
$947$	−48.8444	−1.58723	−0.793615	+	0.608420i	$0.791804\pi$
$948$	0	0
$949$	−52.2389	−1.69575
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−17.5139	−0.567330	−0.283665	−	0.958923i	$-0.591550\pi$
$953$	−17.5139	−0.567330	−0.283665	+	0.958923i	$0.591550\pi$
$954$	0	0
$955$	−47.7250	−1.54434
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.6972	0.410015
$960$	0	0
$961$	−12.4861	−0.402778
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−27.2111	−0.875956
$966$	0	0
$967$	16.5139	0.531051	0.265525	−	0.964104i	$-0.414455\pi$
$967$	16.5139	0.531051	0.265525	+	0.964104i	$0.414455\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−36.8444	−1.18239	−0.591197	−	0.806527i	$-0.701344\pi$
$971$	−36.8444	−1.18239	−0.591197	+	0.806527i	$0.701344\pi$
$972$	0	0
$973$	7.09167	0.227349
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.1472	1.06047	0.530236	−	0.847850i	$-0.322103\pi$
$977$	33.1472	1.06047	0.530236	+	0.847850i	$0.322103\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−23.0917	−0.736510	−0.368255	−	0.929725i	$-0.620045\pi$
$983$	−23.0917	−0.736510	−0.368255	+	0.929725i	$0.620045\pi$
$984$	0	0
$985$	10.6056	0.337921
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	14.0000	0.444725	0.222362	−	0.974964i	$-0.428623\pi$
$991$	14.0000	0.444725	0.222362	+	0.974964i	$0.428623\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	48.6333	1.54178
$996$	0	0
$997$	14.1472	0.448046	0.224023	−	0.974584i	$-0.428081\pi$
$997$	14.1472	0.448046	0.224023	+	0.974584i	$0.428081\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4284.2.a.n.1.2		2
3.2	odd	2		476.2.a.d.1.1	✓	2
12.11	even	2		1904.2.a.h.1.2		2
21.20	even	2		3332.2.a.j.1.2		2
24.5	odd	2		7616.2.a.q.1.2		2
24.11	even	2		7616.2.a.v.1.1		2
51.50	odd	2		8092.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.a.d.1.1	✓	2	3.2	odd	2
1904.2.a.h.1.2		2	12.11	even	2
3332.2.a.j.1.2		2	21.20	even	2
4284.2.a.n.1.2		2	1.1	even	1	trivial
7616.2.a.q.1.2		2	24.5	odd	2
7616.2.a.v.1.1		2	24.11	even	2
8092.2.a.k.1.2		2	51.50	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4284 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4284.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30278 q^{5} +1.00000 q^{7} +6.60555 q^{13} -1.00000 q^{17} +6.60555 q^{19} +0.302776 q^{25} +4.30278 q^{31} +2.30278 q^{35} -2.60555 q^{37} -3.90833 q^{41} +7.30278 q^{43} -4.60555 q^{47} +1.00000 q^{49} +3.69722 q^{53} -9.21110 q^{59} -7.90833 q^{61} +15.2111 q^{65} -1.69722 q^{67} +7.81665 q^{71} -7.90833 q^{73} +12.6056 q^{79} -6.00000 q^{83} -2.30278 q^{85} +16.6056 q^{89} +6.60555 q^{91} +15.2111 q^{95} -6.30278 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} + 2 q^{7} + 6 q^{13} - 2 q^{17} + 6 q^{19} - 3 q^{25} + 5 q^{31} + q^{35} + 2 q^{37} + 3 q^{41} + 11 q^{43} - 2 q^{47} + 2 q^{49} + 11 q^{53} - 4 q^{59} - 5 q^{61} + 16 q^{65} - 7 q^{67} - 6 q^{71}+ \cdots - 9 q^{97}+O(q^{100}) \) 2 * q + q^5 + 2 * q^7 + 6 * q^13 - 2 * q^17 + 6 * q^19 - 3 * q^25 + 5 * q^31 + q^35 + 2 * q^37 + 3 * q^41 + 11 * q^43 - 2 * q^47 + 2 * q^49 + 11 * q^53 - 4 * q^59 - 5 * q^61 + 16 * q^65 - 7 * q^67 - 6 * q^71 - 5 * q^73 + 18 * q^79 - 12 * q^83 - q^85 + 26 * q^89 + 6 * q^91 + 16 * q^95 - 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more