LMFDB - Embedded newform 1792.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(1,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1792.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.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:	$14.3091920422$
Analytic rank:	$1$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 896)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1792.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{5} +1.00000 q^{7} -3.00000 q^{9} +O(q^{10})$	$q+2.82843 q^{5} +1.00000 q^{7} -3.00000 q^{9} -2.82843 q^{11} -2.82843 q^{13} -2.00000 q^{17} -5.65685 q^{19} -8.00000 q^{23} +3.00000 q^{25} -5.65685 q^{29} -8.00000 q^{31} +2.82843 q^{35} +5.65685 q^{37} +6.00000 q^{41} -2.82843 q^{43} -8.48528 q^{45} -8.00000 q^{47} +1.00000 q^{49} +11.3137 q^{53} -8.00000 q^{55} +11.3137 q^{59} +2.82843 q^{61} -3.00000 q^{63} -8.00000 q^{65} +8.48528 q^{67} -8.00000 q^{71} -6.00000 q^{73} -2.82843 q^{77} +8.00000 q^{79} +9.00000 q^{81} +5.65685 q^{83} -5.65685 q^{85} -6.00000 q^{89} -2.82843 q^{91} -16.0000 q^{95} +14.0000 q^{97} +8.48528 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} - 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^7 - 6 * q^9	$ 2 q + 2 q^{7} - 6 q^{9} - 4 q^{17} - 16 q^{23} + 6 q^{25} - 16 q^{31} + 12 q^{41} - 16 q^{47} + 2 q^{49} - 16 q^{55} - 6 q^{63} - 16 q^{65} - 16 q^{71} - 12 q^{73} + 16 q^{79} + 18 q^{81} - 12 q^{89} - 32 q^{95} + 28 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 6 * q^9 - 4 * q^17 - 16 * q^23 + 6 * q^25 - 16 * q^31 + 12 * q^41 - 16 * q^47 + 2 * q^49 - 16 * q^55 - 6 * q^63 - 16 * q^65 - 16 * q^71 - 12 * q^73 + 16 * q^79 + 18 * q^81 - 12 * q^89 - 32 * q^95 + 28 * 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		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−3.00000	−1.00000
$10$	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
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.65685	−1.05045	−0.525226	−	0.850963i	$-0.676019\pi$
$29$	−5.65685	−1.05045	−0.525226	+	0.850963i	$0.676019\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.82843	0.478091
$36$	0	0
$37$	5.65685	0.929981	0.464991	−	0.885316i	$-0.346058\pi$
$37$	5.65685	0.929981	0.464991	+	0.885316i	$0.346058\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−2.82843	−0.431331	−0.215666	−	0.976467i	$-0.569192\pi$
$43$	−2.82843	−0.431331	−0.215666	+	0.976467i	$0.569192\pi$
$44$	0	0
$45$	−8.48528	−1.26491
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.3137	1.55406	0.777029	−	0.629465i	$-0.216726\pi$
$53$	11.3137	1.55406	0.777029	+	0.629465i	$0.216726\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	0	0
$61$	2.82843	0.362143	0.181071	−	0.983470i	$-0.442043\pi$
$61$	2.82843	0.362143	0.181071	+	0.983470i	$0.442043\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	−8.00000	−0.992278
$66$	0	0
$67$	8.48528	1.03664	0.518321	−	0.855186i	$-0.326557\pi$
$67$	8.48528	1.03664	0.518321	+	0.855186i	$0.326557\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.82843	−0.322329
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	5.65685	0.620920	0.310460	−	0.950586i	$-0.399517\pi$
$83$	5.65685	0.620920	0.310460	+	0.950586i	$0.399517\pi$
$84$	0	0
$85$	−5.65685	−0.613572
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.0000	−1.64157
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	8.48528	0.852803
$100$	0	0
$101$	−8.48528	−0.844317	−0.422159	−	0.906522i	$-0.638727\pi$
$101$	−8.48528	−0.844317	−0.422159	+	0.906522i	$0.638727\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.1421	1.36717	0.683586	−	0.729870i	$-0.260419\pi$
$107$	14.1421	1.36717	0.683586	+	0.729870i	$0.260419\pi$
$108$	0	0
$109$	16.9706	1.62549	0.812743	−	0.582623i	$-0.197974\pi$
$109$	16.9706	1.62549	0.812743	+	0.582623i	$0.197974\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−22.6274	−2.11002
$116$	0	0
$117$	8.48528	0.784465
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.65685	0.494242	0.247121	−	0.968985i	$-0.420516\pi$
$131$	5.65685	0.494242	0.247121	+	0.968985i	$0.420516\pi$
$132$	0	0
$133$	−5.65685	−0.490511
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−16.9706	−1.43942	−0.719712	−	0.694273i	$-0.755726\pi$
$139$	−16.9706	−1.43942	−0.719712	+	0.694273i	$0.755726\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	−16.0000	−1.32873
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.3137	−0.926855	−0.463428	−	0.886135i	$-0.653381\pi$
$149$	−11.3137	−0.926855	−0.463428	+	0.886135i	$0.653381\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	−22.6274	−1.81748
$156$	0	0
$157$	8.48528	0.677199	0.338600	−	0.940931i	$-0.390047\pi$
$157$	8.48528	0.677199	0.338600	+	0.940931i	$0.390047\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−19.7990	−1.55078	−0.775388	−	0.631485i	$-0.782446\pi$
$163$	−19.7990	−1.55078	−0.775388	+	0.631485i	$0.782446\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	16.9706	1.29777
$172$	0	0
$173$	8.48528	0.645124	0.322562	−	0.946548i	$-0.395456\pi$
$173$	8.48528	0.645124	0.322562	+	0.946548i	$0.395456\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.48528	0.634220	0.317110	−	0.948389i	$-0.397288\pi$
$179$	8.48528	0.634220	0.317110	+	0.948389i	$0.397288\pi$
$180$	0	0
$181$	−25.4558	−1.89212	−0.946059	−	0.323994i	$-0.894974\pi$
$181$	−25.4558	−1.89212	−0.946059	+	0.323994i	$0.894974\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.0000	1.17634
$186$	0	0
$187$	5.65685	0.413670
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.65685	−0.397033
$204$	0	0
$205$	16.9706	1.18528
$206$	0	0
$207$	24.0000	1.66812
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	19.7990	1.36302	0.681509	−	0.731809i	$-0.261324\pi$
$211$	19.7990	1.36302	0.681509	+	0.731809i	$0.261324\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.65685	0.380521
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	−9.00000	−0.600000
$226$	0	0
$227$	−28.2843	−1.87729	−0.938647	−	0.344881i	$-0.887919\pi$
$227$	−28.2843	−1.87729	−0.938647	+	0.344881i	$0.887919\pi$
$228$	0	0
$229$	−25.4558	−1.68217	−0.841085	−	0.540903i	$-0.818082\pi$
$229$	−25.4558	−1.68217	−0.841085	+	0.540903i	$0.818082\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	−22.6274	−1.47605
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	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
$245$	2.82843	0.180702
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.65685	0.357057	0.178529	−	0.983935i	$-0.442866\pi$
$251$	5.65685	0.357057	0.178529	+	0.983935i	$0.442866\pi$
$252$	0	0
$253$	22.6274	1.42257
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	5.65685	0.351500
$260$	0	0
$261$	16.9706	1.05045
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	32.0000	1.96574
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.82843	−0.172452	−0.0862261	−	0.996276i	$-0.527481\pi$
$269$	−2.82843	−0.172452	−0.0862261	+	0.996276i	$0.527481\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.48528	−0.511682
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	24.0000	1.43684
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	22.6274	1.34506	0.672530	−	0.740070i	$-0.265208\pi$
$283$	22.6274	1.34506	0.672530	+	0.740070i	$0.265208\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.1421	0.826192	0.413096	−	0.910687i	$-0.364447\pi$
$293$	14.1421	0.826192	0.413096	+	0.910687i	$0.364447\pi$
$294$	0	0
$295$	32.0000	1.86311
$296$	0	0
$297$	0	0
$298$	0	0
$299$	22.6274	1.30858
$300$	0	0
$301$	−2.82843	−0.163028
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.00000	0.458079
$306$	0	0
$307$	28.2843	1.61427	0.807134	−	0.590368i	$-0.201017\pi$
$307$	28.2843	1.61427	0.807134	+	0.590368i	$0.201017\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	−8.48528	−0.478091
$316$	0	0
$317$	−22.6274	−1.27088	−0.635441	−	0.772149i	$-0.719182\pi$
$317$	−22.6274	−1.27088	−0.635441	+	0.772149i	$0.719182\pi$
$318$	0	0
$319$	16.0000	0.895828
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11.3137	0.629512
$324$	0	0
$325$	−8.48528	−0.470679
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	8.48528	0.466393	0.233197	−	0.972430i	$-0.425081\pi$
$331$	8.48528	0.466393	0.233197	+	0.972430i	$0.425081\pi$
$332$	0	0
$333$	−16.9706	−0.929981
$334$	0	0
$335$	24.0000	1.31126
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	22.6274	1.22534
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−25.4558	−1.36654	−0.683271	−	0.730165i	$-0.739443\pi$
$347$	−25.4558	−1.36654	−0.683271	+	0.730165i	$0.739443\pi$
$348$	0	0
$349$	8.48528	0.454207	0.227103	−	0.973871i	$-0.427074\pi$
$349$	8.48528	0.454207	0.227103	+	0.973871i	$0.427074\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	−22.6274	−1.20094
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−16.9706	−0.888280
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	−18.0000	−0.937043
$370$	0	0
$371$	11.3137	0.587378
$372$	0	0
$373$	−33.9411	−1.75740	−0.878702	−	0.477370i	$-0.841590\pi$
$373$	−33.9411	−1.75740	−0.878702	+	0.477370i	$0.841590\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.0000	0.824042
$378$	0	0
$379$	8.48528	0.435860	0.217930	−	0.975964i	$-0.430070\pi$
$379$	8.48528	0.435860	0.217930	+	0.975964i	$0.430070\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	8.48528	0.431331
$388$	0	0
$389$	−16.9706	−0.860442	−0.430221	−	0.902724i	$-0.641564\pi$
$389$	−16.9706	−0.860442	−0.430221	+	0.902724i	$0.641564\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	0	0
$394$	0	0
$395$	22.6274	1.13851
$396$	0	0
$397$	−25.4558	−1.27759	−0.638796	−	0.769376i	$-0.720567\pi$
$397$	−25.4558	−1.27759	−0.638796	+	0.769376i	$0.720567\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	22.6274	1.12715
$404$	0	0
$405$	25.4558	1.26491
$406$	0	0
$407$	−16.0000	−0.793091
$408$	0	0
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.3137	0.556711
$414$	0	0
$415$	16.0000	0.785409
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	22.6274	1.10279	0.551396	−	0.834243i	$-0.314095\pi$
$421$	22.6274	1.10279	0.551396	+	0.834243i	$0.314095\pi$
$422$	0	0
$423$	24.0000	1.16692
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	2.82843	0.136877
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	45.2548	2.16483
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	2.82843	0.134383	0.0671913	−	0.997740i	$-0.478596\pi$
$443$	2.82843	0.134383	0.0671913	+	0.997740i	$0.478596\pi$
$444$	0	0
$445$	−16.9706	−0.804482
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−16.9706	−0.799113
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8.00000	−0.375046
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.48528	0.395199	0.197599	−	0.980283i	$-0.436685\pi$
$461$	8.48528	0.395199	0.197599	+	0.980283i	$0.436685\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.65685	−0.261768	−0.130884	−	0.991398i	$-0.541782\pi$
$467$	−5.65685	−0.261768	−0.130884	+	0.991398i	$0.541782\pi$
$468$	0	0
$469$	8.48528	0.391814
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	−16.9706	−0.778663
$476$	0	0
$477$	−33.9411	−1.55406
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	39.5980	1.79805
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−31.1127	−1.40410	−0.702048	−	0.712129i	$-0.747731\pi$
$491$	−31.1127	−1.40410	−0.702048	+	0.712129i	$0.747731\pi$
$492$	0	0
$493$	11.3137	0.509544
$494$	0	0
$495$	24.0000	1.07872
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	−25.4558	−1.13956	−0.569780	−	0.821797i	$-0.692972\pi$
$499$	−25.4558	−1.13956	−0.569780	+	0.821797i	$0.692972\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	0	0
$505$	−24.0000	−1.06799
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.82843	−0.125368	−0.0626839	−	0.998033i	$-0.519966\pi$
$509$	−2.82843	−0.125368	−0.0626839	+	0.998033i	$0.519966\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	22.6274	0.995153
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	0	0
$523$	−11.3137	−0.494714	−0.247357	−	0.968924i	$-0.579562\pi$
$523$	−11.3137	−0.494714	−0.247357	+	0.968924i	$0.579562\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−33.9411	−1.47292
$532$	0	0
$533$	−16.9706	−0.735077
$534$	0	0
$535$	40.0000	1.72935
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.82843	−0.121829
$540$	0	0
$541$	−11.3137	−0.486414	−0.243207	−	0.969974i	$-0.578199\pi$
$541$	−11.3137	−0.486414	−0.243207	+	0.969974i	$0.578199\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	48.0000	2.05609
$546$	0	0
$547$	2.82843	0.120935	0.0604674	−	0.998170i	$-0.480741\pi$
$547$	2.82843	0.120935	0.0604674	+	0.998170i	$0.480741\pi$
$548$	0	0
$549$	−8.48528	−0.362143
$550$	0	0
$551$	32.0000	1.36325
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−45.2548	−1.91751	−0.958754	−	0.284236i	$-0.908260\pi$
$557$	−45.2548	−1.91751	−0.958754	+	0.284236i	$0.908260\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11.3137	−0.476816	−0.238408	−	0.971165i	$-0.576626\pi$
$563$	−11.3137	−0.476816	−0.238408	+	0.971165i	$0.576626\pi$
$564$	0	0
$565$	−16.9706	−0.713957
$566$	0	0
$567$	9.00000	0.377964
$568$	0	0
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	0	0
$571$	19.7990	0.828562	0.414281	−	0.910149i	$-0.364033\pi$
$571$	19.7990	0.828562	0.414281	+	0.910149i	$0.364033\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−24.0000	−1.00087
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.65685	0.234686
$582$	0	0
$583$	−32.0000	−1.32530
$584$	0	0
$585$	24.0000	0.992278
$586$	0	0
$587$	−5.65685	−0.233483	−0.116742	−	0.993162i	$-0.537245\pi$
$587$	−5.65685	−0.233483	−0.116742	+	0.993162i	$0.537245\pi$
$588$	0	0
$589$	45.2548	1.86469
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	−5.65685	−0.231908
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	−25.4558	−1.03664
$604$	0	0
$605$	−8.48528	−0.344976
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	22.6274	0.915407
$612$	0	0
$613$	−28.2843	−1.14239	−0.571195	−	0.820814i	$-0.693520\pi$
$613$	−28.2843	−1.14239	−0.571195	+	0.820814i	$0.693520\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	0	0
$619$	−11.3137	−0.454736	−0.227368	−	0.973809i	$-0.573012\pi$
$619$	−11.3137	−0.454736	−0.227368	+	0.973809i	$0.573012\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−11.3137	−0.451107
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−22.6274	−0.897942
$636$	0	0
$637$	−2.82843	−0.112066
$638$	0	0
$639$	24.0000	0.949425
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	33.9411	1.33851	0.669254	−	0.743034i	$-0.266614\pi$
$643$	33.9411	1.33851	0.669254	+	0.743034i	$0.266614\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	0	0
$649$	−32.0000	−1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.65685	0.221370	0.110685	−	0.993856i	$-0.464696\pi$
$653$	5.65685	0.221370	0.110685	+	0.993856i	$0.464696\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	0	0
$657$	18.0000	0.702247
$658$	0	0
$659$	−8.48528	−0.330540	−0.165270	−	0.986248i	$-0.552849\pi$
$659$	−8.48528	−0.330540	−0.165270	+	0.986248i	$0.552849\pi$
$660$	0	0
$661$	2.82843	0.110013	0.0550065	−	0.998486i	$-0.482482\pi$
$661$	2.82843	0.110013	0.0550065	+	0.998486i	$0.482482\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.0000	−0.620453
$666$	0	0
$667$	45.2548	1.75227
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.82843	−0.108705	−0.0543526	−	0.998522i	$-0.517310\pi$
$677$	−2.82843	−0.108705	−0.0543526	+	0.998522i	$0.517310\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.7696	1.40695	0.703474	−	0.710721i	$-0.251631\pi$
$683$	36.7696	1.40695	0.703474	+	0.710721i	$0.251631\pi$
$684$	0	0
$685$	16.9706	0.648412
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−32.0000	−1.21910
$690$	0	0
$691$	16.9706	0.645591	0.322795	−	0.946469i	$-0.395377\pi$
$691$	16.9706	0.645591	0.322795	+	0.946469i	$0.395377\pi$
$692$	0	0
$693$	8.48528	0.322329
$694$	0	0
$695$	−48.0000	−1.82074
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.65685	0.213656	0.106828	−	0.994277i	$-0.465931\pi$
$701$	5.65685	0.213656	0.106828	+	0.994277i	$0.465931\pi$
$702$	0	0
$703$	−32.0000	−1.20690
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.48528	−0.319122
$708$	0	0
$709$	−16.9706	−0.637343	−0.318671	−	0.947865i	$-0.603237\pi$
$709$	−16.9706	−0.637343	−0.318671	+	0.947865i	$0.603237\pi$
$710$	0	0
$711$	−24.0000	−0.900070
$712$	0	0
$713$	64.0000	2.39682
$714$	0	0
$715$	22.6274	0.846217
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−16.9706	−0.630271
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	5.65685	0.209226
$732$	0	0
$733$	14.1421	0.522352	0.261176	−	0.965291i	$-0.415890\pi$
$733$	14.1421	0.522352	0.261176	+	0.965291i	$0.415890\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	25.4558	0.936408	0.468204	−	0.883620i	$-0.344901\pi$
$739$	25.4558	0.936408	0.468204	+	0.883620i	$0.344901\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−32.0000	−1.17239
$746$	0	0
$747$	−16.9706	−0.620920
$748$	0	0
$749$	14.1421	0.516742
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−22.6274	−0.823496
$756$	0	0
$757$	39.5980	1.43921	0.719607	−	0.694382i	$-0.244322\pi$
$757$	39.5980	1.43921	0.719607	+	0.694382i	$0.244322\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	16.9706	0.614376
$764$	0	0
$765$	16.9706	0.613572
$766$	0	0
$767$	−32.0000	−1.15545
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−19.7990	−0.712120	−0.356060	−	0.934463i	$-0.615880\pi$
$773$	−19.7990	−0.712120	−0.356060	+	0.934463i	$0.615880\pi$
$774$	0	0
$775$	−24.0000	−0.862105
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−33.9411	−1.21607
$780$	0	0
$781$	22.6274	0.809673
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24.0000	0.856597
$786$	0	0
$787$	28.2843	1.00823	0.504113	−	0.863638i	$-0.331820\pi$
$787$	28.2843	1.00823	0.504113	+	0.863638i	$0.331820\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.82843	−0.100188	−0.0500940	−	0.998745i	$-0.515952\pi$
$797$	−2.82843	−0.100188	−0.0500940	+	0.998745i	$0.515952\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	16.9706	0.598878
$804$	0	0
$805$	−22.6274	−0.797512
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−56.0000	−1.96159
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	8.48528	0.296500
$820$	0	0
$821$	−45.2548	−1.57940	−0.789702	−	0.613490i	$-0.789765\pi$
$821$	−45.2548	−1.57940	−0.789702	+	0.613490i	$0.789765\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.7696	1.27860	0.639301	−	0.768956i	$-0.279224\pi$
$827$	36.7696	1.27860	0.639301	+	0.768956i	$0.279224\pi$
$828$	0	0
$829$	48.0833	1.67000	0.835000	−	0.550249i	$-0.185467\pi$
$829$	48.0833	1.67000	0.835000	+	0.550249i	$0.185467\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	3.00000	0.103448
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−14.1421	−0.486504
$846$	0	0
$847$	−3.00000	−0.103081
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−45.2548	−1.55132
$852$	0	0
$853$	−25.4558	−0.871592	−0.435796	−	0.900046i	$-0.643533\pi$
$853$	−25.4558	−0.871592	−0.435796	+	0.900046i	$0.643533\pi$
$854$	0	0
$855$	48.0000	1.64157
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	39.5980	1.35107	0.675533	−	0.737330i	$-0.263914\pi$
$859$	39.5980	1.35107	0.675533	+	0.737330i	$0.263914\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	24.0000	0.816024
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−22.6274	−0.767583
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	−42.0000	−1.42148
$874$	0	0
$875$	−5.65685	−0.191237
$876$	0	0
$877$	−16.9706	−0.573055	−0.286528	−	0.958072i	$-0.592501\pi$
$877$	−16.9706	−0.573055	−0.286528	+	0.958072i	$0.592501\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	−25.4558	−0.856657	−0.428329	−	0.903623i	$-0.640897\pi$
$883$	−25.4558	−0.856657	−0.428329	+	0.903623i	$0.640897\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	−25.4558	−0.852803
$892$	0	0
$893$	45.2548	1.51440
$894$	0	0
$895$	24.0000	0.802232
$896$	0	0
$897$	0	0
$898$	0	0
$899$	45.2548	1.50933
$900$	0	0
$901$	−22.6274	−0.753829
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−72.0000	−2.39336
$906$	0	0
$907$	−19.7990	−0.657415	−0.328707	−	0.944432i	$-0.606613\pi$
$907$	−19.7990	−0.657415	−0.328707	+	0.944432i	$0.606613\pi$
$908$	0	0
$909$	25.4558	0.844317
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.65685	0.186806
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	22.6274	0.744791
$924$	0	0
$925$	16.9706	0.557989
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	−5.65685	−0.185396
$932$	0	0
$933$	0	0
$934$	0	0
$935$	16.0000	0.523256
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.82843	0.0922041	0.0461020	−	0.998937i	$-0.485320\pi$
$941$	2.82843	0.0922041	0.0461020	+	0.998937i	$0.485320\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.4558	0.827204	0.413602	−	0.910458i	$-0.364271\pi$
$947$	25.4558	0.827204	0.413602	+	0.910458i	$0.364271\pi$
$948$	0	0
$949$	16.9706	0.550888
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−10.0000	−0.323932	−0.161966	−	0.986796i	$-0.551783\pi$
$953$	−10.0000	−0.323932	−0.161966	+	0.986796i	$0.551783\pi$
$954$	0	0
$955$	−67.8823	−2.19662
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−42.4264	−1.36717
$964$	0	0
$965$	−5.65685	−0.182101
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11.3137	−0.363074	−0.181537	−	0.983384i	$-0.558107\pi$
$971$	−11.3137	−0.363074	−0.181537	+	0.983384i	$0.558107\pi$
$972$	0	0
$973$	−16.9706	−0.544051
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.0000	−0.319928	−0.159964	−	0.987123i	$-0.551138\pi$
$977$	−10.0000	−0.319928	−0.159964	+	0.987123i	$0.551138\pi$
$978$	0	0
$979$	16.9706	0.542382
$980$	0	0
$981$	−50.9117	−1.62549
$982$	0	0
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	22.6274	0.719510
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25.4558	0.806195	0.403097	−	0.915157i	$-0.367934\pi$
$997$	25.4558	0.806195	0.403097	+	0.915157i	$0.367934\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.a.o.1.2		2
4.3	odd	2		1792.2.a.m.1.2		2
8.3	odd	2		1792.2.a.m.1.1		2
8.5	even	2	inner	1792.2.a.o.1.1		2
16.3	odd	4		896.2.b.d.449.1	yes	2
16.5	even	4		896.2.b.b.449.2	yes	2
16.11	odd	4		896.2.b.d.449.2	yes	2
16.13	even	4		896.2.b.b.449.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.b.b.449.1	✓	2	16.13	even	4
896.2.b.b.449.2	yes	2	16.5	even	4
896.2.b.d.449.1	yes	2	16.3	odd	4
896.2.b.d.449.2	yes	2	16.11	odd	4
1792.2.a.m.1.1		2	8.3	odd	2
1792.2.a.m.1.2		2	4.3	odd	2
1792.2.a.o.1.1		2	8.5	even	2	inner
1792.2.a.o.1.2		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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82843 q^{5} +1.00000 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+2.82843 q^{5} +1.00000 q^{7} -3.00000 q^{9} -2.82843 q^{11} -2.82843 q^{13} -2.00000 q^{17} -5.65685 q^{19} -8.00000 q^{23} +3.00000 q^{25} -5.65685 q^{29} -8.00000 q^{31} +2.82843 q^{35} +5.65685 q^{37} +6.00000 q^{41} -2.82843 q^{43} -8.48528 q^{45} -8.00000 q^{47} +1.00000 q^{49} +11.3137 q^{53} -8.00000 q^{55} +11.3137 q^{59} +2.82843 q^{61} -3.00000 q^{63} -8.00000 q^{65} +8.48528 q^{67} -8.00000 q^{71} -6.00000 q^{73} -2.82843 q^{77} +8.00000 q^{79} +9.00000 q^{81} +5.65685 q^{83} -5.65685 q^{85} -6.00000 q^{89} -2.82843 q^{91} -16.0000 q^{95} +14.0000 q^{97} +8.48528 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^7 - 6 * q^9	\( 2 q + 2 q^{7} - 6 q^{9} - 4 q^{17} - 16 q^{23} + 6 q^{25} - 16 q^{31} + 12 q^{41} - 16 q^{47} + 2 q^{49} - 16 q^{55} - 6 q^{63} - 16 q^{65} - 16 q^{71} - 12 q^{73} + 16 q^{79} + 18 q^{81} - 12 q^{89} - 32 q^{95} + 28 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 6 * q^9 - 4 * q^17 - 16 * q^23 + 6 * q^25 - 16 * q^31 + 12 * q^41 - 16 * q^47 + 2 * q^49 - 16 * q^55 - 6 * q^63 - 16 * q^65 - 16 * q^71 - 12 * q^73 + 16 * q^79 + 18 * q^81 - 12 * q^89 - 32 * q^95 + 28 * q^97

Introduction

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