LMFDB - Embedded newform 4680.2.l.f.2809.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(2809,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4680.2809");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.l (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1698758656.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 $ x^8 + 18x^6 + 97x^4 + 176*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2809.7
Root			$-3.16053i$ of defining polynomial
Character	$\chi$	$=$	4680.2809
Dual form			4680.2.l.f.2809.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.52773 - 1.63280i) q^{5} -2.82843i q^{7} +O(q^{10})$	$q+(1.52773 - 1.63280i) q^{5} -2.82843i q^{7} -1.05545 q^{11} -1.00000i q^{13} -7.14949i q^{17} +6.61827 q^{19} -3.49264i q^{23} +(-0.332104 - 4.98896i) q^{25} +4.82843 q^{29} -9.65685 q^{31} +(-4.61827 - 4.32106i) q^{35} +8.11091i q^{37} +3.26561 q^{41} +7.44670i q^{43} -11.3765i q^{47} -1.00000 q^{49} +4.53122i q^{53} +(-1.61244 + 1.72335i) q^{55} +1.05545 q^{59} -7.97792 q^{61} +(-1.63280 - 1.52773i) q^{65} +1.46878i q^{67} -0.845296 q^{71} -9.28248i q^{73} +2.98527i q^{77} -6.85228 q^{79} +7.97093i q^{83} +(-11.6737 - 10.9225i) q^{85} +0.139976 q^{89} -2.82843 q^{91} +(10.1109 - 10.8063i) q^{95} -6.93933i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{11} + 24 q^{19} - 4 q^{25} + 16 q^{29} - 32 q^{31} - 8 q^{35} + 8 q^{41} - 8 q^{49} - 36 q^{55} - 16 q^{59} + 24 q^{61} - 4 q^{65} + 24 q^{71} + 24 q^{79} - 40 q^{85} - 8 q^{89} + 32 q^{95}+O(q^{100}) $ 8 * q + 16 * q^11 + 24 * q^19 - 4 * q^25 + 16 * q^29 - 32 * q^31 - 8 * q^35 + 8 * q^41 - 8 * q^49 - 36 * q^55 - 16 * q^59 + 24 * q^61 - 4 * q^65 + 24 * q^71 + 24 * q^79 - 40 * q^85 - 8 * q^89 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$2081$	$2341$	$3511$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.52773	−	1.63280i	0.683220	−	0.730213i
$5$	1.52773	−	1.63280i	0.683220	−	0.730213i
$6$		0			0
$7$		−	2.82843i		−	1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$		−	2.82843i		−	1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.05545			−0.318231			−0.159115	−	0.987260i	$-0.550864\pi$
$11$	−1.05545			−0.318231			−0.159115	+	0.987260i	$0.550864\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	7.14949i		−	1.73401i	−0.498303	−	0.867003i	$-0.666043\pi$
$17$		−	7.14949i		−	1.73401i	0.498303	−	0.867003i	$-0.333957\pi$
$18$		0			0
$19$	6.61827			1.51834			0.759168	−	0.650895i	$-0.225606\pi$
$19$	6.61827			1.51834			0.759168	+	0.650895i	$0.225606\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.49264i		−	0.728265i	−0.931347	−	0.364132i	$-0.881366\pi$
$23$		−	3.49264i		−	0.728265i	0.931347	−	0.364132i	$-0.118634\pi$
$24$		0			0
$25$	−0.332104	−	4.98896i	−0.0664208	−	0.997792i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.82843			0.896616			0.448308	−	0.893879i	$-0.352027\pi$
$29$	4.82843			0.896616			0.448308	+	0.893879i	$0.352027\pi$
$30$		0			0
$31$	−9.65685			−1.73442			−0.867211	−	0.497941i	$-0.834090\pi$
$31$	−9.65685			−1.73442			−0.867211	+	0.497941i	$0.834090\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.61827	−	4.32106i	−0.780630	−	0.730393i
$36$		0			0
$37$			8.11091i			1.33342i	0.745315	+	0.666712i	$0.232299\pi$
$37$			8.11091i			1.33342i	−0.745315	+	0.666712i	$0.767701\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.26561			0.510003			0.255001	−	0.966941i	$-0.417924\pi$
$41$	3.26561			0.510003			0.255001	+	0.966941i	$0.417924\pi$
$42$		0			0
$43$			7.44670i			1.13561i	0.823163	+	0.567805i	$0.192207\pi$
$43$			7.44670i			1.13561i	−0.823163	+	0.567805i	$0.807793\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	11.3765i		−	1.65944i	−0.558183	−	0.829718i	$-0.688501\pi$
$47$		−	11.3765i		−	1.65944i	0.558183	−	0.829718i	$-0.311499\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4.53122i			0.622411i	0.950343	+	0.311205i	$0.100733\pi$
$53$			4.53122i			0.622411i	−0.950343	+	0.311205i	$0.899267\pi$
$54$		0			0
$55$	−1.61244	+	1.72335i	−0.217422	+	0.232376i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.05545			0.137408			0.0687041	−	0.997637i	$-0.478114\pi$
$59$	1.05545			0.137408			0.0687041	+	0.997637i	$0.478114\pi$
$60$		0			0
$61$	−7.97792			−1.02147			−0.510734	−	0.859739i	$-0.670626\pi$
$61$	−7.97792			−1.02147			−0.510734	+	0.859739i	$0.670626\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.63280	−	1.52773i	−0.202525	−	0.189491i
$66$		0			0
$67$			1.46878i			0.179440i	0.995967	+	0.0897200i	$0.0285972\pi$
$67$			1.46878i			0.179440i	−0.995967	+	0.0897200i	$0.971403\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.845296			−0.100318			−0.0501591	−	0.998741i	$-0.515973\pi$
$71$	−0.845296			−0.100318			−0.0501591	+	0.998741i	$0.515973\pi$
$72$		0			0
$73$		−	9.28248i		−	1.08643i	−0.839593	−	0.543216i	$-0.817206\pi$
$73$		−	9.28248i		−	1.08643i	0.839593	−	0.543216i	$-0.182794\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.98527i			0.340203i
$78$		0			0
$79$	−6.85228			−0.770942			−0.385471	−	0.922720i	$-0.625961\pi$
$79$	−6.85228			−0.770942			−0.385471	+	0.922720i	$0.625961\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			7.97093i			0.874923i	0.899237	+	0.437462i	$0.144122\pi$
$83$			7.97093i			0.874923i	−0.899237	+	0.437462i	$0.855878\pi$
$84$		0			0
$85$	−11.6737	−	10.9225i	−1.26619	−	1.18471i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.139976			0.0148374			0.00741870	−	0.999972i	$-0.497639\pi$
$89$	0.139976			0.0148374			0.00741870	+	0.999972i	$0.497639\pi$
$90$		0			0
$91$	−2.82843			−0.296500
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.1109	−	10.8063i	1.03736	−	1.10871i
$96$		0			0
$97$		−	6.93933i		−	0.704582i	−0.935890	−	0.352291i	$-0.885403\pi$
$97$		−	6.93933i		−	0.704582i	0.935890	−	0.352291i	$-0.114597\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	16.1421			1.60620			0.803101	−	0.595843i	$-0.203182\pi$
$101$	16.1421			1.60620			0.803101	+	0.595843i	$0.203182\pi$
$102$		0			0
$103$			16.7530i			1.65073i	0.564603	+	0.825363i	$0.309029\pi$
$103$			16.7530i			1.65073i	−0.564603	+	0.825363i	$0.690971\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	4.87437i		−	0.471223i	−0.971847	−	0.235611i	$-0.924291\pi$
$107$		−	4.87437i		−	0.471223i	0.971847	−	0.235611i	$-0.0757092\pi$
$108$		0			0
$109$	7.14949			0.684797			0.342398	−	0.939555i	$-0.388761\pi$
$109$	7.14949			0.684797			0.342398	+	0.939555i	$0.388761\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	13.2604i		−	1.24743i	−0.781651	−	0.623717i	$-0.785622\pi$
$113$		−	13.2604i		−	1.24743i	0.781651	−	0.623717i	$-0.214378\pi$
$114$		0			0
$115$	−5.70279	−	5.33579i	−0.531788	−	0.497565i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−20.2218			−1.85373
$120$		0			0
$121$	−9.88602			−0.898729
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.65336	−	7.07950i	−0.773980	−	0.633210i
$126$		0			0
$127$			13.3063i			1.18075i	0.807130	+	0.590373i	$0.201019\pi$
$127$			13.3063i			1.18075i	−0.807130	+	0.590373i	$0.798981\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.06497			−0.704639			−0.352320	−	0.935880i	$-0.614607\pi$
$131$	−8.06497			−0.704639			−0.352320	+	0.935880i	$0.614607\pi$
$132$		0			0
$133$		−	18.7193i		−	1.62317i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	10.8453i		−	0.926576i	−0.886208	−	0.463288i	$-0.846670\pi$
$137$		−	10.8453i		−	0.926576i	0.886208	−	0.463288i	$-0.153330\pi$
$138$		0			0
$139$	−9.23654			−0.783433			−0.391717	−	0.920086i	$-0.628119\pi$
$139$	−9.23654			−0.783433			−0.391717	+	0.920086i	$0.628119\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			1.05545i			0.0882614i
$144$		0			0
$145$	7.37652	−	7.88388i	0.612586	−	0.654721i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.60140			−0.540808			−0.270404	−	0.962747i	$-0.587157\pi$
$149$	−6.60140			−0.540808			−0.270404	+	0.962747i	$0.587157\pi$
$150$		0			0
$151$	14.8934			1.21201			0.606004	−	0.795462i	$-0.292772\pi$
$151$	14.8934			1.21201			0.606004	+	0.795462i	$0.292772\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−14.7530	+	15.7678i	−1.18499	+	1.26650i
$156$		0			0
$157$		−	11.3835i		−	0.908502i	−0.890874	−	0.454251i	$-0.849907\pi$
$157$		−	11.3835i		−	0.908502i	0.890874	−	0.454251i	$-0.150093\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.87867			−0.778548
$162$		0			0
$163$			7.31371i			0.572854i	0.958102	+	0.286427i	$0.0924676\pi$
$163$			7.31371i			0.572854i	−0.958102	+	0.286427i	$0.907532\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	15.3428i		−	1.18726i	−0.804738	−	0.593630i	$-0.797694\pi$
$167$		−	15.3428i		−	1.18726i	0.804738	−	0.593630i	$-0.202306\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$			18.9706i			1.44231i	0.692776	+	0.721153i	$0.256387\pi$
$173$			18.9706i			1.44231i	−0.692776	+	0.721153i	$0.743613\pi$
$174$		0			0
$175$	−14.1109	+	0.939333i	−1.06668	+	0.0710069i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.9246			−1.19026			−0.595130	−	0.803629i	$-0.702900\pi$
$179$	−15.9246			−1.19026			−0.595130	+	0.803629i	$0.702900\pi$
$180$		0			0
$181$	16.0772			1.19501			0.597503	−	0.801866i	$-0.296159\pi$
$181$	16.0772			1.19501			0.597503	+	0.801866i	$0.296159\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	13.2435	+	12.3912i	0.973683	+	0.911022i
$186$		0			0
$187$			7.54595i			0.551814i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.29898			−0.455778			−0.227889	−	0.973687i	$-0.573182\pi$
$191$	−6.29898			−0.455778			−0.227889	+	0.973687i	$0.573182\pi$
$192$		0			0
$193$			23.7695i			1.71097i	0.517829	+	0.855484i	$0.326740\pi$
$193$			23.7695i			1.71097i	−0.517829	+	0.855484i	$0.673260\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			7.62778i			0.543457i	0.962374	+	0.271729i	$0.0875953\pi$
$197$			7.62778i			0.543457i	−0.962374	+	0.271729i	$0.912405\pi$
$198$		0			0
$199$	−6.85228			−0.485745			−0.242873	−	0.970058i	$-0.578090\pi$
$199$	−6.85228			−0.485745			−0.242873	+	0.970058i	$0.578090\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	13.6569i		−	0.958523i
$204$		0			0
$205$	4.98896	−	5.33210i	0.348444	−	0.372410i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.98527			−0.483181
$210$		0			0
$211$	15.8670			1.09233			0.546165	−	0.837678i	$-0.316087\pi$
$211$	15.8670			1.09233			0.546165	+	0.837678i	$0.316087\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	12.1590	+	11.3765i	0.829237	+	0.775872i
$216$		0			0
$217$			27.3137i			1.85418i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.14949			−0.480927
$222$		0			0
$223$			9.95406i			0.666573i	0.942826	+	0.333287i	$0.108158\pi$
$223$			9.95406i			0.666573i	−0.942826	+	0.333287i	$0.891842\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	11.1547i		−	0.740364i	−0.928959	−	0.370182i	$-0.879295\pi$
$227$		−	11.1547i		−	0.740364i	0.928959	−	0.370182i	$-0.120705\pi$
$228$		0			0
$229$	−23.9657			−1.58370			−0.791850	−	0.610716i	$-0.790882\pi$
$229$	−23.9657			−1.58370			−0.791850	+	0.610716i	$0.790882\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	15.3376i		−	1.00480i	−0.864636	−	0.502399i	$-0.832451\pi$
$233$		−	15.3376i		−	1.00480i	0.864636	−	0.502399i	$-0.167549\pi$
$234$		0			0
$235$	−18.5756	−	17.3802i	−1.21174	−	1.13376i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−14.7344			−0.953088			−0.476544	−	0.879150i	$-0.658111\pi$
$239$	−14.7344			−0.953088			−0.476544	+	0.879150i	$0.658111\pi$
$240$		0			0
$241$	10.1881			0.656272			0.328136	−	0.944631i	$-0.393580\pi$
$241$	10.1881			0.656272			0.328136	+	0.944631i	$0.393580\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.52773	+	1.63280i	−0.0976029	+	0.104316i
$246$		0			0
$247$		−	6.61827i		−	0.421110i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.8909			−1.00302			−0.501511	−	0.865151i	$-0.667222\pi$
$251$	−15.8909			−1.00302			−0.501511	+	0.865151i	$0.667222\pi$
$252$		0			0
$253$			3.68631i			0.231756i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	20.0239i		−	1.24905i	−0.781003	−	0.624527i	$-0.785292\pi$
$257$		−	20.0239i		−	1.24905i	0.781003	−	0.624527i	$-0.214708\pi$
$258$		0			0
$259$	22.9411			1.42549
$260$		0			0
$261$		0			0
$262$		0			0
$263$			27.0282i			1.66663i	0.552800	+	0.833314i	$0.313559\pi$
$263$			27.0282i			1.66663i	−0.552800	+	0.833314i	$0.686441\pi$
$264$		0			0
$265$	7.39860	+	6.92246i	0.454492	+	0.425244i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	31.1274			1.89787			0.948936	−	0.315469i	$-0.102162\pi$
$269$	31.1274			1.89787			0.948936	+	0.315469i	$0.102162\pi$
$270$		0			0
$271$	21.0043			1.27592			0.637960	−	0.770069i	$-0.279778\pi$
$271$	21.0043			1.27592			0.637960	+	0.770069i	$0.279778\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.350520	+	5.26561i	0.0211372	+	0.317528i
$276$		0			0
$277$			13.2145i			0.793980i	0.917823	+	0.396990i	$0.129945\pi$
$277$			13.2145i			0.793980i	−0.917823	+	0.396990i	$0.870055\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.83057			−0.109202			−0.0546012	−	0.998508i	$-0.517389\pi$
$281$	−1.83057			−0.109202			−0.0546012	+	0.998508i	$0.517389\pi$
$282$		0			0
$283$		−	27.5355i		−	1.63682i	−0.574637	−	0.818408i	$-0.694857\pi$
$283$		−	27.5355i		−	1.63682i	0.574637	−	0.818408i	$-0.305143\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	9.23654i		−	0.545216i
$288$		0			0
$289$	−34.1152			−2.00678
$290$		0			0
$291$		0			0
$292$		0			0
$293$			3.95190i			0.230873i	0.993315	+	0.115436i	$0.0368266\pi$
$293$			3.95190i			0.230873i	−0.993315	+	0.115436i	$0.963173\pi$
$294$		0			0
$295$	1.61244	−	1.72335i	0.0938801	−	0.100337i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.49264			−0.201984
$300$		0			0
$301$	21.0624			1.21402
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.1881	+	13.0264i	−0.697887	+	0.745888i
$306$		0			0
$307$		−	14.9271i		−	0.851936i	−0.904738	−	0.425968i	$-0.859934\pi$
$307$		−	14.9271i		−	0.851936i	0.904738	−	0.425968i	$-0.140066\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.87437			−0.276400			−0.138200	−	0.990404i	$-0.544132\pi$
$311$	−4.87437			−0.276400			−0.138200	+	0.990404i	$0.544132\pi$
$312$		0			0
$313$		−	27.2145i		−	1.53825i	−0.639097	−	0.769126i	$-0.720692\pi$
$313$		−	27.2145i		−	1.53825i	0.639097	−	0.769126i	$-0.279308\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	3.68592i		−	0.207022i	−0.994628	−	0.103511i	$-0.966992\pi$
$317$		−	3.68592i		−	0.207022i	0.994628	−	0.103511i	$-0.0330077\pi$
$318$		0			0
$319$	−5.09618			−0.285331
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	47.3173i		−	2.63280i
$324$		0			0
$325$	−4.98896	+	0.332104i	−0.276738	+	0.0184218i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−32.1776			−1.77401
$330$		0			0
$331$	−24.6850			−1.35681			−0.678405	−	0.734688i	$-0.737329\pi$
$331$	−24.6850			−1.35681			−0.678405	+	0.734688i	$0.737329\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.39823	+	2.24389i	0.131029	+	0.122597i
$336$		0			0
$337$		−	30.5282i		−	1.66298i	−0.555543	−	0.831488i	$-0.687490\pi$
$337$		−	30.5282i		−	1.66298i	0.555543	−	0.831488i	$-0.312510\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	10.1924			0.551947
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			8.32844i			0.447094i	0.974693	+	0.223547i	$0.0717636\pi$
$347$			8.32844i			0.447094i	−0.974693	+	0.223547i	$0.928236\pi$
$348$		0			0
$349$	8.57410			0.458961			0.229481	−	0.973313i	$-0.426297\pi$
$349$	8.57410			0.458961			0.229481	+	0.973313i	$0.426297\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			2.42498i			0.129069i	0.997915	+	0.0645344i	$0.0205562\pi$
$353$			2.42498i			0.129069i	−0.997915	+	0.0645344i	$0.979444\pi$
$354$		0			0
$355$	−1.29138	+	1.38020i	−0.0685393	+	0.0732536i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−18.5499			−0.979024			−0.489512	−	0.871997i	$-0.662825\pi$
$359$	−18.5499			−0.979024			−0.489512	+	0.871997i	$0.662825\pi$
$360$		0			0
$361$	24.8015			1.30534
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−15.1565	−	14.1811i	−0.793326	−	0.742272i
$366$		0			0
$367$			5.29973i			0.276644i	0.990387	+	0.138322i	$0.0441709\pi$
$367$			5.29973i			0.276644i	−0.990387	+	0.138322i	$0.955829\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.8162			0.665385
$372$		0			0
$373$		−	29.2071i		−	1.51229i	−0.654407	−	0.756143i	$-0.727082\pi$
$373$		−	29.2071i		−	1.51229i	0.654407	−	0.756143i	$-0.272918\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	4.82843i		−	0.248677i
$378$		0			0
$379$	−10.3523			−0.531762			−0.265881	−	0.964006i	$-0.585663\pi$
$379$	−10.3523			−0.531762			−0.265881	+	0.964006i	$0.585663\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			30.0818i			1.53711i	0.639784	+	0.768555i	$0.279024\pi$
$383$			30.0818i			1.53711i	−0.639784	+	0.768555i	$0.720976\pi$
$384$		0			0
$385$	4.87437	+	4.56068i	0.248421	+	0.232434i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	33.2383			1.68525			0.842625	−	0.538501i	$-0.181009\pi$
$389$	33.2383			1.68525			0.842625	+	0.538501i	$0.181009\pi$
$390$		0			0
$391$	−24.9706			−1.26282
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−10.4684	+	11.1884i	−0.526723	+	0.562952i
$396$		0			0
$397$		−	13.5018i		−	0.677635i	−0.940852	−	0.338818i	$-0.889973\pi$
$397$		−	13.5018i		−	0.677635i	0.940852	−	0.338818i	$-0.110027\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.53159			−0.176359			−0.0881795	−	0.996105i	$-0.528105\pi$
$401$	−3.53159			−0.176359			−0.0881795	+	0.996105i	$0.528105\pi$
$402$		0			0
$403$			9.65685i			0.481042i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	8.56068i		−	0.424337i
$408$		0			0
$409$	24.7530			1.22396			0.611979	−	0.790874i	$-0.290374\pi$
$409$	24.7530			1.22396			0.611979	+	0.790874i	$0.290374\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	2.98527i		−	0.146896i
$414$		0			0
$415$	13.0150	+	12.1774i	0.638880	+	0.597765i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.39338			0.263484			0.131742	−	0.991284i	$-0.457943\pi$
$419$	5.39338			0.263484			0.131742	+	0.991284i	$0.457943\pi$
$420$		0			0
$421$	32.4337			1.58072			0.790362	−	0.612640i	$-0.209893\pi$
$421$	32.4337			1.58072			0.790362	+	0.612640i	$0.209893\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−35.6685	+	2.37438i	−1.73018	+	0.115174i
$426$		0			0
$427$			22.5650i			1.09199i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	31.5983			1.52204			0.761019	−	0.648730i	$-0.224699\pi$
$431$	31.5983			1.52204			0.761019	+	0.648730i	$0.224699\pi$
$432$		0			0
$433$		−	27.8566i		−	1.33870i	−0.742946	−	0.669351i	$-0.766572\pi$
$433$		−	27.8566i		−	1.33870i	0.742946	−	0.669351i	$-0.233428\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	23.1152i		−	1.10575i
$438$		0			0
$439$	−7.09190			−0.338478			−0.169239	−	0.985575i	$-0.554131\pi$
$439$	−7.09190			−0.338478			−0.169239	+	0.985575i	$0.554131\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			0.386578i			0.0183669i	0.999958	+	0.00918343i	$0.00292322\pi$
$443$			0.386578i			0.0183669i	−0.999958	+	0.00918343i	$0.997077\pi$
$444$		0			0
$445$	0.213845	−	0.228553i	0.0101372	−	0.0108344i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.18844			0.0560860			0.0280430	−	0.999607i	$-0.491072\pi$
$449$	1.18844			0.0560860			0.0280430	+	0.999607i	$0.491072\pi$
$450$		0			0
$451$	−3.44670			−0.162299
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.32106	+	4.61827i	−0.202575	+	0.216508i
$456$		0			0
$457$			10.5434i			0.493200i	0.969117	+	0.246600i	$0.0793135\pi$
$457$			10.5434i			0.493200i	−0.969117	+	0.246600i	$0.920687\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11.8961			−0.554056			−0.277028	−	0.960862i	$-0.589349\pi$
$461$	−11.8961			−0.554056			−0.277028	+	0.960862i	$0.589349\pi$
$462$		0			0
$463$		−	1.92032i		−	0.0892451i	−0.999004	−	0.0446225i	$-0.985791\pi$
$463$		−	1.92032i		−	0.0892451i	0.999004	−	0.0446225i	$-0.0142085\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	19.1256i		−	0.885029i	−0.896761	−	0.442514i	$-0.854087\pi$
$467$		−	19.1256i		−	0.885029i	0.896761	−	0.442514i	$-0.145913\pi$
$468$		0			0
$469$	4.15434			0.191829
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	7.85964i		−	0.361386i
$474$		0			0
$475$	−2.19796	−	33.0183i	−0.100849	−	1.51498i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.5793			0.574764			0.287382	−	0.957816i	$-0.407215\pi$
$479$	12.5793			0.574764			0.287382	+	0.957816i	$0.407215\pi$
$480$		0			0
$481$	8.11091			0.369825
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11.3306	−	10.6014i	−0.514495	−	0.481385i
$486$		0			0
$487$			22.8180i			1.03398i	0.855991	+	0.516991i	$0.172948\pi$
$487$			22.8180i			1.03398i	−0.855991	+	0.516991i	$0.827052\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−38.6924			−1.74616			−0.873081	−	0.487574i	$-0.837882\pi$
$491$	−38.6924			−1.74616			−0.873081	+	0.487574i	$0.837882\pi$
$492$		0			0
$493$		−	34.5208i		−	1.55474i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			2.39086i			0.107245i
$498$		0			0
$499$	26.5404			1.18811			0.594055	−	0.804424i	$-0.297526\pi$
$499$	26.5404			1.18811			0.594055	+	0.804424i	$0.297526\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	15.4008i		−	0.686686i	−0.939210	−	0.343343i	$-0.888441\pi$
$503$		−	15.4008i		−	0.686686i	0.939210	−	0.343343i	$-0.111559\pi$
$504$		0			0
$505$	24.6608	−	26.3570i	1.09739	−	1.17287i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−30.1846			−1.33791			−0.668955	−	0.743303i	$-0.733258\pi$
$509$	−30.1846			−1.33791			−0.668955	+	0.743303i	$0.733258\pi$
$510$		0			0
$511$	−26.2548			−1.16144
$512$		0			0
$513$		0			0
$514$		0			0
$515$	27.3544	+	25.5940i	1.20538	+	1.12781i
$516$		0			0
$517$			12.0074i			0.528084i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	26.7825			1.17336			0.586681	−	0.809818i	$-0.300434\pi$
$521$	26.7825			1.17336			0.586681	+	0.809818i	$0.300434\pi$
$522$		0			0
$523$		−	27.8670i		−	1.21854i	−0.792963	−	0.609270i	$-0.791463\pi$
$523$		−	27.8670i		−	1.21854i	0.792963	−	0.609270i	$-0.208537\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			69.0416i			3.00750i
$528$		0			0
$529$	10.8015			0.469630
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	3.26561i		−	0.141449i
$534$		0			0
$535$	−7.95889	−	7.44670i	−0.344093	−	0.321949i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.05545			0.0454616
$540$		0			0
$541$	20.8982			0.898485			0.449242	−	0.893410i	$-0.351694\pi$
$541$	20.8982			0.898485			0.449242	+	0.893410i	$0.351694\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.9225	−	11.6737i	0.467867	−	0.500047i
$546$		0			0
$547$			24.3106i			1.03945i	0.854334	+	0.519724i	$0.173965\pi$
$547$			24.3106i			1.03945i	−0.854334	+	0.519724i	$0.826035\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	31.9558			1.36136
$552$		0			0
$553$			19.3812i			0.824172i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			10.7197i			0.454207i	0.973871	+	0.227104i	$0.0729256\pi$
$557$			10.7197i			0.454207i	−0.973871	+	0.227104i	$0.927074\pi$
$558$		0			0
$559$	7.44670			0.314962
$560$		0			0
$561$		0			0
$562$		0			0
$563$			24.6898i			1.04055i	0.853998	+	0.520276i	$0.174171\pi$
$563$			24.6898i			1.04055i	−0.853998	+	0.520276i	$0.825829\pi$
$564$		0			0
$565$	−21.6516	−	20.2583i	−0.910891	−	0.852271i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−35.6023			−1.49252			−0.746262	−	0.665652i	$-0.768153\pi$
$569$	−35.6023			−1.49252			−0.746262	+	0.665652i	$0.768153\pi$
$570$		0			0
$571$	−13.3254			−0.557649			−0.278825	−	0.960342i	$-0.589945\pi$
$571$	−13.3254			−0.557649			−0.278825	+	0.960342i	$0.589945\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−17.4246	+	1.15992i	−0.726657	+	0.0483720i
$576$		0			0
$577$			26.3112i			1.09535i	0.836692	+	0.547674i	$0.184487\pi$
$577$			26.3112i			1.09535i	−0.836692	+	0.547674i	$0.815513\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	22.5452			0.935332
$582$		0			0
$583$		−	4.78249i		−	0.198070i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	28.3808i		−	1.17140i	−0.810527	−	0.585701i	$-0.800819\pi$
$587$		−	28.3808i		−	1.17140i	0.810527	−	0.585701i	$-0.199181\pi$
$588$		0			0
$589$	−63.9117			−2.63343
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6.81156i		−	0.279717i	−0.990171	−	0.139859i	$-0.955335\pi$
$593$		−	6.81156i		−	0.279717i	0.990171	−	0.139859i	$-0.0446648\pi$
$594$		0			0
$595$	−30.8934	+	33.0183i	−1.26651	+	1.35362i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	38.9307			1.59066			0.795332	−	0.606174i	$-0.207296\pi$
$599$	38.9307			1.59066			0.795332	+	0.606174i	$0.207296\pi$
$600$		0			0
$601$	−2.87131			−0.117123			−0.0585616	−	0.998284i	$-0.518651\pi$
$601$	−2.87131			−0.117123			−0.0585616	+	0.998284i	$0.518651\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−15.1031	+	16.1419i	−0.614030	+	0.656263i
$606$		0			0
$607$		−	10.9221i		−	0.443313i	−0.975125	−	0.221657i	$-0.928854\pi$
$607$		−	10.9221i		−	0.443313i	0.975125	−	0.221657i	$-0.0711464\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−11.3765			−0.460245
$612$		0			0
$613$			38.8779i			1.57026i	0.619328	+	0.785132i	$0.287405\pi$
$613$			38.8779i			1.57026i	−0.619328	+	0.785132i	$0.712595\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1.09656i			0.0441460i	0.999756	+	0.0220730i	$0.00702663\pi$
$617$			1.09656i			0.0441460i	−0.999756	+	0.0220730i	$0.992973\pi$
$618$		0			0
$619$	5.52209			0.221952			0.110976	−	0.993823i	$-0.464602\pi$
$619$	5.52209			0.221952			0.110976	+	0.993823i	$0.464602\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	0.395911i		−	0.0158618i
$624$		0			0
$625$	−24.7794	+	3.31371i	−0.991177	+	0.132548i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	57.9888			2.31217
$630$		0			0
$631$	−1.74873			−0.0696159			−0.0348079	−	0.999394i	$-0.511082\pi$
$631$	−1.74873			−0.0696159			−0.0348079	+	0.999394i	$0.511082\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	21.7266	+	20.3284i	0.862196	+	0.806709i
$636$		0			0
$637$			1.00000i			0.0396214i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.73400			−0.107987			−0.0539933	−	0.998541i	$-0.517195\pi$
$641$	−2.73400			−0.107987			−0.0539933	+	0.998541i	$0.517195\pi$
$642$		0			0
$643$			36.3180i			1.43224i	0.697976	+	0.716121i	$0.254084\pi$
$643$			36.3180i			1.43224i	−0.697976	+	0.716121i	$0.745916\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	28.4295i		−	1.11768i	−0.829276	−	0.558839i	$-0.811247\pi$
$647$		−	28.4295i		−	1.11768i	0.829276	−	0.558839i	$-0.188753\pi$
$648$		0			0
$649$	−1.11398			−0.0437276
$650$		0			0
$651$		0			0
$652$		0			0
$653$			7.48275i			0.292823i	0.989224	+	0.146411i	$0.0467723\pi$
$653$			7.48275i			0.292823i	−0.989224	+	0.146411i	$0.953228\pi$
$654$		0			0
$655$	−12.3211	+	13.1685i	−0.481424	+	0.514537i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.90635			−0.0742609			−0.0371304	−	0.999310i	$-0.511822\pi$
$659$	−1.90635			−0.0742609			−0.0371304	+	0.999310i	$0.511822\pi$
$660$		0			0
$661$	23.9761			0.932564			0.466282	−	0.884636i	$-0.345593\pi$
$661$	23.9761			0.932564			0.466282	+	0.884636i	$0.345593\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−30.5650	−	28.5980i	−1.18526	−	1.10898i
$666$		0			0
$667$		−	16.8639i		−	0.652974i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	8.42031			0.325063
$672$		0			0
$673$			18.6127i			0.717466i	0.933440	+	0.358733i	$0.116791\pi$
$673$			18.6127i			0.717466i	−0.933440	+	0.358733i	$0.883209\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			32.7530i			1.25880i	0.777081	+	0.629401i	$0.216700\pi$
$677$			32.7530i			1.25880i	−0.777081	+	0.629401i	$0.783300\pi$
$678$		0			0
$679$	−19.6274			−0.753230
$680$		0			0
$681$		0			0
$682$		0			0
$683$			34.5976i			1.32384i	0.749575	+	0.661920i	$0.230258\pi$
$683$			34.5976i			1.32384i	−0.749575	+	0.661920i	$0.769742\pi$
$684$		0			0
$685$	−17.7083	−	16.5686i	−0.676598	−	0.633055i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.53122			0.172626
$690$		0			0
$691$	3.99517			0.151984			0.0759918	−	0.997108i	$-0.475788\pi$
$691$	3.99517			0.151984			0.0759918	+	0.997108i	$0.475788\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−14.1109	+	15.0815i	−0.535257	+	0.572073i
$696$		0			0
$697$		−	23.3474i		−	0.884348i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	0.267750			0.0101128			0.00505638	−	0.999987i	$-0.498390\pi$
$701$	0.267750			0.0101128			0.00505638	+	0.999987i	$0.498390\pi$
$702$		0			0
$703$			53.6802i			2.02459i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	45.6569i		−	1.71710i
$708$		0			0
$709$	1.13478			0.0426176			0.0213088	−	0.999773i	$-0.493217\pi$
$709$	1.13478			0.0426176			0.0213088	+	0.999773i	$0.493217\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			33.7279i			1.26312i
$714$		0			0
$715$	1.72335	+	1.61244i	0.0644496	+	0.0603019i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−14.0434			−0.523732			−0.261866	−	0.965104i	$-0.584338\pi$
$719$	−14.0434			−0.523732			−0.261866	+	0.965104i	$0.584338\pi$
$720$		0			0
$721$	47.3847			1.76470
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.60354	−	24.0888i	−0.0595540	−	0.894636i
$726$		0			0
$727$		−	23.5099i		−	0.871934i	−0.899963	−	0.435967i	$-0.856407\pi$
$727$		−	23.5099i		−	0.871934i	0.899963	−	0.435967i	$-0.143593\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	53.2401			1.96916
$732$		0			0
$733$			21.6274i			0.798826i	0.916771	+	0.399413i	$0.130786\pi$
$733$			21.6274i			0.798826i	−0.916771	+	0.399413i	$0.869214\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1.55023i		−	0.0571034i
$738$		0			0
$739$	26.8738			0.988569			0.494285	−	0.869300i	$-0.335430\pi$
$739$	26.8738			0.988569			0.494285	+	0.869300i	$0.335430\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			11.5506i			0.423751i	0.977297	+	0.211875i	$0.0679571\pi$
$743$			11.5506i			0.423751i	−0.977297	+	0.211875i	$0.932043\pi$
$744$		0			0
$745$	−10.0851	+	10.7788i	−0.369491	+	0.394905i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−13.7868			−0.503758
$750$		0			0
$751$	52.5980			1.91933			0.959663	−	0.281151i	$-0.0907163\pi$
$751$	52.5980			1.91933			0.959663	+	0.281151i	$0.0907163\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	22.7530	−	24.3180i	0.828068	−	0.885023i
$756$		0			0
$757$			18.8406i			0.684774i	0.939559	+	0.342387i	$0.111235\pi$
$757$			18.8406i			0.684774i	−0.939559	+	0.342387i	$0.888765\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	49.3323			1.78830			0.894148	−	0.447771i	$-0.147782\pi$
$761$	49.3323			1.78830			0.894148	+	0.447771i	$0.147782\pi$
$762$		0			0
$763$		−	20.2218i		−	0.732079i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	1.05545i		−	0.0381102i
$768$		0			0
$769$	−32.6274			−1.17657			−0.588287	−	0.808652i	$-0.700198\pi$
$769$	−32.6274			−1.17657			−0.588287	+	0.808652i	$0.700198\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	19.0229i		−	0.684208i	−0.939662	−	0.342104i	$-0.888861\pi$
$773$		−	19.0229i		−	0.684208i	0.939662	−	0.342104i	$-0.111139\pi$
$774$		0			0
$775$	3.20708	+	48.1776i	0.115202	+	1.73059i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	21.6127			0.774355
$780$		0			0
$781$	0.892170			0.0319243
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−18.5870	−	17.3909i	−0.663400	−	0.620707i
$786$		0			0
$787$		−	3.79216i		−	0.135176i	−0.997713	−	0.0675880i	$-0.978470\pi$
$787$		−	3.79216i		−	0.135176i	0.997713	−	0.0675880i	$-0.0215303\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−37.5061			−1.33356
$792$		0			0
$793$			7.97792i			0.283304i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			31.0287i			1.09909i	0.835463	+	0.549547i	$0.185199\pi$
$797$			31.0287i			1.09909i	−0.835463	+	0.549547i	$0.814801\pi$
$798$		0			0
$799$	−81.3363			−2.87747
$800$		0			0
$801$		0			0
$802$		0			0
$803$			9.79722i			0.345736i
$804$		0			0
$805$	−15.0919	+	16.1299i	−0.531919	+	0.568505i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	49.2211			1.73052			0.865260	−	0.501323i	$-0.167153\pi$
$809$	49.2211			1.73052			0.865260	+	0.501323i	$0.167153\pi$
$810$		0			0
$811$	8.76291			0.307707			0.153854	−	0.988094i	$-0.450832\pi$
$811$	8.76291			0.307707			0.153854	+	0.988094i	$0.450832\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	11.9419	+	11.1733i	0.418305	+	0.391385i
$816$		0			0
$817$			49.2843i			1.72424i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24.3060			−0.848284			−0.424142	−	0.905596i	$-0.639424\pi$
$821$	−24.3060			−0.848284			−0.424142	+	0.905596i	$0.639424\pi$
$822$		0			0
$823$		−	43.1586i		−	1.50442i	−0.658926	−	0.752208i	$-0.728989\pi$
$823$		−	43.1586i		−	1.50442i	0.658926	−	0.752208i	$-0.271011\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			27.7189i			0.963882i	0.876204	+	0.481941i	$0.160068\pi$
$827$			27.7189i			0.963882i	−0.876204	+	0.481941i	$0.839932\pi$
$828$		0			0
$829$	−23.2094			−0.806096			−0.403048	−	0.915179i	$-0.632049\pi$
$829$	−23.2094			−0.806096			−0.403048	+	0.915179i	$0.632049\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			7.14949i			0.247715i
$834$		0			0
$835$	−25.0518	−	23.4396i	−0.866952	−	0.811160i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−25.2796			−0.872748			−0.436374	−	0.899765i	$-0.643738\pi$
$839$	−25.2796			−0.872748			−0.436374	+	0.899765i	$0.643738\pi$
$840$		0			0
$841$	−5.68629			−0.196079
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.52773	+	1.63280i	−0.0525554	+	0.0561702i
$846$		0			0
$847$			27.9619i			0.960782i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	28.3284			0.971086
$852$		0			0
$853$		−	4.43934i		−	0.152000i	−0.997108	−	0.0760001i	$-0.975785\pi$
$853$		−	4.43934i		−	0.152000i	0.997108	−	0.0760001i	$-0.0242149\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	31.3609i		−	1.07127i	−0.844451	−	0.535633i	$-0.820073\pi$
$857$		−	31.3609i		−	1.07127i	0.844451	−	0.535633i	$-0.179927\pi$
$858$		0			0
$859$	28.7457			0.980791			0.490395	−	0.871500i	$-0.336852\pi$
$859$	28.7457			0.980791			0.490395	+	0.871500i	$0.336852\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			36.3471i			1.23727i	0.785679	+	0.618634i	$0.212314\pi$
$863$			36.3471i			1.23727i	−0.785679	+	0.618634i	$0.787686\pi$
$864$		0			0
$865$	30.9752	+	28.9818i	1.05319	+	0.985412i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.23226			0.245338
$870$		0			0
$871$	1.46878			0.0497677
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−20.0239	+	24.4754i	−0.676930	+	0.827420i
$876$		0			0
$877$		−	21.9896i		−	0.742535i	−0.928526	−	0.371268i	$-0.878923\pi$
$877$		−	21.9896i		−	0.742535i	0.928526	−	0.371268i	$-0.121077\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	44.5348			1.50041			0.750207	−	0.661203i	$-0.229954\pi$
$881$	44.5348			1.50041			0.750207	+	0.661203i	$0.229954\pi$
$882$		0			0
$883$		−	6.05277i		−	0.203692i	−0.994800	−	0.101846i	$-0.967525\pi$
$883$		−	6.05277i		−	0.203692i	0.994800	−	0.101846i	$-0.0324749\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	35.9700i		−	1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$887$		−	35.9700i		−	1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$888$		0			0
$889$	37.6360			1.26227
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	75.2929i		−	2.51958i
$894$		0			0
$895$	−24.3284	+	26.0018i	−0.813210	+	0.869143i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−46.6274			−1.55511
$900$		0			0
$901$	32.3959			1.07926
$902$		0			0
$903$		0			0
$904$		0			0
$905$	24.5615	−	26.2509i	0.816452	−	0.872609i
$906$		0			0
$907$			39.2695i			1.30392i	0.758252	+	0.651962i	$0.226054\pi$
$907$			39.2695i			1.30392i	−0.758252	+	0.651962i	$0.773946\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	26.4583			0.876604			0.438302	−	0.898828i	$-0.355580\pi$
$911$	26.4583			0.876604			0.438302	+	0.898828i	$0.355580\pi$
$912$		0			0
$913$		−	8.41294i		−	0.278428i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			22.8112i			0.753291i
$918$		0			0
$919$	37.1513			1.22551			0.612754	−	0.790274i	$-0.290062\pi$
$919$	37.1513			1.22551			0.612754	+	0.790274i	$0.290062\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			0.845296i			0.0278232i
$924$		0			0
$925$	40.4650	−	2.69367i	1.33048	−	0.0885672i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	15.2412			0.500048			0.250024	−	0.968240i	$-0.419562\pi$
$929$	15.2412			0.500048			0.250024	+	0.968240i	$0.419562\pi$
$930$		0			0
$931$	−6.61827			−0.216905
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.3211	+	11.5281i	0.402942	+	0.377011i
$936$		0			0
$937$			28.1850i			0.920764i	0.887721	+	0.460382i	$0.152288\pi$
$937$			28.1850i			0.920764i	−0.887721	+	0.460382i	$0.847712\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−42.8752			−1.39769			−0.698846	−	0.715272i	$-0.746303\pi$
$941$	−42.8752			−1.39769			−0.698846	+	0.715272i	$0.746303\pi$
$942$		0			0
$943$		−	11.4056i		−	0.371417i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	11.3428i		−	0.368591i	−0.982871	−	0.184295i	$-0.941000\pi$
$947$		−	11.3428i		−	0.368591i	0.982871	−	0.184295i	$-0.0590003\pi$
$948$		0			0
$949$	−9.28248			−0.301322
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	0.754313i		−	0.0244346i	−0.999925	−	0.0122173i	$-0.996111\pi$
$953$		−	0.754313i		−	0.0244346i	0.999925	−	0.0122173i	$-0.00388898\pi$
$954$		0			0
$955$	−9.62312	+	10.2850i	−0.311397	+	0.332815i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−30.6751			−0.990552
$960$		0			0
$961$	62.2548			2.00822
$962$		0			0
$963$		0			0
$964$		0			0
$965$	38.8110	+	36.3133i	1.24937	+	1.16897i
$966$		0			0
$967$			17.2530i			0.554820i	0.960752	+	0.277410i	$0.0894760\pi$
$967$			17.2530i			0.554820i	−0.960752	+	0.277410i	$0.910524\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	35.2825			1.13227			0.566134	−	0.824313i	$-0.308438\pi$
$971$	35.2825			1.13227			0.566134	+	0.824313i	$0.308438\pi$
$972$		0			0
$973$			26.1249i			0.837525i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			9.04808i			0.289474i	0.989470	+	0.144737i	$0.0462336\pi$
$977$			9.04808i			0.289474i	−0.989470	+	0.144737i	$0.953766\pi$
$978$		0			0
$979$	−0.147738			−0.00472172
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	30.5255i		−	0.973611i	−0.873510	−	0.486805i	$-0.838162\pi$
$983$		−	30.5255i		−	0.973611i	0.873510	−	0.486805i	$-0.161838\pi$
$984$		0			0
$985$	12.4547	+	11.6532i	0.396839	+	0.371301i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	26.0086			0.827025
$990$		0			0
$991$	−6.10048			−0.193788			−0.0968940	−	0.995295i	$-0.530891\pi$
$991$	−6.10048			−0.193788			−0.0968940	+	0.995295i	$0.530891\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−10.4684	+	11.1884i	−0.331871	+	0.354697i
$996$		0			0
$997$			31.4510i			0.996062i	0.867159	+	0.498031i	$0.165944\pi$
$997$			31.4510i			0.996062i	−0.867159	+	0.498031i	$0.834056\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.l.f.2809.7		8
3.2	odd	2		1560.2.l.e.1249.5	yes	8
5.4	even	2	inner	4680.2.l.f.2809.8		8
12.11	even	2		3120.2.l.o.1249.1		8
15.2	even	4		7800.2.a.bw.1.4		4
15.8	even	4		7800.2.a.bv.1.2		4
15.14	odd	2		1560.2.l.e.1249.1	✓	8
60.59	even	2		3120.2.l.o.1249.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.e.1249.1	✓	8	15.14	odd	2
1560.2.l.e.1249.5	yes	8	3.2	odd	2
3120.2.l.o.1249.1		8	12.11	even	2
3120.2.l.o.1249.5		8	60.59	even	2
4680.2.l.f.2809.7		8	1.1	even	1	trivial
4680.2.l.f.2809.8		8	5.4	even	2	inner
7800.2.a.bv.1.2		4	15.8	even	4
7800.2.a.bw.1.4		4	15.2	even	4

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:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.52773 - 1.63280i) q^{5} -2.82843i q^{7} +O(q^{10})\)	\(q+(1.52773 - 1.63280i) q^{5} -2.82843i q^{7} -1.05545 q^{11} -1.00000i q^{13} -7.14949i q^{17} +6.61827 q^{19} -3.49264i q^{23} +(-0.332104 - 4.98896i) q^{25} +4.82843 q^{29} -9.65685 q^{31} +(-4.61827 - 4.32106i) q^{35} +8.11091i q^{37} +3.26561 q^{41} +7.44670i q^{43} -11.3765i q^{47} -1.00000 q^{49} +4.53122i q^{53} +(-1.61244 + 1.72335i) q^{55} +1.05545 q^{59} -7.97792 q^{61} +(-1.63280 - 1.52773i) q^{65} +1.46878i q^{67} -0.845296 q^{71} -9.28248i q^{73} +2.98527i q^{77} -6.85228 q^{79} +7.97093i q^{83} +(-11.6737 - 10.9225i) q^{85} +0.139976 q^{89} -2.82843 q^{91} +(10.1109 - 10.8063i) q^{95} -6.93933i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{11} + 24 q^{19} - 4 q^{25} + 16 q^{29} - 32 q^{31} - 8 q^{35} + 8 q^{41} - 8 q^{49} - 36 q^{55} - 16 q^{59} + 24 q^{61} - 4 q^{65} + 24 q^{71} + 24 q^{79} - 40 q^{85} - 8 q^{89} + 32 q^{95}+O(q^{100}) \) 8 * q + 16 * q^11 + 24 * q^19 - 4 * q^25 + 16 * q^29 - 32 * q^31 - 8 * q^35 + 8 * q^41 - 8 * q^49 - 36 * q^55 - 16 * q^59 + 24 * q^61 - 4 * q^65 + 24 * q^71 + 24 * q^79 - 40 * q^85 - 8 * q^89 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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