LMFDB - Embedded newform 2340.2.h.e.469.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2340,2,Mod(469,2340)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2340, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2340.469");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.h (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:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			469.2
Root			$1.45161 + 1.45161i$ of defining polynomial
Character	$\chi$	$=$	2340.469
Dual form			2340.2.h.e.469.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.21432 + 0.311108i) q^{5} -0.903212i q^{7} +0.0666765 q^{11} -1.00000i q^{13} -3.37778i q^{17} -5.11753 q^{19} +4.21432i q^{23} +(4.80642 - 1.37778i) q^{25} +1.52543 q^{29} +4.49532 q^{31} +(0.280996 + 2.00000i) q^{35} +11.9541i q^{37} -2.75557 q^{41} +8.77631i q^{43} -8.90321i q^{47} +6.18421 q^{49} +3.57136i q^{53} +(-0.147643 + 0.0207436i) q^{55} -8.16839 q^{59} -11.1985 q^{61} +(0.311108 + 2.21432i) q^{65} +2.14764i q^{67} +5.54617 q^{71} +2.70964i q^{73} -0.0602231i q^{77} +3.18421 q^{79} +9.89384i q^{83} +(1.05086 + 7.47949i) q^{85} -14.4701 q^{89} -0.903212 q^{91} +(11.3319 - 1.59210i) q^{95} +7.93978i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{19} + 2 q^{25} - 4 q^{29} - 12 q^{35} - 16 q^{41} + 10 q^{49} + 12 q^{55} + 4 q^{59} - 28 q^{61} + 2 q^{65} - 20 q^{71} - 8 q^{79} - 20 q^{85} + 20 q^{89} + 8 q^{91} + 28 q^{95}+O(q^{100}) $ 6 * q - 4 * q^19 + 2 * q^25 - 4 * q^29 - 12 * q^35 - 16 * q^41 + 10 * q^49 + 12 * q^55 + 4 * q^59 - 28 * q^61 + 2 * q^65 - 20 * q^71 - 8 * q^79 - 20 * q^85 + 20 * q^89 + 8 * q^91 + 28 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	−2.21432	+	0.311108i	−0.990274	+	0.139132i
$5$	−2.21432	+	0.311108i	−0.990274	+	0.139132i
$6$		0			0
$7$		−	0.903212i		−	0.341382i	−0.985325	−	0.170691i	$-0.945400\pi$
$7$		−	0.903212i		−	0.341382i	0.985325	−	0.170691i	$-0.0546000\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.0666765			0.0201037			0.0100519	−	0.999949i	$-0.496800\pi$
$11$	0.0666765			0.0201037			0.0100519	+	0.999949i	$0.496800\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.37778i		−	0.819233i	−0.912258	−	0.409617i	$-0.865663\pi$
$17$		−	3.37778i		−	0.819233i	0.912258	−	0.409617i	$-0.134337\pi$
$18$		0			0
$19$	−5.11753			−1.17404			−0.587021	−	0.809572i	$-0.699699\pi$
$19$	−5.11753			−1.17404			−0.587021	+	0.809572i	$0.699699\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.21432i			0.878746i	0.898305	+	0.439373i	$0.144799\pi$
$23$			4.21432i			0.878746i	−0.898305	+	0.439373i	$0.855201\pi$
$24$		0			0
$25$	4.80642	−	1.37778i	0.961285	−	0.275557i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.52543			0.283265			0.141632	−	0.989919i	$-0.454765\pi$
$29$	1.52543			0.283265			0.141632	+	0.989919i	$0.454765\pi$
$30$		0			0
$31$	4.49532			0.807383			0.403691	−	0.914895i	$-0.367727\pi$
$31$	4.49532			0.807383			0.403691	+	0.914895i	$0.367727\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.280996	+	2.00000i	0.0474970	+	0.338062i
$36$		0			0
$37$			11.9541i			1.96524i	0.185638	+	0.982618i	$0.440565\pi$
$37$			11.9541i			1.96524i	−0.185638	+	0.982618i	$0.559435\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.75557			−0.430348			−0.215174	−	0.976576i	$-0.569032\pi$
$41$	−2.75557			−0.430348			−0.215174	+	0.976576i	$0.569032\pi$
$42$		0			0
$43$			8.77631i			1.33838i	0.743094	+	0.669188i	$0.233358\pi$
$43$			8.77631i			1.33838i	−0.743094	+	0.669188i	$0.766642\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	8.90321i		−	1.29867i	−0.760504	−	0.649333i	$-0.775048\pi$
$47$		−	8.90321i		−	1.29867i	0.760504	−	0.649333i	$-0.224952\pi$
$48$		0			0
$49$	6.18421			0.883458
$50$		0			0
$51$		0			0
$52$		0			0
$53$			3.57136i			0.490564i	0.969452	+	0.245282i	$0.0788806\pi$
$53$			3.57136i			0.490564i	−0.969452	+	0.245282i	$0.921119\pi$
$54$		0			0
$55$	−0.147643	+	0.0207436i	−0.0199082	+	0.00279707i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.16839			−1.06343			−0.531717	−	0.846922i	$-0.678453\pi$
$59$	−8.16839			−1.06343			−0.531717	+	0.846922i	$0.678453\pi$
$60$		0			0
$61$	−11.1985			−1.43382			−0.716910	−	0.697165i	$-0.754444\pi$
$61$	−11.1985			−1.43382			−0.716910	+	0.697165i	$0.754444\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.311108	+	2.21432i	0.0385882	+	0.274653i
$66$		0			0
$67$			2.14764i			0.262376i	0.991357	+	0.131188i	$0.0418792\pi$
$67$			2.14764i			0.262376i	−0.991357	+	0.131188i	$0.958121\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.54617			0.658209			0.329105	−	0.944293i	$-0.393253\pi$
$71$	5.54617			0.658209			0.329105	+	0.944293i	$0.393253\pi$
$72$		0			0
$73$			2.70964i			0.317139i	0.987348	+	0.158569i	$0.0506882\pi$
$73$			2.70964i			0.317139i	−0.987348	+	0.158569i	$0.949312\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	0.0602231i		−	0.00686305i
$78$		0			0
$79$	3.18421			0.358251			0.179126	−	0.983826i	$-0.442673\pi$
$79$	3.18421			0.358251			0.179126	+	0.983826i	$0.442673\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			9.89384i			1.08599i	0.839736	+	0.542995i	$0.182710\pi$
$83$			9.89384i			1.08599i	−0.839736	+	0.542995i	$0.817290\pi$
$84$		0			0
$85$	1.05086	+	7.47949i	0.113981	+	0.811265i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−14.4701			−1.53383			−0.766915	−	0.641748i	$-0.778209\pi$
$89$	−14.4701			−1.53383			−0.766915	+	0.641748i	$0.778209\pi$
$90$		0			0
$91$	−0.903212			−0.0946823
$92$		0			0
$93$		0			0
$94$		0			0
$95$	11.3319	−	1.59210i	1.16262	−	0.163346i
$96$		0			0
$97$			7.93978i			0.806162i	0.915164	+	0.403081i	$0.132061\pi$
$97$			7.93978i			0.806162i	−0.915164	+	0.403081i	$0.867939\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.18421			0.515848			0.257924	−	0.966165i	$-0.416962\pi$
$101$	5.18421			0.515848			0.257924	+	0.966165i	$0.416962\pi$
$102$		0			0
$103$			7.88739i			0.777168i	0.921413	+	0.388584i	$0.127036\pi$
$103$			7.88739i			0.777168i	−0.921413	+	0.388584i	$0.872964\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			17.3669i			1.67892i	0.543421	+	0.839460i	$0.317129\pi$
$107$			17.3669i			1.67892i	−0.543421	+	0.839460i	$0.682871\pi$
$108$		0			0
$109$	−0.133353			−0.0127729			−0.00638645	−	0.999980i	$-0.502033\pi$
$109$	−0.133353			−0.0127729			−0.00638645	+	0.999980i	$0.502033\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			5.86665i			0.551888i	0.961174	+	0.275944i	$0.0889904\pi$
$113$			5.86665i			0.551888i	−0.961174	+	0.275944i	$0.911010\pi$
$114$		0			0
$115$	−1.31111	−	9.33185i	−0.122261	−	0.870200i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.05086			−0.279671
$120$		0			0
$121$	−10.9956			−0.999596
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.2143	+	4.54617i	−0.913597	+	0.406622i
$126$		0			0
$127$			15.5002i			1.37542i	0.725984	+	0.687712i	$0.241385\pi$
$127$			15.5002i			1.37542i	−0.725984	+	0.687712i	$0.758615\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.80642			0.507310			0.253655	−	0.967295i	$-0.418367\pi$
$131$	5.80642			0.507310			0.253655	+	0.967295i	$0.418367\pi$
$132$		0			0
$133$			4.62222i			0.400797i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			16.2766i			1.39060i	0.718720	+	0.695300i	$0.244728\pi$
$137$			16.2766i			1.39060i	−0.718720	+	0.695300i	$0.755272\pi$
$138$		0			0
$139$	7.18421			0.609357			0.304678	−	0.952455i	$-0.401451\pi$
$139$	7.18421			0.609357			0.304678	+	0.952455i	$0.401451\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	0.0666765i		−	0.00557577i
$144$		0			0
$145$	−3.37778	+	0.474572i	−0.280510	+	0.0394111i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.2859			1.08842			0.544212	−	0.838947i	$-0.316829\pi$
$149$	13.2859			1.08842			0.544212	+	0.838947i	$0.316829\pi$
$150$		0			0
$151$	−15.3111			−1.24600			−0.623000	−	0.782222i	$-0.714086\pi$
$151$	−15.3111			−1.24600			−0.623000	+	0.782222i	$0.714086\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.95407	+	1.39853i	−0.799530	+	0.112332i
$156$		0			0
$157$		−	18.4701i		−	1.47408i	−0.675851	−	0.737038i	$-0.736224\pi$
$157$		−	18.4701i		−	1.47408i	0.675851	−	0.737038i	$-0.263776\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.80642			0.299988
$162$		0			0
$163$			12.7699i			1.00021i	0.865964	+	0.500106i	$0.166706\pi$
$163$			12.7699i			1.00021i	−0.865964	+	0.500106i	$0.833294\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	7.43356i		−	0.575226i	−0.957747	−	0.287613i	$-0.907138\pi$
$167$		−	7.43356i		−	0.575226i	0.957747	−	0.287613i	$-0.0928617\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$			20.4286i			1.55316i	0.630018	+	0.776580i	$0.283047\pi$
$173$			20.4286i			1.55316i	−0.630018	+	0.776580i	$0.716953\pi$
$174$		0			0
$175$	−1.24443	−	4.34122i	−0.0940702	−	0.328165i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	24.6035			1.83895			0.919475	−	0.393148i	$-0.128614\pi$
$179$	24.6035			1.83895			0.919475	+	0.393148i	$0.128614\pi$
$180$		0			0
$181$	13.9956			1.04028			0.520141	−	0.854081i	$-0.325880\pi$
$181$	13.9956			1.04028			0.520141	+	0.854081i	$0.325880\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.71900	−	26.4701i	−0.273427	−	1.94612i
$186$		0			0
$187$		−	0.225219i		−	0.0164696i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.7556			0.778246			0.389123	−	0.921186i	$-0.372778\pi$
$191$	10.7556			0.778246			0.389123	+	0.921186i	$0.372778\pi$
$192$		0			0
$193$		−	15.4193i		−	1.10990i	−0.831883	−	0.554952i	$-0.812737\pi$
$193$		−	15.4193i		−	1.10990i	0.831883	−	0.554952i	$-0.187263\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	21.2257i		−	1.51227i	−0.654417	−	0.756134i	$-0.727086\pi$
$197$		−	21.2257i		−	1.51227i	0.654417	−	0.756134i	$-0.272914\pi$
$198$		0			0
$199$	−20.3368			−1.44164			−0.720818	−	0.693125i	$-0.756234\pi$
$199$	−20.3368			−1.44164			−0.720818	+	0.693125i	$0.756234\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	1.37778i		−	0.0967015i
$204$		0			0
$205$	6.10171	−	0.857279i	0.426162	−	0.0598750i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.341219			−0.0236026
$210$		0			0
$211$	−20.7239			−1.42669			−0.713347	−	0.700811i	$-0.752822\pi$
$211$	−20.7239			−1.42669			−0.713347	+	0.700811i	$0.752822\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.73038	−	19.4336i	−0.186210	−	1.32536i
$216$		0			0
$217$		−	4.06022i		−	0.275626i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.37778			−0.227214
$222$		0			0
$223$			20.2494i			1.35600i	0.735063	+	0.677999i	$0.237152\pi$
$223$			20.2494i			1.35600i	−0.735063	+	0.677999i	$0.762848\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			16.9032i			1.12191i	0.827848	+	0.560953i	$0.189565\pi$
$227$			16.9032i			1.12191i	−0.827848	+	0.560953i	$0.810435\pi$
$228$		0			0
$229$	−3.96836			−0.262236			−0.131118	−	0.991367i	$-0.541857\pi$
$229$	−3.96836			−0.262236			−0.131118	+	0.991367i	$0.541857\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			15.3176i			1.00349i	0.865017	+	0.501743i	$0.167308\pi$
$233$			15.3176i			1.00349i	−0.865017	+	0.501743i	$0.832692\pi$
$234$		0			0
$235$	2.76986	+	19.7146i	0.180686	+	1.28604i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−14.8637			−0.961455			−0.480727	−	0.876870i	$-0.659627\pi$
$239$	−14.8637			−0.961455			−0.480727	+	0.876870i	$0.659627\pi$
$240$		0			0
$241$	20.2953			1.30733			0.653667	−	0.756782i	$-0.273230\pi$
$241$	20.2953			1.30733			0.653667	+	0.756782i	$0.273230\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−13.6938	+	1.92396i	−0.874866	+	0.122917i
$246$		0			0
$247$			5.11753i			0.325621i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−20.5620			−1.29786			−0.648931	−	0.760847i	$-0.724784\pi$
$251$	−20.5620			−1.29786			−0.648931	+	0.760847i	$0.724784\pi$
$252$		0			0
$253$			0.280996i			0.0176661i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	27.5210i		−	1.71671i	−0.513055	−	0.858356i	$-0.671486\pi$
$257$		−	27.5210i		−	1.71671i	0.513055	−	0.858356i	$-0.328514\pi$
$258$		0			0
$259$	10.7971			0.670896
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	8.58274i		−	0.529234i	−0.964354	−	0.264617i	$-0.914754\pi$
$263$		−	8.58274i		−	0.529234i	0.964354	−	0.264617i	$-0.0852456\pi$
$264$		0			0
$265$	−1.11108	−	7.90813i	−0.0682530	−	0.485793i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.8988			0.664510			0.332255	−	0.943190i	$-0.392191\pi$
$269$	10.8988			0.664510			0.332255	+	0.943190i	$0.392191\pi$
$270$		0			0
$271$	−8.46367			−0.514132			−0.257066	−	0.966394i	$-0.582756\pi$
$271$	−8.46367			−0.514132			−0.257066	+	0.966394i	$0.582756\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.320476	−	0.0918659i	0.0193254	−	0.00553972i
$276$		0			0
$277$		−	28.7239i		−	1.72585i	−0.505329	−	0.862927i	$-0.668629\pi$
$277$		−	28.7239i		−	1.72585i	0.505329	−	0.862927i	$-0.331371\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.1936			0.846718			0.423359	−	0.905962i	$-0.360851\pi$
$281$	14.1936			0.846718			0.423359	+	0.905962i	$0.360851\pi$
$282$		0			0
$283$			15.9190i			0.946288i	0.880985	+	0.473144i	$0.156881\pi$
$283$			15.9190i			0.946288i	−0.880985	+	0.473144i	$0.843119\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			2.48886i			0.146913i
$288$		0			0
$289$	5.59057			0.328857
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	20.4242i		−	1.19319i	−0.802541	−	0.596597i	$-0.796519\pi$
$293$		−	20.4242i		−	1.19319i	0.802541	−	0.596597i	$-0.203481\pi$
$294$		0			0
$295$	18.0874	−	2.54125i	1.05309	−	0.147957i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.21432			0.243720
$300$		0			0
$301$	7.92687			0.456897
$302$		0			0
$303$		0			0
$304$		0			0
$305$	24.7971	−	3.48394i	1.41988	−	0.199490i
$306$		0			0
$307$			0.0874201i			0.00498933i	0.999997	+	0.00249467i	$0.000794078\pi$
$307$			0.0874201i			0.00498933i	−0.999997	+	0.00249467i	$0.999206\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	25.7146			1.45814			0.729069	−	0.684440i	$-0.239953\pi$
$311$	25.7146			1.45814			0.729069	+	0.684440i	$0.239953\pi$
$312$		0			0
$313$			7.67307i			0.433708i	0.976204	+	0.216854i	$0.0695795\pi$
$313$			7.67307i			0.433708i	−0.976204	+	0.216854i	$0.930421\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	3.65878i		−	0.205498i	−0.994707	−	0.102749i	$-0.967236\pi$
$317$		−	3.65878i		−	0.205498i	0.994707	−	0.102749i	$-0.0327638\pi$
$318$		0			0
$319$	0.101710			0.00569468
$320$		0			0
$321$		0			0
$322$		0			0
$323$			17.2859i			0.961814i
$324$		0			0
$325$	−1.37778	−	4.80642i	−0.0764257	−	0.266612i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8.04149			−0.443342
$330$		0			0
$331$	−0.403450			−0.0221756			−0.0110878	−	0.999939i	$-0.503529\pi$
$331$	−0.403450			−0.0221756			−0.0110878	+	0.999939i	$0.503529\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.668149	−	4.75557i	−0.0365049	−	0.259824i
$336$		0			0
$337$		−	20.7971i		−	1.13289i	−0.824100	−	0.566444i	$-0.808319\pi$
$337$		−	20.7971i		−	1.13289i	0.824100	−	0.566444i	$-0.191681\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.299732			0.0162314
$342$		0			0
$343$		−	11.9081i		−	0.642979i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			8.28745i			0.444894i	0.974945	+	0.222447i	$0.0714044\pi$
$347$			8.28745i			0.444894i	−0.974945	+	0.222447i	$0.928596\pi$
$348$		0			0
$349$	−21.3590			−1.14332			−0.571662	−	0.820489i	$-0.693701\pi$
$349$	−21.3590			−1.14332			−0.571662	+	0.820489i	$0.693701\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			23.3002i			1.24014i	0.784545	+	0.620072i	$0.212897\pi$
$353$			23.3002i			1.24014i	−0.784545	+	0.620072i	$0.787103\pi$
$354$		0			0
$355$	−12.2810	+	1.72546i	−0.651808	+	0.0915778i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−34.6987			−1.83133			−0.915665	−	0.401943i	$-0.868335\pi$
$359$	−34.6987			−1.83133			−0.915665	+	0.401943i	$0.868335\pi$
$360$		0			0
$361$	7.18913			0.378375
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.842989	−	6.00000i	−0.0441241	−	0.314054i
$366$		0			0
$367$		−	27.1131i		−	1.41529i	−0.706567	−	0.707646i	$-0.749757\pi$
$367$		−	27.1131i		−	1.41529i	0.706567	−	0.707646i	$-0.250243\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.22570			0.167470
$372$		0			0
$373$		−	19.7146i		−	1.02078i	−0.859943	−	0.510391i	$-0.829501\pi$
$373$		−	19.7146i		−	1.02078i	0.859943	−	0.510391i	$-0.170499\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1.52543i		−	0.0785635i
$378$		0			0
$379$	24.8321			1.27554			0.637769	−	0.770227i	$-0.279857\pi$
$379$	24.8321			1.27554			0.637769	+	0.770227i	$0.279857\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	1.62714i		−	0.0831429i	−0.999136	−	0.0415714i	$-0.986764\pi$
$383$		−	1.62714i		−	0.0831429i	0.999136	−	0.0415714i	$-0.0132364\pi$
$384$		0			0
$385$	0.0187359	+	0.133353i	0.000954868	+	0.00679630i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.48886			−0.430402			−0.215201	−	0.976570i	$-0.569041\pi$
$389$	−8.48886			−0.430402			−0.215201	+	0.976570i	$0.569041\pi$
$390$		0			0
$391$	14.2351			0.719898
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.05086	+	0.990632i	−0.354767	+	0.0498441i
$396$		0			0
$397$		−	9.10663i		−	0.457049i	−0.973538	−	0.228524i	$-0.926610\pi$
$397$		−	9.10663i		−	0.457049i	0.973538	−	0.228524i	$-0.0733901\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.1748			−0.907608			−0.453804	−	0.891101i	$-0.649933\pi$
$401$	−18.1748			−0.907608			−0.453804	+	0.891101i	$0.649933\pi$
$402$		0			0
$403$		−	4.49532i		−	0.223928i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.797056i			0.0395086i
$408$		0			0
$409$	11.3461			0.561031			0.280515	−	0.959850i	$-0.409495\pi$
$409$	11.3461			0.561031			0.280515	+	0.959850i	$0.409495\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			7.37778i			0.363037i
$414$		0			0
$415$	−3.07805	−	21.9081i	−0.151096	−	1.07543i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−15.7047			−0.767225			−0.383613	−	0.923494i	$-0.625320\pi$
$419$	−15.7047			−0.767225			−0.383613	+	0.923494i	$0.625320\pi$
$420$		0			0
$421$	−15.0923			−0.735556			−0.367778	−	0.929914i	$-0.619881\pi$
$421$	−15.0923			−0.735556			−0.367778	+	0.929914i	$0.619881\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.65386	−	16.2351i	−0.225745	−	0.787516i
$426$		0			0
$427$			10.1146i			0.489481i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18.9842			−0.914436			−0.457218	−	0.889355i	$-0.651154\pi$
$431$	−18.9842			−0.914436			−0.457218	+	0.889355i	$0.651154\pi$
$432$		0			0
$433$			1.04101i			0.0500278i	0.999687	+	0.0250139i	$0.00796300\pi$
$433$			1.04101i			0.0500278i	−0.999687	+	0.0250139i	$0.992037\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	21.5669i		−	1.03169i
$438$		0			0
$439$	19.3876			0.925321			0.462661	−	0.886536i	$-0.346895\pi$
$439$	19.3876			0.925321			0.462661	+	0.886536i	$0.346895\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			36.1037i			1.71534i	0.514201	+	0.857670i	$0.328089\pi$
$443$			36.1037i			1.71534i	−0.514201	+	0.857670i	$0.671911\pi$
$444$		0			0
$445$	32.0415	−	4.50177i	1.51891	−	0.213404i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.34614			0.346686			0.173343	−	0.984862i	$-0.444543\pi$
$449$	7.34614			0.346686			0.173343	+	0.984862i	$0.444543\pi$
$450$		0			0
$451$	−0.183732			−0.00865159
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.00000	−	0.280996i	0.0937614	−	0.0131733i
$456$		0			0
$457$			3.67307i			0.171819i	0.996303	+	0.0859095i	$0.0273796\pi$
$457$			3.67307i			0.171819i	−0.996303	+	0.0859095i	$0.972620\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.7748			−1.20045			−0.600226	−	0.799831i	$-0.704923\pi$
$461$	−25.7748			−1.20045			−0.600226	+	0.799831i	$0.704923\pi$
$462$		0			0
$463$			10.6681i			0.495791i	0.968787	+	0.247895i	$0.0797389\pi$
$463$			10.6681i			0.495791i	−0.968787	+	0.247895i	$0.920261\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			13.5190i			0.625584i	0.949822	+	0.312792i	$0.101264\pi$
$467$			13.5190i			0.625584i	−0.949822	+	0.312792i	$0.898736\pi$
$468$		0			0
$469$	1.93978			0.0895706
$470$		0			0
$471$		0			0
$472$		0			0
$473$			0.585174i			0.0269063i
$474$		0			0
$475$	−24.5970	+	7.05086i	−1.12859	+	0.323515i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	19.2192			0.878150			0.439075	−	0.898451i	$-0.355306\pi$
$479$	19.2192			0.878150			0.439075	+	0.898451i	$0.355306\pi$
$480$		0			0
$481$	11.9541			0.545059
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.47013	−	17.5812i	−0.112163	−	0.798321i
$486$		0			0
$487$			4.00445i			0.181459i	0.995876	+	0.0907294i	$0.0289198\pi$
$487$			4.00445i			0.181459i	−0.995876	+	0.0907294i	$0.971080\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	17.5812			0.793429			0.396714	−	0.917942i	$-0.370150\pi$
$491$	17.5812			0.793429			0.396714	+	0.917942i	$0.370150\pi$
$492$		0			0
$493$		−	5.15257i		−	0.232060i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	5.00937i		−	0.224701i
$498$		0			0
$499$	1.37133			0.0613892			0.0306946	−	0.999529i	$-0.490228\pi$
$499$	1.37133			0.0613892			0.0306946	+	0.999529i	$0.490228\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			15.1032i			0.673420i	0.941608	+	0.336710i	$0.109314\pi$
$503$			15.1032i			0.673420i	−0.941608	+	0.336710i	$0.890686\pi$
$504$		0			0
$505$	−11.4795	+	1.61285i	−0.510831	+	0.0717708i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	29.7748			1.31974			0.659872	−	0.751378i	$-0.270610\pi$
$509$	29.7748			1.31974			0.659872	+	0.751378i	$0.270610\pi$
$510$		0			0
$511$	2.44738			0.108266
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.45383	−	17.4652i	−0.108129	−	0.769609i
$516$		0			0
$517$		−	0.593635i		−	0.0261081i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5.79213			−0.253758			−0.126879	−	0.991918i	$-0.540496\pi$
$521$	−5.79213			−0.253758			−0.126879	+	0.991918i	$0.540496\pi$
$522$		0			0
$523$			12.3575i			0.540356i	0.962810	+	0.270178i	$0.0870826\pi$
$523$			12.3575i			0.540356i	−0.962810	+	0.270178i	$0.912917\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	15.1842i		−	0.661434i
$528$		0			0
$529$	5.23951			0.227805
$530$		0			0
$531$		0			0
$532$		0			0
$533$			2.75557i			0.119357i
$534$		0			0
$535$	−5.40297	−	38.4558i	−0.233591	−	1.66259i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.412342			0.0177608
$540$		0			0
$541$	−43.5526			−1.87247			−0.936237	−	0.351370i	$-0.885716\pi$
$541$	−43.5526			−1.87247			−0.936237	+	0.351370i	$0.885716\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.295286	−	0.0414872i	0.0126487	−	0.00177712i
$546$		0			0
$547$		−	1.43017i		−	0.0611497i	−0.999532	−	0.0305748i	$-0.990266\pi$
$547$		−	1.43017i		−	0.0611497i	0.999532	−	0.0305748i	$-0.00973379\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.80642			−0.332565
$552$		0			0
$553$		−	2.87601i		−	0.122301i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	27.0968i		−	1.14813i	−0.818811	−	0.574064i	$-0.805366\pi$
$557$		−	27.0968i		−	1.14813i	0.818811	−	0.574064i	$-0.194634\pi$
$558$		0			0
$559$	8.77631			0.371198
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	26.6113i		−	1.12153i	−0.827974	−	0.560767i	$-0.810506\pi$
$563$		−	26.6113i		−	1.12153i	0.827974	−	0.560767i	$-0.189494\pi$
$564$		0			0
$565$	−1.82516	−	12.9906i	−0.0767850	−	0.546520i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.95407			0.249607			0.124804	−	0.992181i	$-0.460170\pi$
$569$	5.95407			0.249607			0.124804	+	0.992181i	$0.460170\pi$
$570$		0			0
$571$	−13.2859			−0.555998			−0.277999	−	0.960581i	$-0.589671\pi$
$571$	−13.2859			−0.555998			−0.277999	+	0.960581i	$0.589671\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.80642	+	20.2558i	0.242145	+	0.844726i
$576$		0			0
$577$		−	28.4242i		−	1.18331i	−0.806190	−	0.591657i	$-0.798474\pi$
$577$		−	28.4242i		−	1.18331i	0.806190	−	0.591657i	$-0.201526\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8.93624			0.370738
$582$		0			0
$583$			0.238126i			0.00986217i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			8.76986i			0.361971i	0.983486	+	0.180985i	$0.0579287\pi$
$587$			8.76986i			0.361971i	−0.983486	+	0.180985i	$0.942071\pi$
$588$		0			0
$589$	−23.0049			−0.947901
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	30.2449i		−	1.24201i	−0.783807	−	0.621005i	$-0.786725\pi$
$593$		−	30.2449i		−	1.24201i	0.783807	−	0.621005i	$-0.213275\pi$
$594$		0			0
$595$	6.75557	−	0.949145i	0.276951	−	0.0389111i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	17.3274			0.707979			0.353989	−	0.935249i	$-0.384825\pi$
$599$	17.3274			0.707979			0.353989	+	0.935249i	$0.384825\pi$
$600$		0			0
$601$	34.2864			1.39857			0.699286	−	0.714842i	$-0.253502\pi$
$601$	34.2864			1.39857			0.699286	+	0.714842i	$0.253502\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.3477	−	3.42080i	0.989874	−	0.139075i
$606$		0			0
$607$			19.6523i			0.797663i	0.917024	+	0.398832i	$0.130584\pi$
$607$			19.6523i			0.797663i	−0.917024	+	0.398832i	$0.869416\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.90321			−0.360185
$612$		0			0
$613$		−	17.9496i		−	0.724978i	−0.931988	−	0.362489i	$-0.881927\pi$
$613$		−	17.9496i		−	0.724978i	0.931988	−	0.362489i	$-0.118073\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	39.6227i		−	1.59515i	−0.603220	−	0.797575i	$-0.706116\pi$
$617$		−	39.6227i		−	1.59515i	0.603220	−	0.797575i	$-0.293884\pi$
$618$		0			0
$619$	−12.5052			−0.502625			−0.251312	−	0.967906i	$-0.580862\pi$
$619$	−12.5052			−0.502625			−0.251312	+	0.967906i	$0.580862\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			13.0696i			0.523622i
$624$		0			0
$625$	21.2034	−	13.2444i	0.848137	−	0.529777i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	40.3783			1.60999
$630$		0			0
$631$	15.2094			0.605477			0.302738	−	0.953074i	$-0.402099\pi$
$631$	15.2094			0.605477			0.302738	+	0.953074i	$0.402099\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.82225	−	34.3225i	−0.191365	−	1.36205i
$636$		0			0
$637$		−	6.18421i		−	0.245027i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−41.5496			−1.64111			−0.820555	−	0.571568i	$-0.806335\pi$
$641$	−41.5496			−1.64111			−0.820555	+	0.571568i	$0.806335\pi$
$642$		0			0
$643$			18.3225i			0.722568i	0.932456	+	0.361284i	$0.117662\pi$
$643$			18.3225i			0.722568i	−0.932456	+	0.361284i	$0.882338\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	9.31555i		−	0.366232i	−0.983091	−	0.183116i	$-0.941382\pi$
$647$		−	9.31555i		−	0.366232i	0.983091	−	0.183116i	$-0.0586184\pi$
$648$		0			0
$649$	−0.544640			−0.0213790
$650$		0			0
$651$		0			0
$652$		0			0
$653$			37.6128i			1.47190i	0.677033	+	0.735952i	$0.263265\pi$
$653$			37.6128i			1.47190i	−0.677033	+	0.735952i	$0.736735\pi$
$654$		0			0
$655$	−12.8573	+	1.80642i	−0.502375	+	0.0705828i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−46.4800			−1.81060			−0.905301	−	0.424770i	$-0.860355\pi$
$659$	−46.4800			−1.81060			−0.905301	+	0.424770i	$0.860355\pi$
$660$		0			0
$661$	−17.1655			−0.667659			−0.333830	−	0.942633i	$-0.608341\pi$
$661$	−17.1655			−0.667659			−0.333830	+	0.942633i	$0.608341\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.43801	−	10.2351i	−0.0557635	−	0.396899i
$666$		0			0
$667$			6.42864i			0.248918i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.746677			−0.0288252
$672$		0			0
$673$		−	5.06376i		−	0.195194i	−0.995226	−	0.0975968i	$-0.968884\pi$
$673$		−	5.06376i		−	0.195194i	0.995226	−	0.0975968i	$-0.0311156\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	16.6035i		−	0.638124i	−0.947734	−	0.319062i	$-0.896632\pi$
$677$		−	16.6035i		−	0.638124i	0.947734	−	0.319062i	$-0.103368\pi$
$678$		0			0
$679$	7.17130			0.275209
$680$		0			0
$681$		0			0
$682$		0			0
$683$			21.4938i			0.822437i	0.911537	+	0.411218i	$0.134897\pi$
$683$			21.4938i			0.822437i	−0.911537	+	0.411218i	$0.865103\pi$
$684$		0			0
$685$	−5.06376	−	36.0415i	−0.193476	−	1.37707i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.57136			0.136058
$690$		0			0
$691$	48.3245			1.83835			0.919175	−	0.393849i	$-0.128857\pi$
$691$	48.3245			1.83835			0.919175	+	0.393849i	$0.128857\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−15.9081	+	2.23506i	−0.603430	+	0.0847808i
$696$		0			0
$697$			9.30772i			0.352555i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	41.6543			1.57326			0.786631	−	0.617423i	$-0.211823\pi$
$701$	41.6543			1.57326			0.786631	+	0.617423i	$0.211823\pi$
$702$		0			0
$703$		−	61.1753i		−	2.30727i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	4.68244i		−	0.176101i
$708$		0			0
$709$	−42.8988			−1.61110			−0.805548	−	0.592530i	$-0.798129\pi$
$709$	−42.8988			−1.61110			−0.805548	+	0.592530i	$0.798129\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			18.9447i			0.709485i
$714$		0			0
$715$	0.0207436	+	0.147643i	0.000775766	+	0.00552154i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.39700			−0.238568			−0.119284	−	0.992860i	$-0.538060\pi$
$719$	−6.39700			−0.238568			−0.119284	+	0.992860i	$0.538060\pi$
$720$		0			0
$721$	7.12399			0.265311
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.33185	−	2.10171i	0.272298	−	0.0780556i
$726$		0			0
$727$		−	14.0493i		−	0.521061i	−0.965466	−	0.260530i	$-0.916103\pi$
$727$		−	14.0493i		−	0.521061i	0.965466	−	0.260530i	$-0.0838974\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	29.6445			1.09644
$732$		0			0
$733$			18.4701i			0.682210i	0.940025	+	0.341105i	$0.110801\pi$
$733$			18.4701i			0.682210i	−0.940025	+	0.341105i	$0.889199\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			0.143197i			0.00527475i
$738$		0			0
$739$	−27.7911			−1.02231			−0.511156	−	0.859488i	$-0.670782\pi$
$739$	−27.7911			−1.02231			−0.511156	+	0.859488i	$0.670782\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	15.3417i		−	0.562832i	−0.959586	−	0.281416i	$-0.909196\pi$
$743$		−	15.3417i		−	0.562832i	0.959586	−	0.281416i	$-0.0908041\pi$
$744$		0			0
$745$	−29.4193	+	4.13335i	−1.07784	+	0.151434i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	15.6860			0.573153
$750$		0			0
$751$	37.3560			1.36314			0.681570	−	0.731753i	$-0.261298\pi$
$751$	37.3560			1.36314			0.681570	+	0.731753i	$0.261298\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	33.9037	−	4.76341i	1.23388	−	0.173358i
$756$		0			0
$757$			11.5081i			0.418268i	0.977887	+	0.209134i	$0.0670645\pi$
$757$			11.5081i			0.418268i	−0.977887	+	0.209134i	$0.932935\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.23459			−0.0447537			−0.0223769	−	0.999750i	$-0.507123\pi$
$761$	−1.23459			−0.0447537			−0.0223769	+	0.999750i	$0.507123\pi$
$762$		0			0
$763$			0.120446i			0.00436044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			8.16839i			0.294943i
$768$		0			0
$769$	−21.3176			−0.768731			−0.384365	−	0.923181i	$-0.625580\pi$
$769$	−21.3176			−0.768731			−0.384365	+	0.923181i	$0.625580\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			27.8336i			1.00111i	0.865706	+	0.500553i	$0.166870\pi$
$773$			27.8336i			1.00111i	−0.865706	+	0.500553i	$0.833130\pi$
$774$		0			0
$775$	21.6064	−	6.19358i	0.776125	−	0.222480i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14.1017			0.505246
$780$		0			0
$781$	0.369800			0.0132325
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5.74620	+	40.8988i	0.205091	+	1.45974i
$786$		0			0
$787$			43.7832i			1.56070i	0.625340	+	0.780352i	$0.284960\pi$
$787$			43.7832i			1.56070i	−0.625340	+	0.780352i	$0.715040\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.29883			0.188405
$792$		0			0
$793$			11.1985i			0.397670i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	0.314022i		−	0.0111232i	−0.999985	−	0.00556162i	$-0.998230\pi$
$797$		−	0.314022i		−	0.0111232i	0.999985	−	0.00556162i	$-0.00177033\pi$
$798$		0			0
$799$	−30.0731			−1.06391
$800$		0			0
$801$		0			0
$802$		0			0
$803$			0.180669i			0.00637568i
$804$		0			0
$805$	−8.42864	+	1.18421i	−0.297071	+	0.0417379i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−17.4050			−0.611927			−0.305963	−	0.952043i	$-0.598978\pi$
$809$	−17.4050			−0.611927			−0.305963	+	0.952043i	$0.598978\pi$
$810$		0			0
$811$	6.99709			0.245701			0.122850	−	0.992425i	$-0.460796\pi$
$811$	6.99709			0.245701			0.122850	+	0.992425i	$0.460796\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.97280	−	28.2766i	−0.139161	−	0.990484i
$816$		0			0
$817$		−	44.9131i		−	1.57131i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	22.6133			0.789210			0.394605	−	0.918851i	$-0.370881\pi$
$821$	22.6133			0.789210			0.394605	+	0.918851i	$0.370881\pi$
$822$		0			0
$823$		−	19.5506i		−	0.681492i	−0.940155	−	0.340746i	$-0.889320\pi$
$823$		−	19.5506i		−	0.681492i	0.940155	−	0.340746i	$-0.110680\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			20.4746i			0.711971i	0.934492	+	0.355985i	$0.115855\pi$
$827$			20.4746i			0.711971i	−0.934492	+	0.355985i	$0.884145\pi$
$828$		0			0
$829$	39.4019			1.36849			0.684243	−	0.729254i	$-0.260133\pi$
$829$	39.4019			1.36849			0.684243	+	0.729254i	$0.260133\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	20.8889i		−	0.723758i
$834$		0			0
$835$	2.31264	+	16.4603i	0.0800322	+	0.569632i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.69826			−0.0586305			−0.0293152	−	0.999570i	$-0.509333\pi$
$839$	−1.69826			−0.0586305			−0.0293152	+	0.999570i	$0.509333\pi$
$840$		0			0
$841$	−26.6731			−0.919761
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.21432	−	0.311108i	0.0761749	−	0.0107024i
$846$		0			0
$847$			9.93132i			0.341244i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−50.3783			−1.72694
$852$		0			0
$853$			35.4465i			1.21366i	0.794830	+	0.606832i	$0.207560\pi$
$853$			35.4465i			1.21366i	−0.794830	+	0.606832i	$0.792440\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			4.85728i			0.165921i	0.996553	+	0.0829607i	$0.0264376\pi$
$857$			4.85728i			0.165921i	−0.996553	+	0.0829607i	$0.973562\pi$
$858$		0			0
$859$	39.4420			1.34574			0.672872	−	0.739759i	$-0.265060\pi$
$859$	39.4420			1.34574			0.672872	+	0.739759i	$0.265060\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			3.68736i			0.125519i	0.998029	+	0.0627596i	$0.0199901\pi$
$863$			3.68736i			0.125519i	−0.998029	+	0.0627596i	$0.980010\pi$
$864$		0			0
$865$	−6.35551	−	45.2355i	−0.216094	−	1.53805i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.212312			0.00720219
$870$		0			0
$871$	2.14764			0.0727701
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.10616	+	9.22570i	0.138813	+	0.311885i
$876$		0			0
$877$		−	47.4608i		−	1.60264i	−0.598239	−	0.801318i	$-0.704133\pi$
$877$		−	47.4608i		−	1.60264i	0.598239	−	0.801318i	$-0.295867\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−7.49378			−0.252472			−0.126236	−	0.992000i	$-0.540290\pi$
$881$	−7.49378			−0.252472			−0.126236	+	0.992000i	$0.540290\pi$
$882$		0			0
$883$			28.8879i			0.972154i	0.873916	+	0.486077i	$0.161573\pi$
$883$			28.8879i			0.972154i	−0.873916	+	0.486077i	$0.838427\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	59.4499i		−	1.99613i	−0.0621718	−	0.998065i	$-0.519803\pi$
$887$		−	59.4499i		−	1.99613i	0.0621718	−	0.998065i	$-0.480197\pi$
$888$		0			0
$889$	14.0000			0.469545
$890$		0			0
$891$		0			0
$892$		0			0
$893$			45.5625i			1.52469i
$894$		0			0
$895$	−54.4800	+	7.65433i	−1.82106	+	0.255856i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	6.85728			0.228703
$900$		0			0
$901$	12.0633			0.401886
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−30.9906	+	4.35413i	−1.03016	+	0.144736i
$906$		0			0
$907$			8.58274i			0.284985i	0.989796	+	0.142493i	$0.0455117\pi$
$907$			8.58274i			0.284985i	−0.989796	+	0.142493i	$0.954488\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.96836			0.330266			0.165133	−	0.986271i	$-0.447195\pi$
$911$	9.96836			0.330266			0.165133	+	0.986271i	$0.447195\pi$
$912$		0			0
$913$			0.659687i			0.0218325i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	5.24443i		−	0.173186i
$918$		0			0
$919$	22.9590			0.757347			0.378674	−	0.925530i	$-0.376380\pi$
$919$	22.9590			0.757347			0.378674	+	0.925530i	$0.376380\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	5.54617i		−	0.182554i
$924$		0			0
$925$	16.4701	+	57.4563i	0.541534	+	1.88915i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−23.0509			−0.756274			−0.378137	−	0.925750i	$-0.623435\pi$
$929$	−23.0509			−0.756274			−0.378137	+	0.925750i	$0.623435\pi$
$930$		0			0
$931$	−31.6479			−1.03722
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.0700674	+	0.498707i	0.00229145	+	0.0163095i
$936$		0			0
$937$			35.2543i			1.15171i	0.817553	+	0.575853i	$0.195330\pi$
$937$			35.2543i			1.15171i	−0.817553	+	0.575853i	$0.804670\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−10.1334			−0.330338			−0.165169	−	0.986265i	$-0.552817\pi$
$941$	−10.1334			−0.330338			−0.165169	+	0.986265i	$0.552817\pi$
$942$		0			0
$943$		−	11.6128i		−	0.378166i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			32.9447i			1.07056i	0.844675	+	0.535279i	$0.179794\pi$
$947$			32.9447i			1.07056i	−0.844675	+	0.535279i	$0.820206\pi$
$948$		0			0
$949$	2.70964			0.0879585
$950$		0			0
$951$		0			0
$952$		0			0
$953$			22.3684i			0.724584i	0.932065	+	0.362292i	$0.118006\pi$
$953$			22.3684i			0.724584i	−0.932065	+	0.362292i	$0.881994\pi$
$954$		0			0
$955$	−23.8163	+	3.34614i	−0.770676	+	0.108279i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.7012			0.474726
$960$		0			0
$961$	−10.7921			−0.348133
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.79706	+	34.1432i	0.154423	+	1.09911i
$966$		0			0
$967$		−	7.12537i		−	0.229136i	−0.993415	−	0.114568i	$-0.963452\pi$
$967$		−	7.12537i		−	0.229136i	0.993415	−	0.114568i	$-0.0365484\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	10.1521			0.325796			0.162898	−	0.986643i	$-0.447916\pi$
$971$	10.1521			0.325796			0.162898	+	0.986643i	$0.447916\pi$
$972$		0			0
$973$		−	6.48886i		−	0.208023i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	22.4973i		−	0.719753i	−0.933000	−	0.359877i	$-0.882819\pi$
$977$		−	22.4973i		−	0.719753i	0.933000	−	0.359877i	$-0.117181\pi$
$978$		0			0
$979$	−0.964818			−0.0308357
$980$		0			0
$981$		0			0
$982$		0			0
$983$			27.7003i			0.883501i	0.897138	+	0.441751i	$0.145642\pi$
$983$			27.7003i			0.883501i	−0.897138	+	0.441751i	$0.854358\pi$
$984$		0			0
$985$	6.60348	+	47.0005i	0.210404	+	1.49756i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−36.9862			−1.17609
$990$		0			0
$991$	0.898766			0.0285502			0.0142751	−	0.999898i	$-0.495456\pi$
$991$	0.898766			0.0285502			0.0142751	+	0.999898i	$0.495456\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	45.0321	−	6.32693i	1.42761	−	0.200577i
$996$		0			0
$997$			19.5625i			0.619550i	0.950810	+	0.309775i	$0.100254\pi$
$997$			19.5625i			0.619550i	−0.950810	+	0.309775i	$0.899746\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.h.e.469.2		6
3.2	odd	2		260.2.c.a.209.1	✓	6
5.4	even	2	inner	2340.2.h.e.469.1		6
12.11	even	2		1040.2.d.d.209.6		6
15.2	even	4		1300.2.a.j.1.1		3
15.8	even	4		1300.2.a.k.1.3		3
15.14	odd	2		260.2.c.a.209.6	yes	6
39.5	even	4		3380.2.d.a.1689.1		6
39.8	even	4		3380.2.d.b.1689.1		6
39.38	odd	2		3380.2.c.c.2029.1		6
60.23	odd	4		5200.2.a.cd.1.1		3
60.47	odd	4		5200.2.a.cg.1.3		3
60.59	even	2		1040.2.d.d.209.1		6
195.44	even	4		3380.2.d.b.1689.6		6
195.164	even	4		3380.2.d.a.1689.6		6
195.194	odd	2		3380.2.c.c.2029.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.c.a.209.1	✓	6	3.2	odd	2
260.2.c.a.209.6	yes	6	15.14	odd	2
1040.2.d.d.209.1		6	60.59	even	2
1040.2.d.d.209.6		6	12.11	even	2
1300.2.a.j.1.1		3	15.2	even	4
1300.2.a.k.1.3		3	15.8	even	4
2340.2.h.e.469.1		6	5.4	even	2	inner
2340.2.h.e.469.2		6	1.1	even	1	trivial
3380.2.c.c.2029.1		6	39.38	odd	2
3380.2.c.c.2029.6		6	195.194	odd	2
3380.2.d.a.1689.1		6	39.5	even	4
3380.2.d.a.1689.6		6	195.164	even	4
3380.2.d.b.1689.1		6	39.8	even	4
3380.2.d.b.1689.6		6	195.44	even	4
5200.2.a.cd.1.1		3	60.23	odd	4
5200.2.a.cg.1.3		3	60.47	odd	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:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.21432 + 0.311108i) q^{5} -0.903212i q^{7} +0.0666765 q^{11} -1.00000i q^{13} -3.37778i q^{17} -5.11753 q^{19} +4.21432i q^{23} +(4.80642 - 1.37778i) q^{25} +1.52543 q^{29} +4.49532 q^{31} +(0.280996 + 2.00000i) q^{35} +11.9541i q^{37} -2.75557 q^{41} +8.77631i q^{43} -8.90321i q^{47} +6.18421 q^{49} +3.57136i q^{53} +(-0.147643 + 0.0207436i) q^{55} -8.16839 q^{59} -11.1985 q^{61} +(0.311108 + 2.21432i) q^{65} +2.14764i q^{67} +5.54617 q^{71} +2.70964i q^{73} -0.0602231i q^{77} +3.18421 q^{79} +9.89384i q^{83} +(1.05086 + 7.47949i) q^{85} -14.4701 q^{89} -0.903212 q^{91} +(11.3319 - 1.59210i) q^{95} +7.93978i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{19} + 2 q^{25} - 4 q^{29} - 12 q^{35} - 16 q^{41} + 10 q^{49} + 12 q^{55} + 4 q^{59} - 28 q^{61} + 2 q^{65} - 20 q^{71} - 8 q^{79} - 20 q^{85} + 20 q^{89} + 8 q^{91} + 28 q^{95}+O(q^{100}) \) 6 * q - 4 * q^19 + 2 * q^25 - 4 * q^29 - 12 * q^35 - 16 * q^41 + 10 * q^49 + 12 * q^55 + 4 * q^59 - 28 * q^61 + 2 * q^65 - 20 * q^71 - 8 * q^79 - 20 * q^85 + 20 * q^89 + 8 * q^91 + 28 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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