LMFDB - Embedded newform 260.2.c.a.209.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [260,2,Mod(209,260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(260, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("260.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 260 = 2^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	260.c (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:	$2.07611045255$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.6
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	260.209
Dual form			260.2.c.a.209.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.21432i q^{3} +(2.21432 + 0.311108i) q^{5} +0.903212i q^{7} -1.90321 q^{9} -0.0666765 q^{11} +1.00000i q^{13} +(-0.688892 + 4.90321i) q^{15} -3.37778i q^{17} -5.11753 q^{19} -2.00000 q^{21} +4.21432i q^{23} +(4.80642 + 1.37778i) q^{25} +2.42864i q^{27} -1.52543 q^{29} +4.49532 q^{31} -0.147643i q^{33} +(-0.280996 + 2.00000i) q^{35} -11.9541i q^{37} -2.21432 q^{39} +2.75557 q^{41} -8.77631i q^{43} +(-4.21432 - 0.592104i) q^{45} -8.90321i q^{47} +6.18421 q^{49} +7.47949 q^{51} +3.57136i q^{53} +(-0.147643 - 0.0207436i) q^{55} -11.3319i q^{57} +8.16839 q^{59} -11.1985 q^{61} -1.71900i q^{63} +(-0.311108 + 2.21432i) q^{65} -2.14764i q^{67} -9.33185 q^{69} -5.54617 q^{71} -2.70964i q^{73} +(-3.05086 + 10.6430i) q^{75} -0.0602231i q^{77} +3.18421 q^{79} -11.0874 q^{81} +9.89384i q^{83} +(1.05086 - 7.47949i) q^{85} -3.37778i q^{87} +14.4701 q^{89} -0.903212 q^{91} +9.95407i q^{93} +(-11.3319 - 1.59210i) q^{95} -7.93978i q^{97} +0.126900 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{9} - 4 q^{15} - 4 q^{19} - 12 q^{21} + 2 q^{25} + 4 q^{29} + 12 q^{35} + 16 q^{41} - 12 q^{45} + 10 q^{49} - 8 q^{51} + 12 q^{55} - 4 q^{59} - 28 q^{61} - 2 q^{65} - 16 q^{69} + 20 q^{71} + 8 q^{75}+ \cdots + 28 q^{99}+O(q^{100}) $ 6 * q + 2 * q^9 - 4 * q^15 - 4 * q^19 - 12 * q^21 + 2 * q^25 + 4 * q^29 + 12 * q^35 + 16 * q^41 - 12 * q^45 + 10 * q^49 - 8 * q^51 + 12 * q^55 - 4 * q^59 - 28 * q^61 - 2 * q^65 - 16 * q^69 + 20 * q^71 + 8 * q^75 - 8 * q^79 - 26 * q^81 - 20 * q^85 - 20 * q^89 + 8 * q^91 - 28 * q^95 + 28 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/260\mathbb{Z}\right)^\times$.

$n$	$41$	$131$	$157$
$\chi(n)$	$1$	$1$	$-1$

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
$3$			2.21432i			1.27844i	0.769025	+	0.639219i	$0.220742\pi$
$3$			2.21432i			1.27844i	−0.769025	+	0.639219i	$0.779258\pi$
$4$		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	$0.0546000\pi$
$7$			0.903212i			0.341382i	−0.985325	+	0.170691i	$0.945400\pi$
$8$		0			0
$9$	−1.90321			−0.634404
$10$		0			0
$11$	−0.0666765			−0.0201037			−0.0100519	−	0.999949i	$-0.503200\pi$
$11$	−0.0666765			−0.0201037			−0.0100519	+	0.999949i	$0.503200\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$		0			0
$15$	−0.688892	+	4.90321i	−0.177871	+	1.26600i
$16$		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
$21$	−2.00000			−0.436436
$22$		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
$27$			2.42864i			0.467392i
$28$		0			0
$29$	−1.52543			−0.283265			−0.141632	−	0.989919i	$-0.545235\pi$
$29$	−1.52543			−0.283265			−0.141632	+	0.989919i	$0.545235\pi$
$30$		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
$33$		−	0.147643i		−	0.0257014i
$34$		0			0
$35$	−0.280996	+	2.00000i	−0.0474970	+	0.338062i
$36$		0			0
$37$		−	11.9541i		−	1.96524i	−0.185638	−	0.982618i	$-0.559435\pi$
$37$		−	11.9541i		−	1.96524i	0.185638	−	0.982618i	$-0.440565\pi$
$38$		0			0
$39$	−2.21432			−0.354575
$40$		0			0
$41$	2.75557			0.430348			0.215174	−	0.976576i	$-0.430968\pi$
$41$	2.75557			0.430348			0.215174	+	0.976576i	$0.430968\pi$
$42$		0			0
$43$		−	8.77631i		−	1.33838i	−0.743094	−	0.669188i	$-0.766642\pi$
$43$		−	8.77631i		−	1.33838i	0.743094	−	0.669188i	$-0.233358\pi$
$44$		0			0
$45$	−4.21432	−	0.592104i	−0.628234	−	0.0882657i
$46$		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
$51$	7.47949			1.04734
$52$		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
$57$		−	11.3319i		−	1.50094i
$58$		0			0
$59$	8.16839			1.06343			0.531717	−	0.846922i	$-0.321547\pi$
$59$	8.16839			1.06343			0.531717	+	0.846922i	$0.321547\pi$
$60$		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
$63$		−	1.71900i		−	0.216574i
$64$		0			0
$65$	−0.311108	+	2.21432i	−0.0385882	+	0.274653i
$66$		0			0
$67$		−	2.14764i		−	0.262376i	−0.991357	−	0.131188i	$-0.958121\pi$
$67$		−	2.14764i		−	0.262376i	0.991357	−	0.131188i	$-0.0418792\pi$
$68$		0			0
$69$	−9.33185			−1.12342
$70$		0			0
$71$	−5.54617			−0.658209			−0.329105	−	0.944293i	$-0.606747\pi$
$71$	−5.54617			−0.658209			−0.329105	+	0.944293i	$0.606747\pi$
$72$		0			0
$73$		−	2.70964i		−	0.317139i	−0.987348	−	0.158569i	$-0.949312\pi$
$73$		−	2.70964i		−	0.317139i	0.987348	−	0.158569i	$-0.0506882\pi$
$74$		0			0
$75$	−3.05086	+	10.6430i	−0.352282	+	1.22894i
$76$		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
$81$	−11.0874			−1.23194
$82$		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
$87$		−	3.37778i		−	0.362136i
$88$		0			0
$89$	14.4701			1.53383			0.766915	−	0.641748i	$-0.221791\pi$
$89$	14.4701			1.53383			0.766915	+	0.641748i	$0.221791\pi$
$90$		0			0
$91$	−0.903212			−0.0946823
$92$		0			0
$93$			9.95407i			1.03219i
$94$		0			0
$95$	−11.3319	−	1.59210i	−1.16262	−	0.163346i
$96$		0			0
$97$		−	7.93978i		−	0.806162i	−0.915164	−	0.403081i	$-0.867939\pi$
$97$		−	7.93978i		−	0.806162i	0.915164	−	0.403081i	$-0.132061\pi$
$98$		0			0
$99$	0.126900			0.0127539
$100$		0			0
$101$	−5.18421			−0.515848			−0.257924	−	0.966165i	$-0.583038\pi$
$101$	−5.18421			−0.515848			−0.257924	+	0.966165i	$0.583038\pi$
$102$		0			0
$103$		−	7.88739i		−	0.777168i	−0.921413	−	0.388584i	$-0.872964\pi$
$103$		−	7.88739i		−	0.777168i	0.921413	−	0.388584i	$-0.127036\pi$
$104$		0			0
$105$	−4.42864	−	0.622216i	−0.432191	−	0.0607220i
$106$		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
$111$	26.4701			2.51243
$112$		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
$117$		−	1.90321i		−	0.175952i
$118$		0			0
$119$	3.05086			0.279671
$120$		0			0
$121$	−10.9956			−0.999596
$122$		0			0
$123$			6.10171i			0.550173i
$124$		0			0
$125$	10.2143	+	4.54617i	0.913597	+	0.406622i
$126$		0			0
$127$		−	15.5002i		−	1.37542i	−0.725984	−	0.687712i	$-0.758615\pi$
$127$		−	15.5002i		−	1.37542i	0.725984	−	0.687712i	$-0.241385\pi$
$128$		0			0
$129$	19.4336			1.71103
$130$		0			0
$131$	−5.80642			−0.507310			−0.253655	−	0.967295i	$-0.581633\pi$
$131$	−5.80642			−0.507310			−0.253655	+	0.967295i	$0.581633\pi$
$132$		0			0
$133$		−	4.62222i		−	0.400797i
$134$		0			0
$135$	−0.755569	+	5.37778i	−0.0650290	+	0.462846i
$136$		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
$141$	19.7146			1.66027
$142$		0			0
$143$		−	0.0666765i		−	0.00557577i
$144$		0			0
$145$	−3.37778	−	0.474572i	−0.280510	−	0.0394111i
$146$		0			0
$147$			13.6938i			1.12945i
$148$		0			0
$149$	−13.2859			−1.08842			−0.544212	−	0.838947i	$-0.683171\pi$
$149$	−13.2859			−1.08842			−0.544212	+	0.838947i	$0.683171\pi$
$150$		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
$153$			6.42864i			0.519725i
$154$		0			0
$155$	9.95407	+	1.39853i	0.799530	+	0.112332i
$156$		0			0
$157$			18.4701i			1.47408i	0.675851	+	0.737038i	$0.263776\pi$
$157$			18.4701i			1.47408i	−0.675851	+	0.737038i	$0.736224\pi$
$158$		0			0
$159$	−7.90813			−0.627156
$160$		0			0
$161$	−3.80642			−0.299988
$162$		0			0
$163$		−	12.7699i		−	1.00021i	−0.865964	−	0.500106i	$-0.833294\pi$
$163$		−	12.7699i		−	1.00021i	0.865964	−	0.500106i	$-0.166706\pi$
$164$		0			0
$165$	0.0459330	−	0.326929i	0.00357588	−	0.0254514i
$166$		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
$171$	9.73975			0.744817
$172$		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
$177$			18.0874i			1.35953i
$178$		0			0
$179$	−24.6035			−1.83895			−0.919475	−	0.393148i	$-0.871386\pi$
$179$	−24.6035			−1.83895			−0.919475	+	0.393148i	$0.871386\pi$
$180$		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
$183$		−	24.7971i		−	1.83305i
$184$		0			0
$185$	3.71900	−	26.4701i	0.273427	−	1.94612i
$186$		0			0
$187$			0.225219i			0.0164696i
$188$		0			0
$189$	−2.19358			−0.159559
$190$		0			0
$191$	−10.7556			−0.778246			−0.389123	−	0.921186i	$-0.627222\pi$
$191$	−10.7556			−0.778246			−0.389123	+	0.921186i	$0.627222\pi$
$192$		0			0
$193$			15.4193i			1.10990i	0.831883	+	0.554952i	$0.187263\pi$
$193$			15.4193i			1.10990i	−0.831883	+	0.554952i	$0.812737\pi$
$194$		0			0
$195$	−4.90321	−	0.688892i	−0.351126	−	0.0493326i
$196$		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
$201$	4.75557			0.335432
$202$		0			0
$203$		−	1.37778i		−	0.0967015i
$204$		0			0
$205$	6.10171	+	0.857279i	0.426162	+	0.0598750i
$206$		0			0
$207$		−	8.02074i		−	0.557480i
$208$		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
$213$		−	12.2810i		−	0.841480i
$214$		0			0
$215$	2.73038	−	19.4336i	0.186210	−	1.32536i
$216$		0			0
$217$			4.06022i			0.275626i
$218$		0			0
$219$	6.00000			0.405442
$220$		0			0
$221$	3.37778			0.227214
$222$		0			0
$223$		−	20.2494i		−	1.35600i	−0.735063	−	0.677999i	$-0.762848\pi$
$223$		−	20.2494i		−	1.35600i	0.735063	−	0.677999i	$-0.237152\pi$
$224$		0			0
$225$	−9.14764	−	2.62222i	−0.609843	−	0.174814i
$226$		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
$231$	0.133353			0.00877399
$232$		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
$237$			7.05086i			0.458002i
$238$		0			0
$239$	14.8637			0.961455			0.480727	−	0.876870i	$-0.340373\pi$
$239$	14.8637			0.961455			0.480727	+	0.876870i	$0.340373\pi$
$240$		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
$243$		−	17.2652i		−	1.10756i
$244$		0			0
$245$	13.6938	+	1.92396i	0.874866	+	0.122917i
$246$		0			0
$247$		−	5.11753i		−	0.325621i
$248$		0			0
$249$	−21.9081			−1.38837
$250$		0			0
$251$	20.5620			1.29786			0.648931	−	0.760847i	$-0.275216\pi$
$251$	20.5620			1.29786			0.648931	+	0.760847i	$0.275216\pi$
$252$		0			0
$253$		−	0.280996i		−	0.0176661i
$254$		0			0
$255$	16.5620	+	2.32693i	1.03715	+	0.145718i
$256$		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
$261$	2.90321			0.179704
$262$		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
$267$			32.0415i			1.96091i
$268$		0			0
$269$	−10.8988			−0.664510			−0.332255	−	0.943190i	$-0.607809\pi$
$269$	−10.8988			−0.664510			−0.332255	+	0.943190i	$0.607809\pi$
$270$		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
$273$		−	2.00000i		−	0.121046i
$274$		0			0
$275$	−0.320476	−	0.0918659i	−0.0193254	−	0.00553972i
$276$		0			0
$277$			28.7239i			1.72585i	0.505329	+	0.862927i	$0.331371\pi$
$277$			28.7239i			1.72585i	−0.505329	+	0.862927i	$0.668629\pi$
$278$		0			0
$279$	−8.55554			−0.512207
$280$		0			0
$281$	−14.1936			−0.846718			−0.423359	−	0.905962i	$-0.639149\pi$
$281$	−14.1936			−0.846718			−0.423359	+	0.905962i	$0.639149\pi$
$282$		0			0
$283$		−	15.9190i		−	0.946288i	−0.880985	−	0.473144i	$-0.843119\pi$
$283$		−	15.9190i		−	0.946288i	0.880985	−	0.473144i	$-0.156881\pi$
$284$		0			0
$285$	3.52543	−	25.0923i	0.208828	−	1.48634i
$286$		0			0
$287$			2.48886i			0.146913i
$288$		0			0
$289$	5.59057			0.328857
$290$		0			0
$291$	17.5812			1.03063
$292$		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
$297$		−	0.161933i		−	0.00939632i
$298$		0			0
$299$	−4.21432			−0.243720
$300$		0			0
$301$	7.92687			0.456897
$302$		0			0
$303$		−	11.4795i		−	0.659480i
$304$		0			0
$305$	−24.7971	−	3.48394i	−1.41988	−	0.199490i
$306$		0			0
$307$		−	0.0874201i		−	0.00498933i	−0.999997	−	0.00249467i	$-0.999206\pi$
$307$		−	0.0874201i		−	0.00498933i	0.999997	−	0.00249467i	$-0.000794078\pi$
$308$		0			0
$309$	17.4652			0.993561
$310$		0			0
$311$	−25.7146			−1.45814			−0.729069	−	0.684440i	$-0.760047\pi$
$311$	−25.7146			−1.45814			−0.729069	+	0.684440i	$0.760047\pi$
$312$		0			0
$313$		−	7.67307i		−	0.433708i	−0.976204	−	0.216854i	$-0.930421\pi$
$313$		−	7.67307i		−	0.433708i	0.976204	−	0.216854i	$-0.0695795\pi$
$314$		0			0
$315$	0.534795	−	3.80642i	0.0301323	−	0.214468i
$316$		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
$321$	−38.4558			−2.14640
$322$		0			0
$323$			17.2859i			0.961814i
$324$		0			0
$325$	−1.37778	+	4.80642i	−0.0764257	+	0.266612i
$326$		0			0
$327$		−	0.295286i		−	0.0163294i
$328$		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
$333$			22.7511i			1.24675i
$334$		0			0
$335$	0.668149	−	4.75557i	0.0365049	−	0.259824i
$336$		0			0
$337$			20.7971i			1.13289i	0.824100	+	0.566444i	$0.191681\pi$
$337$			20.7971i			1.13289i	−0.824100	+	0.566444i	$0.808319\pi$
$338$		0			0
$339$	−12.9906			−0.705554
$340$		0			0
$341$	−0.299732			−0.0162314
$342$		0			0
$343$			11.9081i			0.642979i
$344$		0			0
$345$	−20.6637	−	2.90321i	−1.11250	−	0.156304i
$346$		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
$351$	−2.42864			−0.129631
$352$		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
$357$			6.75557i			0.357543i
$358$		0			0
$359$	34.6987			1.83133			0.915665	−	0.401943i	$-0.131665\pi$
$359$	34.6987			1.83133			0.915665	+	0.401943i	$0.131665\pi$
$360$		0			0
$361$	7.18913			0.378375
$362$		0			0
$363$		−	24.3477i		−	1.27792i
$364$		0			0
$365$	0.842989	−	6.00000i	0.0441241	−	0.314054i
$366$		0			0
$367$			27.1131i			1.41529i	0.706567	+	0.707646i	$0.250243\pi$
$367$			27.1131i			1.41529i	−0.706567	+	0.707646i	$0.749757\pi$
$368$		0			0
$369$	−5.24443			−0.273014
$370$		0			0
$371$	−3.22570			−0.167470
$372$		0			0
$373$			19.7146i			1.02078i	0.859943	+	0.510391i	$0.170499\pi$
$373$			19.7146i			1.02078i	−0.859943	+	0.510391i	$0.829501\pi$
$374$		0			0
$375$	−10.0667	+	22.6178i	−0.519841	+	1.16798i
$376$		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
$381$	34.3225			1.75839
$382$		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
$387$			16.7032i			0.849070i
$388$		0			0
$389$	8.48886			0.430402			0.215201	−	0.976570i	$-0.430959\pi$
$389$	8.48886			0.430402			0.215201	+	0.976570i	$0.430959\pi$
$390$		0			0
$391$	14.2351			0.719898
$392$		0			0
$393$		−	12.8573i		−	0.648564i
$394$		0			0
$395$	7.05086	+	0.990632i	0.354767	+	0.0498441i
$396$		0			0
$397$			9.10663i			0.457049i	0.973538	+	0.228524i	$0.0733901\pi$
$397$			9.10663i			0.457049i	−0.973538	+	0.228524i	$0.926610\pi$
$398$		0			0
$399$	10.2351			0.512394
$400$		0			0
$401$	18.1748			0.907608			0.453804	−	0.891101i	$-0.350067\pi$
$401$	18.1748			0.907608			0.453804	+	0.891101i	$0.350067\pi$
$402$		0			0
$403$			4.49532i			0.223928i
$404$		0			0
$405$	−24.5511	−	3.44938i	−1.21995	−	0.171401i
$406$		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
$411$	−36.0415			−1.77780
$412$		0			0
$413$			7.37778i			0.363037i
$414$		0			0
$415$	−3.07805	+	21.9081i	−0.151096	+	1.07543i
$416$		0			0
$417$			15.9081i			0.779025i
$418$		0			0
$419$	15.7047			0.767225			0.383613	−	0.923494i	$-0.374680\pi$
$419$	15.7047			0.767225			0.383613	+	0.923494i	$0.374680\pi$
$420$		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
$423$			16.9447i			0.823879i
$424$		0			0
$425$	4.65386	−	16.2351i	0.225745	−	0.787516i
$426$		0			0
$427$		−	10.1146i		−	0.489481i
$428$		0			0
$429$	0.147643			0.00712828
$430$		0			0
$431$	18.9842			0.914436			0.457218	−	0.889355i	$-0.348846\pi$
$431$	18.9842			0.914436			0.457218	+	0.889355i	$0.348846\pi$
$432$		0			0
$433$		−	1.04101i		−	0.0500278i	−0.999687	−	0.0250139i	$-0.992037\pi$
$433$		−	1.04101i		−	0.0500278i	0.999687	−	0.0250139i	$-0.00796300\pi$
$434$		0			0
$435$	1.05086	−	7.47949i	0.0503846	−	0.358614i
$436$		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
$441$	−11.7699			−0.560469
$442$		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
$447$		−	29.4193i		−	1.39148i
$448$		0			0
$449$	−7.34614			−0.346686			−0.173343	−	0.984862i	$-0.555457\pi$
$449$	−7.34614			−0.346686			−0.173343	+	0.984862i	$0.555457\pi$
$450$		0			0
$451$	−0.183732			−0.00865159
$452$		0			0
$453$		−	33.9037i		−	1.59293i
$454$		0			0
$455$	−2.00000	−	0.280996i	−0.0937614	−	0.0131733i
$456$		0			0
$457$		−	3.67307i		−	0.171819i	−0.996303	−	0.0859095i	$-0.972620\pi$
$457$		−	3.67307i		−	0.171819i	0.996303	−	0.0859095i	$-0.0273796\pi$
$458$		0			0
$459$	8.20342			0.382903
$460$		0			0
$461$	25.7748			1.20045			0.600226	−	0.799831i	$-0.295077\pi$
$461$	25.7748			1.20045			0.600226	+	0.799831i	$0.295077\pi$
$462$		0			0
$463$		−	10.6681i		−	0.495791i	−0.968787	−	0.247895i	$-0.920261\pi$
$463$		−	10.6681i		−	0.495791i	0.968787	−	0.247895i	$-0.0797389\pi$
$464$		0			0
$465$	−3.09679	+	22.0415i	−0.143610	+	1.02215i
$466$		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
$471$	−40.8988			−1.88452
$472$		0			0
$473$			0.585174i			0.0269063i
$474$		0			0
$475$	−24.5970	−	7.05086i	−1.12859	−	0.323515i
$476$		0			0
$477$		−	6.79706i		−	0.311216i
$478$		0			0
$479$	−19.2192			−0.878150			−0.439075	−	0.898451i	$-0.644694\pi$
$479$	−19.2192			−0.878150			−0.439075	+	0.898451i	$0.644694\pi$
$480$		0			0
$481$	11.9541			0.545059
$482$		0			0
$483$		−	8.42864i		−	0.383516i
$484$		0			0
$485$	2.47013	−	17.5812i	0.112163	−	0.798321i
$486$		0			0
$487$		−	4.00445i		−	0.181459i	−0.995876	−	0.0907294i	$-0.971080\pi$
$487$		−	4.00445i		−	0.181459i	0.995876	−	0.0907294i	$-0.0289198\pi$
$488$		0			0
$489$	28.2766			1.27871
$490$		0			0
$491$	−17.5812			−0.793429			−0.396714	−	0.917942i	$-0.629850\pi$
$491$	−17.5812			−0.793429			−0.396714	+	0.917942i	$0.629850\pi$
$492$		0			0
$493$			5.15257i			0.232060i
$494$		0			0
$495$	0.280996	+	0.0394795i	0.0126298	+	0.00177447i
$496$		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
$501$	16.4603			0.735391
$502$		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
$507$		−	2.21432i		−	0.0983414i
$508$		0			0
$509$	−29.7748			−1.31974			−0.659872	−	0.751378i	$-0.729390\pi$
$509$	−29.7748			−1.31974			−0.659872	+	0.751378i	$0.729390\pi$
$510$		0			0
$511$	2.44738			0.108266
$512$		0			0
$513$		−	12.4286i		−	0.548738i
$514$		0			0
$515$	2.45383	−	17.4652i	0.108129	−	0.769609i
$516$		0			0
$517$			0.593635i			0.0261081i
$518$		0			0
$519$	−45.2355			−1.98562
$520$		0			0
$521$	5.79213			0.253758			0.126879	−	0.991918i	$-0.459504\pi$
$521$	5.79213			0.253758			0.126879	+	0.991918i	$0.459504\pi$
$522$		0			0
$523$		−	12.3575i		−	0.540356i	−0.962810	−	0.270178i	$-0.912917\pi$
$523$		−	12.3575i		−	0.540356i	0.962810	−	0.270178i	$-0.0870826\pi$
$524$		0			0
$525$	−9.61285	−	2.75557i	−0.419539	−	0.120263i
$526$		0			0
$527$		−	15.1842i		−	0.661434i
$528$		0			0
$529$	5.23951			0.227805
$530$		0			0
$531$	−15.5462			−0.674646
$532$		0			0
$533$			2.75557i			0.119357i
$534$		0			0
$535$	−5.40297	+	38.4558i	−0.233591	+	1.66259i
$536$		0			0
$537$		−	54.4800i		−	2.35098i
$538$		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
$543$			30.9906i			1.32994i
$544$		0			0
$545$	−0.295286	−	0.0414872i	−0.0126487	−	0.00177712i
$546$		0			0
$547$			1.43017i			0.0611497i	0.999532	+	0.0305748i	$0.00973379\pi$
$547$			1.43017i			0.0611497i	−0.999532	+	0.0305748i	$0.990266\pi$
$548$		0			0
$549$	21.3131			0.909622
$550$		0			0
$551$	7.80642			0.332565
$552$		0			0
$553$			2.87601i			0.122301i
$554$		0			0
$555$	58.6133	+	8.23506i	2.48800	+	0.349559i
$556$		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
$561$	−0.498707			−0.0210554
$562$		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
$567$		−	10.0143i		−	0.420561i
$568$		0			0
$569$	−5.95407			−0.249607			−0.124804	−	0.992181i	$-0.539830\pi$
$569$	−5.95407			−0.249607			−0.124804	+	0.992181i	$0.539830\pi$
$570$		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
$573$		−	23.8163i		−	0.994939i
$574$		0			0
$575$	−5.80642	+	20.2558i	−0.242145	+	0.844726i
$576$		0			0
$577$			28.4242i			1.18331i	0.806190	+	0.591657i	$0.201526\pi$
$577$			28.4242i			1.18331i	−0.806190	+	0.591657i	$0.798474\pi$
$578$		0			0
$579$	−34.1432			−1.41894
$580$		0			0
$581$	−8.93624			−0.370738
$582$		0			0
$583$		−	0.238126i		−	0.00986217i
$584$		0			0
$585$	0.592104	−	4.21432i	0.0244805	−	0.174241i
$586$		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
$591$	47.0005			1.93334
$592$		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
$597$		−	45.0321i		−	1.84304i
$598$		0			0
$599$	−17.3274			−0.707979			−0.353989	−	0.935249i	$-0.615175\pi$
$599$	−17.3274			−0.707979			−0.353989	+	0.935249i	$0.615175\pi$
$600$		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
$603$			4.08742i			0.166453i
$604$		0			0
$605$	−24.3477	−	3.42080i	−0.989874	−	0.139075i
$606$		0			0
$607$		−	19.6523i		−	0.797663i	−0.917024	−	0.398832i	$-0.869416\pi$
$607$		−	19.6523i		−	0.797663i	0.917024	−	0.398832i	$-0.130584\pi$
$608$		0			0
$609$	3.05086			0.123627
$610$		0			0
$611$	8.90321			0.360185
$612$		0			0
$613$			17.9496i			0.724978i	0.931988	+	0.362489i	$0.118073\pi$
$613$			17.9496i			0.724978i	−0.931988	+	0.362489i	$0.881927\pi$
$614$		0			0
$615$	−1.89829	+	13.5111i	−0.0765464	+	0.544822i
$616$		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
$621$	−10.2351			−0.410719
$622$		0			0
$623$			13.0696i			0.523622i
$624$		0			0
$625$	21.2034	+	13.2444i	0.848137	+	0.529777i
$626$		0			0
$627$			0.755569i			0.0301745i
$628$		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
$633$		−	45.8894i		−	1.82394i
$634$		0			0
$635$	4.82225	−	34.3225i	0.191365	−	1.36205i
$636$		0			0
$637$			6.18421i			0.245027i
$638$		0			0
$639$	10.5555			0.417571
$640$		0			0
$641$	41.5496			1.64111			0.820555	−	0.571568i	$-0.193665\pi$
$641$	41.5496			1.64111			0.820555	+	0.571568i	$0.193665\pi$
$642$		0			0
$643$		−	18.3225i		−	0.722568i	−0.932456	−	0.361284i	$-0.882338\pi$
$643$		−	18.3225i		−	0.722568i	0.932456	−	0.361284i	$-0.117662\pi$
$644$		0			0
$645$	43.0321	+	6.04593i	1.69439	+	0.238058i
$646$		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
$651$	−8.99063			−0.352371
$652$		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
$657$			5.15701i			0.201194i
$658$		0			0
$659$	46.4800			1.81060			0.905301	−	0.424770i	$-0.139645\pi$
$659$	46.4800			1.81060			0.905301	+	0.424770i	$0.139645\pi$
$660$		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
$663$			7.47949i			0.290480i
$664$		0			0
$665$	1.43801	−	10.2351i	0.0557635	−	0.396899i
$666$		0			0
$667$		−	6.42864i		−	0.248918i
$668$		0			0
$669$	44.8385			1.73356
$670$		0			0
$671$	0.746677			0.0288252
$672$		0			0
$673$			5.06376i			0.195194i	0.995226	+	0.0975968i	$0.0311156\pi$
$673$			5.06376i			0.195194i	−0.995226	+	0.0975968i	$0.968884\pi$
$674$		0			0
$675$	−3.34614	+	11.6731i	−0.128793	+	0.449297i
$676$		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
$681$	−37.4291			−1.43429
$682$		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
$687$		−	8.78721i		−	0.335253i
$688$		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
$693$			0.114617i			0.00435395i
$694$		0			0
$695$	15.9081	+	2.23506i	0.603430	+	0.0847808i
$696$		0			0
$697$		−	9.30772i		−	0.352555i
$698$		0			0
$699$	−33.9180			−1.28290
$700$		0			0
$701$	−41.6543			−1.57326			−0.786631	−	0.617423i	$-0.788177\pi$
$701$	−41.6543			−1.57326			−0.786631	+	0.617423i	$0.788177\pi$
$702$		0			0
$703$			61.1753i			2.30727i
$704$		0			0
$705$	43.6543	+	6.13335i	1.64412	+	0.230995i
$706$		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
$711$	−6.06022			−0.227276
$712$		0			0
$713$			18.9447i			0.709485i
$714$		0			0
$715$	0.0207436	−	0.147643i	0.000775766	−	0.00552154i
$716$		0			0
$717$			32.9131i			1.22916i
$718$		0			0
$719$	6.39700			0.238568			0.119284	−	0.992860i	$-0.461940\pi$
$719$	6.39700			0.238568			0.119284	+	0.992860i	$0.461940\pi$
$720$		0			0
$721$	7.12399			0.265311
$722$		0			0
$723$			44.9403i			1.67135i
$724$		0			0
$725$	−7.33185	−	2.10171i	−0.272298	−	0.0780556i
$726$		0			0
$727$			14.0493i			0.521061i	0.965466	+	0.260530i	$0.0838974\pi$
$727$			14.0493i			0.521061i	−0.965466	+	0.260530i	$0.916103\pi$
$728$		0			0
$729$	4.96836			0.184013
$730$		0			0
$731$	−29.6445			−1.09644
$732$		0			0
$733$		−	18.4701i		−	0.682210i	−0.940025	−	0.341105i	$-0.889199\pi$
$733$		−	18.4701i		−	0.682210i	0.940025	−	0.341105i	$-0.110801\pi$
$734$		0			0
$735$	−4.26025	+	30.3225i	−0.157142	+	1.11846i
$736$		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
$741$	11.3319			0.416286
$742$		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
$747$		−	18.8301i		−	0.688957i
$748$		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
$753$			45.5308i			1.65924i
$754$		0			0
$755$	−33.9037	−	4.76341i	−1.23388	−	0.173358i
$756$		0			0
$757$		−	11.5081i		−	0.418268i	−0.977887	−	0.209134i	$-0.932935\pi$
$757$		−	11.5081i		−	0.418268i	0.977887	−	0.209134i	$-0.0670645\pi$
$758$		0			0
$759$	0.622216			0.0225850
$760$		0			0
$761$	1.23459			0.0447537			0.0223769	−	0.999750i	$-0.492877\pi$
$761$	1.23459			0.0447537			0.0223769	+	0.999750i	$0.492877\pi$
$762$		0			0
$763$		−	0.120446i		−	0.00436044i
$764$		0			0
$765$	−2.00000	+	14.2351i	−0.0723102	+	0.514670i
$766$		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
$771$	60.9403			2.19471
$772$		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
$777$			23.9081i			0.857700i
$778$		0			0
$779$	−14.1017			−0.505246
$780$		0			0
$781$	0.369800			0.0132325
$782$		0			0
$783$		−	3.70471i		−	0.132396i
$784$		0			0
$785$	−5.74620	+	40.8988i	−0.205091	+	1.45974i
$786$		0			0
$787$		−	43.7832i		−	1.56070i	−0.625340	−	0.780352i	$-0.715040\pi$
$787$		−	43.7832i		−	1.56070i	0.625340	−	0.780352i	$-0.284960\pi$
$788$		0			0
$789$	19.0049			0.676593
$790$		0			0
$791$	−5.29883			−0.188405
$792$		0			0
$793$		−	11.1985i		−	0.397670i
$794$		0			0
$795$	−17.5111	−	2.46028i	−0.621056	−	0.0872572i
$796$		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
$801$	−27.5397			−0.973068
$802$		0			0
$803$			0.180669i			0.00637568i
$804$		0			0
$805$	−8.42864	−	1.18421i	−0.297071	−	0.0417379i
$806$		0			0
$807$		−	24.1334i		−	0.849534i
$808$		0			0
$809$	17.4050			0.611927			0.305963	−	0.952043i	$-0.401022\pi$
$809$	17.4050			0.611927			0.305963	+	0.952043i	$0.401022\pi$
$810$		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
$813$		−	18.7413i		−	0.657285i
$814$		0			0
$815$	3.97280	−	28.2766i	0.139161	−	0.990484i
$816$		0			0
$817$			44.9131i			1.57131i
$818$		0			0
$819$	1.71900			0.0600669
$820$		0			0
$821$	−22.6133			−0.789210			−0.394605	−	0.918851i	$-0.629119\pi$
$821$	−22.6133			−0.789210			−0.394605	+	0.918851i	$0.629119\pi$
$822$		0			0
$823$			19.5506i			0.681492i	0.940155	+	0.340746i	$0.110680\pi$
$823$			19.5506i			0.681492i	−0.940155	+	0.340746i	$0.889320\pi$
$824$		0			0
$825$	0.203420	−	0.709636i	0.00708219	−	0.0247063i
$826$		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
$831$	−63.6040			−2.20640
$832$		0			0
$833$		−	20.8889i		−	0.723758i
$834$		0			0
$835$	2.31264	−	16.4603i	0.0800322	−	0.569632i
$836$		0			0
$837$			10.9175i			0.377364i
$838$		0			0
$839$	1.69826			0.0586305			0.0293152	−	0.999570i	$-0.490667\pi$
$839$	1.69826			0.0586305			0.0293152	+	0.999570i	$0.490667\pi$
$840$		0			0
$841$	−26.6731			−0.919761
$842$		0			0
$843$		−	31.4291i		−	1.08248i
$844$		0			0
$845$	−2.21432	−	0.311108i	−0.0761749	−	0.0107024i
$846$		0			0
$847$		−	9.93132i		−	0.341244i
$848$		0			0
$849$	35.2498			1.20977
$850$		0			0
$851$	50.3783			1.72694
$852$		0			0
$853$		−	35.4465i		−	1.21366i	−0.794830	−	0.606832i	$-0.792440\pi$
$853$		−	35.4465i		−	1.21366i	0.794830	−	0.606832i	$-0.207560\pi$
$854$		0			0
$855$	21.5669	+	3.03011i	0.737573	+	0.103628i
$856$		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
$861$	−5.51114			−0.187819
$862$		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
$867$			12.3793i			0.420424i
$868$		0			0
$869$	−0.212312			−0.00720219
$870$		0			0
$871$	2.14764			0.0727701
$872$		0			0
$873$			15.1111i			0.511433i
$874$		0			0
$875$	−4.10616	+	9.22570i	−0.138813	+	0.311885i
$876$		0			0
$877$			47.4608i			1.60264i	0.598239	+	0.801318i	$0.295867\pi$
$877$			47.4608i			1.60264i	−0.598239	+	0.801318i	$0.704133\pi$
$878$		0			0
$879$	45.2257			1.52542
$880$		0			0
$881$	7.49378			0.252472			0.126236	−	0.992000i	$-0.459710\pi$
$881$	7.49378			0.252472			0.126236	+	0.992000i	$0.459710\pi$
$882$		0			0
$883$		−	28.8879i		−	0.972154i	−0.873916	−	0.486077i	$-0.838427\pi$
$883$		−	28.8879i		−	0.972154i	0.873916	−	0.486077i	$-0.161573\pi$
$884$		0			0
$885$	−5.62714	+	40.0513i	−0.189154	+	1.34631i
$886$		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
$891$	0.739271			0.0247665
$892$		0			0
$893$			45.5625i			1.52469i
$894$		0			0
$895$	−54.4800	−	7.65433i	−1.82106	−	0.255856i
$896$		0			0
$897$		−	9.33185i		−	0.311581i
$898$		0			0
$899$	−6.85728			−0.228703
$900$		0			0
$901$	12.0633			0.401886
$902$		0			0
$903$			17.5526i			0.584115i
$904$		0			0
$905$	30.9906	+	4.35413i	1.03016	+	0.144736i
$906$		0			0
$907$		−	8.58274i		−	0.284985i	−0.989796	−	0.142493i	$-0.954488\pi$
$907$		−	8.58274i		−	0.284985i	0.989796	−	0.142493i	$-0.0455117\pi$
$908$		0			0
$909$	9.86665			0.327256
$910$		0			0
$911$	−9.96836			−0.330266			−0.165133	−	0.986271i	$-0.552805\pi$
$911$	−9.96836			−0.330266			−0.165133	+	0.986271i	$0.552805\pi$
$912$		0			0
$913$		−	0.659687i		−	0.0218325i
$914$		0			0
$915$	7.71456	−	54.9086i	0.255035	−	1.81522i
$916$		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
$921$	0.193576			0.00637855
$922$		0			0
$923$		−	5.54617i		−	0.182554i
$924$		0			0
$925$	16.4701	−	57.4563i	0.541534	−	1.88915i
$926$		0			0
$927$			15.0114i			0.493038i
$928$		0			0
$929$	23.0509			0.756274			0.378137	−	0.925750i	$-0.376565\pi$
$929$	23.0509			0.756274			0.378137	+	0.925750i	$0.376565\pi$
$930$		0			0
$931$	−31.6479			−1.03722
$932$		0			0
$933$		−	56.9403i		−	1.86414i
$934$		0			0
$935$	−0.0700674	+	0.498707i	−0.00229145	+	0.0163095i
$936$		0			0
$937$		−	35.2543i		−	1.15171i	−0.817553	−	0.575853i	$-0.804670\pi$
$937$		−	35.2543i		−	1.15171i	0.817553	−	0.575853i	$-0.195330\pi$
$938$		0			0
$939$	16.9906			0.554468
$940$		0			0
$941$	10.1334			0.330338			0.165169	−	0.986265i	$-0.447183\pi$
$941$	10.1334			0.330338			0.165169	+	0.986265i	$0.447183\pi$
$942$		0			0
$943$			11.6128i			0.378166i
$944$		0			0
$945$	−4.85728	−	0.682439i	−0.158007	−	0.0221997i
$946$		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
$951$	8.10171			0.262716
$952$		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
$957$			0.225219i			0.00728030i
$958$		0			0
$959$	−14.7012			−0.474726
$960$		0			0
$961$	−10.7921			−0.348133
$962$		0			0
$963$		−	33.0529i		−	1.06511i
$964$		0			0
$965$	−4.79706	+	34.1432i	−0.154423	+	1.09911i
$966$		0			0
$967$			7.12537i			0.229136i	0.993415	+	0.114568i	$0.0365484\pi$
$967$			7.12537i			0.229136i	−0.993415	+	0.114568i	$0.963452\pi$
$968$		0			0
$969$	−38.2766			−1.22962
$970$		0			0
$971$	−10.1521			−0.325796			−0.162898	−	0.986643i	$-0.552084\pi$
$971$	−10.1521			−0.325796			−0.162898	+	0.986643i	$0.552084\pi$
$972$		0			0
$973$			6.48886i			0.208023i
$974$		0			0
$975$	−10.6430	−	3.05086i	−0.340847	−	0.0977056i
$976$		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
$981$	0.253799			0.00810318
$982$		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
$987$			17.8064i			0.566785i
$988$		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
$993$		−	0.893368i		−	0.0283502i
$994$		0			0
$995$	−45.0321	−	6.32693i	−1.42761	−	0.200577i
$996$		0			0
$997$		−	19.5625i		−	0.619550i	−0.950810	−	0.309775i	$-0.899746\pi$
$997$		−	19.5625i		−	0.619550i	0.950810	−	0.309775i	$-0.100254\pi$
$998$		0			0
$999$	29.0321			0.918536

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	260.2.c.a.209.6	yes	6
3.2	odd	2		2340.2.h.e.469.1		6
4.3	odd	2		1040.2.d.d.209.1		6
5.2	odd	4		1300.2.a.k.1.3		3
5.3	odd	4		1300.2.a.j.1.1		3
5.4	even	2	inner	260.2.c.a.209.1	✓	6
13.5	odd	4		3380.2.d.b.1689.6		6
13.8	odd	4		3380.2.d.a.1689.6		6
13.12	even	2		3380.2.c.c.2029.6		6
15.14	odd	2		2340.2.h.e.469.2		6
20.3	even	4		5200.2.a.cg.1.3		3
20.7	even	4		5200.2.a.cd.1.1		3
20.19	odd	2		1040.2.d.d.209.6		6
65.34	odd	4		3380.2.d.b.1689.1		6
65.44	odd	4		3380.2.d.a.1689.1		6
65.64	even	2		3380.2.c.c.2029.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.c.a.209.1	✓	6	5.4	even	2	inner
260.2.c.a.209.6	yes	6	1.1	even	1	trivial
1040.2.d.d.209.1		6	4.3	odd	2
1040.2.d.d.209.6		6	20.19	odd	2
1300.2.a.j.1.1		3	5.3	odd	4
1300.2.a.k.1.3		3	5.2	odd	4
2340.2.h.e.469.1		6	3.2	odd	2
2340.2.h.e.469.2		6	15.14	odd	2
3380.2.c.c.2029.1		6	65.64	even	2
3380.2.c.c.2029.6		6	13.12	even	2
3380.2.d.a.1689.1		6	65.44	odd	4
3380.2.d.a.1689.6		6	13.8	odd	4
3380.2.d.b.1689.1		6	65.34	odd	4
3380.2.d.b.1689.6		6	13.5	odd	4
5200.2.a.cd.1.1		3	20.7	even	4
5200.2.a.cg.1.3		3	20.3	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:	\( N \)	\(=\)	\( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	260.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.21432i q^{3} +(2.21432 + 0.311108i) q^{5} +0.903212i q^{7} -1.90321 q^{9} -0.0666765 q^{11} +1.00000i q^{13} +(-0.688892 + 4.90321i) q^{15} -3.37778i q^{17} -5.11753 q^{19} -2.00000 q^{21} +4.21432i q^{23} +(4.80642 + 1.37778i) q^{25} +2.42864i q^{27} -1.52543 q^{29} +4.49532 q^{31} -0.147643i q^{33} +(-0.280996 + 2.00000i) q^{35} -11.9541i q^{37} -2.21432 q^{39} +2.75557 q^{41} -8.77631i q^{43} +(-4.21432 - 0.592104i) q^{45} -8.90321i q^{47} +6.18421 q^{49} +7.47949 q^{51} +3.57136i q^{53} +(-0.147643 - 0.0207436i) q^{55} -11.3319i q^{57} +8.16839 q^{59} -11.1985 q^{61} -1.71900i q^{63} +(-0.311108 + 2.21432i) q^{65} -2.14764i q^{67} -9.33185 q^{69} -5.54617 q^{71} -2.70964i q^{73} +(-3.05086 + 10.6430i) q^{75} -0.0602231i q^{77} +3.18421 q^{79} -11.0874 q^{81} +9.89384i q^{83} +(1.05086 - 7.47949i) q^{85} -3.37778i q^{87} +14.4701 q^{89} -0.903212 q^{91} +9.95407i q^{93} +(-11.3319 - 1.59210i) q^{95} -7.93978i q^{97} +0.126900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{9} - 4 q^{15} - 4 q^{19} - 12 q^{21} + 2 q^{25} + 4 q^{29} + 12 q^{35} + 16 q^{41} - 12 q^{45} + 10 q^{49} - 8 q^{51} + 12 q^{55} - 4 q^{59} - 28 q^{61} - 2 q^{65} - 16 q^{69} + 20 q^{71} + 8 q^{75}+ \cdots + 28 q^{99}+O(q^{100}) \) 6 * q + 2 * q^9 - 4 * q^15 - 4 * q^19 - 12 * q^21 + 2 * q^25 + 4 * q^29 + 12 * q^35 + 16 * q^41 - 12 * q^45 + 10 * q^49 - 8 * q^51 + 12 * q^55 - 4 * q^59 - 28 * q^61 - 2 * q^65 - 16 * q^69 + 20 * q^71 + 8 * q^75 - 8 * q^79 - 26 * q^81 - 20 * q^85 - 20 * q^89 + 8 * q^91 - 28 * q^95 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more