LMFDB - Embedded newform 1260.2.s.h.541.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1260,2,Mod(361,1260)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1260.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1260, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1260.s (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.0611506547$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.4406832.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 $ x^6 - x^5 + 6x^4 + 7x^3 + 24x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.3
Root			$-0.827721 + 1.43366i$ of defining polynomial
Character	$\chi$	$=$	1260.541
Dual form			1260.2.s.h.361.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(2.35341 + 1.20891i) q^{7} +(2.85341 + 4.94225i) q^{11} -6.70682 q^{13} +(1.72365 + 2.98545i) q^{17} +(-0.629755 + 1.09077i) q^{19} +(-3.98316 + 6.89904i) q^{23} +(-0.500000 - 0.866025i) q^{25} +7.18780 q^{29} +(0.500000 + 0.866025i) q^{31} +(2.22365 - 1.43366i) q^{35} +(4.48316 - 7.76507i) q^{37} -10.2258 q^{41} -4.44731 q^{43} +(0.740489 - 1.28257i) q^{47} +(4.07706 + 5.69013i) q^{49} +(3.44731 + 5.97091i) q^{53} +5.70682 q^{55} +(-0.146592 - 0.253904i) q^{59} +(0.129755 - 0.224743i) q^{61} +(-3.35341 + 5.80827i) q^{65} +(3.35341 + 5.80827i) q^{67} +11.7068 q^{71} +(-0.483164 - 0.836864i) q^{73} +(0.740489 + 15.0806i) q^{77} +(5.81755 - 10.0763i) q^{79} +14.8609 q^{83} +3.44731 q^{85} +(-6.30071 + 10.9132i) q^{89} +(-15.7839 - 8.10795i) q^{91} +(0.629755 + 1.09077i) q^{95} -13.1541 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{5} - q^{7} + 2 q^{11} - 10 q^{13} - 2 q^{17} - q^{19} - 6 q^{23} - 3 q^{25} + 24 q^{29} + 3 q^{31} + q^{35} + 9 q^{37} - 20 q^{41} - 2 q^{43} + 10 q^{47} - 3 q^{49} - 4 q^{53} + 4 q^{55} - 16 q^{59}+ \cdots - 24 q^{97}+O(q^{100}) $ 6 * q + 3 * q^5 - q^7 + 2 * q^11 - 10 * q^13 - 2 * q^17 - q^19 - 6 * q^23 - 3 * q^25 + 24 * q^29 + 3 * q^31 + q^35 + 9 * q^37 - 20 * q^41 - 2 * q^43 + 10 * q^47 - 3 * q^49 - 4 * q^53 + 4 * q^55 - 16 * q^59 - 2 * q^61 - 5 * q^65 + 5 * q^67 + 40 * q^71 + 15 * q^73 + 10 * q^77 + 13 * q^79 + 4 * q^83 - 4 * q^85 + 2 * q^89 - 37 * q^91 + q^95 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$757$	$1081$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	2.35341	+	1.20891i	0.889505	+	0.456926i
$7$	2.35341	+	1.20891i	0.889505	+	0.456926i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.85341	+	4.94225i	0.860335	+	1.49014i	0.871606	+	0.490207i	$0.163079\pi$
$11$	2.85341	+	4.94225i	0.860335	+	1.49014i	−0.0112708	+	0.999936i	$0.503588\pi$
$12$		0			0
$13$	−6.70682			−1.86014			−0.930068	−	0.367387i	$-0.880252\pi$
$13$	−6.70682			−1.86014			−0.930068	+	0.367387i	$0.880252\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.72365	+	2.98545i	0.418047	+	0.724079i	0.995743	−	0.0921731i	$-0.0293813\pi$
$17$	1.72365	+	2.98545i	0.418047	+	0.724079i	−0.577696	+	0.816252i	$0.696048\pi$
$18$		0			0
$19$	−0.629755	+	1.09077i	−0.144476	+	0.250239i	−0.929177	−	0.369634i	$-0.879483\pi$
$19$	−0.629755	+	1.09077i	−0.144476	+	0.250239i	0.784701	+	0.619874i	$0.212816\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.98316	+	6.89904i	−0.830547	+	1.43855i	0.0670581	+	0.997749i	$0.478639\pi$
$23$	−3.98316	+	6.89904i	−0.830547	+	1.43855i	−0.897605	+	0.440801i	$0.854695\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.18780			1.33474			0.667370	−	0.744726i	$-0.267420\pi$
$29$	7.18780			1.33474			0.667370	+	0.744726i	$0.267420\pi$
$30$		0			0
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	0.907428	−	0.420208i	$-0.138043\pi$
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	−0.817625	+	0.575751i	$0.804710\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.22365	−	1.43366i	0.375866	−	0.242332i
$36$		0			0
$37$	4.48316	−	7.76507i	0.737028	−	1.27657i	−0.216800	−	0.976216i	$-0.569562\pi$
$37$	4.48316	−	7.76507i	0.737028	−	1.27657i	0.953828	−	0.300353i	$-0.0971046\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10.2258			−1.59701			−0.798504	−	0.601990i	$-0.794375\pi$
$41$	−10.2258			−1.59701			−0.798504	+	0.601990i	$0.794375\pi$
$42$		0			0
$43$	−4.44731			−0.678208			−0.339104	−	0.940749i	$-0.610124\pi$
$43$	−4.44731			−0.678208			−0.339104	+	0.940749i	$0.610124\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.740489	−	1.28257i	0.108011	−	0.187081i	−0.806953	−	0.590616i	$-0.798885\pi$
$47$	0.740489	−	1.28257i	0.108011	−	0.187081i	0.914965	+	0.403534i	$0.132218\pi$
$48$		0			0
$49$	4.07706	+	5.69013i	0.582437	+	0.812876i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.44731	+	5.97091i	0.473524	+	0.820167i	0.999541	−	0.0303067i	$-0.00964841\pi$
$53$	3.44731	+	5.97091i	0.473524	+	0.820167i	−0.526017	+	0.850474i	$0.676315\pi$
$54$		0			0
$55$	5.70682			0.769507
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.146592	−	0.253904i	−0.0190846	−	0.0330555i	0.856325	−	0.516437i	$-0.172742\pi$
$59$	−0.146592	−	0.253904i	−0.0190846	−	0.0330555i	−0.875410	+	0.483381i	$0.839409\pi$
$60$		0			0
$61$	0.129755	−	0.224743i	0.0166135	−	0.0287754i	−0.857599	−	0.514319i	$-0.828045\pi$
$61$	0.129755	−	0.224743i	0.0166135	−	0.0287754i	0.874213	+	0.485543i	$0.161378\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.35341	+	5.80827i	−0.415939	+	0.720428i
$66$		0			0
$67$	3.35341	+	5.80827i	0.409684	+	0.709594i	0.994854	−	0.101317i	$-0.0323056\pi$
$67$	3.35341	+	5.80827i	0.409684	+	0.709594i	−0.585170	+	0.810911i	$0.698972\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	11.7068			1.38934			0.694672	−	0.719327i	$-0.255550\pi$
$71$	11.7068			1.38934			0.694672	+	0.719327i	$0.255550\pi$
$72$		0			0
$73$	−0.483164	−	0.836864i	−0.0565500	−	0.0979475i	0.836365	−	0.548174i	$-0.184677\pi$
$73$	−0.483164	−	0.836864i	−0.0565500	−	0.0979475i	−0.892915	+	0.450226i	$0.851343\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.740489	+	15.0806i	0.0843866	+	1.71860i
$78$		0			0
$79$	5.81755	−	10.0763i	0.654526	−	1.13367i	−0.327487	−	0.944856i	$-0.606202\pi$
$79$	5.81755	−	10.0763i	0.654526	−	1.13367i	0.982013	−	0.188816i	$-0.0604649\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	14.8609			1.63120			0.815600	−	0.578616i	$-0.196407\pi$
$83$	14.8609			1.63120			0.815600	+	0.578616i	$0.196407\pi$
$84$		0			0
$85$	3.44731			0.373913
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.30071	+	10.9132i	−0.667874	+	1.15679i	0.310623	+	0.950533i	$0.399462\pi$
$89$	−6.30071	+	10.9132i	−0.667874	+	1.15679i	−0.978497	+	0.206259i	$0.933871\pi$
$90$		0			0
$91$	−15.7839	−	8.10795i	−1.65460	−	0.849945i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.629755	+	1.09077i	0.0646115	+	0.111910i
$96$		0			0
$97$	−13.1541			−1.33560			−0.667799	−	0.744341i	$-0.732764\pi$
$97$	−13.1541			−1.33560			−0.667799	+	0.744341i	$0.732764\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.56022	+	9.63059i	0.553263	+	0.958280i	0.998036	+	0.0626366i	$0.0199509\pi$
$101$	5.56022	+	9.63059i	0.553263	+	0.958280i	−0.444773	+	0.895643i	$0.646716\pi$
$102$		0			0
$103$	1.03586	−	1.79416i	0.102066	−	0.176784i	−0.810470	−	0.585781i	$-0.800788\pi$
$103$	1.03586	−	1.79416i	0.102066	−	0.176784i	0.912536	+	0.408997i	$0.134121\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.983164	−	1.70289i	0.0950460	−	0.164625i	−0.814582	−	0.580049i	$-0.803034\pi$
$107$	0.983164	−	1.70289i	0.0950460	−	0.164625i	0.909628	+	0.415424i	$0.136367\pi$
$108$		0			0
$109$	4.33657	+	7.51116i	0.415368	+	0.719439i	0.995467	−	0.0951072i	$-0.0303194\pi$
$109$	4.33657	+	7.51116i	0.415368	+	0.719439i	−0.580099	+	0.814546i	$0.696986\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.586367			0.0551607			0.0275804	−	0.999620i	$-0.491220\pi$
$113$	0.586367			0.0551607			0.0275804	+	0.999620i	$0.491220\pi$
$114$		0			0
$115$	3.98316	+	6.89904i	0.371432	+	0.643339i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.447306	+	9.10974i	0.0410045	+	0.835088i
$120$		0			0
$121$	−10.7839	+	18.6782i	−0.980353	+	1.69802i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−5.22584			−0.463718			−0.231859	−	0.972749i	$-0.574481\pi$
$127$	−5.22584			−0.463718			−0.231859	+	0.972749i	$0.574481\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.70682	−	15.0806i	0.760718	−	1.31760i	−0.181763	−	0.983342i	$-0.558180\pi$
$131$	8.70682	−	15.0806i	0.760718	−	1.31760i	0.942481	−	0.334260i	$-0.108486\pi$
$132$		0			0
$133$	−2.80071	+	1.80570i	−0.242853	+	0.156574i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.98316	−	12.0952i	−0.596612	−	1.03336i	−0.993317	−	0.115416i	$-0.963180\pi$
$137$	−6.98316	−	12.0952i	−0.596612	−	1.03336i	0.396705	−	0.917946i	$-0.370153\pi$
$138$		0			0
$139$	15.0487			1.27642			0.638209	−	0.769864i	$-0.279676\pi$
$139$	15.0487			1.27642			0.638209	+	0.769864i	$0.279676\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−19.1373	−	33.1468i	−1.60034	−	2.77187i
$144$		0			0
$145$	3.59390	−	6.22481i	0.298457	−	0.516943i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	−10.5434	−	18.2617i	−0.858009	−	1.48611i	−0.873825	−	0.486240i	$-0.838368\pi$
$151$	−10.5434	−	18.2617i	−0.858009	−	1.48611i	0.0158166	−	0.999875i	$-0.494965\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.00000			0.0803219
$156$		0			0
$157$	−4.44731	−	7.70296i	−0.354934	−	0.614763i	0.632173	−	0.774827i	$-0.282163\pi$
$157$	−4.44731	−	7.70296i	−0.354934	−	0.614763i	−0.987107	+	0.160064i	$0.948830\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−17.7143	+	11.4210i	−1.39609	+	0.900098i
$162$		0			0
$163$	−5.18780	+	8.98553i	−0.406340	+	0.703801i	−0.994476	−	0.104961i	$-0.966528\pi$
$163$	−5.18780	+	8.98553i	−0.406340	+	0.703801i	0.588137	+	0.808761i	$0.299862\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.96633			0.152159			0.0760795	−	0.997102i	$-0.475760\pi$
$167$	1.96633			0.152159			0.0760795	+	0.997102i	$0.475760\pi$
$168$		0			0
$169$	31.9814			2.46011
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.740489	−	1.28257i	0.0562984	−	0.0975116i	−0.836503	−	0.547963i	$-0.815404\pi$
$173$	0.740489	−	1.28257i	0.0562984	−	0.0975116i	0.892801	+	0.450451i	$0.148737\pi$
$174$		0			0
$175$	−0.129755	−	2.64257i	−0.00980858	−	0.199759i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.15412	−	15.8554i	−0.684211	−	1.18509i	−0.973684	−	0.227902i	$-0.926813\pi$
$179$	−9.15412	−	15.8554i	−0.684211	−	1.18509i	0.289473	−	0.957186i	$-0.406520\pi$
$180$		0			0
$181$	18.6731			1.38796			0.693982	−	0.719992i	$-0.255855\pi$
$181$	18.6731			1.38796			0.693982	+	0.719992i	$0.255855\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.48316	−	7.76507i	−0.329609	−	0.570899i
$186$		0			0
$187$	−9.83657	+	17.0374i	−0.719321	+	1.24590i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.15412	−	5.46310i	0.228224	−	0.395296i	−0.729058	−	0.684452i	$-0.760041\pi$
$191$	3.15412	−	5.46310i	0.228224	−	0.395296i	0.957282	+	0.289156i	$0.0933747\pi$
$192$		0			0
$193$	−11.4495	−	19.8311i	−0.824152	−	1.42747i	−0.902565	−	0.430553i	$-0.858319\pi$
$193$	−11.4495	−	19.8311i	−0.824152	−	1.42747i	0.0784130	−	0.996921i	$-0.475015\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.5527			1.03684			0.518418	−	0.855127i	$-0.326521\pi$
$197$	14.5527			1.03684			0.518418	+	0.855127i	$0.326521\pi$
$198$		0			0
$199$	8.54339	+	14.7976i	0.605625	+	1.04897i	0.991952	+	0.126611i	$0.0404101\pi$
$199$	8.54339	+	14.7976i	0.605625	+	1.04897i	−0.386328	+	0.922362i	$0.626257\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	16.9158	+	8.68942i	1.18726	+	0.609877i
$204$		0			0
$205$	−5.11292	+	8.85584i	−0.357102	+	0.618518i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.18780			−0.497190
$210$		0			0
$211$	3.36490			0.231649			0.115825	−	0.993270i	$-0.463049\pi$
$211$	3.36490			0.231649			0.115825	+	0.993270i	$0.463049\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.22365	+	3.85148i	−0.151652	+	0.262669i
$216$		0			0
$217$	0.129755	+	2.64257i	0.00880836	+	0.179389i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−11.5602	−	20.0229i	−0.777625	−	1.34689i
$222$		0			0
$223$	−17.7892			−1.19125			−0.595627	−	0.803261i	$-0.703096\pi$
$223$	−17.7892			−1.19125			−0.595627	+	0.803261i	$0.703096\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.46414	−	14.6603i	−0.561785	−	0.973040i	−0.997341	−	0.0728783i	$-0.976782\pi$
$227$	−8.46414	−	14.6603i	−0.561785	−	0.973040i	0.435556	−	0.900162i	$-0.356552\pi$
$228$		0			0
$229$	10.3946	−	18.0040i	0.686895	−	1.18974i	−0.285942	−	0.958247i	$-0.592306\pi$
$229$	10.3946	−	18.0040i	0.686895	−	1.18974i	0.972837	−	0.231491i	$-0.0743603\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.1878	+	17.6458i	−0.667425	+	1.15601i	0.311197	+	0.950345i	$0.399270\pi$
$233$	−10.1878	+	17.6458i	−0.667425	+	1.15601i	−0.978622	+	0.205668i	$0.934063\pi$
$234$		0			0
$235$	−0.740489	−	1.28257i	−0.0483042	−	0.0836653i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	15.0380			0.972730			0.486365	−	0.873756i	$-0.338323\pi$
$239$	15.0380			0.972730			0.486365	+	0.873756i	$0.338323\pi$
$240$		0			0
$241$	1.31755	+	2.28206i	0.0848709	+	0.147001i	0.905336	−	0.424696i	$-0.139619\pi$
$241$	1.31755	+	2.28206i	0.0848709	+	0.147001i	−0.820465	+	0.571696i	$0.806286\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.96633	−	0.685774i	0.445062	−	0.0438125i
$246$		0			0
$247$	4.22365	−	7.31558i	0.268745	−	0.465479i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.5634			−0.982352			−0.491176	−	0.871060i	$-0.663433\pi$
$251$	−15.5634			−0.982352			−0.491176	+	0.871060i	$0.663433\pi$
$252$		0			0
$253$	−45.4624			−2.85819
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.50218	−	9.53006i	0.343217	−	0.594469i	−0.641811	−	0.766863i	$-0.721817\pi$
$257$	5.50218	−	9.53006i	0.343217	−	0.594469i	0.985028	+	0.172394i	$0.0551501\pi$
$258$		0			0
$259$	19.9380	−	12.8546i	1.23889	−	0.798747i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.51902	+	2.63102i	0.0936669	+	0.162236i	0.909051	−	0.416684i	$-0.136808\pi$
$263$	1.51902	+	2.63102i	0.0936669	+	0.162236i	−0.815385	+	0.578920i	$0.803474\pi$
$264$		0			0
$265$	6.89461			0.423533
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.9663	−	18.9942i	−0.668629	−	1.15810i	−0.978288	−	0.207251i	$-0.933548\pi$
$269$	−10.9663	−	18.9942i	−0.668629	−	1.15810i	0.309659	−	0.950848i	$-0.399785\pi$
$270$		0			0
$271$	−3.16343	+	5.47922i	−0.192165	+	0.332839i	−0.945967	−	0.324262i	$-0.894884\pi$
$271$	−3.16343	+	5.47922i	−0.192165	+	0.332839i	0.753803	+	0.657101i	$0.228217\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.85341	−	4.94225i	0.172067	−	0.298029i
$276$		0			0
$277$	1.77635	+	3.07672i	0.106730	+	0.184862i	0.914444	−	0.404713i	$-0.132629\pi$
$277$	1.77635	+	3.07672i	0.106730	+	0.184862i	−0.807713	+	0.589575i	$0.799295\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	22.5190			1.34337			0.671686	−	0.740836i	$-0.265570\pi$
$281$	22.5190			1.34337			0.671686	+	0.740836i	$0.265570\pi$
$282$		0			0
$283$	−5.44949	−	9.43879i	−0.323939	−	0.561078i	0.657358	−	0.753578i	$-0.271674\pi$
$283$	−5.44949	−	9.43879i	−0.323939	−	0.561078i	−0.981297	+	0.192500i	$0.938340\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−24.0656	−	12.3621i	−1.42055	−	0.729714i
$288$		0			0
$289$	2.55804	−	4.43066i	0.150473	−	0.260627i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−24.8946			−1.45436			−0.727179	−	0.686447i	$-0.759169\pi$
$293$	−24.8946			−1.45436			−0.727179	+	0.686447i	$0.759169\pi$
$294$		0			0
$295$	−0.293183			−0.0170698
$296$		0			0
$297$		0			0
$298$		0			0
$299$	26.7143	−	46.2706i	1.54493	−	2.67590i
$300$		0			0
$301$	−10.4663	−	5.37640i	−0.603269	−	0.309891i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.129755	−	0.224743i	−0.00742976	−	0.0128687i
$306$		0			0
$307$	0.774162			0.0441838			0.0220919	−	0.999756i	$-0.492967\pi$
$307$	0.774162			0.0441838			0.0220919	+	0.999756i	$0.492967\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.59390	−	16.6171i	−0.544020	−	0.942270i	−0.998668	−	0.0515991i	$-0.983568\pi$
$311$	−9.59390	−	16.6171i	−0.544020	−	0.942270i	0.454648	−	0.890671i	$-0.349765\pi$
$312$		0			0
$313$	1.32904	−	2.30197i	0.0751218	−	0.130115i	−0.826017	−	0.563645i	$-0.809399\pi$
$313$	1.32904	−	2.30197i	0.0751218	−	0.130115i	0.901139	+	0.433530i	$0.142732\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.53586	+	11.3204i	−0.367090	+	0.635819i	−0.989109	−	0.147183i	$-0.952979\pi$
$317$	−6.53586	+	11.3204i	−0.367090	+	0.635819i	0.622019	+	0.783002i	$0.286313\pi$
$318$		0			0
$319$	20.5097	+	35.5239i	1.14832	+	1.98895i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.34192			−0.241591
$324$		0			0
$325$	3.35341	+	5.80827i	0.186014	+	0.322185i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.29318	−	2.12321i	0.181559	−	0.117057i
$330$		0			0
$331$	2.01902	−	3.49705i	0.110975	−	0.192215i	−0.805188	−	0.593019i	$-0.797936\pi$
$331$	2.01902	−	3.49705i	0.110975	−	0.192215i	0.916164	+	0.400804i	$0.131269\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.70682			0.366433
$336$		0			0
$337$	2.33123			0.126990			0.0634950	−	0.997982i	$-0.479775\pi$
$337$	2.33123			0.126990			0.0634950	+	0.997982i	$0.479775\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.85341	+	4.94225i	−0.154521	+	0.267638i
$342$		0			0
$343$	2.71612	+	18.3200i	0.146657	+	0.989187i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.25951	+	14.3059i	0.443394	+	0.767981i	0.997939	−	0.0641732i	$-0.0204410\pi$
$347$	8.25951	+	14.3059i	0.443394	+	0.767981i	−0.554545	+	0.832154i	$0.687108\pi$
$348$		0			0
$349$	8.89461			0.476118			0.238059	−	0.971251i	$-0.423489\pi$
$349$	8.89461			0.476118			0.238059	+	0.971251i	$0.423489\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.98316	−	12.0952i	−0.371676	−	0.643762i	0.618147	−	0.786062i	$-0.287884\pi$
$353$	−6.98316	−	12.0952i	−0.371676	−	0.643762i	−0.989824	+	0.142300i	$0.954550\pi$
$354$		0			0
$355$	5.85341	−	10.1384i	0.310667	−	0.538090i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.88708	+	6.73262i	−0.205152	+	0.355334i	−0.950181	−	0.311698i	$-0.899102\pi$
$359$	−3.88708	+	6.73262i	−0.205152	+	0.355334i	0.745029	+	0.667032i	$0.232436\pi$
$360$		0			0
$361$	8.70682	+	15.0806i	0.458254	+	0.793718i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.966327			−0.0505799
$366$		0			0
$367$	−7.16561	−	12.4112i	−0.374042	−	0.647860i	0.616141	−	0.787636i	$-0.288695\pi$
$367$	−7.16561	−	12.4112i	−0.374042	−	0.647860i	−0.990183	+	0.139776i	$0.955362\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.894612	+	18.2195i	0.0464460	+	0.945908i
$372$		0			0
$373$	−1.12757	+	1.95301i	−0.0583834	+	0.101123i	−0.893740	−	0.448586i	$-0.851928\pi$
$373$	−1.12757	+	1.95301i	−0.0583834	+	0.101123i	0.835356	+	0.549709i	$0.185261\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−48.2072			−2.48280
$378$		0			0
$379$	2.85657			0.146732			0.0733661	−	0.997305i	$-0.476626\pi$
$379$	2.85657			0.146732			0.0733661	+	0.997305i	$0.476626\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13.9663	−	24.1904i	0.713646	−	1.23607i	−0.249833	−	0.968289i	$-0.580376\pi$
$383$	13.9663	−	24.1904i	0.713646	−	1.23607i	0.963479	−	0.267782i	$-0.0862908\pi$
$384$		0			0
$385$	13.4305	+	6.89904i	0.684480	+	0.351608i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.26704	+	9.12278i	0.267050	+	0.462543i	0.968099	−	0.250570i	$-0.0806179\pi$
$389$	5.26704	+	9.12278i	0.267050	+	0.462543i	−0.701049	+	0.713113i	$0.747285\pi$
$390$		0			0
$391$	−27.4624			−1.38883
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5.81755	−	10.0763i	−0.292713	−	0.506993i
$396$		0			0
$397$	3.00218	−	5.19994i	0.150675	−	0.260977i	−0.780801	−	0.624780i	$-0.785189\pi$
$397$	3.00218	−	5.19994i	0.150675	−	0.260977i	0.931476	+	0.363803i	$0.118522\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.92829	+	8.53604i	−0.246107	+	0.426270i	−0.962442	−	0.271487i	$-0.912485\pi$
$401$	−4.92829	+	8.53604i	−0.246107	+	0.426270i	0.716335	+	0.697756i	$0.245818\pi$
$402$		0			0
$403$	−3.35341	−	5.80827i	−0.167045	−	0.289331i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	51.1692			2.53636
$408$		0			0
$409$	−7.42829	−	12.8662i	−0.367305	−	0.636191i	0.621838	−	0.783146i	$-0.286386\pi$
$409$	−7.42829	−	12.8662i	−0.367305	−	0.636191i	−0.989143	+	0.146955i	$0.953053\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.0380421	−	0.774757i	−0.00187193	−	0.0381233i
$414$		0			0
$415$	7.43047	−	12.8700i	0.364747	−	0.631761i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−27.9327			−1.36460			−0.682300	−	0.731073i	$-0.739020\pi$
$419$	−27.9327			−1.36460			−0.682300	+	0.731073i	$0.739020\pi$
$420$		0			0
$421$	−5.71119			−0.278346			−0.139173	−	0.990268i	$-0.544444\pi$
$421$	−5.71119			−0.278346			−0.139173	+	0.990268i	$0.544444\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.72365	−	2.98545i	0.0836095	−	0.144816i
$426$		0			0
$427$	0.577061	−	0.372049i	0.0279260	−	0.0180047i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.96633	−	8.60193i	−0.239220	−	0.414340i	0.721271	−	0.692653i	$-0.243558\pi$
$431$	−4.96633	−	8.60193i	−0.239220	−	0.414340i	−0.960491	+	0.278313i	$0.910225\pi$
$432$		0			0
$433$	−12.1204			−0.582472			−0.291236	−	0.956651i	$-0.594066\pi$
$433$	−12.1204			−0.582472			−0.291236	+	0.956651i	$0.594066\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.01684	−	8.68942i	−0.239988	−	0.415671i
$438$		0			0
$439$	−11.6151	+	20.1179i	−0.554359	+	0.960177i	0.443594	+	0.896228i	$0.353703\pi$
$439$	−11.6151	+	20.1179i	−0.554359	+	0.960177i	−0.997953	+	0.0639497i	$0.979630\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.51902	+	13.0233i	−0.357239	+	0.618757i	−0.987499	−	0.157628i	$-0.949615\pi$
$443$	−7.51902	+	13.0233i	−0.357239	+	0.618757i	0.630259	+	0.776385i	$0.282949\pi$
$444$		0			0
$445$	6.30071	+	10.9132i	0.298683	+	0.517333i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5.69175			0.268611			0.134305	−	0.990940i	$-0.457120\pi$
$449$	5.69175			0.268611			0.134305	+	0.990940i	$0.457120\pi$
$450$		0			0
$451$	−29.1785	−	50.5386i	−1.37396	−	2.37977i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−14.9136	+	9.61526i	−0.699162	+	0.450770i
$456$		0			0
$457$	−0.387081	+	0.670444i	−0.0181069	+	0.0313621i	−0.874937	−	0.484237i	$-0.839097\pi$
$457$	−0.387081	+	0.670444i	−0.0181069	+	0.0313621i	0.856830	+	0.515599i	$0.172431\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	30.9097			1.43961			0.719804	−	0.694178i	$-0.244232\pi$
$461$	30.9097			1.43961			0.719804	+	0.694178i	$0.244232\pi$
$462$		0			0
$463$	−11.8122			−0.548960			−0.274480	−	0.961593i	$-0.588506\pi$
$463$	−11.8122			−0.548960			−0.274480	+	0.961593i	$0.588506\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.55269	−	4.42140i	0.118125	−	0.204598i	−0.800900	−	0.598798i	$-0.795645\pi$
$467$	2.55269	−	4.42140i	0.118125	−	0.204598i	0.919024	+	0.394200i	$0.128978\pi$
$468$		0			0
$469$	0.870245	+	17.7232i	0.0401842	+	0.818382i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−12.6900	−	21.9797i	−0.583486	−	1.01063i
$474$		0			0
$475$	1.25951			0.0577903
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.93265	+	12.0077i	0.316761	+	0.548646i	0.979810	−	0.199930i	$-0.0640713\pi$
$479$	6.93265	+	12.0077i	0.316761	+	0.548646i	−0.663049	+	0.748576i	$0.730738\pi$
$480$		0			0
$481$	−30.0678	+	52.0789i	−1.37097	+	2.37459i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.57706	+	11.3918i	−0.298649	+	0.517275i
$486$		0			0
$487$	2.28169	+	3.95201i	0.103393	+	0.179083i	0.913081	−	0.407779i	$-0.133697\pi$
$487$	2.28169	+	3.95201i	0.103393	+	0.179083i	−0.809687	+	0.586862i	$0.800363\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.93265			−0.448254			−0.224127	−	0.974560i	$-0.571953\pi$
$491$	−9.93265			−0.448254			−0.224127	+	0.974560i	$0.571953\pi$
$492$		0			0
$493$	12.3893	+	21.4588i	0.557984	+	0.966457i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	27.5509	+	14.1525i	1.23583	+	0.634827i
$498$		0			0
$499$	3.94731	−	6.83693i	0.176706	−	0.306063i	−0.764045	−	0.645163i	$-0.776789\pi$
$499$	3.94731	−	6.83693i	0.176706	−	0.306063i	0.940750	+	0.339100i	$0.110123\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.0000			0.535054			0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000			0.535054			0.267527	+	0.963550i	$0.413794\pi$
$504$		0			0
$505$	11.1204			0.494854
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.4853	+	21.6253i	−0.553403	+	0.958523i	0.444623	+	0.895718i	$0.353338\pi$
$509$	−12.4853	+	21.6253i	−0.553403	+	0.958523i	−0.998026	+	0.0628046i	$0.979995\pi$
$510$		0			0
$511$	−0.125386	−	2.55358i	−0.00554675	−	0.112964i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.03586	−	1.79416i	−0.0456453	−	0.0790600i
$516$		0			0
$517$	8.45168			0.371704
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10.2258	−	17.7117i	−0.448002	−	0.775962i	0.550254	−	0.834997i	$-0.314531\pi$
$521$	−10.2258	−	17.7117i	−0.448002	−	0.775962i	−0.998256	+	0.0590351i	$0.981198\pi$
$522$		0			0
$523$	17.6129	−	30.5065i	0.770159	−	1.33395i	−0.167316	−	0.985903i	$-0.553510\pi$
$523$	17.6129	−	30.5065i	0.770159	−	1.33395i	0.937475	−	0.348052i	$-0.113157\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.72365	+	2.98545i	−0.0750835	+	0.130048i
$528$		0			0
$529$	−20.2312	−	35.0414i	−0.879617	−	1.52354i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	68.5828			2.97065
$534$		0			0
$535$	−0.983164	−	1.70289i	−0.0425059	−	0.0736223i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−16.4885	+	36.3861i	−0.710210	+	1.56726i
$540$		0			0
$541$	22.1014	−	38.2808i	0.950215	−	1.64582i	0.205257	−	0.978708i	$-0.434197\pi$
$541$	22.1014	−	38.2808i	0.950215	−	1.64582i	0.744957	−	0.667112i	$-0.232470\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.67314			0.371517
$546$		0			0
$547$	35.3463			1.51130			0.755649	−	0.654977i	$-0.227322\pi$
$547$	35.3463			1.51130			0.755649	+	0.654977i	$0.227322\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.52655	+	7.84022i	−0.192838	+	0.334005i
$552$		0			0
$553$	25.8724	−	16.6807i	1.10021	−	0.709337i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.41363	+	4.18054i	0.102269	+	0.177135i	0.912619	−	0.408811i	$-0.134056\pi$
$557$	2.41363	+	4.18054i	0.102269	+	0.177135i	−0.810350	+	0.585946i	$0.800723\pi$
$558$		0			0
$559$	29.8273			1.26156
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.0151	+	31.2030i	0.759244	+	1.31505i	0.943236	+	0.332122i	$0.107765\pi$
$563$	18.0151	+	31.2030i	0.759244	+	1.31505i	−0.183992	+	0.982928i	$0.558902\pi$
$564$		0			0
$565$	0.293183	−	0.507809i	0.0123343	−	0.0213637i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13.3724	+	23.1617i	−0.560601	+	0.970990i	0.436843	+	0.899538i	$0.356097\pi$
$569$	−13.3724	+	23.1617i	−0.560601	+	0.970990i	−0.997444	+	0.0714522i	$0.977237\pi$
$570$		0			0
$571$	11.3702	+	19.6938i	0.475830	+	0.824162i	0.999617	−	0.0276878i	$-0.00881444\pi$
$571$	11.3702	+	19.6938i	0.475830	+	0.824162i	−0.523787	+	0.851849i	$0.675481\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7.96633			0.332219
$576$		0			0
$577$	1.83439	+	3.17725i	0.0763665	+	0.132271i	0.901680	−	0.432405i	$-0.142335\pi$
$577$	1.83439	+	3.17725i	0.0763665	+	0.132271i	−0.825313	+	0.564675i	$0.809001\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	34.9739	+	17.9656i	1.45096	+	0.745338i
$582$		0			0
$583$	−19.6731	+	34.0749i	−0.814778	+	1.41124i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	23.2365			0.959074			0.479537	−	0.877522i	$-0.340805\pi$
$587$	23.2365			0.959074			0.479537	+	0.877522i	$0.340805\pi$
$588$		0			0
$589$	−1.25951			−0.0518972
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−17.3251	+	30.0079i	−0.711456	+	1.23228i	0.252855	+	0.967504i	$0.418631\pi$
$593$	−17.3251	+	30.0079i	−0.711456	+	1.23228i	−0.964311	+	0.264773i	$0.914703\pi$
$594$		0			0
$595$	8.11292	+	4.16749i	0.332597	+	0.170850i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	19.3419	+	33.5012i	0.790289	+	1.36882i	0.925788	+	0.378044i	$0.123403\pi$
$599$	19.3419	+	33.5012i	0.790289	+	1.36882i	−0.135498	+	0.990778i	$0.543264\pi$
$600$		0			0
$601$	−16.2302			−0.662044			−0.331022	−	0.943623i	$-0.607393\pi$
$601$	−16.2302			−0.662044			−0.331022	+	0.943623i	$0.607393\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.7839	+	18.6782i	0.438427	+	0.759378i
$606$		0			0
$607$	0.800714	−	1.38688i	0.0325000	−	0.0562916i	−0.849318	−	0.527882i	$-0.822986\pi$
$607$	0.800714	−	1.38688i	0.0325000	−	0.0562916i	0.881818	+	0.471590i	$0.156320\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.96633	+	8.60193i	−0.200916	+	0.347997i
$612$		0			0
$613$	−3.16343	−	5.47922i	−0.127770	−	0.221304i	0.795043	−	0.606554i	$-0.207448\pi$
$613$	−3.16343	−	5.47922i	−0.127770	−	0.221304i	−0.922812	+	0.385250i	$0.874115\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−33.6545			−1.35488			−0.677440	−	0.735578i	$-0.736911\pi$
$617$	−33.6545			−1.35488			−0.677440	+	0.735578i	$0.736911\pi$
$618$		0			0
$619$	20.6204	+	35.7157i	0.828806	+	1.43553i	0.898976	+	0.437999i	$0.144313\pi$
$619$	20.6204	+	35.7157i	0.828806	+	1.43553i	−0.0701697	+	0.997535i	$0.522354\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−28.0212	+	18.0661i	−1.12265	+	0.723803i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	30.9097			1.23245
$630$		0			0
$631$	37.4897			1.49244			0.746221	−	0.665698i	$-0.231866\pi$
$631$	37.4897			1.49244			0.746221	+	0.665698i	$0.231866\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.61292	+	4.52571i	−0.103691	+	0.179597i
$636$		0			0
$637$	−27.3441	−	38.1627i	−1.08341	−	1.51206i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.07925	+	1.86931i	0.0426277	+	0.0738333i	0.886552	−	0.462629i	$-0.153094\pi$
$641$	1.07925	+	1.86931i	0.0426277	+	0.0738333i	−0.843924	+	0.536462i	$0.819760\pi$
$642$		0			0
$643$	−29.2258			−1.15255			−0.576277	−	0.817254i	$-0.695495\pi$
$643$	−29.2258			−1.15255			−0.576277	+	0.817254i	$0.695495\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4.31439	−	7.47274i	−0.169616	−	0.293784i	0.768669	−	0.639647i	$-0.220919\pi$
$647$	−4.31439	−	7.47274i	−0.169616	−	0.293784i	−0.938285	+	0.345863i	$0.887586\pi$
$648$		0			0
$649$	0.836572	−	1.44899i	0.0328383	−	0.0568776i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	8.91145	−	15.4351i	0.348732	−	0.604021i	−0.637293	−	0.770622i	$-0.719946\pi$
$653$	8.91145	−	15.4351i	0.348732	−	0.604021i	0.986024	+	0.166601i	$0.0532790\pi$
$654$		0			0
$655$	−8.70682	−	15.0806i	−0.340203	−	0.589250i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.07608			−0.236691			−0.118345	−	0.992973i	$-0.537759\pi$
$659$	−6.07608			−0.236691			−0.118345	+	0.992973i	$0.537759\pi$
$660$		0			0
$661$	5.97167	+	10.3432i	0.232271	+	0.402305i	0.958476	−	0.285173i	$-0.0920510\pi$
$661$	5.97167	+	10.3432i	0.232271	+	0.402305i	−0.726205	+	0.687478i	$0.758718\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.163428	+	3.32834i	0.00633747	+	0.129068i
$666$		0			0
$667$	−28.6302	+	49.5889i	−1.10856	+	1.92009i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.48098			0.0571726
$672$		0			0
$673$	−43.6014			−1.68071			−0.840356	−	0.542035i	$-0.817654\pi$
$673$	−43.6014			−1.68071			−0.840356	+	0.542035i	$0.817654\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.39243	−	7.60791i	0.168815	−	0.292396i	−0.769189	−	0.639022i	$-0.779339\pi$
$677$	4.39243	−	7.60791i	0.168815	−	0.292396i	0.938003	+	0.346626i	$0.112673\pi$
$678$		0			0
$679$	−30.9570	−	15.9022i	−1.18802	−	0.610270i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.39680	−	11.0796i	−0.244767	−	0.423948i	0.717299	−	0.696765i	$-0.245378\pi$
$683$	−6.39680	−	11.0796i	−0.244767	−	0.423948i	−0.962066	+	0.272817i	$0.912045\pi$
$684$		0			0
$685$	−13.9663			−0.533626
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−23.1204	−	40.0458i	−0.880819	−	1.52562i
$690$		0			0
$691$	−0.826856	+	1.43216i	−0.0314551	+	0.0544818i	−0.881324	−	0.472512i	$-0.843347\pi$
$691$	−0.826856	+	1.43216i	−0.0314551	+	0.0544818i	0.849869	+	0.526993i	$0.176681\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.52437	−	13.0326i	0.285416	−	0.494354i
$696$		0			0
$697$	−17.6258	−	30.5288i	−0.667625	−	1.15636i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	23.0444			0.870374			0.435187	−	0.900340i	$-0.356682\pi$
$701$	23.0444			0.870374			0.435187	+	0.900340i	$0.356682\pi$
$702$		0			0
$703$	5.64659	+	9.78018i	0.212965	+	0.368867i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.44294	+	29.3865i	0.0542672	+	1.10519i
$708$		0			0
$709$	−10.2122	+	17.6880i	−0.383526	+	0.664286i	−0.991563	−	0.129622i	$-0.958624\pi$
$709$	−10.2122	+	17.6880i	−0.383526	+	0.664286i	0.608038	+	0.793908i	$0.291957\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7.96633			−0.298341
$714$		0			0
$715$	−38.2746			−1.43139
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.81220	−	3.13883i	0.0675838	−	0.117059i	−0.830253	−	0.557386i	$-0.811804\pi$
$719$	1.81220	−	3.13883i	0.0675838	−	0.117059i	0.897837	+	0.440327i	$0.145138\pi$
$720$		0			0
$721$	4.60678	−	2.97012i	0.171565	−	0.110613i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.59390	−	6.22481i	−0.133474	−	0.231184i
$726$		0			0
$727$	53.1311			1.97053			0.985263	−	0.171049i	$-0.0547157\pi$
$727$	53.1311			1.97053			0.985263	+	0.171049i	$0.0547157\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.66561	−	13.2772i	−0.283523	−	0.491076i
$732$		0			0
$733$	16.8344	−	29.1580i	0.621792	−	1.07698i	−0.367360	−	0.930079i	$-0.619738\pi$
$733$	16.8344	−	29.1580i	0.621792	−	1.07698i	0.989152	−	0.146897i	$-0.0469285\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−19.1373	+	33.1468i	−0.704931	+	1.22098i
$738$		0			0
$739$	−7.11510	−	12.3237i	−0.261733	−	0.453335i	0.704969	−	0.709238i	$-0.250961\pi$
$739$	−7.11510	−	12.3237i	−0.261733	−	0.453335i	−0.966703	+	0.255902i	$0.917627\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.78486			0.102166			0.0510832	−	0.998694i	$-0.483733\pi$
$743$	2.78486			0.102166			0.0510832	+	0.998694i	$0.483733\pi$
$744$		0			0
$745$	3.00000	+	5.19615i	0.109911	+	0.190372i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.37243	−	2.81904i	0.159765	−	0.103005i
$750$		0			0
$751$	14.4663	−	25.0564i	0.527884	−	0.914322i	−0.471588	−	0.881819i	$-0.656319\pi$
$751$	14.4663	−	25.0564i	0.527884	−	0.914322i	0.999472	−	0.0325024i	$-0.0103477\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21.0868			−0.767426
$756$		0			0
$757$	−15.7219			−0.571421			−0.285711	−	0.958316i	$-0.592230\pi$
$757$	−15.7219			−0.571421			−0.285711	+	0.958316i	$0.592230\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.59390	+	16.6171i	−0.347779	+	0.602370i	−0.985855	−	0.167603i	$-0.946397\pi$
$761$	−9.59390	+	16.6171i	−0.347779	+	0.602370i	0.638076	+	0.769973i	$0.279731\pi$
$762$		0			0
$763$	1.12539	+	22.9194i	0.0407417	+	0.829737i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.983164	+	1.70289i	0.0355000	+	0.0614878i
$768$		0			0
$769$	26.2302			0.945885			0.472943	−	0.881093i	$-0.343192\pi$
$769$	26.2302			0.945885			0.472943	+	0.881093i	$0.343192\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	13.9158	+	24.1029i	0.500517	+	0.866921i	1.00000		0.000597291i	$0.000190124\pi$
$773$	13.9158	+	24.1029i	0.500517	+	0.866921i	−0.499483	+	0.866324i	$0.666477\pi$
$774$		0			0
$775$	0.500000	−	0.866025i	0.0179605	−	0.0311086i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.43978	−	11.1540i	0.230729	−	0.399634i
$780$		0			0
$781$	33.4043	+	57.8580i	1.19530	+	2.07032i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8.89461			−0.317462
$786$		0			0
$787$	17.6395	+	30.5525i	0.628779	+	1.08908i	0.987797	+	0.155747i	$0.0497784\pi$
$787$	17.6395	+	30.5525i	0.628779	+	1.08908i	−0.359018	+	0.933331i	$0.616888\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.37996	+	0.708866i	0.0490657	+	0.0252044i
$792$		0			0
$793$	−0.870245	+	1.50731i	−0.0309033	+	0.0535261i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	40.1399			1.42183			0.710914	−	0.703279i	$-0.248282\pi$
$797$	40.1399			1.42183			0.710914	+	0.703279i	$0.248282\pi$
$798$		0			0
$799$	5.10539			0.180616
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.75733	−	4.77583i	0.0973039	−	0.168535i
$804$		0			0
$805$	1.03367	+	21.0516i	0.0364322	+	0.741970i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−9.77853	−	16.9369i	−0.343795	−	0.595470i	0.641339	−	0.767257i	$-0.278379\pi$
$809$	−9.77853	−	16.9369i	−0.343795	−	0.595470i	−0.985134	+	0.171787i	$0.945046\pi$
$810$		0			0
$811$	4.33559			0.152243			0.0761217	−	0.997099i	$-0.475746\pi$
$811$	4.33559			0.152243			0.0761217	+	0.997099i	$0.475746\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.18780	+	8.98553i	0.181721	+	0.314749i
$816$		0			0
$817$	2.80071	−	4.85098i	0.0979846	−	0.169714i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−21.3388	+	36.9598i	−0.744728	+	1.28991i	0.205594	+	0.978637i	$0.434087\pi$
$821$	−21.3388	+	36.9598i	−0.744728	+	1.28991i	−0.950322	+	0.311269i	$0.899246\pi$
$822$		0			0
$823$	−21.0244	−	36.4153i	−0.732863	−	1.26936i	−0.955654	−	0.294490i	$-0.904850\pi$
$823$	−21.0244	−	36.4153i	−0.732863	−	1.26936i	0.222791	−	0.974866i	$-0.428483\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15.8566			−0.551387			−0.275693	−	0.961246i	$-0.588907\pi$
$827$	−15.8566			−0.551387			−0.275693	+	0.961246i	$0.588907\pi$
$828$		0			0
$829$	−15.8219	−	27.4044i	−0.549518	−	0.951793i	−0.998308	−	0.0581556i	$-0.981478\pi$
$829$	−15.8219	−	27.4044i	−0.549518	−	0.951793i	0.448790	−	0.893637i	$-0.351855\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−9.96018	+	21.9797i	−0.345100	+	0.761551i
$834$		0			0
$835$	0.983164	−	1.70289i	0.0340238	−	0.0589309i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	38.0824			1.31475			0.657375	−	0.753563i	$-0.271667\pi$
$839$	38.0824			1.31475			0.657375	+	0.753563i	$0.271667\pi$
$840$		0			0
$841$	22.6644			0.781531
$842$		0			0
$843$		0			0
$844$		0			0
$845$	15.9907	−	27.6967i	0.550097	−	0.952795i
$846$		0			0
$847$	−47.9592	+	30.9207i	−1.64790	+	1.06245i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	35.7143	+	61.8591i	1.22427	+	2.12050i
$852$		0			0
$853$	45.1399			1.54556			0.772780	−	0.634674i	$-0.218866\pi$
$853$	45.1399			1.54556			0.772780	+	0.634674i	$0.218866\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	11.7954	+	20.4302i	0.402922	+	0.697882i	0.994077	−	0.108676i	$-0.0346612\pi$
$857$	11.7954	+	20.4302i	0.402922	+	0.697882i	−0.591155	+	0.806558i	$0.701328\pi$
$858$		0			0
$859$	15.6151	−	27.0462i	0.532780	−	0.922803i	−0.466487	−	0.884528i	$-0.654480\pi$
$859$	15.6151	−	27.0462i	0.532780	−	0.922803i	0.999267	−	0.0382747i	$-0.0121862\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8.00437	−	13.8640i	0.272472	−	0.471935i	−0.697022	−	0.717049i	$-0.745492\pi$
$863$	8.00437	−	13.8640i	0.272472	−	0.471935i	0.969494	+	0.245114i	$0.0788255\pi$
$864$		0			0
$865$	−0.740489	−	1.28257i	−0.0251774	−	0.0436085i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	66.3994			2.25245
$870$		0			0
$871$	−22.4907	−	38.9550i	−0.762068	−	1.31994i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.35341	−	1.20891i	−0.0795597	−	0.0408687i
$876$		0			0
$877$	0.187796	−	0.325271i	0.00634141	−	0.0109836i	−0.862837	−	0.505482i	$-0.831315\pi$
$877$	0.187796	−	0.325271i	0.00634141	−	0.0109836i	0.869179	+	0.494498i	$0.164648\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−31.7892			−1.07101			−0.535503	−	0.844533i	$-0.679878\pi$
$881$	−31.7892			−1.07101			−0.535503	+	0.844533i	$0.679878\pi$
$882$		0			0
$883$	21.0638			0.708853			0.354427	−	0.935084i	$-0.384676\pi$
$883$	21.0638			0.708853			0.354427	+	0.935084i	$0.384676\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−14.3631	+	24.8777i	−0.482267	+	0.835310i	−0.999793	−	0.0203572i	$-0.993520\pi$
$887$	−14.3631	+	24.8777i	−0.482267	+	0.835310i	0.517526	+	0.855667i	$0.326853\pi$
$888$		0			0
$889$	−12.2985	−	6.31758i	−0.412479	−	0.211885i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.932654	+	1.61540i	0.0312101	+	0.0540575i
$894$		0			0
$895$	−18.3082			−0.611977
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.59390	+	6.22481i	0.119863	+	0.207609i
$900$		0			0
$901$	−11.8839	+	20.5836i	−0.395911	+	0.685738i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.33657	−	16.1714i	0.310358	−	0.537556i
$906$		0			0
$907$	1.48316	+	2.56891i	0.0492476	+	0.0852994i	0.889598	−	0.456744i	$-0.150984\pi$
$907$	1.48316	+	2.56891i	0.0492476	+	0.0852994i	−0.840351	+	0.542043i	$0.817651\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−15.5634			−0.515638			−0.257819	−	0.966193i	$-0.583004\pi$
$911$	−15.5634			−0.515638			−0.257819	+	0.966193i	$0.583004\pi$
$912$		0			0
$913$	42.4043	+	73.4465i	1.40338	+	2.43072i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	38.7219	−	24.9651i	1.27871	−	0.824422i
$918$		0			0
$919$	−12.8419	+	22.2429i	−0.423616	+	0.733724i	−0.996290	−	0.0860589i	$-0.972573\pi$
$919$	−12.8419	+	22.2429i	−0.423616	+	0.733724i	0.572674	+	0.819783i	$0.305906\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−78.5155			−2.58437
$924$		0			0
$925$	−8.96633			−0.294811
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.36806	+	4.10160i	−0.0776935	+	0.134569i	−0.902254	−	0.431204i	$-0.858089\pi$
$929$	−2.36806	+	4.10160i	−0.0776935	+	0.134569i	0.824561	+	0.565773i	$0.191422\pi$
$930$		0			0
$931$	−8.77416	+	0.863740i	−0.287562	+	0.0283079i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	9.83657	+	17.0374i	0.321690	+	0.557184i
$936$		0			0
$937$	−23.3419			−0.762547			−0.381274	−	0.924462i	$-0.624514\pi$
$937$	−23.3419			−0.762547			−0.381274	+	0.924462i	$0.624514\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.22900	−	14.2530i	−0.268258	−	0.464636i	0.700154	−	0.713992i	$-0.253115\pi$
$941$	−8.22900	−	14.2530i	−0.268258	−	0.464636i	−0.968412	+	0.249356i	$0.919781\pi$
$942$		0			0
$943$	40.7312	−	70.5485i	1.32639	−	2.29737i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−11.6183	+	20.1234i	−0.377543	+	0.653923i	−0.990704	−	0.136034i	$-0.956564\pi$
$947$	−11.6183	+	20.1234i	−0.377543	+	0.653923i	0.613161	+	0.789958i	$0.289898\pi$
$948$		0			0
$949$	3.24049	+	5.61269i	0.105191	+	0.182196i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.37559			−0.0769529			−0.0384765	−	0.999260i	$-0.512250\pi$
$953$	−2.37559			−0.0769529			−0.0384765	+	0.999260i	$0.512250\pi$
$954$		0			0
$955$	−3.15412	−	5.46310i	−0.102065	−	0.176782i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.81220	−	36.9070i	−0.0585191	−	1.19179i
$960$		0			0
$961$	15.0000	−	25.9808i	0.483871	−	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−22.8990			−0.737144
$966$		0			0
$967$	36.6121			1.17737			0.588683	−	0.808364i	$-0.299647\pi$
$967$	36.6121			1.17737			0.588683	+	0.808364i	$0.299647\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	18.9327	−	32.7923i	0.607578	−	1.05236i	−0.384061	−	0.923308i	$-0.625475\pi$
$971$	18.9327	−	32.7923i	0.607578	−	1.05236i	0.991638	−	0.129048i	$-0.0411920\pi$
$972$		0			0
$973$	35.4158	+	18.1926i	1.13538	+	0.583228i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−29.0699	−	50.3506i	−0.930030	−	1.61086i	−0.783265	−	0.621687i	$-0.786447\pi$
$977$	−29.0699	−	50.3506i	−0.930030	−	1.61086i	−0.146764	−	0.989171i	$-0.546886\pi$
$978$		0			0
$979$	−71.9140			−2.29838
$980$		0			0
$981$		0			0
$982$		0			0
$983$	12.6900	+	21.9797i	0.404748	+	0.701043i	0.994292	−	0.106693i	$-0.0340261\pi$
$983$	12.6900	+	21.9797i	0.404748	+	0.701043i	−0.589545	+	0.807736i	$0.700693\pi$
$984$		0			0
$985$	7.27635	−	12.6030i	0.231844	−	0.401565i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	17.7143	−	30.6822i	0.563283	−	0.975636i
$990$		0			0
$991$	−15.9000	−	27.5395i	−0.505079	−	0.874822i	−0.999983	−	0.00587449i	$-0.998130\pi$
$991$	−15.9000	−	27.5395i	−0.505079	−	0.874822i	0.494904	−	0.868948i	$-0.335203\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	17.0868			0.541687
$996$		0			0
$997$	−27.0129	−	46.7877i	−0.855506	−	1.48178i	−0.876174	−	0.481994i	$-0.839913\pi$
$997$	−27.0129	−	46.7877i	−0.855506	−	1.48178i	0.0206680	−	0.999786i	$-0.493421\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1260.2.s.h.541.3	yes	6
3.2	odd	2		1260.2.s.g.541.3	yes	6
7.2	even	3		8820.2.a.bn.1.1		3
7.4	even	3	inner	1260.2.s.h.361.3	yes	6
7.5	odd	6		8820.2.a.bp.1.1		3
21.2	odd	6		8820.2.a.bq.1.3		3
21.5	even	6		8820.2.a.bo.1.3		3
21.11	odd	6		1260.2.s.g.361.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.s.g.361.3	✓	6	21.11	odd	6
1260.2.s.g.541.3	yes	6	3.2	odd	2
1260.2.s.h.361.3	yes	6	7.4	even	3	inner
1260.2.s.h.541.3	yes	6	1.1	even	1	trivial
8820.2.a.bn.1.1		3	7.2	even	3
8820.2.a.bo.1.3		3	21.5	even	6
8820.2.a.bp.1.1		3	7.5	odd	6
8820.2.a.bq.1.3		3	21.2	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1260.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +(2.35341 + 1.20891i) q^{7} +(2.85341 + 4.94225i) q^{11} -6.70682 q^{13} +(1.72365 + 2.98545i) q^{17} +(-0.629755 + 1.09077i) q^{19} +(-3.98316 + 6.89904i) q^{23} +(-0.500000 - 0.866025i) q^{25} +7.18780 q^{29} +(0.500000 + 0.866025i) q^{31} +(2.22365 - 1.43366i) q^{35} +(4.48316 - 7.76507i) q^{37} -10.2258 q^{41} -4.44731 q^{43} +(0.740489 - 1.28257i) q^{47} +(4.07706 + 5.69013i) q^{49} +(3.44731 + 5.97091i) q^{53} +5.70682 q^{55} +(-0.146592 - 0.253904i) q^{59} +(0.129755 - 0.224743i) q^{61} +(-3.35341 + 5.80827i) q^{65} +(3.35341 + 5.80827i) q^{67} +11.7068 q^{71} +(-0.483164 - 0.836864i) q^{73} +(0.740489 + 15.0806i) q^{77} +(5.81755 - 10.0763i) q^{79} +14.8609 q^{83} +3.44731 q^{85} +(-6.30071 + 10.9132i) q^{89} +(-15.7839 - 8.10795i) q^{91} +(0.629755 + 1.09077i) q^{95} -13.1541 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5} - q^{7} + 2 q^{11} - 10 q^{13} - 2 q^{17} - q^{19} - 6 q^{23} - 3 q^{25} + 24 q^{29} + 3 q^{31} + q^{35} + 9 q^{37} - 20 q^{41} - 2 q^{43} + 10 q^{47} - 3 q^{49} - 4 q^{53} + 4 q^{55} - 16 q^{59}+ \cdots - 24 q^{97}+O(q^{100}) \) 6 * q + 3 * q^5 - q^7 + 2 * q^11 - 10 * q^13 - 2 * q^17 - q^19 - 6 * q^23 - 3 * q^25 + 24 * q^29 + 3 * q^31 + q^35 + 9 * q^37 - 20 * q^41 - 2 * q^43 + 10 * q^47 - 3 * q^49 - 4 * q^53 + 4 * q^55 - 16 * q^59 - 2 * q^61 - 5 * q^65 + 5 * q^67 + 40 * q^71 + 15 * q^73 + 10 * q^77 + 13 * q^79 + 4 * q^83 - 4 * q^85 + 2 * q^89 - 37 * q^91 + q^95 - 24 * q^97

Introduction

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