LMFDB - Embedded newform 567.2.i.a.215.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [567,2,Mod(215,567)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("567.215"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 567 = 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	567.i (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,4,0,0,-5] 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:	$4.52751779461$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 189)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			215.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	567.215
Dual form			567.2.i.a.269.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{4} +(-2.50000 + 0.866025i) q^{7} +(6.00000 + 3.46410i) q^{13} +4.00000 q^{16} +(4.50000 + 2.59808i) q^{19} +(2.50000 - 4.33013i) q^{25} +(-5.00000 + 1.73205i) q^{28} -1.73205i q^{31} +(-5.00000 + 8.66025i) q^{37} +(-6.50000 - 11.2583i) q^{43} +(5.50000 - 4.33013i) q^{49} +(12.0000 + 6.92820i) q^{52} +8.66025i q^{61} +8.00000 q^{64} -16.0000 q^{67} +(13.5000 - 7.79423i) q^{73} +(9.00000 + 5.19615i) q^{76} -4.00000 q^{79} +(-18.0000 - 3.46410i) q^{91} +(-16.5000 + 9.52628i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} - 5 q^{7} + 12 q^{13} + 8 q^{16} + 9 q^{19} + 5 q^{25} - 10 q^{28} - 10 q^{37} - 13 q^{43} + 11 q^{49} + 24 q^{52} + 16 q^{64} - 32 q^{67} + 27 q^{73} + 18 q^{76} - 8 q^{79} - 36 q^{91} - 33 q^{97}+O(q^{100}) $ 2 * q + 4 * q^4 - 5 * q^7 + 12 * q^13 + 8 * q^16 + 9 * q^19 + 5 * q^25 - 10 * q^28 - 10 * q^37 - 13 * q^43 + 11 * q^49 + 24 * q^52 + 16 * q^64 - 32 * q^67 + 27 * q^73 + 18 * q^76 - 8 * q^79 - 36 * q^91 - 33 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/567\mathbb{Z}\right)^\times$.

$n$	$325$	$407$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$

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		1.00000			$0$
$2$		0			0		−1.00000			$\pi$
$3$		0			0
$3$		0			0
$4$	2.00000			1.00000
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	6.00000	+	3.46410i	1.66410	+	0.960769i	0.970725	+	0.240192i	$0.0772105\pi$
$13$	6.00000	+	3.46410i	1.66410	+	0.960769i	0.693375	+	0.720577i	$0.256123\pi$
$14$		0			0
$15$		0			0
$16$	4.00000			1.00000
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	4.50000	+	2.59808i	1.03237	+	0.596040i	0.917663	−	0.397360i	$-0.130073\pi$
$19$	4.50000	+	2.59808i	1.03237	+	0.596040i	0.114708	+	0.993399i	$0.463407\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$	−5.00000	+	1.73205i	−0.944911	+	0.327327i
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$		−	1.73205i		−	0.311086i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$		−	1.73205i		−	0.311086i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.00000	+	8.66025i	−0.821995	+	1.42374i	0.0821995	+	0.996616i	$0.473806\pi$
$37$	−5.00000	+	8.66025i	−0.821995	+	1.42374i	−0.904194	+	0.427121i	$0.859528\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	−6.50000	−	11.2583i	−0.991241	−	1.71688i	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−6.50000	−	11.2583i	−0.991241	−	1.71688i	−0.381246	−	0.924473i	$-0.624505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$		0			0
$52$	12.0000	+	6.92820i	1.66410	+	0.960769i
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$			8.66025i			1.10883i	0.832240	+	0.554416i	$0.187058\pi$
$61$			8.66025i			1.10883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$		0			0
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	−16.0000			−1.95471			−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000			−1.95471			−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	13.5000	−	7.79423i	1.58006	−	0.912245i	0.585206	−	0.810885i	$-0.301014\pi$
$73$	13.5000	−	7.79423i	1.58006	−	0.912245i	0.994850	−	0.101361i	$-0.0323196\pi$
$74$		0			0
$75$		0			0
$76$	9.00000	+	5.19615i	1.03237	+	0.596040i
$77$		0			0
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$83$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	−18.0000	−	3.46410i	−1.88691	−	0.363137i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−16.5000	+	9.52628i	−1.67532	+	0.967247i	−0.710742	+	0.703452i	$0.751641\pi$
$97$	−16.5000	+	9.52628i	−1.67532	+	0.967247i	−0.964579	+	0.263795i	$0.915026\pi$
$98$		0			0
$99$		0			0
$100$	5.00000	−	8.66025i	0.500000	−	0.866025i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−3.00000	+	1.73205i	−0.295599	+	0.170664i	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−3.00000	+	1.73205i	−0.295599	+	0.170664i	0.344865	+	0.938652i	$0.387925\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$	−9.50000	−	16.4545i	−0.909935	−	1.57605i	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−9.50000	−	16.4545i	−0.909935	−	1.57605i	−0.0957826	−	0.995402i	$-0.530535\pi$
$110$		0			0
$111$		0			0
$112$	−10.0000	+	3.46410i	−0.944911	+	0.327327i
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		−	3.46410i		−	0.311086i
$125$		0			0
$126$		0			0
$127$	−1.00000			−0.0887357			−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000			−0.0887357			−0.0443678	+	0.999015i	$0.514127\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	−13.5000	−	2.59808i	−1.17060	−	0.225282i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	−15.0000	−	8.66025i	−1.27228	−	0.734553i	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−15.0000	−	8.66025i	−1.27228	−	0.734553i	−0.975417	+	0.220366i	$0.929275\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−10.0000	+	17.3205i	−0.821995	+	1.42374i
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	11.5000	−	19.9186i	0.935857	−	1.62095i	0.162758	−	0.986666i	$-0.447961\pi$
$151$	11.5000	−	19.9186i	0.935857	−	1.62095i	0.773099	−	0.634285i	$-0.218706\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	20.7846i		−	1.65879i	−0.558661	−	0.829396i	$-0.688685\pi$
$157$		−	20.7846i		−	1.65879i	0.558661	−	0.829396i	$-0.311315\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−8.50000	+	14.7224i	−0.665771	+	1.15315i	0.313304	+	0.949653i	$0.398564\pi$
$163$	−8.50000	+	14.7224i	−0.665771	+	1.15315i	−0.979076	+	0.203497i	$0.934769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	17.5000	+	30.3109i	1.34615	+	2.33161i
$170$		0			0
$171$		0			0
$172$	−13.0000	−	22.5167i	−0.991241	−	1.71688i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−2.50000	+	12.9904i	−0.188982	+	0.981981i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$			25.9808i			1.93113i	0.260153	+	0.965567i	$0.416227\pi$
$181$			25.9808i			1.93113i	−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$	11.0000	−	8.66025i	0.785714	−	0.618590i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	19.5000	−	11.2583i	1.38232	−	0.798082i	0.389885	−	0.920864i	$-0.372515\pi$
$199$	19.5000	−	11.2583i	1.38232	−	0.798082i	0.992434	+	0.122782i	$0.0391815\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	24.0000	+	13.8564i	1.66410	+	0.960769i
$209$		0			0
$210$		0			0
$211$	14.5000	−	25.1147i	0.998221	−	1.72897i	0.447478	−	0.894295i	$-0.352322\pi$
$211$	14.5000	−	25.1147i	0.998221	−	1.72897i	0.550743	−	0.834675i	$-0.314345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.50000	+	4.33013i	0.101827	+	0.293948i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−9.00000	+	5.19615i	−0.602685	+	0.347960i	−0.770097	−	0.637927i	$-0.779792\pi$
$223$	−9.00000	+	5.19615i	−0.602685	+	0.347960i	0.167412	+	0.985887i	$0.446459\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	7.50000	−	4.33013i	0.495614	−	0.286143i	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	7.50000	−	4.33013i	0.495614	−	0.286143i	0.726900	+	0.686743i	$0.240960\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	1.50000	+	0.866025i	0.0966235	+	0.0557856i	0.547533	−	0.836784i	$-0.315567\pi$
$241$	1.50000	+	0.866025i	0.0966235	+	0.0557856i	−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$		0			0
$244$			17.3205i			1.10883i
$245$		0			0
$246$		0			0
$247$	18.0000	+	31.1769i	1.14531	+	1.98374i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	5.00000	−	25.9808i	0.310685	−	1.61437i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−32.0000			−1.95471
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−28.5000	−	16.4545i	−1.73125	−	0.999539i	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−28.5000	−	16.4545i	−1.73125	−	0.999539i	−0.850439	−	0.526073i	$-0.823664\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.50000	+	4.33013i	−0.150210	+	0.260172i	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−2.50000	+	4.33013i	−0.150210	+	0.260172i	0.781094	+	0.624413i	$0.214662\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$			22.5167i			1.33848i	0.743048	+	0.669238i	$0.233379\pi$
$283$			22.5167i			1.33848i	−0.743048	+	0.669238i	$0.766621\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$	27.0000	−	15.5885i	1.58006	−	0.912245i
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	26.0000	+	22.5167i	1.49862	+	1.29784i
$302$		0			0
$303$		0			0
$304$	18.0000	+	10.3923i	1.03237	+	0.596040i
$305$		0			0
$306$		0			0
$307$			29.4449i			1.68051i	0.542194	+	0.840254i	$0.317594\pi$
$307$			29.4449i			1.68051i	−0.542194	+	0.840254i	$0.682406\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	5.19615i		−	0.293704i	−0.989158	−	0.146852i	$-0.953086\pi$
$313$		−	5.19615i		−	0.293704i	0.989158	−	0.146852i	$-0.0469141\pi$
$314$		0			0
$315$		0			0
$316$	−8.00000			−0.450035
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	30.0000	−	17.3205i	1.66410	−	0.960769i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.00000			0.0549650			0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000			0.0549650			0.0274825	+	0.999622i	$0.491251\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.50000	−	4.33013i	0.136184	−	0.235877i	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	2.50000	−	4.33013i	0.136184	−	0.235877i	0.926049	+	0.377403i	$0.123183\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	25.5000	−	14.7224i	1.36498	−	0.788074i	0.374701	−	0.927146i	$-0.377745\pi$
$349$	25.5000	−	14.7224i	1.36498	−	0.788074i	0.990282	+	0.139072i	$0.0444119\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	4.00000	+	6.92820i	0.210526	+	0.364642i
$362$		0			0
$363$		0			0
$364$	−36.0000	−	6.92820i	−1.88691	−	0.363137i
$365$		0			0
$366$		0			0
$367$	−13.5000	−	7.79423i	−0.704694	−	0.406855i	0.104399	−	0.994535i	$-0.466708\pi$
$367$	−13.5000	−	7.79423i	−0.704694	−	0.406855i	−0.809093	+	0.587680i	$0.800041\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−8.00000			−0.410932			−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000			−0.410932			−0.205466	+	0.978664i	$0.565871\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−33.0000	+	19.0526i	−1.67532	+	0.967247i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	34.5000	+	19.9186i	1.73151	+	0.999685i	0.878300	+	0.478110i	$0.158678\pi$
$397$	34.5000	+	19.9186i	1.73151	+	0.999685i	0.853206	+	0.521575i	$0.174655\pi$
$398$		0			0
$399$		0			0
$400$	10.0000	−	17.3205i	0.500000	−	0.866025i
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	6.00000	−	10.3923i	0.298881	−	0.517678i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	13.8564i		−	0.685155i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$		−	13.8564i		−	0.685155i	0.939490	−	0.342578i	$-0.111300\pi$
$410$		0			0
$411$		0			0
$412$	−6.00000	+	3.46410i	−0.295599	+	0.170664i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$	9.50000	+	16.4545i	0.463002	+	0.801942i	0.999109	−	0.0422075i	$-0.0134391\pi$
$421$	9.50000	+	16.4545i	0.463002	+	0.801942i	−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−7.50000	−	21.6506i	−0.362950	−	1.04775i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$		−	19.0526i		−	0.915608i	−0.889053	−	0.457804i	$-0.848636\pi$
$433$		−	19.0526i		−	0.915608i	0.889053	−	0.457804i	$-0.151364\pi$
$434$		0			0
$435$		0			0
$436$	−19.0000	−	32.9090i	−0.909935	−	1.57605i
$437$		0			0
$438$		0			0
$439$			31.1769i			1.48799i	0.668184	+	0.743996i	$0.267072\pi$
$439$			31.1769i			1.48799i	−0.668184	+	0.743996i	$0.732928\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−20.0000	+	6.92820i	−0.944911	+	0.327327i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	31.0000			1.45012			0.725059	−	0.688686i	$-0.241812\pi$
$457$	31.0000			1.45012			0.725059	+	0.688686i	$0.241812\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$	−11.5000	+	19.9186i	−0.534450	+	0.925695i	0.464739	+	0.885448i	$0.346148\pi$
$463$	−11.5000	+	19.9186i	−0.534450	+	0.925695i	−0.999190	+	0.0402476i	$0.987185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$	40.0000	−	13.8564i	1.84703	−	0.639829i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	22.5000	−	12.9904i	1.03237	−	0.596040i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	−60.0000	+	34.6410i	−2.73576	+	1.57949i
$482$		0			0
$483$		0			0
$484$	−11.0000	−	19.0526i	−0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$	−12.5000	−	21.6506i	−0.566429	−	0.981084i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−12.5000	−	21.6506i	−0.566429	−	0.981084i	0.430486	−	0.902597i	$-0.358342\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		−	6.92820i		−	0.311086i
$497$		0			0
$498$		0			0
$499$	5.50000	+	9.52628i	0.246214	+	0.426455i	0.962472	−	0.271380i	$-0.0874801\pi$
$499$	5.50000	+	9.52628i	0.246214	+	0.426455i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−2.00000			−0.0887357
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−27.0000	+	31.1769i	−1.19441	+	1.37919i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	−39.0000	−	22.5167i	−1.70535	−	0.984585i	−0.940129	−	0.340818i	$-0.889296\pi$
$523$	−39.0000	−	22.5167i	−1.70535	−	0.984585i	−0.765222	−	0.643767i	$-0.777371\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	+	19.9186i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−27.0000	−	5.19615i	−1.17060	−	0.225282i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−14.5000	+	25.1147i	−0.623404	+	1.07977i	0.365444	+	0.930834i	$0.380917\pi$
$541$	−14.5000	+	25.1147i	−0.623404	+	1.07977i	−0.988847	+	0.148933i	$0.952416\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−20.5000	−	35.5070i	−0.876517	−	1.51817i	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−20.5000	−	35.5070i	−0.876517	−	1.51817i	−0.0213785	−	0.999771i	$-0.506805\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	10.0000	−	3.46410i	0.425243	−	0.147309i
$554$		0			0
$555$		0			0
$556$	−30.0000	−	17.3205i	−1.27228	−	0.734553i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		−	90.0666i		−	3.80941i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−31.0000			−1.29731			−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000			−1.29731			−0.648655	+	0.761083i	$0.724668\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−12.0000	+	6.92820i	−0.499567	+	0.288425i	−0.728535	−	0.685009i	$-0.759798\pi$
$577$	−12.0000	+	6.92820i	−0.499567	+	0.288425i	0.228968	+	0.973434i	$0.426465\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$587$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$588$		0			0
$589$	4.50000	−	7.79423i	0.185419	−	0.321156i
$590$		0			0
$591$		0			0
$592$	−20.0000	+	34.6410i	−0.821995	+	1.42374i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−37.5000	+	21.6506i	−1.52966	+	0.883148i	−0.530281	+	0.847822i	$0.677914\pi$
$601$	−37.5000	+	21.6506i	−1.52966	+	0.883148i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$		0			0
$603$		0			0
$604$	23.0000	−	39.8372i	0.935857	−	1.62095i
$605$		0			0
$606$		0			0
$607$	−4.50000	+	2.59808i	−0.182649	+	0.105453i	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−4.50000	+	2.59808i	−0.182649	+	0.105453i	0.405887	+	0.913923i	$0.366962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	18.5000	+	32.0429i	0.747208	+	1.29420i	0.949156	+	0.314806i	$0.101939\pi$
$613$	18.5000	+	32.0429i	0.747208	+	1.29420i	−0.201948	+	0.979396i	$0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$617$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$	33.0000	+	19.0526i	1.32638	+	0.765787i	0.984738	−	0.174042i	$-0.0556830\pi$
$619$	33.0000	+	19.0526i	1.32638	+	0.765787i	0.341644	+	0.939829i	$0.389016\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		−	41.5692i		−	1.65879i
$629$		0			0
$630$		0			0
$631$	−1.00000			−0.0398094			−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000			−0.0398094			−0.0199047	+	0.999802i	$0.506336\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	48.0000	−	6.92820i	1.90183	−	0.274505i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	16.5000	+	9.52628i	0.650696	+	0.375680i	0.788723	−	0.614749i	$-0.210743\pi$
$643$	16.5000	+	9.52628i	0.650696	+	0.375680i	−0.138027	+	0.990429i	$0.544076\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−17.0000	+	29.4449i	−0.665771	+	1.15315i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$			15.5885i			0.606321i	0.952940	+	0.303160i	$0.0980418\pi$
$661$			15.5885i			0.606321i	−0.952940	+	0.303160i	$0.901958\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−18.5000	−	32.0429i	−0.713123	−	1.23516i	−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−18.5000	−	32.0429i	−0.713123	−	1.23516i	0.250557	−	0.968102i	$-0.419386\pi$
$674$		0			0
$675$		0			0
$676$	35.0000	+	60.6218i	1.34615	+	2.33161i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	33.0000	−	38.1051i	1.26642	−	1.46234i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−26.0000	−	45.0333i	−0.991241	−	1.71688i
$689$		0			0
$690$		0			0
$691$			32.9090i			1.25192i	0.779857	+	0.625958i	$0.215292\pi$
$691$			32.9090i			1.25192i	−0.779857	+	0.625958i	$0.784708\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−5.00000	+	25.9808i	−0.188982	+	0.981981i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−45.0000	+	25.9808i	−1.69721	+	0.979883i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	31.0000			1.16423			0.582115	−	0.813107i	$-0.302225\pi$
$709$	31.0000			1.16423			0.582115	+	0.813107i	$0.302225\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	6.00000	−	6.92820i	0.223452	−	0.258020i
$722$		0			0
$723$		0			0
$724$			51.9615i			1.93113i
$725$		0			0
$726$		0			0
$727$	46.5000	−	26.8468i	1.72459	−	0.995692i	0.815935	−	0.578144i	$-0.196223\pi$
$727$	46.5000	−	26.8468i	1.72459	−	0.995692i	0.908655	−	0.417548i	$-0.137111\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	28.5000	−	16.4545i	1.05267	−	0.607760i	0.129275	−	0.991609i	$-0.458735\pi$
$733$	28.5000	−	16.4545i	1.05267	−	0.607760i	0.923396	+	0.383849i	$0.125402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	18.5000	+	32.0429i	0.680534	+	1.17872i	0.974818	+	0.223001i	$0.0715853\pi$
$739$	18.5000	+	32.0429i	0.680534	+	1.17872i	−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	20.5000	+	35.5070i	0.748056	+	1.29567i	0.948753	+	0.316017i	$0.102346\pi$
$751$	20.5000	+	35.5070i	0.748056	+	1.29567i	−0.200698	+	0.979653i	$0.564321\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	55.0000			1.99901			0.999505	−	0.0314762i	$-0.0100208\pi$
$757$	55.0000			1.99901			0.999505	+	0.0314762i	$0.0100208\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	38.0000	+	32.9090i	1.37569	+	1.19138i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−22.5000	−	12.9904i	−0.811371	−	0.468445i	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	−22.5000	−	12.9904i	−0.811371	−	0.468445i	−0.847432	+	0.530904i	$0.821852\pi$
$770$		0			0
$771$		0			0
$772$	−4.00000			−0.143963
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	−7.50000	−	4.33013i	−0.269408	−	0.155543i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	22.0000	−	17.3205i	0.785714	−	0.618590i
$785$		0			0
$786$		0			0
$787$		−	50.2295i		−	1.79049i	−0.445577	−	0.895244i	$-0.647001\pi$
$787$		−	50.2295i		−	1.79049i	0.445577	−	0.895244i	$-0.352999\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−30.0000	+	51.9615i	−1.06533	+	1.84521i
$794$		0			0
$795$		0			0
$796$	39.0000	−	22.5167i	1.38232	−	0.798082i
$797$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$797$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$		−	10.3923i		−	0.364923i	−0.983213	−	0.182462i	$-0.941593\pi$
$811$		−	10.3923i		−	0.364923i	0.983213	−	0.182462i	$-0.0584065\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	67.5500i		−	2.36327i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	47.0000			1.63832			0.819159	−	0.573567i	$-0.194441\pi$
$823$	47.0000			1.63832			0.819159	+	0.573567i	$0.194441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−49.5000	+	28.5788i	−1.71921	+	0.992584i	−0.798823	+	0.601566i	$0.794544\pi$
$829$	−49.5000	+	28.5788i	−1.71921	+	0.992584i	−0.920383	+	0.391018i	$0.872123\pi$
$830$		0			0
$831$		0			0
$832$	48.0000	+	27.7128i	1.66410	+	0.960769i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	−14.5000	+	25.1147i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	29.0000	−	50.2295i	0.998221	−	1.72897i
$845$		0			0
$846$		0			0
$847$	22.0000	+	19.0526i	0.755929	+	0.654654i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−40.5000	+	23.3827i	−1.38669	+	0.800608i	−0.992941	−	0.118609i	$-0.962157\pi$
$853$	−40.5000	+	23.3827i	−1.38669	+	0.800608i	−0.393753	+	0.919216i	$0.628823\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	−34.5000	+	19.9186i	−1.17712	+	0.679613i	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−34.5000	+	19.9186i	−1.17712	+	0.679613i	−0.221777	+	0.975097i	$0.571186\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	3.00000	+	8.66025i	0.101827	+	0.293948i
$869$		0			0
$870$		0			0
$871$	−96.0000	−	55.4256i	−3.25284	−	1.87803i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	12.5000	+	21.6506i	0.422095	+	0.731090i	0.996144	−	0.0877308i	$-0.0279615\pi$
$877$	12.5000	+	21.6506i	0.422095	+	0.731090i	−0.574049	+	0.818821i	$0.694628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−47.0000			−1.58168			−0.790838	−	0.612026i	$-0.790355\pi$
$883$	−47.0000			−1.58168			−0.790838	+	0.612026i	$0.790355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	2.50000	−	0.866025i	0.0838473	−	0.0290456i
$890$		0			0
$891$		0			0
$892$	−18.0000	+	10.3923i	−0.602685	+	0.347960i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−20.0000	+	34.6410i	−0.664089	+	1.15024i	0.315442	+	0.948945i	$0.397847\pi$
$907$	−20.0000	+	34.6410i	−0.664089	+	1.15024i	−0.979531	+	0.201291i	$0.935486\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	15.0000	−	8.66025i	0.495614	−	0.286143i
$917$		0			0
$918$		0			0
$919$	26.5000	−	45.8993i	0.874154	−	1.51408i	0.0164935	−	0.999864i	$-0.494750\pi$
$919$	26.5000	−	45.8993i	0.874154	−	1.51408i	0.857661	−	0.514216i	$-0.171917\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	25.0000	+	43.3013i	0.821995	+	1.42374i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	36.0000	−	5.19615i	1.17985	−	0.170297i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	55.4256i		−	1.81068i	−0.424691	−	0.905338i	$-0.639617\pi$
$937$		−	55.4256i		−	1.81068i	0.424691	−	0.905338i	$-0.360383\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	108.000			3.50583
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	28.0000			0.903226
$962$		0			0
$963$		0			0
$964$	3.00000	+	1.73205i	0.0966235	+	0.0557856i
$965$		0			0
$966$		0			0
$967$	−10.0000	+	17.3205i	−0.321578	+	0.556990i	−0.980814	−	0.194946i	$-0.937547\pi$
$967$	−10.0000	+	17.3205i	−0.321578	+	0.556990i	0.659236	+	0.751936i	$0.270880\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$		0			0
$973$	45.0000	+	8.66025i	1.44263	+	0.277635i
$974$		0			0
$975$		0			0
$976$			34.6410i			1.10883i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	36.0000	+	62.3538i	1.14531	+	1.98374i
$989$		0			0
$990$		0			0
$991$	22.0000	+	38.1051i	0.698853	+	1.21045i	0.968864	+	0.247592i	$0.0796392\pi$
$991$	22.0000	+	38.1051i	0.698853	+	1.21045i	−0.270011	+	0.962857i	$0.587027\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−19.5000	−	11.2583i	−0.617571	−	0.356555i	0.158352	−	0.987383i	$-0.449382\pi$
$997$	−19.5000	−	11.2583i	−0.617571	−	0.356555i	−0.775923	+	0.630828i	$0.782715\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.2.i.a.215.1		2
3.2	odd	2	CM	567.2.i.a.215.1		2
7.3	odd	6		567.2.s.b.458.1		2
9.2	odd	6		567.2.s.b.26.1		2
9.4	even	3		189.2.p.a.26.1	✓	2
9.5	odd	6		189.2.p.a.26.1	✓	2
9.7	even	3		567.2.s.b.26.1		2
21.17	even	6		567.2.s.b.458.1		2
63.5	even	6		1323.2.c.a.1322.2		2
63.23	odd	6		1323.2.c.a.1322.1		2
63.31	odd	6		189.2.p.a.80.1	yes	2
63.38	even	6	inner	567.2.i.a.269.1		2
63.40	odd	6		1323.2.c.a.1322.2		2
63.52	odd	6	inner	567.2.i.a.269.1		2
63.58	even	3		1323.2.c.a.1322.1		2
63.59	even	6		189.2.p.a.80.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.p.a.26.1	✓	2	9.4	even	3
189.2.p.a.26.1	✓	2	9.5	odd	6
189.2.p.a.80.1	yes	2	63.31	odd	6
189.2.p.a.80.1	yes	2	63.59	even	6
567.2.i.a.215.1		2	1.1	even	1	trivial
567.2.i.a.215.1		2	3.2	odd	2	CM
567.2.i.a.269.1		2	63.38	even	6	inner
567.2.i.a.269.1		2	63.52	odd	6	inner
567.2.s.b.26.1		2	9.2	odd	6
567.2.s.b.26.1		2	9.7	even	3
567.2.s.b.458.1		2	7.3	odd	6
567.2.s.b.458.1		2	21.17	even	6
1323.2.c.a.1322.1		2	63.23	odd	6
1323.2.c.a.1322.1		2	63.58	even	3
1323.2.c.a.1322.2		2	63.5	even	6
1323.2.c.a.1322.2		2	63.40	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:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	567.i (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000 q^{4} +(-2.50000 + 0.866025i) q^{7} +(6.00000 + 3.46410i) q^{13} +4.00000 q^{16} +(4.50000 + 2.59808i) q^{19} +(2.50000 - 4.33013i) q^{25} +(-5.00000 + 1.73205i) q^{28} -1.73205i q^{31} +(-5.00000 + 8.66025i) q^{37} +(-6.50000 - 11.2583i) q^{43} +(5.50000 - 4.33013i) q^{49} +(12.0000 + 6.92820i) q^{52} +8.66025i q^{61} +8.00000 q^{64} -16.0000 q^{67} +(13.5000 - 7.79423i) q^{73} +(9.00000 + 5.19615i) q^{76} -4.00000 q^{79} +(-18.0000 - 3.46410i) q^{91} +(-16.5000 + 9.52628i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} - 5 q^{7} + 12 q^{13} + 8 q^{16} + 9 q^{19} + 5 q^{25} - 10 q^{28} - 10 q^{37} - 13 q^{43} + 11 q^{49} + 24 q^{52} + 16 q^{64} - 32 q^{67} + 27 q^{73} + 18 q^{76} - 8 q^{79} - 36 q^{91} - 33 q^{97}+O(q^{100}) \) 2 * q + 4 * q^4 - 5 * q^7 + 12 * q^13 + 8 * q^16 + 9 * q^19 + 5 * q^25 - 10 * q^28 - 10 * q^37 - 13 * q^43 + 11 * q^49 + 24 * q^52 + 16 * q^64 - 32 * q^67 + 27 * q^73 + 18 * q^76 - 8 * q^79 - 36 * q^91 - 33 * q^97

Introduction

L-functions

Modular forms

Varieties

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