LMFDB - Embedded newform 936.2.t.a.217.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [936,2,Mod(217,936)] 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("936.217"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	936.t (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-6,0,0] 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:	$7.47399762919$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	936.217
Dual form			936.2.t.a.289.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{5} +(3.50000 + 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{17} +(2.00000 - 3.46410i) q^{23} +4.00000 q^{25} +(1.50000 - 2.59808i) q^{29} +8.00000 q^{31} +(2.50000 - 4.33013i) q^{37} +(1.50000 - 2.59808i) q^{41} +(-2.00000 - 3.46410i) q^{43} +8.00000 q^{47} +(3.50000 - 6.06218i) q^{49} +13.0000 q^{53} +(6.00000 + 10.3923i) q^{59} +(-7.50000 - 12.9904i) q^{61} +(-10.5000 - 2.59808i) q^{65} +(-6.00000 + 10.3923i) q^{67} +(4.00000 + 6.92820i) q^{71} +3.00000 q^{73} -4.00000 q^{79} -12.0000 q^{83} +(1.50000 + 2.59808i) q^{85} +(5.00000 - 8.66025i) q^{89} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} + 7 q^{13} - q^{17} + 4 q^{23} + 8 q^{25} + 3 q^{29} + 16 q^{31} + 5 q^{37} + 3 q^{41} - 4 q^{43} + 16 q^{47} + 7 q^{49} + 26 q^{53} + 12 q^{59} - 15 q^{61} - 21 q^{65} - 12 q^{67} + 8 q^{71}+ \cdots - 2 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 + 7 * q^13 - q^17 + 4 * q^23 + 8 * q^25 + 3 * q^29 + 16 * q^31 + 5 * q^37 + 3 * q^41 - 4 * q^43 + 16 * q^47 + 7 * q^49 + 26 * q^53 + 12 * q^59 - 15 * q^61 - 21 * q^65 - 12 * q^67 + 8 * q^71 + 6 * q^73 - 8 * q^79 - 24 * q^83 + 3 * q^85 + 10 * q^89 - 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.00000			−1.34164			−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000			−1.34164			−0.670820	+	0.741620i	$0.734058\pi$
$6$		0			0
$7$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$7$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	3.50000	+	0.866025i	0.970725	+	0.240192i
$13$	3.50000	+	0.866025i	0.970725	+	0.240192i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.500000	−	0.866025i	−0.121268	−	0.210042i	0.799000	−	0.601331i	$-0.205363\pi$
$17$	−0.500000	−	0.866025i	−0.121268	−	0.210042i	−0.920268	+	0.391289i	$0.872029\pi$
$18$		0			0
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	−0.578610	−	0.815604i	$-0.696405\pi$
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	0.995639	+	0.0932891i	$0.0297381\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.50000	−	2.59808i	0.278543	−	0.482451i	−0.692480	−	0.721437i	$-0.743482\pi$
$29$	1.50000	−	2.59808i	0.278543	−	0.482451i	0.971023	+	0.238987i	$0.0768152\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	−0.584002	−	0.811752i	$-0.698514\pi$
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	0.994999	+	0.0998840i	$0.0318472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.50000	−	2.59808i	0.234261	−	0.405751i	−0.724797	−	0.688963i	$-0.758066\pi$
$41$	1.50000	−	2.59808i	0.234261	−	0.405751i	0.959058	+	0.283211i	$0.0913998\pi$
$42$		0			0
$43$	−2.00000	−	3.46410i	−0.304997	−	0.528271i	0.672264	−	0.740312i	$-0.265322\pi$
$43$	−2.00000	−	3.46410i	−0.304997	−	0.528271i	−0.977261	+	0.212041i	$0.931989\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$	3.50000	−	6.06218i	0.500000	−	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	13.0000			1.78569			0.892844	−	0.450367i	$-0.148707\pi$
$53$	13.0000			1.78569			0.892844	+	0.450367i	$0.148707\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	0.931282	+	0.364299i	$0.118692\pi$
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	−0.150148	+	0.988663i	$0.547975\pi$
$60$		0			0
$61$	−7.50000	−	12.9904i	−0.960277	−	1.66325i	−0.721803	−	0.692099i	$-0.756686\pi$
$61$	−7.50000	−	12.9904i	−0.960277	−	1.66325i	−0.238474	−	0.971149i	$-0.576647\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−10.5000	−	2.59808i	−1.30236	−	0.322252i
$66$		0			0
$67$	−6.00000	+	10.3923i	−0.733017	+	1.26962i	0.222571	+	0.974916i	$0.428555\pi$
$67$	−6.00000	+	10.3923i	−0.733017	+	1.26962i	−0.955588	+	0.294706i	$0.904778\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.00000	+	6.92820i	0.474713	+	0.822226i	0.999581	−	0.0289572i	$-0.00921865\pi$
$71$	4.00000	+	6.92820i	0.474713	+	0.822226i	−0.524868	+	0.851184i	$0.675885\pi$
$72$		0			0
$73$	3.00000			0.351123			0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000			0.351123			0.175562	+	0.984468i	$0.443826\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		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
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$	1.50000	+	2.59808i	0.162698	+	0.281801i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.00000	−	8.66025i	0.529999	−	0.917985i	−0.469389	−	0.882992i	$-0.655526\pi$
$89$	5.00000	−	8.66025i	0.529999	−	0.917985i	0.999388	−	0.0349934i	$-0.0111410\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	−0.781697	−	0.623658i	$-0.785646\pi$
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	0.930953	+	0.365140i	$0.118979\pi$
$102$		0			0
$103$	16.0000			1.57653			0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000			1.57653			0.788263	+	0.615338i	$0.210980\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.0000	+	17.3205i	−0.966736	+	1.67444i	−0.261861	+	0.965106i	$0.584336\pi$
$107$	−10.0000	+	17.3205i	−0.966736	+	1.67444i	−0.704875	+	0.709331i	$0.748997\pi$
$108$		0			0
$109$	−18.0000			−1.72409			−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000			−1.72409			−0.862044	+	0.506834i	$0.830816\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.50000	−	11.2583i	−0.611469	−	1.05909i	−0.990993	−	0.133913i	$-0.957246\pi$
$113$	−6.50000	−	11.2583i	−0.611469	−	1.05909i	0.379525	−	0.925182i	$-0.376088\pi$
$114$		0			0
$115$	−6.00000	+	10.3923i	−0.559503	+	0.969087i
$116$		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
$124$		0			0
$125$	3.00000			0.268328
$126$		0			0
$127$	6.00000	−	10.3923i	0.532414	−	0.922168i	−0.466870	−	0.884326i	$-0.654618\pi$
$127$	6.00000	−	10.3923i	0.532414	−	0.922168i	0.999284	−	0.0378419i	$-0.0120483\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.50000	−	4.33013i	−0.213589	−	0.369948i	0.739246	−	0.673436i	$-0.235182\pi$
$137$	−2.50000	−	4.33013i	−0.213589	−	0.369948i	−0.952835	+	0.303488i	$0.901849\pi$
$138$		0			0
$139$	−4.00000	−	6.92820i	−0.339276	−	0.587643i	0.645021	−	0.764165i	$-0.276849\pi$
$139$	−4.00000	−	6.92820i	−0.339276	−	0.587643i	−0.984297	+	0.176522i	$0.943515\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−4.50000	+	7.79423i	−0.373705	+	0.647275i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.50000	+	16.4545i	0.778270	+	1.34800i	0.932938	+	0.360037i	$0.117236\pi$
$149$	9.50000	+	16.4545i	0.778270	+	1.34800i	−0.154668	+	0.987967i	$0.549431\pi$
$150$		0			0
$151$	−4.00000			−0.325515			−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000			−0.325515			−0.162758	+	0.986666i	$0.552039\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−24.0000			−1.92773
$156$		0			0
$157$	−13.0000			−1.03751			−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000			−1.03751			−0.518756	+	0.854922i	$0.673605\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.00000	+	13.8564i	0.626608	+	1.08532i	0.988227	+	0.152992i	$0.0488907\pi$
$163$	8.00000	+	13.8564i	0.626608	+	1.08532i	−0.361619	+	0.932326i	$0.617776\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	−0.534875	−	0.844931i	$-0.679641\pi$
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	0.999169	+	0.0407502i	$0.0129748\pi$
$168$		0			0
$169$	11.5000	+	6.06218i	0.884615	+	0.466321i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.00000	−	15.5885i	−0.684257	−	1.18517i	−0.973670	−	0.227964i	$-0.926793\pi$
$173$	−9.00000	−	15.5885i	−0.684257	−	1.18517i	0.289412	−	0.957205i	$-0.406540\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.00000	−	3.46410i	0.149487	−	0.258919i	−0.781551	−	0.623841i	$-0.785571\pi$
$179$	2.00000	−	3.46410i	0.149487	−	0.258919i	0.931038	+	0.364922i	$0.118904\pi$
$180$		0			0
$181$	11.0000			0.817624			0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000			0.817624			0.408812	+	0.912619i	$0.365943\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.50000	+	12.9904i	−0.551411	+	0.955072i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.00000	−	6.92820i	−0.289430	−	0.501307i	0.684244	−	0.729253i	$-0.260132\pi$
$191$	−4.00000	−	6.92820i	−0.289430	−	0.501307i	−0.973674	+	0.227946i	$0.926799\pi$
$192$		0			0
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	−0.993215	−	0.116289i	$-0.962900\pi$
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	0.597317	+	0.802005i	$0.296234\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.00000	−	5.19615i	0.213741	−	0.370211i	−0.739141	−	0.673550i	$-0.764768\pi$
$197$	3.00000	−	5.19615i	0.213741	−	0.370211i	0.952882	+	0.303340i	$0.0981018\pi$
$198$		0			0
$199$	−2.00000	−	3.46410i	−0.141776	−	0.245564i	0.786389	−	0.617731i	$-0.211948\pi$
$199$	−2.00000	−	3.46410i	−0.141776	−	0.245564i	−0.928166	+	0.372168i	$0.878615\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−4.50000	+	7.79423i	−0.314294	+	0.544373i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	−0.694857	−	0.719148i	$-0.744533\pi$
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	0.970229	+	0.242190i	$0.0778659\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.00000	+	10.3923i	0.409197	+	0.708749i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.00000	−	3.46410i	−0.0672673	−	0.233021i
$222$		0			0
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	−0.968309	−	0.249756i	$-0.919650\pi$
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	0.700449	+	0.713702i	$0.252983\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.00000	−	6.92820i	−0.265489	−	0.459841i	0.702202	−	0.711977i	$-0.252200\pi$
$227$	−4.00000	−	6.92820i	−0.265489	−	0.459841i	−0.967692	+	0.252136i	$0.918867\pi$
$228$		0			0
$229$	−10.0000			−0.660819			−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000			−0.660819			−0.330409	+	0.943838i	$0.607187\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$	−24.0000			−1.56559
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.00000			0.517477			0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000			0.517477			0.258738	+	0.965947i	$0.416693\pi$
$240$		0			0
$241$	−5.50000	−	9.52628i	−0.354286	−	0.613642i	0.632709	−	0.774389i	$-0.281943\pi$
$241$	−5.50000	−	9.52628i	−0.354286	−	0.613642i	−0.986996	+	0.160748i	$0.948609\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−10.5000	+	18.1865i	−0.670820	+	1.16190i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.50000	+	4.33013i	−0.155946	+	0.270106i	−0.933403	−	0.358830i	$-0.883176\pi$
$257$	−2.50000	+	4.33013i	−0.155946	+	0.270106i	0.777457	+	0.628936i	$0.216509\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	−0.989561	−	0.144112i	$-0.953967\pi$
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	0.619586	+	0.784929i	$0.287301\pi$
$264$		0			0
$265$	−39.0000			−2.39575
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.00000	+	12.1244i	0.426798	+	0.739235i	0.996586	−	0.0825561i	$-0.0263084\pi$
$269$	7.00000	+	12.1244i	0.426798	+	0.739235i	−0.569789	+	0.821791i	$0.692975\pi$
$270$		0			0
$271$	−14.0000	+	24.2487i	−0.850439	+	1.47300i	0.0303728	+	0.999539i	$0.490331\pi$
$271$	−14.0000	+	24.2487i	−0.850439	+	1.47300i	−0.880812	+	0.473466i	$0.843003\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	12.5000	+	21.6506i	0.751052	+	1.30086i	0.947313	+	0.320309i	$0.103787\pi$
$277$	12.5000	+	21.6506i	0.751052	+	1.30086i	−0.196261	+	0.980552i	$0.562880\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−15.0000			−0.894825			−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000			−0.894825			−0.447412	+	0.894328i	$0.647654\pi$
$282$		0			0
$283$	8.00000	−	13.8564i	0.475551	−	0.823678i	−0.524057	−	0.851683i	$-0.675582\pi$
$283$	8.00000	−	13.8564i	0.475551	−	0.823678i	0.999608	+	0.0280052i	$0.00891551\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.00000	−	13.8564i	0.470588	−	0.815083i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.500000	−	0.866025i	−0.0292103	−	0.0505937i	0.851051	−	0.525084i	$-0.175966\pi$
$293$	−0.500000	−	0.866025i	−0.0292103	−	0.0505937i	−0.880261	+	0.474490i	$0.842633\pi$
$294$		0			0
$295$	−18.0000	−	31.1769i	−1.04800	−	1.81519i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.0000	−	10.3923i	0.578315	−	0.601003i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	22.5000	+	38.9711i	1.28835	+	2.23148i
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−31.0000			−1.74113			−0.870567	−	0.492050i	$-0.836248\pi$
$317$	−31.0000			−1.74113			−0.870567	+	0.492050i	$0.836248\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	14.0000	+	3.46410i	0.776580	+	0.192154i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	16.0000	+	27.7128i	0.879440	+	1.52323i	0.851957	+	0.523612i	$0.175416\pi$
$331$	16.0000	+	27.7128i	0.879440	+	1.52323i	0.0274825	+	0.999622i	$0.491251\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	18.0000	−	31.1769i	0.983445	−	1.70338i
$336$		0			0
$337$	−1.00000			−0.0544735			−0.0272367	−	0.999629i	$-0.508671\pi$
$337$	−1.00000			−0.0544735			−0.0272367	+	0.999629i	$0.508671\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.0000	+	17.3205i	0.536828	+	0.929814i	0.999072	+	0.0430610i	$0.0137110\pi$
$347$	10.0000	+	17.3205i	0.536828	+	0.929814i	−0.462244	+	0.886753i	$0.652956\pi$
$348$		0			0
$349$	17.0000	−	29.4449i	0.909989	−	1.57615i	0.0959126	−	0.995390i	$-0.469423\pi$
$349$	17.0000	−	29.4449i	0.909989	−	1.57615i	0.814076	−	0.580758i	$-0.197244\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5.50000	−	9.52628i	0.292735	−	0.507033i	−0.681720	−	0.731613i	$-0.738768\pi$
$353$	5.50000	−	9.52628i	0.292735	−	0.507033i	0.974456	+	0.224580i	$0.0721011\pi$
$354$		0			0
$355$	−12.0000	−	20.7846i	−0.636894	−	1.10313i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	28.0000			1.47778			0.738892	−	0.673824i	$-0.235349\pi$
$359$	28.0000			1.47778			0.738892	+	0.673824i	$0.235349\pi$
$360$		0			0
$361$	9.50000	−	16.4545i	0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.00000			−0.471082
$366$		0			0
$367$	−18.0000	+	31.1769i	−0.939592	+	1.62742i	−0.173360	+	0.984859i	$0.555462\pi$
$367$	−18.0000	+	31.1769i	−0.939592	+	1.62742i	−0.766233	+	0.642563i	$0.777871\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−5.50000	−	9.52628i	−0.284779	−	0.493252i	0.687776	−	0.725923i	$-0.258587\pi$
$373$	−5.50000	−	9.52628i	−0.284779	−	0.493252i	−0.972556	+	0.232671i	$0.925254\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.50000	−	7.79423i	0.386270	−	0.401423i
$378$		0			0
$379$	10.0000	−	17.3205i	0.513665	−	0.889695i	−0.486209	−	0.873843i	$-0.661621\pi$
$379$	10.0000	−	17.3205i	0.513665	−	0.889695i	0.999874	−	0.0158521i	$-0.00504609\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.0000	−	20.7846i	−0.613171	−	1.06204i	−0.990702	−	0.136047i	$-0.956560\pi$
$383$	−12.0000	−	20.7846i	−0.613171	−	1.06204i	0.377531	−	0.925997i	$-0.376773\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	21.0000			1.06474			0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000			1.06474			0.532371	+	0.846511i	$0.321301\pi$
$390$		0			0
$391$	−4.00000			−0.202289
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.0000			0.603786
$396$		0			0
$397$	1.00000	+	1.73205i	0.0501886	+	0.0869291i	0.890028	−	0.455905i	$-0.150684\pi$
$397$	1.00000	+	1.73205i	0.0501886	+	0.0869291i	−0.839840	+	0.542834i	$0.817351\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.50000	+	7.79423i	−0.224719	+	0.389225i	−0.956235	−	0.292599i	$-0.905480\pi$
$401$	−4.50000	+	7.79423i	−0.224719	+	0.389225i	0.731516	+	0.681824i	$0.238813\pi$
$402$		0			0
$403$	28.0000	+	6.92820i	1.39478	+	0.345118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	4.50000	+	7.79423i	0.222511	+	0.385400i	0.955570	−	0.294765i	$-0.0952414\pi$
$409$	4.50000	+	7.79423i	0.222511	+	0.385400i	−0.733059	+	0.680165i	$0.761908\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	36.0000			1.76717
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.0000	−	20.7846i	0.586238	−	1.01539i	−0.408481	−	0.912767i	$-0.633942\pi$
$419$	12.0000	−	20.7846i	0.586238	−	1.01539i	0.994720	−	0.102628i	$-0.0327251\pi$
$420$		0			0
$421$	27.0000			1.31590			0.657950	−	0.753062i	$-0.271424\pi$
$421$	27.0000			1.31590			0.657950	+	0.753062i	$0.271424\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.00000	−	3.46410i	−0.0970143	−	0.168034i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	14.0000	−	24.2487i	0.674356	−	1.16802i	−0.302300	−	0.953213i	$-0.597755\pi$
$431$	14.0000	−	24.2487i	0.674356	−	1.16802i	0.976657	−	0.214807i	$-0.0689121\pi$
$432$		0			0
$433$	10.5000	+	18.1865i	0.504598	+	0.873989i	0.999986	+	0.00531724i	$0.00169254\pi$
$433$	10.5000	+	18.1865i	0.504598	+	0.873989i	−0.495388	+	0.868672i	$0.664974\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	14.0000	−	24.2487i	0.668184	−	1.15733i	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	14.0000	−	24.2487i	0.668184	−	1.15733i	0.978412	−	0.206666i	$-0.0662612\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−40.0000			−1.90046			−0.950229	−	0.311553i	$-0.899151\pi$
$443$	−40.0000			−1.90046			−0.950229	+	0.311553i	$0.899151\pi$
$444$		0			0
$445$	−15.0000	+	25.9808i	−0.711068	+	1.23161i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.00000	+	1.73205i	0.0471929	+	0.0817405i	0.888657	−	0.458573i	$-0.151639\pi$
$449$	1.00000	+	1.73205i	0.0471929	+	0.0817405i	−0.841464	+	0.540313i	$0.818306\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.50000	−	7.79423i	0.210501	−	0.364599i	−0.741370	−	0.671096i	$-0.765824\pi$
$457$	4.50000	−	7.79423i	0.210501	−	0.364599i	0.951871	+	0.306497i	$0.0991571\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.50000	−	4.33013i	−0.116437	−	0.201674i	0.801917	−	0.597436i	$-0.203814\pi$
$461$	−2.50000	−	4.33013i	−0.116437	−	0.201674i	−0.918353	+	0.395762i	$0.870481\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−24.0000			−1.11059			−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000			−1.11059			−0.555294	+	0.831654i	$0.687394\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.00000	−	10.3923i	0.274147	−	0.474837i	−0.695773	−	0.718262i	$-0.744938\pi$
$479$	6.00000	−	10.3923i	0.274147	−	0.474837i	0.969920	+	0.243426i	$0.0782712\pi$
$480$		0			0
$481$	12.5000	−	12.9904i	0.569951	−	0.592310i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.00000	+	5.19615i	0.136223	+	0.235945i
$486$		0			0
$487$	6.00000	+	10.3923i	0.271886	+	0.470920i	0.969345	−	0.245705i	$-0.0790193\pi$
$487$	6.00000	+	10.3923i	0.271886	+	0.470920i	−0.697459	+	0.716625i	$0.745686\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.00000	−	3.46410i	0.0902587	−	0.156333i	−0.817361	−	0.576126i	$-0.804564\pi$
$491$	2.00000	−	3.46410i	0.0902587	−	0.156333i	0.907620	+	0.419793i	$0.137897\pi$
$492$		0			0
$493$	−3.00000			−0.135113
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	12.0000			0.537194			0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000			0.537194			0.268597	+	0.963253i	$0.413440\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6.00000	+	10.3923i	0.267527	+	0.463370i	0.968223	−	0.250090i	$-0.0804603\pi$
$503$	6.00000	+	10.3923i	0.267527	+	0.463370i	−0.700696	+	0.713460i	$0.747127\pi$
$504$		0			0
$505$	−4.50000	+	7.79423i	−0.200247	+	0.346839i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.50000	−	6.06218i	0.155135	−	0.268701i	−0.777973	−	0.628297i	$-0.783752\pi$
$509$	3.50000	−	6.06218i	0.155135	−	0.268701i	0.933108	+	0.359596i	$0.117085\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−48.0000			−2.11513
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.00000			−0.131432			−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000			−0.131432			−0.0657162	+	0.997838i	$0.520933\pi$
$522$		0			0
$523$	16.0000	−	27.7128i	0.699631	−	1.21180i	−0.268963	−	0.963150i	$-0.586681\pi$
$523$	16.0000	−	27.7128i	0.699631	−	1.21180i	0.968594	−	0.248646i	$-0.0799857\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.00000	−	6.92820i	−0.174243	−	0.301797i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	7.50000	−	7.79423i	0.324861	−	0.337606i
$534$		0			0
$535$	30.0000	−	51.9615i	1.29701	−	2.24649i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	7.00000			0.300954			0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000			0.300954			0.150477	+	0.988614i	$0.451919\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	54.0000			2.31311
$546$		0			0
$547$	−24.0000			−1.02617			−0.513083	−	0.858339i	$-0.671497\pi$
$547$	−24.0000			−1.02617			−0.513083	+	0.858339i	$0.671497\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14.5000	+	25.1147i	−0.614385	+	1.06415i	0.376107	+	0.926576i	$0.377262\pi$
$557$	−14.5000	+	25.1147i	−0.614385	+	1.06415i	−0.990492	+	0.137569i	$0.956071\pi$
$558$		0			0
$559$	−4.00000	−	13.8564i	−0.169182	−	0.586064i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.0000	+	31.1769i	0.758610	+	1.31395i	0.943560	+	0.331202i	$0.107454\pi$
$563$	18.0000	+	31.1769i	0.758610	+	1.31395i	−0.184950	+	0.982748i	$0.559212\pi$
$564$		0			0
$565$	19.5000	+	33.7750i	0.820371	+	1.42092i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19.0000	+	32.9090i	−0.796521	+	1.37962i	0.125347	+	0.992113i	$0.459996\pi$
$569$	−19.0000	+	32.9090i	−0.796521	+	1.37962i	−0.921869	+	0.387503i	$0.873338\pi$
$570$		0			0
$571$	−8.00000			−0.334790			−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000			−0.334790			−0.167395	+	0.985890i	$0.553535\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.00000	−	13.8564i	0.333623	−	0.577852i
$576$		0			0
$577$	−41.0000			−1.70685			−0.853426	−	0.521214i	$-0.825479\pi$
$577$	−41.0000			−1.70685			−0.853426	+	0.521214i	$0.825479\pi$
$578$		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
$587$	8.00000	−	13.8564i	0.330195	−	0.571915i	−0.652355	−	0.757914i	$-0.726219\pi$
$587$	8.00000	−	13.8564i	0.330195	−	0.571915i	0.982550	+	0.185999i	$0.0595520\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.00000			−0.123195			−0.0615976	−	0.998101i	$-0.519620\pi$
$593$	−3.00000			−0.123195			−0.0615976	+	0.998101i	$0.519620\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	−0.992114	−	0.125336i	$-0.959999\pi$
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	0.604601	+	0.796528i	$0.293332\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−16.5000	−	28.5788i	−0.670820	−	1.16190i
$606$		0			0
$607$	2.00000	+	3.46410i	0.0811775	+	0.140604i	0.903756	−	0.428048i	$-0.140799\pi$
$607$	2.00000	+	3.46410i	0.0811775	+	0.140604i	−0.822578	+	0.568652i	$0.807465\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	28.0000	+	6.92820i	1.13276	+	0.280285i
$612$		0			0
$613$	−1.50000	+	2.59808i	−0.0605844	+	0.104935i	−0.894727	−	0.446614i	$-0.852630\pi$
$613$	−1.50000	+	2.59808i	−0.0605844	+	0.104935i	0.834142	+	0.551549i	$0.185963\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.50000	+	12.9904i	0.301939	+	0.522973i	0.976575	−	0.215177i	$-0.0690329\pi$
$617$	7.50000	+	12.9904i	0.301939	+	0.522973i	−0.674636	+	0.738150i	$0.735700\pi$
$618$		0			0
$619$	−28.0000			−1.12542			−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000			−1.12542			−0.562708	+	0.826656i	$0.690240\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.00000			−0.199363
$630$		0			0
$631$	4.00000	+	6.92820i	0.159237	+	0.275807i	0.934594	−	0.355716i	$-0.115763\pi$
$631$	4.00000	+	6.92820i	0.159237	+	0.275807i	−0.775356	+	0.631524i	$0.782430\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−18.0000	+	31.1769i	−0.714308	+	1.23722i
$636$		0			0
$637$	17.5000	−	18.1865i	0.693375	−	0.720577i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4.50000	−	7.79423i	−0.177739	−	0.307854i	0.763367	−	0.645966i	$-0.223545\pi$
$641$	−4.50000	−	7.79423i	−0.177739	−	0.307854i	−0.941106	+	0.338112i	$0.890212\pi$
$642$		0			0
$643$	10.0000	+	17.3205i	0.394362	+	0.683054i	0.993019	−	0.117951i	$-0.0376325\pi$
$643$	10.0000	+	17.3205i	0.394362	+	0.683054i	−0.598658	+	0.801005i	$0.704299\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16.0000	−	27.7128i	0.629025	−	1.08950i	−0.358723	−	0.933444i	$-0.616788\pi$
$647$	16.0000	−	27.7128i	0.629025	−	1.08950i	0.987748	−	0.156059i	$-0.0498790\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.00000	−	12.1244i	0.273931	−	0.474463i	−0.695934	−	0.718106i	$-0.745009\pi$
$653$	7.00000	−	12.1244i	0.273931	−	0.474463i	0.969865	+	0.243643i	$0.0783426\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−20.0000	−	34.6410i	−0.779089	−	1.34942i	−0.932467	−	0.361255i	$-0.882348\pi$
$659$	−20.0000	−	34.6410i	−0.779089	−	1.34942i	0.153378	−	0.988168i	$-0.450985\pi$
$660$		0			0
$661$	−3.50000	+	6.06218i	−0.136134	+	0.235791i	−0.926030	−	0.377450i	$-0.876801\pi$
$661$	−3.50000	+	6.06218i	−0.136134	+	0.235791i	0.789896	+	0.613241i	$0.210135\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−6.00000	−	10.3923i	−0.232321	−	0.402392i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	14.5000	−	25.1147i	0.558934	−	0.968102i	−0.438652	−	0.898657i	$-0.644544\pi$
$673$	14.5000	−	25.1147i	0.558934	−	0.968102i	0.997586	−	0.0694449i	$-0.0221228\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.00000			−0.230599			−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000			−0.230599			−0.115299	+	0.993331i	$0.536783\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.0000	+	20.7846i	0.459167	+	0.795301i	0.998917	−	0.0465244i	$-0.0148145\pi$
$683$	12.0000	+	20.7846i	0.459167	+	0.795301i	−0.539750	+	0.841825i	$0.681481\pi$
$684$		0			0
$685$	7.50000	+	12.9904i	0.286560	+	0.496337i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	45.5000	+	11.2583i	1.73341	+	0.428908i
$690$		0			0
$691$	4.00000	−	6.92820i	0.152167	−	0.263561i	−0.779857	−	0.625958i	$-0.784708\pi$
$691$	4.00000	−	6.92820i	0.152167	−	0.263561i	0.932024	+	0.362397i	$0.118041\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	12.0000	+	20.7846i	0.455186	+	0.788405i
$696$		0			0
$697$	−3.00000			−0.113633
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−9.50000	−	16.4545i	−0.356780	−	0.617961i	0.630641	−	0.776075i	$-0.282792\pi$
$709$	−9.50000	−	16.4545i	−0.356780	−	0.617961i	−0.987421	+	0.158114i	$0.949459\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	16.0000	−	27.7128i	0.599205	−	1.03785i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	22.0000	+	38.1051i	0.820462	+	1.42108i	0.905339	+	0.424690i	$0.139617\pi$
$719$	22.0000	+	38.1051i	0.820462	+	1.42108i	−0.0848774	+	0.996391i	$0.527050\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.00000	−	10.3923i	0.222834	−	0.385961i
$726$		0			0
$727$	−40.0000			−1.48352			−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000			−1.48352			−0.741759	+	0.670667i	$0.766008\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.00000	+	3.46410i	−0.0739727	+	0.128124i
$732$		0			0
$733$	−17.0000			−0.627909			−0.313955	−	0.949438i	$-0.601654\pi$
$733$	−17.0000			−0.627909			−0.313955	+	0.949438i	$0.601654\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−16.0000	+	27.7128i	−0.588570	+	1.01943i	0.405851	+	0.913939i	$0.366975\pi$
$739$	−16.0000	+	27.7128i	−0.588570	+	1.01943i	−0.994420	+	0.105493i	$0.966358\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−16.0000	+	27.7128i	−0.586983	+	1.01668i	0.407642	+	0.913142i	$0.366351\pi$
$743$	−16.0000	+	27.7128i	−0.586983	+	1.01668i	−0.994625	+	0.103543i	$0.966982\pi$
$744$		0			0
$745$	−28.5000	−	49.3634i	−1.04416	−	1.80854i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	10.0000	−	17.3205i	0.364905	−	0.632034i	−0.623856	−	0.781540i	$-0.714435\pi$
$751$	10.0000	−	17.3205i	0.364905	−	0.632034i	0.988761	+	0.149505i	$0.0477681\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.0000			0.436725
$756$		0			0
$757$	−11.0000	+	19.0526i	−0.399802	+	0.692477i	−0.993701	−	0.112062i	$-0.964254\pi$
$757$	−11.0000	+	19.0526i	−0.399802	+	0.692477i	0.593899	+	0.804539i	$0.297588\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−19.0000	−	32.9090i	−0.688749	−	1.19295i	−0.972243	−	0.233975i	$-0.924827\pi$
$761$	−19.0000	−	32.9090i	−0.688749	−	1.19295i	0.283493	−	0.958974i	$-0.408507\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12.0000	+	41.5692i	0.433295	+	1.50098i
$768$		0			0
$769$	−9.00000	+	15.5885i	−0.324548	+	0.562134i	−0.981421	−	0.191867i	$-0.938546\pi$
$769$	−9.00000	+	15.5885i	−0.324548	+	0.562134i	0.656873	+	0.754002i	$0.271879\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.00000	−	8.66025i	−0.179838	−	0.311488i	0.761987	−	0.647592i	$-0.224224\pi$
$773$	−5.00000	−	8.66025i	−0.179838	−	0.311488i	−0.941825	+	0.336104i	$0.890891\pi$
$774$		0			0
$775$	32.0000			1.14947
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	39.0000			1.39197
$786$		0			0
$787$	24.0000	+	41.5692i	0.855508	+	1.48178i	0.876173	+	0.481996i	$0.160088\pi$
$787$	24.0000	+	41.5692i	0.855508	+	1.48178i	−0.0206657	+	0.999786i	$0.506579\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−15.0000	−	51.9615i	−0.532666	−	1.84521i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.00000	−	15.5885i	−0.318796	−	0.552171i	0.661441	−	0.749997i	$-0.269945\pi$
$797$	−9.00000	−	15.5885i	−0.318796	−	0.552171i	−0.980237	+	0.197826i	$0.936612\pi$
$798$		0			0
$799$	−4.00000	−	6.92820i	−0.141510	−	0.245102i
$800$		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
$809$	−10.5000	+	18.1865i	−0.369160	+	0.639404i	−0.989434	−	0.144981i	$-0.953688\pi$
$809$	−10.5000	+	18.1865i	−0.369160	+	0.639404i	0.620274	+	0.784385i	$0.287021\pi$
$810$		0			0
$811$	−56.0000			−1.96643			−0.983213	−	0.182462i	$-0.941593\pi$
$811$	−56.0000			−1.96643			−0.983213	+	0.182462i	$0.941593\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−24.0000	−	41.5692i	−0.840683	−	1.45611i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−21.0000	+	36.3731i	−0.732905	+	1.26943i	0.222731	+	0.974880i	$0.428503\pi$
$821$	−21.0000	+	36.3731i	−0.732905	+	1.26943i	−0.955636	+	0.294549i	$0.904831\pi$
$822$		0			0
$823$	18.0000	+	31.1769i	0.627441	+	1.08676i	0.988063	+	0.154047i	$0.0492308\pi$
$823$	18.0000	+	31.1769i	0.627441	+	1.08676i	−0.360623	+	0.932712i	$0.617436\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−28.0000			−0.973655			−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000			−0.973655			−0.486828	+	0.873498i	$0.661846\pi$
$828$		0			0
$829$	20.5000	−	35.5070i	0.711994	−	1.23321i	−0.252113	−	0.967698i	$-0.581125\pi$
$829$	20.5000	−	35.5070i	0.711994	−	1.23321i	0.964107	−	0.265513i	$-0.0855412\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−7.00000			−0.242536
$834$		0			0
$835$	−18.0000	+	31.1769i	−0.622916	+	1.07892i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6.00000	+	10.3923i	0.207143	+	0.358782i	0.950813	−	0.309764i	$-0.100250\pi$
$839$	6.00000	+	10.3923i	0.207143	+	0.358782i	−0.743670	+	0.668546i	$0.766917\pi$
$840$		0			0
$841$	10.0000	+	17.3205i	0.344828	+	0.597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−34.5000	−	18.1865i	−1.18684	−	0.625636i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−10.0000	−	17.3205i	−0.342796	−	0.593739i
$852$		0			0
$853$	−17.0000			−0.582069			−0.291034	−	0.956713i	$-0.593999\pi$
$853$	−17.0000			−0.582069			−0.291034	+	0.956713i	$0.593999\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−27.0000			−0.922302			−0.461151	−	0.887322i	$-0.652563\pi$
$857$	−27.0000			−0.922302			−0.461151	+	0.887322i	$0.652563\pi$
$858$		0			0
$859$	36.0000			1.22830			0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000			1.22830			0.614152	+	0.789188i	$0.289498\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$		0			0
$865$	27.0000	+	46.7654i	0.918028	+	1.59007i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−30.0000	+	31.1769i	−1.01651	+	1.05639i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	20.5000	+	35.5070i	0.692236	+	1.19899i	0.971104	+	0.238658i	$0.0767075\pi$
$877$	20.5000	+	35.5070i	0.692236	+	1.19899i	−0.278868	+	0.960329i	$0.589959\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−10.5000	+	18.1865i	−0.353754	+	0.612720i	−0.986904	−	0.161309i	$-0.948428\pi$
$881$	−10.5000	+	18.1865i	−0.353754	+	0.612720i	0.633150	+	0.774029i	$0.281762\pi$
$882$		0			0
$883$	−44.0000			−1.48072			−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000			−1.48072			−0.740359	+	0.672212i	$0.765344\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.00000	+	10.3923i	−0.201460	+	0.348939i	−0.948999	−	0.315279i	$-0.897902\pi$
$887$	−6.00000	+	10.3923i	−0.201460	+	0.348939i	0.747539	+	0.664218i	$0.231235\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−6.00000	+	10.3923i	−0.200558	+	0.347376i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	12.0000	−	20.7846i	0.400222	−	0.693206i
$900$		0			0
$901$	−6.50000	−	11.2583i	−0.216546	−	0.375069i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−33.0000			−1.09696
$906$		0			0
$907$	−24.0000	+	41.5692i	−0.796907	+	1.38028i	0.124714	+	0.992193i	$0.460199\pi$
$907$	−24.0000	+	41.5692i	−0.796907	+	1.38028i	−0.921621	+	0.388091i	$0.873135\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	6.00000	+	10.3923i	0.197922	+	0.342811i	0.947854	−	0.318704i	$-0.103247\pi$
$919$	6.00000	+	10.3923i	0.197922	+	0.342811i	−0.749933	+	0.661514i	$0.769914\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8.00000	+	27.7128i	0.263323	+	0.912178i
$924$		0			0
$925$	10.0000	−	17.3205i	0.328798	−	0.569495i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−16.5000	−	28.5788i	−0.541347	−	0.937641i	−0.998827	−	0.0484211i	$-0.984581\pi$
$929$	−16.5000	−	28.5788i	−0.541347	−	0.937641i	0.457480	−	0.889220i	$-0.348752\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−37.0000			−1.20874			−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000			−1.20874			−0.604369	+	0.796705i	$0.706575\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−46.0000			−1.49956			−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000			−1.49956			−0.749779	+	0.661689i	$0.769840\pi$
$942$		0			0
$943$	−6.00000	−	10.3923i	−0.195387	−	0.338420i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−14.0000	+	24.2487i	−0.454939	+	0.787977i	−0.998685	−	0.0512727i	$-0.983672\pi$
$947$	−14.0000	+	24.2487i	−0.454939	+	0.787977i	0.543746	+	0.839250i	$0.317006\pi$
$948$		0			0
$949$	10.5000	+	2.59808i	0.340844	+	0.0843371i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−19.0000	−	32.9090i	−0.615470	−	1.06603i	−0.990302	−	0.138933i	$-0.955633\pi$
$953$	−19.0000	−	32.9090i	−0.615470	−	1.06603i	0.374831	−	0.927093i	$-0.377701\pi$
$954$		0			0
$955$	12.0000	+	20.7846i	0.388311	+	0.672574i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.5000	−	28.5788i	0.531154	−	0.919985i
$966$		0			0
$967$	−32.0000			−1.02905			−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000			−1.02905			−0.514525	+	0.857475i	$0.672032\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.00000	+	3.46410i	0.0641831	+	0.111168i	0.896331	−	0.443385i	$-0.146223\pi$
$971$	2.00000	+	3.46410i	0.0641831	+	0.111168i	−0.832148	+	0.554553i	$0.812889\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−28.5000	+	49.3634i	−0.911796	+	1.57928i	−0.100270	+	0.994960i	$0.531971\pi$
$977$	−28.5000	+	49.3634i	−0.911796	+	1.57928i	−0.811526	+	0.584316i	$0.801363\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.0000			0.510321			0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000			0.510321			0.255160	+	0.966899i	$0.417872\pi$
$984$		0			0
$985$	−9.00000	+	15.5885i	−0.286764	+	0.496690i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.0000			−0.508770
$990$		0			0
$991$	−16.0000	+	27.7128i	−0.508257	+	0.880327i	0.491698	+	0.870766i	$0.336377\pi$
$991$	−16.0000	+	27.7128i	−0.508257	+	0.880327i	−0.999954	+	0.00956046i	$0.996957\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6.00000	+	10.3923i	0.190213	+	0.329458i
$996$		0			0
$997$	−23.5000	−	40.7032i	−0.744252	−	1.28908i	−0.950543	−	0.310592i	$-0.899473\pi$
$997$	−23.5000	−	40.7032i	−0.744252	−	1.28908i	0.206291	−	0.978491i	$-0.433861\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	936.2.t.a.217.1		2
3.2	odd	2		312.2.q.a.217.1	✓	2
4.3	odd	2		1872.2.t.b.1153.1		2
12.11	even	2		624.2.q.f.529.1		2
13.3	even	3	inner	936.2.t.a.289.1		2
39.17	odd	6		4056.2.a.l.1.1		1
39.20	even	12		4056.2.c.i.337.2		2
39.29	odd	6		312.2.q.a.289.1	yes	2
39.32	even	12		4056.2.c.i.337.1		2
39.35	odd	6		4056.2.a.q.1.1		1
52.3	odd	6		1872.2.t.b.289.1		2
156.35	even	6		8112.2.a.n.1.1		1
156.95	even	6		8112.2.a.b.1.1		1
156.107	even	6		624.2.q.f.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.q.a.217.1	✓	2	3.2	odd	2
312.2.q.a.289.1	yes	2	39.29	odd	6
624.2.q.f.289.1		2	156.107	even	6
624.2.q.f.529.1		2	12.11	even	2
936.2.t.a.217.1		2	1.1	even	1	trivial
936.2.t.a.289.1		2	13.3	even	3	inner
1872.2.t.b.289.1		2	52.3	odd	6
1872.2.t.b.1153.1		2	4.3	odd	2
4056.2.a.l.1.1		1	39.17	odd	6
4056.2.a.q.1.1		1	39.35	odd	6
4056.2.c.i.337.1		2	39.32	even	12
4056.2.c.i.337.2		2	39.20	even	12
8112.2.a.b.1.1		1	156.95	even	6
8112.2.a.n.1.1		1	156.35	even	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:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	936.t (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{5} +(3.50000 + 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{17} +(2.00000 - 3.46410i) q^{23} +4.00000 q^{25} +(1.50000 - 2.59808i) q^{29} +8.00000 q^{31} +(2.50000 - 4.33013i) q^{37} +(1.50000 - 2.59808i) q^{41} +(-2.00000 - 3.46410i) q^{43} +8.00000 q^{47} +(3.50000 - 6.06218i) q^{49} +13.0000 q^{53} +(6.00000 + 10.3923i) q^{59} +(-7.50000 - 12.9904i) q^{61} +(-10.5000 - 2.59808i) q^{65} +(-6.00000 + 10.3923i) q^{67} +(4.00000 + 6.92820i) q^{71} +3.00000 q^{73} -4.00000 q^{79} -12.0000 q^{83} +(1.50000 + 2.59808i) q^{85} +(5.00000 - 8.66025i) q^{89} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} + 7 q^{13} - q^{17} + 4 q^{23} + 8 q^{25} + 3 q^{29} + 16 q^{31} + 5 q^{37} + 3 q^{41} - 4 q^{43} + 16 q^{47} + 7 q^{49} + 26 q^{53} + 12 q^{59} - 15 q^{61} - 21 q^{65} - 12 q^{67} + 8 q^{71}+ \cdots - 2 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 + 7 * q^13 - q^17 + 4 * q^23 + 8 * q^25 + 3 * q^29 + 16 * q^31 + 5 * q^37 + 3 * q^41 - 4 * q^43 + 16 * q^47 + 7 * q^49 + 26 * q^53 + 12 * q^59 - 15 * q^61 - 21 * q^65 - 12 * q^67 + 8 * q^71 + 6 * q^73 - 8 * q^79 - 24 * q^83 + 3 * q^85 + 10 * q^89 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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