LMFDB - Embedded newform 416.2.e.a.337.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(337,416)] 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("416.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	416.337
Dual form			416.2.e.a.337.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.00000 q^{5} -3.00000i q^{7} +2.00000 q^{9} +2.00000 q^{11} +(3.00000 + 2.00000i) q^{13} -1.00000i q^{15} +3.00000 q^{17} +3.00000 q^{21} +6.00000 q^{23} -4.00000 q^{25} +5.00000i q^{27} +6.00000i q^{29} +2.00000i q^{33} +3.00000i q^{35} -3.00000 q^{37} +(-2.00000 + 3.00000i) q^{39} -10.0000i q^{41} -9.00000i q^{43} -2.00000 q^{45} +7.00000i q^{47} -2.00000 q^{49} +3.00000i q^{51} -6.00000i q^{53} -2.00000 q^{55} +10.0000 q^{59} -10.0000i q^{61} -6.00000i q^{63} +(-3.00000 - 2.00000i) q^{65} -12.0000 q^{67} +6.00000i q^{69} +5.00000i q^{71} +6.00000i q^{73} -4.00000i q^{75} -6.00000i q^{77} +1.00000 q^{81} -16.0000 q^{83} -3.00000 q^{85} -6.00000 q^{87} +4.00000i q^{89} +(6.00000 - 9.00000i) q^{91} +18.0000i q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{9} + 4 q^{11} + 6 q^{13} + 6 q^{17} + 6 q^{21} + 12 q^{23} - 8 q^{25} - 6 q^{37} - 4 q^{39} - 4 q^{45} - 4 q^{49} - 4 q^{55} + 20 q^{59} - 6 q^{65} - 24 q^{67} + 2 q^{81} - 32 q^{83}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^9 + 4 * q^11 + 6 * q^13 + 6 * q^17 + 6 * q^21 + 12 * q^23 - 8 * q^25 - 6 * q^37 - 4 * q^39 - 4 * q^45 - 4 * q^49 - 4 * q^55 + 20 * q^59 - 6 * q^65 - 24 * q^67 + 2 * q^81 - 32 * q^83 - 6 * q^85 - 12 * q^87 + 12 * q^91 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$			1.00000i			0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$		0			0
$5$	−1.00000			−0.447214			−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000			−0.447214			−0.223607	+	0.974679i	$0.571783\pi$
$6$		0			0
$7$		−	3.00000i		−	1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$		−	3.00000i		−	1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$		0			0
$9$	2.00000			0.666667
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	3.00000			0.654654
$22$		0			0
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$			5.00000i			0.962250i
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$			2.00000i			0.348155i
$34$		0			0
$35$			3.00000i			0.507093i
$36$		0			0
$37$	−3.00000			−0.493197			−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000			−0.493197			−0.246598	+	0.969118i	$0.579313\pi$
$38$		0			0
$39$	−2.00000	+	3.00000i	−0.320256	+	0.480384i
$40$		0			0
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$		0			0
$43$		−	9.00000i		−	1.37249i	−0.727372	−	0.686244i	$-0.759258\pi$
$43$		−	9.00000i		−	1.37249i	0.727372	−	0.686244i	$-0.240742\pi$
$44$		0			0
$45$	−2.00000			−0.298142
$46$		0			0
$47$			7.00000i			1.02105i	0.859861	+	0.510527i	$0.170550\pi$
$47$			7.00000i			1.02105i	−0.859861	+	0.510527i	$0.829450\pi$
$48$		0			0
$49$	−2.00000			−0.285714
$50$		0			0
$51$			3.00000i			0.420084i
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	10.0000			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$60$		0			0
$61$		−	10.0000i		−	1.28037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$		−	10.0000i		−	1.28037i	0.768221	−	0.640184i	$-0.221142\pi$
$62$		0			0
$63$		−	6.00000i		−	0.755929i
$64$		0			0
$65$	−3.00000	−	2.00000i	−0.372104	−	0.248069i
$66$		0			0
$67$	−12.0000			−1.46603			−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000			−1.46603			−0.733017	+	0.680211i	$0.761888\pi$
$68$		0			0
$69$			6.00000i			0.722315i
$70$		0			0
$71$			5.00000i			0.593391i	0.954972	+	0.296695i	$0.0958846\pi$
$71$			5.00000i			0.593391i	−0.954972	+	0.296695i	$0.904115\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$		−	4.00000i		−	0.461880i
$76$		0			0
$77$		−	6.00000i		−	0.683763i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$	−3.00000			−0.325396
$86$		0			0
$87$	−6.00000			−0.643268
$88$		0			0
$89$			4.00000i			0.423999i	0.977270	+	0.212000i	$0.0679975\pi$
$89$			4.00000i			0.423999i	−0.977270	+	0.212000i	$0.932002\pi$
$90$		0			0
$91$	6.00000	−	9.00000i	0.628971	−	0.943456i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			18.0000i			1.82762i	0.406138	+	0.913812i	$0.366875\pi$
$97$			18.0000i			1.82762i	−0.406138	+	0.913812i	$0.633125\pi$
$98$		0			0
$99$	4.00000			0.402015
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$	−3.00000			−0.292770
$106$		0			0
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$	−15.0000			−1.43674			−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000			−1.43674			−0.718370	+	0.695662i	$0.755111\pi$
$110$		0			0
$111$		−	3.00000i		−	0.284747i
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	−6.00000			−0.559503
$116$		0			0
$117$	6.00000	+	4.00000i	0.554700	+	0.369800i
$118$		0			0
$119$		−	9.00000i		−	0.825029i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$	10.0000			0.901670
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$	−18.0000			−1.59724			−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000			−1.59724			−0.798621	+	0.601834i	$0.794437\pi$
$128$		0			0
$129$	9.00000			0.792406
$130$		0			0
$131$			15.0000i			1.31056i	0.755388	+	0.655278i	$0.227449\pi$
$131$			15.0000i			1.31056i	−0.755388	+	0.655278i	$0.772551\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		−	5.00000i		−	0.430331i
$136$		0			0
$137$		−	2.00000i		−	0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$		−	2.00000i		−	0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$		0			0
$139$			9.00000i			0.763370i	0.924292	+	0.381685i	$0.124656\pi$
$139$			9.00000i			0.763370i	−0.924292	+	0.381685i	$0.875344\pi$
$140$		0			0
$141$	−7.00000			−0.589506
$142$		0			0
$143$	6.00000	+	4.00000i	0.501745	+	0.334497i
$144$		0			0
$145$		−	6.00000i		−	0.498273i
$146$		0			0
$147$		−	2.00000i		−	0.164957i
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$			15.0000i			1.22068i	0.792139	+	0.610341i	$0.208968\pi$
$151$			15.0000i			1.22068i	−0.792139	+	0.610341i	$0.791032\pi$
$152$		0			0
$153$	6.00000			0.485071
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	18.0000i		−	1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$		−	18.0000i		−	1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$		0			0
$159$	6.00000			0.475831
$160$		0			0
$161$		−	18.0000i		−	1.41860i
$162$		0			0
$163$	−6.00000			−0.469956			−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000			−0.469956			−0.234978	+	0.972001i	$0.575502\pi$
$164$		0			0
$165$		−	2.00000i		−	0.155700i
$166$		0			0
$167$		−	8.00000i		−	0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$		−	8.00000i		−	0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$			24.0000i			1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$			24.0000i			1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$		0			0
$175$			12.0000i			0.907115i
$176$		0			0
$177$			10.0000i			0.751646i
$178$		0			0
$179$			9.00000i			0.672692i	0.941739	+	0.336346i	$0.109191\pi$
$179$			9.00000i			0.672692i	−0.941739	+	0.336346i	$0.890809\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	10.0000			0.739221
$184$		0			0
$185$	3.00000			0.220564
$186$		0			0
$187$	6.00000			0.438763
$188$		0			0
$189$	15.0000			1.09109
$190$		0			0
$191$	18.0000			1.30243			0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000			1.30243			0.651217	+	0.758891i	$0.274259\pi$
$192$		0			0
$193$		−	24.0000i		−	1.72756i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$		−	24.0000i		−	1.72756i	0.503871	−	0.863779i	$-0.331909\pi$
$194$		0			0
$195$	2.00000	−	3.00000i	0.143223	−	0.214834i
$196$		0			0
$197$	−23.0000			−1.63868			−0.819341	−	0.573306i	$-0.805660\pi$
$197$	−23.0000			−1.63868			−0.819341	+	0.573306i	$0.805660\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		−	12.0000i		−	0.846415i
$202$		0			0
$203$	18.0000			1.26335
$204$		0			0
$205$			10.0000i			0.698430i
$206$		0			0
$207$	12.0000			0.834058
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	5.00000i		−	0.344214i	−0.985078	−	0.172107i	$-0.944942\pi$
$211$		−	5.00000i		−	0.344214i	0.985078	−	0.172107i	$-0.0550575\pi$
$212$		0			0
$213$	−5.00000			−0.342594
$214$		0			0
$215$			9.00000i			0.613795i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−6.00000			−0.405442
$220$		0			0
$221$	9.00000	+	6.00000i	0.605406	+	0.403604i
$222$		0			0
$223$		−	21.0000i		−	1.40626i	−0.711059	−	0.703132i	$-0.751784\pi$
$223$		−	21.0000i		−	1.40626i	0.711059	−	0.703132i	$-0.248216\pi$
$224$		0			0
$225$	−8.00000			−0.533333
$226$		0			0
$227$	−2.00000			−0.132745			−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000			−0.132745			−0.0663723	+	0.997795i	$0.521143\pi$
$228$		0			0
$229$	15.0000			0.991228			0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000			0.991228			0.495614	+	0.868543i	$0.334943\pi$
$230$		0			0
$231$	6.00000			0.394771
$232$		0			0
$233$	9.00000			0.589610			0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000			0.589610			0.294805	+	0.955557i	$0.404745\pi$
$234$		0			0
$235$		−	7.00000i		−	0.456630i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	19.0000i		−	1.22901i	−0.788914	−	0.614504i	$-0.789356\pi$
$239$		−	19.0000i		−	1.22901i	0.788914	−	0.614504i	$-0.210644\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$			16.0000i			1.02640i
$244$		0			0
$245$	2.00000			0.127775
$246$		0			0
$247$		0			0
$248$		0			0
$249$		−	16.0000i		−	1.01396i
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$		0			0
$255$		−	3.00000i		−	0.187867i
$256$		0			0
$257$	3.00000			0.187135			0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000			0.187135			0.0935674	+	0.995613i	$0.470173\pi$
$258$		0			0
$259$			9.00000i			0.559233i
$260$		0			0
$261$			12.0000i			0.742781i
$262$		0			0
$263$	6.00000			0.369976			0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000			0.369976			0.184988	+	0.982741i	$0.440775\pi$
$264$		0			0
$265$			6.00000i			0.368577i
$266$		0			0
$267$	−4.00000			−0.244796
$268$		0			0
$269$			6.00000i			0.365826i	0.983129	+	0.182913i	$0.0585527\pi$
$269$			6.00000i			0.365826i	−0.983129	+	0.182913i	$0.941447\pi$
$270$		0			0
$271$			15.0000i			0.911185i	0.890188	+	0.455593i	$0.150573\pi$
$271$			15.0000i			0.911185i	−0.890188	+	0.455593i	$0.849427\pi$
$272$		0			0
$273$	9.00000	+	6.00000i	0.544705	+	0.363137i
$274$		0			0
$275$	−8.00000			−0.482418
$276$		0			0
$277$		−	18.0000i		−	1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$		−	18.0000i		−	1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			20.0000i			1.19310i	0.802576	+	0.596550i	$0.203462\pi$
$281$			20.0000i			1.19310i	−0.802576	+	0.596550i	$0.796538\pi$
$282$		0			0
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−30.0000			−1.77084
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$	−18.0000			−1.05518
$292$		0			0
$293$	1.00000			0.0584206			0.0292103	−	0.999573i	$-0.490701\pi$
$293$	1.00000			0.0584206			0.0292103	+	0.999573i	$0.490701\pi$
$294$		0			0
$295$	−10.0000			−0.582223
$296$		0			0
$297$			10.0000i			0.580259i
$298$		0			0
$299$	18.0000	+	12.0000i	1.04097	+	0.693978i
$300$		0			0
$301$	−27.0000			−1.55625
$302$		0			0
$303$		0			0
$304$		0			0
$305$			10.0000i			0.572598i
$306$		0			0
$307$	18.0000			1.02731			0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000			1.02731			0.513657	+	0.857996i	$0.328290\pi$
$308$		0			0
$309$		−	4.00000i		−	0.227552i
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	9.00000			0.508710			0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000			0.508710			0.254355	+	0.967111i	$0.418137\pi$
$314$		0			0
$315$			6.00000i			0.338062i
$316$		0			0
$317$	2.00000			0.112331			0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000			0.112331			0.0561656	+	0.998421i	$0.482113\pi$
$318$		0			0
$319$			12.0000i			0.671871i
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−12.0000	−	8.00000i	−0.665640	−	0.443760i
$326$		0			0
$327$		−	15.0000i		−	0.829502i
$328$		0			0
$329$	21.0000			1.15777
$330$		0			0
$331$	12.0000			0.659580			0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000			0.659580			0.329790	+	0.944054i	$0.393022\pi$
$332$		0			0
$333$	−6.00000			−0.328798
$334$		0			0
$335$	12.0000			0.655630
$336$		0			0
$337$	−27.0000			−1.47078			−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000			−1.47078			−0.735392	+	0.677642i	$0.763002\pi$
$338$		0			0
$339$		−	6.00000i		−	0.325875i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	15.0000i		−	0.809924i
$344$		0			0
$345$		−	6.00000i		−	0.323029i
$346$		0			0
$347$		−	27.0000i		−	1.44944i	−0.689046	−	0.724718i	$-0.741970\pi$
$347$		−	27.0000i		−	1.44944i	0.689046	−	0.724718i	$-0.258030\pi$
$348$		0			0
$349$	15.0000			0.802932			0.401466	−	0.915874i	$-0.368501\pi$
$349$	15.0000			0.802932			0.401466	+	0.915874i	$0.368501\pi$
$350$		0			0
$351$	−10.0000	+	15.0000i	−0.533761	+	0.800641i
$352$		0			0
$353$			16.0000i			0.851594i	0.904819	+	0.425797i	$0.140006\pi$
$353$			16.0000i			0.851594i	−0.904819	+	0.425797i	$0.859994\pi$
$354$		0			0
$355$		−	5.00000i		−	0.265372i
$356$		0			0
$357$	9.00000			0.476331
$358$		0			0
$359$		−	4.00000i		−	0.211112i	−0.994413	−	0.105556i	$-0.966338\pi$
$359$		−	4.00000i		−	0.211112i	0.994413	−	0.105556i	$-0.0336622\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		−	7.00000i		−	0.367405i
$364$		0			0
$365$		−	6.00000i		−	0.314054i
$366$		0			0
$367$	−28.0000			−1.46159			−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000			−1.46159			−0.730794	+	0.682598i	$0.760850\pi$
$368$		0			0
$369$		−	20.0000i		−	1.04116i
$370$		0			0
$371$	−18.0000			−0.934513
$372$		0			0
$373$			4.00000i			0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$			4.00000i			0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$		0			0
$375$			9.00000i			0.464758i
$376$		0			0
$377$	−12.0000	+	18.0000i	−0.618031	+	0.927047i
$378$		0			0
$379$	−30.0000			−1.54100			−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000			−1.54100			−0.770498	+	0.637442i	$0.779993\pi$
$380$		0			0
$381$		−	18.0000i		−	0.922168i
$382$		0			0
$383$		−	1.00000i		−	0.0510976i	−0.999674	−	0.0255488i	$-0.991867\pi$
$383$		−	1.00000i		−	0.0510976i	0.999674	−	0.0255488i	$-0.00813332\pi$
$384$		0			0
$385$			6.00000i			0.305788i
$386$		0			0
$387$		−	18.0000i		−	0.914991i
$388$		0			0
$389$			6.00000i			0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$			6.00000i			0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$		0			0
$393$	−15.0000			−0.756650
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−18.0000			−0.903394			−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000			−0.903394			−0.451697	+	0.892171i	$0.649181\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	10.0000i		−	0.499376i	−0.968326	−	0.249688i	$-0.919672\pi$
$401$		−	10.0000i		−	0.499376i	0.968326	−	0.249688i	$-0.0803281\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	−6.00000			−0.297409
$408$		0			0
$409$		−	6.00000i		−	0.296681i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$		−	6.00000i		−	0.296681i	0.988936	−	0.148340i	$-0.0473931\pi$
$410$		0			0
$411$	2.00000			0.0986527
$412$		0			0
$413$		−	30.0000i		−	1.47620i
$414$		0			0
$415$	16.0000			0.785409
$416$		0			0
$417$	−9.00000			−0.440732
$418$		0			0
$419$			9.00000i			0.439679i	0.975536	+	0.219839i	$0.0705533\pi$
$419$			9.00000i			0.439679i	−0.975536	+	0.219839i	$0.929447\pi$
$420$		0			0
$421$	3.00000			0.146211			0.0731055	−	0.997324i	$-0.476709\pi$
$421$	3.00000			0.146211			0.0731055	+	0.997324i	$0.476709\pi$
$422$		0			0
$423$			14.0000i			0.680703i
$424$		0			0
$425$	−12.0000			−0.582086
$426$		0			0
$427$	−30.0000			−1.45180
$428$		0			0
$429$	−4.00000	+	6.00000i	−0.193122	+	0.289683i
$430$		0			0
$431$			5.00000i			0.240842i	0.992723	+	0.120421i	$0.0384244\pi$
$431$			5.00000i			0.240842i	−0.992723	+	0.120421i	$0.961576\pi$
$432$		0			0
$433$	9.00000			0.432512			0.216256	−	0.976337i	$-0.430615\pi$
$433$	9.00000			0.432512			0.216256	+	0.976337i	$0.430615\pi$
$434$		0			0
$435$	6.00000			0.287678
$436$		0			0
$437$		0			0
$438$		0			0
$439$	10.0000			0.477274			0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000			0.477274			0.238637	+	0.971109i	$0.423299\pi$
$440$		0			0
$441$	−4.00000			−0.190476
$442$		0			0
$443$		−	9.00000i		−	0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$		−	9.00000i		−	0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$		0			0
$445$		−	4.00000i		−	0.189618i
$446$		0			0
$447$			10.0000i			0.472984i
$448$		0			0
$449$			14.0000i			0.660701i	0.943858	+	0.330350i	$0.107167\pi$
$449$			14.0000i			0.660701i	−0.943858	+	0.330350i	$0.892833\pi$
$450$		0			0
$451$		−	20.0000i		−	0.941763i
$452$		0			0
$453$	−15.0000			−0.704761
$454$		0			0
$455$	−6.00000	+	9.00000i	−0.281284	+	0.421927i
$456$		0			0
$457$		−	12.0000i		−	0.561336i	−0.959805	−	0.280668i	$-0.909444\pi$
$457$		−	12.0000i		−	0.561336i	0.959805	−	0.280668i	$-0.0905560\pi$
$458$		0			0
$459$			15.0000i			0.700140i
$460$		0			0
$461$	−7.00000			−0.326023			−0.163011	−	0.986624i	$-0.552121\pi$
$461$	−7.00000			−0.326023			−0.163011	+	0.986624i	$0.552121\pi$
$462$		0			0
$463$			24.0000i			1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$			24.0000i			1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	12.0000i		−	0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$		−	12.0000i		−	0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$		0			0
$469$			36.0000i			1.66233i
$470$		0			0
$471$	18.0000			0.829396
$472$		0			0
$473$		−	18.0000i		−	0.827641i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	12.0000i		−	0.549442i
$478$		0			0
$479$		−	29.0000i		−	1.32504i	−0.749043	−	0.662522i	$-0.769486\pi$
$479$		−	29.0000i		−	1.32504i	0.749043	−	0.662522i	$-0.230514\pi$
$480$		0			0
$481$	−9.00000	−	6.00000i	−0.410365	−	0.273576i
$482$		0			0
$483$	18.0000			0.819028
$484$		0			0
$485$		−	18.0000i		−	0.817338i
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		0			0
$489$		−	6.00000i		−	0.271329i
$490$		0			0
$491$			15.0000i			0.676941i	0.940977	+	0.338470i	$0.109909\pi$
$491$			15.0000i			0.676941i	−0.940977	+	0.338470i	$0.890091\pi$
$492$		0			0
$493$			18.0000i			0.810679i
$494$		0			0
$495$	−4.00000			−0.179787
$496$		0			0
$497$	15.0000			0.672842
$498$		0			0
$499$	30.0000			1.34298			0.671492	−	0.741012i	$-0.265654\pi$
$499$	30.0000			1.34298			0.671492	+	0.741012i	$0.265654\pi$
$500$		0			0
$501$	8.00000			0.357414
$502$		0			0
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−12.0000	+	5.00000i	−0.532939	+	0.222058i
$508$		0			0
$509$	10.0000			0.443242			0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000			0.443242			0.221621	+	0.975133i	$0.428865\pi$
$510$		0			0
$511$	18.0000			0.796273
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.00000			0.176261
$516$		0			0
$517$			14.0000i			0.615719i
$518$		0			0
$519$	−24.0000			−1.05348
$520$		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
$523$			36.0000i			1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$			36.0000i			1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$		0			0
$525$	−12.0000			−0.523723
$526$		0			0
$527$		0			0
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$	20.0000			0.867926
$532$		0			0
$533$	20.0000	−	30.0000i	0.866296	−	1.29944i
$534$		0			0
$535$			12.0000i			0.518805i
$536$		0			0
$537$	−9.00000			−0.388379
$538$		0			0
$539$	−4.00000			−0.172292
$540$		0			0
$541$	33.0000			1.41878			0.709390	−	0.704816i	$-0.248970\pi$
$541$	33.0000			1.41878			0.709390	+	0.704816i	$0.248970\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	15.0000			0.642529
$546$		0			0
$547$		−	27.0000i		−	1.15444i	−0.816590	−	0.577218i	$-0.804138\pi$
$547$		−	27.0000i		−	1.15444i	0.816590	−	0.577218i	$-0.195862\pi$
$548$		0			0
$549$		−	20.0000i		−	0.853579i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$			3.00000i			0.127343i
$556$		0			0
$557$	−23.0000			−0.974541			−0.487271	−	0.873251i	$-0.662007\pi$
$557$	−23.0000			−0.974541			−0.487271	+	0.873251i	$0.662007\pi$
$558$		0			0
$559$	18.0000	−	27.0000i	0.761319	−	1.14198i
$560$		0			0
$561$			6.00000i			0.253320i
$562$		0			0
$563$			21.0000i			0.885044i	0.896758	+	0.442522i	$0.145916\pi$
$563$			21.0000i			0.885044i	−0.896758	+	0.442522i	$0.854084\pi$
$564$		0			0
$565$	6.00000			0.252422
$566$		0			0
$567$		−	3.00000i		−	0.125988i
$568$		0			0
$569$	15.0000			0.628833			0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000			0.628833			0.314416	+	0.949285i	$0.398191\pi$
$570$		0			0
$571$			5.00000i			0.209243i	0.994512	+	0.104622i	$0.0333632\pi$
$571$			5.00000i			0.209243i	−0.994512	+	0.104622i	$0.966637\pi$
$572$		0			0
$573$			18.0000i			0.751961i
$574$		0			0
$575$	−24.0000			−1.00087
$576$		0			0
$577$			18.0000i			0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$			18.0000i			0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$		0			0
$579$	24.0000			0.997406
$580$		0			0
$581$			48.0000i			1.99138i
$582$		0			0
$583$		−	12.0000i		−	0.496989i
$584$		0			0
$585$	−6.00000	−	4.00000i	−0.248069	−	0.165380i
$586$		0			0
$587$	−2.00000			−0.0825488			−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000			−0.0825488			−0.0412744	+	0.999148i	$0.513142\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		−	23.0000i		−	0.946094i
$592$		0			0
$593$		−	14.0000i		−	0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$		−	14.0000i		−	0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$		0			0
$595$			9.00000i			0.368964i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	30.0000			1.22577			0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000			1.22577			0.612883	+	0.790173i	$0.290010\pi$
$600$		0			0
$601$	27.0000			1.10135			0.550676	−	0.834719i	$-0.314370\pi$
$601$	27.0000			1.10135			0.550676	+	0.834719i	$0.314370\pi$
$602$		0			0
$603$	−24.0000			−0.977356
$604$		0			0
$605$	7.00000			0.284590
$606$		0			0
$607$	22.0000			0.892952			0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000			0.892952			0.446476	+	0.894795i	$0.352679\pi$
$608$		0			0
$609$			18.0000i			0.729397i
$610$		0			0
$611$	−14.0000	+	21.0000i	−0.566379	+	0.849569i
$612$		0			0
$613$	6.00000			0.242338			0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000			0.242338			0.121169	+	0.992632i	$0.461336\pi$
$614$		0			0
$615$	−10.0000			−0.403239
$616$		0			0
$617$			8.00000i			0.322068i	0.986949	+	0.161034i	$0.0514829\pi$
$617$			8.00000i			0.322068i	−0.986949	+	0.161034i	$0.948517\pi$
$618$		0			0
$619$	−30.0000			−1.20580			−0.602901	−	0.797816i	$-0.705989\pi$
$619$	−30.0000			−1.20580			−0.602901	+	0.797816i	$0.705989\pi$
$620$		0			0
$621$			30.0000i			1.20386i
$622$		0			0
$623$	12.0000			0.480770
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9.00000			−0.358854
$630$		0			0
$631$		−	45.0000i		−	1.79142i	−0.444637	−	0.895711i	$-0.646667\pi$
$631$		−	45.0000i		−	1.79142i	0.444637	−	0.895711i	$-0.353333\pi$
$632$		0			0
$633$	5.00000			0.198732
$634$		0			0
$635$	18.0000			0.714308
$636$		0			0
$637$	−6.00000	−	4.00000i	−0.237729	−	0.158486i
$638$		0			0
$639$			10.0000i			0.395594i
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$	24.0000			0.946468			0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000			0.946468			0.473234	+	0.880937i	$0.343087\pi$
$644$		0			0
$645$	−9.00000			−0.354375
$646$		0			0
$647$	−18.0000			−0.707653			−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000			−0.707653			−0.353827	+	0.935311i	$0.615120\pi$
$648$		0			0
$649$	20.0000			0.785069
$650$		0			0
$651$		0			0
$652$		0			0
$653$			24.0000i			0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$			24.0000i			0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$		0			0
$655$		−	15.0000i		−	0.586098i
$656$		0			0
$657$			12.0000i			0.468165i
$658$		0			0
$659$		−	36.0000i		−	1.40236i	−0.712984	−	0.701180i	$-0.752657\pi$
$659$		−	36.0000i		−	1.40236i	0.712984	−	0.701180i	$-0.247343\pi$
$660$		0			0
$661$	18.0000			0.700119			0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000			0.700119			0.350059	+	0.936727i	$0.386161\pi$
$662$		0			0
$663$	−6.00000	+	9.00000i	−0.233021	+	0.349531i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			36.0000i			1.39393i
$668$		0			0
$669$	21.0000			0.811907
$670$		0			0
$671$		−	20.0000i		−	0.772091i
$672$		0			0
$673$	19.0000			0.732396			0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000			0.732396			0.366198	+	0.930537i	$0.380659\pi$
$674$		0			0
$675$		−	20.0000i		−	0.769800i
$676$		0			0
$677$			12.0000i			0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$			12.0000i			0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$		0			0
$679$	54.0000			2.07233
$680$		0			0
$681$		−	2.00000i		−	0.0766402i
$682$		0			0
$683$	4.00000			0.153056			0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000			0.153056			0.0765279	+	0.997067i	$0.475617\pi$
$684$		0			0
$685$			2.00000i			0.0764161i
$686$		0			0
$687$			15.0000i			0.572286i
$688$		0			0
$689$	12.0000	−	18.0000i	0.457164	−	0.685745i
$690$		0			0
$691$	12.0000			0.456502			0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000			0.456502			0.228251	+	0.973602i	$0.426699\pi$
$692$		0			0
$693$		−	12.0000i		−	0.455842i
$694$		0			0
$695$		−	9.00000i		−	0.341389i
$696$		0			0
$697$		−	30.0000i		−	1.13633i
$698$		0			0
$699$			9.00000i			0.340411i
$700$		0			0
$701$		−	30.0000i		−	1.13308i	−0.824033	−	0.566542i	$-0.808281\pi$
$701$		−	30.0000i		−	1.13308i	0.824033	−	0.566542i	$-0.191719\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	7.00000			0.263635
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−30.0000			−1.12667			−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000			−1.12667			−0.563337	+	0.826227i	$0.690483\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−6.00000	−	4.00000i	−0.224387	−	0.149592i
$716$		0			0
$717$	19.0000			0.709568
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$			12.0000i			0.446903i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	24.0000i		−	0.891338i
$726$		0			0
$727$	−18.0000			−0.667583			−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000			−0.667583			−0.333792	+	0.942647i	$0.608328\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$		−	27.0000i		−	0.998631i
$732$		0			0
$733$	51.0000			1.88373			0.941864	−	0.335994i	$-0.109072\pi$
$733$	51.0000			1.88373			0.941864	+	0.335994i	$0.109072\pi$
$734$		0			0
$735$			2.00000i			0.0737711i
$736$		0			0
$737$	−24.0000			−0.884051
$738$		0			0
$739$	30.0000			1.10357			0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000			1.10357			0.551784	+	0.833987i	$0.313947\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			19.0000i			0.697042i	0.937301	+	0.348521i	$0.113316\pi$
$743$			19.0000i			0.697042i	−0.937301	+	0.348521i	$0.886684\pi$
$744$		0			0
$745$	−10.0000			−0.366372
$746$		0			0
$747$	−32.0000			−1.17082
$748$		0			0
$749$	−36.0000			−1.31541
$750$		0			0
$751$	−22.0000			−0.802791			−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000			−0.802791			−0.401396	+	0.915905i	$0.631475\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	15.0000i		−	0.545906i
$756$		0			0
$757$		−	38.0000i		−	1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$		−	38.0000i		−	1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$		0			0
$759$			12.0000i			0.435572i
$760$		0			0
$761$			10.0000i			0.362500i	0.983437	+	0.181250i	$0.0580143\pi$
$761$			10.0000i			0.362500i	−0.983437	+	0.181250i	$0.941986\pi$
$762$		0			0
$763$			45.0000i			1.62911i
$764$		0			0
$765$	−6.00000			−0.216930
$766$		0			0
$767$	30.0000	+	20.0000i	1.08324	+	0.722158i
$768$		0			0
$769$		−	36.0000i		−	1.29819i	−0.760706	−	0.649097i	$-0.775147\pi$
$769$		−	36.0000i		−	1.29819i	0.760706	−	0.649097i	$-0.224853\pi$
$770$		0			0
$771$			3.00000i			0.108042i
$772$		0			0
$773$	−49.0000			−1.76241			−0.881204	−	0.472737i	$-0.843266\pi$
$773$	−49.0000			−1.76241			−0.881204	+	0.472737i	$0.843266\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−9.00000			−0.322873
$778$		0			0
$779$		0			0
$780$		0			0
$781$			10.0000i			0.357828i
$782$		0			0
$783$	−30.0000			−1.07211
$784$		0			0
$785$			18.0000i			0.642448i
$786$		0			0
$787$	18.0000			0.641631			0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000			0.641631			0.320815	+	0.947142i	$0.396043\pi$
$788$		0			0
$789$			6.00000i			0.213606i
$790$		0			0
$791$			18.0000i			0.640006i
$792$		0			0
$793$	20.0000	−	30.0000i	0.710221	−	1.06533i
$794$		0			0
$795$	−6.00000			−0.212798
$796$		0			0
$797$			12.0000i			0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$			12.0000i			0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$		0			0
$799$			21.0000i			0.742927i
$800$		0			0
$801$			8.00000i			0.282666i
$802$		0			0
$803$			12.0000i			0.423471i
$804$		0			0
$805$			18.0000i			0.634417i
$806$		0			0
$807$	−6.00000			−0.211210
$808$		0			0
$809$	−15.0000			−0.527372			−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000			−0.527372			−0.263686	+	0.964609i	$0.584938\pi$
$810$		0			0
$811$	12.0000			0.421377			0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000			0.421377			0.210688	+	0.977553i	$0.432429\pi$
$812$		0			0
$813$	−15.0000			−0.526073
$814$		0			0
$815$	6.00000			0.210171
$816$		0			0
$817$		0			0
$818$		0			0
$819$	12.0000	−	18.0000i	0.419314	−	0.628971i
$820$		0			0
$821$	43.0000			1.50071			0.750355	−	0.661035i	$-0.229882\pi$
$821$	43.0000			1.50071			0.750355	+	0.661035i	$0.229882\pi$
$822$		0			0
$823$	−14.0000			−0.488009			−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000			−0.488009			−0.244005	+	0.969774i	$0.578461\pi$
$824$		0			0
$825$		−	8.00000i		−	0.278524i
$826$		0			0
$827$	−32.0000			−1.11275			−0.556375	−	0.830932i	$-0.687808\pi$
$827$	−32.0000			−1.11275			−0.556375	+	0.830932i	$0.687808\pi$
$828$		0			0
$829$			16.0000i			0.555703i	0.960624	+	0.277851i	$0.0896223\pi$
$829$			16.0000i			0.555703i	−0.960624	+	0.277851i	$0.910378\pi$
$830$		0			0
$831$	18.0000			0.624413
$832$		0			0
$833$	−6.00000			−0.207888
$834$		0			0
$835$			8.00000i			0.276851i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			16.0000i			0.552381i	0.961103	+	0.276191i	$0.0890721\pi$
$839$			16.0000i			0.552381i	−0.961103	+	0.276191i	$0.910928\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$	−20.0000			−0.688837
$844$		0			0
$845$	−5.00000	−	12.0000i	−0.172005	−	0.412813i
$846$		0			0
$847$			21.0000i			0.721569i
$848$		0			0
$849$	4.00000			0.137280
$850$		0			0
$851$	−18.0000			−0.617032
$852$		0			0
$853$	−9.00000			−0.308154			−0.154077	−	0.988059i	$-0.549240\pi$
$853$	−9.00000			−0.308154			−0.154077	+	0.988059i	$0.549240\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−42.0000			−1.43469			−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000			−1.43469			−0.717346	+	0.696717i	$0.754643\pi$
$858$		0			0
$859$			4.00000i			0.136478i	0.997669	+	0.0682391i	$0.0217381\pi$
$859$			4.00000i			0.136478i	−0.997669	+	0.0682391i	$0.978262\pi$
$860$		0			0
$861$		−	30.0000i		−	1.02240i
$862$		0			0
$863$			49.0000i			1.66798i	0.551780	+	0.833990i	$0.313949\pi$
$863$			49.0000i			1.66798i	−0.551780	+	0.833990i	$0.686051\pi$
$864$		0			0
$865$		−	24.0000i		−	0.816024i
$866$		0			0
$867$		−	8.00000i		−	0.271694i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−36.0000	−	24.0000i	−1.21981	−	0.813209i
$872$		0			0
$873$			36.0000i			1.21842i
$874$		0			0
$875$		−	27.0000i		−	0.912767i
$876$		0			0
$877$	−33.0000			−1.11433			−0.557165	−	0.830402i	$-0.688111\pi$
$877$	−33.0000			−1.11433			−0.557165	+	0.830402i	$0.688111\pi$
$878$		0			0
$879$			1.00000i			0.0337292i
$880$		0			0
$881$	−33.0000			−1.11180			−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000			−1.11180			−0.555899	+	0.831250i	$0.687626\pi$
$882$		0			0
$883$		−	29.0000i		−	0.975928i	−0.872864	−	0.487964i	$-0.837740\pi$
$883$		−	29.0000i		−	0.975928i	0.872864	−	0.487964i	$-0.162260\pi$
$884$		0			0
$885$		−	10.0000i		−	0.336146i
$886$		0			0
$887$	42.0000			1.41022			0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000			1.41022			0.705111	+	0.709097i	$0.250897\pi$
$888$		0			0
$889$			54.0000i			1.81110i
$890$		0			0
$891$	2.00000			0.0670025
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	9.00000i		−	0.300837i
$896$		0			0
$897$	−12.0000	+	18.0000i	−0.400668	+	0.601003i
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	18.0000i		−	0.599667i
$902$		0			0
$903$		−	27.0000i		−	0.898504i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	17.0000i		−	0.564476i	−0.959344	−	0.282238i	$-0.908923\pi$
$907$		−	17.0000i		−	0.564476i	0.959344	−	0.282238i	$-0.0910767\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−42.0000			−1.39152			−0.695761	−	0.718273i	$-0.744933\pi$
$911$	−42.0000			−1.39152			−0.695761	+	0.718273i	$0.744933\pi$
$912$		0			0
$913$	−32.0000			−1.05905
$914$		0			0
$915$	−10.0000			−0.330590
$916$		0			0
$917$	45.0000			1.48603
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$			18.0000i			0.593120i
$922$		0			0
$923$	−10.0000	+	15.0000i	−0.329154	+	0.493731i
$924$		0			0
$925$	12.0000			0.394558
$926$		0			0
$927$	−8.00000			−0.262754
$928$		0			0
$929$		−	56.0000i		−	1.83730i	−0.395072	−	0.918650i	$-0.629280\pi$
$929$		−	56.0000i		−	1.83730i	0.395072	−	0.918650i	$-0.370720\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		−	12.0000i		−	0.392862i
$934$		0			0
$935$	−6.00000			−0.196221
$936$		0			0
$937$	18.0000			0.588034			0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000			0.588034			0.294017	+	0.955800i	$0.405008\pi$
$938$		0			0
$939$			9.00000i			0.293704i
$940$		0			0
$941$	−17.0000			−0.554184			−0.277092	−	0.960843i	$-0.589371\pi$
$941$	−17.0000			−0.554184			−0.277092	+	0.960843i	$0.589371\pi$
$942$		0			0
$943$		−	60.0000i		−	1.95387i
$944$		0			0
$945$	−15.0000			−0.487950
$946$		0			0
$947$	−2.00000			−0.0649913			−0.0324956	−	0.999472i	$-0.510346\pi$
$947$	−2.00000			−0.0649913			−0.0324956	+	0.999472i	$0.510346\pi$
$948$		0			0
$949$	−12.0000	+	18.0000i	−0.389536	+	0.584305i
$950$		0			0
$951$			2.00000i			0.0648544i
$952$		0			0
$953$	−21.0000			−0.680257			−0.340128	−	0.940379i	$-0.610471\pi$
$953$	−21.0000			−0.680257			−0.340128	+	0.940379i	$0.610471\pi$
$954$		0			0
$955$	−18.0000			−0.582466
$956$		0			0
$957$	−12.0000			−0.387905
$958$		0			0
$959$	−6.00000			−0.193750
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		−	24.0000i		−	0.773389i
$964$		0			0
$965$			24.0000i			0.772587i
$966$		0			0
$967$			57.0000i			1.83300i	0.400039	+	0.916498i	$0.368997\pi$
$967$			57.0000i			1.83300i	−0.400039	+	0.916498i	$0.631003\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	45.0000i		−	1.44412i	−0.691831	−	0.722059i	$-0.743196\pi$
$971$		−	45.0000i		−	1.44412i	0.691831	−	0.722059i	$-0.256804\pi$
$972$		0			0
$973$	27.0000			0.865580
$974$		0			0
$975$	8.00000	−	12.0000i	0.256205	−	0.384308i
$976$		0			0
$977$			28.0000i			0.895799i	0.894084	+	0.447900i	$0.147828\pi$
$977$			28.0000i			0.895799i	−0.894084	+	0.447900i	$0.852172\pi$
$978$		0			0
$979$			8.00000i			0.255681i
$980$		0			0
$981$	−30.0000			−0.957826
$982$		0			0
$983$		−	11.0000i		−	0.350846i	−0.984493	−	0.175423i	$-0.943871\pi$
$983$		−	11.0000i		−	0.350846i	0.984493	−	0.175423i	$-0.0561292\pi$
$984$		0			0
$985$	23.0000			0.732841
$986$		0			0
$987$			21.0000i			0.668437i
$988$		0			0
$989$		−	54.0000i		−	1.71710i
$990$		0			0
$991$	−52.0000			−1.65183			−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000			−1.65183			−0.825917	+	0.563791i	$0.809342\pi$
$992$		0			0
$993$			12.0000i			0.380808i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	18.0000i		−	0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$		−	18.0000i		−	0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$		0			0
$999$		−	15.0000i		−	0.474579i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.e.a.337.2		2
3.2	odd	2		3744.2.m.c.1585.1		2
4.3	odd	2		104.2.e.b.77.1	yes	2
8.3	odd	2		104.2.e.a.77.1	✓	2
8.5	even	2		416.2.e.b.337.1		2
12.11	even	2		936.2.m.b.181.2		2
13.12	even	2		416.2.e.b.337.2		2
24.5	odd	2		3744.2.m.b.1585.1		2
24.11	even	2		936.2.m.c.181.2		2
39.38	odd	2		3744.2.m.b.1585.2		2
52.51	odd	2		104.2.e.a.77.2	yes	2
104.51	odd	2		104.2.e.b.77.2	yes	2
104.77	even	2	inner	416.2.e.a.337.1		2
156.155	even	2		936.2.m.c.181.1		2
312.77	odd	2		3744.2.m.c.1585.2		2
312.155	even	2		936.2.m.b.181.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.e.a.77.1	✓	2	8.3	odd	2
104.2.e.a.77.2	yes	2	52.51	odd	2
104.2.e.b.77.1	yes	2	4.3	odd	2
104.2.e.b.77.2	yes	2	104.51	odd	2
416.2.e.a.337.1		2	104.77	even	2	inner
416.2.e.a.337.2		2	1.1	even	1	trivial
416.2.e.b.337.1		2	8.5	even	2
416.2.e.b.337.2		2	13.12	even	2
936.2.m.b.181.1		2	312.155	even	2
936.2.m.b.181.2		2	12.11	even	2
936.2.m.c.181.1		2	156.155	even	2
936.2.m.c.181.2		2	24.11	even	2
3744.2.m.b.1585.1		2	24.5	odd	2
3744.2.m.b.1585.2		2	39.38	odd	2
3744.2.m.c.1585.1		2	3.2	odd	2
3744.2.m.c.1585.2		2	312.77	odd	2

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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.00000 q^{5} -3.00000i q^{7} +2.00000 q^{9} +2.00000 q^{11} +(3.00000 + 2.00000i) q^{13} -1.00000i q^{15} +3.00000 q^{17} +3.00000 q^{21} +6.00000 q^{23} -4.00000 q^{25} +5.00000i q^{27} +6.00000i q^{29} +2.00000i q^{33} +3.00000i q^{35} -3.00000 q^{37} +(-2.00000 + 3.00000i) q^{39} -10.0000i q^{41} -9.00000i q^{43} -2.00000 q^{45} +7.00000i q^{47} -2.00000 q^{49} +3.00000i q^{51} -6.00000i q^{53} -2.00000 q^{55} +10.0000 q^{59} -10.0000i q^{61} -6.00000i q^{63} +(-3.00000 - 2.00000i) q^{65} -12.0000 q^{67} +6.00000i q^{69} +5.00000i q^{71} +6.00000i q^{73} -4.00000i q^{75} -6.00000i q^{77} +1.00000 q^{81} -16.0000 q^{83} -3.00000 q^{85} -6.00000 q^{87} +4.00000i q^{89} +(6.00000 - 9.00000i) q^{91} +18.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{9} + 4 q^{11} + 6 q^{13} + 6 q^{17} + 6 q^{21} + 12 q^{23} - 8 q^{25} - 6 q^{37} - 4 q^{39} - 4 q^{45} - 4 q^{49} - 4 q^{55} + 20 q^{59} - 6 q^{65} - 24 q^{67} + 2 q^{81} - 32 q^{83}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^9 + 4 * q^11 + 6 * q^13 + 6 * q^17 + 6 * q^21 + 12 * q^23 - 8 * q^25 - 6 * q^37 - 4 * q^39 - 4 * q^45 - 4 * q^49 - 4 * q^55 + 20 * q^59 - 6 * q^65 - 24 * q^67 + 2 * q^81 - 32 * q^83 - 6 * q^85 - 12 * q^87 + 12 * q^91 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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