LMFDB - Embedded newform 416.2.b.a.209.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(209,416)] 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("416.209"); 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, 0])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == 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			209.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	416.209
Dual form			416.2.b.a.209.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +3.00000i q^{5} -3.00000 q^{7} +2.00000 q^{9} +1.00000i q^{13} -3.00000 q^{15} -7.00000 q^{17} +4.00000i q^{19} -3.00000i q^{21} -4.00000 q^{23} -4.00000 q^{25} +5.00000i q^{27} -4.00000i q^{29} +8.00000 q^{31} -9.00000i q^{35} +7.00000i q^{37} -1.00000 q^{39} +2.00000 q^{41} +1.00000i q^{43} +6.00000i q^{45} +7.00000 q^{47} +2.00000 q^{49} -7.00000i q^{51} +4.00000i q^{53} -4.00000 q^{57} +14.0000i q^{59} -10.0000i q^{61} -6.00000 q^{63} -3.00000 q^{65} -2.00000i q^{67} -4.00000i q^{69} +3.00000 q^{71} +14.0000 q^{73} -4.00000i q^{75} +10.0000 q^{79} +1.00000 q^{81} -14.0000i q^{83} -21.0000i q^{85} +4.00000 q^{87} -3.00000i q^{91} +8.00000i q^{93} -12.0000 q^{95} +8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{7} + 4 q^{9} - 6 q^{15} - 14 q^{17} - 8 q^{23} - 8 q^{25} + 16 q^{31} - 2 q^{39} + 4 q^{41} + 14 q^{47} + 4 q^{49} - 8 q^{57} - 12 q^{63} - 6 q^{65} + 6 q^{71} + 28 q^{73} + 20 q^{79} + 2 q^{81}+ \cdots + 16 q^{97}+O(q^{100}) $ 2 * q - 6 * q^7 + 4 * q^9 - 6 * q^15 - 14 * q^17 - 8 * q^23 - 8 * q^25 + 16 * q^31 - 2 * q^39 + 4 * q^41 + 14 * q^47 + 4 * q^49 - 8 * q^57 - 12 * q^63 - 6 * q^65 + 6 * q^71 + 28 * q^73 + 20 * q^79 + 2 * q^81 + 8 * q^87 - 24 * q^95 + 16 * q^97

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
$5$	3.00000i	1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$	3.00000i	1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	− 3.00000i	− 0.654654i
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	− 4.00000i	− 0.742781i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	− 4.00000i	− 0.742781i	0.928477	−	0.371391i	$-0.121119\pi$
$30$	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
$35$	− 9.00000i	− 1.52128i
$36$	0	0
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	0	0
$45$	6.00000i	0.894427i
$46$	0	0
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	− 7.00000i	− 0.980196i
$52$	0	0
$53$	4.00000i	0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$	4.00000i	0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	14.0000i	1.82264i	0.411693	+	0.911322i	$0.364937\pi$
$59$	14.0000i	1.82264i	−0.411693	+	0.911322i	$0.635063\pi$
$60$	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
$63$	−6.00000	−0.755929
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	0	0
$69$	− 4.00000i	− 0.481543i
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	− 4.00000i	− 0.461880i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 14.0000i	− 1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$	− 14.0000i	− 1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$	0	0
$85$	− 21.0000i	− 2.27777i
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	− 3.00000i	− 0.314485i
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	−12.0000	−1.23117
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	9.00000	0.878310
$106$	0	0
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	1.00000i	0.0957826i	0.998853	+	0.0478913i	$0.0152501\pi$
$109$	1.00000i	0.0957826i	−0.998853	+	0.0478913i	$0.984750\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	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
$115$	− 12.0000i	− 1.11901i
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	21.0000	1.92507
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	2.00000i	0.180334i
$124$	0	0
$125$	3.00000i	0.268328i
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	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
$133$	− 12.0000i	− 1.04053i
$134$	0	0
$135$	−15.0000	−1.29099
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	− 11.0000i	− 0.933008i	−0.884519	−	0.466504i	$-0.845513\pi$
$139$	− 11.0000i	− 0.933008i	0.884519	−	0.466504i	$-0.154487\pi$
$140$	0	0
$141$	7.00000i	0.589506i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	12.0000	0.996546
$146$	0	0
$147$	2.00000i	0.164957i
$148$	0	0
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	0	0
$153$	−14.0000	−1.13183
$154$	0	0
$155$	24.0000i	1.92773i
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	− 24.0000i	− 1.87983i	−0.341415	−	0.939913i	$-0.610906\pi$
$163$	− 24.0000i	− 1.87983i	0.341415	−	0.939913i	$-0.389094\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	8.00000i	0.611775i
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	0	0
$177$	−14.0000	−1.05230
$178$	0	0
$179$	− 21.0000i	− 1.56961i	−0.619740	−	0.784807i	$-0.712762\pi$
$179$	− 21.0000i	− 1.56961i	0.619740	−	0.784807i	$-0.287238\pi$
$180$	0	0
$181$	10.0000i	0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$	10.0000i	0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	−21.0000	−1.54395
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 15.0000i	− 1.09109i
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	− 3.00000i	− 0.214834i
$196$	0	0
$197$	17.0000i	1.21120i	0.795769	+	0.605600i	$0.207067\pi$
$197$	17.0000i	1.21120i	−0.795769	+	0.605600i	$0.792933\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	12.0000i	0.842235i
$204$	0	0
$205$	6.00000i	0.419058i
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	0	0
$210$	0	0
$211$	15.0000i	1.03264i	0.856395	+	0.516321i	$0.172699\pi$
$211$	15.0000i	1.03264i	−0.856395	+	0.516321i	$0.827301\pi$
$212$	0	0
$213$	3.00000i	0.205557i
$214$	0	0
$215$	−3.00000	−0.204598
$216$	0	0
$217$	−24.0000	−1.62923
$218$	0	0
$219$	14.0000i	0.946032i
$220$	0	0
$221$	− 7.00000i	− 0.470871i
$222$	0	0
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	0	0
$225$	−8.00000	−0.533333
$226$	0	0
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	0	0
$229$	21.0000i	1.38772i	0.720110	+	0.693860i	$0.244091\pi$
$229$	21.0000i	1.38772i	−0.720110	+	0.693860i	$0.755909\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.0000	1.24473	0.622366	−	0.782727i	$-0.286172\pi$
$233$	19.0000	1.24473	0.622366	+	0.782727i	$0.286172\pi$
$234$	0	0
$235$	21.0000i	1.36989i
$236$	0	0
$237$	10.0000i	0.649570i
$238$	0	0
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	6.00000i	0.383326i
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	14.0000	0.887214
$250$	0	0
$251$	− 20.0000i	− 1.26239i	−0.775625	−	0.631194i	$-0.782565\pi$
$251$	− 20.0000i	− 1.26239i	0.775625	−	0.631194i	$-0.217435\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	21.0000	1.31507
$256$	0	0
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	0	0
$259$	− 21.0000i	− 1.30488i
$260$	0	0
$261$	− 8.00000i	− 0.495188i
$262$	0	0
$263$	−14.0000	−0.863277	−0.431638	−	0.902047i	$-0.642064\pi$
$263$	−14.0000	−0.863277	−0.431638	+	0.902047i	$0.642064\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	0	0
$268$	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
$271$	3.00000	0.182237	0.0911185	−	0.995840i	$-0.470956\pi$
$271$	3.00000	0.182237	0.0911185	+	0.995840i	$0.470956\pi$
$272$	0	0
$273$	3.00000	0.181568
$274$	0	0
$275$	0	0
$276$	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
$279$	16.0000	0.957895
$280$	0	0
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	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
$285$	− 12.0000i	− 0.710819i
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	8.00000i	0.468968i
$292$	0	0
$293$	− 1.00000i	− 0.0584206i	−0.999573	−	0.0292103i	$-0.990701\pi$
$293$	− 1.00000i	− 0.0584206i	0.999573	−	0.0292103i	$-0.00929925\pi$
$294$	0	0
$295$	−42.0000	−2.44533
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 4.00000i	− 0.231326i
$300$	0	0
$301$	− 3.00000i	− 0.172917i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	30.0000	1.71780
$306$	0	0
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	0	0
$309$	6.00000i	0.341328i
$310$	0	0
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	0	0
$313$	29.0000	1.63918	0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000	1.63918	0.819588	+	0.572953i	$0.194202\pi$
$314$	0	0
$315$	− 18.0000i	− 1.01419i
$316$	0	0
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	− 28.0000i	− 1.55796i
$324$	0	0
$325$	− 4.00000i	− 0.221880i
$326$	0	0
$327$	−1.00000	−0.0553001
$328$	0	0
$329$	−21.0000	−1.15777
$330$	0	0
$331$	− 10.0000i	− 0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$	− 10.0000i	− 0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$	0	0
$333$	14.0000i	0.767195i
$334$	0	0
$335$	6.00000	0.327815
$336$	0	0
$337$	−17.0000	−0.926049	−0.463025	−	0.886345i	$-0.653236\pi$
$337$	−17.0000	−0.926049	−0.463025	+	0.886345i	$0.653236\pi$
$338$	0	0
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	− 7.00000i	− 0.375780i	−0.982190	−	0.187890i	$-0.939835\pi$
$347$	− 7.00000i	− 0.375780i	0.982190	−	0.187890i	$-0.0601648\pi$
$348$	0	0
$349$	− 19.0000i	− 1.01705i	−0.861048	−	0.508523i	$-0.830192\pi$
$349$	− 19.0000i	− 1.01705i	0.861048	−	0.508523i	$-0.169808\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	9.00000i	0.477670i
$356$	0	0
$357$	21.0000i	1.11144i
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	11.0000i	0.577350i
$364$	0	0
$365$	42.0000i	2.19838i
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	− 12.0000i	− 0.623009i
$372$	0	0
$373$	− 36.0000i	− 1.86401i	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	− 36.0000i	− 1.86401i	0.362446	−	0.932005i	$-0.381942\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	4.00000i	0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$	4.00000i	0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	0	0
$381$	− 8.00000i	− 0.409852i
$382$	0	0
$383$	1.00000	0.0510976	0.0255488	−	0.999674i	$-0.491867\pi$
$383$	1.00000	0.0510976	0.0255488	+	0.999674i	$0.491867\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.00000i	0.101666i
$388$	0	0
$389$	− 34.0000i	− 1.72387i	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	− 34.0000i	− 1.72387i	0.507020	−	0.861934i	$-0.330747\pi$
$390$	0	0
$391$	28.0000	1.41602
$392$	0	0
$393$	−15.0000	−0.756650
$394$	0	0
$395$	30.0000i	1.50946i
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	0	0
$399$	12.0000	0.600751
$400$	0	0
$401$	32.0000	1.59800	0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000	1.59800	0.799002	+	0.601329i	$0.205362\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	3.00000i	0.149071i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	− 12.0000i	− 0.591916i
$412$	0	0
$413$	− 42.0000i	− 2.06668i
$414$	0	0
$415$	42.0000	2.06170
$416$	0	0
$417$	11.0000	0.538672
$418$	0	0
$419$	− 11.0000i	− 0.537385i	−0.963226	−	0.268693i	$-0.913408\pi$
$419$	− 11.0000i	− 0.537385i	0.963226	−	0.268693i	$-0.0865916\pi$
$420$	0	0
$421$	15.0000i	0.731055i	0.930800	+	0.365528i	$0.119111\pi$
$421$	15.0000i	0.731055i	−0.930800	+	0.365528i	$0.880889\pi$
$422$	0	0
$423$	14.0000	0.680703
$424$	0	0
$425$	28.0000	1.35820
$426$	0	0
$427$	30.0000i	1.45180i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.00000	0.144505	0.0722525	−	0.997386i	$-0.476981\pi$
$431$	3.00000	0.144505	0.0722525	+	0.997386i	$0.476981\pi$
$432$	0	0
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	0	0
$435$	12.0000i	0.575356i
$436$	0	0
$437$	− 16.0000i	− 0.765384i
$438$	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
$445$	0	0
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 17.0000i	− 0.798730i
$454$	0	0
$455$	9.00000	0.421927
$456$	0	0
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	0	0
$459$	− 35.0000i	− 1.63366i
$460$	0	0
$461$	5.00000i	0.232873i	0.993198	+	0.116437i	$0.0371472\pi$
$461$	5.00000i	0.232873i	−0.993198	+	0.116437i	$0.962853\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	−24.0000	−1.11297
$466$	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
$469$	6.00000i	0.277054i
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 16.0000i	− 0.734130i
$476$	0	0
$477$	8.00000i	0.366295i
$478$	0	0
$479$	−5.00000	−0.228456	−0.114228	−	0.993455i	$-0.536439\pi$
$479$	−5.00000	−0.228456	−0.114228	+	0.993455i	$0.536439\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	0	0
$483$	12.0000i	0.546019i
$484$	0	0
$485$	24.0000i	1.08978i
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	24.0000	1.08532
$490$	0	0
$491$	− 15.0000i	− 0.676941i	−0.940977	−	0.338470i	$-0.890091\pi$
$491$	− 15.0000i	− 0.676941i	0.940977	−	0.338470i	$-0.109909\pi$
$492$	0	0
$493$	28.0000i	1.26106i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.00000	−0.403705
$498$	0	0
$499$	− 6.00000i	− 0.268597i	−0.990941	−	0.134298i	$-0.957122\pi$
$499$	− 6.00000i	− 0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$	0	0
$501$	12.0000i	0.536120i
$502$	0	0
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.00000i	− 0.0444116i
$508$	0	0
$509$	− 14.0000i	− 0.620539i	−0.950649	−	0.310270i	$-0.899581\pi$
$509$	− 14.0000i	− 0.620539i	0.950649	−	0.310270i	$-0.100419\pi$
$510$	0	0
$511$	−42.0000	−1.85797
$512$	0	0
$513$	−20.0000	−0.883022
$514$	0	0
$515$	18.0000i	0.793175i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	6.00000	0.263371
$520$	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
$523$	− 44.0000i	− 1.92399i	−0.273075	−	0.961993i	$-0.588041\pi$
$523$	− 44.0000i	− 1.92399i	0.273075	−	0.961993i	$-0.411959\pi$
$524$	0	0
$525$	12.0000i	0.523723i
$526$	0	0
$527$	−56.0000	−2.43940
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	28.0000i	1.21510i
$532$	0	0
$533$	2.00000i	0.0866296i
$534$	0	0
$535$	−24.0000	−1.03761
$536$	0	0
$537$	21.0000	0.906217
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 25.0000i	− 1.07483i	−0.843317	−	0.537417i	$-0.819400\pi$
$541$	− 25.0000i	− 1.07483i	0.843317	−	0.537417i	$-0.180600\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−3.00000	−0.128506
$546$	0	0
$547$	13.0000i	0.555840i	0.960604	+	0.277920i	$0.0896450\pi$
$547$	13.0000i	0.555840i	−0.960604	+	0.277920i	$0.910355\pi$
$548$	0	0
$549$	− 20.0000i	− 0.853579i
$550$	0	0
$551$	16.0000	0.681623
$552$	0	0
$553$	−30.0000	−1.27573
$554$	0	0
$555$	− 21.0000i	− 0.891400i
$556$	0	0
$557$	− 13.0000i	− 0.550828i	−0.961326	−	0.275414i	$-0.911185\pi$
$557$	− 13.0000i	− 0.550828i	0.961326	−	0.275414i	$-0.0888149\pi$
$558$	0	0
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	0	0
$562$	0	0
$563$	31.0000i	1.30649i	0.757145	+	0.653247i	$0.226594\pi$
$563$	31.0000i	1.30649i	−0.757145	+	0.653247i	$0.773406\pi$
$564$	0	0
$565$	− 18.0000i	− 0.757266i
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	25.0000	1.04805	0.524027	−	0.851701i	$-0.324429\pi$
$569$	25.0000	1.04805	0.524027	+	0.851701i	$0.324429\pi$
$570$	0	0
$571$	− 5.00000i	− 0.209243i	−0.994512	−	0.104622i	$-0.966637\pi$
$571$	− 5.00000i	− 0.209243i	0.994512	−	0.104622i	$-0.0333632\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	0	0
$575$	16.0000	0.667246
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	− 6.00000i	− 0.249351i
$580$	0	0
$581$	42.0000i	1.74245i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	28.0000i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$	28.0000i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	0	0
$589$	32.0000i	1.31854i
$590$	0	0
$591$	−17.0000	−0.699287
$592$	0	0
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	0	0
$595$	63.0000i	2.58275i
$596$	0	0
$597$	10.0000i	0.409273i
$598$	0	0
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	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
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	33.0000i	1.34164i
$606$	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
$609$	−12.0000	−0.486265
$610$	0	0
$611$	7.00000i	0.283190i
$612$	0	0
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	− 6.00000i	− 0.241160i	−0.992704	−	0.120580i	$-0.961525\pi$
$619$	− 6.00000i	− 0.241160i	0.992704	−	0.120580i	$-0.0384755\pi$
$620$	0	0
$621$	− 20.0000i	− 0.802572i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 49.0000i	− 1.95376i
$630$	0	0
$631$	23.0000	0.915616	0.457808	−	0.889051i	$-0.348635\pi$
$631$	23.0000	0.915616	0.457808	+	0.889051i	$0.348635\pi$
$632$	0	0
$633$	−15.0000	−0.596196
$634$	0	0
$635$	− 24.0000i	− 0.952411i
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	26.0000i	1.02534i	0.858586	+	0.512670i	$0.171344\pi$
$643$	26.0000i	1.02534i	−0.858586	+	0.512670i	$0.828656\pi$
$644$	0	0
$645$	− 3.00000i	− 0.118125i
$646$	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
$649$	0	0
$650$	0	0
$651$	− 24.0000i	− 0.940634i
$652$	0	0
$653$	44.0000i	1.72185i	0.508729	+	0.860927i	$0.330115\pi$
$653$	44.0000i	1.72185i	−0.508729	+	0.860927i	$0.669885\pi$
$654$	0	0
$655$	−45.0000	−1.75830
$656$	0	0
$657$	28.0000	1.09238
$658$	0	0
$659$	4.00000i	0.155818i	0.996960	+	0.0779089i	$0.0248243\pi$
$659$	4.00000i	0.155818i	−0.996960	+	0.0779089i	$0.975176\pi$
$660$	0	0
$661$	30.0000i	1.16686i	0.812162	+	0.583432i	$0.198291\pi$
$661$	30.0000i	1.16686i	−0.812162	+	0.583432i	$0.801709\pi$
$662$	0	0
$663$	7.00000	0.271857
$664$	0	0
$665$	36.0000	1.39602
$666$	0	0
$667$	16.0000i	0.619522i
$668$	0	0
$669$	1.00000i	0.0386622i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	0	0
$675$	− 20.0000i	− 0.769800i
$676$	0	0
$677$	22.0000i	0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$	22.0000i	0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$	0	0
$679$	−24.0000	−0.921035
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	26.0000i	0.994862i	0.867503	+	0.497431i	$0.165723\pi$
$683$	26.0000i	0.994862i	−0.867503	+	0.497431i	$0.834277\pi$
$684$	0	0
$685$	− 36.0000i	− 1.37549i
$686$	0	0
$687$	−21.0000	−0.801200
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	20.0000i	0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$	20.0000i	0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	33.0000	1.25176
$696$	0	0
$697$	−14.0000	−0.530288
$698$	0	0
$699$	19.0000i	0.718646i
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−28.0000	−1.05604
$704$	0	0
$705$	−21.0000	−0.790906
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6.00000i	0.225335i	0.993633	+	0.112667i	$0.0359394\pi$
$709$	6.00000i	0.225335i	−0.993633	+	0.112667i	$0.964061\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 5.00000i	− 0.186728i
$718$	0	0
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	0	0
$723$	− 8.00000i	− 0.297523i
$724$	0	0
$725$	16.0000i	0.594225i
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 7.00000i	− 0.258904i
$732$	0	0
$733$	− 1.00000i	− 0.0369358i	−0.999829	−	0.0184679i	$-0.994121\pi$
$733$	− 1.00000i	− 0.0369358i	0.999829	−	0.0184679i	$-0.00587886\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	0	0
$738$	0	0
$739$	34.0000i	1.25071i	0.780340	+	0.625355i	$0.215046\pi$
$739$	34.0000i	1.25071i	−0.780340	+	0.625355i	$0.784954\pi$
$740$	0	0
$741$	− 4.00000i	− 0.146944i
$742$	0	0
$743$	21.0000	0.770415	0.385208	−	0.922830i	$-0.374130\pi$
$743$	21.0000	0.770415	0.385208	+	0.922830i	$0.374130\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	− 28.0000i	− 1.02447i
$748$	0	0
$749$	− 24.0000i	− 0.876941i
$750$	0	0
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	− 51.0000i	− 1.85608i
$756$	0	0
$757$	32.0000i	1.16306i	0.813525	+	0.581530i	$0.197546\pi$
$757$	32.0000i	1.16306i	−0.813525	+	0.581530i	$0.802454\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	− 3.00000i	− 0.108607i
$764$	0	0
$765$	− 42.0000i	− 1.51851i
$766$	0	0
$767$	−14.0000	−0.505511
$768$	0	0
$769$	−20.0000	−0.721218	−0.360609	−	0.932717i	$-0.617431\pi$
$769$	−20.0000	−0.721218	−0.360609	+	0.932717i	$0.617431\pi$
$770$	0	0
$771$	− 7.00000i	− 0.252099i
$772$	0	0
$773$	− 21.0000i	− 0.755318i	−0.925945	−	0.377659i	$-0.876729\pi$
$773$	− 21.0000i	− 0.755318i	0.925945	−	0.377659i	$-0.123271\pi$
$774$	0	0
$775$	−32.0000	−1.14947
$776$	0	0
$777$	21.0000	0.753371
$778$	0	0
$779$	8.00000i	0.286630i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	20.0000	0.714742
$784$	0	0
$785$	−6.00000	−0.214149
$786$	0	0
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	0	0
$789$	− 14.0000i	− 0.498413i
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	− 12.0000i	− 0.425596i
$796$	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
$799$	−49.0000	−1.73350
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	36.0000i	1.26883i
$806$	0	0
$807$	−6.00000	−0.211210
$808$	0	0
$809$	25.0000	0.878953	0.439477	−	0.898254i	$-0.355164\pi$
$809$	25.0000	0.878953	0.439477	+	0.898254i	$0.355164\pi$
$810$	0	0
$811$	− 40.0000i	− 1.40459i	−0.711886	−	0.702295i	$-0.752159\pi$
$811$	− 40.0000i	− 1.40459i	0.711886	−	0.702295i	$-0.247841\pi$
$812$	0	0
$813$	3.00000i	0.105215i
$814$	0	0
$815$	72.0000	2.52205
$816$	0	0
$817$	−4.00000	−0.139942
$818$	0	0
$819$	− 6.00000i	− 0.209657i
$820$	0	0
$821$	− 45.0000i	− 1.57051i	−0.619172	−	0.785255i	$-0.712532\pi$
$821$	− 45.0000i	− 1.57051i	0.619172	−	0.785255i	$-0.287468\pi$
$822$	0	0
$823$	46.0000	1.60346	0.801730	−	0.597687i	$-0.203913\pi$
$823$	46.0000	1.60346	0.801730	+	0.597687i	$0.203913\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.00000i	0.278187i	0.990279	+	0.139094i	$0.0444189\pi$
$827$	8.00000i	0.278187i	−0.990279	+	0.139094i	$0.955581\pi$
$828$	0	0
$829$	− 14.0000i	− 0.486240i	−0.969996	−	0.243120i	$-0.921829\pi$
$829$	− 14.0000i	− 0.486240i	0.969996	−	0.243120i	$-0.0781709\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	0	0
$833$	−14.0000	−0.485071
$834$	0	0
$835$	36.0000i	1.24583i
$836$	0	0
$837$	40.0000i	1.38260i
$838$	0	0
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	13.0000	0.448276
$842$	0	0
$843$	− 8.00000i	− 0.275535i
$844$	0	0
$845$	− 3.00000i	− 0.103203i
$846$	0	0
$847$	−33.0000	−1.13389
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	− 28.0000i	− 0.959828i
$852$	0	0
$853$	9.00000i	0.308154i	0.988059	+	0.154077i	$0.0492404\pi$
$853$	9.00000i	0.308154i	−0.988059	+	0.154077i	$0.950760\pi$
$854$	0	0
$855$	−24.0000	−0.820783
$856$	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
$859$	− 36.0000i	− 1.22830i	−0.789188	−	0.614152i	$-0.789498\pi$
$859$	− 36.0000i	− 1.22830i	0.789188	−	0.614152i	$-0.210502\pi$
$860$	0	0
$861$	− 6.00000i	− 0.204479i
$862$	0	0
$863$	51.0000	1.73606	0.868030	−	0.496512i	$-0.165386\pi$
$863$	51.0000	1.73606	0.868030	+	0.496512i	$0.165386\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	0	0
$867$	32.0000i	1.08678i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	2.00000	0.0677674
$872$	0	0
$873$	16.0000	0.541518
$874$	0	0
$875$	− 9.00000i	− 0.304256i
$876$	0	0
$877$	7.00000i	0.236373i	0.992991	+	0.118187i	$0.0377081\pi$
$877$	7.00000i	0.236373i	−0.992991	+	0.118187i	$0.962292\pi$
$878$	0	0
$879$	1.00000	0.0337292
$880$	0	0
$881$	−3.00000	−0.101073	−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000	−0.101073	−0.0505363	+	0.998722i	$0.516093\pi$
$882$	0	0
$883$	41.0000i	1.37976i	0.723924	+	0.689880i	$0.242337\pi$
$883$	41.0000i	1.37976i	−0.723924	+	0.689880i	$0.757663\pi$
$884$	0	0
$885$	− 42.0000i	− 1.41181i
$886$	0	0
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	0	0
$892$	0	0
$893$	28.0000i	0.936984i
$894$	0	0
$895$	63.0000	2.10586
$896$	0	0
$897$	4.00000	0.133556
$898$	0	0
$899$	− 32.0000i	− 1.06726i
$900$	0	0
$901$	− 28.0000i	− 0.932815i
$902$	0	0
$903$	3.00000	0.0998337
$904$	0	0
$905$	−30.0000	−0.997234
$906$	0	0
$907$	13.0000i	0.431658i	0.976431	+	0.215829i	$0.0692454\pi$
$907$	13.0000i	0.431658i	−0.976431	+	0.215829i	$0.930755\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	30.0000i	0.991769i
$916$	0	0
$917$	− 45.0000i	− 1.48603i
$918$	0	0
$919$	−60.0000	−1.97922	−0.989609	−	0.143787i	$-0.954072\pi$
$919$	−60.0000	−1.97922	−0.989609	+	0.143787i	$0.954072\pi$
$920$	0	0
$921$	2.00000	0.0659022
$922$	0	0
$923$	3.00000i	0.0987462i
$924$	0	0
$925$	− 28.0000i	− 0.920634i
$926$	0	0
$927$	12.0000	0.394132
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	8.00000i	0.262189i
$932$	0	0
$933$	− 32.0000i	− 1.04763i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	29.0000i	0.946379i
$940$	0	0
$941$	− 25.0000i	− 0.814977i	−0.913210	−	0.407488i	$-0.866405\pi$
$941$	− 25.0000i	− 0.814977i	0.913210	−	0.407488i	$-0.133595\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	45.0000	1.46385
$946$	0	0
$947$	− 32.0000i	− 1.03986i	−0.854209	−	0.519930i	$-0.825958\pi$
$947$	− 32.0000i	− 1.03986i	0.854209	−	0.519930i	$-0.174042\pi$
$948$	0	0
$949$	14.0000i	0.454459i
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	0	0
$955$	− 36.0000i	− 1.16493i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	16.0000i	0.515593i
$964$	0	0
$965$	− 18.0000i	− 0.579441i
$966$	0	0
$967$	7.00000	0.225105	0.112552	−	0.993646i	$-0.464097\pi$
$967$	7.00000	0.225105	0.112552	+	0.993646i	$0.464097\pi$
$968$	0	0
$969$	28.0000	0.899490
$970$	0	0
$971$	15.0000i	0.481373i	0.970603	+	0.240686i	$0.0773725\pi$
$971$	15.0000i	0.481373i	−0.970603	+	0.240686i	$0.922627\pi$
$972$	0	0
$973$	33.0000i	1.05793i
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	2.00000i	0.0638551i
$982$	0	0
$983$	−59.0000	−1.88181	−0.940904	−	0.338674i	$-0.890022\pi$
$983$	−59.0000	−1.88181	−0.940904	+	0.338674i	$0.890022\pi$
$984$	0	0
$985$	−51.0000	−1.62500
$986$	0	0
$987$	− 21.0000i	− 0.668437i
$988$	0	0
$989$	− 4.00000i	− 0.127193i
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	10.0000	0.317340
$994$	0	0
$995$	30.0000i	0.951064i
$996$	0	0
$997$	52.0000i	1.64686i	0.567420	+	0.823428i	$0.307941\pi$
$997$	52.0000i	1.64686i	−0.567420	+	0.823428i	$0.692059\pi$
$998$	0	0
$999$	−35.0000	−1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.b.a.209.2		2
3.2	odd	2		3744.2.g.a.1873.1		2
4.3	odd	2		104.2.b.a.53.2	yes	2
8.3	odd	2		104.2.b.a.53.1	✓	2
8.5	even	2	inner	416.2.b.a.209.1		2
12.11	even	2		936.2.g.a.469.1		2
16.3	odd	4		3328.2.a.j.1.1		1
16.5	even	4		3328.2.a.g.1.1		1
16.11	odd	4		3328.2.a.c.1.1		1
16.13	even	4		3328.2.a.f.1.1		1
24.5	odd	2		3744.2.g.a.1873.2		2
24.11	even	2		936.2.g.a.469.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.b.a.53.1	✓	2	8.3	odd	2
104.2.b.a.53.2	yes	2	4.3	odd	2
416.2.b.a.209.1		2	8.5	even	2	inner
416.2.b.a.209.2		2	1.1	even	1	trivial
936.2.g.a.469.1		2	12.11	even	2
936.2.g.a.469.2		2	24.11	even	2
3328.2.a.c.1.1		1	16.11	odd	4
3328.2.a.f.1.1		1	16.13	even	4
3328.2.a.g.1.1		1	16.5	even	4
3328.2.a.j.1.1		1	16.3	odd	4
3744.2.g.a.1873.1		2	3.2	odd	2
3744.2.g.a.1873.2		2	24.5	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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +3.00000i q^{5} -3.00000 q^{7} +2.00000 q^{9} +1.00000i q^{13} -3.00000 q^{15} -7.00000 q^{17} +4.00000i q^{19} -3.00000i q^{21} -4.00000 q^{23} -4.00000 q^{25} +5.00000i q^{27} -4.00000i q^{29} +8.00000 q^{31} -9.00000i q^{35} +7.00000i q^{37} -1.00000 q^{39} +2.00000 q^{41} +1.00000i q^{43} +6.00000i q^{45} +7.00000 q^{47} +2.00000 q^{49} -7.00000i q^{51} +4.00000i q^{53} -4.00000 q^{57} +14.0000i q^{59} -10.0000i q^{61} -6.00000 q^{63} -3.00000 q^{65} -2.00000i q^{67} -4.00000i q^{69} +3.00000 q^{71} +14.0000 q^{73} -4.00000i q^{75} +10.0000 q^{79} +1.00000 q^{81} -14.0000i q^{83} -21.0000i q^{85} +4.00000 q^{87} -3.00000i q^{91} +8.00000i q^{93} -12.0000 q^{95} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{7} + 4 q^{9} - 6 q^{15} - 14 q^{17} - 8 q^{23} - 8 q^{25} + 16 q^{31} - 2 q^{39} + 4 q^{41} + 14 q^{47} + 4 q^{49} - 8 q^{57} - 12 q^{63} - 6 q^{65} + 6 q^{71} + 28 q^{73} + 20 q^{79} + 2 q^{81}+ \cdots + 16 q^{97}+O(q^{100}) \) 2 * q - 6 * q^7 + 4 * q^9 - 6 * q^15 - 14 * q^17 - 8 * q^23 - 8 * q^25 + 16 * q^31 - 2 * q^39 + 4 * q^41 + 14 * q^47 + 4 * q^49 - 8 * q^57 - 12 * q^63 - 6 * q^65 + 6 * q^71 + 28 * q^73 + 20 * q^79 + 2 * q^81 + 8 * q^87 - 24 * q^95 + 16 * q^97

Introduction

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