LMFDB - Embedded newform 1664.2.b.h.833.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1664,2,Mod(833,1664)] 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("1664.833"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1664 = 2^{7} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1664.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$13.2871068963$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			833.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1664.833
Dual form			1664.2.b.h.833.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{5} +4.24264 q^{7} +3.00000 q^{9} +4.24264i q^{11} -1.00000i q^{13} +4.00000 q^{17} -4.24264i q^{19} +1.00000 q^{25} -2.00000i q^{29} -4.24264 q^{31} +8.48528i q^{35} +6.00000i q^{37} -2.00000 q^{41} -8.48528i q^{43} +6.00000i q^{45} -12.7279 q^{47} +11.0000 q^{49} +8.00000i q^{53} -8.48528 q^{55} -4.24264i q^{59} +12.7279 q^{63} +2.00000 q^{65} -4.24264i q^{67} +4.24264 q^{71} -6.00000 q^{73} +18.0000i q^{77} +8.48528 q^{79} +9.00000 q^{81} -12.7279i q^{83} +8.00000i q^{85} -10.0000 q^{89} -4.24264i q^{91} +8.48528 q^{95} +18.0000 q^{97} +12.7279i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9} + 16 q^{17} + 4 q^{25} - 8 q^{41} + 44 q^{49} + 8 q^{65} - 24 q^{73} + 36 q^{81} - 40 q^{89} + 72 q^{97}+O(q^{100}) $ 4 * q + 12 * q^9 + 16 * q^17 + 4 * q^25 - 8 * q^41 + 44 * q^49 + 8 * q^65 - 24 * q^73 + 36 * q^81 - 40 * q^89 + 72 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1664\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	4.24264	1.60357	0.801784	−	0.597614i	$-0.203885\pi$
$7$	4.24264	1.60357	0.801784	+	0.597614i	$0.203885\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	4.24264i	1.27920i	0.768706	+	0.639602i	$0.220901\pi$
$11$	4.24264i	1.27920i	−0.768706	+	0.639602i	$0.779099\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	− 4.24264i	− 0.973329i	−0.873589	−	0.486664i	$-0.838214\pi$
$19$	− 4.24264i	− 0.973329i	0.873589	−	0.486664i	$-0.161786\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	−4.24264	−0.762001	−0.381000	−	0.924575i	$-0.624420\pi$
$31$	−4.24264	−0.762001	−0.381000	+	0.924575i	$0.624420\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.48528i	1.43427i
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	− 8.48528i	− 1.29399i	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	− 8.48528i	− 1.29399i	0.762493	−	0.646997i	$-0.223975\pi$
$44$	0	0
$45$	6.00000i	0.894427i
$46$	0	0
$47$	−12.7279	−1.85656	−0.928279	−	0.371884i	$-0.878712\pi$
$47$	−12.7279	−1.85656	−0.928279	+	0.371884i	$0.878712\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.00000i	1.09888i	0.835532	+	0.549442i	$0.185160\pi$
$53$	8.00000i	1.09888i	−0.835532	+	0.549442i	$0.814840\pi$
$54$	0	0
$55$	−8.48528	−1.14416
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4.24264i	− 0.552345i	−0.961108	−	0.276172i	$-0.910934\pi$
$59$	− 4.24264i	− 0.552345i	0.961108	−	0.276172i	$-0.0890661\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	12.7279	1.60357
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	− 4.24264i	− 0.518321i	−0.965834	−	0.259161i	$-0.916554\pi$
$67$	− 4.24264i	− 0.518321i	0.965834	−	0.259161i	$-0.0834459\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.24264	0.503509	0.251754	−	0.967791i	$-0.418992\pi$
$71$	4.24264	0.503509	0.251754	+	0.967791i	$0.418992\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.0000i	2.05129i
$78$	0	0
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 12.7279i	− 1.39707i	−0.715575	−	0.698535i	$-0.753835\pi$
$83$	− 12.7279i	− 1.39707i	0.715575	−	0.698535i	$-0.246165\pi$
$84$	0	0
$85$	8.00000i	0.867722i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	− 4.24264i	− 0.444750i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.48528	0.870572
$96$	0	0
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	0	0
$99$	12.7279i	1.27920i
$100$	0	0
$101$	4.00000i	0.398015i	0.979998	+	0.199007i	$0.0637718\pi$
$101$	4.00000i	0.398015i	−0.979998	+	0.199007i	$0.936228\pi$
$102$	0	0
$103$	−16.9706	−1.67216	−0.836080	−	0.548608i	$-0.815158\pi$
$103$	−16.9706	−1.67216	−0.836080	+	0.548608i	$0.815158\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.9706i	1.64061i	0.571929	+	0.820303i	$0.306195\pi$
$107$	16.9706i	1.64061i	−0.571929	+	0.820303i	$0.693805\pi$
$108$	0	0
$109$	2.00000i	0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$	2.00000i	0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.00000i	− 0.277350i
$118$	0	0
$119$	16.9706	1.55569
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.9706i	1.48272i	0.671105	+	0.741362i	$0.265820\pi$
$131$	16.9706i	1.48272i	−0.671105	+	0.741362i	$0.734180\pi$
$132$	0	0
$133$	− 18.0000i	− 1.56080i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	16.9706i	1.43942i	0.694273	+	0.719712i	$0.255726\pi$
$139$	16.9706i	1.43942i	−0.694273	+	0.719712i	$0.744274\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.24264	0.354787
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.0000i	0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$	10.0000i	0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$	0	0
$151$	−12.7279	−1.03578	−0.517892	−	0.855446i	$-0.673283\pi$
$151$	−12.7279	−1.03578	−0.517892	+	0.855446i	$0.673283\pi$
$152$	0	0
$153$	12.0000	0.970143
$154$	0	0
$155$	− 8.48528i	− 0.681554i
$156$	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
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 21.2132i	− 1.66155i	−0.556611	−	0.830773i	$-0.687899\pi$
$163$	− 21.2132i	− 1.66155i	0.556611	−	0.830773i	$-0.312101\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.24264	−0.328305	−0.164153	−	0.986435i	$-0.552489\pi$
$167$	−4.24264	−0.328305	−0.164153	+	0.986435i	$0.552489\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 12.7279i	− 0.973329i
$172$	0	0
$173$	4.00000i	0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	4.00000i	0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$	0	0
$175$	4.24264	0.320713
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.48528i	0.634220i	0.948389	+	0.317110i	$0.102712\pi$
$179$	8.48528i	0.634220i	−0.948389	+	0.317110i	$0.897288\pi$
$180$	0	0
$181$	− 2.00000i	− 0.148659i	−0.997234	−	0.0743294i	$-0.976318\pi$
$181$	− 2.00000i	− 0.148659i	0.997234	−	0.0743294i	$-0.0236816\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	16.9706i	1.24101i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−25.4558	−1.84192	−0.920960	−	0.389657i	$-0.872594\pi$
$191$	−25.4558	−1.84192	−0.920960	+	0.389657i	$0.872594\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 26.0000i	− 1.85242i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$	− 26.0000i	− 1.85242i	0.377004	−	0.926212i	$-0.376954\pi$
$198$	0	0
$199$	16.9706	1.20301	0.601506	−	0.798869i	$-0.294568\pi$
$199$	16.9706	1.20301	0.601506	+	0.798869i	$0.294568\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 8.48528i	− 0.595550i
$204$	0	0
$205$	− 4.00000i	− 0.279372i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	18.0000	1.24509
$210$	0	0
$211$	− 16.9706i	− 1.16830i	−0.811645	−	0.584151i	$-0.801428\pi$
$211$	− 16.9706i	− 1.16830i	0.811645	−	0.584151i	$-0.198572\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.9706	1.15738
$216$	0	0
$217$	−18.0000	−1.22192
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 4.00000i	− 0.269069i
$222$	0	0
$223$	−12.7279	−0.852325	−0.426162	−	0.904647i	$-0.640135\pi$
$223$	−12.7279	−0.852325	−0.426162	+	0.904647i	$0.640135\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	− 21.2132i	− 1.40797i	−0.710215	−	0.703985i	$-0.751402\pi$
$227$	− 21.2132i	− 1.40797i	0.710215	−	0.703985i	$-0.248598\pi$
$228$	0	0
$229$	− 22.0000i	− 1.45380i	−0.686743	−	0.726900i	$-0.740960\pi$
$229$	− 22.0000i	− 1.45380i	0.686743	−	0.726900i	$-0.259040\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	− 25.4558i	− 1.66056i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.24264	0.274434	0.137217	−	0.990541i	$-0.456184\pi$
$239$	4.24264	0.274434	0.137217	+	0.990541i	$0.456184\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	22.0000i	1.40553i
$246$	0	0
$247$	−4.24264	−0.269953
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 8.48528i	− 0.535586i	−0.963476	−	0.267793i	$-0.913706\pi$
$251$	− 8.48528i	− 0.535586i	0.963476	−	0.267793i	$-0.0862944\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	25.4558i	1.58175i
$260$	0	0
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	−8.48528	−0.523225	−0.261612	−	0.965173i	$-0.584254\pi$
$263$	−8.48528	−0.523225	−0.261612	+	0.965173i	$0.584254\pi$
$264$	0	0
$265$	−16.0000	−0.982872
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.00000i	0.121942i	0.998140	+	0.0609711i	$0.0194197\pi$
$269$	2.00000i	0.121942i	−0.998140	+	0.0609711i	$0.980580\pi$
$270$	0	0
$271$	−12.7279	−0.773166	−0.386583	−	0.922255i	$-0.626345\pi$
$271$	−12.7279	−0.773166	−0.386583	+	0.922255i	$0.626345\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.24264i	0.255841i
$276$	0	0
$277$	28.0000i	1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$	28.0000i	1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$	0	0
$279$	−12.7279	−0.762001
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	8.48528i	0.504398i	0.967675	+	0.252199i	$0.0811537\pi$
$283$	8.48528i	0.504398i	−0.967675	+	0.252199i	$0.918846\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.48528	−0.500870
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	0	0
$295$	8.48528	0.494032
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 36.0000i	− 2.07501i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	29.6985i	1.69498i	0.530810	+	0.847491i	$0.321888\pi$
$307$	29.6985i	1.69498i	−0.530810	+	0.847491i	$0.678112\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.9706	0.962312	0.481156	−	0.876635i	$-0.340217\pi$
$311$	16.9706	0.962312	0.481156	+	0.876635i	$0.340217\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	25.4558i	1.43427i
$316$	0	0
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	0	0
$319$	8.48528	0.475085
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 16.9706i	− 0.944267i
$324$	0	0
$325$	− 1.00000i	− 0.0554700i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−54.0000	−2.97712
$330$	0	0
$331$	12.7279i	0.699590i	0.936826	+	0.349795i	$0.113749\pi$
$331$	12.7279i	0.699590i	−0.936826	+	0.349795i	$0.886251\pi$
$332$	0	0
$333$	18.0000i	0.986394i
$334$	0	0
$335$	8.48528	0.463600
$336$	0	0
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 18.0000i	− 0.974755i
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 33.9411i	− 1.82206i	−0.412346	−	0.911028i	$-0.635290\pi$
$347$	− 33.9411i	− 1.82206i	0.412346	−	0.911028i	$-0.364710\pi$
$348$	0	0
$349$	− 18.0000i	− 0.963518i	−0.876304	−	0.481759i	$-0.839998\pi$
$349$	− 18.0000i	− 0.963518i	0.876304	−	0.481759i	$-0.160002\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	8.48528i	0.450352i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.24264	0.223918	0.111959	−	0.993713i	$-0.464287\pi$
$359$	4.24264	0.223918	0.111959	+	0.993713i	$0.464287\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 12.0000i	− 0.628109i
$366$	0	0
$367$	8.48528	0.442928	0.221464	−	0.975169i	$-0.428916\pi$
$367$	8.48528	0.442928	0.221464	+	0.975169i	$0.428916\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	33.9411i	1.76214i
$372$	0	0
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	12.7279i	0.653789i	0.945061	+	0.326895i	$0.106002\pi$
$379$	12.7279i	0.653789i	−0.945061	+	0.326895i	$0.893998\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.24264	0.216789	0.108394	−	0.994108i	$-0.465429\pi$
$383$	4.24264	0.216789	0.108394	+	0.994108i	$0.465429\pi$
$384$	0	0
$385$	−36.0000	−1.83473
$386$	0	0
$387$	− 25.4558i	− 1.29399i
$388$	0	0
$389$	− 26.0000i	− 1.31825i	−0.752032	−	0.659126i	$-0.770926\pi$
$389$	− 26.0000i	− 1.31825i	0.752032	−	0.659126i	$-0.229074\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	16.9706i	0.853882i
$396$	0	0
$397$	− 30.0000i	− 1.50566i	−0.658217	−	0.752828i	$-0.728689\pi$
$397$	− 30.0000i	− 1.50566i	0.658217	−	0.752828i	$-0.271311\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	0	0
$403$	4.24264i	0.211341i
$404$	0	0
$405$	18.0000i	0.894427i
$406$	0	0
$407$	−25.4558	−1.26180
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 18.0000i	− 0.885722i
$414$	0	0
$415$	25.4558	1.24958
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 16.9706i	− 0.829066i	−0.910034	−	0.414533i	$-0.863945\pi$
$419$	− 16.9706i	− 0.829066i	0.910034	−	0.414533i	$-0.136055\pi$
$420$	0	0
$421$	− 26.0000i	− 1.26716i	−0.773676	−	0.633581i	$-0.781584\pi$
$421$	− 26.0000i	− 1.26716i	0.773676	−	0.633581i	$-0.218416\pi$
$422$	0	0
$423$	−38.1838	−1.85656
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.7279	0.613082	0.306541	−	0.951857i	$-0.400828\pi$
$431$	12.7279	0.613082	0.306541	+	0.951857i	$0.400828\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	0	0
$441$	33.0000	1.57143
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	− 20.0000i	− 0.948091i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	− 8.48528i	− 0.399556i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8.48528	0.397796
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.0000i	0.652045i	0.945362	+	0.326023i	$0.105709\pi$
$461$	14.0000i	0.652045i	−0.945362	+	0.326023i	$0.894291\pi$
$462$	0	0
$463$	38.1838	1.77455	0.887275	−	0.461241i	$-0.152596\pi$
$463$	38.1838	1.77455	0.887275	+	0.461241i	$0.152596\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8.48528i	− 0.392652i	−0.980539	−	0.196326i	$-0.937099\pi$
$467$	− 8.48528i	− 0.392652i	0.980539	−	0.196326i	$-0.0629011\pi$
$468$	0	0
$469$	− 18.0000i	− 0.831163i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	36.0000	1.65528
$474$	0	0
$475$	− 4.24264i	− 0.194666i
$476$	0	0
$477$	24.0000i	1.09888i
$478$	0	0
$479$	−4.24264	−0.193851	−0.0969256	−	0.995292i	$-0.530901\pi$
$479$	−4.24264	−0.193851	−0.0969256	+	0.995292i	$0.530901\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	36.0000i	1.63468i
$486$	0	0
$487$	29.6985	1.34577	0.672883	−	0.739749i	$-0.265056\pi$
$487$	29.6985	1.34577	0.672883	+	0.739749i	$0.265056\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 25.4558i	− 1.14881i	−0.818573	−	0.574403i	$-0.805234\pi$
$491$	− 25.4558i	− 1.14881i	0.818573	−	0.574403i	$-0.194766\pi$
$492$	0	0
$493$	− 8.00000i	− 0.360302i
$494$	0	0
$495$	−25.4558	−1.14416
$496$	0	0
$497$	18.0000	0.807410
$498$	0	0
$499$	− 21.2132i	− 0.949633i	−0.880085	−	0.474817i	$-0.842514\pi$
$499$	− 21.2132i	− 0.949633i	0.880085	−	0.474817i	$-0.157486\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−8.48528	−0.378340	−0.189170	−	0.981944i	$-0.560580\pi$
$503$	−8.48528	−0.378340	−0.189170	+	0.981944i	$0.560580\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 38.0000i	− 1.68432i	−0.539227	−	0.842160i	$-0.681284\pi$
$509$	− 38.0000i	− 1.68432i	0.539227	−	0.842160i	$-0.318716\pi$
$510$	0	0
$511$	−25.4558	−1.12610
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 33.9411i	− 1.49562i
$516$	0	0
$517$	− 54.0000i	− 2.37492i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.00000	0.350486	0.175243	−	0.984525i	$-0.443929\pi$
$521$	8.00000	0.350486	0.175243	+	0.984525i	$0.443929\pi$
$522$	0	0
$523$	− 16.9706i	− 0.742071i	−0.928619	−	0.371035i	$-0.879003\pi$
$523$	− 16.9706i	− 0.742071i	0.928619	−	0.371035i	$-0.120997\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.9706	−0.739249
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 12.7279i	− 0.552345i
$532$	0	0
$533$	2.00000i	0.0866296i
$534$	0	0
$535$	−33.9411	−1.46740
$536$	0	0
$537$	0	0
$538$	0	0
$539$	46.6690i	2.01018i
$540$	0	0
$541$	38.0000i	1.63375i	0.576816	+	0.816874i	$0.304295\pi$
$541$	38.0000i	1.63375i	−0.576816	+	0.816874i	$0.695705\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.48528	−0.361485
$552$	0	0
$553$	36.0000	1.53088
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.0000i	1.61011i	0.593199	+	0.805056i	$0.297865\pi$
$557$	38.0000i	1.61011i	−0.593199	+	0.805056i	$0.702135\pi$
$558$	0	0
$559$	−8.48528	−0.358889
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.4558i	1.07284i	0.843952	+	0.536418i	$0.180223\pi$
$563$	25.4558i	1.07284i	−0.843952	+	0.536418i	$0.819777\pi$
$564$	0	0
$565$	32.0000i	1.34625i
$566$	0	0
$567$	38.1838	1.60357
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	− 8.48528i	− 0.355098i	−0.984112	−	0.177549i	$-0.943183\pi$
$571$	− 8.48528i	− 0.355098i	0.984112	−	0.177549i	$-0.0568168\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 54.0000i	− 2.24030i
$582$	0	0
$583$	−33.9411	−1.40570
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	12.7279i	0.525338i	0.964886	+	0.262669i	$0.0846027\pi$
$587$	12.7279i	0.525338i	−0.964886	+	0.262669i	$0.915397\pi$
$588$	0	0
$589$	18.0000i	0.741677i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	33.9411i	1.39145i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.48528	−0.346699	−0.173350	−	0.984860i	$-0.555459\pi$
$599$	−8.48528	−0.346699	−0.173350	+	0.984860i	$0.555459\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	− 12.7279i	− 0.518321i
$604$	0	0
$605$	− 14.0000i	− 0.569181i
$606$	0	0
$607$	42.4264	1.72203	0.861017	−	0.508576i	$-0.169828\pi$
$607$	42.4264	1.72203	0.861017	+	0.508576i	$0.169828\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.7279i	0.514917i
$612$	0	0
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	− 29.6985i	− 1.19368i	−0.802359	−	0.596841i	$-0.796422\pi$
$619$	− 29.6985i	− 1.19368i	0.802359	−	0.596841i	$-0.203578\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−42.4264	−1.69978
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24.0000i	0.956943i
$630$	0	0
$631$	21.2132	0.844484	0.422242	−	0.906483i	$-0.361243\pi$
$631$	21.2132	0.844484	0.422242	+	0.906483i	$0.361243\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 11.0000i	− 0.435836i
$638$	0	0
$639$	12.7279	0.503509
$640$	0	0
$641$	−20.0000	−0.789953	−0.394976	−	0.918691i	$-0.629247\pi$
$641$	−20.0000	−0.789953	−0.394976	+	0.918691i	$0.629247\pi$
$642$	0	0
$643$	38.1838i	1.50582i	0.658123	+	0.752910i	$0.271351\pi$
$643$	38.1838i	1.50582i	−0.658123	+	0.752910i	$0.728649\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	0	0
$651$	0	0
$652$	0	0
$653$	46.0000i	1.80012i	0.435767	+	0.900060i	$0.356477\pi$
$653$	46.0000i	1.80012i	−0.435767	+	0.900060i	$0.643523\pi$
$654$	0	0
$655$	−33.9411	−1.32619
$656$	0	0
$657$	−18.0000	−0.702247
$658$	0	0
$659$	− 16.9706i	− 0.661079i	−0.943792	−	0.330540i	$-0.892769\pi$
$659$	− 16.9706i	− 0.661079i	0.943792	−	0.330540i	$-0.107231\pi$
$660$	0	0
$661$	18.0000i	0.700119i	0.936727	+	0.350059i	$0.113839\pi$
$661$	18.0000i	0.700119i	−0.936727	+	0.350059i	$0.886161\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	36.0000	1.39602
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 16.0000i	− 0.614930i	−0.951559	−	0.307465i	$-0.900519\pi$
$677$	− 16.0000i	− 0.614930i	0.951559	−	0.307465i	$-0.0994807\pi$
$678$	0	0
$679$	76.3675	2.93072
$680$	0	0
$681$	0	0
$682$	0	0
$683$	38.1838i	1.46106i	0.682880	+	0.730531i	$0.260727\pi$
$683$	38.1838i	1.46106i	−0.682880	+	0.730531i	$0.739273\pi$
$684$	0	0
$685$	4.00000i	0.152832i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.00000	0.304776
$690$	0	0
$691$	12.7279i	0.484193i	0.970252	+	0.242096i	$0.0778351\pi$
$691$	12.7279i	0.484193i	−0.970252	+	0.242096i	$0.922165\pi$
$692$	0	0
$693$	54.0000i	2.05129i
$694$	0	0
$695$	−33.9411	−1.28746
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 38.0000i	− 1.43524i	−0.696435	−	0.717620i	$-0.745231\pi$
$701$	− 38.0000i	− 1.43524i	0.696435	−	0.717620i	$-0.254769\pi$
$702$	0	0
$703$	25.4558	0.960085
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.9706i	0.638244i
$708$	0	0
$709$	− 10.0000i	− 0.375558i	−0.982211	−	0.187779i	$-0.939871\pi$
$709$	− 10.0000i	− 0.375558i	0.982211	−	0.187779i	$-0.0601289\pi$
$710$	0	0
$711$	25.4558	0.954669
$712$	0	0
$713$	0	0
$714$	0	0
$715$	8.48528i	0.317332i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16.9706	0.632895	0.316448	−	0.948610i	$-0.397510\pi$
$719$	16.9706	0.632895	0.316448	+	0.948610i	$0.397510\pi$
$720$	0	0
$721$	−72.0000	−2.68142
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.00000i	− 0.0742781i
$726$	0	0
$727$	50.9117	1.88821	0.944105	−	0.329645i	$-0.106929\pi$
$727$	50.9117	1.88821	0.944105	+	0.329645i	$0.106929\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	− 33.9411i	− 1.25536i
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.0000	0.663039
$738$	0	0
$739$	38.1838i	1.40461i	0.711875	+	0.702306i	$0.247846\pi$
$739$	38.1838i	1.40461i	−0.711875	+	0.702306i	$0.752154\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12.7279	−0.466942	−0.233471	−	0.972364i	$-0.575008\pi$
$743$	−12.7279	−0.466942	−0.233471	+	0.972364i	$0.575008\pi$
$744$	0	0
$745$	−20.0000	−0.732743
$746$	0	0
$747$	− 38.1838i	− 1.39707i
$748$	0	0
$749$	72.0000i	2.63082i
$750$	0	0
$751$	−16.9706	−0.619265	−0.309632	−	0.950856i	$-0.600206\pi$
$751$	−16.9706	−0.619265	−0.309632	+	0.950856i	$0.600206\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 25.4558i	− 0.926433i
$756$	0	0
$757$	− 16.0000i	− 0.581530i	−0.956795	−	0.290765i	$-0.906090\pi$
$757$	− 16.0000i	− 0.581530i	0.956795	−	0.290765i	$-0.0939098\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	0	0
$763$	8.48528i	0.307188i
$764$	0	0
$765$	24.0000i	0.867722i
$766$	0	0
$767$	−4.24264	−0.153193
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26.0000i	0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$	26.0000i	0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$	0	0
$775$	−4.24264	−0.152400
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.48528i	0.304017i
$780$	0	0
$781$	18.0000i	0.644091i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	36.0000	1.28490
$786$	0	0
$787$	− 21.2132i	− 0.756169i	−0.925771	−	0.378085i	$-0.876583\pi$
$787$	− 21.2132i	− 0.756169i	0.925771	−	0.378085i	$-0.123417\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	67.8823	2.41361
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	0	0
$799$	−50.9117	−1.80113
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	− 25.4558i	− 0.898317i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	40.0000	1.40633	0.703163	−	0.711029i	$-0.251771\pi$
$809$	40.0000	1.40633	0.703163	+	0.711029i	$0.251771\pi$
$810$	0	0
$811$	4.24264i	0.148979i	0.997222	+	0.0744896i	$0.0237328\pi$
$811$	4.24264i	0.148979i	−0.997222	+	0.0744896i	$0.976267\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	42.4264	1.48613
$816$	0	0
$817$	−36.0000	−1.25948
$818$	0	0
$819$	− 12.7279i	− 0.444750i
$820$	0	0
$821$	10.0000i	0.349002i	0.984657	+	0.174501i	$0.0558313\pi$
$821$	10.0000i	0.349002i	−0.984657	+	0.174501i	$0.944169\pi$
$822$	0	0
$823$	−33.9411	−1.18311	−0.591557	−	0.806263i	$-0.701486\pi$
$823$	−33.9411	−1.18311	−0.591557	+	0.806263i	$0.701486\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 4.24264i	− 0.147531i	−0.997276	−	0.0737655i	$-0.976498\pi$
$827$	− 4.24264i	− 0.147531i	0.997276	−	0.0737655i	$-0.0235016\pi$
$828$	0	0
$829$	− 52.0000i	− 1.80603i	−0.429604	−	0.903017i	$-0.641347\pi$
$829$	− 52.0000i	− 1.80603i	0.429604	−	0.903017i	$-0.358653\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	44.0000	1.52451
$834$	0	0
$835$	− 8.48528i	− 0.293645i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−46.6690	−1.61119	−0.805597	−	0.592464i	$-0.798155\pi$
$839$	−46.6690	−1.61119	−0.805597	+	0.592464i	$0.798155\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 2.00000i	− 0.0688021i
$846$	0	0
$847$	−29.6985	−1.02045
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	6.00000i	0.205436i	0.994711	+	0.102718i	$0.0327539\pi$
$853$	6.00000i	0.205436i	−0.994711	+	0.102718i	$0.967246\pi$
$854$	0	0
$855$	25.4558	0.870572
$856$	0	0
$857$	−26.0000	−0.888143	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000	−0.888143	−0.444072	+	0.895991i	$0.646466\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	46.6690	1.58863	0.794316	−	0.607504i	$-0.207829\pi$
$863$	46.6690	1.58863	0.794316	+	0.607504i	$0.207829\pi$
$864$	0	0
$865$	−8.00000	−0.272008
$866$	0	0
$867$	0	0
$868$	0	0
$869$	36.0000i	1.22122i
$870$	0	0
$871$	−4.24264	−0.143756
$872$	0	0
$873$	54.0000	1.82762
$874$	0	0
$875$	50.9117i	1.72113i
$876$	0	0
$877$	− 54.0000i	− 1.82345i	−0.410801	−	0.911725i	$-0.634751\pi$
$877$	− 54.0000i	− 1.82345i	0.410801	−	0.911725i	$-0.365249\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	− 42.4264i	− 1.42776i	−0.700267	−	0.713881i	$-0.746936\pi$
$883$	− 42.4264i	− 1.42776i	0.700267	−	0.713881i	$-0.253064\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−42.4264	−1.42454	−0.712270	−	0.701906i	$-0.752333\pi$
$887$	−42.4264	−1.42454	−0.712270	+	0.701906i	$0.752333\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	38.1838i	1.27920i
$892$	0	0
$893$	54.0000i	1.80704i
$894$	0	0
$895$	−16.9706	−0.567263
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.48528i	0.283000i
$900$	0	0
$901$	32.0000i	1.06607i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.00000	0.132964
$906$	0	0
$907$	42.4264i	1.40875i	0.709830	+	0.704373i	$0.248772\pi$
$907$	42.4264i	1.40875i	−0.709830	+	0.704373i	$0.751228\pi$
$908$	0	0
$909$	12.0000i	0.398015i
$910$	0	0
$911$	8.48528	0.281130	0.140565	−	0.990071i	$-0.455108\pi$
$911$	8.48528	0.281130	0.140565	+	0.990071i	$0.455108\pi$
$912$	0	0
$913$	54.0000	1.78714
$914$	0	0
$915$	0	0
$916$	0	0
$917$	72.0000i	2.37765i
$918$	0	0
$919$	−42.4264	−1.39952	−0.699759	−	0.714379i	$-0.746709\pi$
$919$	−42.4264	−1.39952	−0.699759	+	0.714379i	$0.746709\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 4.24264i	− 0.139648i
$924$	0	0
$925$	6.00000i	0.197279i
$926$	0	0
$927$	−50.9117	−1.67216
$928$	0	0
$929$	22.0000	0.721797	0.360898	−	0.932605i	$-0.382470\pi$
$929$	22.0000	0.721797	0.360898	+	0.932605i	$0.382470\pi$
$930$	0	0
$931$	− 46.6690i	− 1.52952i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−33.9411	−1.10999
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 34.0000i	− 1.10837i	−0.832394	−	0.554184i	$-0.813030\pi$
$941$	− 34.0000i	− 1.10837i	0.832394	−	0.554184i	$-0.186970\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.24264i	0.137867i	0.997621	+	0.0689336i	$0.0219597\pi$
$947$	4.24264i	0.137867i	−0.997621	+	0.0689336i	$0.978040\pi$
$948$	0	0
$949$	6.00000i	0.194768i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−10.0000	−0.323932	−0.161966	−	0.986796i	$-0.551783\pi$
$953$	−10.0000	−0.323932	−0.161966	+	0.986796i	$0.551783\pi$
$954$	0	0
$955$	− 50.9117i	− 1.64746i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.48528	0.274004
$960$	0	0
$961$	−13.0000	−0.419355
$962$	0	0
$963$	50.9117i	1.64061i
$964$	0	0
$965$	− 44.0000i	− 1.41641i
$966$	0	0
$967$	−46.6690	−1.50078	−0.750388	−	0.660998i	$-0.770133\pi$
$967$	−46.6690	−1.50078	−0.750388	+	0.660998i	$0.770133\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 33.9411i	− 1.08922i	−0.838689	−	0.544611i	$-0.816677\pi$
$971$	− 33.9411i	− 1.08922i	0.838689	−	0.544611i	$-0.183323\pi$
$972$	0	0
$973$	72.0000i	2.30821i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50.0000	1.59964	0.799821	−	0.600239i	$-0.204928\pi$
$977$	50.0000	1.59964	0.799821	+	0.600239i	$0.204928\pi$
$978$	0	0
$979$	− 42.4264i	− 1.35595i
$980$	0	0
$981$	6.00000i	0.191565i
$982$	0	0
$983$	−21.2132	−0.676596	−0.338298	−	0.941039i	$-0.609851\pi$
$983$	−21.2132	−0.676596	−0.338298	+	0.941039i	$0.609851\pi$
$984$	0	0
$985$	52.0000	1.65686
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−25.4558	−0.808632	−0.404316	−	0.914619i	$-0.632490\pi$
$991$	−25.4558	−0.808632	−0.404316	+	0.914619i	$0.632490\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	33.9411i	1.07601i
$996$	0	0
$997$	− 48.0000i	− 1.52018i	−0.649821	−	0.760088i	$-0.725156\pi$
$997$	− 48.0000i	− 1.52018i	0.649821	−	0.760088i	$-0.274844\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1664.2.b.h.833.4	yes	4
4.3	odd	2	inner	1664.2.b.h.833.3	yes	4
8.3	odd	2	inner	1664.2.b.h.833.1	✓	4
8.5	even	2	inner	1664.2.b.h.833.2	yes	4
16.3	odd	4		3328.2.a.w.1.2		2
16.5	even	4		3328.2.a.s.1.1		2
16.11	odd	4		3328.2.a.s.1.2		2
16.13	even	4		3328.2.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.h.833.1	✓	4	8.3	odd	2	inner
1664.2.b.h.833.2	yes	4	8.5	even	2	inner
1664.2.b.h.833.3	yes	4	4.3	odd	2	inner
1664.2.b.h.833.4	yes	4	1.1	even	1	trivial
3328.2.a.s.1.1		2	16.5	even	4
3328.2.a.s.1.2		2	16.11	odd	4
3328.2.a.w.1.1		2	16.13	even	4
3328.2.a.w.1.2		2	16.3	odd	4

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:	\( N \)	\(=\)	\( 1664 = 2^{7} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1664.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{5} +4.24264 q^{7} +3.00000 q^{9} +4.24264i q^{11} -1.00000i q^{13} +4.00000 q^{17} -4.24264i q^{19} +1.00000 q^{25} -2.00000i q^{29} -4.24264 q^{31} +8.48528i q^{35} +6.00000i q^{37} -2.00000 q^{41} -8.48528i q^{43} +6.00000i q^{45} -12.7279 q^{47} +11.0000 q^{49} +8.00000i q^{53} -8.48528 q^{55} -4.24264i q^{59} +12.7279 q^{63} +2.00000 q^{65} -4.24264i q^{67} +4.24264 q^{71} -6.00000 q^{73} +18.0000i q^{77} +8.48528 q^{79} +9.00000 q^{81} -12.7279i q^{83} +8.00000i q^{85} -10.0000 q^{89} -4.24264i q^{91} +8.48528 q^{95} +18.0000 q^{97} +12.7279i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9} + 16 q^{17} + 4 q^{25} - 8 q^{41} + 44 q^{49} + 8 q^{65} - 24 q^{73} + 36 q^{81} - 40 q^{89} + 72 q^{97}+O(q^{100}) \) 4 * q + 12 * q^9 + 16 * q^17 + 4 * q^25 - 8 * q^41 + 44 * q^49 + 8 * q^65 - 24 * q^73 + 36 * q^81 - 40 * q^89 + 72 * q^97

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