LMFDB - Embedded newform 5376.2.c.bc.2689.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5376,2,Mod(2689,5376)] 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("5376.2689"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5376, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5376 = 2^{8} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5376.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,2,0,-2,0,0,0,0,0,4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$42.9275761266$
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 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2689.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	5376.2689
Dual form			5376.2.c.bc.2689.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} +6.00000i q^{13} +2.00000 q^{15} +2.00000 q^{17} -4.00000i q^{19} -1.00000i q^{21} -8.00000 q^{23} +1.00000 q^{25} +1.00000i q^{27} -2.00000i q^{29} +4.00000 q^{33} +2.00000i q^{35} +10.0000i q^{37} +6.00000 q^{39} +6.00000 q^{41} +4.00000i q^{43} -2.00000i q^{45} +1.00000 q^{49} -2.00000i q^{51} -6.00000i q^{53} -8.00000 q^{55} -4.00000 q^{57} -4.00000i q^{59} +6.00000i q^{61} -1.00000 q^{63} -12.0000 q^{65} +4.00000i q^{67} +8.00000i q^{69} -8.00000 q^{71} -10.0000 q^{73} -1.00000i q^{75} +4.00000i q^{77} +1.00000 q^{81} -4.00000i q^{83} +4.00000i q^{85} -2.00000 q^{87} +6.00000 q^{89} +6.00000i q^{91} +8.00000 q^{95} -14.0000 q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} - 2 q^{9} + 4 q^{15} + 4 q^{17} - 16 q^{23} + 2 q^{25} + 8 q^{33} + 12 q^{39} + 12 q^{41} + 2 q^{49} - 16 q^{55} - 8 q^{57} - 2 q^{63} - 24 q^{65} - 16 q^{71} - 20 q^{73} + 2 q^{81} - 4 q^{87}+ \cdots - 28 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 2 * q^9 + 4 * q^15 + 4 * q^17 - 16 * q^23 + 2 * q^25 + 8 * q^33 + 12 * q^39 + 12 * q^41 + 2 * q^49 - 16 * q^55 - 8 * q^57 - 2 * q^63 - 24 * q^65 - 16 * q^71 - 20 * q^73 + 2 * q^81 - 4 * q^87 + 12 * q^89 + 16 * q^95 - 28 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times$.

$n$	$1793$	$2815$	$4609$	$5125$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$4$	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
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	− 1.00000i	− 0.218218i
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	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
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	2.00000i	0.338062i
$36$	0	0
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	− 2.00000i	− 0.298142i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 2.00000i	− 0.280056i
$52$	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
$55$	−8.00000	−1.07872
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	− 4.00000i	− 0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$	− 4.00000i	− 0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$	0	0
$61$	6.00000i	0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$	6.00000i	0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−12.0000	−1.48842
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	8.00000i	0.963087i
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	4.00000i	0.455842i
$78$	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
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	4.00000i	0.433861i
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	6.00000i	0.628971i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	− 4.00000i	− 0.402015i
$100$	0	0
$101$	2.00000i	0.199007i	0.995037	+	0.0995037i	$0.0317255\pi$
$101$	2.00000i	0.199007i	−0.995037	+	0.0995037i	$0.968274\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	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
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	− 16.0000i	− 1.49201i
$116$	0	0
$117$	− 6.00000i	− 0.554700i
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	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
$129$	4.00000	0.352180
$130$	0	0
$131$	− 20.0000i	− 1.74741i	−0.486458	−	0.873704i	$-0.661711\pi$
$131$	− 20.0000i	− 1.74741i	0.486458	−	0.873704i	$-0.338289\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	−2.00000	−0.172133
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	− 4.00000i	− 0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$	− 4.00000i	− 0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−24.0000	−2.00698
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	0	0
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	0	0
$165$	8.00000i	0.622799i
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	22.0000i	1.67263i	0.548250	+	0.836315i	$0.315294\pi$
$173$	22.0000i	1.67263i	−0.548250	+	0.836315i	$0.684706\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	− 12.0000i	− 0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$	− 12.0000i	− 0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$	0	0
$181$	18.0000i	1.33793i	0.743294	+	0.668965i	$0.233262\pi$
$181$	18.0000i	1.33793i	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	−20.0000	−1.47043
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	1.00000i	0.0727393i
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	12.0000i	0.859338i
$196$	0	0
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	12.0000i	0.838116i
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	20.0000i	1.37686i	0.725304	+	0.688428i	$0.241699\pi$
$211$	20.0000i	1.37686i	−0.725304	+	0.688428i	$0.758301\pi$
$212$	0	0
$213$	8.00000i	0.548151i
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.0000i	0.675737i
$220$	0	0
$221$	12.0000i	0.807207i
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	2.00000i	0.132164i	0.997814	+	0.0660819i	$0.0210498\pi$
$229$	2.00000i	0.132164i	−0.997814	+	0.0660819i	$0.978950\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	2.00000i	0.127775i
$246$	0	0
$247$	24.0000	1.52708
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	12.0000i	0.757433i	0.925513	+	0.378717i	$0.123635\pi$
$251$	12.0000i	0.757433i	−0.925513	+	0.378717i	$0.876365\pi$
$252$	0	0
$253$	− 32.0000i	− 2.01182i
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	0	0
$259$	10.0000i	0.621370i
$260$	0	0
$261$	2.00000i	0.123797i
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	0	0
$269$	22.0000i	1.34136i	0.741745	+	0.670682i	$0.233998\pi$
$269$	22.0000i	1.34136i	−0.741745	+	0.670682i	$0.766002\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	4.00000i	0.241209i
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	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
$285$	− 8.00000i	− 0.473879i
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	14.0000i	0.820695i
$292$	0	0
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	− 48.0000i	− 2.77591i
$300$	0	0
$301$	4.00000i	0.230556i
$302$	0	0
$303$	2.00000	0.114897
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	28.0000i	1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$	28.0000i	1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$	0	0
$309$	8.00000i	0.455104i
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	− 2.00000i	− 0.112687i
$316$	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
$319$	8.00000	0.447914
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	6.00000i	0.332820i
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.00000i	0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$	4.00000i	0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	0	0
$333$	− 10.0000i	− 0.547997i
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	14.0000i	0.760376i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−16.0000	−0.861411
$346$	0	0
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	22.0000i	1.17763i	0.808267	+	0.588817i	$0.200406\pi$
$349$	22.0000i	1.17763i	−0.808267	+	0.588817i	$0.799594\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	− 16.0000i	− 0.849192i
$356$	0	0
$357$	− 2.00000i	− 0.105851i
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	5.00000i	0.262432i
$364$	0	0
$365$	− 20.0000i	− 1.04685i
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	− 6.00000i	− 0.311504i
$372$	0	0
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	20.0000i	1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$	20.0000i	1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	− 4.00000i	− 0.203331i
$388$	0	0
$389$	26.0000i	1.31825i	0.752032	+	0.659126i	$0.229074\pi$
$389$	26.0000i	1.31825i	−0.752032	+	0.659126i	$0.770926\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	−20.0000	−1.00887
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.00000i	0.0993808i
$406$	0	0
$407$	−40.0000	−1.98273
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	10.0000i	0.493264i
$412$	0	0
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	− 36.0000i	− 1.75872i	−0.476162	−	0.879358i	$-0.657972\pi$
$419$	− 36.0000i	− 1.75872i	0.476162	−	0.879358i	$-0.342028\pi$
$420$	0	0
$421$	− 6.00000i	− 0.292422i	−0.989253	−	0.146211i	$-0.953292\pi$
$421$	− 6.00000i	− 0.292422i	0.989253	−	0.146211i	$-0.0467079\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	6.00000i	0.290360i
$428$	0	0
$429$	24.0000i	1.15873i
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	− 4.00000i	− 0.191785i
$436$	0	0
$437$	32.0000i	1.53077i
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	12.0000i	0.568855i
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	0	0
$451$	24.0000i	1.13012i
$452$	0	0
$453$	− 8.00000i	− 0.375873i
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	2.00000i	0.0933520i
$460$	0	0
$461$	22.0000i	1.02464i	0.858794	+	0.512321i	$0.171214\pi$
$461$	22.0000i	1.02464i	−0.858794	+	0.512321i	$0.828786\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	0	0
$469$	4.00000i	0.184703i
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	6.00000i	0.274721i
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−60.0000	−2.73576
$482$	0	0
$483$	8.00000i	0.364013i
$484$	0	0
$485$	− 28.0000i	− 1.27141i
$486$	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
$489$	20.0000	0.904431
$490$	0	0
$491$	− 12.0000i	− 0.541552i	−0.962642	−	0.270776i	$-0.912720\pi$
$491$	− 12.0000i	− 0.541552i	0.962642	−	0.270776i	$-0.0872803\pi$
$492$	0	0
$493$	− 4.00000i	− 0.180151i
$494$	0	0
$495$	8.00000	0.359573
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	− 44.0000i	− 1.96971i	−0.173379	−	0.984855i	$-0.555468\pi$
$499$	− 44.0000i	− 1.96971i	0.173379	−	0.984855i	$-0.444532\pi$
$500$	0	0
$501$	− 8.00000i	− 0.357414i
$502$	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
$505$	−4.00000	−0.177998
$506$	0	0
$507$	23.0000i	1.02147i
$508$	0	0
$509$	6.00000i	0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$	6.00000i	0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	− 16.0000i	− 0.705044i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	22.0000	0.965693
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	0	0
$525$	− 1.00000i	− 0.0436436i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	4.00000i	0.173585i
$532$	0	0
$533$	36.0000i	1.55933i
$534$	0	0
$535$	24.0000	1.03761
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	4.00000i	0.172292i
$540$	0	0
$541$	30.0000i	1.28980i	0.764267	+	0.644900i	$0.223101\pi$
$541$	30.0000i	1.28980i	−0.764267	+	0.644900i	$0.776899\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	− 12.0000i	− 0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$	− 12.0000i	− 0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$	0	0
$549$	− 6.00000i	− 0.256074i
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	0	0
$555$	20.0000i	0.848953i
$556$	0	0
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	44.0000i	1.85438i	0.374593	+	0.927189i	$0.377783\pi$
$563$	44.0000i	1.85438i	−0.374593	+	0.927189i	$0.622217\pi$
$564$	0	0
$565$	− 28.0000i	− 1.17797i
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	− 12.0000i	− 0.502184i	−0.967963	−	0.251092i	$-0.919210\pi$
$571$	− 12.0000i	− 0.502184i	0.967963	−	0.251092i	$-0.0807897\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.00000	−0.333623
$576$	0	0
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	0	0
$579$	− 2.00000i	− 0.0831172i
$580$	0	0
$581$	− 4.00000i	− 0.165948i
$582$	0	0
$583$	24.0000	0.993978
$584$	0	0
$585$	12.0000	0.496139
$586$	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
$589$	0	0
$590$	0	0
$591$	10.0000	0.411345
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	4.00000i	0.163984i
$596$	0	0
$597$	8.00000i	0.327418i
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	− 10.0000i	− 0.406558i
$606$	0	0
$607$	48.0000	1.94826	0.974130	−	0.225989i	$-0.0725612\pi$
$607$	48.0000	1.94826	0.974130	+	0.225989i	$0.0725612\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	0	0
$612$	0	0
$613$	42.0000i	1.69636i	0.529705	+	0.848182i	$0.322303\pi$
$613$	42.0000i	1.69636i	−0.529705	+	0.848182i	$0.677697\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	44.0000i	1.76851i	0.467005	+	0.884255i	$0.345333\pi$
$619$	44.0000i	1.76851i	−0.467005	+	0.884255i	$0.654667\pi$
$620$	0	0
$621$	− 8.00000i	− 0.321029i
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	− 16.0000i	− 0.638978i
$628$	0	0
$629$	20.0000i	0.797452i
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	8.00000	0.316475
$640$	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
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	8.00000i	0.315000i
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	0	0
$655$	40.0000	1.56293
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	− 28.0000i	− 1.09073i	−0.838200	−	0.545363i	$-0.816392\pi$
$659$	− 28.0000i	− 1.09073i	0.838200	−	0.545363i	$-0.183608\pi$
$660$	0	0
$661$	2.00000i	0.0777910i	0.999243	+	0.0388955i	$0.0123839\pi$
$661$	2.00000i	0.0777910i	−0.999243	+	0.0388955i	$0.987616\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	16.0000i	0.619522i
$668$	0	0
$669$	16.0000i	0.618596i
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	0	0
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	− 20.0000i	− 0.764161i
$686$	0	0
$687$	2.00000	0.0763048
$688$	0	0
$689$	36.0000	1.37149
$690$	0	0
$691$	− 4.00000i	− 0.152167i	−0.997101	−	0.0760836i	$-0.975758\pi$
$691$	− 4.00000i	− 0.152167i	0.997101	−	0.0760836i	$-0.0242416\pi$
$692$	0	0
$693$	− 4.00000i	− 0.151947i
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	− 22.0000i	− 0.832116i
$700$	0	0
$701$	− 2.00000i	− 0.0755390i	−0.999286	−	0.0377695i	$-0.987975\pi$
$701$	− 2.00000i	− 0.0755390i	0.999286	−	0.0377695i	$-0.0120253\pi$
$702$	0	0
$703$	40.0000	1.50863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.00000i	0.0752177i
$708$	0	0
$709$	10.0000i	0.375558i	0.982211	+	0.187779i	$0.0601289\pi$
$709$	10.0000i	0.375558i	−0.982211	+	0.187779i	$0.939871\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 48.0000i	− 1.79510i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	− 2.00000i	− 0.0743808i
$724$	0	0
$725$	− 2.00000i	− 0.0742781i
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	8.00000i	0.295891i
$732$	0	0
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	− 12.0000i	− 0.441427i	−0.975339	−	0.220714i	$-0.929161\pi$
$739$	− 12.0000i	− 0.441427i	0.975339	−	0.220714i	$-0.0708386\pi$
$740$	0	0
$741$	− 24.0000i	− 0.881662i
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	0	0
$747$	4.00000i	0.146352i
$748$	0	0
$749$	− 12.0000i	− 0.438470i
$750$	0	0
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	16.0000i	0.582300i
$756$	0	0
$757$	− 6.00000i	− 0.218074i	−0.994038	−	0.109037i	$-0.965223\pi$
$757$	− 6.00000i	− 0.218074i	0.994038	−	0.109037i	$-0.0347767\pi$
$758$	0	0
$759$	−32.0000	−1.16153
$760$	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
$763$	− 2.00000i	− 0.0724049i
$764$	0	0
$765$	− 4.00000i	− 0.144620i
$766$	0	0
$767$	24.0000	0.866590
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	30.0000i	1.08042i
$772$	0	0
$773$	2.00000i	0.0719350i	0.999353	+	0.0359675i	$0.0114513\pi$
$773$	2.00000i	0.0719350i	−0.999353	+	0.0359675i	$0.988549\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	10.0000	0.358748
$778$	0	0
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	− 32.0000i	− 1.14505i
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	20.0000	0.713831
$786$	0	0
$787$	− 36.0000i	− 1.28326i	−0.767014	−	0.641631i	$-0.778258\pi$
$787$	− 36.0000i	− 1.28326i	0.767014	−	0.641631i	$-0.221742\pi$
$788$	0	0
$789$	− 24.0000i	− 0.854423i
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	−36.0000	−1.27840
$794$	0	0
$795$	− 12.0000i	− 0.425596i
$796$	0	0
$797$	6.00000i	0.212531i	0.994338	+	0.106265i	$0.0338893\pi$
$797$	6.00000i	0.212531i	−0.994338	+	0.106265i	$0.966111\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	− 40.0000i	− 1.41157i
$804$	0	0
$805$	− 16.0000i	− 0.563926i
$806$	0	0
$807$	22.0000	0.774437
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	44.0000i	1.54505i	0.634985	+	0.772524i	$0.281006\pi$
$811$	44.0000i	1.54505i	−0.634985	+	0.772524i	$0.718994\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−40.0000	−1.40114
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	− 6.00000i	− 0.209657i
$820$	0	0
$821$	− 38.0000i	− 1.32621i	−0.748527	−	0.663105i	$-0.769238\pi$
$821$	− 38.0000i	− 1.32621i	0.748527	−	0.663105i	$-0.230762\pi$
$822$	0	0
$823$	56.0000	1.95204	0.976019	−	0.217687i	$-0.0698512\pi$
$823$	56.0000	1.95204	0.976019	+	0.217687i	$0.0698512\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	0	0
$829$	− 26.0000i	− 0.903017i	−0.892267	−	0.451509i	$-0.850886\pi$
$829$	− 26.0000i	− 0.903017i	0.892267	−	0.451509i	$-0.149114\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	16.0000i	0.553703i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−56.0000	−1.93333	−0.966667	−	0.256036i	$-0.917584\pi$
$839$	−56.0000	−1.93333	−0.966667	+	0.256036i	$0.917584\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	26.0000i	0.895488i
$844$	0	0
$845$	− 46.0000i	− 1.58245i
$846$	0	0
$847$	−5.00000	−0.171802
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	− 80.0000i	− 2.74236i
$852$	0	0
$853$	− 14.0000i	− 0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$	− 14.0000i	− 0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	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
$859$	− 20.0000i	− 0.682391i	−0.939992	−	0.341196i	$-0.889168\pi$
$859$	− 20.0000i	− 0.682391i	0.939992	−	0.341196i	$-0.110832\pi$
$860$	0	0
$861$	− 6.00000i	− 0.204479i
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	−44.0000	−1.49604
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	14.0000	0.473828
$874$	0	0
$875$	12.0000i	0.405674i
$876$	0	0
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	0	0
$879$	−30.0000	−1.01187
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	0	0
$885$	− 8.00000i	− 0.268917i
$886$	0	0
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.00000i	0.134005i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	24.0000	0.802232
$896$	0	0
$897$	−48.0000	−1.60267
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 12.0000i	− 0.399778i
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	−36.0000	−1.19668
$906$	0	0
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	0	0
$909$	− 2.00000i	− 0.0663358i
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	0	0
$915$	12.0000i	0.396708i
$916$	0	0
$917$	− 20.0000i	− 0.660458i
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	0	0
$923$	− 48.0000i	− 1.57994i
$924$	0	0
$925$	10.0000i	0.328798i
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	− 8.00000i	− 0.261908i
$934$	0	0
$935$	−16.0000	−0.523256
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	10.0000i	0.326338i
$940$	0	0
$941$	− 26.0000i	− 0.847576i	−0.905761	−	0.423788i	$-0.860700\pi$
$941$	− 26.0000i	− 0.847576i	0.905761	−	0.423788i	$-0.139300\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	0	0
$949$	− 60.0000i	− 1.94768i
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 8.00000i	− 0.258603i
$958$	0	0
$959$	−10.0000	−0.322917
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	12.0000i	0.386695i
$964$	0	0
$965$	4.00000i	0.128765i
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	12.0000i	0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$971$	12.0000i	0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$972$	0	0
$973$	− 4.00000i	− 0.128234i
$974$	0	0
$975$	6.00000	0.192154
$976$	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
$979$	24.0000i	0.767043i
$980$	0	0
$981$	2.00000i	0.0638551i
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	−20.0000	−0.637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 32.0000i	− 1.01754i
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	4.00000	0.126936
$994$	0	0
$995$	− 16.0000i	− 0.507234i
$996$	0	0
$997$	− 14.0000i	− 0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$	− 14.0000i	− 0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$	0	0
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5376.2.c.bc.2689.1		2
4.3	odd	2		5376.2.c.e.2689.2		2
8.3	odd	2		5376.2.c.e.2689.1		2
8.5	even	2	inner	5376.2.c.bc.2689.2		2
16.3	odd	4		1344.2.a.i.1.1		1
16.5	even	4		42.2.a.a.1.1	✓	1
16.11	odd	4		336.2.a.d.1.1		1
16.13	even	4		1344.2.a.q.1.1		1
48.5	odd	4		126.2.a.a.1.1		1
48.11	even	4		1008.2.a.j.1.1		1
48.29	odd	4		4032.2.a.e.1.1		1
48.35	even	4		4032.2.a.m.1.1		1
80.37	odd	4		1050.2.g.a.799.2		2
80.53	odd	4		1050.2.g.a.799.1		2
80.59	odd	4		8400.2.a.k.1.1		1
80.69	even	4		1050.2.a.i.1.1		1
112.5	odd	12		294.2.e.a.67.1		2
112.11	odd	12		2352.2.q.i.961.1		2
112.13	odd	4		9408.2.a.n.1.1		1
112.27	even	4		2352.2.a.l.1.1		1
112.37	even	12		294.2.e.c.67.1		2
112.53	even	12		294.2.e.c.79.1		2
112.59	even	12		2352.2.q.n.961.1		2
112.69	odd	4		294.2.a.g.1.1		1
112.75	even	12		2352.2.q.n.1537.1		2
112.83	even	4		9408.2.a.bw.1.1		1
112.101	odd	12		294.2.e.a.79.1		2
112.107	odd	12		2352.2.q.i.1537.1		2
144.5	odd	12		1134.2.f.j.379.1		2
144.85	even	12		1134.2.f.g.379.1		2
144.101	odd	12		1134.2.f.j.757.1		2
144.133	even	12		1134.2.f.g.757.1		2
176.21	odd	4		5082.2.a.d.1.1		1
208.181	even	4		7098.2.a.f.1.1		1
240.53	even	4		3150.2.g.r.2899.2		2
240.149	odd	4		3150.2.a.bo.1.1		1
240.197	even	4		3150.2.g.r.2899.1		2
336.5	even	12		882.2.g.j.361.1		2
336.53	odd	12		882.2.g.h.667.1		2
336.101	even	12		882.2.g.j.667.1		2
336.149	odd	12		882.2.g.h.361.1		2
336.251	odd	4		7056.2.a.k.1.1		1
336.293	even	4		882.2.a.b.1.1		1
560.69	odd	4		7350.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.a.a.1.1	✓	1	16.5	even	4
126.2.a.a.1.1		1	48.5	odd	4
294.2.a.g.1.1		1	112.69	odd	4
294.2.e.a.67.1		2	112.5	odd	12
294.2.e.a.79.1		2	112.101	odd	12
294.2.e.c.67.1		2	112.37	even	12
294.2.e.c.79.1		2	112.53	even	12
336.2.a.d.1.1		1	16.11	odd	4
882.2.a.b.1.1		1	336.293	even	4
882.2.g.h.361.1		2	336.149	odd	12
882.2.g.h.667.1		2	336.53	odd	12
882.2.g.j.361.1		2	336.5	even	12
882.2.g.j.667.1		2	336.101	even	12
1008.2.a.j.1.1		1	48.11	even	4
1050.2.a.i.1.1		1	80.69	even	4
1050.2.g.a.799.1		2	80.53	odd	4
1050.2.g.a.799.2		2	80.37	odd	4
1134.2.f.g.379.1		2	144.85	even	12
1134.2.f.g.757.1		2	144.133	even	12
1134.2.f.j.379.1		2	144.5	odd	12
1134.2.f.j.757.1		2	144.101	odd	12
1344.2.a.i.1.1		1	16.3	odd	4
1344.2.a.q.1.1		1	16.13	even	4
2352.2.a.l.1.1		1	112.27	even	4
2352.2.q.i.961.1		2	112.11	odd	12
2352.2.q.i.1537.1		2	112.107	odd	12
2352.2.q.n.961.1		2	112.59	even	12
2352.2.q.n.1537.1		2	112.75	even	12
3150.2.a.bo.1.1		1	240.149	odd	4
3150.2.g.r.2899.1		2	240.197	even	4
3150.2.g.r.2899.2		2	240.53	even	4
4032.2.a.e.1.1		1	48.29	odd	4
4032.2.a.m.1.1		1	48.35	even	4
5082.2.a.d.1.1		1	176.21	odd	4
5376.2.c.e.2689.1		2	8.3	odd	2
5376.2.c.e.2689.2		2	4.3	odd	2
5376.2.c.bc.2689.1		2	1.1	even	1	trivial
5376.2.c.bc.2689.2		2	8.5	even	2	inner
7056.2.a.k.1.1		1	336.251	odd	4
7098.2.a.f.1.1		1	208.181	even	4
7350.2.a.f.1.1		1	560.69	odd	4
8400.2.a.k.1.1		1	80.59	odd	4
9408.2.a.n.1.1		1	112.13	odd	4
9408.2.a.bw.1.1		1	112.83	even	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:	\( N \)	\(=\)	\( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5376.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} +6.00000i q^{13} +2.00000 q^{15} +2.00000 q^{17} -4.00000i q^{19} -1.00000i q^{21} -8.00000 q^{23} +1.00000 q^{25} +1.00000i q^{27} -2.00000i q^{29} +4.00000 q^{33} +2.00000i q^{35} +10.0000i q^{37} +6.00000 q^{39} +6.00000 q^{41} +4.00000i q^{43} -2.00000i q^{45} +1.00000 q^{49} -2.00000i q^{51} -6.00000i q^{53} -8.00000 q^{55} -4.00000 q^{57} -4.00000i q^{59} +6.00000i q^{61} -1.00000 q^{63} -12.0000 q^{65} +4.00000i q^{67} +8.00000i q^{69} -8.00000 q^{71} -10.0000 q^{73} -1.00000i q^{75} +4.00000i q^{77} +1.00000 q^{81} -4.00000i q^{83} +4.00000i q^{85} -2.00000 q^{87} +6.00000 q^{89} +6.00000i q^{91} +8.00000 q^{95} -14.0000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 2 q^{9} + 4 q^{15} + 4 q^{17} - 16 q^{23} + 2 q^{25} + 8 q^{33} + 12 q^{39} + 12 q^{41} + 2 q^{49} - 16 q^{55} - 8 q^{57} - 2 q^{63} - 24 q^{65} - 16 q^{71} - 20 q^{73} + 2 q^{81} - 4 q^{87}+ \cdots - 28 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 2 * q^9 + 4 * q^15 + 4 * q^17 - 16 * q^23 + 2 * q^25 + 8 * q^33 + 12 * q^39 + 12 * q^41 + 2 * q^49 - 16 * q^55 - 8 * q^57 - 2 * q^63 - 24 * q^65 - 16 * q^71 - 20 * q^73 + 2 * q^81 - 4 * q^87 + 12 * q^89 + 16 * q^95 - 28 * q^97

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