LMFDB - Embedded newform 4800.2.f.bg.3649.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4800,2,Mod(3649,4800)] 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("4800.3649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4800 = 2^{6} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4800.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-2,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,12, 0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$38.3281929702$
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 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4800.3649
Dual form			4800.2.f.bg.3649.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.00000 q^{9} +4.00000 q^{11} -2.00000i q^{13} +2.00000i q^{17} +4.00000 q^{19} -8.00000i q^{23} -1.00000i q^{27} +6.00000 q^{29} -8.00000 q^{31} +4.00000i q^{33} -6.00000i q^{37} +2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +7.00000 q^{49} -2.00000 q^{51} -2.00000i q^{53} +4.00000i q^{57} -4.00000 q^{59} +2.00000 q^{61} -4.00000i q^{67} +8.00000 q^{69} -8.00000 q^{71} -10.0000i q^{73} -8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +6.00000i q^{87} +6.00000 q^{89} -8.00000i q^{93} +2.00000i q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} + 8 q^{11} + 8 q^{19} + 12 q^{29} - 16 q^{31} + 4 q^{39} - 12 q^{41} + 14 q^{49} - 4 q^{51} - 8 q^{59} + 4 q^{61} + 16 q^{69} - 16 q^{71} - 16 q^{79} + 2 q^{81} + 12 q^{89} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 8 * q^11 + 8 * q^19 + 12 * q^29 - 16 * q^31 + 4 * q^39 - 12 * q^41 + 14 * q^49 - 4 * q^51 - 8 * q^59 + 4 * q^61 + 16 * q^69 - 16 * q^71 - 16 * q^79 + 2 * q^81 + 12 * q^89 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$4351$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 8.00000i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	− 8.00000i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	4.00000i	0.696311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	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
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000i	0.643268i
$88$	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
$91$	0	0
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	0	0
$105$	0	0
$106$	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
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	− 18.0000i	− 1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	− 18.0000i	− 1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 8.00000i	− 0.668994i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7.00000i	0.577350i
$148$	0	0
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	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
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.00000i	− 0.300658i
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	2.00000i	0.147844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.00000i	0.556038i
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	− 8.00000i	− 0.548151i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	0	0
$225$	0	0
$226$	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
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 10.0000i	− 0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$	− 10.0000i	− 0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.00000i	− 0.509028i
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	− 32.0000i	− 2.01182i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	− 8.00000i	− 0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$	− 8.00000i	− 0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	28.0000i	1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$	28.0000i	1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.00000i	− 0.232104i
$298$	0	0
$299$	−16.0000	−0.925304
$300$	0	0
$301$	0	0
$302$	0	0
$303$	18.0000i	1.03407i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.00000i	− 0.110600i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	6.00000i	0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.0000i	0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$	18.0000i	0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	−32.0000	−1.73290
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	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
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	− 2.00000i	− 0.106449i	−0.998583	−	0.0532246i	$-0.983050\pi$
$353$	− 2.00000i	− 0.106449i	0.998583	−	0.0532246i	$-0.0169499\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	5.00000i	0.262432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000i	0.203331i
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	− 4.00000i	− 0.201773i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	16.0000i	0.797017i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 24.0000i	− 1.18964i
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.0000i	0.587643i
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	8.00000	0.386244
$430$	0	0
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	0	0
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 32.0000i	− 1.53077i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	− 20.0000i	− 0.950229i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$	− 20.0000i	− 0.950229i	0.879924	−	0.475114i	$-0.157593\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	14.0000i	0.662177i
$448$	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
$451$	−24.0000	−1.13012
$452$	0	0
$453$	16.0000i	0.751746i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	0	0
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	− 16.0000i	− 0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.00000i	0.0915737i
$478$	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
$481$	−12.0000	−0.547153
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	32.0000i	1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$	32.0000i	1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	0	0
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.00000i	0.399704i
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 4.00000i	− 0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	0	0
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 16.0000i	− 0.696971i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 12.0000i	− 0.517838i
$538$	0	0
$539$	28.0000	1.20605
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	− 6.00000i	− 0.257485i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	44.0000i	1.88130i	0.339372	+	0.940652i	$0.389785\pi$
$547$	44.0000i	1.88130i	−0.339372	+	0.940652i	$0.610215\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.0000i	1.10166i	0.834619	+	0.550828i	$0.185688\pi$
$557$	26.0000i	1.10166i	−0.834619	+	0.550828i	$0.814312\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	− 28.0000i	− 1.18006i	−0.807382	−	0.590030i	$-0.799116\pi$
$563$	− 28.0000i	− 1.18006i	0.807382	−	0.590030i	$-0.200884\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	0	0
$579$	2.00000	0.0831172
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 8.00000i	− 0.331326i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 44.0000i	− 1.81607i	−0.418890	−	0.908037i	$-0.637581\pi$
$587$	− 44.0000i	− 1.81607i	0.418890	−	0.908037i	$-0.362419\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	14.0000i	0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$	14.0000i	0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.0000i	0.654836i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	40.0000i	1.62355i	0.583970	+	0.811775i	$0.301498\pi$
$607$	40.0000i	1.62355i	−0.583970	+	0.811775i	$0.698502\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	38.0000i	1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$	38.0000i	1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	0	0
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16.0000i	0.638978i
$628$	0	0
$629$	12.0000	0.478471
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	− 20.0000i	− 0.794929i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 14.0000i	− 0.554700i
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	0	0
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8.00000i	− 0.314512i	−0.987558	−	0.157256i	$-0.949735\pi$
$647$	− 8.00000i	− 0.314512i	0.987558	−	0.157256i	$-0.0502649\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10.0000i	0.390137i
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	4.00000i	0.155347i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 48.0000i	− 1.85857i
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.00000i	0.0768662i	0.999261	+	0.0384331i	$0.0122367\pi$
$677$	2.00000i	0.0768662i	−0.999261	+	0.0384331i	$0.987763\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	22.0000i	0.839352i
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 12.0000i	− 0.454532i
$698$	0	0
$699$	10.0000	0.378235
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	− 24.0000i	− 0.905177i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	64.0000i	2.39682i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 16.0000i	− 0.597531i
$718$	0	0
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	18.0000i	0.669427i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 48.0000i	− 1.78022i	−0.455744	−	0.890111i	$-0.650627\pi$
$727$	− 48.0000i	− 1.78022i	0.455744	−	0.890111i	$-0.349373\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 16.0000i	− 0.589368i
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	0	0
$743$	− 8.00000i	− 0.293492i	−0.989174	−	0.146746i	$-0.953120\pi$
$743$	− 8.00000i	− 0.293492i	0.989174	−	0.146746i	$-0.0468799\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	20.0000i	0.728841i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	0	0
$759$	32.0000	1.16153
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00000i	0.288863i
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	− 6.00000i	− 0.214423i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	0	0
$789$	8.00000	0.284808
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 22.0000i	− 0.779280i	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	− 22.0000i	− 0.779280i	0.920967	−	0.389640i	$-0.127401\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	− 40.0000i	− 1.41157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 10.0000i	− 0.352017i
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	− 8.00000i	− 0.280572i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 16.0000i	− 0.559769i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	0	0
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	0	0
$833$	14.0000i	0.485071i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.00000i	0.276520i
$838$	0	0
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	26.0000i	0.895488i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	−48.0000	−1.64542
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.0000i	1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$	42.0000i	1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$	0	0
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 32.0000i	− 1.08929i	−0.838666	−	0.544646i	$-0.816664\pi$
$863$	− 32.0000i	− 1.08929i	0.838666	−	0.544646i	$-0.183336\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	0	0
$879$	18.0000	0.607125
$880$	0	0
$881$	50.0000	1.68454	0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000	1.68454	0.842271	+	0.539054i	$0.181218\pi$
$882$	0	0
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 16.0000i	− 0.534224i
$898$	0	0
$899$	−48.0000	−1.60089
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.00000i	0.132818i	0.997792	+	0.0664089i	$0.0211542\pi$
$907$	4.00000i	0.132818i	−0.997792	+	0.0664089i	$0.978846\pi$
$908$	0	0
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	16.0000i	0.529523i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	16.0000i	0.526646i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 16.0000i	− 0.525509i
$928$	0	0
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	28.0000	0.917663
$932$	0	0
$933$	24.0000i	0.785725i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.0000i	1.37208i	0.727564	+	0.686040i	$0.240653\pi$
$937$	42.0000i	1.37208i	−0.727564	+	0.686040i	$0.759347\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	48.0000i	1.56310i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	54.0000i	1.74923i	0.484817	+	0.874616i	$0.338886\pi$
$953$	54.0000i	1.74923i	−0.484817	+	0.874616i	$0.661114\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	24.0000i	0.775810i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	12.0000i	0.386695i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	16.0000i	0.514525i	0.966342	+	0.257263i	$0.0828206\pi$
$967$	16.0000i	0.514525i	−0.966342	+	0.257263i	$0.917179\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 30.0000i	− 0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$	− 30.0000i	− 0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	20.0000i	0.634681i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	26.0000i	0.823428i	0.911313	+	0.411714i	$0.135070\pi$
$997$	26.0000i	0.823428i	−0.911313	+	0.411714i	$0.864930\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4800.2.f.bg.3649.2		2
4.3	odd	2		4800.2.f.d.3649.1		2
5.2	odd	4		4800.2.a.cc.1.1		1
5.3	odd	4		192.2.a.b.1.1		1
5.4	even	2	inner	4800.2.f.bg.3649.1		2
8.3	odd	2		600.2.f.e.49.2		2
8.5	even	2		1200.2.f.b.49.1		2
15.8	even	4		576.2.a.b.1.1		1
20.3	even	4		192.2.a.d.1.1		1
20.7	even	4		4800.2.a.q.1.1		1
20.19	odd	2		4800.2.f.d.3649.2		2
24.5	odd	2		3600.2.f.r.2449.2		2
24.11	even	2		1800.2.f.c.649.2		2
35.13	even	4		9408.2.a.cc.1.1		1
40.3	even	4		24.2.a.a.1.1	✓	1
40.13	odd	4		48.2.a.a.1.1		1
40.19	odd	2		600.2.f.e.49.1		2
40.27	even	4		600.2.a.h.1.1		1
40.29	even	2		1200.2.f.b.49.2		2
40.37	odd	4		1200.2.a.d.1.1		1
60.23	odd	4		576.2.a.d.1.1		1
80.3	even	4		768.2.d.e.385.2		2
80.13	odd	4		768.2.d.d.385.1		2
80.43	even	4		768.2.d.e.385.1		2
80.53	odd	4		768.2.d.d.385.2		2
120.29	odd	2		3600.2.f.r.2449.1		2
120.53	even	4		144.2.a.b.1.1		1
120.59	even	2		1800.2.f.c.649.1		2
120.77	even	4		3600.2.a.v.1.1		1
120.83	odd	4		72.2.a.a.1.1		1
120.107	odd	4		1800.2.a.m.1.1		1
140.83	odd	4		9408.2.a.h.1.1		1
240.53	even	4		2304.2.d.k.1153.1		2
240.83	odd	4		2304.2.d.i.1153.2		2
240.173	even	4		2304.2.d.k.1153.2		2
240.203	odd	4		2304.2.d.i.1153.1		2
280.3	odd	12		1176.2.q.a.961.1		2
280.13	even	4		2352.2.a.i.1.1		1
280.53	odd	12		2352.2.q.l.961.1		2
280.83	odd	4		1176.2.a.i.1.1		1
280.93	odd	12		2352.2.q.l.1537.1		2
280.123	even	12		1176.2.q.i.961.1		2
280.163	even	12		1176.2.q.i.361.1		2
280.173	even	12		2352.2.q.r.1537.1		2
280.213	even	12		2352.2.q.r.961.1		2
280.243	odd	12		1176.2.q.a.361.1		2
360.13	odd	12		1296.2.i.m.865.1		2
360.43	even	12		648.2.i.g.433.1		2
360.83	odd	12		648.2.i.b.433.1		2
360.133	odd	12		1296.2.i.m.433.1		2
360.173	even	12		1296.2.i.e.433.1		2
360.203	odd	12		648.2.i.b.217.1		2
360.283	even	12		648.2.i.g.217.1		2
360.293	even	12		1296.2.i.e.865.1		2
440.43	odd	4		2904.2.a.c.1.1		1
440.373	even	4		5808.2.a.s.1.1		1
520.83	odd	4		4056.2.c.e.337.2		2
520.203	odd	4		4056.2.c.e.337.1		2
520.363	even	4		4056.2.a.i.1.1		1
520.493	odd	4		8112.2.a.be.1.1		1
680.203	even	4		6936.2.a.p.1.1		1
760.683	odd	4		8664.2.a.j.1.1		1
840.83	even	4		3528.2.a.d.1.1		1
840.293	odd	4		7056.2.a.q.1.1		1
840.443	odd	12		3528.2.s.j.361.1		2
840.563	even	12		3528.2.s.y.3313.1		2
840.683	odd	12		3528.2.s.j.3313.1		2
840.803	even	12		3528.2.s.y.361.1		2
1320.923	even	4		8712.2.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.2.a.a.1.1	✓	1	40.3	even	4
48.2.a.a.1.1		1	40.13	odd	4
72.2.a.a.1.1		1	120.83	odd	4
144.2.a.b.1.1		1	120.53	even	4
192.2.a.b.1.1		1	5.3	odd	4
192.2.a.d.1.1		1	20.3	even	4
576.2.a.b.1.1		1	15.8	even	4
576.2.a.d.1.1		1	60.23	odd	4
600.2.a.h.1.1		1	40.27	even	4
600.2.f.e.49.1		2	40.19	odd	2
600.2.f.e.49.2		2	8.3	odd	2
648.2.i.b.217.1		2	360.203	odd	12
648.2.i.b.433.1		2	360.83	odd	12
648.2.i.g.217.1		2	360.283	even	12
648.2.i.g.433.1		2	360.43	even	12
768.2.d.d.385.1		2	80.13	odd	4
768.2.d.d.385.2		2	80.53	odd	4
768.2.d.e.385.1		2	80.43	even	4
768.2.d.e.385.2		2	80.3	even	4
1176.2.a.i.1.1		1	280.83	odd	4
1176.2.q.a.361.1		2	280.243	odd	12
1176.2.q.a.961.1		2	280.3	odd	12
1176.2.q.i.361.1		2	280.163	even	12
1176.2.q.i.961.1		2	280.123	even	12
1200.2.a.d.1.1		1	40.37	odd	4
1200.2.f.b.49.1		2	8.5	even	2
1200.2.f.b.49.2		2	40.29	even	2
1296.2.i.e.433.1		2	360.173	even	12
1296.2.i.e.865.1		2	360.293	even	12
1296.2.i.m.433.1		2	360.133	odd	12
1296.2.i.m.865.1		2	360.13	odd	12
1800.2.a.m.1.1		1	120.107	odd	4
1800.2.f.c.649.1		2	120.59	even	2
1800.2.f.c.649.2		2	24.11	even	2
2304.2.d.i.1153.1		2	240.203	odd	4
2304.2.d.i.1153.2		2	240.83	odd	4
2304.2.d.k.1153.1		2	240.53	even	4
2304.2.d.k.1153.2		2	240.173	even	4
2352.2.a.i.1.1		1	280.13	even	4
2352.2.q.l.961.1		2	280.53	odd	12
2352.2.q.l.1537.1		2	280.93	odd	12
2352.2.q.r.961.1		2	280.213	even	12
2352.2.q.r.1537.1		2	280.173	even	12
2904.2.a.c.1.1		1	440.43	odd	4
3528.2.a.d.1.1		1	840.83	even	4
3528.2.s.j.361.1		2	840.443	odd	12
3528.2.s.j.3313.1		2	840.683	odd	12
3528.2.s.y.361.1		2	840.803	even	12
3528.2.s.y.3313.1		2	840.563	even	12
3600.2.a.v.1.1		1	120.77	even	4
3600.2.f.r.2449.1		2	120.29	odd	2
3600.2.f.r.2449.2		2	24.5	odd	2
4056.2.a.i.1.1		1	520.363	even	4
4056.2.c.e.337.1		2	520.203	odd	4
4056.2.c.e.337.2		2	520.83	odd	4
4800.2.a.q.1.1		1	20.7	even	4
4800.2.a.cc.1.1		1	5.2	odd	4
4800.2.f.d.3649.1		2	4.3	odd	2
4800.2.f.d.3649.2		2	20.19	odd	2
4800.2.f.bg.3649.1		2	5.4	even	2	inner
4800.2.f.bg.3649.2		2	1.1	even	1	trivial
5808.2.a.s.1.1		1	440.373	even	4
6936.2.a.p.1.1		1	680.203	even	4
7056.2.a.q.1.1		1	840.293	odd	4
8112.2.a.be.1.1		1	520.493	odd	4
8664.2.a.j.1.1		1	760.683	odd	4
8712.2.a.u.1.1		1	1320.923	even	4
9408.2.a.h.1.1		1	140.83	odd	4
9408.2.a.cc.1.1		1	35.13	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:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.00000 q^{9} +4.00000 q^{11} -2.00000i q^{13} +2.00000i q^{17} +4.00000 q^{19} -8.00000i q^{23} -1.00000i q^{27} +6.00000 q^{29} -8.00000 q^{31} +4.00000i q^{33} -6.00000i q^{37} +2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +7.00000 q^{49} -2.00000 q^{51} -2.00000i q^{53} +4.00000i q^{57} -4.00000 q^{59} +2.00000 q^{61} -4.00000i q^{67} +8.00000 q^{69} -8.00000 q^{71} -10.0000i q^{73} -8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +6.00000i q^{87} +6.00000 q^{89} -8.00000i q^{93} +2.00000i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} + 8 q^{11} + 8 q^{19} + 12 q^{29} - 16 q^{31} + 4 q^{39} - 12 q^{41} + 14 q^{49} - 4 q^{51} - 8 q^{59} + 4 q^{61} + 16 q^{69} - 16 q^{71} - 16 q^{79} + 2 q^{81} + 12 q^{89} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 8 * q^11 + 8 * q^19 + 12 * q^29 - 16 * q^31 + 4 * q^39 - 12 * q^41 + 14 * q^49 - 4 * q^51 - 8 * q^59 + 4 * q^61 + 16 * q^69 - 16 * q^71 - 16 * q^79 + 2 * q^81 + 12 * q^89 - 8 * q^99

Introduction

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