LMFDB - Embedded newform 4056.2.c.e.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(337,4056)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4056.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$32.3873230598$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4056.337
Dual form			4056.2.c.e.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.00000i q^{5} +1.00000 q^{9} +4.00000i q^{11} -2.00000i q^{15} -2.00000 q^{17} +4.00000i q^{19} +8.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.00000 q^{29} -8.00000i q^{31} -4.00000i q^{33} +6.00000i q^{37} +6.00000i q^{41} -4.00000 q^{43} +2.00000i q^{45} +7.00000 q^{49} +2.00000 q^{51} -2.00000 q^{53} -8.00000 q^{55} -4.00000i q^{57} +4.00000i q^{59} -2.00000 q^{61} +4.00000i q^{67} -8.00000 q^{69} -8.00000i q^{71} +10.0000i q^{73} -1.00000 q^{75} -8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -4.00000i q^{85} -6.00000 q^{87} -6.00000i q^{89} +8.00000i q^{93} -8.00000 q^{95} -2.00000i q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9} - 4 q^{17} + 16 q^{23} + 2 q^{25} - 2 q^{27} + 12 q^{29} - 8 q^{43} + 14 q^{49} + 4 q^{51} - 4 q^{53} - 16 q^{55} - 4 q^{61} - 16 q^{69} - 2 q^{75} - 16 q^{79} + 2 q^{81} - 12 q^{87}+ \cdots - 16 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 - 4 * q^17 + 16 * q^23 + 2 * q^25 - 2 * q^27 + 12 * q^29 - 8 * q^43 + 14 * q^49 + 4 * q^51 - 4 * q^53 - 16 * q^55 - 4 * q^61 - 16 * q^69 - 2 * q^75 - 16 * q^79 + 2 * q^81 - 12 * q^87 - 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$.

$n$	$1015$	$2029$	$2705$	$3889$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	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
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 2.00000i	− 0.516398i
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	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
$31$	− 8.00000i	− 1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$	− 8.00000i	− 1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$	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	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000i	0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$	6.00000i	0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	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
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	0	0
$57$	− 4.00000i	− 0.529813i
$58$	0	0
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	− 8.00000i	− 0.949425i	−0.880141	−	0.474713i	$-0.842552\pi$
$71$	− 8.00000i	− 0.949425i	0.880141	−	0.474713i	$-0.157448\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	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
$85$	− 4.00000i	− 0.433861i
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	− 6.00000i	− 0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$	− 6.00000i	− 0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	0	0
$99$	4.00000i	0.402015i
$100$	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
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	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
$111$	− 6.00000i	− 0.569495i
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	16.0000i	1.49201i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	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
$135$	− 2.00000i	− 0.172133i
$136$	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	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	12.0000i	0.996546i
$146$	0	0
$147$	−7.00000	−0.577350
$148$	0	0
$149$	− 14.0000i	− 1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	− 14.0000i	− 1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$	0	0
$151$	− 16.0000i	− 1.30206i	−0.759051	−	0.651031i	$-0.774337\pi$
$151$	− 16.0000i	− 1.30206i	0.759051	−	0.651031i	$-0.225663\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	16.0000	1.28515
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	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	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	0	0
$165$	8.00000	0.622799
$166$	0	0
$167$	24.0000i	1.85718i	0.371113	+	0.928588i	$0.378976\pi$
$167$	24.0000i	1.85718i	−0.371113	+	0.928588i	$0.621024\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	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
$183$	2.00000	0.147844
$184$	0	0
$185$	−12.0000	−0.882258
$186$	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	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	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	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	− 4.00000i	− 0.282138i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	8.00000	0.556038
$208$	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
$215$	− 8.00000i	− 0.545595i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 10.0000i	− 0.675737i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	22.0000i	1.45380i	0.686743	+	0.726900i	$0.259040\pi$
$229$	22.0000i	1.45380i	−0.686743	+	0.726900i	$0.740960\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	16.0000i	1.03495i	0.855697	+	0.517477i	$0.173129\pi$
$239$	16.0000i	1.03495i	−0.855697	+	0.517477i	$0.826871\pi$
$240$	0	0
$241$	18.0000i	1.15948i	0.814801	+	0.579741i	$0.196846\pi$
$241$	18.0000i	1.15948i	−0.814801	+	0.579741i	$0.803154\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	14.0000i	0.894427i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 4.00000i	− 0.253490i
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	32.0000i	2.01182i
$254$	0	0
$255$	4.00000i	0.250490i
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	− 4.00000i	− 0.245718i
$266$	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
$271$	8.00000i	0.485965i	0.970031	+	0.242983i	$0.0781258\pi$
$271$	8.00000i	0.485965i	−0.970031	+	0.242983i	$0.921874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.00000i	0.241209i
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	0	0
$279$	− 8.00000i	− 0.478947i
$280$	0	0
$281$	26.0000i	1.55103i	0.631329	+	0.775515i	$0.282510\pi$
$281$	26.0000i	1.55103i	−0.631329	+	0.775515i	$0.717490\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	2.00000i	0.117242i
$292$	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
$295$	−8.00000	−0.465778
$296$	0	0
$297$	− 4.00000i	− 0.232104i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−18.0000	−1.03407
$304$	0	0
$305$	− 4.00000i	− 0.229039i
$306$	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
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	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
$319$	24.0000i	1.34374i
$320$	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
$331$	− 20.0000i	− 1.09930i	−0.835395	−	0.549650i	$-0.814761\pi$
$331$	− 20.0000i	− 1.09930i	0.835395	−	0.549650i	$-0.185239\pi$
$332$	0	0
$333$	6.00000i	0.328798i
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	32.0000	1.73290
$342$	0	0
$343$	0	0
$344$	0	0
$345$	− 16.0000i	− 0.861411i
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	30.0000i	1.60586i	0.596071	+	0.802932i	$0.296728\pi$
$349$	30.0000i	1.60586i	−0.596071	+	0.802932i	$0.703272\pi$
$350$	0	0
$351$	0	0
$352$	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
$355$	16.0000	0.849192
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 24.0000i	− 1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$	− 24.0000i	− 1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	−20.0000	−1.04685
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	6.00000i	0.312348i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	− 12.0000i	− 0.619677i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 20.0000i	− 1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$	− 20.0000i	− 1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$	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
$387$	−4.00000	−0.203331
$388$	0	0
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	− 16.0000i	− 0.805047i
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 30.0000i	− 1.49813i	−0.662497	−	0.749064i	$-0.730503\pi$
$401$	− 30.0000i	− 1.49813i	0.662497	−	0.749064i	$-0.269497\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.00000i	0.0993808i
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	6.00000i	0.296681i	0.988936	+	0.148340i	$0.0473931\pi$
$409$	6.00000i	0.296681i	−0.988936	+	0.148340i	$0.952607\pi$
$410$	0	0
$411$	6.00000i	0.295958i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 32.0000i	− 1.54139i	−0.637207	−	0.770693i	$-0.719910\pi$
$431$	− 32.0000i	− 1.54139i	0.637207	−	0.770693i	$-0.280090\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	− 12.0000i	− 0.575356i
$436$	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
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	0	0
$447$	14.0000i	0.662177i
$448$	0	0
$449$	− 14.0000i	− 0.660701i	−0.943858	−	0.330350i	$-0.892833\pi$
$449$	− 14.0000i	− 0.660701i	0.943858	−	0.330350i	$-0.107167\pi$
$450$	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	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	26.0000i	1.21094i	0.795868	+	0.605470i	$0.207015\pi$
$461$	26.0000i	1.21094i	−0.795868	+	0.605470i	$0.792985\pi$
$462$	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
$465$	−16.0000	−0.741982
$466$	0	0
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	− 16.0000i	− 0.735681i
$474$	0	0
$475$	4.00000i	0.183533i
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	− 16.0000i	− 0.731059i	−0.930800	−	0.365529i	$-0.880888\pi$
$479$	− 16.0000i	− 0.731059i	0.930800	−	0.365529i	$-0.119112\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.00000	0.181631
$486$	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
$489$	− 12.0000i	− 0.542659i
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	−8.00000	−0.359573
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 12.0000i	− 0.537194i	−0.963253	−	0.268597i	$-0.913440\pi$
$499$	− 12.0000i	− 0.537194i	0.963253	−	0.268597i	$-0.0865599\pi$
$500$	0	0
$501$	− 24.0000i	− 1.07224i
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	36.0000i	1.60198i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 6.00000i	− 0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$	− 6.00000i	− 0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 4.00000i	− 0.176604i
$514$	0	0
$515$	− 32.0000i	− 1.41009i
$516$	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
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.0000i	0.696971i
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	4.00000i	0.173585i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 24.0000i	− 1.03761i
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	28.0000i	1.20605i
$540$	0	0
$541$	− 18.0000i	− 0.773880i	−0.922105	−	0.386940i	$-0.873532\pi$
$541$	− 18.0000i	− 0.773880i	0.922105	−	0.386940i	$-0.126468\pi$
$542$	0	0
$543$	6.00000	0.257485
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	44.0000	1.88130	0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	24.0000i	1.02243i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	12.0000	0.509372
$556$	0	0
$557$	− 26.0000i	− 1.10166i	−0.834619	−	0.550828i	$-0.814312\pi$
$557$	− 26.0000i	− 1.10166i	0.834619	−	0.550828i	$-0.185688\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	8.00000i	0.337760i
$562$	0	0
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	0	0
$565$	36.0000i	1.51453i
$566$	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	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	0	0
$579$	− 2.00000i	− 0.0831172i
$580$	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	$0.362419\pi$
$587$	44.0000i	1.81607i	−0.418890	+	0.908037i	$0.637581\pi$
$588$	0	0
$589$	32.0000	1.31854
$590$	0	0
$591$	− 18.0000i	− 0.740421i
$592$	0	0
$593$	− 14.0000i	− 0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$	− 14.0000i	− 0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.0000	0.654836
$598$	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
$605$	− 10.0000i	− 0.406558i
$606$	0	0
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 38.0000i	− 1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$	− 38.0000i	− 1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	− 42.0000i	− 1.69086i	−0.534089	−	0.845428i	$-0.679345\pi$
$617$	− 42.0000i	− 1.69086i	0.534089	−	0.845428i	$-0.320655\pi$
$618$	0	0
$619$	− 44.0000i	− 1.76851i	−0.467005	−	0.884255i	$-0.654667\pi$
$619$	− 44.0000i	− 1.76851i	0.467005	−	0.884255i	$-0.345333\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	16.0000	0.638978
$628$	0	0
$629$	− 12.0000i	− 0.478471i
$630$	0	0
$631$	16.0000i	0.636950i	0.947931	+	0.318475i	$0.103171\pi$
$631$	16.0000i	0.636950i	−0.947931	+	0.318475i	$0.896829\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	0	0
$635$	16.0000i	0.634941i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 8.00000i	− 0.316475i
$640$	0	0
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	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
$645$	8.00000i	0.315000i
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	− 8.00000i	− 0.312586i
$656$	0	0
$657$	10.0000i	0.390137i
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	− 10.0000i	− 0.388955i	−0.980907	−	0.194477i	$-0.937699\pi$
$661$	− 10.0000i	− 0.388955i	0.980907	−	0.194477i	$-0.0623011\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	48.0000	1.85857
$668$	0	0
$669$	− 8.00000i	− 0.309298i
$670$	0	0
$671$	− 8.00000i	− 0.308837i
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	12.0000i	0.459841i
$682$	0	0
$683$	4.00000i	0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$	4.00000i	0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	− 22.0000i	− 0.839352i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	4.00000i	0.152167i	0.997101	+	0.0760836i	$0.0242416\pi$
$691$	4.00000i	0.152167i	−0.997101	+	0.0760836i	$0.975758\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 24.0000i	− 0.910372i
$696$	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
$703$	−24.0000	−0.905177
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	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
$711$	−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	$-0.296479\pi$
$719$	32.0000	1.19340	0.596699	+	0.802465i	$0.296479\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 18.0000i	− 0.669427i
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	−48.0000	−1.78022	−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000	−1.78022	−0.890111	+	0.455744i	$0.849373\pi$
$728$	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	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	0	0
$735$	− 14.0000i	− 0.516398i
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	− 4.00000i	− 0.147142i	−0.997290	−	0.0735712i	$-0.976560\pi$
$739$	− 4.00000i	− 0.147142i	0.997290	−	0.0735712i	$-0.0234396\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.00000i	0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$	8.00000i	0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$	0	0
$745$	28.0000	1.02584
$746$	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
$753$	20.0000	0.728841
$754$	0	0
$755$	32.0000	1.16460
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	0	0
$759$	− 32.0000i	− 1.16153i
$760$	0	0
$761$	− 22.0000i	− 0.797499i	−0.917060	−	0.398750i	$-0.869444\pi$
$761$	− 22.0000i	− 0.797499i	0.917060	−	0.398750i	$-0.130556\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 4.00000i	− 0.144620i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 2.00000i	− 0.0721218i	−0.999350	−	0.0360609i	$-0.988519\pi$
$769$	− 2.00000i	− 0.0721218i	0.999350	−	0.0360609i	$-0.0114810\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	0	0
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	− 8.00000i	− 0.287368i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	− 4.00000i	− 0.142766i
$786$	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
$793$	0	0
$794$	0	0
$795$	4.00000i	0.141865i
$796$	0	0
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	− 6.00000i	− 0.212000i
$802$	0	0
$803$	−40.0000	−1.41157
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	− 4.00000i	− 0.140459i	−0.997531	−	0.0702295i	$-0.977627\pi$
$811$	− 4.00000i	− 0.140459i	0.997531	−	0.0702295i	$-0.0223732\pi$
$812$	0	0
$813$	− 8.00000i	− 0.280572i
$814$	0	0
$815$	−24.0000	−0.840683
$816$	0	0
$817$	− 16.0000i	− 0.559769i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 30.0000i	− 1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$	− 30.0000i	− 1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	− 4.00000i	− 0.139262i
$826$	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	$-0.165222\pi$
$829$	50.0000	1.73657	0.868286	+	0.496064i	$0.165222\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	0	0
$833$	−14.0000	−0.485071
$834$	0	0
$835$	−48.0000	−1.66111
$836$	0	0
$837$	8.00000i	0.276520i
$838$	0	0
$839$	− 24.0000i	− 0.828572i	−0.910147	−	0.414286i	$-0.864031\pi$
$839$	− 24.0000i	− 0.828572i	0.910147	−	0.414286i	$-0.135969\pi$
$840$	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
$851$	48.0000i	1.64542i
$852$	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
$855$	−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
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	0	0
$865$	− 12.0000i	− 0.408012i
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	− 32.0000i	− 1.08553i
$870$	0	0
$871$	0	0
$872$	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
$879$	18.0000i	0.607125i
$880$	0	0
$881$	−50.0000	−1.68454	−0.842271	−	0.539054i	$-0.818782\pi$
$881$	−50.0000	−1.68454	−0.842271	+	0.539054i	$0.818782\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.00000i	0.134005i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 24.0000i	− 0.802232i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 48.0000i	− 1.60089i
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 12.0000i	− 0.398893i
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	18.0000	0.597022
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	4.00000i	0.132236i
$916$	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
$921$	− 12.0000i	− 0.395413i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	6.00000i	0.197279i
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	− 50.0000i	− 1.64045i	−0.572043	−	0.820223i	$-0.693849\pi$
$929$	− 50.0000i	− 1.64045i	0.572043	−	0.820223i	$-0.306151\pi$
$930$	0	0
$931$	28.0000i	0.917663i
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	16.0000	0.523256
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	0	0
$941$	− 6.00000i	− 0.195594i	−0.995206	−	0.0977972i	$-0.968820\pi$
$941$	− 6.00000i	− 0.195594i	0.995206	−	0.0977972i	$-0.0311797\pi$
$942$	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
$949$	0	0
$950$	0	0
$951$	6.00000i	0.194563i
$952$	0	0
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	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
$963$	−12.0000	−0.386695
$964$	0	0
$965$	−4.00000	−0.128765
$966$	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
$969$	8.00000i	0.256997i
$970$	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	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	2.00000i	0.0638551i
$982$	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
$985$	−36.0000	−1.14706
$986$	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	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	20.0000i	0.634681i
$994$	0	0
$995$	− 32.0000i	− 1.01447i
$996$	0	0
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	0	0
$999$	− 6.00000i	− 0.189832i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.e.337.2		2
13.5	odd	4		4056.2.a.i.1.1		1
13.8	odd	4		24.2.a.a.1.1	✓	1
13.12	even	2	inner	4056.2.c.e.337.1		2
39.8	even	4		72.2.a.a.1.1		1
52.31	even	4		8112.2.a.be.1.1		1
52.47	even	4		48.2.a.a.1.1		1
65.8	even	4		600.2.f.e.49.1		2
65.34	odd	4		600.2.a.h.1.1		1
65.47	even	4		600.2.f.e.49.2		2
91.34	even	4		1176.2.a.i.1.1		1
91.47	even	12		1176.2.q.a.361.1		2
91.60	odd	12		1176.2.q.i.961.1		2
91.73	even	12		1176.2.q.a.961.1		2
91.86	odd	12		1176.2.q.i.361.1		2
104.21	odd	4		192.2.a.d.1.1		1
104.99	even	4		192.2.a.b.1.1		1
117.34	odd	12		648.2.i.g.433.1		2
117.47	even	12		648.2.i.b.433.1		2
117.86	even	12		648.2.i.b.217.1		2
117.112	odd	12		648.2.i.g.217.1		2
143.21	even	4		2904.2.a.c.1.1		1
156.47	odd	4		144.2.a.b.1.1		1
195.8	odd	4		1800.2.f.c.649.1		2
195.47	odd	4		1800.2.f.c.649.2		2
195.164	even	4		1800.2.a.m.1.1		1
208.21	odd	4		768.2.d.e.385.2		2
208.99	even	4		768.2.d.d.385.2		2
208.125	odd	4		768.2.d.e.385.1		2
208.203	even	4		768.2.d.d.385.1		2
221.203	odd	4		6936.2.a.p.1.1		1
247.151	even	4		8664.2.a.j.1.1		1
260.47	odd	4		1200.2.f.b.49.1		2
260.99	even	4		1200.2.a.d.1.1		1
260.203	odd	4		1200.2.f.b.49.2		2
273.47	odd	12		3528.2.s.y.361.1		2
273.86	even	12		3528.2.s.j.361.1		2
273.125	odd	4		3528.2.a.d.1.1		1
273.164	odd	12		3528.2.s.y.3313.1		2
273.242	even	12		3528.2.s.j.3313.1		2
312.125	even	4		576.2.a.d.1.1		1
312.203	odd	4		576.2.a.b.1.1		1
364.47	odd	12		2352.2.q.r.1537.1		2
364.151	even	12		2352.2.q.l.961.1		2
364.255	odd	12		2352.2.q.r.961.1		2
364.307	odd	4		2352.2.a.i.1.1		1
364.359	even	12		2352.2.q.l.1537.1		2
429.164	odd	4		8712.2.a.u.1.1		1
468.47	odd	12		1296.2.i.e.433.1		2
468.151	even	12		1296.2.i.m.433.1		2
468.203	odd	12		1296.2.i.e.865.1		2
468.463	even	12		1296.2.i.m.865.1		2
520.99	even	4		4800.2.a.cc.1.1		1
520.203	odd	4		4800.2.f.bg.3649.1		2
520.229	odd	4		4800.2.a.q.1.1		1
520.307	odd	4		4800.2.f.bg.3649.2		2
520.333	even	4		4800.2.f.d.3649.2		2
520.437	even	4		4800.2.f.d.3649.1		2
572.307	odd	4		5808.2.a.s.1.1		1
624.125	even	4		2304.2.d.i.1153.1		2
624.203	odd	4		2304.2.d.k.1153.2		2
624.437	even	4		2304.2.d.i.1153.2		2
624.515	odd	4		2304.2.d.k.1153.1		2
728.125	even	4		9408.2.a.h.1.1		1
728.307	odd	4		9408.2.a.cc.1.1		1
780.47	even	4		3600.2.f.r.2449.2		2
780.203	even	4		3600.2.f.r.2449.1		2
780.359	odd	4		3600.2.a.v.1.1		1
1092.671	even	4		7056.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.2.a.a.1.1	✓	1	13.8	odd	4
48.2.a.a.1.1		1	52.47	even	4
72.2.a.a.1.1		1	39.8	even	4
144.2.a.b.1.1		1	156.47	odd	4
192.2.a.b.1.1		1	104.99	even	4
192.2.a.d.1.1		1	104.21	odd	4
576.2.a.b.1.1		1	312.203	odd	4
576.2.a.d.1.1		1	312.125	even	4
600.2.a.h.1.1		1	65.34	odd	4
600.2.f.e.49.1		2	65.8	even	4
600.2.f.e.49.2		2	65.47	even	4
648.2.i.b.217.1		2	117.86	even	12
648.2.i.b.433.1		2	117.47	even	12
648.2.i.g.217.1		2	117.112	odd	12
648.2.i.g.433.1		2	117.34	odd	12
768.2.d.d.385.1		2	208.203	even	4
768.2.d.d.385.2		2	208.99	even	4
768.2.d.e.385.1		2	208.125	odd	4
768.2.d.e.385.2		2	208.21	odd	4
1176.2.a.i.1.1		1	91.34	even	4
1176.2.q.a.361.1		2	91.47	even	12
1176.2.q.a.961.1		2	91.73	even	12
1176.2.q.i.361.1		2	91.86	odd	12
1176.2.q.i.961.1		2	91.60	odd	12
1200.2.a.d.1.1		1	260.99	even	4
1200.2.f.b.49.1		2	260.47	odd	4
1200.2.f.b.49.2		2	260.203	odd	4
1296.2.i.e.433.1		2	468.47	odd	12
1296.2.i.e.865.1		2	468.203	odd	12
1296.2.i.m.433.1		2	468.151	even	12
1296.2.i.m.865.1		2	468.463	even	12
1800.2.a.m.1.1		1	195.164	even	4
1800.2.f.c.649.1		2	195.8	odd	4
1800.2.f.c.649.2		2	195.47	odd	4
2304.2.d.i.1153.1		2	624.125	even	4
2304.2.d.i.1153.2		2	624.437	even	4
2304.2.d.k.1153.1		2	624.515	odd	4
2304.2.d.k.1153.2		2	624.203	odd	4
2352.2.a.i.1.1		1	364.307	odd	4
2352.2.q.l.961.1		2	364.151	even	12
2352.2.q.l.1537.1		2	364.359	even	12
2352.2.q.r.961.1		2	364.255	odd	12
2352.2.q.r.1537.1		2	364.47	odd	12
2904.2.a.c.1.1		1	143.21	even	4
3528.2.a.d.1.1		1	273.125	odd	4
3528.2.s.j.361.1		2	273.86	even	12
3528.2.s.j.3313.1		2	273.242	even	12
3528.2.s.y.361.1		2	273.47	odd	12
3528.2.s.y.3313.1		2	273.164	odd	12
3600.2.a.v.1.1		1	780.359	odd	4
3600.2.f.r.2449.1		2	780.203	even	4
3600.2.f.r.2449.2		2	780.47	even	4
4056.2.a.i.1.1		1	13.5	odd	4
4056.2.c.e.337.1		2	13.12	even	2	inner
4056.2.c.e.337.2		2	1.1	even	1	trivial
4800.2.a.q.1.1		1	520.229	odd	4
4800.2.a.cc.1.1		1	520.99	even	4
4800.2.f.d.3649.1		2	520.437	even	4
4800.2.f.d.3649.2		2	520.333	even	4
4800.2.f.bg.3649.1		2	520.203	odd	4
4800.2.f.bg.3649.2		2	520.307	odd	4
5808.2.a.s.1.1		1	572.307	odd	4
6936.2.a.p.1.1		1	221.203	odd	4
7056.2.a.q.1.1		1	1092.671	even	4
8112.2.a.be.1.1		1	52.31	even	4
8664.2.a.j.1.1		1	247.151	even	4
8712.2.a.u.1.1		1	429.164	odd	4
9408.2.a.h.1.1		1	728.125	even	4
9408.2.a.cc.1.1		1	728.307	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.00000i q^{5} +1.00000 q^{9} +4.00000i q^{11} -2.00000i q^{15} -2.00000 q^{17} +4.00000i q^{19} +8.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.00000 q^{29} -8.00000i q^{31} -4.00000i q^{33} +6.00000i q^{37} +6.00000i q^{41} -4.00000 q^{43} +2.00000i q^{45} +7.00000 q^{49} +2.00000 q^{51} -2.00000 q^{53} -8.00000 q^{55} -4.00000i q^{57} +4.00000i q^{59} -2.00000 q^{61} +4.00000i q^{67} -8.00000 q^{69} -8.00000i q^{71} +10.0000i q^{73} -1.00000 q^{75} -8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -4.00000i q^{85} -6.00000 q^{87} -6.00000i q^{89} +8.00000i q^{93} -8.00000 q^{95} -2.00000i q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9} - 4 q^{17} + 16 q^{23} + 2 q^{25} - 2 q^{27} + 12 q^{29} - 8 q^{43} + 14 q^{49} + 4 q^{51} - 4 q^{53} - 16 q^{55} - 4 q^{61} - 16 q^{69} - 2 q^{75} - 16 q^{79} + 2 q^{81} - 12 q^{87}+ \cdots - 16 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 - 4 * q^17 + 16 * q^23 + 2 * q^25 - 2 * q^27 + 12 * q^29 - 8 * q^43 + 14 * q^49 + 4 * q^51 - 4 * q^53 - 16 * q^55 - 4 * q^61 - 16 * q^69 - 2 * q^75 - 16 * q^79 + 2 * q^81 - 12 * q^87 - 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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