LMFDB - Embedded newform 4056.2.c.i.337.1

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:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.00000i q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.00000i q^{5} +1.00000 q^{9} -3.00000i q^{15} +1.00000 q^{17} -4.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} +3.00000 q^{29} -8.00000i q^{31} -5.00000i q^{37} -3.00000i q^{41} -4.00000 q^{43} -3.00000i q^{45} -8.00000i q^{47} +7.00000 q^{49} +1.00000 q^{51} -13.0000 q^{53} +12.0000i q^{59} +15.0000 q^{61} -12.0000i q^{67} -4.00000 q^{69} -8.00000i q^{71} +3.00000i q^{73} -4.00000 q^{75} -4.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -3.00000i q^{85} +3.00000 q^{87} +10.0000i q^{89} -8.00000i q^{93} -2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{9} + 2 q^{17} - 8 q^{23} - 8 q^{25} + 2 q^{27} + 6 q^{29} - 8 q^{43} + 14 q^{49} + 2 q^{51} - 26 q^{53} + 30 q^{61} - 8 q^{69} - 8 q^{75} - 8 q^{79} + 2 q^{81} + 6 q^{87}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 + 2 * q^17 - 8 * q^23 - 8 * q^25 + 2 * q^27 + 6 * q^29 - 8 * q^43 + 14 * q^49 + 2 * q^51 - 26 * q^53 + 30 * q^61 - 8 * q^69 - 8 * q^75 - 8 * q^79 + 2 * q^81 + 6 * q^87

Display 100 coefficients 10 coefficients

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
$5$	− 3.00000i	− 1.34164i	−0.741620	−	0.670820i	$-0.765942\pi$
$5$	− 3.00000i	− 1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$	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
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 3.00000i	− 0.774597i
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	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
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5.00000i	− 0.821995i	−0.911636	−	0.410997i	$-0.865181\pi$
$37$	− 5.00000i	− 0.821995i	0.911636	−	0.410997i	$-0.134819\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.00000i	− 0.468521i	−0.972174	−	0.234261i	$-0.924733\pi$
$41$	− 3.00000i	− 0.468521i	0.972174	−	0.234261i	$-0.0752669\pi$
$42$	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
$45$	− 3.00000i	− 0.447214i
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	−13.0000	−1.78569	−0.892844	−	0.450367i	$-0.851293\pi$
$53$	−13.0000	−1.78569	−0.892844	+	0.450367i	$0.851293\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.0000i	1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$	12.0000i	1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$	0	0
$61$	15.0000	1.92055	0.960277	−	0.279050i	$-0.0900195\pi$
$61$	15.0000	1.92055	0.960277	+	0.279050i	$0.0900195\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	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
$73$	3.00000i	0.351123i	0.984468	+	0.175562i	$0.0561742\pi$
$73$	3.00000i	0.351123i	−0.984468	+	0.175562i	$0.943826\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	− 3.00000i	− 0.325396i
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	10.0000i	1.06000i	0.847998	+	0.529999i	$0.177808\pi$
$89$	10.0000i	1.06000i	−0.847998	+	0.529999i	$0.822192\pi$
$90$	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	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	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
$107$	−20.0000	−1.93347	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000	−1.93347	−0.966736	+	0.255774i	$0.917670\pi$
$108$	0	0
$109$	18.0000i	1.72409i	0.506834	+	0.862044i	$0.330816\pi$
$109$	18.0000i	1.72409i	−0.506834	+	0.862044i	$0.669184\pi$
$110$	0	0
$111$	− 5.00000i	− 0.474579i
$112$	0	0
$113$	−13.0000	−1.22294	−0.611469	−	0.791269i	$-0.709421\pi$
$113$	−13.0000	−1.22294	−0.611469	+	0.791269i	$0.709421\pi$
$114$	0	0
$115$	12.0000i	1.11901i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	− 3.00000i	− 0.270501i
$124$	0	0
$125$	− 3.00000i	− 0.268328i
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 3.00000i	− 0.258199i
$136$	0	0
$137$	− 5.00000i	− 0.427179i	−0.976924	−	0.213589i	$-0.931485\pi$
$137$	− 5.00000i	− 0.427179i	0.976924	−	0.213589i	$-0.0685155\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	− 8.00000i	− 0.673722i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 9.00000i	− 0.747409i
$146$	0	0
$147$	7.00000	0.577350
$148$	0	0
$149$	− 19.0000i	− 1.55654i	−0.627929	−	0.778270i	$-0.716097\pi$
$149$	− 19.0000i	− 1.55654i	0.627929	−	0.778270i	$-0.283903\pi$
$150$	0	0
$151$	− 4.00000i	− 0.325515i	−0.986666	−	0.162758i	$-0.947961\pi$
$151$	− 4.00000i	− 0.325515i	0.986666	−	0.162758i	$-0.0520389\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	−24.0000	−1.92773
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	−13.0000	−1.03097
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0000i	0.901975i
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−11.0000	−0.817624	−0.408812	−	0.912619i	$-0.634057\pi$
$181$	−11.0000	−0.817624	−0.408812	+	0.912619i	$0.634057\pi$
$182$	0	0
$183$	15.0000	1.10883
$184$	0	0
$185$	−15.0000	−1.10282
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	11.0000i	0.791797i	0.918294	+	0.395899i	$0.129567\pi$
$193$	11.0000i	0.791797i	−0.918294	+	0.395899i	$0.870433\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	− 12.0000i	− 0.846415i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	− 8.00000i	− 0.548151i
$214$	0	0
$215$	12.0000i	0.818393i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.00000i	0.202721i
$220$	0	0
$221$	0	0
$222$	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
$225$	−4.00000	−0.266667
$226$	0	0
$227$	8.00000i	0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$	8.00000i	0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$	0	0
$229$	− 10.0000i	− 0.660819i	−0.943838	−	0.330409i	$-0.892813\pi$
$229$	− 10.0000i	− 0.660819i	0.943838	−	0.330409i	$-0.107187\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−24.0000	−1.56559
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	8.00000i	0.517477i	0.965947	+	0.258738i	$0.0833068\pi$
$239$	8.00000i	0.517477i	−0.965947	+	0.258738i	$0.916693\pi$
$240$	0	0
$241$	11.0000i	0.708572i	0.935137	+	0.354286i	$0.115276\pi$
$241$	11.0000i	0.708572i	−0.935137	+	0.354286i	$0.884724\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	− 21.0000i	− 1.34164i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 12.0000i	− 0.760469i
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	− 3.00000i	− 0.187867i
$256$	0	0
$257$	5.00000	0.311891	0.155946	−	0.987766i	$-0.450158\pi$
$257$	5.00000	0.311891	0.155946	+	0.987766i	$0.450158\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3.00000	0.185695
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	39.0000i	2.39575i
$266$	0	0
$267$	10.0000i	0.611990i
$268$	0	0
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	28.0000i	1.70088i	0.526073	+	0.850439i	$0.323664\pi$
$271$	28.0000i	1.70088i	−0.526073	+	0.850439i	$0.676336\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	25.0000	1.50210	0.751052	−	0.660243i	$-0.229547\pi$
$277$	25.0000	1.50210	0.751052	+	0.660243i	$0.229547\pi$
$278$	0	0
$279$	− 8.00000i	− 0.478947i
$280$	0	0
$281$	15.0000i	0.894825i	0.894328	+	0.447412i	$0.147654\pi$
$281$	15.0000i	0.894825i	−0.894328	+	0.447412i	$0.852346\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	− 2.00000i	− 0.117242i
$292$	0	0
$293$	− 1.00000i	− 0.0584206i	−0.999573	−	0.0292103i	$-0.990701\pi$
$293$	− 1.00000i	− 0.0584206i	0.999573	−	0.0292103i	$-0.00929925\pi$
$294$	0	0
$295$	36.0000	2.09600
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3.00000	−0.172345
$304$	0	0
$305$	− 45.0000i	− 2.57669i
$306$	0	0
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 31.0000i	− 1.74113i	−0.492050	−	0.870567i	$-0.663752\pi$
$317$	− 31.0000i	− 1.74113i	0.492050	−	0.870567i	$-0.336248\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−20.0000	−1.11629
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	18.0000i	0.995402i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	32.0000i	1.75888i	0.476011	+	0.879440i	$0.342082\pi$
$331$	32.0000i	1.75888i	−0.476011	+	0.879440i	$0.657918\pi$
$332$	0	0
$333$	− 5.00000i	− 0.273998i
$334$	0	0
$335$	−36.0000	−1.96689
$336$	0	0
$337$	1.00000	0.0544735	0.0272367	−	0.999629i	$-0.491329\pi$
$337$	1.00000	0.0544735	0.0272367	+	0.999629i	$0.491329\pi$
$338$	0	0
$339$	−13.0000	−0.706063
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	12.0000i	0.646058i
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	− 34.0000i	− 1.81998i	−0.414632	−	0.909989i	$-0.636090\pi$
$349$	− 34.0000i	− 1.81998i	0.414632	−	0.909989i	$-0.363910\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 11.0000i	− 0.585471i	−0.956193	−	0.292735i	$-0.905434\pi$
$353$	− 11.0000i	− 0.585471i	0.956193	−	0.292735i	$-0.0945655\pi$
$354$	0	0
$355$	−24.0000	−1.27379
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 28.0000i	− 1.47778i	−0.673824	−	0.738892i	$-0.735349\pi$
$359$	− 28.0000i	− 1.47778i	0.673824	−	0.738892i	$-0.264651\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	9.00000	0.471082
$366$	0	0
$367$	36.0000	1.87918	0.939592	−	0.342296i	$-0.111204\pi$
$367$	36.0000	1.87918	0.939592	+	0.342296i	$0.111204\pi$
$368$	0	0
$369$	− 3.00000i	− 0.156174i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	− 3.00000i	− 0.154919i
$376$	0	0
$377$	0	0
$378$	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
$381$	12.0000	0.614779
$382$	0	0
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.0000i	0.603786i
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 9.00000i	− 0.449439i	−0.974424	−	0.224719i	$-0.927853\pi$
$401$	− 9.00000i	− 0.449439i	0.974424	−	0.224719i	$-0.0721465\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 3.00000i	− 0.149071i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	9.00000i	0.445021i	0.974930	+	0.222511i	$0.0714252\pi$
$409$	9.00000i	0.445021i	−0.974930	+	0.222511i	$0.928575\pi$
$410$	0	0
$411$	− 5.00000i	− 0.246632i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−36.0000	−1.76717
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	− 27.0000i	− 1.31590i	−0.753062	−	0.657950i	$-0.771424\pi$
$421$	− 27.0000i	− 1.31590i	0.753062	−	0.657950i	$-0.228576\pi$
$422$	0	0
$423$	− 8.00000i	− 0.388973i
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 28.0000i	− 1.34871i	−0.738406	−	0.674356i	$-0.764421\pi$
$431$	− 28.0000i	− 1.34871i	0.738406	−	0.674356i	$-0.235579\pi$
$432$	0	0
$433$	21.0000	1.00920	0.504598	−	0.863355i	$-0.331641\pi$
$433$	21.0000	1.00920	0.504598	+	0.863355i	$0.331641\pi$
$434$	0	0
$435$	− 9.00000i	− 0.431517i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	7.00000	0.333333
$442$	0	0
$443$	40.0000	1.90046	0.950229	−	0.311553i	$-0.100849\pi$
$443$	40.0000	1.90046	0.950229	+	0.311553i	$0.100849\pi$
$444$	0	0
$445$	30.0000	1.42214
$446$	0	0
$447$	− 19.0000i	− 0.898669i
$448$	0	0
$449$	2.00000i	0.0943858i	0.998886	+	0.0471929i	$0.0150276\pi$
$449$	2.00000i	0.0943858i	−0.998886	+	0.0471929i	$0.984972\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 4.00000i	− 0.187936i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.00000i	0.421002i	0.977594	+	0.210501i	$0.0675096\pi$
$457$	9.00000i	0.421002i	−0.977594	+	0.210501i	$0.932490\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	5.00000i	0.232873i	0.993198	+	0.116437i	$0.0371472\pi$
$461$	5.00000i	0.232873i	−0.993198	+	0.116437i	$0.962853\pi$
$462$	0	0
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	0	0
$465$	−24.0000	−1.11297
$466$	0	0
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−13.0000	−0.599008
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−13.0000	−0.595229
$478$	0	0
$479$	12.0000i	0.548294i	0.961688	+	0.274147i	$0.0883955\pi$
$479$	12.0000i	0.548294i	−0.961688	+	0.274147i	$0.911605\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.00000	−0.272446
$486$	0	0
$487$	12.0000i	0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$	12.0000i	0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$	0	0
$489$	− 16.0000i	− 0.723545i
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	0	0
$493$	3.00000	0.135113
$494$	0	0
$495$	0	0
$496$	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
$501$	12.0000i	0.536120i
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	9.00000i	0.400495i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 7.00000i	− 0.310270i	−0.987893	−	0.155135i	$-0.950419\pi$
$509$	− 7.00000i	− 0.310270i	0.987893	−	0.155135i	$-0.0495812\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	48.0000i	2.11513i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	−32.0000	−1.39926	−0.699631	−	0.714504i	$-0.746652\pi$
$523$	−32.0000	−1.39926	−0.699631	+	0.714504i	$0.746652\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8.00000i	− 0.348485i
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	12.0000i	0.520756i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	60.0000i	2.59403i
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.00000i	0.300954i	0.988614	+	0.150477i	$0.0480809\pi$
$541$	7.00000i	0.300954i	−0.988614	+	0.150477i	$0.951919\pi$
$542$	0	0
$543$	−11.0000	−0.472055
$544$	0	0
$545$	54.0000	2.31311
$546$	0	0
$547$	−24.0000	−1.02617	−0.513083	−	0.858339i	$-0.671497\pi$
$547$	−24.0000	−1.02617	−0.513083	+	0.858339i	$0.671497\pi$
$548$	0	0
$549$	15.0000	0.640184
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−15.0000	−0.636715
$556$	0	0
$557$	− 29.0000i	− 1.22877i	−0.789007	−	0.614385i	$-0.789404\pi$
$557$	− 29.0000i	− 1.22877i	0.789007	−	0.614385i	$-0.210596\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	39.0000i	1.64074i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	16.0000	0.667246
$576$	0	0
$577$	41.0000i	1.70685i	0.521214	+	0.853426i	$0.325479\pi$
$577$	41.0000i	1.70685i	−0.521214	+	0.853426i	$0.674521\pi$
$578$	0	0
$579$	11.0000i	0.457144i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 16.0000i	− 0.660391i	−0.943913	−	0.330195i	$-0.892885\pi$
$587$	− 16.0000i	− 0.660391i	0.943913	−	0.330195i	$-0.107115\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	− 6.00000i	− 0.246807i
$592$	0	0
$593$	3.00000i	0.123195i	0.998101	+	0.0615976i	$0.0196196\pi$
$593$	3.00000i	0.123195i	−0.998101	+	0.0615976i	$0.980380\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.00000	−0.163709
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	0	0
$603$	− 12.0000i	− 0.488678i
$604$	0	0
$605$	− 33.0000i	− 1.34164i
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 3.00000i	− 0.121169i	−0.998163	−	0.0605844i	$-0.980704\pi$
$613$	− 3.00000i	− 0.121169i	0.998163	−	0.0605844i	$-0.0192964\pi$
$614$	0	0
$615$	−9.00000	−0.362915
$616$	0	0
$617$	− 15.0000i	− 0.603877i	−0.953327	−	0.301939i	$-0.902366\pi$
$617$	− 15.0000i	− 0.603877i	0.953327	−	0.301939i	$-0.0976338\pi$
$618$	0	0
$619$	− 28.0000i	− 1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$	− 28.0000i	− 1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 5.00000i	− 0.199363i
$630$	0	0
$631$	− 8.00000i	− 0.318475i	−0.987240	−	0.159237i	$-0.949096\pi$
$631$	− 8.00000i	− 0.318475i	0.987240	−	0.159237i	$-0.0509036\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	− 36.0000i	− 1.42862i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 8.00000i	− 0.316475i
$640$	0	0
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	0	0
$643$	20.0000i	0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$	20.0000i	0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$	0	0
$645$	12.0000i	0.472500i
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.00000i	0.117041i
$658$	0	0
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	0	0
$661$	7.00000i	0.272268i	0.990690	+	0.136134i	$0.0434678\pi$
$661$	7.00000i	0.272268i	−0.990690	+	0.136134i	$0.956532\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.0000	−0.464642
$668$	0	0
$669$	− 8.00000i	− 0.309298i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	8.00000i	0.306561i
$682$	0	0
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	0	0
$685$	−15.0000	−0.573121
$686$	0	0
$687$	− 10.0000i	− 0.381524i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	8.00000i	0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$	8.00000i	0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 24.0000i	− 0.910372i
$696$	0	0
$697$	− 3.00000i	− 0.113633i
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−24.0000	−0.903892
$706$	0	0
$707$	0	0
$708$	0	0
$709$	19.0000i	0.713560i	0.934188	+	0.356780i	$0.116125\pi$
$709$	19.0000i	0.713560i	−0.934188	+	0.356780i	$0.883875\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	32.0000i	1.19841i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.00000i	0.298765i
$718$	0	0
$719$	−44.0000	−1.64092	−0.820462	−	0.571702i	$-0.806283\pi$
$719$	−44.0000	−1.64092	−0.820462	+	0.571702i	$0.806283\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	11.0000i	0.409094i
$724$	0	0
$725$	−12.0000	−0.445669
$726$	0	0
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.00000	−0.147945
$732$	0	0
$733$	17.0000i	0.627909i	0.949438	+	0.313955i	$0.101654\pi$
$733$	17.0000i	0.627909i	−0.949438	+	0.313955i	$0.898346\pi$
$734$	0	0
$735$	− 21.0000i	− 0.774597i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	32.0000i	1.17714i	0.808447	+	0.588570i	$0.200309\pi$
$739$	32.0000i	1.17714i	−0.808447	+	0.588570i	$0.799691\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32.0000i	1.17397i	0.809599	+	0.586983i	$0.199684\pi$
$743$	32.0000i	1.17397i	−0.809599	+	0.586983i	$0.800316\pi$
$744$	0	0
$745$	−57.0000	−2.08832
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 38.0000i	− 1.37750i	−0.724999	−	0.688749i	$-0.758160\pi$
$761$	− 38.0000i	− 1.37750i	0.724999	−	0.688749i	$-0.241840\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 3.00000i	− 0.108465i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 18.0000i	− 0.649097i	−0.945869	−	0.324548i	$-0.894788\pi$
$769$	− 18.0000i	− 0.649097i	0.945869	−	0.324548i	$-0.105212\pi$
$770$	0	0
$771$	5.00000	0.180071
$772$	0	0
$773$	10.0000i	0.359675i	0.983696	+	0.179838i	$0.0575572\pi$
$773$	10.0000i	0.359675i	−0.983696	+	0.179838i	$0.942443\pi$
$774$	0	0
$775$	32.0000i	1.14947i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	3.00000	0.107211
$784$	0	0
$785$	39.0000i	1.39197i
$786$	0	0
$787$	− 48.0000i	− 1.71102i	−0.517790	−	0.855508i	$-0.673245\pi$
$787$	− 48.0000i	− 1.71102i	0.517790	−	0.855508i	$-0.326755\pi$
$788$	0	0
$789$	−12.0000	−0.427211
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	39.0000i	1.38319i
$796$	0	0
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	− 8.00000i	− 0.283020i
$800$	0	0
$801$	10.0000i	0.353333i
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.0000	0.492823
$808$	0	0
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	0	0
$811$	56.0000i	1.96643i	0.182462	+	0.983213i	$0.441593\pi$
$811$	56.0000i	1.96643i	−0.182462	+	0.983213i	$0.558407\pi$
$812$	0	0
$813$	28.0000i	0.982003i
$814$	0	0
$815$	−48.0000	−1.68137
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42.0000i	1.46581i	0.680331	+	0.732905i	$0.261836\pi$
$821$	42.0000i	1.46581i	−0.680331	+	0.732905i	$0.738164\pi$
$822$	0	0
$823$	36.0000	1.25488	0.627441	−	0.778664i	$-0.284103\pi$
$823$	36.0000	1.25488	0.627441	+	0.778664i	$0.284103\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.0000i	0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$	28.0000i	0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$	0	0
$829$	41.0000	1.42399	0.711994	−	0.702185i	$-0.247792\pi$
$829$	41.0000	1.42399	0.711994	+	0.702185i	$0.247792\pi$
$830$	0	0
$831$	25.0000	0.867240
$832$	0	0
$833$	7.00000	0.242536
$834$	0	0
$835$	36.0000	1.24583
$836$	0	0
$837$	− 8.00000i	− 0.276520i
$838$	0	0
$839$	12.0000i	0.414286i	0.978311	+	0.207143i	$0.0664165\pi$
$839$	12.0000i	0.414286i	−0.978311	+	0.207143i	$0.933583\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	15.0000i	0.516627i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	16.0000	0.549119
$850$	0	0
$851$	20.0000i	0.685591i
$852$	0	0
$853$	− 17.0000i	− 0.582069i	−0.956713	−	0.291034i	$-0.906001\pi$
$853$	− 17.0000i	− 0.582069i	0.956713	−	0.291034i	$-0.0939994\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.0000	−0.922302	−0.461151	−	0.887322i	$-0.652563\pi$
$857$	−27.0000	−0.922302	−0.461151	+	0.887322i	$0.652563\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	− 54.0000i	− 1.83606i
$866$	0	0
$867$	−16.0000	−0.543388
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41.0000i	1.38447i	0.721671	+	0.692236i	$0.243374\pi$
$877$	41.0000i	1.38447i	−0.721671	+	0.692236i	$0.756626\pi$
$878$	0	0
$879$	− 1.00000i	− 0.0337292i
$880$	0	0
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	36.0000	1.21013
$886$	0	0
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	12.0000i	0.401116i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 24.0000i	− 0.800445i
$900$	0	0
$901$	−13.0000	−0.433093
$902$	0	0
$903$	0	0
$904$	0	0
$905$	33.0000i	1.09696i
$906$	0	0
$907$	−48.0000	−1.59381	−0.796907	−	0.604102i	$-0.793532\pi$
$907$	−48.0000	−1.59381	−0.796907	+	0.604102i	$0.793532\pi$
$908$	0	0
$909$	−3.00000	−0.0995037
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 45.0000i	− 1.48765i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	0	0
$921$	− 20.0000i	− 0.659022i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	20.0000i	0.657596i
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	33.0000i	1.08269i	0.840799	+	0.541347i	$0.182086\pi$
$929$	33.0000i	1.08269i	−0.840799	+	0.541347i	$0.817914\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−37.0000	−1.20874	−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000	−1.20874	−0.604369	+	0.796705i	$0.706575\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	− 46.0000i	− 1.49956i	−0.661689	−	0.749779i	$-0.730160\pi$
$941$	− 46.0000i	− 1.49956i	0.661689	−	0.749779i	$-0.269840\pi$
$942$	0	0
$943$	12.0000i	0.390774i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 28.0000i	− 0.909878i	−0.890523	−	0.454939i	$-0.849661\pi$
$947$	− 28.0000i	− 0.909878i	0.890523	−	0.454939i	$-0.150339\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 31.0000i	− 1.00524i
$952$	0	0
$953$	38.0000	1.23094	0.615470	−	0.788160i	$-0.288966\pi$
$953$	38.0000	1.23094	0.615470	+	0.788160i	$0.288966\pi$
$954$	0	0
$955$	24.0000i	0.776622i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−33.0000	−1.06452
$962$	0	0
$963$	−20.0000	−0.644491
$964$	0	0
$965$	33.0000	1.06231
$966$	0	0
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.00000	0.128366	0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000	0.128366	0.0641831	+	0.997938i	$0.479556\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	57.0000i	1.82359i	0.410644	+	0.911796i	$0.365304\pi$
$977$	57.0000i	1.82359i	−0.410644	+	0.911796i	$0.634696\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	18.0000i	0.574696i
$982$	0	0
$983$	− 16.0000i	− 0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$	− 16.0000i	− 0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.0000	0.508770
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	32.0000i	1.01549i
$994$	0	0
$995$	12.0000i	0.380426i
$996$	0	0
$997$	47.0000	1.48850	0.744252	−	0.667898i	$-0.232806\pi$
$997$	47.0000	1.48850	0.744252	+	0.667898i	$0.232806\pi$
$998$	0	0
$999$	− 5.00000i	− 0.158193i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.i.337.1		2
13.5	odd	4		4056.2.a.l.1.1		1
13.7	odd	12		312.2.q.a.289.1	yes	2
13.8	odd	4		4056.2.a.q.1.1		1
13.11	odd	12		312.2.q.a.217.1	✓	2
13.12	even	2	inner	4056.2.c.i.337.2		2
39.11	even	12		936.2.t.a.217.1		2
39.20	even	12		936.2.t.a.289.1		2
52.7	even	12		624.2.q.f.289.1		2
52.11	even	12		624.2.q.f.529.1		2
52.31	even	4		8112.2.a.b.1.1		1
52.47	even	4		8112.2.a.n.1.1		1
156.11	odd	12		1872.2.t.b.1153.1		2
156.59	odd	12		1872.2.t.b.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.q.a.217.1	✓	2	13.11	odd	12
312.2.q.a.289.1	yes	2	13.7	odd	12
624.2.q.f.289.1		2	52.7	even	12
624.2.q.f.529.1		2	52.11	even	12
936.2.t.a.217.1		2	39.11	even	12
936.2.t.a.289.1		2	39.20	even	12
1872.2.t.b.289.1		2	156.59	odd	12
1872.2.t.b.1153.1		2	156.11	odd	12
4056.2.a.l.1.1		1	13.5	odd	4
4056.2.a.q.1.1		1	13.8	odd	4
4056.2.c.i.337.1		2	1.1	even	1	trivial
4056.2.c.i.337.2		2	13.12	even	2	inner
8112.2.a.b.1.1		1	52.31	even	4
8112.2.a.n.1.1		1	52.47	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.00000i q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.00000i q^{5} +1.00000 q^{9} -3.00000i q^{15} +1.00000 q^{17} -4.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} +3.00000 q^{29} -8.00000i q^{31} -5.00000i q^{37} -3.00000i q^{41} -4.00000 q^{43} -3.00000i q^{45} -8.00000i q^{47} +7.00000 q^{49} +1.00000 q^{51} -13.0000 q^{53} +12.0000i q^{59} +15.0000 q^{61} -12.0000i q^{67} -4.00000 q^{69} -8.00000i q^{71} +3.00000i q^{73} -4.00000 q^{75} -4.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -3.00000i q^{85} +3.00000 q^{87} +10.0000i q^{89} -8.00000i q^{93} -2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} + 2 q^{17} - 8 q^{23} - 8 q^{25} + 2 q^{27} + 6 q^{29} - 8 q^{43} + 14 q^{49} + 2 q^{51} - 26 q^{53} + 30 q^{61} - 8 q^{69} - 8 q^{75} - 8 q^{79} + 2 q^{81} + 6 q^{87}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 + 2 * q^17 - 8 * q^23 - 8 * q^25 + 2 * q^27 + 6 * q^29 - 8 * q^43 + 14 * q^49 + 2 * q^51 - 26 * q^53 + 30 * q^61 - 8 * q^69 - 8 * q^75 - 8 * q^79 + 2 * q^81 + 6 * q^87

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