LMFDB - Embedded newform 4056.2.c.c.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 312)
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.c.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.00000i q^{5} +4.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.00000i q^{5} +4.00000i q^{7} +1.00000 q^{9} -2.00000i q^{15} -2.00000 q^{17} -8.00000i q^{19} -4.00000i q^{21} -8.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} -4.00000i q^{31} -8.00000 q^{35} -10.0000i q^{37} -2.00000i q^{41} +4.00000 q^{43} +2.00000i q^{45} -12.0000i q^{47} -9.00000 q^{49} +2.00000 q^{51} +6.00000 q^{53} +8.00000i q^{57} -2.00000 q^{61} +4.00000i q^{63} -8.00000i q^{67} +8.00000 q^{69} +12.0000i q^{71} +10.0000i q^{73} -1.00000 q^{75} -8.00000 q^{79} +1.00000 q^{81} -4.00000i q^{85} +2.00000 q^{87} -14.0000i q^{89} +4.00000i q^{93} +16.0000 q^{95} -2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{17} - 16 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 16 q^{35} + 8 q^{43} - 18 q^{49} + 4 q^{51} + 12 q^{53} - 4 q^{61} + 16 q^{69} - 2 q^{75} - 16 q^{79} + 2 q^{81} + 4 q^{87} + 32 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 - 4 * q^29 - 16 * q^35 + 8 * q^43 - 18 * q^49 + 4 * q^51 + 12 * q^53 - 4 * q^61 + 16 * q^69 - 2 * q^75 - 16 * q^79 + 2 * q^81 + 4 * q^87 + 32 * q^95

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
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	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
$15$	− 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
$19$	− 8.00000i	− 1.83533i	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	− 8.00000i	− 1.83533i	0.397360	−	0.917663i	$-0.369927\pi$
$20$	0	0
$21$	− 4.00000i	− 0.872872i
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	− 4.00000i	− 0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$	− 4.00000i	− 0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−8.00000	−1.35225
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.00000i	− 0.312348i	−0.987730	−	0.156174i	$-0.950084\pi$
$41$	− 2.00000i	− 0.312348i	0.987730	−	0.156174i	$-0.0499160\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	0	0
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.00000i	1.05963i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	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
$63$	4.00000i	0.503953i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	12.0000i	1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$	12.0000i	1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$	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
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	− 4.00000i	− 0.433861i
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	− 14.0000i	− 1.48400i	−0.670402	−	0.741999i	$-0.733878\pi$
$89$	− 14.0000i	− 1.48400i	0.670402	−	0.741999i	$-0.266122\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.00000i	0.414781i
$94$	0	0
$95$	16.0000	1.64157
$96$	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
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	8.00000	0.780720
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	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
$111$	10.0000i	0.949158i
$112$	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
$119$	− 8.00000i	− 0.733359i
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	2.00000i	0.180334i
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	32.0000	2.77475
$134$	0	0
$135$	− 2.00000i	− 0.172133i
$136$	0	0
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	12.0000i	1.01058i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 4.00000i	− 0.332182i
$146$	0	0
$147$	9.00000	0.742307
$148$	0	0
$149$	18.0000i	1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$	18.0000i	1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$	0	0
$151$	− 12.0000i	− 0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$	− 12.0000i	− 0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	8.00000	0.642575
$156$	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
$159$	−6.00000	−0.475831
$160$	0	0
$161$	− 32.0000i	− 2.52195i
$162$	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
$171$	− 8.00000i	− 0.611775i
$172$	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
$175$	4.00000i	0.302372i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	20.0000	1.47043
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 4.00000i	− 0.290957i
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 14.0000i	− 0.997459i	−0.866758	−	0.498729i	$-0.833800\pi$
$197$	− 14.0000i	− 0.997459i	0.866758	−	0.498729i	$-0.166200\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	8.00000i	0.564276i
$202$	0	0
$203$	− 8.00000i	− 0.561490i
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	− 12.0000i	− 0.822226i
$214$	0	0
$215$	8.00000i	0.545595i
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	− 10.0000i	− 0.675737i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 20.0000i	− 1.33930i	−0.742677	−	0.669650i	$-0.766444\pi$
$223$	− 20.0000i	− 1.33930i	0.742677	−	0.669650i	$-0.233556\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	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
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	20.0000i	1.29369i	0.762620	+	0.646846i	$0.223912\pi$
$239$	20.0000i	1.29369i	−0.762620	+	0.646846i	$0.776088\pi$
$240$	0	0
$241$	− 14.0000i	− 0.901819i	−0.892570	−	0.450910i	$-0.851100\pi$
$241$	− 14.0000i	− 0.901819i	0.892570	−	0.450910i	$-0.148900\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	− 18.0000i	− 1.14998i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	4.00000i	0.250490i
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	40.0000	2.48548
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	12.0000i	0.737154i
$266$	0	0
$267$	14.0000i	0.856786i
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	− 20.0000i	− 1.21491i	−0.794353	−	0.607457i	$-0.792190\pi$
$271$	− 20.0000i	− 1.21491i	0.794353	−	0.607457i	$-0.207810\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	− 4.00000i	− 0.239474i
$280$	0	0
$281$	− 30.0000i	− 1.78965i	−0.446417	−	0.894825i	$-0.647300\pi$
$281$	− 30.0000i	− 1.78965i	0.446417	−	0.894825i	$-0.352700\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	−16.0000	−0.947758
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	2.00000i	0.117242i
$292$	0	0
$293$	− 2.00000i	− 0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$	− 2.00000i	− 0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	16.0000i	0.922225i
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	− 4.00000i	− 0.229039i
$306$	0	0
$307$	− 32.0000i	− 1.82634i	−0.407583	−	0.913168i	$-0.633628\pi$
$307$	− 32.0000i	− 1.82634i	0.407583	−	0.913168i	$-0.366372\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	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
$315$	−8.00000	−0.450749
$316$	0	0
$317$	10.0000i	0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$	10.0000i	0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	16.0000i	0.890264i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 18.0000i	− 0.995402i
$328$	0	0
$329$	48.0000	2.64633
$330$	0	0
$331$	− 16.0000i	− 0.879440i	−0.898135	−	0.439720i	$-0.855078\pi$
$331$	− 16.0000i	− 0.879440i	0.898135	−	0.439720i	$-0.144922\pi$
$332$	0	0
$333$	− 10.0000i	− 0.547997i
$334$	0	0
$335$	16.0000	0.874173
$336$	0	0
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	0	0
$345$	16.0000i	0.861411i
$346$	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
$349$	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
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	−24.0000	−1.27379
$356$	0	0
$357$	8.00000i	0.423405i
$358$	0	0
$359$	− 20.0000i	− 1.05556i	−0.849381	−	0.527780i	$-0.823025\pi$
$359$	− 20.0000i	− 1.05556i	0.849381	−	0.527780i	$-0.176975\pi$
$360$	0	0
$361$	−45.0000	−2.36842
$362$	0	0
$363$	−11.0000	−0.577350
$364$	0	0
$365$	−20.0000	−1.04685
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	− 2.00000i	− 0.104116i
$370$	0	0
$371$	24.0000i	1.24602i
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	0	0
$375$	− 12.0000i	− 0.619677i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	16.0000i	0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$	16.0000i	0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	− 16.0000i	− 0.805047i
$396$	0	0
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	0	0
$399$	−32.0000	−1.60200
$400$	0	0
$401$	− 22.0000i	− 1.09863i	−0.835616	−	0.549314i	$-0.814889\pi$
$401$	− 22.0000i	− 1.09863i	0.835616	−	0.549314i	$-0.185111\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.00000i	0.0993808i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 10.0000i	− 0.494468i	−0.968956	−	0.247234i	$-0.920478\pi$
$409$	− 10.0000i	− 0.494468i	0.968956	−	0.247234i	$-0.0795217\pi$
$410$	0	0
$411$	− 2.00000i	− 0.0986527i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	− 22.0000i	− 1.07221i	−0.844150	−	0.536107i	$-0.819894\pi$
$421$	− 22.0000i	− 1.07221i	0.844150	−	0.536107i	$-0.180106\pi$
$422$	0	0
$423$	− 12.0000i	− 0.583460i
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 12.0000i	− 0.578020i	−0.957326	−	0.289010i	$-0.906674\pi$
$431$	− 12.0000i	− 0.578020i	0.957326	−	0.289010i	$-0.0933260\pi$
$432$	0	0
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	4.00000i	0.191785i
$436$	0	0
$437$	64.0000i	3.06154i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	28.0000	1.32733
$446$	0	0
$447$	− 18.0000i	− 0.851371i
$448$	0	0
$449$	− 6.00000i	− 0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$	− 6.00000i	− 0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	12.0000i	0.563809i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.00000i	0.280668i	0.990104	+	0.140334i	$0.0448177\pi$
$457$	6.00000i	0.280668i	−0.990104	+	0.140334i	$0.955182\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	10.0000i	0.465746i	0.972507	+	0.232873i	$0.0748127\pi$
$461$	10.0000i	0.465746i	−0.972507	+	0.232873i	$0.925187\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 8.00000i	− 0.367065i
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	4.00000i	0.182765i	0.995816	+	0.0913823i	$0.0291285\pi$
$479$	4.00000i	0.182765i	−0.995816	+	0.0913823i	$0.970871\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	32.0000i	1.45605i
$484$	0	0
$485$	4.00000	0.181631
$486$	0	0
$487$	− 28.0000i	− 1.26880i	−0.773004	−	0.634401i	$-0.781247\pi$
$487$	− 28.0000i	− 1.26880i	0.773004	−	0.634401i	$-0.218753\pi$
$488$	0	0
$489$	16.0000i	0.723545i
$490$	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
$493$	4.00000	0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−48.0000	−2.15309
$498$	0	0
$499$	24.0000i	1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$	24.0000i	1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$	0	0
$501$	− 12.0000i	− 0.536120i
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	− 12.0000i	− 0.533993i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.0000i	0.443242i	0.975133	+	0.221621i	$0.0711348\pi$
$509$	10.0000i	0.443242i	−0.975133	+	0.221621i	$0.928865\pi$
$510$	0	0
$511$	−40.0000	−1.76950
$512$	0	0
$513$	8.00000i	0.353209i
$514$	0	0
$515$	− 16.0000i	− 0.705044i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	− 4.00000i	− 0.174574i
$526$	0	0
$527$	8.00000i	0.348485i
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	8.00000i	0.345870i
$536$	0	0
$537$	20.0000	0.863064
$538$	0	0
$539$	0	0
$540$	0	0
$541$	30.0000i	1.28980i	0.764267	+	0.644900i	$0.223101\pi$
$541$	30.0000i	1.28980i	−0.764267	+	0.644900i	$0.776899\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−36.0000	−1.54207
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	16.0000i	0.681623i
$552$	0	0
$553$	− 32.0000i	− 1.36078i
$554$	0	0
$555$	−20.0000	−0.848953
$556$	0	0
$557$	38.0000i	1.61011i	0.593199	+	0.805056i	$0.297865\pi$
$557$	38.0000i	1.61011i	−0.593199	+	0.805056i	$0.702135\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	44.0000	1.85438	0.927189	−	0.374593i	$-0.122217\pi$
$563$	44.0000	1.85438	0.927189	+	0.374593i	$0.122217\pi$
$564$	0	0
$565$	36.0000i	1.51453i
$566$	0	0
$567$	4.00000i	0.167984i
$568$	0	0
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	24.0000	1.00261
$574$	0	0
$575$	−8.00000	−0.333623
$576$	0	0
$577$	30.0000i	1.24892i	0.781058	+	0.624458i	$0.214680\pi$
$577$	30.0000i	1.24892i	−0.781058	+	0.624458i	$0.785320\pi$
$578$	0	0
$579$	14.0000i	0.581820i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8.00000i	− 0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$	− 8.00000i	− 0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	14.0000i	0.575883i
$592$	0	0
$593$	26.0000i	1.06769i	0.845582	+	0.533846i	$0.179254\pi$
$593$	26.0000i	1.06769i	−0.845582	+	0.533846i	$0.820746\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	0	0
$597$	8.00000	0.327418
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	− 8.00000i	− 0.325785i
$604$	0	0
$605$	22.0000i	0.894427i
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	8.00000i	0.324176i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 22.0000i	− 0.888572i	−0.895885	−	0.444286i	$-0.853457\pi$
$613$	− 22.0000i	− 0.888572i	0.895885	−	0.444286i	$-0.146543\pi$
$614$	0	0
$615$	−4.00000	−0.161296
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	40.0000i	1.60774i	0.594808	+	0.803868i	$0.297228\pi$
$619$	40.0000i	1.60774i	−0.594808	+	0.803868i	$0.702772\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	56.0000	2.24359
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.0000i	0.797452i
$630$	0	0
$631$	− 12.0000i	− 0.477712i	−0.971055	−	0.238856i	$-0.923228\pi$
$631$	− 12.0000i	− 0.477712i	0.971055	−	0.238856i	$-0.0767725\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	12.0000i	0.474713i
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	− 40.0000i	− 1.57745i	−0.614749	−	0.788723i	$-0.710743\pi$
$643$	− 40.0000i	− 1.57745i	0.614749	−	0.788723i	$-0.289257\pi$
$644$	0	0
$645$	− 8.00000i	− 0.315000i
$646$	0	0
$647$	40.0000	1.57256	0.786281	−	0.617869i	$-0.212004\pi$
$647$	40.0000	1.57256	0.786281	+	0.617869i	$0.212004\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−16.0000	−0.627089
$652$	0	0
$653$	46.0000	1.80012	0.900060	−	0.435767i	$-0.143523\pi$
$653$	46.0000	1.80012	0.900060	+	0.435767i	$0.143523\pi$
$654$	0	0
$655$	24.0000i	0.937758i
$656$	0	0
$657$	10.0000i	0.390137i
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	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
$665$	64.0000i	2.48181i
$666$	0	0
$667$	16.0000	0.619522
$668$	0	0
$669$	20.0000i	0.773245i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.0000i	1.22445i	0.790685	+	0.612223i	$0.209725\pi$
$683$	32.0000i	1.22445i	−0.790685	+	0.612223i	$0.790275\pi$
$684$	0	0
$685$	−4.00000	−0.152832
$686$	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	$-0.951375\pi$
$691$	− 8.00000i	− 0.304334i	0.988355	−	0.152167i	$-0.0486252\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.00000i	0.303457i
$696$	0	0
$697$	4.00000i	0.151511i
$698$	0	0
$699$	26.0000	0.983410
$700$	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
$703$	−80.0000	−3.01726
$704$	0	0
$705$	−24.0000	−0.903892
$706$	0	0
$707$	− 24.0000i	− 0.902613i
$708$	0	0
$709$	− 42.0000i	− 1.57734i	−0.614815	−	0.788672i	$-0.710769\pi$
$709$	− 42.0000i	− 1.57734i	0.614815	−	0.788672i	$-0.289231\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	32.0000i	1.19841i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 20.0000i	− 0.746914i
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	− 32.0000i	− 1.19174i
$722$	0	0
$723$	14.0000i	0.520666i
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	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
$731$	−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
$735$	18.0000i	0.663940i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	16.0000i	0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$	16.0000i	0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 4.00000i	− 0.146746i	−0.997305	−	0.0733729i	$-0.976624\pi$
$743$	− 4.00000i	− 0.146746i	0.997305	−	0.0733729i	$-0.0233763\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	0	0
$747$	0	0
$748$	0	0
$749$	16.0000i	0.584627i
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	28.0000	1.02038
$754$	0	0
$755$	24.0000	0.873449
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 46.0000i	− 1.66750i	−0.552143	−	0.833749i	$-0.686190\pi$
$761$	− 46.0000i	− 1.66750i	0.552143	−	0.833749i	$-0.313810\pi$
$762$	0	0
$763$	−72.0000	−2.60658
$764$	0	0
$765$	− 4.00000i	− 0.144620i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	14.0000i	0.504853i	0.967616	+	0.252426i	$0.0812286\pi$
$769$	14.0000i	0.504853i	−0.967616	+	0.252426i	$0.918771\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	− 14.0000i	− 0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$	− 14.0000i	− 0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$	0	0
$775$	− 4.00000i	− 0.143684i
$776$	0	0
$777$	−40.0000	−1.43499
$778$	0	0
$779$	−16.0000	−0.573259
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	− 4.00000i	− 0.142766i
$786$	0	0
$787$	16.0000i	0.570338i	0.958477	+	0.285169i	$0.0920498\pi$
$787$	16.0000i	0.570338i	−0.958477	+	0.285169i	$0.907950\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	72.0000i	2.56003i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 12.0000i	− 0.425596i
$796$	0	0
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	0	0
$799$	24.0000i	0.849059i
$800$	0	0
$801$	− 14.0000i	− 0.494666i
$802$	0	0
$803$	0	0
$804$	0	0
$805$	64.0000	2.25570
$806$	0	0
$807$	18.0000	0.633630
$808$	0	0
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	0	0
$811$	− 48.0000i	− 1.68551i	−0.538299	−	0.842754i	$-0.680933\pi$
$811$	− 48.0000i	− 1.68551i	0.538299	−	0.842754i	$-0.319067\pi$
$812$	0	0
$813$	20.0000i	0.701431i
$814$	0	0
$815$	32.0000	1.12091
$816$	0	0
$817$	− 32.0000i	− 1.11954i
$818$	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
$823$	−56.0000	−1.95204	−0.976019	−	0.217687i	$-0.930149\pi$
$823$	−56.0000	−1.95204	−0.976019	+	0.217687i	$0.930149\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 16.0000i	− 0.556375i	−0.960527	−	0.278187i	$-0.910266\pi$
$827$	− 16.0000i	− 0.556375i	0.960527	−	0.278187i	$-0.0897336\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	−24.0000	−0.830554
$836$	0	0
$837$	4.00000i	0.138260i
$838$	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
$841$	−25.0000	−0.862069
$842$	0	0
$843$	30.0000i	1.03325i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	44.0000i	1.51186i
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	80.0000i	2.74236i
$852$	0	0
$853$	22.0000i	0.753266i	0.926363	+	0.376633i	$0.122918\pi$
$853$	22.0000i	0.753266i	−0.926363	+	0.376633i	$0.877082\pi$
$854$	0	0
$855$	16.0000	0.547188
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	36.0000i	1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$	36.0000i	1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$	0	0
$865$	36.0000i	1.22404i
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	−48.0000	−1.62270
$876$	0	0
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	0	0
$879$	2.00000i	0.0674583i
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−96.0000	−3.21252
$894$	0	0
$895$	− 40.0000i	− 1.33705i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.00000i	0.266815i
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	− 16.0000i	− 0.532447i
$904$	0	0
$905$	20.0000i	0.664822i
$906$	0	0
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	4.00000i	0.132236i
$916$	0	0
$917$	48.0000i	1.58510i
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	32.0000i	1.05444i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 10.0000i	− 0.328798i
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	− 42.0000i	− 1.37798i	−0.724773	−	0.688988i	$-0.758055\pi$
$929$	− 42.0000i	− 1.37798i	0.724773	−	0.688988i	$-0.241945\pi$
$930$	0	0
$931$	72.0000i	2.35970i
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	− 22.0000i	− 0.717180i	−0.933495	−	0.358590i	$-0.883258\pi$
$941$	− 22.0000i	− 0.717180i	0.933495	−	0.358590i	$-0.116742\pi$
$942$	0	0
$943$	16.0000i	0.521032i
$944$	0	0
$945$	8.00000	0.260240
$946$	0	0
$947$	40.0000i	1.29983i	0.760009	+	0.649913i	$0.225195\pi$
$947$	40.0000i	1.29983i	−0.760009	+	0.649913i	$0.774805\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 10.0000i	− 0.324272i
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	0	0
$955$	− 48.0000i	− 1.55324i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.00000	−0.258333
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	28.0000	0.901352
$966$	0	0
$967$	− 12.0000i	− 0.385894i	−0.981209	−	0.192947i	$-0.938195\pi$
$967$	− 12.0000i	− 0.385894i	0.981209	−	0.192947i	$-0.0618045\pi$
$968$	0	0
$969$	− 16.0000i	− 0.513994i
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	16.0000i	0.512936i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 10.0000i	− 0.319928i	−0.987123	−	0.159964i	$-0.948862\pi$
$977$	− 10.0000i	− 0.319928i	0.987123	−	0.159964i	$-0.0511379\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	18.0000i	0.574696i
$982$	0	0
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	0	0
$985$	28.0000	0.892154
$986$	0	0
$987$	−48.0000	−1.52786
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	0	0
$993$	16.0000i	0.507745i
$994$	0	0
$995$	− 16.0000i	− 0.507234i
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	0	0
$999$	10.0000i	0.316386i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.c.337.2		2
13.5	odd	4		4056.2.a.g.1.1		1
13.8	odd	4		312.2.a.a.1.1	✓	1
13.12	even	2	inner	4056.2.c.c.337.1		2
39.8	even	4		936.2.a.h.1.1		1
52.31	even	4		8112.2.a.bg.1.1		1
52.47	even	4		624.2.a.f.1.1		1
65.34	odd	4		7800.2.a.n.1.1		1
104.21	odd	4		2496.2.a.bb.1.1		1
104.99	even	4		2496.2.a.k.1.1		1
156.47	odd	4		1872.2.a.n.1.1		1
312.125	even	4		7488.2.a.t.1.1		1
312.203	odd	4		7488.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.a.a.1.1	✓	1	13.8	odd	4
624.2.a.f.1.1		1	52.47	even	4
936.2.a.h.1.1		1	39.8	even	4
1872.2.a.n.1.1		1	156.47	odd	4
2496.2.a.k.1.1		1	104.99	even	4
2496.2.a.bb.1.1		1	104.21	odd	4
4056.2.a.g.1.1		1	13.5	odd	4
4056.2.c.c.337.1		2	13.12	even	2	inner
4056.2.c.c.337.2		2	1.1	even	1	trivial
7488.2.a.i.1.1		1	312.203	odd	4
7488.2.a.t.1.1		1	312.125	even	4
7800.2.a.n.1.1		1	65.34	odd	4
8112.2.a.bg.1.1		1	52.31	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.00000i q^{5} +4.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.00000i q^{5} +4.00000i q^{7} +1.00000 q^{9} -2.00000i q^{15} -2.00000 q^{17} -8.00000i q^{19} -4.00000i q^{21} -8.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} -4.00000i q^{31} -8.00000 q^{35} -10.0000i q^{37} -2.00000i q^{41} +4.00000 q^{43} +2.00000i q^{45} -12.0000i q^{47} -9.00000 q^{49} +2.00000 q^{51} +6.00000 q^{53} +8.00000i q^{57} -2.00000 q^{61} +4.00000i q^{63} -8.00000i q^{67} +8.00000 q^{69} +12.0000i q^{71} +10.0000i q^{73} -1.00000 q^{75} -8.00000 q^{79} +1.00000 q^{81} -4.00000i q^{85} +2.00000 q^{87} -14.0000i q^{89} +4.00000i q^{93} +16.0000 q^{95} -2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} - 4 q^{17} - 16 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 16 q^{35} + 8 q^{43} - 18 q^{49} + 4 q^{51} + 12 q^{53} - 4 q^{61} + 16 q^{69} - 2 q^{75} - 16 q^{79} + 2 q^{81} + 4 q^{87} + 32 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 - 4 * q^17 - 16 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 - 16 * q^35 + 8 * q^43 - 18 * q^49 + 4 * q^51 + 12 * q^53 - 4 * q^61 + 16 * q^69 - 2 * q^75 - 16 * q^79 + 2 * q^81 + 4 * q^87 + 32 * q^95

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