LMFDB - Embedded newform 3042.2.b.f.1351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1351,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3042.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.b (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:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.f.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{10} +2.00000i q^{11} -1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -6.00000i q^{19} +1.00000i q^{20} -2.00000 q^{22} -4.00000 q^{23} +4.00000 q^{25} -1.00000i q^{28} -2.00000 q^{29} -4.00000i q^{31} +1.00000i q^{32} -3.00000i q^{34} +1.00000 q^{35} +3.00000i q^{37} +6.00000 q^{38} -1.00000 q^{40} +5.00000 q^{43} -2.00000i q^{44} -4.00000i q^{46} -13.0000i q^{47} +6.00000 q^{49} +4.00000i q^{50} -12.0000 q^{53} +2.00000 q^{55} +1.00000 q^{56} -2.00000i q^{58} +10.0000i q^{59} -8.00000 q^{61} +4.00000 q^{62} -1.00000 q^{64} +2.00000i q^{67} +3.00000 q^{68} +1.00000i q^{70} -5.00000i q^{71} -10.0000i q^{73} -3.00000 q^{74} +6.00000i q^{76} -2.00000 q^{77} -4.00000 q^{79} -1.00000i q^{80} +3.00000i q^{85} +5.00000i q^{86} +2.00000 q^{88} -6.00000i q^{89} +4.00000 q^{92} +13.0000 q^{94} -6.00000 q^{95} -14.0000i q^{97} +6.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 2 q^{10} - 2 q^{14} + 2 q^{16} - 6 q^{17} - 4 q^{22} - 8 q^{23} + 8 q^{25} - 4 q^{29} + 2 q^{35} + 12 q^{38} - 2 q^{40} + 10 q^{43} + 12 q^{49} - 24 q^{53} + 4 q^{55} + 2 q^{56} - 16 q^{61} + 8 q^{62} - 2 q^{64} + 6 q^{68} - 6 q^{74} - 4 q^{77} - 8 q^{79} + 4 q^{88} + 8 q^{92} + 26 q^{94} - 12 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^10 - 2 * q^14 + 2 * q^16 - 6 * q^17 - 4 * q^22 - 8 * q^23 + 8 * q^25 - 4 * q^29 + 2 * q^35 + 12 * q^38 - 2 * q^40 + 10 * q^43 + 12 * q^49 - 24 * q^53 + 4 * q^55 + 2 * q^56 - 16 * q^61 + 8 * q^62 - 2 * q^64 + 6 * q^68 - 6 * q^74 - 4 * q^77 - 8 * q^79 + 4 * q^88 + 8 * q^92 + 26 * q^94 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i	−0.974679	−	0.223607i	$-0.928217\pi$
$5$	− 1.00000i	− 0.447214i	0.974679	−	0.223607i	$-0.0717831\pi$
$6$	0	0
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	1.00000	0.316228
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	− 6.00000i	− 1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$	− 6.00000i	− 1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$	1.00000i	0.223607i
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−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
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	−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	$-0.116954\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	− 3.00000i	− 0.514496i
$35$	1.00000	0.169031
$36$	0	0
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	− 2.00000i	− 0.301511i
$45$	0	0
$46$	− 4.00000i	− 0.589768i
$47$	− 13.0000i	− 1.89624i	−0.317905	−	0.948122i	$-0.602979\pi$
$47$	− 13.0000i	− 1.89624i	0.317905	−	0.948122i	$-0.397021\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	4.00000i	0.565685i
$51$	0	0
$52$	0	0
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	1.00000	0.133631
$57$	0	0
$58$	− 2.00000i	− 0.262613i
$59$	10.0000i	1.30189i	0.759125	+	0.650945i	$0.225627\pi$
$59$	10.0000i	1.30189i	−0.759125	+	0.650945i	$0.774373\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	1.00000i	0.119523i
$71$	− 5.00000i	− 0.593391i	−0.954972	−	0.296695i	$-0.904115\pi$
$71$	− 5.00000i	− 0.593391i	0.954972	−	0.296695i	$-0.0958846\pi$
$72$	0	0
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	6.00000i	0.688247i
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	− 1.00000i	− 0.111803i
$81$	0	0
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	3.00000i	0.325396i
$86$	5.00000i	0.539164i
$87$	0	0
$88$	2.00000	0.213201
$89$	− 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
$92$	4.00000	0.417029
$93$	0	0
$94$	13.0000	1.34085
$95$	−6.00000	−0.615587
$96$	0	0
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	6.00000i	0.606092i
$99$	0	0
$100$	−4.00000	−0.400000
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	− 12.0000i	− 1.16554i
$107$	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
$109$	− 19.0000i	− 1.81987i	−0.414751	−	0.909935i	$-0.636131\pi$
$109$	− 19.0000i	− 1.81987i	0.414751	−	0.909935i	$-0.363869\pi$
$110$	2.00000i	0.190693i
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	4.00000i	0.373002i
$116$	2.00000	0.185695
$117$	0	0
$118$	−10.0000	−0.920575
$119$	− 3.00000i	− 0.275010i
$120$	0	0
$121$	7.00000	0.636364
$122$	− 8.00000i	− 0.724286i
$123$	0	0
$124$	4.00000i	0.359211i
$125$	− 9.00000i	− 0.804984i
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	−2.00000	−0.172774
$135$	0	0
$136$	3.00000i	0.257248i
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	5.00000	0.419591
$143$	0	0
$144$	0	0
$145$	2.00000i	0.166091i
$146$	10.0000	0.827606
$147$	0	0
$148$	− 3.00000i	− 0.246598i
$149$	− 18.0000i	− 1.47462i	−0.675556	−	0.737309i	$-0.736096\pi$
$149$	− 18.0000i	− 1.47462i	0.675556	−	0.737309i	$-0.263904\pi$
$150$	0	0
$151$	− 9.00000i	− 0.732410i	−0.930534	−	0.366205i	$-0.880657\pi$
$151$	− 9.00000i	− 0.732410i	0.930534	−	0.366205i	$-0.119343\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	− 2.00000i	− 0.161165i
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	− 4.00000i	− 0.318223i
$159$	0	0
$160$	1.00000	0.0790569
$161$	− 4.00000i	− 0.315244i
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	0	0
$170$	−3.00000	−0.230089
$171$	0	0
$172$	−5.00000	−0.381246
$173$	20.0000	1.52057	0.760286	−	0.649589i	$-0.225059\pi$
$173$	20.0000	1.52057	0.760286	+	0.649589i	$0.225059\pi$
$174$	0	0
$175$	4.00000i	0.302372i
$176$	2.00000i	0.150756i
$177$	0	0
$178$	6.00000	0.449719
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	4.00000i	0.294884i
$185$	3.00000	0.220564
$186$	0	0
$187$	− 6.00000i	− 0.438763i
$188$	13.0000i	0.948122i
$189$	0	0
$190$	− 6.00000i	− 0.435286i
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	0	0
$193$	− 16.0000i	− 1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$	− 16.0000i	− 1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−6.00000	−0.428571
$197$	9.00000i	0.641223i	0.947211	+	0.320612i	$0.103888\pi$
$197$	9.00000i	0.641223i	−0.947211	+	0.320612i	$0.896112\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	− 4.00000i	− 0.282843i
$201$	0	0
$202$	4.00000i	0.281439i
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	0	0
$206$	8.00000i	0.557386i
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	12.0000	0.824163
$213$	0	0
$214$	4.00000i	0.273434i
$215$	− 5.00000i	− 0.340997i
$216$	0	0
$217$	4.00000	0.271538
$218$	19.0000	1.28684
$219$	0	0
$220$	−2.00000	−0.134840
$221$	0	0
$222$	0	0
$223$	21.0000i	1.40626i	0.711059	+	0.703132i	$0.248216\pi$
$223$	21.0000i	1.40626i	−0.711059	+	0.703132i	$0.751784\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	− 2.00000i	− 0.133038i
$227$	− 24.0000i	− 1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$	− 24.0000i	− 1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$	0	0
$229$	− 15.0000i	− 0.991228i	−0.868543	−	0.495614i	$-0.834943\pi$
$229$	− 15.0000i	− 0.991228i	0.868543	−	0.495614i	$-0.165057\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	2.00000i	0.131306i
$233$	−11.0000	−0.720634	−0.360317	−	0.932830i	$-0.617331\pi$
$233$	−11.0000	−0.720634	−0.360317	+	0.932830i	$0.617331\pi$
$234$	0	0
$235$	−13.0000	−0.848026
$236$	− 10.0000i	− 0.650945i
$237$	0	0
$238$	3.00000	0.194461
$239$	9.00000i	0.582162i	0.956698	+	0.291081i	$0.0940149\pi$
$239$	9.00000i	0.582162i	−0.956698	+	0.291081i	$0.905985\pi$
$240$	0	0
$241$	18.0000i	1.15948i	0.814801	+	0.579741i	$0.196846\pi$
$241$	18.0000i	1.15948i	−0.814801	+	0.579741i	$0.803154\pi$
$242$	7.00000i	0.449977i
$243$	0	0
$244$	8.00000	0.512148
$245$	− 6.00000i	− 0.383326i
$246$	0	0
$247$	0	0
$248$	−4.00000	−0.254000
$249$	0	0
$250$	9.00000	0.569210
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	− 16.0000i	− 1.00393i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	0	0
$262$	1.00000i	0.0617802i
$263$	−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
$265$	12.0000i	0.737154i
$266$	6.00000i	0.367884i
$267$	0	0
$268$	− 2.00000i	− 0.122169i
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	13.0000i	0.789694i	0.918747	+	0.394847i	$0.129202\pi$
$271$	13.0000i	0.789694i	−0.918747	+	0.394847i	$0.870798\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	12.0000	0.724947
$275$	8.00000i	0.482418i
$276$	0	0
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	7.00000i	0.419832i
$279$	0	0
$280$	− 1.00000i	− 0.0597614i
$281$	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
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	5.00000i	0.296695i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−2.00000	−0.117444
$291$	0	0
$292$	10.0000i	0.585206i
$293$	− 7.00000i	− 0.408944i	−0.978872	−	0.204472i	$-0.934452\pi$
$293$	− 7.00000i	− 0.408944i	0.978872	−	0.204472i	$-0.0655478\pi$
$294$	0	0
$295$	10.0000	0.582223
$296$	3.00000	0.174371
$297$	0	0
$298$	18.0000	1.04271
$299$	0	0
$300$	0	0
$301$	5.00000i	0.288195i
$302$	9.00000	0.517892
$303$	0	0
$304$	− 6.00000i	− 0.344124i
$305$	8.00000i	0.458079i
$306$	0	0
$307$	14.0000i	0.799022i	0.916728	+	0.399511i	$0.130820\pi$
$307$	14.0000i	0.799022i	−0.916728	+	0.399511i	$0.869180\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	− 4.00000i	− 0.227185i
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	− 10.0000i	− 0.564333i
$315$	0	0
$316$	4.00000	0.225018
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	− 4.00000i	− 0.223957i
$320$	1.00000i	0.0559017i
$321$	0	0
$322$	4.00000	0.222911
$323$	18.0000i	1.00155i
$324$	0	0
$325$	0	0
$326$	4.00000	0.221540
$327$	0	0
$328$	0	0
$329$	13.0000	0.716713
$330$	0	0
$331$	4.00000i	0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$	4.00000i	0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.00000	0.109272
$336$	0	0
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	0	0
$339$	0	0
$340$	− 3.00000i	− 0.162698i
$341$	8.00000	0.433224
$342$	0	0
$343$	13.0000i	0.701934i
$344$	− 5.00000i	− 0.269582i
$345$	0	0
$346$	20.0000i	1.07521i
$347$	9.00000	0.483145	0.241573	−	0.970383i	$-0.422337\pi$
$347$	9.00000	0.483145	0.241573	+	0.970383i	$0.422337\pi$
$348$	0	0
$349$	7.00000i	0.374701i	0.982293	+	0.187351i	$0.0599901\pi$
$349$	7.00000i	0.374701i	−0.982293	+	0.187351i	$0.940010\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	−2.00000	−0.106600
$353$	4.00000i	0.212899i	0.994318	+	0.106449i	$0.0339482\pi$
$353$	4.00000i	0.212899i	−0.994318	+	0.106449i	$0.966052\pi$
$354$	0	0
$355$	−5.00000	−0.265372
$356$	6.00000i	0.317999i
$357$	0	0
$358$	− 9.00000i	− 0.475665i
$359$	− 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
$361$	−17.0000	−0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	3.00000i	0.155963i
$371$	− 12.0000i	− 0.623009i
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	−13.0000	−0.670424
$377$	0	0
$378$	0	0
$379$	− 16.0000i	− 0.821865i	−0.911666	−	0.410932i	$-0.865203\pi$
$379$	− 16.0000i	− 0.821865i	0.911666	−	0.410932i	$-0.134797\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	− 10.0000i	− 0.511645i
$383$	27.0000i	1.37964i	0.723983	+	0.689818i	$0.242309\pi$
$383$	27.0000i	1.37964i	−0.723983	+	0.689818i	$0.757691\pi$
$384$	0	0
$385$	2.00000i	0.101929i
$386$	16.0000	0.814379
$387$	0	0
$388$	14.0000i	0.710742i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	− 6.00000i	− 0.303046i
$393$	0	0
$394$	−9.00000	−0.453413
$395$	4.00000i	0.201262i
$396$	0	0
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	10.0000i	0.501255i
$399$	0	0
$400$	4.00000	0.200000
$401$	− 24.0000i	− 1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$	− 24.0000i	− 1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$	0	0
$403$	0	0
$404$	−4.00000	−0.199007
$405$	0	0
$406$	2.00000	0.0992583
$407$	−6.00000	−0.297409
$408$	0	0
$409$	− 4.00000i	− 0.197787i	−0.995098	−	0.0988936i	$-0.968470\pi$
$409$	− 4.00000i	− 0.197787i	0.995098	−	0.0988936i	$-0.0315304\pi$
$410$	0	0
$411$	0	0
$412$	−8.00000	−0.394132
$413$	−10.0000	−0.492068
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	12.0000i	0.586939i
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	5.00000i	0.243685i	0.992549	+	0.121843i	$0.0388803\pi$
$421$	5.00000i	0.243685i	−0.992549	+	0.121843i	$0.961120\pi$
$422$	23.0000i	1.11962i
$423$	0	0
$424$	12.0000i	0.582772i
$425$	−12.0000	−0.582086
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	−4.00000	−0.193347
$429$	0	0
$430$	5.00000	0.241121
$431$	33.0000i	1.58955i	0.606902	+	0.794777i	$0.292412\pi$
$431$	33.0000i	1.58955i	−0.606902	+	0.794777i	$0.707588\pi$
$432$	0	0
$433$	−7.00000	−0.336399	−0.168199	−	0.985753i	$-0.553795\pi$
$433$	−7.00000	−0.336399	−0.168199	+	0.985753i	$0.553795\pi$
$434$	4.00000i	0.192006i
$435$	0	0
$436$	19.0000i	0.909935i
$437$	24.0000i	1.14808i
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	− 2.00000i	− 0.0953463i
$441$	0	0
$442$	0	0
$443$	39.0000	1.85295	0.926473	−	0.376361i	$-0.122825\pi$
$443$	39.0000	1.85295	0.926473	+	0.376361i	$0.122825\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	−21.0000	−0.994379
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	26.0000i	1.22702i	0.789689	+	0.613508i	$0.210242\pi$
$449$	26.0000i	1.22702i	−0.789689	+	0.613508i	$0.789758\pi$
$450$	0	0
$451$	0	0
$452$	2.00000	0.0940721
$453$	0	0
$454$	24.0000	1.12638
$455$	0	0
$456$	0	0
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	15.0000	0.700904
$459$	0	0
$460$	− 4.00000i	− 0.186501i
$461$	− 21.0000i	− 0.978068i	−0.872265	−	0.489034i	$-0.837349\pi$
$461$	− 21.0000i	− 0.978068i	0.872265	−	0.489034i	$-0.162651\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	− 11.0000i	− 0.509565i
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	− 13.0000i	− 0.599645i
$471$	0	0
$472$	10.0000	0.460287
$473$	10.0000i	0.459800i
$474$	0	0
$475$	− 24.0000i	− 1.10120i
$476$	3.00000i	0.137505i
$477$	0	0
$478$	−9.00000	−0.411650
$479$	3.00000i	0.137073i	0.997649	+	0.0685367i	$0.0218330\pi$
$479$	3.00000i	0.137073i	−0.997649	+	0.0685367i	$0.978167\pi$
$480$	0	0
$481$	0	0
$482$	−18.0000	−0.819878
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−14.0000	−0.635707
$486$	0	0
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	8.00000i	0.362143i
$489$	0	0
$490$	6.00000	0.271052
$491$	−5.00000	−0.225647	−0.112823	−	0.993615i	$-0.535989\pi$
$491$	−5.00000	−0.225647	−0.112823	+	0.993615i	$0.535989\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	0	0
$495$	0	0
$496$	− 4.00000i	− 0.179605i
$497$	5.00000	0.224281
$498$	0	0
$499$	32.0000i	1.43252i	0.697835	+	0.716258i	$0.254147\pi$
$499$	32.0000i	1.43252i	−0.697835	+	0.716258i	$0.745853\pi$
$500$	9.00000i	0.402492i
$501$	0	0
$502$	0	0
$503$	14.0000	0.624229	0.312115	−	0.950044i	$-0.398963\pi$
$503$	14.0000	0.624229	0.312115	+	0.950044i	$0.398963\pi$
$504$	0	0
$505$	− 4.00000i	− 0.177998i
$506$	8.00000	0.355643
$507$	0	0
$508$	16.0000	0.709885
$509$	34.0000i	1.50702i	0.657434	+	0.753512i	$0.271642\pi$
$509$	34.0000i	1.50702i	−0.657434	+	0.753512i	$0.728358\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 15.0000i	− 0.661622i
$515$	− 8.00000i	− 0.352522i
$516$	0	0
$517$	26.0000	1.14348
$518$	− 3.00000i	− 0.131812i
$519$	0	0
$520$	0	0
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	0	0
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	−1.00000	−0.0436852
$525$	0	0
$526$	− 12.0000i	− 0.523225i
$527$	12.0000i	0.522728i
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−12.0000	−0.521247
$531$	0	0
$532$	−6.00000	−0.260133
$533$	0	0
$534$	0	0
$535$	− 4.00000i	− 0.172935i
$536$	2.00000	0.0863868
$537$	0	0
$538$	24.0000i	1.03471i
$539$	12.0000i	0.516877i
$540$	0	0
$541$	17.0000i	0.730887i	0.930834	+	0.365444i	$0.119083\pi$
$541$	17.0000i	0.730887i	−0.930834	+	0.365444i	$0.880917\pi$
$542$	−13.0000	−0.558398
$543$	0	0
$544$	− 3.00000i	− 0.128624i
$545$	−19.0000	−0.813871
$546$	0	0
$547$	37.0000	1.58201	0.791003	−	0.611812i	$-0.209559\pi$
$547$	37.0000	1.58201	0.791003	+	0.611812i	$0.209559\pi$
$548$	12.0000i	0.512615i
$549$	0	0
$550$	−8.00000	−0.341121
$551$	12.0000i	0.511217i
$552$	0	0
$553$	− 4.00000i	− 0.170097i
$554$	− 12.0000i	− 0.509831i
$555$	0	0
$556$	−7.00000	−0.296866
$557$	− 33.0000i	− 1.39825i	−0.714997	−	0.699127i	$-0.753572\pi$
$557$	− 33.0000i	− 1.39825i	0.714997	−	0.699127i	$-0.246428\pi$
$558$	0	0
$559$	0	0
$560$	1.00000	0.0422577
$561$	0	0
$562$	−26.0000	−1.09674
$563$	11.0000	0.463595	0.231797	−	0.972764i	$-0.425539\pi$
$563$	11.0000	0.463595	0.231797	+	0.972764i	$0.425539\pi$
$564$	0	0
$565$	2.00000i	0.0841406i
$566$	− 4.00000i	− 0.168133i
$567$	0	0
$568$	−5.00000	−0.209795
$569$	31.0000	1.29959	0.649794	−	0.760111i	$-0.274855\pi$
$569$	31.0000	1.29959	0.649794	+	0.760111i	$0.274855\pi$
$570$	0	0
$571$	−33.0000	−1.38101	−0.690504	−	0.723329i	$-0.742611\pi$
$571$	−33.0000	−1.38101	−0.690504	+	0.723329i	$0.742611\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−16.0000	−0.667246
$576$	0	0
$577$	− 18.0000i	− 0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$	− 18.0000i	− 0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	− 2.00000i	− 0.0830455i
$581$	0	0
$582$	0	0
$583$	− 24.0000i	− 0.993978i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	7.00000	0.289167
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	10.0000i	0.411693i
$591$	0	0
$592$	3.00000i	0.123299i
$593$	22.0000i	0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$	22.0000i	0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	18.0000i	0.737309i
$597$	0	0
$598$	0	0
$599$	2.00000	0.0817178	0.0408589	−	0.999165i	$-0.486991\pi$
$599$	2.00000	0.0817178	0.0408589	+	0.999165i	$0.486991\pi$
$600$	0	0
$601$	−35.0000	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$602$	−5.00000	−0.203785
$603$	0	0
$604$	9.00000i	0.366205i
$605$	− 7.00000i	− 0.284590i
$606$	0	0
$607$	6.00000	0.243532	0.121766	−	0.992559i	$-0.461144\pi$
$607$	6.00000	0.243532	0.121766	+	0.992559i	$0.461144\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	−8.00000	−0.323911
$611$	0	0
$612$	0	0
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	−14.0000	−0.564994
$615$	0	0
$616$	2.00000i	0.0805823i
$617$	16.0000i	0.644136i	0.946717	+	0.322068i	$0.104378\pi$
$617$	16.0000i	0.644136i	−0.946717	+	0.322068i	$0.895622\pi$
$618$	0	0
$619$	4.00000i	0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$	4.00000i	0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	18.0000i	0.721734i
$623$	6.00000	0.240385
$624$	0	0
$625$	11.0000	0.440000
$626$	− 1.00000i	− 0.0399680i
$627$	0	0
$628$	10.0000	0.399043
$629$	− 9.00000i	− 0.358854i
$630$	0	0
$631$	− 5.00000i	− 0.199047i	−0.995035	−	0.0995234i	$-0.968268\pi$
$631$	− 5.00000i	− 0.199047i	0.995035	−	0.0995234i	$-0.0317318\pi$
$632$	4.00000i	0.159111i
$633$	0	0
$634$	18.0000	0.714871
$635$	16.0000i	0.634941i
$636$	0	0
$637$	0	0
$638$	4.00000	0.158362
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	4.00000i	0.157622i
$645$	0	0
$646$	−18.0000	−0.708201
$647$	−38.0000	−1.49393	−0.746967	−	0.664861i	$-0.768491\pi$
$647$	−38.0000	−1.49393	−0.746967	+	0.664861i	$0.768491\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	0	0
$655$	− 1.00000i	− 0.0390732i
$656$	0	0
$657$	0	0
$658$	13.0000i	0.506793i
$659$	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	$-0.0623011\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	0	0
$665$	− 6.00000i	− 0.232670i
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	0	0
$670$	2.00000i	0.0772667i
$671$	− 16.0000i	− 0.617673i
$672$	0	0
$673$	−37.0000	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$673$	−37.0000	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$674$	− 23.0000i	− 0.885927i
$675$	0	0
$676$	0	0
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	3.00000	0.115045
$681$	0	0
$682$	8.00000i	0.306336i
$683$	44.0000i	1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$	44.0000i	1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	−13.0000	−0.496342
$687$	0	0
$688$	5.00000	0.190623
$689$	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	$0.951375\pi$
$692$	−20.0000	−0.760286
$693$	0	0
$694$	9.00000i	0.341635i
$695$	− 7.00000i	− 0.265525i
$696$	0	0
$697$	0	0
$698$	−7.00000	−0.264954
$699$	0	0
$700$	− 4.00000i	− 0.151186i
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	18.0000	0.678883
$704$	− 2.00000i	− 0.0753778i
$705$	0	0
$706$	−4.00000	−0.150542
$707$	4.00000i	0.150435i
$708$	0	0
$709$	38.0000i	1.42712i	0.700594	+	0.713560i	$0.252918\pi$
$709$	38.0000i	1.42712i	−0.700594	+	0.713560i	$0.747082\pi$
$710$	− 5.00000i	− 0.187647i
$711$	0	0
$712$	−6.00000	−0.224860
$713$	16.0000i	0.599205i
$714$	0	0
$715$	0	0
$716$	9.00000	0.336346
$717$	0	0
$718$	24.0000	0.895672
$719$	−22.0000	−0.820462	−0.410231	−	0.911982i	$-0.634552\pi$
$719$	−22.0000	−0.820462	−0.410231	+	0.911982i	$0.634552\pi$
$720$	0	0
$721$	8.00000i	0.297936i
$722$	− 17.0000i	− 0.632674i
$723$	0	0
$724$	0	0
$725$	−8.00000	−0.297113
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	0	0
$730$	− 10.0000i	− 0.370117i
$731$	−15.0000	−0.554795
$732$	0	0
$733$	43.0000i	1.58824i	0.607760	+	0.794121i	$0.292068\pi$
$733$	43.0000i	1.58824i	−0.607760	+	0.794121i	$0.707932\pi$
$734$	− 10.0000i	− 0.369107i
$735$	0	0
$736$	− 4.00000i	− 0.147442i
$737$	−4.00000	−0.147342
$738$	0	0
$739$	12.0000i	0.441427i	0.975339	+	0.220714i	$0.0708386\pi$
$739$	12.0000i	0.441427i	−0.975339	+	0.220714i	$0.929161\pi$
$740$	−3.00000	−0.110282
$741$	0	0
$742$	12.0000	0.440534
$743$	− 47.0000i	− 1.72426i	−0.506685	−	0.862131i	$-0.669129\pi$
$743$	− 47.0000i	− 1.72426i	0.506685	−	0.862131i	$-0.330871\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	− 4.00000i	− 0.146450i
$747$	0	0
$748$	6.00000i	0.219382i
$749$	4.00000i	0.146157i
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	− 13.0000i	− 0.474061i
$753$	0	0
$754$	0	0
$755$	−9.00000	−0.327544
$756$	0	0
$757$	−12.0000	−0.436147	−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000	−0.436147	−0.218074	+	0.975932i	$0.569977\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	6.00000i	0.217643i
$761$	− 6.00000i	− 0.217500i	−0.994069	−	0.108750i	$-0.965315\pi$
$761$	− 6.00000i	− 0.217500i	0.994069	−	0.108750i	$-0.0346848\pi$
$762$	0	0
$763$	19.0000	0.687846
$764$	10.0000	0.361787
$765$	0	0
$766$	−27.0000	−0.975550
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	−2.00000	−0.0720750
$771$	0	0
$772$	16.0000i	0.575853i
$773$	11.0000i	0.395643i	0.980238	+	0.197821i	$0.0633866\pi$
$773$	11.0000i	0.395643i	−0.980238	+	0.197821i	$0.936613\pi$
$774$	0	0
$775$	− 16.0000i	− 0.574737i
$776$	−14.0000	−0.502571
$777$	0	0
$778$	− 30.0000i	− 1.07555i
$779$	0	0
$780$	0	0
$781$	10.0000	0.357828
$782$	12.0000i	0.429119i
$783$	0	0
$784$	6.00000	0.214286
$785$	10.0000i	0.356915i
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	− 9.00000i	− 0.320612i
$789$	0	0
$790$	−4.00000	−0.142314
$791$	− 2.00000i	− 0.0711118i
$792$	0	0
$793$	0	0
$794$	22.0000	0.780751
$795$	0	0
$796$	−10.0000	−0.354441
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	39.0000i	1.37972i
$800$	4.00000i	0.141421i
$801$	0	0
$802$	24.0000	0.847469
$803$	20.0000	0.705785
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	0	0
$808$	− 4.00000i	− 0.140720i
$809$	9.00000	0.316423	0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000	0.316423	0.158212	+	0.987405i	$0.449427\pi$
$810$	0	0
$811$	28.0000i	0.983213i	0.870817	+	0.491606i	$0.163590\pi$
$811$	28.0000i	0.983213i	−0.870817	+	0.491606i	$0.836410\pi$
$812$	2.00000i	0.0701862i
$813$	0	0
$814$	− 6.00000i	− 0.210300i
$815$	−4.00000	−0.140114
$816$	0	0
$817$	− 30.0000i	− 1.04957i
$818$	4.00000	0.139857
$819$	0	0
$820$	0	0
$821$	− 25.0000i	− 0.872506i	−0.899824	−	0.436253i	$-0.856305\pi$
$821$	− 25.0000i	− 0.872506i	0.899824	−	0.436253i	$-0.143695\pi$
$822$	0	0
$823$	−54.0000	−1.88232	−0.941161	−	0.337959i	$-0.890263\pi$
$823$	−54.0000	−1.88232	−0.941161	+	0.337959i	$0.890263\pi$
$824$	− 8.00000i	− 0.278693i
$825$	0	0
$826$	− 10.0000i	− 0.347945i
$827$	− 30.0000i	− 1.04320i	−0.853189	−	0.521601i	$-0.825335\pi$
$827$	− 30.0000i	− 1.04320i	0.853189	−	0.521601i	$-0.174665\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	0	0
$838$	− 21.0000i	− 0.725433i
$839$	− 56.0000i	− 1.93333i	−0.256036	−	0.966667i	$-0.582416\pi$
$839$	− 56.0000i	− 1.93333i	0.256036	−	0.966667i	$-0.417584\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−5.00000	−0.172311
$843$	0	0
$844$	−23.0000	−0.791693
$845$	0	0
$846$	0	0
$847$	7.00000i	0.240523i
$848$	−12.0000	−0.412082
$849$	0	0
$850$	− 12.0000i	− 0.411597i
$851$	− 12.0000i	− 0.411355i
$852$	0	0
$853$	49.0000i	1.67773i	0.544341	+	0.838864i	$0.316780\pi$
$853$	49.0000i	1.67773i	−0.544341	+	0.838864i	$0.683220\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	− 4.00000i	− 0.136717i
$857$	46.0000	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$857$	46.0000	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	5.00000i	0.170499i
$861$	0	0
$862$	−33.0000	−1.12398
$863$	− 11.0000i	− 0.374444i	−0.982318	−	0.187222i	$-0.940052\pi$
$863$	− 11.0000i	− 0.374444i	0.982318	−	0.187222i	$-0.0599484\pi$
$864$	0	0
$865$	− 20.0000i	− 0.680020i
$866$	− 7.00000i	− 0.237870i
$867$	0	0
$868$	−4.00000	−0.135769
$869$	− 8.00000i	− 0.271381i
$870$	0	0
$871$	0	0
$872$	−19.0000	−0.643421
$873$	0	0
$874$	−24.0000	−0.811812
$875$	9.00000	0.304256
$876$	0	0
$877$	39.0000i	1.31694i	0.752609	+	0.658468i	$0.228795\pi$
$877$	39.0000i	1.31694i	−0.752609	+	0.658468i	$0.771205\pi$
$878$	22.0000i	0.742464i
$879$	0	0
$880$	2.00000	0.0674200
$881$	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
$883$	47.0000	1.58168	0.790838	−	0.612026i	$-0.209645\pi$
$883$	47.0000	1.58168	0.790838	+	0.612026i	$0.209645\pi$
$884$	0	0
$885$	0	0
$886$	39.0000i	1.31023i
$887$	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
$889$	− 16.0000i	− 0.536623i
$890$	− 6.00000i	− 0.201120i
$891$	0	0
$892$	− 21.0000i	− 0.703132i
$893$	−78.0000	−2.61017
$894$	0	0
$895$	9.00000i	0.300837i
$896$	1.00000	0.0334077
$897$	0	0
$898$	−26.0000	−0.867631
$899$	8.00000i	0.266815i
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	2.00000i	0.0665190i
$905$	0	0
$906$	0	0
$907$	9.00000	0.298840	0.149420	−	0.988774i	$-0.452259\pi$
$907$	9.00000	0.298840	0.149420	+	0.988774i	$0.452259\pi$
$908$	24.0000i	0.796468i
$909$	0	0
$910$	0	0
$911$	54.0000	1.78910	0.894550	−	0.446968i	$-0.147496\pi$
$911$	54.0000	1.78910	0.894550	+	0.446968i	$0.147496\pi$
$912$	0	0
$913$	0	0
$914$	10.0000	0.330771
$915$	0	0
$916$	15.0000i	0.495614i
$917$	1.00000i	0.0330229i
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	4.00000	0.131876
$921$	0	0
$922$	21.0000	0.691598
$923$	0	0
$924$	0	0
$925$	12.0000i	0.394558i
$926$	−16.0000	−0.525793
$927$	0	0
$928$	− 2.00000i	− 0.0656532i
$929$	− 36.0000i	− 1.18112i	−0.806993	−	0.590561i	$-0.798907\pi$
$929$	− 36.0000i	− 1.18112i	0.806993	−	0.590561i	$-0.201093\pi$
$930$	0	0
$931$	− 36.0000i	− 1.17985i
$932$	11.0000	0.360317
$933$	0	0
$934$	20.0000i	0.654420i
$935$	−6.00000	−0.196221
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	− 2.00000i	− 0.0653023i
$939$	0	0
$940$	13.0000	0.424013
$941$	25.0000i	0.814977i	0.913210	+	0.407488i	$0.133595\pi$
$941$	25.0000i	0.814977i	−0.913210	+	0.407488i	$0.866405\pi$
$942$	0	0
$943$	0	0
$944$	10.0000i	0.325472i
$945$	0	0
$946$	−10.0000	−0.325128
$947$	18.0000i	0.584921i	0.956278	+	0.292461i	$0.0944741\pi$
$947$	18.0000i	0.584921i	−0.956278	+	0.292461i	$0.905526\pi$
$948$	0	0
$949$	0	0
$950$	24.0000	0.778663
$951$	0	0
$952$	−3.00000	−0.0972306
$953$	23.0000	0.745043	0.372522	−	0.928024i	$-0.378493\pi$
$953$	23.0000	0.745043	0.372522	+	0.928024i	$0.378493\pi$
$954$	0	0
$955$	10.0000i	0.323592i
$956$	− 9.00000i	− 0.291081i
$957$	0	0
$958$	−3.00000	−0.0969256
$959$	12.0000	0.387500
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	0	0
$964$	− 18.0000i	− 0.579741i
$965$	−16.0000	−0.515058
$966$	0	0
$967$	− 23.0000i	− 0.739630i	−0.929105	−	0.369815i	$-0.879421\pi$
$967$	− 23.0000i	− 0.739630i	0.929105	−	0.369815i	$-0.120579\pi$
$968$	− 7.00000i	− 0.224989i
$969$	0	0
$970$	− 14.0000i	− 0.449513i
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	0	0
$973$	7.00000i	0.224410i
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−8.00000	−0.256074
$977$	− 30.0000i	− 0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$	− 30.0000i	− 0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	6.00000i	0.191663i
$981$	0	0
$982$	− 5.00000i	− 0.159556i
$983$	31.0000i	0.988746i	0.869250	+	0.494373i	$0.164602\pi$
$983$	31.0000i	0.988746i	−0.869250	+	0.494373i	$0.835398\pi$
$984$	0	0
$985$	9.00000	0.286764
$986$	6.00000i	0.191079i
$987$	0	0
$988$	0	0
$989$	−20.0000	−0.635963
$990$	0	0
$991$	−30.0000	−0.952981	−0.476491	−	0.879180i	$-0.658091\pi$
$991$	−30.0000	−0.952981	−0.476491	+	0.879180i	$0.658091\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	5.00000i	0.158590i
$995$	− 10.0000i	− 0.317021i
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−32.0000	−1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.f.1351.2		2
3.2	odd	2		338.2.b.a.337.1		2
12.11	even	2		2704.2.f.j.337.2		2
13.5	odd	4		3042.2.a.l.1.1		1
13.8	odd	4		234.2.a.b.1.1		1
13.12	even	2	inner	3042.2.b.f.1351.1		2
39.2	even	12		338.2.c.g.191.1		2
39.5	even	4		338.2.a.a.1.1		1
39.8	even	4		26.2.a.b.1.1	✓	1
39.11	even	12		338.2.c.c.191.1		2
39.17	odd	6		338.2.e.d.23.2		4
39.20	even	12		338.2.c.c.315.1		2
39.23	odd	6		338.2.e.d.147.1		4
39.29	odd	6		338.2.e.d.147.2		4
39.32	even	12		338.2.c.g.315.1		2
39.35	odd	6		338.2.e.d.23.1		4
39.38	odd	2		338.2.b.a.337.2		2
52.47	even	4		1872.2.a.m.1.1		1
65.8	even	4		5850.2.e.v.5149.2		2
65.34	odd	4		5850.2.a.bn.1.1		1
65.47	even	4		5850.2.e.v.5149.1		2
104.21	odd	4		7488.2.a.w.1.1		1
104.99	even	4		7488.2.a.v.1.1		1
117.34	odd	12		2106.2.e.t.1405.1		2
117.47	even	12		2106.2.e.h.1405.1		2
117.86	even	12		2106.2.e.h.703.1		2
117.112	odd	12		2106.2.e.t.703.1		2
156.47	odd	4		208.2.a.d.1.1		1
156.83	odd	4		2704.2.a.n.1.1		1
156.155	even	2		2704.2.f.j.337.1		2
195.8	odd	4		650.2.b.a.599.1		2
195.44	even	4		8450.2.a.y.1.1		1
195.47	odd	4		650.2.b.a.599.2		2
195.164	even	4		650.2.a.g.1.1		1
273.47	odd	12		1274.2.f.a.1145.1		2
273.86	even	12		1274.2.f.l.1145.1		2
273.125	odd	4		1274.2.a.o.1.1		1
273.164	odd	12		1274.2.f.a.79.1		2
273.242	even	12		1274.2.f.l.79.1		2
312.125	even	4		832.2.a.j.1.1		1
312.203	odd	4		832.2.a.a.1.1		1
429.164	odd	4		3146.2.a.a.1.1		1
624.125	even	4		3328.2.b.g.1665.1		2
624.203	odd	4		3328.2.b.k.1665.1		2
624.437	even	4		3328.2.b.g.1665.2		2
624.515	odd	4		3328.2.b.k.1665.2		2
663.203	even	4		7514.2.a.i.1.1		1
741.398	odd	4		9386.2.a.f.1.1		1
780.359	odd	4		5200.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.b.1.1	✓	1	39.8	even	4
208.2.a.d.1.1		1	156.47	odd	4
234.2.a.b.1.1		1	13.8	odd	4
338.2.a.a.1.1		1	39.5	even	4
338.2.b.a.337.1		2	3.2	odd	2
338.2.b.a.337.2		2	39.38	odd	2
338.2.c.c.191.1		2	39.11	even	12
338.2.c.c.315.1		2	39.20	even	12
338.2.c.g.191.1		2	39.2	even	12
338.2.c.g.315.1		2	39.32	even	12
338.2.e.d.23.1		4	39.35	odd	6
338.2.e.d.23.2		4	39.17	odd	6
338.2.e.d.147.1		4	39.23	odd	6
338.2.e.d.147.2		4	39.29	odd	6
650.2.a.g.1.1		1	195.164	even	4
650.2.b.a.599.1		2	195.8	odd	4
650.2.b.a.599.2		2	195.47	odd	4
832.2.a.a.1.1		1	312.203	odd	4
832.2.a.j.1.1		1	312.125	even	4
1274.2.a.o.1.1		1	273.125	odd	4
1274.2.f.a.79.1		2	273.164	odd	12
1274.2.f.a.1145.1		2	273.47	odd	12
1274.2.f.l.79.1		2	273.242	even	12
1274.2.f.l.1145.1		2	273.86	even	12
1872.2.a.m.1.1		1	52.47	even	4
2106.2.e.h.703.1		2	117.86	even	12
2106.2.e.h.1405.1		2	117.47	even	12
2106.2.e.t.703.1		2	117.112	odd	12
2106.2.e.t.1405.1		2	117.34	odd	12
2704.2.a.n.1.1		1	156.83	odd	4
2704.2.f.j.337.1		2	156.155	even	2
2704.2.f.j.337.2		2	12.11	even	2
3042.2.a.l.1.1		1	13.5	odd	4
3042.2.b.f.1351.1		2	13.12	even	2	inner
3042.2.b.f.1351.2		2	1.1	even	1	trivial
3146.2.a.a.1.1		1	429.164	odd	4
3328.2.b.g.1665.1		2	624.125	even	4
3328.2.b.g.1665.2		2	624.437	even	4
3328.2.b.k.1665.1		2	624.203	odd	4
3328.2.b.k.1665.2		2	624.515	odd	4
5200.2.a.c.1.1		1	780.359	odd	4
5850.2.a.bn.1.1		1	65.34	odd	4
5850.2.e.v.5149.1		2	65.47	even	4
5850.2.e.v.5149.2		2	65.8	even	4
7488.2.a.v.1.1		1	104.99	even	4
7488.2.a.w.1.1		1	104.21	odd	4
7514.2.a.i.1.1		1	663.203	even	4
8450.2.a.y.1.1		1	195.44	even	4
9386.2.a.f.1.1		1	741.398	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{10} +2.00000i q^{11} -1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -6.00000i q^{19} +1.00000i q^{20} -2.00000 q^{22} -4.00000 q^{23} +4.00000 q^{25} -1.00000i q^{28} -2.00000 q^{29} -4.00000i q^{31} +1.00000i q^{32} -3.00000i q^{34} +1.00000 q^{35} +3.00000i q^{37} +6.00000 q^{38} -1.00000 q^{40} +5.00000 q^{43} -2.00000i q^{44} -4.00000i q^{46} -13.0000i q^{47} +6.00000 q^{49} +4.00000i q^{50} -12.0000 q^{53} +2.00000 q^{55} +1.00000 q^{56} -2.00000i q^{58} +10.0000i q^{59} -8.00000 q^{61} +4.00000 q^{62} -1.00000 q^{64} +2.00000i q^{67} +3.00000 q^{68} +1.00000i q^{70} -5.00000i q^{71} -10.0000i q^{73} -3.00000 q^{74} +6.00000i q^{76} -2.00000 q^{77} -4.00000 q^{79} -1.00000i q^{80} +3.00000i q^{85} +5.00000i q^{86} +2.00000 q^{88} -6.00000i q^{89} +4.00000 q^{92} +13.0000 q^{94} -6.00000 q^{95} -14.0000i q^{97} +6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 2 q^{10} - 2 q^{14} + 2 q^{16} - 6 q^{17} - 4 q^{22} - 8 q^{23} + 8 q^{25} - 4 q^{29} + 2 q^{35} + 12 q^{38} - 2 q^{40} + 10 q^{43} + 12 q^{49} - 24 q^{53} + 4 q^{55} + 2 q^{56} - 16 q^{61} + 8 q^{62} - 2 q^{64} + 6 q^{68} - 6 q^{74} - 4 q^{77} - 8 q^{79} + 4 q^{88} + 8 q^{92} + 26 q^{94} - 12 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^10 - 2 * q^14 + 2 * q^16 - 6 * q^17 - 4 * q^22 - 8 * q^23 + 8 * q^25 - 4 * q^29 + 2 * q^35 + 12 * q^38 - 2 * q^40 + 10 * q^43 + 12 * q^49 - 24 * q^53 + 4 * q^55 + 2 * q^56 - 16 * q^61 + 8 * q^62 - 2 * q^64 + 6 * q^68 - 6 * q^74 - 4 * q^77 - 8 * q^79 + 4 * q^88 + 8 * q^92 + 26 * q^94 - 12 * q^95

Introduction

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