LMFDB - Embedded newform 3042.2.b.d.1351.1

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 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} -2.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{10} -2.00000i q^{11} -2.00000 q^{14} +1.00000 q^{16} +5.00000 q^{17} +2.00000i q^{19} +1.00000i q^{20} -2.00000 q^{22} +6.00000 q^{23} +4.00000 q^{25} +2.00000i q^{28} +9.00000 q^{29} +4.00000i q^{31} -1.00000i q^{32} -5.00000i q^{34} -2.00000 q^{35} -11.0000i q^{37} +2.00000 q^{38} +1.00000 q^{40} +5.00000i q^{41} -10.0000 q^{43} +2.00000i q^{44} -6.00000i q^{46} -2.00000i q^{47} +3.00000 q^{49} -4.00000i q^{50} +1.00000 q^{53} -2.00000 q^{55} +2.00000 q^{56} -9.00000i q^{58} +8.00000i q^{59} -11.0000 q^{61} +4.00000 q^{62} -1.00000 q^{64} -2.00000i q^{67} -5.00000 q^{68} +2.00000i q^{70} -14.0000i q^{71} -13.0000i q^{73} -11.0000 q^{74} -2.00000i q^{76} -4.00000 q^{77} -4.00000 q^{79} -1.00000i q^{80} +5.00000 q^{82} +6.00000i q^{83} -5.00000i q^{85} +10.0000i q^{86} +2.00000 q^{88} -2.00000i q^{89} -6.00000 q^{92} -2.00000 q^{94} +2.00000 q^{95} +2.00000i q^{97} -3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 2 q^{10} - 4 q^{14} + 2 q^{16} + 10 q^{17} - 4 q^{22} + 12 q^{23} + 8 q^{25} + 18 q^{29} - 4 q^{35} + 4 q^{38} + 2 q^{40} - 20 q^{43} + 6 q^{49} + 2 q^{53} - 4 q^{55} + 4 q^{56} - 22 q^{61} + 8 q^{62} - 2 q^{64} - 10 q^{68} - 22 q^{74} - 8 q^{77} - 8 q^{79} + 10 q^{82} + 4 q^{88} - 12 q^{92} - 4 q^{94} + 4 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^10 - 4 * q^14 + 2 * q^16 + 10 * q^17 - 4 * q^22 + 12 * q^23 + 8 * q^25 + 18 * q^29 - 4 * q^35 + 4 * q^38 + 2 * q^40 - 20 * q^43 + 6 * q^49 + 2 * q^53 - 4 * q^55 + 4 * q^56 - 22 * q^61 + 8 * q^62 - 2 * q^64 - 10 * q^68 - 22 * q^74 - 8 * q^77 - 8 * q^79 + 10 * q^82 + 4 * q^88 - 12 * q^92 - 4 * q^94 + 4 * 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
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	−1.00000	−0.316228
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	2.00000i	0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$	2.00000i	0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	1.00000i	0.223607i
$21$	0	0
$22$	−2.00000	−0.426401
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	0	0
$28$	2.00000i	0.377964i
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 5.00000i	− 0.857493i
$35$	−2.00000	−0.338062
$36$	0	0
$37$	− 11.0000i	− 1.80839i	−0.427121	−	0.904194i	$-0.640472\pi$
$37$	− 11.0000i	− 1.80839i	0.427121	−	0.904194i	$-0.359528\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	1.00000	0.158114
$41$	5.00000i	0.780869i	0.920631	+	0.390434i	$0.127675\pi$
$41$	5.00000i	0.780869i	−0.920631	+	0.390434i	$0.872325\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	2.00000i	0.301511i
$45$	0	0
$46$	− 6.00000i	− 0.884652i
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	− 4.00000i	− 0.565685i
$51$	0	0
$52$	0	0
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	2.00000	0.267261
$57$	0	0
$58$	− 9.00000i	− 1.18176i
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	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	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	−5.00000	−0.606339
$69$	0	0
$70$	2.00000i	0.239046i
$71$	− 14.0000i	− 1.66149i	−0.556650	−	0.830747i	$-0.687914\pi$
$71$	− 14.0000i	− 1.66149i	0.556650	−	0.830747i	$-0.312086\pi$
$72$	0	0
$73$	− 13.0000i	− 1.52153i	−0.649025	−	0.760767i	$-0.724823\pi$
$73$	− 13.0000i	− 1.52153i	0.649025	−	0.760767i	$-0.275177\pi$
$74$	−11.0000	−1.27872
$75$	0	0
$76$	− 2.00000i	− 0.229416i
$77$	−4.00000	−0.455842
$78$	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
$82$	5.00000	0.552158
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	− 5.00000i	− 0.542326i
$86$	10.0000i	1.07833i
$87$	0	0
$88$	2.00000	0.213201
$89$	− 2.00000i	− 0.212000i	−0.994366	−	0.106000i	$-0.966196\pi$
$89$	− 2.00000i	− 0.212000i	0.994366	−	0.106000i	$-0.0338043\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	0	0
$94$	−2.00000	−0.206284
$95$	2.00000	0.205196
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−5.00000	−0.497519	−0.248759	−	0.968565i	$-0.580023\pi$
$101$	−5.00000	−0.497519	−0.248759	+	0.968565i	$0.580023\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	0	0
$106$	− 1.00000i	− 0.0971286i
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	2.00000i	0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$	2.00000i	0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$	2.00000i	0.190693i
$111$	0	0
$112$	− 2.00000i	− 0.188982i
$113$	3.00000	0.282216	0.141108	−	0.989994i	$-0.454933\pi$
$113$	3.00000	0.282216	0.141108	+	0.989994i	$0.454933\pi$
$114$	0	0
$115$	− 6.00000i	− 0.559503i
$116$	−9.00000	−0.835629
$117$	0	0
$118$	8.00000	0.736460
$119$	− 10.0000i	− 0.916698i
$120$	0	0
$121$	7.00000	0.636364
$122$	11.0000i	0.995893i
$123$	0	0
$124$	− 4.00000i	− 0.359211i
$125$	− 9.00000i	− 0.804984i
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−2.00000	−0.172774
$135$	0	0
$136$	5.00000i	0.428746i
$137$	− 17.0000i	− 1.45241i	−0.687479	−	0.726204i	$-0.741283\pi$
$137$	− 17.0000i	− 1.45241i	0.687479	−	0.726204i	$-0.258717\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	−14.0000	−1.17485
$143$	0	0
$144$	0	0
$145$	− 9.00000i	− 0.747409i
$146$	−13.0000	−1.07589
$147$	0	0
$148$	11.0000i	0.904194i
$149$	3.00000i	0.245770i	0.992421	+	0.122885i	$0.0392146\pi$
$149$	3.00000i	0.245770i	−0.992421	+	0.122885i	$0.960785\pi$
$150$	0	0
$151$	− 6.00000i	− 0.488273i	−0.969741	−	0.244137i	$-0.921495\pi$
$151$	− 6.00000i	− 0.488273i	0.969741	−	0.244137i	$-0.0785045\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	4.00000i	0.322329i
$155$	4.00000	0.321288
$156$	0	0
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	4.00000i	0.318223i
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	− 12.0000i	− 0.945732i
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	− 5.00000i	− 0.390434i
$165$	0	0
$166$	6.00000	0.465690
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	0	0
$169$	0	0
$170$	−5.00000	−0.383482
$171$	0	0
$172$	10.0000	0.762493
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	− 8.00000i	− 0.604743i
$176$	− 2.00000i	− 0.150756i
$177$	0	0
$178$	−2.00000	−0.149906
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	0	0
$183$	0	0
$184$	6.00000i	0.442326i
$185$	−11.0000	−0.808736
$186$	0	0
$187$	− 10.0000i	− 0.731272i
$188$	2.00000i	0.145865i
$189$	0	0
$190$	− 2.00000i	− 0.145095i
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	− 17.0000i	− 1.22369i	−0.790979	−	0.611843i	$-0.790428\pi$
$193$	− 17.0000i	− 1.22369i	0.790979	−	0.611843i	$-0.209572\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	4.00000i	0.282843i
$201$	0	0
$202$	5.00000i	0.351799i
$203$	− 18.0000i	− 1.26335i
$204$	0	0
$205$	5.00000	0.349215
$206$	10.0000i	0.696733i
$207$	0	0
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	24.0000	1.65223	0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000	1.65223	0.826114	+	0.563503i	$0.190547\pi$
$212$	−1.00000	−0.0686803
$213$	0	0
$214$	− 18.0000i	− 1.23045i
$215$	10.0000i	0.681994i
$216$	0	0
$217$	8.00000	0.543075
$218$	2.00000	0.135457
$219$	0	0
$220$	2.00000	0.134840
$221$	0	0
$222$	0	0
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	− 3.00000i	− 0.199557i
$227$	14.0000i	0.929213i	0.885517	+	0.464606i	$0.153804\pi$
$227$	14.0000i	0.929213i	−0.885517	+	0.464606i	$0.846196\pi$
$228$	0	0
$229$	10.0000i	0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$	10.0000i	0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	9.00000i	0.590879i
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−2.00000	−0.130466
$236$	− 8.00000i	− 0.520756i
$237$	0	0
$238$	−10.0000	−0.648204
$239$	− 6.00000i	− 0.388108i	−0.980991	−	0.194054i	$-0.937836\pi$
$239$	− 6.00000i	− 0.388108i	0.980991	−	0.194054i	$-0.0621637\pi$
$240$	0	0
$241$	7.00000i	0.450910i	0.974254	+	0.225455i	$0.0723868\pi$
$241$	7.00000i	0.450910i	−0.974254	+	0.225455i	$0.927613\pi$
$242$	− 7.00000i	− 0.449977i
$243$	0	0
$244$	11.0000	0.704203
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	0	0
$248$	−4.00000	−0.254000
$249$	0	0
$250$	−9.00000	−0.569210
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	− 12.0000i	− 0.754434i
$254$	− 12.0000i	− 0.752947i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	0	0
$259$	−22.0000	−1.36701
$260$	0	0
$261$	0	0
$262$	− 8.00000i	− 0.494242i
$263$	−14.0000	−0.863277	−0.431638	−	0.902047i	$-0.642064\pi$
$263$	−14.0000	−0.863277	−0.431638	+	0.902047i	$0.642064\pi$
$264$	0	0
$265$	− 1.00000i	− 0.0614295i
$266$	− 4.00000i	− 0.245256i
$267$	0	0
$268$	2.00000i	0.122169i
$269$	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
$271$	8.00000i	0.485965i	0.970031	+	0.242983i	$0.0781258\pi$
$271$	8.00000i	0.485965i	−0.970031	+	0.242983i	$0.921874\pi$
$272$	5.00000	0.303170
$273$	0	0
$274$	−17.0000	−1.02701
$275$	− 8.00000i	− 0.482418i
$276$	0	0
$277$	11.0000	0.660926	0.330463	−	0.943819i	$-0.392795\pi$
$277$	11.0000	0.660926	0.330463	+	0.943819i	$0.392795\pi$
$278$	12.0000i	0.719712i
$279$	0	0
$280$	− 2.00000i	− 0.119523i
$281$	− 25.0000i	− 1.49137i	−0.666296	−	0.745687i	$-0.732121\pi$
$281$	− 25.0000i	− 1.49137i	0.666296	−	0.745687i	$-0.267879\pi$
$282$	0	0
$283$	−26.0000	−1.54554	−0.772770	−	0.634686i	$-0.781129\pi$
$283$	−26.0000	−1.54554	−0.772770	+	0.634686i	$0.781129\pi$
$284$	14.0000i	0.830747i
$285$	0	0
$286$	0	0
$287$	10.0000	0.590281
$288$	0	0
$289$	8.00000	0.470588
$290$	−9.00000	−0.528498
$291$	0	0
$292$	13.0000i	0.760767i
$293$	1.00000i	0.0584206i	0.999573	+	0.0292103i	$0.00929925\pi$
$293$	1.00000i	0.0584206i	−0.999573	+	0.0292103i	$0.990701\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	11.0000	0.639362
$297$	0	0
$298$	3.00000	0.173785
$299$	0	0
$300$	0	0
$301$	20.0000i	1.15278i
$302$	−6.00000	−0.345261
$303$	0	0
$304$	2.00000i	0.114708i
$305$	11.0000i	0.629858i
$306$	0	0
$307$	− 14.0000i	− 0.799022i	−0.916728	−	0.399511i	$-0.869180\pi$
$307$	− 14.0000i	− 0.799022i	0.916728	−	0.399511i	$-0.130820\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	− 4.00000i	− 0.227185i
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	7.00000i	0.395033i
$315$	0	0
$316$	4.00000	0.225018
$317$	− 33.0000i	− 1.85346i	−0.375722	−	0.926732i	$-0.622605\pi$
$317$	− 33.0000i	− 1.85346i	0.375722	−	0.926732i	$-0.377395\pi$
$318$	0	0
$319$	− 18.0000i	− 1.00781i
$320$	1.00000i	0.0559017i
$321$	0	0
$322$	−12.0000	−0.668734
$323$	10.0000i	0.556415i
$324$	0	0
$325$	0	0
$326$	20.0000	1.10770
$327$	0	0
$328$	−5.00000	−0.276079
$329$	−4.00000	−0.220527
$330$	0	0
$331$	28.0000i	1.53902i	0.638635	+	0.769510i	$0.279499\pi$
$331$	28.0000i	1.53902i	−0.638635	+	0.769510i	$0.720501\pi$
$332$	− 6.00000i	− 0.329293i
$333$	0	0
$334$	−24.0000	−1.31322
$335$	−2.00000	−0.109272
$336$	0	0
$337$	9.00000	0.490261	0.245131	−	0.969490i	$-0.421169\pi$
$337$	9.00000	0.490261	0.245131	+	0.969490i	$0.421169\pi$
$338$	0	0
$339$	0	0
$340$	5.00000i	0.271163i
$341$	8.00000	0.433224
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	− 10.0000i	− 0.539164i
$345$	0	0
$346$	22.0000i	1.18273i
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	6.00000i	0.321173i	0.987022	+	0.160586i	$0.0513385\pi$
$349$	6.00000i	0.321173i	−0.987022	+	0.160586i	$0.948662\pi$
$350$	−8.00000	−0.427618
$351$	0	0
$352$	−2.00000	−0.106600
$353$	17.0000i	0.904819i	0.891810	+	0.452409i	$0.149435\pi$
$353$	17.0000i	0.904819i	−0.891810	+	0.452409i	$0.850565\pi$
$354$	0	0
$355$	−14.0000	−0.743043
$356$	2.00000i	0.106000i
$357$	0	0
$358$	6.00000i	0.317110i
$359$	30.0000i	1.58334i	0.610949	+	0.791670i	$0.290788\pi$
$359$	30.0000i	1.58334i	−0.610949	+	0.791670i	$0.709212\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	5.00000i	0.262794i
$363$	0	0
$364$	0	0
$365$	−13.0000	−0.680451
$366$	0	0
$367$	−2.00000	−0.104399	−0.0521996	−	0.998637i	$-0.516623\pi$
$367$	−2.00000	−0.104399	−0.0521996	+	0.998637i	$0.516623\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	11.0000i	0.571863i
$371$	− 2.00000i	− 0.103835i
$372$	0	0
$373$	9.00000	0.466002	0.233001	−	0.972476i	$-0.425145\pi$
$373$	9.00000	0.466002	0.233001	+	0.972476i	$0.425145\pi$
$374$	−10.0000	−0.517088
$375$	0	0
$376$	2.00000	0.103142
$377$	0	0
$378$	0	0
$379$	− 12.0000i	− 0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$	− 12.0000i	− 0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$	−2.00000	−0.102598
$381$	0	0
$382$	4.00000i	0.204658i
$383$	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
$385$	4.00000i	0.203859i
$386$	−17.0000	−0.865277
$387$	0	0
$388$	− 2.00000i	− 0.101535i
$389$	19.0000	0.963338	0.481669	−	0.876353i	$-0.340031\pi$
$389$	19.0000	0.963338	0.481669	+	0.876353i	$0.340031\pi$
$390$	0	0
$391$	30.0000	1.51717
$392$	3.00000i	0.151523i
$393$	0	0
$394$	6.00000	0.302276
$395$	4.00000i	0.201262i
$396$	0	0
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	10.0000i	0.501255i
$399$	0	0
$400$	4.00000	0.200000
$401$	27.0000i	1.34832i	0.738587	+	0.674158i	$0.235493\pi$
$401$	27.0000i	1.34832i	−0.738587	+	0.674158i	$0.764507\pi$
$402$	0	0
$403$	0	0
$404$	5.00000	0.248759
$405$	0	0
$406$	−18.0000	−0.893325
$407$	−22.0000	−1.09050
$408$	0	0
$409$	− 23.0000i	− 1.13728i	−0.822588	−	0.568638i	$-0.807470\pi$
$409$	− 23.0000i	− 1.13728i	0.822588	−	0.568638i	$-0.192530\pi$
$410$	− 5.00000i	− 0.246932i
$411$	0	0
$412$	10.0000	0.492665
$413$	16.0000	0.787309
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	0	0
$418$	− 4.00000i	− 0.195646i
$419$	32.0000	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$419$	32.0000	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$420$	0	0
$421$	23.0000i	1.12095i	0.828171	+	0.560476i	$0.189382\pi$
$421$	23.0000i	1.12095i	−0.828171	+	0.560476i	$0.810618\pi$
$422$	− 24.0000i	− 1.16830i
$423$	0	0
$424$	1.00000i	0.0485643i
$425$	20.0000	0.970143
$426$	0	0
$427$	22.0000i	1.06465i
$428$	−18.0000	−0.870063
$429$	0	0
$430$	10.0000	0.482243
$431$	− 2.00000i	− 0.0963366i	−0.998839	−	0.0481683i	$-0.984662\pi$
$431$	− 2.00000i	− 0.0963366i	0.998839	−	0.0481683i	$-0.0153384\pi$
$432$	0	0
$433$	21.0000	1.00920	0.504598	−	0.863355i	$-0.331641\pi$
$433$	21.0000	1.00920	0.504598	+	0.863355i	$0.331641\pi$
$434$	− 8.00000i	− 0.384012i
$435$	0	0
$436$	− 2.00000i	− 0.0957826i
$437$	12.0000i	0.574038i
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	− 2.00000i	− 0.0953463i
$441$	0	0
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	16.0000	0.757622
$447$	0	0
$448$	2.00000i	0.0944911i
$449$	30.0000i	1.41579i	0.706319	+	0.707894i	$0.250354\pi$
$449$	30.0000i	1.41579i	−0.706319	+	0.707894i	$0.749646\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	−3.00000	−0.141108
$453$	0	0
$454$	14.0000	0.657053
$455$	0	0
$456$	0	0
$457$	− 3.00000i	− 0.140334i	−0.997535	−	0.0701670i	$-0.977647\pi$
$457$	− 3.00000i	− 0.140334i	0.997535	−	0.0701670i	$-0.0223532\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	6.00000i	0.279751i
$461$	3.00000i	0.139724i	0.997557	+	0.0698620i	$0.0222559\pi$
$461$	3.00000i	0.139724i	−0.997557	+	0.0698620i	$0.977744\pi$
$462$	0	0
$463$	− 14.0000i	− 0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$	− 14.0000i	− 0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	6.00000i	0.277945i
$467$	−22.0000	−1.01804	−0.509019	−	0.860755i	$-0.669992\pi$
$467$	−22.0000	−1.01804	−0.509019	+	0.860755i	$0.669992\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	2.00000i	0.0922531i
$471$	0	0
$472$	−8.00000	−0.368230
$473$	20.0000i	0.919601i
$474$	0	0
$475$	8.00000i	0.367065i
$476$	10.0000i	0.458349i
$477$	0	0
$478$	−6.00000	−0.274434
$479$	− 32.0000i	− 1.46212i	−0.682315	−	0.731059i	$-0.739027\pi$
$479$	− 32.0000i	− 1.46212i	0.682315	−	0.731059i	$-0.260973\pi$
$480$	0	0
$481$	0	0
$482$	7.00000	0.318841
$483$	0	0
$484$	−7.00000	−0.318182
$485$	2.00000	0.0908153
$486$	0	0
$487$	26.0000i	1.17817i	0.808070	+	0.589086i	$0.200512\pi$
$487$	26.0000i	1.17817i	−0.808070	+	0.589086i	$0.799488\pi$
$488$	− 11.0000i	− 0.497947i
$489$	0	0
$490$	−3.00000	−0.135526
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	0	0
$493$	45.0000	2.02670
$494$	0	0
$495$	0	0
$496$	4.00000i	0.179605i
$497$	−28.0000	−1.25597
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	9.00000i	0.402492i
$501$	0	0
$502$	− 4.00000i	− 0.178529i
$503$	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
$505$	5.00000i	0.222497i
$506$	−12.0000	−0.533465
$507$	0	0
$508$	−12.0000	−0.532414
$509$	15.0000i	0.664863i	0.943127	+	0.332432i	$0.107869\pi$
$509$	15.0000i	0.664863i	−0.943127	+	0.332432i	$0.892131\pi$
$510$	0	0
$511$	−26.0000	−1.15017
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	3.00000i	0.132324i
$515$	10.0000i	0.440653i
$516$	0	0
$517$	−4.00000	−0.175920
$518$	22.0000i	0.966625i
$519$	0	0
$520$	0	0
$521$	−25.0000	−1.09527	−0.547635	−	0.836717i	$-0.684472\pi$
$521$	−25.0000	−1.09527	−0.547635	+	0.836717i	$0.684472\pi$
$522$	0	0
$523$	−38.0000	−1.66162	−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−38.0000	−1.66162	−0.830812	+	0.556553i	$0.812124\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	14.0000i	0.610429i
$527$	20.0000i	0.871214i
$528$	0	0
$529$	13.0000	0.565217
$530$	−1.00000	−0.0434372
$531$	0	0
$532$	−4.00000	−0.173422
$533$	0	0
$534$	0	0
$535$	− 18.0000i	− 0.778208i
$536$	2.00000	0.0863868
$537$	0	0
$538$	− 14.0000i	− 0.603583i
$539$	− 6.00000i	− 0.258438i
$540$	0	0
$541$	− 7.00000i	− 0.300954i	−0.988614	−	0.150477i	$-0.951919\pi$
$541$	− 7.00000i	− 0.300954i	0.988614	−	0.150477i	$-0.0480809\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	− 5.00000i	− 0.214373i
$545$	2.00000	0.0856706
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	17.0000i	0.726204i
$549$	0	0
$550$	−8.00000	−0.341121
$551$	18.0000i	0.766826i
$552$	0	0
$553$	8.00000i	0.340195i
$554$	− 11.0000i	− 0.467345i
$555$	0	0
$556$	12.0000	0.508913
$557$	9.00000i	0.381342i	0.981654	+	0.190671i	$0.0610664\pi$
$557$	9.00000i	0.381342i	−0.981654	+	0.190671i	$0.938934\pi$
$558$	0	0
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	−25.0000	−1.05456
$563$	40.0000	1.68580	0.842900	−	0.538071i	$-0.180847\pi$
$563$	40.0000	1.68580	0.842900	+	0.538071i	$0.180847\pi$
$564$	0	0
$565$	− 3.00000i	− 0.126211i
$566$	26.0000i	1.09286i
$567$	0	0
$568$	14.0000	0.587427
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	2.00000	0.0836974	0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000	0.0836974	0.0418487	+	0.999124i	$0.486675\pi$
$572$	0	0
$573$	0	0
$574$	− 10.0000i	− 0.417392i
$575$	24.0000	1.00087
$576$	0	0
$577$	− 27.0000i	− 1.12402i	−0.827129	−	0.562012i	$-0.810027\pi$
$577$	− 27.0000i	− 1.12402i	0.827129	−	0.562012i	$-0.189973\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	9.00000i	0.373705i
$581$	12.0000	0.497844
$582$	0	0
$583$	− 2.00000i	− 0.0828315i
$584$	13.0000	0.537944
$585$	0	0
$586$	1.00000	0.0413096
$587$	− 32.0000i	− 1.32078i	−0.750922	−	0.660391i	$-0.770391\pi$
$587$	− 32.0000i	− 1.32078i	0.750922	−	0.660391i	$-0.229609\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	− 8.00000i	− 0.329355i
$591$	0	0
$592$	− 11.0000i	− 0.452097i
$593$	39.0000i	1.60154i	0.598973	+	0.800769i	$0.295576\pi$
$593$	39.0000i	1.60154i	−0.598973	+	0.800769i	$0.704424\pi$
$594$	0	0
$595$	−10.0000	−0.409960
$596$	− 3.00000i	− 0.122885i
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	11.0000	0.448699	0.224350	−	0.974509i	$-0.427974\pi$
$601$	11.0000	0.448699	0.224350	+	0.974509i	$0.427974\pi$
$602$	20.0000	0.815139
$603$	0	0
$604$	6.00000i	0.244137i
$605$	− 7.00000i	− 0.284590i
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	11.0000	0.445377
$611$	0	0
$612$	0	0
$613$	− 13.0000i	− 0.525065i	−0.964923	−	0.262533i	$-0.915442\pi$
$613$	− 13.0000i	− 0.525065i	0.964923	−	0.262533i	$-0.0845577\pi$
$614$	−14.0000	−0.564994
$615$	0	0
$616$	− 4.00000i	− 0.161165i
$617$	− 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
$619$	32.0000i	1.28619i	0.765787	+	0.643094i	$0.222350\pi$
$619$	32.0000i	1.28619i	−0.765787	+	0.643094i	$0.777650\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	− 6.00000i	− 0.240578i
$623$	−4.00000	−0.160257
$624$	0	0
$625$	11.0000	0.440000
$626$	− 6.00000i	− 0.239808i
$627$	0	0
$628$	7.00000	0.279330
$629$	− 55.0000i	− 2.19299i
$630$	0	0
$631$	12.0000i	0.477712i	0.971055	+	0.238856i	$0.0767725\pi$
$631$	12.0000i	0.477712i	−0.971055	+	0.238856i	$0.923228\pi$
$632$	− 4.00000i	− 0.159111i
$633$	0	0
$634$	−33.0000	−1.31060
$635$	− 12.0000i	− 0.476205i
$636$	0	0
$637$	0	0
$638$	−18.0000	−0.712627
$639$	0	0
$640$	1.00000	0.0395285
$641$	5.00000	0.197488	0.0987441	−	0.995113i	$-0.468517\pi$
$641$	5.00000	0.197488	0.0987441	+	0.995113i	$0.468517\pi$
$642$	0	0
$643$	− 8.00000i	− 0.315489i	−0.987480	−	0.157745i	$-0.949578\pi$
$643$	− 8.00000i	− 0.315489i	0.987480	−	0.157745i	$-0.0504223\pi$
$644$	12.0000i	0.472866i
$645$	0	0
$646$	10.0000	0.393445
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	− 20.0000i	− 0.783260i
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	0	0
$655$	− 8.00000i	− 0.312586i
$656$	5.00000i	0.195217i
$657$	0	0
$658$	4.00000i	0.155936i
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	25.0000i	0.972387i	0.873851	+	0.486194i	$0.161615\pi$
$661$	25.0000i	0.972387i	−0.873851	+	0.486194i	$0.838385\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	−6.00000	−0.232845
$665$	− 4.00000i	− 0.155113i
$666$	0	0
$667$	54.0000	2.09089
$668$	24.0000i	0.928588i
$669$	0	0
$670$	2.00000i	0.0772667i
$671$	22.0000i	0.849301i
$672$	0	0
$673$	−43.0000	−1.65753	−0.828764	−	0.559598i	$-0.810955\pi$
$673$	−43.0000	−1.65753	−0.828764	+	0.559598i	$0.810955\pi$
$674$	− 9.00000i	− 0.346667i
$675$	0	0
$676$	0	0
$677$	46.0000	1.76792	0.883962	−	0.467559i	$-0.154866\pi$
$677$	46.0000	1.76792	0.883962	+	0.467559i	$0.154866\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	5.00000	0.191741
$681$	0	0
$682$	− 8.00000i	− 0.306336i
$683$	40.0000i	1.53056i	0.643699	+	0.765279i	$0.277399\pi$
$683$	40.0000i	1.53056i	−0.643699	+	0.765279i	$0.722601\pi$
$684$	0	0
$685$	−17.0000	−0.649537
$686$	−20.0000	−0.763604
$687$	0	0
$688$	−10.0000	−0.381246
$689$	0	0
$690$	0	0
$691$	2.00000i	0.0760836i	0.999276	+	0.0380418i	$0.0121120\pi$
$691$	2.00000i	0.0760836i	−0.999276	+	0.0380418i	$0.987888\pi$
$692$	22.0000	0.836315
$693$	0	0
$694$	6.00000i	0.227757i
$695$	12.0000i	0.455186i
$696$	0	0
$697$	25.0000i	0.946943i
$698$	6.00000	0.227103
$699$	0	0
$700$	8.00000i	0.302372i
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	0	0
$703$	22.0000	0.829746
$704$	2.00000i	0.0753778i
$705$	0	0
$706$	17.0000	0.639803
$707$	10.0000i	0.376089i
$708$	0	0
$709$	− 15.0000i	− 0.563337i	−0.959512	−	0.281668i	$-0.909112\pi$
$709$	− 15.0000i	− 0.563337i	0.959512	−	0.281668i	$-0.0908878\pi$
$710$	14.0000i	0.525411i
$711$	0	0
$712$	2.00000	0.0749532
$713$	24.0000i	0.898807i
$714$	0	0
$715$	0	0
$716$	6.00000	0.224231
$717$	0	0
$718$	30.0000	1.11959
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	20.0000i	0.744839i
$722$	− 15.0000i	− 0.558242i
$723$	0	0
$724$	5.00000	0.185824
$725$	36.0000	1.33701
$726$	0	0
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	0	0
$729$	0	0
$730$	13.0000i	0.481152i
$731$	−50.0000	−1.84932
$732$	0	0
$733$	− 13.0000i	− 0.480166i	−0.970752	−	0.240083i	$-0.922825\pi$
$733$	− 13.0000i	− 0.480166i	0.970752	−	0.240083i	$-0.0771747\pi$
$734$	2.00000i	0.0738213i
$735$	0	0
$736$	− 6.00000i	− 0.221163i
$737$	−4.00000	−0.147342
$738$	0	0
$739$	16.0000i	0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$	16.0000i	0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$	11.0000	0.404368
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	− 12.0000i	− 0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$	− 12.0000i	− 0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$	0	0
$745$	3.00000	0.109911
$746$	− 9.00000i	− 0.329513i
$747$	0	0
$748$	10.0000i	0.365636i
$749$	− 36.0000i	− 1.31541i
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	− 2.00000i	− 0.0729325i
$753$	0	0
$754$	0	0
$755$	−6.00000	−0.218362
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−12.0000	−0.435860
$759$	0	0
$760$	2.00000i	0.0725476i
$761$	− 34.0000i	− 1.23250i	−0.787551	−	0.616250i	$-0.788651\pi$
$761$	− 34.0000i	− 1.23250i	0.787551	−	0.616250i	$-0.211349\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	4.00000	0.144715
$765$	0	0
$766$	24.0000	0.867155
$767$	0	0
$768$	0	0
$769$	34.0000i	1.22607i	0.790055	+	0.613036i	$0.210052\pi$
$769$	34.0000i	1.22607i	−0.790055	+	0.613036i	$0.789948\pi$
$770$	4.00000	0.144150
$771$	0	0
$772$	17.0000i	0.611843i
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	16.0000i	0.574737i
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	− 19.0000i	− 0.681183i
$779$	−10.0000	−0.358287
$780$	0	0
$781$	−28.0000	−1.00192
$782$	− 30.0000i	− 1.07280i
$783$	0	0
$784$	3.00000	0.107143
$785$	7.00000i	0.249841i
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	− 6.00000i	− 0.213741i
$789$	0	0
$790$	4.00000	0.142314
$791$	− 6.00000i	− 0.213335i
$792$	0	0
$793$	0	0
$794$	−18.0000	−0.638796
$795$	0	0
$796$	10.0000	0.354441
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	− 10.0000i	− 0.353775i
$800$	− 4.00000i	− 0.141421i
$801$	0	0
$802$	27.0000	0.953403
$803$	−26.0000	−0.917520
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	0	0
$808$	− 5.00000i	− 0.175899i
$809$	−5.00000	−0.175791	−0.0878953	−	0.996130i	$-0.528014\pi$
$809$	−5.00000	−0.175791	−0.0878953	+	0.996130i	$0.528014\pi$
$810$	0	0
$811$	36.0000i	1.26413i	0.774915	+	0.632065i	$0.217793\pi$
$811$	36.0000i	1.26413i	−0.774915	+	0.632065i	$0.782207\pi$
$812$	18.0000i	0.631676i
$813$	0	0
$814$	22.0000i	0.771100i
$815$	20.0000	0.700569
$816$	0	0
$817$	− 20.0000i	− 0.699711i
$818$	−23.0000	−0.804176
$819$	0	0
$820$	−5.00000	−0.174608
$821$	− 30.0000i	− 1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$	− 30.0000i	− 1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	− 10.0000i	− 0.348367i
$825$	0	0
$826$	− 16.0000i	− 0.556711i
$827$	8.00000i	0.278187i	0.990279	+	0.139094i	$0.0444189\pi$
$827$	8.00000i	0.278187i	−0.990279	+	0.139094i	$0.955581\pi$
$828$	0	0
$829$	35.0000	1.21560	0.607800	−	0.794090i	$-0.292052\pi$
$829$	35.0000	1.21560	0.607800	+	0.794090i	$0.292052\pi$
$830$	− 6.00000i	− 0.208263i
$831$	0	0
$832$	0	0
$833$	15.0000	0.519719
$834$	0	0
$835$	−24.0000	−0.830554
$836$	−4.00000	−0.138343
$837$	0	0
$838$	− 32.0000i	− 1.10542i
$839$	− 44.0000i	− 1.51905i	−0.650479	−	0.759524i	$-0.725432\pi$
$839$	− 44.0000i	− 1.51905i	0.650479	−	0.759524i	$-0.274568\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	23.0000	0.792632
$843$	0	0
$844$	−24.0000	−0.826114
$845$	0	0
$846$	0	0
$847$	− 14.0000i	− 0.481046i
$848$	1.00000	0.0343401
$849$	0	0
$850$	− 20.0000i	− 0.685994i
$851$	− 66.0000i	− 2.26245i
$852$	0	0
$853$	49.0000i	1.67773i	0.544341	+	0.838864i	$0.316780\pi$
$853$	49.0000i	1.67773i	−0.544341	+	0.838864i	$0.683220\pi$
$854$	22.0000	0.752825
$855$	0	0
$856$	18.0000i	0.615227i
$857$	45.0000	1.53717	0.768585	−	0.639747i	$-0.220961\pi$
$857$	45.0000	1.53717	0.768585	+	0.639747i	$0.220961\pi$
$858$	0	0
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	− 10.0000i	− 0.340997i
$861$	0	0
$862$	−2.00000	−0.0681203
$863$	46.0000i	1.56586i	0.622111	+	0.782929i	$0.286275\pi$
$863$	46.0000i	1.56586i	−0.622111	+	0.782929i	$0.713725\pi$
$864$	0	0
$865$	22.0000i	0.748022i
$866$	− 21.0000i	− 0.713609i
$867$	0	0
$868$	−8.00000	−0.271538
$869$	8.00000i	0.271381i
$870$	0	0
$871$	0	0
$872$	−2.00000	−0.0677285
$873$	0	0
$874$	12.0000	0.405906
$875$	−18.0000	−0.608511
$876$	0	0
$877$	− 37.0000i	− 1.24940i	−0.780864	−	0.624701i	$-0.785221\pi$
$877$	− 37.0000i	− 1.24940i	0.780864	−	0.624701i	$-0.214779\pi$
$878$	10.0000i	0.337484i
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	17.0000	0.572745	0.286372	−	0.958118i	$-0.407551\pi$
$881$	17.0000	0.572745	0.286372	+	0.958118i	$0.407551\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	0	0
$886$	20.0000i	0.671913i
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	− 24.0000i	− 0.804934i
$890$	2.00000i	0.0670402i
$891$	0	0
$892$	− 16.0000i	− 0.535720i
$893$	4.00000	0.133855
$894$	0	0
$895$	6.00000i	0.200558i
$896$	2.00000	0.0668153
$897$	0	0
$898$	30.0000	1.00111
$899$	36.0000i	1.20067i
$900$	0	0
$901$	5.00000	0.166574
$902$	− 10.0000i	− 0.332964i
$903$	0	0
$904$	3.00000i	0.0997785i
$905$	5.00000i	0.166206i
$906$	0	0
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	− 14.0000i	− 0.464606i
$909$	0	0
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	−3.00000	−0.0992312
$915$	0	0
$916$	− 10.0000i	− 0.330409i
$917$	− 16.0000i	− 0.528367i
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	6.00000	0.197814
$921$	0	0
$922$	3.00000	0.0987997
$923$	0	0
$924$	0	0
$925$	− 44.0000i	− 1.44671i
$926$	−14.0000	−0.460069
$927$	0	0
$928$	− 9.00000i	− 0.295439i
$929$	− 23.0000i	− 0.754606i	−0.926090	−	0.377303i	$-0.876852\pi$
$929$	− 23.0000i	− 0.754606i	0.926090	−	0.377303i	$-0.123148\pi$
$930$	0	0
$931$	6.00000i	0.196642i
$932$	6.00000	0.196537
$933$	0	0
$934$	22.0000i	0.719862i
$935$	−10.0000	−0.327035
$936$	0	0
$937$	−1.00000	−0.0326686	−0.0163343	−	0.999867i	$-0.505200\pi$
$937$	−1.00000	−0.0326686	−0.0163343	+	0.999867i	$0.505200\pi$
$938$	4.00000i	0.130605i
$939$	0	0
$940$	2.00000	0.0652328
$941$	− 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
$943$	30.0000i	0.976934i
$944$	8.00000i	0.260378i
$945$	0	0
$946$	20.0000	0.650256
$947$	− 8.00000i	− 0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$	− 8.00000i	− 0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$	0	0
$949$	0	0
$950$	8.00000	0.259554
$951$	0	0
$952$	10.0000	0.324102
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	0	0
$955$	4.00000i	0.129437i
$956$	6.00000i	0.194054i
$957$	0	0
$958$	−32.0000	−1.03387
$959$	−34.0000	−1.09792
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	0	0
$964$	− 7.00000i	− 0.225455i
$965$	−17.0000	−0.547249
$966$	0	0
$967$	50.0000i	1.60789i	0.594703	+	0.803946i	$0.297270\pi$
$967$	50.0000i	1.60789i	−0.594703	+	0.803946i	$0.702730\pi$
$968$	7.00000i	0.224989i
$969$	0	0
$970$	− 2.00000i	− 0.0642161i
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	24.0000i	0.769405i
$974$	26.0000	0.833094
$975$	0	0
$976$	−11.0000	−0.352101
$977$	21.0000i	0.671850i	0.941889	+	0.335925i	$0.109049\pi$
$977$	21.0000i	0.671850i	−0.941889	+	0.335925i	$0.890951\pi$
$978$	0	0
$979$	−4.00000	−0.127841
$980$	3.00000i	0.0958315i
$981$	0	0
$982$	30.0000i	0.957338i
$983$	− 60.0000i	− 1.91370i	−0.290578	−	0.956851i	$-0.593847\pi$
$983$	− 60.0000i	− 1.91370i	0.290578	−	0.956851i	$-0.406153\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	− 45.0000i	− 1.43309i
$987$	0	0
$988$	0	0
$989$	−60.0000	−1.90789
$990$	0	0
$991$	18.0000	0.571789	0.285894	−	0.958261i	$-0.407709\pi$
$991$	18.0000	0.571789	0.285894	+	0.958261i	$0.407709\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	28.0000i	0.888106i
$995$	10.0000i	0.317021i
$996$	0	0
$997$	−23.0000	−0.728417	−0.364209	−	0.931317i	$-0.618661\pi$
$997$	−23.0000	−0.728417	−0.364209	+	0.931317i	$0.618661\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.d.1351.1		2
3.2	odd	2		1014.2.b.a.337.2		2
13.5	odd	4		3042.2.a.d.1.1		1
13.7	odd	12		234.2.h.b.55.1		2
13.8	odd	4		3042.2.a.m.1.1		1
13.11	odd	12		234.2.h.b.217.1		2
13.12	even	2	inner	3042.2.b.d.1351.2		2
39.2	even	12		1014.2.e.d.529.1		2
39.5	even	4		1014.2.a.e.1.1		1
39.8	even	4		1014.2.a.a.1.1		1
39.11	even	12		78.2.e.b.61.1	yes	2
39.17	odd	6		1014.2.i.e.361.1		4
39.20	even	12		78.2.e.b.55.1	✓	2
39.23	odd	6		1014.2.i.e.823.2		4
39.29	odd	6		1014.2.i.e.823.1		4
39.32	even	12		1014.2.e.d.991.1		2
39.35	odd	6		1014.2.i.e.361.2		4
39.38	odd	2		1014.2.b.a.337.1		2
52.7	even	12		1872.2.t.i.289.1		2
52.11	even	12		1872.2.t.i.1153.1		2
156.11	odd	12		624.2.q.b.529.1		2
156.47	odd	4		8112.2.a.x.1.1		1
156.59	odd	12		624.2.q.b.289.1		2
156.83	odd	4		8112.2.a.bb.1.1		1
195.59	even	12		1950.2.i.b.601.1		2
195.89	even	12		1950.2.i.b.451.1		2
195.98	odd	12		1950.2.z.b.1849.2		4
195.128	odd	12		1950.2.z.b.1699.1		4
195.137	odd	12		1950.2.z.b.1849.1		4
195.167	odd	12		1950.2.z.b.1699.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.e.b.55.1	✓	2	39.20	even	12
78.2.e.b.61.1	yes	2	39.11	even	12
234.2.h.b.55.1		2	13.7	odd	12
234.2.h.b.217.1		2	13.11	odd	12
624.2.q.b.289.1		2	156.59	odd	12
624.2.q.b.529.1		2	156.11	odd	12
1014.2.a.a.1.1		1	39.8	even	4
1014.2.a.e.1.1		1	39.5	even	4
1014.2.b.a.337.1		2	39.38	odd	2
1014.2.b.a.337.2		2	3.2	odd	2
1014.2.e.d.529.1		2	39.2	even	12
1014.2.e.d.991.1		2	39.32	even	12
1014.2.i.e.361.1		4	39.17	odd	6
1014.2.i.e.361.2		4	39.35	odd	6
1014.2.i.e.823.1		4	39.29	odd	6
1014.2.i.e.823.2		4	39.23	odd	6
1872.2.t.i.289.1		2	52.7	even	12
1872.2.t.i.1153.1		2	52.11	even	12
1950.2.i.b.451.1		2	195.89	even	12
1950.2.i.b.601.1		2	195.59	even	12
1950.2.z.b.1699.1		4	195.128	odd	12
1950.2.z.b.1699.2		4	195.167	odd	12
1950.2.z.b.1849.1		4	195.137	odd	12
1950.2.z.b.1849.2		4	195.98	odd	12
3042.2.a.d.1.1		1	13.5	odd	4
3042.2.a.m.1.1		1	13.8	odd	4
3042.2.b.d.1351.1		2	1.1	even	1	trivial
3042.2.b.d.1351.2		2	13.12	even	2	inner
8112.2.a.x.1.1		1	156.47	odd	4
8112.2.a.bb.1.1		1	156.83	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} -2.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{10} -2.00000i q^{11} -2.00000 q^{14} +1.00000 q^{16} +5.00000 q^{17} +2.00000i q^{19} +1.00000i q^{20} -2.00000 q^{22} +6.00000 q^{23} +4.00000 q^{25} +2.00000i q^{28} +9.00000 q^{29} +4.00000i q^{31} -1.00000i q^{32} -5.00000i q^{34} -2.00000 q^{35} -11.0000i q^{37} +2.00000 q^{38} +1.00000 q^{40} +5.00000i q^{41} -10.0000 q^{43} +2.00000i q^{44} -6.00000i q^{46} -2.00000i q^{47} +3.00000 q^{49} -4.00000i q^{50} +1.00000 q^{53} -2.00000 q^{55} +2.00000 q^{56} -9.00000i q^{58} +8.00000i q^{59} -11.0000 q^{61} +4.00000 q^{62} -1.00000 q^{64} -2.00000i q^{67} -5.00000 q^{68} +2.00000i q^{70} -14.0000i q^{71} -13.0000i q^{73} -11.0000 q^{74} -2.00000i q^{76} -4.00000 q^{77} -4.00000 q^{79} -1.00000i q^{80} +5.00000 q^{82} +6.00000i q^{83} -5.00000i q^{85} +10.0000i q^{86} +2.00000 q^{88} -2.00000i q^{89} -6.00000 q^{92} -2.00000 q^{94} +2.00000 q^{95} +2.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 2 q^{10} - 4 q^{14} + 2 q^{16} + 10 q^{17} - 4 q^{22} + 12 q^{23} + 8 q^{25} + 18 q^{29} - 4 q^{35} + 4 q^{38} + 2 q^{40} - 20 q^{43} + 6 q^{49} + 2 q^{53} - 4 q^{55} + 4 q^{56} - 22 q^{61} + 8 q^{62} - 2 q^{64} - 10 q^{68} - 22 q^{74} - 8 q^{77} - 8 q^{79} + 10 q^{82} + 4 q^{88} - 12 q^{92} - 4 q^{94} + 4 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^10 - 4 * q^14 + 2 * q^16 + 10 * q^17 - 4 * q^22 + 12 * q^23 + 8 * q^25 + 18 * q^29 - 4 * q^35 + 4 * q^38 + 2 * q^40 - 20 * q^43 + 6 * q^49 + 2 * q^53 - 4 * q^55 + 4 * q^56 - 22 * q^61 + 8 * q^62 - 2 * q^64 - 10 * q^68 - 22 * q^74 - 8 * q^77 - 8 * q^79 + 10 * q^82 + 4 * q^88 - 12 * q^92 - 4 * q^94 + 4 * q^95

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