LMFDB - Embedded newform 1014.2.b.a.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1014,2,Mod(337,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, 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("1014.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1014 = 2 \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1014.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:	$8.09683076496$
Analytic rank:	$1$
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			337.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1014.337
Dual form			1014.2.b.a.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -5.00000 q^{17} +1.00000i q^{18} +2.00000i q^{19} -1.00000i q^{20} +2.00000i q^{21} -2.00000 q^{22} -6.00000 q^{23} +1.00000i q^{24} +4.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} -9.00000 q^{29} +1.00000 q^{30} +4.00000i q^{31} +1.00000i q^{32} -2.00000i q^{33} -5.00000i q^{34} +2.00000 q^{35} -1.00000 q^{36} -11.0000i q^{37} -2.00000 q^{38} +1.00000 q^{40} -5.00000i q^{41} -2.00000 q^{42} -10.0000 q^{43} -2.00000i q^{44} +1.00000i q^{45} -6.00000i q^{46} +2.00000i q^{47} -1.00000 q^{48} +3.00000 q^{49} +4.00000i q^{50} +5.00000 q^{51} -1.00000 q^{53} -1.00000i q^{54} -2.00000 q^{55} -2.00000 q^{56} -2.00000i q^{57} -9.00000i q^{58} -8.00000i q^{59} +1.00000i q^{60} -11.0000 q^{61} -4.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -2.00000i q^{67} +5.00000 q^{68} +6.00000 q^{69} +2.00000i q^{70} +14.0000i q^{71} -1.00000i q^{72} -13.0000i q^{73} +11.0000 q^{74} -4.00000 q^{75} -2.00000i q^{76} +4.00000 q^{77} -4.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +5.00000 q^{82} -6.00000i q^{83} -2.00000i q^{84} -5.00000i q^{85} -10.0000i q^{86} +9.00000 q^{87} +2.00000 q^{88} +2.00000i q^{89} -1.00000 q^{90} +6.00000 q^{92} -4.00000i q^{93} -2.00000 q^{94} -2.00000 q^{95} -1.00000i q^{96} +2.00000i q^{97} +3.00000i q^{98} +2.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} + 4 q^{14} + 2 q^{16} - 10 q^{17} - 4 q^{22} - 12 q^{23} + 8 q^{25} - 2 q^{27} - 18 q^{29} + 2 q^{30} + 4 q^{35} - 2 q^{36} - 4 q^{38} + 2 q^{40} - 4 q^{42} - 20 q^{43} - 2 q^{48} + 6 q^{49} + 10 q^{51} - 2 q^{53} - 4 q^{55} - 4 q^{56} - 22 q^{61} - 8 q^{62} - 2 q^{64} + 4 q^{66} + 10 q^{68} + 12 q^{69} + 22 q^{74} - 8 q^{75} + 8 q^{77} - 8 q^{79} + 2 q^{81} + 10 q^{82} + 18 q^{87} + 4 q^{88} - 2 q^{90} + 12 q^{92} - 4 q^{94} - 4 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^12 + 4 * q^14 + 2 * q^16 - 10 * q^17 - 4 * q^22 - 12 * q^23 + 8 * q^25 - 2 * q^27 - 18 * q^29 + 2 * q^30 + 4 * q^35 - 2 * q^36 - 4 * q^38 + 2 * q^40 - 4 * q^42 - 20 * q^43 - 2 * q^48 + 6 * q^49 + 10 * q^51 - 2 * q^53 - 4 * q^55 - 4 * q^56 - 22 * q^61 - 8 * q^62 - 2 * q^64 + 4 * q^66 + 10 * q^68 + 12 * q^69 + 22 * q^74 - 8 * q^75 + 8 * q^77 - 8 * q^79 + 2 * q^81 + 10 * q^82 + 18 * q^87 + 4 * q^88 - 2 * q^90 + 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 $\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$.

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

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$	1.00000i	0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$	− 1.00000i	− 0.408248i
$7$	− 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
$9$	1.00000	0.333333
$10$	−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	$0.902509\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	1.00000i	0.235702i
$19$	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
$21$	2.00000i	0.436436i
$22$	−2.00000	−0.426401
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000i	0.204124i
$25$	4.00000	0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	2.00000i	0.377964i
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	1.00000	0.182574
$31$	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
$33$	− 2.00000i	− 0.348155i
$34$	− 5.00000i	− 0.857493i
$35$	2.00000	0.338062
$36$	−1.00000	−0.166667
$37$	− 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	$-0.872325\pi$
$41$	− 5.00000i	− 0.780869i	0.920631	−	0.390434i	$-0.127675\pi$
$42$	−2.00000	−0.308607
$43$	−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
$45$	1.00000i	0.149071i
$46$	− 6.00000i	− 0.884652i
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	−1.00000	−0.144338
$49$	3.00000	0.428571
$50$	4.00000i	0.565685i
$51$	5.00000	0.700140
$52$	0	0
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	− 1.00000i	− 0.136083i
$55$	−2.00000	−0.269680
$56$	−2.00000	−0.267261
$57$	− 2.00000i	− 0.264906i
$58$	− 9.00000i	− 1.18176i
$59$	− 8.00000i	− 1.04151i	−0.853706	−	0.520756i	$-0.825650\pi$
$59$	− 8.00000i	− 1.04151i	0.853706	−	0.520756i	$-0.174350\pi$
$60$	1.00000i	0.129099i
$61$	−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
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	− 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
$69$	6.00000	0.722315
$70$	2.00000i	0.239046i
$71$	14.0000i	1.66149i	0.556650	+	0.830747i	$0.312086\pi$
$71$	14.0000i	1.66149i	−0.556650	+	0.830747i	$0.687914\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 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
$75$	−4.00000	−0.461880
$76$	− 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
$81$	1.00000	0.111111
$82$	5.00000	0.552158
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	− 2.00000i	− 0.218218i
$85$	− 5.00000i	− 0.542326i
$86$	− 10.0000i	− 1.07833i
$87$	9.00000	0.964901
$88$	2.00000	0.213201
$89$	2.00000i	0.212000i	0.994366	+	0.106000i	$0.0338043\pi$
$89$	2.00000i	0.212000i	−0.994366	+	0.106000i	$0.966196\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	6.00000	0.625543
$93$	− 4.00000i	− 0.414781i
$94$	−2.00000	−0.206284
$95$	−2.00000	−0.205196
$96$	− 1.00000i	− 0.102062i
$97$	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
$99$	2.00000i	0.201008i
$100$	−4.00000	−0.400000
$101$	5.00000	0.497519	0.248759	−	0.968565i	$-0.419977\pi$
$101$	5.00000	0.497519	0.248759	+	0.968565i	$0.419977\pi$
$102$	5.00000i	0.495074i
$103$	−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
$105$	−2.00000	−0.195180
$106$	− 1.00000i	− 0.0971286i
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	1.00000	0.0962250
$109$	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
$111$	11.0000i	1.04407i
$112$	− 2.00000i	− 0.188982i
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	2.00000	0.187317
$115$	− 6.00000i	− 0.559503i
$116$	9.00000	0.835629
$117$	0	0
$118$	8.00000	0.736460
$119$	10.0000i	0.916698i
$120$	−1.00000	−0.0912871
$121$	7.00000	0.636364
$122$	− 11.0000i	− 0.995893i
$123$	5.00000i	0.450835i
$124$	− 4.00000i	− 0.359211i
$125$	9.00000i	0.804984i
$126$	2.00000	0.178174
$127$	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
$129$	10.0000	0.880451
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	2.00000i	0.174078i
$133$	4.00000	0.346844
$134$	2.00000	0.172774
$135$	− 1.00000i	− 0.0860663i
$136$	5.00000i	0.428746i
$137$	17.0000i	1.45241i	0.687479	+	0.726204i	$0.258717\pi$
$137$	17.0000i	1.45241i	−0.687479	+	0.726204i	$0.741283\pi$
$138$	6.00000i	0.510754i
$139$	−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
$141$	− 2.00000i	− 0.168430i
$142$	−14.0000	−1.17485
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 9.00000i	− 0.747409i
$146$	13.0000	1.07589
$147$	−3.00000	−0.247436
$148$	11.0000i	0.904194i
$149$	− 3.00000i	− 0.245770i	−0.992421	−	0.122885i	$-0.960785\pi$
$149$	− 3.00000i	− 0.245770i	0.992421	−	0.122885i	$-0.0392146\pi$
$150$	− 4.00000i	− 0.326599i
$151$	− 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
$153$	−5.00000	−0.404226
$154$	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
$159$	1.00000	0.0793052
$160$	−1.00000	−0.0790569
$161$	12.0000i	0.945732i
$162$	1.00000i	0.0785674i
$163$	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
$165$	2.00000	0.155700
$166$	6.00000	0.465690
$167$	24.0000i	1.85718i	0.371113	+	0.928588i	$0.378976\pi$
$167$	24.0000i	1.85718i	−0.371113	+	0.928588i	$0.621024\pi$
$168$	2.00000	0.154303
$169$	0	0
$170$	5.00000	0.383482
$171$	2.00000i	0.152944i
$172$	10.0000	0.762493
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	9.00000i	0.682288i
$175$	− 8.00000i	− 0.604743i
$176$	2.00000i	0.150756i
$177$	8.00000i	0.601317i
$178$	−2.00000	−0.149906
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−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
$183$	11.0000	0.813143
$184$	6.00000i	0.442326i
$185$	11.0000	0.808736
$186$	4.00000	0.293294
$187$	− 10.0000i	− 0.731272i
$188$	− 2.00000i	− 0.145865i
$189$	2.00000i	0.145479i
$190$	− 2.00000i	− 0.145095i
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	1.00000	0.0721688
$193$	− 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	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	−2.00000	−0.142134
$199$	−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
$201$	2.00000i	0.141069i
$202$	5.00000i	0.351799i
$203$	18.0000i	1.26335i
$204$	−5.00000	−0.350070
$205$	5.00000	0.349215
$206$	− 10.0000i	− 0.696733i
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−4.00000	−0.276686
$210$	− 2.00000i	− 0.138013i
$211$	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
$213$	− 14.0000i	− 0.959264i
$214$	− 18.0000i	− 1.23045i
$215$	− 10.0000i	− 0.681994i
$216$	1.00000i	0.0680414i
$217$	8.00000	0.543075
$218$	−2.00000	−0.135457
$219$	13.0000i	0.878459i
$220$	2.00000	0.134840
$221$	0	0
$222$	−11.0000	−0.738272
$223$	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
$225$	4.00000	0.266667
$226$	− 3.00000i	− 0.199557i
$227$	− 14.0000i	− 0.929213i	−0.885517	−	0.464606i	$-0.846196\pi$
$227$	− 14.0000i	− 0.929213i	0.885517	−	0.464606i	$-0.153804\pi$
$228$	2.00000i	0.132453i
$229$	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
$231$	−4.00000	−0.263181
$232$	9.00000i	0.590879i
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−2.00000	−0.130466
$236$	8.00000i	0.520756i
$237$	4.00000	0.259828
$238$	−10.0000	−0.648204
$239$	6.00000i	0.388108i	0.980991	+	0.194054i	$0.0621637\pi$
$239$	6.00000i	0.388108i	−0.980991	+	0.194054i	$0.937836\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	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
$243$	−1.00000	−0.0641500
$244$	11.0000	0.704203
$245$	3.00000i	0.191663i
$246$	−5.00000	−0.318788
$247$	0	0
$248$	4.00000	0.254000
$249$	6.00000i	0.380235i
$250$	−9.00000	−0.569210
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	2.00000i	0.125988i
$253$	− 12.0000i	− 0.754434i
$254$	12.0000i	0.752947i
$255$	5.00000i	0.313112i
$256$	1.00000	0.0625000
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	10.0000i	0.622573i
$259$	−22.0000	−1.36701
$260$	0	0
$261$	−9.00000	−0.557086
$262$	− 8.00000i	− 0.494242i
$263$	14.0000	0.863277	0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000	0.863277	0.431638	+	0.902047i	$0.357936\pi$
$264$	−2.00000	−0.123091
$265$	− 1.00000i	− 0.0614295i
$266$	4.00000i	0.245256i
$267$	− 2.00000i	− 0.122398i
$268$	2.00000i	0.122169i
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	1.00000	0.0608581
$271$	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
$276$	−6.00000	−0.361158
$277$	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
$279$	4.00000i	0.239474i
$280$	− 2.00000i	− 0.119523i
$281$	25.0000i	1.49137i	0.666296	+	0.745687i	$0.267879\pi$
$281$	25.0000i	1.49137i	−0.666296	+	0.745687i	$0.732121\pi$
$282$	2.00000	0.119098
$283$	−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
$285$	2.00000	0.118470
$286$	0	0
$287$	−10.0000	−0.590281
$288$	1.00000i	0.0589256i
$289$	8.00000	0.470588
$290$	9.00000	0.528498
$291$	− 2.00000i	− 0.117242i
$292$	13.0000i	0.760767i
$293$	− 1.00000i	− 0.0584206i	−0.999573	−	0.0292103i	$-0.990701\pi$
$293$	− 1.00000i	− 0.0584206i	0.999573	−	0.0292103i	$-0.00929925\pi$
$294$	− 3.00000i	− 0.174964i
$295$	8.00000	0.465778
$296$	−11.0000	−0.639362
$297$	− 2.00000i	− 0.116052i
$298$	3.00000	0.173785
$299$	0	0
$300$	4.00000	0.230940
$301$	20.0000i	1.15278i
$302$	6.00000	0.345261
$303$	−5.00000	−0.287242
$304$	2.00000i	0.114708i
$305$	− 11.0000i	− 0.629858i
$306$	− 5.00000i	− 0.285831i
$307$	− 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
$309$	10.0000	0.568880
$310$	− 4.00000i	− 0.227185i
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	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
$315$	2.00000	0.112687
$316$	4.00000	0.225018
$317$	33.0000i	1.85346i	0.375722	+	0.926732i	$0.377395\pi$
$317$	33.0000i	1.85346i	−0.375722	+	0.926732i	$0.622605\pi$
$318$	1.00000i	0.0560772i
$319$	− 18.0000i	− 1.00781i
$320$	− 1.00000i	− 0.0559017i
$321$	18.0000	1.00466
$322$	−12.0000	−0.668734
$323$	− 10.0000i	− 0.556415i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	− 2.00000i	− 0.110600i
$328$	−5.00000	−0.276079
$329$	4.00000	0.220527
$330$	2.00000i	0.110096i
$331$	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
$333$	− 11.0000i	− 0.602796i
$334$	−24.0000	−1.31322
$335$	2.00000	0.109272
$336$	2.00000i	0.109109i
$337$	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
$339$	3.00000	0.162938
$340$	5.00000i	0.271163i
$341$	−8.00000	−0.433224
$342$	−2.00000	−0.108148
$343$	− 20.0000i	− 1.07990i
$344$	10.0000i	0.539164i
$345$	6.00000i	0.323029i
$346$	22.0000i	1.18273i
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	−9.00000	−0.482451
$349$	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	$-0.850565\pi$
$353$	− 17.0000i	− 0.904819i	0.891810	−	0.452409i	$-0.149435\pi$
$354$	−8.00000	−0.425195
$355$	−14.0000	−0.743043
$356$	− 2.00000i	− 0.106000i
$357$	− 10.0000i	− 0.529256i
$358$	6.00000i	0.317110i
$359$	− 30.0000i	− 1.58334i	−0.610949	−	0.791670i	$-0.709212\pi$
$359$	− 30.0000i	− 1.58334i	0.610949	−	0.791670i	$-0.290788\pi$
$360$	1.00000	0.0527046
$361$	15.0000	0.789474
$362$	− 5.00000i	− 0.262794i
$363$	−7.00000	−0.367405
$364$	0	0
$365$	13.0000	0.680451
$366$	11.0000i	0.574979i
$367$	−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
$369$	− 5.00000i	− 0.260290i
$370$	11.0000i	0.571863i
$371$	2.00000i	0.103835i
$372$	4.00000i	0.207390i
$373$	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
$375$	− 9.00000i	− 0.464758i
$376$	2.00000	0.103142
$377$	0	0
$378$	−2.00000	−0.102869
$379$	− 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
$381$	−12.0000	−0.614779
$382$	4.00000i	0.204658i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	1.00000i	0.0510310i
$385$	4.00000i	0.203859i
$386$	17.0000	0.865277
$387$	−10.0000	−0.508329
$388$	− 2.00000i	− 0.101535i
$389$	−19.0000	−0.963338	−0.481669	−	0.876353i	$-0.659969\pi$
$389$	−19.0000	−0.963338	−0.481669	+	0.876353i	$0.659969\pi$
$390$	0	0
$391$	30.0000	1.51717
$392$	− 3.00000i	− 0.151523i
$393$	8.00000	0.403547
$394$	6.00000	0.302276
$395$	− 4.00000i	− 0.201262i
$396$	− 2.00000i	− 0.100504i
$397$	− 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
$399$	−4.00000	−0.200250
$400$	4.00000	0.200000
$401$	− 27.0000i	− 1.34832i	−0.738587	−	0.674158i	$-0.764507\pi$
$401$	− 27.0000i	− 1.34832i	0.738587	−	0.674158i	$-0.235493\pi$
$402$	−2.00000	−0.0997509
$403$	0	0
$404$	−5.00000	−0.248759
$405$	1.00000i	0.0496904i
$406$	−18.0000	−0.893325
$407$	22.0000	1.09050
$408$	− 5.00000i	− 0.247537i
$409$	− 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
$411$	− 17.0000i	− 0.838548i
$412$	10.0000	0.492665
$413$	−16.0000	−0.787309
$414$	− 6.00000i	− 0.294884i
$415$	6.00000	0.294528
$416$	0	0
$417$	12.0000	0.587643
$418$	− 4.00000i	− 0.195646i
$419$	−32.0000	−1.56330	−0.781651	−	0.623716i	$-0.785622\pi$
$419$	−32.0000	−1.56330	−0.781651	+	0.623716i	$0.785622\pi$
$420$	2.00000	0.0975900
$421$	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
$423$	2.00000i	0.0972433i
$424$	1.00000i	0.0485643i
$425$	−20.0000	−0.970143
$426$	14.0000	0.678302
$427$	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	$0.0153384\pi$
$431$	2.00000i	0.0963366i	−0.998839	+	0.0481683i	$0.984662\pi$
$432$	−1.00000	−0.0481125
$433$	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
$435$	9.00000i	0.431517i
$436$	− 2.00000i	− 0.0957826i
$437$	− 12.0000i	− 0.574038i
$438$	−13.0000	−0.621164
$439$	−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
$441$	3.00000	0.142857
$442$	0	0
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	− 11.0000i	− 0.522037i
$445$	−2.00000	−0.0948091
$446$	−16.0000	−0.757622
$447$	3.00000i	0.141895i
$448$	2.00000i	0.0944911i
$449$	− 30.0000i	− 1.41579i	−0.706319	−	0.707894i	$-0.749646\pi$
$449$	− 30.0000i	− 1.41579i	0.706319	−	0.707894i	$-0.250354\pi$
$450$	4.00000i	0.188562i
$451$	10.0000	0.470882
$452$	3.00000	0.141108
$453$	6.00000i	0.281905i
$454$	14.0000	0.657053
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	− 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
$459$	5.00000	0.233380
$460$	6.00000i	0.279751i
$461$	− 3.00000i	− 0.139724i	−0.997557	−	0.0698620i	$-0.977744\pi$
$461$	− 3.00000i	− 0.139724i	0.997557	−	0.0698620i	$-0.0222559\pi$
$462$	− 4.00000i	− 0.186097i
$463$	− 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
$465$	4.00000	0.185496
$466$	6.00000i	0.277945i
$467$	22.0000	1.01804	0.509019	−	0.860755i	$-0.330008\pi$
$467$	22.0000	1.01804	0.509019	+	0.860755i	$0.330008\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	− 2.00000i	− 0.0922531i
$471$	7.00000	0.322543
$472$	−8.00000	−0.368230
$473$	− 20.0000i	− 0.919601i
$474$	4.00000i	0.183726i
$475$	8.00000i	0.367065i
$476$	− 10.0000i	− 0.458349i
$477$	−1.00000	−0.0457869
$478$	−6.00000	−0.274434
$479$	32.0000i	1.46212i	0.682315	+	0.731059i	$0.260973\pi$
$479$	32.0000i	1.46212i	−0.682315	+	0.731059i	$0.739027\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−7.00000	−0.318841
$483$	− 12.0000i	− 0.546019i
$484$	−7.00000	−0.318182
$485$	−2.00000	−0.0908153
$486$	− 1.00000i	− 0.0453609i
$487$	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
$489$	− 20.0000i	− 0.904431i
$490$	−3.00000	−0.135526
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	− 5.00000i	− 0.225417i
$493$	45.0000	2.02670
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	4.00000i	0.179605i
$497$	28.0000	1.25597
$498$	−6.00000	−0.268866
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 9.00000i	− 0.402492i
$501$	− 24.0000i	− 1.07224i
$502$	− 4.00000i	− 0.178529i
$503$	−14.0000	−0.624229	−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000	−0.624229	−0.312115	+	0.950044i	$0.601037\pi$
$504$	−2.00000	−0.0890871
$505$	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	$-0.892131\pi$
$509$	− 15.0000i	− 0.664863i	0.943127	−	0.332432i	$-0.107869\pi$
$510$	−5.00000	−0.221404
$511$	−26.0000	−1.15017
$512$	1.00000i	0.0441942i
$513$	− 2.00000i	− 0.0883022i
$514$	3.00000i	0.132324i
$515$	− 10.0000i	− 0.440653i
$516$	−10.0000	−0.440225
$517$	−4.00000	−0.175920
$518$	− 22.0000i	− 0.966625i
$519$	−22.0000	−0.965693
$520$	0	0
$521$	25.0000	1.09527	0.547635	−	0.836717i	$-0.315528\pi$
$521$	25.0000	1.09527	0.547635	+	0.836717i	$0.315528\pi$
$522$	− 9.00000i	− 0.393919i
$523$	−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
$525$	8.00000i	0.349149i
$526$	14.0000i	0.610429i
$527$	− 20.0000i	− 0.871214i
$528$	− 2.00000i	− 0.0870388i
$529$	13.0000	0.565217
$530$	1.00000	0.0434372
$531$	− 8.00000i	− 0.347170i
$532$	−4.00000	−0.173422
$533$	0	0
$534$	2.00000	0.0865485
$535$	− 18.0000i	− 0.778208i
$536$	−2.00000	−0.0863868
$537$	−6.00000	−0.258919
$538$	− 14.0000i	− 0.603583i
$539$	6.00000i	0.258438i
$540$	1.00000i	0.0430331i
$541$	− 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
$543$	5.00000	0.214571
$544$	− 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
$549$	−11.0000	−0.469469
$550$	−8.00000	−0.341121
$551$	− 18.0000i	− 0.766826i
$552$	− 6.00000i	− 0.255377i
$553$	8.00000i	0.340195i
$554$	11.0000i	0.467345i
$555$	−11.0000	−0.466924
$556$	12.0000	0.508913
$557$	− 9.00000i	− 0.381342i	−0.981654	−	0.190671i	$-0.938934\pi$
$557$	− 9.00000i	− 0.381342i	0.981654	−	0.190671i	$-0.0610664\pi$
$558$	−4.00000	−0.169334
$559$	0	0
$560$	2.00000	0.0845154
$561$	10.0000i	0.422200i
$562$	−25.0000	−1.05456
$563$	−40.0000	−1.68580	−0.842900	−	0.538071i	$-0.819153\pi$
$563$	−40.0000	−1.68580	−0.842900	+	0.538071i	$0.819153\pi$
$564$	2.00000i	0.0842152i
$565$	− 3.00000i	− 0.126211i
$566$	− 26.0000i	− 1.09286i
$567$	− 2.00000i	− 0.0839921i
$568$	14.0000	0.587427
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	2.00000i	0.0837708i
$571$	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
$573$	−4.00000	−0.167102
$574$	− 10.0000i	− 0.417392i
$575$	−24.0000	−1.00087
$576$	−1.00000	−0.0416667
$577$	− 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
$579$	17.0000i	0.706496i
$580$	9.00000i	0.373705i
$581$	−12.0000	−0.497844
$582$	2.00000	0.0829027
$583$	− 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	$0.229609\pi$
$587$	32.0000i	1.32078i	−0.750922	+	0.660391i	$0.770391\pi$
$588$	3.00000	0.123718
$589$	−8.00000	−0.329634
$590$	8.00000i	0.329355i
$591$	6.00000i	0.246807i
$592$	− 11.0000i	− 0.452097i
$593$	− 39.0000i	− 1.60154i	−0.598973	−	0.800769i	$-0.704424\pi$
$593$	− 39.0000i	− 1.60154i	0.598973	−	0.800769i	$-0.295576\pi$
$594$	2.00000	0.0820610
$595$	−10.0000	−0.409960
$596$	3.00000i	0.122885i
$597$	10.0000	0.409273
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	4.00000i	0.163299i
$601$	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
$603$	− 2.00000i	− 0.0814463i
$604$	6.00000i	0.244137i
$605$	7.00000i	0.284590i
$606$	− 5.00000i	− 0.203111i
$607$	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
$609$	− 18.0000i	− 0.729397i
$610$	11.0000	0.445377
$611$	0	0
$612$	5.00000	0.202113
$613$	− 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
$615$	−5.00000	−0.201619
$616$	− 4.00000i	− 0.161165i
$617$	15.0000i	0.603877i	0.953327	+	0.301939i	$0.0976338\pi$
$617$	15.0000i	0.603877i	−0.953327	+	0.301939i	$0.902366\pi$
$618$	10.0000i	0.402259i
$619$	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
$621$	6.00000	0.240772
$622$	− 6.00000i	− 0.240578i
$623$	4.00000	0.160257
$624$	0	0
$625$	11.0000	0.440000
$626$	6.00000i	0.239808i
$627$	4.00000	0.159745
$628$	7.00000	0.279330
$629$	55.0000i	2.19299i
$630$	2.00000i	0.0796819i
$631$	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
$633$	−24.0000	−0.953914
$634$	−33.0000	−1.31060
$635$	12.0000i	0.476205i
$636$	−1.00000	−0.0396526
$637$	0	0
$638$	18.0000	0.712627
$639$	14.0000i	0.553831i
$640$	1.00000	0.0395285
$641$	−5.00000	−0.197488	−0.0987441	−	0.995113i	$-0.531483\pi$
$641$	−5.00000	−0.197488	−0.0987441	+	0.995113i	$0.531483\pi$
$642$	18.0000i	0.710403i
$643$	− 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
$645$	10.0000i	0.393750i
$646$	10.0000	0.393445
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	16.0000	0.628055
$650$	0	0
$651$	−8.00000	−0.313545
$652$	− 20.0000i	− 0.783260i
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	2.00000	0.0782062
$655$	− 8.00000i	− 0.312586i
$656$	− 5.00000i	− 0.195217i
$657$	− 13.0000i	− 0.507178i
$658$	4.00000i	0.155936i
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	−2.00000	−0.0778499
$661$	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
$666$	11.0000	0.426241
$667$	54.0000	2.09089
$668$	− 24.0000i	− 0.928588i
$669$	− 16.0000i	− 0.618596i
$670$	2.00000i	0.0772667i
$671$	− 22.0000i	− 0.849301i
$672$	−2.00000	−0.0771517
$673$	−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
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−46.0000	−1.76792	−0.883962	−	0.467559i	$-0.845134\pi$
$677$	−46.0000	−1.76792	−0.883962	+	0.467559i	$0.845134\pi$
$678$	3.00000i	0.115214i
$679$	4.00000	0.153506
$680$	−5.00000	−0.191741
$681$	14.0000i	0.536481i
$682$	− 8.00000i	− 0.306336i
$683$	− 40.0000i	− 1.53056i	−0.643699	−	0.765279i	$-0.722601\pi$
$683$	− 40.0000i	− 1.53056i	0.643699	−	0.765279i	$-0.277399\pi$
$684$	− 2.00000i	− 0.0764719i
$685$	−17.0000	−0.649537
$686$	20.0000	0.763604
$687$	− 10.0000i	− 0.381524i
$688$	−10.0000	−0.381246
$689$	0	0
$690$	−6.00000	−0.228416
$691$	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
$693$	4.00000	0.151947
$694$	6.00000i	0.227757i
$695$	− 12.0000i	− 0.455186i
$696$	− 9.00000i	− 0.341144i
$697$	25.0000i	0.946943i
$698$	−6.00000	−0.227103
$699$	−6.00000	−0.226941
$700$	8.00000i	0.302372i
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	0	0
$703$	22.0000	0.829746
$704$	− 2.00000i	− 0.0753778i
$705$	2.00000	0.0753244
$706$	17.0000	0.639803
$707$	− 10.0000i	− 0.376089i
$708$	− 8.00000i	− 0.300658i
$709$	− 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
$711$	−4.00000	−0.150012
$712$	2.00000	0.0749532
$713$	− 24.0000i	− 0.898807i
$714$	10.0000	0.374241
$715$	0	0
$716$	−6.00000	−0.224231
$717$	− 6.00000i	− 0.224074i
$718$	30.0000	1.11959
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	1.00000i	0.0372678i
$721$	20.0000i	0.744839i
$722$	15.0000i	0.558242i
$723$	− 7.00000i	− 0.260333i
$724$	5.00000	0.185824
$725$	−36.0000	−1.33701
$726$	− 7.00000i	− 0.259794i
$727$	−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
$729$	1.00000	0.0370370
$730$	13.0000i	0.481152i
$731$	50.0000	1.84932
$732$	−11.0000	−0.406572
$733$	− 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
$735$	− 3.00000i	− 0.110657i
$736$	− 6.00000i	− 0.221163i
$737$	4.00000	0.147342
$738$	5.00000	0.184053
$739$	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	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	−4.00000	−0.146647
$745$	3.00000	0.109911
$746$	9.00000i	0.329513i
$747$	− 6.00000i	− 0.219529i
$748$	10.0000i	0.365636i
$749$	36.0000i	1.31541i
$750$	9.00000	0.328634
$751$	−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
$753$	4.00000	0.145768
$754$	0	0
$755$	6.00000	0.218362
$756$	− 2.00000i	− 0.0727393i
$757$	−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
$759$	12.0000i	0.435572i
$760$	2.00000i	0.0725476i
$761$	34.0000i	1.23250i	0.787551	+	0.616250i	$0.211349\pi$
$761$	34.0000i	1.23250i	−0.787551	+	0.616250i	$0.788651\pi$
$762$	− 12.0000i	− 0.434714i
$763$	4.00000	0.144810
$764$	−4.00000	−0.144715
$765$	− 5.00000i	− 0.180775i
$766$	24.0000	0.867155
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	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
$771$	−3.00000	−0.108042
$772$	17.0000i	0.611843i
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	− 10.0000i	− 0.359443i
$775$	16.0000i	0.574737i
$776$	2.00000	0.0717958
$777$	22.0000	0.789246
$778$	− 19.0000i	− 0.681183i
$779$	10.0000	0.358287
$780$	0	0
$781$	−28.0000	−1.00192
$782$	30.0000i	1.07280i
$783$	9.00000	0.321634
$784$	3.00000	0.107143
$785$	− 7.00000i	− 0.249841i
$786$	8.00000i	0.285351i
$787$	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
$789$	−14.0000	−0.498413
$790$	4.00000	0.142314
$791$	6.00000i	0.213335i
$792$	2.00000	0.0710669
$793$	0	0
$794$	18.0000	0.638796
$795$	1.00000i	0.0354663i
$796$	10.0000	0.354441
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	− 4.00000i	− 0.141598i
$799$	− 10.0000i	− 0.353775i
$800$	4.00000i	0.141421i
$801$	2.00000i	0.0706665i
$802$	27.0000	0.953403
$803$	26.0000	0.917520
$804$	− 2.00000i	− 0.0705346i
$805$	−12.0000	−0.422944
$806$	0	0
$807$	14.0000	0.492823
$808$	− 5.00000i	− 0.175899i
$809$	5.00000	0.175791	0.0878953	−	0.996130i	$-0.471986\pi$
$809$	5.00000	0.175791	0.0878953	+	0.996130i	$0.471986\pi$
$810$	−1.00000	−0.0351364
$811$	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
$813$	− 8.00000i	− 0.280572i
$814$	22.0000i	0.771100i
$815$	−20.0000	−0.700569
$816$	5.00000	0.175035
$817$	− 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	$0.175375\pi$
$821$	30.0000i	1.04701i	−0.852023	+	0.523504i	$0.824625\pi$
$822$	17.0000	0.592943
$823$	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
$825$	− 8.00000i	− 0.278524i
$826$	− 16.0000i	− 0.556711i
$827$	− 8.00000i	− 0.278187i	−0.990279	−	0.139094i	$-0.955581\pi$
$827$	− 8.00000i	− 0.278187i	0.990279	−	0.139094i	$-0.0444189\pi$
$828$	6.00000	0.208514
$829$	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
$831$	−11.0000	−0.381586
$832$	0	0
$833$	−15.0000	−0.519719
$834$	12.0000i	0.415526i
$835$	−24.0000	−0.830554
$836$	4.00000	0.138343
$837$	− 4.00000i	− 0.138260i
$838$	− 32.0000i	− 1.10542i
$839$	44.0000i	1.51905i	0.650479	+	0.759524i	$0.274568\pi$
$839$	44.0000i	1.51905i	−0.650479	+	0.759524i	$0.725432\pi$
$840$	2.00000i	0.0690066i
$841$	52.0000	1.79310
$842$	−23.0000	−0.792632
$843$	− 25.0000i	− 0.861046i
$844$	−24.0000	−0.826114
$845$	0	0
$846$	−2.00000	−0.0687614
$847$	− 14.0000i	− 0.481046i
$848$	−1.00000	−0.0343401
$849$	26.0000	0.892318
$850$	− 20.0000i	− 0.685994i
$851$	66.0000i	2.26245i
$852$	14.0000i	0.479632i
$853$	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
$855$	−2.00000	−0.0683986
$856$	18.0000i	0.615227i
$857$	−45.0000	−1.53717	−0.768585	−	0.639747i	$-0.779039\pi$
$857$	−45.0000	−1.53717	−0.768585	+	0.639747i	$0.779039\pi$
$858$	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
$861$	10.0000	0.340799
$862$	−2.00000	−0.0681203
$863$	− 46.0000i	− 1.56586i	−0.622111	−	0.782929i	$-0.713725\pi$
$863$	− 46.0000i	− 1.56586i	0.622111	−	0.782929i	$-0.286275\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	22.0000i	0.748022i
$866$	21.0000i	0.713609i
$867$	−8.00000	−0.271694
$868$	−8.00000	−0.271538
$869$	− 8.00000i	− 0.271381i
$870$	−9.00000	−0.305129
$871$	0	0
$872$	2.00000	0.0677285
$873$	2.00000i	0.0676897i
$874$	12.0000	0.405906
$875$	18.0000	0.608511
$876$	− 13.0000i	− 0.439229i
$877$	− 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
$879$	1.00000i	0.0337292i
$880$	−2.00000	−0.0674200
$881$	−17.0000	−0.572745	−0.286372	−	0.958118i	$-0.592449\pi$
$881$	−17.0000	−0.572745	−0.286372	+	0.958118i	$0.592449\pi$
$882$	3.00000i	0.101015i
$883$	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
$885$	−8.00000	−0.268917
$886$	20.0000i	0.671913i
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	11.0000	0.369136
$889$	− 24.0000i	− 0.804934i
$890$	− 2.00000i	− 0.0670402i
$891$	2.00000i	0.0670025i
$892$	− 16.0000i	− 0.535720i
$893$	−4.00000	−0.133855
$894$	−3.00000	−0.100335
$895$	6.00000i	0.200558i
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	30.0000	1.00111
$899$	− 36.0000i	− 1.20067i
$900$	−4.00000	−0.133333
$901$	5.00000	0.166574
$902$	10.0000i	0.332964i
$903$	− 20.0000i	− 0.665558i
$904$	3.00000i	0.0997785i
$905$	− 5.00000i	− 0.166206i
$906$	−6.00000	−0.199337
$907$	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
$909$	5.00000	0.165840
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	− 2.00000i	− 0.0662266i
$913$	12.0000	0.397142
$914$	3.00000	0.0992312
$915$	11.0000i	0.363649i
$916$	− 10.0000i	− 0.330409i
$917$	16.0000i	0.528367i
$918$	5.00000i	0.165025i
$919$	−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
$921$	14.0000i	0.461316i
$922$	3.00000	0.0987997
$923$	0	0
$924$	4.00000	0.131590
$925$	− 44.0000i	− 1.44671i
$926$	14.0000	0.460069
$927$	−10.0000	−0.328443
$928$	− 9.00000i	− 0.295439i
$929$	23.0000i	0.754606i	0.926090	+	0.377303i	$0.123148\pi$
$929$	23.0000i	0.754606i	−0.926090	+	0.377303i	$0.876852\pi$
$930$	4.00000i	0.131165i
$931$	6.00000i	0.196642i
$932$	−6.00000	−0.196537
$933$	6.00000	0.196431
$934$	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
$939$	−6.00000	−0.195803
$940$	2.00000	0.0652328
$941$	22.0000i	0.717180i	0.933495	+	0.358590i	$0.116742\pi$
$941$	22.0000i	0.717180i	−0.933495	+	0.358590i	$0.883258\pi$
$942$	7.00000i	0.228072i
$943$	30.0000i	0.976934i
$944$	− 8.00000i	− 0.260378i
$945$	−2.00000	−0.0650600
$946$	20.0000	0.650256
$947$	8.00000i	0.259965i	0.991516	+	0.129983i	$0.0414921\pi$
$947$	8.00000i	0.259965i	−0.991516	+	0.129983i	$0.958508\pi$
$948$	−4.00000	−0.129914
$949$	0	0
$950$	−8.00000	−0.259554
$951$	− 33.0000i	− 1.07010i
$952$	10.0000	0.324102
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	− 1.00000i	− 0.0323762i
$955$	4.00000i	0.129437i
$956$	− 6.00000i	− 0.194054i
$957$	18.0000i	0.581857i
$958$	−32.0000	−1.03387
$959$	34.0000	1.09792
$960$	1.00000i	0.0322749i
$961$	15.0000	0.483871
$962$	0	0
$963$	−18.0000	−0.580042
$964$	− 7.00000i	− 0.225455i
$965$	17.0000	0.547249
$966$	12.0000	0.386094
$967$	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
$969$	10.0000i	0.321246i
$970$	− 2.00000i	− 0.0642161i
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	1.00000	0.0320750
$973$	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	$-0.890951\pi$
$977$	− 21.0000i	− 0.671850i	0.941889	−	0.335925i	$-0.109049\pi$
$978$	20.0000	0.639529
$979$	−4.00000	−0.127841
$980$	− 3.00000i	− 0.0958315i
$981$	2.00000i	0.0638551i
$982$	30.0000i	0.957338i
$983$	60.0000i	1.91370i	0.290578	+	0.956851i	$0.406153\pi$
$983$	60.0000i	1.91370i	−0.290578	+	0.956851i	$0.593847\pi$
$984$	5.00000	0.159394
$985$	6.00000	0.191176
$986$	45.0000i	1.43309i
$987$	−4.00000	−0.127321
$988$	0	0
$989$	60.0000	1.90789
$990$	− 2.00000i	− 0.0635642i
$991$	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
$993$	− 28.0000i	− 0.888553i
$994$	28.0000i	0.888106i
$995$	− 10.0000i	− 0.317021i
$996$	− 6.00000i	− 0.190117i
$997$	−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
$999$	11.0000i	0.348025i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1014.2.b.a.337.2		2
3.2	odd	2		3042.2.b.d.1351.1		2
13.2	odd	12		1014.2.e.d.529.1		2
13.3	even	3		1014.2.i.e.823.1		4
13.4	even	6		1014.2.i.e.361.1		4
13.5	odd	4		1014.2.a.e.1.1		1
13.6	odd	12		1014.2.e.d.991.1		2
13.7	odd	12		78.2.e.b.55.1	✓	2
13.8	odd	4		1014.2.a.a.1.1		1
13.9	even	3		1014.2.i.e.361.2		4
13.10	even	6		1014.2.i.e.823.2		4
13.11	odd	12		78.2.e.b.61.1	yes	2
13.12	even	2	inner	1014.2.b.a.337.1		2
39.5	even	4		3042.2.a.d.1.1		1
39.8	even	4		3042.2.a.m.1.1		1
39.11	even	12		234.2.h.b.217.1		2
39.20	even	12		234.2.h.b.55.1		2
39.38	odd	2		3042.2.b.d.1351.2		2
52.7	even	12		624.2.q.b.289.1		2
52.11	even	12		624.2.q.b.529.1		2
52.31	even	4		8112.2.a.bb.1.1		1
52.47	even	4		8112.2.a.x.1.1		1
65.7	even	12		1950.2.z.b.1849.1		4
65.24	odd	12		1950.2.i.b.451.1		2
65.33	even	12		1950.2.z.b.1849.2		4
65.37	even	12		1950.2.z.b.1699.2		4
65.59	odd	12		1950.2.i.b.601.1		2
65.63	even	12		1950.2.z.b.1699.1		4
156.11	odd	12		1872.2.t.i.1153.1		2
156.59	odd	12		1872.2.t.i.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.e.b.55.1	✓	2	13.7	odd	12
78.2.e.b.61.1	yes	2	13.11	odd	12
234.2.h.b.55.1		2	39.20	even	12
234.2.h.b.217.1		2	39.11	even	12
624.2.q.b.289.1		2	52.7	even	12
624.2.q.b.529.1		2	52.11	even	12
1014.2.a.a.1.1		1	13.8	odd	4
1014.2.a.e.1.1		1	13.5	odd	4
1014.2.b.a.337.1		2	13.12	even	2	inner
1014.2.b.a.337.2		2	1.1	even	1	trivial
1014.2.e.d.529.1		2	13.2	odd	12
1014.2.e.d.991.1		2	13.6	odd	12
1014.2.i.e.361.1		4	13.4	even	6
1014.2.i.e.361.2		4	13.9	even	3
1014.2.i.e.823.1		4	13.3	even	3
1014.2.i.e.823.2		4	13.10	even	6
1872.2.t.i.289.1		2	156.59	odd	12
1872.2.t.i.1153.1		2	156.11	odd	12
1950.2.i.b.451.1		2	65.24	odd	12
1950.2.i.b.601.1		2	65.59	odd	12
1950.2.z.b.1699.1		4	65.63	even	12
1950.2.z.b.1699.2		4	65.37	even	12
1950.2.z.b.1849.1		4	65.7	even	12
1950.2.z.b.1849.2		4	65.33	even	12
3042.2.a.d.1.1		1	39.5	even	4
3042.2.a.m.1.1		1	39.8	even	4
3042.2.b.d.1351.1		2	3.2	odd	2
3042.2.b.d.1351.2		2	39.38	odd	2
8112.2.a.x.1.1		1	52.47	even	4
8112.2.a.bb.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:	\( N \)	\(=\)	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1014.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -5.00000 q^{17} +1.00000i q^{18} +2.00000i q^{19} -1.00000i q^{20} +2.00000i q^{21} -2.00000 q^{22} -6.00000 q^{23} +1.00000i q^{24} +4.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} -9.00000 q^{29} +1.00000 q^{30} +4.00000i q^{31} +1.00000i q^{32} -2.00000i q^{33} -5.00000i q^{34} +2.00000 q^{35} -1.00000 q^{36} -11.0000i q^{37} -2.00000 q^{38} +1.00000 q^{40} -5.00000i q^{41} -2.00000 q^{42} -10.0000 q^{43} -2.00000i q^{44} +1.00000i q^{45} -6.00000i q^{46} +2.00000i q^{47} -1.00000 q^{48} +3.00000 q^{49} +4.00000i q^{50} +5.00000 q^{51} -1.00000 q^{53} -1.00000i q^{54} -2.00000 q^{55} -2.00000 q^{56} -2.00000i q^{57} -9.00000i q^{58} -8.00000i q^{59} +1.00000i q^{60} -11.0000 q^{61} -4.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -2.00000i q^{67} +5.00000 q^{68} +6.00000 q^{69} +2.00000i q^{70} +14.0000i q^{71} -1.00000i q^{72} -13.0000i q^{73} +11.0000 q^{74} -4.00000 q^{75} -2.00000i q^{76} +4.00000 q^{77} -4.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +5.00000 q^{82} -6.00000i q^{83} -2.00000i q^{84} -5.00000i q^{85} -10.0000i q^{86} +9.00000 q^{87} +2.00000 q^{88} +2.00000i q^{89} -1.00000 q^{90} +6.00000 q^{92} -4.00000i q^{93} -2.00000 q^{94} -2.00000 q^{95} -1.00000i q^{96} +2.00000i q^{97} +3.00000i q^{98} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} + 4 q^{14} + 2 q^{16} - 10 q^{17} - 4 q^{22} - 12 q^{23} + 8 q^{25} - 2 q^{27} - 18 q^{29} + 2 q^{30} + 4 q^{35} - 2 q^{36} - 4 q^{38} + 2 q^{40} - 4 q^{42} - 20 q^{43} - 2 q^{48} + 6 q^{49} + 10 q^{51} - 2 q^{53} - 4 q^{55} - 4 q^{56} - 22 q^{61} - 8 q^{62} - 2 q^{64} + 4 q^{66} + 10 q^{68} + 12 q^{69} + 22 q^{74} - 8 q^{75} + 8 q^{77} - 8 q^{79} + 2 q^{81} + 10 q^{82} + 18 q^{87} + 4 q^{88} - 2 q^{90} + 12 q^{92} - 4 q^{94} - 4 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^12 + 4 * q^14 + 2 * q^16 - 10 * q^17 - 4 * q^22 - 12 * q^23 + 8 * q^25 - 2 * q^27 - 18 * q^29 + 2 * q^30 + 4 * q^35 - 2 * q^36 - 4 * q^38 + 2 * q^40 - 4 * q^42 - 20 * q^43 - 2 * q^48 + 6 * q^49 + 10 * q^51 - 2 * q^53 - 4 * q^55 - 4 * q^56 - 22 * q^61 - 8 * q^62 - 2 * q^64 + 4 * q^66 + 10 * q^68 + 12 * q^69 + 22 * q^74 - 8 * q^75 + 8 * q^77 - 8 * q^79 + 2 * q^81 + 10 * q^82 + 18 * q^87 + 4 * q^88 - 2 * q^90 + 12 * q^92 - 4 * q^94 - 4 * q^95

Introduction

L-functions

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Database

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