LMFDB - Embedded newform 2366.2.d.e.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2366,2,Mod(337,2366)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2366, 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("2366.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2366 = 2 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2366.d (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:	$18.8926051182$
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 182)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2366.337
Dual form			2366.2.d.e.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{6} -1.00000i q^{7} -1.00000i q^{8} -2.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{6} -1.00000i q^{7} -1.00000i q^{8} -2.00000 q^{9} +3.00000i q^{11} -1.00000 q^{12} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{18} +2.00000i q^{19} -1.00000i q^{21} -3.00000 q^{22} +3.00000 q^{23} -1.00000i q^{24} +5.00000 q^{25} -5.00000 q^{27} +1.00000i q^{28} +5.00000i q^{31} +1.00000i q^{32} +3.00000i q^{33} +2.00000 q^{36} +7.00000i q^{37} -2.00000 q^{38} +3.00000i q^{41} +1.00000 q^{42} -8.00000 q^{43} -3.00000i q^{44} +3.00000i q^{46} +3.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} +5.00000i q^{50} -12.0000 q^{53} -5.00000i q^{54} -1.00000 q^{56} +2.00000i q^{57} -6.00000i q^{59} -1.00000 q^{61} -5.00000 q^{62} +2.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +5.00000i q^{67} +3.00000 q^{69} +12.0000i q^{71} +2.00000i q^{72} -11.0000i q^{73} -7.00000 q^{74} +5.00000 q^{75} -2.00000i q^{76} +3.00000 q^{77} -1.00000 q^{79} +1.00000 q^{81} -3.00000 q^{82} +12.0000i q^{83} +1.00000i q^{84} -8.00000i q^{86} +3.00000 q^{88} +18.0000i q^{89} -3.00000 q^{92} +5.00000i q^{93} -3.00000 q^{94} +1.00000i q^{96} +17.0000i q^{97} -1.00000i q^{98} -6.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{4} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^4 - 4 * q^9	$ 2 q + 2 q^{3} - 2 q^{4} - 4 q^{9} - 2 q^{12} + 2 q^{14} + 2 q^{16} - 6 q^{22} + 6 q^{23} + 10 q^{25} - 10 q^{27} + 4 q^{36} - 4 q^{38} + 2 q^{42} - 16 q^{43} + 2 q^{48} - 2 q^{49} - 24 q^{53} - 2 q^{56} - 2 q^{61} - 10 q^{62} - 2 q^{64} - 6 q^{66} + 6 q^{69} - 14 q^{74} + 10 q^{75} + 6 q^{77} - 2 q^{79} + 2 q^{81} - 6 q^{82} + 6 q^{88} - 6 q^{92} - 6 q^{94}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^4 - 4 * q^9 - 2 * q^12 + 2 * q^14 + 2 * q^16 - 6 * q^22 + 6 * q^23 + 10 * q^25 - 10 * q^27 + 4 * q^36 - 4 * q^38 + 2 * q^42 - 16 * q^43 + 2 * q^48 - 2 * q^49 - 24 * q^53 - 2 * q^56 - 2 * q^61 - 10 * q^62 - 2 * q^64 - 6 * q^66 + 6 * q^69 - 14 * q^74 + 10 * q^75 + 6 * q^77 - 2 * q^79 + 2 * q^81 - 6 * q^82 + 6 * q^88 - 6 * q^92 - 6 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times$.

$n$	$339$	$2199$
$\chi(n)$	$1$	$-1$

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	−1.00000	−0.500000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	1.00000i	0.408248i
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	− 1.00000i	− 0.353553i
$9$	−2.00000	−0.666667
$10$	0	0
$11$	3.00000i	0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$	3.00000i	0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 2.00000i	− 0.471405i
$19$	2.00000i	0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$	2.00000i	0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	0	0
$21$	− 1.00000i	− 0.218218i
$22$	−3.00000	−0.639602
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	− 1.00000i	− 0.204124i
$25$	5.00000	1.00000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	1.00000i	0.188982i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	5.00000i	0.898027i	0.893525	+	0.449013i	$0.148224\pi$
$31$	5.00000i	0.898027i	−0.893525	+	0.449013i	$0.851776\pi$
$32$	1.00000i	0.176777i
$33$	3.00000i	0.522233i
$34$	0	0
$35$	0	0
$36$	2.00000	0.333333
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	0	0
$41$	3.00000i	0.468521i	0.972174	+	0.234261i	$0.0752669\pi$
$41$	3.00000i	0.468521i	−0.972174	+	0.234261i	$0.924733\pi$
$42$	1.00000	0.154303
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	− 3.00000i	− 0.452267i
$45$	0	0
$46$	3.00000i	0.442326i
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	1.00000	0.144338
$49$	−1.00000	−0.142857
$50$	5.00000i	0.707107i
$51$	0	0
$52$	0	0
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	− 5.00000i	− 0.680414i
$55$	0	0
$56$	−1.00000	−0.133631
$57$	2.00000i	0.264906i
$58$	0	0
$59$	− 6.00000i	− 0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$	− 6.00000i	− 0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−5.00000	−0.635001
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	5.00000i	0.610847i	0.952217	+	0.305424i	$0.0987981\pi$
$67$	5.00000i	0.610847i	−0.952217	+	0.305424i	$0.901202\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	12.0000i	1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$	12.0000i	1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$	2.00000i	0.235702i
$73$	− 11.0000i	− 1.28745i	−0.765256	−	0.643726i	$-0.777388\pi$
$73$	− 11.0000i	− 1.28745i	0.765256	−	0.643726i	$-0.222612\pi$
$74$	−7.00000	−0.813733
$75$	5.00000	0.577350
$76$	− 2.00000i	− 0.229416i
$77$	3.00000	0.341882
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.00000	−0.331295
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	1.00000i	0.109109i
$85$	0	0
$86$	− 8.00000i	− 0.862662i
$87$	0	0
$88$	3.00000	0.319801
$89$	18.0000i	1.90800i	0.299813	+	0.953998i	$0.403076\pi$
$89$	18.0000i	1.90800i	−0.299813	+	0.953998i	$0.596924\pi$
$90$	0	0
$91$	0	0
$92$	−3.00000	−0.312772
$93$	5.00000i	0.518476i
$94$	−3.00000	−0.309426
$95$	0	0
$96$	1.00000i	0.102062i
$97$	17.0000i	1.72609i	0.505128	+	0.863044i	$0.331445\pi$
$97$	17.0000i	1.72609i	−0.505128	+	0.863044i	$0.668555\pi$
$98$	− 1.00000i	− 0.101015i
$99$	− 6.00000i	− 0.603023i
$100$	−5.00000	−0.500000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	0	0
$106$	− 12.0000i	− 1.16554i
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	5.00000	0.481125
$109$	2.00000i	0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$	2.00000i	0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$	0	0
$111$	7.00000i	0.664411i
$112$	− 1.00000i	− 0.0944911i
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	0	0
$117$	0	0
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	2.00000	0.181818
$122$	− 1.00000i	− 0.0905357i
$123$	3.00000i	0.270501i
$124$	− 5.00000i	− 0.449013i
$125$	0	0
$126$	−2.00000	−0.178174
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	− 3.00000i	− 0.261116i
$133$	2.00000	0.173422
$134$	−5.00000	−0.431934
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	3.00000i	0.255377i
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	3.00000i	0.252646i
$142$	−12.0000	−1.00702
$143$	0	0
$144$	−2.00000	−0.166667
$145$	0	0
$146$	11.0000	0.910366
$147$	−1.00000	−0.0824786
$148$	− 7.00000i	− 0.575396i
$149$	− 15.0000i	− 1.22885i	−0.788976	−	0.614424i	$-0.789388\pi$
$149$	− 15.0000i	− 1.22885i	0.788976	−	0.614424i	$-0.210612\pi$
$150$	5.00000i	0.408248i
$151$	− 8.00000i	− 0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$	− 8.00000i	− 0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	3.00000i	0.241747i
$155$	0	0
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	− 1.00000i	− 0.0795557i
$159$	−12.0000	−0.951662
$160$	0	0
$161$	− 3.00000i	− 0.236433i
$162$	1.00000i	0.0785674i
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	− 3.00000i	− 0.234261i
$165$	0	0
$166$	−12.0000	−0.931381
$167$	24.0000i	1.85718i	0.371113	+	0.928588i	$0.378976\pi$
$167$	24.0000i	1.85718i	−0.371113	+	0.928588i	$0.621024\pi$
$168$	−1.00000	−0.0771517
$169$	0	0
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	8.00000	0.609994
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	− 5.00000i	− 0.377964i
$176$	3.00000i	0.226134i
$177$	− 6.00000i	− 0.450988i
$178$	−18.0000	−1.34916
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	− 3.00000i	− 0.221163i
$185$	0	0
$186$	−5.00000	−0.366618
$187$	0	0
$188$	− 3.00000i	− 0.218797i
$189$	5.00000i	0.363696i
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1.00000	−0.0721688
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	−17.0000	−1.22053
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 3.00000i	− 0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$	− 3.00000i	− 0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$	6.00000	0.426401
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	− 5.00000i	− 0.353553i
$201$	5.00000i	0.352673i
$202$	3.00000i	0.211079i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	− 14.0000i	− 0.975426i
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	12.0000	0.824163
$213$	12.0000i	0.822226i
$214$	0	0
$215$	0	0
$216$	5.00000i	0.340207i
$217$	5.00000	0.339422
$218$	−2.00000	−0.135457
$219$	− 11.0000i	− 0.743311i
$220$	0	0
$221$	0	0
$222$	−7.00000	−0.469809
$223$	− 1.00000i	− 0.0669650i	−0.999439	−	0.0334825i	$-0.989340\pi$
$223$	− 1.00000i	− 0.0669650i	0.999439	−	0.0334825i	$-0.0106598\pi$
$224$	1.00000	0.0668153
$225$	−10.0000	−0.666667
$226$	9.00000i	0.598671i
$227$	− 24.0000i	− 1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$	− 24.0000i	− 1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$	− 2.00000i	− 0.132453i
$229$	4.00000i	0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	4.00000i	0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	−3.00000	−0.196537	−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000	−0.196537	−0.0982683	+	0.995160i	$0.531330\pi$
$234$	0	0
$235$	0	0
$236$	6.00000i	0.390567i
$237$	−1.00000	−0.0649570
$238$	0	0
$239$	6.00000i	0.388108i	0.980991	+	0.194054i	$0.0621637\pi$
$239$	6.00000i	0.388108i	−0.980991	+	0.194054i	$0.937836\pi$
$240$	0	0
$241$	− 26.0000i	− 1.67481i	−0.546585	−	0.837404i	$-0.684072\pi$
$241$	− 26.0000i	− 1.67481i	0.546585	−	0.837404i	$-0.315928\pi$
$242$	2.00000i	0.128565i
$243$	16.0000	1.02640
$244$	1.00000	0.0640184
$245$	0	0
$246$	−3.00000	−0.191273
$247$	0	0
$248$	5.00000	0.317500
$249$	12.0000i	0.760469i
$250$	0	0
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	− 2.00000i	− 0.125988i
$253$	9.00000i	0.565825i
$254$	7.00000i	0.439219i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	− 8.00000i	− 0.498058i
$259$	7.00000	0.434959
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	2.00000i	0.122628i
$267$	18.0000i	1.10158i
$268$	− 5.00000i	− 0.305424i
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	− 11.0000i	− 0.668202i	−0.942537	−	0.334101i	$-0.891567\pi$
$271$	− 11.0000i	− 0.668202i	0.942537	−	0.334101i	$-0.108433\pi$
$272$	0	0
$273$	0	0
$274$	−6.00000	−0.362473
$275$	15.0000i	0.904534i
$276$	−3.00000	−0.180579
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	− 4.00000i	− 0.239904i
$279$	− 10.0000i	− 0.598684i
$280$	0	0
$281$	− 12.0000i	− 0.715860i	−0.933748	−	0.357930i	$-0.883483\pi$
$281$	− 12.0000i	− 0.715860i	0.933748	−	0.357930i	$-0.116517\pi$
$282$	−3.00000	−0.178647
$283$	31.0000	1.84276	0.921379	−	0.388664i	$-0.127063\pi$
$283$	31.0000	1.84276	0.921379	+	0.388664i	$0.127063\pi$
$284$	− 12.0000i	− 0.712069i
$285$	0	0
$286$	0	0
$287$	3.00000	0.177084
$288$	− 2.00000i	− 0.117851i
$289$	−17.0000	−1.00000
$290$	0	0
$291$	17.0000i	0.996558i
$292$	11.0000i	0.643726i
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	− 1.00000i	− 0.0583212i
$295$	0	0
$296$	7.00000	0.406867
$297$	− 15.0000i	− 0.870388i
$298$	15.0000	0.868927
$299$	0	0
$300$	−5.00000	−0.288675
$301$	8.00000i	0.461112i
$302$	8.00000	0.460348
$303$	3.00000	0.172345
$304$	2.00000i	0.114708i
$305$	0	0
$306$	0	0
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	−3.00000	−0.170941
$309$	−14.0000	−0.796432
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	− 13.0000i	− 0.733632i
$315$	0	0
$316$	1.00000	0.0562544
$317$	− 3.00000i	− 0.168497i	−0.996445	−	0.0842484i	$-0.973151\pi$
$317$	− 3.00000i	− 0.168497i	0.996445	−	0.0842484i	$-0.0268489\pi$
$318$	− 12.0000i	− 0.672927i
$319$	0	0
$320$	0	0
$321$	0	0
$322$	3.00000	0.167183
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	20.0000	1.10770
$327$	2.00000i	0.110600i
$328$	3.00000	0.165647
$329$	3.00000	0.165395
$330$	0	0
$331$	− 19.0000i	− 1.04433i	−0.852843	−	0.522167i	$-0.825124\pi$
$331$	− 19.0000i	− 1.04433i	0.852843	−	0.522167i	$-0.174876\pi$
$332$	− 12.0000i	− 0.658586i
$333$	− 14.0000i	− 0.767195i
$334$	−24.0000	−1.31322
$335$	0	0
$336$	− 1.00000i	− 0.0545545i
$337$	31.0000	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$338$	0	0
$339$	9.00000	0.488813
$340$	0	0
$341$	−15.0000	−0.812296
$342$	4.00000	0.216295
$343$	1.00000i	0.0539949i
$344$	8.00000i	0.431331i
$345$	0	0
$346$	18.0000i	0.967686i
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	10.0000i	0.535288i	0.963518	+	0.267644i	$0.0862451\pi$
$349$	10.0000i	0.535288i	−0.963518	+	0.267644i	$0.913755\pi$
$350$	5.00000	0.267261
$351$	0	0
$352$	−3.00000	−0.159901
$353$	15.0000i	0.798369i	0.916871	+	0.399185i	$0.130707\pi$
$353$	15.0000i	0.798369i	−0.916871	+	0.399185i	$0.869293\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	− 18.0000i	− 0.953998i
$357$	0	0
$358$	6.00000i	0.317110i
$359$	24.0000i	1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$	24.0000i	1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	7.00000i	0.367912i
$363$	2.00000	0.104973
$364$	0	0
$365$	0	0
$366$	− 1.00000i	− 0.0522708i
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	3.00000	0.156386
$369$	− 6.00000i	− 0.312348i
$370$	0	0
$371$	12.0000i	0.623009i
$372$	− 5.00000i	− 0.259238i
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	0	0
$376$	3.00000	0.154713
$377$	0	0
$378$	−5.00000	−0.257172
$379$	− 16.0000i	− 0.821865i	−0.911666	−	0.410932i	$-0.865203\pi$
$379$	− 16.0000i	− 0.821865i	0.911666	−	0.410932i	$-0.134797\pi$
$380$	0	0
$381$	7.00000	0.358621
$382$	0	0
$383$	3.00000i	0.153293i	0.997058	+	0.0766464i	$0.0244213\pi$
$383$	3.00000i	0.153293i	−0.997058	+	0.0766464i	$0.975579\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	0	0
$386$	−4.00000	−0.203595
$387$	16.0000	0.813326
$388$	− 17.0000i	− 0.863044i
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	0	0
$392$	1.00000i	0.0505076i
$393$	0	0
$394$	3.00000	0.151138
$395$	0	0
$396$	6.00000i	0.301511i
$397$	− 20.0000i	− 1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$	− 20.0000i	− 1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$	16.0000i	0.802008i
$399$	2.00000	0.100125
$400$	5.00000	0.250000
$401$	− 24.0000i	− 1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$	− 24.0000i	− 1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$	−5.00000	−0.249377
$403$	0	0
$404$	−3.00000	−0.149256
$405$	0	0
$406$	0	0
$407$	−21.0000	−1.04093
$408$	0	0
$409$	− 22.0000i	− 1.08783i	−0.839140	−	0.543915i	$-0.816941\pi$
$409$	− 22.0000i	− 1.08783i	0.839140	−	0.543915i	$-0.183059\pi$
$410$	0	0
$411$	6.00000i	0.295958i
$412$	14.0000	0.689730
$413$	−6.00000	−0.295241
$414$	− 6.00000i	− 0.294884i
$415$	0	0
$416$	0	0
$417$	−4.00000	−0.195881
$418$	− 6.00000i	− 0.293470i
$419$	21.0000	1.02592	0.512959	−	0.858413i	$-0.328549\pi$
$419$	21.0000	1.02592	0.512959	+	0.858413i	$0.328549\pi$
$420$	0	0
$421$	− 37.0000i	− 1.80327i	−0.432498	−	0.901635i	$-0.642368\pi$
$421$	− 37.0000i	− 1.80327i	0.432498	−	0.901635i	$-0.357632\pi$
$422$	− 22.0000i	− 1.07094i
$423$	− 6.00000i	− 0.291730i
$424$	12.0000i	0.582772i
$425$	0	0
$426$	−12.0000	−0.581402
$427$	1.00000i	0.0483934i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 6.00000i	− 0.289010i	−0.989504	−	0.144505i	$-0.953841\pi$
$431$	− 6.00000i	− 0.289010i	0.989504	−	0.144505i	$-0.0461589\pi$
$432$	−5.00000	−0.240563
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	5.00000i	0.240008i
$435$	0	0
$436$	− 2.00000i	− 0.0957826i
$437$	6.00000i	0.287019i
$438$	11.0000	0.525600
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	− 7.00000i	− 0.332205i
$445$	0	0
$446$	1.00000	0.0473514
$447$	− 15.0000i	− 0.709476i
$448$	1.00000i	0.0472456i
$449$	6.00000i	0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$	6.00000i	0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$	− 10.0000i	− 0.471405i
$451$	−9.00000	−0.423793
$452$	−9.00000	−0.423324
$453$	− 8.00000i	− 0.375873i
$454$	24.0000	1.12638
$455$	0	0
$456$	2.00000	0.0936586
$457$	26.0000i	1.21623i	0.793849	+	0.608114i	$0.208074\pi$
$457$	26.0000i	1.21623i	−0.793849	+	0.608114i	$0.791926\pi$
$458$	−4.00000	−0.186908
$459$	0	0
$460$	0	0
$461$	24.0000i	1.11779i	0.829238	+	0.558896i	$0.188775\pi$
$461$	24.0000i	1.11779i	−0.829238	+	0.558896i	$0.811225\pi$
$462$	3.00000i	0.139573i
$463$	40.0000i	1.85896i	0.368875	+	0.929479i	$0.379743\pi$
$463$	40.0000i	1.85896i	−0.368875	+	0.929479i	$0.620257\pi$
$464$	0	0
$465$	0	0
$466$	− 3.00000i	− 0.138972i
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	5.00000	0.230879
$470$	0	0
$471$	−13.0000	−0.599008
$472$	−6.00000	−0.276172
$473$	− 24.0000i	− 1.10352i
$474$	− 1.00000i	− 0.0459315i
$475$	10.0000i	0.458831i
$476$	0	0
$477$	24.0000	1.09888
$478$	−6.00000	−0.274434
$479$	− 24.0000i	− 1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$	− 24.0000i	− 1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$	0	0
$481$	0	0
$482$	26.0000	1.18427
$483$	− 3.00000i	− 0.136505i
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	16.0000i	0.725775i
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	1.00000i	0.0452679i
$489$	− 20.0000i	− 0.904431i
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	− 3.00000i	− 0.135250i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	5.00000i	0.224507i
$497$	12.0000	0.538274
$498$	−12.0000	−0.537733
$499$	23.0000i	1.02962i	0.857304	+	0.514811i	$0.172138\pi$
$499$	23.0000i	1.02962i	−0.857304	+	0.514811i	$0.827862\pi$
$500$	0	0
$501$	24.0000i	1.07224i
$502$	15.0000i	0.669483i
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	−9.00000	−0.400099
$507$	0	0
$508$	−7.00000	−0.310575
$509$	− 30.0000i	− 1.32973i	−0.746965	−	0.664863i	$-0.768490\pi$
$509$	− 30.0000i	− 1.32973i	0.746965	−	0.664863i	$-0.231510\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	1.00000i	0.0441942i
$513$	− 10.0000i	− 0.441511i
$514$	− 24.0000i	− 1.05859i
$515$	0	0
$516$	8.00000	0.352180
$517$	−9.00000	−0.395820
$518$	7.00000i	0.307562i
$519$	18.0000	0.790112
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	−7.00000	−0.306089	−0.153044	−	0.988219i	$-0.548908\pi$
$523$	−7.00000	−0.306089	−0.153044	+	0.988219i	$0.548908\pi$
$524$	0	0
$525$	− 5.00000i	− 0.218218i
$526$	24.0000i	1.04645i
$527$	0	0
$528$	3.00000i	0.130558i
$529$	−14.0000	−0.608696
$530$	0	0
$531$	12.0000i	0.520756i
$532$	−2.00000	−0.0867110
$533$	0	0
$534$	−18.0000	−0.778936
$535$	0	0
$536$	5.00000	0.215967
$537$	6.00000	0.258919
$538$	− 9.00000i	− 0.388018i
$539$	− 3.00000i	− 0.129219i
$540$	0	0
$541$	34.0000i	1.46177i	0.682498	+	0.730887i	$0.260893\pi$
$541$	34.0000i	1.46177i	−0.682498	+	0.730887i	$0.739107\pi$
$542$	11.0000	0.472490
$543$	7.00000	0.300399
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	− 6.00000i	− 0.256307i
$549$	2.00000	0.0853579
$550$	−15.0000	−0.639602
$551$	0	0
$552$	− 3.00000i	− 0.127688i
$553$	1.00000i	0.0425243i
$554$	− 26.0000i	− 1.10463i
$555$	0	0
$556$	4.00000	0.169638
$557$	15.0000i	0.635570i	0.948163	+	0.317785i	$0.102939\pi$
$557$	15.0000i	0.635570i	−0.948163	+	0.317785i	$0.897061\pi$
$558$	10.0000	0.423334
$559$	0	0
$560$	0	0
$561$	0	0
$562$	12.0000	0.506189
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	− 3.00000i	− 0.126323i
$565$	0	0
$566$	31.0000i	1.30303i
$567$	− 1.00000i	− 0.0419961i
$568$	12.0000	0.503509
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	0	0
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	0	0
$573$	0	0
$574$	3.00000i	0.125218i
$575$	15.0000	0.625543
$576$	2.00000	0.0833333
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	− 17.0000i	− 0.707107i
$579$	4.00000i	0.166234i
$580$	0	0
$581$	12.0000	0.497844
$582$	−17.0000	−0.704673
$583$	− 36.0000i	− 1.49097i
$584$	−11.0000	−0.455183
$585$	0	0
$586$	18.0000	0.743573
$587$	18.0000i	0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$	18.0000i	0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$	1.00000	0.0412393
$589$	−10.0000	−0.412043
$590$	0	0
$591$	− 3.00000i	− 0.123404i
$592$	7.00000i	0.287698i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	15.0000	0.615457
$595$	0	0
$596$	15.0000i	0.614424i
$597$	16.0000	0.654836
$598$	0	0
$599$	39.0000	1.59350	0.796748	−	0.604311i	$-0.206552\pi$
$599$	39.0000	1.59350	0.796748	+	0.604311i	$0.206552\pi$
$600$	− 5.00000i	− 0.204124i
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−8.00000	−0.326056
$603$	− 10.0000i	− 0.407231i
$604$	8.00000i	0.325515i
$605$	0	0
$606$	3.00000i	0.121867i
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 25.0000i	− 1.00974i	−0.863195	−	0.504870i	$-0.831540\pi$
$613$	− 25.0000i	− 1.00974i	0.863195	−	0.504870i	$-0.168460\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	− 3.00000i	− 0.120873i
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	− 14.0000i	− 0.563163i
$619$	10.0000i	0.401934i	0.979598	+	0.200967i	$0.0644084\pi$
$619$	10.0000i	0.401934i	−0.979598	+	0.200967i	$0.935592\pi$
$620$	0	0
$621$	−15.0000	−0.601929
$622$	− 30.0000i	− 1.20289i
$623$	18.0000	0.721155
$624$	0	0
$625$	25.0000	1.00000
$626$	26.0000i	1.03917i
$627$	−6.00000	−0.239617
$628$	13.0000	0.518756
$629$	0	0
$630$	0	0
$631$	34.0000i	1.35352i	0.736204	+	0.676759i	$0.236616\pi$
$631$	34.0000i	1.35352i	−0.736204	+	0.676759i	$0.763384\pi$
$632$	1.00000i	0.0397779i
$633$	−22.0000	−0.874421
$634$	3.00000	0.119145
$635$	0	0
$636$	12.0000	0.475831
$637$	0	0
$638$	0	0
$639$	− 24.0000i	− 0.949425i
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	32.0000i	1.26196i	0.775800	+	0.630978i	$0.217346\pi$
$643$	32.0000i	1.26196i	−0.775800	+	0.630978i	$0.782654\pi$
$644$	3.00000i	0.118217i
$645$	0	0
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	18.0000	0.706562
$650$	0	0
$651$	5.00000	0.195965
$652$	20.0000i	0.783260i
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	3.00000i	0.117130i
$657$	22.0000i	0.858302i
$658$	3.00000i	0.116952i
$659$	42.0000	1.63609	0.818044	−	0.575156i	$-0.195059\pi$
$659$	42.0000	1.63609	0.818044	+	0.575156i	$0.195059\pi$
$660$	0	0
$661$	− 32.0000i	− 1.24466i	−0.782757	−	0.622328i	$-0.786187\pi$
$661$	− 32.0000i	− 1.24466i	0.782757	−	0.622328i	$-0.213813\pi$
$662$	19.0000	0.738456
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	14.0000	0.542489
$667$	0	0
$668$	− 24.0000i	− 0.928588i
$669$	− 1.00000i	− 0.0386622i
$670$	0	0
$671$	− 3.00000i	− 0.115814i
$672$	1.00000	0.0385758
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	31.0000i	1.19408i
$675$	−25.0000	−0.962250
$676$	0	0
$677$	−39.0000	−1.49889	−0.749446	−	0.662066i	$-0.769680\pi$
$677$	−39.0000	−1.49889	−0.749446	+	0.662066i	$0.769680\pi$
$678$	9.00000i	0.345643i
$679$	17.0000	0.652400
$680$	0	0
$681$	− 24.0000i	− 0.919682i
$682$	− 15.0000i	− 0.574380i
$683$	27.0000i	1.03313i	0.856249	+	0.516563i	$0.172789\pi$
$683$	27.0000i	1.03313i	−0.856249	+	0.516563i	$0.827211\pi$
$684$	4.00000i	0.152944i
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	4.00000i	0.152610i
$688$	−8.00000	−0.304997
$689$	0	0
$690$	0	0
$691$	8.00000i	0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$	8.00000i	0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$	−18.0000	−0.684257
$693$	−6.00000	−0.227921
$694$	24.0000i	0.911028i
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−10.0000	−0.378506
$699$	−3.00000	−0.113470
$700$	5.00000i	0.188982i
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	− 3.00000i	− 0.113067i
$705$	0	0
$706$	−15.0000	−0.564532
$707$	− 3.00000i	− 0.112827i
$708$	6.00000i	0.225494i
$709$	− 35.0000i	− 1.31445i	−0.753693	−	0.657226i	$-0.771730\pi$
$709$	− 35.0000i	− 1.31445i	0.753693	−	0.657226i	$-0.228270\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	18.0000	0.674579
$713$	15.0000i	0.561754i
$714$	0	0
$715$	0	0
$716$	−6.00000	−0.224231
$717$	6.00000i	0.224074i
$718$	−24.0000	−0.895672
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	14.0000i	0.521387i
$722$	15.0000i	0.558242i
$723$	− 26.0000i	− 0.966950i
$724$	−7.00000	−0.260153
$725$	0	0
$726$	2.00000i	0.0742270i
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	1.00000	0.0369611
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	− 28.0000i	− 1.03350i
$735$	0	0
$736$	3.00000i	0.110581i
$737$	−15.0000	−0.552532
$738$	6.00000	0.220863
$739$	16.0000i	0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$	16.0000i	0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$	0	0
$741$	0	0
$742$	−12.0000	−0.440534
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	5.00000	0.183309
$745$	0	0
$746$	− 4.00000i	− 0.146450i
$747$	− 24.0000i	− 0.878114i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	31.0000	1.13121	0.565603	−	0.824678i	$-0.308643\pi$
$751$	31.0000	1.13121	0.565603	+	0.824678i	$0.308643\pi$
$752$	3.00000i	0.109399i
$753$	15.0000	0.546630
$754$	0	0
$755$	0	0
$756$	− 5.00000i	− 0.181848i
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	16.0000	0.581146
$759$	9.00000i	0.326679i
$760$	0	0
$761$	− 3.00000i	− 0.108750i	−0.998521	−	0.0543750i	$-0.982683\pi$
$761$	− 3.00000i	− 0.108750i	0.998521	−	0.0543750i	$-0.0173166\pi$
$762$	7.00000i	0.253583i
$763$	2.00000	0.0724049
$764$	0	0
$765$	0	0
$766$	−3.00000	−0.108394
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 13.0000i	− 0.468792i	−0.972141	−	0.234396i	$-0.924689\pi$
$769$	− 13.0000i	− 0.468792i	0.972141	−	0.234396i	$-0.0753112\pi$
$770$	0	0
$771$	−24.0000	−0.864339
$772$	− 4.00000i	− 0.143963i
$773$	− 54.0000i	− 1.94225i	−0.238581	−	0.971123i	$-0.576682\pi$
$773$	− 54.0000i	− 1.94225i	0.238581	−	0.971123i	$-0.423318\pi$
$774$	16.0000i	0.575108i
$775$	25.0000i	0.898027i
$776$	17.0000	0.610264
$777$	7.00000	0.251124
$778$	36.0000i	1.29066i
$779$	−6.00000	−0.214972
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	3.00000i	0.106871i
$789$	24.0000	0.854423
$790$	0	0
$791$	− 9.00000i	− 0.320003i
$792$	−6.00000	−0.213201
$793$	0	0
$794$	20.0000	0.709773
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−9.00000	−0.318796	−0.159398	−	0.987214i	$-0.550955\pi$
$797$	−9.00000	−0.318796	−0.159398	+	0.987214i	$0.550955\pi$
$798$	2.00000i	0.0707992i
$799$	0	0
$800$	5.00000i	0.176777i
$801$	− 36.0000i	− 1.27200i
$802$	24.0000	0.847469
$803$	33.0000	1.16454
$804$	− 5.00000i	− 0.176336i
$805$	0	0
$806$	0	0
$807$	−9.00000	−0.316815
$808$	− 3.00000i	− 0.105540i
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	20.0000i	0.702295i	0.936320	+	0.351147i	$0.114208\pi$
$811$	20.0000i	0.702295i	−0.936320	+	0.351147i	$0.885792\pi$
$812$	0	0
$813$	− 11.0000i	− 0.385787i
$814$	− 21.0000i	− 0.736050i
$815$	0	0
$816$	0	0
$817$	− 16.0000i	− 0.559769i
$818$	22.0000	0.769212
$819$	0	0
$820$	0	0
$821$	− 6.00000i	− 0.209401i	−0.994504	−	0.104701i	$-0.966612\pi$
$821$	− 6.00000i	− 0.209401i	0.994504	−	0.104701i	$-0.0333885\pi$
$822$	−6.00000	−0.209274
$823$	−41.0000	−1.42917	−0.714585	−	0.699549i	$-0.753384\pi$
$823$	−41.0000	−1.42917	−0.714585	+	0.699549i	$0.753384\pi$
$824$	14.0000i	0.487713i
$825$	15.0000i	0.522233i
$826$	− 6.00000i	− 0.208767i
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	6.00000	0.208514
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	0	0
$833$	0	0
$834$	− 4.00000i	− 0.138509i
$835$	0	0
$836$	6.00000	0.207514
$837$	− 25.0000i	− 0.864126i
$838$	21.0000i	0.725433i
$839$	21.0000i	0.725001i	0.931984	+	0.362500i	$0.118077\pi$
$839$	21.0000i	0.725001i	−0.931984	+	0.362500i	$0.881923\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	37.0000	1.27510
$843$	− 12.0000i	− 0.413302i
$844$	22.0000	0.757271
$845$	0	0
$846$	6.00000	0.206284
$847$	− 2.00000i	− 0.0687208i
$848$	−12.0000	−0.412082
$849$	31.0000	1.06392
$850$	0	0
$851$	21.0000i	0.719871i
$852$	− 12.0000i	− 0.411113i
$853$	− 44.0000i	− 1.50653i	−0.657716	−	0.753266i	$-0.728477\pi$
$853$	− 44.0000i	− 1.50653i	0.657716	−	0.753266i	$-0.271523\pi$
$854$	−1.00000	−0.0342193
$855$	0	0
$856$	0	0
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	23.0000	0.784750	0.392375	−	0.919805i	$-0.371654\pi$
$859$	23.0000	0.784750	0.392375	+	0.919805i	$0.371654\pi$
$860$	0	0
$861$	3.00000	0.102240
$862$	6.00000	0.204361
$863$	12.0000i	0.408485i	0.978920	+	0.204242i	$0.0654731\pi$
$863$	12.0000i	0.408485i	−0.978920	+	0.204242i	$0.934527\pi$
$864$	− 5.00000i	− 0.170103i
$865$	0	0
$866$	− 38.0000i	− 1.29129i
$867$	−17.0000	−0.577350
$868$	−5.00000	−0.169711
$869$	− 3.00000i	− 0.101768i
$870$	0	0
$871$	0	0
$872$	2.00000	0.0677285
$873$	− 34.0000i	− 1.15073i
$874$	−6.00000	−0.202953
$875$	0	0
$876$	11.0000i	0.371656i
$877$	− 13.0000i	− 0.438979i	−0.975615	−	0.219489i	$-0.929561\pi$
$877$	− 13.0000i	− 0.438979i	0.975615	−	0.219489i	$-0.0704391\pi$
$878$	− 26.0000i	− 0.877457i
$879$	− 18.0000i	− 0.607125i
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	2.00000i	0.0673435i
$883$	−2.00000	−0.0673054	−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000	−0.0673054	−0.0336527	+	0.999434i	$0.510714\pi$
$884$	0	0
$885$	0	0
$886$	24.0000i	0.806296i
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	7.00000	0.234905
$889$	− 7.00000i	− 0.234772i
$890$	0	0
$891$	3.00000i	0.100504i
$892$	1.00000i	0.0334825i
$893$	−6.00000	−0.200782
$894$	15.0000	0.501675
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	0	0
$900$	10.0000	0.333333
$901$	0	0
$902$	− 9.00000i	− 0.299667i
$903$	8.00000i	0.266223i
$904$	− 9.00000i	− 0.299336i
$905$	0	0
$906$	8.00000	0.265782
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	24.0000i	0.796468i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	2.00000i	0.0662266i
$913$	−36.0000	−1.19143
$914$	−26.0000	−0.860004
$915$	0	0
$916$	− 4.00000i	− 0.132164i
$917$	0	0
$918$	0	0
$919$	−25.0000	−0.824674	−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000	−0.824674	−0.412337	+	0.911031i	$0.635287\pi$
$920$	0	0
$921$	16.0000i	0.527218i
$922$	−24.0000	−0.790398
$923$	0	0
$924$	−3.00000	−0.0986928
$925$	35.0000i	1.15079i
$926$	−40.0000	−1.31448
$927$	28.0000	0.919641
$928$	0	0
$929$	− 9.00000i	− 0.295280i	−0.989041	−	0.147640i	$-0.952832\pi$
$929$	− 9.00000i	− 0.295280i	0.989041	−	0.147640i	$-0.0471678\pi$
$930$	0	0
$931$	− 2.00000i	− 0.0655474i
$932$	3.00000	0.0982683
$933$	−30.0000	−0.982156
$934$	− 12.0000i	− 0.392652i
$935$	0	0
$936$	0	0
$937$	−52.0000	−1.69877	−0.849383	−	0.527777i	$-0.823026\pi$
$937$	−52.0000	−1.69877	−0.849383	+	0.527777i	$0.823026\pi$
$938$	5.00000i	0.163256i
$939$	26.0000	0.848478
$940$	0	0
$941$	30.0000i	0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$	30.0000i	0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$	− 13.0000i	− 0.423563i
$943$	9.00000i	0.293080i
$944$	− 6.00000i	− 0.195283i
$945$	0	0
$946$	24.0000	0.780307
$947$	− 48.0000i	− 1.55979i	−0.625910	−	0.779895i	$-0.715272\pi$
$947$	− 48.0000i	− 1.55979i	0.625910	−	0.779895i	$-0.284728\pi$
$948$	1.00000	0.0324785
$949$	0	0
$950$	−10.0000	−0.324443
$951$	− 3.00000i	− 0.0972817i
$952$	0	0
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	24.0000i	0.777029i
$955$	0	0
$956$	− 6.00000i	− 0.194054i
$957$	0	0
$958$	24.0000	0.775405
$959$	6.00000	0.193750
$960$	0	0
$961$	6.00000	0.193548
$962$	0	0
$963$	0	0
$964$	26.0000i	0.837404i
$965$	0	0
$966$	3.00000	0.0965234
$967$	− 22.0000i	− 0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$	− 22.0000i	− 0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	0	0
$970$	0	0
$971$	−21.0000	−0.673922	−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000	−0.673922	−0.336961	+	0.941519i	$0.609399\pi$
$972$	−16.0000	−0.513200
$973$	4.00000i	0.128234i
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	20.0000	0.639529
$979$	−54.0000	−1.72585
$980$	0	0
$981$	− 4.00000i	− 0.127710i
$982$	− 12.0000i	− 0.382935i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	3.00000	0.0956365
$985$	0	0
$986$	0	0
$987$	3.00000	0.0954911
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	11.0000	0.349427	0.174713	−	0.984619i	$-0.444100\pi$
$991$	11.0000	0.349427	0.174713	+	0.984619i	$0.444100\pi$
$992$	−5.00000	−0.158750
$993$	− 19.0000i	− 0.602947i
$994$	12.0000i	0.380617i
$995$	0	0
$996$	− 12.0000i	− 0.380235i
$997$	−37.0000	−1.17180	−0.585901	−	0.810383i	$-0.699259\pi$
$997$	−37.0000	−1.17180	−0.585901	+	0.810383i	$0.699259\pi$
$998$	−23.0000	−0.728052
$999$	− 35.0000i	− 1.10735i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.2.d.e.337.2		2
13.5	odd	4		182.2.a.d.1.1	✓	1
13.8	odd	4		2366.2.a.e.1.1		1
13.12	even	2	inner	2366.2.d.e.337.1		2
39.5	even	4		1638.2.a.f.1.1		1
52.31	even	4		1456.2.a.d.1.1		1
65.44	odd	4		4550.2.a.c.1.1		1
91.5	even	12		1274.2.f.i.1145.1		2
91.18	odd	12		1274.2.f.d.79.1		2
91.31	even	12		1274.2.f.i.79.1		2
91.44	odd	12		1274.2.f.d.1145.1		2
91.83	even	4		1274.2.a.j.1.1		1
104.5	odd	4		5824.2.a.k.1.1		1
104.83	even	4		5824.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.a.d.1.1	✓	1	13.5	odd	4
1274.2.a.j.1.1		1	91.83	even	4
1274.2.f.d.79.1		2	91.18	odd	12
1274.2.f.d.1145.1		2	91.44	odd	12
1274.2.f.i.79.1		2	91.31	even	12
1274.2.f.i.1145.1		2	91.5	even	12
1456.2.a.d.1.1		1	52.31	even	4
1638.2.a.f.1.1		1	39.5	even	4
2366.2.a.e.1.1		1	13.8	odd	4
2366.2.d.e.337.1		2	13.12	even	2	inner
2366.2.d.e.337.2		2	1.1	even	1	trivial
4550.2.a.c.1.1		1	65.44	odd	4
5824.2.a.k.1.1		1	104.5	odd	4
5824.2.a.x.1.1		1	104.83	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:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2366.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{6} -1.00000i q^{7} -1.00000i q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{6} -1.00000i q^{7} -1.00000i q^{8} -2.00000 q^{9} +3.00000i q^{11} -1.00000 q^{12} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{18} +2.00000i q^{19} -1.00000i q^{21} -3.00000 q^{22} +3.00000 q^{23} -1.00000i q^{24} +5.00000 q^{25} -5.00000 q^{27} +1.00000i q^{28} +5.00000i q^{31} +1.00000i q^{32} +3.00000i q^{33} +2.00000 q^{36} +7.00000i q^{37} -2.00000 q^{38} +3.00000i q^{41} +1.00000 q^{42} -8.00000 q^{43} -3.00000i q^{44} +3.00000i q^{46} +3.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} +5.00000i q^{50} -12.0000 q^{53} -5.00000i q^{54} -1.00000 q^{56} +2.00000i q^{57} -6.00000i q^{59} -1.00000 q^{61} -5.00000 q^{62} +2.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +5.00000i q^{67} +3.00000 q^{69} +12.0000i q^{71} +2.00000i q^{72} -11.0000i q^{73} -7.00000 q^{74} +5.00000 q^{75} -2.00000i q^{76} +3.00000 q^{77} -1.00000 q^{79} +1.00000 q^{81} -3.00000 q^{82} +12.0000i q^{83} +1.00000i q^{84} -8.00000i q^{86} +3.00000 q^{88} +18.0000i q^{89} -3.00000 q^{92} +5.00000i q^{93} -3.00000 q^{94} +1.00000i q^{96} +17.0000i q^{97} -1.00000i q^{98} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^4 - 4 * q^9	\( 2 q + 2 q^{3} - 2 q^{4} - 4 q^{9} - 2 q^{12} + 2 q^{14} + 2 q^{16} - 6 q^{22} + 6 q^{23} + 10 q^{25} - 10 q^{27} + 4 q^{36} - 4 q^{38} + 2 q^{42} - 16 q^{43} + 2 q^{48} - 2 q^{49} - 24 q^{53} - 2 q^{56} - 2 q^{61} - 10 q^{62} - 2 q^{64} - 6 q^{66} + 6 q^{69} - 14 q^{74} + 10 q^{75} + 6 q^{77} - 2 q^{79} + 2 q^{81} - 6 q^{82} + 6 q^{88} - 6 q^{92} - 6 q^{94}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^4 - 4 * q^9 - 2 * q^12 + 2 * q^14 + 2 * q^16 - 6 * q^22 + 6 * q^23 + 10 * q^25 - 10 * q^27 + 4 * q^36 - 4 * q^38 + 2 * q^42 - 16 * q^43 + 2 * q^48 - 2 * q^49 - 24 * q^53 - 2 * q^56 - 2 * q^61 - 10 * q^62 - 2 * q^64 - 6 * q^66 + 6 * q^69 - 14 * q^74 + 10 * q^75 + 6 * q^77 - 2 * q^79 + 2 * q^81 - 6 * q^82 + 6 * q^88 - 6 * q^92 - 6 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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