LMFDB - Embedded newform 1690.2.d.c.1351.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1690,2,Mod(1351,1690)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1690, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1690.1351"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,0,0,0,-6,-2,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.4947179416$
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 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1690.1351
Dual form			1690.2.d.c.1351.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} -3.00000i q^{7} +1.00000i q^{8} -3.00000 q^{9} -1.00000 q^{10} -3.00000i q^{11} -3.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +3.00000i q^{18} -7.00000i q^{19} +1.00000i q^{20} -3.00000 q^{22} +4.00000 q^{23} -1.00000 q^{25} +3.00000i q^{28} -8.00000 q^{29} +10.0000i q^{31} -1.00000i q^{32} -4.00000i q^{34} -3.00000 q^{35} +3.00000 q^{36} +3.00000i q^{37} -7.00000 q^{38} +1.00000 q^{40} +2.00000i q^{41} -6.00000 q^{43} +3.00000i q^{44} +3.00000i q^{45} -4.00000i q^{46} -1.00000i q^{47} -2.00000 q^{49} +1.00000i q^{50} -9.00000 q^{53} -3.00000 q^{55} +3.00000 q^{56} +8.00000i q^{58} -4.00000i q^{59} -14.0000 q^{61} +10.0000 q^{62} +9.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} -4.00000 q^{68} +3.00000i q^{70} -6.00000i q^{71} -3.00000i q^{72} +4.00000i q^{73} +3.00000 q^{74} +7.00000i q^{76} -9.00000 q^{77} +10.0000 q^{79} -1.00000i q^{80} +9.00000 q^{81} +2.00000 q^{82} -4.00000i q^{85} +6.00000i q^{86} +3.00000 q^{88} +1.00000i q^{89} +3.00000 q^{90} -4.00000 q^{92} -1.00000 q^{94} -7.00000 q^{95} -16.0000i q^{97} +2.00000i q^{98} +9.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 6 q^{9} - 2 q^{10} - 6 q^{14} + 2 q^{16} + 8 q^{17} - 6 q^{22} + 8 q^{23} - 2 q^{25} - 16 q^{29} - 6 q^{35} + 6 q^{36} - 14 q^{38} + 2 q^{40} - 12 q^{43} - 4 q^{49} - 18 q^{53} - 6 q^{55}+ \cdots - 14 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 6 * q^9 - 2 * q^10 - 6 * q^14 + 2 * q^16 + 8 * q^17 - 6 * q^22 + 8 * q^23 - 2 * q^25 - 16 * q^29 - 6 * q^35 + 6 * q^36 - 14 * q^38 + 2 * q^40 - 12 * q^43 - 4 * q^49 - 18 * q^53 - 6 * q^55 + 6 * q^56 - 28 * q^61 + 20 * q^62 - 2 * q^64 - 8 * q^68 + 6 * q^74 - 18 * q^77 + 20 * q^79 + 18 * q^81 + 4 * q^82 + 6 * q^88 + 6 * q^90 - 8 * q^92 - 2 * q^94 - 14 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	1.00000i	0.353553i
$9$	−3.00000	−1.00000
$10$	−1.00000	−0.316228
$11$	− 3.00000i	− 0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$	− 3.00000i	− 0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	3.00000i	0.707107i
$19$	− 7.00000i	− 1.60591i	−0.596040	−	0.802955i	$-0.703260\pi$
$19$	− 7.00000i	− 1.60591i	0.596040	−	0.802955i	$-0.296740\pi$
$20$	1.00000i	0.223607i
$21$	0	0
$22$	−3.00000	−0.639602
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	3.00000i	0.566947i
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	10.0000i	1.79605i	0.439941	+	0.898027i	$0.354999\pi$
$31$	10.0000i	1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 4.00000i	− 0.685994i
$35$	−3.00000	−0.507093
$36$	3.00000	0.500000
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	−7.00000	−1.13555
$39$	0	0
$40$	1.00000	0.158114
$41$	2.00000i	0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$	2.00000i	0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	3.00000i	0.452267i
$45$	3.00000i	0.447214i
$46$	− 4.00000i	− 0.589768i
$47$	− 1.00000i	− 0.145865i	−0.997337	−	0.0729325i	$-0.976764\pi$
$47$	− 1.00000i	− 0.145865i	0.997337	−	0.0729325i	$-0.0232358\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	1.00000i	0.141421i
$51$	0	0
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	3.00000	0.400892
$57$	0	0
$58$	8.00000i	1.05045i
$59$	− 4.00000i	− 0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$	− 4.00000i	− 0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	10.0000	1.27000
$63$	9.00000i	1.13389i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	3.00000i	0.358569i
$71$	− 6.00000i	− 0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$	− 6.00000i	− 0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$	− 3.00000i	− 0.353553i
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	7.00000i	0.802955i
$77$	−9.00000	−1.02565
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	− 1.00000i	− 0.111803i
$81$	9.00000	1.00000
$82$	2.00000	0.220863
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	− 4.00000i	− 0.433861i
$86$	6.00000i	0.646997i
$87$	0	0
$88$	3.00000	0.319801
$89$	1.00000i	0.106000i	0.998595	+	0.0529999i	$0.0168783\pi$
$89$	1.00000i	0.106000i	−0.998595	+	0.0529999i	$0.983122\pi$
$90$	3.00000	0.316228
$91$	0	0
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−1.00000	−0.103142
$95$	−7.00000	−0.718185
$96$	0	0
$97$	− 16.0000i	− 1.62455i	−0.583272	−	0.812277i	$-0.698228\pi$
$97$	− 16.0000i	− 1.62455i	0.583272	−	0.812277i	$-0.301772\pi$
$98$	2.00000i	0.202031i
$99$	9.00000i	0.904534i
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	0	0
$106$	9.00000i	0.874157i
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	2.00000i	0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$	2.00000i	0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$	3.00000i	0.286039i
$111$	0	0
$112$	− 3.00000i	− 0.283473i
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	− 4.00000i	− 0.373002i
$116$	8.00000	0.742781
$117$	0	0
$118$	−4.00000	−0.368230
$119$	− 12.0000i	− 1.10004i
$120$	0	0
$121$	2.00000	0.181818
$122$	14.0000i	1.26750i
$123$	0	0
$124$	− 10.0000i	− 0.898027i
$125$	1.00000i	0.0894427i
$126$	9.00000	0.801784
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	13.0000	1.13582	0.567908	−	0.823092i	$-0.307753\pi$
$131$	13.0000	1.13582	0.567908	+	0.823092i	$0.307753\pi$
$132$	0	0
$133$	−21.0000	−1.82093
$134$	−4.00000	−0.345547
$135$	0	0
$136$	4.00000i	0.342997i
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	3.00000	0.253546
$141$	0	0
$142$	−6.00000	−0.503509
$143$	0	0
$144$	−3.00000	−0.250000
$145$	8.00000i	0.664364i
$146$	4.00000	0.331042
$147$	0	0
$148$	− 3.00000i	− 0.246598i
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	− 2.00000i	− 0.162758i	−0.996683	−	0.0813788i	$-0.974068\pi$
$151$	− 2.00000i	− 0.162758i	0.996683	−	0.0813788i	$-0.0259324\pi$
$152$	7.00000	0.567775
$153$	−12.0000	−0.970143
$154$	9.00000i	0.725241i
$155$	10.0000	0.803219
$156$	0	0
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	− 10.0000i	− 0.795557i
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	− 12.0000i	− 0.945732i
$162$	− 9.00000i	− 0.707107i
$163$	8.00000i	0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$	8.00000i	0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$	− 2.00000i	− 0.156174i
$165$	0	0
$166$	0	0
$167$	19.0000i	1.47026i	0.677924	+	0.735132i	$0.262880\pi$
$167$	19.0000i	1.47026i	−0.677924	+	0.735132i	$0.737120\pi$
$168$	0	0
$169$	0	0
$170$	−4.00000	−0.306786
$171$	21.0000i	1.60591i
$172$	6.00000	0.457496
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	0	0
$175$	3.00000i	0.226779i
$176$	− 3.00000i	− 0.226134i
$177$	0	0
$178$	1.00000	0.0749532
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	− 3.00000i	− 0.223607i
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	0	0
$184$	4.00000i	0.294884i
$185$	3.00000	0.220564
$186$	0	0
$187$	− 12.0000i	− 0.877527i
$188$	1.00000i	0.0729325i
$189$	0	0
$190$	7.00000i	0.507833i
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	− 24.0000i	− 1.72756i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	− 24.0000i	− 1.72756i	0.503871	−	0.863779i	$-0.331909\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	2.00000	0.142857
$197$	− 13.0000i	− 0.926212i	−0.886303	−	0.463106i	$-0.846735\pi$
$197$	− 13.0000i	− 0.926212i	0.886303	−	0.463106i	$-0.153265\pi$
$198$	9.00000	0.639602
$199$	18.0000	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	0	0
$202$	0	0
$203$	24.0000i	1.68447i
$204$	0	0
$205$	2.00000	0.139686
$206$	− 1.00000i	− 0.0696733i
$207$	−12.0000	−0.834058
$208$	0	0
$209$	−21.0000	−1.45260
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	18.0000i	1.23045i
$215$	6.00000i	0.409197i
$216$	0	0
$217$	30.0000	2.03653
$218$	2.00000	0.135457
$219$	0	0
$220$	3.00000	0.202260
$221$	0	0
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	−3.00000	−0.200446
$225$	3.00000	0.200000
$226$	8.00000i	0.532152i
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	0	0
$229$	− 22.0000i	− 1.45380i	−0.686743	−	0.726900i	$-0.740960\pi$
$229$	− 22.0000i	− 1.45380i	0.686743	−	0.726900i	$-0.259040\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	− 8.00000i	− 0.525226i
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	−1.00000	−0.0652328
$236$	4.00000i	0.260378i
$237$	0	0
$238$	−12.0000	−0.777844
$239$	− 2.00000i	− 0.129369i	−0.997906	−	0.0646846i	$-0.979396\pi$
$239$	− 2.00000i	− 0.129369i	0.997906	−	0.0646846i	$-0.0206041\pi$
$240$	0	0
$241$	− 25.0000i	− 1.61039i	−0.593009	−	0.805196i	$-0.702060\pi$
$241$	− 25.0000i	− 1.61039i	0.593009	−	0.805196i	$-0.297940\pi$
$242$	− 2.00000i	− 0.128565i
$243$	0	0
$244$	14.0000	0.896258
$245$	2.00000i	0.127775i
$246$	0	0
$247$	0	0
$248$	−10.0000	−0.635001
$249$	0	0
$250$	1.00000	0.0632456
$251$	−23.0000	−1.45175	−0.725874	−	0.687828i	$-0.758564\pi$
$251$	−23.0000	−1.45175	−0.725874	+	0.687828i	$0.758564\pi$
$252$	− 9.00000i	− 0.566947i
$253$	− 12.0000i	− 0.754434i
$254$	13.0000i	0.815693i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	9.00000	0.559233
$260$	0	0
$261$	24.0000	1.48556
$262$	− 13.0000i	− 0.803143i
$263$	11.0000	0.678289	0.339145	−	0.940734i	$-0.389862\pi$
$263$	11.0000	0.678289	0.339145	+	0.940734i	$0.389862\pi$
$264$	0	0
$265$	9.00000i	0.552866i
$266$	21.0000i	1.28759i
$267$	0	0
$268$	4.00000i	0.244339i
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	8.00000i	0.485965i	0.970031	+	0.242983i	$0.0781258\pi$
$271$	8.00000i	0.485965i	−0.970031	+	0.242983i	$0.921874\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	12.0000	0.724947
$275$	3.00000i	0.180907i
$276$	0	0
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	5.00000i	0.299880i
$279$	− 30.0000i	− 1.79605i
$280$	− 3.00000i	− 0.179284i
$281$	2.00000i	0.119310i	0.998219	+	0.0596550i	$0.0190001\pi$
$281$	2.00000i	0.119310i	−0.998219	+	0.0596550i	$0.981000\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	6.00000i	0.356034i
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	3.00000i	0.176777i
$289$	−1.00000	−0.0588235
$290$	8.00000	0.469776
$291$	0	0
$292$	− 4.00000i	− 0.234082i
$293$	9.00000i	0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$	9.00000i	0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	−3.00000	−0.174371
$297$	0	0
$298$	6.00000	0.347571
$299$	0	0
$300$	0	0
$301$	18.0000i	1.03750i
$302$	−2.00000	−0.115087
$303$	0	0
$304$	− 7.00000i	− 0.401478i
$305$	14.0000i	0.801638i
$306$	12.0000i	0.685994i
$307$	− 14.0000i	− 0.799022i	−0.916728	−	0.399511i	$-0.869180\pi$
$307$	− 14.0000i	− 0.799022i	0.916728	−	0.399511i	$-0.130820\pi$
$308$	9.00000	0.512823
$309$	0	0
$310$	− 10.0000i	− 0.567962i
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	3.00000i	0.169300i
$315$	9.00000	0.507093
$316$	−10.0000	−0.562544
$317$	− 31.0000i	− 1.74113i	−0.492050	−	0.870567i	$-0.663752\pi$
$317$	− 31.0000i	− 1.74113i	0.492050	−	0.870567i	$-0.336248\pi$
$318$	0	0
$319$	24.0000i	1.34374i
$320$	1.00000i	0.0559017i
$321$	0	0
$322$	−12.0000	−0.668734
$323$	− 28.0000i	− 1.55796i
$324$	−9.00000	−0.500000
$325$	0	0
$326$	8.00000	0.443079
$327$	0	0
$328$	−2.00000	−0.110432
$329$	−3.00000	−0.165395
$330$	0	0
$331$	20.0000i	1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$	20.0000i	1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$	0	0
$333$	− 9.00000i	− 0.493197i
$334$	19.0000	1.03963
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	0	0
$339$	0	0
$340$	4.00000i	0.216930i
$341$	30.0000	1.62459
$342$	21.0000	1.13555
$343$	− 15.0000i	− 0.809924i
$344$	− 6.00000i	− 0.323498i
$345$	0	0
$346$	− 21.0000i	− 1.12897i
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	0	0
$349$	− 16.0000i	− 0.856460i	−0.903670	−	0.428230i	$-0.859137\pi$
$349$	− 16.0000i	− 0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$	3.00000	0.160357
$351$	0	0
$352$	−3.00000	−0.159901
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	− 1.00000i	− 0.0529999i
$357$	0	0
$358$	4.00000i	0.211407i
$359$	10.0000i	0.527780i	0.964553	+	0.263890i	$0.0850056\pi$
$359$	10.0000i	0.527780i	−0.964553	+	0.263890i	$0.914994\pi$
$360$	−3.00000	−0.158114
$361$	−30.0000	−1.57895
$362$	− 14.0000i	− 0.735824i
$363$	0	0
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	4.00000	0.208514
$369$	− 6.00000i	− 0.312348i
$370$	− 3.00000i	− 0.155963i
$371$	27.0000i	1.40177i
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	1.00000	0.0515711
$377$	0	0
$378$	0	0
$379$	− 27.0000i	− 1.38690i	−0.720506	−	0.693448i	$-0.756091\pi$
$379$	− 27.0000i	− 1.38690i	0.720506	−	0.693448i	$-0.243909\pi$
$380$	7.00000	0.359092
$381$	0	0
$382$	6.00000i	0.306987i
$383$	28.0000i	1.43073i	0.698749	+	0.715367i	$0.253740\pi$
$383$	28.0000i	1.43073i	−0.698749	+	0.715367i	$0.746260\pi$
$384$	0	0
$385$	9.00000i	0.458682i
$386$	−24.0000	−1.22157
$387$	18.0000	0.914991
$388$	16.0000i	0.812277i
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	− 2.00000i	− 0.101015i
$393$	0	0
$394$	−13.0000	−0.654931
$395$	− 10.0000i	− 0.503155i
$396$	− 9.00000i	− 0.452267i
$397$	− 21.0000i	− 1.05396i	−0.849878	−	0.526980i	$-0.823324\pi$
$397$	− 21.0000i	− 1.05396i	0.849878	−	0.526980i	$-0.176676\pi$
$398$	− 18.0000i	− 0.902258i
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	− 39.0000i	− 1.94757i	−0.227477	−	0.973784i	$-0.573048\pi$
$401$	− 39.0000i	− 1.94757i	0.227477	−	0.973784i	$-0.426952\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 9.00000i	− 0.447214i
$406$	24.0000	1.19110
$407$	9.00000	0.446113
$408$	0	0
$409$	− 17.0000i	− 0.840596i	−0.907386	−	0.420298i	$-0.861926\pi$
$409$	− 17.0000i	− 0.840596i	0.907386	−	0.420298i	$-0.138074\pi$
$410$	− 2.00000i	− 0.0987730i
$411$	0	0
$412$	−1.00000	−0.0492665
$413$	−12.0000	−0.590481
$414$	12.0000i	0.589768i
$415$	0	0
$416$	0	0
$417$	0	0
$418$	21.0000i	1.02714i
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	− 16.0000i	− 0.779792i	−0.920859	−	0.389896i	$-0.872511\pi$
$421$	− 16.0000i	− 0.779792i	0.920859	−	0.389896i	$-0.127489\pi$
$422$	− 13.0000i	− 0.632830i
$423$	3.00000i	0.145865i
$424$	− 9.00000i	− 0.437079i
$425$	−4.00000	−0.194029
$426$	0	0
$427$	42.0000i	2.03252i
$428$	18.0000	0.870063
$429$	0	0
$430$	6.00000	0.289346
$431$	24.0000i	1.15604i	0.816023	+	0.578020i	$0.196174\pi$
$431$	24.0000i	1.15604i	−0.816023	+	0.578020i	$0.803826\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	− 30.0000i	− 1.44005i
$435$	0	0
$436$	− 2.00000i	− 0.0957826i
$437$	− 28.0000i	− 1.33942i
$438$	0	0
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	− 3.00000i	− 0.143019i
$441$	6.00000	0.285714
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−19.0000	−0.899676
$447$	0	0
$448$	3.00000i	0.141737i
$449$	− 15.0000i	− 0.707894i	−0.935266	−	0.353947i	$-0.884839\pi$
$449$	− 15.0000i	− 0.707894i	0.935266	−	0.353947i	$-0.115161\pi$
$450$	− 3.00000i	− 0.141421i
$451$	6.00000	0.282529
$452$	8.00000	0.376288
$453$	0	0
$454$	4.00000	0.187729
$455$	0	0
$456$	0	0
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	−22.0000	−1.02799
$459$	0	0
$460$	4.00000i	0.186501i
$461$	40.0000i	1.86299i	0.363760	+	0.931493i	$0.381493\pi$
$461$	40.0000i	1.86299i	−0.363760	+	0.931493i	$0.618507\pi$
$462$	0	0
$463$	− 40.0000i	− 1.85896i	−0.368875	−	0.929479i	$-0.620257\pi$
$463$	− 40.0000i	− 1.85896i	0.368875	−	0.929479i	$-0.379743\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	− 18.0000i	− 0.833834i
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	1.00000i	0.0461266i
$471$	0	0
$472$	4.00000	0.184115
$473$	18.0000i	0.827641i
$474$	0	0
$475$	7.00000i	0.321182i
$476$	12.0000i	0.550019i
$477$	27.0000	1.23625
$478$	−2.00000	−0.0914779
$479$	− 28.0000i	− 1.27935i	−0.768644	−	0.639676i	$-0.779068\pi$
$479$	− 28.0000i	− 1.27935i	0.768644	−	0.639676i	$-0.220932\pi$
$480$	0	0
$481$	0	0
$482$	−25.0000	−1.13872
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−16.0000	−0.726523
$486$	0	0
$487$	− 19.0000i	− 0.860972i	−0.902597	−	0.430486i	$-0.858342\pi$
$487$	− 19.0000i	− 0.860972i	0.902597	−	0.430486i	$-0.141658\pi$
$488$	− 14.0000i	− 0.633750i
$489$	0	0
$490$	2.00000	0.0903508
$491$	43.0000	1.94056	0.970281	−	0.241979i	$-0.0777966\pi$
$491$	43.0000	1.94056	0.970281	+	0.241979i	$0.0777966\pi$
$492$	0	0
$493$	−32.0000	−1.44121
$494$	0	0
$495$	9.00000	0.404520
$496$	10.0000i	0.449013i
$497$	−18.0000	−0.807410
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	0	0
$502$	23.0000i	1.02654i
$503$	23.0000	1.02552	0.512760	−	0.858532i	$-0.328623\pi$
$503$	23.0000	1.02552	0.512760	+	0.858532i	$0.328623\pi$
$504$	−9.00000	−0.400892
$505$	0	0
$506$	−12.0000	−0.533465
$507$	0	0
$508$	13.0000	0.576782
$509$	6.00000i	0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$	6.00000i	0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	8.00000i	0.352865i
$515$	− 1.00000i	− 0.0440653i
$516$	0	0
$517$	−3.00000	−0.131940
$518$	− 9.00000i	− 0.395437i
$519$	0	0
$520$	0	0
$521$	−31.0000	−1.35813	−0.679067	−	0.734076i	$-0.737616\pi$
$521$	−31.0000	−1.35813	−0.679067	+	0.734076i	$0.737616\pi$
$522$	− 24.0000i	− 1.05045i
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	−13.0000	−0.567908
$525$	0	0
$526$	− 11.0000i	− 0.479623i
$527$	40.0000i	1.74243i
$528$	0	0
$529$	−7.00000	−0.304348
$530$	9.00000	0.390935
$531$	12.0000i	0.520756i
$532$	21.0000	0.910465
$533$	0	0
$534$	0	0
$535$	18.0000i	0.778208i
$536$	4.00000	0.172774
$537$	0	0
$538$	− 24.0000i	− 1.03471i
$539$	6.00000i	0.258438i
$540$	0	0
$541$	10.0000i	0.429934i	0.976621	+	0.214967i	$0.0689643\pi$
$541$	10.0000i	0.429934i	−0.976621	+	0.214967i	$0.931036\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	− 4.00000i	− 0.171499i
$545$	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	$0.486386\pi$
$548$	− 12.0000i	− 0.512615i
$549$	42.0000	1.79252
$550$	3.00000	0.127920
$551$	56.0000i	2.38568i
$552$	0	0
$553$	− 30.0000i	− 1.27573i
$554$	19.0000i	0.807233i
$555$	0	0
$556$	5.00000	0.212047
$557$	− 31.0000i	− 1.31351i	−0.754103	−	0.656756i	$-0.771928\pi$
$557$	− 31.0000i	− 1.31351i	0.754103	−	0.656756i	$-0.228072\pi$
$558$	−30.0000	−1.27000
$559$	0	0
$560$	−3.00000	−0.126773
$561$	0	0
$562$	2.00000	0.0843649
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	8.00000i	0.336563i
$566$	− 14.0000i	− 0.588464i
$567$	− 27.0000i	− 1.13389i
$568$	6.00000	0.251754
$569$	11.0000	0.461144	0.230572	−	0.973055i	$-0.425940\pi$
$569$	11.0000	0.461144	0.230572	+	0.973055i	$0.425940\pi$
$570$	0	0
$571$	37.0000	1.54840	0.774201	−	0.632940i	$-0.218152\pi$
$571$	37.0000	1.54840	0.774201	+	0.632940i	$0.218152\pi$
$572$	0	0
$573$	0	0
$574$	− 6.00000i	− 0.250435i
$575$	−4.00000	−0.166812
$576$	3.00000	0.125000
$577$	10.0000i	0.416305i	0.978096	+	0.208153i	$0.0667451\pi$
$577$	10.0000i	0.416305i	−0.978096	+	0.208153i	$0.933255\pi$
$578$	1.00000i	0.0415945i
$579$	0	0
$580$	− 8.00000i	− 0.332182i
$581$	0	0
$582$	0	0
$583$	27.0000i	1.11823i
$584$	−4.00000	−0.165521
$585$	0	0
$586$	9.00000	0.371787
$587$	2.00000i	0.0825488i	0.999148	+	0.0412744i	$0.0131418\pi$
$587$	2.00000i	0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$	0	0
$589$	70.0000	2.88430
$590$	4.00000i	0.164677i
$591$	0	0
$592$	3.00000i	0.123299i
$593$	20.0000i	0.821302i	0.911793	+	0.410651i	$0.134698\pi$
$593$	20.0000i	0.821302i	−0.911793	+	0.410651i	$0.865302\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	− 6.00000i	− 0.245770i
$597$	0	0
$598$	0	0
$599$	26.0000	1.06233	0.531166	−	0.847268i	$-0.321754\pi$
$599$	26.0000	1.06233	0.531166	+	0.847268i	$0.321754\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	18.0000	0.733625
$603$	12.0000i	0.488678i
$604$	2.00000i	0.0813788i
$605$	− 2.00000i	− 0.0813116i
$606$	0	0
$607$	21.0000	0.852364	0.426182	−	0.904638i	$-0.359858\pi$
$607$	21.0000	0.852364	0.426182	+	0.904638i	$0.359858\pi$
$608$	−7.00000	−0.283887
$609$	0	0
$610$	14.0000	0.566843
$611$	0	0
$612$	12.0000	0.485071
$613$	− 11.0000i	− 0.444286i	−0.975014	−	0.222143i	$-0.928695\pi$
$613$	− 11.0000i	− 0.444286i	0.975014	−	0.222143i	$-0.0713052\pi$
$614$	−14.0000	−0.564994
$615$	0	0
$616$	− 9.00000i	− 0.362620i
$617$	22.0000i	0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$	22.0000i	0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$	0	0
$619$	− 3.00000i	− 0.120580i	−0.998181	−	0.0602901i	$-0.980797\pi$
$619$	− 3.00000i	− 0.120580i	0.998181	−	0.0602901i	$-0.0192026\pi$
$620$	−10.0000	−0.401610
$621$	0	0
$622$	− 24.0000i	− 0.962312i
$623$	3.00000	0.120192
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 6.00000i	− 0.239808i
$627$	0	0
$628$	3.00000	0.119713
$629$	12.0000i	0.478471i
$630$	− 9.00000i	− 0.358569i
$631$	− 36.0000i	− 1.43314i	−0.697517	−	0.716569i	$-0.745712\pi$
$631$	− 36.0000i	− 1.43314i	0.697517	−	0.716569i	$-0.254288\pi$
$632$	10.0000i	0.397779i
$633$	0	0
$634$	−31.0000	−1.23117
$635$	13.0000i	0.515889i
$636$	0	0
$637$	0	0
$638$	24.0000	0.950169
$639$	18.0000i	0.712069i
$640$	1.00000	0.0395285
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	0	0
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	12.0000i	0.472866i
$645$	0	0
$646$	−28.0000	−1.10165
$647$	7.00000	0.275198	0.137599	−	0.990488i	$-0.456061\pi$
$647$	7.00000	0.275198	0.137599	+	0.990488i	$0.456061\pi$
$648$	9.00000i	0.353553i
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	− 8.00000i	− 0.313304i
$653$	−11.0000	−0.430463	−0.215232	−	0.976563i	$-0.569051\pi$
$653$	−11.0000	−0.430463	−0.215232	+	0.976563i	$0.569051\pi$
$654$	0	0
$655$	− 13.0000i	− 0.507952i
$656$	2.00000i	0.0780869i
$657$	− 12.0000i	− 0.468165i
$658$	3.00000i	0.116952i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	− 14.0000i	− 0.544537i	−0.962221	−	0.272268i	$-0.912226\pi$
$661$	− 14.0000i	− 0.544537i	0.962221	−	0.272268i	$-0.0877739\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	0	0
$665$	21.0000i	0.814345i
$666$	−9.00000	−0.348743
$667$	−32.0000	−1.23904
$668$	− 19.0000i	− 0.735132i
$669$	0	0
$670$	4.00000i	0.154533i
$671$	42.0000i	1.62139i
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	30.0000i	1.15556i
$675$	0	0
$676$	0	0
$677$	−50.0000	−1.92166	−0.960828	−	0.277145i	$-0.910612\pi$
$677$	−50.0000	−1.92166	−0.960828	+	0.277145i	$0.910612\pi$
$678$	0	0
$679$	−48.0000	−1.84207
$680$	4.00000	0.153393
$681$	0	0
$682$	− 30.0000i	− 1.14876i
$683$	− 2.00000i	− 0.0765279i	−0.999268	−	0.0382639i	$-0.987817\pi$
$683$	− 2.00000i	− 0.0765279i	0.999268	−	0.0382639i	$-0.0121828\pi$
$684$	− 21.0000i	− 0.802955i
$685$	12.0000	0.458496
$686$	−15.0000	−0.572703
$687$	0	0
$688$	−6.00000	−0.228748
$689$	0	0
$690$	0	0
$691$	− 29.0000i	− 1.10321i	−0.834105	−	0.551606i	$-0.814015\pi$
$691$	− 29.0000i	− 1.10321i	0.834105	−	0.551606i	$-0.185985\pi$
$692$	−21.0000	−0.798300
$693$	27.0000	1.02565
$694$	16.0000i	0.607352i
$695$	5.00000i	0.189661i
$696$	0	0
$697$	8.00000i	0.303022i
$698$	−16.0000	−0.605609
$699$	0	0
$700$	− 3.00000i	− 0.113389i
$701$	40.0000	1.51078	0.755390	−	0.655276i	$-0.227448\pi$
$701$	40.0000	1.51078	0.755390	+	0.655276i	$0.227448\pi$
$702$	0	0
$703$	21.0000	0.792030
$704$	3.00000i	0.113067i
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8.00000i	0.300446i	0.988652	+	0.150223i	$0.0479992\pi$
$709$	8.00000i	0.300446i	−0.988652	+	0.150223i	$0.952001\pi$
$710$	6.00000i	0.225176i
$711$	−30.0000	−1.12509
$712$	−1.00000	−0.0374766
$713$	40.0000i	1.49801i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	10.0000	0.373197
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	3.00000i	0.111803i
$721$	− 3.00000i	− 0.111726i
$722$	30.0000i	1.11648i
$723$	0	0
$724$	−14.0000	−0.520306
$725$	8.00000	0.297113
$726$	0	0
$727$	−35.0000	−1.29808	−0.649039	−	0.760755i	$-0.724829\pi$
$727$	−35.0000	−1.29808	−0.649039	+	0.760755i	$0.724829\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	− 4.00000i	− 0.148047i
$731$	−24.0000	−0.887672
$732$	0	0
$733$	1.00000i	0.0369358i	0.999829	+	0.0184679i	$0.00587886\pi$
$733$	1.00000i	0.0369358i	−0.999829	+	0.0184679i	$0.994121\pi$
$734$	0	0
$735$	0	0
$736$	− 4.00000i	− 0.147442i
$737$	−12.0000	−0.442026
$738$	−6.00000	−0.220863
$739$	39.0000i	1.43464i	0.696745	+	0.717319i	$0.254631\pi$
$739$	39.0000i	1.43464i	−0.696745	+	0.717319i	$0.745369\pi$
$740$	−3.00000	−0.110282
$741$	0	0
$742$	27.0000	0.991201
$743$	8.00000i	0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$	8.00000i	0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	6.00000i	0.219676i
$747$	0	0
$748$	12.0000i	0.438763i
$749$	54.0000i	1.97312i
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	− 1.00000i	− 0.0364662i
$753$	0	0
$754$	0	0
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	−29.0000	−1.05402	−0.527011	−	0.849858i	$-0.676688\pi$
$757$	−29.0000	−1.05402	−0.527011	+	0.849858i	$0.676688\pi$
$758$	−27.0000	−0.980684
$759$	0	0
$760$	− 7.00000i	− 0.253917i
$761$	21.0000i	0.761249i	0.924730	+	0.380625i	$0.124291\pi$
$761$	21.0000i	0.761249i	−0.924730	+	0.380625i	$0.875709\pi$
$762$	0	0
$763$	6.00000	0.217215
$764$	6.00000	0.217072
$765$	12.0000i	0.433861i
$766$	28.0000	1.01168
$767$	0	0
$768$	0	0
$769$	2.00000i	0.0721218i	0.999350	+	0.0360609i	$0.0114810\pi$
$769$	2.00000i	0.0721218i	−0.999350	+	0.0360609i	$0.988519\pi$
$770$	9.00000	0.324337
$771$	0	0
$772$	24.0000i	0.863779i
$773$	− 31.0000i	− 1.11499i	−0.830179	−	0.557496i	$-0.811762\pi$
$773$	− 31.0000i	− 1.11499i	0.830179	−	0.557496i	$-0.188238\pi$
$774$	− 18.0000i	− 0.646997i
$775$	− 10.0000i	− 0.359211i
$776$	16.0000	0.574367
$777$	0	0
$778$	12.0000i	0.430221i
$779$	14.0000	0.501602
$780$	0	0
$781$	−18.0000	−0.644091
$782$	− 16.0000i	− 0.572159i
$783$	0	0
$784$	−2.00000	−0.0714286
$785$	3.00000i	0.107075i
$786$	0	0
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	13.0000i	0.463106i
$789$	0	0
$790$	−10.0000	−0.355784
$791$	24.0000i	0.853342i
$792$	−9.00000	−0.319801
$793$	0	0
$794$	−21.0000	−0.745262
$795$	0	0
$796$	−18.0000	−0.637993
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	− 4.00000i	− 0.141510i
$800$	1.00000i	0.0353553i
$801$	− 3.00000i	− 0.106000i
$802$	−39.0000	−1.37714
$803$	12.0000	0.423471
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	−9.00000	−0.316228
$811$	− 21.0000i	− 0.737410i	−0.929547	−	0.368705i	$-0.879801\pi$
$811$	− 21.0000i	− 0.737410i	0.929547	−	0.368705i	$-0.120199\pi$
$812$	− 24.0000i	− 0.842235i
$813$	0	0
$814$	− 9.00000i	− 0.315450i
$815$	8.00000	0.280228
$816$	0	0
$817$	42.0000i	1.46939i
$818$	−17.0000	−0.594391
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	18.0000i	0.628204i	0.949389	+	0.314102i	$0.101703\pi$
$821$	18.0000i	0.628204i	−0.949389	+	0.314102i	$0.898297\pi$
$822$	0	0
$823$	37.0000	1.28974	0.644869	−	0.764293i	$-0.276912\pi$
$823$	37.0000	1.28974	0.644869	+	0.764293i	$0.276912\pi$
$824$	1.00000i	0.0348367i
$825$	0	0
$826$	12.0000i	0.417533i
$827$	30.0000i	1.04320i	0.853189	+	0.521601i	$0.174665\pi$
$827$	30.0000i	1.04320i	−0.853189	+	0.521601i	$0.825335\pi$
$828$	12.0000	0.417029
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8.00000	−0.277184
$834$	0	0
$835$	19.0000	0.657522
$836$	21.0000	0.726300
$837$	0	0
$838$	− 12.0000i	− 0.414533i
$839$	− 26.0000i	− 0.897620i	−0.893627	−	0.448810i	$-0.851848\pi$
$839$	− 26.0000i	− 0.897620i	0.893627	−	0.448810i	$-0.148152\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−16.0000	−0.551396
$843$	0	0
$844$	−13.0000	−0.447478
$845$	0	0
$846$	3.00000	0.103142
$847$	− 6.00000i	− 0.206162i
$848$	−9.00000	−0.309061
$849$	0	0
$850$	4.00000i	0.137199i
$851$	12.0000i	0.411355i
$852$	0	0
$853$	− 14.0000i	− 0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$	− 14.0000i	− 0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	42.0000	1.43721
$855$	21.0000	0.718185
$856$	− 18.0000i	− 0.615227i
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−1.00000	−0.0341196	−0.0170598	−	0.999854i	$-0.505431\pi$
$859$	−1.00000	−0.0341196	−0.0170598	+	0.999854i	$0.505431\pi$
$860$	− 6.00000i	− 0.204598i
$861$	0	0
$862$	24.0000	0.817443
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	0	0
$865$	− 21.0000i	− 0.714021i
$866$	16.0000i	0.543702i
$867$	0	0
$868$	−30.0000	−1.01827
$869$	− 30.0000i	− 1.01768i
$870$	0	0
$871$	0	0
$872$	−2.00000	−0.0677285
$873$	48.0000i	1.62455i
$874$	−28.0000	−0.947114
$875$	3.00000	0.101419
$876$	0	0
$877$	34.0000i	1.14810i	0.818821	+	0.574049i	$0.194628\pi$
$877$	34.0000i	1.14810i	−0.818821	+	0.574049i	$0.805372\pi$
$878$	− 6.00000i	− 0.202490i
$879$	0	0
$880$	−3.00000	−0.101130
$881$	−5.00000	−0.168454	−0.0842271	−	0.996447i	$-0.526842\pi$
$881$	−5.00000	−0.168454	−0.0842271	+	0.996447i	$0.526842\pi$
$882$	− 6.00000i	− 0.202031i
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	0	0
$885$	0	0
$886$	18.0000i	0.604722i
$887$	−13.0000	−0.436497	−0.218249	−	0.975893i	$-0.570034\pi$
$887$	−13.0000	−0.436497	−0.218249	+	0.975893i	$0.570034\pi$
$888$	0	0
$889$	39.0000i	1.30802i
$890$	− 1.00000i	− 0.0335201i
$891$	− 27.0000i	− 0.904534i
$892$	19.0000i	0.636167i
$893$	−7.00000	−0.234246
$894$	0	0
$895$	4.00000i	0.133705i
$896$	3.00000	0.100223
$897$	0	0
$898$	−15.0000	−0.500556
$899$	− 80.0000i	− 2.66815i
$900$	−3.00000	−0.100000
$901$	−36.0000	−1.19933
$902$	− 6.00000i	− 0.199778i
$903$	0	0
$904$	− 8.00000i	− 0.266076i
$905$	− 14.0000i	− 0.465376i
$906$	0	0
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	− 4.00000i	− 0.132745i
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	22.0000i	0.726900i
$917$	− 39.0000i	− 1.28789i
$918$	0	0
$919$	50.0000	1.64935	0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000	1.64935	0.824674	+	0.565608i	$0.191359\pi$
$920$	4.00000	0.131876
$921$	0	0
$922$	40.0000	1.31733
$923$	0	0
$924$	0	0
$925$	− 3.00000i	− 0.0986394i
$926$	−40.0000	−1.31448
$927$	−3.00000	−0.0985329
$928$	8.00000i	0.262613i
$929$	14.0000i	0.459325i	0.973270	+	0.229663i	$0.0737623\pi$
$929$	14.0000i	0.459325i	−0.973270	+	0.229663i	$0.926238\pi$
$930$	0	0
$931$	14.0000i	0.458831i
$932$	−18.0000	−0.589610
$933$	0	0
$934$	20.0000i	0.654420i
$935$	−12.0000	−0.392442
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	12.0000i	0.391814i
$939$	0	0
$940$	1.00000	0.0326164
$941$	24.0000i	0.782378i	0.920310	+	0.391189i	$0.127936\pi$
$941$	24.0000i	0.782378i	−0.920310	+	0.391189i	$0.872064\pi$
$942$	0	0
$943$	8.00000i	0.260516i
$944$	− 4.00000i	− 0.130189i
$945$	0	0
$946$	18.0000	0.585230
$947$	38.0000i	1.23483i	0.786636	+	0.617417i	$0.211821\pi$
$947$	38.0000i	1.23483i	−0.786636	+	0.617417i	$0.788179\pi$
$948$	0	0
$949$	0	0
$950$	7.00000	0.227110
$951$	0	0
$952$	12.0000	0.388922
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	− 27.0000i	− 0.874157i
$955$	6.00000i	0.194155i
$956$	2.00000i	0.0646846i
$957$	0	0
$958$	−28.0000	−0.904639
$959$	36.0000	1.16250
$960$	0	0
$961$	−69.0000	−2.22581
$962$	0	0
$963$	54.0000	1.74013
$964$	25.0000i	0.805196i
$965$	−24.0000	−0.772587
$966$	0	0
$967$	33.0000i	1.06121i	0.847620	+	0.530604i	$0.178035\pi$
$967$	33.0000i	1.06121i	−0.847620	+	0.530604i	$0.821965\pi$
$968$	2.00000i	0.0642824i
$969$	0	0
$970$	16.0000i	0.513729i
$971$	−49.0000	−1.57248	−0.786242	−	0.617918i	$-0.787976\pi$
$971$	−49.0000	−1.57248	−0.786242	+	0.617918i	$0.787976\pi$
$972$	0	0
$973$	15.0000i	0.480878i
$974$	−19.0000	−0.608799
$975$	0	0
$976$	−14.0000	−0.448129
$977$	24.0000i	0.767828i	0.923369	+	0.383914i	$0.125424\pi$
$977$	24.0000i	0.767828i	−0.923369	+	0.383914i	$0.874576\pi$
$978$	0	0
$979$	3.00000	0.0958804
$980$	− 2.00000i	− 0.0638877i
$981$	− 6.00000i	− 0.191565i
$982$	− 43.0000i	− 1.37219i
$983$	− 3.00000i	− 0.0956851i	−0.998855	−	0.0478426i	$-0.984765\pi$
$983$	− 3.00000i	− 0.0956851i	0.998855	−	0.0478426i	$-0.0152346\pi$
$984$	0	0
$985$	−13.0000	−0.414214
$986$	32.0000i	1.01909i
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	− 9.00000i	− 0.286039i
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	10.0000	0.317500
$993$	0	0
$994$	18.0000i	0.570925i
$995$	− 18.0000i	− 0.570638i
$996$	0	0
$997$	−11.0000	−0.348373	−0.174187	−	0.984713i	$-0.555730\pi$
$997$	−11.0000	−0.348373	−0.174187	+	0.984713i	$0.555730\pi$
$998$	4.00000	0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.d.c.1351.1		2
13.2	odd	12		1690.2.e.h.191.1		2
13.3	even	3		1690.2.l.e.1161.2		4
13.4	even	6		1690.2.l.e.361.2		4
13.5	odd	4		1690.2.a.c.1.1		1
13.6	odd	12		1690.2.e.h.991.1		2
13.7	odd	12		130.2.e.a.81.1	yes	2
13.8	odd	4		1690.2.a.h.1.1		1
13.9	even	3		1690.2.l.e.361.1		4
13.10	even	6		1690.2.l.e.1161.1		4
13.11	odd	12		130.2.e.a.61.1	✓	2
13.12	even	2	inner	1690.2.d.c.1351.2		2
39.11	even	12		1170.2.i.i.451.1		2
39.20	even	12		1170.2.i.i.991.1		2
52.7	even	12		1040.2.q.h.81.1		2
52.11	even	12		1040.2.q.h.321.1		2
65.7	even	12		650.2.o.d.549.2		4
65.24	odd	12		650.2.e.b.451.1		2
65.33	even	12		650.2.o.d.549.1		4
65.34	odd	4		8450.2.a.g.1.1		1
65.37	even	12		650.2.o.d.399.1		4
65.44	odd	4		8450.2.a.q.1.1		1
65.59	odd	12		650.2.e.b.601.1		2
65.63	even	12		650.2.o.d.399.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.e.a.61.1	✓	2	13.11	odd	12
130.2.e.a.81.1	yes	2	13.7	odd	12
650.2.e.b.451.1		2	65.24	odd	12
650.2.e.b.601.1		2	65.59	odd	12
650.2.o.d.399.1		4	65.37	even	12
650.2.o.d.399.2		4	65.63	even	12
650.2.o.d.549.1		4	65.33	even	12
650.2.o.d.549.2		4	65.7	even	12
1040.2.q.h.81.1		2	52.7	even	12
1040.2.q.h.321.1		2	52.11	even	12
1170.2.i.i.451.1		2	39.11	even	12
1170.2.i.i.991.1		2	39.20	even	12
1690.2.a.c.1.1		1	13.5	odd	4
1690.2.a.h.1.1		1	13.8	odd	4
1690.2.d.c.1351.1		2	1.1	even	1	trivial
1690.2.d.c.1351.2		2	13.12	even	2	inner
1690.2.e.h.191.1		2	13.2	odd	12
1690.2.e.h.991.1		2	13.6	odd	12
1690.2.l.e.361.1		4	13.9	even	3
1690.2.l.e.361.2		4	13.4	even	6
1690.2.l.e.1161.1		4	13.10	even	6
1690.2.l.e.1161.2		4	13.3	even	3
8450.2.a.g.1.1		1	65.34	odd	4
8450.2.a.q.1.1		1	65.44	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} -3.00000i q^{7} +1.00000i q^{8} -3.00000 q^{9} -1.00000 q^{10} -3.00000i q^{11} -3.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +3.00000i q^{18} -7.00000i q^{19} +1.00000i q^{20} -3.00000 q^{22} +4.00000 q^{23} -1.00000 q^{25} +3.00000i q^{28} -8.00000 q^{29} +10.0000i q^{31} -1.00000i q^{32} -4.00000i q^{34} -3.00000 q^{35} +3.00000 q^{36} +3.00000i q^{37} -7.00000 q^{38} +1.00000 q^{40} +2.00000i q^{41} -6.00000 q^{43} +3.00000i q^{44} +3.00000i q^{45} -4.00000i q^{46} -1.00000i q^{47} -2.00000 q^{49} +1.00000i q^{50} -9.00000 q^{53} -3.00000 q^{55} +3.00000 q^{56} +8.00000i q^{58} -4.00000i q^{59} -14.0000 q^{61} +10.0000 q^{62} +9.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} -4.00000 q^{68} +3.00000i q^{70} -6.00000i q^{71} -3.00000i q^{72} +4.00000i q^{73} +3.00000 q^{74} +7.00000i q^{76} -9.00000 q^{77} +10.0000 q^{79} -1.00000i q^{80} +9.00000 q^{81} +2.00000 q^{82} -4.00000i q^{85} +6.00000i q^{86} +3.00000 q^{88} +1.00000i q^{89} +3.00000 q^{90} -4.00000 q^{92} -1.00000 q^{94} -7.00000 q^{95} -16.0000i q^{97} +2.00000i q^{98} +9.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 6 q^{9} - 2 q^{10} - 6 q^{14} + 2 q^{16} + 8 q^{17} - 6 q^{22} + 8 q^{23} - 2 q^{25} - 16 q^{29} - 6 q^{35} + 6 q^{36} - 14 q^{38} + 2 q^{40} - 12 q^{43} - 4 q^{49} - 18 q^{53} - 6 q^{55}+ \cdots - 14 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 6 * q^9 - 2 * q^10 - 6 * q^14 + 2 * q^16 + 8 * q^17 - 6 * q^22 + 8 * q^23 - 2 * q^25 - 16 * q^29 - 6 * q^35 + 6 * q^36 - 14 * q^38 + 2 * q^40 - 12 * q^43 - 4 * q^49 - 18 * q^53 - 6 * q^55 + 6 * q^56 - 28 * q^61 + 20 * q^62 - 2 * q^64 - 8 * q^68 + 6 * q^74 - 18 * q^77 + 20 * q^79 + 18 * q^81 + 4 * q^82 + 6 * q^88 + 6 * q^90 - 8 * q^92 - 2 * q^94 - 14 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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