LMFDB - Embedded newform 1690.2.d.a.1351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1690,2,Mod(1351,1690)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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			1351.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1690.1351
Dual form			1690.2.d.a.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -2.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{6} -1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{6} -1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000i q^{11} +2.00000 q^{12} +1.00000 q^{14} -2.00000i q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000i q^{18} -5.00000i q^{19} -1.00000i q^{20} +2.00000i q^{21} -3.00000 q^{22} +2.00000i q^{24} -1.00000 q^{25} +4.00000 q^{27} +1.00000i q^{28} +2.00000 q^{30} +4.00000i q^{31} +1.00000i q^{32} -6.00000i q^{33} +6.00000i q^{34} +1.00000 q^{35} -1.00000 q^{36} +11.0000i q^{37} +5.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} -2.00000 q^{42} -2.00000 q^{43} -3.00000i q^{44} +1.00000i q^{45} -3.00000i q^{47} -2.00000 q^{48} +6.00000 q^{49} -1.00000i q^{50} -12.0000 q^{51} -9.00000 q^{53} +4.00000i q^{54} -3.00000 q^{55} -1.00000 q^{56} +10.0000i q^{57} +2.00000i q^{60} +8.00000 q^{61} -4.00000 q^{62} -1.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} +16.0000i q^{67} -6.00000 q^{68} +1.00000i q^{70} -6.00000i q^{71} -1.00000i q^{72} +14.0000i q^{73} -11.0000 q^{74} +2.00000 q^{75} +5.00000i q^{76} +3.00000 q^{77} -16.0000 q^{79} +1.00000i q^{80} -11.0000 q^{81} +6.00000 q^{82} +6.00000i q^{83} -2.00000i q^{84} +6.00000i q^{85} -2.00000i q^{86} +3.00000 q^{88} +9.00000i q^{89} -1.00000 q^{90} -8.00000i q^{93} +3.00000 q^{94} +5.00000 q^{95} -2.00000i q^{96} +10.0000i q^{97} +6.00000i q^{98} +3.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^3 - 2 * q^4 + 2 * q^9	$ 2 q - 4 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 4 q^{12} + 2 q^{14} + 2 q^{16} + 12 q^{17} - 6 q^{22} - 2 q^{25} + 8 q^{27} + 4 q^{30} + 2 q^{35} - 2 q^{36} + 10 q^{38} + 2 q^{40} - 4 q^{42} - 4 q^{43} - 4 q^{48} + 12 q^{49} - 24 q^{51} - 18 q^{53} - 6 q^{55} - 2 q^{56} + 16 q^{61} - 8 q^{62} - 2 q^{64} + 12 q^{66} - 12 q^{68} - 22 q^{74} + 4 q^{75} + 6 q^{77} - 32 q^{79} - 22 q^{81} + 12 q^{82} + 6 q^{88} - 2 q^{90} + 6 q^{94} + 10 q^{95}+O(q^{100}) $ 2 * q - 4 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 4 * q^12 + 2 * q^14 + 2 * q^16 + 12 * q^17 - 6 * q^22 - 2 * q^25 + 8 * q^27 + 4 * q^30 + 2 * q^35 - 2 * q^36 + 10 * q^38 + 2 * q^40 - 4 * q^42 - 4 * q^43 - 4 * q^48 + 12 * q^49 - 24 * q^51 - 18 * q^53 - 6 * q^55 - 2 * q^56 + 16 * q^61 - 8 * q^62 - 2 * q^64 + 12 * q^66 - 12 * q^68 - 22 * q^74 + 4 * q^75 + 6 * q^77 - 32 * q^79 - 22 * q^81 + 12 * q^82 + 6 * q^88 - 2 * q^90 + 6 * q^94 + 10 * q^95

Display 100 coefficients 10 coefficients

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	− 2.00000i	− 0.816497i
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	3.00000i	0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$	3.00000i	0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	2.00000	0.577350
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	− 2.00000i	− 0.516398i
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000i	0.235702i
$19$	− 5.00000i	− 1.14708i	−0.819178	−	0.573539i	$-0.805570\pi$
$19$	− 5.00000i	− 1.14708i	0.819178	−	0.573539i	$-0.194430\pi$
$20$	− 1.00000i	− 0.223607i
$21$	2.00000i	0.436436i
$22$	−3.00000	−0.639602
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	2.00000i	0.408248i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	1.00000i	0.188982i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	2.00000	0.365148
$31$	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
$33$	− 6.00000i	− 1.04447i
$34$	6.00000i	1.02899i
$35$	1.00000	0.169031
$36$	−1.00000	−0.166667
$37$	11.0000i	1.80839i	0.427121	+	0.904194i	$0.359528\pi$
$37$	11.0000i	1.80839i	−0.427121	+	0.904194i	$0.640472\pi$
$38$	5.00000	0.811107
$39$	0	0
$40$	1.00000	0.158114
$41$	− 6.00000i	− 0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$	− 6.00000i	− 0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$	−2.00000	−0.308607
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	− 3.00000i	− 0.452267i
$45$	1.00000i	0.149071i
$46$	0	0
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	−2.00000	−0.288675
$49$	6.00000	0.857143
$50$	− 1.00000i	− 0.141421i
$51$	−12.0000	−1.68034
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	4.00000i	0.544331i
$55$	−3.00000	−0.404520
$56$	−1.00000	−0.133631
$57$	10.0000i	1.32453i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	2.00000i	0.258199i
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−4.00000	−0.508001
$63$	− 1.00000i	− 0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	16.0000i	1.95471i	0.211604	+	0.977356i	$0.432131\pi$
$67$	16.0000i	1.95471i	−0.211604	+	0.977356i	$0.567869\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	1.00000i	0.119523i
$71$	− 6.00000i	− 0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$	− 6.00000i	− 0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$	− 1.00000i	− 0.117851i
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−11.0000	−1.27872
$75$	2.00000	0.230940
$76$	5.00000i	0.573539i
$77$	3.00000	0.341882
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	1.00000i	0.111803i
$81$	−11.0000	−1.22222
$82$	6.00000	0.662589
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	− 2.00000i	− 0.218218i
$85$	6.00000i	0.650791i
$86$	− 2.00000i	− 0.215666i
$87$	0	0
$88$	3.00000	0.319801
$89$	9.00000i	0.953998i	0.878904	+	0.476999i	$0.158275\pi$
$89$	9.00000i	0.953998i	−0.878904	+	0.476999i	$0.841725\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	3.00000	0.309426
$95$	5.00000	0.512989
$96$	− 2.00000i	− 0.204124i
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	6.00000i	0.606092i
$99$	3.00000i	0.301511i
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	− 12.0000i	− 1.18818i
$103$	−5.00000	−0.492665	−0.246332	−	0.969185i	$-0.579225\pi$
$103$	−5.00000	−0.492665	−0.246332	+	0.969185i	$0.579225\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	− 9.00000i	− 0.874157i
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−4.00000	−0.384900
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	− 3.00000i	− 0.286039i
$111$	− 22.0000i	− 2.08815i
$112$	− 1.00000i	− 0.0944911i
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	−10.0000	−0.936586
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 6.00000i	− 0.550019i
$120$	−2.00000	−0.182574
$121$	2.00000	0.181818
$122$	8.00000i	0.724286i
$123$	12.0000i	1.08200i
$124$	− 4.00000i	− 0.359211i
$125$	− 1.00000i	− 0.0894427i
$126$	1.00000	0.0890871
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	6.00000i	0.522233i
$133$	−5.00000	−0.433555
$134$	−16.0000	−1.38219
$135$	4.00000i	0.344265i
$136$	− 6.00000i	− 0.514496i
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−19.0000	−1.61156	−0.805779	−	0.592216i	$-0.798253\pi$
$139$	−19.0000	−1.61156	−0.805779	+	0.592216i	$0.798253\pi$
$140$	−1.00000	−0.0845154
$141$	6.00000i	0.505291i
$142$	6.00000	0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	−12.0000	−0.989743
$148$	− 11.0000i	− 0.904194i
$149$	18.0000i	1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$	18.0000i	1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$	2.00000i	0.163299i
$151$	− 16.0000i	− 1.30206i	−0.759051	−	0.651031i	$-0.774337\pi$
$151$	− 16.0000i	− 1.30206i	0.759051	−	0.651031i	$-0.225663\pi$
$152$	−5.00000	−0.405554
$153$	6.00000	0.485071
$154$	3.00000i	0.241747i
$155$	−4.00000	−0.321288
$156$	0	0
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	− 16.0000i	− 1.27289i
$159$	18.0000	1.42749
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	− 11.0000i	− 0.864242i
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	6.00000i	0.468521i
$165$	6.00000	0.467099
$166$	−6.00000	−0.465690
$167$	− 15.0000i	− 1.16073i	−0.814355	−	0.580367i	$-0.802909\pi$
$167$	− 15.0000i	− 1.16073i	0.814355	−	0.580367i	$-0.197091\pi$
$168$	2.00000	0.154303
$169$	0	0
$170$	−6.00000	−0.460179
$171$	− 5.00000i	− 0.382360i
$172$	2.00000	0.152499
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	0	0
$175$	1.00000i	0.0755929i
$176$	3.00000i	0.226134i
$177$	0	0
$178$	−9.00000	−0.674579
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	−16.0000	−1.18275
$184$	0	0
$185$	−11.0000	−0.808736
$186$	8.00000	0.586588
$187$	18.0000i	1.31629i
$188$	3.00000i	0.218797i
$189$	− 4.00000i	− 0.290957i
$190$	5.00000i	0.362738i
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	2.00000	0.144338
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	27.0000i	1.92367i	0.273629	+	0.961835i	$0.411776\pi$
$197$	27.0000i	1.92367i	−0.273629	+	0.961835i	$0.588224\pi$
$198$	−3.00000	−0.213201
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	1.00000i	0.0707107i
$201$	− 32.0000i	− 2.25711i
$202$	6.00000i	0.422159i
$203$	0	0
$204$	12.0000	0.840168
$205$	6.00000	0.419058
$206$	− 5.00000i	− 0.348367i
$207$	0	0
$208$	0	0
$209$	15.0000	1.03757
$210$	− 2.00000i	− 0.138013i
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	9.00000	0.618123
$213$	12.0000i	0.822226i
$214$	− 12.0000i	− 0.820303i
$215$	− 2.00000i	− 0.136399i
$216$	− 4.00000i	− 0.272166i
$217$	4.00000	0.271538
$218$	2.00000	0.135457
$219$	− 28.0000i	− 1.89206i
$220$	3.00000	0.202260
$221$	0	0
$222$	22.0000	1.47654
$223$	19.0000i	1.27233i	0.771551	+	0.636167i	$0.219481\pi$
$223$	19.0000i	1.27233i	−0.771551	+	0.636167i	$0.780519\pi$
$224$	1.00000	0.0668153
$225$	−1.00000	−0.0666667
$226$	− 12.0000i	− 0.798228i
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	− 10.0000i	− 0.662266i
$229$	− 4.00000i	− 0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	− 4.00000i	− 0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	0	0
$237$	32.0000	2.07862
$238$	6.00000	0.388922
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	− 2.00000i	− 0.129099i
$241$	23.0000i	1.48156i	0.671748	+	0.740780i	$0.265544\pi$
$241$	23.0000i	1.48156i	−0.671748	+	0.740780i	$0.734456\pi$
$242$	2.00000i	0.128565i
$243$	10.0000	0.641500
$244$	−8.00000	−0.512148
$245$	6.00000i	0.383326i
$246$	−12.0000	−0.765092
$247$	0	0
$248$	4.00000	0.254000
$249$	− 12.0000i	− 0.760469i
$250$	1.00000	0.0632456
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	1.00000i	0.0629941i
$253$	0	0
$254$	1.00000i	0.0627456i
$255$	− 12.0000i	− 0.751469i
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	4.00000i	0.249029i
$259$	11.0000	0.683507
$260$	0	0
$261$	0	0
$262$	− 9.00000i	− 0.556022i
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	−6.00000	−0.369274
$265$	− 9.00000i	− 0.552866i
$266$	− 5.00000i	− 0.306570i
$267$	− 18.0000i	− 1.10158i
$268$	− 16.0000i	− 0.977356i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	−4.00000	−0.243432
$271$	20.0000i	1.21491i	0.794353	+	0.607457i	$0.207810\pi$
$271$	20.0000i	1.21491i	−0.794353	+	0.607457i	$0.792190\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	− 3.00000i	− 0.180907i
$276$	0	0
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	− 19.0000i	− 1.13954i
$279$	4.00000i	0.239474i
$280$	− 1.00000i	− 0.0597614i
$281$	− 6.00000i	− 0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$	− 6.00000i	− 0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$	−6.00000	−0.357295
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	6.00000i	0.356034i
$285$	−10.0000	−0.592349
$286$	0	0
$287$	−6.00000	−0.354169
$288$	1.00000i	0.0589256i
$289$	19.0000	1.11765
$290$	0	0
$291$	− 20.0000i	− 1.17242i
$292$	− 14.0000i	− 0.819288i
$293$	9.00000i	0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$	9.00000i	0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$	− 12.0000i	− 0.699854i
$295$	0	0
$296$	11.0000	0.639362
$297$	12.0000i	0.696311i
$298$	−18.0000	−1.04271
$299$	0	0
$300$	−2.00000	−0.115470
$301$	2.00000i	0.115278i
$302$	16.0000	0.920697
$303$	−12.0000	−0.689382
$304$	− 5.00000i	− 0.286770i
$305$	8.00000i	0.458079i
$306$	6.00000i	0.342997i
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	−3.00000	−0.170941
$309$	10.0000	0.568880
$310$	− 4.00000i	− 0.227185i
$311$	−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
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	17.0000i	0.959366i
$315$	1.00000	0.0563436
$316$	16.0000	0.900070
$317$	− 15.0000i	− 0.842484i	−0.906948	−	0.421242i	$-0.861594\pi$
$317$	− 15.0000i	− 0.842484i	0.906948	−	0.421242i	$-0.138406\pi$
$318$	18.0000i	1.00939i
$319$	0	0
$320$	− 1.00000i	− 0.0559017i
$321$	24.0000	1.33955
$322$	0	0
$323$	− 30.0000i	− 1.66924i
$324$	11.0000	0.611111
$325$	0	0
$326$	−2.00000	−0.110770
$327$	4.00000i	0.221201i
$328$	−6.00000	−0.331295
$329$	−3.00000	−0.165395
$330$	6.00000i	0.330289i
$331$	− 20.0000i	− 1.09930i	−0.835395	−	0.549650i	$-0.814761\pi$
$331$	− 20.0000i	− 1.09930i	0.835395	−	0.549650i	$-0.185239\pi$
$332$	− 6.00000i	− 0.329293i
$333$	11.0000i	0.602796i
$334$	15.0000	0.820763
$335$	−16.0000	−0.874173
$336$	2.00000i	0.109109i
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	0	0
$339$	24.0000	1.30350
$340$	− 6.00000i	− 0.325396i
$341$	−12.0000	−0.649836
$342$	5.00000	0.270369
$343$	− 13.0000i	− 0.701934i
$344$	2.00000i	0.107833i
$345$	0	0
$346$	− 15.0000i	− 0.806405i
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	0	0
$349$	2.00000i	0.107058i	0.998566	+	0.0535288i	$0.0170469\pi$
$349$	2.00000i	0.107058i	−0.998566	+	0.0535288i	$0.982953\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−3.00000	−0.159901
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	− 9.00000i	− 0.476999i
$357$	12.0000i	0.635107i
$358$	24.0000i	1.26844i
$359$	− 6.00000i	− 0.316668i	−0.987386	−	0.158334i	$-0.949388\pi$
$359$	− 6.00000i	− 0.316668i	0.987386	−	0.158334i	$-0.0506123\pi$
$360$	1.00000	0.0527046
$361$	−6.00000	−0.315789
$362$	− 8.00000i	− 0.420471i
$363$	−4.00000	−0.209946
$364$	0	0
$365$	−14.0000	−0.732793
$366$	− 16.0000i	− 0.836333i
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	− 6.00000i	− 0.312348i
$370$	− 11.0000i	− 0.571863i
$371$	9.00000i	0.467257i
$372$	8.00000i	0.414781i
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−18.0000	−0.930758
$375$	2.00000i	0.103280i
$376$	−3.00000	−0.154713
$377$	0	0
$378$	4.00000	0.205738
$379$	19.0000i	0.975964i	0.872854	+	0.487982i	$0.162267\pi$
$379$	19.0000i	0.975964i	−0.872854	+	0.487982i	$0.837733\pi$
$380$	−5.00000	−0.256495
$381$	−2.00000	−0.102463
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	2.00000i	0.102062i
$385$	3.00000i	0.152894i
$386$	4.00000	0.203595
$387$	−2.00000	−0.101666
$388$	− 10.0000i	− 0.507673i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	0	0
$392$	− 6.00000i	− 0.303046i
$393$	18.0000	0.907980
$394$	−27.0000	−1.36024
$395$	− 16.0000i	− 0.805047i
$396$	− 3.00000i	− 0.150756i
$397$	− 13.0000i	− 0.652451i	−0.945292	−	0.326226i	$-0.894223\pi$
$397$	− 13.0000i	− 0.652451i	0.945292	−	0.326226i	$-0.105777\pi$
$398$	10.0000i	0.501255i
$399$	10.0000	0.500626
$400$	−1.00000	−0.0500000
$401$	− 15.0000i	− 0.749064i	−0.927214	−	0.374532i	$-0.877803\pi$
$401$	− 15.0000i	− 0.749064i	0.927214	−	0.374532i	$-0.122197\pi$
$402$	32.0000	1.59601
$403$	0	0
$404$	−6.00000	−0.298511
$405$	− 11.0000i	− 0.546594i
$406$	0	0
$407$	−33.0000	−1.63575
$408$	12.0000i	0.594089i
$409$	− 5.00000i	− 0.247234i	−0.992330	−	0.123617i	$-0.960551\pi$
$409$	− 5.00000i	− 0.247234i	0.992330	−	0.123617i	$-0.0394494\pi$
$410$	6.00000i	0.296319i
$411$	− 12.0000i	− 0.591916i
$412$	5.00000	0.246332
$413$	0	0
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	38.0000	1.86087
$418$	15.0000i	0.733674i
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	2.00000	0.0975900
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	23.0000i	1.11962i
$423$	− 3.00000i	− 0.145865i
$424$	9.00000i	0.437079i
$425$	−6.00000	−0.291043
$426$	−12.0000	−0.581402
$427$	− 8.00000i	− 0.387147i
$428$	12.0000	0.580042
$429$	0	0
$430$	2.00000	0.0964486
$431$	− 30.0000i	− 1.44505i	−0.691345	−	0.722525i	$-0.742982\pi$
$431$	− 30.0000i	− 1.44505i	0.691345	−	0.722525i	$-0.257018\pi$
$432$	4.00000	0.192450
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	4.00000i	0.192006i
$435$	0	0
$436$	2.00000i	0.0957826i
$437$	0	0
$438$	28.0000	1.33789
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	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	$0.640642\pi$
$444$	22.0000i	1.04407i
$445$	−9.00000	−0.426641
$446$	−19.0000	−0.899676
$447$	− 36.0000i	− 1.70274i
$448$	1.00000i	0.0472456i
$449$	9.00000i	0.424736i	0.977190	+	0.212368i	$0.0681176\pi$
$449$	9.00000i	0.424736i	−0.977190	+	0.212368i	$0.931882\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	18.0000	0.847587
$452$	12.0000	0.564433
$453$	32.0000i	1.50349i
$454$	−24.0000	−1.12638
$455$	0	0
$456$	10.0000	0.468293
$457$	4.00000i	0.187112i	0.995614	+	0.0935561i	$0.0298234\pi$
$457$	4.00000i	0.187112i	−0.995614	+	0.0935561i	$0.970177\pi$
$458$	4.00000	0.186908
$459$	24.0000	1.12022
$460$	0	0
$461$	42.0000i	1.95614i	0.208288	+	0.978068i	$0.433211\pi$
$461$	42.0000i	1.95614i	−0.208288	+	0.978068i	$0.566789\pi$
$462$	− 6.00000i	− 0.279145i
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	24.0000i	1.11178i
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	3.00000i	0.138380i
$471$	−34.0000	−1.56664
$472$	0	0
$473$	− 6.00000i	− 0.275880i
$474$	32.0000i	1.46981i
$475$	5.00000i	0.229416i
$476$	6.00000i	0.275010i
$477$	−9.00000	−0.412082
$478$	0	0
$479$	− 30.0000i	− 1.37073i	−0.728197	−	0.685367i	$-0.759642\pi$
$479$	− 30.0000i	− 1.37073i	0.728197	−	0.685367i	$-0.240358\pi$
$480$	2.00000	0.0912871
$481$	0	0
$482$	−23.0000	−1.04762
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−10.0000	−0.454077
$486$	10.0000i	0.453609i
$487$	19.0000i	0.860972i	0.902597	+	0.430486i	$0.141658\pi$
$487$	19.0000i	0.860972i	−0.902597	+	0.430486i	$0.858342\pi$
$488$	− 8.00000i	− 0.362143i
$489$	− 4.00000i	− 0.180886i
$490$	−6.00000	−0.271052
$491$	−27.0000	−1.21849	−0.609246	−	0.792981i	$-0.708528\pi$
$491$	−27.0000	−1.21849	−0.609246	+	0.792981i	$0.708528\pi$
$492$	− 12.0000i	− 0.541002i
$493$	0	0
$494$	0	0
$495$	−3.00000	−0.134840
$496$	4.00000i	0.179605i
$497$	−6.00000	−0.269137
$498$	12.0000	0.537733
$499$	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
$501$	30.0000i	1.34030i
$502$	15.0000i	0.669483i
$503$	9.00000	0.401290	0.200645	−	0.979664i	$-0.435696\pi$
$503$	9.00000	0.401290	0.200645	+	0.979664i	$0.435696\pi$
$504$	−1.00000	−0.0445435
$505$	6.00000i	0.266996i
$506$	0	0
$507$	0	0
$508$	−1.00000	−0.0443678
$509$	− 6.00000i	− 0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$	− 6.00000i	− 0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	12.0000	0.531369
$511$	14.0000	0.619324
$512$	1.00000i	0.0441942i
$513$	− 20.0000i	− 0.883022i
$514$	− 12.0000i	− 0.529297i
$515$	− 5.00000i	− 0.220326i
$516$	−4.00000	−0.176090
$517$	9.00000	0.395820
$518$	11.0000i	0.483312i
$519$	30.0000	1.31685
$520$	0	0
$521$	−27.0000	−1.18289	−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000	−1.18289	−0.591446	+	0.806345i	$0.701443\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	9.00000	0.393167
$525$	− 2.00000i	− 0.0872872i
$526$	9.00000i	0.392419i
$527$	24.0000i	1.04546i
$528$	− 6.00000i	− 0.261116i
$529$	−23.0000	−1.00000
$530$	9.00000	0.390935
$531$	0	0
$532$	5.00000	0.216777
$533$	0	0
$534$	18.0000	0.778936
$535$	− 12.0000i	− 0.518805i
$536$	16.0000	0.691095
$537$	−48.0000	−2.07135
$538$	6.00000i	0.258678i
$539$	18.0000i	0.775315i
$540$	− 4.00000i	− 0.172133i
$541$	20.0000i	0.859867i	0.902861	+	0.429934i	$0.141463\pi$
$541$	20.0000i	0.859867i	−0.902861	+	0.429934i	$0.858537\pi$
$542$	−20.0000	−0.859074
$543$	16.0000	0.686626
$544$	6.00000i	0.257248i
$545$	2.00000	0.0856706
$546$	0	0
$547$	−34.0000	−1.45374	−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000	−1.45374	−0.726868	+	0.686778i	$0.759025\pi$
$548$	− 6.00000i	− 0.256307i
$549$	8.00000	0.341432
$550$	3.00000	0.127920
$551$	0	0
$552$	0	0
$553$	16.0000i	0.680389i
$554$	1.00000i	0.0424859i
$555$	22.0000	0.933848
$556$	19.0000	0.805779
$557$	21.0000i	0.889799i	0.895581	+	0.444899i	$0.146761\pi$
$557$	21.0000i	0.889799i	−0.895581	+	0.444899i	$0.853239\pi$
$558$	−4.00000	−0.169334
$559$	0	0
$560$	1.00000	0.0422577
$561$	− 36.0000i	− 1.51992i
$562$	6.00000	0.253095
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	− 6.00000i	− 0.252646i
$565$	− 12.0000i	− 0.504844i
$566$	− 14.0000i	− 0.588464i
$567$	11.0000i	0.461957i
$568$	−6.00000	−0.251754
$569$	−9.00000	−0.377300	−0.188650	−	0.982044i	$-0.560411\pi$
$569$	−9.00000	−0.377300	−0.188650	+	0.982044i	$0.560411\pi$
$570$	− 10.0000i	− 0.418854i
$571$	−17.0000	−0.711428	−0.355714	−	0.934595i	$-0.615762\pi$
$571$	−17.0000	−0.711428	−0.355714	+	0.934595i	$0.615762\pi$
$572$	0	0
$573$	0	0
$574$	− 6.00000i	− 0.250435i
$575$	0	0
$576$	−1.00000	−0.0416667
$577$	− 32.0000i	− 1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$	− 32.0000i	− 1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$	19.0000i	0.790296i
$579$	8.00000i	0.332469i
$580$	0	0
$581$	6.00000	0.248922
$582$	20.0000	0.829027
$583$	− 27.0000i	− 1.11823i
$584$	14.0000	0.579324
$585$	0	0
$586$	−9.00000	−0.371787
$587$	6.00000i	0.247647i	0.992304	+	0.123823i	$0.0395156\pi$
$587$	6.00000i	0.247647i	−0.992304	+	0.123823i	$0.960484\pi$
$588$	12.0000	0.494872
$589$	20.0000	0.824086
$590$	0	0
$591$	− 54.0000i	− 2.22126i
$592$	11.0000i	0.452097i
$593$	24.0000i	0.985562i	0.870153	+	0.492781i	$0.164020\pi$
$593$	24.0000i	0.985562i	−0.870153	+	0.492781i	$0.835980\pi$
$594$	−12.0000	−0.492366
$595$	6.00000	0.245976
$596$	− 18.0000i	− 0.737309i
$597$	−20.0000	−0.818546
$598$	0	0
$599$	−18.0000	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−18.0000	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	− 2.00000i	− 0.0816497i
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	−2.00000	−0.0815139
$603$	16.0000i	0.651570i
$604$	16.0000i	0.651031i
$605$	2.00000i	0.0813116i
$606$	− 12.0000i	− 0.487467i
$607$	−13.0000	−0.527654	−0.263827	−	0.964570i	$-0.584985\pi$
$607$	−13.0000	−0.527654	−0.263827	+	0.964570i	$0.584985\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	−8.00000	−0.323911
$611$	0	0
$612$	−6.00000	−0.242536
$613$	13.0000i	0.525065i	0.964923	+	0.262533i	$0.0845577\pi$
$613$	13.0000i	0.525065i	−0.964923	+	0.262533i	$0.915442\pi$
$614$	−2.00000	−0.0807134
$615$	−12.0000	−0.483887
$616$	− 3.00000i	− 0.120873i
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	10.0000i	0.402259i
$619$	− 1.00000i	− 0.0401934i	−0.999798	−	0.0200967i	$-0.993603\pi$
$619$	− 1.00000i	− 0.0401934i	0.999798	−	0.0200967i	$-0.00639741\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	− 30.0000i	− 1.20289i
$623$	9.00000	0.360577
$624$	0	0
$625$	1.00000	0.0400000
$626$	14.0000i	0.559553i
$627$	−30.0000	−1.19808
$628$	−17.0000	−0.678374
$629$	66.0000i	2.63159i
$630$	1.00000i	0.0398410i
$631$	− 28.0000i	− 1.11466i	−0.830290	−	0.557331i	$-0.811825\pi$
$631$	− 28.0000i	− 1.11466i	0.830290	−	0.557331i	$-0.188175\pi$
$632$	16.0000i	0.636446i
$633$	−46.0000	−1.82834
$634$	15.0000	0.595726
$635$	1.00000i	0.0396838i
$636$	−18.0000	−0.713746
$637$	0	0
$638$	0	0
$639$	− 6.00000i	− 0.237356i
$640$	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	$0.443122\pi$
$642$	24.0000i	0.947204i
$643$	− 2.00000i	− 0.0788723i	−0.999222	−	0.0394362i	$-0.987444\pi$
$643$	− 2.00000i	− 0.0788723i	0.999222	−	0.0394362i	$-0.0125562\pi$
$644$	0	0
$645$	4.00000i	0.157500i
$646$	30.0000	1.18033
$647$	9.00000	0.353827	0.176913	−	0.984226i	$-0.443389\pi$
$647$	9.00000	0.353827	0.176913	+	0.984226i	$0.443389\pi$
$648$	11.0000i	0.432121i
$649$	0	0
$650$	0	0
$651$	−8.00000	−0.313545
$652$	− 2.00000i	− 0.0783260i
$653$	−27.0000	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−27.0000	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$654$	−4.00000	−0.156412
$655$	− 9.00000i	− 0.351659i
$656$	− 6.00000i	− 0.234261i
$657$	14.0000i	0.546192i
$658$	− 3.00000i	− 0.116952i
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	−6.00000	−0.233550
$661$	− 4.00000i	− 0.155582i	−0.996970	−	0.0777910i	$-0.975213\pi$
$661$	− 4.00000i	− 0.155582i	0.996970	−	0.0777910i	$-0.0247867\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	6.00000	0.232845
$665$	− 5.00000i	− 0.193892i
$666$	−11.0000	−0.426241
$667$	0	0
$668$	15.0000i	0.580367i
$669$	− 38.0000i	− 1.46916i
$670$	− 16.0000i	− 0.618134i
$671$	24.0000i	0.926510i
$672$	−2.00000	−0.0771517
$673$	−20.0000	−0.770943	−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000	−0.770943	−0.385472	+	0.922720i	$0.625961\pi$
$674$	16.0000i	0.616297i
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	24.0000i	0.921714i
$679$	10.0000	0.383765
$680$	6.00000	0.230089
$681$	− 48.0000i	− 1.83936i
$682$	− 12.0000i	− 0.459504i
$683$	− 6.00000i	− 0.229584i	−0.993390	−	0.114792i	$-0.963380\pi$
$683$	− 6.00000i	− 0.229584i	0.993390	−	0.114792i	$-0.0366201\pi$
$684$	5.00000i	0.191180i
$685$	−6.00000	−0.229248
$686$	13.0000	0.496342
$687$	8.00000i	0.305219i
$688$	−2.00000	−0.0762493
$689$	0	0
$690$	0	0
$691$	13.0000i	0.494543i	0.968946	+	0.247272i	$0.0795340\pi$
$691$	13.0000i	0.494543i	−0.968946	+	0.247272i	$0.920466\pi$
$692$	15.0000	0.570214
$693$	3.00000	0.113961
$694$	6.00000i	0.227757i
$695$	− 19.0000i	− 0.720711i
$696$	0	0
$697$	− 36.0000i	− 1.36360i
$698$	−2.00000	−0.0757011
$699$	−48.0000	−1.81553
$700$	− 1.00000i	− 0.0377964i
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	55.0000	2.07436
$704$	− 3.00000i	− 0.113067i
$705$	−6.00000	−0.225973
$706$	6.00000	0.225813
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	− 10.0000i	− 0.375558i	−0.982211	−	0.187779i	$-0.939871\pi$
$709$	− 10.0000i	− 0.375558i	0.982211	−	0.187779i	$-0.0601289\pi$
$710$	6.00000i	0.225176i
$711$	−16.0000	−0.600047
$712$	9.00000	0.337289
$713$	0	0
$714$	−12.0000	−0.449089
$715$	0	0
$716$	−24.0000	−0.896922
$717$	0	0
$718$	6.00000	0.223918
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	1.00000i	0.0372678i
$721$	5.00000i	0.186210i
$722$	− 6.00000i	− 0.223297i
$723$	− 46.0000i	− 1.71076i
$724$	8.00000	0.297318
$725$	0	0
$726$	− 4.00000i	− 0.148454i
$727$	−29.0000	−1.07555	−0.537775	−	0.843088i	$-0.680735\pi$
$727$	−29.0000	−1.07555	−0.537775	+	0.843088i	$0.680735\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	− 14.0000i	− 0.518163i
$731$	−12.0000	−0.443836
$732$	16.0000	0.591377
$733$	25.0000i	0.923396i	0.887037	+	0.461698i	$0.152760\pi$
$733$	25.0000i	0.923396i	−0.887037	+	0.461698i	$0.847240\pi$
$734$	32.0000i	1.18114i
$735$	− 12.0000i	− 0.442627i
$736$	0	0
$737$	−48.0000	−1.76810
$738$	6.00000	0.220863
$739$	− 7.00000i	− 0.257499i	−0.991677	−	0.128750i	$-0.958904\pi$
$739$	− 7.00000i	− 0.257499i	0.991677	−	0.128750i	$-0.0410963\pi$
$740$	11.0000	0.404368
$741$	0	0
$742$	−9.00000	−0.330400
$743$	− 12.0000i	− 0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$	− 12.0000i	− 0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$	−8.00000	−0.293294
$745$	−18.0000	−0.659469
$746$	14.0000i	0.512576i
$747$	6.00000i	0.219529i
$748$	− 18.0000i	− 0.658145i
$749$	12.0000i	0.438470i
$750$	−2.00000	−0.0730297
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	− 3.00000i	− 0.109399i
$753$	−30.0000	−1.09326
$754$	0	0
$755$	16.0000	0.582300
$756$	4.00000i	0.145479i
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	−19.0000	−0.690111
$759$	0	0
$760$	− 5.00000i	− 0.181369i
$761$	− 3.00000i	− 0.108750i	−0.998521	−	0.0543750i	$-0.982683\pi$
$761$	− 3.00000i	− 0.108750i	0.998521	−	0.0543750i	$-0.0173166\pi$
$762$	− 2.00000i	− 0.0724524i
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	6.00000i	0.216930i
$766$	0	0
$767$	0	0
$768$	−2.00000	−0.0721688
$769$	34.0000i	1.22607i	0.790055	+	0.613036i	$0.210052\pi$
$769$	34.0000i	1.22607i	−0.790055	+	0.613036i	$0.789948\pi$
$770$	−3.00000	−0.108112
$771$	24.0000	0.864339
$772$	4.00000i	0.143963i
$773$	− 39.0000i	− 1.40273i	−0.712801	−	0.701366i	$-0.752574\pi$
$773$	− 39.0000i	− 1.40273i	0.712801	−	0.701366i	$-0.247426\pi$
$774$	− 2.00000i	− 0.0718885i
$775$	− 4.00000i	− 0.143684i
$776$	10.0000	0.358979
$777$	−22.0000	−0.789246
$778$	30.0000i	1.07555i
$779$	−30.0000	−1.07486
$780$	0	0
$781$	18.0000	0.644091
$782$	0	0
$783$	0	0
$784$	6.00000	0.214286
$785$	17.0000i	0.606756i
$786$	18.0000i	0.642039i
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	− 27.0000i	− 0.961835i
$789$	−18.0000	−0.640817
$790$	16.0000	0.569254
$791$	12.0000i	0.426671i
$792$	3.00000	0.106600
$793$	0	0
$794$	13.0000	0.461353
$795$	18.0000i	0.638394i
$796$	−10.0000	−0.354441
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	10.0000i	0.353996i
$799$	− 18.0000i	− 0.636794i
$800$	− 1.00000i	− 0.0353553i
$801$	9.00000i	0.317999i
$802$	15.0000	0.529668
$803$	−42.0000	−1.48215
$804$	32.0000i	1.12855i
$805$	0	0
$806$	0	0
$807$	−12.0000	−0.422420
$808$	− 6.00000i	− 0.211079i
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	11.0000	0.386501
$811$	49.0000i	1.72062i	0.509769	+	0.860311i	$0.329731\pi$
$811$	49.0000i	1.72062i	−0.509769	+	0.860311i	$0.670269\pi$
$812$	0	0
$813$	− 40.0000i	− 1.40286i
$814$	− 33.0000i	− 1.15665i
$815$	−2.00000	−0.0700569
$816$	−12.0000	−0.420084
$817$	10.0000i	0.349856i
$818$	5.00000	0.174821
$819$	0	0
$820$	−6.00000	−0.209529
$821$	− 36.0000i	− 1.25641i	−0.778048	−	0.628204i	$-0.783790\pi$
$821$	− 36.0000i	− 1.25641i	0.778048	−	0.628204i	$-0.216210\pi$
$822$	12.0000	0.418548
$823$	43.0000	1.49889	0.749443	−	0.662069i	$-0.230321\pi$
$823$	43.0000	1.49889	0.749443	+	0.662069i	$0.230321\pi$
$824$	5.00000i	0.174183i
$825$	6.00000i	0.208893i
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	0	0
$829$	40.0000	1.38926	0.694629	−	0.719368i	$-0.255569\pi$
$829$	40.0000	1.38926	0.694629	+	0.719368i	$0.255569\pi$
$830$	− 6.00000i	− 0.208263i
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	36.0000	1.24733
$834$	38.0000i	1.31583i
$835$	15.0000	0.519096
$836$	−15.0000	−0.518786
$837$	16.0000i	0.553041i
$838$	− 36.0000i	− 1.24360i
$839$	− 24.0000i	− 0.828572i	−0.910147	−	0.414286i	$-0.864031\pi$
$839$	− 24.0000i	− 0.828572i	0.910147	−	0.414286i	$-0.135969\pi$
$840$	2.00000i	0.0690066i
$841$	−29.0000	−1.00000
$842$	−10.0000	−0.344623
$843$	12.0000i	0.413302i
$844$	−23.0000	−0.791693
$845$	0	0
$846$	3.00000	0.103142
$847$	− 2.00000i	− 0.0687208i
$848$	−9.00000	−0.309061
$849$	28.0000	0.960958
$850$	− 6.00000i	− 0.205798i
$851$	0	0
$852$	− 12.0000i	− 0.411113i
$853$	− 46.0000i	− 1.57501i	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	− 46.0000i	− 1.57501i	0.616308	−	0.787505i	$-0.288628\pi$
$854$	8.00000	0.273754
$855$	5.00000	0.170996
$856$	12.0000i	0.410152i
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	−31.0000	−1.05771	−0.528853	−	0.848713i	$-0.677378\pi$
$859$	−31.0000	−1.05771	−0.528853	+	0.848713i	$0.677378\pi$
$860$	2.00000i	0.0681994i
$861$	12.0000	0.408959
$862$	30.0000	1.02180
$863$	− 36.0000i	− 1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$	− 36.0000i	− 1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$	4.00000i	0.136083i
$865$	− 15.0000i	− 0.510015i
$866$	16.0000i	0.543702i
$867$	−38.0000	−1.29055
$868$	−4.00000	−0.135769
$869$	− 48.0000i	− 1.62829i
$870$	0	0
$871$	0	0
$872$	−2.00000	−0.0677285
$873$	10.0000i	0.338449i
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	28.0000i	0.946032i
$877$	10.0000i	0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$	10.0000i	0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	− 20.0000i	− 0.674967i
$879$	− 18.0000i	− 0.607125i
$880$	−3.00000	−0.101130
$881$	15.0000	0.505363	0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000	0.505363	0.252681	+	0.967550i	$0.418688\pi$
$882$	6.00000i	0.202031i
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	0	0
$885$	0	0
$886$	− 18.0000i	− 0.604722i
$887$	−27.0000	−0.906571	−0.453286	−	0.891365i	$-0.649748\pi$
$887$	−27.0000	−0.906571	−0.453286	+	0.891365i	$0.649748\pi$
$888$	−22.0000	−0.738272
$889$	− 1.00000i	− 0.0335389i
$890$	− 9.00000i	− 0.301681i
$891$	− 33.0000i	− 1.10554i
$892$	− 19.0000i	− 0.636167i
$893$	−15.0000	−0.501956
$894$	36.0000	1.20402
$895$	24.0000i	0.802232i
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−9.00000	−0.300334
$899$	0	0
$900$	1.00000	0.0333333
$901$	−54.0000	−1.79900
$902$	18.0000i	0.599334i
$903$	− 4.00000i	− 0.133112i
$904$	12.0000i	0.399114i
$905$	− 8.00000i	− 0.265929i
$906$	−32.0000	−1.06313
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	− 24.0000i	− 0.796468i
$909$	6.00000	0.199007
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	10.0000i	0.331133i
$913$	−18.0000	−0.595713
$914$	−4.00000	−0.132308
$915$	− 16.0000i	− 0.528944i
$916$	4.00000i	0.132164i
$917$	9.00000i	0.297206i
$918$	24.0000i	0.792118i
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	0	0
$921$	− 4.00000i	− 0.131804i
$922$	−42.0000	−1.38320
$923$	0	0
$924$	6.00000	0.197386
$925$	− 11.0000i	− 0.361678i
$926$	−8.00000	−0.262896
$927$	−5.00000	−0.164222
$928$	0	0
$929$	− 42.0000i	− 1.37798i	−0.724773	−	0.688988i	$-0.758055\pi$
$929$	− 42.0000i	− 1.37798i	0.724773	−	0.688988i	$-0.241945\pi$
$930$	8.00000i	0.262330i
$931$	− 30.0000i	− 0.983210i
$932$	−24.0000	−0.786146
$933$	60.0000	1.96431
$934$	12.0000i	0.392652i
$935$	−18.0000	−0.588663
$936$	0	0
$937$	50.0000	1.63343	0.816714	−	0.577042i	$-0.195793\pi$
$937$	50.0000	1.63343	0.816714	+	0.577042i	$0.195793\pi$
$938$	16.0000i	0.522419i
$939$	−28.0000	−0.913745
$940$	−3.00000	−0.0978492
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	− 34.0000i	− 1.10778i
$943$	0	0
$944$	0	0
$945$	4.00000	0.130120
$946$	6.00000	0.195077
$947$	48.0000i	1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$	48.0000i	1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$	−32.0000	−1.03931
$949$	0	0
$950$	−5.00000	−0.162221
$951$	30.0000i	0.972817i
$952$	−6.00000	−0.194461
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	− 9.00000i	− 0.291386i
$955$	0	0
$956$	0	0
$957$	0	0
$958$	30.0000	0.969256
$959$	6.00000	0.193750
$960$	2.00000i	0.0645497i
$961$	15.0000	0.483871
$962$	0	0
$963$	−12.0000	−0.386695
$964$	− 23.0000i	− 0.740780i
$965$	4.00000	0.128765
$966$	0	0
$967$	− 5.00000i	− 0.160789i	−0.996763	−	0.0803946i	$-0.974382\pi$
$967$	− 5.00000i	− 0.160789i	0.996763	−	0.0803946i	$-0.0256180\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	60.0000i	1.92748i
$970$	− 10.0000i	− 0.321081i
$971$	33.0000	1.05902	0.529510	−	0.848304i	$-0.322376\pi$
$971$	33.0000	1.05902	0.529510	+	0.848304i	$0.322376\pi$
$972$	−10.0000	−0.320750
$973$	19.0000i	0.609112i
$974$	−19.0000	−0.608799
$975$	0	0
$976$	8.00000	0.256074
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	4.00000	0.127906
$979$	−27.0000	−0.862924
$980$	− 6.00000i	− 0.191663i
$981$	− 2.00000i	− 0.0638551i
$982$	− 27.0000i	− 0.861605i
$983$	39.0000i	1.24391i	0.783054	+	0.621953i	$0.213661\pi$
$983$	39.0000i	1.24391i	−0.783054	+	0.621953i	$0.786339\pi$
$984$	12.0000	0.382546
$985$	−27.0000	−0.860292
$986$	0	0
$987$	6.00000	0.190982
$988$	0	0
$989$	0	0
$990$	− 3.00000i	− 0.0953463i
$991$	−22.0000	−0.698853	−0.349427	−	0.936964i	$-0.613624\pi$
$991$	−22.0000	−0.698853	−0.349427	+	0.936964i	$0.613624\pi$
$992$	−4.00000	−0.127000
$993$	40.0000i	1.26936i
$994$	− 6.00000i	− 0.190308i
$995$	10.0000i	0.317021i
$996$	12.0000i	0.380235i
$997$	41.0000	1.29848	0.649242	−	0.760582i	$-0.275086\pi$
$997$	41.0000	1.29848	0.649242	+	0.760582i	$0.275086\pi$
$998$	−4.00000	−0.126618
$999$	44.0000i	1.39210i

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{6} -1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{6} -1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000i q^{11} +2.00000 q^{12} +1.00000 q^{14} -2.00000i q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000i q^{18} -5.00000i q^{19} -1.00000i q^{20} +2.00000i q^{21} -3.00000 q^{22} +2.00000i q^{24} -1.00000 q^{25} +4.00000 q^{27} +1.00000i q^{28} +2.00000 q^{30} +4.00000i q^{31} +1.00000i q^{32} -6.00000i q^{33} +6.00000i q^{34} +1.00000 q^{35} -1.00000 q^{36} +11.0000i q^{37} +5.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} -2.00000 q^{42} -2.00000 q^{43} -3.00000i q^{44} +1.00000i q^{45} -3.00000i q^{47} -2.00000 q^{48} +6.00000 q^{49} -1.00000i q^{50} -12.0000 q^{51} -9.00000 q^{53} +4.00000i q^{54} -3.00000 q^{55} -1.00000 q^{56} +10.0000i q^{57} +2.00000i q^{60} +8.00000 q^{61} -4.00000 q^{62} -1.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} +16.0000i q^{67} -6.00000 q^{68} +1.00000i q^{70} -6.00000i q^{71} -1.00000i q^{72} +14.0000i q^{73} -11.0000 q^{74} +2.00000 q^{75} +5.00000i q^{76} +3.00000 q^{77} -16.0000 q^{79} +1.00000i q^{80} -11.0000 q^{81} +6.00000 q^{82} +6.00000i q^{83} -2.00000i q^{84} +6.00000i q^{85} -2.00000i q^{86} +3.00000 q^{88} +9.00000i q^{89} -1.00000 q^{90} -8.00000i q^{93} +3.00000 q^{94} +5.00000 q^{95} -2.00000i q^{96} +10.0000i q^{97} +6.00000i q^{98} +3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q - 4 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 4 q^{12} + 2 q^{14} + 2 q^{16} + 12 q^{17} - 6 q^{22} - 2 q^{25} + 8 q^{27} + 4 q^{30} + 2 q^{35} - 2 q^{36} + 10 q^{38} + 2 q^{40} - 4 q^{42} - 4 q^{43} - 4 q^{48} + 12 q^{49} - 24 q^{51} - 18 q^{53} - 6 q^{55} - 2 q^{56} + 16 q^{61} - 8 q^{62} - 2 q^{64} + 12 q^{66} - 12 q^{68} - 22 q^{74} + 4 q^{75} + 6 q^{77} - 32 q^{79} - 22 q^{81} + 12 q^{82} + 6 q^{88} - 2 q^{90} + 6 q^{94} + 10 q^{95}+O(q^{100}) \) 2 * q - 4 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 4 * q^12 + 2 * q^14 + 2 * q^16 + 12 * q^17 - 6 * q^22 - 2 * q^25 + 8 * q^27 + 4 * q^30 + 2 * q^35 - 2 * q^36 + 10 * q^38 + 2 * q^40 - 4 * q^42 - 4 * q^43 - 4 * q^48 + 12 * q^49 - 24 * q^51 - 18 * q^53 - 6 * q^55 - 2 * q^56 + 16 * q^61 - 8 * q^62 - 2 * q^64 + 12 * q^66 - 12 * q^68 - 22 * q^74 + 4 * q^75 + 6 * q^77 - 32 * q^79 - 22 * q^81 + 12 * q^82 + 6 * q^88 - 2 * q^90 + 6 * q^94 + 10 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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