LMFDB - Embedded newform 1690.2.d.f.1351.3

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:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.3
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1690.1351
Dual form			1690.2.d.f.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -2.73205 q^{3} -1.00000 q^{4} -1.00000i q^{5} -2.73205i q^{6} +3.00000i q^{7} -1.00000i q^{8} +4.46410 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.73205 q^{3} -1.00000 q^{4} -1.00000i q^{5} -2.73205i q^{6} +3.00000i q^{7} -1.00000i q^{8} +4.46410 q^{9} +1.00000 q^{10} -3.00000i q^{11} +2.73205 q^{12} -3.00000 q^{14} +2.73205i q^{15} +1.00000 q^{16} -2.19615 q^{17} +4.46410i q^{18} +6.46410i q^{19} +1.00000i q^{20} -8.19615i q^{21} +3.00000 q^{22} +2.53590 q^{23} +2.73205i q^{24} -1.00000 q^{25} -4.00000 q^{27} -3.00000i q^{28} -9.46410 q^{29} -2.73205 q^{30} +1.26795i q^{31} +1.00000i q^{32} +8.19615i q^{33} -2.19615i q^{34} +3.00000 q^{35} -4.46410 q^{36} +11.1962i q^{37} -6.46410 q^{38} -1.00000 q^{40} -10.3923i q^{41} +8.19615 q^{42} +2.00000 q^{43} +3.00000i q^{44} -4.46410i q^{45} +2.53590i q^{46} +3.00000i q^{47} -2.73205 q^{48} -2.00000 q^{49} -1.00000i q^{50} +6.00000 q^{51} -6.46410 q^{53} -4.00000i q^{54} -3.00000 q^{55} +3.00000 q^{56} -17.6603i q^{57} -9.46410i q^{58} -10.3923i q^{59} -2.73205i q^{60} -4.19615 q^{61} -1.26795 q^{62} +13.3923i q^{63} -1.00000 q^{64} -8.19615 q^{66} +2.19615 q^{68} -6.92820 q^{69} +3.00000i q^{70} -6.00000i q^{71} -4.46410i q^{72} -5.66025i q^{73} -11.1962 q^{74} +2.73205 q^{75} -6.46410i q^{76} +9.00000 q^{77} +6.19615 q^{79} -1.00000i q^{80} -2.46410 q^{81} +10.3923 q^{82} +2.19615i q^{83} +8.19615i q^{84} +2.19615i q^{85} +2.00000i q^{86} +25.8564 q^{87} -3.00000 q^{88} -17.1962i q^{89} +4.46410 q^{90} -2.53590 q^{92} -3.46410i q^{93} -3.00000 q^{94} +6.46410 q^{95} -2.73205i q^{96} -15.1244i q^{97} -2.00000i q^{98} -13.3923i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9	$ 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} + 4 q^{10} + 4 q^{12} - 12 q^{14} + 4 q^{16} + 12 q^{17} + 12 q^{22} + 24 q^{23} - 4 q^{25} - 16 q^{27} - 24 q^{29} - 4 q^{30} + 12 q^{35} - 4 q^{36} - 12 q^{38} - 4 q^{40} + 12 q^{42} + 8 q^{43} - 4 q^{48} - 8 q^{49} + 24 q^{51} - 12 q^{53} - 12 q^{55} + 12 q^{56} + 4 q^{61} - 12 q^{62} - 4 q^{64} - 12 q^{66} - 12 q^{68} - 24 q^{74} + 4 q^{75} + 36 q^{77} + 4 q^{79} + 4 q^{81} + 48 q^{87} - 12 q^{88} + 4 q^{90} - 24 q^{92} - 12 q^{94} + 12 q^{95}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 + 4 * q^10 + 4 * q^12 - 12 * q^14 + 4 * q^16 + 12 * q^17 + 12 * q^22 + 24 * q^23 - 4 * q^25 - 16 * q^27 - 24 * q^29 - 4 * q^30 + 12 * q^35 - 4 * q^36 - 12 * q^38 - 4 * q^40 + 12 * q^42 + 8 * q^43 - 4 * q^48 - 8 * q^49 + 24 * q^51 - 12 * q^53 - 12 * q^55 + 12 * q^56 + 4 * q^61 - 12 * q^62 - 4 * q^64 - 12 * q^66 - 12 * q^68 - 24 * q^74 + 4 * q^75 + 36 * q^77 + 4 * q^79 + 4 * q^81 + 48 * q^87 - 12 * q^88 + 4 * q^90 - 24 * q^92 - 12 * q^94 + 12 * 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
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	− 2.73205i	− 1.11536i
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	− 1.00000i	− 0.353553i
$9$	4.46410	1.48803
$10$	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	$-0.149385\pi$
$12$	2.73205	0.788675
$13$	0	0
$13$	0	0
$14$	−3.00000	−0.801784
$15$	2.73205i	0.705412i
$16$	1.00000	0.250000
$17$	−2.19615	−0.532645	−0.266323	−	0.963884i	$-0.585809\pi$
$17$	−2.19615	−0.532645	−0.266323	+	0.963884i	$0.585809\pi$
$18$	4.46410i	1.05220i
$19$	6.46410i	1.48297i	0.670971	+	0.741483i	$0.265877\pi$
$19$	6.46410i	1.48297i	−0.670971	+	0.741483i	$0.734123\pi$
$20$	1.00000i	0.223607i
$21$	− 8.19615i	− 1.78855i
$22$	3.00000	0.639602
$23$	2.53590	0.528771	0.264386	−	0.964417i	$-0.414831\pi$
$23$	2.53590	0.528771	0.264386	+	0.964417i	$0.414831\pi$
$24$	2.73205i	0.557678i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	− 3.00000i	− 0.566947i
$29$	−9.46410	−1.75744	−0.878720	−	0.477338i	$-0.841602\pi$
$29$	−9.46410	−1.75744	−0.878720	+	0.477338i	$0.841602\pi$
$30$	−2.73205	−0.498802
$31$	1.26795i	0.227730i	0.993496	+	0.113865i	$0.0363232\pi$
$31$	1.26795i	0.227730i	−0.993496	+	0.113865i	$0.963677\pi$
$32$	1.00000i	0.176777i
$33$	8.19615i	1.42677i
$34$	− 2.19615i	− 0.376637i
$35$	3.00000	0.507093
$36$	−4.46410	−0.744017
$37$	11.1962i	1.84064i	0.391171	+	0.920318i	$0.372070\pi$
$37$	11.1962i	1.84064i	−0.391171	+	0.920318i	$0.627930\pi$
$38$	−6.46410	−1.04862
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 10.3923i	− 1.62301i	−0.584349	−	0.811503i	$-0.698650\pi$
$41$	− 10.3923i	− 1.62301i	0.584349	−	0.811503i	$-0.301350\pi$
$42$	8.19615	1.26469
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	3.00000i	0.452267i
$45$	− 4.46410i	− 0.665469i
$46$	2.53590i	0.373898i
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	−2.73205	−0.394338
$49$	−2.00000	−0.285714
$50$	− 1.00000i	− 0.141421i
$51$	6.00000	0.840168
$52$	0	0
$53$	−6.46410	−0.887913	−0.443956	−	0.896048i	$-0.646425\pi$
$53$	−6.46410	−0.887913	−0.443956	+	0.896048i	$0.646425\pi$
$54$	− 4.00000i	− 0.544331i
$55$	−3.00000	−0.404520
$56$	3.00000	0.400892
$57$	− 17.6603i	− 2.33916i
$58$	− 9.46410i	− 1.24270i
$59$	− 10.3923i	− 1.35296i	−0.736460	−	0.676481i	$-0.763504\pi$
$59$	− 10.3923i	− 1.35296i	0.736460	−	0.676481i	$-0.236496\pi$
$60$	− 2.73205i	− 0.352706i
$61$	−4.19615	−0.537262	−0.268631	−	0.963243i	$-0.586571\pi$
$61$	−4.19615	−0.537262	−0.268631	+	0.963243i	$0.586571\pi$
$62$	−1.26795	−0.161030
$63$	13.3923i	1.68727i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−8.19615	−1.00888
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	2.19615	0.266323
$69$	−6.92820	−0.834058
$70$	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	$-0.115871\pi$
$72$	− 4.46410i	− 0.526099i
$73$	− 5.66025i	− 0.662483i	−0.943546	−	0.331241i	$-0.892533\pi$
$73$	− 5.66025i	− 0.662483i	0.943546	−	0.331241i	$-0.107467\pi$
$74$	−11.1962	−1.30153
$75$	2.73205	0.315470
$76$	− 6.46410i	− 0.741483i
$77$	9.00000	1.02565
$78$	0	0
$79$	6.19615	0.697122	0.348561	−	0.937286i	$-0.386670\pi$
$79$	6.19615	0.697122	0.348561	+	0.937286i	$0.386670\pi$
$80$	− 1.00000i	− 0.111803i
$81$	−2.46410	−0.273789
$82$	10.3923	1.14764
$83$	2.19615i	0.241059i	0.992710	+	0.120530i	$0.0384592\pi$
$83$	2.19615i	0.241059i	−0.992710	+	0.120530i	$0.961541\pi$
$84$	8.19615i	0.894274i
$85$	2.19615i	0.238206i
$86$	2.00000i	0.215666i
$87$	25.8564	2.77210
$88$	−3.00000	−0.319801
$89$	− 17.1962i	− 1.82279i	−0.411534	−	0.911394i	$-0.635007\pi$
$89$	− 17.1962i	− 1.82279i	0.411534	−	0.911394i	$-0.364993\pi$
$90$	4.46410	0.470558
$91$	0	0
$92$	−2.53590	−0.264386
$93$	− 3.46410i	− 0.359211i
$94$	−3.00000	−0.309426
$95$	6.46410	0.663203
$96$	− 2.73205i	− 0.278839i
$97$	− 15.1244i	− 1.53565i	−0.640662	−	0.767823i	$-0.721340\pi$
$97$	− 15.1244i	− 1.53565i	0.640662	−	0.767823i	$-0.278660\pi$
$98$	− 2.00000i	− 0.202031i
$99$	− 13.3923i	− 1.34598i
$100$	1.00000	0.100000
$101$	7.26795	0.723188	0.361594	−	0.932336i	$-0.382233\pi$
$101$	7.26795	0.723188	0.361594	+	0.932336i	$0.382233\pi$
$102$	6.00000i	0.594089i
$103$	1.19615	0.117860	0.0589302	−	0.998262i	$-0.481231\pi$
$103$	1.19615	0.117860	0.0589302	+	0.998262i	$0.481231\pi$
$104$	0	0
$105$	−8.19615	−0.799863
$106$	− 6.46410i	− 0.627849i
$107$	0.339746	0.0328445	0.0164222	−	0.999865i	$-0.494772\pi$
$107$	0.339746	0.0328445	0.0164222	+	0.999865i	$0.494772\pi$
$108$	4.00000	0.384900
$109$	− 15.4641i	− 1.48119i	−0.671950	−	0.740596i	$-0.734543\pi$
$109$	− 15.4641i	− 1.48119i	0.671950	−	0.740596i	$-0.265457\pi$
$110$	− 3.00000i	− 0.286039i
$111$	− 30.5885i	− 2.90333i
$112$	3.00000i	0.283473i
$113$	−6.92820	−0.651751	−0.325875	−	0.945413i	$-0.605659\pi$
$113$	−6.92820	−0.651751	−0.325875	+	0.945413i	$0.605659\pi$
$114$	17.6603	1.65403
$115$	− 2.53590i	− 0.236474i
$116$	9.46410	0.878720
$117$	0	0
$118$	10.3923	0.956689
$119$	− 6.58846i	− 0.603963i
$120$	2.73205	0.249401
$121$	2.00000	0.181818
$122$	− 4.19615i	− 0.379902i
$123$	28.3923i	2.56005i
$124$	− 1.26795i	− 0.113865i
$125$	1.00000i	0.0894427i
$126$	−13.3923	−1.19308
$127$	21.1962	1.88085	0.940427	−	0.339995i	$-0.110425\pi$
$127$	21.1962	1.88085	0.940427	+	0.339995i	$0.110425\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−5.46410	−0.481087
$130$	0	0
$131$	−18.1244	−1.58353	−0.791766	−	0.610824i	$-0.790838\pi$
$131$	−18.1244	−1.58353	−0.791766	+	0.610824i	$0.790838\pi$
$132$	− 8.19615i	− 0.713384i
$133$	−19.3923	−1.68153
$134$	0	0
$135$	4.00000i	0.344265i
$136$	2.19615i	0.188319i
$137$	− 8.19615i	− 0.700245i	−0.936704	−	0.350122i	$-0.886140\pi$
$137$	− 8.19615i	− 0.700245i	0.936704	−	0.350122i	$-0.113860\pi$
$138$	− 6.92820i	− 0.589768i
$139$	−9.19615	−0.780007	−0.390004	−	0.920813i	$-0.627526\pi$
$139$	−9.19615	−0.780007	−0.390004	+	0.920813i	$0.627526\pi$
$140$	−3.00000	−0.253546
$141$	− 8.19615i	− 0.690241i
$142$	6.00000	0.503509
$143$	0	0
$144$	4.46410	0.372008
$145$	9.46410i	0.785951i
$146$	5.66025	0.468446
$147$	5.46410	0.450672
$148$	− 11.1962i	− 0.920318i
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	2.73205i	0.223071i
$151$	− 6.33975i	− 0.515921i	−0.966155	−	0.257961i	$-0.916950\pi$
$151$	− 6.33975i	− 0.515921i	0.966155	−	0.257961i	$-0.0830505\pi$
$152$	6.46410	0.524308
$153$	−9.80385	−0.792594
$154$	9.00000i	0.725241i
$155$	1.26795	0.101844
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	6.19615i	0.492939i
$159$	17.6603	1.40055
$160$	1.00000	0.0790569
$161$	7.60770i	0.599570i
$162$	− 2.46410i	− 0.193598i
$163$	− 7.26795i	− 0.569270i	−0.958636	−	0.284635i	$-0.908128\pi$
$163$	− 7.26795i	− 0.569270i	0.958636	−	0.284635i	$-0.0918724\pi$
$164$	10.3923i	0.811503i
$165$	8.19615	0.638070
$166$	−2.19615	−0.170454
$167$	− 3.00000i	− 0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$	− 3.00000i	− 0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$	−8.19615	−0.632347
$169$	0	0
$170$	−2.19615	−0.168437
$171$	28.8564i	2.20670i
$172$	−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	$0.693140\pi$
$174$	25.8564i	1.96017i
$175$	− 3.00000i	− 0.226779i
$176$	− 3.00000i	− 0.226134i
$177$	28.3923i	2.13410i
$178$	17.1962	1.28891
$179$	−2.53590	−0.189542	−0.0947710	−	0.995499i	$-0.530212\pi$
$179$	−2.53590	−0.189542	−0.0947710	+	0.995499i	$0.530212\pi$
$180$	4.46410i	0.332734i
$181$	−16.5885	−1.23301	−0.616505	−	0.787351i	$-0.711452\pi$
$181$	−16.5885	−1.23301	−0.616505	+	0.787351i	$0.711452\pi$
$182$	0	0
$183$	11.4641	0.847451
$184$	− 2.53590i	− 0.186949i
$185$	11.1962	0.823157
$186$	3.46410	0.254000
$187$	6.58846i	0.481796i
$188$	− 3.00000i	− 0.218797i
$189$	− 12.0000i	− 0.872872i
$190$	6.46410i	0.468955i
$191$	−19.2679	−1.39418	−0.697090	−	0.716984i	$-0.745522\pi$
$191$	−19.2679	−1.39418	−0.697090	+	0.716984i	$0.745522\pi$
$192$	2.73205	0.197169
$193$	− 4.39230i	− 0.316165i	−0.987426	−	0.158083i	$-0.949469\pi$
$193$	− 4.39230i	− 0.316165i	0.987426	−	0.158083i	$-0.0505312\pi$
$194$	15.1244	1.08587
$195$	0	0
$196$	2.00000	0.142857
$197$	9.58846i	0.683149i	0.939855	+	0.341575i	$0.110960\pi$
$197$	9.58846i	0.683149i	−0.939855	+	0.341575i	$0.889040\pi$
$198$	13.3923	0.951750
$199$	14.3923	1.02024	0.510122	−	0.860102i	$-0.329600\pi$
$199$	14.3923	1.02024	0.510122	+	0.860102i	$0.329600\pi$
$200$	1.00000i	0.0707107i
$201$	0	0
$202$	7.26795i	0.511371i
$203$	− 28.3923i	− 1.99275i
$204$	−6.00000	−0.420084
$205$	−10.3923	−0.725830
$206$	1.19615i	0.0833399i
$207$	11.3205	0.786830
$208$	0	0
$209$	19.3923	1.34139
$210$	− 8.19615i	− 0.565588i
$211$	13.5885	0.935468	0.467734	−	0.883869i	$-0.345071\pi$
$211$	13.5885	0.935468	0.467734	+	0.883869i	$0.345071\pi$
$212$	6.46410	0.443956
$213$	16.3923i	1.12318i
$214$	0.339746i	0.0232246i
$215$	− 2.00000i	− 0.136399i
$216$	4.00000i	0.272166i
$217$	−3.80385	−0.258222
$218$	15.4641	1.04736
$219$	15.4641i	1.04497i
$220$	3.00000	0.202260
$221$	0	0
$222$	30.5885	2.05296
$223$	− 0.464102i	− 0.0310785i	−0.999879	−	0.0155393i	$-0.995053\pi$
$223$	− 0.464102i	− 0.0310785i	0.999879	−	0.0155393i	$-0.00494650\pi$
$224$	−3.00000	−0.200446
$225$	−4.46410	−0.297607
$226$	− 6.92820i	− 0.460857i
$227$	− 4.39230i	− 0.291528i	−0.989319	−	0.145764i	$-0.953436\pi$
$227$	− 4.39230i	− 0.291528i	0.989319	−	0.145764i	$-0.0465639\pi$
$228$	17.6603i	1.16958i
$229$	− 4.73205i	− 0.312703i	−0.987701	−	0.156351i	$-0.950027\pi$
$229$	− 4.73205i	− 0.312703i	0.987701	−	0.156351i	$-0.0499732\pi$
$230$	2.53590	0.167212
$231$	−24.5885	−1.61780
$232$	9.46410i	0.621349i
$233$	1.26795	0.0830661	0.0415331	−	0.999137i	$-0.486776\pi$
$233$	1.26795	0.0830661	0.0415331	+	0.999137i	$0.486776\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	10.3923i	0.676481i
$237$	−16.9282	−1.09960
$238$	6.58846	0.427066
$239$	− 8.19615i	− 0.530165i	−0.964226	−	0.265083i	$-0.914601\pi$
$239$	− 8.19615i	− 0.530165i	0.964226	−	0.265083i	$-0.0853992\pi$
$240$	2.73205i	0.176353i
$241$	8.66025i	0.557856i	0.960312	+	0.278928i	$0.0899791\pi$
$241$	8.66025i	0.557856i	−0.960312	+	0.278928i	$0.910021\pi$
$242$	2.00000i	0.128565i
$243$	18.7321	1.20166
$244$	4.19615	0.268631
$245$	2.00000i	0.127775i
$246$	−28.3923	−1.81023
$247$	0	0
$248$	1.26795	0.0805149
$249$	− 6.00000i	− 0.380235i
$250$	−1.00000	−0.0632456
$251$	6.80385	0.429455	0.214728	−	0.976674i	$-0.431114\pi$
$251$	6.80385	0.429455	0.214728	+	0.976674i	$0.431114\pi$
$252$	− 13.3923i	− 0.843636i
$253$	− 7.60770i	− 0.478292i
$254$	21.1962i	1.32996i
$255$	− 6.00000i	− 0.375735i
$256$	1.00000	0.0625000
$257$	13.8564	0.864339	0.432169	−	0.901792i	$-0.357748\pi$
$257$	13.8564	0.864339	0.432169	+	0.901792i	$0.357748\pi$
$258$	− 5.46410i	− 0.340180i
$259$	−33.5885	−2.08709
$260$	0	0
$261$	−42.2487	−2.61513
$262$	− 18.1244i	− 1.11973i
$263$	−27.5885	−1.70118	−0.850589	−	0.525832i	$-0.823754\pi$
$263$	−27.5885	−1.70118	−0.850589	+	0.525832i	$0.823754\pi$
$264$	8.19615	0.504438
$265$	6.46410i	0.397087i
$266$	− 19.3923i	− 1.18902i
$267$	46.9808i	2.87518i
$268$	0	0
$269$	−2.87564	−0.175331	−0.0876656	−	0.996150i	$-0.527941\pi$
$269$	−2.87564	−0.175331	−0.0876656	+	0.996150i	$0.527941\pi$
$270$	−4.00000	−0.243432
$271$	− 2.53590i	− 0.154045i	−0.997029	−	0.0770224i	$-0.975459\pi$
$271$	− 2.53590i	− 0.154045i	0.997029	−	0.0770224i	$-0.0245413\pi$
$272$	−2.19615	−0.133161
$273$	0	0
$274$	8.19615	0.495148
$275$	3.00000i	0.180907i
$276$	6.92820	0.417029
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	− 9.19615i	− 0.551549i
$279$	5.66025i	0.338871i
$280$	− 3.00000i	− 0.179284i
$281$	10.3923i	0.619953i	0.950744	+	0.309976i	$0.100321\pi$
$281$	10.3923i	0.619953i	−0.950744	+	0.309976i	$0.899679\pi$
$282$	8.19615	0.488074
$283$	−30.3923	−1.80663	−0.903317	−	0.428973i	$-0.858876\pi$
$283$	−30.3923	−1.80663	−0.903317	+	0.428973i	$0.858876\pi$
$284$	6.00000i	0.356034i
$285$	−17.6603	−1.04610
$286$	0	0
$287$	31.1769	1.84032
$288$	4.46410i	0.263050i
$289$	−12.1769	−0.716289
$290$	−9.46410	−0.555751
$291$	41.3205i	2.42225i
$292$	5.66025i	0.331241i
$293$	0.803848i	0.0469613i	0.999724	+	0.0234806i	$0.00747481\pi$
$293$	0.803848i	0.0469613i	−0.999724	+	0.0234806i	$0.992525\pi$
$294$	5.46410i	0.318673i
$295$	−10.3923	−0.605063
$296$	11.1962	0.650763
$297$	12.0000i	0.696311i
$298$	−6.00000	−0.347571
$299$	0	0
$300$	−2.73205	−0.157735
$301$	6.00000i	0.345834i
$302$	6.33975	0.364811
$303$	−19.8564	−1.14072
$304$	6.46410i	0.370742i
$305$	4.19615i	0.240271i
$306$	− 9.80385i	− 0.560449i
$307$	− 20.5359i	− 1.17205i	−0.810295	−	0.586023i	$-0.800693\pi$
$307$	− 20.5359i	− 1.17205i	0.810295	−	0.586023i	$-0.199307\pi$
$308$	−9.00000	−0.512823
$309$	−3.26795	−0.185907
$310$	1.26795i	0.0720147i
$311$	15.1244	0.857624	0.428812	−	0.903394i	$-0.358932\pi$
$311$	15.1244	0.857624	0.428812	+	0.903394i	$0.358932\pi$
$312$	0	0
$313$	5.60770	0.316966	0.158483	−	0.987362i	$-0.449340\pi$
$313$	5.60770	0.316966	0.158483	+	0.987362i	$0.449340\pi$
$314$	− 13.0000i	− 0.733632i
$315$	13.3923	0.754571
$316$	−6.19615	−0.348561
$317$	− 12.8038i	− 0.719136i	−0.933119	−	0.359568i	$-0.882924\pi$
$317$	− 12.8038i	− 0.719136i	0.933119	−	0.359568i	$-0.117076\pi$
$318$	17.6603i	0.990338i
$319$	28.3923i	1.58966i
$320$	1.00000i	0.0559017i
$321$	−0.928203	−0.0518073
$322$	−7.60770	−0.423960
$323$	− 14.1962i	− 0.789895i
$324$	2.46410	0.136895
$325$	0	0
$326$	7.26795	0.402534
$327$	42.2487i	2.33636i
$328$	−10.3923	−0.573819
$329$	−9.00000	−0.496186
$330$	8.19615i	0.451183i
$331$	− 0.928203i	− 0.0510187i	−0.999675	−	0.0255093i	$-0.991879\pi$
$331$	− 0.928203i	− 0.0510187i	0.999675	−	0.0255093i	$-0.00812075\pi$
$332$	− 2.19615i	− 0.120530i
$333$	49.9808i	2.73893i
$334$	3.00000	0.164153
$335$	0	0
$336$	− 8.19615i	− 0.447137i
$337$	−4.19615	−0.228579	−0.114289	−	0.993447i	$-0.536459\pi$
$337$	−4.19615	−0.228579	−0.114289	+	0.993447i	$0.536459\pi$
$338$	0	0
$339$	18.9282	1.02804
$340$	− 2.19615i	− 0.119103i
$341$	3.80385	0.205990
$342$	−28.8564	−1.56038
$343$	15.0000i	0.809924i
$344$	− 2.00000i	− 0.107833i
$345$	6.92820i	0.373002i
$346$	− 15.0000i	− 0.806405i
$347$	−25.2679	−1.35645	−0.678227	−	0.734852i	$-0.737252\pi$
$347$	−25.2679	−1.35645	−0.678227	+	0.734852i	$0.737252\pi$
$348$	−25.8564	−1.38605
$349$	− 34.0526i	− 1.82279i	−0.411531	−	0.911396i	$-0.635006\pi$
$349$	− 34.0526i	− 1.82279i	0.411531	−	0.911396i	$-0.364994\pi$
$350$	3.00000	0.160357
$351$	0	0
$352$	3.00000	0.159901
$353$	− 20.1962i	− 1.07493i	−0.843285	−	0.537466i	$-0.819382\pi$
$353$	− 20.1962i	− 1.07493i	0.843285	−	0.537466i	$-0.180618\pi$
$354$	−28.3923	−1.50903
$355$	−6.00000	−0.318447
$356$	17.1962i	0.911394i
$357$	18.0000i	0.952661i
$358$	− 2.53590i	− 0.134026i
$359$	22.3923i	1.18182i	0.806737	+	0.590910i	$0.201231\pi$
$359$	22.3923i	1.18182i	−0.806737	+	0.590910i	$0.798769\pi$
$360$	−4.46410	−0.235279
$361$	−22.7846	−1.19919
$362$	− 16.5885i	− 0.871870i
$363$	−5.46410	−0.286791
$364$	0	0
$365$	−5.66025	−0.296271
$366$	11.4641i	0.599238i
$367$	−26.3923	−1.37767	−0.688834	−	0.724920i	$-0.741877\pi$
$367$	−26.3923	−1.37767	−0.688834	+	0.724920i	$0.741877\pi$
$368$	2.53590	0.132193
$369$	− 46.3923i	− 2.41509i
$370$	11.1962i	0.582060i
$371$	− 19.3923i	− 1.00680i
$372$	3.46410i	0.179605i
$373$	20.3923	1.05587	0.527937	−	0.849284i	$-0.322966\pi$
$373$	20.3923	1.05587	0.527937	+	0.849284i	$0.322966\pi$
$374$	−6.58846	−0.340681
$375$	− 2.73205i	− 0.141082i
$376$	3.00000	0.154713
$377$	0	0
$378$	12.0000	0.617213
$379$	11.7846i	0.605335i	0.953096	+	0.302667i	$0.0978771\pi$
$379$	11.7846i	0.605335i	−0.953096	+	0.302667i	$0.902123\pi$
$380$	−6.46410	−0.331601
$381$	−57.9090	−2.96677
$382$	− 19.2679i	− 0.985834i
$383$	1.60770i	0.0821494i	0.999156	+	0.0410747i	$0.0130782\pi$
$383$	1.60770i	0.0821494i	−0.999156	+	0.0410747i	$0.986922\pi$
$384$	2.73205i	0.139419i
$385$	− 9.00000i	− 0.458682i
$386$	4.39230	0.223562
$387$	8.92820	0.453846
$388$	15.1244i	0.767823i
$389$	19.2679	0.976924	0.488462	−	0.872585i	$-0.337558\pi$
$389$	19.2679	0.976924	0.488462	+	0.872585i	$0.337558\pi$
$390$	0	0
$391$	−5.56922	−0.281648
$392$	2.00000i	0.101015i
$393$	49.5167	2.49779
$394$	−9.58846	−0.483059
$395$	− 6.19615i	− 0.311762i
$396$	13.3923i	0.672989i
$397$	− 0.803848i	− 0.0403440i	−0.999797	−	0.0201720i	$-0.993579\pi$
$397$	− 0.803848i	− 0.0403440i	0.999797	−	0.0201720i	$-0.00642138\pi$
$398$	14.3923i	0.721421i
$399$	52.9808	2.65236
$400$	−1.00000	−0.0500000
$401$	− 5.19615i	− 0.259483i	−0.991548	−	0.129742i	$-0.958585\pi$
$401$	− 5.19615i	− 0.259483i	0.991548	−	0.129742i	$-0.0414148\pi$
$402$	0	0
$403$	0	0
$404$	−7.26795	−0.361594
$405$	2.46410i	0.122442i
$406$	28.3923	1.40909
$407$	33.5885	1.66492
$408$	− 6.00000i	− 0.297044i
$409$	− 19.7321i	− 0.975687i	−0.872931	−	0.487844i	$-0.837784\pi$
$409$	− 19.7321i	− 0.975687i	0.872931	−	0.487844i	$-0.162216\pi$
$410$	− 10.3923i	− 0.513239i
$411$	22.3923i	1.10453i
$412$	−1.19615	−0.0589302
$413$	31.1769	1.53412
$414$	11.3205i	0.556373i
$415$	2.19615	0.107805
$416$	0	0
$417$	25.1244	1.23034
$418$	19.3923i	0.948509i
$419$	17.3205	0.846162	0.423081	−	0.906092i	$-0.360949\pi$
$419$	17.3205	0.846162	0.423081	+	0.906092i	$0.360949\pi$
$420$	8.19615	0.399931
$421$	27.1244i	1.32196i	0.750403	+	0.660980i	$0.229859\pi$
$421$	27.1244i	1.32196i	−0.750403	+	0.660980i	$0.770141\pi$
$422$	13.5885i	0.661476i
$423$	13.3923i	0.651156i
$424$	6.46410i	0.313925i
$425$	2.19615	0.106529
$426$	−16.3923	−0.794210
$427$	− 12.5885i	− 0.609198i
$428$	−0.339746	−0.0164222
$429$	0	0
$430$	2.00000	0.0964486
$431$	− 2.19615i	− 0.105785i	−0.998600	−	0.0528925i	$-0.983156\pi$
$431$	− 2.19615i	− 0.105785i	0.998600	−	0.0528925i	$-0.0168441\pi$
$432$	−4.00000	−0.192450
$433$	−12.3923	−0.595536	−0.297768	−	0.954638i	$-0.596242\pi$
$433$	−12.3923	−0.595536	−0.297768	+	0.954638i	$0.596242\pi$
$434$	− 3.80385i	− 0.182591i
$435$	− 25.8564i	− 1.23972i
$436$	15.4641i	0.740596i
$437$	16.3923i	0.784150i
$438$	−15.4641	−0.738903
$439$	−34.5885	−1.65082	−0.825408	−	0.564536i	$-0.809055\pi$
$439$	−34.5885	−1.65082	−0.825408	+	0.564536i	$0.809055\pi$
$440$	3.00000i	0.143019i
$441$	−8.92820	−0.425153
$442$	0	0
$443$	−1.60770	−0.0763839	−0.0381920	−	0.999270i	$-0.512160\pi$
$443$	−1.60770	−0.0763839	−0.0381920	+	0.999270i	$0.512160\pi$
$444$	30.5885i	1.45166i
$445$	−17.1962	−0.815176
$446$	0.464102	0.0219758
$447$	− 16.3923i	− 0.775329i
$448$	− 3.00000i	− 0.141737i
$449$	− 15.5885i	− 0.735665i	−0.929892	−	0.367832i	$-0.880100\pi$
$449$	− 15.5885i	− 0.735665i	0.929892	−	0.367832i	$-0.119900\pi$
$450$	− 4.46410i	− 0.210440i
$451$	−31.1769	−1.46806
$452$	6.92820	0.325875
$453$	17.3205i	0.813788i
$454$	4.39230	0.206141
$455$	0	0
$456$	−17.6603	−0.827017
$457$	− 35.6603i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$457$	− 35.6603i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$458$	4.73205	0.221114
$459$	8.78461	0.410030
$460$	2.53590i	0.118237i
$461$	0.588457i	0.0274072i	0.999906	+	0.0137036i	$0.00436213\pi$
$461$	0.588457i	0.0274072i	−0.999906	+	0.0137036i	$0.995638\pi$
$462$	− 24.5885i	− 1.14396i
$463$	− 0.928203i	− 0.0431373i	−0.999767	−	0.0215686i	$-0.993134\pi$
$463$	− 0.928203i	− 0.0431373i	0.999767	−	0.0215686i	$-0.00686604\pi$
$464$	−9.46410	−0.439360
$465$	−3.46410	−0.160644
$466$	1.26795i	0.0587366i
$467$	−10.1436	−0.469390	−0.234695	−	0.972069i	$-0.575409\pi$
$467$	−10.1436	−0.469390	−0.234695	+	0.972069i	$0.575409\pi$
$468$	0	0
$469$	0	0
$470$	3.00000i	0.138380i
$471$	35.5167	1.63652
$472$	−10.3923	−0.478345
$473$	− 6.00000i	− 0.275880i
$474$	− 16.9282i	− 0.777538i
$475$	− 6.46410i	− 0.296593i
$476$	6.58846i	0.301981i
$477$	−28.8564	−1.32124
$478$	8.19615	0.374883
$479$	− 10.9808i	− 0.501724i	−0.968023	−	0.250862i	$-0.919286\pi$
$479$	− 10.9808i	− 0.501724i	0.968023	−	0.250862i	$-0.0807140\pi$
$480$	−2.73205	−0.124700
$481$	0	0
$482$	−8.66025	−0.394464
$483$	− 20.7846i	− 0.945732i
$484$	−2.00000	−0.0909091
$485$	−15.1244	−0.686762
$486$	18.7321i	0.849703i
$487$	33.2487i	1.50664i	0.657652	+	0.753321i	$0.271550\pi$
$487$	33.2487i	1.50664i	−0.657652	+	0.753321i	$0.728450\pi$
$488$	4.19615i	0.189951i
$489$	19.8564i	0.897938i
$490$	−2.00000	−0.0903508
$491$	3.33975	0.150721	0.0753603	−	0.997156i	$-0.475989\pi$
$491$	3.33975	0.150721	0.0753603	+	0.997156i	$0.475989\pi$
$492$	− 28.3923i	− 1.28002i
$493$	20.7846	0.936092
$494$	0	0
$495$	−13.3923	−0.601939
$496$	1.26795i	0.0569326i
$497$	18.0000	0.807410
$498$	6.00000	0.268866
$499$	− 1.85641i	− 0.0831042i	−0.999136	−	0.0415521i	$-0.986770\pi$
$499$	− 1.85641i	− 0.0831042i	0.999136	−	0.0415521i	$-0.0132302\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	8.19615i	0.366177i
$502$	6.80385i	0.303671i
$503$	−9.33975	−0.416439	−0.208219	−	0.978082i	$-0.566767\pi$
$503$	−9.33975	−0.416439	−0.208219	+	0.978082i	$0.566767\pi$
$504$	13.3923	0.596541
$505$	− 7.26795i	− 0.323419i
$506$	7.60770	0.338203
$507$	0	0
$508$	−21.1962	−0.940427
$509$	1.60770i	0.0712598i	0.999365	+	0.0356299i	$0.0113438\pi$
$509$	1.60770i	0.0712598i	−0.999365	+	0.0356299i	$0.988656\pi$
$510$	6.00000	0.265684
$511$	16.9808	0.751185
$512$	1.00000i	0.0441942i
$513$	− 25.8564i	− 1.14159i
$514$	13.8564i	0.611180i
$515$	− 1.19615i	− 0.0527088i
$516$	5.46410	0.240544
$517$	9.00000	0.395820
$518$	− 33.5885i	− 1.47579i
$519$	40.9808	1.79886
$520$	0	0
$521$	−0.464102	−0.0203327	−0.0101663	−	0.999948i	$-0.503236\pi$
$521$	−0.464102	−0.0203327	−0.0101663	+	0.999948i	$0.503236\pi$
$522$	− 42.2487i	− 1.84918i
$523$	18.3923	0.804239	0.402120	−	0.915587i	$-0.368274\pi$
$523$	18.3923	0.804239	0.402120	+	0.915587i	$0.368274\pi$
$524$	18.1244	0.791766
$525$	8.19615i	0.357709i
$526$	− 27.5885i	− 1.20291i
$527$	− 2.78461i	− 0.121300i
$528$	8.19615i	0.356692i
$529$	−16.5692	−0.720401
$530$	−6.46410	−0.280783
$531$	− 46.3923i	− 2.01325i
$532$	19.3923	0.840763
$533$	0	0
$534$	−46.9808	−2.03306
$535$	− 0.339746i	− 0.0146885i
$536$	0	0
$537$	6.92820	0.298974
$538$	− 2.87564i	− 0.123978i
$539$	6.00000i	0.258438i
$540$	− 4.00000i	− 0.172133i
$541$	− 16.0526i	− 0.690153i	−0.938574	−	0.345077i	$-0.887853\pi$
$541$	− 16.0526i	− 0.690153i	0.938574	−	0.345077i	$-0.112147\pi$
$542$	2.53590	0.108926
$543$	45.3205	1.94489
$544$	− 2.19615i	− 0.0941593i
$545$	−15.4641	−0.662409
$546$	0	0
$547$	34.7846	1.48728	0.743641	−	0.668579i	$-0.233097\pi$
$547$	34.7846	1.48728	0.743641	+	0.668579i	$0.233097\pi$
$548$	8.19615i	0.350122i
$549$	−18.7321	−0.799464
$550$	−3.00000	−0.127920
$551$	− 61.1769i	− 2.60622i
$552$	6.92820i	0.294884i
$553$	18.5885i	0.790462i
$554$	− 1.00000i	− 0.0424859i
$555$	−30.5885	−1.29841
$556$	9.19615	0.390004
$557$	17.1962i	0.728624i	0.931277	+	0.364312i	$0.118696\pi$
$557$	17.1962i	0.728624i	−0.931277	+	0.364312i	$0.881304\pi$
$558$	−5.66025	−0.239618
$559$	0	0
$560$	3.00000	0.126773
$561$	− 18.0000i	− 0.759961i
$562$	−10.3923	−0.438373
$563$	21.4641	0.904604	0.452302	−	0.891865i	$-0.350603\pi$
$563$	21.4641	0.904604	0.452302	+	0.891865i	$0.350603\pi$
$564$	8.19615i	0.345120i
$565$	6.92820i	0.291472i
$566$	− 30.3923i	− 1.27748i
$567$	− 7.39230i	− 0.310448i
$568$	−6.00000	−0.251754
$569$	21.2487	0.890792	0.445396	−	0.895334i	$-0.353063\pi$
$569$	21.2487	0.890792	0.445396	+	0.895334i	$0.353063\pi$
$570$	− 17.6603i	− 0.739707i
$571$	34.3731	1.43847	0.719234	−	0.694768i	$-0.244493\pi$
$571$	34.3731	1.43847	0.719234	+	0.694768i	$0.244493\pi$
$572$	0	0
$573$	52.6410	2.19911
$574$	31.1769i	1.30130i
$575$	−2.53590	−0.105754
$576$	−4.46410	−0.186004
$577$	32.4449i	1.35070i	0.737499	+	0.675349i	$0.236007\pi$
$577$	32.4449i	1.35070i	−0.737499	+	0.675349i	$0.763993\pi$
$578$	− 12.1769i	− 0.506493i
$579$	12.0000i	0.498703i
$580$	− 9.46410i	− 0.392975i
$581$	−6.58846	−0.273335
$582$	−41.3205	−1.71279
$583$	19.3923i	0.803147i
$584$	−5.66025	−0.234223
$585$	0	0
$586$	−0.803848	−0.0332066
$587$	− 30.0000i	− 1.23823i	−0.785299	−	0.619116i	$-0.787491\pi$
$587$	− 30.0000i	− 1.23823i	0.785299	−	0.619116i	$-0.212509\pi$
$588$	−5.46410	−0.225336
$589$	−8.19615	−0.337717
$590$	− 10.3923i	− 0.427844i
$591$	− 26.1962i	− 1.07757i
$592$	11.1962i	0.460159i
$593$	20.7846i	0.853522i	0.904365	+	0.426761i	$0.140345\pi$
$593$	20.7846i	0.853522i	−0.904365	+	0.426761i	$0.859655\pi$
$594$	−12.0000	−0.492366
$595$	−6.58846	−0.270100
$596$	− 6.00000i	− 0.245770i
$597$	−39.3205	−1.60928
$598$	0	0
$599$	7.85641	0.321004	0.160502	−	0.987036i	$-0.448689\pi$
$599$	7.85641	0.321004	0.160502	+	0.987036i	$0.448689\pi$
$600$	− 2.73205i	− 0.111536i
$601$	3.78461	0.154377	0.0771887	−	0.997016i	$-0.475406\pi$
$601$	3.78461	0.154377	0.0771887	+	0.997016i	$0.475406\pi$
$602$	−6.00000	−0.244542
$603$	0	0
$604$	6.33975i	0.257961i
$605$	− 2.00000i	− 0.0813116i
$606$	− 19.8564i	− 0.806611i
$607$	−6.41154	−0.260236	−0.130118	−	0.991498i	$-0.541536\pi$
$607$	−6.41154	−0.260236	−0.130118	+	0.991498i	$0.541536\pi$
$608$	−6.46410	−0.262154
$609$	77.5692i	3.14326i
$610$	−4.19615	−0.169897
$611$	0	0
$612$	9.80385	0.396297
$613$	26.9090i	1.08684i	0.839460	+	0.543421i	$0.182871\pi$
$613$	26.9090i	1.08684i	−0.839460	+	0.543421i	$0.817129\pi$
$614$	20.5359	0.828761
$615$	28.3923	1.14489
$616$	− 9.00000i	− 0.362620i
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	− 3.26795i	− 0.131456i
$619$	− 10.6077i	− 0.426359i	−0.977013	−	0.213180i	$-0.931618\pi$
$619$	− 10.6077i	− 0.426359i	0.977013	−	0.213180i	$-0.0683820\pi$
$620$	−1.26795	−0.0509221
$621$	−10.1436	−0.407048
$622$	15.1244i	0.606431i
$623$	51.5885	2.06685
$624$	0	0
$625$	1.00000	0.0400000
$626$	5.60770i	0.224129i
$627$	−52.9808	−2.11585
$628$	13.0000	0.518756
$629$	− 24.5885i	− 0.980406i
$630$	13.3923i	0.533562i
$631$	20.7846i	0.827422i	0.910408	+	0.413711i	$0.135768\pi$
$631$	20.7846i	0.827422i	−0.910408	+	0.413711i	$0.864232\pi$
$632$	− 6.19615i	− 0.246470i
$633$	−37.1244	−1.47556
$634$	12.8038	0.508506
$635$	− 21.1962i	− 0.841144i
$636$	−17.6603	−0.700275
$637$	0	0
$638$	−28.3923	−1.12406
$639$	− 26.7846i	− 1.05958i
$640$	−1.00000	−0.0395285
$641$	45.9282	1.81405	0.907027	−	0.421071i	$-0.138346\pi$
$641$	45.9282	1.81405	0.907027	+	0.421071i	$0.138346\pi$
$642$	− 0.928203i	− 0.0366333i
$643$	− 7.26795i	− 0.286620i	−0.989678	−	0.143310i	$-0.954225\pi$
$643$	− 7.26795i	− 0.286620i	0.989678	−	0.143310i	$-0.0457746\pi$
$644$	− 7.60770i	− 0.299785i
$645$	5.46410i	0.215149i
$646$	14.1962	0.558540
$647$	−11.1962	−0.440166	−0.220083	−	0.975481i	$-0.570633\pi$
$647$	−11.1962	−0.440166	−0.220083	+	0.975481i	$0.570633\pi$
$648$	2.46410i	0.0967991i
$649$	−31.1769	−1.22380
$650$	0	0
$651$	10.3923	0.407307
$652$	7.26795i	0.284635i
$653$	−19.3923	−0.758880	−0.379440	−	0.925216i	$-0.623883\pi$
$653$	−19.3923	−0.758880	−0.379440	+	0.925216i	$0.623883\pi$
$654$	−42.2487	−1.65206
$655$	18.1244i	0.708177i
$656$	− 10.3923i	− 0.405751i
$657$	− 25.2679i	− 0.985797i
$658$	− 9.00000i	− 0.350857i
$659$	29.3205	1.14216	0.571082	−	0.820893i	$-0.306524\pi$
$659$	29.3205	1.14216	0.571082	+	0.820893i	$0.306524\pi$
$660$	−8.19615	−0.319035
$661$	29.9090i	1.16332i	0.813431	+	0.581662i	$0.197597\pi$
$661$	29.9090i	1.16332i	−0.813431	+	0.581662i	$0.802403\pi$
$662$	0.928203	0.0360756
$663$	0	0
$664$	2.19615	0.0852272
$665$	19.3923i	0.752001i
$666$	−49.9808	−1.93672
$667$	−24.0000	−0.929284
$668$	3.00000i	0.116073i
$669$	1.26795i	0.0490217i
$670$	0	0
$671$	12.5885i	0.485972i
$672$	8.19615	0.316173
$673$	3.60770	0.139066	0.0695332	−	0.997580i	$-0.477849\pi$
$673$	3.60770	0.139066	0.0695332	+	0.997580i	$0.477849\pi$
$674$	− 4.19615i	− 0.161630i
$675$	4.00000	0.153960
$676$	0	0
$677$	1.85641	0.0713475	0.0356737	−	0.999363i	$-0.488642\pi$
$677$	1.85641	0.0713475	0.0356737	+	0.999363i	$0.488642\pi$
$678$	18.9282i	0.726933i
$679$	45.3731	1.74126
$680$	2.19615	0.0842186
$681$	12.0000i	0.459841i
$682$	3.80385i	0.145657i
$683$	− 31.1769i	− 1.19295i	−0.802631	−	0.596476i	$-0.796567\pi$
$683$	− 31.1769i	− 1.19295i	0.802631	−	0.596476i	$-0.203433\pi$
$684$	− 28.8564i	− 1.10335i
$685$	−8.19615	−0.313159
$686$	−15.0000	−0.572703
$687$	12.9282i	0.493242i
$688$	2.00000	0.0762493
$689$	0	0
$690$	−6.92820	−0.263752
$691$	− 18.7128i	− 0.711869i	−0.934511	−	0.355934i	$-0.884163\pi$
$691$	− 18.7128i	− 0.711869i	0.934511	−	0.355934i	$-0.115837\pi$
$692$	15.0000	0.570214
$693$	40.1769	1.52619
$694$	− 25.2679i	− 0.959158i
$695$	9.19615i	0.348830i
$696$	− 25.8564i	− 0.980085i
$697$	22.8231i	0.864486i
$698$	34.0526	1.28891
$699$	−3.46410	−0.131024
$700$	3.00000i	0.113389i
$701$	−29.9090	−1.12965	−0.564823	−	0.825212i	$-0.691056\pi$
$701$	−29.9090	−1.12965	−0.564823	+	0.825212i	$0.691056\pi$
$702$	0	0
$703$	−72.3731	−2.72960
$704$	3.00000i	0.113067i
$705$	−8.19615	−0.308685
$706$	20.1962	0.760092
$707$	21.8038i	0.820018i
$708$	− 28.3923i	− 1.06705i
$709$	− 40.9808i	− 1.53906i	−0.638608	−	0.769532i	$-0.720489\pi$
$709$	− 40.9808i	− 1.53906i	0.638608	−	0.769532i	$-0.279511\pi$
$710$	− 6.00000i	− 0.225176i
$711$	27.6603	1.03734
$712$	−17.1962	−0.644453
$713$	3.21539i	0.120417i
$714$	−18.0000	−0.673633
$715$	0	0
$716$	2.53590	0.0947710
$717$	22.3923i	0.836256i
$718$	−22.3923	−0.835673
$719$	−1.85641	−0.0692323	−0.0346161	−	0.999401i	$-0.511021\pi$
$719$	−1.85641	−0.0692323	−0.0346161	+	0.999401i	$0.511021\pi$
$720$	− 4.46410i	− 0.166367i
$721$	3.58846i	0.133641i
$722$	− 22.7846i	− 0.847955i
$723$	− 23.6603i	− 0.879934i
$724$	16.5885	0.616505
$725$	9.46410	0.351488
$726$	− 5.46410i	− 0.202792i
$727$	−38.3731	−1.42318	−0.711589	−	0.702596i	$-0.752024\pi$
$727$	−38.3731	−1.42318	−0.711589	+	0.702596i	$0.752024\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	− 5.66025i	− 0.209495i
$731$	−4.39230	−0.162455
$732$	−11.4641	−0.423725
$733$	− 50.9090i	− 1.88037i	−0.340670	−	0.940183i	$-0.610654\pi$
$733$	− 50.9090i	− 1.88037i	0.340670	−	0.940183i	$-0.389346\pi$
$734$	− 26.3923i	− 0.974158i
$735$	− 5.46410i	− 0.201546i
$736$	2.53590i	0.0934745i
$737$	0	0
$738$	46.3923	1.70772
$739$	50.5692i	1.86022i	0.367283	+	0.930109i	$0.380288\pi$
$739$	50.5692i	1.86022i	−0.367283	+	0.930109i	$0.619712\pi$
$740$	−11.1962	−0.411579
$741$	0	0
$742$	19.3923	0.711914
$743$	34.3923i	1.26173i	0.775892	+	0.630866i	$0.217300\pi$
$743$	34.3923i	1.26173i	−0.775892	+	0.630866i	$0.782700\pi$
$744$	−3.46410	−0.127000
$745$	6.00000	0.219823
$746$	20.3923i	0.746615i
$747$	9.80385i	0.358704i
$748$	− 6.58846i	− 0.240898i
$749$	1.01924i	0.0372421i
$750$	2.73205	0.0997604
$751$	0.392305	0.0143154	0.00715770	−	0.999974i	$-0.497722\pi$
$751$	0.392305	0.0143154	0.00715770	+	0.999974i	$0.497722\pi$
$752$	3.00000i	0.109399i
$753$	−18.5885	−0.677401
$754$	0	0
$755$	−6.33975	−0.230727
$756$	12.0000i	0.436436i
$757$	3.78461	0.137554	0.0687770	−	0.997632i	$-0.478090\pi$
$757$	3.78461	0.137554	0.0687770	+	0.997632i	$0.478090\pi$
$758$	−11.7846	−0.428036
$759$	20.7846i	0.754434i
$760$	− 6.46410i	− 0.234478i
$761$	29.1962i	1.05836i	0.848510	+	0.529180i	$0.177500\pi$
$761$	29.1962i	1.05836i	−0.848510	+	0.529180i	$0.822500\pi$
$762$	− 57.9090i	− 2.09782i
$763$	46.3923	1.67951
$764$	19.2679	0.697090
$765$	9.80385i	0.354459i
$766$	−1.60770	−0.0580884
$767$	0	0
$768$	−2.73205	−0.0985844
$769$	− 38.1051i	− 1.37411i	−0.726607	−	0.687053i	$-0.758904\pi$
$769$	− 38.1051i	− 1.37411i	0.726607	−	0.687053i	$-0.241096\pi$
$770$	9.00000	0.324337
$771$	−37.8564	−1.36337
$772$	4.39230i	0.158083i
$773$	33.5885i	1.20809i	0.796949	+	0.604046i	$0.206446\pi$
$773$	33.5885i	1.20809i	−0.796949	+	0.604046i	$0.793554\pi$
$774$	8.92820i	0.320918i
$775$	− 1.26795i	− 0.0455461i
$776$	−15.1244	−0.542933
$777$	91.7654	3.29206
$778$	19.2679i	0.690789i
$779$	67.1769	2.40686
$780$	0	0
$781$	−18.0000	−0.644091
$782$	− 5.56922i	− 0.199155i
$783$	37.8564	1.35288
$784$	−2.00000	−0.0714286
$785$	13.0000i	0.463990i
$786$	49.5167i	1.76620i
$787$	36.8372i	1.31310i	0.754281	+	0.656552i	$0.227986\pi$
$787$	36.8372i	1.31310i	−0.754281	+	0.656552i	$0.772014\pi$
$788$	− 9.58846i	− 0.341575i
$789$	75.3731	2.68335
$790$	6.19615	0.220449
$791$	− 20.7846i	− 0.739016i
$792$	−13.3923	−0.475875
$793$	0	0
$794$	0.803848	0.0285275
$795$	− 17.6603i	− 0.626345i
$796$	−14.3923	−0.510122
$797$	0.928203	0.0328786	0.0164393	−	0.999865i	$-0.494767\pi$
$797$	0.928203	0.0328786	0.0164393	+	0.999865i	$0.494767\pi$
$798$	52.9808i	1.87550i
$799$	− 6.58846i	− 0.233083i
$800$	− 1.00000i	− 0.0353553i
$801$	− 76.7654i	− 2.71237i
$802$	5.19615	0.183483
$803$	−16.9808	−0.599238
$804$	0	0
$805$	7.60770	0.268136
$806$	0	0
$807$	7.85641	0.276559
$808$	− 7.26795i	− 0.255686i
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	−2.46410	−0.0865797
$811$	22.6077i	0.793864i	0.917848	+	0.396932i	$0.129925\pi$
$811$	22.6077i	0.793864i	−0.917848	+	0.396932i	$0.870075\pi$
$812$	28.3923i	0.996375i
$813$	6.92820i	0.242983i
$814$	33.5885i	1.17727i
$815$	−7.26795	−0.254585
$816$	6.00000	0.210042
$817$	12.9282i	0.452301i
$818$	19.7321	0.689915
$819$	0	0
$820$	10.3923	0.362915
$821$	− 9.80385i	− 0.342157i	−0.985257	−	0.171078i	$-0.945275\pi$
$821$	− 9.80385i	− 0.342157i	0.985257	−	0.171078i	$-0.0547251\pi$
$822$	−22.3923	−0.781021
$823$	−16.8038	−0.585745	−0.292873	−	0.956151i	$-0.594611\pi$
$823$	−16.8038	−0.585745	−0.292873	+	0.956151i	$0.594611\pi$
$824$	− 1.19615i	− 0.0416699i
$825$	− 8.19615i	− 0.285353i
$826$	31.1769i	1.08478i
$827$	− 11.4115i	− 0.396818i	−0.980119	−	0.198409i	$-0.936423\pi$
$827$	− 11.4115i	− 0.396818i	0.980119	−	0.198409i	$-0.0635775\pi$
$828$	−11.3205	−0.393415
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	2.19615i	0.0762296i
$831$	2.73205	0.0947738
$832$	0	0
$833$	4.39230	0.152184
$834$	25.1244i	0.869985i
$835$	−3.00000	−0.103819
$836$	−19.3923	−0.670697
$837$	− 5.07180i	− 0.175307i
$838$	17.3205i	0.598327i
$839$	32.1962i	1.11153i	0.831338	+	0.555767i	$0.187575\pi$
$839$	32.1962i	1.11153i	−0.831338	+	0.555767i	$0.812425\pi$
$840$	8.19615i	0.282794i
$841$	60.5692	2.08859
$842$	−27.1244	−0.934767
$843$	− 28.3923i	− 0.977883i
$844$	−13.5885	−0.467734
$845$	0	0
$846$	−13.3923	−0.460437
$847$	6.00000i	0.206162i
$848$	−6.46410	−0.221978
$849$	83.0333	2.84970
$850$	2.19615i	0.0753274i
$851$	28.3923i	0.973276i
$852$	− 16.3923i	− 0.561591i
$853$	− 13.8564i	− 0.474434i	−0.971457	−	0.237217i	$-0.923765\pi$
$853$	− 13.8564i	− 0.474434i	0.971457	−	0.237217i	$-0.0762353\pi$
$854$	12.5885	0.430768
$855$	28.8564	0.986868
$856$	− 0.339746i	− 0.0116123i
$857$	−13.2679	−0.453225	−0.226612	−	0.973985i	$-0.572765\pi$
$857$	−13.2679	−0.453225	−0.226612	+	0.973985i	$0.572765\pi$
$858$	0	0
$859$	52.3731	1.78695	0.893473	−	0.449117i	$-0.148261\pi$
$859$	52.3731	1.78695	0.893473	+	0.449117i	$0.148261\pi$
$860$	2.00000i	0.0681994i
$861$	−85.1769	−2.90282
$862$	2.19615	0.0748012
$863$	− 19.1769i	− 0.652790i	−0.945234	−	0.326395i	$-0.894166\pi$
$863$	− 19.1769i	− 0.652790i	0.945234	−	0.326395i	$-0.105834\pi$
$864$	− 4.00000i	− 0.136083i
$865$	15.0000i	0.510015i
$866$	− 12.3923i	− 0.421108i
$867$	33.2679	1.12984
$868$	3.80385	0.129111
$869$	− 18.5885i	− 0.630570i
$870$	25.8564	0.876614
$871$	0	0
$872$	−15.4641	−0.523681
$873$	− 67.5167i	− 2.28509i
$874$	−16.3923	−0.554478
$875$	−3.00000	−0.101419
$876$	− 15.4641i	− 0.522484i
$877$	− 20.5359i	− 0.693448i	−0.937967	−	0.346724i	$-0.887294\pi$
$877$	− 20.5359i	− 0.693448i	0.937967	−	0.346724i	$-0.112706\pi$
$878$	− 34.5885i	− 1.16730i
$879$	− 2.19615i	− 0.0740744i
$880$	−3.00000	−0.101130
$881$	−8.32051	−0.280325	−0.140163	−	0.990129i	$-0.544763\pi$
$881$	−8.32051	−0.280325	−0.140163	+	0.990129i	$0.544763\pi$
$882$	− 8.92820i	− 0.300628i
$883$	−26.5885	−0.894773	−0.447386	−	0.894341i	$-0.647645\pi$
$883$	−26.5885	−0.894773	−0.447386	+	0.894341i	$0.647645\pi$
$884$	0	0
$885$	28.3923	0.954397
$886$	− 1.60770i	− 0.0540116i
$887$	52.7654	1.77169	0.885844	−	0.463983i	$-0.153580\pi$
$887$	52.7654	1.77169	0.885844	+	0.463983i	$0.153580\pi$
$888$	−30.5885	−1.02648
$889$	63.5885i	2.13269i
$890$	− 17.1962i	− 0.576416i
$891$	7.39230i	0.247652i
$892$	0.464102i	0.0155393i
$893$	−19.3923	−0.648939
$894$	16.3923	0.548241
$895$	2.53590i	0.0847657i
$896$	3.00000	0.100223
$897$	0	0
$898$	15.5885	0.520194
$899$	− 12.0000i	− 0.400222i
$900$	4.46410	0.148803
$901$	14.1962	0.472942
$902$	− 31.1769i	− 1.03808i
$903$	− 16.3923i	− 0.545502i
$904$	6.92820i	0.230429i
$905$	16.5885i	0.551419i
$906$	−17.3205	−0.575435
$907$	−26.5885	−0.882855	−0.441428	−	0.897297i	$-0.645528\pi$
$907$	−26.5885	−0.882855	−0.441428	+	0.897297i	$0.645528\pi$
$908$	4.39230i	0.145764i
$909$	32.4449	1.07613
$910$	0	0
$911$	−14.5359	−0.481596	−0.240798	−	0.970575i	$-0.577409\pi$
$911$	−14.5359	−0.481596	−0.240798	+	0.970575i	$0.577409\pi$
$912$	− 17.6603i	− 0.584789i
$913$	6.58846	0.218046
$914$	35.6603	1.17954
$915$	− 11.4641i	− 0.378992i
$916$	4.73205i	0.156351i
$917$	− 54.3731i	− 1.79556i
$918$	8.78461i	0.289935i
$919$	−10.7846	−0.355751	−0.177876	−	0.984053i	$-0.556923\pi$
$919$	−10.7846	−0.355751	−0.177876	+	0.984053i	$0.556923\pi$
$920$	−2.53590	−0.0836061
$921$	56.1051i	1.84873i
$922$	−0.588457	−0.0193798
$923$	0	0
$924$	24.5885	0.808901
$925$	− 11.1962i	− 0.368127i
$926$	0.928203	0.0305027
$927$	5.33975	0.175380
$928$	− 9.46410i	− 0.310674i
$929$	− 44.7846i	− 1.46934i	−0.678427	−	0.734668i	$-0.737338\pi$
$929$	− 44.7846i	− 1.46934i	0.678427	−	0.734668i	$-0.262662\pi$
$930$	− 3.46410i	− 0.113592i
$931$	− 12.9282i	− 0.423705i
$932$	−1.26795	−0.0415331
$933$	−41.3205	−1.35277
$934$	− 10.1436i	− 0.331909i
$935$	6.58846	0.215466
$936$	0	0
$937$	−30.3923	−0.992873	−0.496437	−	0.868073i	$-0.665359\pi$
$937$	−30.3923	−0.992873	−0.496437	+	0.868073i	$0.665359\pi$
$938$	0	0
$939$	−15.3205	−0.499966
$940$	−3.00000	−0.0978492
$941$	44.7846i	1.45994i	0.683481	+	0.729968i	$0.260465\pi$
$941$	44.7846i	1.45994i	−0.683481	+	0.729968i	$0.739535\pi$
$942$	35.5167i	1.15720i
$943$	− 26.3538i	− 0.858199i
$944$	− 10.3923i	− 0.338241i
$945$	−12.0000	−0.390360
$946$	6.00000	0.195077
$947$	57.3731i	1.86437i	0.361977	+	0.932187i	$0.382102\pi$
$947$	57.3731i	1.86437i	−0.361977	+	0.932187i	$0.617898\pi$
$948$	16.9282	0.549802
$949$	0	0
$950$	6.46410	0.209723
$951$	34.9808i	1.13433i
$952$	−6.58846	−0.213533
$953$	24.5885	0.796498	0.398249	−	0.917277i	$-0.369618\pi$
$953$	24.5885	0.796498	0.398249	+	0.917277i	$0.369618\pi$
$954$	− 28.8564i	− 0.934261i
$955$	19.2679i	0.623496i
$956$	8.19615i	0.265083i
$957$	− 77.5692i	− 2.50746i
$958$	10.9808	0.354772
$959$	24.5885	0.794003
$960$	− 2.73205i	− 0.0881766i
$961$	29.3923	0.948139
$962$	0	0
$963$	1.51666	0.0488737
$964$	− 8.66025i	− 0.278928i
$965$	−4.39230	−0.141393
$966$	20.7846	0.668734
$967$	38.5692i	1.24030i	0.784482	+	0.620151i	$0.212929\pi$
$967$	38.5692i	1.24030i	−0.784482	+	0.620151i	$0.787071\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	38.7846i	1.24594i
$970$	− 15.1244i	− 0.485614i
$971$	−46.7654	−1.50077	−0.750386	−	0.661000i	$-0.770132\pi$
$971$	−46.7654	−1.50077	−0.750386	+	0.661000i	$0.770132\pi$
$972$	−18.7321	−0.600831
$973$	− 27.5885i	− 0.884445i
$974$	−33.2487	−1.06536
$975$	0	0
$976$	−4.19615	−0.134316
$977$	4.39230i	0.140522i	0.997529	+	0.0702611i	$0.0223833\pi$
$977$	4.39230i	0.140522i	−0.997529	+	0.0702611i	$0.977617\pi$
$978$	−19.8564	−0.634938
$979$	−51.5885	−1.64877
$980$	− 2.00000i	− 0.0638877i
$981$	− 69.0333i	− 2.20406i
$982$	3.33975i	0.106576i
$983$	− 38.5692i	− 1.23017i	−0.788462	−	0.615084i	$-0.789122\pi$
$983$	− 38.5692i	− 1.23017i	0.788462	−	0.615084i	$-0.210878\pi$
$984$	28.3923	0.905114
$985$	9.58846	0.305514
$986$	20.7846i	0.661917i
$987$	24.5885	0.782659
$988$	0	0
$989$	5.07180	0.161274
$990$	− 13.3923i	− 0.425635i
$991$	51.1769	1.62569	0.812844	−	0.582481i	$-0.197918\pi$
$991$	51.1769	1.62569	0.812844	+	0.582481i	$0.197918\pi$
$992$	−1.26795	−0.0402574
$993$	2.53590i	0.0804743i
$994$	18.0000i	0.570925i
$995$	− 14.3923i	− 0.456267i
$996$	6.00000i	0.190117i
$997$	−42.5692	−1.34818	−0.674090	−	0.738649i	$-0.735464\pi$
$997$	−42.5692	−1.34818	−0.674090	+	0.738649i	$0.735464\pi$
$998$	1.85641	0.0587635
$999$	− 44.7846i	− 1.41692i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.d.f.1351.3		4
13.2	odd	12		1690.2.e.l.191.2		4
13.3	even	3		1690.2.l.g.1161.1		4
13.4	even	6		1690.2.l.g.361.1		4
13.5	odd	4		1690.2.a.m.1.1		2
13.6	odd	12		1690.2.e.l.991.2		4
13.7	odd	12		1690.2.e.n.991.2		4
13.8	odd	4		1690.2.a.j.1.1		2
13.9	even	3		130.2.l.a.101.2	✓	4
13.10	even	6		130.2.l.a.121.2	yes	4
13.11	odd	12		1690.2.e.n.191.2		4
13.12	even	2	inner	1690.2.d.f.1351.1		4
39.23	odd	6		1170.2.bs.c.901.1		4
39.35	odd	6		1170.2.bs.c.361.1		4
52.23	odd	6		1040.2.da.a.641.1		4
52.35	odd	6		1040.2.da.a.881.1		4
65.9	even	6		650.2.m.a.101.1		4
65.22	odd	12		650.2.n.b.49.2		4
65.23	odd	12		650.2.n.b.199.2		4
65.34	odd	4		8450.2.a.bm.1.2		2
65.44	odd	4		8450.2.a.bf.1.2		2
65.48	odd	12		650.2.n.a.49.1		4
65.49	even	6		650.2.m.a.251.1		4
65.62	odd	12		650.2.n.a.199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.l.a.101.2	✓	4	13.9	even	3
130.2.l.a.121.2	yes	4	13.10	even	6
650.2.m.a.101.1		4	65.9	even	6
650.2.m.a.251.1		4	65.49	even	6
650.2.n.a.49.1		4	65.48	odd	12
650.2.n.a.199.1		4	65.62	odd	12
650.2.n.b.49.2		4	65.22	odd	12
650.2.n.b.199.2		4	65.23	odd	12
1040.2.da.a.641.1		4	52.23	odd	6
1040.2.da.a.881.1		4	52.35	odd	6
1170.2.bs.c.361.1		4	39.35	odd	6
1170.2.bs.c.901.1		4	39.23	odd	6
1690.2.a.j.1.1		2	13.8	odd	4
1690.2.a.m.1.1		2	13.5	odd	4
1690.2.d.f.1351.1		4	13.12	even	2	inner
1690.2.d.f.1351.3		4	1.1	even	1	trivial
1690.2.e.l.191.2		4	13.2	odd	12
1690.2.e.l.991.2		4	13.6	odd	12
1690.2.e.n.191.2		4	13.11	odd	12
1690.2.e.n.991.2		4	13.7	odd	12
1690.2.l.g.361.1		4	13.4	even	6
1690.2.l.g.1161.1		4	13.3	even	3
8450.2.a.bf.1.2		2	65.44	odd	4
8450.2.a.bm.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.73205 q^{3} -1.00000 q^{4} -1.00000i q^{5} -2.73205i q^{6} +3.00000i q^{7} -1.00000i q^{8} +4.46410 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.73205 q^{3} -1.00000 q^{4} -1.00000i q^{5} -2.73205i q^{6} +3.00000i q^{7} -1.00000i q^{8} +4.46410 q^{9} +1.00000 q^{10} -3.00000i q^{11} +2.73205 q^{12} -3.00000 q^{14} +2.73205i q^{15} +1.00000 q^{16} -2.19615 q^{17} +4.46410i q^{18} +6.46410i q^{19} +1.00000i q^{20} -8.19615i q^{21} +3.00000 q^{22} +2.53590 q^{23} +2.73205i q^{24} -1.00000 q^{25} -4.00000 q^{27} -3.00000i q^{28} -9.46410 q^{29} -2.73205 q^{30} +1.26795i q^{31} +1.00000i q^{32} +8.19615i q^{33} -2.19615i q^{34} +3.00000 q^{35} -4.46410 q^{36} +11.1962i q^{37} -6.46410 q^{38} -1.00000 q^{40} -10.3923i q^{41} +8.19615 q^{42} +2.00000 q^{43} +3.00000i q^{44} -4.46410i q^{45} +2.53590i q^{46} +3.00000i q^{47} -2.73205 q^{48} -2.00000 q^{49} -1.00000i q^{50} +6.00000 q^{51} -6.46410 q^{53} -4.00000i q^{54} -3.00000 q^{55} +3.00000 q^{56} -17.6603i q^{57} -9.46410i q^{58} -10.3923i q^{59} -2.73205i q^{60} -4.19615 q^{61} -1.26795 q^{62} +13.3923i q^{63} -1.00000 q^{64} -8.19615 q^{66} +2.19615 q^{68} -6.92820 q^{69} +3.00000i q^{70} -6.00000i q^{71} -4.46410i q^{72} -5.66025i q^{73} -11.1962 q^{74} +2.73205 q^{75} -6.46410i q^{76} +9.00000 q^{77} +6.19615 q^{79} -1.00000i q^{80} -2.46410 q^{81} +10.3923 q^{82} +2.19615i q^{83} +8.19615i q^{84} +2.19615i q^{85} +2.00000i q^{86} +25.8564 q^{87} -3.00000 q^{88} -17.1962i q^{89} +4.46410 q^{90} -2.53590 q^{92} -3.46410i q^{93} -3.00000 q^{94} +6.46410 q^{95} -2.73205i q^{96} -15.1244i q^{97} -2.00000i q^{98} -13.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9	\( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} + 4 q^{10} + 4 q^{12} - 12 q^{14} + 4 q^{16} + 12 q^{17} + 12 q^{22} + 24 q^{23} - 4 q^{25} - 16 q^{27} - 24 q^{29} - 4 q^{30} + 12 q^{35} - 4 q^{36} - 12 q^{38} - 4 q^{40} + 12 q^{42} + 8 q^{43} - 4 q^{48} - 8 q^{49} + 24 q^{51} - 12 q^{53} - 12 q^{55} + 12 q^{56} + 4 q^{61} - 12 q^{62} - 4 q^{64} - 12 q^{66} - 12 q^{68} - 24 q^{74} + 4 q^{75} + 36 q^{77} + 4 q^{79} + 4 q^{81} + 48 q^{87} - 12 q^{88} + 4 q^{90} - 24 q^{92} - 12 q^{94} + 12 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 + 4 * q^10 + 4 * q^12 - 12 * q^14 + 4 * q^16 + 12 * q^17 + 12 * q^22 + 24 * q^23 - 4 * q^25 - 16 * q^27 - 24 * q^29 - 4 * q^30 + 12 * q^35 - 4 * q^36 - 12 * q^38 - 4 * q^40 + 12 * q^42 + 8 * q^43 - 4 * q^48 - 8 * q^49 + 24 * q^51 - 12 * q^53 - 12 * q^55 + 12 * q^56 + 4 * q^61 - 12 * q^62 - 4 * q^64 - 12 * q^66 - 12 * q^68 - 24 * q^74 + 4 * q^75 + 36 * q^77 + 4 * q^79 + 4 * q^81 + 48 * q^87 - 12 * q^88 + 4 * q^90 - 24 * q^92 - 12 * q^94 + 12 * q^95

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