LMFDB - Embedded newform 845.2.c.d.506.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(506,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, 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("845.506");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.c (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:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			506.2
Root			$-1.30278i$ of defining polynomial
Character	$\chi$	$=$	845.506
Dual form			845.2.c.d.506.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30278i q^{2} +1.00000 q^{3} +0.302776 q^{4} +1.00000i q^{5} -1.30278i q^{6} -1.00000i q^{7} -3.00000i q^{8} -2.00000 q^{9} +O(q^{10})$	\(q-1.30278i q^{2} +1.00000 q^{3} +0.302776 q^{4} +1.00000i q^{5} -1.30278i q^{6} -1.00000i q^{7} -3.00000i q^{8} -2.00000 q^{9} +1.30278 q^{10} -5.60555i q^{11} +0.302776 q^{12} -1.30278 q^{14} +1.00000i q^{15} -3.30278 q^{16} -0.394449 q^{17} +2.60555i q^{18} -1.60555i q^{19} +0.302776i q^{20} -1.00000i q^{21} -7.30278 q^{22} +3.00000 q^{23} -3.00000i q^{24} -1.00000 q^{25} -5.00000 q^{27} -0.302776i q^{28} +8.21110 q^{29} +1.30278 q^{30} +4.00000i q^{31} -1.69722i q^{32} -5.60555i q^{33} +0.513878i q^{34} +1.00000 q^{35} -0.605551 q^{36} -3.60555i q^{37} -2.09167 q^{38} +3.00000 q^{40} -3.00000i q^{41} -1.30278 q^{42} -4.21110 q^{43} -1.69722i q^{44} -2.00000i q^{45} -3.90833i q^{46} -5.21110i q^{47} -3.30278 q^{48} +6.00000 q^{49} +1.30278i q^{50} -0.394449 q^{51} +11.2111 q^{53} +6.51388i q^{54} +5.60555 q^{55} -3.00000 q^{56} -1.60555i q^{57} -10.6972i q^{58} +10.8167i q^{59} +0.302776i q^{60} -1.00000 q^{61} +5.21110 q^{62} +2.00000i q^{63} -8.81665 q^{64} -7.30278 q^{66} +7.00000i q^{67} -0.119429 q^{68} +3.00000 q^{69} -1.30278i q^{70} +16.8167i q^{71} +6.00000i q^{72} -15.2111i q^{73} -4.69722 q^{74} -1.00000 q^{75} -0.486122i q^{76} -5.60555 q^{77} -9.21110 q^{79} -3.30278i q^{80} +1.00000 q^{81} -3.90833 q^{82} -5.21110i q^{83} -0.302776i q^{84} -0.394449i q^{85} +5.48612i q^{86} +8.21110 q^{87} -16.8167 q^{88} +8.21110i q^{89} -2.60555 q^{90} +0.908327 q^{92} +4.00000i q^{93} -6.78890 q^{94} +1.60555 q^{95} -1.69722i q^{96} +15.6056i q^{97} -7.81665i q^{98} +11.2111i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 6 q^{4} - 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 6 * q^4 - 8 * q^9	$ 4 q + 4 q^{3} - 6 q^{4} - 8 q^{9} - 2 q^{10} - 6 q^{12} + 2 q^{14} - 6 q^{16} - 16 q^{17} - 22 q^{22} + 12 q^{23} - 4 q^{25} - 20 q^{27} + 4 q^{29} - 2 q^{30} + 4 q^{35} + 12 q^{36} - 30 q^{38} + 12 q^{40} + 2 q^{42} + 12 q^{43} - 6 q^{48} + 24 q^{49} - 16 q^{51} + 16 q^{53} + 8 q^{55} - 12 q^{56} - 4 q^{61} - 8 q^{62} + 8 q^{64} - 22 q^{66} + 50 q^{68} + 12 q^{69} - 26 q^{74} - 4 q^{75} - 8 q^{77} - 8 q^{79} + 4 q^{81} + 6 q^{82} + 4 q^{87} - 24 q^{88} + 4 q^{90} - 18 q^{92} - 56 q^{94} - 8 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 6 * q^4 - 8 * q^9 - 2 * q^10 - 6 * q^12 + 2 * q^14 - 6 * q^16 - 16 * q^17 - 22 * q^22 + 12 * q^23 - 4 * q^25 - 20 * q^27 + 4 * q^29 - 2 * q^30 + 4 * q^35 + 12 * q^36 - 30 * q^38 + 12 * q^40 + 2 * q^42 + 12 * q^43 - 6 * q^48 + 24 * q^49 - 16 * q^51 + 16 * q^53 + 8 * q^55 - 12 * q^56 - 4 * q^61 - 8 * q^62 + 8 * q^64 - 22 * q^66 + 50 * q^68 + 12 * q^69 - 26 * q^74 - 4 * q^75 - 8 * q^77 - 8 * q^79 + 4 * q^81 + 6 * q^82 + 4 * q^87 - 24 * q^88 + 4 * q^90 - 18 * q^92 - 56 * q^94 - 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$.

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

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 $

$2$	− 1.30278i	− 0.921201i	−0.887607	−	0.460601i	$-0.847634\pi$
$2$	− 1.30278i	− 0.921201i	0.887607	−	0.460601i	$-0.152366\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0.302776	0.151388
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	− 1.30278i	− 0.531856i
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	− 3.00000i	− 1.06066i
$9$	−2.00000	−0.666667
$10$	1.30278	0.411974
$11$	− 5.60555i	− 1.69014i	−0.534658	−	0.845069i	$-0.679559\pi$
$11$	− 5.60555i	− 1.69014i	0.534658	−	0.845069i	$-0.320441\pi$
$12$	0.302776	0.0874038
$13$	0	0
$13$	0	0
$14$	−1.30278	−0.348181
$15$	1.00000i	0.258199i
$16$	−3.30278	−0.825694
$17$	−0.394449	−0.0956679	−0.0478339	−	0.998855i	$-0.515232\pi$
$17$	−0.394449	−0.0956679	−0.0478339	+	0.998855i	$0.515232\pi$
$18$	2.60555i	0.614134i
$19$	− 1.60555i	− 0.368339i	−0.982895	−	0.184169i	$-0.941041\pi$
$19$	− 1.60555i	− 0.368339i	0.982895	−	0.184169i	$-0.0589595\pi$
$20$	0.302776i	0.0677027i
$21$	− 1.00000i	− 0.218218i
$22$	−7.30278	−1.55696
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	− 3.00000i	− 0.612372i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	− 0.302776i	− 0.0572192i
$29$	8.21110	1.52476	0.762382	−	0.647128i	$-0.224030\pi$
$29$	8.21110	1.52476	0.762382	+	0.647128i	$0.224030\pi$
$30$	1.30278	0.237853
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	− 1.69722i	− 0.300030i
$33$	− 5.60555i	− 0.975801i
$34$	0.513878i	0.0881294i
$35$	1.00000	0.169031
$36$	−0.605551	−0.100925
$37$	− 3.60555i	− 0.592749i	−0.955072	−	0.296374i	$-0.904222\pi$
$37$	− 3.60555i	− 0.592749i	0.955072	−	0.296374i	$-0.0957776\pi$
$38$	−2.09167	−0.339314
$39$	0	0
$40$	3.00000	0.474342
$41$	− 3.00000i	− 0.468521i	−0.972174	−	0.234261i	$-0.924733\pi$
$41$	− 3.00000i	− 0.468521i	0.972174	−	0.234261i	$-0.0752669\pi$
$42$	−1.30278	−0.201023
$43$	−4.21110	−0.642187	−0.321094	−	0.947047i	$-0.604050\pi$
$43$	−4.21110	−0.642187	−0.321094	+	0.947047i	$0.604050\pi$
$44$	− 1.69722i	− 0.255866i
$45$	− 2.00000i	− 0.298142i
$46$	− 3.90833i	− 0.576251i
$47$	− 5.21110i	− 0.760117i	−0.924962	−	0.380059i	$-0.875904\pi$
$47$	− 5.21110i	− 0.760117i	0.924962	−	0.380059i	$-0.124096\pi$
$48$	−3.30278	−0.476715
$49$	6.00000	0.857143
$50$	1.30278i	0.184240i
$51$	−0.394449	−0.0552339
$52$	0	0
$53$	11.2111	1.53996	0.769982	−	0.638066i	$-0.220265\pi$
$53$	11.2111	1.53996	0.769982	+	0.638066i	$0.220265\pi$
$54$	6.51388i	0.886427i
$55$	5.60555	0.755852
$56$	−3.00000	−0.400892
$57$	− 1.60555i	− 0.212660i
$58$	− 10.6972i	− 1.40461i
$59$	10.8167i	1.40821i	0.710097	+	0.704104i	$0.248651\pi$
$59$	10.8167i	1.40821i	−0.710097	+	0.704104i	$0.751349\pi$
$60$	0.302776i	0.0390882i
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	5.21110	0.661811
$63$	2.00000i	0.251976i
$64$	−8.81665	−1.10208
$65$	0	0
$66$	−7.30278	−0.898910
$67$	7.00000i	0.855186i	0.903971	+	0.427593i	$0.140638\pi$
$67$	7.00000i	0.855186i	−0.903971	+	0.427593i	$0.859362\pi$
$68$	−0.119429	−0.0144829
$69$	3.00000	0.361158
$70$	− 1.30278i	− 0.155711i
$71$	16.8167i	1.99577i	0.0650069	+	0.997885i	$0.479293\pi$
$71$	16.8167i	1.99577i	−0.0650069	+	0.997885i	$0.520707\pi$
$72$	6.00000i	0.707107i
$73$	− 15.2111i	− 1.78032i	−0.455643	−	0.890162i	$-0.650591\pi$
$73$	− 15.2111i	− 1.78032i	0.455643	−	0.890162i	$-0.349409\pi$
$74$	−4.69722	−0.546041
$75$	−1.00000	−0.115470
$76$	− 0.486122i	− 0.0557620i
$77$	−5.60555	−0.638812
$78$	0	0
$79$	−9.21110	−1.03633	−0.518165	−	0.855281i	$-0.673385\pi$
$79$	−9.21110	−1.03633	−0.518165	+	0.855281i	$0.673385\pi$
$80$	− 3.30278i	− 0.369262i
$81$	1.00000	0.111111
$82$	−3.90833	−0.431603
$83$	− 5.21110i	− 0.571993i	−0.958231	−	0.285996i	$-0.907675\pi$
$83$	− 5.21110i	− 0.571993i	0.958231	−	0.285996i	$-0.0923245\pi$
$84$	− 0.302776i	− 0.0330355i
$85$	− 0.394449i	− 0.0427840i
$86$	5.48612i	0.591584i
$87$	8.21110	0.880323
$88$	−16.8167	−1.79266
$89$	8.21110i	0.870375i	0.900340	+	0.435188i	$0.143318\pi$
$89$	8.21110i	0.870375i	−0.900340	+	0.435188i	$0.856682\pi$
$90$	−2.60555	−0.274649
$91$	0	0
$92$	0.908327	0.0946996
$93$	4.00000i	0.414781i
$94$	−6.78890	−0.700221
$95$	1.60555	0.164726
$96$	− 1.69722i	− 0.173222i
$97$	15.6056i	1.58450i	0.610194	+	0.792252i	$0.291091\pi$
$97$	15.6056i	1.58450i	−0.610194	+	0.792252i	$0.708909\pi$
$98$	− 7.81665i	− 0.789601i
$99$	11.2111i	1.12676i
$100$	−0.302776	−0.0302776
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	0.513878i	0.0508815i
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	− 14.6056i	− 1.41862i
$107$	−8.21110	−0.793797	−0.396899	−	0.917862i	$-0.629914\pi$
$107$	−8.21110	−0.793797	−0.396899	+	0.917862i	$0.629914\pi$
$108$	−1.51388	−0.145673
$109$	4.78890i	0.458693i	0.973345	+	0.229347i	$0.0736589\pi$
$109$	4.78890i	0.458693i	−0.973345	+	0.229347i	$0.926341\pi$
$110$	− 7.30278i	− 0.696292i
$111$	− 3.60555i	− 0.342224i
$112$	3.30278i	0.312083i
$113$	5.60555	0.527326	0.263663	−	0.964615i	$-0.415069\pi$
$113$	5.60555	0.527326	0.263663	+	0.964615i	$0.415069\pi$
$114$	−2.09167	−0.195903
$115$	3.00000i	0.279751i
$116$	2.48612	0.230831
$117$	0	0
$118$	14.0917	1.29724
$119$	0.394449i	0.0361591i
$120$	3.00000	0.273861
$121$	−20.4222	−1.85656
$122$	1.30278i	0.117948i
$123$	− 3.00000i	− 0.270501i
$124$	1.21110i	0.108760i
$125$	− 1.00000i	− 0.0894427i
$126$	2.60555	0.232121
$127$	−10.2111	−0.906089	−0.453044	−	0.891488i	$-0.649662\pi$
$127$	−10.2111	−0.906089	−0.453044	+	0.891488i	$0.649662\pi$
$128$	8.09167i	0.715210i
$129$	−4.21110	−0.370767
$130$	0	0
$131$	−6.78890	−0.593149	−0.296574	−	0.955010i	$-0.595844\pi$
$131$	−6.78890	−0.593149	−0.296574	+	0.955010i	$0.595844\pi$
$132$	− 1.69722i	− 0.147724i
$133$	−1.60555	−0.139219
$134$	9.11943	0.787799
$135$	− 5.00000i	− 0.430331i
$136$	1.18335i	0.101471i
$137$	5.60555i	0.478915i	0.970907	+	0.239457i	$0.0769695\pi$
$137$	5.60555i	0.478915i	−0.970907	+	0.239457i	$0.923030\pi$
$138$	− 3.90833i	− 0.332699i
$139$	13.6056	1.15401	0.577004	−	0.816741i	$-0.304222\pi$
$139$	13.6056	1.15401	0.577004	+	0.816741i	$0.304222\pi$
$140$	0.302776	0.0255892
$141$	− 5.21110i	− 0.438854i
$142$	21.9083	1.83851
$143$	0	0
$144$	6.60555	0.550463
$145$	8.21110i	0.681895i
$146$	−19.8167	−1.64004
$147$	6.00000	0.494872
$148$	− 1.09167i	− 0.0897350i
$149$	− 3.00000i	− 0.245770i	−0.992421	−	0.122885i	$-0.960785\pi$
$149$	− 3.00000i	− 0.245770i	0.992421	−	0.122885i	$-0.0392146\pi$
$150$	1.30278i	0.106371i
$151$	13.2111i	1.07510i	0.843231	+	0.537552i	$0.180651\pi$
$151$	13.2111i	1.07510i	−0.843231	+	0.537552i	$0.819349\pi$
$152$	−4.81665	−0.390682
$153$	0.788897	0.0637786
$154$	7.30278i	0.588474i
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−3.21110	−0.256274	−0.128137	−	0.991756i	$-0.540900\pi$
$157$	−3.21110	−0.256274	−0.128137	+	0.991756i	$0.540900\pi$
$158$	12.0000i	0.954669i
$159$	11.2111	0.889098
$160$	1.69722	0.134177
$161$	− 3.00000i	− 0.236433i
$162$	− 1.30278i	− 0.102356i
$163$	− 18.2111i	− 1.42640i	−0.700959	−	0.713202i	$-0.747244\pi$
$163$	− 18.2111i	− 1.42640i	0.700959	−	0.713202i	$-0.252756\pi$
$164$	− 0.908327i	− 0.0709284i
$165$	5.60555	0.436392
$166$	−6.78890	−0.526921
$167$	9.00000i	0.696441i	0.937413	+	0.348220i	$0.113214\pi$
$167$	9.00000i	0.696441i	−0.937413	+	0.348220i	$0.886786\pi$
$168$	−3.00000	−0.231455
$169$	0	0
$170$	−0.513878	−0.0394127
$171$	3.21110i	0.245559i
$172$	−1.27502	−0.0972193
$173$	16.8167	1.27855	0.639273	−	0.768980i	$-0.279235\pi$
$173$	16.8167	1.27855	0.639273	+	0.768980i	$0.279235\pi$
$174$	− 10.6972i	− 0.810954i
$175$	1.00000i	0.0755929i
$176$	18.5139i	1.39554i
$177$	10.8167i	0.813029i
$178$	10.6972	0.801791
$179$	−1.18335	−0.0884474	−0.0442237	−	0.999022i	$-0.514081\pi$
$179$	−1.18335	−0.0884474	−0.0442237	+	0.999022i	$0.514081\pi$
$180$	− 0.605551i	− 0.0451351i
$181$	25.6333	1.90531	0.952654	−	0.304055i	$-0.0983408\pi$
$181$	25.6333	1.90531	0.952654	+	0.304055i	$0.0983408\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	− 9.00000i	− 0.663489i
$185$	3.60555	0.265085
$186$	5.21110	0.382097
$187$	2.21110i	0.161692i
$188$	− 1.57779i	− 0.115073i
$189$	5.00000i	0.363696i
$190$	− 2.09167i	− 0.151746i
$191$	−4.81665	−0.348521	−0.174260	−	0.984700i	$-0.555753\pi$
$191$	−4.81665	−0.348521	−0.174260	+	0.984700i	$0.555753\pi$
$192$	−8.81665	−0.636287
$193$	8.39445i	0.604246i	0.953269	+	0.302123i	$0.0976953\pi$
$193$	8.39445i	0.604246i	−0.953269	+	0.302123i	$0.902305\pi$
$194$	20.3305	1.45965
$195$	0	0
$196$	1.81665	0.129761
$197$	− 22.8167i	− 1.62562i	−0.582529	−	0.812810i	$-0.697937\pi$
$197$	− 22.8167i	− 1.62562i	0.582529	−	0.812810i	$-0.302063\pi$
$198$	14.6056	1.03797
$199$	8.81665	0.624996	0.312498	−	0.949918i	$-0.398834\pi$
$199$	8.81665	0.624996	0.312498	+	0.949918i	$0.398834\pi$
$200$	3.00000i	0.212132i
$201$	7.00000i	0.493742i
$202$	− 11.7250i	− 0.824967i
$203$	− 8.21110i	− 0.576306i
$204$	−0.119429	−0.00836174
$205$	3.00000	0.209529
$206$	− 5.21110i	− 0.363075i
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−9.00000	−0.622543
$210$	− 1.30278i	− 0.0899001i
$211$	−16.3944	−1.12864	−0.564320	−	0.825556i	$-0.690862\pi$
$211$	−16.3944	−1.12864	−0.564320	+	0.825556i	$0.690862\pi$
$212$	3.39445	0.233132
$213$	16.8167i	1.15226i
$214$	10.6972i	0.731247i
$215$	− 4.21110i	− 0.287195i
$216$	15.0000i	1.02062i
$217$	4.00000	0.271538
$218$	6.23886	0.422549
$219$	− 15.2111i	− 1.02787i
$220$	1.69722	0.114427
$221$	0	0
$222$	−4.69722	−0.315257
$223$	− 10.2111i	− 0.683786i	−0.939739	−	0.341893i	$-0.888932\pi$
$223$	− 10.2111i	− 0.683786i	0.939739	−	0.341893i	$-0.111068\pi$
$224$	−1.69722	−0.113401
$225$	2.00000	0.133333
$226$	− 7.30278i	− 0.485773i
$227$	− 1.42221i	− 0.0943951i	−0.998886	−	0.0471975i	$-0.984971\pi$
$227$	− 1.42221i	− 0.0943951i	0.998886	−	0.0471975i	$-0.0150290\pi$
$228$	− 0.486122i	− 0.0321942i
$229$	14.0000i	0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$	14.0000i	0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$	3.90833	0.257707
$231$	−5.60555	−0.368818
$232$	− 24.6333i	− 1.61726i
$233$	−0.788897	−0.0516824	−0.0258412	−	0.999666i	$-0.508226\pi$
$233$	−0.788897	−0.0516824	−0.0258412	+	0.999666i	$0.508226\pi$
$234$	0	0
$235$	5.21110	0.339935
$236$	3.27502i	0.213186i
$237$	−9.21110	−0.598325
$238$	0.513878	0.0333098
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	− 3.30278i	− 0.213193i
$241$	16.2111i	1.04425i	0.852869	+	0.522124i	$0.174860\pi$
$241$	16.2111i	1.04425i	−0.852869	+	0.522124i	$0.825140\pi$
$242$	26.6056i	1.71027i
$243$	16.0000	1.02640
$244$	−0.302776	−0.0193832
$245$	6.00000i	0.383326i
$246$	−3.90833	−0.249186
$247$	0	0
$248$	12.0000	0.762001
$249$	− 5.21110i	− 0.330240i
$250$	−1.30278	−0.0823948
$251$	28.8167	1.81889	0.909446	−	0.415823i	$-0.136506\pi$
$251$	28.8167	1.81889	0.909446	+	0.415823i	$0.136506\pi$
$252$	0.605551i	0.0381461i
$253$	− 16.8167i	− 1.05725i
$254$	13.3028i	0.834690i
$255$	− 0.394449i	− 0.0247013i
$256$	−7.09167	−0.443230
$257$	23.6056	1.47247	0.736237	−	0.676724i	$-0.236601\pi$
$257$	23.6056	1.47247	0.736237	+	0.676724i	$0.236601\pi$
$258$	5.48612i	0.341551i
$259$	−3.60555	−0.224038
$260$	0	0
$261$	−16.4222	−1.01651
$262$	8.84441i	0.546409i
$263$	26.2111	1.61625	0.808123	−	0.589014i	$-0.200484\pi$
$263$	26.2111	1.61625	0.808123	+	0.589014i	$0.200484\pi$
$264$	−16.8167	−1.03499
$265$	11.2111i	0.688693i
$266$	2.09167i	0.128249i
$267$	8.21110i	0.502511i
$268$	2.11943i	0.129465i
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	−6.51388	−0.396422
$271$	0.816654i	0.0496082i	0.999692	+	0.0248041i	$0.00789620\pi$
$271$	0.816654i	0.0496082i	−0.999692	+	0.0248041i	$0.992104\pi$
$272$	1.30278	0.0789924
$273$	0	0
$274$	7.30278	0.441177
$275$	5.60555i	0.338027i
$276$	0.908327	0.0546749
$277$	−20.3944	−1.22538	−0.612692	−	0.790322i	$-0.709913\pi$
$277$	−20.3944	−1.22538	−0.612692	+	0.790322i	$0.709913\pi$
$278$	− 17.7250i	− 1.06307i
$279$	− 8.00000i	− 0.478947i
$280$	− 3.00000i	− 0.179284i
$281$	− 6.00000i	− 0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$	− 6.00000i	− 0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$	−6.78890	−0.404273
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	5.09167i	0.302135i
$285$	1.60555	0.0951046
$286$	0	0
$287$	−3.00000	−0.177084
$288$	3.39445i	0.200020i
$289$	−16.8444	−0.990848
$290$	10.6972	0.628163
$291$	15.6056i	0.914814i
$292$	− 4.60555i	− 0.269520i
$293$	17.6056i	1.02853i	0.857632	+	0.514264i	$0.171935\pi$
$293$	17.6056i	1.02853i	−0.857632	+	0.514264i	$0.828065\pi$
$294$	− 7.81665i	− 0.455877i
$295$	−10.8167	−0.629770
$296$	−10.8167	−0.628705
$297$	28.0278i	1.62634i
$298$	−3.90833	−0.226403
$299$	0	0
$300$	−0.302776	−0.0174808
$301$	4.21110i	0.242724i
$302$	17.2111	0.990388
$303$	9.00000	0.517036
$304$	5.30278i	0.304135i
$305$	− 1.00000i	− 0.0572598i
$306$	− 1.02776i	− 0.0587529i
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	−1.69722	−0.0967083
$309$	4.00000	0.227552
$310$	5.21110i	0.295971i
$311$	−5.21110	−0.295495	−0.147747	−	0.989025i	$-0.547202\pi$
$311$	−5.21110	−0.295495	−0.147747	+	0.989025i	$0.547202\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	4.18335i	0.236080i
$315$	−2.00000	−0.112687
$316$	−2.78890	−0.156888
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	− 14.6056i	− 0.819039i
$319$	− 46.0278i	− 2.57706i
$320$	− 8.81665i	− 0.492866i
$321$	−8.21110	−0.458299
$322$	−3.90833	−0.217803
$323$	0.633308i	0.0352382i
$324$	0.302776	0.0168209
$325$	0	0
$326$	−23.7250	−1.31401
$327$	4.78890i	0.264827i
$328$	−9.00000	−0.496942
$329$	−5.21110	−0.287297
$330$	− 7.30278i	− 0.402005i
$331$	26.0278i	1.43061i	0.698810	+	0.715307i	$0.253713\pi$
$331$	26.0278i	1.43061i	−0.698810	+	0.715307i	$0.746287\pi$
$332$	− 1.57779i	− 0.0865927i
$333$	7.21110i	0.395166i
$334$	11.7250	0.641562
$335$	−7.00000	−0.382451
$336$	3.30278i	0.180181i
$337$	−17.6333	−0.960547	−0.480274	−	0.877119i	$-0.659463\pi$
$337$	−17.6333	−0.960547	−0.480274	+	0.877119i	$0.659463\pi$
$338$	0	0
$339$	5.60555	0.304452
$340$	− 0.119429i	− 0.00647697i
$341$	22.4222	1.21423
$342$	4.18335	0.226209
$343$	− 13.0000i	− 0.701934i
$344$	12.6333i	0.681142i
$345$	3.00000i	0.161515i
$346$	− 21.9083i	− 1.17780i
$347$	20.2111	1.08499	0.542494	−	0.840059i	$-0.317480\pi$
$347$	20.2111	1.08499	0.542494	+	0.840059i	$0.317480\pi$
$348$	2.48612	0.133270
$349$	− 18.2111i	− 0.974818i	−0.873174	−	0.487409i	$-0.837942\pi$
$349$	− 18.2111i	− 0.974818i	0.873174	−	0.487409i	$-0.162058\pi$
$350$	1.30278	0.0696363
$351$	0	0
$352$	−9.51388	−0.507091
$353$	4.81665i	0.256365i	0.991751	+	0.128182i	$0.0409143\pi$
$353$	4.81665i	0.256365i	−0.991751	+	0.128182i	$0.959086\pi$
$354$	14.0917	0.748964
$355$	−16.8167	−0.892535
$356$	2.48612i	0.131764i
$357$	0.394449i	0.0208764i
$358$	1.54163i	0.0814779i
$359$	− 10.4222i	− 0.550063i	−0.961435	−	0.275031i	$-0.911312\pi$
$359$	− 10.4222i	− 0.550063i	0.961435	−	0.275031i	$-0.0886883\pi$
$360$	−6.00000	−0.316228
$361$	16.4222	0.864327
$362$	− 33.3944i	− 1.75517i
$363$	−20.4222	−1.07189
$364$	0	0
$365$	15.2111	0.796185
$366$	1.30278i	0.0680972i
$367$	−17.4222	−0.909432	−0.454716	−	0.890637i	$-0.650259\pi$
$367$	−17.4222	−0.909432	−0.454716	+	0.890637i	$0.650259\pi$
$368$	−9.90833	−0.516507
$369$	6.00000i	0.312348i
$370$	− 4.69722i	− 0.244197i
$371$	− 11.2111i	− 0.582051i
$372$	1.21110i	0.0627927i
$373$	−27.6056	−1.42936	−0.714681	−	0.699451i	$-0.753428\pi$
$373$	−27.6056	−1.42936	−0.714681	+	0.699451i	$0.753428\pi$
$374$	2.88057	0.148951
$375$	− 1.00000i	− 0.0516398i
$376$	−15.6333	−0.806226
$377$	0	0
$378$	6.51388	0.335038
$379$	− 2.39445i	− 0.122995i	−0.998107	−	0.0614973i	$-0.980412\pi$
$379$	− 2.39445i	− 0.122995i	0.998107	−	0.0614973i	$-0.0195876\pi$
$380$	0.486122	0.0249375
$381$	−10.2111	−0.523131
$382$	6.27502i	0.321058i
$383$	− 18.6333i	− 0.952118i	−0.879413	−	0.476059i	$-0.842065\pi$
$383$	− 18.6333i	− 0.952118i	0.879413	−	0.476059i	$-0.157935\pi$
$384$	8.09167i	0.412926i
$385$	− 5.60555i	− 0.285685i
$386$	10.9361	0.556632
$387$	8.42221	0.428125
$388$	4.72498i	0.239875i
$389$	0.788897	0.0399987	0.0199993	−	0.999800i	$-0.493634\pi$
$389$	0.788897	0.0399987	0.0199993	+	0.999800i	$0.493634\pi$
$390$	0	0
$391$	−1.18335	−0.0598444
$392$	− 18.0000i	− 0.909137i
$393$	−6.78890	−0.342455
$394$	−29.7250	−1.49752
$395$	− 9.21110i	− 0.463461i
$396$	3.39445i	0.170577i
$397$	− 14.0278i	− 0.704033i	−0.935994	−	0.352016i	$-0.885496\pi$
$397$	− 14.0278i	− 0.704033i	0.935994	−	0.352016i	$-0.114504\pi$
$398$	− 11.4861i	− 0.575747i
$399$	−1.60555	−0.0803781
$400$	3.30278	0.165139
$401$	− 2.21110i	− 0.110417i	−0.998475	−	0.0552086i	$-0.982418\pi$
$401$	− 2.21110i	− 0.110417i	0.998475	−	0.0552086i	$-0.0175824\pi$
$402$	9.11943	0.454836
$403$	0	0
$404$	2.72498	0.135573
$405$	1.00000i	0.0496904i
$406$	−10.6972	−0.530894
$407$	−20.2111	−1.00183
$408$	1.18335i	0.0585844i
$409$	6.21110i	0.307119i	0.988139	+	0.153560i	$0.0490737\pi$
$409$	6.21110i	0.307119i	−0.988139	+	0.153560i	$0.950926\pi$
$410$	− 3.90833i	− 0.193019i
$411$	5.60555i	0.276501i
$412$	1.21110	0.0596667
$413$	10.8167	0.532253
$414$	7.81665i	0.384168i
$415$	5.21110	0.255803
$416$	0	0
$417$	13.6056	0.666267
$418$	11.7250i	0.573488i
$419$	−33.2389	−1.62382	−0.811912	−	0.583779i	$-0.801573\pi$
$419$	−33.2389	−1.62382	−0.811912	+	0.583779i	$0.801573\pi$
$420$	0.302776	0.0147739
$421$	− 3.57779i	− 0.174371i	−0.996192	−	0.0871855i	$-0.972213\pi$
$421$	− 3.57779i	− 0.174371i	0.996192	−	0.0871855i	$-0.0277873\pi$
$422$	21.3583i	1.03971i
$423$	10.4222i	0.506745i
$424$	− 33.6333i	− 1.63338i
$425$	0.394449	0.0191336
$426$	21.9083	1.06146
$427$	1.00000i	0.0483934i
$428$	−2.48612	−0.120171
$429$	0	0
$430$	−5.48612	−0.264564
$431$	21.2389i	1.02304i	0.859271	+	0.511520i	$0.170917\pi$
$431$	21.2389i	1.02304i	−0.859271	+	0.511520i	$0.829083\pi$
$432$	16.5139	0.794524
$433$	3.60555	0.173272	0.0866359	−	0.996240i	$-0.472388\pi$
$433$	3.60555	0.173272	0.0866359	+	0.996240i	$0.472388\pi$
$434$	− 5.21110i	− 0.250141i
$435$	8.21110i	0.393692i
$436$	1.44996i	0.0694406i
$437$	− 4.81665i	− 0.230412i
$438$	−19.8167	−0.946876
$439$	−23.2389	−1.10913	−0.554565	−	0.832140i	$-0.687115\pi$
$439$	−23.2389	−1.10913	−0.554565	+	0.832140i	$0.687115\pi$
$440$	− 16.8167i	− 0.801703i
$441$	−12.0000	−0.571429
$442$	0	0
$443$	−22.4222	−1.06531	−0.532656	−	0.846332i	$-0.678806\pi$
$443$	−22.4222	−1.06531	−0.532656	+	0.846332i	$0.678806\pi$
$444$	− 1.09167i	− 0.0518085i
$445$	−8.21110	−0.389244
$446$	−13.3028	−0.629905
$447$	− 3.00000i	− 0.141895i
$448$	8.81665i	0.416548i
$449$	− 12.6333i	− 0.596203i	−0.954534	−	0.298101i	$-0.903647\pi$
$449$	− 12.6333i	− 0.596203i	0.954534	−	0.298101i	$-0.0963533\pi$
$450$	− 2.60555i	− 0.122827i
$451$	−16.8167	−0.791865
$452$	1.69722	0.0798307
$453$	13.2111i	0.620712i
$454$	−1.85281	−0.0869569
$455$	0	0
$456$	−4.81665	−0.225560
$457$	5.18335i	0.242467i	0.992624	+	0.121233i	$0.0386849\pi$
$457$	5.18335i	0.242467i	−0.992624	+	0.121233i	$0.961315\pi$
$458$	18.2389	0.852246
$459$	1.97224	0.0920564
$460$	0.908327i	0.0423510i
$461$	− 21.7889i	− 1.01481i	−0.861708	−	0.507405i	$-0.830605\pi$
$461$	− 21.7889i	− 1.01481i	0.861708	−	0.507405i	$-0.169395\pi$
$462$	7.30278i	0.339756i
$463$	− 5.57779i	− 0.259222i	−0.991565	−	0.129611i	$-0.958627\pi$
$463$	− 5.57779i	− 0.259222i	0.991565	−	0.129611i	$-0.0413729\pi$
$464$	−27.1194	−1.25899
$465$	−4.00000	−0.185496
$466$	1.02776i	0.0476099i
$467$	−17.2111	−0.796435	−0.398217	−	0.917291i	$-0.630371\pi$
$467$	−17.2111	−0.796435	−0.398217	+	0.917291i	$0.630371\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	− 6.78890i	− 0.313148i
$471$	−3.21110	−0.147960
$472$	32.4500	1.49363
$473$	23.6056i	1.08538i
$474$	12.0000i	0.551178i
$475$	1.60555i	0.0736677i
$476$	0.119429i	0.00547404i
$477$	−22.4222	−1.02664
$478$	0	0
$479$	− 7.18335i	− 0.328215i	−0.986442	−	0.164108i	$-0.947526\pi$
$479$	− 7.18335i	− 0.328215i	0.986442	−	0.164108i	$-0.0524745\pi$
$480$	1.69722	0.0774673
$481$	0	0
$482$	21.1194	0.961964
$483$	− 3.00000i	− 0.136505i
$484$	−6.18335	−0.281061
$485$	−15.6056	−0.708612
$486$	− 20.8444i	− 0.945522i
$487$	1.00000i	0.0453143i	0.999743	+	0.0226572i	$0.00721262\pi$
$487$	1.00000i	0.0453143i	−0.999743	+	0.0226572i	$0.992787\pi$
$488$	3.00000i	0.135804i
$489$	− 18.2111i	− 0.823535i
$490$	7.81665	0.353120
$491$	−4.81665	−0.217373	−0.108686	−	0.994076i	$-0.534664\pi$
$491$	−4.81665	−0.217373	−0.108686	+	0.994076i	$0.534664\pi$
$492$	− 0.908327i	− 0.0409505i
$493$	−3.23886	−0.145871
$494$	0	0
$495$	−11.2111	−0.503902
$496$	− 13.2111i	− 0.593196i
$497$	16.8167	0.754330
$498$	−6.78890	−0.304218
$499$	26.4222i	1.18282i	0.806371	+	0.591410i	$0.201429\pi$
$499$	26.4222i	1.18282i	−0.806371	+	0.591410i	$0.798571\pi$
$500$	− 0.302776i	− 0.0135405i
$501$	9.00000i	0.402090i
$502$	− 37.5416i	− 1.67557i
$503$	3.00000	0.133763	0.0668817	−	0.997761i	$-0.478695\pi$
$503$	3.00000	0.133763	0.0668817	+	0.997761i	$0.478695\pi$
$504$	6.00000	0.267261
$505$	9.00000i	0.400495i
$506$	−21.9083	−0.973944
$507$	0	0
$508$	−3.09167	−0.137171
$509$	− 3.00000i	− 0.132973i	−0.997787	−	0.0664863i	$-0.978821\pi$
$509$	− 3.00000i	− 0.132973i	0.997787	−	0.0664863i	$-0.0211789\pi$
$510$	−0.513878	−0.0227549
$511$	−15.2111	−0.672900
$512$	25.4222i	1.12351i
$513$	8.02776i	0.354434i
$514$	− 30.7527i	− 1.35645i
$515$	4.00000i	0.176261i
$516$	−1.27502	−0.0561296
$517$	−29.2111	−1.28470
$518$	4.69722i	0.206384i
$519$	16.8167	0.738169
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	21.3944i	0.936410i
$523$	27.4222	1.19909	0.599545	−	0.800341i	$-0.295348\pi$
$523$	27.4222	1.19909	0.599545	+	0.800341i	$0.295348\pi$
$524$	−2.05551	−0.0897955
$525$	1.00000i	0.0436436i
$526$	− 34.1472i	− 1.48889i
$527$	− 1.57779i	− 0.0687298i
$528$	18.5139i	0.805713i
$529$	−14.0000	−0.608696
$530$	14.6056	0.634425
$531$	− 21.6333i	− 0.938806i
$532$	−0.486122	−0.0210761
$533$	0	0
$534$	10.6972	0.462914
$535$	− 8.21110i	− 0.354997i
$536$	21.0000	0.907062
$537$	−1.18335	−0.0510652
$538$	11.7250i	0.505500i
$539$	− 33.6333i	− 1.44869i
$540$	− 1.51388i	− 0.0651469i
$541$	17.6333i	0.758115i	0.925373	+	0.379058i	$0.123752\pi$
$541$	17.6333i	0.758115i	−0.925373	+	0.379058i	$0.876248\pi$
$542$	1.06392	0.0456991
$543$	25.6333	1.10003
$544$	0.669468i	0.0287032i
$545$	−4.78890	−0.205134
$546$	0	0
$547$	−24.8444	−1.06227	−0.531135	−	0.847287i	$-0.678234\pi$
$547$	−24.8444	−1.06227	−0.531135	+	0.847287i	$0.678234\pi$
$548$	1.69722i	0.0725018i
$549$	2.00000	0.0853579
$550$	7.30278	0.311391
$551$	− 13.1833i	− 0.561629i
$552$	− 9.00000i	− 0.383065i
$553$	9.21110i	0.391696i
$554$	26.5694i	1.12883i
$555$	3.60555	0.153047
$556$	4.11943	0.174703
$557$	5.60555i	0.237515i	0.992923	+	0.118757i	$0.0378911\pi$
$557$	5.60555i	0.237515i	−0.992923	+	0.118757i	$0.962109\pi$
$558$	−10.4222	−0.441207
$559$	0	0
$560$	−3.30278	−0.139568
$561$	2.21110i	0.0933528i
$562$	−7.81665	−0.329726
$563$	19.4222	0.818548	0.409274	−	0.912411i	$-0.365782\pi$
$563$	19.4222	0.818548	0.409274	+	0.912411i	$0.365782\pi$
$564$	− 1.57779i	− 0.0664372i
$565$	5.60555i	0.235827i
$566$	6.51388i	0.273799i
$567$	− 1.00000i	− 0.0419961i
$568$	50.4500	2.11683
$569$	−1.42221	−0.0596219	−0.0298110	−	0.999556i	$-0.509491\pi$
$569$	−1.42221	−0.0596219	−0.0298110	+	0.999556i	$0.509491\pi$
$570$	− 2.09167i	− 0.0876105i
$571$	36.8444	1.54189	0.770945	−	0.636901i	$-0.219784\pi$
$571$	36.8444	1.54189	0.770945	+	0.636901i	$0.219784\pi$
$572$	0	0
$573$	−4.81665	−0.201219
$574$	3.90833i	0.163130i
$575$	−3.00000	−0.125109
$576$	17.6333	0.734721
$577$	− 29.6333i	− 1.23365i	−0.787100	−	0.616825i	$-0.788418\pi$
$577$	− 29.6333i	− 1.23365i	0.787100	−	0.616825i	$-0.211582\pi$
$578$	21.9445i	0.912770i
$579$	8.39445i	0.348861i
$580$	2.48612i	0.103231i
$581$	−5.21110	−0.216193
$582$	20.3305	0.842728
$583$	− 62.8444i	− 2.60275i
$584$	−45.6333	−1.88832
$585$	0	0
$586$	22.9361	0.947481
$587$	4.57779i	0.188946i	0.995527	+	0.0944729i	$0.0301166\pi$
$587$	4.57779i	0.188946i	−0.995527	+	0.0944729i	$0.969883\pi$
$588$	1.81665	0.0749175
$589$	6.42221	0.264622
$590$	14.0917i	0.580145i
$591$	− 22.8167i	− 0.938552i
$592$	11.9083i	0.489429i
$593$	35.2111i	1.44595i	0.690876	+	0.722973i	$0.257225\pi$
$593$	35.2111i	1.44595i	−0.690876	+	0.722973i	$0.742775\pi$
$594$	36.5139	1.49818
$595$	−0.394449	−0.0161708
$596$	− 0.908327i	− 0.0372065i
$597$	8.81665	0.360842
$598$	0	0
$599$	−6.78890	−0.277387	−0.138693	−	0.990335i	$-0.544290\pi$
$599$	−6.78890	−0.277387	−0.138693	+	0.990335i	$0.544290\pi$
$600$	3.00000i	0.122474i
$601$	28.2111	1.15075	0.575377	−	0.817888i	$-0.304855\pi$
$601$	28.2111	1.15075	0.575377	+	0.817888i	$0.304855\pi$
$602$	5.48612	0.223598
$603$	− 14.0000i	− 0.570124i
$604$	4.00000i	0.162758i
$605$	− 20.4222i	− 0.830281i
$606$	− 11.7250i	− 0.476295i
$607$	−19.7889	−0.803207	−0.401603	−	0.915814i	$-0.631547\pi$
$607$	−19.7889	−0.803207	−0.401603	+	0.915814i	$0.631547\pi$
$608$	−2.72498	−0.110513
$609$	− 8.21110i	− 0.332731i
$610$	−1.30278	−0.0527478
$611$	0	0
$612$	0.238859	0.00965530
$613$	− 1.60555i	− 0.0648476i	−0.999474	−	0.0324238i	$-0.989677\pi$
$613$	− 1.60555i	− 0.0648476i	0.999474	−	0.0324238i	$-0.0103226\pi$
$614$	−20.8444	−0.841212
$615$	3.00000	0.120972
$616$	16.8167i	0.677562i
$617$	− 26.4500i	− 1.06484i	−0.846482	−	0.532418i	$-0.821284\pi$
$617$	− 26.4500i	− 1.06484i	0.846482	−	0.532418i	$-0.178716\pi$
$618$	− 5.21110i	− 0.209621i
$619$	− 14.4222i	− 0.579677i	−0.957076	−	0.289839i	$-0.906398\pi$
$619$	− 14.4222i	− 0.579677i	0.957076	−	0.289839i	$-0.0936017\pi$
$620$	−1.21110	−0.0486390
$621$	−15.0000	−0.601929
$622$	6.78890i	0.272210i
$623$	8.21110	0.328971
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 18.2389i	− 0.728971i
$627$	−9.00000	−0.359425
$628$	−0.972244	−0.0387967
$629$	1.42221i	0.0567070i
$630$	2.60555i	0.103808i
$631$	0.0277564i	0.00110496i	1.00000		0.000552482i	$0.000175860\pi$
$631$	0.0277564i	0.00110496i	−1.00000		0.000552482i	$0.999824\pi$
$632$	27.6333i	1.09919i
$633$	−16.3944	−0.651621
$634$	−7.81665	−0.310439
$635$	− 10.2111i	− 0.405215i
$636$	3.39445	0.134599
$637$	0	0
$638$	−59.9638	−2.37399
$639$	− 33.6333i	− 1.33051i
$640$	−8.09167	−0.319851
$641$	19.4222	0.767131	0.383565	−	0.923514i	$-0.374696\pi$
$641$	19.4222	0.767131	0.383565	+	0.923514i	$0.374696\pi$
$642$	10.6972i	0.422186i
$643$	40.6333i	1.60242i	0.598382	+	0.801211i	$0.295810\pi$
$643$	40.6333i	1.60242i	−0.598382	+	0.801211i	$0.704190\pi$
$644$	− 0.908327i	− 0.0357931i
$645$	− 4.21110i	− 0.165812i
$646$	0.825058	0.0324615
$647$	−10.5778	−0.415856	−0.207928	−	0.978144i	$-0.566672\pi$
$647$	−10.5778	−0.415856	−0.207928	+	0.978144i	$0.566672\pi$
$648$	− 3.00000i	− 0.117851i
$649$	60.6333	2.38007
$650$	0	0
$651$	4.00000	0.156772
$652$	− 5.51388i	− 0.215940i
$653$	−28.8167	−1.12768	−0.563841	−	0.825883i	$-0.690677\pi$
$653$	−28.8167	−1.12768	−0.563841	+	0.825883i	$0.690677\pi$
$654$	6.23886	0.243959
$655$	− 6.78890i	− 0.265264i
$656$	9.90833i	0.386855i
$657$	30.4222i	1.18688i
$658$	6.78890i	0.264659i
$659$	−13.1833	−0.513550	−0.256775	−	0.966471i	$-0.582660\pi$
$659$	−13.1833	−0.513550	−0.256775	+	0.966471i	$0.582660\pi$
$660$	1.69722	0.0660644
$661$	38.6333i	1.50266i	0.659926	+	0.751331i	$0.270588\pi$
$661$	38.6333i	1.50266i	−0.659926	+	0.751331i	$0.729412\pi$
$662$	33.9083	1.31788
$663$	0	0
$664$	−15.6333	−0.606690
$665$	− 1.60555i	− 0.0622606i
$666$	9.39445	0.364027
$667$	24.6333	0.953805
$668$	2.72498i	0.105433i
$669$	− 10.2111i	− 0.394784i
$670$	9.11943i	0.352314i
$671$	5.60555i	0.216400i
$672$	−1.69722	−0.0654719
$673$	10.3944	0.400677	0.200338	−	0.979727i	$-0.435796\pi$
$673$	10.3944	0.400677	0.200338	+	0.979727i	$0.435796\pi$
$674$	22.9722i	0.884858i
$675$	5.00000	0.192450
$676$	0	0
$677$	33.6333	1.29263	0.646317	−	0.763069i	$-0.276309\pi$
$677$	33.6333	1.29263	0.646317	+	0.763069i	$0.276309\pi$
$678$	− 7.30278i	− 0.280461i
$679$	15.6056	0.598886
$680$	−1.18335	−0.0453793
$681$	− 1.42221i	− 0.0544990i
$682$	− 29.2111i	− 1.11855i
$683$	21.7889i	0.833729i	0.908969	+	0.416864i	$0.136871\pi$
$683$	21.7889i	0.833729i	−0.908969	+	0.416864i	$0.863129\pi$
$684$	0.972244i	0.0371747i
$685$	−5.60555	−0.214177
$686$	−16.9361	−0.646623
$687$	14.0000i	0.534133i
$688$	13.9083	0.530250
$689$	0	0
$690$	3.90833	0.148787
$691$	− 6.02776i	− 0.229307i	−0.993406	−	0.114653i	$-0.963424\pi$
$691$	− 6.02776i	− 0.229307i	0.993406	−	0.114653i	$-0.0365757\pi$
$692$	5.09167	0.193556
$693$	11.2111	0.425875
$694$	− 26.3305i	− 0.999493i
$695$	13.6056i	0.516088i
$696$	− 24.6333i	− 0.933723i
$697$	1.18335i	0.0448224i
$698$	−23.7250	−0.898004
$699$	−0.788897	−0.0298388
$700$	0.302776i	0.0114438i
$701$	7.57779	0.286209	0.143105	−	0.989708i	$-0.454291\pi$
$701$	7.57779	0.286209	0.143105	+	0.989708i	$0.454291\pi$
$702$	0	0
$703$	−5.78890	−0.218332
$704$	49.4222i	1.86267i
$705$	5.21110	0.196261
$706$	6.27502	0.236163
$707$	− 9.00000i	− 0.338480i
$708$	3.27502i	0.123083i
$709$	43.8444i	1.64661i	0.567598	+	0.823306i	$0.307873\pi$
$709$	43.8444i	1.64661i	−0.567598	+	0.823306i	$0.692127\pi$
$710$	21.9083i	0.822205i
$711$	18.4222	0.690887
$712$	24.6333	0.923172
$713$	12.0000i	0.449404i
$714$	0.513878	0.0192314
$715$	0	0
$716$	−0.358288	−0.0133899
$717$	0	0
$718$	−13.5778	−0.506719
$719$	18.3944	0.685997	0.342999	−	0.939336i	$-0.388557\pi$
$719$	18.3944	0.685997	0.342999	+	0.939336i	$0.388557\pi$
$720$	6.60555i	0.246174i
$721$	− 4.00000i	− 0.148968i
$722$	− 21.3944i	− 0.796219i
$723$	16.2111i	0.602897i
$724$	7.76114	0.288441
$725$	−8.21110	−0.304953
$726$	26.6056i	0.987425i
$727$	−42.4222	−1.57335	−0.786676	−	0.617366i	$-0.788200\pi$
$727$	−42.4222	−1.57335	−0.786676	+	0.617366i	$0.788200\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	− 19.8167i	− 0.733447i
$731$	1.66106	0.0614367
$732$	−0.302776	−0.0111909
$733$	− 10.8444i	− 0.400547i	−0.979740	−	0.200274i	$-0.935817\pi$
$733$	− 10.8444i	− 0.400547i	0.979740	−	0.200274i	$-0.0641831\pi$
$734$	22.6972i	0.837770i
$735$	6.00000i	0.221313i
$736$	− 5.09167i	− 0.187682i
$737$	39.2389	1.44538
$738$	7.81665	0.287735
$739$	− 28.3944i	− 1.04451i	−0.852790	−	0.522253i	$-0.825092\pi$
$739$	− 28.3944i	− 1.04451i	0.852790	−	0.522253i	$-0.174908\pi$
$740$	1.09167	0.0401307
$741$	0	0
$742$	−14.6056	−0.536187
$743$	6.63331i	0.243352i	0.992570	+	0.121676i	$0.0388270\pi$
$743$	6.63331i	0.243352i	−0.992570	+	0.121676i	$0.961173\pi$
$744$	12.0000	0.439941
$745$	3.00000	0.109911
$746$	35.9638i	1.31673i
$747$	10.4222i	0.381329i
$748$	0.669468i	0.0244782i
$749$	8.21110i	0.300027i
$750$	−1.30278	−0.0475706
$751$	18.4500	0.673249	0.336624	−	0.941639i	$-0.390715\pi$
$751$	18.4500	0.673249	0.336624	+	0.941639i	$0.390715\pi$
$752$	17.2111i	0.627624i
$753$	28.8167	1.05014
$754$	0	0
$755$	−13.2111	−0.480801
$756$	1.51388i	0.0550592i
$757$	−20.8167	−0.756594	−0.378297	−	0.925684i	$-0.623490\pi$
$757$	−20.8167	−0.756594	−0.378297	+	0.925684i	$0.623490\pi$
$758$	−3.11943	−0.113303
$759$	− 16.8167i	− 0.610406i
$760$	− 4.81665i	− 0.174718i
$761$	− 24.6333i	− 0.892957i	−0.894794	−	0.446478i	$-0.852678\pi$
$761$	− 24.6333i	− 0.892957i	0.894794	−	0.446478i	$-0.147322\pi$
$762$	13.3028i	0.481909i
$763$	4.78890	0.173370
$764$	−1.45837	−0.0527618
$765$	0.788897i	0.0285226i
$766$	−24.2750	−0.877092
$767$	0	0
$768$	−7.09167	−0.255899
$769$	− 11.0000i	− 0.396670i	−0.980134	−	0.198335i	$-0.936447\pi$
$769$	− 11.0000i	− 0.396670i	0.980134	−	0.198335i	$-0.0635534\pi$
$770$	−7.30278	−0.263174
$771$	23.6056	0.850133
$772$	2.54163i	0.0914754i
$773$	− 29.6056i	− 1.06484i	−0.846481	−	0.532419i	$-0.821283\pi$
$773$	− 29.6056i	− 1.06484i	0.846481	−	0.532419i	$-0.178717\pi$
$774$	− 10.9722i	− 0.394389i
$775$	− 4.00000i	− 0.143684i
$776$	46.8167	1.68062
$777$	−3.60555	−0.129348
$778$	− 1.02776i	− 0.0368469i
$779$	−4.81665	−0.172575
$780$	0	0
$781$	94.2666	3.37312
$782$	1.54163i	0.0551287i
$783$	−41.0555	−1.46720
$784$	−19.8167	−0.707738
$785$	− 3.21110i	− 0.114609i
$786$	8.84441i	0.315470i
$787$	− 28.6333i	− 1.02067i	−0.859977	−	0.510334i	$-0.829522\pi$
$787$	− 28.6333i	− 1.02067i	0.859977	−	0.510334i	$-0.170478\pi$
$788$	− 6.90833i	− 0.246099i
$789$	26.2111	0.933140
$790$	−12.0000	−0.426941
$791$	− 5.60555i	− 0.199310i
$792$	33.6333	1.19511
$793$	0	0
$794$	−18.2750	−0.648556
$795$	11.2111i	0.397617i
$796$	2.66947	0.0946168
$797$	−50.4500	−1.78703	−0.893515	−	0.449034i	$-0.851768\pi$
$797$	−50.4500	−1.78703	−0.893515	+	0.449034i	$0.851768\pi$
$798$	2.09167i	0.0740444i
$799$	2.05551i	0.0727188i
$800$	1.69722i	0.0600059i
$801$	− 16.4222i	− 0.580250i
$802$	−2.88057	−0.101716
$803$	−85.2666	−3.00899
$804$	2.11943i	0.0747465i
$805$	3.00000	0.105736
$806$	0	0
$807$	−9.00000	−0.316815
$808$	− 27.0000i	− 0.949857i
$809$	17.0555	0.599640	0.299820	−	0.953996i	$-0.403073\pi$
$809$	17.0555	0.599640	0.299820	+	0.953996i	$0.403073\pi$
$810$	1.30278	0.0457749
$811$	17.5778i	0.617240i	0.951185	+	0.308620i	$0.0998671\pi$
$811$	17.5778i	0.617240i	−0.951185	+	0.308620i	$0.900133\pi$
$812$	− 2.48612i	− 0.0872458i
$813$	0.816654i	0.0286413i
$814$	26.3305i	0.922885i
$815$	18.2111	0.637907
$816$	1.30278	0.0456063
$817$	6.76114i	0.236542i
$818$	8.09167	0.282919
$819$	0	0
$820$	0.908327	0.0317202
$821$	7.42221i	0.259037i	0.991577	+	0.129518i	$0.0413431\pi$
$821$	7.42221i	0.259037i	−0.991577	+	0.129518i	$0.958657\pi$
$822$	7.30278	0.254714
$823$	−26.6333	−0.928379	−0.464189	−	0.885736i	$-0.653654\pi$
$823$	−26.6333	−0.928379	−0.464189	+	0.885736i	$0.653654\pi$
$824$	− 12.0000i	− 0.418040i
$825$	5.60555i	0.195160i
$826$	− 14.0917i	− 0.490312i
$827$	− 13.5778i	− 0.472146i	−0.971735	−	0.236073i	$-0.924140\pi$
$827$	− 13.5778i	− 0.472146i	0.971735	−	0.236073i	$-0.0758605\pi$
$828$	−1.81665	−0.0631331
$829$	−0.577795	−0.0200676	−0.0100338	−	0.999950i	$-0.503194\pi$
$829$	−0.577795	−0.0200676	−0.0100338	+	0.999950i	$0.503194\pi$
$830$	− 6.78890i	− 0.235646i
$831$	−20.3944	−0.707476
$832$	0	0
$833$	−2.36669	−0.0820010
$834$	− 17.7250i	− 0.613766i
$835$	−9.00000	−0.311458
$836$	−2.72498	−0.0942454
$837$	− 20.0000i	− 0.691301i
$838$	43.3028i	1.49587i
$839$	16.0278i	0.553340i	0.960965	+	0.276670i	$0.0892308\pi$
$839$	16.0278i	0.553340i	−0.960965	+	0.276670i	$0.910769\pi$
$840$	− 3.00000i	− 0.103510i
$841$	38.4222	1.32490
$842$	−4.66106	−0.160631
$843$	− 6.00000i	− 0.206651i
$844$	−4.96384	−0.170862
$845$	0	0
$846$	13.5778	0.466814
$847$	20.4222i	0.701715i
$848$	−37.0278	−1.27154
$849$	−5.00000	−0.171600
$850$	− 0.513878i	− 0.0176259i
$851$	− 10.8167i	− 0.370790i
$852$	5.09167i	0.174438i
$853$	32.7889i	1.12267i	0.827589	+	0.561335i	$0.189712\pi$
$853$	32.7889i	1.12267i	−0.827589	+	0.561335i	$0.810288\pi$
$854$	1.30278	0.0445801
$855$	−3.21110	−0.109817
$856$	24.6333i	0.841949i
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	25.2111	0.860192	0.430096	−	0.902783i	$-0.358480\pi$
$859$	25.2111	0.860192	0.430096	+	0.902783i	$0.358480\pi$
$860$	− 1.27502i	− 0.0434778i
$861$	−3.00000	−0.102240
$862$	27.6695	0.942426
$863$	− 36.0000i	− 1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$	− 36.0000i	− 1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$	8.48612i	0.288704i
$865$	16.8167i	0.571783i
$866$	− 4.69722i	− 0.159618i
$867$	−16.8444	−0.572066
$868$	1.21110	0.0411075
$869$	51.6333i	1.75154i
$870$	10.6972	0.362670
$871$	0	0
$872$	14.3667	0.486518
$873$	− 31.2111i	− 1.05634i
$874$	−6.27502	−0.212256
$875$	−1.00000	−0.0338062
$876$	− 4.60555i	− 0.155607i
$877$	38.0278i	1.28411i	0.766660	+	0.642053i	$0.221917\pi$
$877$	38.0278i	1.28411i	−0.766660	+	0.642053i	$0.778083\pi$
$878$	30.2750i	1.02173i
$879$	17.6056i	0.593821i
$880$	−18.5139	−0.624103
$881$	−35.8444	−1.20763	−0.603814	−	0.797125i	$-0.706353\pi$
$881$	−35.8444	−1.20763	−0.603814	+	0.797125i	$0.706353\pi$
$882$	15.6333i	0.526401i
$883$	31.6333	1.06455	0.532273	−	0.846573i	$-0.321338\pi$
$883$	31.6333	1.06455	0.532273	+	0.846573i	$0.321338\pi$
$884$	0	0
$885$	−10.8167	−0.363598
$886$	29.2111i	0.981366i
$887$	35.0555	1.17705	0.588524	−	0.808479i	$-0.299709\pi$
$887$	35.0555	1.17705	0.588524	+	0.808479i	$0.299709\pi$
$888$	−10.8167	−0.362983
$889$	10.2111i	0.342469i
$890$	10.6972i	0.358572i
$891$	− 5.60555i	− 0.187793i
$892$	− 3.09167i	− 0.103517i
$893$	−8.36669	−0.279981
$894$	−3.90833	−0.130714
$895$	− 1.18335i	− 0.0395549i
$896$	8.09167	0.270324
$897$	0	0
$898$	−16.4584	−0.549223
$899$	32.8444i	1.09542i
$900$	0.605551	0.0201850
$901$	−4.42221	−0.147325
$902$	21.9083i	0.729467i
$903$	4.21110i	0.140137i
$904$	− 16.8167i	− 0.559314i
$905$	25.6333i	0.852080i
$906$	17.2111	0.571801
$907$	−48.2666	−1.60267	−0.801333	−	0.598218i	$-0.795876\pi$
$907$	−48.2666	−1.60267	−0.801333	+	0.598218i	$0.795876\pi$
$908$	− 0.430609i	− 0.0142903i
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	5.30278i	0.175592i
$913$	−29.2111	−0.966746
$914$	6.75274	0.223361
$915$	− 1.00000i	− 0.0330590i
$916$	4.23886i	0.140056i
$917$	6.78890i	0.224189i
$918$	− 2.56939i	− 0.0848025i
$919$	−17.1833	−0.566826	−0.283413	−	0.958998i	$-0.591467\pi$
$919$	−17.1833	−0.566826	−0.283413	+	0.958998i	$0.591467\pi$
$920$	9.00000	0.296721
$921$	− 16.0000i	− 0.527218i
$922$	−28.3860	−0.934845
$923$	0	0
$924$	−1.69722	−0.0558346
$925$	3.60555i	0.118550i
$926$	−7.26662	−0.238796
$927$	−8.00000	−0.262754
$928$	− 13.9361i	− 0.457474i
$929$	− 13.4222i	− 0.440368i	−0.975458	−	0.220184i	$-0.929334\pi$
$929$	− 13.4222i	− 0.440368i	0.975458	−	0.220184i	$-0.0706658\pi$
$930$	5.21110i	0.170879i
$931$	− 9.63331i	− 0.315719i
$932$	−0.238859	−0.00782408
$933$	−5.21110	−0.170604
$934$	22.4222i	0.733677i
$935$	−2.21110	−0.0723108
$936$	0	0
$937$	−46.4777	−1.51836	−0.759180	−	0.650880i	$-0.774400\pi$
$937$	−46.4777	−1.51836	−0.759180	+	0.650880i	$0.774400\pi$
$938$	− 9.11943i	− 0.297760i
$939$	14.0000	0.456873
$940$	1.57779	0.0514620
$941$	− 33.6333i	− 1.09641i	−0.836343	−	0.548207i	$-0.815311\pi$
$941$	− 33.6333i	− 1.09641i	0.836343	−	0.548207i	$-0.184689\pi$
$942$	4.18335i	0.136301i
$943$	− 9.00000i	− 0.293080i
$944$	− 35.7250i	− 1.16275i
$945$	−5.00000	−0.162650
$946$	30.7527	0.999858
$947$	24.6333i	0.800475i	0.916411	+	0.400237i	$0.131072\pi$
$947$	24.6333i	0.800475i	−0.916411	+	0.400237i	$0.868928\pi$
$948$	−2.78890	−0.0905792
$949$	0	0
$950$	2.09167	0.0678628
$951$	− 6.00000i	− 0.194563i
$952$	1.18335	0.0383525
$953$	−50.4500	−1.63423	−0.817117	−	0.576471i	$-0.804429\pi$
$953$	−50.4500	−1.63423	−0.817117	+	0.576471i	$0.804429\pi$
$954$	29.2111i	0.945744i
$955$	− 4.81665i	− 0.155863i
$956$	0	0
$957$	− 46.0278i	− 1.48787i
$958$	−9.35829	−0.302353
$959$	5.60555	0.181013
$960$	− 8.81665i	− 0.284556i
$961$	15.0000	0.483871
$962$	0	0
$963$	16.4222	0.529198
$964$	4.90833i	0.158087i
$965$	−8.39445	−0.270227
$966$	−3.90833	−0.125748
$967$	− 56.4777i	− 1.81620i	−0.418752	−	0.908100i	$-0.637532\pi$
$967$	− 56.4777i	− 1.81620i	0.418752	−	0.908100i	$-0.362468\pi$
$968$	61.2666i	1.96918i
$969$	0.633308i	0.0203448i
$970$	20.3305i	0.652774i
$971$	−7.97224	−0.255841	−0.127921	−	0.991784i	$-0.540830\pi$
$971$	−7.97224	−0.255841	−0.127921	+	0.991784i	$0.540830\pi$
$972$	4.84441	0.155385
$973$	− 13.6056i	− 0.436174i
$974$	1.30278	0.0417436
$975$	0	0
$976$	3.30278	0.105719
$977$	− 7.18335i	− 0.229816i	−0.993376	−	0.114908i	$-0.963343\pi$
$977$	− 7.18335i	− 0.229816i	0.993376	−	0.114908i	$-0.0366573\pi$
$978$	−23.7250	−0.758641
$979$	46.0278	1.47105
$980$	1.81665i	0.0580309i
$981$	− 9.57779i	− 0.305795i
$982$	6.27502i	0.200244i
$983$	− 10.4222i	− 0.332417i	−0.986091	−	0.166208i	$-0.946848\pi$
$983$	− 10.4222i	− 0.332417i	0.986091	−	0.166208i	$-0.0531524\pi$
$984$	−9.00000	−0.286910
$985$	22.8167	0.726999
$986$	4.21951i	0.134376i
$987$	−5.21110	−0.165871
$988$	0	0
$989$	−12.6333	−0.401716
$990$	14.6056i	0.464195i
$991$	3.97224	0.126182	0.0630912	−	0.998008i	$-0.479904\pi$
$991$	3.97224	0.126182	0.0630912	+	0.998008i	$0.479904\pi$
$992$	6.78890	0.215548
$993$	26.0278i	0.825966i
$994$	− 21.9083i	− 0.694890i
$995$	8.81665i	0.279507i
$996$	− 1.57779i	− 0.0499943i
$997$	46.4500	1.47109	0.735543	−	0.677479i	$-0.236927\pi$
$997$	46.4500	1.47109	0.735543	+	0.677479i	$0.236927\pi$
$998$	34.4222	1.08962
$999$	18.0278i	0.570373i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.c.d.506.2		4
13.2	odd	12		845.2.e.d.191.2		4
13.3	even	3		845.2.m.d.316.3		8
13.4	even	6		845.2.m.d.361.3		8
13.5	odd	4		845.2.a.f.1.1		2
13.6	odd	12		845.2.e.d.146.2		4
13.7	odd	12		65.2.e.b.16.1	✓	4
13.8	odd	4		845.2.a.c.1.2		2
13.9	even	3		845.2.m.d.361.2		8
13.10	even	6		845.2.m.d.316.2		8
13.11	odd	12		65.2.e.b.61.1	yes	4
13.12	even	2	inner	845.2.c.d.506.3		4
39.5	even	4		7605.2.a.bb.1.2		2
39.8	even	4		7605.2.a.bg.1.1		2
39.11	even	12		585.2.j.d.451.2		4
39.20	even	12		585.2.j.d.406.2		4
52.7	even	12		1040.2.q.o.81.1		4
52.11	even	12		1040.2.q.o.321.1		4
65.7	even	12		325.2.o.b.224.3		8
65.24	odd	12		325.2.e.a.126.2		4
65.33	even	12		325.2.o.b.224.2		8
65.34	odd	4		4225.2.a.x.1.1		2
65.37	even	12		325.2.o.b.74.2		8
65.44	odd	4		4225.2.a.t.1.2		2
65.59	odd	12		325.2.e.a.276.2		4
65.63	even	12		325.2.o.b.74.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.e.b.16.1	✓	4	13.7	odd	12
65.2.e.b.61.1	yes	4	13.11	odd	12
325.2.e.a.126.2		4	65.24	odd	12
325.2.e.a.276.2		4	65.59	odd	12
325.2.o.b.74.2		8	65.37	even	12
325.2.o.b.74.3		8	65.63	even	12
325.2.o.b.224.2		8	65.33	even	12
325.2.o.b.224.3		8	65.7	even	12
585.2.j.d.406.2		4	39.20	even	12
585.2.j.d.451.2		4	39.11	even	12
845.2.a.c.1.2		2	13.8	odd	4
845.2.a.f.1.1		2	13.5	odd	4
845.2.c.d.506.2		4	1.1	even	1	trivial
845.2.c.d.506.3		4	13.12	even	2	inner
845.2.e.d.146.2		4	13.6	odd	12
845.2.e.d.191.2		4	13.2	odd	12
845.2.m.d.316.2		8	13.10	even	6
845.2.m.d.316.3		8	13.3	even	3
845.2.m.d.361.2		8	13.9	even	3
845.2.m.d.361.3		8	13.4	even	6
1040.2.q.o.81.1		4	52.7	even	12
1040.2.q.o.321.1		4	52.11	even	12
4225.2.a.t.1.2		2	65.44	odd	4
4225.2.a.x.1.1		2	65.34	odd	4
7605.2.a.bb.1.2		2	39.5	even	4
7605.2.a.bg.1.1		2	39.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.30278i q^{2} +1.00000 q^{3} +0.302776 q^{4} +1.00000i q^{5} -1.30278i q^{6} -1.00000i q^{7} -3.00000i q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q-1.30278i q^{2} +1.00000 q^{3} +0.302776 q^{4} +1.00000i q^{5} -1.30278i q^{6} -1.00000i q^{7} -3.00000i q^{8} -2.00000 q^{9} +1.30278 q^{10} -5.60555i q^{11} +0.302776 q^{12} -1.30278 q^{14} +1.00000i q^{15} -3.30278 q^{16} -0.394449 q^{17} +2.60555i q^{18} -1.60555i q^{19} +0.302776i q^{20} -1.00000i q^{21} -7.30278 q^{22} +3.00000 q^{23} -3.00000i q^{24} -1.00000 q^{25} -5.00000 q^{27} -0.302776i q^{28} +8.21110 q^{29} +1.30278 q^{30} +4.00000i q^{31} -1.69722i q^{32} -5.60555i q^{33} +0.513878i q^{34} +1.00000 q^{35} -0.605551 q^{36} -3.60555i q^{37} -2.09167 q^{38} +3.00000 q^{40} -3.00000i q^{41} -1.30278 q^{42} -4.21110 q^{43} -1.69722i q^{44} -2.00000i q^{45} -3.90833i q^{46} -5.21110i q^{47} -3.30278 q^{48} +6.00000 q^{49} +1.30278i q^{50} -0.394449 q^{51} +11.2111 q^{53} +6.51388i q^{54} +5.60555 q^{55} -3.00000 q^{56} -1.60555i q^{57} -10.6972i q^{58} +10.8167i q^{59} +0.302776i q^{60} -1.00000 q^{61} +5.21110 q^{62} +2.00000i q^{63} -8.81665 q^{64} -7.30278 q^{66} +7.00000i q^{67} -0.119429 q^{68} +3.00000 q^{69} -1.30278i q^{70} +16.8167i q^{71} +6.00000i q^{72} -15.2111i q^{73} -4.69722 q^{74} -1.00000 q^{75} -0.486122i q^{76} -5.60555 q^{77} -9.21110 q^{79} -3.30278i q^{80} +1.00000 q^{81} -3.90833 q^{82} -5.21110i q^{83} -0.302776i q^{84} -0.394449i q^{85} +5.48612i q^{86} +8.21110 q^{87} -16.8167 q^{88} +8.21110i q^{89} -2.60555 q^{90} +0.908327 q^{92} +4.00000i q^{93} -6.78890 q^{94} +1.60555 q^{95} -1.69722i q^{96} +15.6056i q^{97} -7.81665i q^{98} +11.2111i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 6 q^{4} - 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 6 * q^4 - 8 * q^9	\( 4 q + 4 q^{3} - 6 q^{4} - 8 q^{9} - 2 q^{10} - 6 q^{12} + 2 q^{14} - 6 q^{16} - 16 q^{17} - 22 q^{22} + 12 q^{23} - 4 q^{25} - 20 q^{27} + 4 q^{29} - 2 q^{30} + 4 q^{35} + 12 q^{36} - 30 q^{38} + 12 q^{40} + 2 q^{42} + 12 q^{43} - 6 q^{48} + 24 q^{49} - 16 q^{51} + 16 q^{53} + 8 q^{55} - 12 q^{56} - 4 q^{61} - 8 q^{62} + 8 q^{64} - 22 q^{66} + 50 q^{68} + 12 q^{69} - 26 q^{74} - 4 q^{75} - 8 q^{77} - 8 q^{79} + 4 q^{81} + 6 q^{82} + 4 q^{87} - 24 q^{88} + 4 q^{90} - 18 q^{92} - 56 q^{94} - 8 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 6 * q^4 - 8 * q^9 - 2 * q^10 - 6 * q^12 + 2 * q^14 - 6 * q^16 - 16 * q^17 - 22 * q^22 + 12 * q^23 - 4 * q^25 - 20 * q^27 + 4 * q^29 - 2 * q^30 + 4 * q^35 + 12 * q^36 - 30 * q^38 + 12 * q^40 + 2 * q^42 + 12 * q^43 - 6 * q^48 + 24 * q^49 - 16 * q^51 + 16 * q^53 + 8 * q^55 - 12 * q^56 - 4 * q^61 - 8 * q^62 + 8 * q^64 - 22 * q^66 + 50 * q^68 + 12 * q^69 - 26 * q^74 - 4 * q^75 - 8 * q^77 - 8 * q^79 + 4 * q^81 + 6 * q^82 + 4 * q^87 - 24 * q^88 + 4 * q^90 - 18 * q^92 - 56 * q^94 - 8 * q^95

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