LMFDB - Embedded newform 5070.2.b.ba.1351.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.b (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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.17284886784.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 30x^{5} + 185x^{4} + 36x^{3} + 8x^{2} + 208x + 2704 $ x^8 - 2x^7 + 2x^6 + 30x^5 + 185x^4 + 36x^3 + 8x^2 + 208*x + 2704
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.6
Root			$3.17270 - 3.17270i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.ba.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -2.32258i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -2.32258i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.34541i q^{11} -1.00000 q^{12} +2.32258 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000i q^{18} -4.02283i q^{19} +1.00000i q^{20} -2.32258i q^{21} +5.34541 q^{22} +4.93593 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +2.32258i q^{28} +4.29078 q^{29} +1.00000 q^{30} +3.47183i q^{31} +1.00000i q^{32} -5.34541i q^{33} -4.00000i q^{34} -2.32258 q^{35} -1.00000 q^{36} +3.14925i q^{37} +4.02283 q^{38} -1.00000 q^{40} -2.64516i q^{41} +2.32258 q^{42} -12.2508 q^{43} +5.34541i q^{44} -1.00000i q^{45} +4.93593i q^{46} -1.81894i q^{47} +1.00000 q^{48} +1.60562 q^{49} -1.00000i q^{50} -4.00000 q^{51} -5.48693 q^{53} +1.00000i q^{54} -5.34541 q^{55} -2.32258 q^{56} -4.02283i q^{57} +4.29078i q^{58} +6.78668i q^{59} +1.00000i q^{60} +0.535898 q^{61} -3.47183 q^{62} -2.32258i q^{63} -1.00000 q^{64} +5.34541 q^{66} -4.10926i q^{67} +4.00000 q^{68} +4.93593 q^{69} -2.32258i q^{70} +15.8719i q^{71} -1.00000i q^{72} -13.5734i q^{73} -3.14925 q^{74} -1.00000 q^{75} +4.02283i q^{76} -12.4151 q^{77} -7.96774 q^{79} -1.00000i q^{80} +1.00000 q^{81} +2.64516 q^{82} -11.3360i q^{83} +2.32258i q^{84} +4.00000i q^{85} -12.2508i q^{86} +4.29078 q^{87} -5.34541 q^{88} +1.73978i q^{89} +1.00000 q^{90} -4.93593 q^{92} +3.47183i q^{93} +1.81894 q^{94} -4.02283 q^{95} +1.00000i q^{96} -16.1093i q^{97} +1.60562i q^{98} -5.34541i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 8 q^{4} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 - 8 * q^4 + 8 * q^9	$ 8 q + 8 q^{3} - 8 q^{4} + 8 q^{9} + 8 q^{10} - 8 q^{12} + 4 q^{14} + 8 q^{16} - 32 q^{17} - 4 q^{22} - 8 q^{23} - 8 q^{25} + 8 q^{27} + 16 q^{29} + 8 q^{30} - 4 q^{35} - 8 q^{36} - 8 q^{40} + 4 q^{42} - 28 q^{43} + 8 q^{48} - 28 q^{49} - 32 q^{51} + 16 q^{53} + 4 q^{55} - 4 q^{56} + 32 q^{61} - 8 q^{62} - 8 q^{64} - 4 q^{66} + 32 q^{68} - 8 q^{69} - 20 q^{74} - 8 q^{75} + 16 q^{77} - 20 q^{79} + 8 q^{81} - 8 q^{82} + 16 q^{87} + 4 q^{88} + 8 q^{90} + 8 q^{92} + 16 q^{94}+O(q^{100}) $ 8 * q + 8 * q^3 - 8 * q^4 + 8 * q^9 + 8 * q^10 - 8 * q^12 + 4 * q^14 + 8 * q^16 - 32 * q^17 - 4 * q^22 - 8 * q^23 - 8 * q^25 + 8 * q^27 + 16 * q^29 + 8 * q^30 - 4 * q^35 - 8 * q^36 - 8 * q^40 + 4 * q^42 - 28 * q^43 + 8 * q^48 - 28 * q^49 - 32 * q^51 + 16 * q^53 + 4 * q^55 - 4 * q^56 + 32 * q^61 - 8 * q^62 - 8 * q^64 - 4 * q^66 + 32 * q^68 - 8 * q^69 - 20 * q^74 - 8 * q^75 + 16 * q^77 - 20 * q^79 + 8 * q^81 - 8 * q^82 + 16 * q^87 + 4 * q^88 + 8 * q^90 + 8 * q^92 + 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	1.00000i	0.408248i
$7$	− 2.32258i	− 0.877853i	−0.898523	−	0.438926i	$-0.855359\pi$
$7$	− 2.32258i	− 0.877853i	0.898523	−	0.438926i	$-0.144641\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	− 5.34541i	− 1.61170i	−0.592119	−	0.805850i	$-0.701709\pi$
$11$	− 5.34541i	− 1.61170i	0.592119	−	0.805850i	$-0.298291\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	2.32258	0.620736
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	1.00000i	0.235702i
$19$	− 4.02283i	− 0.922900i	−0.887166	−	0.461450i	$-0.847329\pi$
$19$	− 4.02283i	− 0.922900i	0.887166	−	0.461450i	$-0.152671\pi$
$20$	1.00000i	0.223607i
$21$	− 2.32258i	− 0.506828i
$22$	5.34541	1.13964
$23$	4.93593	1.02921	0.514607	−	0.857426i	$-0.327938\pi$
$23$	4.93593	1.02921	0.514607	+	0.857426i	$0.327938\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	2.32258i	0.438926i
$29$	4.29078	0.796777	0.398388	−	0.917217i	$-0.369570\pi$
$29$	4.29078	0.796777	0.398388	+	0.917217i	$0.369570\pi$
$30$	1.00000	0.182574
$31$	3.47183i	0.623560i	0.950154	+	0.311780i	$0.100925\pi$
$31$	3.47183i	0.623560i	−0.950154	+	0.311780i	$0.899075\pi$
$32$	1.00000i	0.176777i
$33$	− 5.34541i	− 0.930516i
$34$	− 4.00000i	− 0.685994i
$35$	−2.32258	−0.392588
$36$	−1.00000	−0.166667
$37$	3.14925i	0.517734i	0.965913	+	0.258867i	$0.0833491\pi$
$37$	3.14925i	0.517734i	−0.965913	+	0.258867i	$0.916651\pi$
$38$	4.02283	0.652589
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 2.64516i	− 0.413104i	−0.978436	−	0.206552i	$-0.933776\pi$
$41$	− 2.64516i	− 0.413104i	0.978436	−	0.206552i	$-0.0662243\pi$
$42$	2.32258	0.358382
$43$	−12.2508	−1.86823	−0.934113	−	0.356976i	$-0.883808\pi$
$43$	−12.2508	−1.86823	−0.934113	+	0.356976i	$0.883808\pi$
$44$	5.34541i	0.805850i
$45$	− 1.00000i	− 0.149071i
$46$	4.93593i	0.727764i
$47$	− 1.81894i	− 0.265320i	−0.991162	−	0.132660i	$-0.957648\pi$
$47$	− 1.81894i	− 0.265320i	0.991162	−	0.132660i	$-0.0423518\pi$
$48$	1.00000	0.144338
$49$	1.60562	0.229375
$50$	− 1.00000i	− 0.141421i
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−5.48693	−0.753687	−0.376844	−	0.926277i	$-0.622991\pi$
$53$	−5.48693	−0.753687	−0.376844	+	0.926277i	$0.622991\pi$
$54$	1.00000i	0.136083i
$55$	−5.34541	−0.720774
$56$	−2.32258	−0.310368
$57$	− 4.02283i	− 0.532836i
$58$	4.29078i	0.563406i
$59$	6.78668i	0.883551i	0.897126	+	0.441775i	$0.145651\pi$
$59$	6.78668i	0.883551i	−0.897126	+	0.441775i	$0.854349\pi$
$60$	1.00000i	0.129099i
$61$	0.535898	0.0686148	0.0343074	−	0.999411i	$-0.489077\pi$
$61$	0.535898	0.0686148	0.0343074	+	0.999411i	$0.489077\pi$
$62$	−3.47183	−0.440923
$63$	− 2.32258i	− 0.292618i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	5.34541	0.657974
$67$	− 4.10926i	− 0.502026i	−0.967984	−	0.251013i	$-0.919236\pi$
$67$	− 4.10926i	− 0.502026i	0.967984	−	0.251013i	$-0.0807637\pi$
$68$	4.00000	0.485071
$69$	4.93593	0.594217
$70$	− 2.32258i	− 0.277601i
$71$	15.8719i	1.88364i	0.336112	+	0.941822i	$0.390888\pi$
$71$	15.8719i	1.88364i	−0.336112	+	0.941822i	$0.609112\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 13.5734i	− 1.58864i	−0.607498	−	0.794321i	$-0.707827\pi$
$73$	− 13.5734i	− 1.58864i	0.607498	−	0.794321i	$-0.292173\pi$
$74$	−3.14925	−0.366093
$75$	−1.00000	−0.115470
$76$	4.02283i	0.461450i
$77$	−12.4151	−1.41484
$78$	0	0
$79$	−7.96774	−0.896441	−0.448220	−	0.893923i	$-0.647942\pi$
$79$	−7.96774	−0.896441	−0.448220	+	0.893923i	$0.647942\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	2.64516	0.292109
$83$	− 11.3360i	− 1.24428i	−0.782904	−	0.622142i	$-0.786263\pi$
$83$	− 11.3360i	− 1.24428i	0.782904	−	0.622142i	$-0.213737\pi$
$84$	2.32258i	0.253414i
$85$	4.00000i	0.433861i
$86$	− 12.2508i	− 1.32104i
$87$	4.29078	0.460019
$88$	−5.34541	−0.569822
$89$	1.73978i	0.184417i	0.995740	+	0.0922083i	$0.0293926\pi$
$89$	1.73978i	0.184417i	−0.995740	+	0.0922083i	$0.970607\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−4.93593	−0.514607
$93$	3.47183i	0.360012i
$94$	1.81894	0.187610
$95$	−4.02283	−0.412733
$96$	1.00000i	0.102062i
$97$	− 16.1093i	− 1.63565i	−0.575469	−	0.817824i	$-0.695180\pi$
$97$	− 16.1093i	− 1.63565i	0.575469	−	0.817824i	$-0.304820\pi$
$98$	1.60562i	0.162192i
$99$	− 5.34541i	− 0.537233i
$100$	1.00000	0.100000
$101$	−12.6908	−1.26278	−0.631391	−	0.775464i	$-0.717516\pi$
$101$	−12.6908	−1.26278	−0.631391	+	0.775464i	$0.717516\pi$
$102$	− 4.00000i	− 0.396059i
$103$	−4.79612	−0.472575	−0.236288	−	0.971683i	$-0.575931\pi$
$103$	−4.79612	−0.472575	−0.236288	+	0.971683i	$0.575931\pi$
$104$	0	0
$105$	−2.32258	−0.226661
$106$	− 5.48693i	− 0.532938i
$107$	3.07180	0.296962	0.148481	−	0.988915i	$-0.452562\pi$
$107$	3.07180	0.296962	0.148481	+	0.988915i	$0.452562\pi$
$108$	−1.00000	−0.0962250
$109$	− 6.69081i	− 0.640864i	−0.947272	−	0.320432i	$-0.896172\pi$
$109$	− 6.69081i	− 0.640864i	0.947272	−	0.320432i	$-0.103828\pi$
$110$	− 5.34541i	− 0.509664i
$111$	3.14925i	0.298914i
$112$	− 2.32258i	− 0.219463i
$113$	−7.10972	−0.668826	−0.334413	−	0.942427i	$-0.608538\pi$
$113$	−7.10972	−0.668826	−0.334413	+	0.942427i	$0.608538\pi$
$114$	4.02283	0.376772
$115$	− 4.93593i	− 0.460278i
$116$	−4.29078	−0.398388
$117$	0	0
$118$	−6.78668	−0.624765
$119$	9.29032i	0.851642i
$120$	−1.00000	−0.0912871
$121$	−17.5734	−1.59758
$122$	0.535898i	0.0485180i
$123$	− 2.64516i	− 0.238506i
$124$	− 3.47183i	− 0.311780i
$125$	1.00000i	0.0894427i
$126$	2.32258	0.206912
$127$	2.13209	0.189192	0.0945961	−	0.995516i	$-0.469844\pi$
$127$	2.13209	0.189192	0.0945961	+	0.995516i	$0.469844\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−12.2508	−1.07862
$130$	0	0
$131$	1.25851	0.109957	0.0549785	−	0.998488i	$-0.482491\pi$
$131$	1.25851	0.109957	0.0549785	+	0.998488i	$0.482491\pi$
$132$	5.34541i	0.465258i
$133$	−9.34333	−0.810170
$134$	4.10926	0.354986
$135$	− 1.00000i	− 0.0860663i
$136$	4.00000i	0.342997i
$137$	19.7149i	1.68436i	0.539199	+	0.842178i	$0.318727\pi$
$137$	19.7149i	1.68436i	−0.539199	+	0.842178i	$0.681273\pi$
$138$	4.93593i	0.420175i
$139$	5.67742	0.481553	0.240776	−	0.970581i	$-0.422598\pi$
$139$	5.67742	0.481553	0.240776	+	0.970581i	$0.422598\pi$
$140$	2.32258	0.196294
$141$	− 1.81894i	− 0.153183i
$142$	−15.8719	−1.33194
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 4.29078i	− 0.356329i
$146$	13.5734	1.12334
$147$	1.60562	0.132430
$148$	− 3.14925i	− 0.258867i
$149$	− 19.4775i	− 1.59566i	−0.602884	−	0.797829i	$-0.705982\pi$
$149$	− 19.4775i	− 1.59566i	0.602884	−	0.797829i	$-0.294018\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	14.5170i	1.18138i	0.806899	+	0.590690i	$0.201144\pi$
$151$	14.5170i	1.18138i	−0.806899	+	0.590690i	$0.798856\pi$
$152$	−4.02283	−0.326294
$153$	−4.00000	−0.323381
$154$	− 12.4151i	− 1.00044i
$155$	3.47183	0.278864
$156$	0	0
$157$	24.3829	1.94596	0.972982	−	0.230879i	$-0.0741602\pi$
$157$	24.3829	1.94596	0.972982	+	0.230879i	$0.0741602\pi$
$158$	− 7.96774i	− 0.633879i
$159$	−5.48693	−0.435142
$160$	1.00000	0.0790569
$161$	− 11.4641i	− 0.903498i
$162$	1.00000i	0.0785674i
$163$	23.6267i	1.85059i	0.379249	+	0.925295i	$0.376182\pi$
$163$	23.6267i	1.85059i	−0.379249	+	0.925295i	$0.623818\pi$
$164$	2.64516i	0.206552i
$165$	−5.34541	−0.416139
$166$	11.3360	0.879842
$167$	− 16.7471i	− 1.29593i	−0.761669	−	0.647967i	$-0.775620\pi$
$167$	− 16.7471i	− 1.29593i	0.761669	−	0.647967i	$-0.224380\pi$
$168$	−2.32258	−0.179191
$169$	0	0
$170$	−4.00000	−0.306786
$171$	− 4.02283i	− 0.307633i
$172$	12.2508	0.934113
$173$	−16.1321	−1.22650	−0.613250	−	0.789889i	$-0.710138\pi$
$173$	−16.1321	−1.22650	−0.613250	+	0.789889i	$0.710138\pi$
$174$	4.29078i	0.325283i
$175$	2.32258i	0.175571i
$176$	− 5.34541i	− 0.402925i
$177$	6.78668i	0.510118i
$178$	−1.73978	−0.130402
$179$	−7.32824	−0.547738	−0.273869	−	0.961767i	$-0.588304\pi$
$179$	−7.32824	−0.547738	−0.273869	+	0.961767i	$0.588304\pi$
$180$	1.00000i	0.0745356i
$181$	−19.5734	−1.45488	−0.727438	−	0.686173i	$-0.759289\pi$
$181$	−19.5734	−1.45488	−0.727438	+	0.686173i	$0.759289\pi$
$182$	0	0
$183$	0.535898	0.0396147
$184$	− 4.93593i	− 0.363882i
$185$	3.14925	0.231538
$186$	−3.47183	−0.254567
$187$	21.3816i	1.56358i
$188$	1.81894i	0.132660i
$189$	− 2.32258i	− 0.168943i
$190$	− 4.02283i	− 0.291846i
$191$	−17.0375	−1.23279	−0.616394	−	0.787438i	$-0.711407\pi$
$191$	−17.0375	−1.23279	−0.616394	+	0.787438i	$0.711407\pi$
$192$	−1.00000	−0.0721688
$193$	− 6.15491i	− 0.443040i	−0.975156	−	0.221520i	$-0.928898\pi$
$193$	− 6.15491i	− 0.443040i	0.975156	−	0.221520i	$-0.0711019\pi$
$194$	16.1093	1.15658
$195$	0	0
$196$	−1.60562	−0.114687
$197$	13.7867i	0.982260i	0.871086	+	0.491130i	$0.163416\pi$
$197$	13.7867i	0.982260i	−0.871086	+	0.491130i	$0.836584\pi$
$198$	5.34541	0.379881
$199$	−2.28304	−0.161841	−0.0809203	−	0.996721i	$-0.525786\pi$
$199$	−2.28304	−0.161841	−0.0809203	+	0.996721i	$0.525786\pi$
$200$	1.00000i	0.0707107i
$201$	− 4.10926i	− 0.289845i
$202$	− 12.6908i	− 0.892922i
$203$	− 9.96567i	− 0.699453i
$204$	4.00000	0.280056
$205$	−2.64516	−0.184746
$206$	− 4.79612i	− 0.334161i
$207$	4.93593	0.343071
$208$	0	0
$209$	−21.5036	−1.48744
$210$	− 2.32258i	− 0.160273i
$211$	22.4775	1.54741	0.773707	−	0.633543i	$-0.218400\pi$
$211$	22.4775	1.54741	0.773707	+	0.633543i	$0.218400\pi$
$212$	5.48693	0.376844
$213$	15.8719i	1.08752i
$214$	3.07180i	0.209984i
$215$	12.2508i	0.835496i
$216$	− 1.00000i	− 0.0680414i
$217$	8.06361	0.547393
$218$	6.69081	0.453159
$219$	− 13.5734i	− 0.917203i
$220$	5.34541	0.360387
$221$	0	0
$222$	−3.14925	−0.211364
$223$	− 1.37891i	− 0.0923389i	−0.998934	−	0.0461694i	$-0.985299\pi$
$223$	− 1.37891i	− 0.0923389i	0.998934	−	0.0461694i	$-0.0147014\pi$
$224$	2.32258	0.155184
$225$	−1.00000	−0.0666667
$226$	− 7.10972i	− 0.472931i
$227$	− 19.3205i	− 1.28235i	−0.767396	−	0.641174i	$-0.778448\pi$
$227$	− 19.3205i	− 1.28235i	0.767396	−	0.641174i	$-0.221552\pi$
$228$	4.02283i	0.266418i
$229$	− 15.7626i	− 1.04162i	−0.853672	−	0.520811i	$-0.825630\pi$
$229$	− 15.7626i	− 1.04162i	0.853672	−	0.520811i	$-0.174370\pi$
$230$	4.93593	0.325466
$231$	−12.4151	−0.816856
$232$	− 4.29078i	− 0.281703i
$233$	−16.5549	−1.08455	−0.542275	−	0.840201i	$-0.682437\pi$
$233$	−16.5549	−1.08455	−0.542275	+	0.840201i	$0.682437\pi$
$234$	0	0
$235$	−1.81894	−0.118655
$236$	− 6.78668i	− 0.441775i
$237$	−7.96774	−0.517560
$238$	−9.29032	−0.602202
$239$	26.2006i	1.69477i	0.530976	+	0.847387i	$0.321825\pi$
$239$	26.2006i	1.69477i	−0.530976	+	0.847387i	$0.678175\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	− 15.6496i	− 1.00808i	−0.863681	−	0.504039i	$-0.831847\pi$
$241$	− 15.6496i	− 1.00808i	0.863681	−	0.504039i	$-0.168153\pi$
$242$	− 17.5734i	− 1.12966i
$243$	1.00000	0.0641500
$244$	−0.535898	−0.0343074
$245$	− 1.60562i	− 0.102580i
$246$	2.64516	0.168649
$247$	0	0
$248$	3.47183	0.220462
$249$	− 11.3360i	− 0.718388i
$250$	−1.00000	−0.0632456
$251$	28.3416	1.78891	0.894454	−	0.447160i	$-0.147565\pi$
$251$	28.3416	1.78891	0.894454	+	0.447160i	$0.147565\pi$
$252$	2.32258i	0.146309i
$253$	− 26.3846i	− 1.65878i
$254$	2.13209i	0.133779i
$255$	4.00000i	0.250490i
$256$	1.00000	0.0625000
$257$	12.5093	0.780309	0.390154	−	0.920750i	$-0.372422\pi$
$257$	12.5093	0.780309	0.390154	+	0.920750i	$0.372422\pi$
$258$	− 12.2508i	− 0.762700i
$259$	7.31439	0.454494
$260$	0	0
$261$	4.29078	0.265592
$262$	1.25851i	0.0777513i
$263$	−10.7059	−0.660154	−0.330077	−	0.943954i	$-0.607075\pi$
$263$	−10.7059	−0.660154	−0.330077	+	0.943954i	$0.607075\pi$
$264$	−5.34541	−0.328987
$265$	5.48693i	0.337059i
$266$	− 9.34333i	− 0.572877i
$267$	1.73978i	0.106473i
$268$	4.10926i	0.251013i
$269$	7.52522	0.458821	0.229410	−	0.973330i	$-0.426320\pi$
$269$	7.52522	0.458821	0.229410	+	0.973330i	$0.426320\pi$
$270$	1.00000	0.0608581
$271$	− 9.84868i	− 0.598264i	−0.954212	−	0.299132i	$-0.903303\pi$
$271$	− 9.84868i	− 0.598264i	0.954212	−	0.299132i	$-0.0966972\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−19.7149	−1.19102
$275$	5.34541i	0.322340i
$276$	−4.93593	−0.297108
$277$	−0.953101	−0.0572663	−0.0286331	−	0.999590i	$-0.509115\pi$
$277$	−0.953101	−0.0572663	−0.0286331	+	0.999590i	$0.509115\pi$
$278$	5.67742i	0.340509i
$279$	3.47183i	0.207853i
$280$	2.32258i	0.138801i
$281$	− 5.57336i	− 0.332479i	−0.986085	−	0.166239i	$-0.946838\pi$
$281$	− 5.57336i	− 0.332479i	0.986085	−	0.166239i	$-0.0531625\pi$
$282$	1.81894	0.108316
$283$	−22.0134	−1.30856	−0.654280	−	0.756252i	$-0.727028\pi$
$283$	−22.0134	−1.30856	−0.654280	+	0.756252i	$0.727028\pi$
$284$	− 15.8719i	− 0.941822i
$285$	−4.02283	−0.238292
$286$	0	0
$287$	−6.14359	−0.362645
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	4.29078	0.251963
$291$	− 16.1093i	− 0.944342i
$292$	13.5734i	0.794321i
$293$	6.84302i	0.399773i	0.979819	+	0.199887i	$0.0640574\pi$
$293$	6.84302i	0.399773i	−0.979819	+	0.199887i	$0.935943\pi$
$294$	1.60562i	0.0936419i
$295$	6.78668	0.395136
$296$	3.14925	0.183047
$297$	− 5.34541i	− 0.310172i
$298$	19.4775	1.12830
$299$	0	0
$300$	1.00000	0.0577350
$301$	28.4534i	1.64003i
$302$	−14.5170	−0.835361
$303$	−12.6908	−0.729068
$304$	− 4.02283i	− 0.230725i
$305$	− 0.535898i	− 0.0306855i
$306$	− 4.00000i	− 0.228665i
$307$	− 4.75442i	− 0.271349i	−0.990753	−	0.135675i	$-0.956680\pi$
$307$	− 4.75442i	− 0.271349i	0.990753	−	0.135675i	$-0.0433202\pi$
$308$	12.4151	0.707418
$309$	−4.79612	−0.272842
$310$	3.47183i	0.197187i
$311$	1.93639	0.109803	0.0549013	−	0.998492i	$-0.482516\pi$
$311$	1.93639	0.109803	0.0549013	+	0.998492i	$0.482516\pi$
$312$	0	0
$313$	25.5545	1.44443	0.722213	−	0.691671i	$-0.243125\pi$
$313$	25.5545	1.44443	0.722213	+	0.691671i	$0.243125\pi$
$314$	24.3829i	1.37600i
$315$	−2.32258	−0.130863
$316$	7.96774	0.448220
$317$	− 12.6667i	− 0.711435i	−0.934594	−	0.355717i	$-0.884237\pi$
$317$	− 12.6667i	− 0.711435i	0.934594	−	0.355717i	$-0.115763\pi$
$318$	− 5.48693i	− 0.307692i
$319$	− 22.9359i	− 1.28417i
$320$	1.00000i	0.0559017i
$321$	3.07180	0.171451
$322$	11.4641	0.638869
$323$	16.0913i	0.895344i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−23.6267	−1.30856
$327$	− 6.69081i	− 0.370003i
$328$	−2.64516	−0.146054
$329$	−4.22464	−0.232912
$330$	− 5.34541i	− 0.294255i
$331$	− 23.6981i	− 1.30256i	−0.758836	−	0.651282i	$-0.774231\pi$
$331$	− 23.6981i	− 1.30256i	0.758836	−	0.651282i	$-0.225769\pi$
$332$	11.3360i	0.622142i
$333$	3.14925i	0.172578i
$334$	16.7471	0.916363
$335$	−4.10926	−0.224513
$336$	− 2.32258i	− 0.126707i
$337$	19.5554	1.06525	0.532625	−	0.846351i	$-0.321205\pi$
$337$	19.5554	1.06525	0.532625	+	0.846351i	$0.321205\pi$
$338$	0	0
$339$	−7.10972	−0.386147
$340$	− 4.00000i	− 0.216930i
$341$	18.5584	1.00499
$342$	4.02283	0.217530
$343$	− 19.9872i	− 1.07921i
$344$	12.2508i	0.660518i
$345$	− 4.93593i	− 0.265742i
$346$	− 16.1321i	− 0.867266i
$347$	−23.0375	−1.23672	−0.618358	−	0.785897i	$-0.712202\pi$
$347$	−23.0375	−1.23672	−0.618358	+	0.785897i	$0.712202\pi$
$348$	−4.29078	−0.230010
$349$	− 15.3205i	− 0.820088i	−0.912066	−	0.410044i	$-0.865513\pi$
$349$	− 15.3205i	− 0.820088i	0.912066	−	0.410044i	$-0.134487\pi$
$350$	−2.32258	−0.124147
$351$	0	0
$352$	5.34541	0.284911
$353$	− 28.4078i	− 1.51199i	−0.654576	−	0.755996i	$-0.727153\pi$
$353$	− 28.4078i	− 1.51199i	0.654576	−	0.755996i	$-0.272847\pi$
$354$	−6.78668	−0.360708
$355$	15.8719	0.842391
$356$	− 1.73978i	− 0.0922083i
$357$	9.29032i	0.491696i
$358$	− 7.32824i	− 0.387309i
$359$	23.5734i	1.24415i	0.782956	+	0.622077i	$0.213711\pi$
$359$	23.5734i	1.24415i	−0.782956	+	0.622077i	$0.786289\pi$
$360$	−1.00000	−0.0527046
$361$	2.81687	0.148256
$362$	− 19.5734i	− 1.02875i
$363$	−17.5734	−0.922362
$364$	0	0
$365$	−13.5734	−0.710462
$366$	0.535898i	0.0280119i
$367$	27.4909	1.43501	0.717506	−	0.696552i	$-0.245283\pi$
$367$	27.4909	1.43501	0.717506	+	0.696552i	$0.245283\pi$
$368$	4.93593	0.257303
$369$	− 2.64516i	− 0.137701i
$370$	3.14925i	0.163722i
$371$	12.7438i	0.661627i
$372$	− 3.47183i	− 0.180006i
$373$	26.3421	1.36394	0.681971	−	0.731379i	$-0.261123\pi$
$373$	26.3421	1.36394	0.681971	+	0.731379i	$0.261123\pi$
$374$	−21.3816	−1.10562
$375$	1.00000i	0.0516398i
$376$	−1.81894	−0.0938048
$377$	0	0
$378$	2.32258	0.119461
$379$	− 0.448551i	− 0.0230405i	−0.999934	−	0.0115202i	$-0.996333\pi$
$379$	− 0.448551i	− 0.0230405i	0.999934	−	0.0115202i	$-0.00366709\pi$
$380$	4.02283	0.206367
$381$	2.13209	0.109230
$382$	− 17.0375i	− 0.871713i
$383$	12.1227i	0.619439i	0.950828	+	0.309719i	$0.100235\pi$
$383$	12.1227i	0.619439i	−0.950828	+	0.309719i	$0.899765\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	12.4151i	0.632734i
$386$	6.15491	0.313277
$387$	−12.2508	−0.622742
$388$	16.1093i	0.817824i
$389$	18.6195	0.944045	0.472022	−	0.881587i	$-0.343524\pi$
$389$	18.6195	0.944045	0.472022	+	0.881587i	$0.343524\pi$
$390$	0	0
$391$	−19.7437	−0.998484
$392$	− 1.60562i	− 0.0810962i
$393$	1.25851	0.0634837
$394$	−13.7867	−0.694563
$395$	7.96774i	0.400900i
$396$	5.34541i	0.268617i
$397$	11.3042i	0.567340i	0.958922	+	0.283670i	$0.0915520\pi$
$397$	11.3042i	0.567340i	−0.958922	+	0.283670i	$0.908448\pi$
$398$	− 2.28304i	− 0.114439i
$399$	−9.34333	−0.467752
$400$	−1.00000	−0.0500000
$401$	1.50580i	0.0751959i	0.999293	+	0.0375980i	$0.0119706\pi$
$401$	1.50580i	0.0751959i	−0.999293	+	0.0375980i	$0.988029\pi$
$402$	4.10926	0.204951
$403$	0	0
$404$	12.6908	0.631391
$405$	− 1.00000i	− 0.0496904i
$406$	9.96567	0.494588
$407$	16.8340	0.834432
$408$	4.00000i	0.198030i
$409$	11.1394i	0.550806i	0.961329	+	0.275403i	$0.0888113\pi$
$409$	11.1394i	0.550806i	−0.961329	+	0.275403i	$0.911189\pi$
$410$	− 2.64516i	− 0.130635i
$411$	19.7149i	0.972464i
$412$	4.79612	0.236288
$413$	15.7626	0.775627
$414$	4.93593i	0.242588i
$415$	−11.3360	−0.556461
$416$	0	0
$417$	5.67742	0.278024
$418$	− 21.5036i	− 1.05178i
$419$	−15.5098	−0.757701	−0.378851	−	0.925458i	$-0.623681\pi$
$419$	−15.5098	−0.757701	−0.378851	+	0.925458i	$0.623681\pi$
$420$	2.32258	0.113330
$421$	− 39.4452i	− 1.92244i	−0.275778	−	0.961221i	$-0.588935\pi$
$421$	− 39.4452i	− 1.92244i	0.275778	−	0.961221i	$-0.411065\pi$
$422$	22.4775i	1.09419i
$423$	− 1.81894i	− 0.0884400i
$424$	5.48693i	0.266469i
$425$	4.00000	0.194029
$426$	−15.8719	−0.768995
$427$	− 1.24467i	− 0.0602336i
$428$	−3.07180	−0.148481
$429$	0	0
$430$	−12.2508	−0.590785
$431$	− 2.15491i	− 0.103799i	−0.998652	−	0.0518993i	$-0.983473\pi$
$431$	− 2.15491i	− 0.103799i	0.998652	−	0.0518993i	$-0.0165275\pi$
$432$	1.00000	0.0481125
$433$	−1.33938	−0.0643664	−0.0321832	−	0.999482i	$-0.510246\pi$
$433$	−1.33938	−0.0643664	−0.0321832	+	0.999482i	$0.510246\pi$
$434$	8.06361i	0.387066i
$435$	− 4.29078i	− 0.205727i
$436$	6.69081i	0.320432i
$437$	− 19.8564i	− 0.949861i
$438$	13.5734	0.648560
$439$	31.6981	1.51287	0.756434	−	0.654071i	$-0.226940\pi$
$439$	31.6981	1.51287	0.756434	+	0.654071i	$0.226940\pi$
$440$	5.34541i	0.254832i
$441$	1.60562	0.0764583
$442$	0	0
$443$	−28.0904	−1.33461	−0.667307	−	0.744782i	$-0.732553\pi$
$443$	−28.0904	−1.33461	−0.667307	+	0.744782i	$0.732553\pi$
$444$	− 3.14925i	− 0.149457i
$445$	1.73978	0.0824736
$446$	1.37891	0.0652935
$447$	− 19.4775i	− 0.921254i
$448$	2.32258i	0.109732i
$449$	28.9167i	1.36466i	0.731043	+	0.682332i	$0.239034\pi$
$449$	28.9167i	1.36466i	−0.731043	+	0.682332i	$0.760966\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	−14.1395	−0.665801
$452$	7.10972	0.334413
$453$	14.5170i	0.682070i
$454$	19.3205	0.906756
$455$	0	0
$456$	−4.02283	−0.188386
$457$	− 9.69900i	− 0.453700i	−0.973930	−	0.226850i	$-0.927157\pi$
$457$	− 9.69900i	− 0.453700i	0.973930	−	0.226850i	$-0.0728427\pi$
$458$	15.7626	0.736538
$459$	−4.00000	−0.186704
$460$	4.93593i	0.230139i
$461$	11.9979i	0.558799i	0.960175	+	0.279400i	$0.0901354\pi$
$461$	11.9979i	0.558799i	−0.960175	+	0.279400i	$0.909865\pi$
$462$	− 12.4151i	− 0.577604i
$463$	6.42664i	0.298671i	0.988787	+	0.149336i	$0.0477135\pi$
$463$	6.42664i	0.298671i	−0.988787	+	0.149336i	$0.952287\pi$
$464$	4.29078	0.199194
$465$	3.47183	0.161002
$466$	− 16.5549i	− 0.766893i
$467$	26.2642	1.21536	0.607681	−	0.794182i	$-0.292100\pi$
$467$	26.2642	1.21536	0.607681	+	0.794182i	$0.292100\pi$
$468$	0	0
$469$	−9.54409	−0.440705
$470$	− 1.81894i	− 0.0839015i
$471$	24.3829	1.12350
$472$	6.78668	0.312382
$473$	65.4854i	3.01102i
$474$	− 7.96774i	− 0.365970i
$475$	4.02283i	0.184580i
$476$	− 9.29032i	− 0.425821i
$477$	−5.48693	−0.251229
$478$	−26.2006	−1.19839
$479$	25.6826i	1.17347i	0.809779	+	0.586735i	$0.199587\pi$
$479$	25.6826i	1.17347i	−0.809779	+	0.586735i	$0.800413\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	15.6496	0.712818
$483$	− 11.4641i	− 0.521635i
$484$	17.5734	0.798789
$485$	−16.1093	−0.731484
$486$	1.00000i	0.0453609i
$487$	− 9.37891i	− 0.424999i	−0.977161	−	0.212500i	$-0.931840\pi$
$487$	− 9.37891i	− 0.424999i	0.977161	−	0.212500i	$-0.0681604\pi$
$488$	− 0.535898i	− 0.0242590i
$489$	23.6267i	1.06844i
$490$	1.60562	0.0725347
$491$	7.73854	0.349235	0.174618	−	0.984636i	$-0.444131\pi$
$491$	7.73854	0.349235	0.174618	+	0.984636i	$0.444131\pi$
$492$	2.64516i	0.119253i
$493$	−17.1631	−0.772987
$494$	0	0
$495$	−5.34541	−0.240258
$496$	3.47183i	0.155890i
$497$	36.8637	1.65356
$498$	11.3360	0.507977
$499$	3.18106i	0.142404i	0.997462	+	0.0712019i	$0.0226834\pi$
$499$	3.18106i	0.142404i	−0.997462	+	0.0712019i	$0.977317\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 16.7471i	− 0.748207i
$502$	28.3416i	1.26495i
$503$	−35.0758	−1.56395	−0.781975	−	0.623309i	$-0.785788\pi$
$503$	−35.0758	−1.56395	−0.781975	+	0.623309i	$0.785788\pi$
$504$	−2.32258	−0.103456
$505$	12.6908i	0.564734i
$506$	26.3846	1.17294
$507$	0	0
$508$	−2.13209	−0.0945961
$509$	18.7867i	0.832705i	0.909203	+	0.416353i	$0.136692\pi$
$509$	18.7867i	0.832705i	−0.909203	+	0.416353i	$0.863308\pi$
$510$	−4.00000	−0.177123
$511$	−31.5252	−1.39459
$512$	1.00000i	0.0441942i
$513$	− 4.02283i	− 0.177612i
$514$	12.5093i	0.551761i
$515$	4.79612i	0.211342i
$516$	12.2508	0.539311
$517$	−9.72298	−0.427616
$518$	7.31439i	0.321376i
$519$	−16.1321	−0.708120
$520$	0	0
$521$	5.28512	0.231545	0.115773	−	0.993276i	$-0.463066\pi$
$521$	5.28512	0.231545	0.115773	+	0.993276i	$0.463066\pi$
$522$	4.29078i	0.187802i
$523$	−20.7523	−0.907437	−0.453718	−	0.891145i	$-0.649903\pi$
$523$	−20.7523	−0.907437	−0.453718	+	0.891145i	$0.649903\pi$
$524$	−1.25851	−0.0549785
$525$	2.32258i	0.101366i
$526$	− 10.7059i	− 0.466800i
$527$	− 13.8873i	− 0.604942i
$528$	− 5.34541i	− 0.232629i
$529$	1.36345	0.0592805
$530$	−5.48693	−0.238337
$531$	6.78668i	0.294517i
$532$	9.34333	0.405085
$533$	0	0
$534$	−1.73978	−0.0752877
$535$	− 3.07180i	− 0.132805i
$536$	−4.10926	−0.177493
$537$	−7.32824	−0.316237
$538$	7.52522i	0.324435i
$539$	− 8.58271i	− 0.369683i
$540$	1.00000i	0.0430331i
$541$	− 28.7365i	− 1.23548i	−0.786384	−	0.617739i	$-0.788049\pi$
$541$	− 28.7365i	− 1.23548i	0.786384	−	0.617739i	$-0.211951\pi$
$542$	9.84868	0.423037
$543$	−19.5734	−0.839973
$544$	− 4.00000i	− 0.171499i
$545$	−6.69081	−0.286603
$546$	0	0
$547$	18.3768	0.785737	0.392869	−	0.919595i	$-0.371483\pi$
$547$	18.3768	0.785737	0.392869	+	0.919595i	$0.371483\pi$
$548$	− 19.7149i	− 0.842178i
$549$	0.535898	0.0228716
$550$	−5.34541	−0.227929
$551$	− 17.2610i	− 0.735345i
$552$	− 4.93593i	− 0.210087i
$553$	18.5057i	0.786943i
$554$	− 0.953101i	− 0.0404934i
$555$	3.14925	0.133678
$556$	−5.67742	−0.240776
$557$	− 26.5386i	− 1.12448i	−0.826975	−	0.562238i	$-0.809940\pi$
$557$	− 26.5386i	− 1.12448i	0.826975	−	0.562238i	$-0.190060\pi$
$558$	−3.47183	−0.146974
$559$	0	0
$560$	−2.32258	−0.0981469
$561$	21.3816i	0.902733i
$562$	5.57336	0.235098
$563$	6.06111	0.255446	0.127723	−	0.991810i	$-0.459233\pi$
$563$	6.06111	0.255446	0.127723	+	0.991810i	$0.459233\pi$
$564$	1.81894i	0.0765913i
$565$	7.10972i	0.299108i
$566$	− 22.0134i	− 0.925292i
$567$	− 2.32258i	− 0.0975392i
$568$	15.8719	0.665969
$569$	14.4964	0.607719	0.303860	−	0.952717i	$-0.401725\pi$
$569$	14.4964	0.607719	0.303860	+	0.952717i	$0.401725\pi$
$570$	− 4.02283i	− 0.168498i
$571$	−2.90413	−0.121534	−0.0607670	−	0.998152i	$-0.519355\pi$
$571$	−2.90413	−0.121534	−0.0607670	+	0.998152i	$0.519355\pi$
$572$	0	0
$573$	−17.0375	−0.711750
$574$	− 6.14359i	− 0.256429i
$575$	−4.93593	−0.205843
$576$	−1.00000	−0.0416667
$577$	− 18.8180i	− 0.783405i	−0.920092	−	0.391702i	$-0.871886\pi$
$577$	− 18.8180i	− 0.783405i	0.920092	−	0.391702i	$-0.128114\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	− 6.15491i	− 0.255789i
$580$	4.29078i	0.178165i
$581$	−26.3287	−1.09230
$582$	16.1093	0.667750
$583$	29.3299i	1.21472i
$584$	−13.5734	−0.561670
$585$	0	0
$586$	−6.84302	−0.282682
$587$	3.93639i	0.162472i	0.996695	+	0.0812361i	$0.0258868\pi$
$587$	3.93639i	0.162472i	−0.996695	+	0.0812361i	$0.974113\pi$
$588$	−1.60562	−0.0662148
$589$	13.9666	0.575483
$590$	6.78668i	0.279403i
$591$	13.7867i	0.567108i
$592$	3.14925i	0.129434i
$593$	20.1227i	0.826338i	0.910654	+	0.413169i	$0.135578\pi$
$593$	20.1227i	0.826338i	−0.910654	+	0.413169i	$0.864422\pi$
$594$	5.34541	0.219325
$595$	9.29032	0.380866
$596$	19.4775i	0.797829i
$597$	−2.28304	−0.0934388
$598$	0	0
$599$	48.1172	1.96601	0.983007	−	0.183567i	$-0.0587644\pi$
$599$	48.1172	1.96601	0.983007	+	0.183567i	$0.0587644\pi$
$600$	1.00000i	0.0408248i
$601$	−4.24217	−0.173042	−0.0865209	−	0.996250i	$-0.527575\pi$
$601$	−4.24217	−0.173042	−0.0865209	+	0.996250i	$0.527575\pi$
$602$	−28.4534	−1.15967
$603$	− 4.10926i	− 0.167342i
$604$	− 14.5170i	− 0.590690i
$605$	17.5734i	0.714459i
$606$	− 12.6908i	− 0.515529i
$607$	12.0685	0.489844	0.244922	−	0.969543i	$-0.421238\pi$
$607$	12.0685	0.489844	0.244922	+	0.969543i	$0.421238\pi$
$608$	4.02283	0.163147
$609$	− 9.96567i	− 0.403829i
$610$	0.535898	0.0216979
$611$	0	0
$612$	4.00000	0.161690
$613$	− 41.7421i	− 1.68595i	−0.537954	−	0.842974i	$-0.680803\pi$
$613$	− 41.7421i	− 1.68595i	0.537954	−	0.842974i	$-0.319197\pi$
$614$	4.75442	0.191873
$615$	−2.64516	−0.106663
$616$	12.4151i	0.500220i
$617$	− 33.4586i	− 1.34699i	−0.739190	−	0.673497i	$-0.764792\pi$
$617$	− 33.4586i	− 1.34699i	0.739190	−	0.673497i	$-0.235208\pi$
$618$	− 4.79612i	− 0.192928i
$619$	38.0978i	1.53128i	0.643270	+	0.765639i	$0.277577\pi$
$619$	38.0978i	1.53128i	−0.643270	+	0.765639i	$0.722423\pi$
$620$	−3.47183	−0.139432
$621$	4.93593	0.198072
$622$	1.93639i	0.0776422i
$623$	4.04078	0.161891
$624$	0	0
$625$	1.00000	0.0400000
$626$	25.5545i	1.02136i
$627$	−21.5036	−0.858773
$628$	−24.3829	−0.972982
$629$	− 12.5970i	− 0.502276i
$630$	− 2.32258i	− 0.0925338i
$631$	− 24.8001i	− 0.987275i	−0.869668	−	0.493638i	$-0.835667\pi$
$631$	− 24.8001i	− 0.987275i	0.869668	−	0.493638i	$-0.164333\pi$
$632$	7.96774i	0.316940i
$633$	22.4775	0.893400
$634$	12.6667	0.503060
$635$	− 2.13209i	− 0.0846093i
$636$	5.48693	0.217571
$637$	0	0
$638$	22.9359	0.908042
$639$	15.8719i	0.627881i
$640$	−1.00000	−0.0395285
$641$	−4.08519	−0.161355	−0.0806776	−	0.996740i	$-0.525708\pi$
$641$	−4.08519	−0.161355	−0.0806776	+	0.996740i	$0.525708\pi$
$642$	3.07180i	0.121234i
$643$	− 36.5816i	− 1.44264i	−0.692604	−	0.721318i	$-0.743537\pi$
$643$	− 36.5816i	− 1.44264i	0.692604	−	0.721318i	$-0.256463\pi$
$644$	11.4641i	0.451749i
$645$	12.2508i	0.482374i
$646$	−16.0913	−0.633104
$647$	−17.2341	−0.677541	−0.338771	−	0.940869i	$-0.610011\pi$
$647$	−17.2341	−0.677541	−0.338771	+	0.940869i	$0.610011\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	36.2776	1.42402
$650$	0	0
$651$	8.06361	0.316038
$652$	− 23.6267i	− 0.925295i
$653$	−21.9274	−0.858085	−0.429042	−	0.903284i	$-0.641149\pi$
$653$	−21.9274	−0.858085	−0.429042	+	0.903284i	$0.641149\pi$
$654$	6.69081	0.261631
$655$	− 1.25851i	− 0.0491742i
$656$	− 2.64516i	− 0.103276i
$657$	− 13.5734i	− 0.529547i
$658$	− 4.22464i	− 0.164694i
$659$	−48.4759	−1.88835	−0.944176	−	0.329441i	$-0.893140\pi$
$659$	−48.4759	−1.88835	−0.944176	+	0.329441i	$0.893140\pi$
$660$	5.34541	0.208070
$661$	29.8221i	1.15994i	0.814636	+	0.579972i	$0.196937\pi$
$661$	29.8221i	1.15994i	−0.814636	+	0.579972i	$0.803063\pi$
$662$	23.6981	0.921052
$663$	0	0
$664$	−11.3360	−0.439921
$665$	9.34333i	0.362319i
$666$	−3.14925	−0.122031
$667$	21.1790	0.820054
$668$	16.7471i	0.647967i
$669$	− 1.37891i	− 0.0533119i
$670$	− 4.10926i	− 0.158755i
$671$	− 2.86459i	− 0.110586i
$672$	2.32258	0.0895955
$673$	30.6719	1.18232	0.591158	−	0.806556i	$-0.298671\pi$
$673$	30.6719	1.18232	0.591158	+	0.806556i	$0.298671\pi$
$674$	19.5554i	0.753246i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	21.0831	0.810290	0.405145	−	0.914252i	$-0.367221\pi$
$677$	21.0831	0.810290	0.405145	+	0.914252i	$0.367221\pi$
$678$	− 7.10972i	− 0.273047i
$679$	−37.4150	−1.43586
$680$	4.00000	0.153393
$681$	− 19.3205i	− 0.740363i
$682$	18.5584i	0.710636i
$683$	− 13.3360i	− 0.510287i	−0.966903	−	0.255143i	$-0.917877\pi$
$683$	− 13.3360i	− 0.510287i	0.966903	−	0.255143i	$-0.0821227\pi$
$684$	4.02283i	0.153817i
$685$	19.7149	0.753267
$686$	19.9872	0.763117
$687$	− 15.7626i	− 0.601381i
$688$	−12.2508	−0.467057
$689$	0	0
$690$	4.93593	0.187908
$691$	34.9054i	1.32786i	0.747793	+	0.663932i	$0.231113\pi$
$691$	34.9054i	1.32786i	−0.747793	+	0.663932i	$0.768887\pi$
$692$	16.1321	0.613250
$693$	−12.4151	−0.471612
$694$	− 23.0375i	− 0.874490i
$695$	− 5.67742i	− 0.215357i
$696$	− 4.29078i	− 0.162641i
$697$	10.5806i	0.400770i
$698$	15.3205	0.579890
$699$	−16.5549	−0.626166
$700$	− 2.32258i	− 0.0877853i
$701$	39.6715	1.49837	0.749186	−	0.662360i	$-0.230445\pi$
$701$	39.6715	1.49837	0.749186	+	0.662360i	$0.230445\pi$
$702$	0	0
$703$	12.6689	0.477817
$704$	5.34541i	0.201463i
$705$	−1.81894	−0.0685053
$706$	28.4078	1.06914
$707$	29.4754i	1.10854i
$708$	− 6.78668i	− 0.255059i
$709$	2.83441i	0.106448i	0.998583	+	0.0532242i	$0.0169498\pi$
$709$	2.83441i	0.106448i	−0.998583	+	0.0532242i	$0.983050\pi$
$710$	15.8719i	0.595661i
$711$	−7.96774	−0.298814
$712$	1.73978	0.0652011
$713$	17.1367i	0.641776i
$714$	−9.29032	−0.347681
$715$	0	0
$716$	7.32824	0.273869
$717$	26.2006i	0.978478i
$718$	−23.5734	−0.879750
$719$	43.7128	1.63021	0.815106	−	0.579311i	$-0.196678\pi$
$719$	43.7128	1.63021	0.815106	+	0.579311i	$0.196678\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	11.1394i	0.414852i
$722$	2.81687i	0.104833i
$723$	− 15.6496i	− 0.582014i
$724$	19.5734	0.727438
$725$	−4.29078	−0.159355
$726$	− 17.5734i	− 0.652209i
$727$	−16.2568	−0.602932	−0.301466	−	0.953477i	$-0.597476\pi$
$727$	−16.2568	−0.602932	−0.301466	+	0.953477i	$0.597476\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 13.5734i	− 0.502373i
$731$	49.0031	1.81245
$732$	−0.535898	−0.0198074
$733$	4.96774i	0.183488i	0.995783	+	0.0917438i	$0.0292441\pi$
$733$	4.96774i	0.183488i	−0.995783	+	0.0917438i	$0.970756\pi$
$734$	27.4909i	1.01471i
$735$	− 1.60562i	− 0.0592243i
$736$	4.93593i	0.181941i
$737$	−21.9657	−0.809116
$738$	2.64516	0.0973697
$739$	− 49.6875i	− 1.82778i	−0.405957	−	0.913892i	$-0.633062\pi$
$739$	− 49.6875i	− 1.82778i	0.405957	−	0.913892i	$-0.366938\pi$
$740$	−3.14925	−0.115769
$741$	0	0
$742$	−12.7438	−0.467841
$743$	− 21.5420i	− 0.790300i	−0.918617	−	0.395150i	$-0.870693\pi$
$743$	− 21.5420i	− 0.790300i	0.918617	−	0.395150i	$-0.129307\pi$
$744$	3.47183	0.127284
$745$	−19.4775	−0.713600
$746$	26.3421i	0.964452i
$747$	− 11.3360i	− 0.414761i
$748$	− 21.3816i	− 0.781790i
$749$	− 7.13449i	− 0.260689i
$750$	−1.00000	−0.0365148
$751$	6.34872	0.231668	0.115834	−	0.993269i	$-0.463046\pi$
$751$	6.34872	0.231668	0.115834	+	0.993269i	$0.463046\pi$
$752$	− 1.81894i	− 0.0663300i
$753$	28.3416	1.03283
$754$	0	0
$755$	14.5170	0.528329
$756$	2.32258i	0.0844714i
$757$	−48.1107	−1.74861	−0.874307	−	0.485373i	$-0.838684\pi$
$757$	−48.1107	−1.74861	−0.874307	+	0.485373i	$0.838684\pi$
$758$	0.448551	0.0162921
$759$	− 26.3846i	− 0.957699i
$760$	4.02283i	0.145923i
$761$	33.6866i	1.22114i	0.791963	+	0.610569i	$0.209059\pi$
$761$	33.6866i	1.22114i	−0.791963	+	0.610569i	$0.790941\pi$
$762$	2.13209i	0.0772374i
$763$	−15.5399	−0.562584
$764$	17.0375	0.616394
$765$	4.00000i	0.144620i
$766$	−12.1227	−0.438009
$767$	0	0
$768$	1.00000	0.0360844
$769$	5.98886i	0.215964i	0.994153	+	0.107982i	$0.0344389\pi$
$769$	5.98886i	0.215964i	−0.994153	+	0.107982i	$0.965561\pi$
$770$	−12.4151	−0.447410
$771$	12.5093	0.450511
$772$	6.15491i	0.221520i
$773$	13.4882i	0.485136i	0.970134	+	0.242568i	$0.0779897\pi$
$773$	13.4882i	0.485136i	−0.970134	+	0.242568i	$0.922010\pi$
$774$	− 12.2508i	− 0.440345i
$775$	− 3.47183i	− 0.124712i
$776$	−16.1093	−0.578289
$777$	7.31439	0.262402
$778$	18.6195i	0.667540i
$779$	−10.6410	−0.381254
$780$	0	0
$781$	84.8416	3.03587
$782$	− 19.7437i	− 0.706035i
$783$	4.29078	0.153340
$784$	1.60562	0.0573437
$785$	− 24.3829i	− 0.870262i
$786$	1.25851i	0.0448897i
$787$	− 54.6487i	− 1.94802i	−0.226510	−	0.974009i	$-0.572732\pi$
$787$	− 54.6487i	− 1.94802i	0.226510	−	0.974009i	$-0.427268\pi$
$788$	− 13.7867i	− 0.491130i
$789$	−10.7059	−0.381140
$790$	−7.96774	−0.283479
$791$	16.5129i	0.587131i
$792$	−5.34541	−0.189941
$793$	0	0
$794$	−11.3042	−0.401170
$795$	5.48693i	0.194601i
$796$	2.28304	0.0809203
$797$	38.3244	1.35752	0.678760	−	0.734361i	$-0.262518\pi$
$797$	38.3244	1.35752	0.678760	+	0.734361i	$0.262518\pi$
$798$	− 9.34333i	− 0.330750i
$799$	7.27577i	0.257398i
$800$	− 1.00000i	− 0.0353553i
$801$	1.73978i	0.0614722i
$802$	−1.50580	−0.0531716
$803$	−72.5551	−2.56041
$804$	4.10926i	0.144922i
$805$	−11.4641	−0.404056
$806$	0	0
$807$	7.52522	0.264900
$808$	12.6908i	0.446461i
$809$	11.3816	0.400157	0.200078	−	0.979780i	$-0.435880\pi$
$809$	11.3816	0.400157	0.200078	+	0.979780i	$0.435880\pi$
$810$	1.00000	0.0351364
$811$	− 10.8899i	− 0.382397i	−0.981551	−	0.191198i	$-0.938763\pi$
$811$	− 10.8899i	− 0.382397i	0.981551	−	0.191198i	$-0.0612374\pi$
$812$	9.96567i	0.349726i
$813$	− 9.84868i	− 0.345408i
$814$	16.8340i	0.590033i
$815$	23.6267	0.827609
$816$	−4.00000	−0.140028
$817$	49.2828i	1.72419i
$818$	−11.1394	−0.389479
$819$	0	0
$820$	2.64516	0.0923730
$821$	1.68810i	0.0589152i	0.999566	+	0.0294576i	$0.00937800\pi$
$821$	1.68810i	0.0589152i	−0.999566	+	0.0294576i	$0.990622\pi$
$822$	−19.7149	−0.687636
$823$	−6.77816	−0.236272	−0.118136	−	0.992997i	$-0.537692\pi$
$823$	−6.77816	−0.236272	−0.118136	+	0.992997i	$0.537692\pi$
$824$	4.79612i	0.167081i
$825$	5.34541i	0.186103i
$826$	15.7626i	0.548451i
$827$	− 30.2298i	− 1.05119i	−0.850733	−	0.525597i	$-0.823842\pi$
$827$	− 30.2298i	− 1.05119i	0.850733	−	0.525597i	$-0.176158\pi$
$828$	−4.93593	−0.171536
$829$	10.4405	0.362612	0.181306	−	0.983427i	$-0.441968\pi$
$829$	10.4405	0.362612	0.181306	+	0.983427i	$0.441968\pi$
$830$	− 11.3360i	− 0.393477i
$831$	−0.953101	−0.0330627
$832$	0	0
$833$	−6.42249	−0.222526
$834$	5.67742i	0.196593i
$835$	−16.7471	−0.579559
$836$	21.5036	0.743719
$837$	3.47183i	0.120004i
$838$	− 15.5098i	− 0.535776i
$839$	11.1965i	0.386547i	0.981145	+	0.193273i	$0.0619104\pi$
$839$	11.1965i	0.386547i	−0.981145	+	0.193273i	$0.938090\pi$
$840$	2.32258i	0.0801366i
$841$	−10.5892	−0.365146
$842$	39.4452	1.35937
$843$	− 5.57336i	− 0.191957i
$844$	−22.4775	−0.773707
$845$	0	0
$846$	1.81894	0.0625365
$847$	40.8155i	1.40244i
$848$	−5.48693	−0.188422
$849$	−22.0134	−0.755498
$850$	4.00000i	0.137199i
$851$	15.5445i	0.532859i
$852$	− 15.8719i	− 0.543761i
$853$	21.8185i	0.747051i	0.927620	+	0.373525i	$0.121851\pi$
$853$	21.8185i	0.747051i	−0.927620	+	0.373525i	$0.878149\pi$
$854$	1.24467	0.0425916
$855$	−4.02283	−0.137578
$856$	− 3.07180i	− 0.104992i
$857$	−9.42369	−0.321907	−0.160954	−	0.986962i	$-0.551457\pi$
$857$	−9.42369	−0.321907	−0.160954	+	0.986962i	$0.551457\pi$
$858$	0	0
$859$	24.4775	0.835161	0.417581	−	0.908640i	$-0.362878\pi$
$859$	24.4775	0.835161	0.417581	+	0.908640i	$0.362878\pi$
$860$	− 12.2508i	− 0.417748i
$861$	−6.14359	−0.209373
$862$	2.15491	0.0733966
$863$	− 29.7873i	− 1.01397i	−0.861954	−	0.506986i	$-0.830760\pi$
$863$	− 29.7873i	− 1.01397i	0.861954	−	0.506986i	$-0.169240\pi$
$864$	1.00000i	0.0340207i
$865$	16.1321i	0.548507i
$866$	− 1.33938i	− 0.0455139i
$867$	−1.00000	−0.0339618
$868$	−8.06361	−0.273697
$869$	42.5908i	1.44479i
$870$	4.29078	0.145471
$871$	0	0
$872$	−6.69081	−0.226579
$873$	− 16.1093i	− 0.545216i
$874$	19.8564	0.671653
$875$	2.32258	0.0785175
$876$	13.5734i	0.458601i
$877$	1.90979i	0.0644890i	0.999480	+	0.0322445i	$0.0102655\pi$
$877$	1.90979i	0.0644890i	−0.999480	+	0.0322445i	$0.989734\pi$
$878$	31.6981i	1.06976i
$879$	6.84302i	0.230809i
$880$	−5.34541	−0.180194
$881$	−6.36823	−0.214551	−0.107276	−	0.994229i	$-0.534213\pi$
$881$	−6.36823	−0.214551	−0.107276	+	0.994229i	$0.534213\pi$
$882$	1.60562i	0.0540642i
$883$	−34.9897	−1.17750	−0.588749	−	0.808316i	$-0.700379\pi$
$883$	−34.9897	−1.17750	−0.588749	+	0.808316i	$0.700379\pi$
$884$	0	0
$885$	6.78668	0.228132
$886$	− 28.0904i	− 0.943715i
$887$	21.4001	0.718546	0.359273	−	0.933233i	$-0.383025\pi$
$887$	21.4001	0.718546	0.359273	+	0.933233i	$0.383025\pi$
$888$	3.14925	0.105682
$889$	− 4.95194i	− 0.166083i
$890$	1.73978i	0.0583176i
$891$	− 5.34541i	− 0.179078i
$892$	1.37891i	0.0461694i
$893$	−7.31729	−0.244864
$894$	19.4775	0.651425
$895$	7.32824i	0.244956i
$896$	−2.32258	−0.0775919
$897$	0	0
$898$	−28.9167	−0.964963
$899$	14.8969i	0.496838i
$900$	1.00000	0.0333333
$901$	21.9477	0.731184
$902$	− 14.1395i	− 0.470792i
$903$	28.4534i	0.946871i
$904$	7.10972i	0.236466i
$905$	19.5734i	0.650641i
$906$	−14.5170	−0.482296
$907$	−32.4693	−1.07813	−0.539063	−	0.842266i	$-0.681221\pi$
$907$	−32.4693	−1.07813	−0.539063	+	0.842266i	$0.681221\pi$
$908$	19.3205i	0.641174i
$909$	−12.6908	−0.420928
$910$	0	0
$911$	34.4380	1.14098	0.570490	−	0.821304i	$-0.306753\pi$
$911$	34.4380	1.14098	0.570490	+	0.821304i	$0.306753\pi$
$912$	− 4.02283i	− 0.133209i
$913$	−60.5954	−2.00541
$914$	9.69900	0.320814
$915$	− 0.535898i	− 0.0177163i
$916$	15.7626i	0.520811i
$917$	− 2.92300i	− 0.0965260i
$918$	− 4.00000i	− 0.132020i
$919$	36.4827	1.20345	0.601727	−	0.798702i	$-0.294480\pi$
$919$	36.4827	1.20345	0.601727	+	0.798702i	$0.294480\pi$
$920$	−4.93593	−0.162733
$921$	− 4.75442i	− 0.156663i
$922$	−11.9979	−0.395131
$923$	0	0
$924$	12.4151	0.408428
$925$	− 3.14925i	− 0.103547i
$926$	−6.42664	−0.211192
$927$	−4.79612	−0.157525
$928$	4.29078i	0.140852i
$929$	49.9100i	1.63749i	0.574155	+	0.818747i	$0.305331\pi$
$929$	49.9100i	1.63749i	−0.574155	+	0.818747i	$0.694669\pi$
$930$	3.47183i	0.113846i
$931$	− 6.45914i	− 0.211690i
$932$	16.5549	0.542275
$933$	1.93639	0.0633946
$934$	26.2642i	0.859390i
$935$	21.3816	0.699254
$936$	0	0
$937$	−15.8873	−0.519017	−0.259508	−	0.965741i	$-0.583560\pi$
$937$	−15.8873	−0.519017	−0.259508	+	0.965741i	$0.583560\pi$
$938$	− 9.54409i	− 0.311625i
$939$	25.5545	0.833939
$940$	1.81894	0.0593274
$941$	6.99273i	0.227956i	0.993483	+	0.113978i	$0.0363594\pi$
$941$	6.99273i	0.227956i	−0.993483	+	0.113978i	$0.963641\pi$
$942$	24.3829i	0.794437i
$943$	− 13.0563i	− 0.425173i
$944$	6.78668i	0.220888i
$945$	−2.32258	−0.0755535
$946$	−65.4854	−2.12911
$947$	52.3064i	1.69973i	0.527000	+	0.849865i	$0.323317\pi$
$947$	52.3064i	1.69973i	−0.527000	+	0.849865i	$0.676683\pi$
$948$	7.96774	0.258780
$949$	0	0
$950$	−4.02283	−0.130518
$951$	− 12.6667i	− 0.410747i
$952$	9.29032	0.301101
$953$	−28.8827	−0.935603	−0.467802	−	0.883833i	$-0.654954\pi$
$953$	−28.8827	−0.935603	−0.467802	+	0.883833i	$0.654954\pi$
$954$	− 5.48693i	− 0.177646i
$955$	17.0375i	0.551319i
$956$	− 26.2006i	− 0.847387i
$957$	− 22.9359i	− 0.741413i
$958$	−25.6826	−0.829768
$959$	45.7894	1.47862
$960$	1.00000i	0.0322749i
$961$	18.9464	0.611173
$962$	0	0
$963$	3.07180	0.0989873
$964$	15.6496i	0.504039i
$965$	−6.15491	−0.198134
$966$	11.4641	0.368851
$967$	10.8610i	0.349265i	0.984634	+	0.174633i	$0.0558738\pi$
$967$	10.8610i	0.349265i	−0.984634	+	0.174633i	$0.944126\pi$
$968$	17.5734i	0.564829i
$969$	16.0913i	0.516927i
$970$	− 16.1093i	− 0.517237i
$971$	−24.9979	−0.802222	−0.401111	−	0.916030i	$-0.631376\pi$
$971$	−24.9979	−0.802222	−0.401111	+	0.916030i	$0.631376\pi$
$972$	−1.00000	−0.0320750
$973$	− 13.1863i	− 0.422732i
$974$	9.37891	0.300520
$975$	0	0
$976$	0.535898	0.0171537
$977$	42.7653i	1.36818i	0.729396	+	0.684092i	$0.239801\pi$
$977$	42.7653i	1.36818i	−0.729396	+	0.684092i	$0.760199\pi$
$978$	−23.6267	−0.755500
$979$	9.29984	0.297224
$980$	1.60562i	0.0512898i
$981$	− 6.69081i	− 0.213621i
$982$	7.73854i	0.246947i
$983$	− 10.1288i	− 0.323058i	−0.986868	−	0.161529i	$-0.948358\pi$
$983$	− 10.1288i	− 0.323058i	0.986868	−	0.161529i	$-0.0516425\pi$
$984$	−2.64516	−0.0843246
$985$	13.7867	0.439280
$986$	− 17.1631i	− 0.546584i
$987$	−4.22464	−0.134472
$988$	0	0
$989$	−60.4691	−1.92280
$990$	− 5.34541i	− 0.169888i
$991$	20.2483	0.643208	0.321604	−	0.946874i	$-0.395778\pi$
$991$	20.2483	0.643208	0.321604	+	0.946874i	$0.395778\pi$
$992$	−3.47183	−0.110231
$993$	− 23.6981i	− 0.752036i
$994$	36.8637i	1.16924i
$995$	2.28304i	0.0723774i
$996$	11.3360i	0.359194i
$997$	−20.7961	−0.658620	−0.329310	−	0.944222i	$-0.606816\pi$
$997$	−20.7961	−0.658620	−0.329310	+	0.944222i	$0.606816\pi$
$998$	−3.18106	−0.100695
$999$	3.14925i	0.0996380i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.ba.1351.6		8
13.3	even	3		390.2.bb.c.121.1	✓	8
13.4	even	6		390.2.bb.c.361.1	yes	8
13.5	odd	4		5070.2.a.ca.1.3		4
13.8	odd	4		5070.2.a.bz.1.2		4
13.12	even	2	inner	5070.2.b.ba.1351.3		8
39.17	odd	6		1170.2.bs.f.361.3		8
39.29	odd	6		1170.2.bs.f.901.3		8
65.3	odd	12		1950.2.y.j.199.2		8
65.4	even	6		1950.2.bc.g.751.4		8
65.17	odd	12		1950.2.y.j.49.2		8
65.29	even	6		1950.2.bc.g.901.4		8
65.42	odd	12		1950.2.y.k.199.3		8
65.43	odd	12		1950.2.y.k.49.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bb.c.121.1	✓	8	13.3	even	3
390.2.bb.c.361.1	yes	8	13.4	even	6
1170.2.bs.f.361.3		8	39.17	odd	6
1170.2.bs.f.901.3		8	39.29	odd	6
1950.2.y.j.49.2		8	65.17	odd	12
1950.2.y.j.199.2		8	65.3	odd	12
1950.2.y.k.49.3		8	65.43	odd	12
1950.2.y.k.199.3		8	65.42	odd	12
1950.2.bc.g.751.4		8	65.4	even	6
1950.2.bc.g.901.4		8	65.29	even	6
5070.2.a.bz.1.2		4	13.8	odd	4
5070.2.a.ca.1.3		4	13.5	odd	4
5070.2.b.ba.1351.3		8	13.12	even	2	inner
5070.2.b.ba.1351.6		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -2.32258i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -2.32258i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.34541i q^{11} -1.00000 q^{12} +2.32258 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000i q^{18} -4.02283i q^{19} +1.00000i q^{20} -2.32258i q^{21} +5.34541 q^{22} +4.93593 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +2.32258i q^{28} +4.29078 q^{29} +1.00000 q^{30} +3.47183i q^{31} +1.00000i q^{32} -5.34541i q^{33} -4.00000i q^{34} -2.32258 q^{35} -1.00000 q^{36} +3.14925i q^{37} +4.02283 q^{38} -1.00000 q^{40} -2.64516i q^{41} +2.32258 q^{42} -12.2508 q^{43} +5.34541i q^{44} -1.00000i q^{45} +4.93593i q^{46} -1.81894i q^{47} +1.00000 q^{48} +1.60562 q^{49} -1.00000i q^{50} -4.00000 q^{51} -5.48693 q^{53} +1.00000i q^{54} -5.34541 q^{55} -2.32258 q^{56} -4.02283i q^{57} +4.29078i q^{58} +6.78668i q^{59} +1.00000i q^{60} +0.535898 q^{61} -3.47183 q^{62} -2.32258i q^{63} -1.00000 q^{64} +5.34541 q^{66} -4.10926i q^{67} +4.00000 q^{68} +4.93593 q^{69} -2.32258i q^{70} +15.8719i q^{71} -1.00000i q^{72} -13.5734i q^{73} -3.14925 q^{74} -1.00000 q^{75} +4.02283i q^{76} -12.4151 q^{77} -7.96774 q^{79} -1.00000i q^{80} +1.00000 q^{81} +2.64516 q^{82} -11.3360i q^{83} +2.32258i q^{84} +4.00000i q^{85} -12.2508i q^{86} +4.29078 q^{87} -5.34541 q^{88} +1.73978i q^{89} +1.00000 q^{90} -4.93593 q^{92} +3.47183i q^{93} +1.81894 q^{94} -4.02283 q^{95} +1.00000i q^{96} -16.1093i q^{97} +1.60562i q^{98} -5.34541i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 8 q^{4} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 - 8 * q^4 + 8 * q^9	\( 8 q + 8 q^{3} - 8 q^{4} + 8 q^{9} + 8 q^{10} - 8 q^{12} + 4 q^{14} + 8 q^{16} - 32 q^{17} - 4 q^{22} - 8 q^{23} - 8 q^{25} + 8 q^{27} + 16 q^{29} + 8 q^{30} - 4 q^{35} - 8 q^{36} - 8 q^{40} + 4 q^{42} - 28 q^{43} + 8 q^{48} - 28 q^{49} - 32 q^{51} + 16 q^{53} + 4 q^{55} - 4 q^{56} + 32 q^{61} - 8 q^{62} - 8 q^{64} - 4 q^{66} + 32 q^{68} - 8 q^{69} - 20 q^{74} - 8 q^{75} + 16 q^{77} - 20 q^{79} + 8 q^{81} - 8 q^{82} + 16 q^{87} + 4 q^{88} + 8 q^{90} + 8 q^{92} + 16 q^{94}+O(q^{100}) \) 8 * q + 8 * q^3 - 8 * q^4 + 8 * q^9 + 8 * q^10 - 8 * q^12 + 4 * q^14 + 8 * q^16 - 32 * q^17 - 4 * q^22 - 8 * q^23 - 8 * q^25 + 8 * q^27 + 16 * q^29 + 8 * q^30 - 4 * q^35 - 8 * q^36 - 8 * q^40 + 4 * q^42 - 28 * q^43 + 8 * q^48 - 28 * q^49 - 32 * q^51 + 16 * q^53 + 4 * q^55 - 4 * q^56 + 32 * q^61 - 8 * q^62 - 8 * q^64 - 4 * q^66 + 32 * q^68 - 8 * q^69 - 20 * q^74 - 8 * q^75 + 16 * q^77 - 20 * q^79 + 8 * q^81 - 8 * q^82 + 16 * q^87 + 4 * q^88 + 8 * q^90 + 8 * q^92 + 16 * q^94

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