LMFDB - Embedded newform 5070.2.b.ba.1351.7

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			1351.7
Root			$-1.80668 + 1.80668i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.ba.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.32258i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.32258i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.61335i q^{11} -1.00000 q^{12} -1.32258 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000i q^{18} +2.29078i q^{19} +1.00000i q^{20} +1.32258i q^{21} -4.61335 q^{22} -8.66799 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -1.32258i q^{28} -2.02283 q^{29} +1.00000 q^{30} -10.1321i q^{31} +1.00000i q^{32} +4.61335i q^{33} -4.00000i q^{34} +1.32258 q^{35} -1.00000 q^{36} -6.80951i q^{37} -2.29078 q^{38} -1.00000 q^{40} +4.64516i q^{41} -1.32258 q^{42} -8.60562 q^{43} -4.61335i q^{44} -1.00000i q^{45} -8.66799i q^{46} -9.10926i q^{47} +1.00000 q^{48} +5.25078 q^{49} -1.00000i q^{50} -4.00000 q^{51} +0.826674 q^{53} +1.00000i q^{54} +4.61335 q^{55} +1.32258 q^{56} +2.29078i q^{57} -2.02283i q^{58} +3.14152i q^{59} +1.00000i q^{60} +0.535898 q^{61} +10.1321 q^{62} +1.32258i q^{63} -1.00000 q^{64} -4.61335 q^{66} +3.18106i q^{67} +4.00000 q^{68} -8.66799 q^{69} +1.32258i q^{70} -11.3360i q^{71} -1.00000i q^{72} -6.28304i q^{73} +6.80951 q^{74} -1.00000 q^{75} -2.29078i q^{76} -6.10153 q^{77} +2.96774 q^{79} -1.00000i q^{80} +1.00000 q^{81} -4.64516 q^{82} +15.8719i q^{83} -1.32258i q^{84} +4.00000i q^{85} -8.60562i q^{86} -2.02283 q^{87} +4.61335 q^{88} -11.8641i q^{89} +1.00000 q^{90} +8.66799 q^{92} -10.1321i q^{93} +9.10926 q^{94} +2.29078 q^{95} +1.00000i q^{96} -8.81894i q^{97} +5.25078i q^{98} +4.61335i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	1.32258i	0.499888i	0.968260	+	0.249944i	$0.0804122\pi$
$7$	1.32258i	0.499888i	−0.968260	+	0.249944i	$0.919588\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	4.61335i	1.39098i	0.718536	+	0.695489i	$0.244812\pi$
$11$	4.61335i	1.39098i	−0.718536	+	0.695489i	$0.755188\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−1.32258	−0.353474
$15$	− 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
$19$	2.29078i	0.525540i	0.964859	+	0.262770i	$0.0846360\pi$
$19$	2.29078i	0.525540i	−0.964859	+	0.262770i	$0.915364\pi$
$20$	1.00000i	0.223607i
$21$	1.32258i	0.288611i
$22$	−4.61335	−0.983571
$23$	−8.66799	−1.80740	−0.903700	−	0.428166i	$-0.859160\pi$
$23$	−8.66799	−1.80740	−0.903700	+	0.428166i	$0.859160\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	− 1.32258i	− 0.249944i
$29$	−2.02283	−0.375629	−0.187815	−	0.982204i	$-0.560140\pi$
$29$	−2.02283	−0.375629	−0.187815	+	0.982204i	$0.560140\pi$
$30$	1.00000	0.182574
$31$	− 10.1321i	− 1.81978i	−0.414853	−	0.909888i	$-0.636167\pi$
$31$	− 10.1321i	− 1.81978i	0.414853	−	0.909888i	$-0.363833\pi$
$32$	1.00000i	0.176777i
$33$	4.61335i	0.803082i
$34$	− 4.00000i	− 0.685994i
$35$	1.32258	0.223557
$36$	−1.00000	−0.166667
$37$	− 6.80951i	− 1.11948i	−0.828670	−	0.559738i	$-0.810902\pi$
$37$	− 6.80951i	− 1.11948i	0.828670	−	0.559738i	$-0.189098\pi$
$38$	−2.29078	−0.371613
$39$	0	0
$40$	−1.00000	−0.158114
$41$	4.64516i	0.725452i	0.931896	+	0.362726i	$0.118154\pi$
$41$	4.64516i	0.725452i	−0.931896	+	0.362726i	$0.881846\pi$
$42$	−1.32258	−0.204078
$43$	−8.60562	−1.31235	−0.656173	−	0.754611i	$-0.727826\pi$
$43$	−8.60562	−1.31235	−0.656173	+	0.754611i	$0.727826\pi$
$44$	− 4.61335i	− 0.695489i
$45$	− 1.00000i	− 0.149071i
$46$	− 8.66799i	− 1.27802i
$47$	− 9.10926i	− 1.32872i	−0.747412	−	0.664361i	$-0.768704\pi$
$47$	− 9.10926i	− 1.32872i	0.747412	−	0.664361i	$-0.231296\pi$
$48$	1.00000	0.144338
$49$	5.25078	0.750112
$50$	− 1.00000i	− 0.141421i
$51$	−4.00000	−0.560112
$52$	0	0
$53$	0.826674	0.113552	0.0567762	−	0.998387i	$-0.481918\pi$
$53$	0.826674	0.113552	0.0567762	+	0.998387i	$0.481918\pi$
$54$	1.00000i	0.136083i
$55$	4.61335	0.622065
$56$	1.32258	0.176737
$57$	2.29078i	0.303421i
$58$	− 2.02283i	− 0.265610i
$59$	3.14152i	0.408991i	0.978867	+	0.204496i	$0.0655554\pi$
$59$	3.14152i	0.408991i	−0.978867	+	0.204496i	$0.934445\pi$
$60$	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	$0.489077\pi$
$62$	10.1321	1.28678
$63$	1.32258i	0.166629i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.61335	−0.567865
$67$	3.18106i	0.388628i	0.980939	+	0.194314i	$0.0622481\pi$
$67$	3.18106i	0.388628i	−0.980939	+	0.194314i	$0.937752\pi$
$68$	4.00000	0.485071
$69$	−8.66799	−1.04350
$70$	1.32258i	0.158079i
$71$	− 11.3360i	− 1.34533i	−0.739946	−	0.672666i	$-0.765149\pi$
$71$	− 11.3360i	− 1.34533i	0.739946	−	0.672666i	$-0.234851\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 6.28304i	− 0.735375i	−0.929949	−	0.367687i	$-0.880150\pi$
$73$	− 6.28304i	− 0.735375i	0.929949	−	0.367687i	$-0.119850\pi$
$74$	6.80951	0.791589
$75$	−1.00000	−0.115470
$76$	− 2.29078i	− 0.262770i
$77$	−6.10153	−0.695334
$78$	0	0
$79$	2.96774	0.333897	0.166948	−	0.985966i	$-0.446609\pi$
$79$	2.96774	0.333897	0.166948	+	0.985966i	$0.446609\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	−4.64516	−0.512972
$83$	15.8719i	1.74216i	0.491138	+	0.871082i	$0.336581\pi$
$83$	15.8719i	1.74216i	−0.491138	+	0.871082i	$0.663419\pi$
$84$	− 1.32258i	− 0.144305i
$85$	4.00000i	0.433861i
$86$	− 8.60562i	− 0.927968i
$87$	−2.02283	−0.216870
$88$	4.61335	0.491785
$89$	− 11.8641i	− 1.25760i	−0.777569	−	0.628798i	$-0.783547\pi$
$89$	− 11.8641i	− 1.25760i	0.777569	−	0.628798i	$-0.216453\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	8.66799	0.903700
$93$	− 10.1321i	− 1.05065i
$94$	9.10926	0.939549
$95$	2.29078	0.235029
$96$	1.00000i	0.102062i
$97$	− 8.81894i	− 0.895428i	−0.894177	−	0.447714i	$-0.852238\pi$
$97$	− 8.81894i	− 0.895428i	0.894177	−	0.447714i	$-0.147762\pi$
$98$	5.25078i	0.530409i
$99$	4.61335i	0.463660i
$100$	1.00000	0.100000
$101$	7.22671	0.719085	0.359542	−	0.933129i	$-0.382933\pi$
$101$	7.22671	0.719085	0.359542	+	0.933129i	$0.382933\pi$
$102$	− 4.00000i	− 0.396059i
$103$	−18.4000	−1.81301	−0.906505	−	0.422196i	$-0.861260\pi$
$103$	−18.4000	−1.81301	−0.906505	+	0.422196i	$0.861260\pi$
$104$	0	0
$105$	1.32258	0.129071
$106$	0.826674i	0.0802936i
$107$	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
$109$	13.2267i	1.26689i	0.773788	+	0.633445i	$0.218360\pi$
$109$	13.2267i	1.26689i	−0.773788	+	0.633445i	$0.781640\pi$
$110$	4.61335i	0.439866i
$111$	− 6.80951i	− 0.646330i
$112$	1.32258i	0.124972i
$113$	−8.08643	−0.760708	−0.380354	−	0.924841i	$-0.624198\pi$
$113$	−8.08643	−0.760708	−0.380354	+	0.924841i	$0.624198\pi$
$114$	−2.29078	−0.214551
$115$	8.66799i	0.808294i
$116$	2.02283	0.187815
$117$	0	0
$118$	−3.14152	−0.289201
$119$	− 5.29032i	− 0.484963i
$120$	−1.00000	−0.0912871
$121$	−10.2830	−0.934822
$122$	0.535898i	0.0485180i
$123$	4.64516i	0.418840i
$124$	10.1321i	0.909888i
$125$	1.00000i	0.0894427i
$126$	−1.32258	−0.117825
$127$	−11.4718	−1.01796	−0.508980	−	0.860778i	$-0.669977\pi$
$127$	−11.4718	−1.01796	−0.508980	+	0.860778i	$0.669977\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−8.60562	−0.757683
$130$	0	0
$131$	−15.9906	−1.39710	−0.698551	−	0.715560i	$-0.746172\pi$
$131$	−15.9906	−1.39710	−0.698551	+	0.715560i	$0.746172\pi$
$132$	− 4.61335i	− 0.401541i
$133$	−3.02973	−0.262711
$134$	−3.18106	−0.274802
$135$	− 1.00000i	− 0.0860663i
$136$	4.00000i	0.342997i
$137$	16.0697i	1.37293i	0.727163	+	0.686465i	$0.240838\pi$
$137$	16.0697i	1.37293i	−0.727163	+	0.686465i	$0.759162\pi$
$138$	− 8.66799i	− 0.737868i
$139$	9.32258	0.790731	0.395365	−	0.918524i	$-0.370618\pi$
$139$	9.32258	0.790731	0.395365	+	0.918524i	$0.370618\pi$
$140$	−1.32258	−0.111778
$141$	− 9.10926i	− 0.767138i
$142$	11.3360	0.951294
$143$	0	0
$144$	1.00000	0.0833333
$145$	2.02283i	0.167987i
$146$	6.28304	0.519988
$147$	5.25078	0.433077
$148$	6.80951i	0.559738i
$149$	4.08519i	0.334672i	0.985900	+	0.167336i	$0.0535164\pi$
$149$	4.08519i	0.334672i	−0.985900	+	0.167336i	$0.946484\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	− 19.9811i	− 1.62604i	−0.582235	−	0.813021i	$-0.697822\pi$
$151$	− 19.9811i	− 1.62604i	0.582235	−	0.813021i	$-0.302178\pi$
$152$	2.29078	0.185806
$153$	−4.00000	−0.323381
$154$	− 6.10153i	− 0.491675i
$155$	−10.1321	−0.813829
$156$	0	0
$157$	7.13379	0.569338	0.284669	−	0.958626i	$-0.408116\pi$
$157$	7.13379	0.569338	0.284669	+	0.958626i	$0.408116\pi$
$158$	2.96774i	0.236101i
$159$	0.826674	0.0655595
$160$	1.00000	0.0790569
$161$	− 11.4641i	− 0.903498i
$162$	1.00000i	0.0785674i
$163$	− 9.89470i	− 0.775012i	−0.921867	−	0.387506i	$-0.873337\pi$
$163$	− 9.89470i	− 0.775012i	0.921867	−	0.387506i	$-0.126663\pi$
$164$	− 4.64516i	− 0.362726i
$165$	4.61335	0.359149
$166$	−15.8719	−1.23190
$167$	− 24.0375i	− 1.86007i	−0.367465	−	0.930037i	$-0.619774\pi$
$167$	− 24.0375i	− 1.86007i	0.367465	−	0.930037i	$-0.380226\pi$
$168$	1.32258	0.102039
$169$	0	0
$170$	−4.00000	−0.306786
$171$	2.29078i	0.175180i
$172$	8.60562	0.656173
$173$	−2.52817	−0.192213	−0.0961065	−	0.995371i	$-0.530639\pi$
$173$	−2.52817	−0.192213	−0.0961065	+	0.995371i	$0.530639\pi$
$174$	− 2.02283i	− 0.153350i
$175$	− 1.32258i	− 0.0999776i
$176$	4.61335i	0.347745i
$177$	3.14152i	0.236131i
$178$	11.8641	0.889255
$179$	6.27568	0.469066	0.234533	−	0.972108i	$-0.424644\pi$
$179$	6.27568	0.469066	0.234533	+	0.972108i	$0.424644\pi$
$180$	1.00000i	0.0745356i
$181$	−12.2830	−0.912991	−0.456496	−	0.889726i	$-0.650896\pi$
$181$	−12.2830	−0.912991	−0.456496	+	0.889726i	$0.650896\pi$
$182$	0	0
$183$	0.535898	0.0396147
$184$	8.66799i	0.639012i
$185$	−6.80951	−0.500645
$186$	10.1321	0.742921
$187$	− 18.4534i	− 1.34945i
$188$	9.10926i	0.664361i
$189$	1.32258i	0.0962035i
$190$	2.29078i	0.166190i
$191$	−9.74715	−0.705279	−0.352639	−	0.935759i	$-0.614716\pi$
$191$	−9.74715	−0.705279	−0.352639	+	0.935759i	$0.614716\pi$
$192$	−1.00000	−0.0721688
$193$	13.7626i	0.990654i	0.868707	+	0.495327i	$0.164952\pi$
$193$	13.7626i	0.990654i	−0.868707	+	0.495327i	$0.835048\pi$
$194$	8.81894	0.633163
$195$	0	0
$196$	−5.25078	−0.375056
$197$	10.1415i	0.722554i	0.932459	+	0.361277i	$0.117659\pi$
$197$	10.1415i	0.722554i	−0.932459	+	0.361277i	$0.882341\pi$
$198$	−4.61335	−0.327857
$199$	−9.57336	−0.678638	−0.339319	−	0.940671i	$-0.610197\pi$
$199$	−9.57336	−0.678638	−0.339319	+	0.940671i	$0.610197\pi$
$200$	1.00000i	0.0707107i
$201$	3.18106i	0.224375i
$202$	7.22671i	0.508470i
$203$	− 2.67535i	− 0.187773i
$204$	4.00000	0.280056
$205$	4.64516	0.324432
$206$	− 18.4000i	− 1.28199i
$207$	−8.66799	−0.602467
$208$	0	0
$209$	−10.5682	−0.731015
$210$	1.32258i	0.0912667i
$211$	−1.08519	−0.0747074	−0.0373537	−	0.999302i	$-0.511893\pi$
$211$	−1.08519	−0.0747074	−0.0373537	+	0.999302i	$0.511893\pi$
$212$	−0.826674	−0.0567762
$213$	− 11.3360i	− 0.776728i
$214$	3.07180i	0.209984i
$215$	8.60562i	0.586899i
$216$	− 1.00000i	− 0.0680414i
$217$	13.4005	0.909685
$218$	−13.2267	−0.895826
$219$	− 6.28304i	− 0.424569i
$220$	−4.61335	−0.311032
$221$	0	0
$222$	6.80951	0.457024
$223$	− 24.9416i	− 1.67021i	−0.550089	−	0.835106i	$-0.685406\pi$
$223$	− 24.9416i	− 1.67021i	0.550089	−	0.835106i	$-0.314594\pi$
$224$	−1.32258	−0.0883686
$225$	−1.00000	−0.0666667
$226$	− 8.08643i	− 0.537902i
$227$	− 19.3205i	− 1.28235i	−0.767396	−	0.641174i	$-0.778448\pi$
$227$	− 19.3205i	− 1.28235i	0.767396	−	0.641174i	$-0.221552\pi$
$228$	− 2.29078i	− 0.151710i
$229$	4.15491i	0.274564i	0.990532	+	0.137282i	$0.0438367\pi$
$229$	4.15491i	0.274564i	−0.990532	+	0.137282i	$0.956163\pi$
$230$	−8.66799	−0.571550
$231$	−6.10153	−0.401451
$232$	2.02283i	0.132805i
$233$	16.9665	1.11151	0.555756	−	0.831346i	$-0.312429\pi$
$233$	16.9665	1.11151	0.555756	+	0.831346i	$0.312429\pi$
$234$	0	0
$235$	−9.10926	−0.594223
$236$	− 3.14152i	− 0.204496i
$237$	2.96774	0.192775
$238$	5.29032	0.342920
$239$	− 6.34416i	− 0.410370i	−0.978723	−	0.205185i	$-0.934220\pi$
$239$	− 6.34416i	− 0.410370i	0.978723	−	0.205185i	$-0.0657795\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	24.1855i	1.55792i	0.627072	+	0.778962i	$0.284253\pi$
$241$	24.1855i	1.55792i	−0.627072	+	0.778962i	$0.715747\pi$
$242$	− 10.2830i	− 0.661019i
$243$	1.00000	0.0641500
$244$	−0.535898	−0.0343074
$245$	− 5.25078i	− 0.335460i
$246$	−4.64516	−0.296165
$247$	0	0
$248$	−10.1321	−0.643388
$249$	15.8719i	1.00584i
$250$	−1.00000	−0.0632456
$251$	−8.82497	−0.557027	−0.278514	−	0.960432i	$-0.589842\pi$
$251$	−8.82497	−0.557027	−0.278514	+	0.960432i	$0.589842\pi$
$252$	− 1.32258i	− 0.0833147i
$253$	− 39.9885i	− 2.51406i
$254$	− 11.4718i	− 0.719807i
$255$	4.00000i	0.250490i
$256$	1.00000	0.0625000
$257$	−8.38494	−0.523038	−0.261519	−	0.965198i	$-0.584223\pi$
$257$	−8.38494	−0.523038	−0.261519	+	0.965198i	$0.584223\pi$
$258$	− 8.60562i	− 0.535763i
$259$	9.00612	0.559613
$260$	0	0
$261$	−2.02283	−0.125210
$262$	− 15.9906i	− 0.987900i
$263$	1.92130	0.118472	0.0592361	−	0.998244i	$-0.481134\pi$
$263$	1.92130	0.118472	0.0592361	+	0.998244i	$0.481134\pi$
$264$	4.61335	0.283932
$265$	− 0.826674i	− 0.0507822i
$266$	− 3.02973i	− 0.185765i
$267$	− 11.8641i	− 0.726073i
$268$	− 3.18106i	− 0.194314i
$269$	−32.3098	−1.96996	−0.984982	−	0.172655i	$-0.944766\pi$
$269$	−32.3098	−1.96996	−0.984982	+	0.172655i	$0.944766\pi$
$270$	1.00000	0.0608581
$271$	− 23.4526i	− 1.42464i	−0.701853	−	0.712322i	$-0.747644\pi$
$271$	− 23.4526i	− 1.42464i	0.701853	−	0.712322i	$-0.252356\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−16.0697	−0.970808
$275$	− 4.61335i	− 0.278196i
$276$	8.66799	0.521751
$277$	9.00566	0.541098	0.270549	−	0.962706i	$-0.412795\pi$
$277$	9.00566	0.541098	0.270549	+	0.962706i	$0.412795\pi$
$278$	9.32258i	0.559131i
$279$	− 10.1321i	− 0.606592i
$280$	− 1.32258i	− 0.0790393i
$281$	1.71696i	0.102425i	0.998688	+	0.0512125i	$0.0163086\pi$
$281$	1.71696i	0.102425i	−0.998688	+	0.0512125i	$0.983691\pi$
$282$	9.10926	0.542449
$283$	1.54929	0.0920957	0.0460479	−	0.998939i	$-0.485337\pi$
$283$	1.54929	0.0920957	0.0460479	+	0.998939i	$0.485337\pi$
$284$	11.3360i	0.672666i
$285$	2.29078	0.135694
$286$	0	0
$287$	−6.14359	−0.362645
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	−2.02283	−0.118784
$291$	− 8.81894i	− 0.516976i
$292$	6.28304i	0.367687i
$293$	30.4057i	1.77632i	0.459535	+	0.888160i	$0.348016\pi$
$293$	30.4057i	1.77632i	−0.459535	+	0.888160i	$0.651984\pi$
$294$	5.25078i	0.306232i
$295$	3.14152	0.182906
$296$	−6.80951	−0.395795
$297$	4.61335i	0.267694i
$298$	−4.08519	−0.236649
$299$	0	0
$300$	1.00000	0.0577350
$301$	− 11.3816i	− 0.656026i
$302$	19.9811	1.14978
$303$	7.22671	0.415164
$304$	2.29078i	0.131385i
$305$	− 0.535898i	− 0.0306855i
$306$	− 4.00000i	− 0.228665i
$307$	9.82622i	0.560812i	0.959882	+	0.280406i	$0.0904691\pi$
$307$	9.82622i	0.560812i	−0.959882	+	0.280406i	$0.909531\pi$
$308$	6.10153	0.347667
$309$	−18.4000	−1.04674
$310$	− 10.1321i	− 0.575464i
$311$	−3.40049	−0.192824	−0.0964121	−	0.995342i	$-0.530737\pi$
$311$	−3.40049	−0.192824	−0.0964121	+	0.995342i	$0.530737\pi$
$312$	0	0
$313$	−16.2340	−0.917599	−0.458800	−	0.888540i	$-0.651720\pi$
$313$	−16.2340	−0.917599	−0.458800	+	0.888540i	$0.651720\pi$
$314$	7.13379i	0.402583i
$315$	1.32258	0.0745189
$316$	−2.96774	−0.166948
$317$	23.5231i	1.32119i	0.750742	+	0.660596i	$0.229696\pi$
$317$	23.5231i	1.32119i	−0.750742	+	0.660596i	$0.770304\pi$
$318$	0.826674i	0.0463576i
$319$	− 9.33201i	− 0.522493i
$320$	1.00000i	0.0559017i
$321$	3.07180	0.171451
$322$	11.4641	0.638869
$323$	− 9.16310i	− 0.509849i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	9.89470	0.548016
$327$	13.2267i	0.731439i
$328$	4.64516	0.256486
$329$	12.0477	0.664213
$330$	4.61335i	0.253957i
$331$	18.0904i	0.994338i	0.867654	+	0.497169i	$0.165627\pi$
$331$	18.0904i	0.994338i	−0.867654	+	0.497169i	$0.834373\pi$
$332$	− 15.8719i	− 0.871082i
$333$	− 6.80951i	− 0.373159i
$334$	24.0375	1.31527
$335$	3.18106	0.173800
$336$	1.32258i	0.0721526i
$337$	−5.69900	−0.310444	−0.155222	−	0.987880i	$-0.549609\pi$
$337$	−5.69900	−0.310444	−0.155222	+	0.987880i	$0.549609\pi$
$338$	0	0
$339$	−8.08643	−0.439195
$340$	− 4.00000i	− 0.216930i
$341$	46.7429	2.53127
$342$	−2.29078	−0.123871
$343$	16.2026i	0.874860i
$344$	8.60562i	0.463984i
$345$	8.66799i	0.466669i
$346$	− 2.52817i	− 0.135915i
$347$	−15.7471	−0.845351	−0.422676	−	0.906281i	$-0.638909\pi$
$347$	−15.7471	−0.845351	−0.422676	+	0.906281i	$0.638909\pi$
$348$	2.02283	0.108435
$349$	− 15.3205i	− 0.820088i	−0.912066	−	0.410044i	$-0.865513\pi$
$349$	− 15.3205i	− 0.820088i	0.912066	−	0.410044i	$-0.134487\pi$
$350$	1.32258	0.0706949
$351$	0	0
$352$	−4.61335	−0.245893
$353$	− 1.19993i	− 0.0638657i	−0.999490	−	0.0319328i	$-0.989834\pi$
$353$	− 1.19993i	− 0.0638657i	0.999490	−	0.0319328i	$-0.0101663\pi$
$354$	−3.14152	−0.166970
$355$	−11.3360	−0.601651
$356$	11.8641i	0.628798i
$357$	− 5.29032i	− 0.279993i
$358$	6.27568i	0.331680i
$359$	16.2830i	0.859386i	0.902975	+	0.429693i	$0.141378\pi$
$359$	16.2830i	0.859386i	−0.902975	+	0.429693i	$0.858622\pi$
$360$	−1.00000	−0.0527046
$361$	13.7523	0.723808
$362$	− 12.2830i	− 0.645582i
$363$	−10.2830	−0.539720
$364$	0	0
$365$	−6.28304	−0.328870
$366$	0.535898i	0.0280119i
$367$	−19.6345	−1.02491	−0.512456	−	0.858714i	$-0.671264\pi$
$367$	−19.6345	−1.02491	−0.512456	+	0.858714i	$0.671264\pi$
$368$	−8.66799	−0.451850
$369$	4.64516i	0.241817i
$370$	− 6.80951i	− 0.354009i
$371$	1.09334i	0.0567635i
$372$	10.1321i	0.525324i
$373$	−2.55748	−0.132421	−0.0662106	−	0.997806i	$-0.521091\pi$
$373$	−2.55748	−0.132421	−0.0662106	+	0.997806i	$0.521091\pi$
$374$	18.4534	0.954204
$375$	1.00000i	0.0516398i
$376$	−9.10926	−0.469774
$377$	0	0
$378$	−1.32258	−0.0680262
$379$	15.1088i	0.776087i	0.921641	+	0.388044i	$0.126849\pi$
$379$	15.1088i	0.776087i	−0.921641	+	0.388044i	$0.873151\pi$
$380$	−2.29078	−0.117514
$381$	−11.4718	−0.587720
$382$	− 9.74715i	− 0.498707i
$383$	− 18.7303i	− 0.957076i	−0.878067	−	0.478538i	$-0.841167\pi$
$383$	− 18.7303i	− 0.957076i	0.878067	−	0.478538i	$-0.158833\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	6.10153i	0.310963i
$386$	−13.7626	−0.700498
$387$	−8.60562	−0.437448
$388$	8.81894i	0.447714i
$389$	6.96899	0.353342	0.176671	−	0.984270i	$-0.443467\pi$
$389$	6.96899	0.353342	0.176671	+	0.984270i	$0.443467\pi$
$390$	0	0
$391$	34.6719	1.75344
$392$	− 5.25078i	− 0.265205i
$393$	−15.9906	−0.806617
$394$	−10.1415	−0.510922
$395$	− 2.96774i	− 0.149323i
$396$	− 4.61335i	− 0.231830i
$397$	− 18.5721i	− 0.932108i	−0.884756	−	0.466054i	$-0.845675\pi$
$397$	− 18.5721i	− 0.932108i	0.884756	−	0.466054i	$-0.154325\pi$
$398$	− 9.57336i	− 0.479869i
$399$	−3.02973	−0.151676
$400$	−1.00000	−0.0500000
$401$	29.6904i	1.48267i	0.671138	+	0.741333i	$0.265806\pi$
$401$	29.6904i	1.48267i	−0.671138	+	0.741333i	$0.734194\pi$
$402$	−3.18106	−0.158657
$403$	0	0
$404$	−7.22671	−0.359542
$405$	− 1.00000i	− 0.0496904i
$406$	2.67535	0.132775
$407$	31.4147	1.55717
$408$	4.00000i	0.198030i
$409$	− 24.3355i	− 1.20331i	−0.798755	−	0.601657i	$-0.794507\pi$
$409$	− 24.3355i	− 1.20331i	0.798755	−	0.601657i	$-0.205493\pi$
$410$	4.64516i	0.229408i
$411$	16.0697i	0.792661i
$412$	18.4000	0.906505
$413$	−4.15491	−0.204450
$414$	− 8.66799i	− 0.426008i
$415$	15.8719	0.779119
$416$	0	0
$417$	9.32258	0.456529
$418$	− 10.5682i	− 0.516906i
$419$	−2.88255	−0.140822	−0.0704109	−	0.997518i	$-0.522431\pi$
$419$	−2.88255	−0.140822	−0.0704109	+	0.997518i	$0.522431\pi$
$420$	−1.32258	−0.0645353
$421$	− 4.94707i	− 0.241106i	−0.992707	−	0.120553i	$-0.961533\pi$
$421$	− 4.94707i	− 0.241106i	0.992707	−	0.120553i	$-0.0384667\pi$
$422$	− 1.08519i	− 0.0528261i
$423$	− 9.10926i	− 0.442907i
$424$	− 0.826674i	− 0.0401468i
$425$	4.00000	0.194029
$426$	11.3360	0.549230
$427$	0.708768i	0.0342997i
$428$	−3.07180	−0.148481
$429$	0	0
$430$	−8.60562	−0.415000
$431$	17.7626i	0.855595i	0.903875	+	0.427797i	$0.140710\pi$
$431$	17.7626i	0.855595i	−0.903875	+	0.427797i	$0.859290\pi$
$432$	1.00000	0.0481125
$433$	−35.8375	−1.72224	−0.861121	−	0.508400i	$-0.830237\pi$
$433$	−35.8375	−1.72224	−0.861121	+	0.508400i	$0.830237\pi$
$434$	13.4005i	0.643244i
$435$	2.02283i	0.0969871i
$436$	− 13.2267i	− 0.633445i
$437$	− 19.8564i	− 0.949861i
$438$	6.28304	0.300215
$439$	−10.0904	−0.481588	−0.240794	−	0.970576i	$-0.577408\pi$
$439$	−10.0904	−0.481588	−0.240794	+	0.970576i	$0.577408\pi$
$440$	− 4.61335i	− 0.219933i
$441$	5.25078	0.250037
$442$	0	0
$443$	13.6981	0.650816	0.325408	−	0.945574i	$-0.394498\pi$
$443$	13.6981	0.650816	0.325408	+	0.945574i	$0.394498\pi$
$444$	6.80951i	0.323165i
$445$	−11.8641	−0.562414
$446$	24.9416	1.18102
$447$	4.08519i	0.193223i
$448$	− 1.32258i	− 0.0624860i
$449$	15.3128i	0.722655i	0.932439	+	0.361327i	$0.117676\pi$
$449$	15.3128i	0.722655i	−0.932439	+	0.361327i	$0.882324\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	−21.4298	−1.00909
$452$	8.08643	0.380354
$453$	− 19.9811i	− 0.938795i
$454$	19.3205	0.906756
$455$	0	0
$456$	2.29078	0.107275
$457$	15.5554i	0.727651i	0.931467	+	0.363826i	$0.118530\pi$
$457$	15.5554i	0.727651i	−0.931467	+	0.363826i	$0.881470\pi$
$458$	−4.15491	−0.194146
$459$	−4.00000	−0.186704
$460$	− 8.66799i	− 0.404147i
$461$	15.6431i	0.728571i	0.931287	+	0.364286i	$0.118687\pi$
$461$	15.6431i	0.728571i	−0.931287	+	0.364286i	$0.881313\pi$
$462$	− 6.10153i	− 0.283869i
$463$	13.7170i	0.637481i	0.947842	+	0.318741i	$0.103260\pi$
$463$	13.7170i	0.637481i	−0.947842	+	0.318741i	$0.896740\pi$
$464$	−2.02283	−0.0939073
$465$	−10.1321	−0.469864
$466$	16.9665i	0.785958i
$467$	−0.943666	−0.0436677	−0.0218338	−	0.999762i	$-0.506950\pi$
$467$	−0.943666	−0.0436677	−0.0218338	+	0.999762i	$0.506950\pi$
$468$	0	0
$469$	−4.20720	−0.194271
$470$	− 9.10926i	− 0.420179i
$471$	7.13379	0.328708
$472$	3.14152	0.144600
$473$	− 39.7008i	− 1.82544i
$474$	2.96774i	0.136313i
$475$	− 2.29078i	− 0.105108i
$476$	5.29032i	0.242481i
$477$	0.826674	0.0378508
$478$	6.34416	0.290175
$479$	11.1020i	0.507263i	0.967301	+	0.253631i	$0.0816250\pi$
$479$	11.1020i	0.507263i	−0.967301	+	0.253631i	$0.918375\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−24.1855	−1.10162
$483$	− 11.4641i	− 0.521635i
$484$	10.2830	0.467411
$485$	−8.81894	−0.400448
$486$	1.00000i	0.0453609i
$487$	− 32.9416i	− 1.49273i	−0.665539	−	0.746363i	$-0.731798\pi$
$487$	− 32.9416i	− 1.49273i	0.665539	−	0.746363i	$-0.268202\pi$
$488$	− 0.535898i	− 0.0242590i
$489$	− 9.89470i	− 0.447454i
$490$	5.25078	0.237206
$491$	−28.4513	−1.28399	−0.641996	−	0.766708i	$-0.721893\pi$
$491$	−28.4513	−1.28399	−0.641996	+	0.766708i	$0.721893\pi$
$492$	− 4.64516i	− 0.209420i
$493$	8.09130	0.364414
$494$	0	0
$495$	4.61335	0.207355
$496$	− 10.1321i	− 0.454944i
$497$	14.9927	0.672516
$498$	−15.8719	−0.711235
$499$	− 4.10926i	− 0.183956i	−0.995761	−	0.0919779i	$-0.970681\pi$
$499$	− 4.10926i	− 0.183956i	0.995761	−	0.0919779i	$-0.0293189\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 24.0375i	− 1.07391i
$502$	− 8.82497i	− 0.393878i
$503$	5.73601	0.255756	0.127878	−	0.991790i	$-0.459183\pi$
$503$	5.73601	0.255756	0.127878	+	0.991790i	$0.459183\pi$
$504$	1.32258	0.0589124
$505$	− 7.22671i	− 0.321584i
$506$	39.9885	1.77771
$507$	0	0
$508$	11.4718	0.508980
$509$	15.1415i	0.671136i	0.942016	+	0.335568i	$0.108928\pi$
$509$	15.1415i	0.671136i	−0.942016	+	0.335568i	$0.891072\pi$
$510$	−4.00000	−0.177123
$511$	8.30983	0.367605
$512$	1.00000i	0.0441942i
$513$	2.29078i	0.101140i
$514$	− 8.38494i	− 0.369844i
$515$	18.4000i	0.810802i
$516$	8.60562	0.378841
$517$	42.0243	1.84822
$518$	9.00612i	0.395706i
$519$	−2.52817	−0.110974
$520$	0	0
$521$	8.93027	0.391242	0.195621	−	0.980680i	$-0.437328\pi$
$521$	8.93027	0.391242	0.195621	+	0.980680i	$0.437328\pi$
$522$	− 2.02283i	− 0.0885367i
$523$	−9.81687	−0.429262	−0.214631	−	0.976695i	$-0.568855\pi$
$523$	−9.81687	−0.429262	−0.214631	+	0.976695i	$0.568855\pi$
$524$	15.9906	0.698551
$525$	− 1.32258i	− 0.0577221i
$526$	1.92130i	0.0837725i
$527$	40.5283i	1.76544i
$528$	4.61335i	0.200771i
$529$	52.1340	2.26669
$530$	0.826674	0.0359084
$531$	3.14152i	0.136330i
$532$	3.02973	0.131356
$533$	0	0
$534$	11.8641	0.513411
$535$	− 3.07180i	− 0.132805i
$536$	3.18106	0.137401
$537$	6.27568	0.270816
$538$	− 32.3098i	− 1.39298i
$539$	24.2237i	1.04339i
$540$	1.00000i	0.0430331i
$541$	3.80826i	0.163730i	0.996643	+	0.0818650i	$0.0260876\pi$
$541$	3.80826i	0.163730i	−0.996643	+	0.0818650i	$0.973912\pi$
$542$	23.4526	1.00738
$543$	−12.2830	−0.527116
$544$	− 4.00000i	− 0.171499i
$545$	13.2267	0.566570
$546$	0	0
$547$	45.5847	1.94906	0.974530	−	0.224257i	$-0.0719954\pi$
$547$	45.5847	1.94906	0.974530	+	0.224257i	$0.0719954\pi$
$548$	− 16.0697i	− 0.686465i
$549$	0.535898	0.0228716
$550$	4.61335	0.196714
$551$	− 4.63384i	− 0.197408i
$552$	8.66799i	0.368934i
$553$	3.92507i	0.166911i
$554$	9.00566i	0.382614i
$555$	−6.80951	−0.289047
$556$	−9.32258	−0.395365
$557$	36.8591i	1.56177i	0.624674	+	0.780885i	$0.285232\pi$
$557$	36.8591i	1.56177i	−0.624674	+	0.780885i	$0.714768\pi$
$558$	10.1321	0.428925
$559$	0	0
$560$	1.32258	0.0558892
$561$	− 18.4534i	− 0.779104i
$562$	−1.71696	−0.0724254
$563$	−33.7739	−1.42340	−0.711701	−	0.702483i	$-0.752075\pi$
$563$	−33.7739	−1.42340	−0.711701	+	0.702483i	$0.752075\pi$
$564$	9.10926i	0.383569i
$565$	8.08643i	0.340199i
$566$	1.54929i	0.0651215i
$567$	1.32258i	0.0555431i
$568$	−11.3360	−0.475647
$569$	25.4318	1.06616	0.533079	−	0.846065i	$-0.321035\pi$
$569$	25.4318	1.06616	0.533079	+	0.846065i	$0.321035\pi$
$570$	2.29078i	0.0959500i
$571$	13.3682	0.559443	0.279722	−	0.960081i	$-0.409758\pi$
$571$	13.3682	0.559443	0.279722	+	0.960081i	$0.409758\pi$
$572$	0	0
$573$	−9.74715	−0.407193
$574$	− 6.14359i	− 0.256429i
$575$	8.66799	0.361480
$576$	−1.00000	−0.0416667
$577$	− 9.57428i	− 0.398582i	−0.979940	−	0.199291i	$-0.936136\pi$
$577$	− 9.57428i	− 0.398582i	0.979940	−	0.199291i	$-0.0638639\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	13.7626i	0.571954i
$580$	− 2.02283i	− 0.0839933i
$581$	−20.9918	−0.870887
$582$	8.81894	0.365557
$583$	3.81374i	0.157949i
$584$	−6.28304	−0.259994
$585$	0	0
$586$	−30.4057	−1.25605
$587$	− 1.40049i	− 0.0578045i	−0.999582	−	0.0289023i	$-0.990799\pi$
$587$	− 1.40049i	− 0.0578045i	0.999582	−	0.0289023i	$-0.00920116\pi$
$588$	−5.25078	−0.216539
$589$	23.2103	0.956365
$590$	3.14152i	0.129334i
$591$	10.1415i	0.417166i
$592$	− 6.80951i	− 0.279869i
$593$	− 10.7303i	− 0.440643i	−0.975427	−	0.220321i	$-0.929289\pi$
$593$	− 10.7303i	− 0.440643i	0.975427	−	0.220321i	$-0.0707106\pi$
$594$	−4.61335	−0.189288
$595$	−5.29032	−0.216882
$596$	− 4.08519i	− 0.167336i
$597$	−9.57336	−0.391812
$598$	0	0
$599$	−40.7967	−1.66691	−0.833453	−	0.552590i	$-0.813640\pi$
$599$	−40.7967	−1.66691	−0.833453	+	0.552590i	$0.813640\pi$
$600$	1.00000i	0.0408248i
$601$	42.8832	1.74924	0.874621	−	0.484808i	$-0.161110\pi$
$601$	42.8832	1.74924	0.874621	+	0.484808i	$0.161110\pi$
$602$	11.3816	0.463880
$603$	3.18106i	0.129543i
$604$	19.9811i	0.813021i
$605$	10.2830i	0.418065i
$606$	7.22671i	0.293565i
$607$	−6.87233	−0.278939	−0.139470	−	0.990226i	$-0.544540\pi$
$607$	−6.87233	−0.278939	−0.139470	+	0.990226i	$0.544540\pi$
$608$	−2.29078	−0.0929032
$609$	− 2.67535i	− 0.108411i
$610$	0.535898	0.0216979
$611$	0	0
$612$	4.00000	0.161690
$613$	0.761361i	0.0307511i	0.999882	+	0.0153755i	$0.00489438\pi$
$613$	0.761361i	0.0307511i	−0.999882	+	0.0153755i	$0.995106\pi$
$614$	−9.82622	−0.396554
$615$	4.64516	0.187311
$616$	6.10153i	0.245838i
$617$	24.6022i	0.990448i	0.868765	+	0.495224i	$0.164914\pi$
$617$	24.6022i	0.990448i	−0.868765	+	0.495224i	$0.835086\pi$
$618$	− 18.4000i	− 0.740158i
$619$	17.2035i	0.691468i	0.938333	+	0.345734i	$0.112370\pi$
$619$	17.2035i	0.691468i	−0.938333	+	0.345734i	$0.887630\pi$
$620$	10.1321	0.406914
$621$	−8.66799	−0.347834
$622$	− 3.40049i	− 0.136347i
$623$	15.6913	0.628657
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 16.2340i	− 0.648841i
$627$	−10.5682	−0.422052
$628$	−7.13379	−0.284669
$629$	27.2380i	1.08605i
$630$	1.32258i	0.0526928i
$631$	2.40777i	0.0958517i	0.998851	+	0.0479259i	$0.0152611\pi$
$631$	2.40777i	0.0958517i	−0.998851	+	0.0479259i	$0.984739\pi$
$632$	− 2.96774i	− 0.118050i
$633$	−1.08519	−0.0431323
$634$	−23.5231	−0.934223
$635$	11.4718i	0.455246i
$636$	−0.826674	−0.0327797
$637$	0	0
$638$	9.33201	0.369458
$639$	− 11.3360i	− 0.448444i
$640$	−1.00000	−0.0395285
$641$	19.4775	0.769315	0.384657	−	0.923059i	$-0.374320\pi$
$641$	19.4775	0.769315	0.384657	+	0.923059i	$0.374320\pi$
$642$	3.07180i	0.121234i
$643$	− 23.9543i	− 0.944667i	−0.881420	−	0.472334i	$-0.843412\pi$
$643$	− 23.9543i	− 0.944667i	0.881420	−	0.472334i	$-0.156588\pi$
$644$	11.4641i	0.451749i
$645$	8.60562i	0.338846i
$646$	9.16310	0.360517
$647$	−18.2108	−0.715940	−0.357970	−	0.933733i	$-0.616531\pi$
$647$	−18.2108	−0.715940	−0.357970	+	0.933733i	$0.616531\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−14.4930	−0.568898
$650$	0	0
$651$	13.4005	0.525207
$652$	9.89470i	0.387506i
$653$	−48.1585	−1.88459	−0.942294	−	0.334787i	$-0.891336\pi$
$653$	−48.1585	−1.88459	−0.942294	+	0.334787i	$0.891336\pi$
$654$	−13.2267	−0.517205
$655$	15.9906i	0.624803i
$656$	4.64516i	0.181363i
$657$	− 6.28304i	− 0.245125i
$658$	12.0477i	0.469669i
$659$	−36.8254	−1.43451	−0.717257	−	0.696809i	$-0.754603\pi$
$659$	−36.8254	−1.43451	−0.717257	+	0.696809i	$0.754603\pi$
$660$	−4.61335	−0.179575
$661$	22.5318i	0.876384i	0.898881	+	0.438192i	$0.144381\pi$
$661$	22.5318i	0.876384i	−0.898881	+	0.438192i	$0.855619\pi$
$662$	−18.0904	−0.703103
$663$	0	0
$664$	15.8719	0.615948
$665$	3.02973i	0.117488i
$666$	6.80951	0.263863
$667$	17.5338	0.678912
$668$	24.0375i	0.930037i
$669$	− 24.9416i	− 0.964298i
$670$	3.18106i	0.122895i
$671$	2.47229i	0.0954417i
$672$	−1.32258	−0.0510196
$673$	−23.7437	−0.915254	−0.457627	−	0.889144i	$-0.651300\pi$
$673$	−23.7437	−0.915254	−0.457627	+	0.889144i	$0.651300\pi$
$674$	− 5.69900i	− 0.219517i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	1.16559	0.0447975	0.0223987	−	0.999749i	$-0.492870\pi$
$677$	1.16559	0.0447975	0.0223987	+	0.999749i	$0.492870\pi$
$678$	− 8.08643i	− 0.310558i
$679$	11.6638	0.447614
$680$	4.00000	0.153393
$681$	− 19.3205i	− 0.740363i
$682$	46.7429i	1.78988i
$683$	13.8719i	0.530792i	0.964139	+	0.265396i	$0.0855027\pi$
$683$	13.8719i	0.530792i	−0.964139	+	0.265396i	$0.914497\pi$
$684$	− 2.29078i	− 0.0875900i
$685$	16.0697	0.613993
$686$	−16.2026	−0.618620
$687$	4.15491i	0.158520i
$688$	−8.60562	−0.328086
$689$	0	0
$690$	−8.66799	−0.329985
$691$	41.2190i	1.56804i	0.620733	+	0.784022i	$0.286835\pi$
$691$	41.2190i	1.56804i	−0.620733	+	0.784022i	$0.713165\pi$
$692$	2.52817	0.0961065
$693$	−6.10153	−0.231778
$694$	− 15.7471i	− 0.597753i
$695$	− 9.32258i	− 0.353626i
$696$	2.02283i	0.0766750i
$697$	− 18.5806i	− 0.703792i
$698$	15.3205	0.579890
$699$	16.9665	0.641732
$700$	1.32258i	0.0499888i
$701$	−23.0112	−0.869122	−0.434561	−	0.900642i	$-0.643096\pi$
$701$	−23.0112	−0.869122	−0.434561	+	0.900642i	$0.643096\pi$
$702$	0	0
$703$	15.5991	0.588329
$704$	− 4.61335i	− 0.173872i
$705$	−9.10926	−0.343075
$706$	1.19993	0.0451599
$707$	9.55790i	0.359462i
$708$	− 3.14152i	− 0.118066i
$709$	− 17.0831i	− 0.641570i	−0.947152	−	0.320785i	$-0.896053\pi$
$709$	− 17.0831i	− 0.641570i	0.947152	−	0.320785i	$-0.103947\pi$
$710$	− 11.3360i	− 0.425431i
$711$	2.96774	0.111299
$712$	−11.8641	−0.444627
$713$	87.8248i	3.28906i
$714$	5.29032	0.197985
$715$	0	0
$716$	−6.27568	−0.234533
$717$	− 6.34416i	− 0.236927i
$718$	−16.2830	−0.607678
$719$	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
$721$	− 24.3355i	− 0.906302i
$722$	13.7523i	0.511809i
$723$	24.1855i	0.899467i
$724$	12.2830	0.456496
$725$	2.02283	0.0751259
$726$	− 10.2830i	− 0.381640i
$727$	31.8453	1.18108	0.590538	−	0.807010i	$-0.298916\pi$
$727$	31.8453	1.18108	0.590538	+	0.807010i	$0.298916\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 6.28304i	− 0.232546i
$731$	34.4225	1.27316
$732$	−0.535898	−0.0198074
$733$	− 5.96774i	− 0.220423i	−0.993908	−	0.110212i	$-0.964847\pi$
$733$	− 5.96774i	− 0.220423i	0.993908	−	0.110212i	$-0.0351529\pi$
$734$	− 19.6345i	− 0.724722i
$735$	− 5.25078i	− 0.193678i
$736$	− 8.66799i	− 0.319506i
$737$	−14.6753	−0.540573
$738$	−4.64516	−0.170991
$739$	− 10.8292i	− 0.398357i	−0.979963	−	0.199179i	$-0.936173\pi$
$739$	− 10.8292i	− 0.398357i	0.979963	−	0.199179i	$-0.0638274\pi$
$740$	6.80951	0.250322
$741$	0	0
$742$	−1.09334	−0.0401378
$743$	− 19.8503i	− 0.728237i	−0.931353	−	0.364118i	$-0.881370\pi$
$743$	− 19.8503i	− 0.728237i	0.931353	−	0.364118i	$-0.118630\pi$
$744$	−10.1321	−0.371460
$745$	4.08519	0.149670
$746$	− 2.55748i	− 0.0936359i
$747$	15.8719i	0.580721i
$748$	18.4534i	0.674724i
$749$	4.06270i	0.148448i
$750$	−1.00000	−0.0365148
$751$	15.3308	0.559428	0.279714	−	0.960083i	$-0.409760\pi$
$751$	15.3308	0.559428	0.279714	+	0.960083i	$0.409760\pi$
$752$	− 9.10926i	− 0.332181i
$753$	−8.82497	−0.321600
$754$	0	0
$755$	−19.9811	−0.727188
$756$	− 1.32258i	− 0.0481018i
$757$	45.1633	1.64149	0.820744	−	0.571297i	$-0.193559\pi$
$757$	45.1633	1.64149	0.820744	+	0.571297i	$0.193559\pi$
$758$	−15.1088	−0.548776
$759$	− 39.9885i	− 1.45149i
$760$	− 2.29078i	− 0.0830952i
$761$	− 21.7058i	− 0.786835i	−0.919360	−	0.393418i	$-0.871293\pi$
$761$	− 21.7058i	− 0.786835i	0.919360	−	0.393418i	$-0.128707\pi$
$762$	− 11.4718i	− 0.415581i
$763$	−17.4934	−0.633303
$764$	9.74715	0.352639
$765$	4.00000i	0.144620i
$766$	18.7303	0.676755
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 42.1132i	− 1.51864i	−0.650717	−	0.759321i	$-0.725531\pi$
$769$	− 42.1132i	− 1.51864i	0.650717	−	0.759321i	$-0.274469\pi$
$770$	−6.10153	−0.219884
$771$	−8.38494	−0.301976
$772$	− 13.7626i	− 0.495327i
$773$	29.7605i	1.07041i	0.844722	+	0.535206i	$0.179766\pi$
$773$	29.7605i	1.07041i	−0.844722	+	0.535206i	$0.820234\pi$
$774$	− 8.60562i	− 0.309323i
$775$	10.1321i	0.363955i
$776$	−8.81894	−0.316582
$777$	9.00612	0.323093
$778$	6.96899i	0.249850i
$779$	−10.6410	−0.381254
$780$	0	0
$781$	52.2969	1.87133
$782$	34.6719i	1.23987i
$783$	−2.02283	−0.0722899
$784$	5.25078	0.187528
$785$	− 7.13379i	− 0.254616i
$786$	− 15.9906i	− 0.570365i
$787$	− 41.0448i	− 1.46309i	−0.681793	−	0.731545i	$-0.738800\pi$
$787$	− 41.0448i	− 1.46309i	0.681793	−	0.731545i	$-0.261200\pi$
$788$	− 10.1415i	− 0.361277i
$789$	1.92130	0.0684000
$790$	2.96774	0.105587
$791$	− 10.6950i	− 0.380269i
$792$	4.61335	0.163928
$793$	0	0
$794$	18.5721	0.659100
$795$	− 0.826674i	− 0.0293191i
$796$	9.57336	0.339319
$797$	−45.2526	−1.60293	−0.801464	−	0.598043i	$-0.795945\pi$
$797$	−45.2526	−1.60293	−0.801464	+	0.598043i	$0.795945\pi$
$798$	− 3.02973i	− 0.107251i
$799$	36.4370i	1.28905i
$800$	− 1.00000i	− 0.0353553i
$801$	− 11.8641i	− 0.419199i
$802$	−29.6904	−1.04840
$803$	28.9859	1.02289
$804$	− 3.18106i	− 0.112187i
$805$	−11.4641	−0.404056
$806$	0	0
$807$	−32.3098	−1.13736
$808$	− 7.22671i	− 0.254235i
$809$	−28.4534	−1.00037	−0.500184	−	0.865919i	$-0.666734\pi$
$809$	−28.4534	−1.00037	−0.500184	+	0.865919i	$0.666734\pi$
$810$	1.00000	0.0351364
$811$	− 44.4114i	− 1.55949i	−0.626095	−	0.779747i	$-0.715348\pi$
$811$	− 44.4114i	− 1.55949i	0.626095	−	0.779747i	$-0.284652\pi$
$812$	2.67535i	0.0938863i
$813$	− 23.4526i	− 0.822518i
$814$	31.4147i	1.10108i
$815$	−9.89470	−0.346596
$816$	−4.00000	−0.140028
$817$	− 19.7135i	− 0.689690i
$818$	24.3355	0.850871
$819$	0	0
$820$	−4.64516	−0.162216
$821$	45.1683i	1.57638i	0.615429	+	0.788192i	$0.288983\pi$
$821$	45.1683i	1.57638i	−0.615429	+	0.788192i	$0.711017\pi$
$822$	−16.0697	−0.560496
$823$	−2.41799	−0.0842859	−0.0421430	−	0.999112i	$-0.513418\pi$
$823$	−2.41799	−0.0842859	−0.0421430	+	0.999112i	$0.513418\pi$
$824$	18.4000i	0.640996i
$825$	− 4.61335i	− 0.160616i
$826$	− 4.15491i	− 0.144568i
$827$	4.26832i	0.148424i	0.997242	+	0.0742120i	$0.0236441\pi$
$827$	4.26832i	0.148424i	−0.997242	+	0.0742120i	$0.976356\pi$
$828$	8.66799	0.301233
$829$	42.9852	1.49294	0.746468	−	0.665421i	$-0.231748\pi$
$829$	42.9852	1.49294	0.746468	+	0.665421i	$0.231748\pi$
$830$	15.8719i	0.550921i
$831$	9.00566	0.312403
$832$	0	0
$833$	−21.0031	−0.727715
$834$	9.32258i	0.322815i
$835$	−24.0375	−0.831851
$836$	10.5682	0.365507
$837$	− 10.1321i	− 0.350216i
$838$	− 2.88255i	− 0.0995761i
$839$	− 23.3016i	− 0.804462i	−0.915538	−	0.402231i	$-0.868235\pi$
$839$	− 23.3016i	− 0.804462i	0.915538	−	0.402231i	$-0.131765\pi$
$840$	− 1.32258i	− 0.0456333i
$841$	−24.9082	−0.858903
$842$	4.94707	0.170487
$843$	1.71696i	0.0591351i
$844$	1.08519	0.0373537
$845$	0	0
$846$	9.10926	0.313183
$847$	− 13.6001i	− 0.467307i
$848$	0.826674	0.0283881
$849$	1.54929	0.0531715
$850$	4.00000i	0.137199i
$851$	59.0247i	2.02334i
$852$	11.3360i	0.388364i
$853$	20.8418i	0.713609i	0.934179	+	0.356804i	$0.116134\pi$
$853$	20.8418i	0.713609i	−0.934179	+	0.356804i	$0.883866\pi$
$854$	−0.708768	−0.0242536
$855$	2.29078	0.0783429
$856$	− 3.07180i	− 0.104992i
$857$	36.7250	1.25450	0.627250	−	0.778818i	$-0.284180\pi$
$857$	36.7250	1.25450	0.627250	+	0.778818i	$0.284180\pi$
$858$	0	0
$859$	0.914812	0.0312130	0.0156065	−	0.999878i	$-0.495032\pi$
$859$	0.914812	0.0312130	0.0156065	+	0.999878i	$0.495032\pi$
$860$	− 8.60562i	− 0.293449i
$861$	−6.14359	−0.209373
$862$	−17.7626	−0.604997
$863$	33.6104i	1.14411i	0.820215	+	0.572056i	$0.193854\pi$
$863$	33.6104i	1.14411i	−0.820215	+	0.572056i	$0.806146\pi$
$864$	1.00000i	0.0340207i
$865$	2.52817i	0.0859603i
$866$	− 35.8375i	− 1.21781i
$867$	−1.00000	−0.0339618
$868$	−13.4005	−0.454842
$869$	13.6912i	0.464443i
$870$	−2.02283	−0.0685802
$871$	0	0
$872$	13.2267	0.447913
$873$	− 8.81894i	− 0.298476i
$874$	19.8564	0.671653
$875$	−1.32258	−0.0447114
$876$	6.28304i	0.212284i
$877$	− 24.3213i	− 0.821273i	−0.911799	−	0.410637i	$-0.865307\pi$
$877$	− 24.3213i	− 0.821273i	0.911799	−	0.410637i	$-0.134693\pi$
$878$	− 10.0904i	− 0.340534i
$879$	30.4057i	1.02556i
$880$	4.61335	0.155516
$881$	9.90413	0.333679	0.166839	−	0.985984i	$-0.446644\pi$
$881$	9.90413	0.333679	0.166839	+	0.985984i	$0.446644\pi$
$882$	5.25078i	0.176803i
$883$	−43.9718	−1.47977	−0.739884	−	0.672734i	$-0.765120\pi$
$883$	−43.9718	−1.47977	−0.739884	+	0.672734i	$0.765120\pi$
$884$	0	0
$885$	3.14152	0.105601
$886$	13.6981i	0.460196i
$887$	50.5614	1.69769	0.848843	−	0.528645i	$-0.177300\pi$
$887$	50.5614	1.69769	0.848843	+	0.528645i	$0.177300\pi$
$888$	−6.80951	−0.228512
$889$	− 15.1724i	− 0.508866i
$890$	− 11.8641i	− 0.397687i
$891$	4.61335i	0.154553i
$892$	24.9416i	0.835106i
$893$	20.8673	0.698297
$894$	−4.08519	−0.136629
$895$	− 6.27568i	− 0.209773i
$896$	1.32258	0.0441843
$897$	0	0
$898$	−15.3128	−0.510994
$899$	20.4954i	0.683562i
$900$	1.00000	0.0333333
$901$	−3.30669	−0.110162
$902$	− 21.4298i	− 0.713533i
$903$	− 11.3816i	− 0.378757i
$904$	8.08643i	0.268951i
$905$	12.2830i	0.408302i
$906$	19.9811	0.663829
$907$	−14.2435	−0.472948	−0.236474	−	0.971638i	$-0.575992\pi$
$907$	−14.2435	−0.472948	−0.236474	+	0.971638i	$0.575992\pi$
$908$	19.3205i	0.641174i
$909$	7.22671	0.239695
$910$	0	0
$911$	21.8108	0.722623	0.361311	−	0.932445i	$-0.382329\pi$
$911$	21.8108	0.722623	0.361311	+	0.932445i	$0.382329\pi$
$912$	2.29078i	0.0758551i
$913$	−73.2226	−2.42331
$914$	−15.5554	−0.514527
$915$	− 0.535898i	− 0.0177163i
$916$	− 4.15491i	− 0.137282i
$917$	− 21.1488i	− 0.698395i
$918$	− 4.00000i	− 0.132020i
$919$	−5.30578	−0.175022	−0.0875108	−	0.996164i	$-0.527891\pi$
$919$	−5.30578	−0.175022	−0.0875108	+	0.996164i	$0.527891\pi$
$920$	8.66799	0.285775
$921$	9.82622i	0.323785i
$922$	−15.6431	−0.515178
$923$	0	0
$924$	6.10153	0.200726
$925$	6.80951i	0.223895i
$926$	−13.7170	−0.450767
$927$	−18.4000	−0.604336
$928$	− 2.02283i	− 0.0664025i
$929$	− 44.3408i	− 1.45477i	−0.686228	−	0.727386i	$-0.740735\pi$
$929$	− 44.3408i	− 1.45477i	0.686228	−	0.727386i	$-0.259265\pi$
$930$	− 10.1321i	− 0.332244i
$931$	12.0284i	0.394214i
$932$	−16.9665	−0.555756
$933$	−3.40049	−0.111327
$934$	− 0.943666i	− 0.0308777i
$935$	−18.4534	−0.603491
$936$	0	0
$937$	38.5283	1.25867	0.629333	−	0.777136i	$-0.283328\pi$
$937$	38.5283	1.25867	0.629333	+	0.777136i	$0.283328\pi$
$938$	− 4.20720i	− 0.137370i
$939$	−16.2340	−0.529776
$940$	9.10926	0.297111
$941$	28.8637i	0.940929i	0.882419	+	0.470465i	$0.155914\pi$
$941$	28.8637i	0.940929i	−0.882419	+	0.470465i	$0.844086\pi$
$942$	7.13379i	0.232431i
$943$	− 40.2642i	− 1.31118i
$944$	3.14152i	0.102248i
$945$	1.32258	0.0430235
$946$	39.7008	1.29078
$947$	− 49.2346i	− 1.59991i	−0.600060	−	0.799955i	$-0.704857\pi$
$947$	− 49.2346i	− 1.59991i	0.600060	−	0.799955i	$-0.295143\pi$
$948$	−2.96774	−0.0963877
$949$	0	0
$950$	2.29078	0.0743226
$951$	23.5231i	0.762790i
$952$	−5.29032	−0.171460
$953$	26.5097	0.858732	0.429366	−	0.903131i	$-0.358737\pi$
$953$	26.5097	0.858732	0.429366	+	0.903131i	$0.358737\pi$
$954$	0.826674i	0.0267645i
$955$	9.74715i	0.315410i
$956$	6.34416i	0.205185i
$957$	− 9.33201i	− 0.301661i
$958$	−11.1020	−0.358689
$959$	−21.2535	−0.686311
$960$	1.00000i	0.0322749i
$961$	−71.6592	−2.31159
$962$	0	0
$963$	3.07180	0.0989873
$964$	− 24.1855i	− 0.778962i
$965$	13.7626	0.443034
$966$	11.4641	0.368851
$967$	52.3877i	1.68468i	0.538950	+	0.842338i	$0.318821\pi$
$967$	52.3877i	1.68468i	−0.538950	+	0.842338i	$0.681179\pi$
$968$	10.2830i	0.330510i
$969$	− 9.16310i	− 0.294361i
$970$	− 8.81894i	− 0.283159i
$971$	−28.6431	−0.919200	−0.459600	−	0.888126i	$-0.652007\pi$
$971$	−28.6431	−0.919200	−0.459600	+	0.888126i	$0.652007\pi$
$972$	−1.00000	−0.0320750
$973$	12.3299i	0.395277i
$974$	32.9416	1.05552
$975$	0	0
$976$	0.535898	0.0171537
$977$	− 40.5499i	− 1.29731i	−0.761084	−	0.648654i	$-0.775332\pi$
$977$	− 40.5499i	− 1.29731i	0.761084	−	0.648654i	$-0.224668\pi$
$978$	9.89470	0.316397
$979$	54.7335	1.74929
$980$	5.25078i	0.167730i
$981$	13.2267i	0.422296i
$982$	− 28.4513i	− 0.907919i
$983$	22.4160i	0.714958i	0.933921	+	0.357479i	$0.116364\pi$
$983$	22.4160i	0.714958i	−0.933921	+	0.357479i	$0.883636\pi$
$984$	4.64516	0.148082
$985$	10.1415	0.323136
$986$	8.09130i	0.257680i
$987$	12.0477	0.383483
$988$	0	0
$989$	74.5934	2.37193
$990$	4.61335i	0.146622i
$991$	−28.5688	−0.907518	−0.453759	−	0.891125i	$-0.649917\pi$
$991$	−28.5688	−0.907518	−0.453759	+	0.891125i	$0.649917\pi$
$992$	10.1321	0.321694
$993$	18.0904i	0.574081i
$994$	14.9927i	0.475540i
$995$	9.57336i	0.303496i
$996$	− 15.8719i	− 0.502919i
$997$	−34.4000	−1.08946	−0.544730	−	0.838611i	$-0.683368\pi$
$997$	−34.4000	−1.08946	−0.544730	+	0.838611i	$0.683368\pi$
$998$	4.10926	0.130076
$999$	− 6.80951i	− 0.215443i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.ba.1351.7		8
13.3	even	3		390.2.bb.c.121.2	✓	8
13.4	even	6		390.2.bb.c.361.2	yes	8
13.5	odd	4		5070.2.a.ca.1.2		4
13.8	odd	4		5070.2.a.bz.1.3		4
13.12	even	2	inner	5070.2.b.ba.1351.2		8
39.17	odd	6		1170.2.bs.f.361.4		8
39.29	odd	6		1170.2.bs.f.901.4		8
65.3	odd	12		1950.2.y.j.199.1		8
65.4	even	6		1950.2.bc.g.751.3		8
65.17	odd	12		1950.2.y.j.49.1		8
65.29	even	6		1950.2.bc.g.901.3		8
65.42	odd	12		1950.2.y.k.199.4		8
65.43	odd	12		1950.2.y.k.49.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bb.c.121.2	✓	8	13.3	even	3
390.2.bb.c.361.2	yes	8	13.4	even	6
1170.2.bs.f.361.4		8	39.17	odd	6
1170.2.bs.f.901.4		8	39.29	odd	6
1950.2.y.j.49.1		8	65.17	odd	12
1950.2.y.j.199.1		8	65.3	odd	12
1950.2.y.k.49.4		8	65.43	odd	12
1950.2.y.k.199.4		8	65.42	odd	12
1950.2.bc.g.751.3		8	65.4	even	6
1950.2.bc.g.901.3		8	65.29	even	6
5070.2.a.bz.1.3		4	13.8	odd	4
5070.2.a.ca.1.2		4	13.5	odd	4
5070.2.b.ba.1351.2		8	13.12	even	2	inner
5070.2.b.ba.1351.7		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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.32258i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.32258i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.61335i q^{11} -1.00000 q^{12} -1.32258 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000i q^{18} +2.29078i q^{19} +1.00000i q^{20} +1.32258i q^{21} -4.61335 q^{22} -8.66799 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -1.32258i q^{28} -2.02283 q^{29} +1.00000 q^{30} -10.1321i q^{31} +1.00000i q^{32} +4.61335i q^{33} -4.00000i q^{34} +1.32258 q^{35} -1.00000 q^{36} -6.80951i q^{37} -2.29078 q^{38} -1.00000 q^{40} +4.64516i q^{41} -1.32258 q^{42} -8.60562 q^{43} -4.61335i q^{44} -1.00000i q^{45} -8.66799i q^{46} -9.10926i q^{47} +1.00000 q^{48} +5.25078 q^{49} -1.00000i q^{50} -4.00000 q^{51} +0.826674 q^{53} +1.00000i q^{54} +4.61335 q^{55} +1.32258 q^{56} +2.29078i q^{57} -2.02283i q^{58} +3.14152i q^{59} +1.00000i q^{60} +0.535898 q^{61} +10.1321 q^{62} +1.32258i q^{63} -1.00000 q^{64} -4.61335 q^{66} +3.18106i q^{67} +4.00000 q^{68} -8.66799 q^{69} +1.32258i q^{70} -11.3360i q^{71} -1.00000i q^{72} -6.28304i q^{73} +6.80951 q^{74} -1.00000 q^{75} -2.29078i q^{76} -6.10153 q^{77} +2.96774 q^{79} -1.00000i q^{80} +1.00000 q^{81} -4.64516 q^{82} +15.8719i q^{83} -1.32258i q^{84} +4.00000i q^{85} -8.60562i q^{86} -2.02283 q^{87} +4.61335 q^{88} -11.8641i q^{89} +1.00000 q^{90} +8.66799 q^{92} -10.1321i q^{93} +9.10926 q^{94} +2.29078 q^{95} +1.00000i q^{96} -8.81894i q^{97} +5.25078i q^{98} +4.61335i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 8 q^{4} + 8 q^{9}+O(q^{10}) \) 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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