LMFDB - Embedded newform 5070.2.b.o.1351.4

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:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.4
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.o.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +0.464102i q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000i q^{18} +0.535898i q^{19} -1.00000i q^{20} +2.00000i q^{21} -0.464102 q^{22} +0.267949 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} +3.73205 q^{29} +1.00000 q^{30} +1.73205i q^{31} +1.00000i q^{32} -0.464102i q^{33} -4.00000i q^{34} +2.00000 q^{35} -1.00000 q^{36} +1.19615i q^{37} -0.535898 q^{38} +1.00000 q^{40} -2.00000i q^{41} -2.00000 q^{42} -1.92820 q^{43} -0.464102i q^{44} +1.00000i q^{45} +0.267949i q^{46} -10.4641i q^{47} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} +4.00000 q^{51} -12.9282 q^{53} -1.00000i q^{54} -0.464102 q^{55} -2.00000 q^{56} -0.535898i q^{57} +3.73205i q^{58} +1.53590i q^{59} +1.00000i q^{60} +10.3923 q^{61} -1.73205 q^{62} -2.00000i q^{63} -1.00000 q^{64} +0.464102 q^{66} +4.53590i q^{67} +4.00000 q^{68} -0.267949 q^{69} +2.00000i q^{70} +8.39230i q^{71} -1.00000i q^{72} +2.00000i q^{73} -1.19615 q^{74} +1.00000 q^{75} -0.535898i q^{76} +0.928203 q^{77} -0.0717968 q^{79} +1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} -4.92820i q^{83} -2.00000i q^{84} -4.00000i q^{85} -1.92820i q^{86} -3.73205 q^{87} +0.464102 q^{88} +7.46410i q^{89} -1.00000 q^{90} -0.267949 q^{92} -1.73205i q^{93} +10.4641 q^{94} -0.535898 q^{95} -1.00000i q^{96} +7.46410i q^{97} +3.00000i q^{98} +0.464102i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9	$ 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} + 4 q^{12} + 8 q^{14} + 4 q^{16} - 16 q^{17} + 12 q^{22} + 8 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{29} + 4 q^{30} + 8 q^{35} - 4 q^{36} - 16 q^{38} + 4 q^{40} - 8 q^{42} + 20 q^{43} - 4 q^{48} + 12 q^{49} + 16 q^{51} - 24 q^{53} + 12 q^{55} - 8 q^{56} - 4 q^{64} - 12 q^{66} + 16 q^{68} - 8 q^{69} + 16 q^{74} + 4 q^{75} - 24 q^{77} - 28 q^{79} + 4 q^{81} + 8 q^{82} - 8 q^{87} - 12 q^{88} - 4 q^{90} - 8 q^{92} + 28 q^{94} - 16 q^{95}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 + 4 * q^12 + 8 * q^14 + 4 * q^16 - 16 * q^17 + 12 * q^22 + 8 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^29 + 4 * q^30 + 8 * q^35 - 4 * q^36 - 16 * q^38 + 4 * q^40 - 8 * q^42 + 20 * q^43 - 4 * q^48 + 12 * q^49 + 16 * q^51 - 24 * q^53 + 12 * q^55 - 8 * q^56 - 4 * q^64 - 12 * q^66 + 16 * q^68 - 8 * q^69 + 16 * q^74 + 4 * q^75 - 24 * q^77 - 28 * q^79 + 4 * q^81 + 8 * q^82 - 8 * q^87 - 12 * q^88 - 4 * q^90 - 8 * q^92 + 28 * q^94 - 16 * q^95

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
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0.464102i	0.139932i	0.997549	+	0.0699660i	$0.0222891\pi$
$11$	0.464102i	0.139932i	−0.997549	+	0.0699660i	$0.977711\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	− 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
$19$	0.535898i	0.122944i	0.998109	+	0.0614718i	$0.0195794\pi$
$19$	0.535898i	0.122944i	−0.998109	+	0.0614718i	$0.980421\pi$
$20$	− 1.00000i	− 0.223607i
$21$	2.00000i	0.436436i
$22$	−0.464102	−0.0989468
$23$	0.267949	0.0558713	0.0279356	−	0.999610i	$-0.491107\pi$
$23$	0.267949	0.0558713	0.0279356	+	0.999610i	$0.491107\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	2.00000i	0.377964i
$29$	3.73205	0.693024	0.346512	−	0.938045i	$-0.387366\pi$
$29$	3.73205	0.693024	0.346512	+	0.938045i	$0.387366\pi$
$30$	1.00000	0.182574
$31$	1.73205i	0.311086i	0.987829	+	0.155543i	$0.0497126\pi$
$31$	1.73205i	0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$32$	1.00000i	0.176777i
$33$	− 0.464102i	− 0.0807897i
$34$	− 4.00000i	− 0.685994i
$35$	2.00000	0.338062
$36$	−1.00000	−0.166667
$37$	1.19615i	0.196646i	0.995155	+	0.0983231i	$0.0313479\pi$
$37$	1.19615i	0.196646i	−0.995155	+	0.0983231i	$0.968652\pi$
$38$	−0.535898	−0.0869342
$39$	0	0
$40$	1.00000	0.158114
$41$	− 2.00000i	− 0.312348i	−0.987730	−	0.156174i	$-0.950084\pi$
$41$	− 2.00000i	− 0.312348i	0.987730	−	0.156174i	$-0.0499160\pi$
$42$	−2.00000	−0.308607
$43$	−1.92820	−0.294048	−0.147024	−	0.989133i	$-0.546969\pi$
$43$	−1.92820	−0.294048	−0.147024	+	0.989133i	$0.546969\pi$
$44$	− 0.464102i	− 0.0699660i
$45$	1.00000i	0.149071i
$46$	0.267949i	0.0395070i
$47$	− 10.4641i	− 1.52635i	−0.646194	−	0.763173i	$-0.723640\pi$
$47$	− 10.4641i	− 1.52635i	0.646194	−	0.763173i	$-0.276360\pi$
$48$	−1.00000	−0.144338
$49$	3.00000	0.428571
$50$	− 1.00000i	− 0.141421i
$51$	4.00000	0.560112
$52$	0	0
$53$	−12.9282	−1.77583	−0.887913	−	0.460012i	$-0.847845\pi$
$53$	−12.9282	−1.77583	−0.887913	+	0.460012i	$0.847845\pi$
$54$	− 1.00000i	− 0.136083i
$55$	−0.464102	−0.0625794
$56$	−2.00000	−0.267261
$57$	− 0.535898i	− 0.0709815i
$58$	3.73205i	0.490042i
$59$	1.53590i	0.199957i	0.994990	+	0.0999785i	$0.0318774\pi$
$59$	1.53590i	0.199957i	−0.994990	+	0.0999785i	$0.968123\pi$
$60$	1.00000i	0.129099i
$61$	10.3923	1.33060	0.665299	−	0.746577i	$-0.268304\pi$
$61$	10.3923	1.33060	0.665299	+	0.746577i	$0.268304\pi$
$62$	−1.73205	−0.219971
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0.464102	0.0571270
$67$	4.53590i	0.554148i	0.960849	+	0.277074i	$0.0893647\pi$
$67$	4.53590i	0.554148i	−0.960849	+	0.277074i	$0.910635\pi$
$68$	4.00000	0.485071
$69$	−0.267949	−0.0322573
$70$	2.00000i	0.239046i
$71$	8.39230i	0.995983i	0.867182	+	0.497992i	$0.165929\pi$
$71$	8.39230i	0.995983i	−0.867182	+	0.497992i	$0.834071\pi$
$72$	− 1.00000i	− 0.117851i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	−1.19615	−0.139050
$75$	1.00000	0.115470
$76$	− 0.535898i	− 0.0614718i
$77$	0.928203	0.105779
$78$	0	0
$79$	−0.0717968	−0.00807777	−0.00403888	−	0.999992i	$-0.501286\pi$
$79$	−0.0717968	−0.00807777	−0.00403888	+	0.999992i	$0.501286\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	− 4.92820i	− 0.540941i	−0.962728	−	0.270470i	$-0.912821\pi$
$83$	− 4.92820i	− 0.540941i	0.962728	−	0.270470i	$-0.0871792\pi$
$84$	− 2.00000i	− 0.218218i
$85$	− 4.00000i	− 0.433861i
$86$	− 1.92820i	− 0.207924i
$87$	−3.73205	−0.400118
$88$	0.464102	0.0494734
$89$	7.46410i	0.791193i	0.918424	+	0.395597i	$0.129462\pi$
$89$	7.46410i	0.791193i	−0.918424	+	0.395597i	$0.870538\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−0.267949	−0.0279356
$93$	− 1.73205i	− 0.179605i
$94$	10.4641	1.07929
$95$	−0.535898	−0.0549820
$96$	− 1.00000i	− 0.102062i
$97$	7.46410i	0.757865i	0.925424	+	0.378932i	$0.123709\pi$
$97$	7.46410i	0.757865i	−0.925424	+	0.378932i	$0.876291\pi$
$98$	3.00000i	0.303046i
$99$	0.464102i	0.0466440i
$100$	1.00000	0.100000
$101$	−10.9282	−1.08740	−0.543698	−	0.839281i	$-0.682976\pi$
$101$	−10.9282	−1.08740	−0.543698	+	0.839281i	$0.682976\pi$
$102$	4.00000i	0.396059i
$103$	15.8564	1.56238	0.781189	−	0.624295i	$-0.214613\pi$
$103$	15.8564	1.56238	0.781189	+	0.624295i	$0.214613\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	− 12.9282i	− 1.25570i
$107$	19.8564	1.91959	0.959796	−	0.280700i	$-0.0905665\pi$
$107$	19.8564	1.91959	0.959796	+	0.280700i	$0.0905665\pi$
$108$	1.00000	0.0962250
$109$	11.8564i	1.13564i	0.823154	+	0.567819i	$0.192213\pi$
$109$	11.8564i	1.13564i	−0.823154	+	0.567819i	$0.807787\pi$
$110$	− 0.464102i	− 0.0442504i
$111$	− 1.19615i	− 0.113534i
$112$	− 2.00000i	− 0.188982i
$113$	11.1962	1.05325	0.526623	−	0.850099i	$-0.323458\pi$
$113$	11.1962	1.05325	0.526623	+	0.850099i	$0.323458\pi$
$114$	0.535898	0.0501915
$115$	0.267949i	0.0249864i
$116$	−3.73205	−0.346512
$117$	0	0
$118$	−1.53590	−0.141391
$119$	8.00000i	0.733359i
$120$	−1.00000	−0.0912871
$121$	10.7846	0.980419
$122$	10.3923i	0.940875i
$123$	2.00000i	0.180334i
$124$	− 1.73205i	− 0.155543i
$125$	− 1.00000i	− 0.0894427i
$126$	2.00000	0.178174
$127$	8.92820	0.792250	0.396125	−	0.918197i	$-0.370355\pi$
$127$	8.92820	0.792250	0.396125	+	0.918197i	$0.370355\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	1.92820	0.169769
$130$	0	0
$131$	−1.33975	−0.117054	−0.0585271	−	0.998286i	$-0.518640\pi$
$131$	−1.33975	−0.117054	−0.0585271	+	0.998286i	$0.518640\pi$
$132$	0.464102i	0.0403949i
$133$	1.07180	0.0929366
$134$	−4.53590	−0.391842
$135$	− 1.00000i	− 0.0860663i
$136$	4.00000i	0.342997i
$137$	4.46410i	0.381394i	0.981649	+	0.190697i	$0.0610748\pi$
$137$	4.46410i	0.381394i	−0.981649	+	0.190697i	$0.938925\pi$
$138$	− 0.267949i	− 0.0228093i
$139$	0.928203	0.0787292	0.0393646	−	0.999225i	$-0.487467\pi$
$139$	0.928203	0.0787292	0.0393646	+	0.999225i	$0.487467\pi$
$140$	−2.00000	−0.169031
$141$	10.4641i	0.881236i
$142$	−8.39230	−0.704267
$143$	0	0
$144$	1.00000	0.0833333
$145$	3.73205i	0.309930i
$146$	−2.00000	−0.165521
$147$	−3.00000	−0.247436
$148$	− 1.19615i	− 0.0983231i
$149$	− 20.4641i	− 1.67648i	−0.545298	−	0.838242i	$-0.683584\pi$
$149$	− 20.4641i	− 1.67648i	0.545298	−	0.838242i	$-0.316416\pi$
$150$	1.00000i	0.0816497i
$151$	10.3923i	0.845714i	0.906196	+	0.422857i	$0.138973\pi$
$151$	10.3923i	0.845714i	−0.906196	+	0.422857i	$0.861027\pi$
$152$	0.535898	0.0434671
$153$	−4.00000	−0.323381
$154$	0.928203i	0.0747967i
$155$	−1.73205	−0.139122
$156$	0	0
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	− 0.0717968i	− 0.00571184i
$159$	12.9282	1.02527
$160$	−1.00000	−0.0790569
$161$	− 0.535898i	− 0.0422347i
$162$	1.00000i	0.0785674i
$163$	− 23.0526i	− 1.80562i	−0.430044	−	0.902808i	$-0.641502\pi$
$163$	− 23.0526i	− 1.80562i	0.430044	−	0.902808i	$-0.358498\pi$
$164$	2.00000i	0.156174i
$165$	0.464102	0.0361303
$166$	4.92820	0.382503
$167$	18.3205i	1.41768i	0.705368	+	0.708842i	$0.250782\pi$
$167$	18.3205i	1.41768i	−0.705368	+	0.708842i	$0.749218\pi$
$168$	2.00000	0.154303
$169$	0	0
$170$	4.00000	0.306786
$171$	0.535898i	0.0409812i
$172$	1.92820	0.147024
$173$	−2.92820	−0.222627	−0.111314	−	0.993785i	$-0.535506\pi$
$173$	−2.92820	−0.222627	−0.111314	+	0.993785i	$0.535506\pi$
$174$	− 3.73205i	− 0.282926i
$175$	2.00000i	0.151186i
$176$	0.464102i	0.0349830i
$177$	− 1.53590i	− 0.115445i
$178$	−7.46410	−0.559458
$179$	−16.2679	−1.21592	−0.607962	−	0.793966i	$-0.708013\pi$
$179$	−16.2679	−1.21592	−0.607962	+	0.793966i	$0.708013\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	10.9282	0.812287	0.406143	−	0.913809i	$-0.366873\pi$
$181$	10.9282	0.812287	0.406143	+	0.913809i	$0.366873\pi$
$182$	0	0
$183$	−10.3923	−0.768221
$184$	− 0.267949i	− 0.0197535i
$185$	−1.19615	−0.0879429
$186$	1.73205	0.127000
$187$	− 1.85641i	− 0.135754i
$188$	10.4641i	0.763173i
$189$	2.00000i	0.145479i
$190$	− 0.535898i	− 0.0388782i
$191$	14.5359	1.05178	0.525890	−	0.850552i	$-0.323732\pi$
$191$	14.5359	1.05178	0.525890	+	0.850552i	$0.323732\pi$
$192$	1.00000	0.0721688
$193$	23.3205i	1.67865i	0.543632	+	0.839323i	$0.317049\pi$
$193$	23.3205i	1.67865i	−0.543632	+	0.839323i	$0.682951\pi$
$194$	−7.46410	−0.535891
$195$	0	0
$196$	−3.00000	−0.214286
$197$	16.3923i	1.16790i	0.811788	+	0.583952i	$0.198494\pi$
$197$	16.3923i	1.16790i	−0.811788	+	0.583952i	$0.801506\pi$
$198$	−0.464102	−0.0329823
$199$	−18.9282	−1.34178	−0.670892	−	0.741555i	$-0.734089\pi$
$199$	−18.9282	−1.34178	−0.670892	+	0.741555i	$0.734089\pi$
$200$	1.00000i	0.0707107i
$201$	− 4.53590i	− 0.319938i
$202$	− 10.9282i	− 0.768906i
$203$	− 7.46410i	− 0.523877i
$204$	−4.00000	−0.280056
$205$	2.00000	0.139686
$206$	15.8564i	1.10477i
$207$	0.267949	0.0186238
$208$	0	0
$209$	−0.248711	−0.0172037
$210$	− 2.00000i	− 0.138013i
$211$	23.3205	1.60545	0.802725	−	0.596349i	$-0.203383\pi$
$211$	23.3205	1.60545	0.802725	+	0.596349i	$0.203383\pi$
$212$	12.9282	0.887913
$213$	− 8.39230i	− 0.575031i
$214$	19.8564i	1.35736i
$215$	− 1.92820i	− 0.131502i
$216$	1.00000i	0.0680414i
$217$	3.46410	0.235159
$218$	−11.8564	−0.803017
$219$	− 2.00000i	− 0.135147i
$220$	0.464102	0.0312897
$221$	0	0
$222$	1.19615	0.0802805
$223$	27.4641i	1.83913i	0.392935	+	0.919566i	$0.371460\pi$
$223$	27.4641i	1.83913i	−0.392935	+	0.919566i	$0.628540\pi$
$224$	2.00000	0.133631
$225$	−1.00000	−0.0666667
$226$	11.1962i	0.744757i
$227$	4.39230i	0.291528i	0.989319	+	0.145764i	$0.0465639\pi$
$227$	4.39230i	0.291528i	−0.989319	+	0.145764i	$0.953436\pi$
$228$	0.535898i	0.0354907i
$229$	19.8564i	1.31215i	0.754696	+	0.656074i	$0.227784\pi$
$229$	19.8564i	1.31215i	−0.754696	+	0.656074i	$0.772216\pi$
$230$	−0.267949	−0.0176680
$231$	−0.928203	−0.0610713
$232$	− 3.73205i	− 0.245021i
$233$	−18.1244	−1.18737	−0.593683	−	0.804699i	$-0.702327\pi$
$233$	−18.1244	−1.18737	−0.593683	+	0.804699i	$0.702327\pi$
$234$	0	0
$235$	10.4641	0.682603
$236$	− 1.53590i	− 0.0999785i
$237$	0.0717968	0.00466370
$238$	−8.00000	−0.518563
$239$	4.39230i	0.284115i	0.989858	+	0.142057i	$0.0453717\pi$
$239$	4.39230i	0.284115i	−0.989858	+	0.142057i	$0.954628\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	14.2679i	0.919079i	0.888157	+	0.459540i	$0.151986\pi$
$241$	14.2679i	0.919079i	−0.888157	+	0.459540i	$0.848014\pi$
$242$	10.7846i	0.693261i
$243$	−1.00000	−0.0641500
$244$	−10.3923	−0.665299
$245$	3.00000i	0.191663i
$246$	−2.00000	−0.127515
$247$	0	0
$248$	1.73205	0.109985
$249$	4.92820i	0.312312i
$250$	1.00000	0.0632456
$251$	12.2679	0.774346	0.387173	−	0.922007i	$-0.373452\pi$
$251$	12.2679	0.774346	0.387173	+	0.922007i	$0.373452\pi$
$252$	2.00000i	0.125988i
$253$	0.124356i	0.00781817i
$254$	8.92820i	0.560205i
$255$	4.00000i	0.250490i
$256$	1.00000	0.0625000
$257$	−22.6603	−1.41351	−0.706754	−	0.707459i	$-0.749841\pi$
$257$	−22.6603	−1.41351	−0.706754	+	0.707459i	$0.749841\pi$
$258$	1.92820i	0.120045i
$259$	2.39230	0.148651
$260$	0	0
$261$	3.73205	0.231008
$262$	− 1.33975i	− 0.0827698i
$263$	−18.1244	−1.11760	−0.558798	−	0.829304i	$-0.688737\pi$
$263$	−18.1244	−1.11760	−0.558798	+	0.829304i	$0.688737\pi$
$264$	−0.464102	−0.0285635
$265$	− 12.9282i	− 0.794173i
$266$	1.07180i	0.0657161i
$267$	− 7.46410i	− 0.456796i
$268$	− 4.53590i	− 0.277074i
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	1.00000	0.0608581
$271$	9.19615i	0.558626i	0.960200	+	0.279313i	$0.0901068\pi$
$271$	9.19615i	0.558626i	−0.960200	+	0.279313i	$0.909893\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−4.46410	−0.269686
$275$	− 0.464102i	− 0.0279864i
$276$	0.267949	0.0161286
$277$	9.92820	0.596528	0.298264	−	0.954483i	$-0.403592\pi$
$277$	9.92820	0.596528	0.298264	+	0.954483i	$0.403592\pi$
$278$	0.928203i	0.0556699i
$279$	1.73205i	0.103695i
$280$	− 2.00000i	− 0.119523i
$281$	4.92820i	0.293992i	0.989137	+	0.146996i	$0.0469604\pi$
$281$	4.92820i	0.293992i	−0.989137	+	0.146996i	$0.953040\pi$
$282$	−10.4641	−0.623128
$283$	3.92820	0.233507	0.116754	−	0.993161i	$-0.462751\pi$
$283$	3.92820	0.233507	0.116754	+	0.993161i	$0.462751\pi$
$284$	− 8.39230i	− 0.497992i
$285$	0.535898	0.0317439
$286$	0	0
$287$	−4.00000	−0.236113
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	−3.73205	−0.219154
$291$	− 7.46410i	− 0.437553i
$292$	− 2.00000i	− 0.117041i
$293$	− 4.14359i	− 0.242071i	−0.992648	−	0.121036i	$-0.961378\pi$
$293$	− 4.14359i	− 0.242071i	0.992648	−	0.121036i	$-0.0386215\pi$
$294$	− 3.00000i	− 0.174964i
$295$	−1.53590	−0.0894235
$296$	1.19615	0.0695249
$297$	− 0.464102i	− 0.0269299i
$298$	20.4641	1.18545
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	3.85641i	0.222280i
$302$	−10.3923	−0.598010
$303$	10.9282	0.627809
$304$	0.535898i	0.0307359i
$305$	10.3923i	0.595062i
$306$	− 4.00000i	− 0.228665i
$307$	12.5359i	0.715462i	0.933825	+	0.357731i	$0.116449\pi$
$307$	12.5359i	0.715462i	−0.933825	+	0.357731i	$0.883551\pi$
$308$	−0.928203	−0.0528893
$309$	−15.8564	−0.902039
$310$	− 1.73205i	− 0.0983739i
$311$	−7.60770	−0.431393	−0.215696	−	0.976460i	$-0.569202\pi$
$311$	−7.60770	−0.431393	−0.215696	+	0.976460i	$0.569202\pi$
$312$	0	0
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	− 5.00000i	− 0.282166i
$315$	2.00000	0.112687
$316$	0.0717968	0.00403888
$317$	21.4641i	1.20554i	0.797913	+	0.602772i	$0.205937\pi$
$317$	21.4641i	1.20554i	−0.797913	+	0.602772i	$0.794063\pi$
$318$	12.9282i	0.724978i
$319$	1.73205i	0.0969762i
$320$	− 1.00000i	− 0.0559017i
$321$	−19.8564	−1.10828
$322$	0.535898	0.0298644
$323$	− 2.14359i	− 0.119273i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	23.0526	1.27676
$327$	− 11.8564i	− 0.655661i
$328$	−2.00000	−0.110432
$329$	−20.9282	−1.15381
$330$	0.464102i	0.0255480i
$331$	24.7846i	1.36229i	0.732151	+	0.681143i	$0.238517\pi$
$331$	24.7846i	1.36229i	−0.732151	+	0.681143i	$0.761483\pi$
$332$	4.92820i	0.270470i
$333$	1.19615i	0.0655487i
$334$	−18.3205	−1.00245
$335$	−4.53590	−0.247823
$336$	2.00000i	0.109109i
$337$	−25.3205	−1.37930	−0.689648	−	0.724145i	$-0.742235\pi$
$337$	−25.3205	−1.37930	−0.689648	+	0.724145i	$0.742235\pi$
$338$	0	0
$339$	−11.1962	−0.608092
$340$	4.00000i	0.216930i
$341$	−0.803848	−0.0435308
$342$	−0.535898	−0.0289781
$343$	− 20.0000i	− 1.07990i
$344$	1.92820i	0.103962i
$345$	− 0.267949i	− 0.0144259i
$346$	− 2.92820i	− 0.157421i
$347$	22.3923	1.20208	0.601041	−	0.799218i	$-0.294753\pi$
$347$	22.3923	1.20208	0.601041	+	0.799218i	$0.294753\pi$
$348$	3.73205	0.200059
$349$	− 14.5359i	− 0.778089i	−0.921219	−	0.389044i	$-0.872805\pi$
$349$	− 14.5359i	− 0.778089i	0.921219	−	0.389044i	$-0.127195\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	−0.464102	−0.0247367
$353$	− 2.00000i	− 0.106449i	−0.998583	−	0.0532246i	$-0.983050\pi$
$353$	− 2.00000i	− 0.106449i	0.998583	−	0.0532246i	$-0.0169499\pi$
$354$	1.53590	0.0816321
$355$	−8.39230	−0.445417
$356$	− 7.46410i	− 0.395597i
$357$	− 8.00000i	− 0.423405i
$358$	− 16.2679i	− 0.859788i
$359$	18.9282i	0.998992i	0.866316	+	0.499496i	$0.166482\pi$
$359$	18.9282i	0.998992i	−0.866316	+	0.499496i	$0.833518\pi$
$360$	1.00000	0.0527046
$361$	18.7128	0.984885
$362$	10.9282i	0.574374i
$363$	−10.7846	−0.566045
$364$	0	0
$365$	−2.00000	−0.104685
$366$	− 10.3923i	− 0.543214i
$367$	36.3923	1.89966	0.949831	−	0.312762i	$-0.101254\pi$
$367$	36.3923	1.89966	0.949831	+	0.312762i	$0.101254\pi$
$368$	0.267949	0.0139678
$369$	− 2.00000i	− 0.104116i
$370$	− 1.19615i	− 0.0621850i
$371$	25.8564i	1.34240i
$372$	1.73205i	0.0898027i
$373$	25.7846	1.33508	0.667538	−	0.744576i	$-0.267348\pi$
$373$	25.7846	1.33508	0.667538	+	0.744576i	$0.267348\pi$
$374$	1.85641	0.0959925
$375$	1.00000i	0.0516398i
$376$	−10.4641	−0.539645
$377$	0	0
$378$	−2.00000	−0.102869
$379$	− 0.143594i	− 0.00737590i	−0.999993	−	0.00368795i	$-0.998826\pi$
$379$	− 0.143594i	− 0.00737590i	0.999993	−	0.00368795i	$-0.00117391\pi$
$380$	0.535898	0.0274910
$381$	−8.92820	−0.457406
$382$	14.5359i	0.743721i
$383$	4.60770i	0.235442i	0.993047	+	0.117721i	$0.0375589\pi$
$383$	4.60770i	0.235442i	−0.993047	+	0.117721i	$0.962441\pi$
$384$	1.00000i	0.0510310i
$385$	0.928203i	0.0473056i
$386$	−23.3205	−1.18698
$387$	−1.92820	−0.0980161
$388$	− 7.46410i	− 0.378932i
$389$	−20.2679	−1.02763	−0.513813	−	0.857902i	$-0.671767\pi$
$389$	−20.2679	−1.02763	−0.513813	+	0.857902i	$0.671767\pi$
$390$	0	0
$391$	−1.07180	−0.0542031
$392$	− 3.00000i	− 0.151523i
$393$	1.33975	0.0675812
$394$	−16.3923	−0.825832
$395$	− 0.0717968i	− 0.00361249i
$396$	− 0.464102i	− 0.0233220i
$397$	− 12.1244i	− 0.608504i	−0.952592	−	0.304252i	$-0.901594\pi$
$397$	− 12.1244i	− 0.608504i	0.952592	−	0.304252i	$-0.0984065\pi$
$398$	− 18.9282i	− 0.948785i
$399$	−1.07180	−0.0536570
$400$	−1.00000	−0.0500000
$401$	32.0000i	1.59800i	0.601329	+	0.799002i	$0.294638\pi$
$401$	32.0000i	1.59800i	−0.601329	+	0.799002i	$0.705362\pi$
$402$	4.53590	0.226230
$403$	0	0
$404$	10.9282	0.543698
$405$	1.00000i	0.0496904i
$406$	7.46410	0.370437
$407$	−0.555136	−0.0275171
$408$	− 4.00000i	− 0.198030i
$409$	4.00000i	0.197787i	0.995098	+	0.0988936i	$0.0315304\pi$
$409$	4.00000i	0.197787i	−0.995098	+	0.0988936i	$0.968470\pi$
$410$	2.00000i	0.0987730i
$411$	− 4.46410i	− 0.220198i
$412$	−15.8564	−0.781189
$413$	3.07180	0.151153
$414$	0.267949i	0.0131690i
$415$	4.92820	0.241916
$416$	0	0
$417$	−0.928203	−0.0454543
$418$	− 0.248711i	− 0.0121649i
$419$	1.60770	0.0785410	0.0392705	−	0.999229i	$-0.487497\pi$
$419$	1.60770	0.0785410	0.0392705	+	0.999229i	$0.487497\pi$
$420$	2.00000	0.0975900
$421$	− 16.3923i	− 0.798912i	−0.916752	−	0.399456i	$-0.869199\pi$
$421$	− 16.3923i	− 0.798912i	0.916752	−	0.399456i	$-0.130801\pi$
$422$	23.3205i	1.13522i
$423$	− 10.4641i	− 0.508782i
$424$	12.9282i	0.627849i
$425$	4.00000	0.194029
$426$	8.39230	0.406608
$427$	− 20.7846i	− 1.00584i
$428$	−19.8564	−0.959796
$429$	0	0
$430$	1.92820	0.0929862
$431$	− 7.60770i	− 0.366450i	−0.983071	−	0.183225i	$-0.941346\pi$
$431$	− 7.60770i	− 0.366450i	0.983071	−	0.183225i	$-0.0586537\pi$
$432$	−1.00000	−0.0481125
$433$	−15.3205	−0.736257	−0.368128	−	0.929775i	$-0.620001\pi$
$433$	−15.3205	−0.736257	−0.368128	+	0.929775i	$0.620001\pi$
$434$	3.46410i	0.166282i
$435$	− 3.73205i	− 0.178938i
$436$	− 11.8564i	− 0.567819i
$437$	0.143594i	0.00686901i
$438$	2.00000	0.0955637
$439$	9.85641	0.470421	0.235210	−	0.971945i	$-0.424422\pi$
$439$	9.85641	0.470421	0.235210	+	0.971945i	$0.424422\pi$
$440$	0.464102i	0.0221252i
$441$	3.00000	0.142857
$442$	0	0
$443$	−4.39230	−0.208685	−0.104342	−	0.994541i	$-0.533274\pi$
$443$	−4.39230	−0.208685	−0.104342	+	0.994541i	$0.533274\pi$
$444$	1.19615i	0.0567669i
$445$	−7.46410	−0.353832
$446$	−27.4641	−1.30046
$447$	20.4641i	0.967919i
$448$	2.00000i	0.0944911i
$449$	39.7128i	1.87416i	0.349110	+	0.937082i	$0.386484\pi$
$449$	39.7128i	1.87416i	−0.349110	+	0.937082i	$0.613516\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	0.928203	0.0437074
$452$	−11.1962	−0.526623
$453$	− 10.3923i	− 0.488273i
$454$	−4.39230	−0.206141
$455$	0	0
$456$	−0.535898	−0.0250957
$457$	− 31.4641i	− 1.47183i	−0.677075	−	0.735914i	$-0.736753\pi$
$457$	− 31.4641i	− 1.47183i	0.677075	−	0.735914i	$-0.263247\pi$
$458$	−19.8564	−0.927829
$459$	4.00000	0.186704
$460$	− 0.267949i	− 0.0124932i
$461$	− 6.46410i	− 0.301063i	−0.988605	−	0.150532i	$-0.951901\pi$
$461$	− 6.46410i	− 0.301063i	0.988605	−	0.150532i	$-0.0480985\pi$
$462$	− 0.928203i	− 0.0431839i
$463$	20.9282i	0.972616i	0.873787	+	0.486308i	$0.161657\pi$
$463$	20.9282i	0.972616i	−0.873787	+	0.486308i	$0.838343\pi$
$464$	3.73205	0.173256
$465$	1.73205	0.0803219
$466$	− 18.1244i	− 0.839595i
$467$	11.8564	0.548649	0.274325	−	0.961637i	$-0.411546\pi$
$467$	11.8564	0.548649	0.274325	+	0.961637i	$0.411546\pi$
$468$	0	0
$469$	9.07180	0.418897
$470$	10.4641i	0.482673i
$471$	5.00000	0.230388
$472$	1.53590	0.0706955
$473$	− 0.894882i	− 0.0411467i
$474$	0.0717968i	0.00329773i
$475$	− 0.535898i	− 0.0245887i
$476$	− 8.00000i	− 0.366679i
$477$	−12.9282	−0.591942
$478$	−4.39230	−0.200899
$479$	1.46410i	0.0668965i	0.999440	+	0.0334483i	$0.0106489\pi$
$479$	1.46410i	0.0668965i	−0.999440	+	0.0334483i	$0.989351\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−14.2679	−0.649887
$483$	0.535898i	0.0243842i
$484$	−10.7846	−0.490210
$485$	−7.46410	−0.338927
$486$	− 1.00000i	− 0.0453609i
$487$	− 39.1769i	− 1.77528i	−0.460542	−	0.887638i	$-0.652345\pi$
$487$	− 39.1769i	− 1.77528i	0.460542	−	0.887638i	$-0.347655\pi$
$488$	− 10.3923i	− 0.470438i
$489$	23.0526i	1.04247i
$490$	−3.00000	−0.135526
$491$	17.3205	0.781664	0.390832	−	0.920462i	$-0.372187\pi$
$491$	17.3205	0.781664	0.390832	+	0.920462i	$0.372187\pi$
$492$	− 2.00000i	− 0.0901670i
$493$	−14.9282	−0.672332
$494$	0	0
$495$	−0.464102	−0.0208598
$496$	1.73205i	0.0777714i
$497$	16.7846	0.752893
$498$	−4.92820	−0.220838
$499$	13.4641i	0.602736i	0.953508	+	0.301368i	$0.0974433\pi$
$499$	13.4641i	0.602736i	−0.953508	+	0.301368i	$0.902557\pi$
$500$	1.00000i	0.0447214i
$501$	− 18.3205i	− 0.818500i
$502$	12.2679i	0.547545i
$503$	31.1769	1.39011	0.695055	−	0.718957i	$-0.255380\pi$
$503$	31.1769	1.39011	0.695055	+	0.718957i	$0.255380\pi$
$504$	−2.00000	−0.0890871
$505$	− 10.9282i	− 0.486299i
$506$	−0.124356	−0.00552828
$507$	0	0
$508$	−8.92820	−0.396125
$509$	19.3923i	0.859549i	0.902936	+	0.429774i	$0.141407\pi$
$509$	19.3923i	0.859549i	−0.902936	+	0.429774i	$0.858593\pi$
$510$	−4.00000	−0.177123
$511$	4.00000	0.176950
$512$	1.00000i	0.0441942i
$513$	− 0.535898i	− 0.0236605i
$514$	− 22.6603i	− 0.999501i
$515$	15.8564i	0.698717i
$516$	−1.92820	−0.0848844
$517$	4.85641	0.213585
$518$	2.39230i	0.105112i
$519$	2.92820	0.128534
$520$	0	0
$521$	17.3205	0.758825	0.379413	−	0.925228i	$-0.376126\pi$
$521$	17.3205	0.758825	0.379413	+	0.925228i	$0.376126\pi$
$522$	3.73205i	0.163347i
$523$	−29.7846	−1.30239	−0.651195	−	0.758910i	$-0.725732\pi$
$523$	−29.7846	−1.30239	−0.651195	+	0.758910i	$0.725732\pi$
$524$	1.33975	0.0585271
$525$	− 2.00000i	− 0.0872872i
$526$	− 18.1244i	− 0.790259i
$527$	− 6.92820i	− 0.301797i
$528$	− 0.464102i	− 0.0201974i
$529$	−22.9282	−0.996878
$530$	12.9282	0.561565
$531$	1.53590i	0.0666523i
$532$	−1.07180	−0.0464683
$533$	0	0
$534$	7.46410	0.323003
$535$	19.8564i	0.858467i
$536$	4.53590	0.195921
$537$	16.2679	0.702014
$538$	12.0000i	0.517357i
$539$	1.39230i	0.0599708i
$540$	1.00000i	0.0430331i
$541$	13.0718i	0.562000i	0.959708	+	0.281000i	$0.0906662\pi$
$541$	13.0718i	0.562000i	−0.959708	+	0.281000i	$0.909334\pi$
$542$	−9.19615	−0.395009
$543$	−10.9282	−0.468974
$544$	− 4.00000i	− 0.171499i
$545$	−11.8564	−0.507873
$546$	0	0
$547$	9.07180	0.387882	0.193941	−	0.981013i	$-0.437873\pi$
$547$	9.07180	0.387882	0.193941	+	0.981013i	$0.437873\pi$
$548$	− 4.46410i	− 0.190697i
$549$	10.3923	0.443533
$550$	0.464102	0.0197894
$551$	2.00000i	0.0852029i
$552$	0.267949i	0.0114047i
$553$	0.143594i	0.00610622i
$554$	9.92820i	0.421809i
$555$	1.19615	0.0507738
$556$	−0.928203	−0.0393646
$557$	37.7128i	1.59794i	0.601369	+	0.798972i	$0.294622\pi$
$557$	37.7128i	1.59794i	−0.601369	+	0.798972i	$0.705378\pi$
$558$	−1.73205	−0.0733236
$559$	0	0
$560$	2.00000	0.0845154
$561$	1.85641i	0.0783775i
$562$	−4.92820	−0.207884
$563$	−39.3205	−1.65716	−0.828581	−	0.559869i	$-0.810851\pi$
$563$	−39.3205	−1.65716	−0.828581	+	0.559869i	$0.810851\pi$
$564$	− 10.4641i	− 0.440618i
$565$	11.1962i	0.471026i
$566$	3.92820i	0.165115i
$567$	− 2.00000i	− 0.0839921i
$568$	8.39230	0.352133
$569$	−5.32051	−0.223047	−0.111524	−	0.993762i	$-0.535573\pi$
$569$	−5.32051	−0.223047	−0.111524	+	0.993762i	$0.535573\pi$
$570$	0.535898i	0.0224463i
$571$	45.1769	1.89060	0.945298	−	0.326209i	$-0.105771\pi$
$571$	45.1769	1.89060	0.945298	+	0.326209i	$0.105771\pi$
$572$	0	0
$573$	−14.5359	−0.607246
$574$	− 4.00000i	− 0.166957i
$575$	−0.267949	−0.0111743
$576$	−1.00000	−0.0416667
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	− 23.3205i	− 0.969167i
$580$	− 3.73205i	− 0.154965i
$581$	−9.85641	−0.408913
$582$	7.46410	0.309397
$583$	− 6.00000i	− 0.248495i
$584$	2.00000	0.0827606
$585$	0	0
$586$	4.14359	0.171170
$587$	− 18.3923i	− 0.759132i	−0.925165	−	0.379566i	$-0.876073\pi$
$587$	− 18.3923i	− 0.759132i	0.925165	−	0.379566i	$-0.123927\pi$
$588$	3.00000	0.123718
$589$	−0.928203	−0.0382459
$590$	− 1.53590i	− 0.0632319i
$591$	− 16.3923i	− 0.674289i
$592$	1.19615i	0.0491616i
$593$	− 31.1051i	− 1.27733i	−0.769483	−	0.638667i	$-0.779486\pi$
$593$	− 31.1051i	− 1.27733i	0.769483	−	0.638667i	$-0.220514\pi$
$594$	0.464102	0.0190423
$595$	−8.00000	−0.327968
$596$	20.4641i	0.838242i
$597$	18.9282	0.774680
$598$	0	0
$599$	10.3923	0.424618	0.212309	−	0.977203i	$-0.431902\pi$
$599$	10.3923	0.424618	0.212309	+	0.977203i	$0.431902\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	21.7846	0.888613	0.444306	−	0.895875i	$-0.353450\pi$
$601$	21.7846	0.888613	0.444306	+	0.895875i	$0.353450\pi$
$602$	−3.85641	−0.157175
$603$	4.53590i	0.184716i
$604$	− 10.3923i	− 0.422857i
$605$	10.7846i	0.438457i
$606$	10.9282i	0.443928i
$607$	43.1769	1.75250	0.876248	−	0.481860i	$-0.160038\pi$
$607$	43.1769	1.75250	0.876248	+	0.481860i	$0.160038\pi$
$608$	−0.535898	−0.0217335
$609$	7.46410i	0.302461i
$610$	−10.3923	−0.420772
$611$	0	0
$612$	4.00000	0.161690
$613$	0.947441i	0.0382668i	0.999817	+	0.0191334i	$0.00609072\pi$
$613$	0.947441i	0.0382668i	−0.999817	+	0.0191334i	$0.993909\pi$
$614$	−12.5359	−0.505908
$615$	−2.00000	−0.0806478
$616$	− 0.928203i	− 0.0373984i
$617$	− 15.5359i	− 0.625452i	−0.949843	−	0.312726i	$-0.898758\pi$
$617$	− 15.5359i	− 0.625452i	0.949843	−	0.312726i	$-0.101242\pi$
$618$	− 15.8564i	− 0.637838i
$619$	24.2487i	0.974638i	0.873224	+	0.487319i	$0.162025\pi$
$619$	24.2487i	0.974638i	−0.873224	+	0.487319i	$0.837975\pi$
$620$	1.73205	0.0695608
$621$	−0.267949	−0.0107524
$622$	− 7.60770i	− 0.305041i
$623$	14.9282	0.598086
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 28.0000i	− 1.11911i
$627$	0.248711	0.00993257
$628$	5.00000	0.199522
$629$	− 4.78461i	− 0.190775i
$630$	2.00000i	0.0796819i
$631$	24.5359i	0.976759i	0.872631	+	0.488379i	$0.162412\pi$
$631$	24.5359i	0.976759i	−0.872631	+	0.488379i	$0.837588\pi$
$632$	0.0717968i	0.00285592i
$633$	−23.3205	−0.926907
$634$	−21.4641	−0.852448
$635$	8.92820i	0.354305i
$636$	−12.9282	−0.512637
$637$	0	0
$638$	−1.73205	−0.0685725
$639$	8.39230i	0.331994i
$640$	1.00000	0.0395285
$641$	27.8564	1.10026	0.550131	−	0.835078i	$-0.314578\pi$
$641$	27.8564	1.10026	0.550131	+	0.835078i	$0.314578\pi$
$642$	− 19.8564i	− 0.783670i
$643$	− 27.4641i	− 1.08308i	−0.840675	−	0.541539i	$-0.817842\pi$
$643$	− 27.4641i	− 1.08308i	0.840675	−	0.541539i	$-0.182158\pi$
$644$	0.535898i	0.0211174i
$645$	1.92820i	0.0759229i
$646$	2.14359	0.0843386
$647$	21.3205	0.838196	0.419098	−	0.907941i	$-0.362346\pi$
$647$	21.3205	0.838196	0.419098	+	0.907941i	$0.362346\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−0.712813	−0.0279804
$650$	0	0
$651$	−3.46410	−0.135769
$652$	23.0526i	0.902808i
$653$	−44.2487	−1.73159	−0.865793	−	0.500402i	$-0.833185\pi$
$653$	−44.2487	−1.73159	−0.865793	+	0.500402i	$0.833185\pi$
$654$	11.8564	0.463622
$655$	− 1.33975i	− 0.0523482i
$656$	− 2.00000i	− 0.0780869i
$657$	2.00000i	0.0780274i
$658$	− 20.9282i	− 0.815866i
$659$	−3.73205	−0.145380	−0.0726900	−	0.997355i	$-0.523158\pi$
$659$	−3.73205	−0.145380	−0.0726900	+	0.997355i	$0.523158\pi$
$660$	−0.464102	−0.0180651
$661$	43.3205i	1.68497i	0.538718	+	0.842486i	$0.318909\pi$
$661$	43.3205i	1.68497i	−0.538718	+	0.842486i	$0.681091\pi$
$662$	−24.7846	−0.963281
$663$	0	0
$664$	−4.92820	−0.191251
$665$	1.07180i	0.0415625i
$666$	−1.19615	−0.0463500
$667$	1.00000	0.0387202
$668$	− 18.3205i	− 0.708842i
$669$	− 27.4641i	− 1.06182i
$670$	− 4.53590i	− 0.175237i
$671$	4.82309i	0.186193i
$672$	−2.00000	−0.0771517
$673$	32.0000	1.23351	0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000	1.23351	0.616755	+	0.787155i	$0.288447\pi$
$674$	− 25.3205i	− 0.975310i
$675$	1.00000	0.0384900
$676$	0	0
$677$	11.6077	0.446120	0.223060	−	0.974805i	$-0.428395\pi$
$677$	11.6077	0.446120	0.223060	+	0.974805i	$0.428395\pi$
$678$	− 11.1962i	− 0.429986i
$679$	14.9282	0.572892
$680$	−4.00000	−0.153393
$681$	− 4.39230i	− 0.168313i
$682$	− 0.803848i	− 0.0307809i
$683$	0.784610i	0.0300223i	0.999887	+	0.0150111i	$0.00477837\pi$
$683$	0.784610i	0.0300223i	−0.999887	+	0.0150111i	$0.995222\pi$
$684$	− 0.535898i	− 0.0204906i
$685$	−4.46410	−0.170565
$686$	20.0000	0.763604
$687$	− 19.8564i	− 0.757569i
$688$	−1.92820	−0.0735121
$689$	0	0
$690$	0.267949	0.0102007
$691$	− 35.1769i	− 1.33819i	−0.743176	−	0.669096i	$-0.766681\pi$
$691$	− 35.1769i	− 1.33819i	0.743176	−	0.669096i	$-0.233319\pi$
$692$	2.92820	0.111314
$693$	0.928203	0.0352595
$694$	22.3923i	0.850000i
$695$	0.928203i	0.0352088i
$696$	3.73205i	0.141463i
$697$	8.00000i	0.303022i
$698$	14.5359	0.550192
$699$	18.1244	0.685526
$700$	− 2.00000i	− 0.0755929i
$701$	−3.73205	−0.140958	−0.0704788	−	0.997513i	$-0.522453\pi$
$701$	−3.73205	−0.140958	−0.0704788	+	0.997513i	$0.522453\pi$
$702$	0	0
$703$	−0.641016	−0.0241764
$704$	− 0.464102i	− 0.0174915i
$705$	−10.4641	−0.394101
$706$	2.00000	0.0752710
$707$	21.8564i	0.821995i
$708$	1.53590i	0.0577226i
$709$	− 9.07180i	− 0.340698i	−0.985384	−	0.170349i	$-0.945510\pi$
$709$	− 9.07180i	− 0.340698i	0.985384	−	0.170349i	$-0.0544896\pi$
$710$	− 8.39230i	− 0.314958i
$711$	−0.0717968	−0.00269259
$712$	7.46410	0.279729
$713$	0.464102i	0.0173807i
$714$	8.00000	0.299392
$715$	0	0
$716$	16.2679	0.607962
$717$	− 4.39230i	− 0.164034i
$718$	−18.9282	−0.706394
$719$	34.6410	1.29189	0.645946	−	0.763383i	$-0.276463\pi$
$719$	34.6410	1.29189	0.645946	+	0.763383i	$0.276463\pi$
$720$	1.00000i	0.0372678i
$721$	− 31.7128i	− 1.18105i
$722$	18.7128i	0.696419i
$723$	− 14.2679i	− 0.530631i
$724$	−10.9282	−0.406143
$725$	−3.73205	−0.138605
$726$	− 10.7846i	− 0.400254i
$727$	23.7128	0.879460	0.439730	−	0.898130i	$-0.355074\pi$
$727$	23.7128	0.879460	0.439730	+	0.898130i	$0.355074\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 2.00000i	− 0.0740233i
$731$	7.71281	0.285269
$732$	10.3923	0.384111
$733$	− 37.0718i	− 1.36928i	−0.728882	−	0.684639i	$-0.759960\pi$
$733$	− 37.0718i	− 1.36928i	0.728882	−	0.684639i	$-0.240040\pi$
$734$	36.3923i	1.34326i
$735$	− 3.00000i	− 0.110657i
$736$	0.267949i	0.00987674i
$737$	−2.10512	−0.0775430
$738$	2.00000	0.0736210
$739$	− 15.3205i	− 0.563574i	−0.959477	−	0.281787i	$-0.909073\pi$
$739$	− 15.3205i	− 0.563574i	0.959477	−	0.281787i	$-0.0909272\pi$
$740$	1.19615	0.0439714
$741$	0	0
$742$	−25.8564	−0.949219
$743$	33.5359i	1.23031i	0.788405	+	0.615156i	$0.210907\pi$
$743$	33.5359i	1.23031i	−0.788405	+	0.615156i	$0.789093\pi$
$744$	−1.73205	−0.0635001
$745$	20.4641	0.749747
$746$	25.7846i	0.944042i
$747$	− 4.92820i	− 0.180314i
$748$	1.85641i	0.0678769i
$749$	− 39.7128i	− 1.45107i
$750$	−1.00000	−0.0365148
$751$	27.9282	1.01911	0.509557	−	0.860437i	$-0.329809\pi$
$751$	27.9282	1.01911	0.509557	+	0.860437i	$0.329809\pi$
$752$	− 10.4641i	− 0.381587i
$753$	−12.2679	−0.447069
$754$	0	0
$755$	−10.3923	−0.378215
$756$	− 2.00000i	− 0.0727393i
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	0.143594	0.00521555
$759$	− 0.124356i	− 0.00451382i
$760$	0.535898i	0.0194391i
$761$	18.9282i	0.686147i	0.939309	+	0.343073i	$0.111468\pi$
$761$	18.9282i	0.686147i	−0.939309	+	0.343073i	$0.888532\pi$
$762$	− 8.92820i	− 0.323435i
$763$	23.7128	0.858461
$764$	−14.5359	−0.525890
$765$	− 4.00000i	− 0.144620i
$766$	−4.60770	−0.166483
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 19.5885i	− 0.706378i	−0.935552	−	0.353189i	$-0.885097\pi$
$769$	− 19.5885i	− 0.706378i	0.935552	−	0.353189i	$-0.114903\pi$
$770$	−0.928203	−0.0334501
$771$	22.6603	0.816089
$772$	− 23.3205i	− 0.839323i
$773$	− 27.7128i	− 0.996761i	−0.866959	−	0.498380i	$-0.833928\pi$
$773$	− 27.7128i	− 0.996761i	0.866959	−	0.498380i	$-0.166072\pi$
$774$	− 1.92820i	− 0.0693078i
$775$	− 1.73205i	− 0.0622171i
$776$	7.46410	0.267946
$777$	−2.39230	−0.0858235
$778$	− 20.2679i	− 0.726641i
$779$	1.07180	0.0384011
$780$	0	0
$781$	−3.89488	−0.139370
$782$	− 1.07180i	− 0.0383274i
$783$	−3.73205	−0.133373
$784$	3.00000	0.107143
$785$	− 5.00000i	− 0.178458i
$786$	1.33975i	0.0477872i
$787$	− 1.73205i	− 0.0617409i	−0.999523	−	0.0308705i	$-0.990172\pi$
$787$	− 1.73205i	− 0.0617409i	0.999523	−	0.0308705i	$-0.00982794\pi$
$788$	− 16.3923i	− 0.583952i
$789$	18.1244	0.645244
$790$	0.0717968	0.00255441
$791$	− 22.3923i	− 0.796179i
$792$	0.464102	0.0164911
$793$	0	0
$794$	12.1244	0.430277
$795$	12.9282i	0.458516i
$796$	18.9282	0.670892
$797$	−37.8564	−1.34094	−0.670471	−	0.741935i	$-0.733908\pi$
$797$	−37.8564	−1.34094	−0.670471	+	0.741935i	$0.733908\pi$
$798$	− 1.07180i	− 0.0379412i
$799$	41.8564i	1.48077i
$800$	− 1.00000i	− 0.0353553i
$801$	7.46410i	0.263731i
$802$	−32.0000	−1.12996
$803$	−0.928203	−0.0327556
$804$	4.53590i	0.159969i
$805$	0.535898	0.0188879
$806$	0	0
$807$	−12.0000	−0.422420
$808$	10.9282i	0.384453i
$809$	−1.21539	−0.0427308	−0.0213654	−	0.999772i	$-0.506801\pi$
$809$	−1.21539	−0.0427308	−0.0213654	+	0.999772i	$0.506801\pi$
$810$	−1.00000	−0.0351364
$811$	− 0.784610i	− 0.0275514i	−0.999905	−	0.0137757i	$-0.995615\pi$
$811$	− 0.784610i	− 0.0275514i	0.999905	−	0.0137757i	$-0.00438508\pi$
$812$	7.46410i	0.261939i
$813$	− 9.19615i	− 0.322523i
$814$	− 0.555136i	− 0.0194575i
$815$	23.0526	0.807496
$816$	4.00000	0.140028
$817$	− 1.03332i	− 0.0361513i
$818$	−4.00000	−0.139857
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	− 35.3923i	− 1.23520i	−0.786492	−	0.617600i	$-0.788105\pi$
$821$	− 35.3923i	− 1.23520i	0.786492	−	0.617600i	$-0.211895\pi$
$822$	4.46410	0.155703
$823$	−23.1769	−0.807896	−0.403948	−	0.914782i	$-0.632362\pi$
$823$	−23.1769	−0.807896	−0.403948	+	0.914782i	$0.632362\pi$
$824$	− 15.8564i	− 0.552384i
$825$	0.464102i	0.0161579i
$826$	3.07180i	0.106881i
$827$	− 17.3205i	− 0.602293i	−0.953578	−	0.301147i	$-0.902631\pi$
$827$	− 17.3205i	− 0.602293i	0.953578	−	0.301147i	$-0.0973693\pi$
$828$	−0.267949	−0.00931188
$829$	23.4641	0.814942	0.407471	−	0.913218i	$-0.366411\pi$
$829$	23.4641	0.814942	0.407471	+	0.913218i	$0.366411\pi$
$830$	4.92820i	0.171060i
$831$	−9.92820	−0.344406
$832$	0	0
$833$	−12.0000	−0.415775
$834$	− 0.928203i	− 0.0321410i
$835$	−18.3205	−0.634007
$836$	0.248711	0.00860186
$837$	− 1.73205i	− 0.0598684i
$838$	1.60770i	0.0555369i
$839$	39.5692i	1.36608i	0.730380	+	0.683041i	$0.239343\pi$
$839$	39.5692i	1.36608i	−0.730380	+	0.683041i	$0.760657\pi$
$840$	2.00000i	0.0690066i
$841$	−15.0718	−0.519717
$842$	16.3923	0.564916
$843$	− 4.92820i	− 0.169736i
$844$	−23.3205	−0.802725
$845$	0	0
$846$	10.4641	0.359763
$847$	− 21.5692i	− 0.741127i
$848$	−12.9282	−0.443956
$849$	−3.92820	−0.134816
$850$	4.00000i	0.137199i
$851$	0.320508i	0.0109869i
$852$	8.39230i	0.287516i
$853$	− 35.8372i	− 1.22704i	−0.789679	−	0.613521i	$-0.789753\pi$
$853$	− 35.8372i	− 1.22704i	0.789679	−	0.613521i	$-0.210247\pi$
$854$	20.7846	0.711235
$855$	−0.535898	−0.0183273
$856$	− 19.8564i	− 0.678678i
$857$	−20.5167	−0.700836	−0.350418	−	0.936593i	$-0.613960\pi$
$857$	−20.5167	−0.700836	−0.350418	+	0.936593i	$0.613960\pi$
$858$	0	0
$859$	27.1769	0.927264	0.463632	−	0.886028i	$-0.346546\pi$
$859$	27.1769	0.927264	0.463632	+	0.886028i	$0.346546\pi$
$860$	1.92820i	0.0657512i
$861$	4.00000	0.136320
$862$	7.60770	0.259119
$863$	42.4641i	1.44549i	0.691112	+	0.722747i	$0.257121\pi$
$863$	42.4641i	1.44549i	−0.691112	+	0.722747i	$0.742879\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	− 2.92820i	− 0.0995619i
$866$	− 15.3205i	− 0.520612i
$867$	1.00000	0.0339618
$868$	−3.46410	−0.117579
$869$	− 0.0333210i	− 0.00113034i
$870$	3.73205	0.126528
$871$	0	0
$872$	11.8564	0.401509
$873$	7.46410i	0.252622i
$874$	−0.143594	−0.00485712
$875$	−2.00000	−0.0676123
$876$	2.00000i	0.0675737i
$877$	− 28.1244i	− 0.949692i	−0.880069	−	0.474846i	$-0.842504\pi$
$877$	− 28.1244i	− 0.949692i	0.880069	−	0.474846i	$-0.157496\pi$
$878$	9.85641i	0.332638i
$879$	4.14359i	0.139760i
$880$	−0.464102	−0.0156449
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	3.00000i	0.101015i
$883$	16.0718	0.540859	0.270430	−	0.962740i	$-0.412834\pi$
$883$	16.0718	0.540859	0.270430	+	0.962740i	$0.412834\pi$
$884$	0	0
$885$	1.53590	0.0516287
$886$	− 4.39230i	− 0.147562i
$887$	−10.1244	−0.339943	−0.169971	−	0.985449i	$-0.554368\pi$
$887$	−10.1244	−0.339943	−0.169971	+	0.985449i	$0.554368\pi$
$888$	−1.19615	−0.0401402
$889$	− 17.8564i	− 0.598885i
$890$	− 7.46410i	− 0.250197i
$891$	0.464102i	0.0155480i
$892$	− 27.4641i	− 0.919566i
$893$	5.60770	0.187654
$894$	−20.4641	−0.684422
$895$	− 16.2679i	− 0.543778i
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−39.7128	−1.32523
$899$	6.46410i	0.215590i
$900$	1.00000	0.0333333
$901$	51.7128	1.72280
$902$	0.928203i	0.0309058i
$903$	− 3.85641i	− 0.128333i
$904$	− 11.1962i	− 0.372378i
$905$	10.9282i	0.363266i
$906$	10.3923	0.345261
$907$	6.85641	0.227663	0.113832	−	0.993500i	$-0.463688\pi$
$907$	6.85641	0.227663	0.113832	+	0.993500i	$0.463688\pi$
$908$	− 4.39230i	− 0.145764i
$909$	−10.9282	−0.362466
$910$	0	0
$911$	−24.2487	−0.803396	−0.401698	−	0.915772i	$-0.631580\pi$
$911$	−24.2487	−0.803396	−0.401698	+	0.915772i	$0.631580\pi$
$912$	− 0.535898i	− 0.0177454i
$913$	2.28719	0.0756948
$914$	31.4641	1.04074
$915$	− 10.3923i	− 0.343559i
$916$	− 19.8564i	− 0.656074i
$917$	2.67949i	0.0884846i
$918$	4.00000i	0.132020i
$919$	−54.6410	−1.80244	−0.901220	−	0.433361i	$-0.857328\pi$
$919$	−54.6410	−1.80244	−0.901220	+	0.433361i	$0.857328\pi$
$920$	0.267949	0.00883402
$921$	− 12.5359i	− 0.413072i
$922$	6.46410	0.212884
$923$	0	0
$924$	0.928203	0.0305356
$925$	− 1.19615i	− 0.0393292i
$926$	−20.9282	−0.687743
$927$	15.8564	0.520793
$928$	3.73205i	0.122511i
$929$	− 33.8564i	− 1.11079i	−0.831586	−	0.555396i	$-0.812567\pi$
$929$	− 33.8564i	− 1.11079i	0.831586	−	0.555396i	$-0.187433\pi$
$930$	1.73205i	0.0567962i
$931$	1.60770i	0.0526901i
$932$	18.1244	0.593683
$933$	7.60770	0.249065
$934$	11.8564i	0.387953i
$935$	1.85641	0.0607110
$936$	0	0
$937$	44.6410	1.45836	0.729179	−	0.684323i	$-0.239902\pi$
$937$	44.6410	1.45836	0.729179	+	0.684323i	$0.239902\pi$
$938$	9.07180i	0.296205i
$939$	28.0000	0.913745
$940$	−10.4641	−0.341301
$941$	26.7846i	0.873153i	0.899667	+	0.436577i	$0.143809\pi$
$941$	26.7846i	0.873153i	−0.899667	+	0.436577i	$0.856191\pi$
$942$	5.00000i	0.162909i
$943$	− 0.535898i	− 0.0174513i
$944$	1.53590i	0.0499892i
$945$	−2.00000	−0.0650600
$946$	0.894882	0.0290951
$947$	− 2.53590i	− 0.0824056i	−0.999151	−	0.0412028i	$-0.986881\pi$
$947$	− 2.53590i	− 0.0824056i	0.999151	−	0.0412028i	$-0.0131190\pi$
$948$	−0.0717968	−0.00233185
$949$	0	0
$950$	0.535898	0.0173868
$951$	− 21.4641i	− 0.696021i
$952$	8.00000	0.259281
$953$	−15.7321	−0.509611	−0.254806	−	0.966992i	$-0.582011\pi$
$953$	−15.7321	−0.509611	−0.254806	+	0.966992i	$0.582011\pi$
$954$	− 12.9282i	− 0.418566i
$955$	14.5359i	0.470371i
$956$	− 4.39230i	− 0.142057i
$957$	− 1.73205i	− 0.0559893i
$958$	−1.46410	−0.0473030
$959$	8.92820	0.288307
$960$	1.00000i	0.0322749i
$961$	28.0000	0.903226
$962$	0	0
$963$	19.8564	0.639864
$964$	− 14.2679i	− 0.459540i
$965$	−23.3205	−0.750714
$966$	−0.535898	−0.0172422
$967$	− 34.5359i	− 1.11060i	−0.831650	−	0.555300i	$-0.812604\pi$
$967$	− 34.5359i	− 1.11060i	0.831650	−	0.555300i	$-0.187396\pi$
$968$	− 10.7846i	− 0.346630i
$969$	2.14359i	0.0688621i
$970$	− 7.46410i	− 0.239658i
$971$	11.4641	0.367901	0.183950	−	0.982936i	$-0.441111\pi$
$971$	11.4641	0.367901	0.183950	+	0.982936i	$0.441111\pi$
$972$	1.00000	0.0320750
$973$	− 1.85641i	− 0.0595137i
$974$	39.1769	1.25531
$975$	0	0
$976$	10.3923	0.332650
$977$	− 33.3923i	− 1.06831i	−0.845385	−	0.534157i	$-0.820629\pi$
$977$	− 33.3923i	− 1.06831i	0.845385	−	0.534157i	$-0.179371\pi$
$978$	−23.0526	−0.737140
$979$	−3.46410	−0.110713
$980$	− 3.00000i	− 0.0958315i
$981$	11.8564i	0.378546i
$982$	17.3205i	0.552720i
$983$	− 57.3923i	− 1.83053i	−0.402852	−	0.915265i	$-0.631981\pi$
$983$	− 57.3923i	− 1.83053i	0.402852	−	0.915265i	$-0.368019\pi$
$984$	2.00000	0.0637577
$985$	−16.3923	−0.522302
$986$	− 14.9282i	− 0.475411i
$987$	20.9282	0.666152
$988$	0	0
$989$	−0.516660	−0.0164288
$990$	− 0.464102i	− 0.0147501i
$991$	−25.1436	−0.798713	−0.399356	−	0.916796i	$-0.630766\pi$
$991$	−25.1436	−0.798713	−0.399356	+	0.916796i	$0.630766\pi$
$992$	−1.73205	−0.0549927
$993$	− 24.7846i	− 0.786516i
$994$	16.7846i	0.532375i
$995$	− 18.9282i	− 0.600064i
$996$	− 4.92820i	− 0.156156i
$997$	−47.5692	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$997$	−47.5692	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$998$	−13.4641	−0.426199
$999$	− 1.19615i	− 0.0378446i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.o.1351.4		4
13.5	odd	4		5070.2.a.bg.1.1		2
13.8	odd	4		5070.2.a.y.1.2		2
13.9	even	3		390.2.bb.b.361.2	yes	4
13.10	even	6		390.2.bb.b.121.2	✓	4
13.12	even	2	inner	5070.2.b.o.1351.1		4
39.23	odd	6		1170.2.bs.e.901.1		4
39.35	odd	6		1170.2.bs.e.361.1		4
65.9	even	6		1950.2.bc.b.751.1		4
65.22	odd	12		1950.2.y.f.49.2		4
65.23	odd	12		1950.2.y.f.199.2		4
65.48	odd	12		1950.2.y.c.49.1		4
65.49	even	6		1950.2.bc.b.901.1		4
65.62	odd	12		1950.2.y.c.199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bb.b.121.2	✓	4	13.10	even	6
390.2.bb.b.361.2	yes	4	13.9	even	3
1170.2.bs.e.361.1		4	39.35	odd	6
1170.2.bs.e.901.1		4	39.23	odd	6
1950.2.y.c.49.1		4	65.48	odd	12
1950.2.y.c.199.1		4	65.62	odd	12
1950.2.y.f.49.2		4	65.22	odd	12
1950.2.y.f.199.2		4	65.23	odd	12
1950.2.bc.b.751.1		4	65.9	even	6
1950.2.bc.b.901.1		4	65.49	even	6
5070.2.a.y.1.2		2	13.8	odd	4
5070.2.a.bg.1.1		2	13.5	odd	4
5070.2.b.o.1351.1		4	13.12	even	2	inner
5070.2.b.o.1351.4		4	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +0.464102i q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000i q^{18} +0.535898i q^{19} -1.00000i q^{20} +2.00000i q^{21} -0.464102 q^{22} +0.267949 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} +3.73205 q^{29} +1.00000 q^{30} +1.73205i q^{31} +1.00000i q^{32} -0.464102i q^{33} -4.00000i q^{34} +2.00000 q^{35} -1.00000 q^{36} +1.19615i q^{37} -0.535898 q^{38} +1.00000 q^{40} -2.00000i q^{41} -2.00000 q^{42} -1.92820 q^{43} -0.464102i q^{44} +1.00000i q^{45} +0.267949i q^{46} -10.4641i q^{47} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} +4.00000 q^{51} -12.9282 q^{53} -1.00000i q^{54} -0.464102 q^{55} -2.00000 q^{56} -0.535898i q^{57} +3.73205i q^{58} +1.53590i q^{59} +1.00000i q^{60} +10.3923 q^{61} -1.73205 q^{62} -2.00000i q^{63} -1.00000 q^{64} +0.464102 q^{66} +4.53590i q^{67} +4.00000 q^{68} -0.267949 q^{69} +2.00000i q^{70} +8.39230i q^{71} -1.00000i q^{72} +2.00000i q^{73} -1.19615 q^{74} +1.00000 q^{75} -0.535898i q^{76} +0.928203 q^{77} -0.0717968 q^{79} +1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} -4.92820i q^{83} -2.00000i q^{84} -4.00000i q^{85} -1.92820i q^{86} -3.73205 q^{87} +0.464102 q^{88} +7.46410i q^{89} -1.00000 q^{90} -0.267949 q^{92} -1.73205i q^{93} +10.4641 q^{94} -0.535898 q^{95} -1.00000i q^{96} +7.46410i q^{97} +3.00000i q^{98} +0.464102i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9	\( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} + 4 q^{12} + 8 q^{14} + 4 q^{16} - 16 q^{17} + 12 q^{22} + 8 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{29} + 4 q^{30} + 8 q^{35} - 4 q^{36} - 16 q^{38} + 4 q^{40} - 8 q^{42} + 20 q^{43} - 4 q^{48} + 12 q^{49} + 16 q^{51} - 24 q^{53} + 12 q^{55} - 8 q^{56} - 4 q^{64} - 12 q^{66} + 16 q^{68} - 8 q^{69} + 16 q^{74} + 4 q^{75} - 24 q^{77} - 28 q^{79} + 4 q^{81} + 8 q^{82} - 8 q^{87} - 12 q^{88} - 4 q^{90} - 8 q^{92} + 28 q^{94} - 16 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 + 4 * q^12 + 8 * q^14 + 4 * q^16 - 16 * q^17 + 12 * q^22 + 8 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^29 + 4 * q^30 + 8 * q^35 - 4 * q^36 - 16 * q^38 + 4 * q^40 - 8 * q^42 + 20 * q^43 - 4 * q^48 + 12 * q^49 + 16 * q^51 - 24 * q^53 + 12 * q^55 - 8 * q^56 - 4 * q^64 - 12 * q^66 + 16 * q^68 - 8 * q^69 + 16 * q^74 + 4 * q^75 - 24 * q^77 - 28 * q^79 + 4 * q^81 + 8 * q^82 - 8 * q^87 - 12 * q^88 - 4 * q^90 - 8 * q^92 + 28 * q^94 - 16 * q^95

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