LMFDB - Embedded newform 5070.2.b.r.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:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.4
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.r.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +0.561553i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +0.561553i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.12311i q^{11} -1.00000 q^{12} -0.561553 q^{14} +1.00000i q^{15} +1.00000 q^{16} +3.12311 q^{17} +1.00000i q^{18} +0.561553i q^{19} -1.00000i q^{20} +0.561553i q^{21} +4.12311 q^{22} -4.68466 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -0.561553i q^{28} +2.43845 q^{29} -1.00000 q^{30} -6.68466i q^{31} +1.00000i q^{32} -4.12311i q^{33} +3.12311i q^{34} -0.561553 q^{35} -1.00000 q^{36} +4.12311i q^{37} -0.561553 q^{38} +1.00000 q^{40} +12.2462i q^{41} -0.561553 q^{42} -0.438447 q^{43} +4.12311i q^{44} +1.00000i q^{45} -4.68466i q^{46} -7.00000i q^{47} +1.00000 q^{48} +6.68466 q^{49} -1.00000i q^{50} +3.12311 q^{51} +8.56155 q^{53} +1.00000i q^{54} +4.12311 q^{55} +0.561553 q^{56} +0.561553i q^{57} +2.43845i q^{58} -6.43845i q^{59} -1.00000i q^{60} +6.00000 q^{61} +6.68466 q^{62} +0.561553i q^{63} -1.00000 q^{64} +4.12311 q^{66} -2.24621i q^{67} -3.12311 q^{68} -4.68466 q^{69} -0.561553i q^{70} -13.1231i q^{71} -1.00000i q^{72} -9.36932i q^{73} -4.12311 q^{74} -1.00000 q^{75} -0.561553i q^{76} +2.31534 q^{77} +11.5616 q^{79} +1.00000i q^{80} +1.00000 q^{81} -12.2462 q^{82} -7.12311i q^{83} -0.561553i q^{84} +3.12311i q^{85} -0.438447i q^{86} +2.43845 q^{87} -4.12311 q^{88} +18.8078i q^{89} -1.00000 q^{90} +4.68466 q^{92} -6.68466i q^{93} +7.00000 q^{94} -0.561553 q^{95} +1.00000i q^{96} -7.12311i q^{97} +6.68466i q^{98} -4.12311i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 6 q^{14} + 4 q^{16} - 4 q^{17} + 6 q^{23} - 4 q^{25} + 4 q^{27} + 18 q^{29} - 4 q^{30} + 6 q^{35} - 4 q^{36} + 6 q^{38} + 4 q^{40} + 6 q^{42} - 10 q^{43} + 4 q^{48} + 2 q^{49} - 4 q^{51} + 26 q^{53} - 6 q^{56} + 24 q^{61} + 2 q^{62} - 4 q^{64} + 4 q^{68} + 6 q^{69} - 4 q^{75} + 34 q^{77} + 38 q^{79} + 4 q^{81} - 16 q^{82} + 18 q^{87} - 4 q^{90} - 6 q^{92} + 28 q^{94} + 6 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 - 4 * q^12 + 6 * q^14 + 4 * q^16 - 4 * q^17 + 6 * q^23 - 4 * q^25 + 4 * q^27 + 18 * q^29 - 4 * q^30 + 6 * q^35 - 4 * q^36 + 6 * q^38 + 4 * q^40 + 6 * q^42 - 10 * q^43 + 4 * q^48 + 2 * q^49 - 4 * q^51 + 26 * q^53 - 6 * q^56 + 24 * q^61 + 2 * q^62 - 4 * q^64 + 4 * q^68 + 6 * q^69 - 4 * q^75 + 34 * q^77 + 38 * q^79 + 4 * q^81 - 16 * q^82 + 18 * q^87 - 4 * q^90 - 6 * q^92 + 28 * q^94 + 6 * 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
$7$	0.561553i	0.212247i	0.994353	+	0.106124i	$0.0338439\pi$
$7$	0.561553i	0.212247i	−0.994353	+	0.106124i	$0.966156\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	− 4.12311i	− 1.24316i	−0.783349	−	0.621582i	$-0.786490\pi$
$11$	− 4.12311i	− 1.24316i	0.783349	−	0.621582i	$-0.213510\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−0.561553	−0.150081
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	3.12311	0.757464	0.378732	−	0.925506i	$-0.376360\pi$
$17$	3.12311	0.757464	0.378732	+	0.925506i	$0.376360\pi$
$18$	1.00000i	0.235702i
$19$	0.561553i	0.128829i	0.997923	+	0.0644145i	$0.0205180\pi$
$19$	0.561553i	0.128829i	−0.997923	+	0.0644145i	$0.979482\pi$
$20$	− 1.00000i	− 0.223607i
$21$	0.561553i	0.122541i
$22$	4.12311	0.879049
$23$	−4.68466	−0.976819	−0.488409	−	0.872615i	$-0.662423\pi$
$23$	−4.68466	−0.976819	−0.488409	+	0.872615i	$0.662423\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	− 0.561553i	− 0.106124i
$29$	2.43845	0.452808	0.226404	−	0.974033i	$-0.427303\pi$
$29$	2.43845	0.452808	0.226404	+	0.974033i	$0.427303\pi$
$30$	−1.00000	−0.182574
$31$	− 6.68466i	− 1.20060i	−0.799775	−	0.600300i	$-0.795048\pi$
$31$	− 6.68466i	− 1.20060i	0.799775	−	0.600300i	$-0.204952\pi$
$32$	1.00000i	0.176777i
$33$	− 4.12311i	− 0.717741i
$34$	3.12311i	0.535608i
$35$	−0.561553	−0.0949197
$36$	−1.00000	−0.166667
$37$	4.12311i	0.677834i	0.940816	+	0.338917i	$0.110061\pi$
$37$	4.12311i	0.677834i	−0.940816	+	0.338917i	$0.889939\pi$
$38$	−0.561553	−0.0910959
$39$	0	0
$40$	1.00000	0.158114
$41$	12.2462i	1.91254i	0.292490	+	0.956268i	$0.405516\pi$
$41$	12.2462i	1.91254i	−0.292490	+	0.956268i	$0.594484\pi$
$42$	−0.561553	−0.0866495
$43$	−0.438447	−0.0668626	−0.0334313	−	0.999441i	$-0.510643\pi$
$43$	−0.438447	−0.0668626	−0.0334313	+	0.999441i	$0.510643\pi$
$44$	4.12311i	0.621582i
$45$	1.00000i	0.149071i
$46$	− 4.68466i	− 0.690715i
$47$	− 7.00000i	− 1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$	− 7.00000i	− 1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$	1.00000	0.144338
$49$	6.68466	0.954951
$50$	− 1.00000i	− 0.141421i
$51$	3.12311	0.437322
$52$	0	0
$53$	8.56155	1.17602	0.588010	−	0.808854i	$-0.299912\pi$
$53$	8.56155	1.17602	0.588010	+	0.808854i	$0.299912\pi$
$54$	1.00000i	0.136083i
$55$	4.12311	0.555959
$56$	0.561553	0.0750407
$57$	0.561553i	0.0743795i
$58$	2.43845i	0.320184i
$59$	− 6.43845i	− 0.838214i	−0.907937	−	0.419107i	$-0.862343\pi$
$59$	− 6.43845i	− 0.838214i	0.907937	−	0.419107i	$-0.137657\pi$
$60$	− 1.00000i	− 0.129099i
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	6.68466	0.848952
$63$	0.561553i	0.0707490i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	4.12311	0.507519
$67$	− 2.24621i	− 0.274418i	−0.990542	−	0.137209i	$-0.956187\pi$
$67$	− 2.24621i	− 0.274418i	0.990542	−	0.137209i	$-0.0438133\pi$
$68$	−3.12311	−0.378732
$69$	−4.68466	−0.563967
$70$	− 0.561553i	− 0.0671184i
$71$	− 13.1231i	− 1.55743i	−0.627380	−	0.778713i	$-0.715873\pi$
$71$	− 13.1231i	− 1.55743i	0.627380	−	0.778713i	$-0.284127\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 9.36932i	− 1.09660i	−0.836283	−	0.548298i	$-0.815276\pi$
$73$	− 9.36932i	− 1.09660i	0.836283	−	0.548298i	$-0.184724\pi$
$74$	−4.12311	−0.479301
$75$	−1.00000	−0.115470
$76$	− 0.561553i	− 0.0644145i
$77$	2.31534	0.263858
$78$	0	0
$79$	11.5616	1.30078	0.650388	−	0.759602i	$-0.274606\pi$
$79$	11.5616	1.30078	0.650388	+	0.759602i	$0.274606\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	−12.2462	−1.35237
$83$	− 7.12311i	− 0.781862i	−0.920420	−	0.390931i	$-0.872153\pi$
$83$	− 7.12311i	− 0.781862i	0.920420	−	0.390931i	$-0.127847\pi$
$84$	− 0.561553i	− 0.0612704i
$85$	3.12311i	0.338748i
$86$	− 0.438447i	− 0.0472790i
$87$	2.43845	0.261429
$88$	−4.12311	−0.439525
$89$	18.8078i	1.99362i	0.0798174	+	0.996810i	$0.474566\pi$
$89$	18.8078i	1.99362i	−0.0798174	+	0.996810i	$0.525434\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	4.68466	0.488409
$93$	− 6.68466i	− 0.693167i
$94$	7.00000	0.721995
$95$	−0.561553	−0.0576141
$96$	1.00000i	0.102062i
$97$	− 7.12311i	− 0.723242i	−0.932325	−	0.361621i	$-0.882223\pi$
$97$	− 7.12311i	− 0.723242i	0.932325	−	0.361621i	$-0.117777\pi$
$98$	6.68466i	0.675252i
$99$	− 4.12311i	− 0.414388i
$100$	1.00000	0.100000
$101$	8.87689	0.883284	0.441642	−	0.897191i	$-0.354396\pi$
$101$	8.87689	0.883284	0.441642	+	0.897191i	$0.354396\pi$
$102$	3.12311i	0.309234i
$103$	−4.56155	−0.449463	−0.224732	−	0.974421i	$-0.572151\pi$
$103$	−4.56155	−0.449463	−0.224732	+	0.974421i	$0.572151\pi$
$104$	0	0
$105$	−0.561553	−0.0548019
$106$	8.56155i	0.831572i
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	−1.00000	−0.0962250
$109$	− 3.75379i	− 0.359548i	−0.983708	−	0.179774i	$-0.942463\pi$
$109$	− 3.75379i	− 0.359548i	0.983708	−	0.179774i	$-0.0575366\pi$
$110$	4.12311i	0.393123i
$111$	4.12311i	0.391348i
$112$	0.561553i	0.0530618i
$113$	8.68466	0.816984	0.408492	−	0.912762i	$-0.366055\pi$
$113$	8.68466	0.816984	0.408492	+	0.912762i	$0.366055\pi$
$114$	−0.561553	−0.0525942
$115$	− 4.68466i	− 0.436847i
$116$	−2.43845	−0.226404
$117$	0	0
$118$	6.43845	0.592707
$119$	1.75379i	0.160770i
$120$	1.00000	0.0912871
$121$	−6.00000	−0.545455
$122$	6.00000i	0.543214i
$123$	12.2462i	1.10420i
$124$	6.68466i	0.600300i
$125$	− 1.00000i	− 0.0894427i
$126$	−0.561553	−0.0500271
$127$	8.56155	0.759715	0.379857	−	0.925045i	$-0.375973\pi$
$127$	8.56155	0.759715	0.379857	+	0.925045i	$0.375973\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−0.438447	−0.0386031
$130$	0	0
$131$	−2.12311	−0.185497	−0.0927483	−	0.995690i	$-0.529565\pi$
$131$	−2.12311	−0.185497	−0.0927483	+	0.995690i	$0.529565\pi$
$132$	4.12311i	0.358870i
$133$	−0.315342	−0.0273436
$134$	2.24621	0.194043
$135$	1.00000i	0.0860663i
$136$	− 3.12311i	− 0.267804i
$137$	− 11.8078i	− 1.00881i	−0.863469	−	0.504403i	$-0.831713\pi$
$137$	− 11.8078i	− 1.00881i	0.863469	−	0.504403i	$-0.168287\pi$
$138$	− 4.68466i	− 0.398785i
$139$	12.5616	1.06546	0.532729	−	0.846286i	$-0.321167\pi$
$139$	12.5616	1.06546	0.532729	+	0.846286i	$0.321167\pi$
$140$	0.561553	0.0474599
$141$	− 7.00000i	− 0.589506i
$142$	13.1231	1.10127
$143$	0	0
$144$	1.00000	0.0833333
$145$	2.43845i	0.202502i
$146$	9.36932	0.775410
$147$	6.68466	0.551341
$148$	− 4.12311i	− 0.338917i
$149$	− 2.19224i	− 0.179595i	−0.995960	−	0.0897975i	$-0.971378\pi$
$149$	− 2.19224i	− 0.179595i	0.995960	−	0.0897975i	$-0.0286220\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	15.3693i	1.25074i	0.780329	+	0.625369i	$0.215051\pi$
$151$	15.3693i	1.25074i	−0.780329	+	0.625369i	$0.784949\pi$
$152$	0.561553	0.0455479
$153$	3.12311	0.252488
$154$	2.31534i	0.186576i
$155$	6.68466	0.536925
$156$	0	0
$157$	13.8769	1.10750	0.553748	−	0.832684i	$-0.313197\pi$
$157$	13.8769	1.10750	0.553748	+	0.832684i	$0.313197\pi$
$158$	11.5616i	0.919788i
$159$	8.56155	0.678975
$160$	−1.00000	−0.0790569
$161$	− 2.63068i	− 0.207327i
$162$	1.00000i	0.0785674i
$163$	14.4384i	1.13091i	0.824780	+	0.565453i	$0.191299\pi$
$163$	14.4384i	1.13091i	−0.824780	+	0.565453i	$0.808701\pi$
$164$	− 12.2462i	− 0.956268i
$165$	4.12311	0.320983
$166$	7.12311	0.552860
$167$	− 4.36932i	− 0.338108i	−0.985607	−	0.169054i	$-0.945929\pi$
$167$	− 4.36932i	− 0.338108i	0.985607	−	0.169054i	$-0.0540712\pi$
$168$	0.561553	0.0433247
$169$	0	0
$170$	−3.12311	−0.239531
$171$	0.561553i	0.0429430i
$172$	0.438447	0.0334313
$173$	−20.1771	−1.53404	−0.767018	−	0.641626i	$-0.778260\pi$
$173$	−20.1771	−1.53404	−0.767018	+	0.641626i	$0.778260\pi$
$174$	2.43845i	0.184858i
$175$	− 0.561553i	− 0.0424494i
$176$	− 4.12311i	− 0.310791i
$177$	− 6.43845i	− 0.483943i
$178$	−18.8078	−1.40970
$179$	12.6847	0.948096	0.474048	−	0.880499i	$-0.342792\pi$
$179$	12.6847	0.948096	0.474048	+	0.880499i	$0.342792\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−2.87689	−0.213838	−0.106919	−	0.994268i	$-0.534099\pi$
$181$	−2.87689	−0.213838	−0.106919	+	0.994268i	$0.534099\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	4.68466i	0.345358i
$185$	−4.12311	−0.303137
$186$	6.68466	0.490143
$187$	− 12.8769i	− 0.941652i
$188$	7.00000i	0.510527i
$189$	0.561553i	0.0408470i
$190$	− 0.561553i	− 0.0407393i
$191$	10.8769	0.787024	0.393512	−	0.919319i	$-0.371260\pi$
$191$	10.8769	0.787024	0.393512	+	0.919319i	$0.371260\pi$
$192$	−1.00000	−0.0721688
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	7.12311	0.511409
$195$	0	0
$196$	−6.68466	−0.477476
$197$	12.8078i	0.912515i	0.889848	+	0.456258i	$0.150810\pi$
$197$	12.8078i	0.912515i	−0.889848	+	0.456258i	$0.849190\pi$
$198$	4.12311	0.293016
$199$	−2.87689	−0.203938	−0.101969	−	0.994788i	$-0.532514\pi$
$199$	−2.87689	−0.203938	−0.101969	+	0.994788i	$0.532514\pi$
$200$	1.00000i	0.0707107i
$201$	− 2.24621i	− 0.158436i
$202$	8.87689i	0.624576i
$203$	1.36932i	0.0961072i
$204$	−3.12311	−0.218661
$205$	−12.2462	−0.855312
$206$	− 4.56155i	− 0.317818i
$207$	−4.68466	−0.325606
$208$	0	0
$209$	2.31534	0.160156
$210$	− 0.561553i	− 0.0387508i
$211$	−21.9309	−1.50978	−0.754892	−	0.655850i	$-0.772311\pi$
$211$	−21.9309	−1.50978	−0.754892	+	0.655850i	$0.772311\pi$
$212$	−8.56155	−0.588010
$213$	− 13.1231i	− 0.899180i
$214$	2.00000i	0.136717i
$215$	− 0.438447i	− 0.0299018i
$216$	− 1.00000i	− 0.0680414i
$217$	3.75379	0.254824
$218$	3.75379	0.254239
$219$	− 9.36932i	− 0.633120i
$220$	−4.12311	−0.277980
$221$	0	0
$222$	−4.12311	−0.276725
$223$	27.3002i	1.82816i	0.405539	+	0.914078i	$0.367084\pi$
$223$	27.3002i	1.82816i	−0.405539	+	0.914078i	$0.632916\pi$
$224$	−0.561553	−0.0375203
$225$	−1.00000	−0.0666667
$226$	8.68466i	0.577695i
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	− 0.561553i	− 0.0371897i
$229$	24.2462i	1.60223i	0.598507	+	0.801117i	$0.295761\pi$
$229$	24.2462i	1.60223i	−0.598507	+	0.801117i	$0.704239\pi$
$230$	4.68466	0.308897
$231$	2.31534	0.152338
$232$	− 2.43845i	− 0.160092i
$233$	5.31534	0.348220	0.174110	−	0.984726i	$-0.444295\pi$
$233$	5.31534	0.348220	0.174110	+	0.984726i	$0.444295\pi$
$234$	0	0
$235$	7.00000	0.456630
$236$	6.43845i	0.419107i
$237$	11.5616	0.751004
$238$	−1.75379	−0.113681
$239$	− 11.3693i	− 0.735420i	−0.929941	−	0.367710i	$-0.880142\pi$
$239$	− 11.3693i	− 0.735420i	0.929941	−	0.367710i	$-0.119858\pi$
$240$	1.00000i	0.0645497i
$241$	28.1231i	1.81157i	0.423739	+	0.905784i	$0.360717\pi$
$241$	28.1231i	1.81157i	−0.423739	+	0.905784i	$0.639283\pi$
$242$	− 6.00000i	− 0.385695i
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	6.68466i	0.427067i
$246$	−12.2462	−0.780790
$247$	0	0
$248$	−6.68466	−0.424476
$249$	− 7.12311i	− 0.451408i
$250$	1.00000	0.0632456
$251$	8.12311	0.512726	0.256363	−	0.966581i	$-0.417476\pi$
$251$	8.12311	0.512726	0.256363	+	0.966581i	$0.417476\pi$
$252$	− 0.561553i	− 0.0353745i
$253$	19.3153i	1.21435i
$254$	8.56155i	0.537200i
$255$	3.12311i	0.195576i
$256$	1.00000	0.0625000
$257$	5.56155	0.346920	0.173460	−	0.984841i	$-0.444505\pi$
$257$	5.56155	0.346920	0.173460	+	0.984841i	$0.444505\pi$
$258$	− 0.438447i	− 0.0272965i
$259$	−2.31534	−0.143868
$260$	0	0
$261$	2.43845	0.150936
$262$	− 2.12311i	− 0.131166i
$263$	−13.0000	−0.801614	−0.400807	−	0.916162i	$-0.631270\pi$
$263$	−13.0000	−0.801614	−0.400807	+	0.916162i	$0.631270\pi$
$264$	−4.12311	−0.253760
$265$	8.56155i	0.525932i
$266$	− 0.315342i	− 0.0193348i
$267$	18.8078i	1.15102i
$268$	2.24621i	0.137209i
$269$	21.3693	1.30291	0.651455	−	0.758687i	$-0.274159\pi$
$269$	21.3693	1.30291	0.651455	+	0.758687i	$0.274159\pi$
$270$	−1.00000	−0.0608581
$271$	13.1771i	0.800451i	0.916417	+	0.400225i	$0.131068\pi$
$271$	13.1771i	0.800451i	−0.916417	+	0.400225i	$0.868932\pi$
$272$	3.12311	0.189366
$273$	0	0
$274$	11.8078	0.713333
$275$	4.12311i	0.248633i
$276$	4.68466	0.281983
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	12.5616i	0.753392i
$279$	− 6.68466i	− 0.400200i
$280$	0.561553i	0.0335592i
$281$	− 16.2462i	− 0.969168i	−0.874745	−	0.484584i	$-0.838971\pi$
$281$	− 16.2462i	− 0.969168i	0.874745	−	0.484584i	$-0.161029\pi$
$282$	7.00000	0.416844
$283$	−15.5616	−0.925038	−0.462519	−	0.886609i	$-0.653054\pi$
$283$	−15.5616	−0.925038	−0.462519	+	0.886609i	$0.653054\pi$
$284$	13.1231i	0.778713i
$285$	−0.561553	−0.0332635
$286$	0	0
$287$	−6.87689	−0.405930
$288$	1.00000i	0.0589256i
$289$	−7.24621	−0.426248
$290$	−2.43845	−0.143191
$291$	− 7.12311i	− 0.417564i
$292$	9.36932i	0.548298i
$293$	− 3.93087i	− 0.229644i	−0.993386	−	0.114822i	$-0.963370\pi$
$293$	− 3.93087i	− 0.229644i	0.993386	−	0.114822i	$-0.0366297\pi$
$294$	6.68466i	0.389857i
$295$	6.43845	0.374861
$296$	4.12311	0.239651
$297$	− 4.12311i	− 0.239247i
$298$	2.19224	0.126993
$299$	0	0
$300$	1.00000	0.0577350
$301$	− 0.246211i	− 0.0141914i
$302$	−15.3693	−0.884405
$303$	8.87689	0.509964
$304$	0.561553i	0.0322073i
$305$	6.00000i	0.343559i
$306$	3.12311i	0.178536i
$307$	− 25.6155i	− 1.46196i	−0.682401	−	0.730978i	$-0.739064\pi$
$307$	− 25.6155i	− 1.46196i	0.682401	−	0.730978i	$-0.260936\pi$
$308$	−2.31534	−0.131929
$309$	−4.56155	−0.259498
$310$	6.68466i	0.379663i
$311$	−30.7386	−1.74303	−0.871514	−	0.490371i	$-0.836861\pi$
$311$	−30.7386	−1.74303	−0.871514	+	0.490371i	$0.836861\pi$
$312$	0	0
$313$	31.3693	1.77310	0.886549	−	0.462634i	$-0.153096\pi$
$313$	31.3693	1.77310	0.886549	+	0.462634i	$0.153096\pi$
$314$	13.8769i	0.783118i
$315$	−0.561553	−0.0316399
$316$	−11.5616	−0.650388
$317$	− 24.8078i	− 1.39334i	−0.717390	−	0.696671i	$-0.754664\pi$
$317$	− 24.8078i	− 1.39334i	0.717390	−	0.696671i	$-0.245336\pi$
$318$	8.56155i	0.480108i
$319$	− 10.0540i	− 0.562915i
$320$	− 1.00000i	− 0.0559017i
$321$	2.00000	0.111629
$322$	2.63068	0.146602
$323$	1.75379i	0.0975834i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−14.4384	−0.799672
$327$	− 3.75379i	− 0.207585i
$328$	12.2462	0.676184
$329$	3.93087	0.216716
$330$	4.12311i	0.226969i
$331$	30.7386i	1.68955i	0.535123	+	0.844774i	$0.320265\pi$
$331$	30.7386i	1.68955i	−0.535123	+	0.844774i	$0.679735\pi$
$332$	7.12311i	0.390931i
$333$	4.12311i	0.225945i
$334$	4.36932	0.239078
$335$	2.24621	0.122724
$336$	0.561553i	0.0306352i
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	0	0
$339$	8.68466	0.471686
$340$	− 3.12311i	− 0.169374i
$341$	−27.5616	−1.49254
$342$	−0.561553	−0.0303653
$343$	7.68466i	0.414933i
$344$	0.438447i	0.0236395i
$345$	− 4.68466i	− 0.252214i
$346$	− 20.1771i	− 1.08473i
$347$	0.876894	0.0470742	0.0235371	−	0.999723i	$-0.492507\pi$
$347$	0.876894	0.0470742	0.0235371	+	0.999723i	$0.492507\pi$
$348$	−2.43845	−0.130714
$349$	− 8.49242i	− 0.454589i	−0.973826	−	0.227294i	$-0.927012\pi$
$349$	− 8.49242i	− 0.454589i	0.973826	−	0.227294i	$-0.0729880\pi$
$350$	0.561553	0.0300163
$351$	0	0
$352$	4.12311	0.219762
$353$	− 25.8617i	− 1.37648i	−0.725482	−	0.688241i	$-0.758383\pi$
$353$	− 25.8617i	− 1.37648i	0.725482	−	0.688241i	$-0.241617\pi$
$354$	6.43845	0.342200
$355$	13.1231	0.696502
$356$	− 18.8078i	− 0.996810i
$357$	1.75379i	0.0928203i
$358$	12.6847i	0.670405i
$359$	− 13.1231i	− 0.692611i	−0.938122	−	0.346306i	$-0.887436\pi$
$359$	− 13.1231i	− 0.692611i	0.938122	−	0.346306i	$-0.112564\pi$
$360$	1.00000	0.0527046
$361$	18.6847	0.983403
$362$	− 2.87689i	− 0.151206i
$363$	−6.00000	−0.314918
$364$	0	0
$365$	9.36932	0.490412
$366$	6.00000i	0.313625i
$367$	12.4924	0.652099	0.326050	−	0.945353i	$-0.394282\pi$
$367$	12.4924	0.652099	0.326050	+	0.945353i	$0.394282\pi$
$368$	−4.68466	−0.244205
$369$	12.2462i	0.637512i
$370$	− 4.12311i	− 0.214350i
$371$	4.80776i	0.249607i
$372$	6.68466i	0.346583i
$373$	−25.8078	−1.33628	−0.668138	−	0.744038i	$-0.732908\pi$
$373$	−25.8078	−1.33628	−0.668138	+	0.744038i	$0.732908\pi$
$374$	12.8769	0.665848
$375$	− 1.00000i	− 0.0516398i
$376$	−7.00000	−0.360997
$377$	0	0
$378$	−0.561553	−0.0288832
$379$	− 17.6847i	− 0.908400i	−0.890900	−	0.454200i	$-0.849925\pi$
$379$	− 17.6847i	− 0.908400i	0.890900	−	0.454200i	$-0.150075\pi$
$380$	0.561553	0.0288071
$381$	8.56155	0.438622
$382$	10.8769i	0.556510i
$383$	0.192236i	0.00982280i	0.999988	+	0.00491140i	$0.00156335\pi$
$383$	0.192236i	0.00982280i	−0.999988	+	0.00491140i	$0.998437\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	2.31534i	0.118001i
$386$	0	0
$387$	−0.438447	−0.0222875
$388$	7.12311i	0.361621i
$389$	18.0540	0.915373	0.457686	−	0.889114i	$-0.348678\pi$
$389$	18.0540	0.915373	0.457686	+	0.889114i	$0.348678\pi$
$390$	0	0
$391$	−14.6307	−0.739905
$392$	− 6.68466i	− 0.337626i
$393$	−2.12311	−0.107097
$394$	−12.8078	−0.645246
$395$	11.5616i	0.581725i
$396$	4.12311i	0.207194i
$397$	20.1231i	1.00995i	0.863134	+	0.504975i	$0.168498\pi$
$397$	20.1231i	1.00995i	−0.863134	+	0.504975i	$0.831502\pi$
$398$	− 2.87689i	− 0.144206i
$399$	−0.315342	−0.0157868
$400$	−1.00000	−0.0500000
$401$	− 0.315342i	− 0.0157474i	−0.999969	−	0.00787370i	$-0.997494\pi$
$401$	− 0.315342i	− 0.0157474i	0.999969	−	0.00787370i	$-0.00250630\pi$
$402$	2.24621	0.112031
$403$	0	0
$404$	−8.87689	−0.441642
$405$	1.00000i	0.0496904i
$406$	−1.36932	−0.0679581
$407$	17.0000	0.842659
$408$	− 3.12311i	− 0.154617i
$409$	− 18.8078i	− 0.929984i	−0.885315	−	0.464992i	$-0.846057\pi$
$409$	− 18.8078i	− 0.929984i	0.885315	−	0.464992i	$-0.153943\pi$
$410$	− 12.2462i	− 0.604797i
$411$	− 11.8078i	− 0.582434i
$412$	4.56155	0.224732
$413$	3.61553	0.177909
$414$	− 4.68466i	− 0.230238i
$415$	7.12311	0.349660
$416$	0	0
$417$	12.5616	0.615142
$418$	2.31534i	0.113247i
$419$	32.4924	1.58736	0.793679	−	0.608336i	$-0.208163\pi$
$419$	32.4924	1.58736	0.793679	+	0.608336i	$0.208163\pi$
$420$	0.561553	0.0274010
$421$	− 32.4924i	− 1.58358i	−0.610791	−	0.791792i	$-0.709148\pi$
$421$	− 32.4924i	− 1.58358i	0.610791	−	0.791792i	$-0.290852\pi$
$422$	− 21.9309i	− 1.06758i
$423$	− 7.00000i	− 0.340352i
$424$	− 8.56155i	− 0.415786i
$425$	−3.12311	−0.151493
$426$	13.1231	0.635817
$427$	3.36932i	0.163053i
$428$	−2.00000	−0.0966736
$429$	0	0
$430$	0.438447	0.0211438
$431$	− 22.7386i	− 1.09528i	−0.836714	−	0.547641i	$-0.815526\pi$
$431$	− 22.7386i	− 1.09528i	0.836714	−	0.547641i	$-0.184474\pi$
$432$	1.00000	0.0481125
$433$	18.7386	0.900521	0.450261	−	0.892897i	$-0.351331\pi$
$433$	18.7386	0.900521	0.450261	+	0.892897i	$0.351331\pi$
$434$	3.75379i	0.180188i
$435$	2.43845i	0.116915i
$436$	3.75379i	0.179774i
$437$	− 2.63068i	− 0.125843i
$438$	9.36932	0.447683
$439$	−25.1231	−1.19906	−0.599530	−	0.800352i	$-0.704646\pi$
$439$	−25.1231	−1.19906	−0.599530	+	0.800352i	$0.704646\pi$
$440$	− 4.12311i	− 0.196561i
$441$	6.68466	0.318317
$442$	0	0
$443$	−13.1231	−0.623498	−0.311749	−	0.950165i	$-0.600915\pi$
$443$	−13.1231	−0.623498	−0.311749	+	0.950165i	$0.600915\pi$
$444$	− 4.12311i	− 0.195674i
$445$	−18.8078	−0.891574
$446$	−27.3002	−1.29270
$447$	− 2.19224i	− 0.103689i
$448$	− 0.561553i	− 0.0265309i
$449$	20.1771i	0.952215i	0.879387	+	0.476108i	$0.157953\pi$
$449$	20.1771i	0.952215i	−0.879387	+	0.476108i	$0.842047\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	50.4924	2.37760
$452$	−8.68466	−0.408492
$453$	15.3693i	0.722113i
$454$	20.0000	0.938647
$455$	0	0
$456$	0.561553	0.0262971
$457$	20.2462i	0.947078i	0.880773	+	0.473539i	$0.157024\pi$
$457$	20.2462i	0.947078i	−0.880773	+	0.473539i	$0.842976\pi$
$458$	−24.2462	−1.13295
$459$	3.12311	0.145774
$460$	4.68466i	0.218423i
$461$	− 30.0540i	− 1.39975i	−0.714264	−	0.699877i	$-0.753238\pi$
$461$	− 30.0540i	− 1.39975i	0.714264	−	0.699877i	$-0.246762\pi$
$462$	2.31534i	0.107719i
$463$	7.61553i	0.353924i	0.984218	+	0.176962i	$0.0566269\pi$
$463$	7.61553i	0.353924i	−0.984218	+	0.176962i	$0.943373\pi$
$464$	2.43845	0.113202
$465$	6.68466	0.309994
$466$	5.31534i	0.246228i
$467$	17.8617	0.826543	0.413271	−	0.910608i	$-0.364386\pi$
$467$	17.8617	0.826543	0.413271	+	0.910608i	$0.364386\pi$
$468$	0	0
$469$	1.26137	0.0582445
$470$	7.00000i	0.322886i
$471$	13.8769	0.639414
$472$	−6.43845	−0.296354
$473$	1.80776i	0.0831211i
$474$	11.5616i	0.531040i
$475$	− 0.561553i	− 0.0257658i
$476$	− 1.75379i	− 0.0803848i
$477$	8.56155	0.392007
$478$	11.3693	0.520020
$479$	38.2462i	1.74751i	0.486363	+	0.873757i	$0.338323\pi$
$479$	38.2462i	1.74751i	−0.486363	+	0.873757i	$0.661677\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−28.1231	−1.28097
$483$	− 2.63068i	− 0.119700i
$484$	6.00000	0.272727
$485$	7.12311	0.323444
$486$	1.00000i	0.0453609i
$487$	13.0540i	0.591532i	0.955260	+	0.295766i	$0.0955749\pi$
$487$	13.0540i	0.591532i	−0.955260	+	0.295766i	$0.904425\pi$
$488$	− 6.00000i	− 0.271607i
$489$	14.4384i	0.652929i
$490$	−6.68466	−0.301982
$491$	17.4384	0.786986	0.393493	−	0.919328i	$-0.371267\pi$
$491$	17.4384	0.786986	0.393493	+	0.919328i	$0.371267\pi$
$492$	− 12.2462i	− 0.552102i
$493$	7.61553	0.342986
$494$	0	0
$495$	4.12311	0.185320
$496$	− 6.68466i	− 0.300150i
$497$	7.36932	0.330559
$498$	7.12311	0.319194
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	1.00000i	0.0447214i
$501$	− 4.36932i	− 0.195207i
$502$	8.12311i	0.362552i
$503$	−43.0540	−1.91968	−0.959841	−	0.280545i	$-0.909485\pi$
$503$	−43.0540	−1.91968	−0.959841	+	0.280545i	$0.909485\pi$
$504$	0.561553	0.0250136
$505$	8.87689i	0.395017i
$506$	−19.3153	−0.858672
$507$	0	0
$508$	−8.56155	−0.379857
$509$	40.5464i	1.79719i	0.438782	+	0.898594i	$0.355410\pi$
$509$	40.5464i	1.79719i	−0.438782	+	0.898594i	$0.644590\pi$
$510$	−3.12311	−0.138293
$511$	5.26137	0.232749
$512$	1.00000i	0.0441942i
$513$	0.561553i	0.0247932i
$514$	5.56155i	0.245310i
$515$	− 4.56155i	− 0.201006i
$516$	0.438447	0.0193016
$517$	−28.8617	−1.26934
$518$	− 2.31534i	− 0.101730i
$519$	−20.1771	−0.885676
$520$	0	0
$521$	−6.31534	−0.276680	−0.138340	−	0.990385i	$-0.544177\pi$
$521$	−6.31534	−0.276680	−0.138340	+	0.990385i	$0.544177\pi$
$522$	2.43845i	0.106728i
$523$	−29.8078	−1.30340	−0.651701	−	0.758476i	$-0.725944\pi$
$523$	−29.8078	−1.30340	−0.651701	+	0.758476i	$0.725944\pi$
$524$	2.12311	0.0927483
$525$	− 0.561553i	− 0.0245082i
$526$	− 13.0000i	− 0.566827i
$527$	− 20.8769i	− 0.909412i
$528$	− 4.12311i	− 0.179435i
$529$	−1.05398	−0.0458250
$530$	−8.56155	−0.371890
$531$	− 6.43845i	− 0.279405i
$532$	0.315342	0.0136718
$533$	0	0
$534$	−18.8078	−0.813892
$535$	2.00000i	0.0864675i
$536$	−2.24621	−0.0970215
$537$	12.6847	0.547383
$538$	21.3693i	0.921297i
$539$	− 27.5616i	− 1.18716i
$540$	− 1.00000i	− 0.0430331i
$541$	27.3693i	1.17670i	0.808607	+	0.588349i	$0.200222\pi$
$541$	27.3693i	1.17670i	−0.808607	+	0.588349i	$0.799778\pi$
$542$	−13.1771	−0.566004
$543$	−2.87689	−0.123459
$544$	3.12311i	0.133902i
$545$	3.75379	0.160795
$546$	0	0
$547$	−5.61553	−0.240103	−0.120051	−	0.992768i	$-0.538306\pi$
$547$	−5.61553	−0.240103	−0.120051	+	0.992768i	$0.538306\pi$
$548$	11.8078i	0.504403i
$549$	6.00000	0.256074
$550$	−4.12311	−0.175810
$551$	1.36932i	0.0583349i
$552$	4.68466i	0.199392i
$553$	6.49242i	0.276086i
$554$	1.00000i	0.0424859i
$555$	−4.12311	−0.175016
$556$	−12.5616	−0.532729
$557$	8.56155i	0.362765i	0.983413	+	0.181382i	$0.0580571\pi$
$557$	8.56155i	0.362765i	−0.983413	+	0.181382i	$0.941943\pi$
$558$	6.68466	0.282984
$559$	0	0
$560$	−0.561553	−0.0237299
$561$	− 12.8769i	− 0.543663i
$562$	16.2462	0.685305
$563$	32.9848	1.39015	0.695073	−	0.718939i	$-0.255372\pi$
$563$	32.9848	1.39015	0.695073	+	0.718939i	$0.255372\pi$
$564$	7.00000i	0.294753i
$565$	8.68466i	0.365366i
$566$	− 15.5616i	− 0.654101i
$567$	0.561553i	0.0235830i
$568$	−13.1231	−0.550633
$569$	−26.1771	−1.09740	−0.548700	−	0.836019i	$-0.684877\pi$
$569$	−26.1771	−1.09740	−0.548700	+	0.836019i	$0.684877\pi$
$570$	− 0.561553i	− 0.0235209i
$571$	−23.6847	−0.991172	−0.495586	−	0.868559i	$-0.665047\pi$
$571$	−23.6847	−0.991172	−0.495586	+	0.868559i	$0.665047\pi$
$572$	0	0
$573$	10.8769	0.454389
$574$	− 6.87689i	− 0.287036i
$575$	4.68466	0.195364
$576$	−1.00000	−0.0416667
$577$	− 40.7386i	− 1.69597i	−0.530019	−	0.847986i	$-0.677815\pi$
$577$	− 40.7386i	− 1.69597i	0.530019	−	0.847986i	$-0.322185\pi$
$578$	− 7.24621i	− 0.301403i
$579$	0	0
$580$	− 2.43845i	− 0.101251i
$581$	4.00000	0.165948
$582$	7.12311	0.295262
$583$	− 35.3002i	− 1.46198i
$584$	−9.36932	−0.387705
$585$	0	0
$586$	3.93087	0.162383
$587$	− 40.2462i	− 1.66114i	−0.556915	−	0.830569i	$-0.688015\pi$
$587$	− 40.2462i	− 1.66114i	0.556915	−	0.830569i	$-0.311985\pi$
$588$	−6.68466	−0.275671
$589$	3.75379	0.154672
$590$	6.43845i	0.265067i
$591$	12.8078i	0.526841i
$592$	4.12311i	0.169459i
$593$	− 33.1771i	− 1.36242i	−0.732088	−	0.681210i	$-0.761454\pi$
$593$	− 33.1771i	− 1.36242i	0.732088	−	0.681210i	$-0.238546\pi$
$594$	4.12311	0.169173
$595$	−1.75379	−0.0718983
$596$	2.19224i	0.0897975i
$597$	−2.87689	−0.117743
$598$	0	0
$599$	−14.0000	−0.572024	−0.286012	−	0.958226i	$-0.592330\pi$
$599$	−14.0000	−0.572024	−0.286012	+	0.958226i	$0.592330\pi$
$600$	1.00000i	0.0408248i
$601$	29.9848	1.22311	0.611554	−	0.791203i	$-0.290545\pi$
$601$	29.9848	1.22311	0.611554	+	0.791203i	$0.290545\pi$
$602$	0.246211	0.0100348
$603$	− 2.24621i	− 0.0914728i
$604$	− 15.3693i	− 0.625369i
$605$	− 6.00000i	− 0.243935i
$606$	8.87689i	0.360599i
$607$	32.4233	1.31602	0.658010	−	0.753009i	$-0.271398\pi$
$607$	32.4233	1.31602	0.658010	+	0.753009i	$0.271398\pi$
$608$	−0.561553	−0.0227740
$609$	1.36932i	0.0554875i
$610$	−6.00000	−0.242933
$611$	0	0
$612$	−3.12311	−0.126244
$613$	19.8769i	0.802820i	0.915898	+	0.401410i	$0.131480\pi$
$613$	19.8769i	0.802820i	−0.915898	+	0.401410i	$0.868520\pi$
$614$	25.6155	1.03376
$615$	−12.2462	−0.493815
$616$	− 2.31534i	− 0.0932878i
$617$	− 29.3153i	− 1.18019i	−0.807333	−	0.590096i	$-0.799090\pi$
$617$	− 29.3153i	− 1.18019i	0.807333	−	0.590096i	$-0.200910\pi$
$618$	− 4.56155i	− 0.183493i
$619$	20.5616i	0.826439i	0.910632	+	0.413219i	$0.135596\pi$
$619$	20.5616i	0.826439i	−0.910632	+	0.413219i	$0.864404\pi$
$620$	−6.68466	−0.268462
$621$	−4.68466	−0.187989
$622$	− 30.7386i	− 1.23251i
$623$	−10.5616	−0.423140
$624$	0	0
$625$	1.00000	0.0400000
$626$	31.3693i	1.25377i
$627$	2.31534	0.0924658
$628$	−13.8769	−0.553748
$629$	12.8769i	0.513435i
$630$	− 0.561553i	− 0.0223728i
$631$	13.7538i	0.547530i	0.961797	+	0.273765i	$0.0882690\pi$
$631$	13.7538i	0.547530i	−0.961797	+	0.273765i	$0.911731\pi$
$632$	− 11.5616i	− 0.459894i
$633$	−21.9309	−0.871674
$634$	24.8078	0.985242
$635$	8.56155i	0.339755i
$636$	−8.56155	−0.339488
$637$	0	0
$638$	10.0540	0.398041
$639$	− 13.1231i	− 0.519142i
$640$	1.00000	0.0395285
$641$	11.4384	0.451792	0.225896	−	0.974151i	$-0.427469\pi$
$641$	11.4384	0.451792	0.225896	+	0.974151i	$0.427469\pi$
$642$	2.00000i	0.0789337i
$643$	21.7538i	0.857886i	0.903332	+	0.428943i	$0.141114\pi$
$643$	21.7538i	0.857886i	−0.903332	+	0.428943i	$0.858886\pi$
$644$	2.63068i	0.103663i
$645$	− 0.438447i	− 0.0172638i
$646$	−1.75379	−0.0690019
$647$	−35.0540	−1.37811	−0.689057	−	0.724707i	$-0.741975\pi$
$647$	−35.0540	−1.37811	−0.689057	+	0.724707i	$0.741975\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−26.5464	−1.04204
$650$	0	0
$651$	3.75379	0.147123
$652$	− 14.4384i	− 0.565453i
$653$	−44.5616	−1.74383	−0.871914	−	0.489659i	$-0.837121\pi$
$653$	−44.5616	−1.74383	−0.871914	+	0.489659i	$0.837121\pi$
$654$	3.75379	0.146785
$655$	− 2.12311i	− 0.0829566i
$656$	12.2462i	0.478134i
$657$	− 9.36932i	− 0.365532i
$658$	3.93087i	0.153241i
$659$	−6.05398	−0.235829	−0.117915	−	0.993024i	$-0.537621\pi$
$659$	−6.05398	−0.235829	−0.117915	+	0.993024i	$0.537621\pi$
$660$	−4.12311	−0.160492
$661$	− 29.6155i	− 1.15191i	−0.817481	−	0.575955i	$-0.804630\pi$
$661$	− 29.6155i	− 1.15191i	0.817481	−	0.575955i	$-0.195370\pi$
$662$	−30.7386	−1.19469
$663$	0	0
$664$	−7.12311	−0.276430
$665$	− 0.315342i	− 0.0122284i
$666$	−4.12311	−0.159767
$667$	−11.4233	−0.442312
$668$	4.36932i	0.169054i
$669$	27.3002i	1.05549i
$670$	2.24621i	0.0867787i
$671$	− 24.7386i	− 0.955024i
$672$	−0.561553	−0.0216624
$673$	−25.7538	−0.992736	−0.496368	−	0.868112i	$-0.665333\pi$
$673$	−25.7538	−0.992736	−0.496368	+	0.868112i	$0.665333\pi$
$674$	6.00000i	0.231111i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−37.1231	−1.42676	−0.713378	−	0.700779i	$-0.752836\pi$
$677$	−37.1231	−1.42676	−0.713378	+	0.700779i	$0.752836\pi$
$678$	8.68466i	0.333532i
$679$	4.00000	0.153506
$680$	3.12311	0.119766
$681$	− 20.0000i	− 0.766402i
$682$	− 27.5616i	− 1.05539i
$683$	− 17.1231i	− 0.655197i	−0.944817	−	0.327599i	$-0.893761\pi$
$683$	− 17.1231i	− 0.655197i	0.944817	−	0.327599i	$-0.106239\pi$
$684$	− 0.561553i	− 0.0214715i
$685$	11.8078	0.451151
$686$	−7.68466	−0.293402
$687$	24.2462i	0.925051i
$688$	−0.438447	−0.0167156
$689$	0	0
$690$	4.68466	0.178342
$691$	− 20.5616i	− 0.782198i	−0.920349	−	0.391099i	$-0.872095\pi$
$691$	− 20.5616i	− 0.782198i	0.920349	−	0.391099i	$-0.127905\pi$
$692$	20.1771	0.767018
$693$	2.31534	0.0879526
$694$	0.876894i	0.0332865i
$695$	12.5616i	0.476487i
$696$	− 2.43845i	− 0.0924291i
$697$	38.2462i	1.44868i
$698$	8.49242	0.321443
$699$	5.31534	0.201045
$700$	0.561553i	0.0212247i
$701$	36.3002	1.37104	0.685520	−	0.728054i	$-0.259575\pi$
$701$	36.3002	1.37104	0.685520	+	0.728054i	$0.259575\pi$
$702$	0	0
$703$	−2.31534	−0.0873248
$704$	4.12311i	0.155395i
$705$	7.00000	0.263635
$706$	25.8617	0.973319
$707$	4.98485i	0.187474i
$708$	6.43845i	0.241972i
$709$	− 34.2462i	− 1.28614i	−0.765806	−	0.643072i	$-0.777660\pi$
$709$	− 34.2462i	− 1.28614i	0.765806	−	0.643072i	$-0.222340\pi$
$710$	13.1231i	0.492501i
$711$	11.5616	0.433592
$712$	18.8078	0.704851
$713$	31.3153i	1.17277i
$714$	−1.75379	−0.0656339
$715$	0	0
$716$	−12.6847	−0.474048
$717$	− 11.3693i	− 0.424595i
$718$	13.1231	0.489750
$719$	−44.9848	−1.67765	−0.838826	−	0.544400i	$-0.816757\pi$
$719$	−44.9848	−1.67765	−0.838826	+	0.544400i	$0.816757\pi$
$720$	1.00000i	0.0372678i
$721$	− 2.56155i	− 0.0953972i
$722$	18.6847i	0.695371i
$723$	28.1231i	1.04591i
$724$	2.87689	0.106919
$725$	−2.43845	−0.0905617
$726$	− 6.00000i	− 0.222681i
$727$	−38.4233	−1.42504	−0.712521	−	0.701651i	$-0.752446\pi$
$727$	−38.4233	−1.42504	−0.712521	+	0.701651i	$0.752446\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	9.36932i	0.346774i
$731$	−1.36932	−0.0506460
$732$	−6.00000	−0.221766
$733$	20.0691i	0.741270i	0.928779	+	0.370635i	$0.120860\pi$
$733$	20.0691i	0.741270i	−0.928779	+	0.370635i	$0.879140\pi$
$734$	12.4924i	0.461104i
$735$	6.68466i	0.246567i
$736$	− 4.68466i	− 0.172679i
$737$	−9.26137	−0.341147
$738$	−12.2462	−0.450789
$739$	27.5464i	1.01331i	0.862149	+	0.506655i	$0.169118\pi$
$739$	27.5464i	1.01331i	−0.862149	+	0.506655i	$0.830882\pi$
$740$	4.12311	0.151568
$741$	0	0
$742$	−4.80776	−0.176499
$743$	13.1771i	0.483420i	0.970349	+	0.241710i	$0.0777083\pi$
$743$	13.1771i	0.483420i	−0.970349	+	0.241710i	$0.922292\pi$
$744$	−6.68466	−0.245071
$745$	2.19224	0.0803173
$746$	− 25.8078i	− 0.944889i
$747$	− 7.12311i	− 0.260621i
$748$	12.8769i	0.470826i
$749$	1.12311i	0.0410374i
$750$	1.00000	0.0365148
$751$	23.8078	0.868758	0.434379	−	0.900730i	$-0.356968\pi$
$751$	23.8078	0.868758	0.434379	+	0.900730i	$0.356968\pi$
$752$	− 7.00000i	− 0.255264i
$753$	8.12311	0.296022
$754$	0	0
$755$	−15.3693	−0.559347
$756$	− 0.561553i	− 0.0204235i
$757$	−28.4233	−1.03306	−0.516531	−	0.856268i	$-0.672777\pi$
$757$	−28.4233	−1.03306	−0.516531	+	0.856268i	$0.672777\pi$
$758$	17.6847	0.642336
$759$	19.3153i	0.701102i
$760$	0.561553i	0.0203697i
$761$	17.9309i	0.649994i	0.945715	+	0.324997i	$0.105363\pi$
$761$	17.9309i	0.649994i	−0.945715	+	0.324997i	$0.894637\pi$
$762$	8.56155i	0.310152i
$763$	2.10795	0.0763129
$764$	−10.8769	−0.393512
$765$	3.12311i	0.112916i
$766$	−0.192236	−0.00694577
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 33.3153i	− 1.20138i	−0.799481	−	0.600691i	$-0.794892\pi$
$769$	− 33.3153i	− 1.20138i	0.799481	−	0.600691i	$-0.205108\pi$
$770$	−2.31534	−0.0834391
$771$	5.56155	0.200294
$772$	0	0
$773$	24.1771i	0.869589i	0.900530	+	0.434795i	$0.143179\pi$
$773$	24.1771i	0.869589i	−0.900530	+	0.434795i	$0.856821\pi$
$774$	− 0.438447i	− 0.0157597i
$775$	6.68466i	0.240120i
$776$	−7.12311	−0.255705
$777$	−2.31534	−0.0830624
$778$	18.0540i	0.647266i
$779$	−6.87689	−0.246390
$780$	0	0
$781$	−54.1080	−1.93613
$782$	− 14.6307i	− 0.523192i
$783$	2.43845	0.0871430
$784$	6.68466	0.238738
$785$	13.8769i	0.495288i
$786$	− 2.12311i	− 0.0757287i
$787$	− 31.3153i	− 1.11627i	−0.829750	−	0.558136i	$-0.811517\pi$
$787$	− 31.3153i	− 1.11627i	0.829750	−	0.558136i	$-0.188483\pi$
$788$	− 12.8078i	− 0.456258i
$789$	−13.0000	−0.462812
$790$	−11.5616	−0.411342
$791$	4.87689i	0.173402i
$792$	−4.12311	−0.146508
$793$	0	0
$794$	−20.1231	−0.714142
$795$	8.56155i	0.303647i
$796$	2.87689	0.101969
$797$	−13.1231	−0.464844	−0.232422	−	0.972615i	$-0.574665\pi$
$797$	−13.1231	−0.464844	−0.232422	+	0.972615i	$0.574665\pi$
$798$	− 0.315342i	− 0.0111630i
$799$	− 21.8617i	− 0.773413i
$800$	− 1.00000i	− 0.0353553i
$801$	18.8078i	0.664540i
$802$	0.315342	0.0111351
$803$	−38.6307	−1.36325
$804$	2.24621i	0.0792178i
$805$	2.63068	0.0927194
$806$	0	0
$807$	21.3693	0.752236
$808$	− 8.87689i	− 0.312288i
$809$	−14.4924	−0.509526	−0.254763	−	0.967003i	$-0.581998\pi$
$809$	−14.4924	−0.509526	−0.254763	+	0.967003i	$0.581998\pi$
$810$	−1.00000	−0.0351364
$811$	47.5464i	1.66958i	0.550569	+	0.834790i	$0.314411\pi$
$811$	47.5464i	1.66958i	−0.550569	+	0.834790i	$0.685589\pi$
$812$	− 1.36932i	− 0.0480536i
$813$	13.1771i	0.462140i
$814$	17.0000i	0.595850i
$815$	−14.4384	−0.505757
$816$	3.12311	0.109331
$817$	− 0.246211i	− 0.00861384i
$818$	18.8078	0.657598
$819$	0	0
$820$	12.2462	0.427656
$821$	− 8.43845i	− 0.294504i	−0.989099	−	0.147252i	$-0.952957\pi$
$821$	− 8.43845i	− 0.294504i	0.989099	−	0.147252i	$-0.0470428\pi$
$822$	11.8078	0.411843
$823$	−5.05398	−0.176171	−0.0880853	−	0.996113i	$-0.528075\pi$
$823$	−5.05398	−0.176171	−0.0880853	+	0.996113i	$0.528075\pi$
$824$	4.56155i	0.158909i
$825$	4.12311i	0.143548i
$826$	3.61553i	0.125800i
$827$	28.2462i	0.982217i	0.871098	+	0.491109i	$0.163408\pi$
$827$	28.2462i	0.982217i	−0.871098	+	0.491109i	$0.836592\pi$
$828$	4.68466	0.162803
$829$	−12.3845	−0.430130	−0.215065	−	0.976600i	$-0.568996\pi$
$829$	−12.3845	−0.430130	−0.215065	+	0.976600i	$0.568996\pi$
$830$	7.12311i	0.247247i
$831$	1.00000	0.0346896
$832$	0	0
$833$	20.8769	0.723342
$834$	12.5616i	0.434971i
$835$	4.36932	0.151206
$836$	−2.31534	−0.0800778
$837$	− 6.68466i	− 0.231056i
$838$	32.4924i	1.12243i
$839$	− 36.7386i	− 1.26836i	−0.773186	−	0.634179i	$-0.781338\pi$
$839$	− 36.7386i	− 1.26836i	0.773186	−	0.634179i	$-0.218662\pi$
$840$	0.561553i	0.0193754i
$841$	−23.0540	−0.794965
$842$	32.4924	1.11976
$843$	− 16.2462i	− 0.559549i
$844$	21.9309	0.754892
$845$	0	0
$846$	7.00000	0.240665
$847$	− 3.36932i	− 0.115771i
$848$	8.56155	0.294005
$849$	−15.5616	−0.534071
$850$	− 3.12311i	− 0.107122i
$851$	− 19.3153i	− 0.662121i
$852$	13.1231i	0.449590i
$853$	− 2.30019i	− 0.0787569i	−0.999224	−	0.0393784i	$-0.987462\pi$
$853$	− 2.30019i	− 0.0787569i	0.999224	−	0.0393784i	$-0.0125378\pi$
$854$	−3.36932	−0.115296
$855$	−0.561553	−0.0192047
$856$	− 2.00000i	− 0.0683586i
$857$	6.19224	0.211523	0.105761	−	0.994392i	$-0.466272\pi$
$857$	6.19224	0.211523	0.105761	+	0.994392i	$0.466272\pi$
$858$	0	0
$859$	39.7926	1.35771	0.678853	−	0.734274i	$-0.262477\pi$
$859$	39.7926	1.35771	0.678853	+	0.734274i	$0.262477\pi$
$860$	0.438447i	0.0149509i
$861$	−6.87689	−0.234364
$862$	22.7386	0.774481
$863$	2.43845i	0.0830057i	0.999138	+	0.0415029i	$0.0132146\pi$
$863$	2.43845i	0.0830057i	−0.999138	+	0.0415029i	$0.986785\pi$
$864$	1.00000i	0.0340207i
$865$	− 20.1771i	− 0.686041i
$866$	18.7386i	0.636765i
$867$	−7.24621	−0.246094
$868$	−3.75379	−0.127412
$869$	− 47.6695i	− 1.61708i
$870$	−2.43845	−0.0826711
$871$	0	0
$872$	−3.75379	−0.127119
$873$	− 7.12311i	− 0.241081i
$874$	2.63068	0.0889842
$875$	0.561553	0.0189839
$876$	9.36932i	0.316560i
$877$	8.43845i	0.284946i	0.989799	+	0.142473i	$0.0455054\pi$
$877$	8.43845i	0.284946i	−0.989799	+	0.142473i	$0.954495\pi$
$878$	− 25.1231i	− 0.847864i
$879$	− 3.93087i	− 0.132585i
$880$	4.12311	0.138990
$881$	24.5616	0.827500	0.413750	−	0.910391i	$-0.364219\pi$
$881$	24.5616	0.827500	0.413750	+	0.910391i	$0.364219\pi$
$882$	6.68466i	0.225084i
$883$	19.1771	0.645360	0.322680	−	0.946508i	$-0.395416\pi$
$883$	19.1771	0.645360	0.322680	+	0.946508i	$0.395416\pi$
$884$	0	0
$885$	6.43845	0.216426
$886$	− 13.1231i	− 0.440879i
$887$	−33.1080	−1.11166	−0.555828	−	0.831297i	$-0.687599\pi$
$887$	−33.1080	−1.11166	−0.555828	+	0.831297i	$0.687599\pi$
$888$	4.12311	0.138362
$889$	4.80776i	0.161247i
$890$	− 18.8078i	− 0.630438i
$891$	− 4.12311i	− 0.138129i
$892$	− 27.3002i	− 0.914078i
$893$	3.93087	0.131542
$894$	2.19224	0.0733193
$895$	12.6847i	0.424001i
$896$	0.561553	0.0187602
$897$	0	0
$898$	−20.1771	−0.673318
$899$	− 16.3002i	− 0.543642i
$900$	1.00000	0.0333333
$901$	26.7386	0.890793
$902$	50.4924i	1.68121i
$903$	− 0.246211i	− 0.00819340i
$904$	− 8.68466i	− 0.288847i
$905$	− 2.87689i	− 0.0956312i
$906$	−15.3693	−0.510611
$907$	53.9157	1.79024	0.895121	−	0.445823i	$-0.147089\pi$
$907$	53.9157	1.79024	0.895121	+	0.445823i	$0.147089\pi$
$908$	20.0000i	0.663723i
$909$	8.87689	0.294428
$910$	0	0
$911$	−18.6307	−0.617262	−0.308631	−	0.951182i	$-0.599871\pi$
$911$	−18.6307	−0.617262	−0.308631	+	0.951182i	$0.599871\pi$
$912$	0.561553i	0.0185949i
$913$	−29.3693	−0.971983
$914$	−20.2462	−0.669685
$915$	6.00000i	0.198354i
$916$	− 24.2462i	− 0.801117i
$917$	− 1.19224i	− 0.0393711i
$918$	3.12311i	0.103078i
$919$	16.6307	0.548596	0.274298	−	0.961645i	$-0.411555\pi$
$919$	16.6307	0.548596	0.274298	+	0.961645i	$0.411555\pi$
$920$	−4.68466	−0.154449
$921$	− 25.6155i	− 0.844060i
$922$	30.0540	0.989775
$923$	0	0
$924$	−2.31534	−0.0761691
$925$	− 4.12311i	− 0.135567i
$926$	−7.61553	−0.250262
$927$	−4.56155	−0.149821
$928$	2.43845i	0.0800460i
$929$	− 25.1231i	− 0.824262i	−0.911125	−	0.412131i	$-0.864785\pi$
$929$	− 25.1231i	− 0.824262i	0.911125	−	0.412131i	$-0.135215\pi$
$930$	6.68466i	0.219199i
$931$	3.75379i	0.123025i
$932$	−5.31534	−0.174110
$933$	−30.7386	−1.00634
$934$	17.8617i	0.584454i
$935$	12.8769	0.421119
$936$	0	0
$937$	53.2311	1.73898	0.869491	−	0.493948i	$-0.164447\pi$
$937$	53.2311	1.73898	0.869491	+	0.493948i	$0.164447\pi$
$938$	1.26137i	0.0411851i
$939$	31.3693	1.02370
$940$	−7.00000	−0.228315
$941$	− 25.3693i	− 0.827016i	−0.910500	−	0.413508i	$-0.864303\pi$
$941$	− 25.3693i	− 0.827016i	0.910500	−	0.413508i	$-0.135697\pi$
$942$	13.8769i	0.452134i
$943$	− 57.3693i	− 1.86820i
$944$	− 6.43845i	− 0.209554i
$945$	−0.561553	−0.0182673
$946$	−1.80776	−0.0587755
$947$	43.8617i	1.42532i	0.701512	+	0.712658i	$0.252509\pi$
$947$	43.8617i	1.42532i	−0.701512	+	0.712658i	$0.747491\pi$
$948$	−11.5616	−0.375502
$949$	0	0
$950$	0.561553	0.0182192
$951$	− 24.8078i	− 0.804447i
$952$	1.75379	0.0568406
$953$	−51.1771	−1.65779	−0.828894	−	0.559406i	$-0.811029\pi$
$953$	−51.1771	−1.65779	−0.828894	+	0.559406i	$0.811029\pi$
$954$	8.56155i	0.277191i
$955$	10.8769i	0.351968i
$956$	11.3693i	0.367710i
$957$	− 10.0540i	− 0.324999i
$958$	−38.2462	−1.23568
$959$	6.63068	0.214116
$960$	− 1.00000i	− 0.0322749i
$961$	−13.6847	−0.441441
$962$	0	0
$963$	2.00000	0.0644491
$964$	− 28.1231i	− 0.905784i
$965$	0	0
$966$	2.63068	0.0846408
$967$	15.6847i	0.504385i	0.967677	+	0.252192i	$0.0811516\pi$
$967$	15.6847i	0.504385i	−0.967677	+	0.252192i	$0.918848\pi$
$968$	6.00000i	0.192847i
$969$	1.75379i	0.0563398i
$970$	7.12311i	0.228709i
$971$	−4.31534	−0.138486	−0.0692430	−	0.997600i	$-0.522058\pi$
$971$	−4.31534	−0.138486	−0.0692430	+	0.997600i	$0.522058\pi$
$972$	−1.00000	−0.0320750
$973$	7.05398i	0.226140i
$974$	−13.0540	−0.418276
$975$	0	0
$976$	6.00000	0.192055
$977$	− 48.3002i	− 1.54526i	−0.634857	−	0.772630i	$-0.718941\pi$
$977$	− 48.3002i	− 1.54526i	0.634857	−	0.772630i	$-0.281059\pi$
$978$	−14.4384	−0.461691
$979$	77.5464	2.47839
$980$	− 6.68466i	− 0.213534i
$981$	− 3.75379i	− 0.119849i
$982$	17.4384i	0.556483i
$983$	− 32.1231i	− 1.02457i	−0.858816	−	0.512284i	$-0.828800\pi$
$983$	− 32.1231i	− 1.02457i	0.858816	−	0.512284i	$-0.171200\pi$
$984$	12.2462	0.390395
$985$	−12.8078	−0.408089
$986$	7.61553i	0.242528i
$987$	3.93087	0.125121
$988$	0	0
$989$	2.05398	0.0653126
$990$	4.12311i	0.131041i
$991$	49.4233	1.56998	0.784991	−	0.619507i	$-0.212667\pi$
$991$	49.4233	1.56998	0.784991	+	0.619507i	$0.212667\pi$
$992$	6.68466	0.212238
$993$	30.7386i	0.975461i
$994$	7.36932i	0.233741i
$995$	− 2.87689i	− 0.0912037i
$996$	7.12311i	0.225704i
$997$	−37.5464	−1.18911	−0.594553	−	0.804056i	$-0.702671\pi$
$997$	−37.5464	−1.18911	−0.594553	+	0.804056i	$0.702671\pi$
$998$	−4.00000	−0.126618
$999$	4.12311i	0.130449i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.r.1351.4		4
13.2	odd	12		390.2.i.g.61.2	✓	4
13.5	odd	4		5070.2.a.bi.1.1		2
13.6	odd	12		390.2.i.g.211.2	yes	4
13.8	odd	4		5070.2.a.bb.1.2		2
13.12	even	2	inner	5070.2.b.r.1351.1		4
39.2	even	12		1170.2.i.o.451.2		4
39.32	even	12		1170.2.i.o.991.2		4
65.2	even	12		1950.2.z.n.1699.1		8
65.19	odd	12		1950.2.i.bi.601.1		4
65.28	even	12		1950.2.z.n.1699.4		8
65.32	even	12		1950.2.z.n.1849.4		8
65.54	odd	12		1950.2.i.bi.451.1		4
65.58	even	12		1950.2.z.n.1849.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.g.61.2	✓	4	13.2	odd	12
390.2.i.g.211.2	yes	4	13.6	odd	12
1170.2.i.o.451.2		4	39.2	even	12
1170.2.i.o.991.2		4	39.32	even	12
1950.2.i.bi.451.1		4	65.54	odd	12
1950.2.i.bi.601.1		4	65.19	odd	12
1950.2.z.n.1699.1		8	65.2	even	12
1950.2.z.n.1699.4		8	65.28	even	12
1950.2.z.n.1849.1		8	65.58	even	12
1950.2.z.n.1849.4		8	65.32	even	12
5070.2.a.bb.1.2		2	13.8	odd	4
5070.2.a.bi.1.1		2	13.5	odd	4
5070.2.b.r.1351.1		4	13.12	even	2	inner
5070.2.b.r.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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +0.561553i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +0.561553i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.12311i q^{11} -1.00000 q^{12} -0.561553 q^{14} +1.00000i q^{15} +1.00000 q^{16} +3.12311 q^{17} +1.00000i q^{18} +0.561553i q^{19} -1.00000i q^{20} +0.561553i q^{21} +4.12311 q^{22} -4.68466 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -0.561553i q^{28} +2.43845 q^{29} -1.00000 q^{30} -6.68466i q^{31} +1.00000i q^{32} -4.12311i q^{33} +3.12311i q^{34} -0.561553 q^{35} -1.00000 q^{36} +4.12311i q^{37} -0.561553 q^{38} +1.00000 q^{40} +12.2462i q^{41} -0.561553 q^{42} -0.438447 q^{43} +4.12311i q^{44} +1.00000i q^{45} -4.68466i q^{46} -7.00000i q^{47} +1.00000 q^{48} +6.68466 q^{49} -1.00000i q^{50} +3.12311 q^{51} +8.56155 q^{53} +1.00000i q^{54} +4.12311 q^{55} +0.561553 q^{56} +0.561553i q^{57} +2.43845i q^{58} -6.43845i q^{59} -1.00000i q^{60} +6.00000 q^{61} +6.68466 q^{62} +0.561553i q^{63} -1.00000 q^{64} +4.12311 q^{66} -2.24621i q^{67} -3.12311 q^{68} -4.68466 q^{69} -0.561553i q^{70} -13.1231i q^{71} -1.00000i q^{72} -9.36932i q^{73} -4.12311 q^{74} -1.00000 q^{75} -0.561553i q^{76} +2.31534 q^{77} +11.5616 q^{79} +1.00000i q^{80} +1.00000 q^{81} -12.2462 q^{82} -7.12311i q^{83} -0.561553i q^{84} +3.12311i q^{85} -0.438447i q^{86} +2.43845 q^{87} -4.12311 q^{88} +18.8078i q^{89} -1.00000 q^{90} +4.68466 q^{92} -6.68466i q^{93} +7.00000 q^{94} -0.561553 q^{95} +1.00000i q^{96} -7.12311i q^{97} +6.68466i q^{98} -4.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} - 4 q^{12} + 6 q^{14} + 4 q^{16} - 4 q^{17} + 6 q^{23} - 4 q^{25} + 4 q^{27} + 18 q^{29} - 4 q^{30} + 6 q^{35} - 4 q^{36} + 6 q^{38} + 4 q^{40} + 6 q^{42} - 10 q^{43} + 4 q^{48} + 2 q^{49} - 4 q^{51} + 26 q^{53} - 6 q^{56} + 24 q^{61} + 2 q^{62} - 4 q^{64} + 4 q^{68} + 6 q^{69} - 4 q^{75} + 34 q^{77} + 38 q^{79} + 4 q^{81} - 16 q^{82} + 18 q^{87} - 4 q^{90} - 6 q^{92} + 28 q^{94} + 6 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 - 4 * q^12 + 6 * q^14 + 4 * q^16 - 4 * q^17 + 6 * q^23 - 4 * q^25 + 4 * q^27 + 18 * q^29 - 4 * q^30 + 6 * q^35 - 4 * q^36 + 6 * q^38 + 4 * q^40 + 6 * q^42 - 10 * q^43 + 4 * q^48 + 2 * q^49 - 4 * q^51 + 26 * q^53 - 6 * q^56 + 24 * q^61 + 2 * q^62 - 4 * q^64 + 4 * q^68 + 6 * q^69 - 4 * q^75 + 34 * q^77 + 38 * q^79 + 4 * q^81 - 16 * q^82 + 18 * q^87 - 4 * q^90 - 6 * q^92 + 28 * q^94 + 6 * q^95

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