LMFDB - Embedded newform 5070.2.b.x.1351.5

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:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.5
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.x.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -0.554958i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -0.554958i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.35690i q^{11} -1.00000 q^{12} +0.554958 q^{14} +1.00000i q^{15} +1.00000 q^{16} -1.80194 q^{17} +1.00000i q^{18} +3.33513i q^{19} -1.00000i q^{20} -0.554958i q^{21} +1.35690 q^{22} +0.692021 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +0.554958i q^{28} +5.04892 q^{29} -1.00000 q^{30} +1.75302i q^{31} +1.00000i q^{32} -1.35690i q^{33} -1.80194i q^{34} +0.554958 q^{35} -1.00000 q^{36} +0.158834i q^{37} -3.33513 q^{38} +1.00000 q^{40} -0.664874i q^{41} +0.554958 q^{42} +3.96077 q^{43} +1.35690i q^{44} +1.00000i q^{45} +0.692021i q^{46} +2.10992i q^{47} +1.00000 q^{48} +6.69202 q^{49} -1.00000i q^{50} -1.80194 q^{51} +1.15883 q^{53} +1.00000i q^{54} +1.35690 q^{55} -0.554958 q^{56} +3.33513i q^{57} +5.04892i q^{58} +8.72886i q^{59} -1.00000i q^{60} -5.56465 q^{61} -1.75302 q^{62} -0.554958i q^{63} -1.00000 q^{64} +1.35690 q^{66} +3.58211i q^{67} +1.80194 q^{68} +0.692021 q^{69} +0.554958i q^{70} +4.96615i q^{71} -1.00000i q^{72} -3.70410i q^{73} -0.158834 q^{74} -1.00000 q^{75} -3.33513i q^{76} -0.753020 q^{77} +3.25667 q^{79} +1.00000i q^{80} +1.00000 q^{81} +0.664874 q^{82} +3.85086i q^{83} +0.554958i q^{84} -1.80194i q^{85} +3.96077i q^{86} +5.04892 q^{87} -1.35690 q^{88} -3.13169i q^{89} -1.00000 q^{90} -0.692021 q^{92} +1.75302i q^{93} -2.10992 q^{94} -3.33513 q^{95} +1.00000i q^{96} +15.8170i q^{97} +6.69202i q^{98} -1.35690i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9	$ 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{10} - 6 q^{12} + 4 q^{14} + 6 q^{16} - 2 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} + 12 q^{29} - 6 q^{30} + 4 q^{35} - 6 q^{36} - 18 q^{38} + 6 q^{40} + 4 q^{42} - 2 q^{43} + 6 q^{48} + 30 q^{49} - 2 q^{51} - 10 q^{53} - 4 q^{56} + 10 q^{61} - 20 q^{62} - 6 q^{64} + 2 q^{68} - 6 q^{69} + 16 q^{74} - 6 q^{75} - 14 q^{77} - 34 q^{79} + 6 q^{81} + 6 q^{82} + 12 q^{87} - 6 q^{90} + 6 q^{92} - 14 q^{94} - 18 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 - 6 * q^12 + 4 * q^14 + 6 * q^16 - 2 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 + 12 * q^29 - 6 * q^30 + 4 * q^35 - 6 * q^36 - 18 * q^38 + 6 * q^40 + 4 * q^42 - 2 * q^43 + 6 * q^48 + 30 * q^49 - 2 * q^51 - 10 * q^53 - 4 * q^56 + 10 * q^61 - 20 * q^62 - 6 * q^64 + 2 * q^68 - 6 * q^69 + 16 * q^74 - 6 * q^75 - 14 * q^77 - 34 * q^79 + 6 * q^81 + 6 * q^82 + 12 * q^87 - 6 * q^90 + 6 * q^92 - 14 * q^94 - 18 * 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
$7$	− 0.554958i	− 0.209754i	−0.994485	−	0.104877i	$-0.966555\pi$
$7$	− 0.554958i	− 0.209754i	0.994485	−	0.104877i	$-0.0334450\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	− 1.35690i	− 0.409119i	−0.978854	−	0.204560i	$-0.934424\pi$
$11$	− 1.35690i	− 0.409119i	0.978854	−	0.204560i	$-0.0655763\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	0.554958	0.148319
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	−1.80194	−0.437034	−0.218517	−	0.975833i	$-0.570122\pi$
$17$	−1.80194	−0.437034	−0.218517	+	0.975833i	$0.570122\pi$
$18$	1.00000i	0.235702i
$19$	3.33513i	0.765130i	0.923929	+	0.382565i	$0.124959\pi$
$19$	3.33513i	0.765130i	−0.923929	+	0.382565i	$0.875041\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 0.554958i	− 0.121102i
$22$	1.35690	0.289291
$23$	0.692021	0.144296	0.0721482	−	0.997394i	$-0.477015\pi$
$23$	0.692021	0.144296	0.0721482	+	0.997394i	$0.477015\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0.554958i	0.104877i
$29$	5.04892	0.937560	0.468780	−	0.883315i	$-0.344694\pi$
$29$	5.04892	0.937560	0.468780	+	0.883315i	$0.344694\pi$
$30$	−1.00000	−0.182574
$31$	1.75302i	0.314852i	0.987531	+	0.157426i	$0.0503195\pi$
$31$	1.75302i	0.314852i	−0.987531	+	0.157426i	$0.949680\pi$
$32$	1.00000i	0.176777i
$33$	− 1.35690i	− 0.236205i
$34$	− 1.80194i	− 0.309030i
$35$	0.554958	0.0938050
$36$	−1.00000	−0.166667
$37$	0.158834i	0.0261121i	0.999915	+	0.0130560i	$0.00415599\pi$
$37$	0.158834i	0.0261121i	−0.999915	+	0.0130560i	$0.995844\pi$
$38$	−3.33513	−0.541029
$39$	0	0
$40$	1.00000	0.158114
$41$	− 0.664874i	− 0.103836i	−0.998651	−	0.0519180i	$-0.983467\pi$
$41$	− 0.664874i	− 0.103836i	0.998651	−	0.0519180i	$-0.0165334\pi$
$42$	0.554958	0.0856319
$43$	3.96077	0.604012	0.302006	−	0.953306i	$-0.402344\pi$
$43$	3.96077	0.604012	0.302006	+	0.953306i	$0.402344\pi$
$44$	1.35690i	0.204560i
$45$	1.00000i	0.149071i
$46$	0.692021i	0.102033i
$47$	2.10992i	0.307763i	0.988089	+	0.153881i	$0.0491774\pi$
$47$	2.10992i	0.307763i	−0.988089	+	0.153881i	$0.950823\pi$
$48$	1.00000	0.144338
$49$	6.69202	0.956003
$50$	− 1.00000i	− 0.141421i
$51$	−1.80194	−0.252322
$52$	0	0
$53$	1.15883	0.159178	0.0795890	−	0.996828i	$-0.474639\pi$
$53$	1.15883	0.159178	0.0795890	+	0.996828i	$0.474639\pi$
$54$	1.00000i	0.136083i
$55$	1.35690	0.182964
$56$	−0.554958	−0.0741594
$57$	3.33513i	0.441748i
$58$	5.04892i	0.662955i
$59$	8.72886i	1.13640i	0.822890	+	0.568200i	$0.192360\pi$
$59$	8.72886i	1.13640i	−0.822890	+	0.568200i	$0.807640\pi$
$60$	− 1.00000i	− 0.129099i
$61$	−5.56465	−0.712480	−0.356240	−	0.934394i	$-0.615941\pi$
$61$	−5.56465	−0.712480	−0.356240	+	0.934394i	$0.615941\pi$
$62$	−1.75302	−0.222634
$63$	− 0.554958i	− 0.0699182i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	1.35690	0.167022
$67$	3.58211i	0.437624i	0.975767	+	0.218812i	$0.0702181\pi$
$67$	3.58211i	0.437624i	−0.975767	+	0.218812i	$0.929782\pi$
$68$	1.80194	0.218517
$69$	0.692021	0.0833096
$70$	0.554958i	0.0663302i
$71$	4.96615i	0.589373i	0.955594	+	0.294687i	$0.0952153\pi$
$71$	4.96615i	0.589373i	−0.955594	+	0.294687i	$0.904785\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 3.70410i	− 0.433532i	−0.976224	−	0.216766i	$-0.930449\pi$
$73$	− 3.70410i	− 0.433532i	0.976224	−	0.216766i	$-0.0695509\pi$
$74$	−0.158834	−0.0184640
$75$	−1.00000	−0.115470
$76$	− 3.33513i	− 0.382565i
$77$	−0.753020	−0.0858146
$78$	0	0
$79$	3.25667	0.366404	0.183202	−	0.983075i	$-0.441354\pi$
$79$	3.25667	0.366404	0.183202	+	0.983075i	$0.441354\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	0.664874	0.0734231
$83$	3.85086i	0.422686i	0.977412	+	0.211343i	$0.0677837\pi$
$83$	3.85086i	0.422686i	−0.977412	+	0.211343i	$0.932216\pi$
$84$	0.554958i	0.0605509i
$85$	− 1.80194i	− 0.195448i
$86$	3.96077i	0.427101i
$87$	5.04892	0.541301
$88$	−1.35690	−0.144646
$89$	− 3.13169i	− 0.331958i	−0.986129	−	0.165979i	$-0.946922\pi$
$89$	− 3.13169i	− 0.331958i	0.986129	−	0.165979i	$-0.0530784\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−0.692021	−0.0721482
$93$	1.75302i	0.181780i
$94$	−2.10992	−0.217621
$95$	−3.33513	−0.342177
$96$	1.00000i	0.102062i
$97$	15.8170i	1.60597i	0.595997	+	0.802987i	$0.296757\pi$
$97$	15.8170i	1.60597i	−0.595997	+	0.802987i	$0.703243\pi$
$98$	6.69202i	0.675996i
$99$	− 1.35690i	− 0.136373i
$100$	1.00000	0.100000
$101$	14.6896	1.46167	0.730836	−	0.682553i	$-0.239130\pi$
$101$	14.6896	1.46167	0.730836	+	0.682553i	$0.239130\pi$
$102$	− 1.80194i	− 0.178418i
$103$	−12.4819	−1.22988	−0.614938	−	0.788575i	$-0.710819\pi$
$103$	−12.4819	−1.22988	−0.614938	+	0.788575i	$0.710819\pi$
$104$	0	0
$105$	0.554958	0.0541584
$106$	1.15883i	0.112556i
$107$	12.2349	1.18279	0.591396	−	0.806381i	$-0.298577\pi$
$107$	12.2349	1.18279	0.591396	+	0.806381i	$0.298577\pi$
$108$	−1.00000	−0.0962250
$109$	10.1317i	0.970440i	0.874392	+	0.485220i	$0.161260\pi$
$109$	10.1317i	0.970440i	−0.874392	+	0.485220i	$0.838740\pi$
$110$	1.35690i	0.129375i
$111$	0.158834i	0.0150758i
$112$	− 0.554958i	− 0.0524386i
$113$	−5.56465	−0.523478	−0.261739	−	0.965139i	$-0.584296\pi$
$113$	−5.56465	−0.523478	−0.261739	+	0.965139i	$0.584296\pi$
$114$	−3.33513	−0.312363
$115$	0.692021i	0.0645313i
$116$	−5.04892	−0.468780
$117$	0	0
$118$	−8.72886	−0.803556
$119$	1.00000i	0.0916698i
$120$	1.00000	0.0912871
$121$	9.15883	0.832621
$122$	− 5.56465i	− 0.503799i
$123$	− 0.664874i	− 0.0599497i
$124$	− 1.75302i	− 0.157426i
$125$	− 1.00000i	− 0.0894427i
$126$	0.554958	0.0494396
$127$	13.8485	1.22885	0.614426	−	0.788974i	$-0.289388\pi$
$127$	13.8485	1.22885	0.614426	+	0.788974i	$0.289388\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	3.96077	0.348726
$130$	0	0
$131$	19.1444	1.67265	0.836325	−	0.548234i	$-0.184700\pi$
$131$	19.1444	1.67265	0.836325	+	0.548234i	$0.184700\pi$
$132$	1.35690i	0.118103i
$133$	1.85086	0.160489
$134$	−3.58211	−0.309447
$135$	1.00000i	0.0860663i
$136$	1.80194i	0.154515i
$137$	10.9933i	0.939221i	0.882874	+	0.469610i	$0.155606\pi$
$137$	10.9933i	0.939221i	−0.882874	+	0.469610i	$0.844394\pi$
$138$	0.692021i	0.0589088i
$139$	−11.4940	−0.974905	−0.487452	−	0.873150i	$-0.662074\pi$
$139$	−11.4940	−0.974905	−0.487452	+	0.873150i	$0.662074\pi$
$140$	−0.554958	−0.0469025
$141$	2.10992i	0.177687i
$142$	−4.96615	−0.416750
$143$	0	0
$144$	1.00000	0.0833333
$145$	5.04892i	0.419290i
$146$	3.70410	0.306554
$147$	6.69202	0.551949
$148$	− 0.158834i	− 0.0130560i
$149$	15.0911i	1.23631i	0.786055	+	0.618157i	$0.212120\pi$
$149$	15.0911i	1.23631i	−0.786055	+	0.618157i	$0.787880\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	11.7168i	0.953498i	0.879040	+	0.476749i	$0.158185\pi$
$151$	11.7168i	0.953498i	−0.879040	+	0.476749i	$0.841815\pi$
$152$	3.33513	0.270514
$153$	−1.80194	−0.145678
$154$	− 0.753020i	− 0.0606801i
$155$	−1.75302	−0.140806
$156$	0	0
$157$	−2.88040	−0.229881	−0.114940	−	0.993372i	$-0.536668\pi$
$157$	−2.88040	−0.229881	−0.114940	+	0.993372i	$0.536668\pi$
$158$	3.25667i	0.259087i
$159$	1.15883	0.0919015
$160$	−1.00000	−0.0790569
$161$	− 0.384043i	− 0.0302668i
$162$	1.00000i	0.0785674i
$163$	− 9.83877i	− 0.770632i	−0.922785	−	0.385316i	$-0.874092\pi$
$163$	− 9.83877i	− 0.770632i	0.922785	−	0.385316i	$-0.125908\pi$
$164$	0.664874i	0.0519180i
$165$	1.35690	0.105634
$166$	−3.85086	−0.298884
$167$	− 8.16852i	− 0.632099i	−0.948743	−	0.316050i	$-0.897643\pi$
$167$	− 8.16852i	− 0.632099i	0.948743	−	0.316050i	$-0.102357\pi$
$168$	−0.554958	−0.0428159
$169$	0	0
$170$	1.80194	0.138202
$171$	3.33513i	0.255043i
$172$	−3.96077	−0.302006
$173$	16.0043	1.21679	0.608393	−	0.793636i	$-0.291815\pi$
$173$	16.0043	1.21679	0.608393	+	0.793636i	$0.291815\pi$
$174$	5.04892i	0.382757i
$175$	0.554958i	0.0419509i
$176$	− 1.35690i	− 0.102280i
$177$	8.72886i	0.656101i
$178$	3.13169	0.234730
$179$	14.4819	1.08243	0.541213	−	0.840885i	$-0.317965\pi$
$179$	14.4819	1.08243	0.541213	+	0.840885i	$0.317965\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−20.5502	−1.52748	−0.763741	−	0.645523i	$-0.776640\pi$
$181$	−20.5502	−1.52748	−0.763741	+	0.645523i	$0.776640\pi$
$182$	0	0
$183$	−5.56465	−0.411351
$184$	− 0.692021i	− 0.0510165i
$185$	−0.158834	−0.0116777
$186$	−1.75302	−0.128538
$187$	2.44504i	0.178799i
$188$	− 2.10992i	− 0.153881i
$189$	− 0.554958i	− 0.0403673i
$190$	− 3.33513i	− 0.241955i
$191$	−14.4101	−1.04268	−0.521340	−	0.853349i	$-0.674568\pi$
$191$	−14.4101	−1.04268	−0.521340	+	0.853349i	$0.674568\pi$
$192$	−1.00000	−0.0721688
$193$	7.43727i	0.535347i	0.963510	+	0.267673i	$0.0862548\pi$
$193$	7.43727i	0.535347i	−0.963510	+	0.267673i	$0.913745\pi$
$194$	−15.8170	−1.13559
$195$	0	0
$196$	−6.69202	−0.478002
$197$	1.89977i	0.135353i	0.997707	+	0.0676766i	$0.0215586\pi$
$197$	1.89977i	0.135353i	−0.997707	+	0.0676766i	$0.978441\pi$
$198$	1.35690	0.0964304
$199$	12.3394	0.874720	0.437360	−	0.899287i	$-0.355914\pi$
$199$	12.3394	0.874720	0.437360	+	0.899287i	$0.355914\pi$
$200$	1.00000i	0.0707107i
$201$	3.58211i	0.252662i
$202$	14.6896i	1.03356i
$203$	− 2.80194i	− 0.196657i
$204$	1.80194	0.126161
$205$	0.664874	0.0464368
$206$	− 12.4819i	− 0.869654i
$207$	0.692021	0.0480988
$208$	0	0
$209$	4.52542	0.313030
$210$	0.554958i	0.0382957i
$211$	−17.9323	−1.23451	−0.617255	−	0.786763i	$-0.711755\pi$
$211$	−17.9323	−1.23451	−0.617255	+	0.786763i	$0.711755\pi$
$212$	−1.15883	−0.0795890
$213$	4.96615i	0.340275i
$214$	12.2349i	0.836360i
$215$	3.96077i	0.270122i
$216$	− 1.00000i	− 0.0680414i
$217$	0.972853	0.0660416
$218$	−10.1317	−0.686204
$219$	− 3.70410i	− 0.250300i
$220$	−1.35690	−0.0914819
$221$	0	0
$222$	−0.158834	−0.0106602
$223$	9.95539i	0.666663i	0.942810	+	0.333331i	$0.108173\pi$
$223$	9.95539i	0.666663i	−0.942810	+	0.333331i	$0.891827\pi$
$224$	0.554958	0.0370797
$225$	−1.00000	−0.0666667
$226$	− 5.56465i	− 0.370155i
$227$	− 15.6353i	− 1.03775i	−0.854849	−	0.518877i	$-0.826350\pi$
$227$	− 15.6353i	− 1.03775i	0.854849	−	0.518877i	$-0.173650\pi$
$228$	− 3.33513i	− 0.220874i
$229$	− 13.5985i	− 0.898614i	−0.893377	−	0.449307i	$-0.851671\pi$
$229$	− 13.5985i	− 0.898614i	0.893377	−	0.449307i	$-0.148329\pi$
$230$	−0.692021	−0.0456305
$231$	−0.753020	−0.0495451
$232$	− 5.04892i	− 0.331478i
$233$	−23.0737	−1.51161	−0.755803	−	0.654799i	$-0.772753\pi$
$233$	−23.0737	−1.51161	−0.755803	+	0.654799i	$0.772753\pi$
$234$	0	0
$235$	−2.10992	−0.137636
$236$	− 8.72886i	− 0.568200i
$237$	3.25667	0.211543
$238$	−1.00000	−0.0648204
$239$	16.6431i	1.07655i	0.842768	+	0.538276i	$0.180924\pi$
$239$	16.6431i	1.07655i	−0.842768	+	0.538276i	$0.819076\pi$
$240$	1.00000i	0.0645497i
$241$	− 22.2054i	− 1.43037i	−0.698934	−	0.715186i	$-0.746342\pi$
$241$	− 22.2054i	− 1.43037i	0.698934	−	0.715186i	$-0.253658\pi$
$242$	9.15883i	0.588752i
$243$	1.00000	0.0641500
$244$	5.56465	0.356240
$245$	6.69202i	0.427538i
$246$	0.664874	0.0423908
$247$	0	0
$248$	1.75302	0.111317
$249$	3.85086i	0.244038i
$250$	1.00000	0.0632456
$251$	11.4058	0.719929	0.359964	−	0.932966i	$-0.382789\pi$
$251$	11.4058	0.719929	0.359964	+	0.932966i	$0.382789\pi$
$252$	0.554958i	0.0349591i
$253$	− 0.939001i	− 0.0590345i
$254$	13.8485i	0.868930i
$255$	− 1.80194i	− 0.112842i
$256$	1.00000	0.0625000
$257$	25.4480	1.58740	0.793702	−	0.608307i	$-0.208151\pi$
$257$	25.4480	1.58740	0.793702	+	0.608307i	$0.208151\pi$
$258$	3.96077i	0.246587i
$259$	0.0881460	0.00547713
$260$	0	0
$261$	5.04892	0.312520
$262$	19.1444i	1.18274i
$263$	17.5080	1.07959	0.539794	−	0.841797i	$-0.318502\pi$
$263$	17.5080	1.07959	0.539794	+	0.841797i	$0.318502\pi$
$264$	−1.35690	−0.0835112
$265$	1.15883i	0.0711866i
$266$	1.85086i	0.113483i
$267$	− 3.13169i	− 0.191656i
$268$	− 3.58211i	− 0.218812i
$269$	−17.8388	−1.08765	−0.543825	−	0.839199i	$-0.683024\pi$
$269$	−17.8388	−1.08765	−0.543825	+	0.839199i	$0.683024\pi$
$270$	−1.00000	−0.0608581
$271$	0.936017i	0.0568590i	0.999596	+	0.0284295i	$0.00905061\pi$
$271$	0.936017i	0.0568590i	−0.999596	+	0.0284295i	$0.990949\pi$
$272$	−1.80194	−0.109259
$273$	0	0
$274$	−10.9933	−0.664129
$275$	1.35690i	0.0818239i
$276$	−0.692021	−0.0416548
$277$	10.9390	0.657261	0.328630	−	0.944459i	$-0.393413\pi$
$277$	10.9390	0.657261	0.328630	+	0.944459i	$0.393413\pi$
$278$	− 11.4940i	− 0.689362i
$279$	1.75302i	0.104951i
$280$	− 0.554958i	− 0.0331651i
$281$	19.1618i	1.14310i	0.820568	+	0.571549i	$0.193657\pi$
$281$	19.1618i	1.14310i	−0.820568	+	0.571549i	$0.806343\pi$
$282$	−2.10992	−0.125644
$283$	−24.0411	−1.42910	−0.714549	−	0.699585i	$-0.753368\pi$
$283$	−24.0411	−1.42910	−0.714549	+	0.699585i	$0.753368\pi$
$284$	− 4.96615i	− 0.294687i
$285$	−3.33513	−0.197556
$286$	0	0
$287$	−0.368977	−0.0217801
$288$	1.00000i	0.0589256i
$289$	−13.7530	−0.809001
$290$	−5.04892	−0.296483
$291$	15.8170i	0.927209i
$292$	3.70410i	0.216766i
$293$	9.50604i	0.555349i	0.960675	+	0.277674i	$0.0895636\pi$
$293$	9.50604i	0.555349i	−0.960675	+	0.277674i	$0.910436\pi$
$294$	6.69202i	0.390287i
$295$	−8.72886	−0.508214
$296$	0.158834	0.00923202
$297$	− 1.35690i	− 0.0787351i
$298$	−15.0911	−0.874206
$299$	0	0
$300$	1.00000	0.0577350
$301$	− 2.19806i	− 0.126694i
$302$	−11.7168	−0.674225
$303$	14.6896	0.843897
$304$	3.33513i	0.191283i
$305$	− 5.56465i	− 0.318631i
$306$	− 1.80194i	− 0.103010i
$307$	15.5894i	0.889734i	0.895597	+	0.444867i	$0.146749\pi$
$307$	15.5894i	0.889734i	−0.895597	+	0.444867i	$0.853251\pi$
$308$	0.753020	0.0429073
$309$	−12.4819	−0.710069
$310$	− 1.75302i	− 0.0995649i
$311$	−4.97285	−0.281985	−0.140992	−	0.990011i	$-0.545029\pi$
$311$	−4.97285	−0.281985	−0.140992	+	0.990011i	$0.545029\pi$
$312$	0	0
$313$	−28.1444	−1.59081	−0.795407	−	0.606076i	$-0.792743\pi$
$313$	−28.1444	−1.59081	−0.795407	+	0.606076i	$0.792743\pi$
$314$	− 2.88040i	− 0.162550i
$315$	0.554958	0.0312683
$316$	−3.25667	−0.183202
$317$	− 4.43535i	− 0.249114i	−0.992212	−	0.124557i	$-0.960249\pi$
$317$	− 4.43535i	− 0.249114i	0.992212	−	0.124557i	$-0.0397510\pi$
$318$	1.15883i	0.0649842i
$319$	− 6.85086i	− 0.383574i
$320$	− 1.00000i	− 0.0559017i
$321$	12.2349	0.682885
$322$	0.384043	0.0214019
$323$	− 6.00969i	− 0.334388i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	9.83877	0.544919
$327$	10.1317i	0.560284i
$328$	−0.664874	−0.0367115
$329$	1.17092	0.0645546
$330$	1.35690i	0.0746947i
$331$	− 11.7855i	− 0.647792i	−0.946093	−	0.323896i	$-0.895007\pi$
$331$	− 11.7855i	− 0.647792i	0.946093	−	0.323896i	$-0.104993\pi$
$332$	− 3.85086i	− 0.211343i
$333$	0.158834i	0.00870403i
$334$	8.16852	0.446962
$335$	−3.58211	−0.195711
$336$	− 0.554958i	− 0.0302754i
$337$	−8.18359	−0.445788	−0.222894	−	0.974843i	$-0.571550\pi$
$337$	−8.18359	−0.445788	−0.222894	+	0.974843i	$0.571550\pi$
$338$	0	0
$339$	−5.56465	−0.302230
$340$	1.80194i	0.0977238i
$341$	2.37867	0.128812
$342$	−3.33513	−0.180343
$343$	− 7.59850i	− 0.410280i
$344$	− 3.96077i	− 0.213550i
$345$	0.692021i	0.0372572i
$346$	16.0043i	0.860397i
$347$	−3.14377	−0.168766	−0.0843832	−	0.996433i	$-0.526892\pi$
$347$	−3.14377	−0.168766	−0.0843832	+	0.996433i	$0.526892\pi$
$348$	−5.04892	−0.270650
$349$	18.1444i	0.971245i	0.874169	+	0.485623i	$0.161407\pi$
$349$	18.1444i	0.971245i	−0.874169	+	0.485623i	$0.838593\pi$
$350$	−0.554958	−0.0296638
$351$	0	0
$352$	1.35690	0.0723228
$353$	17.0261i	0.906207i	0.891458	+	0.453103i	$0.149683\pi$
$353$	17.0261i	0.906207i	−0.891458	+	0.453103i	$0.850317\pi$
$354$	−8.72886	−0.463934
$355$	−4.96615	−0.263576
$356$	3.13169i	0.165979i
$357$	1.00000i	0.0529256i
$358$	14.4819i	0.765391i
$359$	− 0.851920i	− 0.0449626i	−0.999747	−	0.0224813i	$-0.992843\pi$
$359$	− 0.851920i	− 0.0449626i	0.999747	−	0.0224813i	$-0.00715663\pi$
$360$	1.00000	0.0527046
$361$	7.87694	0.414576
$362$	− 20.5502i	− 1.08009i
$363$	9.15883	0.480714
$364$	0	0
$365$	3.70410	0.193882
$366$	− 5.56465i	− 0.290869i
$367$	20.9366	1.09288	0.546441	−	0.837498i	$-0.315982\pi$
$367$	20.9366	1.09288	0.546441	+	0.837498i	$0.315982\pi$
$368$	0.692021	0.0360741
$369$	− 0.664874i	− 0.0346120i
$370$	− 0.158834i	− 0.00825737i
$371$	− 0.643104i	− 0.0333883i
$372$	− 1.75302i	− 0.0908899i
$373$	−3.85384	−0.199544	−0.0997721	−	0.995010i	$-0.531811\pi$
$373$	−3.85384	−0.199544	−0.0997721	+	0.995010i	$0.531811\pi$
$374$	−2.44504	−0.126430
$375$	− 1.00000i	− 0.0516398i
$376$	2.10992	0.108811
$377$	0	0
$378$	0.554958	0.0285440
$379$	27.2252i	1.39847i	0.714894	+	0.699233i	$0.246475\pi$
$379$	27.2252i	1.39847i	−0.714894	+	0.699233i	$0.753525\pi$
$380$	3.33513	0.171088
$381$	13.8485	0.709478
$382$	− 14.4101i	− 0.737286i
$383$	− 38.2194i	− 1.95292i	−0.215703	−	0.976459i	$-0.569204\pi$
$383$	− 38.2194i	− 1.95292i	0.215703	−	0.976459i	$-0.430796\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	− 0.753020i	− 0.0383775i
$386$	−7.43727	−0.378547
$387$	3.96077	0.201337
$388$	− 15.8170i	− 0.802987i
$389$	1.42327	0.0721627	0.0360814	−	0.999349i	$-0.488512\pi$
$389$	1.42327	0.0721627	0.0360814	+	0.999349i	$0.488512\pi$
$390$	0	0
$391$	−1.24698	−0.0630625
$392$	− 6.69202i	− 0.337998i
$393$	19.1444	0.965705
$394$	−1.89977	−0.0957092
$395$	3.25667i	0.163861i
$396$	1.35690i	0.0681866i
$397$	− 16.2121i	− 0.813660i	−0.913504	−	0.406830i	$-0.866634\pi$
$397$	− 16.2121i	− 0.813660i	0.913504	−	0.406830i	$-0.133366\pi$
$398$	12.3394i	0.618520i
$399$	1.85086	0.0926586
$400$	−1.00000	−0.0500000
$401$	− 1.25428i	− 0.0626355i	−0.999509	−	0.0313178i	$-0.990030\pi$
$401$	− 1.25428i	− 0.0626355i	0.999509	−	0.0313178i	$-0.00997038\pi$
$402$	−3.58211	−0.178659
$403$	0	0
$404$	−14.6896	−0.730836
$405$	1.00000i	0.0496904i
$406$	2.80194	0.139058
$407$	0.215521	0.0106830
$408$	1.80194i	0.0892092i
$409$	16.4276i	0.812292i	0.913808	+	0.406146i	$0.133128\pi$
$409$	16.4276i	0.812292i	−0.913808	+	0.406146i	$0.866872\pi$
$410$	0.664874i	0.0328358i
$411$	10.9933i	0.542259i
$412$	12.4819	0.614938
$413$	4.84415	0.238365
$414$	0.692021i	0.0340110i
$415$	−3.85086	−0.189031
$416$	0	0
$417$	−11.4940	−0.562862
$418$	4.52542i	0.221345i
$419$	1.78687	0.0872944	0.0436472	−	0.999047i	$-0.486102\pi$
$419$	1.78687	0.0872944	0.0436472	+	0.999047i	$0.486102\pi$
$420$	−0.554958	−0.0270792
$421$	0.445042i	0.0216900i	0.999941	+	0.0108450i	$0.00345214\pi$
$421$	0.445042i	0.0216900i	−0.999941	+	0.0108450i	$0.996548\pi$
$422$	− 17.9323i	− 0.872931i
$423$	2.10992i	0.102588i
$424$	− 1.15883i	− 0.0562779i
$425$	1.80194	0.0874068
$426$	−4.96615	−0.240611
$427$	3.08815i	0.149446i
$428$	−12.2349	−0.591396
$429$	0	0
$430$	−3.96077	−0.191005
$431$	− 14.4547i	− 0.696260i	−0.937446	−	0.348130i	$-0.886817\pi$
$431$	− 14.4547i	− 0.696260i	0.937446	−	0.348130i	$-0.113183\pi$
$432$	1.00000	0.0481125
$433$	−5.69740	−0.273799	−0.136900	−	0.990585i	$-0.543714\pi$
$433$	−5.69740	−0.273799	−0.136900	+	0.990585i	$0.543714\pi$
$434$	0.972853i	0.0466984i
$435$	5.04892i	0.242077i
$436$	− 10.1317i	− 0.485220i
$437$	2.30798i	0.110406i
$438$	3.70410	0.176989
$439$	3.77479	0.180161	0.0900805	−	0.995934i	$-0.471288\pi$
$439$	3.77479	0.180161	0.0900805	+	0.995934i	$0.471288\pi$
$440$	− 1.35690i	− 0.0646875i
$441$	6.69202	0.318668
$442$	0	0
$443$	−7.98685	−0.379467	−0.189733	−	0.981836i	$-0.560762\pi$
$443$	−7.98685	−0.379467	−0.189733	+	0.981836i	$0.560762\pi$
$444$	− 0.158834i	− 0.00753791i
$445$	3.13169	0.148456
$446$	−9.95539	−0.471402
$447$	15.0911i	0.713786i
$448$	0.554958i	0.0262193i
$449$	19.4131i	0.916161i	0.888911	+	0.458081i	$0.151463\pi$
$449$	19.4131i	0.916161i	−0.888911	+	0.458081i	$0.848537\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	−0.902165	−0.0424813
$452$	5.56465	0.261739
$453$	11.7168i	0.550502i
$454$	15.6353	0.733803
$455$	0	0
$456$	3.33513	0.156182
$457$	2.64848i	0.123891i	0.998080	+	0.0619454i	$0.0197305\pi$
$457$	2.64848i	0.123891i	−0.998080	+	0.0619454i	$0.980270\pi$
$458$	13.5985	0.635416
$459$	−1.80194	−0.0841073
$460$	− 0.692021i	− 0.0322657i
$461$	− 22.6649i	− 1.05561i	−0.849366	−	0.527804i	$-0.823015\pi$
$461$	− 22.6649i	− 1.05561i	0.849366	−	0.527804i	$-0.176985\pi$
$462$	− 0.753020i	− 0.0350337i
$463$	− 0.685317i	− 0.0318494i	−0.999873	−	0.0159247i	$-0.994931\pi$
$463$	− 0.685317i	− 0.0318494i	0.999873	−	0.0159247i	$-0.00506920\pi$
$464$	5.04892	0.234390
$465$	−1.75302	−0.0812944
$466$	− 23.0737i	− 1.06887i
$467$	34.4553	1.59440	0.797201	−	0.603714i	$-0.206313\pi$
$467$	34.4553	1.59440	0.797201	+	0.603714i	$0.206313\pi$
$468$	0	0
$469$	1.98792	0.0917935
$470$	− 2.10992i	− 0.0973232i
$471$	−2.88040	−0.132722
$472$	8.72886	0.401778
$473$	− 5.37435i	− 0.247113i
$474$	3.25667i	0.149584i
$475$	− 3.33513i	− 0.153026i
$476$	− 1.00000i	− 0.0458349i
$477$	1.15883	0.0530593
$478$	−16.6431	−0.761238
$479$	− 39.9724i	− 1.82638i	−0.407529	−	0.913192i	$-0.633609\pi$
$479$	− 39.9724i	− 1.82638i	0.407529	−	0.913192i	$-0.366391\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	22.2054	1.01143
$483$	− 0.384043i	− 0.0174746i
$484$	−9.15883	−0.416311
$485$	−15.8170	−0.718213
$486$	1.00000i	0.0453609i
$487$	10.8442i	0.491395i	0.969347	+	0.245698i	$0.0790170\pi$
$487$	10.8442i	0.491395i	−0.969347	+	0.245698i	$0.920983\pi$
$488$	5.56465i	0.251900i
$489$	− 9.83877i	− 0.444925i
$490$	−6.69202	−0.302315
$491$	18.0543	0.814779	0.407389	−	0.913255i	$-0.366439\pi$
$491$	18.0543	0.814779	0.407389	+	0.913255i	$0.366439\pi$
$492$	0.664874i	0.0299749i
$493$	−9.09783	−0.409746
$494$	0	0
$495$	1.35690	0.0609879
$496$	1.75302i	0.0787129i
$497$	2.75600	0.123624
$498$	−3.85086	−0.172561
$499$	13.2271i	0.592128i	0.955168	+	0.296064i	$0.0956741\pi$
$499$	13.2271i	0.592128i	−0.955168	+	0.296064i	$0.904326\pi$
$500$	1.00000i	0.0447214i
$501$	− 8.16852i	− 0.364943i
$502$	11.4058i	0.509066i
$503$	−21.4523	−0.956512	−0.478256	−	0.878220i	$-0.658731\pi$
$503$	−21.4523	−0.956512	−0.478256	+	0.878220i	$0.658731\pi$
$504$	−0.554958	−0.0247198
$505$	14.6896i	0.653680i
$506$	0.939001	0.0417437
$507$	0	0
$508$	−13.8485	−0.614426
$509$	− 5.96077i	− 0.264207i	−0.991236	−	0.132103i	$-0.957827\pi$
$509$	− 5.96077i	− 0.264207i	0.991236	−	0.132103i	$-0.0421731\pi$
$510$	1.80194	0.0797911
$511$	−2.05562	−0.0909354
$512$	1.00000i	0.0441942i
$513$	3.33513i	0.147249i
$514$	25.4480i	1.12246i
$515$	− 12.4819i	− 0.550017i
$516$	−3.96077	−0.174363
$517$	2.86294	0.125912
$518$	0.0881460i	0.00387291i
$519$	16.0043	0.702511
$520$	0	0
$521$	6.14436	0.269189	0.134595	−	0.990901i	$-0.457027\pi$
$521$	6.14436	0.269189	0.134595	+	0.990901i	$0.457027\pi$
$522$	5.04892i	0.220985i
$523$	26.6353	1.16468	0.582341	−	0.812945i	$-0.302137\pi$
$523$	26.6353	1.16468	0.582341	+	0.812945i	$0.302137\pi$
$524$	−19.1444	−0.836325
$525$	0.554958i	0.0242204i
$526$	17.5080i	0.763383i
$527$	− 3.15883i	− 0.137601i
$528$	− 1.35690i	− 0.0590513i
$529$	−22.5211	−0.979179
$530$	−1.15883	−0.0503365
$531$	8.72886i	0.378800i
$532$	−1.85086	−0.0802447
$533$	0	0
$534$	3.13169	0.135521
$535$	12.2349i	0.528961i
$536$	3.58211	0.154723
$537$	14.4819	0.624939
$538$	− 17.8388i	− 0.769084i
$539$	− 9.08038i	− 0.391119i
$540$	− 1.00000i	− 0.0430331i
$541$	41.0200i	1.76359i	0.471636	+	0.881793i	$0.343663\pi$
$541$	41.0200i	1.76359i	−0.471636	+	0.881793i	$0.656337\pi$
$542$	−0.936017	−0.0402054
$543$	−20.5502	−0.881892
$544$	− 1.80194i	− 0.0772574i
$545$	−10.1317	−0.433994
$546$	0	0
$547$	−0.270078	−0.0115477	−0.00577385	−	0.999983i	$-0.501838\pi$
$547$	−0.270078	−0.0115477	−0.00577385	+	0.999983i	$0.501838\pi$
$548$	− 10.9933i	− 0.469610i
$549$	−5.56465	−0.237493
$550$	−1.35690	−0.0578582
$551$	16.8388i	0.717356i
$552$	− 0.692021i	− 0.0294544i
$553$	− 1.80731i	− 0.0768548i
$554$	10.9390i	0.464754i
$555$	−0.158834	−0.00674211
$556$	11.4940	0.487452
$557$	7.28083i	0.308499i	0.988032	+	0.154249i	$0.0492959\pi$
$557$	7.28083i	0.308499i	−0.988032	+	0.154249i	$0.950704\pi$
$558$	−1.75302	−0.0742113
$559$	0	0
$560$	0.554958	0.0234513
$561$	2.44504i	0.103230i
$562$	−19.1618	−0.808292
$563$	7.11769	0.299975	0.149987	−	0.988688i	$-0.452077\pi$
$563$	7.11769	0.299975	0.149987	+	0.988688i	$0.452077\pi$
$564$	− 2.10992i	− 0.0888435i
$565$	− 5.56465i	− 0.234106i
$566$	− 24.0411i	− 1.01052i
$567$	− 0.554958i	− 0.0233061i
$568$	4.96615	0.208375
$569$	25.9661	1.08856	0.544279	−	0.838904i	$-0.316803\pi$
$569$	25.9661	1.08856	0.544279	+	0.838904i	$0.316803\pi$
$570$	− 3.33513i	− 0.139693i
$571$	28.1618	1.17854	0.589268	−	0.807938i	$-0.299416\pi$
$571$	28.1618	1.17854	0.589268	+	0.807938i	$0.299416\pi$
$572$	0	0
$573$	−14.4101	−0.601992
$574$	− 0.368977i	− 0.0154008i
$575$	−0.692021	−0.0288593
$576$	−1.00000	−0.0416667
$577$	5.58450i	0.232486i	0.993221	+	0.116243i	$0.0370851\pi$
$577$	5.58450i	0.232486i	−0.993221	+	0.116243i	$0.962915\pi$
$578$	− 13.7530i	− 0.572050i
$579$	7.43727i	0.309082i
$580$	− 5.04892i	− 0.209645i
$581$	2.13706	0.0886603
$582$	−15.8170	−0.655636
$583$	− 1.57242i	− 0.0651228i
$584$	−3.70410	−0.153277
$585$	0	0
$586$	−9.50604	−0.392691
$587$	− 30.4989i	− 1.25882i	−0.777072	−	0.629411i	$-0.783296\pi$
$587$	− 30.4989i	− 1.25882i	0.777072	−	0.629411i	$-0.216704\pi$
$588$	−6.69202	−0.275974
$589$	−5.84654	−0.240903
$590$	− 8.72886i	− 0.359361i
$591$	1.89977i	0.0781462i
$592$	0.158834i	0.00652802i
$593$	− 42.1473i	− 1.73078i	−0.501096	−	0.865392i	$-0.667069\pi$
$593$	− 42.1473i	− 1.73078i	0.501096	−	0.865392i	$-0.332931\pi$
$594$	1.35690	0.0556741
$595$	−1.00000	−0.0409960
$596$	− 15.0911i	− 0.618157i
$597$	12.3394	0.505020
$598$	0	0
$599$	15.9748	0.652711	0.326356	−	0.945247i	$-0.394179\pi$
$599$	15.9748	0.652711	0.326356	+	0.945247i	$0.394179\pi$
$600$	1.00000i	0.0408248i
$601$	−33.8485	−1.38071	−0.690354	−	0.723472i	$-0.742545\pi$
$601$	−33.8485	−1.38071	−0.690354	+	0.723472i	$0.742545\pi$
$602$	2.19806	0.0895863
$603$	3.58211i	0.145875i
$604$	− 11.7168i	− 0.476749i
$605$	9.15883i	0.372360i
$606$	14.6896i	0.596725i
$607$	0.0827692	0.00335950	0.00167975	−	0.999999i	$-0.499465\pi$
$607$	0.0827692	0.00335950	0.00167975	+	0.999999i	$0.499465\pi$
$608$	−3.33513	−0.135257
$609$	− 2.80194i	− 0.113540i
$610$	5.56465	0.225306
$611$	0	0
$612$	1.80194	0.0728390
$613$	5.54181i	0.223832i	0.993718	+	0.111916i	$0.0356987\pi$
$613$	5.54181i	0.223832i	−0.993718	+	0.111916i	$0.964301\pi$
$614$	−15.5894	−0.629137
$615$	0.664874	0.0268103
$616$	0.753020i	0.0303401i
$617$	− 4.43296i	− 0.178464i	−0.996011	−	0.0892321i	$-0.971559\pi$
$617$	− 4.43296i	− 0.178464i	0.996011	−	0.0892321i	$-0.0284413\pi$
$618$	− 12.4819i	− 0.502095i
$619$	− 21.4179i	− 0.860858i	−0.902624	−	0.430429i	$-0.858362\pi$
$619$	− 21.4179i	− 0.860858i	0.902624	−	0.430429i	$-0.141638\pi$
$620$	1.75302	0.0704030
$621$	0.692021	0.0277699
$622$	− 4.97285i	− 0.199393i
$623$	−1.73795	−0.0696297
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 28.1444i	− 1.12487i
$627$	4.52542	0.180728
$628$	2.88040	0.114940
$629$	− 0.286208i	− 0.0114119i
$630$	0.554958i	0.0221101i
$631$	− 20.6485i	− 0.822003i	−0.911635	−	0.411002i	$-0.865179\pi$
$631$	− 20.6485i	− 0.822003i	0.911635	−	0.411002i	$-0.134821\pi$
$632$	− 3.25667i	− 0.129543i
$633$	−17.9323	−0.712745
$634$	4.43535	0.176150
$635$	13.8485i	0.549559i
$636$	−1.15883	−0.0459507
$637$	0	0
$638$	6.85086	0.271228
$639$	4.96615i	0.196458i
$640$	1.00000	0.0395285
$641$	18.1763	0.717920	0.358960	−	0.933353i	$-0.383131\pi$
$641$	18.1763	0.717920	0.358960	+	0.933353i	$0.383131\pi$
$642$	12.2349i	0.482873i
$643$	− 30.3532i	− 1.19701i	−0.801118	−	0.598506i	$-0.795761\pi$
$643$	− 30.3532i	− 1.19701i	0.801118	−	0.598506i	$-0.204239\pi$
$644$	0.384043i	0.0151334i
$645$	3.96077i	0.155955i
$646$	6.00969	0.236448
$647$	14.4692	0.568843	0.284422	−	0.958699i	$-0.408198\pi$
$647$	14.4692	0.568843	0.284422	+	0.958699i	$0.408198\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	11.8442	0.464924
$650$	0	0
$651$	0.972853	0.0381291
$652$	9.83877i	0.385316i
$653$	21.2185	0.830344	0.415172	−	0.909743i	$-0.363721\pi$
$653$	21.2185	0.830344	0.415172	+	0.909743i	$0.363721\pi$
$654$	−10.1317	−0.396180
$655$	19.1444i	0.748032i
$656$	− 0.664874i	− 0.0259590i
$657$	− 3.70410i	− 0.144511i
$658$	1.17092i	0.0456470i
$659$	−11.8226	−0.460545	−0.230272	−	0.973126i	$-0.573962\pi$
$659$	−11.8226	−0.460545	−0.230272	+	0.973126i	$0.573962\pi$
$660$	−1.35690	−0.0528171
$661$	10.4789i	0.407582i	0.979014	+	0.203791i	$0.0653263\pi$
$661$	10.4789i	0.407582i	−0.979014	+	0.203791i	$0.934674\pi$
$662$	11.7855	0.458058
$663$	0	0
$664$	3.85086	0.149442
$665$	1.85086i	0.0717731i
$666$	−0.158834	−0.00615468
$667$	3.49396	0.135287
$668$	8.16852i	0.316050i
$669$	9.95539i	0.384898i
$670$	− 3.58211i	− 0.138389i
$671$	7.55065i	0.291489i
$672$	0.554958	0.0214080
$673$	−30.5013	−1.17574	−0.587868	−	0.808957i	$-0.700033\pi$
$673$	−30.5013	−1.17574	−0.587868	+	0.808957i	$0.700033\pi$
$674$	− 8.18359i	− 0.315220i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−14.9651	−0.575155	−0.287577	−	0.957757i	$-0.592850\pi$
$677$	−14.9651	−0.575155	−0.287577	+	0.957757i	$0.592850\pi$
$678$	− 5.56465i	− 0.213709i
$679$	8.77777	0.336860
$680$	−1.80194	−0.0691012
$681$	− 15.6353i	− 0.599147i
$682$	2.37867i	0.0910838i
$683$	18.0285i	0.689840i	0.938632	+	0.344920i	$0.112094\pi$
$683$	18.0285i	0.689840i	−0.938632	+	0.344920i	$0.887906\pi$
$684$	− 3.33513i	− 0.127522i
$685$	−10.9933	−0.420032
$686$	7.59850	0.290112
$687$	− 13.5985i	− 0.518815i
$688$	3.96077	0.151003
$689$	0	0
$690$	−0.692021	−0.0263448
$691$	− 5.06829i	− 0.192807i	−0.995342	−	0.0964035i	$-0.969266\pi$
$691$	− 5.06829i	− 0.192807i	0.995342	−	0.0964035i	$-0.0307339\pi$
$692$	−16.0043	−0.608393
$693$	−0.753020	−0.0286049
$694$	− 3.14377i	− 0.119336i
$695$	− 11.4940i	− 0.435991i
$696$	− 5.04892i	− 0.191379i
$697$	1.19806i	0.0453798i
$698$	−18.1444	−0.686774
$699$	−23.0737	−0.872726
$700$	− 0.554958i	− 0.0209754i
$701$	−9.58402	−0.361984	−0.180992	−	0.983485i	$-0.557931\pi$
$701$	−9.58402	−0.361984	−0.180992	+	0.983485i	$0.557931\pi$
$702$	0	0
$703$	−0.529730	−0.0199791
$704$	1.35690i	0.0511399i
$705$	−2.10992	−0.0794640
$706$	−17.0261	−0.640785
$707$	− 8.15213i	− 0.306592i
$708$	− 8.72886i	− 0.328051i
$709$	− 52.3212i	− 1.96497i	−0.186354	−	0.982483i	$-0.559667\pi$
$709$	− 52.3212i	− 1.96497i	0.186354	−	0.982483i	$-0.440333\pi$
$710$	− 4.96615i	− 0.186376i
$711$	3.25667	0.122135
$712$	−3.13169	−0.117365
$713$	1.21313i	0.0454320i
$714$	−1.00000	−0.0374241
$715$	0	0
$716$	−14.4819	−0.541213
$717$	16.6431i	0.621548i
$718$	0.851920	0.0317934
$719$	−32.9439	−1.22860	−0.614300	−	0.789072i	$-0.710562\pi$
$719$	−32.9439	−1.22860	−0.614300	+	0.789072i	$0.710562\pi$
$720$	1.00000i	0.0372678i
$721$	6.92692i	0.257972i
$722$	7.87694i	0.293149i
$723$	− 22.2054i	− 0.825826i
$724$	20.5502	0.763741
$725$	−5.04892	−0.187512
$726$	9.15883i	0.339916i
$727$	24.5894	0.911970	0.455985	−	0.889987i	$-0.349287\pi$
$727$	24.5894	0.911970	0.455985	+	0.889987i	$0.349287\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	3.70410i	0.137095i
$731$	−7.13706	−0.263974
$732$	5.56465	0.205675
$733$	− 43.4196i	− 1.60374i	−0.597500	−	0.801869i	$-0.703839\pi$
$733$	− 43.4196i	− 1.60374i	0.597500	−	0.801869i	$-0.296161\pi$
$734$	20.9366i	0.772784i
$735$	6.69202i	0.246839i
$736$	0.692021i	0.0255082i
$737$	4.86054	0.179040
$738$	0.664874	0.0244744
$739$	31.4902i	1.15839i	0.815190	+	0.579194i	$0.196632\pi$
$739$	31.4902i	1.15839i	−0.815190	+	0.579194i	$0.803368\pi$
$740$	0.158834	0.00583884
$741$	0	0
$742$	0.643104	0.0236091
$743$	− 21.3120i	− 0.781862i	−0.920420	−	0.390931i	$-0.872153\pi$
$743$	− 21.3120i	− 0.781862i	0.920420	−	0.390931i	$-0.127847\pi$
$744$	1.75302	0.0642688
$745$	−15.0911	−0.552896
$746$	− 3.85384i	− 0.141099i
$747$	3.85086i	0.140895i
$748$	− 2.44504i	− 0.0893996i
$749$	− 6.78986i	− 0.248096i
$750$	1.00000	0.0365148
$751$	−19.4370	−0.709267	−0.354633	−	0.935005i	$-0.615394\pi$
$751$	−19.4370	−0.709267	−0.354633	+	0.935005i	$0.615394\pi$
$752$	2.10992i	0.0769407i
$753$	11.4058	0.415651
$754$	0	0
$755$	−11.7168	−0.426417
$756$	0.554958i	0.0201836i
$757$	−9.11529	−0.331301	−0.165650	−	0.986185i	$-0.552972\pi$
$757$	−9.11529	−0.331301	−0.165650	+	0.986185i	$0.552972\pi$
$758$	−27.2252	−0.988864
$759$	− 0.939001i	− 0.0340836i
$760$	3.33513i	0.120978i
$761$	7.89785i	0.286297i	0.989701	+	0.143148i	$0.0457226\pi$
$761$	7.89785i	0.286297i	−0.989701	+	0.143148i	$0.954277\pi$
$762$	13.8485i	0.501677i
$763$	5.62266	0.203554
$764$	14.4101	0.521340
$765$	− 1.80194i	− 0.0651492i
$766$	38.2194	1.38092
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 12.6856i	− 0.457454i	−0.973491	−	0.228727i	$-0.926544\pi$
$769$	− 12.6856i	− 0.457454i	0.973491	−	0.228727i	$-0.0734563\pi$
$770$	0.753020	0.0271370
$771$	25.4480	0.916488
$772$	− 7.43727i	− 0.267673i
$773$	− 31.0974i	− 1.11849i	−0.829001	−	0.559247i	$-0.811090\pi$
$773$	− 31.0974i	− 1.11849i	0.829001	−	0.559247i	$-0.188910\pi$
$774$	3.96077i	0.142367i
$775$	− 1.75302i	− 0.0629704i
$776$	15.8170	0.567797
$777$	0.0881460	0.00316222
$778$	1.42327i	0.0510268i
$779$	2.21744	0.0794480
$780$	0	0
$781$	6.73855	0.241124
$782$	− 1.24698i	− 0.0445919i
$783$	5.04892	0.180434
$784$	6.69202	0.239001
$785$	− 2.88040i	− 0.102806i
$786$	19.1444i	0.682857i
$787$	− 24.7700i	− 0.882955i	−0.897272	−	0.441478i	$-0.854454\pi$
$787$	− 24.7700i	− 0.882955i	0.897272	−	0.441478i	$-0.145546\pi$
$788$	− 1.89977i	− 0.0676766i
$789$	17.5080	0.623300
$790$	−3.25667	−0.115867
$791$	3.08815i	0.109802i
$792$	−1.35690	−0.0482152
$793$	0	0
$794$	16.2121	0.575345
$795$	1.15883i	0.0410996i
$796$	−12.3394	−0.437360
$797$	−7.19029	−0.254693	−0.127347	−	0.991858i	$-0.540646\pi$
$797$	−7.19029	−0.254693	−0.127347	+	0.991858i	$0.540646\pi$
$798$	1.85086i	0.0655196i
$799$	− 3.80194i	− 0.134503i
$800$	− 1.00000i	− 0.0353553i
$801$	− 3.13169i	− 0.110653i
$802$	1.25428	0.0442900
$803$	−5.02608	−0.177367
$804$	− 3.58211i	− 0.126331i
$805$	0.384043	0.0135357
$806$	0	0
$807$	−17.8388	−0.627955
$808$	− 14.6896i	− 0.516779i
$809$	−18.9995	−0.667988	−0.333994	−	0.942575i	$-0.608396\pi$
$809$	−18.9995	−0.667988	−0.333994	+	0.942575i	$0.608396\pi$
$810$	−1.00000	−0.0351364
$811$	42.2016i	1.48190i	0.671560	+	0.740950i	$0.265624\pi$
$811$	42.2016i	1.48190i	−0.671560	+	0.740950i	$0.734376\pi$
$812$	2.80194i	0.0983287i
$813$	0.936017i	0.0328276i
$814$	0.215521i	0.00755400i
$815$	9.83877	0.344637
$816$	−1.80194	−0.0630804
$817$	13.2097i	0.462148i
$818$	−16.4276	−0.574377
$819$	0	0
$820$	−0.664874	−0.0232184
$821$	− 14.7423i	− 0.514509i	−0.966344	−	0.257254i	$-0.917182\pi$
$821$	− 14.7423i	− 0.514509i	0.966344	−	0.257254i	$-0.0828178\pi$
$822$	−10.9933	−0.383435
$823$	−40.6866	−1.41825	−0.709124	−	0.705084i	$-0.750909\pi$
$823$	−40.6866	−1.41825	−0.709124	+	0.705084i	$0.750909\pi$
$824$	12.4819i	0.434827i
$825$	1.35690i	0.0472411i
$826$	4.84415i	0.168550i
$827$	− 31.7120i	− 1.10273i	−0.834263	−	0.551367i	$-0.814106\pi$
$827$	− 31.7120i	− 1.10273i	0.834263	−	0.551367i	$-0.185894\pi$
$828$	−0.692021	−0.0240494
$829$	18.7162	0.650040	0.325020	−	0.945707i	$-0.394629\pi$
$829$	18.7162	0.650040	0.325020	+	0.945707i	$0.394629\pi$
$830$	− 3.85086i	− 0.133665i
$831$	10.9390	0.379470
$832$	0	0
$833$	−12.0586	−0.417806
$834$	− 11.4940i	− 0.398003i
$835$	8.16852	0.282683
$836$	−4.52542	−0.156515
$837$	1.75302i	0.0605932i
$838$	1.78687i	0.0617265i
$839$	29.8611i	1.03092i	0.856913	+	0.515460i	$0.172379\pi$
$839$	29.8611i	1.03092i	−0.856913	+	0.515460i	$0.827621\pi$
$840$	− 0.554958i	− 0.0191479i
$841$	−3.50843	−0.120980
$842$	−0.445042	−0.0153372
$843$	19.1618i	0.659968i
$844$	17.9323	0.617255
$845$	0	0
$846$	−2.10992	−0.0725404
$847$	− 5.08277i	− 0.174646i
$848$	1.15883	0.0397945
$849$	−24.0411	−0.825090
$850$	1.80194i	0.0618060i
$851$	0.109916i	0.00376788i
$852$	− 4.96615i	− 0.170137i
$853$	− 38.4413i	− 1.31621i	−0.752928	−	0.658103i	$-0.771359\pi$
$853$	− 38.4413i	− 1.31621i	0.752928	−	0.658103i	$-0.228641\pi$
$854$	−3.08815	−0.105674
$855$	−3.33513	−0.114059
$856$	− 12.2349i	− 0.418180i
$857$	50.1866	1.71434	0.857170	−	0.515033i	$-0.172220\pi$
$857$	50.1866	1.71434	0.857170	+	0.515033i	$0.172220\pi$
$858$	0	0
$859$	25.2916	0.862938	0.431469	−	0.902128i	$-0.357995\pi$
$859$	25.2916	0.862938	0.431469	+	0.902128i	$0.357995\pi$
$860$	− 3.96077i	− 0.135061i
$861$	−0.368977	−0.0125747
$862$	14.4547	0.492330
$863$	− 14.9347i	− 0.508383i	−0.967154	−	0.254191i	$-0.918191\pi$
$863$	− 14.9347i	− 0.508383i	0.967154	−	0.254191i	$-0.0818093\pi$
$864$	1.00000i	0.0340207i
$865$	16.0043i	0.544163i
$866$	− 5.69740i	− 0.193605i
$867$	−13.7530	−0.467077
$868$	−0.972853	−0.0330208
$869$	− 4.41896i	− 0.149903i
$870$	−5.04892	−0.171174
$871$	0	0
$872$	10.1317	0.343102
$873$	15.8170i	0.535324i
$874$	−2.30798	−0.0780685
$875$	−0.554958	−0.0187610
$876$	3.70410i	0.125150i
$877$	24.9627i	0.842930i	0.906845	+	0.421465i	$0.138484\pi$
$877$	24.9627i	0.842930i	−0.906845	+	0.421465i	$0.861516\pi$
$878$	3.77479i	0.127393i
$879$	9.50604i	0.320631i
$880$	1.35690	0.0457410
$881$	38.1148	1.28412	0.642060	−	0.766654i	$-0.278080\pi$
$881$	38.1148	1.28412	0.642060	+	0.766654i	$0.278080\pi$
$882$	6.69202i	0.225332i
$883$	−30.5700	−1.02876	−0.514382	−	0.857561i	$-0.671978\pi$
$883$	−30.5700	−1.02876	−0.514382	+	0.857561i	$0.671978\pi$
$884$	0	0
$885$	−8.72886	−0.293417
$886$	− 7.98685i	− 0.268324i
$887$	2.77884	0.0933043	0.0466521	−	0.998911i	$-0.485145\pi$
$887$	2.77884	0.0933043	0.0466521	+	0.998911i	$0.485145\pi$
$888$	0.158834	0.00533011
$889$	− 7.68532i	− 0.257757i
$890$	3.13169i	0.104974i
$891$	− 1.35690i	− 0.0454577i
$892$	− 9.95539i	− 0.333331i
$893$	−7.03684	−0.235479
$894$	−15.0911	−0.504723
$895$	14.4819i	0.484076i
$896$	−0.554958	−0.0185398
$897$	0	0
$898$	−19.4131	−0.647824
$899$	8.85086i	0.295193i
$900$	1.00000	0.0333333
$901$	−2.08815	−0.0695662
$902$	− 0.902165i	− 0.0300388i
$903$	− 2.19806i	− 0.0731469i
$904$	5.56465i	0.185077i
$905$	− 20.5502i	− 0.683111i
$906$	−11.7168	−0.389264
$907$	−4.57540	−0.151924	−0.0759618	−	0.997111i	$-0.524203\pi$
$907$	−4.57540	−0.151924	−0.0759618	+	0.997111i	$0.524203\pi$
$908$	15.6353i	0.518877i
$909$	14.6896	0.487224
$910$	0	0
$911$	27.6896	0.917398	0.458699	−	0.888592i	$-0.348316\pi$
$911$	27.6896	0.917398	0.458699	+	0.888592i	$0.348316\pi$
$912$	3.33513i	0.110437i
$913$	5.22521	0.172929
$914$	−2.64848	−0.0876040
$915$	− 5.56465i	− 0.183962i
$916$	13.5985i	0.449307i
$917$	− 10.6243i	− 0.350846i
$918$	− 1.80194i	− 0.0594728i
$919$	8.27652	0.273017	0.136509	−	0.990639i	$-0.456412\pi$
$919$	8.27652	0.273017	0.136509	+	0.990639i	$0.456412\pi$
$920$	0.692021	0.0228153
$921$	15.5894i	0.513688i
$922$	22.6649	0.746428
$923$	0	0
$924$	0.753020	0.0247726
$925$	− 0.158834i	− 0.00522242i
$926$	0.685317	0.0225209
$927$	−12.4819	−0.409959
$928$	5.04892i	0.165739i
$929$	− 19.9463i	− 0.654417i	−0.944952	−	0.327208i	$-0.893892\pi$
$929$	− 19.9463i	− 0.654417i	0.944952	−	0.327208i	$-0.106108\pi$
$930$	− 1.75302i	− 0.0574838i
$931$	22.3187i	0.731467i
$932$	23.0737	0.755803
$933$	−4.97285	−0.162804
$934$	34.4553i	1.12741i
$935$	−2.44504	−0.0799614
$936$	0	0
$937$	−12.2510	−0.400224	−0.200112	−	0.979773i	$-0.564131\pi$
$937$	−12.2510	−0.400224	−0.200112	+	0.979773i	$0.564131\pi$
$938$	1.98792i	0.0649078i
$939$	−28.1444	−0.918456
$940$	2.10992	0.0688179
$941$	− 42.5555i	− 1.38727i	−0.720326	−	0.693636i	$-0.756008\pi$
$941$	− 42.5555i	− 1.38727i	0.720326	−	0.693636i	$-0.243992\pi$
$942$	− 2.88040i	− 0.0938483i
$943$	− 0.460107i	− 0.0149832i
$944$	8.72886i	0.284100i
$945$	0.554958	0.0180528
$946$	5.37435	0.174735
$947$	6.78554i	0.220501i	0.993904	+	0.110250i	$0.0351652\pi$
$947$	6.78554i	0.220501i	−0.993904	+	0.110250i	$0.964835\pi$
$948$	−3.25667	−0.105772
$949$	0	0
$950$	3.33513	0.108206
$951$	− 4.43535i	− 0.143826i
$952$	1.00000	0.0324102
$953$	−0.862674	−0.0279447	−0.0139724	−	0.999902i	$-0.504448\pi$
$953$	−0.862674	−0.0279447	−0.0139724	+	0.999902i	$0.504448\pi$
$954$	1.15883i	0.0375186i
$955$	− 14.4101i	− 0.466301i
$956$	− 16.6431i	− 0.538276i
$957$	− 6.85086i	− 0.221457i
$958$	39.9724	1.29145
$959$	6.10082	0.197006
$960$	− 1.00000i	− 0.0322749i
$961$	27.9269	0.900868
$962$	0	0
$963$	12.2349	0.394264
$964$	22.2054i	0.715186i
$965$	−7.43727	−0.239414
$966$	0.384043	0.0123564
$967$	− 2.02715i	− 0.0651887i	−0.999469	−	0.0325943i	$-0.989623\pi$
$967$	− 2.02715i	− 0.0651887i	0.999469	−	0.0325943i	$-0.0103769\pi$
$968$	− 9.15883i	− 0.294376i
$969$	− 6.00969i	− 0.193059i
$970$	− 15.8170i	− 0.507853i
$971$	9.07905	0.291361	0.145680	−	0.989332i	$-0.453463\pi$
$971$	9.07905	0.291361	0.145680	+	0.989332i	$0.453463\pi$
$972$	−1.00000	−0.0320750
$973$	6.37867i	0.204491i
$974$	−10.8442	−0.347469
$975$	0	0
$976$	−5.56465	−0.178120
$977$	− 48.7593i	− 1.55995i	−0.625813	−	0.779973i	$-0.715233\pi$
$977$	− 48.7593i	− 1.55995i	0.625813	−	0.779973i	$-0.284767\pi$
$978$	9.83877	0.314609
$979$	−4.24937	−0.135811
$980$	− 6.69202i	− 0.213769i
$981$	10.1317i	0.323480i
$982$	18.0543i	0.576136i
$983$	− 49.7717i	− 1.58747i	−0.608264	−	0.793735i	$-0.708134\pi$
$983$	− 49.7717i	− 1.58747i	0.608264	−	0.793735i	$-0.291866\pi$
$984$	−0.664874	−0.0211954
$985$	−1.89977	−0.0605318
$986$	− 9.09783i	− 0.289734i
$987$	1.17092	0.0372706
$988$	0	0
$989$	2.74094	0.0871568
$990$	1.35690i	0.0431250i
$991$	−44.8273	−1.42399	−0.711993	−	0.702187i	$-0.752207\pi$
$991$	−44.8273	−1.42399	−0.711993	+	0.702187i	$0.752207\pi$
$992$	−1.75302	−0.0556585
$993$	− 11.7855i	− 0.374003i
$994$	2.75600i	0.0874151i
$995$	12.3394i	0.391186i
$996$	− 3.85086i	− 0.122019i
$997$	−14.7458	−0.467005	−0.233503	−	0.972356i	$-0.575019\pi$
$997$	−14.7458	−0.467005	−0.233503	+	0.972356i	$0.575019\pi$
$998$	−13.2271	−0.418697
$999$	0.158834i	0.00502527i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.x.1351.5		6
13.5	odd	4		5070.2.a.by.1.2	yes	3
13.8	odd	4		5070.2.a.bn.1.2	✓	3
13.12	even	2	inner	5070.2.b.x.1351.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bn.1.2	✓	3	13.8	odd	4
5070.2.a.by.1.2	yes	3	13.5	odd	4
5070.2.b.x.1351.2		6	13.12	even	2	inner
5070.2.b.x.1351.5		6	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -0.554958i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -0.554958i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.35690i q^{11} -1.00000 q^{12} +0.554958 q^{14} +1.00000i q^{15} +1.00000 q^{16} -1.80194 q^{17} +1.00000i q^{18} +3.33513i q^{19} -1.00000i q^{20} -0.554958i q^{21} +1.35690 q^{22} +0.692021 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +0.554958i q^{28} +5.04892 q^{29} -1.00000 q^{30} +1.75302i q^{31} +1.00000i q^{32} -1.35690i q^{33} -1.80194i q^{34} +0.554958 q^{35} -1.00000 q^{36} +0.158834i q^{37} -3.33513 q^{38} +1.00000 q^{40} -0.664874i q^{41} +0.554958 q^{42} +3.96077 q^{43} +1.35690i q^{44} +1.00000i q^{45} +0.692021i q^{46} +2.10992i q^{47} +1.00000 q^{48} +6.69202 q^{49} -1.00000i q^{50} -1.80194 q^{51} +1.15883 q^{53} +1.00000i q^{54} +1.35690 q^{55} -0.554958 q^{56} +3.33513i q^{57} +5.04892i q^{58} +8.72886i q^{59} -1.00000i q^{60} -5.56465 q^{61} -1.75302 q^{62} -0.554958i q^{63} -1.00000 q^{64} +1.35690 q^{66} +3.58211i q^{67} +1.80194 q^{68} +0.692021 q^{69} +0.554958i q^{70} +4.96615i q^{71} -1.00000i q^{72} -3.70410i q^{73} -0.158834 q^{74} -1.00000 q^{75} -3.33513i q^{76} -0.753020 q^{77} +3.25667 q^{79} +1.00000i q^{80} +1.00000 q^{81} +0.664874 q^{82} +3.85086i q^{83} +0.554958i q^{84} -1.80194i q^{85} +3.96077i q^{86} +5.04892 q^{87} -1.35690 q^{88} -3.13169i q^{89} -1.00000 q^{90} -0.692021 q^{92} +1.75302i q^{93} -2.10992 q^{94} -3.33513 q^{95} +1.00000i q^{96} +15.8170i q^{97} +6.69202i q^{98} -1.35690i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{10} - 6 q^{12} + 4 q^{14} + 6 q^{16} - 2 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} + 12 q^{29} - 6 q^{30} + 4 q^{35} - 6 q^{36} - 18 q^{38} + 6 q^{40} + 4 q^{42} - 2 q^{43} + 6 q^{48} + 30 q^{49} - 2 q^{51} - 10 q^{53} - 4 q^{56} + 10 q^{61} - 20 q^{62} - 6 q^{64} + 2 q^{68} - 6 q^{69} + 16 q^{74} - 6 q^{75} - 14 q^{77} - 34 q^{79} + 6 q^{81} + 6 q^{82} + 12 q^{87} - 6 q^{90} + 6 q^{92} - 14 q^{94} - 18 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 - 6 * q^12 + 4 * q^14 + 6 * q^16 - 2 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 + 12 * q^29 - 6 * q^30 + 4 * q^35 - 6 * q^36 - 18 * q^38 + 6 * q^40 + 4 * q^42 - 2 * q^43 + 6 * q^48 + 30 * q^49 - 2 * q^51 - 10 * q^53 - 4 * q^56 + 10 * q^61 - 20 * q^62 - 6 * q^64 + 2 * q^68 - 6 * q^69 + 16 * q^74 - 6 * q^75 - 14 * q^77 - 34 * q^79 + 6 * q^81 + 6 * q^82 + 12 * q^87 - 6 * q^90 + 6 * q^92 - 14 * q^94 - 18 * q^95

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