LMFDB - Embedded newform 5070.2.b.t.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:	$6$
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			1351.4
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.t.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -3.04892i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -3.04892i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.13706i q^{11} +1.00000 q^{12} +3.04892 q^{14} -1.00000i q^{15} +1.00000 q^{16} -2.69202 q^{17} +1.00000i q^{18} +1.55496i q^{19} -1.00000i q^{20} +3.04892i q^{21} +1.13706 q^{22} -5.40581 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +3.04892i q^{28} -6.15883 q^{29} +1.00000 q^{30} -0.246980i q^{31} +1.00000i q^{32} +1.13706i q^{33} -2.69202i q^{34} +3.04892 q^{35} -1.00000 q^{36} +0.554958i q^{37} -1.55496 q^{38} +1.00000 q^{40} +8.26875i q^{41} -3.04892 q^{42} -2.03923 q^{43} +1.13706i q^{44} +1.00000i q^{45} -5.40581i q^{46} +6.70171i q^{47} -1.00000 q^{48} -2.29590 q^{49} -1.00000i q^{50} +2.69202 q^{51} +5.77479 q^{53} -1.00000i q^{54} +1.13706 q^{55} -3.04892 q^{56} -1.55496i q^{57} -6.15883i q^{58} -1.36898i q^{59} +1.00000i q^{60} +10.9487 q^{61} +0.246980 q^{62} -3.04892i q^{63} -1.00000 q^{64} -1.13706 q^{66} -2.47219i q^{67} +2.69202 q^{68} +5.40581 q^{69} +3.04892i q^{70} -8.57002i q^{71} -1.00000i q^{72} +14.8334i q^{73} -0.554958 q^{74} +1.00000 q^{75} -1.55496i q^{76} -3.46681 q^{77} +5.33513 q^{79} +1.00000i q^{80} +1.00000 q^{81} -8.26875 q^{82} -5.45473i q^{83} -3.04892i q^{84} -2.69202i q^{85} -2.03923i q^{86} +6.15883 q^{87} -1.13706 q^{88} -16.8877i q^{89} -1.00000 q^{90} +5.40581 q^{92} +0.246980i q^{93} -6.70171 q^{94} -1.55496 q^{95} -1.00000i q^{96} +1.93900i q^{97} -2.29590i q^{98} -1.13706i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 6 q^{16} - 6 q^{17} - 4 q^{22} - 6 q^{23} - 6 q^{25} - 6 q^{27} - 20 q^{29} + 6 q^{30} - 6 q^{36} - 10 q^{38} + 6 q^{40} - 38 q^{43} - 6 q^{48} + 14 q^{49} + 6 q^{51} + 38 q^{53} - 4 q^{55} + 2 q^{61} - 8 q^{62} - 6 q^{64} + 4 q^{66} + 6 q^{68} + 6 q^{69} - 4 q^{74} + 6 q^{75} - 14 q^{77} + 30 q^{79} + 6 q^{81} - 34 q^{82} + 20 q^{87} + 4 q^{88} - 6 q^{90} + 6 q^{92} + 14 q^{94} - 10 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 + 6 * q^12 + 6 * q^16 - 6 * q^17 - 4 * q^22 - 6 * q^23 - 6 * q^25 - 6 * q^27 - 20 * q^29 + 6 * q^30 - 6 * q^36 - 10 * q^38 + 6 * q^40 - 38 * q^43 - 6 * q^48 + 14 * q^49 + 6 * q^51 + 38 * q^53 - 4 * q^55 + 2 * q^61 - 8 * q^62 - 6 * q^64 + 4 * q^66 + 6 * q^68 + 6 * q^69 - 4 * q^74 + 6 * q^75 - 14 * q^77 + 30 * q^79 + 6 * q^81 - 34 * q^82 + 20 * q^87 + 4 * q^88 - 6 * q^90 + 6 * q^92 + 14 * q^94 - 10 * 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
$7$	− 3.04892i	− 1.15238i	−0.817315	−	0.576191i	$-0.804538\pi$
$7$	− 3.04892i	− 1.15238i	0.817315	−	0.576191i	$-0.195462\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	− 1.13706i	− 0.342837i	−0.985198	−	0.171419i	$-0.945165\pi$
$11$	− 1.13706i	− 0.342837i	0.985198	−	0.171419i	$-0.0548351\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	3.04892	0.814857
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	−2.69202	−0.652911	−0.326456	−	0.945213i	$-0.605854\pi$
$17$	−2.69202	−0.652911	−0.326456	+	0.945213i	$0.605854\pi$
$18$	1.00000i	0.235702i
$19$	1.55496i	0.356732i	0.983964	+	0.178366i	$0.0570811\pi$
$19$	1.55496i	0.356732i	−0.983964	+	0.178366i	$0.942919\pi$
$20$	− 1.00000i	− 0.223607i
$21$	3.04892i	0.665328i
$22$	1.13706	0.242423
$23$	−5.40581	−1.12719	−0.563595	−	0.826051i	$-0.690582\pi$
$23$	−5.40581	−1.12719	−0.563595	+	0.826051i	$0.690582\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	3.04892i	0.576191i
$29$	−6.15883	−1.14367	−0.571833	−	0.820370i	$-0.693768\pi$
$29$	−6.15883	−1.14367	−0.571833	+	0.820370i	$0.693768\pi$
$30$	1.00000	0.182574
$31$	− 0.246980i	− 0.0443588i	−0.999754	−	0.0221794i	$-0.992939\pi$
$31$	− 0.246980i	− 0.0443588i	0.999754	−	0.0221794i	$-0.00706051\pi$
$32$	1.00000i	0.176777i
$33$	1.13706i	0.197937i
$34$	− 2.69202i	− 0.461678i
$35$	3.04892	0.515361
$36$	−1.00000	−0.166667
$37$	0.554958i	0.0912346i	0.998959	+	0.0456173i	$0.0145255\pi$
$37$	0.554958i	0.0912346i	−0.998959	+	0.0456173i	$0.985475\pi$
$38$	−1.55496	−0.252248
$39$	0	0
$40$	1.00000	0.158114
$41$	8.26875i	1.29136i	0.763607	+	0.645681i	$0.223426\pi$
$41$	8.26875i	1.29136i	−0.763607	+	0.645681i	$0.776574\pi$
$42$	−3.04892	−0.470458
$43$	−2.03923	−0.310979	−0.155490	−	0.987838i	$-0.549696\pi$
$43$	−2.03923	−0.310979	−0.155490	+	0.987838i	$0.549696\pi$
$44$	1.13706i	0.171419i
$45$	1.00000i	0.149071i
$46$	− 5.40581i	− 0.797044i
$47$	6.70171i	0.977545i	0.872411	+	0.488772i	$0.162555\pi$
$47$	6.70171i	0.977545i	−0.872411	+	0.488772i	$0.837445\pi$
$48$	−1.00000	−0.144338
$49$	−2.29590	−0.327985
$50$	− 1.00000i	− 0.141421i
$51$	2.69202	0.376958
$52$	0	0
$53$	5.77479	0.793229	0.396614	−	0.917985i	$-0.370185\pi$
$53$	5.77479	0.793229	0.396614	+	0.917985i	$0.370185\pi$
$54$	− 1.00000i	− 0.136083i
$55$	1.13706	0.153322
$56$	−3.04892	−0.407429
$57$	− 1.55496i	− 0.205959i
$58$	− 6.15883i	− 0.808694i
$59$	− 1.36898i	− 0.178226i	−0.996022	−	0.0891128i	$-0.971597\pi$
$59$	− 1.36898i	− 0.178226i	0.996022	−	0.0891128i	$-0.0284032\pi$
$60$	1.00000i	0.129099i
$61$	10.9487	1.40184	0.700918	−	0.713242i	$-0.252774\pi$
$61$	10.9487	1.40184	0.700918	+	0.713242i	$0.252774\pi$
$62$	0.246980	0.0313664
$63$	− 3.04892i	− 0.384127i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.13706	−0.139963
$67$	− 2.47219i	− 0.302026i	−0.988532	−	0.151013i	$-0.951746\pi$
$67$	− 2.47219i	− 0.302026i	0.988532	−	0.151013i	$-0.0482535\pi$
$68$	2.69202	0.326456
$69$	5.40581	0.650783
$70$	3.04892i	0.364415i
$71$	− 8.57002i	− 1.01707i	−0.861040	−	0.508537i	$-0.830186\pi$
$71$	− 8.57002i	− 1.01707i	0.861040	−	0.508537i	$-0.169814\pi$
$72$	− 1.00000i	− 0.117851i
$73$	14.8334i	1.73612i	0.496461	+	0.868059i	$0.334632\pi$
$73$	14.8334i	1.73612i	−0.496461	+	0.868059i	$0.665368\pi$
$74$	−0.554958	−0.0645126
$75$	1.00000	0.115470
$76$	− 1.55496i	− 0.178366i
$77$	−3.46681	−0.395080
$78$	0	0
$79$	5.33513	0.600249	0.300124	−	0.953900i	$-0.402972\pi$
$79$	5.33513	0.600249	0.300124	+	0.953900i	$0.402972\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	−8.26875	−0.913131
$83$	− 5.45473i	− 0.598734i	−0.954138	−	0.299367i	$-0.903224\pi$
$83$	− 5.45473i	− 0.598734i	0.954138	−	0.299367i	$-0.0967756\pi$
$84$	− 3.04892i	− 0.332664i
$85$	− 2.69202i	− 0.291991i
$86$	− 2.03923i	− 0.219896i
$87$	6.15883	0.660296
$88$	−1.13706	−0.121211
$89$	− 16.8877i	− 1.79009i	−0.445974	−	0.895046i	$-0.647143\pi$
$89$	− 16.8877i	− 1.79009i	0.445974	−	0.895046i	$-0.352857\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	5.40581	0.563595
$93$	0.246980i	0.0256106i
$94$	−6.70171	−0.691229
$95$	−1.55496	−0.159535
$96$	− 1.00000i	− 0.102062i
$97$	1.93900i	0.196876i	0.995143	+	0.0984379i	$0.0313846\pi$
$97$	1.93900i	0.196876i	−0.995143	+	0.0984379i	$0.968615\pi$
$98$	− 2.29590i	− 0.231921i
$99$	− 1.13706i	− 0.114279i
$100$	1.00000	0.100000
$101$	14.4940	1.44220	0.721101	−	0.692830i	$-0.243636\pi$
$101$	14.4940	1.44220	0.721101	+	0.692830i	$0.243636\pi$
$102$	2.69202i	0.266550i
$103$	16.1836	1.59462	0.797308	−	0.603572i	$-0.206257\pi$
$103$	16.1836	1.59462	0.797308	+	0.603572i	$0.206257\pi$
$104$	0	0
$105$	−3.04892	−0.297544
$106$	5.77479i	0.560897i
$107$	4.92931	0.476535	0.238267	−	0.971200i	$-0.423421\pi$
$107$	4.92931	0.476535	0.238267	+	0.971200i	$0.423421\pi$
$108$	1.00000	0.0962250
$109$	0.801938i	0.0768117i	0.999262	+	0.0384059i	$0.0122280\pi$
$109$	0.801938i	0.0768117i	−0.999262	+	0.0384059i	$0.987772\pi$
$110$	1.13706i	0.108415i
$111$	− 0.554958i	− 0.0526743i
$112$	− 3.04892i	− 0.288096i
$113$	−11.5211	−1.08381	−0.541907	−	0.840438i	$-0.682298\pi$
$113$	−11.5211	−1.08381	−0.541907	+	0.840438i	$0.682298\pi$
$114$	1.55496	0.145635
$115$	− 5.40581i	− 0.504095i
$116$	6.15883	0.571833
$117$	0	0
$118$	1.36898	0.126025
$119$	8.20775i	0.752403i
$120$	−1.00000	−0.0912871
$121$	9.70709	0.882462
$122$	10.9487i	0.991248i
$123$	− 8.26875i	− 0.745568i
$124$	0.246980i	0.0221794i
$125$	− 1.00000i	− 0.0894427i
$126$	3.04892	0.271619
$127$	−6.22521	−0.552398	−0.276199	−	0.961100i	$-0.589075\pi$
$127$	−6.22521	−0.552398	−0.276199	+	0.961100i	$0.589075\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	2.03923	0.179544
$130$	0	0
$131$	3.34481	0.292238	0.146119	−	0.989267i	$-0.453322\pi$
$131$	3.34481	0.292238	0.146119	+	0.989267i	$0.453322\pi$
$132$	− 1.13706i	− 0.0989687i
$133$	4.74094	0.411092
$134$	2.47219	0.213565
$135$	− 1.00000i	− 0.0860663i
$136$	2.69202i	0.230839i
$137$	9.53079i	0.814271i	0.913368	+	0.407135i	$0.133472\pi$
$137$	9.53079i	0.814271i	−0.913368	+	0.407135i	$0.866528\pi$
$138$	5.40581i	0.460173i
$139$	18.7017	1.58626	0.793129	−	0.609053i	$-0.208451\pi$
$139$	18.7017	1.58626	0.793129	+	0.609053i	$0.208451\pi$
$140$	−3.04892	−0.257681
$141$	− 6.70171i	− 0.564386i
$142$	8.57002	0.719180
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 6.15883i	− 0.511463i
$146$	−14.8334	−1.22762
$147$	2.29590	0.189362
$148$	− 0.554958i	− 0.0456173i
$149$	16.4209i	1.34525i	0.739983	+	0.672625i	$0.234833\pi$
$149$	16.4209i	1.34525i	−0.739983	+	0.672625i	$0.765167\pi$
$150$	1.00000i	0.0816497i
$151$	9.26636i	0.754085i	0.926196	+	0.377043i	$0.123059\pi$
$151$	9.26636i	0.754085i	−0.926196	+	0.377043i	$0.876941\pi$
$152$	1.55496	0.126124
$153$	−2.69202	−0.217637
$154$	− 3.46681i	− 0.279364i
$155$	0.246980	0.0198379
$156$	0	0
$157$	17.0761	1.36282	0.681409	−	0.731903i	$-0.261367\pi$
$157$	17.0761	1.36282	0.681409	+	0.731903i	$0.261367\pi$
$158$	5.33513i	0.424440i
$159$	−5.77479	−0.457971
$160$	−1.00000	−0.0790569
$161$	16.4819i	1.29895i
$162$	1.00000i	0.0785674i
$163$	9.61894i	0.753414i	0.926333	+	0.376707i	$0.122944\pi$
$163$	9.61894i	0.753414i	−0.926333	+	0.376707i	$0.877056\pi$
$164$	− 8.26875i	− 0.645681i
$165$	−1.13706	−0.0885203
$166$	5.45473	0.423369
$167$	− 21.8509i	− 1.69087i	−0.534078	−	0.845435i	$-0.679341\pi$
$167$	− 21.8509i	− 1.69087i	0.534078	−	0.845435i	$-0.320659\pi$
$168$	3.04892	0.235229
$169$	0	0
$170$	2.69202	0.206469
$171$	1.55496i	0.118911i
$172$	2.03923	0.155490
$173$	−16.6853	−1.26856	−0.634281	−	0.773103i	$-0.718704\pi$
$173$	−16.6853	−1.26856	−0.634281	+	0.773103i	$0.718704\pi$
$174$	6.15883i	0.466900i
$175$	3.04892i	0.230476i
$176$	− 1.13706i	− 0.0857094i
$177$	1.36898i	0.102899i
$178$	16.8877	1.26579
$179$	3.94438	0.294817	0.147408	−	0.989076i	$-0.452907\pi$
$179$	3.94438	0.294817	0.147408	+	0.989076i	$0.452907\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	20.5308	1.52604	0.763021	−	0.646374i	$-0.223715\pi$
$181$	20.5308	1.52604	0.763021	+	0.646374i	$0.223715\pi$
$182$	0	0
$183$	−10.9487	−0.809350
$184$	5.40581i	0.398522i
$185$	−0.554958	−0.0408013
$186$	−0.246980	−0.0181094
$187$	3.06100i	0.223842i
$188$	− 6.70171i	− 0.488772i
$189$	3.04892i	0.221776i
$190$	− 1.55496i	− 0.112809i
$191$	−6.66487	−0.482253	−0.241127	−	0.970494i	$-0.577517\pi$
$191$	−6.66487	−0.482253	−0.241127	+	0.970494i	$0.577517\pi$
$192$	1.00000	0.0721688
$193$	− 16.6558i	− 1.19891i	−0.800409	−	0.599455i	$-0.795384\pi$
$193$	− 16.6558i	− 1.19891i	0.800409	−	0.599455i	$-0.204616\pi$
$194$	−1.93900	−0.139212
$195$	0	0
$196$	2.29590	0.163993
$197$	19.4494i	1.38571i	0.721077	+	0.692855i	$0.243647\pi$
$197$	19.4494i	1.38571i	−0.721077	+	0.692855i	$0.756353\pi$
$198$	1.13706	0.0808076
$199$	21.6558	1.53514	0.767569	−	0.640967i	$-0.221466\pi$
$199$	21.6558	1.53514	0.767569	+	0.640967i	$0.221466\pi$
$200$	1.00000i	0.0707107i
$201$	2.47219i	0.174375i
$202$	14.4940i	1.01979i
$203$	18.7778i	1.31794i
$204$	−2.69202	−0.188479
$205$	−8.26875	−0.577515
$206$	16.1836i	1.12756i
$207$	−5.40581	−0.375730
$208$	0	0
$209$	1.76809	0.122301
$210$	− 3.04892i	− 0.210395i
$211$	12.1521	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$211$	12.1521	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$212$	−5.77479	−0.396614
$213$	8.57002i	0.587208i
$214$	4.92931i	0.336961i
$215$	− 2.03923i	− 0.139074i
$216$	1.00000i	0.0680414i
$217$	−0.753020	−0.0511184
$218$	−0.801938	−0.0543141
$219$	− 14.8334i	− 1.00235i
$220$	−1.13706	−0.0766608
$221$	0	0
$222$	0.554958	0.0372464
$223$	1.11960i	0.0749743i	0.999297	+	0.0374871i	$0.0119353\pi$
$223$	1.11960i	0.0749743i	−0.999297	+	0.0374871i	$0.988065\pi$
$224$	3.04892	0.203714
$225$	−1.00000	−0.0666667
$226$	− 11.5211i	− 0.766373i
$227$	4.06638i	0.269895i	0.990853	+	0.134947i	$0.0430866\pi$
$227$	4.06638i	0.269895i	−0.990853	+	0.134947i	$0.956913\pi$
$228$	1.55496i	0.102980i
$229$	16.5187i	1.09159i	0.837920	+	0.545794i	$0.183772\pi$
$229$	16.5187i	1.09159i	−0.837920	+	0.545794i	$0.816228\pi$
$230$	5.40581	0.356449
$231$	3.46681	0.228099
$232$	6.15883i	0.404347i
$233$	6.16421	0.403831	0.201915	−	0.979403i	$-0.435283\pi$
$233$	6.16421	0.403831	0.201915	+	0.979403i	$0.435283\pi$
$234$	0	0
$235$	−6.70171	−0.437171
$236$	1.36898i	0.0891128i
$237$	−5.33513	−0.346554
$238$	−8.20775	−0.532029
$239$	18.1793i	1.17592i	0.808890	+	0.587960i	$0.200069\pi$
$239$	18.1793i	1.17592i	−0.808890	+	0.587960i	$0.799931\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	9.21744i	0.593747i	0.954917	+	0.296874i	$0.0959441\pi$
$241$	9.21744i	0.593747i	−0.954917	+	0.296874i	$0.904056\pi$
$242$	9.70709i	0.623995i
$243$	−1.00000	−0.0641500
$244$	−10.9487	−0.700918
$245$	− 2.29590i	− 0.146679i
$246$	8.26875	0.527196
$247$	0	0
$248$	−0.246980	−0.0156832
$249$	5.45473i	0.345680i
$250$	1.00000	0.0632456
$251$	−17.2054	−1.08599	−0.542996	−	0.839735i	$-0.682710\pi$
$251$	−17.2054	−1.08599	−0.542996	+	0.839735i	$0.682710\pi$
$252$	3.04892i	0.192064i
$253$	6.14675i	0.386443i
$254$	− 6.22521i	− 0.390604i
$255$	2.69202i	0.168581i
$256$	1.00000	0.0625000
$257$	19.5157	1.21736	0.608679	−	0.793417i	$-0.291700\pi$
$257$	19.5157	1.21736	0.608679	+	0.793417i	$0.291700\pi$
$258$	2.03923i	0.126957i
$259$	1.69202	0.105137
$260$	0	0
$261$	−6.15883	−0.381222
$262$	3.34481i	0.206643i
$263$	12.8834	0.794423	0.397212	−	0.917727i	$-0.369978\pi$
$263$	12.8834	0.794423	0.397212	+	0.917727i	$0.369978\pi$
$264$	1.13706	0.0699814
$265$	5.77479i	0.354743i
$266$	4.74094i	0.290686i
$267$	16.8877i	1.03351i
$268$	2.47219i	0.151013i
$269$	−30.9138	−1.88485	−0.942423	−	0.334423i	$-0.891459\pi$
$269$	−30.9138	−1.88485	−0.942423	+	0.334423i	$0.891459\pi$
$270$	1.00000	0.0608581
$271$	− 7.24027i	− 0.439815i	−0.975521	−	0.219908i	$-0.929424\pi$
$271$	− 7.24027i	− 0.439815i	0.975521	−	0.219908i	$-0.0705756\pi$
$272$	−2.69202	−0.163228
$273$	0	0
$274$	−9.53079	−0.575776
$275$	1.13706i	0.0685675i
$276$	−5.40581	−0.325392
$277$	−12.2446	−0.735706	−0.367853	−	0.929884i	$-0.619907\pi$
$277$	−12.2446	−0.735706	−0.367853	+	0.929884i	$0.619907\pi$
$278$	18.7017i	1.12165i
$279$	− 0.246980i	− 0.0147863i
$280$	− 3.04892i	− 0.182208i
$281$	2.74632i	0.163831i	0.996639	+	0.0819157i	$0.0261038\pi$
$281$	2.74632i	0.163831i	−0.996639	+	0.0819157i	$0.973896\pi$
$282$	6.70171	0.399081
$283$	−14.9312	−0.887570	−0.443785	−	0.896133i	$-0.646365\pi$
$283$	−14.9312	−0.887570	−0.443785	+	0.896133i	$0.646365\pi$
$284$	8.57002i	0.508537i
$285$	1.55496	0.0921078
$286$	0	0
$287$	25.2107	1.48814
$288$	1.00000i	0.0589256i
$289$	−9.75302	−0.573707
$290$	6.15883	0.361659
$291$	− 1.93900i	− 0.113666i
$292$	− 14.8334i	− 0.868059i
$293$	− 18.5810i	− 1.08552i	−0.839889	−	0.542758i	$-0.817380\pi$
$293$	− 18.5810i	− 1.08552i	0.839889	−	0.542758i	$-0.182620\pi$
$294$	2.29590i	0.133899i
$295$	1.36898	0.0797049
$296$	0.554958	0.0322563
$297$	1.13706i	0.0659791i
$298$	−16.4209	−0.951236
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	6.21744i	0.358367i
$302$	−9.26636	−0.533219
$303$	−14.4940	−0.832656
$304$	1.55496i	0.0891830i
$305$	10.9487i	0.626920i
$306$	− 2.69202i	− 0.153893i
$307$	− 32.4359i	− 1.85122i	−0.378482	−	0.925609i	$-0.623554\pi$
$307$	− 32.4359i	− 1.85122i	0.378482	−	0.925609i	$-0.376446\pi$
$308$	3.46681	0.197540
$309$	−16.1836	−0.920652
$310$	0.246980i	0.0140275i
$311$	21.2771	1.20651	0.603257	−	0.797547i	$-0.293869\pi$
$311$	21.2771	1.20651	0.603257	+	0.797547i	$0.293869\pi$
$312$	0	0
$313$	−18.7060	−1.05733	−0.528663	−	0.848832i	$-0.677307\pi$
$313$	−18.7060	−1.05733	−0.528663	+	0.848832i	$0.677307\pi$
$314$	17.0761i	0.963658i
$315$	3.04892	0.171787
$316$	−5.33513	−0.300124
$317$	17.4776i	0.981638i	0.871262	+	0.490819i	$0.163302\pi$
$317$	17.4776i	0.981638i	−0.871262	+	0.490819i	$0.836698\pi$
$318$	− 5.77479i	− 0.323834i
$319$	7.00298i	0.392092i
$320$	− 1.00000i	− 0.0559017i
$321$	−4.92931	−0.275127
$322$	−16.4819	−0.918499
$323$	− 4.18598i	− 0.232914i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−9.61894	−0.532744
$327$	− 0.801938i	− 0.0443473i
$328$	8.26875	0.456565
$329$	20.4330	1.12651
$330$	− 1.13706i	− 0.0625933i
$331$	− 23.1890i	− 1.27458i	−0.770624	−	0.637290i	$-0.780055\pi$
$331$	− 23.1890i	− 1.27458i	0.770624	−	0.637290i	$-0.219945\pi$
$332$	5.45473i	0.299367i
$333$	0.554958i	0.0304115i
$334$	21.8509	1.19563
$335$	2.47219	0.135070
$336$	3.04892i	0.166332i
$337$	26.1400	1.42394	0.711970	−	0.702210i	$-0.247803\pi$
$337$	26.1400	1.42394	0.711970	+	0.702210i	$0.247803\pi$
$338$	0	0
$339$	11.5211	0.625741
$340$	2.69202i	0.145995i
$341$	−0.280831	−0.0152079
$342$	−1.55496	−0.0840825
$343$	− 14.3424i	− 0.774418i
$344$	2.03923i	0.109948i
$345$	5.40581i	0.291039i
$346$	− 16.6853i	− 0.897008i
$347$	15.2959	0.821127	0.410563	−	0.911832i	$-0.365332\pi$
$347$	15.2959	0.821127	0.410563	+	0.911832i	$0.365332\pi$
$348$	−6.15883	−0.330148
$349$	19.2107i	1.02833i	0.857692	+	0.514164i	$0.171898\pi$
$349$	19.2107i	1.02833i	−0.857692	+	0.514164i	$0.828102\pi$
$350$	−3.04892	−0.162971
$351$	0	0
$352$	1.13706	0.0606057
$353$	− 5.35450i	− 0.284991i	−0.989795	−	0.142496i	$-0.954487\pi$
$353$	− 5.35450i	− 0.284991i	0.989795	−	0.142496i	$-0.0455127\pi$
$354$	−1.36898	−0.0727603
$355$	8.57002	0.454850
$356$	16.8877i	0.895046i
$357$	− 8.20775i	− 0.434400i
$358$	3.94438i	0.208467i
$359$	− 8.49934i	− 0.448578i	−0.974523	−	0.224289i	$-0.927994\pi$
$359$	− 8.49934i	− 0.448578i	0.974523	−	0.224289i	$-0.0720059\pi$
$360$	1.00000	0.0527046
$361$	16.5821	0.872742
$362$	20.5308i	1.07907i
$363$	−9.70709	−0.509490
$364$	0	0
$365$	−14.8334	−0.776415
$366$	− 10.9487i	− 0.572297i
$367$	4.76032	0.248486	0.124243	−	0.992252i	$-0.460350\pi$
$367$	4.76032	0.248486	0.124243	+	0.992252i	$0.460350\pi$
$368$	−5.40581	−0.281797
$369$	8.26875i	0.430454i
$370$	− 0.554958i	− 0.0288509i
$371$	− 17.6069i	− 0.914103i
$372$	− 0.246980i	− 0.0128053i
$373$	−20.2064	−1.04625	−0.523124	−	0.852256i	$-0.675234\pi$
$373$	−20.2064	−1.04625	−0.523124	+	0.852256i	$0.675234\pi$
$374$	−3.06100	−0.158280
$375$	1.00000i	0.0516398i
$376$	6.70171	0.345614
$377$	0	0
$378$	−3.04892	−0.156819
$379$	24.4088i	1.25380i	0.779101	+	0.626898i	$0.215676\pi$
$379$	24.4088i	1.25380i	−0.779101	+	0.626898i	$0.784324\pi$
$380$	1.55496	0.0797677
$381$	6.22521	0.318927
$382$	− 6.66487i	− 0.341005i
$383$	32.0237i	1.63633i	0.574981	+	0.818167i	$0.305010\pi$
$383$	32.0237i	1.63633i	−0.574981	+	0.818167i	$0.694990\pi$
$384$	1.00000i	0.0510310i
$385$	− 3.46681i	− 0.176685i
$386$	16.6558	0.847757
$387$	−2.03923	−0.103660
$388$	− 1.93900i	− 0.0984379i
$389$	−21.1879	−1.07427	−0.537135	−	0.843497i	$-0.680493\pi$
$389$	−21.1879	−1.07427	−0.537135	+	0.843497i	$0.680493\pi$
$390$	0	0
$391$	14.5526	0.735955
$392$	2.29590i	0.115960i
$393$	−3.34481	−0.168724
$394$	−19.4494	−0.979844
$395$	5.33513i	0.268439i
$396$	1.13706i	0.0571396i
$397$	− 10.9608i	− 0.550105i	−0.961429	−	0.275053i	$-0.911305\pi$
$397$	− 10.9608i	− 0.550105i	0.961429	−	0.275053i	$-0.0886953\pi$
$398$	21.6558i	1.08551i
$399$	−4.74094	−0.237344
$400$	−1.00000	−0.0500000
$401$	29.2301i	1.45968i	0.683617	+	0.729841i	$0.260406\pi$
$401$	29.2301i	1.45968i	−0.683617	+	0.729841i	$0.739594\pi$
$402$	−2.47219	−0.123302
$403$	0	0
$404$	−14.4940	−0.721101
$405$	1.00000i	0.0496904i
$406$	−18.7778	−0.931925
$407$	0.631023	0.0312786
$408$	− 2.69202i	− 0.133275i
$409$	36.3163i	1.79573i	0.440274	+	0.897864i	$0.354881\pi$
$409$	36.3163i	1.79573i	−0.440274	+	0.897864i	$0.645119\pi$
$410$	− 8.26875i	− 0.408364i
$411$	− 9.53079i	− 0.470119i
$412$	−16.1836	−0.797308
$413$	−4.17390	−0.205384
$414$	− 5.40581i	− 0.265681i
$415$	5.45473	0.267762
$416$	0	0
$417$	−18.7017	−0.915827
$418$	1.76809i	0.0864799i
$419$	11.0382	0.539250	0.269625	−	0.962965i	$-0.413100\pi$
$419$	11.0382	0.539250	0.269625	+	0.962965i	$0.413100\pi$
$420$	3.04892	0.148772
$421$	4.58642i	0.223528i	0.993735	+	0.111764i	$0.0356501\pi$
$421$	4.58642i	0.223528i	−0.993735	+	0.111764i	$0.964350\pi$
$422$	12.1521i	0.591556i
$423$	6.70171i	0.325848i
$424$	− 5.77479i	− 0.280449i
$425$	2.69202	0.130582
$426$	−8.57002	−0.415219
$427$	− 33.3817i	− 1.61545i
$428$	−4.92931	−0.238267
$429$	0	0
$430$	2.03923	0.0983403
$431$	19.3642i	0.932740i	0.884590	+	0.466370i	$0.154439\pi$
$431$	19.3642i	0.932740i	−0.884590	+	0.466370i	$0.845561\pi$
$432$	−1.00000	−0.0481125
$433$	−4.41119	−0.211988	−0.105994	−	0.994367i	$-0.533802\pi$
$433$	−4.41119	−0.211988	−0.105994	+	0.994367i	$0.533802\pi$
$434$	− 0.753020i	− 0.0361461i
$435$	6.15883i	0.295293i
$436$	− 0.801938i	− 0.0384059i
$437$	− 8.40581i	− 0.402105i
$438$	14.8334	0.708767
$439$	−39.5260	−1.88647	−0.943237	−	0.332121i	$-0.892236\pi$
$439$	−39.5260	−1.88647	−0.943237	+	0.332121i	$0.892236\pi$
$440$	− 1.13706i	− 0.0542074i
$441$	−2.29590	−0.109328
$442$	0	0
$443$	−7.16182	−0.340268	−0.170134	−	0.985421i	$-0.554420\pi$
$443$	−7.16182	−0.340268	−0.170134	+	0.985421i	$0.554420\pi$
$444$	0.554958i	0.0263371i
$445$	16.8877	0.800553
$446$	−1.11960	−0.0530148
$447$	− 16.4209i	− 0.776681i
$448$	3.04892i	0.144048i
$449$	4.46011i	0.210485i	0.994447	+	0.105243i	$0.0335620\pi$
$449$	4.46011i	0.210485i	−0.994447	+	0.105243i	$0.966438\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	9.40209	0.442727
$452$	11.5211	0.541907
$453$	− 9.26636i	− 0.435371i
$454$	−4.06638	−0.190844
$455$	0	0
$456$	−1.55496	−0.0728176
$457$	− 8.04593i	− 0.376373i	−0.982133	−	0.188186i	$-0.939739\pi$
$457$	− 8.04593i	− 0.376373i	0.982133	−	0.188186i	$-0.0602609\pi$
$458$	−16.5187	−0.771869
$459$	2.69202	0.125653
$460$	5.40581i	0.252047i
$461$	4.00538i	0.186549i	0.995640	+	0.0932745i	$0.0297334\pi$
$461$	4.00538i	0.186549i	−0.995640	+	0.0932745i	$0.970267\pi$
$462$	3.46681i	0.161291i
$463$	37.6142i	1.74808i	0.485856	+	0.874039i	$0.338508\pi$
$463$	37.6142i	1.74808i	−0.485856	+	0.874039i	$0.661492\pi$
$464$	−6.15883	−0.285917
$465$	−0.246980	−0.0114534
$466$	6.16421i	0.285552i
$467$	32.8079	1.51817	0.759084	−	0.650992i	$-0.225647\pi$
$467$	32.8079	1.51817	0.759084	+	0.650992i	$0.225647\pi$
$468$	0	0
$469$	−7.53750	−0.348049
$470$	− 6.70171i	− 0.309127i
$471$	−17.0761	−0.786824
$472$	−1.36898	−0.0630123
$473$	2.31873i	0.106615i
$474$	− 5.33513i	− 0.245050i
$475$	− 1.55496i	− 0.0713464i
$476$	− 8.20775i	− 0.376202i
$477$	5.77479	0.264410
$478$	−18.1793	−0.831501
$479$	9.88530i	0.451671i	0.974165	+	0.225835i	$0.0725112\pi$
$479$	9.88530i	0.451671i	−0.974165	+	0.225835i	$0.927489\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−9.21744	−0.419843
$483$	− 16.4819i	− 0.749951i
$484$	−9.70709	−0.441231
$485$	−1.93900	−0.0880455
$486$	− 1.00000i	− 0.0453609i
$487$	20.2610i	0.918113i	0.888407	+	0.459056i	$0.151812\pi$
$487$	20.2610i	0.918113i	−0.888407	+	0.459056i	$0.848188\pi$
$488$	− 10.9487i	− 0.495624i
$489$	− 9.61894i	− 0.434984i
$490$	2.29590	0.103718
$491$	22.4940	1.01514	0.507569	−	0.861611i	$-0.330544\pi$
$491$	22.4940	1.01514	0.507569	+	0.861611i	$0.330544\pi$
$492$	8.26875i	0.372784i
$493$	16.5797	0.746713
$494$	0	0
$495$	1.13706	0.0511072
$496$	− 0.246980i	− 0.0110897i
$497$	−26.1293	−1.17206
$498$	−5.45473	−0.244432
$499$	− 15.2728i	− 0.683704i	−0.939754	−	0.341852i	$-0.888946\pi$
$499$	− 15.2728i	− 0.683704i	0.939754	−	0.341852i	$-0.111054\pi$
$500$	1.00000i	0.0447214i
$501$	21.8509i	0.976224i
$502$	− 17.2054i	− 0.767913i
$503$	−2.95838	−0.131908	−0.0659538	−	0.997823i	$-0.521009\pi$
$503$	−2.95838	−0.131908	−0.0659538	+	0.997823i	$0.521009\pi$
$504$	−3.04892	−0.135810
$505$	14.4940i	0.644973i
$506$	−6.14675	−0.273256
$507$	0	0
$508$	6.22521	0.276199
$509$	− 4.34614i	− 0.192639i	−0.995350	−	0.0963197i	$-0.969293\pi$
$509$	− 4.34614i	− 0.192639i	0.995350	−	0.0963197i	$-0.0307071\pi$
$510$	−2.69202	−0.119205
$511$	45.2258	2.00067
$512$	1.00000i	0.0441942i
$513$	− 1.55496i	− 0.0686531i
$514$	19.5157i	0.860802i
$515$	16.1836i	0.713134i
$516$	−2.03923	−0.0897720
$517$	7.62027	0.335139
$518$	1.69202i	0.0743432i
$519$	16.6853	0.732404
$520$	0	0
$521$	9.13228	0.400092	0.200046	−	0.979786i	$-0.435891\pi$
$521$	9.13228	0.400092	0.200046	+	0.979786i	$0.435891\pi$
$522$	− 6.15883i	− 0.269565i
$523$	10.2828	0.449633	0.224817	−	0.974401i	$-0.427822\pi$
$523$	10.2828	0.449633	0.224817	+	0.974401i	$0.427822\pi$
$524$	−3.34481	−0.146119
$525$	− 3.04892i	− 0.133066i
$526$	12.8834i	0.561742i
$527$	0.664874i	0.0289624i
$528$	1.13706i	0.0494843i
$529$	6.22282	0.270557
$530$	−5.77479	−0.250841
$531$	− 1.36898i	− 0.0594086i
$532$	−4.74094	−0.205546
$533$	0	0
$534$	−16.8877	−0.730802
$535$	4.92931i	0.213113i
$536$	−2.47219	−0.106782
$537$	−3.94438	−0.170212
$538$	− 30.9138i	− 1.33279i
$539$	2.61058i	0.112446i
$540$	1.00000i	0.0430331i
$541$	− 1.47517i	− 0.0634226i	−0.999497	−	0.0317113i	$-0.989904\pi$
$541$	− 1.47517i	− 0.0634226i	0.999497	−	0.0317113i	$-0.0100957\pi$
$542$	7.24027	0.310996
$543$	−20.5308	−0.881061
$544$	− 2.69202i	− 0.115419i
$545$	−0.801938	−0.0343512
$546$	0	0
$547$	−4.82908	−0.206477	−0.103238	−	0.994657i	$-0.532920\pi$
$547$	−4.82908	−0.206477	−0.103238	+	0.994657i	$0.532920\pi$
$548$	− 9.53079i	− 0.407135i
$549$	10.9487	0.467279
$550$	−1.13706	−0.0484845
$551$	− 9.57673i	− 0.407982i
$552$	− 5.40581i	− 0.230087i
$553$	− 16.2664i	− 0.691716i
$554$	− 12.2446i	− 0.520223i
$555$	0.554958	0.0235567
$556$	−18.7017	−0.793129
$557$	− 3.26145i	− 0.138192i	−0.997610	−	0.0690961i	$-0.977988\pi$
$557$	− 3.26145i	− 0.138192i	0.997610	−	0.0690961i	$-0.0220115\pi$
$558$	0.246980	0.0104555
$559$	0	0
$560$	3.04892	0.128840
$561$	− 3.06100i	− 0.129235i
$562$	−2.74632	−0.115846
$563$	−18.1551	−0.765147	−0.382573	−	0.923925i	$-0.624962\pi$
$563$	−18.1551	−0.765147	−0.382573	+	0.923925i	$0.624962\pi$
$564$	6.70171i	0.282193i
$565$	− 11.5211i	− 0.484697i
$566$	− 14.9312i	− 0.627606i
$567$	− 3.04892i	− 0.128042i
$568$	−8.57002	−0.359590
$569$	39.2669	1.64616	0.823078	−	0.567928i	$-0.192255\pi$
$569$	39.2669	1.64616	0.823078	+	0.567928i	$0.192255\pi$
$570$	1.55496i	0.0651300i
$571$	36.8649	1.54275	0.771373	−	0.636383i	$-0.219570\pi$
$571$	36.8649	1.54275	0.771373	+	0.636383i	$0.219570\pi$
$572$	0	0
$573$	6.66487	0.278429
$574$	25.2107i	1.05228i
$575$	5.40581	0.225438
$576$	−1.00000	−0.0416667
$577$	− 20.3043i	− 0.845277i	−0.906298	−	0.422639i	$-0.861104\pi$
$577$	− 20.3043i	− 0.845277i	0.906298	−	0.422639i	$-0.138896\pi$
$578$	− 9.75302i	− 0.405672i
$579$	16.6558i	0.692190i
$580$	6.15883i	0.255732i
$581$	−16.6310	−0.689971
$582$	1.93900	0.0803742
$583$	− 6.56630i	− 0.271948i
$584$	14.8334	0.613810
$585$	0	0
$586$	18.5810	0.767576
$587$	24.5472i	1.01317i	0.862190	+	0.506585i	$0.169093\pi$
$587$	24.5472i	1.01317i	−0.862190	+	0.506585i	$0.830907\pi$
$588$	−2.29590	−0.0946812
$589$	0.384043	0.0158242
$590$	1.36898i	0.0563599i
$591$	− 19.4494i	− 0.800040i
$592$	0.554958i	0.0228086i
$593$	24.0844i	0.989029i	0.869169	+	0.494514i	$0.164654\pi$
$593$	24.0844i	0.989029i	−0.869169	+	0.494514i	$0.835346\pi$
$594$	−1.13706	−0.0466543
$595$	−8.20775	−0.336485
$596$	− 16.4209i	− 0.672625i
$597$	−21.6558	−0.886312
$598$	0	0
$599$	1.84415	0.0753499	0.0376750	−	0.999290i	$-0.488005\pi$
$599$	1.84415	0.0753499	0.0376750	+	0.999290i	$0.488005\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	−15.6770	−0.639476	−0.319738	−	0.947506i	$-0.603595\pi$
$601$	−15.6770	−0.639476	−0.319738	+	0.947506i	$0.603595\pi$
$602$	−6.21744	−0.253404
$603$	− 2.47219i	− 0.100675i
$604$	− 9.26636i	− 0.377043i
$605$	9.70709i	0.394649i
$606$	− 14.4940i	− 0.588777i
$607$	−29.6866	−1.20494	−0.602472	−	0.798140i	$-0.705817\pi$
$607$	−29.6866	−1.20494	−0.602472	+	0.798140i	$0.705817\pi$
$608$	−1.55496	−0.0630619
$609$	− 18.7778i	− 0.760914i
$610$	−10.9487	−0.443299
$611$	0	0
$612$	2.69202	0.108819
$613$	− 42.5163i	− 1.71722i	−0.512631	−	0.858609i	$-0.671329\pi$
$613$	− 42.5163i	− 1.71722i	0.512631	−	0.858609i	$-0.328671\pi$
$614$	32.4359	1.30901
$615$	8.26875	0.333428
$616$	3.46681i	0.139682i
$617$	27.3666i	1.10174i	0.834592	+	0.550869i	$0.185703\pi$
$617$	27.3666i	1.10174i	−0.834592	+	0.550869i	$0.814297\pi$
$618$	− 16.1836i	− 0.650999i
$619$	− 27.9420i	− 1.12308i	−0.827449	−	0.561542i	$-0.810208\pi$
$619$	− 27.9420i	− 1.12308i	0.827449	−	0.561542i	$-0.189792\pi$
$620$	−0.246980	−0.00991894
$621$	5.40581	0.216928
$622$	21.2771i	0.853134i
$623$	−51.4892	−2.06287
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 18.7060i	− 0.747643i
$627$	−1.76809	−0.0706105
$628$	−17.0761	−0.681409
$629$	− 1.49396i	− 0.0595681i
$630$	3.04892i	0.121472i
$631$	− 7.43594i	− 0.296020i	−0.988986	−	0.148010i	$-0.952713\pi$
$631$	− 7.43594i	− 0.296020i	0.988986	−	0.148010i	$-0.0472868\pi$
$632$	− 5.33513i	− 0.212220i
$633$	−12.1521	−0.483004
$634$	−17.4776	−0.694123
$635$	− 6.22521i	− 0.247040i
$636$	5.77479	0.228985
$637$	0	0
$638$	−7.00298	−0.277251
$639$	− 8.57002i	− 0.339025i
$640$	1.00000	0.0395285
$641$	24.9444	0.985244	0.492622	−	0.870243i	$-0.336039\pi$
$641$	24.9444	0.985244	0.492622	+	0.870243i	$0.336039\pi$
$642$	− 4.92931i	− 0.194544i
$643$	30.1661i	1.18964i	0.803860	+	0.594818i	$0.202776\pi$
$643$	30.1661i	1.18964i	−0.803860	+	0.594818i	$0.797224\pi$
$644$	− 16.4819i	− 0.649477i
$645$	2.03923i	0.0802946i
$646$	4.18598	0.164695
$647$	44.4053	1.74575	0.872877	−	0.487940i	$-0.162252\pi$
$647$	44.4053	1.74575	0.872877	+	0.487940i	$0.162252\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−1.55661	−0.0611024
$650$	0	0
$651$	0.753020	0.0295132
$652$	− 9.61894i	− 0.376707i
$653$	−29.2573	−1.14493	−0.572463	−	0.819931i	$-0.694012\pi$
$653$	−29.2573	−1.14493	−0.572463	+	0.819931i	$0.694012\pi$
$654$	0.801938	0.0313582
$655$	3.34481i	0.130693i
$656$	8.26875i	0.322840i
$657$	14.8334i	0.578706i
$658$	20.4330i	0.796560i
$659$	−11.4808	−0.447229	−0.223614	−	0.974678i	$-0.571786\pi$
$659$	−11.4808	−0.447229	−0.223614	+	0.974678i	$0.571786\pi$
$660$	1.13706	0.0442601
$661$	− 26.1129i	− 1.01567i	−0.861453	−	0.507837i	$-0.830445\pi$
$661$	− 26.1129i	− 1.01567i	0.861453	−	0.507837i	$-0.169555\pi$
$662$	23.1890	0.901265
$663$	0	0
$664$	−5.45473	−0.211685
$665$	4.74094i	0.183846i
$666$	−0.554958	−0.0215042
$667$	33.2935	1.28913
$668$	21.8509i	0.845435i
$669$	− 1.11960i	− 0.0432864i
$670$	2.47219i	0.0955090i
$671$	− 12.4494i	− 0.480602i
$672$	−3.04892	−0.117615
$673$	12.4190	0.478716	0.239358	−	0.970931i	$-0.423063\pi$
$673$	12.4190	0.478716	0.239358	+	0.970931i	$0.423063\pi$
$674$	26.1400i	1.00688i
$675$	1.00000	0.0384900
$676$	0	0
$677$	12.7646	0.490585	0.245292	−	0.969449i	$-0.421116\pi$
$677$	12.7646	0.490585	0.245292	+	0.969449i	$0.421116\pi$
$678$	11.5211i	0.442465i
$679$	5.91185	0.226876
$680$	−2.69202	−0.103234
$681$	− 4.06638i	− 0.155824i
$682$	− 0.280831i	− 0.0107536i
$683$	− 31.6142i	− 1.20968i	−0.796346	−	0.604841i	$-0.793237\pi$
$683$	− 31.6142i	− 1.20968i	0.796346	−	0.604841i	$-0.206763\pi$
$684$	− 1.55496i	− 0.0594553i
$685$	−9.53079	−0.364153
$686$	14.3424	0.547596
$687$	− 16.5187i	− 0.630228i
$688$	−2.03923	−0.0777449
$689$	0	0
$690$	−5.40581	−0.205796
$691$	2.06962i	0.0787322i	0.999225	+	0.0393661i	$0.0125339\pi$
$691$	2.06962i	0.0787322i	−0.999225	+	0.0393661i	$0.987466\pi$
$692$	16.6853	0.634281
$693$	−3.46681	−0.131693
$694$	15.2959i	0.580624i
$695$	18.7017i	0.709396i
$696$	− 6.15883i	− 0.233450i
$697$	− 22.2597i	− 0.843144i
$698$	−19.2107	−0.727137
$699$	−6.16421	−0.233152
$700$	− 3.04892i	− 0.115238i
$701$	23.0030	0.868811	0.434405	−	0.900717i	$-0.356959\pi$
$701$	23.0030	0.868811	0.434405	+	0.900717i	$0.356959\pi$
$702$	0	0
$703$	−0.862937	−0.0325463
$704$	1.13706i	0.0428547i
$705$	6.70171	0.252401
$706$	5.35450	0.201519
$707$	− 44.1909i	− 1.66197i
$708$	− 1.36898i	− 0.0514493i
$709$	32.4373i	1.21821i	0.793091	+	0.609104i	$0.208471\pi$
$709$	32.4373i	1.21821i	−0.793091	+	0.609104i	$0.791529\pi$
$710$	8.57002i	0.321627i
$711$	5.33513	0.200083
$712$	−16.8877	−0.632893
$713$	1.33513i	0.0500008i
$714$	8.20775	0.307167
$715$	0	0
$716$	−3.94438	−0.147408
$717$	− 18.1793i	− 0.678917i
$718$	8.49934	0.317192
$719$	13.7845	0.514074	0.257037	−	0.966402i	$-0.417254\pi$
$719$	13.7845	0.514074	0.257037	+	0.966402i	$0.417254\pi$
$720$	1.00000i	0.0372678i
$721$	− 49.3424i	− 1.83761i
$722$	16.5821i	0.617122i
$723$	− 9.21744i	− 0.342800i
$724$	−20.5308	−0.763021
$725$	6.15883	0.228733
$726$	− 9.70709i	− 0.360264i
$727$	−4.64848	−0.172403	−0.0862013	−	0.996278i	$-0.527473\pi$
$727$	−4.64848	−0.172403	−0.0862013	+	0.996278i	$0.527473\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 14.8334i	− 0.549009i
$731$	5.48965	0.203042
$732$	10.9487	0.404675
$733$	− 27.4717i	− 1.01469i	−0.861743	−	0.507345i	$-0.830627\pi$
$733$	− 27.4717i	− 1.01469i	0.861743	−	0.507345i	$-0.169373\pi$
$734$	4.76032i	0.175706i
$735$	2.29590i	0.0846854i
$736$	− 5.40581i	− 0.199261i
$737$	−2.81104	−0.103546
$738$	−8.26875	−0.304377
$739$	48.0170i	1.76633i	0.469059	+	0.883167i	$0.344593\pi$
$739$	48.0170i	1.76633i	−0.469059	+	0.883167i	$0.655407\pi$
$740$	0.554958	0.0204007
$741$	0	0
$742$	17.6069	0.646368
$743$	− 23.6517i	− 0.867698i	−0.900986	−	0.433849i	$-0.857155\pi$
$743$	− 23.6517i	− 0.867698i	0.900986	−	0.433849i	$-0.142845\pi$
$744$	0.246980	0.00905471
$745$	−16.4209	−0.601614
$746$	− 20.2064i	− 0.739810i
$747$	− 5.45473i	− 0.199578i
$748$	− 3.06100i	− 0.111921i
$749$	− 15.0291i	− 0.549150i
$750$	−1.00000	−0.0365148
$751$	39.5133	1.44186	0.720931	−	0.693007i	$-0.243714\pi$
$751$	39.5133	1.44186	0.720931	+	0.693007i	$0.243714\pi$
$752$	6.70171i	0.244386i
$753$	17.2054	0.626998
$754$	0	0
$755$	−9.26636	−0.337237
$756$	− 3.04892i	− 0.110888i
$757$	−11.2239	−0.407939	−0.203969	−	0.978977i	$-0.565384\pi$
$757$	−11.2239	−0.407939	−0.203969	+	0.978977i	$0.565384\pi$
$758$	−24.4088	−0.886567
$759$	− 6.14675i	− 0.223113i
$760$	1.55496i	0.0564043i
$761$	5.80864i	0.210563i	0.994442	+	0.105282i	$0.0335744\pi$
$761$	5.80864i	0.210563i	−0.994442	+	0.105282i	$0.966426\pi$
$762$	6.22521i	0.225516i
$763$	2.44504	0.0885165
$764$	6.66487	0.241127
$765$	− 2.69202i	− 0.0973302i
$766$	−32.0237	−1.15706
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 31.8316i	− 1.14788i	−0.818899	−	0.573938i	$-0.805415\pi$
$769$	− 31.8316i	− 1.14788i	0.818899	−	0.573938i	$-0.194585\pi$
$770$	3.46681	0.124935
$771$	−19.5157	−0.702842
$772$	16.6558i	0.599455i
$773$	39.7434i	1.42947i	0.699394	+	0.714736i	$0.253453\pi$
$773$	39.7434i	1.42947i	−0.699394	+	0.714736i	$0.746547\pi$
$774$	− 2.03923i	− 0.0732986i
$775$	0.246980i	0.00887177i
$776$	1.93900	0.0696061
$777$	−1.69202	−0.0607009
$778$	− 21.1879i	− 0.759623i
$779$	−12.8576	−0.460670
$780$	0	0
$781$	−9.74466	−0.348691
$782$	14.5526i	0.520399i
$783$	6.15883	0.220099
$784$	−2.29590	−0.0819963
$785$	17.0761i	0.609471i
$786$	− 3.34481i	− 0.119306i
$787$	− 19.3183i	− 0.688622i	−0.938856	−	0.344311i	$-0.888113\pi$
$787$	− 19.3183i	− 0.688622i	0.938856	−	0.344311i	$-0.111887\pi$
$788$	− 19.4494i	− 0.692855i
$789$	−12.8834	−0.458660
$790$	−5.33513	−0.189815
$791$	35.1269i	1.24897i
$792$	−1.13706	−0.0404038
$793$	0	0
$794$	10.9608	0.388983
$795$	− 5.77479i	− 0.204811i
$796$	−21.6558	−0.767569
$797$	−29.4946	−1.04475	−0.522375	−	0.852716i	$-0.674954\pi$
$797$	−29.4946	−1.04475	−0.522375	+	0.852716i	$0.674954\pi$
$798$	− 4.74094i	− 0.167827i
$799$	− 18.0411i	− 0.638250i
$800$	− 1.00000i	− 0.0353553i
$801$	− 16.8877i	− 0.596697i
$802$	−29.2301	−1.03215
$803$	16.8665	0.595206
$804$	− 2.47219i	− 0.0871874i
$805$	−16.4819	−0.580910
$806$	0	0
$807$	30.9138	1.08822
$808$	− 14.4940i	− 0.509896i
$809$	14.0901	0.495380	0.247690	−	0.968839i	$-0.420329\pi$
$809$	14.0901	0.495380	0.247690	+	0.968839i	$0.420329\pi$
$810$	−1.00000	−0.0351364
$811$	− 45.5060i	− 1.59793i	−0.601375	−	0.798967i	$-0.705380\pi$
$811$	− 45.5060i	− 1.59793i	0.601375	−	0.798967i	$-0.294620\pi$
$812$	− 18.7778i	− 0.658971i
$813$	7.24027i	0.253928i
$814$	0.631023i	0.0221173i
$815$	−9.61894	−0.336937
$816$	2.69202	0.0942396
$817$	− 3.17092i	− 0.110936i
$818$	−36.3163	−1.26977
$819$	0	0
$820$	8.26875	0.288757
$821$	− 14.6853i	− 0.512521i	−0.966608	−	0.256261i	$-0.917509\pi$
$821$	− 14.6853i	− 0.512521i	0.966608	−	0.256261i	$-0.0824905\pi$
$822$	9.53079	0.332425
$823$	32.0551	1.11737	0.558686	−	0.829379i	$-0.311306\pi$
$823$	32.0551	1.11737	0.558686	+	0.829379i	$0.311306\pi$
$824$	− 16.1836i	− 0.563782i
$825$	− 1.13706i	− 0.0395875i
$826$	− 4.17390i	− 0.145229i
$827$	− 8.09352i	− 0.281439i	−0.990049	−	0.140720i	$-0.955058\pi$
$827$	− 8.09352i	− 0.281439i	0.990049	−	0.140720i	$-0.0449416\pi$
$828$	5.40581	0.187865
$829$	23.8708	0.829068	0.414534	−	0.910034i	$-0.363945\pi$
$829$	23.8708	0.829068	0.414534	+	0.910034i	$0.363945\pi$
$830$	5.45473i	0.189336i
$831$	12.2446	0.424760
$832$	0	0
$833$	6.18060	0.214145
$834$	− 18.7017i	− 0.647587i
$835$	21.8509	0.756180
$836$	−1.76809	−0.0611505
$837$	0.246980i	0.00853686i
$838$	11.0382i	0.381307i
$839$	17.5569i	0.606131i	0.952970	+	0.303065i	$0.0980100\pi$
$839$	17.5569i	0.606131i	−0.952970	+	0.303065i	$0.901990\pi$
$840$	3.04892i	0.105198i
$841$	8.93123	0.307973
$842$	−4.58642	−0.158058
$843$	− 2.74632i	− 0.0945881i
$844$	−12.1521	−0.418294
$845$	0	0
$846$	−6.70171	−0.230410
$847$	− 29.5961i	− 1.01693i
$848$	5.77479	0.198307
$849$	14.9312	0.512439
$850$	2.69202i	0.0923356i
$851$	− 3.00000i	− 0.102839i
$852$	− 8.57002i	− 0.293604i
$853$	40.2892i	1.37948i	0.724059	+	0.689738i	$0.242274\pi$
$853$	40.2892i	1.37948i	−0.724059	+	0.689738i	$0.757726\pi$
$854$	33.3817	1.14230
$855$	−1.55496	−0.0531784
$856$	− 4.92931i	− 0.168480i
$857$	−44.5913	−1.52321	−0.761605	−	0.648041i	$-0.775588\pi$
$857$	−44.5913	−1.52321	−0.761605	+	0.648041i	$0.775588\pi$
$858$	0	0
$859$	−40.6467	−1.38685	−0.693423	−	0.720530i	$-0.743898\pi$
$859$	−40.6467	−1.38685	−0.693423	+	0.720530i	$0.743898\pi$
$860$	2.03923i	0.0695371i
$861$	−25.2107	−0.859180
$862$	−19.3642	−0.659547
$863$	− 31.9748i	− 1.08843i	−0.838945	−	0.544217i	$-0.816827\pi$
$863$	− 31.9748i	− 1.08843i	0.838945	−	0.544217i	$-0.183173\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	− 16.6853i	− 0.567318i
$866$	− 4.41119i	− 0.149898i
$867$	9.75302	0.331230
$868$	0.753020	0.0255592
$869$	− 6.06638i	− 0.205788i
$870$	−6.15883	−0.208804
$871$	0	0
$872$	0.801938	0.0271570
$873$	1.93900i	0.0656252i
$874$	8.40581	0.284331
$875$	−3.04892	−0.103072
$876$	14.8334i	0.501174i
$877$	− 39.2476i	− 1.32530i	−0.748931	−	0.662648i	$-0.769433\pi$
$877$	− 39.2476i	− 1.32530i	0.748931	−	0.662648i	$-0.230567\pi$
$878$	− 39.5260i	− 1.33394i
$879$	18.5810i	0.626723i
$880$	1.13706	0.0383304
$881$	24.3153	0.819202	0.409601	−	0.912265i	$-0.365668\pi$
$881$	24.3153	0.819202	0.409601	+	0.912265i	$0.365668\pi$
$882$	− 2.29590i	− 0.0773069i
$883$	−51.8756	−1.74575	−0.872877	−	0.487941i	$-0.837748\pi$
$883$	−51.8756	−1.74575	−0.872877	+	0.487941i	$0.837748\pi$
$884$	0	0
$885$	−1.36898	−0.0460177
$886$	− 7.16182i	− 0.240606i
$887$	−13.4383	−0.451215	−0.225608	−	0.974218i	$-0.572437\pi$
$887$	−13.4383	−0.451215	−0.225608	+	0.974218i	$0.572437\pi$
$888$	−0.554958	−0.0186232
$889$	18.9801i	0.636574i
$890$	16.8877i	0.566077i
$891$	− 1.13706i	− 0.0380931i
$892$	− 1.11960i	− 0.0374871i
$893$	−10.4209	−0.348721
$894$	16.4209	0.549196
$895$	3.94438i	0.131846i
$896$	−3.04892	−0.101857
$897$	0	0
$898$	−4.46011	−0.148836
$899$	1.52111i	0.0507317i
$900$	1.00000	0.0333333
$901$	−15.5459	−0.517908
$902$	9.40209i	0.313055i
$903$	− 6.21744i	− 0.206903i
$904$	11.5211i	0.383186i
$905$	20.5308i	0.682467i
$906$	9.26636	0.307854
$907$	6.58881	0.218778	0.109389	−	0.993999i	$-0.465111\pi$
$907$	6.58881	0.218778	0.109389	+	0.993999i	$0.465111\pi$
$908$	− 4.06638i	− 0.134947i
$909$	14.4940	0.480734
$910$	0	0
$911$	51.2646	1.69847	0.849235	−	0.528015i	$-0.177063\pi$
$911$	51.2646	1.69847	0.849235	+	0.528015i	$0.177063\pi$
$912$	− 1.55496i	− 0.0514898i
$913$	−6.20237	−0.205269
$914$	8.04593	0.266136
$915$	− 10.9487i	− 0.361953i
$916$	− 16.5187i	− 0.545794i
$917$	− 10.1981i	− 0.336770i
$918$	2.69202i	0.0888499i
$919$	50.7066	1.67266	0.836328	−	0.548229i	$-0.184698\pi$
$919$	50.7066	1.67266	0.836328	+	0.548229i	$0.184698\pi$
$920$	−5.40581	−0.178224
$921$	32.4359i	1.06880i
$922$	−4.00538	−0.131910
$923$	0	0
$924$	−3.46681	−0.114050
$925$	− 0.554958i	− 0.0182469i
$926$	−37.6142	−1.23608
$927$	16.1836	0.531539
$928$	− 6.15883i	− 0.202174i
$929$	− 18.2903i	− 0.600084i	−0.953926	−	0.300042i	$-0.902999\pi$
$929$	− 18.2903i	− 0.600084i	0.953926	−	0.300042i	$-0.0970007\pi$
$930$	− 0.246980i	− 0.00809878i
$931$	− 3.57002i	− 0.117003i
$932$	−6.16421	−0.201915
$933$	−21.2771	−0.696581
$934$	32.8079i	1.07351i
$935$	−3.06100	−0.100105
$936$	0	0
$937$	−59.6679	−1.94926	−0.974632	−	0.223814i	$-0.928149\pi$
$937$	−59.6679	−1.94926	−0.974632	+	0.223814i	$0.928149\pi$
$938$	− 7.53750i	− 0.246108i
$939$	18.7060	0.610448
$940$	6.70171	0.218586
$941$	− 36.2513i	− 1.18176i	−0.806760	−	0.590879i	$-0.798781\pi$
$941$	− 36.2513i	− 1.18176i	0.806760	−	0.590879i	$-0.201219\pi$
$942$	− 17.0761i	− 0.556368i
$943$	− 44.6993i	− 1.45561i
$944$	− 1.36898i	− 0.0445564i
$945$	−3.04892	−0.0991813
$946$	−2.31873	−0.0753885
$947$	− 7.02475i	− 0.228274i	−0.993465	−	0.114137i	$-0.963590\pi$
$947$	− 7.02475i	− 0.228274i	0.993465	−	0.114137i	$-0.0364103\pi$
$948$	5.33513	0.173277
$949$	0	0
$950$	1.55496	0.0504495
$951$	− 17.4776i	− 0.566749i
$952$	8.20775	0.266015
$953$	−12.5730	−0.407280	−0.203640	−	0.979046i	$-0.565277\pi$
$953$	−12.5730	−0.407280	−0.203640	+	0.979046i	$0.565277\pi$
$954$	5.77479i	0.186966i
$955$	− 6.66487i	− 0.215670i
$956$	− 18.1793i	− 0.587960i
$957$	− 7.00298i	− 0.226374i
$958$	−9.88530	−0.319379
$959$	29.0586	0.938351
$960$	1.00000i	0.0322749i
$961$	30.9390	0.998032
$962$	0	0
$963$	4.92931	0.158845
$964$	− 9.21744i	− 0.296874i
$965$	16.6558	0.536168
$966$	16.4819	0.530296
$967$	− 23.1371i	− 0.744038i	−0.928225	−	0.372019i	$-0.878666\pi$
$967$	− 23.1371i	− 0.744038i	0.928225	−	0.372019i	$-0.121334\pi$
$968$	− 9.70709i	− 0.311998i
$969$	4.18598i	0.134473i
$970$	− 1.93900i	− 0.0622576i
$971$	−26.5394	−0.851690	−0.425845	−	0.904796i	$-0.640023\pi$
$971$	−26.5394	−0.851690	−0.425845	+	0.904796i	$0.640023\pi$
$972$	1.00000	0.0320750
$973$	− 57.0200i	− 1.82798i
$974$	−20.2610	−0.649204
$975$	0	0
$976$	10.9487	0.350459
$977$	19.5670i	0.626005i	0.949752	+	0.313003i	$0.101335\pi$
$977$	19.5670i	0.626005i	−0.949752	+	0.313003i	$0.898665\pi$
$978$	9.61894	0.307580
$979$	−19.2024	−0.613711
$980$	2.29590i	0.0733397i
$981$	0.801938i	0.0256039i
$982$	22.4940i	0.717811i
$983$	8.82610i	0.281509i	0.990045	+	0.140754i	$0.0449528\pi$
$983$	8.82610i	0.281509i	−0.990045	+	0.140754i	$0.955047\pi$
$984$	−8.26875	−0.263598
$985$	−19.4494	−0.619708
$986$	16.5797i	0.528006i
$987$	−20.4330	−0.650388
$988$	0	0
$989$	11.0237	0.350533
$990$	1.13706i	0.0361382i
$991$	0.601483	0.0191067	0.00955336	−	0.999954i	$-0.496959\pi$
$991$	0.601483	0.0191067	0.00955336	+	0.999954i	$0.496959\pi$
$992$	0.246980	0.00784161
$993$	23.1890i	0.735880i
$994$	− 26.1293i	− 0.828771i
$995$	21.6558i	0.686534i
$996$	− 5.45473i	− 0.172840i
$997$	13.4101	0.424703	0.212351	−	0.977193i	$-0.431888\pi$
$997$	13.4101	0.424703	0.212351	+	0.977193i	$0.431888\pi$
$998$	15.2728	0.483452
$999$	− 0.554958i	− 0.0175581i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.t.1351.4		6
13.5	odd	4		5070.2.a.bu.1.3	yes	3
13.8	odd	4		5070.2.a.bj.1.1	✓	3
13.12	even	2	inner	5070.2.b.t.1351.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bj.1.1	✓	3	13.8	odd	4
5070.2.a.bu.1.3	yes	3	13.5	odd	4
5070.2.b.t.1351.3		6	13.12	even	2	inner
5070.2.b.t.1351.4		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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -3.04892i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -3.04892i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.13706i q^{11} +1.00000 q^{12} +3.04892 q^{14} -1.00000i q^{15} +1.00000 q^{16} -2.69202 q^{17} +1.00000i q^{18} +1.55496i q^{19} -1.00000i q^{20} +3.04892i q^{21} +1.13706 q^{22} -5.40581 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +3.04892i q^{28} -6.15883 q^{29} +1.00000 q^{30} -0.246980i q^{31} +1.00000i q^{32} +1.13706i q^{33} -2.69202i q^{34} +3.04892 q^{35} -1.00000 q^{36} +0.554958i q^{37} -1.55496 q^{38} +1.00000 q^{40} +8.26875i q^{41} -3.04892 q^{42} -2.03923 q^{43} +1.13706i q^{44} +1.00000i q^{45} -5.40581i q^{46} +6.70171i q^{47} -1.00000 q^{48} -2.29590 q^{49} -1.00000i q^{50} +2.69202 q^{51} +5.77479 q^{53} -1.00000i q^{54} +1.13706 q^{55} -3.04892 q^{56} -1.55496i q^{57} -6.15883i q^{58} -1.36898i q^{59} +1.00000i q^{60} +10.9487 q^{61} +0.246980 q^{62} -3.04892i q^{63} -1.00000 q^{64} -1.13706 q^{66} -2.47219i q^{67} +2.69202 q^{68} +5.40581 q^{69} +3.04892i q^{70} -8.57002i q^{71} -1.00000i q^{72} +14.8334i q^{73} -0.554958 q^{74} +1.00000 q^{75} -1.55496i q^{76} -3.46681 q^{77} +5.33513 q^{79} +1.00000i q^{80} +1.00000 q^{81} -8.26875 q^{82} -5.45473i q^{83} -3.04892i q^{84} -2.69202i q^{85} -2.03923i q^{86} +6.15883 q^{87} -1.13706 q^{88} -16.8877i q^{89} -1.00000 q^{90} +5.40581 q^{92} +0.246980i q^{93} -6.70171 q^{94} -1.55496 q^{95} -1.00000i q^{96} +1.93900i q^{97} -2.29590i q^{98} -1.13706i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{10} + 6 q^{12} + 6 q^{16} - 6 q^{17} - 4 q^{22} - 6 q^{23} - 6 q^{25} - 6 q^{27} - 20 q^{29} + 6 q^{30} - 6 q^{36} - 10 q^{38} + 6 q^{40} - 38 q^{43} - 6 q^{48} + 14 q^{49} + 6 q^{51} + 38 q^{53} - 4 q^{55} + 2 q^{61} - 8 q^{62} - 6 q^{64} + 4 q^{66} + 6 q^{68} + 6 q^{69} - 4 q^{74} + 6 q^{75} - 14 q^{77} + 30 q^{79} + 6 q^{81} - 34 q^{82} + 20 q^{87} + 4 q^{88} - 6 q^{90} + 6 q^{92} + 14 q^{94} - 10 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 + 6 * q^12 + 6 * q^16 - 6 * q^17 - 4 * q^22 - 6 * q^23 - 6 * q^25 - 6 * q^27 - 20 * q^29 + 6 * q^30 - 6 * q^36 - 10 * q^38 + 6 * q^40 - 38 * q^43 - 6 * q^48 + 14 * q^49 + 6 * q^51 + 38 * q^53 - 4 * q^55 + 2 * q^61 - 8 * q^62 - 6 * q^64 + 4 * q^66 + 6 * q^68 + 6 * q^69 - 4 * q^74 + 6 * q^75 - 14 * q^77 + 30 * q^79 + 6 * q^81 - 34 * q^82 + 20 * q^87 + 4 * q^88 - 6 * q^90 + 6 * q^92 + 14 * q^94 - 10 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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