LMFDB - Embedded newform 5070.2.a.bq.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5070.1");

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.a (trivial)

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:	yes
Analytic conductor:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +5.04892 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +5.04892 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.74094 q^{11} +1.00000 q^{12} -5.04892 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.08815 q^{17} -1.00000 q^{18} +4.44504 q^{19} +1.00000 q^{20} +5.04892 q^{21} +4.74094 q^{22} +2.69202 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +5.04892 q^{28} -7.93900 q^{29} -1.00000 q^{30} +7.85086 q^{31} -1.00000 q^{32} -4.74094 q^{33} +1.08815 q^{34} +5.04892 q^{35} +1.00000 q^{36} -7.14675 q^{37} -4.44504 q^{38} -1.00000 q^{40} -0.664874 q^{41} -5.04892 q^{42} +4.35690 q^{43} -4.74094 q^{44} +1.00000 q^{45} -2.69202 q^{46} +7.00000 q^{47} +1.00000 q^{48} +18.4916 q^{49} -1.00000 q^{50} -1.08815 q^{51} -10.8605 q^{53} -1.00000 q^{54} -4.74094 q^{55} -5.04892 q^{56} +4.44504 q^{57} +7.93900 q^{58} +12.2784 q^{59} +1.00000 q^{60} -11.0707 q^{61} -7.85086 q^{62} +5.04892 q^{63} +1.00000 q^{64} +4.74094 q^{66} +9.58211 q^{67} -1.08815 q^{68} +2.69202 q^{69} -5.04892 q^{70} +4.47219 q^{71} -1.00000 q^{72} -8.17390 q^{73} +7.14675 q^{74} +1.00000 q^{75} +4.44504 q^{76} -23.9366 q^{77} +14.1468 q^{79} +1.00000 q^{80} +1.00000 q^{81} +0.664874 q^{82} +4.86294 q^{83} +5.04892 q^{84} -1.08815 q^{85} -4.35690 q^{86} -7.93900 q^{87} +4.74094 q^{88} +14.8877 q^{89} -1.00000 q^{90} +2.69202 q^{92} +7.85086 q^{93} -7.00000 q^{94} +4.44504 q^{95} -1.00000 q^{96} +14.9269 q^{97} -18.4916 q^{98} -4.74094 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{12} - 6 q^{14} + 3 q^{15} + 3 q^{16} - 7 q^{17} - 3 q^{18} + 13 q^{19} + 3 q^{20} + 6 q^{21} + 3 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} + 6 q^{28} - 14 q^{29} - 3 q^{30} + 10 q^{31} - 3 q^{32} + 7 q^{34} + 6 q^{35} + 3 q^{36} + 6 q^{37} - 13 q^{38} - 3 q^{40} - 3 q^{41} - 6 q^{42} + 9 q^{43} + 3 q^{45} - 3 q^{46} + 21 q^{47} + 3 q^{48} + 5 q^{49} - 3 q^{50} - 7 q^{51} + 3 q^{53} - 3 q^{54} - 6 q^{56} + 13 q^{57} + 14 q^{58} + 7 q^{59} + 3 q^{60} - 21 q^{61} - 10 q^{62} + 6 q^{63} + 3 q^{64} + 23 q^{67} - 7 q^{68} + 3 q^{69} - 6 q^{70} + 7 q^{71} - 3 q^{72} + 9 q^{73} - 6 q^{74} + 3 q^{75} + 13 q^{76} - 21 q^{77} + 15 q^{79} + 3 q^{80} + 3 q^{81} + 3 q^{82} + 20 q^{83} + 6 q^{84} - 7 q^{85} - 9 q^{86} - 14 q^{87} + 3 q^{89} - 3 q^{90} + 3 q^{92} + 10 q^{93} - 21 q^{94} + 13 q^{95} - 3 q^{96} + 16 q^{97} - 5 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^12 - 6 * q^14 + 3 * q^15 + 3 * q^16 - 7 * q^17 - 3 * q^18 + 13 * q^19 + 3 * q^20 + 6 * q^21 + 3 * q^23 - 3 * q^24 + 3 * q^25 + 3 * q^27 + 6 * q^28 - 14 * q^29 - 3 * q^30 + 10 * q^31 - 3 * q^32 + 7 * q^34 + 6 * q^35 + 3 * q^36 + 6 * q^37 - 13 * q^38 - 3 * q^40 - 3 * q^41 - 6 * q^42 + 9 * q^43 + 3 * q^45 - 3 * q^46 + 21 * q^47 + 3 * q^48 + 5 * q^49 - 3 * q^50 - 7 * q^51 + 3 * q^53 - 3 * q^54 - 6 * q^56 + 13 * q^57 + 14 * q^58 + 7 * q^59 + 3 * q^60 - 21 * q^61 - 10 * q^62 + 6 * q^63 + 3 * q^64 + 23 * q^67 - 7 * q^68 + 3 * q^69 - 6 * q^70 + 7 * q^71 - 3 * q^72 + 9 * q^73 - 6 * q^74 + 3 * q^75 + 13 * q^76 - 21 * q^77 + 15 * q^79 + 3 * q^80 + 3 * q^81 + 3 * q^82 + 20 * q^83 + 6 * q^84 - 7 * q^85 - 9 * q^86 - 14 * q^87 + 3 * q^89 - 3 * q^90 + 3 * q^92 + 10 * q^93 - 21 * q^94 + 13 * q^95 - 3 * q^96 + 16 * q^97 - 5 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	5.04892	1.90831	0.954156	−	0.299311i	$-0.0967567\pi$
$7$	5.04892	1.90831	0.954156	+	0.299311i	$0.0967567\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−4.74094	−1.42945	−0.714723	−	0.699407i	$-0.753447\pi$
$11$	−4.74094	−1.42945	−0.714723	+	0.699407i	$0.753447\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−5.04892	−1.34938
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−1.08815	−0.263914	−0.131957	−	0.991255i	$-0.542126\pi$
$17$	−1.08815	−0.263914	−0.131957	+	0.991255i	$0.542126\pi$
$18$	−1.00000	−0.235702
$19$	4.44504	1.01976	0.509881	−	0.860245i	$-0.329689\pi$
$19$	4.44504	1.01976	0.509881	+	0.860245i	$0.329689\pi$
$20$	1.00000	0.223607
$21$	5.04892	1.10176
$22$	4.74094	1.01077
$23$	2.69202	0.561325	0.280663	−	0.959806i	$-0.409446\pi$
$23$	2.69202	0.561325	0.280663	+	0.959806i	$0.409446\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	5.04892	0.954156
$29$	−7.93900	−1.47424	−0.737118	−	0.675764i	$-0.763814\pi$
$29$	−7.93900	−1.47424	−0.737118	+	0.675764i	$0.763814\pi$
$30$	−1.00000	−0.182574
$31$	7.85086	1.41006	0.705028	−	0.709180i	$-0.250935\pi$
$31$	7.85086	1.41006	0.705028	+	0.709180i	$0.250935\pi$
$32$	−1.00000	−0.176777
$33$	−4.74094	−0.825292
$34$	1.08815	0.186615
$35$	5.04892	0.853423
$36$	1.00000	0.166667
$37$	−7.14675	−1.17492	−0.587459	−	0.809254i	$-0.699872\pi$
$37$	−7.14675	−1.17492	−0.587459	+	0.809254i	$0.699872\pi$
$38$	−4.44504	−0.721081
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−0.664874	−0.103836	−0.0519180	−	0.998651i	$-0.516533\pi$
$41$	−0.664874	−0.103836	−0.0519180	+	0.998651i	$0.516533\pi$
$42$	−5.04892	−0.779065
$43$	4.35690	0.664420	0.332210	−	0.943205i	$-0.392206\pi$
$43$	4.35690	0.664420	0.332210	+	0.943205i	$0.392206\pi$
$44$	−4.74094	−0.714723
$45$	1.00000	0.149071
$46$	−2.69202	−0.396917
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	1.00000	0.144338
$49$	18.4916	2.64165
$50$	−1.00000	−0.141421
$51$	−1.08815	−0.152371
$52$	0	0
$53$	−10.8605	−1.49181	−0.745905	−	0.666052i	$-0.767983\pi$
$53$	−10.8605	−1.49181	−0.745905	+	0.666052i	$0.767983\pi$
$54$	−1.00000	−0.136083
$55$	−4.74094	−0.639268
$56$	−5.04892	−0.674690
$57$	4.44504	0.588760
$58$	7.93900	1.04244
$59$	12.2784	1.59852	0.799258	−	0.600988i	$-0.205226\pi$
$59$	12.2784	1.59852	0.799258	+	0.600988i	$0.205226\pi$
$60$	1.00000	0.129099
$61$	−11.0707	−1.41746	−0.708728	−	0.705482i	$-0.750731\pi$
$61$	−11.0707	−1.41746	−0.708728	+	0.705482i	$0.750731\pi$
$62$	−7.85086	−0.997060
$63$	5.04892	0.636104
$64$	1.00000	0.125000
$65$	0	0
$66$	4.74094	0.583569
$67$	9.58211	1.17064	0.585320	−	0.810802i	$-0.300969\pi$
$67$	9.58211	1.17064	0.585320	+	0.810802i	$0.300969\pi$
$68$	−1.08815	−0.131957
$69$	2.69202	0.324081
$70$	−5.04892	−0.603461
$71$	4.47219	0.530751	0.265376	−	0.964145i	$-0.414504\pi$
$71$	4.47219	0.530751	0.265376	+	0.964145i	$0.414504\pi$
$72$	−1.00000	−0.117851
$73$	−8.17390	−0.956683	−0.478341	−	0.878174i	$-0.658762\pi$
$73$	−8.17390	−0.956683	−0.478341	+	0.878174i	$0.658762\pi$
$74$	7.14675	0.830793
$75$	1.00000	0.115470
$76$	4.44504	0.509881
$77$	−23.9366	−2.72783
$78$	0	0
$79$	14.1468	1.59163	0.795817	−	0.605537i	$-0.207042\pi$
$79$	14.1468	1.59163	0.795817	+	0.605537i	$0.207042\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	0.664874	0.0734231
$83$	4.86294	0.533777	0.266888	−	0.963727i	$-0.414005\pi$
$83$	4.86294	0.533777	0.266888	+	0.963727i	$0.414005\pi$
$84$	5.04892	0.550882
$85$	−1.08815	−0.118026
$86$	−4.35690	−0.469816
$87$	−7.93900	−0.851150
$88$	4.74094	0.505386
$89$	14.8877	1.57809	0.789046	−	0.614334i	$-0.210575\pi$
$89$	14.8877	1.57809	0.789046	+	0.614334i	$0.210575\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	2.69202	0.280663
$93$	7.85086	0.814096
$94$	−7.00000	−0.721995
$95$	4.44504	0.456052
$96$	−1.00000	−0.102062
$97$	14.9269	1.51560	0.757800	−	0.652487i	$-0.226274\pi$
$97$	14.9269	1.51560	0.757800	+	0.652487i	$0.226274\pi$
$98$	−18.4916	−1.86793
$99$	−4.74094	−0.476482
$100$	1.00000	0.100000
$101$	11.4819	1.14249	0.571245	−	0.820780i	$-0.306461\pi$
$101$	11.4819	1.14249	0.571245	+	0.820780i	$0.306461\pi$
$102$	1.08815	0.107743
$103$	4.16421	0.410312	0.205156	−	0.978729i	$-0.434230\pi$
$103$	4.16421	0.410312	0.205156	+	0.978729i	$0.434230\pi$
$104$	0	0
$105$	5.04892	0.492724
$106$	10.8605	1.05487
$107$	−0.796561	−0.0770065	−0.0385032	−	0.999258i	$-0.512259\pi$
$107$	−0.796561	−0.0770065	−0.0385032	+	0.999258i	$0.512259\pi$
$108$	1.00000	0.0962250
$109$	0.704103	0.0674408	0.0337204	−	0.999431i	$-0.489264\pi$
$109$	0.704103	0.0674408	0.0337204	+	0.999431i	$0.489264\pi$
$110$	4.74094	0.452031
$111$	−7.14675	−0.678340
$112$	5.04892	0.477078
$113$	14.4983	1.36388	0.681941	−	0.731407i	$-0.261136\pi$
$113$	14.4983	1.36388	0.681941	+	0.731407i	$0.261136\pi$
$114$	−4.44504	−0.416316
$115$	2.69202	0.251032
$116$	−7.93900	−0.737118
$117$	0	0
$118$	−12.2784	−1.13032
$119$	−5.49396	−0.503630
$120$	−1.00000	−0.0912871
$121$	11.4765	1.04332
$122$	11.0707	1.00229
$123$	−0.664874	−0.0599497
$124$	7.85086	0.705028
$125$	1.00000	0.0894427
$126$	−5.04892	−0.449793
$127$	−2.93900	−0.260794	−0.130397	−	0.991462i	$-0.541625\pi$
$127$	−2.93900	−0.260794	−0.130397	+	0.991462i	$0.541625\pi$
$128$	−1.00000	−0.0883883
$129$	4.35690	0.383603
$130$	0	0
$131$	−3.56465	−0.311445	−0.155722	−	0.987801i	$-0.549771\pi$
$131$	−3.56465	−0.311445	−0.155722	+	0.987801i	$0.549771\pi$
$132$	−4.74094	−0.412646
$133$	22.4426	1.94602
$134$	−9.58211	−0.827768
$135$	1.00000	0.0860663
$136$	1.08815	0.0933077
$137$	−9.47650	−0.809632	−0.404816	−	0.914398i	$-0.632664\pi$
$137$	−9.47650	−0.809632	−0.404816	+	0.914398i	$0.632664\pi$
$138$	−2.69202	−0.229160
$139$	−14.2078	−1.20509	−0.602543	−	0.798087i	$-0.705846\pi$
$139$	−14.2078	−1.20509	−0.602543	+	0.798087i	$0.705846\pi$
$140$	5.04892	0.426711
$141$	7.00000	0.589506
$142$	−4.47219	−0.375298
$143$	0	0
$144$	1.00000	0.0833333
$145$	−7.93900	−0.659298
$146$	8.17390	0.676477
$147$	18.4916	1.52516
$148$	−7.14675	−0.587459
$149$	10.2010	0.835702	0.417851	−	0.908516i	$-0.362783\pi$
$149$	10.2010	0.835702	0.417851	+	0.908516i	$0.362783\pi$
$150$	−1.00000	−0.0816497
$151$	3.86294	0.314361	0.157181	−	0.987570i	$-0.449760\pi$
$151$	3.86294	0.314361	0.157181	+	0.987570i	$0.449760\pi$
$152$	−4.44504	−0.360541
$153$	−1.08815	−0.0879714
$154$	23.9366	1.92887
$155$	7.85086	0.630596
$156$	0	0
$157$	−11.2174	−0.895249	−0.447625	−	0.894222i	$-0.647730\pi$
$157$	−11.2174	−0.895249	−0.447625	+	0.894222i	$0.647730\pi$
$158$	−14.1468	−1.12546
$159$	−10.8605	−0.861297
$160$	−1.00000	−0.0790569
$161$	13.5918	1.07118
$162$	−1.00000	−0.0785674
$163$	−6.50902	−0.509826	−0.254913	−	0.966964i	$-0.582047\pi$
$163$	−6.50902	−0.509826	−0.254913	+	0.966964i	$0.582047\pi$
$164$	−0.664874	−0.0519180
$165$	−4.74094	−0.369082
$166$	−4.86294	−0.377437
$167$	9.67456	0.748640	0.374320	−	0.927300i	$-0.377876\pi$
$167$	9.67456	0.748640	0.374320	+	0.927300i	$0.377876\pi$
$168$	−5.04892	−0.389532
$169$	0	0
$170$	1.08815	0.0834570
$171$	4.44504	0.339921
$172$	4.35690	0.332210
$173$	−13.8388	−1.05214	−0.526071	−	0.850441i	$-0.676335\pi$
$173$	−13.8388	−1.05214	−0.526071	+	0.850441i	$0.676335\pi$
$174$	7.93900	0.601854
$175$	5.04892	0.381662
$176$	−4.74094	−0.357362
$177$	12.2784	0.922904
$178$	−14.8877	−1.11588
$179$	−14.2078	−1.06194	−0.530969	−	0.847392i	$-0.678172\pi$
$179$	−14.2078	−1.06194	−0.530969	+	0.847392i	$0.678172\pi$
$180$	1.00000	0.0745356
$181$	−24.9028	−1.85101	−0.925504	−	0.378739i	$-0.876358\pi$
$181$	−24.9028	−1.85101	−0.925504	+	0.378739i	$0.876358\pi$
$182$	0	0
$183$	−11.0707	−0.818369
$184$	−2.69202	−0.198458
$185$	−7.14675	−0.525440
$186$	−7.85086	−0.575653
$187$	5.15883	0.377251
$188$	7.00000	0.510527
$189$	5.04892	0.367255
$190$	−4.44504	−0.322477
$191$	2.44504	0.176917	0.0884585	−	0.996080i	$-0.471806\pi$
$191$	2.44504	0.176917	0.0884585	+	0.996080i	$0.471806\pi$
$192$	1.00000	0.0721688
$193$	−15.2959	−1.10102	−0.550511	−	0.834828i	$-0.685567\pi$
$193$	−15.2959	−1.10102	−0.550511	+	0.834828i	$0.685567\pi$
$194$	−14.9269	−1.07169
$195$	0	0
$196$	18.4916	1.32083
$197$	3.44935	0.245756	0.122878	−	0.992422i	$-0.460788\pi$
$197$	3.44935	0.245756	0.122878	+	0.992422i	$0.460788\pi$
$198$	4.74094	0.336924
$199$	−9.18598	−0.651177	−0.325588	−	0.945512i	$-0.605562\pi$
$199$	−9.18598	−0.651177	−0.325588	+	0.945512i	$0.605562\pi$
$200$	−1.00000	−0.0707107
$201$	9.58211	0.675870
$202$	−11.4819	−0.807862
$203$	−40.0834	−2.81330
$204$	−1.08815	−0.0761855
$205$	−0.664874	−0.0464368
$206$	−4.16421	−0.290134
$207$	2.69202	0.187108
$208$	0	0
$209$	−21.0737	−1.45770
$210$	−5.04892	−0.348408
$211$	−20.7875	−1.43107	−0.715534	−	0.698578i	$-0.753817\pi$
$211$	−20.7875	−1.43107	−0.715534	+	0.698578i	$0.753817\pi$
$212$	−10.8605	−0.745905
$213$	4.47219	0.306429
$214$	0.796561	0.0544518
$215$	4.35690	0.297138
$216$	−1.00000	−0.0680414
$217$	39.6383	2.69082
$218$	−0.704103	−0.0476879
$219$	−8.17390	−0.552341
$220$	−4.74094	−0.319634
$221$	0	0
$222$	7.14675	0.479659
$223$	17.3502	1.16185	0.580927	−	0.813955i	$-0.302690\pi$
$223$	17.3502	1.16185	0.580927	+	0.813955i	$0.302690\pi$
$224$	−5.04892	−0.337345
$225$	1.00000	0.0666667
$226$	−14.4983	−0.964411
$227$	−13.7573	−0.913106	−0.456553	−	0.889696i	$-0.650916\pi$
$227$	−13.7573	−0.913106	−0.456553	+	0.889696i	$0.650916\pi$
$228$	4.44504	0.294380
$229$	1.75541	0.116001	0.0580005	−	0.998317i	$-0.481528\pi$
$229$	1.75541	0.116001	0.0580005	+	0.998317i	$0.481528\pi$
$230$	−2.69202	−0.177507
$231$	−23.9366	−1.57491
$232$	7.93900	0.521221
$233$	−5.49396	−0.359921	−0.179961	−	0.983674i	$-0.557597\pi$
$233$	−5.49396	−0.359921	−0.179961	+	0.983674i	$0.557597\pi$
$234$	0	0
$235$	7.00000	0.456630
$236$	12.2784	0.799258
$237$	14.1468	0.918930
$238$	5.49396	0.356120
$239$	−17.4004	−1.12554	−0.562770	−	0.826613i	$-0.690264\pi$
$239$	−17.4004	−1.12554	−0.562770	+	0.826613i	$0.690264\pi$
$240$	1.00000	0.0645497
$241$	−3.15452	−0.203201	−0.101600	−	0.994825i	$-0.532396\pi$
$241$	−3.15452	−0.203201	−0.101600	+	0.994825i	$0.532396\pi$
$242$	−11.4765	−0.737737
$243$	1.00000	0.0641500
$244$	−11.0707	−0.708728
$245$	18.4916	1.18138
$246$	0.664874	0.0423908
$247$	0	0
$248$	−7.85086	−0.498530
$249$	4.86294	0.308176
$250$	−1.00000	−0.0632456
$251$	23.9433	1.51129	0.755644	−	0.654982i	$-0.227324\pi$
$251$	23.9433	1.51129	0.755644	+	0.654982i	$0.227324\pi$
$252$	5.04892	0.318052
$253$	−12.7627	−0.802385
$254$	2.93900	0.184409
$255$	−1.08815	−0.0681423
$256$	1.00000	0.0625000
$257$	12.3381	0.769630	0.384815	−	0.922994i	$-0.374265\pi$
$257$	12.3381	0.769630	0.384815	+	0.922994i	$0.374265\pi$
$258$	−4.35690	−0.271248
$259$	−36.0834	−2.24211
$260$	0	0
$261$	−7.93900	−0.491412
$262$	3.56465	0.220225
$263$	2.70709	0.166926	0.0834631	−	0.996511i	$-0.473402\pi$
$263$	2.70709	0.166926	0.0834631	+	0.996511i	$0.473402\pi$
$264$	4.74094	0.291785
$265$	−10.8605	−0.667158
$266$	−22.4426	−1.37605
$267$	14.8877	0.911112
$268$	9.58211	0.585320
$269$	20.9681	1.27845	0.639223	−	0.769022i	$-0.279256\pi$
$269$	20.9681	1.27845	0.639223	+	0.769022i	$0.279256\pi$
$270$	−1.00000	−0.0608581
$271$	−7.99761	−0.485820	−0.242910	−	0.970049i	$-0.578102\pi$
$271$	−7.99761	−0.485820	−0.242910	+	0.970049i	$0.578102\pi$
$272$	−1.08815	−0.0659785
$273$	0	0
$274$	9.47650	0.572496
$275$	−4.74094	−0.285889
$276$	2.69202	0.162041
$277$	−5.78554	−0.347620	−0.173810	−	0.984779i	$-0.555608\pi$
$277$	−5.78554	−0.347620	−0.173810	+	0.984779i	$0.555608\pi$
$278$	14.2078	0.852124
$279$	7.85086	0.470018
$280$	−5.04892	−0.301731
$281$	19.5579	1.16673	0.583365	−	0.812210i	$-0.301736\pi$
$281$	19.5579	1.16673	0.583365	+	0.812210i	$0.301736\pi$
$282$	−7.00000	−0.416844
$283$	31.0726	1.84707	0.923537	−	0.383508i	$-0.125284\pi$
$283$	31.0726	1.84707	0.923537	+	0.383508i	$0.125284\pi$
$284$	4.47219	0.265376
$285$	4.44504	0.263302
$286$	0	0
$287$	−3.35690	−0.198151
$288$	−1.00000	−0.0589256
$289$	−15.8159	−0.930349
$290$	7.93900	0.466194
$291$	14.9269	0.875032
$292$	−8.17390	−0.478341
$293$	−10.0543	−0.587378	−0.293689	−	0.955901i	$-0.594883\pi$
$293$	−10.0543	−0.587378	−0.293689	+	0.955901i	$0.594883\pi$
$294$	−18.4916	−1.07845
$295$	12.2784	0.714878
$296$	7.14675	0.415397
$297$	−4.74094	−0.275097
$298$	−10.2010	−0.590931
$299$	0	0
$300$	1.00000	0.0577350
$301$	21.9976	1.26792
$302$	−3.86294	−0.222287
$303$	11.4819	0.659617
$304$	4.44504	0.254941
$305$	−11.0707	−0.633906
$306$	1.08815	0.0622052
$307$	−23.6872	−1.35190	−0.675951	−	0.736947i	$-0.736267\pi$
$307$	−23.6872	−1.35190	−0.675951	+	0.736947i	$0.736267\pi$
$308$	−23.9366	−1.36391
$309$	4.16421	0.236894
$310$	−7.85086	−0.445899
$311$	6.77240	0.384027	0.192014	−	0.981392i	$-0.438498\pi$
$311$	6.77240	0.384027	0.192014	+	0.981392i	$0.438498\pi$
$312$	0	0
$313$	7.16123	0.404776	0.202388	−	0.979305i	$-0.435130\pi$
$313$	7.16123	0.404776	0.202388	+	0.979305i	$0.435130\pi$
$314$	11.2174	0.633037
$315$	5.04892	0.284474
$316$	14.1468	0.795817
$317$	−1.20344	−0.0675919	−0.0337959	−	0.999429i	$-0.510760\pi$
$317$	−1.20344	−0.0675919	−0.0337959	+	0.999429i	$0.510760\pi$
$318$	10.8605	0.609029
$319$	37.6383	2.10734
$320$	1.00000	0.0559017
$321$	−0.796561	−0.0444597
$322$	−13.5918	−0.757441
$323$	−4.83685	−0.269130
$324$	1.00000	0.0555556
$325$	0	0
$326$	6.50902	0.360502
$327$	0.704103	0.0389370
$328$	0.664874	0.0367115
$329$	35.3424	1.94849
$330$	4.74094	0.260980
$331$	20.9933	1.15390	0.576948	−	0.816781i	$-0.304244\pi$
$331$	20.9933	1.15390	0.576948	+	0.816781i	$0.304244\pi$
$332$	4.86294	0.266888
$333$	−7.14675	−0.391640
$334$	−9.67456	−0.529369
$335$	9.58211	0.523526
$336$	5.04892	0.275441
$337$	−3.01938	−0.164476	−0.0822380	−	0.996613i	$-0.526207\pi$
$337$	−3.01938	−0.164476	−0.0822380	+	0.996613i	$0.526207\pi$
$338$	0	0
$339$	14.4983	0.787438
$340$	−1.08815	−0.0590130
$341$	−37.2204	−2.01560
$342$	−4.44504	−0.240360
$343$	58.0200	3.13278
$344$	−4.35690	−0.234908
$345$	2.69202	0.144934
$346$	13.8388	0.743977
$347$	11.2610	0.604521	0.302261	−	0.953225i	$-0.402259\pi$
$347$	11.2610	0.604521	0.302261	+	0.953225i	$0.402259\pi$
$348$	−7.93900	−0.425575
$349$	−5.84010	−0.312613	−0.156307	−	0.987709i	$-0.549959\pi$
$349$	−5.84010	−0.312613	−0.156307	+	0.987709i	$0.549959\pi$
$350$	−5.04892	−0.269876
$351$	0	0
$352$	4.74094	0.252693
$353$	0.675628	0.0359601	0.0179800	−	0.999838i	$-0.494276\pi$
$353$	0.675628	0.0359601	0.0179800	+	0.999838i	$0.494276\pi$
$354$	−12.2784	−0.652592
$355$	4.47219	0.237359
$356$	14.8877	0.789046
$357$	−5.49396	−0.290771
$358$	14.2078	0.750903
$359$	−8.52350	−0.449853	−0.224927	−	0.974376i	$-0.572214\pi$
$359$	−8.52350	−0.449853	−0.224927	+	0.974376i	$0.572214\pi$
$360$	−1.00000	−0.0527046
$361$	0.758397	0.0399156
$362$	24.9028	1.30886
$363$	11.4765	0.602360
$364$	0	0
$365$	−8.17390	−0.427841
$366$	11.0707	0.578674
$367$	−9.40044	−0.490699	−0.245349	−	0.969435i	$-0.578903\pi$
$367$	−9.40044	−0.490699	−0.245349	+	0.969435i	$0.578903\pi$
$368$	2.69202	0.140331
$369$	−0.664874	−0.0346120
$370$	7.14675	0.371542
$371$	−54.8340	−2.84684
$372$	7.85086	0.407048
$373$	23.6582	1.22497	0.612487	−	0.790481i	$-0.290169\pi$
$373$	23.6582	1.22497	0.612487	+	0.790481i	$0.290169\pi$
$374$	−5.15883	−0.266757
$375$	1.00000	0.0516398
$376$	−7.00000	−0.360997
$377$	0	0
$378$	−5.04892	−0.259688
$379$	16.3437	0.839522	0.419761	−	0.907635i	$-0.362114\pi$
$379$	16.3437	0.839522	0.419761	+	0.907635i	$0.362114\pi$
$380$	4.44504	0.228026
$381$	−2.93900	−0.150570
$382$	−2.44504	−0.125099
$383$	−4.79656	−0.245093	−0.122546	−	0.992463i	$-0.539106\pi$
$383$	−4.79656	−0.245093	−0.122546	+	0.992463i	$0.539106\pi$
$384$	−1.00000	−0.0510310
$385$	−23.9366	−1.21992
$386$	15.2959	0.778541
$387$	4.35690	0.221473
$388$	14.9269	0.757800
$389$	21.4534	1.08773	0.543865	−	0.839173i	$-0.316960\pi$
$389$	21.4534	1.08773	0.543865	+	0.839173i	$0.316960\pi$
$390$	0	0
$391$	−2.92931	−0.148142
$392$	−18.4916	−0.933965
$393$	−3.56465	−0.179813
$394$	−3.44935	−0.173776
$395$	14.1468	0.711800
$396$	−4.74094	−0.238241
$397$	1.36765	0.0686404	0.0343202	−	0.999411i	$-0.489073\pi$
$397$	1.36765	0.0686404	0.0343202	+	0.999411i	$0.489073\pi$
$398$	9.18598	0.460452
$399$	22.4426	1.12354
$400$	1.00000	0.0500000
$401$	−18.8931	−0.943475	−0.471737	−	0.881739i	$-0.656373\pi$
$401$	−18.8931	−0.943475	−0.471737	+	0.881739i	$0.656373\pi$
$402$	−9.58211	−0.477912
$403$	0	0
$404$	11.4819	0.571245
$405$	1.00000	0.0496904
$406$	40.0834	1.98930
$407$	33.8823	1.67948
$408$	1.08815	0.0538713
$409$	−25.9638	−1.28383	−0.641913	−	0.766778i	$-0.721859\pi$
$409$	−25.9638	−1.28383	−0.641913	+	0.766778i	$0.721859\pi$
$410$	0.664874	0.0328358
$411$	−9.47650	−0.467441
$412$	4.16421	0.205156
$413$	61.9928	3.05047
$414$	−2.69202	−0.132306
$415$	4.86294	0.238712
$416$	0	0
$417$	−14.2078	−0.695757
$418$	21.0737	1.03075
$419$	−32.4282	−1.58422	−0.792110	−	0.610378i	$-0.791017\pi$
$419$	−32.4282	−1.58422	−0.792110	+	0.610378i	$0.791017\pi$
$420$	5.04892	0.246362
$421$	−33.2868	−1.62230	−0.811150	−	0.584839i	$-0.801158\pi$
$421$	−33.2868	−1.62230	−0.811150	+	0.584839i	$0.801158\pi$
$422$	20.7875	1.01192
$423$	7.00000	0.340352
$424$	10.8605	0.527435
$425$	−1.08815	−0.0527828
$426$	−4.47219	−0.216678
$427$	−55.8950	−2.70495
$428$	−0.796561	−0.0385032
$429$	0	0
$430$	−4.35690	−0.210108
$431$	11.1578	0.537451	0.268725	−	0.963217i	$-0.413398\pi$
$431$	11.1578	0.537451	0.268725	+	0.963217i	$0.413398\pi$
$432$	1.00000	0.0481125
$433$	14.0828	0.676775	0.338387	−	0.941007i	$-0.390119\pi$
$433$	14.0828	0.676775	0.338387	+	0.941007i	$0.390119\pi$
$434$	−39.6383	−1.90270
$435$	−7.93900	−0.380646
$436$	0.704103	0.0337204
$437$	11.9661	0.572418
$438$	8.17390	0.390564
$439$	−28.9933	−1.38377	−0.691887	−	0.722006i	$-0.743220\pi$
$439$	−28.9933	−1.38377	−0.691887	+	0.722006i	$0.743220\pi$
$440$	4.74094	0.226015
$441$	18.4916	0.880551
$442$	0	0
$443$	33.3080	1.58251	0.791255	−	0.611486i	$-0.209428\pi$
$443$	33.3080	1.58251	0.791255	+	0.611486i	$0.209428\pi$
$444$	−7.14675	−0.339170
$445$	14.8877	0.705744
$446$	−17.3502	−0.821555
$447$	10.2010	0.482493
$448$	5.04892	0.238539
$449$	−37.7899	−1.78341	−0.891707	−	0.452614i	$-0.850492\pi$
$449$	−37.7899	−1.78341	−0.891707	+	0.452614i	$0.850492\pi$
$450$	−1.00000	−0.0471405
$451$	3.15213	0.148428
$452$	14.4983	0.681941
$453$	3.86294	0.181497
$454$	13.7573	0.645664
$455$	0	0
$456$	−4.44504	−0.208158
$457$	−10.5700	−0.494445	−0.247222	−	0.968959i	$-0.579518\pi$
$457$	−10.5700	−0.494445	−0.247222	+	0.968959i	$0.579518\pi$
$458$	−1.75541	−0.0820251
$459$	−1.08815	−0.0507903
$460$	2.69202	0.125516
$461$	−20.1420	−0.938105	−0.469052	−	0.883170i	$-0.655404\pi$
$461$	−20.1420	−0.938105	−0.469052	+	0.883170i	$0.655404\pi$
$462$	23.9366	1.11363
$463$	15.9065	0.739237	0.369618	−	0.929184i	$-0.379488\pi$
$463$	15.9065	0.739237	0.369618	+	0.929184i	$0.379488\pi$
$464$	−7.93900	−0.368559
$465$	7.85086	0.364075
$466$	5.49396	0.254503
$467$	−25.6136	−1.18525	−0.592627	−	0.805477i	$-0.701909\pi$
$467$	−25.6136	−1.18525	−0.592627	+	0.805477i	$0.701909\pi$
$468$	0	0
$469$	48.3793	2.23395
$470$	−7.00000	−0.322886
$471$	−11.2174	−0.516872
$472$	−12.2784	−0.565161
$473$	−20.6558	−0.949754
$474$	−14.1468	−0.649782
$475$	4.44504	0.203953
$476$	−5.49396	−0.251815
$477$	−10.8605	−0.497270
$478$	17.4004	0.795877
$479$	−7.59658	−0.347097	−0.173548	−	0.984825i	$-0.555523\pi$
$479$	−7.59658	−0.347097	−0.173548	+	0.984825i	$0.555523\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	3.15452	0.143685
$483$	13.5918	0.618448
$484$	11.4765	0.521659
$485$	14.9269	0.677796
$486$	−1.00000	−0.0453609
$487$	0.779103	0.0353045	0.0176523	−	0.999844i	$-0.494381\pi$
$487$	0.779103	0.0353045	0.0176523	+	0.999844i	$0.494381\pi$
$488$	11.0707	0.501146
$489$	−6.50902	−0.294348
$490$	−18.4916	−0.835364
$491$	11.3948	0.514240	0.257120	−	0.966379i	$-0.417226\pi$
$491$	11.3948	0.514240	0.257120	+	0.966379i	$0.417226\pi$
$492$	−0.664874	−0.0299749
$493$	8.63879	0.389072
$494$	0	0
$495$	−4.74094	−0.213089
$496$	7.85086	0.352514
$497$	22.5797	1.01284
$498$	−4.86294	−0.217913
$499$	23.5555	1.05449	0.527246	−	0.849713i	$-0.323225\pi$
$499$	23.5555	1.05449	0.527246	+	0.849713i	$0.323225\pi$
$500$	1.00000	0.0447214
$501$	9.67456	0.432228
$502$	−23.9433	−1.06864
$503$	−12.6165	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$503$	−12.6165	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$504$	−5.04892	−0.224897
$505$	11.4819	0.510937
$506$	12.7627	0.567372
$507$	0	0
$508$	−2.93900	−0.130397
$509$	−8.45473	−0.374749	−0.187375	−	0.982289i	$-0.559998\pi$
$509$	−8.45473	−0.374749	−0.187375	+	0.982289i	$0.559998\pi$
$510$	1.08815	0.0481839
$511$	−41.2693	−1.82565
$512$	−1.00000	−0.0441942
$513$	4.44504	0.196253
$514$	−12.3381	−0.544211
$515$	4.16421	0.183497
$516$	4.35690	0.191802
$517$	−33.1866	−1.45954
$518$	36.0834	1.58541
$519$	−13.8388	−0.607455
$520$	0	0
$521$	34.7254	1.52135	0.760674	−	0.649134i	$-0.224869\pi$
$521$	34.7254	1.52135	0.760674	+	0.649134i	$0.224869\pi$
$522$	7.93900	0.347481
$523$	29.3163	1.28191	0.640957	−	0.767577i	$-0.278538\pi$
$523$	29.3163	1.28191	0.640957	+	0.767577i	$0.278538\pi$
$524$	−3.56465	−0.155722
$525$	5.04892	0.220353
$526$	−2.70709	−0.118035
$527$	−8.54288	−0.372134
$528$	−4.74094	−0.206323
$529$	−15.7530	−0.684914
$530$	10.8605	0.471752
$531$	12.2784	0.532839
$532$	22.4426	0.973012
$533$	0	0
$534$	−14.8877	−0.644253
$535$	−0.796561	−0.0344383
$536$	−9.58211	−0.413884
$537$	−14.2078	−0.613110
$538$	−20.9681	−0.903998
$539$	−87.6674	−3.77610
$540$	1.00000	0.0430331
$541$	29.1032	1.25124	0.625622	−	0.780126i	$-0.284845\pi$
$541$	29.1032	1.25124	0.625622	+	0.780126i	$0.284845\pi$
$542$	7.99761	0.343527
$543$	−24.9028	−1.06868
$544$	1.08815	0.0466539
$545$	0.704103	0.0301605
$546$	0	0
$547$	−2.37004	−0.101336	−0.0506678	−	0.998716i	$-0.516135\pi$
$547$	−2.37004	−0.101336	−0.0506678	+	0.998716i	$0.516135\pi$
$548$	−9.47650	−0.404816
$549$	−11.0707	−0.472485
$550$	4.74094	0.202154
$551$	−35.2892	−1.50337
$552$	−2.69202	−0.114580
$553$	71.4258	3.03733
$554$	5.78554	0.245804
$555$	−7.14675	−0.303363
$556$	−14.2078	−0.602543
$557$	28.8364	1.22184	0.610918	−	0.791694i	$-0.290800\pi$
$557$	28.8364	1.22184	0.610918	+	0.791694i	$0.290800\pi$
$558$	−7.85086	−0.332353
$559$	0	0
$560$	5.04892	0.213356
$561$	5.15883	0.217806
$562$	−19.5579	−0.825002
$563$	−35.7133	−1.50514	−0.752568	−	0.658514i	$-0.771185\pi$
$563$	−35.7133	−1.50514	−0.752568	+	0.658514i	$0.771185\pi$
$564$	7.00000	0.294753
$565$	14.4983	0.609947
$566$	−31.0726	−1.30608
$567$	5.04892	0.212035
$568$	−4.47219	−0.187649
$569$	42.1618	1.76752	0.883758	−	0.467945i	$-0.155005\pi$
$569$	42.1618	1.76752	0.883758	+	0.467945i	$0.155005\pi$
$570$	−4.44504	−0.186182
$571$	23.1497	0.968786	0.484393	−	0.874850i	$-0.339040\pi$
$571$	23.1497	0.968786	0.484393	+	0.874850i	$0.339040\pi$
$572$	0	0
$573$	2.44504	0.102143
$574$	3.35690	0.140114
$575$	2.69202	0.112265
$576$	1.00000	0.0416667
$577$	−41.7318	−1.73732	−0.868660	−	0.495409i	$-0.835018\pi$
$577$	−41.7318	−1.73732	−0.868660	+	0.495409i	$0.835018\pi$
$578$	15.8159	0.657856
$579$	−15.2959	−0.635676
$580$	−7.93900	−0.329649
$581$	24.5526	1.01861
$582$	−14.9269	−0.618741
$583$	51.4892	2.13246
$584$	8.17390	0.338238
$585$	0	0
$586$	10.0543	0.415339
$587$	−3.52648	−0.145554	−0.0727768	−	0.997348i	$-0.523186\pi$
$587$	−3.52648	−0.145554	−0.0727768	+	0.997348i	$0.523186\pi$
$588$	18.4916	0.762579
$589$	34.8974	1.43792
$590$	−12.2784	−0.505495
$591$	3.44935	0.141887
$592$	−7.14675	−0.293730
$593$	7.53617	0.309473	0.154737	−	0.987956i	$-0.450547\pi$
$593$	7.53617	0.309473	0.154737	+	0.987956i	$0.450547\pi$
$594$	4.74094	0.194523
$595$	−5.49396	−0.225230
$596$	10.2010	0.417851
$597$	−9.18598	−0.375957
$598$	0	0
$599$	−44.2161	−1.80662	−0.903311	−	0.428987i	$-0.858871\pi$
$599$	−44.2161	−1.80662	−0.903311	+	0.428987i	$0.858871\pi$
$600$	−1.00000	−0.0408248
$601$	−40.7251	−1.66121	−0.830606	−	0.556860i	$-0.812006\pi$
$601$	−40.7251	−1.66121	−0.830606	+	0.556860i	$0.812006\pi$
$602$	−21.9976	−0.896556
$603$	9.58211	0.390213
$604$	3.86294	0.157181
$605$	11.4765	0.466586
$606$	−11.4819	−0.466419
$607$	40.9788	1.66328	0.831640	−	0.555316i	$-0.187403\pi$
$607$	40.9788	1.66328	0.831640	+	0.555316i	$0.187403\pi$
$608$	−4.44504	−0.180270
$609$	−40.0834	−1.62426
$610$	11.0707	0.448239
$611$	0	0
$612$	−1.08815	−0.0439857
$613$	−40.5491	−1.63776	−0.818882	−	0.573963i	$-0.805405\pi$
$613$	−40.5491	−1.63776	−0.818882	+	0.573963i	$0.805405\pi$
$614$	23.6872	0.955939
$615$	−0.664874	−0.0268103
$616$	23.9366	0.964433
$617$	−3.06829	−0.123525	−0.0617624	−	0.998091i	$-0.519672\pi$
$617$	−3.06829	−0.123525	−0.0617624	+	0.998091i	$0.519672\pi$
$618$	−4.16421	−0.167509
$619$	−31.0847	−1.24940	−0.624700	−	0.780865i	$-0.714779\pi$
$619$	−31.0847	−1.24940	−0.624700	+	0.780865i	$0.714779\pi$
$620$	7.85086	0.315298
$621$	2.69202	0.108027
$622$	−6.77240	−0.271548
$623$	75.1667	3.01149
$624$	0	0
$625$	1.00000	0.0400000
$626$	−7.16123	−0.286220
$627$	−21.0737	−0.841601
$628$	−11.2174	−0.447625
$629$	7.77671	0.310078
$630$	−5.04892	−0.201154
$631$	23.7101	0.943883	0.471942	−	0.881630i	$-0.343553\pi$
$631$	23.7101	0.943883	0.471942	+	0.881630i	$0.343553\pi$
$632$	−14.1468	−0.562728
$633$	−20.7875	−0.826227
$634$	1.20344	0.0477947
$635$	−2.93900	−0.116631
$636$	−10.8605	−0.430649
$637$	0	0
$638$	−37.6383	−1.49012
$639$	4.47219	0.176917
$640$	−1.00000	−0.0395285
$641$	−48.3866	−1.91115	−0.955577	−	0.294742i	$-0.904766\pi$
$641$	−48.3866	−1.91115	−0.955577	+	0.294742i	$0.904766\pi$
$642$	0.796561	0.0314378
$643$	20.4209	0.805321	0.402660	−	0.915349i	$-0.368086\pi$
$643$	20.4209	0.805321	0.402660	+	0.915349i	$0.368086\pi$
$644$	13.5918	0.535592
$645$	4.35690	0.171553
$646$	4.83685	0.190303
$647$	−0.292913	−0.0115156	−0.00575780	−	0.999983i	$-0.501833\pi$
$647$	−0.292913	−0.0115156	−0.00575780	+	0.999983i	$0.501833\pi$
$648$	−1.00000	−0.0392837
$649$	−58.2113	−2.28499
$650$	0	0
$651$	39.6383	1.55355
$652$	−6.50902	−0.254913
$653$	21.8323	0.854365	0.427183	−	0.904165i	$-0.359506\pi$
$653$	21.8323	0.854365	0.427183	+	0.904165i	$0.359506\pi$
$654$	−0.704103	−0.0275326
$655$	−3.56465	−0.139282
$656$	−0.664874	−0.0259590
$657$	−8.17390	−0.318894
$658$	−35.3424	−1.37779
$659$	−17.0519	−0.664248	−0.332124	−	0.943236i	$-0.607765\pi$
$659$	−17.0519	−0.664248	−0.332124	+	0.943236i	$0.607765\pi$
$660$	−4.74094	−0.184541
$661$	−31.8098	−1.23726	−0.618629	−	0.785683i	$-0.712312\pi$
$661$	−31.8098	−1.23726	−0.618629	+	0.785683i	$0.712312\pi$
$662$	−20.9933	−0.815928
$663$	0	0
$664$	−4.86294	−0.188719
$665$	22.4426	0.870289
$666$	7.14675	0.276931
$667$	−21.3720	−0.827526
$668$	9.67456	0.374320
$669$	17.3502	0.670797
$670$	−9.58211	−0.370189
$671$	52.4855	2.02618
$672$	−5.04892	−0.194766
$673$	−4.10513	−0.158241	−0.0791206	−	0.996865i	$-0.525211\pi$
$673$	−4.10513	−0.158241	−0.0791206	+	0.996865i	$0.525211\pi$
$674$	3.01938	0.116302
$675$	1.00000	0.0384900
$676$	0	0
$677$	11.0543	0.424851	0.212426	−	0.977177i	$-0.431864\pi$
$677$	11.0543	0.424851	0.212426	+	0.977177i	$0.431864\pi$
$678$	−14.4983	−0.556803
$679$	75.3648	2.89223
$680$	1.08815	0.0417285
$681$	−13.7573	−0.527182
$682$	37.2204	1.42524
$683$	−9.06590	−0.346897	−0.173449	−	0.984843i	$-0.555491\pi$
$683$	−9.06590	−0.346897	−0.173449	+	0.984843i	$0.555491\pi$
$684$	4.44504	0.169960
$685$	−9.47650	−0.362078
$686$	−58.0200	−2.21521
$687$	1.75541	0.0669732
$688$	4.35690	0.166105
$689$	0	0
$690$	−2.69202	−0.102484
$691$	−12.4282	−0.472790	−0.236395	−	0.971657i	$-0.575966\pi$
$691$	−12.4282	−0.472790	−0.236395	+	0.971657i	$0.575966\pi$
$692$	−13.8388	−0.526071
$693$	−23.9366	−0.909277
$694$	−11.2610	−0.427461
$695$	−14.2078	−0.538931
$696$	7.93900	0.300927
$697$	0.723480	0.0274038
$698$	5.84010	0.221051
$699$	−5.49396	−0.207801
$700$	5.04892	0.190831
$701$	−38.5418	−1.45570	−0.727852	−	0.685734i	$-0.759481\pi$
$701$	−38.5418	−1.45570	−0.727852	+	0.685734i	$0.759481\pi$
$702$	0	0
$703$	−31.7676	−1.19814
$704$	−4.74094	−0.178681
$705$	7.00000	0.263635
$706$	−0.675628	−0.0254276
$707$	57.9711	2.18023
$708$	12.2784	0.461452
$709$	38.7851	1.45660	0.728302	−	0.685256i	$-0.240310\pi$
$709$	38.7851	1.45660	0.728302	+	0.685256i	$0.240310\pi$
$710$	−4.47219	−0.167838
$711$	14.1468	0.530545
$712$	−14.8877	−0.557940
$713$	21.1347	0.791500
$714$	5.49396	0.205606
$715$	0	0
$716$	−14.2078	−0.530969
$717$	−17.4004	−0.649831
$718$	8.52350	0.318094
$719$	−12.1021	−0.451334	−0.225667	−	0.974205i	$-0.572456\pi$
$719$	−12.1021	−0.451334	−0.225667	+	0.974205i	$0.572456\pi$
$720$	1.00000	0.0372678
$721$	21.0248	0.783003
$722$	−0.758397	−0.0282246
$723$	−3.15452	−0.117318
$724$	−24.9028	−0.925504
$725$	−7.93900	−0.294847
$726$	−11.4765	−0.425933
$727$	40.6088	1.50610	0.753048	−	0.657965i	$-0.228583\pi$
$727$	40.6088	1.50610	0.753048	+	0.657965i	$0.228583\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	8.17390	0.302530
$731$	−4.74094	−0.175350
$732$	−11.0707	−0.409184
$733$	32.6601	1.20633	0.603164	−	0.797617i	$-0.293906\pi$
$733$	32.6601	1.20633	0.603164	+	0.797617i	$0.293906\pi$
$734$	9.40044	0.346976
$735$	18.4916	0.682072
$736$	−2.69202	−0.0992292
$737$	−45.4282	−1.67337
$738$	0.664874	0.0244744
$739$	3.72348	0.136970	0.0684852	−	0.997652i	$-0.478183\pi$
$739$	3.72348	0.136970	0.0684852	+	0.997652i	$0.478183\pi$
$740$	−7.14675	−0.262720
$741$	0	0
$742$	54.8340	2.01302
$743$	−31.3099	−1.14865	−0.574324	−	0.818628i	$-0.694735\pi$
$743$	−31.3099	−1.14865	−0.574324	+	0.818628i	$0.694735\pi$
$744$	−7.85086	−0.287826
$745$	10.2010	0.373737
$746$	−23.6582	−0.866187
$747$	4.86294	0.177926
$748$	5.15883	0.188626
$749$	−4.02177	−0.146952
$750$	−1.00000	−0.0365148
$751$	2.66296	0.0971726	0.0485863	−	0.998819i	$-0.484528\pi$
$751$	2.66296	0.0971726	0.0485863	+	0.998819i	$0.484528\pi$
$752$	7.00000	0.255264
$753$	23.9433	0.872543
$754$	0	0
$755$	3.86294	0.140587
$756$	5.04892	0.183627
$757$	32.2553	1.17234	0.586170	−	0.810188i	$-0.300635\pi$
$757$	32.2553	1.17234	0.586170	+	0.810188i	$0.300635\pi$
$758$	−16.3437	−0.593632
$759$	−12.7627	−0.463257
$760$	−4.44504	−0.161239
$761$	27.2228	0.986826	0.493413	−	0.869795i	$-0.335749\pi$
$761$	27.2228	0.986826	0.493413	+	0.869795i	$0.335749\pi$
$762$	2.93900	0.106469
$763$	3.55496	0.128698
$764$	2.44504	0.0884585
$765$	−1.08815	−0.0393420
$766$	4.79656	0.173307
$767$	0	0
$768$	1.00000	0.0360844
$769$	−5.62133	−0.202710	−0.101355	−	0.994850i	$-0.532318\pi$
$769$	−5.62133	−0.202710	−0.101355	+	0.994850i	$0.532318\pi$
$770$	23.9366	0.862615
$771$	12.3381	0.444346
$772$	−15.2959	−0.550511
$773$	35.7918	1.28734	0.643670	−	0.765303i	$-0.277411\pi$
$773$	35.7918	1.28734	0.643670	+	0.765303i	$0.277411\pi$
$774$	−4.35690	−0.156605
$775$	7.85086	0.282011
$776$	−14.9269	−0.535845
$777$	−36.0834	−1.29448
$778$	−21.4534	−0.769142
$779$	−2.95539	−0.105888
$780$	0	0
$781$	−21.2024	−0.758681
$782$	2.92931	0.104752
$783$	−7.93900	−0.283717
$784$	18.4916	0.660413
$785$	−11.2174	−0.400368
$786$	3.56465	0.127147
$787$	−18.5442	−0.661030	−0.330515	−	0.943801i	$-0.607222\pi$
$787$	−18.5442	−0.661030	−0.330515	+	0.943801i	$0.607222\pi$
$788$	3.44935	0.122878
$789$	2.70709	0.0963748
$790$	−14.1468	−0.503319
$791$	73.2006	2.60271
$792$	4.74094	0.168462
$793$	0	0
$794$	−1.36765	−0.0485361
$795$	−10.8605	−0.385184
$796$	−9.18598	−0.325588
$797$	−33.0898	−1.17210	−0.586050	−	0.810275i	$-0.699318\pi$
$797$	−33.0898	−1.17210	−0.586050	+	0.810275i	$0.699318\pi$
$798$	−22.4426	−0.794461
$799$	−7.61702	−0.269471
$800$	−1.00000	−0.0353553
$801$	14.8877	0.526031
$802$	18.8931	0.667137
$803$	38.7520	1.36753
$804$	9.58211	0.337935
$805$	13.5918	0.479048
$806$	0	0
$807$	20.9681	0.738111
$808$	−11.4819	−0.403931
$809$	11.4004	0.400818	0.200409	−	0.979712i	$-0.435773\pi$
$809$	11.4004	0.400818	0.200409	+	0.979712i	$0.435773\pi$
$810$	−1.00000	−0.0351364
$811$	38.9724	1.36851	0.684253	−	0.729245i	$-0.260129\pi$
$811$	38.9724	1.36851	0.684253	+	0.729245i	$0.260129\pi$
$812$	−40.0834	−1.40665
$813$	−7.99761	−0.280488
$814$	−33.8823	−1.18757
$815$	−6.50902	−0.228001
$816$	−1.08815	−0.0380927
$817$	19.3666	0.677551
$818$	25.9638	0.907801
$819$	0	0
$820$	−0.664874	−0.0232184
$821$	−7.12498	−0.248664	−0.124332	−	0.992241i	$-0.539679\pi$
$821$	−7.12498	−0.248664	−0.124332	+	0.992241i	$0.539679\pi$
$822$	9.47650	0.330531
$823$	19.1323	0.666909	0.333455	−	0.942766i	$-0.391786\pi$
$823$	19.1323	0.666909	0.333455	+	0.942766i	$0.391786\pi$
$824$	−4.16421	−0.145067
$825$	−4.74094	−0.165058
$826$	−61.9928	−2.15701
$827$	42.0538	1.46235	0.731177	−	0.682187i	$-0.238971\pi$
$827$	42.0538	1.46235	0.731177	+	0.682187i	$0.238971\pi$
$828$	2.69202	0.0935542
$829$	−17.6364	−0.612537	−0.306269	−	0.951945i	$-0.599081\pi$
$829$	−17.6364	−0.612537	−0.306269	+	0.951945i	$0.599081\pi$
$830$	−4.86294	−0.168795
$831$	−5.78554	−0.200698
$832$	0	0
$833$	−20.1215	−0.697169
$834$	14.2078	0.491974
$835$	9.67456	0.334802
$836$	−21.0737	−0.728848
$837$	7.85086	0.271365
$838$	32.4282	1.12021
$839$	−44.4905	−1.53598	−0.767991	−	0.640460i	$-0.778744\pi$
$839$	−44.4905	−1.53598	−0.767991	+	0.640460i	$0.778744\pi$
$840$	−5.04892	−0.174204
$841$	34.0277	1.17337
$842$	33.2868	1.14714
$843$	19.5579	0.673611
$844$	−20.7875	−0.715534
$845$	0	0
$846$	−7.00000	−0.240665
$847$	57.9439	1.99098
$848$	−10.8605	−0.372953
$849$	31.0726	1.06641
$850$	1.08815	0.0373231
$851$	−19.2392	−0.659512
$852$	4.47219	0.153215
$853$	7.59743	0.260131	0.130066	−	0.991505i	$-0.458481\pi$
$853$	7.59743	0.260131	0.130066	+	0.991505i	$0.458481\pi$
$854$	55.8950	1.91269
$855$	4.44504	0.152017
$856$	0.796561	0.0272259
$857$	−34.2978	−1.17159	−0.585796	−	0.810459i	$-0.699218\pi$
$857$	−34.2978	−1.17159	−0.585796	+	0.810459i	$0.699218\pi$
$858$	0	0
$859$	−30.8412	−1.05229	−0.526144	−	0.850396i	$-0.676363\pi$
$859$	−30.8412	−1.05229	−0.526144	+	0.850396i	$0.676363\pi$
$860$	4.35690	0.148569
$861$	−3.35690	−0.114403
$862$	−11.1578	−0.380035
$863$	−16.1075	−0.548306	−0.274153	−	0.961686i	$-0.588398\pi$
$863$	−16.1075	−0.548306	−0.274153	+	0.961686i	$0.588398\pi$
$864$	−1.00000	−0.0340207
$865$	−13.8388	−0.470532
$866$	−14.0828	−0.478552
$867$	−15.8159	−0.537137
$868$	39.6383	1.34541
$869$	−67.0689	−2.27516
$870$	7.93900	0.269157
$871$	0	0
$872$	−0.704103	−0.0238439
$873$	14.9269	0.505200
$874$	−11.9661	−0.404761
$875$	5.04892	0.170685
$876$	−8.17390	−0.276170
$877$	−32.7342	−1.10536	−0.552678	−	0.833395i	$-0.686394\pi$
$877$	−32.7342	−1.10536	−0.552678	+	0.833395i	$0.686394\pi$
$878$	28.9933	0.978476
$879$	−10.0543	−0.339123
$880$	−4.74094	−0.159817
$881$	−9.32736	−0.314247	−0.157123	−	0.987579i	$-0.550222\pi$
$881$	−9.32736	−0.314247	−0.157123	+	0.987579i	$0.550222\pi$
$882$	−18.4916	−0.622643
$883$	45.1135	1.51819	0.759095	−	0.650980i	$-0.225642\pi$
$883$	45.1135	1.51819	0.759095	+	0.650980i	$0.225642\pi$
$884$	0	0
$885$	12.2784	0.412735
$886$	−33.3080	−1.11900
$887$	−37.4383	−1.25706	−0.628528	−	0.777787i	$-0.716342\pi$
$887$	−37.4383	−1.25706	−0.628528	+	0.777787i	$0.716342\pi$
$888$	7.14675	0.239829
$889$	−14.8388	−0.497676
$890$	−14.8877	−0.499037
$891$	−4.74094	−0.158827
$892$	17.3502	0.580927
$893$	31.1153	1.04123
$894$	−10.2010	−0.341174
$895$	−14.2078	−0.474913
$896$	−5.04892	−0.168672
$897$	0	0
$898$	37.7899	1.26106
$899$	−62.3279	−2.07875
$900$	1.00000	0.0333333
$901$	11.8179	0.393710
$902$	−3.15213	−0.104954
$903$	21.9976	0.732035
$904$	−14.4983	−0.482205
$905$	−24.9028	−0.827796
$906$	−3.86294	−0.128337
$907$	−35.9004	−1.19205	−0.596026	−	0.802965i	$-0.703254\pi$
$907$	−35.9004	−1.19205	−0.596026	+	0.802965i	$0.703254\pi$
$908$	−13.7573	−0.456553
$909$	11.4819	0.380830
$910$	0	0
$911$	34.0025	1.12655	0.563277	−	0.826268i	$-0.309541\pi$
$911$	34.0025	1.12655	0.563277	+	0.826268i	$0.309541\pi$
$912$	4.44504	0.147190
$913$	−23.0549	−0.763005
$914$	10.5700	0.349625
$915$	−11.0707	−0.365986
$916$	1.75541	0.0580005
$917$	−17.9976	−0.594333
$918$	1.08815	0.0359142
$919$	−27.7778	−0.916304	−0.458152	−	0.888874i	$-0.651489\pi$
$919$	−27.7778	−0.916304	−0.458152	+	0.888874i	$0.651489\pi$
$920$	−2.69202	−0.0887533
$921$	−23.6872	−0.780521
$922$	20.1420	0.663340
$923$	0	0
$924$	−23.9366	−0.787457
$925$	−7.14675	−0.234984
$926$	−15.9065	−0.522719
$927$	4.16421	0.136771
$928$	7.93900	0.260610
$929$	−44.9342	−1.47424	−0.737122	−	0.675760i	$-0.763816\pi$
$929$	−44.9342	−1.47424	−0.737122	+	0.675760i	$0.763816\pi$
$930$	−7.85086	−0.257440
$931$	82.1958	2.69386
$932$	−5.49396	−0.179961
$933$	6.77240	0.221718
$934$	25.6136	0.838101
$935$	5.15883	0.168712
$936$	0	0
$937$	−4.50843	−0.147284	−0.0736421	−	0.997285i	$-0.523462\pi$
$937$	−4.50843	−0.147284	−0.0736421	+	0.997285i	$0.523462\pi$
$938$	−48.3793	−1.57964
$939$	7.16123	0.233698
$940$	7.00000	0.228315
$941$	−18.4926	−0.602843	−0.301421	−	0.953491i	$-0.597461\pi$
$941$	−18.4926	−0.602843	−0.301421	+	0.953491i	$0.597461\pi$
$942$	11.2174	0.365484
$943$	−1.78986	−0.0582857
$944$	12.2784	0.399629
$945$	5.04892	0.164241
$946$	20.6558	0.671577
$947$	−60.3062	−1.95969	−0.979844	−	0.199766i	$-0.935982\pi$
$947$	−60.3062	−1.95969	−0.979844	+	0.199766i	$0.935982\pi$
$948$	14.1468	0.459465
$949$	0	0
$950$	−4.44504	−0.144216
$951$	−1.20344	−0.0390242
$952$	5.49396	0.178060
$953$	23.6394	0.765755	0.382877	−	0.923799i	$-0.374933\pi$
$953$	23.6394	0.765755	0.382877	+	0.923799i	$0.374933\pi$
$954$	10.8605	0.351623
$955$	2.44504	0.0791197
$956$	−17.4004	−0.562770
$957$	37.6383	1.21667
$958$	7.59658	0.245434
$959$	−47.8461	−1.54503
$960$	1.00000	0.0322749
$961$	30.6359	0.988256
$962$	0	0
$963$	−0.796561	−0.0256688
$964$	−3.15452	−0.101600
$965$	−15.2959	−0.492392
$966$	−13.5918	−0.437309
$967$	−22.4910	−0.723261	−0.361631	−	0.932321i	$-0.617780\pi$
$967$	−22.4910	−0.723261	−0.361631	+	0.932321i	$0.617780\pi$
$968$	−11.4765	−0.368869
$969$	−4.83685	−0.155382
$970$	−14.9269	−0.479275
$971$	−15.4257	−0.495033	−0.247517	−	0.968884i	$-0.579614\pi$
$971$	−15.4257	−0.495033	−0.247517	+	0.968884i	$0.579614\pi$
$972$	1.00000	0.0320750
$973$	−71.7338	−2.29968
$974$	−0.779103	−0.0249641
$975$	0	0
$976$	−11.0707	−0.354364
$977$	38.5018	1.23178	0.615892	−	0.787831i	$-0.288796\pi$
$977$	38.5018	1.23178	0.615892	+	0.787831i	$0.288796\pi$
$978$	6.50902	0.208136
$979$	−70.5816	−2.25580
$980$	18.4916	0.590691
$981$	0.704103	0.0224803
$982$	−11.3948	−0.363623
$983$	3.83340	0.122266	0.0611332	−	0.998130i	$-0.480529\pi$
$983$	3.83340	0.122266	0.0611332	+	0.998130i	$0.480529\pi$
$984$	0.664874	0.0211954
$985$	3.44935	0.109906
$986$	−8.63879	−0.275115
$987$	35.3424	1.12496
$988$	0	0
$989$	11.7289	0.372956
$990$	4.74094	0.150677
$991$	−46.7670	−1.48560	−0.742802	−	0.669512i	$-0.766503\pi$
$991$	−46.7670	−1.48560	−0.742802	+	0.669512i	$0.766503\pi$
$992$	−7.85086	−0.249265
$993$	20.9933	0.666202
$994$	−22.5797	−0.716185
$995$	−9.18598	−0.291215
$996$	4.86294	0.154088
$997$	−42.8920	−1.35840	−0.679202	−	0.733952i	$-0.737674\pi$
$997$	−42.8920	−1.35840	−0.679202	+	0.733952i	$0.737674\pi$
$998$	−23.5555	−0.745638
$999$	−7.14675	−0.226113

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bq.1.3	✓	3
13.5	odd	4		5070.2.b.y.1351.6		6
13.8	odd	4		5070.2.b.y.1351.1		6
13.12	even	2		5070.2.a.bv.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bq.1.3	✓	3	1.1	even	1	trivial
5070.2.a.bv.1.1	yes	3	13.12	even	2
5070.2.b.y.1351.1		6	13.8	odd	4
5070.2.b.y.1351.6		6	13.5	odd	4

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.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +5.04892 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +5.04892 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.74094 q^{11} +1.00000 q^{12} -5.04892 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.08815 q^{17} -1.00000 q^{18} +4.44504 q^{19} +1.00000 q^{20} +5.04892 q^{21} +4.74094 q^{22} +2.69202 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +5.04892 q^{28} -7.93900 q^{29} -1.00000 q^{30} +7.85086 q^{31} -1.00000 q^{32} -4.74094 q^{33} +1.08815 q^{34} +5.04892 q^{35} +1.00000 q^{36} -7.14675 q^{37} -4.44504 q^{38} -1.00000 q^{40} -0.664874 q^{41} -5.04892 q^{42} +4.35690 q^{43} -4.74094 q^{44} +1.00000 q^{45} -2.69202 q^{46} +7.00000 q^{47} +1.00000 q^{48} +18.4916 q^{49} -1.00000 q^{50} -1.08815 q^{51} -10.8605 q^{53} -1.00000 q^{54} -4.74094 q^{55} -5.04892 q^{56} +4.44504 q^{57} +7.93900 q^{58} +12.2784 q^{59} +1.00000 q^{60} -11.0707 q^{61} -7.85086 q^{62} +5.04892 q^{63} +1.00000 q^{64} +4.74094 q^{66} +9.58211 q^{67} -1.08815 q^{68} +2.69202 q^{69} -5.04892 q^{70} +4.47219 q^{71} -1.00000 q^{72} -8.17390 q^{73} +7.14675 q^{74} +1.00000 q^{75} +4.44504 q^{76} -23.9366 q^{77} +14.1468 q^{79} +1.00000 q^{80} +1.00000 q^{81} +0.664874 q^{82} +4.86294 q^{83} +5.04892 q^{84} -1.08815 q^{85} -4.35690 q^{86} -7.93900 q^{87} +4.74094 q^{88} +14.8877 q^{89} -1.00000 q^{90} +2.69202 q^{92} +7.85086 q^{93} -7.00000 q^{94} +4.44504 q^{95} -1.00000 q^{96} +14.9269 q^{97} -18.4916 q^{98} -4.74094 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{12} - 6 q^{14} + 3 q^{15} + 3 q^{16} - 7 q^{17} - 3 q^{18} + 13 q^{19} + 3 q^{20} + 6 q^{21} + 3 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} + 6 q^{28} - 14 q^{29} - 3 q^{30} + 10 q^{31} - 3 q^{32} + 7 q^{34} + 6 q^{35} + 3 q^{36} + 6 q^{37} - 13 q^{38} - 3 q^{40} - 3 q^{41} - 6 q^{42} + 9 q^{43} + 3 q^{45} - 3 q^{46} + 21 q^{47} + 3 q^{48} + 5 q^{49} - 3 q^{50} - 7 q^{51} + 3 q^{53} - 3 q^{54} - 6 q^{56} + 13 q^{57} + 14 q^{58} + 7 q^{59} + 3 q^{60} - 21 q^{61} - 10 q^{62} + 6 q^{63} + 3 q^{64} + 23 q^{67} - 7 q^{68} + 3 q^{69} - 6 q^{70} + 7 q^{71} - 3 q^{72} + 9 q^{73} - 6 q^{74} + 3 q^{75} + 13 q^{76} - 21 q^{77} + 15 q^{79} + 3 q^{80} + 3 q^{81} + 3 q^{82} + 20 q^{83} + 6 q^{84} - 7 q^{85} - 9 q^{86} - 14 q^{87} + 3 q^{89} - 3 q^{90} + 3 q^{92} + 10 q^{93} - 21 q^{94} + 13 q^{95} - 3 q^{96} + 16 q^{97} - 5 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^12 - 6 * q^14 + 3 * q^15 + 3 * q^16 - 7 * q^17 - 3 * q^18 + 13 * q^19 + 3 * q^20 + 6 * q^21 + 3 * q^23 - 3 * q^24 + 3 * q^25 + 3 * q^27 + 6 * q^28 - 14 * q^29 - 3 * q^30 + 10 * q^31 - 3 * q^32 + 7 * q^34 + 6 * q^35 + 3 * q^36 + 6 * q^37 - 13 * q^38 - 3 * q^40 - 3 * q^41 - 6 * q^42 + 9 * q^43 + 3 * q^45 - 3 * q^46 + 21 * q^47 + 3 * q^48 + 5 * q^49 - 3 * q^50 - 7 * q^51 + 3 * q^53 - 3 * q^54 - 6 * q^56 + 13 * q^57 + 14 * q^58 + 7 * q^59 + 3 * q^60 - 21 * q^61 - 10 * q^62 + 6 * q^63 + 3 * q^64 + 23 * q^67 - 7 * q^68 + 3 * q^69 - 6 * q^70 + 7 * q^71 - 3 * q^72 + 9 * q^73 - 6 * q^74 + 3 * q^75 + 13 * q^76 - 21 * q^77 + 15 * q^79 + 3 * q^80 + 3 * q^81 + 3 * q^82 + 20 * q^83 + 6 * q^84 - 7 * q^85 - 9 * q^86 - 14 * q^87 + 3 * q^89 - 3 * q^90 + 3 * q^92 + 10 * q^93 - 21 * q^94 + 13 * q^95 - 3 * q^96 + 16 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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