LMFDB - Embedded newform 5070.2.a.bw.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			$-1.24698$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.04892 q^{11} +1.00000 q^{12} +3.69202 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.85086 q^{17} +1.00000 q^{18} +0.911854 q^{19} -1.00000 q^{20} +3.69202 q^{21} -3.04892 q^{22} -0.356896 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +3.69202 q^{28} +10.5036 q^{29} -1.00000 q^{30} +2.06100 q^{31} +1.00000 q^{32} -3.04892 q^{33} -6.85086 q^{34} -3.69202 q^{35} +1.00000 q^{36} -0.899772 q^{37} +0.911854 q^{38} -1.00000 q^{40} +10.2959 q^{41} +3.69202 q^{42} +9.43296 q^{43} -3.04892 q^{44} -1.00000 q^{45} -0.356896 q^{46} +11.5918 q^{47} +1.00000 q^{48} +6.63102 q^{49} +1.00000 q^{50} -6.85086 q^{51} -3.40581 q^{53} +1.00000 q^{54} +3.04892 q^{55} +3.69202 q^{56} +0.911854 q^{57} +10.5036 q^{58} +8.54288 q^{59} -1.00000 q^{60} +1.55496 q^{61} +2.06100 q^{62} +3.69202 q^{63} +1.00000 q^{64} -3.04892 q^{66} +1.14914 q^{67} -6.85086 q^{68} -0.356896 q^{69} -3.69202 q^{70} -4.13706 q^{71} +1.00000 q^{72} +11.1685 q^{73} -0.899772 q^{74} +1.00000 q^{75} +0.911854 q^{76} -11.2567 q^{77} +9.62565 q^{79} -1.00000 q^{80} +1.00000 q^{81} +10.2959 q^{82} -9.86054 q^{83} +3.69202 q^{84} +6.85086 q^{85} +9.43296 q^{86} +10.5036 q^{87} -3.04892 q^{88} -6.41119 q^{89} -1.00000 q^{90} -0.356896 q^{92} +2.06100 q^{93} +11.5918 q^{94} -0.911854 q^{95} +1.00000 q^{96} +6.97823 q^{97} +6.63102 q^{98} -3.04892 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - q^{19} - 3 q^{20} + 6 q^{21} + 3 q^{23} + 3 q^{24} + 3 q^{25} + 3 q^{27} + 6 q^{28} - 3 q^{30} + 16 q^{31} + 3 q^{32} - 7 q^{34} - 6 q^{35} + 3 q^{36} + 20 q^{37} - q^{38} - 3 q^{40} + 17 q^{41} + 6 q^{42} + 9 q^{43} - 3 q^{45} + 3 q^{46} + 7 q^{47} + 3 q^{48} + 5 q^{49} + 3 q^{50} - 7 q^{51} + 3 q^{53} + 3 q^{54} + 6 q^{56} - q^{57} + 7 q^{59} - 3 q^{60} + 5 q^{61} + 16 q^{62} + 6 q^{63} + 3 q^{64} + 17 q^{67} - 7 q^{68} + 3 q^{69} - 6 q^{70} - 7 q^{71} + 3 q^{72} + 3 q^{73} + 20 q^{74} + 3 q^{75} - q^{76} - 7 q^{77} + 17 q^{79} - 3 q^{80} + 3 q^{81} + 17 q^{82} + 6 q^{83} + 6 q^{84} + 7 q^{85} + 9 q^{86} - 3 q^{89} - 3 q^{90} + 3 q^{92} + 16 q^{93} + 7 q^{94} + q^{95} + 3 q^{96} + 24 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 - q^19 - 3 * q^20 + 6 * q^21 + 3 * q^23 + 3 * q^24 + 3 * q^25 + 3 * q^27 + 6 * q^28 - 3 * q^30 + 16 * q^31 + 3 * q^32 - 7 * q^34 - 6 * q^35 + 3 * q^36 + 20 * q^37 - q^38 - 3 * q^40 + 17 * q^41 + 6 * q^42 + 9 * q^43 - 3 * q^45 + 3 * q^46 + 7 * q^47 + 3 * q^48 + 5 * q^49 + 3 * q^50 - 7 * q^51 + 3 * q^53 + 3 * q^54 + 6 * q^56 - q^57 + 7 * q^59 - 3 * q^60 + 5 * q^61 + 16 * q^62 + 6 * q^63 + 3 * q^64 + 17 * q^67 - 7 * q^68 + 3 * q^69 - 6 * q^70 - 7 * q^71 + 3 * q^72 + 3 * q^73 + 20 * q^74 + 3 * q^75 - q^76 - 7 * q^77 + 17 * q^79 - 3 * q^80 + 3 * q^81 + 17 * q^82 + 6 * q^83 + 6 * q^84 + 7 * q^85 + 9 * q^86 - 3 * q^89 - 3 * q^90 + 3 * q^92 + 16 * q^93 + 7 * q^94 + q^95 + 3 * q^96 + 24 * 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
$7$	3.69202	1.39545	0.697726	−	0.716364i	$-0.254195\pi$
$7$	3.69202	1.39545	0.697726	+	0.716364i	$0.254195\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−3.04892	−0.919283	−0.459642	−	0.888105i	$-0.652022\pi$
$11$	−3.04892	−0.919283	−0.459642	+	0.888105i	$0.652022\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	3.69202	0.986734
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−6.85086	−1.66158	−0.830788	−	0.556589i	$-0.812110\pi$
$17$	−6.85086	−1.66158	−0.830788	+	0.556589i	$0.812110\pi$
$18$	1.00000	0.235702
$19$	0.911854	0.209194	0.104597	−	0.994515i	$-0.466645\pi$
$19$	0.911854	0.209194	0.104597	+	0.994515i	$0.466645\pi$
$20$	−1.00000	−0.223607
$21$	3.69202	0.805665
$22$	−3.04892	−0.650031
$23$	−0.356896	−0.0744179	−0.0372090	−	0.999308i	$-0.511847\pi$
$23$	−0.356896	−0.0744179	−0.0372090	+	0.999308i	$0.511847\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	3.69202	0.697726
$29$	10.5036	1.95048	0.975239	−	0.221153i	$-0.0709819\pi$
$29$	10.5036	1.95048	0.975239	+	0.221153i	$0.0709819\pi$
$30$	−1.00000	−0.182574
$31$	2.06100	0.370166	0.185083	−	0.982723i	$-0.440745\pi$
$31$	2.06100	0.370166	0.185083	+	0.982723i	$0.440745\pi$
$32$	1.00000	0.176777
$33$	−3.04892	−0.530748
$34$	−6.85086	−1.17491
$35$	−3.69202	−0.624066
$36$	1.00000	0.166667
$37$	−0.899772	−0.147922	−0.0739608	−	0.997261i	$-0.523564\pi$
$37$	−0.899772	−0.147922	−0.0739608	+	0.997261i	$0.523564\pi$
$38$	0.911854	0.147922
$39$	0	0
$40$	−1.00000	−0.158114
$41$	10.2959	1.60795	0.803974	−	0.594664i	$-0.202715\pi$
$41$	10.2959	1.60795	0.803974	+	0.594664i	$0.202715\pi$
$42$	3.69202	0.569691
$43$	9.43296	1.43851	0.719256	−	0.694745i	$-0.244483\pi$
$43$	9.43296	1.43851	0.719256	+	0.694745i	$0.244483\pi$
$44$	−3.04892	−0.459642
$45$	−1.00000	−0.149071
$46$	−0.356896	−0.0526214
$47$	11.5918	1.69084	0.845418	−	0.534105i	$-0.179351\pi$
$47$	11.5918	1.69084	0.845418	+	0.534105i	$0.179351\pi$
$48$	1.00000	0.144338
$49$	6.63102	0.947289
$50$	1.00000	0.141421
$51$	−6.85086	−0.959312
$52$	0	0
$53$	−3.40581	−0.467824	−0.233912	−	0.972258i	$-0.575153\pi$
$53$	−3.40581	−0.467824	−0.233912	+	0.972258i	$0.575153\pi$
$54$	1.00000	0.136083
$55$	3.04892	0.411116
$56$	3.69202	0.493367
$57$	0.911854	0.120778
$58$	10.5036	1.37920
$59$	8.54288	1.11219	0.556094	−	0.831119i	$-0.312299\pi$
$59$	8.54288	1.11219	0.556094	+	0.831119i	$0.312299\pi$
$60$	−1.00000	−0.129099
$61$	1.55496	0.199092	0.0995460	−	0.995033i	$-0.468261\pi$
$61$	1.55496	0.199092	0.0995460	+	0.995033i	$0.468261\pi$
$62$	2.06100	0.261747
$63$	3.69202	0.465151
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.04892	−0.375296
$67$	1.14914	0.140390	0.0701952	−	0.997533i	$-0.477638\pi$
$67$	1.14914	0.140390	0.0701952	+	0.997533i	$0.477638\pi$
$68$	−6.85086	−0.830788
$69$	−0.356896	−0.0429652
$70$	−3.69202	−0.441281
$71$	−4.13706	−0.490979	−0.245490	−	0.969399i	$-0.578949\pi$
$71$	−4.13706	−0.490979	−0.245490	+	0.969399i	$0.578949\pi$
$72$	1.00000	0.117851
$73$	11.1685	1.30718	0.653588	−	0.756850i	$-0.273263\pi$
$73$	11.1685	1.30718	0.653588	+	0.756850i	$0.273263\pi$
$74$	−0.899772	−0.104596
$75$	1.00000	0.115470
$76$	0.911854	0.104597
$77$	−11.2567	−1.28282
$78$	0	0
$79$	9.62565	1.08297	0.541485	−	0.840710i	$-0.317862\pi$
$79$	9.62565	1.08297	0.541485	+	0.840710i	$0.317862\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	10.2959	1.13699
$83$	−9.86054	−1.08234	−0.541168	−	0.840915i	$-0.682018\pi$
$83$	−9.86054	−1.08234	−0.541168	+	0.840915i	$0.682018\pi$
$84$	3.69202	0.402833
$85$	6.85086	0.743080
$86$	9.43296	1.01718
$87$	10.5036	1.12611
$88$	−3.04892	−0.325016
$89$	−6.41119	−0.679585	−0.339792	−	0.940500i	$-0.610357\pi$
$89$	−6.41119	−0.679585	−0.339792	+	0.940500i	$0.610357\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−0.356896	−0.0372090
$93$	2.06100	0.213716
$94$	11.5918	1.19560
$95$	−0.911854	−0.0935542
$96$	1.00000	0.102062
$97$	6.97823	0.708532	0.354266	−	0.935145i	$-0.384731\pi$
$97$	6.97823	0.708532	0.354266	+	0.935145i	$0.384731\pi$
$98$	6.63102	0.669834
$99$	−3.04892	−0.306428
$100$	1.00000	0.100000
$101$	−4.09783	−0.407750	−0.203875	−	0.978997i	$-0.565354\pi$
$101$	−4.09783	−0.407750	−0.203875	+	0.978997i	$0.565354\pi$
$102$	−6.85086	−0.678336
$103$	−16.9215	−1.66733	−0.833665	−	0.552271i	$-0.813761\pi$
$103$	−16.9215	−1.66733	−0.833665	+	0.552271i	$0.813761\pi$
$104$	0	0
$105$	−3.69202	−0.360304
$106$	−3.40581	−0.330802
$107$	−12.7192	−1.22961	−0.614804	−	0.788680i	$-0.710765\pi$
$107$	−12.7192	−1.22961	−0.614804	+	0.788680i	$0.710765\pi$
$108$	1.00000	0.0962250
$109$	19.9922	1.91491	0.957454	−	0.288585i	$-0.0931849\pi$
$109$	19.9922	1.91491	0.957454	+	0.288585i	$0.0931849\pi$
$110$	3.04892	0.290703
$111$	−0.899772	−0.0854026
$112$	3.69202	0.348863
$113$	−4.49934	−0.423262	−0.211631	−	0.977350i	$-0.567877\pi$
$113$	−4.49934	−0.423262	−0.211631	+	0.977350i	$0.567877\pi$
$114$	0.911854	0.0854030
$115$	0.356896	0.0332807
$116$	10.5036	0.975239
$117$	0	0
$118$	8.54288	0.786436
$119$	−25.2935	−2.31865
$120$	−1.00000	−0.0912871
$121$	−1.70410	−0.154918
$122$	1.55496	0.140779
$123$	10.2959	0.928350
$124$	2.06100	0.185083
$125$	−1.00000	−0.0894427
$126$	3.69202	0.328911
$127$	−7.08815	−0.628971	−0.314486	−	0.949262i	$-0.601832\pi$
$127$	−7.08815	−0.628971	−0.314486	+	0.949262i	$0.601832\pi$
$128$	1.00000	0.0883883
$129$	9.43296	0.830526
$130$	0	0
$131$	19.7506	1.72562	0.862810	−	0.505528i	$-0.168702\pi$
$131$	19.7506	1.72562	0.862810	+	0.505528i	$0.168702\pi$
$132$	−3.04892	−0.265374
$133$	3.36658	0.291920
$134$	1.14914	0.0992710
$135$	−1.00000	−0.0860663
$136$	−6.85086	−0.587456
$137$	−7.94331	−0.678643	−0.339322	−	0.940670i	$-0.610197\pi$
$137$	−7.94331	−0.678643	−0.339322	+	0.940670i	$0.610197\pi$
$138$	−0.356896	−0.0303810
$139$	−1.71379	−0.145362	−0.0726810	−	0.997355i	$-0.523155\pi$
$139$	−1.71379	−0.145362	−0.0726810	+	0.997355i	$0.523155\pi$
$140$	−3.69202	−0.312033
$141$	11.5918	0.976205
$142$	−4.13706	−0.347175
$143$	0	0
$144$	1.00000	0.0833333
$145$	−10.5036	−0.872280
$146$	11.1685	0.924313
$147$	6.63102	0.546918
$148$	−0.899772	−0.0739608
$149$	−6.91185	−0.566241	−0.283121	−	0.959084i	$-0.591370\pi$
$149$	−6.91185	−0.566241	−0.283121	+	0.959084i	$0.591370\pi$
$150$	1.00000	0.0816497
$151$	5.43296	0.442128	0.221064	−	0.975259i	$-0.429047\pi$
$151$	5.43296	0.442128	0.221064	+	0.975259i	$0.429047\pi$
$152$	0.911854	0.0739611
$153$	−6.85086	−0.553859
$154$	−11.2567	−0.907088
$155$	−2.06100	−0.165543
$156$	0	0
$157$	−9.13706	−0.729217	−0.364609	−	0.931161i	$-0.618797\pi$
$157$	−9.13706	−0.729217	−0.364609	+	0.931161i	$0.618797\pi$
$158$	9.62565	0.765775
$159$	−3.40581	−0.270099
$160$	−1.00000	−0.0790569
$161$	−1.31767	−0.103847
$162$	1.00000	0.0785674
$163$	14.3773	1.12612	0.563060	−	0.826416i	$-0.309624\pi$
$163$	14.3773	1.12612	0.563060	+	0.826416i	$0.309624\pi$
$164$	10.2959	0.803974
$165$	3.04892	0.237358
$166$	−9.86054	−0.765327
$167$	−3.00538	−0.232563	−0.116282	−	0.993216i	$-0.537097\pi$
$167$	−3.00538	−0.232563	−0.116282	+	0.993216i	$0.537097\pi$
$168$	3.69202	0.284846
$169$	0	0
$170$	6.85086	0.525437
$171$	0.911854	0.0697312
$172$	9.43296	0.719256
$173$	6.89546	0.524252	0.262126	−	0.965034i	$-0.415576\pi$
$173$	6.89546	0.524252	0.262126	+	0.965034i	$0.415576\pi$
$174$	10.5036	0.796279
$175$	3.69202	0.279091
$176$	−3.04892	−0.229821
$177$	8.54288	0.642122
$178$	−6.41119	−0.480539
$179$	−25.6461	−1.91688	−0.958439	−	0.285296i	$-0.907908\pi$
$179$	−25.6461	−1.91688	−0.958439	+	0.285296i	$0.907908\pi$
$180$	−1.00000	−0.0745356
$181$	−25.8092	−1.91839	−0.959193	−	0.282753i	$-0.908752\pi$
$181$	−25.8092	−1.91839	−0.959193	+	0.282753i	$0.908752\pi$
$182$	0	0
$183$	1.55496	0.114946
$184$	−0.356896	−0.0263107
$185$	0.899772	0.0661526
$186$	2.06100	0.151120
$187$	20.8877	1.52746
$188$	11.5918	0.845418
$189$	3.69202	0.268555
$190$	−0.911854	−0.0661528
$191$	−14.5700	−1.05425	−0.527125	−	0.849788i	$-0.676730\pi$
$191$	−14.5700	−1.05425	−0.527125	+	0.849788i	$0.676730\pi$
$192$	1.00000	0.0721688
$193$	8.32544	0.599278	0.299639	−	0.954053i	$-0.403134\pi$
$193$	8.32544	0.599278	0.299639	+	0.954053i	$0.403134\pi$
$194$	6.97823	0.501008
$195$	0	0
$196$	6.63102	0.473644
$197$	−26.8810	−1.91519	−0.957595	−	0.288116i	$-0.906971\pi$
$197$	−26.8810	−1.91519	−0.957595	+	0.288116i	$0.906971\pi$
$198$	−3.04892	−0.216677
$199$	−18.7047	−1.32594	−0.662970	−	0.748646i	$-0.730704\pi$
$199$	−18.7047	−1.32594	−0.662970	+	0.748646i	$0.730704\pi$
$200$	1.00000	0.0707107
$201$	1.14914	0.0810544
$202$	−4.09783	−0.288323
$203$	38.7797	2.72180
$204$	−6.85086	−0.479656
$205$	−10.2959	−0.719097
$206$	−16.9215	−1.17898
$207$	−0.356896	−0.0248060
$208$	0	0
$209$	−2.78017	−0.192308
$210$	−3.69202	−0.254774
$211$	2.93362	0.201959	0.100980	−	0.994889i	$-0.467802\pi$
$211$	2.93362	0.201959	0.100980	+	0.994889i	$0.467802\pi$
$212$	−3.40581	−0.233912
$213$	−4.13706	−0.283467
$214$	−12.7192	−0.869464
$215$	−9.43296	−0.643323
$216$	1.00000	0.0680414
$217$	7.60925	0.516550
$218$	19.9922	1.35404
$219$	11.1685	0.754699
$220$	3.04892	0.205558
$221$	0	0
$222$	−0.899772	−0.0603888
$223$	−18.1444	−1.21504	−0.607518	−	0.794306i	$-0.707835\pi$
$223$	−18.1444	−1.21504	−0.607518	+	0.794306i	$0.707835\pi$
$224$	3.69202	0.246684
$225$	1.00000	0.0666667
$226$	−4.49934	−0.299291
$227$	12.3056	0.816750	0.408375	−	0.912814i	$-0.366096\pi$
$227$	12.3056	0.816750	0.408375	+	0.912814i	$0.366096\pi$
$228$	0.911854	0.0603890
$229$	4.64848	0.307180	0.153590	−	0.988135i	$-0.450916\pi$
$229$	4.64848	0.307180	0.153590	+	0.988135i	$0.450916\pi$
$230$	0.356896	0.0235330
$231$	−11.2567	−0.740634
$232$	10.5036	0.689598
$233$	0.658170	0.0431181	0.0215591	−	0.999768i	$-0.493137\pi$
$233$	0.658170	0.0431181	0.0215591	+	0.999768i	$0.493137\pi$
$234$	0	0
$235$	−11.5918	−0.756165
$236$	8.54288	0.556094
$237$	9.62565	0.625253
$238$	−25.2935	−1.63953
$239$	−18.7748	−1.21444	−0.607220	−	0.794534i	$-0.707715\pi$
$239$	−18.7748	−1.21444	−0.607220	+	0.794534i	$0.707715\pi$
$240$	−1.00000	−0.0645497
$241$	18.0271	1.16123	0.580615	−	0.814178i	$-0.302812\pi$
$241$	18.0271	1.16123	0.580615	+	0.814178i	$0.302812\pi$
$242$	−1.70410	−0.109544
$243$	1.00000	0.0641500
$244$	1.55496	0.0995460
$245$	−6.63102	−0.423640
$246$	10.2959	0.656442
$247$	0	0
$248$	2.06100	0.130874
$249$	−9.86054	−0.624887
$250$	−1.00000	−0.0632456
$251$	−2.18060	−0.137638	−0.0688192	−	0.997629i	$-0.521923\pi$
$251$	−2.18060	−0.137638	−0.0688192	+	0.997629i	$0.521923\pi$
$252$	3.69202	0.232575
$253$	1.08815	0.0684112
$254$	−7.08815	−0.444750
$255$	6.85086	0.429017
$256$	1.00000	0.0625000
$257$	−9.58748	−0.598051	−0.299025	−	0.954245i	$-0.596662\pi$
$257$	−9.58748	−0.598051	−0.299025	+	0.954245i	$0.596662\pi$
$258$	9.43296	0.587270
$259$	−3.32198	−0.206418
$260$	0	0
$261$	10.5036	0.650159
$262$	19.7506	1.22020
$263$	−1.41789	−0.0874311	−0.0437156	−	0.999044i	$-0.513920\pi$
$263$	−1.41789	−0.0874311	−0.0437156	+	0.999044i	$0.513920\pi$
$264$	−3.04892	−0.187648
$265$	3.40581	0.209217
$266$	3.36658	0.206419
$267$	−6.41119	−0.392358
$268$	1.14914	0.0701952
$269$	−4.14675	−0.252832	−0.126416	−	0.991977i	$-0.540347\pi$
$269$	−4.14675	−0.252832	−0.126416	+	0.991977i	$0.540347\pi$
$270$	−1.00000	−0.0608581
$271$	6.40044	0.388799	0.194399	−	0.980922i	$-0.437724\pi$
$271$	6.40044	0.388799	0.194399	+	0.980922i	$0.437724\pi$
$272$	−6.85086	−0.415394
$273$	0	0
$274$	−7.94331	−0.479873
$275$	−3.04892	−0.183857
$276$	−0.356896	−0.0214826
$277$	10.3394	0.621237	0.310618	−	0.950535i	$-0.399464\pi$
$277$	10.3394	0.621237	0.310618	+	0.950535i	$0.399464\pi$
$278$	−1.71379	−0.102786
$279$	2.06100	0.123389
$280$	−3.69202	−0.220640
$281$	5.08144	0.303133	0.151567	−	0.988447i	$-0.451568\pi$
$281$	5.08144	0.303133	0.151567	+	0.988447i	$0.451568\pi$
$282$	11.5918	0.690281
$283$	24.5284	1.45806	0.729031	−	0.684481i	$-0.239971\pi$
$283$	24.5284	1.45806	0.729031	+	0.684481i	$0.239971\pi$
$284$	−4.13706	−0.245490
$285$	−0.911854	−0.0540136
$286$	0	0
$287$	38.0127	2.24382
$288$	1.00000	0.0589256
$289$	29.9342	1.76084
$290$	−10.5036	−0.616795
$291$	6.97823	0.409071
$292$	11.1685	0.653588
$293$	15.8538	0.926191	0.463096	−	0.886308i	$-0.346739\pi$
$293$	15.8538	0.926191	0.463096	+	0.886308i	$0.346739\pi$
$294$	6.63102	0.386729
$295$	−8.54288	−0.497386
$296$	−0.899772	−0.0522982
$297$	−3.04892	−0.176916
$298$	−6.91185	−0.400393
$299$	0	0
$300$	1.00000	0.0577350
$301$	34.8267	2.00738
$302$	5.43296	0.312632
$303$	−4.09783	−0.235414
$304$	0.911854	0.0522984
$305$	−1.55496	−0.0890366
$306$	−6.85086	−0.391637
$307$	−14.2121	−0.811125	−0.405563	−	0.914067i	$-0.632924\pi$
$307$	−14.2121	−0.811125	−0.405563	+	0.914067i	$0.632924\pi$
$308$	−11.2567	−0.641408
$309$	−16.9215	−0.962633
$310$	−2.06100	−0.117057
$311$	15.0127	0.851291	0.425645	−	0.904890i	$-0.360047\pi$
$311$	15.0127	0.851291	0.425645	+	0.904890i	$0.360047\pi$
$312$	0	0
$313$	−13.4993	−0.763028	−0.381514	−	0.924363i	$-0.624597\pi$
$313$	−13.4993	−0.763028	−0.381514	+	0.924363i	$0.624597\pi$
$314$	−9.13706	−0.515634
$315$	−3.69202	−0.208022
$316$	9.62565	0.541485
$317$	33.9463	1.90661	0.953307	−	0.302003i	$-0.0976552\pi$
$317$	33.9463	1.90661	0.953307	+	0.302003i	$0.0976552\pi$
$318$	−3.40581	−0.190989
$319$	−32.0248	−1.79304
$320$	−1.00000	−0.0559017
$321$	−12.7192	−0.709915
$322$	−1.31767	−0.0734307
$323$	−6.24698	−0.347591
$324$	1.00000	0.0555556
$325$	0	0
$326$	14.3773	0.796287
$327$	19.9922	1.10557
$328$	10.2959	0.568496
$329$	42.7972	2.35948
$330$	3.04892	0.167837
$331$	−6.84415	−0.376189	−0.188094	−	0.982151i	$-0.560231\pi$
$331$	−6.84415	−0.376189	−0.188094	+	0.982151i	$0.560231\pi$
$332$	−9.86054	−0.541168
$333$	−0.899772	−0.0493072
$334$	−3.00538	−0.164447
$335$	−1.14914	−0.0627845
$336$	3.69202	0.201416
$337$	27.7332	1.51072	0.755361	−	0.655309i	$-0.227462\pi$
$337$	27.7332	1.51072	0.755361	+	0.655309i	$0.227462\pi$
$338$	0	0
$339$	−4.49934	−0.244370
$340$	6.85086	0.371540
$341$	−6.28382	−0.340288
$342$	0.911854	0.0493074
$343$	−1.36227	−0.0735558
$344$	9.43296	0.508591
$345$	0.356896	0.0192146
$346$	6.89546	0.370702
$347$	−10.4862	−0.562928	−0.281464	−	0.959572i	$-0.590820\pi$
$347$	−10.4862	−0.562928	−0.281464	+	0.959572i	$0.590820\pi$
$348$	10.5036	0.563055
$349$	−8.13467	−0.435439	−0.217719	−	0.976011i	$-0.569862\pi$
$349$	−8.13467	−0.435439	−0.217719	+	0.976011i	$0.569862\pi$
$350$	3.69202	0.197347
$351$	0	0
$352$	−3.04892	−0.162508
$353$	−25.8649	−1.37665	−0.688324	−	0.725404i	$-0.741653\pi$
$353$	−25.8649	−1.37665	−0.688324	+	0.725404i	$0.741653\pi$
$354$	8.54288	0.454049
$355$	4.13706	0.219573
$356$	−6.41119	−0.339792
$357$	−25.2935	−1.33867
$358$	−25.6461	−1.35544
$359$	−30.2959	−1.59896	−0.799478	−	0.600695i	$-0.794891\pi$
$359$	−30.2959	−1.59896	−0.799478	+	0.600695i	$0.794891\pi$
$360$	−1.00000	−0.0527046
$361$	−18.1685	−0.956238
$362$	−25.8092	−1.35650
$363$	−1.70410	−0.0894422
$364$	0	0
$365$	−11.1685	−0.584587
$366$	1.55496	0.0812790
$367$	−3.41013	−0.178007	−0.0890035	−	0.996031i	$-0.528368\pi$
$367$	−3.41013	−0.178007	−0.0890035	+	0.996031i	$0.528368\pi$
$368$	−0.356896	−0.0186045
$369$	10.2959	0.535983
$370$	0.899772	0.0467769
$371$	−12.5743	−0.652827
$372$	2.06100	0.106858
$373$	3.35988	0.173968	0.0869840	−	0.996210i	$-0.472277\pi$
$373$	3.35988	0.173968	0.0869840	+	0.996210i	$0.472277\pi$
$374$	20.8877	1.08008
$375$	−1.00000	−0.0516398
$376$	11.5918	0.597801
$377$	0	0
$378$	3.69202	0.189897
$379$	−31.4916	−1.61761	−0.808807	−	0.588075i	$-0.799886\pi$
$379$	−31.4916	−1.61761	−0.808807	+	0.588075i	$0.799886\pi$
$380$	−0.911854	−0.0467771
$381$	−7.08815	−0.363137
$382$	−14.5700	−0.745467
$383$	21.8485	1.11640	0.558202	−	0.829705i	$-0.311491\pi$
$383$	21.8485	1.11640	0.558202	+	0.829705i	$0.311491\pi$
$384$	1.00000	0.0510310
$385$	11.2567	0.573693
$386$	8.32544	0.423754
$387$	9.43296	0.479504
$388$	6.97823	0.354266
$389$	−11.3787	−0.576921	−0.288461	−	0.957492i	$-0.593143\pi$
$389$	−11.3787	−0.576921	−0.288461	+	0.957492i	$0.593143\pi$
$390$	0	0
$391$	2.44504	0.123651
$392$	6.63102	0.334917
$393$	19.7506	0.996287
$394$	−26.8810	−1.35424
$395$	−9.62565	−0.484319
$396$	−3.04892	−0.153214
$397$	22.0476	1.10654	0.553268	−	0.833003i	$-0.313380\pi$
$397$	22.0476	1.10654	0.553268	+	0.833003i	$0.313380\pi$
$398$	−18.7047	−0.937582
$399$	3.36658	0.168540
$400$	1.00000	0.0500000
$401$	10.2459	0.511657	0.255828	−	0.966722i	$-0.417652\pi$
$401$	10.2459	0.511657	0.255828	+	0.966722i	$0.417652\pi$
$402$	1.14914	0.0573141
$403$	0	0
$404$	−4.09783	−0.203875
$405$	−1.00000	−0.0496904
$406$	38.7797	1.92460
$407$	2.74333	0.135982
$408$	−6.85086	−0.339168
$409$	22.8189	1.12832	0.564162	−	0.825664i	$-0.309200\pi$
$409$	22.8189	1.12832	0.564162	+	0.825664i	$0.309200\pi$
$410$	−10.2959	−0.508478
$411$	−7.94331	−0.391815
$412$	−16.9215	−0.833665
$413$	31.5405	1.55201
$414$	−0.356896	−0.0175405
$415$	9.86054	0.484035
$416$	0	0
$417$	−1.71379	−0.0839247
$418$	−2.78017	−0.135982
$419$	−28.5773	−1.39609	−0.698047	−	0.716052i	$-0.745947\pi$
$419$	−28.5773	−1.39609	−0.698047	+	0.716052i	$0.745947\pi$
$420$	−3.69202	−0.180152
$421$	32.3937	1.57877	0.789387	−	0.613896i	$-0.210399\pi$
$421$	32.3937	1.57877	0.789387	+	0.613896i	$0.210399\pi$
$422$	2.93362	0.142807
$423$	11.5918	0.563612
$424$	−3.40581	−0.165401
$425$	−6.85086	−0.332315
$426$	−4.13706	−0.200441
$427$	5.74094	0.277824
$428$	−12.7192	−0.614804
$429$	0	0
$430$	−9.43296	−0.454898
$431$	−14.6455	−0.705449	−0.352724	−	0.935727i	$-0.614745\pi$
$431$	−14.6455	−0.705449	−0.352724	+	0.935727i	$0.614745\pi$
$432$	1.00000	0.0481125
$433$	−35.8485	−1.72277	−0.861384	−	0.507955i	$-0.830402\pi$
$433$	−35.8485	−1.72277	−0.861384	+	0.507955i	$0.830402\pi$
$434$	7.60925	0.365256
$435$	−10.5036	−0.503611
$436$	19.9922	0.957454
$437$	−0.325437	−0.0155678
$438$	11.1685	0.533653
$439$	2.55927	0.122147	0.0610736	−	0.998133i	$-0.480548\pi$
$439$	2.55927	0.122147	0.0610736	+	0.998133i	$0.480548\pi$
$440$	3.04892	0.145351
$441$	6.63102	0.315763
$442$	0	0
$443$	23.2857	1.10634	0.553169	−	0.833069i	$-0.313418\pi$
$443$	23.2857	1.10634	0.553169	+	0.833069i	$0.313418\pi$
$444$	−0.899772	−0.0427013
$445$	6.41119	0.303920
$446$	−18.1444	−0.859160
$447$	−6.91185	−0.326919
$448$	3.69202	0.174432
$449$	−27.3327	−1.28991	−0.644956	−	0.764220i	$-0.723124\pi$
$449$	−27.3327	−1.28991	−0.644956	+	0.764220i	$0.723124\pi$
$450$	1.00000	0.0471405
$451$	−31.3913	−1.47816
$452$	−4.49934	−0.211631
$453$	5.43296	0.255263
$454$	12.3056	0.577530
$455$	0	0
$456$	0.911854	0.0427015
$457$	−7.74094	−0.362106	−0.181053	−	0.983473i	$-0.557951\pi$
$457$	−7.74094	−0.362106	−0.181053	+	0.983473i	$0.557951\pi$
$458$	4.64848	0.217209
$459$	−6.85086	−0.319771
$460$	0.356896	0.0166404
$461$	17.1511	0.798805	0.399402	−	0.916776i	$-0.369218\pi$
$461$	17.1511	0.798805	0.399402	+	0.916776i	$0.369218\pi$
$462$	−11.2567	−0.523708
$463$	−34.2747	−1.59288	−0.796441	−	0.604717i	$-0.793286\pi$
$463$	−34.2747	−1.59288	−0.796441	+	0.604717i	$0.793286\pi$
$464$	10.5036	0.487620
$465$	−2.06100	−0.0955765
$466$	0.658170	0.0304891
$467$	20.9474	0.969328	0.484664	−	0.874700i	$-0.338942\pi$
$467$	20.9474	0.969328	0.484664	+	0.874700i	$0.338942\pi$
$468$	0	0
$469$	4.24267	0.195908
$470$	−11.5918	−0.534690
$471$	−9.13706	−0.421014
$472$	8.54288	0.393218
$473$	−28.7603	−1.32240
$474$	9.62565	0.442121
$475$	0.911854	0.0418387
$476$	−25.2935	−1.15933
$477$	−3.40581	−0.155941
$478$	−18.7748	−0.858739
$479$	15.8310	0.723337	0.361669	−	0.932307i	$-0.382207\pi$
$479$	15.8310	0.723337	0.361669	+	0.932307i	$0.382207\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	18.0271	0.821114
$483$	−1.31767	−0.0599559
$484$	−1.70410	−0.0774592
$485$	−6.97823	−0.316865
$486$	1.00000	0.0453609
$487$	33.7060	1.52737	0.763683	−	0.645592i	$-0.223389\pi$
$487$	33.7060	1.52737	0.763683	+	0.645592i	$0.223389\pi$
$488$	1.55496	0.0703896
$489$	14.3773	0.650166
$490$	−6.63102	−0.299559
$491$	3.98925	0.180032	0.0900161	−	0.995940i	$-0.471308\pi$
$491$	3.98925	0.180032	0.0900161	+	0.995940i	$0.471308\pi$
$492$	10.2959	0.464175
$493$	−71.9590	−3.24087
$494$	0	0
$495$	3.04892	0.137039
$496$	2.06100	0.0925416
$497$	−15.2741	−0.685138
$498$	−9.86054	−0.441862
$499$	20.4698	0.916354	0.458177	−	0.888861i	$-0.348503\pi$
$499$	20.4698	0.916354	0.458177	+	0.888861i	$0.348503\pi$
$500$	−1.00000	−0.0447214
$501$	−3.00538	−0.134270
$502$	−2.18060	−0.0973251
$503$	22.5351	1.00479	0.502395	−	0.864638i	$-0.332452\pi$
$503$	22.5351	1.00479	0.502395	+	0.864638i	$0.332452\pi$
$504$	3.69202	0.164456
$505$	4.09783	0.182351
$506$	1.08815	0.0483740
$507$	0	0
$508$	−7.08815	−0.314486
$509$	−10.9718	−0.486316	−0.243158	−	0.969987i	$-0.578183\pi$
$509$	−10.9718	−0.486316	−0.243158	+	0.969987i	$0.578183\pi$
$510$	6.85086	0.303361
$511$	41.2344	1.82410
$512$	1.00000	0.0441942
$513$	0.911854	0.0402593
$514$	−9.58748	−0.422886
$515$	16.9215	0.745652
$516$	9.43296	0.415263
$517$	−35.3424	−1.55436
$518$	−3.32198	−0.145959
$519$	6.89546	0.302677
$520$	0	0
$521$	15.9691	0.699620	0.349810	−	0.936821i	$-0.386246\pi$
$521$	15.9691	0.699620	0.349810	+	0.936821i	$0.386246\pi$
$522$	10.5036	0.459732
$523$	−24.7681	−1.08303	−0.541516	−	0.840690i	$-0.682150\pi$
$523$	−24.7681	−1.08303	−0.541516	+	0.840690i	$0.682150\pi$
$524$	19.7506	0.862810
$525$	3.69202	0.161133
$526$	−1.41789	−0.0618232
$527$	−14.1196	−0.615060
$528$	−3.04892	−0.132687
$529$	−22.8726	−0.994462
$530$	3.40581	0.147939
$531$	8.54288	0.370729
$532$	3.36658	0.145960
$533$	0	0
$534$	−6.41119	−0.277439
$535$	12.7192	0.549898
$536$	1.14914	0.0496355
$537$	−25.6461	−1.10671
$538$	−4.14675	−0.178779
$539$	−20.2174	−0.870827
$540$	−1.00000	−0.0430331
$541$	0.460107	0.0197816	0.00989078	−	0.999951i	$-0.496852\pi$
$541$	0.460107	0.0197816	0.00989078	+	0.999951i	$0.496852\pi$
$542$	6.40044	0.274922
$543$	−25.8092	−1.10758
$544$	−6.85086	−0.293728
$545$	−19.9922	−0.856373
$546$	0	0
$547$	−8.70410	−0.372161	−0.186080	−	0.982535i	$-0.559578\pi$
$547$	−8.70410	−0.372161	−0.186080	+	0.982535i	$0.559578\pi$
$548$	−7.94331	−0.339322
$549$	1.55496	0.0663640
$550$	−3.04892	−0.130006
$551$	9.57779	0.408028
$552$	−0.356896	−0.0151905
$553$	35.5381	1.51123
$554$	10.3394	0.439281
$555$	0.899772	0.0381932
$556$	−1.71379	−0.0726810
$557$	−36.7415	−1.55679	−0.778394	−	0.627776i	$-0.783966\pi$
$557$	−36.7415	−1.55679	−0.778394	+	0.627776i	$0.783966\pi$
$558$	2.06100	0.0872490
$559$	0	0
$560$	−3.69202	−0.156016
$561$	20.8877	0.881879
$562$	5.08144	0.214348
$563$	−27.7713	−1.17042	−0.585211	−	0.810881i	$-0.698988\pi$
$563$	−27.7713	−1.17042	−0.585211	+	0.810881i	$0.698988\pi$
$564$	11.5918	0.488103
$565$	4.49934	0.189288
$566$	24.5284	1.03101
$567$	3.69202	0.155050
$568$	−4.13706	−0.173587
$569$	38.4379	1.61140	0.805700	−	0.592324i	$-0.201790\pi$
$569$	38.4379	1.61140	0.805700	+	0.592324i	$0.201790\pi$
$570$	−0.911854	−0.0381934
$571$	10.8086	0.452328	0.226164	−	0.974089i	$-0.427382\pi$
$571$	10.8086	0.452328	0.226164	+	0.974089i	$0.427382\pi$
$572$	0	0
$573$	−14.5700	−0.608671
$574$	38.0127	1.58662
$575$	−0.356896	−0.0148836
$576$	1.00000	0.0416667
$577$	35.5991	1.48201	0.741005	−	0.671500i	$-0.234350\pi$
$577$	35.5991	1.48201	0.741005	+	0.671500i	$0.234350\pi$
$578$	29.9342	1.24510
$579$	8.32544	0.345993
$580$	−10.5036	−0.436140
$581$	−36.4053	−1.51035
$582$	6.97823	0.289257
$583$	10.3840	0.430063
$584$	11.1685	0.462157
$585$	0	0
$586$	15.8538	0.654916
$587$	11.4741	0.473587	0.236794	−	0.971560i	$-0.423903\pi$
$587$	11.4741	0.473587	0.236794	+	0.971560i	$0.423903\pi$
$588$	6.63102	0.273459
$589$	1.87933	0.0774364
$590$	−8.54288	−0.351705
$591$	−26.8810	−1.10574
$592$	−0.899772	−0.0369804
$593$	15.5147	0.637111	0.318555	−	0.947904i	$-0.396802\pi$
$593$	15.5147	0.637111	0.318555	+	0.947904i	$0.396802\pi$
$594$	−3.04892	−0.125099
$595$	25.2935	1.03693
$596$	−6.91185	−0.283121
$597$	−18.7047	−0.765532
$598$	0	0
$599$	−14.2034	−0.580337	−0.290168	−	0.956976i	$-0.593711\pi$
$599$	−14.2034	−0.580337	−0.290168	+	0.956976i	$0.593711\pi$
$600$	1.00000	0.0408248
$601$	−39.5579	−1.61360	−0.806801	−	0.590823i	$-0.798803\pi$
$601$	−39.5579	−1.61360	−0.806801	+	0.590823i	$0.798803\pi$
$602$	34.8267	1.41943
$603$	1.14914	0.0467968
$604$	5.43296	0.221064
$605$	1.70410	0.0692816
$606$	−4.09783	−0.166463
$607$	4.77346	0.193749	0.0968744	−	0.995297i	$-0.469115\pi$
$607$	4.77346	0.193749	0.0968744	+	0.995297i	$0.469115\pi$
$608$	0.911854	0.0369806
$609$	38.7797	1.57143
$610$	−1.55496	−0.0629584
$611$	0	0
$612$	−6.85086	−0.276929
$613$	5.43429	0.219489	0.109744	−	0.993960i	$-0.464997\pi$
$613$	5.43429	0.219489	0.109744	+	0.993960i	$0.464997\pi$
$614$	−14.2121	−0.573552
$615$	−10.2959	−0.415171
$616$	−11.2567	−0.453544
$617$	−20.6692	−0.832110	−0.416055	−	0.909339i	$-0.636588\pi$
$617$	−20.6692	−0.832110	−0.416055	+	0.909339i	$0.636588\pi$
$618$	−16.9215	−0.680684
$619$	24.4077	0.981030	0.490515	−	0.871433i	$-0.336809\pi$
$619$	24.4077	0.981030	0.490515	+	0.871433i	$0.336809\pi$
$620$	−2.06100	−0.0827717
$621$	−0.356896	−0.0143217
$622$	15.0127	0.601953
$623$	−23.6703	−0.948329
$624$	0	0
$625$	1.00000	0.0400000
$626$	−13.4993	−0.539542
$627$	−2.78017	−0.111029
$628$	−9.13706	−0.364609
$629$	6.16421	0.245783
$630$	−3.69202	−0.147094
$631$	−32.0538	−1.27604	−0.638021	−	0.770019i	$-0.720247\pi$
$631$	−32.0538	−1.27604	−0.638021	+	0.770019i	$0.720247\pi$
$632$	9.62565	0.382888
$633$	2.93362	0.116601
$634$	33.9463	1.34818
$635$	7.08815	0.281284
$636$	−3.40581	−0.135049
$637$	0	0
$638$	−32.0248	−1.26787
$639$	−4.13706	−0.163660
$640$	−1.00000	−0.0395285
$641$	18.5327	0.731998	0.365999	−	0.930615i	$-0.380727\pi$
$641$	18.5327	0.731998	0.365999	+	0.930615i	$0.380727\pi$
$642$	−12.7192	−0.501986
$643$	−22.3454	−0.881217	−0.440608	−	0.897699i	$-0.645237\pi$
$643$	−22.3454	−0.881217	−0.440608	+	0.897699i	$0.645237\pi$
$644$	−1.31767	−0.0519234
$645$	−9.43296	−0.371422
$646$	−6.24698	−0.245784
$647$	7.86964	0.309388	0.154694	−	0.987962i	$-0.450561\pi$
$647$	7.86964	0.309388	0.154694	+	0.987962i	$0.450561\pi$
$648$	1.00000	0.0392837
$649$	−26.0465	−1.02242
$650$	0	0
$651$	7.60925	0.298230
$652$	14.3773	0.563060
$653$	−47.2271	−1.84814	−0.924070	−	0.382223i	$-0.875159\pi$
$653$	−47.2271	−1.84814	−0.924070	+	0.382223i	$0.875159\pi$
$654$	19.9922	0.781758
$655$	−19.7506	−0.771721
$656$	10.2959	0.401987
$657$	11.1685	0.435726
$658$	42.7972	1.66841
$659$	−46.6276	−1.81635	−0.908176	−	0.418588i	$-0.862525\pi$
$659$	−46.6276	−1.81635	−0.908176	+	0.418588i	$0.862525\pi$
$660$	3.04892	0.118679
$661$	−39.9415	−1.55354	−0.776772	−	0.629781i	$-0.783144\pi$
$661$	−39.9415	−1.55354	−0.776772	+	0.629781i	$0.783144\pi$
$662$	−6.84415	−0.266005
$663$	0	0
$664$	−9.86054	−0.382663
$665$	−3.36658	−0.130551
$666$	−0.899772	−0.0348655
$667$	−3.74871	−0.145151
$668$	−3.00538	−0.116282
$669$	−18.1444	−0.701501
$670$	−1.14914	−0.0443953
$671$	−4.74094	−0.183022
$672$	3.69202	0.142423
$673$	−35.3612	−1.36307	−0.681537	−	0.731783i	$-0.738688\pi$
$673$	−35.3612	−1.36307	−0.681537	+	0.731783i	$0.738688\pi$
$674$	27.7332	1.06824
$675$	1.00000	0.0384900
$676$	0	0
$677$	19.1715	0.736821	0.368410	−	0.929663i	$-0.379902\pi$
$677$	19.1715	0.736821	0.368410	+	0.929663i	$0.379902\pi$
$678$	−4.49934	−0.172796
$679$	25.7638	0.988723
$680$	6.85086	0.262718
$681$	12.3056	0.471551
$682$	−6.28382	−0.240620
$683$	−1.13946	−0.0436001	−0.0218000	−	0.999762i	$-0.506940\pi$
$683$	−1.13946	−0.0436001	−0.0218000	+	0.999762i	$0.506940\pi$
$684$	0.911854	0.0348656
$685$	7.94331	0.303498
$686$	−1.36227	−0.0520118
$687$	4.64848	0.177351
$688$	9.43296	0.359628
$689$	0	0
$690$	0.356896	0.0135868
$691$	11.9071	0.452966	0.226483	−	0.974015i	$-0.427277\pi$
$691$	11.9071	0.452966	0.226483	+	0.974015i	$0.427277\pi$
$692$	6.89546	0.262126
$693$	−11.2567	−0.427605
$694$	−10.4862	−0.398050
$695$	1.71379	0.0650078
$696$	10.5036	0.398140
$697$	−70.5357	−2.67173
$698$	−8.13467	−0.307902
$699$	0.658170	0.0248943
$700$	3.69202	0.139545
$701$	38.0575	1.43741	0.718707	−	0.695313i	$-0.244734\pi$
$701$	38.0575	1.43741	0.718707	+	0.695313i	$0.244734\pi$
$702$	0	0
$703$	−0.820461	−0.0309443
$704$	−3.04892	−0.114910
$705$	−11.5918	−0.436572
$706$	−25.8649	−0.973437
$707$	−15.1293	−0.568996
$708$	8.54288	0.321061
$709$	−26.1215	−0.981014	−0.490507	−	0.871437i	$-0.663188\pi$
$709$	−26.1215	−0.981014	−0.490507	+	0.871437i	$0.663188\pi$
$710$	4.13706	0.155261
$711$	9.62565	0.360990
$712$	−6.41119	−0.240270
$713$	−0.735562	−0.0275470
$714$	−25.2935	−0.946586
$715$	0	0
$716$	−25.6461	−0.958439
$717$	−18.7748	−0.701157
$718$	−30.2959	−1.13063
$719$	21.3739	0.797111	0.398556	−	0.917144i	$-0.369512\pi$
$719$	21.3739	0.797111	0.398556	+	0.917144i	$0.369512\pi$
$720$	−1.00000	−0.0372678
$721$	−62.4747	−2.32668
$722$	−18.1685	−0.676162
$723$	18.0271	0.670437
$724$	−25.8092	−0.959193
$725$	10.5036	0.390096
$726$	−1.70410	−0.0632452
$727$	−17.9124	−0.664336	−0.332168	−	0.943220i	$-0.607780\pi$
$727$	−17.9124	−0.664336	−0.332168	+	0.943220i	$0.607780\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−11.1685	−0.413366
$731$	−64.6238	−2.39020
$732$	1.55496	0.0574729
$733$	−27.5386	−1.01716	−0.508580	−	0.861015i	$-0.669829\pi$
$733$	−27.5386	−1.01716	−0.508580	+	0.861015i	$0.669829\pi$
$734$	−3.41013	−0.125870
$735$	−6.63102	−0.244589
$736$	−0.356896	−0.0131554
$737$	−3.50365	−0.129059
$738$	10.2959	0.378997
$739$	7.99090	0.293950	0.146975	−	0.989140i	$-0.453046\pi$
$739$	7.99090	0.293950	0.146975	+	0.989140i	$0.453046\pi$
$740$	0.899772	0.0330763
$741$	0	0
$742$	−12.5743	−0.461618
$743$	−19.8398	−0.727853	−0.363927	−	0.931428i	$-0.618564\pi$
$743$	−19.8398	−0.727853	−0.363927	+	0.931428i	$0.618564\pi$
$744$	2.06100	0.0755599
$745$	6.91185	0.253231
$746$	3.35988	0.123014
$747$	−9.86054	−0.360778
$748$	20.8877	0.763730
$749$	−46.9594	−1.71586
$750$	−1.00000	−0.0365148
$751$	−25.7525	−0.939724	−0.469862	−	0.882740i	$-0.655696\pi$
$751$	−25.7525	−0.939724	−0.469862	+	0.882740i	$0.655696\pi$
$752$	11.5918	0.422709
$753$	−2.18060	−0.0794656
$754$	0	0
$755$	−5.43296	−0.197726
$756$	3.69202	0.134278
$757$	−51.5096	−1.87215	−0.936074	−	0.351802i	$-0.885569\pi$
$757$	−51.5096	−1.87215	−0.936074	+	0.351802i	$0.885569\pi$
$758$	−31.4916	−1.14383
$759$	1.08815	0.0394972
$760$	−0.911854	−0.0330764
$761$	31.4765	1.14102	0.570511	−	0.821290i	$-0.306745\pi$
$761$	31.4765	1.14102	0.570511	+	0.821290i	$0.306745\pi$
$762$	−7.08815	−0.256776
$763$	73.8117	2.67216
$764$	−14.5700	−0.527125
$765$	6.85086	0.247693
$766$	21.8485	0.789417
$767$	0	0
$768$	1.00000	0.0360844
$769$	47.1584	1.70057	0.850287	−	0.526319i	$-0.176428\pi$
$769$	47.1584	1.70057	0.850287	+	0.526319i	$0.176428\pi$
$770$	11.2567	0.405662
$771$	−9.58748	−0.345285
$772$	8.32544	0.299639
$773$	3.71917	0.133769	0.0668846	−	0.997761i	$-0.478694\pi$
$773$	3.71917	0.133769	0.0668846	+	0.997761i	$0.478694\pi$
$774$	9.43296	0.339061
$775$	2.06100	0.0740333
$776$	6.97823	0.250504
$777$	−3.32198	−0.119175
$778$	−11.3787	−0.407945
$779$	9.38835	0.336373
$780$	0	0
$781$	12.6136	0.451349
$782$	2.44504	0.0874345
$783$	10.5036	0.375370
$784$	6.63102	0.236822
$785$	9.13706	0.326116
$786$	19.7506	0.704482
$787$	−47.2282	−1.68350	−0.841752	−	0.539865i	$-0.818475\pi$
$787$	−47.2282	−1.68350	−0.841752	+	0.539865i	$0.818475\pi$
$788$	−26.8810	−0.957595
$789$	−1.41789	−0.0504784
$790$	−9.62565	−0.342465
$791$	−16.6116	−0.590642
$792$	−3.04892	−0.108339
$793$	0	0
$794$	22.0476	0.782440
$795$	3.40581	0.120792
$796$	−18.7047	−0.662970
$797$	34.4359	1.21978	0.609892	−	0.792485i	$-0.291213\pi$
$797$	34.4359	1.21978	0.609892	+	0.792485i	$0.291213\pi$
$798$	3.36658	0.119176
$799$	−79.4137	−2.80945
$800$	1.00000	0.0353553
$801$	−6.41119	−0.226528
$802$	10.2459	0.361796
$803$	−34.0519	−1.20167
$804$	1.14914	0.0405272
$805$	1.31767	0.0464417
$806$	0	0
$807$	−4.14675	−0.145973
$808$	−4.09783	−0.144161
$809$	−32.3846	−1.13858	−0.569292	−	0.822136i	$-0.692782\pi$
$809$	−32.3846	−1.13858	−0.569292	+	0.822136i	$0.692782\pi$
$810$	−1.00000	−0.0351364
$811$	37.8297	1.32838	0.664190	−	0.747564i	$-0.268777\pi$
$811$	37.8297	1.32838	0.664190	+	0.747564i	$0.268777\pi$
$812$	38.7797	1.36090
$813$	6.40044	0.224473
$814$	2.74333	0.0961537
$815$	−14.3773	−0.503616
$816$	−6.85086	−0.239828
$817$	8.60148	0.300928
$818$	22.8189	0.797845
$819$	0	0
$820$	−10.2959	−0.359548
$821$	43.2820	1.51055	0.755276	−	0.655406i	$-0.227503\pi$
$821$	43.2820	1.51055	0.755276	+	0.655406i	$0.227503\pi$
$822$	−7.94331	−0.277055
$823$	−26.7399	−0.932093	−0.466047	−	0.884760i	$-0.654322\pi$
$823$	−26.7399	−0.932093	−0.466047	+	0.884760i	$0.654322\pi$
$824$	−16.9215	−0.589490
$825$	−3.04892	−0.106150
$826$	31.5405	1.09743
$827$	21.4437	0.745671	0.372835	−	0.927897i	$-0.378386\pi$
$827$	21.4437	0.745671	0.372835	+	0.927897i	$0.378386\pi$
$828$	−0.356896	−0.0124030
$829$	22.4547	0.779885	0.389943	−	0.920839i	$-0.372495\pi$
$829$	22.4547	0.779885	0.389943	+	0.920839i	$0.372495\pi$
$830$	9.86054	0.342264
$831$	10.3394	0.358671
$832$	0	0
$833$	−45.4282	−1.57399
$834$	−1.71379	−0.0593438
$835$	3.00538	0.104005
$836$	−2.78017	−0.0961541
$837$	2.06100	0.0712385
$838$	−28.5773	−0.987187
$839$	24.4034	0.842500	0.421250	−	0.906945i	$-0.361592\pi$
$839$	24.4034	0.842500	0.421250	+	0.906945i	$0.361592\pi$
$840$	−3.69202	−0.127387
$841$	81.3266	2.80437
$842$	32.3937	1.11636
$843$	5.08144	0.175014
$844$	2.93362	0.100980
$845$	0	0
$846$	11.5918	0.398534
$847$	−6.29159	−0.216181
$848$	−3.40581	−0.116956
$849$	24.5284	0.841813
$850$	−6.85086	−0.234982
$851$	0.321125	0.0110080
$852$	−4.13706	−0.141733
$853$	−7.96184	−0.272608	−0.136304	−	0.990667i	$-0.543522\pi$
$853$	−7.96184	−0.272608	−0.136304	+	0.990667i	$0.543522\pi$
$854$	5.74094	0.196451
$855$	−0.911854	−0.0311847
$856$	−12.7192	−0.434732
$857$	−8.12737	−0.277626	−0.138813	−	0.990319i	$-0.544329\pi$
$857$	−8.12737	−0.277626	−0.138813	+	0.990319i	$0.544329\pi$
$858$	0	0
$859$	17.1666	0.585717	0.292858	−	0.956156i	$-0.405394\pi$
$859$	17.1666	0.585717	0.292858	+	0.956156i	$0.405394\pi$
$860$	−9.43296	−0.321661
$861$	38.0127	1.29547
$862$	−14.6455	−0.498828
$863$	−46.2857	−1.57558	−0.787792	−	0.615941i	$-0.788776\pi$
$863$	−46.2857	−1.57558	−0.787792	+	0.615941i	$0.788776\pi$
$864$	1.00000	0.0340207
$865$	−6.89546	−0.234453
$866$	−35.8485	−1.21818
$867$	29.9342	1.01662
$868$	7.60925	0.258275
$869$	−29.3478	−0.995556
$870$	−10.5036	−0.356107
$871$	0	0
$872$	19.9922	0.677022
$873$	6.97823	0.236177
$874$	−0.325437	−0.0110081
$875$	−3.69202	−0.124813
$876$	11.1685	0.377349
$877$	2.72156	0.0919006	0.0459503	−	0.998944i	$-0.485368\pi$
$877$	2.72156	0.0919006	0.0459503	+	0.998944i	$0.485368\pi$
$878$	2.55927	0.0863712
$879$	15.8538	0.534737
$880$	3.04892	0.102779
$881$	−16.7241	−0.563448	−0.281724	−	0.959495i	$-0.590906\pi$
$881$	−16.7241	−0.563448	−0.281724	+	0.959495i	$0.590906\pi$
$882$	6.63102	0.223278
$883$	−2.88364	−0.0970423	−0.0485211	−	0.998822i	$-0.515451\pi$
$883$	−2.88364	−0.0970423	−0.0485211	+	0.998822i	$0.515451\pi$
$884$	0	0
$885$	−8.54288	−0.287166
$886$	23.2857	0.782300
$887$	55.8866	1.87649	0.938245	−	0.345973i	$-0.112451\pi$
$887$	55.8866	1.87649	0.938245	+	0.345973i	$0.112451\pi$
$888$	−0.899772	−0.0301944
$889$	−26.1696	−0.877700
$890$	6.41119	0.214904
$891$	−3.04892	−0.102143
$892$	−18.1444	−0.607518
$893$	10.5700	0.353712
$894$	−6.91185	−0.231167
$895$	25.6461	0.857254
$896$	3.69202	0.123342
$897$	0	0
$898$	−27.3327	−0.912105
$899$	21.6480	0.722001
$900$	1.00000	0.0333333
$901$	23.3327	0.777326
$902$	−31.3913	−1.04522
$903$	34.8267	1.15896
$904$	−4.49934	−0.149646
$905$	25.8092	0.857928
$906$	5.43296	0.180498
$907$	49.5381	1.64489	0.822443	−	0.568848i	$-0.192611\pi$
$907$	49.5381	1.64489	0.822443	+	0.568848i	$0.192611\pi$
$908$	12.3056	0.408375
$909$	−4.09783	−0.135917
$910$	0	0
$911$	37.0603	1.22786	0.613931	−	0.789360i	$-0.289587\pi$
$911$	37.0603	1.22786	0.613931	+	0.789360i	$0.289587\pi$
$912$	0.911854	0.0301945
$913$	30.0640	0.994973
$914$	−7.74094	−0.256047
$915$	−1.55496	−0.0514053
$916$	4.64848	0.153590
$917$	72.9197	2.40802
$918$	−6.85086	−0.226112
$919$	53.6228	1.76885	0.884426	−	0.466680i	$-0.154550\pi$
$919$	53.6228	1.76885	0.884426	+	0.466680i	$0.154550\pi$
$920$	0.356896	0.0117665
$921$	−14.2121	−0.468303
$922$	17.1511	0.564840
$923$	0	0
$924$	−11.2567	−0.370317
$925$	−0.899772	−0.0295843
$926$	−34.2747	−1.12634
$927$	−16.9215	−0.555776
$928$	10.5036	0.344799
$929$	−10.6726	−0.350158	−0.175079	−	0.984554i	$-0.556018\pi$
$929$	−10.6726	−0.350158	−0.175079	+	0.984554i	$0.556018\pi$
$930$	−2.06100	−0.0675828
$931$	6.04652	0.198167
$932$	0.658170	0.0215591
$933$	15.0127	0.491493
$934$	20.9474	0.685419
$935$	−20.8877	−0.683101
$936$	0	0
$937$	17.0538	0.557124	0.278562	−	0.960418i	$-0.410142\pi$
$937$	17.0538	0.557124	0.278562	+	0.960418i	$0.410142\pi$
$938$	4.24267	0.138528
$939$	−13.4993	−0.440534
$940$	−11.5918	−0.378083
$941$	−46.7706	−1.52468	−0.762339	−	0.647178i	$-0.775949\pi$
$941$	−46.7706	−1.52468	−0.762339	+	0.647178i	$0.775949\pi$
$942$	−9.13706	−0.297702
$943$	−3.67456	−0.119660
$944$	8.54288	0.278047
$945$	−3.69202	−0.120101
$946$	−28.7603	−0.935079
$947$	−30.0398	−0.976163	−0.488081	−	0.872798i	$-0.662303\pi$
$947$	−30.0398	−0.976163	−0.488081	+	0.872798i	$0.662303\pi$
$948$	9.62565	0.312626
$949$	0	0
$950$	0.911854	0.0295845
$951$	33.9463	1.10078
$952$	−25.2935	−0.819767
$953$	−29.4252	−0.953175	−0.476588	−	0.879127i	$-0.658127\pi$
$953$	−29.4252	−0.953175	−0.476588	+	0.879127i	$0.658127\pi$
$954$	−3.40581	−0.110267
$955$	14.5700	0.471475
$956$	−18.7748	−0.607220
$957$	−32.0248	−1.03521
$958$	15.8310	0.511477
$959$	−29.3269	−0.947014
$960$	−1.00000	−0.0322749
$961$	−26.7523	−0.862977
$962$	0	0
$963$	−12.7192	−0.409869
$964$	18.0271	0.580615
$965$	−8.32544	−0.268005
$966$	−1.31767	−0.0423952
$967$	12.7023	0.408478	0.204239	−	0.978921i	$-0.434528\pi$
$967$	12.7023	0.408478	0.204239	+	0.978921i	$0.434528\pi$
$968$	−1.70410	−0.0547719
$969$	−6.24698	−0.200682
$970$	−6.97823	−0.224057
$971$	22.9105	0.735234	0.367617	−	0.929977i	$-0.380174\pi$
$971$	22.9105	0.735234	0.367617	+	0.929977i	$0.380174\pi$
$972$	1.00000	0.0320750
$973$	−6.32736	−0.202846
$974$	33.7060	1.08001
$975$	0	0
$976$	1.55496	0.0497730
$977$	−25.1390	−0.804267	−0.402134	−	0.915581i	$-0.631731\pi$
$977$	−25.1390	−0.804267	−0.402134	+	0.915581i	$0.631731\pi$
$978$	14.3773	0.459737
$979$	19.5472	0.624731
$980$	−6.63102	−0.211820
$981$	19.9922	0.638303
$982$	3.98925	0.127302
$983$	−50.3895	−1.60718	−0.803588	−	0.595186i	$-0.797079\pi$
$983$	−50.3895	−1.60718	−0.803588	+	0.595186i	$0.797079\pi$
$984$	10.2959	0.328221
$985$	26.8810	0.856499
$986$	−71.9590	−2.29164
$987$	42.7972	1.36225
$988$	0	0
$989$	−3.36658	−0.107051
$990$	3.04892	0.0969010
$991$	23.3375	0.741341	0.370670	−	0.928764i	$-0.379128\pi$
$991$	23.3375	0.741341	0.370670	+	0.928764i	$0.379128\pi$
$992$	2.06100	0.0654368
$993$	−6.84415	−0.217193
$994$	−15.2741	−0.484466
$995$	18.7047	0.592979
$996$	−9.86054	−0.312443
$997$	22.8127	0.722485	0.361243	−	0.932472i	$-0.382353\pi$
$997$	22.8127	0.722485	0.361243	+	0.932472i	$0.382353\pi$
$998$	20.4698	0.647960
$999$	−0.899772	−0.0284675

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bp.1.1	✓	3	13.12	even	2
5070.2.a.bw.1.3	yes	3	1.1	even	1	trivial
5070.2.b.z.1351.3		6	13.5	odd	4
5070.2.b.z.1351.4		6	13.8	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.04892 q^{11} +1.00000 q^{12} +3.69202 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.85086 q^{17} +1.00000 q^{18} +0.911854 q^{19} -1.00000 q^{20} +3.69202 q^{21} -3.04892 q^{22} -0.356896 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +3.69202 q^{28} +10.5036 q^{29} -1.00000 q^{30} +2.06100 q^{31} +1.00000 q^{32} -3.04892 q^{33} -6.85086 q^{34} -3.69202 q^{35} +1.00000 q^{36} -0.899772 q^{37} +0.911854 q^{38} -1.00000 q^{40} +10.2959 q^{41} +3.69202 q^{42} +9.43296 q^{43} -3.04892 q^{44} -1.00000 q^{45} -0.356896 q^{46} +11.5918 q^{47} +1.00000 q^{48} +6.63102 q^{49} +1.00000 q^{50} -6.85086 q^{51} -3.40581 q^{53} +1.00000 q^{54} +3.04892 q^{55} +3.69202 q^{56} +0.911854 q^{57} +10.5036 q^{58} +8.54288 q^{59} -1.00000 q^{60} +1.55496 q^{61} +2.06100 q^{62} +3.69202 q^{63} +1.00000 q^{64} -3.04892 q^{66} +1.14914 q^{67} -6.85086 q^{68} -0.356896 q^{69} -3.69202 q^{70} -4.13706 q^{71} +1.00000 q^{72} +11.1685 q^{73} -0.899772 q^{74} +1.00000 q^{75} +0.911854 q^{76} -11.2567 q^{77} +9.62565 q^{79} -1.00000 q^{80} +1.00000 q^{81} +10.2959 q^{82} -9.86054 q^{83} +3.69202 q^{84} +6.85086 q^{85} +9.43296 q^{86} +10.5036 q^{87} -3.04892 q^{88} -6.41119 q^{89} -1.00000 q^{90} -0.356896 q^{92} +2.06100 q^{93} +11.5918 q^{94} -0.911854 q^{95} +1.00000 q^{96} +6.97823 q^{97} +6.63102 q^{98} -3.04892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 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	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{12} + 6 q^{14} - 3 q^{15} + 3 q^{16} - 7 q^{17} + 3 q^{18} - q^{19} - 3 q^{20} + 6 q^{21} + 3 q^{23} + 3 q^{24} + 3 q^{25} + 3 q^{27} + 6 q^{28} - 3 q^{30} + 16 q^{31} + 3 q^{32} - 7 q^{34} - 6 q^{35} + 3 q^{36} + 20 q^{37} - q^{38} - 3 q^{40} + 17 q^{41} + 6 q^{42} + 9 q^{43} - 3 q^{45} + 3 q^{46} + 7 q^{47} + 3 q^{48} + 5 q^{49} + 3 q^{50} - 7 q^{51} + 3 q^{53} + 3 q^{54} + 6 q^{56} - q^{57} + 7 q^{59} - 3 q^{60} + 5 q^{61} + 16 q^{62} + 6 q^{63} + 3 q^{64} + 17 q^{67} - 7 q^{68} + 3 q^{69} - 6 q^{70} - 7 q^{71} + 3 q^{72} + 3 q^{73} + 20 q^{74} + 3 q^{75} - q^{76} - 7 q^{77} + 17 q^{79} - 3 q^{80} + 3 q^{81} + 17 q^{82} + 6 q^{83} + 6 q^{84} + 7 q^{85} + 9 q^{86} - 3 q^{89} - 3 q^{90} + 3 q^{92} + 16 q^{93} + 7 q^{94} + q^{95} + 3 q^{96} + 24 q^{97} + 5 q^{98}+O(q^{100}) \) 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 - q^19 - 3 * q^20 + 6 * q^21 + 3 * q^23 + 3 * q^24 + 3 * q^25 + 3 * q^27 + 6 * q^28 - 3 * q^30 + 16 * q^31 + 3 * q^32 - 7 * q^34 - 6 * q^35 + 3 * q^36 + 20 * q^37 - q^38 - 3 * q^40 + 17 * q^41 + 6 * q^42 + 9 * q^43 - 3 * q^45 + 3 * q^46 + 7 * q^47 + 3 * q^48 + 5 * q^49 + 3 * q^50 - 7 * q^51 + 3 * q^53 + 3 * q^54 + 6 * q^56 - q^57 + 7 * q^59 - 3 * q^60 + 5 * q^61 + 16 * q^62 + 6 * q^63 + 3 * q^64 + 17 * q^67 - 7 * q^68 + 3 * q^69 - 6 * q^70 - 7 * q^71 + 3 * q^72 + 3 * q^73 + 20 * q^74 + 3 * q^75 - q^76 - 7 * q^77 + 17 * q^79 - 3 * q^80 + 3 * q^81 + 17 * q^82 + 6 * q^83 + 6 * q^84 + 7 * q^85 + 9 * q^86 - 3 * q^89 - 3 * q^90 + 3 * q^92 + 16 * q^93 + 7 * q^94 + q^95 + 3 * q^96 + 24 * q^97 + 5 * q^98

Introduction

L-functions

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