LMFDB - Embedded newform 65.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [65,2,Mod(1,65)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("65.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 65 = 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	65.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:	$0.519027613138$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	65.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.414214 q^{2} +1.41421 q^{3} -1.82843 q^{4} +1.00000 q^{5} +0.585786 q^{6} -0.828427 q^{7} -1.58579 q^{8} -1.00000 q^{9} +0.414214 q^{10} +0.585786 q^{11} -2.58579 q^{12} -1.00000 q^{13} -0.343146 q^{14} +1.41421 q^{15} +3.00000 q^{16} -4.82843 q^{17} -0.414214 q^{18} +3.41421 q^{19} -1.82843 q^{20} -1.17157 q^{21} +0.242641 q^{22} -1.41421 q^{23} -2.24264 q^{24} +1.00000 q^{25} -0.414214 q^{26} -5.65685 q^{27} +1.51472 q^{28} +5.65685 q^{29} +0.585786 q^{30} +10.2426 q^{31} +4.41421 q^{32} +0.828427 q^{33} -2.00000 q^{34} -0.828427 q^{35} +1.82843 q^{36} +8.48528 q^{37} +1.41421 q^{38} -1.41421 q^{39} -1.58579 q^{40} -8.82843 q^{41} -0.485281 q^{42} +3.07107 q^{43} -1.07107 q^{44} -1.00000 q^{45} -0.585786 q^{46} +0.828427 q^{47} +4.24264 q^{48} -6.31371 q^{49} +0.414214 q^{50} -6.82843 q^{51} +1.82843 q^{52} -14.4853 q^{53} -2.34315 q^{54} +0.585786 q^{55} +1.31371 q^{56} +4.82843 q^{57} +2.34315 q^{58} +10.2426 q^{59} -2.58579 q^{60} -8.00000 q^{61} +4.24264 q^{62} +0.828427 q^{63} -4.17157 q^{64} -1.00000 q^{65} +0.343146 q^{66} -2.00000 q^{67} +8.82843 q^{68} -2.00000 q^{69} -0.343146 q^{70} -7.89949 q^{71} +1.58579 q^{72} -8.48528 q^{73} +3.51472 q^{74} +1.41421 q^{75} -6.24264 q^{76} -0.485281 q^{77} -0.585786 q^{78} +8.48528 q^{79} +3.00000 q^{80} -5.00000 q^{81} -3.65685 q^{82} -8.82843 q^{83} +2.14214 q^{84} -4.82843 q^{85} +1.27208 q^{86} +8.00000 q^{87} -0.928932 q^{88} +6.00000 q^{89} -0.414214 q^{90} +0.828427 q^{91} +2.58579 q^{92} +14.4853 q^{93} +0.343146 q^{94} +3.41421 q^{95} +6.24264 q^{96} +3.65685 q^{97} -2.61522 q^{98} -0.585786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{7} - 6 q^{8} - 2 q^{9} - 2 q^{10} + 4 q^{11} - 8 q^{12} - 2 q^{13} - 12 q^{14} + 6 q^{16} - 4 q^{17} + 2 q^{18} + 4 q^{19} + 2 q^{20} - 8 q^{21} - 8 q^{22}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^6 + 4 * q^7 - 6 * q^8 - 2 * q^9 - 2 * q^10 + 4 * q^11 - 8 * q^12 - 2 * q^13 - 12 * q^14 + 6 * q^16 - 4 * q^17 + 2 * q^18 + 4 * q^19 + 2 * q^20 - 8 * q^21 - 8 * q^22 + 4 * q^24 + 2 * q^25 + 2 * q^26 + 20 * q^28 + 4 * q^30 + 12 * q^31 + 6 * q^32 - 4 * q^33 - 4 * q^34 + 4 * q^35 - 2 * q^36 - 6 * q^40 - 12 * q^41 + 16 * q^42 - 8 * q^43 + 12 * q^44 - 2 * q^45 - 4 * q^46 - 4 * q^47 + 10 * q^49 - 2 * q^50 - 8 * q^51 - 2 * q^52 - 12 * q^53 - 16 * q^54 + 4 * q^55 - 20 * q^56 + 4 * q^57 + 16 * q^58 + 12 * q^59 - 8 * q^60 - 16 * q^61 - 4 * q^63 - 14 * q^64 - 2 * q^65 + 12 * q^66 - 4 * q^67 + 12 * q^68 - 4 * q^69 - 12 * q^70 + 4 * q^71 + 6 * q^72 + 24 * q^74 - 4 * q^76 + 16 * q^77 - 4 * q^78 + 6 * q^80 - 10 * q^81 + 4 * q^82 - 12 * q^83 - 24 * q^84 - 4 * q^85 + 28 * q^86 + 16 * q^87 - 16 * q^88 + 12 * q^89 + 2 * q^90 - 4 * q^91 + 8 * q^92 + 12 * q^93 + 12 * q^94 + 4 * q^95 + 4 * q^96 - 4 * q^97 - 42 * q^98 - 4 * q^99

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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	−1.82843	−0.914214
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.585786	0.239146
$7$	−0.828427	−0.313116	−0.156558	−	0.987669i	$-0.550040\pi$
$7$	−0.828427	−0.313116	−0.156558	+	0.987669i	$0.550040\pi$
$8$	−1.58579	−0.560660
$9$	−1.00000	−0.333333
$10$	0.414214	0.130986
$11$	0.585786	0.176621	0.0883106	−	0.996093i	$-0.471853\pi$
$11$	0.585786	0.176621	0.0883106	+	0.996093i	$0.471853\pi$
$12$	−2.58579	−0.746452
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.343146	−0.0917096
$15$	1.41421	0.365148
$16$	3.00000	0.750000
$17$	−4.82843	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$17$	−4.82843	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$18$	−0.414214	−0.0976311
$19$	3.41421	0.783274	0.391637	−	0.920120i	$-0.371909\pi$
$19$	3.41421	0.783274	0.391637	+	0.920120i	$0.371909\pi$
$20$	−1.82843	−0.408849
$21$	−1.17157	−0.255658
$22$	0.242641	0.0517312
$23$	−1.41421	−0.294884	−0.147442	−	0.989071i	$-0.547104\pi$
$23$	−1.41421	−0.294884	−0.147442	+	0.989071i	$0.547104\pi$
$24$	−2.24264	−0.457777
$25$	1.00000	0.200000
$26$	−0.414214	−0.0812340
$27$	−5.65685	−1.08866
$28$	1.51472	0.286255
$29$	5.65685	1.05045	0.525226	−	0.850963i	$-0.323981\pi$
$29$	5.65685	1.05045	0.525226	+	0.850963i	$0.323981\pi$
$30$	0.585786	0.106949
$31$	10.2426	1.83963	0.919816	−	0.392349i	$-0.128338\pi$
$31$	10.2426	1.83963	0.919816	+	0.392349i	$0.128338\pi$
$32$	4.41421	0.780330
$33$	0.828427	0.144211
$34$	−2.00000	−0.342997
$35$	−0.828427	−0.140030
$36$	1.82843	0.304738
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	1.41421	0.229416
$39$	−1.41421	−0.226455
$40$	−1.58579	−0.250735
$41$	−8.82843	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$41$	−8.82843	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$42$	−0.485281	−0.0748805
$43$	3.07107	0.468333	0.234167	−	0.972196i	$-0.424764\pi$
$43$	3.07107	0.468333	0.234167	+	0.972196i	$0.424764\pi$
$44$	−1.07107	−0.161470
$45$	−1.00000	−0.149071
$46$	−0.585786	−0.0863695
$47$	0.828427	0.120839	0.0604193	−	0.998173i	$-0.480756\pi$
$47$	0.828427	0.120839	0.0604193	+	0.998173i	$0.480756\pi$
$48$	4.24264	0.612372
$49$	−6.31371	−0.901958
$50$	0.414214	0.0585786
$51$	−6.82843	−0.956171
$52$	1.82843	0.253557
$53$	−14.4853	−1.98971	−0.994853	−	0.101327i	$-0.967691\pi$
$53$	−14.4853	−1.98971	−0.994853	+	0.101327i	$0.967691\pi$
$54$	−2.34315	−0.318862
$55$	0.585786	0.0789874
$56$	1.31371	0.175552
$57$	4.82843	0.639541
$58$	2.34315	0.307670
$59$	10.2426	1.33348	0.666739	−	0.745291i	$-0.267690\pi$
$59$	10.2426	1.33348	0.666739	+	0.745291i	$0.267690\pi$
$60$	−2.58579	−0.333824
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	4.24264	0.538816
$63$	0.828427	0.104372
$64$	−4.17157	−0.521447
$65$	−1.00000	−0.124035
$66$	0.343146	0.0422383
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	8.82843	1.07060
$69$	−2.00000	−0.240772
$70$	−0.343146	−0.0410138
$71$	−7.89949	−0.937498	−0.468749	−	0.883332i	$-0.655295\pi$
$71$	−7.89949	−0.937498	−0.468749	+	0.883332i	$0.655295\pi$
$72$	1.58579	0.186887
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	3.51472	0.408578
$75$	1.41421	0.163299
$76$	−6.24264	−0.716080
$77$	−0.485281	−0.0553029
$78$	−0.585786	−0.0663273
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	3.00000	0.335410
$81$	−5.00000	−0.555556
$82$	−3.65685	−0.403832
$83$	−8.82843	−0.969046	−0.484523	−	0.874779i	$-0.661007\pi$
$83$	−8.82843	−0.969046	−0.484523	+	0.874779i	$0.661007\pi$
$84$	2.14214	0.233726
$85$	−4.82843	−0.523716
$86$	1.27208	0.137172
$87$	8.00000	0.857690
$88$	−0.928932	−0.0990245
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−0.414214	−0.0436619
$91$	0.828427	0.0868428
$92$	2.58579	0.269587
$93$	14.4853	1.50205
$94$	0.343146	0.0353928
$95$	3.41421	0.350291
$96$	6.24264	0.637137
$97$	3.65685	0.371297	0.185649	−	0.982616i	$-0.440561\pi$
$97$	3.65685	0.371297	0.185649	+	0.982616i	$0.440561\pi$
$98$	−2.61522	−0.264177
$99$	−0.585786	−0.0588738
$100$	−1.82843	−0.182843
$101$	7.65685	0.761885	0.380943	−	0.924599i	$-0.375599\pi$
$101$	7.65685	0.761885	0.380943	+	0.924599i	$0.375599\pi$
$102$	−2.82843	−0.280056
$103$	17.4142	1.71587	0.857937	−	0.513755i	$-0.171746\pi$
$103$	17.4142	1.71587	0.857937	+	0.513755i	$0.171746\pi$
$104$	1.58579	0.155499
$105$	−1.17157	−0.114334
$106$	−6.00000	−0.582772
$107$	−6.58579	−0.636672	−0.318336	−	0.947978i	$-0.603124\pi$
$107$	−6.58579	−0.636672	−0.318336	+	0.947978i	$0.603124\pi$
$108$	10.3431	0.995270
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0.242641	0.0231349
$111$	12.0000	1.13899
$112$	−2.48528	−0.234837
$113$	−3.17157	−0.298356	−0.149178	−	0.988810i	$-0.547663\pi$
$113$	−3.17157	−0.298356	−0.149178	+	0.988810i	$0.547663\pi$
$114$	2.00000	0.187317
$115$	−1.41421	−0.131876
$116$	−10.3431	−0.960337
$117$	1.00000	0.0924500
$118$	4.24264	0.390567
$119$	4.00000	0.366679
$120$	−2.24264	−0.204724
$121$	−10.6569	−0.968805
$122$	−3.31371	−0.300009
$123$	−12.4853	−1.12576
$124$	−18.7279	−1.68182
$125$	1.00000	0.0894427
$126$	0.343146	0.0305699
$127$	−9.41421	−0.835376	−0.417688	−	0.908590i	$-0.637160\pi$
$127$	−9.41421	−0.835376	−0.417688	+	0.908590i	$0.637160\pi$
$128$	−10.5563	−0.933058
$129$	4.34315	0.382393
$130$	−0.414214	−0.0363289
$131$	16.9706	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$132$	−1.51472	−0.131839
$133$	−2.82843	−0.245256
$134$	−0.828427	−0.0715652
$135$	−5.65685	−0.486864
$136$	7.65685	0.656570
$137$	5.31371	0.453981	0.226990	−	0.973897i	$-0.427111\pi$
$137$	5.31371	0.453981	0.226990	+	0.973897i	$0.427111\pi$
$138$	−0.828427	−0.0705204
$139$	−12.4853	−1.05899	−0.529494	−	0.848314i	$-0.677618\pi$
$139$	−12.4853	−1.05899	−0.529494	+	0.848314i	$0.677618\pi$
$140$	1.51472	0.128017
$141$	1.17157	0.0986642
$142$	−3.27208	−0.274587
$143$	−0.585786	−0.0489859
$144$	−3.00000	−0.250000
$145$	5.65685	0.469776
$146$	−3.51472	−0.290880
$147$	−8.92893	−0.736446
$148$	−15.5147	−1.27530
$149$	−0.343146	−0.0281116	−0.0140558	−	0.999901i	$-0.504474\pi$
$149$	−0.343146	−0.0281116	−0.0140558	+	0.999901i	$0.504474\pi$
$150$	0.585786	0.0478293
$151$	18.2426	1.48457	0.742283	−	0.670087i	$-0.233743\pi$
$151$	18.2426	1.48457	0.742283	+	0.670087i	$0.233743\pi$
$152$	−5.41421	−0.439151
$153$	4.82843	0.390355
$154$	−0.201010	−0.0161979
$155$	10.2426	0.822709
$156$	2.58579	0.207029
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	3.51472	0.279616
$159$	−20.4853	−1.62459
$160$	4.41421	0.348974
$161$	1.17157	0.0923329
$162$	−2.07107	−0.162718
$163$	−14.9706	−1.17258	−0.586292	−	0.810099i	$-0.699413\pi$
$163$	−14.9706	−1.17258	−0.586292	+	0.810099i	$0.699413\pi$
$164$	16.1421	1.26049
$165$	0.828427	0.0644930
$166$	−3.65685	−0.283827
$167$	−8.82843	−0.683164	−0.341582	−	0.939852i	$-0.610963\pi$
$167$	−8.82843	−0.683164	−0.341582	+	0.939852i	$0.610963\pi$
$168$	1.85786	0.143337
$169$	1.00000	0.0769231
$170$	−2.00000	−0.153393
$171$	−3.41421	−0.261091
$172$	−5.61522	−0.428157
$173$	11.1716	0.849359	0.424679	−	0.905344i	$-0.360387\pi$
$173$	11.1716	0.849359	0.424679	+	0.905344i	$0.360387\pi$
$174$	3.31371	0.251212
$175$	−0.828427	−0.0626232
$176$	1.75736	0.132466
$177$	14.4853	1.08878
$178$	2.48528	0.186280
$179$	−5.65685	−0.422813	−0.211407	−	0.977398i	$-0.567804\pi$
$179$	−5.65685	−0.422813	−0.211407	+	0.977398i	$0.567804\pi$
$180$	1.82843	0.136283
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0.343146	0.0254357
$183$	−11.3137	−0.836333
$184$	2.24264	0.165330
$185$	8.48528	0.623850
$186$	6.00000	0.439941
$187$	−2.82843	−0.206835
$188$	−1.51472	−0.110472
$189$	4.68629	0.340878
$190$	1.41421	0.102598
$191$	13.6569	0.988175	0.494088	−	0.869412i	$-0.335502\pi$
$191$	13.6569	0.988175	0.494088	+	0.869412i	$0.335502\pi$
$192$	−5.89949	−0.425759
$193$	15.6569	1.12701	0.563503	−	0.826114i	$-0.309454\pi$
$193$	15.6569	1.12701	0.563503	+	0.826114i	$0.309454\pi$
$194$	1.51472	0.108750
$195$	−1.41421	−0.101274
$196$	11.5442	0.824583
$197$	−22.9706	−1.63658	−0.818292	−	0.574802i	$-0.805079\pi$
$197$	−22.9706	−1.63658	−0.818292	+	0.574802i	$0.805079\pi$
$198$	−0.242641	−0.0172437
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	−1.58579	−0.112132
$201$	−2.82843	−0.199502
$202$	3.17157	0.223151
$203$	−4.68629	−0.328913
$204$	12.4853	0.874145
$205$	−8.82843	−0.616604
$206$	7.21320	0.502568
$207$	1.41421	0.0982946
$208$	−3.00000	−0.208013
$209$	2.00000	0.138343
$210$	−0.485281	−0.0334876
$211$	−19.3137	−1.32961	−0.664805	−	0.747017i	$-0.731485\pi$
$211$	−19.3137	−1.32961	−0.664805	+	0.747017i	$0.731485\pi$
$212$	26.4853	1.81902
$213$	−11.1716	−0.765464
$214$	−2.72792	−0.186477
$215$	3.07107	0.209445
$216$	8.97056	0.610369
$217$	−8.48528	−0.576018
$218$	−0.828427	−0.0561082
$219$	−12.0000	−0.810885
$220$	−1.07107	−0.0722114
$221$	4.82843	0.324795
$222$	4.97056	0.333602
$223$	26.4853	1.77359	0.886793	−	0.462167i	$-0.152928\pi$
$223$	26.4853	1.77359	0.886793	+	0.462167i	$0.152928\pi$
$224$	−3.65685	−0.244334
$225$	−1.00000	−0.0666667
$226$	−1.31371	−0.0873866
$227$	−27.6569	−1.83565	−0.917825	−	0.396985i	$-0.870056\pi$
$227$	−27.6569	−1.83565	−0.917825	+	0.396985i	$0.870056\pi$
$228$	−8.82843	−0.584677
$229$	0.828427	0.0547440	0.0273720	−	0.999625i	$-0.491286\pi$
$229$	0.828427	0.0547440	0.0273720	+	0.999625i	$0.491286\pi$
$230$	−0.585786	−0.0386256
$231$	−0.686292	−0.0451547
$232$	−8.97056	−0.588946
$233$	24.6274	1.61340	0.806698	−	0.590964i	$-0.201253\pi$
$233$	24.6274	1.61340	0.806698	+	0.590964i	$0.201253\pi$
$234$	0.414214	0.0270780
$235$	0.828427	0.0540406
$236$	−18.7279	−1.21908
$237$	12.0000	0.779484
$238$	1.65685	0.107398
$239$	−0.585786	−0.0378914	−0.0189457	−	0.999821i	$-0.506031\pi$
$239$	−0.585786	−0.0378914	−0.0189457	+	0.999821i	$0.506031\pi$
$240$	4.24264	0.273861
$241$	2.48528	0.160091	0.0800455	−	0.996791i	$-0.474493\pi$
$241$	2.48528	0.160091	0.0800455	+	0.996791i	$0.474493\pi$
$242$	−4.41421	−0.283756
$243$	9.89949	0.635053
$244$	14.6274	0.936424
$245$	−6.31371	−0.403368
$246$	−5.17157	−0.329727
$247$	−3.41421	−0.217241
$248$	−16.2426	−1.03141
$249$	−12.4853	−0.791223
$250$	0.414214	0.0261972
$251$	−19.7990	−1.24970	−0.624851	−	0.780744i	$-0.714840\pi$
$251$	−19.7990	−1.24970	−0.624851	+	0.780744i	$0.714840\pi$
$252$	−1.51472	−0.0954183
$253$	−0.828427	−0.0520828
$254$	−3.89949	−0.244676
$255$	−6.82843	−0.427613
$256$	3.97056	0.248160
$257$	16.3431	1.01946	0.509729	−	0.860335i	$-0.329746\pi$
$257$	16.3431	1.01946	0.509729	+	0.860335i	$0.329746\pi$
$258$	1.79899	0.112000
$259$	−7.02944	−0.436788
$260$	1.82843	0.113394
$261$	−5.65685	−0.350150
$262$	7.02944	0.434280
$263$	−13.4142	−0.827156	−0.413578	−	0.910469i	$-0.635721\pi$
$263$	−13.4142	−0.827156	−0.413578	+	0.910469i	$0.635721\pi$
$264$	−1.31371	−0.0808532
$265$	−14.4853	−0.889824
$266$	−1.17157	−0.0718337
$267$	8.48528	0.519291
$268$	3.65685	0.223378
$269$	−2.68629	−0.163786	−0.0818930	−	0.996641i	$-0.526097\pi$
$269$	−2.68629	−0.163786	−0.0818930	+	0.996641i	$0.526097\pi$
$270$	−2.34315	−0.142599
$271$	1.27208	0.0772732	0.0386366	−	0.999253i	$-0.487699\pi$
$271$	1.27208	0.0772732	0.0386366	+	0.999253i	$0.487699\pi$
$272$	−14.4853	−0.878299
$273$	1.17157	0.0709068
$274$	2.20101	0.132968
$275$	0.585786	0.0353243
$276$	3.65685	0.220117
$277$	−7.17157	−0.430898	−0.215449	−	0.976515i	$-0.569122\pi$
$277$	−7.17157	−0.430898	−0.215449	+	0.976515i	$0.569122\pi$
$278$	−5.17157	−0.310170
$279$	−10.2426	−0.613211
$280$	1.31371	0.0785091
$281$	−17.7990	−1.06180	−0.530899	−	0.847435i	$-0.678146\pi$
$281$	−17.7990	−1.06180	−0.530899	+	0.847435i	$0.678146\pi$
$282$	0.485281	0.0288981
$283$	8.72792	0.518821	0.259411	−	0.965767i	$-0.416472\pi$
$283$	8.72792	0.518821	0.259411	+	0.965767i	$0.416472\pi$
$284$	14.4437	0.857073
$285$	4.82843	0.286011
$286$	−0.242641	−0.0143476
$287$	7.31371	0.431715
$288$	−4.41421	−0.260110
$289$	6.31371	0.371395
$290$	2.34315	0.137594
$291$	5.17157	0.303163
$292$	15.5147	0.907930
$293$	−2.14214	−0.125145	−0.0625724	−	0.998040i	$-0.519930\pi$
$293$	−2.14214	−0.125145	−0.0625724	+	0.998040i	$0.519930\pi$
$294$	−3.69848	−0.215700
$295$	10.2426	0.596350
$296$	−13.4558	−0.782105
$297$	−3.31371	−0.192281
$298$	−0.142136	−0.00823370
$299$	1.41421	0.0817861
$300$	−2.58579	−0.149290
$301$	−2.54416	−0.146643
$302$	7.55635	0.434819
$303$	10.8284	0.622077
$304$	10.2426	0.587456
$305$	−8.00000	−0.458079
$306$	2.00000	0.114332
$307$	19.1716	1.09418	0.547090	−	0.837074i	$-0.315736\pi$
$307$	19.1716	1.09418	0.547090	+	0.837074i	$0.315736\pi$
$308$	0.887302	0.0505587
$309$	24.6274	1.40100
$310$	4.24264	0.240966
$311$	−8.48528	−0.481156	−0.240578	−	0.970630i	$-0.577337\pi$
$311$	−8.48528	−0.481156	−0.240578	+	0.970630i	$0.577337\pi$
$312$	2.24264	0.126965
$313$	0.828427	0.0468255	0.0234127	−	0.999726i	$-0.492547\pi$
$313$	0.828427	0.0468255	0.0234127	+	0.999726i	$0.492547\pi$
$314$	7.45584	0.420758
$315$	0.828427	0.0466766
$316$	−15.5147	−0.872771
$317$	26.1421	1.46829	0.734144	−	0.678993i	$-0.237584\pi$
$317$	26.1421	1.46829	0.734144	+	0.678993i	$0.237584\pi$
$318$	−8.48528	−0.475831
$319$	3.31371	0.185532
$320$	−4.17157	−0.233198
$321$	−9.31371	−0.519841
$322$	0.485281	0.0270437
$323$	−16.4853	−0.917266
$324$	9.14214	0.507896
$325$	−1.00000	−0.0554700
$326$	−6.20101	−0.343442
$327$	−2.82843	−0.156412
$328$	14.0000	0.773021
$329$	−0.686292	−0.0378365
$330$	0.343146	0.0188896
$331$	22.0416	1.21152	0.605759	−	0.795648i	$-0.292870\pi$
$331$	22.0416	1.21152	0.605759	+	0.795648i	$0.292870\pi$
$332$	16.1421	0.885915
$333$	−8.48528	−0.464991
$334$	−3.65685	−0.200094
$335$	−2.00000	−0.109272
$336$	−3.51472	−0.191744
$337$	7.17157	0.390660	0.195330	−	0.980738i	$-0.437422\pi$
$337$	7.17157	0.390660	0.195330	+	0.980738i	$0.437422\pi$
$338$	0.414214	0.0225302
$339$	−4.48528	−0.243607
$340$	8.82843	0.478789
$341$	6.00000	0.324918
$342$	−1.41421	−0.0764719
$343$	11.0294	0.595534
$344$	−4.87006	−0.262576
$345$	−2.00000	−0.107676
$346$	4.62742	0.248771
$347$	4.24264	0.227757	0.113878	−	0.993495i	$-0.463673\pi$
$347$	4.24264	0.227757	0.113878	+	0.993495i	$0.463673\pi$
$348$	−14.6274	−0.784112
$349$	1.51472	0.0810810	0.0405405	−	0.999178i	$-0.487092\pi$
$349$	1.51472	0.0810810	0.0405405	+	0.999178i	$0.487092\pi$
$350$	−0.343146	−0.0183419
$351$	5.65685	0.301941
$352$	2.58579	0.137823
$353$	9.17157	0.488154	0.244077	−	0.969756i	$-0.421515\pi$
$353$	9.17157	0.488154	0.244077	+	0.969756i	$0.421515\pi$
$354$	6.00000	0.318896
$355$	−7.89949	−0.419262
$356$	−10.9706	−0.581439
$357$	5.65685	0.299392
$358$	−2.34315	−0.123839
$359$	−27.8995	−1.47248	−0.736240	−	0.676721i	$-0.763400\pi$
$359$	−27.8995	−1.47248	−0.736240	+	0.676721i	$0.763400\pi$
$360$	1.58579	0.0835783
$361$	−7.34315	−0.386481
$362$	0	0
$363$	−15.0711	−0.791026
$364$	−1.51472	−0.0793928
$365$	−8.48528	−0.444140
$366$	−4.68629	−0.244956
$367$	4.44365	0.231957	0.115978	−	0.993252i	$-0.463000\pi$
$367$	4.44365	0.231957	0.115978	+	0.993252i	$0.463000\pi$
$368$	−4.24264	−0.221163
$369$	8.82843	0.459590
$370$	3.51472	0.182722
$371$	12.0000	0.623009
$372$	−26.4853	−1.37320
$373$	−25.3137	−1.31069	−0.655347	−	0.755328i	$-0.727478\pi$
$373$	−25.3137	−1.31069	−0.655347	+	0.755328i	$0.727478\pi$
$374$	−1.17157	−0.0605806
$375$	1.41421	0.0730297
$376$	−1.31371	−0.0677493
$377$	−5.65685	−0.291343
$378$	1.94113	0.0998407
$379$	14.9289	0.766848	0.383424	−	0.923572i	$-0.374745\pi$
$379$	14.9289	0.766848	0.383424	+	0.923572i	$0.374745\pi$
$380$	−6.24264	−0.320241
$381$	−13.3137	−0.682082
$382$	5.65685	0.289430
$383$	33.1127	1.69198	0.845990	−	0.533199i	$-0.179010\pi$
$383$	33.1127	1.69198	0.845990	+	0.533199i	$0.179010\pi$
$384$	−14.9289	−0.761839
$385$	−0.485281	−0.0247322
$386$	6.48528	0.330092
$387$	−3.07107	−0.156111
$388$	−6.68629	−0.339445
$389$	−16.6274	−0.843044	−0.421522	−	0.906818i	$-0.638504\pi$
$389$	−16.6274	−0.843044	−0.421522	+	0.906818i	$0.638504\pi$
$390$	−0.585786	−0.0296624
$391$	6.82843	0.345328
$392$	10.0122	0.505692
$393$	24.0000	1.21064
$394$	−9.51472	−0.479345
$395$	8.48528	0.426941
$396$	1.07107	0.0538232
$397$	−27.7990	−1.39519	−0.697596	−	0.716492i	$-0.745747\pi$
$397$	−27.7990	−1.39519	−0.697596	+	0.716492i	$0.745747\pi$
$398$	1.65685	0.0830506
$399$	−4.00000	−0.200250
$400$	3.00000	0.150000
$401$	17.3137	0.864605	0.432303	−	0.901729i	$-0.357701\pi$
$401$	17.3137	0.864605	0.432303	+	0.901729i	$0.357701\pi$
$402$	−1.17157	−0.0584327
$403$	−10.2426	−0.510222
$404$	−14.0000	−0.696526
$405$	−5.00000	−0.248452
$406$	−1.94113	−0.0963364
$407$	4.97056	0.246382
$408$	10.8284	0.536087
$409$	12.8284	0.634325	0.317162	−	0.948371i	$-0.397270\pi$
$409$	12.8284	0.634325	0.317162	+	0.948371i	$0.397270\pi$
$410$	−3.65685	−0.180599
$411$	7.51472	0.370674
$412$	−31.8406	−1.56867
$413$	−8.48528	−0.417533
$414$	0.585786	0.0287898
$415$	−8.82843	−0.433370
$416$	−4.41421	−0.216425
$417$	−17.6569	−0.864660
$418$	0.828427	0.0405197
$419$	5.17157	0.252648	0.126324	−	0.991989i	$-0.459682\pi$
$419$	5.17157	0.252648	0.126324	+	0.991989i	$0.459682\pi$
$420$	2.14214	0.104526
$421$	−1.02944	−0.0501717	−0.0250859	−	0.999685i	$-0.507986\pi$
$421$	−1.02944	−0.0501717	−0.0250859	+	0.999685i	$0.507986\pi$
$422$	−8.00000	−0.389434
$423$	−0.828427	−0.0402795
$424$	22.9706	1.11555
$425$	−4.82843	−0.234213
$426$	−4.62742	−0.224199
$427$	6.62742	0.320723
$428$	12.0416	0.582054
$429$	−0.828427	−0.0399968
$430$	1.27208	0.0613450
$431$	3.61522	0.174139	0.0870696	−	0.996202i	$-0.472250\pi$
$431$	3.61522	0.174139	0.0870696	+	0.996202i	$0.472250\pi$
$432$	−16.9706	−0.816497
$433$	−3.65685	−0.175737	−0.0878686	−	0.996132i	$-0.528006\pi$
$433$	−3.65685	−0.175737	−0.0878686	+	0.996132i	$0.528006\pi$
$434$	−3.51472	−0.168712
$435$	8.00000	0.383571
$436$	3.65685	0.175132
$437$	−4.82843	−0.230975
$438$	−4.97056	−0.237503
$439$	−32.9706	−1.57360	−0.786800	−	0.617209i	$-0.788263\pi$
$439$	−32.9706	−1.57360	−0.786800	+	0.617209i	$0.788263\pi$
$440$	−0.928932	−0.0442851
$441$	6.31371	0.300653
$442$	2.00000	0.0951303
$443$	−6.58579	−0.312900	−0.156450	−	0.987686i	$-0.550005\pi$
$443$	−6.58579	−0.312900	−0.156450	+	0.987686i	$0.550005\pi$
$444$	−21.9411	−1.04128
$445$	6.00000	0.284427
$446$	10.9706	0.519471
$447$	−0.485281	−0.0229530
$448$	3.45584	0.163273
$449$	29.1127	1.37391	0.686957	−	0.726698i	$-0.258946\pi$
$449$	29.1127	1.37391	0.686957	+	0.726698i	$0.258946\pi$
$450$	−0.414214	−0.0195262
$451$	−5.17157	−0.243520
$452$	5.79899	0.272762
$453$	25.7990	1.21214
$454$	−11.4558	−0.537649
$455$	0.828427	0.0388373
$456$	−7.65685	−0.358565
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	0.343146	0.0160341
$459$	27.3137	1.27489
$460$	2.58579	0.120563
$461$	26.4853	1.23354	0.616771	−	0.787142i	$-0.288440\pi$
$461$	26.4853	1.23354	0.616771	+	0.787142i	$0.288440\pi$
$462$	−0.284271	−0.0132255
$463$	−15.6569	−0.727636	−0.363818	−	0.931470i	$-0.618527\pi$
$463$	−15.6569	−0.727636	−0.363818	+	0.931470i	$0.618527\pi$
$464$	16.9706	0.787839
$465$	14.4853	0.671739
$466$	10.2010	0.472553
$467$	−10.5858	−0.489852	−0.244926	−	0.969542i	$-0.578764\pi$
$467$	−10.5858	−0.489852	−0.244926	+	0.969542i	$0.578764\pi$
$468$	−1.82843	−0.0845191
$469$	1.65685	0.0765064
$470$	0.343146	0.0158281
$471$	25.4558	1.17294
$472$	−16.2426	−0.747628
$473$	1.79899	0.0827176
$474$	4.97056	0.228306
$475$	3.41421	0.156655
$476$	−7.31371	−0.335223
$477$	14.4853	0.663235
$478$	−0.242641	−0.0110981
$479$	−5.27208	−0.240887	−0.120444	−	0.992720i	$-0.538432\pi$
$479$	−5.27208	−0.240887	−0.120444	+	0.992720i	$0.538432\pi$
$480$	6.24264	0.284936
$481$	−8.48528	−0.386896
$482$	1.02944	0.0468896
$483$	1.65685	0.0753895
$484$	19.4853	0.885695
$485$	3.65685	0.166049
$486$	4.10051	0.186003
$487$	22.9706	1.04090	0.520448	−	0.853894i	$-0.325765\pi$
$487$	22.9706	1.04090	0.520448	+	0.853894i	$0.325765\pi$
$488$	12.6863	0.574281
$489$	−21.1716	−0.957412
$490$	−2.61522	−0.118144
$491$	10.8284	0.488680	0.244340	−	0.969690i	$-0.421429\pi$
$491$	10.8284	0.488680	0.244340	+	0.969690i	$0.421429\pi$
$492$	22.8284	1.02918
$493$	−27.3137	−1.23015
$494$	−1.41421	−0.0636285
$495$	−0.585786	−0.0263291
$496$	30.7279	1.37972
$497$	6.54416	0.293546
$498$	−5.17157	−0.231744
$499$	10.4437	0.467522	0.233761	−	0.972294i	$-0.424897\pi$
$499$	10.4437	0.467522	0.233761	+	0.972294i	$0.424897\pi$
$500$	−1.82843	−0.0817697
$501$	−12.4853	−0.557801
$502$	−8.20101	−0.366029
$503$	18.1005	0.807062	0.403531	−	0.914966i	$-0.367783\pi$
$503$	18.1005	0.807062	0.403531	+	0.914966i	$0.367783\pi$
$504$	−1.31371	−0.0585172
$505$	7.65685	0.340726
$506$	−0.343146	−0.0152547
$507$	1.41421	0.0628074
$508$	17.2132	0.763712
$509$	−21.1127	−0.935804	−0.467902	−	0.883780i	$-0.654990\pi$
$509$	−21.1127	−0.935804	−0.467902	+	0.883780i	$0.654990\pi$
$510$	−2.82843	−0.125245
$511$	7.02944	0.310964
$512$	22.7574	1.00574
$513$	−19.3137	−0.852721
$514$	6.76955	0.298592
$515$	17.4142	0.767362
$516$	−7.94113	−0.349589
$517$	0.485281	0.0213427
$518$	−2.91169	−0.127932
$519$	15.7990	0.693499
$520$	1.58579	0.0695413
$521$	−6.34315	−0.277898	−0.138949	−	0.990300i	$-0.544372\pi$
$521$	−6.34315	−0.277898	−0.138949	+	0.990300i	$0.544372\pi$
$522$	−2.34315	−0.102557
$523$	−28.2426	−1.23496	−0.617482	−	0.786585i	$-0.711847\pi$
$523$	−28.2426	−1.23496	−0.617482	+	0.786585i	$0.711847\pi$
$524$	−31.0294	−1.35553
$525$	−1.17157	−0.0511316
$526$	−5.55635	−0.242268
$527$	−49.4558	−2.15433
$528$	2.48528	0.108158
$529$	−21.0000	−0.913043
$530$	−6.00000	−0.260623
$531$	−10.2426	−0.444493
$532$	5.17157	0.224216
$533$	8.82843	0.382402
$534$	3.51472	0.152097
$535$	−6.58579	−0.284728
$536$	3.17157	0.136991
$537$	−8.00000	−0.345225
$538$	−1.11270	−0.0479718
$539$	−3.69848	−0.159305
$540$	10.3431	0.445098
$541$	−12.8284	−0.551537	−0.275769	−	0.961224i	$-0.588932\pi$
$541$	−12.8284	−0.551537	−0.275769	+	0.961224i	$0.588932\pi$
$542$	0.526912	0.0226328
$543$	0	0
$544$	−21.3137	−0.913818
$545$	−2.00000	−0.0856706
$546$	0.485281	0.0207681
$547$	−29.2132	−1.24907	−0.624533	−	0.780998i	$-0.714711\pi$
$547$	−29.2132	−1.24907	−0.624533	+	0.780998i	$0.714711\pi$
$548$	−9.71573	−0.415035
$549$	8.00000	0.341432
$550$	0.242641	0.0103462
$551$	19.3137	0.822792
$552$	3.17157	0.134991
$553$	−7.02944	−0.298922
$554$	−2.97056	−0.126207
$555$	12.0000	0.509372
$556$	22.8284	0.968141
$557$	3.79899	0.160968	0.0804842	−	0.996756i	$-0.474353\pi$
$557$	3.79899	0.160968	0.0804842	+	0.996756i	$0.474353\pi$
$558$	−4.24264	−0.179605
$559$	−3.07107	−0.129892
$560$	−2.48528	−0.105022
$561$	−4.00000	−0.168880
$562$	−7.37258	−0.310994
$563$	−16.2426	−0.684546	−0.342273	−	0.939601i	$-0.611197\pi$
$563$	−16.2426	−0.684546	−0.342273	+	0.939601i	$0.611197\pi$
$564$	−2.14214	−0.0902002
$565$	−3.17157	−0.133429
$566$	3.61522	0.151959
$567$	4.14214	0.173953
$568$	12.5269	0.525618
$569$	−21.6569	−0.907903	−0.453951	−	0.891027i	$-0.649986\pi$
$569$	−21.6569	−0.907903	−0.453951	+	0.891027i	$0.649986\pi$
$570$	2.00000	0.0837708
$571$	−28.4853	−1.19207	−0.596036	−	0.802958i	$-0.703258\pi$
$571$	−28.4853	−1.19207	−0.596036	+	0.802958i	$0.703258\pi$
$572$	1.07107	0.0447836
$573$	19.3137	0.806842
$574$	3.02944	0.126446
$575$	−1.41421	−0.0589768
$576$	4.17157	0.173816
$577$	−29.1716	−1.21443	−0.607214	−	0.794538i	$-0.707713\pi$
$577$	−29.1716	−1.21443	−0.607214	+	0.794538i	$0.707713\pi$
$578$	2.61522	0.108779
$579$	22.1421	0.920196
$580$	−10.3431	−0.429476
$581$	7.31371	0.303424
$582$	2.14214	0.0887944
$583$	−8.48528	−0.351424
$584$	13.4558	0.556807
$585$	1.00000	0.0413449
$586$	−0.887302	−0.0366541
$587$	31.6569	1.30662	0.653309	−	0.757091i	$-0.273380\pi$
$587$	31.6569	1.30662	0.653309	+	0.757091i	$0.273380\pi$
$588$	16.3259	0.673269
$589$	34.9706	1.44094
$590$	4.24264	0.174667
$591$	−32.4853	−1.33627
$592$	25.4558	1.04623
$593$	20.6274	0.847066	0.423533	−	0.905881i	$-0.360790\pi$
$593$	20.6274	0.847066	0.423533	+	0.905881i	$0.360790\pi$
$594$	−1.37258	−0.0563178
$595$	4.00000	0.163984
$596$	0.627417	0.0257000
$597$	5.65685	0.231520
$598$	0.585786	0.0239546
$599$	−25.4558	−1.04010	−0.520049	−	0.854137i	$-0.674086\pi$
$599$	−25.4558	−1.04010	−0.520049	+	0.854137i	$0.674086\pi$
$600$	−2.24264	−0.0915554
$601$	0.627417	0.0255929	0.0127964	−	0.999918i	$-0.495927\pi$
$601$	0.627417	0.0255929	0.0127964	+	0.999918i	$0.495927\pi$
$602$	−1.05382	−0.0429507
$603$	2.00000	0.0814463
$604$	−33.3553	−1.35721
$605$	−10.6569	−0.433263
$606$	4.48528	0.182202
$607$	40.2426	1.63340	0.816699	−	0.577064i	$-0.195802\pi$
$607$	40.2426	1.63340	0.816699	+	0.577064i	$0.195802\pi$
$608$	15.0711	0.611213
$609$	−6.62742	−0.268556
$610$	−3.31371	−0.134168
$611$	−0.828427	−0.0335146
$612$	−8.82843	−0.356868
$613$	−37.3137	−1.50709	−0.753543	−	0.657398i	$-0.771657\pi$
$613$	−37.3137	−1.50709	−0.753543	+	0.657398i	$0.771657\pi$
$614$	7.94113	0.320478
$615$	−12.4853	−0.503455
$616$	0.769553	0.0310062
$617$	−22.9706	−0.924760	−0.462380	−	0.886682i	$-0.653004\pi$
$617$	−22.9706	−0.924760	−0.462380	+	0.886682i	$0.653004\pi$
$618$	10.2010	0.410345
$619$	10.2426	0.411686	0.205843	−	0.978585i	$-0.434006\pi$
$619$	10.2426	0.411686	0.205843	+	0.978585i	$0.434006\pi$
$620$	−18.7279	−0.752131
$621$	8.00000	0.321029
$622$	−3.51472	−0.140927
$623$	−4.97056	−0.199141
$624$	−4.24264	−0.169842
$625$	1.00000	0.0400000
$626$	0.343146	0.0137149
$627$	2.82843	0.112956
$628$	−32.9117	−1.31332
$629$	−40.9706	−1.63360
$630$	0.343146	0.0136713
$631$	−18.2426	−0.726228	−0.363114	−	0.931745i	$-0.618286\pi$
$631$	−18.2426	−0.726228	−0.363114	+	0.931745i	$0.618286\pi$
$632$	−13.4558	−0.535245
$633$	−27.3137	−1.08562
$634$	10.8284	0.430052
$635$	−9.41421	−0.373592
$636$	37.4558	1.48522
$637$	6.31371	0.250158
$638$	1.37258	0.0543411
$639$	7.89949	0.312499
$640$	−10.5563	−0.417276
$641$	36.3431	1.43547	0.717734	−	0.696317i	$-0.245179\pi$
$641$	36.3431	1.43547	0.717734	+	0.696317i	$0.245179\pi$
$642$	−3.85786	−0.152258
$643$	26.4853	1.04448	0.522239	−	0.852799i	$-0.325097\pi$
$643$	26.4853	1.04448	0.522239	+	0.852799i	$0.325097\pi$
$644$	−2.14214	−0.0844120
$645$	4.34315	0.171011
$646$	−6.82843	−0.268661
$647$	6.58579	0.258914	0.129457	−	0.991585i	$-0.458677\pi$
$647$	6.58579	0.258914	0.129457	+	0.991585i	$0.458677\pi$
$648$	7.92893	0.311478
$649$	6.00000	0.235521
$650$	−0.414214	−0.0162468
$651$	−12.0000	−0.470317
$652$	27.3726	1.07199
$653$	13.0294	0.509881	0.254941	−	0.966957i	$-0.417944\pi$
$653$	13.0294	0.509881	0.254941	+	0.966957i	$0.417944\pi$
$654$	−1.17157	−0.0458121
$655$	16.9706	0.663095
$656$	−26.4853	−1.03408
$657$	8.48528	0.331042
$658$	−0.284271	−0.0110820
$659$	46.1421	1.79744	0.898721	−	0.438520i	$-0.144497\pi$
$659$	46.1421	1.79744	0.898721	+	0.438520i	$0.144497\pi$
$660$	−1.51472	−0.0589603
$661$	−49.5980	−1.92914	−0.964569	−	0.263831i	$-0.915014\pi$
$661$	−49.5980	−1.92914	−0.964569	+	0.263831i	$0.915014\pi$
$662$	9.12994	0.354845
$663$	6.82843	0.265194
$664$	14.0000	0.543305
$665$	−2.82843	−0.109682
$666$	−3.51472	−0.136193
$667$	−8.00000	−0.309761
$668$	16.1421	0.624558
$669$	37.4558	1.44813
$670$	−0.828427	−0.0320049
$671$	−4.68629	−0.180912
$672$	−5.17157	−0.199498
$673$	−10.4853	−0.404178	−0.202089	−	0.979367i	$-0.564773\pi$
$673$	−10.4853	−0.404178	−0.202089	+	0.979367i	$0.564773\pi$
$674$	2.97056	0.114422
$675$	−5.65685	−0.217732
$676$	−1.82843	−0.0703241
$677$	8.14214	0.312928	0.156464	−	0.987684i	$-0.449991\pi$
$677$	8.14214	0.312928	0.156464	+	0.987684i	$0.449991\pi$
$678$	−1.85786	−0.0713509
$679$	−3.02944	−0.116259
$680$	7.65685	0.293627
$681$	−39.1127	−1.49880
$682$	2.48528	0.0951663
$683$	33.3137	1.27471	0.637357	−	0.770569i	$-0.280028\pi$
$683$	33.3137	1.27471	0.637357	+	0.770569i	$0.280028\pi$
$684$	6.24264	0.238693
$685$	5.31371	0.203026
$686$	4.56854	0.174428
$687$	1.17157	0.0446983
$688$	9.21320	0.351250
$689$	14.4853	0.551845
$690$	−0.828427	−0.0315377
$691$	21.0711	0.801581	0.400791	−	0.916170i	$-0.368735\pi$
$691$	21.0711	0.801581	0.400791	+	0.916170i	$0.368735\pi$
$692$	−20.4264	−0.776495
$693$	0.485281	0.0184343
$694$	1.75736	0.0667084
$695$	−12.4853	−0.473594
$696$	−12.6863	−0.480873
$697$	42.6274	1.61463
$698$	0.627417	0.0237481
$699$	34.8284	1.31733
$700$	1.51472	0.0572510
$701$	37.3137	1.40932	0.704660	−	0.709545i	$-0.251100\pi$
$701$	37.3137	1.40932	0.704660	+	0.709545i	$0.251100\pi$
$702$	2.34315	0.0884363
$703$	28.9706	1.09265
$704$	−2.44365	−0.0920986
$705$	1.17157	0.0441240
$706$	3.79899	0.142977
$707$	−6.34315	−0.238559
$708$	−26.4853	−0.995378
$709$	−17.1127	−0.642681	−0.321340	−	0.946964i	$-0.604133\pi$
$709$	−17.1127	−0.642681	−0.321340	+	0.946964i	$0.604133\pi$
$710$	−3.27208	−0.122799
$711$	−8.48528	−0.318223
$712$	−9.51472	−0.356579
$713$	−14.4853	−0.542478
$714$	2.34315	0.0876900
$715$	−0.585786	−0.0219072
$716$	10.3431	0.386542
$717$	−0.828427	−0.0309382
$718$	−11.5563	−0.431279
$719$	−4.97056	−0.185371	−0.0926854	−	0.995695i	$-0.529545\pi$
$719$	−4.97056	−0.185371	−0.0926854	+	0.995695i	$0.529545\pi$
$720$	−3.00000	−0.111803
$721$	−14.4264	−0.537267
$722$	−3.04163	−0.113198
$723$	3.51472	0.130714
$724$	0	0
$725$	5.65685	0.210090
$726$	−6.24264	−0.231686
$727$	19.3553	0.717850	0.358925	−	0.933366i	$-0.383143\pi$
$727$	19.3553	0.717850	0.358925	+	0.933366i	$0.383143\pi$
$728$	−1.31371	−0.0486893
$729$	29.0000	1.07407
$730$	−3.51472	−0.130086
$731$	−14.8284	−0.548449
$732$	20.6863	0.764587
$733$	−1.31371	−0.0485229	−0.0242615	−	0.999706i	$-0.507723\pi$
$733$	−1.31371	−0.0485229	−0.0242615	+	0.999706i	$0.507723\pi$
$734$	1.84062	0.0679385
$735$	−8.92893	−0.329349
$736$	−6.24264	−0.230107
$737$	−1.17157	−0.0431554
$738$	3.65685	0.134611
$739$	30.7279	1.13034	0.565172	−	0.824973i	$-0.308810\pi$
$739$	30.7279	1.13034	0.565172	+	0.824973i	$0.308810\pi$
$740$	−15.5147	−0.570332
$741$	−4.82843	−0.177377
$742$	4.97056	0.182475
$743$	−38.4853	−1.41189	−0.705944	−	0.708268i	$-0.749477\pi$
$743$	−38.4853	−1.41189	−0.705944	+	0.708268i	$0.749477\pi$
$744$	−22.9706	−0.842142
$745$	−0.343146	−0.0125719
$746$	−10.4853	−0.383893
$747$	8.82843	0.323015
$748$	5.17157	0.189091
$749$	5.45584	0.199352
$750$	0.585786	0.0213899
$751$	−44.4853	−1.62329	−0.811645	−	0.584150i	$-0.801428\pi$
$751$	−44.4853	−1.62329	−0.811645	+	0.584150i	$0.801428\pi$
$752$	2.48528	0.0906289
$753$	−28.0000	−1.02038
$754$	−2.34315	−0.0853323
$755$	18.2426	0.663918
$756$	−8.56854	−0.311635
$757$	4.14214	0.150548	0.0752742	−	0.997163i	$-0.476017\pi$
$757$	4.14214	0.150548	0.0752742	+	0.997163i	$0.476017\pi$
$758$	6.18377	0.224605
$759$	−1.17157	−0.0425254
$760$	−5.41421	−0.196394
$761$	−36.6274	−1.32774	−0.663871	−	0.747847i	$-0.731088\pi$
$761$	−36.6274	−1.32774	−0.663871	+	0.747847i	$0.731088\pi$
$762$	−5.51472	−0.199777
$763$	1.65685	0.0599822
$764$	−24.9706	−0.903403
$765$	4.82843	0.174572
$766$	13.7157	0.495569
$767$	−10.2426	−0.369840
$768$	5.61522	0.202622
$769$	10.9706	0.395609	0.197804	−	0.980242i	$-0.436619\pi$
$769$	10.9706	0.395609	0.197804	+	0.980242i	$0.436619\pi$
$770$	−0.201010	−0.00724390
$771$	23.1127	0.832383
$772$	−28.6274	−1.03032
$773$	6.14214	0.220917	0.110459	−	0.993881i	$-0.464768\pi$
$773$	6.14214	0.220917	0.110459	+	0.993881i	$0.464768\pi$
$774$	−1.27208	−0.0457239
$775$	10.2426	0.367927
$776$	−5.79899	−0.208172
$777$	−9.94113	−0.356636
$778$	−6.88730	−0.246922
$779$	−30.1421	−1.07995
$780$	2.58579	0.0925860
$781$	−4.62742	−0.165582
$782$	2.82843	0.101144
$783$	−32.0000	−1.14359
$784$	−18.9411	−0.676469
$785$	18.0000	0.642448
$786$	9.94113	0.354588
$787$	5.51472	0.196578	0.0982892	−	0.995158i	$-0.468663\pi$
$787$	5.51472	0.196578	0.0982892	+	0.995158i	$0.468663\pi$
$788$	42.0000	1.49619
$789$	−18.9706	−0.675370
$790$	3.51472	0.125048
$791$	2.62742	0.0934202
$792$	0.928932	0.0330082
$793$	8.00000	0.284088
$794$	−11.5147	−0.408642
$795$	−20.4853	−0.726538
$796$	−7.31371	−0.259228
$797$	10.9706	0.388597	0.194299	−	0.980942i	$-0.437757\pi$
$797$	10.9706	0.388597	0.194299	+	0.980942i	$0.437757\pi$
$798$	−1.65685	−0.0586520
$799$	−4.00000	−0.141510
$800$	4.41421	0.156066
$801$	−6.00000	−0.212000
$802$	7.17157	0.253237
$803$	−4.97056	−0.175407
$804$	5.17157	0.182387
$805$	1.17157	0.0412925
$806$	−4.24264	−0.149441
$807$	−3.79899	−0.133731
$808$	−12.1421	−0.427159
$809$	−45.2548	−1.59108	−0.795538	−	0.605904i	$-0.792811\pi$
$809$	−45.2548	−1.59108	−0.795538	+	0.605904i	$0.792811\pi$
$810$	−2.07107	−0.0727699
$811$	8.38478	0.294429	0.147215	−	0.989105i	$-0.452969\pi$
$811$	8.38478	0.294429	0.147215	+	0.989105i	$0.452969\pi$
$812$	8.56854	0.300697
$813$	1.79899	0.0630933
$814$	2.05887	0.0721635
$815$	−14.9706	−0.524396
$816$	−20.4853	−0.717128
$817$	10.4853	0.366834
$818$	5.31371	0.185789
$819$	−0.828427	−0.0289476
$820$	16.1421	0.563708
$821$	39.2548	1.37000	0.685002	−	0.728542i	$-0.259801\pi$
$821$	39.2548	1.37000	0.685002	+	0.728542i	$0.259801\pi$
$822$	3.11270	0.108568
$823$	34.3848	1.19858	0.599289	−	0.800533i	$-0.295450\pi$
$823$	34.3848	1.19858	0.599289	+	0.800533i	$0.295450\pi$
$824$	−27.6152	−0.962022
$825$	0.828427	0.0288421
$826$	−3.51472	−0.122293
$827$	27.8579	0.968713	0.484356	−	0.874871i	$-0.339054\pi$
$827$	27.8579	0.968713	0.484356	+	0.874871i	$0.339054\pi$
$828$	−2.58579	−0.0898623
$829$	7.02944	0.244142	0.122071	−	0.992521i	$-0.461046\pi$
$829$	7.02944	0.244142	0.122071	+	0.992521i	$0.461046\pi$
$830$	−3.65685	−0.126931
$831$	−10.1421	−0.351827
$832$	4.17157	0.144623
$833$	30.4853	1.05625
$834$	−7.31371	−0.253253
$835$	−8.82843	−0.305520
$836$	−3.65685	−0.126475
$837$	−57.9411	−2.00274
$838$	2.14214	0.0739988
$839$	−18.7279	−0.646560	−0.323280	−	0.946303i	$-0.604786\pi$
$839$	−18.7279	−0.646560	−0.323280	+	0.946303i	$0.604786\pi$
$840$	1.85786	0.0641024
$841$	3.00000	0.103448
$842$	−0.426407	−0.0146950
$843$	−25.1716	−0.866955
$844$	35.3137	1.21555
$845$	1.00000	0.0344010
$846$	−0.343146	−0.0117976
$847$	8.82843	0.303348
$848$	−43.4558	−1.49228
$849$	12.3431	0.423616
$850$	−2.00000	−0.0685994
$851$	−12.0000	−0.411355
$852$	20.4264	0.699797
$853$	37.4558	1.28246	0.641232	−	0.767347i	$-0.278424\pi$
$853$	37.4558	1.28246	0.641232	+	0.767347i	$0.278424\pi$
$854$	2.74517	0.0939376
$855$	−3.41421	−0.116764
$856$	10.4437	0.356957
$857$	0.343146	0.0117216	0.00586082	−	0.999983i	$-0.498134\pi$
$857$	0.343146	0.0117216	0.00586082	+	0.999983i	$0.498134\pi$
$858$	−0.343146	−0.0117148
$859$	11.7990	0.402576	0.201288	−	0.979532i	$-0.435487\pi$
$859$	11.7990	0.402576	0.201288	+	0.979532i	$0.435487\pi$
$860$	−5.61522	−0.191478
$861$	10.3431	0.352493
$862$	1.49747	0.0510042
$863$	19.4558	0.662285	0.331142	−	0.943581i	$-0.392566\pi$
$863$	19.4558	0.662285	0.331142	+	0.943581i	$0.392566\pi$
$864$	−24.9706	−0.849516
$865$	11.1716	0.379845
$866$	−1.51472	−0.0514722
$867$	8.92893	0.303242
$868$	15.5147	0.526604
$869$	4.97056	0.168615
$870$	3.31371	0.112345
$871$	2.00000	0.0677674
$872$	3.17157	0.107403
$873$	−3.65685	−0.123766
$874$	−2.00000	−0.0676510
$875$	−0.828427	−0.0280059
$876$	21.9411	0.741322
$877$	2.68629	0.0907096	0.0453548	−	0.998971i	$-0.485558\pi$
$877$	2.68629	0.0907096	0.0453548	+	0.998971i	$0.485558\pi$
$878$	−13.6569	−0.460897
$879$	−3.02944	−0.102180
$880$	1.75736	0.0592406
$881$	52.9706	1.78462	0.892312	−	0.451420i	$-0.149082\pi$
$881$	52.9706	1.78462	0.892312	+	0.451420i	$0.149082\pi$
$882$	2.61522	0.0880592
$883$	−32.2426	−1.08505	−0.542526	−	0.840039i	$-0.682532\pi$
$883$	−32.2426	−1.08505	−0.542526	+	0.840039i	$0.682532\pi$
$884$	−8.82843	−0.296932
$885$	14.4853	0.486917
$886$	−2.72792	−0.0916463
$887$	14.3848	0.482994	0.241497	−	0.970402i	$-0.422362\pi$
$887$	14.3848	0.482994	0.241497	+	0.970402i	$0.422362\pi$
$888$	−19.0294	−0.638586
$889$	7.79899	0.261570
$890$	2.48528	0.0833068
$891$	−2.92893	−0.0981229
$892$	−48.4264	−1.62144
$893$	2.82843	0.0946497
$894$	−0.201010	−0.00672278
$895$	−5.65685	−0.189088
$896$	8.74517	0.292155
$897$	2.00000	0.0667781
$898$	12.0589	0.402410
$899$	57.9411	1.93244
$900$	1.82843	0.0609476
$901$	69.9411	2.33008
$902$	−2.14214	−0.0713253
$903$	−3.59798	−0.119733
$904$	5.02944	0.167277
$905$	0	0
$906$	10.6863	0.355028
$907$	−33.2132	−1.10283	−0.551413	−	0.834232i	$-0.685911\pi$
$907$	−33.2132	−1.10283	−0.551413	+	0.834232i	$0.685911\pi$
$908$	50.5685	1.67818
$909$	−7.65685	−0.253962
$910$	0.343146	0.0113752
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	14.4853	0.479656
$913$	−5.17157	−0.171154
$914$	−7.45584	−0.246617
$915$	−11.3137	−0.374020
$916$	−1.51472	−0.0500477
$917$	−14.0589	−0.464265
$918$	11.3137	0.373408
$919$	−16.4853	−0.543799	−0.271900	−	0.962326i	$-0.587652\pi$
$919$	−16.4853	−0.543799	−0.271900	+	0.962326i	$0.587652\pi$
$920$	2.24264	0.0739377
$921$	27.1127	0.893394
$922$	10.9706	0.361296
$923$	7.89949	0.260015
$924$	1.25483	0.0412810
$925$	8.48528	0.278994
$926$	−6.48528	−0.213120
$927$	−17.4142	−0.571958
$928$	24.9706	0.819699
$929$	11.1716	0.366527	0.183264	−	0.983064i	$-0.441334\pi$
$929$	11.1716	0.366527	0.183264	+	0.983064i	$0.441334\pi$
$930$	6.00000	0.196748
$931$	−21.5563	−0.706481
$932$	−45.0294	−1.47499
$933$	−12.0000	−0.392862
$934$	−4.38478	−0.143474
$935$	−2.82843	−0.0924995
$936$	−1.58579	−0.0518331
$937$	10.9706	0.358393	0.179196	−	0.983813i	$-0.442650\pi$
$937$	10.9706	0.358393	0.179196	+	0.983813i	$0.442650\pi$
$938$	0.686292	0.0224082
$939$	1.17157	0.0382328
$940$	−1.51472	−0.0494047
$941$	−54.7696	−1.78544	−0.892718	−	0.450615i	$-0.851205\pi$
$941$	−54.7696	−1.78544	−0.892718	+	0.450615i	$0.851205\pi$
$942$	10.5442	0.343547
$943$	12.4853	0.406577
$944$	30.7279	1.00011
$945$	4.68629	0.152445
$946$	0.745166	0.0242274
$947$	−45.1127	−1.46597	−0.732983	−	0.680247i	$-0.761872\pi$
$947$	−45.1127	−1.46597	−0.732983	+	0.680247i	$0.761872\pi$
$948$	−21.9411	−0.712615
$949$	8.48528	0.275444
$950$	1.41421	0.0458831
$951$	36.9706	1.19885
$952$	−6.34315	−0.205583
$953$	−55.2548	−1.78988	−0.894940	−	0.446187i	$-0.852782\pi$
$953$	−55.2548	−1.78988	−0.894940	+	0.446187i	$0.852782\pi$
$954$	6.00000	0.194257
$955$	13.6569	0.441925
$956$	1.07107	0.0346408
$957$	4.68629	0.151486
$958$	−2.18377	−0.0705543
$959$	−4.40202	−0.142149
$960$	−5.89949	−0.190405
$961$	73.9117	2.38425
$962$	−3.51472	−0.113319
$963$	6.58579	0.212224
$964$	−4.54416	−0.146357
$965$	15.6569	0.504012
$966$	0.686292	0.0220811
$967$	−19.9411	−0.641263	−0.320632	−	0.947204i	$-0.603895\pi$
$967$	−19.9411	−0.641263	−0.320632	+	0.947204i	$0.603895\pi$
$968$	16.8995	0.543170
$969$	−23.3137	−0.748944
$970$	1.51472	0.0486347
$971$	−12.2843	−0.394221	−0.197111	−	0.980381i	$-0.563156\pi$
$971$	−12.2843	−0.394221	−0.197111	+	0.980381i	$0.563156\pi$
$972$	−18.1005	−0.580574
$973$	10.3431	0.331586
$974$	9.51472	0.304871
$975$	−1.41421	−0.0452911
$976$	−24.0000	−0.768221
$977$	−56.4853	−1.80712	−0.903562	−	0.428457i	$-0.859057\pi$
$977$	−56.4853	−1.80712	−0.903562	+	0.428457i	$0.859057\pi$
$978$	−8.76955	−0.280419
$979$	3.51472	0.112331
$980$	11.5442	0.368765
$981$	2.00000	0.0638551
$982$	4.48528	0.143131
$983$	−34.9706	−1.11539	−0.557694	−	0.830047i	$-0.688314\pi$
$983$	−34.9706	−1.11539	−0.557694	+	0.830047i	$0.688314\pi$
$984$	19.7990	0.631169
$985$	−22.9706	−0.731903
$986$	−11.3137	−0.360302
$987$	−0.970563	−0.0308934
$988$	6.24264	0.198605
$989$	−4.34315	−0.138104
$990$	−0.242641	−0.00771163
$991$	15.0294	0.477426	0.238713	−	0.971090i	$-0.423275\pi$
$991$	15.0294	0.477426	0.238713	+	0.971090i	$0.423275\pi$
$992$	45.2132	1.43552
$993$	31.1716	0.989200
$994$	2.71068	0.0859775
$995$	4.00000	0.126809
$996$	22.8284	0.723346
$997$	23.1716	0.733851	0.366926	−	0.930250i	$-0.380410\pi$
$997$	23.1716	0.733851	0.366926	+	0.930250i	$0.380410\pi$
$998$	4.32590	0.136934
$999$	−48.0000	−1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	65.2.a.b.1.2	✓	2
3.2	odd	2		585.2.a.m.1.1		2
4.3	odd	2		1040.2.a.j.1.1		2
5.2	odd	4		325.2.b.f.274.3		4
5.3	odd	4		325.2.b.f.274.2		4
5.4	even	2		325.2.a.i.1.1		2
7.6	odd	2		3185.2.a.j.1.2		2
8.3	odd	2		4160.2.a.z.1.2		2
8.5	even	2		4160.2.a.bf.1.1		2
11.10	odd	2		7865.2.a.j.1.1		2
12.11	even	2		9360.2.a.cd.1.2		2
13.2	odd	12		845.2.m.f.316.3		8
13.3	even	3		845.2.e.h.191.1		4
13.4	even	6		845.2.e.c.146.2		4
13.5	odd	4		845.2.c.b.506.2		4
13.6	odd	12		845.2.m.f.361.2		8
13.7	odd	12		845.2.m.f.361.3		8
13.8	odd	4		845.2.c.b.506.3		4
13.9	even	3		845.2.e.h.146.1		4
13.10	even	6		845.2.e.c.191.2		4
13.11	odd	12		845.2.m.f.316.2		8
13.12	even	2		845.2.a.g.1.1		2
15.2	even	4		2925.2.c.r.2224.2		4
15.8	even	4		2925.2.c.r.2224.3		4
15.14	odd	2		2925.2.a.u.1.2		2
20.19	odd	2		5200.2.a.bu.1.2		2
39.38	odd	2		7605.2.a.x.1.2		2
65.64	even	2		4225.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.b.1.2	✓	2	1.1	even	1	trivial
325.2.a.i.1.1		2	5.4	even	2
325.2.b.f.274.2		4	5.3	odd	4
325.2.b.f.274.3		4	5.2	odd	4
585.2.a.m.1.1		2	3.2	odd	2
845.2.a.g.1.1		2	13.12	even	2
845.2.c.b.506.2		4	13.5	odd	4
845.2.c.b.506.3		4	13.8	odd	4
845.2.e.c.146.2		4	13.4	even	6
845.2.e.c.191.2		4	13.10	even	6
845.2.e.h.146.1		4	13.9	even	3
845.2.e.h.191.1		4	13.3	even	3
845.2.m.f.316.2		8	13.11	odd	12
845.2.m.f.316.3		8	13.2	odd	12
845.2.m.f.361.2		8	13.6	odd	12
845.2.m.f.361.3		8	13.7	odd	12
1040.2.a.j.1.1		2	4.3	odd	2
2925.2.a.u.1.2		2	15.14	odd	2
2925.2.c.r.2224.2		4	15.2	even	4
2925.2.c.r.2224.3		4	15.8	even	4
3185.2.a.j.1.2		2	7.6	odd	2
4160.2.a.z.1.2		2	8.3	odd	2
4160.2.a.bf.1.1		2	8.5	even	2
4225.2.a.r.1.2		2	65.64	even	2
5200.2.a.bu.1.2		2	20.19	odd	2
7605.2.a.x.1.2		2	39.38	odd	2
7865.2.a.j.1.1		2	11.10	odd	2
9360.2.a.cd.1.2		2	12.11	even	2

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:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	65.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} +1.41421 q^{3} -1.82843 q^{4} +1.00000 q^{5} +0.585786 q^{6} -0.828427 q^{7} -1.58579 q^{8} -1.00000 q^{9} +0.414214 q^{10} +0.585786 q^{11} -2.58579 q^{12} -1.00000 q^{13} -0.343146 q^{14} +1.41421 q^{15} +3.00000 q^{16} -4.82843 q^{17} -0.414214 q^{18} +3.41421 q^{19} -1.82843 q^{20} -1.17157 q^{21} +0.242641 q^{22} -1.41421 q^{23} -2.24264 q^{24} +1.00000 q^{25} -0.414214 q^{26} -5.65685 q^{27} +1.51472 q^{28} +5.65685 q^{29} +0.585786 q^{30} +10.2426 q^{31} +4.41421 q^{32} +0.828427 q^{33} -2.00000 q^{34} -0.828427 q^{35} +1.82843 q^{36} +8.48528 q^{37} +1.41421 q^{38} -1.41421 q^{39} -1.58579 q^{40} -8.82843 q^{41} -0.485281 q^{42} +3.07107 q^{43} -1.07107 q^{44} -1.00000 q^{45} -0.585786 q^{46} +0.828427 q^{47} +4.24264 q^{48} -6.31371 q^{49} +0.414214 q^{50} -6.82843 q^{51} +1.82843 q^{52} -14.4853 q^{53} -2.34315 q^{54} +0.585786 q^{55} +1.31371 q^{56} +4.82843 q^{57} +2.34315 q^{58} +10.2426 q^{59} -2.58579 q^{60} -8.00000 q^{61} +4.24264 q^{62} +0.828427 q^{63} -4.17157 q^{64} -1.00000 q^{65} +0.343146 q^{66} -2.00000 q^{67} +8.82843 q^{68} -2.00000 q^{69} -0.343146 q^{70} -7.89949 q^{71} +1.58579 q^{72} -8.48528 q^{73} +3.51472 q^{74} +1.41421 q^{75} -6.24264 q^{76} -0.485281 q^{77} -0.585786 q^{78} +8.48528 q^{79} +3.00000 q^{80} -5.00000 q^{81} -3.65685 q^{82} -8.82843 q^{83} +2.14214 q^{84} -4.82843 q^{85} +1.27208 q^{86} +8.00000 q^{87} -0.928932 q^{88} +6.00000 q^{89} -0.414214 q^{90} +0.828427 q^{91} +2.58579 q^{92} +14.4853 q^{93} +0.343146 q^{94} +3.41421 q^{95} +6.24264 q^{96} +3.65685 q^{97} -2.61522 q^{98} -0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{7} - 6 q^{8} - 2 q^{9} - 2 q^{10} + 4 q^{11} - 8 q^{12} - 2 q^{13} - 12 q^{14} + 6 q^{16} - 4 q^{17} + 2 q^{18} + 4 q^{19} + 2 q^{20} - 8 q^{21} - 8 q^{22}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^6 + 4 * q^7 - 6 * q^8 - 2 * q^9 - 2 * q^10 + 4 * q^11 - 8 * q^12 - 2 * q^13 - 12 * q^14 + 6 * q^16 - 4 * q^17 + 2 * q^18 + 4 * q^19 + 2 * q^20 - 8 * q^21 - 8 * q^22 + 4 * q^24 + 2 * q^25 + 2 * q^26 + 20 * q^28 + 4 * q^30 + 12 * q^31 + 6 * q^32 - 4 * q^33 - 4 * q^34 + 4 * q^35 - 2 * q^36 - 6 * q^40 - 12 * q^41 + 16 * q^42 - 8 * q^43 + 12 * q^44 - 2 * q^45 - 4 * q^46 - 4 * q^47 + 10 * q^49 - 2 * q^50 - 8 * q^51 - 2 * q^52 - 12 * q^53 - 16 * q^54 + 4 * q^55 - 20 * q^56 + 4 * q^57 + 16 * q^58 + 12 * q^59 - 8 * q^60 - 16 * q^61 - 4 * q^63 - 14 * q^64 - 2 * q^65 + 12 * q^66 - 4 * q^67 + 12 * q^68 - 4 * q^69 - 12 * q^70 + 4 * q^71 + 6 * q^72 + 24 * q^74 - 4 * q^76 + 16 * q^77 - 4 * q^78 + 6 * q^80 - 10 * q^81 + 4 * q^82 - 12 * q^83 - 24 * q^84 - 4 * q^85 + 28 * q^86 + 16 * q^87 - 16 * q^88 + 12 * q^89 + 2 * q^90 - 4 * q^91 + 8 * q^92 + 12 * q^93 + 12 * q^94 + 4 * q^95 + 4 * q^96 - 4 * q^97 - 42 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more