LMFDB - Embedded newform 3381.2.a.bj.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, 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("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 13x^{8} + 41x^{7} + 47x^{6} - 165x^{5} - 45x^{4} + 207x^{3} + 12x^{2} - 76x - 2 $ x^10 - 3x^9 - 13x^8 + 41x^7 + 47x^6 - 165x^5 - 45x^4 + 207x^3 + 12x^2 - 76*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.864859$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.864859 q^{2} +1.00000 q^{3} -1.25202 q^{4} +2.49557 q^{5} +0.864859 q^{6} -2.81254 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.864859 q^{2} +1.00000 q^{3} -1.25202 q^{4} +2.49557 q^{5} +0.864859 q^{6} -2.81254 q^{8} +1.00000 q^{9} +2.15832 q^{10} -3.98310 q^{11} -1.25202 q^{12} -2.80699 q^{13} +2.49557 q^{15} +0.0715855 q^{16} +6.98770 q^{17} +0.864859 q^{18} +5.98058 q^{19} -3.12450 q^{20} -3.44482 q^{22} +1.00000 q^{23} -2.81254 q^{24} +1.22789 q^{25} -2.42765 q^{26} +1.00000 q^{27} -4.06198 q^{29} +2.15832 q^{30} +3.44346 q^{31} +5.68699 q^{32} -3.98310 q^{33} +6.04338 q^{34} -1.25202 q^{36} -1.47693 q^{37} +5.17236 q^{38} -2.80699 q^{39} -7.01890 q^{40} +7.65339 q^{41} +7.07993 q^{43} +4.98691 q^{44} +2.49557 q^{45} +0.864859 q^{46} +3.04531 q^{47} +0.0715855 q^{48} +1.06195 q^{50} +6.98770 q^{51} +3.51440 q^{52} +9.57104 q^{53} +0.864859 q^{54} -9.94011 q^{55} +5.98058 q^{57} -3.51304 q^{58} +8.30618 q^{59} -3.12450 q^{60} +3.71282 q^{61} +2.97811 q^{62} +4.77528 q^{64} -7.00504 q^{65} -3.44482 q^{66} +4.46957 q^{67} -8.74872 q^{68} +1.00000 q^{69} -9.49028 q^{71} -2.81254 q^{72} -12.9021 q^{73} -1.27733 q^{74} +1.22789 q^{75} -7.48779 q^{76} -2.42765 q^{78} +7.12304 q^{79} +0.178647 q^{80} +1.00000 q^{81} +6.61911 q^{82} -1.05716 q^{83} +17.4383 q^{85} +6.12315 q^{86} -4.06198 q^{87} +11.2026 q^{88} +3.30053 q^{89} +2.15832 q^{90} -1.25202 q^{92} +3.44346 q^{93} +2.63377 q^{94} +14.9250 q^{95} +5.68699 q^{96} +15.9312 q^{97} -3.98310 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{2} + 10 q^{3} + 15 q^{4} + 5 q^{5} + 3 q^{6} + 9 q^{8} + 10 q^{9}+O(q^{10}) $ 10 * q + 3 * q^2 + 10 * q^3 + 15 * q^4 + 5 * q^5 + 3 * q^6 + 9 * q^8 + 10 * q^9	$ 10 q + 3 q^{2} + 10 q^{3} + 15 q^{4} + 5 q^{5} + 3 q^{6} + 9 q^{8} + 10 q^{9} - 11 q^{10} + 8 q^{11} + 15 q^{12} + 5 q^{15} + 37 q^{16} + 11 q^{17} + 3 q^{18} - q^{19} + 15 q^{20} + 6 q^{22} + 10 q^{23} + 9 q^{24} + 21 q^{25} - q^{26} + 10 q^{27} + 22 q^{29} - 11 q^{30} + 3 q^{31} + 11 q^{32} + 8 q^{33} + 3 q^{34} + 15 q^{36} - 3 q^{37} - 16 q^{38} - 39 q^{40} + 26 q^{41} + 27 q^{43} + 16 q^{44} + 5 q^{45} + 3 q^{46} - 11 q^{47} + 37 q^{48} + 2 q^{50} + 11 q^{51} - 29 q^{52} + 5 q^{53} + 3 q^{54} + 18 q^{55} - q^{57} + 16 q^{58} + 10 q^{59} + 15 q^{60} - 22 q^{61} + 32 q^{62} + 69 q^{64} - 11 q^{65} + 6 q^{66} - 2 q^{67} + 21 q^{68} + 10 q^{69} + 27 q^{71} + 9 q^{72} + 8 q^{73} + 14 q^{74} + 21 q^{75} + 22 q^{76} - q^{78} + 21 q^{79} + 53 q^{80} + 10 q^{81} - 36 q^{82} + 12 q^{83} + 23 q^{85} + 18 q^{86} + 22 q^{87} - 10 q^{88} - 6 q^{89} - 11 q^{90} + 15 q^{92} + 3 q^{93} - 35 q^{94} + 44 q^{95} + 11 q^{96} - 6 q^{97} + 8 q^{99}+O(q^{100}) $ 10 * q + 3 * q^2 + 10 * q^3 + 15 * q^4 + 5 * q^5 + 3 * q^6 + 9 * q^8 + 10 * q^9 - 11 * q^10 + 8 * q^11 + 15 * q^12 + 5 * q^15 + 37 * q^16 + 11 * q^17 + 3 * q^18 - q^19 + 15 * q^20 + 6 * q^22 + 10 * q^23 + 9 * q^24 + 21 * q^25 - q^26 + 10 * q^27 + 22 * q^29 - 11 * q^30 + 3 * q^31 + 11 * q^32 + 8 * q^33 + 3 * q^34 + 15 * q^36 - 3 * q^37 - 16 * q^38 - 39 * q^40 + 26 * q^41 + 27 * q^43 + 16 * q^44 + 5 * q^45 + 3 * q^46 - 11 * q^47 + 37 * q^48 + 2 * q^50 + 11 * q^51 - 29 * q^52 + 5 * q^53 + 3 * q^54 + 18 * q^55 - q^57 + 16 * q^58 + 10 * q^59 + 15 * q^60 - 22 * q^61 + 32 * q^62 + 69 * q^64 - 11 * q^65 + 6 * q^66 - 2 * q^67 + 21 * q^68 + 10 * q^69 + 27 * q^71 + 9 * q^72 + 8 * q^73 + 14 * q^74 + 21 * q^75 + 22 * q^76 - q^78 + 21 * q^79 + 53 * q^80 + 10 * q^81 - 36 * q^82 + 12 * q^83 + 23 * q^85 + 18 * q^86 + 22 * q^87 - 10 * q^88 - 6 * q^89 - 11 * q^90 + 15 * q^92 + 3 * q^93 - 35 * q^94 + 44 * q^95 + 11 * q^96 - 6 * q^97 + 8 * q^99

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 $

$2$	0.864859	0.611548	0.305774	−	0.952104i	$-0.401085\pi$
$2$	0.864859	0.611548	0.305774	+	0.952104i	$0.401085\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.25202	−0.626009
$5$	2.49557	1.11605	0.558027	−	0.829823i	$-0.311559\pi$
$5$	2.49557	1.11605	0.558027	+	0.829823i	$0.311559\pi$
$6$	0.864859	0.353077
$7$	0	0
$7$	0	0
$8$	−2.81254	−0.994383
$9$	1.00000	0.333333
$10$	2.15832	0.682521
$11$	−3.98310	−1.20095	−0.600475	−	0.799644i	$-0.705022\pi$
$11$	−3.98310	−1.20095	−0.600475	+	0.799644i	$0.705022\pi$
$12$	−1.25202	−0.361426
$13$	−2.80699	−0.778518	−0.389259	−	0.921128i	$-0.627269\pi$
$13$	−2.80699	−0.778518	−0.389259	+	0.921128i	$0.627269\pi$
$14$	0	0
$15$	2.49557	0.644354
$16$	0.0715855	0.0178964
$17$	6.98770	1.69477	0.847383	−	0.530983i	$-0.178177\pi$
$17$	6.98770	1.69477	0.847383	+	0.530983i	$0.178177\pi$
$18$	0.864859	0.203849
$19$	5.98058	1.37204	0.686019	−	0.727583i	$-0.259357\pi$
$19$	5.98058	1.37204	0.686019	+	0.727583i	$0.259357\pi$
$20$	−3.12450	−0.698660
$21$	0	0
$22$	−3.44482	−0.734438
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−2.81254	−0.574107
$25$	1.22789	0.245577
$26$	−2.42765	−0.476101
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.06198	−0.754291	−0.377145	−	0.926154i	$-0.623094\pi$
$29$	−4.06198	−0.754291	−0.377145	+	0.926154i	$0.623094\pi$
$30$	2.15832	0.394054
$31$	3.44346	0.618464	0.309232	−	0.950987i	$-0.399928\pi$
$31$	3.44346	0.618464	0.309232	+	0.950987i	$0.399928\pi$
$32$	5.68699	1.00533
$33$	−3.98310	−0.693368
$34$	6.04338	1.03643
$35$	0	0
$36$	−1.25202	−0.208670
$37$	−1.47693	−0.242805	−0.121402	−	0.992603i	$-0.538739\pi$
$37$	−1.47693	−0.242805	−0.121402	+	0.992603i	$0.538739\pi$
$38$	5.17236	0.839067
$39$	−2.80699	−0.449477
$40$	−7.01890	−1.10978
$41$	7.65339	1.19526	0.597630	−	0.801772i	$-0.296109\pi$
$41$	7.65339	1.19526	0.597630	+	0.801772i	$0.296109\pi$
$42$	0	0
$43$	7.07993	1.07968	0.539840	−	0.841768i	$-0.318485\pi$
$43$	7.07993	1.07968	0.539840	+	0.841768i	$0.318485\pi$
$44$	4.98691	0.751805
$45$	2.49557	0.372018
$46$	0.864859	0.127517
$47$	3.04531	0.444204	0.222102	−	0.975023i	$-0.428708\pi$
$47$	3.04531	0.444204	0.222102	+	0.975023i	$0.428708\pi$
$48$	0.0715855	0.0103325
$49$	0	0
$50$	1.06195	0.150182
$51$	6.98770	0.978473
$52$	3.51440	0.487359
$53$	9.57104	1.31468	0.657342	−	0.753593i	$-0.271681\pi$
$53$	9.57104	1.31468	0.657342	+	0.753593i	$0.271681\pi$
$54$	0.864859	0.117692
$55$	−9.94011	−1.34032
$56$	0	0
$57$	5.98058	0.792147
$58$	−3.51304	−0.461285
$59$	8.30618	1.08137	0.540686	−	0.841224i	$-0.318165\pi$
$59$	8.30618	1.08137	0.540686	+	0.841224i	$0.318165\pi$
$60$	−3.12450	−0.403372
$61$	3.71282	0.475377	0.237689	−	0.971341i	$-0.423610\pi$
$61$	3.71282	0.475377	0.237689	+	0.971341i	$0.423610\pi$
$62$	2.97811	0.378220
$63$	0	0
$64$	4.77528	0.596909
$65$	−7.00504	−0.868868
$66$	−3.44482	−0.424028
$67$	4.46957	0.546045	0.273023	−	0.962008i	$-0.411977\pi$
$67$	4.46957	0.546045	0.273023	+	0.962008i	$0.411977\pi$
$68$	−8.74872	−1.06094
$69$	1.00000	0.120386
$70$	0	0
$71$	−9.49028	−1.12629	−0.563144	−	0.826359i	$-0.690408\pi$
$71$	−9.49028	−1.12629	−0.563144	+	0.826359i	$0.690408\pi$
$72$	−2.81254	−0.331461
$73$	−12.9021	−1.51008	−0.755039	−	0.655680i	$-0.772382\pi$
$73$	−12.9021	−1.51008	−0.755039	+	0.655680i	$0.772382\pi$
$74$	−1.27733	−0.148487
$75$	1.22789	0.141784
$76$	−7.48779	−0.858908
$77$	0	0
$78$	−2.42765	−0.274877
$79$	7.12304	0.801405	0.400702	−	0.916208i	$-0.368766\pi$
$79$	7.12304	0.801405	0.400702	+	0.916208i	$0.368766\pi$
$80$	0.178647	0.0199733
$81$	1.00000	0.111111
$82$	6.61911	0.730958
$83$	−1.05716	−0.116038	−0.0580192	−	0.998315i	$-0.518478\pi$
$83$	−1.05716	−0.116038	−0.0580192	+	0.998315i	$0.518478\pi$
$84$	0	0
$85$	17.4383	1.89145
$86$	6.12315	0.660276
$87$	−4.06198	−0.435490
$88$	11.2026	1.19420
$89$	3.30053	0.349856	0.174928	−	0.984581i	$-0.444031\pi$
$89$	3.30053	0.349856	0.174928	+	0.984581i	$0.444031\pi$
$90$	2.15832	0.227507
$91$	0	0
$92$	−1.25202	−0.130532
$93$	3.44346	0.357070
$94$	2.63377	0.271652
$95$	14.9250	1.53127
$96$	5.68699	0.580426
$97$	15.9312	1.61757	0.808786	−	0.588103i	$-0.200125\pi$
$97$	15.9312	1.61757	0.808786	+	0.588103i	$0.200125\pi$
$98$	0	0
$99$	−3.98310	−0.400316
$100$	−1.53733	−0.153733
$101$	−6.96429	−0.692972	−0.346486	−	0.938055i	$-0.612625\pi$
$101$	−6.96429	−0.692972	−0.346486	+	0.938055i	$0.612625\pi$
$102$	6.04338	0.598383
$103$	−6.00488	−0.591678	−0.295839	−	0.955238i	$-0.595599\pi$
$103$	−6.00488	−0.591678	−0.295839	+	0.955238i	$0.595599\pi$
$104$	7.89475	0.774144
$105$	0	0
$106$	8.27760	0.803992
$107$	5.71797	0.552777	0.276388	−	0.961046i	$-0.410862\pi$
$107$	5.71797	0.552777	0.276388	+	0.961046i	$0.410862\pi$
$108$	−1.25202	−0.120475
$109$	6.25586	0.599202	0.299601	−	0.954065i	$-0.403146\pi$
$109$	6.25586	0.599202	0.299601	+	0.954065i	$0.403146\pi$
$110$	−8.59680	−0.819673
$111$	−1.47693	−0.140184
$112$	0	0
$113$	−3.79544	−0.357045	−0.178523	−	0.983936i	$-0.557132\pi$
$113$	−3.79544	−0.357045	−0.178523	+	0.983936i	$0.557132\pi$
$114$	5.17236	0.484436
$115$	2.49557	0.232713
$116$	5.08567	0.472193
$117$	−2.80699	−0.259506
$118$	7.18368	0.661311
$119$	0	0
$120$	−7.01890	−0.640735
$121$	4.86506	0.442279
$122$	3.21106	0.290716
$123$	7.65339	0.690083
$124$	−4.31127	−0.387164
$125$	−9.41359	−0.841977
$126$	0	0
$127$	20.8819	1.85297	0.926484	−	0.376333i	$-0.122815\pi$
$127$	20.8819	1.85297	0.926484	+	0.376333i	$0.122815\pi$
$128$	−7.24404	−0.640288
$129$	7.07993	0.623353
$130$	−6.05837	−0.531354
$131$	−12.1488	−1.06144	−0.530722	−	0.847546i	$-0.678079\pi$
$131$	−12.1488	−1.06144	−0.530722	+	0.847546i	$0.678079\pi$
$132$	4.98691	0.434055
$133$	0	0
$134$	3.86555	0.333933
$135$	2.49557	0.214785
$136$	−19.6532	−1.68525
$137$	−0.828633	−0.0707949	−0.0353974	−	0.999373i	$-0.511270\pi$
$137$	−0.828633	−0.0707949	−0.0353974	+	0.999373i	$0.511270\pi$
$138$	0.864859	0.0736217
$139$	−0.546099	−0.0463195	−0.0231598	−	0.999732i	$-0.507373\pi$
$139$	−0.546099	−0.0463195	−0.0231598	+	0.999732i	$0.507373\pi$
$140$	0	0
$141$	3.04531	0.256461
$142$	−8.20775	−0.688779
$143$	11.1805	0.934960
$144$	0.0715855	0.00596546
$145$	−10.1370	−0.841830
$146$	−11.1585	−0.923485
$147$	0	0
$148$	1.84914	0.151998
$149$	−18.5487	−1.51957	−0.759783	−	0.650176i	$-0.774695\pi$
$149$	−18.5487	−1.51957	−0.759783	+	0.650176i	$0.774695\pi$
$150$	1.06195	0.0867077
$151$	−5.77251	−0.469760	−0.234880	−	0.972024i	$-0.575470\pi$
$151$	−5.77251	−0.469760	−0.234880	+	0.972024i	$0.575470\pi$
$152$	−16.8206	−1.36433
$153$	6.98770	0.564922
$154$	0	0
$155$	8.59340	0.690239
$156$	3.51440	0.281377
$157$	−14.1290	−1.12761	−0.563807	−	0.825906i	$-0.690664\pi$
$157$	−14.1290	−1.12761	−0.563807	+	0.825906i	$0.690664\pi$
$158$	6.16043	0.490097
$159$	9.57104	0.759033
$160$	14.1923	1.12200
$161$	0	0
$162$	0.864859	0.0679498
$163$	−16.7496	−1.31193	−0.655966	−	0.754790i	$-0.727739\pi$
$163$	−16.7496	−1.31193	−0.655966	+	0.754790i	$0.727739\pi$
$164$	−9.58219	−0.748243
$165$	−9.94011	−0.773837
$166$	−0.914295	−0.0709631
$167$	−17.8518	−1.38141	−0.690707	−	0.723135i	$-0.742701\pi$
$167$	−17.8518	−1.38141	−0.690707	+	0.723135i	$0.742701\pi$
$168$	0	0
$169$	−5.12084	−0.393910
$170$	15.0817	1.15671
$171$	5.98058	0.457346
$172$	−8.86420	−0.675889
$173$	−2.65359	−0.201749	−0.100874	−	0.994899i	$-0.532164\pi$
$173$	−2.65359	−0.201749	−0.100874	+	0.994899i	$0.532164\pi$
$174$	−3.51304	−0.266323
$175$	0	0
$176$	−0.285132	−0.0214926
$177$	8.30618	0.624331
$178$	2.85450	0.213954
$179$	11.4696	0.857281	0.428640	−	0.903475i	$-0.358993\pi$
$179$	11.4696	0.857281	0.428640	+	0.903475i	$0.358993\pi$
$180$	−3.12450	−0.232887
$181$	15.8393	1.17732	0.588662	−	0.808380i	$-0.299655\pi$
$181$	15.8393	1.17732	0.588662	+	0.808380i	$0.299655\pi$
$182$	0	0
$183$	3.71282	0.274459
$184$	−2.81254	−0.207343
$185$	−3.68577	−0.270984
$186$	2.97811	0.218365
$187$	−27.8327	−2.03533
$188$	−3.81278	−0.278076
$189$	0	0
$190$	12.9080	0.936445
$191$	21.7951	1.57704	0.788518	−	0.615012i	$-0.210849\pi$
$191$	21.7951	1.57704	0.788518	+	0.615012i	$0.210849\pi$
$192$	4.77528	0.344626
$193$	16.0333	1.15411	0.577053	−	0.816707i	$-0.304203\pi$
$193$	16.0333	1.15411	0.577053	+	0.816707i	$0.304203\pi$
$194$	13.7783	0.989223
$195$	−7.00504	−0.501641
$196$	0	0
$197$	10.6653	0.759870	0.379935	−	0.925013i	$-0.375946\pi$
$197$	10.6653	0.759870	0.379935	+	0.925013i	$0.375946\pi$
$198$	−3.44482	−0.244813
$199$	3.65905	0.259383	0.129691	−	0.991554i	$-0.458601\pi$
$199$	3.65905	0.259383	0.129691	+	0.991554i	$0.458601\pi$
$200$	−3.45347	−0.244198
$201$	4.46957	0.315259
$202$	−6.02313	−0.423786
$203$	0	0
$204$	−8.74872	−0.612533
$205$	19.0996	1.33397
$206$	−5.19338	−0.361840
$207$	1.00000	0.0695048
$208$	−0.200939	−0.0139326
$209$	−23.8212	−1.64775
$210$	0	0
$211$	−28.1857	−1.94039	−0.970193	−	0.242332i	$-0.922088\pi$
$211$	−28.1857	−1.94039	−0.970193	+	0.242332i	$0.922088\pi$
$212$	−11.9831	−0.823004
$213$	−9.49028	−0.650263
$214$	4.94524	0.338050
$215$	17.6685	1.20498
$216$	−2.81254	−0.191369
$217$	0	0
$218$	5.41044	0.366441
$219$	−12.9021	−0.871844
$220$	12.4452	0.839055
$221$	−19.6144	−1.31940
$222$	−1.27733	−0.0857290
$223$	−24.2280	−1.62243	−0.811215	−	0.584748i	$-0.801193\pi$
$223$	−24.2280	−1.62243	−0.811215	+	0.584748i	$0.801193\pi$
$224$	0	0
$225$	1.22789	0.0818590
$226$	−3.28252	−0.218350
$227$	8.03654	0.533404	0.266702	−	0.963779i	$-0.414066\pi$
$227$	8.03654	0.533404	0.266702	+	0.963779i	$0.414066\pi$
$228$	−7.48779	−0.495891
$229$	−13.7790	−0.910542	−0.455271	−	0.890353i	$-0.650458\pi$
$229$	−13.7790	−0.910542	−0.455271	+	0.890353i	$0.650458\pi$
$230$	2.15832	0.142315
$231$	0	0
$232$	11.4245	0.750054
$233$	−1.83601	−0.120281	−0.0601405	−	0.998190i	$-0.519155\pi$
$233$	−1.83601	−0.120281	−0.0601405	+	0.998190i	$0.519155\pi$
$234$	−2.42765	−0.158700
$235$	7.59979	0.495756
$236$	−10.3995	−0.676949
$237$	7.12304	0.462691
$238$	0	0
$239$	−4.42206	−0.286039	−0.143020	−	0.989720i	$-0.545681\pi$
$239$	−4.42206	−0.286039	−0.143020	+	0.989720i	$0.545681\pi$
$240$	0.178647	0.0115316
$241$	28.4589	1.83320	0.916598	−	0.399810i	$-0.130924\pi$
$241$	28.4589	1.83320	0.916598	+	0.399810i	$0.130924\pi$
$242$	4.20760	0.270475
$243$	1.00000	0.0641500
$244$	−4.64851	−0.297591
$245$	0	0
$246$	6.61911	0.422019
$247$	−16.7874	−1.06816
$248$	−9.68486	−0.614989
$249$	−1.05716	−0.0669948
$250$	−8.14143	−0.514909
$251$	26.7064	1.68569	0.842846	−	0.538156i	$-0.180879\pi$
$251$	26.7064	1.68569	0.842846	+	0.538156i	$0.180879\pi$
$252$	0	0
$253$	−3.98310	−0.250415
$254$	18.0599	1.13318
$255$	17.4383	1.09203
$256$	−15.8156	−0.988476
$257$	−4.36432	−0.272239	−0.136119	−	0.990692i	$-0.543463\pi$
$257$	−4.36432	−0.272239	−0.136119	+	0.990692i	$0.543463\pi$
$258$	6.12315	0.381210
$259$	0	0
$260$	8.77043	0.543919
$261$	−4.06198	−0.251430
$262$	−10.5070	−0.649124
$263$	1.54914	0.0955242	0.0477621	−	0.998859i	$-0.484791\pi$
$263$	1.54914	0.0955242	0.0477621	+	0.998859i	$0.484791\pi$
$264$	11.2026	0.689473
$265$	23.8852	1.46726
$266$	0	0
$267$	3.30053	0.201989
$268$	−5.59599	−0.341829
$269$	3.71532	0.226527	0.113264	−	0.993565i	$-0.463870\pi$
$269$	3.71532	0.226527	0.113264	+	0.993565i	$0.463870\pi$
$270$	2.15832	0.131351
$271$	−4.02009	−0.244203	−0.122101	−	0.992518i	$-0.538963\pi$
$271$	−4.02009	−0.244203	−0.122101	+	0.992518i	$0.538963\pi$
$272$	0.500218	0.0303302
$273$	0	0
$274$	−0.716651	−0.0432945
$275$	−4.89079	−0.294925
$276$	−1.25202	−0.0753626
$277$	−11.3927	−0.684523	−0.342261	−	0.939605i	$-0.611193\pi$
$277$	−11.3927	−0.684523	−0.342261	+	0.939605i	$0.611193\pi$
$278$	−0.472299	−0.0283266
$279$	3.44346	0.206155
$280$	0	0
$281$	−27.9503	−1.66738	−0.833689	−	0.552235i	$-0.813775\pi$
$281$	−27.9503	−1.66738	−0.833689	+	0.552235i	$0.813775\pi$
$282$	2.63377	0.156838
$283$	29.9754	1.78185	0.890925	−	0.454150i	$-0.150057\pi$
$283$	29.9754	1.78185	0.890925	+	0.454150i	$0.150057\pi$
$284$	11.8820	0.705067
$285$	14.9250	0.884079
$286$	9.66956	0.571773
$287$	0	0
$288$	5.68699	0.335109
$289$	31.8279	1.87223
$290$	−8.76706	−0.514819
$291$	15.9312	0.933906
$292$	16.1537	0.945322
$293$	4.87346	0.284711	0.142355	−	0.989816i	$-0.454532\pi$
$293$	4.87346	0.284711	0.142355	+	0.989816i	$0.454532\pi$
$294$	0	0
$295$	20.7287	1.20687
$296$	4.15391	0.241441
$297$	−3.98310	−0.231123
$298$	−16.0420	−0.929288
$299$	−2.80699	−0.162332
$300$	−1.53733	−0.0887580
$301$	0	0
$302$	−4.99241	−0.287281
$303$	−6.96429	−0.400088
$304$	0.428122	0.0245545
$305$	9.26560	0.530547
$306$	6.04338	0.345477
$307$	27.9064	1.59270	0.796351	−	0.604835i	$-0.206761\pi$
$307$	27.9064	1.59270	0.796351	+	0.604835i	$0.206761\pi$
$308$	0	0
$309$	−6.00488	−0.341606
$310$	7.43209	0.422114
$311$	−27.0086	−1.53152	−0.765758	−	0.643129i	$-0.777636\pi$
$311$	−27.0086	−1.53152	−0.765758	+	0.643129i	$0.777636\pi$
$312$	7.89475	0.446952
$313$	−4.21402	−0.238190	−0.119095	−	0.992883i	$-0.537999\pi$
$313$	−4.21402	−0.238190	−0.119095	+	0.992883i	$0.537999\pi$
$314$	−12.2196	−0.689590
$315$	0	0
$316$	−8.91818	−0.501687
$317$	−7.99795	−0.449210	−0.224605	−	0.974450i	$-0.572109\pi$
$317$	−7.99795	−0.449210	−0.224605	+	0.974450i	$0.572109\pi$
$318$	8.27760	0.464185
$319$	16.1793	0.905865
$320$	11.9170	0.666183
$321$	5.71797	0.319146
$322$	0	0
$323$	41.7905	2.32528
$324$	−1.25202	−0.0695566
$325$	−3.44666	−0.191186
$326$	−14.4861	−0.802309
$327$	6.25586	0.345950
$328$	−21.5255	−1.18854
$329$	0	0
$330$	−8.59680	−0.473238
$331$	24.0871	1.32395	0.661974	−	0.749527i	$-0.269719\pi$
$331$	24.0871	1.32395	0.661974	+	0.749527i	$0.269719\pi$
$332$	1.32358	0.0726411
$333$	−1.47693	−0.0809350
$334$	−15.4393	−0.844801
$335$	11.1541	0.609416
$336$	0	0
$337$	−6.91298	−0.376574	−0.188287	−	0.982114i	$-0.560294\pi$
$337$	−6.91298	−0.376574	−0.188287	+	0.982114i	$0.560294\pi$
$338$	−4.42880	−0.240895
$339$	−3.79544	−0.206140
$340$	−21.8331	−1.18406
$341$	−13.7156	−0.742743
$342$	5.17236	0.279689
$343$	0	0
$344$	−19.9126	−1.07361
$345$	2.49557	0.134357
$346$	−2.29499	−0.123379
$347$	12.1876	0.654266	0.327133	−	0.944978i	$-0.393917\pi$
$347$	12.1876	0.654266	0.327133	+	0.944978i	$0.393917\pi$
$348$	5.08567	0.272621
$349$	−27.0701	−1.44903	−0.724515	−	0.689259i	$-0.757936\pi$
$349$	−27.0701	−1.44903	−0.724515	+	0.689259i	$0.757936\pi$
$350$	0	0
$351$	−2.80699	−0.149826
$352$	−22.6518	−1.20735
$353$	28.5772	1.52101	0.760505	−	0.649332i	$-0.224951\pi$
$353$	28.5772	1.52101	0.760505	+	0.649332i	$0.224951\pi$
$354$	7.18368	0.381808
$355$	−23.6837	−1.25700
$356$	−4.13233	−0.219013
$357$	0	0
$358$	9.91962	0.524268
$359$	−19.4627	−1.02720	−0.513601	−	0.858029i	$-0.671689\pi$
$359$	−19.4627	−1.02720	−0.513601	+	0.858029i	$0.671689\pi$
$360$	−7.01890	−0.369928
$361$	16.7673	0.882489
$362$	13.6987	0.719990
$363$	4.86506	0.255350
$364$	0	0
$365$	−32.1982	−1.68533
$366$	3.21106	0.167845
$367$	−4.81840	−0.251518	−0.125759	−	0.992061i	$-0.540137\pi$
$367$	−4.81840	−0.251518	−0.125759	+	0.992061i	$0.540137\pi$
$368$	0.0715855	0.00373165
$369$	7.65339	0.398420
$370$	−3.18768	−0.165719
$371$	0	0
$372$	−4.31127	−0.223529
$373$	−24.6352	−1.27556	−0.637782	−	0.770217i	$-0.720148\pi$
$373$	−24.6352	−1.27556	−0.637782	+	0.770217i	$0.720148\pi$
$374$	−24.0714	−1.24470
$375$	−9.41359	−0.486116
$376$	−8.56505	−0.441709
$377$	11.4019	0.587229
$378$	0	0
$379$	31.7983	1.63337	0.816684	−	0.577085i	$-0.195810\pi$
$379$	31.7983	1.63337	0.816684	+	0.577085i	$0.195810\pi$
$380$	−18.6863	−0.958588
$381$	20.8819	1.06981
$382$	18.8497	0.964433
$383$	22.3654	1.14282	0.571409	−	0.820666i	$-0.306397\pi$
$383$	22.3654	1.14282	0.571409	+	0.820666i	$0.306397\pi$
$384$	−7.24404	−0.369671
$385$	0	0
$386$	13.8666	0.705791
$387$	7.07993	0.359893
$388$	−19.9462	−1.01262
$389$	19.9097	1.00946	0.504730	−	0.863277i	$-0.331592\pi$
$389$	19.9097	1.00946	0.504730	+	0.863277i	$0.331592\pi$
$390$	−6.05837	−0.306778
$391$	6.98770	0.353383
$392$	0	0
$393$	−12.1488	−0.612825
$394$	9.22397	0.464697
$395$	17.7761	0.894411
$396$	4.98691	0.250602
$397$	−27.1066	−1.36044	−0.680222	−	0.733006i	$-0.738117\pi$
$397$	−27.1066	−1.36044	−0.680222	+	0.733006i	$0.738117\pi$
$398$	3.16456	0.158625
$399$	0	0
$400$	0.0878987	0.00439494
$401$	−34.1782	−1.70678	−0.853390	−	0.521274i	$-0.825457\pi$
$401$	−34.1782	−1.70678	−0.853390	+	0.521274i	$0.825457\pi$
$402$	3.86555	0.192796
$403$	−9.66574	−0.481485
$404$	8.71941	0.433807
$405$	2.49557	0.124006
$406$	0	0
$407$	5.88274	0.291596
$408$	−19.6532	−0.972977
$409$	−15.1477	−0.749006	−0.374503	−	0.927226i	$-0.622187\pi$
$409$	−15.1477	−0.749006	−0.374503	+	0.927226i	$0.622187\pi$
$410$	16.5185	0.815789
$411$	−0.828633	−0.0408734
$412$	7.51822	0.370396
$413$	0	0
$414$	0.864859	0.0425055
$415$	−2.63822	−0.129505
$416$	−15.9633	−0.782665
$417$	−0.546099	−0.0267426
$418$	−20.6020	−1.00768
$419$	−14.0504	−0.686405	−0.343202	−	0.939262i	$-0.611512\pi$
$419$	−14.0504	−0.686405	−0.343202	+	0.939262i	$0.611512\pi$
$420$	0	0
$421$	−26.0914	−1.27161	−0.635807	−	0.771848i	$-0.719333\pi$
$421$	−26.0914	−1.27161	−0.635807	+	0.771848i	$0.719333\pi$
$422$	−24.3767	−1.18664
$423$	3.04531	0.148068
$424$	−26.9189	−1.30730
$425$	8.58009	0.416195
$426$	−8.20775	−0.397667
$427$	0	0
$428$	−7.15900	−0.346043
$429$	11.1805	0.539799
$430$	15.2808	0.736904
$431$	3.09446	0.149055	0.0745274	−	0.997219i	$-0.476255\pi$
$431$	3.09446	0.149055	0.0745274	+	0.997219i	$0.476255\pi$
$432$	0.0715855	0.00344416
$433$	−18.9836	−0.912293	−0.456146	−	0.889905i	$-0.650771\pi$
$433$	−18.9836	−0.912293	−0.456146	+	0.889905i	$0.650771\pi$
$434$	0	0
$435$	−10.1370	−0.486031
$436$	−7.83244	−0.375106
$437$	5.98058	0.286090
$438$	−11.1585	−0.533174
$439$	11.9712	0.571355	0.285677	−	0.958326i	$-0.407781\pi$
$439$	11.9712	0.571355	0.285677	+	0.958326i	$0.407781\pi$
$440$	27.9569	1.33280
$441$	0	0
$442$	−16.9637	−0.806879
$443$	19.4820	0.925616	0.462808	−	0.886458i	$-0.346842\pi$
$443$	19.4820	0.925616	0.462808	+	0.886458i	$0.346842\pi$
$444$	1.84914	0.0877562
$445$	8.23672	0.390458
$446$	−20.9539	−0.992194
$447$	−18.5487	−0.877322
$448$	0	0
$449$	−25.9358	−1.22399	−0.611993	−	0.790864i	$-0.709632\pi$
$449$	−25.9358	−1.22399	−0.611993	+	0.790864i	$0.709632\pi$
$450$	1.06195	0.0500607
$451$	−30.4842	−1.43545
$452$	4.75196	0.223513
$453$	−5.77251	−0.271216
$454$	6.95048	0.326202
$455$	0	0
$456$	−16.8206	−0.787697
$457$	4.61332	0.215802	0.107901	−	0.994162i	$-0.465587\pi$
$457$	4.61332	0.215802	0.107901	+	0.994162i	$0.465587\pi$
$458$	−11.9169	−0.556840
$459$	6.98770	0.326158
$460$	−3.12450	−0.145681
$461$	−13.8181	−0.643571	−0.321786	−	0.946813i	$-0.604283\pi$
$461$	−13.8181	−0.643571	−0.321786	+	0.946813i	$0.604283\pi$
$462$	0	0
$463$	−34.1291	−1.58611	−0.793057	−	0.609147i	$-0.791512\pi$
$463$	−34.1291	−1.58611	−0.793057	+	0.609147i	$0.791512\pi$
$464$	−0.290779	−0.0134991
$465$	8.59340	0.398510
$466$	−1.58789	−0.0735576
$467$	27.9716	1.29437	0.647185	−	0.762333i	$-0.275946\pi$
$467$	27.9716	1.29437	0.647185	+	0.762333i	$0.275946\pi$
$468$	3.51440	0.162453
$469$	0	0
$470$	6.57275	0.303179
$471$	−14.1290	−0.651029
$472$	−23.3614	−1.07530
$473$	−28.2001	−1.29664
$474$	6.16043	0.282958
$475$	7.34346	0.336941
$476$	0	0
$477$	9.57104	0.438228
$478$	−3.82446	−0.174927
$479$	32.6269	1.49076	0.745380	−	0.666639i	$-0.232268\pi$
$479$	32.6269	1.49076	0.745380	+	0.666639i	$0.232268\pi$
$480$	14.1923	0.647787
$481$	4.14571	0.189028
$482$	24.6129	1.12109
$483$	0	0
$484$	−6.09115	−0.276870
$485$	39.7576	1.80530
$486$	0.864859	0.0392308
$487$	−25.9421	−1.17555	−0.587775	−	0.809025i	$-0.699996\pi$
$487$	−25.9421	−1.17555	−0.587775	+	0.809025i	$0.699996\pi$
$488$	−10.4424	−0.472707
$489$	−16.7496	−0.757444
$490$	0	0
$491$	26.0005	1.17339	0.586693	−	0.809810i	$-0.300430\pi$
$491$	26.0005	1.17339	0.586693	+	0.809810i	$0.300430\pi$
$492$	−9.58219	−0.431998
$493$	−28.3839	−1.27835
$494$	−14.5187	−0.653229
$495$	−9.94011	−0.446775
$496$	0.246502	0.0110683
$497$	0	0
$498$	−0.914295	−0.0409705
$499$	−23.2072	−1.03890	−0.519448	−	0.854502i	$-0.673862\pi$
$499$	−23.2072	−1.03890	−0.519448	+	0.854502i	$0.673862\pi$
$500$	11.7860	0.527085
$501$	−17.8518	−0.797560
$502$	23.0973	1.03088
$503$	−12.5204	−0.558256	−0.279128	−	0.960254i	$-0.590045\pi$
$503$	−12.5204	−0.558256	−0.279128	+	0.960254i	$0.590045\pi$
$504$	0	0
$505$	−17.3799	−0.773395
$506$	−3.44482	−0.153141
$507$	−5.12084	−0.227424
$508$	−26.1445	−1.15998
$509$	24.2118	1.07317	0.536585	−	0.843847i	$-0.319714\pi$
$509$	24.2118	1.07317	0.536585	+	0.843847i	$0.319714\pi$
$510$	15.0817	0.667828
$511$	0	0
$512$	0.809780	0.0357875
$513$	5.98058	0.264049
$514$	−3.77452	−0.166487
$515$	−14.9856	−0.660345
$516$	−8.86420	−0.390225
$517$	−12.1298	−0.533466
$518$	0	0
$519$	−2.65359	−0.116480
$520$	19.7019	0.863987
$521$	6.82821	0.299149	0.149575	−	0.988750i	$-0.452210\pi$
$521$	6.82821	0.299149	0.149575	+	0.988750i	$0.452210\pi$
$522$	−3.51304	−0.153762
$523$	9.54004	0.417157	0.208579	−	0.978006i	$-0.433116\pi$
$523$	9.54004	0.417157	0.208579	+	0.978006i	$0.433116\pi$
$524$	15.2105	0.664474
$525$	0	0
$526$	1.33979	0.0584177
$527$	24.0619	1.04815
$528$	−0.285132	−0.0124088
$529$	1.00000	0.0434783
$530$	20.6574	0.897298
$531$	8.30618	0.360457
$532$	0	0
$533$	−21.4830	−0.930530
$534$	2.85450	0.123526
$535$	14.2696	0.616929
$536$	−12.5708	−0.542978
$537$	11.4696	0.494951
$538$	3.21323	0.138532
$539$	0	0
$540$	−3.12450	−0.134457
$541$	33.1748	1.42629	0.713147	−	0.701014i	$-0.247269\pi$
$541$	33.1748	1.42629	0.713147	+	0.701014i	$0.247269\pi$
$542$	−3.47681	−0.149342
$543$	15.8393	0.679728
$544$	39.7390	1.70379
$545$	15.6119	0.668742
$546$	0	0
$547$	17.2100	0.735848	0.367924	−	0.929856i	$-0.380069\pi$
$547$	17.2100	0.735848	0.367924	+	0.929856i	$0.380069\pi$
$548$	1.03746	0.0443182
$549$	3.71282	0.158459
$550$	−4.22984	−0.180361
$551$	−24.2930	−1.03492
$552$	−2.81254	−0.119710
$553$	0	0
$554$	−9.85311	−0.418619
$555$	−3.68577	−0.156452
$556$	0.683726	0.0289964
$557$	17.8924	0.758125	0.379062	−	0.925371i	$-0.376247\pi$
$557$	17.8924	0.758125	0.379062	+	0.925371i	$0.376247\pi$
$558$	2.97811	0.126073
$559$	−19.8733	−0.840550
$560$	0	0
$561$	−27.8327	−1.17510
$562$	−24.1731	−1.01968
$563$	−16.4160	−0.691852	−0.345926	−	0.938262i	$-0.612435\pi$
$563$	−16.4160	−0.691852	−0.345926	+	0.938262i	$0.612435\pi$
$564$	−3.81278	−0.160547
$565$	−9.47180	−0.398482
$566$	25.9245	1.08969
$567$	0	0
$568$	26.6918	1.11996
$569$	4.94134	0.207152	0.103576	−	0.994622i	$-0.466972\pi$
$569$	4.94134	0.207152	0.103576	+	0.994622i	$0.466972\pi$
$570$	12.9080	0.540657
$571$	−42.8618	−1.79371	−0.896855	−	0.442325i	$-0.854154\pi$
$571$	−42.8618	−1.79371	−0.896855	+	0.442325i	$0.854154\pi$
$572$	−13.9982	−0.585293
$573$	21.7951	0.910502
$574$	0	0
$575$	1.22789	0.0512063
$576$	4.77528	0.198970
$577$	−13.8130	−0.575043	−0.287522	−	0.957774i	$-0.592831\pi$
$577$	−13.8130	−0.575043	−0.287522	+	0.957774i	$0.592831\pi$
$578$	27.5267	1.14496
$579$	16.0333	0.666323
$580$	12.6917	0.526993
$581$	0	0
$582$	13.7783	0.571128
$583$	−38.1224	−1.57887
$584$	36.2877	1.50160
$585$	−7.00504	−0.289623
$586$	4.21486	0.174114
$587$	−19.9885	−0.825012	−0.412506	−	0.910955i	$-0.635346\pi$
$587$	−19.9885	−0.825012	−0.412506	+	0.910955i	$0.635346\pi$
$588$	0	0
$589$	20.5939	0.848556
$590$	17.9274	0.738059
$591$	10.6653	0.438711
$592$	−0.105726	−0.00434533
$593$	2.72068	0.111725	0.0558625	−	0.998438i	$-0.482209\pi$
$593$	2.72068	0.111725	0.0558625	+	0.998438i	$0.482209\pi$
$594$	−3.44482	−0.141343
$595$	0	0
$596$	23.2233	0.951262
$597$	3.65905	0.149755
$598$	−2.42765	−0.0992739
$599$	17.4698	0.713799	0.356899	−	0.934143i	$-0.383834\pi$
$599$	17.4698	0.713799	0.356899	+	0.934143i	$0.383834\pi$
$600$	−3.45347	−0.140987
$601$	−27.6015	−1.12589	−0.562944	−	0.826495i	$-0.690331\pi$
$601$	−27.6015	−1.12589	−0.562944	+	0.826495i	$0.690331\pi$
$602$	0	0
$603$	4.46957	0.182015
$604$	7.22728	0.294074
$605$	12.1411	0.493607
$606$	−6.02313	−0.244673
$607$	12.9173	0.524295	0.262148	−	0.965028i	$-0.415569\pi$
$607$	12.9173	0.524295	0.262148	+	0.965028i	$0.415569\pi$
$608$	34.0115	1.37935
$609$	0	0
$610$	8.01345	0.324455
$611$	−8.54814	−0.345821
$612$	−8.74872	−0.353646
$613$	−38.8541	−1.56930	−0.784651	−	0.619937i	$-0.787158\pi$
$613$	−38.8541	−1.56930	−0.784651	+	0.619937i	$0.787158\pi$
$614$	24.1351	0.974013
$615$	19.0996	0.770170
$616$	0	0
$617$	−25.3417	−1.02022	−0.510108	−	0.860110i	$-0.670395\pi$
$617$	−25.3417	−1.02022	−0.510108	+	0.860110i	$0.670395\pi$
$618$	−5.19338	−0.208908
$619$	−35.5852	−1.43029	−0.715145	−	0.698976i	$-0.753639\pi$
$619$	−35.5852	−1.43029	−0.715145	+	0.698976i	$0.753639\pi$
$620$	−10.7591	−0.432096
$621$	1.00000	0.0401286
$622$	−23.3586	−0.936596
$623$	0	0
$624$	−0.200939	−0.00804401
$625$	−29.6317	−1.18527
$626$	−3.64453	−0.145665
$627$	−23.8212	−0.951328
$628$	17.6897	0.705897
$629$	−10.3203	−0.411498
$630$	0	0
$631$	−12.5559	−0.499842	−0.249921	−	0.968266i	$-0.580405\pi$
$631$	−12.5559	−0.499842	−0.249921	+	0.968266i	$0.580405\pi$
$632$	−20.0338	−0.796903
$633$	−28.1857	−1.12028
$634$	−6.91710	−0.274713
$635$	52.1123	2.06801
$636$	−11.9831	−0.475161
$637$	0	0
$638$	13.9928	0.553980
$639$	−9.49028	−0.375429
$640$	−18.0780	−0.714596
$641$	−44.0488	−1.73982	−0.869912	−	0.493207i	$-0.835825\pi$
$641$	−44.0488	−1.73982	−0.869912	+	0.493207i	$0.835825\pi$
$642$	4.94524	0.195173
$643$	−27.1797	−1.07186	−0.535931	−	0.844262i	$-0.680039\pi$
$643$	−27.1797	−1.07186	−0.535931	+	0.844262i	$0.680039\pi$
$644$	0	0
$645$	17.6685	0.695696
$646$	36.1429	1.42202
$647$	−33.7648	−1.32743	−0.663716	−	0.747985i	$-0.731022\pi$
$647$	−33.7648	−1.32743	−0.663716	+	0.747985i	$0.731022\pi$
$648$	−2.81254	−0.110487
$649$	−33.0843	−1.29867
$650$	−2.98087	−0.116919
$651$	0	0
$652$	20.9708	0.821281
$653$	31.9330	1.24963	0.624817	−	0.780771i	$-0.285174\pi$
$653$	31.9330	1.24963	0.624817	+	0.780771i	$0.285174\pi$
$654$	5.41044	0.211565
$655$	−30.3182	−1.18463
$656$	0.547872	0.0213908
$657$	−12.9021	−0.503359
$658$	0	0
$659$	47.4986	1.85028	0.925142	−	0.379622i	$-0.123946\pi$
$659$	47.4986	1.85028	0.925142	+	0.379622i	$0.123946\pi$
$660$	12.4452	0.484429
$661$	29.4292	1.14466	0.572332	−	0.820022i	$-0.306039\pi$
$661$	29.4292	1.14466	0.572332	+	0.820022i	$0.306039\pi$
$662$	20.8320	0.809658
$663$	−19.6144	−0.761759
$664$	2.97330	0.115387
$665$	0	0
$666$	−1.27733	−0.0494956
$667$	−4.06198	−0.157281
$668$	22.3508	0.864778
$669$	−24.2280	−0.936710
$670$	9.64677	0.372687
$671$	−14.7885	−0.570904
$672$	0	0
$673$	−21.8311	−0.841528	−0.420764	−	0.907170i	$-0.638238\pi$
$673$	−21.8311	−0.841528	−0.420764	+	0.907170i	$0.638238\pi$
$674$	−5.97876	−0.230293
$675$	1.22789	0.0472613
$676$	6.41138	0.246591
$677$	−31.9593	−1.22830	−0.614148	−	0.789191i	$-0.710500\pi$
$677$	−31.9593	−1.22830	−0.614148	+	0.789191i	$0.710500\pi$
$678$	−3.28252	−0.126065
$679$	0	0
$680$	−49.0459	−1.88083
$681$	8.03654	0.307961
$682$	−11.8621	−0.454223
$683$	5.80239	0.222022	0.111011	−	0.993819i	$-0.464591\pi$
$683$	5.80239	0.222022	0.111011	+	0.993819i	$0.464591\pi$
$684$	−7.48779	−0.286303
$685$	−2.06791	−0.0790109
$686$	0	0
$687$	−13.7790	−0.525702
$688$	0.506820	0.0193223
$689$	−26.8658	−1.02350
$690$	2.15832	0.0821658
$691$	−50.4258	−1.91829	−0.959145	−	0.282916i	$-0.908698\pi$
$691$	−50.4258	−1.91829	−0.959145	+	0.282916i	$0.908698\pi$
$692$	3.32235	0.126297
$693$	0	0
$694$	10.5406	0.400115
$695$	−1.36283	−0.0516951
$696$	11.4245	0.433044
$697$	53.4796	2.02568
$698$	−23.4118	−0.886151
$699$	−1.83601	−0.0694443
$700$	0	0
$701$	28.8283	1.08883	0.544414	−	0.838817i	$-0.316752\pi$
$701$	28.8283	1.08883	0.544414	+	0.838817i	$0.316752\pi$
$702$	−2.42765	−0.0916257
$703$	−8.83286	−0.333138
$704$	−19.0204	−0.716858
$705$	7.59979	0.286225
$706$	24.7153	0.930171
$707$	0	0
$708$	−10.3995	−0.390837
$709$	32.4556	1.21889	0.609447	−	0.792826i	$-0.291391\pi$
$709$	32.4556	1.21889	0.609447	+	0.792826i	$0.291391\pi$
$710$	−20.4831	−0.768715
$711$	7.12304	0.267135
$712$	−9.28288	−0.347891
$713$	3.44346	0.128959
$714$	0	0
$715$	27.9017	1.04347
$716$	−14.3602	−0.536665
$717$	−4.42206	−0.165145
$718$	−16.8325	−0.628184
$719$	16.0351	0.598009	0.299004	−	0.954252i	$-0.403345\pi$
$719$	16.0351	0.598009	0.299004	+	0.954252i	$0.403345\pi$
$720$	0.178647	0.00665777
$721$	0	0
$722$	14.5014	0.539685
$723$	28.4589	1.05840
$724$	−19.8310	−0.737015
$725$	−4.98765	−0.185237
$726$	4.20760	0.156159
$727$	−40.6150	−1.50633	−0.753164	−	0.657833i	$-0.771473\pi$
$727$	−40.6150	−1.50633	−0.753164	+	0.657833i	$0.771473\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−27.8469	−1.03066
$731$	49.4724	1.82980
$732$	−4.64851	−0.171814
$733$	2.41457	0.0891843	0.0445922	−	0.999005i	$-0.485801\pi$
$733$	2.41457	0.0891843	0.0445922	+	0.999005i	$0.485801\pi$
$734$	−4.16724	−0.153816
$735$	0	0
$736$	5.68699	0.209625
$737$	−17.8027	−0.655772
$738$	6.61911	0.243653
$739$	2.79635	0.102865	0.0514327	−	0.998676i	$-0.483621\pi$
$739$	2.79635	0.102865	0.0514327	+	0.998676i	$0.483621\pi$
$740$	4.61466	0.169638
$741$	−16.7874	−0.616700
$742$	0	0
$743$	3.56515	0.130793	0.0653964	−	0.997859i	$-0.479169\pi$
$743$	3.56515	0.130793	0.0653964	+	0.997859i	$0.479169\pi$
$744$	−9.68486	−0.355064
$745$	−46.2896	−1.69592
$746$	−21.3060	−0.780069
$747$	−1.05716	−0.0386795
$748$	34.8470	1.27413
$749$	0	0
$750$	−8.14143	−0.297283
$751$	−3.79314	−0.138413	−0.0692067	−	0.997602i	$-0.522047\pi$
$751$	−3.79314	−0.138413	−0.0692067	+	0.997602i	$0.522047\pi$
$752$	0.218000	0.00794964
$753$	26.7064	0.973234
$754$	9.86106	0.359119
$755$	−14.4057	−0.524278
$756$	0	0
$757$	−6.45659	−0.234669	−0.117334	−	0.993092i	$-0.537435\pi$
$757$	−6.45659	−0.234669	−0.117334	+	0.993092i	$0.537435\pi$
$758$	27.5010	0.998883
$759$	−3.98310	−0.144577
$760$	−41.9770	−1.52267
$761$	22.4004	0.812014	0.406007	−	0.913870i	$-0.366921\pi$
$761$	22.4004	0.812014	0.406007	+	0.913870i	$0.366921\pi$
$762$	18.0599	0.654241
$763$	0	0
$764$	−27.2878	−0.987239
$765$	17.4383	0.630483
$766$	19.3429	0.698888
$767$	−23.3153	−0.841867
$768$	−15.8156	−0.570697
$769$	−53.5732	−1.93190	−0.965950	−	0.258731i	$-0.916696\pi$
$769$	−53.5732	−1.93190	−0.965950	+	0.258731i	$0.916696\pi$
$770$	0	0
$771$	−4.36432	−0.157177
$772$	−20.0740	−0.722481
$773$	44.5244	1.60143	0.800715	−	0.599045i	$-0.204453\pi$
$773$	44.5244	1.60143	0.800715	+	0.599045i	$0.204453\pi$
$774$	6.12315	0.220092
$775$	4.22817	0.151880
$776$	−44.8072	−1.60849
$777$	0	0
$778$	17.2191	0.617333
$779$	45.7717	1.63994
$780$	8.77043	0.314032
$781$	37.8007	1.35262
$782$	6.04338	0.216111
$783$	−4.06198	−0.145163
$784$	0	0
$785$	−35.2599	−1.25848
$786$	−10.5070	−0.374772
$787$	15.2713	0.544364	0.272182	−	0.962246i	$-0.412255\pi$
$787$	15.2713	0.544364	0.272182	+	0.962246i	$0.412255\pi$
$788$	−13.3531	−0.475685
$789$	1.54914	0.0551509
$790$	15.3738	0.546975
$791$	0	0
$792$	11.2026	0.398068
$793$	−10.4218	−0.370090
$794$	−23.4434	−0.831977
$795$	23.8852	0.847122
$796$	−4.58119	−0.162376
$797$	−38.5369	−1.36505	−0.682524	−	0.730863i	$-0.739118\pi$
$797$	−38.5369	−1.36505	−0.682524	+	0.730863i	$0.739118\pi$
$798$	0	0
$799$	21.2797	0.752822
$800$	6.98297	0.246885
$801$	3.30053	0.116619
$802$	−29.5594	−1.04378
$803$	51.3904	1.81353
$804$	−5.59599	−0.197355
$805$	0	0
$806$	−8.35951	−0.294451
$807$	3.71532	0.130786
$808$	19.5873	0.689080
$809$	−36.3518	−1.27806	−0.639031	−	0.769181i	$-0.720664\pi$
$809$	−36.3518	−1.27806	−0.639031	+	0.769181i	$0.720664\pi$
$810$	2.15832	0.0758356
$811$	21.9469	0.770661	0.385330	−	0.922779i	$-0.374088\pi$
$811$	21.9469	0.770661	0.385330	+	0.922779i	$0.374088\pi$
$812$	0	0
$813$	−4.02009	−0.140991
$814$	5.08774	0.178325
$815$	−41.7999	−1.46419
$816$	0.500218	0.0175111
$817$	42.3421	1.48136
$818$	−13.1006	−0.458053
$819$	0	0
$820$	−23.9130	−0.835080
$821$	−13.0796	−0.456481	−0.228241	−	0.973605i	$-0.573297\pi$
$821$	−13.0796	−0.456481	−0.228241	+	0.973605i	$0.573297\pi$
$822$	−0.716651	−0.0249961
$823$	31.6053	1.10169	0.550845	−	0.834608i	$-0.314306\pi$
$823$	31.6053	1.10169	0.550845	+	0.834608i	$0.314306\pi$
$824$	16.8890	0.588355
$825$	−4.89079	−0.170275
$826$	0	0
$827$	21.5412	0.749060	0.374530	−	0.927215i	$-0.377804\pi$
$827$	21.5412	0.749060	0.374530	+	0.927215i	$0.377804\pi$
$828$	−1.25202	−0.0435106
$829$	−33.3139	−1.15704	−0.578519	−	0.815669i	$-0.696369\pi$
$829$	−33.3139	−1.15704	−0.578519	+	0.815669i	$0.696369\pi$
$830$	−2.28169	−0.0791986
$831$	−11.3927	−0.395209
$832$	−13.4041	−0.464704
$833$	0	0
$834$	−0.472299	−0.0163544
$835$	−44.5505	−1.54173
$836$	29.8246	1.03151
$837$	3.44346	0.119023
$838$	−12.1516	−0.419769
$839$	−4.75994	−0.164331	−0.0821656	−	0.996619i	$-0.526184\pi$
$839$	−4.75994	−0.164331	−0.0821656	+	0.996619i	$0.526184\pi$
$840$	0	0
$841$	−12.5003	−0.431045
$842$	−22.5654	−0.777653
$843$	−27.9503	−0.962661
$844$	35.2891	1.21470
$845$	−12.7794	−0.439625
$846$	2.63377	0.0905507
$847$	0	0
$848$	0.685147	0.0235281
$849$	29.9754	1.02875
$850$	7.42057	0.254524
$851$	−1.47693	−0.0506283
$852$	11.8820	0.407070
$853$	−34.8330	−1.19266	−0.596330	−	0.802740i	$-0.703375\pi$
$853$	−34.8330	−1.19266	−0.596330	+	0.802740i	$0.703375\pi$
$854$	0	0
$855$	14.9250	0.510423
$856$	−16.0820	−0.549672
$857$	−19.8576	−0.678323	−0.339161	−	0.940728i	$-0.610143\pi$
$857$	−19.8576	−0.678323	−0.339161	+	0.940728i	$0.610143\pi$
$858$	9.66956	0.330113
$859$	−33.7997	−1.15323	−0.576615	−	0.817016i	$-0.695627\pi$
$859$	−33.7997	−1.15323	−0.576615	+	0.817016i	$0.695627\pi$
$860$	−22.1213	−0.754329
$861$	0	0
$862$	2.67627	0.0911541
$863$	35.6001	1.21184	0.605921	−	0.795525i	$-0.292805\pi$
$863$	35.6001	1.21184	0.605921	+	0.795525i	$0.292805\pi$
$864$	5.68699	0.193475
$865$	−6.62224	−0.225163
$866$	−16.4181	−0.557911
$867$	31.8279	1.08093
$868$	0	0
$869$	−28.3718	−0.962446
$870$	−8.76706	−0.297231
$871$	−12.5460	−0.425106
$872$	−17.5948	−0.595836
$873$	15.9312	0.539191
$874$	5.17236	0.174958
$875$	0	0
$876$	16.1537	0.545782
$877$	14.8802	0.502470	0.251235	−	0.967926i	$-0.419163\pi$
$877$	14.8802	0.502470	0.251235	+	0.967926i	$0.419163\pi$
$878$	10.3534	0.349411
$879$	4.87346	0.164378
$880$	−0.711568	−0.0239869
$881$	25.2288	0.849980	0.424990	−	0.905198i	$-0.360278\pi$
$881$	25.2288	0.849980	0.424990	+	0.905198i	$0.360278\pi$
$882$	0	0
$883$	−5.13080	−0.172665	−0.0863325	−	0.996266i	$-0.527515\pi$
$883$	−5.13080	−0.172665	−0.0863325	+	0.996266i	$0.527515\pi$
$884$	24.5575	0.825959
$885$	20.7287	0.696787
$886$	16.8492	0.566059
$887$	4.27219	0.143446	0.0717231	−	0.997425i	$-0.477150\pi$
$887$	4.27219	0.143446	0.0717231	+	0.997425i	$0.477150\pi$
$888$	4.15391	0.139396
$889$	0	0
$890$	7.12361	0.238784
$891$	−3.98310	−0.133439
$892$	30.3340	1.01566
$893$	18.2127	0.609465
$894$	−16.0420	−0.536525
$895$	28.6233	0.956772
$896$	0	0
$897$	−2.80699	−0.0937225
$898$	−22.4308	−0.748526
$899$	−13.9873	−0.466501
$900$	−1.53733	−0.0512445
$901$	66.8795	2.22808
$902$	−26.3646	−0.877844
$903$	0	0
$904$	10.6748	0.355039
$905$	39.5280	1.31396
$906$	−4.99241	−0.165862
$907$	−29.2234	−0.970346	−0.485173	−	0.874418i	$-0.661243\pi$
$907$	−29.2234	−0.970346	−0.485173	+	0.874418i	$0.661243\pi$
$908$	−10.0619	−0.333916
$909$	−6.96429	−0.230991
$910$	0	0
$911$	−48.3422	−1.60165	−0.800824	−	0.598900i	$-0.795605\pi$
$911$	−48.3422	−1.60165	−0.800824	+	0.598900i	$0.795605\pi$
$912$	0.428122	0.0141766
$913$	4.21077	0.139356
$914$	3.98987	0.131973
$915$	9.26560	0.306311
$916$	17.2516	0.570008
$917$	0	0
$918$	6.04338	0.199461
$919$	30.7928	1.01576	0.507881	−	0.861427i	$-0.330429\pi$
$919$	30.7928	1.01576	0.507881	+	0.861427i	$0.330429\pi$
$920$	−7.01890	−0.231406
$921$	27.9064	0.919547
$922$	−11.9507	−0.393575
$923$	26.6391	0.876835
$924$	0	0
$925$	−1.81349	−0.0596273
$926$	−29.5169	−0.969985
$927$	−6.00488	−0.197226
$928$	−23.1004	−0.758309
$929$	24.6571	0.808975	0.404487	−	0.914544i	$-0.367450\pi$
$929$	24.6571	0.808975	0.404487	+	0.914544i	$0.367450\pi$
$930$	7.43209	0.243708
$931$	0	0
$932$	2.29872	0.0752970
$933$	−27.0086	−0.884221
$934$	24.1915	0.791569
$935$	−69.4585	−2.27154
$936$	7.89475	0.258048
$937$	−35.2046	−1.15008	−0.575042	−	0.818124i	$-0.695014\pi$
$937$	−35.2046	−1.15008	−0.575042	+	0.818124i	$0.695014\pi$
$938$	0	0
$939$	−4.21402	−0.137519
$940$	−9.51508	−0.310348
$941$	22.0830	0.719885	0.359942	−	0.932975i	$-0.382796\pi$
$941$	22.0830	0.719885	0.359942	+	0.932975i	$0.382796\pi$
$942$	−12.2196	−0.398135
$943$	7.65339	0.249229
$944$	0.594602	0.0193526
$945$	0	0
$946$	−24.3891	−0.792958
$947$	5.85753	0.190344	0.0951720	−	0.995461i	$-0.469660\pi$
$947$	5.85753	0.190344	0.0951720	+	0.995461i	$0.469660\pi$
$948$	−8.91818	−0.289649
$949$	36.2160	1.17562
$950$	6.35106	0.206056
$951$	−7.99795	−0.259351
$952$	0	0
$953$	−48.8701	−1.58306	−0.791530	−	0.611131i	$-0.790715\pi$
$953$	−48.8701	−1.58306	−0.791530	+	0.611131i	$0.790715\pi$
$954$	8.27760	0.267997
$955$	54.3912	1.76006
$956$	5.53650	0.179063
$957$	16.1793	0.523001
$958$	28.2177	0.911672
$959$	0	0
$960$	11.9170	0.384621
$961$	−19.1426	−0.617503
$962$	3.58545	0.115600
$963$	5.71797	0.184259
$964$	−35.6310	−1.14760
$965$	40.0124	1.28804
$966$	0	0
$967$	11.7300	0.377213	0.188606	−	0.982053i	$-0.439603\pi$
$967$	11.7300	0.377213	0.188606	+	0.982053i	$0.439603\pi$
$968$	−13.6832	−0.439794
$969$	41.7905	1.34250
$970$	34.3847	1.10403
$971$	24.3527	0.781516	0.390758	−	0.920493i	$-0.372213\pi$
$971$	24.3527	0.781516	0.390758	+	0.920493i	$0.372213\pi$
$972$	−1.25202	−0.0401585
$973$	0	0
$974$	−22.4363	−0.718905
$975$	−3.44666	−0.110381
$976$	0.265784	0.00850753
$977$	50.9094	1.62874	0.814368	−	0.580348i	$-0.197084\pi$
$977$	50.9094	1.62874	0.814368	+	0.580348i	$0.197084\pi$
$978$	−14.4861	−0.463214
$979$	−13.1463	−0.420159
$980$	0	0
$981$	6.25586	0.199734
$982$	22.4868	0.717581
$983$	33.5958	1.07154	0.535770	−	0.844364i	$-0.320021\pi$
$983$	33.5958	1.07154	0.535770	+	0.844364i	$0.320021\pi$
$984$	−21.5255	−0.686207
$985$	26.6160	0.848056
$986$	−24.5481	−0.781770
$987$	0	0
$988$	21.0181	0.668675
$989$	7.07993	0.225129
$990$	−8.59680	−0.273224
$991$	10.4350	0.331480	0.165740	−	0.986169i	$-0.446999\pi$
$991$	10.4350	0.331480	0.165740	+	0.986169i	$0.446999\pi$
$992$	19.5829	0.621758
$993$	24.0871	0.764382
$994$	0	0
$995$	9.13142	0.289485
$996$	1.32358	0.0419394
$997$	−15.1736	−0.480553	−0.240277	−	0.970704i	$-0.577238\pi$
$997$	−15.1736	−0.480553	−0.240277	+	0.970704i	$0.577238\pi$
$998$	−20.0710	−0.635335
$999$	−1.47693	−0.0467278

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bj.1.6		10
7.3	odd	6		483.2.i.h.415.5	yes	20
7.5	odd	6		483.2.i.h.277.5	✓	20
7.6	odd	2		3381.2.a.bi.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.i.h.277.5	✓	20	7.5	odd	6
483.2.i.h.415.5	yes	20	7.3	odd	6
3381.2.a.bi.1.6		10	7.6	odd	2
3381.2.a.bj.1.6		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q+0.864859 q^{2} +1.00000 q^{3} -1.25202 q^{4} +2.49557 q^{5} +0.864859 q^{6} -2.81254 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.864859 q^{2} +1.00000 q^{3} -1.25202 q^{4} +2.49557 q^{5} +0.864859 q^{6} -2.81254 q^{8} +1.00000 q^{9} +2.15832 q^{10} -3.98310 q^{11} -1.25202 q^{12} -2.80699 q^{13} +2.49557 q^{15} +0.0715855 q^{16} +6.98770 q^{17} +0.864859 q^{18} +5.98058 q^{19} -3.12450 q^{20} -3.44482 q^{22} +1.00000 q^{23} -2.81254 q^{24} +1.22789 q^{25} -2.42765 q^{26} +1.00000 q^{27} -4.06198 q^{29} +2.15832 q^{30} +3.44346 q^{31} +5.68699 q^{32} -3.98310 q^{33} +6.04338 q^{34} -1.25202 q^{36} -1.47693 q^{37} +5.17236 q^{38} -2.80699 q^{39} -7.01890 q^{40} +7.65339 q^{41} +7.07993 q^{43} +4.98691 q^{44} +2.49557 q^{45} +0.864859 q^{46} +3.04531 q^{47} +0.0715855 q^{48} +1.06195 q^{50} +6.98770 q^{51} +3.51440 q^{52} +9.57104 q^{53} +0.864859 q^{54} -9.94011 q^{55} +5.98058 q^{57} -3.51304 q^{58} +8.30618 q^{59} -3.12450 q^{60} +3.71282 q^{61} +2.97811 q^{62} +4.77528 q^{64} -7.00504 q^{65} -3.44482 q^{66} +4.46957 q^{67} -8.74872 q^{68} +1.00000 q^{69} -9.49028 q^{71} -2.81254 q^{72} -12.9021 q^{73} -1.27733 q^{74} +1.22789 q^{75} -7.48779 q^{76} -2.42765 q^{78} +7.12304 q^{79} +0.178647 q^{80} +1.00000 q^{81} +6.61911 q^{82} -1.05716 q^{83} +17.4383 q^{85} +6.12315 q^{86} -4.06198 q^{87} +11.2026 q^{88} +3.30053 q^{89} +2.15832 q^{90} -1.25202 q^{92} +3.44346 q^{93} +2.63377 q^{94} +14.9250 q^{95} +5.68699 q^{96} +15.9312 q^{97} -3.98310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{2} + 10 q^{3} + 15 q^{4} + 5 q^{5} + 3 q^{6} + 9 q^{8} + 10 q^{9}+O(q^{10}) \) 10 * q + 3 * q^2 + 10 * q^3 + 15 * q^4 + 5 * q^5 + 3 * q^6 + 9 * q^8 + 10 * q^9	\( 10 q + 3 q^{2} + 10 q^{3} + 15 q^{4} + 5 q^{5} + 3 q^{6} + 9 q^{8} + 10 q^{9} - 11 q^{10} + 8 q^{11} + 15 q^{12} + 5 q^{15} + 37 q^{16} + 11 q^{17} + 3 q^{18} - q^{19} + 15 q^{20} + 6 q^{22} + 10 q^{23} + 9 q^{24} + 21 q^{25} - q^{26} + 10 q^{27} + 22 q^{29} - 11 q^{30} + 3 q^{31} + 11 q^{32} + 8 q^{33} + 3 q^{34} + 15 q^{36} - 3 q^{37} - 16 q^{38} - 39 q^{40} + 26 q^{41} + 27 q^{43} + 16 q^{44} + 5 q^{45} + 3 q^{46} - 11 q^{47} + 37 q^{48} + 2 q^{50} + 11 q^{51} - 29 q^{52} + 5 q^{53} + 3 q^{54} + 18 q^{55} - q^{57} + 16 q^{58} + 10 q^{59} + 15 q^{60} - 22 q^{61} + 32 q^{62} + 69 q^{64} - 11 q^{65} + 6 q^{66} - 2 q^{67} + 21 q^{68} + 10 q^{69} + 27 q^{71} + 9 q^{72} + 8 q^{73} + 14 q^{74} + 21 q^{75} + 22 q^{76} - q^{78} + 21 q^{79} + 53 q^{80} + 10 q^{81} - 36 q^{82} + 12 q^{83} + 23 q^{85} + 18 q^{86} + 22 q^{87} - 10 q^{88} - 6 q^{89} - 11 q^{90} + 15 q^{92} + 3 q^{93} - 35 q^{94} + 44 q^{95} + 11 q^{96} - 6 q^{97} + 8 q^{99}+O(q^{100}) \) 10 * q + 3 * q^2 + 10 * q^3 + 15 * q^4 + 5 * q^5 + 3 * q^6 + 9 * q^8 + 10 * q^9 - 11 * q^10 + 8 * q^11 + 15 * q^12 + 5 * q^15 + 37 * q^16 + 11 * q^17 + 3 * q^18 - q^19 + 15 * q^20 + 6 * q^22 + 10 * q^23 + 9 * q^24 + 21 * q^25 - q^26 + 10 * q^27 + 22 * q^29 - 11 * q^30 + 3 * q^31 + 11 * q^32 + 8 * q^33 + 3 * q^34 + 15 * q^36 - 3 * q^37 - 16 * q^38 - 39 * q^40 + 26 * q^41 + 27 * q^43 + 16 * q^44 + 5 * q^45 + 3 * q^46 - 11 * q^47 + 37 * q^48 + 2 * q^50 + 11 * q^51 - 29 * q^52 + 5 * q^53 + 3 * q^54 + 18 * q^55 - q^57 + 16 * q^58 + 10 * q^59 + 15 * q^60 - 22 * q^61 + 32 * q^62 + 69 * q^64 - 11 * q^65 + 6 * q^66 - 2 * q^67 + 21 * q^68 + 10 * q^69 + 27 * q^71 + 9 * q^72 + 8 * q^73 + 14 * q^74 + 21 * q^75 + 22 * q^76 - q^78 + 21 * q^79 + 53 * q^80 + 10 * q^81 - 36 * q^82 + 12 * q^83 + 23 * q^85 + 18 * q^86 + 22 * q^87 - 10 * q^88 - 6 * q^89 - 11 * q^90 + 15 * q^92 + 3 * q^93 - 35 * q^94 + 44 * q^95 + 11 * q^96 - 6 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more