LMFDB - Embedded newform 3381.2.a.bi.1.8

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			1.8
Root			$2.27502$ 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)

$f(q)$	$=$	$q+2.27502 q^{2} -1.00000 q^{3} +3.17573 q^{4} -1.23869 q^{5} -2.27502 q^{6} +2.67481 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.27502 q^{2} -1.00000 q^{3} +3.17573 q^{4} -1.23869 q^{5} -2.27502 q^{6} +2.67481 q^{8} +1.00000 q^{9} -2.81806 q^{10} +6.12669 q^{11} -3.17573 q^{12} +0.910400 q^{13} +1.23869 q^{15} -0.266205 q^{16} +4.90873 q^{17} +2.27502 q^{18} -6.90370 q^{19} -3.93376 q^{20} +13.9384 q^{22} +1.00000 q^{23} -2.67481 q^{24} -3.46564 q^{25} +2.07118 q^{26} -1.00000 q^{27} +3.36004 q^{29} +2.81806 q^{30} +5.43735 q^{31} -5.95524 q^{32} -6.12669 q^{33} +11.1675 q^{34} +3.17573 q^{36} +6.54665 q^{37} -15.7061 q^{38} -0.910400 q^{39} -3.31327 q^{40} -2.52207 q^{41} -0.231369 q^{43} +19.4567 q^{44} -1.23869 q^{45} +2.27502 q^{46} +13.3578 q^{47} +0.266205 q^{48} -7.88440 q^{50} -4.90873 q^{51} +2.89118 q^{52} +4.22681 q^{53} -2.27502 q^{54} -7.58909 q^{55} +6.90370 q^{57} +7.64417 q^{58} -0.754201 q^{59} +3.93376 q^{60} +3.57664 q^{61} +12.3701 q^{62} -13.0159 q^{64} -1.12771 q^{65} -13.9384 q^{66} -4.49406 q^{67} +15.5888 q^{68} -1.00000 q^{69} +15.8575 q^{71} +2.67481 q^{72} -4.27776 q^{73} +14.8938 q^{74} +3.46564 q^{75} -21.9243 q^{76} -2.07118 q^{78} -0.222015 q^{79} +0.329747 q^{80} +1.00000 q^{81} -5.73777 q^{82} +15.5197 q^{83} -6.08041 q^{85} -0.526369 q^{86} -3.36004 q^{87} +16.3877 q^{88} +0.559363 q^{89} -2.81806 q^{90} +3.17573 q^{92} -5.43735 q^{93} +30.3893 q^{94} +8.55157 q^{95} +5.95524 q^{96} +17.6476 q^{97} +6.12669 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.27502	1.60868	0.804342	−	0.594167i	$-0.202518\pi$
$2$	2.27502	1.60868	0.804342	+	0.594167i	$0.202518\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.17573	1.58786
$5$	−1.23869	−0.553961	−0.276980	−	0.960876i	$-0.589334\pi$
$5$	−1.23869	−0.553961	−0.276980	+	0.960876i	$0.589334\pi$
$6$	−2.27502	−0.928774
$7$	0	0
$7$	0	0
$8$	2.67481	0.945688
$9$	1.00000	0.333333
$10$	−2.81806	−0.891148
$11$	6.12669	1.84727	0.923633	−	0.383277i	$-0.125205\pi$
$11$	6.12669	1.84727	0.923633	+	0.383277i	$0.125205\pi$
$12$	−3.17573	−0.916754
$13$	0.910400	0.252500	0.126250	−	0.991998i	$-0.459706\pi$
$13$	0.910400	0.252500	0.126250	+	0.991998i	$0.459706\pi$
$14$	0	0
$15$	1.23869	0.319829
$16$	−0.266205	−0.0665513
$17$	4.90873	1.19054	0.595271	−	0.803525i	$-0.297045\pi$
$17$	4.90873	1.19054	0.595271	+	0.803525i	$0.297045\pi$
$18$	2.27502	0.536228
$19$	−6.90370	−1.58382	−0.791909	−	0.610639i	$-0.790913\pi$
$19$	−6.90370	−1.58382	−0.791909	+	0.610639i	$0.790913\pi$
$20$	−3.93376	−0.879614
$21$	0	0
$22$	13.9384	2.97167
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−2.67481	−0.545993
$25$	−3.46564	−0.693128
$26$	2.07118	0.406192
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.36004	0.623944	0.311972	−	0.950091i	$-0.399010\pi$
$29$	3.36004	0.623944	0.311972	+	0.950091i	$0.399010\pi$
$30$	2.81806	0.514504
$31$	5.43735	0.976577	0.488289	−	0.872682i	$-0.337621\pi$
$31$	5.43735	0.976577	0.488289	+	0.872682i	$0.337621\pi$
$32$	−5.95524	−1.05275
$33$	−6.12669	−1.06652
$34$	11.1675	1.91521
$35$	0	0
$36$	3.17573	0.529288
$37$	6.54665	1.07626	0.538132	−	0.842861i	$-0.319130\pi$
$37$	6.54665	1.07626	0.538132	+	0.842861i	$0.319130\pi$
$38$	−15.7061	−2.54786
$39$	−0.910400	−0.145781
$40$	−3.31327	−0.523874
$41$	−2.52207	−0.393882	−0.196941	−	0.980415i	$-0.563101\pi$
$41$	−2.52207	−0.393882	−0.196941	+	0.980415i	$0.563101\pi$
$42$	0	0
$43$	−0.231369	−0.0352834	−0.0176417	−	0.999844i	$-0.505616\pi$
$43$	−0.231369	−0.0352834	−0.0176417	+	0.999844i	$0.505616\pi$
$44$	19.4567	2.93321
$45$	−1.23869	−0.184654
$46$	2.27502	0.335434
$47$	13.3578	1.94843	0.974217	−	0.225614i	$-0.0724388\pi$
$47$	13.3578	1.94843	0.974217	+	0.225614i	$0.0724388\pi$
$48$	0.266205	0.0384234
$49$	0	0
$50$	−7.88440	−1.11502
$51$	−4.90873	−0.687360
$52$	2.89118	0.400935
$53$	4.22681	0.580597	0.290298	−	0.956936i	$-0.406245\pi$
$53$	4.22681	0.580597	0.290298	+	0.956936i	$0.406245\pi$
$54$	−2.27502	−0.309591
$55$	−7.58909	−1.02331
$56$	0	0
$57$	6.90370	0.914418
$58$	7.64417	1.00373
$59$	−0.754201	−0.0981886	−0.0490943	−	0.998794i	$-0.515633\pi$
$59$	−0.754201	−0.0981886	−0.0490943	+	0.998794i	$0.515633\pi$
$60$	3.93376	0.507846
$61$	3.57664	0.457942	0.228971	−	0.973433i	$-0.426464\pi$
$61$	3.57664	0.457942	0.228971	+	0.973433i	$0.426464\pi$
$62$	12.3701	1.57100
$63$	0	0
$64$	−13.0159	−1.62699
$65$	−1.12771	−0.139875
$66$	−13.9384	−1.71569
$67$	−4.49406	−0.549036	−0.274518	−	0.961582i	$-0.588518\pi$
$67$	−4.49406	−0.549036	−0.274518	+	0.961582i	$0.588518\pi$
$68$	15.5888	1.89042
$69$	−1.00000	−0.120386
$70$	0	0
$71$	15.8575	1.88194	0.940968	−	0.338495i	$-0.109918\pi$
$71$	15.8575	1.88194	0.940968	+	0.338495i	$0.109918\pi$
$72$	2.67481	0.315229
$73$	−4.27776	−0.500674	−0.250337	−	0.968159i	$-0.580541\pi$
$73$	−4.27776	−0.500674	−0.250337	+	0.968159i	$0.580541\pi$
$74$	14.8938	1.73137
$75$	3.46564	0.400177
$76$	−21.9243	−2.51489
$77$	0	0
$78$	−2.07118	−0.234515
$79$	−0.222015	−0.0249787	−0.0124893	−	0.999922i	$-0.503976\pi$
$79$	−0.222015	−0.0249787	−0.0124893	+	0.999922i	$0.503976\pi$
$80$	0.329747	0.0368668
$81$	1.00000	0.111111
$82$	−5.73777	−0.633631
$83$	15.5197	1.70351	0.851753	−	0.523944i	$-0.175540\pi$
$83$	15.5197	1.70351	0.851753	+	0.523944i	$0.175540\pi$
$84$	0	0
$85$	−6.08041	−0.659513
$86$	−0.526369	−0.0567598
$87$	−3.36004	−0.360234
$88$	16.3877	1.74694
$89$	0.559363	0.0592924	0.0296462	−	0.999560i	$-0.490562\pi$
$89$	0.559363	0.0592924	0.0296462	+	0.999560i	$0.490562\pi$
$90$	−2.81806	−0.297049
$91$	0	0
$92$	3.17573	0.331093
$93$	−5.43735	−0.563827
$94$	30.3893	3.13441
$95$	8.55157	0.877373
$96$	5.95524	0.607804
$97$	17.6476	1.79185	0.895923	−	0.444209i	$-0.146515\pi$
$97$	17.6476	1.79185	0.895923	+	0.444209i	$0.146515\pi$
$98$	0	0
$99$	6.12669	0.615756
$100$	−11.0059	−1.10059
$101$	−8.13075	−0.809040	−0.404520	−	0.914529i	$-0.632561\pi$
$101$	−8.13075	−0.809040	−0.404520	+	0.914529i	$0.632561\pi$
$102$	−11.1675	−1.10574
$103$	−14.2795	−1.40700	−0.703500	−	0.710695i	$-0.748381\pi$
$103$	−14.2795	−1.40700	−0.703500	+	0.710695i	$0.748381\pi$
$104$	2.43515	0.238786
$105$	0	0
$106$	9.61609	0.933997
$107$	−6.55764	−0.633951	−0.316975	−	0.948434i	$-0.602667\pi$
$107$	−6.55764	−0.633951	−0.316975	+	0.948434i	$0.602667\pi$
$108$	−3.17573	−0.305585
$109$	18.2358	1.74668	0.873338	−	0.487114i	$-0.161950\pi$
$109$	18.2358	1.74668	0.873338	+	0.487114i	$0.161950\pi$
$110$	−17.2654	−1.64619
$111$	−6.54665	−0.621381
$112$	0	0
$113$	−18.5945	−1.74922	−0.874610	−	0.484828i	$-0.838882\pi$
$113$	−18.5945	−1.74922	−0.874610	+	0.484828i	$0.838882\pi$
$114$	15.7061	1.47101
$115$	−1.23869	−0.115509
$116$	10.6706	0.990738
$117$	0.910400	0.0841665
$118$	−1.71582	−0.157954
$119$	0	0
$120$	3.31327	0.302459
$121$	26.5363	2.41239
$122$	8.13695	0.736685
$123$	2.52207	0.227408
$124$	17.2676	1.55067
$125$	10.4863	0.937926
$126$	0	0
$127$	−4.19807	−0.372518	−0.186259	−	0.982501i	$-0.559636\pi$
$127$	−4.19807	−0.372518	−0.186259	+	0.982501i	$0.559636\pi$
$128$	−17.7010	−1.56456
$129$	0.231369	0.0203709
$130$	−2.56556	−0.225014
$131$	3.83542	0.335102	0.167551	−	0.985863i	$-0.446414\pi$
$131$	3.83542	0.335102	0.167551	+	0.985863i	$0.446414\pi$
$132$	−19.4567	−1.69349
$133$	0	0
$134$	−10.2241	−0.883226
$135$	1.23869	0.106610
$136$	13.1299	1.12588
$137$	7.25647	0.619962	0.309981	−	0.950743i	$-0.399677\pi$
$137$	7.25647	0.619962	0.309981	+	0.950743i	$0.399677\pi$
$138$	−2.27502	−0.193663
$139$	3.52170	0.298707	0.149353	−	0.988784i	$-0.452281\pi$
$139$	3.52170	0.298707	0.149353	+	0.988784i	$0.452281\pi$
$140$	0	0
$141$	−13.3578	−1.12493
$142$	36.0761	3.02744
$143$	5.57774	0.466434
$144$	−0.266205	−0.0221838
$145$	−4.16206	−0.345640
$146$	−9.73201	−0.805427
$147$	0	0
$148$	20.7904	1.70896
$149$	−10.9243	−0.894951	−0.447475	−	0.894296i	$-0.647677\pi$
$149$	−10.9243	−0.894951	−0.447475	+	0.894296i	$0.647677\pi$
$150$	7.88440	0.643759
$151$	−15.8535	−1.29014	−0.645071	−	0.764122i	$-0.723172\pi$
$151$	−15.8535	−1.29014	−0.645071	+	0.764122i	$0.723172\pi$
$152$	−18.4661	−1.49780
$153$	4.90873	0.396847
$154$	0	0
$155$	−6.73521	−0.540985
$156$	−2.89118	−0.231480
$157$	−9.19996	−0.734237	−0.367118	−	0.930174i	$-0.619656\pi$
$157$	−9.19996	−0.734237	−0.367118	+	0.930174i	$0.619656\pi$
$158$	−0.505090	−0.0401828
$159$	−4.22681	−0.335208
$160$	7.37672	0.583181
$161$	0	0
$162$	2.27502	0.178743
$163$	−1.64877	−0.129142	−0.0645709	−	0.997913i	$-0.520568\pi$
$163$	−1.64877	−0.129142	−0.0645709	+	0.997913i	$0.520568\pi$
$164$	−8.00942	−0.625431
$165$	7.58909	0.590810
$166$	35.3076	2.74040
$167$	5.46655	0.423015	0.211507	−	0.977376i	$-0.432163\pi$
$167$	5.46655	0.423015	0.211507	+	0.977376i	$0.432163\pi$
$168$	0	0
$169$	−12.1712	−0.936244
$170$	−13.8331	−1.06095
$171$	−6.90370	−0.527939
$172$	−0.734764	−0.0560252
$173$	−0.509172	−0.0387117	−0.0193558	−	0.999813i	$-0.506162\pi$
$173$	−0.509172	−0.0387117	−0.0193558	+	0.999813i	$0.506162\pi$
$174$	−7.64417	−0.579503
$175$	0	0
$176$	−1.63096	−0.122938
$177$	0.754201	0.0566892
$178$	1.27256	0.0953827
$179$	8.35087	0.624173	0.312087	−	0.950054i	$-0.398972\pi$
$179$	8.35087	0.624173	0.312087	+	0.950054i	$0.398972\pi$
$180$	−3.93376	−0.293205
$181$	−2.42861	−0.180517	−0.0902584	−	0.995918i	$-0.528769\pi$
$181$	−2.42861	−0.180517	−0.0902584	+	0.995918i	$0.528769\pi$
$182$	0	0
$183$	−3.57664	−0.264393
$184$	2.67481	0.197190
$185$	−8.10930	−0.596208
$186$	−12.3701	−0.907020
$187$	30.0743	2.19925
$188$	42.4207	3.09385
$189$	0	0
$190$	19.4550	1.41142
$191$	−14.2931	−1.03421	−0.517107	−	0.855921i	$-0.672991\pi$
$191$	−14.2931	−1.03421	−0.517107	+	0.855921i	$0.672991\pi$
$192$	13.0159	0.939342
$193$	2.09005	0.150445	0.0752225	−	0.997167i	$-0.476033\pi$
$193$	2.09005	0.150445	0.0752225	+	0.997167i	$0.476033\pi$
$194$	40.1488	2.88251
$195$	1.12771	0.0807568
$196$	0	0
$197$	10.8863	0.775614	0.387807	−	0.921741i	$-0.373233\pi$
$197$	10.8863	0.775614	0.387807	+	0.921741i	$0.373233\pi$
$198$	13.9384	0.990556
$199$	3.44053	0.243893	0.121946	−	0.992537i	$-0.461086\pi$
$199$	3.44053	0.243893	0.121946	+	0.992537i	$0.461086\pi$
$200$	−9.26992	−0.655482
$201$	4.49406	0.316986
$202$	−18.4976	−1.30149
$203$	0	0
$204$	−15.5888	−1.09143
$205$	3.12408	0.218195
$206$	−32.4862	−2.26342
$207$	1.00000	0.0695048
$208$	−0.242353	−0.0168042
$209$	−42.2968	−2.92573
$210$	0	0
$211$	11.6690	0.803324	0.401662	−	0.915788i	$-0.368433\pi$
$211$	11.6690	0.803324	0.401662	+	0.915788i	$0.368433\pi$
$212$	13.4232	0.921909
$213$	−15.8575	−1.08654
$214$	−14.9188	−1.01983
$215$	0.286595	0.0195456
$216$	−2.67481	−0.181998
$217$	0	0
$218$	41.4869	2.80985
$219$	4.27776	0.289065
$220$	−24.1009	−1.62488
$221$	4.46891	0.300611
$222$	−14.8938	−0.999605
$223$	2.48168	0.166185	0.0830926	−	0.996542i	$-0.473520\pi$
$223$	2.48168	0.166185	0.0830926	+	0.996542i	$0.473520\pi$
$224$	0	0
$225$	−3.46564	−0.231043
$226$	−42.3028	−2.81394
$227$	−22.0495	−1.46348	−0.731738	−	0.681586i	$-0.761290\pi$
$227$	−22.0495	−1.46348	−0.731738	+	0.681586i	$0.761290\pi$
$228$	21.9243	1.45197
$229$	14.7467	0.974488	0.487244	−	0.873266i	$-0.338002\pi$
$229$	14.7467	0.974488	0.487244	+	0.873266i	$0.338002\pi$
$230$	−2.81806	−0.185817
$231$	0	0
$232$	8.98747	0.590056
$233$	−3.41615	−0.223799	−0.111900	−	0.993720i	$-0.535694\pi$
$233$	−3.41615	−0.223799	−0.111900	+	0.993720i	$0.535694\pi$
$234$	2.07118	0.135397
$235$	−16.5462	−1.07936
$236$	−2.39514	−0.155910
$237$	0.222015	0.0144215
$238$	0	0
$239$	−10.8751	−0.703453	−0.351727	−	0.936103i	$-0.614405\pi$
$239$	−10.8751	−0.703453	−0.351727	+	0.936103i	$0.614405\pi$
$240$	−0.329747	−0.0212851
$241$	−21.3965	−1.37827	−0.689134	−	0.724634i	$-0.742009\pi$
$241$	−21.3965	−1.37827	−0.689134	+	0.724634i	$0.742009\pi$
$242$	60.3708	3.88078
$243$	−1.00000	−0.0641500
$244$	11.3585	0.727150
$245$	0	0
$246$	5.73777	0.365827
$247$	−6.28513	−0.399913
$248$	14.5439	0.923537
$249$	−15.5197	−0.983520
$250$	23.8566	1.50883
$251$	4.68560	0.295752	0.147876	−	0.989006i	$-0.452756\pi$
$251$	4.68560	0.295752	0.147876	+	0.989006i	$0.452756\pi$
$252$	0	0
$253$	6.12669	0.385182
$254$	−9.55070	−0.599264
$255$	6.08041	0.380770
$256$	−14.2383	−0.889897
$257$	−27.0174	−1.68530	−0.842650	−	0.538462i	$-0.819006\pi$
$257$	−27.0174	−1.68530	−0.842650	+	0.538462i	$0.819006\pi$
$258$	0.526369	0.0327703
$259$	0	0
$260$	−3.58129	−0.222102
$261$	3.36004	0.207981
$262$	8.72566	0.539073
$263$	−20.0063	−1.23364	−0.616822	−	0.787103i	$-0.711580\pi$
$263$	−20.0063	−1.23364	−0.616822	+	0.787103i	$0.711580\pi$
$264$	−16.3877	−1.00859
$265$	−5.23572	−0.321628
$266$	0	0
$267$	−0.559363	−0.0342325
$268$	−14.2719	−0.871795
$269$	−6.28072	−0.382942	−0.191471	−	0.981498i	$-0.561326\pi$
$269$	−6.28072	−0.382942	−0.191471	+	0.981498i	$0.561326\pi$
$270$	2.81806	0.171501
$271$	25.4203	1.54417	0.772085	−	0.635519i	$-0.219214\pi$
$271$	25.4203	1.54417	0.772085	+	0.635519i	$0.219214\pi$
$272$	−1.30673	−0.0792321
$273$	0	0
$274$	16.5086	0.997323
$275$	−21.2329	−1.28039
$276$	−3.17573	−0.191156
$277$	−15.1411	−0.909739	−0.454869	−	0.890558i	$-0.650314\pi$
$277$	−15.1411	−0.909739	−0.454869	+	0.890558i	$0.650314\pi$
$278$	8.01195	0.480525
$279$	5.43735	0.325526
$280$	0	0
$281$	−27.3149	−1.62947	−0.814736	−	0.579833i	$-0.803118\pi$
$281$	−27.3149	−1.62947	−0.814736	+	0.579833i	$0.803118\pi$
$282$	−30.3893	−1.80965
$283$	8.54261	0.507805	0.253903	−	0.967230i	$-0.418286\pi$
$283$	8.54261	0.507805	0.253903	+	0.967230i	$0.418286\pi$
$284$	50.3590	2.98826
$285$	−8.55157	−0.506551
$286$	12.6895	0.750345
$287$	0	0
$288$	−5.95524	−0.350916
$289$	7.09562	0.417390
$290$	−9.46879	−0.556026
$291$	−17.6476	−1.03452
$292$	−13.5850	−0.795003
$293$	13.6153	0.795413	0.397707	−	0.917513i	$-0.369806\pi$
$293$	13.6153	0.795413	0.397707	+	0.917513i	$0.369806\pi$
$294$	0	0
$295$	0.934224	0.0543926
$296$	17.5110	1.01781
$297$	−6.12669	−0.355507
$298$	−24.8530	−1.43969
$299$	0.910400	0.0526498
$300$	11.0059	0.635427
$301$	0	0
$302$	−36.0672	−2.07543
$303$	8.13075	0.467100
$304$	1.83780	0.105405
$305$	−4.43037	−0.253682
$306$	11.1675	0.638402
$307$	−1.94494	−0.111003	−0.0555017	−	0.998459i	$-0.517676\pi$
$307$	−1.94494	−0.111003	−0.0555017	+	0.998459i	$0.517676\pi$
$308$	0	0
$309$	14.2795	0.812332
$310$	−15.3228	−0.870275
$311$	−14.2020	−0.805319	−0.402660	−	0.915350i	$-0.631914\pi$
$311$	−14.2020	−0.805319	−0.402660	+	0.915350i	$0.631914\pi$
$312$	−2.43515	−0.137863
$313$	−4.20457	−0.237656	−0.118828	−	0.992915i	$-0.537914\pi$
$313$	−4.20457	−0.237656	−0.118828	+	0.992915i	$0.537914\pi$
$314$	−20.9301	−1.18115
$315$	0	0
$316$	−0.705061	−0.0396628
$317$	0.738433	0.0414745	0.0207373	−	0.999785i	$-0.493399\pi$
$317$	0.738433	0.0414745	0.0207373	+	0.999785i	$0.493399\pi$
$318$	−9.61609	−0.539243
$319$	20.5859	1.15259
$320$	16.1227	0.901287
$321$	6.55764	0.366012
$322$	0	0
$323$	−33.8884	−1.88560
$324$	3.17573	0.176429
$325$	−3.15512	−0.175014
$326$	−3.75100	−0.207748
$327$	−18.2358	−1.00844
$328$	−6.74607	−0.372489
$329$	0	0
$330$	17.2654	0.950427
$331$	−20.5989	−1.13222	−0.566108	−	0.824331i	$-0.691552\pi$
$331$	−20.5989	−1.13222	−0.566108	+	0.824331i	$0.691552\pi$
$332$	49.2863	2.70494
$333$	6.54665	0.358754
$334$	12.4365	0.680497
$335$	5.56676	0.304145
$336$	0	0
$337$	−27.9032	−1.51999	−0.759993	−	0.649932i	$-0.774797\pi$
$337$	−27.9032	−1.51999	−0.759993	+	0.649932i	$0.774797\pi$
$338$	−27.6897	−1.50612
$339$	18.5945	1.00991
$340$	−19.3097	−1.04722
$341$	33.3130	1.80400
$342$	−15.7061	−0.849287
$343$	0	0
$344$	−0.618867	−0.0333671
$345$	1.23869	0.0666890
$346$	−1.15838	−0.0622748
$347$	−2.08533	−0.111946	−0.0559732	−	0.998432i	$-0.517826\pi$
$347$	−2.08533	−0.111946	−0.0559732	+	0.998432i	$0.517826\pi$
$348$	−10.6706	−0.572003
$349$	−4.04322	−0.216429	−0.108214	−	0.994128i	$-0.534513\pi$
$349$	−4.04322	−0.216429	−0.108214	+	0.994128i	$0.534513\pi$
$350$	0	0
$351$	−0.910400	−0.0485936
$352$	−36.4859	−1.94471
$353$	−4.50793	−0.239933	−0.119966	−	0.992778i	$-0.538279\pi$
$353$	−4.50793	−0.239933	−0.119966	+	0.992778i	$0.538279\pi$
$354$	1.71582	0.0911950
$355$	−19.6426	−1.04252
$356$	1.77639	0.0941483
$357$	0	0
$358$	18.9984	1.00410
$359$	−20.8138	−1.09851	−0.549254	−	0.835656i	$-0.685088\pi$
$359$	−20.8138	−1.09851	−0.549254	+	0.835656i	$0.685088\pi$
$360$	−3.31327	−0.174625
$361$	28.6611	1.50848
$362$	−5.52513	−0.290394
$363$	−26.5363	−1.39280
$364$	0	0
$365$	5.29884	0.277354
$366$	−8.13695	−0.425325
$367$	8.57218	0.447464	0.223732	−	0.974651i	$-0.428176\pi$
$367$	8.57218	0.447464	0.223732	+	0.974651i	$0.428176\pi$
$368$	−0.266205	−0.0138769
$369$	−2.52207	−0.131294
$370$	−18.4488	−0.959109
$371$	0	0
$372$	−17.2676	−0.895281
$373$	−1.45507	−0.0753407	−0.0376703	−	0.999290i	$-0.511994\pi$
$373$	−1.45507	−0.0753407	−0.0376703	+	0.999290i	$0.511994\pi$
$374$	68.4196	3.53790
$375$	−10.4863	−0.541512
$376$	35.7295	1.84261
$377$	3.05898	0.157546
$378$	0	0
$379$	−2.61370	−0.134257	−0.0671283	−	0.997744i	$-0.521384\pi$
$379$	−2.61370	−0.134257	−0.0671283	+	0.997744i	$0.521384\pi$
$380$	27.1575	1.39315
$381$	4.19807	0.215074
$382$	−32.5172	−1.66372
$383$	−11.3203	−0.578438	−0.289219	−	0.957263i	$-0.593396\pi$
$383$	−11.3203	−0.578438	−0.289219	+	0.957263i	$0.593396\pi$
$384$	17.7010	0.903300
$385$	0	0
$386$	4.75491	0.242018
$387$	−0.231369	−0.0117611
$388$	56.0441	2.84521
$389$	−30.9710	−1.57029	−0.785147	−	0.619310i	$-0.787412\pi$
$389$	−30.9710	−1.57029	−0.785147	+	0.619310i	$0.787412\pi$
$390$	2.56556	0.129912
$391$	4.90873	0.248245
$392$	0	0
$393$	−3.83542	−0.193471
$394$	24.7665	1.24772
$395$	0.275009	0.0138372
$396$	19.4567	0.977736
$397$	25.0396	1.25670	0.628350	−	0.777931i	$-0.283731\pi$
$397$	25.0396	1.25670	0.628350	+	0.777931i	$0.283731\pi$
$398$	7.82728	0.392346
$399$	0	0
$400$	0.922571	0.0461285
$401$	34.9745	1.74654	0.873272	−	0.487234i	$-0.161994\pi$
$401$	34.9745	1.74654	0.873272	+	0.487234i	$0.161994\pi$
$402$	10.2241	0.509931
$403$	4.95017	0.246585
$404$	−25.8211	−1.28465
$405$	−1.23869	−0.0615512
$406$	0	0
$407$	40.1093	1.98815
$408$	−13.1299	−0.650028
$409$	−24.1704	−1.19515	−0.597576	−	0.801812i	$-0.703869\pi$
$409$	−24.1704	−1.19515	−0.597576	+	0.801812i	$0.703869\pi$
$410$	7.10735	0.351007
$411$	−7.25647	−0.357935
$412$	−45.3478	−2.23413
$413$	0	0
$414$	2.27502	0.111811
$415$	−19.2241	−0.943675
$416$	−5.42165	−0.265818
$417$	−3.52170	−0.172458
$418$	−96.2263	−4.70658
$419$	−19.9385	−0.974059	−0.487029	−	0.873386i	$-0.661920\pi$
$419$	−19.9385	−0.974059	−0.487029	+	0.873386i	$0.661920\pi$
$420$	0	0
$421$	−36.1003	−1.75942	−0.879711	−	0.475508i	$-0.842264\pi$
$421$	−36.1003	−1.75942	−0.879711	+	0.475508i	$0.842264\pi$
$422$	26.5471	1.29229
$423$	13.3578	0.649478
$424$	11.3059	0.549063
$425$	−17.0119	−0.825197
$426$	−36.0761	−1.74789
$427$	0	0
$428$	−20.8253	−1.00663
$429$	−5.57774	−0.269296
$430$	0.652010	0.0314427
$431$	10.8677	0.523479	0.261739	−	0.965139i	$-0.415704\pi$
$431$	10.8677	0.523479	0.261739	+	0.965139i	$0.415704\pi$
$432$	0.266205	0.0128078
$433$	−26.1898	−1.25860	−0.629301	−	0.777161i	$-0.716659\pi$
$433$	−26.1898	−1.25860	−0.629301	+	0.777161i	$0.716659\pi$
$434$	0	0
$435$	4.16206	0.199556
$436$	57.9121	2.77349
$437$	−6.90370	−0.330249
$438$	9.73201	0.465013
$439$	−31.4062	−1.49894	−0.749468	−	0.662041i	$-0.769690\pi$
$439$	−31.4062	−1.49894	−0.749468	+	0.662041i	$0.769690\pi$
$440$	−20.2994	−0.967735
$441$	0	0
$442$	10.1669	0.483589
$443$	−9.07886	−0.431350	−0.215675	−	0.976465i	$-0.569195\pi$
$443$	−9.07886	−0.431350	−0.215675	+	0.976465i	$0.569195\pi$
$444$	−20.7904	−0.986668
$445$	−0.692880	−0.0328457
$446$	5.64587	0.267340
$447$	10.9243	0.516700
$448$	0	0
$449$	23.9369	1.12965	0.564825	−	0.825211i	$-0.308944\pi$
$449$	23.9369	1.12965	0.564825	+	0.825211i	$0.308944\pi$
$450$	−7.88440	−0.371674
$451$	−15.4520	−0.727604
$452$	−59.0510	−2.77752
$453$	15.8535	0.744864
$454$	−50.1631	−2.35427
$455$	0	0
$456$	18.4661	0.864754
$457$	−4.81036	−0.225019	−0.112509	−	0.993651i	$-0.535889\pi$
$457$	−4.81036	−0.225019	−0.112509	+	0.993651i	$0.535889\pi$
$458$	33.5490	1.56764
$459$	−4.90873	−0.229120
$460$	−3.93376	−0.183412
$461$	23.4694	1.09308	0.546540	−	0.837433i	$-0.315945\pi$
$461$	23.4694	1.09308	0.546540	+	0.837433i	$0.315945\pi$
$462$	0	0
$463$	23.4343	1.08908	0.544542	−	0.838733i	$-0.316703\pi$
$463$	23.4343	1.08908	0.544542	+	0.838733i	$0.316703\pi$
$464$	−0.894460	−0.0415243
$465$	6.73521	0.312338
$466$	−7.77182	−0.360022
$467$	13.4506	0.622417	0.311209	−	0.950342i	$-0.399266\pi$
$467$	13.4506	0.622417	0.311209	+	0.950342i	$0.399266\pi$
$468$	2.89118	0.133645
$469$	0	0
$470$	−37.6430	−1.73634
$471$	9.19996	0.423912
$472$	−2.01734	−0.0928557
$473$	−1.41752	−0.0651778
$474$	0.505090	0.0231996
$475$	23.9257	1.09779
$476$	0	0
$477$	4.22681	0.193532
$478$	−24.7411	−1.13163
$479$	6.76164	0.308947	0.154474	−	0.987997i	$-0.450632\pi$
$479$	6.76164	0.308947	0.154474	+	0.987997i	$0.450632\pi$
$480$	−7.37672	−0.336700
$481$	5.96007	0.271756
$482$	−48.6774	−2.21720
$483$	0	0
$484$	84.2722	3.83055
$485$	−21.8600	−0.992612
$486$	−2.27502	−0.103197
$487$	19.8666	0.900241	0.450121	−	0.892968i	$-0.351381\pi$
$487$	19.8666	0.900241	0.450121	+	0.892968i	$0.351381\pi$
$488$	9.56684	0.433071
$489$	1.64877	0.0745601
$490$	0	0
$491$	4.20340	0.189697	0.0948483	−	0.995492i	$-0.469763\pi$
$491$	4.20340	0.189697	0.0948483	+	0.995492i	$0.469763\pi$
$492$	8.00942	0.361093
$493$	16.4935	0.742831
$494$	−14.2988	−0.643334
$495$	−7.58909	−0.341104
$496$	−1.44745	−0.0649925
$497$	0	0
$498$	−35.3076	−1.58217
$499$	6.94840	0.311053	0.155527	−	0.987832i	$-0.450293\pi$
$499$	6.94840	0.311053	0.155527	+	0.987832i	$0.450293\pi$
$500$	33.3017	1.48930
$501$	−5.46655	−0.244228
$502$	10.6598	0.475772
$503$	5.90346	0.263222	0.131611	−	0.991301i	$-0.457985\pi$
$503$	5.90346	0.263222	0.131611	+	0.991301i	$0.457985\pi$
$504$	0	0
$505$	10.0715	0.448176
$506$	13.9384	0.619636
$507$	12.1712	0.540541
$508$	−13.3319	−0.591508
$509$	−26.6647	−1.18189	−0.590947	−	0.806711i	$-0.701246\pi$
$509$	−26.6647	−1.18189	−0.590947	+	0.806711i	$0.701246\pi$
$510$	13.8331	0.612539
$511$	0	0
$512$	3.00941	0.132998
$513$	6.90370	0.304806
$514$	−61.4652	−2.71111
$515$	17.6879	0.779423
$516$	0.734764	0.0323462
$517$	81.8390	3.59928
$518$	0	0
$519$	0.509172	0.0223502
$520$	−3.01640	−0.132278
$521$	5.31311	0.232771	0.116386	−	0.993204i	$-0.462869\pi$
$521$	5.31311	0.232771	0.116386	+	0.993204i	$0.462869\pi$
$522$	7.64417	0.334576
$523$	17.0386	0.745047	0.372524	−	0.928023i	$-0.378493\pi$
$523$	17.0386	0.745047	0.372524	+	0.928023i	$0.378493\pi$
$524$	12.1802	0.532096
$525$	0	0
$526$	−45.5149	−1.98454
$527$	26.6905	1.16266
$528$	1.63096	0.0709783
$529$	1.00000	0.0434783
$530$	−11.9114	−0.517398
$531$	−0.754201	−0.0327295
$532$	0	0
$533$	−2.29610	−0.0994550
$534$	−1.27256	−0.0550692
$535$	8.12290	0.351184
$536$	−12.0207	−0.519217
$537$	−8.35087	−0.360367
$538$	−14.2888	−0.616033
$539$	0	0
$540$	3.93376	0.169282
$541$	38.7960	1.66797	0.833986	−	0.551786i	$-0.186053\pi$
$541$	38.7960	1.66797	0.833986	+	0.551786i	$0.186053\pi$
$542$	57.8317	2.48408
$543$	2.42861	0.104221
$544$	−29.2327	−1.25334
$545$	−22.5886	−0.967590
$546$	0	0
$547$	29.7356	1.27140	0.635701	−	0.771935i	$-0.280711\pi$
$547$	29.7356	1.27140	0.635701	+	0.771935i	$0.280711\pi$
$548$	23.0446	0.984415
$549$	3.57664	0.152647
$550$	−48.3053	−2.05975
$551$	−23.1967	−0.988214
$552$	−2.67481	−0.113847
$553$	0	0
$554$	−34.4463	−1.46348
$555$	8.10930	0.344221
$556$	11.1840	0.474306
$557$	−33.5645	−1.42217	−0.711086	−	0.703105i	$-0.751796\pi$
$557$	−33.5645	−1.42217	−0.711086	+	0.703105i	$0.751796\pi$
$558$	12.3701	0.523668
$559$	−0.210638	−0.00890904
$560$	0	0
$561$	−30.0743	−1.26974
$562$	−62.1421	−2.62130
$563$	−30.5532	−1.28766	−0.643831	−	0.765167i	$-0.722656\pi$
$563$	−30.5532	−1.28766	−0.643831	+	0.765167i	$0.722656\pi$
$564$	−42.4207	−1.78623
$565$	23.0328	0.968999
$566$	19.4346	0.816898
$567$	0	0
$568$	42.4157	1.77972
$569$	42.1160	1.76559	0.882797	−	0.469754i	$-0.155657\pi$
$569$	42.1160	1.76559	0.882797	+	0.469754i	$0.155657\pi$
$570$	−19.4550	−0.814881
$571$	6.51703	0.272729	0.136365	−	0.990659i	$-0.456458\pi$
$571$	6.51703	0.272729	0.136365	+	0.990659i	$0.456458\pi$
$572$	17.7134	0.740634
$573$	14.2931	0.597104
$574$	0	0
$575$	−3.46564	−0.144527
$576$	−13.0159	−0.542329
$577$	41.5356	1.72915	0.864575	−	0.502503i	$-0.167588\pi$
$577$	41.5356	1.72915	0.864575	+	0.502503i	$0.167588\pi$
$578$	16.1427	0.671448
$579$	−2.09005	−0.0868594
$580$	−13.2176	−0.548830
$581$	0	0
$582$	−40.1488	−1.66422
$583$	25.8963	1.07252
$584$	−11.4422	−0.473482
$585$	−1.12771	−0.0466249
$586$	30.9751	1.27957
$587$	12.6109	0.520506	0.260253	−	0.965540i	$-0.416194\pi$
$587$	12.6109	0.520506	0.260253	+	0.965540i	$0.416194\pi$
$588$	0	0
$589$	−37.5379	−1.54672
$590$	2.12538	0.0875005
$591$	−10.8863	−0.447801
$592$	−1.74275	−0.0716267
$593$	10.3859	0.426498	0.213249	−	0.976998i	$-0.431596\pi$
$593$	10.3859	0.426498	0.213249	+	0.976998i	$0.431596\pi$
$594$	−13.9384	−0.571898
$595$	0	0
$596$	−34.6925	−1.42106
$597$	−3.44053	−0.140812
$598$	2.07118	0.0846969
$599$	28.0358	1.14551	0.572757	−	0.819725i	$-0.305874\pi$
$599$	28.0358	1.14551	0.572757	+	0.819725i	$0.305874\pi$
$600$	9.26992	0.378443
$601$	3.81190	0.155491	0.0777454	−	0.996973i	$-0.475228\pi$
$601$	3.81190	0.155491	0.0777454	+	0.996973i	$0.475228\pi$
$602$	0	0
$603$	−4.49406	−0.183012
$604$	−50.3465	−2.04857
$605$	−32.8704	−1.33637
$606$	18.4976	0.751416
$607$	0.159930	0.00649136	0.00324568	−	0.999995i	$-0.498967\pi$
$607$	0.159930	0.00649136	0.00324568	+	0.999995i	$0.498967\pi$
$608$	41.1132	1.66736
$609$	0	0
$610$	−10.0792	−0.408094
$611$	12.1609	0.491979
$612$	15.5888	0.630140
$613$	−16.5339	−0.667797	−0.333899	−	0.942609i	$-0.608364\pi$
$613$	−16.5339	−0.667797	−0.333899	+	0.942609i	$0.608364\pi$
$614$	−4.42477	−0.178569
$615$	−3.12408	−0.125975
$616$	0	0
$617$	35.8727	1.44418	0.722091	−	0.691798i	$-0.243181\pi$
$617$	35.8727	1.44418	0.722091	+	0.691798i	$0.243181\pi$
$618$	32.4862	1.30679
$619$	−36.6950	−1.47490	−0.737448	−	0.675404i	$-0.763969\pi$
$619$	−36.6950	−1.47490	−0.737448	+	0.675404i	$0.763969\pi$
$620$	−21.3892	−0.859011
$621$	−1.00000	−0.0401286
$622$	−32.3098	−1.29550
$623$	0	0
$624$	0.242353	0.00970189
$625$	4.33883	0.173553
$626$	−9.56548	−0.382314
$627$	42.2968	1.68917
$628$	−29.2166	−1.16587
$629$	32.1358	1.28134
$630$	0	0
$631$	12.3865	0.493098	0.246549	−	0.969130i	$-0.420703\pi$
$631$	12.3865	0.493098	0.246549	+	0.969130i	$0.420703\pi$
$632$	−0.593849	−0.0236220
$633$	−11.6690	−0.463799
$634$	1.67995	0.0667194
$635$	5.20012	0.206360
$636$	−13.4232	−0.532264
$637$	0	0
$638$	46.8335	1.85415
$639$	15.8575	0.627312
$640$	21.9261	0.866705
$641$	27.0383	1.06795	0.533974	−	0.845501i	$-0.320698\pi$
$641$	27.0383	1.06795	0.533974	+	0.845501i	$0.320698\pi$
$642$	14.9188	0.588797
$643$	−27.4371	−1.08201	−0.541006	−	0.841019i	$-0.681956\pi$
$643$	−27.4371	−1.08201	−0.541006	+	0.841019i	$0.681956\pi$
$644$	0	0
$645$	−0.286595	−0.0112847
$646$	−77.0969	−3.03334
$647$	0.875536	0.0344209	0.0172104	−	0.999852i	$-0.494521\pi$
$647$	0.875536	0.0344209	0.0172104	+	0.999852i	$0.494521\pi$
$648$	2.67481	0.105076
$649$	−4.62075	−0.181380
$650$	−7.17796	−0.281543
$651$	0	0
$652$	−5.23606	−0.205060
$653$	−15.5487	−0.608466	−0.304233	−	0.952598i	$-0.598400\pi$
$653$	−15.5487	−0.608466	−0.304233	+	0.952598i	$0.598400\pi$
$654$	−41.4869	−1.62227
$655$	−4.75091	−0.185633
$656$	0.671389	0.0262133
$657$	−4.27776	−0.166891
$658$	0	0
$659$	21.1989	0.825794	0.412897	−	0.910778i	$-0.364517\pi$
$659$	21.1989	0.825794	0.412897	+	0.910778i	$0.364517\pi$
$660$	24.1009	0.938126
$661$	−8.65093	−0.336482	−0.168241	−	0.985746i	$-0.553809\pi$
$661$	−8.65093	−0.336482	−0.168241	+	0.985746i	$0.553809\pi$
$662$	−46.8629	−1.82138
$663$	−4.46891	−0.173558
$664$	41.5122	1.61099
$665$	0	0
$666$	14.8938	0.577122
$667$	3.36004	0.130101
$668$	17.3603	0.671690
$669$	−2.48168	−0.0959471
$670$	12.6645	0.489273
$671$	21.9130	0.845942
$672$	0	0
$673$	−9.05137	−0.348905	−0.174452	−	0.984666i	$-0.555816\pi$
$673$	−9.05137	−0.348905	−0.174452	+	0.984666i	$0.555816\pi$
$674$	−63.4805	−2.44518
$675$	3.46564	0.133392
$676$	−38.6523	−1.48663
$677$	−16.8996	−0.649503	−0.324751	−	0.945799i	$-0.605281\pi$
$677$	−16.8996	−0.649503	−0.324751	+	0.945799i	$0.605281\pi$
$678$	42.3028	1.62463
$679$	0	0
$680$	−16.2639	−0.623694
$681$	22.0495	0.844938
$682$	75.7878	2.90206
$683$	22.7419	0.870194	0.435097	−	0.900384i	$-0.356714\pi$
$683$	22.7419	0.870194	0.435097	+	0.900384i	$0.356714\pi$
$684$	−21.9243	−0.838296
$685$	−8.98854	−0.343434
$686$	0	0
$687$	−14.7467	−0.562621
$688$	0.0615915	0.00234815
$689$	3.84809	0.146600
$690$	2.81806	0.107282
$691$	35.1619	1.33762	0.668810	−	0.743433i	$-0.266804\pi$
$691$	35.1619	1.33762	0.668810	+	0.743433i	$0.266804\pi$
$692$	−1.61699	−0.0614689
$693$	0	0
$694$	−4.74418	−0.180086
$695$	−4.36231	−0.165472
$696$	−8.98747	−0.340669
$697$	−12.3802	−0.468933
$698$	−9.19842	−0.348165
$699$	3.41615	0.129211
$700$	0	0
$701$	−35.0402	−1.32345	−0.661725	−	0.749746i	$-0.730175\pi$
$701$	−35.0402	−1.32345	−0.661725	+	0.749746i	$0.730175\pi$
$702$	−2.07118	−0.0781717
$703$	−45.1961	−1.70460
$704$	−79.7444	−3.00548
$705$	16.5462	0.623166
$706$	−10.2556	−0.385976
$707$	0	0
$708$	2.39514	0.0900147
$709$	−49.6644	−1.86519	−0.932593	−	0.360930i	$-0.882459\pi$
$709$	−49.6644	−1.86519	−0.932593	+	0.360930i	$0.882459\pi$
$710$	−44.6873	−1.67708
$711$	−0.222015	−0.00832623
$712$	1.49619	0.0560721
$713$	5.43735	0.203630
$714$	0	0
$715$	−6.90911	−0.258386
$716$	26.5201	0.991103
$717$	10.8751	0.406139
$718$	−47.3518	−1.76715
$719$	3.54087	0.132052	0.0660261	−	0.997818i	$-0.478968\pi$
$719$	3.54087	0.132052	0.0660261	+	0.997818i	$0.478968\pi$
$720$	0.329747	0.0122889
$721$	0	0
$722$	65.2046	2.42667
$723$	21.3965	0.795743
$724$	−7.71259	−0.286636
$725$	−11.6447	−0.432473
$726$	−60.3708	−2.24057
$727$	7.64097	0.283388	0.141694	−	0.989911i	$-0.454745\pi$
$727$	7.64097	0.283388	0.141694	+	0.989911i	$0.454745\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	12.0550	0.446175
$731$	−1.13573	−0.0420063
$732$	−11.3585	−0.419820
$733$	20.1456	0.744094	0.372047	−	0.928214i	$-0.378656\pi$
$733$	20.1456	0.744094	0.372047	+	0.928214i	$0.378656\pi$
$734$	19.5019	0.719829
$735$	0	0
$736$	−5.95524	−0.219513
$737$	−27.5337	−1.01422
$738$	−5.73777	−0.211210
$739$	38.9079	1.43125	0.715626	−	0.698484i	$-0.246142\pi$
$739$	38.9079	1.43125	0.715626	+	0.698484i	$0.246142\pi$
$740$	−25.7529	−0.946697
$741$	6.28513	0.230890
$742$	0	0
$743$	−12.5828	−0.461617	−0.230809	−	0.972999i	$-0.574137\pi$
$743$	−12.5828	−0.461617	−0.230809	+	0.972999i	$0.574137\pi$
$744$	−14.5439	−0.533205
$745$	13.5318	0.495768
$746$	−3.31032	−0.121199
$747$	15.5197	0.567835
$748$	95.5077	3.49211
$749$	0	0
$750$	−23.8566	−0.871122
$751$	−23.2255	−0.847511	−0.423756	−	0.905777i	$-0.639288\pi$
$751$	−23.2255	−0.847511	−0.423756	+	0.905777i	$0.639288\pi$
$752$	−3.55591	−0.129671
$753$	−4.68560	−0.170753
$754$	6.95925	0.253441
$755$	19.6377	0.714688
$756$	0	0
$757$	−31.3210	−1.13838	−0.569191	−	0.822205i	$-0.692744\pi$
$757$	−31.3210	−1.13838	−0.569191	+	0.822205i	$0.692744\pi$
$758$	−5.94622	−0.215976
$759$	−6.12669	−0.222385
$760$	22.8738	0.829721
$761$	−25.8770	−0.938040	−0.469020	−	0.883188i	$-0.655393\pi$
$761$	−25.8770	−0.938040	−0.469020	+	0.883188i	$0.655393\pi$
$762$	9.55070	0.345985
$763$	0	0
$764$	−45.3911	−1.64219
$765$	−6.08041	−0.219838
$766$	−25.7538	−0.930524
$767$	−0.686624	−0.0247926
$768$	14.2383	0.513782
$769$	2.66864	0.0962338	0.0481169	−	0.998842i	$-0.484678\pi$
$769$	2.66864	0.0962338	0.0481169	+	0.998842i	$0.484678\pi$
$770$	0	0
$771$	27.0174	0.973008
$772$	6.63742	0.238886
$773$	−5.78286	−0.207995	−0.103998	−	0.994578i	$-0.533163\pi$
$773$	−5.78286	−0.207995	−0.103998	+	0.994578i	$0.533163\pi$
$774$	−0.526369	−0.0189199
$775$	−18.8439	−0.676893
$776$	47.2041	1.69453
$777$	0	0
$778$	−70.4598	−2.52611
$779$	17.4116	0.623837
$780$	3.58129	0.128231
$781$	97.1539	3.47644
$782$	11.1675	0.399348
$783$	−3.36004	−0.120078
$784$	0	0
$785$	11.3959	0.406738
$786$	−8.72566	−0.311234
$787$	−7.55455	−0.269291	−0.134645	−	0.990894i	$-0.542990\pi$
$787$	−7.55455	−0.269291	−0.134645	+	0.990894i	$0.542990\pi$
$788$	34.5718	1.23157
$789$	20.0063	0.712244
$790$	0.625652	0.0222597
$791$	0	0
$792$	16.3877	0.582313
$793$	3.25618	0.115630
$794$	56.9656	2.02163
$795$	5.23572	0.185692
$796$	10.9262	0.387269
$797$	−17.6173	−0.624035	−0.312018	−	0.950076i	$-0.601005\pi$
$797$	−17.6173	−0.624035	−0.312018	+	0.950076i	$0.601005\pi$
$798$	0	0
$799$	65.5698	2.31969
$800$	20.6387	0.729689
$801$	0.559363	0.0197641
$802$	79.5678	2.80964
$803$	−26.2085	−0.924879
$804$	14.2719	0.503331
$805$	0	0
$806$	11.2617	0.396678
$807$	6.28072	0.221092
$808$	−21.7482	−0.765099
$809$	23.2841	0.818624	0.409312	−	0.912394i	$-0.365769\pi$
$809$	23.2841	0.818624	0.409312	+	0.912394i	$0.365769\pi$
$810$	−2.81806	−0.0990164
$811$	35.2946	1.23936	0.619680	−	0.784855i	$-0.287262\pi$
$811$	35.2946	1.23936	0.619680	+	0.784855i	$0.287262\pi$
$812$	0	0
$813$	−25.4203	−0.891527
$814$	91.2496	3.19830
$815$	2.04232	0.0715395
$816$	1.30673	0.0457447
$817$	1.59730	0.0558824
$818$	−54.9883	−1.92262
$819$	0	0
$820$	9.92122	0.346464
$821$	1.86839	0.0652072	0.0326036	−	0.999468i	$-0.489620\pi$
$821$	1.86839	0.0652072	0.0326036	+	0.999468i	$0.489620\pi$
$822$	−16.5086	−0.575804
$823$	−19.4504	−0.677997	−0.338999	−	0.940787i	$-0.610088\pi$
$823$	−19.4504	−0.677997	−0.338999	+	0.940787i	$0.610088\pi$
$824$	−38.1949	−1.33058
$825$	21.2329	0.739234
$826$	0	0
$827$	11.1883	0.389055	0.194527	−	0.980897i	$-0.437683\pi$
$827$	11.1883	0.389055	0.194527	+	0.980897i	$0.437683\pi$
$828$	3.17573	0.110364
$829$	0.598968	0.0208030	0.0104015	−	0.999946i	$-0.496689\pi$
$829$	0.598968	0.0208030	0.0104015	+	0.999946i	$0.496689\pi$
$830$	−43.7353	−1.51808
$831$	15.1411	0.525238
$832$	−11.8497	−0.410814
$833$	0	0
$834$	−8.01195	−0.277431
$835$	−6.77139	−0.234333
$836$	−134.323	−4.64567
$837$	−5.43735	−0.187942
$838$	−45.3605	−1.56695
$839$	−43.0684	−1.48689	−0.743443	−	0.668799i	$-0.766809\pi$
$839$	−43.0684	−1.48689	−0.743443	+	0.668799i	$0.766809\pi$
$840$	0	0
$841$	−17.7101	−0.610694
$842$	−82.1291	−2.83036
$843$	27.3149	0.940776
$844$	37.0574	1.27557
$845$	15.0764	0.518642
$846$	30.3893	1.04480
$847$	0	0
$848$	−1.12520	−0.0386395
$849$	−8.54261	−0.293182
$850$	−38.7024	−1.32748
$851$	6.54665	0.224416
$852$	−50.3590	−1.72527
$853$	32.0125	1.09609	0.548043	−	0.836450i	$-0.315373\pi$
$853$	32.0125	1.09609	0.548043	+	0.836450i	$0.315373\pi$
$854$	0	0
$855$	8.55157	0.292458
$856$	−17.5404	−0.599520
$857$	−45.7684	−1.56342	−0.781710	−	0.623642i	$-0.785652\pi$
$857$	−45.7684	−1.56342	−0.781710	+	0.623642i	$0.785652\pi$
$858$	−12.6895	−0.433212
$859$	−43.2491	−1.47564	−0.737820	−	0.674998i	$-0.764145\pi$
$859$	−43.2491	−1.47564	−0.737820	+	0.674998i	$0.764145\pi$
$860$	0.910147	0.0310358
$861$	0	0
$862$	24.7243	0.842112
$863$	−36.0384	−1.22676	−0.613380	−	0.789788i	$-0.710191\pi$
$863$	−36.0384	−1.22676	−0.613380	+	0.789788i	$0.710191\pi$
$864$	5.95524	0.202601
$865$	0.630709	0.0214447
$866$	−59.5825	−2.02469
$867$	−7.09562	−0.240980
$868$	0	0
$869$	−1.36022	−0.0461423
$870$	9.46879	0.321022
$871$	−4.09139	−0.138631
$872$	48.7774	1.65181
$873$	17.6476	0.597282
$874$	−15.7061	−0.531266
$875$	0	0
$876$	13.5850	0.458995
$877$	26.9432	0.909806	0.454903	−	0.890541i	$-0.349674\pi$
$877$	26.9432	0.909806	0.454903	+	0.890541i	$0.349674\pi$
$878$	−71.4498	−2.41131
$879$	−13.6153	−0.459232
$880$	2.02026	0.0681028
$881$	−36.5231	−1.23050	−0.615248	−	0.788334i	$-0.710944\pi$
$881$	−36.5231	−1.23050	−0.615248	+	0.788334i	$0.710944\pi$
$882$	0	0
$883$	−54.7823	−1.84357	−0.921785	−	0.387702i	$-0.873269\pi$
$883$	−54.7823	−1.84357	−0.921785	+	0.387702i	$0.873269\pi$
$884$	14.1920	0.477330
$885$	−0.934224	−0.0314036
$886$	−20.6546	−0.693906
$887$	−13.7438	−0.461470	−0.230735	−	0.973017i	$-0.574113\pi$
$887$	−13.7438	−0.461470	−0.230735	+	0.973017i	$0.574113\pi$
$888$	−17.5110	−0.587632
$889$	0	0
$890$	−1.57632	−0.0528383
$891$	6.12669	0.205252
$892$	7.88113	0.263880
$893$	−92.2182	−3.08596
$894$	24.8530	0.831207
$895$	−10.3442	−0.345767
$896$	0	0
$897$	−0.910400	−0.0303974
$898$	54.4569	1.81725
$899$	18.2697	0.609330
$900$	−11.0059	−0.366864
$901$	20.7483	0.691225
$902$	−35.1536	−1.17049
$903$	0	0
$904$	−49.7366	−1.65422
$905$	3.00830	0.0999992
$906$	36.0672	1.19825
$907$	−21.9841	−0.729970	−0.364985	−	0.931013i	$-0.618926\pi$
$907$	−21.9841	−0.729970	−0.364985	+	0.931013i	$0.618926\pi$
$908$	−70.0232	−2.32380
$909$	−8.13075	−0.269680
$910$	0	0
$911$	−17.3602	−0.575170	−0.287585	−	0.957755i	$-0.592852\pi$
$911$	−17.3602	−0.575170	−0.287585	+	0.957755i	$0.592852\pi$
$912$	−1.83780	−0.0608557
$913$	95.0843	3.14683
$914$	−10.9437	−0.361984
$915$	4.43037	0.146463
$916$	46.8315	1.54735
$917$	0	0
$918$	−11.1675	−0.368581
$919$	5.14845	0.169832	0.0849159	−	0.996388i	$-0.472938\pi$
$919$	5.14845	0.169832	0.0849159	+	0.996388i	$0.472938\pi$
$920$	−3.31327	−0.109235
$921$	1.94494	0.0640878
$922$	53.3935	1.75842
$923$	14.4367	0.475188
$924$	0	0
$925$	−22.6883	−0.745988
$926$	53.3136	1.75199
$927$	−14.2795	−0.469000
$928$	−20.0099	−0.656856
$929$	12.4649	0.408961	0.204480	−	0.978871i	$-0.434450\pi$
$929$	12.4649	0.408961	0.204480	+	0.978871i	$0.434450\pi$
$930$	15.3228	0.502453
$931$	0	0
$932$	−10.8488	−0.355363
$933$	14.2020	0.464951
$934$	30.6003	1.00127
$935$	−37.2528	−1.21830
$936$	2.43515	0.0795953
$937$	−1.66532	−0.0544038	−0.0272019	−	0.999630i	$-0.508660\pi$
$937$	−1.66532	−0.0544038	−0.0272019	+	0.999630i	$0.508660\pi$
$938$	0	0
$939$	4.20457	0.137211
$940$	−52.5463	−1.71387
$941$	−34.1661	−1.11378	−0.556892	−	0.830585i	$-0.688006\pi$
$941$	−34.1661	−1.11378	−0.556892	+	0.830585i	$0.688006\pi$
$942$	20.9301	0.681940
$943$	−2.52207	−0.0821300
$944$	0.200772	0.00653457
$945$	0	0
$946$	−3.22490	−0.104851
$947$	52.3162	1.70005	0.850024	−	0.526744i	$-0.176587\pi$
$947$	52.3162	1.70005	0.850024	+	0.526744i	$0.176587\pi$
$948$	0.705061	0.0228993
$949$	−3.89448	−0.126420
$950$	54.4316	1.76599
$951$	−0.738433	−0.0239453
$952$	0	0
$953$	−35.3873	−1.14631	−0.573154	−	0.819448i	$-0.694280\pi$
$953$	−35.3873	−1.14631	−0.573154	+	0.819448i	$0.694280\pi$
$954$	9.61609	0.311332
$955$	17.7048	0.572914
$956$	−34.5364	−1.11699
$957$	−20.5859	−0.665449
$958$	15.3829	0.496998
$959$	0	0
$960$	−16.1227	−0.520358
$961$	−1.43520	−0.0462967
$962$	13.5593	0.437170
$963$	−6.55764	−0.211317
$964$	−67.9494	−2.18850
$965$	−2.58893	−0.0833406
$966$	0	0
$967$	−1.97543	−0.0635254	−0.0317627	−	0.999495i	$-0.510112\pi$
$967$	−1.97543	−0.0635254	−0.0317627	+	0.999495i	$0.510112\pi$
$968$	70.9796	2.28137
$969$	33.8884	1.08865
$970$	−49.7320	−1.59680
$971$	−28.7536	−0.922746	−0.461373	−	0.887206i	$-0.652643\pi$
$971$	−28.7536	−0.922746	−0.461373	+	0.887206i	$0.652643\pi$
$972$	−3.17573	−0.101862
$973$	0	0
$974$	45.1969	1.44820
$975$	3.15512	0.101045
$976$	−0.952121	−0.0304767
$977$	−10.7374	−0.343520	−0.171760	−	0.985139i	$-0.554945\pi$
$977$	−10.7374	−0.343520	−0.171760	+	0.985139i	$0.554945\pi$
$978$	3.75100	0.119944
$979$	3.42705	0.109529
$980$	0	0
$981$	18.2358	0.582226
$982$	9.56282	0.305162
$983$	−46.2766	−1.47599	−0.737997	−	0.674804i	$-0.764228\pi$
$983$	−46.2766	−1.47599	−0.737997	+	0.674804i	$0.764228\pi$
$984$	6.74607	0.215057
$985$	−13.4848	−0.429660
$986$	37.5232	1.19498
$987$	0	0
$988$	−19.9599	−0.635008
$989$	−0.231369	−0.00735709
$990$	−17.2654	−0.548729
$991$	−42.9752	−1.36515	−0.682576	−	0.730814i	$-0.739141\pi$
$991$	−42.9752	−1.36515	−0.682576	+	0.730814i	$0.739141\pi$
$992$	−32.3807	−1.02809
$993$	20.5989	0.653686
$994$	0	0
$995$	−4.26176	−0.135107
$996$	−49.2863	−1.56170
$997$	−12.0816	−0.382628	−0.191314	−	0.981529i	$-0.561275\pi$
$997$	−12.0816	−0.382628	−0.191314	+	0.981529i	$0.561275\pi$
$998$	15.8078	0.500386
$999$	−6.54665	−0.207127

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bi.1.8		10
7.2	even	3		483.2.i.h.277.3	✓	20
7.4	even	3		483.2.i.h.415.3	yes	20
7.6	odd	2		3381.2.a.bj.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.i.h.277.3	✓	20	7.2	even	3
483.2.i.h.415.3	yes	20	7.4	even	3
3381.2.a.bi.1.8		10	1.1	even	1	trivial
3381.2.a.bj.1.8		10	7.6	odd	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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q+2.27502 q^{2} -1.00000 q^{3} +3.17573 q^{4} -1.23869 q^{5} -2.27502 q^{6} +2.67481 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.27502 q^{2} -1.00000 q^{3} +3.17573 q^{4} -1.23869 q^{5} -2.27502 q^{6} +2.67481 q^{8} +1.00000 q^{9} -2.81806 q^{10} +6.12669 q^{11} -3.17573 q^{12} +0.910400 q^{13} +1.23869 q^{15} -0.266205 q^{16} +4.90873 q^{17} +2.27502 q^{18} -6.90370 q^{19} -3.93376 q^{20} +13.9384 q^{22} +1.00000 q^{23} -2.67481 q^{24} -3.46564 q^{25} +2.07118 q^{26} -1.00000 q^{27} +3.36004 q^{29} +2.81806 q^{30} +5.43735 q^{31} -5.95524 q^{32} -6.12669 q^{33} +11.1675 q^{34} +3.17573 q^{36} +6.54665 q^{37} -15.7061 q^{38} -0.910400 q^{39} -3.31327 q^{40} -2.52207 q^{41} -0.231369 q^{43} +19.4567 q^{44} -1.23869 q^{45} +2.27502 q^{46} +13.3578 q^{47} +0.266205 q^{48} -7.88440 q^{50} -4.90873 q^{51} +2.89118 q^{52} +4.22681 q^{53} -2.27502 q^{54} -7.58909 q^{55} +6.90370 q^{57} +7.64417 q^{58} -0.754201 q^{59} +3.93376 q^{60} +3.57664 q^{61} +12.3701 q^{62} -13.0159 q^{64} -1.12771 q^{65} -13.9384 q^{66} -4.49406 q^{67} +15.5888 q^{68} -1.00000 q^{69} +15.8575 q^{71} +2.67481 q^{72} -4.27776 q^{73} +14.8938 q^{74} +3.46564 q^{75} -21.9243 q^{76} -2.07118 q^{78} -0.222015 q^{79} +0.329747 q^{80} +1.00000 q^{81} -5.73777 q^{82} +15.5197 q^{83} -6.08041 q^{85} -0.526369 q^{86} -3.36004 q^{87} +16.3877 q^{88} +0.559363 q^{89} -2.81806 q^{90} +3.17573 q^{92} -5.43735 q^{93} +30.3893 q^{94} +8.55157 q^{95} +5.95524 q^{96} +17.6476 q^{97} +6.12669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{2} - 10 q^{3} + 15 q^{4} - 5 q^{5} - 3 q^{6} + 9 q^{8} + 10 q^{9}+O(q^{10}) \) 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