LMFDB - Embedded newform 6027.2.a.bm.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, 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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 14 x^{14} + 68 x^{13} + 64 x^{12} - 456 x^{11} - 54 x^{10} + 1532 x^{9} - 400 x^{8} + \cdots - 49 $ x^16 - 4x^15 - 14x^14 + 68x^13 + 64x^12 - 456x^11 - 54x^10 + 1532x^9 - 400x^8 - 2708x^7 + 1218x^6 + 2424x^5 - 1276x^4 - 960x^3 + 500x^2 + 112*x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-1.10456$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.10456 q^{2} +1.00000 q^{3} -0.779950 q^{4} -3.67747 q^{5} +1.10456 q^{6} -3.07062 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.10456 q^{2} +1.00000 q^{3} -0.779950 q^{4} -3.67747 q^{5} +1.10456 q^{6} -3.07062 q^{8} +1.00000 q^{9} -4.06198 q^{10} +1.52221 q^{11} -0.779950 q^{12} +3.67604 q^{13} -3.67747 q^{15} -1.83178 q^{16} -4.16206 q^{17} +1.10456 q^{18} +5.23036 q^{19} +2.86824 q^{20} +1.68137 q^{22} +1.53784 q^{23} -3.07062 q^{24} +8.52379 q^{25} +4.06040 q^{26} +1.00000 q^{27} -8.04051 q^{29} -4.06198 q^{30} +8.26086 q^{31} +4.11793 q^{32} +1.52221 q^{33} -4.59724 q^{34} -0.779950 q^{36} -9.47010 q^{37} +5.77725 q^{38} +3.67604 q^{39} +11.2921 q^{40} +1.00000 q^{41} +5.19544 q^{43} -1.18725 q^{44} -3.67747 q^{45} +1.69863 q^{46} -2.92116 q^{47} -1.83178 q^{48} +9.41503 q^{50} -4.16206 q^{51} -2.86712 q^{52} -9.52691 q^{53} +1.10456 q^{54} -5.59789 q^{55} +5.23036 q^{57} -8.88122 q^{58} +1.95320 q^{59} +2.86824 q^{60} -10.9140 q^{61} +9.12461 q^{62} +8.21205 q^{64} -13.5185 q^{65} +1.68137 q^{66} +7.95894 q^{67} +3.24619 q^{68} +1.53784 q^{69} -12.3213 q^{71} -3.07062 q^{72} +13.8614 q^{73} -10.4603 q^{74} +8.52379 q^{75} -4.07942 q^{76} +4.06040 q^{78} -5.81290 q^{79} +6.73631 q^{80} +1.00000 q^{81} +1.10456 q^{82} -11.6164 q^{83} +15.3058 q^{85} +5.73867 q^{86} -8.04051 q^{87} -4.67413 q^{88} +18.1860 q^{89} -4.06198 q^{90} -1.19944 q^{92} +8.26086 q^{93} -3.22659 q^{94} -19.2345 q^{95} +4.11793 q^{96} +4.22436 q^{97} +1.52221 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9	$ 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9} - 4 q^{10} - 4 q^{11} + 12 q^{12} - 12 q^{15} - 8 q^{17} - 4 q^{18} + 4 q^{19} - 20 q^{20} - 16 q^{22} - 12 q^{23} - 12 q^{24} - 8 q^{25} - 8 q^{26} + 16 q^{27} - 16 q^{29} - 4 q^{30} - 4 q^{31} - 48 q^{32} - 4 q^{33} + 16 q^{34} + 12 q^{36} - 48 q^{37} - 4 q^{38} + 56 q^{40} + 16 q^{41} - 16 q^{43} - 12 q^{45} - 4 q^{46} - 36 q^{47} - 8 q^{50} - 8 q^{51} - 60 q^{53} - 4 q^{54} + 8 q^{55} + 4 q^{57} - 36 q^{58} - 36 q^{59} - 20 q^{60} - 4 q^{61} - 12 q^{62} + 52 q^{64} - 36 q^{65} - 16 q^{66} - 52 q^{67} - 8 q^{68} - 12 q^{69} - 12 q^{71} - 12 q^{72} - 16 q^{73} + 4 q^{74} - 8 q^{75} + 16 q^{76} - 8 q^{78} - 36 q^{79} - 68 q^{80} + 16 q^{81} - 4 q^{82} - 32 q^{83} - 28 q^{85} - 8 q^{86} - 16 q^{87} - 36 q^{88} - 12 q^{89} - 4 q^{90} - 36 q^{92} - 4 q^{93} + 24 q^{94} - 20 q^{95} - 48 q^{96} + 48 q^{97} - 4 q^{99}+O(q^{100}) $ 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9 - 4 * q^10 - 4 * q^11 + 12 * q^12 - 12 * q^15 - 8 * q^17 - 4 * q^18 + 4 * q^19 - 20 * q^20 - 16 * q^22 - 12 * q^23 - 12 * q^24 - 8 * q^25 - 8 * q^26 + 16 * q^27 - 16 * q^29 - 4 * q^30 - 4 * q^31 - 48 * q^32 - 4 * q^33 + 16 * q^34 + 12 * q^36 - 48 * q^37 - 4 * q^38 + 56 * q^40 + 16 * q^41 - 16 * q^43 - 12 * q^45 - 4 * q^46 - 36 * q^47 - 8 * q^50 - 8 * q^51 - 60 * q^53 - 4 * q^54 + 8 * q^55 + 4 * q^57 - 36 * q^58 - 36 * q^59 - 20 * q^60 - 4 * q^61 - 12 * q^62 + 52 * q^64 - 36 * q^65 - 16 * q^66 - 52 * q^67 - 8 * q^68 - 12 * q^69 - 12 * q^71 - 12 * q^72 - 16 * q^73 + 4 * q^74 - 8 * q^75 + 16 * q^76 - 8 * q^78 - 36 * q^79 - 68 * q^80 + 16 * q^81 - 4 * q^82 - 32 * q^83 - 28 * q^85 - 8 * q^86 - 16 * q^87 - 36 * q^88 - 12 * q^89 - 4 * q^90 - 36 * q^92 - 4 * q^93 + 24 * q^94 - 20 * q^95 - 48 * q^96 + 48 * q^97 - 4 * 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 $

$2$	1.10456	0.781041	0.390521	−	0.920594i	$-0.372295\pi$
$2$	1.10456	0.781041	0.390521	+	0.920594i	$0.372295\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.779950	−0.389975
$5$	−3.67747	−1.64461	−0.822307	−	0.569044i	$-0.807314\pi$
$5$	−3.67747	−1.64461	−0.822307	+	0.569044i	$0.807314\pi$
$6$	1.10456	0.450934
$7$	0	0
$7$	0	0
$8$	−3.07062	−1.08563
$9$	1.00000	0.333333
$10$	−4.06198	−1.28451
$11$	1.52221	0.458964	0.229482	−	0.973313i	$-0.426297\pi$
$11$	1.52221	0.458964	0.229482	+	0.973313i	$0.426297\pi$
$12$	−0.779950	−0.225152
$13$	3.67604	1.01955	0.509774	−	0.860308i	$-0.329729\pi$
$13$	3.67604	1.01955	0.509774	+	0.860308i	$0.329729\pi$
$14$	0	0
$15$	−3.67747	−0.949519
$16$	−1.83178	−0.457945
$17$	−4.16206	−1.00945	−0.504724	−	0.863281i	$-0.668406\pi$
$17$	−4.16206	−1.00945	−0.504724	+	0.863281i	$0.668406\pi$
$18$	1.10456	0.260347
$19$	5.23036	1.19993	0.599964	−	0.800027i	$-0.295182\pi$
$19$	5.23036	1.19993	0.599964	+	0.800027i	$0.295182\pi$
$20$	2.86824	0.641358
$21$	0	0
$22$	1.68137	0.358470
$23$	1.53784	0.320662	0.160331	−	0.987063i	$-0.448744\pi$
$23$	1.53784	0.320662	0.160331	+	0.987063i	$0.448744\pi$
$24$	−3.07062	−0.626787
$25$	8.52379	1.70476
$26$	4.06040	0.796309
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.04051	−1.49309	−0.746543	−	0.665337i	$-0.768288\pi$
$29$	−8.04051	−1.49309	−0.746543	+	0.665337i	$0.768288\pi$
$30$	−4.06198	−0.741613
$31$	8.26086	1.48369	0.741847	−	0.670569i	$-0.233950\pi$
$31$	8.26086	1.48369	0.741847	+	0.670569i	$0.233950\pi$
$32$	4.11793	0.727954
$33$	1.52221	0.264983
$34$	−4.59724	−0.788420
$35$	0	0
$36$	−0.779950	−0.129992
$37$	−9.47010	−1.55687	−0.778437	−	0.627723i	$-0.783987\pi$
$37$	−9.47010	−1.55687	−0.778437	+	0.627723i	$0.783987\pi$
$38$	5.77725	0.937193
$39$	3.67604	0.588637
$40$	11.2921	1.78544
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	5.19544	0.792297	0.396148	−	0.918186i	$-0.370347\pi$
$43$	5.19544	0.792297	0.396148	+	0.918186i	$0.370347\pi$
$44$	−1.18725	−0.178984
$45$	−3.67747	−0.548205
$46$	1.69863	0.250450
$47$	−2.92116	−0.426095	−0.213047	−	0.977042i	$-0.568339\pi$
$47$	−2.92116	−0.426095	−0.213047	+	0.977042i	$0.568339\pi$
$48$	−1.83178	−0.264395
$49$	0	0
$50$	9.41503	1.33149
$51$	−4.16206	−0.582805
$52$	−2.86712	−0.397598
$53$	−9.52691	−1.30862	−0.654311	−	0.756226i	$-0.727041\pi$
$53$	−9.52691	−1.30862	−0.654311	+	0.756226i	$0.727041\pi$
$54$	1.10456	0.150311
$55$	−5.59789	−0.754819
$56$	0	0
$57$	5.23036	0.692779
$58$	−8.88122	−1.16616
$59$	1.95320	0.254285	0.127142	−	0.991884i	$-0.459419\pi$
$59$	1.95320	0.254285	0.127142	+	0.991884i	$0.459419\pi$
$60$	2.86824	0.370288
$61$	−10.9140	−1.39739	−0.698695	−	0.715420i	$-0.746236\pi$
$61$	−10.9140	−1.39739	−0.698695	+	0.715420i	$0.746236\pi$
$62$	9.12461	1.15883
$63$	0	0
$64$	8.21205	1.02651
$65$	−13.5185	−1.67676
$66$	1.68137	0.206963
$67$	7.95894	0.972339	0.486169	−	0.873865i	$-0.338394\pi$
$67$	7.95894	0.972339	0.486169	+	0.873865i	$0.338394\pi$
$68$	3.24619	0.393659
$69$	1.53784	0.185134
$70$	0	0
$71$	−12.3213	−1.46227	−0.731134	−	0.682233i	$-0.761009\pi$
$71$	−12.3213	−1.46227	−0.731134	+	0.682233i	$0.761009\pi$
$72$	−3.07062	−0.361876
$73$	13.8614	1.62235	0.811175	−	0.584803i	$-0.198828\pi$
$73$	13.8614	1.62235	0.811175	+	0.584803i	$0.198828\pi$
$74$	−10.4603	−1.21598
$75$	8.52379	0.984242
$76$	−4.07942	−0.467942
$77$	0	0
$78$	4.06040	0.459749
$79$	−5.81290	−0.654003	−0.327001	−	0.945024i	$-0.606038\pi$
$79$	−5.81290	−0.654003	−0.327001	+	0.945024i	$0.606038\pi$
$80$	6.73631	0.753143
$81$	1.00000	0.111111
$82$	1.10456	0.121978
$83$	−11.6164	−1.27506	−0.637530	−	0.770425i	$-0.720044\pi$
$83$	−11.6164	−1.27506	−0.637530	+	0.770425i	$0.720044\pi$
$84$	0	0
$85$	15.3058	1.66015
$86$	5.73867	0.618816
$87$	−8.04051	−0.862033
$88$	−4.67413	−0.498264
$89$	18.1860	1.92771	0.963854	−	0.266432i	$-0.0858447\pi$
$89$	18.1860	1.92771	0.963854	+	0.266432i	$0.0858447\pi$
$90$	−4.06198	−0.428171
$91$	0	0
$92$	−1.19944	−0.125050
$93$	8.26086	0.856612
$94$	−3.22659	−0.332798
$95$	−19.2345	−1.97342
$96$	4.11793	0.420284
$97$	4.22436	0.428919	0.214459	−	0.976733i	$-0.431201\pi$
$97$	4.22436	0.428919	0.214459	+	0.976733i	$0.431201\pi$
$98$	0	0
$99$	1.52221	0.152988
$100$	−6.64813	−0.664813
$101$	−12.4856	−1.24236	−0.621182	−	0.783666i	$-0.713347\pi$
$101$	−12.4856	−1.24236	−0.621182	+	0.783666i	$0.713347\pi$
$102$	−4.59724	−0.455194
$103$	−17.5997	−1.73415	−0.867073	−	0.498182i	$-0.834001\pi$
$103$	−17.5997	−1.73415	−0.867073	+	0.498182i	$0.834001\pi$
$104$	−11.2877	−1.10685
$105$	0	0
$106$	−10.5230	−1.02209
$107$	−8.93227	−0.863515	−0.431757	−	0.901990i	$-0.642106\pi$
$107$	−8.93227	−0.863515	−0.431757	+	0.901990i	$0.642106\pi$
$108$	−0.779950	−0.0750507
$109$	−13.0978	−1.25454	−0.627272	−	0.778801i	$-0.715828\pi$
$109$	−13.0978	−1.25454	−0.627272	+	0.778801i	$0.715828\pi$
$110$	−6.18320	−0.589545
$111$	−9.47010	−0.898862
$112$	0	0
$113$	0.658461	0.0619428	0.0309714	−	0.999520i	$-0.490140\pi$
$113$	0.658461	0.0619428	0.0309714	+	0.999520i	$0.490140\pi$
$114$	5.77725	0.541089
$115$	−5.65536	−0.527365
$116$	6.27119	0.582266
$117$	3.67604	0.339850
$118$	2.15742	0.198607
$119$	0	0
$120$	11.2921	1.03082
$121$	−8.68287	−0.789352
$122$	−12.0551	−1.09142
$123$	1.00000	0.0901670
$124$	−6.44306	−0.578604
$125$	−12.9586	−1.15906
$126$	0	0
$127$	−8.74772	−0.776235	−0.388117	−	0.921610i	$-0.626874\pi$
$127$	−8.74772	−0.776235	−0.388117	+	0.921610i	$0.626874\pi$
$128$	0.834840	0.0737901
$129$	5.19544	0.457433
$130$	−14.9320	−1.30962
$131$	−18.4044	−1.60800	−0.804002	−	0.594627i	$-0.797300\pi$
$131$	−18.4044	−1.60800	−0.804002	+	0.594627i	$0.797300\pi$
$132$	−1.18725	−0.103337
$133$	0	0
$134$	8.79112	0.759437
$135$	−3.67747	−0.316506
$136$	12.7801	1.09588
$137$	−2.50601	−0.214103	−0.107052	−	0.994253i	$-0.534141\pi$
$137$	−2.50601	−0.214103	−0.107052	+	0.994253i	$0.534141\pi$
$138$	1.69863	0.144597
$139$	4.25917	0.361258	0.180629	−	0.983551i	$-0.442187\pi$
$139$	4.25917	0.361258	0.180629	+	0.983551i	$0.442187\pi$
$140$	0	0
$141$	−2.92116	−0.246006
$142$	−13.6096	−1.14209
$143$	5.59570	0.467936
$144$	−1.83178	−0.152648
$145$	29.5687	2.45555
$146$	15.3107	1.26712
$147$	0	0
$148$	7.38620	0.607142
$149$	−4.22166	−0.345852	−0.172926	−	0.984935i	$-0.555322\pi$
$149$	−4.22166	−0.345852	−0.172926	+	0.984935i	$0.555322\pi$
$150$	9.41503	0.768734
$151$	15.5047	1.26175	0.630876	−	0.775884i	$-0.282696\pi$
$151$	15.5047	1.26175	0.630876	+	0.775884i	$0.282696\pi$
$152$	−16.0605	−1.30267
$153$	−4.16206	−0.336482
$154$	0	0
$155$	−30.3791	−2.44011
$156$	−2.86712	−0.229554
$157$	−3.46926	−0.276877	−0.138438	−	0.990371i	$-0.544208\pi$
$157$	−3.46926	−0.276877	−0.138438	+	0.990371i	$0.544208\pi$
$158$	−6.42069	−0.510803
$159$	−9.52691	−0.755533
$160$	−15.1436	−1.19720
$161$	0	0
$162$	1.10456	0.0867823
$163$	−18.7787	−1.47086	−0.735429	−	0.677602i	$-0.763019\pi$
$163$	−18.7787	−1.47086	−0.735429	+	0.677602i	$0.763019\pi$
$164$	−0.779950	−0.0609038
$165$	−5.59789	−0.435795
$166$	−12.8309	−0.995875
$167$	10.2446	0.792751	0.396376	−	0.918088i	$-0.370268\pi$
$167$	10.2446	0.792751	0.396376	+	0.918088i	$0.370268\pi$
$168$	0	0
$169$	0.513236	0.0394797
$170$	16.9062	1.29665
$171$	5.23036	0.399976
$172$	−4.05218	−0.308976
$173$	14.0080	1.06501	0.532504	−	0.846428i	$-0.321251\pi$
$173$	14.0080	1.06501	0.532504	+	0.846428i	$0.321251\pi$
$174$	−8.88122	−0.673283
$175$	0	0
$176$	−2.78836	−0.210180
$177$	1.95320	0.146811
$178$	20.0875	1.50562
$179$	6.20988	0.464148	0.232074	−	0.972698i	$-0.425449\pi$
$179$	6.20988	0.464148	0.232074	+	0.972698i	$0.425449\pi$
$180$	2.86824	0.213786
$181$	−21.9328	−1.63025	−0.815127	−	0.579283i	$-0.803333\pi$
$181$	−21.9328	−1.63025	−0.815127	+	0.579283i	$0.803333\pi$
$182$	0	0
$183$	−10.9140	−0.806784
$184$	−4.72212	−0.348119
$185$	34.8260	2.56046
$186$	9.12461	0.669049
$187$	−6.33553	−0.463300
$188$	2.27836	0.166166
$189$	0	0
$190$	−21.2457	−1.54132
$191$	−5.12001	−0.370471	−0.185235	−	0.982694i	$-0.559305\pi$
$191$	−5.12001	−0.370471	−0.185235	+	0.982694i	$0.559305\pi$
$192$	8.21205	0.592654
$193$	11.1033	0.799234	0.399617	−	0.916682i	$-0.369143\pi$
$193$	11.1033	0.799234	0.399617	+	0.916682i	$0.369143\pi$
$194$	4.66606	0.335003
$195$	−13.5185	−0.968081
$196$	0	0
$197$	−3.72992	−0.265746	−0.132873	−	0.991133i	$-0.542420\pi$
$197$	−3.72992	−0.265746	−0.132873	+	0.991133i	$0.542420\pi$
$198$	1.68137	0.119490
$199$	14.4265	1.02267	0.511334	−	0.859382i	$-0.329151\pi$
$199$	14.4265	1.02267	0.511334	+	0.859382i	$0.329151\pi$
$200$	−26.1733	−1.85073
$201$	7.95894	0.561380
$202$	−13.7911	−0.970338
$203$	0	0
$204$	3.24619	0.227279
$205$	−3.67747	−0.256846
$206$	−19.4399	−1.35444
$207$	1.53784	0.106887
$208$	−6.73369	−0.466897
$209$	7.96172	0.550724
$210$	0	0
$211$	−13.3208	−0.917041	−0.458521	−	0.888684i	$-0.651621\pi$
$211$	−13.3208	−0.917041	−0.458521	+	0.888684i	$0.651621\pi$
$212$	7.43051	0.510329
$213$	−12.3213	−0.844241
$214$	−9.86621	−0.674441
$215$	−19.1061	−1.30302
$216$	−3.07062	−0.208929
$217$	0	0
$218$	−14.4673	−0.979850
$219$	13.8614	0.936664
$220$	4.36607	0.294360
$221$	−15.2999	−1.02918
$222$	−10.4603	−0.702048
$223$	−23.8810	−1.59919	−0.799594	−	0.600540i	$-0.794952\pi$
$223$	−23.8810	−1.59919	−0.799594	+	0.600540i	$0.794952\pi$
$224$	0	0
$225$	8.52379	0.568253
$226$	0.727309	0.0483799
$227$	3.77010	0.250230	0.125115	−	0.992142i	$-0.460070\pi$
$227$	3.77010	0.250230	0.125115	+	0.992142i	$0.460070\pi$
$228$	−4.07942	−0.270166
$229$	−6.51600	−0.430589	−0.215295	−	0.976549i	$-0.569071\pi$
$229$	−6.51600	−0.430589	−0.215295	+	0.976549i	$0.569071\pi$
$230$	−6.24668	−0.411894
$231$	0	0
$232$	24.6893	1.62093
$233$	−8.18148	−0.535987	−0.267993	−	0.963421i	$-0.586361\pi$
$233$	−8.18148	−0.535987	−0.267993	+	0.963421i	$0.586361\pi$
$234$	4.06040	0.265436
$235$	10.7425	0.700762
$236$	−1.52340	−0.0991646
$237$	−5.81290	−0.377589
$238$	0	0
$239$	−13.7032	−0.886387	−0.443193	−	0.896426i	$-0.646154\pi$
$239$	−13.7032	−0.886387	−0.443193	+	0.896426i	$0.646154\pi$
$240$	6.73631	0.434827
$241$	−13.2872	−0.855903	−0.427952	−	0.903802i	$-0.640765\pi$
$241$	−13.2872	−0.855903	−0.427952	+	0.903802i	$0.640765\pi$
$242$	−9.59074	−0.616516
$243$	1.00000	0.0641500
$244$	8.51234	0.544947
$245$	0	0
$246$	1.10456	0.0704241
$247$	19.2270	1.22339
$248$	−25.3660	−1.61074
$249$	−11.6164	−0.736156
$250$	−14.3136	−0.905270
$251$	−7.91598	−0.499653	−0.249826	−	0.968291i	$-0.580374\pi$
$251$	−7.91598	−0.499653	−0.249826	+	0.968291i	$0.580374\pi$
$252$	0	0
$253$	2.34092	0.147172
$254$	−9.66237	−0.606271
$255$	15.3058	0.958489
$256$	−15.5020	−0.968873
$257$	5.83445	0.363943	0.181971	−	0.983304i	$-0.441752\pi$
$257$	5.83445	0.363943	0.181971	+	0.983304i	$0.441752\pi$
$258$	5.73867	0.357274
$259$	0	0
$260$	10.5438	0.653896
$261$	−8.04051	−0.497695
$262$	−20.3288	−1.25592
$263$	−1.97786	−0.121960	−0.0609800	−	0.998139i	$-0.519423\pi$
$263$	−1.97786	−0.121960	−0.0609800	+	0.998139i	$0.519423\pi$
$264$	−4.67413	−0.287673
$265$	35.0349	2.15218
$266$	0	0
$267$	18.1860	1.11296
$268$	−6.20757	−0.379188
$269$	23.3647	1.42457	0.712287	−	0.701889i	$-0.247660\pi$
$269$	23.3647	1.42457	0.712287	+	0.701889i	$0.247660\pi$
$270$	−4.06198	−0.247204
$271$	23.7310	1.44155	0.720776	−	0.693168i	$-0.243786\pi$
$271$	23.7310	1.44155	0.720776	+	0.693168i	$0.243786\pi$
$272$	7.62397	0.462271
$273$	0	0
$274$	−2.76804	−0.167223
$275$	12.9750	0.782423
$276$	−1.19944	−0.0721977
$277$	21.0539	1.26501	0.632504	−	0.774557i	$-0.282027\pi$
$277$	21.0539	1.26501	0.632504	+	0.774557i	$0.282027\pi$
$278$	4.70450	0.282157
$279$	8.26086	0.494565
$280$	0	0
$281$	−25.8960	−1.54483	−0.772413	−	0.635120i	$-0.780950\pi$
$281$	−25.8960	−1.54483	−0.772413	+	0.635120i	$0.780950\pi$
$282$	−3.22659	−0.192141
$283$	−25.7797	−1.53245	−0.766223	−	0.642575i	$-0.777866\pi$
$283$	−25.7797	−1.53245	−0.766223	+	0.642575i	$0.777866\pi$
$284$	9.60999	0.570248
$285$	−19.2345	−1.13935
$286$	6.18078	0.365477
$287$	0	0
$288$	4.11793	0.242651
$289$	0.322719	0.0189835
$290$	32.6604	1.91789
$291$	4.22436	0.247636
$292$	−10.8112	−0.632676
$293$	20.8476	1.21793	0.608964	−	0.793198i	$-0.291585\pi$
$293$	20.8476	1.21793	0.608964	+	0.793198i	$0.291585\pi$
$294$	0	0
$295$	−7.18283	−0.418200
$296$	29.0790	1.69019
$297$	1.52221	0.0883277
$298$	−4.66307	−0.270125
$299$	5.65315	0.326930
$300$	−6.64813	−0.383830
$301$	0	0
$302$	17.1258	0.985480
$303$	−12.4856	−0.717279
$304$	−9.58087	−0.549501
$305$	40.1358	2.29817
$306$	−4.59724	−0.262807
$307$	−9.56642	−0.545984	−0.272992	−	0.962016i	$-0.588013\pi$
$307$	−9.56642	−0.545984	−0.272992	+	0.962016i	$0.588013\pi$
$308$	0	0
$309$	−17.5997	−1.00121
$310$	−33.5555	−1.90582
$311$	−16.1256	−0.914400	−0.457200	−	0.889364i	$-0.651148\pi$
$311$	−16.1256	−0.914400	−0.457200	+	0.889364i	$0.651148\pi$
$312$	−11.2877	−0.639040
$313$	21.2132	1.19904	0.599520	−	0.800360i	$-0.295358\pi$
$313$	21.2132	1.19904	0.599520	+	0.800360i	$0.295358\pi$
$314$	−3.83200	−0.216252
$315$	0	0
$316$	4.53377	0.255045
$317$	13.4949	0.757951	0.378975	−	0.925407i	$-0.376276\pi$
$317$	13.4949	0.757951	0.378975	+	0.925407i	$0.376276\pi$
$318$	−10.5230	−0.590102
$319$	−12.2394	−0.685273
$320$	−30.1996	−1.68821
$321$	−8.93227	−0.498551
$322$	0	0
$323$	−21.7691	−1.21126
$324$	−0.779950	−0.0433305
$325$	31.3337	1.73808
$326$	−20.7421	−1.14880
$327$	−13.0978	−0.724311
$328$	−3.07062	−0.169547
$329$	0	0
$330$	−6.18320	−0.340374
$331$	5.17282	0.284324	0.142162	−	0.989843i	$-0.454595\pi$
$331$	5.17282	0.284324	0.142162	+	0.989843i	$0.454595\pi$
$332$	9.06017	0.497241
$333$	−9.47010	−0.518958
$334$	11.3158	0.619171
$335$	−29.2688	−1.59912
$336$	0	0
$337$	−0.770683	−0.0419818	−0.0209909	−	0.999780i	$-0.506682\pi$
$337$	−0.770683	−0.0419818	−0.0209909	+	0.999780i	$0.506682\pi$
$338$	0.566900	0.0308353
$339$	0.658461	0.0357627
$340$	−11.9378	−0.647417
$341$	12.5748	0.680963
$342$	5.77725	0.312398
$343$	0	0
$344$	−15.9532	−0.860139
$345$	−5.65536	−0.304474
$346$	15.4726	0.831814
$347$	−7.00830	−0.376225	−0.188113	−	0.982147i	$-0.560237\pi$
$347$	−7.00830	−0.376225	−0.188113	+	0.982147i	$0.560237\pi$
$348$	6.27119	0.336171
$349$	22.4246	1.20036	0.600181	−	0.799864i	$-0.295095\pi$
$349$	22.4246	1.20036	0.600181	+	0.799864i	$0.295095\pi$
$350$	0	0
$351$	3.67604	0.196212
$352$	6.26836	0.334105
$353$	23.2159	1.23566	0.617828	−	0.786313i	$-0.288013\pi$
$353$	23.2159	1.23566	0.617828	+	0.786313i	$0.288013\pi$
$354$	2.15742	0.114666
$355$	45.3112	2.40487
$356$	−14.1841	−0.751757
$357$	0	0
$358$	6.85918	0.362519
$359$	−9.57666	−0.505437	−0.252719	−	0.967540i	$-0.581325\pi$
$359$	−9.57666	−0.505437	−0.252719	+	0.967540i	$0.581325\pi$
$360$	11.2921	0.595146
$361$	8.35671	0.439827
$362$	−24.2261	−1.27330
$363$	−8.68287	−0.455733
$364$	0	0
$365$	−50.9748	−2.66814
$366$	−12.0551	−0.630131
$367$	4.64264	0.242344	0.121172	−	0.992632i	$-0.461335\pi$
$367$	4.64264	0.242344	0.121172	+	0.992632i	$0.461335\pi$
$368$	−2.81698	−0.146845
$369$	1.00000	0.0520579
$370$	38.4674	1.99982
$371$	0	0
$372$	−6.44306	−0.334057
$373$	−16.1711	−0.837307	−0.418653	−	0.908146i	$-0.637498\pi$
$373$	−16.1711	−0.837307	−0.418653	+	0.908146i	$0.637498\pi$
$374$	−6.99797	−0.361856
$375$	−12.9586	−0.669181
$376$	8.96976	0.462580
$377$	−29.5572	−1.52227
$378$	0	0
$379$	0.105888	0.00543911	0.00271956	−	0.999996i	$-0.499134\pi$
$379$	0.105888	0.00543911	0.00271956	+	0.999996i	$0.499134\pi$
$380$	15.0019	0.769584
$381$	−8.74772	−0.448159
$382$	−5.65535	−0.289353
$383$	−9.96830	−0.509356	−0.254678	−	0.967026i	$-0.581970\pi$
$383$	−9.96830	−0.509356	−0.254678	+	0.967026i	$0.581970\pi$
$384$	0.834840	0.0426027
$385$	0	0
$386$	12.2643	0.624234
$387$	5.19544	0.264099
$388$	−3.29479	−0.167268
$389$	−27.7414	−1.40654	−0.703271	−	0.710921i	$-0.748278\pi$
$389$	−27.7414	−1.40654	−0.703271	+	0.710921i	$0.748278\pi$
$390$	−14.9320	−0.756111
$391$	−6.40058	−0.323691
$392$	0	0
$393$	−18.4044	−0.928381
$394$	−4.11991	−0.207558
$395$	21.3768	1.07558
$396$	−1.18725	−0.0596615
$397$	6.99697	0.351168	0.175584	−	0.984464i	$-0.443819\pi$
$397$	6.99697	0.351168	0.175584	+	0.984464i	$0.443819\pi$
$398$	15.9349	0.798746
$399$	0	0
$400$	−15.6137	−0.780685
$401$	−12.0665	−0.602570	−0.301285	−	0.953534i	$-0.597416\pi$
$401$	−12.0665	−0.602570	−0.301285	+	0.953534i	$0.597416\pi$
$402$	8.79112	0.438461
$403$	30.3672	1.51270
$404$	9.73814	0.484491
$405$	−3.67747	−0.182735
$406$	0	0
$407$	−14.4155	−0.714549
$408$	12.7801	0.632709
$409$	−12.8745	−0.636603	−0.318302	−	0.947989i	$-0.603112\pi$
$409$	−12.8745	−0.636603	−0.318302	+	0.947989i	$0.603112\pi$
$410$	−4.06198	−0.200607
$411$	−2.50601	−0.123612
$412$	13.7268	0.676273
$413$	0	0
$414$	1.69863	0.0834833
$415$	42.7188	2.09698
$416$	15.1376	0.742184
$417$	4.25917	0.208572
$418$	8.79419	0.430138
$419$	28.8557	1.40969	0.704847	−	0.709359i	$-0.251015\pi$
$419$	28.8557	1.40969	0.704847	+	0.709359i	$0.251015\pi$
$420$	0	0
$421$	6.97023	0.339708	0.169854	−	0.985469i	$-0.445670\pi$
$421$	6.97023	0.339708	0.169854	+	0.985469i	$0.445670\pi$
$422$	−14.7136	−0.716247
$423$	−2.92116	−0.142032
$424$	29.2535	1.42068
$425$	−35.4765	−1.72086
$426$	−13.6096	−0.659387
$427$	0	0
$428$	6.96672	0.336749
$429$	5.59570	0.270163
$430$	−21.1038	−1.01771
$431$	34.6761	1.67029	0.835145	−	0.550031i	$-0.185384\pi$
$431$	34.6761	1.67029	0.835145	+	0.550031i	$0.185384\pi$
$432$	−1.83178	−0.0881315
$433$	−38.1831	−1.83496	−0.917480	−	0.397782i	$-0.869780\pi$
$433$	−38.1831	−1.83496	−0.917480	+	0.397782i	$0.869780\pi$
$434$	0	0
$435$	29.5687	1.41771
$436$	10.2156	0.489240
$437$	8.04346	0.384771
$438$	15.3107	0.731573
$439$	−30.6755	−1.46406	−0.732031	−	0.681271i	$-0.761428\pi$
$439$	−30.6755	−1.46406	−0.732031	+	0.681271i	$0.761428\pi$
$440$	17.1890	0.819452
$441$	0	0
$442$	−16.8996	−0.803832
$443$	−19.6537	−0.933773	−0.466887	−	0.884317i	$-0.654624\pi$
$443$	−19.6537	−0.933773	−0.466887	+	0.884317i	$0.654624\pi$
$444$	7.38620	0.350533
$445$	−66.8783	−3.17034
$446$	−26.3779	−1.24903
$447$	−4.22166	−0.199678
$448$	0	0
$449$	33.1895	1.56631	0.783155	−	0.621827i	$-0.213609\pi$
$449$	33.1895	1.56631	0.783155	+	0.621827i	$0.213609\pi$
$450$	9.41503	0.443829
$451$	1.52221	0.0716782
$452$	−0.513566	−0.0241561
$453$	15.5047	0.728472
$454$	4.16430	0.195440
$455$	0	0
$456$	−16.0605	−0.752100
$457$	−18.7154	−0.875470	−0.437735	−	0.899104i	$-0.644219\pi$
$457$	−18.7154	−0.875470	−0.437735	+	0.899104i	$0.644219\pi$
$458$	−7.19731	−0.336308
$459$	−4.16206	−0.194268
$460$	4.41090	0.205659
$461$	−0.256027	−0.0119243	−0.00596217	−	0.999982i	$-0.501898\pi$
$461$	−0.256027	−0.0119243	−0.00596217	+	0.999982i	$0.501898\pi$
$462$	0	0
$463$	−21.1023	−0.980709	−0.490355	−	0.871523i	$-0.663133\pi$
$463$	−21.1023	−0.980709	−0.490355	+	0.871523i	$0.663133\pi$
$464$	14.7284	0.683751
$465$	−30.3791	−1.40880
$466$	−9.03693	−0.418628
$467$	−1.02605	−0.0474801	−0.0237400	−	0.999718i	$-0.507557\pi$
$467$	−1.02605	−0.0474801	−0.0237400	+	0.999718i	$0.507557\pi$
$468$	−2.86712	−0.132533
$469$	0	0
$470$	11.8657	0.547324
$471$	−3.46926	−0.159855
$472$	−5.99753	−0.276059
$473$	7.90856	0.363636
$474$	−6.42069	−0.294912
$475$	44.5825	2.04559
$476$	0	0
$477$	−9.52691	−0.436207
$478$	−15.1360	−0.692304
$479$	27.2043	1.24299	0.621497	−	0.783417i	$-0.286525\pi$
$479$	27.2043	1.24299	0.621497	+	0.783417i	$0.286525\pi$
$480$	−15.1436	−0.691206
$481$	−34.8124	−1.58731
$482$	−14.6765	−0.668496
$483$	0	0
$484$	6.77220	0.307827
$485$	−15.5350	−0.705406
$486$	1.10456	0.0501038
$487$	12.1352	0.549898	0.274949	−	0.961459i	$-0.411339\pi$
$487$	12.1352	0.549898	0.274949	+	0.961459i	$0.411339\pi$
$488$	33.5126	1.51705
$489$	−18.7787	−0.849200
$490$	0	0
$491$	26.9411	1.21584	0.607919	−	0.793999i	$-0.292005\pi$
$491$	26.9411	1.21584	0.607919	+	0.793999i	$0.292005\pi$
$492$	−0.779950	−0.0351628
$493$	33.4651	1.50719
$494$	21.2374	0.955514
$495$	−5.59789	−0.251606
$496$	−15.1321	−0.679450
$497$	0	0
$498$	−12.8309	−0.574968
$499$	−7.71740	−0.345478	−0.172739	−	0.984968i	$-0.555262\pi$
$499$	−7.71740	−0.345478	−0.172739	+	0.984968i	$0.555262\pi$
$500$	10.1071	0.452002
$501$	10.2446	0.457695
$502$	−8.74367	−0.390249
$503$	−25.1080	−1.11951	−0.559754	−	0.828659i	$-0.689104\pi$
$503$	−25.1080	−1.11951	−0.559754	+	0.828659i	$0.689104\pi$
$504$	0	0
$505$	45.9154	2.04321
$506$	2.58568	0.114948
$507$	0.513236	0.0227936
$508$	6.82278	0.302712
$509$	5.41173	0.239871	0.119935	−	0.992782i	$-0.461731\pi$
$509$	5.41173	0.239871	0.119935	+	0.992782i	$0.461731\pi$
$510$	16.9062	0.748619
$511$	0	0
$512$	−18.7925	−0.830520
$513$	5.23036	0.230926
$514$	6.44449	0.284254
$515$	64.7222	2.85200
$516$	−4.05218	−0.178387
$517$	−4.44662	−0.195562
$518$	0	0
$519$	14.0080	0.614882
$520$	41.5102	1.82034
$521$	20.0868	0.880017	0.440008	−	0.897994i	$-0.354975\pi$
$521$	20.0868	0.880017	0.440008	+	0.897994i	$0.354975\pi$
$522$	−8.88122	−0.388720
$523$	6.29814	0.275398	0.137699	−	0.990474i	$-0.456029\pi$
$523$	6.29814	0.275398	0.137699	+	0.990474i	$0.456029\pi$
$524$	14.3545	0.627081
$525$	0	0
$526$	−2.18466	−0.0952558
$527$	−34.3822	−1.49771
$528$	−2.78836	−0.121348
$529$	−20.6350	−0.897176
$530$	38.6981	1.68094
$531$	1.95320	0.0847616
$532$	0	0
$533$	3.67604	0.159227
$534$	20.0875	0.869269
$535$	32.8481	1.42015
$536$	−24.4389	−1.05560
$537$	6.20988	0.267976
$538$	25.8077	1.11265
$539$	0	0
$540$	2.86824	0.123429
$541$	−27.1030	−1.16525	−0.582624	−	0.812742i	$-0.697974\pi$
$541$	−27.1030	−1.16525	−0.582624	+	0.812742i	$0.697974\pi$
$542$	26.2122	1.12591
$543$	−21.9328	−0.941227
$544$	−17.1391	−0.734831
$545$	48.1668	2.06324
$546$	0	0
$547$	18.9156	0.808771	0.404386	−	0.914589i	$-0.367485\pi$
$547$	18.9156	0.808771	0.404386	+	0.914589i	$0.367485\pi$
$548$	1.95456	0.0834948
$549$	−10.9140	−0.465797
$550$	14.3317	0.611104
$551$	−42.0548	−1.79160
$552$	−4.72212	−0.200987
$553$	0	0
$554$	23.2553	0.988023
$555$	34.8260	1.47828
$556$	−3.32194	−0.140881
$557$	−20.9212	−0.886458	−0.443229	−	0.896408i	$-0.646167\pi$
$557$	−20.9212	−0.886458	−0.443229	+	0.896408i	$0.646167\pi$
$558$	9.12461	0.386276
$559$	19.0986	0.807785
$560$	0	0
$561$	−6.33553	−0.267486
$562$	−28.6037	−1.20657
$563$	−21.0926	−0.888949	−0.444474	−	0.895792i	$-0.646609\pi$
$563$	−21.0926	−0.888949	−0.444474	+	0.895792i	$0.646609\pi$
$564$	2.27836	0.0959361
$565$	−2.42147	−0.101872
$566$	−28.4752	−1.19690
$567$	0	0
$568$	37.8340	1.58748
$569$	14.0507	0.589036	0.294518	−	0.955646i	$-0.404841\pi$
$569$	14.0507	0.589036	0.294518	+	0.955646i	$0.404841\pi$
$570$	−21.2457	−0.889882
$571$	20.6691	0.864973	0.432486	−	0.901640i	$-0.357636\pi$
$571$	20.6691	0.864973	0.432486	+	0.901640i	$0.357636\pi$
$572$	−4.36437	−0.182483
$573$	−5.12001	−0.213891
$574$	0	0
$575$	13.1082	0.546651
$576$	8.21205	0.342169
$577$	−39.7417	−1.65447	−0.827234	−	0.561858i	$-0.810087\pi$
$577$	−39.7417	−1.65447	−0.827234	+	0.561858i	$0.810087\pi$
$578$	0.356462	0.0148269
$579$	11.1033	0.461438
$580$	−23.0621	−0.957603
$581$	0	0
$582$	4.66606	0.193414
$583$	−14.5020	−0.600610
$584$	−42.5630	−1.76127
$585$	−13.5185	−0.558922
$586$	23.0274	0.951252
$587$	−12.9442	−0.534266	−0.267133	−	0.963660i	$-0.586076\pi$
$587$	−12.9442	−0.534266	−0.267133	+	0.963660i	$0.586076\pi$
$588$	0	0
$589$	43.2073	1.78033
$590$	−7.93386	−0.326632
$591$	−3.72992	−0.153428
$592$	17.3471	0.712962
$593$	8.11030	0.333050	0.166525	−	0.986037i	$-0.446745\pi$
$593$	8.11030	0.333050	0.166525	+	0.986037i	$0.446745\pi$
$594$	1.68137	0.0689876
$595$	0	0
$596$	3.29268	0.134874
$597$	14.4265	0.590438
$598$	6.24424	0.255346
$599$	33.7505	1.37901	0.689504	−	0.724282i	$-0.257829\pi$
$599$	33.7505	1.37901	0.689504	+	0.724282i	$0.257829\pi$
$600$	−26.1733	−1.06852
$601$	20.2822	0.827327	0.413664	−	0.910430i	$-0.364249\pi$
$601$	20.2822	0.827327	0.413664	+	0.910430i	$0.364249\pi$
$602$	0	0
$603$	7.95894	0.324113
$604$	−12.0929	−0.492051
$605$	31.9310	1.29818
$606$	−13.7911	−0.560225
$607$	30.2667	1.22849	0.614243	−	0.789117i	$-0.289461\pi$
$607$	30.2667	1.22849	0.614243	+	0.789117i	$0.289461\pi$
$608$	21.5383	0.873492
$609$	0	0
$610$	44.3323	1.79496
$611$	−10.7383	−0.434424
$612$	3.24619	0.131220
$613$	1.31409	0.0530755	0.0265378	−	0.999648i	$-0.491552\pi$
$613$	1.31409	0.0530755	0.0265378	+	0.999648i	$0.491552\pi$
$614$	−10.5667	−0.426436
$615$	−3.67747	−0.148290
$616$	0	0
$617$	29.5191	1.18839	0.594196	−	0.804320i	$-0.297470\pi$
$617$	29.5191	1.18839	0.594196	+	0.804320i	$0.297470\pi$
$618$	−19.4399	−0.781986
$619$	18.3832	0.738882	0.369441	−	0.929254i	$-0.379549\pi$
$619$	18.3832	0.738882	0.369441	+	0.929254i	$0.379549\pi$
$620$	23.6942	0.951580
$621$	1.53784	0.0617114
$622$	−17.8117	−0.714184
$623$	0	0
$624$	−6.73369	−0.269563
$625$	5.03603	0.201441
$626$	23.4312	0.936499
$627$	7.96172	0.317961
$628$	2.70584	0.107975
$629$	39.4151	1.57158
$630$	0	0
$631$	−19.1228	−0.761268	−0.380634	−	0.924726i	$-0.624294\pi$
$631$	−19.1228	−0.761268	−0.380634	+	0.924726i	$0.624294\pi$
$632$	17.8492	0.710003
$633$	−13.3208	−0.529454
$634$	14.9059	0.591991
$635$	32.1695	1.27661
$636$	7.43051	0.294639
$637$	0	0
$638$	−13.5191	−0.535226
$639$	−12.3213	−0.487423
$640$	−3.07010	−0.121356
$641$	21.9048	0.865187	0.432594	−	0.901589i	$-0.357599\pi$
$641$	21.9048	0.865187	0.432594	+	0.901589i	$0.357599\pi$
$642$	−9.86621	−0.389388
$643$	−29.6162	−1.16795	−0.583974	−	0.811772i	$-0.698503\pi$
$643$	−29.6162	−1.16795	−0.583974	+	0.811772i	$0.698503\pi$
$644$	0	0
$645$	−19.1061	−0.752301
$646$	−24.0452	−0.946047
$647$	−11.5650	−0.454669	−0.227334	−	0.973817i	$-0.573001\pi$
$647$	−11.5650	−0.454669	−0.227334	+	0.973817i	$0.573001\pi$
$648$	−3.07062	−0.120625
$649$	2.97318	0.116708
$650$	34.6100	1.35751
$651$	0	0
$652$	14.6464	0.573598
$653$	−6.59821	−0.258208	−0.129104	−	0.991631i	$-0.541210\pi$
$653$	−6.59821	−0.258208	−0.129104	+	0.991631i	$0.541210\pi$
$654$	−14.4673	−0.565717
$655$	67.6818	2.64455
$656$	−1.83178	−0.0715190
$657$	13.8614	0.540783
$658$	0	0
$659$	−14.7272	−0.573691	−0.286846	−	0.957977i	$-0.592607\pi$
$659$	−14.7272	−0.573691	−0.286846	+	0.957977i	$0.592607\pi$
$660$	4.36607	0.169949
$661$	8.44705	0.328552	0.164276	−	0.986414i	$-0.447471\pi$
$661$	8.44705	0.328552	0.164276	+	0.986414i	$0.447471\pi$
$662$	5.71369	0.222069
$663$	−15.2999	−0.594198
$664$	35.6694	1.38424
$665$	0	0
$666$	−10.4603	−0.405328
$667$	−12.3650	−0.478775
$668$	−7.99027	−0.309153
$669$	−23.8810	−0.923292
$670$	−32.3291	−1.24898
$671$	−16.6134	−0.641352
$672$	0	0
$673$	37.3476	1.43964	0.719822	−	0.694159i	$-0.244223\pi$
$673$	37.3476	1.43964	0.719822	+	0.694159i	$0.244223\pi$
$674$	−0.851265	−0.0327895
$675$	8.52379	0.328081
$676$	−0.400298	−0.0153961
$677$	7.84359	0.301454	0.150727	−	0.988575i	$-0.451839\pi$
$677$	7.84359	0.301454	0.150727	+	0.988575i	$0.451839\pi$
$678$	0.727309	0.0279321
$679$	0	0
$680$	−46.9984	−1.80231
$681$	3.77010	0.144471
$682$	13.8896	0.531860
$683$	1.10391	0.0422400	0.0211200	−	0.999777i	$-0.493277\pi$
$683$	1.10391	0.0422400	0.0211200	+	0.999777i	$0.493277\pi$
$684$	−4.07942	−0.155981
$685$	9.21578	0.352117
$686$	0	0
$687$	−6.51600	−0.248601
$688$	−9.51690	−0.362828
$689$	−35.0212	−1.33420
$690$	−6.24668	−0.237807
$691$	−14.7201	−0.559978	−0.279989	−	0.960003i	$-0.590331\pi$
$691$	−14.7201	−0.559978	−0.279989	+	0.960003i	$0.590331\pi$
$692$	−10.9255	−0.415326
$693$	0	0
$694$	−7.74108	−0.293848
$695$	−15.6630	−0.594130
$696$	24.6893	0.935847
$697$	−4.16206	−0.157649
$698$	24.7693	0.937532
$699$	−8.18148	−0.309452
$700$	0	0
$701$	−15.8325	−0.597987	−0.298993	−	0.954255i	$-0.596651\pi$
$701$	−15.8325	−0.597987	−0.298993	+	0.954255i	$0.596651\pi$
$702$	4.06040	0.153250
$703$	−49.5320	−1.86814
$704$	12.5005	0.471130
$705$	10.7425	0.404585
$706$	25.6433	0.965098
$707$	0	0
$708$	−1.52340	−0.0572527
$709$	24.3093	0.912955	0.456478	−	0.889735i	$-0.349111\pi$
$709$	24.3093	0.912955	0.456478	+	0.889735i	$0.349111\pi$
$710$	50.0489	1.87830
$711$	−5.81290	−0.218001
$712$	−55.8421	−2.09277
$713$	12.7039	0.475764
$714$	0	0
$715$	−20.5780	−0.769575
$716$	−4.84339	−0.181006
$717$	−13.7032	−0.511756
$718$	−10.5780	−0.394767
$719$	25.4055	0.947464	0.473732	−	0.880669i	$-0.342906\pi$
$719$	25.4055	0.947464	0.473732	+	0.880669i	$0.342906\pi$
$720$	6.73631	0.251048
$721$	0	0
$722$	9.23048	0.343523
$723$	−13.2872	−0.494156
$724$	17.1065	0.635758
$725$	−68.5356	−2.54535
$726$	−9.59074	−0.355946
$727$	25.5947	0.949255	0.474628	−	0.880187i	$-0.342583\pi$
$727$	25.5947	0.949255	0.474628	+	0.880187i	$0.342583\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−56.3046	−2.08393
$731$	−21.6237	−0.799782
$732$	8.51234	0.314625
$733$	−21.1670	−0.781821	−0.390911	−	0.920429i	$-0.627840\pi$
$733$	−21.1670	−0.781821	−0.390911	+	0.920429i	$0.627840\pi$
$734$	5.12806	0.189280
$735$	0	0
$736$	6.33271	0.233427
$737$	12.1152	0.446269
$738$	1.10456	0.0406594
$739$	−38.7709	−1.42621	−0.713106	−	0.701056i	$-0.752712\pi$
$739$	−38.7709	−1.42621	−0.713106	+	0.701056i	$0.752712\pi$
$740$	−27.1625	−0.998514
$741$	19.2270	0.706322
$742$	0	0
$743$	34.8769	1.27951	0.639755	−	0.768579i	$-0.279036\pi$
$743$	34.8769	1.27951	0.639755	+	0.768579i	$0.279036\pi$
$744$	−25.3660	−0.929961
$745$	15.5250	0.568793
$746$	−17.8619	−0.653971
$747$	−11.6164	−0.425020
$748$	4.94140	0.180675
$749$	0	0
$750$	−14.3136	−0.522658
$751$	15.7762	0.575680	0.287840	−	0.957678i	$-0.407063\pi$
$751$	15.7762	0.575680	0.287840	+	0.957678i	$0.407063\pi$
$752$	5.35092	0.195128
$753$	−7.91598	−0.288475
$754$	−32.6477	−1.18896
$755$	−57.0179	−2.07509
$756$	0	0
$757$	−14.9534	−0.543491	−0.271746	−	0.962369i	$-0.587601\pi$
$757$	−14.9534	−0.543491	−0.271746	+	0.962369i	$0.587601\pi$
$758$	0.116960	0.00424817
$759$	2.34092	0.0849699
$760$	59.0618	2.14240
$761$	−25.1974	−0.913404	−0.456702	−	0.889620i	$-0.650969\pi$
$761$	−25.1974	−0.913404	−0.456702	+	0.889620i	$0.650969\pi$
$762$	−9.66237	−0.350031
$763$	0	0
$764$	3.99335	0.144474
$765$	15.3058	0.553384
$766$	−11.0106	−0.397828
$767$	7.18003	0.259256
$768$	−15.5020	−0.559379
$769$	−9.10070	−0.328180	−0.164090	−	0.986445i	$-0.552469\pi$
$769$	−9.10070	−0.328180	−0.164090	+	0.986445i	$0.552469\pi$
$770$	0	0
$771$	5.83445	0.210122
$772$	−8.66002	−0.311681
$773$	13.8184	0.497015	0.248507	−	0.968630i	$-0.420060\pi$
$773$	13.8184	0.497015	0.248507	+	0.968630i	$0.420060\pi$
$774$	5.73867	0.206272
$775$	70.4138	2.52934
$776$	−12.9714	−0.465646
$777$	0	0
$778$	−30.6420	−1.09857
$779$	5.23036	0.187397
$780$	10.5438	0.377527
$781$	−18.7556	−0.671129
$782$	−7.06981	−0.252816
$783$	−8.04051	−0.287344
$784$	0	0
$785$	12.7581	0.455356
$786$	−20.3288	−0.725104
$787$	−24.4151	−0.870306	−0.435153	−	0.900357i	$-0.643306\pi$
$787$	−24.4151	−0.870306	−0.435153	+	0.900357i	$0.643306\pi$
$788$	2.90915	0.103634
$789$	−1.97786	−0.0704136
$790$	23.6119	0.840074
$791$	0	0
$792$	−4.67413	−0.166088
$793$	−40.1201	−1.42471
$794$	7.72856	0.274276
$795$	35.0349	1.24256
$796$	−11.2520	−0.398815
$797$	−33.4142	−1.18359	−0.591796	−	0.806088i	$-0.701581\pi$
$797$	−33.4142	−1.18359	−0.591796	+	0.806088i	$0.701581\pi$
$798$	0	0
$799$	12.1580	0.430120
$800$	35.1004	1.24098
$801$	18.1860	0.642569
$802$	−13.3281	−0.470632
$803$	21.0999	0.744601
$804$	−6.20757	−0.218924
$805$	0	0
$806$	33.5424	1.18148
$807$	23.3647	0.822478
$808$	38.3385	1.34874
$809$	−19.3112	−0.678947	−0.339473	−	0.940616i	$-0.610249\pi$
$809$	−19.3112	−0.678947	−0.339473	+	0.940616i	$0.610249\pi$
$810$	−4.06198	−0.142724
$811$	39.8362	1.39884	0.699418	−	0.714713i	$-0.253442\pi$
$811$	39.8362	1.39884	0.699418	+	0.714713i	$0.253442\pi$
$812$	0	0
$813$	23.7310	0.832281
$814$	−15.9228	−0.558092
$815$	69.0580	2.41900
$816$	7.62397	0.266892
$817$	27.1740	0.950699
$818$	−14.2206	−0.497213
$819$	0	0
$820$	2.86824	0.100163
$821$	39.6593	1.38412	0.692059	−	0.721841i	$-0.256704\pi$
$821$	39.6593	1.38412	0.692059	+	0.721841i	$0.256704\pi$
$822$	−2.76804	−0.0965464
$823$	−19.0595	−0.664373	−0.332186	−	0.943214i	$-0.607786\pi$
$823$	−19.0595	−0.664373	−0.332186	+	0.943214i	$0.607786\pi$
$824$	54.0418	1.88264
$825$	12.9750	0.451732
$826$	0	0
$827$	17.1040	0.594763	0.297381	−	0.954759i	$-0.403887\pi$
$827$	17.1040	0.594763	0.297381	+	0.954759i	$0.403887\pi$
$828$	−1.19944	−0.0416833
$829$	−10.2235	−0.355079	−0.177539	−	0.984114i	$-0.556814\pi$
$829$	−10.2235	−0.355079	−0.177539	+	0.984114i	$0.556814\pi$
$830$	47.1854	1.63783
$831$	21.0539	0.730353
$832$	30.1878	1.04657
$833$	0	0
$834$	4.70450	0.162904
$835$	−37.6742	−1.30377
$836$	−6.20974	−0.214768
$837$	8.26086	0.285537
$838$	31.8729	1.10103
$839$	−46.7936	−1.61549	−0.807747	−	0.589530i	$-0.799313\pi$
$839$	−46.7936	−1.61549	−0.807747	+	0.589530i	$0.799313\pi$
$840$	0	0
$841$	35.6498	1.22930
$842$	7.69903	0.265326
$843$	−25.8960	−0.891906
$844$	10.3896	0.357623
$845$	−1.88741	−0.0649289
$846$	−3.22659	−0.110933
$847$	0	0
$848$	17.4512	0.599276
$849$	−25.7797	−0.884758
$850$	−39.1859	−1.34406
$851$	−14.5635	−0.499230
$852$	9.60999	0.329233
$853$	−5.07317	−0.173702	−0.0868509	−	0.996221i	$-0.527680\pi$
$853$	−5.07317	−0.173702	−0.0868509	+	0.996221i	$0.527680\pi$
$854$	0	0
$855$	−19.2345	−0.657806
$856$	27.4276	0.937455
$857$	11.5246	0.393674	0.196837	−	0.980436i	$-0.436933\pi$
$857$	11.5246	0.393674	0.196837	+	0.980436i	$0.436933\pi$
$858$	6.18078	0.211009
$859$	−58.4061	−1.99279	−0.996395	−	0.0848356i	$-0.972963\pi$
$859$	−58.4061	−1.99279	−0.996395	+	0.0848356i	$0.972963\pi$
$860$	14.9018	0.508146
$861$	0	0
$862$	38.3018	1.30456
$863$	22.5392	0.767244	0.383622	−	0.923490i	$-0.374677\pi$
$863$	22.5392	0.767244	0.383622	+	0.923490i	$0.374677\pi$
$864$	4.11793	0.140095
$865$	−51.5139	−1.75153
$866$	−42.1754	−1.43318
$867$	0.322719	0.0109601
$868$	0	0
$869$	−8.84847	−0.300164
$870$	32.6604	1.10729
$871$	29.2573	0.991347
$872$	40.2184	1.36197
$873$	4.22436	0.142973
$874$	8.88448	0.300522
$875$	0	0
$876$	−10.8112	−0.365276
$877$	4.63609	0.156550	0.0782749	−	0.996932i	$-0.475059\pi$
$877$	4.63609	0.156550	0.0782749	+	0.996932i	$0.475059\pi$
$878$	−33.8829	−1.14349
$879$	20.8476	0.703171
$880$	10.2541	0.345666
$881$	−34.1449	−1.15037	−0.575185	−	0.818023i	$-0.695070\pi$
$881$	−34.1449	−1.15037	−0.575185	+	0.818023i	$0.695070\pi$
$882$	0	0
$883$	−1.63476	−0.0550141	−0.0275071	−	0.999622i	$-0.508757\pi$
$883$	−1.63476	−0.0550141	−0.0275071	+	0.999622i	$0.508757\pi$
$884$	11.9331	0.401355
$885$	−7.18283	−0.241448
$886$	−21.7086	−0.729315
$887$	2.28069	0.0765781	0.0382890	−	0.999267i	$-0.487809\pi$
$887$	2.28069	0.0765781	0.0382890	+	0.999267i	$0.487809\pi$
$888$	29.0790	0.975829
$889$	0	0
$890$	−73.8710	−2.47616
$891$	1.52221	0.0509960
$892$	18.6260	0.623643
$893$	−15.2787	−0.511283
$894$	−4.66307	−0.155956
$895$	−22.8366	−0.763345
$896$	0	0
$897$	5.65315	0.188753
$898$	36.6598	1.22335
$899$	−66.4216	−2.21528
$900$	−6.64813	−0.221604
$901$	39.6515	1.32098
$902$	1.68137	0.0559836
$903$	0	0
$904$	−2.02188	−0.0672468
$905$	80.6573	2.68114
$906$	17.1258	0.568967
$907$	−4.61452	−0.153223	−0.0766113	−	0.997061i	$-0.524410\pi$
$907$	−4.61452	−0.153223	−0.0766113	+	0.997061i	$0.524410\pi$
$908$	−2.94049	−0.0975836
$909$	−12.4856	−0.414121
$910$	0	0
$911$	37.2131	1.23292	0.616462	−	0.787385i	$-0.288565\pi$
$911$	37.2131	1.23292	0.616462	+	0.787385i	$0.288565\pi$
$912$	−9.58087	−0.317254
$913$	−17.6825	−0.585207
$914$	−20.6723	−0.683778
$915$	40.1358	1.32685
$916$	5.08215	0.167919
$917$	0	0
$918$	−4.59724	−0.151731
$919$	−25.5836	−0.843926	−0.421963	−	0.906613i	$-0.638659\pi$
$919$	−25.5836	−0.843926	−0.421963	+	0.906613i	$0.638659\pi$
$920$	17.3655	0.572522
$921$	−9.56642	−0.315224
$922$	−0.282796	−0.00931341
$923$	−45.2935	−1.49085
$924$	0	0
$925$	−80.7211	−2.65409
$926$	−23.3088	−0.765974
$927$	−17.5997	−0.578048
$928$	−33.1103	−1.08690
$929$	−59.7088	−1.95898	−0.979491	−	0.201488i	$-0.935422\pi$
$929$	−59.7088	−1.95898	−0.979491	+	0.201488i	$0.935422\pi$
$930$	−33.5555	−1.10033
$931$	0	0
$932$	6.38115	0.209021
$933$	−16.1256	−0.527929
$934$	−1.13334	−0.0370839
$935$	23.2987	0.761950
$936$	−11.2877	−0.368950
$937$	−2.29189	−0.0748728	−0.0374364	−	0.999299i	$-0.511919\pi$
$937$	−2.29189	−0.0748728	−0.0374364	+	0.999299i	$0.511919\pi$
$938$	0	0
$939$	21.2132	0.692266
$940$	−8.37859	−0.273279
$941$	31.1495	1.01544	0.507722	−	0.861521i	$-0.330488\pi$
$941$	31.1495	1.01544	0.507722	+	0.861521i	$0.330488\pi$
$942$	−3.83200	−0.124853
$943$	1.53784	0.0500790
$944$	−3.57783	−0.116448
$945$	0	0
$946$	8.73547	0.284015
$947$	−18.6973	−0.607580	−0.303790	−	0.952739i	$-0.598252\pi$
$947$	−18.6973	−0.607580	−0.303790	+	0.952739i	$0.598252\pi$
$948$	4.53377	0.147250
$949$	50.9549	1.65407
$950$	49.2440	1.59769
$951$	13.4949	0.437603
$952$	0	0
$953$	15.2900	0.495291	0.247646	−	0.968851i	$-0.420343\pi$
$953$	15.2900	0.495291	0.247646	+	0.968851i	$0.420343\pi$
$954$	−10.5230	−0.340696
$955$	18.8287	0.609282
$956$	10.6878	0.345668
$957$	−12.2394	−0.395642
$958$	30.0487	0.970829
$959$	0	0
$960$	−30.1996	−0.974687
$961$	37.2419	1.20135
$962$	−38.4524	−1.23975
$963$	−8.93227	−0.287838
$964$	10.3633	0.333781
$965$	−40.8321	−1.31443
$966$	0	0
$967$	−36.5656	−1.17587	−0.587934	−	0.808909i	$-0.700059\pi$
$967$	−36.5656	−1.17587	−0.587934	+	0.808909i	$0.700059\pi$
$968$	26.6618	0.856942
$969$	−21.7691	−0.699323
$970$	−17.1593	−0.550951
$971$	8.08521	0.259467	0.129733	−	0.991549i	$-0.458588\pi$
$971$	8.08521	0.259467	0.129733	+	0.991549i	$0.458588\pi$
$972$	−0.779950	−0.0250169
$973$	0	0
$974$	13.4040	0.429493
$975$	31.3337	1.00348
$976$	19.9920	0.639928
$977$	50.6243	1.61962	0.809808	−	0.586695i	$-0.199571\pi$
$977$	50.6243	1.61962	0.809808	+	0.586695i	$0.199571\pi$
$978$	−20.7421	−0.663260
$979$	27.6829	0.884749
$980$	0	0
$981$	−13.0978	−0.418181
$982$	29.7581	0.949619
$983$	−41.8918	−1.33614	−0.668070	−	0.744098i	$-0.732879\pi$
$983$	−41.8918	−1.33614	−0.668070	+	0.744098i	$0.732879\pi$
$984$	−3.07062	−0.0978877
$985$	13.7167	0.437049
$986$	36.9641	1.17718
$987$	0	0
$988$	−14.9961	−0.477089
$989$	7.98975	0.254059
$990$	−6.18320	−0.196515
$991$	−29.9239	−0.950564	−0.475282	−	0.879833i	$-0.657654\pi$
$991$	−29.9239	−0.950564	−0.475282	+	0.879833i	$0.657654\pi$
$992$	34.0176	1.08006
$993$	5.17282	0.164155
$994$	0	0
$995$	−53.0531	−1.68190
$996$	9.06017	0.287082
$997$	11.3961	0.360919	0.180459	−	0.983582i	$-0.442242\pi$
$997$	11.3961	0.360919	0.180459	+	0.983582i	$0.442242\pi$
$998$	−8.52433	−0.269833
$999$	−9.47010	−0.299621

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.bm.1.12	yes	16
7.6	odd	2		6027.2.a.bl.1.12	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.bl.1.12	✓	16	7.6	odd	2
6027.2.a.bm.1.12	yes	16	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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q+1.10456 q^{2} +1.00000 q^{3} -0.779950 q^{4} -3.67747 q^{5} +1.10456 q^{6} -3.07062 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.10456 q^{2} +1.00000 q^{3} -0.779950 q^{4} -3.67747 q^{5} +1.10456 q^{6} -3.07062 q^{8} +1.00000 q^{9} -4.06198 q^{10} +1.52221 q^{11} -0.779950 q^{12} +3.67604 q^{13} -3.67747 q^{15} -1.83178 q^{16} -4.16206 q^{17} +1.10456 q^{18} +5.23036 q^{19} +2.86824 q^{20} +1.68137 q^{22} +1.53784 q^{23} -3.07062 q^{24} +8.52379 q^{25} +4.06040 q^{26} +1.00000 q^{27} -8.04051 q^{29} -4.06198 q^{30} +8.26086 q^{31} +4.11793 q^{32} +1.52221 q^{33} -4.59724 q^{34} -0.779950 q^{36} -9.47010 q^{37} +5.77725 q^{38} +3.67604 q^{39} +11.2921 q^{40} +1.00000 q^{41} +5.19544 q^{43} -1.18725 q^{44} -3.67747 q^{45} +1.69863 q^{46} -2.92116 q^{47} -1.83178 q^{48} +9.41503 q^{50} -4.16206 q^{51} -2.86712 q^{52} -9.52691 q^{53} +1.10456 q^{54} -5.59789 q^{55} +5.23036 q^{57} -8.88122 q^{58} +1.95320 q^{59} +2.86824 q^{60} -10.9140 q^{61} +9.12461 q^{62} +8.21205 q^{64} -13.5185 q^{65} +1.68137 q^{66} +7.95894 q^{67} +3.24619 q^{68} +1.53784 q^{69} -12.3213 q^{71} -3.07062 q^{72} +13.8614 q^{73} -10.4603 q^{74} +8.52379 q^{75} -4.07942 q^{76} +4.06040 q^{78} -5.81290 q^{79} +6.73631 q^{80} +1.00000 q^{81} +1.10456 q^{82} -11.6164 q^{83} +15.3058 q^{85} +5.73867 q^{86} -8.04051 q^{87} -4.67413 q^{88} +18.1860 q^{89} -4.06198 q^{90} -1.19944 q^{92} +8.26086 q^{93} -3.22659 q^{94} -19.2345 q^{95} +4.11793 q^{96} +4.22436 q^{97} +1.52221 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9	\( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9} - 4 q^{10} - 4 q^{11} + 12 q^{12} - 12 q^{15} - 8 q^{17} - 4 q^{18} + 4 q^{19} - 20 q^{20} - 16 q^{22} - 12 q^{23} - 12 q^{24} - 8 q^{25} - 8 q^{26} + 16 q^{27} - 16 q^{29} - 4 q^{30} - 4 q^{31} - 48 q^{32} - 4 q^{33} + 16 q^{34} + 12 q^{36} - 48 q^{37} - 4 q^{38} + 56 q^{40} + 16 q^{41} - 16 q^{43} - 12 q^{45} - 4 q^{46} - 36 q^{47} - 8 q^{50} - 8 q^{51} - 60 q^{53} - 4 q^{54} + 8 q^{55} + 4 q^{57} - 36 q^{58} - 36 q^{59} - 20 q^{60} - 4 q^{61} - 12 q^{62} + 52 q^{64} - 36 q^{65} - 16 q^{66} - 52 q^{67} - 8 q^{68} - 12 q^{69} - 12 q^{71} - 12 q^{72} - 16 q^{73} + 4 q^{74} - 8 q^{75} + 16 q^{76} - 8 q^{78} - 36 q^{79} - 68 q^{80} + 16 q^{81} - 4 q^{82} - 32 q^{83} - 28 q^{85} - 8 q^{86} - 16 q^{87} - 36 q^{88} - 12 q^{89} - 4 q^{90} - 36 q^{92} - 4 q^{93} + 24 q^{94} - 20 q^{95} - 48 q^{96} + 48 q^{97} - 4 q^{99}+O(q^{100}) \) 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9 - 4 * q^10 - 4 * q^11 + 12 * q^12 - 12 * q^15 - 8 * q^17 - 4 * q^18 + 4 * q^19 - 20 * q^20 - 16 * q^22 - 12 * q^23 - 12 * q^24 - 8 * q^25 - 8 * q^26 + 16 * q^27 - 16 * q^29 - 4 * q^30 - 4 * q^31 - 48 * q^32 - 4 * q^33 + 16 * q^34 + 12 * q^36 - 48 * q^37 - 4 * q^38 + 56 * q^40 + 16 * q^41 - 16 * q^43 - 12 * q^45 - 4 * q^46 - 36 * q^47 - 8 * q^50 - 8 * q^51 - 60 * q^53 - 4 * q^54 + 8 * q^55 + 4 * q^57 - 36 * q^58 - 36 * q^59 - 20 * q^60 - 4 * q^61 - 12 * q^62 + 52 * q^64 - 36 * q^65 - 16 * q^66 - 52 * q^67 - 8 * q^68 - 12 * q^69 - 12 * q^71 - 12 * q^72 - 16 * q^73 + 4 * q^74 - 8 * q^75 + 16 * q^76 - 8 * q^78 - 36 * q^79 - 68 * q^80 + 16 * q^81 - 4 * q^82 - 32 * q^83 - 28 * q^85 - 8 * q^86 - 16 * q^87 - 36 * q^88 - 12 * q^89 - 4 * q^90 - 36 * q^92 - 4 * q^93 + 24 * q^94 - 20 * q^95 - 48 * q^96 + 48 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

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Database

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