LMFDB - Embedded newform 6027.2.a.ba.1.6

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 8x^{6} + 14x^{5} + 18x^{4} - 24x^{3} - 10x^{2} + 10x - 1 $ x^8 - 2x^7 - 8x^6 + 14x^5 + 18x^4 - 24x^3 - 10x^2 + 10*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.896239$ 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)

$f(q)$	$=$	$q+0.896239 q^{2} +1.00000 q^{3} -1.19676 q^{4} -1.54330 q^{5} +0.896239 q^{6} -2.86506 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.896239 q^{2} +1.00000 q^{3} -1.19676 q^{4} -1.54330 q^{5} +0.896239 q^{6} -2.86506 q^{8} +1.00000 q^{9} -1.38317 q^{10} +4.37007 q^{11} -1.19676 q^{12} -4.62531 q^{13} -1.54330 q^{15} -0.174261 q^{16} +3.03264 q^{17} +0.896239 q^{18} +5.57781 q^{19} +1.84696 q^{20} +3.91662 q^{22} -8.66757 q^{23} -2.86506 q^{24} -2.61822 q^{25} -4.14538 q^{26} +1.00000 q^{27} -0.974240 q^{29} -1.38317 q^{30} +2.60877 q^{31} +5.57393 q^{32} +4.37007 q^{33} +2.71797 q^{34} -1.19676 q^{36} +1.30928 q^{37} +4.99905 q^{38} -4.62531 q^{39} +4.42165 q^{40} -1.00000 q^{41} +3.47112 q^{43} -5.22991 q^{44} -1.54330 q^{45} -7.76821 q^{46} -9.89136 q^{47} -0.174261 q^{48} -2.34655 q^{50} +3.03264 q^{51} +5.53537 q^{52} -10.8467 q^{53} +0.896239 q^{54} -6.74434 q^{55} +5.57781 q^{57} -0.873151 q^{58} +2.06228 q^{59} +1.84696 q^{60} +11.1606 q^{61} +2.33808 q^{62} +5.34410 q^{64} +7.13825 q^{65} +3.91662 q^{66} -4.68999 q^{67} -3.62934 q^{68} -8.66757 q^{69} +8.11541 q^{71} -2.86506 q^{72} -4.54207 q^{73} +1.17343 q^{74} -2.61822 q^{75} -6.67528 q^{76} -4.14538 q^{78} -10.0347 q^{79} +0.268937 q^{80} +1.00000 q^{81} -0.896239 q^{82} -3.48090 q^{83} -4.68029 q^{85} +3.11095 q^{86} -0.974240 q^{87} -12.5205 q^{88} +12.9147 q^{89} -1.38317 q^{90} +10.3730 q^{92} +2.60877 q^{93} -8.86502 q^{94} -8.60825 q^{95} +5.57393 q^{96} -14.9037 q^{97} +4.37007 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 8 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 2 * q^2 + 8 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 6 * q^8 + 8 * q^9	$ 8 q - 2 q^{2} + 8 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{8} + 8 q^{9} + 2 q^{10} - 2 q^{11} + 4 q^{12} - 4 q^{13} - 2 q^{15} - 8 q^{17} - 2 q^{18} - 6 q^{19} - 4 q^{20} - 14 q^{22} - 12 q^{23} - 6 q^{24} - 4 q^{25} - 4 q^{26} + 8 q^{27} - 4 q^{29} + 2 q^{30} + 10 q^{31} - 4 q^{32} - 2 q^{33} - 4 q^{34} + 4 q^{36} - 20 q^{37} + 18 q^{38} - 4 q^{39} - 12 q^{40} - 8 q^{41} - 8 q^{43} + 20 q^{44} - 2 q^{45} - 12 q^{46} - 24 q^{47} - 22 q^{50} - 8 q^{51} + 30 q^{52} - 36 q^{53} - 2 q^{54} - 4 q^{55} - 6 q^{57} + 14 q^{58} - 10 q^{59} - 4 q^{60} + 22 q^{61} - 30 q^{62} - 24 q^{64} + 8 q^{65} - 14 q^{66} - 14 q^{67} - 38 q^{68} - 12 q^{69} - 10 q^{71} - 6 q^{72} + 12 q^{73} - 2 q^{74} - 4 q^{75} - 32 q^{76} - 4 q^{78} + 16 q^{79} + 14 q^{80} + 8 q^{81} + 2 q^{82} - 24 q^{83} - 44 q^{85} + 36 q^{86} - 4 q^{87} - 34 q^{88} - 2 q^{89} + 2 q^{90} - 48 q^{92} + 10 q^{93} + 34 q^{94} - 24 q^{95} - 4 q^{96} - 16 q^{97} - 2 q^{99}+O(q^{100}) $ 8 * q - 2 * q^2 + 8 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 6 * q^8 + 8 * q^9 + 2 * q^10 - 2 * q^11 + 4 * q^12 - 4 * q^13 - 2 * q^15 - 8 * q^17 - 2 * q^18 - 6 * q^19 - 4 * q^20 - 14 * q^22 - 12 * q^23 - 6 * q^24 - 4 * q^25 - 4 * q^26 + 8 * q^27 - 4 * q^29 + 2 * q^30 + 10 * q^31 - 4 * q^32 - 2 * q^33 - 4 * q^34 + 4 * q^36 - 20 * q^37 + 18 * q^38 - 4 * q^39 - 12 * q^40 - 8 * q^41 - 8 * q^43 + 20 * q^44 - 2 * q^45 - 12 * q^46 - 24 * q^47 - 22 * q^50 - 8 * q^51 + 30 * q^52 - 36 * q^53 - 2 * q^54 - 4 * q^55 - 6 * q^57 + 14 * q^58 - 10 * q^59 - 4 * q^60 + 22 * q^61 - 30 * q^62 - 24 * q^64 + 8 * q^65 - 14 * q^66 - 14 * q^67 - 38 * q^68 - 12 * q^69 - 10 * q^71 - 6 * q^72 + 12 * q^73 - 2 * q^74 - 4 * q^75 - 32 * q^76 - 4 * q^78 + 16 * q^79 + 14 * q^80 + 8 * q^81 + 2 * q^82 - 24 * q^83 - 44 * q^85 + 36 * q^86 - 4 * q^87 - 34 * q^88 - 2 * q^89 + 2 * q^90 - 48 * q^92 + 10 * q^93 + 34 * q^94 - 24 * q^95 - 4 * q^96 - 16 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.896239	0.633736	0.316868	−	0.948470i	$-0.397369\pi$
$2$	0.896239	0.633736	0.316868	+	0.948470i	$0.397369\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.19676	−0.598378
$5$	−1.54330	−0.690186	−0.345093	−	0.938568i	$-0.612153\pi$
$5$	−1.54330	−0.690186	−0.345093	+	0.938568i	$0.612153\pi$
$6$	0.896239	0.365888
$7$	0	0
$7$	0	0
$8$	−2.86506	−1.01295
$9$	1.00000	0.333333
$10$	−1.38317	−0.437396
$11$	4.37007	1.31762	0.658812	−	0.752307i	$-0.271059\pi$
$11$	4.37007	1.31762	0.658812	+	0.752307i	$0.271059\pi$
$12$	−1.19676	−0.345474
$13$	−4.62531	−1.28283	−0.641415	−	0.767194i	$-0.721652\pi$
$13$	−4.62531	−1.28283	−0.641415	+	0.767194i	$0.721652\pi$
$14$	0	0
$15$	−1.54330	−0.398479
$16$	−0.174261	−0.0435652
$17$	3.03264	0.735524	0.367762	−	0.929920i	$-0.380124\pi$
$17$	3.03264	0.735524	0.367762	+	0.929920i	$0.380124\pi$
$18$	0.896239	0.211245
$19$	5.57781	1.27964	0.639819	−	0.768526i	$-0.279009\pi$
$19$	5.57781	1.27964	0.639819	+	0.768526i	$0.279009\pi$
$20$	1.84696	0.412992
$21$	0	0
$22$	3.91662	0.835027
$23$	−8.66757	−1.80731	−0.903657	−	0.428257i	$-0.859128\pi$
$23$	−8.66757	−1.80731	−0.903657	+	0.428257i	$0.859128\pi$
$24$	−2.86506	−0.584827
$25$	−2.61822	−0.523643
$26$	−4.14538	−0.812976
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.974240	−0.180912	−0.0904559	−	0.995900i	$-0.528832\pi$
$29$	−0.974240	−0.180912	−0.0904559	+	0.995900i	$0.528832\pi$
$30$	−1.38317	−0.252531
$31$	2.60877	0.468549	0.234275	−	0.972170i	$-0.424729\pi$
$31$	2.60877	0.468549	0.234275	+	0.972170i	$0.424729\pi$
$32$	5.57393	0.985342
$33$	4.37007	0.760731
$34$	2.71797	0.466128
$35$	0	0
$36$	−1.19676	−0.199459
$37$	1.30928	0.215245	0.107622	−	0.994192i	$-0.465676\pi$
$37$	1.30928	0.215245	0.107622	+	0.994192i	$0.465676\pi$
$38$	4.99905	0.810953
$39$	−4.62531	−0.740642
$40$	4.42165	0.699124
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	3.47112	0.529341	0.264671	−	0.964339i	$-0.414737\pi$
$43$	3.47112	0.529341	0.264671	+	0.964339i	$0.414737\pi$
$44$	−5.22991	−0.788438
$45$	−1.54330	−0.230062
$46$	−7.76821	−1.14536
$47$	−9.89136	−1.44280	−0.721402	−	0.692517i	$-0.756502\pi$
$47$	−9.89136	−1.44280	−0.721402	+	0.692517i	$0.756502\pi$
$48$	−0.174261	−0.0251524
$49$	0	0
$50$	−2.34655	−0.331852
$51$	3.03264	0.424655
$52$	5.53537	0.767618
$53$	−10.8467	−1.48991	−0.744955	−	0.667115i	$-0.767529\pi$
$53$	−10.8467	−1.48991	−0.744955	+	0.667115i	$0.767529\pi$
$54$	0.896239	0.121963
$55$	−6.74434	−0.909406
$56$	0	0
$57$	5.57781	0.738799
$58$	−0.873151	−0.114650
$59$	2.06228	0.268486	0.134243	−	0.990948i	$-0.457140\pi$
$59$	2.06228	0.268486	0.134243	+	0.990948i	$0.457140\pi$
$60$	1.84696	0.238441
$61$	11.1606	1.42896	0.714481	−	0.699655i	$-0.246663\pi$
$61$	11.1606	1.42896	0.714481	+	0.699655i	$0.246663\pi$
$62$	2.33808	0.296937
$63$	0	0
$64$	5.34410	0.668012
$65$	7.13825	0.885391
$66$	3.91662	0.482103
$67$	−4.68999	−0.572973	−0.286487	−	0.958084i	$-0.592487\pi$
$67$	−4.68999	−0.572973	−0.286487	+	0.958084i	$0.592487\pi$
$68$	−3.62934	−0.440122
$69$	−8.66757	−1.04345
$70$	0	0
$71$	8.11541	0.963121	0.481561	−	0.876413i	$-0.340070\pi$
$71$	8.11541	0.963121	0.481561	+	0.876413i	$0.340070\pi$
$72$	−2.86506	−0.337650
$73$	−4.54207	−0.531609	−0.265804	−	0.964027i	$-0.585638\pi$
$73$	−4.54207	−0.531609	−0.265804	+	0.964027i	$0.585638\pi$
$74$	1.17343	0.136409
$75$	−2.61822	−0.302326
$76$	−6.67528	−0.765707
$77$	0	0
$78$	−4.14538	−0.469372
$79$	−10.0347	−1.12899	−0.564494	−	0.825437i	$-0.690929\pi$
$79$	−10.0347	−1.12899	−0.564494	+	0.825437i	$0.690929\pi$
$80$	0.268937	0.0300681
$81$	1.00000	0.111111
$82$	−0.896239	−0.0989730
$83$	−3.48090	−0.382078	−0.191039	−	0.981582i	$-0.561186\pi$
$83$	−3.48090	−0.382078	−0.191039	+	0.981582i	$0.561186\pi$
$84$	0	0
$85$	−4.68029	−0.507649
$86$	3.11095	0.335463
$87$	−0.974240	−0.104449
$88$	−12.5205	−1.33469
$89$	12.9147	1.36896	0.684479	−	0.729032i	$-0.260030\pi$
$89$	12.9147	1.36896	0.684479	+	0.729032i	$0.260030\pi$
$90$	−1.38317	−0.145799
$91$	0	0
$92$	10.3730	1.08146
$93$	2.60877	0.270517
$94$	−8.86502	−0.914357
$95$	−8.60825	−0.883188
$96$	5.57393	0.568887
$97$	−14.9037	−1.51324	−0.756621	−	0.653853i	$-0.773151\pi$
$97$	−14.9037	−1.51324	−0.756621	+	0.653853i	$0.773151\pi$
$98$	0	0
$99$	4.37007	0.439208
$100$	3.13337	0.313337
$101$	−13.2863	−1.32204	−0.661018	−	0.750370i	$-0.729875\pi$
$101$	−13.2863	−1.32204	−0.661018	+	0.750370i	$0.729875\pi$
$102$	2.71797	0.269119
$103$	−10.3002	−1.01491	−0.507453	−	0.861680i	$-0.669413\pi$
$103$	−10.3002	−1.01491	−0.507453	+	0.861680i	$0.669413\pi$
$104$	13.2518	1.29944
$105$	0	0
$106$	−9.72124	−0.944210
$107$	18.8200	1.81939	0.909697	−	0.415273i	$-0.136314\pi$
$107$	18.8200	1.81939	0.909697	+	0.415273i	$0.136314\pi$
$108$	−1.19676	−0.115158
$109$	−14.0229	−1.34315	−0.671575	−	0.740936i	$-0.734382\pi$
$109$	−14.0229	−1.34315	−0.671575	+	0.740936i	$0.734382\pi$
$110$	−6.04453	−0.576324
$111$	1.30928	0.124272
$112$	0	0
$113$	−21.1570	−1.99029	−0.995143	−	0.0984424i	$-0.968614\pi$
$113$	−21.1570	−1.99029	−0.995143	+	0.0984424i	$0.968614\pi$
$114$	4.99905	0.468204
$115$	13.3767	1.24738
$116$	1.16593	0.108254
$117$	−4.62531	−0.427610
$118$	1.84829	0.170149
$119$	0	0
$120$	4.42165	0.403640
$121$	8.09748	0.736135
$122$	10.0025	0.905585
$123$	−1.00000	−0.0901670
$124$	−3.12206	−0.280370
$125$	11.7572	1.05160
$126$	0	0
$127$	−3.91184	−0.347120	−0.173560	−	0.984823i	$-0.555527\pi$
$127$	−3.91184	−0.347120	−0.173560	+	0.984823i	$0.555527\pi$
$128$	−6.35828	−0.561998
$129$	3.47112	0.305615
$130$	6.39758	0.561105
$131$	15.5731	1.36063	0.680315	−	0.732920i	$-0.261843\pi$
$131$	15.5731	1.36063	0.680315	+	0.732920i	$0.261843\pi$
$132$	−5.22991	−0.455205
$133$	0	0
$134$	−4.20335	−0.363114
$135$	−1.54330	−0.132826
$136$	−8.68870	−0.745050
$137$	−11.0741	−0.946121	−0.473061	−	0.881030i	$-0.656851\pi$
$137$	−11.0741	−0.946121	−0.473061	+	0.881030i	$0.656851\pi$
$138$	−7.76821	−0.661274
$139$	−15.7862	−1.33897	−0.669485	−	0.742825i	$-0.733485\pi$
$139$	−15.7862	−1.33897	−0.669485	+	0.742825i	$0.733485\pi$
$140$	0	0
$141$	−9.89136	−0.833003
$142$	7.27334	0.610365
$143$	−20.2129	−1.69029
$144$	−0.174261	−0.0145217
$145$	1.50355	0.124863
$146$	−4.07078	−0.336900
$147$	0	0
$148$	−1.56689	−0.128798
$149$	0.987946	0.0809357	0.0404679	−	0.999181i	$-0.487115\pi$
$149$	0.987946	0.0809357	0.0404679	+	0.999181i	$0.487115\pi$
$150$	−2.34655	−0.191595
$151$	−20.1158	−1.63700	−0.818502	−	0.574504i	$-0.805195\pi$
$151$	−20.1158	−1.63700	−0.818502	+	0.574504i	$0.805195\pi$
$152$	−15.9807	−1.29621
$153$	3.03264	0.245175
$154$	0	0
$155$	−4.02613	−0.323386
$156$	5.53537	0.443184
$157$	6.11461	0.487999	0.244000	−	0.969775i	$-0.421540\pi$
$157$	6.11461	0.487999	0.244000	+	0.969775i	$0.421540\pi$
$158$	−8.99345	−0.715480
$159$	−10.8467	−0.860200
$160$	−8.60227	−0.680069
$161$	0	0
$162$	0.896239	0.0704151
$163$	−4.13189	−0.323634	−0.161817	−	0.986821i	$-0.551735\pi$
$163$	−4.13189	−0.323634	−0.161817	+	0.986821i	$0.551735\pi$
$164$	1.19676	0.0934510
$165$	−6.74434	−0.525046
$166$	−3.11972	−0.242137
$167$	−9.87327	−0.764016	−0.382008	−	0.924159i	$-0.624767\pi$
$167$	−9.87327	−0.764016	−0.382008	+	0.924159i	$0.624767\pi$
$168$	0	0
$169$	8.39349	0.645653
$170$	−4.19465	−0.321715
$171$	5.57781	0.426546
$172$	−4.15409	−0.316746
$173$	2.98061	0.226612	0.113306	−	0.993560i	$-0.463856\pi$
$173$	2.98061	0.226612	0.113306	+	0.993560i	$0.463856\pi$
$174$	−0.873151	−0.0661934
$175$	0	0
$176$	−0.761531	−0.0574025
$177$	2.06228	0.155010
$178$	11.5747	0.867559
$179$	−20.1915	−1.50919	−0.754593	−	0.656193i	$-0.772166\pi$
$179$	−20.1915	−1.50919	−0.754593	+	0.656193i	$0.772166\pi$
$180$	1.84696	0.137664
$181$	22.0185	1.63662	0.818310	−	0.574777i	$-0.194911\pi$
$181$	22.0185	1.63662	0.818310	+	0.574777i	$0.194911\pi$
$182$	0	0
$183$	11.1606	0.825012
$184$	24.8331	1.83072
$185$	−2.02062	−0.148559
$186$	2.33808	0.171436
$187$	13.2529	0.969145
$188$	11.8375	0.863342
$189$	0	0
$190$	−7.71505	−0.559708
$191$	−11.2438	−0.813570	−0.406785	−	0.913524i	$-0.633350\pi$
$191$	−11.2438	−0.813570	−0.406785	+	0.913524i	$0.633350\pi$
$192$	5.34410	0.385677
$193$	−9.65346	−0.694871	−0.347436	−	0.937704i	$-0.612948\pi$
$193$	−9.65346	−0.694871	−0.347436	+	0.937704i	$0.612948\pi$
$194$	−13.3573	−0.958997
$195$	7.13825	0.511181
$196$	0	0
$197$	−8.98705	−0.640301	−0.320150	−	0.947367i	$-0.603733\pi$
$197$	−8.98705	−0.640301	−0.320150	+	0.947367i	$0.603733\pi$
$198$	3.91662	0.278342
$199$	−0.603687	−0.0427942	−0.0213971	−	0.999771i	$-0.506811\pi$
$199$	−0.603687	−0.0427942	−0.0213971	+	0.999771i	$0.506811\pi$
$200$	7.50134	0.530425
$201$	−4.68999	−0.330806
$202$	−11.9077	−0.837822
$203$	0	0
$204$	−3.62934	−0.254104
$205$	1.54330	0.107789
$206$	−9.23141	−0.643183
$207$	−8.66757	−0.602438
$208$	0.806009	0.0558867
$209$	24.3754	1.68608
$210$	0	0
$211$	−3.85847	−0.265628	−0.132814	−	0.991141i	$-0.542401\pi$
$211$	−3.85847	−0.265628	−0.132814	+	0.991141i	$0.542401\pi$
$212$	12.9809	0.891530
$213$	8.11541	0.556058
$214$	16.8672	1.15302
$215$	−5.35699	−0.365344
$216$	−2.86506	−0.194942
$217$	0	0
$218$	−12.5679	−0.851203
$219$	−4.54207	−0.306925
$220$	8.07133	0.544169
$221$	−14.0269	−0.943553
$222$	1.17343	0.0787555
$223$	5.49765	0.368150	0.184075	−	0.982912i	$-0.441071\pi$
$223$	5.49765	0.368150	0.184075	+	0.982912i	$0.441071\pi$
$224$	0	0
$225$	−2.61822	−0.174548
$226$	−18.9617	−1.26132
$227$	1.67191	0.110969	0.0554843	−	0.998460i	$-0.482330\pi$
$227$	1.67191	0.110969	0.0554843	+	0.998460i	$0.482330\pi$
$228$	−6.67528	−0.442081
$229$	−26.9495	−1.78087	−0.890436	−	0.455108i	$-0.849601\pi$
$229$	−26.9495	−1.78087	−0.890436	+	0.455108i	$0.849601\pi$
$230$	11.9887	0.790512
$231$	0	0
$232$	2.79125	0.183255
$233$	−20.7406	−1.35876	−0.679380	−	0.733786i	$-0.737751\pi$
$233$	−20.7406	−1.35876	−0.679380	+	0.733786i	$0.737751\pi$
$234$	−4.14538	−0.270992
$235$	15.2654	0.995803
$236$	−2.46804	−0.160656
$237$	−10.0347	−0.651821
$238$	0	0
$239$	−4.11267	−0.266026	−0.133013	−	0.991114i	$-0.542465\pi$
$239$	−4.11267	−0.266026	−0.133013	+	0.991114i	$0.542465\pi$
$240$	0.268937	0.0173598
$241$	0.365350	0.0235343	0.0117671	−	0.999931i	$-0.496254\pi$
$241$	0.365350	0.0235343	0.0117671	+	0.999931i	$0.496254\pi$
$242$	7.25728	0.466515
$243$	1.00000	0.0641500
$244$	−13.3565	−0.855060
$245$	0	0
$246$	−0.896239	−0.0571421
$247$	−25.7991	−1.64156
$248$	−7.47428	−0.474617
$249$	−3.48090	−0.220593
$250$	10.5373	0.666435
$251$	−15.1723	−0.957669	−0.478834	−	0.877905i	$-0.658941\pi$
$251$	−15.1723	−0.957669	−0.478834	+	0.877905i	$0.658941\pi$
$252$	0	0
$253$	−37.8779	−2.38136
$254$	−3.50594	−0.219982
$255$	−4.68029	−0.293091
$256$	−16.3867	−1.02417
$257$	−5.35154	−0.333820	−0.166910	−	0.985972i	$-0.553379\pi$
$257$	−5.35154	−0.333820	−0.166910	+	0.985972i	$0.553379\pi$
$258$	3.11095	0.193680
$259$	0	0
$260$	−8.54275	−0.529799
$261$	−0.974240	−0.0603039
$262$	13.9572	0.862280
$263$	−5.79651	−0.357428	−0.178714	−	0.983901i	$-0.557194\pi$
$263$	−5.79651	−0.357428	−0.178714	+	0.983901i	$0.557194\pi$
$264$	−12.5205	−0.770583
$265$	16.7398	1.02832
$266$	0	0
$267$	12.9147	0.790368
$268$	5.61277	0.342855
$269$	15.4409	0.941449	0.470724	−	0.882280i	$-0.343993\pi$
$269$	15.4409	0.941449	0.470724	+	0.882280i	$0.343993\pi$
$270$	−1.38317	−0.0841769
$271$	16.2731	0.988519	0.494260	−	0.869314i	$-0.335439\pi$
$271$	16.2731	0.988519	0.494260	+	0.869314i	$0.335439\pi$
$272$	−0.528471	−0.0320432
$273$	0	0
$274$	−9.92500	−0.599591
$275$	−11.4418	−0.689965
$276$	10.3730	0.624380
$277$	19.0510	1.14467	0.572333	−	0.820022i	$-0.306039\pi$
$277$	19.0510	1.14467	0.572333	+	0.820022i	$0.306039\pi$
$278$	−14.1482	−0.848554
$279$	2.60877	0.156183
$280$	0	0
$281$	−2.91985	−0.174184	−0.0870919	−	0.996200i	$-0.527757\pi$
$281$	−2.91985	−0.174184	−0.0870919	+	0.996200i	$0.527757\pi$
$282$	−8.86502	−0.527904
$283$	−25.0265	−1.48767	−0.743836	−	0.668362i	$-0.766996\pi$
$283$	−25.0265	−1.48767	−0.743836	+	0.668362i	$0.766996\pi$
$284$	−9.71216	−0.576311
$285$	−8.60825	−0.509909
$286$	−18.1156	−1.07120
$287$	0	0
$288$	5.57393	0.328447
$289$	−7.80307	−0.459004
$290$	1.34754	0.0791301
$291$	−14.9037	−0.873671
$292$	5.43575	0.318103
$293$	25.0951	1.46607	0.733036	−	0.680190i	$-0.238103\pi$
$293$	25.0951	1.46607	0.733036	+	0.680190i	$0.238103\pi$
$294$	0	0
$295$	−3.18272	−0.185305
$296$	−3.75117	−0.218032
$297$	4.37007	0.253577
$298$	0.885435	0.0512919
$299$	40.0902	2.31848
$300$	3.13337	0.180905
$301$	0	0
$302$	−18.0286	−1.03743
$303$	−13.2863	−0.763278
$304$	−0.971993	−0.0557476
$305$	−17.2241	−0.986250
$306$	2.71797	0.155376
$307$	22.5538	1.28722	0.643608	−	0.765356i	$-0.277437\pi$
$307$	22.5538	1.28722	0.643608	+	0.765356i	$0.277437\pi$
$308$	0	0
$309$	−10.3002	−0.585956
$310$	−3.60837	−0.204942
$311$	−7.47367	−0.423793	−0.211897	−	0.977292i	$-0.567964\pi$
$311$	−7.47367	−0.423793	−0.211897	+	0.977292i	$0.567964\pi$
$312$	13.2518	0.750234
$313$	22.4306	1.26785	0.633926	−	0.773394i	$-0.281442\pi$
$313$	22.4306	1.26785	0.633926	+	0.773394i	$0.281442\pi$
$314$	5.48015	0.309263
$315$	0	0
$316$	12.0090	0.675562
$317$	−19.9396	−1.11992	−0.559960	−	0.828520i	$-0.689184\pi$
$317$	−19.9396	−1.11992	−0.559960	+	0.828520i	$0.689184\pi$
$318$	−9.72124	−0.545140
$319$	−4.25749	−0.238374
$320$	−8.24756	−0.461052
$321$	18.8200	1.05043
$322$	0	0
$323$	16.9155	0.941205
$324$	−1.19676	−0.0664865
$325$	12.1101	0.671745
$326$	−3.70316	−0.205099
$327$	−14.0229	−0.775468
$328$	2.86506	0.158196
$329$	0	0
$330$	−6.04453	−0.332741
$331$	34.1797	1.87869	0.939343	−	0.342978i	$-0.111436\pi$
$331$	34.1797	1.87869	0.939343	+	0.342978i	$0.111436\pi$
$332$	4.16579	0.228627
$333$	1.30928	0.0717483
$334$	−8.84880	−0.484185
$335$	7.23807	0.395458
$336$	0	0
$337$	−14.5677	−0.793554	−0.396777	−	0.917915i	$-0.629871\pi$
$337$	−14.5677	−0.793554	−0.396777	+	0.917915i	$0.629871\pi$
$338$	7.52257	0.409174
$339$	−21.1570	−1.14909
$340$	5.60117	0.303766
$341$	11.4005	0.617372
$342$	4.99905	0.270318
$343$	0	0
$344$	−9.94496	−0.536196
$345$	13.3767	0.720177
$346$	2.67134	0.143612
$347$	3.80110	0.204054	0.102027	−	0.994782i	$-0.467467\pi$
$347$	3.80110	0.204054	0.102027	+	0.994782i	$0.467467\pi$
$348$	1.16593	0.0625003
$349$	−17.6840	−0.946605	−0.473302	−	0.880900i	$-0.656938\pi$
$349$	−17.6840	−0.946605	−0.473302	+	0.880900i	$0.656938\pi$
$350$	0	0
$351$	−4.62531	−0.246881
$352$	24.3585	1.29831
$353$	−14.2570	−0.758822	−0.379411	−	0.925228i	$-0.623873\pi$
$353$	−14.2570	−0.758822	−0.379411	+	0.925228i	$0.623873\pi$
$354$	1.84829	0.0982356
$355$	−12.5245	−0.664733
$356$	−15.4558	−0.819155
$357$	0	0
$358$	−18.0964	−0.956426
$359$	−0.560299	−0.0295715	−0.0147857	−	0.999891i	$-0.504707\pi$
$359$	−0.560299	−0.0295715	−0.0147857	+	0.999891i	$0.504707\pi$
$360$	4.42165	0.233041
$361$	12.1120	0.637473
$362$	19.7338	1.03719
$363$	8.09748	0.425008
$364$	0	0
$365$	7.00979	0.366909
$366$	10.0025	0.522840
$367$	−10.4769	−0.546891	−0.273445	−	0.961888i	$-0.588163\pi$
$367$	−10.4769	−0.546891	−0.273445	+	0.961888i	$0.588163\pi$
$368$	1.51042	0.0787359
$369$	−1.00000	−0.0520579
$370$	−1.81096	−0.0941473
$371$	0	0
$372$	−3.12206	−0.161872
$373$	−9.67483	−0.500944	−0.250472	−	0.968124i	$-0.580586\pi$
$373$	−9.67483	−0.500944	−0.250472	+	0.968124i	$0.580586\pi$
$374$	11.8777	0.614182
$375$	11.7572	0.607140
$376$	28.3393	1.46149
$377$	4.50616	0.232079
$378$	0	0
$379$	4.55757	0.234107	0.117053	−	0.993126i	$-0.462655\pi$
$379$	4.55757	0.234107	0.117053	+	0.993126i	$0.462655\pi$
$380$	10.3020	0.528481
$381$	−3.91184	−0.200410
$382$	−10.0771	−0.515589
$383$	10.3658	0.529669	0.264834	−	0.964294i	$-0.414683\pi$
$383$	10.3658	0.529669	0.264834	+	0.964294i	$0.414683\pi$
$384$	−6.35828	−0.324470
$385$	0	0
$386$	−8.65181	−0.440365
$387$	3.47112	0.176447
$388$	17.8361	0.905492
$389$	19.2488	0.975952	0.487976	−	0.872857i	$-0.337735\pi$
$389$	19.2488	0.975952	0.487976	+	0.872857i	$0.337735\pi$
$390$	6.39758	0.323954
$391$	−26.2857	−1.32932
$392$	0	0
$393$	15.5731	0.785560
$394$	−8.05454	−0.405782
$395$	15.4865	0.779211
$396$	−5.22991	−0.262813
$397$	21.4235	1.07521	0.537606	−	0.843196i	$-0.319329\pi$
$397$	21.4235	1.07521	0.537606	+	0.843196i	$0.319329\pi$
$398$	−0.541047	−0.0271203
$399$	0	0
$400$	0.456252	0.0228126
$401$	38.2370	1.90947	0.954733	−	0.297463i	$-0.0961405\pi$
$401$	38.2370	1.90947	0.954733	+	0.297463i	$0.0961405\pi$
$402$	−4.20335	−0.209644
$403$	−12.0664	−0.601069
$404$	15.9005	0.791078
$405$	−1.54330	−0.0766873
$406$	0	0
$407$	5.72166	0.283612
$408$	−8.68870	−0.430155
$409$	23.3734	1.15574	0.577870	−	0.816128i	$-0.303884\pi$
$409$	23.3734	1.15574	0.577870	+	0.816128i	$0.303884\pi$
$410$	1.38317	0.0683098
$411$	−11.0741	−0.546243
$412$	12.3268	0.607298
$413$	0	0
$414$	−7.76821	−0.381787
$415$	5.37208	0.263705
$416$	−25.7812	−1.26403
$417$	−15.7862	−0.773055
$418$	21.8462	1.06853
$419$	−28.6130	−1.39784	−0.698919	−	0.715201i	$-0.746335\pi$
$419$	−28.6130	−1.39784	−0.698919	+	0.715201i	$0.746335\pi$
$420$	0	0
$421$	40.1048	1.95459	0.977294	−	0.211886i	$-0.0679606\pi$
$421$	40.1048	1.95459	0.977294	+	0.211886i	$0.0679606\pi$
$422$	−3.45811	−0.168338
$423$	−9.89136	−0.480934
$424$	31.0764	1.50921
$425$	−7.94012	−0.385152
$426$	7.27334	0.352394
$427$	0	0
$428$	−22.5229	−1.08869
$429$	−20.2129	−0.975889
$430$	−4.80114	−0.231532
$431$	−26.3938	−1.27135	−0.635673	−	0.771959i	$-0.719277\pi$
$431$	−26.3938	−1.27135	−0.635673	+	0.771959i	$0.719277\pi$
$432$	−0.174261	−0.00838412
$433$	34.9483	1.67951	0.839755	−	0.542966i	$-0.182699\pi$
$433$	34.9483	1.67951	0.839755	+	0.542966i	$0.182699\pi$
$434$	0	0
$435$	1.50355	0.0720896
$436$	16.7820	0.803712
$437$	−48.3461	−2.31271
$438$	−4.07078	−0.194509
$439$	17.9289	0.855699	0.427849	−	0.903850i	$-0.359271\pi$
$439$	17.9289	0.855699	0.427849	+	0.903850i	$0.359271\pi$
$440$	19.3229	0.921183
$441$	0	0
$442$	−12.5715	−0.597964
$443$	6.54341	0.310887	0.155443	−	0.987845i	$-0.450319\pi$
$443$	6.54341	0.310887	0.155443	+	0.987845i	$0.450319\pi$
$444$	−1.56689	−0.0743615
$445$	−19.9313	−0.944836
$446$	4.92720	0.233310
$447$	0.987946	0.0467283
$448$	0	0
$449$	−15.1410	−0.714547	−0.357274	−	0.934000i	$-0.616294\pi$
$449$	−15.1410	−0.714547	−0.357274	+	0.934000i	$0.616294\pi$
$450$	−2.34655	−0.110617
$451$	−4.37007	−0.205778
$452$	25.3198	1.19094
$453$	−20.1158	−0.945124
$454$	1.49843	0.0703248
$455$	0	0
$456$	−15.9807	−0.748367
$457$	27.7133	1.29638	0.648188	−	0.761481i	$-0.275527\pi$
$457$	27.7133	1.29638	0.648188	+	0.761481i	$0.275527\pi$
$458$	−24.1532	−1.12860
$459$	3.03264	0.141552
$460$	−16.0086	−0.746407
$461$	14.2264	0.662590	0.331295	−	0.943527i	$-0.392514\pi$
$461$	14.2264	0.662590	0.331295	+	0.943527i	$0.392514\pi$
$462$	0	0
$463$	9.46878	0.440051	0.220026	−	0.975494i	$-0.429386\pi$
$463$	9.46878	0.440051	0.220026	+	0.975494i	$0.429386\pi$
$464$	0.169772	0.00788145
$465$	−4.02613	−0.186707
$466$	−18.5885	−0.861096
$467$	−21.5136	−0.995532	−0.497766	−	0.867311i	$-0.665846\pi$
$467$	−21.5136	−0.995532	−0.497766	+	0.867311i	$0.665846\pi$
$468$	5.53537	0.255873
$469$	0	0
$470$	13.6814	0.631076
$471$	6.11461	0.281746
$472$	−5.90854	−0.271963
$473$	15.1690	0.697473
$474$	−8.99345	−0.413083
$475$	−14.6039	−0.670074
$476$	0	0
$477$	−10.8467	−0.496637
$478$	−3.68593	−0.168591
$479$	−18.0570	−0.825045	−0.412522	−	0.910947i	$-0.635352\pi$
$479$	−18.0570	−0.825045	−0.412522	+	0.910947i	$0.635352\pi$
$480$	−8.60227	−0.392638
$481$	−6.05584	−0.276123
$482$	0.327441	0.0149145
$483$	0	0
$484$	−9.69072	−0.440487
$485$	23.0009	1.04442
$486$	0.896239	0.0406542
$487$	−36.3039	−1.64509	−0.822543	−	0.568703i	$-0.807445\pi$
$487$	−36.3039	−1.64509	−0.822543	+	0.568703i	$0.807445\pi$
$488$	−31.9756	−1.44747
$489$	−4.13189	−0.186850
$490$	0	0
$491$	−20.8236	−0.939758	−0.469879	−	0.882731i	$-0.655702\pi$
$491$	−20.8236	−0.939758	−0.469879	+	0.882731i	$0.655702\pi$
$492$	1.19676	0.0539540
$493$	−2.95452	−0.133065
$494$	−23.1222	−1.04031
$495$	−6.74434	−0.303135
$496$	−0.454606	−0.0204124
$497$	0	0
$498$	−3.11972	−0.139798
$499$	−11.0100	−0.492877	−0.246439	−	0.969158i	$-0.579260\pi$
$499$	−11.0100	−0.492877	−0.246439	+	0.969158i	$0.579260\pi$
$500$	−14.0705	−0.629253
$501$	−9.87327	−0.441105
$502$	−13.5980	−0.606909
$503$	−32.6045	−1.45376	−0.726881	−	0.686763i	$-0.759031\pi$
$503$	−32.6045	−1.45376	−0.726881	+	0.686763i	$0.759031\pi$
$504$	0	0
$505$	20.5048	0.912451
$506$	−33.9476	−1.50915
$507$	8.39349	0.372768
$508$	4.68152	0.207709
$509$	−9.05908	−0.401537	−0.200768	−	0.979639i	$-0.564344\pi$
$509$	−9.05908	−0.401537	−0.200768	+	0.979639i	$0.564344\pi$
$510$	−4.19465	−0.185742
$511$	0	0
$512$	−1.96985	−0.0870559
$513$	5.57781	0.246266
$514$	−4.79626	−0.211554
$515$	15.8963	0.700474
$516$	−4.15409	−0.182874
$517$	−43.2259	−1.90107
$518$	0	0
$519$	2.98061	0.130834
$520$	−20.4515	−0.896858
$521$	0.758356	0.0332242	0.0166121	−	0.999862i	$-0.494712\pi$
$521$	0.758356	0.0332242	0.0166121	+	0.999862i	$0.494712\pi$
$522$	−0.873151	−0.0382168
$523$	−38.3928	−1.67880	−0.839400	−	0.543514i	$-0.817093\pi$
$523$	−38.3928	−1.67880	−0.839400	+	0.543514i	$0.817093\pi$
$524$	−18.6372	−0.814171
$525$	0	0
$526$	−5.19505	−0.226515
$527$	7.91148	0.344629
$528$	−0.761531	−0.0331414
$529$	52.1268	2.26638
$530$	15.0028	0.651681
$531$	2.06228	0.0894952
$532$	0	0
$533$	4.62531	0.200344
$534$	11.5747	0.500885
$535$	−29.0449	−1.25572
$536$	13.4371	0.580393
$537$	−20.1915	−0.871329
$538$	13.8387	0.596630
$539$	0	0
$540$	1.84696	0.0794804
$541$	13.4662	0.578959	0.289479	−	0.957184i	$-0.406518\pi$
$541$	13.4662	0.578959	0.289479	+	0.957184i	$0.406518\pi$
$542$	14.5846	0.626461
$543$	22.0185	0.944903
$544$	16.9038	0.724743
$545$	21.6416	0.927024
$546$	0	0
$547$	41.9335	1.79295	0.896474	−	0.443097i	$-0.146120\pi$
$547$	41.9335	1.79295	0.896474	+	0.443097i	$0.146120\pi$
$548$	13.2530	0.566138
$549$	11.1606	0.476321
$550$	−10.2546	−0.437256
$551$	−5.43413	−0.231502
$552$	24.8331	1.05697
$553$	0	0
$554$	17.0743	0.725416
$555$	−2.02062	−0.0857706
$556$	18.8923	0.801211
$557$	−18.8127	−0.797121	−0.398560	−	0.917142i	$-0.630490\pi$
$557$	−18.8127	−0.797121	−0.398560	+	0.917142i	$0.630490\pi$
$558$	2.33808	0.0989789
$559$	−16.0550	−0.679055
$560$	0	0
$561$	13.2529	0.559536
$562$	−2.61688	−0.110387
$563$	3.57171	0.150530	0.0752649	−	0.997164i	$-0.476020\pi$
$563$	3.57171	0.150530	0.0752649	+	0.997164i	$0.476020\pi$
$564$	11.8375	0.498451
$565$	32.6517	1.37367
$566$	−22.4297	−0.942792
$567$	0	0
$568$	−23.2511	−0.975594
$569$	−18.4542	−0.773641	−0.386821	−	0.922155i	$-0.626427\pi$
$569$	−18.4542	−0.773641	−0.386821	+	0.922155i	$0.626427\pi$
$570$	−7.71505	−0.323148
$571$	3.92800	0.164381	0.0821907	−	0.996617i	$-0.473808\pi$
$571$	3.92800	0.164381	0.0821907	+	0.996617i	$0.473808\pi$
$572$	24.1899	1.01143
$573$	−11.2438	−0.469715
$574$	0	0
$575$	22.6936	0.946388
$576$	5.34410	0.222671
$577$	43.9872	1.83121	0.915605	−	0.402078i	$-0.131712\pi$
$577$	43.9872	1.83121	0.915605	+	0.402078i	$0.131712\pi$
$578$	−6.99341	−0.290888
$579$	−9.65346	−0.401184
$580$	−1.79938	−0.0747152
$581$	0	0
$582$	−13.3573	−0.553677
$583$	−47.4008	−1.96314
$584$	13.0133	0.538493
$585$	7.13825	0.295130
$586$	22.4912	0.929103
$587$	−23.4504	−0.967901	−0.483950	−	0.875096i	$-0.660798\pi$
$587$	−23.4504	−0.967901	−0.483950	+	0.875096i	$0.660798\pi$
$588$	0	0
$589$	14.5512	0.599573
$590$	−2.85247	−0.117435
$591$	−8.98705	−0.369678
$592$	−0.228157	−0.00937718
$593$	25.5681	1.04996	0.524978	−	0.851116i	$-0.324073\pi$
$593$	25.5681	1.04996	0.524978	+	0.851116i	$0.324073\pi$
$594$	3.91662	0.160701
$595$	0	0
$596$	−1.18233	−0.0484302
$597$	−0.603687	−0.0247073
$598$	35.9304	1.46930
$599$	17.8531	0.729458	0.364729	−	0.931114i	$-0.381162\pi$
$599$	17.8531	0.729458	0.364729	+	0.931114i	$0.381162\pi$
$600$	7.50134	0.306241
$601$	−1.79806	−0.0733446	−0.0366723	−	0.999327i	$-0.511676\pi$
$601$	−1.79806	−0.0733446	−0.0366723	+	0.999327i	$0.511676\pi$
$602$	0	0
$603$	−4.68999	−0.190991
$604$	24.0738	0.979547
$605$	−12.4969	−0.508070
$606$	−11.9077	−0.483717
$607$	9.45273	0.383674	0.191837	−	0.981427i	$-0.438555\pi$
$607$	9.45273	0.383674	0.191837	+	0.981427i	$0.438555\pi$
$608$	31.0904	1.26088
$609$	0	0
$610$	−15.4369	−0.625022
$611$	45.7506	1.85087
$612$	−3.62934	−0.146707
$613$	−15.1934	−0.613655	−0.306828	−	0.951765i	$-0.599267\pi$
$613$	−15.1934	−0.613655	−0.306828	+	0.951765i	$0.599267\pi$
$614$	20.2136	0.815755
$615$	1.54330	0.0622320
$616$	0	0
$617$	7.82109	0.314865	0.157433	−	0.987530i	$-0.449678\pi$
$617$	7.82109	0.314865	0.157433	+	0.987530i	$0.449678\pi$
$618$	−9.23141	−0.371342
$619$	17.8977	0.719370	0.359685	−	0.933074i	$-0.382884\pi$
$619$	17.8977	0.719370	0.359685	+	0.933074i	$0.382884\pi$
$620$	4.81829	0.193507
$621$	−8.66757	−0.347818
$622$	−6.69819	−0.268573
$623$	0	0
$624$	0.806009	0.0322662
$625$	−5.05386	−0.202154
$626$	20.1032	0.803484
$627$	24.3754	0.973460
$628$	−7.31770	−0.292008
$629$	3.97059	0.158318
$630$	0	0
$631$	−14.9500	−0.595152	−0.297576	−	0.954698i	$-0.596178\pi$
$631$	−14.9500	−0.595152	−0.297576	+	0.954698i	$0.596178\pi$
$632$	28.7499	1.14361
$633$	−3.85847	−0.153360
$634$	−17.8706	−0.709734
$635$	6.03715	0.239577
$636$	12.9809	0.514725
$637$	0	0
$638$	−3.81573	−0.151066
$639$	8.11541	0.321040
$640$	9.81276	0.387883
$641$	2.42823	0.0959093	0.0479547	−	0.998850i	$-0.484730\pi$
$641$	2.42823	0.0959093	0.0479547	+	0.998850i	$0.484730\pi$
$642$	16.8672	0.665694
$643$	−10.9098	−0.430239	−0.215120	−	0.976588i	$-0.569014\pi$
$643$	−10.9098	−0.430239	−0.215120	+	0.976588i	$0.569014\pi$
$644$	0	0
$645$	−5.35699	−0.210931
$646$	15.1603	0.596476
$647$	21.3386	0.838907	0.419453	−	0.907777i	$-0.362222\pi$
$647$	21.3386	0.838907	0.419453	+	0.907777i	$0.362222\pi$
$648$	−2.86506	−0.112550
$649$	9.01229	0.353763
$650$	10.8535	0.425709
$651$	0	0
$652$	4.94486	0.193656
$653$	8.35303	0.326879	0.163440	−	0.986553i	$-0.447741\pi$
$653$	8.35303	0.326879	0.163440	+	0.986553i	$0.447741\pi$
$654$	−12.5679	−0.491442
$655$	−24.0340	−0.939088
$656$	0.174261	0.00680374
$657$	−4.54207	−0.177203
$658$	0	0
$659$	25.7751	1.00405	0.502027	−	0.864852i	$-0.332588\pi$
$659$	25.7751	1.00405	0.502027	+	0.864852i	$0.332588\pi$
$660$	8.07133	0.314176
$661$	14.0300	0.545705	0.272852	−	0.962056i	$-0.412033\pi$
$661$	14.0300	0.545705	0.272852	+	0.962056i	$0.412033\pi$
$662$	30.6332	1.19059
$663$	−14.0269	−0.544760
$664$	9.97297	0.387026
$665$	0	0
$666$	1.17343	0.0454695
$667$	8.44429	0.326964
$668$	11.8159	0.457171
$669$	5.49765	0.212551
$670$	6.48704	0.250616
$671$	48.7724	1.88284
$672$	0	0
$673$	−44.2346	−1.70512	−0.852560	−	0.522629i	$-0.824951\pi$
$673$	−44.2346	−1.70512	−0.852560	+	0.522629i	$0.824951\pi$
$674$	−13.0562	−0.502904
$675$	−2.61822	−0.100775
$676$	−10.0450	−0.386345
$677$	22.1926	0.852932	0.426466	−	0.904504i	$-0.359758\pi$
$677$	22.1926	0.852932	0.426466	+	0.904504i	$0.359758\pi$
$678$	−18.9617	−0.728221
$679$	0	0
$680$	13.4093	0.514223
$681$	1.67191	0.0640677
$682$	10.2176	0.391251
$683$	−43.3099	−1.65721	−0.828604	−	0.559835i	$-0.810865\pi$
$683$	−43.3099	−1.65721	−0.828604	+	0.559835i	$0.810865\pi$
$684$	−6.67528	−0.255236
$685$	17.0906	0.653000
$686$	0	0
$687$	−26.9495	−1.02819
$688$	−0.604880	−0.0230608
$689$	50.1694	1.91130
$690$	11.9887	0.456402
$691$	38.2465	1.45497	0.727483	−	0.686126i	$-0.240690\pi$
$691$	38.2465	1.45497	0.727483	+	0.686126i	$0.240690\pi$
$692$	−3.56706	−0.135599
$693$	0	0
$694$	3.40669	0.129316
$695$	24.3629	0.924139
$696$	2.79125	0.105802
$697$	−3.03264	−0.114870
$698$	−15.8491	−0.599898
$699$	−20.7406	−0.784481
$700$	0	0
$701$	−15.5095	−0.585786	−0.292893	−	0.956145i	$-0.594618\pi$
$701$	−15.5095	−0.585786	−0.292893	+	0.956145i	$0.594618\pi$
$702$	−4.14538	−0.156457
$703$	7.30294	0.275436
$704$	23.3541	0.880189
$705$	15.2654	0.574927
$706$	−12.7776	−0.480893
$707$	0	0
$708$	−2.46804	−0.0927547
$709$	43.0615	1.61721	0.808604	−	0.588353i	$-0.200223\pi$
$709$	43.0615	1.61721	0.808604	+	0.588353i	$0.200223\pi$
$710$	−11.2250	−0.421265
$711$	−10.0347	−0.376329
$712$	−37.0014	−1.38669
$713$	−22.6117	−0.846815
$714$	0	0
$715$	31.1946	1.16661
$716$	24.1643	0.903064
$717$	−4.11267	−0.153590
$718$	−0.502162	−0.0187405
$719$	−3.02420	−0.112784	−0.0563918	−	0.998409i	$-0.517960\pi$
$719$	−3.02420	−0.112784	−0.0563918	+	0.998409i	$0.517960\pi$
$720$	0.268937	0.0100227
$721$	0	0
$722$	10.8552	0.403990
$723$	0.365350	0.0135875
$724$	−26.3508	−0.979318
$725$	2.55077	0.0947333
$726$	7.25728	0.269343
$727$	17.8331	0.661395	0.330697	−	0.943737i	$-0.392716\pi$
$727$	17.8331	0.661395	0.330697	+	0.943737i	$0.392716\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	6.28244	0.232524
$731$	10.5267	0.389343
$732$	−13.3565	−0.493669
$733$	30.9233	1.14218	0.571088	−	0.820889i	$-0.306521\pi$
$733$	30.9233	1.14218	0.571088	+	0.820889i	$0.306521\pi$
$734$	−9.38982	−0.346585
$735$	0	0
$736$	−48.3125	−1.78082
$737$	−20.4956	−0.754964
$738$	−0.896239	−0.0329910
$739$	16.5434	0.608559	0.304279	−	0.952583i	$-0.401584\pi$
$739$	16.5434	0.608559	0.304279	+	0.952583i	$0.401584\pi$
$740$	2.41819	0.0888945
$741$	−25.7991	−0.947754
$742$	0	0
$743$	36.2897	1.33134	0.665670	−	0.746246i	$-0.268146\pi$
$743$	36.2897	1.33134	0.665670	+	0.746246i	$0.268146\pi$
$744$	−7.47428	−0.274020
$745$	−1.52470	−0.0558607
$746$	−8.67096	−0.317466
$747$	−3.48090	−0.127359
$748$	−15.8604	−0.579915
$749$	0	0
$750$	10.5373	0.384767
$751$	16.4972	0.601992	0.300996	−	0.953625i	$-0.402681\pi$
$751$	16.4972	0.601992	0.300996	+	0.953625i	$0.402681\pi$
$752$	1.72367	0.0628560
$753$	−15.1723	−0.552910
$754$	4.03860	0.147077
$755$	31.0448	1.12984
$756$	0	0
$757$	9.72580	0.353490	0.176745	−	0.984257i	$-0.443443\pi$
$757$	9.72580	0.353490	0.176745	+	0.984257i	$0.443443\pi$
$758$	4.08467	0.148362
$759$	−37.8779	−1.37488
$760$	24.6631	0.894626
$761$	1.23561	0.0447909	0.0223955	−	0.999749i	$-0.492871\pi$
$761$	1.23561	0.0447909	0.0223955	+	0.999749i	$0.492871\pi$
$762$	−3.50594	−0.127007
$763$	0	0
$764$	13.4560	0.486823
$765$	−4.68029	−0.169216
$766$	9.29025	0.335670
$767$	−9.53867	−0.344421
$768$	−16.3867	−0.591305
$769$	−7.08833	−0.255612	−0.127806	−	0.991799i	$-0.540793\pi$
$769$	−7.08833	−0.255612	−0.127806	+	0.991799i	$0.540793\pi$
$770$	0	0
$771$	−5.35154	−0.192731
$772$	11.5528	0.415796
$773$	−34.9550	−1.25724	−0.628622	−	0.777711i	$-0.716381\pi$
$773$	−34.9550	−1.25724	−0.628622	+	0.777711i	$0.716381\pi$
$774$	3.11095	0.111821
$775$	−6.83033	−0.245353
$776$	42.7000	1.53284
$777$	0	0
$778$	17.2515	0.618496
$779$	−5.57781	−0.199846
$780$	−8.54275	−0.305880
$781$	35.4649	1.26903
$782$	−23.5582	−0.842440
$783$	−0.974240	−0.0348165
$784$	0	0
$785$	−9.43670	−0.336810
$786$	13.9572	0.497838
$787$	37.3757	1.33230	0.666150	−	0.745818i	$-0.267941\pi$
$787$	37.3757	1.33230	0.666150	+	0.745818i	$0.267941\pi$
$788$	10.7553	0.383142
$789$	−5.79651	−0.206361
$790$	13.8796	0.493815
$791$	0	0
$792$	−12.5205	−0.444896
$793$	−51.6210	−1.83312
$794$	19.2005	0.681401
$795$	16.7398	0.593698
$796$	0.722466	0.0256071
$797$	44.0532	1.56044	0.780222	−	0.625503i	$-0.215106\pi$
$797$	44.0532	1.56044	0.780222	+	0.625503i	$0.215106\pi$
$798$	0	0
$799$	−29.9970	−1.06122
$800$	−14.5938	−0.515967
$801$	12.9147	0.456319
$802$	34.2695	1.21010
$803$	−19.8491	−0.700461
$804$	5.61277	0.197947
$805$	0	0
$806$	−10.8144	−0.380919
$807$	15.4409	0.543546
$808$	38.0660	1.33916
$809$	15.7165	0.552563	0.276282	−	0.961077i	$-0.410898\pi$
$809$	15.7165	0.552563	0.276282	+	0.961077i	$0.410898\pi$
$810$	−1.38317	−0.0485995
$811$	−1.62960	−0.0572229	−0.0286114	−	0.999591i	$-0.509109\pi$
$811$	−1.62960	−0.0572229	−0.0286114	+	0.999591i	$0.509109\pi$
$812$	0	0
$813$	16.2731	0.570722
$814$	5.12797	0.179735
$815$	6.37675	0.223368
$816$	−0.528471	−0.0185002
$817$	19.3613	0.677365
$818$	20.9482	0.732435
$819$	0	0
$820$	−1.84696	−0.0644986
$821$	29.6012	1.03309	0.516545	−	0.856260i	$-0.327218\pi$
$821$	29.6012	1.03309	0.516545	+	0.856260i	$0.327218\pi$
$822$	−9.92500	−0.346174
$823$	−32.8568	−1.14532	−0.572659	−	0.819794i	$-0.694088\pi$
$823$	−32.8568	−1.14532	−0.572659	+	0.819794i	$0.694088\pi$
$824$	29.5106	1.02805
$825$	−11.4418	−0.398352
$826$	0	0
$827$	−8.55421	−0.297459	−0.148730	−	0.988878i	$-0.547518\pi$
$827$	−8.55421	−0.297459	−0.148730	+	0.988878i	$0.547518\pi$
$828$	10.3730	0.360486
$829$	−11.4157	−0.396483	−0.198241	−	0.980153i	$-0.563523\pi$
$829$	−11.4157	−0.396483	−0.198241	+	0.980153i	$0.563523\pi$
$830$	4.81467	0.167119
$831$	19.0510	0.660873
$832$	−24.7181	−0.856946
$833$	0	0
$834$	−14.1482	−0.489913
$835$	15.2374	0.527313
$836$	−29.1714	−1.00892
$837$	2.60877	0.0901724
$838$	−25.6441	−0.885861
$839$	39.9662	1.37979	0.689893	−	0.723911i	$-0.257658\pi$
$839$	39.9662	1.37979	0.689893	+	0.723911i	$0.257658\pi$
$840$	0	0
$841$	−28.0509	−0.967271
$842$	35.9435	1.23869
$843$	−2.91985	−0.100565
$844$	4.61764	0.158946
$845$	−12.9537	−0.445621
$846$	−8.86502	−0.304786
$847$	0	0
$848$	1.89015	0.0649082
$849$	−25.0265	−0.858908
$850$	−7.11624	−0.244085
$851$	−11.3483	−0.389015
$852$	−9.71216	−0.332733
$853$	−39.0009	−1.33537	−0.667684	−	0.744445i	$-0.732714\pi$
$853$	−39.0009	−1.33537	−0.667684	+	0.744445i	$0.732714\pi$
$854$	0	0
$855$	−8.60825	−0.294396
$856$	−53.9202	−1.84296
$857$	−16.6309	−0.568101	−0.284050	−	0.958809i	$-0.591678\pi$
$857$	−16.6309	−0.568101	−0.284050	+	0.958809i	$0.591678\pi$
$858$	−18.1156	−0.618456
$859$	11.6894	0.398836	0.199418	−	0.979915i	$-0.436095\pi$
$859$	11.6894	0.398836	0.199418	+	0.979915i	$0.436095\pi$
$860$	6.41102	0.218614
$861$	0	0
$862$	−23.6551	−0.805698
$863$	4.52677	0.154093	0.0770466	−	0.997027i	$-0.475451\pi$
$863$	4.52677	0.154093	0.0770466	+	0.997027i	$0.475451\pi$
$864$	5.57393	0.189629
$865$	−4.59998	−0.156404
$866$	31.3220	1.06437
$867$	−7.80307	−0.265006
$868$	0	0
$869$	−43.8521	−1.48758
$870$	1.34754	0.0456858
$871$	21.6926	0.735027
$872$	40.1764	1.36054
$873$	−14.9037	−0.504414
$874$	−43.3296	−1.46565
$875$	0	0
$876$	5.43575	0.183657
$877$	6.06758	0.204888	0.102444	−	0.994739i	$-0.467334\pi$
$877$	6.06758	0.204888	0.102444	+	0.994739i	$0.467334\pi$
$878$	16.0686	0.542287
$879$	25.0951	0.846437
$880$	1.17527	0.0396184
$881$	−17.1347	−0.577283	−0.288641	−	0.957437i	$-0.593204\pi$
$881$	−17.1347	−0.577283	−0.288641	+	0.957437i	$0.593204\pi$
$882$	0	0
$883$	12.9481	0.435739	0.217869	−	0.975978i	$-0.430089\pi$
$883$	12.9481	0.435739	0.217869	+	0.975978i	$0.430089\pi$
$884$	16.7868	0.564601
$885$	−3.18272	−0.106986
$886$	5.86445	0.197020
$887$	−40.9952	−1.37648	−0.688242	−	0.725481i	$-0.741617\pi$
$887$	−40.9952	−1.37648	−0.688242	+	0.725481i	$0.741617\pi$
$888$	−3.75117	−0.125881
$889$	0	0
$890$	−17.8632	−0.598777
$891$	4.37007	0.146403
$892$	−6.57935	−0.220293
$893$	−55.1721	−1.84627
$894$	0.885435	0.0296134
$895$	31.1616	1.04162
$896$	0	0
$897$	40.0902	1.33857
$898$	−13.5699	−0.452835
$899$	−2.54157	−0.0847661
$900$	3.13337	0.104446
$901$	−32.8942	−1.09587
$902$	−3.91662	−0.130409
$903$	0	0
$904$	60.6161	2.01606
$905$	−33.9812	−1.12957
$906$	−18.0286	−0.598960
$907$	−44.7755	−1.48674	−0.743372	−	0.668878i	$-0.766775\pi$
$907$	−44.7755	−1.48674	−0.743372	+	0.668878i	$0.766775\pi$
$908$	−2.00087	−0.0664012
$909$	−13.2863	−0.440679
$910$	0	0
$911$	36.7332	1.21703	0.608513	−	0.793544i	$-0.291766\pi$
$911$	36.7332	1.21703	0.608513	+	0.793544i	$0.291766\pi$
$912$	−0.971993	−0.0321859
$913$	−15.2118	−0.503436
$914$	24.8378	0.821560
$915$	−17.2241	−0.569412
$916$	32.2520	1.06564
$917$	0	0
$918$	2.71797	0.0897065
$919$	−37.6230	−1.24107	−0.620535	−	0.784179i	$-0.713084\pi$
$919$	−37.6230	−1.24107	−0.620535	+	0.784179i	$0.713084\pi$
$920$	−38.3250	−1.26354
$921$	22.5538	0.743174
$922$	12.7503	0.419907
$923$	−37.5363	−1.23552
$924$	0	0
$925$	−3.42799	−0.112712
$926$	8.48628	0.278877
$927$	−10.3002	−0.338302
$928$	−5.43035	−0.178260
$929$	−47.1159	−1.54582	−0.772912	−	0.634513i	$-0.781201\pi$
$929$	−47.1159	−1.54582	−0.772912	+	0.634513i	$0.781201\pi$
$930$	−3.60837	−0.118323
$931$	0	0
$932$	24.8214	0.813053
$933$	−7.47367	−0.244677
$934$	−19.2813	−0.630905
$935$	−20.4532	−0.668890
$936$	13.2518	0.433148
$937$	9.28037	0.303177	0.151588	−	0.988444i	$-0.451561\pi$
$937$	9.28037	0.303177	0.151588	+	0.988444i	$0.451561\pi$
$938$	0	0
$939$	22.4306	0.731994
$940$	−18.2689	−0.595867
$941$	−14.5639	−0.474771	−0.237385	−	0.971416i	$-0.576290\pi$
$941$	−14.5639	−0.474771	−0.237385	+	0.971416i	$0.576290\pi$
$942$	5.48015	0.178553
$943$	8.66757	0.282255
$944$	−0.359374	−0.0116966
$945$	0	0
$946$	13.5951	0.442014
$947$	−1.21392	−0.0394470	−0.0197235	−	0.999805i	$-0.506279\pi$
$947$	−1.21392	−0.0394470	−0.0197235	+	0.999805i	$0.506279\pi$
$948$	12.0090	0.390036
$949$	21.0085	0.681964
$950$	−13.0886	−0.424650
$951$	−19.9396	−0.646586
$952$	0	0
$953$	28.1528	0.911958	0.455979	−	0.889990i	$-0.349289\pi$
$953$	28.1528	0.911958	0.455979	+	0.889990i	$0.349289\pi$
$954$	−9.72124	−0.314737
$955$	17.3525	0.561515
$956$	4.92186	0.159184
$957$	−4.25749	−0.137625
$958$	−16.1834	−0.522861
$959$	0	0
$960$	−8.24756	−0.266189
$961$	−24.1943	−0.780462
$962$	−5.42748	−0.174989
$963$	18.8200	0.606464
$964$	−0.437235	−0.0140824
$965$	14.8982	0.479591
$966$	0	0
$967$	−43.8216	−1.40921	−0.704604	−	0.709601i	$-0.748875\pi$
$967$	−43.8216	−1.40921	−0.704604	+	0.709601i	$0.748875\pi$
$968$	−23.1997	−0.745668
$969$	16.9155	0.543405
$970$	20.6143	0.661886
$971$	7.01150	0.225010	0.112505	−	0.993651i	$-0.464113\pi$
$971$	7.01150	0.225010	0.112505	+	0.993651i	$0.464113\pi$
$972$	−1.19676	−0.0383860
$973$	0	0
$974$	−32.5369	−1.04255
$975$	12.1101	0.387832
$976$	−1.94485	−0.0622530
$977$	−9.54378	−0.305333	−0.152666	−	0.988278i	$-0.548786\pi$
$977$	−9.54378	−0.305333	−0.152666	+	0.988278i	$0.548786\pi$
$978$	−3.70316	−0.118414
$979$	56.4382	1.80377
$980$	0	0
$981$	−14.0229	−0.447717
$982$	−18.6629	−0.595559
$983$	2.76328	0.0881351	0.0440675	−	0.999029i	$-0.485968\pi$
$983$	2.76328	0.0881351	0.0440675	+	0.999029i	$0.485968\pi$
$984$	2.86506	0.0913347
$985$	13.8697	0.441927
$986$	−2.64796	−0.0843281
$987$	0	0
$988$	30.8752	0.982273
$989$	−30.0862	−0.956686
$990$	−6.04453	−0.192108
$991$	31.5003	1.00064	0.500320	−	0.865840i	$-0.333216\pi$
$991$	31.5003	1.00064	0.500320	+	0.865840i	$0.333216\pi$
$992$	14.5411	0.461681
$993$	34.1797	1.08466
$994$	0	0
$995$	0.931671	0.0295360
$996$	4.16579	0.131998
$997$	−9.19338	−0.291157	−0.145579	−	0.989347i	$-0.546504\pi$
$997$	−9.19338	−0.291157	−0.145579	+	0.989347i	$0.546504\pi$
$998$	−9.86763	−0.312354
$999$	1.30928	0.0414239

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.ba.1.6	yes	8
7.6	odd	2		6027.2.a.z.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.z.1.6	✓	8	7.6	odd	2
6027.2.a.ba.1.6	yes	8	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)

\(f(q)\)	\(=\)	\(q+0.896239 q^{2} +1.00000 q^{3} -1.19676 q^{4} -1.54330 q^{5} +0.896239 q^{6} -2.86506 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.896239 q^{2} +1.00000 q^{3} -1.19676 q^{4} -1.54330 q^{5} +0.896239 q^{6} -2.86506 q^{8} +1.00000 q^{9} -1.38317 q^{10} +4.37007 q^{11} -1.19676 q^{12} -4.62531 q^{13} -1.54330 q^{15} -0.174261 q^{16} +3.03264 q^{17} +0.896239 q^{18} +5.57781 q^{19} +1.84696 q^{20} +3.91662 q^{22} -8.66757 q^{23} -2.86506 q^{24} -2.61822 q^{25} -4.14538 q^{26} +1.00000 q^{27} -0.974240 q^{29} -1.38317 q^{30} +2.60877 q^{31} +5.57393 q^{32} +4.37007 q^{33} +2.71797 q^{34} -1.19676 q^{36} +1.30928 q^{37} +4.99905 q^{38} -4.62531 q^{39} +4.42165 q^{40} -1.00000 q^{41} +3.47112 q^{43} -5.22991 q^{44} -1.54330 q^{45} -7.76821 q^{46} -9.89136 q^{47} -0.174261 q^{48} -2.34655 q^{50} +3.03264 q^{51} +5.53537 q^{52} -10.8467 q^{53} +0.896239 q^{54} -6.74434 q^{55} +5.57781 q^{57} -0.873151 q^{58} +2.06228 q^{59} +1.84696 q^{60} +11.1606 q^{61} +2.33808 q^{62} +5.34410 q^{64} +7.13825 q^{65} +3.91662 q^{66} -4.68999 q^{67} -3.62934 q^{68} -8.66757 q^{69} +8.11541 q^{71} -2.86506 q^{72} -4.54207 q^{73} +1.17343 q^{74} -2.61822 q^{75} -6.67528 q^{76} -4.14538 q^{78} -10.0347 q^{79} +0.268937 q^{80} +1.00000 q^{81} -0.896239 q^{82} -3.48090 q^{83} -4.68029 q^{85} +3.11095 q^{86} -0.974240 q^{87} -12.5205 q^{88} +12.9147 q^{89} -1.38317 q^{90} +10.3730 q^{92} +2.60877 q^{93} -8.86502 q^{94} -8.60825 q^{95} +5.57393 q^{96} -14.9037 q^{97} +4.37007 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 8 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 2 * q^2 + 8 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 6 * q^8 + 8 * q^9	\( 8 q - 2 q^{2} + 8 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{8} + 8 q^{9} + 2 q^{10} - 2 q^{11} + 4 q^{12} - 4 q^{13} - 2 q^{15} - 8 q^{17} - 2 q^{18} - 6 q^{19} - 4 q^{20} - 14 q^{22} - 12 q^{23} - 6 q^{24} - 4 q^{25} - 4 q^{26} + 8 q^{27} - 4 q^{29} + 2 q^{30} + 10 q^{31} - 4 q^{32} - 2 q^{33} - 4 q^{34} + 4 q^{36} - 20 q^{37} + 18 q^{38} - 4 q^{39} - 12 q^{40} - 8 q^{41} - 8 q^{43} + 20 q^{44} - 2 q^{45} - 12 q^{46} - 24 q^{47} - 22 q^{50} - 8 q^{51} + 30 q^{52} - 36 q^{53} - 2 q^{54} - 4 q^{55} - 6 q^{57} + 14 q^{58} - 10 q^{59} - 4 q^{60} + 22 q^{61} - 30 q^{62} - 24 q^{64} + 8 q^{65} - 14 q^{66} - 14 q^{67} - 38 q^{68} - 12 q^{69} - 10 q^{71} - 6 q^{72} + 12 q^{73} - 2 q^{74} - 4 q^{75} - 32 q^{76} - 4 q^{78} + 16 q^{79} + 14 q^{80} + 8 q^{81} + 2 q^{82} - 24 q^{83} - 44 q^{85} + 36 q^{86} - 4 q^{87} - 34 q^{88} - 2 q^{89} + 2 q^{90} - 48 q^{92} + 10 q^{93} + 34 q^{94} - 24 q^{95} - 4 q^{96} - 16 q^{97} - 2 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 + 8 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 6 * q^8 + 8 * q^9 + 2 * q^10 - 2 * q^11 + 4 * q^12 - 4 * q^13 - 2 * q^15 - 8 * q^17 - 2 * q^18 - 6 * q^19 - 4 * q^20 - 14 * q^22 - 12 * q^23 - 6 * q^24 - 4 * q^25 - 4 * q^26 + 8 * q^27 - 4 * q^29 + 2 * q^30 + 10 * q^31 - 4 * q^32 - 2 * q^33 - 4 * q^34 + 4 * q^36 - 20 * q^37 + 18 * q^38 - 4 * q^39 - 12 * q^40 - 8 * q^41 - 8 * q^43 + 20 * q^44 - 2 * q^45 - 12 * q^46 - 24 * q^47 - 22 * q^50 - 8 * q^51 + 30 * q^52 - 36 * q^53 - 2 * q^54 - 4 * q^55 - 6 * q^57 + 14 * q^58 - 10 * q^59 - 4 * q^60 + 22 * q^61 - 30 * q^62 - 24 * q^64 + 8 * q^65 - 14 * q^66 - 14 * q^67 - 38 * q^68 - 12 * q^69 - 10 * q^71 - 6 * q^72 + 12 * q^73 - 2 * q^74 - 4 * q^75 - 32 * q^76 - 4 * q^78 + 16 * q^79 + 14 * q^80 + 8 * q^81 + 2 * q^82 - 24 * q^83 - 44 * q^85 + 36 * q^86 - 4 * q^87 - 34 * q^88 - 2 * q^89 + 2 * q^90 - 48 * q^92 + 10 * q^93 + 34 * q^94 - 24 * q^95 - 4 * q^96 - 16 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

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