LMFDB - Embedded newform 6027.2.a.bc.1.3

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:	$0$
Dimension:	$8$
Coefficient field:	8.8.7457527933.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 6x^{6} + 23x^{5} - 4x^{4} - 27x^{3} + 8x^{2} + 8x + 1 $ x^8 - 3x^7 - 6x^6 + 23x^5 - 4x^4 - 27x^3 + 8x^2 + 8*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 861)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.21768$ 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)

$f(q)$	$=$	$q-1.51387 q^{2} +1.00000 q^{3} +0.291794 q^{4} -3.80505 q^{5} -1.51387 q^{6} +2.58600 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.51387 q^{2} +1.00000 q^{3} +0.291794 q^{4} -3.80505 q^{5} -1.51387 q^{6} +2.58600 q^{8} +1.00000 q^{9} +5.76034 q^{10} +1.15660 q^{11} +0.291794 q^{12} +4.56081 q^{13} -3.80505 q^{15} -4.49844 q^{16} +6.92527 q^{17} -1.51387 q^{18} -2.40606 q^{19} -1.11029 q^{20} -1.75093 q^{22} +3.28488 q^{23} +2.58600 q^{24} +9.47840 q^{25} -6.90445 q^{26} +1.00000 q^{27} +1.28140 q^{29} +5.76034 q^{30} +0.334671 q^{31} +1.63805 q^{32} +1.15660 q^{33} -10.4839 q^{34} +0.291794 q^{36} +1.13227 q^{37} +3.64245 q^{38} +4.56081 q^{39} -9.83985 q^{40} +1.00000 q^{41} +2.12116 q^{43} +0.337487 q^{44} -3.80505 q^{45} -4.97287 q^{46} +7.39779 q^{47} -4.49844 q^{48} -14.3490 q^{50} +6.92527 q^{51} +1.33082 q^{52} -2.25189 q^{53} -1.51387 q^{54} -4.40090 q^{55} -2.40606 q^{57} -1.93987 q^{58} +12.9132 q^{59} -1.11029 q^{60} -2.57480 q^{61} -0.506648 q^{62} +6.51709 q^{64} -17.3541 q^{65} -1.75093 q^{66} -6.21525 q^{67} +2.02075 q^{68} +3.28488 q^{69} +2.30854 q^{71} +2.58600 q^{72} -11.9079 q^{73} -1.71411 q^{74} +9.47840 q^{75} -0.702072 q^{76} -6.90445 q^{78} -12.7517 q^{79} +17.1168 q^{80} +1.00000 q^{81} -1.51387 q^{82} -2.58930 q^{83} -26.3510 q^{85} -3.21115 q^{86} +1.28140 q^{87} +2.99095 q^{88} +9.21039 q^{89} +5.76034 q^{90} +0.958507 q^{92} +0.334671 q^{93} -11.1993 q^{94} +9.15516 q^{95} +1.63805 q^{96} +12.8974 q^{97} +1.15660 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 8 q^{3} + 13 q^{4} + 7 q^{5} + q^{6} + 6 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + q^2 + 8 * q^3 + 13 * q^4 + 7 * q^5 + q^6 + 6 * q^8 + 8 * q^9	$ 8 q + q^{2} + 8 q^{3} + 13 q^{4} + 7 q^{5} + q^{6} + 6 q^{8} + 8 q^{9} + 8 q^{10} + 11 q^{11} + 13 q^{12} + 10 q^{13} + 7 q^{15} - 17 q^{16} + 3 q^{17} + q^{18} + 6 q^{19} + 11 q^{20} + 15 q^{22} + 14 q^{23} + 6 q^{24} + 25 q^{25} + 24 q^{26} + 8 q^{27} + 2 q^{29} + 8 q^{30} + 16 q^{31} + 3 q^{32} + 11 q^{33} - 4 q^{34} + 13 q^{36} - 20 q^{37} + 10 q^{38} + 10 q^{39} - 3 q^{40} + 8 q^{41} + 7 q^{43} + 7 q^{45} - 5 q^{46} + 14 q^{47} - 17 q^{48} - 5 q^{50} + 3 q^{51} + 23 q^{52} + 7 q^{53} + q^{54} + 48 q^{55} + 6 q^{57} - 20 q^{58} + 22 q^{59} + 11 q^{60} - 33 q^{62} - 10 q^{64} - 14 q^{65} + 15 q^{66} + 12 q^{67} - 27 q^{68} + 14 q^{69} - 5 q^{71} + 6 q^{72} + 2 q^{73} + 6 q^{74} + 25 q^{75} + 43 q^{76} + 24 q^{78} - 15 q^{79} - 7 q^{80} + 8 q^{81} + q^{82} + 15 q^{83} - 43 q^{85} + 31 q^{86} + 2 q^{87} + 17 q^{88} + 29 q^{89} + 8 q^{90} + 19 q^{92} + 16 q^{93} + 20 q^{94} + 14 q^{95} + 3 q^{96} + 19 q^{97} + 11 q^{99}+O(q^{100}) $ 8 * q + q^2 + 8 * q^3 + 13 * q^4 + 7 * q^5 + q^6 + 6 * q^8 + 8 * q^9 + 8 * q^10 + 11 * q^11 + 13 * q^12 + 10 * q^13 + 7 * q^15 - 17 * q^16 + 3 * q^17 + q^18 + 6 * q^19 + 11 * q^20 + 15 * q^22 + 14 * q^23 + 6 * q^24 + 25 * q^25 + 24 * q^26 + 8 * q^27 + 2 * q^29 + 8 * q^30 + 16 * q^31 + 3 * q^32 + 11 * q^33 - 4 * q^34 + 13 * q^36 - 20 * q^37 + 10 * q^38 + 10 * q^39 - 3 * q^40 + 8 * q^41 + 7 * q^43 + 7 * q^45 - 5 * q^46 + 14 * q^47 - 17 * q^48 - 5 * q^50 + 3 * q^51 + 23 * q^52 + 7 * q^53 + q^54 + 48 * q^55 + 6 * q^57 - 20 * q^58 + 22 * q^59 + 11 * q^60 - 33 * q^62 - 10 * q^64 - 14 * q^65 + 15 * q^66 + 12 * q^67 - 27 * q^68 + 14 * q^69 - 5 * q^71 + 6 * q^72 + 2 * q^73 + 6 * q^74 + 25 * q^75 + 43 * q^76 + 24 * q^78 - 15 * q^79 - 7 * q^80 + 8 * q^81 + q^82 + 15 * q^83 - 43 * q^85 + 31 * q^86 + 2 * q^87 + 17 * q^88 + 29 * q^89 + 8 * q^90 + 19 * q^92 + 16 * q^93 + 20 * q^94 + 14 * q^95 + 3 * q^96 + 19 * q^97 + 11 * 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 $

$2$	−1.51387	−1.07047	−0.535233	−	0.844705i	$-0.679776\pi$
$2$	−1.51387	−1.07047	−0.535233	+	0.844705i	$0.679776\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.291794	0.145897
$5$	−3.80505	−1.70167	−0.850835	−	0.525433i	$-0.823903\pi$
$5$	−3.80505	−1.70167	−0.850835	+	0.525433i	$0.823903\pi$
$6$	−1.51387	−0.618034
$7$	0	0
$7$	0	0
$8$	2.58600	0.914288
$9$	1.00000	0.333333
$10$	5.76034	1.82158
$11$	1.15660	0.348727	0.174363	−	0.984681i	$-0.444213\pi$
$11$	1.15660	0.348727	0.174363	+	0.984681i	$0.444213\pi$
$12$	0.291794	0.0842336
$13$	4.56081	1.26494	0.632470	−	0.774585i	$-0.282041\pi$
$13$	4.56081	1.26494	0.632470	+	0.774585i	$0.282041\pi$
$14$	0	0
$15$	−3.80505	−0.982460
$16$	−4.49844	−1.12461
$17$	6.92527	1.67963	0.839813	−	0.542876i	$-0.182665\pi$
$17$	6.92527	1.67963	0.839813	+	0.542876i	$0.182665\pi$
$18$	−1.51387	−0.356822
$19$	−2.40606	−0.551987	−0.275994	−	0.961159i	$-0.589007\pi$
$19$	−2.40606	−0.551987	−0.275994	+	0.961159i	$0.589007\pi$
$20$	−1.11029	−0.248268
$21$	0	0
$22$	−1.75093	−0.373300
$23$	3.28488	0.684944	0.342472	−	0.939528i	$-0.388736\pi$
$23$	3.28488	0.684944	0.342472	+	0.939528i	$0.388736\pi$
$24$	2.58600	0.527864
$25$	9.47840	1.89568
$26$	−6.90445	−1.35407
$27$	1.00000	0.192450
$28$	0	0
$29$	1.28140	0.237950	0.118975	−	0.992897i	$-0.462039\pi$
$29$	1.28140	0.237950	0.118975	+	0.992897i	$0.462039\pi$
$30$	5.76034	1.05169
$31$	0.334671	0.0601087	0.0300544	−	0.999548i	$-0.490432\pi$
$31$	0.334671	0.0601087	0.0300544	+	0.999548i	$0.490432\pi$
$32$	1.63805	0.289569
$33$	1.15660	0.201337
$34$	−10.4839	−1.79798
$35$	0	0
$36$	0.291794	0.0486323
$37$	1.13227	0.186145	0.0930724	−	0.995659i	$-0.470331\pi$
$37$	1.13227	0.186145	0.0930724	+	0.995659i	$0.470331\pi$
$38$	3.64245	0.590883
$39$	4.56081	0.730313
$40$	−9.83985	−1.55582
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	2.12116	0.323474	0.161737	−	0.986834i	$-0.448290\pi$
$43$	2.12116	0.323474	0.161737	+	0.986834i	$0.448290\pi$
$44$	0.337487	0.0508781
$45$	−3.80505	−0.567223
$46$	−4.97287	−0.733209
$47$	7.39779	1.07908	0.539540	−	0.841960i	$-0.318598\pi$
$47$	7.39779	1.07908	0.539540	+	0.841960i	$0.318598\pi$
$48$	−4.49844	−0.649294
$49$	0	0
$50$	−14.3490	−2.02926
$51$	6.92527	0.969732
$52$	1.33082	0.184551
$53$	−2.25189	−0.309322	−0.154661	−	0.987968i	$-0.549428\pi$
$53$	−2.25189	−0.309322	−0.154661	+	0.987968i	$0.549428\pi$
$54$	−1.51387	−0.206011
$55$	−4.40090	−0.593418
$56$	0	0
$57$	−2.40606	−0.318690
$58$	−1.93987	−0.254717
$59$	12.9132	1.68115	0.840575	−	0.541696i	$-0.182218\pi$
$59$	12.9132	1.68115	0.840575	+	0.541696i	$0.182218\pi$
$60$	−1.11029	−0.143338
$61$	−2.57480	−0.329669	−0.164834	−	0.986321i	$-0.552709\pi$
$61$	−2.57480	−0.329669	−0.164834	+	0.986321i	$0.552709\pi$
$62$	−0.506648	−0.0643443
$63$	0	0
$64$	6.51709	0.814637
$65$	−17.3541	−2.15251
$66$	−1.75093	−0.215525
$67$	−6.21525	−0.759313	−0.379657	−	0.925127i	$-0.623958\pi$
$67$	−6.21525	−0.759313	−0.379657	+	0.925127i	$0.623958\pi$
$68$	2.02075	0.245052
$69$	3.28488	0.395453
$70$	0	0
$71$	2.30854	0.273973	0.136986	−	0.990573i	$-0.456258\pi$
$71$	2.30854	0.273973	0.136986	+	0.990573i	$0.456258\pi$
$72$	2.58600	0.304763
$73$	−11.9079	−1.39371	−0.696855	−	0.717212i	$-0.745418\pi$
$73$	−11.9079	−1.39371	−0.696855	+	0.717212i	$0.745418\pi$
$74$	−1.71411	−0.199262
$75$	9.47840	1.09447
$76$	−0.702072	−0.0805332
$77$	0	0
$78$	−6.90445	−0.781775
$79$	−12.7517	−1.43467	−0.717337	−	0.696727i	$-0.754639\pi$
$79$	−12.7517	−1.43467	−0.717337	+	0.696727i	$0.754639\pi$
$80$	17.1168	1.91372
$81$	1.00000	0.111111
$82$	−1.51387	−0.167179
$83$	−2.58930	−0.284213	−0.142106	−	0.989851i	$-0.545388\pi$
$83$	−2.58930	−0.284213	−0.142106	+	0.989851i	$0.545388\pi$
$84$	0	0
$85$	−26.3510	−2.85817
$86$	−3.21115	−0.346267
$87$	1.28140	0.137380
$88$	2.99095	0.318837
$89$	9.21039	0.976299	0.488150	−	0.872760i	$-0.337672\pi$
$89$	9.21039	0.976299	0.488150	+	0.872760i	$0.337672\pi$
$90$	5.76034	0.607193
$91$	0	0
$92$	0.958507	0.0999312
$93$	0.334671	0.0347038
$94$	−11.1993	−1.15512
$95$	9.15516	0.939300
$96$	1.63805	0.167183
$97$	12.8974	1.30953	0.654766	−	0.755832i	$-0.272767\pi$
$97$	12.8974	1.30953	0.654766	+	0.755832i	$0.272767\pi$
$98$	0	0
$99$	1.15660	0.116242
$100$	2.76574	0.276574
$101$	−0.177513	−0.0176632	−0.00883162	−	0.999961i	$-0.502811\pi$
$101$	−0.177513	−0.0176632	−0.00883162	+	0.999961i	$0.502811\pi$
$102$	−10.4839	−1.03807
$103$	14.5341	1.43209	0.716046	−	0.698053i	$-0.245950\pi$
$103$	14.5341	1.43209	0.716046	+	0.698053i	$0.245950\pi$
$104$	11.7942	1.15652
$105$	0	0
$106$	3.40907	0.331118
$107$	−12.0732	−1.16716	−0.583579	−	0.812056i	$-0.698348\pi$
$107$	−12.0732	−1.16716	−0.583579	+	0.812056i	$0.698348\pi$
$108$	0.291794	0.0280779
$109$	17.0080	1.62907	0.814537	−	0.580111i	$-0.196991\pi$
$109$	17.0080	1.62907	0.814537	+	0.580111i	$0.196991\pi$
$110$	6.66238	0.635233
$111$	1.13227	0.107471
$112$	0	0
$113$	1.63814	0.154103	0.0770516	−	0.997027i	$-0.475449\pi$
$113$	1.63814	0.154103	0.0770516	+	0.997027i	$0.475449\pi$
$114$	3.64245	0.341147
$115$	−12.4991	−1.16555
$116$	0.373905	0.0347162
$117$	4.56081	0.421647
$118$	−19.5488	−1.79961
$119$	0	0
$120$	−9.83985	−0.898251
$121$	−9.66229	−0.878390
$122$	3.89790	0.352899
$123$	1.00000	0.0901670
$124$	0.0976550	0.00876968
$125$	−17.0405	−1.52415
$126$	0	0
$127$	−20.7696	−1.84300	−0.921502	−	0.388374i	$-0.873037\pi$
$127$	−20.7696	−1.84300	−0.921502	+	0.388374i	$0.873037\pi$
$128$	−13.1421	−1.16161
$129$	2.12116	0.186758
$130$	26.2718	2.30419
$131$	−12.0351	−1.05151	−0.525756	−	0.850636i	$-0.676217\pi$
$131$	−12.0351	−1.05151	−0.525756	+	0.850636i	$0.676217\pi$
$132$	0.337487	0.0293745
$133$	0	0
$134$	9.40906	0.812819
$135$	−3.80505	−0.327487
$136$	17.9087	1.53566
$137$	−6.50063	−0.555386	−0.277693	−	0.960670i	$-0.589570\pi$
$137$	−6.50063	−0.555386	−0.277693	+	0.960670i	$0.589570\pi$
$138$	−4.97287	−0.423318
$139$	−23.1305	−1.96191	−0.980954	−	0.194243i	$-0.937775\pi$
$139$	−23.1305	−1.96191	−0.980954	+	0.194243i	$0.937775\pi$
$140$	0	0
$141$	7.39779	0.623007
$142$	−3.49482	−0.293279
$143$	5.27501	0.441118
$144$	−4.49844	−0.374870
$145$	−4.87579	−0.404912
$146$	18.0269	1.49192
$147$	0	0
$148$	0.330391	0.0271580
$149$	−5.20043	−0.426036	−0.213018	−	0.977048i	$-0.568329\pi$
$149$	−5.20043	−0.426036	−0.213018	+	0.977048i	$0.568329\pi$
$150$	−14.3490	−1.17159
$151$	−10.2898	−0.837370	−0.418685	−	0.908131i	$-0.637509\pi$
$151$	−10.2898	−0.837370	−0.418685	+	0.908131i	$0.637509\pi$
$152$	−6.22205	−0.504675
$153$	6.92527	0.559875
$154$	0	0
$155$	−1.27344	−0.102285
$156$	1.33082	0.106550
$157$	−4.03587	−0.322097	−0.161049	−	0.986946i	$-0.551488\pi$
$157$	−4.03587	−0.322097	−0.161049	+	0.986946i	$0.551488\pi$
$158$	19.3043	1.53577
$159$	−2.25189	−0.178587
$160$	−6.23287	−0.492752
$161$	0	0
$162$	−1.51387	−0.118941
$163$	6.84917	0.536468	0.268234	−	0.963354i	$-0.413560\pi$
$163$	6.84917	0.536468	0.268234	+	0.963354i	$0.413560\pi$
$164$	0.291794	0.0227853
$165$	−4.40090	−0.342610
$166$	3.91986	0.304240
$167$	−2.72776	−0.211080	−0.105540	−	0.994415i	$-0.533657\pi$
$167$	−2.72776	−0.211080	−0.105540	+	0.994415i	$0.533657\pi$
$168$	0	0
$169$	7.80095	0.600073
$170$	39.8919	3.05957
$171$	−2.40606	−0.183996
$172$	0.618941	0.0471938
$173$	3.57517	0.271815	0.135908	−	0.990722i	$-0.456605\pi$
$173$	3.57517	0.271815	0.135908	+	0.990722i	$0.456605\pi$
$174$	−1.93987	−0.147061
$175$	0	0
$176$	−5.20288	−0.392182
$177$	12.9132	0.970612
$178$	−13.9433	−1.04510
$179$	13.3401	0.997084	0.498542	−	0.866865i	$-0.333869\pi$
$179$	13.3401	0.997084	0.498542	+	0.866865i	$0.333869\pi$
$180$	−1.11029	−0.0827561
$181$	6.15975	0.457850	0.228925	−	0.973444i	$-0.426479\pi$
$181$	6.15975	0.457850	0.228925	+	0.973444i	$0.426479\pi$
$182$	0	0
$183$	−2.57480	−0.190334
$184$	8.49468	0.626236
$185$	−4.30836	−0.316757
$186$	−0.506648	−0.0371492
$187$	8.00974	0.585730
$188$	2.15863	0.157434
$189$	0	0
$190$	−13.8597	−1.00549
$191$	23.8691	1.72711	0.863554	−	0.504256i	$-0.168233\pi$
$191$	23.8691	1.72711	0.863554	+	0.504256i	$0.168233\pi$
$192$	6.51709	0.470331
$193$	−21.6807	−1.56061	−0.780304	−	0.625400i	$-0.784936\pi$
$193$	−21.6807	−1.56061	−0.780304	+	0.625400i	$0.784936\pi$
$194$	−19.5249	−1.40181
$195$	−17.3541	−1.24275
$196$	0	0
$197$	22.1707	1.57959	0.789797	−	0.613368i	$-0.210186\pi$
$197$	22.1707	1.57959	0.789797	+	0.613368i	$0.210186\pi$
$198$	−1.75093	−0.124433
$199$	7.79167	0.552337	0.276168	−	0.961109i	$-0.410935\pi$
$199$	7.79167	0.552337	0.276168	+	0.961109i	$0.410935\pi$
$200$	24.5111	1.73320
$201$	−6.21525	−0.438390
$202$	0.268732	0.0189079
$203$	0	0
$204$	2.02075	0.141481
$205$	−3.80505	−0.265756
$206$	−22.0028	−1.53301
$207$	3.28488	0.228315
$208$	−20.5165	−1.42257
$209$	−2.78283	−0.192493
$210$	0	0
$211$	7.81577	0.538060	0.269030	−	0.963132i	$-0.413297\pi$
$211$	7.81577	0.538060	0.269030	+	0.963132i	$0.413297\pi$
$212$	−0.657089	−0.0451291
$213$	2.30854	0.158178
$214$	18.2772	1.24940
$215$	−8.07111	−0.550445
$216$	2.58600	0.175955
$217$	0	0
$218$	−25.7479	−1.74387
$219$	−11.9079	−0.804658
$220$	−1.28416	−0.0865778
$221$	31.5848	2.12463
$222$	−1.71411	−0.115044
$223$	−22.6963	−1.51986	−0.759929	−	0.650006i	$-0.774766\pi$
$223$	−22.6963	−1.51986	−0.759929	+	0.650006i	$0.774766\pi$
$224$	0	0
$225$	9.47840	0.631893
$226$	−2.47992	−0.164962
$227$	1.16540	0.0773504	0.0386752	−	0.999252i	$-0.487686\pi$
$227$	1.16540	0.0773504	0.0386752	+	0.999252i	$0.487686\pi$
$228$	−0.702072	−0.0464959
$229$	24.8081	1.63936	0.819682	−	0.572819i	$-0.194150\pi$
$229$	24.8081	1.63936	0.819682	+	0.572819i	$0.194150\pi$
$230$	18.9220	1.24768
$231$	0	0
$232$	3.31370	0.217555
$233$	−18.9135	−1.23906	−0.619531	−	0.784972i	$-0.712677\pi$
$233$	−18.9135	−1.23906	−0.619531	+	0.784972i	$0.712677\pi$
$234$	−6.90445	−0.451358
$235$	−28.1490	−1.83624
$236$	3.76798	0.245275
$237$	−12.7517	−0.828309
$238$	0	0
$239$	5.69526	0.368396	0.184198	−	0.982889i	$-0.441031\pi$
$239$	5.69526	0.368396	0.184198	+	0.982889i	$0.441031\pi$
$240$	17.1168	1.10488
$241$	−12.4796	−0.803884	−0.401942	−	0.915665i	$-0.631665\pi$
$241$	−12.4796	−0.803884	−0.401942	+	0.915665i	$0.631665\pi$
$242$	14.6274	0.940286
$243$	1.00000	0.0641500
$244$	−0.751310	−0.0480977
$245$	0	0
$246$	−1.51387	−0.0965206
$247$	−10.9736	−0.698230
$248$	0.865459	0.0549567
$249$	−2.58930	−0.164090
$250$	25.7971	1.63155
$251$	29.7508	1.87785	0.938926	−	0.344120i	$-0.111823\pi$
$251$	29.7508	1.87785	0.938926	+	0.344120i	$0.111823\pi$
$252$	0	0
$253$	3.79927	0.238858
$254$	31.4424	1.97287
$255$	−26.3510	−1.65016
$256$	6.86124	0.428827
$257$	−4.66090	−0.290739	−0.145369	−	0.989377i	$-0.546437\pi$
$257$	−4.66090	−0.290739	−0.145369	+	0.989377i	$0.546437\pi$
$258$	−3.21115	−0.199918
$259$	0	0
$260$	−5.06382	−0.314045
$261$	1.28140	0.0793166
$262$	18.2195	1.12561
$263$	24.2173	1.49330	0.746650	−	0.665217i	$-0.231661\pi$
$263$	24.2173	1.49330	0.746650	+	0.665217i	$0.231661\pi$
$264$	2.99095	0.184080
$265$	8.56857	0.526363
$266$	0	0
$267$	9.21039	0.563667
$268$	−1.81357	−0.110782
$269$	26.7174	1.62899	0.814495	−	0.580170i	$-0.197014\pi$
$269$	26.7174	1.62899	0.814495	+	0.580170i	$0.197014\pi$
$270$	5.76034	0.350563
$271$	21.2293	1.28959	0.644795	−	0.764356i	$-0.276943\pi$
$271$	21.2293	1.28959	0.644795	+	0.764356i	$0.276943\pi$
$272$	−31.1530	−1.88893
$273$	0	0
$274$	9.84108	0.594522
$275$	10.9627	0.661074
$276$	0.958507	0.0576953
$277$	−8.30807	−0.499184	−0.249592	−	0.968351i	$-0.580296\pi$
$277$	−8.30807	−0.499184	−0.249592	+	0.968351i	$0.580296\pi$
$278$	35.0166	2.10015
$279$	0.334671	0.0200362
$280$	0	0
$281$	−19.0193	−1.13460	−0.567300	−	0.823511i	$-0.692012\pi$
$281$	−19.0193	−1.13460	−0.567300	+	0.823511i	$0.692012\pi$
$282$	−11.1993	−0.666907
$283$	−8.80867	−0.523621	−0.261811	−	0.965119i	$-0.584320\pi$
$283$	−8.80867	−0.523621	−0.261811	+	0.965119i	$0.584320\pi$
$284$	0.673617	0.0399718
$285$	9.15516	0.542305
$286$	−7.98566	−0.472202
$287$	0	0
$288$	1.63805	0.0965232
$289$	30.9594	1.82114
$290$	7.38130	0.433445
$291$	12.8974	0.756058
$292$	−3.47464	−0.203338
$293$	30.0047	1.75289	0.876446	−	0.481500i	$-0.159908\pi$
$293$	30.0047	1.75289	0.876446	+	0.481500i	$0.159908\pi$
$294$	0	0
$295$	−49.1352	−2.86076
$296$	2.92806	0.170190
$297$	1.15660	0.0671125
$298$	7.87276	0.456056
$299$	14.9817	0.866413
$300$	2.76574	0.159680
$301$	0	0
$302$	15.5774	0.896376
$303$	−0.177513	−0.0101979
$304$	10.8235	0.620771
$305$	9.79723	0.560988
$306$	−10.4839	−0.599327
$307$	26.3241	1.50240	0.751199	−	0.660076i	$-0.229476\pi$
$307$	26.3241	1.50240	0.751199	+	0.660076i	$0.229476\pi$
$308$	0	0
$309$	14.5341	0.826819
$310$	1.92782	0.109493
$311$	−21.2895	−1.20721	−0.603607	−	0.797282i	$-0.706270\pi$
$311$	−21.2895	−1.20721	−0.603607	+	0.797282i	$0.706270\pi$
$312$	11.7942	0.667717
$313$	14.8092	0.837065	0.418532	−	0.908202i	$-0.362545\pi$
$313$	14.8092	0.837065	0.418532	+	0.908202i	$0.362545\pi$
$314$	6.10977	0.344794
$315$	0	0
$316$	−3.72085	−0.209314
$317$	3.95277	0.222010	0.111005	−	0.993820i	$-0.464593\pi$
$317$	3.95277	0.222010	0.111005	+	0.993820i	$0.464593\pi$
$318$	3.40907	0.191171
$319$	1.48206	0.0829795
$320$	−24.7979	−1.38624
$321$	−12.0732	−0.673859
$322$	0	0
$323$	−16.6626	−0.927132
$324$	0.291794	0.0162108
$325$	43.2291	2.39792
$326$	−10.3687	−0.574271
$327$	17.0080	0.940547
$328$	2.58600	0.142788
$329$	0	0
$330$	6.66238	0.366752
$331$	30.9459	1.70094	0.850470	−	0.526023i	$-0.176317\pi$
$331$	30.9459	1.70094	0.850470	+	0.526023i	$0.176317\pi$
$332$	−0.755542	−0.0414658
$333$	1.13227	0.0620483
$334$	4.12946	0.225954
$335$	23.6493	1.29210
$336$	0	0
$337$	32.5965	1.77564	0.887822	−	0.460188i	$-0.152218\pi$
$337$	32.5965	1.77564	0.887822	+	0.460188i	$0.152218\pi$
$338$	−11.8096	−0.642357
$339$	1.63814	0.0889715
$340$	−7.68906	−0.416998
$341$	0.387079	0.0209615
$342$	3.64245	0.196961
$343$	0	0
$344$	5.48531	0.295748
$345$	−12.4991	−0.672930
$346$	−5.41234	−0.290969
$347$	−24.1324	−1.29549	−0.647747	−	0.761855i	$-0.724289\pi$
$347$	−24.1324	−1.29549	−0.647747	+	0.761855i	$0.724289\pi$
$348$	0.373905	0.0200434
$349$	24.9455	1.33530	0.667651	−	0.744474i	$-0.267300\pi$
$349$	24.9455	1.33530	0.667651	+	0.744474i	$0.267300\pi$
$350$	0	0
$351$	4.56081	0.243438
$352$	1.89456	0.100981
$353$	35.1182	1.86916	0.934578	−	0.355759i	$-0.115778\pi$
$353$	35.1182	1.86916	0.934578	+	0.355759i	$0.115778\pi$
$354$	−19.5488	−1.03901
$355$	−8.78410	−0.466211
$356$	2.68754	0.142439
$357$	0	0
$358$	−20.1951	−1.06734
$359$	−2.09790	−0.110723	−0.0553614	−	0.998466i	$-0.517631\pi$
$359$	−2.09790	−0.110723	−0.0553614	+	0.998466i	$0.517631\pi$
$360$	−9.83985	−0.518605
$361$	−13.2109	−0.695310
$362$	−9.32504	−0.490113
$363$	−9.66229	−0.507139
$364$	0	0
$365$	45.3100	2.37163
$366$	3.89790	0.203746
$367$	−15.7507	−0.822180	−0.411090	−	0.911595i	$-0.634852\pi$
$367$	−15.7507	−0.822180	−0.411090	+	0.911595i	$0.634852\pi$
$368$	−14.7768	−0.770295
$369$	1.00000	0.0520579
$370$	6.52229	0.339077
$371$	0	0
$372$	0.0976550	0.00506318
$373$	−10.4347	−0.540289	−0.270144	−	0.962820i	$-0.587071\pi$
$373$	−10.4347	−0.540289	−0.270144	+	0.962820i	$0.587071\pi$
$374$	−12.1257	−0.627004
$375$	−17.0405	−0.879969
$376$	19.1307	0.986589
$377$	5.84421	0.300992
$378$	0	0
$379$	−13.0415	−0.669895	−0.334947	−	0.942237i	$-0.608719\pi$
$379$	−13.0415	−0.669895	−0.334947	+	0.942237i	$0.608719\pi$
$380$	2.67142	0.137041
$381$	−20.7696	−1.06406
$382$	−36.1347	−1.84881
$383$	−13.8502	−0.707714	−0.353857	−	0.935299i	$-0.615130\pi$
$383$	−13.8502	−0.707714	−0.353857	+	0.935299i	$0.615130\pi$
$384$	−13.1421	−0.670656
$385$	0	0
$386$	32.8216	1.67058
$387$	2.12116	0.107825
$388$	3.76338	0.191057
$389$	29.4776	1.49458	0.747288	−	0.664500i	$-0.231356\pi$
$389$	29.4776	1.49458	0.747288	+	0.664500i	$0.231356\pi$
$390$	26.2718	1.33032
$391$	22.7487	1.15045
$392$	0	0
$393$	−12.0351	−0.607090
$394$	−33.5634	−1.69090
$395$	48.5207	2.44134
$396$	0.337487	0.0169594
$397$	25.5717	1.28341	0.641703	−	0.766954i	$-0.278228\pi$
$397$	25.5717	1.28341	0.641703	+	0.766954i	$0.278228\pi$
$398$	−11.7955	−0.591257
$399$	0	0
$400$	−42.6381	−2.13190
$401$	6.69303	0.334234	0.167117	−	0.985937i	$-0.446554\pi$
$401$	6.69303	0.334234	0.167117	+	0.985937i	$0.446554\pi$
$402$	9.40906	0.469281
$403$	1.52637	0.0760339
$404$	−0.0517973	−0.00257701
$405$	−3.80505	−0.189074
$406$	0	0
$407$	1.30958	0.0649136
$408$	17.9087	0.886615
$409$	18.9785	0.938425	0.469212	−	0.883085i	$-0.344538\pi$
$409$	18.9785	0.938425	0.469212	+	0.883085i	$0.344538\pi$
$410$	5.76034	0.284483
$411$	−6.50063	−0.320652
$412$	4.24097	0.208938
$413$	0	0
$414$	−4.97287	−0.244403
$415$	9.85242	0.483636
$416$	7.47084	0.366288
$417$	−23.1305	−1.13271
$418$	4.21284	0.206057
$419$	−38.3019	−1.87117	−0.935586	−	0.353099i	$-0.885128\pi$
$419$	−38.3019	−1.87117	−0.935586	+	0.353099i	$0.885128\pi$
$420$	0	0
$421$	13.7293	0.669124	0.334562	−	0.942374i	$-0.391412\pi$
$421$	13.7293	0.669124	0.334562	+	0.942374i	$0.391412\pi$
$422$	−11.8320	−0.575974
$423$	7.39779	0.359693
$424$	−5.82339	−0.282809
$425$	65.6405	3.18403
$426$	−3.49482	−0.169325
$427$	0	0
$428$	−3.52288	−0.170285
$429$	5.27501	0.254680
$430$	12.2186	0.589233
$431$	−1.39769	−0.0673244	−0.0336622	−	0.999433i	$-0.510717\pi$
$431$	−1.39769	−0.0673244	−0.0336622	+	0.999433i	$0.510717\pi$
$432$	−4.49844	−0.216431
$433$	−19.3022	−0.927605	−0.463802	−	0.885939i	$-0.653515\pi$
$433$	−19.3022	−0.927605	−0.463802	+	0.885939i	$0.653515\pi$
$434$	0	0
$435$	−4.87579	−0.233776
$436$	4.96284	0.237677
$437$	−7.90359	−0.378080
$438$	18.0269	0.861359
$439$	2.98535	0.142483	0.0712416	−	0.997459i	$-0.477304\pi$
$439$	2.98535	0.142483	0.0712416	+	0.997459i	$0.477304\pi$
$440$	−11.3807	−0.542555
$441$	0	0
$442$	−47.8152	−2.27434
$443$	35.3757	1.68075	0.840374	−	0.542007i	$-0.182335\pi$
$443$	35.3757	1.68075	0.840374	+	0.542007i	$0.182335\pi$
$444$	0.330391	0.0156797
$445$	−35.0460	−1.66134
$446$	34.3592	1.62696
$447$	−5.20043	−0.245972
$448$	0	0
$449$	−24.7278	−1.16698	−0.583489	−	0.812121i	$-0.698313\pi$
$449$	−24.7278	−1.16698	−0.583489	+	0.812121i	$0.698313\pi$
$450$	−14.3490	−0.676420
$451$	1.15660	0.0544619
$452$	0.477999	0.0224832
$453$	−10.2898	−0.483456
$454$	−1.76426	−0.0828010
$455$	0	0
$456$	−6.22205	−0.291374
$457$	−9.46161	−0.442595	−0.221298	−	0.975206i	$-0.571029\pi$
$457$	−9.46161	−0.442595	−0.221298	+	0.975206i	$0.571029\pi$
$458$	−37.5561	−1.75488
$459$	6.92527	0.323244
$460$	−3.64716	−0.170050
$461$	20.5811	0.958557	0.479279	−	0.877663i	$-0.340898\pi$
$461$	20.5811	0.958557	0.479279	+	0.877663i	$0.340898\pi$
$462$	0	0
$463$	−27.1195	−1.26035	−0.630175	−	0.776453i	$-0.717017\pi$
$463$	−27.1195	−1.26035	−0.630175	+	0.776453i	$0.717017\pi$
$464$	−5.76430	−0.267601
$465$	−1.27344	−0.0590544
$466$	28.6325	1.32637
$467$	−6.72518	−0.311204	−0.155602	−	0.987820i	$-0.549732\pi$
$467$	−6.72518	−0.311204	−0.155602	+	0.987820i	$0.549732\pi$
$468$	1.33082	0.0615169
$469$	0	0
$470$	42.6138	1.96563
$471$	−4.03587	−0.185963
$472$	33.3934	1.53705
$473$	2.45332	0.112804
$474$	19.3043	0.886676
$475$	−22.8056	−1.04639
$476$	0	0
$477$	−2.25189	−0.103107
$478$	−8.62186	−0.394355
$479$	−18.3927	−0.840382	−0.420191	−	0.907436i	$-0.638037\pi$
$479$	−18.3927	−0.840382	−0.420191	+	0.907436i	$0.638037\pi$
$480$	−6.23287	−0.284490
$481$	5.16408	0.235462
$482$	18.8925	0.860531
$483$	0	0
$484$	−2.81940	−0.128154
$485$	−49.0752	−2.22839
$486$	−1.51387	−0.0686704
$487$	8.31877	0.376960	0.188480	−	0.982077i	$-0.439644\pi$
$487$	8.31877	0.376960	0.188480	+	0.982077i	$0.439644\pi$
$488$	−6.65842	−0.301412
$489$	6.84917	0.309730
$490$	0	0
$491$	−0.703140	−0.0317323	−0.0158661	−	0.999874i	$-0.505051\pi$
$491$	−0.703140	−0.0317323	−0.0158661	+	0.999874i	$0.505051\pi$
$492$	0.291794	0.0131551
$493$	8.87404	0.399667
$494$	16.6125	0.747432
$495$	−4.40090	−0.197806
$496$	−1.50550	−0.0675989
$497$	0	0
$498$	3.91986	0.175653
$499$	−25.1798	−1.12720	−0.563601	−	0.826047i	$-0.690584\pi$
$499$	−25.1798	−1.12720	−0.563601	+	0.826047i	$0.690584\pi$
$500$	−4.97232	−0.222369
$501$	−2.72776	−0.121867
$502$	−45.0387	−2.01018
$503$	9.10380	0.405918	0.202959	−	0.979187i	$-0.434944\pi$
$503$	9.10380	0.405918	0.202959	+	0.979187i	$0.434944\pi$
$504$	0	0
$505$	0.675447	0.0300570
$506$	−5.75159	−0.255689
$507$	7.80095	0.346452
$508$	−6.06044	−0.268889
$509$	−8.52853	−0.378021	−0.189010	−	0.981975i	$-0.560528\pi$
$509$	−8.52853	−0.378021	−0.189010	+	0.981975i	$0.560528\pi$
$510$	39.8919	1.76644
$511$	0	0
$512$	15.8972	0.702565
$513$	−2.40606	−0.106230
$514$	7.05598	0.311226
$515$	−55.3031	−2.43695
$516$	0.618941	0.0272474
$517$	8.55625	0.376304
$518$	0	0
$519$	3.57517	0.156933
$520$	−44.8776	−1.96801
$521$	−29.7200	−1.30206	−0.651029	−	0.759053i	$-0.725662\pi$
$521$	−29.7200	−1.30206	−0.651029	+	0.759053i	$0.725662\pi$
$522$	−1.93987	−0.0849058
$523$	−23.3081	−1.01919	−0.509597	−	0.860413i	$-0.670205\pi$
$523$	−23.3081	−1.01919	−0.509597	+	0.860413i	$0.670205\pi$
$524$	−3.51177	−0.153412
$525$	0	0
$526$	−36.6617	−1.59853
$527$	2.31769	0.100960
$528$	−5.20288	−0.226426
$529$	−12.2096	−0.530852
$530$	−12.9717	−0.563454
$531$	12.9132	0.560383
$532$	0	0
$533$	4.56081	0.197550
$534$	−13.9433	−0.603386
$535$	45.9390	1.98612
$536$	−16.0726	−0.694231
$537$	13.3401	0.575667
$538$	−40.4466	−1.74378
$539$	0	0
$540$	−1.11029	−0.0477793
$541$	22.9665	0.987407	0.493704	−	0.869630i	$-0.335643\pi$
$541$	22.9665	0.987407	0.493704	+	0.869630i	$0.335643\pi$
$542$	−32.1384	−1.38046
$543$	6.15975	0.264340
$544$	11.3440	0.486368
$545$	−64.7164	−2.77215
$546$	0	0
$547$	−19.0779	−0.815711	−0.407856	−	0.913046i	$-0.633723\pi$
$547$	−19.0779	−0.815711	−0.407856	+	0.913046i	$0.633723\pi$
$548$	−1.89684	−0.0810291
$549$	−2.57480	−0.109890
$550$	−16.5960	−0.707657
$551$	−3.08312	−0.131345
$552$	8.49468	0.361558
$553$	0	0
$554$	12.5773	0.534359
$555$	−4.30836	−0.182880
$556$	−6.74935	−0.286236
$557$	−1.60892	−0.0681723	−0.0340861	−	0.999419i	$-0.510852\pi$
$557$	−1.60892	−0.0681723	−0.0340861	+	0.999419i	$0.510852\pi$
$558$	−0.506648	−0.0214481
$559$	9.67419	0.409175
$560$	0	0
$561$	8.00974	0.338171
$562$	28.7928	1.21455
$563$	−40.0650	−1.68854	−0.844268	−	0.535921i	$-0.819965\pi$
$563$	−40.0650	−1.68854	−0.844268	+	0.535921i	$0.819965\pi$
$564$	2.15863	0.0908948
$565$	−6.23320	−0.262233
$566$	13.3352	0.560519
$567$	0	0
$568$	5.96987	0.250490
$569$	0.910623	0.0381753	0.0190876	−	0.999818i	$-0.493924\pi$
$569$	0.910623	0.0381753	0.0190876	+	0.999818i	$0.493924\pi$
$570$	−13.8597	−0.580519
$571$	0.733915	0.0307134	0.0153567	−	0.999882i	$-0.495112\pi$
$571$	0.733915	0.0307134	0.0153567	+	0.999882i	$0.495112\pi$
$572$	1.53921	0.0643578
$573$	23.8691	0.997146
$574$	0	0
$575$	31.1354	1.29843
$576$	6.51709	0.271546
$577$	31.6908	1.31930	0.659652	−	0.751571i	$-0.270704\pi$
$577$	31.6908	1.31930	0.659652	+	0.751571i	$0.270704\pi$
$578$	−46.8684	−1.94947
$579$	−21.6807	−0.901017
$580$	−1.42273	−0.0590754
$581$	0	0
$582$	−19.5249	−0.809334
$583$	−2.60453	−0.107869
$584$	−30.7937	−1.27425
$585$	−17.3541	−0.717503
$586$	−45.4231	−1.87641
$587$	20.9045	0.862823	0.431411	−	0.902155i	$-0.358016\pi$
$587$	20.9045	0.862823	0.431411	+	0.902155i	$0.358016\pi$
$588$	0	0
$589$	−0.805238	−0.0331792
$590$	74.3841	3.06235
$591$	22.1707	0.911979
$592$	−5.09347	−0.209340
$593$	−1.44402	−0.0592987	−0.0296494	−	0.999560i	$-0.509439\pi$
$593$	−1.44402	−0.0592987	−0.0296494	+	0.999560i	$0.509439\pi$
$594$	−1.75093	−0.0718416
$595$	0	0
$596$	−1.51745	−0.0621573
$597$	7.79167	0.318892
$598$	−22.6803	−0.927465
$599$	10.3458	0.422719	0.211360	−	0.977408i	$-0.432211\pi$
$599$	10.3458	0.422719	0.211360	+	0.977408i	$0.432211\pi$
$600$	24.5111	1.00066
$601$	−8.35641	−0.340865	−0.170432	−	0.985369i	$-0.554516\pi$
$601$	−8.35641	−0.340865	−0.170432	+	0.985369i	$0.554516\pi$
$602$	0	0
$603$	−6.21525	−0.253104
$604$	−3.00249	−0.122170
$605$	36.7655	1.49473
$606$	0.268732	0.0109165
$607$	0.600544	0.0243753	0.0121877	−	0.999926i	$-0.496120\pi$
$607$	0.600544	0.0243753	0.0121877	+	0.999926i	$0.496120\pi$
$608$	−3.94125	−0.159839
$609$	0	0
$610$	−14.8317	−0.600518
$611$	33.7399	1.36497
$612$	2.02075	0.0816841
$613$	−20.2750	−0.818899	−0.409449	−	0.912333i	$-0.634279\pi$
$613$	−20.2750	−0.818899	−0.409449	+	0.912333i	$0.634279\pi$
$614$	−39.8512	−1.60827
$615$	−3.80505	−0.153434
$616$	0	0
$617$	−23.4448	−0.943854	−0.471927	−	0.881638i	$-0.656441\pi$
$617$	−23.4448	−0.943854	−0.471927	+	0.881638i	$0.656441\pi$
$618$	−22.0028	−0.885081
$619$	−10.3180	−0.414716	−0.207358	−	0.978265i	$-0.566487\pi$
$619$	−10.3180	−0.414716	−0.207358	+	0.978265i	$0.566487\pi$
$620$	−0.371582	−0.0149231
$621$	3.28488	0.131818
$622$	32.2294	1.29228
$623$	0	0
$624$	−20.5165	−0.821318
$625$	17.4481	0.697923
$626$	−22.4191	−0.896049
$627$	−2.78283	−0.111136
$628$	−1.17764	−0.0469930
$629$	7.84131	0.312654
$630$	0	0
$631$	5.95019	0.236873	0.118437	−	0.992962i	$-0.462212\pi$
$631$	5.95019	0.236873	0.118437	+	0.992962i	$0.462212\pi$
$632$	−32.9757	−1.31170
$633$	7.81577	0.310649
$634$	−5.98397	−0.237654
$635$	79.0293	3.13618
$636$	−0.657089	−0.0260553
$637$	0	0
$638$	−2.24364	−0.0888267
$639$	2.30854	0.0913243
$640$	50.0064	1.97668
$641$	−18.7783	−0.741697	−0.370848	−	0.928693i	$-0.620933\pi$
$641$	−18.7783	−0.741697	−0.370848	+	0.928693i	$0.620933\pi$
$642$	18.2772	0.721343
$643$	26.6022	1.04909	0.524543	−	0.851384i	$-0.324236\pi$
$643$	26.6022	1.04909	0.524543	+	0.851384i	$0.324236\pi$
$644$	0	0
$645$	−8.07111	−0.317800
$646$	25.2250	0.992463
$647$	23.5755	0.926847	0.463424	−	0.886137i	$-0.346621\pi$
$647$	23.5755	0.926847	0.463424	+	0.886137i	$0.346621\pi$
$648$	2.58600	0.101588
$649$	14.9353	0.586261
$650$	−65.4432	−2.56689
$651$	0	0
$652$	1.99855	0.0782691
$653$	−17.2768	−0.676093	−0.338046	−	0.941129i	$-0.609766\pi$
$653$	−17.2768	−0.676093	−0.338046	+	0.941129i	$0.609766\pi$
$654$	−25.7479	−1.00682
$655$	45.7941	1.78933
$656$	−4.49844	−0.175635
$657$	−11.9079	−0.464570
$658$	0	0
$659$	32.3358	1.25963	0.629813	−	0.776747i	$-0.283132\pi$
$659$	32.3358	1.25963	0.629813	+	0.776747i	$0.283132\pi$
$660$	−1.28416	−0.0499857
$661$	41.8828	1.62905	0.814525	−	0.580128i	$-0.196998\pi$
$661$	41.8828	1.62905	0.814525	+	0.580128i	$0.196998\pi$
$662$	−46.8480	−1.82080
$663$	31.5848	1.22665
$664$	−6.69593	−0.259852
$665$	0	0
$666$	−1.71411	−0.0664205
$667$	4.20924	0.162982
$668$	−0.795943	−0.0307960
$669$	−22.6963	−0.877490
$670$	−35.8019	−1.38315
$671$	−2.97800	−0.114964
$672$	0	0
$673$	−32.0621	−1.23590	−0.617951	−	0.786217i	$-0.712037\pi$
$673$	−32.0621	−1.23590	−0.617951	+	0.786217i	$0.712037\pi$
$674$	−49.3467	−1.90077
$675$	9.47840	0.364824
$676$	2.27627	0.0875488
$677$	−39.2517	−1.50857	−0.754283	−	0.656549i	$-0.772016\pi$
$677$	−39.2517	−1.50857	−0.754283	+	0.656549i	$0.772016\pi$
$678$	−2.47992	−0.0952409
$679$	0	0
$680$	−68.1436	−2.61319
$681$	1.16540	0.0446583
$682$	−0.585987	−0.0224386
$683$	−15.5806	−0.596176	−0.298088	−	0.954538i	$-0.596349\pi$
$683$	−15.5806	−0.596176	−0.298088	+	0.954538i	$0.596349\pi$
$684$	−0.702072	−0.0268444
$685$	24.7352	0.945084
$686$	0	0
$687$	24.8081	0.946487
$688$	−9.54191	−0.363782
$689$	−10.2705	−0.391273
$690$	18.9220	0.720348
$691$	18.0215	0.685569	0.342784	−	0.939414i	$-0.388630\pi$
$691$	18.0215	0.685569	0.342784	+	0.939414i	$0.388630\pi$
$692$	1.04321	0.0396570
$693$	0	0
$694$	36.5332	1.38678
$695$	88.0129	3.33852
$696$	3.31370	0.125605
$697$	6.92527	0.262313
$698$	−37.7642	−1.42940
$699$	−18.9135	−0.715373
$700$	0	0
$701$	−5.16472	−0.195069	−0.0975344	−	0.995232i	$-0.531096\pi$
$701$	−5.16472	−0.195069	−0.0975344	+	0.995232i	$0.531096\pi$
$702$	−6.90445	−0.260592
$703$	−2.72432	−0.102750
$704$	7.53764	0.284086
$705$	−28.1490	−1.06015
$706$	−53.1643	−2.00087
$707$	0	0
$708$	3.76798	0.141609
$709$	11.3643	0.426796	0.213398	−	0.976965i	$-0.431547\pi$
$709$	11.3643	0.426796	0.213398	+	0.976965i	$0.431547\pi$
$710$	13.2980	0.499063
$711$	−12.7517	−0.478224
$712$	23.8180	0.892619
$713$	1.09935	0.0411711
$714$	0	0
$715$	−20.0717	−0.750637
$716$	3.89255	0.145472
$717$	5.69526	0.212693
$718$	3.17594	0.118525
$719$	30.5843	1.14060	0.570302	−	0.821435i	$-0.306826\pi$
$719$	30.5843	1.14060	0.570302	+	0.821435i	$0.306826\pi$
$720$	17.1168	0.637906
$721$	0	0
$722$	19.9995	0.744306
$723$	−12.4796	−0.464123
$724$	1.79738	0.0667990
$725$	12.1456	0.451077
$726$	14.6274	0.542874
$727$	18.0711	0.670221	0.335111	−	0.942179i	$-0.391226\pi$
$727$	18.0711	0.670221	0.335111	+	0.942179i	$0.391226\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−68.5933	−2.53875
$731$	14.6896	0.543315
$732$	−0.751310	−0.0277692
$733$	−49.3640	−1.82330	−0.911651	−	0.410966i	$-0.865192\pi$
$733$	−49.3640	−1.82330	−0.911651	+	0.410966i	$0.865192\pi$
$734$	23.8445	0.880115
$735$	0	0
$736$	5.38080	0.198339
$737$	−7.18853	−0.264793
$738$	−1.51387	−0.0557262
$739$	−9.21630	−0.339027	−0.169514	−	0.985528i	$-0.554220\pi$
$739$	−9.21630	−0.339027	−0.169514	+	0.985528i	$0.554220\pi$
$740$	−1.25715	−0.0462139
$741$	−10.9736	−0.403124
$742$	0	0
$743$	−47.8911	−1.75696	−0.878478	−	0.477783i	$-0.841440\pi$
$743$	−47.8911	−1.75696	−0.878478	+	0.477783i	$0.841440\pi$
$744$	0.865459	0.0317293
$745$	19.7879	0.724972
$746$	15.7968	0.578361
$747$	−2.58930	−0.0947376
$748$	2.33719	0.0854562
$749$	0	0
$750$	25.7971	0.941977
$751$	−0.992350	−0.0362114	−0.0181057	−	0.999836i	$-0.505764\pi$
$751$	−0.992350	−0.0362114	−0.0181057	+	0.999836i	$0.505764\pi$
$752$	−33.2786	−1.21354
$753$	29.7508	1.08418
$754$	−8.84736	−0.322202
$755$	39.1531	1.42493
$756$	0	0
$757$	17.2370	0.626490	0.313245	−	0.949672i	$-0.398584\pi$
$757$	17.2370	0.626490	0.313245	+	0.949672i	$0.398584\pi$
$758$	19.7430	0.717100
$759$	3.79927	0.137905
$760$	23.6752	0.858791
$761$	−25.0256	−0.907177	−0.453589	−	0.891211i	$-0.649856\pi$
$761$	−25.0256	−0.907177	−0.453589	+	0.891211i	$0.649856\pi$
$762$	31.4424	1.13904
$763$	0	0
$764$	6.96486	0.251980
$765$	−26.3510	−0.952723
$766$	20.9674	0.757584
$767$	58.8944	2.12655
$768$	6.86124	0.247584
$769$	12.7073	0.458239	0.229119	−	0.973398i	$-0.426415\pi$
$769$	12.7073	0.458239	0.229119	+	0.973398i	$0.426415\pi$
$770$	0	0
$771$	−4.66090	−0.167858
$772$	−6.32628	−0.227688
$773$	47.3022	1.70134	0.850671	−	0.525699i	$-0.176196\pi$
$773$	47.3022	1.70134	0.850671	+	0.525699i	$0.176196\pi$
$774$	−3.21115	−0.115422
$775$	3.17215	0.113947
$776$	33.3526	1.19729
$777$	0	0
$778$	−44.6252	−1.59989
$779$	−2.40606	−0.0862059
$780$	−5.06382	−0.181314
$781$	2.67004	0.0955417
$782$	−34.4385	−1.23152
$783$	1.28140	0.0457935
$784$	0	0
$785$	15.3567	0.548103
$786$	18.2195	0.649869
$787$	41.5945	1.48269	0.741343	−	0.671127i	$-0.234189\pi$
$787$	41.5945	1.48269	0.741343	+	0.671127i	$0.234189\pi$
$788$	6.46927	0.230458
$789$	24.2173	0.862158
$790$	−73.4539	−2.61337
$791$	0	0
$792$	2.99095	0.106279
$793$	−11.7431	−0.417011
$794$	−38.7121	−1.37384
$795$	8.56857	0.303896
$796$	2.27356	0.0805842
$797$	50.4009	1.78529	0.892645	−	0.450760i	$-0.148847\pi$
$797$	50.4009	1.78529	0.892645	+	0.450760i	$0.148847\pi$
$798$	0	0
$799$	51.2317	1.81245
$800$	15.5261	0.548931
$801$	9.21039	0.325433
$802$	−10.1324	−0.357786
$803$	−13.7726	−0.486024
$804$	−1.81357	−0.0639597
$805$	0	0
$806$	−2.31072	−0.0813917
$807$	26.7174	0.940498
$808$	−0.459049	−0.0161493
$809$	18.2063	0.640098	0.320049	−	0.947401i	$-0.396301\pi$
$809$	18.2063	0.640098	0.320049	+	0.947401i	$0.396301\pi$
$810$	5.76034	0.202398
$811$	47.0365	1.65168	0.825838	−	0.563908i	$-0.190703\pi$
$811$	47.0365	1.65168	0.825838	+	0.563908i	$0.190703\pi$
$812$	0	0
$813$	21.2293	0.744545
$814$	−1.98254	−0.0694878
$815$	−26.0614	−0.912892
$816$	−31.1530	−1.09057
$817$	−5.10363	−0.178553
$818$	−28.7309	−1.00455
$819$	0	0
$820$	−1.11029	−0.0387730
$821$	−52.1811	−1.82113	−0.910566	−	0.413364i	$-0.864354\pi$
$821$	−52.1811	−1.82113	−0.910566	+	0.413364i	$0.864354\pi$
$822$	9.84108	0.343247
$823$	−4.29085	−0.149570	−0.0747848	−	0.997200i	$-0.523827\pi$
$823$	−4.29085	−0.149570	−0.0747848	+	0.997200i	$0.523827\pi$
$824$	37.5853	1.30934
$825$	10.9627	0.381671
$826$	0	0
$827$	46.0862	1.60257	0.801287	−	0.598280i	$-0.204149\pi$
$827$	46.0862	1.60257	0.801287	+	0.598280i	$0.204149\pi$
$828$	0.958507	0.0333104
$829$	12.1577	0.422256	0.211128	−	0.977458i	$-0.432286\pi$
$829$	12.1577	0.422256	0.211128	+	0.977458i	$0.432286\pi$
$830$	−14.9153	−0.517716
$831$	−8.30807	−0.288204
$832$	29.7232	1.03047
$833$	0	0
$834$	35.0166	1.21252
$835$	10.3793	0.359189
$836$	−0.812014	−0.0280841
$837$	0.334671	0.0115679
$838$	57.9840	2.00303
$839$	4.79517	0.165548	0.0827739	−	0.996568i	$-0.473622\pi$
$839$	4.79517	0.165548	0.0827739	+	0.996568i	$0.473622\pi$
$840$	0	0
$841$	−27.3580	−0.943380
$842$	−20.7843	−0.716274
$843$	−19.0193	−0.655061
$844$	2.28059	0.0785012
$845$	−29.6830	−1.02113
$846$	−11.1993	−0.385039
$847$	0	0
$848$	10.1300	0.347866
$849$	−8.80867	−0.302313
$850$	−99.3710	−3.40840
$851$	3.71938	0.127499
$852$	0.673617	0.0230777
$853$	5.26286	0.180197	0.0900984	−	0.995933i	$-0.471282\pi$
$853$	5.26286	0.180197	0.0900984	+	0.995933i	$0.471282\pi$
$854$	0	0
$855$	9.15516	0.313100
$856$	−31.2212	−1.06712
$857$	0.907168	0.0309883	0.0154941	−	0.999880i	$-0.495068\pi$
$857$	0.907168	0.0309883	0.0154941	+	0.999880i	$0.495068\pi$
$858$	−7.98566	−0.272626
$859$	−28.1280	−0.959715	−0.479858	−	0.877346i	$-0.659312\pi$
$859$	−28.1280	−0.959715	−0.479858	+	0.877346i	$0.659312\pi$
$860$	−2.35510	−0.0803083
$861$	0	0
$862$	2.11592	0.0720685
$863$	4.62233	0.157346	0.0786730	−	0.996900i	$-0.474932\pi$
$863$	4.62233	0.157346	0.0786730	+	0.996900i	$0.474932\pi$
$864$	1.63805	0.0557277
$865$	−13.6037	−0.462540
$866$	29.2210	0.992969
$867$	30.9594	1.05144
$868$	0	0
$869$	−14.7485	−0.500309
$870$	7.38130	0.250249
$871$	−28.3465	−0.960486
$872$	43.9827	1.48944
$873$	12.8974	0.436510
$874$	11.9650	0.404722
$875$	0	0
$876$	−3.47464	−0.117397
$877$	−33.5121	−1.13162	−0.565812	−	0.824534i	$-0.691437\pi$
$877$	−33.5121	−1.13162	−0.565812	+	0.824534i	$0.691437\pi$
$878$	−4.51943	−0.152523
$879$	30.0047	1.01203
$880$	19.7972	0.667364
$881$	−5.55431	−0.187129	−0.0935647	−	0.995613i	$-0.529826\pi$
$881$	−5.55431	−0.187129	−0.0935647	+	0.995613i	$0.529826\pi$
$882$	0	0
$883$	33.3288	1.12160	0.560802	−	0.827950i	$-0.310493\pi$
$883$	33.3288	1.12160	0.560802	+	0.827950i	$0.310493\pi$
$884$	9.21626	0.309976
$885$	−49.1352	−1.65166
$886$	−53.5541	−1.79918
$887$	31.4901	1.05733	0.528667	−	0.848829i	$-0.322692\pi$
$887$	31.4901	1.05733	0.528667	+	0.848829i	$0.322692\pi$
$888$	2.92806	0.0982592
$889$	0	0
$890$	53.0550	1.77841
$891$	1.15660	0.0387474
$892$	−6.62265	−0.221743
$893$	−17.7995	−0.595638
$894$	7.87276	0.263304
$895$	−50.7597	−1.69671
$896$	0	0
$897$	14.9817	0.500224
$898$	37.4346	1.24921
$899$	0.428848	0.0143029
$900$	2.76574	0.0921913
$901$	−15.5950	−0.519544
$902$	−1.75093	−0.0582996
$903$	0	0
$904$	4.23622	0.140895
$905$	−23.4381	−0.779110
$906$	15.5774	0.517523
$907$	−24.1540	−0.802021	−0.401010	−	0.916074i	$-0.631341\pi$
$907$	−24.1540	−0.802021	−0.401010	+	0.916074i	$0.631341\pi$
$908$	0.340057	0.0112852
$909$	−0.177513	−0.00588775
$910$	0	0
$911$	13.5342	0.448407	0.224203	−	0.974542i	$-0.428022\pi$
$911$	13.5342	0.448407	0.224203	+	0.974542i	$0.428022\pi$
$912$	10.8235	0.358402
$913$	−2.99477	−0.0991126
$914$	14.3236	0.473783
$915$	9.79723	0.323886
$916$	7.23884	0.239178
$917$	0	0
$918$	−10.4839	−0.346022
$919$	−6.10358	−0.201339	−0.100669	−	0.994920i	$-0.532098\pi$
$919$	−6.10358	−0.201339	−0.100669	+	0.994920i	$0.532098\pi$
$920$	−32.3227	−1.06565
$921$	26.3241	0.867410
$922$	−31.1571	−1.02610
$923$	10.5288	0.346559
$924$	0	0
$925$	10.7322	0.352871
$926$	41.0553	1.34916
$927$	14.5341	0.477364
$928$	2.09900	0.0689030
$929$	34.5625	1.13396	0.566979	−	0.823732i	$-0.308112\pi$
$929$	34.5625	1.13396	0.566979	+	0.823732i	$0.308112\pi$
$930$	1.92782	0.0632157
$931$	0	0
$932$	−5.51883	−0.180775
$933$	−21.2895	−0.696985
$934$	10.1810	0.333133
$935$	−30.4775	−0.996719
$936$	11.7942	0.385506
$937$	−3.45162	−0.112759	−0.0563797	−	0.998409i	$-0.517956\pi$
$937$	−3.45162	−0.112759	−0.0563797	+	0.998409i	$0.517956\pi$
$938$	0	0
$939$	14.8092	0.483280
$940$	−8.21370	−0.267901
$941$	−33.2147	−1.08277	−0.541384	−	0.840775i	$-0.682100\pi$
$941$	−33.2147	−1.08277	−0.541384	+	0.840775i	$0.682100\pi$
$942$	6.10977	0.199067
$943$	3.28488	0.106970
$944$	−58.0891	−1.89064
$945$	0	0
$946$	−3.71400	−0.120753
$947$	29.1330	0.946694	0.473347	−	0.880876i	$-0.343046\pi$
$947$	29.1330	0.946694	0.473347	+	0.880876i	$0.343046\pi$
$948$	−3.72085	−0.120848
$949$	−54.3094	−1.76296
$950$	34.5246	1.12013
$951$	3.95277	0.128177
$952$	0	0
$953$	37.7997	1.22445	0.612226	−	0.790683i	$-0.290274\pi$
$953$	37.7997	1.22445	0.612226	+	0.790683i	$0.290274\pi$
$954$	3.40907	0.110373
$955$	−90.8231	−2.93897
$956$	1.66184	0.0537478
$957$	1.48206	0.0479082
$958$	27.8440	0.899600
$959$	0	0
$960$	−24.7979	−0.800348
$961$	−30.8880	−0.996387
$962$	−7.81774	−0.252054
$963$	−12.0732	−0.389053
$964$	−3.64148	−0.117284
$965$	82.4960	2.65564
$966$	0	0
$967$	17.1460	0.551379	0.275690	−	0.961247i	$-0.411094\pi$
$967$	17.1460	0.551379	0.275690	+	0.961247i	$0.411094\pi$
$968$	−24.9866	−0.803101
$969$	−16.6626	−0.535280
$970$	74.2933	2.38541
$971$	10.5979	0.340104	0.170052	−	0.985435i	$-0.445606\pi$
$971$	10.5979	0.340104	0.170052	+	0.985435i	$0.445606\pi$
$972$	0.291794	0.00935929
$973$	0	0
$974$	−12.5935	−0.403522
$975$	43.2291	1.38444
$976$	11.5826	0.370749
$977$	26.8876	0.860209	0.430105	−	0.902779i	$-0.358477\pi$
$977$	26.8876	0.860209	0.430105	+	0.902779i	$0.358477\pi$
$978$	−10.3687	−0.331555
$979$	10.6527	0.340462
$980$	0	0
$981$	17.0080	0.543025
$982$	1.06446	0.0339683
$983$	27.7596	0.885392	0.442696	−	0.896672i	$-0.354022\pi$
$983$	27.7596	0.885392	0.442696	+	0.896672i	$0.354022\pi$
$984$	2.58600	0.0824386
$985$	−84.3605	−2.68795
$986$	−13.4341	−0.427830
$987$	0	0
$988$	−3.20202	−0.101870
$989$	6.96774	0.221561
$990$	6.66238	0.211744
$991$	12.1510	0.385988	0.192994	−	0.981200i	$-0.438180\pi$
$991$	12.1510	0.385988	0.192994	+	0.981200i	$0.438180\pi$
$992$	0.548209	0.0174057
$993$	30.9459	0.982039
$994$	0	0
$995$	−29.6477	−0.939894
$996$	−0.755542	−0.0239403
$997$	−15.7997	−0.500382	−0.250191	−	0.968197i	$-0.580493\pi$
$997$	−15.7997	−0.500382	−0.250191	+	0.968197i	$0.580493\pi$
$998$	38.1188	1.20663
$999$	1.13227	0.0358236

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.bc.1.3		8
7.3	odd	6		861.2.i.d.247.6	✓	16
7.5	odd	6		861.2.i.d.739.6	yes	16
7.6	odd	2		6027.2.a.bb.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.i.d.247.6	✓	16	7.3	odd	6
861.2.i.d.739.6	yes	16	7.5	odd	6
6027.2.a.bb.1.3		8	7.6	odd	2
6027.2.a.bc.1.3		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)

\(f(q)\)	\(=\)	\(q-1.51387 q^{2} +1.00000 q^{3} +0.291794 q^{4} -3.80505 q^{5} -1.51387 q^{6} +2.58600 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.51387 q^{2} +1.00000 q^{3} +0.291794 q^{4} -3.80505 q^{5} -1.51387 q^{6} +2.58600 q^{8} +1.00000 q^{9} +5.76034 q^{10} +1.15660 q^{11} +0.291794 q^{12} +4.56081 q^{13} -3.80505 q^{15} -4.49844 q^{16} +6.92527 q^{17} -1.51387 q^{18} -2.40606 q^{19} -1.11029 q^{20} -1.75093 q^{22} +3.28488 q^{23} +2.58600 q^{24} +9.47840 q^{25} -6.90445 q^{26} +1.00000 q^{27} +1.28140 q^{29} +5.76034 q^{30} +0.334671 q^{31} +1.63805 q^{32} +1.15660 q^{33} -10.4839 q^{34} +0.291794 q^{36} +1.13227 q^{37} +3.64245 q^{38} +4.56081 q^{39} -9.83985 q^{40} +1.00000 q^{41} +2.12116 q^{43} +0.337487 q^{44} -3.80505 q^{45} -4.97287 q^{46} +7.39779 q^{47} -4.49844 q^{48} -14.3490 q^{50} +6.92527 q^{51} +1.33082 q^{52} -2.25189 q^{53} -1.51387 q^{54} -4.40090 q^{55} -2.40606 q^{57} -1.93987 q^{58} +12.9132 q^{59} -1.11029 q^{60} -2.57480 q^{61} -0.506648 q^{62} +6.51709 q^{64} -17.3541 q^{65} -1.75093 q^{66} -6.21525 q^{67} +2.02075 q^{68} +3.28488 q^{69} +2.30854 q^{71} +2.58600 q^{72} -11.9079 q^{73} -1.71411 q^{74} +9.47840 q^{75} -0.702072 q^{76} -6.90445 q^{78} -12.7517 q^{79} +17.1168 q^{80} +1.00000 q^{81} -1.51387 q^{82} -2.58930 q^{83} -26.3510 q^{85} -3.21115 q^{86} +1.28140 q^{87} +2.99095 q^{88} +9.21039 q^{89} +5.76034 q^{90} +0.958507 q^{92} +0.334671 q^{93} -11.1993 q^{94} +9.15516 q^{95} +1.63805 q^{96} +12.8974 q^{97} +1.15660 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 8 q^{3} + 13 q^{4} + 7 q^{5} + q^{6} + 6 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + q^2 + 8 * q^3 + 13 * q^4 + 7 * q^5 + q^6 + 6 * q^8 + 8 * q^9	\( 8 q + q^{2} + 8 q^{3} + 13 q^{4} + 7 q^{5} + q^{6} + 6 q^{8} + 8 q^{9} + 8 q^{10} + 11 q^{11} + 13 q^{12} + 10 q^{13} + 7 q^{15} - 17 q^{16} + 3 q^{17} + q^{18} + 6 q^{19} + 11 q^{20} + 15 q^{22} + 14 q^{23} + 6 q^{24} + 25 q^{25} + 24 q^{26} + 8 q^{27} + 2 q^{29} + 8 q^{30} + 16 q^{31} + 3 q^{32} + 11 q^{33} - 4 q^{34} + 13 q^{36} - 20 q^{37} + 10 q^{38} + 10 q^{39} - 3 q^{40} + 8 q^{41} + 7 q^{43} + 7 q^{45} - 5 q^{46} + 14 q^{47} - 17 q^{48} - 5 q^{50} + 3 q^{51} + 23 q^{52} + 7 q^{53} + q^{54} + 48 q^{55} + 6 q^{57} - 20 q^{58} + 22 q^{59} + 11 q^{60} - 33 q^{62} - 10 q^{64} - 14 q^{65} + 15 q^{66} + 12 q^{67} - 27 q^{68} + 14 q^{69} - 5 q^{71} + 6 q^{72} + 2 q^{73} + 6 q^{74} + 25 q^{75} + 43 q^{76} + 24 q^{78} - 15 q^{79} - 7 q^{80} + 8 q^{81} + q^{82} + 15 q^{83} - 43 q^{85} + 31 q^{86} + 2 q^{87} + 17 q^{88} + 29 q^{89} + 8 q^{90} + 19 q^{92} + 16 q^{93} + 20 q^{94} + 14 q^{95} + 3 q^{96} + 19 q^{97} + 11 q^{99}+O(q^{100}) \) 8 * q + q^2 + 8 * q^3 + 13 * q^4 + 7 * q^5 + q^6 + 6 * q^8 + 8 * q^9 + 8 * q^10 + 11 * q^11 + 13 * q^12 + 10 * q^13 + 7 * q^15 - 17 * q^16 + 3 * q^17 + q^18 + 6 * q^19 + 11 * q^20 + 15 * q^22 + 14 * q^23 + 6 * q^24 + 25 * q^25 + 24 * q^26 + 8 * q^27 + 2 * q^29 + 8 * q^30 + 16 * q^31 + 3 * q^32 + 11 * q^33 - 4 * q^34 + 13 * q^36 - 20 * q^37 + 10 * q^38 + 10 * q^39 - 3 * q^40 + 8 * q^41 + 7 * q^43 + 7 * q^45 - 5 * q^46 + 14 * q^47 - 17 * q^48 - 5 * q^50 + 3 * q^51 + 23 * q^52 + 7 * q^53 + q^54 + 48 * q^55 + 6 * q^57 - 20 * q^58 + 22 * q^59 + 11 * q^60 - 33 * q^62 - 10 * q^64 - 14 * q^65 + 15 * q^66 + 12 * q^67 - 27 * q^68 + 14 * q^69 - 5 * q^71 + 6 * q^72 + 2 * q^73 + 6 * q^74 + 25 * q^75 + 43 * q^76 + 24 * q^78 - 15 * q^79 - 7 * q^80 + 8 * q^81 + q^82 + 15 * q^83 - 43 * q^85 + 31 * q^86 + 2 * q^87 + 17 * q^88 + 29 * q^89 + 8 * q^90 + 19 * q^92 + 16 * q^93 + 20 * q^94 + 14 * q^95 + 3 * q^96 + 19 * q^97 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more