LMFDB - Embedded newform 1335.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1335,2,Mod(1,1335)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1335, 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("1335.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1335 = 3 \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1335.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:	$10.6600286698$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.10407557.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 2x^{3} + 18x^{2} + 7x - 2 $ x^6 - x^5 - 8x^4 + 2x^3 + 18x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.43605$ of defining polynomial
Character	$\chi$	$=$	1335.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.49828 q^{2} -1.00000 q^{3} +0.244835 q^{4} -1.00000 q^{5} +1.49828 q^{6} -3.23110 q^{7} +2.62972 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.49828 q^{2} -1.00000 q^{3} +0.244835 q^{4} -1.00000 q^{5} +1.49828 q^{6} -3.23110 q^{7} +2.62972 q^{8} +1.00000 q^{9} +1.49828 q^{10} -0.244835 q^{12} +2.15654 q^{13} +4.84109 q^{14} +1.00000 q^{15} -4.42973 q^{16} -2.54656 q^{17} -1.49828 q^{18} +2.00000 q^{19} -0.244835 q^{20} +3.23110 q^{21} +2.98282 q^{23} -2.62972 q^{24} +1.00000 q^{25} -3.23110 q^{26} -1.00000 q^{27} -0.791087 q^{28} +1.21629 q^{29} -1.49828 q^{30} +8.46220 q^{31} +1.37751 q^{32} +3.81545 q^{34} +3.23110 q^{35} +0.244835 q^{36} -0.546560 q^{37} -2.99655 q^{38} -2.15654 q^{39} -2.62972 q^{40} +2.84484 q^{41} -4.84109 q^{42} +6.19911 q^{43} -1.00000 q^{45} -4.46909 q^{46} -9.73799 q^{47} +4.42973 q^{48} +3.44001 q^{49} -1.49828 q^{50} +2.54656 q^{51} +0.527997 q^{52} -11.0249 q^{53} +1.49828 q^{54} -8.49690 q^{56} -2.00000 q^{57} -1.82234 q^{58} +2.82628 q^{59} +0.244835 q^{60} +2.28347 q^{61} -12.6787 q^{62} -3.23110 q^{63} +6.79556 q^{64} -2.15654 q^{65} -1.95665 q^{67} -0.623487 q^{68} -2.98282 q^{69} -4.84109 q^{70} -5.79380 q^{71} +2.62972 q^{72} -12.9406 q^{73} +0.818898 q^{74} -1.00000 q^{75} +0.489670 q^{76} +3.23110 q^{78} +8.06481 q^{79} +4.42973 q^{80} +1.00000 q^{81} -4.26236 q^{82} -3.47938 q^{83} +0.791087 q^{84} +2.54656 q^{85} -9.28799 q^{86} -1.21629 q^{87} -1.00000 q^{89} +1.49828 q^{90} -6.96801 q^{91} +0.730299 q^{92} -8.46220 q^{93} +14.5902 q^{94} -2.00000 q^{95} -1.37751 q^{96} +3.30280 q^{97} -5.15410 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} - 6 q^{3} + 8 q^{4} - 6 q^{5} + 4 q^{6} + q^{7} - 9 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 - 6 * q^5 + 4 * q^6 + q^7 - 9 * q^8 + 6 * q^9	$ 6 q - 4 q^{2} - 6 q^{3} + 8 q^{4} - 6 q^{5} + 4 q^{6} + q^{7} - 9 q^{8} + 6 q^{9} + 4 q^{10} - 8 q^{12} - 3 q^{13} - 5 q^{14} + 6 q^{15} + 12 q^{16} - 13 q^{17} - 4 q^{18} + 12 q^{19} - 8 q^{20} - q^{21} - 19 q^{23} + 9 q^{24} + 6 q^{25} + q^{26} - 6 q^{27} - 6 q^{28} - 4 q^{30} + 10 q^{31} - 17 q^{32} - 2 q^{34} - q^{35} + 8 q^{36} - q^{37} - 8 q^{38} + 3 q^{39} + 9 q^{40} - 4 q^{41} + 5 q^{42} - 7 q^{43} - 6 q^{45} - 6 q^{46} - 15 q^{47} - 12 q^{48} - q^{49} - 4 q^{50} + 13 q^{51} + q^{52} - 27 q^{53} + 4 q^{54} - 14 q^{56} - 12 q^{57} - 6 q^{58} - 4 q^{59} + 8 q^{60} + 8 q^{61} + 2 q^{62} + q^{63} - q^{64} + 3 q^{65} - 11 q^{67} - 47 q^{68} + 19 q^{69} + 5 q^{70} - 16 q^{71} - 9 q^{72} + q^{73} - 10 q^{74} - 6 q^{75} + 16 q^{76} - q^{78} - 12 q^{80} + 6 q^{81} + q^{82} - 17 q^{83} + 6 q^{84} + 13 q^{85} - 20 q^{86} - 6 q^{89} + 4 q^{90} - 18 q^{91} - 36 q^{92} - 10 q^{93} + 17 q^{94} - 12 q^{95} + 17 q^{96} - 29 q^{97} - 23 q^{98}+O(q^{100}) $ 6 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 - 6 * q^5 + 4 * q^6 + q^7 - 9 * q^8 + 6 * q^9 + 4 * q^10 - 8 * q^12 - 3 * q^13 - 5 * q^14 + 6 * q^15 + 12 * q^16 - 13 * q^17 - 4 * q^18 + 12 * q^19 - 8 * q^20 - q^21 - 19 * q^23 + 9 * q^24 + 6 * q^25 + q^26 - 6 * q^27 - 6 * q^28 - 4 * q^30 + 10 * q^31 - 17 * q^32 - 2 * q^34 - q^35 + 8 * q^36 - q^37 - 8 * q^38 + 3 * q^39 + 9 * q^40 - 4 * q^41 + 5 * q^42 - 7 * q^43 - 6 * q^45 - 6 * q^46 - 15 * q^47 - 12 * q^48 - q^49 - 4 * q^50 + 13 * q^51 + q^52 - 27 * q^53 + 4 * q^54 - 14 * q^56 - 12 * q^57 - 6 * q^58 - 4 * q^59 + 8 * q^60 + 8 * q^61 + 2 * q^62 + q^63 - q^64 + 3 * q^65 - 11 * q^67 - 47 * q^68 + 19 * q^69 + 5 * q^70 - 16 * q^71 - 9 * q^72 + q^73 - 10 * q^74 - 6 * q^75 + 16 * q^76 - q^78 - 12 * q^80 + 6 * q^81 + q^82 - 17 * q^83 + 6 * q^84 + 13 * q^85 - 20 * q^86 - 6 * q^89 + 4 * q^90 - 18 * q^91 - 36 * q^92 - 10 * q^93 + 17 * q^94 - 12 * q^95 + 17 * q^96 - 29 * q^97 - 23 * q^98

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 $

$2$	−1.49828	−1.05944	−0.529721	−	0.848172i	$-0.677703\pi$
$2$	−1.49828	−1.05944	−0.529721	+	0.848172i	$0.677703\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.244835	0.122418
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.49828	0.611669
$7$	−3.23110	−1.22124	−0.610621	−	0.791923i	$-0.709080\pi$
$7$	−3.23110	−1.22124	−0.610621	+	0.791923i	$0.709080\pi$
$8$	2.62972	0.929748
$9$	1.00000	0.333333
$10$	1.49828	0.473797
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−0.244835	−0.0706778
$13$	2.15654	0.598118	0.299059	−	0.954235i	$-0.403327\pi$
$13$	2.15654	0.598118	0.299059	+	0.954235i	$0.403327\pi$
$14$	4.84109	1.29383
$15$	1.00000	0.258199
$16$	−4.42973	−1.10743
$17$	−2.54656	−0.617631	−0.308816	−	0.951122i	$-0.599933\pi$
$17$	−2.54656	−0.617631	−0.308816	+	0.951122i	$0.599933\pi$
$18$	−1.49828	−0.353147
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−0.244835	−0.0547468
$21$	3.23110	0.705084
$22$	0	0
$23$	2.98282	0.621961	0.310981	−	0.950416i	$-0.399343\pi$
$23$	2.98282	0.621961	0.310981	+	0.950416i	$0.399343\pi$
$24$	−2.62972	−0.536790
$25$	1.00000	0.200000
$26$	−3.23110	−0.633671
$27$	−1.00000	−0.192450
$28$	−0.791087	−0.149501
$29$	1.21629	0.225860	0.112930	−	0.993603i	$-0.463976\pi$
$29$	1.21629	0.225860	0.112930	+	0.993603i	$0.463976\pi$
$30$	−1.49828	−0.273547
$31$	8.46220	1.51986	0.759928	−	0.650007i	$-0.225234\pi$
$31$	8.46220	1.51986	0.759928	+	0.650007i	$0.225234\pi$
$32$	1.37751	0.243512
$33$	0	0
$34$	3.81545	0.654345
$35$	3.23110	0.546156
$36$	0.244835	0.0408058
$37$	−0.546560	−0.0898538	−0.0449269	−	0.998990i	$-0.514305\pi$
$37$	−0.546560	−0.0898538	−0.0449269	+	0.998990i	$0.514305\pi$
$38$	−2.99655	−0.486105
$39$	−2.15654	−0.345323
$40$	−2.62972	−0.415796
$41$	2.84484	0.444289	0.222145	−	0.975014i	$-0.428694\pi$
$41$	2.84484	0.444289	0.222145	+	0.975014i	$0.428694\pi$
$42$	−4.84109	−0.746996
$43$	6.19911	0.945356	0.472678	−	0.881235i	$-0.343287\pi$
$43$	6.19911	0.945356	0.472678	+	0.881235i	$0.343287\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	−4.46909	−0.658932
$47$	−9.73799	−1.42043	−0.710216	−	0.703984i	$-0.751403\pi$
$47$	−9.73799	−1.42043	−0.710216	+	0.703984i	$0.751403\pi$
$48$	4.42973	0.639376
$49$	3.44001	0.491431
$50$	−1.49828	−0.211888
$51$	2.54656	0.356590
$52$	0.527997	0.0732201
$53$	−11.0249	−1.51439	−0.757193	−	0.653191i	$-0.773430\pi$
$53$	−11.0249	−1.51439	−0.757193	+	0.653191i	$0.773430\pi$
$54$	1.49828	0.203890
$55$	0	0
$56$	−8.49690	−1.13545
$57$	−2.00000	−0.264906
$58$	−1.82234	−0.239285
$59$	2.82628	0.367950	0.183975	−	0.982931i	$-0.441103\pi$
$59$	2.82628	0.367950	0.183975	+	0.982931i	$0.441103\pi$
$60$	0.244835	0.0316081
$61$	2.28347	0.292368	0.146184	−	0.989257i	$-0.453301\pi$
$61$	2.28347	0.292368	0.146184	+	0.989257i	$0.453301\pi$
$62$	−12.6787	−1.61020
$63$	−3.23110	−0.407080
$64$	6.79556	0.849445
$65$	−2.15654	−0.267486
$66$	0	0
$67$	−1.95665	−0.239043	−0.119521	−	0.992832i	$-0.538136\pi$
$67$	−1.95665	−0.239043	−0.119521	+	0.992832i	$0.538136\pi$
$68$	−0.623487	−0.0756089
$69$	−2.98282	−0.359089
$70$	−4.84109	−0.578620
$71$	−5.79380	−0.687598	−0.343799	−	0.939043i	$-0.611714\pi$
$71$	−5.79380	−0.687598	−0.343799	+	0.939043i	$0.611714\pi$
$72$	2.62972	0.309916
$73$	−12.9406	−1.51458	−0.757291	−	0.653078i	$-0.773478\pi$
$73$	−12.9406	−1.51458	−0.757291	+	0.653078i	$0.773478\pi$
$74$	0.818898	0.0951949
$75$	−1.00000	−0.115470
$76$	0.489670	0.0561690
$77$	0	0
$78$	3.23110	0.365850
$79$	8.06481	0.907362	0.453681	−	0.891164i	$-0.350111\pi$
$79$	8.06481	0.907362	0.453681	+	0.891164i	$0.350111\pi$
$80$	4.42973	0.495258
$81$	1.00000	0.111111
$82$	−4.26236	−0.470699
$83$	−3.47938	−0.381912	−0.190956	−	0.981599i	$-0.561159\pi$
$83$	−3.47938	−0.381912	−0.190956	+	0.981599i	$0.561159\pi$
$84$	0.791087	0.0863146
$85$	2.54656	0.276213
$86$	−9.28799	−1.00155
$87$	−1.21629	−0.130400
$88$	0	0
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	1.49828	0.157932
$91$	−6.96801	−0.730446
$92$	0.730299	0.0761389
$93$	−8.46220	−0.877489
$94$	14.5902	1.50486
$95$	−2.00000	−0.205196
$96$	−1.37751	−0.140592
$97$	3.30280	0.335348	0.167674	−	0.985842i	$-0.446374\pi$
$97$	3.30280	0.335348	0.167674	+	0.985842i	$0.446374\pi$
$98$	−5.15410	−0.520642
$99$	0	0
$100$	0.244835	0.0244835
$101$	−11.6600	−1.16021	−0.580106	−	0.814541i	$-0.696989\pi$
$101$	−11.6600	−1.16021	−0.580106	+	0.814541i	$0.696989\pi$
$102$	−3.81545	−0.377786
$103$	9.18569	0.905093	0.452546	−	0.891741i	$-0.350516\pi$
$103$	9.18569	0.905093	0.452546	+	0.891741i	$0.350516\pi$
$104$	5.67112	0.556099
$105$	−3.23110	−0.315323
$106$	16.5184	1.60440
$107$	−9.81908	−0.949246	−0.474623	−	0.880189i	$-0.657416\pi$
$107$	−9.81908	−0.949246	−0.474623	+	0.880189i	$0.657416\pi$
$108$	−0.244835	−0.0235593
$109$	−7.49468	−0.717860	−0.358930	−	0.933364i	$-0.616858\pi$
$109$	−7.49468	−0.717860	−0.358930	+	0.933364i	$0.616858\pi$
$110$	0	0
$111$	0.546560	0.0518771
$112$	14.3129	1.35244
$113$	−14.6423	−1.37743	−0.688714	−	0.725033i	$-0.741825\pi$
$113$	−14.6423	−1.37743	−0.688714	+	0.725033i	$0.741825\pi$
$114$	2.99655	0.280653
$115$	−2.98282	−0.278149
$116$	0.297791	0.0276492
$117$	2.15654	0.199373
$118$	−4.23455	−0.389822
$119$	8.22819	0.754277
$120$	2.62972	0.240060
$121$	−11.0000	−1.00000
$122$	−3.42127	−0.309747
$123$	−2.84484	−0.256510
$124$	2.07184	0.186057
$125$	−1.00000	−0.0894427
$126$	4.84109	0.431278
$127$	2.21289	0.196363	0.0981813	−	0.995169i	$-0.468697\pi$
$127$	2.21289	0.196363	0.0981813	+	0.995169i	$0.468697\pi$
$128$	−12.9367	−1.14345
$129$	−6.19911	−0.545801
$130$	3.23110	0.283386
$131$	−14.7034	−1.28465	−0.642323	−	0.766434i	$-0.722029\pi$
$131$	−14.7034	−1.28465	−0.642323	+	0.766434i	$0.722029\pi$
$132$	0	0
$133$	−6.46220	−0.560344
$134$	2.93160	0.253252
$135$	1.00000	0.0860663
$136$	−6.69675	−0.574241
$137$	−2.17658	−0.185958	−0.0929790	−	0.995668i	$-0.529639\pi$
$137$	−2.17658	−0.185958	−0.0929790	+	0.995668i	$0.529639\pi$
$138$	4.46909	0.380434
$139$	4.98228	0.422592	0.211296	−	0.977422i	$-0.432232\pi$
$139$	4.98228	0.422592	0.211296	+	0.977422i	$0.432232\pi$
$140$	0.791087	0.0668590
$141$	9.73799	0.820086
$142$	8.68072	0.728470
$143$	0	0
$144$	−4.42973	−0.369144
$145$	−1.21629	−0.101008
$146$	19.3886	1.60461
$147$	−3.44001	−0.283728
$148$	−0.133817	−0.0109997
$149$	8.34404	0.683570	0.341785	−	0.939778i	$-0.388969\pi$
$149$	8.34404	0.683570	0.341785	+	0.939778i	$0.388969\pi$
$150$	1.49828	0.122334
$151$	1.51722	0.123470	0.0617348	−	0.998093i	$-0.480337\pi$
$151$	1.51722	0.123470	0.0617348	+	0.998093i	$0.480337\pi$
$152$	5.25945	0.426598
$153$	−2.54656	−0.205877
$154$	0	0
$155$	−8.46220	−0.679700
$156$	−0.527997	−0.0422736
$157$	−11.6296	−0.928141	−0.464071	−	0.885798i	$-0.653612\pi$
$157$	−11.6296	−0.928141	−0.464071	+	0.885798i	$0.653612\pi$
$158$	−12.0833	−0.961297
$159$	11.0249	0.874332
$160$	−1.37751	−0.108902
$161$	−9.63780	−0.759565
$162$	−1.49828	−0.117716
$163$	−7.66687	−0.600516	−0.300258	−	0.953858i	$-0.597073\pi$
$163$	−7.66687	−0.600516	−0.300258	+	0.953858i	$0.597073\pi$
$164$	0.696516	0.0543888
$165$	0	0
$166$	5.21308	0.404613
$167$	−0.0521124	−0.00403258	−0.00201629	−	0.999998i	$-0.500642\pi$
$167$	−0.0521124	−0.00403258	−0.00201629	+	0.999998i	$0.500642\pi$
$168$	8.49690	0.655550
$169$	−8.34932	−0.642255
$170$	−3.81545	−0.292632
$171$	2.00000	0.152944
$172$	1.51776	0.115728
$173$	−1.99257	−0.151492	−0.0757461	−	0.997127i	$-0.524134\pi$
$173$	−1.99257	−0.151492	−0.0757461	+	0.997127i	$0.524134\pi$
$174$	1.82234	0.138151
$175$	−3.23110	−0.244248
$176$	0	0
$177$	−2.82628	−0.212436
$178$	1.49828	0.112301
$179$	−11.4982	−0.859418	−0.429709	−	0.902967i	$-0.641384\pi$
$179$	−11.4982	−0.859418	−0.429709	+	0.902967i	$0.641384\pi$
$180$	−0.244835	−0.0182489
$181$	−10.0381	−0.746126	−0.373063	−	0.927806i	$-0.621692\pi$
$181$	−10.0381	−0.746126	−0.373063	+	0.927806i	$0.621692\pi$
$182$	10.4400	0.773865
$183$	−2.28347	−0.168799
$184$	7.84400	0.578267
$185$	0.546560	0.0401839
$186$	12.6787	0.929649
$187$	0	0
$188$	−2.38420	−0.173886
$189$	3.23110	0.235028
$190$	2.99655	0.217393
$191$	3.72009	0.269176	0.134588	−	0.990902i	$-0.457029\pi$
$191$	3.72009	0.269176	0.134588	+	0.990902i	$0.457029\pi$
$192$	−6.79556	−0.490427
$193$	6.31733	0.454731	0.227366	−	0.973809i	$-0.426989\pi$
$193$	6.31733	0.454731	0.227366	+	0.973809i	$0.426989\pi$
$194$	−4.94851	−0.355282
$195$	2.15654	0.154433
$196$	0.842236	0.0601597
$197$	−23.1725	−1.65098	−0.825488	−	0.564420i	$-0.809100\pi$
$197$	−23.1725	−1.65098	−0.825488	+	0.564420i	$0.809100\pi$
$198$	0	0
$199$	−5.96196	−0.422632	−0.211316	−	0.977418i	$-0.567775\pi$
$199$	−5.96196	−0.422632	−0.211316	+	0.977418i	$0.567775\pi$
$200$	2.62972	0.185950
$201$	1.95665	0.138011
$202$	17.4699	1.22918
$203$	−3.92996	−0.275829
$204$	0.623487	0.0436528
$205$	−2.84484	−0.198692
$206$	−13.7627	−0.958893
$207$	2.98282	0.207320
$208$	−9.55290	−0.662374
$209$	0	0
$210$	4.84109	0.334067
$211$	16.0794	1.10695	0.553476	−	0.832865i	$-0.313301\pi$
$211$	16.0794	1.10695	0.553476	+	0.832865i	$0.313301\pi$
$212$	−2.69928	−0.185387
$213$	5.79380	0.396985
$214$	14.7117	1.00567
$215$	−6.19911	−0.422776
$216$	−2.62972	−0.178930
$217$	−27.3422	−1.85611
$218$	11.2291	0.760531
$219$	12.9406	0.874444
$220$	0	0
$221$	−5.49177	−0.369416
$222$	−0.818898	−0.0549608
$223$	17.8324	1.19415	0.597073	−	0.802187i	$-0.296330\pi$
$223$	17.8324	1.19415	0.597073	+	0.802187i	$0.296330\pi$
$224$	−4.45087	−0.297387
$225$	1.00000	0.0666667
$226$	21.9382	1.45931
$227$	12.9380	0.858725	0.429362	−	0.903132i	$-0.358738\pi$
$227$	12.9380	0.858725	0.429362	+	0.903132i	$0.358738\pi$
$228$	−0.489670	−0.0324292
$229$	−6.12854	−0.404985	−0.202493	−	0.979284i	$-0.564904\pi$
$229$	−6.12854	−0.404985	−0.202493	+	0.979284i	$0.564904\pi$
$230$	4.46909	0.294683
$231$	0	0
$232$	3.19851	0.209993
$233$	15.6325	1.02412	0.512060	−	0.858950i	$-0.328883\pi$
$233$	15.6325	1.02412	0.512060	+	0.858950i	$0.328883\pi$
$234$	−3.23110	−0.211224
$235$	9.73799	0.635236
$236$	0.691972	0.0450435
$237$	−8.06481	−0.523866
$238$	−12.3281	−0.799113
$239$	14.7740	0.955649	0.477824	−	0.878455i	$-0.341426\pi$
$239$	14.7740	0.955649	0.477824	+	0.878455i	$0.341426\pi$
$240$	−4.42973	−0.285938
$241$	−3.26281	−0.210176	−0.105088	−	0.994463i	$-0.533512\pi$
$241$	−3.26281	−0.210176	−0.105088	+	0.994463i	$0.533512\pi$
$242$	16.4811	1.05944
$243$	−1.00000	−0.0641500
$244$	0.559073	0.0357910
$245$	−3.44001	−0.219774
$246$	4.26236	0.271758
$247$	4.31309	0.274435
$248$	22.2533	1.41308
$249$	3.47938	0.220497
$250$	1.49828	0.0947594
$251$	20.2434	1.27775	0.638876	−	0.769310i	$-0.279400\pi$
$251$	20.2434	1.27775	0.638876	+	0.769310i	$0.279400\pi$
$252$	−0.791087	−0.0498338
$253$	0	0
$254$	−3.31553	−0.208035
$255$	−2.54656	−0.159472
$256$	5.79158	0.361973
$257$	−1.50609	−0.0939472	−0.0469736	−	0.998896i	$-0.514958\pi$
$257$	−1.50609	−0.0939472	−0.0469736	+	0.998896i	$0.514958\pi$
$258$	9.28799	0.578245
$259$	1.76599	0.109733
$260$	−0.527997	−0.0327450
$261$	1.21629	0.0752866
$262$	22.0298	1.36101
$263$	−22.0466	−1.35945	−0.679727	−	0.733465i	$-0.737902\pi$
$263$	−22.0466	−1.35945	−0.679727	+	0.733465i	$0.737902\pi$
$264$	0	0
$265$	11.0249	0.677254
$266$	9.68217	0.593652
$267$	1.00000	0.0611990
$268$	−0.479056	−0.0292630
$269$	22.2582	1.35710	0.678552	−	0.734553i	$-0.262608\pi$
$269$	22.2582	1.35710	0.678552	+	0.734553i	$0.262608\pi$
$270$	−1.49828	−0.0911823
$271$	−21.3624	−1.29767	−0.648837	−	0.760928i	$-0.724744\pi$
$271$	−21.3624	−1.29767	−0.648837	+	0.760928i	$0.724744\pi$
$272$	11.2806	0.683984
$273$	6.96801	0.421723
$274$	3.26112	0.197012
$275$	0	0
$276$	−0.730299	−0.0439588
$277$	13.3948	0.804817	0.402409	−	0.915460i	$-0.368173\pi$
$277$	13.3948	0.804817	0.402409	+	0.915460i	$0.368173\pi$
$278$	−7.46484	−0.447711
$279$	8.46220	0.506619
$280$	8.49690	0.507787
$281$	10.9173	0.651273	0.325636	−	0.945495i	$-0.394422\pi$
$281$	10.9173	0.651273	0.325636	+	0.945495i	$0.394422\pi$
$282$	−14.5902	−0.868834
$283$	−16.4451	−0.977557	−0.488779	−	0.872408i	$-0.662557\pi$
$283$	−16.4451	−0.977557	−0.488779	+	0.872408i	$0.662557\pi$
$284$	−1.41853	−0.0841740
$285$	2.00000	0.118470
$286$	0	0
$287$	−9.19196	−0.542584
$288$	1.37751	0.0811706
$289$	−10.5150	−0.618531
$290$	1.82234	0.107012
$291$	−3.30280	−0.193614
$292$	−3.16831	−0.185411
$293$	11.4085	0.666493	0.333246	−	0.942840i	$-0.391856\pi$
$293$	11.4085	0.666493	0.333246	+	0.942840i	$0.391856\pi$
$294$	5.15410	0.300593
$295$	−2.82628	−0.164552
$296$	−1.43730	−0.0835414
$297$	0	0
$298$	−12.5017	−0.724203
$299$	6.43258	0.372006
$300$	−0.244835	−0.0141356
$301$	−20.0300	−1.15451
$302$	−2.27322	−0.130809
$303$	11.6600	0.669849
$304$	−8.85945	−0.508124
$305$	−2.28347	−0.130751
$306$	3.81545	0.218115
$307$	−9.24434	−0.527602	−0.263801	−	0.964577i	$-0.584976\pi$
$307$	−9.24434	−0.527602	−0.263801	+	0.964577i	$0.584976\pi$
$308$	0	0
$309$	−9.18569	−0.522555
$310$	12.6787	0.720103
$311$	−19.6701	−1.11539	−0.557694	−	0.830047i	$-0.688314\pi$
$311$	−19.6701	−1.11539	−0.557694	+	0.830047i	$0.688314\pi$
$312$	−5.67112	−0.321064
$313$	−6.82421	−0.385727	−0.192864	−	0.981226i	$-0.561777\pi$
$313$	−6.82421	−0.385727	−0.192864	+	0.981226i	$0.561777\pi$
$314$	17.4243	0.983312
$315$	3.23110	0.182052
$316$	1.97455	0.111077
$317$	−5.59469	−0.314229	−0.157114	−	0.987580i	$-0.550219\pi$
$317$	−5.59469	−0.314229	−0.157114	+	0.987580i	$0.550219\pi$
$318$	−16.5184	−0.926304
$319$	0	0
$320$	−6.79556	−0.379883
$321$	9.81908	0.548048
$322$	14.4401	0.804715
$323$	−5.09312	−0.283389
$324$	0.244835	0.0136019
$325$	2.15654	0.119624
$326$	11.4871	0.636212
$327$	7.49468	0.414457
$328$	7.48114	0.413077
$329$	31.4644	1.73469
$330$	0	0
$331$	27.8906	1.53301	0.766503	−	0.642241i	$-0.221995\pi$
$331$	27.8906	1.53301	0.766503	+	0.642241i	$0.221995\pi$
$332$	−0.851874	−0.0467527
$333$	−0.546560	−0.0299513
$334$	0.0780788	0.00427228
$335$	1.95665	0.106903
$336$	−14.3129	−0.780832
$337$	0.472821	0.0257562	0.0128781	−	0.999917i	$-0.495901\pi$
$337$	0.472821	0.0257562	0.0128781	+	0.999917i	$0.495901\pi$
$338$	12.5096	0.680432
$339$	14.6423	0.795259
$340$	0.623487	0.0338133
$341$	0	0
$342$	−2.99655	−0.162035
$343$	11.5027	0.621086
$344$	16.3020	0.878943
$345$	2.98282	0.160590
$346$	2.98542	0.160497
$347$	0.470376	0.0252511	0.0126256	−	0.999920i	$-0.495981\pi$
$347$	0.470376	0.0252511	0.0126256	+	0.999920i	$0.495981\pi$
$348$	−0.297791	−0.0159633
$349$	−9.95257	−0.532749	−0.266374	−	0.963870i	$-0.585826\pi$
$349$	−9.95257	−0.532749	−0.266374	+	0.963870i	$0.585826\pi$
$350$	4.84109	0.258767
$351$	−2.15654	−0.115108
$352$	0	0
$353$	−4.51311	−0.240209	−0.120104	−	0.992761i	$-0.538323\pi$
$353$	−4.51311	−0.240209	−0.120104	+	0.992761i	$0.538323\pi$
$354$	4.23455	0.225064
$355$	5.79380	0.307503
$356$	−0.244835	−0.0129762
$357$	−8.22819	−0.435482
$358$	17.2275	0.910504
$359$	−8.77166	−0.462951	−0.231475	−	0.972841i	$-0.574355\pi$
$359$	−8.77166	−0.462951	−0.231475	+	0.972841i	$0.574355\pi$
$360$	−2.62972	−0.138599
$361$	−15.0000	−0.789474
$362$	15.0399	0.790477
$363$	11.0000	0.577350
$364$	−1.70601	−0.0894194
$365$	12.9406	0.677342
$366$	3.42127	0.178833
$367$	23.2320	1.21270	0.606350	−	0.795198i	$-0.292633\pi$
$367$	23.2320	1.21270	0.606350	+	0.795198i	$0.292633\pi$
$368$	−13.2131	−0.688779
$369$	2.84484	0.148096
$370$	−0.818898	−0.0425725
$371$	35.6226	1.84943
$372$	−2.07184	−0.107420
$373$	3.45253	0.178765	0.0893826	−	0.995997i	$-0.471511\pi$
$373$	3.45253	0.178765	0.0893826	+	0.995997i	$0.471511\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−25.6082	−1.32064
$377$	2.62299	0.135091
$378$	−4.84109	−0.248999
$379$	−9.71751	−0.499155	−0.249577	−	0.968355i	$-0.580292\pi$
$379$	−9.71751	−0.499155	−0.249577	+	0.968355i	$0.580292\pi$
$380$	−0.489670	−0.0251195
$381$	−2.21289	−0.113370
$382$	−5.57372	−0.285176
$383$	10.5389	0.538510	0.269255	−	0.963069i	$-0.413222\pi$
$383$	10.5389	0.538510	0.269255	+	0.963069i	$0.413222\pi$
$384$	12.9367	0.660171
$385$	0	0
$386$	−9.46511	−0.481762
$387$	6.19911	0.315119
$388$	0.808641	0.0410525
$389$	8.60815	0.436451	0.218225	−	0.975898i	$-0.429973\pi$
$389$	8.60815	0.436451	0.218225	+	0.975898i	$0.429973\pi$
$390$	−3.23110	−0.163613
$391$	−7.59593	−0.384143
$392$	9.04629	0.456907
$393$	14.7034	0.741691
$394$	34.7189	1.74911
$395$	−8.06481	−0.405785
$396$	0	0
$397$	11.5589	0.580127	0.290063	−	0.957007i	$-0.406324\pi$
$397$	11.5589	0.580127	0.290063	+	0.957007i	$0.406324\pi$
$398$	8.93268	0.447755
$399$	6.46220	0.323515
$400$	−4.42973	−0.221486
$401$	−16.5326	−0.825598	−0.412799	−	0.910822i	$-0.635449\pi$
$401$	−16.5326	−0.825598	−0.412799	+	0.910822i	$0.635449\pi$
$402$	−2.93160	−0.146215
$403$	18.2491	0.909053
$404$	−2.85477	−0.142030
$405$	−1.00000	−0.0496904
$406$	5.88817	0.292225
$407$	0	0
$408$	6.69675	0.331538
$409$	−26.8607	−1.32817	−0.664087	−	0.747655i	$-0.731180\pi$
$409$	−26.8607	−1.32817	−0.664087	+	0.747655i	$0.731180\pi$
$410$	4.26236	0.210503
$411$	2.17658	0.107363
$412$	2.24898	0.110799
$413$	−9.13199	−0.449356
$414$	−4.46909	−0.219644
$415$	3.47938	0.170796
$416$	2.97066	0.145649
$417$	−4.98228	−0.243983
$418$	0	0
$419$	14.7573	0.720942	0.360471	−	0.932770i	$-0.382616\pi$
$419$	14.7573	0.720942	0.360471	+	0.932770i	$0.382616\pi$
$420$	−0.791087	−0.0386011
$421$	−1.69539	−0.0826285	−0.0413142	−	0.999146i	$-0.513154\pi$
$421$	−1.69539	−0.0826285	−0.0413142	+	0.999146i	$0.513154\pi$
$422$	−24.0914	−1.17275
$423$	−9.73799	−0.473477
$424$	−28.9924	−1.40800
$425$	−2.54656	−0.123526
$426$	−8.68072	−0.420582
$427$	−7.37812	−0.357052
$428$	−2.40405	−0.116204
$429$	0	0
$430$	9.28799	0.447907
$431$	−40.1685	−1.93485	−0.967424	−	0.253161i	$-0.918530\pi$
$431$	−40.1685	−1.93485	−0.967424	+	0.253161i	$0.918530\pi$
$432$	4.42973	0.213125
$433$	22.4284	1.07784	0.538920	−	0.842357i	$-0.318833\pi$
$433$	22.4284	1.07784	0.538920	+	0.842357i	$0.318833\pi$
$434$	40.9662	1.96644
$435$	1.21629	0.0583167
$436$	−1.83496	−0.0878786
$437$	5.96564	0.285375
$438$	−19.3886	−0.926423
$439$	−0.580012	−0.0276824	−0.0138412	−	0.999904i	$-0.504406\pi$
$439$	−0.580012	−0.0276824	−0.0138412	+	0.999904i	$0.504406\pi$
$440$	0	0
$441$	3.44001	0.163810
$442$	8.22819	0.391375
$443$	−4.28962	−0.203806	−0.101903	−	0.994794i	$-0.532493\pi$
$443$	−4.28962	−0.203806	−0.101903	+	0.994794i	$0.532493\pi$
$444$	0.133817	0.00635067
$445$	1.00000	0.0474045
$446$	−26.7179	−1.26513
$447$	−8.34404	−0.394659
$448$	−21.9571	−1.03738
$449$	22.9873	1.08484	0.542418	−	0.840109i	$-0.317509\pi$
$449$	22.9873	1.08484	0.542418	+	0.840109i	$0.317509\pi$
$450$	−1.49828	−0.0706295
$451$	0	0
$452$	−3.58494	−0.168621
$453$	−1.51722	−0.0712852
$454$	−19.3847	−0.909769
$455$	6.96801	0.326665
$456$	−5.25945	−0.246296
$457$	−26.7097	−1.24943	−0.624714	−	0.780854i	$-0.714784\pi$
$457$	−26.7097	−1.24943	−0.624714	+	0.780854i	$0.714784\pi$
$458$	9.18225	0.429058
$459$	2.54656	0.118863
$460$	−0.730299	−0.0340504
$461$	−24.1175	−1.12326	−0.561632	−	0.827387i	$-0.689826\pi$
$461$	−24.1175	−1.12326	−0.561632	+	0.827387i	$0.689826\pi$
$462$	0	0
$463$	−30.8578	−1.43408	−0.717041	−	0.697031i	$-0.754504\pi$
$463$	−30.8578	−1.43408	−0.717041	+	0.697031i	$0.754504\pi$
$464$	−5.38784	−0.250124
$465$	8.46220	0.392425
$466$	−23.4218	−1.08500
$467$	−30.4542	−1.40925	−0.704626	−	0.709579i	$-0.748885\pi$
$467$	−30.4542	−1.40925	−0.704626	+	0.709579i	$0.748885\pi$
$468$	0.527997	0.0244067
$469$	6.32213	0.291929
$470$	−14.5902	−0.672996
$471$	11.6296	0.535862
$472$	7.43233	0.342101
$473$	0	0
$474$	12.0833	0.555005
$475$	2.00000	0.0917663
$476$	2.01455	0.0923367
$477$	−11.0249	−0.504796
$478$	−22.1355	−1.01245
$479$	−8.27076	−0.377901	−0.188950	−	0.981987i	$-0.560509\pi$
$479$	−8.27076	−0.377901	−0.188950	+	0.981987i	$0.560509\pi$
$480$	1.37751	0.0628744
$481$	−1.17868	−0.0537432
$482$	4.88859	0.222669
$483$	9.63780	0.438535
$484$	−2.69319	−0.122418
$485$	−3.30280	−0.149972
$486$	1.49828	0.0679632
$487$	6.33163	0.286914	0.143457	−	0.989657i	$-0.454178\pi$
$487$	6.33163	0.286914	0.143457	+	0.989657i	$0.454178\pi$
$488$	6.00490	0.271829
$489$	7.66687	0.346708
$490$	5.15410	0.232838
$491$	−43.0296	−1.94190	−0.970949	−	0.239288i	$-0.923086\pi$
$491$	−43.0296	−1.94190	−0.970949	+	0.239288i	$0.923086\pi$
$492$	−0.696516	−0.0314014
$493$	−3.09736	−0.139498
$494$	−6.46220	−0.290748
$495$	0	0
$496$	−37.4852	−1.68314
$497$	18.7204	0.839723
$498$	−5.21308	−0.233604
$499$	5.39064	0.241318	0.120659	−	0.992694i	$-0.461499\pi$
$499$	5.39064	0.241318	0.120659	+	0.992694i	$0.461499\pi$
$500$	−0.244835	−0.0109494
$501$	0.0521124	0.00232821
$502$	−30.3302	−1.35370
$503$	15.9822	0.712611	0.356305	−	0.934370i	$-0.384036\pi$
$503$	15.9822	0.712611	0.356305	+	0.934370i	$0.384036\pi$
$504$	−8.49690	−0.378482
$505$	11.6600	0.518862
$506$	0	0
$507$	8.34932	0.370806
$508$	0.541794	0.0240382
$509$	25.9239	1.14906	0.574529	−	0.818484i	$-0.305185\pi$
$509$	25.9239	1.14906	0.574529	+	0.818484i	$0.305185\pi$
$510$	3.81545	0.168951
$511$	41.8124	1.84967
$512$	17.1959	0.759960
$513$	−2.00000	−0.0883022
$514$	2.25654	0.0995316
$515$	−9.18569	−0.404770
$516$	−1.51776	−0.0668157
$517$	0	0
$518$	−2.64594	−0.116256
$519$	1.99257	0.0874641
$520$	−5.67112	−0.248695
$521$	−32.8737	−1.44022	−0.720112	−	0.693858i	$-0.755909\pi$
$521$	−32.8737	−1.44022	−0.720112	+	0.693858i	$0.755909\pi$
$522$	−1.82234	−0.0797618
$523$	36.7804	1.60830	0.804148	−	0.594430i	$-0.202622\pi$
$523$	36.7804	1.60830	0.804148	+	0.594430i	$0.202622\pi$
$524$	−3.59992	−0.157263
$525$	3.23110	0.141017
$526$	33.0320	1.44026
$527$	−21.5495	−0.938711
$528$	0	0
$529$	−14.1028	−0.613164
$530$	−16.5184	−0.717512
$531$	2.82628	0.122650
$532$	−1.58217	−0.0685959
$533$	6.13502	0.265737
$534$	−1.49828	−0.0648368
$535$	9.81908	0.424516
$536$	−5.14545	−0.222249
$537$	11.4982	0.496185
$538$	−33.3489	−1.43777
$539$	0	0
$540$	0.244835	0.0105360
$541$	14.0677	0.604818	0.302409	−	0.953178i	$-0.402209\pi$
$541$	14.0677	0.604818	0.302409	+	0.953178i	$0.402209\pi$
$542$	32.0068	1.37481
$543$	10.0381	0.430776
$544$	−3.50791	−0.150400
$545$	7.49468	0.321037
$546$	−10.4400	−0.446791
$547$	−43.7363	−1.87003	−0.935014	−	0.354612i	$-0.884613\pi$
$547$	−43.7363	−1.87003	−0.935014	+	0.354612i	$0.884613\pi$
$548$	−0.532903	−0.0227645
$549$	2.28347	0.0974561
$550$	0	0
$551$	2.43258	0.103632
$552$	−7.84400	−0.333863
$553$	−26.0582	−1.10811
$554$	−20.0692	−0.852657
$555$	−0.546560	−0.0232002
$556$	1.21984	0.0517326
$557$	6.47140	0.274202	0.137101	−	0.990557i	$-0.456222\pi$
$557$	6.47140	0.274202	0.137101	+	0.990557i	$0.456222\pi$
$558$	−12.6787	−0.536733
$559$	13.3687	0.565434
$560$	−14.3129	−0.604830
$561$	0	0
$562$	−16.3572	−0.689986
$563$	−11.4507	−0.482591	−0.241296	−	0.970452i	$-0.577572\pi$
$563$	−11.4507	−0.482591	−0.241296	+	0.970452i	$0.577572\pi$
$564$	2.38420	0.100393
$565$	14.6423	0.616005
$566$	24.6393	1.03567
$567$	−3.23110	−0.135693
$568$	−15.2361	−0.639292
$569$	−38.6520	−1.62037	−0.810187	−	0.586171i	$-0.800635\pi$
$569$	−38.6520	−1.62037	−0.810187	+	0.586171i	$0.800635\pi$
$570$	−2.99655	−0.125512
$571$	39.0392	1.63374	0.816869	−	0.576823i	$-0.195708\pi$
$571$	39.0392	1.63374	0.816869	+	0.576823i	$0.195708\pi$
$572$	0	0
$573$	−3.72009	−0.155409
$574$	13.7721	0.574837
$575$	2.98282	0.124392
$576$	6.79556	0.283148
$577$	19.7589	0.822576	0.411288	−	0.911506i	$-0.365079\pi$
$577$	19.7589	0.822576	0.411288	+	0.911506i	$0.365079\pi$
$578$	15.7544	0.655298
$579$	−6.31733	−0.262539
$580$	−0.297791	−0.0123651
$581$	11.2422	0.466406
$582$	4.94851	0.205122
$583$	0	0
$584$	−34.0302	−1.40818
$585$	−2.15654	−0.0891621
$586$	−17.0931	−0.706110
$587$	9.55716	0.394466	0.197233	−	0.980357i	$-0.436804\pi$
$587$	9.55716	0.394466	0.197233	+	0.980357i	$0.436804\pi$
$588$	−0.842236	−0.0347332
$589$	16.9244	0.697358
$590$	4.23455	0.174334
$591$	23.1725	0.953191
$592$	2.42111	0.0995070
$593$	−19.2958	−0.792384	−0.396192	−	0.918168i	$-0.629669\pi$
$593$	−19.2958	−0.792384	−0.396192	+	0.918168i	$0.629669\pi$
$594$	0	0
$595$	−8.22819	−0.337323
$596$	2.04291	0.0836809
$597$	5.96196	0.244007
$598$	−9.63780	−0.394119
$599$	25.9498	1.06028	0.530139	−	0.847910i	$-0.322140\pi$
$599$	25.9498	1.06028	0.530139	+	0.847910i	$0.322140\pi$
$600$	−2.62972	−0.107358
$601$	2.99846	0.122310	0.0611549	−	0.998128i	$-0.480522\pi$
$601$	2.99846	0.122310	0.0611549	+	0.998128i	$0.480522\pi$
$602$	30.0104	1.22313
$603$	−1.95665	−0.0796809
$604$	0.371469	0.0151148
$605$	11.0000	0.447214
$606$	−17.4699	−0.709666
$607$	−27.2889	−1.10762	−0.553812	−	0.832642i	$-0.686827\pi$
$607$	−27.2889	−1.10762	−0.553812	+	0.832642i	$0.686827\pi$
$608$	2.75502	0.111731
$609$	3.92996	0.159250
$610$	3.42127	0.138523
$611$	−21.0004	−0.849585
$612$	−0.623487	−0.0252030
$613$	−24.6000	−0.993586	−0.496793	−	0.867869i	$-0.665489\pi$
$613$	−24.6000	−0.993586	−0.496793	+	0.867869i	$0.665489\pi$
$614$	13.8506	0.558964
$615$	2.84484	0.114715
$616$	0	0
$617$	−29.5894	−1.19122	−0.595612	−	0.803272i	$-0.703090\pi$
$617$	−29.5894	−1.19122	−0.595612	+	0.803272i	$0.703090\pi$
$618$	13.7627	0.553617
$619$	−24.2403	−0.974299	−0.487150	−	0.873318i	$-0.661963\pi$
$619$	−24.2403	−0.974299	−0.487150	+	0.873318i	$0.661963\pi$
$620$	−2.07184	−0.0832072
$621$	−2.98282	−0.119696
$622$	29.4712	1.18169
$623$	3.23110	0.129451
$624$	9.55290	0.382422
$625$	1.00000	0.0400000
$626$	10.2246	0.408656
$627$	0	0
$628$	−2.84733	−0.113621
$629$	1.39185	0.0554966
$630$	−4.84109	−0.192873
$631$	−1.07165	−0.0426616	−0.0213308	−	0.999772i	$-0.506790\pi$
$631$	−1.07165	−0.0426616	−0.0213308	+	0.999772i	$0.506790\pi$
$632$	21.2082	0.843618
$633$	−16.0794	−0.639100
$634$	8.38239	0.332907
$635$	−2.21289	−0.0878160
$636$	2.69928	0.107033
$637$	7.41854	0.293933
$638$	0	0
$639$	−5.79380	−0.229199
$640$	12.9367	0.511366
$641$	−31.4849	−1.24358	−0.621790	−	0.783184i	$-0.713594\pi$
$641$	−31.4849	−1.24358	−0.621790	+	0.783184i	$0.713594\pi$
$642$	−14.7117	−0.580625
$643$	−16.0382	−0.632486	−0.316243	−	0.948678i	$-0.602421\pi$
$643$	−16.0382	−0.632486	−0.316243	+	0.948678i	$0.602421\pi$
$644$	−2.35967	−0.0929840
$645$	6.19911	0.244090
$646$	7.63091	0.300234
$647$	9.45360	0.371659	0.185830	−	0.982582i	$-0.440503\pi$
$647$	9.45360	0.371659	0.185830	+	0.982582i	$0.440503\pi$
$648$	2.62972	0.103305
$649$	0	0
$650$	−3.23110	−0.126734
$651$	27.3422	1.07163
$652$	−1.87712	−0.0735137
$653$	7.23401	0.283089	0.141544	−	0.989932i	$-0.454793\pi$
$653$	7.23401	0.283089	0.141544	+	0.989932i	$0.454793\pi$
$654$	−11.2291	−0.439093
$655$	14.7034	0.574511
$656$	−12.6019	−0.492020
$657$	−12.9406	−0.504861
$658$	−47.1424	−1.83780
$659$	−29.3533	−1.14344	−0.571722	−	0.820448i	$-0.693724\pi$
$659$	−29.3533	−1.14344	−0.571722	+	0.820448i	$0.693724\pi$
$660$	0	0
$661$	−16.8668	−0.656043	−0.328021	−	0.944670i	$-0.606382\pi$
$661$	−16.8668	−0.656043	−0.328021	+	0.944670i	$0.606382\pi$
$662$	−41.7879	−1.62413
$663$	5.49177	0.213283
$664$	−9.14981	−0.355082
$665$	6.46220	0.250593
$666$	0.818898	0.0317316
$667$	3.62798	0.140476
$668$	−0.0127589	−0.000493658 0
$669$	−17.8324	−0.689441
$670$	−2.93160	−0.113258
$671$	0	0
$672$	4.45087	0.171696
$673$	49.9471	1.92532	0.962659	−	0.270716i	$-0.0872604\pi$
$673$	49.9471	1.92532	0.962659	+	0.270716i	$0.0872604\pi$
$674$	−0.708416	−0.0272872
$675$	−1.00000	−0.0384900
$676$	−2.04421	−0.0786233
$677$	−10.3125	−0.396341	−0.198170	−	0.980168i	$-0.563500\pi$
$677$	−10.3125	−0.396341	−0.198170	+	0.980168i	$0.563500\pi$
$678$	−21.9382	−0.842530
$679$	−10.6717	−0.409541
$680$	6.69675	0.256809
$681$	−12.9380	−0.495785
$682$	0	0
$683$	−29.1314	−1.11468	−0.557342	−	0.830283i	$-0.688179\pi$
$683$	−29.1314	−1.11468	−0.557342	+	0.830283i	$0.688179\pi$
$684$	0.489670	0.0187230
$685$	2.17658	0.0831629
$686$	−17.2342	−0.658005
$687$	6.12854	0.233818
$688$	−27.4604	−1.04692
$689$	−23.7757	−0.905781
$690$	−4.46909	−0.170135
$691$	−22.7023	−0.863635	−0.431817	−	0.901961i	$-0.642128\pi$
$691$	−22.7023	−0.863635	−0.431817	+	0.901961i	$0.642128\pi$
$692$	−0.487851	−0.0185453
$693$	0	0
$694$	−0.704754	−0.0267521
$695$	−4.98228	−0.188989
$696$	−3.19851	−0.121239
$697$	−7.24455	−0.274407
$698$	14.9117	0.564417
$699$	−15.6325	−0.591276
$700$	−0.791087	−0.0299003
$701$	42.0114	1.58675	0.793375	−	0.608734i	$-0.208322\pi$
$701$	42.0114	1.58675	0.793375	+	0.608734i	$0.208322\pi$
$702$	3.23110	0.121950
$703$	−1.09312	−0.0412278
$704$	0	0
$705$	−9.73799	−0.366754
$706$	6.76189	0.254487
$707$	37.6746	1.41690
$708$	−0.691972	−0.0260059
$709$	−35.4923	−1.33294	−0.666471	−	0.745531i	$-0.732196\pi$
$709$	−35.4923	−1.33294	−0.666471	+	0.745531i	$0.732196\pi$
$710$	−8.68072	−0.325782
$711$	8.06481	0.302454
$712$	−2.62972	−0.0985531
$713$	25.2412	0.945292
$714$	12.3281	0.461368
$715$	0	0
$716$	−2.81517	−0.105208
$717$	−14.7740	−0.551744
$718$	13.1424	0.490470
$719$	−19.8978	−0.742064	−0.371032	−	0.928620i	$-0.620996\pi$
$719$	−19.8978	−0.742064	−0.371032	+	0.928620i	$0.620996\pi$
$720$	4.42973	0.165086
$721$	−29.6799	−1.10534
$722$	22.4742	0.836402
$723$	3.26281	0.121345
$724$	−2.45768	−0.0913389
$725$	1.21629	0.0451720
$726$	−16.4811	−0.611669
$727$	−20.3167	−0.753506	−0.376753	−	0.926314i	$-0.622959\pi$
$727$	−20.3167	−0.753506	−0.376753	+	0.926314i	$0.622959\pi$
$728$	−18.3239	−0.679131
$729$	1.00000	0.0370370
$730$	−19.3886	−0.717604
$731$	−15.7864	−0.583881
$732$	−0.559073	−0.0206640
$733$	−20.3832	−0.752871	−0.376435	−	0.926443i	$-0.622850\pi$
$733$	−20.3832	−0.752871	−0.376435	+	0.926443i	$0.622850\pi$
$734$	−34.8079	−1.28478
$735$	3.44001	0.126887
$736$	4.10887	0.151455
$737$	0	0
$738$	−4.26236	−0.156900
$739$	−27.2671	−1.00304	−0.501518	−	0.865147i	$-0.667225\pi$
$739$	−27.2671	−1.00304	−0.501518	+	0.865147i	$0.667225\pi$
$740$	0.133817	0.00491921
$741$	−4.31309	−0.158445
$742$	−53.3725	−1.95937
$743$	24.7332	0.907374	0.453687	−	0.891161i	$-0.350108\pi$
$743$	24.7332	0.907374	0.453687	+	0.891161i	$0.350108\pi$
$744$	−22.2533	−0.815844
$745$	−8.34404	−0.305702
$746$	−5.17285	−0.189391
$747$	−3.47938	−0.127304
$748$	0	0
$749$	31.7264	1.15926
$750$	−1.49828	−0.0547094
$751$	9.20976	0.336069	0.168034	−	0.985781i	$-0.446258\pi$
$751$	9.20976	0.336069	0.168034	+	0.985781i	$0.446258\pi$
$752$	43.1366	1.57303
$753$	−20.2434	−0.737710
$754$	−3.92996	−0.143121
$755$	−1.51722	−0.0552173
$756$	0.791087	0.0287715
$757$	−46.1220	−1.67633	−0.838166	−	0.545415i	$-0.816372\pi$
$757$	−46.1220	−1.67633	−0.838166	+	0.545415i	$0.816372\pi$
$758$	14.5595	0.528826
$759$	0	0
$760$	−5.25945	−0.190780
$761$	17.2495	0.625293	0.312647	−	0.949869i	$-0.398784\pi$
$761$	17.2495	0.625293	0.312647	+	0.949869i	$0.398784\pi$
$762$	3.31553	0.120109
$763$	24.2161	0.876680
$764$	0.910808	0.0329519
$765$	2.54656	0.0920711
$766$	−15.7901	−0.570521
$767$	6.09499	0.220077
$768$	−5.79158	−0.208985
$769$	−26.5353	−0.956887	−0.478443	−	0.878118i	$-0.658799\pi$
$769$	−26.5353	−0.956887	−0.478443	+	0.878118i	$0.658799\pi$
$770$	0	0
$771$	1.50609	0.0542405
$772$	1.54670	0.0556671
$773$	4.11066	0.147850	0.0739251	−	0.997264i	$-0.476447\pi$
$773$	4.11066	0.147850	0.0739251	+	0.997264i	$0.476447\pi$
$774$	−9.28799	−0.333850
$775$	8.46220	0.303971
$776$	8.68545	0.311790
$777$	−1.76599	−0.0633545
$778$	−12.8974	−0.462394
$779$	5.68968	0.203854
$780$	0.527997	0.0189053
$781$	0	0
$782$	11.3808	0.406977
$783$	−1.21629	−0.0434667
$784$	−15.2383	−0.544226
$785$	11.6296	0.415077
$786$	−22.0298	−0.785778
$787$	16.2423	0.578976	0.289488	−	0.957182i	$-0.406515\pi$
$787$	16.2423	0.578976	0.289488	+	0.957182i	$0.406515\pi$
$788$	−5.67345	−0.202108
$789$	22.0466	0.784881
$790$	12.0833	0.429905
$791$	47.3106	1.68217
$792$	0	0
$793$	4.92440	0.174871
$794$	−17.3185	−0.614611
$795$	−11.0249	−0.391013
$796$	−1.45970	−0.0517376
$797$	40.7251	1.44256	0.721279	−	0.692644i	$-0.243554\pi$
$797$	40.7251	1.44256	0.721279	+	0.692644i	$0.243554\pi$
$798$	−9.68217	−0.342745
$799$	24.7984	0.877303
$800$	1.37751	0.0487023
$801$	−1.00000	−0.0353333
$802$	24.7704	0.874673
$803$	0	0
$804$	0.479056	0.0168950
$805$	9.63780	0.339688
$806$	−27.3422	−0.963089
$807$	−22.2582	−0.783524
$808$	−30.6625	−1.07870
$809$	−18.2512	−0.641678	−0.320839	−	0.947134i	$-0.603965\pi$
$809$	−18.2512	−0.641678	−0.320839	+	0.947134i	$0.603965\pi$
$810$	1.49828	0.0526441
$811$	43.8706	1.54051	0.770253	−	0.637738i	$-0.220130\pi$
$811$	43.8706	1.54051	0.770253	+	0.637738i	$0.220130\pi$
$812$	−0.962193	−0.0337663
$813$	21.3624	0.749212
$814$	0	0
$815$	7.66687	0.268559
$816$	−11.2806	−0.394899
$817$	12.3982	0.433759
$818$	40.2447	1.40712
$819$	−6.96801	−0.243482
$820$	−0.696516	−0.0243234
$821$	−14.0303	−0.489661	−0.244831	−	0.969566i	$-0.578732\pi$
$821$	−14.0303	−0.489661	−0.244831	+	0.969566i	$0.578732\pi$
$822$	−3.26112	−0.113745
$823$	−5.50462	−0.191879	−0.0959394	−	0.995387i	$-0.530586\pi$
$823$	−5.50462	−0.191879	−0.0959394	+	0.995387i	$0.530586\pi$
$824$	24.1558	0.841508
$825$	0	0
$826$	13.6822	0.476066
$827$	43.1206	1.49945	0.749726	−	0.661749i	$-0.230185\pi$
$827$	43.1206	1.49945	0.749726	+	0.661749i	$0.230185\pi$
$828$	0.730299	0.0253796
$829$	−28.8526	−1.00209	−0.501047	−	0.865420i	$-0.667052\pi$
$829$	−28.8526	−1.00209	−0.501047	+	0.865420i	$0.667052\pi$
$830$	−5.21308	−0.180949
$831$	−13.3948	−0.464661
$832$	14.6549	0.508068
$833$	−8.76020	−0.303523
$834$	7.46484	0.258486
$835$	0.0521124	0.00180342
$836$	0	0
$837$	−8.46220	−0.292496
$838$	−22.1106	−0.763796
$839$	21.9011	0.756108	0.378054	−	0.925783i	$-0.376593\pi$
$839$	21.9011	0.756108	0.378054	+	0.925783i	$0.376593\pi$
$840$	−8.49690	−0.293171
$841$	−27.5206	−0.948987
$842$	2.54017	0.0875401
$843$	−10.9173	−0.376012
$844$	3.93681	0.135510
$845$	8.34932	0.287225
$846$	14.5902	0.501622
$847$	35.5421	1.22124
$848$	48.8373	1.67708
$849$	16.4451	0.564393
$850$	3.81545	0.130869
$851$	−1.63029	−0.0558856
$852$	1.41853	0.0485979
$853$	−1.93424	−0.0662270	−0.0331135	−	0.999452i	$-0.510542\pi$
$853$	−1.93424	−0.0662270	−0.0331135	+	0.999452i	$0.510542\pi$
$854$	11.0545	0.378276
$855$	−2.00000	−0.0683986
$856$	−25.8215	−0.882560
$857$	−58.4401	−1.99627	−0.998137	−	0.0610065i	$-0.980569\pi$
$857$	−58.4401	−1.99627	−0.998137	+	0.0610065i	$0.980569\pi$
$858$	0	0
$859$	−28.3839	−0.968447	−0.484223	−	0.874944i	$-0.660898\pi$
$859$	−28.3839	−0.968447	−0.484223	+	0.874944i	$0.660898\pi$
$860$	−1.51776	−0.0517552
$861$	9.19196	0.313261
$862$	60.1835	2.04986
$863$	24.2940	0.826977	0.413489	−	0.910509i	$-0.364310\pi$
$863$	24.2940	0.826977	0.413489	+	0.910509i	$0.364310\pi$
$864$	−1.37751	−0.0468638
$865$	1.99257	0.0677494
$866$	−33.6039	−1.14191
$867$	10.5150	0.357109
$868$	−6.69434	−0.227221
$869$	0	0
$870$	−1.82234	−0.0617832
$871$	−4.21960	−0.142976
$872$	−19.7089	−0.667429
$873$	3.30280	0.111783
$874$	−8.93819	−0.302339
$875$	3.23110	0.109231
$876$	3.16831	0.107047
$877$	50.8099	1.71573	0.857863	−	0.513878i	$-0.171792\pi$
$877$	50.8099	1.71573	0.857863	+	0.513878i	$0.171792\pi$
$878$	0.869018	0.0293279
$879$	−11.4085	−0.384800
$880$	0	0
$881$	3.05502	0.102926	0.0514631	−	0.998675i	$-0.483612\pi$
$881$	3.05502	0.102926	0.0514631	+	0.998675i	$0.483612\pi$
$882$	−5.15410	−0.173547
$883$	−16.0936	−0.541593	−0.270797	−	0.962637i	$-0.587287\pi$
$883$	−16.0936	−0.541593	−0.270797	+	0.962637i	$0.587287\pi$
$884$	−1.34458	−0.0452230
$885$	2.82628	0.0950043
$886$	6.42704	0.215920
$887$	1.16665	0.0391724	0.0195862	−	0.999808i	$-0.493765\pi$
$887$	1.16665	0.0391724	0.0195862	+	0.999808i	$0.493765\pi$
$888$	1.43730	0.0482327
$889$	−7.15009	−0.239806
$890$	−1.49828	−0.0502224
$891$	0	0
$892$	4.36600	0.146184
$893$	−19.4760	−0.651739
$894$	12.5017	0.418119
$895$	11.4982	0.384344
$896$	41.7996	1.39643
$897$	−6.43258	−0.214778
$898$	−34.4413	−1.14932
$899$	10.2925	0.343274
$900$	0.244835	0.00816117
$901$	28.0756	0.935333
$902$	0	0
$903$	20.0300	0.666555
$904$	−38.5051	−1.28066
$905$	10.0381	0.333678
$906$	2.27322	0.0755226
$907$	−5.56857	−0.184901	−0.0924506	−	0.995717i	$-0.529470\pi$
$907$	−5.56857	−0.184901	−0.0924506	+	0.995717i	$0.529470\pi$
$908$	3.16767	0.105123
$909$	−11.6600	−0.386737
$910$	−10.4400	−0.346083
$911$	−25.5924	−0.847915	−0.423957	−	0.905682i	$-0.639359\pi$
$911$	−25.5924	−0.847915	−0.423957	+	0.905682i	$0.639359\pi$
$912$	8.85945	0.293366
$913$	0	0
$914$	40.0186	1.32370
$915$	2.28347	0.0754892
$916$	−1.50048	−0.0495773
$917$	47.5083	1.56886
$918$	−3.81545	−0.125929
$919$	−58.7341	−1.93746	−0.968730	−	0.248118i	$-0.920188\pi$
$919$	−58.7341	−1.93746	−0.968730	+	0.248118i	$0.920188\pi$
$920$	−7.84400	−0.258609
$921$	9.24434	0.304611
$922$	36.1347	1.19003
$923$	−12.4946	−0.411264
$924$	0	0
$925$	−0.546560	−0.0179708
$926$	46.2335	1.51933
$927$	9.18569	0.301698
$928$	1.67545	0.0549995
$929$	−29.9339	−0.982101	−0.491050	−	0.871131i	$-0.663387\pi$
$929$	−29.9339	−0.982101	−0.491050	+	0.871131i	$0.663387\pi$
$930$	−12.6787	−0.415752
$931$	6.88003	0.225484
$932$	3.82739	0.125370
$933$	19.6701	0.643969
$934$	45.6288	1.49302
$935$	0	0
$936$	5.67112	0.185366
$937$	38.4314	1.25550	0.627750	−	0.778415i	$-0.283976\pi$
$937$	38.4314	1.25550	0.627750	+	0.778415i	$0.283976\pi$
$938$	−9.47230	−0.309282
$939$	6.82421	0.222700
$940$	2.38420	0.0777640
$941$	38.6236	1.25909	0.629546	−	0.776963i	$-0.283241\pi$
$941$	38.6236	1.25909	0.629546	+	0.776963i	$0.283241\pi$
$942$	−17.4243	−0.567715
$943$	8.48564	0.276331
$944$	−12.5196	−0.407479
$945$	−3.23110	−0.105108
$946$	0	0
$947$	−8.44416	−0.274398	−0.137199	−	0.990543i	$-0.543810\pi$
$947$	−8.44416	−0.274398	−0.137199	+	0.990543i	$0.543810\pi$
$948$	−1.97455	−0.0641303
$949$	−27.9070	−0.905898
$950$	−2.99655	−0.0972211
$951$	5.59469	0.181420
$952$	21.6379	0.701287
$953$	−40.1285	−1.29989	−0.649945	−	0.759981i	$-0.725208\pi$
$953$	−40.1285	−1.29989	−0.649945	+	0.759981i	$0.725208\pi$
$954$	16.5184	0.534802
$955$	−3.72009	−0.120379
$956$	3.61719	0.116988
$957$	0	0
$958$	12.3919	0.400364
$959$	7.03276	0.227100
$960$	6.79556	0.219326
$961$	40.6089	1.30996
$962$	1.76599	0.0569378
$963$	−9.81908	−0.316415
$964$	−0.798850	−0.0257292
$965$	−6.31733	−0.203362
$966$	−14.4401	−0.464602
$967$	4.32318	0.139024	0.0695120	−	0.997581i	$-0.477856\pi$
$967$	4.32318	0.139024	0.0695120	+	0.997581i	$0.477856\pi$
$968$	−28.9270	−0.929748
$969$	5.09312	0.163615
$970$	4.94851	0.158887
$971$	−3.48751	−0.111919	−0.0559597	−	0.998433i	$-0.517822\pi$
$971$	−3.48751	−0.111919	−0.0559597	+	0.998433i	$0.517822\pi$
$972$	−0.244835	−0.00785309
$973$	−16.0983	−0.516086
$974$	−9.48654	−0.303968
$975$	−2.15654	−0.0690647
$976$	−10.1151	−0.323778
$977$	19.8788	0.635980	0.317990	−	0.948094i	$-0.396992\pi$
$977$	19.8788	0.635980	0.317990	+	0.948094i	$0.396992\pi$
$978$	−11.4871	−0.367317
$979$	0	0
$980$	−0.842236	−0.0269042
$981$	−7.49468	−0.239287
$982$	64.4702	2.05733
$983$	52.3297	1.66906	0.834529	−	0.550963i	$-0.185740\pi$
$983$	52.3297	1.66906	0.834529	+	0.550963i	$0.185740\pi$
$984$	−7.48114	−0.238490
$985$	23.1725	0.738339
$986$	4.64071	0.147790
$987$	−31.4644	−1.00152
$988$	1.05599	0.0335957
$989$	18.4908	0.587975
$990$	0	0
$991$	−16.8109	−0.534016	−0.267008	−	0.963694i	$-0.586035\pi$
$991$	−16.8109	−0.534016	−0.267008	+	0.963694i	$0.586035\pi$
$992$	11.6568	0.370103
$993$	−27.8906	−0.885081
$994$	−28.0483	−0.889638
$995$	5.96196	0.189007
$996$	0.851874	0.0269927
$997$	−14.5992	−0.462362	−0.231181	−	0.972911i	$-0.574259\pi$
$997$	−14.5992	−0.462362	−0.231181	+	0.972911i	$0.574259\pi$
$998$	−8.07667	−0.255662
$999$	0.546560	0.0172924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1335.2.a.g.1.3	✓	6
3.2	odd	2		4005.2.a.n.1.4		6
5.4	even	2		6675.2.a.u.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.g.1.3	✓	6	1.1	even	1	trivial
4005.2.a.n.1.4		6	3.2	odd	2
6675.2.a.u.1.4		6	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1335 = 3 \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1335.a (trivial)

\(f(q)\)	\(=\)	\(q-1.49828 q^{2} -1.00000 q^{3} +0.244835 q^{4} -1.00000 q^{5} +1.49828 q^{6} -3.23110 q^{7} +2.62972 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.49828 q^{2} -1.00000 q^{3} +0.244835 q^{4} -1.00000 q^{5} +1.49828 q^{6} -3.23110 q^{7} +2.62972 q^{8} +1.00000 q^{9} +1.49828 q^{10} -0.244835 q^{12} +2.15654 q^{13} +4.84109 q^{14} +1.00000 q^{15} -4.42973 q^{16} -2.54656 q^{17} -1.49828 q^{18} +2.00000 q^{19} -0.244835 q^{20} +3.23110 q^{21} +2.98282 q^{23} -2.62972 q^{24} +1.00000 q^{25} -3.23110 q^{26} -1.00000 q^{27} -0.791087 q^{28} +1.21629 q^{29} -1.49828 q^{30} +8.46220 q^{31} +1.37751 q^{32} +3.81545 q^{34} +3.23110 q^{35} +0.244835 q^{36} -0.546560 q^{37} -2.99655 q^{38} -2.15654 q^{39} -2.62972 q^{40} +2.84484 q^{41} -4.84109 q^{42} +6.19911 q^{43} -1.00000 q^{45} -4.46909 q^{46} -9.73799 q^{47} +4.42973 q^{48} +3.44001 q^{49} -1.49828 q^{50} +2.54656 q^{51} +0.527997 q^{52} -11.0249 q^{53} +1.49828 q^{54} -8.49690 q^{56} -2.00000 q^{57} -1.82234 q^{58} +2.82628 q^{59} +0.244835 q^{60} +2.28347 q^{61} -12.6787 q^{62} -3.23110 q^{63} +6.79556 q^{64} -2.15654 q^{65} -1.95665 q^{67} -0.623487 q^{68} -2.98282 q^{69} -4.84109 q^{70} -5.79380 q^{71} +2.62972 q^{72} -12.9406 q^{73} +0.818898 q^{74} -1.00000 q^{75} +0.489670 q^{76} +3.23110 q^{78} +8.06481 q^{79} +4.42973 q^{80} +1.00000 q^{81} -4.26236 q^{82} -3.47938 q^{83} +0.791087 q^{84} +2.54656 q^{85} -9.28799 q^{86} -1.21629 q^{87} -1.00000 q^{89} +1.49828 q^{90} -6.96801 q^{91} +0.730299 q^{92} -8.46220 q^{93} +14.5902 q^{94} -2.00000 q^{95} -1.37751 q^{96} +3.30280 q^{97} -5.15410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} - 6 q^{3} + 8 q^{4} - 6 q^{5} + 4 q^{6} + q^{7} - 9 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 - 6 * q^5 + 4 * q^6 + q^7 - 9 * q^8 + 6 * q^9	\( 6 q - 4 q^{2} - 6 q^{3} + 8 q^{4} - 6 q^{5} + 4 q^{6} + q^{7} - 9 q^{8} + 6 q^{9} + 4 q^{10} - 8 q^{12} - 3 q^{13} - 5 q^{14} + 6 q^{15} + 12 q^{16} - 13 q^{17} - 4 q^{18} + 12 q^{19} - 8 q^{20} - q^{21} - 19 q^{23} + 9 q^{24} + 6 q^{25} + q^{26} - 6 q^{27} - 6 q^{28} - 4 q^{30} + 10 q^{31} - 17 q^{32} - 2 q^{34} - q^{35} + 8 q^{36} - q^{37} - 8 q^{38} + 3 q^{39} + 9 q^{40} - 4 q^{41} + 5 q^{42} - 7 q^{43} - 6 q^{45} - 6 q^{46} - 15 q^{47} - 12 q^{48} - q^{49} - 4 q^{50} + 13 q^{51} + q^{52} - 27 q^{53} + 4 q^{54} - 14 q^{56} - 12 q^{57} - 6 q^{58} - 4 q^{59} + 8 q^{60} + 8 q^{61} + 2 q^{62} + q^{63} - q^{64} + 3 q^{65} - 11 q^{67} - 47 q^{68} + 19 q^{69} + 5 q^{70} - 16 q^{71} - 9 q^{72} + q^{73} - 10 q^{74} - 6 q^{75} + 16 q^{76} - q^{78} - 12 q^{80} + 6 q^{81} + q^{82} - 17 q^{83} + 6 q^{84} + 13 q^{85} - 20 q^{86} - 6 q^{89} + 4 q^{90} - 18 q^{91} - 36 q^{92} - 10 q^{93} + 17 q^{94} - 12 q^{95} + 17 q^{96} - 29 q^{97} - 23 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 - 6 * q^5 + 4 * q^6 + q^7 - 9 * q^8 + 6 * q^9 + 4 * q^10 - 8 * q^12 - 3 * q^13 - 5 * q^14 + 6 * q^15 + 12 * q^16 - 13 * q^17 - 4 * q^18 + 12 * q^19 - 8 * q^20 - q^21 - 19 * q^23 + 9 * q^24 + 6 * q^25 + q^26 - 6 * q^27 - 6 * q^28 - 4 * q^30 + 10 * q^31 - 17 * q^32 - 2 * q^34 - q^35 + 8 * q^36 - q^37 - 8 * q^38 + 3 * q^39 + 9 * q^40 - 4 * q^41 + 5 * q^42 - 7 * q^43 - 6 * q^45 - 6 * q^46 - 15 * q^47 - 12 * q^48 - q^49 - 4 * q^50 + 13 * q^51 + q^52 - 27 * q^53 + 4 * q^54 - 14 * q^56 - 12 * q^57 - 6 * q^58 - 4 * q^59 + 8 * q^60 + 8 * q^61 + 2 * q^62 + q^63 - q^64 + 3 * q^65 - 11 * q^67 - 47 * q^68 + 19 * q^69 + 5 * q^70 - 16 * q^71 - 9 * q^72 + q^73 - 10 * q^74 - 6 * q^75 + 16 * q^76 - q^78 - 12 * q^80 + 6 * q^81 + q^82 - 17 * q^83 + 6 * q^84 + 13 * q^85 - 20 * q^86 - 6 * q^89 + 4 * q^90 - 18 * q^91 - 36 * q^92 - 10 * q^93 + 17 * q^94 - 12 * q^95 + 17 * q^96 - 29 * q^97 - 23 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more