LMFDB - Embedded newform 3381.2.a.bf.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.9974209234$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 15x^{6} + 11x^{5} + 75x^{4} - 35x^{3} - 141x^{2} + 37x + 80 $ x^8 - x^7 - 15x^6 + 11x^5 + 75x^4 - 35x^3 - 141x^2 + 37x + 80
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.47391$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.47391 q^{2} +1.00000 q^{3} +0.172415 q^{4} +3.62341 q^{5} +1.47391 q^{6} -2.69370 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.47391 q^{2} +1.00000 q^{3} +0.172415 q^{4} +3.62341 q^{5} +1.47391 q^{6} -2.69370 q^{8} +1.00000 q^{9} +5.34058 q^{10} +1.81102 q^{11} +0.172415 q^{12} +3.13385 q^{13} +3.62341 q^{15} -4.31510 q^{16} +3.47040 q^{17} +1.47391 q^{18} +1.56480 q^{19} +0.624731 q^{20} +2.66929 q^{22} -1.00000 q^{23} -2.69370 q^{24} +8.12908 q^{25} +4.61901 q^{26} +1.00000 q^{27} -4.13548 q^{29} +5.34058 q^{30} -1.63993 q^{31} -0.972685 q^{32} +1.81102 q^{33} +5.11506 q^{34} +0.172415 q^{36} +0.807580 q^{37} +2.30638 q^{38} +3.13385 q^{39} -9.76037 q^{40} +1.16765 q^{41} -10.3803 q^{43} +0.312248 q^{44} +3.62341 q^{45} -1.47391 q^{46} +7.98144 q^{47} -4.31510 q^{48} +11.9815 q^{50} +3.47040 q^{51} +0.540323 q^{52} +2.16670 q^{53} +1.47391 q^{54} +6.56207 q^{55} +1.56480 q^{57} -6.09533 q^{58} -9.03647 q^{59} +0.624731 q^{60} -10.0787 q^{61} -2.41711 q^{62} +7.19656 q^{64} +11.3552 q^{65} +2.66929 q^{66} +6.40398 q^{67} +0.598350 q^{68} -1.00000 q^{69} +13.3495 q^{71} -2.69370 q^{72} -6.92778 q^{73} +1.19030 q^{74} +8.12908 q^{75} +0.269796 q^{76} +4.61901 q^{78} +16.7678 q^{79} -15.6354 q^{80} +1.00000 q^{81} +1.72101 q^{82} -5.16584 q^{83} +12.5747 q^{85} -15.2997 q^{86} -4.13548 q^{87} -4.87835 q^{88} -1.33374 q^{89} +5.34058 q^{90} -0.172415 q^{92} -1.63993 q^{93} +11.7639 q^{94} +5.66992 q^{95} -0.972685 q^{96} -3.85363 q^{97} +1.81102 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + q^2 + 8 * q^3 + 15 * q^4 + 5 * q^5 + q^6 + 9 * q^8 + 8 * q^9	$ 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9} - 3 q^{10} + 10 q^{11} + 15 q^{12} - 6 q^{13} + 5 q^{15} + 13 q^{16} + 21 q^{17} + q^{18} + 5 q^{19} - q^{20} + 18 q^{22} - 8 q^{23} + 9 q^{24} + 27 q^{25} + 3 q^{26} + 8 q^{27} + 2 q^{29} - 3 q^{30} - 13 q^{31} + 29 q^{32} + 10 q^{33} - 19 q^{34} + 15 q^{36} + 13 q^{37} - 6 q^{38} - 6 q^{39} + 7 q^{40} + 16 q^{41} + 15 q^{43} + 24 q^{44} + 5 q^{45} - q^{46} - q^{47} + 13 q^{48} + 16 q^{50} + 21 q^{51} - 19 q^{52} + 3 q^{53} + q^{54} - 10 q^{55} + 5 q^{57} - 40 q^{58} + 26 q^{59} - q^{60} - 14 q^{61} - 14 q^{62} + 49 q^{64} - 3 q^{65} + 18 q^{66} + 38 q^{67} + 43 q^{68} - 8 q^{69} + 9 q^{71} + 9 q^{72} - 6 q^{73} + 32 q^{74} + 27 q^{75} - 14 q^{76} + 3 q^{78} + 23 q^{79} - 17 q^{80} + 8 q^{81} + 20 q^{82} + 30 q^{83} - 37 q^{85} - 28 q^{86} + 2 q^{87} + 86 q^{88} + 12 q^{89} - 3 q^{90} - 15 q^{92} - 13 q^{93} - 45 q^{94} + 16 q^{95} + 29 q^{96} + 14 q^{97} + 10 q^{99}+O(q^{100}) $ 8 * q + q^2 + 8 * q^3 + 15 * q^4 + 5 * q^5 + q^6 + 9 * q^8 + 8 * q^9 - 3 * q^10 + 10 * q^11 + 15 * q^12 - 6 * q^13 + 5 * q^15 + 13 * q^16 + 21 * q^17 + q^18 + 5 * q^19 - q^20 + 18 * q^22 - 8 * q^23 + 9 * q^24 + 27 * q^25 + 3 * q^26 + 8 * q^27 + 2 * q^29 - 3 * q^30 - 13 * q^31 + 29 * q^32 + 10 * q^33 - 19 * q^34 + 15 * q^36 + 13 * q^37 - 6 * q^38 - 6 * q^39 + 7 * q^40 + 16 * q^41 + 15 * q^43 + 24 * q^44 + 5 * q^45 - q^46 - q^47 + 13 * q^48 + 16 * q^50 + 21 * q^51 - 19 * q^52 + 3 * q^53 + q^54 - 10 * q^55 + 5 * q^57 - 40 * q^58 + 26 * q^59 - q^60 - 14 * q^61 - 14 * q^62 + 49 * q^64 - 3 * q^65 + 18 * q^66 + 38 * q^67 + 43 * q^68 - 8 * q^69 + 9 * q^71 + 9 * q^72 - 6 * q^73 + 32 * q^74 + 27 * q^75 - 14 * q^76 + 3 * q^78 + 23 * q^79 - 17 * q^80 + 8 * q^81 + 20 * q^82 + 30 * q^83 - 37 * q^85 - 28 * q^86 + 2 * q^87 + 86 * q^88 + 12 * q^89 - 3 * q^90 - 15 * q^92 - 13 * q^93 - 45 * q^94 + 16 * q^95 + 29 * q^96 + 14 * q^97 + 10 * 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 $

$2$	1.47391	1.04221	0.521106	−	0.853492i	$-0.325519\pi$
$2$	1.47391	1.04221	0.521106	+	0.853492i	$0.325519\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.172415	0.0862077
$5$	3.62341	1.62044	0.810219	−	0.586128i	$-0.199348\pi$
$5$	3.62341	1.62044	0.810219	+	0.586128i	$0.199348\pi$
$6$	1.47391	0.601722
$7$	0	0
$7$	0	0
$8$	−2.69370	−0.952366
$9$	1.00000	0.333333
$10$	5.34058	1.68884
$11$	1.81102	0.546044	0.273022	−	0.962008i	$-0.411977\pi$
$11$	1.81102	0.546044	0.273022	+	0.962008i	$0.411977\pi$
$12$	0.172415	0.0497721
$13$	3.13385	0.869173	0.434586	−	0.900630i	$-0.356895\pi$
$13$	3.13385	0.869173	0.434586	+	0.900630i	$0.356895\pi$
$14$	0	0
$15$	3.62341	0.935560
$16$	−4.31510	−1.07878
$17$	3.47040	0.841696	0.420848	−	0.907131i	$-0.361733\pi$
$17$	3.47040	0.841696	0.420848	+	0.907131i	$0.361733\pi$
$18$	1.47391	0.347404
$19$	1.56480	0.358991	0.179495	−	0.983759i	$-0.442554\pi$
$19$	1.56480	0.358991	0.179495	+	0.983759i	$0.442554\pi$
$20$	0.624731	0.139694
$21$	0	0
$22$	2.66929	0.569094
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−2.69370	−0.549849
$25$	8.12908	1.62582
$26$	4.61901	0.905863
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.13548	−0.767939	−0.383969	−	0.923346i	$-0.625443\pi$
$29$	−4.13548	−0.767939	−0.383969	+	0.923346i	$0.625443\pi$
$30$	5.34058	0.975052
$31$	−1.63993	−0.294540	−0.147270	−	0.989096i	$-0.547049\pi$
$31$	−1.63993	−0.294540	−0.147270	+	0.989096i	$0.547049\pi$
$32$	−0.972685	−0.171948
$33$	1.81102	0.315258
$34$	5.11506	0.877226
$35$	0	0
$36$	0.172415	0.0287359
$37$	0.807580	0.132765	0.0663827	−	0.997794i	$-0.478854\pi$
$37$	0.807580	0.132765	0.0663827	+	0.997794i	$0.478854\pi$
$38$	2.30638	0.374145
$39$	3.13385	0.501817
$40$	−9.76037	−1.54325
$41$	1.16765	0.182356	0.0911781	−	0.995835i	$-0.470937\pi$
$41$	1.16765	0.182356	0.0911781	+	0.995835i	$0.470937\pi$
$42$	0	0
$43$	−10.3803	−1.58299	−0.791493	−	0.611178i	$-0.790696\pi$
$43$	−10.3803	−1.58299	−0.791493	+	0.611178i	$0.790696\pi$
$44$	0.312248	0.0470732
$45$	3.62341	0.540146
$46$	−1.47391	−0.217316
$47$	7.98144	1.16421	0.582106	−	0.813113i	$-0.302229\pi$
$47$	7.98144	1.16421	0.582106	+	0.813113i	$0.302229\pi$
$48$	−4.31510	−0.622832
$49$	0	0
$50$	11.9815	1.69445
$51$	3.47040	0.485953
$52$	0.540323	0.0749294
$53$	2.16670	0.297619	0.148809	−	0.988866i	$-0.452456\pi$
$53$	2.16670	0.297619	0.148809	+	0.988866i	$0.452456\pi$
$54$	1.47391	0.200574
$55$	6.56207	0.884829
$56$	0	0
$57$	1.56480	0.207263
$58$	−6.09533	−0.800356
$59$	−9.03647	−1.17645	−0.588224	−	0.808698i	$-0.700173\pi$
$59$	−9.03647	−1.17645	−0.588224	+	0.808698i	$0.700173\pi$
$60$	0.624731	0.0806525
$61$	−10.0787	−1.29045	−0.645224	−	0.763994i	$-0.723236\pi$
$61$	−10.0787	−1.29045	−0.645224	+	0.763994i	$0.723236\pi$
$62$	−2.41711	−0.306974
$63$	0	0
$64$	7.19656	0.899569
$65$	11.3552	1.40844
$66$	2.66929	0.328566
$67$	6.40398	0.782370	0.391185	−	0.920312i	$-0.372065\pi$
$67$	6.40398	0.782370	0.391185	+	0.920312i	$0.372065\pi$
$68$	0.598350	0.0725607
$69$	−1.00000	−0.120386
$70$	0	0
$71$	13.3495	1.58429	0.792145	−	0.610333i	$-0.208964\pi$
$71$	13.3495	1.58429	0.792145	+	0.610333i	$0.208964\pi$
$72$	−2.69370	−0.317455
$73$	−6.92778	−0.810835	−0.405418	−	0.914132i	$-0.632874\pi$
$73$	−6.92778	−0.810835	−0.405418	+	0.914132i	$0.632874\pi$
$74$	1.19030	0.138370
$75$	8.12908	0.938665
$76$	0.269796	0.0309478
$77$	0	0
$78$	4.61901	0.523000
$79$	16.7678	1.88652	0.943261	−	0.332053i	$-0.107741\pi$
$79$	16.7678	1.88652	0.943261	+	0.332053i	$0.107741\pi$
$80$	−15.6354	−1.74809
$81$	1.00000	0.111111
$82$	1.72101	0.190054
$83$	−5.16584	−0.567024	−0.283512	−	0.958969i	$-0.591500\pi$
$83$	−5.16584	−0.567024	−0.283512	+	0.958969i	$0.591500\pi$
$84$	0	0
$85$	12.5747	1.36391
$86$	−15.2997	−1.64981
$87$	−4.13548	−0.443370
$88$	−4.87835	−0.520033
$89$	−1.33374	−0.141376	−0.0706882	−	0.997498i	$-0.522520\pi$
$89$	−1.33374	−0.141376	−0.0706882	+	0.997498i	$0.522520\pi$
$90$	5.34058	0.562947
$91$	0	0
$92$	−0.172415	−0.0179756
$93$	−1.63993	−0.170053
$94$	11.7639	1.21336
$95$	5.66992	0.581722
$96$	−0.972685	−0.0992743
$97$	−3.85363	−0.391277	−0.195638	−	0.980676i	$-0.562678\pi$
$97$	−3.85363	−0.391277	−0.195638	+	0.980676i	$0.562678\pi$
$98$	0	0
$99$	1.81102	0.182015
$100$	1.40158	0.140158
$101$	−13.5698	−1.35025	−0.675124	−	0.737704i	$-0.735910\pi$
$101$	−13.5698	−1.35025	−0.675124	+	0.737704i	$0.735910\pi$
$102$	5.11506	0.506467
$103$	15.1002	1.48787	0.743933	−	0.668254i	$-0.232958\pi$
$103$	15.1002	1.48787	0.743933	+	0.668254i	$0.232958\pi$
$104$	−8.44164	−0.827770
$105$	0	0
$106$	3.19352	0.310182
$107$	10.0846	0.974917	0.487459	−	0.873146i	$-0.337924\pi$
$107$	10.0846	0.974917	0.487459	+	0.873146i	$0.337924\pi$
$108$	0.172415	0.0165907
$109$	20.0766	1.92299	0.961494	−	0.274825i	$-0.0886200\pi$
$109$	20.0766	1.92299	0.961494	+	0.274825i	$0.0886200\pi$
$110$	9.67191	0.922181
$111$	0.807580	0.0766521
$112$	0	0
$113$	−19.6637	−1.84980	−0.924901	−	0.380207i	$-0.875853\pi$
$113$	−19.6637	−1.84980	−0.924901	+	0.380207i	$0.875853\pi$
$114$	2.30638	0.216012
$115$	−3.62341	−0.337884
$116$	−0.713020	−0.0662023
$117$	3.13385	0.289724
$118$	−13.3190	−1.22611
$119$	0	0
$120$	−9.76037	−0.890995
$121$	−7.72020	−0.701836
$122$	−14.8551	−1.34492
$123$	1.16765	0.105283
$124$	−0.282749	−0.0253916
$125$	11.3379	1.01410
$126$	0	0
$127$	−9.57658	−0.849784	−0.424892	−	0.905244i	$-0.639688\pi$
$127$	−9.57658	−0.849784	−0.424892	+	0.905244i	$0.639688\pi$
$128$	12.5525	1.10949
$129$	−10.3803	−0.913937
$130$	16.7366	1.46789
$131$	6.21403	0.542922	0.271461	−	0.962449i	$-0.412493\pi$
$131$	6.21403	0.542922	0.271461	+	0.962449i	$0.412493\pi$
$132$	0.312248	0.0271777
$133$	0	0
$134$	9.43890	0.815396
$135$	3.62341	0.311853
$136$	−9.34821	−0.801602
$137$	−21.6128	−1.84650	−0.923252	−	0.384196i	$-0.874479\pi$
$137$	−21.6128	−1.84650	−0.923252	+	0.384196i	$0.874479\pi$
$138$	−1.47391	−0.125468
$139$	15.2669	1.29492	0.647460	−	0.762099i	$-0.275831\pi$
$139$	15.2669	1.29492	0.647460	+	0.762099i	$0.275831\pi$
$140$	0	0
$141$	7.98144	0.672158
$142$	19.6759	1.65117
$143$	5.67546	0.474606
$144$	−4.31510	−0.359592
$145$	−14.9845	−1.24440
$146$	−10.2109	−0.845063
$147$	0	0
$148$	0.139239	0.0114454
$149$	−10.3089	−0.844535	−0.422267	−	0.906471i	$-0.638766\pi$
$149$	−10.3089	−0.844535	−0.422267	+	0.906471i	$0.638766\pi$
$150$	11.9815	0.978289
$151$	3.19142	0.259714	0.129857	−	0.991533i	$-0.458548\pi$
$151$	3.19142	0.259714	0.129857	+	0.991533i	$0.458548\pi$
$152$	−4.21511	−0.341890
$153$	3.47040	0.280565
$154$	0	0
$155$	−5.94214	−0.477284
$156$	0.540323	0.0432605
$157$	−17.2028	−1.37294	−0.686468	−	0.727160i	$-0.740840\pi$
$157$	−17.2028	−1.37294	−0.686468	+	0.727160i	$0.740840\pi$
$158$	24.7142	1.96616
$159$	2.16670	0.171830
$160$	−3.52444	−0.278631
$161$	0	0
$162$	1.47391	0.115801
$163$	13.7343	1.07576	0.537878	−	0.843023i	$-0.319226\pi$
$163$	13.7343	1.07576	0.537878	+	0.843023i	$0.319226\pi$
$164$	0.201321	0.0157205
$165$	6.56207	0.510856
$166$	−7.61398	−0.590960
$167$	9.47839	0.733460	0.366730	−	0.930327i	$-0.380477\pi$
$167$	9.47839	0.733460	0.366730	+	0.930327i	$0.380477\pi$
$168$	0	0
$169$	−3.17901	−0.244539
$170$	18.5340	1.42149
$171$	1.56480	0.119664
$172$	−1.78973	−0.136466
$173$	10.7801	0.819598	0.409799	−	0.912176i	$-0.365599\pi$
$173$	10.7801	0.819598	0.409799	+	0.912176i	$0.365599\pi$
$174$	−6.09533	−0.462086
$175$	0	0
$176$	−7.81475	−0.589059
$177$	−9.03647	−0.679223
$178$	−1.96582	−0.147344
$179$	4.24218	0.317076	0.158538	−	0.987353i	$-0.449322\pi$
$179$	4.24218	0.317076	0.158538	+	0.987353i	$0.449322\pi$
$180$	0.624731	0.0465647
$181$	−9.69306	−0.720479	−0.360240	−	0.932860i	$-0.617305\pi$
$181$	−9.69306	−0.720479	−0.360240	+	0.932860i	$0.617305\pi$
$182$	0	0
$183$	−10.0787	−0.745040
$184$	2.69370	0.198582
$185$	2.92619	0.215138
$186$	−2.41711	−0.177231
$187$	6.28497	0.459603
$188$	1.37612	0.100364
$189$	0	0
$190$	8.35696	0.606278
$191$	−24.5608	−1.77716	−0.888580	−	0.458722i	$-0.848308\pi$
$191$	−24.5608	−1.77716	−0.888580	+	0.458722i	$0.848308\pi$
$192$	7.19656	0.519367
$193$	5.56746	0.400755	0.200377	−	0.979719i	$-0.435783\pi$
$193$	5.56746	0.400755	0.200377	+	0.979719i	$0.435783\pi$
$194$	−5.67991	−0.407793
$195$	11.3552	0.813163
$196$	0	0
$197$	14.4658	1.03064	0.515322	−	0.856997i	$-0.327672\pi$
$197$	14.4658	1.03064	0.515322	+	0.856997i	$0.327672\pi$
$198$	2.66929	0.189698
$199$	−16.1477	−1.14468	−0.572341	−	0.820016i	$-0.693965\pi$
$199$	−16.1477	−1.14468	−0.572341	+	0.820016i	$0.693965\pi$
$200$	−21.8973	−1.54837
$201$	6.40398	0.451702
$202$	−20.0007	−1.40725
$203$	0	0
$204$	0.598350	0.0418929
$205$	4.23087	0.295497
$206$	22.2564	1.55067
$207$	−1.00000	−0.0695048
$208$	−13.5229	−0.937642
$209$	2.83389	0.196025
$210$	0	0
$211$	−9.44925	−0.650513	−0.325257	−	0.945626i	$-0.605451\pi$
$211$	−9.44925	−0.650513	−0.325257	+	0.945626i	$0.605451\pi$
$212$	0.373572	0.0256570
$213$	13.3495	0.914690
$214$	14.8638	1.01607
$215$	−37.6122	−2.56513
$216$	−2.69370	−0.183283
$217$	0	0
$218$	29.5911	2.00416
$219$	−6.92778	−0.468136
$220$	1.13140	0.0762791
$221$	10.8757	0.731579
$222$	1.19030	0.0798878
$223$	1.03843	0.0695381	0.0347691	−	0.999395i	$-0.488930\pi$
$223$	1.03843	0.0695381	0.0347691	+	0.999395i	$0.488930\pi$
$224$	0	0
$225$	8.12908	0.541939
$226$	−28.9825	−1.92789
$227$	15.7608	1.04608	0.523040	−	0.852308i	$-0.324798\pi$
$227$	15.7608	1.04608	0.523040	+	0.852308i	$0.324798\pi$
$228$	0.269796	0.0178677
$229$	−0.707720	−0.0467674	−0.0233837	−	0.999727i	$-0.507444\pi$
$229$	−0.707720	−0.0467674	−0.0233837	+	0.999727i	$0.507444\pi$
$230$	−5.34058	−0.352148
$231$	0	0
$232$	11.1397	0.731359
$233$	0.902898	0.0591508	0.0295754	−	0.999563i	$-0.490584\pi$
$233$	0.902898	0.0591508	0.0295754	+	0.999563i	$0.490584\pi$
$234$	4.61901	0.301954
$235$	28.9200	1.88653
$236$	−1.55803	−0.101419
$237$	16.7678	1.08918
$238$	0	0
$239$	−28.2947	−1.83023	−0.915116	−	0.403190i	$-0.867901\pi$
$239$	−28.2947	−1.83023	−0.915116	+	0.403190i	$0.867901\pi$
$240$	−15.6354	−1.00926
$241$	−25.5304	−1.64456	−0.822280	−	0.569083i	$-0.807298\pi$
$241$	−25.5304	−1.64456	−0.822280	+	0.569083i	$0.807298\pi$
$242$	−11.3789	−0.731463
$243$	1.00000	0.0641500
$244$	−1.73773	−0.111247
$245$	0	0
$246$	1.72101	0.109728
$247$	4.90385	0.312025
$248$	4.41748	0.280510
$249$	−5.16584	−0.327372
$250$	16.7111	1.05690
$251$	−13.0414	−0.823165	−0.411582	−	0.911373i	$-0.635024\pi$
$251$	−13.0414	−0.823165	−0.411582	+	0.911373i	$0.635024\pi$
$252$	0	0
$253$	−1.81102	−0.113858
$254$	−14.1150	−0.885656
$255$	12.5747	0.787456
$256$	4.10810	0.256756
$257$	−27.2803	−1.70170	−0.850850	−	0.525409i	$-0.823912\pi$
$257$	−27.2803	−1.70170	−0.850850	+	0.525409i	$0.823912\pi$
$258$	−15.2997	−0.952517
$259$	0	0
$260$	1.95781	0.121418
$261$	−4.13548	−0.255980
$262$	9.15893	0.565840
$263$	−21.1162	−1.30208	−0.651040	−	0.759044i	$-0.725667\pi$
$263$	−21.1162	−1.30208	−0.651040	+	0.759044i	$0.725667\pi$
$264$	−4.87835	−0.300241
$265$	7.85082	0.482272
$266$	0	0
$267$	−1.33374	−0.0816237
$268$	1.10414	0.0674464
$269$	2.62903	0.160295	0.0801473	−	0.996783i	$-0.474461\pi$
$269$	2.62903	0.160295	0.0801473	+	0.996783i	$0.474461\pi$
$270$	5.34058	0.325017
$271$	14.3468	0.871506	0.435753	−	0.900066i	$-0.356482\pi$
$271$	14.3468	0.871506	0.435753	+	0.900066i	$0.356482\pi$
$272$	−14.9751	−0.908001
$273$	0	0
$274$	−31.8553	−1.92445
$275$	14.7219	0.887767
$276$	−0.172415	−0.0103782
$277$	−3.07129	−0.184536	−0.0922679	−	0.995734i	$-0.529412\pi$
$277$	−3.07129	−0.184536	−0.0922679	+	0.995734i	$0.529412\pi$
$278$	22.5020	1.34958
$279$	−1.63993	−0.0981801
$280$	0	0
$281$	17.2232	1.02745	0.513724	−	0.857955i	$-0.328266\pi$
$281$	17.2232	1.02745	0.513724	+	0.857955i	$0.328266\pi$
$282$	11.7639	0.700532
$283$	6.86290	0.407957	0.203979	−	0.978975i	$-0.434613\pi$
$283$	6.86290	0.407957	0.203979	+	0.978975i	$0.434613\pi$
$284$	2.30165	0.136578
$285$	5.66992	0.335857
$286$	8.36513	0.494641
$287$	0	0
$288$	−0.972685	−0.0573160
$289$	−4.95633	−0.291549
$290$	−22.0859	−1.29693
$291$	−3.85363	−0.225904
$292$	−1.19446	−0.0699003
$293$	−22.3607	−1.30633	−0.653163	−	0.757217i	$-0.726558\pi$
$293$	−22.3607	−1.30633	−0.653163	+	0.757217i	$0.726558\pi$
$294$	0	0
$295$	−32.7428	−1.90636
$296$	−2.17538	−0.126441
$297$	1.81102	0.105086
$298$	−15.1943	−0.880185
$299$	−3.13385	−0.181235
$300$	1.40158	0.0809202
$301$	0	0
$302$	4.70387	0.270677
$303$	−13.5698	−0.779566
$304$	−6.75229	−0.387270
$305$	−36.5193	−2.09109
$306$	5.11506	0.292409
$307$	−29.2905	−1.67170	−0.835849	−	0.548960i	$-0.815024\pi$
$307$	−29.2905	−1.67170	−0.835849	+	0.548960i	$0.815024\pi$
$308$	0	0
$309$	15.1002	0.859020
$310$	−8.75818	−0.497431
$311$	20.2305	1.14717	0.573583	−	0.819148i	$-0.305553\pi$
$311$	20.2305	1.14717	0.573583	+	0.819148i	$0.305553\pi$
$312$	−8.44164	−0.477914
$313$	19.3901	1.09599	0.547997	−	0.836480i	$-0.315390\pi$
$313$	19.3901	1.09599	0.547997	+	0.836480i	$0.315390\pi$
$314$	−25.3555	−1.43089
$315$	0	0
$316$	2.89102	0.162633
$317$	−2.89880	−0.162813	−0.0814065	−	0.996681i	$-0.525941\pi$
$317$	−2.89880	−0.162813	−0.0814065	+	0.996681i	$0.525941\pi$
$318$	3.19352	0.179084
$319$	−7.48944	−0.419328
$320$	26.0761	1.45770
$321$	10.0846	0.562869
$322$	0	0
$323$	5.43049	0.302161
$324$	0.172415	0.00957864
$325$	25.4753	1.41311
$326$	20.2432	1.12117
$327$	20.0766	1.11024
$328$	−3.14530	−0.173670
$329$	0	0
$330$	9.67191	0.532421
$331$	23.2662	1.27882	0.639412	−	0.768864i	$-0.279178\pi$
$331$	23.2662	1.27882	0.639412	+	0.768864i	$0.279178\pi$
$332$	−0.890670	−0.0488819
$333$	0.807580	0.0442551
$334$	13.9703	0.764421
$335$	23.2042	1.26778
$336$	0	0
$337$	12.4951	0.680651	0.340326	−	0.940308i	$-0.389463\pi$
$337$	12.4951	0.680651	0.340326	+	0.940308i	$0.389463\pi$
$338$	−4.68558	−0.254862
$339$	−19.6637	−1.06798
$340$	2.16807	0.117580
$341$	−2.96995	−0.160832
$342$	2.30638	0.124715
$343$	0	0
$344$	27.9615	1.50758
$345$	−3.62341	−0.195078
$346$	15.8890	0.854196
$347$	−1.21519	−0.0652348	−0.0326174	−	0.999468i	$-0.510384\pi$
$347$	−1.21519	−0.0652348	−0.0326174	+	0.999468i	$0.510384\pi$
$348$	−0.713020	−0.0382219
$349$	−8.27647	−0.443030	−0.221515	−	0.975157i	$-0.571100\pi$
$349$	−8.27647	−0.443030	−0.221515	+	0.975157i	$0.571100\pi$
$350$	0	0
$351$	3.13385	0.167272
$352$	−1.76155	−0.0938912
$353$	−6.36310	−0.338674	−0.169337	−	0.985558i	$-0.554163\pi$
$353$	−6.36310	−0.338674	−0.169337	+	0.985558i	$0.554163\pi$
$354$	−13.3190	−0.707894
$355$	48.3706	2.56724
$356$	−0.229958	−0.0121877
$357$	0	0
$358$	6.25260	0.330460
$359$	18.0851	0.954493	0.477246	−	0.878770i	$-0.341635\pi$
$359$	18.0851	0.954493	0.477246	+	0.878770i	$0.341635\pi$
$360$	−9.76037	−0.514416
$361$	−16.5514	−0.871126
$362$	−14.2867	−0.750893
$363$	−7.72020	−0.405205
$364$	0	0
$365$	−25.1022	−1.31391
$366$	−14.8551	−0.776490
$367$	9.22439	0.481509	0.240755	−	0.970586i	$-0.422605\pi$
$367$	9.22439	0.481509	0.240755	+	0.970586i	$0.422605\pi$
$368$	4.31510	0.224940
$369$	1.16765	0.0607854
$370$	4.31295	0.224220
$371$	0	0
$372$	−0.282749	−0.0146599
$373$	−18.1967	−0.942189	−0.471095	−	0.882083i	$-0.656141\pi$
$373$	−18.1967	−0.942189	−0.471095	+	0.882083i	$0.656141\pi$
$374$	9.26349	0.479004
$375$	11.3379	0.585488
$376$	−21.4996	−1.10876
$377$	−12.9600	−0.667471
$378$	0	0
$379$	20.8172	1.06931	0.534655	−	0.845070i	$-0.320442\pi$
$379$	20.8172	1.06931	0.534655	+	0.845070i	$0.320442\pi$
$380$	0.977582	0.0501489
$381$	−9.57658	−0.490623
$382$	−36.2005	−1.85218
$383$	−11.4489	−0.585011	−0.292506	−	0.956264i	$-0.594489\pi$
$383$	−11.4489	−0.585011	−0.292506	+	0.956264i	$0.594489\pi$
$384$	12.5525	0.640565
$385$	0	0
$386$	8.20595	0.417672
$387$	−10.3803	−0.527662
$388$	−0.664425	−0.0337311
$389$	0.787249	0.0399151	0.0199576	−	0.999801i	$-0.493647\pi$
$389$	0.787249	0.0399151	0.0199576	+	0.999801i	$0.493647\pi$
$390$	16.7366	0.847489
$391$	−3.47040	−0.175506
$392$	0	0
$393$	6.21403	0.313456
$394$	21.3213	1.07415
$395$	60.7565	3.05699
$396$	0.312248	0.0156911
$397$	−15.6765	−0.786781	−0.393391	−	0.919371i	$-0.628698\pi$
$397$	−15.6765	−0.786781	−0.393391	+	0.919371i	$0.628698\pi$
$398$	−23.8003	−1.19300
$399$	0	0
$400$	−35.0778	−1.75389
$401$	−9.88573	−0.493670	−0.246835	−	0.969058i	$-0.579391\pi$
$401$	−9.88573	−0.493670	−0.246835	+	0.969058i	$0.579391\pi$
$402$	9.43890	0.470769
$403$	−5.13929	−0.256006
$404$	−2.33965	−0.116402
$405$	3.62341	0.180049
$406$	0	0
$407$	1.46255	0.0724957
$408$	−9.34821	−0.462805
$409$	−14.5025	−0.717103	−0.358552	−	0.933510i	$-0.616729\pi$
$409$	−14.5025	−0.717103	−0.358552	+	0.933510i	$0.616729\pi$
$410$	6.23593	0.307971
$411$	−21.6128	−1.06608
$412$	2.60351	0.128266
$413$	0	0
$414$	−1.47391	−0.0724388
$415$	−18.7179	−0.918827
$416$	−3.04825	−0.149453
$417$	15.2669	0.747623
$418$	4.17691	0.204299
$419$	−27.6663	−1.35159	−0.675795	−	0.737090i	$-0.736199\pi$
$419$	−27.6663	−1.35159	−0.675795	+	0.737090i	$0.736199\pi$
$420$	0	0
$421$	−2.36548	−0.115286	−0.0576432	−	0.998337i	$-0.518359\pi$
$421$	−2.36548	−0.115286	−0.0576432	+	0.998337i	$0.518359\pi$
$422$	−13.9274	−0.677973
$423$	7.98144	0.388071
$424$	−5.83643	−0.283442
$425$	28.2112	1.36844
$426$	19.6759	0.953302
$427$	0	0
$428$	1.73874	0.0840454
$429$	5.67546	0.274014
$430$	−55.4370	−2.67341
$431$	−17.6627	−0.850780	−0.425390	−	0.905010i	$-0.639863\pi$
$431$	−17.6627	−0.850780	−0.425390	+	0.905010i	$0.639863\pi$
$432$	−4.31510	−0.207611
$433$	23.2741	1.11848	0.559240	−	0.829006i	$-0.311093\pi$
$433$	23.2741	1.11848	0.559240	+	0.829006i	$0.311093\pi$
$434$	0	0
$435$	−14.9845	−0.718453
$436$	3.46151	0.165776
$437$	−1.56480	−0.0748547
$438$	−10.2109	−0.487897
$439$	−12.0788	−0.576488	−0.288244	−	0.957557i	$-0.593071\pi$
$439$	−12.0788	−0.576488	−0.288244	+	0.957557i	$0.593071\pi$
$440$	−17.6762	−0.842681
$441$	0	0
$442$	16.0298	0.762461
$443$	36.5974	1.73879	0.869396	−	0.494115i	$-0.164508\pi$
$443$	36.5974	1.73879	0.869396	+	0.494115i	$0.164508\pi$
$444$	0.139239	0.00660801
$445$	−4.83269	−0.229091
$446$	1.53055	0.0724735
$447$	−10.3089	−0.487592
$448$	0	0
$449$	23.9113	1.12844	0.564221	−	0.825623i	$-0.309176\pi$
$449$	23.9113	1.12844	0.564221	+	0.825623i	$0.309176\pi$
$450$	11.9815	0.564816
$451$	2.11464	0.0995745
$452$	−3.39032	−0.159467
$453$	3.19142	0.149946
$454$	23.2300	1.09024
$455$	0	0
$456$	−4.21511	−0.197391
$457$	40.8007	1.90858	0.954288	−	0.298890i	$-0.0966163\pi$
$457$	40.8007	1.90858	0.954288	+	0.298890i	$0.0966163\pi$
$458$	−1.04312	−0.0487416
$459$	3.47040	0.161984
$460$	−0.624731	−0.0291283
$461$	28.7994	1.34132	0.670661	−	0.741764i	$-0.266011\pi$
$461$	28.7994	1.34132	0.670661	+	0.741764i	$0.266011\pi$
$462$	0	0
$463$	4.93852	0.229512	0.114756	−	0.993394i	$-0.463391\pi$
$463$	4.93852	0.229512	0.114756	+	0.993394i	$0.463391\pi$
$464$	17.8450	0.828434
$465$	−5.94214	−0.275560
$466$	1.33079	0.0616477
$467$	−7.61272	−0.352275	−0.176137	−	0.984366i	$-0.556360\pi$
$467$	−7.61272	−0.352275	−0.176137	+	0.984366i	$0.556360\pi$
$468$	0.540323	0.0249765
$469$	0	0
$470$	42.6255	1.96617
$471$	−17.2028	−0.792665
$472$	24.3415	1.12041
$473$	−18.7990	−0.864379
$474$	24.7142	1.13516
$475$	12.7204	0.583653
$476$	0	0
$477$	2.16670	0.0992062
$478$	−41.7039	−1.90749
$479$	−0.220014	−0.0100527	−0.00502634	−	0.999987i	$-0.501600\pi$
$479$	−0.220014	−0.0100527	−0.00502634	+	0.999987i	$0.501600\pi$
$480$	−3.52444	−0.160868
$481$	2.53083	0.115396
$482$	−37.6296	−1.71398
$483$	0	0
$484$	−1.33108	−0.0605037
$485$	−13.9633	−0.634039
$486$	1.47391	0.0668580
$487$	−13.9599	−0.632586	−0.316293	−	0.948662i	$-0.602438\pi$
$487$	−13.9599	−0.632586	−0.316293	+	0.948662i	$0.602438\pi$
$488$	27.1490	1.22898
$489$	13.7343	0.621088
$490$	0	0
$491$	16.5329	0.746117	0.373059	−	0.927808i	$-0.378309\pi$
$491$	16.5329	0.746117	0.373059	+	0.927808i	$0.378309\pi$
$492$	0.201321	0.00907625
$493$	−14.3518	−0.646371
$494$	7.22785	0.325196
$495$	6.56207	0.294943
$496$	7.07647	0.317743
$497$	0	0
$498$	−7.61398	−0.341191
$499$	−20.4818	−0.916891	−0.458445	−	0.888723i	$-0.651594\pi$
$499$	−20.4818	−0.916891	−0.458445	+	0.888723i	$0.651594\pi$
$500$	1.95483	0.0874229
$501$	9.47839	0.423463
$502$	−19.2218	−0.857913
$503$	41.9044	1.86843	0.934213	−	0.356715i	$-0.116103\pi$
$503$	41.9044	1.86843	0.934213	+	0.356715i	$0.116103\pi$
$504$	0	0
$505$	−49.1690	−2.18799
$506$	−2.66929	−0.118664
$507$	−3.17901	−0.141185
$508$	−1.65115	−0.0732579
$509$	10.9768	0.486539	0.243270	−	0.969959i	$-0.421780\pi$
$509$	10.9768	0.486539	0.243270	+	0.969959i	$0.421780\pi$
$510$	18.5340	0.820697
$511$	0	0
$512$	−19.0499	−0.841896
$513$	1.56480	0.0690878
$514$	−40.2088	−1.77353
$515$	54.7142	2.41099
$516$	−1.78973	−0.0787884
$517$	14.4546	0.635711
$518$	0	0
$519$	10.7801	0.473195
$520$	−30.5875	−1.34135
$521$	−25.4539	−1.11515	−0.557577	−	0.830125i	$-0.688269\pi$
$521$	−25.4539	−1.11515	−0.557577	+	0.830125i	$0.688269\pi$
$522$	−6.09533	−0.266785
$523$	−37.8104	−1.65333	−0.826667	−	0.562691i	$-0.809766\pi$
$523$	−37.8104	−1.65333	−0.826667	+	0.562691i	$0.809766\pi$
$524$	1.07139	0.0468041
$525$	0	0
$526$	−31.1234	−1.35704
$527$	−5.69121	−0.247913
$528$	−7.81475	−0.340093
$529$	1.00000	0.0434783
$530$	11.5714	0.502630
$531$	−9.03647	−0.392149
$532$	0	0
$533$	3.65924	0.158499
$534$	−1.96582	−0.0850692
$535$	36.5407	1.57979
$536$	−17.2504	−0.745103
$537$	4.24218	0.183064
$538$	3.87495	0.167061
$539$	0	0
$540$	0.624731	0.0268842
$541$	−31.0014	−1.33286	−0.666428	−	0.745570i	$-0.732177\pi$
$541$	−31.0014	−1.33286	−0.666428	+	0.745570i	$0.732177\pi$
$542$	21.1459	0.908295
$543$	−9.69306	−0.415969
$544$	−3.37561	−0.144728
$545$	72.7457	3.11608
$546$	0	0
$547$	−15.8705	−0.678572	−0.339286	−	0.940683i	$-0.610185\pi$
$547$	−15.8705	−0.678572	−0.339286	+	0.940683i	$0.610185\pi$
$548$	−3.72638	−0.159183
$549$	−10.0787	−0.430149
$550$	21.6988	0.925242
$551$	−6.47121	−0.275683
$552$	2.69370	0.114651
$553$	0	0
$554$	−4.52681	−0.192326
$555$	2.92619	0.124210
$556$	2.63225	0.111632
$557$	22.5830	0.956872	0.478436	−	0.878123i	$-0.341204\pi$
$557$	22.5830	0.956872	0.478436	+	0.878123i	$0.341204\pi$
$558$	−2.41711	−0.102325
$559$	−32.5304	−1.37589
$560$	0	0
$561$	6.28497	0.265352
$562$	25.3854	1.07082
$563$	−13.5197	−0.569787	−0.284893	−	0.958559i	$-0.591958\pi$
$563$	−13.5197	−0.569787	−0.284893	+	0.958559i	$0.591958\pi$
$564$	1.37612	0.0579452
$565$	−71.2495	−2.99749
$566$	10.1153	0.425178
$567$	0	0
$568$	−35.9594	−1.50882
$569$	−2.67976	−0.112342	−0.0561708	−	0.998421i	$-0.517889\pi$
$569$	−2.67976	−0.112342	−0.0561708	+	0.998421i	$0.517889\pi$
$570$	8.35696	0.350035
$571$	3.48717	0.145933	0.0729667	−	0.997334i	$-0.476753\pi$
$571$	3.48717	0.145933	0.0729667	+	0.997334i	$0.476753\pi$
$572$	0.978538	0.0409147
$573$	−24.5608	−1.02604
$574$	0	0
$575$	−8.12908	−0.339006
$576$	7.19656	0.299856
$577$	3.79951	0.158176	0.0790878	−	0.996868i	$-0.474799\pi$
$577$	3.79951	0.158176	0.0790878	+	0.996868i	$0.474799\pi$
$578$	−7.30519	−0.303856
$579$	5.56746	0.231376
$580$	−2.58356	−0.107277
$581$	0	0
$582$	−5.67991	−0.235440
$583$	3.92393	0.162513
$584$	18.6613	0.772212
$585$	11.3552	0.469480
$586$	−32.9577	−1.36147
$587$	−6.93134	−0.286087	−0.143043	−	0.989716i	$-0.545689\pi$
$587$	−6.93134	−0.286087	−0.143043	+	0.989716i	$0.545689\pi$
$588$	0	0
$589$	−2.56617	−0.105737
$590$	−48.2600	−1.98683
$591$	14.4658	0.595043
$592$	−3.48479	−0.143224
$593$	18.3246	0.752499	0.376250	−	0.926518i	$-0.377214\pi$
$593$	18.3246	0.752499	0.376250	+	0.926518i	$0.377214\pi$
$594$	2.66929	0.109522
$595$	0	0
$596$	−1.77741	−0.0728054
$597$	−16.1477	−0.660882
$598$	−4.61901	−0.188885
$599$	−26.4787	−1.08189	−0.540945	−	0.841058i	$-0.681933\pi$
$599$	−26.4787	−1.08189	−0.540945	+	0.841058i	$0.681933\pi$
$600$	−21.8973	−0.893953
$601$	−17.0913	−0.697168	−0.348584	−	0.937278i	$-0.613337\pi$
$601$	−17.0913	−0.697168	−0.348584	+	0.937278i	$0.613337\pi$
$602$	0	0
$603$	6.40398	0.260790
$604$	0.550249	0.0223893
$605$	−27.9734	−1.13728
$606$	−20.0007	−0.812474
$607$	−43.0275	−1.74643	−0.873215	−	0.487335i	$-0.837969\pi$
$607$	−43.0275	−1.74643	−0.873215	+	0.487335i	$0.837969\pi$
$608$	−1.52206	−0.0617278
$609$	0	0
$610$	−53.8262	−2.17936
$611$	25.0126	1.01190
$612$	0.598350	0.0241869
$613$	−14.2073	−0.573826	−0.286913	−	0.957957i	$-0.592629\pi$
$613$	−14.2073	−0.573826	−0.286913	+	0.957957i	$0.592629\pi$
$614$	−43.1716	−1.74226
$615$	4.23087	0.170605
$616$	0	0
$617$	29.4369	1.18509	0.592543	−	0.805539i	$-0.298124\pi$
$617$	29.4369	1.18509	0.592543	+	0.805539i	$0.298124\pi$
$618$	22.2564	0.895282
$619$	−19.3370	−0.777220	−0.388610	−	0.921402i	$-0.627045\pi$
$619$	−19.3370	−0.777220	−0.388610	+	0.921402i	$0.627045\pi$
$620$	−1.02452	−0.0411456
$621$	−1.00000	−0.0401286
$622$	29.8179	1.19559
$623$	0	0
$624$	−13.5229	−0.541348
$625$	0.436550	0.0174620
$626$	28.5793	1.14226
$627$	2.83389	0.113175
$628$	−2.96604	−0.118358
$629$	2.80263	0.111748
$630$	0	0
$631$	6.33012	0.251998	0.125999	−	0.992030i	$-0.459786\pi$
$631$	6.33012	0.251998	0.125999	+	0.992030i	$0.459786\pi$
$632$	−45.1673	−1.79666
$633$	−9.44925	−0.375574
$634$	−4.27258	−0.169686
$635$	−34.6998	−1.37702
$636$	0.373572	0.0148131
$637$	0	0
$638$	−11.0388	−0.437029
$639$	13.3495	0.528097
$640$	45.4827	1.79786
$641$	−18.8029	−0.742669	−0.371335	−	0.928499i	$-0.621100\pi$
$641$	−18.8029	−0.742669	−0.371335	+	0.928499i	$0.621100\pi$
$642$	14.8638	0.586629
$643$	9.01232	0.355411	0.177706	−	0.984084i	$-0.443133\pi$
$643$	9.01232	0.355411	0.177706	+	0.984084i	$0.443133\pi$
$644$	0	0
$645$	−37.6122	−1.48098
$646$	8.00407	0.314916
$647$	−39.4516	−1.55100	−0.775502	−	0.631345i	$-0.782503\pi$
$647$	−39.4516	−1.55100	−0.775502	+	0.631345i	$0.782503\pi$
$648$	−2.69370	−0.105818
$649$	−16.3652	−0.642392
$650$	37.5483	1.47277
$651$	0	0
$652$	2.36801	0.0927385
$653$	−45.8575	−1.79454	−0.897272	−	0.441479i	$-0.854454\pi$
$653$	−45.8575	−1.79454	−0.897272	+	0.441479i	$0.854454\pi$
$654$	29.5911	1.15710
$655$	22.5160	0.879771
$656$	−5.03853	−0.196722
$657$	−6.92778	−0.270278
$658$	0	0
$659$	−8.63338	−0.336309	−0.168154	−	0.985761i	$-0.553781\pi$
$659$	−8.63338	−0.336309	−0.168154	+	0.985761i	$0.553781\pi$
$660$	1.13140	0.0440398
$661$	−32.7107	−1.27230	−0.636148	−	0.771567i	$-0.719473\pi$
$661$	−32.7107	−1.27230	−0.636148	+	0.771567i	$0.719473\pi$
$662$	34.2923	1.33281
$663$	10.8757	0.422377
$664$	13.9152	0.540015
$665$	0	0
$666$	1.19030	0.0461233
$667$	4.13548	0.160126
$668$	1.63422	0.0632299
$669$	1.03843	0.0401479
$670$	34.2010	1.32130
$671$	−18.2528	−0.704640
$672$	0	0
$673$	15.2454	0.587668	0.293834	−	0.955857i	$-0.405069\pi$
$673$	15.2454	0.587668	0.293834	+	0.955857i	$0.405069\pi$
$674$	18.4167	0.709384
$675$	8.12908	0.312888
$676$	−0.548110	−0.0210812
$677$	44.8173	1.72247	0.861235	−	0.508208i	$-0.169692\pi$
$677$	44.8173	1.72247	0.861235	+	0.508208i	$0.169692\pi$
$678$	−28.9825	−1.11307
$679$	0	0
$680$	−33.8724	−1.29895
$681$	15.7608	0.603955
$682$	−4.37744	−0.167621
$683$	−11.3680	−0.434985	−0.217492	−	0.976062i	$-0.569788\pi$
$683$	−11.3680	−0.434985	−0.217492	+	0.976062i	$0.569788\pi$
$684$	0.269796	0.0103159
$685$	−78.3119	−2.99214
$686$	0	0
$687$	−0.707720	−0.0270012
$688$	44.7922	1.70769
$689$	6.79009	0.258682
$690$	−5.34058	−0.203312
$691$	−34.2565	−1.30318	−0.651590	−	0.758571i	$-0.725898\pi$
$691$	−34.2565	−1.30318	−0.651590	+	0.758571i	$0.725898\pi$
$692$	1.85866	0.0706557
$693$	0	0
$694$	−1.79108	−0.0679886
$695$	55.3182	2.09834
$696$	11.1397	0.422250
$697$	4.05221	0.153488
$698$	−12.1988	−0.461731
$699$	0.902898	0.0341507
$700$	0	0
$701$	13.4739	0.508902	0.254451	−	0.967086i	$-0.418105\pi$
$701$	13.4739	0.508902	0.254451	+	0.967086i	$0.418105\pi$
$702$	4.61901	0.174333
$703$	1.26370	0.0476615
$704$	13.0331	0.491204
$705$	28.9200	1.08919
$706$	−9.37865	−0.352970
$707$	0	0
$708$	−1.55803	−0.0585542
$709$	48.1654	1.80889	0.904445	−	0.426591i	$-0.140286\pi$
$709$	48.1654	1.80889	0.904445	+	0.426591i	$0.140286\pi$
$710$	71.2939	2.67561
$711$	16.7678	0.628840
$712$	3.59270	0.134642
$713$	1.63993	0.0614159
$714$	0	0
$715$	20.5645	0.769069
$716$	0.731418	0.0273344
$717$	−28.2947	−1.05669
$718$	26.6558	0.994785
$719$	16.7845	0.625957	0.312979	−	0.949760i	$-0.398673\pi$
$719$	16.7845	0.625957	0.312979	+	0.949760i	$0.398673\pi$
$720$	−15.6354	−0.582696
$721$	0	0
$722$	−24.3953	−0.907898
$723$	−25.5304	−0.949487
$724$	−1.67123	−0.0621109
$725$	−33.6176	−1.24853
$726$	−11.3789	−0.422310
$727$	−12.8214	−0.475518	−0.237759	−	0.971324i	$-0.576413\pi$
$727$	−12.8214	−0.475518	−0.237759	+	0.971324i	$0.576413\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−36.9984	−1.36937
$731$	−36.0239	−1.33239
$732$	−1.73773	−0.0642282
$733$	−2.07207	−0.0765336	−0.0382668	−	0.999268i	$-0.512184\pi$
$733$	−2.07207	−0.0765336	−0.0382668	+	0.999268i	$0.512184\pi$
$734$	13.5959	0.501835
$735$	0	0
$736$	0.972685	0.0358537
$737$	11.5977	0.427208
$738$	1.72101	0.0633514
$739$	−10.0740	−0.370578	−0.185289	−	0.982684i	$-0.559322\pi$
$739$	−10.0740	−0.370578	−0.185289	+	0.982684i	$0.559322\pi$
$740$	0.504521	0.0185466
$741$	4.90385	0.180148
$742$	0	0
$743$	10.5218	0.386006	0.193003	−	0.981198i	$-0.438177\pi$
$743$	10.5218	0.386006	0.193003	+	0.981198i	$0.438177\pi$
$744$	4.41748	0.161953
$745$	−37.3532	−1.36852
$746$	−26.8203	−0.981962
$747$	−5.16584	−0.189008
$748$	1.08363	0.0396213
$749$	0	0
$750$	16.7111	0.610204
$751$	38.7361	1.41350	0.706750	−	0.707464i	$-0.250161\pi$
$751$	38.7361	1.41350	0.706750	+	0.707464i	$0.250161\pi$
$752$	−34.4407	−1.25592
$753$	−13.0414	−0.475255
$754$	−19.1018	−0.695647
$755$	11.5638	0.420850
$756$	0	0
$757$	−38.1123	−1.38522	−0.692608	−	0.721314i	$-0.743539\pi$
$757$	−38.1123	−1.38522	−0.692608	+	0.721314i	$0.743539\pi$
$758$	30.6828	1.11445
$759$	−1.81102	−0.0657359
$760$	−15.2731	−0.554012
$761$	−9.42576	−0.341684	−0.170842	−	0.985298i	$-0.554649\pi$
$761$	−9.42576	−0.341684	−0.170842	+	0.985298i	$0.554649\pi$
$762$	−14.1150	−0.511334
$763$	0	0
$764$	−4.23467	−0.153205
$765$	12.5747	0.454638
$766$	−16.8747	−0.609706
$767$	−28.3189	−1.02254
$768$	4.10810	0.148238
$769$	−17.7859	−0.641375	−0.320688	−	0.947185i	$-0.603914\pi$
$769$	−17.7859	−0.641375	−0.320688	+	0.947185i	$0.603914\pi$
$770$	0	0
$771$	−27.2803	−0.982477
$772$	0.959917	0.0345482
$773$	33.4227	1.20213	0.601065	−	0.799200i	$-0.294743\pi$
$773$	33.4227	1.20213	0.601065	+	0.799200i	$0.294743\pi$
$774$	−15.2997	−0.549936
$775$	−13.3311	−0.478868
$776$	10.3805	0.372639
$777$	0	0
$778$	1.16034	0.0416000
$779$	1.82714	0.0654642
$780$	1.95781	0.0701009
$781$	24.1762	0.865092
$782$	−5.11506	−0.182914
$783$	−4.13548	−0.147790
$784$	0	0
$785$	−62.3329	−2.22476
$786$	9.15893	0.326688
$787$	3.38432	0.120638	0.0603191	−	0.998179i	$-0.480788\pi$
$787$	3.38432	0.120638	0.0603191	+	0.998179i	$0.480788\pi$
$788$	2.49412	0.0888495
$789$	−21.1162	−0.751756
$790$	89.5496	3.18603
$791$	0	0
$792$	−4.87835	−0.173344
$793$	−31.5851	−1.12162
$794$	−23.1058	−0.819993
$795$	7.85082	0.278440
$796$	−2.78412	−0.0986804
$797$	−17.7625	−0.629180	−0.314590	−	0.949228i	$-0.601867\pi$
$797$	−17.7625	−0.629180	−0.314590	+	0.949228i	$0.601867\pi$
$798$	0	0
$799$	27.6988	0.979912
$800$	−7.90704	−0.279556
$801$	−1.33374	−0.0471254
$802$	−14.5707	−0.514509
$803$	−12.5464	−0.442751
$804$	1.10414	0.0389402
$805$	0	0
$806$	−7.57486	−0.266813
$807$	2.62903	0.0925462
$808$	36.5530	1.28593
$809$	32.6033	1.14627	0.573135	−	0.819461i	$-0.305727\pi$
$809$	32.6033	1.14627	0.573135	+	0.819461i	$0.305727\pi$
$810$	5.34058	0.187649
$811$	30.1405	1.05838	0.529188	−	0.848504i	$-0.322497\pi$
$811$	30.1405	1.05838	0.529188	+	0.848504i	$0.322497\pi$
$812$	0	0
$813$	14.3468	0.503164
$814$	2.15566	0.0755560
$815$	49.7651	1.74319
$816$	−14.9751	−0.524235
$817$	−16.2432	−0.568277
$818$	−21.3754	−0.747374
$819$	0	0
$820$	0.729468	0.0254741
$821$	−36.1163	−1.26047	−0.630233	−	0.776406i	$-0.717041\pi$
$821$	−36.1163	−1.26047	−0.630233	+	0.776406i	$0.717041\pi$
$822$	−31.8553	−1.11108
$823$	−34.6824	−1.20895	−0.604476	−	0.796623i	$-0.706617\pi$
$823$	−34.6824	−1.20895	−0.604476	+	0.796623i	$0.706617\pi$
$824$	−40.6754	−1.41699
$825$	14.7219	0.512552
$826$	0	0
$827$	−1.91220	−0.0664937	−0.0332468	−	0.999447i	$-0.510585\pi$
$827$	−1.91220	−0.0664937	−0.0332468	+	0.999447i	$0.510585\pi$
$828$	−0.172415	−0.00599185
$829$	25.2687	0.877618	0.438809	−	0.898580i	$-0.355400\pi$
$829$	25.2687	0.877618	0.438809	+	0.898580i	$0.355400\pi$
$830$	−27.5886	−0.957613
$831$	−3.07129	−0.106542
$832$	22.5529	0.781881
$833$	0	0
$834$	22.5020	0.779182
$835$	34.3441	1.18853
$836$	0.488607	0.0168988
$837$	−1.63993	−0.0566843
$838$	−40.7778	−1.40864
$839$	32.3127	1.11556	0.557779	−	0.829989i	$-0.311654\pi$
$839$	32.3127	1.11556	0.557779	+	0.829989i	$0.311654\pi$
$840$	0	0
$841$	−11.8978	−0.410270
$842$	−3.48651	−0.120153
$843$	17.2232	0.593197
$844$	−1.62920	−0.0560793
$845$	−11.5188	−0.396260
$846$	11.7639	0.404452
$847$	0	0
$848$	−9.34952	−0.321064
$849$	6.86290	0.235534
$850$	41.5808	1.42621
$851$	−0.807580	−0.0276835
$852$	2.30165	0.0788534
$853$	13.5399	0.463596	0.231798	−	0.972764i	$-0.425539\pi$
$853$	13.5399	0.463596	0.231798	+	0.972764i	$0.425539\pi$
$854$	0	0
$855$	5.66992	0.193907
$856$	−27.1649	−0.928478
$857$	10.7806	0.368258	0.184129	−	0.982902i	$-0.441054\pi$
$857$	10.7806	0.368258	0.184129	+	0.982902i	$0.441054\pi$
$858$	8.36513	0.285581
$859$	−30.7020	−1.04754	−0.523770	−	0.851860i	$-0.675475\pi$
$859$	−30.7020	−1.04754	−0.523770	+	0.851860i	$0.675475\pi$
$860$	−6.48492	−0.221134
$861$	0	0
$862$	−26.0332	−0.886694
$863$	37.8134	1.28718	0.643592	−	0.765369i	$-0.277443\pi$
$863$	37.8134	1.28718	0.643592	+	0.765369i	$0.277443\pi$
$864$	−0.972685	−0.0330914
$865$	39.0608	1.32811
$866$	34.3039	1.16569
$867$	−4.95633	−0.168326
$868$	0	0
$869$	30.3668	1.03012
$870$	−22.0859	−0.748781
$871$	20.0691	0.680015
$872$	−54.0803	−1.83139
$873$	−3.85363	−0.130426
$874$	−2.30638	−0.0780145
$875$	0	0
$876$	−1.19446	−0.0403569
$877$	−31.2193	−1.05420	−0.527100	−	0.849804i	$-0.676721\pi$
$877$	−31.2193	−1.05420	−0.527100	+	0.849804i	$0.676721\pi$
$878$	−17.8030	−0.600824
$879$	−22.3607	−0.754208
$880$	−28.3160	−0.954533
$881$	42.8695	1.44431	0.722155	−	0.691731i	$-0.243152\pi$
$881$	42.8695	1.44431	0.722155	+	0.691731i	$0.243152\pi$
$882$	0	0
$883$	31.4959	1.05992	0.529960	−	0.848023i	$-0.322207\pi$
$883$	31.4959	1.05992	0.529960	+	0.848023i	$0.322207\pi$
$884$	1.87514	0.0630677
$885$	−32.7428	−1.10064
$886$	53.9413	1.81219
$887$	43.4216	1.45795	0.728977	−	0.684539i	$-0.239996\pi$
$887$	43.4216	1.45795	0.728977	+	0.684539i	$0.239996\pi$
$888$	−2.17538	−0.0730009
$889$	0	0
$890$	−7.12296	−0.238762
$891$	1.81102	0.0606715
$892$	0.179041	0.00599472
$893$	12.4894	0.417941
$894$	−15.1943	−0.508175
$895$	15.3712	0.513801
$896$	0	0
$897$	−3.13385	−0.104636
$898$	35.2431	1.17608
$899$	6.78190	0.226189
$900$	1.40158	0.0467193
$901$	7.51930	0.250504
$902$	3.11679	0.103778
$903$	0	0
$904$	52.9680	1.76169
$905$	−35.1219	−1.16749
$906$	4.70387	0.156275
$907$	44.8538	1.48935	0.744673	−	0.667430i	$-0.232606\pi$
$907$	44.8538	1.48935	0.744673	+	0.667430i	$0.232606\pi$
$908$	2.71740	0.0901802
$909$	−13.5698	−0.450083
$910$	0	0
$911$	14.8143	0.490821	0.245410	−	0.969419i	$-0.421077\pi$
$911$	14.8143	0.490821	0.245410	+	0.969419i	$0.421077\pi$
$912$	−6.75229	−0.223591
$913$	−9.35544	−0.309620
$914$	60.1366	1.98914
$915$	−36.5193	−1.20729
$916$	−0.122022	−0.00403171
$917$	0	0
$918$	5.11506	0.168822
$919$	20.6533	0.681291	0.340645	−	0.940192i	$-0.389354\pi$
$919$	20.6533	0.681291	0.340645	+	0.940192i	$0.389354\pi$
$920$	9.76037	0.321790
$921$	−29.2905	−0.965155
$922$	42.4478	1.39794
$923$	41.8352	1.37702
$924$	0	0
$925$	6.56489	0.215852
$926$	7.27894	0.239201
$927$	15.1002	0.495955
$928$	4.02252	0.132046
$929$	53.9942	1.77149	0.885746	−	0.464170i	$-0.153647\pi$
$929$	53.9942	1.77149	0.885746	+	0.464170i	$0.153647\pi$
$930$	−8.75818	−0.287192
$931$	0	0
$932$	0.155674	0.00509926
$933$	20.2305	0.662316
$934$	−11.2205	−0.367145
$935$	22.7730	0.744757
$936$	−8.44164	−0.275923
$937$	−38.9987	−1.27403	−0.637016	−	0.770850i	$-0.719832\pi$
$937$	−38.9987	−1.27403	−0.637016	+	0.770850i	$0.719832\pi$
$938$	0	0
$939$	19.3901	0.632773
$940$	4.98625	0.162634
$941$	−9.26719	−0.302102	−0.151051	−	0.988526i	$-0.548266\pi$
$941$	−9.26719	−0.302102	−0.151051	+	0.988526i	$0.548266\pi$
$942$	−25.3555	−0.826126
$943$	−1.16765	−0.0380239
$944$	38.9933	1.26912
$945$	0	0
$946$	−27.7081	−0.900867
$947$	34.9305	1.13509	0.567544	−	0.823343i	$-0.307894\pi$
$947$	34.9305	1.13509	0.567544	+	0.823343i	$0.307894\pi$
$948$	2.89102	0.0938960
$949$	−21.7106	−0.704756
$950$	18.7488	0.608290
$951$	−2.89880	−0.0940001
$952$	0	0
$953$	−1.73312	−0.0561413	−0.0280707	−	0.999606i	$-0.508936\pi$
$953$	−1.73312	−0.0561413	−0.0280707	+	0.999606i	$0.508936\pi$
$954$	3.19352	0.103394
$955$	−88.9939	−2.87978
$956$	−4.87845	−0.157780
$957$	−7.48944	−0.242099
$958$	−0.324281	−0.0104770
$959$	0	0
$960$	26.0761	0.841601
$961$	−28.3106	−0.913246
$962$	3.73022	0.120267
$963$	10.0846	0.324972
$964$	−4.40184	−0.141774
$965$	20.1732	0.649398
$966$	0	0
$967$	37.3373	1.20069	0.600344	−	0.799742i	$-0.295030\pi$
$967$	37.3373	1.20069	0.600344	+	0.799742i	$0.295030\pi$
$968$	20.7959	0.668405
$969$	5.43049	0.174453
$970$	−20.5806	−0.660804
$971$	−12.6245	−0.405139	−0.202570	−	0.979268i	$-0.564929\pi$
$971$	−12.6245	−0.405139	−0.202570	+	0.979268i	$0.564929\pi$
$972$	0.172415	0.00553023
$973$	0	0
$974$	−20.5757	−0.659289
$975$	25.4753	0.815862
$976$	43.4907	1.39210
$977$	11.0037	0.352039	0.176019	−	0.984387i	$-0.443678\pi$
$977$	11.0037	0.352039	0.176019	+	0.984387i	$0.443678\pi$
$978$	20.2432	0.647306
$979$	−2.41544	−0.0771977
$980$	0	0
$981$	20.0766	0.640996
$982$	24.3680	0.777613
$983$	0.758657	0.0241974	0.0120987	−	0.999927i	$-0.496149\pi$
$983$	0.758657	0.0241974	0.0120987	+	0.999927i	$0.496149\pi$
$984$	−3.14530	−0.100268
$985$	52.4154	1.67009
$986$	−21.1532	−0.673656
$987$	0	0
$988$	0.845500	0.0268989
$989$	10.3803	0.330075
$990$	9.67191	0.307394
$991$	−33.8685	−1.07587	−0.537934	−	0.842987i	$-0.680795\pi$
$991$	−33.8685	−1.07587	−0.537934	+	0.842987i	$0.680795\pi$
$992$	1.59514	0.0506456
$993$	23.2662	0.738330
$994$	0	0
$995$	−58.5098	−1.85488
$996$	−0.890670	−0.0282220
$997$	40.6123	1.28621	0.643103	−	0.765780i	$-0.277647\pi$
$997$	40.6123	1.28621	0.643103	+	0.765780i	$0.277647\pi$
$998$	−30.1883	−0.955595
$999$	0.807580	0.0255507

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bf.1.6		8
7.2	even	3		483.2.i.g.277.3	✓	16
7.4	even	3		483.2.i.g.415.3	yes	16
7.6	odd	2		3381.2.a.be.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.i.g.277.3	✓	16	7.2	even	3
483.2.i.g.415.3	yes	16	7.4	even	3
3381.2.a.be.1.6		8	7.6	odd	2
3381.2.a.bf.1.6		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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q+1.47391 q^{2} +1.00000 q^{3} +0.172415 q^{4} +3.62341 q^{5} +1.47391 q^{6} -2.69370 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.47391 q^{2} +1.00000 q^{3} +0.172415 q^{4} +3.62341 q^{5} +1.47391 q^{6} -2.69370 q^{8} +1.00000 q^{9} +5.34058 q^{10} +1.81102 q^{11} +0.172415 q^{12} +3.13385 q^{13} +3.62341 q^{15} -4.31510 q^{16} +3.47040 q^{17} +1.47391 q^{18} +1.56480 q^{19} +0.624731 q^{20} +2.66929 q^{22} -1.00000 q^{23} -2.69370 q^{24} +8.12908 q^{25} +4.61901 q^{26} +1.00000 q^{27} -4.13548 q^{29} +5.34058 q^{30} -1.63993 q^{31} -0.972685 q^{32} +1.81102 q^{33} +5.11506 q^{34} +0.172415 q^{36} +0.807580 q^{37} +2.30638 q^{38} +3.13385 q^{39} -9.76037 q^{40} +1.16765 q^{41} -10.3803 q^{43} +0.312248 q^{44} +3.62341 q^{45} -1.47391 q^{46} +7.98144 q^{47} -4.31510 q^{48} +11.9815 q^{50} +3.47040 q^{51} +0.540323 q^{52} +2.16670 q^{53} +1.47391 q^{54} +6.56207 q^{55} +1.56480 q^{57} -6.09533 q^{58} -9.03647 q^{59} +0.624731 q^{60} -10.0787 q^{61} -2.41711 q^{62} +7.19656 q^{64} +11.3552 q^{65} +2.66929 q^{66} +6.40398 q^{67} +0.598350 q^{68} -1.00000 q^{69} +13.3495 q^{71} -2.69370 q^{72} -6.92778 q^{73} +1.19030 q^{74} +8.12908 q^{75} +0.269796 q^{76} +4.61901 q^{78} +16.7678 q^{79} -15.6354 q^{80} +1.00000 q^{81} +1.72101 q^{82} -5.16584 q^{83} +12.5747 q^{85} -15.2997 q^{86} -4.13548 q^{87} -4.87835 q^{88} -1.33374 q^{89} +5.34058 q^{90} -0.172415 q^{92} -1.63993 q^{93} +11.7639 q^{94} +5.66992 q^{95} -0.972685 q^{96} -3.85363 q^{97} +1.81102 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + q^2 + 8 * q^3 + 15 * q^4 + 5 * q^5 + q^6 + 9 * q^8 + 8 * q^9	\( 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9} - 3 q^{10} + 10 q^{11} + 15 q^{12} - 6 q^{13} + 5 q^{15} + 13 q^{16} + 21 q^{17} + q^{18} + 5 q^{19} - q^{20} + 18 q^{22} - 8 q^{23} + 9 q^{24} + 27 q^{25} + 3 q^{26} + 8 q^{27} + 2 q^{29} - 3 q^{30} - 13 q^{31} + 29 q^{32} + 10 q^{33} - 19 q^{34} + 15 q^{36} + 13 q^{37} - 6 q^{38} - 6 q^{39} + 7 q^{40} + 16 q^{41} + 15 q^{43} + 24 q^{44} + 5 q^{45} - q^{46} - q^{47} + 13 q^{48} + 16 q^{50} + 21 q^{51} - 19 q^{52} + 3 q^{53} + q^{54} - 10 q^{55} + 5 q^{57} - 40 q^{58} + 26 q^{59} - q^{60} - 14 q^{61} - 14 q^{62} + 49 q^{64} - 3 q^{65} + 18 q^{66} + 38 q^{67} + 43 q^{68} - 8 q^{69} + 9 q^{71} + 9 q^{72} - 6 q^{73} + 32 q^{74} + 27 q^{75} - 14 q^{76} + 3 q^{78} + 23 q^{79} - 17 q^{80} + 8 q^{81} + 20 q^{82} + 30 q^{83} - 37 q^{85} - 28 q^{86} + 2 q^{87} + 86 q^{88} + 12 q^{89} - 3 q^{90} - 15 q^{92} - 13 q^{93} - 45 q^{94} + 16 q^{95} + 29 q^{96} + 14 q^{97} + 10 q^{99}+O(q^{100}) \) 8 * q + q^2 + 8 * q^3 + 15 * q^4 + 5 * q^5 + q^6 + 9 * q^8 + 8 * q^9 - 3 * q^10 + 10 * q^11 + 15 * q^12 - 6 * q^13 + 5 * q^15 + 13 * q^16 + 21 * q^17 + q^18 + 5 * q^19 - q^20 + 18 * q^22 - 8 * q^23 + 9 * q^24 + 27 * q^25 + 3 * q^26 + 8 * q^27 + 2 * q^29 - 3 * q^30 - 13 * q^31 + 29 * q^32 + 10 * q^33 - 19 * q^34 + 15 * q^36 + 13 * q^37 - 6 * q^38 - 6 * q^39 + 7 * q^40 + 16 * q^41 + 15 * q^43 + 24 * q^44 + 5 * q^45 - q^46 - q^47 + 13 * q^48 + 16 * q^50 + 21 * q^51 - 19 * q^52 + 3 * q^53 + q^54 - 10 * q^55 + 5 * q^57 - 40 * q^58 + 26 * q^59 - q^60 - 14 * q^61 - 14 * q^62 + 49 * q^64 - 3 * q^65 + 18 * q^66 + 38 * q^67 + 43 * q^68 - 8 * q^69 + 9 * q^71 + 9 * q^72 - 6 * q^73 + 32 * q^74 + 27 * q^75 - 14 * q^76 + 3 * q^78 + 23 * q^79 - 17 * q^80 + 8 * q^81 + 20 * q^82 + 30 * q^83 - 37 * q^85 - 28 * q^86 + 2 * q^87 + 86 * q^88 + 12 * q^89 - 3 * q^90 - 15 * q^92 - 13 * q^93 - 45 * q^94 + 16 * q^95 + 29 * q^96 + 14 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

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