LMFDB - Embedded newform 2667.2.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2667,2,Mod(1,2667)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2667, 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("2667.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2667 = 3 \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2667.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:	$21.2961022191$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} + \cdots - 8 $ x^13 - 4x^12 - 10x^11 + 53x^10 + 19x^9 - 242x^8 + 61x^7 + 467x^6 - 211x^5 - 372x^4 + 146x^3 + 116x^2 - 12x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.910949$ of defining polynomial
Character	$\chi$	$=$	2667.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.910949 q^{2} -1.00000 q^{3} -1.17017 q^{4} -2.06751 q^{5} +0.910949 q^{6} +1.00000 q^{7} +2.88787 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.910949 q^{2} -1.00000 q^{3} -1.17017 q^{4} -2.06751 q^{5} +0.910949 q^{6} +1.00000 q^{7} +2.88787 q^{8} +1.00000 q^{9} +1.88339 q^{10} +4.76209 q^{11} +1.17017 q^{12} +2.03773 q^{13} -0.910949 q^{14} +2.06751 q^{15} -0.290355 q^{16} +0.854179 q^{17} -0.910949 q^{18} +2.23714 q^{19} +2.41934 q^{20} -1.00000 q^{21} -4.33802 q^{22} +5.99633 q^{23} -2.88787 q^{24} -0.725414 q^{25} -1.85627 q^{26} -1.00000 q^{27} -1.17017 q^{28} -5.46515 q^{29} -1.88339 q^{30} +3.43070 q^{31} -5.51123 q^{32} -4.76209 q^{33} -0.778113 q^{34} -2.06751 q^{35} -1.17017 q^{36} +1.22426 q^{37} -2.03792 q^{38} -2.03773 q^{39} -5.97068 q^{40} -7.05781 q^{41} +0.910949 q^{42} -5.15064 q^{43} -5.57246 q^{44} -2.06751 q^{45} -5.46235 q^{46} +7.66065 q^{47} +0.290355 q^{48} +1.00000 q^{49} +0.660815 q^{50} -0.854179 q^{51} -2.38450 q^{52} -1.39018 q^{53} +0.910949 q^{54} -9.84565 q^{55} +2.88787 q^{56} -2.23714 q^{57} +4.97848 q^{58} +8.77497 q^{59} -2.41934 q^{60} +6.04889 q^{61} -3.12519 q^{62} +1.00000 q^{63} +5.60116 q^{64} -4.21303 q^{65} +4.33802 q^{66} -5.42409 q^{67} -0.999536 q^{68} -5.99633 q^{69} +1.88339 q^{70} -1.57089 q^{71} +2.88787 q^{72} -9.67049 q^{73} -1.11524 q^{74} +0.725414 q^{75} -2.61784 q^{76} +4.76209 q^{77} +1.85627 q^{78} -4.49693 q^{79} +0.600311 q^{80} +1.00000 q^{81} +6.42930 q^{82} +8.88129 q^{83} +1.17017 q^{84} -1.76602 q^{85} +4.69197 q^{86} +5.46515 q^{87} +13.7523 q^{88} +10.1548 q^{89} +1.88339 q^{90} +2.03773 q^{91} -7.01673 q^{92} -3.43070 q^{93} -6.97846 q^{94} -4.62530 q^{95} +5.51123 q^{96} +1.09551 q^{97} -0.910949 q^{98} +4.76209 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 4 q^{2} - 13 q^{3} + 10 q^{4} + 12 q^{5} - 4 q^{6} + 13 q^{7} + 9 q^{8} + 13 q^{9}+O(q^{10}) $ 13 * q + 4 * q^2 - 13 * q^3 + 10 * q^4 + 12 * q^5 - 4 * q^6 + 13 * q^7 + 9 * q^8 + 13 * q^9	\( 13 q + 4 q^{2} - 13 q^{3} + 10 q^{4} + 12 q^{5} - 4 q^{6} + 13 q^{7} + 9 q^{8} + 13 q^{9} + 6 q^{10} + 3 q^{11} - 10 q^{12} + 21 q^{13} + 4 q^{14} - 12 q^{15} + 8 q^{16} + 17 q^{17} + 4 q^{18} + 5 q^{19} + 29 q^{20} - 13 q^{21} + q^{22} + 4 q^{23} - 9 q^{24} + q^{25} + 22 q^{26} - 13 q^{27} + 10 q^{28} + 21 q^{29} - 6 q^{30} - 7 q^{31} + 12 q^{32} - 3 q^{33} + 2 q^{34} + 12 q^{35} + 10 q^{36} + 7 q^{37} - 9 q^{38} - 21 q^{39} + 29 q^{40} + 21 q^{41} - 4 q^{42} - 9 q^{43} - 2 q^{44} + 12 q^{45} - 28 q^{46} + 23 q^{47} - 8 q^{48} + 13 q^{49} + 15 q^{50} - 17 q^{51} + 15 q^{52} + 31 q^{53} - 4 q^{54} - 8 q^{55} + 9 q^{56} - 5 q^{57} - 25 q^{58} + 28 q^{59} - 29 q^{60} + 29 q^{61} - 3 q^{62} + 13 q^{63} + 9 q^{64} + 30 q^{65} - q^{66} - 18 q^{67} + 34 q^{68} - 4 q^{69} + 6 q^{70} + 10 q^{71} + 9 q^{72} + 24 q^{73} - 19 q^{74} - q^{75} + 3 q^{77} - 22 q^{78} - 28 q^{79} + 26 q^{80} + 13 q^{81} + 18 q^{82} + 26 q^{83} - 10 q^{84} + 20 q^{85} - 2 q^{86} - 21 q^{87} - 17 q^{88} + 44 q^{89} + 6 q^{90} + 21 q^{91} + 6 q^{92} + 7 q^{93} - 9 q^{94} - 2 q^{95} - 12 q^{96} + 17 q^{97} + 4 q^{98} + 3 q^{99}+O(q^{100}) \) 13 * q + 4 * q^2 - 13 * q^3 + 10 * q^4 + 12 * q^5 - 4 * q^6 + 13 * q^7 + 9 * q^8 + 13 * q^9 + 6 * q^10 + 3 * q^11 - 10 * q^12 + 21 * q^13 + 4 * q^14 - 12 * q^15 + 8 * q^16 + 17 * q^17 + 4 * q^18 + 5 * q^19 + 29 * q^20 - 13 * q^21 + q^22 + 4 * q^23 - 9 * q^24 + q^25 + 22 * q^26 - 13 * q^27 + 10 * q^28 + 21 * q^29 - 6 * q^30 - 7 * q^31 + 12 * q^32 - 3 * q^33 + 2 * q^34 + 12 * q^35 + 10 * q^36 + 7 * q^37 - 9 * q^38 - 21 * q^39 + 29 * q^40 + 21 * q^41 - 4 * q^42 - 9 * q^43 - 2 * q^44 + 12 * q^45 - 28 * q^46 + 23 * q^47 - 8 * q^48 + 13 * q^49 + 15 * q^50 - 17 * q^51 + 15 * q^52 + 31 * q^53 - 4 * q^54 - 8 * q^55 + 9 * q^56 - 5 * q^57 - 25 * q^58 + 28 * q^59 - 29 * q^60 + 29 * q^61 - 3 * q^62 + 13 * q^63 + 9 * q^64 + 30 * q^65 - q^66 - 18 * q^67 + 34 * q^68 - 4 * q^69 + 6 * q^70 + 10 * q^71 + 9 * q^72 + 24 * q^73 - 19 * q^74 - q^75 + 3 * q^77 - 22 * q^78 - 28 * q^79 + 26 * q^80 + 13 * q^81 + 18 * q^82 + 26 * q^83 - 10 * q^84 + 20 * q^85 - 2 * q^86 - 21 * q^87 - 17 * q^88 + 44 * q^89 + 6 * q^90 + 21 * q^91 + 6 * q^92 + 7 * q^93 - 9 * q^94 - 2 * q^95 - 12 * q^96 + 17 * q^97 + 4 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.910949	−0.644138	−0.322069	−	0.946716i	$-0.604378\pi$
$2$	−0.910949	−0.644138	−0.322069	+	0.946716i	$0.604378\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.17017	−0.585086
$5$	−2.06751	−0.924617	−0.462309	−	0.886719i	$-0.652979\pi$
$5$	−2.06751	−0.924617	−0.462309	+	0.886719i	$0.652979\pi$
$6$	0.910949	0.371893
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.88787	1.02101
$9$	1.00000	0.333333
$10$	1.88339	0.595581
$11$	4.76209	1.43582	0.717912	−	0.696134i	$-0.245098\pi$
$11$	4.76209	1.43582	0.717912	+	0.696134i	$0.245098\pi$
$12$	1.17017	0.337799
$13$	2.03773	0.565166	0.282583	−	0.959243i	$-0.408809\pi$
$13$	2.03773	0.565166	0.282583	+	0.959243i	$0.408809\pi$
$14$	−0.910949	−0.243461
$15$	2.06751	0.533828
$16$	−0.290355	−0.0725888
$17$	0.854179	0.207169	0.103584	−	0.994621i	$-0.466969\pi$
$17$	0.854179	0.207169	0.103584	+	0.994621i	$0.466969\pi$
$18$	−0.910949	−0.214713
$19$	2.23714	0.513235	0.256617	−	0.966513i	$-0.417392\pi$
$19$	2.23714	0.513235	0.256617	+	0.966513i	$0.417392\pi$
$20$	2.41934	0.540980
$21$	−1.00000	−0.218218
$22$	−4.33802	−0.924869
$23$	5.99633	1.25032	0.625160	−	0.780496i	$-0.285034\pi$
$23$	5.99633	1.25032	0.625160	+	0.780496i	$0.285034\pi$
$24$	−2.88787	−0.589483
$25$	−0.725414	−0.145083
$26$	−1.85627	−0.364045
$27$	−1.00000	−0.192450
$28$	−1.17017	−0.221142
$29$	−5.46515	−1.01485	−0.507427	−	0.861695i	$-0.669403\pi$
$29$	−5.46515	−1.01485	−0.507427	+	0.861695i	$0.669403\pi$
$30$	−1.88339	−0.343859
$31$	3.43070	0.616172	0.308086	−	0.951358i	$-0.400312\pi$
$31$	3.43070	0.616172	0.308086	+	0.951358i	$0.400312\pi$
$32$	−5.51123	−0.974257
$33$	−4.76209	−0.828973
$34$	−0.778113	−0.133445
$35$	−2.06751	−0.349472
$36$	−1.17017	−0.195029
$37$	1.22426	0.201268	0.100634	−	0.994924i	$-0.467913\pi$
$37$	1.22426	0.201268	0.100634	+	0.994924i	$0.467913\pi$
$38$	−2.03792	−0.330594
$39$	−2.03773	−0.326299
$40$	−5.97068	−0.944048
$41$	−7.05781	−1.10224	−0.551122	−	0.834425i	$-0.685800\pi$
$41$	−7.05781	−1.10224	−0.551122	+	0.834425i	$0.685800\pi$
$42$	0.910949	0.140563
$43$	−5.15064	−0.785465	−0.392732	−	0.919653i	$-0.628470\pi$
$43$	−5.15064	−0.785465	−0.392732	+	0.919653i	$0.628470\pi$
$44$	−5.57246	−0.840080
$45$	−2.06751	−0.308206
$46$	−5.46235	−0.805380
$47$	7.66065	1.11742	0.558710	−	0.829363i	$-0.311296\pi$
$47$	7.66065	1.11742	0.558710	+	0.829363i	$0.311296\pi$
$48$	0.290355	0.0419091
$49$	1.00000	0.142857
$50$	0.660815	0.0934534
$51$	−0.854179	−0.119609
$52$	−2.38450	−0.330670
$53$	−1.39018	−0.190956	−0.0954779	−	0.995432i	$-0.530438\pi$
$53$	−1.39018	−0.190956	−0.0954779	+	0.995432i	$0.530438\pi$
$54$	0.910949	0.123964
$55$	−9.84565	−1.32759
$56$	2.88787	0.385907
$57$	−2.23714	−0.296316
$58$	4.97848	0.653706
$59$	8.77497	1.14240	0.571202	−	0.820809i	$-0.306477\pi$
$59$	8.77497	1.14240	0.571202	+	0.820809i	$0.306477\pi$
$60$	−2.41934	−0.312335
$61$	6.04889	0.774481	0.387240	−	0.921979i	$-0.373428\pi$
$61$	6.04889	0.774481	0.387240	+	0.921979i	$0.373428\pi$
$62$	−3.12519	−0.396900
$63$	1.00000	0.125988
$64$	5.60116	0.700145
$65$	−4.21303	−0.522562
$66$	4.33802	0.533973
$67$	−5.42409	−0.662658	−0.331329	−	0.943515i	$-0.607497\pi$
$67$	−5.42409	−0.662658	−0.331329	+	0.943515i	$0.607497\pi$
$68$	−0.999536	−0.121212
$69$	−5.99633	−0.721873
$70$	1.88339	0.225109
$71$	−1.57089	−0.186431	−0.0932153	−	0.995646i	$-0.529714\pi$
$71$	−1.57089	−0.186431	−0.0932153	+	0.995646i	$0.529714\pi$
$72$	2.88787	0.340338
$73$	−9.67049	−1.13185	−0.565923	−	0.824458i	$-0.691480\pi$
$73$	−9.67049	−1.13185	−0.565923	+	0.824458i	$0.691480\pi$
$74$	−1.11524	−0.129644
$75$	0.725414	0.0837636
$76$	−2.61784	−0.300287
$77$	4.76209	0.542690
$78$	1.85627	0.210181
$79$	−4.49693	−0.505944	−0.252972	−	0.967474i	$-0.581408\pi$
$79$	−4.49693	−0.505944	−0.252972	+	0.967474i	$0.581408\pi$
$80$	0.600311	0.0671168
$81$	1.00000	0.111111
$82$	6.42930	0.709998
$83$	8.88129	0.974848	0.487424	−	0.873165i	$-0.337937\pi$
$83$	8.88129	0.974848	0.487424	+	0.873165i	$0.337937\pi$
$84$	1.17017	0.127676
$85$	−1.76602	−0.191552
$86$	4.69197	0.505948
$87$	5.46515	0.585926
$88$	13.7523	1.46600
$89$	10.1548	1.07641	0.538204	−	0.842815i	$-0.319103\pi$
$89$	10.1548	1.07641	0.538204	+	0.842815i	$0.319103\pi$
$90$	1.88339	0.198527
$91$	2.03773	0.213613
$92$	−7.01673	−0.731545
$93$	−3.43070	−0.355747
$94$	−6.97846	−0.719773
$95$	−4.62530	−0.474546
$96$	5.51123	0.562488
$97$	1.09551	0.111233	0.0556163	−	0.998452i	$-0.482288\pi$
$97$	1.09551	0.111233	0.0556163	+	0.998452i	$0.482288\pi$
$98$	−0.910949	−0.0920198
$99$	4.76209	0.478608
$100$	0.848859	0.0848859
$101$	3.53703	0.351948	0.175974	−	0.984395i	$-0.443693\pi$
$101$	3.53703	0.351948	0.175974	+	0.984395i	$0.443693\pi$
$102$	0.778113	0.0770447
$103$	1.77000	0.174404	0.0872018	−	0.996191i	$-0.472208\pi$
$103$	1.77000	0.174404	0.0872018	+	0.996191i	$0.472208\pi$
$104$	5.88470	0.577042
$105$	2.06751	0.201768
$106$	1.26638	0.123002
$107$	11.4134	1.10338	0.551689	−	0.834050i	$-0.313983\pi$
$107$	11.4134	1.10338	0.551689	+	0.834050i	$0.313983\pi$
$108$	1.17017	0.112600
$109$	−15.2946	−1.46496	−0.732480	−	0.680788i	$-0.761637\pi$
$109$	−15.2946	−1.46496	−0.732480	+	0.680788i	$0.761637\pi$
$110$	8.96889	0.855150
$111$	−1.22426	−0.116202
$112$	−0.290355	−0.0274360
$113$	−15.9081	−1.49651	−0.748254	−	0.663413i	$-0.769107\pi$
$113$	−15.9081	−1.49651	−0.748254	+	0.663413i	$0.769107\pi$
$114$	2.03792	0.190869
$115$	−12.3975	−1.15607
$116$	6.39517	0.593776
$117$	2.03773	0.188389
$118$	−7.99355	−0.735866
$119$	0.854179	0.0783024
$120$	5.97068	0.545046
$121$	11.6775	1.06159
$122$	−5.51023	−0.498873
$123$	7.05781	0.636381
$124$	−4.01451	−0.360514
$125$	11.8373	1.05876
$126$	−0.910949	−0.0811538
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	5.92009	0.523267
$129$	5.15064	0.453488
$130$	3.83786	0.336602
$131$	14.7361	1.28750	0.643749	−	0.765237i	$-0.277378\pi$
$131$	14.7361	1.28750	0.643749	+	0.765237i	$0.277378\pi$
$132$	5.57246	0.485020
$133$	2.23714	0.193985
$134$	4.94107	0.426844
$135$	2.06751	0.177943
$136$	2.46675	0.211522
$137$	7.61686	0.650752	0.325376	−	0.945585i	$-0.394509\pi$
$137$	7.61686	0.650752	0.325376	+	0.945585i	$0.394509\pi$
$138$	5.46235	0.464986
$139$	5.10352	0.432875	0.216437	−	0.976296i	$-0.430556\pi$
$139$	5.10352	0.432875	0.216437	+	0.976296i	$0.430556\pi$
$140$	2.41934	0.204471
$141$	−7.66065	−0.645143
$142$	1.43100	0.120087
$143$	9.70387	0.811478
$144$	−0.290355	−0.0241963
$145$	11.2992	0.938351
$146$	8.80933	0.729065
$147$	−1.00000	−0.0824786
$148$	−1.43260	−0.117759
$149$	−0.345030	−0.0282659	−0.0141330	−	0.999900i	$-0.504499\pi$
$149$	−0.345030	−0.0282659	−0.0141330	+	0.999900i	$0.504499\pi$
$150$	−0.660815	−0.0539553
$151$	6.54380	0.532527	0.266264	−	0.963900i	$-0.414211\pi$
$151$	6.54380	0.532527	0.266264	+	0.963900i	$0.414211\pi$
$152$	6.46056	0.524020
$153$	0.854179	0.0690563
$154$	−4.33802	−0.349568
$155$	−7.09300	−0.569724
$156$	2.38450	0.190913
$157$	5.25736	0.419583	0.209792	−	0.977746i	$-0.432721\pi$
$157$	5.25736	0.419583	0.209792	+	0.977746i	$0.432721\pi$
$158$	4.09647	0.325898
$159$	1.39018	0.110248
$160$	11.3945	0.900815
$161$	5.99633	0.472577
$162$	−0.910949	−0.0715709
$163$	−4.27474	−0.334824	−0.167412	−	0.985887i	$-0.553541\pi$
$163$	−4.27474	−0.334824	−0.167412	+	0.985887i	$0.553541\pi$
$164$	8.25885	0.644907
$165$	9.84565	0.766483
$166$	−8.09040	−0.627937
$167$	15.9971	1.23790	0.618948	−	0.785432i	$-0.287559\pi$
$167$	15.9971	1.23790	0.618948	+	0.785432i	$0.287559\pi$
$168$	−2.88787	−0.222804
$169$	−8.84764	−0.680588
$170$	1.60876	0.123386
$171$	2.23714	0.171078
$172$	6.02713	0.459564
$173$	7.16537	0.544773	0.272387	−	0.962188i	$-0.412187\pi$
$173$	7.16537	0.544773	0.272387	+	0.962188i	$0.412187\pi$
$174$	−4.97848	−0.377417
$175$	−0.725414	−0.0548361
$176$	−1.38270	−0.104225
$177$	−8.77497	−0.659567
$178$	−9.25051	−0.693355
$179$	0.409220	0.0305866	0.0152933	−	0.999883i	$-0.495132\pi$
$179$	0.409220	0.0305866	0.0152933	+	0.999883i	$0.495132\pi$
$180$	2.41934	0.180327
$181$	−10.5918	−0.787279	−0.393639	−	0.919265i	$-0.628784\pi$
$181$	−10.5918	−0.787279	−0.393639	+	0.919265i	$0.628784\pi$
$182$	−1.85627	−0.137596
$183$	−6.04889	−0.447147
$184$	17.3166	1.27660
$185$	−2.53117	−0.186096
$186$	3.12519	0.229150
$187$	4.06767	0.297458
$188$	−8.96427	−0.653787
$189$	−1.00000	−0.0727393
$190$	4.21341	0.305673
$191$	14.3830	1.04072	0.520358	−	0.853948i	$-0.325799\pi$
$191$	14.3830	1.04072	0.520358	+	0.853948i	$0.325799\pi$
$192$	−5.60116	−0.404229
$193$	4.91728	0.353954	0.176977	−	0.984215i	$-0.443368\pi$
$193$	4.91728	0.353954	0.176977	+	0.984215i	$0.443368\pi$
$194$	−0.997957	−0.0716491
$195$	4.21303	0.301701
$196$	−1.17017	−0.0835837
$197$	−18.4666	−1.31569	−0.657846	−	0.753152i	$-0.728532\pi$
$197$	−18.4666	−1.31569	−0.657846	+	0.753152i	$0.728532\pi$
$198$	−4.33802	−0.308290
$199$	−15.1069	−1.07090	−0.535450	−	0.844567i	$-0.679858\pi$
$199$	−15.1069	−1.07090	−0.535450	+	0.844567i	$0.679858\pi$
$200$	−2.09490	−0.148132
$201$	5.42409	0.382586
$202$	−3.22206	−0.226703
$203$	−5.46515	−0.383578
$204$	0.999536	0.0699815
$205$	14.5921	1.01915
$206$	−1.61238	−0.112340
$207$	5.99633	0.416774
$208$	−0.591666	−0.0410247
$209$	10.6535	0.736915
$210$	−1.88339	−0.129967
$211$	8.27452	0.569641	0.284821	−	0.958581i	$-0.408066\pi$
$211$	8.27452	0.569641	0.284821	+	0.958581i	$0.408066\pi$
$212$	1.62675	0.111726
$213$	1.57089	0.107636
$214$	−10.3971	−0.710728
$215$	10.6490	0.726254
$216$	−2.88787	−0.196494
$217$	3.43070	0.232891
$218$	13.9326	0.943637
$219$	9.67049	0.653471
$220$	11.5211	0.776753
$221$	1.74059	0.117085
$222$	1.11524	0.0748501
$223$	9.20017	0.616089	0.308044	−	0.951372i	$-0.400325\pi$
$223$	9.20017	0.616089	0.308044	+	0.951372i	$0.400325\pi$
$224$	−5.51123	−0.368235
$225$	−0.725414	−0.0483609
$226$	14.4915	0.963958
$227$	4.61206	0.306113	0.153057	−	0.988217i	$-0.451088\pi$
$227$	4.61206	0.306113	0.153057	+	0.988217i	$0.451088\pi$
$228$	2.61784	0.173370
$229$	8.47798	0.560240	0.280120	−	0.959965i	$-0.409626\pi$
$229$	8.47798	0.560240	0.280120	+	0.959965i	$0.409626\pi$
$230$	11.2934	0.744668
$231$	−4.76209	−0.313322
$232$	−15.7826	−1.03618
$233$	−11.9704	−0.784209	−0.392104	−	0.919921i	$-0.628253\pi$
$233$	−11.9704	−0.784209	−0.392104	+	0.919921i	$0.628253\pi$
$234$	−1.85627	−0.121348
$235$	−15.8384	−1.03319
$236$	−10.2682	−0.668404
$237$	4.49693	0.292107
$238$	−0.778113	−0.0504376
$239$	−20.9024	−1.35206	−0.676031	−	0.736873i	$-0.736301\pi$
$239$	−20.9024	−1.35206	−0.676031	+	0.736873i	$0.736301\pi$
$240$	−0.600311	−0.0387499
$241$	5.81959	0.374873	0.187436	−	0.982277i	$-0.439982\pi$
$241$	5.81959	0.374873	0.187436	+	0.982277i	$0.439982\pi$
$242$	−10.6376	−0.683810
$243$	−1.00000	−0.0641500
$244$	−7.07824	−0.453138
$245$	−2.06751	−0.132088
$246$	−6.42930	−0.409917
$247$	4.55869	0.290063
$248$	9.90740	0.629121
$249$	−8.88129	−0.562829
$250$	−10.7832	−0.681990
$251$	0.00650335	0.000410488 0	0.000205244	−	1.00000i	$-0.499935\pi$
$251$	0.00650335	0.000410488 0	0.000205244		1.00000i	$0.499935\pi$
$252$	−1.17017	−0.0737139
$253$	28.5550	1.79524
$254$	0.910949	0.0571580
$255$	1.76602	0.110593
$256$	−16.5952	−1.03720
$257$	21.6261	1.34900	0.674498	−	0.738276i	$-0.264360\pi$
$257$	21.6261	1.34900	0.674498	+	0.738276i	$0.264360\pi$
$258$	−4.69197	−0.292109
$259$	1.22426	0.0760721
$260$	4.92997	0.305744
$261$	−5.46515	−0.338284
$262$	−13.4238	−0.829326
$263$	6.14267	0.378773	0.189386	−	0.981903i	$-0.439350\pi$
$263$	6.14267	0.378773	0.189386	+	0.981903i	$0.439350\pi$
$264$	−13.7523	−0.846394
$265$	2.87421	0.176561
$266$	−2.03792	−0.124953
$267$	−10.1548	−0.621464
$268$	6.34712	0.387712
$269$	−20.1163	−1.22651	−0.613255	−	0.789885i	$-0.710140\pi$
$269$	−20.1163	−1.22651	−0.613255	+	0.789885i	$0.710140\pi$
$270$	−1.88339	−0.114620
$271$	−13.3909	−0.813440	−0.406720	−	0.913553i	$-0.633327\pi$
$271$	−13.3909	−0.813440	−0.406720	+	0.913553i	$0.633327\pi$
$272$	−0.248015	−0.0150381
$273$	−2.03773	−0.123329
$274$	−6.93857	−0.419174
$275$	−3.45449	−0.208313
$276$	7.01673	0.422358
$277$	8.05697	0.484096	0.242048	−	0.970264i	$-0.422181\pi$
$277$	8.05697	0.484096	0.242048	+	0.970264i	$0.422181\pi$
$278$	−4.64905	−0.278831
$279$	3.43070	0.205391
$280$	−5.97068	−0.356816
$281$	27.8734	1.66279	0.831395	−	0.555682i	$-0.187543\pi$
$281$	27.8734	1.66279	0.831395	+	0.555682i	$0.187543\pi$
$282$	6.97846	0.415561
$283$	−10.1043	−0.600639	−0.300320	−	0.953839i	$-0.597093\pi$
$283$	−10.1043	−0.600639	−0.300320	+	0.953839i	$0.597093\pi$
$284$	1.83821	0.109078
$285$	4.62530	0.273979
$286$	−8.83973	−0.522704
$287$	−7.05781	−0.416609
$288$	−5.51123	−0.324752
$289$	−16.2704	−0.957081
$290$	−10.2930	−0.604428
$291$	−1.09551	−0.0642201
$292$	11.3161	0.662227
$293$	5.16465	0.301722	0.150861	−	0.988555i	$-0.451795\pi$
$293$	5.16465	0.301722	0.150861	+	0.988555i	$0.451795\pi$
$294$	0.910949	0.0531276
$295$	−18.1423	−1.05629
$296$	3.53551	0.205497
$297$	−4.76209	−0.276324
$298$	0.314305	0.0182072
$299$	12.2189	0.706638
$300$	−0.848859	−0.0490089
$301$	−5.15064	−0.296878
$302$	−5.96107	−0.343021
$303$	−3.53703	−0.203197
$304$	−0.649565	−0.0372551
$305$	−12.5061	−0.716098
$306$	−0.778113	−0.0444818
$307$	3.03374	0.173144	0.0865722	−	0.996246i	$-0.472409\pi$
$307$	3.03374	0.173144	0.0865722	+	0.996246i	$0.472409\pi$
$308$	−5.57246	−0.317520
$309$	−1.77000	−0.100692
$310$	6.46136	0.366981
$311$	22.1371	1.25528	0.627640	−	0.778504i	$-0.284021\pi$
$311$	22.1371	1.25528	0.627640	+	0.778504i	$0.284021\pi$
$312$	−5.88470	−0.333156
$313$	9.66850	0.546496	0.273248	−	0.961944i	$-0.411902\pi$
$313$	9.66850	0.546496	0.273248	+	0.961944i	$0.411902\pi$
$314$	−4.78919	−0.270270
$315$	−2.06751	−0.116491
$316$	5.26218	0.296021
$317$	−9.91891	−0.557101	−0.278551	−	0.960422i	$-0.589854\pi$
$317$	−9.91891	−0.557101	−0.278551	+	0.960422i	$0.589854\pi$
$318$	−1.26638	−0.0710152
$319$	−26.0255	−1.45715
$320$	−11.5804	−0.647366
$321$	−11.4134	−0.637036
$322$	−5.46235	−0.304405
$323$	1.91092	0.106326
$324$	−1.17017	−0.0650095
$325$	−1.47820	−0.0819958
$326$	3.89408	0.215673
$327$	15.2946	0.845795
$328$	−20.3820	−1.12541
$329$	7.66065	0.422345
$330$	−8.96889	−0.493721
$331$	22.7606	1.25104	0.625518	−	0.780210i	$-0.284888\pi$
$331$	22.7606	1.25104	0.625518	+	0.780210i	$0.284888\pi$
$332$	−10.3926	−0.570370
$333$	1.22426	0.0670892
$334$	−14.5726	−0.797376
$335$	11.2143	0.612705
$336$	0.290355	0.0158402
$337$	16.9411	0.922840	0.461420	−	0.887182i	$-0.347340\pi$
$337$	16.9411	0.922840	0.461420	+	0.887182i	$0.347340\pi$
$338$	8.05975	0.438393
$339$	15.9081	0.864009
$340$	2.06655	0.112074
$341$	16.3373	0.884715
$342$	−2.03792	−0.110198
$343$	1.00000	0.0539949
$344$	−14.8743	−0.801971
$345$	12.3975	0.667456
$346$	−6.52729	−0.350909
$347$	35.6135	1.91183	0.955917	−	0.293636i	$-0.0948653\pi$
$347$	35.6135	1.91183	0.955917	+	0.293636i	$0.0948653\pi$
$348$	−6.39517	−0.342817
$349$	−0.202587	−0.0108443	−0.00542213	−	0.999985i	$-0.501726\pi$
$349$	−0.202587	−0.0108443	−0.00542213	+	0.999985i	$0.501726\pi$
$350$	0.660815	0.0353221
$351$	−2.03773	−0.108766
$352$	−26.2450	−1.39886
$353$	14.0659	0.748653	0.374327	−	0.927297i	$-0.377874\pi$
$353$	14.0659	0.748653	0.374327	+	0.927297i	$0.377874\pi$
$354$	7.99355	0.424853
$355$	3.24783	0.172377
$356$	−11.8829	−0.629791
$357$	−0.854179	−0.0452079
$358$	−0.372779	−0.0197020
$359$	−9.61467	−0.507443	−0.253722	−	0.967277i	$-0.581655\pi$
$359$	−9.61467	−0.507443	−0.253722	+	0.967277i	$0.581655\pi$
$360$	−5.97068	−0.314683
$361$	−13.9952	−0.736590
$362$	9.64855	0.507116
$363$	−11.6775	−0.612909
$364$	−2.38450	−0.124982
$365$	19.9938	1.04652
$366$	5.51023	0.288024
$367$	−23.8796	−1.24651	−0.623254	−	0.782020i	$-0.714190\pi$
$367$	−23.8796	−1.24651	−0.623254	+	0.782020i	$0.714190\pi$
$368$	−1.74106	−0.0907592
$369$	−7.05781	−0.367415
$370$	2.30577	0.119871
$371$	−1.39018	−0.0721745
$372$	4.01451	0.208143
$373$	19.0393	0.985816	0.492908	−	0.870081i	$-0.335934\pi$
$373$	19.0393	0.985816	0.492908	+	0.870081i	$0.335934\pi$
$374$	−3.70544	−0.191604
$375$	−11.8373	−0.611277
$376$	22.1229	1.14090
$377$	−11.1365	−0.573560
$378$	0.910949	0.0468542
$379$	−8.43150	−0.433097	−0.216548	−	0.976272i	$-0.569480\pi$
$379$	−8.43150	−0.433097	−0.216548	+	0.976272i	$0.569480\pi$
$380$	5.41240	0.277650
$381$	1.00000	0.0512316
$382$	−13.1022	−0.670365
$383$	−22.7597	−1.16296	−0.581482	−	0.813559i	$-0.697527\pi$
$383$	−22.7597	−1.16296	−0.581482	+	0.813559i	$0.697527\pi$
$384$	−5.92009	−0.302108
$385$	−9.84565	−0.501781
$386$	−4.47939	−0.227995
$387$	−5.15064	−0.261822
$388$	−1.28194	−0.0650806
$389$	15.4243	0.782041	0.391021	−	0.920382i	$-0.372122\pi$
$389$	15.4243	0.782041	0.391021	+	0.920382i	$0.372122\pi$
$390$	−3.83786	−0.194337
$391$	5.12194	0.259027
$392$	2.88787	0.145859
$393$	−14.7361	−0.743337
$394$	16.8222	0.847488
$395$	9.29743	0.467804
$396$	−5.57246	−0.280027
$397$	−7.28576	−0.365662	−0.182831	−	0.983144i	$-0.558526\pi$
$397$	−7.28576	−0.365662	−0.182831	+	0.983144i	$0.558526\pi$
$398$	13.7616	0.689807
$399$	−2.23714	−0.111997
$400$	0.210628	0.0105314
$401$	25.8458	1.29068	0.645340	−	0.763896i	$-0.276716\pi$
$401$	25.8458	1.29068	0.645340	+	0.763896i	$0.276716\pi$
$402$	−4.94107	−0.246438
$403$	6.99086	0.348239
$404$	−4.13894	−0.205920
$405$	−2.06751	−0.102735
$406$	4.97848	0.247078
$407$	5.83005	0.288985
$408$	−2.46675	−0.122122
$409$	−18.8712	−0.933120	−0.466560	−	0.884490i	$-0.654507\pi$
$409$	−18.8712	−0.933120	−0.466560	+	0.884490i	$0.654507\pi$
$410$	−13.2926	−0.656476
$411$	−7.61686	−0.375712
$412$	−2.07121	−0.102041
$413$	8.77497	0.431788
$414$	−5.46235	−0.268460
$415$	−18.3621	−0.901361
$416$	−11.2304	−0.550617
$417$	−5.10352	−0.249920
$418$	−9.70476	−0.474675
$419$	36.8364	1.79958	0.899788	−	0.436328i	$-0.143721\pi$
$419$	36.8364	1.79958	0.899788	+	0.436328i	$0.143721\pi$
$420$	−2.41934	−0.118052
$421$	1.66960	0.0813711	0.0406855	−	0.999172i	$-0.487046\pi$
$421$	1.66960	0.0813711	0.0406855	+	0.999172i	$0.487046\pi$
$422$	−7.53767	−0.366928
$423$	7.66065	0.372473
$424$	−4.01465	−0.194969
$425$	−0.619633	−0.0300566
$426$	−1.43100	−0.0693323
$427$	6.04889	0.292726
$428$	−13.3557	−0.645571
$429$	−9.70387	−0.468507
$430$	−9.70068	−0.467808
$431$	22.4575	1.08174	0.540869	−	0.841107i	$-0.318095\pi$
$431$	22.4575	1.08174	0.540869	+	0.841107i	$0.318095\pi$
$432$	0.290355	0.0139697
$433$	34.2436	1.64564	0.822822	−	0.568299i	$-0.192398\pi$
$433$	34.2436	1.64564	0.822822	+	0.568299i	$0.192398\pi$
$434$	−3.12519	−0.150014
$435$	−11.2992	−0.541757
$436$	17.8974	0.857128
$437$	13.4146	0.641708
$438$	−8.80933	−0.420926
$439$	21.2700	1.01516	0.507581	−	0.861604i	$-0.330540\pi$
$439$	21.2700	1.01516	0.507581	+	0.861604i	$0.330540\pi$
$440$	−28.4329	−1.35549
$441$	1.00000	0.0476190
$442$	−1.58559	−0.0754187
$443$	−30.7912	−1.46294	−0.731468	−	0.681876i	$-0.761165\pi$
$443$	−30.7912	−1.46294	−0.731468	+	0.681876i	$0.761165\pi$
$444$	1.43260	0.0679881
$445$	−20.9951	−0.995265
$446$	−8.38089	−0.396846
$447$	0.345030	0.0163194
$448$	5.60116	0.264630
$449$	26.8456	1.26692	0.633462	−	0.773774i	$-0.281633\pi$
$449$	26.8456	1.26692	0.633462	+	0.773774i	$0.281633\pi$
$450$	0.660815	0.0311511
$451$	−33.6099	−1.58263
$452$	18.6152	0.875585
$453$	−6.54380	−0.307455
$454$	−4.20135	−0.197179
$455$	−4.21303	−0.197510
$456$	−6.46056	−0.302543
$457$	2.07317	0.0969789	0.0484895	−	0.998824i	$-0.484559\pi$
$457$	2.07317	0.0969789	0.0484895	+	0.998824i	$0.484559\pi$
$458$	−7.72300	−0.360872
$459$	−0.854179	−0.0398697
$460$	14.5071	0.676399
$461$	37.8084	1.76091	0.880455	−	0.474129i	$-0.157237\pi$
$461$	37.8084	1.76091	0.880455	+	0.474129i	$0.157237\pi$
$462$	4.33802	0.201823
$463$	30.4977	1.41735	0.708674	−	0.705536i	$-0.249294\pi$
$463$	30.4977	1.41735	0.708674	+	0.705536i	$0.249294\pi$
$464$	1.58683	0.0736669
$465$	7.09300	0.328930
$466$	10.9045	0.505139
$467$	−7.92034	−0.366509	−0.183255	−	0.983065i	$-0.558663\pi$
$467$	−7.92034	−0.366509	−0.183255	+	0.983065i	$0.558663\pi$
$468$	−2.38450	−0.110223
$469$	−5.42409	−0.250461
$470$	14.4280	0.665515
$471$	−5.25736	−0.242247
$472$	25.3409	1.16641
$473$	−24.5278	−1.12779
$474$	−4.09647	−0.188157
$475$	−1.62285	−0.0744616
$476$	−0.999536	−0.0458137
$477$	−1.39018	−0.0636520
$478$	19.0410	0.870915
$479$	−34.0782	−1.55707	−0.778537	−	0.627599i	$-0.784038\pi$
$479$	−34.0782	−1.55707	−0.778537	+	0.627599i	$0.784038\pi$
$480$	−11.3945	−0.520086
$481$	2.49472	0.113750
$482$	−5.30135	−0.241470
$483$	−5.99633	−0.272842
$484$	−13.6647	−0.621121
$485$	−2.26498	−0.102848
$486$	0.910949	0.0413215
$487$	−10.7347	−0.486436	−0.243218	−	0.969972i	$-0.578203\pi$
$487$	−10.7347	−0.486436	−0.243218	+	0.969972i	$0.578203\pi$
$488$	17.4684	0.790756
$489$	4.27474	0.193311
$490$	1.88339	0.0850831
$491$	−35.6179	−1.60741	−0.803707	−	0.595025i	$-0.797142\pi$
$491$	−35.6179	−1.60741	−0.803707	+	0.595025i	$0.797142\pi$
$492$	−8.25885	−0.372337
$493$	−4.66822	−0.210246
$494$	−4.15274	−0.186841
$495$	−9.84565	−0.442529
$496$	−0.996122	−0.0447272
$497$	−1.57089	−0.0704641
$498$	8.09040	0.362539
$499$	−29.5279	−1.32185	−0.660925	−	0.750452i	$-0.729836\pi$
$499$	−29.5279	−1.32185	−0.660925	+	0.750452i	$0.729836\pi$
$500$	−13.8517	−0.619467
$501$	−15.9971	−0.714700
$502$	−0.00592422	−0.000264411 0
$503$	−21.5076	−0.958976	−0.479488	−	0.877549i	$-0.659178\pi$
$503$	−21.5076	−0.958976	−0.479488	+	0.877549i	$0.659178\pi$
$504$	2.88787	0.128636
$505$	−7.31284	−0.325417
$506$	−26.0122	−1.15638
$507$	8.84764	0.392938
$508$	1.17017	0.0519180
$509$	−20.3805	−0.903348	−0.451674	−	0.892183i	$-0.649173\pi$
$509$	−20.3805	−0.903348	−0.451674	+	0.892183i	$0.649173\pi$
$510$	−1.60876	−0.0712369
$511$	−9.67049	−0.427797
$512$	3.27723	0.144834
$513$	−2.23714	−0.0987721
$514$	−19.7002	−0.868941
$515$	−3.65949	−0.161257
$516$	−6.02713	−0.265330
$517$	36.4807	1.60442
$518$	−1.11524	−0.0490009
$519$	−7.16537	−0.314525
$520$	−12.1667	−0.533543
$521$	−35.3393	−1.54824	−0.774122	−	0.633037i	$-0.781808\pi$
$521$	−35.3393	−1.54824	−0.774122	+	0.633037i	$0.781808\pi$
$522$	4.97848	0.217902
$523$	41.3525	1.80822	0.904110	−	0.427300i	$-0.140535\pi$
$523$	41.3525	1.80822	0.904110	+	0.427300i	$0.140535\pi$
$524$	−17.2437	−0.753297
$525$	0.725414	0.0316597
$526$	−5.59566	−0.243982
$527$	2.93043	0.127652
$528$	1.38270	0.0601741
$529$	12.9559	0.563302
$530$	−2.61826	−0.113730
$531$	8.77497	0.380801
$532$	−2.61784	−0.113498
$533$	−14.3819	−0.622951
$534$	9.25051	0.400309
$535$	−23.5974	−1.02020
$536$	−15.6640	−0.676584
$537$	−0.409220	−0.0176592
$538$	18.3249	0.790042
$539$	4.76209	0.205118
$540$	−2.41934	−0.104112
$541$	25.4880	1.09582	0.547908	−	0.836539i	$-0.315424\pi$
$541$	25.4880	1.09582	0.547908	+	0.836539i	$0.315424\pi$
$542$	12.1984	0.523968
$543$	10.5918	0.454536
$544$	−4.70758	−0.201836
$545$	31.6218	1.35453
$546$	1.85627	0.0794411
$547$	30.4992	1.30405	0.652025	−	0.758198i	$-0.273920\pi$
$547$	30.4992	1.30405	0.652025	+	0.758198i	$0.273920\pi$
$548$	−8.91303	−0.380746
$549$	6.04889	0.258160
$550$	3.14686	0.134183
$551$	−12.2263	−0.520858
$552$	−17.3166	−0.737043
$553$	−4.49693	−0.191229
$554$	−7.33949	−0.311825
$555$	2.53117	0.107442
$556$	−5.97199	−0.253269
$557$	36.9759	1.56672	0.783361	−	0.621568i	$-0.213504\pi$
$557$	36.9759	1.56672	0.783361	+	0.621568i	$0.213504\pi$
$558$	−3.12519	−0.132300
$559$	−10.4956	−0.443918
$560$	0.600311	0.0253678
$561$	−4.06767	−0.171737
$562$	−25.3913	−1.07107
$563$	36.8641	1.55363	0.776817	−	0.629726i	$-0.216833\pi$
$563$	36.8641	1.55363	0.776817	+	0.629726i	$0.216833\pi$
$564$	8.96427	0.377464
$565$	32.8901	1.38370
$566$	9.20452	0.386895
$567$	1.00000	0.0419961
$568$	−4.53652	−0.190348
$569$	12.5246	0.525057	0.262529	−	0.964924i	$-0.415444\pi$
$569$	12.5246	0.525057	0.262529	+	0.964924i	$0.415444\pi$
$570$	−4.21341	−0.176481
$571$	13.9998	0.585875	0.292937	−	0.956132i	$-0.405367\pi$
$571$	13.9998	0.585875	0.292937	+	0.956132i	$0.405367\pi$
$572$	−11.3552	−0.474784
$573$	−14.3830	−0.600857
$574$	6.42930	0.268354
$575$	−4.34982	−0.181400
$576$	5.60116	0.233382
$577$	−31.7993	−1.32382	−0.661911	−	0.749582i	$-0.730254\pi$
$577$	−31.7993	−1.32382	−0.661911	+	0.749582i	$0.730254\pi$
$578$	14.8215	0.616493
$579$	−4.91728	−0.204355
$580$	−13.2221	−0.549016
$581$	8.88129	0.368458
$582$	0.997957	0.0413666
$583$	−6.62016	−0.274179
$584$	−27.9271	−1.15563
$585$	−4.21303	−0.174187
$586$	−4.70474	−0.194351
$587$	0.697457	0.0287871	0.0143936	−	0.999896i	$-0.495418\pi$
$587$	0.697457	0.0287871	0.0143936	+	0.999896i	$0.495418\pi$
$588$	1.17017	0.0482571
$589$	7.67496	0.316241
$590$	16.5267	0.680395
$591$	18.4666	0.759616
$592$	−0.355471	−0.0146098
$593$	8.07973	0.331795	0.165897	−	0.986143i	$-0.446948\pi$
$593$	8.07973	0.331795	0.165897	+	0.986143i	$0.446948\pi$
$594$	4.33802	0.177991
$595$	−1.76602	−0.0723998
$596$	0.403744	0.0165380
$597$	15.1069	0.618284
$598$	−11.1308	−0.455173
$599$	15.7153	0.642108	0.321054	−	0.947061i	$-0.395963\pi$
$599$	15.7153	0.642108	0.321054	+	0.947061i	$0.395963\pi$
$600$	2.09490	0.0855239
$601$	−40.1903	−1.63939	−0.819697	−	0.572797i	$-0.805858\pi$
$601$	−40.1903	−1.63939	−0.819697	+	0.572797i	$0.805858\pi$
$602$	4.69197	0.191230
$603$	−5.42409	−0.220886
$604$	−7.65737	−0.311574
$605$	−24.1433	−0.981564
$606$	3.22206	0.130887
$607$	−4.34490	−0.176354	−0.0881770	−	0.996105i	$-0.528104\pi$
$607$	−4.34490	−0.176354	−0.0881770	+	0.996105i	$0.528104\pi$
$608$	−12.3294	−0.500023
$609$	5.46515	0.221459
$610$	11.3924	0.461266
$611$	15.6104	0.631528
$612$	−0.999536	−0.0404038
$613$	16.5086	0.666775	0.333388	−	0.942790i	$-0.391808\pi$
$613$	16.5086	0.666775	0.333388	+	0.942790i	$0.391808\pi$
$614$	−2.76358	−0.111529
$615$	−14.5921	−0.588409
$616$	13.7523	0.554095
$617$	−35.3853	−1.42456	−0.712280	−	0.701896i	$-0.752337\pi$
$617$	−35.3853	−1.42456	−0.712280	+	0.701896i	$0.752337\pi$
$618$	1.61238	0.0648595
$619$	5.41450	0.217627	0.108814	−	0.994062i	$-0.465295\pi$
$619$	5.41450	0.217627	0.108814	+	0.994062i	$0.465295\pi$
$620$	8.30003	0.333337
$621$	−5.99633	−0.240624
$622$	−20.1658	−0.808574
$623$	10.1548	0.406844
$624$	0.591666	0.0236856
$625$	−20.8467	−0.833868
$626$	−8.80751	−0.352019
$627$	−10.6535	−0.425458
$628$	−6.15202	−0.245492
$629$	1.04574	0.0416964
$630$	1.88339	0.0750362
$631$	−4.17145	−0.166063	−0.0830315	−	0.996547i	$-0.526460\pi$
$631$	−4.17145	−0.166063	−0.0830315	+	0.996547i	$0.526460\pi$
$632$	−12.9865	−0.516576
$633$	−8.27452	−0.328883
$634$	9.03562	0.358850
$635$	2.06751	0.0820465
$636$	−1.62675	−0.0645048
$637$	2.03773	0.0807380
$638$	23.7079	0.938606
$639$	−1.57089	−0.0621435
$640$	−12.2398	−0.483822
$641$	4.95692	0.195787	0.0978933	−	0.995197i	$-0.468790\pi$
$641$	4.95692	0.195787	0.0978933	+	0.995197i	$0.468790\pi$
$642$	10.3971	0.410339
$643$	−14.0285	−0.553230	−0.276615	−	0.960981i	$-0.589213\pi$
$643$	−14.0285	−0.553230	−0.276615	+	0.960981i	$0.589213\pi$
$644$	−7.01673	−0.276498
$645$	−10.6490	−0.419303
$646$	−1.74075	−0.0684888
$647$	−15.8380	−0.622658	−0.311329	−	0.950302i	$-0.600774\pi$
$647$	−15.8380	−0.622658	−0.311329	+	0.950302i	$0.600774\pi$
$648$	2.88787	0.113446
$649$	41.7872	1.64029
$650$	1.34657	0.0528167
$651$	−3.43070	−0.134460
$652$	5.00219	0.195901
$653$	29.1768	1.14178	0.570888	−	0.821028i	$-0.306599\pi$
$653$	29.1768	1.14178	0.570888	+	0.821028i	$0.306599\pi$
$654$	−13.9326	−0.544809
$655$	−30.4670	−1.19044
$656$	2.04927	0.0800105
$657$	−9.67049	−0.377282
$658$	−6.97846	−0.272049
$659$	−3.36477	−0.131073	−0.0655364	−	0.997850i	$-0.520876\pi$
$659$	−3.36477	−0.131073	−0.0655364	+	0.997850i	$0.520876\pi$
$660$	−11.5211	−0.448458
$661$	−3.56055	−0.138489	−0.0692447	−	0.997600i	$-0.522059\pi$
$661$	−3.56055	−0.138489	−0.0692447	+	0.997600i	$0.522059\pi$
$662$	−20.7337	−0.805840
$663$	−1.74059	−0.0675989
$664$	25.6480	0.995334
$665$	−4.62530	−0.179362
$666$	−1.11524	−0.0432148
$667$	−32.7708	−1.26889
$668$	−18.7194	−0.724275
$669$	−9.20017	−0.355699
$670$	−10.2157	−0.394667
$671$	28.8053	1.11202
$672$	5.51123	0.212600
$673$	−6.79286	−0.261845	−0.130923	−	0.991393i	$-0.541794\pi$
$673$	−6.79286	−0.261845	−0.130923	+	0.991393i	$0.541794\pi$
$674$	−15.4325	−0.594437
$675$	0.725414	0.0279212
$676$	10.3533	0.398202
$677$	3.52953	0.135651	0.0678254	−	0.997697i	$-0.478394\pi$
$677$	3.52953	0.135651	0.0678254	+	0.997697i	$0.478394\pi$
$678$	−14.4915	−0.556541
$679$	1.09551	0.0420419
$680$	−5.10003	−0.195577
$681$	−4.61206	−0.176735
$682$	−14.8825	−0.569879
$683$	36.8948	1.41174	0.705871	−	0.708341i	$-0.250556\pi$
$683$	36.8948	1.41174	0.705871	+	0.708341i	$0.250556\pi$
$684$	−2.61784	−0.100096
$685$	−15.7479	−0.601696
$686$	−0.910949	−0.0347802
$687$	−8.47798	−0.323455
$688$	1.49551	0.0570159
$689$	−2.83282	−0.107922
$690$	−11.2934	−0.429934
$691$	−0.939809	−0.0357520	−0.0178760	−	0.999840i	$-0.505690\pi$
$691$	−0.939809	−0.0357520	−0.0178760	+	0.999840i	$0.505690\pi$
$692$	−8.38472	−0.318739
$693$	4.76209	0.180897
$694$	−32.4421	−1.23149
$695$	−10.5516	−0.400244
$696$	15.7826	0.598239
$697$	−6.02863	−0.228351
$698$	0.184547	0.00698520
$699$	11.9704	0.452763
$700$	0.848859	0.0320839
$701$	−15.3128	−0.578356	−0.289178	−	0.957275i	$-0.593382\pi$
$701$	−15.3128	−0.578356	−0.289178	+	0.957275i	$0.593382\pi$
$702$	1.85627	0.0700605
$703$	2.73885	0.103298
$704$	26.6732	1.00529
$705$	15.8384	0.596510
$706$	−12.8133	−0.482236
$707$	3.53703	0.133024
$708$	10.2682	0.385903
$709$	−20.5964	−0.773515	−0.386757	−	0.922182i	$-0.626405\pi$
$709$	−20.5964	−0.773515	−0.386757	+	0.922182i	$0.626405\pi$
$710$	−2.95861	−0.111035
$711$	−4.49693	−0.168648
$712$	29.3257	1.09903
$713$	20.5716	0.770413
$714$	0.778113	0.0291202
$715$	−20.0628	−0.750307
$716$	−0.478858	−0.0178958
$717$	20.9024	0.780613
$718$	8.75848	0.326864
$719$	25.9093	0.966255	0.483128	−	0.875550i	$-0.339501\pi$
$719$	25.9093	0.966255	0.483128	+	0.875550i	$0.339501\pi$
$720$	0.600311	0.0223723
$721$	1.77000	0.0659184
$722$	12.7489	0.474466
$723$	−5.81959	−0.216433
$724$	12.3942	0.460626
$725$	3.96450	0.147238
$726$	10.6376	0.394798
$727$	33.5208	1.24322	0.621609	−	0.783328i	$-0.286479\pi$
$727$	33.5208	1.24322	0.621609	+	0.783328i	$0.286479\pi$
$728$	5.88470	0.218102
$729$	1.00000	0.0370370
$730$	−18.2133	−0.674106
$731$	−4.39956	−0.162724
$732$	7.07824	0.261619
$733$	8.91851	0.329413	0.164706	−	0.986343i	$-0.447332\pi$
$733$	8.91851	0.329413	0.164706	+	0.986343i	$0.447332\pi$
$734$	21.7531	0.802923
$735$	2.06751	0.0762612
$736$	−33.0472	−1.21813
$737$	−25.8300	−0.951460
$738$	6.42930	0.236666
$739$	−17.5441	−0.645368	−0.322684	−	0.946507i	$-0.604585\pi$
$739$	−17.5441	−0.645368	−0.322684	+	0.946507i	$0.604585\pi$
$740$	2.96191	0.108882
$741$	−4.55869	−0.167468
$742$	1.26638	0.0464904
$743$	−31.3395	−1.14973	−0.574867	−	0.818247i	$-0.694946\pi$
$743$	−31.3395	−1.14973	−0.574867	+	0.818247i	$0.694946\pi$
$744$	−9.90740	−0.363223
$745$	0.713352	0.0261352
$746$	−17.3438	−0.635002
$747$	8.88129	0.324949
$748$	−4.75988	−0.174038
$749$	11.4134	0.417038
$750$	10.7832	0.393747
$751$	44.1522	1.61114	0.805568	−	0.592503i	$-0.201860\pi$
$751$	44.1522	1.61114	0.805568	+	0.592503i	$0.201860\pi$
$752$	−2.22431	−0.0811122
$753$	−0.00650335	−0.000236995 0
$754$	10.1448	0.369452
$755$	−13.5294	−0.492384
$756$	1.17017	0.0425587
$757$	−17.5802	−0.638964	−0.319482	−	0.947592i	$-0.603509\pi$
$757$	−17.5802	−0.638964	−0.319482	+	0.947592i	$0.603509\pi$
$758$	7.68066	0.278974
$759$	−28.5550	−1.03648
$760$	−13.3572	−0.484518
$761$	−15.5785	−0.564720	−0.282360	−	0.959308i	$-0.591117\pi$
$761$	−15.5785	−0.564720	−0.282360	+	0.959308i	$0.591117\pi$
$762$	−0.910949	−0.0330002
$763$	−15.2946	−0.553703
$764$	−16.8305	−0.608908
$765$	−1.76602	−0.0638506
$766$	20.7329	0.749110
$767$	17.8811	0.645648
$768$	16.5952	0.598829
$769$	−26.4832	−0.955009	−0.477505	−	0.878629i	$-0.658459\pi$
$769$	−26.4832	−0.955009	−0.477505	+	0.878629i	$0.658459\pi$
$770$	8.96889	0.323216
$771$	−21.6261	−0.778844
$772$	−5.75406	−0.207093
$773$	−4.85564	−0.174645	−0.0873226	−	0.996180i	$-0.527831\pi$
$773$	−4.85564	−0.174645	−0.0873226	+	0.996180i	$0.527831\pi$
$774$	4.69197	0.168649
$775$	−2.48868	−0.0893960
$776$	3.16369	0.113570
$777$	−1.22426	−0.0439202
$778$	−14.0507	−0.503743
$779$	−15.7893	−0.565710
$780$	−4.92997	−0.176521
$781$	−7.48072	−0.267681
$782$	−4.66582	−0.166850
$783$	5.46515	0.195309
$784$	−0.290355	−0.0103698
$785$	−10.8696	−0.387954
$786$	13.4238	0.478812
$787$	−6.95194	−0.247810	−0.123905	−	0.992294i	$-0.539542\pi$
$787$	−6.95194	−0.247810	−0.123905	+	0.992294i	$0.539542\pi$
$788$	21.6091	0.769793
$789$	−6.14267	−0.218685
$790$	−8.46949	−0.301331
$791$	−15.9081	−0.565627
$792$	13.7523	0.488666
$793$	12.3260	0.437710
$794$	6.63695	0.235537
$795$	−2.87421	−0.101938
$796$	17.6777	0.626568
$797$	26.4161	0.935705	0.467852	−	0.883807i	$-0.345028\pi$
$797$	26.4161	0.935705	0.467852	+	0.883807i	$0.345028\pi$
$798$	2.03792	0.0721416
$799$	6.54356	0.231495
$800$	3.99792	0.141348
$801$	10.1548	0.358802
$802$	−23.5442	−0.831376
$803$	−46.0517	−1.62513
$804$	−6.34712	−0.223846
$805$	−12.3975	−0.436953
$806$	−6.36832	−0.224314
$807$	20.1163	0.708126
$808$	10.2145	0.359344
$809$	−20.0526	−0.705010	−0.352505	−	0.935810i	$-0.614670\pi$
$809$	−20.0526	−0.705010	−0.352505	+	0.935810i	$0.614670\pi$
$810$	1.88339	0.0661757
$811$	−6.59746	−0.231668	−0.115834	−	0.993269i	$-0.536954\pi$
$811$	−6.59746	−0.231668	−0.115834	+	0.993269i	$0.536954\pi$
$812$	6.39517	0.224426
$813$	13.3909	0.469640
$814$	−5.31088	−0.186146
$815$	8.83807	0.309584
$816$	0.248015	0.00868227
$817$	−11.5227	−0.403128
$818$	17.1907	0.601058
$819$	2.03773	0.0712042
$820$	−17.0752	−0.596293
$821$	32.1085	1.12060	0.560298	−	0.828291i	$-0.310687\pi$
$821$	32.1085	1.12060	0.560298	+	0.828291i	$0.310687\pi$
$822$	6.93857	0.242010
$823$	53.0767	1.85014	0.925069	−	0.379799i	$-0.124007\pi$
$823$	53.0767	1.85014	0.925069	+	0.379799i	$0.124007\pi$
$824$	5.11153	0.178069
$825$	3.45449	0.120270
$826$	−7.99355	−0.278131
$827$	−5.72183	−0.198968	−0.0994838	−	0.995039i	$-0.531719\pi$
$827$	−5.72183	−0.198968	−0.0994838	+	0.995039i	$0.531719\pi$
$828$	−7.01673	−0.243848
$829$	36.9131	1.28204	0.641022	−	0.767522i	$-0.278511\pi$
$829$	36.9131	1.28204	0.641022	+	0.767522i	$0.278511\pi$
$830$	16.7270	0.580601
$831$	−8.05697	−0.279493
$832$	11.4137	0.395698
$833$	0.854179	0.0295955
$834$	4.64905	0.160983
$835$	−33.0742	−1.14458
$836$	−12.4664	−0.431158
$837$	−3.43070	−0.118582
$838$	−33.5561	−1.15918
$839$	−35.8828	−1.23881	−0.619406	−	0.785071i	$-0.712626\pi$
$839$	−35.8828	−1.23881	−0.619406	+	0.785071i	$0.712626\pi$
$840$	5.97068	0.206008
$841$	0.867890	0.0299272
$842$	−1.52092	−0.0524142
$843$	−27.8734	−0.960012
$844$	−9.68261	−0.333289
$845$	18.2926	0.629283
$846$	−6.97846	−0.239924
$847$	11.6775	0.401243
$848$	0.403646	0.0138612
$849$	10.1043	0.346779
$850$	0.564454	0.0193606
$851$	7.34109	0.251649
$852$	−1.83821	−0.0629761
$853$	38.1872	1.30750	0.653752	−	0.756709i	$-0.273194\pi$
$853$	38.1872	1.30750	0.653752	+	0.756709i	$0.273194\pi$
$854$	−5.51023	−0.188556
$855$	−4.62530	−0.158182
$856$	32.9605	1.12657
$857$	−5.39552	−0.184308	−0.0921538	−	0.995745i	$-0.529375\pi$
$857$	−5.39552	−0.184308	−0.0921538	+	0.995745i	$0.529375\pi$
$858$	8.83973	0.301783
$859$	−22.8095	−0.778250	−0.389125	−	0.921185i	$-0.627223\pi$
$859$	−22.8095	−0.778250	−0.389125	+	0.921185i	$0.627223\pi$
$860$	−12.4611	−0.424921
$861$	7.05781	0.240529
$862$	−20.4576	−0.696789
$863$	15.7818	0.537220	0.268610	−	0.963249i	$-0.413436\pi$
$863$	15.7818	0.537220	0.268610	+	0.963249i	$0.413436\pi$
$864$	5.51123	0.187496
$865$	−14.8145	−0.503707
$866$	−31.1942	−1.06002
$867$	16.2704	0.552571
$868$	−4.01451	−0.136261
$869$	−21.4148	−0.726446
$870$	10.2930	0.348967
$871$	−11.0529	−0.374512
$872$	−44.1689	−1.49575
$873$	1.09551	0.0370775
$874$	−12.2200	−0.413349
$875$	11.8373	0.400175
$876$	−11.3161	−0.382337
$877$	−27.3621	−0.923952	−0.461976	−	0.886892i	$-0.652859\pi$
$877$	−27.3621	−0.923952	−0.461976	+	0.886892i	$0.652859\pi$
$878$	−19.3759	−0.653905
$879$	−5.16465	−0.174199
$880$	2.85873	0.0963679
$881$	−39.0186	−1.31457	−0.657284	−	0.753643i	$-0.728295\pi$
$881$	−39.0186	−1.31457	−0.657284	+	0.753643i	$0.728295\pi$
$882$	−0.910949	−0.0306733
$883$	−47.1651	−1.58723	−0.793616	−	0.608419i	$-0.791804\pi$
$883$	−47.1651	−1.58723	−0.793616	+	0.608419i	$0.791804\pi$
$884$	−2.03679	−0.0685046
$885$	18.1423	0.609847
$886$	28.0493	0.942333
$887$	1.26669	0.0425313	0.0212656	−	0.999774i	$-0.493230\pi$
$887$	1.26669	0.0425313	0.0212656	+	0.999774i	$0.493230\pi$
$888$	−3.53551	−0.118644
$889$	−1.00000	−0.0335389
$890$	19.1255	0.641088
$891$	4.76209	0.159536
$892$	−10.7658	−0.360465
$893$	17.1379	0.573499
$894$	−0.314305	−0.0105119
$895$	−0.846066	−0.0282809
$896$	5.92009	0.197776
$897$	−12.2189	−0.407978
$898$	−24.4550	−0.816074
$899$	−18.7493	−0.625324
$900$	0.848859	0.0282953
$901$	−1.18746	−0.0395601
$902$	30.6169	1.01943
$903$	5.15064	0.171402
$904$	−45.9404	−1.52796
$905$	21.8985	0.727931
$906$	5.96107	0.198043
$907$	−14.3655	−0.476999	−0.238500	−	0.971143i	$-0.576656\pi$
$907$	−14.3655	−0.476999	−0.238500	+	0.971143i	$0.576656\pi$
$908$	−5.39690	−0.179102
$909$	3.53703	0.117316
$910$	3.83786	0.127224
$911$	−15.5385	−0.514812	−0.257406	−	0.966303i	$-0.582868\pi$
$911$	−15.5385	−0.514812	−0.257406	+	0.966303i	$0.582868\pi$
$912$	0.649565	0.0215092
$913$	42.2935	1.39971
$914$	−1.88855	−0.0624678
$915$	12.5061	0.413440
$916$	−9.92069	−0.327789
$917$	14.7361	0.486628
$918$	0.778113	0.0256816
$919$	0.672694	0.0221901	0.0110951	−	0.999938i	$-0.496468\pi$
$919$	0.672694	0.0221901	0.0110951	+	0.999938i	$0.496468\pi$
$920$	−35.8022	−1.18036
$921$	−3.03374	−0.0999650
$922$	−34.4415	−1.13427
$923$	−3.20106	−0.105364
$924$	5.57246	0.183320
$925$	−0.888098	−0.0292005
$926$	−27.7819	−0.912968
$927$	1.77000	0.0581345
$928$	30.1197	0.988728
$929$	49.6218	1.62804	0.814020	−	0.580837i	$-0.197275\pi$
$929$	49.6218	1.62804	0.814020	+	0.580837i	$0.197275\pi$
$930$	−6.46136	−0.211876
$931$	2.23714	0.0733193
$932$	14.0075	0.458829
$933$	−22.1371	−0.724736
$934$	7.21502	0.236083
$935$	−8.40995	−0.275035
$936$	5.88470	0.192347
$937$	−26.7103	−0.872586	−0.436293	−	0.899805i	$-0.643709\pi$
$937$	−26.7103	−0.872586	−0.436293	+	0.899805i	$0.643709\pi$
$938$	4.94107	0.161332
$939$	−9.66850	−0.315520
$940$	18.5337	0.604503
$941$	−12.6965	−0.413895	−0.206948	−	0.978352i	$-0.566353\pi$
$941$	−12.6965	−0.413895	−0.206948	+	0.978352i	$0.566353\pi$
$942$	4.78919	0.156040
$943$	−42.3209	−1.37816
$944$	−2.54786	−0.0829257
$945$	2.06751	0.0672560
$946$	22.3436	0.726452
$947$	−38.5444	−1.25253	−0.626263	−	0.779612i	$-0.715416\pi$
$947$	−38.5444	−1.25253	−0.626263	+	0.779612i	$0.715416\pi$
$948$	−5.26218	−0.170908
$949$	−19.7059	−0.639680
$950$	1.47834	0.0479636
$951$	9.91891	0.321643
$952$	2.46675	0.0799479
$953$	−20.6289	−0.668235	−0.334118	−	0.942531i	$-0.608438\pi$
$953$	−20.6289	−0.668235	−0.334118	+	0.942531i	$0.608438\pi$
$954$	1.26638	0.0410007
$955$	−29.7369	−0.962263
$956$	24.4594	0.791072
$957$	26.0255	0.841286
$958$	31.0435	1.00297
$959$	7.61686	0.245961
$960$	11.5804	0.373757
$961$	−19.2303	−0.620332
$962$	−2.27257	−0.0732705
$963$	11.4134	0.367793
$964$	−6.80992	−0.219333
$965$	−10.1665	−0.327272
$966$	5.46235	0.175748
$967$	−12.4906	−0.401669	−0.200835	−	0.979625i	$-0.564365\pi$
$967$	−12.4906	−0.401669	−0.200835	+	0.979625i	$0.564365\pi$
$968$	33.7230	1.08390
$969$	−1.91092	−0.0613875
$970$	2.06328	0.0662480
$971$	38.8901	1.24804	0.624022	−	0.781407i	$-0.285498\pi$
$971$	38.8901	1.24804	0.624022	+	0.781407i	$0.285498\pi$
$972$	1.17017	0.0375333
$973$	5.10352	0.163611
$974$	9.77877	0.313332
$975$	1.47820	0.0473403
$976$	−1.75633	−0.0562186
$977$	−17.5486	−0.561428	−0.280714	−	0.959791i	$-0.590571\pi$
$977$	−17.5486	−0.561428	−0.280714	+	0.959791i	$0.590571\pi$
$978$	−3.89408	−0.124519
$979$	48.3581	1.54553
$980$	2.41934	0.0772829
$981$	−15.2946	−0.488320
$982$	32.4461	1.03540
$983$	35.7325	1.13969	0.569845	−	0.821752i	$-0.307003\pi$
$983$	35.7325	1.13969	0.569845	+	0.821752i	$0.307003\pi$
$984$	20.3820	0.649754
$985$	38.1799	1.21651
$986$	4.25251	0.135427
$987$	−7.66065	−0.243841
$988$	−5.33446	−0.169712
$989$	−30.8849	−0.982083
$990$	8.96889	0.285050
$991$	38.7774	1.23180	0.615902	−	0.787822i	$-0.288792\pi$
$991$	38.7774	1.23180	0.615902	+	0.787822i	$0.288792\pi$
$992$	−18.9074	−0.600310
$993$	−22.7606	−0.722286
$994$	1.43100	0.0453886
$995$	31.2336	0.990172
$996$	10.3926	0.329303
$997$	58.7906	1.86192	0.930958	−	0.365127i	$-0.118974\pi$
$997$	58.7906	1.86192	0.930958	+	0.365127i	$0.118974\pi$
$998$	26.8984	0.851455
$999$	−1.22426	−0.0387340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2667.2.a.l.1.4	✓	13
3.2	odd	2		8001.2.a.o.1.10		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.l.1.4	✓	13	1.1	even	1	trivial
8001.2.a.o.1.10		13	3.2	odd	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:	\( N \)	\(=\)	\( 2667 = 3 \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2667.a (trivial)

\(f(q)\)	\(=\)	\(q-0.910949 q^{2} -1.00000 q^{3} -1.17017 q^{4} -2.06751 q^{5} +0.910949 q^{6} +1.00000 q^{7} +2.88787 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.910949 q^{2} -1.00000 q^{3} -1.17017 q^{4} -2.06751 q^{5} +0.910949 q^{6} +1.00000 q^{7} +2.88787 q^{8} +1.00000 q^{9} +1.88339 q^{10} +4.76209 q^{11} +1.17017 q^{12} +2.03773 q^{13} -0.910949 q^{14} +2.06751 q^{15} -0.290355 q^{16} +0.854179 q^{17} -0.910949 q^{18} +2.23714 q^{19} +2.41934 q^{20} -1.00000 q^{21} -4.33802 q^{22} +5.99633 q^{23} -2.88787 q^{24} -0.725414 q^{25} -1.85627 q^{26} -1.00000 q^{27} -1.17017 q^{28} -5.46515 q^{29} -1.88339 q^{30} +3.43070 q^{31} -5.51123 q^{32} -4.76209 q^{33} -0.778113 q^{34} -2.06751 q^{35} -1.17017 q^{36} +1.22426 q^{37} -2.03792 q^{38} -2.03773 q^{39} -5.97068 q^{40} -7.05781 q^{41} +0.910949 q^{42} -5.15064 q^{43} -5.57246 q^{44} -2.06751 q^{45} -5.46235 q^{46} +7.66065 q^{47} +0.290355 q^{48} +1.00000 q^{49} +0.660815 q^{50} -0.854179 q^{51} -2.38450 q^{52} -1.39018 q^{53} +0.910949 q^{54} -9.84565 q^{55} +2.88787 q^{56} -2.23714 q^{57} +4.97848 q^{58} +8.77497 q^{59} -2.41934 q^{60} +6.04889 q^{61} -3.12519 q^{62} +1.00000 q^{63} +5.60116 q^{64} -4.21303 q^{65} +4.33802 q^{66} -5.42409 q^{67} -0.999536 q^{68} -5.99633 q^{69} +1.88339 q^{70} -1.57089 q^{71} +2.88787 q^{72} -9.67049 q^{73} -1.11524 q^{74} +0.725414 q^{75} -2.61784 q^{76} +4.76209 q^{77} +1.85627 q^{78} -4.49693 q^{79} +0.600311 q^{80} +1.00000 q^{81} +6.42930 q^{82} +8.88129 q^{83} +1.17017 q^{84} -1.76602 q^{85} +4.69197 q^{86} +5.46515 q^{87} +13.7523 q^{88} +10.1548 q^{89} +1.88339 q^{90} +2.03773 q^{91} -7.01673 q^{92} -3.43070 q^{93} -6.97846 q^{94} -4.62530 q^{95} +5.51123 q^{96} +1.09551 q^{97} -0.910949 q^{98} +4.76209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{2} - 13 q^{3} + 10 q^{4} + 12 q^{5} - 4 q^{6} + 13 q^{7} + 9 q^{8} + 13 q^{9}+O(q^{10}) \) 13 * q + 4 * q^2 - 13 * q^3 + 10 * q^4 + 12 * q^5 - 4 * q^6 + 13 * q^7 + 9 * q^8 + 13 * q^9	\( 13 q + 4 q^{2} - 13 q^{3} + 10 q^{4} + 12 q^{5} - 4 q^{6} + 13 q^{7} + 9 q^{8} + 13 q^{9} + 6 q^{10} + 3 q^{11} - 10 q^{12} + 21 q^{13} + 4 q^{14} - 12 q^{15} + 8 q^{16} + 17 q^{17} + 4 q^{18} + 5 q^{19} + 29 q^{20} - 13 q^{21} + q^{22} + 4 q^{23} - 9 q^{24} + q^{25} + 22 q^{26} - 13 q^{27} + 10 q^{28} + 21 q^{29} - 6 q^{30} - 7 q^{31} + 12 q^{32} - 3 q^{33} + 2 q^{34} + 12 q^{35} + 10 q^{36} + 7 q^{37} - 9 q^{38} - 21 q^{39} + 29 q^{40} + 21 q^{41} - 4 q^{42} - 9 q^{43} - 2 q^{44} + 12 q^{45} - 28 q^{46} + 23 q^{47} - 8 q^{48} + 13 q^{49} + 15 q^{50} - 17 q^{51} + 15 q^{52} + 31 q^{53} - 4 q^{54} - 8 q^{55} + 9 q^{56} - 5 q^{57} - 25 q^{58} + 28 q^{59} - 29 q^{60} + 29 q^{61} - 3 q^{62} + 13 q^{63} + 9 q^{64} + 30 q^{65} - q^{66} - 18 q^{67} + 34 q^{68} - 4 q^{69} + 6 q^{70} + 10 q^{71} + 9 q^{72} + 24 q^{73} - 19 q^{74} - q^{75} + 3 q^{77} - 22 q^{78} - 28 q^{79} + 26 q^{80} + 13 q^{81} + 18 q^{82} + 26 q^{83} - 10 q^{84} + 20 q^{85} - 2 q^{86} - 21 q^{87} - 17 q^{88} + 44 q^{89} + 6 q^{90} + 21 q^{91} + 6 q^{92} + 7 q^{93} - 9 q^{94} - 2 q^{95} - 12 q^{96} + 17 q^{97} + 4 q^{98} + 3 q^{99}+O(q^{100}) \) 13 * q + 4 * q^2 - 13 * q^3 + 10 * q^4 + 12 * q^5 - 4 * q^6 + 13 * q^7 + 9 * q^8 + 13 * q^9 + 6 * q^10 + 3 * q^11 - 10 * q^12 + 21 * q^13 + 4 * q^14 - 12 * q^15 + 8 * q^16 + 17 * q^17 + 4 * q^18 + 5 * q^19 + 29 * q^20 - 13 * q^21 + q^22 + 4 * q^23 - 9 * q^24 + q^25 + 22 * q^26 - 13 * q^27 + 10 * q^28 + 21 * q^29 - 6 * q^30 - 7 * q^31 + 12 * q^32 - 3 * q^33 + 2 * q^34 + 12 * q^35 + 10 * q^36 + 7 * q^37 - 9 * q^38 - 21 * q^39 + 29 * q^40 + 21 * q^41 - 4 * q^42 - 9 * q^43 - 2 * q^44 + 12 * q^45 - 28 * q^46 + 23 * q^47 - 8 * q^48 + 13 * q^49 + 15 * q^50 - 17 * q^51 + 15 * q^52 + 31 * q^53 - 4 * q^54 - 8 * q^55 + 9 * q^56 - 5 * q^57 - 25 * q^58 + 28 * q^59 - 29 * q^60 + 29 * q^61 - 3 * q^62 + 13 * q^63 + 9 * q^64 + 30 * q^65 - q^66 - 18 * q^67 + 34 * q^68 - 4 * q^69 + 6 * q^70 + 10 * q^71 + 9 * q^72 + 24 * q^73 - 19 * q^74 - q^75 + 3 * q^77 - 22 * q^78 - 28 * q^79 + 26 * q^80 + 13 * q^81 + 18 * q^82 + 26 * q^83 - 10 * q^84 + 20 * q^85 - 2 * q^86 - 21 * q^87 - 17 * q^88 + 44 * q^89 + 6 * q^90 + 21 * q^91 + 6 * q^92 + 7 * q^93 - 9 * q^94 - 2 * q^95 - 12 * q^96 + 17 * q^97 + 4 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more