LMFDB - Embedded newform 2006.2.a.w.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2006,2,Mod(1,2006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2006 = 2 \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2006.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:	$16.0179906455$
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} - 6 x^{12} - 9 x^{11} + 91 x^{10} + 14 x^{9} - 517 x^{8} + 70 x^{7} + 1296 x^{6} - 300 x^{5} + \cdots + 16 $ x^13 - 6x^12 - 9x^11 + 91x^10 + 14x^9 - 517x^8 + 70x^7 + 1296x^6 - 300x^5 - 1312x^4 + 472x^3 + 368x^2 - 176x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-2.05797$ of defining polynomial
Character	$\chi$	$=$	2006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +3.05797 q^{3} +1.00000 q^{4} -4.28893 q^{5} +3.05797 q^{6} -2.80349 q^{7} +1.00000 q^{8} +6.35118 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +3.05797 q^{3} +1.00000 q^{4} -4.28893 q^{5} +3.05797 q^{6} -2.80349 q^{7} +1.00000 q^{8} +6.35118 q^{9} -4.28893 q^{10} +6.06569 q^{11} +3.05797 q^{12} +3.99253 q^{13} -2.80349 q^{14} -13.1154 q^{15} +1.00000 q^{16} +1.00000 q^{17} +6.35118 q^{18} -3.85756 q^{19} -4.28893 q^{20} -8.57298 q^{21} +6.06569 q^{22} +3.83097 q^{23} +3.05797 q^{24} +13.3949 q^{25} +3.99253 q^{26} +10.2478 q^{27} -2.80349 q^{28} -4.39341 q^{29} -13.1154 q^{30} +5.70843 q^{31} +1.00000 q^{32} +18.5487 q^{33} +1.00000 q^{34} +12.0240 q^{35} +6.35118 q^{36} -3.02783 q^{37} -3.85756 q^{38} +12.2090 q^{39} -4.28893 q^{40} +10.8572 q^{41} -8.57298 q^{42} -5.06977 q^{43} +6.06569 q^{44} -27.2398 q^{45} +3.83097 q^{46} +7.89683 q^{47} +3.05797 q^{48} +0.859534 q^{49} +13.3949 q^{50} +3.05797 q^{51} +3.99253 q^{52} -13.7758 q^{53} +10.2478 q^{54} -26.0153 q^{55} -2.80349 q^{56} -11.7963 q^{57} -4.39341 q^{58} +1.00000 q^{59} -13.1154 q^{60} +13.2241 q^{61} +5.70843 q^{62} -17.8054 q^{63} +1.00000 q^{64} -17.1237 q^{65} +18.5487 q^{66} -1.85961 q^{67} +1.00000 q^{68} +11.7150 q^{69} +12.0240 q^{70} -6.80271 q^{71} +6.35118 q^{72} -3.44145 q^{73} -3.02783 q^{74} +40.9613 q^{75} -3.85756 q^{76} -17.0051 q^{77} +12.2090 q^{78} +16.8040 q^{79} -4.28893 q^{80} +12.2839 q^{81} +10.8572 q^{82} -7.52359 q^{83} -8.57298 q^{84} -4.28893 q^{85} -5.06977 q^{86} -13.4349 q^{87} +6.06569 q^{88} +4.65810 q^{89} -27.2398 q^{90} -11.1930 q^{91} +3.83097 q^{92} +17.4562 q^{93} +7.89683 q^{94} +16.5448 q^{95} +3.05797 q^{96} -7.21113 q^{97} +0.859534 q^{98} +38.5243 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{2} + 7 q^{3} + 13 q^{4} + q^{5} + 7 q^{6} + 8 q^{7} + 13 q^{8} + 16 q^{9}+O(q^{10}) $ 13 * q + 13 * q^2 + 7 * q^3 + 13 * q^4 + q^5 + 7 * q^6 + 8 * q^7 + 13 * q^8 + 16 * q^9	\( 13 q + 13 q^{2} + 7 q^{3} + 13 q^{4} + q^{5} + 7 q^{6} + 8 q^{7} + 13 q^{8} + 16 q^{9} + q^{10} + 13 q^{11} + 7 q^{12} + 9 q^{13} + 8 q^{14} - 9 q^{15} + 13 q^{16} + 13 q^{17} + 16 q^{18} + 16 q^{19} + q^{20} - 3 q^{21} + 13 q^{22} + 18 q^{23} + 7 q^{24} + 14 q^{25} + 9 q^{26} + 10 q^{27} + 8 q^{28} + 8 q^{29} - 9 q^{30} + 32 q^{31} + 13 q^{32} + 13 q^{33} + 13 q^{34} - q^{35} + 16 q^{36} - 3 q^{37} + 16 q^{38} + 20 q^{39} + q^{40} + 24 q^{41} - 3 q^{42} - 2 q^{43} + 13 q^{44} - 3 q^{45} + 18 q^{46} + 26 q^{47} + 7 q^{48} + 23 q^{49} + 14 q^{50} + 7 q^{51} + 9 q^{52} - 27 q^{53} + 10 q^{54} - 7 q^{55} + 8 q^{56} - 21 q^{57} + 8 q^{58} + 13 q^{59} - 9 q^{60} + 20 q^{61} + 32 q^{62} - 3 q^{63} + 13 q^{64} + 16 q^{65} + 13 q^{66} + 4 q^{67} + 13 q^{68} - 2 q^{69} - q^{70} - 8 q^{71} + 16 q^{72} + 10 q^{73} - 3 q^{74} + 39 q^{75} + 16 q^{76} - 15 q^{77} + 20 q^{78} + 35 q^{79} + q^{80} + 29 q^{81} + 24 q^{82} - q^{83} - 3 q^{84} + q^{85} - 2 q^{86} - 32 q^{87} + 13 q^{88} - 20 q^{89} - 3 q^{90} - 15 q^{91} + 18 q^{92} - 7 q^{93} + 26 q^{94} - 3 q^{95} + 7 q^{96} - 3 q^{97} + 23 q^{98} + 65 q^{99}+O(q^{100}) \) 13 * q + 13 * q^2 + 7 * q^3 + 13 * q^4 + q^5 + 7 * q^6 + 8 * q^7 + 13 * q^8 + 16 * q^9 + q^10 + 13 * q^11 + 7 * q^12 + 9 * q^13 + 8 * q^14 - 9 * q^15 + 13 * q^16 + 13 * q^17 + 16 * q^18 + 16 * q^19 + q^20 - 3 * q^21 + 13 * q^22 + 18 * q^23 + 7 * q^24 + 14 * q^25 + 9 * q^26 + 10 * q^27 + 8 * q^28 + 8 * q^29 - 9 * q^30 + 32 * q^31 + 13 * q^32 + 13 * q^33 + 13 * q^34 - q^35 + 16 * q^36 - 3 * q^37 + 16 * q^38 + 20 * q^39 + q^40 + 24 * q^41 - 3 * q^42 - 2 * q^43 + 13 * q^44 - 3 * q^45 + 18 * q^46 + 26 * q^47 + 7 * q^48 + 23 * q^49 + 14 * q^50 + 7 * q^51 + 9 * q^52 - 27 * q^53 + 10 * q^54 - 7 * q^55 + 8 * q^56 - 21 * q^57 + 8 * q^58 + 13 * q^59 - 9 * q^60 + 20 * q^61 + 32 * q^62 - 3 * q^63 + 13 * q^64 + 16 * q^65 + 13 * q^66 + 4 * q^67 + 13 * q^68 - 2 * q^69 - q^70 - 8 * q^71 + 16 * q^72 + 10 * q^73 - 3 * q^74 + 39 * q^75 + 16 * q^76 - 15 * q^77 + 20 * q^78 + 35 * q^79 + q^80 + 29 * q^81 + 24 * q^82 - q^83 - 3 * q^84 + q^85 - 2 * q^86 - 32 * q^87 + 13 * q^88 - 20 * q^89 - 3 * q^90 - 15 * q^91 + 18 * q^92 - 7 * q^93 + 26 * q^94 - 3 * q^95 + 7 * q^96 - 3 * q^97 + 23 * q^98 + 65 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	3.05797	1.76552	0.882760	−	0.469824i	$-0.155683\pi$
$3$	3.05797	1.76552	0.882760	+	0.469824i	$0.155683\pi$
$4$	1.00000	0.500000
$5$	−4.28893	−1.91807	−0.959034	−	0.283290i	$-0.908574\pi$
$5$	−4.28893	−1.91807	−0.959034	+	0.283290i	$0.908574\pi$
$6$	3.05797	1.24841
$7$	−2.80349	−1.05962	−0.529809	−	0.848117i	$-0.677736\pi$
$7$	−2.80349	−1.05962	−0.529809	+	0.848117i	$0.677736\pi$
$8$	1.00000	0.353553
$9$	6.35118	2.11706
$10$	−4.28893	−1.35628
$11$	6.06569	1.82887	0.914437	−	0.404728i	$-0.132634\pi$
$11$	6.06569	1.82887	0.914437	+	0.404728i	$0.132634\pi$
$12$	3.05797	0.882760
$13$	3.99253	1.10733	0.553665	−	0.832740i	$-0.313229\pi$
$13$	3.99253	1.10733	0.553665	+	0.832740i	$0.313229\pi$
$14$	−2.80349	−0.749263
$15$	−13.1154	−3.38639
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	6.35118	1.49699
$19$	−3.85756	−0.884985	−0.442492	−	0.896772i	$-0.645906\pi$
$19$	−3.85756	−0.884985	−0.442492	+	0.896772i	$0.645906\pi$
$20$	−4.28893	−0.959034
$21$	−8.57298	−1.87078
$22$	6.06569	1.29321
$23$	3.83097	0.798813	0.399406	−	0.916774i	$-0.369216\pi$
$23$	3.83097	0.798813	0.399406	+	0.916774i	$0.369216\pi$
$24$	3.05797	0.624205
$25$	13.3949	2.67899
$26$	3.99253	0.783000
$27$	10.2478	1.97219
$28$	−2.80349	−0.529809
$29$	−4.39341	−0.815835	−0.407918	−	0.913019i	$-0.633745\pi$
$29$	−4.39341	−0.815835	−0.407918	+	0.913019i	$0.633745\pi$
$30$	−13.1154	−2.39454
$31$	5.70843	1.02526	0.512632	−	0.858608i	$-0.328671\pi$
$31$	5.70843	1.02526	0.512632	+	0.858608i	$0.328671\pi$
$32$	1.00000	0.176777
$33$	18.5487	3.22891
$34$	1.00000	0.171499
$35$	12.0240	2.03242
$36$	6.35118	1.05853
$37$	−3.02783	−0.497772	−0.248886	−	0.968533i	$-0.580064\pi$
$37$	−3.02783	−0.497772	−0.248886	+	0.968533i	$0.580064\pi$
$38$	−3.85756	−0.625779
$39$	12.2090	1.95501
$40$	−4.28893	−0.678140
$41$	10.8572	1.69561	0.847803	−	0.530312i	$-0.177925\pi$
$41$	10.8572	1.69561	0.847803	+	0.530312i	$0.177925\pi$
$42$	−8.57298	−1.32284
$43$	−5.06977	−0.773132	−0.386566	−	0.922262i	$-0.626339\pi$
$43$	−5.06977	−0.773132	−0.386566	+	0.922262i	$0.626339\pi$
$44$	6.06569	0.914437
$45$	−27.2398	−4.06067
$46$	3.83097	0.564846
$47$	7.89683	1.15187	0.575936	−	0.817495i	$-0.304638\pi$
$47$	7.89683	1.15187	0.575936	+	0.817495i	$0.304638\pi$
$48$	3.05797	0.441380
$49$	0.859534	0.122791
$50$	13.3949	1.89433
$51$	3.05797	0.428201
$52$	3.99253	0.553665
$53$	−13.7758	−1.89225	−0.946126	−	0.323799i	$-0.895040\pi$
$53$	−13.7758	−1.89225	−0.946126	+	0.323799i	$0.895040\pi$
$54$	10.2478	1.39455
$55$	−26.0153	−3.50791
$56$	−2.80349	−0.374632
$57$	−11.7963	−1.56246
$58$	−4.39341	−0.576883
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	−13.1154	−1.69319
$61$	13.2241	1.69317	0.846585	−	0.532254i	$-0.178655\pi$
$61$	13.2241	1.69317	0.846585	+	0.532254i	$0.178655\pi$
$62$	5.70843	0.724971
$63$	−17.8054	−2.24328
$64$	1.00000	0.125000
$65$	−17.1237	−2.12393
$66$	18.5487	2.28319
$67$	−1.85961	−0.227188	−0.113594	−	0.993527i	$-0.536236\pi$
$67$	−1.85961	−0.227188	−0.113594	+	0.993527i	$0.536236\pi$
$68$	1.00000	0.121268
$69$	11.7150	1.41032
$70$	12.0240	1.43714
$71$	−6.80271	−0.807334	−0.403667	−	0.914906i	$-0.632264\pi$
$71$	−6.80271	−0.807334	−0.403667	+	0.914906i	$0.632264\pi$
$72$	6.35118	0.748494
$73$	−3.44145	−0.402791	−0.201396	−	0.979510i	$-0.564548\pi$
$73$	−3.44145	−0.402791	−0.201396	+	0.979510i	$0.564548\pi$
$74$	−3.02783	−0.351978
$75$	40.9613	4.72981
$76$	−3.85756	−0.442492
$77$	−17.0051	−1.93791
$78$	12.2090	1.38240
$79$	16.8040	1.89060	0.945301	−	0.326200i	$-0.105768\pi$
$79$	16.8040	1.89060	0.945301	+	0.326200i	$0.105768\pi$
$80$	−4.28893	−0.479517
$81$	12.2839	1.36488
$82$	10.8572	1.19897
$83$	−7.52359	−0.825821	−0.412911	−	0.910772i	$-0.635488\pi$
$83$	−7.52359	−0.825821	−0.412911	+	0.910772i	$0.635488\pi$
$84$	−8.57298	−0.935388
$85$	−4.28893	−0.465200
$86$	−5.06977	−0.546687
$87$	−13.4349	−1.44037
$88$	6.06569	0.646605
$89$	4.65810	0.493757	0.246879	−	0.969046i	$-0.420595\pi$
$89$	4.65810	0.493757	0.246879	+	0.969046i	$0.420595\pi$
$90$	−27.2398	−2.87132
$91$	−11.1930	−1.17335
$92$	3.83097	0.399406
$93$	17.4562	1.81012
$94$	7.89683	0.814496
$95$	16.5448	1.69746
$96$	3.05797	0.312103
$97$	−7.21113	−0.732179	−0.366089	−	0.930580i	$-0.619304\pi$
$97$	−7.21113	−0.732179	−0.366089	+	0.930580i	$0.619304\pi$
$98$	0.859534	0.0868261
$99$	38.5243	3.87184
$100$	13.3949	1.33949
$101$	−8.13131	−0.809096	−0.404548	−	0.914517i	$-0.632571\pi$
$101$	−8.13131	−0.809096	−0.404548	+	0.914517i	$0.632571\pi$
$102$	3.05797	0.302784
$103$	2.55237	0.251493	0.125746	−	0.992062i	$-0.459867\pi$
$103$	2.55237	0.251493	0.125746	+	0.992062i	$0.459867\pi$
$104$	3.99253	0.391500
$105$	36.7689	3.58828
$106$	−13.7758	−1.33802
$107$	−6.01436	−0.581430	−0.290715	−	0.956810i	$-0.593893\pi$
$107$	−6.01436	−0.581430	−0.290715	+	0.956810i	$0.593893\pi$
$108$	10.2478	0.986096
$109$	−16.8743	−1.61627	−0.808134	−	0.588998i	$-0.799522\pi$
$109$	−16.8743	−1.61627	−0.808134	+	0.588998i	$0.799522\pi$
$110$	−26.0153	−2.48046
$111$	−9.25901	−0.878826
$112$	−2.80349	−0.264905
$113$	9.57163	0.900423	0.450212	−	0.892922i	$-0.351349\pi$
$113$	9.57163	0.900423	0.450212	+	0.892922i	$0.351349\pi$
$114$	−11.7963	−1.10482
$115$	−16.4308	−1.53218
$116$	−4.39341	−0.407918
$117$	25.3573	2.34428
$118$	1.00000	0.0920575
$119$	−2.80349	−0.256995
$120$	−13.1154	−1.19727
$121$	25.7926	2.34478
$122$	13.2241	1.19725
$123$	33.2009	2.99362
$124$	5.70843	0.512632
$125$	−36.0053	−3.22041
$126$	−17.8054	−1.58624
$127$	3.04743	0.270415	0.135208	−	0.990817i	$-0.456830\pi$
$127$	3.04743	0.270415	0.135208	+	0.990817i	$0.456830\pi$
$128$	1.00000	0.0883883
$129$	−15.5032	−1.36498
$130$	−17.1237	−1.50185
$131$	−2.21845	−0.193827	−0.0969135	−	0.995293i	$-0.530897\pi$
$131$	−2.21845	−0.193827	−0.0969135	+	0.995293i	$0.530897\pi$
$132$	18.5487	1.61446
$133$	10.8146	0.937746
$134$	−1.85961	−0.160646
$135$	−43.9521	−3.78280
$136$	1.00000	0.0857493
$137$	−3.02880	−0.258768	−0.129384	−	0.991595i	$-0.541300\pi$
$137$	−3.02880	−0.258768	−0.129384	+	0.991595i	$0.541300\pi$
$138$	11.7150	0.997246
$139$	−2.74627	−0.232935	−0.116468	−	0.993194i	$-0.537157\pi$
$139$	−2.74627	−0.232935	−0.116468	+	0.993194i	$0.537157\pi$
$140$	12.0240	1.01621
$141$	24.1483	2.03365
$142$	−6.80271	−0.570871
$143$	24.2175	2.02517
$144$	6.35118	0.529265
$145$	18.8430	1.56483
$146$	−3.44145	−0.284817
$147$	2.62843	0.216789
$148$	−3.02783	−0.248886
$149$	−9.94122	−0.814416	−0.407208	−	0.913335i	$-0.633498\pi$
$149$	−9.94122	−0.814416	−0.407208	+	0.913335i	$0.633498\pi$
$150$	40.9613	3.34448
$151$	−4.94695	−0.402577	−0.201289	−	0.979532i	$-0.564513\pi$
$151$	−4.94695	−0.402577	−0.201289	+	0.979532i	$0.564513\pi$
$152$	−3.85756	−0.312889
$153$	6.35118	0.513462
$154$	−17.0051	−1.37031
$155$	−24.4831	−1.96653
$156$	12.2090	0.977506
$157$	−19.1076	−1.52495	−0.762477	−	0.647015i	$-0.776017\pi$
$157$	−19.1076	−1.52495	−0.762477	+	0.647015i	$0.776017\pi$
$158$	16.8040	1.33686
$159$	−42.1260	−3.34081
$160$	−4.28893	−0.339070
$161$	−10.7401	−0.846436
$162$	12.2839	0.965118
$163$	−19.7693	−1.54845	−0.774227	−	0.632908i	$-0.781861\pi$
$163$	−19.7693	−1.54845	−0.774227	+	0.632908i	$0.781861\pi$
$164$	10.8572	0.847803
$165$	−79.5541	−6.19328
$166$	−7.52359	−0.583944
$167$	−13.0858	−1.01261	−0.506306	−	0.862354i	$-0.668989\pi$
$167$	−13.0858	−1.01261	−0.506306	+	0.862354i	$0.668989\pi$
$168$	−8.57298	−0.661419
$169$	2.94031	0.226178
$170$	−4.28893	−0.328946
$171$	−24.5001	−1.87357
$172$	−5.06977	−0.386566
$173$	2.79497	0.212497	0.106249	−	0.994340i	$-0.466116\pi$
$173$	2.79497	0.212497	0.106249	+	0.994340i	$0.466116\pi$
$174$	−13.4349	−1.01850
$175$	−37.5525	−2.83870
$176$	6.06569	0.457219
$177$	3.05797	0.229851
$178$	4.65810	0.349139
$179$	−11.6805	−0.873041	−0.436521	−	0.899694i	$-0.643789\pi$
$179$	−11.6805	−0.873041	−0.436521	+	0.899694i	$0.643789\pi$
$180$	−27.2398	−2.03033
$181$	14.7710	1.09792	0.548960	−	0.835849i	$-0.315024\pi$
$181$	14.7710	1.09792	0.548960	+	0.835849i	$0.315024\pi$
$182$	−11.1930	−0.829681
$183$	40.4388	2.98933
$184$	3.83097	0.282423
$185$	12.9862	0.954761
$186$	17.4562	1.27995
$187$	6.06569	0.443567
$188$	7.89683	0.575936
$189$	−28.7296	−2.08977
$190$	16.5448	1.20029
$191$	−8.07995	−0.584645	−0.292322	−	0.956320i	$-0.594428\pi$
$191$	−8.07995	−0.584645	−0.292322	+	0.956320i	$0.594428\pi$
$192$	3.05797	0.220690
$193$	8.00222	0.576012	0.288006	−	0.957629i	$-0.407008\pi$
$193$	8.00222	0.576012	0.288006	+	0.957629i	$0.407008\pi$
$194$	−7.21113	−0.517729
$195$	−52.3638	−3.74985
$196$	0.859534	0.0613953
$197$	10.0336	0.714861	0.357431	−	0.933940i	$-0.383653\pi$
$197$	10.0336	0.714861	0.357431	+	0.933940i	$0.383653\pi$
$198$	38.5243	2.73780
$199$	−0.774708	−0.0549176	−0.0274588	−	0.999623i	$-0.508742\pi$
$199$	−0.774708	−0.0549176	−0.0274588	+	0.999623i	$0.508742\pi$
$200$	13.3949	0.947165
$201$	−5.68663	−0.401104
$202$	−8.13131	−0.572117
$203$	12.3169	0.864474
$204$	3.05797	0.214101
$205$	−46.5657	−3.25229
$206$	2.55237	0.177832
$207$	24.3312	1.69113
$208$	3.99253	0.276832
$209$	−23.3988	−1.61853
$210$	36.7689	2.53730
$211$	3.84503	0.264703	0.132351	−	0.991203i	$-0.457747\pi$
$211$	3.84503	0.264703	0.132351	+	0.991203i	$0.457747\pi$
$212$	−13.7758	−0.946126
$213$	−20.8025	−1.42536
$214$	−6.01436	−0.411133
$215$	21.7439	1.48292
$216$	10.2478	0.697275
$217$	−16.0035	−1.08639
$218$	−16.8743	−1.14287
$219$	−10.5239	−0.711136
$220$	−26.0153	−1.75395
$221$	3.99253	0.268567
$222$	−9.25901	−0.621424
$223$	−17.9845	−1.20433	−0.602167	−	0.798370i	$-0.705696\pi$
$223$	−17.9845	−1.20433	−0.602167	+	0.798370i	$0.705696\pi$
$224$	−2.80349	−0.187316
$225$	85.0737	5.67158
$226$	9.57163	0.636695
$227$	−21.5278	−1.42885	−0.714424	−	0.699713i	$-0.753311\pi$
$227$	−21.5278	−1.42885	−0.714424	+	0.699713i	$0.753311\pi$
$228$	−11.7963	−0.781229
$229$	−3.55606	−0.234991	−0.117495	−	0.993073i	$-0.537487\pi$
$229$	−3.55606	−0.234991	−0.117495	+	0.993073i	$0.537487\pi$
$230$	−16.4308	−1.08341
$231$	−52.0010	−3.42142
$232$	−4.39341	−0.288441
$233$	7.40038	0.484815	0.242408	−	0.970174i	$-0.422063\pi$
$233$	7.40038	0.484815	0.242408	+	0.970174i	$0.422063\pi$
$234$	25.3573	1.65766
$235$	−33.8690	−2.20937
$236$	1.00000	0.0650945
$237$	51.3862	3.33789
$238$	−2.80349	−0.181723
$239$	−14.9412	−0.966466	−0.483233	−	0.875492i	$-0.660538\pi$
$239$	−14.9412	−0.966466	−0.483233	+	0.875492i	$0.660538\pi$
$240$	−13.1154	−0.846597
$241$	18.9134	1.21832	0.609160	−	0.793048i	$-0.291507\pi$
$241$	18.9134	1.21832	0.609160	+	0.793048i	$0.291507\pi$
$242$	25.7926	1.65801
$243$	6.82051	0.437536
$244$	13.2241	0.846585
$245$	−3.68649	−0.235521
$246$	33.2009	2.11681
$247$	−15.4014	−0.979969
$248$	5.70843	0.362486
$249$	−23.0069	−1.45800
$250$	−36.0053	−2.27718
$251$	10.0114	0.631915	0.315958	−	0.948773i	$-0.397674\pi$
$251$	10.0114	0.631915	0.315958	+	0.948773i	$0.397674\pi$
$252$	−17.8054	−1.12164
$253$	23.2375	1.46093
$254$	3.04743	0.191213
$255$	−13.1154	−0.821320
$256$	1.00000	0.0625000
$257$	11.1102	0.693036	0.346518	−	0.938043i	$-0.387364\pi$
$257$	11.1102	0.693036	0.346518	+	0.938043i	$0.387364\pi$
$258$	−15.5032	−0.965187
$259$	8.48847	0.527448
$260$	−17.1237	−1.06197
$261$	−27.9033	−1.72717
$262$	−2.21845	−0.137056
$263$	−20.2625	−1.24944	−0.624721	−	0.780848i	$-0.714787\pi$
$263$	−20.2625	−1.24944	−0.624721	+	0.780848i	$0.714787\pi$
$264$	18.5487	1.14159
$265$	59.0835	3.62947
$266$	10.8146	0.663086
$267$	14.2443	0.871738
$268$	−1.85961	−0.113594
$269$	1.65867	0.101131	0.0505653	−	0.998721i	$-0.483898\pi$
$269$	1.65867	0.101131	0.0505653	+	0.998721i	$0.483898\pi$
$270$	−43.9521	−2.67484
$271$	−6.01641	−0.365471	−0.182736	−	0.983162i	$-0.558495\pi$
$271$	−6.01641	−0.365471	−0.182736	+	0.983162i	$0.558495\pi$
$272$	1.00000	0.0606339
$273$	−34.2279	−2.07157
$274$	−3.02880	−0.182976
$275$	81.2496	4.89953
$276$	11.7150	0.705160
$277$	26.2806	1.57905	0.789524	−	0.613720i	$-0.210328\pi$
$277$	26.2806	1.57905	0.789524	+	0.613720i	$0.210328\pi$
$278$	−2.74627	−0.164710
$279$	36.2552	2.17054
$280$	12.0240	0.718569
$281$	−7.18624	−0.428695	−0.214348	−	0.976757i	$-0.568762\pi$
$281$	−7.18624	−0.428695	−0.214348	+	0.976757i	$0.568762\pi$
$282$	24.1483	1.43801
$283$	4.32647	0.257182	0.128591	−	0.991698i	$-0.458955\pi$
$283$	4.32647	0.257182	0.128591	+	0.991698i	$0.458955\pi$
$284$	−6.80271	−0.403667
$285$	50.5935	2.99690
$286$	24.2175	1.43201
$287$	−30.4379	−1.79669
$288$	6.35118	0.374247
$289$	1.00000	0.0588235
$290$	18.8430	1.10650
$291$	−22.0514	−1.29268
$292$	−3.44145	−0.201396
$293$	1.12583	0.0657715	0.0328857	−	0.999459i	$-0.489530\pi$
$293$	1.12583	0.0657715	0.0328857	+	0.999459i	$0.489530\pi$
$294$	2.62843	0.153293
$295$	−4.28893	−0.249711
$296$	−3.02783	−0.175989
$297$	62.1600	3.60689
$298$	−9.94122	−0.575879
$299$	15.2953	0.884548
$300$	40.9613	2.36490
$301$	14.2130	0.819225
$302$	−4.94695	−0.284665
$303$	−24.8653	−1.42847
$304$	−3.85756	−0.221246
$305$	−56.7172	−3.24762
$306$	6.35118	0.363073
$307$	9.10129	0.519438	0.259719	−	0.965684i	$-0.416370\pi$
$307$	9.10129	0.519438	0.259719	+	0.965684i	$0.416370\pi$
$308$	−17.0051	−0.968954
$309$	7.80508	0.444015
$310$	−24.4831	−1.39054
$311$	15.2886	0.866939	0.433469	−	0.901168i	$-0.357289\pi$
$311$	15.2886	0.866939	0.433469	+	0.901168i	$0.357289\pi$
$312$	12.2090	0.691201
$313$	13.5736	0.767227	0.383614	−	0.923494i	$-0.374679\pi$
$313$	13.5736	0.767227	0.383614	+	0.923494i	$0.374679\pi$
$314$	−19.1076	−1.07831
$315$	76.3663	4.30276
$316$	16.8040	0.945301
$317$	−24.4328	−1.37228	−0.686142	−	0.727467i	$-0.740697\pi$
$317$	−24.4328	−1.37228	−0.686142	+	0.727467i	$0.740697\pi$
$318$	−42.1260	−2.36231
$319$	−26.6491	−1.49206
$320$	−4.28893	−0.239759
$321$	−18.3917	−1.02653
$322$	−10.7401	−0.598521
$323$	−3.85756	−0.214640
$324$	12.2839	0.682441
$325$	53.4797	2.96652
$326$	−19.7693	−1.09492
$327$	−51.6012	−2.85355
$328$	10.8572	0.599487
$329$	−22.1387	−1.22054
$330$	−79.5541	−4.37931
$331$	−8.84398	−0.486109	−0.243055	−	0.970013i	$-0.578149\pi$
$331$	−8.84398	−0.486109	−0.243055	+	0.970013i	$0.578149\pi$
$332$	−7.52359	−0.412911
$333$	−19.2303	−1.05381
$334$	−13.0858	−0.716025
$335$	7.97574	0.435761
$336$	−8.57298	−0.467694
$337$	−6.82292	−0.371668	−0.185834	−	0.982581i	$-0.559499\pi$
$337$	−6.82292	−0.371668	−0.185834	+	0.982581i	$0.559499\pi$
$338$	2.94031	0.159932
$339$	29.2698	1.58971
$340$	−4.28893	−0.232600
$341$	34.6256	1.87508
$342$	−24.5001	−1.32481
$343$	17.2147	0.929507
$344$	−5.06977	−0.273344
$345$	−50.2448	−2.70509
$346$	2.79497	0.150258
$347$	−16.2861	−0.874286	−0.437143	−	0.899392i	$-0.644010\pi$
$347$	−16.2861	−0.874286	−0.437143	+	0.899392i	$0.644010\pi$
$348$	−13.4349	−0.720187
$349$	2.39663	0.128289	0.0641444	−	0.997941i	$-0.479568\pi$
$349$	2.39663	0.128289	0.0641444	+	0.997941i	$0.479568\pi$
$350$	−37.5525	−2.00727
$351$	40.9147	2.18386
$352$	6.06569	0.323302
$353$	−28.4366	−1.51353	−0.756763	−	0.653689i	$-0.773220\pi$
$353$	−28.4366	−1.51353	−0.756763	+	0.653689i	$0.773220\pi$
$354$	3.05797	0.162529
$355$	29.1764	1.54852
$356$	4.65810	0.246879
$357$	−8.57298	−0.453730
$358$	−11.6805	−0.617333
$359$	−30.9337	−1.63262	−0.816310	−	0.577614i	$-0.803984\pi$
$359$	−30.9337	−1.63262	−0.816310	+	0.577614i	$0.803984\pi$
$360$	−27.2398	−1.43566
$361$	−4.11924	−0.216802
$362$	14.7710	0.776347
$363$	78.8730	4.13976
$364$	−11.1930	−0.586673
$365$	14.7602	0.772582
$366$	40.4388	2.11377
$367$	23.5977	1.23179	0.615894	−	0.787829i	$-0.288795\pi$
$367$	23.5977	1.23179	0.615894	+	0.787829i	$0.288795\pi$
$368$	3.83097	0.199703
$369$	68.9558	3.58970
$370$	12.9862	0.675118
$371$	38.6203	2.00506
$372$	17.4562	0.905062
$373$	24.5002	1.26857	0.634287	−	0.773097i	$-0.281294\pi$
$373$	24.5002	1.26857	0.634287	+	0.773097i	$0.281294\pi$
$374$	6.06569	0.313649
$375$	−110.103	−5.68570
$376$	7.89683	0.407248
$377$	−17.5408	−0.903398
$378$	−28.7296	−1.47769
$379$	26.5562	1.36410	0.682050	−	0.731306i	$-0.261089\pi$
$379$	26.5562	1.36410	0.682050	+	0.731306i	$0.261089\pi$
$380$	16.5448	0.848731
$381$	9.31894	0.477424
$382$	−8.07995	−0.413406
$383$	−1.63551	−0.0835705	−0.0417852	−	0.999127i	$-0.513305\pi$
$383$	−1.63551	−0.0835705	−0.0417852	+	0.999127i	$0.513305\pi$
$384$	3.05797	0.156051
$385$	72.9336	3.71704
$386$	8.00222	0.407302
$387$	−32.1990	−1.63677
$388$	−7.21113	−0.366089
$389$	23.2404	1.17833	0.589167	−	0.808011i	$-0.299456\pi$
$389$	23.2404	1.17833	0.589167	+	0.808011i	$0.299456\pi$
$390$	−52.3638	−2.65154
$391$	3.83097	0.193741
$392$	0.859534	0.0434130
$393$	−6.78396	−0.342205
$394$	10.0336	0.505483
$395$	−72.0714	−3.62630
$396$	38.5243	1.93592
$397$	−25.2396	−1.26674	−0.633370	−	0.773849i	$-0.718329\pi$
$397$	−25.2396	−1.26674	−0.633370	+	0.773849i	$0.718329\pi$
$398$	−0.774708	−0.0388326
$399$	33.0708	1.65561
$400$	13.3949	0.669747
$401$	−13.3824	−0.668285	−0.334142	−	0.942523i	$-0.608447\pi$
$401$	−13.3824	−0.668285	−0.334142	+	0.942523i	$0.608447\pi$
$402$	−5.68663	−0.283624
$403$	22.7911	1.13530
$404$	−8.13131	−0.404548
$405$	−52.6850	−2.61794
$406$	12.3169	0.611275
$407$	−18.3659	−0.910362
$408$	3.05797	0.151392
$409$	−6.95907	−0.344104	−0.172052	−	0.985088i	$-0.555040\pi$
$409$	−6.95907	−0.344104	−0.172052	+	0.985088i	$0.555040\pi$
$410$	−46.5657	−2.29971
$411$	−9.26197	−0.456859
$412$	2.55237	0.125746
$413$	−2.80349	−0.137951
$414$	24.3312	1.19581
$415$	32.2682	1.58398
$416$	3.99253	0.195750
$417$	−8.39800	−0.411252
$418$	−23.3988	−1.14447
$419$	−15.1412	−0.739696	−0.369848	−	0.929092i	$-0.620590\pi$
$419$	−15.1412	−0.739696	−0.369848	+	0.929092i	$0.620590\pi$
$420$	36.7689	1.79414
$421$	3.24418	0.158111	0.0790557	−	0.996870i	$-0.474809\pi$
$421$	3.24418	0.158111	0.0790557	+	0.996870i	$0.474809\pi$
$422$	3.84503	0.187173
$423$	50.1542	2.43858
$424$	−13.7758	−0.669012
$425$	13.3949	0.649750
$426$	−20.8025	−1.00788
$427$	−37.0735	−1.79411
$428$	−6.01436	−0.290715
$429$	74.0563	3.57547
$430$	21.7439	1.04858
$431$	40.0755	1.93037	0.965185	−	0.261568i	$-0.0842397\pi$
$431$	40.0755	1.93037	0.965185	+	0.261568i	$0.0842397\pi$
$432$	10.2478	0.493048
$433$	8.65215	0.415796	0.207898	−	0.978151i	$-0.433338\pi$
$433$	8.65215	0.415796	0.207898	+	0.978151i	$0.433338\pi$
$434$	−16.0035	−0.768192
$435$	57.6214	2.76274
$436$	−16.8743	−0.808134
$437$	−14.7782	−0.706937
$438$	−10.5239	−0.502849
$439$	33.5814	1.60275	0.801376	−	0.598161i	$-0.204102\pi$
$439$	33.5814	1.60275	0.801376	+	0.598161i	$0.204102\pi$
$440$	−26.0153	−1.24023
$441$	5.45906	0.259955
$442$	3.99253	0.189905
$443$	−3.87539	−0.184125	−0.0920627	−	0.995753i	$-0.529346\pi$
$443$	−3.87539	−0.184125	−0.0920627	+	0.995753i	$0.529346\pi$
$444$	−9.25901	−0.439413
$445$	−19.9783	−0.947061
$446$	−17.9845	−0.851592
$447$	−30.3999	−1.43787
$448$	−2.80349	−0.132452
$449$	−9.05625	−0.427391	−0.213695	−	0.976900i	$-0.568550\pi$
$449$	−9.05625	−0.427391	−0.213695	+	0.976900i	$0.568550\pi$
$450$	85.0737	4.01041
$451$	65.8562	3.10105
$452$	9.57163	0.450212
$453$	−15.1276	−0.710758
$454$	−21.5278	−1.01035
$455$	48.0061	2.25056
$456$	−11.7963	−0.552412
$457$	17.1172	0.800709	0.400355	−	0.916360i	$-0.368887\pi$
$457$	17.1172	0.800709	0.400355	+	0.916360i	$0.368887\pi$
$458$	−3.55606	−0.166164
$459$	10.2478	0.478327
$460$	−16.4308	−0.766089
$461$	9.04227	0.421141	0.210570	−	0.977579i	$-0.432468\pi$
$461$	9.04227	0.421141	0.210570	+	0.977579i	$0.432468\pi$
$462$	−52.0010	−2.41931
$463$	32.2414	1.49839	0.749193	−	0.662351i	$-0.230441\pi$
$463$	32.2414	1.49839	0.749193	+	0.662351i	$0.230441\pi$
$464$	−4.39341	−0.203959
$465$	−74.8685	−3.47194
$466$	7.40038	0.342816
$467$	−0.712792	−0.0329841	−0.0164920	−	0.999864i	$-0.505250\pi$
$467$	−0.712792	−0.0329841	−0.0164920	+	0.999864i	$0.505250\pi$
$468$	25.3573	1.17214
$469$	5.21339	0.240732
$470$	−33.8690	−1.56226
$471$	−58.4305	−2.69234
$472$	1.00000	0.0460287
$473$	−30.7516	−1.41396
$474$	51.3862	2.36025
$475$	−51.6718	−2.37086
$476$	−2.80349	−0.128498
$477$	−87.4926	−4.00601
$478$	−14.9412	−0.683395
$479$	37.9374	1.73340	0.866701	−	0.498828i	$-0.166236\pi$
$479$	37.9374	1.73340	0.866701	+	0.498828i	$0.166236\pi$
$480$	−13.1154	−0.598635
$481$	−12.0887	−0.551197
$482$	18.9134	0.861482
$483$	−32.8428	−1.49440
$484$	25.7926	1.17239
$485$	30.9280	1.40437
$486$	6.82051	0.309385
$487$	4.47524	0.202792	0.101396	−	0.994846i	$-0.467669\pi$
$487$	4.47524	0.202792	0.101396	+	0.994846i	$0.467669\pi$
$488$	13.2241	0.598626
$489$	−60.4540	−2.73383
$490$	−3.68649	−0.166538
$491$	7.39586	0.333770	0.166885	−	0.985976i	$-0.446629\pi$
$491$	7.39586	0.333770	0.166885	+	0.985976i	$0.446629\pi$
$492$	33.2009	1.49681
$493$	−4.39341	−0.197869
$494$	−15.4014	−0.692943
$495$	−165.228	−7.42645
$496$	5.70843	0.256316
$497$	19.0713	0.855465
$498$	−23.0069	−1.03096
$499$	−29.1625	−1.30549	−0.652745	−	0.757577i	$-0.726383\pi$
$499$	−29.1625	−1.30549	−0.652745	+	0.757577i	$0.726383\pi$
$500$	−36.0053	−1.61021
$501$	−40.0161	−1.78779
$502$	10.0114	0.446832
$503$	−26.4069	−1.17742	−0.588712	−	0.808343i	$-0.700365\pi$
$503$	−26.4069	−1.17742	−0.588712	+	0.808343i	$0.700365\pi$
$504$	−17.8054	−0.793118
$505$	34.8747	1.55190
$506$	23.2375	1.03303
$507$	8.99138	0.399321
$508$	3.04743	0.135208
$509$	11.0050	0.487789	0.243895	−	0.969802i	$-0.421575\pi$
$509$	11.0050	0.487789	0.243895	+	0.969802i	$0.421575\pi$
$510$	−13.1154	−0.580761
$511$	9.64806	0.426805
$512$	1.00000	0.0441942
$513$	−39.5315	−1.74536
$514$	11.1102	0.490050
$515$	−10.9470	−0.482380
$516$	−15.5032	−0.682490
$517$	47.8997	2.10663
$518$	8.48847	0.372962
$519$	8.54692	0.375168
$520$	−17.1237	−0.750924
$521$	−10.8811	−0.476708	−0.238354	−	0.971178i	$-0.576608\pi$
$521$	−10.8811	−0.476708	−0.238354	+	0.971178i	$0.576608\pi$
$522$	−27.9033	−1.22130
$523$	13.9127	0.608362	0.304181	−	0.952614i	$-0.401617\pi$
$523$	13.9127	0.608362	0.304181	+	0.952614i	$0.401617\pi$
$524$	−2.21845	−0.0969135
$525$	−114.834	−5.01179
$526$	−20.2625	−0.883489
$527$	5.70843	0.248663
$528$	18.5487	0.807228
$529$	−8.32366	−0.361898
$530$	59.0835	2.56642
$531$	6.35118	0.275618
$532$	10.8146	0.468873
$533$	43.3476	1.87759
$534$	14.2443	0.616412
$535$	25.7952	1.11522
$536$	−1.85961	−0.0803230
$537$	−35.7186	−1.54137
$538$	1.65867	0.0715101
$539$	5.21367	0.224569
$540$	−43.9521	−1.89140
$541$	10.5101	0.451864	0.225932	−	0.974143i	$-0.427457\pi$
$541$	10.5101	0.451864	0.225932	+	0.974143i	$0.427457\pi$
$542$	−6.01641	−0.258427
$543$	45.1693	1.93840
$544$	1.00000	0.0428746
$545$	72.3729	3.10011
$546$	−34.2279	−1.46482
$547$	−8.57677	−0.366716	−0.183358	−	0.983046i	$-0.558697\pi$
$547$	−8.57677	−0.366716	−0.183358	+	0.983046i	$0.558697\pi$
$548$	−3.02880	−0.129384
$549$	83.9885	3.58454
$550$	81.2496	3.46449
$551$	16.9478	0.722002
$552$	11.7150	0.498623
$553$	−47.1099	−2.00332
$554$	26.2806	1.11655
$555$	39.7113	1.68565
$556$	−2.74627	−0.116468
$557$	8.31886	0.352481	0.176241	−	0.984347i	$-0.443606\pi$
$557$	8.31886	0.352481	0.176241	+	0.984347i	$0.443606\pi$
$558$	36.2552	1.53481
$559$	−20.2412	−0.856112
$560$	12.0240	0.508105
$561$	18.5487	0.783127
$562$	−7.18624	−0.303133
$563$	25.3924	1.07016	0.535080	−	0.844801i	$-0.320281\pi$
$563$	25.3924	1.07016	0.535080	+	0.844801i	$0.320281\pi$
$564$	24.1483	1.01683
$565$	−41.0521	−1.72707
$566$	4.32647	0.181855
$567$	−34.4379	−1.44625
$568$	−6.80271	−0.285436
$569$	13.3611	0.560128	0.280064	−	0.959981i	$-0.409644\pi$
$569$	13.3611	0.560128	0.280064	+	0.959981i	$0.409644\pi$
$570$	50.5935	2.11913
$571$	16.7260	0.699961	0.349980	−	0.936757i	$-0.386188\pi$
$571$	16.7260	0.699961	0.349980	+	0.936757i	$0.386188\pi$
$572$	24.2175	1.01258
$573$	−24.7082	−1.03220
$574$	−30.4379	−1.27045
$575$	51.3156	2.14001
$576$	6.35118	0.264632
$577$	2.07819	0.0865163	0.0432582	−	0.999064i	$-0.486226\pi$
$577$	2.07819	0.0865163	0.0432582	+	0.999064i	$0.486226\pi$
$578$	1.00000	0.0415945
$579$	24.4705	1.01696
$580$	18.8430	0.782414
$581$	21.0923	0.875055
$582$	−22.0514	−0.914060
$583$	−83.5597	−3.46069
$584$	−3.44145	−0.142408
$585$	−108.756	−4.49649
$586$	1.12583	0.0465074
$587$	25.3334	1.04562	0.522811	−	0.852448i	$-0.324883\pi$
$587$	25.3334	1.04562	0.522811	+	0.852448i	$0.324883\pi$
$588$	2.62843	0.108395
$589$	−22.0206	−0.907343
$590$	−4.28893	−0.176573
$591$	30.6823	1.26210
$592$	−3.02783	−0.124443
$593$	7.04701	0.289386	0.144693	−	0.989477i	$-0.453781\pi$
$593$	7.04701	0.289386	0.144693	+	0.989477i	$0.453781\pi$
$594$	62.1600	2.55046
$595$	12.0240	0.492934
$596$	−9.94122	−0.407208
$597$	−2.36903	−0.0969581
$598$	15.2953	0.625470
$599$	−31.3748	−1.28194	−0.640969	−	0.767567i	$-0.721467\pi$
$599$	−31.3748	−1.28194	−0.640969	+	0.767567i	$0.721467\pi$
$600$	40.9613	1.67224
$601$	8.48175	0.345978	0.172989	−	0.984924i	$-0.444658\pi$
$601$	8.48175	0.345978	0.172989	+	0.984924i	$0.444658\pi$
$602$	14.2130	0.579279
$603$	−11.8107	−0.480970
$604$	−4.94695	−0.201289
$605$	−110.623	−4.49745
$606$	−24.8653	−1.01008
$607$	4.76466	0.193391	0.0966957	−	0.995314i	$-0.469173\pi$
$607$	4.76466	0.193391	0.0966957	+	0.995314i	$0.469173\pi$
$608$	−3.85756	−0.156445
$609$	37.6646	1.52625
$610$	−56.7172	−2.29641
$611$	31.5284	1.27550
$612$	6.35118	0.256731
$613$	29.8370	1.20511	0.602553	−	0.798079i	$-0.294150\pi$
$613$	29.8370	1.20511	0.602553	+	0.798079i	$0.294150\pi$
$614$	9.10129	0.367298
$615$	−142.396	−5.74198
$616$	−17.0051	−0.685154
$617$	−26.2341	−1.05615	−0.528073	−	0.849199i	$-0.677085\pi$
$617$	−26.2341	−1.05615	−0.528073	+	0.849199i	$0.677085\pi$
$618$	7.80508	0.313966
$619$	0.0302691	0.00121662	0.000608309	−	1.00000i	$-0.499806\pi$
$619$	0.0302691	0.00121662	0.000608309		1.00000i	$0.499806\pi$
$620$	−24.4831	−0.983263
$621$	39.2590	1.57541
$622$	15.2886	0.613018
$623$	−13.0589	−0.523194
$624$	12.2090	0.488753
$625$	87.4497	3.49799
$626$	13.5736	0.542511
$627$	−71.5527	−2.85754
$628$	−19.1076	−0.762477
$629$	−3.02783	−0.120727
$630$	76.3663	3.04251
$631$	−6.41636	−0.255431	−0.127716	−	0.991811i	$-0.540765\pi$
$631$	−6.41636	−0.255431	−0.127716	+	0.991811i	$0.540765\pi$
$632$	16.8040	0.668429
$633$	11.7580	0.467338
$634$	−24.4328	−0.970352
$635$	−13.0702	−0.518675
$636$	−42.1260	−1.67040
$637$	3.43172	0.135970
$638$	−26.6491	−1.05505
$639$	−43.2053	−1.70917
$640$	−4.28893	−0.169535
$641$	10.3195	0.407597	0.203798	−	0.979013i	$-0.434671\pi$
$641$	10.3195	0.407597	0.203798	+	0.979013i	$0.434671\pi$
$642$	−18.3917	−0.725863
$643$	31.9204	1.25882	0.629409	−	0.777074i	$-0.283297\pi$
$643$	31.9204	1.25882	0.629409	+	0.777074i	$0.283297\pi$
$644$	−10.7401	−0.423218
$645$	66.4921	2.61813
$646$	−3.85756	−0.151774
$647$	11.4467	0.450017	0.225008	−	0.974357i	$-0.427759\pi$
$647$	11.4467	0.450017	0.225008	+	0.974357i	$0.427759\pi$
$648$	12.2839	0.482559
$649$	6.06569	0.238099
$650$	53.4797	2.09765
$651$	−48.9382	−1.91804
$652$	−19.7693	−0.774227
$653$	23.9881	0.938726	0.469363	−	0.883005i	$-0.344484\pi$
$653$	23.9881	0.938726	0.469363	+	0.883005i	$0.344484\pi$
$654$	−51.6012	−2.01777
$655$	9.51479	0.371773
$656$	10.8572	0.423901
$657$	−21.8573	−0.852734
$658$	−22.1387	−0.863055
$659$	−32.9941	−1.28527	−0.642633	−	0.766174i	$-0.722158\pi$
$659$	−32.9941	−1.28527	−0.642633	+	0.766174i	$0.722158\pi$
$660$	−79.5541	−3.09664
$661$	−27.6139	−1.07406	−0.537029	−	0.843564i	$-0.680453\pi$
$661$	−27.6139	−1.07406	−0.537029	+	0.843564i	$0.680453\pi$
$662$	−8.84398	−0.343731
$663$	12.2090	0.474160
$664$	−7.52359	−0.291972
$665$	−46.3831	−1.79866
$666$	−19.2303	−0.745158
$667$	−16.8310	−0.651700
$668$	−13.0858	−0.506306
$669$	−54.9961	−2.12627
$670$	7.97574	0.308130
$671$	80.2132	3.09660
$672$	−8.57298	−0.330710
$673$	−28.6365	−1.10385	−0.551927	−	0.833892i	$-0.686107\pi$
$673$	−28.6365	−1.10385	−0.551927	+	0.833892i	$0.686107\pi$
$674$	−6.82292	−0.262809
$675$	137.269	5.28348
$676$	2.94031	0.113089
$677$	−21.1356	−0.812307	−0.406154	−	0.913805i	$-0.633130\pi$
$677$	−21.1356	−0.812307	−0.406154	+	0.913805i	$0.633130\pi$
$678$	29.2698	1.12410
$679$	20.2163	0.775830
$680$	−4.28893	−0.164473
$681$	−65.8313	−2.52266
$682$	34.6256	1.32588
$683$	−40.4316	−1.54707	−0.773536	−	0.633752i	$-0.781514\pi$
$683$	−40.4316	−1.54707	−0.773536	+	0.633752i	$0.781514\pi$
$684$	−24.5001	−0.936783
$685$	12.9903	0.496334
$686$	17.2147	0.657261
$687$	−10.8743	−0.414881
$688$	−5.06977	−0.193283
$689$	−55.0003	−2.09535
$690$	−50.2448	−1.91279
$691$	22.4583	0.854355	0.427178	−	0.904168i	$-0.359508\pi$
$691$	22.4583	0.854355	0.427178	+	0.904168i	$0.359508\pi$
$692$	2.79497	0.106249
$693$	−108.002	−4.10267
$694$	−16.2861	−0.618214
$695$	11.7786	0.446786
$696$	−13.4349	−0.509249
$697$	10.8572	0.411245
$698$	2.39663	0.0907139
$699$	22.6301	0.855951
$700$	−37.5525	−1.41935
$701$	−30.7910	−1.16296	−0.581479	−	0.813561i	$-0.697526\pi$
$701$	−30.7910	−1.16296	−0.581479	+	0.813561i	$0.697526\pi$
$702$	40.9147	1.54423
$703$	11.6800	0.440521
$704$	6.06569	0.228609
$705$	−103.570	−3.90068
$706$	−28.4366	−1.07022
$707$	22.7960	0.857333
$708$	3.05797	0.114926
$709$	−18.8029	−0.706158	−0.353079	−	0.935593i	$-0.614865\pi$
$709$	−18.8029	−0.706158	−0.353079	+	0.935593i	$0.614865\pi$
$710$	29.1764	1.09497
$711$	106.725	4.00252
$712$	4.65810	0.174570
$713$	21.8688	0.818994
$714$	−8.57298	−0.320836
$715$	−103.867	−3.88441
$716$	−11.6805	−0.436521
$717$	−45.6897	−1.70631
$718$	−30.9337	−1.15444
$719$	−25.2986	−0.943480	−0.471740	−	0.881738i	$-0.656374\pi$
$719$	−25.2986	−0.943480	−0.471740	+	0.881738i	$0.656374\pi$
$720$	−27.2398	−1.01517
$721$	−7.15554	−0.266486
$722$	−4.11924	−0.153302
$723$	57.8366	2.15097
$724$	14.7710	0.548960
$725$	−58.8494	−2.18561
$726$	78.8730	2.92725
$727$	−11.3521	−0.421025	−0.210512	−	0.977591i	$-0.567513\pi$
$727$	−11.3521	−0.421025	−0.210512	+	0.977591i	$0.567513\pi$
$728$	−11.1930	−0.414840
$729$	−15.9949	−0.592404
$730$	14.7602	0.546298
$731$	−5.06977	−0.187512
$732$	40.4388	1.49466
$733$	46.0187	1.69974	0.849869	−	0.526994i	$-0.176681\pi$
$733$	46.0187	1.69974	0.849869	+	0.526994i	$0.176681\pi$
$734$	23.5977	0.871006
$735$	−11.2732	−0.415817
$736$	3.83097	0.141211
$737$	−11.2798	−0.415498
$738$	68.9558	2.53830
$739$	6.26823	0.230580	0.115290	−	0.993332i	$-0.463220\pi$
$739$	6.26823	0.230580	0.115290	+	0.993332i	$0.463220\pi$
$740$	12.9862	0.477380
$741$	−47.0971	−1.73016
$742$	38.6203	1.41779
$743$	−36.5670	−1.34151	−0.670756	−	0.741678i	$-0.734030\pi$
$743$	−36.5670	−1.34151	−0.670756	+	0.741678i	$0.734030\pi$
$744$	17.4562	0.639975
$745$	42.6372	1.56211
$746$	24.5002	0.897018
$747$	−47.7837	−1.74831
$748$	6.06569	0.221784
$749$	16.8612	0.616094
$750$	−110.103	−4.02040
$751$	−32.6307	−1.19071	−0.595355	−	0.803463i	$-0.702989\pi$
$751$	−32.6307	−1.19071	−0.595355	+	0.803463i	$0.702989\pi$
$752$	7.89683	0.287968
$753$	30.6146	1.11566
$754$	−17.5408	−0.638799
$755$	21.2171	0.772171
$756$	−28.7296	−1.04488
$757$	−3.29506	−0.119761	−0.0598805	−	0.998206i	$-0.519072\pi$
$757$	−3.29506	−0.119761	−0.0598805	+	0.998206i	$0.519072\pi$
$758$	26.5562	0.964564
$759$	71.0595	2.57930
$760$	16.5448	0.600143
$761$	−46.5275	−1.68662	−0.843310	−	0.537427i	$-0.819396\pi$
$761$	−46.5275	−1.68662	−0.843310	+	0.537427i	$0.819396\pi$
$762$	9.31894	0.337590
$763$	47.3070	1.71263
$764$	−8.07995	−0.292322
$765$	−27.2398	−0.984856
$766$	−1.63551	−0.0590933
$767$	3.99253	0.144162
$768$	3.05797	0.110345
$769$	17.0303	0.614127	0.307064	−	0.951689i	$-0.400654\pi$
$769$	17.0303	0.614127	0.307064	+	0.951689i	$0.400654\pi$
$770$	72.9336	2.62835
$771$	33.9747	1.22357
$772$	8.00222	0.288006
$773$	−23.2274	−0.835433	−0.417717	−	0.908577i	$-0.637170\pi$
$773$	−23.2274	−0.835433	−0.417717	+	0.908577i	$0.637170\pi$
$774$	−32.1990	−1.15737
$775$	76.4640	2.74667
$776$	−7.21113	−0.258864
$777$	25.9575	0.931220
$778$	23.2404	0.833208
$779$	−41.8822	−1.50058
$780$	−52.3638	−1.87492
$781$	−41.2632	−1.47651
$782$	3.83097	0.136995
$783$	−45.0228	−1.60898
$784$	0.859534	0.0306977
$785$	81.9513	2.92497
$786$	−6.78396	−0.241976
$787$	−20.6328	−0.735481	−0.367741	−	0.929928i	$-0.619869\pi$
$787$	−20.6328	−0.735481	−0.367741	+	0.929928i	$0.619869\pi$
$788$	10.0336	0.357431
$789$	−61.9622	−2.20591
$790$	−72.0714	−2.56418
$791$	−26.8339	−0.954105
$792$	38.5243	1.36890
$793$	52.7976	1.87490
$794$	−25.2396	−0.895720
$795$	180.675	6.40790
$796$	−0.774708	−0.0274588
$797$	−52.2626	−1.85124	−0.925618	−	0.378458i	$-0.876454\pi$
$797$	−52.2626	−1.85124	−0.925618	+	0.378458i	$0.876454\pi$
$798$	33.0708	1.17069
$799$	7.89683	0.279370
$800$	13.3949	0.473583
$801$	29.5844	1.04531
$802$	−13.3824	−0.472549
$803$	−20.8748	−0.736655
$804$	−5.68663	−0.200552
$805$	46.0634	1.62352
$806$	22.7911	0.802782
$807$	5.07215	0.178548
$808$	−8.13131	−0.286059
$809$	−15.0075	−0.527636	−0.263818	−	0.964573i	$-0.584982\pi$
$809$	−15.0075	−0.527636	−0.263818	+	0.964573i	$0.584982\pi$
$810$	−52.6850	−1.85116
$811$	−7.63924	−0.268250	−0.134125	−	0.990964i	$-0.542822\pi$
$811$	−7.63924	−0.268250	−0.134125	+	0.990964i	$0.542822\pi$
$812$	12.3169	0.432237
$813$	−18.3980	−0.645246
$814$	−18.3659	−0.643723
$815$	84.7893	2.97004
$816$	3.05797	0.107050
$817$	19.5569	0.684210
$818$	−6.95907	−0.243318
$819$	−71.0888	−2.48404
$820$	−46.5657	−1.62614
$821$	27.1717	0.948300	0.474150	−	0.880444i	$-0.342755\pi$
$821$	27.1717	0.948300	0.474150	+	0.880444i	$0.342755\pi$
$822$	−9.26197	−0.323048
$823$	−23.0250	−0.802601	−0.401301	−	0.915946i	$-0.631442\pi$
$823$	−23.0250	−0.802601	−0.401301	+	0.915946i	$0.631442\pi$
$824$	2.55237	0.0889161
$825$	248.459	8.65022
$826$	−2.80349	−0.0975458
$827$	−15.9716	−0.555388	−0.277694	−	0.960670i	$-0.589570\pi$
$827$	−15.9716	−0.555388	−0.277694	+	0.960670i	$0.589570\pi$
$828$	24.3312	0.845567
$829$	26.0039	0.903152	0.451576	−	0.892233i	$-0.350862\pi$
$829$	26.0039	0.903152	0.451576	+	0.892233i	$0.350862\pi$
$830$	32.2682	1.12004
$831$	80.3652	2.78784
$832$	3.99253	0.138416
$833$	0.859534	0.0297811
$834$	−8.39800	−0.290799
$835$	56.1242	1.94226
$836$	−23.3988	−0.809263
$837$	58.4989	2.02202
$838$	−15.1412	−0.523044
$839$	12.0911	0.417431	0.208715	−	0.977976i	$-0.433072\pi$
$839$	12.0911	0.417431	0.208715	+	0.977976i	$0.433072\pi$
$840$	36.7689	1.26865
$841$	−9.69796	−0.334412
$842$	3.24418	0.111802
$843$	−21.9753	−0.756870
$844$	3.84503	0.132351
$845$	−12.6108	−0.433824
$846$	50.1542	1.72434
$847$	−72.3092	−2.48457
$848$	−13.7758	−0.473063
$849$	13.2302	0.454060
$850$	13.3949	0.459443
$851$	−11.5995	−0.397626
$852$	−20.8025	−0.712682
$853$	−30.6705	−1.05014	−0.525070	−	0.851059i	$-0.675961\pi$
$853$	−30.6705	−1.05014	−0.525070	+	0.851059i	$0.675961\pi$
$854$	−37.0735	−1.26863
$855$	105.079	3.59363
$856$	−6.01436	−0.205566
$857$	24.9531	0.852383	0.426191	−	0.904633i	$-0.359855\pi$
$857$	24.9531	0.852383	0.426191	+	0.904633i	$0.359855\pi$
$858$	74.0563	2.52824
$859$	14.5779	0.497391	0.248696	−	0.968582i	$-0.419998\pi$
$859$	14.5779	0.497391	0.248696	+	0.968582i	$0.419998\pi$
$860$	21.7439	0.741460
$861$	−93.0783	−3.17210
$862$	40.0755	1.36498
$863$	10.1487	0.345466	0.172733	−	0.984969i	$-0.444740\pi$
$863$	10.1487	0.345466	0.172733	+	0.984969i	$0.444740\pi$
$864$	10.2478	0.348637
$865$	−11.9874	−0.407585
$866$	8.65215	0.294012
$867$	3.05797	0.103854
$868$	−16.0035	−0.543194
$869$	101.928	3.45767
$870$	57.6214	1.95355
$871$	−7.42456	−0.251571
$872$	−16.8743	−0.571437
$873$	−45.7992	−1.55007
$874$	−14.7782	−0.499880
$875$	100.940	3.41241
$876$	−10.5239	−0.355568
$877$	−15.3749	−0.519173	−0.259586	−	0.965720i	$-0.583586\pi$
$877$	−15.3749	−0.519173	−0.259586	+	0.965720i	$0.583586\pi$
$878$	33.5814	1.13332
$879$	3.44274	0.116121
$880$	−26.0153	−0.876977
$881$	22.3986	0.754626	0.377313	−	0.926086i	$-0.376848\pi$
$881$	22.3986	0.754626	0.377313	+	0.926086i	$0.376848\pi$
$882$	5.45906	0.183816
$883$	20.8231	0.700754	0.350377	−	0.936609i	$-0.386054\pi$
$883$	20.8231	0.700754	0.350377	+	0.936609i	$0.386054\pi$
$884$	3.99253	0.134283
$885$	−13.1154	−0.440870
$886$	−3.87539	−0.130196
$887$	47.7426	1.60304	0.801519	−	0.597969i	$-0.204025\pi$
$887$	47.7426	1.60304	0.801519	+	0.597969i	$0.204025\pi$
$888$	−9.25901	−0.310712
$889$	−8.54342	−0.286537
$890$	−19.9783	−0.669673
$891$	74.5106	2.49620
$892$	−17.9845	−0.602167
$893$	−30.4625	−1.01939
$894$	−30.3999	−1.01673
$895$	50.0969	1.67455
$896$	−2.80349	−0.0936579
$897$	46.7725	1.56169
$898$	−9.05625	−0.302211
$899$	−25.0795	−0.836447
$900$	85.0737	2.83579
$901$	−13.7758	−0.458938
$902$	65.8562	2.19277
$903$	43.4630	1.44636
$904$	9.57163	0.318348
$905$	−63.3518	−2.10589
$906$	−15.1276	−0.502582
$907$	−20.1859	−0.670263	−0.335132	−	0.942171i	$-0.608781\pi$
$907$	−20.1859	−0.670263	−0.335132	+	0.942171i	$0.608781\pi$
$908$	−21.5278	−0.714424
$909$	−51.6434	−1.71290
$910$	48.0061	1.59139
$911$	40.5173	1.34240	0.671199	−	0.741277i	$-0.265779\pi$
$911$	40.5173	1.34240	0.671199	+	0.741277i	$0.265779\pi$
$912$	−11.7963	−0.390615
$913$	−45.6358	−1.51032
$914$	17.1172	0.566187
$915$	−173.439	−5.73373
$916$	−3.55606	−0.117495
$917$	6.21940	0.205383
$918$	10.2478	0.338228
$919$	−11.6929	−0.385713	−0.192856	−	0.981227i	$-0.561775\pi$
$919$	−11.6929	−0.385713	−0.192856	+	0.981227i	$0.561775\pi$
$920$	−16.4308	−0.541707
$921$	27.8315	0.917078
$922$	9.04227	0.297791
$923$	−27.1601	−0.893984
$924$	−52.0010	−1.71071
$925$	−40.5576	−1.33352
$926$	32.2414	1.05952
$927$	16.2106	0.532425
$928$	−4.39341	−0.144221
$929$	−10.7560	−0.352892	−0.176446	−	0.984310i	$-0.556460\pi$
$929$	−10.7560	−0.352892	−0.176446	+	0.984310i	$0.556460\pi$
$930$	−74.8685	−2.45503
$931$	−3.31571	−0.108668
$932$	7.40038	0.242408
$933$	46.7522	1.53060
$934$	−0.712792	−0.0233233
$935$	−26.0153	−0.850792
$936$	25.3573	0.828829
$937$	26.3048	0.859340	0.429670	−	0.902986i	$-0.358630\pi$
$937$	26.3048	0.859340	0.429670	+	0.902986i	$0.358630\pi$
$938$	5.21339	0.170223
$939$	41.5078	1.35455
$940$	−33.8690	−1.10468
$941$	44.4495	1.44901	0.724507	−	0.689268i	$-0.242068\pi$
$941$	44.4495	1.44901	0.724507	+	0.689268i	$0.242068\pi$
$942$	−58.4305	−1.90377
$943$	41.5935	1.35447
$944$	1.00000	0.0325472
$945$	123.219	4.00832
$946$	−30.7516	−0.999822
$947$	26.7059	0.867826	0.433913	−	0.900955i	$-0.357132\pi$
$947$	26.7059	0.867826	0.433913	+	0.900955i	$0.357132\pi$
$948$	51.3862	1.66895
$949$	−13.7401	−0.446023
$950$	−51.6718	−1.67645
$951$	−74.7149	−2.42280
$952$	−2.80349	−0.0908615
$953$	18.8956	0.612088	0.306044	−	0.952017i	$-0.400995\pi$
$953$	18.8956	0.612088	0.306044	+	0.952017i	$0.400995\pi$
$954$	−87.4926	−2.83268
$955$	34.6544	1.12139
$956$	−14.9412	−0.483233
$957$	−81.4920	−2.63426
$958$	37.9374	1.22570
$959$	8.49119	0.274195
$960$	−13.1154	−0.423299
$961$	1.58615	0.0511660
$962$	−12.0887	−0.389755
$963$	−38.1983	−1.23092
$964$	18.9134	0.609160
$965$	−34.3210	−1.10483
$966$	−32.8428	−1.05670
$967$	−29.5500	−0.950265	−0.475132	−	0.879914i	$-0.657600\pi$
$967$	−29.5500	−0.950265	−0.475132	+	0.879914i	$0.657600\pi$
$968$	25.7926	0.829006
$969$	−11.7963	−0.378952
$970$	30.9280	0.993039
$971$	−2.41304	−0.0774381	−0.0387191	−	0.999250i	$-0.512328\pi$
$971$	−2.41304	−0.0774381	−0.0387191	+	0.999250i	$0.512328\pi$
$972$	6.82051	0.218768
$973$	7.69912	0.246823
$974$	4.47524	0.143396
$975$	163.539	5.23745
$976$	13.2241	0.423293
$977$	−19.1483	−0.612609	−0.306305	−	0.951934i	$-0.599093\pi$
$977$	−19.1483	−0.612609	−0.306305	+	0.951934i	$0.599093\pi$
$978$	−60.4540	−1.93311
$979$	28.2546	0.903020
$980$	−3.68649	−0.117760
$981$	−107.172	−3.42174
$982$	7.39586	0.236011
$983$	−45.7179	−1.45817	−0.729087	−	0.684421i	$-0.760055\pi$
$983$	−45.7179	−1.45817	−0.729087	+	0.684421i	$0.760055\pi$
$984$	33.2009	1.05841
$985$	−43.0332	−1.37115
$986$	−4.39341	−0.139915
$987$	−67.6994	−2.15489
$988$	−15.4014	−0.489985
$989$	−19.4221	−0.617588
$990$	−165.228	−5.25129
$991$	20.8674	0.662874	0.331437	−	0.943477i	$-0.392467\pi$
$991$	20.8674	0.662874	0.331437	+	0.943477i	$0.392467\pi$
$992$	5.70843	0.181243
$993$	−27.0446	−0.858236
$994$	19.0713	0.604905
$995$	3.32267	0.105336
$996$	−23.0069	−0.729002
$997$	16.7988	0.532023	0.266011	−	0.963970i	$-0.414294\pi$
$997$	16.7988	0.532023	0.266011	+	0.963970i	$0.414294\pi$
$998$	−29.1625	−0.923121
$999$	−31.0286	−0.981701

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2006.2.a.w.1.12	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2006.2.a.w.1.12	✓	13	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:	\( N \)	\(=\)	\( 2006 = 2 \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.05797 q^{3} +1.00000 q^{4} -4.28893 q^{5} +3.05797 q^{6} -2.80349 q^{7} +1.00000 q^{8} +6.35118 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +3.05797 q^{3} +1.00000 q^{4} -4.28893 q^{5} +3.05797 q^{6} -2.80349 q^{7} +1.00000 q^{8} +6.35118 q^{9} -4.28893 q^{10} +6.06569 q^{11} +3.05797 q^{12} +3.99253 q^{13} -2.80349 q^{14} -13.1154 q^{15} +1.00000 q^{16} +1.00000 q^{17} +6.35118 q^{18} -3.85756 q^{19} -4.28893 q^{20} -8.57298 q^{21} +6.06569 q^{22} +3.83097 q^{23} +3.05797 q^{24} +13.3949 q^{25} +3.99253 q^{26} +10.2478 q^{27} -2.80349 q^{28} -4.39341 q^{29} -13.1154 q^{30} +5.70843 q^{31} +1.00000 q^{32} +18.5487 q^{33} +1.00000 q^{34} +12.0240 q^{35} +6.35118 q^{36} -3.02783 q^{37} -3.85756 q^{38} +12.2090 q^{39} -4.28893 q^{40} +10.8572 q^{41} -8.57298 q^{42} -5.06977 q^{43} +6.06569 q^{44} -27.2398 q^{45} +3.83097 q^{46} +7.89683 q^{47} +3.05797 q^{48} +0.859534 q^{49} +13.3949 q^{50} +3.05797 q^{51} +3.99253 q^{52} -13.7758 q^{53} +10.2478 q^{54} -26.0153 q^{55} -2.80349 q^{56} -11.7963 q^{57} -4.39341 q^{58} +1.00000 q^{59} -13.1154 q^{60} +13.2241 q^{61} +5.70843 q^{62} -17.8054 q^{63} +1.00000 q^{64} -17.1237 q^{65} +18.5487 q^{66} -1.85961 q^{67} +1.00000 q^{68} +11.7150 q^{69} +12.0240 q^{70} -6.80271 q^{71} +6.35118 q^{72} -3.44145 q^{73} -3.02783 q^{74} +40.9613 q^{75} -3.85756 q^{76} -17.0051 q^{77} +12.2090 q^{78} +16.8040 q^{79} -4.28893 q^{80} +12.2839 q^{81} +10.8572 q^{82} -7.52359 q^{83} -8.57298 q^{84} -4.28893 q^{85} -5.06977 q^{86} -13.4349 q^{87} +6.06569 q^{88} +4.65810 q^{89} -27.2398 q^{90} -11.1930 q^{91} +3.83097 q^{92} +17.4562 q^{93} +7.89683 q^{94} +16.5448 q^{95} +3.05797 q^{96} -7.21113 q^{97} +0.859534 q^{98} +38.5243 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{2} + 7 q^{3} + 13 q^{4} + q^{5} + 7 q^{6} + 8 q^{7} + 13 q^{8} + 16 q^{9}+O(q^{10}) \) 13 * q + 13 * q^2 + 7 * q^3 + 13 * q^4 + q^5 + 7 * q^6 + 8 * q^7 + 13 * q^8 + 16 * q^9	\( 13 q + 13 q^{2} + 7 q^{3} + 13 q^{4} + q^{5} + 7 q^{6} + 8 q^{7} + 13 q^{8} + 16 q^{9} + q^{10} + 13 q^{11} + 7 q^{12} + 9 q^{13} + 8 q^{14} - 9 q^{15} + 13 q^{16} + 13 q^{17} + 16 q^{18} + 16 q^{19} + q^{20} - 3 q^{21} + 13 q^{22} + 18 q^{23} + 7 q^{24} + 14 q^{25} + 9 q^{26} + 10 q^{27} + 8 q^{28} + 8 q^{29} - 9 q^{30} + 32 q^{31} + 13 q^{32} + 13 q^{33} + 13 q^{34} - q^{35} + 16 q^{36} - 3 q^{37} + 16 q^{38} + 20 q^{39} + q^{40} + 24 q^{41} - 3 q^{42} - 2 q^{43} + 13 q^{44} - 3 q^{45} + 18 q^{46} + 26 q^{47} + 7 q^{48} + 23 q^{49} + 14 q^{50} + 7 q^{51} + 9 q^{52} - 27 q^{53} + 10 q^{54} - 7 q^{55} + 8 q^{56} - 21 q^{57} + 8 q^{58} + 13 q^{59} - 9 q^{60} + 20 q^{61} + 32 q^{62} - 3 q^{63} + 13 q^{64} + 16 q^{65} + 13 q^{66} + 4 q^{67} + 13 q^{68} - 2 q^{69} - q^{70} - 8 q^{71} + 16 q^{72} + 10 q^{73} - 3 q^{74} + 39 q^{75} + 16 q^{76} - 15 q^{77} + 20 q^{78} + 35 q^{79} + q^{80} + 29 q^{81} + 24 q^{82} - q^{83} - 3 q^{84} + q^{85} - 2 q^{86} - 32 q^{87} + 13 q^{88} - 20 q^{89} - 3 q^{90} - 15 q^{91} + 18 q^{92} - 7 q^{93} + 26 q^{94} - 3 q^{95} + 7 q^{96} - 3 q^{97} + 23 q^{98} + 65 q^{99}+O(q^{100}) \) 13 * q + 13 * q^2 + 7 * q^3 + 13 * q^4 + q^5 + 7 * q^6 + 8 * q^7 + 13 * q^8 + 16 * q^9 + q^10 + 13 * q^11 + 7 * q^12 + 9 * q^13 + 8 * q^14 - 9 * q^15 + 13 * q^16 + 13 * q^17 + 16 * q^18 + 16 * q^19 + q^20 - 3 * q^21 + 13 * q^22 + 18 * q^23 + 7 * q^24 + 14 * q^25 + 9 * q^26 + 10 * q^27 + 8 * q^28 + 8 * q^29 - 9 * q^30 + 32 * q^31 + 13 * q^32 + 13 * q^33 + 13 * q^34 - q^35 + 16 * q^36 - 3 * q^37 + 16 * q^38 + 20 * q^39 + q^40 + 24 * q^41 - 3 * q^42 - 2 * q^43 + 13 * q^44 - 3 * q^45 + 18 * q^46 + 26 * q^47 + 7 * q^48 + 23 * q^49 + 14 * q^50 + 7 * q^51 + 9 * q^52 - 27 * q^53 + 10 * q^54 - 7 * q^55 + 8 * q^56 - 21 * q^57 + 8 * q^58 + 13 * q^59 - 9 * q^60 + 20 * q^61 + 32 * q^62 - 3 * q^63 + 13 * q^64 + 16 * q^65 + 13 * q^66 + 4 * q^67 + 13 * q^68 - 2 * q^69 - q^70 - 8 * q^71 + 16 * q^72 + 10 * q^73 - 3 * q^74 + 39 * q^75 + 16 * q^76 - 15 * q^77 + 20 * q^78 + 35 * q^79 + q^80 + 29 * q^81 + 24 * q^82 - q^83 - 3 * q^84 + q^85 - 2 * q^86 - 32 * q^87 + 13 * q^88 - 20 * q^89 - 3 * q^90 - 15 * q^91 + 18 * q^92 - 7 * q^93 + 26 * q^94 - 3 * q^95 + 7 * q^96 - 3 * q^97 + 23 * q^98 + 65 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more