LMFDB - Embedded newform 4013.2.a.b.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4013,2,Mod(1,4013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4013, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4013 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4013.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:	$32.0439663311$
Analytic rank:	$1$
Dimension:	$157$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51500 q^{2} -3.29902 q^{3} +4.32520 q^{4} -0.150822 q^{5} +8.29703 q^{6} -3.52495 q^{7} -5.84787 q^{8} +7.88356 q^{9} +O(q^{10})$	\(q-2.51500 q^{2} -3.29902 q^{3} +4.32520 q^{4} -0.150822 q^{5} +8.29703 q^{6} -3.52495 q^{7} -5.84787 q^{8} +7.88356 q^{9} +0.379318 q^{10} +1.90159 q^{11} -14.2689 q^{12} +3.94141 q^{13} +8.86522 q^{14} +0.497567 q^{15} +6.05697 q^{16} +7.17750 q^{17} -19.8271 q^{18} -0.695126 q^{19} -0.652337 q^{20} +11.6289 q^{21} -4.78248 q^{22} -7.98884 q^{23} +19.2923 q^{24} -4.97725 q^{25} -9.91263 q^{26} -16.1110 q^{27} -15.2461 q^{28} -3.37498 q^{29} -1.25138 q^{30} +5.48811 q^{31} -3.53750 q^{32} -6.27338 q^{33} -18.0514 q^{34} +0.531641 q^{35} +34.0980 q^{36} -1.10392 q^{37} +1.74824 q^{38} -13.0028 q^{39} +0.881990 q^{40} +4.65892 q^{41} -29.2466 q^{42} +2.81935 q^{43} +8.22474 q^{44} -1.18902 q^{45} +20.0919 q^{46} +2.50183 q^{47} -19.9821 q^{48} +5.42525 q^{49} +12.5178 q^{50} -23.6788 q^{51} +17.0474 q^{52} -7.70124 q^{53} +40.5191 q^{54} -0.286802 q^{55} +20.6134 q^{56} +2.29324 q^{57} +8.48807 q^{58} -12.7793 q^{59} +2.15208 q^{60} +4.56957 q^{61} -13.8026 q^{62} -27.7891 q^{63} -3.21714 q^{64} -0.594453 q^{65} +15.7775 q^{66} -12.5575 q^{67} +31.0442 q^{68} +26.3554 q^{69} -1.33707 q^{70} -0.590444 q^{71} -46.1021 q^{72} -8.48233 q^{73} +2.77634 q^{74} +16.4201 q^{75} -3.00656 q^{76} -6.70299 q^{77} +32.7020 q^{78} +9.09185 q^{79} -0.913526 q^{80} +29.4999 q^{81} -11.7172 q^{82} +1.99877 q^{83} +50.2973 q^{84} -1.08253 q^{85} -7.09065 q^{86} +11.1342 q^{87} -11.1202 q^{88} +6.75561 q^{89} +2.99037 q^{90} -13.8933 q^{91} -34.5534 q^{92} -18.1054 q^{93} -6.29208 q^{94} +0.104841 q^{95} +11.6703 q^{96} -9.29359 q^{97} -13.6445 q^{98} +14.9913 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9}+O(q^{10}) $ 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9	\( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9} - 61 q^{10} - 27 q^{11} - 93 q^{12} - 97 q^{13} - 12 q^{14} - 36 q^{15} + 105 q^{16} - 45 q^{17} - 68 q^{18} - 128 q^{19} - 30 q^{20} - 26 q^{21} - 68 q^{22} - 41 q^{23} - 40 q^{24} + 102 q^{25} - 5 q^{26} - 189 q^{27} - 115 q^{28} - 26 q^{29} - 12 q^{30} - 88 q^{31} - 89 q^{32} - 52 q^{33} - 61 q^{34} - 87 q^{35} + 110 q^{36} - 62 q^{37} - 37 q^{38} - 20 q^{39} - 161 q^{40} - 34 q^{41} - 53 q^{42} - 254 q^{43} - 19 q^{44} - 46 q^{45} - 52 q^{46} - 76 q^{47} - 162 q^{48} + 96 q^{49} - 54 q^{50} - 76 q^{51} - 259 q^{52} - 48 q^{53} - 12 q^{54} - 194 q^{55} - 10 q^{56} - 30 q^{57} - 52 q^{58} - 64 q^{59} - 31 q^{60} - 107 q^{61} - 51 q^{62} - 106 q^{63} + 54 q^{64} - 17 q^{65} - 13 q^{66} - 193 q^{67} - 118 q^{68} - 55 q^{69} - 86 q^{70} - 11 q^{71} - 172 q^{72} - 173 q^{73} - 11 q^{74} - 209 q^{75} - 213 q^{76} - 84 q^{77} - 30 q^{78} - 111 q^{79} - 6 q^{80} + 157 q^{81} - 117 q^{82} - 154 q^{83} - 6 q^{84} - 91 q^{85} + 28 q^{86} - 165 q^{87} - 165 q^{88} - 32 q^{89} - 103 q^{90} - 200 q^{91} - 86 q^{92} - 39 q^{93} - 118 q^{94} - 22 q^{95} - 28 q^{96} - 151 q^{97} - 38 q^{98} - 91 q^{99}+O(q^{100}) \) 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9 - 61 * q^10 - 27 * q^11 - 93 * q^12 - 97 * q^13 - 12 * q^14 - 36 * q^15 + 105 * q^16 - 45 * q^17 - 68 * q^18 - 128 * q^19 - 30 * q^20 - 26 * q^21 - 68 * q^22 - 41 * q^23 - 40 * q^24 + 102 * q^25 - 5 * q^26 - 189 * q^27 - 115 * q^28 - 26 * q^29 - 12 * q^30 - 88 * q^31 - 89 * q^32 - 52 * q^33 - 61 * q^34 - 87 * q^35 + 110 * q^36 - 62 * q^37 - 37 * q^38 - 20 * q^39 - 161 * q^40 - 34 * q^41 - 53 * q^42 - 254 * q^43 - 19 * q^44 - 46 * q^45 - 52 * q^46 - 76 * q^47 - 162 * q^48 + 96 * q^49 - 54 * q^50 - 76 * q^51 - 259 * q^52 - 48 * q^53 - 12 * q^54 - 194 * q^55 - 10 * q^56 - 30 * q^57 - 52 * q^58 - 64 * q^59 - 31 * q^60 - 107 * q^61 - 51 * q^62 - 106 * q^63 + 54 * q^64 - 17 * q^65 - 13 * q^66 - 193 * q^67 - 118 * q^68 - 55 * q^69 - 86 * q^70 - 11 * q^71 - 172 * q^72 - 173 * q^73 - 11 * q^74 - 209 * q^75 - 213 * q^76 - 84 * q^77 - 30 * q^78 - 111 * q^79 - 6 * q^80 + 157 * q^81 - 117 * q^82 - 154 * q^83 - 6 * q^84 - 91 * q^85 + 28 * q^86 - 165 * q^87 - 165 * q^88 - 32 * q^89 - 103 * q^90 - 200 * q^91 - 86 * q^92 - 39 * q^93 - 118 * q^94 - 22 * q^95 - 28 * q^96 - 151 * q^97 - 38 * q^98 - 91 * 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 $

$2$	−2.51500	−1.77837	−0.889185	−	0.457548i	$-0.848728\pi$
$2$	−2.51500	−1.77837	−0.889185	+	0.457548i	$0.848728\pi$
$3$	−3.29902	−1.90469	−0.952346	−	0.305019i	$-0.901337\pi$
$3$	−3.29902	−1.90469	−0.952346	+	0.305019i	$0.901337\pi$
$4$	4.32520	2.16260
$5$	−0.150822	−0.0674498	−0.0337249	−	0.999431i	$-0.510737\pi$
$5$	−0.150822	−0.0674498	−0.0337249	+	0.999431i	$0.510737\pi$
$6$	8.29703	3.38725
$7$	−3.52495	−1.33230	−0.666152	−	0.745816i	$-0.732060\pi$
$7$	−3.52495	−1.33230	−0.666152	+	0.745816i	$0.732060\pi$
$8$	−5.84787	−2.06753
$9$	7.88356	2.62785
$10$	0.379318	0.119951
$11$	1.90159	0.573350	0.286675	−	0.958028i	$-0.407450\pi$
$11$	1.90159	0.573350	0.286675	+	0.958028i	$0.407450\pi$
$12$	−14.2689	−4.11909
$13$	3.94141	1.09315	0.546575	−	0.837410i	$-0.315931\pi$
$13$	3.94141	1.09315	0.546575	+	0.837410i	$0.315931\pi$
$14$	8.86522	2.36933
$15$	0.497567	0.128471
$16$	6.05697	1.51424
$17$	7.17750	1.74080	0.870400	−	0.492345i	$-0.163860\pi$
$17$	7.17750	1.74080	0.870400	+	0.492345i	$0.163860\pi$
$18$	−19.8271	−4.67330
$19$	−0.695126	−0.159473	−0.0797365	−	0.996816i	$-0.525408\pi$
$19$	−0.695126	−0.159473	−0.0797365	+	0.996816i	$0.525408\pi$
$20$	−0.652337	−0.145867
$21$	11.6289	2.53763
$22$	−4.78248	−1.01963
$23$	−7.98884	−1.66579	−0.832894	−	0.553432i	$-0.813318\pi$
$23$	−7.98884	−1.66579	−0.832894	+	0.553432i	$0.813318\pi$
$24$	19.2923	3.93802
$25$	−4.97725	−0.995451
$26$	−9.91263	−1.94403
$27$	−16.1110	−3.10056
$28$	−15.2461	−2.88124
$29$	−3.37498	−0.626719	−0.313359	−	0.949635i	$-0.601454\pi$
$29$	−3.37498	−0.626719	−0.313359	+	0.949635i	$0.601454\pi$
$30$	−1.25138	−0.228469
$31$	5.48811	0.985693	0.492847	−	0.870116i	$-0.335956\pi$
$31$	5.48811	0.985693	0.492847	+	0.870116i	$0.335956\pi$
$32$	−3.53750	−0.625348
$33$	−6.27338	−1.09206
$34$	−18.0514	−3.09579
$35$	0.531641	0.0898637
$36$	34.0980	5.68300
$37$	−1.10392	−0.181483	−0.0907413	−	0.995874i	$-0.528924\pi$
$37$	−1.10392	−0.181483	−0.0907413	+	0.995874i	$0.528924\pi$
$38$	1.74824	0.283602
$39$	−13.0028	−2.08212
$40$	0.881990	0.139455
$41$	4.65892	0.727602	0.363801	−	0.931477i	$-0.381479\pi$
$41$	4.65892	0.727602	0.363801	+	0.931477i	$0.381479\pi$
$42$	−29.2466	−4.51285
$43$	2.81935	0.429947	0.214973	−	0.976620i	$-0.431034\pi$
$43$	2.81935	0.429947	0.214973	+	0.976620i	$0.431034\pi$
$44$	8.22474	1.23993
$45$	−1.18902	−0.177248
$46$	20.0919	2.96239
$47$	2.50183	0.364929	0.182464	−	0.983212i	$-0.441593\pi$
$47$	2.50183	0.364929	0.182464	+	0.983212i	$0.441593\pi$
$48$	−19.9821	−2.88417
$49$	5.42525	0.775036
$50$	12.5178	1.77028
$51$	−23.6788	−3.31569
$52$	17.0474	2.36405
$53$	−7.70124	−1.05785	−0.528923	−	0.848670i	$-0.677404\pi$
$53$	−7.70124	−1.05785	−0.528923	+	0.848670i	$0.677404\pi$
$54$	40.5191	5.51395
$55$	−0.286802	−0.0386723
$56$	20.6134	2.75459
$57$	2.29324	0.303747
$58$	8.48807	1.11454
$59$	−12.7793	−1.66372	−0.831861	−	0.554983i	$-0.812725\pi$
$59$	−12.7793	−1.66372	−0.831861	+	0.554983i	$0.812725\pi$
$60$	2.15208	0.277832
$61$	4.56957	0.585074	0.292537	−	0.956254i	$-0.405501\pi$
$61$	4.56957	0.585074	0.292537	+	0.956254i	$0.405501\pi$
$62$	−13.8026	−1.75293
$63$	−27.7891	−3.50110
$64$	−3.21714	−0.402142
$65$	−0.594453	−0.0737328
$66$	15.7775	1.94208
$67$	−12.5575	−1.53415	−0.767074	−	0.641559i	$-0.778288\pi$
$67$	−12.5575	−1.53415	−0.767074	+	0.641559i	$0.778288\pi$
$68$	31.0442	3.76466
$69$	26.3554	3.17282
$70$	−1.33707	−0.159811
$71$	−0.590444	−0.0700728	−0.0350364	−	0.999386i	$-0.511155\pi$
$71$	−0.590444	−0.0700728	−0.0350364	+	0.999386i	$0.511155\pi$
$72$	−46.1021	−5.43318
$73$	−8.48233	−0.992782	−0.496391	−	0.868099i	$-0.665342\pi$
$73$	−8.48233	−0.992782	−0.496391	+	0.868099i	$0.665342\pi$
$74$	2.77634	0.322743
$75$	16.4201	1.89603
$76$	−3.00656	−0.344876
$77$	−6.70299	−0.763877
$78$	32.7020	3.70277
$79$	9.09185	1.02291	0.511457	−	0.859309i	$-0.329106\pi$
$79$	9.09185	1.02291	0.511457	+	0.859309i	$0.329106\pi$
$80$	−0.913526	−0.102135
$81$	29.4999	3.27776
$82$	−11.7172	−1.29395
$83$	1.99877	0.219393	0.109697	−	0.993965i	$-0.465012\pi$
$83$	1.99877	0.219393	0.109697	+	0.993965i	$0.465012\pi$
$84$	50.2973	5.48788
$85$	−1.08253	−0.117417
$86$	−7.09065	−0.764605
$87$	11.1342	1.19371
$88$	−11.1202	−1.18542
$89$	6.75561	0.716093	0.358047	−	0.933704i	$-0.383443\pi$
$89$	6.75561	0.716093	0.358047	+	0.933704i	$0.383443\pi$
$90$	2.99037	0.315213
$91$	−13.8933	−1.45641
$92$	−34.5534	−3.60244
$93$	−18.1054	−1.87744
$94$	−6.29208	−0.648978
$95$	0.104841	0.0107564
$96$	11.6703	1.19110
$97$	−9.29359	−0.943621	−0.471810	−	0.881700i	$-0.656399\pi$
$97$	−9.29359	−0.943621	−0.471810	+	0.881700i	$0.656399\pi$
$98$	−13.6445	−1.37830
$99$	14.9913	1.50668
$100$	−21.5276	−2.15276
$101$	18.0037	1.79143	0.895715	−	0.444628i	$-0.146664\pi$
$101$	18.0037	1.79143	0.895715	+	0.444628i	$0.146664\pi$
$102$	59.5520	5.89652
$103$	7.29629	0.718925	0.359463	−	0.933160i	$-0.382960\pi$
$103$	7.29629	0.718925	0.359463	+	0.933160i	$0.382960\pi$
$104$	−23.0489	−2.26013
$105$	−1.75390	−0.171163
$106$	19.3686	1.88124
$107$	12.2465	1.18391	0.591957	−	0.805970i	$-0.298356\pi$
$107$	12.2465	1.18391	0.591957	+	0.805970i	$0.298356\pi$
$108$	−69.6833	−6.70528
$109$	2.85671	0.273623	0.136811	−	0.990597i	$-0.456315\pi$
$109$	2.85671	0.273623	0.136811	+	0.990597i	$0.456315\pi$
$110$	0.721305	0.0687737
$111$	3.64185	0.345669
$112$	−21.3505	−2.01743
$113$	−8.85220	−0.832745	−0.416372	−	0.909194i	$-0.636699\pi$
$113$	−8.85220	−0.832745	−0.416372	+	0.909194i	$0.636699\pi$
$114$	−5.76749	−0.540175
$115$	1.20490	0.112357
$116$	−14.5975	−1.35534
$117$	31.0723	2.87264
$118$	32.1399	2.95872
$119$	−25.3003	−2.31928
$120$	−2.90971	−0.265619
$121$	−7.38397	−0.671270
$122$	−11.4925	−1.04048
$123$	−15.3699	−1.38586
$124$	23.7372	2.13166
$125$	1.50479	0.134593
$126$	69.8895	6.22626
$127$	−18.9318	−1.67992	−0.839961	−	0.542646i	$-0.817422\pi$
$127$	−18.9318	−1.67992	−0.839961	+	0.542646i	$0.817422\pi$
$128$	15.1661	1.34051
$129$	−9.30110	−0.818917
$130$	1.49505	0.131124
$131$	14.3516	1.25391	0.626953	−	0.779057i	$-0.284302\pi$
$131$	14.3516	1.25391	0.626953	+	0.779057i	$0.284302\pi$
$132$	−27.1336	−2.36168
$133$	2.45028	0.212467
$134$	31.5822	2.72828
$135$	2.42990	0.209132
$136$	−41.9731	−3.59917
$137$	12.0624	1.03056	0.515282	−	0.857021i	$-0.327687\pi$
$137$	12.0624	1.03056	0.515282	+	0.857021i	$0.327687\pi$
$138$	−66.2837	−5.64244
$139$	−9.77838	−0.829391	−0.414696	−	0.909960i	$-0.636112\pi$
$139$	−9.77838	−0.829391	−0.414696	+	0.909960i	$0.636112\pi$
$140$	2.29945	0.194339
$141$	−8.25358	−0.695077
$142$	1.48496	0.124615
$143$	7.49493	0.626758
$144$	47.7505	3.97921
$145$	0.509023	0.0422720
$146$	21.3330	1.76553
$147$	−17.8980	−1.47620
$148$	−4.77466	−0.392475
$149$	−15.8608	−1.29937	−0.649683	−	0.760205i	$-0.725098\pi$
$149$	−15.8608	−1.29937	−0.649683	+	0.760205i	$0.725098\pi$
$150$	−41.2964	−3.37184
$151$	8.02664	0.653199	0.326599	−	0.945163i	$-0.394097\pi$
$151$	8.02664	0.653199	0.326599	+	0.945163i	$0.394097\pi$
$152$	4.06501	0.329716
$153$	56.5843	4.57457
$154$	16.8580	1.35846
$155$	−0.827729	−0.0664848
$156$	−56.2398	−4.50278
$157$	−2.95198	−0.235594	−0.117797	−	0.993038i	$-0.537583\pi$
$157$	−2.95198	−0.235594	−0.117797	+	0.993038i	$0.537583\pi$
$158$	−22.8660	−1.81912
$159$	25.4066	2.01487
$160$	0.533534	0.0421796
$161$	28.1602	2.21934
$162$	−74.1920	−5.82908
$163$	12.5471	0.982766	0.491383	−	0.870943i	$-0.336491\pi$
$163$	12.5471	0.982766	0.491383	+	0.870943i	$0.336491\pi$
$164$	20.1508	1.57351
$165$	0.946166	0.0736589
$166$	−5.02689	−0.390163
$167$	12.8231	0.992280	0.496140	−	0.868243i	$-0.334750\pi$
$167$	12.8231	0.992280	0.496140	+	0.868243i	$0.334750\pi$
$168$	−68.0042	−5.24664
$169$	2.53471	0.194978
$170$	2.72255	0.208810
$171$	−5.48007	−0.419072
$172$	12.1943	0.929803
$173$	24.5434	1.86600	0.933000	−	0.359877i	$-0.117181\pi$
$173$	24.5434	1.86600	0.933000	+	0.359877i	$0.117181\pi$
$174$	−28.0023	−2.12285
$175$	17.5446	1.32624
$176$	11.5178	0.868190
$177$	42.1592	3.16888
$178$	−16.9903	−1.27348
$179$	−19.7541	−1.47649	−0.738247	−	0.674531i	$-0.764346\pi$
$179$	−19.7541	−1.47649	−0.738247	+	0.674531i	$0.764346\pi$
$180$	−5.14274	−0.383317
$181$	2.31328	0.171945	0.0859723	−	0.996298i	$-0.472600\pi$
$181$	2.31328	0.171945	0.0859723	+	0.996298i	$0.472600\pi$
$182$	34.9415	2.59004
$183$	−15.0751	−1.11439
$184$	46.7177	3.44408
$185$	0.166495	0.0122410
$186$	45.5350	3.33879
$187$	13.6486	0.998088
$188$	10.8209	0.789195
$189$	56.7904	4.13089
$190$	−0.263674	−0.0191289
$191$	−0.932938	−0.0675050	−0.0337525	−	0.999430i	$-0.510746\pi$
$191$	−0.932938	−0.0675050	−0.0337525	+	0.999430i	$0.510746\pi$
$192$	10.6134	0.765957
$193$	12.2883	0.884532	0.442266	−	0.896884i	$-0.354175\pi$
$193$	12.2883	0.884532	0.442266	+	0.896884i	$0.354175\pi$
$194$	23.3733	1.67811
$195$	1.96111	0.140438
$196$	23.4653	1.67609
$197$	20.0658	1.42963	0.714816	−	0.699313i	$-0.246511\pi$
$197$	20.0658	1.42963	0.714816	+	0.699313i	$0.246511\pi$
$198$	−37.7030	−2.67943
$199$	−13.2050	−0.936078	−0.468039	−	0.883708i	$-0.655039\pi$
$199$	−13.2050	−0.936078	−0.468039	+	0.883708i	$0.655039\pi$
$200$	29.1063	2.05813
$201$	41.4276	2.92208
$202$	−45.2791	−3.18583
$203$	11.8966	0.834980
$204$	−102.415	−7.17051
$205$	−0.702670	−0.0490766
$206$	−18.3501	−1.27852
$207$	−62.9805	−4.37745
$208$	23.8730	1.65529
$209$	−1.32184	−0.0914338
$210$	4.41104	0.304391
$211$	0.133147	0.00916625	0.00458313	−	0.999989i	$-0.498541\pi$
$211$	0.133147	0.00916625	0.00458313	+	0.999989i	$0.498541\pi$
$212$	−33.3094	−2.28770
$213$	1.94789	0.133467
$214$	−30.7999	−2.10544
$215$	−0.425221	−0.0289998
$216$	94.2150	6.41052
$217$	−19.3453	−1.31324
$218$	−7.18461	−0.486603
$219$	27.9834	1.89094
$220$	−1.24048	−0.0836328
$221$	28.2895	1.90296
$222$	−9.15922	−0.614727
$223$	6.17458	0.413480	0.206740	−	0.978396i	$-0.433715\pi$
$223$	6.17458	0.413480	0.206740	+	0.978396i	$0.433715\pi$
$224$	12.4695	0.833154
$225$	−39.2385	−2.61590
$226$	22.2632	1.48093
$227$	−5.79378	−0.384546	−0.192273	−	0.981341i	$-0.561586\pi$
$227$	−5.79378	−0.384546	−0.192273	+	0.981341i	$0.561586\pi$
$228$	9.91872	0.656883
$229$	−16.6592	−1.10087	−0.550435	−	0.834878i	$-0.685538\pi$
$229$	−16.6592	−1.10087	−0.550435	+	0.834878i	$0.685538\pi$
$230$	−3.03031	−0.199813
$231$	22.1133	1.45495
$232$	19.7365	1.29576
$233$	7.63997	0.500511	0.250255	−	0.968180i	$-0.419485\pi$
$233$	7.63997	0.500511	0.250255	+	0.968180i	$0.419485\pi$
$234$	−78.1468	−5.10862
$235$	−0.377331	−0.0246144
$236$	−55.2730	−3.59797
$237$	−29.9942	−1.94834
$238$	63.6302	4.12453
$239$	11.8748	0.768120	0.384060	−	0.923308i	$-0.374526\pi$
$239$	11.8748	0.768120	0.384060	+	0.923308i	$0.374526\pi$
$240$	3.01374	0.194536
$241$	−4.94514	−0.318544	−0.159272	−	0.987235i	$-0.550915\pi$
$241$	−4.94514	−0.318544	−0.159272	+	0.987235i	$0.550915\pi$
$242$	18.5706	1.19377
$243$	−48.9878	−3.14257
$244$	19.7643	1.26528
$245$	−0.818249	−0.0522760
$246$	38.6552	2.46457
$247$	−2.73978	−0.174328
$248$	−32.0937	−2.03795
$249$	−6.59399	−0.417877
$250$	−3.78455	−0.239356
$251$	27.7390	1.75087	0.875435	−	0.483336i	$-0.160575\pi$
$251$	27.7390	1.75087	0.875435	+	0.483336i	$0.160575\pi$
$252$	−120.194	−7.57149
$253$	−15.1915	−0.955080
$254$	47.6133	2.98752
$255$	3.57129	0.223643
$256$	−31.7084	−1.98177
$257$	18.2638	1.13927	0.569634	−	0.821899i	$-0.307085\pi$
$257$	18.2638	1.13927	0.569634	+	0.821899i	$0.307085\pi$
$258$	23.3922	1.45634
$259$	3.89124	0.241790
$260$	−2.57113	−0.159455
$261$	−26.6069	−1.64692
$262$	−36.0942	−2.22991
$263$	13.2174	0.815018	0.407509	−	0.913201i	$-0.366398\pi$
$263$	13.2174	0.815018	0.407509	+	0.913201i	$0.366398\pi$
$264$	36.6859	2.25786
$265$	1.16152	0.0713515
$266$	−6.16245	−0.377844
$267$	−22.2869	−1.36394
$268$	−54.3139	−3.31775
$269$	0.281881	0.0171866	0.00859329	−	0.999963i	$-0.497265\pi$
$269$	0.281881	0.0171866	0.00859329	+	0.999963i	$0.497265\pi$
$270$	−6.11118	−0.371915
$271$	−18.6029	−1.13004	−0.565021	−	0.825076i	$-0.691132\pi$
$271$	−18.6029	−1.13004	−0.565021	+	0.825076i	$0.691132\pi$
$272$	43.4739	2.63599
$273$	45.8342	2.77401
$274$	−30.3370	−1.83272
$275$	−9.46468	−0.570741
$276$	113.992	6.86153
$277$	−24.9290	−1.49784	−0.748919	−	0.662662i	$-0.769427\pi$
$277$	−24.9290	−1.49784	−0.748919	+	0.662662i	$0.769427\pi$
$278$	24.5926	1.47496
$279$	43.2658	2.59026
$280$	−3.10897	−0.185796
$281$	−12.3799	−0.738520	−0.369260	−	0.929326i	$-0.620389\pi$
$281$	−12.3799	−0.738520	−0.369260	+	0.929326i	$0.620389\pi$
$282$	20.7577	1.23610
$283$	−21.4059	−1.27245	−0.636224	−	0.771505i	$-0.719504\pi$
$283$	−21.4059	−1.27245	−0.636224	+	0.771505i	$0.719504\pi$
$284$	−2.55379	−0.151539
$285$	−0.345872	−0.0204877
$286$	−18.8497	−1.11461
$287$	−16.4225	−0.969387
$288$	−27.8881	−1.64332
$289$	34.5166	2.03039
$290$	−1.28019	−0.0751754
$291$	30.6598	1.79731
$292$	−36.6878	−2.14699
$293$	−25.0848	−1.46547	−0.732736	−	0.680513i	$-0.761757\pi$
$293$	−25.0848	−1.46547	−0.732736	+	0.680513i	$0.761757\pi$
$294$	45.0135	2.62524
$295$	1.92740	0.112218
$296$	6.45556	0.375222
$297$	−30.6364	−1.77771
$298$	39.8898	2.31075
$299$	−31.4873	−1.82096
$300$	71.0201	4.10035
$301$	−9.93806	−0.572820
$302$	−20.1870	−1.16163
$303$	−59.3945	−3.41213
$304$	−4.21036	−0.241481
$305$	−0.689194	−0.0394631
$306$	−142.309	−8.13528
$307$	11.0033	0.627990	0.313995	−	0.949425i	$-0.398333\pi$
$307$	11.0033	0.627990	0.313995	+	0.949425i	$0.398333\pi$
$308$	−28.9918	−1.65196
$309$	−24.0706	−1.36933
$310$	2.08174	0.118235
$311$	−14.2091	−0.805726	−0.402863	−	0.915260i	$-0.631985\pi$
$311$	−14.2091	−0.805726	−0.402863	+	0.915260i	$0.631985\pi$
$312$	76.0387	4.30485
$313$	−19.4290	−1.09819	−0.549096	−	0.835759i	$-0.685028\pi$
$313$	−19.4290	−1.09819	−0.549096	+	0.835759i	$0.685028\pi$
$314$	7.42422	0.418973
$315$	4.19122	0.236149
$316$	39.3241	2.21215
$317$	−0.199190	−0.0111876	−0.00559382	−	0.999984i	$-0.501781\pi$
$317$	−0.199190	−0.0111876	−0.00559382	+	0.999984i	$0.501781\pi$
$318$	−63.8974	−3.58319
$319$	−6.41782	−0.359329
$320$	0.485216	0.0271244
$321$	−40.4015	−2.25499
$322$	−70.8229	−3.94680
$323$	−4.98927	−0.277611
$324$	127.593	7.08849
$325$	−19.6174	−1.08818
$326$	−31.5560	−1.74772
$327$	−9.42435	−0.521168
$328$	−27.2448	−1.50434
$329$	−8.81880	−0.486196
$330$	−2.37960	−0.130993
$331$	−22.5383	−1.23882	−0.619408	−	0.785069i	$-0.712627\pi$
$331$	−22.5383	−1.23882	−0.619408	+	0.785069i	$0.712627\pi$
$332$	8.64508	0.474460
$333$	−8.70279	−0.476910
$334$	−32.2500	−1.76464
$335$	1.89396	0.103478
$336$	70.4358	3.84259
$337$	10.9611	0.597090	0.298545	−	0.954395i	$-0.403499\pi$
$337$	10.9611	0.597090	0.298545	+	0.954395i	$0.403499\pi$
$338$	−6.37479	−0.346743
$339$	29.2036	1.58612
$340$	−4.68215	−0.253925
$341$	10.4361	0.565147
$342$	13.7824	0.745264
$343$	5.55091	0.299721
$344$	−16.4872	−0.888930
$345$	−3.97498	−0.214006
$346$	−61.7265	−3.31844
$347$	−3.92998	−0.210972	−0.105486	−	0.994421i	$-0.533640\pi$
$347$	−3.92998	−0.210972	−0.105486	+	0.994421i	$0.533640\pi$
$348$	48.1574	2.58151
$349$	29.5659	1.58263	0.791313	−	0.611411i	$-0.209398\pi$
$349$	29.5659	1.58263	0.791313	+	0.611411i	$0.209398\pi$
$350$	−44.1245	−2.35855
$351$	−63.5000	−3.38938
$352$	−6.72686	−0.358543
$353$	−6.12162	−0.325821	−0.162910	−	0.986641i	$-0.552088\pi$
$353$	−6.12162	−0.325821	−0.162910	+	0.986641i	$0.552088\pi$
$354$	−106.030	−5.63544
$355$	0.0890521	0.00472640
$356$	29.2194	1.54862
$357$	83.4664	4.41751
$358$	49.6816	2.62575
$359$	−6.36050	−0.335694	−0.167847	−	0.985813i	$-0.553682\pi$
$359$	−6.36050	−0.335694	−0.167847	+	0.985813i	$0.553682\pi$
$360$	6.95322	0.366467
$361$	−18.5168	−0.974568
$362$	−5.81788	−0.305781
$363$	24.3599	1.27856
$364$	−60.0912	−3.14963
$365$	1.27933	0.0669630
$366$	37.9139	1.98179
$367$	23.5872	1.23124	0.615620	−	0.788043i	$-0.288906\pi$
$367$	23.5872	1.23124	0.615620	+	0.788043i	$0.288906\pi$
$368$	−48.3882	−2.52241
$369$	36.7289	1.91203
$370$	−0.418735	−0.0217690
$371$	27.1465	1.40937
$372$	−78.3095	−4.06016
$373$	−37.9909	−1.96709	−0.983547	−	0.180653i	$-0.942179\pi$
$373$	−37.9909	−1.96709	−0.983547	+	0.180653i	$0.942179\pi$
$374$	−34.3263	−1.77497
$375$	−4.96435	−0.256358
$376$	−14.6304	−0.754503
$377$	−13.3022	−0.685098
$378$	−142.828	−7.34626
$379$	1.60110	0.0822427	0.0411214	−	0.999154i	$-0.486907\pi$
$379$	1.60110	0.0822427	0.0411214	+	0.999154i	$0.486907\pi$
$380$	0.453457	0.0232618
$381$	62.4564	3.19974
$382$	2.34633	0.120049
$383$	−14.6006	−0.746057	−0.373028	−	0.927820i	$-0.621681\pi$
$383$	−14.6006	−0.746057	−0.373028	+	0.927820i	$0.621681\pi$
$384$	−50.0333	−2.55325
$385$	1.01096	0.0515233
$386$	−30.9051	−1.57303
$387$	22.2265	1.12984
$388$	−40.1966	−2.04068
$389$	−18.0605	−0.915705	−0.457853	−	0.889028i	$-0.651381\pi$
$389$	−18.0605	−0.915705	−0.457853	+	0.889028i	$0.651381\pi$
$390$	−4.93219	−0.249751
$391$	−57.3399	−2.89981
$392$	−31.7262	−1.60241
$393$	−47.3463	−2.38831
$394$	−50.4655	−2.54241
$395$	−1.37125	−0.0689953
$396$	64.8403	3.25835
$397$	2.61309	0.131147	0.0655736	−	0.997848i	$-0.479112\pi$
$397$	2.61309	0.131147	0.0655736	+	0.997848i	$0.479112\pi$
$398$	33.2105	1.66469
$399$	−8.08355	−0.404683
$400$	−30.1471	−1.50735
$401$	−16.4845	−0.823195	−0.411597	−	0.911366i	$-0.635029\pi$
$401$	−16.4845	−0.823195	−0.411597	+	0.911366i	$0.635029\pi$
$402$	−104.190	−5.19654
$403$	21.6309	1.07751
$404$	77.8695	3.87415
$405$	−4.44924	−0.221084
$406$	−29.9200	−1.48490
$407$	−2.09919	−0.104053
$408$	138.470	6.85530
$409$	−16.3719	−0.809539	−0.404770	−	0.914419i	$-0.632648\pi$
$409$	−16.3719	−0.809539	−0.404770	+	0.914419i	$0.632648\pi$
$410$	1.76721	0.0872763
$411$	−39.7943	−1.96291
$412$	31.5579	1.55475
$413$	45.0463	2.21659
$414$	158.396	7.78473
$415$	−0.301459	−0.0147980
$416$	−13.9427	−0.683599
$417$	32.2591	1.57974
$418$	3.32443	0.162603
$419$	31.5030	1.53902	0.769510	−	0.638635i	$-0.220500\pi$
$419$	31.5030	1.53902	0.769510	+	0.638635i	$0.220500\pi$
$420$	−7.58595	−0.370157
$421$	−8.98468	−0.437887	−0.218943	−	0.975738i	$-0.570261\pi$
$421$	−8.98468	−0.437887	−0.218943	+	0.975738i	$0.570261\pi$
$422$	−0.334865	−0.0163010
$423$	19.7233	0.958979
$424$	45.0358	2.18713
$425$	−35.7243	−1.73288
$426$	−4.89893	−0.237354
$427$	−16.1075	−0.779497
$428$	52.9686	2.56033
$429$	−24.7260	−1.19378
$430$	1.06943	0.0515724
$431$	36.4786	1.75711	0.878555	−	0.477641i	$-0.158508\pi$
$431$	36.4786	1.75711	0.878555	+	0.477641i	$0.158508\pi$
$432$	−97.5837	−4.69500
$433$	−9.56452	−0.459642	−0.229821	−	0.973233i	$-0.573814\pi$
$433$	−9.56452	−0.459642	−0.229821	+	0.973233i	$0.573814\pi$
$434$	48.6533	2.33543
$435$	−1.67928	−0.0805153
$436$	12.3558	0.591737
$437$	5.55326	0.265648
$438$	−70.3782	−3.36280
$439$	−7.89468	−0.376793	−0.188396	−	0.982093i	$-0.560329\pi$
$439$	−7.89468	−0.376793	−0.188396	+	0.982093i	$0.560329\pi$
$440$	1.67718	0.0799564
$441$	42.7703	2.03668
$442$	−71.1479	−3.38416
$443$	37.6642	1.78948	0.894740	−	0.446587i	$-0.147361\pi$
$443$	37.6642	1.78948	0.894740	+	0.446587i	$0.147361\pi$
$444$	15.7517	0.747543
$445$	−1.01890	−0.0483004
$446$	−15.5290	−0.735321
$447$	52.3251	2.47489
$448$	11.3402	0.535776
$449$	4.67544	0.220648	0.110324	−	0.993896i	$-0.464811\pi$
$449$	4.67544	0.220648	0.110324	+	0.993896i	$0.464811\pi$
$450$	98.6846	4.65204
$451$	8.85935	0.417170
$452$	−38.2875	−1.80089
$453$	−26.4801	−1.24414
$454$	14.5713	0.683866
$455$	2.09541	0.0982345
$456$	−13.4106	−0.628007
$457$	32.9275	1.54028	0.770142	−	0.637873i	$-0.220186\pi$
$457$	32.9275	1.54028	0.770142	+	0.637873i	$0.220186\pi$
$458$	41.8978	1.95775
$459$	−115.637	−5.39746
$460$	5.21142	0.242984
$461$	−25.4711	−1.18631	−0.593154	−	0.805089i	$-0.702117\pi$
$461$	−25.4711	−1.18631	−0.593154	+	0.805089i	$0.702117\pi$
$462$	−55.6149	−2.58744
$463$	14.0628	0.653553	0.326777	−	0.945102i	$-0.394038\pi$
$463$	14.0628	0.653553	0.326777	+	0.945102i	$0.394038\pi$
$464$	−20.4422	−0.949003
$465$	2.73070	0.126633
$466$	−19.2145	−0.890094
$467$	−38.4864	−1.78094	−0.890469	−	0.455043i	$-0.849624\pi$
$467$	−38.4864	−1.78094	−0.890469	+	0.455043i	$0.849624\pi$
$468$	134.394	6.21237
$469$	44.2647	2.04395
$470$	0.948986	0.0437735
$471$	9.73865	0.448733
$472$	74.7317	3.43981
$473$	5.36124	0.246510
$474$	75.4354	3.46486
$475$	3.45982	0.158747
$476$	−109.429	−5.01567
$477$	−60.7132	−2.77987
$478$	−29.8652	−1.36600
$479$	0.628161	0.0287014	0.0143507	−	0.999897i	$-0.495432\pi$
$479$	0.628161	0.0287014	0.0143507	+	0.999897i	$0.495432\pi$
$480$	−1.76014	−0.0803391
$481$	−4.35098	−0.198388
$482$	12.4370	0.566490
$483$	−92.9013	−4.22716
$484$	−31.9372	−1.45169
$485$	1.40168	0.0636470
$486$	123.204	5.58865
$487$	−43.7780	−1.98377	−0.991885	−	0.127138i	$-0.959421\pi$
$487$	−43.7780	−1.98377	−0.991885	+	0.127138i	$0.959421\pi$
$488$	−26.7223	−1.20966
$489$	−41.3933	−1.87187
$490$	2.05789	0.0929661
$491$	−35.8852	−1.61947	−0.809737	−	0.586793i	$-0.800390\pi$
$491$	−35.8852	−1.61947	−0.809737	+	0.586793i	$0.800390\pi$
$492$	−66.4779	−2.99706
$493$	−24.2240	−1.09099
$494$	6.89053	0.310020
$495$	−2.26102	−0.101625
$496$	33.2413	1.49258
$497$	2.08128	0.0933583
$498$	16.5838	0.743140
$499$	−25.9599	−1.16213	−0.581063	−	0.813859i	$-0.697363\pi$
$499$	−25.9599	−1.16213	−0.581063	+	0.813859i	$0.697363\pi$
$500$	6.50853	0.291070
$501$	−42.3037	−1.88999
$502$	−69.7634	−3.11369
$503$	−5.06465	−0.225822	−0.112911	−	0.993605i	$-0.536017\pi$
$503$	−5.06465	−0.225822	−0.112911	+	0.993605i	$0.536017\pi$
$504$	162.507	7.23865
$505$	−2.71535	−0.120832
$506$	38.2065	1.69849
$507$	−8.36207	−0.371373
$508$	−81.8837	−3.63300
$509$	34.6317	1.53502	0.767511	−	0.641035i	$-0.221495\pi$
$509$	34.6317	1.53502	0.767511	+	0.641035i	$0.221495\pi$
$510$	−8.98177	−0.397719
$511$	29.8998	1.32269
$512$	49.4142	2.18382
$513$	11.1992	0.494456
$514$	−45.9335	−2.02604
$515$	−1.10044	−0.0484914
$516$	−40.2292	−1.77099
$517$	4.75744	0.209232
$518$	−9.78646	−0.429992
$519$	−80.9692	−3.55415
$520$	3.47628	0.152445
$521$	−34.7811	−1.52379	−0.761894	−	0.647702i	$-0.775730\pi$
$521$	−34.7811	−1.52379	−0.761894	+	0.647702i	$0.775730\pi$
$522$	66.9162	2.92884
$523$	−36.7470	−1.60683	−0.803416	−	0.595418i	$-0.796987\pi$
$523$	−36.7470	−1.60683	−0.803416	+	0.595418i	$0.796987\pi$
$524$	62.0736	2.71170
$525$	−57.8799	−2.52609
$526$	−33.2416	−1.44940
$527$	39.3909	1.71590
$528$	−37.9977	−1.65364
$529$	40.8216	1.77485
$530$	−2.92121	−0.126889
$531$	−100.746	−4.37202
$532$	10.5980	0.459480
$533$	18.3627	0.795378
$534$	56.0515	2.42559
$535$	−1.84705	−0.0798547
$536$	73.4349	3.17190
$537$	65.1694	2.81227
$538$	−0.708929	−0.0305641
$539$	10.3166	0.444367
$540$	10.5098	0.452270
$541$	35.5884	1.53007	0.765033	−	0.643991i	$-0.222723\pi$
$541$	35.5884	1.53007	0.765033	+	0.643991i	$0.222723\pi$
$542$	46.7861	2.00963
$543$	−7.63156	−0.327502
$544$	−25.3904	−1.08861
$545$	−0.430855	−0.0184558
$546$	−115.273	−4.93322
$547$	1.74655	0.0746773	0.0373386	−	0.999303i	$-0.488112\pi$
$547$	1.74655	0.0746773	0.0373386	+	0.999303i	$0.488112\pi$
$548$	52.1725	2.22870
$549$	36.0245	1.53749
$550$	23.8036	1.01499
$551$	2.34604	0.0999447
$552$	−154.123	−6.55991
$553$	−32.0483	−1.36283
$554$	62.6963	2.66371
$555$	−0.549272	−0.0233153
$556$	−42.2935	−1.79364
$557$	2.15745	0.0914139	0.0457070	−	0.998955i	$-0.485446\pi$
$557$	2.15745	0.0914139	0.0457070	+	0.998955i	$0.485446\pi$
$558$	−108.813	−4.60644
$559$	11.1122	0.469997
$560$	3.22013	0.136075
$561$	−45.0272	−1.90105
$562$	31.1353	1.31336
$563$	−19.5302	−0.823101	−0.411550	−	0.911387i	$-0.635013\pi$
$563$	−19.5302	−0.823101	−0.411550	+	0.911387i	$0.635013\pi$
$564$	−35.6984	−1.50317
$565$	1.33511	0.0561685
$566$	53.8357	2.26288
$567$	−103.985	−4.36698
$568$	3.45284	0.144878
$569$	25.4911	1.06864	0.534320	−	0.845282i	$-0.320568\pi$
$569$	25.4911	1.06864	0.534320	+	0.845282i	$0.320568\pi$
$570$	0.869866	0.0364347
$571$	18.0382	0.754877	0.377439	−	0.926035i	$-0.376805\pi$
$571$	18.0382	0.754877	0.377439	+	0.926035i	$0.376805\pi$
$572$	32.4171	1.35543
$573$	3.07778	0.128576
$574$	41.3024	1.72393
$575$	39.7625	1.65821
$576$	−25.3625	−1.05677
$577$	−9.93437	−0.413573	−0.206787	−	0.978386i	$-0.566301\pi$
$577$	−9.93437	−0.413573	−0.206787	+	0.978386i	$0.566301\pi$
$578$	−86.8090	−3.61078
$579$	−40.5395	−1.68476
$580$	2.20163	0.0914176
$581$	−7.04555	−0.292299
$582$	−77.1092	−3.19628
$583$	−14.6446	−0.606516
$584$	49.6036	2.05261
$585$	−4.68640	−0.193759
$586$	63.0883	2.60615
$587$	−18.6128	−0.768234	−0.384117	−	0.923284i	$-0.625494\pi$
$587$	−18.6128	−0.768234	−0.384117	+	0.923284i	$0.625494\pi$
$588$	−77.4126	−3.19244
$589$	−3.81493	−0.157191
$590$	−4.84741	−0.199565
$591$	−66.1977	−2.72301
$592$	−6.68638	−0.274809
$593$	18.8547	0.774268	0.387134	−	0.922023i	$-0.373465\pi$
$593$	18.8547	0.774268	0.387134	+	0.922023i	$0.373465\pi$
$594$	77.0505	3.16142
$595$	3.81585	0.156435
$596$	−68.6011	−2.81001
$597$	43.5636	1.78294
$598$	79.1904	3.23834
$599$	−10.3390	−0.422438	−0.211219	−	0.977439i	$-0.567743\pi$
$599$	−10.3390	−0.422438	−0.211219	+	0.977439i	$0.567743\pi$
$600$	−96.0225	−3.92010
$601$	23.9834	0.978303	0.489152	−	0.872199i	$-0.337306\pi$
$601$	23.9834	0.978303	0.489152	+	0.872199i	$0.337306\pi$
$602$	24.9942	1.01869
$603$	−98.9982	−4.03152
$604$	34.7168	1.41261
$605$	1.11367	0.0452770
$606$	149.377	6.06802
$607$	−18.2857	−0.742194	−0.371097	−	0.928594i	$-0.621018\pi$
$607$	−18.2857	−0.742194	−0.371097	+	0.928594i	$0.621018\pi$
$608$	2.45901	0.0997260
$609$	−39.2473	−1.59038
$610$	1.73332	0.0701800
$611$	9.86072	0.398922
$612$	244.739	9.89297
$613$	−35.8606	−1.44839	−0.724197	−	0.689593i	$-0.757789\pi$
$613$	−35.8606	−1.44839	−0.724197	+	0.689593i	$0.757789\pi$
$614$	−27.6732	−1.11680
$615$	2.31812	0.0934758
$616$	39.1982	1.57934
$617$	29.4296	1.18479	0.592395	−	0.805648i	$-0.298182\pi$
$617$	29.4296	1.18479	0.592395	+	0.805648i	$0.298182\pi$
$618$	60.5376	2.43518
$619$	6.62785	0.266396	0.133198	−	0.991089i	$-0.457475\pi$
$619$	6.62785	0.266396	0.133198	+	0.991089i	$0.457475\pi$
$620$	−3.58010	−0.143780
$621$	128.708	5.16488
$622$	35.7359	1.43288
$623$	−23.8132	−0.954055
$624$	−78.7576	−3.15283
$625$	24.6593	0.986372
$626$	48.8638	1.95299
$627$	4.36079	0.174153
$628$	−12.7679	−0.509495
$629$	−7.92336	−0.315925
$630$	−10.5409	−0.419960
$631$	45.2096	1.79976	0.899882	−	0.436133i	$-0.143652\pi$
$631$	45.2096	1.79976	0.899882	+	0.436133i	$0.143652\pi$
$632$	−53.1680	−2.11491
$633$	−0.439257	−0.0174589
$634$	0.500962	0.0198958
$635$	2.85533	0.113310
$636$	109.889	4.35736
$637$	21.3831	0.847230
$638$	16.1408	0.639020
$639$	−4.65480	−0.184141
$640$	−2.28738	−0.0904168
$641$	−7.96162	−0.314465	−0.157232	−	0.987562i	$-0.550257\pi$
$641$	−7.96162	−0.314465	−0.157232	+	0.987562i	$0.550257\pi$
$642$	101.610	4.01021
$643$	47.4334	1.87059	0.935295	−	0.353870i	$-0.115134\pi$
$643$	47.4334	1.87059	0.935295	+	0.353870i	$0.115134\pi$
$644$	121.799	4.79954
$645$	1.40281	0.0552358
$646$	12.5480	0.493694
$647$	−1.27384	−0.0500797	−0.0250398	−	0.999686i	$-0.507971\pi$
$647$	−1.27384	−0.0500797	−0.0250398	+	0.999686i	$0.507971\pi$
$648$	−172.511	−6.77689
$649$	−24.3009	−0.953895
$650$	49.3377	1.93518
$651$	63.8206	2.50133
$652$	54.2688	2.12533
$653$	−5.96234	−0.233324	−0.116662	−	0.993172i	$-0.537219\pi$
$653$	−5.96234	−0.233324	−0.116662	+	0.993172i	$0.537219\pi$
$654$	23.7022	0.926829
$655$	−2.16454	−0.0845757
$656$	28.2189	1.10176
$657$	−66.8710	−2.60889
$658$	22.1792	0.864637
$659$	8.90616	0.346935	0.173467	−	0.984840i	$-0.444503\pi$
$659$	8.90616	0.346935	0.173467	+	0.984840i	$0.444503\pi$
$660$	4.09236	0.159295
$661$	−17.6859	−0.687903	−0.343951	−	0.938987i	$-0.611766\pi$
$661$	−17.6859	−0.687903	−0.343951	+	0.938987i	$0.611766\pi$
$662$	56.6837	2.20307
$663$	−93.3277	−3.62455
$664$	−11.6885	−0.453603
$665$	−0.369558	−0.0143308
$666$	21.8875	0.848122
$667$	26.9622	1.04398
$668$	55.4624	2.14591
$669$	−20.3701	−0.787553
$670$	−4.76330	−0.184022
$671$	8.68944	0.335452
$672$	−41.1372	−1.58690
$673$	30.3014	1.16803	0.584017	−	0.811741i	$-0.301480\pi$
$673$	30.3014	1.16803	0.584017	+	0.811741i	$0.301480\pi$
$674$	−27.5672	−1.06185
$675$	80.1885	3.08646
$676$	10.9631	0.421659
$677$	−40.9957	−1.57559	−0.787797	−	0.615936i	$-0.788778\pi$
$677$	−40.9957	−1.57559	−0.787797	+	0.615936i	$0.788778\pi$
$678$	−73.4469	−2.82071
$679$	32.7594	1.25719
$680$	6.33049	0.242763
$681$	19.1138	0.732443
$682$	−26.2468	−1.00504
$683$	16.8091	0.643182	0.321591	−	0.946879i	$-0.395782\pi$
$683$	16.8091	0.643182	0.321591	+	0.946879i	$0.395782\pi$
$684$	−23.7024	−0.906285
$685$	−1.81929	−0.0695114
$686$	−13.9605	−0.533015
$687$	54.9591	2.09682
$688$	17.0767	0.651043
$689$	−30.3537	−1.15639
$690$	9.99706	0.380582
$691$	−6.08506	−0.231487	−0.115743	−	0.993279i	$-0.536925\pi$
$691$	−6.08506	−0.231487	−0.115743	+	0.993279i	$0.536925\pi$
$692$	106.155	4.03541
$693$	−52.8434	−2.00736
$694$	9.88389	0.375187
$695$	1.47480	0.0559423
$696$	−65.1111	−2.46803
$697$	33.4394	1.26661
$698$	−74.3581	−2.81450
$699$	−25.2044	−0.953319
$700$	75.8837	2.86814
$701$	−1.13901	−0.0430199	−0.0215100	−	0.999769i	$-0.506847\pi$
$701$	−1.13901	−0.0430199	−0.0215100	+	0.999769i	$0.506847\pi$
$702$	159.702	6.02757
$703$	0.767361	0.0289416
$704$	−6.11766	−0.230568
$705$	1.24482	0.0468828
$706$	15.3958	0.579430
$707$	−63.4619	−2.38673
$708$	182.347	6.85302
$709$	−23.6107	−0.886718	−0.443359	−	0.896344i	$-0.646213\pi$
$709$	−23.6107	−0.886718	−0.443359	+	0.896344i	$0.646213\pi$
$710$	−0.223966	−0.00840528
$711$	71.6762	2.68807
$712$	−39.5060	−1.48055
$713$	−43.8436	−1.64196
$714$	−209.918	−7.85597
$715$	−1.13040	−0.0422747
$716$	−85.4406	−3.19307
$717$	−39.1754	−1.46303
$718$	15.9966	0.596989
$719$	−30.2433	−1.12789	−0.563943	−	0.825813i	$-0.690716\pi$
$719$	−30.2433	−1.12789	−0.563943	+	0.825813i	$0.690716\pi$
$720$	−7.20184	−0.268397
$721$	−25.7190	−0.957827
$722$	46.5697	1.73314
$723$	16.3141	0.606729
$724$	10.0054	0.371847
$725$	16.7981	0.623867
$726$	−61.2650	−2.27376
$727$	−21.8659	−0.810961	−0.405480	−	0.914104i	$-0.632896\pi$
$727$	−21.8659	−0.810961	−0.405480	+	0.914104i	$0.632896\pi$
$728$	81.2460	3.01118
$729$	73.1123	2.70786
$730$	−3.21750	−0.119085
$731$	20.2359	0.748452
$732$	−65.2030	−2.40997
$733$	−5.06688	−0.187149	−0.0935747	−	0.995612i	$-0.529829\pi$
$733$	−5.06688	−0.187149	−0.0935747	+	0.995612i	$0.529829\pi$
$734$	−59.3216	−2.18960
$735$	2.69942	0.0995697
$736$	28.2605	1.04170
$737$	−23.8793	−0.879604
$738$	−92.3730	−3.40030
$739$	39.1473	1.44006	0.720028	−	0.693945i	$-0.244129\pi$
$739$	39.1473	1.44006	0.720028	+	0.693945i	$0.244129\pi$
$740$	0.720125	0.0264723
$741$	9.03859	0.332041
$742$	−68.2732	−2.50639
$743$	−1.86043	−0.0682525	−0.0341263	−	0.999418i	$-0.510865\pi$
$743$	−1.86043	−0.0682525	−0.0341263	+	0.999418i	$0.510865\pi$
$744$	105.878	3.88168
$745$	2.39216	0.0876420
$746$	95.5469	3.49822
$747$	15.7574	0.576534
$748$	59.0331	2.15847
$749$	−43.1682	−1.57733
$750$	12.4853	0.455899
$751$	−20.2295	−0.738185	−0.369092	−	0.929393i	$-0.620331\pi$
$751$	−20.2295	−0.738185	−0.369092	+	0.929393i	$0.620331\pi$
$752$	15.1535	0.552590
$753$	−91.5116	−3.33487
$754$	33.4549	1.21836
$755$	−1.21060	−0.0440581
$756$	245.630	8.93347
$757$	38.7283	1.40761	0.703803	−	0.710396i	$-0.251484\pi$
$757$	38.7283	1.40761	0.703803	+	0.710396i	$0.251484\pi$
$758$	−4.02675	−0.146258
$759$	50.1170	1.81913
$760$	−0.613094	−0.0222393
$761$	5.68803	0.206191	0.103095	−	0.994671i	$-0.467125\pi$
$761$	5.68803	0.206191	0.103095	+	0.994671i	$0.467125\pi$
$762$	−157.077	−5.69032
$763$	−10.0697	−0.364549
$764$	−4.03514	−0.145986
$765$	−8.53418	−0.308554
$766$	36.7205	1.32677
$767$	−50.3684	−1.81870
$768$	104.607	3.77467
$769$	18.5730	0.669758	0.334879	−	0.942261i	$-0.391305\pi$
$769$	18.5730	0.669758	0.334879	+	0.942261i	$0.391305\pi$
$770$	−2.54256	−0.0916276
$771$	−60.2529	−2.16995
$772$	53.1494	1.91289
$773$	−28.3628	−1.02014	−0.510069	−	0.860133i	$-0.670380\pi$
$773$	−28.3628	−1.02014	−0.510069	+	0.860133i	$0.670380\pi$
$774$	−55.8996	−2.00927
$775$	−27.3157	−0.981209
$776$	54.3477	1.95097
$777$	−12.8373	−0.460536
$778$	45.4222	1.62846
$779$	−3.23854	−0.116033
$780$	8.48221	0.303712
$781$	−1.12278	−0.0401762
$782$	144.210	5.15693
$783$	54.3743	1.94318
$784$	32.8606	1.17359
$785$	0.445225	0.0158907
$786$	119.076	4.24729
$787$	12.7290	0.453740	0.226870	−	0.973925i	$-0.427151\pi$
$787$	12.7290	0.453740	0.226870	+	0.973925i	$0.427151\pi$
$788$	86.7888	3.09172
$789$	−43.6044	−1.55236
$790$	3.44870	0.122699
$791$	31.2035	1.10947
$792$	−87.6670	−3.11511
$793$	18.0106	0.639574
$794$	−6.57191	−0.233228
$795$	−3.83188	−0.135903
$796$	−57.1143	−2.02436
$797$	−1.95301	−0.0691791	−0.0345895	−	0.999402i	$-0.511012\pi$
$797$	−1.95301	−0.0691791	−0.0345895	+	0.999402i	$0.511012\pi$
$798$	20.3301	0.719677
$799$	17.9569	0.635268
$800$	17.6070	0.622503
$801$	53.2583	1.88179
$802$	41.4584	1.46395
$803$	−16.1299	−0.569211
$804$	179.183	6.31929
$805$	−4.24719	−0.149694
$806$	−54.4016	−1.91621
$807$	−0.929932	−0.0327352
$808$	−105.283	−3.70385
$809$	−17.4774	−0.614471	−0.307236	−	0.951633i	$-0.599404\pi$
$809$	−17.4774	−0.614471	−0.307236	+	0.951633i	$0.599404\pi$
$810$	11.1898	0.393170
$811$	−52.0046	−1.82613	−0.913064	−	0.407815i	$-0.866291\pi$
$811$	−52.0046	−1.82613	−0.913064	+	0.407815i	$0.866291\pi$
$812$	51.4553	1.80573
$813$	61.3713	2.15238
$814$	5.27946	0.185045
$815$	−1.89239	−0.0662874
$816$	−143.421	−5.02076
$817$	−1.95980	−0.0685649
$818$	41.1753	1.43966
$819$	−109.528	−3.82723
$820$	−3.03919	−0.106133
$821$	−39.9260	−1.39343	−0.696713	−	0.717350i	$-0.745355\pi$
$821$	−39.9260	−1.39343	−0.696713	+	0.717350i	$0.745355\pi$
$822$	100.082	3.49078
$823$	−48.5794	−1.69337	−0.846686	−	0.532093i	$-0.821406\pi$
$823$	−48.5794	−1.69337	−0.846686	+	0.532093i	$0.821406\pi$
$824$	−42.6678	−1.48640
$825$	31.2242	1.08709
$826$	−113.291	−3.94191
$827$	29.6403	1.03069	0.515347	−	0.856982i	$-0.327663\pi$
$827$	29.6403	1.03069	0.515347	+	0.856982i	$0.327663\pi$
$828$	−272.404	−9.46668
$829$	−18.4186	−0.639704	−0.319852	−	0.947468i	$-0.603633\pi$
$829$	−18.4186	−0.639704	−0.319852	+	0.947468i	$0.603633\pi$
$830$	0.758168	0.0263164
$831$	82.2413	2.85292
$832$	−12.6801	−0.439602
$833$	38.9398	1.34918
$834$	−81.1315	−2.80935
$835$	−1.93401	−0.0669291
$836$	−5.71724	−0.197735
$837$	−88.4188	−3.05620
$838$	−79.2298	−2.73695
$839$	−37.7087	−1.30185	−0.650925	−	0.759142i	$-0.725619\pi$
$839$	−37.7087	−1.30185	−0.650925	+	0.759142i	$0.725619\pi$
$840$	10.2566	0.353885
$841$	−17.6095	−0.607224
$842$	22.5964	0.778724
$843$	40.8414	1.40665
$844$	0.575890	0.0198229
$845$	−0.382291	−0.0131512
$846$	−49.6040	−1.70542
$847$	26.0281	0.894336
$848$	−46.6461	−1.60183
$849$	70.6185	2.42362
$850$	89.8463	3.08170
$851$	8.81901	0.302312
$852$	8.42501	0.288636
$853$	−13.2943	−0.455188	−0.227594	−	0.973756i	$-0.573086\pi$
$853$	−13.2943	−0.455188	−0.227594	+	0.973756i	$0.573086\pi$
$854$	40.5103	1.38623
$855$	0.826517	0.0282663
$856$	−71.6159	−2.44778
$857$	−9.90941	−0.338499	−0.169250	−	0.985573i	$-0.554134\pi$
$857$	−9.90941	−0.338499	−0.169250	+	0.985573i	$0.554134\pi$
$858$	62.1857	2.12298
$859$	23.5268	0.802725	0.401362	−	0.915919i	$-0.368537\pi$
$859$	23.5268	0.802725	0.401362	+	0.915919i	$0.368537\pi$
$860$	−1.83917	−0.0627151
$861$	54.1781	1.84638
$862$	−91.7434	−3.12479
$863$	42.0676	1.43200	0.715999	−	0.698101i	$-0.245971\pi$
$863$	42.0676	1.43200	0.715999	+	0.698101i	$0.245971\pi$
$864$	56.9926	1.93893
$865$	−3.70169	−0.125861
$866$	24.0547	0.817413
$867$	−113.871	−3.86726
$868$	−83.6723	−2.84002
$869$	17.2889	0.586487
$870$	4.22338	0.143186
$871$	−49.4944	−1.67705
$872$	−16.7057	−0.565725
$873$	−73.2666	−2.47970
$874$	−13.9664	−0.472421
$875$	−5.30431	−0.179319
$876$	121.034	4.08936
$877$	−24.8528	−0.839219	−0.419610	−	0.907705i	$-0.637833\pi$
$877$	−24.8528	−0.839219	−0.419610	+	0.907705i	$0.637833\pi$
$878$	19.8551	0.670077
$879$	82.7555	2.79127
$880$	−1.73715	−0.0585593
$881$	35.8262	1.20702	0.603508	−	0.797357i	$-0.293769\pi$
$881$	35.8262	1.20702	0.603508	+	0.797357i	$0.293769\pi$
$882$	−107.567	−3.62197
$883$	−6.43434	−0.216533	−0.108266	−	0.994122i	$-0.534530\pi$
$883$	−6.43434	−0.216533	−0.108266	+	0.994122i	$0.534530\pi$
$884$	122.358	4.11534
$885$	−6.35855	−0.213740
$886$	−94.7253	−3.18236
$887$	29.2088	0.980735	0.490367	−	0.871516i	$-0.336863\pi$
$887$	29.2088	0.980735	0.490367	+	0.871516i	$0.336863\pi$
$888$	−21.2970	−0.714682
$889$	66.7335	2.23817
$890$	2.56252	0.0858959
$891$	56.0965	1.87930
$892$	26.7063	0.894193
$893$	−1.73908	−0.0581963
$894$	−131.597	−4.40128
$895$	2.97936	0.0995892
$896$	−53.4596	−1.78596
$897$	103.877	3.46836
$898$	−11.7587	−0.392393
$899$	−18.5223	−0.617752
$900$	−169.714	−5.65714
$901$	−55.2757	−1.84150
$902$	−22.2812	−0.741883
$903$	32.7859	1.09105
$904$	51.7665	1.72173
$905$	−0.348894	−0.0115976
$906$	66.5973	2.21255
$907$	53.5598	1.77842	0.889212	−	0.457496i	$-0.151254\pi$
$907$	53.5598	1.77842	0.889212	+	0.457496i	$0.151254\pi$
$908$	−25.0593	−0.831621
$909$	141.933	4.70762
$910$	−5.26996	−0.174697
$911$	−2.01259	−0.0666799	−0.0333400	−	0.999444i	$-0.510614\pi$
$911$	−2.01259	−0.0666799	−0.0333400	+	0.999444i	$0.510614\pi$
$912$	13.8901	0.459946
$913$	3.80083	0.125789
$914$	−82.8125	−2.73919
$915$	2.27367	0.0751651
$916$	−72.0543	−2.38074
$917$	−50.5887	−1.67059
$918$	290.826	9.59868
$919$	57.3266	1.89103	0.945515	−	0.325579i	$-0.105559\pi$
$919$	57.3266	1.89103	0.945515	+	0.325579i	$0.105559\pi$
$920$	−7.04608	−0.232302
$921$	−36.3001	−1.19613
$922$	64.0597	2.10969
$923$	−2.32718	−0.0766001
$924$	95.6446	3.14648
$925$	5.49447	0.180657
$926$	−35.3678	−1.16226
$927$	57.5208	1.88923
$928$	11.9390	0.391917
$929$	4.28569	0.140609	0.0703044	−	0.997526i	$-0.477603\pi$
$929$	4.28569	0.140609	0.0703044	+	0.997526i	$0.477603\pi$
$930$	−6.86770	−0.225201
$931$	−3.77123	−0.123597
$932$	33.0444	1.08241
$933$	46.8763	1.53466
$934$	96.7931	3.16717
$935$	−2.05852	−0.0673208
$936$	−181.707	−5.93928
$937$	−6.00355	−0.196127	−0.0980637	−	0.995180i	$-0.531265\pi$
$937$	−6.00355	−0.196127	−0.0980637	+	0.995180i	$0.531265\pi$
$938$	−111.325	−3.63490
$939$	64.0967	2.09172
$940$	−1.63203	−0.0532311
$941$	−19.1884	−0.625524	−0.312762	−	0.949831i	$-0.601254\pi$
$941$	−19.1884	−0.625524	−0.312762	+	0.949831i	$0.601254\pi$
$942$	−24.4927	−0.798014
$943$	−37.2194	−1.21203
$944$	−77.4038	−2.51928
$945$	−8.56526	−0.278628
$946$	−13.4835	−0.438386
$947$	−10.2630	−0.333503	−0.166751	−	0.985999i	$-0.553328\pi$
$947$	−10.2630	−0.333503	−0.166751	+	0.985999i	$0.553328\pi$
$948$	−129.731	−4.21347
$949$	−33.4324	−1.08526
$950$	−8.70143	−0.282312
$951$	0.657133	0.0213090
$952$	147.953	4.79519
$953$	−37.5809	−1.21737	−0.608683	−	0.793414i	$-0.708302\pi$
$953$	−37.5809	−1.21737	−0.608683	+	0.793414i	$0.708302\pi$
$954$	152.693	4.94363
$955$	0.140708	0.00455320
$956$	51.3611	1.66114
$957$	21.1725	0.684411
$958$	−1.57982	−0.0510418
$959$	−42.5195	−1.37303
$960$	−1.60074	−0.0516637
$961$	−0.880679	−0.0284090
$962$	10.9427	0.352807
$963$	96.5460	3.11115
$964$	−21.3887	−0.688884
$965$	−1.85335	−0.0596615
$966$	233.646	7.51745
$967$	−9.60114	−0.308752	−0.154376	−	0.988012i	$-0.549337\pi$
$967$	−9.60114	−0.308752	−0.154376	+	0.988012i	$0.549337\pi$
$968$	43.1805	1.38787
$969$	16.4597	0.528763
$970$	−3.52522	−0.113188
$971$	−17.3263	−0.556027	−0.278013	−	0.960577i	$-0.589676\pi$
$971$	−17.3263	−0.556027	−0.278013	+	0.960577i	$0.589676\pi$
$972$	−211.882	−6.79612
$973$	34.4683	1.10500
$974$	110.101	3.52788
$975$	64.7183	2.07264
$976$	27.6777	0.885943
$977$	−3.98944	−0.127633	−0.0638167	−	0.997962i	$-0.520327\pi$
$977$	−3.98944	−0.127633	−0.0638167	+	0.997962i	$0.520327\pi$
$978$	104.104	3.32887
$979$	12.8464	0.410572
$980$	−3.53909	−0.113052
$981$	22.5210	0.719041
$982$	90.2510	2.88003
$983$	46.5343	1.48421	0.742106	−	0.670282i	$-0.233827\pi$
$983$	46.5343	1.48421	0.742106	+	0.670282i	$0.233827\pi$
$984$	89.8812	2.86531
$985$	−3.02638	−0.0964284
$986$	60.9231	1.94019
$987$	29.0934	0.926054
$988$	−11.8501	−0.377002
$989$	−22.5233	−0.716201
$990$	5.68645	0.180727
$991$	13.8831	0.441010	0.220505	−	0.975386i	$-0.429230\pi$
$991$	13.8831	0.441010	0.220505	+	0.975386i	$0.429230\pi$
$992$	−19.4142	−0.616401
$993$	74.3544	2.35956
$994$	−5.23442	−0.166026
$995$	1.99161	0.0631383
$996$	−28.5203	−0.903701
$997$	16.2041	0.513190	0.256595	−	0.966519i	$-0.417399\pi$
$997$	16.2041	0.513190	0.256595	+	0.966519i	$0.417399\pi$
$998$	65.2891	2.06669
$999$	17.7852	0.562698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4013.2.a.b.1.13	✓	157

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4013.2.a.b.1.13	✓	157	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:	\( N \)	\(=\)	\( 4013 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51500 q^{2} -3.29902 q^{3} +4.32520 q^{4} -0.150822 q^{5} +8.29703 q^{6} -3.52495 q^{7} -5.84787 q^{8} +7.88356 q^{9} +O(q^{10})\)	\(q-2.51500 q^{2} -3.29902 q^{3} +4.32520 q^{4} -0.150822 q^{5} +8.29703 q^{6} -3.52495 q^{7} -5.84787 q^{8} +7.88356 q^{9} +0.379318 q^{10} +1.90159 q^{11} -14.2689 q^{12} +3.94141 q^{13} +8.86522 q^{14} +0.497567 q^{15} +6.05697 q^{16} +7.17750 q^{17} -19.8271 q^{18} -0.695126 q^{19} -0.652337 q^{20} +11.6289 q^{21} -4.78248 q^{22} -7.98884 q^{23} +19.2923 q^{24} -4.97725 q^{25} -9.91263 q^{26} -16.1110 q^{27} -15.2461 q^{28} -3.37498 q^{29} -1.25138 q^{30} +5.48811 q^{31} -3.53750 q^{32} -6.27338 q^{33} -18.0514 q^{34} +0.531641 q^{35} +34.0980 q^{36} -1.10392 q^{37} +1.74824 q^{38} -13.0028 q^{39} +0.881990 q^{40} +4.65892 q^{41} -29.2466 q^{42} +2.81935 q^{43} +8.22474 q^{44} -1.18902 q^{45} +20.0919 q^{46} +2.50183 q^{47} -19.9821 q^{48} +5.42525 q^{49} +12.5178 q^{50} -23.6788 q^{51} +17.0474 q^{52} -7.70124 q^{53} +40.5191 q^{54} -0.286802 q^{55} +20.6134 q^{56} +2.29324 q^{57} +8.48807 q^{58} -12.7793 q^{59} +2.15208 q^{60} +4.56957 q^{61} -13.8026 q^{62} -27.7891 q^{63} -3.21714 q^{64} -0.594453 q^{65} +15.7775 q^{66} -12.5575 q^{67} +31.0442 q^{68} +26.3554 q^{69} -1.33707 q^{70} -0.590444 q^{71} -46.1021 q^{72} -8.48233 q^{73} +2.77634 q^{74} +16.4201 q^{75} -3.00656 q^{76} -6.70299 q^{77} +32.7020 q^{78} +9.09185 q^{79} -0.913526 q^{80} +29.4999 q^{81} -11.7172 q^{82} +1.99877 q^{83} +50.2973 q^{84} -1.08253 q^{85} -7.09065 q^{86} +11.1342 q^{87} -11.1202 q^{88} +6.75561 q^{89} +2.99037 q^{90} -13.8933 q^{91} -34.5534 q^{92} -18.1054 q^{93} -6.29208 q^{94} +0.104841 q^{95} +11.6703 q^{96} -9.29359 q^{97} -13.6445 q^{98} +14.9913 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9}+O(q^{10}) \) 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9	\( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9} - 61 q^{10} - 27 q^{11} - 93 q^{12} - 97 q^{13} - 12 q^{14} - 36 q^{15} + 105 q^{16} - 45 q^{17} - 68 q^{18} - 128 q^{19} - 30 q^{20} - 26 q^{21} - 68 q^{22} - 41 q^{23} - 40 q^{24} + 102 q^{25} - 5 q^{26} - 189 q^{27} - 115 q^{28} - 26 q^{29} - 12 q^{30} - 88 q^{31} - 89 q^{32} - 52 q^{33} - 61 q^{34} - 87 q^{35} + 110 q^{36} - 62 q^{37} - 37 q^{38} - 20 q^{39} - 161 q^{40} - 34 q^{41} - 53 q^{42} - 254 q^{43} - 19 q^{44} - 46 q^{45} - 52 q^{46} - 76 q^{47} - 162 q^{48} + 96 q^{49} - 54 q^{50} - 76 q^{51} - 259 q^{52} - 48 q^{53} - 12 q^{54} - 194 q^{55} - 10 q^{56} - 30 q^{57} - 52 q^{58} - 64 q^{59} - 31 q^{60} - 107 q^{61} - 51 q^{62} - 106 q^{63} + 54 q^{64} - 17 q^{65} - 13 q^{66} - 193 q^{67} - 118 q^{68} - 55 q^{69} - 86 q^{70} - 11 q^{71} - 172 q^{72} - 173 q^{73} - 11 q^{74} - 209 q^{75} - 213 q^{76} - 84 q^{77} - 30 q^{78} - 111 q^{79} - 6 q^{80} + 157 q^{81} - 117 q^{82} - 154 q^{83} - 6 q^{84} - 91 q^{85} + 28 q^{86} - 165 q^{87} - 165 q^{88} - 32 q^{89} - 103 q^{90} - 200 q^{91} - 86 q^{92} - 39 q^{93} - 118 q^{94} - 22 q^{95} - 28 q^{96} - 151 q^{97} - 38 q^{98} - 91 q^{99}+O(q^{100}) \) 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9 - 61 * q^10 - 27 * q^11 - 93 * q^12 - 97 * q^13 - 12 * q^14 - 36 * q^15 + 105 * q^16 - 45 * q^17 - 68 * q^18 - 128 * q^19 - 30 * q^20 - 26 * q^21 - 68 * q^22 - 41 * q^23 - 40 * q^24 + 102 * q^25 - 5 * q^26 - 189 * q^27 - 115 * q^28 - 26 * q^29 - 12 * q^30 - 88 * q^31 - 89 * q^32 - 52 * q^33 - 61 * q^34 - 87 * q^35 + 110 * q^36 - 62 * q^37 - 37 * q^38 - 20 * q^39 - 161 * q^40 - 34 * q^41 - 53 * q^42 - 254 * q^43 - 19 * q^44 - 46 * q^45 - 52 * q^46 - 76 * q^47 - 162 * q^48 + 96 * q^49 - 54 * q^50 - 76 * q^51 - 259 * q^52 - 48 * q^53 - 12 * q^54 - 194 * q^55 - 10 * q^56 - 30 * q^57 - 52 * q^58 - 64 * q^59 - 31 * q^60 - 107 * q^61 - 51 * q^62 - 106 * q^63 + 54 * q^64 - 17 * q^65 - 13 * q^66 - 193 * q^67 - 118 * q^68 - 55 * q^69 - 86 * q^70 - 11 * q^71 - 172 * q^72 - 173 * q^73 - 11 * q^74 - 209 * q^75 - 213 * q^76 - 84 * q^77 - 30 * q^78 - 111 * q^79 - 6 * q^80 + 157 * q^81 - 117 * q^82 - 154 * q^83 - 6 * q^84 - 91 * q^85 + 28 * q^86 - 165 * q^87 - 165 * q^88 - 32 * q^89 - 103 * q^90 - 200 * q^91 - 86 * q^92 - 39 * q^93 - 118 * q^94 - 22 * q^95 - 28 * q^96 - 151 * q^97 - 38 * q^98 - 91 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more