LMFDB - Embedded newform 1441.2.a.d.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1441,2,Mod(1,1441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1441, 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("1441.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1441 = 11 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1441.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:	$11.5064429313$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	1441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.465389 q^{2} +1.44715 q^{3} -1.78341 q^{4} +0.0108050 q^{5} +0.673486 q^{6} +0.934440 q^{7} -1.76076 q^{8} -0.905766 q^{9} +O(q^{10})$	\(q+0.465389 q^{2} +1.44715 q^{3} -1.78341 q^{4} +0.0108050 q^{5} +0.673486 q^{6} +0.934440 q^{7} -1.76076 q^{8} -0.905766 q^{9} +0.00502854 q^{10} -1.00000 q^{11} -2.58086 q^{12} -4.66944 q^{13} +0.434878 q^{14} +0.0156365 q^{15} +2.74739 q^{16} +3.30753 q^{17} -0.421533 q^{18} -5.70011 q^{19} -0.0192698 q^{20} +1.35227 q^{21} -0.465389 q^{22} +5.06048 q^{23} -2.54808 q^{24} -4.99988 q^{25} -2.17311 q^{26} -5.65222 q^{27} -1.66649 q^{28} +1.85224 q^{29} +0.00727703 q^{30} -7.24926 q^{31} +4.80012 q^{32} -1.44715 q^{33} +1.53929 q^{34} +0.0100967 q^{35} +1.61536 q^{36} -4.26049 q^{37} -2.65277 q^{38} -6.75737 q^{39} -0.0190250 q^{40} -1.24030 q^{41} +0.629332 q^{42} -12.9497 q^{43} +1.78341 q^{44} -0.00978682 q^{45} +2.35509 q^{46} +0.479276 q^{47} +3.97588 q^{48} -6.12682 q^{49} -2.32689 q^{50} +4.78648 q^{51} +8.32754 q^{52} +2.14066 q^{53} -2.63048 q^{54} -0.0108050 q^{55} -1.64532 q^{56} -8.24889 q^{57} +0.862011 q^{58} -7.72748 q^{59} -0.0278863 q^{60} +13.4961 q^{61} -3.37373 q^{62} -0.846384 q^{63} -3.26086 q^{64} -0.0504534 q^{65} -0.673486 q^{66} +9.59809 q^{67} -5.89870 q^{68} +7.32325 q^{69} +0.00469887 q^{70} +1.68786 q^{71} +1.59483 q^{72} -9.63211 q^{73} -1.98279 q^{74} -7.23557 q^{75} +10.1656 q^{76} -0.934440 q^{77} -3.14480 q^{78} +7.87574 q^{79} +0.0296856 q^{80} -5.46229 q^{81} -0.577223 q^{82} +12.8744 q^{83} -2.41166 q^{84} +0.0357380 q^{85} -6.02664 q^{86} +2.68046 q^{87} +1.76076 q^{88} -1.25742 q^{89} -0.00455468 q^{90} -4.36332 q^{91} -9.02492 q^{92} -10.4908 q^{93} +0.223050 q^{94} -0.0615898 q^{95} +6.94648 q^{96} +10.1108 q^{97} -2.85135 q^{98} +0.905766 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 7 q^{2} - 3 q^{3} + 19 q^{4} - 9 q^{5} - 5 q^{6} - 8 q^{7} - 15 q^{8} + 12 q^{9}+O(q^{10}) $ 23 * q - 7 * q^2 - 3 * q^3 + 19 * q^4 - 9 * q^5 - 5 * q^6 - 8 * q^7 - 15 * q^8 + 12 * q^9	\( 23 q - 7 q^{2} - 3 q^{3} + 19 q^{4} - 9 q^{5} - 5 q^{6} - 8 q^{7} - 15 q^{8} + 12 q^{9} - 2 q^{10} - 23 q^{11} - 12 q^{12} + 6 q^{13} - 13 q^{14} - 27 q^{15} - q^{16} - 11 q^{17} - 16 q^{19} - 24 q^{20} - 7 q^{21} + 7 q^{22} - 30 q^{23} - 20 q^{24} + 10 q^{25} - 22 q^{26} - 15 q^{27} + 11 q^{28} - 15 q^{29} + 17 q^{30} - 31 q^{31} - 22 q^{32} + 3 q^{33} - 18 q^{34} - 34 q^{35} - 18 q^{36} - 26 q^{37} - 32 q^{39} + 16 q^{40} - 21 q^{41} - 37 q^{42} - 2 q^{43} - 19 q^{44} - 18 q^{45} - 6 q^{46} - 25 q^{47} + 14 q^{48} + 7 q^{49} - 27 q^{50} - 7 q^{51} + 23 q^{52} - 34 q^{53} + 26 q^{54} + 9 q^{55} - 42 q^{56} + 6 q^{57} - 5 q^{58} - 31 q^{59} - 7 q^{60} + 19 q^{61} + 24 q^{62} - 34 q^{63} - 17 q^{64} - 13 q^{65} + 5 q^{66} - 39 q^{67} - 59 q^{68} - 12 q^{69} + 17 q^{70} - 103 q^{71} + 19 q^{72} + q^{73} - 9 q^{74} - 19 q^{75} + 10 q^{76} + 8 q^{77} - 43 q^{78} - 14 q^{79} - q^{80} - 29 q^{81} + 20 q^{82} - 14 q^{83} + 57 q^{84} + 20 q^{85} - 54 q^{86} - 23 q^{87} + 15 q^{88} - 71 q^{89} - 52 q^{90} - 18 q^{91} - 24 q^{92} - q^{93} + q^{94} - 38 q^{95} - 31 q^{96} - 26 q^{97} + 19 q^{98} - 12 q^{99}+O(q^{100}) \) 23 * q - 7 * q^2 - 3 * q^3 + 19 * q^4 - 9 * q^5 - 5 * q^6 - 8 * q^7 - 15 * q^8 + 12 * q^9 - 2 * q^10 - 23 * q^11 - 12 * q^12 + 6 * q^13 - 13 * q^14 - 27 * q^15 - q^16 - 11 * q^17 - 16 * q^19 - 24 * q^20 - 7 * q^21 + 7 * q^22 - 30 * q^23 - 20 * q^24 + 10 * q^25 - 22 * q^26 - 15 * q^27 + 11 * q^28 - 15 * q^29 + 17 * q^30 - 31 * q^31 - 22 * q^32 + 3 * q^33 - 18 * q^34 - 34 * q^35 - 18 * q^36 - 26 * q^37 - 32 * q^39 + 16 * q^40 - 21 * q^41 - 37 * q^42 - 2 * q^43 - 19 * q^44 - 18 * q^45 - 6 * q^46 - 25 * q^47 + 14 * q^48 + 7 * q^49 - 27 * q^50 - 7 * q^51 + 23 * q^52 - 34 * q^53 + 26 * q^54 + 9 * q^55 - 42 * q^56 + 6 * q^57 - 5 * q^58 - 31 * q^59 - 7 * q^60 + 19 * q^61 + 24 * q^62 - 34 * q^63 - 17 * q^64 - 13 * q^65 + 5 * q^66 - 39 * q^67 - 59 * q^68 - 12 * q^69 + 17 * q^70 - 103 * q^71 + 19 * q^72 + q^73 - 9 * q^74 - 19 * q^75 + 10 * q^76 + 8 * q^77 - 43 * q^78 - 14 * q^79 - q^80 - 29 * q^81 + 20 * q^82 - 14 * q^83 + 57 * q^84 + 20 * q^85 - 54 * q^86 - 23 * q^87 + 15 * q^88 - 71 * q^89 - 52 * q^90 - 18 * q^91 - 24 * q^92 - q^93 + q^94 - 38 * q^95 - 31 * q^96 - 26 * q^97 + 19 * q^98 - 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.465389	0.329079	0.164540	−	0.986370i	$-0.447386\pi$
$2$	0.465389	0.329079	0.164540	+	0.986370i	$0.447386\pi$
$3$	1.44715	0.835511	0.417755	−	0.908560i	$-0.362817\pi$
$3$	1.44715	0.835511	0.417755	+	0.908560i	$0.362817\pi$
$4$	−1.78341	−0.891707
$5$	0.0108050	0.00483215	0.00241608	−	0.999997i	$-0.499231\pi$
$5$	0.0108050	0.00483215	0.00241608	+	0.999997i	$0.499231\pi$
$6$	0.673486	0.274949
$7$	0.934440	0.353185	0.176593	−	0.984284i	$-0.443492\pi$
$7$	0.934440	0.353185	0.176593	+	0.984284i	$0.443492\pi$
$8$	−1.76076	−0.622522
$9$	−0.905766	−0.301922
$10$	0.00502854	0.00159016
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.58086	−0.745030
$13$	−4.66944	−1.29507	−0.647535	−	0.762036i	$-0.724200\pi$
$13$	−4.66944	−1.29507	−0.647535	+	0.762036i	$0.724200\pi$
$14$	0.434878	0.116226
$15$	0.0156365	0.00403732
$16$	2.74739	0.686848
$17$	3.30753	0.802194	0.401097	−	0.916036i	$-0.368629\pi$
$17$	3.30753	0.802194	0.401097	+	0.916036i	$0.368629\pi$
$18$	−0.421533	−0.0993563
$19$	−5.70011	−1.30769	−0.653847	−	0.756627i	$-0.726846\pi$
$19$	−5.70011	−1.30769	−0.653847	+	0.756627i	$0.726846\pi$
$20$	−0.0192698	−0.00430886
$21$	1.35227	0.295090
$22$	−0.465389	−0.0992212
$23$	5.06048	1.05518	0.527591	−	0.849498i	$-0.323095\pi$
$23$	5.06048	1.05518	0.527591	+	0.849498i	$0.323095\pi$
$24$	−2.54808	−0.520124
$25$	−4.99988	−0.999977
$26$	−2.17311	−0.426181
$27$	−5.65222	−1.08777
$28$	−1.66649	−0.314938
$29$	1.85224	0.343952	0.171976	−	0.985101i	$-0.444985\pi$
$29$	1.85224	0.343952	0.171976	+	0.985101i	$0.444985\pi$
$30$	0.00727703	0.00132860
$31$	−7.24926	−1.30201	−0.651003	−	0.759075i	$-0.725652\pi$
$31$	−7.24926	−1.30201	−0.651003	+	0.759075i	$0.725652\pi$
$32$	4.80012	0.848549
$33$	−1.44715	−0.251916
$34$	1.53929	0.263986
$35$	0.0100967	0.00170665
$36$	1.61536	0.269226
$37$	−4.26049	−0.700421	−0.350210	−	0.936671i	$-0.613890\pi$
$37$	−4.26049	−0.700421	−0.350210	+	0.936671i	$0.613890\pi$
$38$	−2.65277	−0.430335
$39$	−6.75737	−1.08204
$40$	−0.0190250	−0.00300812
$41$	−1.24030	−0.193703	−0.0968515	−	0.995299i	$-0.530877\pi$
$41$	−1.24030	−0.193703	−0.0968515	+	0.995299i	$0.530877\pi$
$42$	0.629332	0.0971081
$43$	−12.9497	−1.97481	−0.987404	−	0.158216i	$-0.949426\pi$
$43$	−12.9497	−1.97481	−0.987404	+	0.158216i	$0.949426\pi$
$44$	1.78341	0.268860
$45$	−0.00978682	−0.00145893
$46$	2.35509	0.347239
$47$	0.479276	0.0699096	0.0349548	−	0.999389i	$-0.488871\pi$
$47$	0.479276	0.0699096	0.0349548	+	0.999389i	$0.488871\pi$
$48$	3.97588	0.573868
$49$	−6.12682	−0.875260
$50$	−2.32689	−0.329072
$51$	4.78648	0.670242
$52$	8.32754	1.15482
$53$	2.14066	0.294043	0.147021	−	0.989133i	$-0.453031\pi$
$53$	2.14066	0.294043	0.147021	+	0.989133i	$0.453031\pi$
$54$	−2.63048	−0.357963
$55$	−0.0108050	−0.00145695
$56$	−1.64532	−0.219866
$57$	−8.24889	−1.09259
$58$	0.862011	0.113188
$59$	−7.72748	−1.00603	−0.503016	−	0.864277i	$-0.667776\pi$
$59$	−7.72748	−1.00603	−0.503016	+	0.864277i	$0.667776\pi$
$60$	−0.0278863	−0.00360010
$61$	13.4961	1.72800	0.863998	−	0.503495i	$-0.167953\pi$
$61$	13.4961	1.72800	0.863998	+	0.503495i	$0.167953\pi$
$62$	−3.37373	−0.428464
$63$	−0.846384	−0.106634
$64$	−3.26086	−0.407607
$65$	−0.0504534	−0.00625798
$66$	−0.673486	−0.0829004
$67$	9.59809	1.17259	0.586297	−	0.810096i	$-0.300585\pi$
$67$	9.59809	1.17259	0.586297	+	0.810096i	$0.300585\pi$
$68$	−5.89870	−0.715322
$69$	7.32325	0.881616
$70$	0.00469887	0.000561622 0
$71$	1.68786	0.200313	0.100156	−	0.994972i	$-0.468066\pi$
$71$	1.68786	0.200313	0.100156	+	0.994972i	$0.468066\pi$
$72$	1.59483	0.187953
$73$	−9.63211	−1.12735	−0.563676	−	0.825996i	$-0.690613\pi$
$73$	−9.63211	−1.12735	−0.563676	+	0.825996i	$0.690613\pi$
$74$	−1.98279	−0.230494
$75$	−7.23557	−0.835491
$76$	10.1656	1.16608
$77$	−0.934440	−0.106489
$78$	−3.14480	−0.356079
$79$	7.87574	0.886090	0.443045	−	0.896499i	$-0.353898\pi$
$79$	7.87574	0.886090	0.443045	+	0.896499i	$0.353898\pi$
$80$	0.0296856	0.00331895
$81$	−5.46229	−0.606921
$82$	−0.577223	−0.0637437
$83$	12.8744	1.41315	0.706574	−	0.707639i	$-0.250240\pi$
$83$	12.8744	1.41315	0.706574	+	0.707639i	$0.250240\pi$
$84$	−2.41166	−0.263134
$85$	0.0357380	0.00387633
$86$	−6.02664	−0.649869
$87$	2.68046	0.287376
$88$	1.76076	0.187697
$89$	−1.25742	−0.133286	−0.0666432	−	0.997777i	$-0.521229\pi$
$89$	−1.25742	−0.133286	−0.0666432	+	0.997777i	$0.521229\pi$
$90$	−0.00455468	−0.000480105 0
$91$	−4.36332	−0.457400
$92$	−9.02492	−0.940913
$93$	−10.4908	−1.08784
$94$	0.223050	0.0230058
$95$	−0.0615898	−0.00631898
$96$	6.94648	0.708972
$97$	10.1108	1.02660	0.513299	−	0.858210i	$-0.328423\pi$
$97$	10.1108	1.02660	0.513299	+	0.858210i	$0.328423\pi$
$98$	−2.85135	−0.288030
$99$	0.905766	0.0910329
$100$	8.91686	0.891686
$101$	−9.73083	−0.968253	−0.484127	−	0.874998i	$-0.660863\pi$
$101$	−9.73083	−0.968253	−0.484127	+	0.874998i	$0.660863\pi$
$102$	2.22758	0.220563
$103$	−2.30467	−0.227086	−0.113543	−	0.993533i	$-0.536220\pi$
$103$	−2.30467	−0.227086	−0.113543	+	0.993533i	$0.536220\pi$
$104$	8.22175	0.806209
$105$	0.0146113	0.00142592
$106$	0.996241	0.0967635
$107$	−12.9622	−1.25310	−0.626552	−	0.779380i	$-0.715534\pi$
$107$	−12.9622	−1.25310	−0.626552	+	0.779380i	$0.715534\pi$
$108$	10.0802	0.969971
$109$	−6.99601	−0.670096	−0.335048	−	0.942201i	$-0.608753\pi$
$109$	−6.99601	−0.670096	−0.335048	+	0.942201i	$0.608753\pi$
$110$	−0.00502854	−0.000479452 0
$111$	−6.16556	−0.585209
$112$	2.56727	0.242584
$113$	−12.2981	−1.15691	−0.578456	−	0.815714i	$-0.696344\pi$
$113$	−12.2981	−1.15691	−0.578456	+	0.815714i	$0.696344\pi$
$114$	−3.83894	−0.359550
$115$	0.0546786	0.00509880
$116$	−3.30331	−0.306705
$117$	4.22942	0.391010
$118$	−3.59628	−0.331065
$119$	3.09069	0.283323
$120$	−0.0275320	−0.00251332
$121$	1.00000	0.0909091
$122$	6.28092	0.568648
$123$	−1.79490	−0.161841
$124$	12.9284	1.16101
$125$	−0.108049	−0.00966419
$126$	−0.393898	−0.0350912
$127$	−0.278103	−0.0246777	−0.0123388	−	0.999924i	$-0.503928\pi$
$127$	−0.278103	−0.0246777	−0.0123388	+	0.999924i	$0.503928\pi$
$128$	−11.1178	−0.982685
$129$	−18.7401	−1.64997
$130$	−0.0234804	−0.00205937
$131$	−1.00000	−0.0873704
$131$	−1.00000	−0.0873704
$132$	2.58086	0.224635
$133$	−5.32641	−0.461858
$134$	4.46684	0.385876
$135$	−0.0610723	−0.00525627
$136$	−5.82376	−0.499384
$137$	2.99500	0.255880	0.127940	−	0.991782i	$-0.459163\pi$
$137$	2.99500	0.255880	0.127940	+	0.991782i	$0.459163\pi$
$138$	3.40816	0.290122
$139$	4.96246	0.420911	0.210455	−	0.977603i	$-0.432505\pi$
$139$	4.96246	0.420911	0.210455	+	0.977603i	$0.432505\pi$
$140$	−0.0180065	−0.00152183
$141$	0.693583	0.0584102
$142$	0.785513	0.0659188
$143$	4.66944	0.390478
$144$	−2.48849	−0.207374
$145$	0.0200135	0.00166203
$146$	−4.48267	−0.370989
$147$	−8.86641	−0.731289
$148$	7.59822	0.624570
$149$	0.706526	0.0578809	0.0289404	−	0.999581i	$-0.490787\pi$
$149$	0.706526	0.0578809	0.0289404	+	0.999581i	$0.490787\pi$
$150$	−3.36735	−0.274943
$151$	6.83710	0.556396	0.278198	−	0.960524i	$-0.410263\pi$
$151$	6.83710	0.556396	0.278198	+	0.960524i	$0.410263\pi$
$152$	10.0365	0.814068
$153$	−2.99585	−0.242200
$154$	−0.434878	−0.0350435
$155$	−0.0783285	−0.00629149
$156$	12.0512	0.964867
$157$	−19.4057	−1.54875	−0.774374	−	0.632729i	$-0.781935\pi$
$157$	−19.4057	−1.54875	−0.774374	+	0.632729i	$0.781935\pi$
$158$	3.66528	0.291594
$159$	3.09786	0.245676
$160$	0.0518654	0.00410032
$161$	4.72871	0.372675
$162$	−2.54209	−0.199725
$163$	7.97629	0.624752	0.312376	−	0.949959i	$-0.398875\pi$
$163$	7.97629	0.624752	0.312376	+	0.949959i	$0.398875\pi$
$164$	2.21197	0.172726
$165$	−0.0156365	−0.00121730
$166$	5.99160	0.465038
$167$	8.90398	0.689011	0.344505	−	0.938784i	$-0.388047\pi$
$167$	8.90398	0.689011	0.344505	+	0.938784i	$0.388047\pi$
$168$	−2.38102	−0.183700
$169$	8.80368	0.677206
$170$	0.0166320	0.00127562
$171$	5.16296	0.394822
$172$	23.0946	1.76095
$173$	21.6938	1.64935	0.824676	−	0.565605i	$-0.191357\pi$
$173$	21.6938	1.64935	0.824676	+	0.565605i	$0.191357\pi$
$174$	1.24746	0.0945695
$175$	−4.67209	−0.353177
$176$	−2.74739	−0.207092
$177$	−11.1828	−0.840551
$178$	−0.585190	−0.0438618
$179$	14.9607	1.11821	0.559106	−	0.829096i	$-0.311145\pi$
$179$	14.9607	1.11821	0.559106	+	0.829096i	$0.311145\pi$
$180$	0.0174539	0.00130094
$181$	14.7536	1.09662	0.548312	−	0.836274i	$-0.315271\pi$
$181$	14.7536	1.09662	0.548312	+	0.836274i	$0.315271\pi$
$182$	−2.03064	−0.150521
$183$	19.5308	1.44376
$184$	−8.91027	−0.656874
$185$	−0.0460347	−0.00338454
$186$	−4.88228	−0.357986
$187$	−3.30753	−0.241871
$188$	−0.854748	−0.0623389
$189$	−5.28166	−0.384184
$190$	−0.0286632	−0.00207945
$191$	−14.9321	−1.08045	−0.540226	−	0.841520i	$-0.681661\pi$
$191$	−14.9321	−1.08045	−0.540226	+	0.841520i	$0.681661\pi$
$192$	−4.71894	−0.340560
$193$	0.686514	0.0494164	0.0247082	−	0.999695i	$-0.492134\pi$
$193$	0.686514	0.0494164	0.0247082	+	0.999695i	$0.492134\pi$
$194$	4.70546	0.337832
$195$	−0.0730135	−0.00522861
$196$	10.9267	0.780475
$197$	3.82409	0.272455	0.136228	−	0.990678i	$-0.456502\pi$
$197$	3.82409	0.272455	0.136228	+	0.990678i	$0.456502\pi$
$198$	0.421533	0.0299571
$199$	−2.10973	−0.149555	−0.0747774	−	0.997200i	$-0.523825\pi$
$199$	−2.10973	−0.149555	−0.0747774	+	0.997200i	$0.523825\pi$
$200$	8.80358	0.622507
$201$	13.8898	0.979714
$202$	−4.52862	−0.318632
$203$	1.73081	0.121479
$204$	−8.53628	−0.597659
$205$	−0.0134015	−0.000936002 0
$206$	−1.07257	−0.0747294
$207$	−4.58361	−0.318583
$208$	−12.8288	−0.889516
$209$	5.70011	0.394285
$210$	0.00679995	0.000469241 0
$211$	21.6788	1.49243	0.746216	−	0.665704i	$-0.231869\pi$
$211$	21.6788	1.49243	0.746216	+	0.665704i	$0.231869\pi$
$212$	−3.81769	−0.262200
$213$	2.44259	0.167363
$214$	−6.03246	−0.412371
$215$	−0.139922	−0.00954258
$216$	9.95218	0.677160
$217$	−6.77401	−0.459850
$218$	−3.25586	−0.220515
$219$	−13.9391	−0.941915
$220$	0.0192698	0.00129917
$221$	−15.4443	−1.03890
$222$	−2.86938	−0.192580
$223$	−17.6401	−1.18127	−0.590634	−	0.806940i	$-0.701122\pi$
$223$	−17.6401	−1.18127	−0.590634	+	0.806940i	$0.701122\pi$
$224$	4.48543	0.299695
$225$	4.52872	0.301915
$226$	−5.72341	−0.380716
$227$	18.6868	1.24029	0.620144	−	0.784488i	$-0.287074\pi$
$227$	18.6868	1.24029	0.620144	+	0.784488i	$0.287074\pi$
$228$	14.7112	0.974272
$229$	1.90686	0.126009	0.0630046	−	0.998013i	$-0.479932\pi$
$229$	1.90686	0.126009	0.0630046	+	0.998013i	$0.479932\pi$
$230$	0.0254468	0.00167791
$231$	−1.35227	−0.0889730
$232$	−3.26135	−0.214118
$233$	−17.7611	−1.16357	−0.581785	−	0.813342i	$-0.697646\pi$
$233$	−17.7611	−1.16357	−0.581785	+	0.813342i	$0.697646\pi$
$234$	1.96832	0.128673
$235$	0.00517859	0.000337814 0
$236$	13.7813	0.897086
$237$	11.3974	0.740338
$238$	1.43837	0.0932359
$239$	−23.8113	−1.54023	−0.770114	−	0.637907i	$-0.779801\pi$
$239$	−23.8113	−1.54023	−0.770114	+	0.637907i	$0.779801\pi$
$240$	0.0429594	0.00277302
$241$	−8.63576	−0.556278	−0.278139	−	0.960541i	$-0.589718\pi$
$241$	−8.63576	−0.556278	−0.278139	+	0.960541i	$0.589718\pi$
$242$	0.465389	0.0299163
$243$	9.05191	0.580681
$244$	−24.0691	−1.54087
$245$	−0.0662004	−0.00422939
$246$	−0.835327	−0.0532585
$247$	26.6163	1.69356
$248$	12.7642	0.810527
$249$	18.6311	1.18070
$250$	−0.0502848	−0.00318029
$251$	11.1021	0.700761	0.350381	−	0.936607i	$-0.386052\pi$
$251$	11.1021	0.700761	0.350381	+	0.936607i	$0.386052\pi$
$252$	1.50945	0.0950866
$253$	−5.06048	−0.318149
$254$	−0.129426	−0.00812092
$255$	0.0517181	0.00323871
$256$	1.34762	0.0842260
$257$	16.8730	1.05251	0.526254	−	0.850328i	$-0.323596\pi$
$257$	16.8730	1.05251	0.526254	+	0.850328i	$0.323596\pi$
$258$	−8.72143	−0.542973
$259$	−3.98118	−0.247378
$260$	0.0899793	0.00558028
$261$	−1.67770	−0.103847
$262$	−0.465389	−0.0287518
$263$	−23.6213	−1.45655	−0.728275	−	0.685285i	$-0.759678\pi$
$263$	−23.6213	−1.45655	−0.728275	+	0.685285i	$0.759678\pi$
$264$	2.54808	0.156823
$265$	0.0231299	0.00142086
$266$	−2.47885	−0.151988
$267$	−1.81967	−0.111362
$268$	−17.1174	−1.04561
$269$	27.3700	1.66878	0.834388	−	0.551178i	$-0.185821\pi$
$269$	27.3700	1.66878	0.834388	+	0.551178i	$0.185821\pi$
$270$	−0.0284224	−0.00172973
$271$	4.13087	0.250932	0.125466	−	0.992098i	$-0.459957\pi$
$271$	4.13087	0.250932	0.125466	+	0.992098i	$0.459957\pi$
$272$	9.08708	0.550985
$273$	−6.31436	−0.382162
$274$	1.39384	0.0842049
$275$	4.99988	0.301504
$276$	−13.0604	−0.786143
$277$	−1.81116	−0.108822	−0.0544111	−	0.998519i	$-0.517328\pi$
$277$	−1.81116	−0.108822	−0.0544111	+	0.998519i	$0.517328\pi$
$278$	2.30947	0.138513
$279$	6.56614	0.393104
$280$	−0.0177778	−0.00106242
$281$	−9.74242	−0.581184	−0.290592	−	0.956847i	$-0.593852\pi$
$281$	−9.74242	−0.581184	−0.290592	+	0.956847i	$0.593852\pi$
$282$	0.322786	0.0192216
$283$	7.97245	0.473913	0.236957	−	0.971520i	$-0.423850\pi$
$283$	7.97245	0.473913	0.236957	+	0.971520i	$0.423850\pi$
$284$	−3.01016	−0.178620
$285$	−0.0891295	−0.00527957
$286$	2.17311	0.128498
$287$	−1.15899	−0.0684130
$288$	−4.34778	−0.256196
$289$	−6.06023	−0.356484
$290$	0.00931405	0.000546940 0
$291$	14.6318	0.857733
$292$	17.1780	1.00527
$293$	−9.14825	−0.534447	−0.267223	−	0.963635i	$-0.586106\pi$
$293$	−9.14825	−0.534447	−0.267223	+	0.963635i	$0.586106\pi$
$294$	−4.12633	−0.240652
$295$	−0.0834956	−0.00486130
$296$	7.50170	0.436027
$297$	5.65222	0.327975
$298$	0.328809	0.0190474
$299$	−23.6296	−1.36653
$300$	12.9040	0.745013
$301$	−12.1007	−0.697473
$302$	3.18191	0.183098
$303$	−14.0819	−0.808986
$304$	−15.6604	−0.898187
$305$	0.145825	0.00834994
$306$	−1.39423	−0.0797031
$307$	4.59304	0.262139	0.131069	−	0.991373i	$-0.458159\pi$
$307$	4.59304	0.262139	0.131069	+	0.991373i	$0.458159\pi$
$308$	1.66649	0.0949573
$309$	−3.33520	−0.189733
$310$	−0.0364532	−0.00207040
$311$	−19.5012	−1.10581	−0.552905	−	0.833244i	$-0.686481\pi$
$311$	−19.5012	−1.10581	−0.552905	+	0.833244i	$0.686481\pi$
$312$	11.8981	0.673597
$313$	7.84789	0.443589	0.221794	−	0.975093i	$-0.428809\pi$
$313$	7.84789	0.443589	0.221794	+	0.975093i	$0.428809\pi$
$314$	−9.03121	−0.509661
$315$	−0.00914520	−0.000515274 0
$316$	−14.0457	−0.790133
$317$	−18.1285	−1.01820	−0.509098	−	0.860709i	$-0.670021\pi$
$317$	−18.1285	−1.01820	−0.509098	+	0.860709i	$0.670021\pi$
$318$	1.44171	0.0808469
$319$	−1.85224	−0.103706
$320$	−0.0352337	−0.00196962
$321$	−18.7582	−1.04698
$322$	2.20069	0.122640
$323$	−18.8533	−1.04902
$324$	9.74152	0.541196
$325$	23.3467	1.29504
$326$	3.71208	0.205593
$327$	−10.1243	−0.559872
$328$	2.18387	0.120584
$329$	0.447855	0.0246911
$330$	−0.00727703	−0.000400587 0
$331$	−1.53696	−0.0844789	−0.0422394	−	0.999108i	$-0.513449\pi$
$331$	−1.53696	−0.0844789	−0.0422394	+	0.999108i	$0.513449\pi$
$332$	−22.9604	−1.26011
$333$	3.85901	0.211472
$334$	4.14381	0.226739
$335$	0.103708	0.00566615
$336$	3.71522	0.202682
$337$	13.3491	0.727173	0.363587	−	0.931560i	$-0.381552\pi$
$337$	13.3491	0.727173	0.363587	+	0.931560i	$0.381552\pi$
$338$	4.09713	0.222855
$339$	−17.7972	−0.966611
$340$	−0.0637356	−0.00345655
$341$	7.24926	0.392570
$342$	2.40278	0.129928
$343$	−12.2662	−0.662314
$344$	22.8013	1.22936
$345$	0.0791279	0.00426010
$346$	10.0961	0.542768
$347$	34.5077	1.85247	0.926236	−	0.376944i	$-0.123025\pi$
$347$	34.5077	1.85247	0.926236	+	0.376944i	$0.123025\pi$
$348$	−4.78037	−0.256255
$349$	25.1379	1.34560	0.672801	−	0.739824i	$-0.265091\pi$
$349$	25.1379	1.34560	0.672801	+	0.739824i	$0.265091\pi$
$350$	−2.17434	−0.116223
$351$	26.3927	1.40874
$352$	−4.80012	−0.255847
$353$	−1.11157	−0.0591631	−0.0295816	−	0.999562i	$-0.509417\pi$
$353$	−1.11157	−0.0591631	−0.0295816	+	0.999562i	$0.509417\pi$
$354$	−5.20435	−0.276608
$355$	0.0182374	0.000967941 0
$356$	2.24250	0.118852
$357$	4.47268	0.236720
$358$	6.96252	0.367981
$359$	4.29249	0.226549	0.113275	−	0.993564i	$-0.463866\pi$
$359$	4.29249	0.226549	0.113275	+	0.993564i	$0.463866\pi$
$360$	0.0172322	0.000908218 0
$361$	13.4912	0.710064
$362$	6.86614	0.360876
$363$	1.44715	0.0759555
$364$	7.78159	0.407866
$365$	−0.104075	−0.00544754
$366$	9.08942	0.475112
$367$	−28.1890	−1.47146	−0.735728	−	0.677277i	$-0.763160\pi$
$367$	−28.1890	−1.47146	−0.735728	+	0.677277i	$0.763160\pi$
$368$	13.9031	0.724749
$369$	1.12342	0.0584832
$370$	−0.0214240	−0.00111378
$371$	2.00032	0.103852
$372$	18.7093	0.970034
$373$	13.1366	0.680186	0.340093	−	0.940392i	$-0.389541\pi$
$373$	13.1366	0.680186	0.340093	+	0.940392i	$0.389541\pi$
$374$	−1.53929	−0.0795947
$375$	−0.156363	−0.00807454
$376$	−0.843889	−0.0435203
$377$	−8.64892	−0.445442
$378$	−2.45802	−0.126427
$379$	−15.3943	−0.790751	−0.395375	−	0.918520i	$-0.629385\pi$
$379$	−15.3943	−0.790751	−0.395375	+	0.918520i	$0.629385\pi$
$380$	0.109840	0.00563468
$381$	−0.402457	−0.0206185
$382$	−6.94925	−0.355555
$383$	−28.6017	−1.46148	−0.730740	−	0.682656i	$-0.760825\pi$
$383$	−28.6017	−1.46148	−0.730740	+	0.682656i	$0.760825\pi$
$384$	−16.0891	−0.821043
$385$	−0.0100967	−0.000514573 0
$386$	0.319496	0.0162619
$387$	11.7294	0.596238
$388$	−18.0318	−0.915424
$389$	5.01999	0.254523	0.127262	−	0.991869i	$-0.459381\pi$
$389$	5.01999	0.254523	0.127262	+	0.991869i	$0.459381\pi$
$390$	−0.0339797	−0.00172063
$391$	16.7377	0.846461
$392$	10.7878	0.544869
$393$	−1.44715	−0.0729989
$394$	1.77969	0.0896594
$395$	0.0850976	0.00428172
$396$	−1.61536	−0.0811746
$397$	−19.7122	−0.989325	−0.494662	−	0.869085i	$-0.664708\pi$
$397$	−19.7122	−0.989325	−0.494662	+	0.869085i	$0.664708\pi$
$398$	−0.981844	−0.0492154
$399$	−7.70810	−0.385888
$400$	−13.7366	−0.686831
$401$	−14.2749	−0.712856	−0.356428	−	0.934323i	$-0.616005\pi$
$401$	−14.2749	−0.712856	−0.356428	+	0.934323i	$0.616005\pi$
$402$	6.46418	0.322404
$403$	33.8500	1.68619
$404$	17.3541	0.863398
$405$	−0.0590202	−0.00293274
$406$	0.805498	0.0399762
$407$	4.26049	0.211185
$408$	−8.42784	−0.417240
$409$	5.00745	0.247603	0.123801	−	0.992307i	$-0.460491\pi$
$409$	5.00745	0.247603	0.123801	+	0.992307i	$0.460491\pi$
$410$	−0.00623691	−0.000308019 0
$411$	4.33420	0.213791
$412$	4.11019	0.202494
$413$	−7.22087	−0.355316
$414$	−2.13316	−0.104839
$415$	0.139108	0.00682855
$416$	−22.4139	−1.09893
$417$	7.18142	0.351675
$418$	2.65277	0.129751
$419$	−10.3755	−0.506875	−0.253438	−	0.967352i	$-0.581561\pi$
$419$	−10.3755	−0.506875	−0.253438	+	0.967352i	$0.581561\pi$
$420$	−0.0260581	−0.00127150
$421$	29.3738	1.43159	0.715795	−	0.698311i	$-0.246065\pi$
$421$	29.3738	1.43159	0.715795	+	0.698311i	$0.246065\pi$
$422$	10.0891	0.491129
$423$	−0.434112	−0.0211073
$424$	−3.76919	−0.183048
$425$	−16.5373	−0.802176
$426$	1.13675	0.0550758
$427$	12.6113	0.610303
$428$	23.1170	1.11740
$429$	6.75737	0.326249
$430$	−0.0651180	−0.00314027
$431$	34.9010	1.68112	0.840562	−	0.541716i	$-0.182225\pi$
$431$	34.9010	1.68112	0.840562	+	0.541716i	$0.182225\pi$
$432$	−15.5288	−0.747132
$433$	−31.6709	−1.52201	−0.761003	−	0.648748i	$-0.775293\pi$
$433$	−31.6709	−1.52201	−0.761003	+	0.648748i	$0.775293\pi$
$434$	−3.15255	−0.151327
$435$	0.0289625	0.00138864
$436$	12.4768	0.597529
$437$	−28.8453	−1.37986
$438$	−6.48709	−0.309965
$439$	−19.0653	−0.909935	−0.454967	−	0.890508i	$-0.650349\pi$
$439$	−19.0653	−0.909935	−0.454967	+	0.890508i	$0.650349\pi$
$440$	0.0190250	0.000906983 0
$441$	5.54947	0.264260
$442$	−7.18761	−0.341880
$443$	30.0545	1.42793	0.713967	−	0.700180i	$-0.246897\pi$
$443$	30.0545	1.42793	0.713967	+	0.700180i	$0.246897\pi$
$444$	10.9957	0.521835
$445$	−0.0135865	−0.000644061 0
$446$	−8.20950	−0.388731
$447$	1.02245	0.0483601
$448$	−3.04708	−0.143961
$449$	−14.1359	−0.667113	−0.333557	−	0.942730i	$-0.608249\pi$
$449$	−14.1359	−0.667113	−0.333557	+	0.942730i	$0.608249\pi$
$450$	2.10762	0.0993540
$451$	1.24030	0.0584036
$452$	21.9327	1.03163
$453$	9.89429	0.464874
$454$	8.69664	0.408153
$455$	−0.0471457	−0.00221023
$456$	14.5243	0.680163
$457$	−18.4051	−0.860956	−0.430478	−	0.902601i	$-0.641655\pi$
$457$	−18.4051	−0.860956	−0.430478	+	0.902601i	$0.641655\pi$
$458$	0.887433	0.0414670
$459$	−18.6949	−0.872603
$460$	−0.0975145	−0.00454664
$461$	−1.25439	−0.0584230	−0.0292115	−	0.999573i	$-0.509300\pi$
$461$	−1.25439	−0.0584230	−0.0292115	+	0.999573i	$0.509300\pi$
$462$	−0.629332	−0.0292792
$463$	−41.1481	−1.91231	−0.956157	−	0.292856i	$-0.905394\pi$
$463$	−41.1481	−1.91231	−0.956157	+	0.292856i	$0.905394\pi$
$464$	5.08882	0.236243
$465$	−0.113353	−0.00525661
$466$	−8.26583	−0.382907
$467$	−17.2036	−0.796089	−0.398045	−	0.917366i	$-0.630311\pi$
$467$	−17.2036	−0.796089	−0.398045	+	0.917366i	$0.630311\pi$
$468$	−7.54281	−0.348666
$469$	8.96884	0.414143
$470$	0.00241006	0.000111168 0
$471$	−28.0830	−1.29399
$472$	13.6062	0.626277
$473$	12.9497	0.595427
$474$	5.30420	0.243630
$475$	28.4999	1.30766
$476$	−5.51198	−0.252641
$477$	−1.93894	−0.0887780
$478$	−11.0815	−0.506857
$479$	−24.6428	−1.12596	−0.562979	−	0.826471i	$-0.690345\pi$
$479$	−24.6428	−1.12596	−0.562979	+	0.826471i	$0.690345\pi$
$480$	0.0750569	0.00342586
$481$	19.8941	0.907094
$482$	−4.01899	−0.183060
$483$	6.84314	0.311374
$484$	−1.78341	−0.0810642
$485$	0.109248	0.00496068
$486$	4.21266	0.191090
$487$	−14.5246	−0.658171	−0.329085	−	0.944300i	$-0.606740\pi$
$487$	−14.5246	−0.658171	−0.329085	+	0.944300i	$0.606740\pi$
$488$	−23.7633	−1.07572
$489$	11.5429	0.521987
$490$	−0.0308089	−0.00139181
$491$	−19.3877	−0.874954	−0.437477	−	0.899230i	$-0.644128\pi$
$491$	−19.3877	−0.874954	−0.437477	+	0.899230i	$0.644128\pi$
$492$	3.20105	0.144315
$493$	6.12634	0.275917
$494$	12.3869	0.557314
$495$	0.00978682	0.000439885 0
$496$	−19.9166	−0.894280
$497$	1.57721	0.0707475
$498$	8.67072	0.388544
$499$	1.76485	0.0790054	0.0395027	−	0.999219i	$-0.487423\pi$
$499$	1.76485	0.0790054	0.0395027	+	0.999219i	$0.487423\pi$
$500$	0.192696	0.00861763
$501$	12.8854	0.575676
$502$	5.16681	0.230606
$503$	20.6046	0.918715	0.459358	−	0.888251i	$-0.348080\pi$
$503$	20.6046	0.918715	0.459358	+	0.888251i	$0.348080\pi$
$504$	1.49028	0.0663822
$505$	−0.105142	−0.00467875
$506$	−2.35509	−0.104696
$507$	12.7402	0.565813
$508$	0.495973	0.0220053
$509$	1.29257	0.0572923	0.0286462	−	0.999590i	$-0.490880\pi$
$509$	1.29257	0.0572923	0.0286462	+	0.999590i	$0.490880\pi$
$510$	0.0240690	0.00106579
$511$	−9.00063	−0.398164
$512$	22.8628	1.01040
$513$	32.2182	1.42247
$514$	7.85249	0.346359
$515$	−0.0249021	−0.00109732
$516$	33.4213	1.47129
$517$	−0.479276	−0.0210785
$518$	−1.85280	−0.0814072
$519$	31.3942	1.37805
$520$	0.0888362	0.00389573
$521$	20.3215	0.890300	0.445150	−	0.895456i	$-0.353150\pi$
$521$	20.3215	0.890300	0.445150	+	0.895456i	$0.353150\pi$
$522$	−0.780780	−0.0341738
$523$	−18.6868	−0.817116	−0.408558	−	0.912732i	$-0.633968\pi$
$523$	−18.6868	−0.817116	−0.408558	+	0.912732i	$0.633968\pi$
$524$	1.78341	0.0779088
$525$	−6.76121	−0.295083
$526$	−10.9931	−0.479321
$527$	−23.9772	−1.04446
$528$	−3.97588	−0.173028
$529$	2.60841	0.113409
$530$	0.0107644	0.000467576 0
$531$	6.99929	0.303743
$532$	9.49919	0.411842
$533$	5.79153	0.250859
$534$	−0.846856	−0.0366470
$535$	−0.140057	−0.00605519
$536$	−16.8999	−0.729965
$537$	21.6503	0.934278
$538$	12.7377	0.549160
$539$	6.12682	0.263901
$540$	0.108917	0.00468705
$541$	−13.9190	−0.598424	−0.299212	−	0.954187i	$-0.596724\pi$
$541$	−13.9190	−0.598424	−0.299212	+	0.954187i	$0.596724\pi$
$542$	1.92246	0.0825767
$543$	21.3506	0.916241
$544$	15.8765	0.680701
$545$	−0.0755920	−0.00323801
$546$	−2.93863	−0.125762
$547$	26.1733	1.11909	0.559545	−	0.828800i	$-0.310976\pi$
$547$	26.1733	1.11909	0.559545	+	0.828800i	$0.310976\pi$
$548$	−5.34132	−0.228170
$549$	−12.2243	−0.521720
$550$	2.32689	0.0992189
$551$	−10.5580	−0.449784
$552$	−12.8945	−0.548825
$553$	7.35941	0.312954
$554$	−0.842894	−0.0358112
$555$	−0.0666190	−0.00282782
$556$	−8.85013	−0.375329
$557$	−10.4716	−0.443698	−0.221849	−	0.975081i	$-0.571209\pi$
$557$	−10.4716	−0.443698	−0.221849	+	0.975081i	$0.571209\pi$
$558$	3.05581	0.129363
$559$	60.4678	2.55752
$560$	0.0277394	0.00117221
$561$	−4.78648	−0.202086
$562$	−4.53401	−0.191256
$563$	0.912327	0.0384500	0.0192250	−	0.999815i	$-0.493880\pi$
$563$	0.912327	0.0384500	0.0192250	+	0.999815i	$0.493880\pi$
$564$	−1.23695	−0.0520848
$565$	−0.132882	−0.00559037
$566$	3.71029	0.155955
$567$	−5.10419	−0.214356
$568$	−2.97192	−0.124699
$569$	34.8406	1.46059	0.730297	−	0.683130i	$-0.239382\pi$
$569$	34.8406	1.46059	0.730297	+	0.683130i	$0.239382\pi$
$570$	−0.0414798	−0.00173740
$571$	20.9810	0.878029	0.439015	−	0.898480i	$-0.355328\pi$
$571$	20.9810	0.878029	0.439015	+	0.898480i	$0.355328\pi$
$572$	−8.32754	−0.348192
$573$	−21.6090	−0.902729
$574$	−0.539381	−0.0225133
$575$	−25.3018	−1.05516
$576$	2.95357	0.123066
$577$	30.9594	1.28886	0.644428	−	0.764665i	$-0.277096\pi$
$577$	30.9594	1.28886	0.644428	+	0.764665i	$0.277096\pi$
$578$	−2.82036	−0.117312
$579$	0.993487	0.0412879
$580$	−0.0356923	−0.00148204
$581$	12.0304	0.499103
$582$	6.80949	0.282262
$583$	−2.14066	−0.0886573
$584$	16.9598	0.701802
$585$	0.0456990	0.00188942
$586$	−4.25749	−0.175875
$587$	−15.1056	−0.623474	−0.311737	−	0.950168i	$-0.600911\pi$
$587$	−15.1056	−0.623474	−0.311737	+	0.950168i	$0.600911\pi$
$588$	15.8125	0.652095
$589$	41.3216	1.70263
$590$	−0.0388579	−0.00159976
$591$	5.53402	0.227639
$592$	−11.7052	−0.481082
$593$	−40.0309	−1.64387	−0.821936	−	0.569580i	$-0.807106\pi$
$593$	−40.0309	−1.64387	−0.821936	+	0.569580i	$0.807106\pi$
$594$	2.63048	0.107930
$595$	0.0333950	0.00136906
$596$	−1.26003	−0.0516127
$597$	−3.05309	−0.124955
$598$	−10.9969	−0.449699
$599$	−4.72039	−0.192870	−0.0964350	−	0.995339i	$-0.530744\pi$
$599$	−4.72039	−0.192870	−0.0964350	+	0.995339i	$0.530744\pi$
$600$	12.7401	0.520111
$601$	14.2992	0.583275	0.291637	−	0.956529i	$-0.405800\pi$
$601$	14.2992	0.583275	0.291637	+	0.956529i	$0.405800\pi$
$602$	−5.63153	−0.229524
$603$	−8.69362	−0.354032
$604$	−12.1934	−0.496142
$605$	0.0108050	0.000439287 0
$606$	−6.55357	−0.266221
$607$	−14.0445	−0.570051	−0.285025	−	0.958520i	$-0.592002\pi$
$607$	−14.0445	−0.570051	−0.285025	+	0.958520i	$0.592002\pi$
$608$	−27.3612	−1.10964
$609$	2.50473	0.101497
$610$	0.0678655	0.00274779
$611$	−2.23795	−0.0905379
$612$	5.34284	0.215971
$613$	4.16095	0.168059	0.0840296	−	0.996463i	$-0.473221\pi$
$613$	4.16095	0.168059	0.0840296	+	0.996463i	$0.473221\pi$
$614$	2.13755	0.0862644
$615$	−0.0193940	−0.000782040 0
$616$	1.64532	0.0662920
$617$	−25.6848	−1.03403	−0.517016	−	0.855976i	$-0.672957\pi$
$617$	−25.6848	−1.03403	−0.517016	+	0.855976i	$0.672957\pi$
$618$	−1.55217	−0.0624372
$619$	−22.8485	−0.918357	−0.459178	−	0.888344i	$-0.651856\pi$
$619$	−22.8485	−0.918357	−0.459178	+	0.888344i	$0.651856\pi$
$620$	0.139692	0.00561017
$621$	−28.6029	−1.14780
$622$	−9.07563	−0.363900
$623$	−1.17499	−0.0470748
$624$	−18.5651	−0.743200
$625$	24.9982	0.999930
$626$	3.65232	0.145976
$627$	8.24889	0.329429
$628$	34.6085	1.38103
$629$	−14.0917	−0.561874
$630$	−0.00425607	−0.000169566 0
$631$	−32.0657	−1.27652	−0.638258	−	0.769823i	$-0.720344\pi$
$631$	−32.0657	−1.27652	−0.638258	+	0.769823i	$0.720344\pi$
$632$	−13.8673	−0.551611
$633$	31.3724	1.24694
$634$	−8.43678	−0.335067
$635$	−0.00300491	−0.000119246 0
$636$	−5.52476	−0.219071
$637$	28.6088	1.13352
$638$	−0.862011	−0.0341274
$639$	−1.52881	−0.0604788
$640$	−0.120128	−0.00474848
$641$	−35.3526	−1.39634	−0.698171	−	0.715931i	$-0.746003\pi$
$641$	−35.3526	−1.39634	−0.698171	+	0.715931i	$0.746003\pi$
$642$	−8.72986	−0.344540
$643$	−49.3401	−1.94579	−0.972893	−	0.231258i	$-0.925716\pi$
$643$	−49.3401	−1.94579	−0.972893	+	0.231258i	$0.925716\pi$
$644$	−8.43325	−0.332317
$645$	−0.202487	−0.00797293
$646$	−8.77411	−0.345213
$647$	−41.9019	−1.64733	−0.823666	−	0.567075i	$-0.808075\pi$
$647$	−41.9019	−1.64733	−0.823666	+	0.567075i	$0.808075\pi$
$648$	9.61777	0.377822
$649$	7.72748	0.303330
$650$	10.8653	0.426171
$651$	−9.80298	−0.384209
$652$	−14.2250	−0.557095
$653$	−3.75059	−0.146772	−0.0733859	−	0.997304i	$-0.523380\pi$
$653$	−3.75059	−0.146772	−0.0733859	+	0.997304i	$0.523380\pi$
$654$	−4.71171	−0.184243
$655$	−0.0108050	−0.000422187 0
$656$	−3.40760	−0.133044
$657$	8.72443	0.340373
$658$	0.208427	0.00812532
$659$	−0.962475	−0.0374927	−0.0187464	−	0.999824i	$-0.505967\pi$
$659$	−0.962475	−0.0374927	−0.0187464	+	0.999824i	$0.505967\pi$
$660$	0.0278863	0.00108547
$661$	−3.25592	−0.126641	−0.0633203	−	0.997993i	$-0.520169\pi$
$661$	−3.25592	−0.126641	−0.0633203	+	0.997993i	$0.520169\pi$
$662$	−0.715283	−0.0278003
$663$	−22.3502	−0.868010
$664$	−22.6687	−0.879716
$665$	−0.0575520	−0.00223177
$666$	1.79594	0.0695913
$667$	9.37321	0.362932
$668$	−15.8795	−0.614395
$669$	−25.5278	−0.986962
$670$	0.0482643	0.00186461
$671$	−13.4961	−0.521010
$672$	6.49107	0.250398
$673$	−30.9648	−1.19360	−0.596802	−	0.802389i	$-0.703562\pi$
$673$	−30.9648	−1.19360	−0.596802	+	0.802389i	$0.703562\pi$
$674$	6.21253	0.239298
$675$	28.2604	1.08774
$676$	−15.7006	−0.603869
$677$	5.91106	0.227181	0.113590	−	0.993528i	$-0.463765\pi$
$677$	5.91106	0.227181	0.113590	+	0.993528i	$0.463765\pi$
$678$	−8.28262	−0.318092
$679$	9.44795	0.362579
$680$	−0.0629259	−0.00241310
$681$	27.0426	1.03627
$682$	3.37373	0.129187
$683$	−29.5994	−1.13259	−0.566294	−	0.824203i	$-0.691623\pi$
$683$	−29.5994	−1.13259	−0.566294	+	0.824203i	$0.691623\pi$
$684$	−9.20770	−0.352065
$685$	0.0323610	0.00123645
$686$	−5.70857	−0.217954
$687$	2.75951	0.105282
$688$	−35.5778	−1.35639
$689$	−9.99571	−0.380806
$690$	0.0368252	0.00140191
$691$	−37.8027	−1.43808	−0.719041	−	0.694968i	$-0.755419\pi$
$691$	−37.8027	−1.43808	−0.719041	+	0.694968i	$0.755419\pi$
$692$	−38.6891	−1.47074
$693$	0.846384	0.0321515
$694$	16.0595	0.609611
$695$	0.0536195	0.00203391
$696$	−4.71965	−0.178898
$697$	−4.10234	−0.155387
$698$	11.6989	0.442810
$699$	−25.7030	−0.972176
$700$	8.33227	0.314930
$701$	−27.9021	−1.05385	−0.526924	−	0.849913i	$-0.676655\pi$
$701$	−27.9021	−1.05385	−0.526924	+	0.849913i	$0.676655\pi$
$702$	12.2829	0.463587
$703$	24.2853	0.915936
$704$	3.26086	0.122898
$705$	0.00749418	0.000282247 0
$706$	−0.517314	−0.0194694
$707$	−9.09288	−0.341973
$708$	19.9436	0.749525
$709$	25.6303	0.962567	0.481283	−	0.876565i	$-0.340171\pi$
$709$	25.6303	0.962567	0.481283	+	0.876565i	$0.340171\pi$
$710$	0.00848748	0.000318530 0
$711$	−7.13358	−0.267530
$712$	2.21402	0.0829737
$713$	−36.6847	−1.37385
$714$	2.08154	0.0778996
$715$	0.0504534	0.00188685
$716$	−26.6810	−0.997117
$717$	−34.4585	−1.28688
$718$	1.99768	0.0745527
$719$	−19.1151	−0.712872	−0.356436	−	0.934320i	$-0.616008\pi$
$719$	−19.1151	−0.712872	−0.356436	+	0.934320i	$0.616008\pi$
$720$	−0.0268882	−0.00100206
$721$	−2.15358	−0.0802035
$722$	6.27866	0.233668
$723$	−12.4972	−0.464776
$724$	−26.3117	−0.977867
$725$	−9.26098	−0.343944
$726$	0.673486	0.0249954
$727$	20.1584	0.747635	0.373818	−	0.927502i	$-0.378049\pi$
$727$	20.1584	0.747635	0.373818	+	0.927502i	$0.378049\pi$
$728$	7.68274	0.284741
$729$	29.4863	1.09209
$730$	−0.0484354	−0.00179267
$731$	−42.8315	−1.58418
$732$	−34.8315	−1.28741
$733$	−23.4444	−0.865938	−0.432969	−	0.901409i	$-0.642534\pi$
$733$	−23.4444	−0.865938	−0.432969	+	0.901409i	$0.642534\pi$
$734$	−13.1189	−0.484226
$735$	−0.0958018	−0.00353370
$736$	24.2909	0.895374
$737$	−9.59809	−0.353550
$738$	0.522829	0.0192456
$739$	5.70360	0.209810	0.104905	−	0.994482i	$-0.466546\pi$
$739$	5.70360	0.209810	0.104905	+	0.994482i	$0.466546\pi$
$740$	0.0820990	0.00301802
$741$	38.5177	1.41498
$742$	0.930928	0.0341754
$743$	13.1418	0.482126	0.241063	−	0.970509i	$-0.422504\pi$
$743$	13.1418	0.482126	0.241063	+	0.970509i	$0.422504\pi$
$744$	18.4717	0.677204
$745$	0.00763403	0.000279689 0
$746$	6.11362	0.223835
$747$	−11.6612	−0.426660
$748$	5.89870	0.215678
$749$	−12.1124	−0.442578
$750$	−0.0727694	−0.00265716
$751$	4.35611	0.158957	0.0794783	−	0.996837i	$-0.474675\pi$
$751$	4.35611	0.158957	0.0794783	+	0.996837i	$0.474675\pi$
$752$	1.31676	0.0480173
$753$	16.0664	0.585493
$754$	−4.02511	−0.146586
$755$	0.0738751	0.00268859
$756$	9.41938	0.342580
$757$	−38.5601	−1.40149	−0.700746	−	0.713411i	$-0.747149\pi$
$757$	−38.5601	−1.40149	−0.700746	+	0.713411i	$0.747149\pi$
$758$	−7.16432	−0.260220
$759$	−7.32325	−0.265817
$760$	0.108445	0.00393370
$761$	40.6600	1.47392	0.736961	−	0.675935i	$-0.236260\pi$
$761$	40.6600	1.47392	0.736961	+	0.675935i	$0.236260\pi$
$762$	−0.187299	−0.00678512
$763$	−6.53735	−0.236668
$764$	26.6302	0.963446
$765$	−0.0323702	−0.00117035
$766$	−13.3109	−0.480943
$767$	36.0830	1.30288
$768$	1.95020	0.0703718
$769$	15.3932	0.555093	0.277547	−	0.960712i	$-0.410479\pi$
$769$	15.3932	0.555093	0.277547	+	0.960712i	$0.410479\pi$
$770$	−0.00469887	−0.000169335 0
$771$	24.4177	0.879381
$772$	−1.22434	−0.0440649
$773$	−48.7493	−1.75339	−0.876695	−	0.481047i	$-0.840257\pi$
$773$	−48.7493	−1.75339	−0.876695	+	0.481047i	$0.840257\pi$
$774$	5.45872	0.196210
$775$	36.2455	1.30198
$776$	−17.8027	−0.639079
$777$	−5.76135	−0.206687
$778$	2.33624	0.0837584
$779$	7.06986	0.253304
$780$	0.130213	0.00466238
$781$	−1.68786	−0.0603965
$782$	7.78953	0.278553
$783$	−10.4693	−0.374141
$784$	−16.8328	−0.601170
$785$	−0.209680	−0.00748378
$786$	−0.673486	−0.0240224
$787$	4.54659	0.162068	0.0810342	−	0.996711i	$-0.474178\pi$
$787$	4.54659	0.162068	0.0810342	+	0.996711i	$0.474178\pi$
$788$	−6.81993	−0.242950
$789$	−34.1834	−1.21696
$790$	0.0396035	0.00140903
$791$	−11.4919	−0.408604
$792$	−1.59483	−0.0566700
$793$	−63.0192	−2.23788
$794$	−9.17381	−0.325566
$795$	0.0334724	0.00118714
$796$	3.76252	0.133359
$797$	22.7234	0.804903	0.402451	−	0.915441i	$-0.368158\pi$
$797$	22.7234	0.804903	0.402451	+	0.915441i	$0.368158\pi$
$798$	−3.58726	−0.126988
$799$	1.58522	0.0560811
$800$	−24.0000	−0.848529
$801$	1.13893	0.0402421
$802$	−6.64339	−0.234586
$803$	9.63211	0.339910
$804$	−24.7713	−0.873618
$805$	0.0510939	0.00180082
$806$	15.7534	0.554890
$807$	39.6083	1.39428
$808$	17.1336	0.602759
$809$	14.9141	0.524352	0.262176	−	0.965020i	$-0.415560\pi$
$809$	14.9141	0.524352	0.262176	+	0.965020i	$0.415560\pi$
$810$	−0.0274673	−0.000965103 0
$811$	8.53015	0.299534	0.149767	−	0.988721i	$-0.452148\pi$
$811$	8.53015	0.299534	0.149767	+	0.988721i	$0.452148\pi$
$812$	−3.08675	−0.108324
$813$	5.97797	0.209657
$814$	1.98279	0.0694966
$815$	0.0861840	0.00301890
$816$	13.1503	0.460354
$817$	73.8146	2.58245
$818$	2.33041	0.0814809
$819$	3.95214	0.138099
$820$	0.0239004	0.000834639 0
$821$	48.7740	1.70222	0.851112	−	0.524984i	$-0.175929\pi$
$821$	48.7740	1.70222	0.851112	+	0.524984i	$0.175929\pi$
$822$	2.01709	0.0703541
$823$	−25.1329	−0.876078	−0.438039	−	0.898956i	$-0.644327\pi$
$823$	−25.1329	−0.876078	−0.438039	+	0.898956i	$0.644327\pi$
$824$	4.05797	0.141366
$825$	7.23557	0.251910
$826$	−3.36051	−0.116927
$827$	−31.5290	−1.09637	−0.548185	−	0.836357i	$-0.684681\pi$
$827$	−31.5290	−1.09637	−0.548185	+	0.836357i	$0.684681\pi$
$828$	8.17446	0.284082
$829$	−0.137377	−0.00477129	−0.00238564	−	0.999997i	$-0.500759\pi$
$829$	−0.137377	−0.00477129	−0.00238564	+	0.999997i	$0.500759\pi$
$830$	0.0647393	0.00224713
$831$	−2.62102	−0.0909221
$832$	15.2264	0.527880
$833$	−20.2647	−0.702129
$834$	3.34215	0.115729
$835$	0.0962077	0.00332940
$836$	−10.1656	−0.351586
$837$	40.9744	1.41628
$838$	−4.82863	−0.166802
$839$	−8.06689	−0.278500	−0.139250	−	0.990257i	$-0.544469\pi$
$839$	−8.06689	−0.278500	−0.139250	+	0.990257i	$0.544469\pi$
$840$	−0.0257270	−0.000887667 0
$841$	−25.5692	−0.881697
$842$	13.6702	0.471107
$843$	−14.0987	−0.485585
$844$	−38.6623	−1.33081
$845$	0.0951240	0.00327236
$846$	−0.202031	−0.00694596
$847$	0.934440	0.0321078
$848$	5.88124	0.201963
$849$	11.5373	0.395960
$850$	−7.69626	−0.263980
$851$	−21.5601	−0.739072
$852$	−4.35614	−0.149239
$853$	−4.47079	−0.153077	−0.0765384	−	0.997067i	$-0.524387\pi$
$853$	−4.47079	−0.153077	−0.0765384	+	0.997067i	$0.524387\pi$
$854$	5.86915	0.200838
$855$	0.0557859	0.00190784
$856$	22.8233	0.780085
$857$	32.1880	1.09952	0.549761	−	0.835322i	$-0.314719\pi$
$857$	32.1880	1.09952	0.549761	+	0.835322i	$0.314719\pi$
$858$	3.14480	0.107362
$859$	37.4370	1.27733	0.638666	−	0.769484i	$-0.279486\pi$
$859$	37.4370	1.27733	0.638666	+	0.769484i	$0.279486\pi$
$860$	0.249538	0.00850918
$861$	−1.67723	−0.0571598
$862$	16.2425	0.553223
$863$	20.4883	0.697429	0.348714	−	0.937229i	$-0.386618\pi$
$863$	20.4883	0.697429	0.348714	+	0.937229i	$0.386618\pi$
$864$	−27.1313	−0.923026
$865$	0.234402	0.00796992
$866$	−14.7393	−0.500861
$867$	−8.77005	−0.297846
$868$	12.0809	0.410051
$869$	−7.87574	−0.267166
$870$	0.0134788	0.000456974 0
$871$	−44.8177	−1.51859
$872$	12.3183	0.417149
$873$	−9.15803	−0.309952
$874$	−13.4243	−0.454082
$875$	−0.100965	−0.00341325
$876$	24.8591	0.839912
$877$	−21.9122	−0.739923	−0.369961	−	0.929047i	$-0.620629\pi$
$877$	−21.9122	−0.739923	−0.369961	+	0.929047i	$0.620629\pi$
$878$	−8.87275	−0.299441
$879$	−13.2389	−0.446536
$880$	−0.0296856	−0.00100070
$881$	−45.1103	−1.51980	−0.759902	−	0.650037i	$-0.774753\pi$
$881$	−45.1103	−1.51980	−0.759902	+	0.650037i	$0.774753\pi$
$882$	2.58266	0.0869626
$883$	−19.0553	−0.641263	−0.320631	−	0.947204i	$-0.603895\pi$
$883$	−19.0553	−0.641263	−0.320631	+	0.947204i	$0.603895\pi$
$884$	27.5436	0.926392
$885$	−0.120830	−0.00406167
$886$	13.9870	0.469904
$887$	−43.2133	−1.45096	−0.725481	−	0.688242i	$-0.758383\pi$
$887$	−43.2133	−1.45096	−0.725481	+	0.688242i	$0.758383\pi$
$888$	10.8561	0.364306
$889$	−0.259871	−0.00871580
$890$	−0.00632299	−0.000211947 0
$891$	5.46229	0.182994
$892$	31.4596	1.05334
$893$	−2.73193	−0.0914204
$894$	0.475835	0.0159143
$895$	0.161650	0.00540337
$896$	−10.3889	−0.347070
$897$	−34.1955	−1.14175
$898$	−6.57868	−0.219533
$899$	−13.4274	−0.447828
$900$	−8.07659	−0.269220
$901$	7.08032	0.235880
$902$	0.577223	0.0192194
$903$	−17.5115	−0.582747
$904$	21.6540	0.720202
$905$	0.159413	0.00529906
$906$	4.60469	0.152981
$907$	8.28201	0.275000	0.137500	−	0.990502i	$-0.456093\pi$
$907$	8.28201	0.275000	0.137500	+	0.990502i	$0.456093\pi$
$908$	−33.3263	−1.10597
$909$	8.81385	0.292337
$910$	−0.0219411	−0.000727340 0
$911$	−12.1412	−0.402257	−0.201129	−	0.979565i	$-0.564461\pi$
$911$	−12.1412	−0.402257	−0.201129	+	0.979565i	$0.564461\pi$
$912$	−22.6629	−0.750444
$913$	−12.8744	−0.426080
$914$	−8.56554	−0.283323
$915$	0.211031	0.00697647
$916$	−3.40073	−0.112363
$917$	−0.934440	−0.0308579
$918$	−8.70039	−0.287156
$919$	−27.0401	−0.891969	−0.445985	−	0.895041i	$-0.647146\pi$
$919$	−27.0401	−0.891969	−0.445985	+	0.895041i	$0.647146\pi$
$920$	−0.0962757	−0.00317412
$921$	6.64680	0.219020
$922$	−0.583781	−0.0192258
$923$	−7.88138	−0.259419
$924$	2.41166	0.0793378
$925$	21.3020	0.700405
$926$	−19.1498	−0.629303
$927$	2.08750	0.0685623
$928$	8.89097	0.291860
$929$	21.1308	0.693280	0.346640	−	0.937998i	$-0.387323\pi$
$929$	21.1308	0.693280	0.346640	+	0.937998i	$0.387323\pi$
$930$	−0.0527531	−0.00172984
$931$	34.9235	1.14457
$932$	31.6754	1.03756
$933$	−28.2211	−0.923917
$934$	−8.00638	−0.261977
$935$	−0.0357380	−0.00116876
$936$	−7.44699	−0.243412
$937$	−18.7312	−0.611921	−0.305961	−	0.952044i	$-0.598978\pi$
$937$	−18.7312	−0.611921	−0.305961	+	0.952044i	$0.598978\pi$
$938$	4.17400	0.136286
$939$	11.3570	0.370623
$940$	−0.00923557	−0.000301231 0
$941$	3.28042	0.106939	0.0534694	−	0.998569i	$-0.482972\pi$
$941$	3.28042	0.106939	0.0534694	+	0.998569i	$0.482972\pi$
$942$	−13.0695	−0.425827
$943$	−6.27653	−0.204392
$944$	−21.2304	−0.690991
$945$	−0.0570685	−0.00185644
$946$	6.02664	0.195943
$947$	−24.5289	−0.797084	−0.398542	−	0.917150i	$-0.630484\pi$
$947$	−24.5289	−0.797084	−0.398542	+	0.917150i	$0.630484\pi$
$948$	−20.3262	−0.660164
$949$	44.9766	1.46000
$950$	13.2635	0.430325
$951$	−26.2345	−0.850713
$952$	−5.44196	−0.176375
$953$	12.9157	0.418382	0.209191	−	0.977875i	$-0.432917\pi$
$953$	12.9157	0.418382	0.209191	+	0.977875i	$0.432917\pi$
$954$	−0.902361	−0.0292150
$955$	−0.161342	−0.00522091
$956$	42.4655	1.37343
$957$	−2.68046	−0.0866471
$958$	−11.4685	−0.370530
$959$	2.79865	0.0903731
$960$	−0.0509883	−0.00164564
$961$	21.5518	0.695220
$962$	9.25850	0.298506
$963$	11.7407	0.378340
$964$	15.4011	0.496037
$965$	0.00741780	0.000238787 0
$966$	3.18472	0.102467
$967$	40.6049	1.30577	0.652883	−	0.757459i	$-0.273559\pi$
$967$	40.6049	1.30577	0.652883	+	0.757459i	$0.273559\pi$
$968$	−1.76076	−0.0565929
$969$	−27.2835	−0.876471
$970$	0.0508426	0.00163246
$971$	5.52846	0.177417	0.0887084	−	0.996058i	$-0.471726\pi$
$971$	5.52846	0.177417	0.0887084	+	0.996058i	$0.471726\pi$
$972$	−16.1433	−0.517797
$973$	4.63713	0.148659
$974$	−6.75957	−0.216590
$975$	33.7860	1.08202
$976$	37.0790	1.18687
$977$	52.8708	1.69149	0.845743	−	0.533591i	$-0.179158\pi$
$977$	52.8708	1.69149	0.845743	+	0.533591i	$0.179158\pi$
$978$	5.37192	0.171775
$979$	1.25742	0.0401874
$980$	0.118063	0.00377138
$981$	6.33675	0.202317
$982$	−9.02281	−0.287929
$983$	49.2651	1.57131	0.785656	−	0.618664i	$-0.212326\pi$
$983$	49.2651	1.57131	0.785656	+	0.618664i	$0.212326\pi$
$984$	3.16039	0.100749
$985$	0.0413194	0.00131655
$986$	2.85113	0.0907985
$987$	0.648112	0.0206296
$988$	−47.4679	−1.51015
$989$	−65.5316	−2.08378
$990$	0.00455468	0.000144757 0
$991$	−19.2740	−0.612260	−0.306130	−	0.951990i	$-0.599034\pi$
$991$	−19.2740	−0.612260	−0.306130	+	0.951990i	$0.599034\pi$
$992$	−34.7973	−1.10482
$993$	−2.22420	−0.0705830
$994$	0.734015	0.0232815
$995$	−0.0227957	−0.000722671 0
$996$	−33.2270	−1.05284
$997$	38.4680	1.21830	0.609148	−	0.793057i	$-0.291512\pi$
$997$	38.4680	1.21830	0.609148	+	0.793057i	$0.291512\pi$
$998$	0.821340	0.0259991
$999$	24.0812	0.761897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1441.2.a.d.1.15	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1441.2.a.d.1.15	✓	23	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:	\( N \)	\(=\)	\( 1441 = 11 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1441.a (trivial)

\(f(q)\)	\(=\)	\(q+0.465389 q^{2} +1.44715 q^{3} -1.78341 q^{4} +0.0108050 q^{5} +0.673486 q^{6} +0.934440 q^{7} -1.76076 q^{8} -0.905766 q^{9} +O(q^{10})\)	\(q+0.465389 q^{2} +1.44715 q^{3} -1.78341 q^{4} +0.0108050 q^{5} +0.673486 q^{6} +0.934440 q^{7} -1.76076 q^{8} -0.905766 q^{9} +0.00502854 q^{10} -1.00000 q^{11} -2.58086 q^{12} -4.66944 q^{13} +0.434878 q^{14} +0.0156365 q^{15} +2.74739 q^{16} +3.30753 q^{17} -0.421533 q^{18} -5.70011 q^{19} -0.0192698 q^{20} +1.35227 q^{21} -0.465389 q^{22} +5.06048 q^{23} -2.54808 q^{24} -4.99988 q^{25} -2.17311 q^{26} -5.65222 q^{27} -1.66649 q^{28} +1.85224 q^{29} +0.00727703 q^{30} -7.24926 q^{31} +4.80012 q^{32} -1.44715 q^{33} +1.53929 q^{34} +0.0100967 q^{35} +1.61536 q^{36} -4.26049 q^{37} -2.65277 q^{38} -6.75737 q^{39} -0.0190250 q^{40} -1.24030 q^{41} +0.629332 q^{42} -12.9497 q^{43} +1.78341 q^{44} -0.00978682 q^{45} +2.35509 q^{46} +0.479276 q^{47} +3.97588 q^{48} -6.12682 q^{49} -2.32689 q^{50} +4.78648 q^{51} +8.32754 q^{52} +2.14066 q^{53} -2.63048 q^{54} -0.0108050 q^{55} -1.64532 q^{56} -8.24889 q^{57} +0.862011 q^{58} -7.72748 q^{59} -0.0278863 q^{60} +13.4961 q^{61} -3.37373 q^{62} -0.846384 q^{63} -3.26086 q^{64} -0.0504534 q^{65} -0.673486 q^{66} +9.59809 q^{67} -5.89870 q^{68} +7.32325 q^{69} +0.00469887 q^{70} +1.68786 q^{71} +1.59483 q^{72} -9.63211 q^{73} -1.98279 q^{74} -7.23557 q^{75} +10.1656 q^{76} -0.934440 q^{77} -3.14480 q^{78} +7.87574 q^{79} +0.0296856 q^{80} -5.46229 q^{81} -0.577223 q^{82} +12.8744 q^{83} -2.41166 q^{84} +0.0357380 q^{85} -6.02664 q^{86} +2.68046 q^{87} +1.76076 q^{88} -1.25742 q^{89} -0.00455468 q^{90} -4.36332 q^{91} -9.02492 q^{92} -10.4908 q^{93} +0.223050 q^{94} -0.0615898 q^{95} +6.94648 q^{96} +10.1108 q^{97} -2.85135 q^{98} +0.905766 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 7 q^{2} - 3 q^{3} + 19 q^{4} - 9 q^{5} - 5 q^{6} - 8 q^{7} - 15 q^{8} + 12 q^{9}+O(q^{10}) \) 23 * q - 7 * q^2 - 3 * q^3 + 19 * q^4 - 9 * q^5 - 5 * q^6 - 8 * q^7 - 15 * q^8 + 12 * q^9	\( 23 q - 7 q^{2} - 3 q^{3} + 19 q^{4} - 9 q^{5} - 5 q^{6} - 8 q^{7} - 15 q^{8} + 12 q^{9} - 2 q^{10} - 23 q^{11} - 12 q^{12} + 6 q^{13} - 13 q^{14} - 27 q^{15} - q^{16} - 11 q^{17} - 16 q^{19} - 24 q^{20} - 7 q^{21} + 7 q^{22} - 30 q^{23} - 20 q^{24} + 10 q^{25} - 22 q^{26} - 15 q^{27} + 11 q^{28} - 15 q^{29} + 17 q^{30} - 31 q^{31} - 22 q^{32} + 3 q^{33} - 18 q^{34} - 34 q^{35} - 18 q^{36} - 26 q^{37} - 32 q^{39} + 16 q^{40} - 21 q^{41} - 37 q^{42} - 2 q^{43} - 19 q^{44} - 18 q^{45} - 6 q^{46} - 25 q^{47} + 14 q^{48} + 7 q^{49} - 27 q^{50} - 7 q^{51} + 23 q^{52} - 34 q^{53} + 26 q^{54} + 9 q^{55} - 42 q^{56} + 6 q^{57} - 5 q^{58} - 31 q^{59} - 7 q^{60} + 19 q^{61} + 24 q^{62} - 34 q^{63} - 17 q^{64} - 13 q^{65} + 5 q^{66} - 39 q^{67} - 59 q^{68} - 12 q^{69} + 17 q^{70} - 103 q^{71} + 19 q^{72} + q^{73} - 9 q^{74} - 19 q^{75} + 10 q^{76} + 8 q^{77} - 43 q^{78} - 14 q^{79} - q^{80} - 29 q^{81} + 20 q^{82} - 14 q^{83} + 57 q^{84} + 20 q^{85} - 54 q^{86} - 23 q^{87} + 15 q^{88} - 71 q^{89} - 52 q^{90} - 18 q^{91} - 24 q^{92} - q^{93} + q^{94} - 38 q^{95} - 31 q^{96} - 26 q^{97} + 19 q^{98} - 12 q^{99}+O(q^{100}) \) 23 * q - 7 * q^2 - 3 * q^3 + 19 * q^4 - 9 * q^5 - 5 * q^6 - 8 * q^7 - 15 * q^8 + 12 * q^9 - 2 * q^10 - 23 * q^11 - 12 * q^12 + 6 * q^13 - 13 * q^14 - 27 * q^15 - q^16 - 11 * q^17 - 16 * q^19 - 24 * q^20 - 7 * q^21 + 7 * q^22 - 30 * q^23 - 20 * q^24 + 10 * q^25 - 22 * q^26 - 15 * q^27 + 11 * q^28 - 15 * q^29 + 17 * q^30 - 31 * q^31 - 22 * q^32 + 3 * q^33 - 18 * q^34 - 34 * q^35 - 18 * q^36 - 26 * q^37 - 32 * q^39 + 16 * q^40 - 21 * q^41 - 37 * q^42 - 2 * q^43 - 19 * q^44 - 18 * q^45 - 6 * q^46 - 25 * q^47 + 14 * q^48 + 7 * q^49 - 27 * q^50 - 7 * q^51 + 23 * q^52 - 34 * q^53 + 26 * q^54 + 9 * q^55 - 42 * q^56 + 6 * q^57 - 5 * q^58 - 31 * q^59 - 7 * q^60 + 19 * q^61 + 24 * q^62 - 34 * q^63 - 17 * q^64 - 13 * q^65 + 5 * q^66 - 39 * q^67 - 59 * q^68 - 12 * q^69 + 17 * q^70 - 103 * q^71 + 19 * q^72 + q^73 - 9 * q^74 - 19 * q^75 + 10 * q^76 + 8 * q^77 - 43 * q^78 - 14 * q^79 - q^80 - 29 * q^81 + 20 * q^82 - 14 * q^83 + 57 * q^84 + 20 * q^85 - 54 * q^86 - 23 * q^87 + 15 * q^88 - 71 * q^89 - 52 * q^90 - 18 * q^91 - 24 * q^92 - q^93 + q^94 - 38 * q^95 - 31 * q^96 - 26 * q^97 + 19 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more