LMFDB - Embedded newform 8045.2.a.d.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, 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("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$0$
Dimension:	$141$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68753 q^{2} -2.90926 q^{3} +5.22280 q^{4} -1.00000 q^{5} +7.81872 q^{6} -1.56079 q^{7} -8.66137 q^{8} +5.46381 q^{9} +O(q^{10})$	\(q-2.68753 q^{2} -2.90926 q^{3} +5.22280 q^{4} -1.00000 q^{5} +7.81872 q^{6} -1.56079 q^{7} -8.66137 q^{8} +5.46381 q^{9} +2.68753 q^{10} -5.41942 q^{11} -15.1945 q^{12} -2.54347 q^{13} +4.19467 q^{14} +2.90926 q^{15} +12.8321 q^{16} +1.41493 q^{17} -14.6841 q^{18} +6.79597 q^{19} -5.22280 q^{20} +4.54076 q^{21} +14.5648 q^{22} -1.01891 q^{23} +25.1982 q^{24} +1.00000 q^{25} +6.83563 q^{26} -7.16786 q^{27} -8.15171 q^{28} +6.60136 q^{29} -7.81872 q^{30} +9.95792 q^{31} -17.1638 q^{32} +15.7665 q^{33} -3.80266 q^{34} +1.56079 q^{35} +28.5364 q^{36} +11.1938 q^{37} -18.2643 q^{38} +7.39961 q^{39} +8.66137 q^{40} +2.60847 q^{41} -12.2034 q^{42} -0.960865 q^{43} -28.3046 q^{44} -5.46381 q^{45} +2.73834 q^{46} +2.99608 q^{47} -37.3318 q^{48} -4.56393 q^{49} -2.68753 q^{50} -4.11640 q^{51} -13.2840 q^{52} +6.33397 q^{53} +19.2638 q^{54} +5.41942 q^{55} +13.5186 q^{56} -19.7712 q^{57} -17.7413 q^{58} -1.89910 q^{59} +15.1945 q^{60} +2.14996 q^{61} -26.7622 q^{62} -8.52787 q^{63} +20.4640 q^{64} +2.54347 q^{65} -42.3730 q^{66} +11.7442 q^{67} +7.38990 q^{68} +2.96426 q^{69} -4.19467 q^{70} +13.1287 q^{71} -47.3241 q^{72} -2.17969 q^{73} -30.0837 q^{74} -2.90926 q^{75} +35.4940 q^{76} +8.45860 q^{77} -19.8866 q^{78} -0.477179 q^{79} -12.8321 q^{80} +4.46177 q^{81} -7.01034 q^{82} +0.118768 q^{83} +23.7155 q^{84} -1.41493 q^{85} +2.58235 q^{86} -19.2051 q^{87} +46.9396 q^{88} -16.2714 q^{89} +14.6841 q^{90} +3.96982 q^{91} -5.32154 q^{92} -28.9702 q^{93} -8.05205 q^{94} -6.79597 q^{95} +49.9339 q^{96} -4.98740 q^{97} +12.2657 q^{98} -29.6107 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9}+O(q^{10}) $ 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9	\( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9} + 8 q^{10} + 32 q^{11} + 17 q^{12} + 35 q^{13} + 18 q^{14} - 11 q^{15} + 188 q^{16} - 13 q^{17} - 16 q^{18} + 152 q^{19} - 158 q^{20} + 40 q^{21} + 14 q^{22} - 77 q^{23} + 69 q^{24} + 141 q^{25} + 27 q^{26} + 38 q^{27} + 67 q^{28} + 22 q^{29} - 23 q^{30} + 86 q^{31} - 65 q^{32} + 51 q^{33} + 79 q^{34} - 29 q^{35} + 191 q^{36} + 45 q^{37} - 9 q^{38} + 55 q^{39} + 21 q^{40} + 36 q^{41} + 6 q^{42} + 132 q^{43} + 74 q^{44} - 160 q^{45} + 72 q^{46} - 16 q^{47} + 22 q^{48} + 212 q^{49} - 8 q^{50} + 82 q^{51} + 106 q^{52} - 28 q^{53} + 86 q^{54} - 32 q^{55} + 27 q^{56} + 10 q^{57} + 23 q^{58} + 71 q^{59} - 17 q^{60} + 116 q^{61} - 36 q^{62} + 45 q^{63} + 237 q^{64} - 35 q^{65} + 69 q^{66} + 99 q^{67} - 7 q^{68} + 45 q^{69} - 18 q^{70} + 34 q^{71} - 53 q^{72} + 125 q^{73} + 50 q^{74} + 11 q^{75} + 271 q^{76} - 31 q^{77} + 2 q^{78} + 101 q^{79} - 188 q^{80} + 221 q^{81} + 67 q^{82} + 67 q^{83} + 141 q^{84} + 13 q^{85} + 48 q^{86} - 21 q^{87} + 71 q^{88} + 79 q^{89} + 16 q^{90} + 228 q^{91} - 198 q^{92} - 12 q^{93} + 114 q^{94} - 152 q^{95} + 129 q^{96} + 98 q^{97} - 31 q^{98} + 195 q^{99}+O(q^{100}) \) 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9 + 8 * q^10 + 32 * q^11 + 17 * q^12 + 35 * q^13 + 18 * q^14 - 11 * q^15 + 188 * q^16 - 13 * q^17 - 16 * q^18 + 152 * q^19 - 158 * q^20 + 40 * q^21 + 14 * q^22 - 77 * q^23 + 69 * q^24 + 141 * q^25 + 27 * q^26 + 38 * q^27 + 67 * q^28 + 22 * q^29 - 23 * q^30 + 86 * q^31 - 65 * q^32 + 51 * q^33 + 79 * q^34 - 29 * q^35 + 191 * q^36 + 45 * q^37 - 9 * q^38 + 55 * q^39 + 21 * q^40 + 36 * q^41 + 6 * q^42 + 132 * q^43 + 74 * q^44 - 160 * q^45 + 72 * q^46 - 16 * q^47 + 22 * q^48 + 212 * q^49 - 8 * q^50 + 82 * q^51 + 106 * q^52 - 28 * q^53 + 86 * q^54 - 32 * q^55 + 27 * q^56 + 10 * q^57 + 23 * q^58 + 71 * q^59 - 17 * q^60 + 116 * q^61 - 36 * q^62 + 45 * q^63 + 237 * q^64 - 35 * q^65 + 69 * q^66 + 99 * q^67 - 7 * q^68 + 45 * q^69 - 18 * q^70 + 34 * q^71 - 53 * q^72 + 125 * q^73 + 50 * q^74 + 11 * q^75 + 271 * q^76 - 31 * q^77 + 2 * q^78 + 101 * q^79 - 188 * q^80 + 221 * q^81 + 67 * q^82 + 67 * q^83 + 141 * q^84 + 13 * q^85 + 48 * q^86 - 21 * q^87 + 71 * q^88 + 79 * q^89 + 16 * q^90 + 228 * q^91 - 198 * q^92 - 12 * q^93 + 114 * q^94 - 152 * q^95 + 129 * q^96 + 98 * q^97 - 31 * q^98 + 195 * 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 $

$2$	−2.68753	−1.90037	−0.950184	−	0.311688i	$-0.899105\pi$
$2$	−2.68753	−1.90037	−0.950184	+	0.311688i	$0.899105\pi$
$3$	−2.90926	−1.67966	−0.839832	−	0.542847i	$-0.817346\pi$
$3$	−2.90926	−1.67966	−0.839832	+	0.542847i	$0.817346\pi$
$4$	5.22280	2.61140
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	7.81872	3.19198
$7$	−1.56079	−0.589924	−0.294962	−	0.955509i	$-0.595307\pi$
$7$	−1.56079	−0.589924	−0.294962	+	0.955509i	$0.595307\pi$
$8$	−8.66137	−3.06226
$9$	5.46381	1.82127
$10$	2.68753	0.849871
$11$	−5.41942	−1.63402	−0.817009	−	0.576625i	$-0.804369\pi$
$11$	−5.41942	−1.63402	−0.817009	+	0.576625i	$0.804369\pi$
$12$	−15.1945	−4.38628
$13$	−2.54347	−0.705430	−0.352715	−	0.935731i	$-0.614742\pi$
$13$	−2.54347	−0.705430	−0.352715	+	0.935731i	$0.614742\pi$
$14$	4.19467	1.12107
$15$	2.90926	0.751168
$16$	12.8321	3.20802
$17$	1.41493	0.343171	0.171585	−	0.985169i	$-0.445111\pi$
$17$	1.41493	0.343171	0.171585	+	0.985169i	$0.445111\pi$
$18$	−14.6841	−3.46108
$19$	6.79597	1.55910	0.779551	−	0.626339i	$-0.215447\pi$
$19$	6.79597	1.55910	0.779551	+	0.626339i	$0.215447\pi$
$20$	−5.22280	−1.16785
$21$	4.54076	0.990874
$22$	14.5648	3.10524
$23$	−1.01891	−0.212457	−0.106228	−	0.994342i	$-0.533877\pi$
$23$	−1.01891	−0.212457	−0.106228	+	0.994342i	$0.533877\pi$
$24$	25.1982	5.14356
$25$	1.00000	0.200000
$26$	6.83563	1.34058
$27$	−7.16786	−1.37946
$28$	−8.15171	−1.54053
$29$	6.60136	1.22584	0.612921	−	0.790144i	$-0.289994\pi$
$29$	6.60136	1.22584	0.612921	+	0.790144i	$0.289994\pi$
$30$	−7.81872	−1.42750
$31$	9.95792	1.78850	0.894248	−	0.447572i	$-0.147711\pi$
$31$	9.95792	1.78850	0.894248	+	0.447572i	$0.147711\pi$
$32$	−17.1638	−3.03416
$33$	15.7665	2.74460
$34$	−3.80266	−0.652151
$35$	1.56079	0.263822
$36$	28.5364	4.75606
$37$	11.1938	1.84026	0.920128	−	0.391619i	$-0.128085\pi$
$37$	11.1938	1.84026	0.920128	+	0.391619i	$0.128085\pi$
$38$	−18.2643	−2.96287
$39$	7.39961	1.18489
$40$	8.66137	1.36948
$41$	2.60847	0.407375	0.203687	−	0.979036i	$-0.434707\pi$
$41$	2.60847	0.407375	0.203687	+	0.979036i	$0.434707\pi$
$42$	−12.2034	−1.88303
$43$	−0.960865	−0.146531	−0.0732653	−	0.997312i	$-0.523342\pi$
$43$	−0.960865	−0.146531	−0.0732653	+	0.997312i	$0.523342\pi$
$44$	−28.3046	−4.26708
$45$	−5.46381	−0.814496
$46$	2.73834	0.403746
$47$	2.99608	0.437023	0.218512	−	0.975834i	$-0.429880\pi$
$47$	2.99608	0.437023	0.218512	+	0.975834i	$0.429880\pi$
$48$	−37.3318	−5.38839
$49$	−4.56393	−0.651989
$50$	−2.68753	−0.380074
$51$	−4.11640	−0.576411
$52$	−13.2840	−1.84216
$53$	6.33397	0.870038	0.435019	−	0.900421i	$-0.356742\pi$
$53$	6.33397	0.870038	0.435019	+	0.900421i	$0.356742\pi$
$54$	19.2638	2.62147
$55$	5.41942	0.730755
$56$	13.5186	1.80650
$57$	−19.7712	−2.61877
$58$	−17.7413	−2.32955
$59$	−1.89910	−0.247242	−0.123621	−	0.992330i	$-0.539451\pi$
$59$	−1.89910	−0.247242	−0.123621	+	0.992330i	$0.539451\pi$
$60$	15.1945	1.96160
$61$	2.14996	0.275274	0.137637	−	0.990483i	$-0.456049\pi$
$61$	2.14996	0.275274	0.137637	+	0.990483i	$0.456049\pi$
$62$	−26.7622	−3.39880
$63$	−8.52787	−1.07441
$64$	20.4640	2.55800
$65$	2.54347	0.315478
$66$	−42.3730	−5.21575
$67$	11.7442	1.43478	0.717391	−	0.696671i	$-0.245336\pi$
$67$	11.7442	1.43478	0.717391	+	0.696671i	$0.245336\pi$
$68$	7.38990	0.896157
$69$	2.96426	0.356856
$70$	−4.19467	−0.501359
$71$	13.1287	1.55809	0.779047	−	0.626965i	$-0.215703\pi$
$71$	13.1287	1.55809	0.779047	+	0.626965i	$0.215703\pi$
$72$	−47.3241	−5.57719
$73$	−2.17969	−0.255114	−0.127557	−	0.991831i	$-0.540714\pi$
$73$	−2.17969	−0.255114	−0.127557	+	0.991831i	$0.540714\pi$
$74$	−30.0837	−3.49716
$75$	−2.90926	−0.335933
$76$	35.4940	4.07144
$77$	8.45860	0.963946
$78$	−19.8866	−2.25172
$79$	−0.477179	−0.0536868	−0.0268434	−	0.999640i	$-0.508546\pi$
$79$	−0.477179	−0.0536868	−0.0268434	+	0.999640i	$0.508546\pi$
$80$	−12.8321	−1.43467
$81$	4.46177	0.495752
$82$	−7.01034	−0.774163
$83$	0.118768	0.0130365	0.00651824	−	0.999979i	$-0.497925\pi$
$83$	0.118768	0.0130365	0.00651824	+	0.999979i	$0.497925\pi$
$84$	23.7155	2.58757
$85$	−1.41493	−0.153471
$86$	2.58235	0.278462
$87$	−19.2051	−2.05900
$88$	46.9396	5.00378
$89$	−16.2714	−1.72476	−0.862381	−	0.506259i	$-0.831028\pi$
$89$	−16.2714	−1.72476	−0.862381	+	0.506259i	$0.831028\pi$
$90$	14.6841	1.54784
$91$	3.96982	0.416150
$92$	−5.32154	−0.554809
$93$	−28.9702	−3.00407
$94$	−8.05205	−0.830505
$95$	−6.79597	−0.697251
$96$	49.9339	5.09636
$97$	−4.98740	−0.506394	−0.253197	−	0.967415i	$-0.581482\pi$
$97$	−4.98740	−0.506394	−0.253197	+	0.967415i	$0.581482\pi$
$98$	12.2657	1.23902
$99$	−29.6107	−2.97599
$100$	5.22280	0.522280
$101$	−5.36581	−0.533918	−0.266959	−	0.963708i	$-0.586019\pi$
$101$	−5.36581	−0.533918	−0.266959	+	0.963708i	$0.586019\pi$
$102$	11.0629	1.09539
$103$	18.7370	1.84621	0.923104	−	0.384550i	$-0.125643\pi$
$103$	18.7370	1.84621	0.923104	+	0.384550i	$0.125643\pi$
$104$	22.0299	2.16021
$105$	−4.54076	−0.443132
$106$	−17.0227	−1.65339
$107$	2.10360	0.203363	0.101681	−	0.994817i	$-0.467578\pi$
$107$	2.10360	0.203363	0.101681	+	0.994817i	$0.467578\pi$
$108$	−37.4363	−3.60231
$109$	−1.09771	−0.105141	−0.0525707	−	0.998617i	$-0.516742\pi$
$109$	−1.09771	−0.105141	−0.0525707	+	0.998617i	$0.516742\pi$
$110$	−14.5648	−1.38870
$111$	−32.5658	−3.09101
$112$	−20.0282	−1.89249
$113$	−3.65869	−0.344180	−0.172090	−	0.985081i	$-0.555052\pi$
$113$	−3.65869	−0.344180	−0.172090	+	0.985081i	$0.555052\pi$
$114$	53.1358	4.97662
$115$	1.01891	0.0950135
$116$	34.4776	3.20116
$117$	−13.8970	−1.28478
$118$	5.10388	0.469850
$119$	−2.20841	−0.202445
$120$	−25.1982	−2.30027
$121$	18.3701	1.67001
$122$	−5.77808	−0.523123
$123$	−7.58873	−0.684253
$124$	52.0083	4.67048
$125$	−1.00000	−0.0894427
$126$	22.9189	2.04178
$127$	−1.18044	−0.104747	−0.0523737	−	0.998628i	$-0.516679\pi$
$127$	−1.18044	−0.104747	−0.0523737	+	0.998628i	$0.516679\pi$
$128$	−20.6700	−1.82699
$129$	2.79541	0.246122
$130$	−6.83563	−0.599525
$131$	15.8606	1.38575	0.692875	−	0.721058i	$-0.256344\pi$
$131$	15.8606	1.38575	0.692875	+	0.721058i	$0.256344\pi$
$132$	82.3454	7.16725
$133$	−10.6071	−0.919752
$134$	−31.5628	−2.72661
$135$	7.16786	0.616911
$136$	−12.2552	−1.05088
$137$	−6.05338	−0.517175	−0.258588	−	0.965988i	$-0.583257\pi$
$137$	−6.05338	−0.517175	−0.258588	+	0.965988i	$0.583257\pi$
$138$	−7.96654	−0.678157
$139$	16.6751	1.41436	0.707182	−	0.707032i	$-0.249966\pi$
$139$	16.6751	1.41436	0.707182	+	0.707032i	$0.249966\pi$
$140$	8.15171	0.688946
$141$	−8.71638	−0.734052
$142$	−35.2838	−2.96095
$143$	13.7841	1.15269
$144$	70.1119	5.84266
$145$	−6.60136	−0.548213
$146$	5.85798	0.484810
$147$	13.2777	1.09512
$148$	58.4632	4.80564
$149$	−6.90639	−0.565794	−0.282897	−	0.959150i	$-0.591295\pi$
$149$	−6.90639	−0.565794	−0.282897	+	0.959150i	$0.591295\pi$
$150$	7.81872	0.638396
$151$	8.29168	0.674767	0.337384	−	0.941367i	$-0.390458\pi$
$151$	8.29168	0.674767	0.337384	+	0.941367i	$0.390458\pi$
$152$	−58.8624	−4.77437
$153$	7.73090	0.625006
$154$	−22.7327	−1.83185
$155$	−9.95792	−0.799839
$156$	38.6467	3.09421
$157$	−5.14783	−0.410841	−0.205421	−	0.978674i	$-0.565856\pi$
$157$	−5.14783	−0.410841	−0.205421	+	0.978674i	$0.565856\pi$
$158$	1.28243	0.102025
$159$	−18.4272	−1.46137
$160$	17.1638	1.35692
$161$	1.59030	0.125333
$162$	−11.9911	−0.942112
$163$	23.4025	1.83302	0.916511	−	0.400009i	$-0.130993\pi$
$163$	23.4025	1.83302	0.916511	+	0.400009i	$0.130993\pi$
$164$	13.6235	1.06382
$165$	−15.7665	−1.22742
$166$	−0.319192	−0.0247741
$167$	19.7076	1.52502	0.762510	−	0.646977i	$-0.223967\pi$
$167$	19.7076	1.52502	0.762510	+	0.646977i	$0.223967\pi$
$168$	−39.3292	−3.03431
$169$	−6.53078	−0.502368
$170$	3.80266	0.291651
$171$	37.1318	2.83954
$172$	−5.01841	−0.382650
$173$	18.9416	1.44010	0.720050	−	0.693922i	$-0.244119\pi$
$173$	18.9416	1.44010	0.720050	+	0.693922i	$0.244119\pi$
$174$	51.6142	3.91286
$175$	−1.56079	−0.117985
$176$	−69.5424	−5.24195
$177$	5.52498	0.415283
$178$	43.7298	3.27769
$179$	−1.50015	−0.112126	−0.0560632	−	0.998427i	$-0.517855\pi$
$179$	−1.50015	−0.112126	−0.0560632	+	0.998427i	$0.517855\pi$
$180$	−28.5364	−2.12698
$181$	−24.8984	−1.85069	−0.925343	−	0.379131i	$-0.876223\pi$
$181$	−24.8984	−1.85069	−0.925343	+	0.379131i	$0.876223\pi$
$182$	−10.6690	−0.790839
$183$	−6.25480	−0.462368
$184$	8.82512	0.650597
$185$	−11.1938	−0.822987
$186$	77.8582	5.70884
$187$	−7.66810	−0.560747
$188$	15.6479	1.14124
$189$	11.1875	0.813774
$190$	18.2643	1.32503
$191$	25.0642	1.81358	0.906790	−	0.421583i	$-0.138525\pi$
$191$	25.0642	1.81358	0.906790	+	0.421583i	$0.138525\pi$
$192$	−59.5351	−4.29658
$193$	−9.76850	−0.703152	−0.351576	−	0.936159i	$-0.614354\pi$
$193$	−9.76850	−0.703152	−0.351576	+	0.936159i	$0.614354\pi$
$194$	13.4038	0.962334
$195$	−7.39961	−0.529897
$196$	−23.8365	−1.70261
$197$	−7.57605	−0.539771	−0.269885	−	0.962892i	$-0.586986\pi$
$197$	−7.57605	−0.539771	−0.269885	+	0.962892i	$0.586986\pi$
$198$	79.5795	5.65547
$199$	3.91711	0.277676	0.138838	−	0.990315i	$-0.455663\pi$
$199$	3.91711	0.277676	0.138838	+	0.990315i	$0.455663\pi$
$200$	−8.66137	−0.612451
$201$	−34.1669	−2.40995
$202$	14.4208	1.01464
$203$	−10.3034	−0.723154
$204$	−21.4992	−1.50524
$205$	−2.60847	−0.182184
$206$	−50.3561	−3.50848
$207$	−5.56711	−0.386941
$208$	−32.6379	−2.26303
$209$	−36.8302	−2.54760
$210$	12.2034	0.842115
$211$	−16.6257	−1.14456	−0.572281	−	0.820057i	$-0.693941\pi$
$211$	−16.6257	−1.14456	−0.572281	+	0.820057i	$0.693941\pi$
$212$	33.0811	2.27202
$213$	−38.1950	−2.61708
$214$	−5.65348	−0.386464
$215$	0.960865	0.0655304
$216$	62.0835	4.22425
$217$	−15.5423	−1.05508
$218$	2.95012	0.199808
$219$	6.34130	0.428505
$220$	28.3046	1.90829
$221$	−3.59882	−0.242083
$222$	87.5215	5.87406
$223$	−4.43302	−0.296857	−0.148428	−	0.988923i	$-0.547421\pi$
$223$	−4.43302	−0.296857	−0.148428	+	0.988923i	$0.547421\pi$
$224$	26.7891	1.78992
$225$	5.46381	0.364254
$226$	9.83282	0.654070
$227$	−14.2988	−0.949044	−0.474522	−	0.880244i	$-0.657379\pi$
$227$	−14.2988	−0.949044	−0.474522	+	0.880244i	$0.657379\pi$
$228$	−103.261	−6.83865
$229$	−0.425247	−0.0281011	−0.0140506	−	0.999901i	$-0.504473\pi$
$229$	−0.425247	−0.0281011	−0.0140506	+	0.999901i	$0.504473\pi$
$230$	−2.73834	−0.180561
$231$	−24.6083	−1.61911
$232$	−57.1768	−3.75384
$233$	−16.9867	−1.11283	−0.556417	−	0.830903i	$-0.687824\pi$
$233$	−16.9867	−1.11283	−0.556417	+	0.830903i	$0.687824\pi$
$234$	37.3486	2.44155
$235$	−2.99608	−0.195443
$236$	−9.91862	−0.645647
$237$	1.38824	0.0901758
$238$	5.93517	0.384720
$239$	−1.40886	−0.0911314	−0.0455657	−	0.998961i	$-0.514509\pi$
$239$	−1.40886	−0.0911314	−0.0455657	+	0.998961i	$0.514509\pi$
$240$	37.3318	2.40976
$241$	22.8035	1.46890	0.734450	−	0.678662i	$-0.237440\pi$
$241$	22.8035	1.46890	0.734450	+	0.678662i	$0.237440\pi$
$242$	−49.3703	−3.17364
$243$	8.52313	0.546759
$244$	11.2288	0.718852
$245$	4.56393	0.291579
$246$	20.3949	1.30033
$247$	−17.2853	−1.09984
$248$	−86.2492	−5.47683
$249$	−0.345527	−0.0218969
$250$	2.68753	0.169974
$251$	22.7585	1.43650	0.718250	−	0.695785i	$-0.244943\pi$
$251$	22.7585	1.43650	0.718250	+	0.695785i	$0.244943\pi$
$252$	−44.5394	−2.80572
$253$	5.52188	0.347158
$254$	3.17247	0.199059
$255$	4.11640	0.257779
$256$	14.6232	0.913948
$257$	1.36137	0.0849198	0.0424599	−	0.999098i	$-0.486481\pi$
$257$	1.36137	0.0849198	0.0424599	+	0.999098i	$0.486481\pi$
$258$	−7.51274	−0.467722
$259$	−17.4713	−1.08561
$260$	13.2840	0.823840
$261$	36.0686	2.23259
$262$	−42.6259	−2.63343
$263$	−31.4029	−1.93639	−0.968193	−	0.250206i	$-0.919502\pi$
$263$	−31.4029	−1.93639	−0.968193	+	0.250206i	$0.919502\pi$
$264$	−136.560	−8.40467
$265$	−6.33397	−0.389093
$266$	28.5069	1.74787
$267$	47.3377	2.89702
$268$	61.3376	3.74679
$269$	−1.71279	−0.104430	−0.0522152	−	0.998636i	$-0.516628\pi$
$269$	−1.71279	−0.104430	−0.0522152	+	0.998636i	$0.516628\pi$
$270$	−19.2638	−1.17236
$271$	−0.245942	−0.0149399	−0.00746997	−	0.999972i	$-0.502378\pi$
$271$	−0.245942	−0.0149399	−0.00746997	+	0.999972i	$0.502378\pi$
$272$	18.1565	1.10090
$273$	−11.5493	−0.698993
$274$	16.2686	0.982824
$275$	−5.41942	−0.326803
$276$	15.4818	0.931893
$277$	14.7658	0.887193	0.443596	−	0.896227i	$-0.353702\pi$
$277$	14.7658	0.887193	0.443596	+	0.896227i	$0.353702\pi$
$278$	−44.8148	−2.68781
$279$	54.4082	3.25733
$280$	−13.5186	−0.807891
$281$	−3.91439	−0.233513	−0.116757	−	0.993161i	$-0.537250\pi$
$281$	−3.91439	−0.233513	−0.116757	+	0.993161i	$0.537250\pi$
$282$	23.4255	1.39497
$283$	16.2510	0.966020	0.483010	−	0.875615i	$-0.339543\pi$
$283$	16.2510	0.966020	0.483010	+	0.875615i	$0.339543\pi$
$284$	68.5688	4.06881
$285$	19.7712	1.17115
$286$	−37.0452	−2.19053
$287$	−4.07129	−0.240320
$288$	−93.7796	−5.52601
$289$	−14.9980	−0.882234
$290$	17.7413	1.04181
$291$	14.5096	0.850571
$292$	−11.3841	−0.666204
$293$	5.76783	0.336960	0.168480	−	0.985705i	$-0.446114\pi$
$293$	5.76783	0.336960	0.168480	+	0.985705i	$0.446114\pi$
$294$	−35.6841	−2.08114
$295$	1.89910	0.110570
$296$	−96.9539	−5.63533
$297$	38.8457	2.25405
$298$	18.5611	1.07522
$299$	2.59155	0.149873
$300$	−15.1945	−0.877255
$301$	1.49971	0.0864419
$302$	−22.2841	−1.28231
$303$	15.6105	0.896802
$304$	87.2063	5.00162
$305$	−2.14996	−0.123106
$306$	−20.7770	−1.18774
$307$	−6.26199	−0.357391	−0.178695	−	0.983904i	$-0.557188\pi$
$307$	−6.26199	−0.357391	−0.178695	+	0.983904i	$0.557188\pi$
$308$	44.1776	2.51725
$309$	−54.5108	−3.10101
$310$	26.7622	1.51999
$311$	−13.3975	−0.759701	−0.379850	−	0.925048i	$-0.624025\pi$
$311$	−13.3975	−0.759701	−0.379850	+	0.925048i	$0.624025\pi$
$312$	−64.0907	−3.62842
$313$	−29.5475	−1.67012	−0.835062	−	0.550156i	$-0.814568\pi$
$313$	−29.5475	−1.67012	−0.835062	+	0.550156i	$0.814568\pi$
$314$	13.8349	0.780750
$315$	8.52787	0.480491
$316$	−2.49221	−0.140198
$317$	−31.0457	−1.74370	−0.871849	−	0.489774i	$-0.837079\pi$
$317$	−31.0457	−1.74370	−0.871849	+	0.489774i	$0.837079\pi$
$318$	49.5236	2.77714
$319$	−35.7756	−2.00305
$320$	−20.4640	−1.14397
$321$	−6.11992	−0.341581
$322$	−4.27398	−0.238179
$323$	9.61581	0.535038
$324$	23.3029	1.29461
$325$	−2.54347	−0.141086
$326$	−62.8948	−3.48342
$327$	3.19352	0.176602
$328$	−22.5929	−1.24749
$329$	−4.67626	−0.257811
$330$	42.3730	2.33255
$331$	−11.8109	−0.649186	−0.324593	−	0.945854i	$-0.605227\pi$
$331$	−11.8109	−0.649186	−0.324593	+	0.945854i	$0.605227\pi$
$332$	0.620301	0.0340435
$333$	61.1610	3.35160
$334$	−52.9647	−2.89810
$335$	−11.7442	−0.641654
$336$	58.2673	3.17874
$337$	17.2148	0.937751	0.468876	−	0.883264i	$-0.344659\pi$
$337$	17.2148	0.937751	0.468876	+	0.883264i	$0.344659\pi$
$338$	17.5517	0.954684
$339$	10.6441	0.578107
$340$	−7.38990	−0.400773
$341$	−53.9662	−2.92243
$342$	−99.7929	−5.39618
$343$	18.0489	0.974549
$344$	8.32241	0.448714
$345$	−2.96426	−0.159591
$346$	−50.9060	−2.73672
$347$	19.5106	1.04738	0.523691	−	0.851908i	$-0.324555\pi$
$347$	19.5106	1.04738	0.523691	+	0.851908i	$0.324555\pi$
$348$	−100.304	−5.37688
$349$	19.0980	1.02229	0.511146	−	0.859494i	$-0.329221\pi$
$349$	19.0980	1.02229	0.511146	+	0.859494i	$0.329221\pi$
$350$	4.19467	0.224215
$351$	18.2312	0.973110
$352$	93.0178	4.95786
$353$	6.52711	0.347403	0.173702	−	0.984798i	$-0.444427\pi$
$353$	6.52711	0.347403	0.173702	+	0.984798i	$0.444427\pi$
$354$	−14.8485	−0.789190
$355$	−13.1287	−0.696801
$356$	−84.9822	−4.50405
$357$	6.42485	0.340039
$358$	4.03169	0.213082
$359$	19.3013	1.01868	0.509342	−	0.860564i	$-0.329889\pi$
$359$	19.3013	1.01868	0.509342	+	0.860564i	$0.329889\pi$
$360$	47.3241	2.49420
$361$	27.1852	1.43080
$362$	66.9152	3.51699
$363$	−53.4436	−2.80506
$364$	20.7336	1.08674
$365$	2.17969	0.114090
$366$	16.8100	0.878670
$367$	17.6788	0.922828	0.461414	−	0.887185i	$-0.347342\pi$
$367$	17.6788	0.922828	0.461414	+	0.887185i	$0.347342\pi$
$368$	−13.0747	−0.681564
$369$	14.2522	0.741939
$370$	30.0837	1.56398
$371$	−9.88602	−0.513256
$372$	−151.306	−7.84483
$373$	25.9717	1.34477	0.672383	−	0.740204i	$-0.265271\pi$
$373$	25.9717	1.34477	0.672383	+	0.740204i	$0.265271\pi$
$374$	20.6082	1.06563
$375$	2.90926	0.150234
$376$	−25.9502	−1.33828
$377$	−16.7903	−0.864746
$378$	−30.0668	−1.54647
$379$	−27.6054	−1.41800	−0.708998	−	0.705211i	$-0.750852\pi$
$379$	−27.6054	−1.41800	−0.708998	+	0.705211i	$0.750852\pi$
$380$	−35.4940	−1.82080
$381$	3.43422	0.175940
$382$	−67.3606	−3.44647
$383$	−35.4949	−1.81370	−0.906852	−	0.421450i	$-0.861521\pi$
$383$	−35.4949	−1.81370	−0.906852	+	0.421450i	$0.861521\pi$
$384$	60.1344	3.06872
$385$	−8.45860	−0.431090
$386$	26.2531	1.33625
$387$	−5.24998	−0.266872
$388$	−26.0482	−1.32240
$389$	20.5819	1.04354	0.521771	−	0.853086i	$-0.325272\pi$
$389$	20.5819	1.04354	0.521771	+	0.853086i	$0.325272\pi$
$390$	19.8866	1.00700
$391$	−1.44168	−0.0729089
$392$	39.5298	1.99656
$393$	−46.1427	−2.32759
$394$	20.3608	1.02576
$395$	0.477179	0.0240095
$396$	−154.651	−7.77149
$397$	3.46791	0.174049	0.0870246	−	0.996206i	$-0.472264\pi$
$397$	3.46791	0.174049	0.0870246	+	0.996206i	$0.472264\pi$
$398$	−10.5273	−0.527687
$399$	30.8588	1.54487
$400$	12.8321	0.641603
$401$	34.5537	1.72553	0.862764	−	0.505607i	$-0.168731\pi$
$401$	34.5537	1.72553	0.862764	+	0.505607i	$0.168731\pi$
$402$	91.8246	4.57979
$403$	−25.3276	−1.26166
$404$	−28.0246	−1.39427
$405$	−4.46177	−0.221707
$406$	27.6905	1.37426
$407$	−60.6641	−3.00701
$408$	35.6537	1.76512
$409$	29.0503	1.43645	0.718223	−	0.695813i	$-0.244956\pi$
$409$	29.0503	1.43645	0.718223	+	0.695813i	$0.244956\pi$
$410$	7.01034	0.346216
$411$	17.6109	0.868680
$412$	97.8595	4.82119
$413$	2.96410	0.145854
$414$	14.9617	0.735330
$415$	−0.118768	−0.00583009
$416$	43.6555	2.14039
$417$	−48.5123	−2.37566
$418$	98.9822	4.84138
$419$	−13.9363	−0.680834	−0.340417	−	0.940275i	$-0.610568\pi$
$419$	−13.9363	−0.680834	−0.340417	+	0.940275i	$0.610568\pi$
$420$	−23.7155	−1.15720
$421$	2.65662	0.129476	0.0647378	−	0.997902i	$-0.479379\pi$
$421$	2.65662	0.129476	0.0647378	+	0.997902i	$0.479379\pi$
$422$	44.6821	2.17509
$423$	16.3700	0.795937
$424$	−54.8609	−2.66428
$425$	1.41493	0.0686342
$426$	102.650	4.97341
$427$	−3.35565	−0.162391
$428$	10.9867	0.531061
$429$	−40.1016	−1.93612
$430$	−2.58235	−0.124532
$431$	4.82907	0.232608	0.116304	−	0.993214i	$-0.462895\pi$
$431$	4.82907	0.232608	0.116304	+	0.993214i	$0.462895\pi$
$432$	−91.9785	−4.42532
$433$	−24.8031	−1.19196	−0.595982	−	0.802998i	$-0.703237\pi$
$433$	−24.8031	−1.19196	−0.595982	+	0.802998i	$0.703237\pi$
$434$	41.7702	2.00504
$435$	19.2051	0.920813
$436$	−5.73312	−0.274567
$437$	−6.92445	−0.331241
$438$	−17.0424	−0.814318
$439$	−19.2088	−0.916785	−0.458393	−	0.888750i	$-0.651575\pi$
$439$	−19.2088	−0.916785	−0.458393	+	0.888750i	$0.651575\pi$
$440$	−46.9396	−2.23776
$441$	−24.9364	−1.18745
$442$	9.67194	0.460047
$443$	3.48151	0.165412	0.0827058	−	0.996574i	$-0.473644\pi$
$443$	3.48151	0.165412	0.0827058	+	0.996574i	$0.473644\pi$
$444$	−170.085	−8.07187
$445$	16.2714	0.771337
$446$	11.9139	0.564138
$447$	20.0925	0.950343
$448$	−31.9401	−1.50903
$449$	−18.0227	−0.850545	−0.425272	−	0.905065i	$-0.639822\pi$
$449$	−18.0227	−0.850545	−0.425272	+	0.905065i	$0.639822\pi$
$450$	−14.6841	−0.692217
$451$	−14.1364	−0.665658
$452$	−19.1086	−0.898793
$453$	−24.1227	−1.13338
$454$	38.4284	1.80353
$455$	−3.96982	−0.186108
$456$	171.246	8.01933
$457$	4.91446	0.229889	0.114944	−	0.993372i	$-0.463331\pi$
$457$	4.91446	0.229889	0.114944	+	0.993372i	$0.463331\pi$
$458$	1.14286	0.0534025
$459$	−10.1420	−0.473389
$460$	5.32154	0.248118
$461$	37.1144	1.72859	0.864294	−	0.502986i	$-0.167765\pi$
$461$	37.1144	1.72859	0.864294	+	0.502986i	$0.167765\pi$
$462$	66.1354	3.07690
$463$	−33.8229	−1.57188	−0.785942	−	0.618300i	$-0.787822\pi$
$463$	−33.8229	−1.57188	−0.785942	+	0.618300i	$0.787822\pi$
$464$	84.7091	3.93252
$465$	28.9702	1.34346
$466$	45.6521	2.11479
$467$	16.1312	0.746464	0.373232	−	0.927738i	$-0.378250\pi$
$467$	16.1312	0.746464	0.373232	+	0.927738i	$0.378250\pi$
$468$	−72.5813	−3.35507
$469$	−18.3303	−0.846412
$470$	8.05205	0.371413
$471$	14.9764	0.690075
$472$	16.4488	0.757117
$473$	5.20733	0.239433
$474$	−3.73093	−0.171367
$475$	6.79597	0.311820
$476$	−11.5341	−0.528665
$477$	34.6076	1.58457
$478$	3.78634	0.173183
$479$	−9.15073	−0.418107	−0.209054	−	0.977904i	$-0.567038\pi$
$479$	−9.15073	−0.418107	−0.209054	+	0.977904i	$0.567038\pi$
$480$	−49.9339	−2.27916
$481$	−28.4711	−1.29817
$482$	−61.2850	−2.79145
$483$	−4.62660	−0.210518
$484$	95.9436	4.36107
$485$	4.98740	0.226466
$486$	−22.9061	−1.03904
$487$	22.7904	1.03273	0.516366	−	0.856368i	$-0.327284\pi$
$487$	22.7904	1.03273	0.516366	+	0.856368i	$0.327284\pi$
$488$	−18.6216	−0.842961
$489$	−68.0839	−3.07886
$490$	−12.2657	−0.554107
$491$	−19.7234	−0.890103	−0.445052	−	0.895505i	$-0.646815\pi$
$491$	−19.7234	−0.890103	−0.445052	+	0.895505i	$0.646815\pi$
$492$	−39.6344	−1.78686
$493$	9.34046	0.420673
$494$	46.4547	2.09010
$495$	29.6107	1.33090
$496$	127.781	5.73752
$497$	−20.4912	−0.919158
$498$	0.928613	0.0416122
$499$	−32.2332	−1.44296	−0.721479	−	0.692436i	$-0.756537\pi$
$499$	−32.2332	−1.44296	−0.721479	+	0.692436i	$0.756537\pi$
$500$	−5.22280	−0.233571
$501$	−57.3346	−2.56152
$502$	−61.1640	−2.72988
$503$	−21.5254	−0.959772	−0.479886	−	0.877331i	$-0.659322\pi$
$503$	−21.5254	−0.959772	−0.479886	+	0.877331i	$0.659322\pi$
$504$	73.8631	3.29012
$505$	5.36581	0.238775
$506$	−14.8402	−0.659728
$507$	18.9998	0.843809
$508$	−6.16522	−0.273538
$509$	−27.0879	−1.20065	−0.600326	−	0.799756i	$-0.704962\pi$
$509$	−27.0879	−1.20065	−0.600326	+	0.799756i	$0.704962\pi$
$510$	−11.0629	−0.489875
$511$	3.40205	0.150498
$512$	2.03982	0.0901480
$513$	−48.7125	−2.15071
$514$	−3.65871	−0.161379
$515$	−18.7370	−0.825649
$516$	14.5999	0.642723
$517$	−16.2370	−0.714104
$518$	46.9545	2.06306
$519$	−55.1060	−2.41888
$520$	−22.0299	−0.966075
$521$	−21.7330	−0.952139	−0.476070	−	0.879408i	$-0.657939\pi$
$521$	−21.7330	−0.952139	−0.476070	+	0.879408i	$0.657939\pi$
$522$	−96.9352	−4.24274
$523$	23.3180	1.01962	0.509812	−	0.860286i	$-0.329715\pi$
$523$	23.3180	1.01962	0.509812	+	0.860286i	$0.329715\pi$
$524$	82.8369	3.61875
$525$	4.54076	0.198175
$526$	84.3961	3.67985
$527$	14.0898	0.613759
$528$	202.317	8.80472
$529$	−21.9618	−0.954862
$530$	17.0227	0.739420
$531$	−10.3763	−0.450293
$532$	−55.3988	−2.40184
$533$	−6.63456	−0.287375
$534$	−127.221	−5.50541
$535$	−2.10360	−0.0909465
$536$	−101.721	−4.39367
$537$	4.36433	0.188335
$538$	4.60316	0.198456
$539$	24.7338	1.06536
$540$	37.4363	1.61100
$541$	8.97799	0.385994	0.192997	−	0.981199i	$-0.438179\pi$
$541$	8.97799	0.385994	0.192997	+	0.981199i	$0.438179\pi$
$542$	0.660977	0.0283914
$543$	72.4361	3.10853
$544$	−24.2855	−1.04123
$545$	1.09771	0.0470207
$546$	31.0389	1.32834
$547$	12.9068	0.551855	0.275928	−	0.961178i	$-0.411015\pi$
$547$	12.9068	0.551855	0.275928	+	0.961178i	$0.411015\pi$
$548$	−31.6156	−1.35055
$549$	11.7470	0.501349
$550$	14.5648	0.621047
$551$	44.8626	1.91121
$552$	−25.6746	−1.09278
$553$	0.744777	0.0316712
$554$	−39.6836	−1.68599
$555$	32.5658	1.38234
$556$	87.0908	3.69347
$557$	45.2972	1.91930	0.959652	−	0.281190i	$-0.0907291\pi$
$557$	45.2972	1.91930	0.959652	+	0.281190i	$0.0907291\pi$
$558$	−146.223	−6.19013
$559$	2.44393	0.103367
$560$	20.0282	0.846346
$561$	22.3085	0.941866
$562$	10.5200	0.443761
$563$	−16.7820	−0.707277	−0.353639	−	0.935382i	$-0.615056\pi$
$563$	−16.7820	−0.707277	−0.353639	+	0.935382i	$0.615056\pi$
$564$	−45.5240	−1.91690
$565$	3.65869	0.153922
$566$	−43.6749	−1.83579
$567$	−6.96390	−0.292456
$568$	−113.713	−4.77129
$569$	−4.12662	−0.172997	−0.0864985	−	0.996252i	$-0.527568\pi$
$569$	−4.12662	−0.172997	−0.0864985	+	0.996252i	$0.527568\pi$
$570$	−53.1358	−2.22561
$571$	−4.54121	−0.190044	−0.0950219	−	0.995475i	$-0.530292\pi$
$571$	−4.54121	−0.190044	−0.0950219	+	0.995475i	$0.530292\pi$
$572$	71.9917	3.01012
$573$	−72.9182	−3.04620
$574$	10.9417	0.456697
$575$	−1.01891	−0.0424913
$576$	111.811	4.65881
$577$	−14.7426	−0.613742	−0.306871	−	0.951751i	$-0.599282\pi$
$577$	−14.7426	−0.613742	−0.306871	+	0.951751i	$0.599282\pi$
$578$	40.3075	1.67657
$579$	28.4191	1.18106
$580$	−34.4776	−1.43160
$581$	−0.185372	−0.00769053
$582$	−38.9951	−1.61640
$583$	−34.3265	−1.42166
$584$	18.8791	0.781223
$585$	13.8970	0.574570
$586$	−15.5012	−0.640349
$587$	−3.76785	−0.155516	−0.0777580	−	0.996972i	$-0.524776\pi$
$587$	−3.76785	−0.155516	−0.0777580	+	0.996972i	$0.524776\pi$
$588$	69.3466	2.85980
$589$	67.6737	2.78845
$590$	−5.10388	−0.210123
$591$	22.0407	0.906633
$592$	143.640	5.90357
$593$	−20.2857	−0.833032	−0.416516	−	0.909128i	$-0.636749\pi$
$593$	−20.2857	−0.833032	−0.416516	+	0.909128i	$0.636749\pi$
$594$	−104.399	−4.28353
$595$	2.20841	0.0905361
$596$	−36.0707	−1.47751
$597$	−11.3959	−0.466403
$598$	−6.96487	−0.284815
$599$	18.4574	0.754149	0.377075	−	0.926183i	$-0.376930\pi$
$599$	18.4574	0.754149	0.377075	+	0.926183i	$0.376930\pi$
$600$	25.1982	1.02871
$601$	−12.0030	−0.489614	−0.244807	−	0.969572i	$-0.578725\pi$
$601$	−12.0030	−0.489614	−0.244807	+	0.969572i	$0.578725\pi$
$602$	−4.03051	−0.164272
$603$	64.1680	2.61312
$604$	43.3058	1.76209
$605$	−18.3701	−0.746852
$606$	−41.9538	−1.70425
$607$	11.8328	0.480279	0.240139	−	0.970738i	$-0.422807\pi$
$607$	11.8328	0.480279	0.240139	+	0.970738i	$0.422807\pi$
$608$	−116.644	−4.73056
$609$	29.9752	1.21465
$610$	5.77808	0.233948
$611$	−7.62043	−0.308290
$612$	40.3770	1.63214
$613$	37.0327	1.49574	0.747868	−	0.663848i	$-0.231078\pi$
$613$	37.0327	1.49574	0.747868	+	0.663848i	$0.231078\pi$
$614$	16.8293	0.679174
$615$	7.58873	0.306007
$616$	−73.2630	−2.95185
$617$	44.5120	1.79199	0.895993	−	0.444067i	$-0.146465\pi$
$617$	44.5120	1.79199	0.895993	+	0.444067i	$0.146465\pi$
$618$	146.499	5.89306
$619$	−29.0998	−1.16962	−0.584809	−	0.811171i	$-0.698831\pi$
$619$	−29.0998	−1.16962	−0.584809	+	0.811171i	$0.698831\pi$
$620$	−52.0083	−2.08870
$621$	7.30338	0.293074
$622$	36.0061	1.44371
$623$	25.3963	1.01748
$624$	94.9522	3.80113
$625$	1.00000	0.0400000
$626$	79.4097	3.17385
$627$	107.149	4.27911
$628$	−26.8861	−1.07287
$629$	15.8385	0.631522
$630$	−22.9189	−0.913110
$631$	0.638027	0.0253994	0.0126997	−	0.999919i	$-0.495957\pi$
$631$	0.638027	0.0253994	0.0126997	+	0.999919i	$0.495957\pi$
$632$	4.13302	0.164403
$633$	48.3686	1.92248
$634$	83.4361	3.31367
$635$	1.18044	0.0468445
$636$	−96.2416	−3.81623
$637$	11.6082	0.459933
$638$	96.1478	3.80653
$639$	71.7329	2.83771
$640$	20.6700	0.817053
$641$	19.4978	0.770115	0.385058	−	0.922892i	$-0.374182\pi$
$641$	19.4978	0.770115	0.385058	+	0.922892i	$0.374182\pi$
$642$	16.4475	0.649129
$643$	17.7721	0.700862	0.350431	−	0.936589i	$-0.386035\pi$
$643$	17.7721	0.700862	0.350431	+	0.936589i	$0.386035\pi$
$644$	8.30583	0.327296
$645$	−2.79541	−0.110069
$646$	−25.8428	−1.01677
$647$	1.01118	0.0397535	0.0198768	−	0.999802i	$-0.493673\pi$
$647$	1.01118	0.0397535	0.0198768	+	0.999802i	$0.493673\pi$
$648$	−38.6450	−1.51812
$649$	10.2920	0.403997
$650$	6.83563	0.268116
$651$	45.2165	1.77217
$652$	122.226	4.78676
$653$	22.0121	0.861400	0.430700	−	0.902495i	$-0.358267\pi$
$653$	22.0121	0.861400	0.430700	+	0.902495i	$0.358267\pi$
$654$	−8.58268	−0.335609
$655$	−15.8606	−0.619726
$656$	33.4721	1.30687
$657$	−11.9094	−0.464631
$658$	12.5676	0.489935
$659$	16.9007	0.658358	0.329179	−	0.944267i	$-0.393228\pi$
$659$	16.9007	0.658358	0.329179	+	0.944267i	$0.393228\pi$
$660$	−82.3454	−3.20529
$661$	28.8934	1.12382	0.561911	−	0.827198i	$-0.310066\pi$
$661$	28.8934	1.12382	0.561911	+	0.827198i	$0.310066\pi$
$662$	31.7421	1.23369
$663$	10.4699	0.406618
$664$	−1.02869	−0.0399210
$665$	10.6071	0.411326
$666$	−164.372	−6.36928
$667$	−6.72616	−0.260438
$668$	102.929	3.98244
$669$	12.8968	0.498620
$670$	31.5628	1.21938
$671$	−11.6516	−0.449803
$672$	−77.9365	−3.00647
$673$	−16.6159	−0.640495	−0.320247	−	0.947334i	$-0.603766\pi$
$673$	−16.6159	−0.640495	−0.320247	+	0.947334i	$0.603766\pi$
$674$	−46.2653	−1.78207
$675$	−7.16786	−0.275891
$676$	−34.1090	−1.31188
$677$	27.2420	1.04700	0.523498	−	0.852027i	$-0.324627\pi$
$677$	27.2420	1.04700	0.523498	+	0.852027i	$0.324627\pi$
$678$	−28.6063	−1.09862
$679$	7.78429	0.298734
$680$	12.2552	0.469967
$681$	41.5989	1.59407
$682$	145.036	5.55370
$683$	−39.2520	−1.50194	−0.750969	−	0.660338i	$-0.770413\pi$
$683$	−39.2520	−1.50194	−0.750969	+	0.660338i	$0.770413\pi$
$684$	193.932	7.41519
$685$	6.05338	0.231288
$686$	−48.5069	−1.85200
$687$	1.23716	0.0472004
$688$	−12.3299	−0.470072
$689$	−16.1102	−0.613751
$690$	7.96654	0.303281
$691$	33.4053	1.27080	0.635399	−	0.772184i	$-0.280836\pi$
$691$	33.4053	1.27080	0.635399	+	0.772184i	$0.280836\pi$
$692$	98.9280	3.76068
$693$	46.2161	1.75561
$694$	−52.4352	−1.99041
$695$	−16.6751	−0.632523
$696$	166.342	6.30519
$697$	3.69080	0.139799
$698$	−51.3264	−1.94273
$699$	49.4187	1.86919
$700$	−8.15171	−0.308106
$701$	−3.33213	−0.125853	−0.0629263	−	0.998018i	$-0.520043\pi$
$701$	−3.33213	−0.125853	−0.0629263	+	0.998018i	$0.520043\pi$
$702$	−48.9969	−1.84927
$703$	76.0729	2.86914
$704$	−110.903	−4.17982
$705$	8.71638	0.328278
$706$	−17.5418	−0.660194
$707$	8.37491	0.314971
$708$	28.8559	1.08447
$709$	−35.2659	−1.32444	−0.662220	−	0.749310i	$-0.730385\pi$
$709$	−35.2659	−1.32444	−0.662220	+	0.749310i	$0.730385\pi$
$710$	35.2838	1.32418
$711$	−2.60721	−0.0977781
$712$	140.932	5.28167
$713$	−10.1462	−0.379978
$714$	−17.2670	−0.646200
$715$	−13.7841	−0.515497
$716$	−7.83498	−0.292807
$717$	4.09874	0.153070
$718$	−51.8728	−1.93588
$719$	22.5042	0.839264	0.419632	−	0.907694i	$-0.362159\pi$
$719$	22.5042	0.839264	0.419632	+	0.907694i	$0.362159\pi$
$720$	−70.1119	−2.61292
$721$	−29.2445	−1.08912
$722$	−73.0608	−2.71904
$723$	−66.3413	−2.46726
$724$	−130.040	−4.83288
$725$	6.60136	0.245168
$726$	143.631	5.33065
$727$	−24.5187	−0.909350	−0.454675	−	0.890658i	$-0.650245\pi$
$727$	−24.5187	−0.909350	−0.454675	+	0.890658i	$0.650245\pi$
$728$	−34.3841	−1.27436
$729$	−38.1813	−1.41412
$730$	−5.85798	−0.216814
$731$	−1.35956	−0.0502850
$732$	−32.6676	−1.20743
$733$	35.8844	1.32542	0.662711	−	0.748875i	$-0.269406\pi$
$733$	35.8844	1.32542	0.662711	+	0.748875i	$0.269406\pi$
$734$	−47.5124	−1.75371
$735$	−13.2777	−0.489754
$736$	17.4883	0.644626
$737$	−63.6467	−2.34446
$738$	−38.3032	−1.40996
$739$	−32.6597	−1.20141	−0.600704	−	0.799472i	$-0.705113\pi$
$739$	−32.6597	−1.20141	−0.600704	+	0.799472i	$0.705113\pi$
$740$	−58.4632	−2.14915
$741$	50.2875	1.84736
$742$	26.5689	0.975377
$743$	7.43074	0.272608	0.136304	−	0.990667i	$-0.456478\pi$
$743$	7.43074	0.272608	0.136304	+	0.990667i	$0.456478\pi$
$744$	250.922	9.19923
$745$	6.90639	0.253031
$746$	−69.7997	−2.55555
$747$	0.648925	0.0237429
$748$	−40.0490	−1.46434
$749$	−3.28328	−0.119969
$750$	−7.81872	−0.285499
$751$	−28.9666	−1.05701	−0.528504	−	0.848931i	$-0.677247\pi$
$751$	−28.9666	−1.05701	−0.528504	+	0.848931i	$0.677247\pi$
$752$	38.4459	1.40198
$753$	−66.2103	−2.41284
$754$	45.1245	1.64334
$755$	−8.29168	−0.301765
$756$	58.4304	2.12509
$757$	25.7866	0.937230	0.468615	−	0.883403i	$-0.344753\pi$
$757$	25.7866	0.937230	0.468615	+	0.883403i	$0.344753\pi$
$758$	74.1904	2.69472
$759$	−16.0646	−0.583108
$760$	58.8624	2.13516
$761$	1.71849	0.0622954	0.0311477	−	0.999515i	$-0.490084\pi$
$761$	1.71849	0.0622954	0.0311477	+	0.999515i	$0.490084\pi$
$762$	−9.22956	−0.334352
$763$	1.71330	0.0620255
$764$	130.905	4.73598
$765$	−7.73090	−0.279511
$766$	95.3934	3.44670
$767$	4.83029	0.174412
$768$	−42.5426	−1.53512
$769$	17.2083	0.620548	0.310274	−	0.950647i	$-0.399579\pi$
$769$	17.2083	0.620548	0.310274	+	0.950647i	$0.399579\pi$
$770$	22.7327	0.819230
$771$	−3.96058	−0.142637
$772$	−51.0189	−1.83621
$773$	−23.4607	−0.843823	−0.421912	−	0.906637i	$-0.638641\pi$
$773$	−23.4607	−0.843823	−0.421912	+	0.906637i	$0.638641\pi$
$774$	14.1095	0.507154
$775$	9.95792	0.357699
$776$	43.1977	1.55071
$777$	50.8285	1.82346
$778$	−55.3143	−1.98311
$779$	17.7271	0.635139
$780$	−38.6467	−1.38377
$781$	−71.1502	−2.54595
$782$	3.87455	0.138554
$783$	−47.3176	−1.69099
$784$	−58.5646	−2.09159
$785$	5.14783	0.183734
$786$	124.010	4.42328
$787$	−8.30490	−0.296038	−0.148019	−	0.988985i	$-0.547290\pi$
$787$	−8.30490	−0.296038	−0.148019	+	0.988985i	$0.547290\pi$
$788$	−39.5682	−1.40956
$789$	91.3593	3.25248
$790$	−1.28243	−0.0456268
$791$	5.71045	0.203040
$792$	256.469	9.11323
$793$	−5.46835	−0.194187
$794$	−9.32009	−0.330758
$795$	18.4272	0.653545
$796$	20.4583	0.725124
$797$	−21.3814	−0.757368	−0.378684	−	0.925526i	$-0.623623\pi$
$797$	−21.3814	−0.757368	−0.378684	+	0.925526i	$0.623623\pi$
$798$	−82.9339	−2.93583
$799$	4.23924	0.149974
$800$	−17.1638	−0.606831
$801$	−88.9037	−3.14126
$802$	−92.8639	−3.27914
$803$	11.8127	0.416860
$804$	−178.447	−6.29335
$805$	−1.59030	−0.0560508
$806$	68.0687	2.39762
$807$	4.98295	0.175408
$808$	46.4752	1.63499
$809$	11.2957	0.397134	0.198567	−	0.980087i	$-0.436371\pi$
$809$	11.2957	0.397134	0.198567	+	0.980087i	$0.436371\pi$
$810$	11.9911	0.421325
$811$	−2.96967	−0.104279	−0.0521396	−	0.998640i	$-0.516604\pi$
$811$	−2.96967	−0.104279	−0.0521396	+	0.998640i	$0.516604\pi$
$812$	−53.8124	−1.88844
$813$	0.715511	0.0250941
$814$	163.036	5.71443
$815$	−23.4025	−0.819753
$816$	−52.8219	−1.84914
$817$	−6.53001	−0.228456
$818$	−78.0735	−2.72978
$819$	21.6903	0.757922
$820$	−13.6235	−0.475755
$821$	−34.6559	−1.20950	−0.604750	−	0.796415i	$-0.706727\pi$
$821$	−34.6559	−1.20950	−0.604750	+	0.796415i	$0.706727\pi$
$822$	−47.3297	−1.65081
$823$	2.35069	0.0819398	0.0409699	−	0.999160i	$-0.486955\pi$
$823$	2.35069	0.0819398	0.0409699	+	0.999160i	$0.486955\pi$
$824$	−162.288	−5.65356
$825$	15.7665	0.548920
$826$	−7.96610	−0.277176
$827$	47.3898	1.64790	0.823952	−	0.566660i	$-0.191765\pi$
$827$	47.3898	1.64790	0.823952	+	0.566660i	$0.191765\pi$
$828$	−29.0759	−1.01046
$829$	28.5618	0.991993	0.495997	−	0.868324i	$-0.334803\pi$
$829$	28.5618	0.991993	0.495997	+	0.868324i	$0.334803\pi$
$830$	0.319192	0.0110793
$831$	−42.9577	−1.49019
$832$	−52.0495	−1.80449
$833$	−6.45763	−0.223744
$834$	130.378	4.51462
$835$	−19.7076	−0.682009
$836$	−192.357	−6.65280
$837$	−71.3770	−2.46715
$838$	37.4543	1.29384
$839$	55.8395	1.92779	0.963896	−	0.266278i	$-0.0857940\pi$
$839$	55.8395	1.92779	0.963896	+	0.266278i	$0.0857940\pi$
$840$	39.3292	1.35699
$841$	14.5779	0.502688
$842$	−7.13973	−0.246051
$843$	11.3880	0.392223
$844$	−86.8329	−2.98891
$845$	6.53078	0.224666
$846$	−43.9948	−1.51257
$847$	−28.6720	−0.985181
$848$	81.2779	2.79110
$849$	−47.2784	−1.62259
$850$	−3.80266	−0.130430
$851$	−11.4055	−0.390974
$852$	−199.485	−6.83423
$853$	43.6203	1.49353	0.746765	−	0.665088i	$-0.231606\pi$
$853$	43.6203	1.49353	0.746765	+	0.665088i	$0.231606\pi$
$854$	9.01839	0.308603
$855$	−37.1318	−1.26988
$856$	−18.2200	−0.622748
$857$	−41.4522	−1.41598	−0.707990	−	0.706223i	$-0.750398\pi$
$857$	−41.4522	−1.41598	−0.707990	+	0.706223i	$0.750398\pi$
$858$	107.774	3.67935
$859$	−13.3030	−0.453891	−0.226945	−	0.973907i	$-0.572874\pi$
$859$	−13.3030	−0.453891	−0.226945	+	0.973907i	$0.572874\pi$
$860$	5.01841	0.171126
$861$	11.8444	0.403657
$862$	−12.9783	−0.442041
$863$	25.6909	0.874527	0.437264	−	0.899333i	$-0.355948\pi$
$863$	25.6909	0.874527	0.437264	+	0.899333i	$0.355948\pi$
$864$	123.028	4.18548
$865$	−18.9416	−0.644032
$866$	66.6591	2.26517
$867$	43.6330	1.48186
$868$	−81.1741	−2.75523
$869$	2.58603	0.0877252
$870$	−51.6142	−1.74988
$871$	−29.8709	−1.01214
$872$	9.50766	0.321970
$873$	−27.2502	−0.922279
$874$	18.6096	0.629481
$875$	1.56079	0.0527644
$876$	33.1193	1.11900
$877$	−10.7807	−0.364039	−0.182019	−	0.983295i	$-0.558263\pi$
$877$	−10.7807	−0.364039	−0.182019	+	0.983295i	$0.558263\pi$
$878$	51.6241	1.74223
$879$	−16.7801	−0.565980
$880$	69.5424	2.34427
$881$	−25.2755	−0.851552	−0.425776	−	0.904829i	$-0.639999\pi$
$881$	−25.2755	−0.851552	−0.425776	+	0.904829i	$0.639999\pi$
$882$	67.0173	2.25659
$883$	7.57725	0.254995	0.127497	−	0.991839i	$-0.459306\pi$
$883$	7.57725	0.254995	0.127497	+	0.991839i	$0.459306\pi$
$884$	−18.7959	−0.632176
$885$	−5.52498	−0.185720
$886$	−9.35666	−0.314343
$887$	44.5119	1.49456	0.747281	−	0.664508i	$-0.231359\pi$
$887$	44.5119	1.49456	0.747281	+	0.664508i	$0.231359\pi$
$888$	282.064	9.46546
$889$	1.84243	0.0617931
$890$	−43.7298	−1.46583
$891$	−24.1802	−0.810068
$892$	−23.1528	−0.775212
$893$	20.3613	0.681364
$894$	−53.9991	−1.80600
$895$	1.50015	0.0501445
$896$	32.2616	1.07778
$897$	−7.53950	−0.251737
$898$	48.4365	1.61635
$899$	65.7358	2.19241
$900$	28.5364	0.951213
$901$	8.96212	0.298572
$902$	37.9920	1.26500
$903$	−4.36305	−0.145193
$904$	31.6893	1.05397
$905$	24.8984	0.827652
$906$	64.8303	2.15384
$907$	−25.5733	−0.849147	−0.424574	−	0.905393i	$-0.639576\pi$
$907$	−25.5733	−0.849147	−0.424574	+	0.905393i	$0.639576\pi$
$908$	−74.6797	−2.47833
$909$	−29.3177	−0.972408
$910$	10.6690	0.353674
$911$	8.69404	0.288046	0.144023	−	0.989574i	$-0.453996\pi$
$911$	8.69404	0.288046	0.144023	+	0.989574i	$0.453996\pi$
$912$	−253.706	−8.40104
$913$	−0.643654	−0.0213018
$914$	−13.2077	−0.436873
$915$	6.25480	0.206777
$916$	−2.22098	−0.0733833
$917$	−24.7551	−0.817487
$918$	27.2570	0.899614
$919$	−36.2089	−1.19442	−0.597211	−	0.802084i	$-0.703724\pi$
$919$	−36.2089	−1.19442	−0.597211	+	0.802084i	$0.703724\pi$
$920$	−8.82512	−0.290956
$921$	18.2178	0.600296
$922$	−99.7459	−3.28496
$923$	−33.3925	−1.09913
$924$	−128.524	−4.22813
$925$	11.1938	0.368051
$926$	90.9000	2.98716
$927$	102.375	3.36244
$928$	−113.304	−3.71939
$929$	32.9400	1.08073	0.540363	−	0.841432i	$-0.318287\pi$
$929$	32.9400	1.08073	0.540363	+	0.841432i	$0.318287\pi$
$930$	−77.8582	−2.55307
$931$	−31.0163	−1.01652
$932$	−88.7180	−2.90606
$933$	38.9768	1.27604
$934$	−43.3531	−1.41856
$935$	7.66810	0.250774
$936$	120.367	3.93432
$937$	−30.0572	−0.981927	−0.490964	−	0.871180i	$-0.663355\pi$
$937$	−30.0572	−0.981927	−0.490964	+	0.871180i	$0.663355\pi$
$938$	49.2631	1.60850
$939$	85.9615	2.80525
$940$	−15.6479	−0.510379
$941$	−11.7725	−0.383772	−0.191886	−	0.981417i	$-0.561460\pi$
$941$	−11.7725	−0.383772	−0.191886	+	0.981417i	$0.561460\pi$
$942$	−40.2494	−1.31140
$943$	−2.65779	−0.0865495
$944$	−24.3694	−0.793155
$945$	−11.1875	−0.363931
$946$	−13.9948	−0.455012
$947$	−46.6011	−1.51433	−0.757166	−	0.653222i	$-0.773417\pi$
$947$	−46.6011	−1.51433	−0.757166	+	0.653222i	$0.773417\pi$
$948$	7.25049	0.235485
$949$	5.54397	0.179965
$950$	−18.2643	−0.592574
$951$	90.3200	2.92883
$952$	19.1279	0.619938
$953$	16.7888	0.543843	0.271921	−	0.962319i	$-0.412341\pi$
$953$	16.7888	0.543843	0.271921	+	0.962319i	$0.412341\pi$
$954$	−93.0089	−3.01127
$955$	−25.0642	−0.811057
$956$	−7.35819	−0.237981
$957$	104.080	3.36444
$958$	24.5928	0.794558
$959$	9.44808	0.305094
$960$	59.5351	1.92149
$961$	68.1602	2.19872
$962$	76.5169	2.46701
$963$	11.4937	0.370378
$964$	119.098	3.83589
$965$	9.76850	0.314459
$966$	12.4341	0.400061
$967$	45.6276	1.46728	0.733642	−	0.679536i	$-0.237819\pi$
$967$	45.6276	1.46728	0.733642	+	0.679536i	$0.237819\pi$
$968$	−159.111	−5.11401
$969$	−27.9749	−0.898684
$970$	−13.4038	−0.430369
$971$	−32.0937	−1.02994	−0.514968	−	0.857210i	$-0.672196\pi$
$971$	−32.0937	−1.02994	−0.514968	+	0.857210i	$0.672196\pi$
$972$	44.5146	1.42781
$973$	−26.0264	−0.834368
$974$	−61.2498	−1.96257
$975$	7.39961	0.236977
$976$	27.5884	0.883085
$977$	4.53160	0.144979	0.0724893	−	0.997369i	$-0.476906\pi$
$977$	4.53160	0.144979	0.0724893	+	0.997369i	$0.476906\pi$
$978$	182.977	5.85097
$979$	88.1815	2.81829
$980$	23.8365	0.761429
$981$	−5.99767	−0.191491
$982$	53.0071	1.69152
$983$	21.5872	0.688524	0.344262	−	0.938874i	$-0.388129\pi$
$983$	21.5872	0.688524	0.344262	+	0.938874i	$0.388129\pi$
$984$	65.7288	2.09536
$985$	7.57605	0.241393
$986$	−25.1027	−0.799434
$987$	13.6045	0.433035
$988$	−90.2777	−2.87212
$989$	0.979031	0.0311314
$990$	−79.5795	−2.52920
$991$	41.3634	1.31395	0.656976	−	0.753911i	$-0.271835\pi$
$991$	41.3634	1.31395	0.656976	+	0.753911i	$0.271835\pi$
$992$	−170.916	−5.42657
$993$	34.3610	1.09041
$994$	55.0708	1.74674
$995$	−3.91711	−0.124181
$996$	−1.80462	−0.0571816
$997$	−57.7101	−1.82770	−0.913849	−	0.406054i	$-0.866904\pi$
$997$	−57.7101	−1.82770	−0.913849	+	0.406054i	$0.866904\pi$
$998$	86.6277	2.74215
$999$	−80.2359	−2.53855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.d.1.8	✓	141

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.d.1.8	✓	141	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.68753 q^{2} -2.90926 q^{3} +5.22280 q^{4} -1.00000 q^{5} +7.81872 q^{6} -1.56079 q^{7} -8.66137 q^{8} +5.46381 q^{9} +O(q^{10})\)	\(q-2.68753 q^{2} -2.90926 q^{3} +5.22280 q^{4} -1.00000 q^{5} +7.81872 q^{6} -1.56079 q^{7} -8.66137 q^{8} +5.46381 q^{9} +2.68753 q^{10} -5.41942 q^{11} -15.1945 q^{12} -2.54347 q^{13} +4.19467 q^{14} +2.90926 q^{15} +12.8321 q^{16} +1.41493 q^{17} -14.6841 q^{18} +6.79597 q^{19} -5.22280 q^{20} +4.54076 q^{21} +14.5648 q^{22} -1.01891 q^{23} +25.1982 q^{24} +1.00000 q^{25} +6.83563 q^{26} -7.16786 q^{27} -8.15171 q^{28} +6.60136 q^{29} -7.81872 q^{30} +9.95792 q^{31} -17.1638 q^{32} +15.7665 q^{33} -3.80266 q^{34} +1.56079 q^{35} +28.5364 q^{36} +11.1938 q^{37} -18.2643 q^{38} +7.39961 q^{39} +8.66137 q^{40} +2.60847 q^{41} -12.2034 q^{42} -0.960865 q^{43} -28.3046 q^{44} -5.46381 q^{45} +2.73834 q^{46} +2.99608 q^{47} -37.3318 q^{48} -4.56393 q^{49} -2.68753 q^{50} -4.11640 q^{51} -13.2840 q^{52} +6.33397 q^{53} +19.2638 q^{54} +5.41942 q^{55} +13.5186 q^{56} -19.7712 q^{57} -17.7413 q^{58} -1.89910 q^{59} +15.1945 q^{60} +2.14996 q^{61} -26.7622 q^{62} -8.52787 q^{63} +20.4640 q^{64} +2.54347 q^{65} -42.3730 q^{66} +11.7442 q^{67} +7.38990 q^{68} +2.96426 q^{69} -4.19467 q^{70} +13.1287 q^{71} -47.3241 q^{72} -2.17969 q^{73} -30.0837 q^{74} -2.90926 q^{75} +35.4940 q^{76} +8.45860 q^{77} -19.8866 q^{78} -0.477179 q^{79} -12.8321 q^{80} +4.46177 q^{81} -7.01034 q^{82} +0.118768 q^{83} +23.7155 q^{84} -1.41493 q^{85} +2.58235 q^{86} -19.2051 q^{87} +46.9396 q^{88} -16.2714 q^{89} +14.6841 q^{90} +3.96982 q^{91} -5.32154 q^{92} -28.9702 q^{93} -8.05205 q^{94} -6.79597 q^{95} +49.9339 q^{96} -4.98740 q^{97} +12.2657 q^{98} -29.6107 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9}+O(q^{10}) \) 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9	\( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9} + 8 q^{10} + 32 q^{11} + 17 q^{12} + 35 q^{13} + 18 q^{14} - 11 q^{15} + 188 q^{16} - 13 q^{17} - 16 q^{18} + 152 q^{19} - 158 q^{20} + 40 q^{21} + 14 q^{22} - 77 q^{23} + 69 q^{24} + 141 q^{25} + 27 q^{26} + 38 q^{27} + 67 q^{28} + 22 q^{29} - 23 q^{30} + 86 q^{31} - 65 q^{32} + 51 q^{33} + 79 q^{34} - 29 q^{35} + 191 q^{36} + 45 q^{37} - 9 q^{38} + 55 q^{39} + 21 q^{40} + 36 q^{41} + 6 q^{42} + 132 q^{43} + 74 q^{44} - 160 q^{45} + 72 q^{46} - 16 q^{47} + 22 q^{48} + 212 q^{49} - 8 q^{50} + 82 q^{51} + 106 q^{52} - 28 q^{53} + 86 q^{54} - 32 q^{55} + 27 q^{56} + 10 q^{57} + 23 q^{58} + 71 q^{59} - 17 q^{60} + 116 q^{61} - 36 q^{62} + 45 q^{63} + 237 q^{64} - 35 q^{65} + 69 q^{66} + 99 q^{67} - 7 q^{68} + 45 q^{69} - 18 q^{70} + 34 q^{71} - 53 q^{72} + 125 q^{73} + 50 q^{74} + 11 q^{75} + 271 q^{76} - 31 q^{77} + 2 q^{78} + 101 q^{79} - 188 q^{80} + 221 q^{81} + 67 q^{82} + 67 q^{83} + 141 q^{84} + 13 q^{85} + 48 q^{86} - 21 q^{87} + 71 q^{88} + 79 q^{89} + 16 q^{90} + 228 q^{91} - 198 q^{92} - 12 q^{93} + 114 q^{94} - 152 q^{95} + 129 q^{96} + 98 q^{97} - 31 q^{98} + 195 q^{99}+O(q^{100}) \) 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9 + 8 * q^10 + 32 * q^11 + 17 * q^12 + 35 * q^13 + 18 * q^14 - 11 * q^15 + 188 * q^16 - 13 * q^17 - 16 * q^18 + 152 * q^19 - 158 * q^20 + 40 * q^21 + 14 * q^22 - 77 * q^23 + 69 * q^24 + 141 * q^25 + 27 * q^26 + 38 * q^27 + 67 * q^28 + 22 * q^29 - 23 * q^30 + 86 * q^31 - 65 * q^32 + 51 * q^33 + 79 * q^34 - 29 * q^35 + 191 * q^36 + 45 * q^37 - 9 * q^38 + 55 * q^39 + 21 * q^40 + 36 * q^41 + 6 * q^42 + 132 * q^43 + 74 * q^44 - 160 * q^45 + 72 * q^46 - 16 * q^47 + 22 * q^48 + 212 * q^49 - 8 * q^50 + 82 * q^51 + 106 * q^52 - 28 * q^53 + 86 * q^54 - 32 * q^55 + 27 * q^56 + 10 * q^57 + 23 * q^58 + 71 * q^59 - 17 * q^60 + 116 * q^61 - 36 * q^62 + 45 * q^63 + 237 * q^64 - 35 * q^65 + 69 * q^66 + 99 * q^67 - 7 * q^68 + 45 * q^69 - 18 * q^70 + 34 * q^71 - 53 * q^72 + 125 * q^73 + 50 * q^74 + 11 * q^75 + 271 * q^76 - 31 * q^77 + 2 * q^78 + 101 * q^79 - 188 * q^80 + 221 * q^81 + 67 * q^82 + 67 * q^83 + 141 * q^84 + 13 * q^85 + 48 * q^86 - 21 * q^87 + 71 * q^88 + 79 * q^89 + 16 * q^90 + 228 * q^91 - 198 * q^92 - 12 * q^93 + 114 * q^94 - 152 * q^95 + 129 * q^96 + 98 * q^97 - 31 * q^98 + 195 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more