LMFDB - Embedded newform 8005.2.a.e.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8005,2,Mod(1,8005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8005, 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("8005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8005 = 5 \cdot 1601 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8005.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:	$63.9202468180$
Analytic rank:	$1$
Dimension:	$126$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.18184 q^{2} -0.884770 q^{3} +2.76044 q^{4} +1.00000 q^{5} +1.93043 q^{6} +1.24169 q^{7} -1.65916 q^{8} -2.21718 q^{9} +O(q^{10})$	\(q-2.18184 q^{2} -0.884770 q^{3} +2.76044 q^{4} +1.00000 q^{5} +1.93043 q^{6} +1.24169 q^{7} -1.65916 q^{8} -2.21718 q^{9} -2.18184 q^{10} +3.52586 q^{11} -2.44236 q^{12} -3.73784 q^{13} -2.70918 q^{14} -0.884770 q^{15} -1.90085 q^{16} +0.0581484 q^{17} +4.83755 q^{18} -0.258911 q^{19} +2.76044 q^{20} -1.09861 q^{21} -7.69288 q^{22} -0.359415 q^{23} +1.46798 q^{24} +1.00000 q^{25} +8.15538 q^{26} +4.61601 q^{27} +3.42762 q^{28} +7.03323 q^{29} +1.93043 q^{30} -5.98642 q^{31} +7.46568 q^{32} -3.11958 q^{33} -0.126871 q^{34} +1.24169 q^{35} -6.12040 q^{36} -10.8394 q^{37} +0.564903 q^{38} +3.30713 q^{39} -1.65916 q^{40} -0.959332 q^{41} +2.39700 q^{42} +0.822082 q^{43} +9.73294 q^{44} -2.21718 q^{45} +0.784188 q^{46} +0.240871 q^{47} +1.68181 q^{48} -5.45820 q^{49} -2.18184 q^{50} -0.0514480 q^{51} -10.3181 q^{52} +5.49046 q^{53} -10.0714 q^{54} +3.52586 q^{55} -2.06017 q^{56} +0.229077 q^{57} -15.3454 q^{58} +4.69641 q^{59} -2.44236 q^{60} +9.25912 q^{61} +13.0614 q^{62} -2.75306 q^{63} -12.4872 q^{64} -3.73784 q^{65} +6.80643 q^{66} +2.06425 q^{67} +0.160515 q^{68} +0.318000 q^{69} -2.70918 q^{70} +4.30967 q^{71} +3.67867 q^{72} -5.75346 q^{73} +23.6499 q^{74} -0.884770 q^{75} -0.714708 q^{76} +4.37804 q^{77} -7.21563 q^{78} +7.06502 q^{79} -1.90085 q^{80} +2.56744 q^{81} +2.09311 q^{82} +9.46945 q^{83} -3.03266 q^{84} +0.0581484 q^{85} -1.79365 q^{86} -6.22279 q^{87} -5.84999 q^{88} -6.16736 q^{89} +4.83755 q^{90} -4.64125 q^{91} -0.992144 q^{92} +5.29660 q^{93} -0.525542 q^{94} -0.258911 q^{95} -6.60541 q^{96} +3.84003 q^{97} +11.9089 q^{98} -7.81748 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 126 q - 15 q^{2} - 46 q^{3} + 119 q^{4} + 126 q^{5} - 10 q^{6} - 60 q^{7} - 57 q^{8} + 112 q^{9}+O(q^{10}) $ 126 * q - 15 * q^2 - 46 * q^3 + 119 * q^4 + 126 * q^5 - 10 * q^6 - 60 * q^7 - 57 * q^8 + 112 * q^9	\( 126 q - 15 q^{2} - 46 q^{3} + 119 q^{4} + 126 q^{5} - 10 q^{6} - 60 q^{7} - 57 q^{8} + 112 q^{9} - 15 q^{10} - 35 q^{11} - 86 q^{12} - 36 q^{13} - 31 q^{14} - 46 q^{15} + 101 q^{16} - 62 q^{17} - 45 q^{18} - 29 q^{19} + 119 q^{20} - 16 q^{21} - 67 q^{22} - 107 q^{23} - 9 q^{24} + 126 q^{25} - 53 q^{26} - 181 q^{27} - 100 q^{28} - 39 q^{29} - 10 q^{30} - 29 q^{31} - 99 q^{32} - 72 q^{33} - 18 q^{34} - 60 q^{35} + 93 q^{36} - 72 q^{37} - 93 q^{38} - 8 q^{39} - 57 q^{40} - 28 q^{41} - 22 q^{42} - 103 q^{43} - 47 q^{44} + 112 q^{45} + q^{46} - 130 q^{47} - 134 q^{48} + 116 q^{49} - 15 q^{50} - 46 q^{51} - 117 q^{52} - 103 q^{53} + 11 q^{54} - 35 q^{55} - 84 q^{56} - 70 q^{57} - 77 q^{58} - 219 q^{59} - 86 q^{60} - 4 q^{61} - 77 q^{62} - 145 q^{63} + 51 q^{64} - 36 q^{65} + 16 q^{66} - 150 q^{67} - 130 q^{68} - 21 q^{69} - 31 q^{70} - 92 q^{71} - 115 q^{72} - 79 q^{73} - 57 q^{74} - 46 q^{75} - 38 q^{76} - 75 q^{77} - 23 q^{78} - 20 q^{79} + 101 q^{80} + 142 q^{81} - 63 q^{82} - 243 q^{83} - 2 q^{84} - 62 q^{85} - 14 q^{86} - 107 q^{87} - 121 q^{88} - 84 q^{89} - 45 q^{90} - 82 q^{91} - 228 q^{92} - 149 q^{93} - 29 q^{95} + 38 q^{96} - 85 q^{97} - 48 q^{98} - 33 q^{99}+O(q^{100}) \) 126 * q - 15 * q^2 - 46 * q^3 + 119 * q^4 + 126 * q^5 - 10 * q^6 - 60 * q^7 - 57 * q^8 + 112 * q^9 - 15 * q^10 - 35 * q^11 - 86 * q^12 - 36 * q^13 - 31 * q^14 - 46 * q^15 + 101 * q^16 - 62 * q^17 - 45 * q^18 - 29 * q^19 + 119 * q^20 - 16 * q^21 - 67 * q^22 - 107 * q^23 - 9 * q^24 + 126 * q^25 - 53 * q^26 - 181 * q^27 - 100 * q^28 - 39 * q^29 - 10 * q^30 - 29 * q^31 - 99 * q^32 - 72 * q^33 - 18 * q^34 - 60 * q^35 + 93 * q^36 - 72 * q^37 - 93 * q^38 - 8 * q^39 - 57 * q^40 - 28 * q^41 - 22 * q^42 - 103 * q^43 - 47 * q^44 + 112 * q^45 + q^46 - 130 * q^47 - 134 * q^48 + 116 * q^49 - 15 * q^50 - 46 * q^51 - 117 * q^52 - 103 * q^53 + 11 * q^54 - 35 * q^55 - 84 * q^56 - 70 * q^57 - 77 * q^58 - 219 * q^59 - 86 * q^60 - 4 * q^61 - 77 * q^62 - 145 * q^63 + 51 * q^64 - 36 * q^65 + 16 * q^66 - 150 * q^67 - 130 * q^68 - 21 * q^69 - 31 * q^70 - 92 * q^71 - 115 * q^72 - 79 * q^73 - 57 * q^74 - 46 * q^75 - 38 * q^76 - 75 * q^77 - 23 * q^78 - 20 * q^79 + 101 * q^80 + 142 * q^81 - 63 * q^82 - 243 * q^83 - 2 * q^84 - 62 * q^85 - 14 * q^86 - 107 * q^87 - 121 * q^88 - 84 * q^89 - 45 * q^90 - 82 * q^91 - 228 * q^92 - 149 * q^93 - 29 * q^95 + 38 * q^96 - 85 * q^97 - 48 * q^98 - 33 * 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 $

$2$	−2.18184	−1.54280	−0.771398	−	0.636353i	$-0.780442\pi$
$2$	−2.18184	−1.54280	−0.771398	+	0.636353i	$0.780442\pi$
$3$	−0.884770	−0.510822	−0.255411	−	0.966833i	$-0.582211\pi$
$3$	−0.884770	−0.510822	−0.255411	+	0.966833i	$0.582211\pi$
$4$	2.76044	1.38022
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.93043	0.788094
$7$	1.24169	0.469316	0.234658	−	0.972078i	$-0.424603\pi$
$7$	1.24169	0.469316	0.234658	+	0.972078i	$0.424603\pi$
$8$	−1.65916	−0.586603
$9$	−2.21718	−0.739061
$10$	−2.18184	−0.689960
$11$	3.52586	1.06309	0.531544	−	0.847031i	$-0.321612\pi$
$11$	3.52586	1.06309	0.531544	+	0.847031i	$0.321612\pi$
$12$	−2.44236	−0.705047
$13$	−3.73784	−1.03669	−0.518345	−	0.855172i	$-0.673452\pi$
$13$	−3.73784	−1.03669	−0.518345	+	0.855172i	$0.673452\pi$
$14$	−2.70918	−0.724059
$15$	−0.884770	−0.228447
$16$	−1.90085	−0.475211
$17$	0.0581484	0.0141031	0.00705153	−	0.999975i	$-0.497755\pi$
$17$	0.0581484	0.0141031	0.00705153	+	0.999975i	$0.497755\pi$
$18$	4.83755	1.14022
$19$	−0.258911	−0.0593982	−0.0296991	−	0.999559i	$-0.509455\pi$
$19$	−0.258911	−0.0593982	−0.0296991	+	0.999559i	$0.509455\pi$
$20$	2.76044	0.617254
$21$	−1.09861	−0.239737
$22$	−7.69288	−1.64013
$23$	−0.359415	−0.0749432	−0.0374716	−	0.999298i	$-0.511930\pi$
$23$	−0.359415	−0.0749432	−0.0374716	+	0.999298i	$0.511930\pi$
$24$	1.46798	0.299650
$25$	1.00000	0.200000
$26$	8.15538	1.59940
$27$	4.61601	0.888351
$28$	3.42762	0.647759
$29$	7.03323	1.30604	0.653019	−	0.757342i	$-0.273502\pi$
$29$	7.03323	1.30604	0.653019	+	0.757342i	$0.273502\pi$
$30$	1.93043	0.352447
$31$	−5.98642	−1.07519	−0.537596	−	0.843203i	$-0.680667\pi$
$31$	−5.98642	−1.07519	−0.537596	+	0.843203i	$0.680667\pi$
$32$	7.46568	1.31976
$33$	−3.11958	−0.543049
$34$	−0.126871	−0.0217582
$35$	1.24169	0.209884
$36$	−6.12040	−1.02007
$37$	−10.8394	−1.78199	−0.890994	−	0.454014i	$-0.849992\pi$
$37$	−10.8394	−1.78199	−0.890994	+	0.454014i	$0.849992\pi$
$38$	0.564903	0.0916394
$39$	3.30713	0.529564
$40$	−1.65916	−0.262337
$41$	−0.959332	−0.149822	−0.0749112	−	0.997190i	$-0.523867\pi$
$41$	−0.959332	−0.149822	−0.0749112	+	0.997190i	$0.523867\pi$
$42$	2.39700	0.369865
$43$	0.822082	0.125366	0.0626832	−	0.998033i	$-0.480034\pi$
$43$	0.822082	0.125366	0.0626832	+	0.998033i	$0.480034\pi$
$44$	9.73294	1.46730
$45$	−2.21718	−0.330518
$46$	0.784188	0.115622
$47$	0.240871	0.0351346	0.0175673	−	0.999846i	$-0.494408\pi$
$47$	0.240871	0.0351346	0.0175673	+	0.999846i	$0.494408\pi$
$48$	1.68181	0.242748
$49$	−5.45820	−0.779743
$50$	−2.18184	−0.308559
$51$	−0.0514480	−0.00720416
$52$	−10.3181	−1.43086
$53$	5.49046	0.754172	0.377086	−	0.926178i	$-0.376926\pi$
$53$	5.49046	0.754172	0.377086	+	0.926178i	$0.376926\pi$
$54$	−10.0714	−1.37054
$55$	3.52586	0.475427
$56$	−2.06017	−0.275302
$57$	0.229077	0.0303419
$58$	−15.3454	−2.01495
$59$	4.69641	0.611421	0.305710	−	0.952125i	$-0.401106\pi$
$59$	4.69641	0.611421	0.305710	+	0.952125i	$0.401106\pi$
$60$	−2.44236	−0.315307
$61$	9.25912	1.18551	0.592755	−	0.805383i	$-0.298040\pi$
$61$	9.25912	1.18551	0.592755	+	0.805383i	$0.298040\pi$
$62$	13.0614	1.65880
$63$	−2.75306	−0.346853
$64$	−12.4872	−1.56091
$65$	−3.73784	−0.463622
$66$	6.80643	0.837814
$67$	2.06425	0.252189	0.126094	−	0.992018i	$-0.459756\pi$
$67$	2.06425	0.252189	0.126094	+	0.992018i	$0.459756\pi$
$68$	0.160515	0.0194653
$69$	0.318000	0.0382827
$70$	−2.70918	−0.323809
$71$	4.30967	0.511463	0.255732	−	0.966748i	$-0.417684\pi$
$71$	4.30967	0.511463	0.255732	+	0.966748i	$0.417684\pi$
$72$	3.67867	0.433535
$73$	−5.75346	−0.673392	−0.336696	−	0.941613i	$-0.609309\pi$
$73$	−5.75346	−0.673392	−0.336696	+	0.941613i	$0.609309\pi$
$74$	23.6499	2.74925
$75$	−0.884770	−0.102164
$76$	−0.714708	−0.0819827
$77$	4.37804	0.498924
$78$	−7.21563	−0.817010
$79$	7.06502	0.794877	0.397439	−	0.917629i	$-0.369899\pi$
$79$	7.06502	0.794877	0.397439	+	0.917629i	$0.369899\pi$
$80$	−1.90085	−0.212521
$81$	2.56744	0.285272
$82$	2.09311	0.231146
$83$	9.46945	1.03941	0.519704	−	0.854347i	$-0.326042\pi$
$83$	9.46945	1.03941	0.519704	+	0.854347i	$0.326042\pi$
$84$	−3.03266	−0.330890
$85$	0.0581484	0.00630708
$86$	−1.79365	−0.193415
$87$	−6.22279	−0.667153
$88$	−5.84999	−0.623611
$89$	−6.16736	−0.653739	−0.326869	−	0.945070i	$-0.605994\pi$
$89$	−6.16736	−0.653739	−0.326869	+	0.945070i	$0.605994\pi$
$90$	4.83755	0.509922
$91$	−4.64125	−0.486535
$92$	−0.992144	−0.103438
$93$	5.29660	0.549232
$94$	−0.525542	−0.0542055
$95$	−0.258911	−0.0265637
$96$	−6.60541	−0.674161
$97$	3.84003	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$97$	3.84003	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$98$	11.9089	1.20298
$99$	−7.81748	−0.785687
$100$	2.76044	0.276044
$101$	9.52787	0.948059	0.474029	−	0.880509i	$-0.342799\pi$
$101$	9.52787	0.948059	0.474029	+	0.880509i	$0.342799\pi$
$102$	0.112251	0.0111145
$103$	0.324964	0.0320197	0.0160098	−	0.999872i	$-0.494904\pi$
$103$	0.324964	0.0320197	0.0160098	+	0.999872i	$0.494904\pi$
$104$	6.20169	0.608126
$105$	−1.09861	−0.107214
$106$	−11.9793	−1.16353
$107$	−17.8670	−1.72726	−0.863632	−	0.504123i	$-0.831816\pi$
$107$	−17.8670	−1.72726	−0.863632	+	0.504123i	$0.831816\pi$
$108$	12.7422	1.22612
$109$	−3.68816	−0.353261	−0.176631	−	0.984277i	$-0.556520\pi$
$109$	−3.68816	−0.353261	−0.176631	+	0.984277i	$0.556520\pi$
$110$	−7.69288	−0.733488
$111$	9.59039	0.910279
$112$	−2.36027	−0.223024
$113$	−18.4758	−1.73806	−0.869030	−	0.494759i	$-0.835256\pi$
$113$	−18.4758	−1.73806	−0.869030	+	0.494759i	$0.835256\pi$
$114$	−0.499809	−0.0468114
$115$	−0.359415	−0.0335156
$116$	19.4148	1.80262
$117$	8.28747	0.766177
$118$	−10.2468	−0.943298
$119$	0.0722025	0.00661879
$120$	1.46798	0.134008
$121$	1.43172	0.130156
$122$	−20.2020	−1.82900
$123$	0.848788	0.0765326
$124$	−16.5251	−1.48400
$125$	1.00000	0.0894427
$126$	6.00675	0.535123
$127$	−17.7127	−1.57175	−0.785873	−	0.618388i	$-0.787786\pi$
$127$	−17.7127	−1.57175	−0.785873	+	0.618388i	$0.787786\pi$
$128$	12.3139	1.08840
$129$	−0.727353	−0.0640399
$130$	8.15538	0.715274
$131$	−19.3051	−1.68670	−0.843348	−	0.537367i	$-0.819419\pi$
$131$	−19.3051	−1.68670	−0.843348	+	0.537367i	$0.819419\pi$
$132$	−8.61141	−0.749527
$133$	−0.321488	−0.0278765
$134$	−4.50388	−0.389076
$135$	4.61601	0.397282
$136$	−0.0964778	−0.00827290
$137$	−18.0050	−1.53827	−0.769135	−	0.639086i	$-0.779313\pi$
$137$	−18.0050	−1.53827	−0.769135	+	0.639086i	$0.779313\pi$
$138$	−0.693825	−0.0590623
$139$	9.11021	0.772718	0.386359	−	0.922349i	$-0.373733\pi$
$139$	9.11021	0.772718	0.386359	+	0.922349i	$0.373733\pi$
$140$	3.42762	0.289687
$141$	−0.213115	−0.0179475
$142$	−9.40302	−0.789084
$143$	−13.1791	−1.10209
$144$	4.21452	0.351210
$145$	7.03323	0.584078
$146$	12.5532	1.03891
$147$	4.82925	0.398310
$148$	−29.9216	−2.45954
$149$	−9.11424	−0.746668	−0.373334	−	0.927697i	$-0.621785\pi$
$149$	−9.11424	−0.746668	−0.373334	+	0.927697i	$0.621785\pi$
$150$	1.93043	0.157619
$151$	4.03437	0.328312	0.164156	−	0.986434i	$-0.447510\pi$
$151$	4.03437	0.328312	0.164156	+	0.986434i	$0.447510\pi$
$152$	0.429576	0.0348432
$153$	−0.128926	−0.0104230
$154$	−9.55220	−0.769738
$155$	−5.98642	−0.480840
$156$	9.12913	0.730915
$157$	3.63396	0.290021	0.145011	−	0.989430i	$-0.453678\pi$
$157$	3.63396	0.290021	0.145011	+	0.989430i	$0.453678\pi$
$158$	−15.4148	−1.22633
$159$	−4.85779	−0.385248
$160$	7.46568	0.590214
$161$	−0.446283	−0.0351720
$162$	−5.60176	−0.440116
$163$	21.8493	1.71137	0.855686	−	0.517495i	$-0.173135\pi$
$163$	21.8493	1.71137	0.855686	+	0.517495i	$0.173135\pi$
$164$	−2.64818	−0.206788
$165$	−3.11958	−0.242859
$166$	−20.6609	−1.60359
$167$	−14.0749	−1.08915	−0.544573	−	0.838713i	$-0.683308\pi$
$167$	−14.0749	−1.08915	−0.544573	+	0.838713i	$0.683308\pi$
$168$	1.82278	0.140630
$169$	0.971439	0.0747260
$170$	−0.126871	−0.00973054
$171$	0.574053	0.0438989
$172$	2.26931	0.173033
$173$	−1.78365	−0.135608	−0.0678040	−	0.997699i	$-0.521599\pi$
$173$	−1.78365	−0.135608	−0.0678040	+	0.997699i	$0.521599\pi$
$174$	13.5772	1.02928
$175$	1.24169	0.0938632
$176$	−6.70212	−0.505192
$177$	−4.15524	−0.312327
$178$	13.4562	1.00859
$179$	19.1943	1.43465	0.717324	−	0.696739i	$-0.245367\pi$
$179$	19.1943	1.43465	0.717324	+	0.696739i	$0.245367\pi$
$180$	−6.12040	−0.456188
$181$	−4.05289	−0.301249	−0.150625	−	0.988591i	$-0.548128\pi$
$181$	−4.05289	−0.301249	−0.150625	+	0.988591i	$0.548128\pi$
$182$	10.1265	0.750624
$183$	−8.19219	−0.605584
$184$	0.596329	0.0439619
$185$	−10.8394	−0.796930
$186$	−11.5564	−0.847353
$187$	0.205023	0.0149928
$188$	0.664910	0.0484935
$189$	5.73166	0.416917
$190$	0.564903	0.0409824
$191$	7.80752	0.564932	0.282466	−	0.959277i	$-0.408847\pi$
$191$	7.80752	0.564932	0.282466	+	0.959277i	$0.408847\pi$
$192$	11.0483	0.797345
$193$	3.45241	0.248510	0.124255	−	0.992250i	$-0.460346\pi$
$193$	3.45241	0.248510	0.124255	+	0.992250i	$0.460346\pi$
$194$	−8.37834	−0.601530
$195$	3.30713	0.236828
$196$	−15.0670	−1.07622
$197$	−21.1791	−1.50895	−0.754474	−	0.656330i	$-0.772108\pi$
$197$	−21.1791	−1.50895	−0.754474	+	0.656330i	$0.772108\pi$
$198$	17.0565	1.21215
$199$	−9.23301	−0.654511	−0.327255	−	0.944936i	$-0.606124\pi$
$199$	−9.23301	−0.654511	−0.327255	+	0.944936i	$0.606124\pi$
$200$	−1.65916	−0.117321
$201$	−1.82639	−0.128824
$202$	−20.7883	−1.46266
$203$	8.73311	0.612944
$204$	−0.142019	−0.00994333
$205$	−0.959332	−0.0670026
$206$	−0.709021	−0.0493999
$207$	0.796889	0.0553876
$208$	7.10505	0.492647
$209$	−0.912885	−0.0631455
$210$	2.39700	0.165409
$211$	14.8939	1.02534	0.512670	−	0.858586i	$-0.328657\pi$
$211$	14.8939	1.02534	0.512670	+	0.858586i	$0.328657\pi$
$212$	15.1561	1.04092
$213$	−3.81306	−0.261267
$214$	38.9829	2.66482
$215$	0.822082	0.0560655
$216$	−7.65871	−0.521109
$217$	−7.43329	−0.504605
$218$	8.04698	0.545010
$219$	5.09049	0.343983
$220$	9.73294	0.656195
$221$	−0.217349	−0.0146205
$222$	−20.9247	−1.40438
$223$	5.46032	0.365650	0.182825	−	0.983145i	$-0.441476\pi$
$223$	5.46032	0.365650	0.182825	+	0.983145i	$0.441476\pi$
$224$	9.27008	0.619383
$225$	−2.21718	−0.147812
$226$	40.3114	2.68147
$227$	6.19510	0.411183	0.205592	−	0.978638i	$-0.434088\pi$
$227$	6.19510	0.411183	0.205592	+	0.978638i	$0.434088\pi$
$228$	0.632352	0.0418786
$229$	19.8806	1.31375	0.656874	−	0.754000i	$-0.271878\pi$
$229$	19.8806	1.31375	0.656874	+	0.754000i	$0.271878\pi$
$230$	0.784188	0.0517078
$231$	−3.87356	−0.254861
$232$	−11.6693	−0.766126
$233$	−4.74297	−0.310723	−0.155361	−	0.987858i	$-0.549654\pi$
$233$	−4.74297	−0.310723	−0.155361	+	0.987858i	$0.549654\pi$
$234$	−18.0820	−1.18205
$235$	0.240871	0.0157127
$236$	12.9642	0.843896
$237$	−6.25092	−0.406041
$238$	−0.157535	−0.0102114
$239$	−7.58978	−0.490942	−0.245471	−	0.969404i	$-0.578943\pi$
$239$	−7.58978	−0.490942	−0.245471	+	0.969404i	$0.578943\pi$
$240$	1.68181	0.108560
$241$	−4.66898	−0.300756	−0.150378	−	0.988629i	$-0.548049\pi$
$241$	−4.66898	−0.300756	−0.150378	+	0.988629i	$0.548049\pi$
$242$	−3.12379	−0.200805
$243$	−16.1196	−1.03407
$244$	25.5593	1.63626
$245$	−5.45820	−0.348712
$246$	−1.85192	−0.118074
$247$	0.967767	0.0615775
$248$	9.93245	0.630711
$249$	−8.37828	−0.530952
$250$	−2.18184	−0.137992
$251$	25.9365	1.63710	0.818548	−	0.574439i	$-0.194780\pi$
$251$	25.9365	1.63710	0.818548	+	0.574439i	$0.194780\pi$
$252$	−7.59966	−0.478734
$253$	−1.26725	−0.0796713
$254$	38.6463	2.42488
$255$	−0.0514480	−0.00322180
$256$	−1.89244	−0.118278
$257$	−1.20303	−0.0750432	−0.0375216	−	0.999296i	$-0.511946\pi$
$257$	−1.20303	−0.0750432	−0.0375216	+	0.999296i	$0.511946\pi$
$258$	1.58697	0.0988005
$259$	−13.4592	−0.836316
$260$	−10.3181	−0.639900
$261$	−15.5940	−0.965241
$262$	42.1208	2.60223
$263$	3.82591	0.235916	0.117958	−	0.993019i	$-0.462365\pi$
$263$	3.82591	0.235916	0.117958	+	0.993019i	$0.462365\pi$
$264$	5.17589	0.318554
$265$	5.49046	0.337276
$266$	0.701436	0.0430078
$267$	5.45669	0.333944
$268$	5.69825	0.348076
$269$	−6.97312	−0.425159	−0.212579	−	0.977144i	$-0.568186\pi$
$269$	−6.97312	−0.425159	−0.212579	+	0.977144i	$0.568186\pi$
$270$	−10.0714	−0.612926
$271$	−18.1882	−1.10486	−0.552428	−	0.833561i	$-0.686298\pi$
$271$	−18.1882	−1.10486	−0.552428	+	0.833561i	$0.686298\pi$
$272$	−0.110531	−0.00670194
$273$	4.10644	0.248533
$274$	39.2841	2.37324
$275$	3.52586	0.212618
$276$	0.877819	0.0528385
$277$	8.87917	0.533498	0.266749	−	0.963766i	$-0.414051\pi$
$277$	8.87917	0.533498	0.266749	+	0.963766i	$0.414051\pi$
$278$	−19.8770	−1.19215
$279$	13.2730	0.794632
$280$	−2.06017	−0.123119
$281$	6.33978	0.378200	0.189100	−	0.981958i	$-0.439443\pi$
$281$	6.33978	0.378200	0.189100	+	0.981958i	$0.439443\pi$
$282$	0.464984	0.0276894
$283$	11.9398	0.709748	0.354874	−	0.934914i	$-0.384524\pi$
$283$	11.9398	0.709748	0.354874	+	0.934914i	$0.384524\pi$
$284$	11.8966	0.705932
$285$	0.229077	0.0135693
$286$	28.7548	1.70030
$287$	−1.19120	−0.0703141
$288$	−16.5528	−0.975381
$289$	−16.9966	−0.999801
$290$	−15.3454	−0.901113
$291$	−3.39754	−0.199167
$292$	−15.8821	−0.929429
$293$	−23.7653	−1.38839	−0.694193	−	0.719789i	$-0.744239\pi$
$293$	−23.7653	−1.38839	−0.694193	+	0.719789i	$0.744239\pi$
$294$	−10.5367	−0.614511
$295$	4.69641	0.273436
$296$	17.9844	1.04532
$297$	16.2754	0.944395
$298$	19.8859	1.15196
$299$	1.34344	0.0776929
$300$	−2.44236	−0.141009
$301$	1.02077	0.0588364
$302$	−8.80236	−0.506519
$303$	−8.42997	−0.484289
$304$	0.492150	0.0282267
$305$	9.25912	0.530176
$306$	0.281296	0.0160806
$307$	1.59412	0.0909813	0.0454907	−	0.998965i	$-0.485515\pi$
$307$	1.59412	0.0909813	0.0454907	+	0.998965i	$0.485515\pi$
$308$	12.0853	0.688625
$309$	−0.287519	−0.0163564
$310$	13.0614	0.741839
$311$	−3.50568	−0.198789	−0.0993945	−	0.995048i	$-0.531691\pi$
$311$	−3.50568	−0.198789	−0.0993945	+	0.995048i	$0.531691\pi$
$312$	−5.48707	−0.310644
$313$	7.53282	0.425780	0.212890	−	0.977076i	$-0.431712\pi$
$313$	7.53282	0.425780	0.212890	+	0.977076i	$0.431712\pi$
$314$	−7.92872	−0.447444
$315$	−2.75306	−0.155117
$316$	19.5026	1.09711
$317$	−0.910966	−0.0511650	−0.0255825	−	0.999673i	$-0.508144\pi$
$317$	−0.910966	−0.0511650	−0.0255825	+	0.999673i	$0.508144\pi$
$318$	10.5989	0.594359
$319$	24.7982	1.38843
$320$	−12.4872	−0.698058
$321$	15.8081	0.882324
$322$	0.973720	0.0542633
$323$	−0.0150553	−0.000837697 0
$324$	7.08728	0.393738
$325$	−3.73784	−0.207338
$326$	−47.6719	−2.64030
$327$	3.26317	0.180454
$328$	1.59169	0.0878864
$329$	0.299087	0.0164892
$330$	6.80643	0.374682
$331$	27.1145	1.49035	0.745175	−	0.666869i	$-0.232366\pi$
$331$	27.1145	1.49035	0.745175	+	0.666869i	$0.232366\pi$
$332$	26.1399	1.43461
$333$	24.0330	1.31700
$334$	30.7092	1.68033
$335$	2.06425	0.112782
$336$	2.08829	0.113926
$337$	−17.8137	−0.970372	−0.485186	−	0.874411i	$-0.661248\pi$
$337$	−17.8137	−0.970372	−0.485186	+	0.874411i	$0.661248\pi$
$338$	−2.11953	−0.115287
$339$	16.3469	0.887840
$340$	0.160515	0.00870517
$341$	−21.1073	−1.14302
$342$	−1.25249	−0.0677271
$343$	−15.4693	−0.835261
$344$	−1.36397	−0.0735403
$345$	0.318000	0.0171205
$346$	3.89164	0.209216
$347$	−24.7923	−1.33092	−0.665461	−	0.746432i	$-0.731765\pi$
$347$	−24.7923	−1.33092	−0.665461	+	0.746432i	$0.731765\pi$
$348$	−17.1776	−0.920818
$349$	−24.4531	−1.30895	−0.654473	−	0.756085i	$-0.727110\pi$
$349$	−24.4531	−1.30895	−0.654473	+	0.756085i	$0.727110\pi$
$350$	−2.70918	−0.144812
$351$	−17.2539	−0.920944
$352$	26.3230	1.40302
$353$	22.2909	1.18642	0.593212	−	0.805047i	$-0.297860\pi$
$353$	22.2909	1.18642	0.593212	+	0.805047i	$0.297860\pi$
$354$	9.06609	0.481857
$355$	4.30967	0.228733
$356$	−17.0246	−0.902304
$357$	−0.0638826	−0.00338103
$358$	−41.8789	−2.21337
$359$	−7.66262	−0.404417	−0.202209	−	0.979342i	$-0.564812\pi$
$359$	−7.66262	−0.404417	−0.202209	+	0.979342i	$0.564812\pi$
$360$	3.67867	0.193883
$361$	−18.9330	−0.996472
$362$	8.84278	0.464766
$363$	−1.26674	−0.0664868
$364$	−12.8119	−0.671526
$365$	−5.75346	−0.301150
$366$	17.8741	0.934293
$367$	4.98433	0.260180	0.130090	−	0.991502i	$-0.458473\pi$
$367$	4.98433	0.260180	0.130090	+	0.991502i	$0.458473\pi$
$368$	0.683193	0.0356139
$369$	2.12701	0.110728
$370$	23.6499	1.22950
$371$	6.81746	0.353945
$372$	14.6210	0.758061
$373$	−20.2870	−1.05042	−0.525210	−	0.850972i	$-0.676013\pi$
$373$	−20.2870	−1.05042	−0.525210	+	0.850972i	$0.676013\pi$
$374$	−0.447329	−0.0231308
$375$	−0.884770	−0.0456893
$376$	−0.399644	−0.0206101
$377$	−26.2891	−1.35396
$378$	−12.5056	−0.643218
$379$	−0.902302	−0.0463481	−0.0231741	−	0.999731i	$-0.507377\pi$
$379$	−0.902302	−0.0463481	−0.0231741	+	0.999731i	$0.507377\pi$
$380$	−0.714708	−0.0366638
$381$	15.6716	0.802883
$382$	−17.0348	−0.871576
$383$	15.1975	0.776558	0.388279	−	0.921542i	$-0.373070\pi$
$383$	15.1975	0.776558	0.388279	+	0.921542i	$0.373070\pi$
$384$	−10.8949	−0.555980
$385$	4.37804	0.223126
$386$	−7.53261	−0.383400
$387$	−1.82271	−0.0926533
$388$	10.6002	0.538142
$389$	−26.7835	−1.35798	−0.678989	−	0.734148i	$-0.737582\pi$
$389$	−26.7835	−1.35798	−0.678989	+	0.734148i	$0.737582\pi$
$390$	−7.21563	−0.365378
$391$	−0.0208994	−0.00105693
$392$	9.05605	0.457400
$393$	17.0806	0.861602
$394$	46.2095	2.32800
$395$	7.06502	0.355480
$396$	−21.5797	−1.08442
$397$	18.4224	0.924594	0.462297	−	0.886725i	$-0.347025\pi$
$397$	18.4224	0.924594	0.462297	+	0.886725i	$0.347025\pi$
$398$	20.1450	1.00978
$399$	0.284443	0.0142399
$400$	−1.90085	−0.0950423
$401$	20.9628	1.04683	0.523416	−	0.852077i	$-0.324657\pi$
$401$	20.9628	1.04683	0.523416	+	0.852077i	$0.324657\pi$
$402$	3.98490	0.198749
$403$	22.3763	1.11464
$404$	26.3011	1.30853
$405$	2.56744	0.127577
$406$	−19.0543	−0.945648
$407$	−38.2183	−1.89441
$408$	0.0853607	0.00422598
$409$	25.2881	1.25042	0.625209	−	0.780458i	$-0.285014\pi$
$409$	25.2881	1.25042	0.625209	+	0.780458i	$0.285014\pi$
$410$	2.09311	0.103371
$411$	15.9303	0.785783
$412$	0.897045	0.0441942
$413$	5.83150	0.286949
$414$	−1.73869	−0.0854518
$415$	9.46945	0.464837
$416$	−27.9055	−1.36818
$417$	−8.06044	−0.394721
$418$	1.99177	0.0974207
$419$	16.3201	0.797288	0.398644	−	0.917106i	$-0.369481\pi$
$419$	16.3201	0.797288	0.398644	+	0.917106i	$0.369481\pi$
$420$	−3.03266	−0.147978
$421$	26.3738	1.28538	0.642689	−	0.766127i	$-0.277819\pi$
$421$	26.3738	1.28538	0.642689	+	0.766127i	$0.277819\pi$
$422$	−32.4962	−1.58189
$423$	−0.534054	−0.0259666
$424$	−9.10957	−0.442400
$425$	0.0581484	0.00282061
$426$	8.31951	0.403081
$427$	11.4970	0.556378
$428$	−49.3207	−2.38401
$429$	11.6605	0.562973
$430$	−1.79365	−0.0864977
$431$	16.7366	0.806175	0.403087	−	0.915162i	$-0.367937\pi$
$431$	16.7366	0.806175	0.403087	+	0.915162i	$0.367937\pi$
$432$	−8.77431	−0.422154
$433$	−29.7875	−1.43149	−0.715747	−	0.698360i	$-0.753914\pi$
$433$	−29.7875	−1.43149	−0.715747	+	0.698360i	$0.753914\pi$
$434$	16.2183	0.778502
$435$	−6.22279	−0.298360
$436$	−10.1809	−0.487579
$437$	0.0930565	0.00445149
$438$	−11.1066	−0.530696
$439$	−7.49400	−0.357669	−0.178835	−	0.983879i	$-0.557233\pi$
$439$	−7.49400	−0.357669	−0.178835	+	0.983879i	$0.557233\pi$
$440$	−5.84999	−0.278887
$441$	12.1018	0.576277
$442$	0.474223	0.0225565
$443$	8.27059	0.392948	0.196474	−	0.980509i	$-0.437051\pi$
$443$	8.27059	0.392948	0.196474	+	0.980509i	$0.437051\pi$
$444$	26.4737	1.25639
$445$	−6.16736	−0.292361
$446$	−11.9136	−0.564123
$447$	8.06401	0.381414
$448$	−15.5053	−0.732558
$449$	22.2195	1.04860	0.524301	−	0.851533i	$-0.324327\pi$
$449$	22.2195	1.04860	0.524301	+	0.851533i	$0.324327\pi$
$450$	4.83755	0.228044
$451$	−3.38247	−0.159274
$452$	−51.0015	−2.39891
$453$	−3.56949	−0.167709
$454$	−13.5167	−0.634372
$455$	−4.64125	−0.217585
$456$	−0.380076	−0.0177987
$457$	−13.9146	−0.650895	−0.325448	−	0.945560i	$-0.605515\pi$
$457$	−13.9146	−0.650895	−0.325448	+	0.945560i	$0.605515\pi$
$458$	−43.3764	−2.02685
$459$	0.268413	0.0125285
$460$	−0.992144	−0.0462590
$461$	36.2536	1.68850	0.844249	−	0.535951i	$-0.180047\pi$
$461$	36.2536	1.68850	0.844249	+	0.535951i	$0.180047\pi$
$462$	8.45150	0.393199
$463$	−30.7576	−1.42943	−0.714713	−	0.699418i	$-0.753443\pi$
$463$	−30.7576	−1.42943	−0.714713	+	0.699418i	$0.753443\pi$
$464$	−13.3691	−0.620644
$465$	5.29660	0.245624
$466$	10.3484	0.479382
$467$	−12.8537	−0.594797	−0.297398	−	0.954754i	$-0.596119\pi$
$467$	−12.8537	−0.594797	−0.297398	+	0.954754i	$0.596119\pi$
$468$	22.8771	1.05749
$469$	2.56317	0.118356
$470$	−0.525542	−0.0242415
$471$	−3.21521	−0.148149
$472$	−7.79212	−0.358661
$473$	2.89855	0.133275
$474$	13.6385	0.626438
$475$	−0.258911	−0.0118796
$476$	0.199311	0.00913539
$477$	−12.1733	−0.557379
$478$	16.5597	0.757424
$479$	5.11819	0.233856	0.116928	−	0.993140i	$-0.462695\pi$
$479$	5.11819	0.233856	0.116928	+	0.993140i	$0.462695\pi$
$480$	−6.60541	−0.301494
$481$	40.5160	1.84737
$482$	10.1870	0.464005
$483$	0.394858	0.0179667
$484$	3.95218	0.179645
$485$	3.84003	0.174367
$486$	35.1705	1.59537
$487$	−1.69724	−0.0769095	−0.0384547	−	0.999260i	$-0.512244\pi$
$487$	−1.69724	−0.0769095	−0.0384547	+	0.999260i	$0.512244\pi$
$488$	−15.3624	−0.695424
$489$	−19.3316	−0.874207
$490$	11.9089	0.537991
$491$	−9.82546	−0.443417	−0.221708	−	0.975113i	$-0.571163\pi$
$491$	−9.82546	−0.443417	−0.221708	+	0.975113i	$0.571163\pi$
$492$	2.34303	0.105632
$493$	0.408971	0.0184191
$494$	−2.11152	−0.0950016
$495$	−7.81748	−0.351370
$496$	11.3793	0.510943
$497$	5.35128	0.240038
$498$	18.2801	0.819151
$499$	−22.0443	−0.986838	−0.493419	−	0.869792i	$-0.664253\pi$
$499$	−22.0443	−0.986838	−0.493419	+	0.869792i	$0.664253\pi$
$500$	2.76044	0.123451
$501$	12.4530	0.556360
$502$	−56.5893	−2.52570
$503$	−3.64798	−0.162655	−0.0813276	−	0.996687i	$-0.525916\pi$
$503$	−3.64798	−0.162655	−0.0813276	+	0.996687i	$0.525916\pi$
$504$	4.56778	0.203465
$505$	9.52787	0.423985
$506$	2.76494	0.122917
$507$	−0.859500	−0.0381717
$508$	−48.8948	−2.16936
$509$	−23.1616	−1.02662	−0.513311	−	0.858203i	$-0.671581\pi$
$509$	−23.1616	−1.02662	−0.513311	+	0.858203i	$0.671581\pi$
$510$	0.112251	0.00497058
$511$	−7.14403	−0.316033
$512$	−20.4987	−0.905924
$513$	−1.19513	−0.0527665
$514$	2.62483	0.115776
$515$	0.324964	0.0143196
$516$	−2.00782	−0.0883892
$517$	0.849277	0.0373512
$518$	29.3659	1.29026
$519$	1.57812	0.0692716
$520$	6.20169	0.271962
$521$	−9.10628	−0.398953	−0.199477	−	0.979903i	$-0.563924\pi$
$521$	−9.10628	−0.398953	−0.199477	+	0.979903i	$0.563924\pi$
$522$	34.0236	1.48917
$523$	−10.4141	−0.455375	−0.227688	−	0.973734i	$-0.573116\pi$
$523$	−10.4141	−0.455375	−0.227688	+	0.973734i	$0.573116\pi$
$524$	−53.2907	−2.32801
$525$	−1.09861	−0.0479474
$526$	−8.34753	−0.363970
$527$	−0.348101	−0.0151635
$528$	5.92984	0.258063
$529$	−22.8708	−0.994384
$530$	−11.9793	−0.520349
$531$	−10.4128	−0.451877
$532$	−0.887448	−0.0384758
$533$	3.58583	0.155319
$534$	−11.9057	−0.515208
$535$	−17.8670	−0.772456
$536$	−3.42494	−0.147935
$537$	−16.9825	−0.732850
$538$	15.2143	0.655933
$539$	−19.2449	−0.828935
$540$	12.7422	0.548338
$541$	−41.8460	−1.79910	−0.899549	−	0.436819i	$-0.856105\pi$
$541$	−41.8460	−1.79910	−0.899549	+	0.436819i	$0.856105\pi$
$542$	39.6838	1.70457
$543$	3.58588	0.153885
$544$	0.434117	0.0186126
$545$	−3.68816	−0.157983
$546$	−8.95960	−0.383436
$547$	−18.8314	−0.805173	−0.402587	−	0.915382i	$-0.631889\pi$
$547$	−18.8314	−0.805173	−0.402587	+	0.915382i	$0.631889\pi$
$548$	−49.7017	−2.12315
$549$	−20.5292	−0.876163
$550$	−7.69288	−0.328026
$551$	−1.82098	−0.0775763
$552$	−0.527614	−0.0224567
$553$	8.77259	0.373048
$554$	−19.3730	−0.823078
$555$	9.59039	0.407089
$556$	25.1482	1.06652
$557$	30.6406	1.29829	0.649143	−	0.760666i	$-0.275128\pi$
$557$	30.6406	1.29829	0.649143	+	0.760666i	$0.275128\pi$
$558$	−28.9596	−1.22596
$559$	−3.07281	−0.129966
$560$	−2.36027	−0.0997395
$561$	−0.181399	−0.00765865
$562$	−13.8324	−0.583485
$563$	−7.26980	−0.306386	−0.153193	−	0.988196i	$-0.548956\pi$
$563$	−7.26980	−0.306386	−0.153193	+	0.988196i	$0.548956\pi$
$564$	−0.588292	−0.0247716
$565$	−18.4758	−0.777284
$566$	−26.0508	−1.09500
$567$	3.18798	0.133882
$568$	−7.15044	−0.300026
$569$	−11.0492	−0.463206	−0.231603	−	0.972810i	$-0.574397\pi$
$569$	−11.0492	−0.463206	−0.231603	+	0.972810i	$0.574397\pi$
$570$	−0.499809	−0.0209347
$571$	23.2929	0.974779	0.487390	−	0.873185i	$-0.337949\pi$
$571$	23.2929	0.974779	0.487390	+	0.873185i	$0.337949\pi$
$572$	−36.3802	−1.52113
$573$	−6.90786	−0.288580
$574$	2.59900	0.108480
$575$	−0.359415	−0.0149886
$576$	27.6865	1.15360
$577$	−1.89494	−0.0788876	−0.0394438	−	0.999222i	$-0.512559\pi$
$577$	−1.89494	−0.0788876	−0.0394438	+	0.999222i	$0.512559\pi$
$578$	37.0840	1.54249
$579$	−3.05458	−0.126944
$580$	19.4148	0.806156
$581$	11.7581	0.487810
$582$	7.41291	0.307275
$583$	19.3586	0.801752
$584$	9.54594	0.395014
$585$	8.28747	0.342645
$586$	51.8523	2.14200
$587$	−14.5219	−0.599384	−0.299692	−	0.954036i	$-0.596884\pi$
$587$	−14.5219	−0.599384	−0.299692	+	0.954036i	$0.596884\pi$
$588$	13.3309	0.549755
$589$	1.54995	0.0638645
$590$	−10.2468	−0.421856
$591$	18.7386	0.770804
$592$	20.6041	0.846821
$593$	−9.83448	−0.403854	−0.201927	−	0.979401i	$-0.564720\pi$
$593$	−9.83448	−0.403854	−0.201927	+	0.979401i	$0.564720\pi$
$594$	−35.5104	−1.45701
$595$	0.0722025	0.00296001
$596$	−25.1593	−1.03057
$597$	8.16909	0.334338
$598$	−2.93117	−0.119864
$599$	−22.3792	−0.914390	−0.457195	−	0.889366i	$-0.651146\pi$
$599$	−22.3792	−0.914390	−0.457195	+	0.889366i	$0.651146\pi$
$600$	1.46798	0.0599300
$601$	10.4178	0.424949	0.212474	−	0.977167i	$-0.431848\pi$
$601$	10.4178	0.424949	0.212474	+	0.977167i	$0.431848\pi$
$602$	−2.22717	−0.0907726
$603$	−4.57683	−0.186383
$604$	11.1366	0.453143
$605$	1.43172	0.0582077
$606$	18.3929	0.747160
$607$	−30.8766	−1.25324	−0.626622	−	0.779323i	$-0.715563\pi$
$607$	−30.8766	−1.25324	−0.626622	+	0.779323i	$0.715563\pi$
$608$	−1.93294	−0.0783913
$609$	−7.72679	−0.313105
$610$	−20.2020	−0.817953
$611$	−0.900336	−0.0364237
$612$	−0.355892	−0.0143861
$613$	11.3517	0.458491	0.229246	−	0.973369i	$-0.426374\pi$
$613$	11.3517	0.458491	0.229246	+	0.973369i	$0.426374\pi$
$614$	−3.47812	−0.140366
$615$	0.848788	0.0342264
$616$	−7.26389	−0.292670
$617$	0.628953	0.0253207	0.0126603	−	0.999920i	$-0.495970\pi$
$617$	0.628953	0.0253207	0.0126603	+	0.999920i	$0.495970\pi$
$618$	0.627321	0.0252345
$619$	19.5103	0.784185	0.392092	−	0.919926i	$-0.371751\pi$
$619$	19.5103	0.784185	0.392092	+	0.919926i	$0.371751\pi$
$620$	−16.5251	−0.663666
$621$	−1.65906	−0.0665759
$622$	7.64885	0.306691
$623$	−7.65797	−0.306810
$624$	−6.28634	−0.251655
$625$	1.00000	0.0400000
$626$	−16.4354	−0.656892
$627$	0.807693	0.0322561
$628$	10.0313	0.400293
$629$	−0.630295	−0.0251315
$630$	6.00675	0.239314
$631$	44.4702	1.77033	0.885165	−	0.465277i	$-0.154045\pi$
$631$	44.4702	1.77033	0.885165	+	0.465277i	$0.154045\pi$
$632$	−11.7220	−0.466277
$633$	−13.1777	−0.523766
$634$	1.98759	0.0789371
$635$	−17.7127	−0.702906
$636$	−13.4097	−0.531727
$637$	20.4019	0.808351
$638$	−54.1058	−2.14207
$639$	−9.55531	−0.378002
$640$	12.3139	0.486748
$641$	19.3075	0.762599	0.381299	−	0.924452i	$-0.375477\pi$
$641$	19.3075	0.762599	0.381299	+	0.924452i	$0.375477\pi$
$642$	−34.4909	−1.36125
$643$	−1.97845	−0.0780224	−0.0390112	−	0.999239i	$-0.512421\pi$
$643$	−1.97845	−0.0780224	−0.0390112	+	0.999239i	$0.512421\pi$
$644$	−1.23194	−0.0485452
$645$	−0.727353	−0.0286395
$646$	0.0328482	0.00129240
$647$	37.3553	1.46859	0.734295	−	0.678831i	$-0.237513\pi$
$647$	37.3553	1.46859	0.734295	+	0.678831i	$0.237513\pi$
$648$	−4.25981	−0.167341
$649$	16.5589	0.649994
$650$	8.15538	0.319880
$651$	6.57675	0.257763
$652$	60.3138	2.36207
$653$	7.10511	0.278045	0.139022	−	0.990289i	$-0.455604\pi$
$653$	7.10511	0.278045	0.139022	+	0.990289i	$0.455604\pi$
$654$	−7.11973	−0.278403
$655$	−19.3051	−0.754314
$656$	1.82354	0.0711973
$657$	12.7565	0.497677
$658$	−0.652562	−0.0254395
$659$	−22.1240	−0.861829	−0.430915	−	0.902393i	$-0.641809\pi$
$659$	−22.1240	−0.861829	−0.430915	+	0.902393i	$0.641809\pi$
$660$	−8.61141	−0.335199
$661$	−29.0298	−1.12913	−0.564565	−	0.825389i	$-0.690956\pi$
$661$	−29.0298	−1.12913	−0.564565	+	0.825389i	$0.690956\pi$
$662$	−59.1597	−2.29931
$663$	0.192304	0.00746848
$664$	−15.7114	−0.609720
$665$	−0.321488	−0.0124668
$666$	−52.4362	−2.03186
$667$	−2.52785	−0.0978787
$668$	−38.8528	−1.50326
$669$	−4.83112	−0.186782
$670$	−4.50388	−0.174000
$671$	32.6464	1.26030
$672$	−8.20189	−0.316395
$673$	−37.2832	−1.43716	−0.718582	−	0.695442i	$-0.755208\pi$
$673$	−37.2832	−1.43716	−0.718582	+	0.695442i	$0.755208\pi$
$674$	38.8666	1.49709
$675$	4.61601	0.177670
$676$	2.68160	0.103138
$677$	22.4767	0.863850	0.431925	−	0.901909i	$-0.357834\pi$
$677$	22.4767	0.863850	0.431925	+	0.901909i	$0.357834\pi$
$678$	−35.6663	−1.36976
$679$	4.76814	0.182984
$680$	−0.0964778	−0.00369976
$681$	−5.48124	−0.210042
$682$	46.0528	1.76345
$683$	−45.1142	−1.72625	−0.863124	−	0.504992i	$-0.831495\pi$
$683$	−45.1142	−1.72625	−0.863124	+	0.504992i	$0.831495\pi$
$684$	1.58464	0.0605902
$685$	−18.0050	−0.687936
$686$	33.7515	1.28864
$687$	−17.5898	−0.671091
$688$	−1.56265	−0.0595755
$689$	−20.5224	−0.781843
$690$	−0.693825	−0.0264135
$691$	5.25027	0.199730	0.0998648	−	0.995001i	$-0.468159\pi$
$691$	5.25027	0.199730	0.0998648	+	0.995001i	$0.468159\pi$
$692$	−4.92365	−0.187169
$693$	−9.70692	−0.368735
$694$	54.0930	2.05334
$695$	9.11021	0.345570
$696$	10.3246	0.391354
$697$	−0.0557836	−0.00211296
$698$	53.3529	2.01944
$699$	4.19644	0.158724
$700$	3.42762	0.129552
$701$	45.1832	1.70655	0.853273	−	0.521465i	$-0.174614\pi$
$701$	45.1832	1.70655	0.853273	+	0.521465i	$0.174614\pi$
$702$	37.6453	1.42083
$703$	2.80644	0.105847
$704$	−44.0283	−1.65938
$705$	−0.213115	−0.00802638
$706$	−48.6352	−1.83041
$707$	11.8307	0.444939
$708$	−11.4703	−0.431081
$709$	21.5824	0.810543	0.405271	−	0.914196i	$-0.367177\pi$
$709$	21.5824	0.810543	0.405271	+	0.914196i	$0.367177\pi$
$710$	−9.40302	−0.352889
$711$	−15.6644	−0.587462
$712$	10.2327	0.383485
$713$	2.15161	0.0805784
$714$	0.139382	0.00521623
$715$	−13.1791	−0.492871
$716$	52.9847	1.98013
$717$	6.71521	0.250784
$718$	16.7186	0.623934
$719$	−46.1385	−1.72068	−0.860338	−	0.509724i	$-0.829747\pi$
$719$	−46.1385	−1.72068	−0.860338	+	0.509724i	$0.829747\pi$
$720$	4.21452	0.157066
$721$	0.403506	0.0150273
$722$	41.3088	1.53735
$723$	4.13098	0.153633
$724$	−11.1878	−0.415791
$725$	7.03323	0.261208
$726$	2.76384	0.102576
$727$	−20.0440	−0.743391	−0.371696	−	0.928355i	$-0.621223\pi$
$727$	−20.0440	−0.743391	−0.371696	+	0.928355i	$0.621223\pi$
$728$	7.70059	0.285403
$729$	6.55981	0.242956
$730$	12.5532	0.464613
$731$	0.0478028	0.00176805
$732$	−22.6141	−0.835840
$733$	0.603764	0.0223005	0.0111503	−	0.999938i	$-0.496451\pi$
$733$	0.603764	0.0223005	0.0111503	+	0.999938i	$0.496451\pi$
$734$	−10.8750	−0.401405
$735$	4.82925	0.178130
$736$	−2.68328	−0.0989069
$737$	7.27828	0.268099
$738$	−4.64081	−0.170831
$739$	−25.6086	−0.942028	−0.471014	−	0.882126i	$-0.656112\pi$
$739$	−25.6086	−0.942028	−0.471014	+	0.882126i	$0.656112\pi$
$740$	−29.9216	−1.09994
$741$	−0.856251	−0.0314552
$742$	−14.8746	−0.546065
$743$	−24.8979	−0.913416	−0.456708	−	0.889617i	$-0.650971\pi$
$743$	−24.8979	−0.913416	−0.456708	+	0.889617i	$0.650971\pi$
$744$	−8.78793	−0.322181
$745$	−9.11424	−0.333920
$746$	44.2630	1.62059
$747$	−20.9955	−0.768185
$748$	0.565955	0.0206934
$749$	−22.1853	−0.810632
$750$	1.93043	0.0704893
$751$	−6.46687	−0.235980	−0.117990	−	0.993015i	$-0.537645\pi$
$751$	−6.46687	−0.235980	−0.117990	+	0.993015i	$0.537645\pi$
$752$	−0.457858	−0.0166964
$753$	−22.9478	−0.836264
$754$	57.3587	2.08888
$755$	4.03437	0.146826
$756$	15.8219	0.575438
$757$	6.80115	0.247192	0.123596	−	0.992333i	$-0.460557\pi$
$757$	6.80115	0.247192	0.123596	+	0.992333i	$0.460557\pi$
$758$	1.96868	0.0715057
$759$	1.12122	0.0406978
$760$	0.429576	0.0155823
$761$	−26.2137	−0.950245	−0.475122	−	0.879920i	$-0.657596\pi$
$761$	−26.2137	−0.950245	−0.475122	+	0.879920i	$0.657596\pi$
$762$	−34.1931	−1.23868
$763$	−4.57956	−0.165791
$764$	21.5522	0.779732
$765$	−0.128926	−0.00466132
$766$	−33.1587	−1.19807
$767$	−17.5544	−0.633854
$768$	1.67437	0.0604188
$769$	−20.5222	−0.740051	−0.370026	−	0.929022i	$-0.620651\pi$
$769$	−20.5222	−0.740051	−0.370026	+	0.929022i	$0.620651\pi$
$770$	−9.55220	−0.344237
$771$	1.06441	0.0383337
$772$	9.53017	0.342998
$773$	−44.6756	−1.60687	−0.803434	−	0.595394i	$-0.796996\pi$
$773$	−44.6756	−1.60687	−0.803434	+	0.595394i	$0.796996\pi$
$774$	3.97686	0.142945
$775$	−5.98642	−0.215038
$776$	−6.37124	−0.228714
$777$	11.9083	0.427208
$778$	58.4374	2.09508
$779$	0.248381	0.00889919
$780$	9.12913	0.326875
$781$	15.1953	0.543730
$782$	0.0455993	0.00163063
$783$	32.4654	1.16022
$784$	10.3752	0.370543
$785$	3.63396	0.129701
$786$	−37.2672	−1.32928
$787$	−45.6116	−1.62588	−0.812939	−	0.582349i	$-0.802134\pi$
$787$	−45.6116	−1.62588	−0.812939	+	0.582349i	$0.802134\pi$
$788$	−58.4636	−2.08268
$789$	−3.38505	−0.120511
$790$	−15.4148	−0.548433
$791$	−22.9413	−0.815699
$792$	12.9705	0.460886
$793$	−34.6091	−1.22901
$794$	−40.1948	−1.42646
$795$	−4.85779	−0.172288
$796$	−25.4872	−0.903369
$797$	10.2810	0.364173	0.182086	−	0.983283i	$-0.441715\pi$
$797$	10.2810	0.364173	0.182086	+	0.983283i	$0.441715\pi$
$798$	−0.620609	−0.0219693
$799$	0.0140063	0.000495506 0
$800$	7.46568	0.263952
$801$	13.6742	0.483153
$802$	−45.7375	−1.61505
$803$	−20.2859	−0.715875
$804$	−5.04164	−0.177805
$805$	−0.446283	−0.0157294
$806$	−48.8215	−1.71966
$807$	6.16961	0.217180
$808$	−15.8083	−0.556134
$809$	−41.9413	−1.47458	−0.737288	−	0.675579i	$-0.763894\pi$
$809$	−41.9413	−1.47458	−0.737288	+	0.675579i	$0.763894\pi$
$810$	−5.60176	−0.196826
$811$	−21.7393	−0.763371	−0.381686	−	0.924292i	$-0.624656\pi$
$811$	−21.7393	−0.763371	−0.381686	+	0.924292i	$0.624656\pi$
$812$	24.1072	0.845998
$813$	16.0924	0.564385
$814$	83.3864	2.92269
$815$	21.8493	0.765349
$816$	0.0977946	0.00342350
$817$	−0.212846	−0.00744654
$818$	−55.1747	−1.92914
$819$	10.2905	0.359579
$820$	−2.64818	−0.0924784
$821$	−56.8815	−1.98518	−0.992589	−	0.121517i	$-0.961224\pi$
$821$	−56.8815	−1.98518	−0.992589	+	0.121517i	$0.961224\pi$
$822$	−34.7574	−1.21230
$823$	−3.98705	−0.138980	−0.0694900	−	0.997583i	$-0.522137\pi$
$823$	−3.98705	−0.138980	−0.0694900	+	0.997583i	$0.522137\pi$
$824$	−0.539169	−0.0187829
$825$	−3.11958	−0.108610
$826$	−12.7234	−0.442705
$827$	−32.2486	−1.12139	−0.560696	−	0.828021i	$-0.689467\pi$
$827$	−32.2486	−1.12139	−0.560696	+	0.828021i	$0.689467\pi$
$828$	2.19977	0.0764471
$829$	−14.4134	−0.500599	−0.250299	−	0.968168i	$-0.580529\pi$
$829$	−14.4134	−0.500599	−0.250299	+	0.968168i	$0.580529\pi$
$830$	−20.6609	−0.717149
$831$	−7.85602	−0.272522
$832$	46.6753	1.61818
$833$	−0.317386	−0.0109968
$834$	17.5866	0.608975
$835$	−14.0749	−0.487081
$836$	−2.51996	−0.0871548
$837$	−27.6333	−0.955147
$838$	−35.6078	−1.23005
$839$	−29.8603	−1.03089	−0.515445	−	0.856922i	$-0.672374\pi$
$839$	−29.8603	−1.03089	−0.515445	+	0.856922i	$0.672374\pi$
$840$	1.82278	0.0628918
$841$	20.4663	0.705735
$842$	−57.5434	−1.98308
$843$	−5.60925	−0.193193
$844$	41.1138	1.41519
$845$	0.971439	0.0334185
$846$	1.16522	0.0400612
$847$	1.77776	0.0610845
$848$	−10.4365	−0.358391
$849$	−10.5640	−0.362555
$850$	−0.126871	−0.00435163
$851$	3.89585	0.133548
$852$	−10.5257	−0.360606
$853$	44.4111	1.52061	0.760303	−	0.649569i	$-0.225051\pi$
$853$	44.4111	1.52061	0.760303	+	0.649569i	$0.225051\pi$
$854$	−25.0846	−0.858378
$855$	0.574053	0.0196322
$856$	29.6442	1.01322
$857$	−48.9838	−1.67326	−0.836628	−	0.547771i	$-0.815476\pi$
$857$	−48.9838	−1.67326	−0.836628	+	0.547771i	$0.815476\pi$
$858$	−25.4413	−0.868553
$859$	−11.1738	−0.381247	−0.190623	−	0.981663i	$-0.561051\pi$
$859$	−11.1738	−0.381247	−0.190623	+	0.981663i	$0.561051\pi$
$860$	2.26931	0.0773828
$861$	1.05393	0.0359180
$862$	−36.5167	−1.24376
$863$	−6.80934	−0.231793	−0.115896	−	0.993261i	$-0.536974\pi$
$863$	−6.80934	−0.231793	−0.115896	+	0.993261i	$0.536974\pi$
$864$	34.4616	1.17241
$865$	−1.78365	−0.0606458
$866$	64.9916	2.20850
$867$	15.0381	0.510720
$868$	−20.5192	−0.696466
$869$	24.9103	0.845024
$870$	13.5772	0.460309
$871$	−7.71585	−0.261442
$872$	6.11926	0.207224
$873$	−8.51405	−0.288157
$874$	−0.203035	−0.00686775
$875$	1.24169	0.0419769
$876$	14.0520	0.474773
$877$	−28.7598	−0.971149	−0.485575	−	0.874195i	$-0.661390\pi$
$877$	−28.7598	−0.971149	−0.485575	+	0.874195i	$0.661390\pi$
$878$	16.3507	0.551811
$879$	21.0269	0.709218
$880$	−6.70212	−0.225929
$881$	−37.8720	−1.27594	−0.637971	−	0.770061i	$-0.720226\pi$
$881$	−37.8720	−1.27594	−0.637971	+	0.770061i	$0.720226\pi$
$882$	−26.4043	−0.889078
$883$	−31.8223	−1.07090	−0.535452	−	0.844565i	$-0.679859\pi$
$883$	−31.8223	−1.07090	−0.535452	+	0.844565i	$0.679859\pi$
$884$	−0.599980	−0.0201795
$885$	−4.15524	−0.139677
$886$	−18.0451	−0.606238
$887$	11.9748	0.402075	0.201038	−	0.979583i	$-0.435569\pi$
$887$	11.9748	0.402075	0.201038	+	0.979583i	$0.435569\pi$
$888$	−15.9120	−0.533973
$889$	−21.9937	−0.737645
$890$	13.4562	0.451053
$891$	9.05246	0.303269
$892$	15.0729	0.504678
$893$	−0.0623640	−0.00208693
$894$	−17.5944	−0.588445
$895$	19.1943	0.641594
$896$	15.2900	0.510805
$897$	−1.18863	−0.0396872
$898$	−48.4795	−1.61778
$899$	−42.1038	−1.40424
$900$	−6.12040	−0.204013
$901$	0.319262	0.0106361
$902$	7.38003	0.245728
$903$	−0.903149	−0.0300549
$904$	30.6545	1.01955
$905$	−4.05289	−0.134723
$906$	7.78806	0.258741
$907$	−24.2767	−0.806095	−0.403048	−	0.915179i	$-0.632049\pi$
$907$	−24.2767	−0.806095	−0.403048	+	0.915179i	$0.632049\pi$
$908$	17.1012	0.567524
$909$	−21.1250	−0.700673
$910$	10.1265	0.335689
$911$	25.0009	0.828318	0.414159	−	0.910205i	$-0.364076\pi$
$911$	25.0009	0.828318	0.414159	+	0.910205i	$0.364076\pi$
$912$	−0.435439	−0.0144188
$913$	33.3880	1.10498
$914$	30.3594	1.00420
$915$	−8.19219	−0.270826
$916$	54.8793	1.81326
$917$	−23.9710	−0.791593
$918$	−0.585636	−0.0193289
$919$	8.02873	0.264843	0.132422	−	0.991193i	$-0.457725\pi$
$919$	8.02873	0.264843	0.132422	+	0.991193i	$0.457725\pi$
$920$	0.596329	0.0196604
$921$	−1.41043	−0.0464753
$922$	−79.0997	−2.60501
$923$	−16.1088	−0.530229
$924$	−10.6927	−0.351765
$925$	−10.8394	−0.356398
$926$	67.1082	2.20531
$927$	−0.720505	−0.0236645
$928$	52.5078	1.72365
$929$	−32.5595	−1.06824	−0.534121	−	0.845408i	$-0.679357\pi$
$929$	−32.5595	−1.06824	−0.534121	+	0.845408i	$0.679357\pi$
$930$	−11.5564	−0.378948
$931$	1.41319	0.0463153
$932$	−13.0927	−0.428866
$933$	3.10172	0.101546
$934$	28.0447	0.917650
$935$	0.205023	0.00670498
$936$	−13.7503	−0.449442
$937$	41.0678	1.34163	0.670814	−	0.741626i	$-0.265945\pi$
$937$	41.0678	1.34163	0.670814	+	0.741626i	$0.265945\pi$
$938$	−5.59244	−0.182599
$939$	−6.66481	−0.217498
$940$	0.664910	0.0216870
$941$	11.7776	0.383938	0.191969	−	0.981401i	$-0.438513\pi$
$941$	11.7776	0.383938	0.191969	+	0.981401i	$0.438513\pi$
$942$	7.01510	0.228564
$943$	0.344798	0.0112282
$944$	−8.92715	−0.290554
$945$	5.73166	0.186451
$946$	−6.32418	−0.205617
$947$	−1.93497	−0.0628782	−0.0314391	−	0.999506i	$-0.510009\pi$
$947$	−1.93497	−0.0628782	−0.0314391	+	0.999506i	$0.510009\pi$
$948$	−17.2553	−0.560426
$949$	21.5055	0.698098
$950$	0.564903	0.0183279
$951$	0.805995	0.0261362
$952$	−0.119796	−0.00388260
$953$	−2.85363	−0.0924383	−0.0462191	−	0.998931i	$-0.514717\pi$
$953$	−2.85363	−0.0924383	−0.0462191	+	0.998931i	$0.514717\pi$
$954$	26.5603	0.859923
$955$	7.80752	0.252645
$956$	−20.9511	−0.677608
$957$	−21.9407	−0.709242
$958$	−11.1671	−0.360792
$959$	−22.3567	−0.721935
$960$	11.0483	0.356584
$961$	4.83717	0.156038
$962$	−88.3996	−2.85012
$963$	39.6143	1.27655
$964$	−12.8885	−0.415109
$965$	3.45241	0.111137
$966$	−0.861518	−0.0277189
$967$	−45.9834	−1.47873	−0.739363	−	0.673307i	$-0.764873\pi$
$967$	−45.9834	−1.47873	−0.739363	+	0.673307i	$0.764873\pi$
$968$	−2.37546	−0.0763502
$969$	0.0133204	0.000427914 0
$970$	−8.37834	−0.269012
$971$	4.00478	0.128519	0.0642597	−	0.997933i	$-0.479531\pi$
$971$	4.00478	0.128519	0.0642597	+	0.997933i	$0.479531\pi$
$972$	−44.4973	−1.42725
$973$	11.3121	0.362649
$974$	3.70312	0.118656
$975$	3.30713	0.105913
$976$	−17.6002	−0.563368
$977$	−2.52124	−0.0806617	−0.0403308	−	0.999186i	$-0.512841\pi$
$977$	−2.52124	−0.0806617	−0.0403308	+	0.999186i	$0.512841\pi$
$978$	42.1786	1.34872
$979$	−21.7453	−0.694982
$980$	−15.0670	−0.481299
$981$	8.17732	0.261082
$982$	21.4376	0.684102
$983$	7.98592	0.254711	0.127356	−	0.991857i	$-0.459351\pi$
$983$	7.98592	0.254711	0.127356	+	0.991857i	$0.459351\pi$
$984$	−1.40828	−0.0448943
$985$	−21.1791	−0.674822
$986$	−0.892311	−0.0284170
$987$	−0.264624	−0.00842306
$988$	2.67146	0.0849906
$989$	−0.295469	−0.00939536
$990$	17.0565	0.542092
$991$	60.7512	1.92982	0.964912	−	0.262572i	$-0.0845706\pi$
$991$	60.7512	1.92982	0.964912	+	0.262572i	$0.0845706\pi$
$992$	−44.6926	−1.41899
$993$	−23.9901	−0.761304
$994$	−11.6757	−0.370329
$995$	−9.23301	−0.292706
$996$	−23.1278	−0.732831
$997$	24.7683	0.784419	0.392209	−	0.919876i	$-0.371711\pi$
$997$	24.7683	0.784419	0.392209	+	0.919876i	$0.371711\pi$
$998$	48.0972	1.52249
$999$	−50.0348	−1.58303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8005.2.a.e.1.20	✓	126

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8005.2.a.e.1.20	✓	126	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:	\( N \)	\(=\)	\( 8005 = 5 \cdot 1601 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.18184 q^{2} -0.884770 q^{3} +2.76044 q^{4} +1.00000 q^{5} +1.93043 q^{6} +1.24169 q^{7} -1.65916 q^{8} -2.21718 q^{9} +O(q^{10})\)	\(q-2.18184 q^{2} -0.884770 q^{3} +2.76044 q^{4} +1.00000 q^{5} +1.93043 q^{6} +1.24169 q^{7} -1.65916 q^{8} -2.21718 q^{9} -2.18184 q^{10} +3.52586 q^{11} -2.44236 q^{12} -3.73784 q^{13} -2.70918 q^{14} -0.884770 q^{15} -1.90085 q^{16} +0.0581484 q^{17} +4.83755 q^{18} -0.258911 q^{19} +2.76044 q^{20} -1.09861 q^{21} -7.69288 q^{22} -0.359415 q^{23} +1.46798 q^{24} +1.00000 q^{25} +8.15538 q^{26} +4.61601 q^{27} +3.42762 q^{28} +7.03323 q^{29} +1.93043 q^{30} -5.98642 q^{31} +7.46568 q^{32} -3.11958 q^{33} -0.126871 q^{34} +1.24169 q^{35} -6.12040 q^{36} -10.8394 q^{37} +0.564903 q^{38} +3.30713 q^{39} -1.65916 q^{40} -0.959332 q^{41} +2.39700 q^{42} +0.822082 q^{43} +9.73294 q^{44} -2.21718 q^{45} +0.784188 q^{46} +0.240871 q^{47} +1.68181 q^{48} -5.45820 q^{49} -2.18184 q^{50} -0.0514480 q^{51} -10.3181 q^{52} +5.49046 q^{53} -10.0714 q^{54} +3.52586 q^{55} -2.06017 q^{56} +0.229077 q^{57} -15.3454 q^{58} +4.69641 q^{59} -2.44236 q^{60} +9.25912 q^{61} +13.0614 q^{62} -2.75306 q^{63} -12.4872 q^{64} -3.73784 q^{65} +6.80643 q^{66} +2.06425 q^{67} +0.160515 q^{68} +0.318000 q^{69} -2.70918 q^{70} +4.30967 q^{71} +3.67867 q^{72} -5.75346 q^{73} +23.6499 q^{74} -0.884770 q^{75} -0.714708 q^{76} +4.37804 q^{77} -7.21563 q^{78} +7.06502 q^{79} -1.90085 q^{80} +2.56744 q^{81} +2.09311 q^{82} +9.46945 q^{83} -3.03266 q^{84} +0.0581484 q^{85} -1.79365 q^{86} -6.22279 q^{87} -5.84999 q^{88} -6.16736 q^{89} +4.83755 q^{90} -4.64125 q^{91} -0.992144 q^{92} +5.29660 q^{93} -0.525542 q^{94} -0.258911 q^{95} -6.60541 q^{96} +3.84003 q^{97} +11.9089 q^{98} -7.81748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 15 q^{2} - 46 q^{3} + 119 q^{4} + 126 q^{5} - 10 q^{6} - 60 q^{7} - 57 q^{8} + 112 q^{9}+O(q^{10}) \) 126 * q - 15 * q^2 - 46 * q^3 + 119 * q^4 + 126 * q^5 - 10 * q^6 - 60 * q^7 - 57 * q^8 + 112 * q^9	\( 126 q - 15 q^{2} - 46 q^{3} + 119 q^{4} + 126 q^{5} - 10 q^{6} - 60 q^{7} - 57 q^{8} + 112 q^{9} - 15 q^{10} - 35 q^{11} - 86 q^{12} - 36 q^{13} - 31 q^{14} - 46 q^{15} + 101 q^{16} - 62 q^{17} - 45 q^{18} - 29 q^{19} + 119 q^{20} - 16 q^{21} - 67 q^{22} - 107 q^{23} - 9 q^{24} + 126 q^{25} - 53 q^{26} - 181 q^{27} - 100 q^{28} - 39 q^{29} - 10 q^{30} - 29 q^{31} - 99 q^{32} - 72 q^{33} - 18 q^{34} - 60 q^{35} + 93 q^{36} - 72 q^{37} - 93 q^{38} - 8 q^{39} - 57 q^{40} - 28 q^{41} - 22 q^{42} - 103 q^{43} - 47 q^{44} + 112 q^{45} + q^{46} - 130 q^{47} - 134 q^{48} + 116 q^{49} - 15 q^{50} - 46 q^{51} - 117 q^{52} - 103 q^{53} + 11 q^{54} - 35 q^{55} - 84 q^{56} - 70 q^{57} - 77 q^{58} - 219 q^{59} - 86 q^{60} - 4 q^{61} - 77 q^{62} - 145 q^{63} + 51 q^{64} - 36 q^{65} + 16 q^{66} - 150 q^{67} - 130 q^{68} - 21 q^{69} - 31 q^{70} - 92 q^{71} - 115 q^{72} - 79 q^{73} - 57 q^{74} - 46 q^{75} - 38 q^{76} - 75 q^{77} - 23 q^{78} - 20 q^{79} + 101 q^{80} + 142 q^{81} - 63 q^{82} - 243 q^{83} - 2 q^{84} - 62 q^{85} - 14 q^{86} - 107 q^{87} - 121 q^{88} - 84 q^{89} - 45 q^{90} - 82 q^{91} - 228 q^{92} - 149 q^{93} - 29 q^{95} + 38 q^{96} - 85 q^{97} - 48 q^{98} - 33 q^{99}+O(q^{100}) \) 126 * q - 15 * q^2 - 46 * q^3 + 119 * q^4 + 126 * q^5 - 10 * q^6 - 60 * q^7 - 57 * q^8 + 112 * q^9 - 15 * q^10 - 35 * q^11 - 86 * q^12 - 36 * q^13 - 31 * q^14 - 46 * q^15 + 101 * q^16 - 62 * q^17 - 45 * q^18 - 29 * q^19 + 119 * q^20 - 16 * q^21 - 67 * q^22 - 107 * q^23 - 9 * q^24 + 126 * q^25 - 53 * q^26 - 181 * q^27 - 100 * q^28 - 39 * q^29 - 10 * q^30 - 29 * q^31 - 99 * q^32 - 72 * q^33 - 18 * q^34 - 60 * q^35 + 93 * q^36 - 72 * q^37 - 93 * q^38 - 8 * q^39 - 57 * q^40 - 28 * q^41 - 22 * q^42 - 103 * q^43 - 47 * q^44 + 112 * q^45 + q^46 - 130 * q^47 - 134 * q^48 + 116 * q^49 - 15 * q^50 - 46 * q^51 - 117 * q^52 - 103 * q^53 + 11 * q^54 - 35 * q^55 - 84 * q^56 - 70 * q^57 - 77 * q^58 - 219 * q^59 - 86 * q^60 - 4 * q^61 - 77 * q^62 - 145 * q^63 + 51 * q^64 - 36 * q^65 + 16 * q^66 - 150 * q^67 - 130 * q^68 - 21 * q^69 - 31 * q^70 - 92 * q^71 - 115 * q^72 - 79 * q^73 - 57 * q^74 - 46 * q^75 - 38 * q^76 - 75 * q^77 - 23 * q^78 - 20 * q^79 + 101 * q^80 + 142 * q^81 - 63 * q^82 - 243 * q^83 - 2 * q^84 - 62 * q^85 - 14 * q^86 - 107 * q^87 - 121 * q^88 - 84 * q^89 - 45 * q^90 - 82 * q^91 - 228 * q^92 - 149 * q^93 - 29 * q^95 + 38 * q^96 - 85 * q^97 - 48 * q^98 - 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more