LMFDB - Embedded newform 8015.2.a.h.1.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8015,2,Mod(1,8015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8015, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8015 = 5 \cdot 7 \cdot 229 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8015.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:	$64.0000972201$
Analytic rank:	$1$
Dimension:	$38$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
Character	$\chi$	$=$	8015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.185341 q^{2} -0.587692 q^{3} -1.96565 q^{4} +1.00000 q^{5} -0.108923 q^{6} +1.00000 q^{7} -0.734998 q^{8} -2.65462 q^{9} +O(q^{10})$	\(q+0.185341 q^{2} -0.587692 q^{3} -1.96565 q^{4} +1.00000 q^{5} -0.108923 q^{6} +1.00000 q^{7} -0.734998 q^{8} -2.65462 q^{9} +0.185341 q^{10} +2.91097 q^{11} +1.15520 q^{12} -1.78576 q^{13} +0.185341 q^{14} -0.587692 q^{15} +3.79507 q^{16} -4.66005 q^{17} -0.492010 q^{18} +5.48579 q^{19} -1.96565 q^{20} -0.587692 q^{21} +0.539523 q^{22} -5.72088 q^{23} +0.431952 q^{24} +1.00000 q^{25} -0.330974 q^{26} +3.32317 q^{27} -1.96565 q^{28} +1.64651 q^{29} -0.108923 q^{30} -3.05170 q^{31} +2.17338 q^{32} -1.71076 q^{33} -0.863699 q^{34} +1.00000 q^{35} +5.21805 q^{36} +10.5438 q^{37} +1.01674 q^{38} +1.04948 q^{39} -0.734998 q^{40} -2.77314 q^{41} -0.108923 q^{42} -7.94015 q^{43} -5.72195 q^{44} -2.65462 q^{45} -1.06031 q^{46} +5.94889 q^{47} -2.23033 q^{48} +1.00000 q^{49} +0.185341 q^{50} +2.73867 q^{51} +3.51017 q^{52} -4.83723 q^{53} +0.615921 q^{54} +2.91097 q^{55} -0.734998 q^{56} -3.22395 q^{57} +0.305166 q^{58} -1.67739 q^{59} +1.15520 q^{60} -6.33576 q^{61} -0.565605 q^{62} -2.65462 q^{63} -7.18733 q^{64} -1.78576 q^{65} -0.317073 q^{66} +12.0831 q^{67} +9.16002 q^{68} +3.36212 q^{69} +0.185341 q^{70} -2.77916 q^{71} +1.95114 q^{72} +3.24737 q^{73} +1.95419 q^{74} -0.587692 q^{75} -10.7831 q^{76} +2.91097 q^{77} +0.194511 q^{78} -10.5249 q^{79} +3.79507 q^{80} +6.01085 q^{81} -0.513977 q^{82} +7.29713 q^{83} +1.15520 q^{84} -4.66005 q^{85} -1.47164 q^{86} -0.967642 q^{87} -2.13956 q^{88} +9.19710 q^{89} -0.492010 q^{90} -1.78576 q^{91} +11.2452 q^{92} +1.79346 q^{93} +1.10257 q^{94} +5.48579 q^{95} -1.27728 q^{96} -7.33877 q^{97} +0.185341 q^{98} -7.72752 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 6 q^{2} - 9 q^{3} + 24 q^{4} + 38 q^{5} - 10 q^{6} + 38 q^{7} - 21 q^{8} + 13 q^{9}+O(q^{10}) $ 38 * q - 6 * q^2 - 9 * q^3 + 24 * q^4 + 38 * q^5 - 10 * q^6 + 38 * q^7 - 21 * q^8 + 13 * q^9	\( 38 q - 6 q^{2} - 9 q^{3} + 24 q^{4} + 38 q^{5} - 10 q^{6} + 38 q^{7} - 21 q^{8} + 13 q^{9} - 6 q^{10} - 18 q^{11} - 20 q^{12} - 25 q^{13} - 6 q^{14} - 9 q^{15} - 21 q^{17} - 7 q^{18} - 14 q^{19} + 24 q^{20} - 9 q^{21} - 11 q^{22} - 16 q^{23} - 10 q^{24} + 38 q^{25} - 21 q^{26} - 15 q^{27} + 24 q^{28} - 52 q^{29} - 10 q^{30} - 30 q^{31} - 8 q^{32} - 13 q^{33} - 9 q^{34} + 38 q^{35} - 16 q^{36} - 47 q^{37} - 10 q^{38} - 50 q^{39} - 21 q^{40} - 35 q^{41} - 10 q^{42} - 18 q^{43} - 22 q^{44} + 13 q^{45} - 17 q^{46} - 23 q^{47} - 13 q^{48} + 38 q^{49} - 6 q^{50} - 5 q^{51} - 41 q^{52} - 12 q^{53} + 35 q^{54} - 18 q^{55} - 21 q^{56} - 3 q^{57} - 16 q^{58} - 16 q^{59} - 20 q^{60} - 47 q^{61} + 14 q^{62} + 13 q^{63} - 55 q^{64} - 25 q^{65} - 6 q^{66} - 54 q^{67} + 31 q^{68} - 95 q^{69} - 6 q^{70} - 41 q^{71} - 59 q^{73} - q^{74} - 9 q^{75} - 23 q^{76} - 18 q^{77} + 19 q^{78} - 117 q^{79} - 38 q^{81} + 19 q^{82} - 21 q^{83} - 20 q^{84} - 21 q^{85} - 12 q^{86} - 14 q^{87} - 40 q^{88} - 96 q^{89} - 7 q^{90} - 25 q^{91} + 53 q^{92} - 53 q^{93} - 28 q^{94} - 14 q^{95} + 40 q^{96} - 70 q^{97} - 6 q^{98} - 52 q^{99}+O(q^{100}) \) 38 * q - 6 * q^2 - 9 * q^3 + 24 * q^4 + 38 * q^5 - 10 * q^6 + 38 * q^7 - 21 * q^8 + 13 * q^9 - 6 * q^10 - 18 * q^11 - 20 * q^12 - 25 * q^13 - 6 * q^14 - 9 * q^15 - 21 * q^17 - 7 * q^18 - 14 * q^19 + 24 * q^20 - 9 * q^21 - 11 * q^22 - 16 * q^23 - 10 * q^24 + 38 * q^25 - 21 * q^26 - 15 * q^27 + 24 * q^28 - 52 * q^29 - 10 * q^30 - 30 * q^31 - 8 * q^32 - 13 * q^33 - 9 * q^34 + 38 * q^35 - 16 * q^36 - 47 * q^37 - 10 * q^38 - 50 * q^39 - 21 * q^40 - 35 * q^41 - 10 * q^42 - 18 * q^43 - 22 * q^44 + 13 * q^45 - 17 * q^46 - 23 * q^47 - 13 * q^48 + 38 * q^49 - 6 * q^50 - 5 * q^51 - 41 * q^52 - 12 * q^53 + 35 * q^54 - 18 * q^55 - 21 * q^56 - 3 * q^57 - 16 * q^58 - 16 * q^59 - 20 * q^60 - 47 * q^61 + 14 * q^62 + 13 * q^63 - 55 * q^64 - 25 * q^65 - 6 * q^66 - 54 * q^67 + 31 * q^68 - 95 * q^69 - 6 * q^70 - 41 * q^71 - 59 * q^73 - q^74 - 9 * q^75 - 23 * q^76 - 18 * q^77 + 19 * q^78 - 117 * q^79 - 38 * q^81 + 19 * q^82 - 21 * q^83 - 20 * q^84 - 21 * q^85 - 12 * q^86 - 14 * q^87 - 40 * q^88 - 96 * q^89 - 7 * q^90 - 25 * q^91 + 53 * q^92 - 53 * q^93 - 28 * q^94 - 14 * q^95 + 40 * q^96 - 70 * q^97 - 6 * q^98 - 52 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.185341	0.131056	0.0655280	−	0.997851i	$-0.479127\pi$
$2$	0.185341	0.131056	0.0655280	+	0.997851i	$0.479127\pi$
$3$	−0.587692	−0.339304	−0.169652	−	0.985504i	$-0.554264\pi$
$3$	−0.587692	−0.339304	−0.169652	+	0.985504i	$0.554264\pi$
$4$	−1.96565	−0.982824
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.108923	−0.0444678
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.734998	−0.259861
$9$	−2.65462	−0.884873
$10$	0.185341	0.0586100
$11$	2.91097	0.877691	0.438846	−	0.898563i	$-0.355388\pi$
$11$	2.91097	0.877691	0.438846	+	0.898563i	$0.355388\pi$
$12$	1.15520	0.333476
$13$	−1.78576	−0.495280	−0.247640	−	0.968852i	$-0.579655\pi$
$13$	−1.78576	−0.495280	−0.247640	+	0.968852i	$0.579655\pi$
$14$	0.185341	0.0495345
$15$	−0.587692	−0.151741
$16$	3.79507	0.948768
$17$	−4.66005	−1.13023	−0.565114	−	0.825013i	$-0.691168\pi$
$17$	−4.66005	−1.13023	−0.565114	+	0.825013i	$0.691168\pi$
$18$	−0.492010	−0.115968
$19$	5.48579	1.25853	0.629263	−	0.777192i	$-0.283357\pi$
$19$	5.48579	1.25853	0.629263	+	0.777192i	$0.283357\pi$
$20$	−1.96565	−0.439532
$21$	−0.587692	−0.128245
$22$	0.539523	0.115027
$23$	−5.72088	−1.19289	−0.596443	−	0.802655i	$-0.703420\pi$
$23$	−5.72088	−1.19289	−0.596443	+	0.802655i	$0.703420\pi$
$24$	0.431952	0.0881719
$25$	1.00000	0.200000
$26$	−0.330974	−0.0649094
$27$	3.32317	0.639545
$28$	−1.96565	−0.371473
$29$	1.64651	0.305749	0.152875	−	0.988246i	$-0.451147\pi$
$29$	1.64651	0.305749	0.152875	+	0.988246i	$0.451147\pi$
$30$	−0.108923	−0.0198866
$31$	−3.05170	−0.548101	−0.274050	−	0.961715i	$-0.588364\pi$
$31$	−3.05170	−0.548101	−0.274050	+	0.961715i	$0.588364\pi$
$32$	2.17338	0.384203
$33$	−1.71076	−0.297804
$34$	−0.863699	−0.148123
$35$	1.00000	0.169031
$36$	5.21805	0.869674
$37$	10.5438	1.73338	0.866691	−	0.498845i	$-0.166242\pi$
$37$	10.5438	1.73338	0.866691	+	0.498845i	$0.166242\pi$
$38$	1.01674	0.164937
$39$	1.04948	0.168051
$40$	−0.734998	−0.116213
$41$	−2.77314	−0.433092	−0.216546	−	0.976272i	$-0.569479\pi$
$41$	−2.77314	−0.433092	−0.216546	+	0.976272i	$0.569479\pi$
$42$	−0.108923	−0.0168073
$43$	−7.94015	−1.21086	−0.605430	−	0.795898i	$-0.706999\pi$
$43$	−7.94015	−1.21086	−0.605430	+	0.795898i	$0.706999\pi$
$44$	−5.72195	−0.862616
$45$	−2.65462	−0.395727
$46$	−1.06031	−0.156335
$47$	5.94889	0.867734	0.433867	−	0.900977i	$-0.357149\pi$
$47$	5.94889	0.867734	0.433867	+	0.900977i	$0.357149\pi$
$48$	−2.23033	−0.321921
$49$	1.00000	0.142857
$50$	0.185341	0.0262112
$51$	2.73867	0.383491
$52$	3.51017	0.486773
$53$	−4.83723	−0.664445	−0.332223	−	0.943201i	$-0.607799\pi$
$53$	−4.83723	−0.664445	−0.332223	+	0.943201i	$0.607799\pi$
$54$	0.615921	0.0838162
$55$	2.91097	0.392515
$56$	−0.734998	−0.0982182
$57$	−3.22395	−0.427023
$58$	0.305166	0.0400703
$59$	−1.67739	−0.218378	−0.109189	−	0.994021i	$-0.534825\pi$
$59$	−1.67739	−0.218378	−0.109189	+	0.994021i	$0.534825\pi$
$60$	1.15520	0.149135
$61$	−6.33576	−0.811211	−0.405606	−	0.914048i	$-0.632939\pi$
$61$	−6.33576	−0.811211	−0.405606	+	0.914048i	$0.632939\pi$
$62$	−0.565605	−0.0718319
$63$	−2.65462	−0.334450
$64$	−7.18733	−0.898416
$65$	−1.78576	−0.221496
$66$	−0.317073	−0.0390290
$67$	12.0831	1.47619	0.738093	−	0.674699i	$-0.235727\pi$
$67$	12.0831	1.47619	0.738093	+	0.674699i	$0.235727\pi$
$68$	9.16002	1.11082
$69$	3.36212	0.404751
$70$	0.185341	0.0221525
$71$	−2.77916	−0.329826	−0.164913	−	0.986308i	$-0.552734\pi$
$71$	−2.77916	−0.329826	−0.164913	+	0.986308i	$0.552734\pi$
$72$	1.95114	0.229944
$73$	3.24737	0.380075	0.190038	−	0.981777i	$-0.439139\pi$
$73$	3.24737	0.380075	0.190038	+	0.981777i	$0.439139\pi$
$74$	1.95419	0.227170
$75$	−0.587692	−0.0678608
$76$	−10.7831	−1.23691
$77$	2.91097	0.331736
$78$	0.194511	0.0220240
$79$	−10.5249	−1.18414	−0.592071	−	0.805886i	$-0.701689\pi$
$79$	−10.5249	−1.18414	−0.592071	+	0.805886i	$0.701689\pi$
$80$	3.79507	0.424302
$81$	6.01085	0.667872
$82$	−0.513977	−0.0567593
$83$	7.29713	0.800964	0.400482	−	0.916305i	$-0.368843\pi$
$83$	7.29713	0.800964	0.400482	+	0.916305i	$0.368843\pi$
$84$	1.15520	0.126042
$85$	−4.66005	−0.505453
$86$	−1.47164	−0.158690
$87$	−0.967642	−0.103742
$88$	−2.13956	−0.228078
$89$	9.19710	0.974891	0.487445	−	0.873153i	$-0.337929\pi$
$89$	9.19710	0.974891	0.487445	+	0.873153i	$0.337929\pi$
$90$	−0.492010	−0.0518624
$91$	−1.78576	−0.187198
$92$	11.2452	1.17240
$93$	1.79346	0.185973
$94$	1.10257	0.113722
$95$	5.48579	0.562830
$96$	−1.27728	−0.130362
$97$	−7.33877	−0.745139	−0.372569	−	0.928004i	$-0.621523\pi$
$97$	−7.33877	−0.745139	−0.372569	+	0.928004i	$0.621523\pi$
$98$	0.185341	0.0187223
$99$	−7.72752	−0.776645
$100$	−1.96565	−0.196565
$101$	1.30671	0.130022	0.0650112	−	0.997885i	$-0.479292\pi$
$101$	1.30671	0.130022	0.0650112	+	0.997885i	$0.479292\pi$
$102$	0.507589	0.0502588
$103$	−2.16120	−0.212950	−0.106475	−	0.994315i	$-0.533956\pi$
$103$	−2.16120	−0.212950	−0.106475	+	0.994315i	$0.533956\pi$
$104$	1.31253	0.128704
$105$	−0.587692	−0.0573529
$106$	−0.896538	−0.0870795
$107$	−1.39554	−0.134911	−0.0674557	−	0.997722i	$-0.521488\pi$
$107$	−1.39554	−0.134911	−0.0674557	+	0.997722i	$0.521488\pi$
$108$	−6.53219	−0.628561
$109$	−5.62634	−0.538905	−0.269453	−	0.963014i	$-0.586843\pi$
$109$	−5.62634	−0.538905	−0.269453	+	0.963014i	$0.586843\pi$
$110$	0.539523	0.0514415
$111$	−6.19648	−0.588144
$112$	3.79507	0.358601
$113$	5.66598	0.533010	0.266505	−	0.963833i	$-0.414131\pi$
$113$	5.66598	0.533010	0.266505	+	0.963833i	$0.414131\pi$
$114$	−0.597531	−0.0559639
$115$	−5.72088	−0.533475
$116$	−3.23646	−0.300498
$117$	4.74050	0.438260
$118$	−0.310890	−0.0286197
$119$	−4.66005	−0.427186
$120$	0.431952	0.0394317
$121$	−2.52624	−0.229658
$122$	−1.17428	−0.106314
$123$	1.62975	0.146950
$124$	5.99856	0.538687
$125$	1.00000	0.0894427
$126$	−0.492010	−0.0438317
$127$	10.4672	0.928811	0.464405	−	0.885623i	$-0.346268\pi$
$127$	10.4672	0.928811	0.464405	+	0.885623i	$0.346268\pi$
$128$	−5.67886	−0.501945
$129$	4.66636	0.410850
$130$	−0.330974	−0.0290284
$131$	−6.82506	−0.596308	−0.298154	−	0.954518i	$-0.596371\pi$
$131$	−6.82506	−0.596308	−0.298154	+	0.954518i	$0.596371\pi$
$132$	3.36274	0.292689
$133$	5.48579	0.475678
$134$	2.23949	0.193463
$135$	3.32317	0.286013
$136$	3.42513	0.293702
$137$	−13.4584	−1.14983	−0.574913	−	0.818214i	$-0.694964\pi$
$137$	−13.4584	−1.14983	−0.574913	+	0.818214i	$0.694964\pi$
$138$	0.623138	0.0530450
$139$	−1.69487	−0.143757	−0.0718787	−	0.997413i	$-0.522899\pi$
$139$	−1.69487	−0.143757	−0.0718787	+	0.997413i	$0.522899\pi$
$140$	−1.96565	−0.166128
$141$	−3.49611	−0.294426
$142$	−0.515093	−0.0432256
$143$	−5.19829	−0.434703
$144$	−10.0745	−0.839539
$145$	1.64651	0.136735
$146$	0.601870	0.0498111
$147$	−0.587692	−0.0484720
$148$	−20.7253	−1.70361
$149$	3.30247	0.270549	0.135274	−	0.990808i	$-0.456808\pi$
$149$	3.30247	0.270549	0.135274	+	0.990808i	$0.456808\pi$
$150$	−0.108923	−0.00889357
$151$	−4.29499	−0.349521	−0.174761	−	0.984611i	$-0.555915\pi$
$151$	−4.29499	−0.349521	−0.174761	+	0.984611i	$0.555915\pi$
$152$	−4.03204	−0.327042
$153$	12.3707	1.00011
$154$	0.539523	0.0434760
$155$	−3.05170	−0.245118
$156$	−2.06290	−0.165164
$157$	17.0859	1.36360	0.681801	−	0.731538i	$-0.261197\pi$
$157$	17.0859	1.36360	0.681801	+	0.731538i	$0.261197\pi$
$158$	−1.95069	−0.155189
$159$	2.84280	0.225449
$160$	2.17338	0.171821
$161$	−5.72088	−0.450868
$162$	1.11406	0.0875286
$163$	−18.0079	−1.41049	−0.705243	−	0.708966i	$-0.749162\pi$
$163$	−18.0079	−1.41049	−0.705243	+	0.708966i	$0.749162\pi$
$164$	5.45102	0.425653
$165$	−1.71076	−0.133182
$166$	1.35246	0.104971
$167$	−10.8305	−0.838090	−0.419045	−	0.907965i	$-0.637635\pi$
$167$	−10.8305	−0.838090	−0.419045	+	0.907965i	$0.637635\pi$
$168$	0.431952	0.0333258
$169$	−9.81107	−0.754698
$170$	−0.863699	−0.0662427
$171$	−14.5627	−1.11364
$172$	15.6075	1.19006
$173$	10.6813	0.812084	0.406042	−	0.913854i	$-0.366909\pi$
$173$	10.6813	0.812084	0.406042	+	0.913854i	$0.366909\pi$
$174$	−0.179344	−0.0135960
$175$	1.00000	0.0755929
$176$	11.0473	0.832725
$177$	0.985790	0.0740965
$178$	1.70460	0.127765
$179$	18.3496	1.37151	0.685757	−	0.727831i	$-0.259471\pi$
$179$	18.3496	1.37151	0.685757	+	0.727831i	$0.259471\pi$
$180$	5.21805	0.388930
$181$	−13.6494	−1.01455	−0.507277	−	0.861783i	$-0.669348\pi$
$181$	−13.6494	−1.01455	−0.507277	+	0.861783i	$0.669348\pi$
$182$	−0.330974	−0.0245334
$183$	3.72348	0.275247
$184$	4.20483	0.309984
$185$	10.5438	0.775192
$186$	0.332401	0.0243729
$187$	−13.5653	−0.991991
$188$	−11.6934	−0.852830
$189$	3.32317	0.241725
$190$	1.01674	0.0737622
$191$	−9.46702	−0.685009	−0.342505	−	0.939516i	$-0.611275\pi$
$191$	−9.46702	−0.685009	−0.342505	+	0.939516i	$0.611275\pi$
$192$	4.22394	0.304836
$193$	17.0677	1.22856	0.614279	−	0.789089i	$-0.289447\pi$
$193$	17.0677	1.22856	0.614279	+	0.789089i	$0.289447\pi$
$194$	−1.36017	−0.0976549
$195$	1.04948	0.0751545
$196$	−1.96565	−0.140403
$197$	−10.9947	−0.783341	−0.391670	−	0.920106i	$-0.628103\pi$
$197$	−10.9947	−0.783341	−0.391670	+	0.920106i	$0.628103\pi$
$198$	−1.43223	−0.101784
$199$	−16.5342	−1.17208	−0.586039	−	0.810283i	$-0.699313\pi$
$199$	−16.5342	−1.17208	−0.586039	+	0.810283i	$0.699313\pi$
$200$	−0.734998	−0.0519722
$201$	−7.10114	−0.500876
$202$	0.242187	0.0170402
$203$	1.64651	0.115562
$204$	−5.38327	−0.376904
$205$	−2.77314	−0.193685
$206$	−0.400560	−0.0279083
$207$	15.1868	1.05555
$208$	−6.77708	−0.469906
$209$	15.9690	1.10460
$210$	−0.108923	−0.00751643
$211$	−9.15217	−0.630061	−0.315031	−	0.949082i	$-0.602015\pi$
$211$	−9.15217	−0.630061	−0.315031	+	0.949082i	$0.602015\pi$
$212$	9.50830	0.653033
$213$	1.63329	0.111911
$214$	−0.258650	−0.0176809
$215$	−7.94015	−0.541513
$216$	−2.44253	−0.166193
$217$	−3.05170	−0.207163
$218$	−1.04279	−0.0706267
$219$	−1.90845	−0.128961
$220$	−5.72195	−0.385774
$221$	8.32172	0.559779
$222$	−1.14846	−0.0770797
$223$	14.5022	0.971141	0.485571	−	0.874197i	$-0.338612\pi$
$223$	14.5022	0.971141	0.485571	+	0.874197i	$0.338612\pi$
$224$	2.17338	0.145215
$225$	−2.65462	−0.176975
$226$	1.05014	0.0698542
$227$	−12.7187	−0.844168	−0.422084	−	0.906557i	$-0.638701\pi$
$227$	−12.7187	−0.844168	−0.422084	+	0.906557i	$0.638701\pi$
$228$	6.33716	0.419689
$229$	−1.00000	−0.0660819
$229$	−1.00000	−0.0660819
$230$	−1.06031	−0.0699150
$231$	−1.71076	−0.112559
$232$	−1.21018	−0.0794523
$233$	26.0534	1.70682	0.853408	−	0.521243i	$-0.174531\pi$
$233$	26.0534	1.70682	0.853408	+	0.521243i	$0.174531\pi$
$234$	0.878610	0.0574365
$235$	5.94889	0.388062
$236$	3.29716	0.214627
$237$	6.18539	0.401784
$238$	−0.863699	−0.0559853
$239$	0.123023	0.00795769	0.00397884	−	0.999992i	$-0.498733\pi$
$239$	0.123023	0.00795769	0.00397884	+	0.999992i	$0.498733\pi$
$240$	−2.23033	−0.143967
$241$	−22.2806	−1.43522	−0.717611	−	0.696444i	$-0.754764\pi$
$241$	−22.2806	−1.43522	−0.717611	+	0.696444i	$0.754764\pi$
$242$	−0.468217	−0.0300981
$243$	−13.5021	−0.866157
$244$	12.4539	0.797278
$245$	1.00000	0.0638877
$246$	0.302060	0.0192587
$247$	−9.79629	−0.623323
$248$	2.24299	0.142430
$249$	−4.28847	−0.271771
$250$	0.185341	0.0117220
$251$	−5.14138	−0.324521	−0.162261	−	0.986748i	$-0.551879\pi$
$251$	−5.14138	−0.324521	−0.162261	+	0.986748i	$0.551879\pi$
$252$	5.21805	0.328706
$253$	−16.6533	−1.04699
$254$	1.94000	0.121726
$255$	2.73867	0.171502
$256$	13.3221	0.832633
$257$	−1.36546	−0.0851752	−0.0425876	−	0.999093i	$-0.513560\pi$
$257$	−1.36546	−0.0851752	−0.0425876	+	0.999093i	$0.513560\pi$
$258$	0.864868	0.0538443
$259$	10.5438	0.655157
$260$	3.51017	0.217692
$261$	−4.37086	−0.270549
$262$	−1.26496	−0.0781497
$263$	9.91475	0.611370	0.305685	−	0.952133i	$-0.401115\pi$
$263$	9.91475	0.611370	0.305685	+	0.952133i	$0.401115\pi$
$264$	1.25740	0.0773877
$265$	−4.83723	−0.297149
$266$	1.01674	0.0623404
$267$	−5.40506	−0.330785
$268$	−23.7511	−1.45083
$269$	13.7922	0.840923	0.420461	−	0.907310i	$-0.361868\pi$
$269$	13.7922	0.840923	0.420461	+	0.907310i	$0.361868\pi$
$270$	0.615921	0.0374837
$271$	−5.91994	−0.359611	−0.179805	−	0.983702i	$-0.557547\pi$
$271$	−5.91994	−0.359611	−0.179805	+	0.983702i	$0.557547\pi$
$272$	−17.6852	−1.07232
$273$	1.04948	0.0635171
$274$	−2.49439	−0.150692
$275$	2.91097	0.175538
$276$	−6.60874	−0.397799
$277$	−28.3969	−1.70620	−0.853101	−	0.521745i	$-0.825281\pi$
$277$	−28.3969	−1.70620	−0.853101	+	0.521745i	$0.825281\pi$
$278$	−0.314130	−0.0188402
$279$	8.10109	0.485000
$280$	−0.734998	−0.0439245
$281$	5.40251	0.322287	0.161143	−	0.986931i	$-0.448482\pi$
$281$	5.40251	0.322287	0.161143	+	0.986931i	$0.448482\pi$
$282$	−0.647973	−0.0385862
$283$	−21.6796	−1.28872	−0.644358	−	0.764724i	$-0.722875\pi$
$283$	−21.6796	−1.28872	−0.644358	+	0.764724i	$0.722875\pi$
$284$	5.46285	0.324161
$285$	−3.22395	−0.190971
$286$	−0.963456	−0.0569704
$287$	−2.77314	−0.163693
$288$	−5.76949	−0.339970
$289$	4.71607	0.277416
$290$	0.305166	0.0179200
$291$	4.31293	0.252829
$292$	−6.38318	−0.373547
$293$	−9.34342	−0.545849	−0.272924	−	0.962036i	$-0.587991\pi$
$293$	−9.34342	−0.545849	−0.272924	+	0.962036i	$0.587991\pi$
$294$	−0.108923	−0.00635255
$295$	−1.67739	−0.0976616
$296$	−7.74963	−0.450438
$297$	9.67367	0.561323
$298$	0.612083	0.0354570
$299$	10.2161	0.590812
$300$	1.15520	0.0666953
$301$	−7.94015	−0.457662
$302$	−0.796038	−0.0458068
$303$	−0.767942	−0.0441171
$304$	20.8190	1.19405
$305$	−6.33576	−0.362785
$306$	2.29279	0.131070
$307$	−20.1228	−1.14847	−0.574235	−	0.818690i	$-0.694701\pi$
$307$	−20.1228	−1.14847	−0.574235	+	0.818690i	$0.694701\pi$
$308$	−5.72195	−0.326038
$309$	1.27012	0.0722548
$310$	−0.565605	−0.0321242
$311$	−5.95975	−0.337946	−0.168973	−	0.985621i	$-0.554045\pi$
$311$	−5.95975	−0.337946	−0.168973	+	0.985621i	$0.554045\pi$
$312$	−0.771362	−0.0436698
$313$	0.540404	0.0305455	0.0152727	−	0.999883i	$-0.495138\pi$
$313$	0.540404	0.0305455	0.0152727	+	0.999883i	$0.495138\pi$
$314$	3.16672	0.178708
$315$	−2.65462	−0.149571
$316$	20.6882	1.16380
$317$	13.2772	0.745724	0.372862	−	0.927887i	$-0.378376\pi$
$317$	13.2772	0.745724	0.372862	+	0.927887i	$0.378376\pi$
$318$	0.526888	0.0295464
$319$	4.79295	0.268354
$320$	−7.18733	−0.401784
$321$	0.820145	0.0457760
$322$	−1.06031	−0.0590890
$323$	−25.5641	−1.42242
$324$	−11.8152	−0.656401
$325$	−1.78576	−0.0990560
$326$	−3.33760	−0.184853
$327$	3.30655	0.182853
$328$	2.03825	0.112544
$329$	5.94889	0.327973
$330$	−0.317073	−0.0174543
$331$	−25.2034	−1.38531	−0.692653	−	0.721271i	$-0.743558\pi$
$331$	−25.2034	−1.38531	−0.692653	+	0.721271i	$0.743558\pi$
$332$	−14.3436	−0.787207
$333$	−27.9896	−1.53382
$334$	−2.00734	−0.109837
$335$	12.0831	0.660170
$336$	−2.23033	−0.121675
$337$	−14.5607	−0.793170	−0.396585	−	0.917998i	$-0.629805\pi$
$337$	−14.5607	−0.793170	−0.396585	+	0.917998i	$0.629805\pi$
$338$	−1.81839	−0.0989076
$339$	−3.32985	−0.180853
$340$	9.16002	0.496772
$341$	−8.88340	−0.481063
$342$	−2.69906	−0.145949
$343$	1.00000	0.0539949
$344$	5.83599	0.314655
$345$	3.36212	0.181010
$346$	1.97968	0.106428
$347$	7.70046	0.413383	0.206691	−	0.978406i	$-0.433730\pi$
$347$	7.70046	0.413383	0.206691	+	0.978406i	$0.433730\pi$
$348$	1.90204	0.101960
$349$	−13.3444	−0.714312	−0.357156	−	0.934045i	$-0.616253\pi$
$349$	−13.3444	−0.714312	−0.357156	+	0.934045i	$0.616253\pi$
$350$	0.185341	0.00990690
$351$	−5.93438	−0.316754
$352$	6.32664	0.337211
$353$	−25.5855	−1.36178	−0.680891	−	0.732385i	$-0.738407\pi$
$353$	−25.5855	−1.36178	−0.680891	+	0.732385i	$0.738407\pi$
$354$	0.182707	0.00971079
$355$	−2.77916	−0.147503
$356$	−18.0783	−0.958147
$357$	2.73867	0.144946
$358$	3.40094	0.179745
$359$	−7.30561	−0.385575	−0.192788	−	0.981240i	$-0.561753\pi$
$359$	−7.30561	−0.385575	−0.192788	+	0.981240i	$0.561753\pi$
$360$	1.95114	0.102834
$361$	11.0939	0.583888
$362$	−2.52980	−0.132963
$363$	1.48465	0.0779241
$364$	3.51017	0.183983
$365$	3.24737	0.169975
$366$	0.690113	0.0360728
$367$	4.81237	0.251204	0.125602	−	0.992081i	$-0.459914\pi$
$367$	4.81237	0.251204	0.125602	+	0.992081i	$0.459914\pi$
$368$	−21.7112	−1.13177
$369$	7.36163	0.383231
$370$	1.95419	0.101594
$371$	−4.83723	−0.251137
$372$	−3.52531	−0.182779
$373$	−4.06164	−0.210304	−0.105152	−	0.994456i	$-0.533533\pi$
$373$	−4.06164	−0.210304	−0.105152	+	0.994456i	$0.533533\pi$
$374$	−2.51420	−0.130006
$375$	−0.587692	−0.0303483
$376$	−4.37242	−0.225490
$377$	−2.94027	−0.151432
$378$	0.615921	0.0316795
$379$	−19.1091	−0.981570	−0.490785	−	0.871281i	$-0.663290\pi$
$379$	−19.1091	−0.981570	−0.490785	+	0.871281i	$0.663290\pi$
$380$	−10.7831	−0.553163
$381$	−6.15147	−0.315149
$382$	−1.75463	−0.0897745
$383$	−6.04628	−0.308951	−0.154475	−	0.987997i	$-0.549369\pi$
$383$	−6.04628	−0.308951	−0.154475	+	0.987997i	$0.549369\pi$
$384$	3.33742	0.170312
$385$	2.91097	0.148357
$386$	3.16334	0.161010
$387$	21.0781	1.07146
$388$	14.4254	0.732341
$389$	−11.9677	−0.606786	−0.303393	−	0.952866i	$-0.598119\pi$
$389$	−11.9677	−0.606786	−0.303393	+	0.952866i	$0.598119\pi$
$390$	0.194511	0.00984944
$391$	26.6596	1.34823
$392$	−0.734998	−0.0371230
$393$	4.01103	0.202330
$394$	−2.03777	−0.102661
$395$	−10.5249	−0.529564
$396$	15.1896	0.763305
$397$	16.1996	0.813036	0.406518	−	0.913643i	$-0.366743\pi$
$397$	16.1996	0.813036	0.406518	+	0.913643i	$0.366743\pi$
$398$	−3.06446	−0.153608
$399$	−3.22395	−0.161400
$400$	3.79507	0.189754
$401$	−29.4077	−1.46855	−0.734276	−	0.678851i	$-0.762478\pi$
$401$	−29.4077	−1.46855	−0.734276	+	0.678851i	$0.762478\pi$
$402$	−1.31613	−0.0656428
$403$	5.44959	0.271463
$404$	−2.56853	−0.127789
$405$	6.01085	0.298682
$406$	0.305166	0.0151451
$407$	30.6926	1.52137
$408$	−2.01292	−0.0996544
$409$	1.44327	0.0713650	0.0356825	−	0.999363i	$-0.488640\pi$
$409$	1.44327	0.0713650	0.0356825	+	0.999363i	$0.488640\pi$
$410$	−0.513977	−0.0253835
$411$	7.90938	0.390141
$412$	4.24817	0.209292
$413$	−1.67739	−0.0825391
$414$	2.81473	0.138336
$415$	7.29713	0.358202
$416$	−3.88112	−0.190288
$417$	0.996064	0.0487775
$418$	2.95971	0.144764
$419$	−20.2551	−0.989524	−0.494762	−	0.869028i	$-0.664745\pi$
$419$	−20.2551	−0.989524	−0.494762	+	0.869028i	$0.664745\pi$
$420$	1.15520	0.0563678
$421$	4.51257	0.219929	0.109965	−	0.993936i	$-0.464926\pi$
$421$	4.51257	0.219929	0.109965	+	0.993936i	$0.464926\pi$
$422$	−1.69627	−0.0825733
$423$	−15.7920	−0.767834
$424$	3.55536	0.172663
$425$	−4.66005	−0.226046
$426$	0.302716	0.0146666
$427$	−6.33576	−0.306609
$428$	2.74313	0.132594
$429$	3.05499	0.147496
$430$	−1.47164	−0.0709685
$431$	−20.9231	−1.00783	−0.503916	−	0.863753i	$-0.668108\pi$
$431$	−20.9231	−1.00783	−0.503916	+	0.863753i	$0.668108\pi$
$432$	12.6117	0.606780
$433$	10.2951	0.494750	0.247375	−	0.968920i	$-0.420432\pi$
$433$	10.2951	0.494750	0.247375	+	0.968920i	$0.420432\pi$
$434$	−0.565605	−0.0271499
$435$	−0.967642	−0.0463949
$436$	11.0594	0.529649
$437$	−31.3835	−1.50128
$438$	−0.353714	−0.0169011
$439$	10.3093	0.492038	0.246019	−	0.969265i	$-0.420877\pi$
$439$	10.3093	0.492038	0.246019	+	0.969265i	$0.420877\pi$
$440$	−2.13956	−0.101999
$441$	−2.65462	−0.126410
$442$	1.54236	0.0733624
$443$	−12.1417	−0.576872	−0.288436	−	0.957499i	$-0.593135\pi$
$443$	−12.1417	−0.576872	−0.288436	+	0.957499i	$0.593135\pi$
$444$	12.1801	0.578042
$445$	9.19710	0.435984
$446$	2.68786	0.127274
$447$	−1.94083	−0.0917983
$448$	−7.18733	−0.339569
$449$	−9.21618	−0.434938	−0.217469	−	0.976067i	$-0.569780\pi$
$449$	−9.21618	−0.434938	−0.217469	+	0.976067i	$0.569780\pi$
$450$	−0.492010	−0.0231936
$451$	−8.07254	−0.380121
$452$	−11.1373	−0.523856
$453$	2.52413	0.118594
$454$	−2.35729	−0.110633
$455$	−1.78576	−0.0837176
$456$	2.36960	0.110967
$457$	5.75431	0.269175	0.134588	−	0.990902i	$-0.457029\pi$
$457$	5.75431	0.269175	0.134588	+	0.990902i	$0.457029\pi$
$458$	−0.185341	−0.00866042
$459$	−15.4862	−0.722832
$460$	11.2452	0.524312
$461$	−35.0029	−1.63025	−0.815124	−	0.579287i	$-0.803331\pi$
$461$	−35.0029	−1.63025	−0.815124	+	0.579287i	$0.803331\pi$
$462$	−0.317073	−0.0147516
$463$	−39.4155	−1.83180	−0.915898	−	0.401412i	$-0.868520\pi$
$463$	−39.4155	−1.83180	−0.915898	+	0.401412i	$0.868520\pi$
$464$	6.24863	0.290085
$465$	1.79346	0.0831696
$466$	4.82877	0.223688
$467$	−15.3539	−0.710493	−0.355246	−	0.934773i	$-0.615603\pi$
$467$	−15.3539	−0.710493	−0.355246	+	0.934773i	$0.615603\pi$
$468$	−9.31816	−0.430732
$469$	12.0831	0.557946
$470$	1.10257	0.0508579
$471$	−10.0412	−0.462676
$472$	1.23288	0.0567479
$473$	−23.1135	−1.06276
$474$	1.14641	0.0526562
$475$	5.48579	0.251705
$476$	9.16002	0.419849
$477$	12.8410	0.587949
$478$	0.0228012	0.00104290
$479$	9.83532	0.449387	0.224694	−	0.974429i	$-0.427862\pi$
$479$	9.83532	0.449387	0.224694	+	0.974429i	$0.427862\pi$
$480$	−1.27728	−0.0582995
$481$	−18.8286	−0.858509
$482$	−4.12952	−0.188094
$483$	3.36212	0.152982
$484$	4.96571	0.225714
$485$	−7.33877	−0.333236
$486$	−2.50248	−0.113515
$487$	25.7231	1.16562	0.582812	−	0.812607i	$-0.301953\pi$
$487$	25.7231	1.16562	0.582812	+	0.812607i	$0.301953\pi$
$488$	4.65677	0.210802
$489$	10.5831	0.478584
$490$	0.185341	0.00837286
$491$	25.1022	1.13285	0.566424	−	0.824114i	$-0.308326\pi$
$491$	25.1022	1.13285	0.566424	+	0.824114i	$0.308326\pi$
$492$	−3.20352	−0.144426
$493$	−7.67283	−0.345567
$494$	−1.81565	−0.0816901
$495$	−7.72752	−0.347326
$496$	−11.5814	−0.520021
$497$	−2.77916	−0.124662
$498$	−0.794829	−0.0356171
$499$	32.3529	1.44832	0.724158	−	0.689634i	$-0.242229\pi$
$499$	32.3529	1.44832	0.724158	+	0.689634i	$0.242229\pi$
$500$	−1.96565	−0.0879065
$501$	6.36501	0.284368
$502$	−0.952910	−0.0425304
$503$	−19.6507	−0.876180	−0.438090	−	0.898931i	$-0.644345\pi$
$503$	−19.6507	−0.876180	−0.438090	+	0.898931i	$0.644345\pi$
$504$	1.95114	0.0869106
$505$	1.30671	0.0581477
$506$	−3.08654	−0.137214
$507$	5.76589	0.256072
$508$	−20.5748	−0.912858
$509$	−19.0581	−0.844736	−0.422368	−	0.906425i	$-0.638801\pi$
$509$	−19.0581	−0.844736	−0.422368	+	0.906425i	$0.638801\pi$
$510$	0.507589	0.0224764
$511$	3.24737	0.143655
$512$	13.8269	0.611067
$513$	18.2302	0.804884
$514$	−0.253076	−0.0111627
$515$	−2.16120	−0.0952341
$516$	−9.17243	−0.403794
$517$	17.3170	0.761602
$518$	1.95419	0.0858622
$519$	−6.27731	−0.275543
$520$	1.31253	0.0575581
$521$	29.5370	1.29404	0.647020	−	0.762473i	$-0.276015\pi$
$521$	29.5370	1.29404	0.647020	+	0.762473i	$0.276015\pi$
$522$	−0.810100	−0.0354571
$523$	−13.3301	−0.582884	−0.291442	−	0.956589i	$-0.594135\pi$
$523$	−13.3301	−0.582884	−0.291442	+	0.956589i	$0.594135\pi$
$524$	13.4157	0.586066
$525$	−0.587692	−0.0256490
$526$	1.83761	0.0801236
$527$	14.2211	0.619479
$528$	−6.49244	−0.282547
$529$	9.72847	0.422977
$530$	−0.896538	−0.0389431
$531$	4.45284	0.193237
$532$	−10.7831	−0.467508
$533$	4.95216	0.214502
$534$	−1.00178	−0.0433513
$535$	−1.39554	−0.0603342
$536$	−8.88105	−0.383603
$537$	−10.7839	−0.465360
$538$	2.55625	0.110208
$539$	2.91097	0.125384
$540$	−6.53219	−0.281101
$541$	−4.24181	−0.182370	−0.0911848	−	0.995834i	$-0.529065\pi$
$541$	−4.24181	−0.182370	−0.0911848	+	0.995834i	$0.529065\pi$
$542$	−1.09721	−0.0471291
$543$	8.02167	0.344243
$544$	−10.1281	−0.434237
$545$	−5.62634	−0.241006
$546$	0.194511	0.00832430
$547$	−24.3779	−1.04232	−0.521162	−	0.853458i	$-0.674501\pi$
$547$	−24.3779	−1.04232	−0.521162	+	0.853458i	$0.674501\pi$
$548$	26.4544	1.13008
$549$	16.8190	0.717819
$550$	0.539523	0.0230053
$551$	9.03241	0.384794
$552$	−2.47115	−0.105179
$553$	−10.5249	−0.447563
$554$	−5.26311	−0.223608
$555$	−6.19648	−0.263026
$556$	3.33153	0.141288
$557$	−2.94906	−0.124956	−0.0624778	−	0.998046i	$-0.519900\pi$
$557$	−2.94906	−0.124956	−0.0624778	+	0.998046i	$0.519900\pi$
$558$	1.50146	0.0635621
$559$	14.1792	0.599715
$560$	3.79507	0.160371
$561$	7.97221	0.336587
$562$	1.00131	0.0422376
$563$	−19.8329	−0.835856	−0.417928	−	0.908480i	$-0.637244\pi$
$563$	−19.8329	−0.835856	−0.417928	+	0.908480i	$0.637244\pi$
$564$	6.87213	0.289369
$565$	5.66598	0.238369
$566$	−4.01811	−0.168894
$567$	6.01085	0.252432
$568$	2.04268	0.0857088
$569$	25.6553	1.07553	0.537763	−	0.843096i	$-0.319269\pi$
$569$	25.6553	1.07553	0.537763	+	0.843096i	$0.319269\pi$
$570$	−0.597531	−0.0250278
$571$	−7.39044	−0.309280	−0.154640	−	0.987971i	$-0.549422\pi$
$571$	−7.39044	−0.309280	−0.154640	+	0.987971i	$0.549422\pi$
$572$	10.2180	0.427236
$573$	5.56369	0.232426
$574$	−0.513977	−0.0214530
$575$	−5.72088	−0.238577
$576$	19.0796	0.794984
$577$	4.11121	0.171152	0.0855760	−	0.996332i	$-0.472727\pi$
$577$	4.11121	0.171152	0.0855760	+	0.996332i	$0.472727\pi$
$578$	0.874082	0.0363570
$579$	−10.0305	−0.416855
$580$	−3.23646	−0.134387
$581$	7.29713	0.302736
$582$	0.799364	0.0331347
$583$	−14.0811	−0.583178
$584$	−2.38681	−0.0987667
$585$	4.74050	0.195996
$586$	−1.73172	−0.0715367
$587$	18.2781	0.754417	0.377208	−	0.926128i	$-0.376884\pi$
$587$	18.2781	0.754417	0.377208	+	0.926128i	$0.376884\pi$
$588$	1.15520	0.0476395
$589$	−16.7410	−0.689799
$590$	−0.310890	−0.0127991
$591$	6.46150	0.265791
$592$	40.0143	1.64458
$593$	−43.3466	−1.78003	−0.890016	−	0.455929i	$-0.849307\pi$
$593$	−43.3466	−1.78003	−0.890016	+	0.455929i	$0.849307\pi$
$594$	1.79293	0.0735647
$595$	−4.66005	−0.191043
$596$	−6.49149	−0.265902
$597$	9.71701	0.397691
$598$	1.89346	0.0774295
$599$	6.58547	0.269075	0.134538	−	0.990908i	$-0.457045\pi$
$599$	6.58547	0.269075	0.134538	+	0.990908i	$0.457045\pi$
$600$	0.431952	0.0176344
$601$	12.7167	0.518726	0.259363	−	0.965780i	$-0.416487\pi$
$601$	12.7167	0.518726	0.259363	+	0.965780i	$0.416487\pi$
$602$	−1.47164	−0.0599794
$603$	−32.0760	−1.30624
$604$	8.44244	0.343518
$605$	−2.52624	−0.102706
$606$	−0.142331	−0.00578181
$607$	−44.7282	−1.81546	−0.907731	−	0.419553i	$-0.862187\pi$
$607$	−44.7282	−1.81546	−0.907731	+	0.419553i	$0.862187\pi$
$608$	11.9227	0.483529
$609$	−0.967642	−0.0392108
$610$	−1.17428	−0.0475451
$611$	−10.6233	−0.429771
$612$	−24.3164	−0.982931
$613$	30.9788	1.25122	0.625611	−	0.780136i	$-0.284850\pi$
$613$	30.9788	1.25122	0.625611	+	0.780136i	$0.284850\pi$
$614$	−3.72959	−0.150514
$615$	1.62975	0.0657180
$616$	−2.13956	−0.0862052
$617$	−34.0383	−1.37033	−0.685166	−	0.728387i	$-0.740270\pi$
$617$	−34.0383	−1.37033	−0.685166	+	0.728387i	$0.740270\pi$
$618$	0.235406	0.00946942
$619$	−8.38339	−0.336957	−0.168478	−	0.985705i	$-0.553885\pi$
$619$	−8.38339	−0.336957	−0.168478	+	0.985705i	$0.553885\pi$
$620$	5.99856	0.240908
$621$	−19.0115	−0.762904
$622$	−1.10459	−0.0442899
$623$	9.19710	0.368474
$624$	3.98283	0.159441
$625$	1.00000	0.0400000
$626$	0.100159	0.00400316
$627$	−9.38484	−0.374794
$628$	−33.5848	−1.34018
$629$	−49.1344	−1.95912
$630$	−0.492010	−0.0196021
$631$	−10.2658	−0.408677	−0.204338	−	0.978900i	$-0.565504\pi$
$631$	−10.2658	−0.408677	−0.204338	+	0.978900i	$0.565504\pi$
$632$	7.73576	0.307712
$633$	5.37866	0.213782
$634$	2.46082	0.0977316
$635$	10.4672	0.415377
$636$	−5.58795	−0.221577
$637$	−1.78576	−0.0707543
$638$	0.888330	0.0351693
$639$	7.37761	0.291854
$640$	−5.67886	−0.224477
$641$	7.09147	0.280096	0.140048	−	0.990145i	$-0.455274\pi$
$641$	7.09147	0.280096	0.140048	+	0.990145i	$0.455274\pi$
$642$	0.152007	0.00599922
$643$	22.5676	0.889979	0.444989	−	0.895536i	$-0.353207\pi$
$643$	22.5676	0.889979	0.444989	+	0.895536i	$0.353207\pi$
$644$	11.2452	0.443125
$645$	4.66636	0.183738
$646$	−4.73807	−0.186417
$647$	17.7874	0.699296	0.349648	−	0.936881i	$-0.386301\pi$
$647$	17.7874	0.699296	0.349648	+	0.936881i	$0.386301\pi$
$648$	−4.41796	−0.173554
$649$	−4.88284	−0.191668
$650$	−0.330974	−0.0129819
$651$	1.79346	0.0702912
$652$	35.3972	1.38626
$653$	−26.1740	−1.02427	−0.512135	−	0.858905i	$-0.671145\pi$
$653$	−26.1740	−1.02427	−0.512135	+	0.858905i	$0.671145\pi$
$654$	0.612840	0.0239640
$655$	−6.82506	−0.266677
$656$	−10.5243	−0.410904
$657$	−8.62052	−0.336318
$658$	1.10257	0.0429828
$659$	−49.3945	−1.92414	−0.962069	−	0.272807i	$-0.912048\pi$
$659$	−49.3945	−1.92414	−0.962069	+	0.272807i	$0.912048\pi$
$660$	3.36274	0.130895
$661$	8.49161	0.330285	0.165143	−	0.986270i	$-0.447192\pi$
$661$	8.49161	0.330285	0.165143	+	0.986270i	$0.447192\pi$
$662$	−4.67123	−0.181553
$663$	−4.89061	−0.189935
$664$	−5.36338	−0.208139
$665$	5.48579	0.212730
$666$	−5.18763	−0.201017
$667$	−9.41949	−0.364724
$668$	21.2890	0.823695
$669$	−8.52285	−0.329512
$670$	2.23949	0.0865192
$671$	−18.4432	−0.711993
$672$	−1.27728	−0.0492720
$673$	22.0695	0.850716	0.425358	−	0.905025i	$-0.360148\pi$
$673$	22.0695	0.850716	0.425358	+	0.905025i	$0.360148\pi$
$674$	−2.69869	−0.103950
$675$	3.32317	0.127909
$676$	19.2851	0.741735
$677$	1.70961	0.0657056	0.0328528	−	0.999460i	$-0.489541\pi$
$677$	1.70961	0.0657056	0.0328528	+	0.999460i	$0.489541\pi$
$678$	−0.617158	−0.0237018
$679$	−7.33877	−0.281636
$680$	3.42513	0.131348
$681$	7.47466	0.286430
$682$	−1.64646	−0.0630462
$683$	−45.2996	−1.73334	−0.866671	−	0.498880i	$-0.833745\pi$
$683$	−45.2996	−1.73334	−0.866671	+	0.498880i	$0.833745\pi$
$684$	28.6251	1.09451
$685$	−13.4584	−0.514218
$686$	0.185341	0.00707635
$687$	0.587692	0.0224219
$688$	−30.1334	−1.14883
$689$	8.63812	0.329086
$690$	0.623138	0.0237225
$691$	−0.396648	−0.0150892	−0.00754459	−	0.999972i	$-0.502402\pi$
$691$	−0.396648	−0.0150892	−0.00754459	+	0.999972i	$0.502402\pi$
$692$	−20.9957	−0.798136
$693$	−7.72752	−0.293544
$694$	1.42721	0.0541762
$695$	−1.69487	−0.0642902
$696$	0.711214	0.0269585
$697$	12.9230	0.489493
$698$	−2.47327	−0.0936148
$699$	−15.3114	−0.579130
$700$	−1.96565	−0.0742945
$701$	0.545017	0.0205850	0.0102925	−	0.999947i	$-0.496724\pi$
$701$	0.545017	0.0205850	0.0102925	+	0.999947i	$0.496724\pi$
$702$	−1.09988	−0.0415125
$703$	57.8408	2.18151
$704$	−20.9221	−0.788532
$705$	−3.49611	−0.131671
$706$	−4.74205	−0.178470
$707$	1.30671	0.0491438
$708$	−1.93772	−0.0728239
$709$	−27.3200	−1.02602	−0.513011	−	0.858382i	$-0.671470\pi$
$709$	−27.3200	−1.02602	−0.513011	+	0.858382i	$0.671470\pi$
$710$	−0.515093	−0.0193311
$711$	27.9395	1.04781
$712$	−6.75985	−0.253336
$713$	17.4584	0.653822
$714$	0.507589	0.0189960
$715$	−5.19829	−0.194405
$716$	−36.0689	−1.34796
$717$	−0.0722995	−0.00270008
$718$	−1.35403	−0.0505319
$719$	9.63078	0.359168	0.179584	−	0.983743i	$-0.442525\pi$
$719$	9.63078	0.359168	0.179584	+	0.983743i	$0.442525\pi$
$720$	−10.0745	−0.375453
$721$	−2.16120	−0.0804875
$722$	2.05615	0.0765220
$723$	13.0941	0.486977
$724$	26.8300	0.997129
$725$	1.64651	0.0611499
$726$	0.275167	0.0102124
$727$	10.9924	0.407684	0.203842	−	0.979004i	$-0.434657\pi$
$727$	10.9924	0.407684	0.203842	+	0.979004i	$0.434657\pi$
$728$	1.31253	0.0486455
$729$	−10.0975	−0.373982
$730$	0.601870	0.0222762
$731$	37.0015	1.36855
$732$	−7.31905	−0.270520
$733$	−17.6793	−0.653001	−0.326500	−	0.945197i	$-0.605869\pi$
$733$	−17.6793	−0.653001	−0.326500	+	0.945197i	$0.605869\pi$
$734$	0.891931	0.0329218
$735$	−0.587692	−0.0216773
$736$	−12.4336	−0.458310
$737$	35.1736	1.29563
$738$	1.36441	0.0502247
$739$	−6.33152	−0.232909	−0.116454	−	0.993196i	$-0.537153\pi$
$739$	−6.33152	−0.232909	−0.116454	+	0.993196i	$0.537153\pi$
$740$	−20.7253	−0.761878
$741$	5.75720	0.211496
$742$	−0.896538	−0.0329129
$743$	13.4053	0.491794	0.245897	−	0.969296i	$-0.420918\pi$
$743$	13.4053	0.491794	0.245897	+	0.969296i	$0.420918\pi$
$744$	−1.31819	−0.0483271
$745$	3.30247	0.120993
$746$	−0.752788	−0.0275615
$747$	−19.3711	−0.708752
$748$	26.6646	0.974953
$749$	−1.39554	−0.0509917
$750$	−0.108923	−0.00397732
$751$	−38.9788	−1.42236	−0.711178	−	0.703012i	$-0.751838\pi$
$751$	−38.9788	−1.42236	−0.711178	+	0.703012i	$0.751838\pi$
$752$	22.5764	0.823278
$753$	3.02155	0.110111
$754$	−0.544953	−0.0198460
$755$	−4.29499	−0.156311
$756$	−6.53219	−0.237574
$757$	−24.9760	−0.907767	−0.453883	−	0.891061i	$-0.649962\pi$
$757$	−24.9760	−0.907767	−0.453883	+	0.891061i	$0.649962\pi$
$758$	−3.54171	−0.128641
$759$	9.78702	0.355246
$760$	−4.03204	−0.146258
$761$	−17.2365	−0.624822	−0.312411	−	0.949947i	$-0.601137\pi$
$761$	−17.2365	−0.624822	−0.312411	+	0.949947i	$0.601137\pi$
$762$	−1.14012	−0.0413022
$763$	−5.62634	−0.203687
$764$	18.6088	0.673244
$765$	12.3707	0.447262
$766$	−1.12062	−0.0404898
$767$	2.99541	0.108158
$768$	−7.82931	−0.282516
$769$	−1.55266	−0.0559903	−0.0279952	−	0.999608i	$-0.508912\pi$
$769$	−1.55266	−0.0559903	−0.0279952	+	0.999608i	$0.508912\pi$
$770$	0.539523	0.0194430
$771$	0.802471	0.0289003
$772$	−33.5490	−1.20746
$773$	3.66633	0.131869	0.0659344	−	0.997824i	$-0.478997\pi$
$773$	3.66633	0.131869	0.0659344	+	0.997824i	$0.478997\pi$
$774$	3.90663	0.140421
$775$	−3.05170	−0.109620
$776$	5.39398	0.193632
$777$	−6.19648	−0.222298
$778$	−2.21810	−0.0795228
$779$	−15.2129	−0.545058
$780$	−2.06290	−0.0738637
$781$	−8.09006	−0.289485
$782$	4.94112	0.176694
$783$	5.47164	0.195541
$784$	3.79507	0.135538
$785$	17.0859	0.609821
$786$	0.743409	0.0265165
$787$	44.7383	1.59475	0.797374	−	0.603485i	$-0.206222\pi$
$787$	44.7383	1.59475	0.797374	+	0.603485i	$0.206222\pi$
$788$	21.6117	0.769886
$789$	−5.82682	−0.207440
$790$	−1.95069	−0.0694025
$791$	5.66598	0.201459
$792$	5.67971	0.201820
$793$	11.3141	0.401777
$794$	3.00246	0.106553
$795$	2.84280	0.100824
$796$	32.5004	1.15195
$797$	10.0921	0.357480	0.178740	−	0.983896i	$-0.442798\pi$
$797$	10.0921	0.357480	0.178740	+	0.983896i	$0.442798\pi$
$798$	−0.597531	−0.0211524
$799$	−27.7221	−0.980738
$800$	2.17338	0.0768405
$801$	−24.4148	−0.862654
$802$	−5.45046	−0.192462
$803$	9.45299	0.333589
$804$	13.9583	0.492273
$805$	−5.72088	−0.201635
$806$	1.01003	0.0355769
$807$	−8.10554	−0.285329
$808$	−0.960427	−0.0337877
$809$	47.2616	1.66163	0.830815	−	0.556549i	$-0.187875\pi$
$809$	47.2616	1.66163	0.830815	+	0.556549i	$0.187875\pi$
$810$	1.11406	0.0391440
$811$	51.2648	1.80015	0.900075	−	0.435735i	$-0.143512\pi$
$811$	51.2648	1.80015	0.900075	+	0.435735i	$0.143512\pi$
$812$	−3.23646	−0.113578
$813$	3.47910	0.122017
$814$	5.68859	0.199385
$815$	−18.0079	−0.630788
$816$	10.3935	0.363844
$817$	−43.5580	−1.52390
$818$	0.267497	0.00935280
$819$	4.74050	0.165647
$820$	5.45102	0.190358
$821$	46.8387	1.63468	0.817340	−	0.576155i	$-0.195448\pi$
$821$	46.8387	1.63468	0.817340	+	0.576155i	$0.195448\pi$
$822$	1.46593	0.0511303
$823$	−52.3571	−1.82506	−0.912528	−	0.409015i	$-0.865872\pi$
$823$	−52.3571	−1.82506	−0.912528	+	0.409015i	$0.865872\pi$
$824$	1.58848	0.0553373
$825$	−1.71076	−0.0595608
$826$	−0.310890	−0.0108172
$827$	−10.0582	−0.349759	−0.174879	−	0.984590i	$-0.555954\pi$
$827$	−10.0582	−0.349759	−0.174879	+	0.984590i	$0.555954\pi$
$828$	−29.8518	−1.03742
$829$	−23.5475	−0.817839	−0.408919	−	0.912571i	$-0.634094\pi$
$829$	−23.5475	−0.817839	−0.408919	+	0.912571i	$0.634094\pi$
$830$	1.35246	0.0469445
$831$	16.6886	0.578922
$832$	12.8348	0.444967
$833$	−4.66005	−0.161461
$834$	0.184612	0.00639257
$835$	−10.8305	−0.374805
$836$	−31.3894	−1.08562
$837$	−10.1413	−0.350535
$838$	−3.75409	−0.129683
$839$	26.3238	0.908800	0.454400	−	0.890798i	$-0.349854\pi$
$839$	26.3238	0.908800	0.454400	+	0.890798i	$0.349854\pi$
$840$	0.431952	0.0149038
$841$	−26.2890	−0.906517
$842$	0.836365	0.0288230
$843$	−3.17501	−0.109353
$844$	17.9899	0.619239
$845$	−9.81107	−0.337511
$846$	−2.92691	−0.100629
$847$	−2.52624	−0.0868027
$848$	−18.3577	−0.630404
$849$	12.7409	0.437267
$850$	−0.863699	−0.0296246
$851$	−60.3196	−2.06773
$852$	−3.21048	−0.109989
$853$	−0.868673	−0.0297428	−0.0148714	−	0.999889i	$-0.504734\pi$
$853$	−0.868673	−0.0297428	−0.0148714	+	0.999889i	$0.504734\pi$
$854$	−1.17428	−0.0401829
$855$	−14.5627	−0.498033
$856$	1.02572	0.0350582
$857$	27.6798	0.945523	0.472761	−	0.881190i	$-0.343257\pi$
$857$	27.6798	0.945523	0.472761	+	0.881190i	$0.343257\pi$
$858$	0.566216	0.0193303
$859$	−48.8412	−1.66644	−0.833221	−	0.552940i	$-0.813506\pi$
$859$	−48.8412	−1.66644	−0.833221	+	0.552940i	$0.813506\pi$
$860$	15.6075	0.532213
$861$	1.62975	0.0555419
$862$	−3.87792	−0.132082
$863$	12.8552	0.437596	0.218798	−	0.975770i	$-0.429786\pi$
$863$	12.8552	0.437596	0.218798	+	0.975770i	$0.429786\pi$
$864$	7.22251	0.245715
$865$	10.6813	0.363175
$866$	1.90810	0.0648400
$867$	−2.77160	−0.0941284
$868$	5.99856	0.203605
$869$	−30.6376	−1.03931
$870$	−0.179344	−0.00608032
$871$	−21.5775	−0.731125
$872$	4.13534	0.140040
$873$	19.4816	0.659353
$874$	−5.81666	−0.196751
$875$	1.00000	0.0338062
$876$	3.75135	0.126746
$877$	29.0333	0.980385	0.490192	−	0.871614i	$-0.336927\pi$
$877$	29.0333	0.980385	0.490192	+	0.871614i	$0.336927\pi$
$878$	1.91074	0.0644845
$879$	5.49106	0.185209
$880$	11.0473	0.372406
$881$	13.3857	0.450975	0.225487	−	0.974246i	$-0.427603\pi$
$881$	13.3857	0.450975	0.225487	+	0.974246i	$0.427603\pi$
$882$	−0.492010	−0.0165668
$883$	27.9678	0.941190	0.470595	−	0.882349i	$-0.344039\pi$
$883$	27.9678	0.941190	0.470595	+	0.882349i	$0.344039\pi$
$884$	−16.3576	−0.550165
$885$	0.985790	0.0331370
$886$	−2.25036	−0.0756025
$887$	36.2657	1.21768	0.608842	−	0.793291i	$-0.291634\pi$
$887$	36.2657	1.21768	0.608842	+	0.793291i	$0.291634\pi$
$888$	4.55440	0.152836
$889$	10.4672	0.351057
$890$	1.70460	0.0571383
$891$	17.4974	0.586186
$892$	−28.5063	−0.954461
$893$	32.6343	1.09207
$894$	−0.359716	−0.0120307
$895$	18.3496	0.613360
$896$	−5.67886	−0.189717
$897$	−6.00392	−0.200465
$898$	−1.70814	−0.0570013
$899$	−5.02465	−0.167582
$900$	5.21805	0.173935
$901$	22.5418	0.750975
$902$	−1.49617	−0.0498171
$903$	4.66636	0.155287
$904$	−4.16448	−0.138509
$905$	−13.6494	−0.453723
$906$	0.467825	0.0155425
$907$	34.2434	1.13703	0.568517	−	0.822671i	$-0.307517\pi$
$907$	34.2434	1.13703	0.568517	+	0.822671i	$0.307517\pi$
$908$	25.0004	0.829668
$909$	−3.46881	−0.115053
$910$	−0.330974	−0.0109717
$911$	−6.06864	−0.201063	−0.100531	−	0.994934i	$-0.532054\pi$
$911$	−6.06864	−0.201063	−0.100531	+	0.994934i	$0.532054\pi$
$912$	−12.2351	−0.405146
$913$	21.2417	0.702999
$914$	1.06651	0.0352770
$915$	3.72348	0.123094
$916$	1.96565	0.0649469
$917$	−6.82506	−0.225383
$918$	−2.87022	−0.0947314
$919$	−18.1552	−0.598884	−0.299442	−	0.954115i	$-0.596800\pi$
$919$	−18.1552	−0.598884	−0.299442	+	0.954115i	$0.596800\pi$
$920$	4.20483	0.138629
$921$	11.8260	0.389681
$922$	−6.48747	−0.213654
$923$	4.96291	0.163356
$924$	3.36274	0.110626
$925$	10.5438	0.346677
$926$	−7.30532	−0.240068
$927$	5.73717	0.188433
$928$	3.57849	0.117470
$929$	−32.1179	−1.05375	−0.526876	−	0.849942i	$-0.676637\pi$
$929$	−32.1179	−1.05375	−0.526876	+	0.849942i	$0.676637\pi$
$930$	0.332401	0.0108999
$931$	5.48579	0.179789
$932$	−51.2119	−1.67750
$933$	3.50250	0.114667
$934$	−2.84570	−0.0931143
$935$	−13.5653	−0.443632
$936$	−3.48426	−0.113887
$937$	19.4328	0.634841	0.317420	−	0.948285i	$-0.397183\pi$
$937$	19.4328	0.634841	0.317420	+	0.948285i	$0.397183\pi$
$938$	2.23949	0.0731221
$939$	−0.317591	−0.0103642
$940$	−11.6934	−0.381397
$941$	48.2985	1.57449	0.787244	−	0.616642i	$-0.211507\pi$
$941$	48.2985	1.57449	0.787244	+	0.616642i	$0.211507\pi$
$942$	−1.86105	−0.0606364
$943$	15.8648	0.516629
$944$	−6.36582	−0.207190
$945$	3.32317	0.108103
$946$	−4.28389	−0.139281
$947$	−36.4902	−1.18577	−0.592886	−	0.805287i	$-0.702011\pi$
$947$	−36.4902	−1.18577	−0.592886	+	0.805287i	$0.702011\pi$
$948$	−12.1583	−0.394883
$949$	−5.79901	−0.188244
$950$	1.01674	0.0329875
$951$	−7.80293	−0.253027
$952$	3.42513	0.111009
$953$	8.07101	0.261446	0.130723	−	0.991419i	$-0.458270\pi$
$953$	8.07101	0.261446	0.130723	+	0.991419i	$0.458270\pi$
$954$	2.37997	0.0770542
$955$	−9.46702	−0.306345
$956$	−0.241820	−0.00782101
$957$	−2.81678	−0.0910535
$958$	1.82289	0.0588948
$959$	−13.4584	−0.434593
$960$	4.22394	0.136327
$961$	−21.6871	−0.699585
$962$	−3.48971	−0.112513
$963$	3.70461	0.119379
$964$	43.7959	1.41057
$965$	17.0677	0.549428
$966$	0.623138	0.0200491
$967$	43.8140	1.40896	0.704482	−	0.709722i	$-0.251179\pi$
$967$	43.8140	1.40896	0.704482	+	0.709722i	$0.251179\pi$
$968$	1.85678	0.0596793
$969$	15.0238	0.482634
$970$	−1.36017	−0.0436726
$971$	−23.8027	−0.763864	−0.381932	−	0.924191i	$-0.624741\pi$
$971$	−23.8027	−0.763864	−0.381932	+	0.924191i	$0.624741\pi$
$972$	26.5403	0.851280
$973$	−1.69487	−0.0543352
$974$	4.76754	0.152762
$975$	1.04948	0.0336101
$976$	−24.0447	−0.769651
$977$	−7.21336	−0.230776	−0.115388	−	0.993321i	$-0.536811\pi$
$977$	−7.21336	−0.230776	−0.115388	+	0.993321i	$0.536811\pi$
$978$	1.96148	0.0627212
$979$	26.7725	0.855653
$980$	−1.96565	−0.0627903
$981$	14.9358	0.476863
$982$	4.65248	0.148467
$983$	46.1446	1.47178	0.735891	−	0.677100i	$-0.236763\pi$
$983$	46.1446	1.47178	0.735891	+	0.677100i	$0.236763\pi$
$984$	−1.19787	−0.0381865
$985$	−10.9947	−0.350321
$986$	−1.42209	−0.0452886
$987$	−3.49611	−0.111282
$988$	19.2561	0.612617
$989$	45.4246	1.44442
$990$	−1.43223	−0.0455191
$991$	7.08633	0.225105	0.112552	−	0.993646i	$-0.464097\pi$
$991$	7.08633	0.225105	0.112552	+	0.993646i	$0.464097\pi$
$992$	−6.63249	−0.210582
$993$	14.8119	0.470040
$994$	−0.515093	−0.0163377
$995$	−16.5342	−0.524169
$996$	8.42962	0.267103
$997$	7.10111	0.224894	0.112447	−	0.993658i	$-0.464131\pi$
$997$	7.10111	0.224894	0.112447	+	0.993658i	$0.464131\pi$
$998$	5.99633	0.189810
$999$	35.0387	1.10858

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8015.2.a.h.1.22	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8015.2.a.h.1.22	✓	38	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:	\( N \)	\(=\)	\( 8015 = 5 \cdot 7 \cdot 229 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8015.a (trivial)

\(f(q)\)	\(=\)	\(q+0.185341 q^{2} -0.587692 q^{3} -1.96565 q^{4} +1.00000 q^{5} -0.108923 q^{6} +1.00000 q^{7} -0.734998 q^{8} -2.65462 q^{9} +O(q^{10})\)	\(q+0.185341 q^{2} -0.587692 q^{3} -1.96565 q^{4} +1.00000 q^{5} -0.108923 q^{6} +1.00000 q^{7} -0.734998 q^{8} -2.65462 q^{9} +0.185341 q^{10} +2.91097 q^{11} +1.15520 q^{12} -1.78576 q^{13} +0.185341 q^{14} -0.587692 q^{15} +3.79507 q^{16} -4.66005 q^{17} -0.492010 q^{18} +5.48579 q^{19} -1.96565 q^{20} -0.587692 q^{21} +0.539523 q^{22} -5.72088 q^{23} +0.431952 q^{24} +1.00000 q^{25} -0.330974 q^{26} +3.32317 q^{27} -1.96565 q^{28} +1.64651 q^{29} -0.108923 q^{30} -3.05170 q^{31} +2.17338 q^{32} -1.71076 q^{33} -0.863699 q^{34} +1.00000 q^{35} +5.21805 q^{36} +10.5438 q^{37} +1.01674 q^{38} +1.04948 q^{39} -0.734998 q^{40} -2.77314 q^{41} -0.108923 q^{42} -7.94015 q^{43} -5.72195 q^{44} -2.65462 q^{45} -1.06031 q^{46} +5.94889 q^{47} -2.23033 q^{48} +1.00000 q^{49} +0.185341 q^{50} +2.73867 q^{51} +3.51017 q^{52} -4.83723 q^{53} +0.615921 q^{54} +2.91097 q^{55} -0.734998 q^{56} -3.22395 q^{57} +0.305166 q^{58} -1.67739 q^{59} +1.15520 q^{60} -6.33576 q^{61} -0.565605 q^{62} -2.65462 q^{63} -7.18733 q^{64} -1.78576 q^{65} -0.317073 q^{66} +12.0831 q^{67} +9.16002 q^{68} +3.36212 q^{69} +0.185341 q^{70} -2.77916 q^{71} +1.95114 q^{72} +3.24737 q^{73} +1.95419 q^{74} -0.587692 q^{75} -10.7831 q^{76} +2.91097 q^{77} +0.194511 q^{78} -10.5249 q^{79} +3.79507 q^{80} +6.01085 q^{81} -0.513977 q^{82} +7.29713 q^{83} +1.15520 q^{84} -4.66005 q^{85} -1.47164 q^{86} -0.967642 q^{87} -2.13956 q^{88} +9.19710 q^{89} -0.492010 q^{90} -1.78576 q^{91} +11.2452 q^{92} +1.79346 q^{93} +1.10257 q^{94} +5.48579 q^{95} -1.27728 q^{96} -7.33877 q^{97} +0.185341 q^{98} -7.72752 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 6 q^{2} - 9 q^{3} + 24 q^{4} + 38 q^{5} - 10 q^{6} + 38 q^{7} - 21 q^{8} + 13 q^{9}+O(q^{10}) \) 38 * q - 6 * q^2 - 9 * q^3 + 24 * q^4 + 38 * q^5 - 10 * q^6 + 38 * q^7 - 21 * q^8 + 13 * q^9	\( 38 q - 6 q^{2} - 9 q^{3} + 24 q^{4} + 38 q^{5} - 10 q^{6} + 38 q^{7} - 21 q^{8} + 13 q^{9} - 6 q^{10} - 18 q^{11} - 20 q^{12} - 25 q^{13} - 6 q^{14} - 9 q^{15} - 21 q^{17} - 7 q^{18} - 14 q^{19} + 24 q^{20} - 9 q^{21} - 11 q^{22} - 16 q^{23} - 10 q^{24} + 38 q^{25} - 21 q^{26} - 15 q^{27} + 24 q^{28} - 52 q^{29} - 10 q^{30} - 30 q^{31} - 8 q^{32} - 13 q^{33} - 9 q^{34} + 38 q^{35} - 16 q^{36} - 47 q^{37} - 10 q^{38} - 50 q^{39} - 21 q^{40} - 35 q^{41} - 10 q^{42} - 18 q^{43} - 22 q^{44} + 13 q^{45} - 17 q^{46} - 23 q^{47} - 13 q^{48} + 38 q^{49} - 6 q^{50} - 5 q^{51} - 41 q^{52} - 12 q^{53} + 35 q^{54} - 18 q^{55} - 21 q^{56} - 3 q^{57} - 16 q^{58} - 16 q^{59} - 20 q^{60} - 47 q^{61} + 14 q^{62} + 13 q^{63} - 55 q^{64} - 25 q^{65} - 6 q^{66} - 54 q^{67} + 31 q^{68} - 95 q^{69} - 6 q^{70} - 41 q^{71} - 59 q^{73} - q^{74} - 9 q^{75} - 23 q^{76} - 18 q^{77} + 19 q^{78} - 117 q^{79} - 38 q^{81} + 19 q^{82} - 21 q^{83} - 20 q^{84} - 21 q^{85} - 12 q^{86} - 14 q^{87} - 40 q^{88} - 96 q^{89} - 7 q^{90} - 25 q^{91} + 53 q^{92} - 53 q^{93} - 28 q^{94} - 14 q^{95} + 40 q^{96} - 70 q^{97} - 6 q^{98} - 52 q^{99}+O(q^{100}) \) 38 * q - 6 * q^2 - 9 * q^3 + 24 * q^4 + 38 * q^5 - 10 * q^6 + 38 * q^7 - 21 * q^8 + 13 * q^9 - 6 * q^10 - 18 * q^11 - 20 * q^12 - 25 * q^13 - 6 * q^14 - 9 * q^15 - 21 * q^17 - 7 * q^18 - 14 * q^19 + 24 * q^20 - 9 * q^21 - 11 * q^22 - 16 * q^23 - 10 * q^24 + 38 * q^25 - 21 * q^26 - 15 * q^27 + 24 * q^28 - 52 * q^29 - 10 * q^30 - 30 * q^31 - 8 * q^32 - 13 * q^33 - 9 * q^34 + 38 * q^35 - 16 * q^36 - 47 * q^37 - 10 * q^38 - 50 * q^39 - 21 * q^40 - 35 * q^41 - 10 * q^42 - 18 * q^43 - 22 * q^44 + 13 * q^45 - 17 * q^46 - 23 * q^47 - 13 * q^48 + 38 * q^49 - 6 * q^50 - 5 * q^51 - 41 * q^52 - 12 * q^53 + 35 * q^54 - 18 * q^55 - 21 * q^56 - 3 * q^57 - 16 * q^58 - 16 * q^59 - 20 * q^60 - 47 * q^61 + 14 * q^62 + 13 * q^63 - 55 * q^64 - 25 * q^65 - 6 * q^66 - 54 * q^67 + 31 * q^68 - 95 * q^69 - 6 * q^70 - 41 * q^71 - 59 * q^73 - q^74 - 9 * q^75 - 23 * q^76 - 18 * q^77 + 19 * q^78 - 117 * q^79 - 38 * q^81 + 19 * q^82 - 21 * q^83 - 20 * q^84 - 21 * q^85 - 12 * q^86 - 14 * q^87 - 40 * q^88 - 96 * q^89 - 7 * q^90 - 25 * q^91 + 53 * q^92 - 53 * q^93 - 28 * q^94 - 14 * q^95 + 40 * q^96 - 70 * q^97 - 6 * q^98 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more