LMFDB - Embedded newform 4015.2.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4015,2,Mod(1,4015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4015, 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("4015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4015 = 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4015.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0599364115$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	4015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.32698 q^{2} -0.285099 q^{3} +3.41485 q^{4} +1.00000 q^{5} +0.663421 q^{6} -4.31195 q^{7} -3.29232 q^{8} -2.91872 q^{9} +O(q^{10})$	\(q-2.32698 q^{2} -0.285099 q^{3} +3.41485 q^{4} +1.00000 q^{5} +0.663421 q^{6} -4.31195 q^{7} -3.29232 q^{8} -2.91872 q^{9} -2.32698 q^{10} +1.00000 q^{11} -0.973570 q^{12} -0.0311482 q^{13} +10.0338 q^{14} -0.285099 q^{15} +0.831478 q^{16} +3.30972 q^{17} +6.79181 q^{18} -8.69443 q^{19} +3.41485 q^{20} +1.22934 q^{21} -2.32698 q^{22} +6.33292 q^{23} +0.938638 q^{24} +1.00000 q^{25} +0.0724814 q^{26} +1.68742 q^{27} -14.7247 q^{28} +0.503184 q^{29} +0.663421 q^{30} +3.04020 q^{31} +4.64980 q^{32} -0.285099 q^{33} -7.70166 q^{34} -4.31195 q^{35} -9.96697 q^{36} -4.88462 q^{37} +20.2318 q^{38} +0.00888034 q^{39} -3.29232 q^{40} +2.69066 q^{41} -2.86064 q^{42} +9.88240 q^{43} +3.41485 q^{44} -2.91872 q^{45} -14.7366 q^{46} +11.1154 q^{47} -0.237054 q^{48} +11.5930 q^{49} -2.32698 q^{50} -0.943599 q^{51} -0.106366 q^{52} -1.80985 q^{53} -3.92660 q^{54} +1.00000 q^{55} +14.1963 q^{56} +2.47878 q^{57} -1.17090 q^{58} +0.636456 q^{59} -0.973570 q^{60} -3.00936 q^{61} -7.07449 q^{62} +12.5854 q^{63} -12.4830 q^{64} -0.0311482 q^{65} +0.663421 q^{66} +5.95480 q^{67} +11.3022 q^{68} -1.80551 q^{69} +10.0338 q^{70} -1.01590 q^{71} +9.60935 q^{72} -1.00000 q^{73} +11.3664 q^{74} -0.285099 q^{75} -29.6901 q^{76} -4.31195 q^{77} -0.0206644 q^{78} -4.42245 q^{79} +0.831478 q^{80} +8.27507 q^{81} -6.26112 q^{82} +2.68419 q^{83} +4.19799 q^{84} +3.30972 q^{85} -22.9962 q^{86} -0.143457 q^{87} -3.29232 q^{88} -8.77823 q^{89} +6.79181 q^{90} +0.134310 q^{91} +21.6259 q^{92} -0.866759 q^{93} -25.8654 q^{94} -8.69443 q^{95} -1.32566 q^{96} +2.58435 q^{97} -26.9766 q^{98} -2.91872 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 5 q^{2} - 3 q^{3} + 15 q^{4} + 23 q^{5} - 15 q^{6} - 10 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) $ 23 * q - 5 * q^2 - 3 * q^3 + 15 * q^4 + 23 * q^5 - 15 * q^6 - 10 * q^7 - 12 * q^8 + 8 * q^9	\( 23 q - 5 q^{2} - 3 q^{3} + 15 q^{4} + 23 q^{5} - 15 q^{6} - 10 q^{7} - 12 q^{8} + 8 q^{9} - 5 q^{10} + 23 q^{11} + 4 q^{12} - 19 q^{13} - 5 q^{14} - 3 q^{15} - q^{16} - 26 q^{17} + 5 q^{18} - 34 q^{19} + 15 q^{20} - 26 q^{21} - 5 q^{22} - 4 q^{23} - 23 q^{24} + 23 q^{25} - 13 q^{26} - 3 q^{27} - 28 q^{28} - 36 q^{29} - 15 q^{30} - 24 q^{31} - 19 q^{32} - 3 q^{33} - 4 q^{34} - 10 q^{35} - 14 q^{36} + 4 q^{37} - 15 q^{38} - 34 q^{39} - 12 q^{40} - 74 q^{41} + 7 q^{42} - 15 q^{43} + 15 q^{44} + 8 q^{45} + 3 q^{46} + q^{47} + 29 q^{48} + q^{49} - 5 q^{50} - 47 q^{51} - 23 q^{52} - 6 q^{53} - 11 q^{54} + 23 q^{55} - 20 q^{56} - 19 q^{57} + 22 q^{58} - 17 q^{59} + 4 q^{60} - 59 q^{61} + 38 q^{62} - 21 q^{63} - 18 q^{64} - 19 q^{65} - 15 q^{66} + 12 q^{67} + 5 q^{68} - 8 q^{69} - 5 q^{70} - 34 q^{71} + 21 q^{72} - 23 q^{73} - 9 q^{74} - 3 q^{75} - 53 q^{76} - 10 q^{77} + 23 q^{78} - 62 q^{79} - q^{80} + 7 q^{81} + 24 q^{82} - 8 q^{83} + 46 q^{84} - 26 q^{85} - 11 q^{86} + q^{87} - 12 q^{88} - 77 q^{89} + 5 q^{90} - 6 q^{91} + 47 q^{92} - 32 q^{94} - 34 q^{95} - 53 q^{96} - 21 q^{97} - 3 q^{98} + 8 q^{99}+O(q^{100}) \) 23 * q - 5 * q^2 - 3 * q^3 + 15 * q^4 + 23 * q^5 - 15 * q^6 - 10 * q^7 - 12 * q^8 + 8 * q^9 - 5 * q^10 + 23 * q^11 + 4 * q^12 - 19 * q^13 - 5 * q^14 - 3 * q^15 - q^16 - 26 * q^17 + 5 * q^18 - 34 * q^19 + 15 * q^20 - 26 * q^21 - 5 * q^22 - 4 * q^23 - 23 * q^24 + 23 * q^25 - 13 * q^26 - 3 * q^27 - 28 * q^28 - 36 * q^29 - 15 * q^30 - 24 * q^31 - 19 * q^32 - 3 * q^33 - 4 * q^34 - 10 * q^35 - 14 * q^36 + 4 * q^37 - 15 * q^38 - 34 * q^39 - 12 * q^40 - 74 * q^41 + 7 * q^42 - 15 * q^43 + 15 * q^44 + 8 * q^45 + 3 * q^46 + q^47 + 29 * q^48 + q^49 - 5 * q^50 - 47 * q^51 - 23 * q^52 - 6 * q^53 - 11 * q^54 + 23 * q^55 - 20 * q^56 - 19 * q^57 + 22 * q^58 - 17 * q^59 + 4 * q^60 - 59 * q^61 + 38 * q^62 - 21 * q^63 - 18 * q^64 - 19 * q^65 - 15 * q^66 + 12 * q^67 + 5 * q^68 - 8 * q^69 - 5 * q^70 - 34 * q^71 + 21 * q^72 - 23 * q^73 - 9 * q^74 - 3 * q^75 - 53 * q^76 - 10 * q^77 + 23 * q^78 - 62 * q^79 - q^80 + 7 * q^81 + 24 * q^82 - 8 * q^83 + 46 * q^84 - 26 * q^85 - 11 * q^86 + q^87 - 12 * q^88 - 77 * q^89 + 5 * q^90 - 6 * q^91 + 47 * q^92 - 32 * q^94 - 34 * q^95 - 53 * q^96 - 21 * q^97 - 3 * q^98 + 8 * 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 $

$2$	−2.32698	−1.64542	−0.822712	−	0.568458i	$-0.807540\pi$
$2$	−2.32698	−1.64542	−0.822712	+	0.568458i	$0.807540\pi$
$3$	−0.285099	−0.164602	−0.0823011	−	0.996608i	$-0.526227\pi$
$3$	−0.285099	−0.164602	−0.0823011	+	0.996608i	$0.526227\pi$
$4$	3.41485	1.70742
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.663421	0.270840
$7$	−4.31195	−1.62977	−0.814883	−	0.579626i	$-0.803199\pi$
$7$	−4.31195	−1.62977	−0.814883	+	0.579626i	$0.803199\pi$
$8$	−3.29232	−1.16401
$9$	−2.91872	−0.972906
$10$	−2.32698	−0.735856
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.973570	−0.281045
$13$	−0.0311482	−0.00863896	−0.00431948	−	0.999991i	$-0.501375\pi$
$13$	−0.0311482	−0.00863896	−0.00431948	+	0.999991i	$0.501375\pi$
$14$	10.0338	2.68166
$15$	−0.285099	−0.0736123
$16$	0.831478	0.207870
$17$	3.30972	0.802725	0.401363	−	0.915919i	$-0.368537\pi$
$17$	3.30972	0.802725	0.401363	+	0.915919i	$0.368537\pi$
$18$	6.79181	1.60084
$19$	−8.69443	−1.99464	−0.997320	−	0.0731694i	$-0.976689\pi$
$19$	−8.69443	−1.99464	−0.997320	+	0.0731694i	$0.976689\pi$
$20$	3.41485	0.763583
$21$	1.22934	0.268263
$22$	−2.32698	−0.496114
$23$	6.33292	1.32050	0.660252	−	0.751044i	$-0.270449\pi$
$23$	6.33292	1.32050	0.660252	+	0.751044i	$0.270449\pi$
$24$	0.938638	0.191599
$25$	1.00000	0.200000
$26$	0.0724814	0.0142148
$27$	1.68742	0.324745
$28$	−14.7247	−2.78270
$29$	0.503184	0.0934389	0.0467194	−	0.998908i	$-0.485123\pi$
$29$	0.503184	0.0934389	0.0467194	+	0.998908i	$0.485123\pi$
$30$	0.663421	0.121124
$31$	3.04020	0.546036	0.273018	−	0.962009i	$-0.411978\pi$
$31$	3.04020	0.546036	0.273018	+	0.962009i	$0.411978\pi$
$32$	4.64980	0.821977
$33$	−0.285099	−0.0496294
$34$	−7.70166	−1.32082
$35$	−4.31195	−0.728853
$36$	−9.96697	−1.66116
$37$	−4.88462	−0.803027	−0.401513	−	0.915853i	$-0.631516\pi$
$37$	−4.88462	−0.803027	−0.401513	+	0.915853i	$0.631516\pi$
$38$	20.2318	3.28203
$39$	0.00888034	0.00142199
$40$	−3.29232	−0.520561
$41$	2.69066	0.420211	0.210105	−	0.977679i	$-0.432619\pi$
$41$	2.69066	0.420211	0.210105	+	0.977679i	$0.432619\pi$
$42$	−2.86064	−0.441407
$43$	9.88240	1.50705	0.753526	−	0.657418i	$-0.228352\pi$
$43$	9.88240	1.50705	0.753526	+	0.657418i	$0.228352\pi$
$44$	3.41485	0.514807
$45$	−2.91872	−0.435097
$46$	−14.7366	−2.17279
$47$	11.1154	1.62135	0.810677	−	0.585494i	$-0.199099\pi$
$47$	11.1154	1.62135	0.810677	+	0.585494i	$0.199099\pi$
$48$	−0.237054	−0.0342158
$49$	11.5930	1.65614
$50$	−2.32698	−0.329085
$51$	−0.943599	−0.132130
$52$	−0.106366	−0.0147504
$53$	−1.80985	−0.248602	−0.124301	−	0.992245i	$-0.539669\pi$
$53$	−1.80985	−0.248602	−0.124301	+	0.992245i	$0.539669\pi$
$54$	−3.92660	−0.534343
$55$	1.00000	0.134840
$56$	14.1963	1.89707
$57$	2.47878	0.328322
$58$	−1.17090	−0.153747
$59$	0.636456	0.0828595	0.0414298	−	0.999141i	$-0.486809\pi$
$59$	0.636456	0.0828595	0.0414298	+	0.999141i	$0.486809\pi$
$60$	−0.973570	−0.125687
$61$	−3.00936	−0.385309	−0.192654	−	0.981267i	$-0.561710\pi$
$61$	−3.00936	−0.385309	−0.192654	+	0.981267i	$0.561710\pi$
$62$	−7.07449	−0.898461
$63$	12.5854	1.58561
$64$	−12.4830	−1.56037
$65$	−0.0311482	−0.00386346
$66$	0.663421	0.0816615
$67$	5.95480	0.727495	0.363747	−	0.931498i	$-0.381497\pi$
$67$	5.95480	0.727495	0.363747	+	0.931498i	$0.381497\pi$
$68$	11.3022	1.37059
$69$	−1.80551	−0.217358
$70$	10.0338	1.19927
$71$	−1.01590	−0.120565	−0.0602824	−	0.998181i	$-0.519200\pi$
$71$	−1.01590	−0.120565	−0.0602824	+	0.998181i	$0.519200\pi$
$72$	9.60935	1.13247
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	11.3664	1.32132
$75$	−0.285099	−0.0329204
$76$	−29.6901	−3.40569
$77$	−4.31195	−0.491393
$78$	−0.0206644	−0.00233978
$79$	−4.42245	−0.497564	−0.248782	−	0.968559i	$-0.580030\pi$
$79$	−4.42245	−0.497564	−0.248782	+	0.968559i	$0.580030\pi$
$80$	0.831478	0.0929621
$81$	8.27507	0.919452
$82$	−6.26112	−0.691425
$83$	2.68419	0.294628	0.147314	−	0.989090i	$-0.452937\pi$
$83$	2.68419	0.294628	0.147314	+	0.989090i	$0.452937\pi$
$84$	4.19799	0.458038
$85$	3.30972	0.358990
$86$	−22.9962	−2.47974
$87$	−0.143457	−0.0153802
$88$	−3.29232	−0.350962
$89$	−8.77823	−0.930490	−0.465245	−	0.885182i	$-0.654034\pi$
$89$	−8.77823	−0.930490	−0.465245	+	0.885182i	$0.654034\pi$
$90$	6.79181	0.715919
$91$	0.134310	0.0140795
$92$	21.6259	2.25466
$93$	−0.866759	−0.0898787
$94$	−25.8654	−2.66782
$95$	−8.69443	−0.892030
$96$	−1.32566	−0.135299
$97$	2.58435	0.262401	0.131201	−	0.991356i	$-0.458117\pi$
$97$	2.58435	0.262401	0.131201	+	0.991356i	$0.458117\pi$
$98$	−26.9766	−2.72505
$99$	−2.91872	−0.293342
$100$	3.41485	0.341485
$101$	4.51971	0.449728	0.224864	−	0.974390i	$-0.427806\pi$
$101$	4.51971	0.449728	0.224864	+	0.974390i	$0.427806\pi$
$102$	2.19574	0.217411
$103$	0.526511	0.0518787	0.0259394	−	0.999664i	$-0.491742\pi$
$103$	0.526511	0.0518787	0.0259394	+	0.999664i	$0.491742\pi$
$104$	0.102550	0.0100558
$105$	1.22934	0.119971
$106$	4.21149	0.409056
$107$	−17.2599	−1.66858	−0.834291	−	0.551325i	$-0.814122\pi$
$107$	−17.2599	−1.66858	−0.834291	+	0.551325i	$0.814122\pi$
$108$	5.76229	0.554476
$109$	−15.4329	−1.47821	−0.739104	−	0.673592i	$-0.764751\pi$
$109$	−15.4329	−1.47821	−0.739104	+	0.673592i	$0.764751\pi$
$110$	−2.32698	−0.221869
$111$	1.39260	0.132180
$112$	−3.58530	−0.338779
$113$	−6.46526	−0.608200	−0.304100	−	0.952640i	$-0.598356\pi$
$113$	−6.46526	−0.608200	−0.304100	+	0.952640i	$0.598356\pi$
$114$	−5.76807	−0.540229
$115$	6.33292	0.590547
$116$	1.71829	0.159540
$117$	0.0909129	0.00840490
$118$	−1.48102	−0.136339
$119$	−14.2714	−1.30825
$120$	0.938638	0.0856856
$121$	1.00000	0.0909091
$122$	7.00272	0.633997
$123$	−0.767106	−0.0691676
$124$	10.3818	0.932314
$125$	1.00000	0.0894427
$126$	−29.2860	−2.60900
$127$	−13.3836	−1.18760	−0.593802	−	0.804612i	$-0.702374\pi$
$127$	−13.3836	−1.18760	−0.593802	+	0.804612i	$0.702374\pi$
$128$	19.7480	1.74550
$129$	−2.81746	−0.248064
$130$	0.0724814	0.00635704
$131$	−11.5661	−1.01053	−0.505266	−	0.862964i	$-0.668606\pi$
$131$	−11.5661	−1.01053	−0.505266	+	0.862964i	$0.668606\pi$
$132$	−0.973570	−0.0847384
$133$	37.4900	3.25079
$134$	−13.8567	−1.19704
$135$	1.68742	0.145230
$136$	−10.8967	−0.934381
$137$	2.38738	0.203967	0.101984	−	0.994786i	$-0.467481\pi$
$137$	2.38738	0.203967	0.101984	+	0.994786i	$0.467481\pi$
$138$	4.20139	0.357646
$139$	12.1895	1.03390	0.516951	−	0.856015i	$-0.327067\pi$
$139$	12.1895	1.03390	0.516951	+	0.856015i	$0.327067\pi$
$140$	−14.7247	−1.24446
$141$	−3.16901	−0.266878
$142$	2.36398	0.198380
$143$	−0.0311482	−0.00260475
$144$	−2.42685	−0.202238
$145$	0.503184	0.0417871
$146$	2.32698	0.192582
$147$	−3.30514	−0.272604
$148$	−16.6802	−1.37111
$149$	−10.6561	−0.872979	−0.436490	−	0.899709i	$-0.643778\pi$
$149$	−10.6561	−0.872979	−0.436490	+	0.899709i	$0.643778\pi$
$150$	0.663421	0.0541681
$151$	−9.91269	−0.806683	−0.403342	−	0.915049i	$-0.632151\pi$
$151$	−9.91269	−0.806683	−0.403342	+	0.915049i	$0.632151\pi$
$152$	28.6248	2.32178
$153$	−9.66014	−0.780976
$154$	10.0338	0.808550
$155$	3.04020	0.244195
$156$	0.0303250	0.00242794
$157$	1.22836	0.0980339	0.0490169	−	0.998798i	$-0.484391\pi$
$157$	1.22836	0.0980339	0.0490169	+	0.998798i	$0.484391\pi$
$158$	10.2910	0.818705
$159$	0.515987	0.0409204
$160$	4.64980	0.367599
$161$	−27.3072	−2.15211
$162$	−19.2559	−1.51289
$163$	5.19811	0.407147	0.203574	−	0.979060i	$-0.434744\pi$
$163$	5.19811	0.407147	0.203574	+	0.979060i	$0.434744\pi$
$164$	9.18820	0.717478
$165$	−0.285099	−0.0221950
$166$	−6.24605	−0.484788
$167$	−16.6738	−1.29026	−0.645128	−	0.764075i	$-0.723196\pi$
$167$	−16.6738	−1.29026	−0.645128	+	0.764075i	$0.723196\pi$
$168$	−4.04737	−0.312261
$169$	−12.9990	−0.999925
$170$	−7.70166	−0.590691
$171$	25.3766	1.94060
$172$	33.7469	2.57317
$173$	8.07285	0.613767	0.306884	−	0.951747i	$-0.400714\pi$
$173$	8.07285	0.613767	0.306884	+	0.951747i	$0.400714\pi$
$174$	0.333823	0.0253070
$175$	−4.31195	−0.325953
$176$	0.831478	0.0626750
$177$	−0.181453	−0.0136389
$178$	20.4268	1.53105
$179$	2.36800	0.176993	0.0884963	−	0.996077i	$-0.471794\pi$
$179$	2.36800	0.176993	0.0884963	+	0.996077i	$0.471794\pi$
$180$	−9.96697	−0.742894
$181$	5.36157	0.398522	0.199261	−	0.979946i	$-0.436146\pi$
$181$	5.36157	0.398522	0.199261	+	0.979946i	$0.436146\pi$
$182$	−0.312536	−0.0231667
$183$	0.857966	0.0634227
$184$	−20.8500	−1.53708
$185$	−4.88462	−0.359125
$186$	2.01693	0.147889
$187$	3.30972	0.242031
$188$	37.9575	2.76834
$189$	−7.27609	−0.529258
$190$	20.2318	1.46777
$191$	−2.87826	−0.208264	−0.104132	−	0.994564i	$-0.533206\pi$
$191$	−2.87826	−0.208264	−0.104132	+	0.994564i	$0.533206\pi$
$192$	3.55889	0.256840
$193$	−3.06281	−0.220466	−0.110233	−	0.993906i	$-0.535160\pi$
$193$	−3.06281	−0.220466	−0.110233	+	0.993906i	$0.535160\pi$
$194$	−6.01374	−0.431761
$195$	0.00888034	0.000635934 0
$196$	39.5882	2.82773
$197$	21.1672	1.50810	0.754049	−	0.656818i	$-0.228098\pi$
$197$	21.1672	1.50810	0.754049	+	0.656818i	$0.228098\pi$
$198$	6.79181	0.482673
$199$	18.6806	1.32423	0.662117	−	0.749400i	$-0.269658\pi$
$199$	18.6806	1.32423	0.662117	+	0.749400i	$0.269658\pi$
$200$	−3.29232	−0.232802
$201$	−1.69771	−0.119747
$202$	−10.5173	−0.739994
$203$	−2.16971	−0.152283
$204$	−3.22225	−0.225602
$205$	2.69066	0.187924
$206$	−1.22518	−0.0853625
$207$	−18.4840	−1.28473
$208$	−0.0258991	−0.00179578
$209$	−8.69443	−0.601406
$210$	−2.86064	−0.197403
$211$	8.79179	0.605252	0.302626	−	0.953109i	$-0.402137\pi$
$211$	8.79179	0.605252	0.302626	+	0.953109i	$0.402137\pi$
$212$	−6.18036	−0.424469
$213$	0.289632	0.0198452
$214$	40.1636	2.74552
$215$	9.88240	0.673974
$216$	−5.55554	−0.378006
$217$	−13.1092	−0.889911
$218$	35.9122	2.43228
$219$	0.285099	0.0192652
$220$	3.41485	0.230229
$221$	−0.103092	−0.00693471
$222$	−3.24056	−0.217492
$223$	22.1826	1.48546	0.742730	−	0.669591i	$-0.233531\pi$
$223$	22.1826	1.48546	0.742730	+	0.669591i	$0.233531\pi$
$224$	−20.0497	−1.33963
$225$	−2.91872	−0.194581
$226$	15.0445	1.00075
$227$	−6.58234	−0.436885	−0.218443	−	0.975850i	$-0.570098\pi$
$227$	−6.58234	−0.436885	−0.218443	+	0.975850i	$0.570098\pi$
$228$	8.46464	0.560584
$229$	−25.8152	−1.70591	−0.852957	−	0.521981i	$-0.825193\pi$
$229$	−25.8152	−1.70591	−0.852957	+	0.521981i	$0.825193\pi$
$230$	−14.7366	−0.971701
$231$	1.22934	0.0808843
$232$	−1.65664	−0.108764
$233$	−26.4103	−1.73020	−0.865099	−	0.501601i	$-0.832744\pi$
$233$	−26.4103	−1.73020	−0.865099	+	0.501601i	$0.832744\pi$
$234$	−0.211553	−0.0138296
$235$	11.1154	0.725092
$236$	2.17340	0.141476
$237$	1.26084	0.0819002
$238$	33.2092	2.15263
$239$	10.9638	0.709191	0.354595	−	0.935020i	$-0.384619\pi$
$239$	10.9638	0.709191	0.354595	+	0.935020i	$0.384619\pi$
$240$	−0.237054	−0.0153018
$241$	4.04011	0.260246	0.130123	−	0.991498i	$-0.458463\pi$
$241$	4.04011	0.260246	0.130123	+	0.991498i	$0.458463\pi$
$242$	−2.32698	−0.149584
$243$	−7.42149	−0.476088
$244$	−10.2765	−0.657885
$245$	11.5930	0.740647
$246$	1.78504	0.113810
$247$	0.270816	0.0172316
$248$	−10.0093	−0.635592
$249$	−0.765260	−0.0484964
$250$	−2.32698	−0.147171
$251$	20.8027	1.31305	0.656527	−	0.754302i	$-0.272025\pi$
$251$	20.8027	1.31305	0.656527	+	0.754302i	$0.272025\pi$
$252$	42.9771	2.70731
$253$	6.33292	0.398147
$254$	31.1434	1.95411
$255$	−0.943599	−0.0590905
$256$	−20.9874	−1.31171
$257$	−9.91509	−0.618486	−0.309243	−	0.950983i	$-0.600076\pi$
$257$	−9.91509	−0.618486	−0.309243	+	0.950983i	$0.600076\pi$
$258$	6.55619	0.408171
$259$	21.0623	1.30875
$260$	−0.106366	−0.00659656
$261$	−1.46865	−0.0909072
$262$	26.9140	1.66275
$263$	28.7008	1.76977	0.884885	−	0.465810i	$-0.154237\pi$
$263$	28.7008	1.76977	0.884885	+	0.465810i	$0.154237\pi$
$264$	0.938638	0.0577692
$265$	−1.80985	−0.111178
$266$	−87.2385	−5.34894
$267$	2.50267	0.153161
$268$	20.3347	1.24214
$269$	−3.41057	−0.207946	−0.103973	−	0.994580i	$-0.533156\pi$
$269$	−3.41057	−0.207946	−0.103973	+	0.994580i	$0.533156\pi$
$270$	−3.92660	−0.238965
$271$	2.78079	0.168921	0.0844603	−	0.996427i	$-0.473083\pi$
$271$	2.78079	0.168921	0.0844603	+	0.996427i	$0.473083\pi$
$272$	2.75196	0.166862
$273$	−0.0382916	−0.00231751
$274$	−5.55539	−0.335613
$275$	1.00000	0.0603023
$276$	−6.16554	−0.371122
$277$	−26.0707	−1.56644	−0.783218	−	0.621747i	$-0.786423\pi$
$277$	−26.0707	−1.56644	−0.783218	+	0.621747i	$0.786423\pi$
$278$	−28.3648	−1.70121
$279$	−8.87349	−0.531242
$280$	14.1963	0.848393
$281$	−11.0617	−0.659886	−0.329943	−	0.944001i	$-0.607029\pi$
$281$	−11.0617	−0.659886	−0.329943	+	0.944001i	$0.607029\pi$
$282$	7.37422	0.439128
$283$	−0.584831	−0.0347646	−0.0173823	−	0.999849i	$-0.505533\pi$
$283$	−0.584831	−0.0347646	−0.0173823	+	0.999849i	$0.505533\pi$
$284$	−3.46913	−0.205855
$285$	2.47878	0.146830
$286$	0.0724814	0.00428591
$287$	−11.6020	−0.684845
$288$	−13.5715	−0.799707
$289$	−6.04574	−0.355632
$290$	−1.17090	−0.0687576
$291$	−0.736797	−0.0431918
$292$	−3.41485	−0.199839
$293$	10.5631	0.617104	0.308552	−	0.951207i	$-0.400156\pi$
$293$	10.5631	0.617104	0.308552	+	0.951207i	$0.400156\pi$
$294$	7.69101	0.448549
$295$	0.636456	0.0370559
$296$	16.0817	0.934732
$297$	1.68742	0.0979142
$298$	24.7965	1.43642
$299$	−0.197259	−0.0114078
$300$	−0.973570	−0.0562091
$301$	−42.6124	−2.45614
$302$	23.0667	1.32734
$303$	−1.28857	−0.0740262
$304$	−7.22923	−0.414625
$305$	−3.00936	−0.172315
$306$	22.4790	1.28504
$307$	−5.46285	−0.311781	−0.155891	−	0.987774i	$-0.549825\pi$
$307$	−5.46285	−0.311781	−0.155891	+	0.987774i	$0.549825\pi$
$308$	−14.7247	−0.839015
$309$	−0.150108	−0.00853935
$310$	−7.07449	−0.401804
$311$	−1.97319	−0.111889	−0.0559447	−	0.998434i	$-0.517817\pi$
$311$	−1.97319	−0.111889	−0.0559447	+	0.998434i	$0.517817\pi$
$312$	−0.0292369	−0.00165521
$313$	−23.2501	−1.31418	−0.657088	−	0.753814i	$-0.728212\pi$
$313$	−23.2501	−1.31418	−0.657088	+	0.753814i	$0.728212\pi$
$314$	−2.85837	−0.161307
$315$	12.5854	0.709106
$316$	−15.1020	−0.849553
$317$	−12.6850	−0.712459	−0.356230	−	0.934398i	$-0.615938\pi$
$317$	−12.6850	−0.712459	−0.356230	+	0.934398i	$0.615938\pi$
$318$	−1.20069	−0.0673315
$319$	0.503184	0.0281729
$320$	−12.4830	−0.697819
$321$	4.92080	0.274652
$322$	63.5435	3.54114
$323$	−28.7761	−1.60115
$324$	28.2581	1.56989
$325$	−0.0311482	−0.00172779
$326$	−12.0959	−0.669930
$327$	4.39992	0.243316
$328$	−8.85852	−0.489130
$329$	−47.9293	−2.64243
$330$	0.663421	0.0365201
$331$	2.32267	0.127666	0.0638328	−	0.997961i	$-0.479668\pi$
$331$	2.32267	0.127666	0.0638328	+	0.997961i	$0.479668\pi$
$332$	9.16608	0.503054
$333$	14.2568	0.781270
$334$	38.7996	2.12302
$335$	5.95480	0.325346
$336$	1.02217	0.0557637
$337$	−21.3542	−1.16324	−0.581620	−	0.813461i	$-0.697581\pi$
$337$	−21.3542	−1.16324	−0.581620	+	0.813461i	$0.697581\pi$
$338$	30.2485	1.64530
$339$	1.84324	0.100111
$340$	11.3022	0.612947
$341$	3.04020	0.164636
$342$	−59.0509	−3.19311
$343$	−19.8046	−1.06935
$344$	−32.5360	−1.75422
$345$	−1.80551	−0.0972054
$346$	−18.7854	−1.00991
$347$	30.6673	1.64631	0.823154	−	0.567818i	$-0.192212\pi$
$347$	30.6673	1.64631	0.823154	+	0.567818i	$0.192212\pi$
$348$	−0.489885	−0.0262606
$349$	−7.12549	−0.381419	−0.190709	−	0.981647i	$-0.561079\pi$
$349$	−7.12549	−0.381419	−0.190709	+	0.981647i	$0.561079\pi$
$350$	10.0338	0.536331
$351$	−0.0525602	−0.00280546
$352$	4.64980	0.247835
$353$	−35.4951	−1.88922	−0.944608	−	0.328202i	$-0.893557\pi$
$353$	−35.4951	−1.88922	−0.944608	+	0.328202i	$0.893557\pi$
$354$	0.422238	0.0224417
$355$	−1.01590	−0.0539183
$356$	−29.9763	−1.58874
$357$	4.06876	0.215341
$358$	−5.51029	−0.291228
$359$	−4.53769	−0.239490	−0.119745	−	0.992805i	$-0.538208\pi$
$359$	−4.53769	−0.239490	−0.119745	+	0.992805i	$0.538208\pi$
$360$	9.60935	0.506457
$361$	56.5931	2.97858
$362$	−12.4763	−0.655738
$363$	−0.285099	−0.0149638
$364$	0.458647	0.0240396
$365$	−1.00000	−0.0523424
$366$	−1.99647	−0.104357
$367$	−16.4472	−0.858536	−0.429268	−	0.903177i	$-0.641228\pi$
$367$	−16.4472	−0.858536	−0.429268	+	0.903177i	$0.641228\pi$
$368$	5.26568	0.274493
$369$	−7.85329	−0.408826
$370$	11.3664	0.590912
$371$	7.80399	0.405163
$372$	−2.95985	−0.153461
$373$	17.7205	0.917530	0.458765	−	0.888558i	$-0.348292\pi$
$373$	17.7205	0.917530	0.458765	+	0.888558i	$0.348292\pi$
$374$	−7.70166	−0.398243
$375$	−0.285099	−0.0147225
$376$	−36.5956	−1.88727
$377$	−0.0156733	−0.000807215 0
$378$	16.9313	0.870854
$379$	−26.8973	−1.38162	−0.690810	−	0.723036i	$-0.742746\pi$
$379$	−26.8973	−1.38162	−0.690810	+	0.723036i	$0.742746\pi$
$380$	−29.6901	−1.52307
$381$	3.81566	0.195482
$382$	6.69766	0.342682
$383$	−29.5927	−1.51211	−0.756057	−	0.654506i	$-0.772877\pi$
$383$	−29.5927	−1.51211	−0.756057	+	0.654506i	$0.772877\pi$
$384$	−5.63015	−0.287312
$385$	−4.31195	−0.219758
$386$	7.12711	0.362760
$387$	−28.8439	−1.46622
$388$	8.82516	0.448030
$389$	14.0634	0.713044	0.356522	−	0.934287i	$-0.383963\pi$
$389$	14.0634	0.713044	0.356522	+	0.934287i	$0.383963\pi$
$390$	−0.0206644	−0.00104638
$391$	20.9602	1.06000
$392$	−38.1677	−1.92776
$393$	3.29748	0.166336
$394$	−49.2556	−2.48146
$395$	−4.42245	−0.222518
$396$	−9.96697	−0.500859
$397$	27.1782	1.36403	0.682017	−	0.731336i	$-0.261103\pi$
$397$	27.1782	1.36403	0.682017	+	0.731336i	$0.261103\pi$
$398$	−43.4695	−2.17893
$399$	−10.6884	−0.535088
$400$	0.831478	0.0415739
$401$	17.0315	0.850512	0.425256	−	0.905073i	$-0.360184\pi$
$401$	17.0315	0.850512	0.425256	+	0.905073i	$0.360184\pi$
$402$	3.95054	0.197035
$403$	−0.0946968	−0.00471719
$404$	15.4341	0.767876
$405$	8.27507	0.411192
$406$	5.04887	0.250571
$407$	−4.88462	−0.242122
$408$	3.10663	0.153801
$409$	−30.9818	−1.53195	−0.765976	−	0.642869i	$-0.777744\pi$
$409$	−30.9818	−1.53195	−0.765976	+	0.642869i	$0.777744\pi$
$410$	−6.26112	−0.309215
$411$	−0.680640	−0.0335735
$412$	1.79796	0.0885789
$413$	−2.74437	−0.135042
$414$	43.0119	2.11392
$415$	2.68419	0.131762
$416$	−0.144833	−0.00710103
$417$	−3.47523	−0.170183
$418$	20.2318	0.989569
$419$	2.31126	0.112913	0.0564563	−	0.998405i	$-0.482020\pi$
$419$	2.31126	0.112913	0.0564563	+	0.998405i	$0.482020\pi$
$420$	4.19799	0.204841
$421$	−25.9390	−1.26419	−0.632096	−	0.774890i	$-0.717805\pi$
$421$	−25.9390	−1.26419	−0.632096	+	0.774890i	$0.717805\pi$
$422$	−20.4583	−0.995896
$423$	−32.4428	−1.57743
$424$	5.95860	0.289375
$425$	3.30972	0.160545
$426$	−0.673968	−0.0326539
$427$	12.9762	0.627963
$428$	−58.9400	−2.84897
$429$	0.00888034	0.000428747 0
$430$	−22.9962	−1.10897
$431$	−32.3963	−1.56048	−0.780238	−	0.625483i	$-0.784902\pi$
$431$	−32.3963	−1.56048	−0.780238	+	0.625483i	$0.784902\pi$
$432$	1.40306	0.0675045
$433$	15.4808	0.743960	0.371980	−	0.928241i	$-0.378679\pi$
$433$	15.4808	0.743960	0.371980	+	0.928241i	$0.378679\pi$
$434$	30.5049	1.46428
$435$	−0.143457	−0.00687825
$436$	−52.7011	−2.52393
$437$	−55.0611	−2.63393
$438$	−0.663421	−0.0316995
$439$	−30.4954	−1.45547	−0.727733	−	0.685860i	$-0.759426\pi$
$439$	−30.4954	−1.45547	−0.727733	+	0.685860i	$0.759426\pi$
$440$	−3.29232	−0.156955
$441$	−33.8366	−1.61127
$442$	0.239893	0.0114106
$443$	−23.1433	−1.09957	−0.549787	−	0.835305i	$-0.685291\pi$
$443$	−23.1433	−1.09957	−0.549787	+	0.835305i	$0.685291\pi$
$444$	4.75552	0.225687
$445$	−8.77823	−0.416128
$446$	−51.6186	−2.44421
$447$	3.03804	0.143694
$448$	53.8260	2.54304
$449$	−38.0179	−1.79417	−0.897087	−	0.441854i	$-0.854321\pi$
$449$	−38.0179	−1.79417	−0.897087	+	0.441854i	$0.854321\pi$
$450$	6.79181	0.320169
$451$	2.69066	0.126698
$452$	−22.0779	−1.03846
$453$	2.82610	0.132782
$454$	15.3170	0.718862
$455$	0.134310	0.00629654
$456$	−8.16092	−0.382170
$457$	−4.42192	−0.206849	−0.103424	−	0.994637i	$-0.532980\pi$
$457$	−4.42192	−0.206849	−0.103424	+	0.994637i	$0.532980\pi$
$458$	60.0714	2.80695
$459$	5.58490	0.260681
$460$	21.6259	1.00831
$461$	−29.7945	−1.38767	−0.693835	−	0.720134i	$-0.744080\pi$
$461$	−29.7945	−1.38767	−0.693835	+	0.720134i	$0.744080\pi$
$462$	−2.86064	−0.133089
$463$	−1.48607	−0.0690637	−0.0345319	−	0.999404i	$-0.510994\pi$
$463$	−1.48607	−0.0690637	−0.0345319	+	0.999404i	$0.510994\pi$
$464$	0.418386	0.0194231
$465$	−0.866759	−0.0401950
$466$	61.4564	2.84691
$467$	−0.245824	−0.0113754	−0.00568770	−	0.999984i	$-0.501810\pi$
$467$	−0.245824	−0.0113754	−0.00568770	+	0.999984i	$0.501810\pi$
$468$	0.310453	0.0143507
$469$	−25.6768	−1.18565
$470$	−25.8654	−1.19308
$471$	−0.350205	−0.0161366
$472$	−2.09542	−0.0964494
$473$	9.88240	0.454393
$474$	−2.93395	−0.134761
$475$	−8.69443	−0.398928
$476$	−48.7345	−2.23374
$477$	5.28244	0.241866
$478$	−25.5126	−1.16692
$479$	10.0841	0.460754	0.230377	−	0.973101i	$-0.426004\pi$
$479$	10.0841	0.460754	0.230377	+	0.973101i	$0.426004\pi$
$480$	−1.32566	−0.0605076
$481$	0.152147	0.00693732
$482$	−9.40126	−0.428215
$483$	7.78528	0.354242
$484$	3.41485	0.155220
$485$	2.58435	0.117349
$486$	17.2697	0.783368
$487$	−29.6086	−1.34169	−0.670847	−	0.741596i	$-0.734069\pi$
$487$	−29.6086	−1.34169	−0.670847	+	0.741596i	$0.734069\pi$
$488$	9.90777	0.448504
$489$	−1.48198	−0.0670173
$490$	−26.9766	−1.21868
$491$	16.1342	0.728126	0.364063	−	0.931374i	$-0.381389\pi$
$491$	16.1342	0.728126	0.364063	+	0.931374i	$0.381389\pi$
$492$	−2.61955	−0.118098
$493$	1.66540	0.0750057
$494$	−0.630184	−0.0283533
$495$	−2.91872	−0.131187
$496$	2.52786	0.113504
$497$	4.38051	0.196493
$498$	1.78075	0.0797971
$499$	42.6668	1.91003	0.955014	−	0.296559i	$-0.0958393\pi$
$499$	42.6668	1.91003	0.955014	+	0.296559i	$0.0958393\pi$
$500$	3.41485	0.152717
$501$	4.75368	0.212379
$502$	−48.4075	−2.16053
$503$	−5.43353	−0.242269	−0.121135	−	0.992636i	$-0.538653\pi$
$503$	−5.43353	−0.242269	−0.121135	+	0.992636i	$0.538653\pi$
$504$	−41.4351	−1.84567
$505$	4.51971	0.201124
$506$	−14.7366	−0.655121
$507$	3.70601	0.164590
$508$	−45.7030	−2.02774
$509$	9.93095	0.440182	0.220091	−	0.975479i	$-0.429365\pi$
$509$	9.93095	0.440182	0.220091	+	0.975479i	$0.429365\pi$
$510$	2.19574	0.0972289
$511$	4.31195	0.190750
$512$	9.34120	0.412827
$513$	−14.6712	−0.647748
$514$	23.0722	1.01767
$515$	0.526511	0.0232009
$516$	−9.62121	−0.423550
$517$	11.1154	0.488857
$518$	−49.0115	−2.15344
$519$	−2.30156	−0.101027
$520$	0.102550	0.00449711
$521$	−24.2208	−1.06113	−0.530566	−	0.847644i	$-0.678021\pi$
$521$	−24.2208	−1.06113	−0.530566	+	0.847644i	$0.678021\pi$
$522$	3.41753	0.149581
$523$	41.0790	1.79626	0.898129	−	0.439731i	$-0.144926\pi$
$523$	41.0790	1.79626	0.898129	+	0.439731i	$0.144926\pi$
$524$	−39.4963	−1.72541
$525$	1.22934	0.0536526
$526$	−66.7863	−2.91202
$527$	10.0622	0.438317
$528$	−0.237054	−0.0103164
$529$	17.1058	0.743731
$530$	4.21149	0.182935
$531$	−1.85764	−0.0806145
$532$	128.023	5.55048
$533$	−0.0838094	−0.00363019
$534$	−5.82366	−0.252015
$535$	−17.2599	−0.746212
$536$	−19.6051	−0.846812
$537$	−0.675115	−0.0291334
$538$	7.93633	0.342160
$539$	11.5930	0.499344
$540$	5.76229	0.247969
$541$	−28.9721	−1.24561	−0.622804	−	0.782378i	$-0.714007\pi$
$541$	−28.9721	−1.24561	−0.622804	+	0.782378i	$0.714007\pi$
$542$	−6.47084	−0.277946
$543$	−1.52858	−0.0655976
$544$	15.3896	0.659822
$545$	−15.4329	−0.661075
$546$	0.0891039	0.00381329
$547$	−33.9141	−1.45006	−0.725032	−	0.688715i	$-0.758175\pi$
$547$	−33.9141	−1.45006	−0.725032	+	0.688715i	$0.758175\pi$
$548$	8.15253	0.348259
$549$	8.78347	0.374869
$550$	−2.32698	−0.0992228
$551$	−4.37490	−0.186377
$552$	5.94432	0.253007
$553$	19.0694	0.810913
$554$	60.6660	2.57745
$555$	1.39260	0.0591127
$556$	41.6253	1.76531
$557$	−23.8151	−1.00908	−0.504538	−	0.863389i	$-0.668337\pi$
$557$	−23.8151	−1.00908	−0.504538	+	0.863389i	$0.668337\pi$
$558$	20.6484	0.874119
$559$	−0.307819	−0.0130194
$560$	−3.58530	−0.151506
$561$	−0.943599	−0.0398388
$562$	25.7404	1.08579
$563$	−25.6042	−1.07909	−0.539544	−	0.841958i	$-0.681403\pi$
$563$	−25.6042	−1.07909	−0.539544	+	0.841958i	$0.681403\pi$
$564$	−10.8217	−0.455674
$565$	−6.46526	−0.271995
$566$	1.36089	0.0572025
$567$	−35.6817	−1.49849
$568$	3.34466	0.140339
$569$	−4.22017	−0.176919	−0.0884593	−	0.996080i	$-0.528194\pi$
$569$	−4.22017	−0.176919	−0.0884593	+	0.996080i	$0.528194\pi$
$570$	−5.76807	−0.241598
$571$	−26.6598	−1.11568	−0.557839	−	0.829949i	$-0.688369\pi$
$571$	−26.6598	−1.11568	−0.557839	+	0.829949i	$0.688369\pi$
$572$	−0.106366	−0.00444740
$573$	0.820590	0.0342806
$574$	26.9977	1.12686
$575$	6.33292	0.264101
$576$	36.4343	1.51809
$577$	33.9188	1.41206	0.706029	−	0.708183i	$-0.250485\pi$
$577$	33.9188	1.41206	0.706029	+	0.708183i	$0.250485\pi$
$578$	14.0683	0.585166
$579$	0.873205	0.0362892
$580$	1.71829	0.0713483
$581$	−11.5741	−0.480174
$582$	1.71451	0.0710689
$583$	−1.80985	−0.0749563
$584$	3.29232	0.136237
$585$	0.0909129	0.00375879
$586$	−24.5802	−1.01540
$587$	−10.4036	−0.429402	−0.214701	−	0.976680i	$-0.568878\pi$
$587$	−10.4036	−0.429402	−0.214701	+	0.976680i	$0.568878\pi$
$588$	−11.2866	−0.465450
$589$	−26.4328	−1.08914
$590$	−1.48102	−0.0609727
$591$	−6.03475	−0.248236
$592$	−4.06146	−0.166925
$593$	12.5566	0.515638	0.257819	−	0.966193i	$-0.416996\pi$
$593$	12.5566	0.515638	0.257819	+	0.966193i	$0.416996\pi$
$594$	−3.92660	−0.161110
$595$	−14.2714	−0.585069
$596$	−36.3888	−1.49054
$597$	−5.32583	−0.217972
$598$	0.459018	0.0187707
$599$	40.0188	1.63512	0.817562	−	0.575840i	$-0.195325\pi$
$599$	40.0188	1.63512	0.817562	+	0.575840i	$0.195325\pi$
$600$	0.938638	0.0383197
$601$	33.2294	1.35546	0.677728	−	0.735312i	$-0.262964\pi$
$601$	33.2294	1.35546	0.677728	+	0.735312i	$0.262964\pi$
$602$	99.1584	4.04139
$603$	−17.3804	−0.707784
$604$	−33.8503	−1.37735
$605$	1.00000	0.0406558
$606$	2.99847	0.121805
$607$	−24.1210	−0.979041	−0.489521	−	0.871992i	$-0.662828\pi$
$607$	−24.1210	−0.979041	−0.489521	+	0.871992i	$0.662828\pi$
$608$	−40.4274	−1.63955
$609$	0.618582	0.0250662
$610$	7.00272	0.283532
$611$	−0.346226	−0.0140068
$612$	−32.9879	−1.33346
$613$	25.7830	1.04137	0.520683	−	0.853750i	$-0.325677\pi$
$613$	25.7830	1.04137	0.520683	+	0.853750i	$0.325677\pi$
$614$	12.7120	0.513013
$615$	−0.767106	−0.0309327
$616$	14.1963	0.571987
$617$	−12.0969	−0.487005	−0.243502	−	0.969900i	$-0.578296\pi$
$617$	−12.0969	−0.487005	−0.243502	+	0.969900i	$0.578296\pi$
$618$	0.349299	0.0140509
$619$	20.8815	0.839298	0.419649	−	0.907686i	$-0.362153\pi$
$619$	20.8815	0.839298	0.419649	+	0.907686i	$0.362153\pi$
$620$	10.3818	0.416944
$621$	10.6863	0.428827
$622$	4.59158	0.184106
$623$	37.8513	1.51648
$624$	0.00738381	0.000295589 0
$625$	1.00000	0.0400000
$626$	54.1027	2.16238
$627$	2.47878	0.0989928
$628$	4.19466	0.167385
$629$	−16.1667	−0.644610
$630$	−29.2860	−1.16678
$631$	−18.1842	−0.723904	−0.361952	−	0.932197i	$-0.617889\pi$
$631$	−18.1842	−0.723904	−0.361952	+	0.932197i	$0.617889\pi$
$632$	14.5601	0.579170
$633$	−2.50653	−0.0996257
$634$	29.5177	1.17230
$635$	−13.3836	−0.531112
$636$	1.76202	0.0698685
$637$	−0.361100	−0.0143073
$638$	−1.17090	−0.0463564
$639$	2.96512	0.117298
$640$	19.7480	0.780610
$641$	26.5476	1.04857	0.524284	−	0.851543i	$-0.324333\pi$
$641$	26.5476	1.04857	0.524284	+	0.851543i	$0.324333\pi$
$642$	−11.4506	−0.451919
$643$	−4.23477	−0.167003	−0.0835014	−	0.996508i	$-0.526610\pi$
$643$	−4.23477	−0.167003	−0.0835014	+	0.996508i	$0.526610\pi$
$644$	−93.2500	−3.67457
$645$	−2.81746	−0.110938
$646$	66.9616	2.63457
$647$	16.8922	0.664100	0.332050	−	0.943262i	$-0.392260\pi$
$647$	16.8922	0.664100	0.332050	+	0.943262i	$0.392260\pi$
$648$	−27.2442	−1.07025
$649$	0.636456	0.0249831
$650$	0.0724814	0.00284295
$651$	3.73743	0.146481
$652$	17.7507	0.695172
$653$	6.71517	0.262785	0.131393	−	0.991330i	$-0.458055\pi$
$653$	6.71517	0.262785	0.131393	+	0.991330i	$0.458055\pi$
$654$	−10.2385	−0.400358
$655$	−11.5661	−0.451924
$656$	2.23723	0.0873491
$657$	2.91872	0.113870
$658$	111.531	4.34792
$659$	31.8960	1.24249	0.621246	−	0.783616i	$-0.286627\pi$
$659$	31.8960	1.24249	0.621246	+	0.783616i	$0.286627\pi$
$660$	−0.973570	−0.0378962
$661$	43.0808	1.67565	0.837825	−	0.545939i	$-0.183827\pi$
$661$	43.0808	1.67565	0.837825	+	0.545939i	$0.183827\pi$
$662$	−5.40482	−0.210064
$663$	0.0293914	0.00114147
$664$	−8.83720	−0.342950
$665$	37.4900	1.45380
$666$	−33.1754	−1.28552
$667$	3.18662	0.123386
$668$	−56.9384	−2.20301
$669$	−6.32425	−0.244510
$670$	−13.8567	−0.535332
$671$	−3.00936	−0.116175
$672$	5.71617	0.220506
$673$	−6.70149	−0.258323	−0.129162	−	0.991624i	$-0.541229\pi$
$673$	−6.70149	−0.258323	−0.129162	+	0.991624i	$0.541229\pi$
$674$	49.6909	1.91402
$675$	1.68742	0.0649489
$676$	−44.3897	−1.70730
$677$	5.93455	0.228083	0.114042	−	0.993476i	$-0.463620\pi$
$677$	5.93455	0.228083	0.114042	+	0.993476i	$0.463620\pi$
$678$	−4.28919	−0.164725
$679$	−11.1436	−0.427653
$680$	−10.8967	−0.417868
$681$	1.87662	0.0719123
$682$	−7.07449	−0.270896
$683$	31.1613	1.19236	0.596178	−	0.802853i	$-0.296685\pi$
$683$	31.1613	1.19236	0.596178	+	0.802853i	$0.296685\pi$
$684$	86.6571	3.31342
$685$	2.38738	0.0912170
$686$	46.0850	1.75953
$687$	7.35989	0.280797
$688$	8.21700	0.313270
$689$	0.0563736	0.00214766
$690$	4.20139	0.159944
$691$	−31.4779	−1.19748	−0.598739	−	0.800945i	$-0.704331\pi$
$691$	−31.4779	−1.19748	−0.598739	+	0.800945i	$0.704331\pi$
$692$	27.5675	1.04796
$693$	12.5854	0.478079
$694$	−71.3623	−2.70888
$695$	12.1895	0.462375
$696$	0.472307	0.0179028
$697$	8.90534	0.337314
$698$	16.5809	0.627596
$699$	7.52957	0.284794
$700$	−14.7247	−0.556540
$701$	−6.65674	−0.251422	−0.125711	−	0.992067i	$-0.540121\pi$
$701$	−6.65674	−0.251422	−0.125711	+	0.992067i	$0.540121\pi$
$702$	0.122307	0.00461617
$703$	42.4690	1.60175
$704$	−12.4830	−0.470470
$705$	−3.16901	−0.119352
$706$	82.5965	3.10856
$707$	−19.4888	−0.732951
$708$	−0.619635	−0.0232873
$709$	29.5240	1.10880	0.554399	−	0.832251i	$-0.312948\pi$
$709$	29.5240	1.10880	0.554399	+	0.832251i	$0.312948\pi$
$710$	2.36398	0.0887184
$711$	12.9079	0.484083
$712$	28.9007	1.08310
$713$	19.2533	0.721043
$714$	−9.46793	−0.354328
$715$	−0.0311482	−0.00116488
$716$	8.08635	0.302201
$717$	−3.12578	−0.116734
$718$	10.5591	0.394063
$719$	7.88109	0.293915	0.146957	−	0.989143i	$-0.453052\pi$
$719$	7.88109	0.293915	0.146957	+	0.989143i	$0.453052\pi$
$720$	−2.42685	−0.0904434
$721$	−2.27029	−0.0845502
$722$	−131.691	−4.90104
$723$	−1.15183	−0.0428371
$724$	18.3089	0.680446
$725$	0.503184	0.0186878
$726$	0.663421	0.0246219
$727$	−15.1654	−0.562455	−0.281227	−	0.959641i	$-0.590742\pi$
$727$	−15.1654	−0.562455	−0.281227	+	0.959641i	$0.590742\pi$
$728$	−0.442191	−0.0163887
$729$	−22.7094	−0.841087
$730$	2.32698	0.0861255
$731$	32.7080	1.20975
$732$	2.92982	0.108289
$733$	−52.7509	−1.94840	−0.974200	−	0.225688i	$-0.927537\pi$
$733$	−52.7509	−1.94840	−0.974200	+	0.225688i	$0.927537\pi$
$734$	38.2723	1.41266
$735$	−3.30514	−0.121912
$736$	29.4468	1.08542
$737$	5.95480	0.219348
$738$	18.2745	0.672692
$739$	−23.4375	−0.862164	−0.431082	−	0.902313i	$-0.641868\pi$
$739$	−23.4375	−0.862164	−0.431082	+	0.902313i	$0.641868\pi$
$740$	−16.6802	−0.613177
$741$	−0.0772095	−0.00283636
$742$	−18.1597	−0.666665
$743$	13.3299	0.489026	0.244513	−	0.969646i	$-0.421372\pi$
$743$	13.3299	0.489026	0.244513	+	0.969646i	$0.421372\pi$
$744$	2.85365	0.104620
$745$	−10.6561	−0.390408
$746$	−41.2352	−1.50973
$747$	−7.83439	−0.286645
$748$	11.3022	0.413249
$749$	74.4241	2.71940
$750$	0.663421	0.0242247
$751$	−5.72608	−0.208948	−0.104474	−	0.994528i	$-0.533316\pi$
$751$	−5.72608	−0.208948	−0.104474	+	0.994528i	$0.533316\pi$
$752$	9.24225	0.337030
$753$	−5.93084	−0.216132
$754$	0.0364714	0.00132821
$755$	−9.91269	−0.360760
$756$	−24.8467	−0.903667
$757$	28.0998	1.02131	0.510653	−	0.859787i	$-0.329404\pi$
$757$	28.0998	1.02131	0.510653	+	0.859787i	$0.329404\pi$
$758$	62.5895	2.27335
$759$	−1.80551	−0.0655359
$760$	28.6248	1.03833
$761$	−25.0360	−0.907553	−0.453777	−	0.891116i	$-0.649924\pi$
$761$	−25.0360	−0.907553	−0.453777	+	0.891116i	$0.649924\pi$
$762$	−8.87897	−0.321651
$763$	66.5461	2.40913
$764$	−9.82881	−0.355594
$765$	−9.66014	−0.349263
$766$	68.8616	2.48807
$767$	−0.0198245	−0.000715820 0
$768$	5.98349	0.215911
$769$	44.3163	1.59809	0.799043	−	0.601274i	$-0.205340\pi$
$769$	44.3163	1.59809	0.799043	+	0.601274i	$0.205340\pi$
$770$	10.0338	0.361595
$771$	2.82679	0.101804
$772$	−10.4590	−0.376429
$773$	−42.6204	−1.53295	−0.766474	−	0.642275i	$-0.777991\pi$
$773$	−42.6204	−1.53295	−0.766474	+	0.642275i	$0.777991\pi$
$774$	67.1193	2.41255
$775$	3.04020	0.109207
$776$	−8.50851	−0.305438
$777$	−6.00484	−0.215422
$778$	−32.7253	−1.17326
$779$	−23.3938	−0.838169
$780$	0.0303250	0.00108581
$781$	−1.01590	−0.0363517
$782$	−48.7740	−1.74415
$783$	0.849084	0.0303438
$784$	9.63929	0.344260
$785$	1.22836	0.0438421
$786$	−7.67317	−0.273693
$787$	27.6841	0.986831	0.493415	−	0.869794i	$-0.335748\pi$
$787$	27.6841	0.986831	0.493415	+	0.869794i	$0.335748\pi$
$788$	72.2826	2.57496
$789$	−8.18259	−0.291308
$790$	10.2910	0.366136
$791$	27.8779	0.991224
$792$	9.60935	0.341454
$793$	0.0937362	0.00332867
$794$	−63.2432	−2.24442
$795$	0.515987	0.0183002
$796$	63.7915	2.26103
$797$	−28.8277	−1.02113	−0.510565	−	0.859839i	$-0.670564\pi$
$797$	−28.8277	−1.02113	−0.510565	+	0.859839i	$0.670564\pi$
$798$	24.8716	0.880447
$799$	36.7890	1.30150
$800$	4.64980	0.164395
$801$	25.6212	0.905280
$802$	−39.6320	−1.39945
$803$	−1.00000	−0.0352892
$804$	−5.79742	−0.204459
$805$	−27.3072	−0.962454
$806$	0.220358	0.00776177
$807$	0.972351	0.0342284
$808$	−14.8803	−0.523488
$809$	17.1902	0.604376	0.302188	−	0.953248i	$-0.402283\pi$
$809$	17.1902	0.604376	0.302188	+	0.953248i	$0.402283\pi$
$810$	−19.2559	−0.676585
$811$	−18.1798	−0.638378	−0.319189	−	0.947691i	$-0.603410\pi$
$811$	−18.1798	−0.638378	−0.319189	+	0.947691i	$0.603410\pi$
$812$	−7.40921	−0.260012
$813$	−0.792800	−0.0278047
$814$	11.3664	0.398393
$815$	5.19811	0.182082
$816$	−0.784583	−0.0274659
$817$	−85.9218	−3.00602
$818$	72.0941	2.52071
$819$	−0.392012	−0.0136980
$820$	9.18820	0.320866
$821$	−10.1403	−0.353898	−0.176949	−	0.984220i	$-0.556623\pi$
$821$	−10.1403	−0.353898	−0.176949	+	0.984220i	$0.556623\pi$
$822$	1.58384	0.0552426
$823$	−23.3458	−0.813784	−0.406892	−	0.913476i	$-0.633387\pi$
$823$	−23.3458	−0.813784	−0.406892	+	0.913476i	$0.633387\pi$
$824$	−1.73344	−0.0603874
$825$	−0.285099	−0.00992588
$826$	6.38610	0.222201
$827$	−46.5285	−1.61796	−0.808978	−	0.587839i	$-0.799979\pi$
$827$	−46.5285	−1.61796	−0.808978	+	0.587839i	$0.799979\pi$
$828$	−63.1200	−2.19357
$829$	−47.1168	−1.63643	−0.818217	−	0.574909i	$-0.805037\pi$
$829$	−47.1168	−1.63643	−0.818217	+	0.574909i	$0.805037\pi$
$830$	−6.24605	−0.216804
$831$	7.43274	0.257839
$832$	0.388822	0.0134800
$833$	38.3695	1.32942
$834$	8.08679	0.280023
$835$	−16.6738	−0.577020
$836$	−29.6901	−1.02685
$837$	5.13010	0.177322
$838$	−5.37827	−0.185789
$839$	42.3637	1.46256	0.731279	−	0.682079i	$-0.238924\pi$
$839$	42.3637	1.46256	0.731279	+	0.682079i	$0.238924\pi$
$840$	−4.04737	−0.139647
$841$	−28.7468	−0.991269
$842$	60.3597	2.08013
$843$	3.15368	0.108619
$844$	30.0226	1.03342
$845$	−12.9990	−0.447180
$846$	75.4939	2.59553
$847$	−4.31195	−0.148161
$848$	−1.50485	−0.0516768
$849$	0.166735	0.00572233
$850$	−7.70166	−0.264165
$851$	−30.9339	−1.06040
$852$	0.989048	0.0338842
$853$	−44.4699	−1.52262	−0.761310	−	0.648388i	$-0.775444\pi$
$853$	−44.4699	−1.52262	−0.761310	+	0.648388i	$0.775444\pi$
$854$	−30.1954	−1.03327
$855$	25.3766	0.867861
$856$	56.8252	1.94225
$857$	0.665174	0.0227219	0.0113610	−	0.999935i	$-0.496384\pi$
$857$	0.665174	0.0227219	0.0113610	+	0.999935i	$0.496384\pi$
$858$	−0.0206644	−0.000705470 0
$859$	−49.7763	−1.69835	−0.849173	−	0.528115i	$-0.822899\pi$
$859$	−49.7763	−1.69835	−0.849173	+	0.528115i	$0.822899\pi$
$860$	33.7469	1.15076
$861$	3.30773	0.112727
$862$	75.3857	2.56765
$863$	0.717804	0.0244343	0.0122172	−	0.999925i	$-0.496111\pi$
$863$	0.717804	0.0244343	0.0122172	+	0.999925i	$0.496111\pi$
$864$	7.84618	0.266933
$865$	8.07285	0.274485
$866$	−36.0235	−1.22413
$867$	1.72364	0.0585378
$868$	−44.7659	−1.51945
$869$	−4.42245	−0.150021
$870$	0.333823	0.0113176
$871$	−0.185482	−0.00628480
$872$	50.8102	1.72065
$873$	−7.54300	−0.255292
$874$	128.126	4.33393
$875$	−4.31195	−0.145771
$876$	0.973570	0.0328939
$877$	−9.11253	−0.307708	−0.153854	−	0.988094i	$-0.549169\pi$
$877$	−9.11253	−0.307708	−0.153854	+	0.988094i	$0.549169\pi$
$878$	70.9623	2.39486
$879$	−3.01154	−0.101577
$880$	0.831478	0.0280291
$881$	−6.73239	−0.226820	−0.113410	−	0.993548i	$-0.536177\pi$
$881$	−6.73239	−0.226820	−0.113410	+	0.993548i	$0.536177\pi$
$882$	78.7371	2.65122
$883$	−2.40618	−0.0809744	−0.0404872	−	0.999180i	$-0.512891\pi$
$883$	−2.40618	−0.0809744	−0.0404872	+	0.999180i	$0.512891\pi$
$884$	−0.352043	−0.0118405
$885$	−0.181453	−0.00609948
$886$	53.8542	1.80927
$887$	33.6807	1.13089	0.565444	−	0.824787i	$-0.308705\pi$
$887$	33.6807	1.13089	0.565444	+	0.824787i	$0.308705\pi$
$888$	−4.58489	−0.153859
$889$	57.7095	1.93551
$890$	20.4268	0.684707
$891$	8.27507	0.277225
$892$	75.7503	2.53631
$893$	−96.6424	−3.23402
$894$	−7.06946	−0.236438
$895$	2.36800	0.0791535
$896$	−85.1526	−2.84475
$897$	0.0562384	0.00187775
$898$	88.4669	2.95218
$899$	1.52978	0.0510210
$900$	−9.96697	−0.332232
$901$	−5.99010	−0.199559
$902$	−6.26112	−0.208473
$903$	12.1488	0.404286
$904$	21.2857	0.707952
$905$	5.36157	0.178225
$906$	−6.57629	−0.218483
$907$	38.1556	1.26693	0.633467	−	0.773770i	$-0.281631\pi$
$907$	38.1556	1.26693	0.633467	+	0.773770i	$0.281631\pi$
$908$	−22.4777	−0.745948
$909$	−13.1918	−0.437543
$910$	−0.312536	−0.0103605
$911$	−49.4756	−1.63920	−0.819600	−	0.572937i	$-0.805804\pi$
$911$	−49.4756	−1.63920	−0.819600	+	0.572937i	$0.805804\pi$
$912$	2.06105	0.0682482
$913$	2.68419	0.0888336
$914$	10.2897	0.340354
$915$	0.857966	0.0283635
$916$	−88.1548	−2.91272
$917$	49.8724	1.64693
$918$	−12.9960	−0.428931
$919$	16.3239	0.538477	0.269239	−	0.963073i	$-0.413228\pi$
$919$	16.3239	0.538477	0.269239	+	0.963073i	$0.413228\pi$
$920$	−20.8500	−0.687404
$921$	1.55745	0.0513199
$922$	69.3314	2.28331
$923$	0.0316434	0.00104156
$924$	4.19799	0.138104
$925$	−4.88462	−0.160605
$926$	3.45807	0.113639
$927$	−1.53674	−0.0504731
$928$	2.33971	0.0768046
$929$	−51.1321	−1.67759	−0.838796	−	0.544446i	$-0.816740\pi$
$929$	−51.1321	−1.67759	−0.838796	+	0.544446i	$0.816740\pi$
$930$	2.01693	0.0661378
$931$	−100.794	−3.30339
$932$	−90.1872	−2.95418
$933$	0.562555	0.0184172
$934$	0.572029	0.0187174
$935$	3.30972	0.108239
$936$	−0.299314	−0.00978339
$937$	46.7466	1.52714	0.763572	−	0.645723i	$-0.223444\pi$
$937$	46.7466	1.52714	0.763572	+	0.645723i	$0.223444\pi$
$938$	59.7495	1.95089
$939$	6.62860	0.216316
$940$	37.9575	1.23804
$941$	−1.75782	−0.0573032	−0.0286516	−	0.999589i	$-0.509121\pi$
$941$	−1.75782	−0.0573032	−0.0286516	+	0.999589i	$0.509121\pi$
$942$	0.814920	0.0265515
$943$	17.0397	0.554890
$944$	0.529200	0.0172240
$945$	−7.27609	−0.236691
$946$	−22.9962	−0.747670
$947$	20.0993	0.653138	0.326569	−	0.945173i	$-0.394107\pi$
$947$	20.0993	0.653138	0.326569	+	0.945173i	$0.394107\pi$
$948$	4.30556	0.139838
$949$	0.0311482	0.00101111
$950$	20.2318	0.656406
$951$	3.61648	0.117272
$952$	46.9859	1.52282
$953$	51.8497	1.67958	0.839788	−	0.542915i	$-0.182679\pi$
$953$	51.8497	1.67958	0.839788	+	0.542915i	$0.182679\pi$
$954$	−12.2921	−0.397973
$955$	−2.87826	−0.0931383
$956$	37.4398	1.21089
$957$	−0.143457	−0.00463732
$958$	−23.4655	−0.758136
$959$	−10.2943	−0.332419
$960$	3.55889	0.114863
$961$	−21.7572	−0.701845
$962$	−0.354044	−0.0114148
$963$	50.3769	1.62337
$964$	13.7963	0.444350
$965$	−3.06281	−0.0985954
$966$	−18.1162	−0.582879
$967$	56.0885	1.80368	0.901842	−	0.432066i	$-0.142215\pi$
$967$	56.0885	1.80368	0.901842	+	0.432066i	$0.142215\pi$
$968$	−3.29232	−0.105819
$969$	8.20406	0.263552
$970$	−6.01374	−0.193090
$971$	−24.3206	−0.780484	−0.390242	−	0.920712i	$-0.627609\pi$
$971$	−24.3206	−0.780484	−0.390242	+	0.920712i	$0.627609\pi$
$972$	−25.3432	−0.812884
$973$	−52.5607	−1.68502
$974$	68.8987	2.20766
$975$	0.00888034	0.000284398 0
$976$	−2.50222	−0.0800940
$977$	35.6862	1.14170	0.570851	−	0.821054i	$-0.306613\pi$
$977$	35.6862	1.14170	0.570851	+	0.821054i	$0.306613\pi$
$978$	3.44853	0.110272
$979$	−8.77823	−0.280553
$980$	39.5882	1.26460
$981$	45.0444	1.43816
$982$	−37.5440	−1.19808
$983$	−25.0732	−0.799711	−0.399855	−	0.916578i	$-0.630940\pi$
$983$	−25.0732	−0.799711	−0.399855	+	0.916578i	$0.630940\pi$
$984$	2.52556	0.0805119
$985$	21.1672	0.674442
$986$	−3.87535	−0.123416
$987$	13.6646	0.434949
$988$	0.924795	0.0294216
$989$	62.5844	1.99007
$990$	6.79181	0.215858
$991$	−51.1987	−1.62638	−0.813189	−	0.581999i	$-0.802271\pi$
$991$	−51.1987	−1.62638	−0.813189	+	0.581999i	$0.802271\pi$
$992$	14.1363	0.448829
$993$	−0.662192	−0.0210140
$994$	−10.1934	−0.323314
$995$	18.6806	0.592216
$996$	−2.61324	−0.0828038
$997$	−3.26215	−0.103313	−0.0516566	−	0.998665i	$-0.516450\pi$
$997$	−3.26215	−0.103313	−0.0516566	+	0.998665i	$0.516450\pi$
$998$	−99.2849	−3.14281
$999$	−8.24242	−0.260779

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.b.1.3	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.b.1.3	✓	23	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4015 = 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4015.a (trivial)

\(f(q)\)	\(=\)	\(q-2.32698 q^{2} -0.285099 q^{3} +3.41485 q^{4} +1.00000 q^{5} +0.663421 q^{6} -4.31195 q^{7} -3.29232 q^{8} -2.91872 q^{9} +O(q^{10})\)	\(q-2.32698 q^{2} -0.285099 q^{3} +3.41485 q^{4} +1.00000 q^{5} +0.663421 q^{6} -4.31195 q^{7} -3.29232 q^{8} -2.91872 q^{9} -2.32698 q^{10} +1.00000 q^{11} -0.973570 q^{12} -0.0311482 q^{13} +10.0338 q^{14} -0.285099 q^{15} +0.831478 q^{16} +3.30972 q^{17} +6.79181 q^{18} -8.69443 q^{19} +3.41485 q^{20} +1.22934 q^{21} -2.32698 q^{22} +6.33292 q^{23} +0.938638 q^{24} +1.00000 q^{25} +0.0724814 q^{26} +1.68742 q^{27} -14.7247 q^{28} +0.503184 q^{29} +0.663421 q^{30} +3.04020 q^{31} +4.64980 q^{32} -0.285099 q^{33} -7.70166 q^{34} -4.31195 q^{35} -9.96697 q^{36} -4.88462 q^{37} +20.2318 q^{38} +0.00888034 q^{39} -3.29232 q^{40} +2.69066 q^{41} -2.86064 q^{42} +9.88240 q^{43} +3.41485 q^{44} -2.91872 q^{45} -14.7366 q^{46} +11.1154 q^{47} -0.237054 q^{48} +11.5930 q^{49} -2.32698 q^{50} -0.943599 q^{51} -0.106366 q^{52} -1.80985 q^{53} -3.92660 q^{54} +1.00000 q^{55} +14.1963 q^{56} +2.47878 q^{57} -1.17090 q^{58} +0.636456 q^{59} -0.973570 q^{60} -3.00936 q^{61} -7.07449 q^{62} +12.5854 q^{63} -12.4830 q^{64} -0.0311482 q^{65} +0.663421 q^{66} +5.95480 q^{67} +11.3022 q^{68} -1.80551 q^{69} +10.0338 q^{70} -1.01590 q^{71} +9.60935 q^{72} -1.00000 q^{73} +11.3664 q^{74} -0.285099 q^{75} -29.6901 q^{76} -4.31195 q^{77} -0.0206644 q^{78} -4.42245 q^{79} +0.831478 q^{80} +8.27507 q^{81} -6.26112 q^{82} +2.68419 q^{83} +4.19799 q^{84} +3.30972 q^{85} -22.9962 q^{86} -0.143457 q^{87} -3.29232 q^{88} -8.77823 q^{89} +6.79181 q^{90} +0.134310 q^{91} +21.6259 q^{92} -0.866759 q^{93} -25.8654 q^{94} -8.69443 q^{95} -1.32566 q^{96} +2.58435 q^{97} -26.9766 q^{98} -2.91872 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 5 q^{2} - 3 q^{3} + 15 q^{4} + 23 q^{5} - 15 q^{6} - 10 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) 23 * q - 5 * q^2 - 3 * q^3 + 15 * q^4 + 23 * q^5 - 15 * q^6 - 10 * q^7 - 12 * q^8 + 8 * q^9	\( 23 q - 5 q^{2} - 3 q^{3} + 15 q^{4} + 23 q^{5} - 15 q^{6} - 10 q^{7} - 12 q^{8} + 8 q^{9} - 5 q^{10} + 23 q^{11} + 4 q^{12} - 19 q^{13} - 5 q^{14} - 3 q^{15} - q^{16} - 26 q^{17} + 5 q^{18} - 34 q^{19} + 15 q^{20} - 26 q^{21} - 5 q^{22} - 4 q^{23} - 23 q^{24} + 23 q^{25} - 13 q^{26} - 3 q^{27} - 28 q^{28} - 36 q^{29} - 15 q^{30} - 24 q^{31} - 19 q^{32} - 3 q^{33} - 4 q^{34} - 10 q^{35} - 14 q^{36} + 4 q^{37} - 15 q^{38} - 34 q^{39} - 12 q^{40} - 74 q^{41} + 7 q^{42} - 15 q^{43} + 15 q^{44} + 8 q^{45} + 3 q^{46} + q^{47} + 29 q^{48} + q^{49} - 5 q^{50} - 47 q^{51} - 23 q^{52} - 6 q^{53} - 11 q^{54} + 23 q^{55} - 20 q^{56} - 19 q^{57} + 22 q^{58} - 17 q^{59} + 4 q^{60} - 59 q^{61} + 38 q^{62} - 21 q^{63} - 18 q^{64} - 19 q^{65} - 15 q^{66} + 12 q^{67} + 5 q^{68} - 8 q^{69} - 5 q^{70} - 34 q^{71} + 21 q^{72} - 23 q^{73} - 9 q^{74} - 3 q^{75} - 53 q^{76} - 10 q^{77} + 23 q^{78} - 62 q^{79} - q^{80} + 7 q^{81} + 24 q^{82} - 8 q^{83} + 46 q^{84} - 26 q^{85} - 11 q^{86} + q^{87} - 12 q^{88} - 77 q^{89} + 5 q^{90} - 6 q^{91} + 47 q^{92} - 32 q^{94} - 34 q^{95} - 53 q^{96} - 21 q^{97} - 3 q^{98} + 8 q^{99}+O(q^{100}) \) 23 * q - 5 * q^2 - 3 * q^3 + 15 * q^4 + 23 * q^5 - 15 * q^6 - 10 * q^7 - 12 * q^8 + 8 * q^9 - 5 * q^10 + 23 * q^11 + 4 * q^12 - 19 * q^13 - 5 * q^14 - 3 * q^15 - q^16 - 26 * q^17 + 5 * q^18 - 34 * q^19 + 15 * q^20 - 26 * q^21 - 5 * q^22 - 4 * q^23 - 23 * q^24 + 23 * q^25 - 13 * q^26 - 3 * q^27 - 28 * q^28 - 36 * q^29 - 15 * q^30 - 24 * q^31 - 19 * q^32 - 3 * q^33 - 4 * q^34 - 10 * q^35 - 14 * q^36 + 4 * q^37 - 15 * q^38 - 34 * q^39 - 12 * q^40 - 74 * q^41 + 7 * q^42 - 15 * q^43 + 15 * q^44 + 8 * q^45 + 3 * q^46 + q^47 + 29 * q^48 + q^49 - 5 * q^50 - 47 * q^51 - 23 * q^52 - 6 * q^53 - 11 * q^54 + 23 * q^55 - 20 * q^56 - 19 * q^57 + 22 * q^58 - 17 * q^59 + 4 * q^60 - 59 * q^61 + 38 * q^62 - 21 * q^63 - 18 * q^64 - 19 * q^65 - 15 * q^66 + 12 * q^67 + 5 * q^68 - 8 * q^69 - 5 * q^70 - 34 * q^71 + 21 * q^72 - 23 * q^73 - 9 * q^74 - 3 * q^75 - 53 * q^76 - 10 * q^77 + 23 * q^78 - 62 * q^79 - q^80 + 7 * q^81 + 24 * q^82 - 8 * q^83 + 46 * q^84 - 26 * q^85 - 11 * q^86 + q^87 - 12 * q^88 - 77 * q^89 + 5 * q^90 - 6 * q^91 + 47 * q^92 - 32 * q^94 - 34 * q^95 - 53 * q^96 - 21 * q^97 - 3 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more