LMFDB - Embedded newform 5054.2.a.bg.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5054,2,Mod(1,5054)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5054, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5054.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5054 = 2 \cdot 7 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5054.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:	$40.3563931816$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.19520000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 $ x^8 - 2x^7 - 12x^6 + 16x^5 + 50x^4 - 24x^3 - 72x^2 - 32*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.236286$ of defining polynomial
Character	$\chi$	$=$	5054.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +3.04815 q^{3} +1.00000 q^{4} +3.90247 q^{5} -3.04815 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.29119 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +3.04815 q^{3} +1.00000 q^{4} +3.90247 q^{5} -3.04815 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.29119 q^{9} -3.90247 q^{10} -5.43763 q^{11} +3.04815 q^{12} +2.24823 q^{13} -1.00000 q^{14} +11.8953 q^{15} +1.00000 q^{16} +0.977224 q^{17} -6.29119 q^{18} +3.90247 q^{20} +3.04815 q^{21} +5.43763 q^{22} +0.640603 q^{23} -3.04815 q^{24} +10.2292 q^{25} -2.24823 q^{26} +10.0320 q^{27} +1.00000 q^{28} +6.85823 q^{29} -11.8953 q^{30} +3.00774 q^{31} -1.00000 q^{32} -16.5747 q^{33} -0.977224 q^{34} +3.90247 q^{35} +6.29119 q^{36} +0.824327 q^{37} +6.85294 q^{39} -3.90247 q^{40} +11.3245 q^{41} -3.04815 q^{42} -12.2093 q^{43} -5.43763 q^{44} +24.5512 q^{45} -0.640603 q^{46} -12.6901 q^{47} +3.04815 q^{48} +1.00000 q^{49} -10.2292 q^{50} +2.97872 q^{51} +2.24823 q^{52} -2.05679 q^{53} -10.0320 q^{54} -21.2202 q^{55} -1.00000 q^{56} -6.85823 q^{58} -3.71241 q^{59} +11.8953 q^{60} +3.56439 q^{61} -3.00774 q^{62} +6.29119 q^{63} +1.00000 q^{64} +8.77365 q^{65} +16.5747 q^{66} +1.14843 q^{67} +0.977224 q^{68} +1.95265 q^{69} -3.90247 q^{70} -4.58975 q^{71} -6.29119 q^{72} -6.31633 q^{73} -0.824327 q^{74} +31.1802 q^{75} -5.43763 q^{77} -6.85294 q^{78} +12.3096 q^{79} +3.90247 q^{80} +11.7055 q^{81} -11.3245 q^{82} -9.84224 q^{83} +3.04815 q^{84} +3.81358 q^{85} +12.2093 q^{86} +20.9049 q^{87} +5.43763 q^{88} -11.8845 q^{89} -24.5512 q^{90} +2.24823 q^{91} +0.640603 q^{92} +9.16802 q^{93} +12.6901 q^{94} -3.04815 q^{96} -7.94295 q^{97} -1.00000 q^{98} -34.2092 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 + 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9} + 2 q^{10} - 12 q^{11} + 4 q^{12} + 10 q^{13} - 8 q^{14} + 24 q^{15} + 8 q^{16} - 6 q^{17} - 8 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 20 q^{23} - 4 q^{24} + 8 q^{25} - 10 q^{26} + 22 q^{27} + 8 q^{28} + 8 q^{29} - 24 q^{30} + 18 q^{31} - 8 q^{32} - 16 q^{33} + 6 q^{34} - 2 q^{35} + 8 q^{36} + 36 q^{37} + 2 q^{40} + 4 q^{41} - 4 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} + 20 q^{46} - 10 q^{47} + 4 q^{48} + 8 q^{49} - 8 q^{50} + 12 q^{51} + 10 q^{52} + 32 q^{53} - 22 q^{54} - 22 q^{55} - 8 q^{56} - 8 q^{58} - 2 q^{59} + 24 q^{60} + 2 q^{61} - 18 q^{62} + 8 q^{63} + 8 q^{64} + 16 q^{66} + 44 q^{67} - 6 q^{68} + 2 q^{70} + 8 q^{71} - 8 q^{72} + 30 q^{73} - 36 q^{74} - 16 q^{75} - 12 q^{77} + 60 q^{79} - 2 q^{80} + 12 q^{81} - 4 q^{82} - 28 q^{83} + 4 q^{84} - 16 q^{85} + 16 q^{86} + 24 q^{87} + 12 q^{88} - 22 q^{89} - 8 q^{90} + 10 q^{91} - 20 q^{92} - 16 q^{93} + 10 q^{94} - 4 q^{96} - 6 q^{97} - 8 q^{98} - 52 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 + 8 * q^7 - 8 * q^8 + 8 * q^9 + 2 * q^10 - 12 * q^11 + 4 * q^12 + 10 * q^13 - 8 * q^14 + 24 * q^15 + 8 * q^16 - 6 * q^17 - 8 * q^18 - 2 * q^20 + 4 * q^21 + 12 * q^22 - 20 * q^23 - 4 * q^24 + 8 * q^25 - 10 * q^26 + 22 * q^27 + 8 * q^28 + 8 * q^29 - 24 * q^30 + 18 * q^31 - 8 * q^32 - 16 * q^33 + 6 * q^34 - 2 * q^35 + 8 * q^36 + 36 * q^37 + 2 * q^40 + 4 * q^41 - 4 * q^42 - 16 * q^43 - 12 * q^44 + 8 * q^45 + 20 * q^46 - 10 * q^47 + 4 * q^48 + 8 * q^49 - 8 * q^50 + 12 * q^51 + 10 * q^52 + 32 * q^53 - 22 * q^54 - 22 * q^55 - 8 * q^56 - 8 * q^58 - 2 * q^59 + 24 * q^60 + 2 * q^61 - 18 * q^62 + 8 * q^63 + 8 * q^64 + 16 * q^66 + 44 * q^67 - 6 * q^68 + 2 * q^70 + 8 * q^71 - 8 * q^72 + 30 * q^73 - 36 * q^74 - 16 * q^75 - 12 * q^77 + 60 * q^79 - 2 * q^80 + 12 * q^81 - 4 * q^82 - 28 * q^83 + 4 * q^84 - 16 * q^85 + 16 * q^86 + 24 * q^87 + 12 * q^88 - 22 * q^89 - 8 * q^90 + 10 * q^91 - 20 * q^92 - 16 * q^93 + 10 * q^94 - 4 * q^96 - 6 * q^97 - 8 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	3.04815	1.75985	0.879924	−	0.475115i	$-0.157593\pi$
$3$	3.04815	1.75985	0.879924	+	0.475115i	$0.157593\pi$
$4$	1.00000	0.500000
$5$	3.90247	1.74524	0.872618	−	0.488404i	$-0.162421\pi$
$5$	3.90247	1.74524	0.872618	+	0.488404i	$0.162421\pi$
$6$	−3.04815	−1.24440
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	6.29119	2.09706
$10$	−3.90247	−1.23407
$11$	−5.43763	−1.63951	−0.819754	−	0.572716i	$-0.805890\pi$
$11$	−5.43763	−1.63951	−0.819754	+	0.572716i	$0.805890\pi$
$12$	3.04815	0.879924
$13$	2.24823	0.623548	0.311774	−	0.950156i	$-0.399077\pi$
$13$	2.24823	0.623548	0.311774	+	0.950156i	$0.399077\pi$
$14$	−1.00000	−0.267261
$15$	11.8953	3.07135
$16$	1.00000	0.250000
$17$	0.977224	0.237012	0.118506	−	0.992953i	$-0.462190\pi$
$17$	0.977224	0.237012	0.118506	+	0.992953i	$0.462190\pi$
$18$	−6.29119	−1.48285
$19$	0	0
$19$	0	0
$20$	3.90247	0.872618
$21$	3.04815	0.665160
$22$	5.43763	1.15931
$23$	0.640603	0.133575	0.0667875	−	0.997767i	$-0.478725\pi$
$23$	0.640603	0.133575	0.0667875	+	0.997767i	$0.478725\pi$
$24$	−3.04815	−0.622200
$25$	10.2292	2.04585
$26$	−2.24823	−0.440915
$27$	10.0320	1.93067
$28$	1.00000	0.188982
$29$	6.85823	1.27354	0.636771	−	0.771053i	$-0.280270\pi$
$29$	6.85823	1.27354	0.636771	+	0.771053i	$0.280270\pi$
$30$	−11.8953	−2.17177
$31$	3.00774	0.540205	0.270103	−	0.962832i	$-0.412942\pi$
$31$	3.00774	0.540205	0.270103	+	0.962832i	$0.412942\pi$
$32$	−1.00000	−0.176777
$33$	−16.5747	−2.88528
$34$	−0.977224	−0.167592
$35$	3.90247	0.659637
$36$	6.29119	1.04853
$37$	0.824327	0.135518	0.0677592	−	0.997702i	$-0.478415\pi$
$37$	0.824327	0.135518	0.0677592	+	0.997702i	$0.478415\pi$
$38$	0	0
$39$	6.85294	1.09735
$40$	−3.90247	−0.617034
$41$	11.3245	1.76859	0.884295	−	0.466928i	$-0.154639\pi$
$41$	11.3245	1.76859	0.884295	+	0.466928i	$0.154639\pi$
$42$	−3.04815	−0.470339
$43$	−12.2093	−1.86190	−0.930952	−	0.365142i	$-0.881020\pi$
$43$	−12.2093	−1.86190	−0.930952	+	0.365142i	$0.881020\pi$
$44$	−5.43763	−0.819754
$45$	24.5512	3.65987
$46$	−0.640603	−0.0944517
$47$	−12.6901	−1.85104	−0.925518	−	0.378704i	$-0.876370\pi$
$47$	−12.6901	−1.85104	−0.925518	+	0.378704i	$0.876370\pi$
$48$	3.04815	0.439962
$49$	1.00000	0.142857
$50$	−10.2292	−1.44663
$51$	2.97872	0.417104
$52$	2.24823	0.311774
$53$	−2.05679	−0.282521	−0.141261	−	0.989972i	$-0.545116\pi$
$53$	−2.05679	−0.282521	−0.141261	+	0.989972i	$0.545116\pi$
$54$	−10.0320	−1.36519
$55$	−21.2202	−2.86133
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−6.85823	−0.900530
$59$	−3.71241	−0.483315	−0.241658	−	0.970362i	$-0.577691\pi$
$59$	−3.71241	−0.483315	−0.241658	+	0.970362i	$0.577691\pi$
$60$	11.8953	1.53567
$61$	3.56439	0.456374	0.228187	−	0.973617i	$-0.426720\pi$
$61$	3.56439	0.456374	0.228187	+	0.973617i	$0.426720\pi$
$62$	−3.00774	−0.381983
$63$	6.29119	0.792616
$64$	1.00000	0.125000
$65$	8.77365	1.08824
$66$	16.5747	2.04020
$67$	1.14843	0.140303	0.0701516	−	0.997536i	$-0.477652\pi$
$67$	1.14843	0.140303	0.0701516	+	0.997536i	$0.477652\pi$
$68$	0.977224	0.118506
$69$	1.95265	0.235072
$70$	−3.90247	−0.466434
$71$	−4.58975	−0.544703	−0.272352	−	0.962198i	$-0.587801\pi$
$71$	−4.58975	−0.544703	−0.272352	+	0.962198i	$0.587801\pi$
$72$	−6.29119	−0.741424
$73$	−6.31633	−0.739271	−0.369635	−	0.929177i	$-0.620517\pi$
$73$	−6.31633	−0.739271	−0.369635	+	0.929177i	$0.620517\pi$
$74$	−0.824327	−0.0958260
$75$	31.1802	3.60038
$76$	0	0
$77$	−5.43763	−0.619675
$78$	−6.85294	−0.775943
$79$	12.3096	1.38494	0.692471	−	0.721446i	$-0.256522\pi$
$79$	12.3096	1.38494	0.692471	+	0.721446i	$0.256522\pi$
$80$	3.90247	0.436309
$81$	11.7055	1.30061
$82$	−11.3245	−1.25058
$83$	−9.84224	−1.08033	−0.540163	−	0.841561i	$-0.681637\pi$
$83$	−9.84224	−1.08033	−0.540163	+	0.841561i	$0.681637\pi$
$84$	3.04815	0.332580
$85$	3.81358	0.413641
$86$	12.2093	1.31656
$87$	20.9049	2.24124
$88$	5.43763	0.579653
$89$	−11.8845	−1.25976	−0.629879	−	0.776693i	$-0.716896\pi$
$89$	−11.8845	−1.25976	−0.629879	+	0.776693i	$0.716896\pi$
$90$	−24.5512	−2.58792
$91$	2.24823	0.235679
$92$	0.640603	0.0667875
$93$	9.16802	0.950679
$94$	12.6901	1.30888
$95$	0	0
$96$	−3.04815	−0.311100
$97$	−7.94295	−0.806485	−0.403242	−	0.915093i	$-0.632117\pi$
$97$	−7.94295	−0.806485	−0.403242	+	0.915093i	$0.632117\pi$
$98$	−1.00000	−0.101015
$99$	−34.2092	−3.43815
$100$	10.2292	1.02292
$101$	12.5382	1.24760	0.623799	−	0.781585i	$-0.285588\pi$
$101$	12.5382	1.24760	0.623799	+	0.781585i	$0.285588\pi$
$102$	−2.97872	−0.294937
$103$	−13.4463	−1.32490	−0.662450	−	0.749106i	$-0.730483\pi$
$103$	−13.4463	−1.32490	−0.662450	+	0.749106i	$0.730483\pi$
$104$	−2.24823	−0.220457
$105$	11.8953	1.16086
$106$	2.05679	0.199773
$107$	−11.5515	−1.11673	−0.558363	−	0.829597i	$-0.688570\pi$
$107$	−11.5515	−1.11673	−0.558363	+	0.829597i	$0.688570\pi$
$108$	10.0320	0.965333
$109$	18.8662	1.80706	0.903528	−	0.428528i	$-0.140968\pi$
$109$	18.8662	1.80706	0.903528	+	0.428528i	$0.140968\pi$
$110$	21.2202	2.02326
$111$	2.51267	0.238492
$112$	1.00000	0.0944911
$113$	−2.61917	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$113$	−2.61917	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$114$	0	0
$115$	2.49993	0.233120
$116$	6.85823	0.636771
$117$	14.1441	1.30762
$118$	3.71241	0.341755
$119$	0.977224	0.0895819
$120$	−11.8953	−1.08589
$121$	18.5678	1.68798
$122$	−3.56439	−0.322705
$123$	34.5188	3.11245
$124$	3.00774	0.270103
$125$	20.4069	1.82525
$126$	−6.29119	−0.560464
$127$	2.91792	0.258924	0.129462	−	0.991584i	$-0.458675\pi$
$127$	2.91792	0.258924	0.129462	+	0.991584i	$0.458675\pi$
$128$	−1.00000	−0.0883883
$129$	−37.2158	−3.27667
$130$	−8.77365	−0.769500
$131$	0.814699	0.0711805	0.0355903	−	0.999366i	$-0.488669\pi$
$131$	0.814699	0.0711805	0.0355903	+	0.999366i	$0.488669\pi$
$132$	−16.5747	−1.44264
$133$	0	0
$134$	−1.14843	−0.0992094
$135$	39.1497	3.36947
$136$	−0.977224	−0.0837962
$137$	−7.30748	−0.624320	−0.312160	−	0.950029i	$-0.601053\pi$
$137$	−7.30748	−0.624320	−0.312160	+	0.950029i	$0.601053\pi$
$138$	−1.95265	−0.166221
$139$	15.1627	1.28608	0.643040	−	0.765833i	$-0.277673\pi$
$139$	15.1627	1.28608	0.643040	+	0.765833i	$0.277673\pi$
$140$	3.90247	0.329819
$141$	−38.6812	−3.25754
$142$	4.58975	0.385164
$143$	−12.2251	−1.02231
$144$	6.29119	0.524266
$145$	26.7640	2.22263
$146$	6.31633	0.522743
$147$	3.04815	0.251407
$148$	0.824327	0.0677592
$149$	−5.81139	−0.476088	−0.238044	−	0.971254i	$-0.576506\pi$
$149$	−5.81139	−0.476088	−0.238044	+	0.971254i	$0.576506\pi$
$150$	−31.1802	−2.54585
$151$	−0.480998	−0.0391430	−0.0195715	−	0.999808i	$-0.506230\pi$
$151$	−0.480998	−0.0391430	−0.0195715	+	0.999808i	$0.506230\pi$
$152$	0	0
$153$	6.14790	0.497028
$154$	5.43763	0.438177
$155$	11.7376	0.942786
$156$	6.85294	0.548675
$157$	−5.53300	−0.441581	−0.220791	−	0.975321i	$-0.570864\pi$
$157$	−5.53300	−0.441581	−0.220791	+	0.975321i	$0.570864\pi$
$158$	−12.3096	−0.979302
$159$	−6.26939	−0.497195
$160$	−3.90247	−0.308517
$161$	0.640603	0.0504866
$162$	−11.7055	−0.919673
$163$	−2.43842	−0.190992	−0.0954958	−	0.995430i	$-0.530444\pi$
$163$	−2.43842	−0.190992	−0.0954958	+	0.995430i	$0.530444\pi$
$164$	11.3245	0.884295
$165$	−64.6822	−5.03550
$166$	9.84224	0.763906
$167$	3.74745	0.289986	0.144993	−	0.989433i	$-0.453684\pi$
$167$	3.74745	0.289986	0.144993	+	0.989433i	$0.453684\pi$
$168$	−3.04815	−0.235170
$169$	−7.94545	−0.611188
$170$	−3.81358	−0.292488
$171$	0	0
$172$	−12.2093	−0.930952
$173$	6.84994	0.520792	0.260396	−	0.965502i	$-0.416147\pi$
$173$	6.84994	0.520792	0.260396	+	0.965502i	$0.416147\pi$
$174$	−20.9049	−1.58480
$175$	10.2292	0.773258
$176$	−5.43763	−0.409877
$177$	−11.3160	−0.850561
$178$	11.8845	0.890784
$179$	13.2215	0.988222	0.494111	−	0.869399i	$-0.335494\pi$
$179$	13.2215	0.988222	0.494111	+	0.869399i	$0.335494\pi$
$180$	24.5512	1.82994
$181$	5.11234	0.379997	0.189998	−	0.981784i	$-0.439152\pi$
$181$	5.11234	0.379997	0.189998	+	0.981784i	$0.439152\pi$
$182$	−2.24823	−0.166650
$183$	10.8648	0.803149
$184$	−0.640603	−0.0472259
$185$	3.21691	0.236512
$186$	−9.16802	−0.672232
$187$	−5.31378	−0.388582
$188$	−12.6901	−0.925518
$189$	10.0320	0.729723
$190$	0	0
$191$	26.4002	1.91025	0.955124	−	0.296206i	$-0.0957216\pi$
$191$	26.4002	1.91025	0.955124	+	0.296206i	$0.0957216\pi$
$192$	3.04815	0.219981
$193$	15.7941	1.13688	0.568442	−	0.822723i	$-0.307546\pi$
$193$	15.7941	1.13688	0.568442	+	0.822723i	$0.307546\pi$
$194$	7.94295	0.570271
$195$	26.7434	1.91513
$196$	1.00000	0.0714286
$197$	−3.73311	−0.265973	−0.132987	−	0.991118i	$-0.542457\pi$
$197$	−3.73311	−0.265973	−0.132987	+	0.991118i	$0.542457\pi$
$198$	34.2092	2.43114
$199$	−16.1988	−1.14830	−0.574151	−	0.818749i	$-0.694668\pi$
$199$	−16.1988	−1.14830	−0.574151	+	0.818749i	$0.694668\pi$
$200$	−10.2292	−0.723317
$201$	3.50059	0.246912
$202$	−12.5382	−0.882185
$203$	6.85823	0.481353
$204$	2.97872	0.208552
$205$	44.1935	3.08661
$206$	13.4463	0.936845
$207$	4.03016	0.280115
$208$	2.24823	0.155887
$209$	0	0
$210$	−11.8953	−0.820853
$211$	13.9798	0.962412	0.481206	−	0.876607i	$-0.340199\pi$
$211$	13.9798	0.962412	0.481206	+	0.876607i	$0.340199\pi$
$212$	−2.05679	−0.141261
$213$	−13.9902	−0.958595
$214$	11.5515	0.789644
$215$	−47.6464	−3.24946
$216$	−10.0320	−0.682594
$217$	3.00774	0.204178
$218$	−18.8662	−1.27778
$219$	−19.2531	−1.30100
$220$	−21.2202	−1.43066
$221$	2.19703	0.147788
$222$	−2.51267	−0.168639
$223$	−11.9087	−0.797466	−0.398733	−	0.917067i	$-0.630550\pi$
$223$	−11.9087	−0.797466	−0.398733	+	0.917067i	$0.630550\pi$
$224$	−1.00000	−0.0668153
$225$	64.3541	4.29028
$226$	2.61917	0.174224
$227$	−13.4459	−0.892438	−0.446219	−	0.894924i	$-0.647230\pi$
$227$	−13.4459	−0.892438	−0.446219	+	0.894924i	$0.647230\pi$
$228$	0	0
$229$	−5.02164	−0.331839	−0.165920	−	0.986139i	$-0.553059\pi$
$229$	−5.02164	−0.331839	−0.165920	+	0.986139i	$0.553059\pi$
$230$	−2.49993	−0.164841
$231$	−16.5747	−1.09053
$232$	−6.85823	−0.450265
$233$	3.80785	0.249460	0.124730	−	0.992191i	$-0.460193\pi$
$233$	3.80785	0.249460	0.124730	+	0.992191i	$0.460193\pi$
$234$	−14.1441	−0.924627
$235$	−49.5225	−3.23049
$236$	−3.71241	−0.241658
$237$	37.5216	2.43729
$238$	−0.977224	−0.0633440
$239$	−2.48796	−0.160933	−0.0804663	−	0.996757i	$-0.525641\pi$
$239$	−2.48796	−0.160933	−0.0804663	+	0.996757i	$0.525641\pi$
$240$	11.8953	0.767837
$241$	−5.53763	−0.356710	−0.178355	−	0.983966i	$-0.557078\pi$
$241$	−5.53763	−0.356710	−0.178355	+	0.983966i	$0.557078\pi$
$242$	−18.5678	−1.19358
$243$	5.58406	0.358218
$244$	3.56439	0.228187
$245$	3.90247	0.249319
$246$	−34.5188	−2.20084
$247$	0	0
$248$	−3.00774	−0.190991
$249$	−30.0006	−1.90121
$250$	−20.4069	−1.29065
$251$	−5.55332	−0.350523	−0.175261	−	0.984522i	$-0.556077\pi$
$251$	−5.55332	−0.350523	−0.175261	+	0.984522i	$0.556077\pi$
$252$	6.29119	0.396308
$253$	−3.48336	−0.218997
$254$	−2.91792	−0.183087
$255$	11.6244	0.727945
$256$	1.00000	0.0625000
$257$	−21.9978	−1.37219	−0.686093	−	0.727514i	$-0.740676\pi$
$257$	−21.9978	−1.37219	−0.686093	+	0.727514i	$0.740676\pi$
$258$	37.2158	2.31695
$259$	0.824327	0.0512212
$260$	8.77365	0.544119
$261$	43.1464	2.67070
$262$	−0.814699	−0.0503322
$263$	−22.9548	−1.41546	−0.707728	−	0.706485i	$-0.750280\pi$
$263$	−22.9548	−1.41546	−0.707728	+	0.706485i	$0.750280\pi$
$264$	16.5747	1.02010
$265$	−8.02654	−0.493066
$266$	0	0
$267$	−36.2258	−2.21698
$268$	1.14843	0.0701516
$269$	18.7189	1.14131	0.570656	−	0.821189i	$-0.306689\pi$
$269$	18.7189	1.14131	0.570656	+	0.821189i	$0.306689\pi$
$270$	−39.1497	−2.38257
$271$	13.1366	0.797994	0.398997	−	0.916952i	$-0.369358\pi$
$271$	13.1366	0.797994	0.398997	+	0.916952i	$0.369358\pi$
$272$	0.977224	0.0592529
$273$	6.85294	0.414759
$274$	7.30748	0.441461
$275$	−55.6228	−3.35418
$276$	1.95265	0.117536
$277$	9.00748	0.541207	0.270604	−	0.962691i	$-0.412777\pi$
$277$	9.00748	0.541207	0.270604	+	0.962691i	$0.412777\pi$
$278$	−15.1627	−0.909396
$279$	18.9223	1.13285
$280$	−3.90247	−0.233217
$281$	−13.8823	−0.828149	−0.414074	−	0.910243i	$-0.635895\pi$
$281$	−13.8823	−0.828149	−0.414074	+	0.910243i	$0.635895\pi$
$282$	38.6812	2.30343
$283$	3.52545	0.209566	0.104783	−	0.994495i	$-0.466585\pi$
$283$	3.52545	0.209566	0.104783	+	0.994495i	$0.466585\pi$
$284$	−4.58975	−0.272352
$285$	0	0
$286$	12.2251	0.722883
$287$	11.3245	0.668465
$288$	−6.29119	−0.370712
$289$	−16.0450	−0.943826
$290$	−26.7640	−1.57164
$291$	−24.2113	−1.41929
$292$	−6.31633	−0.369635
$293$	30.6355	1.78975	0.894873	−	0.446321i	$-0.147266\pi$
$293$	30.6355	1.78975	0.894873	+	0.446321i	$0.147266\pi$
$294$	−3.04815	−0.177771
$295$	−14.4876	−0.843499
$296$	−0.824327	−0.0479130
$297$	−54.5505	−3.16534
$298$	5.81139	0.336645
$299$	1.44022	0.0832903
$300$	31.1802	1.80019
$301$	−12.2093	−0.703733
$302$	0.480998	0.0276783
$303$	38.2183	2.19558
$304$	0	0
$305$	13.9099	0.796480
$306$	−6.14790	−0.351452
$307$	−25.2223	−1.43951	−0.719756	−	0.694227i	$-0.755746\pi$
$307$	−25.2223	−1.43951	−0.719756	+	0.694227i	$0.755746\pi$
$308$	−5.43763	−0.309838
$309$	−40.9862	−2.33162
$310$	−11.7376	−0.666650
$311$	−5.26030	−0.298285	−0.149142	−	0.988816i	$-0.547651\pi$
$311$	−5.26030	−0.298285	−0.149142	+	0.988816i	$0.547651\pi$
$312$	−6.85294	−0.387971
$313$	−3.76787	−0.212972	−0.106486	−	0.994314i	$-0.533960\pi$
$313$	−3.76787	−0.212972	−0.106486	+	0.994314i	$0.533960\pi$
$314$	5.53300	0.312245
$315$	24.5512	1.38330
$316$	12.3096	0.692471
$317$	16.6180	0.933359	0.466679	−	0.884427i	$-0.345450\pi$
$317$	16.6180	0.933359	0.466679	+	0.884427i	$0.345450\pi$
$318$	6.26939	0.351570
$319$	−37.2925	−2.08798
$320$	3.90247	0.218154
$321$	−35.2106	−1.96527
$322$	−0.640603	−0.0356994
$323$	0	0
$324$	11.7055	0.650307
$325$	22.9977	1.27568
$326$	2.43842	0.135052
$327$	57.5070	3.18014
$328$	−11.3245	−0.625291
$329$	−12.6901	−0.699626
$330$	64.6822	3.56064
$331$	12.6506	0.695341	0.347671	−	0.937617i	$-0.386973\pi$
$331$	12.6506	0.695341	0.347671	+	0.937617i	$0.386973\pi$
$332$	−9.84224	−0.540163
$333$	5.18600	0.284191
$334$	−3.74745	−0.205051
$335$	4.48172	0.244862
$336$	3.04815	0.166290
$337$	−9.93403	−0.541141	−0.270570	−	0.962700i	$-0.587212\pi$
$337$	−9.93403	−0.541141	−0.270570	+	0.962700i	$0.587212\pi$
$338$	7.94545	0.432175
$339$	−7.98360	−0.433610
$340$	3.81358	0.206821
$341$	−16.3550	−0.885671
$342$	0	0
$343$	1.00000	0.0539949
$344$	12.2093	0.658282
$345$	7.62015	0.410255
$346$	−6.84994	−0.368255
$347$	−13.2140	−0.709366	−0.354683	−	0.934987i	$-0.615411\pi$
$347$	−13.2140	−0.709366	−0.354683	+	0.934987i	$0.615411\pi$
$348$	20.9049	1.12062
$349$	−2.44779	−0.131027	−0.0655137	−	0.997852i	$-0.520869\pi$
$349$	−2.44779	−0.131027	−0.0655137	+	0.997852i	$0.520869\pi$
$350$	−10.2292	−0.546776
$351$	22.5544	1.20386
$352$	5.43763	0.289827
$353$	35.5373	1.89146	0.945729	−	0.324957i	$-0.105350\pi$
$353$	35.5373	1.89146	0.945729	+	0.324957i	$0.105350\pi$
$354$	11.3160	0.601438
$355$	−17.9114	−0.950636
$356$	−11.8845	−0.629879
$357$	2.97872	0.157651
$358$	−13.2215	−0.698778
$359$	−28.0462	−1.48022	−0.740112	−	0.672483i	$-0.765228\pi$
$359$	−28.0462	−1.48022	−0.740112	+	0.672483i	$0.765228\pi$
$360$	−24.5512	−1.29396
$361$	0	0
$362$	−5.11234	−0.268698
$363$	56.5974	2.97060
$364$	2.24823	0.117839
$365$	−24.6493	−1.29020
$366$	−10.8648	−0.567912
$367$	5.52498	0.288402	0.144201	−	0.989548i	$-0.453939\pi$
$367$	5.52498	0.288402	0.144201	+	0.989548i	$0.453939\pi$
$368$	0.640603	0.0333937
$369$	71.2447	3.70885
$370$	−3.21691	−0.167239
$371$	−2.05679	−0.106783
$372$	9.16802	0.475340
$373$	−1.66659	−0.0862927	−0.0431464	−	0.999069i	$-0.513738\pi$
$373$	−1.66659	−0.0862927	−0.0431464	+	0.999069i	$0.513738\pi$
$374$	5.31378	0.274769
$375$	62.2033	3.21216
$376$	12.6901	0.654440
$377$	15.4189	0.794114
$378$	−10.0320	−0.515992
$379$	24.2234	1.24427	0.622137	−	0.782908i	$-0.286265\pi$
$379$	24.2234	1.24427	0.622137	+	0.782908i	$0.286265\pi$
$380$	0	0
$381$	8.89426	0.455667
$382$	−26.4002	−1.35075
$383$	−5.06432	−0.258774	−0.129387	−	0.991594i	$-0.541301\pi$
$383$	−5.06432	−0.258774	−0.129387	+	0.991594i	$0.541301\pi$
$384$	−3.04815	−0.155550
$385$	−21.2202	−1.08148
$386$	−15.7941	−0.803898
$387$	−76.8112	−3.90453
$388$	−7.94295	−0.403242
$389$	−23.5536	−1.19421	−0.597107	−	0.802162i	$-0.703683\pi$
$389$	−23.5536	−1.19421	−0.597107	+	0.802162i	$0.703683\pi$
$390$	−26.7434	−1.35420
$391$	0.626012	0.0316588
$392$	−1.00000	−0.0505076
$393$	2.48332	0.125267
$394$	3.73311	0.188072
$395$	48.0379	2.41705
$396$	−34.2092	−1.71908
$397$	−16.9499	−0.850691	−0.425345	−	0.905031i	$-0.639847\pi$
$397$	−16.9499	−0.850691	−0.425345	+	0.905031i	$0.639847\pi$
$398$	16.1988	0.811972
$399$	0	0
$400$	10.2292	0.511462
$401$	−22.6365	−1.13041	−0.565207	−	0.824949i	$-0.691204\pi$
$401$	−22.6365	−1.13041	−0.565207	+	0.824949i	$0.691204\pi$
$402$	−3.50059	−0.174593
$403$	6.76209	0.336844
$404$	12.5382	0.623799
$405$	45.6804	2.26988
$406$	−6.85823	−0.340368
$407$	−4.48238	−0.222183
$408$	−2.97872	−0.147469
$409$	−1.54005	−0.0761503	−0.0380752	−	0.999275i	$-0.512123\pi$
$409$	−1.54005	−0.0761503	−0.0380752	+	0.999275i	$0.512123\pi$
$410$	−44.1935	−2.18256
$411$	−22.2743	−1.09871
$412$	−13.4463	−0.662450
$413$	−3.71241	−0.182676
$414$	−4.03016	−0.198071
$415$	−38.4090	−1.88542
$416$	−2.24823	−0.110229
$417$	46.2180	2.26330
$418$	0	0
$419$	−2.33850	−0.114243	−0.0571216	−	0.998367i	$-0.518192\pi$
$419$	−2.33850	−0.114243	−0.0571216	+	0.998367i	$0.518192\pi$
$420$	11.8953	0.580430
$421$	−9.21012	−0.448874	−0.224437	−	0.974489i	$-0.572054\pi$
$421$	−9.21012	−0.448874	−0.224437	+	0.974489i	$0.572054\pi$
$422$	−13.9798	−0.680528
$423$	−79.8356	−3.88174
$424$	2.05679	0.0998864
$425$	9.99625	0.484890
$426$	13.9902	0.677829
$427$	3.56439	0.172493
$428$	−11.5515	−0.558363
$429$	−37.2638	−1.79911
$430$	47.6464	2.29772
$431$	7.77763	0.374635	0.187318	−	0.982299i	$-0.440021\pi$
$431$	7.77763	0.374635	0.187318	+	0.982299i	$0.440021\pi$
$432$	10.0320	0.482667
$433$	−3.43962	−0.165298	−0.0826488	−	0.996579i	$-0.526338\pi$
$433$	−3.43962	−0.165298	−0.0826488	+	0.996579i	$0.526338\pi$
$434$	−3.00774	−0.144376
$435$	81.5806	3.91149
$436$	18.8662	0.903528
$437$	0	0
$438$	19.2531	0.919949
$439$	27.4754	1.31133	0.655664	−	0.755053i	$-0.272389\pi$
$439$	27.4754	1.31133	0.655664	+	0.755053i	$0.272389\pi$
$440$	21.2202	1.01163
$441$	6.29119	0.299581
$442$	−2.19703	−0.104502
$443$	29.7544	1.41367	0.706836	−	0.707377i	$-0.250122\pi$
$443$	29.7544	1.41367	0.706836	+	0.707377i	$0.250122\pi$
$444$	2.51267	0.119246
$445$	−46.3790	−2.19858
$446$	11.9087	0.563894
$447$	−17.7140	−0.837842
$448$	1.00000	0.0472456
$449$	−27.8042	−1.31216	−0.656081	−	0.754690i	$-0.727787\pi$
$449$	−27.8042	−1.31216	−0.656081	+	0.754690i	$0.727787\pi$
$450$	−64.3541	−3.03368
$451$	−61.5785	−2.89962
$452$	−2.61917	−0.123195
$453$	−1.46615	−0.0688858
$454$	13.4459	0.631049
$455$	8.77365	0.411315
$456$	0	0
$457$	−7.02276	−0.328511	−0.164255	−	0.986418i	$-0.552522\pi$
$457$	−7.02276	−0.328511	−0.164255	+	0.986418i	$0.552522\pi$
$458$	5.02164	0.234646
$459$	9.80354	0.457590
$460$	2.49993	0.116560
$461$	−11.0511	−0.514699	−0.257350	−	0.966318i	$-0.582849\pi$
$461$	−11.0511	−0.514699	−0.257350	+	0.966318i	$0.582849\pi$
$462$	16.5747	0.771124
$463$	−11.9667	−0.556138	−0.278069	−	0.960561i	$-0.589694\pi$
$463$	−11.9667	−0.556138	−0.278069	+	0.960561i	$0.589694\pi$
$464$	6.85823	0.318385
$465$	35.7779	1.65916
$466$	−3.80785	−0.176395
$467$	−33.3755	−1.54444	−0.772218	−	0.635358i	$-0.780853\pi$
$467$	−33.3755	−1.54444	−0.772218	+	0.635358i	$0.780853\pi$
$468$	14.1441	0.653810
$469$	1.14843	0.0530297
$470$	49.5225	2.28430
$471$	−16.8654	−0.777116
$472$	3.71241	0.170878
$473$	66.3898	3.05260
$474$	−37.5216	−1.72342
$475$	0	0
$476$	0.977224	0.0447910
$477$	−12.9396	−0.592465
$478$	2.48796	0.113797
$479$	28.5407	1.30406	0.652029	−	0.758194i	$-0.273918\pi$
$479$	28.5407	1.30406	0.652029	+	0.758194i	$0.273918\pi$
$480$	−11.8953	−0.542943
$481$	1.85328	0.0845022
$482$	5.53763	0.252232
$483$	1.95265	0.0888487
$484$	18.5678	0.843992
$485$	−30.9971	−1.40751
$486$	−5.58406	−0.253298
$487$	27.5006	1.24617	0.623087	−	0.782153i	$-0.285878\pi$
$487$	27.5006	1.24617	0.623087	+	0.782153i	$0.285878\pi$
$488$	−3.56439	−0.161353
$489$	−7.43266	−0.336116
$490$	−3.90247	−0.176295
$491$	16.7667	0.756671	0.378336	−	0.925669i	$-0.376497\pi$
$491$	16.7667	0.756671	0.378336	+	0.925669i	$0.376497\pi$
$492$	34.5188	1.55623
$493$	6.70202	0.301844
$494$	0	0
$495$	−133.500	−6.00039
$496$	3.00774	0.135051
$497$	−4.58975	−0.205879
$498$	30.0006	1.34436
$499$	−39.8615	−1.78444	−0.892222	−	0.451597i	$-0.850854\pi$
$499$	−39.8615	−1.78444	−0.892222	+	0.451597i	$0.850854\pi$
$500$	20.4069	0.912626
$501$	11.4228	0.510331
$502$	5.55332	0.247857
$503$	16.5023	0.735800	0.367900	−	0.929865i	$-0.380077\pi$
$503$	16.5023	0.735800	0.367900	+	0.929865i	$0.380077\pi$
$504$	−6.29119	−0.280232
$505$	48.9299	2.17735
$506$	3.48336	0.154854
$507$	−24.2189	−1.07560
$508$	2.91792	0.129462
$509$	−27.8470	−1.23429	−0.617147	−	0.786848i	$-0.711712\pi$
$509$	−27.8470	−1.23429	−0.617147	+	0.786848i	$0.711712\pi$
$510$	−11.6244	−0.514735
$511$	−6.31633	−0.279418
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	21.9978	0.970282
$515$	−52.4736	−2.31226
$516$	−37.2158	−1.63833
$517$	69.0039	3.03479
$518$	−0.824327	−0.0362188
$519$	20.8796	0.916514
$520$	−8.77365	−0.384750
$521$	−28.0944	−1.23084	−0.615418	−	0.788201i	$-0.711013\pi$
$521$	−28.0944	−1.23084	−0.615418	+	0.788201i	$0.711013\pi$
$522$	−43.1464	−1.88847
$523$	−14.6392	−0.640129	−0.320064	−	0.947396i	$-0.603705\pi$
$523$	−14.6392	−0.640129	−0.320064	+	0.947396i	$0.603705\pi$
$524$	0.814699	0.0355903
$525$	31.1802	1.36082
$526$	22.9548	1.00088
$527$	2.93923	0.128035
$528$	−16.5747	−0.721321
$529$	−22.5896	−0.982158
$530$	8.02654	0.348651
$531$	−23.3555	−1.01354
$532$	0	0
$533$	25.4601	1.10280
$534$	36.2258	1.56764
$535$	−45.0793	−1.94895
$536$	−1.14843	−0.0496047
$537$	40.3011	1.73912
$538$	−18.7189	−0.807030
$539$	−5.43763	−0.234215
$540$	39.1497	1.68473
$541$	−10.3054	−0.443064	−0.221532	−	0.975153i	$-0.571106\pi$
$541$	−10.3054	−0.443064	−0.221532	+	0.975153i	$0.571106\pi$
$542$	−13.1366	−0.564267
$543$	15.5831	0.668737
$544$	−0.977224	−0.0418981
$545$	73.6248	3.15374
$546$	−6.85294	−0.293279
$547$	−24.8564	−1.06278	−0.531392	−	0.847126i	$-0.678331\pi$
$547$	−24.8564	−1.06278	−0.531392	+	0.847126i	$0.678331\pi$
$548$	−7.30748	−0.312160
$549$	22.4243	0.957045
$550$	55.6228	2.37177
$551$	0	0
$552$	−1.95265	−0.0831103
$553$	12.3096	0.523459
$554$	−9.00748	−0.382691
$555$	9.80560	0.416225
$556$	15.1627	0.643040
$557$	−36.8917	−1.56315	−0.781575	−	0.623812i	$-0.785583\pi$
$557$	−36.8917	−1.56315	−0.781575	+	0.623812i	$0.785583\pi$
$558$	−18.9223	−0.801043
$559$	−27.4494	−1.16099
$560$	3.90247	0.164909
$561$	−16.1972	−0.683845
$562$	13.8823	0.585589
$563$	7.28154	0.306880	0.153440	−	0.988158i	$-0.450965\pi$
$563$	7.28154	0.306880	0.153440	+	0.988158i	$0.450965\pi$
$564$	−38.6812	−1.62877
$565$	−10.2212	−0.430009
$566$	−3.52545	−0.148186
$567$	11.7055	0.491586
$568$	4.58975	0.192582
$569$	11.4629	0.480551	0.240276	−	0.970705i	$-0.422762\pi$
$569$	11.4629	0.480551	0.240276	+	0.970705i	$0.422762\pi$
$570$	0	0
$571$	20.3517	0.851691	0.425846	−	0.904796i	$-0.359977\pi$
$571$	20.3517	0.851691	0.425846	+	0.904796i	$0.359977\pi$
$572$	−12.2251	−0.511156
$573$	80.4715	3.36175
$574$	−11.3245	−0.472676
$575$	6.55288	0.273274
$576$	6.29119	0.262133
$577$	37.2381	1.55024	0.775122	−	0.631811i	$-0.217688\pi$
$577$	37.2381	1.55024	0.775122	+	0.631811i	$0.217688\pi$
$578$	16.0450	0.667385
$579$	48.1427	2.00074
$580$	26.7640	1.11131
$581$	−9.84224	−0.408325
$582$	24.2113	1.00359
$583$	11.1840	0.463196
$584$	6.31633	0.261372
$585$	55.1968	2.28210
$586$	−30.6355	−1.26554
$587$	11.2034	0.462416	0.231208	−	0.972904i	$-0.425732\pi$
$587$	11.2034	0.462416	0.231208	+	0.972904i	$0.425732\pi$
$588$	3.04815	0.125703
$589$	0	0
$590$	14.4876	0.596444
$591$	−11.3791	−0.468073
$592$	0.824327	0.0338796
$593$	−37.0804	−1.52271	−0.761354	−	0.648337i	$-0.775465\pi$
$593$	−37.0804	−1.52271	−0.761354	+	0.648337i	$0.775465\pi$
$594$	54.5505	2.23823
$595$	3.81358	0.156342
$596$	−5.81139	−0.238044
$597$	−49.3763	−2.02084
$598$	−1.44022	−0.0588952
$599$	31.7534	1.29741	0.648705	−	0.761040i	$-0.275311\pi$
$599$	31.7534	1.29741	0.648705	+	0.761040i	$0.275311\pi$
$600$	−31.1802	−1.27293
$601$	−19.7428	−0.805324	−0.402662	−	0.915349i	$-0.631915\pi$
$601$	−19.7428	−0.805324	−0.402662	+	0.915349i	$0.631915\pi$
$602$	12.2093	0.497615
$603$	7.22501	0.294225
$604$	−0.480998	−0.0195715
$605$	72.4603	2.94593
$606$	−38.2183	−1.55251
$607$	21.2159	0.861128	0.430564	−	0.902560i	$-0.358315\pi$
$607$	21.2159	0.861128	0.430564	+	0.902560i	$0.358315\pi$
$608$	0	0
$609$	20.9049	0.847109
$610$	−13.9099	−0.563196
$611$	−28.5302	−1.15421
$612$	6.14790	0.248514
$613$	19.9161	0.804405	0.402203	−	0.915551i	$-0.368245\pi$
$613$	19.9161	0.804405	0.402203	+	0.915551i	$0.368245\pi$
$614$	25.2223	1.01789
$615$	134.708	5.43196
$616$	5.43763	0.219088
$617$	18.0683	0.727402	0.363701	−	0.931516i	$-0.381513\pi$
$617$	18.0683	0.727402	0.363701	+	0.931516i	$0.381513\pi$
$618$	40.9862	1.64871
$619$	31.5626	1.26861	0.634303	−	0.773084i	$-0.281287\pi$
$619$	31.5626	1.26861	0.634303	+	0.773084i	$0.281287\pi$
$620$	11.7376	0.471393
$621$	6.42655	0.257889
$622$	5.26030	0.210919
$623$	−11.8845	−0.476144
$624$	6.85294	0.274337
$625$	28.4912	1.13965
$626$	3.76787	0.150594
$627$	0	0
$628$	−5.53300	−0.220791
$629$	0.805551	0.0321194
$630$	−24.5512	−0.978142
$631$	25.1232	1.00014	0.500070	−	0.865985i	$-0.333308\pi$
$631$	25.1232	1.00014	0.500070	+	0.865985i	$0.333308\pi$
$632$	−12.3096	−0.489651
$633$	42.6126	1.69370
$634$	−16.6180	−0.659984
$635$	11.3871	0.451883
$636$	−6.26939	−0.248597
$637$	2.24823	0.0890782
$638$	37.2925	1.47642
$639$	−28.8750	−1.14228
$640$	−3.90247	−0.154259
$641$	3.23583	0.127808	0.0639039	−	0.997956i	$-0.479645\pi$
$641$	3.23583	0.127808	0.0639039	+	0.997956i	$0.479645\pi$
$642$	35.2106	1.38965
$643$	−19.3127	−0.761619	−0.380809	−	0.924654i	$-0.624355\pi$
$643$	−19.3127	−0.761619	−0.380809	+	0.924654i	$0.624355\pi$
$644$	0.640603	0.0252433
$645$	−145.233	−5.71856
$646$	0	0
$647$	−11.5212	−0.452943	−0.226472	−	0.974018i	$-0.572719\pi$
$647$	−11.5212	−0.452943	−0.226472	+	0.974018i	$0.572719\pi$
$648$	−11.7055	−0.459837
$649$	20.1867	0.792399
$650$	−22.9977	−0.902045
$651$	9.16802	0.359323
$652$	−2.43842	−0.0954958
$653$	−43.5279	−1.70338	−0.851689	−	0.524047i	$-0.824421\pi$
$653$	−43.5279	−1.70338	−0.851689	+	0.524047i	$0.824421\pi$
$654$	−57.5070	−2.24870
$655$	3.17933	0.124227
$656$	11.3245	0.442148
$657$	−39.7373	−1.55030
$658$	12.6901	0.494710
$659$	−5.02462	−0.195731	−0.0978657	−	0.995200i	$-0.531202\pi$
$659$	−5.02462	−0.195731	−0.0978657	+	0.995200i	$0.531202\pi$
$660$	−64.6822	−2.51775
$661$	16.0652	0.624864	0.312432	−	0.949940i	$-0.398856\pi$
$661$	16.0652	0.624864	0.312432	+	0.949940i	$0.398856\pi$
$662$	−12.6506	−0.491680
$663$	6.69686	0.260084
$664$	9.84224	0.381953
$665$	0	0
$666$	−5.18600	−0.200953
$667$	4.39340	0.170113
$668$	3.74745	0.144993
$669$	−36.2995	−1.40342
$670$	−4.48172	−0.173144
$671$	−19.3819	−0.748228
$672$	−3.04815	−0.117585
$673$	1.32631	0.0511255	0.0255627	−	0.999673i	$-0.491862\pi$
$673$	1.32631	0.0511255	0.0255627	+	0.999673i	$0.491862\pi$
$674$	9.93403	0.382644
$675$	102.620	3.94985
$676$	−7.94545	−0.305594
$677$	−17.6403	−0.677971	−0.338986	−	0.940792i	$-0.610084\pi$
$677$	−17.6403	−0.677971	−0.338986	+	0.940792i	$0.610084\pi$
$678$	7.98360	0.306608
$679$	−7.94295	−0.304823
$680$	−3.81358	−0.146244
$681$	−40.9852	−1.57056
$682$	16.3550	0.626264
$683$	17.9010	0.684965	0.342482	−	0.939524i	$-0.388732\pi$
$683$	17.9010	0.684965	0.342482	+	0.939524i	$0.388732\pi$
$684$	0	0
$685$	−28.5172	−1.08959
$686$	−1.00000	−0.0381802
$687$	−15.3067	−0.583986
$688$	−12.2093	−0.465476
$689$	−4.62414	−0.176166
$690$	−7.62015	−0.290094
$691$	10.4222	0.396480	0.198240	−	0.980153i	$-0.436477\pi$
$691$	10.4222	0.396480	0.198240	+	0.980153i	$0.436477\pi$
$692$	6.84994	0.260396
$693$	−34.2092	−1.29950
$694$	13.2140	0.501598
$695$	59.1718	2.24451
$696$	−20.9049	−0.792398
$697$	11.0666	0.419176
$698$	2.44779	0.0926503
$699$	11.6069	0.439012
$700$	10.2292	0.386629
$701$	−45.9994	−1.73737	−0.868686	−	0.495362i	$-0.835035\pi$
$701$	−45.9994	−1.73737	−0.868686	+	0.495362i	$0.835035\pi$
$702$	−22.5544	−0.851259
$703$	0	0
$704$	−5.43763	−0.204938
$705$	−150.952	−5.68518
$706$	−35.5373	−1.33746
$707$	12.5382	0.471548
$708$	−11.3160	−0.425281
$709$	4.31058	0.161887	0.0809436	−	0.996719i	$-0.474207\pi$
$709$	4.31058	0.161887	0.0809436	+	0.996719i	$0.474207\pi$
$710$	17.9114	0.672201
$711$	77.4423	2.90431
$712$	11.8845	0.445392
$713$	1.92676	0.0721579
$714$	−2.97872	−0.111476
$715$	−47.7079	−1.78417
$716$	13.2215	0.494111
$717$	−7.58366	−0.283217
$718$	28.0462	1.04668
$719$	32.1996	1.20084	0.600421	−	0.799684i	$-0.295000\pi$
$719$	32.1996	1.20084	0.600421	+	0.799684i	$0.295000\pi$
$720$	24.5512	0.914968
$721$	−13.4463	−0.500765
$722$	0	0
$723$	−16.8795	−0.627755
$724$	5.11234	0.189998
$725$	70.1545	2.60547
$726$	−56.5974	−2.10053
$727$	15.3908	0.570814	0.285407	−	0.958406i	$-0.407871\pi$
$727$	15.3908	0.570814	0.285407	+	0.958406i	$0.407871\pi$
$728$	−2.24823	−0.0833251
$729$	−18.0956	−0.670206
$730$	24.6493	0.912310
$731$	−11.9312	−0.441293
$732$	10.8648	0.401574
$733$	−1.87547	−0.0692719	−0.0346360	−	0.999400i	$-0.511027\pi$
$733$	−1.87547	−0.0692719	−0.0346360	+	0.999400i	$0.511027\pi$
$734$	−5.52498	−0.203931
$735$	11.8953	0.438764
$736$	−0.640603	−0.0236129
$737$	−6.24475	−0.230028
$738$	−71.2447	−2.62255
$739$	−23.4196	−0.861502	−0.430751	−	0.902471i	$-0.641751\pi$
$739$	−23.4196	−0.861502	−0.430751	+	0.902471i	$0.641751\pi$
$740$	3.21691	0.118256
$741$	0	0
$742$	2.05679	0.0755070
$743$	−18.5235	−0.679561	−0.339780	−	0.940505i	$-0.610353\pi$
$743$	−18.5235	−0.679561	−0.339780	+	0.940505i	$0.610353\pi$
$744$	−9.16802	−0.336116
$745$	−22.6788	−0.830886
$746$	1.66659	0.0610182
$747$	−61.9194	−2.26551
$748$	−5.31378	−0.194291
$749$	−11.5515	−0.422082
$750$	−62.2033	−2.27134
$751$	−1.49895	−0.0546974	−0.0273487	−	0.999626i	$-0.508706\pi$
$751$	−1.49895	−0.0546974	−0.0273487	+	0.999626i	$0.508706\pi$
$752$	−12.6901	−0.462759
$753$	−16.9273	−0.616867
$754$	−15.4189	−0.561523
$755$	−1.87708	−0.0683138
$756$	10.0320	0.364862
$757$	28.5048	1.03603	0.518013	−	0.855373i	$-0.326672\pi$
$757$	28.5048	1.03603	0.518013	+	0.855373i	$0.326672\pi$
$758$	−24.2234	−0.879835
$759$	−10.6178	−0.385401
$760$	0	0
$761$	37.9783	1.37671	0.688356	−	0.725373i	$-0.258333\pi$
$761$	37.9783	1.37671	0.688356	+	0.725373i	$0.258333\pi$
$762$	−8.89426	−0.322205
$763$	18.8662	0.683003
$764$	26.4002	0.955124
$765$	23.9920	0.867432
$766$	5.06432	0.182981
$767$	−8.34637	−0.301370
$768$	3.04815	0.109990
$769$	−32.6697	−1.17810	−0.589049	−	0.808097i	$-0.700498\pi$
$769$	−32.6697	−1.17810	−0.589049	+	0.808097i	$0.700498\pi$
$770$	21.2202	0.764722
$771$	−67.0526	−2.41484
$772$	15.7941	0.568442
$773$	−19.9190	−0.716436	−0.358218	−	0.933638i	$-0.616615\pi$
$773$	−19.9190	−0.716436	−0.358218	+	0.933638i	$0.616615\pi$
$774$	76.8112	2.76092
$775$	30.7669	1.10518
$776$	7.94295	0.285135
$777$	2.51267	0.0901414
$778$	23.5536	0.844437
$779$	0	0
$780$	26.7434	0.957566
$781$	24.9574	0.893045
$782$	−0.626012	−0.0223861
$783$	68.8020	2.45878
$784$	1.00000	0.0357143
$785$	−21.5923	−0.770664
$786$	−2.48332	−0.0885771
$787$	6.83078	0.243491	0.121745	−	0.992561i	$-0.461151\pi$
$787$	6.83078	0.243491	0.121745	+	0.992561i	$0.461151\pi$
$788$	−3.73311	−0.132987
$789$	−69.9697	−2.49099
$790$	−48.0379	−1.70911
$791$	−2.61917	−0.0931268
$792$	34.2092	1.21557
$793$	8.01359	0.284571
$794$	16.9499	0.601529
$795$	−24.4661	−0.867722
$796$	−16.1988	−0.574151
$797$	14.9421	0.529275	0.264638	−	0.964348i	$-0.414748\pi$
$797$	14.9421	0.529275	0.264638	+	0.964348i	$0.414748\pi$
$798$	0	0
$799$	−12.4010	−0.438717
$800$	−10.2292	−0.361658
$801$	−74.7679	−2.64179
$802$	22.6365	0.799324
$803$	34.3459	1.21204
$804$	3.50059	0.123456
$805$	2.49993	0.0881110
$806$	−6.76209	−0.238185
$807$	57.0580	2.00854
$808$	−12.5382	−0.441092
$809$	14.6592	0.515390	0.257695	−	0.966226i	$-0.417037\pi$
$809$	14.6592	0.515390	0.257695	+	0.966226i	$0.417037\pi$
$810$	−45.6804	−1.60505
$811$	49.7049	1.74537	0.872687	−	0.488280i	$-0.162375\pi$
$811$	49.7049	1.74537	0.872687	+	0.488280i	$0.162375\pi$
$812$	6.85823	0.240677
$813$	40.0424	1.40435
$814$	4.48238	0.157107
$815$	−9.51585	−0.333326
$816$	2.97872	0.104276
$817$	0	0
$818$	1.54005	0.0538464
$819$	14.1441	0.494234
$820$	44.1935	1.54330
$821$	−26.7176	−0.932449	−0.466225	−	0.884666i	$-0.654386\pi$
$821$	−26.7176	−0.932449	−0.466225	+	0.884666i	$0.654386\pi$
$822$	22.2743	0.776905
$823$	3.16141	0.110200	0.0550999	−	0.998481i	$-0.482452\pi$
$823$	3.16141	0.110200	0.0550999	+	0.998481i	$0.482452\pi$
$824$	13.4463	0.468423
$825$	−169.546	−5.90285
$826$	3.71241	0.129171
$827$	−24.2855	−0.844488	−0.422244	−	0.906482i	$-0.638757\pi$
$827$	−24.2855	−0.844488	−0.422244	+	0.906482i	$0.638757\pi$
$828$	4.03016	0.140058
$829$	22.9819	0.798195	0.399098	−	0.916908i	$-0.369323\pi$
$829$	22.9819	0.798195	0.399098	+	0.916908i	$0.369323\pi$
$830$	38.4090	1.33320
$831$	27.4561	0.952442
$832$	2.24823	0.0779435
$833$	0.977224	0.0338588
$834$	−46.2180	−1.60040
$835$	14.6243	0.506094
$836$	0	0
$837$	30.1737	1.04296
$838$	2.33850	0.0807821
$839$	7.80775	0.269553	0.134777	−	0.990876i	$-0.456968\pi$
$839$	7.80775	0.269553	0.134777	+	0.990876i	$0.456968\pi$
$840$	−11.8953	−0.410426
$841$	18.0353	0.621907
$842$	9.21012	0.317402
$843$	−42.3153	−1.45742
$844$	13.9798	0.481206
$845$	−31.0068	−1.06667
$846$	79.8356	2.74481
$847$	18.5678	0.637998
$848$	−2.05679	−0.0706303
$849$	10.7461	0.368805
$850$	−9.99625	−0.342869
$851$	0.528066	0.0181019
$852$	−13.9902	−0.479298
$853$	−0.148294	−0.00507747	−0.00253874	−	0.999997i	$-0.500808\pi$
$853$	−0.148294	−0.00507747	−0.00253874	+	0.999997i	$0.500808\pi$
$854$	−3.56439	−0.121971
$855$	0	0
$856$	11.5515	0.394822
$857$	−17.4431	−0.595845	−0.297923	−	0.954590i	$-0.596294\pi$
$857$	−17.4431	−0.595845	−0.297923	+	0.954590i	$0.596294\pi$
$858$	37.2638	1.27216
$859$	44.9110	1.53234	0.766171	−	0.642636i	$-0.222159\pi$
$859$	44.9110	1.53234	0.766171	+	0.642636i	$0.222159\pi$
$860$	−47.6464	−1.62473
$861$	34.5188	1.17640
$862$	−7.77763	−0.264907
$863$	−4.73021	−0.161018	−0.0805091	−	0.996754i	$-0.525655\pi$
$863$	−4.73021	−0.161018	−0.0805091	+	0.996754i	$0.525655\pi$
$864$	−10.0320	−0.341297
$865$	26.7317	0.908904
$866$	3.43962	0.116883
$867$	−48.9076	−1.66099
$868$	3.00774	0.102089
$869$	−66.9352	−2.27062
$870$	−81.5806	−2.76584
$871$	2.58194	0.0874858
$872$	−18.8662	−0.638891
$873$	−49.9707	−1.69125
$874$	0	0
$875$	20.4069	0.689880
$876$	−19.2531	−0.650502
$877$	−22.3767	−0.755608	−0.377804	−	0.925886i	$-0.623321\pi$
$877$	−22.3767	−0.755608	−0.377804	+	0.925886i	$0.623321\pi$
$878$	−27.4754	−0.927249
$879$	93.3815	3.14968
$880$	−21.2202	−0.715332
$881$	17.3063	0.583064	0.291532	−	0.956561i	$-0.405835\pi$
$881$	17.3063	0.583064	0.291532	+	0.956561i	$0.405835\pi$
$882$	−6.29119	−0.211835
$883$	22.2407	0.748459	0.374230	−	0.927336i	$-0.377907\pi$
$883$	22.2407	0.748459	0.374230	+	0.927336i	$0.377907\pi$
$884$	2.19703	0.0738940
$885$	−44.1602	−1.48443
$886$	−29.7544	−0.999618
$887$	31.6781	1.06365	0.531823	−	0.846856i	$-0.321507\pi$
$887$	31.6781	1.06365	0.531823	+	0.846856i	$0.321507\pi$
$888$	−2.51267	−0.0843196
$889$	2.91792	0.0978640
$890$	46.3790	1.55463
$891$	−63.6504	−2.13237
$892$	−11.9087	−0.398733
$893$	0	0
$894$	17.7140	0.592444
$895$	51.5965	1.72468
$896$	−1.00000	−0.0334077
$897$	4.39001	0.146578
$898$	27.8042	0.927839
$899$	20.6277	0.687974
$900$	64.3541	2.14514
$901$	−2.00994	−0.0669608
$902$	61.5785	2.05034
$903$	−37.2158	−1.23846
$904$	2.61917	0.0871122
$905$	19.9507	0.663184
$906$	1.46615	0.0487096
$907$	−35.7705	−1.18774	−0.593870	−	0.804561i	$-0.702401\pi$
$907$	−35.7705	−1.18774	−0.593870	+	0.804561i	$0.702401\pi$
$908$	−13.4459	−0.446219
$909$	78.8803	2.61629
$910$	−8.77365	−0.290844
$911$	1.00147	0.0331803	0.0165902	−	0.999862i	$-0.494719\pi$
$911$	1.00147	0.0331803	0.0165902	+	0.999862i	$0.494719\pi$
$912$	0	0
$913$	53.5184	1.77120
$914$	7.02276	0.232292
$915$	42.3995	1.40168
$916$	−5.02164	−0.165920
$917$	0.814699	0.0269037
$918$	−9.80354	−0.323565
$919$	27.1273	0.894847	0.447423	−	0.894322i	$-0.352342\pi$
$919$	27.1273	0.894847	0.447423	+	0.894322i	$0.352342\pi$
$920$	−2.49993	−0.0824203
$921$	−76.8812	−2.53332
$922$	11.0511	0.363947
$923$	−10.3188	−0.339649
$924$	−16.5747	−0.545267
$925$	8.43223	0.277250
$926$	11.9667	0.393249
$927$	−84.5930	−2.77840
$928$	−6.85823	−0.225132
$929$	2.19665	0.0720697	0.0360349	−	0.999351i	$-0.488527\pi$
$929$	2.19665	0.0720697	0.0360349	+	0.999351i	$0.488527\pi$
$930$	−35.7779	−1.17320
$931$	0	0
$932$	3.80785	0.124730
$933$	−16.0342	−0.524935
$934$	33.3755	1.09208
$935$	−20.7368	−0.678167
$936$	−14.1441	−0.462313
$937$	−16.3687	−0.534742	−0.267371	−	0.963594i	$-0.586155\pi$
$937$	−16.3687	−0.534742	−0.267371	+	0.963594i	$0.586155\pi$
$938$	−1.14843	−0.0374976
$939$	−11.4850	−0.374799
$940$	−49.5225	−1.61525
$941$	49.5617	1.61566	0.807832	−	0.589413i	$-0.200641\pi$
$941$	49.5617	1.61566	0.807832	+	0.589413i	$0.200641\pi$
$942$	16.8654	0.549504
$943$	7.25451	0.236239
$944$	−3.71241	−0.120829
$945$	39.1497	1.27354
$946$	−66.3898	−2.15852
$947$	32.3241	1.05039	0.525196	−	0.850981i	$-0.323992\pi$
$947$	32.3241	1.05039	0.525196	+	0.850981i	$0.323992\pi$
$948$	37.5216	1.21864
$949$	−14.2006	−0.460971
$950$	0	0
$951$	50.6540	1.64257
$952$	−0.977224	−0.0316720
$953$	−29.6712	−0.961143	−0.480572	−	0.876955i	$-0.659571\pi$
$953$	−29.6712	−0.961143	−0.480572	+	0.876955i	$0.659571\pi$
$954$	12.9396	0.418936
$955$	103.026	3.33383
$956$	−2.48796	−0.0804663
$957$	−113.673	−3.67453
$958$	−28.5407	−0.922109
$959$	−7.30748	−0.235971
$960$	11.8953	0.383919
$961$	−21.9535	−0.708178
$962$	−1.85328	−0.0597521
$963$	−72.6727	−2.34184
$964$	−5.53763	−0.178355
$965$	61.6359	1.98413
$966$	−1.95265	−0.0628255
$967$	23.6909	0.761847	0.380923	−	0.924607i	$-0.375606\pi$
$967$	23.6909	0.761847	0.380923	+	0.924607i	$0.375606\pi$
$968$	−18.5678	−0.596792
$969$	0	0
$970$	30.9971	0.995257
$971$	−29.8444	−0.957753	−0.478876	−	0.877882i	$-0.658956\pi$
$971$	−29.8444	−0.957753	−0.478876	+	0.877882i	$0.658956\pi$
$972$	5.58406	0.179109
$973$	15.1627	0.486093
$974$	−27.5006	−0.881177
$975$	70.1004	2.24501
$976$	3.56439	0.114093
$977$	−55.2708	−1.76827	−0.884135	−	0.467232i	$-0.845251\pi$
$977$	−55.2708	−1.76827	−0.884135	+	0.467232i	$0.845251\pi$
$978$	7.43266	0.237670
$979$	64.6237	2.06538
$980$	3.90247	0.124660
$981$	118.691	3.78951
$982$	−16.7667	−0.535047
$983$	−24.8474	−0.792508	−0.396254	−	0.918141i	$-0.629690\pi$
$983$	−24.8474	−0.792508	−0.396254	+	0.918141i	$0.629690\pi$
$984$	−34.5188	−1.10042
$985$	−14.5683	−0.464186
$986$	−6.70202	−0.213436
$987$	−38.6812	−1.23123
$988$	0	0
$989$	−7.82132	−0.248704
$990$	133.500	4.24291
$991$	−37.4253	−1.18885	−0.594427	−	0.804150i	$-0.702621\pi$
$991$	−37.4253	−1.18885	−0.594427	+	0.804150i	$0.702621\pi$
$992$	−3.00774	−0.0954957
$993$	38.5609	1.22369
$994$	4.58975	0.145578
$995$	−63.2152	−2.00406
$996$	−30.0006	−0.950604
$997$	33.0295	1.04605	0.523027	−	0.852316i	$-0.324803\pi$
$997$	33.0295	1.04605	0.523027	+	0.852316i	$0.324803\pi$
$998$	39.8615	1.26179
$999$	8.26967	0.261641

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5054.2.a.bg.1.7	✓	8
19.18	odd	2		5054.2.a.bh.1.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5054.2.a.bg.1.7	✓	8	1.1	even	1	trivial
5054.2.a.bh.1.2	yes	8	19.18	odd	2

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:	\( N \)	\(=\)	\( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5054.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.04815 q^{3} +1.00000 q^{4} +3.90247 q^{5} -3.04815 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.29119 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +3.04815 q^{3} +1.00000 q^{4} +3.90247 q^{5} -3.04815 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.29119 q^{9} -3.90247 q^{10} -5.43763 q^{11} +3.04815 q^{12} +2.24823 q^{13} -1.00000 q^{14} +11.8953 q^{15} +1.00000 q^{16} +0.977224 q^{17} -6.29119 q^{18} +3.90247 q^{20} +3.04815 q^{21} +5.43763 q^{22} +0.640603 q^{23} -3.04815 q^{24} +10.2292 q^{25} -2.24823 q^{26} +10.0320 q^{27} +1.00000 q^{28} +6.85823 q^{29} -11.8953 q^{30} +3.00774 q^{31} -1.00000 q^{32} -16.5747 q^{33} -0.977224 q^{34} +3.90247 q^{35} +6.29119 q^{36} +0.824327 q^{37} +6.85294 q^{39} -3.90247 q^{40} +11.3245 q^{41} -3.04815 q^{42} -12.2093 q^{43} -5.43763 q^{44} +24.5512 q^{45} -0.640603 q^{46} -12.6901 q^{47} +3.04815 q^{48} +1.00000 q^{49} -10.2292 q^{50} +2.97872 q^{51} +2.24823 q^{52} -2.05679 q^{53} -10.0320 q^{54} -21.2202 q^{55} -1.00000 q^{56} -6.85823 q^{58} -3.71241 q^{59} +11.8953 q^{60} +3.56439 q^{61} -3.00774 q^{62} +6.29119 q^{63} +1.00000 q^{64} +8.77365 q^{65} +16.5747 q^{66} +1.14843 q^{67} +0.977224 q^{68} +1.95265 q^{69} -3.90247 q^{70} -4.58975 q^{71} -6.29119 q^{72} -6.31633 q^{73} -0.824327 q^{74} +31.1802 q^{75} -5.43763 q^{77} -6.85294 q^{78} +12.3096 q^{79} +3.90247 q^{80} +11.7055 q^{81} -11.3245 q^{82} -9.84224 q^{83} +3.04815 q^{84} +3.81358 q^{85} +12.2093 q^{86} +20.9049 q^{87} +5.43763 q^{88} -11.8845 q^{89} -24.5512 q^{90} +2.24823 q^{91} +0.640603 q^{92} +9.16802 q^{93} +12.6901 q^{94} -3.04815 q^{96} -7.94295 q^{97} -1.00000 q^{98} -34.2092 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 + 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9} + 2 q^{10} - 12 q^{11} + 4 q^{12} + 10 q^{13} - 8 q^{14} + 24 q^{15} + 8 q^{16} - 6 q^{17} - 8 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 20 q^{23} - 4 q^{24} + 8 q^{25} - 10 q^{26} + 22 q^{27} + 8 q^{28} + 8 q^{29} - 24 q^{30} + 18 q^{31} - 8 q^{32} - 16 q^{33} + 6 q^{34} - 2 q^{35} + 8 q^{36} + 36 q^{37} + 2 q^{40} + 4 q^{41} - 4 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} + 20 q^{46} - 10 q^{47} + 4 q^{48} + 8 q^{49} - 8 q^{50} + 12 q^{51} + 10 q^{52} + 32 q^{53} - 22 q^{54} - 22 q^{55} - 8 q^{56} - 8 q^{58} - 2 q^{59} + 24 q^{60} + 2 q^{61} - 18 q^{62} + 8 q^{63} + 8 q^{64} + 16 q^{66} + 44 q^{67} - 6 q^{68} + 2 q^{70} + 8 q^{71} - 8 q^{72} + 30 q^{73} - 36 q^{74} - 16 q^{75} - 12 q^{77} + 60 q^{79} - 2 q^{80} + 12 q^{81} - 4 q^{82} - 28 q^{83} + 4 q^{84} - 16 q^{85} + 16 q^{86} + 24 q^{87} + 12 q^{88} - 22 q^{89} - 8 q^{90} + 10 q^{91} - 20 q^{92} - 16 q^{93} + 10 q^{94} - 4 q^{96} - 6 q^{97} - 8 q^{98} - 52 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 + 8 * q^7 - 8 * q^8 + 8 * q^9 + 2 * q^10 - 12 * q^11 + 4 * q^12 + 10 * q^13 - 8 * q^14 + 24 * q^15 + 8 * q^16 - 6 * q^17 - 8 * q^18 - 2 * q^20 + 4 * q^21 + 12 * q^22 - 20 * q^23 - 4 * q^24 + 8 * q^25 - 10 * q^26 + 22 * q^27 + 8 * q^28 + 8 * q^29 - 24 * q^30 + 18 * q^31 - 8 * q^32 - 16 * q^33 + 6 * q^34 - 2 * q^35 + 8 * q^36 + 36 * q^37 + 2 * q^40 + 4 * q^41 - 4 * q^42 - 16 * q^43 - 12 * q^44 + 8 * q^45 + 20 * q^46 - 10 * q^47 + 4 * q^48 + 8 * q^49 - 8 * q^50 + 12 * q^51 + 10 * q^52 + 32 * q^53 - 22 * q^54 - 22 * q^55 - 8 * q^56 - 8 * q^58 - 2 * q^59 + 24 * q^60 + 2 * q^61 - 18 * q^62 + 8 * q^63 + 8 * q^64 + 16 * q^66 + 44 * q^67 - 6 * q^68 + 2 * q^70 + 8 * q^71 - 8 * q^72 + 30 * q^73 - 36 * q^74 - 16 * q^75 - 12 * q^77 + 60 * q^79 - 2 * q^80 + 12 * q^81 - 4 * q^82 - 28 * q^83 + 4 * q^84 - 16 * q^85 + 16 * q^86 + 24 * q^87 + 12 * q^88 - 22 * q^89 - 8 * q^90 + 10 * q^91 - 20 * q^92 - 16 * q^93 + 10 * q^94 - 4 * q^96 - 6 * q^97 - 8 * q^98 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more