LMFDB - Embedded newform 8007.2.a.i.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.9362168984$
Analytic rank:	$0$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.74167 q^{2} +1.00000 q^{3} +1.03340 q^{4} -2.69066 q^{5} -1.74167 q^{6} +0.494465 q^{7} +1.68349 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.74167 q^{2} +1.00000 q^{3} +1.03340 q^{4} -2.69066 q^{5} -1.74167 q^{6} +0.494465 q^{7} +1.68349 q^{8} +1.00000 q^{9} +4.68623 q^{10} +0.873517 q^{11} +1.03340 q^{12} -1.26796 q^{13} -0.861194 q^{14} -2.69066 q^{15} -4.99888 q^{16} +1.00000 q^{17} -1.74167 q^{18} -5.33535 q^{19} -2.78053 q^{20} +0.494465 q^{21} -1.52137 q^{22} -6.99440 q^{23} +1.68349 q^{24} +2.23963 q^{25} +2.20837 q^{26} +1.00000 q^{27} +0.510982 q^{28} -10.1491 q^{29} +4.68623 q^{30} +1.69431 q^{31} +5.33941 q^{32} +0.873517 q^{33} -1.74167 q^{34} -1.33044 q^{35} +1.03340 q^{36} -4.84974 q^{37} +9.29240 q^{38} -1.26796 q^{39} -4.52969 q^{40} -0.151498 q^{41} -0.861194 q^{42} -2.06224 q^{43} +0.902695 q^{44} -2.69066 q^{45} +12.1819 q^{46} +12.1671 q^{47} -4.99888 q^{48} -6.75550 q^{49} -3.90068 q^{50} +1.00000 q^{51} -1.31031 q^{52} -3.63394 q^{53} -1.74167 q^{54} -2.35033 q^{55} +0.832427 q^{56} -5.33535 q^{57} +17.6764 q^{58} +12.3745 q^{59} -2.78053 q^{60} +2.64960 q^{61} -2.95092 q^{62} +0.494465 q^{63} +0.698292 q^{64} +3.41165 q^{65} -1.52137 q^{66} +6.04454 q^{67} +1.03340 q^{68} -6.99440 q^{69} +2.31718 q^{70} +10.6189 q^{71} +1.68349 q^{72} -12.2844 q^{73} +8.44663 q^{74} +2.23963 q^{75} -5.51357 q^{76} +0.431924 q^{77} +2.20837 q^{78} -12.5646 q^{79} +13.4503 q^{80} +1.00000 q^{81} +0.263860 q^{82} +2.35620 q^{83} +0.510982 q^{84} -2.69066 q^{85} +3.59173 q^{86} -10.1491 q^{87} +1.47056 q^{88} +3.25252 q^{89} +4.68623 q^{90} -0.626963 q^{91} -7.22803 q^{92} +1.69431 q^{93} -21.1911 q^{94} +14.3556 q^{95} +5.33941 q^{96} +17.1861 q^{97} +11.7658 q^{98} +0.873517 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9}+O(q^{10}) $ 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9	\( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9} + 4 q^{10} + 23 q^{11} + 70 q^{12} + 10 q^{13} + 18 q^{14} + 19 q^{15} + 72 q^{16} + 63 q^{17} + 10 q^{18} + 6 q^{19} + 48 q^{20} + 11 q^{21} + 21 q^{22} + 44 q^{23} + 27 q^{24} + 110 q^{25} + 41 q^{26} + 63 q^{27} + 26 q^{28} + 35 q^{29} + 4 q^{30} + q^{31} + 54 q^{32} + 23 q^{33} + 10 q^{34} + 47 q^{35} + 70 q^{36} + 40 q^{37} + 38 q^{38} + 10 q^{39} - 10 q^{40} + 35 q^{41} + 18 q^{42} + 27 q^{43} + 46 q^{44} + 19 q^{45} + 8 q^{46} + 29 q^{47} + 72 q^{48} + 114 q^{49} + 27 q^{50} + 63 q^{51} - q^{52} + 75 q^{53} + 10 q^{54} + 5 q^{55} + 24 q^{56} + 6 q^{57} + 41 q^{58} + 105 q^{59} + 48 q^{60} + 5 q^{61} + 22 q^{62} + 11 q^{63} + 61 q^{64} + 49 q^{65} + 21 q^{66} + 4 q^{67} + 70 q^{68} + 44 q^{69} - 16 q^{70} + 16 q^{71} + 27 q^{72} + 39 q^{73} + 54 q^{74} + 110 q^{75} + 6 q^{76} + 88 q^{77} + 41 q^{78} + 16 q^{79} + 102 q^{80} + 63 q^{81} - 29 q^{82} + 73 q^{83} + 26 q^{84} + 19 q^{85} + 46 q^{86} + 35 q^{87} + 18 q^{88} + 88 q^{89} + 4 q^{90} - 15 q^{91} + 110 q^{92} + q^{93} - 8 q^{94} + 28 q^{95} + 54 q^{96} + 70 q^{97} + 33 q^{98} + 23 q^{99}+O(q^{100}) \) 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9 + 4 * q^10 + 23 * q^11 + 70 * q^12 + 10 * q^13 + 18 * q^14 + 19 * q^15 + 72 * q^16 + 63 * q^17 + 10 * q^18 + 6 * q^19 + 48 * q^20 + 11 * q^21 + 21 * q^22 + 44 * q^23 + 27 * q^24 + 110 * q^25 + 41 * q^26 + 63 * q^27 + 26 * q^28 + 35 * q^29 + 4 * q^30 + q^31 + 54 * q^32 + 23 * q^33 + 10 * q^34 + 47 * q^35 + 70 * q^36 + 40 * q^37 + 38 * q^38 + 10 * q^39 - 10 * q^40 + 35 * q^41 + 18 * q^42 + 27 * q^43 + 46 * q^44 + 19 * q^45 + 8 * q^46 + 29 * q^47 + 72 * q^48 + 114 * q^49 + 27 * q^50 + 63 * q^51 - q^52 + 75 * q^53 + 10 * q^54 + 5 * q^55 + 24 * q^56 + 6 * q^57 + 41 * q^58 + 105 * q^59 + 48 * q^60 + 5 * q^61 + 22 * q^62 + 11 * q^63 + 61 * q^64 + 49 * q^65 + 21 * q^66 + 4 * q^67 + 70 * q^68 + 44 * q^69 - 16 * q^70 + 16 * q^71 + 27 * q^72 + 39 * q^73 + 54 * q^74 + 110 * q^75 + 6 * q^76 + 88 * q^77 + 41 * q^78 + 16 * q^79 + 102 * q^80 + 63 * q^81 - 29 * q^82 + 73 * q^83 + 26 * q^84 + 19 * q^85 + 46 * q^86 + 35 * q^87 + 18 * q^88 + 88 * q^89 + 4 * q^90 - 15 * q^91 + 110 * q^92 + q^93 - 8 * q^94 + 28 * q^95 + 54 * q^96 + 70 * q^97 + 33 * q^98 + 23 * 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 $

$2$	−1.74167	−1.23154	−0.615772	−	0.787924i	$-0.711156\pi$
$2$	−1.74167	−1.23154	−0.615772	+	0.787924i	$0.711156\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.03340	0.516702
$5$	−2.69066	−1.20330	−0.601649	−	0.798761i	$-0.705489\pi$
$5$	−2.69066	−1.20330	−0.601649	+	0.798761i	$0.705489\pi$
$6$	−1.74167	−0.711032
$7$	0.494465	0.186890	0.0934451	−	0.995624i	$-0.470212\pi$
$7$	0.494465	0.186890	0.0934451	+	0.995624i	$0.470212\pi$
$8$	1.68349	0.595203
$9$	1.00000	0.333333
$10$	4.68623	1.48191
$11$	0.873517	0.263375	0.131688	−	0.991291i	$-0.457960\pi$
$11$	0.873517	0.263375	0.131688	+	0.991291i	$0.457960\pi$
$12$	1.03340	0.298318
$13$	−1.26796	−0.351669	−0.175835	−	0.984420i	$-0.556262\pi$
$13$	−1.26796	−0.351669	−0.175835	+	0.984420i	$0.556262\pi$
$14$	−0.861194	−0.230164
$15$	−2.69066	−0.694724
$16$	−4.99888	−1.24972
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.74167	−0.410515
$19$	−5.33535	−1.22401	−0.612007	−	0.790852i	$-0.709637\pi$
$19$	−5.33535	−1.22401	−0.612007	+	0.790852i	$0.709637\pi$
$20$	−2.78053	−0.621746
$21$	0.494465	0.107901
$22$	−1.52137	−0.324358
$23$	−6.99440	−1.45843	−0.729216	−	0.684283i	$-0.760115\pi$
$23$	−6.99440	−1.45843	−0.729216	+	0.684283i	$0.760115\pi$
$24$	1.68349	0.343641
$25$	2.23963	0.447925
$26$	2.20837	0.433096
$27$	1.00000	0.192450
$28$	0.510982	0.0965665
$29$	−10.1491	−1.88465	−0.942325	−	0.334700i	$-0.891365\pi$
$29$	−10.1491	−1.88465	−0.942325	+	0.334700i	$0.891365\pi$
$30$	4.68623	0.855584
$31$	1.69431	0.304307	0.152154	−	0.988357i	$-0.451379\pi$
$31$	1.69431	0.304307	0.152154	+	0.988357i	$0.451379\pi$
$32$	5.33941	0.943884
$33$	0.873517	0.152060
$34$	−1.74167	−0.298693
$35$	−1.33044	−0.224885
$36$	1.03340	0.172234
$37$	−4.84974	−0.797292	−0.398646	−	0.917105i	$-0.630520\pi$
$37$	−4.84974	−0.797292	−0.398646	+	0.917105i	$0.630520\pi$
$38$	9.29240	1.50743
$39$	−1.26796	−0.203036
$40$	−4.52969	−0.716207
$41$	−0.151498	−0.0236601	−0.0118300	−	0.999930i	$-0.503766\pi$
$41$	−0.151498	−0.0236601	−0.0118300	+	0.999930i	$0.503766\pi$
$42$	−0.861194	−0.132885
$43$	−2.06224	−0.314489	−0.157244	−	0.987560i	$-0.550261\pi$
$43$	−2.06224	−0.314489	−0.157244	+	0.987560i	$0.550261\pi$
$44$	0.902695	0.136086
$45$	−2.69066	−0.401099
$46$	12.1819	1.79612
$47$	12.1671	1.77476	0.887380	−	0.461038i	$-0.152523\pi$
$47$	12.1671	1.77476	0.887380	+	0.461038i	$0.152523\pi$
$48$	−4.99888	−0.721527
$49$	−6.75550	−0.965072
$50$	−3.90068	−0.551640
$51$	1.00000	0.140028
$52$	−1.31031	−0.181708
$53$	−3.63394	−0.499160	−0.249580	−	0.968354i	$-0.580293\pi$
$53$	−3.63394	−0.499160	−0.249580	+	0.968354i	$0.580293\pi$
$54$	−1.74167	−0.237011
$55$	−2.35033	−0.316919
$56$	0.832427	0.111238
$57$	−5.33535	−0.706685
$58$	17.6764	2.32103
$59$	12.3745	1.61102	0.805511	−	0.592581i	$-0.201891\pi$
$59$	12.3745	1.61102	0.805511	+	0.592581i	$0.201891\pi$
$60$	−2.78053	−0.358965
$61$	2.64960	0.339247	0.169624	−	0.985509i	$-0.445745\pi$
$61$	2.64960	0.339247	0.169624	+	0.985509i	$0.445745\pi$
$62$	−2.95092	−0.374768
$63$	0.494465	0.0622968
$64$	0.698292	0.0872865
$65$	3.41165	0.423163
$66$	−1.52137	−0.187268
$67$	6.04454	0.738459	0.369229	−	0.929338i	$-0.379622\pi$
$67$	6.04454	0.738459	0.369229	+	0.929338i	$0.379622\pi$
$68$	1.03340	0.125319
$69$	−6.99440	−0.842026
$70$	2.31718	0.276955
$71$	10.6189	1.26023	0.630117	−	0.776500i	$-0.283007\pi$
$71$	10.6189	1.26023	0.630117	+	0.776500i	$0.283007\pi$
$72$	1.68349	0.198401
$73$	−12.2844	−1.43778	−0.718892	−	0.695122i	$-0.755350\pi$
$73$	−12.2844	−1.43778	−0.718892	+	0.695122i	$0.755350\pi$
$74$	8.44663	0.981900
$75$	2.23963	0.258610
$76$	−5.51357	−0.632450
$77$	0.431924	0.0492223
$78$	2.20837	0.250048
$79$	−12.5646	−1.41362	−0.706812	−	0.707401i	$-0.749868\pi$
$79$	−12.5646	−1.41362	−0.706812	+	0.707401i	$0.749868\pi$
$80$	13.4503	1.50379
$81$	1.00000	0.111111
$82$	0.263860	0.0291384
$83$	2.35620	0.258627	0.129313	−	0.991604i	$-0.458723\pi$
$83$	2.35620	0.258627	0.129313	+	0.991604i	$0.458723\pi$
$84$	0.510982	0.0557527
$85$	−2.69066	−0.291843
$86$	3.59173	0.387307
$87$	−10.1491	−1.08810
$88$	1.47056	0.156762
$89$	3.25252	0.344766	0.172383	−	0.985030i	$-0.444853\pi$
$89$	3.25252	0.344766	0.172383	+	0.985030i	$0.444853\pi$
$90$	4.68623	0.493972
$91$	−0.626963	−0.0657235
$92$	−7.22803	−0.753574
$93$	1.69431	0.175692
$94$	−21.1911	−2.18570
$95$	14.3556	1.47285
$96$	5.33941	0.544951
$97$	17.1861	1.74499	0.872494	−	0.488625i	$-0.162501\pi$
$97$	17.1861	1.74499	0.872494	+	0.488625i	$0.162501\pi$
$98$	11.7658	1.18853
$99$	0.873517	0.0877917
$100$	2.31444	0.231444
$101$	−7.85531	−0.781633	−0.390816	−	0.920469i	$-0.627807\pi$
$101$	−7.85531	−0.781633	−0.390816	+	0.920469i	$0.627807\pi$
$102$	−1.74167	−0.172451
$103$	3.43177	0.338142	0.169071	−	0.985604i	$-0.445923\pi$
$103$	3.43177	0.338142	0.169071	+	0.985604i	$0.445923\pi$
$104$	−2.13460	−0.209315
$105$	−1.33044	−0.129837
$106$	6.32911	0.614738
$107$	0.0262434	0.00253704	0.00126852	−	0.999999i	$-0.499596\pi$
$107$	0.0262434	0.00253704	0.00126852	+	0.999999i	$0.499596\pi$
$108$	1.03340	0.0994393
$109$	−1.58053	−0.151387	−0.0756936	−	0.997131i	$-0.524117\pi$
$109$	−1.58053	−0.151387	−0.0756936	+	0.997131i	$0.524117\pi$
$110$	4.09350	0.390299
$111$	−4.84974	−0.460317
$112$	−2.47177	−0.233561
$113$	−1.35508	−0.127476	−0.0637378	−	0.997967i	$-0.520302\pi$
$113$	−1.35508	−0.127476	−0.0637378	+	0.997967i	$0.520302\pi$
$114$	9.29240	0.870313
$115$	18.8195	1.75493
$116$	−10.4882	−0.973801
$117$	−1.26796	−0.117223
$118$	−21.5522	−1.98404
$119$	0.494465	0.0453275
$120$	−4.52969	−0.413502
$121$	−10.2370	−0.930634
$122$	−4.61473	−0.417798
$123$	−0.151498	−0.0136601
$124$	1.75091	0.157236
$125$	7.42721	0.664310
$126$	−0.861194	−0.0767212
$127$	−9.70348	−0.861045	−0.430522	−	0.902580i	$-0.641671\pi$
$127$	−9.70348	−0.861045	−0.430522	+	0.902580i	$0.641671\pi$
$128$	−11.8950	−1.05138
$129$	−2.06224	−0.181570
$130$	−5.94195	−0.521144
$131$	17.7561	1.55136	0.775680	−	0.631127i	$-0.217407\pi$
$131$	17.7561	1.55136	0.775680	+	0.631127i	$0.217407\pi$
$132$	0.902695	0.0785695
$133$	−2.63815	−0.228756
$134$	−10.5276	−0.909445
$135$	−2.69066	−0.231575
$136$	1.68349	0.144358
$137$	−5.45133	−0.465738	−0.232869	−	0.972508i	$-0.574811\pi$
$137$	−5.45133	−0.465738	−0.232869	+	0.972508i	$0.574811\pi$
$138$	12.1819	1.03699
$139$	18.8471	1.59859	0.799294	−	0.600940i	$-0.205207\pi$
$139$	18.8471	1.59859	0.799294	+	0.600940i	$0.205207\pi$
$140$	−1.37488	−0.116198
$141$	12.1671	1.02466
$142$	−18.4946	−1.55203
$143$	−1.10758	−0.0926209
$144$	−4.99888	−0.416574
$145$	27.3079	2.26779
$146$	21.3954	1.77069
$147$	−6.75550	−0.557185
$148$	−5.01173	−0.411962
$149$	−9.31491	−0.763107	−0.381553	−	0.924347i	$-0.624611\pi$
$149$	−9.31491	−0.763107	−0.381553	+	0.924347i	$0.624611\pi$
$150$	−3.90068	−0.318490
$151$	10.1466	0.825721	0.412861	−	0.910794i	$-0.364530\pi$
$151$	10.1466	0.825721	0.412861	+	0.910794i	$0.364530\pi$
$152$	−8.98201	−0.728537
$153$	1.00000	0.0808452
$154$	−0.752267	−0.0606194
$155$	−4.55881	−0.366172
$156$	−1.31031	−0.104909
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	21.8833	1.74094
$159$	−3.63394	−0.288190
$160$	−14.3665	−1.13577
$161$	−3.45848	−0.272567
$162$	−1.74167	−0.136838
$163$	−13.7565	−1.07749	−0.538745	−	0.842469i	$-0.681101\pi$
$163$	−13.7565	−1.07749	−0.538745	+	0.842469i	$0.681101\pi$
$164$	−0.156559	−0.0122252
$165$	−2.35033	−0.182973
$166$	−4.10372	−0.318510
$167$	−15.5635	−1.20434	−0.602170	−	0.798368i	$-0.705697\pi$
$167$	−15.5635	−1.20434	−0.602170	+	0.798368i	$0.705697\pi$
$168$	0.832427	0.0642231
$169$	−11.3923	−0.876329
$170$	4.68623	0.359417
$171$	−5.33535	−0.408005
$172$	−2.13113	−0.162497
$173$	0.0150804	0.00114654	0.000573271	−	1.00000i	$-0.499818\pi$
$173$	0.0150804	0.00114654	0.000573271		1.00000i	$0.499818\pi$
$174$	17.6764	1.34005
$175$	1.10742	0.0837129
$176$	−4.36661	−0.329145
$177$	12.3745	0.930124
$178$	−5.66480	−0.424595
$179$	12.6231	0.943496	0.471748	−	0.881733i	$-0.343623\pi$
$179$	12.6231	0.943496	0.471748	+	0.881733i	$0.343623\pi$
$180$	−2.78053	−0.207249
$181$	1.59302	0.118408	0.0592041	−	0.998246i	$-0.481144\pi$
$181$	1.59302	0.118408	0.0592041	+	0.998246i	$0.481144\pi$
$182$	1.09196	0.0809414
$183$	2.64960	0.195864
$184$	−11.7750	−0.868064
$185$	13.0490	0.959379
$186$	−2.95092	−0.216372
$187$	0.873517	0.0638779
$188$	12.5736	0.917022
$189$	0.494465	0.0359671
$190$	−25.0027	−1.81388
$191$	−11.6176	−0.840620	−0.420310	−	0.907381i	$-0.638079\pi$
$191$	−11.6176	−0.840620	−0.420310	+	0.907381i	$0.638079\pi$
$192$	0.698292	0.0503949
$193$	−9.25963	−0.666523	−0.333262	−	0.942834i	$-0.608149\pi$
$193$	−9.25963	−0.666523	−0.333262	+	0.942834i	$0.608149\pi$
$194$	−29.9325	−2.14903
$195$	3.41165	0.244313
$196$	−6.98116	−0.498654
$197$	−17.1486	−1.22179	−0.610894	−	0.791713i	$-0.709190\pi$
$197$	−17.1486	−1.22179	−0.610894	+	0.791713i	$0.709190\pi$
$198$	−1.52137	−0.108119
$199$	5.10637	0.361981	0.180990	−	0.983485i	$-0.442070\pi$
$199$	5.10637	0.361981	0.180990	+	0.983485i	$0.442070\pi$
$200$	3.77039	0.266607
$201$	6.04454	0.426349
$202$	13.6813	0.962616
$203$	−5.01840	−0.352223
$204$	1.03340	0.0723527
$205$	0.407630	0.0284701
$206$	−5.97699	−0.416437
$207$	−6.99440	−0.486144
$208$	6.33839	0.439488
$209$	−4.66052	−0.322375
$210$	2.31718	0.159900
$211$	−15.1392	−1.04222	−0.521112	−	0.853488i	$-0.674483\pi$
$211$	−15.1392	−1.04222	−0.521112	+	0.853488i	$0.674483\pi$
$212$	−3.75533	−0.257917
$213$	10.6189	0.727596
$214$	−0.0457072	−0.00312448
$215$	5.54878	0.378423
$216$	1.68349	0.114547
$217$	0.837777	0.0568720
$218$	2.75275	0.186440
$219$	−12.2844	−0.830105
$220$	−2.42884	−0.163752
$221$	−1.26796	−0.0852923
$222$	8.44663	0.566900
$223$	14.9217	0.999231	0.499616	−	0.866247i	$-0.333475\pi$
$223$	14.9217	0.999231	0.499616	+	0.866247i	$0.333475\pi$
$224$	2.64015	0.176403
$225$	2.23963	0.149308
$226$	2.36010	0.156992
$227$	25.1154	1.66697	0.833485	−	0.552543i	$-0.186342\pi$
$227$	25.1154	1.66697	0.833485	+	0.552543i	$0.186342\pi$
$228$	−5.51357	−0.365145
$229$	−22.5430	−1.48968	−0.744840	−	0.667243i	$-0.767474\pi$
$229$	−22.5430	−1.48968	−0.744840	+	0.667243i	$0.767474\pi$
$230$	−32.7773	−2.16127
$231$	0.431924	0.0284185
$232$	−17.0860	−1.12175
$233$	−26.2459	−1.71943	−0.859714	−	0.510776i	$-0.829358\pi$
$233$	−26.2459	−1.71943	−0.859714	+	0.510776i	$0.829358\pi$
$234$	2.20837	0.144365
$235$	−32.7376	−2.13557
$236$	12.7878	0.832418
$237$	−12.5646	−0.816157
$238$	−0.861194	−0.0558229
$239$	3.65267	0.236272	0.118136	−	0.992997i	$-0.462308\pi$
$239$	3.65267	0.236272	0.118136	+	0.992997i	$0.462308\pi$
$240$	13.4503	0.868212
$241$	7.42013	0.477972	0.238986	−	0.971023i	$-0.423185\pi$
$241$	7.42013	0.477972	0.238986	+	0.971023i	$0.423185\pi$
$242$	17.8294	1.14612
$243$	1.00000	0.0641500
$244$	2.73811	0.175290
$245$	18.1767	1.16127
$246$	0.263860	0.0168231
$247$	6.76502	0.430448
$248$	2.85235	0.181125
$249$	2.35620	0.149318
$250$	−12.9357	−0.818127
$251$	9.18800	0.579942	0.289971	−	0.957035i	$-0.406354\pi$
$251$	9.18800	0.579942	0.289971	+	0.957035i	$0.406354\pi$
$252$	0.510982	0.0321888
$253$	−6.10972	−0.384115
$254$	16.9002	1.06042
$255$	−2.69066	−0.168495
$256$	19.3206	1.20754
$257$	20.0692	1.25188	0.625942	−	0.779870i	$-0.284715\pi$
$257$	20.0692	1.25188	0.625942	+	0.779870i	$0.284715\pi$
$258$	3.59173	0.223612
$259$	−2.39803	−0.149006
$260$	3.52561	0.218649
$261$	−10.1491	−0.628216
$262$	−30.9252	−1.91057
$263$	10.4265	0.642927	0.321463	−	0.946922i	$-0.395825\pi$
$263$	10.4265	0.642927	0.321463	+	0.946922i	$0.395825\pi$
$264$	1.47056	0.0905065
$265$	9.77768	0.600638
$266$	4.59477	0.281723
$267$	3.25252	0.199051
$268$	6.24645	0.381563
$269$	27.9692	1.70531	0.852657	−	0.522472i	$-0.174990\pi$
$269$	27.9692	1.70531	0.852657	+	0.522472i	$0.174990\pi$
$270$	4.68623	0.285195
$271$	−11.1232	−0.675686	−0.337843	−	0.941203i	$-0.609697\pi$
$271$	−11.1232	−0.675686	−0.337843	+	0.941203i	$0.609697\pi$
$272$	−4.99888	−0.303102
$273$	−0.626963	−0.0379455
$274$	9.49440	0.573577
$275$	1.95635	0.117972
$276$	−7.22803	−0.435076
$277$	−25.0583	−1.50561	−0.752803	−	0.658246i	$-0.771299\pi$
$277$	−25.0583	−1.50561	−0.752803	+	0.658246i	$0.771299\pi$
$278$	−32.8253	−1.96873
$279$	1.69431	0.101436
$280$	−2.23977	−0.133852
$281$	12.8839	0.768591	0.384295	−	0.923210i	$-0.374444\pi$
$281$	12.8839	0.768591	0.384295	+	0.923210i	$0.374444\pi$
$282$	−21.1911	−1.26191
$283$	−21.2872	−1.26539	−0.632696	−	0.774400i	$-0.718052\pi$
$283$	−21.2872	−1.26539	−0.632696	+	0.774400i	$0.718052\pi$
$284$	10.9736	0.651165
$285$	14.3556	0.850352
$286$	1.92904	0.114067
$287$	−0.0749106	−0.00442183
$288$	5.33941	0.314628
$289$	1.00000	0.0588235
$290$	−47.5612	−2.79289
$291$	17.1861	1.00747
$292$	−12.6948	−0.742905
$293$	6.82253	0.398576	0.199288	−	0.979941i	$-0.436137\pi$
$293$	6.82253	0.398576	0.199288	+	0.979941i	$0.436137\pi$
$294$	11.7658	0.686198
$295$	−33.2955	−1.93854
$296$	−8.16448	−0.474551
$297$	0.873517	0.0506866
$298$	16.2235	0.939800
$299$	8.86862	0.512886
$300$	2.31444	0.133624
$301$	−1.01971	−0.0587749
$302$	−17.6721	−1.01691
$303$	−7.85531	−0.451276
$304$	26.6708	1.52968
$305$	−7.12917	−0.408215
$306$	−1.74167	−0.0995645
$307$	16.3530	0.933312	0.466656	−	0.884439i	$-0.345459\pi$
$307$	16.3530	0.933312	0.466656	+	0.884439i	$0.345459\pi$
$308$	0.446351	0.0254332
$309$	3.43177	0.195226
$310$	7.93992	0.450957
$311$	31.3304	1.77658	0.888291	−	0.459280i	$-0.151893\pi$
$311$	31.3304	1.77658	0.888291	+	0.459280i	$0.151893\pi$
$312$	−2.13460	−0.120848
$313$	4.10958	0.232287	0.116144	−	0.993232i	$-0.462947\pi$
$313$	4.10958	0.232287	0.116144	+	0.993232i	$0.462947\pi$
$314$	−1.74167	−0.0982879
$315$	−1.33044	−0.0749615
$316$	−12.9843	−0.730422
$317$	25.3654	1.42466	0.712330	−	0.701844i	$-0.247640\pi$
$317$	25.3654	1.42466	0.712330	+	0.701844i	$0.247640\pi$
$318$	6.32911	0.354919
$319$	−8.86545	−0.496370
$320$	−1.87886	−0.105032
$321$	0.0262434	0.00146476
$322$	6.02353	0.335678
$323$	−5.33535	−0.296867
$324$	1.03340	0.0574113
$325$	−2.83976	−0.157522
$326$	23.9592	1.32698
$327$	−1.58053	−0.0874034
$328$	−0.255046	−0.0140825
$329$	6.01623	0.331686
$330$	4.09350	0.225340
$331$	23.9379	1.31575	0.657874	−	0.753128i	$-0.271456\pi$
$331$	23.9379	1.31575	0.657874	+	0.753128i	$0.271456\pi$
$332$	2.43491	0.133633
$333$	−4.84974	−0.265764
$334$	27.1064	1.48320
$335$	−16.2638	−0.888586
$336$	−2.47177	−0.134846
$337$	−14.6238	−0.796609	−0.398304	−	0.917253i	$-0.630401\pi$
$337$	−14.6238	−0.796609	−0.398304	+	0.917253i	$0.630401\pi$
$338$	19.8415	1.07924
$339$	−1.35508	−0.0735980
$340$	−2.78053	−0.150796
$341$	1.48001	0.0801469
$342$	9.29240	0.502476
$343$	−6.80162	−0.367253
$344$	−3.47176	−0.187185
$345$	18.8195	1.01321
$346$	−0.0262651	−0.00141202
$347$	35.5293	1.90731	0.953657	−	0.300897i	$-0.0972862\pi$
$347$	35.5293	1.90731	0.953657	+	0.300897i	$0.0972862\pi$
$348$	−10.4882	−0.562224
$349$	−4.71448	−0.252361	−0.126180	−	0.992007i	$-0.540272\pi$
$349$	−4.71448	−0.252361	−0.126180	+	0.992007i	$0.540272\pi$
$350$	−1.92875	−0.103096
$351$	−1.26796	−0.0676787
$352$	4.66406	0.248595
$353$	15.0723	0.802217	0.401109	−	0.916031i	$-0.368625\pi$
$353$	15.0723	0.802217	0.401109	+	0.916031i	$0.368625\pi$
$354$	−21.5522	−1.14549
$355$	−28.5718	−1.51644
$356$	3.36116	0.178141
$357$	0.494465	0.0261699
$358$	−21.9853	−1.16196
$359$	−3.18038	−0.167854	−0.0839269	−	0.996472i	$-0.526746\pi$
$359$	−3.18038	−0.167854	−0.0839269	+	0.996472i	$0.526746\pi$
$360$	−4.52969	−0.238736
$361$	9.46598	0.498209
$362$	−2.77451	−0.145825
$363$	−10.2370	−0.537302
$364$	−0.647905	−0.0339595
$365$	33.0532	1.73008
$366$	−4.61473	−0.241216
$367$	3.63846	0.189926	0.0949630	−	0.995481i	$-0.469727\pi$
$367$	3.63846	0.189926	0.0949630	+	0.995481i	$0.469727\pi$
$368$	34.9642	1.82263
$369$	−0.151498	−0.00788669
$370$	−22.7270	−1.18152
$371$	−1.79686	−0.0932882
$372$	1.75091	0.0907802
$373$	31.7828	1.64565	0.822826	−	0.568293i	$-0.192396\pi$
$373$	31.7828	1.64565	0.822826	+	0.568293i	$0.192396\pi$
$374$	−1.52137	−0.0786684
$375$	7.42721	0.383540
$376$	20.4833	1.05634
$377$	12.8687	0.662773
$378$	−0.861194	−0.0442950
$379$	16.4009	0.842459	0.421230	−	0.906954i	$-0.361599\pi$
$379$	16.4009	0.842459	0.421230	+	0.906954i	$0.361599\pi$
$380$	14.8351	0.761025
$381$	−9.70348	−0.497125
$382$	20.2340	1.03526
$383$	27.0068	1.37998	0.689990	−	0.723819i	$-0.257615\pi$
$383$	27.0068	1.37998	0.689990	+	0.723819i	$0.257615\pi$
$384$	−11.8950	−0.607015
$385$	−1.16216	−0.0592290
$386$	16.1272	0.820853
$387$	−2.06224	−0.104830
$388$	17.7602	0.901638
$389$	−2.47351	−0.125412	−0.0627059	−	0.998032i	$-0.519973\pi$
$389$	−2.47351	−0.125412	−0.0627059	+	0.998032i	$0.519973\pi$
$390$	−5.94195	−0.300882
$391$	−6.99440	−0.353722
$392$	−11.3728	−0.574414
$393$	17.7561	0.895678
$394$	29.8672	1.50469
$395$	33.8069	1.70101
$396$	0.902695	0.0453621
$397$	−26.5939	−1.33471	−0.667356	−	0.744739i	$-0.732574\pi$
$397$	−26.5939	−1.33471	−0.667356	+	0.744739i	$0.732574\pi$
$398$	−8.89359	−0.445795
$399$	−2.63815	−0.132072
$400$	−11.1956	−0.559782
$401$	8.68040	0.433479	0.216739	−	0.976230i	$-0.430458\pi$
$401$	8.68040	0.433479	0.216739	+	0.976230i	$0.430458\pi$
$402$	−10.5276	−0.525068
$403$	−2.14832	−0.107015
$404$	−8.11771	−0.403871
$405$	−2.69066	−0.133700
$406$	8.74038	0.433778
$407$	−4.23633	−0.209987
$408$	1.68349	0.0833451
$409$	38.2325	1.89048	0.945238	−	0.326381i	$-0.105829\pi$
$409$	38.2325	1.89048	0.945238	+	0.326381i	$0.105829\pi$
$410$	−0.709955	−0.0350622
$411$	−5.45133	−0.268894
$412$	3.54640	0.174719
$413$	6.11876	0.301084
$414$	12.1819	0.598708
$415$	−6.33972	−0.311205
$416$	−6.77017	−0.331935
$417$	18.8471	0.922945
$418$	8.11707	0.397019
$419$	12.5100	0.611151	0.305576	−	0.952168i	$-0.401151\pi$
$419$	12.5100	0.611151	0.305576	+	0.952168i	$0.401151\pi$
$420$	−1.37488	−0.0670871
$421$	36.3411	1.77116	0.885580	−	0.464488i	$-0.153762\pi$
$421$	36.3411	1.77116	0.885580	+	0.464488i	$0.153762\pi$
$422$	26.3674	1.28355
$423$	12.1671	0.591587
$424$	−6.11770	−0.297102
$425$	2.23963	0.108638
$426$	−18.4946	−0.896067
$427$	1.31014	0.0634020
$428$	0.0271200	0.00131089
$429$	−1.10758	−0.0534747
$430$	−9.66412	−0.466045
$431$	−34.1833	−1.64655	−0.823277	−	0.567641i	$-0.807856\pi$
$431$	−34.1833	−1.64655	−0.823277	+	0.567641i	$0.807856\pi$
$432$	−4.99888	−0.240509
$433$	9.90152	0.475837	0.237918	−	0.971285i	$-0.423535\pi$
$433$	9.90152	0.475837	0.237918	+	0.971285i	$0.423535\pi$
$434$	−1.45913	−0.0700404
$435$	27.3079	1.30931
$436$	−1.63332	−0.0782220
$437$	37.3176	1.78514
$438$	21.3954	1.02231
$439$	0.364697	0.0174061	0.00870303	−	0.999962i	$-0.497230\pi$
$439$	0.364697	0.0174061	0.00870303	+	0.999962i	$0.497230\pi$
$440$	−3.95676	−0.188631
$441$	−6.75550	−0.321691
$442$	2.20837	0.105041
$443$	31.9823	1.51952	0.759762	−	0.650201i	$-0.225315\pi$
$443$	31.9823	1.51952	0.759762	+	0.650201i	$0.225315\pi$
$444$	−5.01173	−0.237846
$445$	−8.75140	−0.414856
$446$	−25.9886	−1.23060
$447$	−9.31491	−0.440580
$448$	0.345281	0.0163130
$449$	−24.8794	−1.17413	−0.587066	−	0.809539i	$-0.699717\pi$
$449$	−24.8794	−1.17413	−0.587066	+	0.809539i	$0.699717\pi$
$450$	−3.90068	−0.183880
$451$	−0.132336	−0.00623147
$452$	−1.40035	−0.0658668
$453$	10.1466	0.476730
$454$	−43.7427	−2.05295
$455$	1.68694	0.0790850
$456$	−8.98201	−0.420621
$457$	26.7685	1.25218	0.626088	−	0.779752i	$-0.284655\pi$
$457$	26.7685	1.25218	0.626088	+	0.779752i	$0.284655\pi$
$458$	39.2623	1.83461
$459$	1.00000	0.0466760
$460$	19.4481	0.906774
$461$	−37.8793	−1.76422	−0.882108	−	0.471048i	$-0.843876\pi$
$461$	−37.8793	−1.76422	−0.882108	+	0.471048i	$0.843876\pi$
$462$	−0.752267	−0.0349986
$463$	8.03435	0.373388	0.186694	−	0.982418i	$-0.440223\pi$
$463$	8.03435	0.373388	0.186694	+	0.982418i	$0.440223\pi$
$464$	50.7344	2.35529
$465$	−4.55881	−0.211410
$466$	45.7117	2.11755
$467$	−31.8545	−1.47405	−0.737025	−	0.675866i	$-0.763770\pi$
$467$	−31.8545	−1.47405	−0.737025	+	0.675866i	$0.763770\pi$
$468$	−1.31031	−0.0605693
$469$	2.98882	0.138011
$470$	57.0180	2.63004
$471$	1.00000	0.0460776
$472$	20.8323	0.958886
$473$	−1.80140	−0.0828285
$474$	21.8833	1.00513
$475$	−11.9492	−0.548267
$476$	0.510982	0.0234208
$477$	−3.63394	−0.166387
$478$	−6.36174	−0.290979
$479$	9.16871	0.418929	0.209464	−	0.977816i	$-0.432828\pi$
$479$	9.16871	0.418929	0.209464	+	0.977816i	$0.432828\pi$
$480$	−14.3665	−0.655739
$481$	6.14928	0.280383
$482$	−12.9234	−0.588644
$483$	−3.45848	−0.157367
$484$	−10.5789	−0.480860
$485$	−46.2420	−2.09974
$486$	−1.74167	−0.0790036
$487$	27.6830	1.25443	0.627217	−	0.778844i	$-0.284194\pi$
$487$	27.6830	1.25443	0.627217	+	0.778844i	$0.284194\pi$
$488$	4.46058	0.201921
$489$	−13.7565	−0.622089
$490$	−31.6578	−1.43015
$491$	−36.3366	−1.63985	−0.819923	−	0.572473i	$-0.805984\pi$
$491$	−36.3366	−1.63985	−0.819923	+	0.572473i	$0.805984\pi$
$492$	−0.156559	−0.00705822
$493$	−10.1491	−0.457095
$494$	−11.7824	−0.530116
$495$	−2.35033	−0.105640
$496$	−8.46966	−0.380299
$497$	5.25068	0.235525
$498$	−4.10372	−0.183892
$499$	−29.9108	−1.33899	−0.669496	−	0.742816i	$-0.733490\pi$
$499$	−29.9108	−1.33899	−0.669496	+	0.742816i	$0.733490\pi$
$500$	7.67531	0.343250
$501$	−15.5635	−0.695326
$502$	−16.0024	−0.714224
$503$	4.89980	0.218471	0.109236	−	0.994016i	$-0.465160\pi$
$503$	4.89980	0.218471	0.109236	+	0.994016i	$0.465160\pi$
$504$	0.832427	0.0370792
$505$	21.1359	0.940537
$506$	10.6411	0.473054
$507$	−11.3923	−0.505949
$508$	−10.0276	−0.444903
$509$	−6.78675	−0.300817	−0.150409	−	0.988624i	$-0.548059\pi$
$509$	−6.78675	−0.300817	−0.150409	+	0.988624i	$0.548059\pi$
$510$	4.68623	0.207510
$511$	−6.07422	−0.268708
$512$	−9.85996	−0.435753
$513$	−5.33535	−0.235562
$514$	−34.9539	−1.54175
$515$	−9.23370	−0.406886
$516$	−2.13113	−0.0938176
$517$	10.6282	0.467428
$518$	4.17656	0.183508
$519$	0.0150804	0.000661957 0
$520$	5.74347	0.251868
$521$	4.79667	0.210146	0.105073	−	0.994465i	$-0.466492\pi$
$521$	4.79667	0.210146	0.105073	+	0.994465i	$0.466492\pi$
$522$	17.6764	0.773676
$523$	−39.7654	−1.73882	−0.869411	−	0.494090i	$-0.835501\pi$
$523$	−39.7654	−1.73882	−0.869411	+	0.494090i	$0.835501\pi$
$524$	18.3492	0.801590
$525$	1.10742	0.0483317
$526$	−18.1595	−0.791793
$527$	1.69431	0.0738053
$528$	−4.36661	−0.190032
$529$	25.9216	1.12702
$530$	−17.0295	−0.739713
$531$	12.3745	0.537007
$532$	−2.72627	−0.118199
$533$	0.192094	0.00832051
$534$	−5.66480	−0.245140
$535$	−0.0706119	−0.00305282
$536$	10.1759	0.439533
$537$	12.6231	0.544728
$538$	−48.7131	−2.10017
$539$	−5.90104	−0.254176
$540$	−2.78053	−0.119655
$541$	31.1380	1.33873	0.669364	−	0.742935i	$-0.266567\pi$
$541$	31.1380	1.33873	0.669364	+	0.742935i	$0.266567\pi$
$542$	19.3729	0.832137
$543$	1.59302	0.0683630
$544$	5.33941	0.228925
$545$	4.25266	0.182164
$546$	1.09196	0.0467316
$547$	22.0551	0.943008	0.471504	−	0.881864i	$-0.343711\pi$
$547$	22.0551	0.943008	0.471504	+	0.881864i	$0.343711\pi$
$548$	−5.63342	−0.240648
$549$	2.64960	0.113082
$550$	−3.40731	−0.145288
$551$	54.1493	2.30684
$552$	−11.7750	−0.501177
$553$	−6.21274	−0.264193
$554$	43.6432	1.85422
$555$	13.0490	0.553898
$556$	19.4766	0.825993
$557$	18.8570	0.798997	0.399499	−	0.916734i	$-0.369184\pi$
$557$	18.8570	0.798997	0.399499	+	0.916734i	$0.369184\pi$
$558$	−2.95092	−0.124923
$559$	2.61484	0.110596
$560$	6.65069	0.281043
$561$	0.873517	0.0368799
$562$	−22.4395	−0.946553
$563$	−5.40051	−0.227604	−0.113802	−	0.993503i	$-0.536303\pi$
$563$	−5.40051	−0.227604	−0.113802	+	0.993503i	$0.536303\pi$
$564$	12.5736	0.529443
$565$	3.64606	0.153391
$566$	37.0752	1.55839
$567$	0.494465	0.0207656
$568$	17.8768	0.750095
$569$	30.5877	1.28230	0.641152	−	0.767414i	$-0.278457\pi$
$569$	30.5877	1.28230	0.641152	+	0.767414i	$0.278457\pi$
$570$	−25.0027	−1.04725
$571$	−23.0707	−0.965480	−0.482740	−	0.875764i	$-0.660358\pi$
$571$	−23.0707	−0.965480	−0.482740	+	0.875764i	$0.660358\pi$
$572$	−1.14458	−0.0478574
$573$	−11.6176	−0.485332
$574$	0.130469	0.00544569
$575$	−15.6648	−0.653269
$576$	0.698292	0.0290955
$577$	−13.2448	−0.551388	−0.275694	−	0.961245i	$-0.588908\pi$
$577$	−13.2448	−0.551388	−0.275694	+	0.961245i	$0.588908\pi$
$578$	−1.74167	−0.0724438
$579$	−9.25963	−0.384817
$580$	28.2200	1.17177
$581$	1.16506	0.0483348
$582$	−29.9325	−1.24074
$583$	−3.17431	−0.131466
$584$	−20.6807	−0.855774
$585$	3.41165	0.141054
$586$	−11.8826	−0.490865
$587$	6.56295	0.270882	0.135441	−	0.990785i	$-0.456755\pi$
$587$	6.56295	0.270882	0.135441	+	0.990785i	$0.456755\pi$
$588$	−6.98116	−0.287898
$589$	−9.03974	−0.372476
$590$	57.9897	2.38740
$591$	−17.1486	−0.705399
$592$	24.2433	0.996392
$593$	42.4330	1.74252	0.871258	−	0.490826i	$-0.163305\pi$
$593$	42.4330	1.74252	0.871258	+	0.490826i	$0.163305\pi$
$594$	−1.52137	−0.0624228
$595$	−1.33044	−0.0545425
$596$	−9.62606	−0.394299
$597$	5.10637	0.208990
$598$	−15.4462	−0.631641
$599$	35.8622	1.46529	0.732644	−	0.680612i	$-0.238286\pi$
$599$	35.8622	1.46529	0.732644	+	0.680612i	$0.238286\pi$
$600$	3.77039	0.153925
$601$	−20.4446	−0.833951	−0.416976	−	0.908918i	$-0.636910\pi$
$601$	−20.4446	−0.833951	−0.416976	+	0.908918i	$0.636910\pi$
$602$	1.77599	0.0723839
$603$	6.04454	0.246153
$604$	10.4856	0.426652
$605$	27.5442	1.11983
$606$	13.6813	0.555766
$607$	−40.5760	−1.64693	−0.823465	−	0.567367i	$-0.807962\pi$
$607$	−40.5760	−1.64693	−0.823465	+	0.567367i	$0.807962\pi$
$608$	−28.4876	−1.15533
$609$	−5.01840	−0.203356
$610$	12.4166	0.502735
$611$	−15.4275	−0.624129
$612$	1.03340	0.0417729
$613$	−15.3100	−0.618366	−0.309183	−	0.951003i	$-0.600056\pi$
$613$	−15.3100	−0.618366	−0.309183	+	0.951003i	$0.600056\pi$
$614$	−28.4814	−1.14942
$615$	0.407630	0.0164372
$616$	0.727139	0.0292973
$617$	−45.0504	−1.81366	−0.906831	−	0.421495i	$-0.861505\pi$
$617$	−45.0504	−1.81366	−0.906831	+	0.421495i	$0.861505\pi$
$618$	−5.97699	−0.240430
$619$	10.9512	0.440164	0.220082	−	0.975481i	$-0.429367\pi$
$619$	10.9512	0.440164	0.220082	+	0.975481i	$0.429367\pi$
$620$	−4.71108	−0.189202
$621$	−6.99440	−0.280675
$622$	−54.5671	−2.18794
$623$	1.60826	0.0644334
$624$	6.33839	0.253739
$625$	−31.1822	−1.24729
$626$	−7.15753	−0.286072
$627$	−4.66052	−0.186123
$628$	1.03340	0.0412373
$629$	−4.84974	−0.193372
$630$	2.31718	0.0923185
$631$	−7.73919	−0.308092	−0.154046	−	0.988064i	$-0.549230\pi$
$631$	−7.73919	−0.308092	−0.154046	+	0.988064i	$0.549230\pi$
$632$	−21.1523	−0.841394
$633$	−15.1392	−0.601729
$634$	−44.1780	−1.75453
$635$	26.1087	1.03609
$636$	−3.75533	−0.148908
$637$	8.56572	0.339386
$638$	15.4407	0.611301
$639$	10.6189	0.420078
$640$	32.0054	1.26512
$641$	37.8904	1.49658	0.748290	−	0.663372i	$-0.230875\pi$
$641$	37.8904	1.49658	0.748290	+	0.663372i	$0.230875\pi$
$642$	−0.0457072	−0.00180392
$643$	−10.0802	−0.397523	−0.198762	−	0.980048i	$-0.563692\pi$
$643$	−10.0802	−0.397523	−0.198762	+	0.980048i	$0.563692\pi$
$644$	−3.57401	−0.140836
$645$	5.54878	0.218483
$646$	9.29240	0.365605
$647$	34.2694	1.34727	0.673635	−	0.739065i	$-0.264732\pi$
$647$	34.2694	1.34727	0.673635	+	0.739065i	$0.264732\pi$
$648$	1.68349	0.0661337
$649$	10.8093	0.424303
$650$	4.94592	0.193995
$651$	0.837777	0.0328351
$652$	−14.2160	−0.556741
$653$	−38.3712	−1.50158	−0.750791	−	0.660540i	$-0.770328\pi$
$653$	−38.3712	−1.50158	−0.750791	+	0.660540i	$0.770328\pi$
$654$	2.75275	0.107641
$655$	−47.7756	−1.86675
$656$	0.757322	0.0295685
$657$	−12.2844	−0.479261
$658$	−10.4783	−0.408486
$659$	−20.6938	−0.806115	−0.403058	−	0.915175i	$-0.632053\pi$
$659$	−20.6938	−0.806115	−0.403058	+	0.915175i	$0.632053\pi$
$660$	−2.42884	−0.0945425
$661$	12.7182	0.494682	0.247341	−	0.968928i	$-0.420443\pi$
$661$	12.7182	0.494682	0.247341	+	0.968928i	$0.420443\pi$
$662$	−41.6919	−1.62040
$663$	−1.26796	−0.0492435
$664$	3.96664	0.153935
$665$	7.09834	0.275262
$666$	8.44663	0.327300
$667$	70.9871	2.74863
$668$	−16.0834	−0.622284
$669$	14.9217	0.576906
$670$	28.3261	1.09433
$671$	2.31447	0.0893493
$672$	2.64015	0.101846
$673$	−22.0702	−0.850742	−0.425371	−	0.905019i	$-0.639856\pi$
$673$	−22.0702	−0.850742	−0.425371	+	0.905019i	$0.639856\pi$
$674$	25.4698	0.981059
$675$	2.23963	0.0862033
$676$	−11.7728	−0.452801
$677$	4.20522	0.161620	0.0808100	−	0.996730i	$-0.474249\pi$
$677$	4.20522	0.161620	0.0808100	+	0.996730i	$0.474249\pi$
$678$	2.36010	0.0906392
$679$	8.49795	0.326121
$680$	−4.52969	−0.173706
$681$	25.1154	0.962425
$682$	−2.57768	−0.0987045
$683$	−18.9677	−0.725779	−0.362889	−	0.931832i	$-0.618210\pi$
$683$	−18.9677	−0.725779	−0.362889	+	0.931832i	$0.618210\pi$
$684$	−5.51357	−0.210817
$685$	14.6676	0.560422
$686$	11.8462	0.452288
$687$	−22.5430	−0.860068
$688$	10.3089	0.393023
$689$	4.60769	0.175539
$690$	−32.7773	−1.24781
$691$	−2.48693	−0.0946072	−0.0473036	−	0.998881i	$-0.515063\pi$
$691$	−2.48693	−0.0946072	−0.0473036	+	0.998881i	$0.515063\pi$
$692$	0.0155842	0.000592421 0
$693$	0.431924	0.0164074
$694$	−61.8802	−2.34894
$695$	−50.7110	−1.92358
$696$	−17.0860	−0.647642
$697$	−0.151498	−0.00573841
$698$	8.21106	0.310793
$699$	−26.2459	−0.992712
$700$	1.14441	0.0432546
$701$	−47.8747	−1.80820	−0.904102	−	0.427317i	$-0.859459\pi$
$701$	−47.8747	−1.80820	−0.904102	+	0.427317i	$0.859459\pi$
$702$	2.20837	0.0833494
$703$	25.8751	0.975896
$704$	0.609969	0.0229891
$705$	−32.7376	−1.23297
$706$	−26.2509	−0.987966
$707$	−3.88418	−0.146080
$708$	12.7878	0.480597
$709$	38.4808	1.44518	0.722588	−	0.691279i	$-0.242952\pi$
$709$	38.4808	1.44518	0.722588	+	0.691279i	$0.242952\pi$
$710$	49.7626	1.86756
$711$	−12.5646	−0.471208
$712$	5.47558	0.205206
$713$	−11.8507	−0.443811
$714$	−0.861194	−0.0322294
$715$	2.98013	0.111451
$716$	13.0448	0.487506
$717$	3.65267	0.136412
$718$	5.53916	0.206719
$719$	−25.8954	−0.965735	−0.482867	−	0.875693i	$-0.660405\pi$
$719$	−25.8954	−0.965735	−0.482867	+	0.875693i	$0.660405\pi$
$720$	13.4503	0.501262
$721$	1.69689	0.0631955
$722$	−16.4866	−0.613567
$723$	7.42013	0.275957
$724$	1.64623	0.0611817
$725$	−22.7303	−0.844182
$726$	17.8294	0.661711
$727$	46.9505	1.74130	0.870650	−	0.491904i	$-0.163699\pi$
$727$	46.9505	1.74130	0.870650	+	0.491904i	$0.163699\pi$
$728$	−1.05548	−0.0391189
$729$	1.00000	0.0370370
$730$	−57.5676	−2.13067
$731$	−2.06224	−0.0762747
$732$	2.73811	0.101203
$733$	−40.8942	−1.51046	−0.755231	−	0.655458i	$-0.772475\pi$
$733$	−40.8942	−1.51046	−0.755231	+	0.655458i	$0.772475\pi$
$734$	−6.33698	−0.233902
$735$	18.1767	0.670459
$736$	−37.3460	−1.37659
$737$	5.28001	0.194492
$738$	0.263860	0.00971280
$739$	14.0262	0.515963	0.257981	−	0.966150i	$-0.416943\pi$
$739$	14.0262	0.515963	0.257981	+	0.966150i	$0.416943\pi$
$740$	13.4849	0.495713
$741$	6.76502	0.248519
$742$	3.12953	0.114889
$743$	−0.841671	−0.0308779	−0.0154390	−	0.999881i	$-0.504915\pi$
$743$	−0.841671	−0.0308779	−0.0154390	+	0.999881i	$0.504915\pi$
$744$	2.85235	0.104572
$745$	25.0632	0.918245
$746$	−55.3551	−2.02669
$747$	2.35620	0.0862089
$748$	0.902695	0.0330058
$749$	0.0129764	0.000474149 0
$750$	−12.9357	−0.472346
$751$	18.9900	0.692954	0.346477	−	0.938058i	$-0.387378\pi$
$751$	18.9900	0.692954	0.346477	+	0.938058i	$0.387378\pi$
$752$	−60.8222	−2.21796
$753$	9.18800	0.334829
$754$	−22.4130	−0.816234
$755$	−27.3011	−0.993589
$756$	0.510982	0.0185842
$757$	−52.5568	−1.91021	−0.955104	−	0.296270i	$-0.904257\pi$
$757$	−52.5568	−1.91021	−0.955104	+	0.296270i	$0.904257\pi$
$758$	−28.5649	−1.03753
$759$	−6.10972	−0.221769
$760$	24.1675	0.876647
$761$	46.8763	1.69926	0.849631	−	0.527377i	$-0.176824\pi$
$761$	46.8763	1.69926	0.849631	+	0.527377i	$0.176824\pi$
$762$	16.9002	0.612231
$763$	−0.781516	−0.0282928
$764$	−12.0057	−0.434350
$765$	−2.69066	−0.0972809
$766$	−47.0368	−1.69951
$767$	−15.6904	−0.566547
$768$	19.3206	0.697171
$769$	10.3509	0.373262	0.186631	−	0.982430i	$-0.440243\pi$
$769$	10.3509	0.373262	0.186631	+	0.982430i	$0.440243\pi$
$770$	2.02409	0.0729432
$771$	20.0692	0.722775
$772$	−9.56894	−0.344394
$773$	−13.4204	−0.482698	−0.241349	−	0.970438i	$-0.577590\pi$
$773$	−13.4204	−0.482698	−0.241349	+	0.970438i	$0.577590\pi$
$774$	3.59173	0.129102
$775$	3.79462	0.136307
$776$	28.9327	1.03862
$777$	−2.39803	−0.0860287
$778$	4.30803	0.154450
$779$	0.808297	0.0289602
$780$	3.52561	0.126237
$781$	9.27580	0.331914
$782$	12.1819	0.435624
$783$	−10.1491	−0.362701
$784$	33.7700	1.20607
$785$	−2.69066	−0.0960336
$786$	−30.9252	−1.10307
$787$	50.4898	1.79977	0.899883	−	0.436131i	$-0.143652\pi$
$787$	50.4898	1.79977	0.899883	+	0.436131i	$0.143652\pi$
$788$	−17.7214	−0.631300
$789$	10.4265	0.371194
$790$	−58.8804	−2.09487
$791$	−0.670042	−0.0238239
$792$	1.47056	0.0522539
$793$	−3.35960	−0.119303
$794$	46.3178	1.64376
$795$	9.77768	0.346779
$796$	5.27694	0.187036
$797$	29.2035	1.03444	0.517220	−	0.855852i	$-0.326967\pi$
$797$	29.2035	1.03444	0.517220	+	0.855852i	$0.326967\pi$
$798$	4.59477	0.162653
$799$	12.1671	0.430443
$800$	11.9583	0.422789
$801$	3.25252	0.114922
$802$	−15.1184	−0.533848
$803$	−10.7307	−0.378677
$804$	6.24645	0.220295
$805$	9.30559	0.327979
$806$	3.74166	0.131794
$807$	27.9692	0.984563
$808$	−13.2243	−0.465231
$809$	−7.56079	−0.265823	−0.132912	−	0.991128i	$-0.542433\pi$
$809$	−7.56079	−0.265823	−0.132912	+	0.991128i	$0.542433\pi$
$810$	4.68623	0.164657
$811$	−46.3992	−1.62930	−0.814648	−	0.579956i	$-0.803070\pi$
$811$	−46.3992	−1.62930	−0.814648	+	0.579956i	$0.803070\pi$
$812$	−5.18603	−0.181994
$813$	−11.1232	−0.390107
$814$	7.37827	0.258608
$815$	37.0139	1.29654
$816$	−4.99888	−0.174996
$817$	11.0028	0.384938
$818$	−66.5883	−2.32821
$819$	−0.626963	−0.0219078
$820$	0.421246	0.0147105
$821$	28.0308	0.978281	0.489140	−	0.872205i	$-0.337311\pi$
$821$	28.0308	0.978281	0.489140	+	0.872205i	$0.337311\pi$
$822$	9.49440	0.331155
$823$	52.1661	1.81840	0.909198	−	0.416365i	$-0.136696\pi$
$823$	52.1661	1.81840	0.909198	+	0.416365i	$0.136696\pi$
$824$	5.77734	0.201263
$825$	1.95635	0.0681114
$826$	−10.6568	−0.370799
$827$	21.1776	0.736418	0.368209	−	0.929743i	$-0.379971\pi$
$827$	21.1776	0.736418	0.368209	+	0.929743i	$0.379971\pi$
$828$	−7.22803	−0.251191
$829$	−9.04256	−0.314061	−0.157030	−	0.987594i	$-0.550192\pi$
$829$	−9.04256	−0.314061	−0.157030	+	0.987594i	$0.550192\pi$
$830$	11.0417	0.383263
$831$	−25.0583	−0.869262
$832$	−0.885407	−0.0306960
$833$	−6.75550	−0.234064
$834$	−32.8253	−1.13665
$835$	41.8760	1.44918
$836$	−4.81619	−0.166572
$837$	1.69431	0.0585639
$838$	−21.7882	−0.752660
$839$	24.0591	0.830612	0.415306	−	0.909682i	$-0.363675\pi$
$839$	24.0591	0.830612	0.415306	+	0.909682i	$0.363675\pi$
$840$	−2.23977	−0.0772795
$841$	74.0052	2.55190
$842$	−63.2942	−2.18126
$843$	12.8839	0.443746
$844$	−15.6449	−0.538519
$845$	30.6527	1.05448
$846$	−21.1911	−0.728566
$847$	−5.06182	−0.173926
$848$	18.1656	0.623811
$849$	−21.2872	−0.730574
$850$	−3.90068	−0.133792
$851$	33.9210	1.16280
$852$	10.9736	0.375950
$853$	1.76687	0.0604965	0.0302483	−	0.999542i	$-0.490370\pi$
$853$	1.76687	0.0604965	0.0302483	+	0.999542i	$0.490370\pi$
$854$	−2.28182	−0.0780824
$855$	14.3556	0.490951
$856$	0.0441804	0.00151006
$857$	−8.27000	−0.282498	−0.141249	−	0.989974i	$-0.545112\pi$
$857$	−8.27000	−0.282498	−0.141249	+	0.989974i	$0.545112\pi$
$858$	1.92904	0.0658565
$859$	35.6527	1.21645	0.608227	−	0.793763i	$-0.291881\pi$
$859$	35.6527	1.21645	0.608227	+	0.793763i	$0.291881\pi$
$860$	5.73412	0.195532
$861$	−0.0749106	−0.00255295
$862$	59.5360	2.02780
$863$	−16.6637	−0.567239	−0.283620	−	0.958937i	$-0.591535\pi$
$863$	−16.6637	−0.567239	−0.283620	+	0.958937i	$0.591535\pi$
$864$	5.33941	0.181650
$865$	−0.0405762	−0.00137963
$866$	−17.2451	−0.586014
$867$	1.00000	0.0339618
$868$	0.865762	0.0293859
$869$	−10.9754	−0.372314
$870$	−47.5612	−1.61248
$871$	−7.66425	−0.259693
$872$	−2.66080	−0.0901062
$873$	17.1861	0.581663
$874$	−64.9948	−2.19848
$875$	3.67250	0.124153
$876$	−12.6948	−0.428917
$877$	9.63113	0.325220	0.162610	−	0.986690i	$-0.448009\pi$
$877$	9.63113	0.325220	0.162610	+	0.986690i	$0.448009\pi$
$878$	−0.635181	−0.0214363
$879$	6.82253	0.230118
$880$	11.7490	0.396060
$881$	7.22219	0.243322	0.121661	−	0.992572i	$-0.461178\pi$
$881$	7.22219	0.243322	0.121661	+	0.992572i	$0.461178\pi$
$882$	11.7658	0.396176
$883$	21.9453	0.738519	0.369260	−	0.929326i	$-0.379611\pi$
$883$	21.9453	0.738519	0.369260	+	0.929326i	$0.379611\pi$
$884$	−1.31031	−0.0440707
$885$	−33.2955	−1.11922
$886$	−55.7025	−1.87136
$887$	29.6692	0.996196	0.498098	−	0.867121i	$-0.334032\pi$
$887$	29.6692	0.996196	0.498098	+	0.867121i	$0.334032\pi$
$888$	−8.16448	−0.273982
$889$	−4.79803	−0.160921
$890$	15.2420	0.510914
$891$	0.873517	0.0292639
$892$	15.4201	0.516304
$893$	−64.9160	−2.17233
$894$	16.2235	0.542594
$895$	−33.9645	−1.13531
$896$	−5.88167	−0.196493
$897$	8.86862	0.296115
$898$	43.3317	1.44600
$899$	−17.1958	−0.573512
$900$	2.31444	0.0771479
$901$	−3.63394	−0.121064
$902$	0.230486	0.00767433
$903$	−1.01971	−0.0339337
$904$	−2.28127	−0.0758739
$905$	−4.28627	−0.142480
$906$	−17.6721	−0.587115
$907$	9.62905	0.319727	0.159864	−	0.987139i	$-0.448895\pi$
$907$	9.62905	0.319727	0.159864	+	0.987139i	$0.448895\pi$
$908$	25.9544	0.861326
$909$	−7.85531	−0.260544
$910$	−2.93809	−0.0973967
$911$	−53.9587	−1.78773	−0.893866	−	0.448334i	$-0.852018\pi$
$911$	−53.9587	−1.78773	−0.893866	+	0.448334i	$0.852018\pi$
$912$	26.6708	0.883159
$913$	2.05818	0.0681158
$914$	−46.6218	−1.54211
$915$	−7.12917	−0.235683
$916$	−23.2960	−0.769721
$917$	8.77978	0.289934
$918$	−1.74167	−0.0574836
$919$	−5.92126	−0.195324	−0.0976621	−	0.995220i	$-0.531136\pi$
$919$	−5.92126	−0.195324	−0.0976621	+	0.995220i	$0.531136\pi$
$920$	31.6824	1.04454
$921$	16.3530	0.538848
$922$	65.9731	2.17271
$923$	−13.4644	−0.443185
$924$	0.446351	0.0146839
$925$	−10.8616	−0.357127
$926$	−13.9932	−0.459844
$927$	3.43177	0.112714
$928$	−54.1905	−1.77889
$929$	−29.4130	−0.965009	−0.482504	−	0.875894i	$-0.660273\pi$
$929$	−29.4130	−0.965009	−0.482504	+	0.875894i	$0.660273\pi$
$930$	7.93992	0.260360
$931$	36.0430	1.18126
$932$	−27.1226	−0.888431
$933$	31.3304	1.02571
$934$	55.4799	1.81536
$935$	−2.35033	−0.0768641
$936$	−2.13460	−0.0697715
$937$	−9.04977	−0.295643	−0.147821	−	0.989014i	$-0.547226\pi$
$937$	−9.04977	−0.295643	−0.147821	+	0.989014i	$0.547226\pi$
$938$	−5.20552	−0.169966
$939$	4.10958	0.134111
$940$	−33.8311	−1.10345
$941$	−49.8407	−1.62476	−0.812381	−	0.583127i	$-0.801829\pi$
$941$	−49.8407	−1.62476	−0.812381	+	0.583127i	$0.801829\pi$
$942$	−1.74167	−0.0567466
$943$	1.05964	0.0345066
$944$	−61.8587	−2.01333
$945$	−1.33044	−0.0432791
$946$	3.13744	0.102007
$947$	13.5126	0.439101	0.219551	−	0.975601i	$-0.429541\pi$
$947$	13.5126	0.439101	0.219551	+	0.975601i	$0.429541\pi$
$948$	−12.9843	−0.421710
$949$	15.5762	0.505624
$950$	20.8115	0.675215
$951$	25.3654	0.822528
$952$	0.832427	0.0269791
$953$	42.6814	1.38259	0.691293	−	0.722574i	$-0.257041\pi$
$953$	42.6814	1.38259	0.691293	+	0.722574i	$0.257041\pi$
$954$	6.32911	0.204913
$955$	31.2590	1.01152
$956$	3.77469	0.122082
$957$	−8.86545	−0.286579
$958$	−15.9688	−0.515929
$959$	−2.69549	−0.0870420
$960$	−1.87886	−0.0606400
$961$	−28.1293	−0.907397
$962$	−10.7100	−0.345304
$963$	0.0262434	0.000845681 0
$964$	7.66798	0.246969
$965$	24.9145	0.802026
$966$	6.02353	0.193804
$967$	−22.1002	−0.710695	−0.355347	−	0.934734i	$-0.615637\pi$
$967$	−22.1002	−0.710695	−0.355347	+	0.934734i	$0.615637\pi$
$968$	−17.2338	−0.553916
$969$	−5.33535	−0.171396
$970$	80.5381	2.58592
$971$	−18.9018	−0.606586	−0.303293	−	0.952897i	$-0.598086\pi$
$971$	−18.9018	−0.606586	−0.303293	+	0.952897i	$0.598086\pi$
$972$	1.03340	0.0331464
$973$	9.31922	0.298760
$974$	−48.2145	−1.54489
$975$	−2.83976	−0.0909451
$976$	−13.2451	−0.423964
$977$	−32.7501	−1.04777	−0.523885	−	0.851789i	$-0.675518\pi$
$977$	−32.7501	−1.04777	−0.523885	+	0.851789i	$0.675518\pi$
$978$	23.9592	0.766131
$979$	2.84113	0.0908028
$980$	18.7839	0.600030
$981$	−1.58053	−0.0504624
$982$	63.2862	2.01954
$983$	29.7878	0.950083	0.475042	−	0.879963i	$-0.342433\pi$
$983$	29.7878	0.950083	0.475042	+	0.879963i	$0.342433\pi$
$984$	−0.255046	−0.00813056
$985$	46.1410	1.47017
$986$	17.6764	0.562932
$987$	6.01623	0.191499
$988$	6.99099	0.222413
$989$	14.4241	0.458660
$990$	4.09350	0.130100
$991$	−1.35952	−0.0431864	−0.0215932	−	0.999767i	$-0.506874\pi$
$991$	−1.35952	−0.0431864	−0.0215932	+	0.999767i	$0.506874\pi$
$992$	9.04662	0.287231
$993$	23.9379	0.759648
$994$	−9.14494	−0.290060
$995$	−13.7395	−0.435571
$996$	2.43491	0.0771529
$997$	11.5037	0.364327	0.182163	−	0.983268i	$-0.441690\pi$
$997$	11.5037	0.364327	0.182163	+	0.983268i	$0.441690\pi$
$998$	52.0947	1.64903
$999$	−4.84974	−0.153439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.i.1.15	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.i.1.15	✓	63	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-1.74167 q^{2} +1.00000 q^{3} +1.03340 q^{4} -2.69066 q^{5} -1.74167 q^{6} +0.494465 q^{7} +1.68349 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.74167 q^{2} +1.00000 q^{3} +1.03340 q^{4} -2.69066 q^{5} -1.74167 q^{6} +0.494465 q^{7} +1.68349 q^{8} +1.00000 q^{9} +4.68623 q^{10} +0.873517 q^{11} +1.03340 q^{12} -1.26796 q^{13} -0.861194 q^{14} -2.69066 q^{15} -4.99888 q^{16} +1.00000 q^{17} -1.74167 q^{18} -5.33535 q^{19} -2.78053 q^{20} +0.494465 q^{21} -1.52137 q^{22} -6.99440 q^{23} +1.68349 q^{24} +2.23963 q^{25} +2.20837 q^{26} +1.00000 q^{27} +0.510982 q^{28} -10.1491 q^{29} +4.68623 q^{30} +1.69431 q^{31} +5.33941 q^{32} +0.873517 q^{33} -1.74167 q^{34} -1.33044 q^{35} +1.03340 q^{36} -4.84974 q^{37} +9.29240 q^{38} -1.26796 q^{39} -4.52969 q^{40} -0.151498 q^{41} -0.861194 q^{42} -2.06224 q^{43} +0.902695 q^{44} -2.69066 q^{45} +12.1819 q^{46} +12.1671 q^{47} -4.99888 q^{48} -6.75550 q^{49} -3.90068 q^{50} +1.00000 q^{51} -1.31031 q^{52} -3.63394 q^{53} -1.74167 q^{54} -2.35033 q^{55} +0.832427 q^{56} -5.33535 q^{57} +17.6764 q^{58} +12.3745 q^{59} -2.78053 q^{60} +2.64960 q^{61} -2.95092 q^{62} +0.494465 q^{63} +0.698292 q^{64} +3.41165 q^{65} -1.52137 q^{66} +6.04454 q^{67} +1.03340 q^{68} -6.99440 q^{69} +2.31718 q^{70} +10.6189 q^{71} +1.68349 q^{72} -12.2844 q^{73} +8.44663 q^{74} +2.23963 q^{75} -5.51357 q^{76} +0.431924 q^{77} +2.20837 q^{78} -12.5646 q^{79} +13.4503 q^{80} +1.00000 q^{81} +0.263860 q^{82} +2.35620 q^{83} +0.510982 q^{84} -2.69066 q^{85} +3.59173 q^{86} -10.1491 q^{87} +1.47056 q^{88} +3.25252 q^{89} +4.68623 q^{90} -0.626963 q^{91} -7.22803 q^{92} +1.69431 q^{93} -21.1911 q^{94} +14.3556 q^{95} +5.33941 q^{96} +17.1861 q^{97} +11.7658 q^{98} +0.873517 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9}+O(q^{10}) \) 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9	\( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9} + 4 q^{10} + 23 q^{11} + 70 q^{12} + 10 q^{13} + 18 q^{14} + 19 q^{15} + 72 q^{16} + 63 q^{17} + 10 q^{18} + 6 q^{19} + 48 q^{20} + 11 q^{21} + 21 q^{22} + 44 q^{23} + 27 q^{24} + 110 q^{25} + 41 q^{26} + 63 q^{27} + 26 q^{28} + 35 q^{29} + 4 q^{30} + q^{31} + 54 q^{32} + 23 q^{33} + 10 q^{34} + 47 q^{35} + 70 q^{36} + 40 q^{37} + 38 q^{38} + 10 q^{39} - 10 q^{40} + 35 q^{41} + 18 q^{42} + 27 q^{43} + 46 q^{44} + 19 q^{45} + 8 q^{46} + 29 q^{47} + 72 q^{48} + 114 q^{49} + 27 q^{50} + 63 q^{51} - q^{52} + 75 q^{53} + 10 q^{54} + 5 q^{55} + 24 q^{56} + 6 q^{57} + 41 q^{58} + 105 q^{59} + 48 q^{60} + 5 q^{61} + 22 q^{62} + 11 q^{63} + 61 q^{64} + 49 q^{65} + 21 q^{66} + 4 q^{67} + 70 q^{68} + 44 q^{69} - 16 q^{70} + 16 q^{71} + 27 q^{72} + 39 q^{73} + 54 q^{74} + 110 q^{75} + 6 q^{76} + 88 q^{77} + 41 q^{78} + 16 q^{79} + 102 q^{80} + 63 q^{81} - 29 q^{82} + 73 q^{83} + 26 q^{84} + 19 q^{85} + 46 q^{86} + 35 q^{87} + 18 q^{88} + 88 q^{89} + 4 q^{90} - 15 q^{91} + 110 q^{92} + q^{93} - 8 q^{94} + 28 q^{95} + 54 q^{96} + 70 q^{97} + 33 q^{98} + 23 q^{99}+O(q^{100}) \) 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9 + 4 * q^10 + 23 * q^11 + 70 * q^12 + 10 * q^13 + 18 * q^14 + 19 * q^15 + 72 * q^16 + 63 * q^17 + 10 * q^18 + 6 * q^19 + 48 * q^20 + 11 * q^21 + 21 * q^22 + 44 * q^23 + 27 * q^24 + 110 * q^25 + 41 * q^26 + 63 * q^27 + 26 * q^28 + 35 * q^29 + 4 * q^30 + q^31 + 54 * q^32 + 23 * q^33 + 10 * q^34 + 47 * q^35 + 70 * q^36 + 40 * q^37 + 38 * q^38 + 10 * q^39 - 10 * q^40 + 35 * q^41 + 18 * q^42 + 27 * q^43 + 46 * q^44 + 19 * q^45 + 8 * q^46 + 29 * q^47 + 72 * q^48 + 114 * q^49 + 27 * q^50 + 63 * q^51 - q^52 + 75 * q^53 + 10 * q^54 + 5 * q^55 + 24 * q^56 + 6 * q^57 + 41 * q^58 + 105 * q^59 + 48 * q^60 + 5 * q^61 + 22 * q^62 + 11 * q^63 + 61 * q^64 + 49 * q^65 + 21 * q^66 + 4 * q^67 + 70 * q^68 + 44 * q^69 - 16 * q^70 + 16 * q^71 + 27 * q^72 + 39 * q^73 + 54 * q^74 + 110 * q^75 + 6 * q^76 + 88 * q^77 + 41 * q^78 + 16 * q^79 + 102 * q^80 + 63 * q^81 - 29 * q^82 + 73 * q^83 + 26 * q^84 + 19 * q^85 + 46 * q^86 + 35 * q^87 + 18 * q^88 + 88 * q^89 + 4 * q^90 - 15 * q^91 + 110 * q^92 + q^93 - 8 * q^94 + 28 * q^95 + 54 * q^96 + 70 * q^97 + 33 * q^98 + 23 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more