LMFDB - Embedded newform 4019.2.a.b.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4019,2,Mod(1,4019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4019, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4019.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$32.0918765724$
Analytic rank:	$0$
Dimension:	$186$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	4019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.39579 q^{2} +3.13150 q^{3} +3.73980 q^{4} +0.529965 q^{5} -7.50242 q^{6} -1.46029 q^{7} -4.16820 q^{8} +6.80631 q^{9} +O(q^{10})$	\(q-2.39579 q^{2} +3.13150 q^{3} +3.73980 q^{4} +0.529965 q^{5} -7.50242 q^{6} -1.46029 q^{7} -4.16820 q^{8} +6.80631 q^{9} -1.26968 q^{10} +4.42489 q^{11} +11.7112 q^{12} +6.46275 q^{13} +3.49855 q^{14} +1.65959 q^{15} +2.50652 q^{16} +1.71135 q^{17} -16.3065 q^{18} -1.59807 q^{19} +1.98197 q^{20} -4.57291 q^{21} -10.6011 q^{22} +5.85306 q^{23} -13.0527 q^{24} -4.71914 q^{25} -15.4834 q^{26} +11.9195 q^{27} -5.46120 q^{28} -6.42396 q^{29} -3.97602 q^{30} -2.54140 q^{31} +2.33130 q^{32} +13.8565 q^{33} -4.10004 q^{34} -0.773904 q^{35} +25.4543 q^{36} +7.82017 q^{37} +3.82864 q^{38} +20.2381 q^{39} -2.20900 q^{40} +12.2992 q^{41} +10.9557 q^{42} -2.30731 q^{43} +16.5482 q^{44} +3.60711 q^{45} -14.0227 q^{46} -1.76939 q^{47} +7.84919 q^{48} -4.86755 q^{49} +11.3061 q^{50} +5.35911 q^{51} +24.1694 q^{52} -0.795921 q^{53} -28.5566 q^{54} +2.34504 q^{55} +6.08679 q^{56} -5.00436 q^{57} +15.3904 q^{58} -14.9641 q^{59} +6.20653 q^{60} +12.9958 q^{61} +6.08866 q^{62} -9.93920 q^{63} -10.5984 q^{64} +3.42503 q^{65} -33.1974 q^{66} +12.9765 q^{67} +6.40012 q^{68} +18.3289 q^{69} +1.85411 q^{70} -9.51351 q^{71} -28.3701 q^{72} -0.209957 q^{73} -18.7355 q^{74} -14.7780 q^{75} -5.97647 q^{76} -6.46162 q^{77} -48.4862 q^{78} -5.60830 q^{79} +1.32837 q^{80} +16.9070 q^{81} -29.4663 q^{82} -6.61460 q^{83} -17.1018 q^{84} +0.906958 q^{85} +5.52782 q^{86} -20.1166 q^{87} -18.4438 q^{88} -9.99098 q^{89} -8.64187 q^{90} -9.43749 q^{91} +21.8893 q^{92} -7.95841 q^{93} +4.23909 q^{94} -0.846921 q^{95} +7.30048 q^{96} -5.09962 q^{97} +11.6616 q^{98} +30.1172 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9}+O(q^{10}) $ 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9} + 50 q^{10} + 25 q^{11} + 17 q^{12} + 113 q^{13} + 12 q^{14} + 12 q^{15} + 252 q^{16} + 35 q^{17} + 13 q^{18} + 97 q^{19} + 55 q^{20} + 115 q^{21} + 14 q^{22} + 27 q^{23} + 122 q^{24} + 244 q^{25} + 39 q^{26} + 34 q^{27} + 66 q^{28} + 91 q^{29} + 4 q^{30} + 135 q^{31} + 21 q^{32} + 32 q^{33} + 58 q^{34} + 17 q^{35} + 273 q^{36} + 133 q^{37} - 3 q^{38} + 55 q^{39} + 142 q^{40} + 97 q^{41} - 8 q^{42} + 67 q^{43} + 44 q^{44} + 154 q^{45} + 101 q^{46} + 20 q^{47} - 7 q^{48} + 312 q^{49} + 21 q^{50} + 23 q^{51} + 193 q^{52} + 22 q^{53} + 141 q^{54} + 88 q^{55} + 28 q^{56} + 65 q^{57} + 62 q^{58} + 41 q^{59} + q^{60} + 377 q^{61} + 29 q^{62} + 39 q^{63} + 311 q^{64} + 21 q^{65} + 35 q^{66} + 42 q^{67} + 24 q^{68} + 137 q^{69} + 35 q^{70} + 17 q^{71} - 8 q^{72} + 213 q^{73} - 9 q^{74} + 2 q^{75} + 242 q^{76} + 60 q^{77} + 103 q^{79} + 80 q^{80} + 270 q^{81} + 84 q^{82} + 42 q^{83} + 137 q^{84} + 294 q^{85} - 9 q^{86} + 22 q^{87} - 13 q^{88} + 78 q^{89} + 69 q^{90} + 118 q^{91} + 49 q^{92} + 51 q^{93} + 93 q^{94} + 10 q^{95} + 260 q^{96} + 142 q^{97} - 31 q^{98} + 78 q^{99}+O(q^{100}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9 + 50 * q^10 + 25 * q^11 + 17 * q^12 + 113 * q^13 + 12 * q^14 + 12 * q^15 + 252 * q^16 + 35 * q^17 + 13 * q^18 + 97 * q^19 + 55 * q^20 + 115 * q^21 + 14 * q^22 + 27 * q^23 + 122 * q^24 + 244 * q^25 + 39 * q^26 + 34 * q^27 + 66 * q^28 + 91 * q^29 + 4 * q^30 + 135 * q^31 + 21 * q^32 + 32 * q^33 + 58 * q^34 + 17 * q^35 + 273 * q^36 + 133 * q^37 - 3 * q^38 + 55 * q^39 + 142 * q^40 + 97 * q^41 - 8 * q^42 + 67 * q^43 + 44 * q^44 + 154 * q^45 + 101 * q^46 + 20 * q^47 - 7 * q^48 + 312 * q^49 + 21 * q^50 + 23 * q^51 + 193 * q^52 + 22 * q^53 + 141 * q^54 + 88 * q^55 + 28 * q^56 + 65 * q^57 + 62 * q^58 + 41 * q^59 + q^60 + 377 * q^61 + 29 * q^62 + 39 * q^63 + 311 * q^64 + 21 * q^65 + 35 * q^66 + 42 * q^67 + 24 * q^68 + 137 * q^69 + 35 * q^70 + 17 * q^71 - 8 * q^72 + 213 * q^73 - 9 * q^74 + 2 * q^75 + 242 * q^76 + 60 * q^77 + 103 * q^79 + 80 * q^80 + 270 * q^81 + 84 * q^82 + 42 * q^83 + 137 * q^84 + 294 * q^85 - 9 * q^86 + 22 * q^87 - 13 * q^88 + 78 * q^89 + 69 * q^90 + 118 * q^91 + 49 * q^92 + 51 * q^93 + 93 * q^94 + 10 * q^95 + 260 * q^96 + 142 * q^97 - 31 * q^98 + 78 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.39579	−1.69408	−0.847039	−	0.531530i	$-0.821617\pi$
$2$	−2.39579	−1.69408	−0.847039	+	0.531530i	$0.821617\pi$
$3$	3.13150	1.80797	0.903987	−	0.427560i	$-0.140627\pi$
$3$	3.13150	1.80797	0.903987	+	0.427560i	$0.140627\pi$
$4$	3.73980	1.86990
$5$	0.529965	0.237008	0.118504	−	0.992954i	$-0.462190\pi$
$5$	0.529965	0.237008	0.118504	+	0.992954i	$0.462190\pi$
$6$	−7.50242	−3.06285
$7$	−1.46029	−0.551938	−0.275969	−	0.961167i	$-0.588999\pi$
$7$	−1.46029	−0.551938	−0.275969	+	0.961167i	$0.588999\pi$
$8$	−4.16820	−1.47368
$9$	6.80631	2.26877
$10$	−1.26968	−0.401510
$11$	4.42489	1.33415	0.667077	−	0.744989i	$-0.267545\pi$
$11$	4.42489	1.33415	0.667077	+	0.744989i	$0.267545\pi$
$12$	11.7112	3.38073
$13$	6.46275	1.79244	0.896222	−	0.443606i	$-0.146301\pi$
$13$	6.46275	1.79244	0.896222	+	0.443606i	$0.146301\pi$
$14$	3.49855	0.935027
$15$	1.65959	0.428504
$16$	2.50652	0.626631
$17$	1.71135	0.415064	0.207532	−	0.978228i	$-0.433457\pi$
$17$	1.71135	0.415064	0.207532	+	0.978228i	$0.433457\pi$
$18$	−16.3065	−3.84348
$19$	−1.59807	−0.366622	−0.183311	−	0.983055i	$-0.558682\pi$
$19$	−1.59807	−0.366622	−0.183311	+	0.983055i	$0.558682\pi$
$20$	1.98197	0.443181
$21$	−4.57291	−0.997890
$22$	−10.6011	−2.26016
$23$	5.85306	1.22045	0.610223	−	0.792229i	$-0.291080\pi$
$23$	5.85306	1.22045	0.610223	+	0.792229i	$0.291080\pi$
$24$	−13.0527	−2.66438
$25$	−4.71914	−0.943827
$26$	−15.4834	−3.03654
$27$	11.9195	2.29391
$28$	−5.46120	−1.03207
$29$	−6.42396	−1.19290	−0.596450	−	0.802651i	$-0.703422\pi$
$29$	−6.42396	−1.19290	−0.596450	+	0.802651i	$0.703422\pi$
$30$	−3.97602	−0.725919
$31$	−2.54140	−0.456449	−0.228225	−	0.973608i	$-0.573292\pi$
$31$	−2.54140	−0.456449	−0.228225	+	0.973608i	$0.573292\pi$
$32$	2.33130	0.412120
$33$	13.8565	2.41211
$34$	−4.10004	−0.703151
$35$	−0.773904	−0.130814
$36$	25.4543	4.24238
$37$	7.82017	1.28563	0.642814	−	0.766022i	$-0.277767\pi$
$37$	7.82017	1.28563	0.642814	+	0.766022i	$0.277767\pi$
$38$	3.82864	0.621087
$39$	20.2381	3.24069
$40$	−2.20900	−0.349274
$41$	12.2992	1.92081	0.960405	−	0.278607i	$-0.0898728\pi$
$41$	12.2992	1.92081	0.960405	+	0.278607i	$0.0898728\pi$
$42$	10.9557	1.69050
$43$	−2.30731	−0.351861	−0.175931	−	0.984403i	$-0.556293\pi$
$43$	−2.30731	−0.351861	−0.175931	+	0.984403i	$0.556293\pi$
$44$	16.5482	2.49474
$45$	3.60711	0.537716
$46$	−14.0227	−2.06753
$47$	−1.76939	−0.258092	−0.129046	−	0.991639i	$-0.541192\pi$
$47$	−1.76939	−0.258092	−0.129046	+	0.991639i	$0.541192\pi$
$48$	7.84919	1.13293
$49$	−4.86755	−0.695364
$50$	11.3061	1.59892
$51$	5.35911	0.750425
$52$	24.1694	3.35169
$53$	−0.795921	−0.109328	−0.0546641	−	0.998505i	$-0.517409\pi$
$53$	−0.795921	−0.109328	−0.0546641	+	0.998505i	$0.517409\pi$
$54$	−28.5566	−3.88606
$55$	2.34504	0.316205
$56$	6.08679	0.813381
$57$	−5.00436	−0.662844
$58$	15.3904	2.02086
$59$	−14.9641	−1.94817	−0.974083	−	0.226191i	$-0.927373\pi$
$59$	−14.9641	−1.94817	−0.974083	+	0.226191i	$0.927373\pi$
$60$	6.20653	0.801260
$61$	12.9958	1.66394	0.831968	−	0.554824i	$-0.187214\pi$
$61$	12.9958	1.66394	0.831968	+	0.554824i	$0.187214\pi$
$62$	6.08866	0.773261
$63$	−9.93920	−1.25222
$64$	−10.5984	−1.32479
$65$	3.42503	0.424823
$66$	−33.1974	−4.08631
$67$	12.9765	1.58534	0.792668	−	0.609653i	$-0.208691\pi$
$67$	12.9765	1.58534	0.792668	+	0.609653i	$0.208691\pi$
$68$	6.40012	0.776129
$69$	18.3289	2.20654
$70$	1.85411	0.221608
$71$	−9.51351	−1.12905	−0.564523	−	0.825417i	$-0.690940\pi$
$71$	−9.51351	−1.12905	−0.564523	+	0.825417i	$0.690940\pi$
$72$	−28.3701	−3.34345
$73$	−0.209957	−0.0245737	−0.0122868	−	0.999925i	$-0.503911\pi$
$73$	−0.209957	−0.0245737	−0.0122868	+	0.999925i	$0.503911\pi$
$74$	−18.7355	−2.17795
$75$	−14.7780	−1.70642
$76$	−5.97647	−0.685548
$77$	−6.46162	−0.736370
$78$	−48.4862	−5.48999
$79$	−5.60830	−0.630983	−0.315492	−	0.948928i	$-0.602169\pi$
$79$	−5.60830	−0.630983	−0.315492	+	0.948928i	$0.602169\pi$
$80$	1.32837	0.148516
$81$	16.9070	1.87855
$82$	−29.4663	−3.25400
$83$	−6.61460	−0.726047	−0.363023	−	0.931780i	$-0.618255\pi$
$83$	−6.61460	−0.726047	−0.363023	+	0.931780i	$0.618255\pi$
$84$	−17.1018	−1.86596
$85$	0.906958	0.0983734
$86$	5.52782	0.596080
$87$	−20.1166	−2.15673
$88$	−18.4438	−1.96612
$89$	−9.99098	−1.05904	−0.529521	−	0.848297i	$-0.677628\pi$
$89$	−9.99098	−1.05904	−0.529521	+	0.848297i	$0.677628\pi$
$90$	−8.64187	−0.910933
$91$	−9.43749	−0.989318
$92$	21.8893	2.28212
$93$	−7.95841	−0.825249
$94$	4.23909	0.437228
$95$	−0.846921	−0.0868923
$96$	7.30048	0.745102
$97$	−5.09962	−0.517788	−0.258894	−	0.965906i	$-0.583358\pi$
$97$	−5.09962	−0.517788	−0.258894	+	0.965906i	$0.583358\pi$
$98$	11.6616	1.17800
$99$	30.1172	3.02689
$100$	−17.6486	−1.76486
$101$	−6.38324	−0.635156	−0.317578	−	0.948232i	$-0.602870\pi$
$101$	−6.38324	−0.635156	−0.317578	+	0.948232i	$0.602870\pi$
$102$	−12.8393	−1.27128
$103$	1.54634	0.152365	0.0761826	−	0.997094i	$-0.475727\pi$
$103$	1.54634	0.152365	0.0761826	+	0.997094i	$0.475727\pi$
$104$	−26.9380	−2.64149
$105$	−2.42348	−0.236508
$106$	1.90686	0.185211
$107$	17.1517	1.65812	0.829060	−	0.559160i	$-0.188876\pi$
$107$	17.1517	1.65812	0.829060	+	0.559160i	$0.188876\pi$
$108$	44.5765	4.28938
$109$	15.1280	1.44900	0.724499	−	0.689276i	$-0.242071\pi$
$109$	15.1280	1.44900	0.724499	+	0.689276i	$0.242071\pi$
$110$	−5.61821	−0.535675
$111$	24.4889	2.32438
$112$	−3.66025	−0.345862
$113$	−6.23685	−0.586714	−0.293357	−	0.956003i	$-0.594772\pi$
$113$	−6.23685	−0.586714	−0.293357	+	0.956003i	$0.594772\pi$
$114$	11.9894	1.12291
$115$	3.10192	0.289255
$116$	−24.0243	−2.23060
$117$	43.9875	4.06664
$118$	35.8509	3.30035
$119$	−2.49907	−0.229090
$120$	−6.91750	−0.631478
$121$	8.57962	0.779965
$122$	−31.1351	−2.81884
$123$	38.5149	3.47278
$124$	−9.50435	−0.853515
$125$	−5.15081	−0.460702
$126$	23.8122	2.12136
$127$	−6.93786	−0.615635	−0.307818	−	0.951445i	$-0.599599\pi$
$127$	−6.93786	−0.615635	−0.307818	+	0.951445i	$0.599599\pi$
$128$	20.7288	1.83219
$129$	−7.22534	−0.636156
$130$	−8.20565	−0.719683
$131$	−12.7954	−1.11794	−0.558969	−	0.829188i	$-0.688803\pi$
$131$	−12.7954	−1.11794	−0.558969	+	0.829188i	$0.688803\pi$
$132$	51.8208	4.51042
$133$	2.33365	0.202353
$134$	−31.0890	−2.68568
$135$	6.31691	0.543673
$136$	−7.13326	−0.611672
$137$	−5.68126	−0.485383	−0.242692	−	0.970104i	$-0.578030\pi$
$137$	−5.68126	−0.485383	−0.242692	+	0.970104i	$0.578030\pi$
$138$	−43.9121	−3.73805
$139$	−5.61317	−0.476103	−0.238051	−	0.971253i	$-0.576509\pi$
$139$	−5.61317	−0.476103	−0.238051	+	0.971253i	$0.576509\pi$
$140$	−2.89425	−0.244609
$141$	−5.54085	−0.466624
$142$	22.7924	1.91269
$143$	28.5969	2.39139
$144$	17.0602	1.42168
$145$	−3.40448	−0.282726
$146$	0.503014	0.0416297
$147$	−15.2427	−1.25720
$148$	29.2459	2.40400
$149$	−8.32032	−0.681627	−0.340814	−	0.940131i	$-0.610703\pi$
$149$	−8.32032	−0.681627	−0.340814	+	0.940131i	$0.610703\pi$
$150$	35.4049	2.89080
$151$	−2.37678	−0.193419	−0.0967097	−	0.995313i	$-0.530832\pi$
$151$	−2.37678	−0.193419	−0.0967097	+	0.995313i	$0.530832\pi$
$152$	6.66108	0.540285
$153$	11.6480	0.941685
$154$	15.4807	1.24747
$155$	−1.34686	−0.108182
$156$	75.6866	6.05978
$157$	−19.7733	−1.57808	−0.789042	−	0.614339i	$-0.789423\pi$
$157$	−19.7733	−1.57808	−0.789042	+	0.614339i	$0.789423\pi$
$158$	13.4363	1.06893
$159$	−2.49243	−0.197663
$160$	1.23551	0.0976756
$161$	−8.54717	−0.673611
$162$	−40.5055	−3.18241
$163$	−15.5566	−1.21848	−0.609242	−	0.792984i	$-0.708526\pi$
$163$	−15.5566	−1.21848	−0.609242	+	0.792984i	$0.708526\pi$
$164$	45.9965	3.59173
$165$	7.34349	0.571690
$166$	15.8472	1.22998
$167$	6.56436	0.507965	0.253983	−	0.967209i	$-0.418259\pi$
$167$	6.56436	0.507965	0.253983	+	0.967209i	$0.418259\pi$
$168$	19.0608	1.47057
$169$	28.7671	2.21285
$170$	−2.17288	−0.166652
$171$	−10.8770	−0.831782
$172$	−8.62888	−0.657946
$173$	−8.26455	−0.628342	−0.314171	−	0.949366i	$-0.601727\pi$
$173$	−8.26455	−0.628342	−0.314171	+	0.949366i	$0.601727\pi$
$174$	48.1952	3.65367
$175$	6.89131	0.520934
$176$	11.0911	0.836022
$177$	−46.8603	−3.52223
$178$	23.9363	1.79410
$179$	−19.9138	−1.48843	−0.744214	−	0.667941i	$-0.767176\pi$
$179$	−19.9138	−1.48843	−0.744214	+	0.667941i	$0.767176\pi$
$180$	13.4899	1.00548
$181$	19.8840	1.47797	0.738985	−	0.673722i	$-0.235306\pi$
$181$	19.8840	1.47797	0.738985	+	0.673722i	$0.235306\pi$
$182$	22.6102	1.67598
$183$	40.6963	3.00835
$184$	−24.3967	−1.79855
$185$	4.14442	0.304704
$186$	19.0667	1.39804
$187$	7.57254	0.553759
$188$	−6.61718	−0.482607
$189$	−17.4059	−1.26609
$190$	2.02904	0.147202
$191$	5.33729	0.386193	0.193096	−	0.981180i	$-0.438147\pi$
$191$	5.33729	0.386193	0.193096	+	0.981180i	$0.438147\pi$
$192$	−33.1888	−2.39519
$193$	14.1630	1.01948	0.509738	−	0.860330i	$-0.329742\pi$
$193$	14.1630	1.01948	0.509738	+	0.860330i	$0.329742\pi$
$194$	12.2176	0.877174
$195$	10.7255	0.768069
$196$	−18.2037	−1.30026
$197$	−8.88875	−0.633297	−0.316649	−	0.948543i	$-0.602558\pi$
$197$	−8.88875	−0.633297	−0.316649	+	0.948543i	$0.602558\pi$
$198$	−72.1544	−5.12779
$199$	19.0844	1.35286	0.676429	−	0.736508i	$-0.263527\pi$
$199$	19.0844	1.35286	0.676429	+	0.736508i	$0.263527\pi$
$200$	19.6703	1.39090
$201$	40.6361	2.86625
$202$	15.2929	1.07600
$203$	9.38085	0.658407
$204$	20.0420	1.40322
$205$	6.51814	0.455247
$206$	−3.70470	−0.258119
$207$	39.8377	2.76891
$208$	16.1990	1.12320
$209$	−7.07128	−0.489130
$210$	5.80615	0.400662
$211$	10.0373	0.690994	0.345497	−	0.938420i	$-0.387710\pi$
$211$	10.0373	0.690994	0.345497	+	0.938420i	$0.387710\pi$
$212$	−2.97659	−0.204433
$213$	−29.7916	−2.04129
$214$	−41.0919	−2.80899
$215$	−1.22279	−0.0833938
$216$	−49.6828	−3.38049
$217$	3.71119	0.251932
$218$	−36.2434	−2.45472
$219$	−0.657482	−0.0444285
$220$	8.76997	0.591272
$221$	11.0600	0.743979
$222$	−58.6702	−3.93769
$223$	−1.53730	−0.102945	−0.0514725	−	0.998674i	$-0.516391\pi$
$223$	−1.53730	−0.102945	−0.0514725	+	0.998674i	$0.516391\pi$
$224$	−3.40438	−0.227465
$225$	−32.1199	−2.14133
$226$	14.9422	0.993939
$227$	25.0467	1.66240	0.831202	−	0.555970i	$-0.187653\pi$
$227$	25.0467	1.66240	0.831202	+	0.555970i	$0.187653\pi$
$228$	−18.7153	−1.23945
$229$	−10.0343	−0.663086	−0.331543	−	0.943440i	$-0.607569\pi$
$229$	−10.0343	−0.663086	−0.331543	+	0.943440i	$0.607569\pi$
$230$	−7.43154	−0.490021
$231$	−20.2346	−1.33134
$232$	26.7764	1.75795
$233$	−7.48980	−0.490673	−0.245337	−	0.969438i	$-0.578898\pi$
$233$	−7.48980	−0.490673	−0.245337	+	0.969438i	$0.578898\pi$
$234$	−105.385	−6.88921
$235$	−0.937716	−0.0611698
$236$	−55.9630	−3.64288
$237$	−17.5624	−1.14080
$238$	5.98725	0.388096
$239$	−18.1515	−1.17412	−0.587062	−	0.809542i	$-0.699716\pi$
$239$	−18.1515	−1.17412	−0.587062	+	0.809542i	$0.699716\pi$
$240$	4.15980	0.268514
$241$	22.0854	1.42265	0.711323	−	0.702866i	$-0.248097\pi$
$241$	22.0854	1.42265	0.711323	+	0.702866i	$0.248097\pi$
$242$	−20.5550	−1.32132
$243$	17.1857	1.10247
$244$	48.6016	3.11140
$245$	−2.57963	−0.164807
$246$	−92.2737	−5.88315
$247$	−10.3279	−0.657150
$248$	10.5931	0.672661
$249$	−20.7136	−1.31267
$250$	12.3402	0.780465
$251$	6.25714	0.394947	0.197473	−	0.980308i	$-0.436726\pi$
$251$	6.25714	0.394947	0.197473	+	0.980308i	$0.436726\pi$
$252$	−37.1706	−2.34153
$253$	25.8991	1.62826
$254$	16.6216	1.04293
$255$	2.84014	0.177857
$256$	−28.4652	−1.77907
$257$	2.16060	0.134775	0.0673874	−	0.997727i	$-0.478534\pi$
$257$	2.16060	0.134775	0.0673874	+	0.997727i	$0.478534\pi$
$258$	17.3104	1.07770
$259$	−11.4197	−0.709587
$260$	12.8089	0.794377
$261$	−43.7235	−2.70642
$262$	30.6551	1.89388
$263$	−0.965852	−0.0595570	−0.0297785	−	0.999557i	$-0.509480\pi$
$263$	−0.965852	−0.0595570	−0.0297785	+	0.999557i	$0.509480\pi$
$264$	−57.7569	−3.55469
$265$	−0.421811	−0.0259116
$266$	−5.59093	−0.342802
$267$	−31.2868	−1.91472
$268$	48.5297	2.96442
$269$	−14.7866	−0.901552	−0.450776	−	0.892637i	$-0.648853\pi$
$269$	−14.7866	−0.901552	−0.450776	+	0.892637i	$0.648853\pi$
$270$	−15.1340	−0.921025
$271$	14.8258	0.900602	0.450301	−	0.892877i	$-0.351317\pi$
$271$	14.8258	0.900602	0.450301	+	0.892877i	$0.351317\pi$
$272$	4.28955	0.260092
$273$	−29.5535	−1.78866
$274$	13.6111	0.822277
$275$	−20.8816	−1.25921
$276$	68.5464	4.12601
$277$	12.8112	0.769748	0.384874	−	0.922969i	$-0.374245\pi$
$277$	12.8112	0.769748	0.384874	+	0.922969i	$0.374245\pi$
$278$	13.4480	0.806555
$279$	−17.2976	−1.03558
$280$	3.22579	0.192778
$281$	20.9247	1.24826	0.624131	−	0.781320i	$-0.285453\pi$
$281$	20.9247	1.24826	0.624131	+	0.781320i	$0.285453\pi$
$282$	13.2747	0.790498
$283$	−20.7712	−1.23472	−0.617360	−	0.786680i	$-0.711798\pi$
$283$	−20.7712	−1.23472	−0.617360	+	0.786680i	$0.711798\pi$
$284$	−35.5787	−2.11121
$285$	−2.65214	−0.157099
$286$	−68.5122	−4.05121
$287$	−17.9604	−1.06017
$288$	15.8676	0.935006
$289$	−14.0713	−0.827722
$290$	8.15640	0.478960
$291$	−15.9695	−0.936148
$292$	−0.785200	−0.0459503
$293$	20.2046	1.18036	0.590182	−	0.807270i	$-0.299056\pi$
$293$	20.2046	1.18036	0.590182	+	0.807270i	$0.299056\pi$
$294$	36.5184	2.12980
$295$	−7.93048	−0.461730
$296$	−32.5961	−1.89461
$297$	52.7423	3.06042
$298$	19.9337	1.15473
$299$	37.8268	2.18758
$300$	−55.2668	−3.19083
$301$	3.36934	0.194206
$302$	5.69426	0.327668
$303$	−19.9891	−1.14835
$304$	−4.00560	−0.229737
$305$	6.88730	0.394366
$306$	−27.9062	−1.59529
$307$	−32.6732	−1.86476	−0.932379	−	0.361483i	$-0.882270\pi$
$307$	−32.6732	−1.86476	−0.932379	+	0.361483i	$0.882270\pi$
$308$	−24.1652	−1.37694
$309$	4.84237	0.275473
$310$	3.22678	0.183269
$311$	−16.3018	−0.924393	−0.462197	−	0.886778i	$-0.652939\pi$
$311$	−16.3018	−0.924393	−0.462197	+	0.886778i	$0.652939\pi$
$312$	−84.3566	−4.77575
$313$	−17.3529	−0.980843	−0.490422	−	0.871485i	$-0.663157\pi$
$313$	−17.3529	−0.980843	−0.490422	+	0.871485i	$0.663157\pi$
$314$	47.3727	2.67340
$315$	−5.26743	−0.296786
$316$	−20.9739	−1.17988
$317$	−14.9627	−0.840390	−0.420195	−	0.907434i	$-0.638038\pi$
$317$	−14.9627	−0.840390	−0.420195	+	0.907434i	$0.638038\pi$
$318$	5.97134	0.334856
$319$	−28.4253	−1.59151
$320$	−5.61676	−0.313986
$321$	53.7107	2.99784
$322$	20.4772	1.14115
$323$	−2.73486	−0.152172
$324$	63.2287	3.51271
$325$	−30.4986	−1.69176
$326$	37.2702	2.06421
$327$	47.3733	2.61975
$328$	−51.2655	−2.83066
$329$	2.58383	0.142451
$330$	−17.5934	−0.968487
$331$	25.2940	1.39028	0.695141	−	0.718874i	$-0.255342\pi$
$331$	25.2940	1.39028	0.695141	+	0.718874i	$0.255342\pi$
$332$	−24.7373	−1.35764
$333$	53.2265	2.91680
$334$	−15.7268	−0.860533
$335$	6.87711	0.375737
$336$	−11.4621	−0.625309
$337$	22.1697	1.20766	0.603831	−	0.797112i	$-0.293640\pi$
$337$	22.1697	1.20766	0.603831	+	0.797112i	$0.293640\pi$
$338$	−68.9199	−3.74875
$339$	−19.5307	−1.06076
$340$	3.39184	0.183949
$341$	−11.2454	−0.608973
$342$	26.0589	1.40910
$343$	17.3301	0.935736
$344$	9.61733	0.518531
$345$	9.71366	0.522966
$346$	19.8001	1.06446
$347$	−25.0865	−1.34671	−0.673357	−	0.739318i	$-0.735148\pi$
$347$	−25.0865	−1.34671	−0.673357	+	0.739318i	$0.735148\pi$
$348$	−75.2323	−4.03288
$349$	19.9923	1.07017	0.535083	−	0.844800i	$-0.320281\pi$
$349$	19.9923	1.07017	0.535083	+	0.844800i	$0.320281\pi$
$350$	−16.5101	−0.882504
$351$	77.0326	4.11170
$352$	10.3157	0.549831
$353$	−2.44246	−0.129999	−0.0649995	−	0.997885i	$-0.520705\pi$
$353$	−2.44246	−0.129999	−0.0649995	+	0.997885i	$0.520705\pi$
$354$	112.267	5.96694
$355$	−5.04183	−0.267593
$356$	−37.3643	−1.98030
$357$	−7.82586	−0.414188
$358$	47.7093	2.52151
$359$	−13.4527	−0.710007	−0.355003	−	0.934865i	$-0.615520\pi$
$359$	−13.4527	−0.710007	−0.355003	+	0.934865i	$0.615520\pi$
$360$	−15.0352	−0.792423
$361$	−16.4462	−0.865588
$362$	−47.6380	−2.50380
$363$	26.8671	1.41016
$364$	−35.2944	−1.84993
$365$	−0.111270	−0.00582415
$366$	−97.4996	−5.09639
$367$	7.25487	0.378701	0.189351	−	0.981910i	$-0.439362\pi$
$367$	7.25487	0.378701	0.189351	+	0.981910i	$0.439362\pi$
$368$	14.6708	0.764770
$369$	83.7121	4.35788
$370$	−9.92915	−0.516192
$371$	1.16228	0.0603424
$372$	−29.7629	−1.54313
$373$	1.38444	0.0716837	0.0358419	−	0.999357i	$-0.488589\pi$
$373$	1.38444	0.0716837	0.0358419	+	0.999357i	$0.488589\pi$
$374$	−18.1422	−0.938111
$375$	−16.1298	−0.832937
$376$	7.37518	0.380346
$377$	−41.5164	−2.13820
$378$	41.7009	2.14486
$379$	26.1572	1.34360	0.671802	−	0.740730i	$-0.265520\pi$
$379$	26.1572	1.34360	0.671802	+	0.740730i	$0.265520\pi$
$380$	−3.16732	−0.162480
$381$	−21.7259	−1.11305
$382$	−12.7870	−0.654241
$383$	−10.8836	−0.556125	−0.278063	−	0.960563i	$-0.589692\pi$
$383$	−10.8836	−0.556125	−0.278063	+	0.960563i	$0.589692\pi$
$384$	64.9124	3.31254
$385$	−3.42444	−0.174525
$386$	−33.9316	−1.72707
$387$	−15.7043	−0.798292
$388$	−19.0716	−0.968213
$389$	34.1828	1.73314	0.866569	−	0.499057i	$-0.166320\pi$
$389$	34.1828	1.73314	0.866569	+	0.499057i	$0.166320\pi$
$390$	−25.6960	−1.30117
$391$	10.0166	0.506564
$392$	20.2889	1.02475
$393$	−40.0688	−2.02120
$394$	21.2956	1.07286
$395$	−2.97220	−0.149548
$396$	112.632	5.65998
$397$	24.3851	1.22386	0.611928	−	0.790914i	$-0.290394\pi$
$397$	24.3851	1.22386	0.611928	+	0.790914i	$0.290394\pi$
$398$	−45.7222	−2.29185
$399$	7.30782	0.365849
$400$	−11.8286	−0.591431
$401$	10.9896	0.548796	0.274398	−	0.961616i	$-0.411521\pi$
$401$	10.9896	0.548796	0.274398	+	0.961616i	$0.411521\pi$
$402$	−97.3554	−4.85565
$403$	−16.4244	−0.818160
$404$	−23.8721	−1.18768
$405$	8.96010	0.445231
$406$	−22.4745	−1.11539
$407$	34.6034	1.71523
$408$	−22.3378	−1.10589
$409$	10.3479	0.511670	0.255835	−	0.966720i	$-0.417650\pi$
$409$	10.3479	0.511670	0.255835	+	0.966720i	$0.417650\pi$
$410$	−15.6161	−0.771224
$411$	−17.7909	−0.877560
$412$	5.78300	0.284908
$413$	21.8520	1.07527
$414$	−95.4428	−4.69076
$415$	−3.50551	−0.172079
$416$	15.0666	0.738702
$417$	−17.5776	−0.860781
$418$	16.9413	0.828625
$419$	20.5506	1.00396	0.501982	−	0.864878i	$-0.332604\pi$
$419$	20.5506	1.00396	0.501982	+	0.864878i	$0.332604\pi$
$420$	−9.06335	−0.442246
$421$	30.2571	1.47464	0.737320	−	0.675544i	$-0.236091\pi$
$421$	30.2571	1.47464	0.737320	+	0.675544i	$0.236091\pi$
$422$	−24.0472	−1.17060
$423$	−12.0430	−0.585552
$424$	3.31756	0.161115
$425$	−8.07611	−0.391749
$426$	71.3744	3.45810
$427$	−18.9776	−0.918390
$428$	64.1441	3.10052
$429$	89.5514	4.32358
$430$	2.92955	0.141276
$431$	−31.5229	−1.51841	−0.759203	−	0.650854i	$-0.774411\pi$
$431$	−31.5229	−1.51841	−0.759203	+	0.650854i	$0.774411\pi$
$432$	29.8765	1.43743
$433$	−10.6414	−0.511393	−0.255696	−	0.966757i	$-0.582305\pi$
$433$	−10.6414	−0.511393	−0.255696	+	0.966757i	$0.582305\pi$
$434$	−8.89122	−0.426792
$435$	−10.6611	−0.511162
$436$	56.5757	2.70948
$437$	−9.35359	−0.447443
$438$	1.57519	0.0752654
$439$	3.21336	0.153365	0.0766827	−	0.997056i	$-0.475567\pi$
$439$	3.21336	0.153365	0.0766827	+	0.997056i	$0.475567\pi$
$440$	−9.77458	−0.465985
$441$	−33.1301	−1.57762
$442$	−26.4975	−1.26036
$443$	−9.92140	−0.471380	−0.235690	−	0.971828i	$-0.575735\pi$
$443$	−9.92140	−0.471380	−0.235690	+	0.971828i	$0.575735\pi$
$444$	91.5836	4.34637
$445$	−5.29487	−0.251001
$446$	3.68304	0.174397
$447$	−26.0551	−1.23236
$448$	15.4767	0.731205
$449$	−17.5137	−0.826525	−0.413262	−	0.910612i	$-0.635611\pi$
$449$	−17.5137	−0.826525	−0.413262	+	0.910612i	$0.635611\pi$
$450$	76.9525	3.62758
$451$	54.4225	2.56266
$452$	−23.3246	−1.09710
$453$	−7.44289	−0.349697
$454$	−60.0065	−2.81624
$455$	−5.00154	−0.234476
$456$	20.8592	0.976821
$457$	−12.7970	−0.598618	−0.299309	−	0.954156i	$-0.596756\pi$
$457$	−12.7970	−0.598618	−0.299309	+	0.954156i	$0.596756\pi$
$458$	24.0401	1.12332
$459$	20.3984	0.952117
$460$	11.6006	0.540879
$461$	36.4427	1.69731	0.848653	−	0.528951i	$-0.177414\pi$
$461$	36.4427	1.69731	0.848653	+	0.528951i	$0.177414\pi$
$462$	48.4778	2.25539
$463$	3.63225	0.168805	0.0844025	−	0.996432i	$-0.473102\pi$
$463$	3.63225	0.168805	0.0844025	+	0.996432i	$0.473102\pi$
$464$	−16.1018	−0.747508
$465$	−4.21768	−0.195590
$466$	17.9440	0.831239
$467$	−28.4290	−1.31554	−0.657768	−	0.753221i	$-0.728499\pi$
$467$	−28.4290	−1.31554	−0.657768	+	0.753221i	$0.728499\pi$
$468$	164.505	7.60423
$469$	−18.9495	−0.875008
$470$	2.24657	0.103627
$471$	−61.9203	−2.85313
$472$	62.3736	2.87098
$473$	−10.2096	−0.469437
$474$	42.0758	1.93261
$475$	7.54151	0.346028
$476$	−9.34604	−0.428375
$477$	−5.41729	−0.248041
$478$	43.4872	1.98906
$479$	13.5961	0.621219	0.310610	−	0.950538i	$-0.399467\pi$
$479$	13.5961	0.621219	0.310610	+	0.950538i	$0.399467\pi$
$480$	3.86900	0.176595
$481$	50.5398	2.30442
$482$	−52.9119	−2.41007
$483$	−26.7655	−1.21787
$484$	32.0861	1.45846
$485$	−2.70262	−0.122720
$486$	−41.1734	−1.86766
$487$	−19.2412	−0.871904	−0.435952	−	0.899970i	$-0.643588\pi$
$487$	−19.2412	−0.871904	−0.435952	+	0.899970i	$0.643588\pi$
$488$	−54.1689	−2.45211
$489$	−48.7154	−2.20299
$490$	6.18025	0.279195
$491$	36.7971	1.66063	0.830315	−	0.557295i	$-0.188161\pi$
$491$	36.7971	1.66063	0.830315	+	0.557295i	$0.188161\pi$
$492$	144.038	6.49375
$493$	−10.9937	−0.495130
$494$	24.7435	1.11326
$495$	15.9610	0.717396
$496$	−6.37009	−0.286025
$497$	13.8925	0.623164
$498$	49.6255	2.22377
$499$	−22.3439	−1.00025	−0.500125	−	0.865953i	$-0.666713\pi$
$499$	−22.3439	−1.00025	−0.500125	+	0.865953i	$0.666713\pi$
$500$	−19.2630	−0.861468
$501$	20.5563	0.918388
$502$	−14.9908	−0.669071
$503$	28.7700	1.28279	0.641394	−	0.767211i	$-0.278356\pi$
$503$	28.7700	1.28279	0.641394	+	0.767211i	$0.278356\pi$
$504$	41.4286	1.84538
$505$	−3.38290	−0.150537
$506$	−62.0488	−2.75841
$507$	90.0843	4.00078
$508$	−25.9462	−1.15118
$509$	29.8458	1.32289	0.661446	−	0.749993i	$-0.269943\pi$
$509$	29.8458	1.32289	0.661446	+	0.749993i	$0.269943\pi$
$510$	−6.80438	−0.301303
$511$	0.306599	0.0135631
$512$	26.7389	1.18170
$513$	−19.0482	−0.840997
$514$	−5.17635	−0.228319
$515$	0.819506	0.0361117
$516$	−27.0214	−1.18955
$517$	−7.82935	−0.344335
$518$	27.3593	1.20210
$519$	−25.8805	−1.13603
$520$	−14.2762	−0.626054
$521$	−31.4999	−1.38004	−0.690019	−	0.723792i	$-0.742398\pi$
$521$	−31.4999	−1.38004	−0.690019	+	0.723792i	$0.742398\pi$
$522$	104.752	4.58488
$523$	4.27785	0.187057	0.0935286	−	0.995617i	$-0.470185\pi$
$523$	4.27785	0.187057	0.0935286	+	0.995617i	$0.470185\pi$
$524$	−47.8523	−2.09044
$525$	21.5802	0.941836
$526$	2.31398	0.100894
$527$	−4.34924	−0.189456
$528$	34.7318	1.51151
$529$	11.2583	0.489490
$530$	1.01057	0.0438963
$531$	−101.851	−4.41994
$532$	8.72738	0.378380
$533$	79.4865	3.44294
$534$	74.9565	3.24369
$535$	9.08982	0.392987
$536$	−54.0888	−2.33628
$537$	−62.3602	−2.69104
$538$	35.4255	1.52730
$539$	−21.5384	−0.927723
$540$	23.6240	1.01662
$541$	−6.94259	−0.298485	−0.149243	−	0.988801i	$-0.547684\pi$
$541$	−6.94259	−0.298485	−0.149243	+	0.988801i	$0.547684\pi$
$542$	−35.5194	−1.52569
$543$	62.2670	2.67213
$544$	3.98968	0.171056
$545$	8.01730	0.343424
$546$	70.8040	3.03013
$547$	−7.97016	−0.340779	−0.170390	−	0.985377i	$-0.554503\pi$
$547$	−7.97016	−0.340779	−0.170390	+	0.985377i	$0.554503\pi$
$548$	−21.2468	−0.907619
$549$	88.4532	3.77509
$550$	50.0280	2.13320
$551$	10.2659	0.437343
$552$	−76.3984	−3.25173
$553$	8.18975	0.348264
$554$	−30.6928	−1.30401
$555$	12.9783	0.550897
$556$	−20.9921	−0.890265
$557$	39.7466	1.68412	0.842060	−	0.539385i	$-0.181343\pi$
$557$	39.7466	1.68412	0.842060	+	0.539385i	$0.181343\pi$
$558$	41.4413	1.75435
$559$	−14.9115	−0.630691
$560$	−1.93981	−0.0819718
$561$	23.7134	1.00118
$562$	−50.1311	−2.11465
$563$	16.2836	0.686272	0.343136	−	0.939286i	$-0.388511\pi$
$563$	16.2836	0.686272	0.343136	+	0.939286i	$0.388511\pi$
$564$	−20.7217	−0.872541
$565$	−3.30531	−0.139056
$566$	49.7635	2.09171
$567$	−24.6891	−1.03684
$568$	39.6542	1.66385
$569$	−15.1540	−0.635287	−0.317644	−	0.948210i	$-0.602892\pi$
$569$	−15.1540	−0.635287	−0.317644	+	0.948210i	$0.602892\pi$
$570$	6.35396	0.266138
$571$	−40.6365	−1.70058	−0.850292	−	0.526312i	$-0.823575\pi$
$571$	−40.6365	−1.70058	−0.850292	+	0.526312i	$0.823575\pi$
$572$	106.947	4.47167
$573$	16.7137	0.698226
$574$	43.0293	1.79601
$575$	−27.6214	−1.15189
$576$	−72.1357	−3.00565
$577$	−20.3499	−0.847176	−0.423588	−	0.905855i	$-0.639230\pi$
$577$	−20.3499	−0.847176	−0.423588	+	0.905855i	$0.639230\pi$
$578$	33.7118	1.40223
$579$	44.3515	1.84319
$580$	−12.7321	−0.528670
$581$	9.65924	0.400733
$582$	38.2595	1.58591
$583$	−3.52186	−0.145861
$584$	0.875145	0.0362138
$585$	23.3118	0.963826
$586$	−48.4059	−1.99963
$587$	−34.2532	−1.41378	−0.706890	−	0.707324i	$-0.749902\pi$
$587$	−34.2532	−1.41378	−0.706890	+	0.707324i	$0.749902\pi$
$588$	−57.0049	−2.35084
$589$	4.06134	0.167345
$590$	18.9997	0.782207
$591$	−27.8352	−1.14499
$592$	19.6014	0.805614
$593$	−15.4467	−0.634319	−0.317160	−	0.948372i	$-0.602729\pi$
$593$	−15.4467	−0.634319	−0.317160	+	0.948372i	$0.602729\pi$
$594$	−126.360	−5.18459
$595$	−1.32442	−0.0542960
$596$	−31.1164	−1.27458
$597$	59.7629	2.44593
$598$	−90.6251	−3.70594
$599$	12.2812	0.501796	0.250898	−	0.968013i	$-0.419274\pi$
$599$	12.2812	0.501796	0.250898	+	0.968013i	$0.419274\pi$
$600$	61.5977	2.51471
$601$	6.24490	0.254735	0.127367	−	0.991856i	$-0.459347\pi$
$601$	6.24490	0.254735	0.127367	+	0.991856i	$0.459347\pi$
$602$	−8.07223	−0.329000
$603$	88.3224	3.59677
$604$	−8.88868	−0.361675
$605$	4.54690	0.184858
$606$	47.8898	1.94539
$607$	−34.3688	−1.39499	−0.697493	−	0.716592i	$-0.745701\pi$
$607$	−34.3688	−1.39499	−0.697493	+	0.716592i	$0.745701\pi$
$608$	−3.72558	−0.151092
$609$	29.3762	1.19038
$610$	−16.5005	−0.668086
$611$	−11.4351	−0.462616
$612$	43.5612	1.76086
$613$	−42.5856	−1.72002	−0.860009	−	0.510279i	$-0.829542\pi$
$613$	−42.5856	−1.72002	−0.860009	+	0.510279i	$0.829542\pi$
$614$	78.2781	3.15905
$615$	20.4116	0.823074
$616$	26.9333	1.08518
$617$	30.2193	1.21658	0.608292	−	0.793713i	$-0.291855\pi$
$617$	30.2193	1.21658	0.608292	+	0.793713i	$0.291855\pi$
$618$	−11.6013	−0.466672
$619$	39.6495	1.59365	0.796824	−	0.604211i	$-0.206512\pi$
$619$	39.6495	1.59365	0.796824	+	0.604211i	$0.206512\pi$
$620$	−5.03697	−0.202290
$621$	69.7654	2.79959
$622$	39.0558	1.56599
$623$	14.5897	0.584526
$624$	50.7273	2.03072
$625$	20.8659	0.834637
$626$	41.5739	1.66163
$627$	−22.1437	−0.884335
$628$	−73.9484	−2.95086
$629$	13.3831	0.533618
$630$	12.6196	0.502779
$631$	−26.1083	−1.03935	−0.519677	−	0.854363i	$-0.673948\pi$
$631$	−26.1083	−1.03935	−0.519677	+	0.854363i	$0.673948\pi$
$632$	23.3765	0.929868
$633$	31.4317	1.24930
$634$	35.8475	1.42369
$635$	−3.67682	−0.145910
$636$	−9.32120	−0.369610
$637$	−31.4577	−1.24640
$638$	68.1010	2.69614
$639$	−64.7519	−2.56155
$640$	10.9856	0.434242
$641$	−49.0900	−1.93894	−0.969470	−	0.245210i	$-0.921143\pi$
$641$	−49.0900	−1.93894	−0.969470	+	0.245210i	$0.921143\pi$
$642$	−128.679	−5.07857
$643$	−6.00524	−0.236824	−0.118412	−	0.992965i	$-0.537780\pi$
$643$	−6.00524	−0.236824	−0.118412	+	0.992965i	$0.537780\pi$
$644$	−31.9647	−1.25959
$645$	−3.82918	−0.150774
$646$	6.55215	0.257791
$647$	33.1462	1.30311	0.651555	−	0.758601i	$-0.274117\pi$
$647$	33.1462	1.30311	0.651555	+	0.758601i	$0.274117\pi$
$648$	−70.4716	−2.76839
$649$	−66.2146	−2.59915
$650$	73.0682	2.86597
$651$	11.6216	0.455486
$652$	−58.1785	−2.27845
$653$	−2.85218	−0.111614	−0.0558071	−	0.998442i	$-0.517773\pi$
$653$	−2.85218	−0.111614	−0.0558071	+	0.998442i	$0.517773\pi$
$654$	−113.496	−4.43806
$655$	−6.78111	−0.264960
$656$	30.8282	1.20364
$657$	−1.42904	−0.0557520
$658$	−6.19030	−0.241323
$659$	−30.2362	−1.17784	−0.588918	−	0.808193i	$-0.700446\pi$
$659$	−30.2362	−1.17784	−0.588918	+	0.808193i	$0.700446\pi$
$660$	27.4632	1.06900
$661$	45.7913	1.78107	0.890537	−	0.454910i	$-0.150329\pi$
$661$	45.7913	1.78107	0.890537	+	0.454910i	$0.150329\pi$
$662$	−60.5990	−2.35525
$663$	34.6346	1.34509
$664$	27.5710	1.06996
$665$	1.23675	0.0479592
$666$	−127.520	−4.94128
$667$	−37.5998	−1.45587
$668$	24.5494	0.949845
$669$	−4.81405	−0.186122
$670$	−16.4761	−0.636528
$671$	57.5047	2.21995
$672$	−10.6608	−0.411250
$673$	14.6444	0.564502	0.282251	−	0.959341i	$-0.408919\pi$
$673$	14.6444	0.564502	0.282251	+	0.959341i	$0.408919\pi$
$674$	−53.1140	−2.04587
$675$	−56.2497	−2.16505
$676$	107.583	4.13782
$677$	−41.9309	−1.61154	−0.805768	−	0.592232i	$-0.798247\pi$
$677$	−41.9309	−1.61154	−0.805768	+	0.592232i	$0.798247\pi$
$678$	46.7915	1.79702
$679$	7.44693	0.285787
$680$	−3.78038	−0.144971
$681$	78.4337	3.00559
$682$	26.9416	1.03165
$683$	16.6061	0.635414	0.317707	−	0.948189i	$-0.397087\pi$
$683$	16.6061	0.635414	0.317707	+	0.948189i	$0.397087\pi$
$684$	−40.6777	−1.55535
$685$	−3.01087	−0.115040
$686$	−41.5192	−1.58521
$687$	−31.4225	−1.19884
$688$	−5.78332	−0.220487
$689$	−5.14384	−0.195965
$690$	−23.2719	−0.885946
$691$	13.1873	0.501669	0.250834	−	0.968030i	$-0.419295\pi$
$691$	13.1873	0.501669	0.250834	+	0.968030i	$0.419295\pi$
$692$	−30.9078	−1.17494
$693$	−43.9798	−1.67066
$694$	60.1019	2.28144
$695$	−2.97478	−0.112840
$696$	83.8503	3.17834
$697$	21.0482	0.797259
$698$	−47.8974	−1.81294
$699$	−23.4543	−0.887124
$700$	25.7722	0.974096
$701$	−18.5417	−0.700311	−0.350155	−	0.936692i	$-0.613871\pi$
$701$	−18.5417	−0.700311	−0.350155	+	0.936692i	$0.613871\pi$
$702$	−184.554	−6.96554
$703$	−12.4972	−0.471340
$704$	−46.8965	−1.76748
$705$	−2.93646	−0.110593
$706$	5.85162	0.220229
$707$	9.32139	0.350567
$708$	−175.248	−6.58623
$709$	21.4816	0.806757	0.403378	−	0.915033i	$-0.367836\pi$
$709$	21.4816	0.806757	0.403378	+	0.915033i	$0.367836\pi$
$710$	12.0792	0.453323
$711$	−38.1719	−1.43156
$712$	41.6444	1.56069
$713$	−14.8750	−0.557072
$714$	18.7491	0.701667
$715$	15.1554	0.566779
$716$	−74.4737	−2.78321
$717$	−56.8415	−2.12279
$718$	32.2298	1.20281
$719$	25.9331	0.967140	0.483570	−	0.875306i	$-0.339340\pi$
$719$	25.9331	0.967140	0.483570	+	0.875306i	$0.339340\pi$
$720$	9.04131	0.336950
$721$	−2.25810	−0.0840962
$722$	39.4016	1.46637
$723$	69.1605	2.57211
$724$	74.3624	2.76366
$725$	30.3155	1.12589
$726$	−64.3679	−2.38892
$727$	17.4509	0.647217	0.323608	−	0.946191i	$-0.395104\pi$
$727$	17.4509	0.647217	0.323608	+	0.946191i	$0.395104\pi$
$728$	39.3374	1.45794
$729$	3.09635	0.114680
$730$	0.266580	0.00986656
$731$	−3.94862	−0.146045
$732$	152.196	5.62533
$733$	−33.3952	−1.23348	−0.616740	−	0.787167i	$-0.711547\pi$
$733$	−33.3952	−1.23348	−0.616740	+	0.787167i	$0.711547\pi$
$734$	−17.3811	−0.641550
$735$	−8.07813	−0.297966
$736$	13.6452	0.502970
$737$	57.4197	2.11508
$738$	−200.557	−7.38259
$739$	50.8204	1.86946	0.934730	−	0.355359i	$-0.115641\pi$
$739$	50.8204	1.86946	0.934730	+	0.355359i	$0.115641\pi$
$740$	15.4993	0.569766
$741$	−32.3419	−1.18811
$742$	−2.78457	−0.102225
$743$	−4.00185	−0.146814	−0.0734068	−	0.997302i	$-0.523387\pi$
$743$	−4.00185	−0.146814	−0.0734068	+	0.997302i	$0.523387\pi$
$744$	33.1723	1.21615
$745$	−4.40948	−0.161551
$746$	−3.31683	−0.121438
$747$	−45.0210	−1.64723
$748$	28.3198	1.03548
$749$	−25.0465	−0.915180
$750$	38.6435	1.41106
$751$	49.9720	1.82350	0.911752	−	0.410742i	$-0.134730\pi$
$751$	49.9720	1.82350	0.911752	+	0.410742i	$0.134730\pi$
$752$	−4.43502	−0.161729
$753$	19.5942	0.714054
$754$	99.4646	3.62229
$755$	−1.25961	−0.0458419
$756$	−65.0947	−2.36747
$757$	−52.0610	−1.89219	−0.946094	−	0.323891i	$-0.895009\pi$
$757$	−52.0610	−1.89219	−0.946094	+	0.323891i	$0.895009\pi$
$758$	−62.6671	−2.27617
$759$	81.1032	2.94386
$760$	3.53014	0.128052
$761$	−13.7063	−0.496855	−0.248427	−	0.968651i	$-0.579914\pi$
$761$	−13.7063	−0.496855	−0.248427	+	0.968651i	$0.579914\pi$
$762$	52.0507	1.88560
$763$	−22.0913	−0.799757
$764$	19.9604	0.722142
$765$	6.17304	0.223187
$766$	26.0748	0.942120
$767$	−96.7095	−3.49198
$768$	−89.1387	−3.21652
$769$	4.44679	0.160355	0.0801777	−	0.996781i	$-0.474451\pi$
$769$	4.44679	0.160355	0.0801777	+	0.996781i	$0.474451\pi$
$770$	8.20422	0.295660
$771$	6.76594	0.243669
$772$	52.9669	1.90632
$773$	11.0974	0.399144	0.199572	−	0.979883i	$-0.436045\pi$
$773$	11.0974	0.399144	0.199572	+	0.979883i	$0.436045\pi$
$774$	37.6241	1.35237
$775$	11.9932	0.430809
$776$	21.2563	0.763055
$777$	−35.7609	−1.28292
$778$	−81.8948	−2.93607
$779$	−19.6550	−0.704212
$780$	40.1113	1.43621
$781$	−42.0962	−1.50632
$782$	−23.9978	−0.858158
$783$	−76.5703	−2.73640
$784$	−12.2006	−0.435737
$785$	−10.4792	−0.374018
$786$	95.9964	3.42408
$787$	−1.88559	−0.0672139	−0.0336069	−	0.999435i	$-0.510699\pi$
$787$	−1.88559	−0.0672139	−0.0336069	+	0.999435i	$0.510699\pi$
$788$	−33.2422	−1.18420
$789$	−3.02457	−0.107678
$790$	7.12077	0.253346
$791$	9.10762	0.323830
$792$	−125.534	−4.46067
$793$	83.9883	2.98251
$794$	−58.4217	−2.07331
$795$	−1.32090	−0.0468475
$796$	71.3719	2.52971
$797$	5.85623	0.207438	0.103719	−	0.994607i	$-0.466926\pi$
$797$	5.85623	0.207438	0.103719	+	0.994607i	$0.466926\pi$
$798$	−17.5080	−0.619776
$799$	−3.02805	−0.107125
$800$	−11.0017	−0.388970
$801$	−68.0017	−2.40272
$802$	−26.3288	−0.929704
$803$	−0.929038	−0.0327850
$804$	151.971	5.35960
$805$	−4.52970	−0.159651
$806$	39.3495	1.38603
$807$	−46.3041	−1.62998
$808$	26.6066	0.936019
$809$	−7.49863	−0.263638	−0.131819	−	0.991274i	$-0.542082\pi$
$809$	−7.49863	−0.263638	−0.131819	+	0.991274i	$0.542082\pi$
$810$	−21.4665	−0.754256
$811$	−31.3364	−1.10037	−0.550185	−	0.835043i	$-0.685443\pi$
$811$	−31.3364	−1.10037	−0.550185	+	0.835043i	$0.685443\pi$
$812$	35.0825	1.23116
$813$	46.4270	1.62826
$814$	−82.9024	−2.90573
$815$	−8.24444	−0.288790
$816$	13.4327	0.470240
$817$	3.68724	0.129000
$818$	−24.7914	−0.866810
$819$	−64.2345	−2.24454
$820$	24.3766	0.851267
$821$	−44.4557	−1.55152	−0.775758	−	0.631031i	$-0.782632\pi$
$821$	−44.4557	−1.55152	−0.775758	+	0.631031i	$0.782632\pi$
$822$	42.6232	1.48666
$823$	48.7892	1.70068	0.850342	−	0.526231i	$-0.176395\pi$
$823$	48.7892	1.70068	0.850342	+	0.526231i	$0.176395\pi$
$824$	−6.44545	−0.224538
$825$	−65.3909	−2.27662
$826$	−52.3528	−1.82159
$827$	−7.85896	−0.273283	−0.136641	−	0.990621i	$-0.543631\pi$
$827$	−7.85896	−0.273283	−0.136641	+	0.990621i	$0.543631\pi$
$828$	148.985	5.17760
$829$	−12.8270	−0.445500	−0.222750	−	0.974876i	$-0.571503\pi$
$829$	−12.8270	−0.445500	−0.222750	+	0.974876i	$0.571503\pi$
$830$	8.39846	0.291515
$831$	40.1182	1.39168
$832$	−68.4945	−2.37462
$833$	−8.33009	−0.288621
$834$	42.1123	1.45823
$835$	3.47888	0.120392
$836$	−26.4452	−0.914626
$837$	−30.2922	−1.04705
$838$	−49.2350	−1.70079
$839$	19.3220	0.667070	0.333535	−	0.942738i	$-0.391758\pi$
$839$	19.3220	0.667070	0.333535	+	0.942738i	$0.391758\pi$
$840$	10.1016	0.348537
$841$	12.2672	0.423009
$842$	−72.4896	−2.49816
$843$	65.5257	2.25683
$844$	37.5374	1.29209
$845$	15.2456	0.524464
$846$	28.8526	0.991971
$847$	−12.5287	−0.430493
$848$	−1.99500	−0.0685084
$849$	−65.0451	−2.23234
$850$	19.3486	0.663653
$851$	45.7719	1.56904
$852$	−111.415	−3.81700
$853$	44.8801	1.53667	0.768333	−	0.640050i	$-0.221086\pi$
$853$	44.8801	1.53667	0.768333	+	0.640050i	$0.221086\pi$
$854$	45.4663	1.55582
$855$	−5.76441	−0.197139
$856$	−71.4919	−2.44354
$857$	−11.4945	−0.392645	−0.196323	−	0.980539i	$-0.562900\pi$
$857$	−11.4945	−0.392645	−0.196323	+	0.980539i	$0.562900\pi$
$858$	−214.546	−7.32448
$859$	−12.2624	−0.418388	−0.209194	−	0.977874i	$-0.567084\pi$
$859$	−12.2624	−0.418388	−0.209194	+	0.977874i	$0.567084\pi$
$860$	−4.57301	−0.155938
$861$	−56.2430	−1.91676
$862$	75.5223	2.57230
$863$	14.4831	0.493010	0.246505	−	0.969142i	$-0.420718\pi$
$863$	14.4831	0.493010	0.246505	+	0.969142i	$0.420718\pi$
$864$	27.7879	0.945364
$865$	−4.37992	−0.148922
$866$	25.4945	0.866339
$867$	−44.0642	−1.49650
$868$	13.8791	0.471088
$869$	−24.8161	−0.841828
$870$	25.5418	0.865948
$871$	83.8641	2.84163
$872$	−63.0565	−2.13536
$873$	−34.7096	−1.17474
$874$	22.4092	0.758004
$875$	7.52168	0.254279
$876$	−2.45885	−0.0830770
$877$	−4.57721	−0.154561	−0.0772807	−	0.997009i	$-0.524624\pi$
$877$	−4.57721	−0.154561	−0.0772807	+	0.997009i	$0.524624\pi$
$878$	−7.69854	−0.259813
$879$	63.2707	2.13407
$880$	5.87789	0.198144
$881$	−22.1840	−0.747397	−0.373698	−	0.927550i	$-0.621910\pi$
$881$	−22.1840	−0.747397	−0.373698	+	0.927550i	$0.621910\pi$
$882$	79.3726	2.67262
$883$	17.0892	0.575098	0.287549	−	0.957766i	$-0.407160\pi$
$883$	17.0892	0.575098	0.287549	+	0.957766i	$0.407160\pi$
$884$	41.3624	1.39117
$885$	−24.8343	−0.834796
$886$	23.7696	0.798555
$887$	−5.99375	−0.201250	−0.100625	−	0.994924i	$-0.532084\pi$
$887$	−5.99375	−0.201250	−0.100625	+	0.994924i	$0.532084\pi$
$888$	−102.075	−3.42540
$889$	10.1313	0.339793
$890$	12.6854	0.425215
$891$	74.8114	2.50627
$892$	−5.74918	−0.192497
$893$	2.82761	0.0946224
$894$	62.4226	2.08772
$895$	−10.5536	−0.352769
$896$	−30.2701	−1.01125
$897$	118.455	3.95509
$898$	41.9592	1.40020
$899$	16.3259	0.544498
$900$	−120.122	−4.00407
$901$	−1.36210	−0.0453782
$902$	−130.385	−4.34134
$903$	10.5511	0.351119
$904$	25.9965	0.864629
$905$	10.5379	0.350290
$906$	17.8316	0.592415
$907$	−22.8978	−0.760307	−0.380154	−	0.924923i	$-0.624129\pi$
$907$	−22.8978	−0.760307	−0.380154	+	0.924923i	$0.624129\pi$
$908$	93.6696	3.10853
$909$	−43.4464	−1.44102
$910$	11.9826	0.397221
$911$	−42.6176	−1.41199	−0.705993	−	0.708219i	$-0.749499\pi$
$911$	−42.6176	−1.41199	−0.705993	+	0.708219i	$0.749499\pi$
$912$	−12.5435	−0.415358
$913$	−29.2689	−0.968658
$914$	30.6589	1.01411
$915$	21.5676	0.713003
$916$	−37.5264	−1.23991
$917$	18.6850	0.617033
$918$	−48.8703	−1.61296
$919$	−1.50308	−0.0495821	−0.0247910	−	0.999693i	$-0.507892\pi$
$919$	−1.50308	−0.0495821	−0.0247910	+	0.999693i	$0.507892\pi$
$920$	−12.9294	−0.426270
$921$	−102.316	−3.37143
$922$	−87.3090	−2.87537
$923$	−61.4834	−2.02375
$924$	−75.6734	−2.48947
$925$	−36.9045	−1.21341
$926$	−8.70211	−0.285969
$927$	10.5249	0.345682
$928$	−14.9762	−0.491617
$929$	−28.0069	−0.918878	−0.459439	−	0.888209i	$-0.651950\pi$
$929$	−28.0069	−0.918878	−0.459439	+	0.888209i	$0.651950\pi$
$930$	10.1047	0.331345
$931$	7.77868	0.254936
$932$	−28.0104	−0.917511
$933$	−51.0493	−1.67128
$934$	68.1098	2.22862
$935$	4.01318	0.131245
$936$	−183.349	−5.99294
$937$	−17.3071	−0.565397	−0.282699	−	0.959209i	$-0.591230\pi$
$937$	−17.3071	−0.565397	−0.282699	+	0.959209i	$0.591230\pi$
$938$	45.3991	1.48233
$939$	−54.3406	−1.77334
$940$	−3.50687	−0.114382
$941$	0.476801	0.0155433	0.00777163	−	0.999970i	$-0.497526\pi$
$941$	0.476801	0.0155433	0.00777163	+	0.999970i	$0.497526\pi$
$942$	148.348	4.83343
$943$	71.9878	2.34425
$944$	−37.5080	−1.22078
$945$	−9.22453	−0.300074
$946$	24.4600	0.795263
$947$	23.5709	0.765950	0.382975	−	0.923759i	$-0.374899\pi$
$947$	23.5709	0.765950	0.382975	+	0.923759i	$0.374899\pi$
$948$	−65.6800	−2.13319
$949$	−1.35690	−0.0440469
$950$	−18.0679	−0.586199
$951$	−46.8558	−1.51940
$952$	10.4166	0.337605
$953$	−0.209171	−0.00677570	−0.00338785	−	0.999994i	$-0.501078\pi$
$953$	−0.209171	−0.00677570	−0.00338785	+	0.999994i	$0.501078\pi$
$954$	12.9787	0.420200
$955$	2.82858	0.0915306
$956$	−67.8831	−2.19550
$957$	−89.0139	−2.87741
$958$	−32.5733	−1.05239
$959$	8.29630	0.267901
$960$	−17.5889	−0.567679
$961$	−24.5413	−0.791654
$962$	−121.083	−3.90386
$963$	116.740	3.76189
$964$	82.5950	2.66021
$965$	7.50590	0.241623
$966$	64.1244	2.06317
$967$	−40.5567	−1.30421	−0.652107	−	0.758127i	$-0.726115\pi$
$967$	−40.5567	−1.30421	−0.652107	+	0.758127i	$0.726115\pi$
$968$	−35.7616	−1.14942
$969$	−8.56423	−0.275123
$970$	6.47491	0.207897
$971$	−26.1376	−0.838795	−0.419397	−	0.907803i	$-0.637759\pi$
$971$	−26.1376	−0.838795	−0.419397	+	0.907803i	$0.637759\pi$
$972$	64.2713	2.06150
$973$	8.19686	0.262779
$974$	46.0980	1.47707
$975$	−95.5064	−3.05865
$976$	32.5742	1.04267
$977$	26.7372	0.855400	0.427700	−	0.903921i	$-0.359324\pi$
$977$	26.7372	0.855400	0.427700	+	0.903921i	$0.359324\pi$
$978$	116.712	3.73203
$979$	−44.2090	−1.41292
$980$	−9.64732	−0.308172
$981$	102.966	3.28744
$982$	−88.1580	−2.81324
$983$	−9.80023	−0.312579	−0.156289	−	0.987711i	$-0.549953\pi$
$983$	−9.80023	−0.312579	−0.156289	+	0.987711i	$0.549953\pi$
$984$	−160.538	−5.11777
$985$	−4.71073	−0.150096
$986$	26.3385	0.838788
$987$	8.09126	0.257548
$988$	−38.6244	−1.22881
$989$	−13.5048	−0.429428
$990$	−38.2393	−1.21532
$991$	29.8744	0.948991	0.474495	−	0.880258i	$-0.342631\pi$
$991$	29.8744	0.948991	0.474495	+	0.880258i	$0.342631\pi$
$992$	−5.92478	−0.188112
$993$	79.2081	2.51359
$994$	−33.2835	−1.05569
$995$	10.1141	0.320638
$996$	−77.4650	−2.45457
$997$	23.8507	0.755359	0.377680	−	0.925936i	$-0.376722\pi$
$997$	23.8507	0.755359	0.377680	+	0.925936i	$0.376722\pi$
$998$	53.5312	1.69450
$999$	93.2124	2.94911

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4019.2.a.b.1.19	✓	186

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4019.2.a.b.1.19	✓	186	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:	\( N \)	\(=\)	\( 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4019.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39579 q^{2} +3.13150 q^{3} +3.73980 q^{4} +0.529965 q^{5} -7.50242 q^{6} -1.46029 q^{7} -4.16820 q^{8} +6.80631 q^{9} +O(q^{10})\)	\(q-2.39579 q^{2} +3.13150 q^{3} +3.73980 q^{4} +0.529965 q^{5} -7.50242 q^{6} -1.46029 q^{7} -4.16820 q^{8} +6.80631 q^{9} -1.26968 q^{10} +4.42489 q^{11} +11.7112 q^{12} +6.46275 q^{13} +3.49855 q^{14} +1.65959 q^{15} +2.50652 q^{16} +1.71135 q^{17} -16.3065 q^{18} -1.59807 q^{19} +1.98197 q^{20} -4.57291 q^{21} -10.6011 q^{22} +5.85306 q^{23} -13.0527 q^{24} -4.71914 q^{25} -15.4834 q^{26} +11.9195 q^{27} -5.46120 q^{28} -6.42396 q^{29} -3.97602 q^{30} -2.54140 q^{31} +2.33130 q^{32} +13.8565 q^{33} -4.10004 q^{34} -0.773904 q^{35} +25.4543 q^{36} +7.82017 q^{37} +3.82864 q^{38} +20.2381 q^{39} -2.20900 q^{40} +12.2992 q^{41} +10.9557 q^{42} -2.30731 q^{43} +16.5482 q^{44} +3.60711 q^{45} -14.0227 q^{46} -1.76939 q^{47} +7.84919 q^{48} -4.86755 q^{49} +11.3061 q^{50} +5.35911 q^{51} +24.1694 q^{52} -0.795921 q^{53} -28.5566 q^{54} +2.34504 q^{55} +6.08679 q^{56} -5.00436 q^{57} +15.3904 q^{58} -14.9641 q^{59} +6.20653 q^{60} +12.9958 q^{61} +6.08866 q^{62} -9.93920 q^{63} -10.5984 q^{64} +3.42503 q^{65} -33.1974 q^{66} +12.9765 q^{67} +6.40012 q^{68} +18.3289 q^{69} +1.85411 q^{70} -9.51351 q^{71} -28.3701 q^{72} -0.209957 q^{73} -18.7355 q^{74} -14.7780 q^{75} -5.97647 q^{76} -6.46162 q^{77} -48.4862 q^{78} -5.60830 q^{79} +1.32837 q^{80} +16.9070 q^{81} -29.4663 q^{82} -6.61460 q^{83} -17.1018 q^{84} +0.906958 q^{85} +5.52782 q^{86} -20.1166 q^{87} -18.4438 q^{88} -9.99098 q^{89} -8.64187 q^{90} -9.43749 q^{91} +21.8893 q^{92} -7.95841 q^{93} +4.23909 q^{94} -0.846921 q^{95} +7.30048 q^{96} -5.09962 q^{97} +11.6616 q^{98} +30.1172 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9}+O(q^{10}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9} + 50 q^{10} + 25 q^{11} + 17 q^{12} + 113 q^{13} + 12 q^{14} + 12 q^{15} + 252 q^{16} + 35 q^{17} + 13 q^{18} + 97 q^{19} + 55 q^{20} + 115 q^{21} + 14 q^{22} + 27 q^{23} + 122 q^{24} + 244 q^{25} + 39 q^{26} + 34 q^{27} + 66 q^{28} + 91 q^{29} + 4 q^{30} + 135 q^{31} + 21 q^{32} + 32 q^{33} + 58 q^{34} + 17 q^{35} + 273 q^{36} + 133 q^{37} - 3 q^{38} + 55 q^{39} + 142 q^{40} + 97 q^{41} - 8 q^{42} + 67 q^{43} + 44 q^{44} + 154 q^{45} + 101 q^{46} + 20 q^{47} - 7 q^{48} + 312 q^{49} + 21 q^{50} + 23 q^{51} + 193 q^{52} + 22 q^{53} + 141 q^{54} + 88 q^{55} + 28 q^{56} + 65 q^{57} + 62 q^{58} + 41 q^{59} + q^{60} + 377 q^{61} + 29 q^{62} + 39 q^{63} + 311 q^{64} + 21 q^{65} + 35 q^{66} + 42 q^{67} + 24 q^{68} + 137 q^{69} + 35 q^{70} + 17 q^{71} - 8 q^{72} + 213 q^{73} - 9 q^{74} + 2 q^{75} + 242 q^{76} + 60 q^{77} + 103 q^{79} + 80 q^{80} + 270 q^{81} + 84 q^{82} + 42 q^{83} + 137 q^{84} + 294 q^{85} - 9 q^{86} + 22 q^{87} - 13 q^{88} + 78 q^{89} + 69 q^{90} + 118 q^{91} + 49 q^{92} + 51 q^{93} + 93 q^{94} + 10 q^{95} + 260 q^{96} + 142 q^{97} - 31 q^{98} + 78 q^{99}+O(q^{100}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9 + 50 * q^10 + 25 * q^11 + 17 * q^12 + 113 * q^13 + 12 * q^14 + 12 * q^15 + 252 * q^16 + 35 * q^17 + 13 * q^18 + 97 * q^19 + 55 * q^20 + 115 * q^21 + 14 * q^22 + 27 * q^23 + 122 * q^24 + 244 * q^25 + 39 * q^26 + 34 * q^27 + 66 * q^28 + 91 * q^29 + 4 * q^30 + 135 * q^31 + 21 * q^32 + 32 * q^33 + 58 * q^34 + 17 * q^35 + 273 * q^36 + 133 * q^37 - 3 * q^38 + 55 * q^39 + 142 * q^40 + 97 * q^41 - 8 * q^42 + 67 * q^43 + 44 * q^44 + 154 * q^45 + 101 * q^46 + 20 * q^47 - 7 * q^48 + 312 * q^49 + 21 * q^50 + 23 * q^51 + 193 * q^52 + 22 * q^53 + 141 * q^54 + 88 * q^55 + 28 * q^56 + 65 * q^57 + 62 * q^58 + 41 * q^59 + q^60 + 377 * q^61 + 29 * q^62 + 39 * q^63 + 311 * q^64 + 21 * q^65 + 35 * q^66 + 42 * q^67 + 24 * q^68 + 137 * q^69 + 35 * q^70 + 17 * q^71 - 8 * q^72 + 213 * q^73 - 9 * q^74 + 2 * q^75 + 242 * q^76 + 60 * q^77 + 103 * q^79 + 80 * q^80 + 270 * q^81 + 84 * q^82 + 42 * q^83 + 137 * q^84 + 294 * q^85 - 9 * q^86 + 22 * q^87 - 13 * q^88 + 78 * q^89 + 69 * q^90 + 118 * q^91 + 49 * q^92 + 51 * q^93 + 93 * q^94 + 10 * q^95 + 260 * q^96 + 142 * q^97 - 31 * q^98 + 78 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more