LMFDB - Embedded newform 8007.2.a.i.1.4

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			1.4
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)

$f(q)$	$=$	$q-2.40365 q^{2} +1.00000 q^{3} +3.77751 q^{4} -0.410000 q^{5} -2.40365 q^{6} -0.100135 q^{7} -4.27252 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.40365 q^{2} +1.00000 q^{3} +3.77751 q^{4} -0.410000 q^{5} -2.40365 q^{6} -0.100135 q^{7} -4.27252 q^{8} +1.00000 q^{9} +0.985496 q^{10} +3.98265 q^{11} +3.77751 q^{12} +2.24403 q^{13} +0.240688 q^{14} -0.410000 q^{15} +2.71459 q^{16} +1.00000 q^{17} -2.40365 q^{18} +4.00363 q^{19} -1.54878 q^{20} -0.100135 q^{21} -9.57287 q^{22} +0.228657 q^{23} -4.27252 q^{24} -4.83190 q^{25} -5.39385 q^{26} +1.00000 q^{27} -0.378260 q^{28} +3.26863 q^{29} +0.985496 q^{30} -3.63794 q^{31} +2.02013 q^{32} +3.98265 q^{33} -2.40365 q^{34} +0.0410552 q^{35} +3.77751 q^{36} +9.48759 q^{37} -9.62331 q^{38} +2.24403 q^{39} +1.75173 q^{40} +4.90359 q^{41} +0.240688 q^{42} -12.4998 q^{43} +15.0445 q^{44} -0.410000 q^{45} -0.549611 q^{46} +3.40368 q^{47} +2.71459 q^{48} -6.98997 q^{49} +11.6142 q^{50} +1.00000 q^{51} +8.47685 q^{52} -11.8722 q^{53} -2.40365 q^{54} -1.63289 q^{55} +0.427827 q^{56} +4.00363 q^{57} -7.85663 q^{58} +9.75846 q^{59} -1.54878 q^{60} +12.8274 q^{61} +8.74432 q^{62} -0.100135 q^{63} -10.2848 q^{64} -0.920053 q^{65} -9.57287 q^{66} +0.0189575 q^{67} +3.77751 q^{68} +0.228657 q^{69} -0.0986822 q^{70} +9.14180 q^{71} -4.27252 q^{72} +16.5298 q^{73} -22.8048 q^{74} -4.83190 q^{75} +15.1238 q^{76} -0.398801 q^{77} -5.39385 q^{78} -4.39617 q^{79} -1.11298 q^{80} +1.00000 q^{81} -11.7865 q^{82} +17.7868 q^{83} -0.378260 q^{84} -0.410000 q^{85} +30.0451 q^{86} +3.26863 q^{87} -17.0159 q^{88} -8.21712 q^{89} +0.985496 q^{90} -0.224705 q^{91} +0.863756 q^{92} -3.63794 q^{93} -8.18125 q^{94} -1.64149 q^{95} +2.02013 q^{96} -0.636030 q^{97} +16.8014 q^{98} +3.98265 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.40365	−1.69963	−0.849817	−	0.527077i	$-0.823288\pi$
$2$	−2.40365	−1.69963	−0.849817	+	0.527077i	$0.823288\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.77751	1.88876
$5$	−0.410000	−0.183358	−0.0916789	−	0.995789i	$-0.529223\pi$
$5$	−0.410000	−0.183358	−0.0916789	+	0.995789i	$0.529223\pi$
$6$	−2.40365	−0.981284
$7$	−0.100135	−0.0378473	−0.0189237	−	0.999821i	$-0.506024\pi$
$7$	−0.100135	−0.0378473	−0.0189237	+	0.999821i	$0.506024\pi$
$8$	−4.27252	−1.51056
$9$	1.00000	0.333333
$10$	0.985496	0.311641
$11$	3.98265	1.20081	0.600407	−	0.799695i	$-0.295005\pi$
$11$	3.98265	1.20081	0.600407	+	0.799695i	$0.295005\pi$
$12$	3.77751	1.09047
$13$	2.24403	0.622382	0.311191	−	0.950347i	$-0.399272\pi$
$13$	2.24403	0.622382	0.311191	+	0.950347i	$0.399272\pi$
$14$	0.240688	0.0643266
$15$	−0.410000	−0.105862
$16$	2.71459	0.678647
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.40365	−0.566545
$19$	4.00363	0.918496	0.459248	−	0.888308i	$-0.348119\pi$
$19$	4.00363	0.918496	0.459248	+	0.888308i	$0.348119\pi$
$20$	−1.54878	−0.346318
$21$	−0.100135	−0.0218512
$22$	−9.57287	−2.04094
$23$	0.228657	0.0476783	0.0238392	−	0.999716i	$-0.492411\pi$
$23$	0.228657	0.0476783	0.0238392	+	0.999716i	$0.492411\pi$
$24$	−4.27252	−0.872124
$25$	−4.83190	−0.966380
$26$	−5.39385	−1.05782
$27$	1.00000	0.192450
$28$	−0.378260	−0.0714844
$29$	3.26863	0.606969	0.303485	−	0.952836i	$-0.401850\pi$
$29$	3.26863	0.606969	0.303485	+	0.952836i	$0.401850\pi$
$30$	0.985496	0.179926
$31$	−3.63794	−0.653393	−0.326697	−	0.945129i	$-0.605936\pi$
$31$	−3.63794	−0.653393	−0.326697	+	0.945129i	$0.605936\pi$
$32$	2.02013	0.357111
$33$	3.98265	0.693290
$34$	−2.40365	−0.412222
$35$	0.0410552	0.00693960
$36$	3.77751	0.629586
$37$	9.48759	1.55975	0.779875	−	0.625935i	$-0.215283\pi$
$37$	9.48759	1.55975	0.779875	+	0.625935i	$0.215283\pi$
$38$	−9.62331	−1.56111
$39$	2.24403	0.359332
$40$	1.75173	0.276973
$41$	4.90359	0.765812	0.382906	−	0.923787i	$-0.374923\pi$
$41$	4.90359	0.765812	0.382906	+	0.923787i	$0.374923\pi$
$42$	0.240688	0.0371390
$43$	−12.4998	−1.90620	−0.953101	−	0.302652i	$-0.902128\pi$
$43$	−12.4998	−1.90620	−0.953101	+	0.302652i	$0.902128\pi$
$44$	15.0445	2.26804
$45$	−0.410000	−0.0611192
$46$	−0.549611	−0.0810357
$47$	3.40368	0.496478	0.248239	−	0.968699i	$-0.420148\pi$
$47$	3.40368	0.496478	0.248239	+	0.968699i	$0.420148\pi$
$48$	2.71459	0.391817
$49$	−6.98997	−0.998568
$50$	11.6142	1.64249
$51$	1.00000	0.140028
$52$	8.47685	1.17553
$53$	−11.8722	−1.63077	−0.815386	−	0.578918i	$-0.803475\pi$
$53$	−11.8722	−1.63077	−0.815386	+	0.578918i	$0.803475\pi$
$54$	−2.40365	−0.327095
$55$	−1.63289	−0.220178
$56$	0.427827	0.0571707
$57$	4.00363	0.530294
$58$	−7.85663	−1.03163
$59$	9.75846	1.27044	0.635221	−	0.772330i	$-0.280909\pi$
$59$	9.75846	1.27044	0.635221	+	0.772330i	$0.280909\pi$
$60$	−1.54878	−0.199947
$61$	12.8274	1.64238	0.821192	−	0.570652i	$-0.193309\pi$
$61$	12.8274	1.64238	0.821192	+	0.570652i	$0.193309\pi$
$62$	8.74432	1.11053
$63$	−0.100135	−0.0126158
$64$	−10.2848	−1.28561
$65$	−0.920053	−0.114118
$66$	−9.57287	−1.17834
$67$	0.0189575	0.00231603	0.00115802	−	0.999999i	$-0.499631\pi$
$67$	0.0189575	0.00231603	0.00115802	+	0.999999i	$0.499631\pi$
$68$	3.77751	0.458091
$69$	0.228657	0.0275271
$70$	−0.0986822	−0.0117948
$71$	9.14180	1.08493	0.542466	−	0.840078i	$-0.317491\pi$
$71$	9.14180	1.08493	0.542466	+	0.840078i	$0.317491\pi$
$72$	−4.27252	−0.503521
$73$	16.5298	1.93467	0.967336	−	0.253497i	$-0.0815807\pi$
$73$	16.5298	1.93467	0.967336	+	0.253497i	$0.0815807\pi$
$74$	−22.8048	−2.65100
$75$	−4.83190	−0.557940
$76$	15.1238	1.73482
$77$	−0.398801	−0.0454476
$78$	−5.39385	−0.610733
$79$	−4.39617	−0.494608	−0.247304	−	0.968938i	$-0.579545\pi$
$79$	−4.39617	−0.494608	−0.247304	+	0.968938i	$0.579545\pi$
$80$	−1.11298	−0.124435
$81$	1.00000	0.111111
$82$	−11.7865	−1.30160
$83$	17.7868	1.95235	0.976177	−	0.216978i	$-0.0696199\pi$
$83$	17.7868	1.95235	0.976177	+	0.216978i	$0.0696199\pi$
$84$	−0.378260	−0.0412715
$85$	−0.410000	−0.0444708
$86$	30.0451	3.23985
$87$	3.26863	0.350434
$88$	−17.0159	−1.81390
$89$	−8.21712	−0.871013	−0.435507	−	0.900186i	$-0.643431\pi$
$89$	−8.21712	−0.871013	−0.435507	+	0.900186i	$0.643431\pi$
$90$	0.985496	0.103880
$91$	−0.224705	−0.0235555
$92$	0.863756	0.0900527
$93$	−3.63794	−0.377237
$94$	−8.18125	−0.843832
$95$	−1.64149	−0.168413
$96$	2.02013	0.206178
$97$	−0.636030	−0.0645791	−0.0322895	−	0.999479i	$-0.510280\pi$
$97$	−0.636030	−0.0645791	−0.0322895	+	0.999479i	$0.510280\pi$
$98$	16.8014	1.69720
$99$	3.98265	0.400271
$100$	−18.2526	−1.82526
$101$	−8.16752	−0.812698	−0.406349	−	0.913718i	$-0.633198\pi$
$101$	−8.16752	−0.812698	−0.406349	+	0.913718i	$0.633198\pi$
$102$	−2.40365	−0.237996
$103$	15.3451	1.51199	0.755997	−	0.654575i	$-0.227152\pi$
$103$	15.3451	1.51199	0.755997	+	0.654575i	$0.227152\pi$
$104$	−9.58765	−0.940146
$105$	0.0410552	0.00400658
$106$	28.5366	2.77172
$107$	12.7841	1.23589	0.617943	−	0.786223i	$-0.287966\pi$
$107$	12.7841	1.23589	0.617943	+	0.786223i	$0.287966\pi$
$108$	3.77751	0.363492
$109$	−17.6904	−1.69444	−0.847218	−	0.531245i	$-0.821724\pi$
$109$	−17.6904	−1.69444	−0.847218	+	0.531245i	$0.821724\pi$
$110$	3.92488	0.374223
$111$	9.48759	0.900522
$112$	−0.271824	−0.0256850
$113$	5.11931	0.481585	0.240792	−	0.970577i	$-0.422593\pi$
$113$	5.11931	0.481585	0.240792	+	0.970577i	$0.422593\pi$
$114$	−9.62331	−0.901306
$115$	−0.0937495	−0.00874218
$116$	12.3473	1.14642
$117$	2.24403	0.207461
$118$	−23.4559	−2.15929
$119$	−0.100135	−0.00917932
$120$	1.75173	0.159911
$121$	4.86147	0.441952
$122$	−30.8326	−2.79145
$123$	4.90359	0.442142
$124$	−13.7424	−1.23410
$125$	4.03108	0.360551
$126$	0.240688	0.0214422
$127$	−8.07929	−0.716921	−0.358460	−	0.933545i	$-0.616698\pi$
$127$	−8.07929	−0.716921	−0.358460	+	0.933545i	$0.616698\pi$
$128$	20.6809	1.82795
$129$	−12.4998	−1.10055
$130$	2.21148	0.193960
$131$	−3.62894	−0.317062	−0.158531	−	0.987354i	$-0.550676\pi$
$131$	−3.62894	−0.317062	−0.158531	+	0.987354i	$0.550676\pi$
$132$	15.0445	1.30946
$133$	−0.400902	−0.0347626
$134$	−0.0455672	−0.00393641
$135$	−0.410000	−0.0352872
$136$	−4.27252	−0.366365
$137$	8.98314	0.767482	0.383741	−	0.923441i	$-0.374636\pi$
$137$	8.98314	0.767482	0.383741	+	0.923441i	$0.374636\pi$
$138$	−0.549611	−0.0467860
$139$	−13.2766	−1.12611	−0.563053	−	0.826421i	$-0.690373\pi$
$139$	−13.2766	−1.12611	−0.563053	+	0.826421i	$0.690373\pi$
$140$	0.155087	0.0131072
$141$	3.40368	0.286642
$142$	−21.9737	−1.84399
$143$	8.93717	0.747364
$144$	2.71459	0.226216
$145$	−1.34014	−0.111293
$146$	−39.7319	−3.28824
$147$	−6.98997	−0.576523
$148$	35.8395	2.94599
$149$	2.34999	0.192519	0.0962595	−	0.995356i	$-0.469312\pi$
$149$	2.34999	0.192519	0.0962595	+	0.995356i	$0.469312\pi$
$150$	11.6142	0.948294
$151$	−14.7522	−1.20051	−0.600257	−	0.799807i	$-0.704935\pi$
$151$	−14.7522	−1.20051	−0.600257	+	0.799807i	$0.704935\pi$
$152$	−17.1056	−1.38745
$153$	1.00000	0.0808452
$154$	0.958576	0.0772442
$155$	1.49156	0.119805
$156$	8.47685	0.678691
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	10.5668	0.840653
$159$	−11.8722	−0.941526
$160$	−0.828252	−0.0654791
$161$	−0.0228965	−0.00180450
$162$	−2.40365	−0.188848
$163$	−20.0014	−1.56663	−0.783316	−	0.621624i	$-0.786473\pi$
$163$	−20.0014	−1.56663	−0.783316	+	0.621624i	$0.786473\pi$
$164$	18.5234	1.44643
$165$	−1.63289	−0.127120
$166$	−42.7531	−3.31829
$167$	4.48824	0.347310	0.173655	−	0.984807i	$-0.444442\pi$
$167$	4.48824	0.347310	0.173655	+	0.984807i	$0.444442\pi$
$168$	0.427827	0.0330075
$169$	−7.96433	−0.612641
$170$	0.985496	0.0755841
$171$	4.00363	0.306165
$172$	−47.2182	−3.60035
$173$	−6.26238	−0.476120	−0.238060	−	0.971250i	$-0.576511\pi$
$173$	−6.26238	−0.476120	−0.238060	+	0.971250i	$0.576511\pi$
$174$	−7.85663	−0.595610
$175$	0.483840	0.0365749
$176$	10.8112	0.814928
$177$	9.75846	0.733490
$178$	19.7510	1.48040
$179$	−1.58551	−0.118507	−0.0592533	−	0.998243i	$-0.518872\pi$
$179$	−1.58551	−0.118507	−0.0592533	+	0.998243i	$0.518872\pi$
$180$	−1.54878	−0.115439
$181$	6.00836	0.446598	0.223299	−	0.974750i	$-0.428317\pi$
$181$	6.00836	0.446598	0.223299	+	0.974750i	$0.428317\pi$
$182$	0.540111	0.0400357
$183$	12.8274	0.948231
$184$	−0.976941	−0.0720211
$185$	−3.88991	−0.285992
$186$	8.74432	0.641164
$187$	3.98265	0.291240
$188$	12.8575	0.937727
$189$	−0.100135	−0.00728372
$190$	3.94556	0.286241
$191$	−7.12153	−0.515296	−0.257648	−	0.966239i	$-0.582947\pi$
$191$	−7.12153	−0.515296	−0.257648	+	0.966239i	$0.582947\pi$
$192$	−10.2848	−0.742244
$193$	16.6624	1.19938	0.599692	−	0.800231i	$-0.295290\pi$
$193$	16.6624	1.19938	0.599692	+	0.800231i	$0.295290\pi$
$194$	1.52879	0.109761
$195$	−0.920053	−0.0658863
$196$	−26.4047	−1.88605
$197$	4.87401	0.347259	0.173630	−	0.984811i	$-0.444450\pi$
$197$	4.87401	0.347259	0.173630	+	0.984811i	$0.444450\pi$
$198$	−9.57287	−0.680314
$199$	−4.12510	−0.292421	−0.146210	−	0.989254i	$-0.546708\pi$
$199$	−4.12510	−0.292421	−0.146210	+	0.989254i	$0.546708\pi$
$200$	20.6444	1.45978
$201$	0.0189575	0.00133716
$202$	19.6318	1.38129
$203$	−0.327303	−0.0229722
$204$	3.77751	0.264479
$205$	−2.01047	−0.140418
$206$	−36.8841	−2.56984
$207$	0.228657	0.0158928
$208$	6.09161	0.422377
$209$	15.9450	1.10294
$210$	−0.0986822	−0.00680972
$211$	11.2341	0.773385	0.386693	−	0.922209i	$-0.373617\pi$
$211$	11.2341	0.773385	0.386693	+	0.922209i	$0.373617\pi$
$212$	−44.8474	−3.08013
$213$	9.14180	0.626386
$214$	−30.7284	−2.10055
$215$	5.12492	0.349517
$216$	−4.27252	−0.290708
$217$	0.364283	0.0247292
$218$	42.5216	2.87992
$219$	16.5298	1.11698
$220$	−6.16825	−0.415863
$221$	2.24403	0.150950
$222$	−22.8048	−1.53056
$223$	−22.1560	−1.48368	−0.741839	−	0.670578i	$-0.766046\pi$
$223$	−22.1560	−1.48368	−0.741839	+	0.670578i	$0.766046\pi$
$224$	−0.202284	−0.0135157
$225$	−4.83190	−0.322127
$226$	−12.3050	−0.818518
$227$	11.5034	0.763508	0.381754	−	0.924264i	$-0.375320\pi$
$227$	11.5034	0.763508	0.381754	+	0.924264i	$0.375320\pi$
$228$	15.1238	1.00160
$229$	13.4411	0.888212	0.444106	−	0.895974i	$-0.353521\pi$
$229$	13.4411	0.888212	0.444106	+	0.895974i	$0.353521\pi$
$230$	0.225341	0.0148585
$231$	−0.398801	−0.0262392
$232$	−13.9653	−0.916865
$233$	−2.44740	−0.160334	−0.0801672	−	0.996781i	$-0.525545\pi$
$233$	−2.44740	−0.160334	−0.0801672	+	0.996781i	$0.525545\pi$
$234$	−5.39385	−0.352607
$235$	−1.39551	−0.0910331
$236$	36.8627	2.39956
$237$	−4.39617	−0.285562
$238$	0.240688	0.0156015
$239$	3.77540	0.244210	0.122105	−	0.992517i	$-0.461035\pi$
$239$	3.77540	0.244210	0.122105	+	0.992517i	$0.461035\pi$
$240$	−1.11298	−0.0718426
$241$	15.5358	1.00075	0.500375	−	0.865809i	$-0.333196\pi$
$241$	15.5358	1.00075	0.500375	+	0.865809i	$0.333196\pi$
$242$	−11.6853	−0.751157
$243$	1.00000	0.0641500
$244$	48.4558	3.10206
$245$	2.86589	0.183095
$246$	−11.7865	−0.751480
$247$	8.98426	0.571655
$248$	15.5432	0.986991
$249$	17.7868	1.12719
$250$	−9.68929	−0.612805
$251$	20.3888	1.28693	0.643466	−	0.765474i	$-0.277496\pi$
$251$	20.3888	1.28693	0.643466	+	0.765474i	$0.277496\pi$
$252$	−0.378260	−0.0238281
$253$	0.910660	0.0572527
$254$	19.4197	1.21850
$255$	−0.410000	−0.0256752
$256$	−29.1398	−1.82124
$257$	−7.52823	−0.469598	−0.234799	−	0.972044i	$-0.575443\pi$
$257$	−7.52823	−0.469598	−0.234799	+	0.972044i	$0.575443\pi$
$258$	30.0451	1.87053
$259$	−0.950036	−0.0590324
$260$	−3.47551	−0.215542
$261$	3.26863	0.202323
$262$	8.72269	0.538889
$263$	17.9501	1.10685	0.553425	−	0.832899i	$-0.313321\pi$
$263$	17.9501	1.10685	0.553425	+	0.832899i	$0.313321\pi$
$264$	−17.0159	−1.04726
$265$	4.86760	0.299014
$266$	0.963626	0.0590837
$267$	−8.21712	−0.502880
$268$	0.0716124	0.00437442
$269$	11.7482	0.716303	0.358152	−	0.933663i	$-0.383407\pi$
$269$	11.7482	0.716303	0.358152	+	0.933663i	$0.383407\pi$
$270$	0.985496	0.0599754
$271$	−8.12553	−0.493591	−0.246795	−	0.969068i	$-0.579378\pi$
$271$	−8.12553	−0.493591	−0.246795	+	0.969068i	$0.579378\pi$
$272$	2.71459	0.164596
$273$	−0.224705	−0.0135998
$274$	−21.5923	−1.30444
$275$	−19.2437	−1.16044
$276$	0.863756	0.0519920
$277$	6.58388	0.395587	0.197793	−	0.980244i	$-0.436622\pi$
$277$	6.58388	0.395587	0.197793	+	0.980244i	$0.436622\pi$
$278$	31.9122	1.91397
$279$	−3.63794	−0.217798
$280$	−0.175409	−0.0104827
$281$	−28.5833	−1.70513	−0.852567	−	0.522618i	$-0.824955\pi$
$281$	−28.5833	−1.70513	−0.852567	+	0.522618i	$0.824955\pi$
$282$	−8.18125	−0.487186
$283$	4.26204	0.253352	0.126676	−	0.991944i	$-0.459569\pi$
$283$	4.26204	0.253352	0.126676	+	0.991944i	$0.459569\pi$
$284$	34.5333	2.04917
$285$	−1.64149	−0.0972335
$286$	−21.4818	−1.27025
$287$	−0.491019	−0.0289839
$288$	2.02013	0.119037
$289$	1.00000	0.0588235
$290$	3.22122	0.189157
$291$	−0.636030	−0.0372847
$292$	62.4417	3.65413
$293$	−17.8906	−1.04518	−0.522589	−	0.852585i	$-0.675034\pi$
$293$	−17.8906	−1.04518	−0.522589	+	0.852585i	$0.675034\pi$
$294$	16.8014	0.979879
$295$	−4.00097	−0.232945
$296$	−40.5359	−2.35610
$297$	3.98265	0.231097
$298$	−5.64855	−0.327212
$299$	0.513113	0.0296741
$300$	−18.2526	−1.05381
$301$	1.25166	0.0721446
$302$	35.4590	2.04044
$303$	−8.16752	−0.469212
$304$	10.8682	0.623334
$305$	−5.25925	−0.301144
$306$	−2.40365	−0.137407
$307$	2.45752	0.140258	0.0701290	−	0.997538i	$-0.477659\pi$
$307$	2.45752	0.140258	0.0701290	+	0.997538i	$0.477659\pi$
$308$	−1.50648	−0.0858394
$309$	15.3451	0.872950
$310$	−3.58517	−0.203624
$311$	−29.5833	−1.67752	−0.838758	−	0.544505i	$-0.816718\pi$
$311$	−29.5833	−1.67752	−0.838758	+	0.544505i	$0.816718\pi$
$312$	−9.58765	−0.542794
$313$	25.4457	1.43828	0.719139	−	0.694866i	$-0.244536\pi$
$313$	25.4457	1.43828	0.719139	+	0.694866i	$0.244536\pi$
$314$	−2.40365	−0.135646
$315$	0.0410552	0.00231320
$316$	−16.6066	−0.934195
$317$	22.6641	1.27294	0.636472	−	0.771300i	$-0.280393\pi$
$317$	22.6641	1.27294	0.636472	+	0.771300i	$0.280393\pi$
$318$	28.5366	1.60025
$319$	13.0178	0.728857
$320$	4.21679	0.235726
$321$	12.7841	0.713539
$322$	0.0550350	0.00306698
$323$	4.00363	0.222768
$324$	3.77751	0.209862
$325$	−10.8429	−0.601457
$326$	48.0763	2.66270
$327$	−17.6904	−0.978283
$328$	−20.9507	−1.15681
$329$	−0.340826	−0.0187904
$330$	3.92488	0.216058
$331$	−22.2428	−1.22257	−0.611287	−	0.791409i	$-0.709348\pi$
$331$	−22.2428	−1.22257	−0.611287	+	0.791409i	$0.709348\pi$
$332$	67.1898	3.68752
$333$	9.48759	0.519917
$334$	−10.7881	−0.590300
$335$	−0.00777260	−0.000424663 0
$336$	−0.271824	−0.0148292
$337$	20.0225	1.09069	0.545347	−	0.838210i	$-0.316398\pi$
$337$	20.0225	1.09069	0.545347	+	0.838210i	$0.316398\pi$
$338$	19.1434	1.04127
$339$	5.11931	0.278043
$340$	−1.54878	−0.0839945
$341$	−14.4886	−0.784603
$342$	−9.62331	−0.520369
$343$	1.40088	0.0756404
$344$	53.4056	2.87944
$345$	−0.0937495	−0.00504730
$346$	15.0525	0.809230
$347$	0.860863	0.0462135	0.0231068	−	0.999733i	$-0.492644\pi$
$347$	0.860863	0.0462135	0.0231068	+	0.999733i	$0.492644\pi$
$348$	12.3473	0.661885
$349$	10.2272	0.547447	0.273724	−	0.961808i	$-0.411745\pi$
$349$	10.2272	0.547447	0.273724	+	0.961808i	$0.411745\pi$
$350$	−1.16298	−0.0621639
$351$	2.24403	0.119777
$352$	8.04544	0.428824
$353$	6.20065	0.330028	0.165014	−	0.986291i	$-0.447233\pi$
$353$	6.20065	0.330028	0.165014	+	0.986291i	$0.447233\pi$
$354$	−23.4559	−1.24667
$355$	−3.74814	−0.198931
$356$	−31.0403	−1.64513
$357$	−0.100135	−0.00529968
$358$	3.81100	0.201418
$359$	5.32467	0.281025	0.140513	−	0.990079i	$-0.455125\pi$
$359$	5.32467	0.281025	0.140513	+	0.990079i	$0.455125\pi$
$360$	1.75173	0.0923244
$361$	−2.97094	−0.156365
$362$	−14.4420	−0.759053
$363$	4.86147	0.255161
$364$	−0.848826	−0.0444906
$365$	−6.77724	−0.354737
$366$	−30.8326	−1.61165
$367$	−18.2183	−0.950989	−0.475495	−	0.879719i	$-0.657731\pi$
$367$	−18.2183	−0.950989	−0.475495	+	0.879719i	$0.657731\pi$
$368$	0.620710	0.0323567
$369$	4.90359	0.255271
$370$	9.34998	0.486082
$371$	1.18882	0.0617203
$372$	−13.7424	−0.712508
$373$	−19.6799	−1.01898	−0.509492	−	0.860475i	$-0.670167\pi$
$373$	−19.6799	−1.01898	−0.509492	+	0.860475i	$0.670167\pi$
$374$	−9.57287	−0.495001
$375$	4.03108	0.208164
$376$	−14.5423	−0.749961
$377$	7.33490	0.377767
$378$	0.240688	0.0123797
$379$	30.7337	1.57869	0.789343	−	0.613953i	$-0.210421\pi$
$379$	30.7337	1.57869	0.789343	+	0.613953i	$0.210421\pi$
$380$	−6.20075	−0.318092
$381$	−8.07929	−0.413914
$382$	17.1176	0.875815
$383$	−22.4961	−1.14950	−0.574750	−	0.818329i	$-0.694901\pi$
$383$	−22.4961	−1.14950	−0.574750	+	0.818329i	$0.694901\pi$
$384$	20.6809	1.05537
$385$	0.163508	0.00833316
$386$	−40.0505	−2.03851
$387$	−12.4998	−0.635401
$388$	−2.40261	−0.121974
$389$	−29.2074	−1.48088	−0.740438	−	0.672124i	$-0.765382\pi$
$389$	−29.2074	−1.48088	−0.740438	+	0.672124i	$0.765382\pi$
$390$	2.21148	0.111983
$391$	0.228657	0.0115637
$392$	29.8648	1.50840
$393$	−3.62894	−0.183056
$394$	−11.7154	−0.590214
$395$	1.80243	0.0906902
$396$	15.0445	0.756015
$397$	1.47118	0.0738364	0.0369182	−	0.999318i	$-0.488246\pi$
$397$	1.47118	0.0738364	0.0369182	+	0.999318i	$0.488246\pi$
$398$	9.91528	0.497008
$399$	−0.400902	−0.0200702
$400$	−13.1166	−0.655831
$401$	13.5059	0.674451	0.337225	−	0.941424i	$-0.390512\pi$
$401$	13.5059	0.674451	0.337225	+	0.941424i	$0.390512\pi$
$402$	−0.0455672	−0.00227269
$403$	−8.16364	−0.406660
$404$	−30.8529	−1.53499
$405$	−0.410000	−0.0203731
$406$	0.786720	0.0390443
$407$	37.7857	1.87297
$408$	−4.27252	−0.211521
$409$	3.28629	0.162496	0.0812482	−	0.996694i	$-0.474109\pi$
$409$	3.28629	0.162496	0.0812482	+	0.996694i	$0.474109\pi$
$410$	4.83247	0.238659
$411$	8.98314	0.443106
$412$	57.9662	2.85579
$413$	−0.977159	−0.0480828
$414$	−0.549611	−0.0270119
$415$	−7.29259	−0.357979
$416$	4.53322	0.222259
$417$	−13.2766	−0.650158
$418$	−38.3262	−1.87460
$419$	10.9968	0.537231	0.268615	−	0.963248i	$-0.413434\pi$
$419$	10.9968	0.537231	0.268615	+	0.963248i	$0.413434\pi$
$420$	0.155087	0.00756745
$421$	−11.2012	−0.545911	−0.272956	−	0.962027i	$-0.588001\pi$
$421$	−11.2012	−0.545911	−0.272956	+	0.962027i	$0.588001\pi$
$422$	−27.0027	−1.31447
$423$	3.40368	0.165493
$424$	50.7241	2.46338
$425$	−4.83190	−0.234382
$426$	−21.9737	−1.06463
$427$	−1.28447	−0.0621598
$428$	48.2921	2.33429
$429$	8.93717	0.431491
$430$	−12.3185	−0.594051
$431$	1.71224	0.0824755	0.0412378	−	0.999149i	$-0.486870\pi$
$431$	1.71224	0.0824755	0.0412378	+	0.999149i	$0.486870\pi$
$432$	2.71459	0.130606
$433$	−27.2007	−1.30718	−0.653591	−	0.756848i	$-0.726738\pi$
$433$	−27.2007	−1.30718	−0.653591	+	0.756848i	$0.726738\pi$
$434$	−0.875609	−0.0420306
$435$	−1.34014	−0.0642548
$436$	−66.8259	−3.20038
$437$	0.915459	0.0437923
$438$	−39.7319	−1.89846
$439$	5.11474	0.244113	0.122057	−	0.992523i	$-0.461051\pi$
$439$	5.11474	0.244113	0.122057	+	0.992523i	$0.461051\pi$
$440$	6.97653	0.332593
$441$	−6.98997	−0.332856
$442$	−5.39385	−0.256559
$443$	−1.66258	−0.0789917	−0.0394959	−	0.999220i	$-0.512575\pi$
$443$	−1.66258	−0.0789917	−0.0394959	+	0.999220i	$0.512575\pi$
$444$	35.8395	1.70087
$445$	3.36902	0.159707
$446$	53.2553	2.52171
$447$	2.34999	0.111151
$448$	1.02987	0.0486567
$449$	23.4976	1.10892	0.554460	−	0.832210i	$-0.312925\pi$
$449$	23.4976	1.10892	0.554460	+	0.832210i	$0.312925\pi$
$450$	11.6142	0.547498
$451$	19.5293	0.919597
$452$	19.3383	0.909596
$453$	−14.7522	−0.693118
$454$	−27.6501	−1.29768
$455$	0.0921291	0.00431908
$456$	−17.1056	−0.801042
$457$	−37.0888	−1.73494	−0.867471	−	0.497487i	$-0.834256\pi$
$457$	−37.0888	−1.73494	−0.867471	+	0.497487i	$0.834256\pi$
$458$	−32.3076	−1.50964
$459$	1.00000	0.0466760
$460$	−0.354140	−0.0165119
$461$	27.3071	1.27182	0.635909	−	0.771764i	$-0.280625\pi$
$461$	27.3071	1.27182	0.635909	+	0.771764i	$0.280625\pi$
$462$	0.958576	0.0445970
$463$	15.0754	0.700614	0.350307	−	0.936635i	$-0.386077\pi$
$463$	15.0754	0.700614	0.350307	+	0.936635i	$0.386077\pi$
$464$	8.87298	0.411918
$465$	1.49156	0.0691692
$466$	5.88268	0.272510
$467$	−25.3565	−1.17336	−0.586680	−	0.809819i	$-0.699565\pi$
$467$	−25.3565	−1.17336	−0.586680	+	0.809819i	$0.699565\pi$
$468$	8.47685	0.391843
$469$	−0.00189831	−8.76556e−5 0
$470$	3.35432	0.154723
$471$	1.00000	0.0460776
$472$	−41.6932	−1.91908
$473$	−49.7823	−2.28899
$474$	10.5668	0.485351
$475$	−19.3451	−0.887616
$476$	−0.378260	−0.0173375
$477$	−11.8722	−0.543590
$478$	−9.07472	−0.415068
$479$	23.5656	1.07674	0.538369	−	0.842709i	$-0.319041\pi$
$479$	23.5656	1.07674	0.538369	+	0.842709i	$0.319041\pi$
$480$	−0.828252	−0.0378044
$481$	21.2904	0.970760
$482$	−37.3426	−1.70091
$483$	−0.0228965	−0.00104183
$484$	18.3643	0.834740
$485$	0.260773	0.0118411
$486$	−2.40365	−0.109032
$487$	32.6910	1.48137	0.740685	−	0.671853i	$-0.234501\pi$
$487$	32.6910	1.48137	0.740685	+	0.671853i	$0.234501\pi$
$488$	−54.8054	−2.48092
$489$	−20.0014	−0.904495
$490$	−6.88859	−0.311195
$491$	−25.9273	−1.17008	−0.585042	−	0.811003i	$-0.698922\pi$
$491$	−25.9273	−1.17008	−0.585042	+	0.811003i	$0.698922\pi$
$492$	18.5234	0.835099
$493$	3.26863	0.147212
$494$	−21.5950	−0.971605
$495$	−1.63289	−0.0733928
$496$	−9.87550	−0.443423
$497$	−0.915411	−0.0410618
$498$	−42.7531	−1.91581
$499$	−30.1869	−1.35135	−0.675675	−	0.737200i	$-0.736148\pi$
$499$	−30.1869	−1.35135	−0.675675	+	0.737200i	$0.736148\pi$
$500$	15.2275	0.680993
$501$	4.48824	0.200520
$502$	−49.0076	−2.18731
$503$	9.64232	0.429930	0.214965	−	0.976622i	$-0.431036\pi$
$503$	9.64232	0.429930	0.214965	+	0.976622i	$0.431036\pi$
$504$	0.427827	0.0190569
$505$	3.34868	0.149015
$506$	−2.18891	−0.0973087
$507$	−7.96433	−0.353708
$508$	−30.5196	−1.35409
$509$	32.1609	1.42551	0.712754	−	0.701414i	$-0.247448\pi$
$509$	32.1609	1.42551	0.712754	+	0.701414i	$0.247448\pi$
$510$	0.985496	0.0436385
$511$	−1.65521	−0.0732222
$512$	28.6800	1.26749
$513$	4.00363	0.176765
$514$	18.0952	0.798145
$515$	−6.29148	−0.277236
$516$	−47.2182	−2.07867
$517$	13.5557	0.596178
$518$	2.28355	0.100333
$519$	−6.26238	−0.274888
$520$	3.93094	0.172383
$521$	−24.5101	−1.07381	−0.536903	−	0.843644i	$-0.680406\pi$
$521$	−24.5101	−1.07381	−0.536903	+	0.843644i	$0.680406\pi$
$522$	−7.85663	−0.343875
$523$	7.45953	0.326182	0.163091	−	0.986611i	$-0.447854\pi$
$523$	7.45953	0.326182	0.163091	+	0.986611i	$0.447854\pi$
$524$	−13.7084	−0.598853
$525$	0.483840	0.0211165
$526$	−43.1457	−1.88124
$527$	−3.63794	−0.158471
$528$	10.8112	0.470499
$529$	−22.9477	−0.997727
$530$	−11.7000	−0.508215
$531$	9.75846	0.423481
$532$	−1.51441	−0.0656581
$533$	11.0038	0.476628
$534$	19.7510	0.854711
$535$	−5.24148	−0.226609
$536$	−0.0809964	−0.00349851
$537$	−1.58551	−0.0684198
$538$	−28.2386	−1.21745
$539$	−27.8386	−1.19909
$540$	−1.54878	−0.0666490
$541$	−26.5510	−1.14152	−0.570759	−	0.821118i	$-0.693351\pi$
$541$	−26.5510	−1.14152	−0.570759	+	0.821118i	$0.693351\pi$
$542$	19.5309	0.838924
$543$	6.00836	0.257843
$544$	2.02013	0.0866122
$545$	7.25308	0.310688
$546$	0.540111	0.0231146
$547$	9.80799	0.419359	0.209680	−	0.977770i	$-0.432758\pi$
$547$	9.80799	0.419359	0.209680	+	0.977770i	$0.432758\pi$
$548$	33.9339	1.44959
$549$	12.8274	0.547461
$550$	46.2552	1.97233
$551$	13.0864	0.557499
$552$	−0.976941	−0.0415814
$553$	0.440209	0.0187196
$554$	−15.8253	−0.672353
$555$	−3.88991	−0.165118
$556$	−50.1525	−2.12694
$557$	32.7029	1.38567	0.692833	−	0.721098i	$-0.256362\pi$
$557$	32.7029	1.38567	0.692833	+	0.721098i	$0.256362\pi$
$558$	8.74432	0.370176
$559$	−28.0499	−1.18639
$560$	0.111448	0.00470953
$561$	3.98265	0.168147
$562$	68.7040	2.89810
$563$	27.3597	1.15307	0.576536	−	0.817072i	$-0.304404\pi$
$563$	27.3597	1.15307	0.576536	+	0.817072i	$0.304404\pi$
$564$	12.8575	0.541397
$565$	−2.09892	−0.0883022
$566$	−10.2444	−0.430606
$567$	−0.100135	−0.00420526
$568$	−39.0585	−1.63886
$569$	−23.1783	−0.971684	−0.485842	−	0.874047i	$-0.661487\pi$
$569$	−23.1783	−0.971684	−0.485842	+	0.874047i	$0.661487\pi$
$570$	3.94556	0.165261
$571$	−3.05455	−0.127829	−0.0639145	−	0.997955i	$-0.520358\pi$
$571$	−3.05455	−0.127829	−0.0639145	+	0.997955i	$0.520358\pi$
$572$	33.7603	1.41159
$573$	−7.12153	−0.297506
$574$	1.18024	0.0492621
$575$	−1.10485	−0.0460754
$576$	−10.2848	−0.428535
$577$	−16.9335	−0.704949	−0.352475	−	0.935821i	$-0.614660\pi$
$577$	−16.9335	−0.704949	−0.352475	+	0.935821i	$0.614660\pi$
$578$	−2.40365	−0.0999785
$579$	16.6624	0.692465
$580$	−5.06240	−0.210205
$581$	−1.78107	−0.0738913
$582$	1.52879	0.0633704
$583$	−47.2828	−1.95825
$584$	−70.6240	−2.92244
$585$	−0.920053	−0.0380395
$586$	43.0026	1.77642
$587$	25.5809	1.05584	0.527919	−	0.849295i	$-0.322972\pi$
$587$	25.5809	1.05584	0.527919	+	0.849295i	$0.322972\pi$
$588$	−26.4047	−1.08891
$589$	−14.5650	−0.600139
$590$	9.61692	0.395922
$591$	4.87401	0.200490
$592$	25.7549	1.05852
$593$	9.38275	0.385303	0.192652	−	0.981267i	$-0.438291\pi$
$593$	9.38275	0.385303	0.192652	+	0.981267i	$0.438291\pi$
$594$	−9.57287	−0.392780
$595$	0.0410552	0.00168310
$596$	8.87713	0.363621
$597$	−4.12510	−0.168829
$598$	−1.23334	−0.0504351
$599$	33.1862	1.35595	0.677976	−	0.735084i	$-0.262857\pi$
$599$	33.1862	1.35595	0.677976	+	0.735084i	$0.262857\pi$
$600$	20.6444	0.842803
$601$	−27.5149	−1.12236	−0.561179	−	0.827695i	$-0.689652\pi$
$601$	−27.5149	−1.12236	−0.561179	+	0.827695i	$0.689652\pi$
$602$	−3.00855	−0.122620
$603$	0.0189575	0.000772011 0
$604$	−55.7266	−2.26748
$605$	−1.99321	−0.0810353
$606$	19.6318	0.797488
$607$	26.9451	1.09367	0.546833	−	0.837241i	$-0.315833\pi$
$607$	26.9451	1.09367	0.546833	+	0.837241i	$0.315833\pi$
$608$	8.08784	0.328005
$609$	−0.327303	−0.0132630
$610$	12.6414	0.511834
$611$	7.63797	0.308999
$612$	3.77751	0.152697
$613$	2.55940	0.103373	0.0516866	−	0.998663i	$-0.483540\pi$
$613$	2.55940	0.103373	0.0516866	+	0.998663i	$0.483540\pi$
$614$	−5.90701	−0.238387
$615$	−2.01047	−0.0810701
$616$	1.70388	0.0686514
$617$	−42.2769	−1.70200	−0.851002	−	0.525163i	$-0.824004\pi$
$617$	−42.2769	−1.70200	−0.851002	+	0.525163i	$0.824004\pi$
$618$	−36.8841	−1.48370
$619$	32.0452	1.28800	0.644002	−	0.765023i	$-0.277273\pi$
$619$	32.0452	1.28800	0.644002	+	0.765023i	$0.277273\pi$
$620$	5.63437	0.226282
$621$	0.228657	0.00917569
$622$	71.1078	2.85116
$623$	0.822818	0.0329655
$624$	6.09161	0.243860
$625$	22.5068	0.900270
$626$	−61.1626	−2.44455
$627$	15.9450	0.636784
$628$	3.77751	0.150739
$629$	9.48759	0.378295
$630$	−0.0986822	−0.00393159
$631$	−32.6086	−1.29813	−0.649064	−	0.760734i	$-0.724839\pi$
$631$	−32.6086	−1.29813	−0.649064	+	0.760734i	$0.724839\pi$
$632$	18.7827	0.747137
$633$	11.2341	0.446514
$634$	−54.4765	−2.16354
$635$	3.31251	0.131453
$636$	−44.8474	−1.77831
$637$	−15.6857	−0.621490
$638$	−31.2902	−1.23879
$639$	9.14180	0.361644
$640$	−8.47916	−0.335168
$641$	−4.92316	−0.194453	−0.0972265	−	0.995262i	$-0.530997\pi$
$641$	−4.92316	−0.194453	−0.0972265	+	0.995262i	$0.530997\pi$
$642$	−30.7284	−1.21275
$643$	4.74698	0.187203	0.0936013	−	0.995610i	$-0.470162\pi$
$643$	4.74698	0.187203	0.0936013	+	0.995610i	$0.470162\pi$
$644$	−0.0864918	−0.00340825
$645$	5.12492	0.201794
$646$	−9.62331	−0.378624
$647$	−7.52240	−0.295736	−0.147868	−	0.989007i	$-0.547241\pi$
$647$	−7.52240	−0.295736	−0.147868	+	0.989007i	$0.547241\pi$
$648$	−4.27252	−0.167840
$649$	38.8645	1.52556
$650$	26.0626	1.02226
$651$	0.364283	0.0142774
$652$	−75.5556	−2.95899
$653$	32.5994	1.27571	0.637857	−	0.770155i	$-0.279821\pi$
$653$	32.5994	1.27571	0.637857	+	0.770155i	$0.279821\pi$
$654$	42.5216	1.66272
$655$	1.48787	0.0581358
$656$	13.3112	0.519716
$657$	16.5298	0.644891
$658$	0.819226	0.0319368
$659$	10.8502	0.422664	0.211332	−	0.977414i	$-0.432220\pi$
$659$	10.8502	0.422664	0.211332	+	0.977414i	$0.432220\pi$
$660$	−6.16825	−0.240099
$661$	36.7995	1.43134	0.715668	−	0.698441i	$-0.246123\pi$
$661$	36.7995	1.43134	0.715668	+	0.698441i	$0.246123\pi$
$662$	53.4638	2.07793
$663$	2.24403	0.0871509
$664$	−75.9943	−2.94915
$665$	0.164370	0.00637399
$666$	−22.8048	−0.883668
$667$	0.747396	0.0289393
$668$	16.9544	0.655985
$669$	−22.1560	−0.856602
$670$	0.0186826	0.000721771 0
$671$	51.0871	1.97220
$672$	−0.202284	−0.00780329
$673$	20.8651	0.804289	0.402145	−	0.915576i	$-0.368265\pi$
$673$	20.8651	0.804289	0.402145	+	0.915576i	$0.368265\pi$
$674$	−48.1270	−1.85378
$675$	−4.83190	−0.185980
$676$	−30.0854	−1.15713
$677$	−9.41487	−0.361843	−0.180921	−	0.983498i	$-0.557908\pi$
$677$	−9.41487	−0.361843	−0.180921	+	0.983498i	$0.557908\pi$
$678$	−12.3050	−0.472571
$679$	0.0636886	0.00244414
$680$	1.75173	0.0671759
$681$	11.5034	0.440811
$682$	34.8255	1.33354
$683$	−11.0396	−0.422420	−0.211210	−	0.977441i	$-0.567740\pi$
$683$	−11.0396	−0.422420	−0.211210	+	0.977441i	$0.567740\pi$
$684$	15.1238	0.578272
$685$	−3.68309	−0.140724
$686$	−3.36722	−0.128561
$687$	13.4411	0.512810
$688$	−33.9318	−1.29364
$689$	−26.6415	−1.01496
$690$	0.225341	0.00857857
$691$	38.5997	1.46840	0.734200	−	0.678933i	$-0.237557\pi$
$691$	38.5997	1.46840	0.734200	+	0.678933i	$0.237557\pi$
$692$	−23.6562	−0.899275
$693$	−0.398801	−0.0151492
$694$	−2.06921	−0.0785461
$695$	5.44341	0.206480
$696$	−13.9653	−0.529352
$697$	4.90359	0.185737
$698$	−24.5825	−0.930460
$699$	−2.44740	−0.0925691
$700$	1.82771	0.0690811
$701$	4.90510	0.185263	0.0926315	−	0.995700i	$-0.470472\pi$
$701$	4.90510	0.185263	0.0926315	+	0.995700i	$0.470472\pi$
$702$	−5.39385	−0.203578
$703$	37.9848	1.43262
$704$	−40.9609	−1.54377
$705$	−1.39551	−0.0525580
$706$	−14.9042	−0.560926
$707$	0.817851	0.0307585
$708$	36.8627	1.38539
$709$	8.97937	0.337227	0.168614	−	0.985682i	$-0.446071\pi$
$709$	8.97937	0.337227	0.168614	+	0.985682i	$0.446071\pi$
$710$	9.00921	0.338109
$711$	−4.39617	−0.164869
$712$	35.1078	1.31572
$713$	−0.831841	−0.0311527
$714$	0.240688	0.00900753
$715$	−3.66424	−0.137035
$716$	−5.98928	−0.223830
$717$	3.77540	0.140995
$718$	−12.7986	−0.477640
$719$	37.5424	1.40010	0.700048	−	0.714096i	$-0.253162\pi$
$719$	37.5424	1.40010	0.700048	+	0.714096i	$0.253162\pi$
$720$	−1.11298	−0.0414784
$721$	−1.53657	−0.0572249
$722$	7.14109	0.265764
$723$	15.5358	0.577783
$724$	22.6967	0.843515
$725$	−15.7937	−0.586563
$726$	−11.6853	−0.433681
$727$	12.3840	0.459299	0.229649	−	0.973273i	$-0.426242\pi$
$727$	12.3840	0.459299	0.229649	+	0.973273i	$0.426242\pi$
$728$	0.960055	0.0355820
$729$	1.00000	0.0370370
$730$	16.2901	0.602923
$731$	−12.4998	−0.462322
$732$	48.4558	1.79098
$733$	25.3892	0.937770	0.468885	−	0.883259i	$-0.344656\pi$
$733$	25.3892	0.937770	0.468885	+	0.883259i	$0.344656\pi$
$734$	43.7904	1.61633
$735$	2.86589	0.105710
$736$	0.461916	0.0170265
$737$	0.0755012	0.00278112
$738$	−11.7865	−0.433867
$739$	32.7646	1.20527	0.602633	−	0.798019i	$-0.294118\pi$
$739$	32.7646	1.20527	0.602633	+	0.798019i	$0.294118\pi$
$740$	−14.6942	−0.540170
$741$	8.98426	0.330045
$742$	−2.85750	−0.104902
$743$	−53.0830	−1.94742	−0.973712	−	0.227781i	$-0.926853\pi$
$743$	−53.0830	−1.94742	−0.973712	+	0.227781i	$0.926853\pi$
$744$	15.5432	0.569840
$745$	−0.963498	−0.0352998
$746$	47.3034	1.73190
$747$	17.7868	0.650784
$748$	15.0445	0.550082
$749$	−1.28013	−0.0467749
$750$	−9.68929	−0.353803
$751$	−6.89385	−0.251560	−0.125780	−	0.992058i	$-0.540143\pi$
$751$	−6.89385	−0.251560	−0.125780	+	0.992058i	$0.540143\pi$
$752$	9.23960	0.336933
$753$	20.3888	0.743011
$754$	−17.6305	−0.642065
$755$	6.04840	0.220124
$756$	−0.378260	−0.0137572
$757$	−42.5301	−1.54578	−0.772892	−	0.634538i	$-0.781190\pi$
$757$	−42.5301	−1.54578	−0.772892	+	0.634538i	$0.781190\pi$
$758$	−73.8730	−2.68319
$759$	0.910660	0.0330549
$760$	7.01329	0.254399
$761$	−4.54838	−0.164879	−0.0824393	−	0.996596i	$-0.526271\pi$
$761$	−4.54838	−0.164879	−0.0824393	+	0.996596i	$0.526271\pi$
$762$	19.4197	0.703503
$763$	1.77142	0.0641299
$764$	−26.9017	−0.973269
$765$	−0.410000	−0.0148236
$766$	54.0728	1.95373
$767$	21.8983	0.790700
$768$	−29.1398	−1.05149
$769$	38.7063	1.39578	0.697892	−	0.716203i	$-0.254121\pi$
$769$	38.7063	1.39578	0.697892	+	0.716203i	$0.254121\pi$
$770$	−0.393016	−0.0141633
$771$	−7.52823	−0.271122
$772$	62.9424	2.26535
$773$	9.92555	0.356997	0.178499	−	0.983940i	$-0.442876\pi$
$773$	9.92555	0.356997	0.178499	+	0.983940i	$0.442876\pi$
$774$	30.0451	1.07995
$775$	17.5782	0.631426
$776$	2.71745	0.0975507
$777$	−0.950036	−0.0340823
$778$	70.2044	2.51695
$779$	19.6322	0.703395
$780$	−3.47551	−0.124443
$781$	36.4086	1.30280
$782$	−0.549611	−0.0196540
$783$	3.26863	0.116811
$784$	−18.9749	−0.677675
$785$	−0.410000	−0.0146335
$786$	8.72269	0.311128
$787$	50.7614	1.80945	0.904725	−	0.425996i	$-0.140076\pi$
$787$	50.7614	1.80945	0.904725	+	0.425996i	$0.140076\pi$
$788$	18.4117	0.655888
$789$	17.9501	0.639040
$790$	−4.33241	−0.154140
$791$	−0.512620	−0.0182267
$792$	−17.0159	−0.604634
$793$	28.7851	1.02219
$794$	−3.53619	−0.125495
$795$	4.86760	0.172636
$796$	−15.5826	−0.552312
$797$	−9.36759	−0.331817	−0.165909	−	0.986141i	$-0.553056\pi$
$797$	−9.36759	−0.331817	−0.165909	+	0.986141i	$0.553056\pi$
$798$	0.963626	0.0341120
$799$	3.40368	0.120414
$800$	−9.76104	−0.345105
$801$	−8.21712	−0.290338
$802$	−32.4633	−1.14632
$803$	65.8325	2.32318
$804$	0.0716124	0.00252558
$805$	0.00938757	0.000330868 0
$806$	19.6225	0.691173
$807$	11.7482	0.413558
$808$	34.8959	1.22763
$809$	40.4007	1.42041	0.710207	−	0.703993i	$-0.248601\pi$
$809$	40.4007	1.42041	0.710207	+	0.703993i	$0.248601\pi$
$810$	0.985496	0.0346268
$811$	22.8895	0.803758	0.401879	−	0.915693i	$-0.368357\pi$
$811$	22.8895	0.803758	0.401879	+	0.915693i	$0.368357\pi$
$812$	−1.23639	−0.0433888
$813$	−8.12553	−0.284975
$814$	−90.8235	−3.18336
$815$	8.20058	0.287254
$816$	2.71459	0.0950296
$817$	−50.0446	−1.75084
$818$	−7.89907	−0.276184
$819$	−0.224705	−0.00785183
$820$	−7.59459	−0.265215
$821$	−28.3758	−0.990321	−0.495161	−	0.868801i	$-0.664891\pi$
$821$	−28.3758	−0.990321	−0.495161	+	0.868801i	$0.664891\pi$
$822$	−21.5923	−0.753118
$823$	−34.6693	−1.20850	−0.604248	−	0.796796i	$-0.706527\pi$
$823$	−34.6693	−1.20850	−0.604248	+	0.796796i	$0.706527\pi$
$824$	−65.5620	−2.28396
$825$	−19.2437	−0.669981
$826$	2.34874	0.0817233
$827$	2.74168	0.0953375	0.0476687	−	0.998863i	$-0.484821\pi$
$827$	2.74168	0.0953375	0.0476687	+	0.998863i	$0.484821\pi$
$828$	0.863756	0.0300176
$829$	−21.8288	−0.758147	−0.379073	−	0.925367i	$-0.623757\pi$
$829$	−21.8288	−0.758147	−0.379073	+	0.925367i	$0.623757\pi$
$830$	17.5288	0.608433
$831$	6.58388	0.228392
$832$	−23.0795	−0.800137
$833$	−6.98997	−0.242188
$834$	31.9122	1.10503
$835$	−1.84018	−0.0636820
$836$	60.2326	2.08319
$837$	−3.63794	−0.125746
$838$	−26.4325	−0.913096
$839$	−40.9165	−1.41260	−0.706298	−	0.707915i	$-0.749636\pi$
$839$	−40.9165	−1.41260	−0.706298	+	0.707915i	$0.749636\pi$
$840$	−0.175409	−0.00605219
$841$	−18.3161	−0.631588
$842$	26.9236	0.927849
$843$	−28.5833	−0.984459
$844$	42.4369	1.46074
$845$	3.26538	0.112332
$846$	−8.18125	−0.281277
$847$	−0.486802	−0.0167267
$848$	−32.2281	−1.10672
$849$	4.26204	0.146273
$850$	11.6142	0.398363
$851$	2.16940	0.0743662
$852$	34.5333	1.18309
$853$	4.78186	0.163728	0.0818639	−	0.996644i	$-0.473913\pi$
$853$	4.78186	0.163728	0.0818639	+	0.996644i	$0.473913\pi$
$854$	3.08741	0.105649
$855$	−1.64149	−0.0561378
$856$	−54.6203	−1.86688
$857$	−5.30997	−0.181385	−0.0906926	−	0.995879i	$-0.528908\pi$
$857$	−5.30997	−0.181385	−0.0906926	+	0.995879i	$0.528908\pi$
$858$	−21.4818	−0.733377
$859$	34.0792	1.16277	0.581383	−	0.813630i	$-0.302512\pi$
$859$	34.0792	1.16277	0.581383	+	0.813630i	$0.302512\pi$
$860$	19.3595	0.660153
$861$	−0.491019	−0.0167339
$862$	−4.11561	−0.140178
$863$	32.7447	1.11464	0.557321	−	0.830297i	$-0.311829\pi$
$863$	32.7447	1.11464	0.557321	+	0.830297i	$0.311829\pi$
$864$	2.02013	0.0687261
$865$	2.56758	0.0873002
$866$	65.3808	2.22173
$867$	1.00000	0.0339618
$868$	1.37609	0.0467074
$869$	−17.5084	−0.593932
$870$	3.22122	0.109210
$871$	0.0425413	0.00144146
$872$	75.5827	2.55955
$873$	−0.636030	−0.0215264
$874$	−2.20044	−0.0744309
$875$	−0.403651	−0.0136459
$876$	62.4417	2.10971
$877$	−16.1242	−0.544475	−0.272237	−	0.962230i	$-0.587764\pi$
$877$	−16.1242	−0.544475	−0.272237	+	0.962230i	$0.587764\pi$
$878$	−12.2940	−0.414903
$879$	−17.8906	−0.603434
$880$	−4.43261	−0.149423
$881$	48.6903	1.64042	0.820209	−	0.572063i	$-0.193857\pi$
$881$	48.6903	1.64042	0.820209	+	0.572063i	$0.193857\pi$
$882$	16.8014	0.565733
$883$	8.13627	0.273807	0.136904	−	0.990584i	$-0.456285\pi$
$883$	8.13627	0.273807	0.136904	+	0.990584i	$0.456285\pi$
$884$	8.47685	0.285107
$885$	−4.00097	−0.134491
$886$	3.99626	0.134257
$887$	43.3371	1.45512	0.727559	−	0.686045i	$-0.240655\pi$
$887$	43.3371	1.45512	0.727559	+	0.686045i	$0.240655\pi$
$888$	−40.5359	−1.36029
$889$	0.809016	0.0271335
$890$	−8.09794	−0.271443
$891$	3.98265	0.133424
$892$	−83.6948	−2.80231
$893$	13.6271	0.456013
$894$	−5.64855	−0.188916
$895$	0.650059	0.0217291
$896$	−2.07087	−0.0691829
$897$	0.513113	0.0171324
$898$	−56.4799	−1.88476
$899$	−11.8911	−0.396590
$900$	−18.2526	−0.608419
$901$	−11.8722	−0.395520
$902$	−46.9415	−1.56298
$903$	1.25166	0.0416527
$904$	−21.8723	−0.727463
$905$	−2.46343	−0.0818872
$906$	35.4590	1.17805
$907$	53.8864	1.78927	0.894634	−	0.446800i	$-0.147436\pi$
$907$	53.8864	1.78927	0.894634	+	0.446800i	$0.147436\pi$
$908$	43.4543	1.44208
$909$	−8.16752	−0.270899
$910$	−0.221446	−0.00734085
$911$	13.5394	0.448580	0.224290	−	0.974522i	$-0.427994\pi$
$911$	13.5394	0.448580	0.224290	+	0.974522i	$0.427994\pi$
$912$	10.8682	0.359882
$913$	70.8385	2.34441
$914$	89.1485	2.94877
$915$	−5.25925	−0.173865
$916$	50.7739	1.67762
$917$	0.363382	0.0119999
$918$	−2.40365	−0.0793321
$919$	−29.3919	−0.969548	−0.484774	−	0.874639i	$-0.661098\pi$
$919$	−29.3919	−0.969548	−0.484774	+	0.874639i	$0.661098\pi$
$920$	0.400546	0.0132056
$921$	2.45752	0.0809780
$922$	−65.6366	−2.16163
$923$	20.5145	0.675242
$924$	−1.50648	−0.0495594
$925$	−45.8431	−1.50731
$926$	−36.2359	−1.19079
$927$	15.3451	0.503998
$928$	6.60304	0.216756
$929$	−29.8393	−0.978996	−0.489498	−	0.872004i	$-0.662820\pi$
$929$	−29.8393	−0.978996	−0.489498	+	0.872004i	$0.662820\pi$
$930$	−3.58517	−0.117562
$931$	−27.9853	−0.917180
$932$	−9.24508	−0.302833
$933$	−29.5833	−0.968514
$934$	60.9481	1.99428
$935$	−1.63289	−0.0534011
$936$	−9.58765	−0.313382
$937$	16.2651	0.531359	0.265680	−	0.964061i	$-0.414404\pi$
$937$	16.2651	0.531359	0.265680	+	0.964061i	$0.414404\pi$
$938$	0.00456286	0.000148983 0
$939$	25.4457	0.830391
$940$	−5.27156	−0.171939
$941$	6.86116	0.223668	0.111834	−	0.993727i	$-0.464328\pi$
$941$	6.86116	0.223668	0.111834	+	0.993727i	$0.464328\pi$
$942$	−2.40365	−0.0783150
$943$	1.12124	0.0365126
$944$	26.4902	0.862182
$945$	0.0410552	0.00133553
$946$	119.659	3.89045
$947$	14.3563	0.466515	0.233258	−	0.972415i	$-0.425061\pi$
$947$	14.3563	0.466515	0.233258	+	0.972415i	$0.425061\pi$
$948$	−16.6066	−0.539358
$949$	37.0935	1.20410
$950$	46.4989	1.50862
$951$	22.6641	0.734935
$952$	0.427827	0.0138659
$953$	28.7843	0.932415	0.466208	−	0.884675i	$-0.345620\pi$
$953$	28.7843	0.932415	0.466208	+	0.884675i	$0.345620\pi$
$954$	28.5366	0.923905
$955$	2.91983	0.0944835
$956$	14.2616	0.461254
$957$	13.0178	0.420806
$958$	−56.6433	−1.83006
$959$	−0.899523	−0.0290471
$960$	4.21679	0.136096
$961$	−17.7654	−0.573078
$962$	−51.1746	−1.64994
$963$	12.7841	0.411962
$964$	58.6867	1.89017
$965$	−6.83158	−0.219916
$966$	0.0550350	0.00177072
$967$	−46.3533	−1.49062	−0.745310	−	0.666718i	$-0.767699\pi$
$967$	−46.3533	−1.49062	−0.745310	+	0.666718i	$0.767699\pi$
$968$	−20.7707	−0.667596
$969$	4.00363	0.128615
$970$	−0.626805	−0.0201255
$971$	−45.2146	−1.45100	−0.725502	−	0.688220i	$-0.758392\pi$
$971$	−45.2146	−1.45100	−0.725502	+	0.688220i	$0.758392\pi$
$972$	3.77751	0.121164
$973$	1.32945	0.0426201
$974$	−78.5775	−2.51779
$975$	−10.8429	−0.347251
$976$	34.8212	1.11460
$977$	−43.0871	−1.37848	−0.689239	−	0.724534i	$-0.742055\pi$
$977$	−43.0871	−1.37848	−0.689239	+	0.724534i	$0.742055\pi$
$978$	48.0763	1.53731
$979$	−32.7259	−1.04592
$980$	10.8259	0.345822
$981$	−17.6904	−0.564812
$982$	62.3201	1.98871
$983$	−28.9084	−0.922034	−0.461017	−	0.887391i	$-0.652515\pi$
$983$	−28.9084	−0.922034	−0.461017	+	0.887391i	$0.652515\pi$
$984$	−20.9507	−0.667883
$985$	−1.99835	−0.0636726
$986$	−7.85663	−0.250206
$987$	−0.340826	−0.0108486
$988$	33.9382	1.07972
$989$	−2.85817	−0.0908845
$990$	3.92488	0.124741
$991$	29.9084	0.950073	0.475036	−	0.879966i	$-0.342435\pi$
$991$	29.9084	0.950073	0.475036	+	0.879966i	$0.342435\pi$
$992$	−7.34909	−0.233334
$993$	−22.2428	−0.705854
$994$	2.20032	0.0697900
$995$	1.69129	0.0536176
$996$	67.1898	2.12899
$997$	−9.38292	−0.297160	−0.148580	−	0.988900i	$-0.547470\pi$
$997$	−9.38292	−0.297160	−0.148580	+	0.988900i	$0.547470\pi$
$998$	72.5586	2.29680
$999$	9.48759	0.300174

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.i.1.4	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.i.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-2.40365 q^{2} +1.00000 q^{3} +3.77751 q^{4} -0.410000 q^{5} -2.40365 q^{6} -0.100135 q^{7} -4.27252 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.40365 q^{2} +1.00000 q^{3} +3.77751 q^{4} -0.410000 q^{5} -2.40365 q^{6} -0.100135 q^{7} -4.27252 q^{8} +1.00000 q^{9} +0.985496 q^{10} +3.98265 q^{11} +3.77751 q^{12} +2.24403 q^{13} +0.240688 q^{14} -0.410000 q^{15} +2.71459 q^{16} +1.00000 q^{17} -2.40365 q^{18} +4.00363 q^{19} -1.54878 q^{20} -0.100135 q^{21} -9.57287 q^{22} +0.228657 q^{23} -4.27252 q^{24} -4.83190 q^{25} -5.39385 q^{26} +1.00000 q^{27} -0.378260 q^{28} +3.26863 q^{29} +0.985496 q^{30} -3.63794 q^{31} +2.02013 q^{32} +3.98265 q^{33} -2.40365 q^{34} +0.0410552 q^{35} +3.77751 q^{36} +9.48759 q^{37} -9.62331 q^{38} +2.24403 q^{39} +1.75173 q^{40} +4.90359 q^{41} +0.240688 q^{42} -12.4998 q^{43} +15.0445 q^{44} -0.410000 q^{45} -0.549611 q^{46} +3.40368 q^{47} +2.71459 q^{48} -6.98997 q^{49} +11.6142 q^{50} +1.00000 q^{51} +8.47685 q^{52} -11.8722 q^{53} -2.40365 q^{54} -1.63289 q^{55} +0.427827 q^{56} +4.00363 q^{57} -7.85663 q^{58} +9.75846 q^{59} -1.54878 q^{60} +12.8274 q^{61} +8.74432 q^{62} -0.100135 q^{63} -10.2848 q^{64} -0.920053 q^{65} -9.57287 q^{66} +0.0189575 q^{67} +3.77751 q^{68} +0.228657 q^{69} -0.0986822 q^{70} +9.14180 q^{71} -4.27252 q^{72} +16.5298 q^{73} -22.8048 q^{74} -4.83190 q^{75} +15.1238 q^{76} -0.398801 q^{77} -5.39385 q^{78} -4.39617 q^{79} -1.11298 q^{80} +1.00000 q^{81} -11.7865 q^{82} +17.7868 q^{83} -0.378260 q^{84} -0.410000 q^{85} +30.0451 q^{86} +3.26863 q^{87} -17.0159 q^{88} -8.21712 q^{89} +0.985496 q^{90} -0.224705 q^{91} +0.863756 q^{92} -3.63794 q^{93} -8.18125 q^{94} -1.64149 q^{95} +2.02013 q^{96} -0.636030 q^{97} +16.8014 q^{98} +3.98265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9}+O(q^{10}) \) 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