LMFDB - Embedded newform 8007.2.a.i.1.14

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			1.14
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.76021 q^{2} +1.00000 q^{3} +1.09834 q^{4} +1.37447 q^{5} -1.76021 q^{6} +3.49395 q^{7} +1.58711 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.76021 q^{2} +1.00000 q^{3} +1.09834 q^{4} +1.37447 q^{5} -1.76021 q^{6} +3.49395 q^{7} +1.58711 q^{8} +1.00000 q^{9} -2.41936 q^{10} +3.10297 q^{11} +1.09834 q^{12} +3.72655 q^{13} -6.15008 q^{14} +1.37447 q^{15} -4.99033 q^{16} +1.00000 q^{17} -1.76021 q^{18} -0.330170 q^{19} +1.50964 q^{20} +3.49395 q^{21} -5.46188 q^{22} +5.36334 q^{23} +1.58711 q^{24} -3.11083 q^{25} -6.55952 q^{26} +1.00000 q^{27} +3.83754 q^{28} -0.550988 q^{29} -2.41936 q^{30} +0.867291 q^{31} +5.60981 q^{32} +3.10297 q^{33} -1.76021 q^{34} +4.80233 q^{35} +1.09834 q^{36} +5.37516 q^{37} +0.581168 q^{38} +3.72655 q^{39} +2.18144 q^{40} -6.80587 q^{41} -6.15008 q^{42} +7.82551 q^{43} +3.40812 q^{44} +1.37447 q^{45} -9.44060 q^{46} +1.17414 q^{47} -4.99033 q^{48} +5.20768 q^{49} +5.47572 q^{50} +1.00000 q^{51} +4.09302 q^{52} -4.35990 q^{53} -1.76021 q^{54} +4.26494 q^{55} +5.54528 q^{56} -0.330170 q^{57} +0.969855 q^{58} -10.1558 q^{59} +1.50964 q^{60} -1.22226 q^{61} -1.52661 q^{62} +3.49395 q^{63} +0.106220 q^{64} +5.12203 q^{65} -5.46188 q^{66} -4.23946 q^{67} +1.09834 q^{68} +5.36334 q^{69} -8.45311 q^{70} +6.91234 q^{71} +1.58711 q^{72} -11.5342 q^{73} -9.46141 q^{74} -3.11083 q^{75} -0.362639 q^{76} +10.8416 q^{77} -6.55952 q^{78} +15.5061 q^{79} -6.85906 q^{80} +1.00000 q^{81} +11.9798 q^{82} +14.5814 q^{83} +3.83754 q^{84} +1.37447 q^{85} -13.7745 q^{86} -0.550988 q^{87} +4.92476 q^{88} +0.759803 q^{89} -2.41936 q^{90} +13.0204 q^{91} +5.89077 q^{92} +0.867291 q^{93} -2.06674 q^{94} -0.453809 q^{95} +5.60981 q^{96} -5.41318 q^{97} -9.16661 q^{98} +3.10297 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9}+O(q^{10}) $ 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9	\( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9} + 4 q^{10} + 23 q^{11} + 70 q^{12} + 10 q^{13} + 18 q^{14} + 19 q^{15} + 72 q^{16} + 63 q^{17} + 10 q^{18} + 6 q^{19} + 48 q^{20} + 11 q^{21} + 21 q^{22} + 44 q^{23} + 27 q^{24} + 110 q^{25} + 41 q^{26} + 63 q^{27} + 26 q^{28} + 35 q^{29} + 4 q^{30} + q^{31} + 54 q^{32} + 23 q^{33} + 10 q^{34} + 47 q^{35} + 70 q^{36} + 40 q^{37} + 38 q^{38} + 10 q^{39} - 10 q^{40} + 35 q^{41} + 18 q^{42} + 27 q^{43} + 46 q^{44} + 19 q^{45} + 8 q^{46} + 29 q^{47} + 72 q^{48} + 114 q^{49} + 27 q^{50} + 63 q^{51} - q^{52} + 75 q^{53} + 10 q^{54} + 5 q^{55} + 24 q^{56} + 6 q^{57} + 41 q^{58} + 105 q^{59} + 48 q^{60} + 5 q^{61} + 22 q^{62} + 11 q^{63} + 61 q^{64} + 49 q^{65} + 21 q^{66} + 4 q^{67} + 70 q^{68} + 44 q^{69} - 16 q^{70} + 16 q^{71} + 27 q^{72} + 39 q^{73} + 54 q^{74} + 110 q^{75} + 6 q^{76} + 88 q^{77} + 41 q^{78} + 16 q^{79} + 102 q^{80} + 63 q^{81} - 29 q^{82} + 73 q^{83} + 26 q^{84} + 19 q^{85} + 46 q^{86} + 35 q^{87} + 18 q^{88} + 88 q^{89} + 4 q^{90} - 15 q^{91} + 110 q^{92} + q^{93} - 8 q^{94} + 28 q^{95} + 54 q^{96} + 70 q^{97} + 33 q^{98} + 23 q^{99}+O(q^{100}) \) 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9 + 4 * q^10 + 23 * q^11 + 70 * q^12 + 10 * q^13 + 18 * q^14 + 19 * q^15 + 72 * q^16 + 63 * q^17 + 10 * q^18 + 6 * q^19 + 48 * q^20 + 11 * q^21 + 21 * q^22 + 44 * q^23 + 27 * q^24 + 110 * q^25 + 41 * q^26 + 63 * q^27 + 26 * q^28 + 35 * q^29 + 4 * q^30 + q^31 + 54 * q^32 + 23 * q^33 + 10 * q^34 + 47 * q^35 + 70 * q^36 + 40 * q^37 + 38 * q^38 + 10 * q^39 - 10 * q^40 + 35 * q^41 + 18 * q^42 + 27 * q^43 + 46 * q^44 + 19 * q^45 + 8 * q^46 + 29 * q^47 + 72 * q^48 + 114 * q^49 + 27 * q^50 + 63 * q^51 - q^52 + 75 * q^53 + 10 * q^54 + 5 * q^55 + 24 * q^56 + 6 * q^57 + 41 * q^58 + 105 * q^59 + 48 * q^60 + 5 * q^61 + 22 * q^62 + 11 * q^63 + 61 * q^64 + 49 * q^65 + 21 * q^66 + 4 * q^67 + 70 * q^68 + 44 * q^69 - 16 * q^70 + 16 * q^71 + 27 * q^72 + 39 * q^73 + 54 * q^74 + 110 * q^75 + 6 * q^76 + 88 * q^77 + 41 * q^78 + 16 * q^79 + 102 * q^80 + 63 * q^81 - 29 * q^82 + 73 * q^83 + 26 * q^84 + 19 * q^85 + 46 * q^86 + 35 * q^87 + 18 * q^88 + 88 * q^89 + 4 * q^90 - 15 * q^91 + 110 * q^92 + q^93 - 8 * q^94 + 28 * q^95 + 54 * q^96 + 70 * q^97 + 33 * q^98 + 23 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.76021	−1.24466	−0.622328	−	0.782756i	$-0.713813\pi$
$2$	−1.76021	−1.24466	−0.622328	+	0.782756i	$0.713813\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.09834	0.549170
$5$	1.37447	0.614682	0.307341	−	0.951599i	$-0.400561\pi$
$5$	1.37447	0.614682	0.307341	+	0.951599i	$0.400561\pi$
$6$	−1.76021	−0.718603
$7$	3.49395	1.32059	0.660294	−	0.751007i	$-0.270432\pi$
$7$	3.49395	1.32059	0.660294	+	0.751007i	$0.270432\pi$
$8$	1.58711	0.561129
$9$	1.00000	0.333333
$10$	−2.41936	−0.765068
$11$	3.10297	0.935581	0.467791	−	0.883839i	$-0.345050\pi$
$11$	3.10297	0.935581	0.467791	+	0.883839i	$0.345050\pi$
$12$	1.09834	0.317063
$13$	3.72655	1.03356	0.516780	−	0.856118i	$-0.327131\pi$
$13$	3.72655	1.03356	0.516780	+	0.856118i	$0.327131\pi$
$14$	−6.15008	−1.64368
$15$	1.37447	0.354887
$16$	−4.99033	−1.24758
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.76021	−0.414886
$19$	−0.330170	−0.0757461	−0.0378731	−	0.999283i	$-0.512058\pi$
$19$	−0.330170	−0.0757461	−0.0378731	+	0.999283i	$0.512058\pi$
$20$	1.50964	0.337565
$21$	3.49395	0.762442
$22$	−5.46188	−1.16448
$23$	5.36334	1.11833	0.559166	−	0.829055i	$-0.311121\pi$
$23$	5.36334	1.11833	0.559166	+	0.829055i	$0.311121\pi$
$24$	1.58711	0.323968
$25$	−3.11083	−0.622166
$26$	−6.55952	−1.28643
$27$	1.00000	0.192450
$28$	3.83754	0.725228
$29$	−0.550988	−0.102316	−0.0511579	−	0.998691i	$-0.516291\pi$
$29$	−0.550988	−0.102316	−0.0511579	+	0.998691i	$0.516291\pi$
$30$	−2.41936	−0.441712
$31$	0.867291	0.155770	0.0778850	−	0.996962i	$-0.475183\pi$
$31$	0.867291	0.155770	0.0778850	+	0.996962i	$0.475183\pi$
$32$	5.60981	0.991683
$33$	3.10297	0.540158
$34$	−1.76021	−0.301874
$35$	4.80233	0.811742
$36$	1.09834	0.183057
$37$	5.37516	0.883671	0.441835	−	0.897096i	$-0.354328\pi$
$37$	5.37516	0.883671	0.441835	+	0.897096i	$0.354328\pi$
$38$	0.581168	0.0942779
$39$	3.72655	0.596726
$40$	2.18144	0.344915
$41$	−6.80587	−1.06290	−0.531449	−	0.847090i	$-0.678352\pi$
$41$	−6.80587	−1.06290	−0.531449	+	0.847090i	$0.678352\pi$
$42$	−6.15008	−0.948979
$43$	7.82551	1.19338	0.596690	−	0.802472i	$-0.296482\pi$
$43$	7.82551	1.19338	0.596690	+	0.802472i	$0.296482\pi$
$44$	3.40812	0.513793
$45$	1.37447	0.204894
$46$	−9.44060	−1.39194
$47$	1.17414	0.171267	0.0856333	−	0.996327i	$-0.472709\pi$
$47$	1.17414	0.171267	0.0856333	+	0.996327i	$0.472709\pi$
$48$	−4.99033	−0.720292
$49$	5.20768	0.743954
$50$	5.47572	0.774383
$51$	1.00000	0.140028
$52$	4.09302	0.567600
$53$	−4.35990	−0.598878	−0.299439	−	0.954115i	$-0.596799\pi$
$53$	−4.35990	−0.598878	−0.299439	+	0.954115i	$0.596799\pi$
$54$	−1.76021	−0.239534
$55$	4.26494	0.575085
$56$	5.54528	0.741020
$57$	−0.330170	−0.0437321
$58$	0.969855	0.127348
$59$	−10.1558	−1.32218	−0.661088	−	0.750309i	$-0.729905\pi$
$59$	−10.1558	−1.32218	−0.661088	+	0.750309i	$0.729905\pi$
$60$	1.50964	0.194893
$61$	−1.22226	−0.156494	−0.0782471	−	0.996934i	$-0.524932\pi$
$61$	−1.22226	−0.156494	−0.0782471	+	0.996934i	$0.524932\pi$
$62$	−1.52661	−0.193880
$63$	3.49395	0.440196
$64$	0.106220	0.0132775
$65$	5.12203	0.635310
$66$	−5.46188	−0.672311
$67$	−4.23946	−0.517932	−0.258966	−	0.965886i	$-0.583382\pi$
$67$	−4.23946	−0.517932	−0.258966	+	0.965886i	$0.583382\pi$
$68$	1.09834	0.133193
$69$	5.36334	0.645670
$70$	−8.45311	−1.01034
$71$	6.91234	0.820344	0.410172	−	0.912008i	$-0.365469\pi$
$71$	6.91234	0.820344	0.410172	+	0.912008i	$0.365469\pi$
$72$	1.58711	0.187043
$73$	−11.5342	−1.34998	−0.674988	−	0.737829i	$-0.735851\pi$
$73$	−11.5342	−1.34998	−0.674988	+	0.737829i	$0.735851\pi$
$74$	−9.46141	−1.09987
$75$	−3.11083	−0.359208
$76$	−0.362639	−0.0415975
$77$	10.8416	1.23552
$78$	−6.55952	−0.742719
$79$	15.5061	1.74457	0.872284	−	0.489000i	$-0.162638\pi$
$79$	15.5061	1.74457	0.872284	+	0.489000i	$0.162638\pi$
$80$	−6.85906	−0.766866
$81$	1.00000	0.111111
$82$	11.9798	1.32294
$83$	14.5814	1.60052	0.800261	−	0.599652i	$-0.204694\pi$
$83$	14.5814	1.60052	0.800261	+	0.599652i	$0.204694\pi$
$84$	3.83754	0.418710
$85$	1.37447	0.149082
$86$	−13.7745	−1.48535
$87$	−0.550988	−0.0590721
$88$	4.92476	0.524981
$89$	0.759803	0.0805390	0.0402695	−	0.999189i	$-0.487178\pi$
$89$	0.759803	0.0805390	0.0402695	+	0.999189i	$0.487178\pi$
$90$	−2.41936	−0.255023
$91$	13.0204	1.36491
$92$	5.89077	0.614155
$93$	0.867291	0.0899338
$94$	−2.06674	−0.213168
$95$	−0.453809	−0.0465598
$96$	5.60981	0.572548
$97$	−5.41318	−0.549625	−0.274813	−	0.961498i	$-0.588616\pi$
$97$	−5.41318	−0.549625	−0.274813	+	0.961498i	$0.588616\pi$
$98$	−9.16661	−0.925967
$99$	3.10297	0.311860
$100$	−3.41675	−0.341675
$101$	9.97328	0.992378	0.496189	−	0.868214i	$-0.334732\pi$
$101$	9.97328	0.992378	0.496189	+	0.868214i	$0.334732\pi$
$102$	−1.76021	−0.174287
$103$	−3.56750	−0.351517	−0.175758	−	0.984433i	$-0.556238\pi$
$103$	−3.56750	−0.351517	−0.175758	+	0.984433i	$0.556238\pi$
$104$	5.91445	0.579960
$105$	4.80233	0.468659
$106$	7.67433	0.745397
$107$	18.1078	1.75055	0.875275	−	0.483626i	$-0.160680\pi$
$107$	18.1078	1.75055	0.875275	+	0.483626i	$0.160680\pi$
$108$	1.09834	0.105688
$109$	−8.44301	−0.808694	−0.404347	−	0.914606i	$-0.632501\pi$
$109$	−8.44301	−0.808694	−0.404347	+	0.914606i	$0.632501\pi$
$110$	−7.50719	−0.715783
$111$	5.37516	0.510187
$112$	−17.4360	−1.64754
$113$	−5.58184	−0.525095	−0.262548	−	0.964919i	$-0.584563\pi$
$113$	−5.58184	−0.525095	−0.262548	+	0.964919i	$0.584563\pi$
$114$	0.581168	0.0544314
$115$	7.37174	0.687419
$116$	−0.605172	−0.0561888
$117$	3.72655	0.344520
$118$	17.8764	1.64565
$119$	3.49395	0.320290
$120$	2.18144	0.199137
$121$	−1.37157	−0.124688
$122$	2.15143	0.194781
$123$	−6.80587	−0.613664
$124$	0.952580	0.0855442
$125$	−11.1481	−0.997116
$126$	−6.15008	−0.547893
$127$	−5.97060	−0.529805	−0.264902	−	0.964275i	$-0.585340\pi$
$127$	−5.97060	−0.529805	−0.264902	+	0.964275i	$0.585340\pi$
$128$	−11.4066	−1.00821
$129$	7.82551	0.688998
$130$	−9.01586	−0.790743
$131$	17.6465	1.54178	0.770890	−	0.636969i	$-0.219812\pi$
$131$	17.6465	1.54178	0.770890	+	0.636969i	$0.219812\pi$
$132$	3.40812	0.296639
$133$	−1.15360	−0.100029
$134$	7.46233	0.644648
$135$	1.37447	0.118296
$136$	1.58711	0.136094
$137$	−9.72705	−0.831038	−0.415519	−	0.909585i	$-0.636400\pi$
$137$	−9.72705	−0.831038	−0.415519	+	0.909585i	$0.636400\pi$
$138$	−9.44060	−0.803637
$139$	19.8967	1.68762	0.843810	−	0.536643i	$-0.180308\pi$
$139$	19.8967	1.68762	0.843810	+	0.536643i	$0.180308\pi$
$140$	5.27459	0.445784
$141$	1.17414	0.0988808
$142$	−12.1672	−1.02105
$143$	11.5634	0.966979
$144$	−4.99033	−0.415861
$145$	−0.757316	−0.0628917
$146$	20.3026	1.68026
$147$	5.20768	0.429522
$148$	5.90375	0.485285
$149$	9.51226	0.779274	0.389637	−	0.920968i	$-0.372600\pi$
$149$	9.51226	0.779274	0.389637	+	0.920968i	$0.372600\pi$
$150$	5.47572	0.447090
$151$	−9.97033	−0.811374	−0.405687	−	0.914012i	$-0.632968\pi$
$151$	−9.97033	−0.811374	−0.405687	+	0.914012i	$0.632968\pi$
$152$	−0.524016	−0.0425033
$153$	1.00000	0.0808452
$154$	−19.0835	−1.53779
$155$	1.19206	0.0957490
$156$	4.09302	0.327704
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	−27.2939	−2.17139
$159$	−4.35990	−0.345762
$160$	7.71051	0.609569
$161$	18.7392	1.47686
$162$	−1.76021	−0.138295
$163$	−15.6188	−1.22336	−0.611680	−	0.791105i	$-0.709506\pi$
$163$	−15.6188	−1.22336	−0.611680	+	0.791105i	$0.709506\pi$
$164$	−7.47515	−0.583712
$165$	4.26494	0.332025
$166$	−25.6664	−1.99210
$167$	0.513843	0.0397624	0.0198812	−	0.999802i	$-0.493671\pi$
$167$	0.513843	0.0397624	0.0198812	+	0.999802i	$0.493671\pi$
$168$	5.54528	0.427828
$169$	0.887192	0.0682455
$170$	−2.41936	−0.185556
$171$	−0.330170	−0.0252487
$172$	8.59507	0.655368
$173$	1.37509	0.104546	0.0522731	−	0.998633i	$-0.483353\pi$
$173$	1.37509	0.104546	0.0522731	+	0.998633i	$0.483353\pi$
$174$	0.969855	0.0735245
$175$	−10.8691	−0.821626
$176$	−15.4848	−1.16721
$177$	−10.1558	−0.763358
$178$	−1.33741	−0.100243
$179$	−4.95528	−0.370375	−0.185188	−	0.982703i	$-0.559289\pi$
$179$	−4.95528	−0.370375	−0.185188	+	0.982703i	$0.559289\pi$
$180$	1.50964	0.112522
$181$	4.28644	0.318609	0.159304	−	0.987230i	$-0.449075\pi$
$181$	4.28644	0.318609	0.159304	+	0.987230i	$0.449075\pi$
$182$	−22.9186	−1.69884
$183$	−1.22226	−0.0903519
$184$	8.51221	0.627528
$185$	7.38799	0.543176
$186$	−1.52661	−0.111937
$187$	3.10297	0.226912
$188$	1.28961	0.0940545
$189$	3.49395	0.254147
$190$	0.798798	0.0579509
$191$	3.91519	0.283293	0.141647	−	0.989917i	$-0.454760\pi$
$191$	3.91519	0.283293	0.141647	+	0.989917i	$0.454760\pi$
$192$	0.106220	0.00766579
$193$	−6.47865	−0.466344	−0.233172	−	0.972436i	$-0.574910\pi$
$193$	−6.47865	−0.466344	−0.233172	+	0.972436i	$0.574910\pi$
$194$	9.52834	0.684095
$195$	5.12203	0.366797
$196$	5.71980	0.408557
$197$	5.68002	0.404685	0.202342	−	0.979315i	$-0.435145\pi$
$197$	5.68002	0.404685	0.202342	+	0.979315i	$0.435145\pi$
$198$	−5.46188	−0.388159
$199$	17.5160	1.24168	0.620840	−	0.783938i	$-0.286792\pi$
$199$	17.5160	1.24168	0.620840	+	0.783938i	$0.286792\pi$
$200$	−4.93724	−0.349115
$201$	−4.23946	−0.299028
$202$	−17.5551	−1.23517
$203$	−1.92512	−0.135117
$204$	1.09834	0.0768992
$205$	−9.35446	−0.653344
$206$	6.27956	0.437518
$207$	5.36334	0.372778
$208$	−18.5967	−1.28945
$209$	−1.02451	−0.0708667
$210$	−8.45311	−0.583320
$211$	−19.7919	−1.36253	−0.681264	−	0.732037i	$-0.738570\pi$
$211$	−19.7919	−1.36253	−0.681264	+	0.732037i	$0.738570\pi$
$212$	−4.78865	−0.328886
$213$	6.91234	0.473626
$214$	−31.8736	−2.17883
$215$	10.7559	0.733549
$216$	1.58711	0.107989
$217$	3.03027	0.205708
$218$	14.8615	1.00655
$219$	−11.5342	−0.779409
$220$	4.68436	0.315819
$221$	3.72655	0.250675
$222$	−9.46141	−0.635008
$223$	−25.7399	−1.72367	−0.861836	−	0.507188i	$-0.830685\pi$
$223$	−25.7399	−1.72367	−0.861836	+	0.507188i	$0.830685\pi$
$224$	19.6004	1.30961
$225$	−3.11083	−0.207389
$226$	9.82521	0.653563
$227$	−20.8377	−1.38305	−0.691523	−	0.722355i	$-0.743060\pi$
$227$	−20.8377	−1.38305	−0.691523	+	0.722355i	$0.743060\pi$
$228$	−0.362639	−0.0240163
$229$	−1.41135	−0.0932646	−0.0466323	−	0.998912i	$-0.514849\pi$
$229$	−1.41135	−0.0932646	−0.0466323	+	0.998912i	$0.514849\pi$
$230$	−12.9758	−0.855600
$231$	10.8416	0.713326
$232$	−0.874479	−0.0574124
$233$	−17.9289	−1.17456	−0.587282	−	0.809382i	$-0.699802\pi$
$233$	−17.9289	−1.17456	−0.587282	+	0.809382i	$0.699802\pi$
$234$	−6.55952	−0.428809
$235$	1.61383	0.105274
$236$	−11.1545	−0.726099
$237$	15.5061	1.00723
$238$	−6.15008	−0.398651
$239$	6.47931	0.419112	0.209556	−	0.977797i	$-0.432798\pi$
$239$	6.47931	0.419112	0.209556	+	0.977797i	$0.432798\pi$
$240$	−6.85906	−0.442750
$241$	−19.2669	−1.24109	−0.620546	−	0.784170i	$-0.713089\pi$
$241$	−19.2669	−1.24109	−0.620546	+	0.784170i	$0.713089\pi$
$242$	2.41425	0.155194
$243$	1.00000	0.0641500
$244$	−1.34245	−0.0859419
$245$	7.15780	0.457295
$246$	11.9798	0.763801
$247$	−1.23040	−0.0782882
$248$	1.37649	0.0874070
$249$	14.5814	0.924062
$250$	19.6230	1.24107
$251$	6.80855	0.429752	0.214876	−	0.976641i	$-0.431065\pi$
$251$	6.80855	0.429752	0.214876	+	0.976641i	$0.431065\pi$
$252$	3.83754	0.241743
$253$	16.6423	1.04629
$254$	10.5095	0.659425
$255$	1.37447	0.0860727
$256$	19.8655	1.24160
$257$	−0.836208	−0.0521612	−0.0260806	−	0.999660i	$-0.508303\pi$
$257$	−0.836208	−0.0521612	−0.0260806	+	0.999660i	$0.508303\pi$
$258$	−13.7745	−0.857566
$259$	18.7805	1.16697
$260$	5.62574	0.348893
$261$	−0.550988	−0.0341053
$262$	−31.0615	−1.91899
$263$	−2.22750	−0.137354	−0.0686769	−	0.997639i	$-0.521878\pi$
$263$	−2.22750	−0.137354	−0.0686769	+	0.997639i	$0.521878\pi$
$264$	4.92476	0.303098
$265$	−5.99255	−0.368119
$266$	2.03057	0.124502
$267$	0.759803	0.0464992
$268$	−4.65636	−0.284433
$269$	5.17960	0.315806	0.157903	−	0.987455i	$-0.449527\pi$
$269$	5.17960	0.315806	0.157903	+	0.987455i	$0.449527\pi$
$270$	−2.41936	−0.147237
$271$	−7.23287	−0.439366	−0.219683	−	0.975571i	$-0.570502\pi$
$271$	−7.23287	−0.439366	−0.219683	+	0.975571i	$0.570502\pi$
$272$	−4.99033	−0.302583
$273$	13.0204	0.788029
$274$	17.1217	1.03436
$275$	−9.65282	−0.582087
$276$	5.89077	0.354582
$277$	−8.52934	−0.512478	−0.256239	−	0.966613i	$-0.582483\pi$
$277$	−8.52934	−0.512478	−0.256239	+	0.966613i	$0.582483\pi$
$278$	−35.0224	−2.10051
$279$	0.867291	0.0519233
$280$	7.62183	0.455491
$281$	−1.53320	−0.0914630	−0.0457315	−	0.998954i	$-0.514562\pi$
$281$	−1.53320	−0.0914630	−0.0457315	+	0.998954i	$0.514562\pi$
$282$	−2.06674	−0.123073
$283$	−18.9957	−1.12918	−0.564588	−	0.825373i	$-0.690965\pi$
$283$	−18.9957	−1.12918	−0.564588	+	0.825373i	$0.690965\pi$
$284$	7.59210	0.450509
$285$	−0.453809	−0.0268813
$286$	−20.3540	−1.20356
$287$	−23.7793	−1.40365
$288$	5.60981	0.330561
$289$	1.00000	0.0588235
$290$	1.33304	0.0782786
$291$	−5.41318	−0.317326
$292$	−12.6685	−0.741366
$293$	29.3679	1.71569	0.857846	−	0.513907i	$-0.171802\pi$
$293$	29.3679	1.71569	0.857846	+	0.513907i	$0.171802\pi$
$294$	−9.16661	−0.534607
$295$	−13.9589	−0.812717
$296$	8.53097	0.495853
$297$	3.10297	0.180053
$298$	−16.7436	−0.969929
$299$	19.9868	1.15586
$300$	−3.41675	−0.197266
$301$	27.3419	1.57596
$302$	17.5499	1.00988
$303$	9.97328	0.572950
$304$	1.64766	0.0944996
$305$	−1.67996	−0.0961941
$306$	−1.76021	−0.100625
$307$	10.9824	0.626798	0.313399	−	0.949622i	$-0.398532\pi$
$307$	10.9824	0.626798	0.313399	+	0.949622i	$0.398532\pi$
$308$	11.9078	0.678509
$309$	−3.56750	−0.202948
$310$	−2.09828	−0.119175
$311$	8.98438	0.509457	0.254729	−	0.967013i	$-0.418014\pi$
$311$	8.98438	0.509457	0.254729	+	0.967013i	$0.418014\pi$
$312$	5.91445	0.334840
$313$	−14.5718	−0.823649	−0.411825	−	0.911263i	$-0.635108\pi$
$313$	−14.5718	−0.823649	−0.411825	+	0.911263i	$0.635108\pi$
$314$	−1.76021	−0.0993344
$315$	4.80233	0.270581
$316$	17.0309	0.958064
$317$	−2.34827	−0.131892	−0.0659460	−	0.997823i	$-0.521007\pi$
$317$	−2.34827	−0.131892	−0.0659460	+	0.997823i	$0.521007\pi$
$318$	7.67433	0.430355
$319$	−1.70970	−0.0957248
$320$	0.145997	0.00816146
$321$	18.1078	1.01068
$322$	−32.9850	−1.83818
$323$	−0.330170	−0.0183711
$324$	1.09834	0.0610189
$325$	−11.5927	−0.643046
$326$	27.4924	1.52266
$327$	−8.44301	−0.466900
$328$	−10.8017	−0.596422
$329$	4.10240	0.226173
$330$	−7.50719	−0.413257
$331$	−17.7922	−0.977947	−0.488974	−	0.872299i	$-0.662629\pi$
$331$	−17.7922	−0.977947	−0.488974	+	0.872299i	$0.662629\pi$
$332$	16.0154	0.878958
$333$	5.37516	0.294557
$334$	−0.904472	−0.0494905
$335$	−5.82701	−0.318363
$336$	−17.4360	−0.951209
$337$	−17.3441	−0.944792	−0.472396	−	0.881386i	$-0.656611\pi$
$337$	−17.3441	−0.944792	−0.472396	+	0.881386i	$0.656611\pi$
$338$	−1.56164	−0.0849423
$339$	−5.58184	−0.303164
$340$	1.50964	0.0818715
$341$	2.69118	0.145735
$342$	0.581168	0.0314260
$343$	−6.26228	−0.338132
$344$	12.4200	0.669639
$345$	7.37174	0.396881
$346$	−2.42045	−0.130124
$347$	17.7548	0.953128	0.476564	−	0.879140i	$-0.341882\pi$
$347$	17.7548	0.953128	0.476564	+	0.879140i	$0.341882\pi$
$348$	−0.605172	−0.0324406
$349$	7.85713	0.420583	0.210291	−	0.977639i	$-0.432559\pi$
$349$	7.85713	0.420583	0.210291	+	0.977639i	$0.432559\pi$
$350$	19.1319	1.02264
$351$	3.72655	0.198909
$352$	17.4071	0.927800
$353$	−19.9320	−1.06087	−0.530436	−	0.847725i	$-0.677972\pi$
$353$	−19.9320	−1.06087	−0.530436	+	0.847725i	$0.677972\pi$
$354$	17.8764	0.950119
$355$	9.50081	0.504251
$356$	0.834522	0.0442296
$357$	3.49395	0.184919
$358$	8.72234	0.460990
$359$	−36.9277	−1.94897	−0.974485	−	0.224451i	$-0.927941\pi$
$359$	−36.9277	−1.94897	−0.974485	+	0.224451i	$0.927941\pi$
$360$	2.18144	0.114972
$361$	−18.8910	−0.994263
$362$	−7.54503	−0.396558
$363$	−1.37157	−0.0719887
$364$	14.3008	0.749566
$365$	−15.8534	−0.829805
$366$	2.15143	0.112457
$367$	33.5375	1.75064	0.875321	−	0.483543i	$-0.160650\pi$
$367$	33.5375	1.75064	0.875321	+	0.483543i	$0.160650\pi$
$368$	−26.7648	−1.39521
$369$	−6.80587	−0.354299
$370$	−13.0044	−0.676068
$371$	−15.2333	−0.790871
$372$	0.952580	0.0493890
$373$	−19.4145	−1.00525	−0.502623	−	0.864506i	$-0.667631\pi$
$373$	−19.4145	−1.00525	−0.502623	+	0.864506i	$0.667631\pi$
$374$	−5.46188	−0.282427
$375$	−11.1481	−0.575685
$376$	1.86350	0.0961026
$377$	−2.05329	−0.105750
$378$	−6.15008	−0.316326
$379$	−15.5064	−0.796511	−0.398255	−	0.917275i	$-0.630384\pi$
$379$	−15.5064	−0.796511	−0.398255	+	0.917275i	$0.630384\pi$
$380$	−0.498436	−0.0255692
$381$	−5.97060	−0.305883
$382$	−6.89156	−0.352603
$383$	11.7866	0.602268	0.301134	−	0.953582i	$-0.402635\pi$
$383$	11.7866	0.602268	0.301134	+	0.953582i	$0.402635\pi$
$384$	−11.4066	−0.582090
$385$	14.9015	0.759450
$386$	11.4038	0.580438
$387$	7.82551	0.397793
$388$	−5.94551	−0.301838
$389$	21.7857	1.10458	0.552290	−	0.833652i	$-0.313754\pi$
$389$	21.7857	1.10458	0.552290	+	0.833652i	$0.313754\pi$
$390$	−9.01586	−0.456536
$391$	5.36334	0.271236
$392$	8.26516	0.417454
$393$	17.6465	0.890147
$394$	−9.99803	−0.503693
$395$	21.3126	1.07235
$396$	3.40812	0.171264
$397$	33.1300	1.66275	0.831373	−	0.555716i	$-0.187556\pi$
$397$	33.1300	1.66275	0.831373	+	0.555716i	$0.187556\pi$
$398$	−30.8319	−1.54546
$399$	−1.15360	−0.0577521
$400$	15.5241	0.776204
$401$	5.37014	0.268172	0.134086	−	0.990970i	$-0.457190\pi$
$401$	5.37014	0.268172	0.134086	+	0.990970i	$0.457190\pi$
$402$	7.46233	0.372187
$403$	3.23200	0.160998
$404$	10.9540	0.544984
$405$	1.37447	0.0682980
$406$	3.38862	0.168174
$407$	16.6790	0.826745
$408$	1.58711	0.0785737
$409$	−30.1006	−1.48838	−0.744189	−	0.667969i	$-0.767164\pi$
$409$	−30.1006	−1.48838	−0.744189	+	0.667969i	$0.767164\pi$
$410$	16.4658	0.813189
$411$	−9.72705	−0.479800
$412$	−3.91833	−0.193042
$413$	−35.4839	−1.74605
$414$	−9.44060	−0.463980
$415$	20.0418	0.983811
$416$	20.9052	1.02496
$417$	19.8967	0.974347
$418$	1.80335	0.0882047
$419$	−19.2765	−0.941719	−0.470859	−	0.882208i	$-0.656056\pi$
$419$	−19.2765	−0.941719	−0.470859	+	0.882208i	$0.656056\pi$
$420$	5.27459	0.257374
$421$	−4.57830	−0.223133	−0.111566	−	0.993757i	$-0.535587\pi$
$421$	−4.57830	−0.223133	−0.111566	+	0.993757i	$0.535587\pi$
$422$	34.8379	1.69588
$423$	1.17414	0.0570889
$424$	−6.91964	−0.336047
$425$	−3.11083	−0.150898
$426$	−12.1672	−0.589502
$427$	−4.27051	−0.206664
$428$	19.8885	0.961349
$429$	11.5634	0.558285
$430$	−18.9327	−0.913016
$431$	23.9933	1.15572	0.577858	−	0.816138i	$-0.303889\pi$
$431$	23.9933	1.15572	0.577858	+	0.816138i	$0.303889\pi$
$432$	−4.99033	−0.240097
$433$	−1.06443	−0.0511532	−0.0255766	−	0.999673i	$-0.508142\pi$
$433$	−1.06443	−0.0511532	−0.0255766	+	0.999673i	$0.508142\pi$
$434$	−5.33391	−0.256036
$435$	−0.757316	−0.0363105
$436$	−9.27330	−0.444110
$437$	−1.77081	−0.0847094
$438$	20.3026	0.970096
$439$	14.2364	0.679468	0.339734	−	0.940522i	$-0.389663\pi$
$439$	14.2364	0.679468	0.339734	+	0.940522i	$0.389663\pi$
$440$	6.76894	0.322696
$441$	5.20768	0.247985
$442$	−6.55952	−0.312004
$443$	12.0398	0.572028	0.286014	−	0.958225i	$-0.407670\pi$
$443$	12.0398	0.572028	0.286014	+	0.958225i	$0.407670\pi$
$444$	5.90375	0.280180
$445$	1.04433	0.0495058
$446$	45.3076	2.14538
$447$	9.51226	0.449914
$448$	0.371128	0.0175342
$449$	−4.95298	−0.233745	−0.116873	−	0.993147i	$-0.537287\pi$
$449$	−4.95298	−0.233745	−0.116873	+	0.993147i	$0.537287\pi$
$450$	5.47572	0.258128
$451$	−21.1184	−0.994427
$452$	−6.13076	−0.288367
$453$	−9.97033	−0.468447
$454$	36.6787	1.72142
$455$	17.8961	0.838983
$456$	−0.524016	−0.0245393
$457$	23.9648	1.12103	0.560514	−	0.828145i	$-0.310604\pi$
$457$	23.9648	1.12103	0.560514	+	0.828145i	$0.310604\pi$
$458$	2.48427	0.116082
$459$	1.00000	0.0466760
$460$	8.09668	0.377510
$461$	−17.2243	−0.802217	−0.401109	−	0.916031i	$-0.631375\pi$
$461$	−17.2243	−0.802217	−0.401109	+	0.916031i	$0.631375\pi$
$462$	−19.0835	−0.887846
$463$	−28.9368	−1.34481	−0.672403	−	0.740185i	$-0.734738\pi$
$463$	−28.9368	−1.34481	−0.672403	+	0.740185i	$0.734738\pi$
$464$	2.74961	0.127647
$465$	1.19206	0.0552807
$466$	31.5587	1.46193
$467$	−12.0580	−0.557977	−0.278989	−	0.960294i	$-0.589999\pi$
$467$	−12.0580	−0.557977	−0.278989	+	0.960294i	$0.589999\pi$
$468$	4.09302	0.189200
$469$	−14.8124	−0.683975
$470$	−2.84067	−0.131031
$471$	1.00000	0.0460776
$472$	−16.1184	−0.741910
$473$	24.2823	1.11650
$474$	−27.2939	−1.25365
$475$	1.02710	0.0471267
$476$	3.83754	0.175894
$477$	−4.35990	−0.199626
$478$	−11.4049	−0.521650
$479$	14.6365	0.668761	0.334380	−	0.942438i	$-0.391473\pi$
$479$	14.6365	0.668761	0.334380	+	0.942438i	$0.391473\pi$
$480$	7.71051	0.351935
$481$	20.0308	0.913326
$482$	33.9138	1.54473
$483$	18.7392	0.852664
$484$	−1.50645	−0.0684750
$485$	−7.44026	−0.337845
$486$	−1.76021	−0.0798448
$487$	−5.17780	−0.234628	−0.117314	−	0.993095i	$-0.537428\pi$
$487$	−5.17780	−0.234628	−0.117314	+	0.993095i	$0.537428\pi$
$488$	−1.93986	−0.0878133
$489$	−15.6188	−0.706308
$490$	−12.5992	−0.569175
$491$	−7.52846	−0.339755	−0.169877	−	0.985465i	$-0.554337\pi$
$491$	−7.52846	−0.339755	−0.169877	+	0.985465i	$0.554337\pi$
$492$	−7.47515	−0.337006
$493$	−0.550988	−0.0248152
$494$	2.16575	0.0974419
$495$	4.26494	0.191695
$496$	−4.32807	−0.194336
$497$	24.1514	1.08334
$498$	−25.6664	−1.15014
$499$	−6.29864	−0.281966	−0.140983	−	0.990012i	$-0.545026\pi$
$499$	−6.29864	−0.281966	−0.140983	+	0.990012i	$0.545026\pi$
$500$	−12.2444	−0.547586
$501$	0.513843	0.0229568
$502$	−11.9845	−0.534893
$503$	8.89410	0.396568	0.198284	−	0.980145i	$-0.436463\pi$
$503$	8.89410	0.396568	0.198284	+	0.980145i	$0.436463\pi$
$504$	5.54528	0.247007
$505$	13.7080	0.609997
$506$	−29.2939	−1.30227
$507$	0.887192	0.0394016
$508$	−6.55775	−0.290953
$509$	−35.6904	−1.58195	−0.790973	−	0.611850i	$-0.790425\pi$
$509$	−35.6904	−1.58195	−0.790973	+	0.611850i	$0.790425\pi$
$510$	−2.41936	−0.107131
$511$	−40.2999	−1.78276
$512$	−12.1544	−0.537152
$513$	−0.330170	−0.0145774
$514$	1.47190	0.0649228
$515$	−4.90343	−0.216071
$516$	8.59507	0.378377
$517$	3.64334	0.160234
$518$	−33.0577	−1.45247
$519$	1.37509	0.0603598
$520$	8.12924	0.356491
$521$	−8.12376	−0.355908	−0.177954	−	0.984039i	$-0.556948\pi$
$521$	−8.12376	−0.355908	−0.177954	+	0.984039i	$0.556948\pi$
$522$	0.969855	0.0424494
$523$	17.1569	0.750219	0.375110	−	0.926980i	$-0.377605\pi$
$523$	17.1569	0.750219	0.375110	+	0.926980i	$0.377605\pi$
$524$	19.3818	0.846699
$525$	−10.8691	−0.474366
$526$	3.92087	0.170958
$527$	0.867291	0.0377798
$528$	−15.4848	−0.673892
$529$	5.76537	0.250668
$530$	10.5481	0.458182
$531$	−10.1558	−0.440725
$532$	−1.26704	−0.0549332
$533$	−25.3624	−1.09857
$534$	−1.33741	−0.0578755
$535$	24.8887	1.07603
$536$	−6.72849	−0.290626
$537$	−4.95528	−0.213836
$538$	−9.11719	−0.393070
$539$	16.1593	0.696029
$540$	1.50964	0.0649644
$541$	−21.7332	−0.934382	−0.467191	−	0.884156i	$-0.654734\pi$
$541$	−21.7332	−0.934382	−0.467191	+	0.884156i	$0.654734\pi$
$542$	12.7314	0.546859
$543$	4.28644	0.183949
$544$	5.60981	0.240518
$545$	−11.6047	−0.497089
$546$	−22.9186	−0.980826
$547$	21.3956	0.914808	0.457404	−	0.889259i	$-0.348779\pi$
$547$	21.3956	0.914808	0.457404	+	0.889259i	$0.348779\pi$
$548$	−10.6836	−0.456381
$549$	−1.22226	−0.0521647
$550$	16.9910	0.724498
$551$	0.181920	0.00775003
$552$	8.51221	0.362304
$553$	54.1774	2.30386
$554$	15.0134	0.637860
$555$	7.38799	0.313603
$556$	21.8534	0.926790
$557$	−2.64647	−0.112134	−0.0560672	−	0.998427i	$-0.517856\pi$
$557$	−2.64647	−0.112134	−0.0560672	+	0.998427i	$0.517856\pi$
$558$	−1.52661	−0.0646267
$559$	29.1622	1.23343
$560$	−23.9652	−1.01271
$561$	3.10297	0.131008
$562$	2.69875	0.113840
$563$	0.297574	0.0125412	0.00627062	−	0.999980i	$-0.498004\pi$
$563$	0.297574	0.0125412	0.00627062	+	0.999980i	$0.498004\pi$
$564$	1.28961	0.0543024
$565$	−7.67207	−0.322766
$566$	33.4364	1.40544
$567$	3.49395	0.146732
$568$	10.9707	0.460319
$569$	44.7180	1.87468	0.937338	−	0.348422i	$-0.113282\pi$
$569$	44.7180	1.87468	0.937338	+	0.348422i	$0.113282\pi$
$570$	0.798798	0.0334580
$571$	41.2350	1.72563	0.862816	−	0.505517i	$-0.168698\pi$
$571$	41.2350	1.72563	0.862816	+	0.505517i	$0.168698\pi$
$572$	12.7005	0.531036
$573$	3.91519	0.163560
$574$	41.8566	1.74706
$575$	−16.6844	−0.695789
$576$	0.106220	0.00442585
$577$	14.4534	0.601701	0.300851	−	0.953671i	$-0.402729\pi$
$577$	14.4534	0.601701	0.300851	+	0.953671i	$0.402729\pi$
$578$	−1.76021	−0.0732151
$579$	−6.47865	−0.269244
$580$	−0.831791	−0.0345382
$581$	50.9468	2.11363
$582$	9.52834	0.394962
$583$	−13.5286	−0.560299
$584$	−18.3060	−0.757510
$585$	5.12203	0.211770
$586$	−51.6937	−2.13545
$587$	10.1942	0.420761	0.210380	−	0.977620i	$-0.432530\pi$
$587$	10.1942	0.420761	0.210380	+	0.977620i	$0.432530\pi$
$588$	5.71980	0.235881
$589$	−0.286353	−0.0117990
$590$	24.5706	1.01155
$591$	5.68002	0.233645
$592$	−26.8238	−1.10245
$593$	−28.1230	−1.15487	−0.577436	−	0.816436i	$-0.695947\pi$
$593$	−28.1230	−1.15487	−0.577436	+	0.816436i	$0.695947\pi$
$594$	−5.46188	−0.224104
$595$	4.80233	0.196876
$596$	10.4477	0.427954
$597$	17.5160	0.716884
$598$	−35.1809	−1.43865
$599$	−2.71745	−0.111032	−0.0555159	−	0.998458i	$-0.517680\pi$
$599$	−2.71745	−0.111032	−0.0555159	+	0.998458i	$0.517680\pi$
$600$	−4.93724	−0.201562
$601$	−31.8579	−1.29951	−0.649755	−	0.760144i	$-0.725128\pi$
$601$	−31.8579	−1.29951	−0.649755	+	0.760144i	$0.725128\pi$
$602$	−48.1276	−1.96153
$603$	−4.23946	−0.172644
$604$	−10.9508	−0.445582
$605$	−1.88518	−0.0766435
$606$	−17.5551	−0.713126
$607$	−4.61829	−0.187450	−0.0937252	−	0.995598i	$-0.529878\pi$
$607$	−4.61829	−0.187450	−0.0937252	+	0.995598i	$0.529878\pi$
$608$	−1.85219	−0.0751162
$609$	−1.92512	−0.0780099
$610$	2.95708	0.119729
$611$	4.37551	0.177014
$612$	1.09834	0.0443978
$613$	35.0799	1.41687	0.708433	−	0.705778i	$-0.249403\pi$
$613$	35.0799	1.41687	0.708433	+	0.705778i	$0.249403\pi$
$614$	−19.3313	−0.780148
$615$	−9.35446	−0.377208
$616$	17.2069	0.693284
$617$	−28.5969	−1.15127	−0.575634	−	0.817708i	$-0.695245\pi$
$617$	−28.5969	−1.15127	−0.575634	+	0.817708i	$0.695245\pi$
$618$	6.27956	0.252601
$619$	22.2500	0.894303	0.447152	−	0.894458i	$-0.352438\pi$
$619$	22.2500	0.894303	0.447152	+	0.894458i	$0.352438\pi$
$620$	1.30929	0.0525825
$621$	5.36334	0.215223
$622$	−15.8144	−0.634100
$623$	2.65471	0.106359
$624$	−18.5967	−0.744465
$625$	0.231434	0.00925735
$626$	25.6495	1.02516
$627$	−1.02451	−0.0409149
$628$	1.09834	0.0438285
$629$	5.37516	0.214322
$630$	−8.45311	−0.336780
$631$	3.72909	0.148453	0.0742264	−	0.997241i	$-0.476351\pi$
$631$	3.72909	0.148453	0.0742264	+	0.997241i	$0.476351\pi$
$632$	24.6098	0.978927
$633$	−19.7919	−0.786656
$634$	4.13345	0.164160
$635$	−8.20641	−0.325661
$636$	−4.78865	−0.189882
$637$	19.4067	0.768921
$638$	3.00943	0.119145
$639$	6.91234	0.273448
$640$	−15.6780	−0.619728
$641$	15.0926	0.596122	0.298061	−	0.954547i	$-0.403660\pi$
$641$	15.0926	0.596122	0.298061	+	0.954547i	$0.403660\pi$
$642$	−31.8736	−1.25795
$643$	−22.5413	−0.888942	−0.444471	−	0.895793i	$-0.646608\pi$
$643$	−22.5413	−0.888942	−0.444471	+	0.895793i	$0.646608\pi$
$644$	20.5820	0.811046
$645$	10.7559	0.423514
$646$	0.581168	0.0228658
$647$	8.13659	0.319882	0.159941	−	0.987127i	$-0.448870\pi$
$647$	8.13659	0.319882	0.159941	+	0.987127i	$0.448870\pi$
$648$	1.58711	0.0623476
$649$	−31.5132	−1.23700
$650$	20.4055	0.800371
$651$	3.03027	0.118766
$652$	−17.1548	−0.671833
$653$	33.4575	1.30929	0.654647	−	0.755935i	$-0.272817\pi$
$653$	33.4575	1.30929	0.654647	+	0.755935i	$0.272817\pi$
$654$	14.8615	0.581130
$655$	24.2545	0.947704
$656$	33.9635	1.32605
$657$	−11.5342	−0.449992
$658$	−7.22109	−0.281507
$659$	−11.0796	−0.431598	−0.215799	−	0.976438i	$-0.569236\pi$
$659$	−11.0796	−0.431598	−0.215799	+	0.976438i	$0.569236\pi$
$660$	4.68436	0.182338
$661$	−14.9335	−0.580845	−0.290422	−	0.956899i	$-0.593796\pi$
$661$	−14.9335	−0.580845	−0.290422	+	0.956899i	$0.593796\pi$
$662$	31.3180	1.21721
$663$	3.72655	0.144727
$664$	23.1424	0.898098
$665$	−1.58558	−0.0614863
$666$	−9.46141	−0.366622
$667$	−2.95513	−0.114423
$668$	0.564375	0.0218363
$669$	−25.7399	−0.995162
$670$	10.2568	0.396253
$671$	−3.79263	−0.146413
$672$	19.6004	0.756101
$673$	5.54681	0.213814	0.106907	−	0.994269i	$-0.465905\pi$
$673$	5.54681	0.213814	0.106907	+	0.994269i	$0.465905\pi$
$674$	30.5292	1.17594
$675$	−3.11083	−0.119736
$676$	0.974438	0.0374784
$677$	−28.6176	−1.09987	−0.549933	−	0.835209i	$-0.685347\pi$
$677$	−28.6176	−1.09987	−0.549933	+	0.835209i	$0.685347\pi$
$678$	9.82521	0.377335
$679$	−18.9134	−0.725829
$680$	2.18144	0.0836543
$681$	−20.8377	−0.798501
$682$	−4.73704	−0.181391
$683$	0.0318949	0.00122043	0.000610213	−	1.00000i	$-0.499806\pi$
$683$	0.0318949	0.00122043	0.000610213		1.00000i	$0.499806\pi$
$684$	−0.362639	−0.0138658
$685$	−13.3695	−0.510824
$686$	11.0229	0.420858
$687$	−1.41135	−0.0538463
$688$	−39.0519	−1.48884
$689$	−16.2474	−0.618976
$690$	−12.9758	−0.493981
$691$	−24.3069	−0.924677	−0.462339	−	0.886703i	$-0.652990\pi$
$691$	−24.3069	−0.924677	−0.462339	+	0.886703i	$0.652990\pi$
$692$	1.51032	0.0574136
$693$	10.8416	0.411839
$694$	−31.2522	−1.18632
$695$	27.3475	1.03735
$696$	−0.874479	−0.0331470
$697$	−6.80587	−0.257791
$698$	−13.8302	−0.523481
$699$	−17.9289	−0.678135
$700$	−11.9380	−0.451212
$701$	42.1267	1.59110	0.795552	−	0.605886i	$-0.207181\pi$
$701$	42.1267	1.59110	0.795552	+	0.605886i	$0.207181\pi$
$702$	−6.55952	−0.247573
$703$	−1.77471	−0.0669346
$704$	0.329599	0.0124222
$705$	1.61383	0.0607802
$706$	35.0845	1.32042
$707$	34.8461	1.31052
$708$	−11.1545	−0.419214
$709$	6.40634	0.240595	0.120298	−	0.992738i	$-0.461615\pi$
$709$	6.40634	0.240595	0.120298	+	0.992738i	$0.461615\pi$
$710$	−16.7234	−0.627619
$711$	15.5061	0.581522
$712$	1.20589	0.0451927
$713$	4.65157	0.174203
$714$	−6.15008	−0.230161
$715$	15.8935	0.594384
$716$	−5.44259	−0.203399
$717$	6.47931	0.241974
$718$	65.0006	2.42580
$719$	31.3362	1.16864	0.584322	−	0.811522i	$-0.301361\pi$
$719$	31.3362	1.16864	0.584322	+	0.811522i	$0.301361\pi$
$720$	−6.85906	−0.255622
$721$	−12.4647	−0.464209
$722$	33.2521	1.23752
$723$	−19.2669	−0.716545
$724$	4.70797	0.174970
$725$	1.71403	0.0636575
$726$	2.41425	0.0896012
$727$	−1.01008	−0.0374618	−0.0187309	−	0.999825i	$-0.505963\pi$
$727$	−1.01008	−0.0374618	−0.0187309	+	0.999825i	$0.505963\pi$
$728$	20.6648	0.765888
$729$	1.00000	0.0370370
$730$	27.9053	1.03282
$731$	7.82551	0.289437
$732$	−1.34245	−0.0496186
$733$	14.0592	0.519287	0.259644	−	0.965704i	$-0.416395\pi$
$733$	14.0592	0.519287	0.259644	+	0.965704i	$0.416395\pi$
$734$	−59.0330	−2.17895
$735$	7.15780	0.264019
$736$	30.0873	1.10903
$737$	−13.1549	−0.484567
$738$	11.9798	0.440981
$739$	−8.85859	−0.325869	−0.162934	−	0.986637i	$-0.552096\pi$
$739$	−8.85859	−0.325869	−0.162934	+	0.986637i	$0.552096\pi$
$740$	8.11453	0.298296
$741$	−1.23040	−0.0451997
$742$	26.8137	0.984363
$743$	18.4892	0.678304	0.339152	−	0.940732i	$-0.389860\pi$
$743$	18.4892	0.678304	0.339152	+	0.940732i	$0.389860\pi$
$744$	1.37649	0.0504644
$745$	13.0743	0.479006
$746$	34.1736	1.25119
$747$	14.5814	0.533507
$748$	3.40812	0.124613
$749$	63.2678	2.31176
$750$	19.6230	0.716530
$751$	27.9006	1.01811	0.509053	−	0.860735i	$-0.329996\pi$
$751$	27.9006	1.01811	0.509053	+	0.860735i	$0.329996\pi$
$752$	−5.85937	−0.213669
$753$	6.80855	0.248117
$754$	3.61421	0.131622
$755$	−13.7039	−0.498737
$756$	3.83754	0.139570
$757$	−38.7026	−1.40667	−0.703335	−	0.710858i	$-0.748307\pi$
$757$	−38.7026	−1.40667	−0.703335	+	0.710858i	$0.748307\pi$
$758$	27.2945	0.991382
$759$	16.6423	0.604076
$760$	−0.720245	−0.0261260
$761$	37.6280	1.36401	0.682007	−	0.731346i	$-0.261108\pi$
$761$	37.6280	1.36401	0.682007	+	0.731346i	$0.261108\pi$
$762$	10.5095	0.380719
$763$	−29.4995	−1.06795
$764$	4.30021	0.155576
$765$	1.37447	0.0496941
$766$	−20.7469	−0.749617
$767$	−37.8462	−1.36655
$768$	19.8655	0.716836
$769$	17.8812	0.644814	0.322407	−	0.946601i	$-0.395508\pi$
$769$	17.8812	0.644814	0.322407	+	0.946601i	$0.395508\pi$
$770$	−26.2297	−0.945254
$771$	−0.836208	−0.0301153
$772$	−7.11576	−0.256102
$773$	−51.3768	−1.84790	−0.923948	−	0.382518i	$-0.875057\pi$
$773$	−51.3768	−1.84790	−0.923948	+	0.382518i	$0.875057\pi$
$774$	−13.7745	−0.495116
$775$	−2.69799	−0.0969148
$776$	−8.59132	−0.308410
$777$	18.7805	0.673748
$778$	−38.3475	−1.37482
$779$	2.24709	0.0805104
$780$	5.62574	0.201434
$781$	21.4488	0.767499
$782$	−9.44060	−0.337595
$783$	−0.550988	−0.0196907
$784$	−25.9880	−0.928144
$785$	1.37447	0.0490569
$786$	−31.0615	−1.10793
$787$	−37.9145	−1.35151	−0.675753	−	0.737128i	$-0.736181\pi$
$787$	−37.9145	−1.35151	−0.675753	+	0.737128i	$0.736181\pi$
$788$	6.23859	0.222241
$789$	−2.22750	−0.0793012
$790$	−37.5147	−1.33471
$791$	−19.5027	−0.693435
$792$	4.92476	0.174994
$793$	−4.55481	−0.161746
$794$	−58.3157	−2.06955
$795$	−5.99255	−0.212534
$796$	19.2386	0.681893
$797$	−36.2279	−1.28326	−0.641629	−	0.767015i	$-0.721741\pi$
$797$	−36.2279	−1.28326	−0.641629	+	0.767015i	$0.721741\pi$
$798$	2.03057	0.0718815
$799$	1.17414	0.0415383
$800$	−17.4512	−0.616992
$801$	0.759803	0.0268463
$802$	−9.45257	−0.333782
$803$	−35.7903	−1.26301
$804$	−4.65636	−0.164217
$805$	25.7565	0.907797
$806$	−5.68901	−0.200387
$807$	5.17960	0.182331
$808$	15.8287	0.556852
$809$	45.1625	1.58783	0.793915	−	0.608029i	$-0.208040\pi$
$809$	45.1625	1.58783	0.793915	+	0.608029i	$0.208040\pi$
$810$	−2.41936	−0.0850075
$811$	25.6941	0.902241	0.451121	−	0.892463i	$-0.351024\pi$
$811$	25.6941	0.902241	0.451121	+	0.892463i	$0.351024\pi$
$812$	−2.11444	−0.0742023
$813$	−7.23287	−0.253668
$814$	−29.3585	−1.02901
$815$	−21.4676	−0.751978
$816$	−4.99033	−0.174696
$817$	−2.58375	−0.0903939
$818$	52.9834	1.85252
$819$	13.0204	0.454969
$820$	−10.2744	−0.358797
$821$	20.7046	0.722594	0.361297	−	0.932451i	$-0.382334\pi$
$821$	20.7046	0.722594	0.361297	+	0.932451i	$0.382334\pi$
$822$	17.1217	0.597186
$823$	4.92374	0.171631	0.0858154	−	0.996311i	$-0.472650\pi$
$823$	4.92374	0.171631	0.0858154	+	0.996311i	$0.472650\pi$
$824$	−5.66203	−0.197246
$825$	−9.65282	−0.336068
$826$	62.4592	2.17323
$827$	18.1909	0.632558	0.316279	−	0.948666i	$-0.397566\pi$
$827$	18.1909	0.632558	0.316279	+	0.948666i	$0.397566\pi$
$828$	5.89077	0.204718
$829$	−15.3304	−0.532447	−0.266223	−	0.963911i	$-0.585776\pi$
$829$	−15.3304	−0.532447	−0.266223	+	0.963911i	$0.585776\pi$
$830$	−35.2777	−1.22451
$831$	−8.52934	−0.295880
$832$	0.395836	0.0137231
$833$	5.20768	0.180435
$834$	−35.0224	−1.21273
$835$	0.706262	0.0244412
$836$	−1.12526	−0.0389178
$837$	0.867291	0.0299779
$838$	33.9307	1.17212
$839$	−21.5278	−0.743223	−0.371611	−	0.928388i	$-0.621195\pi$
$839$	−21.5278	−0.743223	−0.371611	+	0.928388i	$0.621195\pi$
$840$	7.62183	0.262978
$841$	−28.6964	−0.989531
$842$	8.05877	0.277723
$843$	−1.53320	−0.0528062
$844$	−21.7382	−0.748260
$845$	1.21942	0.0419493
$846$	−2.06674	−0.0710560
$847$	−4.79219	−0.164662
$848$	21.7573	0.747149
$849$	−18.9957	−0.651930
$850$	5.47572	0.187816
$851$	28.8288	0.988238
$852$	7.59210	0.260101
$853$	−16.3409	−0.559501	−0.279750	−	0.960073i	$-0.590252\pi$
$853$	−16.3409	−0.559501	−0.279750	+	0.960073i	$0.590252\pi$
$854$	7.51699	0.257226
$855$	−0.453809	−0.0155199
$856$	28.7391	0.982283
$857$	−12.1978	−0.416670	−0.208335	−	0.978058i	$-0.566804\pi$
$857$	−12.1978	−0.416670	−0.208335	+	0.978058i	$0.566804\pi$
$858$	−20.3540	−0.694874
$859$	−15.5810	−0.531616	−0.265808	−	0.964026i	$-0.585639\pi$
$859$	−15.5810	−0.531616	−0.265808	+	0.964026i	$0.585639\pi$
$860$	11.8137	0.402843
$861$	−23.7793	−0.810398
$862$	−42.2332	−1.43847
$863$	33.2434	1.13162	0.565810	−	0.824536i	$-0.308564\pi$
$863$	33.2434	1.13162	0.565810	+	0.824536i	$0.308564\pi$
$864$	5.60981	0.190849
$865$	1.89002	0.0642627
$866$	1.87362	0.0636681
$867$	1.00000	0.0339618
$868$	3.32827	0.112969
$869$	48.1148	1.63218
$870$	1.33304	0.0451942
$871$	−15.7986	−0.535314
$872$	−13.4000	−0.453781
$873$	−5.41318	−0.183208
$874$	3.11700	0.105434
$875$	−38.9509	−1.31678
$876$	−12.6685	−0.428028
$877$	22.4736	0.758880	0.379440	−	0.925216i	$-0.376117\pi$
$877$	22.4736	0.758880	0.379440	+	0.925216i	$0.376117\pi$
$878$	−25.0591	−0.845704
$879$	29.3679	0.990555
$880$	−21.2835	−0.717465
$881$	−51.7721	−1.74425	−0.872123	−	0.489287i	$-0.837257\pi$
$881$	−51.7721	−1.74425	−0.872123	+	0.489287i	$0.837257\pi$
$882$	−9.16661	−0.308656
$883$	−46.2396	−1.55609	−0.778044	−	0.628210i	$-0.783788\pi$
$883$	−46.2396	−1.55609	−0.778044	+	0.628210i	$0.783788\pi$
$884$	4.09302	0.137663
$885$	−13.9589	−0.469222
$886$	−21.1926	−0.711978
$887$	42.4190	1.42429	0.712146	−	0.702031i	$-0.247723\pi$
$887$	42.4190	1.42429	0.712146	+	0.702031i	$0.247723\pi$
$888$	8.53097	0.286281
$889$	−20.8610	−0.699654
$890$	−1.83823	−0.0616177
$891$	3.10297	0.103953
$892$	−28.2712	−0.946588
$893$	−0.387667	−0.0129728
$894$	−16.7436	−0.559989
$895$	−6.81089	−0.227663
$896$	−39.8540	−1.33143
$897$	19.9868	0.667338
$898$	8.71828	0.290933
$899$	−0.477867	−0.0159377
$900$	−3.41675	−0.113892
$901$	−4.35990	−0.145249
$902$	37.1728	1.23772
$903$	27.3419	0.909883
$904$	−8.85900	−0.294646
$905$	5.89158	0.195843
$906$	17.5499	0.583056
$907$	−30.1384	−1.00073	−0.500365	−	0.865815i	$-0.666801\pi$
$907$	−30.1384	−1.00073	−0.500365	+	0.865815i	$0.666801\pi$
$908$	−22.8869	−0.759527
$909$	9.97328	0.330793
$910$	−31.5009	−1.04425
$911$	14.7398	0.488352	0.244176	−	0.969731i	$-0.421483\pi$
$911$	14.7398	0.488352	0.244176	+	0.969731i	$0.421483\pi$
$912$	1.64766	0.0545593
$913$	45.2458	1.49742
$914$	−42.1831	−1.39529
$915$	−1.67996	−0.0555377
$916$	−1.55014	−0.0512181
$917$	61.6559	2.03606
$918$	−1.76021	−0.0580956
$919$	23.5558	0.777035	0.388518	−	0.921441i	$-0.372987\pi$
$919$	23.5558	0.777035	0.388518	+	0.921441i	$0.372987\pi$
$920$	11.6998	0.385730
$921$	10.9824	0.361882
$922$	30.3184	0.998485
$923$	25.7592	0.847875
$924$	11.9078	0.391737
$925$	−16.7212	−0.549790
$926$	50.9348	1.67382
$927$	−3.56750	−0.117172
$928$	−3.09094	−0.101465
$929$	−0.985640	−0.0323378	−0.0161689	−	0.999869i	$-0.505147\pi$
$929$	−0.985640	−0.0323378	−0.0161689	+	0.999869i	$0.505147\pi$
$930$	−2.09828	−0.0688055
$931$	−1.71942	−0.0563516
$932$	−19.6921	−0.645036
$933$	8.98438	0.294135
$934$	21.2246	0.694490
$935$	4.26494	0.139478
$936$	5.91445	0.193320
$937$	−23.1960	−0.757779	−0.378889	−	0.925442i	$-0.623694\pi$
$937$	−23.1960	−0.757779	−0.378889	+	0.925442i	$0.623694\pi$
$938$	26.0730	0.851314
$939$	−14.5718	−0.475534
$940$	1.77253	0.0578136
$941$	1.37095	0.0446917	0.0223459	−	0.999750i	$-0.492887\pi$
$941$	1.37095	0.0446917	0.0223459	+	0.999750i	$0.492887\pi$
$942$	−1.76021	−0.0573507
$943$	−36.5021	−1.18867
$944$	50.6809	1.64952
$945$	4.80233	0.156220
$946$	−42.7420	−1.38966
$947$	−15.9049	−0.516839	−0.258419	−	0.966033i	$-0.583202\pi$
$947$	−15.9049	−0.516839	−0.258419	+	0.966033i	$0.583202\pi$
$948$	17.0309	0.553139
$949$	−42.9828	−1.39528
$950$	−1.80792	−0.0586566
$951$	−2.34827	−0.0761479
$952$	5.54528	0.179724
$953$	−43.4516	−1.40753	−0.703767	−	0.710431i	$-0.748500\pi$
$953$	−43.4516	−1.40753	−0.703767	+	0.710431i	$0.748500\pi$
$954$	7.67433	0.248466
$955$	5.38132	0.174135
$956$	7.11648	0.230163
$957$	−1.70970	−0.0552667
$958$	−25.7634	−0.832377
$959$	−33.9858	−1.09746
$960$	0.145997	0.00471202
$961$	−30.2478	−0.975736
$962$	−35.2584	−1.13678
$963$	18.1078	0.583517
$964$	−21.1616	−0.681570
$965$	−8.90471	−0.286653
$966$	−32.9850	−1.06127
$967$	−12.0357	−0.387043	−0.193522	−	0.981096i	$-0.561991\pi$
$967$	−12.0357	−0.387043	−0.193522	+	0.981096i	$0.561991\pi$
$968$	−2.17683	−0.0699661
$969$	−0.330170	−0.0106066
$970$	13.0964	0.420501
$971$	42.0455	1.34930	0.674651	−	0.738136i	$-0.264294\pi$
$971$	42.0455	1.34930	0.674651	+	0.738136i	$0.264294\pi$
$972$	1.09834	0.0352293
$973$	69.5182	2.22865
$974$	9.11401	0.292032
$975$	−11.5927	−0.371263
$976$	6.09947	0.195239
$977$	−10.5647	−0.337995	−0.168998	−	0.985616i	$-0.554053\pi$
$977$	−10.5647	−0.337995	−0.168998	+	0.985616i	$0.554053\pi$
$978$	27.4924	0.879111
$979$	2.35765	0.0753507
$980$	7.86169	0.251133
$981$	−8.44301	−0.269565
$982$	13.2517	0.422878
$983$	12.0180	0.383314	0.191657	−	0.981462i	$-0.438614\pi$
$983$	12.0180	0.383314	0.191657	+	0.981462i	$0.438614\pi$
$984$	−10.8017	−0.344345
$985$	7.80702	0.248752
$986$	0.969855	0.0308865
$987$	4.10240	0.130581
$988$	−1.35139	−0.0429935
$989$	41.9709	1.33460
$990$	−7.50719	−0.238594
$991$	4.05315	0.128753	0.0643763	−	0.997926i	$-0.479494\pi$
$991$	4.05315	0.128753	0.0643763	+	0.997926i	$0.479494\pi$
$992$	4.86533	0.154474
$993$	−17.7922	−0.564618
$994$	−42.5115	−1.34838
$995$	24.0753	0.763237
$996$	16.0154	0.507467
$997$	44.1471	1.39815	0.699076	−	0.715048i	$-0.253595\pi$
$997$	44.1471	1.39815	0.699076	+	0.715048i	$0.253595\pi$
$998$	11.0869	0.350951
$999$	5.37516	0.170062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.i.1.14	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.i.1.14	✓	63	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.76021 q^{2} +1.00000 q^{3} +1.09834 q^{4} +1.37447 q^{5} -1.76021 q^{6} +3.49395 q^{7} +1.58711 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.76021 q^{2} +1.00000 q^{3} +1.09834 q^{4} +1.37447 q^{5} -1.76021 q^{6} +3.49395 q^{7} +1.58711 q^{8} +1.00000 q^{9} -2.41936 q^{10} +3.10297 q^{11} +1.09834 q^{12} +3.72655 q^{13} -6.15008 q^{14} +1.37447 q^{15} -4.99033 q^{16} +1.00000 q^{17} -1.76021 q^{18} -0.330170 q^{19} +1.50964 q^{20} +3.49395 q^{21} -5.46188 q^{22} +5.36334 q^{23} +1.58711 q^{24} -3.11083 q^{25} -6.55952 q^{26} +1.00000 q^{27} +3.83754 q^{28} -0.550988 q^{29} -2.41936 q^{30} +0.867291 q^{31} +5.60981 q^{32} +3.10297 q^{33} -1.76021 q^{34} +4.80233 q^{35} +1.09834 q^{36} +5.37516 q^{37} +0.581168 q^{38} +3.72655 q^{39} +2.18144 q^{40} -6.80587 q^{41} -6.15008 q^{42} +7.82551 q^{43} +3.40812 q^{44} +1.37447 q^{45} -9.44060 q^{46} +1.17414 q^{47} -4.99033 q^{48} +5.20768 q^{49} +5.47572 q^{50} +1.00000 q^{51} +4.09302 q^{52} -4.35990 q^{53} -1.76021 q^{54} +4.26494 q^{55} +5.54528 q^{56} -0.330170 q^{57} +0.969855 q^{58} -10.1558 q^{59} +1.50964 q^{60} -1.22226 q^{61} -1.52661 q^{62} +3.49395 q^{63} +0.106220 q^{64} +5.12203 q^{65} -5.46188 q^{66} -4.23946 q^{67} +1.09834 q^{68} +5.36334 q^{69} -8.45311 q^{70} +6.91234 q^{71} +1.58711 q^{72} -11.5342 q^{73} -9.46141 q^{74} -3.11083 q^{75} -0.362639 q^{76} +10.8416 q^{77} -6.55952 q^{78} +15.5061 q^{79} -6.85906 q^{80} +1.00000 q^{81} +11.9798 q^{82} +14.5814 q^{83} +3.83754 q^{84} +1.37447 q^{85} -13.7745 q^{86} -0.550988 q^{87} +4.92476 q^{88} +0.759803 q^{89} -2.41936 q^{90} +13.0204 q^{91} +5.89077 q^{92} +0.867291 q^{93} -2.06674 q^{94} -0.453809 q^{95} +5.60981 q^{96} -5.41318 q^{97} -9.16661 q^{98} +3.10297 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9}+O(q^{10}) \) 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