LMFDB - Embedded newform 8002.2.a.d.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.38948 q^{3} +1.00000 q^{4} +1.42184 q^{5} -2.38948 q^{6} -5.14183 q^{7} +1.00000 q^{8} +2.70964 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.38948 q^{3} +1.00000 q^{4} +1.42184 q^{5} -2.38948 q^{6} -5.14183 q^{7} +1.00000 q^{8} +2.70964 q^{9} +1.42184 q^{10} +2.90336 q^{11} -2.38948 q^{12} -4.20957 q^{13} -5.14183 q^{14} -3.39746 q^{15} +1.00000 q^{16} -4.01441 q^{17} +2.70964 q^{18} +1.37465 q^{19} +1.42184 q^{20} +12.2863 q^{21} +2.90336 q^{22} +1.34345 q^{23} -2.38948 q^{24} -2.97838 q^{25} -4.20957 q^{26} +0.693819 q^{27} -5.14183 q^{28} +5.22382 q^{29} -3.39746 q^{30} +9.21736 q^{31} +1.00000 q^{32} -6.93752 q^{33} -4.01441 q^{34} -7.31084 q^{35} +2.70964 q^{36} +9.16884 q^{37} +1.37465 q^{38} +10.0587 q^{39} +1.42184 q^{40} +3.42707 q^{41} +12.2863 q^{42} +10.8992 q^{43} +2.90336 q^{44} +3.85266 q^{45} +1.34345 q^{46} -11.4604 q^{47} -2.38948 q^{48} +19.4384 q^{49} -2.97838 q^{50} +9.59236 q^{51} -4.20957 q^{52} -4.25818 q^{53} +0.693819 q^{54} +4.12810 q^{55} -5.14183 q^{56} -3.28471 q^{57} +5.22382 q^{58} -2.10949 q^{59} -3.39746 q^{60} -10.6807 q^{61} +9.21736 q^{62} -13.9325 q^{63} +1.00000 q^{64} -5.98532 q^{65} -6.93752 q^{66} -9.40553 q^{67} -4.01441 q^{68} -3.21015 q^{69} -7.31084 q^{70} -2.34289 q^{71} +2.70964 q^{72} +13.5006 q^{73} +9.16884 q^{74} +7.11679 q^{75} +1.37465 q^{76} -14.9286 q^{77} +10.0587 q^{78} +0.841782 q^{79} +1.42184 q^{80} -9.78678 q^{81} +3.42707 q^{82} +0.945519 q^{83} +12.2863 q^{84} -5.70783 q^{85} +10.8992 q^{86} -12.4822 q^{87} +2.90336 q^{88} -13.0187 q^{89} +3.85266 q^{90} +21.6449 q^{91} +1.34345 q^{92} -22.0247 q^{93} -11.4604 q^{94} +1.95453 q^{95} -2.38948 q^{96} -14.8817 q^{97} +19.4384 q^{98} +7.86704 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.38948	−1.37957	−0.689785	−	0.724014i	$-0.742295\pi$
$3$	−2.38948	−1.37957	−0.689785	+	0.724014i	$0.742295\pi$
$4$	1.00000	0.500000
$5$	1.42184	0.635865	0.317933	−	0.948113i	$-0.397011\pi$
$5$	1.42184	0.635865	0.317933	+	0.948113i	$0.397011\pi$
$6$	−2.38948	−0.975503
$7$	−5.14183	−1.94343	−0.971714	−	0.236161i	$-0.924111\pi$
$7$	−5.14183	−1.94343	−0.971714	+	0.236161i	$0.924111\pi$
$8$	1.00000	0.353553
$9$	2.70964	0.903212
$10$	1.42184	0.449625
$11$	2.90336	0.875395	0.437697	−	0.899122i	$-0.355794\pi$
$11$	2.90336	0.875395	0.437697	+	0.899122i	$0.355794\pi$
$12$	−2.38948	−0.689785
$13$	−4.20957	−1.16752	−0.583762	−	0.811925i	$-0.698420\pi$
$13$	−4.20957	−1.16752	−0.583762	+	0.811925i	$0.698420\pi$
$14$	−5.14183	−1.37421
$15$	−3.39746	−0.877220
$16$	1.00000	0.250000
$17$	−4.01441	−0.973636	−0.486818	−	0.873503i	$-0.661843\pi$
$17$	−4.01441	−0.973636	−0.486818	+	0.873503i	$0.661843\pi$
$18$	2.70964	0.638667
$19$	1.37465	0.315366	0.157683	−	0.987490i	$-0.449598\pi$
$19$	1.37465	0.315366	0.157683	+	0.987490i	$0.449598\pi$
$20$	1.42184	0.317933
$21$	12.2863	2.68109
$22$	2.90336	0.618997
$23$	1.34345	0.280128	0.140064	−	0.990142i	$-0.455269\pi$
$23$	1.34345	0.280128	0.140064	+	0.990142i	$0.455269\pi$
$24$	−2.38948	−0.487751
$25$	−2.97838	−0.595676
$26$	−4.20957	−0.825564
$27$	0.693819	0.133526
$28$	−5.14183	−0.971714
$29$	5.22382	0.970040	0.485020	−	0.874503i	$-0.338812\pi$
$29$	5.22382	0.970040	0.485020	+	0.874503i	$0.338812\pi$
$30$	−3.39746	−0.620288
$31$	9.21736	1.65549	0.827743	−	0.561107i	$-0.189624\pi$
$31$	9.21736	1.65549	0.827743	+	0.561107i	$0.189624\pi$
$32$	1.00000	0.176777
$33$	−6.93752	−1.20767
$34$	−4.01441	−0.688465
$35$	−7.31084	−1.23576
$36$	2.70964	0.451606
$37$	9.16884	1.50735	0.753674	−	0.657249i	$-0.228280\pi$
$37$	9.16884	1.50735	0.753674	+	0.657249i	$0.228280\pi$
$38$	1.37465	0.222998
$39$	10.0587	1.61068
$40$	1.42184	0.224812
$41$	3.42707	0.535218	0.267609	−	0.963528i	$-0.413766\pi$
$41$	3.42707	0.535218	0.267609	+	0.963528i	$0.413766\pi$
$42$	12.2863	1.89582
$43$	10.8992	1.66211	0.831056	−	0.556189i	$-0.187737\pi$
$43$	10.8992	1.66211	0.831056	+	0.556189i	$0.187737\pi$
$44$	2.90336	0.437697
$45$	3.85266	0.574321
$46$	1.34345	0.198081
$47$	−11.4604	−1.67167	−0.835835	−	0.548981i	$-0.815016\pi$
$47$	−11.4604	−1.67167	−0.835835	+	0.548981i	$0.815016\pi$
$48$	−2.38948	−0.344892
$49$	19.4384	2.77691
$50$	−2.97838	−0.421206
$51$	9.59236	1.34320
$52$	−4.20957	−0.583762
$53$	−4.25818	−0.584906	−0.292453	−	0.956280i	$-0.594472\pi$
$53$	−4.25818	−0.584906	−0.292453	+	0.956280i	$0.594472\pi$
$54$	0.693819	0.0944168
$55$	4.12810	0.556633
$56$	−5.14183	−0.687106
$57$	−3.28471	−0.435070
$58$	5.22382	0.685922
$59$	−2.10949	−0.274632	−0.137316	−	0.990527i	$-0.543848\pi$
$59$	−2.10949	−0.274632	−0.137316	+	0.990527i	$0.543848\pi$
$60$	−3.39746	−0.438610
$61$	−10.6807	−1.36753	−0.683764	−	0.729703i	$-0.739658\pi$
$61$	−10.6807	−1.36753	−0.683764	+	0.729703i	$0.739658\pi$
$62$	9.21736	1.17061
$63$	−13.9325	−1.75533
$64$	1.00000	0.125000
$65$	−5.98532	−0.742388
$66$	−6.93752	−0.853950
$67$	−9.40553	−1.14907	−0.574534	−	0.818481i	$-0.694817\pi$
$67$	−9.40553	−1.14907	−0.574534	+	0.818481i	$0.694817\pi$
$68$	−4.01441	−0.486818
$69$	−3.21015	−0.386456
$70$	−7.31084	−0.873813
$71$	−2.34289	−0.278050	−0.139025	−	0.990289i	$-0.544397\pi$
$71$	−2.34289	−0.278050	−0.139025	+	0.990289i	$0.544397\pi$
$72$	2.70964	0.319334
$73$	13.5006	1.58012	0.790062	−	0.613026i	$-0.210048\pi$
$73$	13.5006	1.58012	0.790062	+	0.613026i	$0.210048\pi$
$74$	9.16884	1.06586
$75$	7.11679	0.821776
$76$	1.37465	0.157683
$77$	−14.9286	−1.70127
$78$	10.0587	1.13892
$79$	0.841782	0.0947078	0.0473539	−	0.998878i	$-0.484921\pi$
$79$	0.841782	0.0947078	0.0473539	+	0.998878i	$0.484921\pi$
$80$	1.42184	0.158966
$81$	−9.78678	−1.08742
$82$	3.42707	0.378456
$83$	0.945519	0.103784	0.0518921	−	0.998653i	$-0.483475\pi$
$83$	0.945519	0.103784	0.0518921	+	0.998653i	$0.483475\pi$
$84$	12.2863	1.34055
$85$	−5.70783	−0.619101
$86$	10.8992	1.17529
$87$	−12.4822	−1.33824
$88$	2.90336	0.309499
$89$	−13.0187	−1.37998	−0.689988	−	0.723821i	$-0.742384\pi$
$89$	−13.0187	−1.37998	−0.689988	+	0.723821i	$0.742384\pi$
$90$	3.85266	0.406106
$91$	21.6449	2.26900
$92$	1.34345	0.140064
$93$	−22.0247	−2.28386
$94$	−11.4604	−1.18205
$95$	1.95453	0.200530
$96$	−2.38948	−0.243876
$97$	−14.8817	−1.51100	−0.755502	−	0.655146i	$-0.772607\pi$
$97$	−14.8817	−1.51100	−0.755502	+	0.655146i	$0.772607\pi$
$98$	19.4384	1.96357
$99$	7.86704	0.790667
$100$	−2.97838	−0.297838
$101$	−7.67362	−0.763553	−0.381777	−	0.924255i	$-0.624688\pi$
$101$	−7.67362	−0.763553	−0.381777	+	0.924255i	$0.624688\pi$
$102$	9.59236	0.949785
$103$	−14.9345	−1.47154	−0.735770	−	0.677231i	$-0.763180\pi$
$103$	−14.9345	−1.47154	−0.735770	+	0.677231i	$0.763180\pi$
$104$	−4.20957	−0.412782
$105$	17.4691	1.70481
$106$	−4.25818	−0.413591
$107$	14.5995	1.41138	0.705691	−	0.708519i	$-0.250636\pi$
$107$	14.5995	1.41138	0.705691	+	0.708519i	$0.250636\pi$
$108$	0.693819	0.0667628
$109$	7.70309	0.737822	0.368911	−	0.929465i	$-0.379731\pi$
$109$	7.70309	0.737822	0.368911	+	0.929465i	$0.379731\pi$
$110$	4.12810	0.393599
$111$	−21.9088	−2.07949
$112$	−5.14183	−0.485857
$113$	−16.4504	−1.54752	−0.773762	−	0.633477i	$-0.781627\pi$
$113$	−16.4504	−1.54752	−0.773762	+	0.633477i	$0.781627\pi$
$114$	−3.28471	−0.307641
$115$	1.91016	0.178124
$116$	5.22382	0.485020
$117$	−11.4064	−1.05452
$118$	−2.10949	−0.194194
$119$	20.6414	1.89219
$120$	−3.39746	−0.310144
$121$	−2.57053	−0.233684
$122$	−10.6807	−0.966989
$123$	−8.18892	−0.738370
$124$	9.21736	0.827743
$125$	−11.3440	−1.01463
$126$	−13.9325	−1.24120
$127$	0.453407	0.0402334	0.0201167	−	0.999798i	$-0.493596\pi$
$127$	0.453407	0.0402334	0.0201167	+	0.999798i	$0.493596\pi$
$128$	1.00000	0.0883883
$129$	−26.0435	−2.29300
$130$	−5.98532	−0.524947
$131$	−10.0133	−0.874862	−0.437431	−	0.899252i	$-0.644112\pi$
$131$	−10.0133	−0.874862	−0.437431	+	0.899252i	$0.644112\pi$
$132$	−6.93752	−0.603834
$133$	−7.06821	−0.612892
$134$	−9.40553	−0.812514
$135$	0.986498	0.0849042
$136$	−4.01441	−0.344232
$137$	−11.7423	−1.00321	−0.501605	−	0.865097i	$-0.667257\pi$
$137$	−11.7423	−1.00321	−0.501605	+	0.865097i	$0.667257\pi$
$138$	−3.21015	−0.273266
$139$	2.92705	0.248269	0.124134	−	0.992265i	$-0.460385\pi$
$139$	2.92705	0.248269	0.124134	+	0.992265i	$0.460385\pi$
$140$	−7.31084	−0.617879
$141$	27.3844	2.30618
$142$	−2.34289	−0.196611
$143$	−12.2219	−1.02204
$144$	2.70964	0.225803
$145$	7.42743	0.616814
$146$	13.5006	1.11732
$147$	−46.4477	−3.83094
$148$	9.16884	0.753674
$149$	15.0761	1.23508	0.617539	−	0.786540i	$-0.288130\pi$
$149$	15.0761	1.23508	0.617539	+	0.786540i	$0.288130\pi$
$150$	7.11679	0.581083
$151$	21.2341	1.72801	0.864005	−	0.503483i	$-0.167948\pi$
$151$	21.2341	1.72801	0.864005	+	0.503483i	$0.167948\pi$
$152$	1.37465	0.111499
$153$	−10.8776	−0.879400
$154$	−14.9286	−1.20298
$155$	13.1056	1.05267
$156$	10.0587	0.805340
$157$	5.39203	0.430331	0.215165	−	0.976578i	$-0.430971\pi$
$157$	5.39203	0.430331	0.215165	+	0.976578i	$0.430971\pi$
$158$	0.841782	0.0669686
$159$	10.1749	0.806919
$160$	1.42184	0.112406
$161$	−6.90778	−0.544409
$162$	−9.78678	−0.768922
$163$	−3.59900	−0.281896	−0.140948	−	0.990017i	$-0.545015\pi$
$163$	−3.59900	−0.281896	−0.140948	+	0.990017i	$0.545015\pi$
$164$	3.42707	0.267609
$165$	−9.86403	−0.767914
$166$	0.945519	0.0733865
$167$	10.5908	0.819544	0.409772	−	0.912188i	$-0.365608\pi$
$167$	10.5908	0.819544	0.409772	+	0.912188i	$0.365608\pi$
$168$	12.2863	0.947910
$169$	4.72046	0.363112
$170$	−5.70783	−0.437771
$171$	3.72480	0.284843
$172$	10.8992	0.831056
$173$	0.430848	0.0327568	0.0163784	−	0.999866i	$-0.494786\pi$
$173$	0.430848	0.0327568	0.0163784	+	0.999866i	$0.494786\pi$
$174$	−12.4822	−0.946277
$175$	15.3143	1.15765
$176$	2.90336	0.218849
$177$	5.04058	0.378873
$178$	−13.0187	−0.975790
$179$	19.1538	1.43162	0.715809	−	0.698296i	$-0.246058\pi$
$179$	19.1538	1.43162	0.715809	+	0.698296i	$0.246058\pi$
$180$	3.85266	0.287161
$181$	0.121914	0.00906181	0.00453091	−	0.999990i	$-0.498558\pi$
$181$	0.121914	0.00906181	0.00453091	+	0.999990i	$0.498558\pi$
$182$	21.6449	1.60442
$183$	25.5215	1.88660
$184$	1.34345	0.0990403
$185$	13.0366	0.958470
$186$	−22.0247	−1.61493
$187$	−11.6552	−0.852316
$188$	−11.4604	−0.835835
$189$	−3.56750	−0.259497
$190$	1.95453	0.141796
$191$	2.34992	0.170034	0.0850170	−	0.996379i	$-0.472906\pi$
$191$	2.34992	0.170034	0.0850170	+	0.996379i	$0.472906\pi$
$192$	−2.38948	−0.172446
$193$	−20.8041	−1.49751	−0.748755	−	0.662847i	$-0.769348\pi$
$193$	−20.8041	−1.49751	−0.748755	+	0.662847i	$0.769348\pi$
$194$	−14.8817	−1.06844
$195$	14.3018	1.02418
$196$	19.4384	1.38846
$197$	−13.6446	−0.972140	−0.486070	−	0.873920i	$-0.661570\pi$
$197$	−13.6446	−0.972140	−0.486070	+	0.873920i	$0.661570\pi$
$198$	7.86704	0.559086
$199$	17.0564	1.20910	0.604549	−	0.796568i	$-0.293354\pi$
$199$	17.0564	1.20910	0.604549	+	0.796568i	$0.293354\pi$
$200$	−2.97838	−0.210603
$201$	22.4744	1.58522
$202$	−7.67362	−0.539914
$203$	−26.8600	−1.88520
$204$	9.59236	0.671600
$205$	4.87273	0.340326
$206$	−14.9345	−1.04054
$207$	3.64026	0.253015
$208$	−4.20957	−0.291881
$209$	3.99110	0.276070
$210$	17.4691	1.20549
$211$	2.25313	0.155112	0.0775561	−	0.996988i	$-0.475288\pi$
$211$	2.25313	0.155112	0.0775561	+	0.996988i	$0.475288\pi$
$212$	−4.25818	−0.292453
$213$	5.59830	0.383589
$214$	14.5995	0.997998
$215$	15.4969	1.05688
$216$	0.693819	0.0472084
$217$	−47.3941	−3.21732
$218$	7.70309	0.521719
$219$	−32.2595	−2.17989
$220$	4.12810	0.278316
$221$	16.8989	1.13674
$222$	−21.9088	−1.47042
$223$	−19.8645	−1.33023	−0.665113	−	0.746743i	$-0.731617\pi$
$223$	−19.8645	−1.33023	−0.665113	+	0.746743i	$0.731617\pi$
$224$	−5.14183	−0.343553
$225$	−8.07032	−0.538021
$226$	−16.4504	−1.09426
$227$	3.37445	0.223970	0.111985	−	0.993710i	$-0.464279\pi$
$227$	3.37445	0.223970	0.111985	+	0.993710i	$0.464279\pi$
$228$	−3.28471	−0.217535
$229$	−18.9033	−1.24916	−0.624582	−	0.780959i	$-0.714731\pi$
$229$	−18.9033	−1.24916	−0.624582	+	0.780959i	$0.714731\pi$
$230$	1.91016	0.125953
$231$	35.6715	2.34702
$232$	5.22382	0.342961
$233$	−16.4601	−1.07834	−0.539169	−	0.842197i	$-0.681262\pi$
$233$	−16.4601	−1.07834	−0.539169	+	0.842197i	$0.681262\pi$
$234$	−11.4064	−0.745659
$235$	−16.2948	−1.06296
$236$	−2.10949	−0.137316
$237$	−2.01142	−0.130656
$238$	20.6414	1.33798
$239$	−26.5319	−1.71620	−0.858102	−	0.513479i	$-0.828356\pi$
$239$	−26.5319	−1.71620	−0.858102	+	0.513479i	$0.828356\pi$
$240$	−3.39746	−0.219305
$241$	−7.44610	−0.479646	−0.239823	−	0.970817i	$-0.577089\pi$
$241$	−7.44610	−0.479646	−0.239823	+	0.970817i	$0.577089\pi$
$242$	−2.57053	−0.165240
$243$	21.3039	1.36665
$244$	−10.6807	−0.683764
$245$	27.6382	1.76574
$246$	−8.18892	−0.522107
$247$	−5.78668	−0.368198
$248$	9.21736	0.585303
$249$	−2.25930	−0.143177
$250$	−11.3440	−0.717455
$251$	−1.86646	−0.117810	−0.0589050	−	0.998264i	$-0.518761\pi$
$251$	−1.86646	−0.117810	−0.0589050	+	0.998264i	$0.518761\pi$
$252$	−13.9325	−0.877664
$253$	3.90051	0.245223
$254$	0.453407	0.0284493
$255$	13.6388	0.854093
$256$	1.00000	0.0625000
$257$	7.34741	0.458319	0.229159	−	0.973389i	$-0.426402\pi$
$257$	7.34741	0.458319	0.229159	+	0.973389i	$0.426402\pi$
$258$	−26.0435	−1.62140
$259$	−47.1446	−2.92942
$260$	−5.98532	−0.371194
$261$	14.1547	0.876152
$262$	−10.0133	−0.618621
$263$	−20.6752	−1.27489	−0.637445	−	0.770496i	$-0.720009\pi$
$263$	−20.6752	−1.27489	−0.637445	+	0.770496i	$0.720009\pi$
$264$	−6.93752	−0.426975
$265$	−6.05444	−0.371921
$266$	−7.06821	−0.433380
$267$	31.1079	1.90377
$268$	−9.40553	−0.574534
$269$	−17.4334	−1.06293	−0.531466	−	0.847080i	$-0.678359\pi$
$269$	−17.4334	−1.06293	−0.531466	+	0.847080i	$0.678359\pi$
$270$	0.986498	0.0600364
$271$	−18.9859	−1.15331	−0.576656	−	0.816987i	$-0.695643\pi$
$271$	−18.9859	−1.15331	−0.576656	+	0.816987i	$0.695643\pi$
$272$	−4.01441	−0.243409
$273$	−51.7201	−3.13024
$274$	−11.7423	−0.709376
$275$	−8.64729	−0.521451
$276$	−3.21015	−0.193228
$277$	29.3601	1.76408	0.882040	−	0.471175i	$-0.156170\pi$
$277$	29.3601	1.76408	0.882040	+	0.471175i	$0.156170\pi$
$278$	2.92705	0.175553
$279$	24.9757	1.49526
$280$	−7.31084	−0.436906
$281$	−15.9187	−0.949630	−0.474815	−	0.880086i	$-0.657485\pi$
$281$	−15.9187	−0.949630	−0.474815	+	0.880086i	$0.657485\pi$
$282$	27.3844	1.63072
$283$	24.5215	1.45765	0.728826	−	0.684699i	$-0.240066\pi$
$283$	24.5215	1.45765	0.728826	+	0.684699i	$0.240066\pi$
$284$	−2.34289	−0.139025
$285$	−4.67032	−0.276646
$286$	−12.2219	−0.722694
$287$	−17.6214	−1.04016
$288$	2.70964	0.159667
$289$	−0.884550	−0.0520324
$290$	7.42743	0.436154
$291$	35.5595	2.08454
$292$	13.5006	0.790062
$293$	−9.15367	−0.534763	−0.267382	−	0.963591i	$-0.586158\pi$
$293$	−9.15367	−0.534763	−0.267382	+	0.963591i	$0.586158\pi$
$294$	−46.4477	−2.70889
$295$	−2.99935	−0.174629
$296$	9.16884	0.532928
$297$	2.01440	0.116888
$298$	15.0761	0.873333
$299$	−5.65533	−0.327056
$300$	7.11679	0.410888
$301$	−56.0418	−3.23019
$302$	21.2341	1.22189
$303$	18.3360	1.05338
$304$	1.37465	0.0788416
$305$	−15.1863	−0.869564
$306$	−10.8776	−0.621830
$307$	−1.15471	−0.0659030	−0.0329515	−	0.999457i	$-0.510491\pi$
$307$	−1.15471	−0.0659030	−0.0329515	+	0.999457i	$0.510491\pi$
$308$	−14.9286	−0.850633
$309$	35.6858	2.03009
$310$	13.1056	0.744347
$311$	3.08684	0.175039	0.0875194	−	0.996163i	$-0.472106\pi$
$311$	3.08684	0.175039	0.0875194	+	0.996163i	$0.472106\pi$
$312$	10.0587	0.569462
$313$	−3.44957	−0.194981	−0.0974907	−	0.995236i	$-0.531082\pi$
$313$	−3.44957	−0.194981	−0.0974907	+	0.995236i	$0.531082\pi$
$314$	5.39203	0.304290
$315$	−19.8097	−1.11615
$316$	0.841782	0.0473539
$317$	23.8691	1.34062	0.670311	−	0.742080i	$-0.266160\pi$
$317$	23.8691	1.34062	0.670311	+	0.742080i	$0.266160\pi$
$318$	10.1749	0.570578
$319$	15.1666	0.849167
$320$	1.42184	0.0794831
$321$	−34.8852	−1.94710
$322$	−6.90778	−0.384955
$323$	−5.51840	−0.307052
$324$	−9.78678	−0.543710
$325$	12.5377	0.695465
$326$	−3.59900	−0.199330
$327$	−18.4064	−1.01788
$328$	3.42707	0.189228
$329$	58.9273	3.24877
$330$	−9.86403	−0.542997
$331$	−29.2312	−1.60669	−0.803346	−	0.595512i	$-0.796949\pi$
$331$	−29.2312	−1.60669	−0.803346	+	0.595512i	$0.796949\pi$
$332$	0.945519	0.0518921
$333$	24.8442	1.36145
$334$	10.5908	0.579505
$335$	−13.3731	−0.730652
$336$	12.2863	0.670274
$337$	−5.93720	−0.323420	−0.161710	−	0.986838i	$-0.551701\pi$
$337$	−5.93720	−0.323420	−0.161710	+	0.986838i	$0.551701\pi$
$338$	4.72046	0.256759
$339$	39.3080	2.13492
$340$	−5.70783	−0.309551
$341$	26.7613	1.44920
$342$	3.72480	0.201414
$343$	−63.9560	−3.45330
$344$	10.8992	0.587645
$345$	−4.56431	−0.245734
$346$	0.430848	0.0231625
$347$	16.1106	0.864864	0.432432	−	0.901666i	$-0.357656\pi$
$347$	16.1106	0.864864	0.432432	+	0.901666i	$0.357656\pi$
$348$	−12.4822	−0.669119
$349$	−35.1279	−1.88035	−0.940177	−	0.340687i	$-0.889340\pi$
$349$	−35.1279	−1.88035	−0.940177	+	0.340687i	$0.889340\pi$
$350$	15.3143	0.818584
$351$	−2.92068	−0.155894
$352$	2.90336	0.154749
$353$	15.1299	0.805284	0.402642	−	0.915358i	$-0.368092\pi$
$353$	15.1299	0.805284	0.402642	+	0.915358i	$0.368092\pi$
$354$	5.04058	0.267904
$355$	−3.33121	−0.176802
$356$	−13.0187	−0.689988
$357$	−49.3223	−2.61041
$358$	19.1538	1.01231
$359$	−22.5462	−1.18994	−0.594972	−	0.803746i	$-0.702837\pi$
$359$	−22.5462	−1.18994	−0.594972	+	0.803746i	$0.702837\pi$
$360$	3.85266	0.203053
$361$	−17.1103	−0.900544
$362$	0.121914	0.00640767
$363$	6.14224	0.322384
$364$	21.6449	1.13450
$365$	19.1956	1.00475
$366$	25.5215	1.33403
$367$	6.83473	0.356770	0.178385	−	0.983961i	$-0.442913\pi$
$367$	6.83473	0.356770	0.178385	+	0.983961i	$0.442913\pi$
$368$	1.34345	0.0700321
$369$	9.28611	0.483415
$370$	13.0366	0.677740
$371$	21.8948	1.13672
$372$	−22.0247	−1.14193
$373$	−0.0491684	−0.00254584	−0.00127292	−	0.999999i	$-0.500405\pi$
$373$	−0.0491684	−0.00254584	−0.00127292	+	0.999999i	$0.500405\pi$
$374$	−11.6552	−0.602678
$375$	27.1062	1.39976
$376$	−11.4604	−0.591024
$377$	−21.9900	−1.13254
$378$	−3.56750	−0.183492
$379$	−1.17398	−0.0603035	−0.0301518	−	0.999545i	$-0.509599\pi$
$379$	−1.17398	−0.0603035	−0.0301518	+	0.999545i	$0.509599\pi$
$380$	1.95453	0.100265
$381$	−1.08341	−0.0555048
$382$	2.34992	0.120232
$383$	9.07320	0.463619	0.231810	−	0.972761i	$-0.425535\pi$
$383$	9.07320	0.463619	0.231810	+	0.972761i	$0.425535\pi$
$384$	−2.38948	−0.121938
$385$	−21.2260	−1.08178
$386$	−20.8041	−1.05890
$387$	29.5329	1.50124
$388$	−14.8817	−0.755502
$389$	33.3885	1.69286	0.846432	−	0.532497i	$-0.178746\pi$
$389$	33.3885	1.69286	0.846432	+	0.532497i	$0.178746\pi$
$390$	14.3018	0.724201
$391$	−5.39315	−0.272743
$392$	19.4384	0.981787
$393$	23.9265	1.20693
$394$	−13.6446	−0.687407
$395$	1.19688	0.0602214
$396$	7.86704	0.395333
$397$	−9.96264	−0.500010	−0.250005	−	0.968244i	$-0.580432\pi$
$397$	−9.96264	−0.500010	−0.250005	+	0.968244i	$0.580432\pi$
$398$	17.0564	0.854961
$399$	16.8894	0.845527
$400$	−2.97838	−0.148919
$401$	1.14364	0.0571105	0.0285553	−	0.999592i	$-0.490909\pi$
$401$	1.14364	0.0571105	0.0285553	+	0.999592i	$0.490909\pi$
$402$	22.4744	1.12092
$403$	−38.8011	−1.93282
$404$	−7.67362	−0.381777
$405$	−13.9152	−0.691452
$406$	−26.8600	−1.33304
$407$	26.6204	1.31952
$408$	9.59236	0.474893
$409$	−19.3315	−0.955880	−0.477940	−	0.878392i	$-0.658616\pi$
$409$	−19.3315	−0.955880	−0.477940	+	0.878392i	$0.658616\pi$
$410$	4.87273	0.240647
$411$	28.0580	1.38400
$412$	−14.9345	−0.735770
$413$	10.8466	0.533727
$414$	3.64026	0.178909
$415$	1.34437	0.0659927
$416$	−4.20957	−0.206391
$417$	−6.99414	−0.342504
$418$	3.99110	0.195211
$419$	22.2939	1.08913	0.544563	−	0.838720i	$-0.316695\pi$
$419$	22.2939	1.08913	0.544563	+	0.838720i	$0.316695\pi$
$420$	17.4691	0.852407
$421$	14.8667	0.724556	0.362278	−	0.932070i	$-0.381999\pi$
$421$	14.8667	0.724556	0.362278	+	0.932070i	$0.381999\pi$
$422$	2.25313	0.109681
$423$	−31.0535	−1.50987
$424$	−4.25818	−0.206796
$425$	11.9564	0.579971
$426$	5.59830	0.271238
$427$	54.9185	2.65769
$428$	14.5995	0.705691
$429$	29.2040	1.40998
$430$	15.4969	0.747326
$431$	−24.4479	−1.17761	−0.588806	−	0.808274i	$-0.700402\pi$
$431$	−24.4479	−1.17761	−0.588806	+	0.808274i	$0.700402\pi$
$432$	0.693819	0.0333814
$433$	10.2369	0.491955	0.245977	−	0.969276i	$-0.420891\pi$
$433$	10.2369	0.491955	0.245977	+	0.969276i	$0.420891\pi$
$434$	−47.3941	−2.27499
$435$	−17.7477	−0.850938
$436$	7.70309	0.368911
$437$	1.84677	0.0883431
$438$	−32.2595	−1.54142
$439$	21.3360	1.01831	0.509155	−	0.860675i	$-0.329958\pi$
$439$	21.3360	1.01831	0.509155	+	0.860675i	$0.329958\pi$
$440$	4.12810	0.196799
$441$	52.6710	2.50814
$442$	16.8989	0.803799
$443$	−22.7643	−1.08156	−0.540782	−	0.841163i	$-0.681871\pi$
$443$	−22.7643	−1.08156	−0.540782	+	0.841163i	$0.681871\pi$
$444$	−21.9088	−1.03975
$445$	−18.5104	−0.877479
$446$	−19.8645	−0.940612
$447$	−36.0240	−1.70388
$448$	−5.14183	−0.242928
$449$	−21.7543	−1.02665	−0.513325	−	0.858194i	$-0.671587\pi$
$449$	−21.7543	−1.02665	−0.513325	+	0.858194i	$0.671587\pi$
$450$	−8.07032	−0.380439
$451$	9.94999	0.468527
$452$	−16.4504	−0.773762
$453$	−50.7387	−2.38391
$454$	3.37445	0.158371
$455$	30.7755	1.44278
$456$	−3.28471	−0.153820
$457$	1.50675	0.0704830	0.0352415	−	0.999379i	$-0.488780\pi$
$457$	1.50675	0.0704830	0.0352415	+	0.999379i	$0.488780\pi$
$458$	−18.9033	−0.883293
$459$	−2.78527	−0.130005
$460$	1.91016	0.0890619
$461$	−37.9320	−1.76667	−0.883336	−	0.468741i	$-0.844708\pi$
$461$	−37.9320	−1.76667	−0.883336	+	0.468741i	$0.844708\pi$
$462$	35.6715	1.65959
$463$	6.51234	0.302654	0.151327	−	0.988484i	$-0.451645\pi$
$463$	6.51234	0.302654	0.151327	+	0.988484i	$0.451645\pi$
$464$	5.22382	0.242510
$465$	−31.3156	−1.45223
$466$	−16.4601	−0.762501
$467$	12.8750	0.595785	0.297892	−	0.954599i	$-0.403716\pi$
$467$	12.8750	0.595785	0.297892	+	0.954599i	$0.403716\pi$
$468$	−11.4064	−0.527261
$469$	48.3616	2.23313
$470$	−16.2948	−0.751623
$471$	−12.8842	−0.593671
$472$	−2.10949	−0.0970969
$473$	31.6442	1.45500
$474$	−2.01142	−0.0923878
$475$	−4.09423	−0.187856
$476$	20.6414	0.946096
$477$	−11.5381	−0.528294
$478$	−26.5319	−1.21354
$479$	−14.7058	−0.671925	−0.335963	−	0.941875i	$-0.609062\pi$
$479$	−14.7058	−0.671925	−0.335963	+	0.941875i	$0.609062\pi$
$480$	−3.39746	−0.155072
$481$	−38.5968	−1.75986
$482$	−7.44610	−0.339161
$483$	16.5060	0.751050
$484$	−2.57053	−0.116842
$485$	−21.1593	−0.960795
$486$	21.3039	0.966365
$487$	−3.07533	−0.139357	−0.0696784	−	0.997570i	$-0.522197\pi$
$487$	−3.07533	−0.139357	−0.0696784	+	0.997570i	$0.522197\pi$
$488$	−10.6807	−0.483494
$489$	8.59976	0.388895
$490$	27.6382	1.24857
$491$	14.4147	0.650525	0.325263	−	0.945624i	$-0.394547\pi$
$491$	14.4147	0.650525	0.325263	+	0.945624i	$0.394547\pi$
$492$	−8.18892	−0.369185
$493$	−20.9705	−0.944466
$494$	−5.78668	−0.260355
$495$	11.1856	0.502758
$496$	9.21736	0.413872
$497$	12.0467	0.540370
$498$	−2.25930	−0.101242
$499$	25.0040	1.11933	0.559667	−	0.828718i	$-0.310929\pi$
$499$	25.0040	1.11933	0.559667	+	0.828718i	$0.310929\pi$
$500$	−11.3440	−0.507317
$501$	−25.3067	−1.13062
$502$	−1.86646	−0.0833043
$503$	18.6060	0.829601	0.414801	−	0.909912i	$-0.363851\pi$
$503$	18.6060	0.829601	0.414801	+	0.909912i	$0.363851\pi$
$504$	−13.9325	−0.620602
$505$	−10.9106	−0.485517
$506$	3.90051	0.173399
$507$	−11.2795	−0.500938
$508$	0.453407	0.0201167
$509$	44.4767	1.97139	0.985697	−	0.168526i	$-0.0539008\pi$
$509$	44.4767	1.97139	0.985697	+	0.168526i	$0.0539008\pi$
$510$	13.6388	0.603935
$511$	−69.4177	−3.07086
$512$	1.00000	0.0441942
$513$	0.953759	0.0421095
$514$	7.34741	0.324080
$515$	−21.2344	−0.935701
$516$	−26.0435	−1.14650
$517$	−33.2736	−1.46337
$518$	−47.1446	−2.07141
$519$	−1.02950	−0.0451902
$520$	−5.98532	−0.262474
$521$	−29.5541	−1.29479	−0.647395	−	0.762155i	$-0.724141\pi$
$521$	−29.5541	−1.29479	−0.647395	+	0.762155i	$0.724141\pi$
$522$	14.1547	0.619533
$523$	4.11148	0.179782	0.0898911	−	0.995952i	$-0.471348\pi$
$523$	4.11148	0.179782	0.0898911	+	0.995952i	$0.471348\pi$
$524$	−10.0133	−0.437431
$525$	−36.5933	−1.59706
$526$	−20.6752	−0.901483
$527$	−37.0022	−1.61184
$528$	−6.93752	−0.301917
$529$	−21.1951	−0.921528
$530$	−6.05444	−0.262988
$531$	−5.71594	−0.248051
$532$	−7.06821	−0.306446
$533$	−14.4265	−0.624880
$534$	31.1079	1.34617
$535$	20.7581	0.897449
$536$	−9.40553	−0.406257
$537$	−45.7676	−1.97502
$538$	−17.4334	−0.751607
$539$	56.4365	2.43089
$540$	0.986498	0.0424521
$541$	38.5446	1.65716	0.828582	−	0.559868i	$-0.189148\pi$
$541$	38.5446	1.65716	0.828582	+	0.559868i	$0.189148\pi$
$542$	−18.9859	−0.815514
$543$	−0.291312	−0.0125014
$544$	−4.01441	−0.172116
$545$	10.9525	0.469155
$546$	−51.7201	−2.21341
$547$	−33.7371	−1.44249	−0.721246	−	0.692679i	$-0.756430\pi$
$547$	−33.7371	−1.44249	−0.721246	+	0.692679i	$0.756430\pi$
$548$	−11.7423	−0.501605
$549$	−28.9409	−1.23517
$550$	−8.64729	−0.368722
$551$	7.18093	0.305918
$552$	−3.21015	−0.136633
$553$	−4.32830	−0.184058
$554$	29.3601	1.24739
$555$	−31.1507	−1.32228
$556$	2.92705	0.124134
$557$	18.6700	0.791072	0.395536	−	0.918450i	$-0.370559\pi$
$557$	18.6700	0.791072	0.395536	+	0.918450i	$0.370559\pi$
$558$	24.9757	1.05731
$559$	−45.8809	−1.94056
$560$	−7.31084	−0.308940
$561$	27.8500	1.17583
$562$	−15.9187	−0.671490
$563$	−44.7862	−1.88751	−0.943756	−	0.330643i	$-0.892734\pi$
$563$	−44.7862	−1.88751	−0.943756	+	0.330643i	$0.892734\pi$
$564$	27.3844	1.15309
$565$	−23.3898	−0.984016
$566$	24.5215	1.03072
$567$	50.3219	2.11332
$568$	−2.34289	−0.0983054
$569$	−1.74341	−0.0730874	−0.0365437	−	0.999332i	$-0.511635\pi$
$569$	−1.74341	−0.0730874	−0.0365437	+	0.999332i	$0.511635\pi$
$570$	−4.67032	−0.195618
$571$	−23.9500	−1.00227	−0.501137	−	0.865368i	$-0.667085\pi$
$571$	−23.9500	−1.00227	−0.501137	+	0.865368i	$0.667085\pi$
$572$	−12.2219	−0.511022
$573$	−5.61509	−0.234574
$574$	−17.6214	−0.735503
$575$	−4.00130	−0.166866
$576$	2.70964	0.112902
$577$	−36.6678	−1.52650	−0.763251	−	0.646102i	$-0.776398\pi$
$577$	−36.6678	−1.52650	−0.763251	+	0.646102i	$0.776398\pi$
$578$	−0.884550	−0.0367924
$579$	49.7110	2.06592
$580$	7.42743	0.308407
$581$	−4.86169	−0.201697
$582$	35.5595	1.47399
$583$	−12.3630	−0.512024
$584$	13.5006	0.558658
$585$	−16.2180	−0.670534
$586$	−9.15367	−0.378135
$587$	−5.97899	−0.246779	−0.123390	−	0.992358i	$-0.539376\pi$
$587$	−5.97899	−0.246779	−0.123390	+	0.992358i	$0.539376\pi$
$588$	−46.4477	−1.91547
$589$	12.6706	0.522085
$590$	−2.99935	−0.123481
$591$	32.6036	1.34113
$592$	9.16884	0.376837
$593$	−38.4295	−1.57811	−0.789055	−	0.614322i	$-0.789430\pi$
$593$	−38.4295	−1.57811	−0.789055	+	0.614322i	$0.789430\pi$
$594$	2.01440	0.0826520
$595$	29.3487	1.20318
$596$	15.0761	0.617539
$597$	−40.7560	−1.66803
$598$	−5.65533	−0.231264
$599$	−17.1038	−0.698842	−0.349421	−	0.936966i	$-0.613622\pi$
$599$	−17.1038	−0.698842	−0.349421	+	0.936966i	$0.613622\pi$
$600$	7.11679	0.290542
$601$	16.8252	0.686316	0.343158	−	0.939278i	$-0.388503\pi$
$601$	16.8252	0.686316	0.343158	+	0.939278i	$0.388503\pi$
$602$	−56.0418	−2.28409
$603$	−25.4856	−1.03785
$604$	21.2341	0.864005
$605$	−3.65487	−0.148592
$606$	18.3360	0.744849
$607$	−8.43571	−0.342395	−0.171197	−	0.985237i	$-0.554764\pi$
$607$	−8.43571	−0.342395	−0.171197	+	0.985237i	$0.554764\pi$
$608$	1.37465	0.0557494
$609$	64.1815	2.60077
$610$	−15.1863	−0.614874
$611$	48.2433	1.95171
$612$	−10.8776	−0.439700
$613$	−40.1922	−1.62335	−0.811674	−	0.584111i	$-0.801443\pi$
$613$	−40.1922	−1.62335	−0.811674	+	0.584111i	$0.801443\pi$
$614$	−1.15471	−0.0466005
$615$	−11.6433	−0.469504
$616$	−14.9286	−0.601488
$617$	11.7673	0.473734	0.236867	−	0.971542i	$-0.423879\pi$
$617$	11.7673	0.473734	0.236867	+	0.971542i	$0.423879\pi$
$618$	35.6858	1.43549
$619$	8.17547	0.328600	0.164300	−	0.986410i	$-0.447464\pi$
$619$	8.17547	0.328600	0.164300	+	0.986410i	$0.447464\pi$
$620$	13.1056	0.526333
$621$	0.932110	0.0374043
$622$	3.08684	0.123771
$623$	66.9397	2.68188
$624$	10.0587	0.402670
$625$	−1.23738	−0.0494950
$626$	−3.44957	−0.137873
$627$	−9.53667	−0.380858
$628$	5.39203	0.215165
$629$	−36.8074	−1.46761
$630$	−19.8097	−0.789238
$631$	10.6052	0.422187	0.211093	−	0.977466i	$-0.432298\pi$
$631$	10.6052	0.422187	0.211093	+	0.977466i	$0.432298\pi$
$632$	0.841782	0.0334843
$633$	−5.38383	−0.213988
$634$	23.8691	0.947964
$635$	0.644672	0.0255830
$636$	10.1749	0.403459
$637$	−81.8272	−3.24211
$638$	15.1666	0.600452
$639$	−6.34838	−0.251138
$640$	1.42184	0.0562031
$641$	−10.4993	−0.414698	−0.207349	−	0.978267i	$-0.566484\pi$
$641$	−10.4993	−0.414698	−0.207349	+	0.978267i	$0.566484\pi$
$642$	−34.8852	−1.37681
$643$	44.5720	1.75775	0.878875	−	0.477053i	$-0.158295\pi$
$643$	44.5720	1.75775	0.878875	+	0.477053i	$0.158295\pi$
$644$	−6.90778	−0.272205
$645$	−37.0296	−1.45804
$646$	−5.51840	−0.217119
$647$	37.3445	1.46816	0.734082	−	0.679061i	$-0.237613\pi$
$647$	37.3445	1.46816	0.734082	+	0.679061i	$0.237613\pi$
$648$	−9.78678	−0.384461
$649$	−6.12459	−0.240411
$650$	12.5377	0.491768
$651$	113.247	4.43852
$652$	−3.59900	−0.140948
$653$	13.2176	0.517243	0.258622	−	0.965979i	$-0.416732\pi$
$653$	13.2176	0.517243	0.258622	+	0.965979i	$0.416732\pi$
$654$	−18.4064	−0.719748
$655$	−14.2372	−0.556294
$656$	3.42707	0.133804
$657$	36.5817	1.42719
$658$	58.9273	2.29723
$659$	36.9145	1.43798	0.718992	−	0.695019i	$-0.244604\pi$
$659$	36.9145	1.43798	0.718992	+	0.695019i	$0.244604\pi$
$660$	−9.86403	−0.383957
$661$	22.0881	0.859129	0.429565	−	0.903036i	$-0.358667\pi$
$661$	22.0881	0.859129	0.429565	+	0.903036i	$0.358667\pi$
$662$	−29.2312	−1.13610
$663$	−40.3797	−1.56822
$664$	0.945519	0.0366932
$665$	−10.0499	−0.389717
$666$	24.8442	0.962694
$667$	7.01794	0.271736
$668$	10.5908	0.409772
$669$	47.4659	1.83514
$670$	−13.3731	−0.516649
$671$	−31.0100	−1.19713
$672$	12.2863	0.473955
$673$	22.6873	0.874531	0.437266	−	0.899332i	$-0.355947\pi$
$673$	22.6873	0.874531	0.437266	+	0.899332i	$0.355947\pi$
$674$	−5.93720	−0.228692
$675$	−2.06646	−0.0795379
$676$	4.72046	0.181556
$677$	−27.6032	−1.06088	−0.530439	−	0.847723i	$-0.677973\pi$
$677$	−27.6032	−1.06088	−0.530439	+	0.847723i	$0.677973\pi$
$678$	39.3080	1.50961
$679$	76.5190	2.93653
$680$	−5.70783	−0.218885
$681$	−8.06319	−0.308982
$682$	26.7613	1.02474
$683$	−14.1438	−0.541196	−0.270598	−	0.962692i	$-0.587221\pi$
$683$	−14.1438	−0.541196	−0.270598	+	0.962692i	$0.587221\pi$
$684$	3.72480	0.142421
$685$	−16.6956	−0.637906
$686$	−63.9560	−2.44185
$687$	45.1691	1.72331
$688$	10.8992	0.415528
$689$	17.9251	0.682892
$690$	−4.56431	−0.173760
$691$	−23.2651	−0.885047	−0.442523	−	0.896757i	$-0.645917\pi$
$691$	−23.2651	−0.885047	−0.442523	+	0.896757i	$0.645917\pi$
$692$	0.430848	0.0163784
$693$	−40.4509	−1.53660
$694$	16.1106	0.611551
$695$	4.16179	0.157866
$696$	−12.4822	−0.473138
$697$	−13.7576	−0.521108
$698$	−35.1279	−1.32961
$699$	39.3312	1.48764
$700$	15.3143	0.578826
$701$	−13.7469	−0.519215	−0.259607	−	0.965714i	$-0.583593\pi$
$701$	−13.7469	−0.519215	−0.259607	+	0.965714i	$0.583593\pi$
$702$	−2.92068	−0.110234
$703$	12.6039	0.475367
$704$	2.90336	0.109424
$705$	38.9362	1.46642
$706$	15.1299	0.569421
$707$	39.4564	1.48391
$708$	5.04058	0.189437
$709$	−43.8140	−1.64547	−0.822734	−	0.568427i	$-0.807552\pi$
$709$	−43.8140	−1.64547	−0.822734	+	0.568427i	$0.807552\pi$
$710$	−3.33121	−0.125018
$711$	2.28092	0.0855413
$712$	−13.0187	−0.487895
$713$	12.3830	0.463749
$714$	−49.3223	−1.84584
$715$	−17.3775	−0.649882
$716$	19.1538	0.715809
$717$	63.3975	2.36762
$718$	−22.5462	−0.841418
$719$	−3.99758	−0.149084	−0.0745422	−	0.997218i	$-0.523750\pi$
$719$	−3.99758	−0.149084	−0.0745422	+	0.997218i	$0.523750\pi$
$720$	3.85266	0.143580
$721$	76.7906	2.85983
$722$	−17.1103	−0.636781
$723$	17.7923	0.661704
$724$	0.121914	0.00453091
$725$	−15.5585	−0.577829
$726$	6.14224	0.227960
$727$	23.4184	0.868542	0.434271	−	0.900782i	$-0.357006\pi$
$727$	23.4184	0.868542	0.434271	+	0.900782i	$0.357006\pi$
$728$	21.6449	0.802212
$729$	−21.5450	−0.797963
$730$	19.1956	0.710463
$731$	−43.7538	−1.61829
$732$	25.5215	0.943301
$733$	−27.7628	−1.02544	−0.512721	−	0.858555i	$-0.671363\pi$
$733$	−27.7628	−1.02544	−0.512721	+	0.858555i	$0.671363\pi$
$734$	6.83473	0.252274
$735$	−66.0411	−2.43596
$736$	1.34345	0.0495202
$737$	−27.3076	−1.00589
$738$	9.28611	0.341826
$739$	24.4177	0.898219	0.449110	−	0.893477i	$-0.351741\pi$
$739$	24.4177	0.898219	0.449110	+	0.893477i	$0.351741\pi$
$740$	13.0366	0.479235
$741$	13.8272	0.507954
$742$	21.8948	0.803785
$743$	−33.0279	−1.21168	−0.605839	−	0.795588i	$-0.707162\pi$
$743$	−33.0279	−1.21168	−0.605839	+	0.795588i	$0.707162\pi$
$744$	−22.0247	−0.807466
$745$	21.4357	0.785344
$746$	−0.0491684	−0.00180018
$747$	2.56201	0.0937391
$748$	−11.6552	−0.426158
$749$	−75.0679	−2.74292
$750$	27.1062	0.989779
$751$	−46.3488	−1.69129	−0.845647	−	0.533743i	$-0.820785\pi$
$751$	−46.3488	−1.69129	−0.845647	+	0.533743i	$0.820785\pi$
$752$	−11.4604	−0.417917
$753$	4.45988	0.162527
$754$	−21.9900	−0.800830
$755$	30.1915	1.09878
$756$	−3.56750	−0.129749
$757$	52.5408	1.90963	0.954814	−	0.297204i	$-0.0960540\pi$
$757$	52.5408	1.90963	0.954814	+	0.297204i	$0.0960540\pi$
$758$	−1.17398	−0.0426410
$759$	−9.32020	−0.338302
$760$	1.95453	0.0708982
$761$	36.9872	1.34078	0.670392	−	0.742007i	$-0.266126\pi$
$761$	36.9872	1.34078	0.670392	+	0.742007i	$0.266126\pi$
$762$	−1.08341	−0.0392478
$763$	−39.6080	−1.43390
$764$	2.34992	0.0850170
$765$	−15.4662	−0.559180
$766$	9.07320	0.327828
$767$	8.88002	0.320639
$768$	−2.38948	−0.0862231
$769$	−26.9105	−0.970417	−0.485208	−	0.874399i	$-0.661256\pi$
$769$	−26.9105	−0.970417	−0.485208	+	0.874399i	$0.661256\pi$
$770$	−21.2260	−0.764931
$771$	−17.5565	−0.632283
$772$	−20.8041	−0.748755
$773$	−27.2374	−0.979661	−0.489831	−	0.871818i	$-0.662941\pi$
$773$	−27.2374	−0.979661	−0.489831	+	0.871818i	$0.662941\pi$
$774$	29.5329	1.06154
$775$	−27.4528	−0.986133
$776$	−14.8817	−0.534221
$777$	112.651	4.04134
$778$	33.3885	1.19704
$779$	4.71102	0.168790
$780$	14.3018	0.512088
$781$	−6.80224	−0.243403
$782$	−5.39315	−0.192858
$783$	3.62439	0.129525
$784$	19.4384	0.694228
$785$	7.66659	0.273632
$786$	23.9265	0.853431
$787$	−30.0060	−1.06960	−0.534800	−	0.844979i	$-0.679613\pi$
$787$	−30.0060	−1.06960	−0.534800	+	0.844979i	$0.679613\pi$
$788$	−13.6446	−0.486070
$789$	49.4031	1.75880
$790$	1.19688	0.0425830
$791$	84.5851	3.00750
$792$	7.86704	0.279543
$793$	44.9613	1.59662
$794$	−9.96264	−0.353561
$795$	14.4670	0.513092
$796$	17.0564	0.604549
$797$	−39.2211	−1.38928	−0.694641	−	0.719356i	$-0.744437\pi$
$797$	−39.2211	−1.38928	−0.694641	+	0.719356i	$0.744437\pi$
$798$	16.8894	0.597878
$799$	46.0066	1.62760
$800$	−2.97838	−0.105302
$801$	−35.2759	−1.24641
$802$	1.14364	0.0403832
$803$	39.1970	1.38323
$804$	22.4744	0.792609
$805$	−9.82174	−0.346171
$806$	−38.8011	−1.36671
$807$	41.6568	1.46639
$808$	−7.67362	−0.269957
$809$	−44.4862	−1.56405	−0.782026	−	0.623246i	$-0.785813\pi$
$809$	−44.4862	−1.56405	−0.782026	+	0.623246i	$0.785813\pi$
$810$	−13.9152	−0.488931
$811$	14.8183	0.520341	0.260171	−	0.965563i	$-0.416221\pi$
$811$	14.8183	0.520341	0.260171	+	0.965563i	$0.416221\pi$
$812$	−26.8600	−0.942601
$813$	45.3665	1.59107
$814$	26.6204	0.933044
$815$	−5.11720	−0.179248
$816$	9.59236	0.335800
$817$	14.9826	0.524174
$818$	−19.3315	−0.675909
$819$	58.6497	2.04939
$820$	4.87273	0.170163
$821$	36.7861	1.28384	0.641921	−	0.766771i	$-0.278138\pi$
$821$	36.7861	1.28384	0.641921	+	0.766771i	$0.278138\pi$
$822$	28.0580	0.978634
$823$	3.11744	0.108667	0.0543335	−	0.998523i	$-0.482697\pi$
$823$	3.11744	0.108667	0.0543335	+	0.998523i	$0.482697\pi$
$824$	−14.9345	−0.520268
$825$	20.6626	0.719378
$826$	10.8466	0.377402
$827$	19.8669	0.690838	0.345419	−	0.938448i	$-0.387737\pi$
$827$	19.8669	0.690838	0.345419	+	0.938448i	$0.387737\pi$
$828$	3.64026	0.126508
$829$	−16.7153	−0.580547	−0.290274	−	0.956944i	$-0.593746\pi$
$829$	−16.7153	−0.580547	−0.290274	+	0.956944i	$0.593746\pi$
$830$	1.34437	0.0466639
$831$	−70.1556	−2.43367
$832$	−4.20957	−0.145940
$833$	−78.0336	−2.70370
$834$	−6.99414	−0.242187
$835$	15.0585	0.521120
$836$	3.99110	0.138035
$837$	6.39518	0.221050
$838$	22.2939	0.770129
$839$	−51.2827	−1.77048	−0.885238	−	0.465138i	$-0.846005\pi$
$839$	−51.2827	−1.77048	−0.885238	+	0.465138i	$0.846005\pi$
$840$	17.4691	0.602743
$841$	−1.71167	−0.0590232
$842$	14.8667	0.512339
$843$	38.0375	1.31008
$844$	2.25313	0.0775561
$845$	6.71172	0.230890
$846$	−31.0535	−1.06764
$847$	13.2172	0.454149
$848$	−4.25818	−0.146227
$849$	−58.5938	−2.01093
$850$	11.9564	0.410102
$851$	12.3179	0.422251
$852$	5.59830	0.191794
$853$	22.7006	0.777255	0.388628	−	0.921395i	$-0.372949\pi$
$853$	22.7006	0.777255	0.388628	+	0.921395i	$0.372949\pi$
$854$	54.9185	1.87927
$855$	5.29606	0.181122
$856$	14.5995	0.498999
$857$	−27.1963	−0.929009	−0.464505	−	0.885571i	$-0.653768\pi$
$857$	−27.1963	−0.929009	−0.464505	+	0.885571i	$0.653768\pi$
$858$	29.2040	0.997007
$859$	24.3669	0.831386	0.415693	−	0.909505i	$-0.363539\pi$
$859$	24.3669	0.831386	0.415693	+	0.909505i	$0.363539\pi$
$860$	15.4969	0.528439
$861$	42.1060	1.43497
$862$	−24.4479	−0.832698
$863$	15.6830	0.533856	0.266928	−	0.963717i	$-0.413991\pi$
$863$	15.6830	0.533856	0.266928	+	0.963717i	$0.413991\pi$
$864$	0.693819	0.0236042
$865$	0.612596	0.0208289
$866$	10.2369	0.347865
$867$	2.11362	0.0717823
$868$	−47.3941	−1.60866
$869$	2.44399	0.0829067
$870$	−17.7477	−0.601704
$871$	39.5932	1.34156
$872$	7.70309	0.260860
$873$	−40.3239	−1.36476
$874$	1.84677	0.0624680
$875$	58.3287	1.97187
$876$	−32.2595	−1.08995
$877$	−4.09764	−0.138367	−0.0691837	−	0.997604i	$-0.522039\pi$
$877$	−4.09764	−0.138367	−0.0691837	+	0.997604i	$0.522039\pi$
$878$	21.3360	0.720054
$879$	21.8726	0.737743
$880$	4.12810	0.139158
$881$	45.9598	1.54842	0.774212	−	0.632926i	$-0.218146\pi$
$881$	45.9598	1.54842	0.774212	+	0.632926i	$0.218146\pi$
$882$	52.6710	1.77352
$883$	−23.1347	−0.778544	−0.389272	−	0.921123i	$-0.627273\pi$
$883$	−23.1347	−0.778544	−0.389272	+	0.921123i	$0.627273\pi$
$884$	16.8989	0.568372
$885$	7.16689	0.240912
$886$	−22.7643	−0.764781
$887$	−41.2367	−1.38459	−0.692297	−	0.721613i	$-0.743401\pi$
$887$	−41.2367	−1.38459	−0.692297	+	0.721613i	$0.743401\pi$
$888$	−21.9088	−0.735211
$889$	−2.33134	−0.0781907
$890$	−18.5104	−0.620471
$891$	−28.4145	−0.951921
$892$	−19.8645	−0.665113
$893$	−15.7540	−0.527188
$894$	−36.0240	−1.20482
$895$	27.2335	0.910316
$896$	−5.14183	−0.171776
$897$	13.5133	0.451197
$898$	−21.7543	−0.725952
$899$	48.1499	1.60589
$900$	−8.07032	−0.269011
$901$	17.0941	0.569486
$902$	9.94999	0.331299
$903$	133.911	4.45628
$904$	−16.4504	−0.547132
$905$	0.173342	0.00576209
$906$	−50.7387	−1.68568
$907$	13.1698	0.437296	0.218648	−	0.975804i	$-0.429835\pi$
$907$	13.1698	0.437296	0.218648	+	0.975804i	$0.429835\pi$
$908$	3.37445	0.111985
$909$	−20.7927	−0.689651
$910$	30.7755	1.02020
$911$	42.1216	1.39555	0.697776	−	0.716316i	$-0.254173\pi$
$911$	42.1216	1.39555	0.697776	+	0.716316i	$0.254173\pi$
$912$	−3.28471	−0.108767
$913$	2.74518	0.0908521
$914$	1.50675	0.0498390
$915$	36.2874	1.19962
$916$	−18.9033	−0.624582
$917$	51.4864	1.70023
$918$	−2.78527	−0.0919276
$919$	−25.9396	−0.855669	−0.427835	−	0.903857i	$-0.640723\pi$
$919$	−25.9396	−0.855669	−0.427835	+	0.903857i	$0.640723\pi$
$920$	1.91016	0.0629763
$921$	2.75917	0.0909178
$922$	−37.9320	−1.24923
$923$	9.86255	0.324630
$924$	35.6715	1.17351
$925$	−27.3083	−0.897890
$926$	6.51234	0.214009
$927$	−40.4671	−1.32911
$928$	5.22382	0.171480
$929$	51.6682	1.69518	0.847590	−	0.530652i	$-0.178053\pi$
$929$	51.6682	1.69518	0.847590	+	0.530652i	$0.178053\pi$
$930$	−31.3156	−1.02688
$931$	26.7210	0.875745
$932$	−16.4601	−0.539169
$933$	−7.37596	−0.241478
$934$	12.8750	0.421284
$935$	−16.5719	−0.541958
$936$	−11.4064	−0.372830
$937$	37.0522	1.21044	0.605221	−	0.796057i	$-0.293085\pi$
$937$	37.0522	1.21044	0.605221	+	0.796057i	$0.293085\pi$
$938$	48.3616	1.57906
$939$	8.24271	0.268991
$940$	−16.2948	−0.531478
$941$	18.6478	0.607901	0.303950	−	0.952688i	$-0.401694\pi$
$941$	18.6478	0.607901	0.303950	+	0.952688i	$0.401694\pi$
$942$	−12.8842	−0.419789
$943$	4.60409	0.149930
$944$	−2.10949	−0.0686579
$945$	−5.07240	−0.165005
$946$	31.6442	1.02884
$947$	−33.9415	−1.10295	−0.551475	−	0.834191i	$-0.685935\pi$
$947$	−33.9415	−1.10295	−0.551475	+	0.834191i	$0.685935\pi$
$948$	−2.01142	−0.0653280
$949$	−56.8316	−1.84483
$950$	−4.09423	−0.132834
$951$	−57.0349	−1.84948
$952$	20.6414	0.668991
$953$	−8.99410	−0.291347	−0.145674	−	0.989333i	$-0.546535\pi$
$953$	−8.99410	−0.291347	−0.145674	+	0.989333i	$0.546535\pi$
$954$	−11.5381	−0.373561
$955$	3.34120	0.108119
$956$	−26.5319	−0.858102
$957$	−36.2404	−1.17149
$958$	−14.7058	−0.475123
$959$	60.3767	1.94966
$960$	−3.39746	−0.109653
$961$	53.9597	1.74064
$962$	−38.5968	−1.24441
$963$	39.5592	1.27478
$964$	−7.44610	−0.239823
$965$	−29.5800	−0.952214
$966$	16.5060	0.531073
$967$	4.86864	0.156565	0.0782824	−	0.996931i	$-0.475056\pi$
$967$	4.86864	0.156565	0.0782824	+	0.996931i	$0.475056\pi$
$968$	−2.57053	−0.0826199
$969$	13.1861	0.423600
$970$	−21.1593	−0.679385
$971$	10.6197	0.340802	0.170401	−	0.985375i	$-0.445494\pi$
$971$	10.6197	0.340802	0.170401	+	0.985375i	$0.445494\pi$
$972$	21.3039	0.683323
$973$	−15.0504	−0.482493
$974$	−3.07533	−0.0985401
$975$	−29.9586	−0.959443
$976$	−10.6807	−0.341882
$977$	−43.8560	−1.40308	−0.701538	−	0.712632i	$-0.747503\pi$
$977$	−43.8560	−1.40308	−0.701538	+	0.712632i	$0.747503\pi$
$978$	8.59976	0.274990
$979$	−37.7978	−1.20802
$980$	27.6382	0.882871
$981$	20.8726	0.666410
$982$	14.4147	0.459991
$983$	−38.4462	−1.22624	−0.613122	−	0.789988i	$-0.710087\pi$
$983$	−38.4462	−1.22624	−0.613122	+	0.789988i	$0.710087\pi$
$984$	−8.18892	−0.261053
$985$	−19.4005	−0.618150
$986$	−20.9705	−0.667838
$987$	−140.806	−4.48190
$988$	−5.78668	−0.184099
$989$	14.6425	0.465605
$990$	11.1856	0.355503
$991$	−25.1973	−0.800418	−0.400209	−	0.916424i	$-0.631062\pi$
$991$	−25.1973	−0.800418	−0.400209	+	0.916424i	$0.631062\pi$
$992$	9.21736	0.292651
$993$	69.8475	2.21654
$994$	12.0467	0.382099
$995$	24.2514	0.768823
$996$	−2.25930	−0.0715887
$997$	1.07065	0.0339078	0.0169539	−	0.999856i	$-0.494603\pi$
$997$	1.07065	0.0339078	0.0169539	+	0.999856i	$0.494603\pi$
$998$	25.0040	0.791488
$999$	6.36151	0.201269

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.14	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.14	✓	69	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.38948 q^{3} +1.00000 q^{4} +1.42184 q^{5} -2.38948 q^{6} -5.14183 q^{7} +1.00000 q^{8} +2.70964 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.38948 q^{3} +1.00000 q^{4} +1.42184 q^{5} -2.38948 q^{6} -5.14183 q^{7} +1.00000 q^{8} +2.70964 q^{9} +1.42184 q^{10} +2.90336 q^{11} -2.38948 q^{12} -4.20957 q^{13} -5.14183 q^{14} -3.39746 q^{15} +1.00000 q^{16} -4.01441 q^{17} +2.70964 q^{18} +1.37465 q^{19} +1.42184 q^{20} +12.2863 q^{21} +2.90336 q^{22} +1.34345 q^{23} -2.38948 q^{24} -2.97838 q^{25} -4.20957 q^{26} +0.693819 q^{27} -5.14183 q^{28} +5.22382 q^{29} -3.39746 q^{30} +9.21736 q^{31} +1.00000 q^{32} -6.93752 q^{33} -4.01441 q^{34} -7.31084 q^{35} +2.70964 q^{36} +9.16884 q^{37} +1.37465 q^{38} +10.0587 q^{39} +1.42184 q^{40} +3.42707 q^{41} +12.2863 q^{42} +10.8992 q^{43} +2.90336 q^{44} +3.85266 q^{45} +1.34345 q^{46} -11.4604 q^{47} -2.38948 q^{48} +19.4384 q^{49} -2.97838 q^{50} +9.59236 q^{51} -4.20957 q^{52} -4.25818 q^{53} +0.693819 q^{54} +4.12810 q^{55} -5.14183 q^{56} -3.28471 q^{57} +5.22382 q^{58} -2.10949 q^{59} -3.39746 q^{60} -10.6807 q^{61} +9.21736 q^{62} -13.9325 q^{63} +1.00000 q^{64} -5.98532 q^{65} -6.93752 q^{66} -9.40553 q^{67} -4.01441 q^{68} -3.21015 q^{69} -7.31084 q^{70} -2.34289 q^{71} +2.70964 q^{72} +13.5006 q^{73} +9.16884 q^{74} +7.11679 q^{75} +1.37465 q^{76} -14.9286 q^{77} +10.0587 q^{78} +0.841782 q^{79} +1.42184 q^{80} -9.78678 q^{81} +3.42707 q^{82} +0.945519 q^{83} +12.2863 q^{84} -5.70783 q^{85} +10.8992 q^{86} -12.4822 q^{87} +2.90336 q^{88} -13.0187 q^{89} +3.85266 q^{90} +21.6449 q^{91} +1.34345 q^{92} -22.0247 q^{93} -11.4604 q^{94} +1.95453 q^{95} -2.38948 q^{96} -14.8817 q^{97} +19.4384 q^{98} +7.86704 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more