LMFDB - Embedded newform 47.20.a.b.1.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [47,20,Mod(1,47)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(47, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("47.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 47 $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	47.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:	$107.543847381$
Analytic rank:	$0$
Dimension:	$39$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
Character	$\chi$	$=$	47.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+324.262 q^{2} +26444.0 q^{3} -419142. q^{4} -5.99723e6 q^{5} +8.57479e6 q^{6} +1.98397e8 q^{7} -3.05919e8 q^{8} -4.62977e8 q^{9} +O(q^{10})$	\(q+324.262 q^{2} +26444.0 q^{3} -419142. q^{4} -5.99723e6 q^{5} +8.57479e6 q^{6} +1.98397e8 q^{7} -3.05919e8 q^{8} -4.62977e8 q^{9} -1.94468e9 q^{10} +1.34519e9 q^{11} -1.10838e10 q^{12} +2.69166e10 q^{13} +6.43326e10 q^{14} -1.58591e11 q^{15} +1.20553e11 q^{16} -6.53361e11 q^{17} -1.50126e11 q^{18} +3.36362e11 q^{19} +2.51369e12 q^{20} +5.24640e12 q^{21} +4.36194e11 q^{22} -5.14658e12 q^{23} -8.08971e12 q^{24} +1.68933e13 q^{25} +8.72805e12 q^{26} -4.29778e13 q^{27} -8.31564e13 q^{28} -1.51418e14 q^{29} -5.14250e13 q^{30} +4.38188e13 q^{31} +1.99480e14 q^{32} +3.55722e13 q^{33} -2.11860e14 q^{34} -1.18983e15 q^{35} +1.94053e14 q^{36} -3.06301e14 q^{37} +1.09069e14 q^{38} +7.11783e14 q^{39} +1.83467e15 q^{40} +4.15541e15 q^{41} +1.70121e15 q^{42} +1.98490e15 q^{43} -5.63825e14 q^{44} +2.77658e15 q^{45} -1.66884e15 q^{46} -1.11913e15 q^{47} +3.18791e15 q^{48} +2.79623e16 q^{49} +5.47785e15 q^{50} -1.72775e16 q^{51} -1.12819e16 q^{52} +1.85083e16 q^{53} -1.39361e16 q^{54} -8.06741e15 q^{55} -6.06933e16 q^{56} +8.89475e15 q^{57} -4.90992e16 q^{58} +8.61975e16 q^{59} +6.64720e16 q^{60} +1.37903e17 q^{61} +1.42088e16 q^{62} -9.18531e16 q^{63} +1.47943e15 q^{64} -1.61425e17 q^{65} +1.15347e16 q^{66} +3.07347e17 q^{67} +2.73851e17 q^{68} -1.36096e17 q^{69} -3.85817e17 q^{70} -2.58635e17 q^{71} +1.41633e17 q^{72} -4.30212e16 q^{73} -9.93219e16 q^{74} +4.46726e17 q^{75} -1.40983e17 q^{76} +2.66881e17 q^{77} +2.30805e17 q^{78} -4.21836e17 q^{79} -7.22985e17 q^{80} -5.98404e17 q^{81} +1.34744e18 q^{82} +2.88231e17 q^{83} -2.19899e18 q^{84} +3.91835e18 q^{85} +6.43629e17 q^{86} -4.00410e18 q^{87} -4.11519e17 q^{88} +2.32680e18 q^{89} +9.00340e17 q^{90} +5.34017e18 q^{91} +2.15715e18 q^{92} +1.15874e18 q^{93} -3.62892e17 q^{94} -2.01724e18 q^{95} +5.27506e18 q^{96} -1.89590e18 q^{97} +9.06714e18 q^{98} -6.22791e17 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 39 q + 567 q^{2} + 4180 q^{3} + 11374277 q^{4} + 7841414 q^{5} - 3289088 q^{6} + 280678228 q^{7} + 616397649 q^{8} + 15832291053 q^{9}+O(q^{10}) $ 39 * q + 567 * q^2 + 4180 * q^3 + 11374277 * q^4 + 7841414 * q^5 - 3289088 * q^6 + 280678228 * q^7 + 616397649 * q^8 + 15832291053 * q^9	$ 39 q + 567 q^{2} + 4180 q^{3} + 11374277 q^{4} + 7841414 q^{5} - 3289088 q^{6} + 280678228 q^{7} + 616397649 q^{8} + 15832291053 q^{9} - 197084160 q^{10} + 6183770516 q^{11} - 18595076275 q^{12} + 72670351796 q^{13} - 286195652197 q^{14} + 216978245574 q^{15} + 4395775708833 q^{16} + 1565738603712 q^{17} + 6109717535226 q^{18} + 3193929321662 q^{19} - 5906920535432 q^{20} - 7386396792532 q^{21} - 8877997844072 q^{22} - 24482520509106 q^{23} - 7153616576581 q^{24} + 205574470566045 q^{25} + 29760604099536 q^{26} + 37673737054348 q^{27} + 359478142575004 q^{28} + 236042103421602 q^{29} + 10\!\cdots\!54 q^{30}+ \cdots + 26\!\cdots\!62 q^{99}+O(q^{100}) $ 39 * q + 567 * q^2 + 4180 * q^3 + 11374277 * q^4 + 7841414 * q^5 - 3289088 * q^6 + 280678228 * q^7 + 616397649 * q^8 + 15832291053 * q^9 - 197084160 * q^10 + 6183770516 * q^11 - 18595076275 * q^12 + 72670351796 * q^13 - 286195652197 * q^14 + 216978245574 * q^15 + 4395775708833 * q^16 + 1565738603712 * q^17 + 6109717535226 * q^18 + 3193929321662 * q^19 - 5906920535432 * q^20 - 7386396792532 * q^21 - 8877997844072 * q^22 - 24482520509106 * q^23 - 7153616576581 * q^24 + 205574470566045 * q^25 + 29760604099536 * q^26 + 37673737054348 * q^27 + 359478142575004 * q^28 + 236042103421602 * q^29 + 1030005166297154 * q^30 + 238097499775556 * q^31 + 980418196894414 * q^32 + 1875903246842268 * q^33 + 1649701857955214 * q^34 + 1575167660282324 * q^35 + 7318800726486641 * q^36 + 3601713987410012 * q^37 + 2042887131890906 * q^38 + 712786595145686 * q^39 - 38233381179154 * q^40 + 3933352398746224 * q^41 - 1818473706528187 * q^42 + 9956432184119776 * q^43 - 8374864809587786 * q^44 + 4814834336659374 * q^45 - 21420879452965494 * q^46 - 43646088451007913 * q^47 - 71590477049164572 * q^48 + 118361365545367441 * q^49 - 57666947668617547 * q^50 + 63071710735479788 * q^51 - 1295800694634632 * q^52 - 70710792109226748 * q^53 - 259379096268487409 * q^54 - 138894593750942388 * q^55 - 316931141949206936 * q^56 - 3219688617629350 * q^57 + 33593423193920552 * q^58 - 338601629983507156 * q^59 - 602000063011577726 * q^60 - 19162753603253692 * q^61 + 15505488837731256 * q^62 + 441133809282344796 * q^63 + 1575599177864752889 * q^64 + 991696997985687508 * q^65 - 223449751457078784 * q^66 + 789160046018441716 * q^67 + 793527065771503959 * q^68 + 1895066450986425554 * q^69 + 1371841313181399546 * q^70 + 901635006217520756 * q^71 + 4179441926330901705 * q^72 + 1704470931364824164 * q^73 - 936673704019194682 * q^74 + 2721299115185599300 * q^75 + 902909748895301048 * q^76 + 136895176815000502 * q^77 + 3735106483941260580 * q^78 - 384249994690730008 * q^79 - 1467637000368371240 * q^80 + 8095662645266644679 * q^81 + 6903936236738341456 * q^82 - 5730978816086659996 * q^83 - 10890678559479540867 * q^84 + 8583250605589725902 * q^85 + 15620118301833942618 * q^86 - 6279890617984303996 * q^87 + 17694374817110019346 * q^88 - 6455617251918472412 * q^89 - 170944452240128902 * q^90 - 1869573151265783134 * q^91 - 9347654291170404076 * q^92 + 11297106145565168410 * q^93 - 634546978249268889 * q^94 + 10375725689895196340 * q^95 + 38268359616807245724 * q^96 + 39090934998462984208 * q^97 + 41754136007920530767 * q^98 + 26521922158952497062 * 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 $

$2$	324.262	0.447828	0.223914	−	0.974609i	$-0.428116\pi$
$2$	324.262	0.447828	0.223914	+	0.974609i	$0.428116\pi$
$3$	26444.0	0.775667	0.387833	−	0.921730i	$-0.373224\pi$
$3$	26444.0	0.775667	0.387833	+	0.921730i	$0.373224\pi$
$4$	−419142.	−0.799450
$5$	−5.99723e6	−1.37321	−0.686603	−	0.727033i	$-0.740899\pi$
$5$	−5.99723e6	−1.37321	−0.686603	+	0.727033i	$0.740899\pi$
$6$	8.57479e6	0.347365
$7$	1.98397e8	1.85825	0.929123	−	0.369772i	$-0.120564\pi$
$7$	1.98397e8	1.85825	0.929123	+	0.369772i	$0.120564\pi$
$8$	−3.05919e8	−0.805845
$9$	−4.62977e8	−0.398341
$10$	−1.94468e9	−0.614960
$11$	1.34519e9	0.172010	0.0860048	−	0.996295i	$-0.472590\pi$
$11$	1.34519e9	0.172010	0.0860048	+	0.996295i	$0.472590\pi$
$12$	−1.10838e10	−0.620106
$13$	2.69166e10	0.703978	0.351989	−	0.936004i	$-0.385505\pi$
$13$	2.69166e10	0.703978	0.351989	+	0.936004i	$0.385505\pi$
$14$	6.43326e10	0.832175
$15$	−1.58591e11	−1.06515
$16$	1.20553e11	0.438570
$17$	−6.53361e11	−1.33625	−0.668126	−	0.744048i	$-0.732903\pi$
$17$	−6.53361e11	−1.33625	−0.668126	+	0.744048i	$0.732903\pi$
$18$	−1.50126e11	−0.178389
$19$	3.36362e11	0.239137	0.119569	−	0.992826i	$-0.461849\pi$
$19$	3.36362e11	0.239137	0.119569	+	0.992826i	$0.461849\pi$
$20$	2.51369e12	1.09781
$21$	5.24640e12	1.44138
$22$	4.36194e11	0.0770308
$23$	−5.14658e12	−0.595805	−0.297902	−	0.954596i	$-0.596287\pi$
$23$	−5.14658e12	−0.595805	−0.297902	+	0.954596i	$0.596287\pi$
$24$	−8.08971e12	−0.625067
$25$	1.68933e13	0.885694
$26$	8.72805e12	0.315261
$27$	−4.29778e13	−1.08465
$28$	−8.31564e13	−1.48557
$29$	−1.51418e14	−1.93819	−0.969097	−	0.246681i	$-0.920660\pi$
$29$	−1.51418e14	−1.93819	−0.969097	+	0.246681i	$0.920660\pi$
$30$	−5.14250e13	−0.477004
$31$	4.38188e13	0.297663	0.148831	−	0.988863i	$-0.452449\pi$
$31$	4.38188e13	0.297663	0.148831	+	0.988863i	$0.452449\pi$
$32$	1.99480e14	1.00225
$33$	3.55722e13	0.133422
$34$	−2.11860e14	−0.598411
$35$	−1.18983e15	−2.55175
$36$	1.94053e14	0.318454
$37$	−3.06301e14	−0.387465	−0.193732	−	0.981054i	$-0.562059\pi$
$37$	−3.06301e14	−0.387465	−0.193732	+	0.981054i	$0.562059\pi$
$38$	1.09069e14	0.107092
$39$	7.11783e14	0.546052
$40$	1.83467e15	1.10659
$41$	4.15541e15	1.98229	0.991146	−	0.132779i	$-0.0423899\pi$
$41$	4.15541e15	1.98229	0.991146	+	0.132779i	$0.0423899\pi$
$42$	1.70121e15	0.645490
$43$	1.98490e15	0.602266	0.301133	−	0.953582i	$-0.402635\pi$
$43$	1.98490e15	0.602266	0.301133	+	0.953582i	$0.402635\pi$
$44$	−5.63825e14	−0.137513
$45$	2.77658e15	0.547005
$46$	−1.66884e15	−0.266818
$47$	−1.11913e15	−0.145865
$47$	−1.11913e15	−0.145865
$48$	3.18791e15	0.340184
$49$	2.79623e16	2.45308
$50$	5.47785e15	0.396639
$51$	−1.72775e16	−1.03649
$52$	−1.12819e16	−0.562795
$53$	1.85083e16	0.770452	0.385226	−	0.922822i	$-0.374124\pi$
$53$	1.85083e16	0.770452	0.385226	+	0.922822i	$0.374124\pi$
$54$	−1.39361e16	−0.485736
$55$	−8.06741e15	−0.236205
$56$	−6.06933e16	−1.49746
$57$	8.89475e15	0.185491
$58$	−4.90992e16	−0.867978
$59$	8.61975e16	1.29539	0.647695	−	0.761900i	$-0.275733\pi$
$59$	8.61975e16	1.29539	0.647695	+	0.761900i	$0.275733\pi$
$60$	6.64720e16	0.851534
$61$	1.37903e17	1.50988	0.754938	−	0.655796i	$-0.227667\pi$
$61$	1.37903e17	1.50988	0.754938	+	0.655796i	$0.227667\pi$
$62$	1.42088e16	0.133302
$63$	−9.18531e16	−0.740216
$64$	1.47943e15	0.0102656
$65$	−1.61425e17	−0.966706
$66$	1.15347e16	0.0597502
$67$	3.07347e17	1.38013	0.690063	−	0.723749i	$-0.257583\pi$
$67$	3.07347e17	1.38013	0.690063	+	0.723749i	$0.257583\pi$
$68$	2.73851e17	1.06827
$69$	−1.36096e17	−0.462146
$70$	−3.85817e17	−1.14275
$71$	−2.58635e17	−0.669473	−0.334736	−	0.942312i	$-0.608647\pi$
$71$	−2.58635e17	−0.669473	−0.334736	+	0.942312i	$0.608647\pi$
$72$	1.41633e17	0.321001
$73$	−4.30212e16	−0.0855294	−0.0427647	−	0.999085i	$-0.513617\pi$
$73$	−4.30212e16	−0.0855294	−0.0427647	+	0.999085i	$0.513617\pi$
$74$	−9.93219e16	−0.173518
$75$	4.46726e17	0.687003
$76$	−1.40983e17	−0.191178
$77$	2.66881e17	0.319636
$78$	2.30805e17	0.244538
$79$	−4.21836e17	−0.395992	−0.197996	−	0.980203i	$-0.563443\pi$
$79$	−4.21836e17	−0.395992	−0.197996	+	0.980203i	$0.563443\pi$
$80$	−7.22985e17	−0.602246
$81$	−5.98404e17	−0.442983
$82$	1.34744e18	0.887726
$83$	2.88231e17	0.169238	0.0846192	−	0.996413i	$-0.473033\pi$
$83$	2.88231e17	0.169238	0.0846192	+	0.996413i	$0.473033\pi$
$84$	−2.19899e18	−1.15231
$85$	3.91835e18	1.83495
$86$	6.43629e17	0.269712
$87$	−4.00410e18	−1.50339
$88$	−4.11519e17	−0.138613
$89$	2.32680e18	0.703970	0.351985	−	0.936006i	$-0.385507\pi$
$89$	2.32680e18	0.703970	0.351985	+	0.936006i	$0.385507\pi$
$90$	9.00340e17	0.244964
$91$	5.34017e18	1.30816
$92$	2.15715e18	0.476316
$93$	1.15874e18	0.230887
$94$	−3.62892e17	−0.0653225
$95$	−2.01724e18	−0.328385
$96$	5.27506e18	0.777411
$97$	−1.89590e18	−0.253211	−0.126606	−	0.991953i	$-0.540408\pi$
$97$	−1.89590e18	−0.253211	−0.126606	+	0.991953i	$0.540408\pi$
$98$	9.06714e18	1.09856
$99$	−6.22791e17	−0.0685186
$100$	−7.08068e18	−0.708068
$101$	2.12134e19	1.93000	0.964998	−	0.262256i	$-0.0844664\pi$
$101$	2.12134e19	1.93000	0.964998	+	0.262256i	$0.0844664\pi$
$102$	−5.60243e18	−0.464168
$103$	9.12250e18	0.688906	0.344453	−	0.938804i	$-0.388064\pi$
$103$	9.12250e18	0.688906	0.344453	+	0.938804i	$0.388064\pi$
$104$	−8.23431e18	−0.567297
$105$	−3.14639e19	−1.97931
$106$	6.00154e18	0.345030
$107$	−2.58256e19	−1.35802	−0.679008	−	0.734131i	$-0.737590\pi$
$107$	−2.58256e19	−1.35802	−0.679008	+	0.734131i	$0.737590\pi$
$108$	1.80138e19	0.867121
$109$	1.22729e18	0.0541249	0.0270624	−	0.999634i	$-0.491385\pi$
$109$	1.22729e18	0.0541249	0.0270624	+	0.999634i	$0.491385\pi$
$110$	−2.61596e18	−0.105779
$111$	−8.09983e18	−0.300544
$112$	2.39173e19	0.814970
$113$	3.44078e19	1.07749	0.538743	−	0.842470i	$-0.318899\pi$
$113$	3.44078e19	1.07749	0.538743	+	0.842470i	$0.318899\pi$
$114$	2.88423e18	0.0830681
$115$	3.08652e19	0.818162
$116$	6.34657e19	1.54949
$117$	−1.24618e19	−0.280423
$118$	2.79506e19	0.580112
$119$	−1.29625e20	−2.48308
$120$	4.85159e19	0.858345
$121$	−5.93496e19	−0.970413
$122$	4.47169e19	0.676165
$123$	1.09886e20	1.53760
$124$	−1.83663e19	−0.237966
$125$	1.30752e19	0.156966
$126$	−2.97845e19	−0.331490
$127$	−1.11062e20	−1.14665	−0.573324	−	0.819329i	$-0.694346\pi$
$127$	−1.11062e20	−1.14665	−0.573324	+	0.819329i	$0.694346\pi$
$128$	−1.04105e20	−0.997651
$129$	5.24887e19	0.467158
$130$	−5.23441e19	−0.432919
$131$	−1.54132e19	−0.118527	−0.0592634	−	0.998242i	$-0.518875\pi$
$131$	−1.54132e19	−0.118527	−0.0592634	+	0.998242i	$0.518875\pi$
$132$	−1.49098e19	−0.106664
$133$	6.67331e19	0.444376
$134$	9.96612e19	0.618060
$135$	2.57748e20	1.48944
$136$	1.99875e20	1.07681
$137$	9.24037e19	0.464349	0.232174	−	0.972674i	$-0.425416\pi$
$137$	9.24037e19	0.464349	0.232174	+	0.972674i	$0.425416\pi$
$138$	−4.41308e19	−0.206962
$139$	1.71943e20	0.752913	0.376457	−	0.926434i	$-0.377142\pi$
$139$	1.71943e20	0.752913	0.376457	+	0.926434i	$0.377142\pi$
$140$	4.98708e20	2.04000
$141$	−2.95943e19	−0.113143
$142$	−8.38656e19	−0.299809
$143$	3.62080e19	0.121091
$144$	−5.58133e19	−0.174700
$145$	9.08090e20	2.66154
$146$	−1.39502e19	−0.0383025
$147$	7.39436e20	1.90277
$148$	1.28384e20	0.309759
$149$	−4.91558e20	−1.11251	−0.556256	−	0.831011i	$-0.687763\pi$
$149$	−4.91558e20	−1.11251	−0.556256	+	0.831011i	$0.687763\pi$
$150$	1.44856e20	0.307660
$151$	5.34567e20	1.06591	0.532956	−	0.846143i	$-0.321081\pi$
$151$	5.34567e20	1.06591	0.532956	+	0.846143i	$0.321081\pi$
$152$	−1.02899e20	−0.192708
$153$	3.02491e20	0.532284
$154$	8.65395e19	0.143142
$155$	−2.62791e20	−0.408752
$156$	−2.98338e20	−0.436541
$157$	1.24981e21	1.72106	0.860530	−	0.509399i	$-0.170132\pi$
$157$	1.24981e21	1.72106	0.860530	+	0.509399i	$0.170132\pi$
$158$	−1.36786e20	−0.177336
$159$	4.89433e20	0.597614
$160$	−1.19633e21	−1.37629
$161$	−1.02106e21	−1.10715
$162$	−1.94040e20	−0.198380
$163$	1.65916e21	1.59995	0.799975	−	0.600033i	$-0.204846\pi$
$163$	1.65916e21	1.59995	0.799975	+	0.600033i	$0.204846\pi$
$164$	−1.74171e21	−1.58474
$165$	−2.13334e20	−0.183216
$166$	9.34625e19	0.0757898
$167$	−8.05945e20	−0.617304	−0.308652	−	0.951175i	$-0.599878\pi$
$167$	−8.05945e20	−0.617304	−0.308652	+	0.951175i	$0.599878\pi$
$168$	−1.60497e21	−1.16153
$169$	−7.37415e20	−0.504415
$170$	1.27057e21	0.821742
$171$	−1.55728e20	−0.0952583
$172$	−8.31955e20	−0.481482
$173$	−3.98715e20	−0.218386	−0.109193	−	0.994021i	$-0.534827\pi$
$173$	−3.98715e20	−0.218386	−0.109193	+	0.994021i	$0.534827\pi$
$174$	−1.29838e21	−0.673261
$175$	3.35157e21	1.64584
$176$	1.62167e20	0.0754382
$177$	2.27941e21	1.00479
$178$	7.54494e20	0.315258
$179$	−3.37441e21	−1.33689	−0.668443	−	0.743764i	$-0.733039\pi$
$179$	−3.37441e21	−1.33689	−0.668443	+	0.743764i	$0.733039\pi$
$180$	−1.16378e21	−0.437303
$181$	2.92085e21	1.04127	0.520634	−	0.853780i	$-0.325696\pi$
$181$	2.92085e21	1.04127	0.520634	+	0.853780i	$0.325696\pi$
$182$	1.73162e21	0.585833
$183$	3.64672e21	1.17116
$184$	1.57443e21	0.480126
$185$	1.83696e21	0.532069
$186$	3.75737e20	0.103398
$187$	−8.78893e20	−0.229848
$188$	4.69074e20	0.116612
$189$	−8.52665e21	−2.01554
$190$	−6.54115e20	−0.147060
$191$	−7.70680e21	−1.64838	−0.824190	−	0.566314i	$-0.808369\pi$
$191$	−7.70680e21	−1.64838	−0.824190	+	0.566314i	$0.808369\pi$
$192$	3.91221e19	0.00796270
$193$	6.62243e21	1.28299	0.641495	−	0.767128i	$-0.278315\pi$
$193$	6.62243e21	1.28299	0.641495	+	0.767128i	$0.278315\pi$
$194$	−6.14767e20	−0.113395
$195$	−4.26873e21	−0.749842
$196$	−1.17202e22	−1.96111
$197$	9.29722e21	1.48226	0.741129	−	0.671362i	$-0.234291\pi$
$197$	9.29722e21	1.48226	0.741129	+	0.671362i	$0.234291\pi$
$198$	−2.01948e20	−0.0306845
$199$	6.08539e21	0.881422	0.440711	−	0.897649i	$-0.354726\pi$
$199$	6.08539e21	0.881422	0.440711	+	0.897649i	$0.354726\pi$
$200$	−5.16797e21	−0.713732
$201$	8.12749e21	1.07052
$202$	6.87870e21	0.864307
$203$	−3.00409e22	−3.60164
$204$	7.24171e21	0.828618
$205$	−2.49210e22	−2.72209
$206$	2.95808e21	0.308512
$207$	2.38275e21	0.237334
$208$	3.24488e21	0.308743
$209$	4.52470e20	0.0411339
$210$	−1.02025e22	−0.886391
$211$	4.53964e21	0.376997	0.188499	−	0.982073i	$-0.439638\pi$
$211$	4.53964e21	0.376997	0.188499	+	0.982073i	$0.439638\pi$
$212$	−7.75760e21	−0.615937
$213$	−6.83934e21	−0.519288
$214$	−8.37428e21	−0.608158
$215$	−1.19039e22	−0.827036
$216$	1.31477e22	0.874057
$217$	8.69350e21	0.553130
$218$	3.97965e20	0.0242387
$219$	−1.13765e21	−0.0663423
$220$	3.38139e21	0.188834
$221$	−1.75863e22	−0.940691
$222$	−2.62647e21	−0.134592
$223$	1.66112e22	0.815651	0.407825	−	0.913060i	$-0.366287\pi$
$223$	1.66112e22	0.815651	0.407825	+	0.913060i	$0.366287\pi$
$224$	3.95762e22	1.86242
$225$	−7.82119e21	−0.352809
$226$	1.11572e22	0.482529
$227$	1.12896e22	0.468202	0.234101	−	0.972212i	$-0.424785\pi$
$227$	1.12896e22	0.468202	0.234101	+	0.972212i	$0.424785\pi$
$228$	−3.72816e21	−0.148291
$229$	−1.90868e22	−0.728277	−0.364138	−	0.931345i	$-0.618637\pi$
$229$	−1.90868e22	−0.728277	−0.364138	+	0.931345i	$0.618637\pi$
$230$	1.00084e22	0.366396
$231$	7.05740e21	0.247931
$232$	4.63217e22	1.56188
$233$	−1.95939e22	−0.634219	−0.317110	−	0.948389i	$-0.602712\pi$
$233$	−1.95939e22	−0.634219	−0.317110	+	0.948389i	$0.602712\pi$
$234$	−4.04089e21	−0.125582
$235$	6.71168e21	0.200303
$236$	−3.61290e22	−1.03560
$237$	−1.11550e22	−0.307158
$238$	−4.20324e22	−1.11199
$239$	4.03253e22	1.02517	0.512587	−	0.858635i	$-0.328687\pi$
$239$	4.03253e22	1.02517	0.512587	+	0.858635i	$0.328687\pi$
$240$	−1.91186e22	−0.467142
$241$	−2.29933e22	−0.540056	−0.270028	−	0.962853i	$-0.587033\pi$
$241$	−2.29933e22	−0.540056	−0.270028	+	0.962853i	$0.587033\pi$
$242$	−1.92448e22	−0.434578
$243$	3.41272e22	0.741040
$244$	−5.78011e22	−1.20707
$245$	−1.67697e23	−3.36858
$246$	3.56318e22	0.688580
$247$	9.05373e21	0.168347
$248$	−1.34050e22	−0.239870
$249$	7.62198e21	0.131273
$250$	4.23981e21	0.0702937
$251$	4.17950e21	0.0667150	0.0333575	−	0.999443i	$-0.489380\pi$
$251$	4.17950e21	0.0667150	0.0333575	+	0.999443i	$0.489380\pi$
$252$	3.84995e22	0.591766
$253$	−6.92312e21	−0.102484
$254$	−3.60132e22	−0.513501
$255$	1.03617e23	1.42331
$256$	−3.45331e22	−0.457042
$257$	−1.85406e22	−0.236461	−0.118231	−	0.992986i	$-0.537722\pi$
$257$	−1.85406e22	−0.236461	−0.118231	+	0.992986i	$0.537722\pi$
$258$	1.70201e22	0.209207
$259$	−6.07691e22	−0.720005
$260$	6.76601e22	0.772833
$261$	7.01031e22	0.772063
$262$	−4.99793e21	−0.0530797
$263$	1.55661e23	1.59441	0.797203	−	0.603711i	$-0.206312\pi$
$263$	1.55661e23	1.59441	0.797203	+	0.603711i	$0.206312\pi$
$264$	−1.08822e22	−0.107517
$265$	−1.10998e23	−1.05799
$266$	2.16390e22	0.199004
$267$	6.15299e22	0.546046
$268$	−1.28822e23	−1.10334
$269$	−3.26200e22	−0.269673	−0.134836	−	0.990868i	$-0.543051\pi$
$269$	−3.26200e22	−0.269673	−0.134836	+	0.990868i	$0.543051\pi$
$270$	8.35779e22	0.667015
$271$	−1.97043e22	−0.151829	−0.0759143	−	0.997114i	$-0.524188\pi$
$271$	−1.97043e22	−0.151829	−0.0759143	+	0.997114i	$0.524188\pi$
$272$	−7.87646e22	−0.586039
$273$	1.41215e23	1.01470
$274$	2.99631e22	0.207948
$275$	2.27246e22	0.152348
$276$	5.70435e22	0.369462
$277$	−2.45864e23	−1.53864	−0.769321	−	0.638863i	$-0.779405\pi$
$277$	−2.45864e23	−1.53864	−0.769321	+	0.638863i	$0.779405\pi$
$278$	5.57548e22	0.337176
$279$	−2.02871e22	−0.118571
$280$	3.63991e23	2.05632
$281$	−5.54507e20	−0.00302829	−0.00151414	−	0.999999i	$-0.500482\pi$
$281$	−5.54507e20	−0.00302829	−0.00151414	+	0.999999i	$0.500482\pi$
$282$	−9.59631e21	−0.0506685
$283$	−2.05271e23	−1.04799	−0.523994	−	0.851722i	$-0.675559\pi$
$283$	−2.05271e23	−1.04799	−0.523994	+	0.851722i	$0.675559\pi$
$284$	1.08405e23	0.535210
$285$	−5.33438e22	−0.254717
$286$	1.17409e22	0.0542280
$287$	8.24420e23	3.68358
$288$	−9.23548e22	−0.399237
$289$	1.87808e23	0.785567
$290$	2.94459e23	1.19191
$291$	−5.01350e22	−0.196407
$292$	1.80320e22	0.0683764
$293$	6.79150e22	0.249301	0.124650	−	0.992201i	$-0.460219\pi$
$293$	6.79150e22	0.249301	0.124650	+	0.992201i	$0.460219\pi$
$294$	2.39771e23	0.852114
$295$	−5.16946e23	−1.77884
$296$	9.37033e22	0.312236
$297$	−5.78132e22	−0.186570
$298$	−1.59394e23	−0.498215
$299$	−1.38529e23	−0.419433
$300$	−1.87241e23	−0.549225
$301$	3.93798e23	1.11916
$302$	1.73340e23	0.477346
$303$	5.60966e23	1.49703
$304$	4.05495e22	0.104878
$305$	−8.27038e23	−2.07337
$306$	9.80864e22	0.238372
$307$	5.67899e23	1.33800	0.669001	−	0.743262i	$-0.266722\pi$
$307$	5.67899e23	1.33800	0.669001	+	0.743262i	$0.266722\pi$
$308$	−1.11861e23	−0.255533
$309$	2.41235e23	0.534361
$310$	−8.52133e22	−0.183051
$311$	−3.81835e23	−0.795523	−0.397761	−	0.917489i	$-0.630213\pi$
$311$	−3.81835e23	−0.795523	−0.397761	+	0.917489i	$0.630213\pi$
$312$	−2.17748e23	−0.440033
$313$	1.77224e23	0.347416	0.173708	−	0.984797i	$-0.444425\pi$
$313$	1.77224e23	0.347416	0.173708	+	0.984797i	$0.444425\pi$
$314$	4.05265e23	0.770740
$315$	5.50864e23	1.01647
$316$	1.76809e23	0.316576
$317$	−1.02712e24	−1.78467	−0.892336	−	0.451372i	$-0.850935\pi$
$317$	−1.02712e24	−1.78467	−0.892336	+	0.451372i	$0.850935\pi$
$318$	1.58705e23	0.267628
$319$	−2.03686e23	−0.333388
$320$	−8.87249e21	−0.0140968
$321$	−6.82933e23	−1.05337
$322$	−3.31092e23	−0.495814
$323$	−2.19766e23	−0.319548
$324$	2.50816e23	0.354142
$325$	4.54710e23	0.623509
$326$	5.38004e23	0.716503
$327$	3.24546e22	0.0419829
$328$	−1.27122e24	−1.59742
$329$	−2.22032e23	−0.271053
$330$	−6.91763e22	−0.0820493
$331$	4.92982e23	0.568153	0.284077	−	0.958802i	$-0.408313\pi$
$331$	4.92982e23	0.568153	0.284077	+	0.958802i	$0.408313\pi$
$332$	−1.20810e23	−0.135298
$333$	1.41810e23	0.154343
$334$	−2.61338e23	−0.276446
$335$	−1.84323e24	−1.89520
$336$	6.32470e23	0.632145
$337$	−1.74609e23	−0.169661	−0.0848306	−	0.996395i	$-0.527035\pi$
$337$	−1.74609e23	−0.169661	−0.0848306	+	0.996395i	$0.527035\pi$
$338$	−2.39116e23	−0.225891
$339$	9.09880e23	0.835770
$340$	−1.64235e24	−1.46695
$341$	5.89446e22	0.0512008
$342$	−5.04966e22	−0.0426594
$343$	3.28613e24	2.70017
$344$	−6.07219e23	−0.485333
$345$	8.16199e23	0.634621
$346$	−1.29288e23	−0.0977993
$347$	−1.47198e24	−1.08336	−0.541679	−	0.840585i	$-0.682211\pi$
$347$	−1.47198e24	−1.08336	−0.541679	+	0.840585i	$0.682211\pi$
$348$	1.67829e24	1.20189
$349$	−1.49367e23	−0.104091	−0.0520456	−	0.998645i	$-0.516574\pi$
$349$	−1.49367e23	−0.104091	−0.0520456	+	0.998645i	$0.516574\pi$
$350$	1.08679e24	0.737052
$351$	−1.15682e24	−0.763567
$352$	2.68339e23	0.172396
$353$	2.08207e24	1.30207	0.651035	−	0.759047i	$-0.274335\pi$
$353$	2.08207e24	1.30207	0.651035	+	0.759047i	$0.274335\pi$
$354$	7.39125e23	0.449974
$355$	1.55109e24	0.919324
$356$	−9.75260e23	−0.562789
$357$	−3.42779e24	−1.92604
$358$	−1.09419e24	−0.598695
$359$	2.30226e24	1.22675	0.613377	−	0.789791i	$-0.289811\pi$
$359$	2.30226e24	1.22675	0.613377	+	0.789791i	$0.289811\pi$
$360$	−8.49408e23	−0.440801
$361$	−1.86528e24	−0.942813
$362$	9.47121e23	0.466309
$363$	−1.56944e24	−0.752717
$364$	−2.23829e24	−1.04581
$365$	2.58008e23	0.117449
$366$	1.18249e24	0.524479
$367$	−3.07409e24	−1.32858	−0.664292	−	0.747473i	$-0.731267\pi$
$367$	−3.07409e24	−1.32858	−0.664292	+	0.747473i	$0.731267\pi$
$368$	−6.20436e23	−0.261302
$369$	−1.92386e24	−0.789629
$370$	5.95656e23	0.238276
$371$	3.67198e24	1.43169
$372$	−4.85678e23	−0.184583
$373$	−2.58688e24	−0.958391	−0.479195	−	0.877708i	$-0.659071\pi$
$373$	−2.58688e24	−0.958391	−0.479195	+	0.877708i	$0.659071\pi$
$374$	−2.84992e23	−0.102932
$375$	3.45762e23	0.121753
$376$	3.42363e23	0.117545
$377$	−4.07567e24	−1.36445
$378$	−2.76487e24	−0.902616
$379$	−4.34597e24	−1.38361	−0.691806	−	0.722084i	$-0.743184\pi$
$379$	−4.34597e24	−1.38361	−0.691806	+	0.722084i	$0.743184\pi$
$380$	8.45509e23	0.262527
$381$	−2.93692e24	−0.889416
$382$	−2.49903e24	−0.738191
$383$	2.08282e24	0.600154	0.300077	−	0.953915i	$-0.402988\pi$
$383$	2.08282e24	0.600154	0.300077	+	0.953915i	$0.402988\pi$
$384$	−2.75296e24	−0.773845
$385$	−1.60055e24	−0.438926
$386$	2.14741e24	0.574559
$387$	−9.18964e23	−0.239908
$388$	7.94649e23	0.202430
$389$	1.67027e24	0.415207	0.207603	−	0.978213i	$-0.433434\pi$
$389$	1.67027e24	0.415207	0.207603	+	0.978213i	$0.433434\pi$
$390$	−1.38419e24	−0.335800
$391$	3.36257e24	0.796145
$392$	−8.55421e24	−1.97680
$393$	−4.07588e23	−0.0919373
$394$	3.01474e24	0.663797
$395$	2.52985e24	0.543778
$396$	2.61038e23	0.0547771
$397$	3.38782e24	0.694081	0.347041	−	0.937850i	$-0.387187\pi$
$397$	3.38782e24	0.694081	0.347041	+	0.937850i	$0.387187\pi$
$398$	1.97326e24	0.394726
$399$	1.76469e24	0.344687
$400$	2.03654e24	0.388439
$401$	3.87470e23	0.0721716	0.0360858	−	0.999349i	$-0.488511\pi$
$401$	3.87470e23	0.0721716	0.0360858	+	0.999349i	$0.488511\pi$
$402$	2.63544e24	0.479408
$403$	1.17945e24	0.209548
$404$	−8.89141e24	−1.54294
$405$	3.58877e24	0.608306
$406$	−9.74112e24	−1.61292
$407$	−4.12033e23	−0.0666477
$408$	5.28550e24	0.835246
$409$	6.40361e24	0.988674	0.494337	−	0.869270i	$-0.335411\pi$
$409$	6.40361e24	0.988674	0.494337	+	0.869270i	$0.335411\pi$
$410$	−8.08093e24	−1.21903
$411$	2.44352e24	0.360180
$412$	−3.82362e24	−0.550746
$413$	1.71013e25	2.40715
$414$	7.72635e23	0.106285
$415$	−1.72859e24	−0.232399
$416$	5.36934e24	0.705561
$417$	4.54687e24	0.584010
$418$	1.46719e23	0.0184209
$419$	1.31093e25	1.60896	0.804480	−	0.593979i	$-0.202444\pi$
$419$	1.31093e25	1.60896	0.804480	+	0.593979i	$0.202444\pi$
$420$	1.31878e25	1.58236
$421$	−1.01543e25	−1.19116	−0.595578	−	0.803298i	$-0.703077\pi$
$421$	−1.01543e25	−1.19116	−0.595578	+	0.803298i	$0.703077\pi$
$422$	1.47203e24	0.168830
$423$	5.18132e23	0.0581041
$424$	−5.66203e24	−0.620864
$425$	−1.10374e25	−1.18351
$426$	−2.21774e24	−0.232552
$427$	2.73596e25	2.80572
$428$	1.08246e25	1.08567
$429$	9.57483e23	0.0939262
$430$	−3.85999e24	−0.370370
$431$	−1.16552e24	−0.109392	−0.0546960	−	0.998503i	$-0.517419\pi$
$431$	−1.16552e24	−0.109392	−0.0546960	+	0.998503i	$0.517419\pi$
$432$	−5.18111e24	−0.475693
$433$	−1.77830e25	−1.59724	−0.798619	−	0.601836i	$-0.794436\pi$
$433$	−1.77830e25	−1.59724	−0.798619	+	0.601836i	$0.794436\pi$
$434$	2.81898e24	0.247707
$435$	2.40135e25	2.06447
$436$	−5.14410e23	−0.0432701
$437$	−1.73111e24	−0.142479
$438$	−3.68898e23	−0.0297099
$439$	5.17295e24	0.407685	0.203843	−	0.979004i	$-0.434657\pi$
$439$	5.17295e24	0.407685	0.203843	+	0.979004i	$0.434657\pi$
$440$	2.46797e24	0.190344
$441$	−1.29459e25	−0.977161
$442$	−5.70256e24	−0.421268
$443$	2.19255e25	1.58531	0.792656	−	0.609669i	$-0.208697\pi$
$443$	2.19255e25	1.58531	0.792656	+	0.609669i	$0.208697\pi$
$444$	3.39498e24	0.240269
$445$	−1.39544e25	−0.966696
$446$	5.38638e24	0.365271
$447$	−1.29987e25	−0.862939
$448$	2.93514e23	0.0190760
$449$	−1.03794e24	−0.0660440	−0.0330220	−	0.999455i	$-0.510513\pi$
$449$	−1.03794e24	−0.0660440	−0.0330220	+	0.999455i	$0.510513\pi$
$450$	−2.53612e24	−0.157998
$451$	5.58981e24	0.340973
$452$	−1.44218e25	−0.861397
$453$	1.41361e25	0.826792
$454$	3.66080e24	0.209674
$455$	−3.20262e25	−1.79638
$456$	−2.72107e24	−0.149477
$457$	1.36306e25	0.733351	0.366675	−	0.930349i	$-0.380496\pi$
$457$	1.36306e25	0.733351	0.366675	+	0.930349i	$0.380496\pi$
$458$	−6.18914e24	−0.326143
$459$	2.80800e25	1.44936
$460$	−1.29369e25	−0.654080
$461$	1.25808e25	0.623086	0.311543	−	0.950232i	$-0.399154\pi$
$461$	1.25808e25	0.623086	0.311543	+	0.950232i	$0.399154\pi$
$462$	2.28845e24	0.111031
$463$	1.50429e25	0.715008	0.357504	−	0.933912i	$-0.383628\pi$
$463$	1.50429e25	0.715008	0.357504	+	0.933912i	$0.383628\pi$
$464$	−1.82539e25	−0.850033
$465$	−6.94925e24	−0.317055
$466$	−6.35356e24	−0.284021
$467$	−3.15105e25	−1.38021	−0.690104	−	0.723710i	$-0.742435\pi$
$467$	−3.15105e25	−1.38021	−0.690104	+	0.723710i	$0.742435\pi$
$468$	5.22325e24	0.224185
$469$	6.09767e25	2.56461
$470$	2.17635e24	0.0897012
$471$	3.30499e25	1.33497
$472$	−2.63694e25	−1.04388
$473$	2.67007e24	0.103596
$474$	−3.61716e24	−0.137554
$475$	5.68225e24	0.211803
$476$	5.43311e25	1.98510
$477$	−8.56891e24	−0.306903
$478$	1.30760e25	0.459102
$479$	2.29290e24	0.0789218	0.0394609	−	0.999221i	$-0.487436\pi$
$479$	2.29290e24	0.0789218	0.0394609	+	0.999221i	$0.487436\pi$
$480$	−3.16357e25	−1.06754
$481$	−8.24460e24	−0.272767
$482$	−7.45585e24	−0.241852
$483$	−2.70010e25	−0.858780
$484$	2.48759e25	0.775796
$485$	1.13701e25	0.347711
$486$	1.10662e25	0.331859
$487$	−4.50155e25	−1.32385	−0.661923	−	0.749572i	$-0.730259\pi$
$487$	−4.50155e25	−1.32385	−0.661923	+	0.749572i	$0.730259\pi$
$488$	−4.21872e25	−1.21673
$489$	4.38749e25	1.24103
$490$	−5.43777e25	−1.50854
$491$	−4.88485e24	−0.132916	−0.0664579	−	0.997789i	$-0.521170\pi$
$491$	−4.88485e24	−0.132916	−0.0664579	+	0.997789i	$0.521170\pi$
$492$	−4.60577e25	−1.22923
$493$	9.89307e25	2.58991
$494$	2.93578e24	0.0753907
$495$	3.73502e24	0.0940901
$496$	5.28249e24	0.130546
$497$	−5.13123e25	−1.24404
$498$	2.47152e24	0.0587876
$499$	−7.74230e25	−1.80682	−0.903410	−	0.428778i	$-0.858944\pi$
$499$	−7.74230e25	−1.80682	−0.903410	+	0.428778i	$0.858944\pi$
$500$	−5.48038e24	−0.125486
$501$	−2.13124e25	−0.478822
$502$	1.35525e24	0.0298769
$503$	1.45620e24	0.0315011	0.0157505	−	0.999876i	$-0.494986\pi$
$503$	1.45620e24	0.0315011	0.0157505	+	0.999876i	$0.494986\pi$
$504$	2.80996e25	0.596499
$505$	−1.27221e26	−2.65028
$506$	−2.24491e24	−0.0458953
$507$	−1.95002e25	−0.391258
$508$	4.65507e25	0.916687
$509$	6.81047e25	1.31631	0.658155	−	0.752882i	$-0.271337\pi$
$509$	6.81047e25	1.31631	0.658155	+	0.752882i	$0.271337\pi$
$510$	3.35991e25	0.637398
$511$	−8.53526e24	−0.158935
$512$	4.33834e25	0.792975
$513$	−1.44561e25	−0.259380
$514$	−6.01203e24	−0.105894
$515$	−5.47097e25	−0.946010
$516$	−2.20002e25	−0.373469
$517$	−1.50544e24	−0.0250902
$518$	−1.97051e25	−0.322438
$519$	−1.05436e25	−0.169395
$520$	4.93830e25	0.779015
$521$	−1.03172e25	−0.159810	−0.0799050	−	0.996802i	$-0.525462\pi$
$521$	−1.03172e25	−0.159810	−0.0799050	+	0.996802i	$0.525462\pi$
$522$	2.27318e25	0.345752
$523$	8.44171e25	1.26085	0.630426	−	0.776249i	$-0.282880\pi$
$523$	8.44171e25	1.26085	0.630426	+	0.776249i	$0.282880\pi$
$524$	6.46033e24	0.0947562
$525$	8.86289e25	1.27662
$526$	5.04749e25	0.714020
$527$	−2.86295e25	−0.397752
$528$	4.28833e24	0.0585149
$529$	−4.81282e25	−0.645017
$530$	−3.59926e25	−0.473797
$531$	−3.99074e25	−0.516007
$532$	−2.79706e25	−0.355256
$533$	1.11850e26	1.39549
$534$	1.99518e25	0.244535
$535$	1.54882e26	1.86484
$536$	−9.40234e25	−1.11217
$537$	−8.92329e25	−1.03698
$538$	−1.05774e25	−0.120767
$539$	3.76146e25	0.421953
$540$	−1.08033e26	−1.19073
$541$	−1.19762e25	−0.129702	−0.0648509	−	0.997895i	$-0.520657\pi$
$541$	−1.19762e25	−0.129702	−0.0648509	+	0.997895i	$0.520657\pi$
$542$	−6.38938e24	−0.0679932
$543$	7.72388e25	0.807676
$544$	−1.30333e26	−1.33926
$545$	−7.36036e24	−0.0743246
$546$	4.57909e25	0.454411
$547$	1.36995e26	1.33606	0.668030	−	0.744134i	$-0.267138\pi$
$547$	1.36995e26	1.33606	0.668030	+	0.744134i	$0.267138\pi$
$548$	−3.87303e25	−0.371223
$549$	−6.38461e25	−0.601446
$550$	7.36875e24	0.0682257
$551$	−5.09313e25	−0.463494
$552$	4.16343e25	0.372418
$553$	−8.36909e25	−0.735850
$554$	−7.97245e25	−0.689047
$555$	4.85765e25	0.412708
$556$	−7.20687e25	−0.601916
$557$	−1.87549e26	−1.53989	−0.769945	−	0.638111i	$-0.779716\pi$
$557$	−1.87549e26	−1.53989	−0.769945	+	0.638111i	$0.779716\pi$
$558$	−6.57834e24	−0.0530996
$559$	5.34269e25	0.423982
$560$	−1.43438e26	−1.11912
$561$	−2.32414e25	−0.178285
$562$	−1.79806e23	−0.00135615
$563$	−1.33333e26	−0.988798	−0.494399	−	0.869235i	$-0.664612\pi$
$563$	−1.33333e26	−0.988798	−0.494399	+	0.869235i	$0.664612\pi$
$564$	1.24042e25	0.0904518
$565$	−2.06352e26	−1.47961
$566$	−6.65617e25	−0.469319
$567$	−1.18721e26	−0.823171
$568$	7.91213e25	0.539491
$569$	1.73311e26	1.16214	0.581072	−	0.813852i	$-0.302634\pi$
$569$	1.73311e26	1.16214	0.581072	+	0.813852i	$0.302634\pi$
$570$	−1.72974e25	−0.114070
$571$	1.37925e25	0.0894541	0.0447270	−	0.998999i	$-0.485758\pi$
$571$	1.37925e25	0.0894541	0.0447270	+	0.998999i	$0.485758\pi$
$572$	−1.51763e25	−0.0968061
$573$	−2.03799e26	−1.27859
$574$	2.67328e26	1.64961
$575$	−8.69425e25	−0.527701
$576$	−6.84943e23	−0.00408922
$577$	5.09963e25	0.299481	0.149740	−	0.988725i	$-0.452156\pi$
$577$	5.09963e25	0.299481	0.149740	+	0.988725i	$0.452156\pi$
$578$	6.08989e25	0.351799
$579$	1.75123e26	0.995172
$580$	−3.80618e26	−2.12777
$581$	5.71841e25	0.314487
$582$	−1.62569e25	−0.0879568
$583$	2.48971e25	0.132525
$584$	1.31610e25	0.0689234
$585$	7.47362e25	0.385079
$586$	2.20223e25	0.111644
$587$	−4.01737e25	−0.200392	−0.100196	−	0.994968i	$-0.531947\pi$
$587$	−4.01737e25	−0.200392	−0.100196	+	0.994968i	$0.531947\pi$
$588$	−3.09929e26	−1.52117
$589$	1.47390e25	0.0711823
$590$	−1.67626e26	−0.796614
$591$	2.45856e26	1.14974
$592$	−3.69256e25	−0.169930
$593$	2.02819e26	0.918521	0.459261	−	0.888302i	$-0.348114\pi$
$593$	2.02819e26	0.918521	0.459261	+	0.888302i	$0.348114\pi$
$594$	−1.87467e25	−0.0835512
$595$	7.77388e26	3.40978
$596$	2.06032e26	0.889398
$597$	1.60922e26	0.683690
$598$	−4.49196e25	−0.187834
$599$	3.80758e26	1.56709	0.783545	−	0.621335i	$-0.213410\pi$
$599$	3.80758e26	1.56709	0.783545	+	0.621335i	$0.213410\pi$
$600$	−1.36662e26	−0.553618
$601$	4.40053e26	1.75468	0.877339	−	0.479872i	$-0.159317\pi$
$601$	4.40053e26	1.75468	0.877339	+	0.479872i	$0.159317\pi$
$602$	1.27694e26	0.501191
$603$	−1.42295e26	−0.549762
$604$	−2.24060e26	−0.852143
$605$	3.55933e26	1.33258
$606$	1.81900e26	0.670414
$607$	6.48137e24	0.0235166	0.0117583	−	0.999931i	$-0.496257\pi$
$607$	6.48137e24	0.0235166	0.0117583	+	0.999931i	$0.496257\pi$
$608$	6.70976e25	0.239675
$609$	−7.94400e26	−2.79367
$610$	−2.68177e26	−0.928514
$611$	−3.01232e25	−0.102686
$612$	−1.26787e26	−0.425534
$613$	3.58835e26	1.18582	0.592911	−	0.805268i	$-0.297978\pi$
$613$	3.58835e26	1.18582	0.592911	+	0.805268i	$0.297978\pi$
$614$	1.84148e26	0.599195
$615$	−6.59010e26	−2.11144
$616$	−8.16439e25	−0.257577
$617$	4.88564e26	1.51779	0.758897	−	0.651211i	$-0.225739\pi$
$617$	4.88564e26	1.51779	0.758897	+	0.651211i	$0.225739\pi$
$618$	7.82235e25	0.239302
$619$	−1.85396e26	−0.558522	−0.279261	−	0.960215i	$-0.590089\pi$
$619$	−1.85396e26	−0.558522	−0.279261	+	0.960215i	$0.590089\pi$
$620$	1.10147e26	0.326777
$621$	2.21188e26	0.646238
$622$	−1.23815e26	−0.356258
$623$	4.61630e26	1.30815
$624$	8.58077e25	0.239482
$625$	−4.00629e26	−1.10124
$626$	5.74669e25	0.155583
$627$	1.19651e25	0.0319062
$628$	−5.23846e26	−1.37590
$629$	2.00125e26	0.517750
$630$	1.78624e26	0.455204
$631$	1.24721e26	0.313084	0.156542	−	0.987671i	$-0.449965\pi$
$631$	1.24721e26	0.313084	0.156542	+	0.987671i	$0.449965\pi$
$632$	1.29048e26	0.319108
$633$	1.20046e26	0.292424
$634$	−3.33057e26	−0.799227
$635$	6.66064e26	1.57458
$636$	−2.05142e26	−0.477762
$637$	7.52652e26	1.72691
$638$	−6.60477e25	−0.149301
$639$	1.19742e26	0.266679
$640$	6.24344e26	1.36998
$641$	−1.40139e26	−0.302975	−0.151487	−	0.988459i	$-0.548406\pi$
$641$	−1.40139e26	−0.302975	−0.151487	+	0.988459i	$0.548406\pi$
$642$	−2.21449e26	−0.471728
$643$	−1.17485e26	−0.246592	−0.123296	−	0.992370i	$-0.539347\pi$
$643$	−1.17485e26	−0.246592	−0.123296	+	0.992370i	$0.539347\pi$
$644$	4.27971e26	0.885112
$645$	−3.14787e26	−0.641504
$646$	−7.12617e25	−0.143102
$647$	2.00234e25	0.0396230	0.0198115	−	0.999804i	$-0.493693\pi$
$647$	2.00234e25	0.0396230	0.0198115	+	0.999804i	$0.493693\pi$
$648$	1.83063e26	0.356975
$649$	1.15952e26	0.222820
$650$	1.47445e26	0.279225
$651$	2.29891e26	0.429045
$652$	−6.95424e26	−1.27908
$653$	−5.31362e26	−0.963197	−0.481599	−	0.876392i	$-0.659944\pi$
$653$	−5.31362e26	−0.963197	−0.481599	+	0.876392i	$0.659944\pi$
$654$	1.05238e25	0.0188011
$655$	9.24367e25	0.162762
$656$	5.00948e26	0.869373
$657$	1.99178e25	0.0340699
$658$	−7.19965e25	−0.121385
$659$	−1.96493e26	−0.326539	−0.163270	−	0.986581i	$-0.552204\pi$
$659$	−1.96493e26	−0.326539	−0.163270	+	0.986581i	$0.552204\pi$
$660$	8.94174e25	0.146472
$661$	−7.36438e26	−1.18911	−0.594555	−	0.804055i	$-0.702672\pi$
$661$	−7.36438e26	−1.18911	−0.594555	+	0.804055i	$0.702672\pi$
$662$	1.59856e26	0.254435
$663$	−4.65051e26	−0.729663
$664$	−8.81753e25	−0.136380
$665$	−4.00213e26	−0.610219
$666$	4.59838e25	0.0691193
$667$	7.79285e26	1.15478
$668$	3.37806e26	0.493504
$669$	4.39266e26	0.632673
$670$	−5.97691e26	−0.848724
$671$	1.85506e26	0.259713
$672$	1.04655e27	1.44462
$673$	3.47381e26	0.472784	0.236392	−	0.971658i	$-0.424035\pi$
$673$	3.47381e26	0.472784	0.236392	+	0.971658i	$0.424035\pi$
$674$	−5.66191e25	−0.0759791
$675$	−7.26035e26	−0.960665
$676$	3.09082e26	0.403255
$677$	−5.36278e26	−0.689918	−0.344959	−	0.938618i	$-0.612107\pi$
$677$	−5.36278e26	−0.689918	−0.344959	+	0.938618i	$0.612107\pi$
$678$	2.95040e26	0.374282
$679$	−3.76139e26	−0.470528
$680$	−1.19870e27	−1.47868
$681$	2.98543e26	0.363169
$682$	1.91135e25	0.0229292
$683$	3.11135e26	0.368088	0.184044	−	0.982918i	$-0.441081\pi$
$683$	3.11135e26	0.368088	0.184044	+	0.982918i	$0.441081\pi$
$684$	6.52720e25	0.0761542
$685$	−5.54166e26	−0.637646
$686$	1.06557e27	1.20921
$687$	−5.04732e26	−0.564900
$688$	2.39286e26	0.264136
$689$	4.98181e26	0.542381
$690$	2.64663e26	0.284201
$691$	−8.99392e26	−0.952594	−0.476297	−	0.879285i	$-0.658021\pi$
$691$	−8.99392e26	−0.952594	−0.476297	+	0.879285i	$0.658021\pi$
$692$	1.67118e26	0.174588
$693$	−1.23560e26	−0.127324
$694$	−4.77308e26	−0.485158
$695$	−1.03118e27	−1.03390
$696$	1.22493e27	1.21150
$697$	−2.71498e27	−2.64884
$698$	−4.84342e25	−0.0466150
$699$	−5.18141e26	−0.491943
$700$	−1.40478e27	−1.31576
$701$	−4.03551e26	−0.372887	−0.186444	−	0.982466i	$-0.559696\pi$
$701$	−4.03551e26	−0.372887	−0.186444	+	0.982466i	$0.559696\pi$
$702$	−3.75112e26	−0.341947
$703$	−1.03028e26	−0.0926573
$704$	1.99012e24	0.00176579
$705$	1.77484e26	0.155368
$706$	6.75135e26	0.583104
$707$	4.20866e27	3.58641
$708$	−9.55394e26	−0.803280
$709$	−1.89544e27	−1.57243	−0.786214	−	0.617954i	$-0.787962\pi$
$709$	−1.89544e27	−1.57243	−0.786214	+	0.617954i	$0.787962\pi$
$710$	5.02961e26	0.411699
$711$	1.95300e26	0.157740
$712$	−7.11812e26	−0.567291
$713$	−2.25517e26	−0.177349
$714$	−1.11150e27	−0.862537
$715$	−2.17147e26	−0.166283
$716$	1.41436e27	1.06877
$717$	1.06636e27	0.795193
$718$	7.46537e26	0.549375
$719$	1.43604e27	1.04290	0.521449	−	0.853282i	$-0.325392\pi$
$719$	1.43604e27	1.04290	0.521449	+	0.853282i	$0.325392\pi$
$720$	3.34725e26	0.239900
$721$	1.80987e27	1.28016
$722$	−6.04840e26	−0.422219
$723$	−6.08033e26	−0.418903
$724$	−1.22425e27	−0.832441
$725$	−2.55795e27	−1.71665
$726$	−5.08910e26	−0.337088
$727$	−8.04737e26	−0.526111	−0.263055	−	0.964781i	$-0.584730\pi$
$727$	−8.04737e26	−0.526111	−0.263055	+	0.964781i	$0.584730\pi$
$728$	−1.63366e27	−1.05418
$729$	1.59796e27	1.01778
$730$	8.36623e25	0.0525972
$731$	−1.29686e27	−0.804779
$732$	−1.52849e27	−0.936284
$733$	2.07217e27	1.25296	0.626482	−	0.779436i	$-0.284494\pi$
$733$	2.07217e27	1.25296	0.626482	+	0.779436i	$0.284494\pi$
$734$	−9.96812e26	−0.594978
$735$	−4.43457e27	−2.61289
$736$	−1.02664e27	−0.597144
$737$	4.13440e26	0.237395
$738$	−6.23835e26	−0.353618
$739$	−3.55097e26	−0.198712	−0.0993562	−	0.995052i	$-0.531678\pi$
$739$	−3.55097e26	−0.198712	−0.0993562	+	0.995052i	$0.531678\pi$
$740$	−7.69946e26	−0.425362
$741$	2.39417e26	0.130581
$742$	1.19069e27	0.641150
$743$	9.06935e26	0.482151	0.241075	−	0.970506i	$-0.422500\pi$
$743$	9.06935e26	0.482151	0.241075	+	0.970506i	$0.422500\pi$
$744$	−3.54482e26	−0.186059
$745$	2.94798e27	1.52771
$746$	−8.38828e26	−0.429195
$747$	−1.33444e26	−0.0674147
$748$	3.68381e26	0.183752
$749$	−5.12372e27	−2.52353
$750$	1.12117e26	0.0545245
$751$	3.50539e27	1.68328	0.841641	−	0.540038i	$-0.181590\pi$
$751$	3.50539e27	1.68328	0.841641	+	0.540038i	$0.181590\pi$
$752$	−1.34915e26	−0.0639720
$753$	1.10523e26	0.0517486
$754$	−1.32159e27	−0.611037
$755$	−3.20592e27	−1.46372
$756$	3.57388e27	1.61132
$757$	3.64919e27	1.62475	0.812373	−	0.583138i	$-0.198175\pi$
$757$	3.64919e27	1.62475	0.812373	+	0.583138i	$0.198175\pi$
$758$	−1.40923e27	−0.619620
$759$	−1.83075e26	−0.0794935
$760$	6.17111e26	0.264627
$761$	2.06390e26	0.0874046	0.0437023	−	0.999045i	$-0.486085\pi$
$761$	2.06390e26	0.0874046	0.0437023	+	0.999045i	$0.486085\pi$
$762$	−9.52333e26	−0.398306
$763$	2.43491e26	0.100577
$764$	3.23024e27	1.31780
$765$	−1.81411e27	−0.730936
$766$	6.75379e26	0.268766
$767$	2.32015e27	0.911926
$768$	−9.13194e26	−0.354512
$769$	−1.74431e27	−0.668840	−0.334420	−	0.942424i	$-0.608540\pi$
$769$	−1.74431e27	−0.668840	−0.334420	+	0.942424i	$0.608540\pi$
$770$	−5.18997e26	−0.196564
$771$	−4.90288e26	−0.183415
$772$	−2.77574e27	−1.02569
$773$	2.39941e26	0.0875788	0.0437894	−	0.999041i	$-0.486057\pi$
$773$	2.39941e26	0.0875788	0.0437894	+	0.999041i	$0.486057\pi$
$774$	−2.97985e26	−0.107437
$775$	7.40243e26	0.263638
$776$	5.79990e26	0.204049
$777$	−1.60698e27	−0.558484
$778$	5.41604e26	0.185941
$779$	1.39772e27	0.474040
$780$	1.78920e27	0.599461
$781$	−3.47913e26	−0.115156
$782$	1.09035e27	0.356536
$783$	6.50762e27	2.10225
$784$	3.37095e27	1.07584
$785$	−7.49538e27	−2.36337
$786$	−1.32165e26	−0.0411721
$787$	3.14451e27	0.967816	0.483908	−	0.875119i	$-0.339217\pi$
$787$	3.14451e27	0.967816	0.483908	+	0.875119i	$0.339217\pi$
$788$	−3.89686e27	−1.18499
$789$	4.11629e27	1.23673
$790$	8.20334e26	0.243519
$791$	6.82640e27	2.00223
$792$	1.90524e26	0.0552153
$793$	3.71189e27	1.06292
$794$	1.09854e27	0.310829
$795$	−2.93524e27	−0.820646
$796$	−2.55064e27	−0.704653
$797$	1.88854e27	0.515553	0.257776	−	0.966205i	$-0.417010\pi$
$797$	1.88854e27	0.515553	0.257776	+	0.966205i	$0.417010\pi$
$798$	5.72222e26	0.154361
$799$	7.31196e26	0.194912
$800$	3.36988e27	0.887686
$801$	−1.07726e27	−0.280420
$802$	1.25642e26	0.0323205
$803$	−5.78716e25	−0.0147119
$804$	−3.40657e27	−0.855826
$805$	6.12355e27	1.52035
$806$	3.82453e26	0.0938415
$807$	−8.62603e26	−0.209176
$808$	−6.48957e27	−1.55528
$809$	5.06998e26	0.120087	0.0600434	−	0.998196i	$-0.480876\pi$
$809$	5.06998e26	0.120087	0.0600434	+	0.998196i	$0.480876\pi$
$810$	1.16370e27	0.272417
$811$	−8.30205e27	−1.92082	−0.960412	−	0.278585i	$-0.910135\pi$
$811$	−8.30205e27	−1.92082	−0.960412	+	0.278585i	$0.910135\pi$
$812$	1.25914e28	2.87933
$813$	−5.21061e26	−0.117768
$814$	−1.33607e26	−0.0298467
$815$	−9.95037e27	−2.19706
$816$	−2.08285e27	−0.454571
$817$	6.67645e26	0.144024
$818$	2.07645e27	0.442756
$819$	−2.47238e27	−0.521096
$820$	1.04454e28	2.17618
$821$	−8.30733e27	−1.71081	−0.855404	−	0.517961i	$-0.826691\pi$
$821$	−8.30733e27	−1.71081	−0.855404	+	0.517961i	$0.826691\pi$
$822$	7.92343e26	0.161299
$823$	−3.30254e26	−0.0664584	−0.0332292	−	0.999448i	$-0.510579\pi$
$823$	−3.30254e26	−0.0664584	−0.0332292	+	0.999448i	$0.510579\pi$
$824$	−2.79074e27	−0.555151
$825$	6.00930e26	0.118171
$826$	5.54531e27	1.07799
$827$	−6.92799e27	−1.33139	−0.665694	−	0.746225i	$-0.731864\pi$
$827$	−6.92799e27	−1.33139	−0.665694	+	0.746225i	$0.731864\pi$
$828$	−9.98709e26	−0.189736
$829$	4.53460e27	0.851670	0.425835	−	0.904801i	$-0.359980\pi$
$829$	4.53460e27	0.851670	0.425835	+	0.904801i	$0.359980\pi$
$830$	−5.60516e26	−0.104075
$831$	−6.50163e27	−1.19347
$832$	3.98213e25	0.00722677
$833$	−1.82695e28	−3.27792
$834$	1.47438e27	0.261536
$835$	4.83344e27	0.847686
$836$	−1.89649e26	−0.0328845
$837$	−1.88323e27	−0.322859
$838$	4.25084e27	0.720538
$839$	7.97188e27	1.33605	0.668024	−	0.744140i	$-0.267140\pi$
$839$	7.97188e27	1.33605	0.668024	+	0.744140i	$0.267140\pi$
$840$	9.62539e27	1.59502
$841$	1.68242e28	2.75659
$842$	−3.29265e27	−0.533433
$843$	−1.46634e25	−0.00234894
$844$	−1.90275e27	−0.301390
$845$	4.42245e27	0.692666
$846$	1.68011e26	0.0260206
$847$	−1.17748e28	−1.80326
$848$	2.23123e27	0.337897
$849$	−5.42818e27	−0.812890
$850$	−3.57901e27	−0.530009
$851$	1.57640e27	0.230853
$852$	2.86665e27	0.415144
$853$	−6.88900e27	−0.986598	−0.493299	−	0.869860i	$-0.664209\pi$
$853$	−6.88900e27	−0.986598	−0.493299	+	0.869860i	$0.664209\pi$
$854$	8.87168e27	1.25648
$855$	9.33935e26	0.130809
$856$	7.90055e27	1.09435
$857$	3.15216e27	0.431807	0.215904	−	0.976415i	$-0.430730\pi$
$857$	3.15216e27	0.431807	0.215904	+	0.976415i	$0.430730\pi$
$858$	3.10476e26	0.0420628
$859$	1.00305e28	1.34397	0.671985	−	0.740565i	$-0.265442\pi$
$859$	1.00305e28	1.34397	0.671985	+	0.740565i	$0.265442\pi$
$860$	4.98943e27	0.661174
$861$	2.18010e28	2.85723
$862$	−3.77934e26	−0.0489889
$863$	−7.69546e27	−0.986579	−0.493290	−	0.869865i	$-0.664206\pi$
$863$	−7.69546e27	−0.986579	−0.493290	+	0.869865i	$0.664206\pi$
$864$	−8.57323e27	−1.08709
$865$	2.39118e27	0.299889
$866$	−5.76636e27	−0.715289
$867$	4.96638e27	0.609338
$868$	−3.64381e27	−0.442200
$869$	−5.67449e26	−0.0681144
$870$	7.78668e27	0.924526
$871$	8.27276e27	0.971579
$872$	−3.75452e26	−0.0436162
$873$	8.77756e26	0.100864
$874$	−5.61334e26	−0.0638062
$875$	2.59409e27	0.291681
$876$	4.76838e26	0.0530373
$877$	3.61896e27	0.398187	0.199094	−	0.979980i	$-0.436200\pi$
$877$	3.61896e27	0.398187	0.199094	+	0.979980i	$0.436200\pi$
$878$	1.67739e27	0.182573
$879$	1.79594e27	0.193374
$880$	−9.72551e26	−0.103592
$881$	−9.57973e27	−1.00944	−0.504722	−	0.863282i	$-0.668405\pi$
$881$	−9.57973e27	−1.00944	−0.504722	+	0.863282i	$0.668405\pi$
$882$	−4.19787e27	−0.437601
$883$	−3.59781e27	−0.371032	−0.185516	−	0.982641i	$-0.559396\pi$
$883$	−3.59781e27	−0.371032	−0.185516	+	0.982641i	$0.559396\pi$
$884$	7.37114e27	0.752035
$885$	−1.36701e28	−1.37978
$886$	7.10963e27	0.709948
$887$	−3.60928e27	−0.356572	−0.178286	−	0.983979i	$-0.557055\pi$
$887$	−3.60928e27	−0.356572	−0.178286	+	0.983979i	$0.557055\pi$
$888$	2.47789e27	0.242191
$889$	−2.20343e28	−2.13075
$890$	−4.52487e27	−0.432914
$891$	−8.04966e26	−0.0761973
$892$	−6.96244e27	−0.652072
$893$	−3.76433e26	−0.0348818
$894$	−4.21500e27	−0.386449
$895$	2.02371e28	1.83582
$896$	−2.06542e28	−1.85388
$897$	−3.66325e27	−0.325340
$898$	−3.36566e26	−0.0295764
$899$	−6.63496e27	−0.576928
$900$	3.27819e27	0.282053
$901$	−1.20926e28	−1.02952
$902$	1.81257e27	0.152697
$903$	1.04136e28	0.868094
$904$	−1.05260e28	−0.868287
$905$	−1.75170e28	−1.42987
$906$	4.58380e27	0.370261
$907$	−1.42293e28	−1.13740	−0.568701	−	0.822545i	$-0.692554\pi$
$907$	−1.42293e28	−1.13740	−0.568701	+	0.822545i	$0.692554\pi$
$908$	−4.73195e27	−0.374304
$909$	−9.82130e27	−0.768798
$910$	−1.03849e28	−0.804469
$911$	2.63631e27	0.202103	0.101051	−	0.994881i	$-0.467779\pi$
$911$	2.63631e27	0.202103	0.101051	+	0.994881i	$0.467779\pi$
$912$	1.07229e27	0.0813507
$913$	3.87725e26	0.0291106
$914$	4.41990e27	0.328415
$915$	−2.18702e28	−1.60824
$916$	8.00009e27	0.582221
$917$	−3.05794e27	−0.220252
$918$	9.10528e27	0.649065
$919$	7.18572e27	0.506959	0.253480	−	0.967341i	$-0.418425\pi$
$919$	7.18572e27	0.506959	0.253480	+	0.967341i	$0.418425\pi$
$920$	−9.44224e27	−0.659312
$921$	1.50175e28	1.03784
$922$	4.07947e27	0.279036
$923$	−6.96158e27	−0.471294
$924$	−2.95805e27	−0.198208
$925$	−5.17443e27	−0.343175
$926$	4.87784e27	0.320201
$927$	−4.22350e27	−0.274420
$928$	−3.02050e28	−1.94255
$929$	−9.73331e27	−0.619600	−0.309800	−	0.950802i	$-0.600262\pi$
$929$	−9.73331e27	−0.619600	−0.309800	+	0.950802i	$0.600262\pi$
$930$	−2.25338e27	−0.141986
$931$	9.40547e27	0.586622
$932$	8.21262e27	0.507027
$933$	−1.00973e28	−0.617060
$934$	−1.02177e28	−0.618096
$935$	5.27092e27	0.315629
$936$	3.81229e27	0.225978
$937$	−1.27129e28	−0.745966	−0.372983	−	0.927838i	$-0.621665\pi$
$937$	−1.27129e28	−0.745966	−0.372983	+	0.927838i	$0.621665\pi$
$938$	1.97725e28	1.14851
$939$	4.68650e27	0.269479
$940$	−2.81315e27	−0.160132
$941$	1.65319e28	0.931581	0.465790	−	0.884895i	$-0.345770\pi$
$941$	1.65319e28	0.931581	0.465790	+	0.884895i	$0.345770\pi$
$942$	1.07168e28	0.597837
$943$	−2.13861e28	−1.18106
$944$	1.03914e28	0.568119
$945$	5.11363e28	2.76775
$946$	8.65802e26	0.0463931
$947$	−8.17156e27	−0.433491	−0.216745	−	0.976228i	$-0.569544\pi$
$947$	−8.17156e27	−0.433491	−0.216745	+	0.976228i	$0.569544\pi$
$948$	4.67554e27	0.245557
$949$	−1.15799e27	−0.0602108
$950$	1.84254e27	0.0948512
$951$	−2.71612e28	−1.38431
$952$	3.96546e28	2.00098
$953$	1.25754e28	0.628260	0.314130	−	0.949380i	$-0.398287\pi$
$953$	1.25754e28	0.628260	0.314130	+	0.949380i	$0.398287\pi$
$954$	−2.77857e27	−0.137440
$955$	4.62195e28	2.26356
$956$	−1.69020e28	−0.819575
$957$	−5.38627e27	−0.258598
$958$	7.43500e26	0.0353434
$959$	1.83326e28	0.862874
$960$	−2.34624e26	−0.0109344
$961$	−1.97506e28	−0.911397
$962$	−2.67341e27	−0.122153
$963$	1.19567e28	0.540954
$964$	9.63744e27	0.431747
$965$	−3.97162e28	−1.76181
$966$	−8.75541e27	−0.384586
$967$	−3.86959e28	−1.68312	−0.841558	−	0.540167i	$-0.818361\pi$
$967$	−3.86959e28	−1.68312	−0.841558	+	0.540167i	$0.818361\pi$
$968$	1.81561e28	0.782002
$969$	−5.81148e27	−0.247862
$970$	3.68690e27	0.155715
$971$	1.61790e28	0.676656	0.338328	−	0.941028i	$-0.390139\pi$
$971$	1.61790e28	0.676656	0.338328	+	0.941028i	$0.390139\pi$
$972$	−1.43042e28	−0.592424
$973$	3.41130e28	1.39910
$974$	−1.45968e28	−0.592856
$975$	1.20243e28	0.483635
$976$	1.66247e28	0.662186
$977$	−3.70914e28	−1.46310	−0.731551	−	0.681787i	$-0.761203\pi$
$977$	−3.70914e28	−1.46310	−0.731551	+	0.681787i	$0.761203\pi$
$978$	1.42270e28	0.555767
$979$	3.12999e27	0.121090
$980$	7.02887e28	2.69301
$981$	−5.68209e26	−0.0215602
$982$	−1.58397e27	−0.0595235
$983$	−6.84045e27	−0.254581	−0.127291	−	0.991865i	$-0.540628\pi$
$983$	−6.84045e27	−0.254581	−0.127291	+	0.991865i	$0.540628\pi$
$984$	−3.36161e28	−1.23906
$985$	−5.57576e28	−2.03545
$986$	3.20795e28	1.15984
$987$	−5.87141e27	−0.210247
$988$	−3.79480e27	−0.134585
$989$	−1.02154e28	−0.358833
$990$	1.21113e27	0.0421362
$991$	−3.16338e28	−1.09006	−0.545031	−	0.838416i	$-0.683482\pi$
$991$	−3.16338e28	−1.09006	−0.545031	+	0.838416i	$0.683482\pi$
$992$	8.74099e27	0.298332
$993$	1.30364e28	0.440697
$994$	−1.66387e28	−0.557118
$995$	−3.64955e28	−1.21037
$996$	−3.19469e27	−0.104946
$997$	9.30769e27	0.302857	0.151429	−	0.988468i	$-0.451613\pi$
$997$	9.30769e27	0.302857	0.151429	+	0.988468i	$0.451613\pi$
$998$	−2.51054e28	−0.809145
$999$	1.31641e28	0.420262

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	47.20.a.b.1.23	✓	39

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
47.20.a.b.1.23	✓	39	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:	\( N \)	\(=\)	\( 47 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	47.a (trivial)

\(f(q)\)	\(=\)	\(q+324.262 q^{2} +26444.0 q^{3} -419142. q^{4} -5.99723e6 q^{5} +8.57479e6 q^{6} +1.98397e8 q^{7} -3.05919e8 q^{8} -4.62977e8 q^{9} +O(q^{10})\)	\(q+324.262 q^{2} +26444.0 q^{3} -419142. q^{4} -5.99723e6 q^{5} +8.57479e6 q^{6} +1.98397e8 q^{7} -3.05919e8 q^{8} -4.62977e8 q^{9} -1.94468e9 q^{10} +1.34519e9 q^{11} -1.10838e10 q^{12} +2.69166e10 q^{13} +6.43326e10 q^{14} -1.58591e11 q^{15} +1.20553e11 q^{16} -6.53361e11 q^{17} -1.50126e11 q^{18} +3.36362e11 q^{19} +2.51369e12 q^{20} +5.24640e12 q^{21} +4.36194e11 q^{22} -5.14658e12 q^{23} -8.08971e12 q^{24} +1.68933e13 q^{25} +8.72805e12 q^{26} -4.29778e13 q^{27} -8.31564e13 q^{28} -1.51418e14 q^{29} -5.14250e13 q^{30} +4.38188e13 q^{31} +1.99480e14 q^{32} +3.55722e13 q^{33} -2.11860e14 q^{34} -1.18983e15 q^{35} +1.94053e14 q^{36} -3.06301e14 q^{37} +1.09069e14 q^{38} +7.11783e14 q^{39} +1.83467e15 q^{40} +4.15541e15 q^{41} +1.70121e15 q^{42} +1.98490e15 q^{43} -5.63825e14 q^{44} +2.77658e15 q^{45} -1.66884e15 q^{46} -1.11913e15 q^{47} +3.18791e15 q^{48} +2.79623e16 q^{49} +5.47785e15 q^{50} -1.72775e16 q^{51} -1.12819e16 q^{52} +1.85083e16 q^{53} -1.39361e16 q^{54} -8.06741e15 q^{55} -6.06933e16 q^{56} +8.89475e15 q^{57} -4.90992e16 q^{58} +8.61975e16 q^{59} +6.64720e16 q^{60} +1.37903e17 q^{61} +1.42088e16 q^{62} -9.18531e16 q^{63} +1.47943e15 q^{64} -1.61425e17 q^{65} +1.15347e16 q^{66} +3.07347e17 q^{67} +2.73851e17 q^{68} -1.36096e17 q^{69} -3.85817e17 q^{70} -2.58635e17 q^{71} +1.41633e17 q^{72} -4.30212e16 q^{73} -9.93219e16 q^{74} +4.46726e17 q^{75} -1.40983e17 q^{76} +2.66881e17 q^{77} +2.30805e17 q^{78} -4.21836e17 q^{79} -7.22985e17 q^{80} -5.98404e17 q^{81} +1.34744e18 q^{82} +2.88231e17 q^{83} -2.19899e18 q^{84} +3.91835e18 q^{85} +6.43629e17 q^{86} -4.00410e18 q^{87} -4.11519e17 q^{88} +2.32680e18 q^{89} +9.00340e17 q^{90} +5.34017e18 q^{91} +2.15715e18 q^{92} +1.15874e18 q^{93} -3.62892e17 q^{94} -2.01724e18 q^{95} +5.27506e18 q^{96} -1.89590e18 q^{97} +9.06714e18 q^{98} -6.22791e17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39 q + 567 q^{2} + 4180 q^{3} + 11374277 q^{4} + 7841414 q^{5} - 3289088 q^{6} + 280678228 q^{7} + 616397649 q^{8} + 15832291053 q^{9}+O(q^{10}) \) 39 * q + 567 * q^2 + 4180 * q^3 + 11374277 * q^4 + 7841414 * q^5 - 3289088 * q^6 + 280678228 * q^7 + 616397649 * q^8 + 15832291053 * q^9	\( 39 q + 567 q^{2} + 4180 q^{3} + 11374277 q^{4} + 7841414 q^{5} - 3289088 q^{6} + 280678228 q^{7} + 616397649 q^{8} + 15832291053 q^{9} - 197084160 q^{10} + 6183770516 q^{11} - 18595076275 q^{12} + 72670351796 q^{13} - 286195652197 q^{14} + 216978245574 q^{15} + 4395775708833 q^{16} + 1565738603712 q^{17} + 6109717535226 q^{18} + 3193929321662 q^{19} - 5906920535432 q^{20} - 7386396792532 q^{21} - 8877997844072 q^{22} - 24482520509106 q^{23} - 7153616576581 q^{24} + 205574470566045 q^{25} + 29760604099536 q^{26} + 37673737054348 q^{27} + 359478142575004 q^{28} + 236042103421602 q^{29} + 10\!\cdots\!54 q^{30}+ \cdots + 26\!\cdots\!62 q^{99}+O(q^{100}) \) 39 * q + 567 * q^2 + 4180 * q^3 + 11374277 * q^4 + 7841414 * q^5 - 3289088 * q^6 + 280678228 * q^7 + 616397649 * q^8 + 15832291053 * q^9 - 197084160 * q^10 + 6183770516 * q^11 - 18595076275 * q^12 + 72670351796 * q^13 - 286195652197 * q^14 + 216978245574 * q^15 + 4395775708833 * q^16 + 1565738603712 * q^17 + 6109717535226 * q^18 + 3193929321662 * q^19 - 5906920535432 * q^20 - 7386396792532 * q^21 - 8877997844072 * q^22 - 24482520509106 * q^23 - 7153616576581 * q^24 + 205574470566045 * q^25 + 29760604099536 * q^26 + 37673737054348 * q^27 + 359478142575004 * q^28 + 236042103421602 * q^29 + 1030005166297154 * q^30 + 238097499775556 * q^31 + 980418196894414 * q^32 + 1875903246842268 * q^33 + 1649701857955214 * q^34 + 1575167660282324 * q^35 + 7318800726486641 * q^36 + 3601713987410012 * q^37 + 2042887131890906 * q^38 + 712786595145686 * q^39 - 38233381179154 * q^40 + 3933352398746224 * q^41 - 1818473706528187 * q^42 + 9956432184119776 * q^43 - 8374864809587786 * q^44 + 4814834336659374 * q^45 - 21420879452965494 * q^46 - 43646088451007913 * q^47 - 71590477049164572 * q^48 + 118361365545367441 * q^49 - 57666947668617547 * q^50 + 63071710735479788 * q^51 - 1295800694634632 * q^52 - 70710792109226748 * q^53 - 259379096268487409 * q^54 - 138894593750942388 * q^55 - 316931141949206936 * q^56 - 3219688617629350 * q^57 + 33593423193920552 * q^58 - 338601629983507156 * q^59 - 602000063011577726 * q^60 - 19162753603253692 * q^61 + 15505488837731256 * q^62 + 441133809282344796 * q^63 + 1575599177864752889 * q^64 + 991696997985687508 * q^65 - 223449751457078784 * q^66 + 789160046018441716 * q^67 + 793527065771503959 * q^68 + 1895066450986425554 * q^69 + 1371841313181399546 * q^70 + 901635006217520756 * q^71 + 4179441926330901705 * q^72 + 1704470931364824164 * q^73 - 936673704019194682 * q^74 + 2721299115185599300 * q^75 + 902909748895301048 * q^76 + 136895176815000502 * q^77 + 3735106483941260580 * q^78 - 384249994690730008 * q^79 - 1467637000368371240 * q^80 + 8095662645266644679 * q^81 + 6903936236738341456 * q^82 - 5730978816086659996 * q^83 - 10890678559479540867 * q^84 + 8583250605589725902 * q^85 + 15620118301833942618 * q^86 - 6279890617984303996 * q^87 + 17694374817110019346 * q^88 - 6455617251918472412 * q^89 - 170944452240128902 * q^90 - 1869573151265783134 * q^91 - 9347654291170404076 * q^92 + 11297106145565168410 * q^93 - 634546978249268889 * q^94 + 10375725689895196340 * q^95 + 38268359616807245724 * q^96 + 39090934998462984208 * q^97 + 41754136007920530767 * q^98 + 26521922158952497062 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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