LMFDB - Embedded newform 4006.2.a.h.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4006,2,Mod(1,4006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4006, 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("4006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4006 = 2 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4006.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:	$31.9880710497$
Analytic rank:	$0$
Dimension:	$42$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	4006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.79722 q^{3} +1.00000 q^{4} +2.06703 q^{5} +2.79722 q^{6} +3.37710 q^{7} -1.00000 q^{8} +4.82443 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.79722 q^{3} +1.00000 q^{4} +2.06703 q^{5} +2.79722 q^{6} +3.37710 q^{7} -1.00000 q^{8} +4.82443 q^{9} -2.06703 q^{10} +4.36878 q^{11} -2.79722 q^{12} -4.42507 q^{13} -3.37710 q^{14} -5.78193 q^{15} +1.00000 q^{16} +1.92568 q^{17} -4.82443 q^{18} +3.03848 q^{19} +2.06703 q^{20} -9.44648 q^{21} -4.36878 q^{22} -3.72645 q^{23} +2.79722 q^{24} -0.727393 q^{25} +4.42507 q^{26} -5.10331 q^{27} +3.37710 q^{28} +5.04753 q^{29} +5.78193 q^{30} +10.7665 q^{31} -1.00000 q^{32} -12.2204 q^{33} -1.92568 q^{34} +6.98056 q^{35} +4.82443 q^{36} +10.5217 q^{37} -3.03848 q^{38} +12.3779 q^{39} -2.06703 q^{40} +4.85937 q^{41} +9.44648 q^{42} -7.83453 q^{43} +4.36878 q^{44} +9.97222 q^{45} +3.72645 q^{46} -6.31541 q^{47} -2.79722 q^{48} +4.40481 q^{49} +0.727393 q^{50} -5.38655 q^{51} -4.42507 q^{52} -3.65845 q^{53} +5.10331 q^{54} +9.03040 q^{55} -3.37710 q^{56} -8.49928 q^{57} -5.04753 q^{58} +13.3946 q^{59} -5.78193 q^{60} +3.64557 q^{61} -10.7665 q^{62} +16.2926 q^{63} +1.00000 q^{64} -9.14675 q^{65} +12.2204 q^{66} +8.76486 q^{67} +1.92568 q^{68} +10.4237 q^{69} -6.98056 q^{70} +3.75773 q^{71} -4.82443 q^{72} -5.20592 q^{73} -10.5217 q^{74} +2.03468 q^{75} +3.03848 q^{76} +14.7538 q^{77} -12.3779 q^{78} -5.14654 q^{79} +2.06703 q^{80} -0.198196 q^{81} -4.85937 q^{82} -0.713442 q^{83} -9.44648 q^{84} +3.98044 q^{85} +7.83453 q^{86} -14.1190 q^{87} -4.36878 q^{88} +6.77175 q^{89} -9.97222 q^{90} -14.9439 q^{91} -3.72645 q^{92} -30.1163 q^{93} +6.31541 q^{94} +6.28062 q^{95} +2.79722 q^{96} +12.5802 q^{97} -4.40481 q^{98} +21.0769 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9}+O(q^{10}) $ 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9	\( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9} - 27 q^{10} + 23 q^{11} + 15 q^{13} + 10 q^{14} + 6 q^{15} + 42 q^{16} + 14 q^{17} - 56 q^{18} - 4 q^{19} + 27 q^{20} + 26 q^{21} - 23 q^{22} + 12 q^{23} + 45 q^{25} - 15 q^{26} - 3 q^{27} - 10 q^{28} + 41 q^{29} - 6 q^{30} + 18 q^{31} - 42 q^{32} + 25 q^{33} - 14 q^{34} + 8 q^{35} + 56 q^{36} + 33 q^{37} + 4 q^{38} + 10 q^{39} - 27 q^{40} + 84 q^{41} - 26 q^{42} - 36 q^{43} + 23 q^{44} + 66 q^{45} - 12 q^{46} + 28 q^{47} + 58 q^{49} - 45 q^{50} + 17 q^{51} + 15 q^{52} + 68 q^{53} + 3 q^{54} - 28 q^{55} + 10 q^{56} - 9 q^{57} - 41 q^{58} + 59 q^{59} + 6 q^{60} + 41 q^{61} - 18 q^{62} - 28 q^{63} + 42 q^{64} + 44 q^{65} - 25 q^{66} + 14 q^{68} + 67 q^{69} - 8 q^{70} + 69 q^{71} - 56 q^{72} - 27 q^{73} - 33 q^{74} + 14 q^{75} - 4 q^{76} + 43 q^{77} - 10 q^{78} - 19 q^{79} + 27 q^{80} + 74 q^{81} - 84 q^{82} + 20 q^{83} + 26 q^{84} + 16 q^{85} + 36 q^{86} - 28 q^{87} - 23 q^{88} + 123 q^{89} - 66 q^{90} - 9 q^{91} + 12 q^{92} + 48 q^{93} - 28 q^{94} + 28 q^{95} + 10 q^{97} - 58 q^{98} + 75 q^{99}+O(q^{100}) \) 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9 - 27 * q^10 + 23 * q^11 + 15 * q^13 + 10 * q^14 + 6 * q^15 + 42 * q^16 + 14 * q^17 - 56 * q^18 - 4 * q^19 + 27 * q^20 + 26 * q^21 - 23 * q^22 + 12 * q^23 + 45 * q^25 - 15 * q^26 - 3 * q^27 - 10 * q^28 + 41 * q^29 - 6 * q^30 + 18 * q^31 - 42 * q^32 + 25 * q^33 - 14 * q^34 + 8 * q^35 + 56 * q^36 + 33 * q^37 + 4 * q^38 + 10 * q^39 - 27 * q^40 + 84 * q^41 - 26 * q^42 - 36 * q^43 + 23 * q^44 + 66 * q^45 - 12 * q^46 + 28 * q^47 + 58 * q^49 - 45 * q^50 + 17 * q^51 + 15 * q^52 + 68 * q^53 + 3 * q^54 - 28 * q^55 + 10 * q^56 - 9 * q^57 - 41 * q^58 + 59 * q^59 + 6 * q^60 + 41 * q^61 - 18 * q^62 - 28 * q^63 + 42 * q^64 + 44 * q^65 - 25 * q^66 + 14 * q^68 + 67 * q^69 - 8 * q^70 + 69 * q^71 - 56 * q^72 - 27 * q^73 - 33 * q^74 + 14 * q^75 - 4 * q^76 + 43 * q^77 - 10 * q^78 - 19 * q^79 + 27 * q^80 + 74 * q^81 - 84 * q^82 + 20 * q^83 + 26 * q^84 + 16 * q^85 + 36 * q^86 - 28 * q^87 - 23 * q^88 + 123 * q^89 - 66 * q^90 - 9 * q^91 + 12 * q^92 + 48 * q^93 - 28 * q^94 + 28 * q^95 + 10 * q^97 - 58 * q^98 + 75 * 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
$3$	−2.79722	−1.61497	−0.807487	−	0.589885i	$-0.799173\pi$
$3$	−2.79722	−1.61497	−0.807487	+	0.589885i	$0.799173\pi$
$4$	1.00000	0.500000
$5$	2.06703	0.924403	0.462202	−	0.886775i	$-0.347060\pi$
$5$	2.06703	0.924403	0.462202	+	0.886775i	$0.347060\pi$
$6$	2.79722	1.14196
$7$	3.37710	1.27642	0.638212	−	0.769861i	$-0.279674\pi$
$7$	3.37710	1.27642	0.638212	+	0.769861i	$0.279674\pi$
$8$	−1.00000	−0.353553
$9$	4.82443	1.60814
$10$	−2.06703	−0.653652
$11$	4.36878	1.31724	0.658619	−	0.752477i	$-0.271141\pi$
$11$	4.36878	1.31724	0.658619	+	0.752477i	$0.271141\pi$
$12$	−2.79722	−0.807487
$13$	−4.42507	−1.22729	−0.613647	−	0.789581i	$-0.710298\pi$
$13$	−4.42507	−1.22729	−0.613647	+	0.789581i	$0.710298\pi$
$14$	−3.37710	−0.902568
$15$	−5.78193	−1.49289
$16$	1.00000	0.250000
$17$	1.92568	0.467046	0.233523	−	0.972351i	$-0.424975\pi$
$17$	1.92568	0.467046	0.233523	+	0.972351i	$0.424975\pi$
$18$	−4.82443	−1.13713
$19$	3.03848	0.697074	0.348537	−	0.937295i	$-0.386679\pi$
$19$	3.03848	0.697074	0.348537	+	0.937295i	$0.386679\pi$
$20$	2.06703	0.462202
$21$	−9.44648	−2.06139
$22$	−4.36878	−0.931428
$23$	−3.72645	−0.777019	−0.388510	−	0.921445i	$-0.627010\pi$
$23$	−3.72645	−0.777019	−0.388510	+	0.921445i	$0.627010\pi$
$24$	2.79722	0.570980
$25$	−0.727393	−0.145479
$26$	4.42507	0.867828
$27$	−5.10331	−0.982133
$28$	3.37710	0.638212
$29$	5.04753	0.937303	0.468651	−	0.883383i	$-0.344740\pi$
$29$	5.04753	0.937303	0.468651	+	0.883383i	$0.344740\pi$
$30$	5.78193	1.05563
$31$	10.7665	1.93372	0.966861	−	0.255302i	$-0.0821748\pi$
$31$	10.7665	1.93372	0.966861	+	0.255302i	$0.0821748\pi$
$32$	−1.00000	−0.176777
$33$	−12.2204	−2.12731
$34$	−1.92568	−0.330252
$35$	6.98056	1.17993
$36$	4.82443	0.804071
$37$	10.5217	1.72976	0.864878	−	0.501982i	$-0.167396\pi$
$37$	10.5217	1.72976	0.864878	+	0.501982i	$0.167396\pi$
$38$	−3.03848	−0.492906
$39$	12.3779	1.98205
$40$	−2.06703	−0.326826
$41$	4.85937	0.758906	0.379453	−	0.925211i	$-0.376112\pi$
$41$	4.85937	0.758906	0.379453	+	0.925211i	$0.376112\pi$
$42$	9.44648	1.45762
$43$	−7.83453	−1.19476	−0.597378	−	0.801960i	$-0.703791\pi$
$43$	−7.83453	−1.19476	−0.597378	+	0.801960i	$0.703791\pi$
$44$	4.36878	0.658619
$45$	9.97222	1.48657
$46$	3.72645	0.549436
$47$	−6.31541	−0.921197	−0.460599	−	0.887608i	$-0.652365\pi$
$47$	−6.31541	−0.921197	−0.460599	+	0.887608i	$0.652365\pi$
$48$	−2.79722	−0.403744
$49$	4.40481	0.629258
$50$	0.727393	0.102869
$51$	−5.38655	−0.754268
$52$	−4.42507	−0.613647
$53$	−3.65845	−0.502527	−0.251264	−	0.967919i	$-0.580846\pi$
$53$	−3.65845	−0.502527	−0.251264	+	0.967919i	$0.580846\pi$
$54$	5.10331	0.694473
$55$	9.03040	1.21766
$56$	−3.37710	−0.451284
$57$	−8.49928	−1.12576
$58$	−5.04753	−0.662773
$59$	13.3946	1.74383	0.871916	−	0.489655i	$-0.162877\pi$
$59$	13.3946	1.74383	0.871916	+	0.489655i	$0.162877\pi$
$60$	−5.78193	−0.746444
$61$	3.64557	0.466767	0.233383	−	0.972385i	$-0.425020\pi$
$61$	3.64557	0.466767	0.233383	+	0.972385i	$0.425020\pi$
$62$	−10.7665	−1.36735
$63$	16.2926	2.05267
$64$	1.00000	0.125000
$65$	−9.14675	−1.13451
$66$	12.2204	1.50423
$67$	8.76486	1.07080	0.535399	−	0.844599i	$-0.320161\pi$
$67$	8.76486	1.07080	0.535399	+	0.844599i	$0.320161\pi$
$68$	1.92568	0.233523
$69$	10.4237	1.25487
$70$	−6.98056	−0.834337
$71$	3.75773	0.445961	0.222980	−	0.974823i	$-0.428421\pi$
$71$	3.75773	0.445961	0.222980	+	0.974823i	$0.428421\pi$
$72$	−4.82443	−0.568564
$73$	−5.20592	−0.609306	−0.304653	−	0.952463i	$-0.598541\pi$
$73$	−5.20592	−0.609306	−0.304653	+	0.952463i	$0.598541\pi$
$74$	−10.5217	−1.22312
$75$	2.03468	0.234944
$76$	3.03848	0.348537
$77$	14.7538	1.68135
$78$	−12.3779	−1.40152
$79$	−5.14654	−0.579031	−0.289516	−	0.957173i	$-0.593494\pi$
$79$	−5.14654	−0.579031	−0.289516	+	0.957173i	$0.593494\pi$
$80$	2.06703	0.231101
$81$	−0.198196	−0.0220218
$82$	−4.85937	−0.536628
$83$	−0.713442	−0.0783105	−0.0391552	−	0.999233i	$-0.512467\pi$
$83$	−0.713442	−0.0783105	−0.0391552	+	0.999233i	$0.512467\pi$
$84$	−9.44648	−1.03070
$85$	3.98044	0.431739
$86$	7.83453	0.844820
$87$	−14.1190	−1.51372
$88$	−4.36878	−0.465714
$89$	6.77175	0.717804	0.358902	−	0.933375i	$-0.383151\pi$
$89$	6.77175	0.717804	0.358902	+	0.933375i	$0.383151\pi$
$90$	−9.97222	−1.05116
$91$	−14.9439	−1.56655
$92$	−3.72645	−0.388510
$93$	−30.1163	−3.12291
$94$	6.31541	0.651385
$95$	6.28062	0.644378
$96$	2.79722	0.285490
$97$	12.5802	1.27733	0.638664	−	0.769486i	$-0.279488\pi$
$97$	12.5802	1.27733	0.638664	+	0.769486i	$0.279488\pi$
$98$	−4.40481	−0.444953
$99$	21.0769	2.11831
$100$	−0.727393	−0.0727393
$101$	−13.7178	−1.36498	−0.682488	−	0.730896i	$-0.739102\pi$
$101$	−13.7178	−1.36498	−0.682488	+	0.730896i	$0.739102\pi$
$102$	5.38655	0.533348
$103$	−17.2627	−1.70095	−0.850473	−	0.526019i	$-0.823684\pi$
$103$	−17.2627	−1.70095	−0.850473	+	0.526019i	$0.823684\pi$
$104$	4.42507	0.433914
$105$	−19.5262	−1.90556
$106$	3.65845	0.355341
$107$	−17.8933	−1.72981	−0.864905	−	0.501935i	$-0.832621\pi$
$107$	−17.8933	−1.72981	−0.864905	+	0.501935i	$0.832621\pi$
$108$	−5.10331	−0.491067
$109$	2.35211	0.225292	0.112646	−	0.993635i	$-0.464067\pi$
$109$	2.35211	0.225292	0.112646	+	0.993635i	$0.464067\pi$
$110$	−9.03040	−0.861015
$111$	−29.4315	−2.79351
$112$	3.37710	0.319106
$113$	14.2591	1.34138	0.670692	−	0.741736i	$-0.265997\pi$
$113$	14.2591	1.34138	0.670692	+	0.741736i	$0.265997\pi$
$114$	8.49928	0.796030
$115$	−7.70269	−0.718279
$116$	5.04753	0.468651
$117$	−21.3484	−1.97366
$118$	−13.3946	−1.23308
$119$	6.50322	0.596149
$120$	5.78193	0.527815
$121$	8.08627	0.735115
$122$	−3.64557	−0.330054
$123$	−13.5927	−1.22561
$124$	10.7665	0.966861
$125$	−11.8387	−1.05888
$126$	−16.2926	−1.45146
$127$	−13.1134	−1.16363	−0.581814	−	0.813322i	$-0.697657\pi$
$127$	−13.1134	−1.16363	−0.581814	+	0.813322i	$0.697657\pi$
$128$	−1.00000	−0.0883883
$129$	21.9149	1.92950
$130$	9.14675	0.802223
$131$	−10.9261	−0.954621	−0.477311	−	0.878735i	$-0.658388\pi$
$131$	−10.9261	−0.954621	−0.477311	+	0.878735i	$0.658388\pi$
$132$	−12.2204	−1.06365
$133$	10.2612	0.889762
$134$	−8.76486	−0.757169
$135$	−10.5487	−0.907887
$136$	−1.92568	−0.165126
$137$	9.75182	0.833154	0.416577	−	0.909100i	$-0.363230\pi$
$137$	9.75182	0.833154	0.416577	+	0.909100i	$0.363230\pi$
$138$	−10.4237	−0.887324
$139$	−13.5504	−1.14933	−0.574667	−	0.818388i	$-0.694868\pi$
$139$	−13.5504	−1.14933	−0.574667	+	0.818388i	$0.694868\pi$
$140$	6.98056	0.589965
$141$	17.6656	1.48771
$142$	−3.75773	−0.315342
$143$	−19.3322	−1.61664
$144$	4.82443	0.402035
$145$	10.4334	0.866446
$146$	5.20592	0.430845
$147$	−12.3212	−1.01624
$148$	10.5217	0.864878
$149$	−1.06936	−0.0876052	−0.0438026	−	0.999040i	$-0.513947\pi$
$149$	−1.06936	−0.0876052	−0.0438026	+	0.999040i	$0.513947\pi$
$150$	−2.03468	−0.166131
$151$	−7.37076	−0.599824	−0.299912	−	0.953967i	$-0.596957\pi$
$151$	−7.37076	−0.599824	−0.299912	+	0.953967i	$0.596957\pi$
$152$	−3.03848	−0.246453
$153$	9.29031	0.751077
$154$	−14.7538	−1.18890
$155$	22.2547	1.78754
$156$	12.3779	0.991024
$157$	2.60808	0.208147	0.104074	−	0.994570i	$-0.466812\pi$
$157$	2.60808	0.208147	0.104074	+	0.994570i	$0.466812\pi$
$158$	5.14654	0.409437
$159$	10.2335	0.811569
$160$	−2.06703	−0.163413
$161$	−12.5846	−0.991806
$162$	0.198196	0.0155718
$163$	20.9818	1.64342	0.821712	−	0.569903i	$-0.193019\pi$
$163$	20.9818	1.64342	0.821712	+	0.569903i	$0.193019\pi$
$164$	4.85937	0.379453
$165$	−25.2600	−1.96649
$166$	0.713442	0.0553739
$167$	7.28999	0.564116	0.282058	−	0.959397i	$-0.408983\pi$
$167$	7.28999	0.564116	0.282058	+	0.959397i	$0.408983\pi$
$168$	9.44648	0.728812
$169$	6.58125	0.506250
$170$	−3.98044	−0.305286
$171$	14.6589	1.12099
$172$	−7.83453	−0.597378
$173$	3.52798	0.268227	0.134114	−	0.990966i	$-0.457181\pi$
$173$	3.52798	0.268227	0.134114	+	0.990966i	$0.457181\pi$
$174$	14.1190	1.07036
$175$	−2.45648	−0.185692
$176$	4.36878	0.329309
$177$	−37.4677	−2.81625
$178$	−6.77175	−0.507564
$179$	−21.2401	−1.58756	−0.793782	−	0.608203i	$-0.791891\pi$
$179$	−21.2401	−1.58756	−0.793782	+	0.608203i	$0.791891\pi$
$180$	9.97222	0.743286
$181$	13.5544	1.00749	0.503746	−	0.863852i	$-0.331955\pi$
$181$	13.5544	1.00749	0.503746	+	0.863852i	$0.331955\pi$
$182$	14.9439	1.10772
$183$	−10.1974	−0.753817
$184$	3.72645	0.274718
$185$	21.7486	1.59899
$186$	30.1163	2.20823
$187$	8.41289	0.615211
$188$	−6.31541	−0.460599
$189$	−17.2344	−1.25362
$190$	−6.28062	−0.455644
$191$	13.2775	0.960725	0.480362	−	0.877070i	$-0.340505\pi$
$191$	13.2775	0.960725	0.480362	+	0.877070i	$0.340505\pi$
$192$	−2.79722	−0.201872
$193$	4.53208	0.326226	0.163113	−	0.986607i	$-0.447846\pi$
$193$	4.53208	0.326226	0.163113	+	0.986607i	$0.447846\pi$
$194$	−12.5802	−0.903207
$195$	25.5854	1.83221
$196$	4.40481	0.314629
$197$	21.3221	1.51914	0.759570	−	0.650426i	$-0.225409\pi$
$197$	21.3221	1.51914	0.759570	+	0.650426i	$0.225409\pi$
$198$	−21.0769	−1.49787
$199$	−5.99105	−0.424694	−0.212347	−	0.977194i	$-0.568111\pi$
$199$	−5.99105	−0.424694	−0.212347	+	0.977194i	$0.568111\pi$
$200$	0.727393	0.0514345
$201$	−24.5172	−1.72931
$202$	13.7178	0.965184
$203$	17.0460	1.19640
$204$	−5.38655	−0.377134
$205$	10.0445	0.701535
$206$	17.2627	1.20275
$207$	−17.9780	−1.24956
$208$	−4.42507	−0.306823
$209$	13.2744	0.918213
$210$	19.5262	1.34743
$211$	−12.2269	−0.841732	−0.420866	−	0.907123i	$-0.638274\pi$
$211$	−12.2269	−0.841732	−0.420866	+	0.907123i	$0.638274\pi$
$212$	−3.65845	−0.251264
$213$	−10.5112	−0.720215
$214$	17.8933	1.22316
$215$	−16.1942	−1.10444
$216$	5.10331	0.347237
$217$	36.3596	2.46825
$218$	−2.35211	−0.159305
$219$	14.5621	0.984014
$220$	9.03040	0.608829
$221$	−8.52128	−0.573203
$222$	29.4315	1.97531
$223$	7.43993	0.498214	0.249107	−	0.968476i	$-0.419863\pi$
$223$	7.43993	0.498214	0.249107	+	0.968476i	$0.419863\pi$
$224$	−3.37710	−0.225642
$225$	−3.50925	−0.233950
$226$	−14.2591	−0.948501
$227$	−9.96949	−0.661699	−0.330849	−	0.943684i	$-0.607335\pi$
$227$	−9.96949	−0.661699	−0.330849	+	0.943684i	$0.607335\pi$
$228$	−8.49928	−0.562879
$229$	−4.26985	−0.282160	−0.141080	−	0.989998i	$-0.545057\pi$
$229$	−4.26985	−0.282160	−0.141080	+	0.989998i	$0.545057\pi$
$230$	7.70269	0.507900
$231$	−41.2696	−2.71534
$232$	−5.04753	−0.331387
$233$	−13.0789	−0.856824	−0.428412	−	0.903584i	$-0.640927\pi$
$233$	−13.0789	−0.856824	−0.428412	+	0.903584i	$0.640927\pi$
$234$	21.3484	1.39559
$235$	−13.0541	−0.851558
$236$	13.3946	0.871916
$237$	14.3960	0.935121
$238$	−6.50322	−0.421541
$239$	−11.7040	−0.757066	−0.378533	−	0.925588i	$-0.623571\pi$
$239$	−11.7040	−0.757066	−0.378533	+	0.925588i	$0.623571\pi$
$240$	−5.78193	−0.373222
$241$	−20.8067	−1.34027	−0.670137	−	0.742237i	$-0.733765\pi$
$241$	−20.8067	−1.34027	−0.670137	+	0.742237i	$0.733765\pi$
$242$	−8.08627	−0.519805
$243$	15.8643	1.01770
$244$	3.64557	0.233383
$245$	9.10487	0.581689
$246$	13.5927	0.866640
$247$	−13.4455	−0.855515
$248$	−10.7665	−0.683674
$249$	1.99565	0.126469
$250$	11.8387	0.748744
$251$	−0.0333421	−0.00210453	−0.00105227	−	0.999999i	$-0.500335\pi$
$251$	−0.0333421	−0.00210453	−0.00105227	+	0.999999i	$0.500335\pi$
$252$	16.2926	1.02634
$253$	−16.2801	−1.02352
$254$	13.1134	0.822809
$255$	−11.1342	−0.697248
$256$	1.00000	0.0625000
$257$	22.5321	1.40551	0.702756	−	0.711431i	$-0.251952\pi$
$257$	22.5321	1.40551	0.702756	+	0.711431i	$0.251952\pi$
$258$	−21.9149	−1.36436
$259$	35.5328	2.20790
$260$	−9.14675	−0.567257
$261$	24.3514	1.50732
$262$	10.9261	0.675019
$263$	27.3819	1.68844	0.844219	−	0.535999i	$-0.180065\pi$
$263$	27.3819	1.68844	0.844219	+	0.535999i	$0.180065\pi$
$264$	12.2204	0.752116
$265$	−7.56213	−0.464538
$266$	−10.2612	−0.629157
$267$	−18.9420	−1.15923
$268$	8.76486	0.535399
$269$	26.1036	1.59156	0.795782	−	0.605584i	$-0.207060\pi$
$269$	26.1036	1.59156	0.795782	+	0.605584i	$0.207060\pi$
$270$	10.5487	0.641973
$271$	1.99195	0.121003	0.0605013	−	0.998168i	$-0.480730\pi$
$271$	1.99195	0.121003	0.0605013	+	0.998168i	$0.480730\pi$
$272$	1.92568	0.116762
$273$	41.8014	2.52993
$274$	−9.75182	−0.589129
$275$	−3.17782	−0.191630
$276$	10.4237	0.627433
$277$	22.3367	1.34208	0.671040	−	0.741421i	$-0.265848\pi$
$277$	22.3367	1.34208	0.671040	+	0.741421i	$0.265848\pi$
$278$	13.5504	0.812701
$279$	51.9422	3.10970
$280$	−6.98056	−0.417168
$281$	3.89863	0.232573	0.116286	−	0.993216i	$-0.462901\pi$
$281$	3.89863	0.232573	0.116286	+	0.993216i	$0.462901\pi$
$282$	−17.6656	−1.05197
$283$	19.3017	1.14737	0.573683	−	0.819077i	$-0.305514\pi$
$283$	19.3017	1.14737	0.573683	+	0.819077i	$0.305514\pi$
$284$	3.75773	0.222980
$285$	−17.5683	−1.04065
$286$	19.3322	1.14314
$287$	16.4106	0.968686
$288$	−4.82443	−0.284282
$289$	−13.2917	−0.781868
$290$	−10.4334	−0.612670
$291$	−35.1896	−2.06285
$292$	−5.20592	−0.304653
$293$	−22.8236	−1.33337	−0.666683	−	0.745341i	$-0.732287\pi$
$293$	−22.8236	−1.33337	−0.666683	+	0.745341i	$0.732287\pi$
$294$	12.3212	0.718587
$295$	27.6871	1.61200
$296$	−10.5217	−0.611561
$297$	−22.2953	−1.29370
$298$	1.06936	0.0619463
$299$	16.4898	0.953631
$300$	2.03468	0.117472
$301$	−26.4580	−1.52501
$302$	7.37076	0.424139
$303$	38.3718	2.20440
$304$	3.03848	0.174269
$305$	7.53549	0.431481
$306$	−9.29031	−0.531092
$307$	−6.82058	−0.389271	−0.194635	−	0.980876i	$-0.562352\pi$
$307$	−6.82058	−0.389271	−0.194635	+	0.980876i	$0.562352\pi$
$308$	14.7538	0.840677
$309$	48.2876	2.74698
$310$	−22.2547	−1.26398
$311$	−12.0152	−0.681317	−0.340658	−	0.940187i	$-0.610650\pi$
$311$	−12.0152	−0.681317	−0.340658	+	0.940187i	$0.610650\pi$
$312$	−12.3779	−0.700760
$313$	−15.0406	−0.850146	−0.425073	−	0.905159i	$-0.639752\pi$
$313$	−15.0406	−0.850146	−0.425073	+	0.905159i	$0.639752\pi$
$314$	−2.60808	−0.147182
$315$	33.6772	1.89750
$316$	−5.14654	−0.289516
$317$	18.6627	1.04820	0.524100	−	0.851657i	$-0.324402\pi$
$317$	18.6627	1.04820	0.524100	+	0.851657i	$0.324402\pi$
$318$	−10.2335	−0.573866
$319$	22.0516	1.23465
$320$	2.06703	0.115550
$321$	50.0514	2.79360
$322$	12.5846	0.701313
$323$	5.85114	0.325566
$324$	−0.198196	−0.0110109
$325$	3.21877	0.178545
$326$	−20.9818	−1.16208
$327$	−6.57937	−0.363840
$328$	−4.85937	−0.268314
$329$	−21.3278	−1.17584
$330$	25.2600	1.39052
$331$	−1.78793	−0.0982735	−0.0491368	−	0.998792i	$-0.515647\pi$
$331$	−1.78793	−0.0982735	−0.0491368	+	0.998792i	$0.515647\pi$
$332$	−0.713442	−0.0391552
$333$	50.7611	2.78169
$334$	−7.28999	−0.398890
$335$	18.1172	0.989850
$336$	−9.44648	−0.515348
$337$	3.46854	0.188943	0.0944717	−	0.995528i	$-0.469884\pi$
$337$	3.46854	0.188943	0.0944717	+	0.995528i	$0.469884\pi$
$338$	−6.58125	−0.357973
$339$	−39.8858	−2.16630
$340$	3.98044	0.215870
$341$	47.0366	2.54717
$342$	−14.6589	−0.792663
$343$	−8.76422	−0.473223
$344$	7.83453	0.422410
$345$	21.5461	1.16000
$346$	−3.52798	−0.189665
$347$	−1.97064	−0.105789	−0.0528947	−	0.998600i	$-0.516845\pi$
$347$	−1.97064	−0.105789	−0.0528947	+	0.998600i	$0.516845\pi$
$348$	−14.1190	−0.756860
$349$	5.17492	0.277007	0.138504	−	0.990362i	$-0.455771\pi$
$349$	5.17492	0.277007	0.138504	+	0.990362i	$0.455771\pi$
$350$	2.45648	0.131304
$351$	22.5825	1.20537
$352$	−4.36878	−0.232857
$353$	12.3417	0.656885	0.328442	−	0.944524i	$-0.393476\pi$
$353$	12.3417	0.656885	0.328442	+	0.944524i	$0.393476\pi$
$354$	37.4677	1.99139
$355$	7.76734	0.412248
$356$	6.77175	0.358902
$357$	−18.1909	−0.962766
$358$	21.2401	1.12258
$359$	−36.9904	−1.95228	−0.976140	−	0.217140i	$-0.930327\pi$
$359$	−36.9904	−1.95228	−0.976140	+	0.217140i	$0.930327\pi$
$360$	−9.97222	−0.525582
$361$	−9.76766	−0.514087
$362$	−13.5544	−0.712404
$363$	−22.6191	−1.18719
$364$	−14.9439	−0.783274
$365$	−10.7608	−0.563245
$366$	10.1974	0.533029
$367$	−34.2125	−1.78588	−0.892939	−	0.450178i	$-0.851361\pi$
$367$	−34.2125	−1.78588	−0.892939	+	0.450178i	$0.851361\pi$
$368$	−3.72645	−0.194255
$369$	23.4437	1.22043
$370$	−21.7486	−1.13066
$371$	−12.3550	−0.641438
$372$	−30.1163	−1.56146
$373$	−35.1178	−1.81833	−0.909166	−	0.416434i	$-0.863280\pi$
$373$	−35.1178	−1.81833	−0.909166	+	0.416434i	$0.863280\pi$
$374$	−8.41289	−0.435020
$375$	33.1154	1.71007
$376$	6.31541	0.325692
$377$	−22.3357	−1.15035
$378$	17.2344	0.886442
$379$	4.73949	0.243451	0.121726	−	0.992564i	$-0.461157\pi$
$379$	4.73949	0.243451	0.121726	+	0.992564i	$0.461157\pi$
$380$	6.28062	0.322189
$381$	36.6811	1.87923
$382$	−13.2775	−0.679335
$383$	30.4988	1.55841	0.779207	−	0.626766i	$-0.215622\pi$
$383$	30.4988	1.55841	0.779207	+	0.626766i	$0.215622\pi$
$384$	2.79722	0.142745
$385$	30.4966	1.55425
$386$	−4.53208	−0.230677
$387$	−37.7971	−1.92134
$388$	12.5802	0.638664
$389$	32.1043	1.62775	0.813876	−	0.581039i	$-0.197354\pi$
$389$	32.1043	1.62775	0.813876	+	0.581039i	$0.197354\pi$
$390$	−25.5854	−1.29557
$391$	−7.17596	−0.362904
$392$	−4.40481	−0.222476
$393$	30.5628	1.54169
$394$	−21.3221	−1.07419
$395$	−10.6381	−0.535259
$396$	21.0769	1.05915
$397$	3.21619	0.161416	0.0807079	−	0.996738i	$-0.474282\pi$
$397$	3.21619	0.161416	0.0807079	+	0.996738i	$0.474282\pi$
$398$	5.99105	0.300304
$399$	−28.7029	−1.43694
$400$	−0.727393	−0.0363697
$401$	21.2063	1.05899	0.529496	−	0.848313i	$-0.322381\pi$
$401$	21.2063	1.05899	0.529496	+	0.848313i	$0.322381\pi$
$402$	24.5172	1.22281
$403$	−47.6426	−2.37325
$404$	−13.7178	−0.682488
$405$	−0.409677	−0.0203570
$406$	−17.0460	−0.845979
$407$	45.9670	2.27850
$408$	5.38655	0.266674
$409$	−9.06407	−0.448189	−0.224095	−	0.974567i	$-0.571942\pi$
$409$	−9.06407	−0.448189	−0.224095	+	0.974567i	$0.571942\pi$
$410$	−10.0445	−0.496060
$411$	−27.2780	−1.34552
$412$	−17.2627	−0.850473
$413$	45.2350	2.22587
$414$	17.9780	0.883570
$415$	−1.47471	−0.0723904
$416$	4.42507	0.216957
$417$	37.9035	1.85614
$418$	−13.2744	−0.649274
$419$	24.5531	1.19950	0.599748	−	0.800189i	$-0.295268\pi$
$419$	24.5531	1.19950	0.599748	+	0.800189i	$0.295268\pi$
$420$	−19.5262	−0.952779
$421$	14.3446	0.699113	0.349557	−	0.936915i	$-0.386332\pi$
$421$	14.3446	0.699113	0.349557	+	0.936915i	$0.386332\pi$
$422$	12.2269	0.595194
$423$	−30.4682	−1.48142
$424$	3.65845	0.177670
$425$	−1.40073	−0.0679453
$426$	10.5112	0.509269
$427$	12.3114	0.595793
$428$	−17.8933	−0.864905
$429$	54.0763	2.61083
$430$	16.1942	0.780954
$431$	14.8799	0.716741	0.358371	−	0.933579i	$-0.383332\pi$
$431$	14.8799	0.716741	0.358371	+	0.933579i	$0.383332\pi$
$432$	−5.10331	−0.245533
$433$	−6.71108	−0.322514	−0.161257	−	0.986912i	$-0.551555\pi$
$433$	−6.71108	−0.322514	−0.161257	+	0.986912i	$0.551555\pi$
$434$	−36.3596	−1.74532
$435$	−29.1845	−1.39929
$436$	2.35211	0.112646
$437$	−11.3227	−0.541640
$438$	−14.5621	−0.695803
$439$	−11.9360	−0.569673	−0.284837	−	0.958576i	$-0.591939\pi$
$439$	−11.9360	−0.569673	−0.284837	+	0.958576i	$0.591939\pi$
$440$	−9.03040	−0.430507
$441$	21.2507	1.01194
$442$	8.52128	0.405316
$443$	−21.1850	−1.00653	−0.503265	−	0.864132i	$-0.667868\pi$
$443$	−21.1850	−1.00653	−0.503265	+	0.864132i	$0.667868\pi$
$444$	−29.4315	−1.39676
$445$	13.9974	0.663540
$446$	−7.43993	−0.352291
$447$	2.99123	0.141480
$448$	3.37710	0.159553
$449$	18.5021	0.873170	0.436585	−	0.899663i	$-0.356188\pi$
$449$	18.5021	0.873170	0.436585	+	0.899663i	$0.356188\pi$
$450$	3.50925	0.165428
$451$	21.2295	0.999660
$452$	14.2591	0.670692
$453$	20.6176	0.968700
$454$	9.96949	0.467892
$455$	−30.8895	−1.44812
$456$	8.49928	0.398015
$457$	−33.4717	−1.56574	−0.782870	−	0.622186i	$-0.786245\pi$
$457$	−33.4717	−1.56574	−0.782870	+	0.622186i	$0.786245\pi$
$458$	4.26985	0.199517
$459$	−9.82736	−0.458702
$460$	−7.70269	−0.359140
$461$	30.6207	1.42615	0.713075	−	0.701088i	$-0.247302\pi$
$461$	30.6207	1.42615	0.713075	+	0.701088i	$0.247302\pi$
$462$	41.2696	1.92004
$463$	14.8860	0.691812	0.345906	−	0.938269i	$-0.387572\pi$
$463$	14.8860	0.691812	0.345906	+	0.938269i	$0.387572\pi$
$464$	5.04753	0.234326
$465$	−62.2512	−2.88683
$466$	13.0789	0.605866
$467$	13.4724	0.623430	0.311715	−	0.950176i	$-0.399097\pi$
$467$	13.4724	0.623430	0.311715	+	0.950176i	$0.399097\pi$
$468$	−21.3484	−0.986831
$469$	29.5998	1.36679
$470$	13.0541	0.602142
$471$	−7.29536	−0.336152
$472$	−13.3946	−0.616538
$473$	−34.2274	−1.57378
$474$	−14.3960	−0.661230
$475$	−2.21017	−0.101409
$476$	6.50322	0.298075
$477$	−17.6499	−0.808135
$478$	11.7040	0.535326
$479$	−32.4759	−1.48386	−0.741932	−	0.670475i	$-0.766090\pi$
$479$	−32.4759	−1.48386	−0.741932	+	0.670475i	$0.766090\pi$
$480$	5.78193	0.263908
$481$	−46.5592	−2.12292
$482$	20.8067	0.947717
$483$	35.2019	1.60174
$484$	8.08627	0.367558
$485$	26.0037	1.18077
$486$	−15.8643	−0.719621
$487$	16.2206	0.735028	0.367514	−	0.930018i	$-0.380209\pi$
$487$	16.2206	0.735028	0.367514	+	0.930018i	$0.380209\pi$
$488$	−3.64557	−0.165027
$489$	−58.6908	−2.65409
$490$	−9.10487	−0.411316
$491$	27.7330	1.25157	0.625785	−	0.779995i	$-0.284779\pi$
$491$	27.7330	1.25157	0.625785	+	0.779995i	$0.284779\pi$
$492$	−13.5927	−0.612807
$493$	9.71994	0.437764
$494$	13.4455	0.604941
$495$	43.5665	1.95817
$496$	10.7665	0.483431
$497$	12.6902	0.569235
$498$	−1.99565	−0.0894273
$499$	24.2484	1.08551	0.542755	−	0.839891i	$-0.317381\pi$
$499$	24.2484	1.08551	0.542755	+	0.839891i	$0.317381\pi$
$500$	−11.8387	−0.529442
$501$	−20.3917	−0.911033
$502$	0.0333421	0.00148813
$503$	−18.3994	−0.820388	−0.410194	−	0.911998i	$-0.634539\pi$
$503$	−18.3994	−0.820388	−0.410194	+	0.911998i	$0.634539\pi$
$504$	−16.2926	−0.725729
$505$	−28.3552	−1.26179
$506$	16.2801	0.723737
$507$	−18.4092	−0.817581
$508$	−13.1134	−0.581814
$509$	40.5706	1.79826	0.899129	−	0.437683i	$-0.144201\pi$
$509$	40.5706	1.79826	0.899129	+	0.437683i	$0.144201\pi$
$510$	11.1342	0.493029
$511$	−17.5809	−0.777733
$512$	−1.00000	−0.0441942
$513$	−15.5063	−0.684620
$514$	−22.5321	−0.993847
$515$	−35.6825	−1.57236
$516$	21.9149	0.964749
$517$	−27.5907	−1.21344
$518$	−35.5328	−1.56122
$519$	−9.86852	−0.433180
$520$	9.14675	0.401111
$521$	−31.0192	−1.35898	−0.679488	−	0.733687i	$-0.737798\pi$
$521$	−31.0192	−1.35898	−0.679488	+	0.733687i	$0.737798\pi$
$522$	−24.3514	−1.06583
$523$	−0.124291	−0.00543488	−0.00271744	−	0.999996i	$-0.500865\pi$
$523$	−0.124291	−0.00543488	−0.00271744	+	0.999996i	$0.500865\pi$
$524$	−10.9261	−0.477311
$525$	6.87131	0.299888
$526$	−27.3819	−1.19391
$527$	20.7329	0.903138
$528$	−12.2204	−0.531826
$529$	−9.11354	−0.396241
$530$	7.56213	0.328478
$531$	64.6214	2.80433
$532$	10.2612	0.444881
$533$	−21.5031	−0.931401
$534$	18.9420	0.819703
$535$	−36.9860	−1.59904
$536$	−8.76486	−0.378584
$537$	59.4133	2.56387
$538$	−26.1036	−1.12541
$539$	19.2437	0.828883
$540$	−10.5487	−0.453944
$541$	−12.8240	−0.551348	−0.275674	−	0.961251i	$-0.588901\pi$
$541$	−12.8240	−0.551348	−0.275674	+	0.961251i	$0.588901\pi$
$542$	−1.99195	−0.0855618
$543$	−37.9146	−1.62707
$544$	−1.92568	−0.0825629
$545$	4.86188	0.208260
$546$	−41.8014	−1.78893
$547$	8.65275	0.369965	0.184983	−	0.982742i	$-0.440777\pi$
$547$	8.65275	0.369965	0.184983	+	0.982742i	$0.440777\pi$
$548$	9.75182	0.416577
$549$	17.5878	0.750627
$550$	3.17782	0.135503
$551$	15.3368	0.653370
$552$	−10.4237	−0.443662
$553$	−17.3804	−0.739090
$554$	−22.3367	−0.948994
$555$	−60.8357	−2.58233
$556$	−13.5504	−0.574667
$557$	31.4222	1.33140	0.665701	−	0.746218i	$-0.268133\pi$
$557$	31.4222	1.33140	0.665701	+	0.746218i	$0.268133\pi$
$558$	−51.9422	−2.19889
$559$	34.6684	1.46632
$560$	6.98056	0.294983
$561$	−23.5327	−0.993550
$562$	−3.89863	−0.164454
$563$	26.5013	1.11690	0.558448	−	0.829539i	$-0.311397\pi$
$563$	26.5013	1.11690	0.558448	+	0.829539i	$0.311397\pi$
$564$	17.6656	0.743855
$565$	29.4740	1.23998
$566$	−19.3017	−0.811311
$567$	−0.669329	−0.0281092
$568$	−3.75773	−0.157671
$569$	26.4222	1.10768	0.553838	−	0.832624i	$-0.313163\pi$
$569$	26.4222	1.10768	0.553838	+	0.832624i	$0.313163\pi$
$570$	17.5683	0.735853
$571$	34.4961	1.44362	0.721808	−	0.692093i	$-0.243311\pi$
$571$	34.4961	1.44362	0.721808	+	0.692093i	$0.243311\pi$
$572$	−19.3322	−0.808319
$573$	−37.1400	−1.55155
$574$	−16.4106	−0.684964
$575$	2.71060	0.113040
$576$	4.82443	0.201018
$577$	−3.86894	−0.161066	−0.0805330	−	0.996752i	$-0.525662\pi$
$577$	−3.86894	−0.161066	−0.0805330	+	0.996752i	$0.525662\pi$
$578$	13.2917	0.552864
$579$	−12.6772	−0.526847
$580$	10.4334	0.433223
$581$	−2.40937	−0.0999574
$582$	35.1896	1.45866
$583$	−15.9830	−0.661948
$584$	5.20592	0.215422
$585$	−44.1278	−1.82446
$586$	22.8236	0.942833
$587$	−17.9842	−0.742288	−0.371144	−	0.928575i	$-0.621034\pi$
$587$	−17.9842	−0.742288	−0.371144	+	0.928575i	$0.621034\pi$
$588$	−12.3212	−0.508118
$589$	32.7138	1.34795
$590$	−27.6871	−1.13986
$591$	−59.6427	−2.45337
$592$	10.5217	0.432439
$593$	13.5137	0.554940	0.277470	−	0.960734i	$-0.410504\pi$
$593$	13.5137	0.554940	0.277470	+	0.960734i	$0.410504\pi$
$594$	22.2953	0.914786
$595$	13.4423	0.551082
$596$	−1.06936	−0.0438026
$597$	16.7583	0.685870
$598$	−16.4898	−0.674319
$599$	−32.1505	−1.31363	−0.656816	−	0.754051i	$-0.728097\pi$
$599$	−32.1505	−1.31363	−0.656816	+	0.754051i	$0.728097\pi$
$600$	−2.03468	−0.0830653
$601$	17.1238	0.698494	0.349247	−	0.937031i	$-0.386437\pi$
$601$	17.1238	0.698494	0.349247	+	0.937031i	$0.386437\pi$
$602$	26.4580	1.07835
$603$	42.2854	1.72200
$604$	−7.37076	−0.299912
$605$	16.7146	0.679543
$606$	−38.3718	−1.55875
$607$	25.0597	1.01714	0.508571	−	0.861020i	$-0.330174\pi$
$607$	25.0597	1.01714	0.508571	+	0.861020i	$0.330174\pi$
$608$	−3.03848	−0.123227
$609$	−47.6814	−1.93215
$610$	−7.53549	−0.305103
$611$	27.9461	1.13058
$612$	9.29031	0.375538
$613$	−23.1177	−0.933717	−0.466859	−	0.884332i	$-0.654614\pi$
$613$	−23.1177	−0.933717	−0.466859	+	0.884332i	$0.654614\pi$
$614$	6.82058	0.275256
$615$	−28.0965	−1.13296
$616$	−14.7538	−0.594448
$617$	−34.7812	−1.40024	−0.700119	−	0.714026i	$-0.746870\pi$
$617$	−34.7812	−1.40024	−0.700119	+	0.714026i	$0.746870\pi$
$618$	−48.2876	−1.94241
$619$	19.2977	0.775639	0.387820	−	0.921735i	$-0.373228\pi$
$619$	19.2977	0.775639	0.387820	+	0.921735i	$0.373228\pi$
$620$	22.2547	0.893770
$621$	19.0173	0.763137
$622$	12.0152	0.481764
$623$	22.8689	0.916222
$624$	12.3779	0.495512
$625$	−20.8339	−0.833357
$626$	15.0406	0.601144
$627$	−37.1315	−1.48289
$628$	2.60808	0.104074
$629$	20.2614	0.807876
$630$	−33.6772	−1.34173
$631$	6.77574	0.269738	0.134869	−	0.990863i	$-0.456939\pi$
$631$	6.77574	0.269738	0.134869	+	0.990863i	$0.456939\pi$
$632$	5.14654	0.204719
$633$	34.2012	1.35938
$634$	−18.6627	−0.741190
$635$	−27.1058	−1.07566
$636$	10.2335	0.405784
$637$	−19.4916	−0.772285
$638$	−22.0516	−0.873030
$639$	18.1289	0.717168
$640$	−2.06703	−0.0817065
$641$	−25.2145	−0.995912	−0.497956	−	0.867202i	$-0.665916\pi$
$641$	−25.2145	−0.995912	−0.497956	+	0.867202i	$0.665916\pi$
$642$	−50.0514	−1.97537
$643$	18.7940	0.741161	0.370581	−	0.928800i	$-0.379159\pi$
$643$	18.7940	0.741161	0.370581	+	0.928800i	$0.379159\pi$
$644$	−12.5846	−0.495903
$645$	45.2987	1.78364
$646$	−5.85114	−0.230210
$647$	33.7746	1.32782	0.663908	−	0.747814i	$-0.268897\pi$
$647$	33.7746	1.32782	0.663908	+	0.747814i	$0.268897\pi$
$648$	0.198196	0.00778588
$649$	58.5183	2.29704
$650$	−3.21877	−0.126250
$651$	−101.706	−3.98616
$652$	20.9818	0.821712
$653$	6.81673	0.266759	0.133380	−	0.991065i	$-0.457417\pi$
$653$	6.81673	0.266759	0.133380	+	0.991065i	$0.457417\pi$
$654$	6.57937	0.257274
$655$	−22.5846	−0.882455
$656$	4.85937	0.189727
$657$	−25.1155	−0.979851
$658$	21.3278	0.831443
$659$	−27.9848	−1.09013	−0.545067	−	0.838392i	$-0.683496\pi$
$659$	−27.9848	−1.09013	−0.545067	+	0.838392i	$0.683496\pi$
$660$	−25.2600	−0.983244
$661$	−0.644816	−0.0250804	−0.0125402	−	0.999921i	$-0.503992\pi$
$661$	−0.644816	−0.0250804	−0.0125402	+	0.999921i	$0.503992\pi$
$662$	1.78793	0.0694899
$663$	23.8359	0.925708
$664$	0.713442	0.0276869
$665$	21.2103	0.822499
$666$	−50.7611	−1.96695
$667$	−18.8094	−0.728302
$668$	7.28999	0.282058
$669$	−20.8111	−0.804603
$670$	−18.1172	−0.699929
$671$	15.9267	0.614843
$672$	9.44648	0.364406
$673$	−27.9805	−1.07857	−0.539284	−	0.842124i	$-0.681305\pi$
$673$	−27.9805	−1.07857	−0.539284	+	0.842124i	$0.681305\pi$
$674$	−3.46854	−0.133603
$675$	3.71212	0.142879
$676$	6.58125	0.253125
$677$	41.8735	1.60933	0.804665	−	0.593729i	$-0.202345\pi$
$677$	41.8735	1.60933	0.804665	+	0.593729i	$0.202345\pi$
$678$	39.8858	1.53181
$679$	42.4846	1.63041
$680$	−3.98044	−0.152643
$681$	27.8868	1.06863
$682$	−47.0366	−1.80112
$683$	−4.22495	−0.161663	−0.0808315	−	0.996728i	$-0.525758\pi$
$683$	−4.22495	−0.161663	−0.0808315	+	0.996728i	$0.525758\pi$
$684$	14.6589	0.560497
$685$	20.1573	0.770171
$686$	8.76422	0.334620
$687$	11.9437	0.455681
$688$	−7.83453	−0.298689
$689$	16.1889	0.616749
$690$	−21.5461	−0.820246
$691$	−1.58710	−0.0603761	−0.0301881	−	0.999544i	$-0.509611\pi$
$691$	−1.58710	−0.0603761	−0.0301881	+	0.999544i	$0.509611\pi$
$692$	3.52798	0.134114
$693$	71.1787	2.70386
$694$	1.97064	0.0748044
$695$	−28.0091	−1.06245
$696$	14.1190	0.535181
$697$	9.35760	0.354444
$698$	−5.17492	−0.195874
$699$	36.5844	1.38375
$700$	−2.45648	−0.0928462
$701$	18.5785	0.701699	0.350850	−	0.936432i	$-0.385893\pi$
$701$	18.5785	0.701699	0.350850	+	0.936432i	$0.385893\pi$
$702$	−22.5825	−0.852323
$703$	31.9699	1.20577
$704$	4.36878	0.164655
$705$	36.5153	1.37524
$706$	−12.3417	−0.464488
$707$	−46.3266	−1.74229
$708$	−37.4677	−1.40812
$709$	−16.2903	−0.611794	−0.305897	−	0.952065i	$-0.598956\pi$
$709$	−16.2903	−0.611794	−0.305897	+	0.952065i	$0.598956\pi$
$710$	−7.76734	−0.291503
$711$	−24.8291	−0.931165
$712$	−6.77175	−0.253782
$713$	−40.1209	−1.50254
$714$	18.1909	0.680778
$715$	−39.9602	−1.49443
$716$	−21.2401	−0.793782
$717$	32.7385	1.22264
$718$	36.9904	1.38047
$719$	−28.1701	−1.05057	−0.525283	−	0.850928i	$-0.676040\pi$
$719$	−28.1701	−1.05057	−0.525283	+	0.850928i	$0.676040\pi$
$720$	9.97222	0.371643
$721$	−58.2979	−2.17113
$722$	9.76766	0.363515
$723$	58.2007	2.16451
$724$	13.5544	0.503746
$725$	−3.67154	−0.136357
$726$	22.6191	0.839472
$727$	21.3004	0.789989	0.394994	−	0.918684i	$-0.370747\pi$
$727$	21.3004	0.789989	0.394994	+	0.918684i	$0.370747\pi$
$728$	14.9439	0.553858
$729$	−43.7814	−1.62153
$730$	10.7608	0.398274
$731$	−15.0868	−0.558006
$732$	−10.1974	−0.376908
$733$	−39.4343	−1.45654	−0.728269	−	0.685291i	$-0.759675\pi$
$733$	−39.4343	−1.45654	−0.728269	+	0.685291i	$0.759675\pi$
$734$	34.2125	1.26281
$735$	−25.4683	−0.939412
$736$	3.72645	0.137359
$737$	38.2918	1.41050
$738$	−23.4437	−0.862973
$739$	13.0984	0.481831	0.240916	−	0.970546i	$-0.422552\pi$
$739$	13.0984	0.481831	0.240916	+	0.970546i	$0.422552\pi$
$740$	21.7486	0.799496
$741$	37.6099	1.38163
$742$	12.3550	0.453565
$743$	−19.5026	−0.715481	−0.357741	−	0.933821i	$-0.616453\pi$
$743$	−19.5026	−0.715481	−0.357741	+	0.933821i	$0.616453\pi$
$744$	30.1163	1.10412
$745$	−2.21039	−0.0809826
$746$	35.1178	1.28575
$747$	−3.44195	−0.125934
$748$	8.41289	0.307606
$749$	−60.4275	−2.20797
$750$	−33.1154	−1.20920
$751$	−44.4521	−1.62208	−0.811039	−	0.584992i	$-0.801098\pi$
$751$	−44.4521	−1.62208	−0.811039	+	0.584992i	$0.801098\pi$
$752$	−6.31541	−0.230299
$753$	0.0932650	0.00339877
$754$	22.3357	0.813417
$755$	−15.2356	−0.554479
$756$	−17.2344	−0.626809
$757$	−13.2105	−0.480145	−0.240072	−	0.970755i	$-0.577171\pi$
$757$	−13.2105	−0.480145	−0.240072	+	0.970755i	$0.577171\pi$
$758$	−4.73949	−0.172146
$759$	45.5389	1.65296
$760$	−6.28062	−0.227822
$761$	14.2171	0.515371	0.257685	−	0.966229i	$-0.417040\pi$
$761$	14.2171	0.515371	0.257685	+	0.966229i	$0.417040\pi$
$762$	−36.6811	−1.32882
$763$	7.94332	0.287568
$764$	13.2775	0.480362
$765$	19.2033	0.694298
$766$	−30.4988	−1.10197
$767$	−59.2722	−2.14020
$768$	−2.79722	−0.100936
$769$	32.7454	1.18083	0.590415	−	0.807100i	$-0.298964\pi$
$769$	32.7454	1.18083	0.590415	+	0.807100i	$0.298964\pi$
$770$	−30.4966	−1.09902
$771$	−63.0271	−2.26987
$772$	4.53208	0.163113
$773$	42.2528	1.51973	0.759864	−	0.650082i	$-0.225266\pi$
$773$	42.2528	1.51973	0.759864	+	0.650082i	$0.225266\pi$
$774$	37.7971	1.35859
$775$	−7.83149	−0.281315
$776$	−12.5802	−0.451603
$777$	−99.3930	−3.56570
$778$	−32.1043	−1.15099
$779$	14.7651	0.529014
$780$	25.5854	0.916106
$781$	16.4167	0.587437
$782$	7.17596	0.256612
$783$	−25.7591	−0.920556
$784$	4.40481	0.157315
$785$	5.39097	0.192412
$786$	−30.5628	−1.09014
$787$	15.1890	0.541428	0.270714	−	0.962660i	$-0.412740\pi$
$787$	15.1890	0.541428	0.270714	+	0.962660i	$0.412740\pi$
$788$	21.3221	0.759570
$789$	−76.5930	−2.72678
$790$	10.6381	0.378485
$791$	48.1544	1.71217
$792$	−21.0769	−0.748934
$793$	−16.1319	−0.572860
$794$	−3.21619	−0.114138
$795$	21.1529	0.750217
$796$	−5.99105	−0.212347
$797$	−33.2323	−1.17715	−0.588574	−	0.808444i	$-0.700310\pi$
$797$	−33.2323	−1.17715	−0.588574	+	0.808444i	$0.700310\pi$
$798$	28.7029	1.01607
$799$	−12.1615	−0.430242
$800$	0.727393	0.0257172
$801$	32.6698	1.15433
$802$	−21.2063	−0.748820
$803$	−22.7435	−0.802601
$804$	−24.5172	−0.864656
$805$	−26.0127	−0.916829
$806$	47.6426	1.67814
$807$	−73.0174	−2.57033
$808$	13.7178	0.482592
$809$	14.7491	0.518550	0.259275	−	0.965804i	$-0.416516\pi$
$809$	14.7491	0.518550	0.259275	+	0.965804i	$0.416516\pi$
$810$	0.409677	0.0143946
$811$	16.8322	0.591059	0.295530	−	0.955334i	$-0.404504\pi$
$811$	16.8322	0.591059	0.295530	+	0.955334i	$0.404504\pi$
$812$	17.0460	0.598198
$813$	−5.57193	−0.195416
$814$	−45.9670	−1.61114
$815$	43.3701	1.51919
$816$	−5.38655	−0.188567
$817$	−23.8050	−0.832833
$818$	9.06407	0.316918
$819$	−72.0958	−2.51923
$820$	10.0445	0.350768
$821$	32.3087	1.12758	0.563790	−	0.825918i	$-0.309343\pi$
$821$	32.3087	1.12758	0.563790	+	0.825918i	$0.309343\pi$
$822$	27.2780	0.951428
$823$	−48.2403	−1.68155	−0.840776	−	0.541384i	$-0.817901\pi$
$823$	−48.2403	−1.68155	−0.840776	+	0.541384i	$0.817901\pi$
$824$	17.2627	0.601375
$825$	8.88906	0.309477
$826$	−45.2350	−1.57393
$827$	−25.3274	−0.880720	−0.440360	−	0.897821i	$-0.645149\pi$
$827$	−25.3274	−0.880720	−0.440360	+	0.897821i	$0.645149\pi$
$828$	−17.9780	−0.624779
$829$	−40.6260	−1.41100	−0.705500	−	0.708710i	$-0.749278\pi$
$829$	−40.6260	−1.41100	−0.705500	+	0.708710i	$0.749278\pi$
$830$	1.47471	0.0511878
$831$	−62.4805	−2.16743
$832$	−4.42507	−0.153412
$833$	8.48226	0.293893
$834$	−37.9035	−1.31249
$835$	15.0686	0.521471
$836$	13.2744	0.459106
$837$	−54.9449	−1.89917
$838$	−24.5531	−0.848171
$839$	48.4887	1.67402	0.837009	−	0.547190i	$-0.184302\pi$
$839$	48.4887	1.67402	0.837009	+	0.547190i	$0.184302\pi$
$840$	19.5262	0.673716
$841$	−3.52245	−0.121464
$842$	−14.3446	−0.494348
$843$	−10.9053	−0.375599
$844$	−12.2269	−0.420866
$845$	13.6036	0.467979
$846$	30.4682	1.04752
$847$	27.3081	0.938319
$848$	−3.65845	−0.125632
$849$	−53.9910	−1.85297
$850$	1.40073	0.0480446
$851$	−39.2086	−1.34405
$852$	−10.5112	−0.360108
$853$	50.5119	1.72949	0.864747	−	0.502207i	$-0.167479\pi$
$853$	50.5119	1.72949	0.864747	+	0.502207i	$0.167479\pi$
$854$	−12.3114	−0.421289
$855$	30.3004	1.03625
$856$	17.8933	0.611580
$857$	−10.8812	−0.371693	−0.185847	−	0.982579i	$-0.559503\pi$
$857$	−10.8812	−0.371693	−0.185847	+	0.982579i	$0.559503\pi$
$858$	−54.0763	−1.84613
$859$	−27.6428	−0.943160	−0.471580	−	0.881823i	$-0.656316\pi$
$859$	−27.6428	−0.943160	−0.471580	+	0.881823i	$0.656316\pi$
$860$	−16.1942	−0.552218
$861$	−45.9040	−1.56440
$862$	−14.8799	−0.506812
$863$	38.4476	1.30877	0.654386	−	0.756161i	$-0.272927\pi$
$863$	38.4476	1.30877	0.654386	+	0.756161i	$0.272927\pi$
$864$	5.10331	0.173618
$865$	7.29243	0.247950
$866$	6.71108	0.228052
$867$	37.1799	1.26270
$868$	36.3596	1.23413
$869$	−22.4841	−0.762722
$870$	29.1845	0.989446
$871$	−38.7851	−1.31418
$872$	−2.35211	−0.0796526
$873$	60.6923	2.05412
$874$	11.3227	0.382997
$875$	−39.9804	−1.35159
$876$	14.5621	0.492007
$877$	17.0757	0.576605	0.288302	−	0.957539i	$-0.406909\pi$
$877$	17.0757	0.576605	0.288302	+	0.957539i	$0.406909\pi$
$878$	11.9360	0.402820
$879$	63.8425	2.15335
$880$	9.03040	0.304415
$881$	−53.9179	−1.81654	−0.908270	−	0.418384i	$-0.862597\pi$
$881$	−53.9179	−1.81654	−0.908270	+	0.418384i	$0.862597\pi$
$882$	−21.2507	−0.715547
$883$	−42.2148	−1.42064	−0.710321	−	0.703878i	$-0.751450\pi$
$883$	−42.2148	−1.42064	−0.710321	+	0.703878i	$0.751450\pi$
$884$	−8.52128	−0.286602
$885$	−77.4468	−2.60335
$886$	21.1850	0.711724
$887$	13.8996	0.466702	0.233351	−	0.972393i	$-0.425031\pi$
$887$	13.8996	0.466702	0.233351	+	0.972393i	$0.425031\pi$
$888$	29.4315	0.987655
$889$	−44.2854	−1.48528
$890$	−13.9974	−0.469194
$891$	−0.865876	−0.0290080
$892$	7.43993	0.249107
$893$	−19.1892	−0.642143
$894$	−2.99123	−0.100042
$895$	−43.9040	−1.46755
$896$	−3.37710	−0.112821
$897$	−46.1256	−1.54009
$898$	−18.5021	−0.617424
$899$	54.3443	1.81248
$900$	−3.50925	−0.116975
$901$	−7.04502	−0.234704
$902$	−21.2295	−0.706866
$903$	74.0088	2.46286
$904$	−14.2591	−0.474251
$905$	28.0173	0.931328
$906$	−20.6176	−0.684974
$907$	−45.1865	−1.50039	−0.750196	−	0.661216i	$-0.770041\pi$
$907$	−45.1865	−1.50039	−0.750196	+	0.661216i	$0.770041\pi$
$908$	−9.96949	−0.330849
$909$	−66.1807	−2.19508
$910$	30.8895	1.02398
$911$	−16.4947	−0.546492	−0.273246	−	0.961944i	$-0.588097\pi$
$911$	−16.4947	−0.546492	−0.273246	+	0.961944i	$0.588097\pi$
$912$	−8.49928	−0.281439
$913$	−3.11687	−0.103153
$914$	33.4717	1.10715
$915$	−21.0784	−0.696831
$916$	−4.26985	−0.141080
$917$	−36.8987	−1.21850
$918$	9.82736	0.324351
$919$	9.98949	0.329523	0.164761	−	0.986333i	$-0.447315\pi$
$919$	9.98949	0.329523	0.164761	+	0.986333i	$0.447315\pi$
$920$	7.70269	0.253950
$921$	19.0786	0.628663
$922$	−30.6207	−1.00844
$923$	−16.6282	−0.547325
$924$	−41.2696	−1.35767
$925$	−7.65341	−0.251642
$926$	−14.8860	−0.489185
$927$	−83.2827	−2.73536
$928$	−5.04753	−0.165693
$929$	30.0400	0.985580	0.492790	−	0.870148i	$-0.335977\pi$
$929$	30.0400	0.985580	0.492790	+	0.870148i	$0.335977\pi$
$930$	62.2512	2.04130
$931$	13.3839	0.438640
$932$	−13.0789	−0.428412
$933$	33.6090	1.10031
$934$	−13.4724	−0.440831
$935$	17.3897	0.568703
$936$	21.3484	0.697795
$937$	−0.492754	−0.0160976	−0.00804879	−	0.999968i	$-0.502562\pi$
$937$	−0.492754	−0.0160976	−0.00804879	+	0.999968i	$0.502562\pi$
$938$	−29.5998	−0.966468
$939$	42.0719	1.37296
$940$	−13.0541	−0.425779
$941$	13.7909	0.449571	0.224786	−	0.974408i	$-0.427832\pi$
$941$	13.7909	0.449571	0.224786	+	0.974408i	$0.427832\pi$
$942$	7.29536	0.237696
$943$	−18.1082	−0.589685
$944$	13.3946	0.435958
$945$	−35.6240	−1.15885
$946$	34.2274	1.11283
$947$	−12.3180	−0.400281	−0.200141	−	0.979767i	$-0.564140\pi$
$947$	−12.3180	−0.400281	−0.200141	+	0.979767i	$0.564140\pi$
$948$	14.3960	0.467560
$949$	23.0365	0.747798
$950$	2.21017	0.0717073
$951$	−52.2036	−1.69282
$952$	−6.50322	−0.210771
$953$	18.0843	0.585807	0.292903	−	0.956142i	$-0.405379\pi$
$953$	18.0843	0.585807	0.292903	+	0.956142i	$0.405379\pi$
$954$	17.6499	0.571438
$955$	27.4449	0.888097
$956$	−11.7040	−0.378533
$957$	−61.6830	−1.99393
$958$	32.4759	1.04925
$959$	32.9329	1.06346
$960$	−5.78193	−0.186611
$961$	84.9178	2.73928
$962$	46.5592	1.50113
$963$	−86.3249	−2.78178
$964$	−20.8067	−0.670137
$965$	9.36794	0.301565
$966$	−35.2019	−1.13260
$967$	−33.7367	−1.08490	−0.542450	−	0.840088i	$-0.682503\pi$
$967$	−33.7367	−1.08490	−0.542450	+	0.840088i	$0.682503\pi$
$968$	−8.08627	−0.259903
$969$	−16.3669	−0.525781
$970$	−26.0037	−0.834927
$971$	50.4048	1.61757	0.808784	−	0.588106i	$-0.200126\pi$
$971$	50.4048	1.61757	0.808784	+	0.588106i	$0.200126\pi$
$972$	15.8643	0.508849
$973$	−45.7612	−1.46704
$974$	−16.2206	−0.519743
$975$	−9.00359	−0.288346
$976$	3.64557	0.116692
$977$	−29.9011	−0.956622	−0.478311	−	0.878191i	$-0.658751\pi$
$977$	−29.9011	−0.956622	−0.478311	+	0.878191i	$0.658751\pi$
$978$	58.6908	1.87672
$979$	29.5843	0.945518
$980$	9.10487	0.290844
$981$	11.3476	0.362301
$982$	−27.7330	−0.884994
$983$	2.21261	0.0705712	0.0352856	−	0.999377i	$-0.488766\pi$
$983$	2.21261	0.0705712	0.0352856	+	0.999377i	$0.488766\pi$
$984$	13.5927	0.433320
$985$	44.0735	1.40430
$986$	−9.71994	−0.309546
$987$	59.6584	1.89895
$988$	−13.4455	−0.427758
$989$	29.1950	0.928348
$990$	−43.5665	−1.38463
$991$	58.5087	1.85859	0.929296	−	0.369337i	$-0.120415\pi$
$991$	58.5087	1.85859	0.929296	+	0.369337i	$0.120415\pi$
$992$	−10.7665	−0.341837
$993$	5.00123	0.158709
$994$	−12.6902	−0.402510
$995$	−12.3837	−0.392589
$996$	1.99565	0.0632347
$997$	16.5850	0.525252	0.262626	−	0.964898i	$-0.415411\pi$
$997$	16.5850	0.525252	0.262626	+	0.964898i	$0.415411\pi$
$998$	−24.2484	−0.767571
$999$	−53.6955	−1.69885

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.h.1.6	✓	42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.h.1.6	✓	42	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:	\( N \)	\(=\)	\( 4006 = 2 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.79722 q^{3} +1.00000 q^{4} +2.06703 q^{5} +2.79722 q^{6} +3.37710 q^{7} -1.00000 q^{8} +4.82443 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.79722 q^{3} +1.00000 q^{4} +2.06703 q^{5} +2.79722 q^{6} +3.37710 q^{7} -1.00000 q^{8} +4.82443 q^{9} -2.06703 q^{10} +4.36878 q^{11} -2.79722 q^{12} -4.42507 q^{13} -3.37710 q^{14} -5.78193 q^{15} +1.00000 q^{16} +1.92568 q^{17} -4.82443 q^{18} +3.03848 q^{19} +2.06703 q^{20} -9.44648 q^{21} -4.36878 q^{22} -3.72645 q^{23} +2.79722 q^{24} -0.727393 q^{25} +4.42507 q^{26} -5.10331 q^{27} +3.37710 q^{28} +5.04753 q^{29} +5.78193 q^{30} +10.7665 q^{31} -1.00000 q^{32} -12.2204 q^{33} -1.92568 q^{34} +6.98056 q^{35} +4.82443 q^{36} +10.5217 q^{37} -3.03848 q^{38} +12.3779 q^{39} -2.06703 q^{40} +4.85937 q^{41} +9.44648 q^{42} -7.83453 q^{43} +4.36878 q^{44} +9.97222 q^{45} +3.72645 q^{46} -6.31541 q^{47} -2.79722 q^{48} +4.40481 q^{49} +0.727393 q^{50} -5.38655 q^{51} -4.42507 q^{52} -3.65845 q^{53} +5.10331 q^{54} +9.03040 q^{55} -3.37710 q^{56} -8.49928 q^{57} -5.04753 q^{58} +13.3946 q^{59} -5.78193 q^{60} +3.64557 q^{61} -10.7665 q^{62} +16.2926 q^{63} +1.00000 q^{64} -9.14675 q^{65} +12.2204 q^{66} +8.76486 q^{67} +1.92568 q^{68} +10.4237 q^{69} -6.98056 q^{70} +3.75773 q^{71} -4.82443 q^{72} -5.20592 q^{73} -10.5217 q^{74} +2.03468 q^{75} +3.03848 q^{76} +14.7538 q^{77} -12.3779 q^{78} -5.14654 q^{79} +2.06703 q^{80} -0.198196 q^{81} -4.85937 q^{82} -0.713442 q^{83} -9.44648 q^{84} +3.98044 q^{85} +7.83453 q^{86} -14.1190 q^{87} -4.36878 q^{88} +6.77175 q^{89} -9.97222 q^{90} -14.9439 q^{91} -3.72645 q^{92} -30.1163 q^{93} +6.31541 q^{94} +6.28062 q^{95} +2.79722 q^{96} +12.5802 q^{97} -4.40481 q^{98} +21.0769 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9}+O(q^{10}) \) 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9	\( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9} - 27 q^{10} + 23 q^{11} + 15 q^{13} + 10 q^{14} + 6 q^{15} + 42 q^{16} + 14 q^{17} - 56 q^{18} - 4 q^{19} + 27 q^{20} + 26 q^{21} - 23 q^{22} + 12 q^{23} + 45 q^{25} - 15 q^{26} - 3 q^{27} - 10 q^{28} + 41 q^{29} - 6 q^{30} + 18 q^{31} - 42 q^{32} + 25 q^{33} - 14 q^{34} + 8 q^{35} + 56 q^{36} + 33 q^{37} + 4 q^{38} + 10 q^{39} - 27 q^{40} + 84 q^{41} - 26 q^{42} - 36 q^{43} + 23 q^{44} + 66 q^{45} - 12 q^{46} + 28 q^{47} + 58 q^{49} - 45 q^{50} + 17 q^{51} + 15 q^{52} + 68 q^{53} + 3 q^{54} - 28 q^{55} + 10 q^{56} - 9 q^{57} - 41 q^{58} + 59 q^{59} + 6 q^{60} + 41 q^{61} - 18 q^{62} - 28 q^{63} + 42 q^{64} + 44 q^{65} - 25 q^{66} + 14 q^{68} + 67 q^{69} - 8 q^{70} + 69 q^{71} - 56 q^{72} - 27 q^{73} - 33 q^{74} + 14 q^{75} - 4 q^{76} + 43 q^{77} - 10 q^{78} - 19 q^{79} + 27 q^{80} + 74 q^{81} - 84 q^{82} + 20 q^{83} + 26 q^{84} + 16 q^{85} + 36 q^{86} - 28 q^{87} - 23 q^{88} + 123 q^{89} - 66 q^{90} - 9 q^{91} + 12 q^{92} + 48 q^{93} - 28 q^{94} + 28 q^{95} + 10 q^{97} - 58 q^{98} + 75 q^{99}+O(q^{100}) \) 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9 - 27 * q^10 + 23 * q^11 + 15 * q^13 + 10 * q^14 + 6 * q^15 + 42 * q^16 + 14 * q^17 - 56 * q^18 - 4 * q^19 + 27 * q^20 + 26 * q^21 - 23 * q^22 + 12 * q^23 + 45 * q^25 - 15 * q^26 - 3 * q^27 - 10 * q^28 + 41 * q^29 - 6 * q^30 + 18 * q^31 - 42 * q^32 + 25 * q^33 - 14 * q^34 + 8 * q^35 + 56 * q^36 + 33 * q^37 + 4 * q^38 + 10 * q^39 - 27 * q^40 + 84 * q^41 - 26 * q^42 - 36 * q^43 + 23 * q^44 + 66 * q^45 - 12 * q^46 + 28 * q^47 + 58 * q^49 - 45 * q^50 + 17 * q^51 + 15 * q^52 + 68 * q^53 + 3 * q^54 - 28 * q^55 + 10 * q^56 - 9 * q^57 - 41 * q^58 + 59 * q^59 + 6 * q^60 + 41 * q^61 - 18 * q^62 - 28 * q^63 + 42 * q^64 + 44 * q^65 - 25 * q^66 + 14 * q^68 + 67 * q^69 - 8 * q^70 + 69 * q^71 - 56 * q^72 - 27 * q^73 - 33 * q^74 + 14 * q^75 - 4 * q^76 + 43 * q^77 - 10 * q^78 - 19 * q^79 + 27 * q^80 + 74 * q^81 - 84 * q^82 + 20 * q^83 + 26 * q^84 + 16 * q^85 + 36 * q^86 - 28 * q^87 - 23 * q^88 + 123 * q^89 - 66 * q^90 - 9 * q^91 + 12 * q^92 + 48 * q^93 - 28 * q^94 + 28 * q^95 + 10 * q^97 - 58 * q^98 + 75 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more