LMFDB - Embedded newform 4022.2.a.e.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4022,2,Mod(1,4022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4022, 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("4022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4022 = 2 \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4022.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:	$32.1158316930$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.60420 q^{3} +1.00000 q^{4} -3.64067 q^{5} +1.60420 q^{6} +4.45692 q^{7} -1.00000 q^{8} -0.426530 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.60420 q^{3} +1.00000 q^{4} -3.64067 q^{5} +1.60420 q^{6} +4.45692 q^{7} -1.00000 q^{8} -0.426530 q^{9} +3.64067 q^{10} -2.70607 q^{11} -1.60420 q^{12} +4.50222 q^{13} -4.45692 q^{14} +5.84038 q^{15} +1.00000 q^{16} +6.55326 q^{17} +0.426530 q^{18} +6.91966 q^{19} -3.64067 q^{20} -7.14982 q^{21} +2.70607 q^{22} -5.15246 q^{23} +1.60420 q^{24} +8.25449 q^{25} -4.50222 q^{26} +5.49685 q^{27} +4.45692 q^{28} +7.35997 q^{29} -5.84038 q^{30} -10.6554 q^{31} -1.00000 q^{32} +4.34108 q^{33} -6.55326 q^{34} -16.2262 q^{35} -0.426530 q^{36} +5.79900 q^{37} -6.91966 q^{38} -7.22247 q^{39} +3.64067 q^{40} +1.40147 q^{41} +7.14982 q^{42} -1.43708 q^{43} -2.70607 q^{44} +1.55286 q^{45} +5.15246 q^{46} -8.40919 q^{47} -1.60420 q^{48} +12.8642 q^{49} -8.25449 q^{50} -10.5128 q^{51} +4.50222 q^{52} -6.35597 q^{53} -5.49685 q^{54} +9.85190 q^{55} -4.45692 q^{56} -11.1005 q^{57} -7.35997 q^{58} +6.27851 q^{59} +5.84038 q^{60} -6.69352 q^{61} +10.6554 q^{62} -1.90101 q^{63} +1.00000 q^{64} -16.3911 q^{65} -4.34108 q^{66} -4.01792 q^{67} +6.55326 q^{68} +8.26560 q^{69} +16.2262 q^{70} +14.8580 q^{71} +0.426530 q^{72} -10.5802 q^{73} -5.79900 q^{74} -13.2419 q^{75} +6.91966 q^{76} -12.0607 q^{77} +7.22247 q^{78} +9.06301 q^{79} -3.64067 q^{80} -7.53848 q^{81} -1.40147 q^{82} -2.82824 q^{83} -7.14982 q^{84} -23.8583 q^{85} +1.43708 q^{86} -11.8069 q^{87} +2.70607 q^{88} +0.614680 q^{89} -1.55286 q^{90} +20.0660 q^{91} -5.15246 q^{92} +17.0934 q^{93} +8.40919 q^{94} -25.1922 q^{95} +1.60420 q^{96} -5.73198 q^{97} -12.8642 q^{98} +1.15422 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9}+O(q^{10}) $ 46 * q - 46 * q^2 + 8 * q^3 + 46 * q^4 + 14 * q^5 - 8 * q^6 + 28 * q^7 - 46 * q^8 + 58 * q^9	\( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9} - 14 q^{10} - 6 q^{11} + 8 q^{12} + 37 q^{13} - 28 q^{14} + 9 q^{15} + 46 q^{16} + 6 q^{17} - 58 q^{18} + 18 q^{19} + 14 q^{20} + 19 q^{21} + 6 q^{22} - 4 q^{23} - 8 q^{24} + 86 q^{25} - 37 q^{26} + 32 q^{27} + 28 q^{28} + 15 q^{29} - 9 q^{30} + 18 q^{31} - 46 q^{32} + 37 q^{33} - 6 q^{34} - 2 q^{35} + 58 q^{36} + 74 q^{37} - 18 q^{38} - 3 q^{39} - 14 q^{40} - 18 q^{41} - 19 q^{42} + 25 q^{43} - 6 q^{44} + 94 q^{45} + 4 q^{46} + 18 q^{47} + 8 q^{48} + 92 q^{49} - 86 q^{50} - 10 q^{51} + 37 q^{52} + 17 q^{53} - 32 q^{54} + 37 q^{55} - 28 q^{56} + 43 q^{57} - 15 q^{58} - 24 q^{59} + 9 q^{60} + 46 q^{61} - 18 q^{62} + 80 q^{63} + 46 q^{64} + 24 q^{65} - 37 q^{66} + 61 q^{67} + 6 q^{68} + 59 q^{69} + 2 q^{70} - 8 q^{71} - 58 q^{72} + 101 q^{73} - 74 q^{74} + 34 q^{75} + 18 q^{76} + 40 q^{77} + 3 q^{78} + 9 q^{79} + 14 q^{80} + 58 q^{81} + 18 q^{82} + 18 q^{83} + 19 q^{84} + 60 q^{85} - 25 q^{86} + 20 q^{87} + 6 q^{88} - 25 q^{89} - 94 q^{90} + 51 q^{91} - 4 q^{92} + 63 q^{93} - 18 q^{94} - 31 q^{95} - 8 q^{96} + 76 q^{97} - 92 q^{98} + 21 q^{99}+O(q^{100}) \) 46 * q - 46 * q^2 + 8 * q^3 + 46 * q^4 + 14 * q^5 - 8 * q^6 + 28 * q^7 - 46 * q^8 + 58 * q^9 - 14 * q^10 - 6 * q^11 + 8 * q^12 + 37 * q^13 - 28 * q^14 + 9 * q^15 + 46 * q^16 + 6 * q^17 - 58 * q^18 + 18 * q^19 + 14 * q^20 + 19 * q^21 + 6 * q^22 - 4 * q^23 - 8 * q^24 + 86 * q^25 - 37 * q^26 + 32 * q^27 + 28 * q^28 + 15 * q^29 - 9 * q^30 + 18 * q^31 - 46 * q^32 + 37 * q^33 - 6 * q^34 - 2 * q^35 + 58 * q^36 + 74 * q^37 - 18 * q^38 - 3 * q^39 - 14 * q^40 - 18 * q^41 - 19 * q^42 + 25 * q^43 - 6 * q^44 + 94 * q^45 + 4 * q^46 + 18 * q^47 + 8 * q^48 + 92 * q^49 - 86 * q^50 - 10 * q^51 + 37 * q^52 + 17 * q^53 - 32 * q^54 + 37 * q^55 - 28 * q^56 + 43 * q^57 - 15 * q^58 - 24 * q^59 + 9 * q^60 + 46 * q^61 - 18 * q^62 + 80 * q^63 + 46 * q^64 + 24 * q^65 - 37 * q^66 + 61 * q^67 + 6 * q^68 + 59 * q^69 + 2 * q^70 - 8 * q^71 - 58 * q^72 + 101 * q^73 - 74 * q^74 + 34 * q^75 + 18 * q^76 + 40 * q^77 + 3 * q^78 + 9 * q^79 + 14 * q^80 + 58 * q^81 + 18 * q^82 + 18 * q^83 + 19 * q^84 + 60 * q^85 - 25 * q^86 + 20 * q^87 + 6 * q^88 - 25 * q^89 - 94 * q^90 + 51 * q^91 - 4 * q^92 + 63 * q^93 - 18 * q^94 - 31 * q^95 - 8 * q^96 + 76 * q^97 - 92 * q^98 + 21 * 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
$3$	−1.60420	−0.926188	−0.463094	−	0.886309i	$-0.653261\pi$
$3$	−1.60420	−0.926188	−0.463094	+	0.886309i	$0.653261\pi$
$4$	1.00000	0.500000
$5$	−3.64067	−1.62816	−0.814079	−	0.580754i	$-0.802758\pi$
$5$	−3.64067	−1.62816	−0.814079	+	0.580754i	$0.802758\pi$
$6$	1.60420	0.654913
$7$	4.45692	1.68456	0.842280	−	0.539041i	$-0.181213\pi$
$7$	4.45692	1.68456	0.842280	+	0.539041i	$0.181213\pi$
$8$	−1.00000	−0.353553
$9$	−0.426530	−0.142177
$10$	3.64067	1.15128
$11$	−2.70607	−0.815910	−0.407955	−	0.913002i	$-0.633758\pi$
$11$	−2.70607	−0.815910	−0.407955	+	0.913002i	$0.633758\pi$
$12$	−1.60420	−0.463094
$13$	4.50222	1.24869	0.624345	−	0.781149i	$-0.285366\pi$
$13$	4.50222	1.24869	0.624345	+	0.781149i	$0.285366\pi$
$14$	−4.45692	−1.19116
$15$	5.84038	1.50798
$16$	1.00000	0.250000
$17$	6.55326	1.58940	0.794700	−	0.607003i	$-0.207628\pi$
$17$	6.55326	1.58940	0.794700	+	0.607003i	$0.207628\pi$
$18$	0.426530	0.100534
$19$	6.91966	1.58748	0.793739	−	0.608258i	$-0.208132\pi$
$19$	6.91966	1.58748	0.793739	+	0.608258i	$0.208132\pi$
$20$	−3.64067	−0.814079
$21$	−7.14982	−1.56022
$22$	2.70607	0.576935
$23$	−5.15246	−1.07436	−0.537181	−	0.843467i	$-0.680511\pi$
$23$	−5.15246	−1.07436	−0.537181	+	0.843467i	$0.680511\pi$
$24$	1.60420	0.327457
$25$	8.25449	1.65090
$26$	−4.50222	−0.882957
$27$	5.49685	1.05787
$28$	4.45692	0.842280
$29$	7.35997	1.36671	0.683356	−	0.730086i	$-0.260520\pi$
$29$	7.35997	1.36671	0.683356	+	0.730086i	$0.260520\pi$
$30$	−5.84038	−1.06630
$31$	−10.6554	−1.91376	−0.956879	−	0.290485i	$-0.906183\pi$
$31$	−10.6554	−1.91376	−0.956879	+	0.290485i	$0.906183\pi$
$32$	−1.00000	−0.176777
$33$	4.34108	0.755685
$34$	−6.55326	−1.12387
$35$	−16.2262	−2.74273
$36$	−0.426530	−0.0710883
$37$	5.79900	0.953349	0.476675	−	0.879080i	$-0.341842\pi$
$37$	5.79900	0.953349	0.476675	+	0.879080i	$0.341842\pi$
$38$	−6.91966	−1.12252
$39$	−7.22247	−1.15652
$40$	3.64067	0.575641
$41$	1.40147	0.218872	0.109436	−	0.993994i	$-0.465095\pi$
$41$	1.40147	0.218872	0.109436	+	0.993994i	$0.465095\pi$
$42$	7.14982	1.10324
$43$	−1.43708	−0.219153	−0.109576	−	0.993978i	$-0.534949\pi$
$43$	−1.43708	−0.219153	−0.109576	+	0.993978i	$0.534949\pi$
$44$	−2.70607	−0.407955
$45$	1.55286	0.231486
$46$	5.15246	0.759689
$47$	−8.40919	−1.22661	−0.613304	−	0.789847i	$-0.710160\pi$
$47$	−8.40919	−1.22661	−0.613304	+	0.789847i	$0.710160\pi$
$48$	−1.60420	−0.231547
$49$	12.8642	1.83774
$50$	−8.25449	−1.16736
$51$	−10.5128	−1.47208
$52$	4.50222	0.624345
$53$	−6.35597	−0.873060	−0.436530	−	0.899690i	$-0.643793\pi$
$53$	−6.35597	−0.873060	−0.436530	+	0.899690i	$0.643793\pi$
$54$	−5.49685	−0.748027
$55$	9.85190	1.32843
$56$	−4.45692	−0.595582
$57$	−11.1005	−1.47030
$58$	−7.35997	−0.966411
$59$	6.27851	0.817393	0.408696	−	0.912670i	$-0.365984\pi$
$59$	6.27851	0.817393	0.408696	+	0.912670i	$0.365984\pi$
$60$	5.84038	0.753990
$61$	−6.69352	−0.857018	−0.428509	−	0.903538i	$-0.640961\pi$
$61$	−6.69352	−0.857018	−0.428509	+	0.903538i	$0.640961\pi$
$62$	10.6554	1.35323
$63$	−1.90101	−0.239505
$64$	1.00000	0.125000
$65$	−16.3911	−2.03306
$66$	−4.34108	−0.534350
$67$	−4.01792	−0.490867	−0.245434	−	0.969413i	$-0.578930\pi$
$67$	−4.01792	−0.490867	−0.245434	+	0.969413i	$0.578930\pi$
$68$	6.55326	0.794700
$69$	8.26560	0.995061
$70$	16.2262	1.93940
$71$	14.8580	1.76332	0.881658	−	0.471890i	$-0.156428\pi$
$71$	14.8580	1.76332	0.881658	+	0.471890i	$0.156428\pi$
$72$	0.426530	0.0502671
$73$	−10.5802	−1.23832	−0.619159	−	0.785266i	$-0.712526\pi$
$73$	−10.5802	−1.23832	−0.619159	+	0.785266i	$0.712526\pi$
$74$	−5.79900	−0.674120
$75$	−13.2419	−1.52904
$76$	6.91966	0.793739
$77$	−12.0607	−1.37445
$78$	7.22247	0.817784
$79$	9.06301	1.01967	0.509834	−	0.860273i	$-0.329707\pi$
$79$	9.06301	1.01967	0.509834	+	0.860273i	$0.329707\pi$
$80$	−3.64067	−0.407039
$81$	−7.53848	−0.837609
$82$	−1.40147	−0.154766
$83$	−2.82824	−0.310440	−0.155220	−	0.987880i	$-0.549609\pi$
$83$	−2.82824	−0.310440	−0.155220	+	0.987880i	$0.549609\pi$
$84$	−7.14982	−0.780109
$85$	−23.8583	−2.58779
$86$	1.43708	0.154964
$87$	−11.8069	−1.26583
$88$	2.70607	0.288468
$89$	0.614680	0.0651559	0.0325780	−	0.999469i	$-0.489628\pi$
$89$	0.614680	0.0651559	0.0325780	+	0.999469i	$0.489628\pi$
$90$	−1.55286	−0.163685
$91$	20.0660	2.10349
$92$	−5.15246	−0.537181
$93$	17.0934	1.77250
$94$	8.40919	0.867342
$95$	−25.1922	−2.58466
$96$	1.60420	0.163728
$97$	−5.73198	−0.581994	−0.290997	−	0.956724i	$-0.593987\pi$
$97$	−5.73198	−0.581994	−0.290997	+	0.956724i	$0.593987\pi$
$98$	−12.8642	−1.29948
$99$	1.15422	0.116003
$100$	8.25449	0.825449
$101$	7.16833	0.713275	0.356638	−	0.934243i	$-0.383923\pi$
$101$	7.16833	0.713275	0.356638	+	0.934243i	$0.383923\pi$
$102$	10.5128	1.04092
$103$	9.17116	0.903661	0.451831	−	0.892104i	$-0.350771\pi$
$103$	9.17116	0.903661	0.451831	+	0.892104i	$0.350771\pi$
$104$	−4.50222	−0.441479
$105$	26.0301	2.54028
$106$	6.35597	0.617346
$107$	3.81992	0.369285	0.184643	−	0.982806i	$-0.440887\pi$
$107$	3.81992	0.369285	0.184643	+	0.982806i	$0.440887\pi$
$108$	5.49685	0.528935
$109$	14.9136	1.42847	0.714234	−	0.699907i	$-0.246775\pi$
$109$	14.9136	1.42847	0.714234	+	0.699907i	$0.246775\pi$
$110$	−9.85190	−0.939342
$111$	−9.30278	−0.882980
$112$	4.45692	0.421140
$113$	5.68687	0.534976	0.267488	−	0.963561i	$-0.413806\pi$
$113$	5.68687	0.534976	0.267488	+	0.963561i	$0.413806\pi$
$114$	11.1005	1.03966
$115$	18.7584	1.74923
$116$	7.35997	0.683356
$117$	−1.92033	−0.177535
$118$	−6.27851	−0.577984
$119$	29.2074	2.67744
$120$	−5.84038	−0.533151
$121$	−3.67720	−0.334291
$122$	6.69352	0.606003
$123$	−2.24824	−0.202717
$124$	−10.6554	−0.956879
$125$	−11.8485	−1.05976
$126$	1.90101	0.169356
$127$	−14.8721	−1.31968	−0.659841	−	0.751406i	$-0.729376\pi$
$127$	−14.8721	−1.31968	−0.659841	+	0.751406i	$0.729376\pi$
$128$	−1.00000	−0.0883883
$129$	2.30537	0.202976
$130$	16.3911	1.43759
$131$	−19.3851	−1.69368	−0.846841	−	0.531847i	$-0.821498\pi$
$131$	−19.3851	−1.69368	−0.846841	+	0.531847i	$0.821498\pi$
$132$	4.34108	0.377843
$133$	30.8404	2.67420
$134$	4.01792	0.347095
$135$	−20.0122	−1.72238
$136$	−6.55326	−0.561937
$137$	−12.8889	−1.10118	−0.550588	−	0.834777i	$-0.685596\pi$
$137$	−12.8889	−1.10118	−0.550588	+	0.834777i	$0.685596\pi$
$138$	−8.26560	−0.703615
$139$	−13.0018	−1.10280	−0.551401	−	0.834240i	$-0.685907\pi$
$139$	−13.0018	−1.10280	−0.551401	+	0.834240i	$0.685907\pi$
$140$	−16.2262	−1.37136
$141$	13.4901	1.13607
$142$	−14.8580	−1.24685
$143$	−12.1833	−1.01882
$144$	−0.426530	−0.0355442
$145$	−26.7952	−2.22522
$146$	10.5802	0.875623
$147$	−20.6368	−1.70209
$148$	5.79900	0.476675
$149$	−0.509423	−0.0417336	−0.0208668	−	0.999782i	$-0.506643\pi$
$149$	−0.509423	−0.0417336	−0.0208668	+	0.999782i	$0.506643\pi$
$150$	13.2419	1.08120
$151$	−4.40424	−0.358412	−0.179206	−	0.983812i	$-0.557353\pi$
$151$	−4.40424	−0.358412	−0.179206	+	0.983812i	$0.557353\pi$
$152$	−6.91966	−0.561258
$153$	−2.79516	−0.225976
$154$	12.0607	0.971882
$155$	38.7927	3.11590
$156$	−7.22247	−0.578261
$157$	0.623520	0.0497624	0.0248812	−	0.999690i	$-0.492079\pi$
$157$	0.623520	0.0497624	0.0248812	+	0.999690i	$0.492079\pi$
$158$	−9.06301	−0.721014
$159$	10.1963	0.808617
$160$	3.64067	0.287820
$161$	−22.9641	−1.80983
$162$	7.53848	0.592279
$163$	11.0241	0.863472	0.431736	−	0.902000i	$-0.357901\pi$
$163$	11.0241	0.863472	0.431736	+	0.902000i	$0.357901\pi$
$164$	1.40147	0.109436
$165$	−15.8045	−1.23038
$166$	2.82824	0.219514
$167$	9.28956	0.718848	0.359424	−	0.933174i	$-0.382973\pi$
$167$	9.28956	0.718848	0.359424	+	0.933174i	$0.382973\pi$
$168$	7.14982	0.551620
$169$	7.26996	0.559228
$170$	23.8583	1.82985
$171$	−2.95144	−0.225702
$172$	−1.43708	−0.109576
$173$	13.4824	1.02505	0.512525	−	0.858673i	$-0.328710\pi$
$173$	13.4824	1.02505	0.512525	+	0.858673i	$0.328710\pi$
$174$	11.8069	0.895078
$175$	36.7896	2.78103
$176$	−2.70607	−0.203977
$177$	−10.0720	−0.757059
$178$	−0.614680	−0.0460722
$179$	16.6376	1.24355	0.621776	−	0.783196i	$-0.286412\pi$
$179$	16.6376	1.24355	0.621776	+	0.783196i	$0.286412\pi$
$180$	1.55286	0.115743
$181$	11.4436	0.850598	0.425299	−	0.905053i	$-0.360169\pi$
$181$	11.4436	0.850598	0.425299	+	0.905053i	$0.360169\pi$
$182$	−20.0660	−1.48739
$183$	10.7378	0.793759
$184$	5.15246	0.379845
$185$	−21.1122	−1.55220
$186$	−17.0934	−1.25335
$187$	−17.7336	−1.29681
$188$	−8.40919	−0.613304
$189$	24.4991	1.78204
$190$	25.1922	1.82763
$191$	−22.1453	−1.60237	−0.801187	−	0.598413i	$-0.795798\pi$
$191$	−22.1453	−1.60237	−0.801187	+	0.598413i	$0.795798\pi$
$192$	−1.60420	−0.115773
$193$	9.15160	0.658747	0.329373	−	0.944200i	$-0.393163\pi$
$193$	9.15160	0.658747	0.329373	+	0.944200i	$0.393163\pi$
$194$	5.73198	0.411532
$195$	26.2947	1.88300
$196$	12.8642	0.918870
$197$	−15.4637	−1.10175	−0.550873	−	0.834589i	$-0.685705\pi$
$197$	−15.4637	−1.10175	−0.550873	+	0.834589i	$0.685705\pi$
$198$	−1.15422	−0.0820268
$199$	−5.71470	−0.405104	−0.202552	−	0.979271i	$-0.564924\pi$
$199$	−5.71470	−0.405104	−0.202552	+	0.979271i	$0.564924\pi$
$200$	−8.25449	−0.583680
$201$	6.44556	0.454635
$202$	−7.16833	−0.504362
$203$	32.8028	2.30231
$204$	−10.5128	−0.736041
$205$	−5.10228	−0.356359
$206$	−9.17116	−0.638985
$207$	2.19768	0.152749
$208$	4.50222	0.312173
$209$	−18.7251	−1.29524
$210$	−26.0301	−1.79625
$211$	5.08362	0.349971	0.174985	−	0.984571i	$-0.444012\pi$
$211$	5.08362	0.349971	0.174985	+	0.984571i	$0.444012\pi$
$212$	−6.35597	−0.436530
$213$	−23.8352	−1.63316
$214$	−3.81992	−0.261124
$215$	5.23193	0.356815
$216$	−5.49685	−0.374013
$217$	−47.4901	−3.22384
$218$	−14.9136	−1.01008
$219$	16.9728	1.14691
$220$	9.85190	0.664215
$221$	29.5042	1.98467
$222$	9.30278	0.624361
$223$	16.1397	1.08079	0.540396	−	0.841411i	$-0.318274\pi$
$223$	16.1397	1.08079	0.540396	+	0.841411i	$0.318274\pi$
$224$	−4.45692	−0.297791
$225$	−3.52079	−0.234719
$226$	−5.68687	−0.378285
$227$	25.2563	1.67632	0.838160	−	0.545424i	$-0.183631\pi$
$227$	25.2563	1.67632	0.838160	+	0.545424i	$0.183631\pi$
$228$	−11.1005	−0.735151
$229$	19.7728	1.30663	0.653313	−	0.757088i	$-0.273379\pi$
$229$	19.7728	1.30663	0.653313	+	0.757088i	$0.273379\pi$
$230$	−18.7584	−1.23689
$231$	19.3479	1.27300
$232$	−7.35997	−0.483205
$233$	−6.40640	−0.419697	−0.209849	−	0.977734i	$-0.567297\pi$
$233$	−6.40640	−0.419697	−0.209849	+	0.977734i	$0.567297\pi$
$234$	1.92033	0.125536
$235$	30.6151	1.99711
$236$	6.27851	0.408696
$237$	−14.5389	−0.944404
$238$	−29.2074	−1.89323
$239$	4.12843	0.267046	0.133523	−	0.991046i	$-0.457371\pi$
$239$	4.12843	0.267046	0.133523	+	0.991046i	$0.457371\pi$
$240$	5.84038	0.376995
$241$	−3.70928	−0.238936	−0.119468	−	0.992838i	$-0.538119\pi$
$241$	−3.70928	−0.238936	−0.119468	+	0.992838i	$0.538119\pi$
$242$	3.67720	0.236380
$243$	−4.39730	−0.282087
$244$	−6.69352	−0.428509
$245$	−46.8342	−2.99213
$246$	2.24824	0.143343
$247$	31.1538	1.98227
$248$	10.6554	0.676616
$249$	4.53708	0.287526
$250$	11.8485	0.749366
$251$	27.2873	1.72236	0.861178	−	0.508303i	$-0.169727\pi$
$251$	27.2873	1.72236	0.861178	+	0.508303i	$0.169727\pi$
$252$	−1.90101	−0.119753
$253$	13.9429	0.876583
$254$	14.8721	0.933156
$255$	38.2735	2.39678
$256$	1.00000	0.0625000
$257$	14.8529	0.926499	0.463249	−	0.886228i	$-0.346683\pi$
$257$	14.8529	0.926499	0.463249	+	0.886228i	$0.346683\pi$
$258$	−2.30537	−0.143526
$259$	25.8457	1.60597
$260$	−16.3911	−1.01653
$261$	−3.13925	−0.194314
$262$	19.3851	1.19761
$263$	10.4421	0.643886	0.321943	−	0.946759i	$-0.395664\pi$
$263$	10.4421	0.643886	0.321943	+	0.946759i	$0.395664\pi$
$264$	−4.34108	−0.267175
$265$	23.1400	1.42148
$266$	−30.8404	−1.89095
$267$	−0.986072	−0.0603466
$268$	−4.01792	−0.245434
$269$	−13.2912	−0.810378	−0.405189	−	0.914233i	$-0.632794\pi$
$269$	−13.2912	−0.810378	−0.405189	+	0.914233i	$0.632794\pi$
$270$	20.0122	1.21791
$271$	−20.8683	−1.26766	−0.633828	−	0.773474i	$-0.718517\pi$
$271$	−20.8683	−1.26766	−0.633828	+	0.773474i	$0.718517\pi$
$272$	6.55326	0.397350
$273$	−32.1900	−1.94823
$274$	12.8889	0.778649
$275$	−22.3372	−1.34698
$276$	8.26560	0.497531
$277$	16.1256	0.968896	0.484448	−	0.874820i	$-0.339020\pi$
$277$	16.1256	0.968896	0.484448	+	0.874820i	$0.339020\pi$
$278$	13.0018	0.779799
$279$	4.54483	0.272092
$280$	16.2262	0.969701
$281$	24.2683	1.44773	0.723864	−	0.689943i	$-0.242364\pi$
$281$	24.2683	1.44773	0.723864	+	0.689943i	$0.242364\pi$
$282$	−13.4901	−0.803321
$283$	−23.3600	−1.38861	−0.694303	−	0.719682i	$-0.744287\pi$
$283$	−23.3600	−1.38861	−0.694303	+	0.719682i	$0.744287\pi$
$284$	14.8580	0.881658
$285$	40.4134	2.39388
$286$	12.1833	0.720414
$287$	6.24623	0.368704
$288$	0.426530	0.0251335
$289$	25.9452	1.52619
$290$	26.7952	1.57347
$291$	9.19526	0.539036
$292$	−10.5802	−0.619159
$293$	−3.43044	−0.200408	−0.100204	−	0.994967i	$-0.531950\pi$
$293$	−3.43044	−0.200408	−0.100204	+	0.994967i	$0.531950\pi$
$294$	20.6368	1.20356
$295$	−22.8580	−1.33084
$296$	−5.79900	−0.337060
$297$	−14.8748	−0.863126
$298$	0.509423	0.0295101
$299$	−23.1975	−1.34155
$300$	−13.2419	−0.764520
$301$	−6.40495	−0.369175
$302$	4.40424	0.253436
$303$	−11.4995	−0.660626
$304$	6.91966	0.396870
$305$	24.3689	1.39536
$306$	2.79516	0.159789
$307$	15.6353	0.892351	0.446176	−	0.894945i	$-0.352786\pi$
$307$	15.6353	0.892351	0.446176	+	0.894945i	$0.352786\pi$
$308$	−12.0607	−0.687224
$309$	−14.7124	−0.836960
$310$	−38.7927	−2.20327
$311$	−20.1729	−1.14390	−0.571951	−	0.820288i	$-0.693813\pi$
$311$	−20.1729	−1.14390	−0.571951	+	0.820288i	$0.693813\pi$
$312$	7.22247	0.408892
$313$	22.7636	1.28667	0.643337	−	0.765583i	$-0.277550\pi$
$313$	22.7636	1.28667	0.643337	+	0.765583i	$0.277550\pi$
$314$	−0.623520	−0.0351873
$315$	6.92096	0.389952
$316$	9.06301	0.509834
$317$	25.5988	1.43777	0.718887	−	0.695127i	$-0.244652\pi$
$317$	25.5988	1.43777	0.718887	+	0.695127i	$0.244652\pi$
$318$	−10.1963	−0.571779
$319$	−19.9166	−1.11511
$320$	−3.64067	−0.203520
$321$	−6.12792	−0.342027
$322$	22.9641	1.27974
$323$	45.3463	2.52314
$324$	−7.53848	−0.418805
$325$	37.1635	2.06146
$326$	−11.0241	−0.610567
$327$	−23.9245	−1.32303
$328$	−1.40147	−0.0773831
$329$	−37.4791	−2.06629
$330$	15.8045	0.870007
$331$	−16.0190	−0.880485	−0.440242	−	0.897879i	$-0.645107\pi$
$331$	−16.0190	−0.880485	−0.440242	+	0.897879i	$0.645107\pi$
$332$	−2.82824	−0.155220
$333$	−2.47345	−0.135544
$334$	−9.28956	−0.508302
$335$	14.6279	0.799209
$336$	−7.14982	−0.390054
$337$	13.6466	0.743376	0.371688	−	0.928358i	$-0.378779\pi$
$337$	13.6466	0.743376	0.371688	+	0.928358i	$0.378779\pi$
$338$	−7.26996	−0.395434
$339$	−9.12290	−0.495488
$340$	−23.8583	−1.29390
$341$	28.8341	1.56145
$342$	2.95144	0.159596
$343$	26.1362	1.41122
$344$	1.43708	0.0774821
$345$	−30.0923	−1.62012
$346$	−13.4824	−0.724819
$347$	22.5429	1.21017	0.605084	−	0.796162i	$-0.293139\pi$
$347$	22.5429	1.21017	0.605084	+	0.796162i	$0.293139\pi$
$348$	−11.8069	−0.632915
$349$	−6.01638	−0.322049	−0.161025	−	0.986950i	$-0.551480\pi$
$349$	−6.01638	−0.322049	−0.161025	+	0.986950i	$0.551480\pi$
$350$	−36.7896	−1.96649
$351$	24.7480	1.32095
$352$	2.70607	0.144234
$353$	−25.0424	−1.33287	−0.666436	−	0.745562i	$-0.732181\pi$
$353$	−25.0424	−1.33287	−0.666436	+	0.745562i	$0.732181\pi$
$354$	10.0720	0.535321
$355$	−54.0929	−2.87096
$356$	0.614680	0.0325780
$357$	−46.8546	−2.47981
$358$	−16.6376	−0.879323
$359$	20.3889	1.07609	0.538043	−	0.842918i	$-0.319164\pi$
$359$	20.3889	1.07609	0.538043	+	0.842918i	$0.319164\pi$
$360$	−1.55286	−0.0818427
$361$	28.8817	1.52009
$362$	−11.4436	−0.601464
$363$	5.89898	0.309616
$364$	20.0660	1.05175
$365$	38.5190	2.01618
$366$	−10.7378	−0.561273
$367$	33.6930	1.75876	0.879379	−	0.476123i	$-0.157958\pi$
$367$	33.6930	1.75876	0.879379	+	0.476123i	$0.157958\pi$
$368$	−5.15246	−0.268591
$369$	−0.597768	−0.0311186
$370$	21.1122	1.09757
$371$	−28.3281	−1.47072
$372$	17.0934	0.886250
$373$	17.6279	0.912739	0.456370	−	0.889790i	$-0.349149\pi$
$373$	17.6279	0.912739	0.456370	+	0.889790i	$0.349149\pi$
$374$	17.7336	0.916980
$375$	19.0074	0.981540
$376$	8.40919	0.433671
$377$	33.1362	1.70660
$378$	−24.4991	−1.26010
$379$	−10.7009	−0.549666	−0.274833	−	0.961492i	$-0.588623\pi$
$379$	−10.7009	−0.549666	−0.274833	+	0.961492i	$0.588623\pi$
$380$	−25.1922	−1.29233
$381$	23.8578	1.22227
$382$	22.1453	1.13305
$383$	−13.1558	−0.672231	−0.336115	−	0.941821i	$-0.609113\pi$
$383$	−13.1558	−0.672231	−0.336115	+	0.941821i	$0.609113\pi$
$384$	1.60420	0.0818642
$385$	43.9092	2.23782
$386$	−9.15160	−0.465804
$387$	0.612958	0.0311584
$388$	−5.73198	−0.290997
$389$	−18.8291	−0.954672	−0.477336	−	0.878721i	$-0.658398\pi$
$389$	−18.8291	−0.954672	−0.477336	+	0.878721i	$0.658398\pi$
$390$	−26.2947	−1.33148
$391$	−33.7654	−1.70759
$392$	−12.8642	−0.649739
$393$	31.0976	1.56867
$394$	15.4637	0.779052
$395$	−32.9954	−1.66018
$396$	1.15422	0.0580017
$397$	12.7586	0.640338	0.320169	−	0.947360i	$-0.396260\pi$
$397$	12.7586	0.640338	0.320169	+	0.947360i	$0.396260\pi$
$398$	5.71470	0.286452
$399$	−49.4743	−2.47681
$400$	8.25449	0.412724
$401$	−15.6183	−0.779940	−0.389970	−	0.920828i	$-0.627515\pi$
$401$	−15.6183	−0.779940	−0.389970	+	0.920828i	$0.627515\pi$
$402$	−6.44556	−0.321475
$403$	−47.9727	−2.38969
$404$	7.16833	0.356638
$405$	27.4451	1.36376
$406$	−32.8028	−1.62798
$407$	−15.6925	−0.777847
$408$	10.5128	0.520459
$409$	2.19104	0.108340	0.0541699	−	0.998532i	$-0.482749\pi$
$409$	2.19104	0.108340	0.0541699	+	0.998532i	$0.482749\pi$
$410$	5.10228	0.251984
$411$	20.6765	1.01990
$412$	9.17116	0.451831
$413$	27.9829	1.37695
$414$	−2.19768	−0.108010
$415$	10.2967	0.505445
$416$	−4.50222	−0.220739
$417$	20.8576	1.02140
$418$	18.7251	0.915872
$419$	11.5818	0.565808	0.282904	−	0.959148i	$-0.408702\pi$
$419$	11.5818	0.565808	0.282904	+	0.959148i	$0.408702\pi$
$420$	26.0301	1.27014
$421$	17.5198	0.853865	0.426932	−	0.904284i	$-0.359594\pi$
$421$	17.5198	0.853865	0.426932	+	0.904284i	$0.359594\pi$
$422$	−5.08362	−0.247467
$423$	3.58677	0.174395
$424$	6.35597	0.308673
$425$	54.0938	2.62394
$426$	23.8352	1.15482
$427$	−29.8325	−1.44370
$428$	3.81992	0.184643
$429$	19.5445	0.943617
$430$	−5.23193	−0.252306
$431$	38.7695	1.86746	0.933732	−	0.357974i	$-0.116532\pi$
$431$	38.7695	1.86746	0.933732	+	0.357974i	$0.116532\pi$
$432$	5.49685	0.264467
$433$	−14.1437	−0.679704	−0.339852	−	0.940479i	$-0.610377\pi$
$433$	−14.1437	−0.679704	−0.339852	+	0.940479i	$0.610377\pi$
$434$	47.4901	2.27960
$435$	42.9850	2.06097
$436$	14.9136	0.714234
$437$	−35.6533	−1.70553
$438$	−16.9728	−0.810991
$439$	21.9033	1.04539	0.522694	−	0.852520i	$-0.324927\pi$
$439$	21.9033	1.04539	0.522694	+	0.852520i	$0.324927\pi$
$440$	−9.85190	−0.469671
$441$	−5.48696	−0.261284
$442$	−29.5042	−1.40337
$443$	−7.99835	−0.380013	−0.190007	−	0.981783i	$-0.560851\pi$
$443$	−7.99835	−0.380013	−0.190007	+	0.981783i	$0.560851\pi$
$444$	−9.30278	−0.441490
$445$	−2.23785	−0.106084
$446$	−16.1397	−0.764235
$447$	0.817218	0.0386531
$448$	4.45692	0.210570
$449$	23.0378	1.08722	0.543609	−	0.839338i	$-0.317057\pi$
$449$	23.0378	1.08722	0.543609	+	0.839338i	$0.317057\pi$
$450$	3.52079	0.165972
$451$	−3.79246	−0.178580
$452$	5.68687	0.267488
$453$	7.06530	0.331957
$454$	−25.2563	−1.18534
$455$	−73.0539	−3.42482
$456$	11.1005	0.519830
$457$	−31.4402	−1.47071	−0.735356	−	0.677681i	$-0.762985\pi$
$457$	−31.4402	−1.47071	−0.735356	+	0.677681i	$0.762985\pi$
$458$	−19.7728	−0.923924
$459$	36.0223	1.68138
$460$	18.7584	0.874616
$461$	34.3236	1.59861	0.799304	−	0.600927i	$-0.205202\pi$
$461$	34.3236	1.59861	0.799304	+	0.600927i	$0.205202\pi$
$462$	−19.3479	−0.900145
$463$	0.0339085	0.00157586	0.000787930	−	1.00000i	$-0.499749\pi$
$463$	0.0339085	0.00157586	0.000787930		1.00000i	$0.499749\pi$
$464$	7.35997	0.341678
$465$	−62.2313	−2.88591
$466$	6.40640	0.296771
$467$	−37.4615	−1.73351	−0.866756	−	0.498732i	$-0.833799\pi$
$467$	−37.4615	−1.73351	−0.866756	+	0.498732i	$0.833799\pi$
$468$	−1.92033	−0.0887673
$469$	−17.9076	−0.826895
$470$	−30.6151	−1.41217
$471$	−1.00025	−0.0460893
$472$	−6.27851	−0.288992
$473$	3.88883	0.178809
$474$	14.5389	0.667794
$475$	57.1182	2.62076
$476$	29.2074	1.33872
$477$	2.71101	0.124129
$478$	−4.12843	−0.188830
$479$	−1.75402	−0.0801430	−0.0400715	−	0.999197i	$-0.512759\pi$
$479$	−1.75402	−0.0801430	−0.0400715	+	0.999197i	$0.512759\pi$
$480$	−5.84038	−0.266576
$481$	26.1083	1.19044
$482$	3.70928	0.168953
$483$	36.8392	1.67624
$484$	−3.67720	−0.167146
$485$	20.8683	0.947579
$486$	4.39730	0.199465
$487$	−31.5078	−1.42775	−0.713877	−	0.700271i	$-0.753062\pi$
$487$	−31.5078	−1.42775	−0.713877	+	0.700271i	$0.753062\pi$
$488$	6.69352	0.303002
$489$	−17.6849	−0.799737
$490$	46.8342	2.11576
$491$	−35.1214	−1.58501	−0.792504	−	0.609867i	$-0.791223\pi$
$491$	−35.1214	−1.58501	−0.792504	+	0.609867i	$0.791223\pi$
$492$	−2.24824	−0.101358
$493$	48.2318	2.17225
$494$	−31.1538	−1.40168
$495$	−4.20213	−0.188872
$496$	−10.6554	−0.478440
$497$	66.2208	2.97041
$498$	−4.53708	−0.203311
$499$	3.65271	0.163518	0.0817588	−	0.996652i	$-0.473946\pi$
$499$	3.65271	0.163518	0.0817588	+	0.996652i	$0.473946\pi$
$500$	−11.8485	−0.529882
$501$	−14.9023	−0.665788
$502$	−27.2873	−1.21789
$503$	−0.430490	−0.0191946	−0.00959730	−	0.999954i	$-0.503055\pi$
$503$	−0.430490	−0.0191946	−0.00959730	+	0.999954i	$0.503055\pi$
$504$	1.90101	0.0846778
$505$	−26.0975	−1.16132
$506$	−13.9429	−0.619838
$507$	−11.6625	−0.517950
$508$	−14.8721	−0.659841
$509$	−36.7124	−1.62725	−0.813625	−	0.581390i	$-0.802509\pi$
$509$	−36.7124	−1.62725	−0.813625	+	0.581390i	$0.802509\pi$
$510$	−38.2735	−1.69478
$511$	−47.1551	−2.08602
$512$	−1.00000	−0.0441942
$513$	38.0363	1.67935
$514$	−14.8529	−0.655134
$515$	−33.3892	−1.47130
$516$	2.30537	0.101488
$517$	22.7558	1.00080
$518$	−25.8457	−1.13559
$519$	−21.6285	−0.949388
$520$	16.3911	0.718797
$521$	−25.9706	−1.13779	−0.568897	−	0.822409i	$-0.692630\pi$
$521$	−25.9706	−1.13779	−0.568897	+	0.822409i	$0.692630\pi$
$522$	3.13925	0.137401
$523$	12.3953	0.542010	0.271005	−	0.962578i	$-0.412644\pi$
$523$	12.3953	0.542010	0.271005	+	0.962578i	$0.412644\pi$
$524$	−19.3851	−0.846841
$525$	−59.0181	−2.57576
$526$	−10.4421	−0.455296
$527$	−69.8273	−3.04173
$528$	4.34108	0.188921
$529$	3.54787	0.154255
$530$	−23.1400	−1.00514
$531$	−2.67797	−0.116214
$532$	30.8404	1.33710
$533$	6.30971	0.273304
$534$	0.986072	0.0426715
$535$	−13.9071	−0.601255
$536$	4.01792	0.173548
$537$	−26.6901	−1.15176
$538$	13.2912	0.573024
$539$	−34.8113	−1.49943
$540$	−20.0122	−0.861189
$541$	−1.57879	−0.0678773	−0.0339387	−	0.999424i	$-0.510805\pi$
$541$	−1.57879	−0.0678773	−0.0339387	+	0.999424i	$0.510805\pi$
$542$	20.8683	0.896369
$543$	−18.3579	−0.787814
$544$	−6.55326	−0.280969
$545$	−54.2957	−2.32577
$546$	32.1900	1.37761
$547$	34.8892	1.49175	0.745877	−	0.666084i	$-0.232031\pi$
$547$	34.8892	1.49175	0.745877	+	0.666084i	$0.232031\pi$
$548$	−12.8889	−0.550588
$549$	2.85499	0.121848
$550$	22.3372	0.952461
$551$	50.9284	2.16962
$552$	−8.26560	−0.351807
$553$	40.3931	1.71769
$554$	−16.1256	−0.685113
$555$	33.8683	1.43763
$556$	−13.0018	−0.551401
$557$	22.8361	0.967595	0.483798	−	0.875180i	$-0.339257\pi$
$557$	22.8361	0.967595	0.483798	+	0.875180i	$0.339257\pi$
$558$	−4.54483	−0.192398
$559$	−6.47004	−0.273654
$560$	−16.2262	−0.685682
$561$	28.4482	1.20109
$562$	−24.2683	−1.02370
$563$	6.43807	0.271332	0.135666	−	0.990755i	$-0.456683\pi$
$563$	6.43807	0.271332	0.135666	+	0.990755i	$0.456683\pi$
$564$	13.4901	0.568034
$565$	−20.7040	−0.871025
$566$	23.3600	0.981893
$567$	−33.5984	−1.41100
$568$	−14.8580	−0.623426
$569$	26.8224	1.12445	0.562226	−	0.826984i	$-0.309945\pi$
$569$	26.8224	1.12445	0.562226	+	0.826984i	$0.309945\pi$
$570$	−40.4134	−1.69273
$571$	−10.2036	−0.427005	−0.213503	−	0.976942i	$-0.568487\pi$
$571$	−10.2036	−0.427005	−0.213503	+	0.976942i	$0.568487\pi$
$572$	−12.1833	−0.509409
$573$	35.5255	1.48410
$574$	−6.24623	−0.260713
$575$	−42.5309	−1.77366
$576$	−0.426530	−0.0177721
$577$	24.3501	1.01371	0.506854	−	0.862032i	$-0.330808\pi$
$577$	24.3501	1.01371	0.506854	+	0.862032i	$0.330808\pi$
$578$	−25.9452	−1.07918
$579$	−14.6810	−0.610123
$580$	−26.7952	−1.11261
$581$	−12.6053	−0.522955
$582$	−9.19526	−0.381156
$583$	17.1997	0.712338
$584$	10.5802	0.437811
$585$	6.99129	0.289054
$586$	3.43044	0.141710
$587$	−20.3045	−0.838055	−0.419027	−	0.907974i	$-0.637629\pi$
$587$	−20.3045	−0.838055	−0.419027	+	0.907974i	$0.637629\pi$
$588$	−20.6368	−0.851046
$589$	−73.7314	−3.03805
$590$	22.8580	0.941049
$591$	24.8070	1.02042
$592$	5.79900	0.238337
$593$	21.6530	0.889182	0.444591	−	0.895734i	$-0.353349\pi$
$593$	21.6530	0.889182	0.444591	+	0.895734i	$0.353349\pi$
$594$	14.8748	0.610322
$595$	−106.334	−4.35929
$596$	−0.509423	−0.0208668
$597$	9.16754	0.375203
$598$	23.1975	0.948617
$599$	−23.0591	−0.942171	−0.471085	−	0.882088i	$-0.656138\pi$
$599$	−23.0591	−0.942171	−0.471085	+	0.882088i	$0.656138\pi$
$600$	13.2419	0.540598
$601$	−43.2367	−1.76366	−0.881830	−	0.471567i	$-0.843689\pi$
$601$	−43.2367	−1.76366	−0.881830	+	0.471567i	$0.843689\pi$
$602$	6.40495	0.261046
$603$	1.71376	0.0697899
$604$	−4.40424	−0.179206
$605$	13.3875	0.544279
$606$	11.4995	0.467133
$607$	−18.0064	−0.730858	−0.365429	−	0.930839i	$-0.619078\pi$
$607$	−18.0064	−0.730858	−0.365429	+	0.930839i	$0.619078\pi$
$608$	−6.91966	−0.280629
$609$	−52.6224	−2.13237
$610$	−24.3689	−0.986669
$611$	−37.8600	−1.53165
$612$	−2.79516	−0.112988
$613$	−32.6912	−1.32038	−0.660192	−	0.751097i	$-0.729525\pi$
$613$	−32.6912	−1.32038	−0.660192	+	0.751097i	$0.729525\pi$
$614$	−15.6353	−0.630988
$615$	8.18510	0.330055
$616$	12.0607	0.485941
$617$	47.8584	1.92670	0.963352	−	0.268239i	$-0.0864416\pi$
$617$	47.8584	1.92670	0.963352	+	0.268239i	$0.0864416\pi$
$618$	14.7124	0.591820
$619$	6.92439	0.278315	0.139157	−	0.990270i	$-0.455561\pi$
$619$	6.92439	0.278315	0.139157	+	0.990270i	$0.455561\pi$
$620$	38.7927	1.55795
$621$	−28.3223	−1.13654
$622$	20.1729	0.808861
$623$	2.73958	0.109759
$624$	−7.22247	−0.289130
$625$	1.86414	0.0745655
$626$	−22.7636	−0.909815
$627$	30.0388	1.19963
$628$	0.623520	0.0248812
$629$	38.0023	1.51525
$630$	−6.92096	−0.275738
$631$	−3.25960	−0.129763	−0.0648814	−	0.997893i	$-0.520667\pi$
$631$	−3.25960	−0.129763	−0.0648814	+	0.997893i	$0.520667\pi$
$632$	−9.06301	−0.360507
$633$	−8.15516	−0.324139
$634$	−25.5988	−1.01666
$635$	54.1443	2.14865
$636$	10.1963	0.404308
$637$	57.9173	2.29477
$638$	19.9166	0.788504
$639$	−6.33737	−0.250702
$640$	3.64067	0.143910
$641$	−28.7794	−1.13672	−0.568359	−	0.822781i	$-0.692421\pi$
$641$	−28.7794	−1.13672	−0.568359	+	0.822781i	$0.692421\pi$
$642$	6.12792	0.241850
$643$	20.3907	0.804131	0.402065	−	0.915611i	$-0.368292\pi$
$643$	20.3907	0.804131	0.402065	+	0.915611i	$0.368292\pi$
$644$	−22.9641	−0.904914
$645$	−8.39309	−0.330477
$646$	−45.3463	−1.78413
$647$	44.3256	1.74262	0.871309	−	0.490735i	$-0.163272\pi$
$647$	44.3256	1.74262	0.871309	+	0.490735i	$0.163272\pi$
$648$	7.53848	0.296140
$649$	−16.9901	−0.666919
$650$	−37.1635	−1.45767
$651$	76.1838	2.98588
$652$	11.0241	0.431736
$653$	0.302812	0.0118499	0.00592497	−	0.999982i	$-0.498114\pi$
$653$	0.302812	0.0118499	0.00592497	+	0.999982i	$0.498114\pi$
$654$	23.9245	0.935523
$655$	70.5747	2.75758
$656$	1.40147	0.0547181
$657$	4.51277	0.176060
$658$	37.4791	1.46109
$659$	−11.9328	−0.464836	−0.232418	−	0.972616i	$-0.574664\pi$
$659$	−11.9328	−0.464836	−0.232418	+	0.972616i	$0.574664\pi$
$660$	−15.8045	−0.615188
$661$	23.0528	0.896651	0.448325	−	0.893870i	$-0.352021\pi$
$661$	23.0528	0.896651	0.448325	+	0.893870i	$0.352021\pi$
$662$	16.0190	0.622597
$663$	−47.3307	−1.83817
$664$	2.82824	0.109757
$665$	−112.280	−4.35402
$666$	2.47345	0.0958441
$667$	−37.9219	−1.46834
$668$	9.28956	0.359424
$669$	−25.8913	−1.00102
$670$	−14.6279	−0.565126
$671$	18.1131	0.699249
$672$	7.14982	0.275810
$673$	−25.6374	−0.988251	−0.494125	−	0.869391i	$-0.664512\pi$
$673$	−25.6374	−0.988251	−0.494125	+	0.869391i	$0.664512\pi$
$674$	−13.6466	−0.525646
$675$	45.3737	1.74643
$676$	7.26996	0.279614
$677$	29.0936	1.11816	0.559079	−	0.829114i	$-0.311155\pi$
$677$	29.0936	1.11816	0.559079	+	0.829114i	$0.311155\pi$
$678$	9.12290	0.350363
$679$	−25.5470	−0.980404
$680$	23.8583	0.914923
$681$	−40.5163	−1.55259
$682$	−28.8341	−1.10412
$683$	−7.68278	−0.293974	−0.146987	−	0.989138i	$-0.546957\pi$
$683$	−7.68278	−0.293974	−0.146987	+	0.989138i	$0.546957\pi$
$684$	−2.95144	−0.112851
$685$	46.9244	1.79289
$686$	−26.1362	−0.997884
$687$	−31.7197	−1.21018
$688$	−1.43708	−0.0547881
$689$	−28.6160	−1.09018
$690$	30.0923	1.14560
$691$	2.78742	0.106038	0.0530192	−	0.998593i	$-0.483116\pi$
$691$	2.78742	0.106038	0.0530192	+	0.998593i	$0.483116\pi$
$692$	13.4824	0.512525
$693$	5.14427	0.195415
$694$	−22.5429	−0.855718
$695$	47.3354	1.79554
$696$	11.8069	0.447539
$697$	9.18418	0.347876
$698$	6.01638	0.227723
$699$	10.2772	0.388718
$700$	36.7896	1.39052
$701$	−28.5655	−1.07890	−0.539451	−	0.842017i	$-0.681368\pi$
$701$	−28.5655	−1.07890	−0.539451	+	0.842017i	$0.681368\pi$
$702$	−24.7480	−0.934054
$703$	40.1271	1.51342
$704$	−2.70607	−0.101989
$705$	−49.1129	−1.84970
$706$	25.0424	0.942483
$707$	31.9487	1.20155
$708$	−10.0720	−0.378529
$709$	11.3483	0.426195	0.213097	−	0.977031i	$-0.431645\pi$
$709$	11.3483	0.426195	0.213097	+	0.977031i	$0.431645\pi$
$710$	54.0929	2.03007
$711$	−3.86564	−0.144973
$712$	−0.614680	−0.0230361
$713$	54.9013	2.05607
$714$	46.8546	1.75349
$715$	44.3554	1.65880
$716$	16.6376	0.621776
$717$	−6.62284	−0.247335
$718$	−20.3889	−0.760907
$719$	−16.6106	−0.619472	−0.309736	−	0.950823i	$-0.600241\pi$
$719$	−16.6106	−0.619472	−0.309736	+	0.950823i	$0.600241\pi$
$720$	1.55286	0.0578715
$721$	40.8752	1.52227
$722$	−28.8817	−1.07486
$723$	5.95044	0.221299
$724$	11.4436	0.425299
$725$	60.7528	2.25630
$726$	−5.89898	−0.218932
$727$	−13.3240	−0.494159	−0.247080	−	0.968995i	$-0.579471\pi$
$727$	−13.3240	−0.494159	−0.247080	+	0.968995i	$0.579471\pi$
$728$	−20.0660	−0.743697
$729$	29.6696	1.09887
$730$	−38.5190	−1.42565
$731$	−9.41755	−0.348321
$732$	10.7378	0.396880
$733$	−9.54849	−0.352681	−0.176341	−	0.984329i	$-0.556426\pi$
$733$	−9.54849	−0.352681	−0.176341	+	0.984329i	$0.556426\pi$
$734$	−33.6930	−1.24363
$735$	75.1317	2.77127
$736$	5.15246	0.189922
$737$	10.8728	0.400503
$738$	0.597768	0.0220041
$739$	12.0271	0.442423	0.221211	−	0.975226i	$-0.428999\pi$
$739$	12.0271	0.442423	0.221211	+	0.975226i	$0.428999\pi$
$740$	−21.1122	−0.776102
$741$	−49.9770	−1.83595
$742$	28.3281	1.03996
$743$	28.5581	1.04770	0.523848	−	0.851812i	$-0.324496\pi$
$743$	28.5581	1.04770	0.523848	+	0.851812i	$0.324496\pi$
$744$	−17.0934	−0.626673
$745$	1.85464	0.0679488
$746$	−17.6279	−0.645404
$747$	1.20633	0.0441373
$748$	−17.7336	−0.648403
$749$	17.0251	0.622083
$750$	−19.0074	−0.694054
$751$	−42.2333	−1.54112	−0.770558	−	0.637370i	$-0.780022\pi$
$751$	−42.2333	−1.54112	−0.770558	+	0.637370i	$0.780022\pi$
$752$	−8.40919	−0.306652
$753$	−43.7743	−1.59523
$754$	−33.1362	−1.20675
$755$	16.0344	0.583551
$756$	24.4991	0.891022
$757$	17.6433	0.641255	0.320627	−	0.947205i	$-0.396106\pi$
$757$	17.6433	0.641255	0.320627	+	0.947205i	$0.396106\pi$
$758$	10.7009	0.388672
$759$	−22.3673	−0.811880
$760$	25.1922	0.913817
$761$	47.7628	1.73140	0.865700	−	0.500563i	$-0.166874\pi$
$761$	47.7628	1.73140	0.865700	+	0.500563i	$0.166874\pi$
$762$	−23.8578	−0.864277
$763$	66.4690	2.40634
$764$	−22.1453	−0.801187
$765$	10.1763	0.367924
$766$	13.1558	0.475339
$767$	28.2672	1.02067
$768$	−1.60420	−0.0578867
$769$	−12.0240	−0.433596	−0.216798	−	0.976216i	$-0.569561\pi$
$769$	−12.0240	−0.433596	−0.216798	+	0.976216i	$0.569561\pi$
$770$	−43.9092	−1.58238
$771$	−23.8271	−0.858112
$772$	9.15160	0.329373
$773$	25.4659	0.915944	0.457972	−	0.888967i	$-0.348576\pi$
$773$	25.4659	0.915944	0.457972	+	0.888967i	$0.348576\pi$
$774$	−0.612958	−0.0220323
$775$	−87.9545	−3.15942
$776$	5.73198	0.205766
$777$	−41.4618	−1.48743
$778$	18.8291	0.675055
$779$	9.69767	0.347455
$780$	26.2947	0.941500
$781$	−40.2066	−1.43871
$782$	33.7654	1.20745
$783$	40.4566	1.44580
$784$	12.8642	0.459435
$785$	−2.27003	−0.0810210
$786$	−31.0976	−1.10921
$787$	−47.4708	−1.69215	−0.846076	−	0.533063i	$-0.821041\pi$
$787$	−47.4708	−1.69215	−0.846076	+	0.533063i	$0.821041\pi$
$788$	−15.4637	−0.550873
$789$	−16.7512	−0.596359
$790$	32.9954	1.17392
$791$	25.3460	0.901198
$792$	−1.15422	−0.0410134
$793$	−30.1357	−1.07015
$794$	−12.7586	−0.452787
$795$	−37.1213	−1.31656
$796$	−5.71470	−0.202552
$797$	54.0732	1.91537	0.957686	−	0.287814i	$-0.0929285\pi$
$797$	54.0732	1.91537	0.957686	+	0.287814i	$0.0929285\pi$
$798$	49.4743	1.75137
$799$	−55.1076	−1.94957
$800$	−8.25449	−0.291840
$801$	−0.262180	−0.00926366
$802$	15.6183	0.551501
$803$	28.6307	1.01036
$804$	6.44556	0.227317
$805$	83.6049	2.94668
$806$	47.9727	1.68977
$807$	21.3218	0.750562
$808$	−7.16833	−0.252181
$809$	−31.3503	−1.10222	−0.551108	−	0.834434i	$-0.685795\pi$
$809$	−31.3503	−1.10222	−0.551108	+	0.834434i	$0.685795\pi$
$810$	−27.4451	−0.964324
$811$	35.9869	1.26367	0.631835	−	0.775103i	$-0.282302\pi$
$811$	35.9869	1.26367	0.631835	+	0.775103i	$0.282302\pi$
$812$	32.8028	1.15115
$813$	33.4769	1.17409
$814$	15.6925	0.550021
$815$	−40.1350	−1.40587
$816$	−10.5128	−0.368020
$817$	−9.94409	−0.347900
$818$	−2.19104	−0.0766078
$819$	−8.55877	−0.299068
$820$	−5.10228	−0.178179
$821$	44.0717	1.53811	0.769056	−	0.639182i	$-0.220727\pi$
$821$	44.0717	1.53811	0.769056	+	0.639182i	$0.220727\pi$
$822$	−20.6765	−0.721175
$823$	−45.6582	−1.59155	−0.795773	−	0.605595i	$-0.792935\pi$
$823$	−45.6582	−1.59155	−0.795773	+	0.605595i	$0.792935\pi$
$824$	−9.17116	−0.319492
$825$	35.8334	1.24756
$826$	−27.9829	−0.973648
$827$	54.3264	1.88912	0.944558	−	0.328345i	$-0.106491\pi$
$827$	54.3264	1.88912	0.944558	+	0.328345i	$0.106491\pi$
$828$	2.19768	0.0763747
$829$	−10.6967	−0.371511	−0.185755	−	0.982596i	$-0.559473\pi$
$829$	−10.6967	−0.371511	−0.185755	+	0.982596i	$0.559473\pi$
$830$	−10.2967	−0.357404
$831$	−25.8688	−0.897380
$832$	4.50222	0.156086
$833$	84.3023	2.92090
$834$	−20.8576	−0.722240
$835$	−33.8202	−1.17040
$836$	−18.7251	−0.647619
$837$	−58.5709	−2.02451
$838$	−11.5818	−0.400086
$839$	34.6330	1.19566	0.597832	−	0.801622i	$-0.296029\pi$
$839$	34.6330	1.19566	0.597832	+	0.801622i	$0.296029\pi$
$840$	−26.0301	−0.898125
$841$	25.1691	0.867900
$842$	−17.5198	−0.603774
$843$	−38.9314	−1.34087
$844$	5.08362	0.174985
$845$	−26.4675	−0.910511
$846$	−3.58677	−0.123316
$847$	−16.3890	−0.563133
$848$	−6.35597	−0.218265
$849$	37.4742	1.28611
$850$	−54.0938	−1.85540
$851$	−29.8791	−1.02424
$852$	−23.8352	−0.816580
$853$	20.3768	0.697687	0.348844	−	0.937181i	$-0.386574\pi$
$853$	20.3768	0.697687	0.348844	+	0.937181i	$0.386574\pi$
$854$	29.8325	1.02085
$855$	10.7452	0.367479
$856$	−3.81992	−0.130562
$857$	53.3961	1.82398	0.911988	−	0.410217i	$-0.134547\pi$
$857$	53.3961	1.82398	0.911988	+	0.410217i	$0.134547\pi$
$858$	−19.5445	−0.667238
$859$	40.3055	1.37521	0.687603	−	0.726087i	$-0.258663\pi$
$859$	40.3055	1.37521	0.687603	+	0.726087i	$0.258663\pi$
$860$	5.23193	0.178407
$861$	−10.0202	−0.341489
$862$	−38.7695	−1.32050
$863$	−32.6088	−1.11002	−0.555009	−	0.831844i	$-0.687285\pi$
$863$	−32.6088	−1.11002	−0.555009	+	0.831844i	$0.687285\pi$
$864$	−5.49685	−0.187007
$865$	−49.0851	−1.66894
$866$	14.1437	0.480623
$867$	−41.6214	−1.41354
$868$	−47.4901	−1.61192
$869$	−24.5251	−0.831957
$870$	−42.9850	−1.45733
$871$	−18.0896	−0.612941
$872$	−14.9136	−0.505040
$873$	2.44486	0.0827460
$874$	35.6533	1.20599
$875$	−52.8080	−1.78524
$876$	16.9728	0.573457
$877$	−31.0843	−1.04964	−0.524821	−	0.851213i	$-0.675868\pi$
$877$	−31.0843	−1.04964	−0.524821	+	0.851213i	$0.675868\pi$
$878$	−21.9033	−0.739202
$879$	5.50312	0.185616
$880$	9.85190	0.332107
$881$	17.3831	0.585653	0.292826	−	0.956166i	$-0.405404\pi$
$881$	17.3831	0.585653	0.292826	+	0.956166i	$0.405404\pi$
$882$	5.48696	0.184755
$883$	−50.8551	−1.71141	−0.855706	−	0.517463i	$-0.826877\pi$
$883$	−50.8551	−1.71141	−0.855706	+	0.517463i	$0.826877\pi$
$884$	29.5042	0.992334
$885$	36.6689	1.23261
$886$	7.99835	0.268710
$887$	17.9123	0.601436	0.300718	−	0.953713i	$-0.402774\pi$
$887$	17.9123	0.601436	0.300718	+	0.953713i	$0.402774\pi$
$888$	9.30278	0.312181
$889$	−66.2836	−2.22308
$890$	2.23785	0.0750128
$891$	20.3996	0.683413
$892$	16.1397	0.540396
$893$	−58.1887	−1.94721
$894$	−0.817218	−0.0273319
$895$	−60.5720	−2.02470
$896$	−4.45692	−0.148895
$897$	37.2135	1.24252
$898$	−23.0378	−0.768780
$899$	−78.4231	−2.61556
$900$	−3.52079	−0.117360
$901$	−41.6523	−1.38764
$902$	3.79246	0.126275
$903$	10.2749	0.341926
$904$	−5.68687	−0.189143
$905$	−41.6625	−1.38491
$906$	−7.06530	−0.234729
$907$	−26.3688	−0.875563	−0.437781	−	0.899081i	$-0.644236\pi$
$907$	−26.3688	−0.875563	−0.437781	+	0.899081i	$0.644236\pi$
$908$	25.2563	0.838160
$909$	−3.05751	−0.101411
$910$	73.0539	2.42171
$911$	−31.8035	−1.05370	−0.526848	−	0.849959i	$-0.676626\pi$
$911$	−31.8035	−1.05370	−0.526848	+	0.849959i	$0.676626\pi$
$912$	−11.1005	−0.367576
$913$	7.65341	0.253291
$914$	31.4402	1.03995
$915$	−39.0927	−1.29237
$916$	19.7728	0.653313
$917$	−86.3978	−2.85311
$918$	−36.0223	−1.18891
$919$	51.1667	1.68783	0.843916	−	0.536475i	$-0.180245\pi$
$919$	51.1667	1.68783	0.843916	+	0.536475i	$0.180245\pi$
$920$	−18.7584	−0.618447
$921$	−25.0821	−0.826485
$922$	−34.3236	−1.13039
$923$	66.8938	2.20183
$924$	19.3479	0.636498
$925$	47.8678	1.57388
$926$	−0.0339085	−0.00111430
$927$	−3.91177	−0.128480
$928$	−7.35997	−0.241603
$929$	−36.5045	−1.19767	−0.598837	−	0.800871i	$-0.704370\pi$
$929$	−36.5045	−1.19767	−0.598837	+	0.800871i	$0.704370\pi$
$930$	62.2313	2.04065
$931$	89.0157	2.91737
$932$	−6.40640	−0.209849
$933$	32.3615	1.05947
$934$	37.4615	1.22578
$935$	64.5621	2.11141
$936$	1.92033	0.0627680
$937$	25.5632	0.835112	0.417556	−	0.908651i	$-0.362887\pi$
$937$	25.5632	0.835112	0.417556	+	0.908651i	$0.362887\pi$
$938$	17.9076	0.584703
$939$	−36.5174	−1.19170
$940$	30.6151	0.998555
$941$	1.31146	0.0427525	0.0213763	−	0.999772i	$-0.493195\pi$
$941$	1.31146	0.0427525	0.0213763	+	0.999772i	$0.493195\pi$
$942$	1.00025	0.0325900
$943$	−7.22101	−0.235148
$944$	6.27851	0.204348
$945$	−89.1930	−2.90145
$946$	−3.88883	−0.126437
$947$	32.6903	1.06229	0.531146	−	0.847281i	$-0.321762\pi$
$947$	32.6903	1.06229	0.531146	+	0.847281i	$0.321762\pi$
$948$	−14.5389	−0.472202
$949$	−47.6343	−1.54628
$950$	−57.1182	−1.85316
$951$	−41.0657	−1.33165
$952$	−29.2074	−0.946617
$953$	−24.5444	−0.795070	−0.397535	−	0.917587i	$-0.630134\pi$
$953$	−24.5444	−0.795070	−0.397535	+	0.917587i	$0.630134\pi$
$954$	−2.71101	−0.0877723
$955$	80.6236	2.60892
$956$	4.12843	0.133523
$957$	31.9502	1.03280
$958$	1.75402	0.0566697
$959$	−57.4450	−1.85500
$960$	5.84038	0.188497
$961$	82.5367	2.66247
$962$	−26.1083	−0.841767
$963$	−1.62931	−0.0525038
$964$	−3.70928	−0.119468
$965$	−33.3180	−1.07254
$966$	−36.8392	−1.18528
$967$	−34.3679	−1.10520	−0.552598	−	0.833448i	$-0.686364\pi$
$967$	−34.3679	−1.10520	−0.552598	+	0.833448i	$0.686364\pi$
$968$	3.67720	0.118190
$969$	−72.7447	−2.33690
$970$	−20.8683	−0.670039
$971$	25.4902	0.818019	0.409009	−	0.912530i	$-0.365874\pi$
$971$	25.4902	0.818019	0.409009	+	0.912530i	$0.365874\pi$
$972$	−4.39730	−0.141043
$973$	−57.9482	−1.85773
$974$	31.5078	1.00957
$975$	−59.6178	−1.90930
$976$	−6.69352	−0.214254
$977$	−6.24634	−0.199838	−0.0999191	−	0.994996i	$-0.531858\pi$
$977$	−6.24634	−0.199838	−0.0999191	+	0.994996i	$0.531858\pi$
$978$	17.6849	0.565500
$979$	−1.66336	−0.0531614
$980$	−46.8342	−1.49606
$981$	−6.36112	−0.203095
$982$	35.1214	1.12077
$983$	−12.8237	−0.409012	−0.204506	−	0.978865i	$-0.565559\pi$
$983$	−12.8237	−0.409012	−0.204506	+	0.978865i	$0.565559\pi$
$984$	2.24824	0.0716713
$985$	56.2984	1.79382
$986$	−48.2318	−1.53601
$987$	60.1242	1.91377
$988$	31.1538	0.991134
$989$	7.40450	0.235449
$990$	4.20213	0.133553
$991$	−5.02931	−0.159761	−0.0798807	−	0.996804i	$-0.525454\pi$
$991$	−5.02931	−0.159761	−0.0798807	+	0.996804i	$0.525454\pi$
$992$	10.6554	0.338308
$993$	25.6978	0.815494
$994$	−66.2208	−2.10040
$995$	20.8053	0.659574
$996$	4.53708	0.143763
$997$	28.7540	0.910649	0.455325	−	0.890325i	$-0.349523\pi$
$997$	28.7540	0.910649	0.455325	+	0.890325i	$0.349523\pi$
$998$	−3.65271	−0.115624
$999$	31.8762	1.00852

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4022.2.a.e.1.13	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4022.2.a.e.1.13	✓	46	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:	\( N \)	\(=\)	\( 4022 = 2 \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4022.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.60420 q^{3} +1.00000 q^{4} -3.64067 q^{5} +1.60420 q^{6} +4.45692 q^{7} -1.00000 q^{8} -0.426530 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.60420 q^{3} +1.00000 q^{4} -3.64067 q^{5} +1.60420 q^{6} +4.45692 q^{7} -1.00000 q^{8} -0.426530 q^{9} +3.64067 q^{10} -2.70607 q^{11} -1.60420 q^{12} +4.50222 q^{13} -4.45692 q^{14} +5.84038 q^{15} +1.00000 q^{16} +6.55326 q^{17} +0.426530 q^{18} +6.91966 q^{19} -3.64067 q^{20} -7.14982 q^{21} +2.70607 q^{22} -5.15246 q^{23} +1.60420 q^{24} +8.25449 q^{25} -4.50222 q^{26} +5.49685 q^{27} +4.45692 q^{28} +7.35997 q^{29} -5.84038 q^{30} -10.6554 q^{31} -1.00000 q^{32} +4.34108 q^{33} -6.55326 q^{34} -16.2262 q^{35} -0.426530 q^{36} +5.79900 q^{37} -6.91966 q^{38} -7.22247 q^{39} +3.64067 q^{40} +1.40147 q^{41} +7.14982 q^{42} -1.43708 q^{43} -2.70607 q^{44} +1.55286 q^{45} +5.15246 q^{46} -8.40919 q^{47} -1.60420 q^{48} +12.8642 q^{49} -8.25449 q^{50} -10.5128 q^{51} +4.50222 q^{52} -6.35597 q^{53} -5.49685 q^{54} +9.85190 q^{55} -4.45692 q^{56} -11.1005 q^{57} -7.35997 q^{58} +6.27851 q^{59} +5.84038 q^{60} -6.69352 q^{61} +10.6554 q^{62} -1.90101 q^{63} +1.00000 q^{64} -16.3911 q^{65} -4.34108 q^{66} -4.01792 q^{67} +6.55326 q^{68} +8.26560 q^{69} +16.2262 q^{70} +14.8580 q^{71} +0.426530 q^{72} -10.5802 q^{73} -5.79900 q^{74} -13.2419 q^{75} +6.91966 q^{76} -12.0607 q^{77} +7.22247 q^{78} +9.06301 q^{79} -3.64067 q^{80} -7.53848 q^{81} -1.40147 q^{82} -2.82824 q^{83} -7.14982 q^{84} -23.8583 q^{85} +1.43708 q^{86} -11.8069 q^{87} +2.70607 q^{88} +0.614680 q^{89} -1.55286 q^{90} +20.0660 q^{91} -5.15246 q^{92} +17.0934 q^{93} +8.40919 q^{94} -25.1922 q^{95} +1.60420 q^{96} -5.73198 q^{97} -12.8642 q^{98} +1.15422 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9}+O(q^{10}) \) 46 * q - 46 * q^2 + 8 * q^3 + 46 * q^4 + 14 * q^5 - 8 * q^6 + 28 * q^7 - 46 * q^8 + 58 * q^9	\( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9} - 14 q^{10} - 6 q^{11} + 8 q^{12} + 37 q^{13} - 28 q^{14} + 9 q^{15} + 46 q^{16} + 6 q^{17} - 58 q^{18} + 18 q^{19} + 14 q^{20} + 19 q^{21} + 6 q^{22} - 4 q^{23} - 8 q^{24} + 86 q^{25} - 37 q^{26} + 32 q^{27} + 28 q^{28} + 15 q^{29} - 9 q^{30} + 18 q^{31} - 46 q^{32} + 37 q^{33} - 6 q^{34} - 2 q^{35} + 58 q^{36} + 74 q^{37} - 18 q^{38} - 3 q^{39} - 14 q^{40} - 18 q^{41} - 19 q^{42} + 25 q^{43} - 6 q^{44} + 94 q^{45} + 4 q^{46} + 18 q^{47} + 8 q^{48} + 92 q^{49} - 86 q^{50} - 10 q^{51} + 37 q^{52} + 17 q^{53} - 32 q^{54} + 37 q^{55} - 28 q^{56} + 43 q^{57} - 15 q^{58} - 24 q^{59} + 9 q^{60} + 46 q^{61} - 18 q^{62} + 80 q^{63} + 46 q^{64} + 24 q^{65} - 37 q^{66} + 61 q^{67} + 6 q^{68} + 59 q^{69} + 2 q^{70} - 8 q^{71} - 58 q^{72} + 101 q^{73} - 74 q^{74} + 34 q^{75} + 18 q^{76} + 40 q^{77} + 3 q^{78} + 9 q^{79} + 14 q^{80} + 58 q^{81} + 18 q^{82} + 18 q^{83} + 19 q^{84} + 60 q^{85} - 25 q^{86} + 20 q^{87} + 6 q^{88} - 25 q^{89} - 94 q^{90} + 51 q^{91} - 4 q^{92} + 63 q^{93} - 18 q^{94} - 31 q^{95} - 8 q^{96} + 76 q^{97} - 92 q^{98} + 21 q^{99}+O(q^{100}) \) 46 * q - 46 * q^2 + 8 * q^3 + 46 * q^4 + 14 * q^5 - 8 * q^6 + 28 * q^7 - 46 * q^8 + 58 * q^9 - 14 * q^10 - 6 * q^11 + 8 * q^12 + 37 * q^13 - 28 * q^14 + 9 * q^15 + 46 * q^16 + 6 * q^17 - 58 * q^18 + 18 * q^19 + 14 * q^20 + 19 * q^21 + 6 * q^22 - 4 * q^23 - 8 * q^24 + 86 * q^25 - 37 * q^26 + 32 * q^27 + 28 * q^28 + 15 * q^29 - 9 * q^30 + 18 * q^31 - 46 * q^32 + 37 * q^33 - 6 * q^34 - 2 * q^35 + 58 * q^36 + 74 * q^37 - 18 * q^38 - 3 * q^39 - 14 * q^40 - 18 * q^41 - 19 * q^42 + 25 * q^43 - 6 * q^44 + 94 * q^45 + 4 * q^46 + 18 * q^47 + 8 * q^48 + 92 * q^49 - 86 * q^50 - 10 * q^51 + 37 * q^52 + 17 * q^53 - 32 * q^54 + 37 * q^55 - 28 * q^56 + 43 * q^57 - 15 * q^58 - 24 * q^59 + 9 * q^60 + 46 * q^61 - 18 * q^62 + 80 * q^63 + 46 * q^64 + 24 * q^65 - 37 * q^66 + 61 * q^67 + 6 * q^68 + 59 * q^69 + 2 * q^70 - 8 * q^71 - 58 * q^72 + 101 * q^73 - 74 * q^74 + 34 * q^75 + 18 * q^76 + 40 * q^77 + 3 * q^78 + 9 * q^79 + 14 * q^80 + 58 * q^81 + 18 * q^82 + 18 * q^83 + 19 * q^84 + 60 * q^85 - 25 * q^86 + 20 * q^87 + 6 * q^88 - 25 * q^89 - 94 * q^90 + 51 * q^91 - 4 * q^92 + 63 * q^93 - 18 * q^94 - 31 * q^95 - 8 * q^96 + 76 * q^97 - 92 * q^98 + 21 * q^99

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