LMFDB - Embedded newform 4022.2.a.e.1.9

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			1.9
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.06796 q^{3} +1.00000 q^{4} +2.09562 q^{5} +2.06796 q^{6} +5.11749 q^{7} -1.00000 q^{8} +1.27644 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.06796 q^{3} +1.00000 q^{4} +2.09562 q^{5} +2.06796 q^{6} +5.11749 q^{7} -1.00000 q^{8} +1.27644 q^{9} -2.09562 q^{10} +2.69915 q^{11} -2.06796 q^{12} -4.18150 q^{13} -5.11749 q^{14} -4.33365 q^{15} +1.00000 q^{16} -1.41352 q^{17} -1.27644 q^{18} -1.56553 q^{19} +2.09562 q^{20} -10.5827 q^{21} -2.69915 q^{22} +2.42616 q^{23} +2.06796 q^{24} -0.608375 q^{25} +4.18150 q^{26} +3.56424 q^{27} +5.11749 q^{28} -1.92559 q^{29} +4.33365 q^{30} -7.21072 q^{31} -1.00000 q^{32} -5.58172 q^{33} +1.41352 q^{34} +10.7243 q^{35} +1.27644 q^{36} -4.20063 q^{37} +1.56553 q^{38} +8.64717 q^{39} -2.09562 q^{40} +12.1150 q^{41} +10.5827 q^{42} +12.5535 q^{43} +2.69915 q^{44} +2.67494 q^{45} -2.42616 q^{46} +0.862520 q^{47} -2.06796 q^{48} +19.1887 q^{49} +0.608375 q^{50} +2.92310 q^{51} -4.18150 q^{52} +0.339777 q^{53} -3.56424 q^{54} +5.65639 q^{55} -5.11749 q^{56} +3.23744 q^{57} +1.92559 q^{58} +12.5452 q^{59} -4.33365 q^{60} -2.83884 q^{61} +7.21072 q^{62} +6.53218 q^{63} +1.00000 q^{64} -8.76285 q^{65} +5.58172 q^{66} -6.55585 q^{67} -1.41352 q^{68} -5.01720 q^{69} -10.7243 q^{70} -8.67503 q^{71} -1.27644 q^{72} +11.7817 q^{73} +4.20063 q^{74} +1.25809 q^{75} -1.56553 q^{76} +13.8128 q^{77} -8.64717 q^{78} +2.52870 q^{79} +2.09562 q^{80} -11.2000 q^{81} -12.1150 q^{82} -3.91937 q^{83} -10.5827 q^{84} -2.96220 q^{85} -12.5535 q^{86} +3.98204 q^{87} -2.69915 q^{88} -16.3518 q^{89} -2.67494 q^{90} -21.3988 q^{91} +2.42616 q^{92} +14.9115 q^{93} -0.862520 q^{94} -3.28075 q^{95} +2.06796 q^{96} -1.57199 q^{97} -19.1887 q^{98} +3.44531 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.06796	−1.19394	−0.596968	−	0.802265i	$-0.703628\pi$
$3$	−2.06796	−1.19394	−0.596968	+	0.802265i	$0.703628\pi$
$4$	1.00000	0.500000
$5$	2.09562	0.937190	0.468595	−	0.883413i	$-0.344760\pi$
$5$	2.09562	0.937190	0.468595	+	0.883413i	$0.344760\pi$
$6$	2.06796	0.844240
$7$	5.11749	1.93423	0.967114	−	0.254344i	$-0.0818595\pi$
$7$	5.11749	1.93423	0.967114	+	0.254344i	$0.0818595\pi$
$8$	−1.00000	−0.353553
$9$	1.27644	0.425481
$10$	−2.09562	−0.662693
$11$	2.69915	0.813823	0.406912	−	0.913468i	$-0.366606\pi$
$11$	2.69915	0.813823	0.406912	+	0.913468i	$0.366606\pi$
$12$	−2.06796	−0.596968
$13$	−4.18150	−1.15974	−0.579870	−	0.814709i	$-0.696897\pi$
$13$	−4.18150	−1.15974	−0.579870	+	0.814709i	$0.696897\pi$
$14$	−5.11749	−1.36771
$15$	−4.33365	−1.11894
$16$	1.00000	0.250000
$17$	−1.41352	−0.342829	−0.171414	−	0.985199i	$-0.554834\pi$
$17$	−1.41352	−0.342829	−0.171414	+	0.985199i	$0.554834\pi$
$18$	−1.27644	−0.300861
$19$	−1.56553	−0.359156	−0.179578	−	0.983744i	$-0.557473\pi$
$19$	−1.56553	−0.359156	−0.179578	+	0.983744i	$0.557473\pi$
$20$	2.09562	0.468595
$21$	−10.5827	−2.30934
$22$	−2.69915	−0.575460
$23$	2.42616	0.505890	0.252945	−	0.967481i	$-0.418601\pi$
$23$	2.42616	0.505890	0.252945	+	0.967481i	$0.418601\pi$
$24$	2.06796	0.422120
$25$	−0.608375	−0.121675
$26$	4.18150	0.820060
$27$	3.56424	0.685938
$28$	5.11749	0.967114
$29$	−1.92559	−0.357574	−0.178787	−	0.983888i	$-0.557217\pi$
$29$	−1.92559	−0.357574	−0.178787	+	0.983888i	$0.557217\pi$
$30$	4.33365	0.791213
$31$	−7.21072	−1.29508	−0.647542	−	0.762030i	$-0.724203\pi$
$31$	−7.21072	−1.29508	−0.647542	+	0.762030i	$0.724203\pi$
$32$	−1.00000	−0.176777
$33$	−5.58172	−0.971652
$34$	1.41352	0.242417
$35$	10.7243	1.81274
$36$	1.27644	0.212741
$37$	−4.20063	−0.690579	−0.345289	−	0.938496i	$-0.612219\pi$
$37$	−4.20063	−0.690579	−0.345289	+	0.938496i	$0.612219\pi$
$38$	1.56553	0.253962
$39$	8.64717	1.38466
$40$	−2.09562	−0.331347
$41$	12.1150	1.89205	0.946025	−	0.324094i	$-0.105060\pi$
$41$	12.1150	1.89205	0.946025	+	0.324094i	$0.105060\pi$
$42$	10.5827	1.63295
$43$	12.5535	1.91439	0.957195	−	0.289443i	$-0.0934702\pi$
$43$	12.5535	1.91439	0.957195	+	0.289443i	$0.0934702\pi$
$44$	2.69915	0.406912
$45$	2.67494	0.398757
$46$	−2.42616	−0.357718
$47$	0.862520	0.125811	0.0629057	−	0.998019i	$-0.479963\pi$
$47$	0.862520	0.125811	0.0629057	+	0.998019i	$0.479963\pi$
$48$	−2.06796	−0.298484
$49$	19.1887	2.74124
$50$	0.608375	0.0860372
$51$	2.92310	0.409315
$52$	−4.18150	−0.579870
$53$	0.339777	0.0466720	0.0233360	−	0.999728i	$-0.492571\pi$
$53$	0.339777	0.0466720	0.0233360	+	0.999728i	$0.492571\pi$
$54$	−3.56424	−0.485032
$55$	5.65639	0.762707
$56$	−5.11749	−0.683853
$57$	3.23744	0.428809
$58$	1.92559	0.252843
$59$	12.5452	1.63324	0.816621	−	0.577175i	$-0.195845\pi$
$59$	12.5452	1.63324	0.816621	+	0.577175i	$0.195845\pi$
$60$	−4.33365	−0.559472
$61$	−2.83884	−0.363476	−0.181738	−	0.983347i	$-0.558172\pi$
$61$	−2.83884	−0.363476	−0.181738	+	0.983347i	$0.558172\pi$
$62$	7.21072	0.915763
$63$	6.53218	0.822977
$64$	1.00000	0.125000
$65$	−8.76285	−1.08690
$66$	5.58172	0.687062
$67$	−6.55585	−0.800925	−0.400462	−	0.916313i	$-0.631151\pi$
$67$	−6.55585	−0.800925	−0.400462	+	0.916313i	$0.631151\pi$
$68$	−1.41352	−0.171414
$69$	−5.01720	−0.604000
$70$	−10.7243	−1.28180
$71$	−8.67503	−1.02954	−0.514768	−	0.857330i	$-0.672122\pi$
$71$	−8.67503	−1.02954	−0.514768	+	0.857330i	$0.672122\pi$
$72$	−1.27644	−0.150430
$73$	11.7817	1.37894	0.689471	−	0.724314i	$-0.257843\pi$
$73$	11.7817	1.37894	0.689471	+	0.724314i	$0.257843\pi$
$74$	4.20063	0.488313
$75$	1.25809	0.145272
$76$	−1.56553	−0.179578
$77$	13.8128	1.57412
$78$	−8.64717	−0.979099
$79$	2.52870	0.284501	0.142250	−	0.989831i	$-0.454566\pi$
$79$	2.52870	0.284501	0.142250	+	0.989831i	$0.454566\pi$
$80$	2.09562	0.234297
$81$	−11.2000	−1.24445
$82$	−12.1150	−1.33788
$83$	−3.91937	−0.430206	−0.215103	−	0.976591i	$-0.569009\pi$
$83$	−3.91937	−0.430206	−0.215103	+	0.976591i	$0.569009\pi$
$84$	−10.5827	−1.15467
$85$	−2.96220	−0.321296
$86$	−12.5535	−1.35368
$87$	3.98204	0.426920
$88$	−2.69915	−0.287730
$89$	−16.3518	−1.73329	−0.866645	−	0.498925i	$-0.833728\pi$
$89$	−16.3518	−1.73329	−0.866645	+	0.498925i	$0.833728\pi$
$90$	−2.67494	−0.281963
$91$	−21.3988	−2.24320
$92$	2.42616	0.252945
$93$	14.9115	1.54625
$94$	−0.862520	−0.0889622
$95$	−3.28075	−0.336598
$96$	2.06796	0.211060
$97$	−1.57199	−0.159612	−0.0798058	−	0.996810i	$-0.525430\pi$
$97$	−1.57199	−0.159612	−0.0798058	+	0.996810i	$0.525430\pi$
$98$	−19.1887	−1.93835
$99$	3.44531	0.346266
$100$	−0.608375	−0.0608375
$101$	16.2335	1.61529	0.807645	−	0.589669i	$-0.200742\pi$
$101$	16.2335	1.61529	0.807645	+	0.589669i	$0.200742\pi$
$102$	−2.92310	−0.289430
$103$	10.2103	1.00605	0.503026	−	0.864271i	$-0.332220\pi$
$103$	10.2103	1.00605	0.503026	+	0.864271i	$0.332220\pi$
$104$	4.18150	0.410030
$105$	−22.1774	−2.16429
$106$	−0.339777	−0.0330021
$107$	9.19506	0.888920	0.444460	−	0.895799i	$-0.353396\pi$
$107$	9.19506	0.888920	0.444460	+	0.895799i	$0.353396\pi$
$108$	3.56424	0.342969
$109$	18.7894	1.79970	0.899848	−	0.436203i	$-0.143677\pi$
$109$	18.7894	1.79970	0.899848	+	0.436203i	$0.143677\pi$
$110$	−5.65639	−0.539315
$111$	8.68671	0.824506
$112$	5.11749	0.483557
$113$	3.78051	0.355641	0.177820	−	0.984063i	$-0.443095\pi$
$113$	3.78051	0.355641	0.177820	+	0.984063i	$0.443095\pi$
$114$	−3.23744	−0.303214
$115$	5.08432	0.474115
$116$	−1.92559	−0.178787
$117$	−5.33745	−0.493448
$118$	−12.5452	−1.15488
$119$	−7.23367	−0.663109
$120$	4.33365	0.395606
$121$	−3.71461	−0.337692
$122$	2.83884	0.257016
$123$	−25.0533	−2.25898
$124$	−7.21072	−0.647542
$125$	−11.7530	−1.05122
$126$	−6.53218	−0.581933
$127$	13.3399	1.18373	0.591864	−	0.806037i	$-0.298392\pi$
$127$	13.3399	1.18373	0.591864	+	0.806037i	$0.298392\pi$
$128$	−1.00000	−0.0883883
$129$	−25.9601	−2.28566
$130$	8.76285	0.768552
$131$	−1.39415	−0.121807	−0.0609035	−	0.998144i	$-0.519398\pi$
$131$	−1.39415	−0.121807	−0.0609035	+	0.998144i	$0.519398\pi$
$132$	−5.58172	−0.485826
$133$	−8.01155	−0.694690
$134$	6.55585	0.566339
$135$	7.46929	0.642854
$136$	1.41352	0.121208
$137$	−16.5396	−1.41308	−0.706538	−	0.707675i	$-0.749744\pi$
$137$	−16.5396	−1.41308	−0.706538	+	0.707675i	$0.749744\pi$
$138$	5.01720	0.427093
$139$	19.3595	1.64205	0.821027	−	0.570890i	$-0.193402\pi$
$139$	19.3595	1.64205	0.821027	+	0.570890i	$0.193402\pi$
$140$	10.7243	0.906369
$141$	−1.78365	−0.150211
$142$	8.67503	0.727992
$143$	−11.2865	−0.943824
$144$	1.27644	0.106370
$145$	−4.03531	−0.335114
$146$	−11.7817	−0.975059
$147$	−39.6813	−3.27286
$148$	−4.20063	−0.345289
$149$	15.9247	1.30460	0.652301	−	0.757960i	$-0.273804\pi$
$149$	15.9247	1.30460	0.652301	+	0.757960i	$0.273804\pi$
$150$	−1.25809	−0.102723
$151$	−0.844468	−0.0687218	−0.0343609	−	0.999409i	$-0.510940\pi$
$151$	−0.844468	−0.0687218	−0.0343609	+	0.999409i	$0.510940\pi$
$152$	1.56553	0.126981
$153$	−1.80428	−0.145867
$154$	−13.8128	−1.11307
$155$	−15.1109	−1.21374
$156$	8.64717	0.692328
$157$	2.33193	0.186108	0.0930541	−	0.995661i	$-0.470337\pi$
$157$	2.33193	0.186108	0.0930541	+	0.995661i	$0.470337\pi$
$158$	−2.52870	−0.201172
$159$	−0.702644	−0.0557233
$160$	−2.09562	−0.165673
$161$	12.4159	0.978507
$162$	11.2000	0.879957
$163$	0.929192	0.0727799	0.0363900	−	0.999338i	$-0.488414\pi$
$163$	0.929192	0.0727799	0.0363900	+	0.999338i	$0.488414\pi$
$164$	12.1150	0.946025
$165$	−11.6972	−0.910623
$166$	3.91937	0.304202
$167$	−11.9538	−0.925015	−0.462507	−	0.886615i	$-0.653050\pi$
$167$	−11.9538	−0.925015	−0.462507	+	0.886615i	$0.653050\pi$
$168$	10.5827	0.816476
$169$	4.48498	0.344998
$170$	2.96220	0.227190
$171$	−1.99830	−0.152814
$172$	12.5535	0.957195
$173$	−17.2950	−1.31491	−0.657457	−	0.753492i	$-0.728368\pi$
$173$	−17.2950	−1.31491	−0.657457	+	0.753492i	$0.728368\pi$
$174$	−3.98204	−0.301878
$175$	−3.11335	−0.235347
$176$	2.69915	0.203456
$177$	−25.9429	−1.94998
$178$	16.3518	1.22562
$179$	−14.7303	−1.10099	−0.550497	−	0.834837i	$-0.685562\pi$
$179$	−14.7303	−1.10099	−0.550497	+	0.834837i	$0.685562\pi$
$180$	2.67494	0.199378
$181$	3.05742	0.227256	0.113628	−	0.993523i	$-0.463753\pi$
$181$	3.05742	0.227256	0.113628	+	0.993523i	$0.463753\pi$
$182$	21.3988	1.58618
$183$	5.87059	0.433967
$184$	−2.42616	−0.178859
$185$	−8.80292	−0.647203
$186$	−14.9115	−1.09336
$187$	−3.81530	−0.279002
$188$	0.862520	0.0629057
$189$	18.2399	1.32676
$190$	3.28075	0.238010
$191$	−12.9836	−0.939457	−0.469729	−	0.882811i	$-0.655648\pi$
$191$	−12.9836	−0.939457	−0.469729	+	0.882811i	$0.655648\pi$
$192$	−2.06796	−0.149242
$193$	12.4585	0.896783	0.448391	−	0.893837i	$-0.351997\pi$
$193$	12.4585	0.896783	0.448391	+	0.893837i	$0.351997\pi$
$194$	1.57199	0.112862
$195$	18.1212	1.29768
$196$	19.1887	1.37062
$197$	16.1477	1.15048	0.575239	−	0.817986i	$-0.304909\pi$
$197$	16.1477	1.15048	0.575239	+	0.817986i	$0.304909\pi$
$198$	−3.44531	−0.244847
$199$	−12.6526	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$199$	−12.6526	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$200$	0.608375	0.0430186
$201$	13.5572	0.956252
$202$	−16.2335	−1.14218
$203$	−9.85419	−0.691629
$204$	2.92310	0.204658
$205$	25.3885	1.77321
$206$	−10.2103	−0.711387
$207$	3.09686	0.215247
$208$	−4.18150	−0.289935
$209$	−4.22558	−0.292290
$210$	22.1774	1.53039
$211$	2.97778	0.204999	0.102500	−	0.994733i	$-0.467316\pi$
$211$	2.97778	0.204999	0.102500	+	0.994733i	$0.467316\pi$
$212$	0.339777	0.0233360
$213$	17.9396	1.22920
$214$	−9.19506	−0.628561
$215$	26.3074	1.79415
$216$	−3.56424	−0.242516
$217$	−36.9008	−2.50499
$218$	−18.7894	−1.27258
$219$	−24.3640	−1.64637
$220$	5.65639	0.381353
$221$	5.91064	0.397593
$222$	−8.68671	−0.583014
$223$	11.2347	0.752333	0.376167	−	0.926552i	$-0.377242\pi$
$223$	11.2347	0.752333	0.376167	+	0.926552i	$0.377242\pi$
$224$	−5.11749	−0.341926
$225$	−0.776556	−0.0517704
$226$	−3.78051	−0.251476
$227$	6.97184	0.462737	0.231369	−	0.972866i	$-0.425680\pi$
$227$	6.97184	0.462737	0.231369	+	0.972866i	$0.425680\pi$
$228$	3.23744	0.214405
$229$	19.3042	1.27566	0.637829	−	0.770178i	$-0.279833\pi$
$229$	19.3042	1.27566	0.637829	+	0.770178i	$0.279833\pi$
$230$	−5.08432	−0.335250
$231$	−28.5644	−1.87940
$232$	1.92559	0.126421
$233$	20.6021	1.34969	0.674846	−	0.737959i	$-0.264210\pi$
$233$	20.6021	1.34969	0.674846	+	0.737959i	$0.264210\pi$
$234$	5.33745	0.348920
$235$	1.80751	0.117909
$236$	12.5452	0.816621
$237$	−5.22924	−0.339676
$238$	7.23367	0.468889
$239$	−28.3927	−1.83657	−0.918285	−	0.395921i	$-0.870425\pi$
$239$	−28.3927	−1.83657	−0.918285	+	0.395921i	$0.870425\pi$
$240$	−4.33365	−0.279736
$241$	16.8805	1.08737	0.543684	−	0.839290i	$-0.317029\pi$
$241$	16.8805	1.08737	0.543684	+	0.839290i	$0.317029\pi$
$242$	3.71461	0.238784
$243$	12.4684	0.799851
$244$	−2.83884	−0.181738
$245$	40.2121	2.56906
$246$	25.0533	1.59734
$247$	6.54625	0.416528
$248$	7.21072	0.457881
$249$	8.10508	0.513638
$250$	11.7530	0.743327
$251$	−15.0144	−0.947701	−0.473850	−	0.880605i	$-0.657136\pi$
$251$	−15.0144	−0.947701	−0.473850	+	0.880605i	$0.657136\pi$
$252$	6.53218	0.411489
$253$	6.54857	0.411705
$254$	−13.3399	−0.837023
$255$	6.12570	0.383606
$256$	1.00000	0.0625000
$257$	−14.5963	−0.910491	−0.455245	−	0.890366i	$-0.650448\pi$
$257$	−14.5963	−0.910491	−0.455245	+	0.890366i	$0.650448\pi$
$258$	25.9601	1.61620
$259$	−21.4966	−1.33574
$260$	−8.76285	−0.543449
$261$	−2.45791	−0.152141
$262$	1.39415	0.0861306
$263$	22.1757	1.36741	0.683707	−	0.729757i	$-0.260367\pi$
$263$	22.1757	1.36741	0.683707	+	0.729757i	$0.260367\pi$
$264$	5.58172	0.343531
$265$	0.712044	0.0437405
$266$	8.01155	0.491220
$267$	33.8149	2.06944
$268$	−6.55585	−0.400462
$269$	1.00705	0.0614012	0.0307006	−	0.999529i	$-0.490226\pi$
$269$	1.00705	0.0614012	0.0307006	+	0.999529i	$0.490226\pi$
$270$	−7.46929	−0.454567
$271$	9.20153	0.558953	0.279477	−	0.960152i	$-0.409839\pi$
$271$	9.20153	0.558953	0.279477	+	0.960152i	$0.409839\pi$
$272$	−1.41352	−0.0857072
$273$	44.2518	2.67824
$274$	16.5396	0.999196
$275$	−1.64209	−0.0990219
$276$	−5.01720	−0.302000
$277$	19.0330	1.14358	0.571790	−	0.820400i	$-0.306249\pi$
$277$	19.0330	1.14358	0.571790	+	0.820400i	$0.306249\pi$
$278$	−19.3595	−1.16111
$279$	−9.20408	−0.551034
$280$	−10.7243	−0.640900
$281$	15.6191	0.931758	0.465879	−	0.884848i	$-0.345738\pi$
$281$	15.6191	0.931758	0.465879	+	0.884848i	$0.345738\pi$
$282$	1.78365	0.106215
$283$	11.6682	0.693603	0.346802	−	0.937939i	$-0.387268\pi$
$283$	11.6682	0.693603	0.346802	+	0.937939i	$0.387268\pi$
$284$	−8.67503	−0.514768
$285$	6.78444	0.401876
$286$	11.2865	0.667384
$287$	61.9985	3.65965
$288$	−1.27644	−0.0752151
$289$	−15.0020	−0.882468
$290$	4.03531	0.236962
$291$	3.25081	0.190566
$292$	11.7817	0.689471
$293$	−6.00341	−0.350723	−0.175361	−	0.984504i	$-0.556109\pi$
$293$	−6.00341	−0.350723	−0.175361	+	0.984504i	$0.556109\pi$
$294$	39.6813	2.31426
$295$	26.2899	1.53066
$296$	4.20063	0.244156
$297$	9.62040	0.558232
$298$	−15.9247	−0.922493
$299$	−10.1450	−0.586702
$300$	1.25809	0.0726360
$301$	64.2423	3.70287
$302$	0.844468	0.0485936
$303$	−33.5701	−1.92855
$304$	−1.56553	−0.0897890
$305$	−5.94913	−0.340646
$306$	1.80428	0.103144
$307$	2.34839	0.134029	0.0670147	−	0.997752i	$-0.478653\pi$
$307$	2.34839	0.134029	0.0670147	+	0.997752i	$0.478653\pi$
$308$	13.8128	0.787060
$309$	−21.1145	−1.20116
$310$	15.1109	0.858244
$311$	1.03121	0.0584747	0.0292374	−	0.999572i	$-0.490692\pi$
$311$	1.03121	0.0584747	0.0292374	+	0.999572i	$0.490692\pi$
$312$	−8.64717	−0.489550
$313$	23.2168	1.31229	0.656146	−	0.754634i	$-0.272186\pi$
$313$	23.2168	1.31229	0.656146	+	0.754634i	$0.272186\pi$
$314$	−2.33193	−0.131598
$315$	13.6890	0.771286
$316$	2.52870	0.142250
$317$	−6.94590	−0.390121	−0.195060	−	0.980791i	$-0.562490\pi$
$317$	−6.94590	−0.390121	−0.195060	+	0.980791i	$0.562490\pi$
$318$	0.702644	0.0394023
$319$	−5.19746	−0.291002
$320$	2.09562	0.117149
$321$	−19.0150	−1.06131
$322$	−12.4159	−0.691909
$323$	2.21290	0.123129
$324$	−11.2000	−0.622223
$325$	2.54392	0.141111
$326$	−0.929192	−0.0514632
$327$	−38.8556	−2.14872
$328$	−12.1150	−0.668940
$329$	4.41393	0.243348
$330$	11.6972	0.643907
$331$	−27.4752	−1.51018	−0.755088	−	0.655623i	$-0.772406\pi$
$331$	−27.4752	−1.51018	−0.755088	+	0.655623i	$0.772406\pi$
$332$	−3.91937	−0.215103
$333$	−5.36186	−0.293828
$334$	11.9538	0.654084
$335$	−13.7386	−0.750619
$336$	−10.5827	−0.577336
$337$	−13.5892	−0.740249	−0.370125	−	0.928982i	$-0.620685\pi$
$337$	−13.5892	−0.740249	−0.370125	+	0.928982i	$0.620685\pi$
$338$	−4.48498	−0.243951
$339$	−7.81794	−0.424612
$340$	−2.96220	−0.160648
$341$	−19.4628	−1.05397
$342$	1.99830	0.108056
$343$	62.3753	3.36795
$344$	−12.5535	−0.676839
$345$	−10.5142	−0.566063
$346$	17.2950	0.929785
$347$	10.4464	0.560790	0.280395	−	0.959885i	$-0.409534\pi$
$347$	10.4464	0.560790	0.280395	+	0.959885i	$0.409534\pi$
$348$	3.98204	0.213460
$349$	23.2916	1.24677	0.623386	−	0.781914i	$-0.285757\pi$
$349$	23.2916	1.24677	0.623386	+	0.781914i	$0.285757\pi$
$350$	3.11335	0.166416
$351$	−14.9039	−0.795511
$352$	−2.69915	−0.143865
$353$	−13.0497	−0.694564	−0.347282	−	0.937761i	$-0.612895\pi$
$353$	−13.0497	−0.694564	−0.347282	+	0.937761i	$0.612895\pi$
$354$	25.9429	1.37885
$355$	−18.1796	−0.964871
$356$	−16.3518	−0.866645
$357$	14.9589	0.791709
$358$	14.7303	0.778520
$359$	−6.48303	−0.342161	−0.171081	−	0.985257i	$-0.554726\pi$
$359$	−6.48303	−0.342161	−0.171081	+	0.985257i	$0.554726\pi$
$360$	−2.67494	−0.140982
$361$	−16.5491	−0.871007
$362$	−3.05742	−0.160695
$363$	7.68165	0.403182
$364$	−21.3988	−1.12160
$365$	24.6899	1.29233
$366$	−5.87059	−0.306861
$367$	−12.3562	−0.644989	−0.322495	−	0.946571i	$-0.604521\pi$
$367$	−12.3562	−0.644989	−0.322495	+	0.946571i	$0.604521\pi$
$368$	2.42616	0.126473
$369$	15.4641	0.805031
$370$	8.80292	0.457642
$371$	1.73880	0.0902743
$372$	14.9115	0.773123
$373$	6.22313	0.322222	0.161111	−	0.986936i	$-0.448492\pi$
$373$	6.22313	0.322222	0.161111	+	0.986936i	$0.448492\pi$
$374$	3.81530	0.197284
$375$	24.3047	1.25509
$376$	−0.862520	−0.0444811
$377$	8.05188	0.414693
$378$	−18.2399	−0.938162
$379$	16.4346	0.844189	0.422095	−	0.906552i	$-0.361295\pi$
$379$	16.4346	0.844189	0.422095	+	0.906552i	$0.361295\pi$
$380$	−3.28075	−0.168299
$381$	−27.5864	−1.41330
$382$	12.9836	0.664297
$383$	−22.8580	−1.16799	−0.583993	−	0.811758i	$-0.698511\pi$
$383$	−22.8580	−1.16799	−0.583993	+	0.811758i	$0.698511\pi$
$384$	2.06796	0.105530
$385$	28.9465	1.47525
$386$	−12.4585	−0.634121
$387$	16.0238	0.814537
$388$	−1.57199	−0.0798058
$389$	5.20236	0.263770	0.131885	−	0.991265i	$-0.457897\pi$
$389$	5.20236	0.263770	0.131885	+	0.991265i	$0.457897\pi$
$390$	−18.1212	−0.917602
$391$	−3.42943	−0.173434
$392$	−19.1887	−0.969174
$393$	2.88303	0.145430
$394$	−16.1477	−0.813510
$395$	5.29919	0.266631
$396$	3.44531	0.173133
$397$	37.2950	1.87178	0.935892	−	0.352288i	$-0.114596\pi$
$397$	37.2950	1.87178	0.935892	+	0.352288i	$0.114596\pi$
$398$	12.6526	0.634219
$399$	16.5675	0.829415
$400$	−0.608375	−0.0304187
$401$	8.14505	0.406744	0.203372	−	0.979102i	$-0.434810\pi$
$401$	8.14505	0.406744	0.203372	+	0.979102i	$0.434810\pi$
$402$	−13.5572	−0.676173
$403$	30.1517	1.50196
$404$	16.2335	0.807645
$405$	−23.4710	−1.16628
$406$	9.85419	0.489055
$407$	−11.3381	−0.562009
$408$	−2.92310	−0.144715
$409$	−23.4599	−1.16002	−0.580009	−	0.814610i	$-0.696951\pi$
$409$	−23.4599	−1.16002	−0.580009	+	0.814610i	$0.696951\pi$
$410$	−25.3885	−1.25385
$411$	34.2033	1.68712
$412$	10.2103	0.503026
$413$	64.1997	3.15906
$414$	−3.09686	−0.152202
$415$	−8.21350	−0.403185
$416$	4.18150	0.205015
$417$	−40.0346	−1.96050
$418$	4.22558	0.206680
$419$	−23.9200	−1.16857	−0.584284	−	0.811549i	$-0.698625\pi$
$419$	−23.9200	−1.16857	−0.584284	+	0.811549i	$0.698625\pi$
$420$	−22.1774	−1.08215
$421$	−21.5526	−1.05041	−0.525206	−	0.850975i	$-0.676012\pi$
$421$	−21.5526	−1.05041	−0.525206	+	0.850975i	$0.676012\pi$
$422$	−2.97778	−0.144956
$423$	1.10096	0.0535304
$424$	−0.339777	−0.0165010
$425$	0.859950	0.0417137
$426$	−17.9396	−0.869175
$427$	−14.5277	−0.703045
$428$	9.19506	0.444460
$429$	23.3400	1.12686
$430$	−26.3074	−1.26865
$431$	12.6070	0.607257	0.303628	−	0.952791i	$-0.401802\pi$
$431$	12.6070	0.607257	0.303628	+	0.952791i	$0.401802\pi$
$432$	3.56424	0.171485
$433$	39.9578	1.92025	0.960123	−	0.279576i	$-0.0901940\pi$
$433$	39.9578	1.92025	0.960123	+	0.279576i	$0.0901940\pi$
$434$	36.9008	1.77129
$435$	8.34485	0.400105
$436$	18.7894	0.899848
$437$	−3.79822	−0.181694
$438$	24.3640	1.16416
$439$	−5.36348	−0.255985	−0.127992	−	0.991775i	$-0.540853\pi$
$439$	−5.36348	−0.255985	−0.127992	+	0.991775i	$0.540853\pi$
$440$	−5.65639	−0.269658
$441$	24.4932	1.16634
$442$	−5.91064	−0.281140
$443$	7.48982	0.355852	0.177926	−	0.984044i	$-0.443061\pi$
$443$	7.48982	0.355852	0.177926	+	0.984044i	$0.443061\pi$
$444$	8.68671	0.412253
$445$	−34.2672	−1.62442
$446$	−11.2347	−0.531980
$447$	−32.9316	−1.55761
$448$	5.11749	0.241778
$449$	0.0455603	0.00215012	0.00107506	−	0.999999i	$-0.499658\pi$
$449$	0.0455603	0.00215012	0.00107506	+	0.999999i	$0.499658\pi$
$450$	0.776556	0.0366072
$451$	32.7002	1.53979
$452$	3.78051	0.177820
$453$	1.74632	0.0820494
$454$	−6.97184	−0.327204
$455$	−44.8437	−2.10231
$456$	−3.23744	−0.151607
$457$	6.85601	0.320711	0.160355	−	0.987059i	$-0.448736\pi$
$457$	6.85601	0.320711	0.160355	+	0.987059i	$0.448736\pi$
$458$	−19.3042	−0.902026
$459$	−5.03812	−0.235159
$460$	5.08432	0.237058
$461$	12.2650	0.571237	0.285618	−	0.958343i	$-0.407801\pi$
$461$	12.2650	0.571237	0.285618	+	0.958343i	$0.407801\pi$
$462$	28.5644	1.32893
$463$	20.1015	0.934198	0.467099	−	0.884205i	$-0.345299\pi$
$463$	20.1015	0.934198	0.467099	+	0.884205i	$0.345299\pi$
$464$	−1.92559	−0.0893934
$465$	31.2488	1.44913
$466$	−20.6021	−0.954376
$467$	17.1260	0.792497	0.396249	−	0.918143i	$-0.370312\pi$
$467$	17.1260	0.792497	0.396249	+	0.918143i	$0.370312\pi$
$468$	−5.33745	−0.246724
$469$	−33.5495	−1.54917
$470$	−1.80751	−0.0833744
$471$	−4.82233	−0.222201
$472$	−12.5452	−0.577438
$473$	33.8837	1.55798
$474$	5.22924	0.240187
$475$	0.952427	0.0437003
$476$	−7.23367	−0.331555
$477$	0.433706	0.0198580
$478$	28.3927	1.29865
$479$	−10.5932	−0.484014	−0.242007	−	0.970275i	$-0.577806\pi$
$479$	−10.5932	−0.484014	−0.242007	+	0.970275i	$0.577806\pi$
$480$	4.33365	0.197803
$481$	17.5649	0.800892
$482$	−16.8805	−0.768886
$483$	−25.6755	−1.16827
$484$	−3.71461	−0.168846
$485$	−3.29430	−0.149586
$486$	−12.4684	−0.565580
$487$	−18.4627	−0.836626	−0.418313	−	0.908303i	$-0.637378\pi$
$487$	−18.4627	−0.836626	−0.418313	+	0.908303i	$0.637378\pi$
$488$	2.83884	0.128508
$489$	−1.92153	−0.0868945
$490$	−40.2121	−1.81660
$491$	−34.5177	−1.55776	−0.778882	−	0.627171i	$-0.784213\pi$
$491$	−34.5177	−1.55776	−0.778882	+	0.627171i	$0.784213\pi$
$492$	−25.0533	−1.12949
$493$	2.72186	0.122587
$494$	−6.54625	−0.294530
$495$	7.22006	0.324517
$496$	−7.21072	−0.323771
$497$	−44.3943	−1.99136
$498$	−8.10508	−0.363197
$499$	−0.994535	−0.0445215	−0.0222607	−	0.999752i	$-0.507086\pi$
$499$	−0.994535	−0.0445215	−0.0222607	+	0.999752i	$0.507086\pi$
$500$	−11.7530	−0.525611
$501$	24.7200	1.10441
$502$	15.0144	0.670126
$503$	−23.5820	−1.05147	−0.525734	−	0.850649i	$-0.676209\pi$
$503$	−23.5820	−1.05147	−0.525734	+	0.850649i	$0.676209\pi$
$504$	−6.53218	−0.290966
$505$	34.0192	1.51383
$506$	−6.54857	−0.291120
$507$	−9.27474	−0.411906
$508$	13.3399	0.591864
$509$	25.6010	1.13474	0.567372	−	0.823462i	$-0.307960\pi$
$509$	25.6010	1.13474	0.567372	+	0.823462i	$0.307960\pi$
$510$	−6.12570	−0.271251
$511$	60.2926	2.66719
$512$	−1.00000	−0.0441942
$513$	−5.57991	−0.246359
$514$	14.5963	0.643814
$515$	21.3970	0.942862
$516$	−25.9601	−1.14283
$517$	2.32807	0.102388
$518$	21.4966	0.944508
$519$	35.7653	1.56992
$520$	8.76285	0.384276
$521$	21.3303	0.934497	0.467248	−	0.884126i	$-0.345245\pi$
$521$	21.3303	0.934497	0.467248	+	0.884126i	$0.345245\pi$
$522$	2.45791	0.107580
$523$	26.7258	1.16864	0.584318	−	0.811524i	$-0.301362\pi$
$523$	26.7258	1.16864	0.584318	+	0.811524i	$0.301362\pi$
$524$	−1.39415	−0.0609035
$525$	6.43827	0.280989
$526$	−22.1757	−0.966907
$527$	10.1925	0.443992
$528$	−5.58172	−0.242913
$529$	−17.1137	−0.744075
$530$	−0.712044	−0.0309292
$531$	16.0132	0.694913
$532$	−8.01155	−0.347345
$533$	−50.6590	−2.19429
$534$	−33.8149	−1.46331
$535$	19.2693	0.833087
$536$	6.55585	0.283170
$537$	30.4616	1.31452
$538$	−1.00705	−0.0434172
$539$	51.7930	2.23088
$540$	7.46929	0.321427
$541$	−26.9847	−1.16016	−0.580081	−	0.814559i	$-0.696979\pi$
$541$	−26.9847	−1.16016	−0.580081	+	0.814559i	$0.696979\pi$
$542$	−9.20153	−0.395240
$543$	−6.32262	−0.271329
$544$	1.41352	0.0606042
$545$	39.3754	1.68666
$546$	−44.2518	−1.89380
$547$	−44.8968	−1.91965	−0.959823	−	0.280605i	$-0.909465\pi$
$547$	−44.8968	−1.91965	−0.959823	+	0.280605i	$0.909465\pi$
$548$	−16.5396	−0.706538
$549$	−3.62361	−0.154652
$550$	1.64209	0.0700191
$551$	3.01456	0.128425
$552$	5.01720	0.213546
$553$	12.9406	0.550289
$554$	−19.0330	−0.808633
$555$	18.2041	0.772719
$556$	19.3595	0.821027
$557$	19.5733	0.829348	0.414674	−	0.909970i	$-0.363896\pi$
$557$	19.5733	0.829348	0.414674	+	0.909970i	$0.363896\pi$
$558$	9.20408	0.389640
$559$	−52.4925	−2.22020
$560$	10.7243	0.453185
$561$	7.88987	0.333110
$562$	−15.6191	−0.658852
$563$	13.2142	0.556911	0.278456	−	0.960449i	$-0.410177\pi$
$563$	13.2142	0.556911	0.278456	+	0.960449i	$0.410177\pi$
$564$	−1.78365	−0.0751054
$565$	7.92252	0.333303
$566$	−11.6682	−0.490451
$567$	−57.3160	−2.40704
$568$	8.67503	0.363996
$569$	−43.2373	−1.81260	−0.906302	−	0.422631i	$-0.861107\pi$
$569$	−43.2373	−1.81260	−0.906302	+	0.422631i	$0.861107\pi$
$570$	−6.78444	−0.284169
$571$	−24.6281	−1.03066	−0.515328	−	0.856993i	$-0.672330\pi$
$571$	−24.6281	−1.03066	−0.515328	+	0.856993i	$0.672330\pi$
$572$	−11.2865	−0.471912
$573$	26.8494	1.12165
$574$	−61.9985	−2.58777
$575$	−1.47602	−0.0615542
$576$	1.27644	0.0531851
$577$	21.0033	0.874380	0.437190	−	0.899369i	$-0.355974\pi$
$577$	21.0033	0.874380	0.437190	+	0.899369i	$0.355974\pi$
$578$	15.0020	0.623999
$579$	−25.7636	−1.07070
$580$	−4.03531	−0.167557
$581$	−20.0573	−0.832117
$582$	−3.25081	−0.134750
$583$	0.917108	0.0379827
$584$	−11.7817	−0.487529
$585$	−11.1853	−0.462454
$586$	6.00341	0.247998
$587$	−44.7176	−1.84569	−0.922846	−	0.385169i	$-0.874143\pi$
$587$	−44.7176	−1.84569	−0.922846	+	0.385169i	$0.874143\pi$
$588$	−39.6813	−1.63643
$589$	11.2886	0.465138
$590$	−26.2899	−1.08234
$591$	−33.3928	−1.37360
$592$	−4.20063	−0.172645
$593$	31.2003	1.28124	0.640621	−	0.767857i	$-0.278677\pi$
$593$	31.2003	1.28124	0.640621	+	0.767857i	$0.278677\pi$
$594$	−9.62040	−0.394730
$595$	−15.1590	−0.621459
$596$	15.9247	0.652301
$597$	26.1651	1.07087
$598$	10.1450	0.414861
$599$	8.22556	0.336087	0.168044	−	0.985780i	$-0.446255\pi$
$599$	8.22556	0.336087	0.168044	+	0.985780i	$0.446255\pi$
$600$	−1.25809	−0.0513614
$601$	11.7345	0.478659	0.239329	−	0.970938i	$-0.423072\pi$
$601$	11.7345	0.478659	0.239329	+	0.970938i	$0.423072\pi$
$602$	−64.2423	−2.61832
$603$	−8.36817	−0.340778
$604$	−0.844468	−0.0343609
$605$	−7.78441	−0.316481
$606$	33.5701	1.36369
$607$	−43.0809	−1.74860	−0.874300	−	0.485387i	$-0.838679\pi$
$607$	−43.0809	−1.74860	−0.874300	+	0.485387i	$0.838679\pi$
$608$	1.56553	0.0634904
$609$	20.3780	0.825760
$610$	5.94913	0.240873
$611$	−3.60663	−0.145909
$612$	−1.80428	−0.0729336
$613$	−43.2865	−1.74833	−0.874164	−	0.485632i	$-0.838590\pi$
$613$	−43.2865	−1.74833	−0.874164	+	0.485632i	$0.838590\pi$
$614$	−2.34839	−0.0947731
$615$	−52.5023	−2.11710
$616$	−13.8128	−0.556535
$617$	−18.0217	−0.725527	−0.362763	−	0.931881i	$-0.618167\pi$
$617$	−18.0217	−0.725527	−0.362763	+	0.931881i	$0.618167\pi$
$618$	21.1145	0.849349
$619$	21.4687	0.862898	0.431449	−	0.902137i	$-0.358002\pi$
$619$	21.4687	0.862898	0.431449	+	0.902137i	$0.358002\pi$
$620$	−15.1109	−0.606870
$621$	8.64743	0.347010
$622$	−1.03121	−0.0413479
$623$	−83.6802	−3.35258
$624$	8.64717	0.346164
$625$	−21.5880	−0.863520
$626$	−23.2168	−0.927930
$627$	8.73832	0.348975
$628$	2.33193	0.0930541
$629$	5.93767	0.236750
$630$	−13.6890	−0.545382
$631$	29.7959	1.18616	0.593078	−	0.805145i	$-0.297912\pi$
$631$	29.7959	1.18616	0.593078	+	0.805145i	$0.297912\pi$
$632$	−2.52870	−0.100586
$633$	−6.15793	−0.244756
$634$	6.94590	0.275857
$635$	27.9555	1.10938
$636$	−0.702644	−0.0278617
$637$	−80.2375	−3.17912
$638$	5.19746	0.205769
$639$	−11.0732	−0.438048
$640$	−2.09562	−0.0828367
$641$	−2.95869	−0.116861	−0.0584307	−	0.998291i	$-0.518610\pi$
$641$	−2.95869	−0.116861	−0.0584307	+	0.998291i	$0.518610\pi$
$642$	19.0150	0.750461
$643$	−19.4824	−0.768310	−0.384155	−	0.923269i	$-0.625507\pi$
$643$	−19.4824	−0.768310	−0.384155	+	0.923269i	$0.625507\pi$
$644$	12.4159	0.489254
$645$	−54.4025	−2.14210
$646$	−2.21290	−0.0870654
$647$	44.7305	1.75854	0.879270	−	0.476325i	$-0.158031\pi$
$647$	44.7305	1.75854	0.879270	+	0.476325i	$0.158031\pi$
$648$	11.2000	0.439978
$649$	33.8612	1.32917
$650$	−2.54392	−0.0997809
$651$	76.3092	2.99079
$652$	0.929192	0.0363900
$653$	6.68410	0.261569	0.130785	−	0.991411i	$-0.458250\pi$
$653$	6.68410	0.261569	0.130785	+	0.991411i	$0.458250\pi$
$654$	38.8556	1.51938
$655$	−2.92160	−0.114156
$656$	12.1150	0.473012
$657$	15.0386	0.586714
$658$	−4.41393	−0.172073
$659$	−5.03922	−0.196300	−0.0981500	−	0.995172i	$-0.531292\pi$
$659$	−5.03922	−0.196300	−0.0981500	+	0.995172i	$0.531292\pi$
$660$	−11.6972	−0.455311
$661$	21.0109	0.817230	0.408615	−	0.912707i	$-0.366012\pi$
$661$	21.0109	0.817230	0.408615	+	0.912707i	$0.366012\pi$
$662$	27.4752	1.06786
$663$	−12.2229	−0.474700
$664$	3.91937	0.152101
$665$	−16.7892	−0.651056
$666$	5.36186	0.207768
$667$	−4.67181	−0.180893
$668$	−11.9538	−0.462507
$669$	−23.2329	−0.898237
$670$	13.7386	0.530768
$671$	−7.66244	−0.295805
$672$	10.5827	0.408238
$673$	3.18323	0.122704	0.0613522	−	0.998116i	$-0.480459\pi$
$673$	3.18323	0.122704	0.0613522	+	0.998116i	$0.480459\pi$
$674$	13.5892	0.523435
$675$	−2.16839	−0.0834615
$676$	4.48498	0.172499
$677$	26.6420	1.02394	0.511968	−	0.859005i	$-0.328917\pi$
$677$	26.6420	1.02394	0.511968	+	0.859005i	$0.328917\pi$
$678$	7.81794	0.300246
$679$	−8.04464	−0.308725
$680$	2.96220	0.113595
$681$	−14.4175	−0.552478
$682$	19.4628	0.745269
$683$	43.1979	1.65292	0.826462	−	0.562993i	$-0.190350\pi$
$683$	43.1979	1.65292	0.826462	+	0.562993i	$0.190350\pi$
$684$	−1.99830	−0.0764071
$685$	−34.6608	−1.32432
$686$	−62.3753	−2.38150
$687$	−39.9202	−1.52305
$688$	12.5535	0.478598
$689$	−1.42078	−0.0541274
$690$	10.5142	0.400267
$691$	11.1664	0.424788	0.212394	−	0.977184i	$-0.431874\pi$
$691$	11.1664	0.424788	0.212394	+	0.977184i	$0.431874\pi$
$692$	−17.2950	−0.657457
$693$	17.6313	0.669758
$694$	−10.4464	−0.396539
$695$	40.5702	1.53892
$696$	−3.98204	−0.150939
$697$	−17.1248	−0.648649
$698$	−23.2916	−0.881601
$699$	−42.6043	−1.61144
$700$	−3.11335	−0.117674
$701$	5.08009	0.191872	0.0959362	−	0.995387i	$-0.469416\pi$
$701$	5.08009	0.191872	0.0959362	+	0.995387i	$0.469416\pi$
$702$	14.9039	0.562511
$703$	6.57619	0.248026
$704$	2.69915	0.101728
$705$	−3.73786	−0.140776
$706$	13.0497	0.491131
$707$	83.0745	3.12434
$708$	−25.9429	−0.974992
$709$	11.8468	0.444916	0.222458	−	0.974942i	$-0.428592\pi$
$709$	11.8468	0.444916	0.222458	+	0.974942i	$0.428592\pi$
$710$	18.1796	0.682267
$711$	3.22774	0.121050
$712$	16.3518	0.612811
$713$	−17.4944	−0.655171
$714$	−14.9589	−0.559823
$715$	−23.6522	−0.884542
$716$	−14.7303	−0.550497
$717$	58.7148	2.19274
$718$	6.48303	0.241945
$719$	1.97547	0.0736725	0.0368362	−	0.999321i	$-0.488272\pi$
$719$	1.97547	0.0736725	0.0368362	+	0.999321i	$0.488272\pi$
$720$	2.67494	0.0996891
$721$	52.2512	1.94593
$722$	16.5491	0.615895
$723$	−34.9081	−1.29825
$724$	3.05742	0.113628
$725$	1.17148	0.0435078
$726$	−7.68165	−0.285093
$727$	−37.4881	−1.39036	−0.695179	−	0.718837i	$-0.744675\pi$
$727$	−37.4881	−1.39036	−0.695179	+	0.718837i	$0.744675\pi$
$728$	21.3988	0.793092
$729$	7.81589	0.289477
$730$	−24.6899	−0.913815
$731$	−17.7446	−0.656308
$732$	5.87059	0.216983
$733$	−11.2285	−0.414733	−0.207367	−	0.978263i	$-0.566489\pi$
$733$	−11.2285	−0.414733	−0.207367	+	0.978263i	$0.566489\pi$
$734$	12.3562	0.456076
$735$	−83.1570	−3.06729
$736$	−2.42616	−0.0894296
$737$	−17.6952	−0.651811
$738$	−15.4641	−0.569243
$739$	−18.6856	−0.687360	−0.343680	−	0.939087i	$-0.611674\pi$
$739$	−18.6856	−0.687360	−0.343680	+	0.939087i	$0.611674\pi$
$740$	−8.80292	−0.323602
$741$	−13.5374	−0.497307
$742$	−1.73880	−0.0638335
$743$	4.42146	0.162208	0.0811038	−	0.996706i	$-0.474155\pi$
$743$	4.42146	0.162208	0.0811038	+	0.996706i	$0.474155\pi$
$744$	−14.9115	−0.546681
$745$	33.3721	1.22266
$746$	−6.22313	−0.227845
$747$	−5.00285	−0.183045
$748$	−3.81530	−0.139501
$749$	47.0556	1.71937
$750$	−24.3047	−0.887484
$751$	−13.0638	−0.476705	−0.238352	−	0.971179i	$-0.576607\pi$
$751$	−13.0638	−0.476705	−0.238352	+	0.971179i	$0.576607\pi$
$752$	0.862520	0.0314529
$753$	31.0491	1.13149
$754$	−8.05188	−0.293232
$755$	−1.76968	−0.0644054
$756$	18.2399	0.663380
$757$	−40.7779	−1.48210	−0.741048	−	0.671452i	$-0.765671\pi$
$757$	−40.7779	−1.48210	−0.741048	+	0.671452i	$0.765671\pi$
$758$	−16.4346	−0.596932
$759$	−13.5422	−0.491549
$760$	3.28075	0.119005
$761$	32.2526	1.16915	0.584577	−	0.811338i	$-0.301260\pi$
$761$	32.2526	1.16915	0.584577	+	0.811338i	$0.301260\pi$
$762$	27.5864	0.999351
$763$	96.1544	3.48102
$764$	−12.9836	−0.469729
$765$	−3.78108	−0.136705
$766$	22.8580	0.825892
$767$	−52.4577	−1.89414
$768$	−2.06796	−0.0746209
$769$	5.59127	0.201626	0.100813	−	0.994905i	$-0.467856\pi$
$769$	5.59127	0.201626	0.100813	+	0.994905i	$0.467856\pi$
$770$	−28.9465	−1.04316
$771$	30.1845	1.08707
$772$	12.4585	0.448391
$773$	−5.51404	−0.198326	−0.0991631	−	0.995071i	$-0.531617\pi$
$773$	−5.51404	−0.198326	−0.0991631	+	0.995071i	$0.531617\pi$
$774$	−16.0238	−0.575965
$775$	4.38682	0.157579
$776$	1.57199	0.0564312
$777$	44.4541	1.59478
$778$	−5.20236	−0.186514
$779$	−18.9664	−0.679541
$780$	18.1212	0.648842
$781$	−23.4152	−0.837860
$782$	3.42943	0.122636
$783$	−6.86328	−0.245273
$784$	19.1887	0.685309
$785$	4.88684	0.174419
$786$	−2.88303	−0.102834
$787$	22.0901	0.787428	0.393714	−	0.919233i	$-0.371190\pi$
$787$	22.0901	0.787428	0.393714	+	0.919233i	$0.371190\pi$
$788$	16.1477	0.575239
$789$	−45.8584	−1.63260
$790$	−5.29919	−0.188537
$791$	19.3467	0.687890
$792$	−3.44531	−0.122424
$793$	11.8706	0.421538
$794$	−37.2950	−1.32355
$795$	−1.47248	−0.0522233
$796$	−12.6526	−0.448461
$797$	−7.90653	−0.280064	−0.140032	−	0.990147i	$-0.544720\pi$
$797$	−7.90653	−0.280064	−0.140032	+	0.990147i	$0.544720\pi$
$798$	−16.5675	−0.586485
$799$	−1.21919	−0.0431318
$800$	0.608375	0.0215093
$801$	−20.8722	−0.737482
$802$	−8.14505	−0.287612
$803$	31.8005	1.12221
$804$	13.5572	0.478126
$805$	26.0189	0.917047
$806$	−30.1517	−1.06205
$807$	−2.08255	−0.0733091
$808$	−16.2335	−0.571091
$809$	−45.1770	−1.58834	−0.794170	−	0.607696i	$-0.792094\pi$
$809$	−45.1770	−1.58834	−0.794170	+	0.607696i	$0.792094\pi$
$810$	23.4710	0.824687
$811$	−7.19538	−0.252664	−0.126332	−	0.991988i	$-0.540320\pi$
$811$	−7.19538	−0.252664	−0.126332	+	0.991988i	$0.540320\pi$
$812$	−9.85419	−0.345814
$813$	−19.0284	−0.667354
$814$	11.3381	0.397400
$815$	1.94723	0.0682086
$816$	2.92310	0.102329
$817$	−19.6528	−0.687565
$818$	23.4599	0.820256
$819$	−27.3143	−0.954440
$820$	25.3885	0.886605
$821$	−21.3254	−0.744262	−0.372131	−	0.928180i	$-0.621373\pi$
$821$	−21.3254	−0.744262	−0.372131	+	0.928180i	$0.621373\pi$
$822$	−34.2033	−1.19298
$823$	25.4723	0.887907	0.443954	−	0.896050i	$-0.353576\pi$
$823$	25.4723	0.887907	0.443954	+	0.896050i	$0.353576\pi$
$824$	−10.2103	−0.355693
$825$	3.39578	0.118226
$826$	−64.1997	−2.23379
$827$	−16.6157	−0.577783	−0.288891	−	0.957362i	$-0.593287\pi$
$827$	−16.6157	−0.577783	−0.288891	+	0.957362i	$0.593287\pi$
$828$	3.09686	0.107623
$829$	32.8391	1.14055	0.570275	−	0.821454i	$-0.306837\pi$
$829$	32.8391	1.14055	0.570275	+	0.821454i	$0.306837\pi$
$830$	8.21350	0.285095
$831$	−39.3593	−1.36536
$832$	−4.18150	−0.144968
$833$	−27.1235	−0.939775
$834$	40.0346	1.38629
$835$	−25.0507	−0.866914
$836$	−4.22558	−0.146145
$837$	−25.7008	−0.888348
$838$	23.9200	0.826303
$839$	−0.440914	−0.0152220	−0.00761101	−	0.999971i	$-0.502423\pi$
$839$	−0.440914	−0.0152220	−0.00761101	+	0.999971i	$0.502423\pi$
$840$	22.1774	0.765193
$841$	−25.2921	−0.872141
$842$	21.5526	0.742753
$843$	−32.2996	−1.11246
$844$	2.97778	0.102500
$845$	9.39881	0.323329
$846$	−1.10096	−0.0378517
$847$	−19.0095	−0.653173
$848$	0.339777	0.0116680
$849$	−24.1293	−0.828117
$850$	−0.859950	−0.0294960
$851$	−10.1914	−0.349357
$852$	17.9396	0.614600
$853$	12.4764	0.427182	0.213591	−	0.976923i	$-0.431484\pi$
$853$	12.4764	0.427182	0.213591	+	0.976923i	$0.431484\pi$
$854$	14.5277	0.497128
$855$	−4.18769	−0.143216
$856$	−9.19506	−0.314281
$857$	−8.61992	−0.294451	−0.147225	−	0.989103i	$-0.547034\pi$
$857$	−8.61992	−0.294451	−0.147225	+	0.989103i	$0.547034\pi$
$858$	−23.3400	−0.796813
$859$	0.640155	0.0218418	0.0109209	−	0.999940i	$-0.496524\pi$
$859$	0.640155	0.0218418	0.0109209	+	0.999940i	$0.496524\pi$
$860$	26.3074	0.897074
$861$	−128.210	−4.36939
$862$	−12.6070	−0.429395
$863$	18.4248	0.627186	0.313593	−	0.949557i	$-0.398467\pi$
$863$	18.4248	0.627186	0.313593	+	0.949557i	$0.398467\pi$
$864$	−3.56424	−0.121258
$865$	−36.2437	−1.23232
$866$	−39.9578	−1.35782
$867$	31.0234	1.05361
$868$	−36.9008	−1.25249
$869$	6.82533	0.231533
$870$	−8.34485	−0.282917
$871$	27.4133	0.928865
$872$	−18.7894	−0.636289
$873$	−2.00656	−0.0679117
$874$	3.79822	0.128477
$875$	−60.1459	−2.03330
$876$	−24.3640	−0.823183
$877$	29.1281	0.983584	0.491792	−	0.870713i	$-0.336342\pi$
$877$	29.1281	0.983584	0.491792	+	0.870713i	$0.336342\pi$
$878$	5.36348	0.181009
$879$	12.4148	0.418740
$880$	5.65639	0.190677
$881$	−49.5713	−1.67010	−0.835050	−	0.550174i	$-0.814561\pi$
$881$	−49.5713	−1.67010	−0.835050	+	0.550174i	$0.814561\pi$
$882$	−24.4932	−0.824730
$883$	−24.2945	−0.817576	−0.408788	−	0.912629i	$-0.634048\pi$
$883$	−24.2945	−0.817576	−0.408788	+	0.912629i	$0.634048\pi$
$884$	5.91064	0.198796
$885$	−54.3664	−1.82751
$886$	−7.48982	−0.251626
$887$	48.5000	1.62847	0.814236	−	0.580535i	$-0.197156\pi$
$887$	48.5000	1.62847	0.814236	+	0.580535i	$0.197156\pi$
$888$	−8.68671	−0.291507
$889$	68.2670	2.28960
$890$	34.2672	1.14864
$891$	−30.2305	−1.01276
$892$	11.2347	0.376167
$893$	−1.35030	−0.0451860
$894$	32.9316	1.10140
$895$	−30.8691	−1.03184
$896$	−5.11749	−0.170963
$897$	20.9795	0.700484
$898$	−0.0455603	−0.00152037
$899$	13.8849	0.463088
$900$	−0.776556	−0.0258852
$901$	−0.480282	−0.0160005
$902$	−32.7002	−1.08880
$903$	−132.850	−4.42098
$904$	−3.78051	−0.125738
$905$	6.40720	0.212982
$906$	−1.74632	−0.0580177
$907$	26.7349	0.887718	0.443859	−	0.896097i	$-0.353609\pi$
$907$	26.7349	0.887718	0.443859	+	0.896097i	$0.353609\pi$
$908$	6.97184	0.231369
$909$	20.7211	0.687275
$910$	44.8437	1.48656
$911$	42.3325	1.40254	0.701270	−	0.712896i	$-0.252617\pi$
$911$	42.3325	1.40254	0.701270	+	0.712896i	$0.252617\pi$
$912$	3.23744	0.107202
$913$	−10.5789	−0.350112
$914$	−6.85601	−0.226777
$915$	12.3025	0.406709
$916$	19.3042	0.637829
$917$	−7.13452	−0.235603
$918$	5.03812	0.166283
$919$	−21.1932	−0.699099	−0.349550	−	0.936918i	$-0.613665\pi$
$919$	−21.1932	−0.699099	−0.349550	+	0.936918i	$0.613665\pi$
$920$	−5.08432	−0.167625
$921$	−4.85636	−0.160022
$922$	−12.2650	−0.403925
$923$	36.2747	1.19399
$924$	−28.5644	−0.939698
$925$	2.55556	0.0840262
$926$	−20.1015	−0.660578
$927$	13.0329	0.428056
$928$	1.92559	0.0632107
$929$	−44.7966	−1.46973	−0.734865	−	0.678214i	$-0.762754\pi$
$929$	−44.7966	−1.46973	−0.734865	+	0.678214i	$0.762754\pi$
$930$	−31.2488	−1.02469
$931$	−30.0403	−0.984532
$932$	20.6021	0.674846
$933$	−2.13250	−0.0698150
$934$	−17.1260	−0.560380
$935$	−7.99541	−0.261478
$936$	5.33745	0.174460
$937$	51.3778	1.67844	0.839219	−	0.543793i	$-0.183012\pi$
$937$	51.3778	1.67844	0.839219	+	0.543793i	$0.183012\pi$
$938$	33.5495	1.09543
$939$	−48.0114	−1.56679
$940$	1.80751	0.0589546
$941$	42.5167	1.38601	0.693003	−	0.720935i	$-0.256287\pi$
$941$	42.5167	1.38601	0.693003	+	0.720935i	$0.256287\pi$
$942$	4.82233	0.157120
$943$	29.3931	0.957170
$944$	12.5452	0.408310
$945$	38.2240	1.24343
$946$	−33.8837	−1.10165
$947$	−9.77709	−0.317713	−0.158856	−	0.987302i	$-0.550781\pi$
$947$	−9.77709	−0.317713	−0.158856	+	0.987302i	$0.550781\pi$
$948$	−5.22924	−0.169838
$949$	−49.2652	−1.59921
$950$	−0.952427	−0.0309008
$951$	14.3638	0.465779
$952$	7.23367	0.234444
$953$	43.5069	1.40933	0.704663	−	0.709542i	$-0.251098\pi$
$953$	43.5069	1.40933	0.704663	+	0.709542i	$0.251098\pi$
$954$	−0.433706	−0.0140418
$955$	−27.2086	−0.880450
$956$	−28.3927	−0.918285
$957$	10.7481	0.347437
$958$	10.5932	0.342250
$959$	−84.6414	−2.73321
$960$	−4.33365	−0.139868
$961$	20.9945	0.677243
$962$	−17.5649	−0.566316
$963$	11.7370	0.378219
$964$	16.8805	0.543684
$965$	26.1083	0.840456
$966$	25.6755	0.826094
$967$	−47.3942	−1.52410	−0.762048	−	0.647521i	$-0.775806\pi$
$967$	−47.3942	−1.52410	−0.762048	+	0.647521i	$0.775806\pi$
$968$	3.71461	0.119392
$969$	−4.57618	−0.147008
$970$	3.29430	0.105774
$971$	−5.79391	−0.185935	−0.0929677	−	0.995669i	$-0.529635\pi$
$971$	−5.79391	−0.185935	−0.0929677	+	0.995669i	$0.529635\pi$
$972$	12.4684	0.399925
$973$	99.0721	3.17610
$974$	18.4627	0.591584
$975$	−5.26072	−0.168478
$976$	−2.83884	−0.0908690
$977$	−15.0461	−0.481367	−0.240684	−	0.970604i	$-0.577372\pi$
$977$	−15.0461	−0.481367	−0.240684	+	0.970604i	$0.577372\pi$
$978$	1.92153	0.0614437
$979$	−44.1360	−1.41059
$980$	40.2121	1.28453
$981$	23.9836	0.765737
$982$	34.5177	1.10151
$983$	−23.7900	−0.758782	−0.379391	−	0.925236i	$-0.623866\pi$
$983$	−23.7900	−0.758782	−0.379391	+	0.925236i	$0.623866\pi$
$984$	25.0533	0.798672
$985$	33.8395	1.07822
$986$	−2.72186	−0.0866818
$987$	−9.12782	−0.290542
$988$	6.54625	0.208264
$989$	30.4569	0.968472
$990$	−7.22006	−0.229468
$991$	52.9147	1.68089	0.840446	−	0.541896i	$-0.182293\pi$
$991$	52.9147	1.68089	0.840446	+	0.541896i	$0.182293\pi$
$992$	7.21072	0.228941
$993$	56.8176	1.80305
$994$	44.3943	1.40810
$995$	−26.5151	−0.840586
$996$	8.10508	0.256819
$997$	10.1601	0.321773	0.160887	−	0.986973i	$-0.448565\pi$
$997$	10.1601	0.321773	0.160887	+	0.986973i	$0.448565\pi$
$998$	0.994535	0.0314814
$999$	−14.9720	−0.473694

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4022.2.a.e.1.9	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4022.2.a.e.1.9	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.06796 q^{3} +1.00000 q^{4} +2.09562 q^{5} +2.06796 q^{6} +5.11749 q^{7} -1.00000 q^{8} +1.27644 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.06796 q^{3} +1.00000 q^{4} +2.09562 q^{5} +2.06796 q^{6} +5.11749 q^{7} -1.00000 q^{8} +1.27644 q^{9} -2.09562 q^{10} +2.69915 q^{11} -2.06796 q^{12} -4.18150 q^{13} -5.11749 q^{14} -4.33365 q^{15} +1.00000 q^{16} -1.41352 q^{17} -1.27644 q^{18} -1.56553 q^{19} +2.09562 q^{20} -10.5827 q^{21} -2.69915 q^{22} +2.42616 q^{23} +2.06796 q^{24} -0.608375 q^{25} +4.18150 q^{26} +3.56424 q^{27} +5.11749 q^{28} -1.92559 q^{29} +4.33365 q^{30} -7.21072 q^{31} -1.00000 q^{32} -5.58172 q^{33} +1.41352 q^{34} +10.7243 q^{35} +1.27644 q^{36} -4.20063 q^{37} +1.56553 q^{38} +8.64717 q^{39} -2.09562 q^{40} +12.1150 q^{41} +10.5827 q^{42} +12.5535 q^{43} +2.69915 q^{44} +2.67494 q^{45} -2.42616 q^{46} +0.862520 q^{47} -2.06796 q^{48} +19.1887 q^{49} +0.608375 q^{50} +2.92310 q^{51} -4.18150 q^{52} +0.339777 q^{53} -3.56424 q^{54} +5.65639 q^{55} -5.11749 q^{56} +3.23744 q^{57} +1.92559 q^{58} +12.5452 q^{59} -4.33365 q^{60} -2.83884 q^{61} +7.21072 q^{62} +6.53218 q^{63} +1.00000 q^{64} -8.76285 q^{65} +5.58172 q^{66} -6.55585 q^{67} -1.41352 q^{68} -5.01720 q^{69} -10.7243 q^{70} -8.67503 q^{71} -1.27644 q^{72} +11.7817 q^{73} +4.20063 q^{74} +1.25809 q^{75} -1.56553 q^{76} +13.8128 q^{77} -8.64717 q^{78} +2.52870 q^{79} +2.09562 q^{80} -11.2000 q^{81} -12.1150 q^{82} -3.91937 q^{83} -10.5827 q^{84} -2.96220 q^{85} -12.5535 q^{86} +3.98204 q^{87} -2.69915 q^{88} -16.3518 q^{89} -2.67494 q^{90} -21.3988 q^{91} +2.42616 q^{92} +14.9115 q^{93} -0.862520 q^{94} -3.28075 q^{95} +2.06796 q^{96} -1.57199 q^{97} -19.1887 q^{98} +3.44531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9}+O(q^{10}) \) 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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