LMFDB - Embedded newform 4006.2.a.f.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4006, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4006 = 2 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4006.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.9880710497$
Analytic rank:	$1$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	4006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.466489 q^{3} +1.00000 q^{4} +1.06478 q^{5} -0.466489 q^{6} +2.04789 q^{7} +1.00000 q^{8} -2.78239 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.466489 q^{3} +1.00000 q^{4} +1.06478 q^{5} -0.466489 q^{6} +2.04789 q^{7} +1.00000 q^{8} -2.78239 q^{9} +1.06478 q^{10} -1.55040 q^{11} -0.466489 q^{12} -4.04229 q^{13} +2.04789 q^{14} -0.496709 q^{15} +1.00000 q^{16} -6.58820 q^{17} -2.78239 q^{18} +5.97262 q^{19} +1.06478 q^{20} -0.955318 q^{21} -1.55040 q^{22} -1.66591 q^{23} -0.466489 q^{24} -3.86624 q^{25} -4.04229 q^{26} +2.69742 q^{27} +2.04789 q^{28} -6.46605 q^{29} -0.496709 q^{30} +9.29719 q^{31} +1.00000 q^{32} +0.723247 q^{33} -6.58820 q^{34} +2.18055 q^{35} -2.78239 q^{36} -9.50873 q^{37} +5.97262 q^{38} +1.88568 q^{39} +1.06478 q^{40} -3.87954 q^{41} -0.955318 q^{42} -8.89119 q^{43} -1.55040 q^{44} -2.96263 q^{45} -1.66591 q^{46} -4.19032 q^{47} -0.466489 q^{48} -2.80615 q^{49} -3.86624 q^{50} +3.07332 q^{51} -4.04229 q^{52} +2.20240 q^{53} +2.69742 q^{54} -1.65084 q^{55} +2.04789 q^{56} -2.78616 q^{57} -6.46605 q^{58} -2.29671 q^{59} -0.496709 q^{60} +7.08661 q^{61} +9.29719 q^{62} -5.69802 q^{63} +1.00000 q^{64} -4.30415 q^{65} +0.723247 q^{66} +6.39982 q^{67} -6.58820 q^{68} +0.777129 q^{69} +2.18055 q^{70} +2.46316 q^{71} -2.78239 q^{72} +2.99792 q^{73} -9.50873 q^{74} +1.80356 q^{75} +5.97262 q^{76} -3.17506 q^{77} +1.88568 q^{78} -11.5113 q^{79} +1.06478 q^{80} +7.08885 q^{81} -3.87954 q^{82} -13.6361 q^{83} -0.955318 q^{84} -7.01499 q^{85} -8.89119 q^{86} +3.01634 q^{87} -1.55040 q^{88} -0.000578536 q^{89} -2.96263 q^{90} -8.27816 q^{91} -1.66591 q^{92} -4.33704 q^{93} -4.19032 q^{94} +6.35953 q^{95} -0.466489 q^{96} -5.18607 q^{97} -2.80615 q^{98} +4.31383 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9}+O(q^{10}) $ 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9	\( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9} - 23 q^{10} - 32 q^{11} - 13 q^{12} - 8 q^{13} - 18 q^{14} - 14 q^{15} + 31 q^{16} - 30 q^{17} + 20 q^{18} - 38 q^{19} - 23 q^{20} - 16 q^{21} - 32 q^{22} - 24 q^{23} - 13 q^{24} + 40 q^{25} - 8 q^{26} - 28 q^{27} - 18 q^{28} - 7 q^{29} - 14 q^{30} - 32 q^{31} + 31 q^{32} - 9 q^{33} - 30 q^{34} - 14 q^{35} + 20 q^{36} + 11 q^{37} - 38 q^{38} - 9 q^{39} - 23 q^{40} - 76 q^{41} - 16 q^{42} - 33 q^{43} - 32 q^{44} - 40 q^{45} - 24 q^{46} - 96 q^{47} - 13 q^{48} + 15 q^{49} + 40 q^{50} - 55 q^{51} - 8 q^{52} - 28 q^{53} - 28 q^{54} - 52 q^{55} - 18 q^{56} - 21 q^{57} - 7 q^{58} - 72 q^{59} - 14 q^{60} - 9 q^{61} - 32 q^{62} - 54 q^{63} + 31 q^{64} - 38 q^{65} - 9 q^{66} - 4 q^{67} - 30 q^{68} - 17 q^{69} - 14 q^{70} - 61 q^{71} + 20 q^{72} - 62 q^{73} + 11 q^{74} - 63 q^{75} - 38 q^{76} - 9 q^{77} - 9 q^{78} - 30 q^{79} - 23 q^{80} - 13 q^{81} - 76 q^{82} - 90 q^{83} - 16 q^{84} + 26 q^{85} - 33 q^{86} - 34 q^{87} - 32 q^{88} - 99 q^{89} - 40 q^{90} - 47 q^{91} - 24 q^{92} - 6 q^{93} - 96 q^{94} - 24 q^{95} - 13 q^{96} - 46 q^{97} + 15 q^{98} - 48 q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9 - 23 * q^10 - 32 * q^11 - 13 * q^12 - 8 * q^13 - 18 * q^14 - 14 * q^15 + 31 * q^16 - 30 * q^17 + 20 * q^18 - 38 * q^19 - 23 * q^20 - 16 * q^21 - 32 * q^22 - 24 * q^23 - 13 * q^24 + 40 * q^25 - 8 * q^26 - 28 * q^27 - 18 * q^28 - 7 * q^29 - 14 * q^30 - 32 * q^31 + 31 * q^32 - 9 * q^33 - 30 * q^34 - 14 * q^35 + 20 * q^36 + 11 * q^37 - 38 * q^38 - 9 * q^39 - 23 * q^40 - 76 * q^41 - 16 * q^42 - 33 * q^43 - 32 * q^44 - 40 * q^45 - 24 * q^46 - 96 * q^47 - 13 * q^48 + 15 * q^49 + 40 * q^50 - 55 * q^51 - 8 * q^52 - 28 * q^53 - 28 * q^54 - 52 * q^55 - 18 * q^56 - 21 * q^57 - 7 * q^58 - 72 * q^59 - 14 * q^60 - 9 * q^61 - 32 * q^62 - 54 * q^63 + 31 * q^64 - 38 * q^65 - 9 * q^66 - 4 * q^67 - 30 * q^68 - 17 * q^69 - 14 * q^70 - 61 * q^71 + 20 * q^72 - 62 * q^73 + 11 * q^74 - 63 * q^75 - 38 * q^76 - 9 * q^77 - 9 * q^78 - 30 * q^79 - 23 * q^80 - 13 * q^81 - 76 * q^82 - 90 * q^83 - 16 * q^84 + 26 * q^85 - 33 * q^86 - 34 * q^87 - 32 * q^88 - 99 * q^89 - 40 * q^90 - 47 * q^91 - 24 * q^92 - 6 * q^93 - 96 * q^94 - 24 * q^95 - 13 * q^96 - 46 * q^97 + 15 * q^98 - 48 * 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
$3$	−0.466489	−0.269328	−0.134664	−	0.990891i	$-0.542995\pi$
$3$	−0.466489	−0.269328	−0.134664	+	0.990891i	$0.542995\pi$
$4$	1.00000	0.500000
$5$	1.06478	0.476184	0.238092	−	0.971243i	$-0.423478\pi$
$5$	1.06478	0.476184	0.238092	+	0.971243i	$0.423478\pi$
$6$	−0.466489	−0.190443
$7$	2.04789	0.774030	0.387015	−	0.922073i	$-0.373506\pi$
$7$	2.04789	0.774030	0.387015	+	0.922073i	$0.373506\pi$
$8$	1.00000	0.353553
$9$	−2.78239	−0.927463
$10$	1.06478	0.336713
$11$	−1.55040	−0.467465	−0.233732	−	0.972301i	$-0.575094\pi$
$11$	−1.55040	−0.467465	−0.233732	+	0.972301i	$0.575094\pi$
$12$	−0.466489	−0.134664
$13$	−4.04229	−1.12113	−0.560564	−	0.828111i	$-0.689416\pi$
$13$	−4.04229	−1.12113	−0.560564	+	0.828111i	$0.689416\pi$
$14$	2.04789	0.547322
$15$	−0.496709	−0.128250
$16$	1.00000	0.250000
$17$	−6.58820	−1.59787	−0.798937	−	0.601415i	$-0.794604\pi$
$17$	−6.58820	−1.59787	−0.798937	+	0.601415i	$0.794604\pi$
$18$	−2.78239	−0.655815
$19$	5.97262	1.37021	0.685106	−	0.728443i	$-0.259756\pi$
$19$	5.97262	1.37021	0.685106	+	0.728443i	$0.259756\pi$
$20$	1.06478	0.238092
$21$	−0.955318	−0.208468
$22$	−1.55040	−0.330547
$23$	−1.66591	−0.347366	−0.173683	−	0.984802i	$-0.555567\pi$
$23$	−1.66591	−0.347366	−0.173683	+	0.984802i	$0.555567\pi$
$24$	−0.466489	−0.0952217
$25$	−3.86624	−0.773248
$26$	−4.04229	−0.792758
$27$	2.69742	0.519119
$28$	2.04789	0.387015
$29$	−6.46605	−1.20071	−0.600357	−	0.799732i	$-0.704975\pi$
$29$	−6.46605	−1.20071	−0.600357	+	0.799732i	$0.704975\pi$
$30$	−0.496709	−0.0906862
$31$	9.29719	1.66982	0.834912	−	0.550383i	$-0.185518\pi$
$31$	9.29719	1.66982	0.834912	+	0.550383i	$0.185518\pi$
$32$	1.00000	0.176777
$33$	0.723247	0.125901
$34$	−6.58820	−1.12987
$35$	2.18055	0.368581
$36$	−2.78239	−0.463731
$37$	−9.50873	−1.56323	−0.781613	−	0.623764i	$-0.785603\pi$
$37$	−9.50873	−1.56323	−0.781613	+	0.623764i	$0.785603\pi$
$38$	5.97262	0.968887
$39$	1.88568	0.301951
$40$	1.06478	0.168357
$41$	−3.87954	−0.605883	−0.302941	−	0.953009i	$-0.597969\pi$
$41$	−3.87954	−0.605883	−0.302941	+	0.953009i	$0.597969\pi$
$42$	−0.955318	−0.147409
$43$	−8.89119	−1.35589	−0.677947	−	0.735111i	$-0.737130\pi$
$43$	−8.89119	−1.35589	−0.677947	+	0.735111i	$0.737130\pi$
$44$	−1.55040	−0.233732
$45$	−2.96263	−0.441643
$46$	−1.66591	−0.245625
$47$	−4.19032	−0.611220	−0.305610	−	0.952157i	$-0.598860\pi$
$47$	−4.19032	−0.611220	−0.305610	+	0.952157i	$0.598860\pi$
$48$	−0.466489	−0.0673319
$49$	−2.80615	−0.400878
$50$	−3.86624	−0.546769
$51$	3.07332	0.430352
$52$	−4.04229	−0.560564
$53$	2.20240	0.302522	0.151261	−	0.988494i	$-0.451667\pi$
$53$	2.20240	0.302522	0.151261	+	0.988494i	$0.451667\pi$
$54$	2.69742	0.367072
$55$	−1.65084	−0.222599
$56$	2.04789	0.273661
$57$	−2.78616	−0.369036
$58$	−6.46605	−0.849034
$59$	−2.29671	−0.299006	−0.149503	−	0.988761i	$-0.547767\pi$
$59$	−2.29671	−0.299006	−0.149503	+	0.988761i	$0.547767\pi$
$60$	−0.496709	−0.0641248
$61$	7.08661	0.907348	0.453674	−	0.891168i	$-0.350113\pi$
$61$	7.08661	0.907348	0.453674	+	0.891168i	$0.350113\pi$
$62$	9.29719	1.18074
$63$	−5.69802	−0.717884
$64$	1.00000	0.125000
$65$	−4.30415	−0.533864
$66$	0.723247	0.0890255
$67$	6.39982	0.781863	0.390932	−	0.920420i	$-0.372153\pi$
$67$	6.39982	0.781863	0.390932	+	0.920420i	$0.372153\pi$
$68$	−6.58820	−0.798937
$69$	0.777129	0.0935553
$70$	2.18055	0.260626
$71$	2.46316	0.292323	0.146162	−	0.989261i	$-0.453308\pi$
$71$	2.46316	0.292323	0.146162	+	0.989261i	$0.453308\pi$
$72$	−2.78239	−0.327908
$73$	2.99792	0.350880	0.175440	−	0.984490i	$-0.443865\pi$
$73$	2.99792	0.350880	0.175440	+	0.984490i	$0.443865\pi$
$74$	−9.50873	−1.10537
$75$	1.80356	0.208257
$76$	5.97262	0.685106
$77$	−3.17506	−0.361831
$78$	1.88568	0.213511
$79$	−11.5113	−1.29512	−0.647562	−	0.762013i	$-0.724211\pi$
$79$	−11.5113	−1.29512	−0.647562	+	0.762013i	$0.724211\pi$
$80$	1.06478	0.119046
$81$	7.08885	0.787650
$82$	−3.87954	−0.428424
$83$	−13.6361	−1.49675	−0.748377	−	0.663274i	$-0.769166\pi$
$83$	−13.6361	−1.49675	−0.748377	+	0.663274i	$0.769166\pi$
$84$	−0.955318	−0.104234
$85$	−7.01499	−0.760883
$86$	−8.89119	−0.958761
$87$	3.01634	0.323386
$88$	−1.55040	−0.165274
$89$	−0.000578536 0	−6.13247e−5 0	−3.06623e−5	−	1.00000i	$-0.500010\pi$
$89$	−0.000578536 0	−6.13247e−5 0	−3.06623e−5		1.00000i	$0.500010\pi$
$90$	−2.96263	−0.312289
$91$	−8.27816	−0.867787
$92$	−1.66591	−0.173683
$93$	−4.33704	−0.449730
$94$	−4.19032	−0.432198
$95$	6.35953	0.652474
$96$	−0.466489	−0.0476108
$97$	−5.18607	−0.526566	−0.263283	−	0.964719i	$-0.584805\pi$
$97$	−5.18607	−0.526566	−0.263283	+	0.964719i	$0.584805\pi$
$98$	−2.80615	−0.283464
$99$	4.31383	0.433556
$100$	−3.86624	−0.386624
$101$	−18.1712	−1.80810	−0.904052	−	0.427423i	$-0.859422\pi$
$101$	−18.1712	−1.80810	−0.904052	+	0.427423i	$0.859422\pi$
$102$	3.07332	0.304304
$103$	18.3055	1.80370	0.901850	−	0.432050i	$-0.142210\pi$
$103$	18.3055	1.80370	0.901850	+	0.432050i	$0.142210\pi$
$104$	−4.04229	−0.396379
$105$	−1.01720	−0.0992690
$106$	2.20240	0.213916
$107$	18.0640	1.74631	0.873156	−	0.487441i	$-0.162069\pi$
$107$	18.0640	1.74631	0.873156	+	0.487441i	$0.162069\pi$
$108$	2.69742	0.259559
$109$	−1.28666	−0.123239	−0.0616197	−	0.998100i	$-0.519627\pi$
$109$	−1.28666	−0.123239	−0.0616197	+	0.998100i	$0.519627\pi$
$110$	−1.65084	−0.157402
$111$	4.43572	0.421020
$112$	2.04789	0.193507
$113$	−3.47684	−0.327074	−0.163537	−	0.986537i	$-0.552290\pi$
$113$	−3.47684	−0.327074	−0.163537	+	0.986537i	$0.552290\pi$
$114$	−2.78616	−0.260948
$115$	−1.77383	−0.165410
$116$	−6.46605	−0.600357
$117$	11.2472	1.03980
$118$	−2.29671	−0.211429
$119$	−13.4919	−1.23680
$120$	−0.496709	−0.0453431
$121$	−8.59625	−0.781477
$122$	7.08661	0.641592
$123$	1.80976	0.163181
$124$	9.29719	0.834912
$125$	−9.44060	−0.844393
$126$	−5.69802	−0.507620
$127$	13.5796	1.20500	0.602498	−	0.798120i	$-0.294172\pi$
$127$	13.5796	1.20500	0.602498	+	0.798120i	$0.294172\pi$
$128$	1.00000	0.0883883
$129$	4.14764	0.365179
$130$	−4.30415	−0.377499
$131$	20.5868	1.79868	0.899339	−	0.437252i	$-0.144048\pi$
$131$	20.5868	1.79868	0.899339	+	0.437252i	$0.144048\pi$
$132$	0.723247	0.0629506
$133$	12.2313	1.06059
$134$	6.39982	0.552861
$135$	2.87216	0.247196
$136$	−6.58820	−0.564934
$137$	−15.0941	−1.28957	−0.644786	−	0.764363i	$-0.723053\pi$
$137$	−15.0941	−1.28957	−0.644786	+	0.764363i	$0.723053\pi$
$138$	0.777129	0.0661536
$139$	−0.851224	−0.0721998	−0.0360999	−	0.999348i	$-0.511493\pi$
$139$	−0.851224	−0.0721998	−0.0360999	+	0.999348i	$0.511493\pi$
$140$	2.18055	0.184290
$141$	1.95474	0.164619
$142$	2.46316	0.206704
$143$	6.26718	0.524088
$144$	−2.78239	−0.231866
$145$	−6.88492	−0.571762
$146$	2.99792	0.248110
$147$	1.30904	0.107968
$148$	−9.50873	−0.781613
$149$	−13.0006	−1.06505	−0.532525	−	0.846414i	$-0.678757\pi$
$149$	−13.0006	−1.06505	−0.532525	+	0.846414i	$0.678757\pi$
$150$	1.80356	0.147260
$151$	−5.54007	−0.450845	−0.225422	−	0.974261i	$-0.572376\pi$
$151$	−5.54007	−0.450845	−0.225422	+	0.974261i	$0.572376\pi$
$152$	5.97262	0.484443
$153$	18.3309	1.48197
$154$	−3.17506	−0.255853
$155$	9.89947	0.795144
$156$	1.88568	0.150975
$157$	8.16046	0.651275	0.325638	−	0.945495i	$-0.394421\pi$
$157$	8.16046	0.651275	0.325638	+	0.945495i	$0.394421\pi$
$158$	−11.5113	−0.915791
$159$	−1.02739	−0.0814776
$160$	1.06478	0.0841783
$161$	−3.41160	−0.268872
$162$	7.08885	0.556952
$163$	13.1192	1.02758	0.513789	−	0.857917i	$-0.328241\pi$
$163$	13.1192	1.02758	0.513789	+	0.857917i	$0.328241\pi$
$164$	−3.87954	−0.302941
$165$	0.770099	0.0599521
$166$	−13.6361	−1.05836
$167$	6.43423	0.497896	0.248948	−	0.968517i	$-0.419915\pi$
$167$	6.43423	0.497896	0.248948	+	0.968517i	$0.419915\pi$
$168$	−0.955318	−0.0737044
$169$	3.34008	0.256929
$170$	−7.01499	−0.538025
$171$	−16.6181	−1.27082
$172$	−8.89119	−0.677947
$173$	20.0889	1.52733	0.763665	−	0.645613i	$-0.223398\pi$
$173$	20.0889	1.52733	0.763665	+	0.645613i	$0.223398\pi$
$174$	3.01634	0.228668
$175$	−7.91764	−0.598517
$176$	−1.55040	−0.116866
$177$	1.07139	0.0805305
$178$	−0.000578536 0	−4.33631e−5 0
$179$	−17.7008	−1.32302	−0.661509	−	0.749937i	$-0.730084\pi$
$179$	−17.7008	−1.32302	−0.661509	+	0.749937i	$0.730084\pi$
$180$	−2.96263	−0.220822
$181$	−0.426354	−0.0316906	−0.0158453	−	0.999874i	$-0.505044\pi$
$181$	−0.426354	−0.0316906	−0.0158453	+	0.999874i	$0.505044\pi$
$182$	−8.27816	−0.613618
$183$	−3.30583	−0.244374
$184$	−1.66591	−0.122813
$185$	−10.1247	−0.744384
$186$	−4.33704	−0.318007
$187$	10.2144	0.746950
$188$	−4.19032	−0.305610
$189$	5.52402	0.401813
$190$	6.35953	0.461369
$191$	−26.1783	−1.89420	−0.947098	−	0.320945i	$-0.896000\pi$
$191$	−26.1783	−1.89420	−0.947098	+	0.320945i	$0.896000\pi$
$192$	−0.466489	−0.0336659
$193$	14.6292	1.05304	0.526518	−	0.850164i	$-0.323497\pi$
$193$	14.6292	1.05304	0.526518	+	0.850164i	$0.323497\pi$
$194$	−5.18607	−0.372338
$195$	2.00784	0.143784
$196$	−2.80615	−0.200439
$197$	−7.13353	−0.508243	−0.254121	−	0.967172i	$-0.581786\pi$
$197$	−7.13353	−0.508243	−0.254121	+	0.967172i	$0.581786\pi$
$198$	4.31383	0.306570
$199$	−5.26144	−0.372974	−0.186487	−	0.982457i	$-0.559710\pi$
$199$	−5.26144	−0.372974	−0.186487	+	0.982457i	$0.559710\pi$
$200$	−3.86624	−0.273385
$201$	−2.98545	−0.210577
$202$	−18.1712	−1.27852
$203$	−13.2418	−0.929389
$204$	3.07332	0.215176
$205$	−4.13086	−0.288512
$206$	18.3055	1.27541
$207$	4.63521	0.322169
$208$	−4.04229	−0.280282
$209$	−9.25998	−0.640526
$210$	−1.01720	−0.0701938
$211$	1.70465	0.117353	0.0586763	−	0.998277i	$-0.481312\pi$
$211$	1.70465	0.117353	0.0586763	+	0.998277i	$0.481312\pi$
$212$	2.20240	0.151261
$213$	−1.14904	−0.0787307
$214$	18.0640	1.23483
$215$	−9.46716	−0.645655
$216$	2.69742	0.183536
$217$	19.0396	1.29249
$218$	−1.28666	−0.0871434
$219$	−1.39850	−0.0945017
$220$	−1.65084	−0.111300
$221$	26.6314	1.79142
$222$	4.43572	0.297706
$223$	8.74841	0.585837	0.292918	−	0.956137i	$-0.405374\pi$
$223$	8.74841	0.585837	0.292918	+	0.956137i	$0.405374\pi$
$224$	2.04789	0.136830
$225$	10.7574	0.717159
$226$	−3.47684	−0.231276
$227$	−3.50226	−0.232453	−0.116227	−	0.993223i	$-0.537080\pi$
$227$	−3.50226	−0.232453	−0.116227	+	0.993223i	$0.537080\pi$
$228$	−2.78616	−0.184518
$229$	7.62737	0.504031	0.252016	−	0.967723i	$-0.418907\pi$
$229$	7.62737	0.504031	0.252016	+	0.967723i	$0.418907\pi$
$230$	−1.77383	−0.116963
$231$	1.48113	0.0974512
$232$	−6.46605	−0.424517
$233$	12.0808	0.791439	0.395719	−	0.918372i	$-0.370495\pi$
$233$	12.0808	0.791439	0.395719	+	0.918372i	$0.370495\pi$
$234$	11.2472	0.735253
$235$	−4.46177	−0.291054
$236$	−2.29671	−0.149503
$237$	5.36990	0.348813
$238$	−13.4919	−0.874551
$239$	−7.10961	−0.459883	−0.229941	−	0.973205i	$-0.573853\pi$
$239$	−7.10961	−0.459883	−0.229941	+	0.973205i	$0.573853\pi$
$240$	−0.496709	−0.0320624
$241$	−7.13001	−0.459284	−0.229642	−	0.973275i	$-0.573756\pi$
$241$	−7.13001	−0.459284	−0.229642	+	0.973275i	$0.573756\pi$
$242$	−8.59625	−0.552588
$243$	−11.3991	−0.731255
$244$	7.08661	0.453674
$245$	−2.98793	−0.190892
$246$	1.80976	0.115386
$247$	−24.1430	−1.53618
$248$	9.29719	0.590372
$249$	6.36108	0.403117
$250$	−9.44060	−0.597076
$251$	−27.3738	−1.72782	−0.863909	−	0.503648i	$-0.831991\pi$
$251$	−27.3738	−1.72782	−0.863909	+	0.503648i	$0.831991\pi$
$252$	−5.69802	−0.358942
$253$	2.58283	0.162381
$254$	13.5796	0.852061
$255$	3.27242	0.204927
$256$	1.00000	0.0625000
$257$	27.3506	1.70608	0.853041	−	0.521843i	$-0.174755\pi$
$257$	27.3506	1.70608	0.853041	+	0.521843i	$0.174755\pi$
$258$	4.14764	0.258221
$259$	−19.4728	−1.20998
$260$	−4.30415	−0.266932
$261$	17.9911	1.11362
$262$	20.5868	1.27186
$263$	−4.68244	−0.288731	−0.144366	−	0.989524i	$-0.546114\pi$
$263$	−4.68244	−0.288731	−0.144366	+	0.989524i	$0.546114\pi$
$264$	0.723247	0.0445128
$265$	2.34507	0.144056
$266$	12.2313	0.749947
$267$	0.000269881 0	1.65164e−5 0
$268$	6.39982	0.390932
$269$	11.6457	0.710053	0.355027	−	0.934856i	$-0.384472\pi$
$269$	11.6457	0.710053	0.355027	+	0.934856i	$0.384472\pi$
$270$	2.87216	0.174794
$271$	−10.6490	−0.646883	−0.323442	−	0.946248i	$-0.604840\pi$
$271$	−10.6490	−0.646883	−0.323442	+	0.946248i	$0.604840\pi$
$272$	−6.58820	−0.399469
$273$	3.86167	0.233719
$274$	−15.0941	−0.911865
$275$	5.99424	0.361466
$276$	0.777129	0.0467777
$277$	−24.0027	−1.44218	−0.721092	−	0.692839i	$-0.756359\pi$
$277$	−24.0027	−1.44218	−0.721092	+	0.692839i	$0.756359\pi$
$278$	−0.851224	−0.0510530
$279$	−25.8684	−1.54870
$280$	2.18055	0.130313
$281$	−26.8933	−1.60432	−0.802159	−	0.597111i	$-0.796315\pi$
$281$	−26.8933	−1.60432	−0.802159	+	0.597111i	$0.796315\pi$
$282$	1.95474	0.116403
$283$	1.68514	0.100171	0.0500857	−	0.998745i	$-0.484051\pi$
$283$	1.68514	0.100171	0.0500857	+	0.998745i	$0.484051\pi$
$284$	2.46316	0.146162
$285$	−2.96665	−0.175729
$286$	6.26718	0.370586
$287$	−7.94488	−0.468971
$288$	−2.78239	−0.163954
$289$	26.4044	1.55320
$290$	−6.88492	−0.404297
$291$	2.41924	0.141819
$292$	2.99792	0.175440
$293$	4.76186	0.278191	0.139095	−	0.990279i	$-0.455581\pi$
$293$	4.76186	0.278191	0.139095	+	0.990279i	$0.455581\pi$
$294$	1.30904	0.0763446
$295$	−2.44549	−0.142382
$296$	−9.50873	−0.552684
$297$	−4.18209	−0.242670
$298$	−13.0006	−0.753104
$299$	6.73408	0.389442
$300$	1.80356	0.104129
$301$	−18.2082	−1.04950
$302$	−5.54007	−0.318795
$303$	8.47667	0.486972
$304$	5.97262	0.342553
$305$	7.54569	0.432065
$306$	18.3309	1.04791
$307$	2.06903	0.118086	0.0590429	−	0.998255i	$-0.481195\pi$
$307$	2.06903	0.118086	0.0590429	+	0.998255i	$0.481195\pi$
$308$	−3.17506	−0.180916
$309$	−8.53934	−0.485786
$310$	9.89947	0.562252
$311$	9.75720	0.553280	0.276640	−	0.960974i	$-0.410779\pi$
$311$	9.75720	0.553280	0.276640	+	0.960974i	$0.410779\pi$
$312$	1.88568	0.106756
$313$	−16.9023	−0.955377	−0.477689	−	0.878529i	$-0.658525\pi$
$313$	−16.9023	−0.955377	−0.477689	+	0.878529i	$0.658525\pi$
$314$	8.16046	0.460521
$315$	−6.06715	−0.341845
$316$	−11.5113	−0.647562
$317$	−17.4359	−0.979295	−0.489648	−	0.871920i	$-0.662875\pi$
$317$	−17.4359	−0.979295	−0.489648	+	0.871920i	$0.662875\pi$
$318$	−1.02739	−0.0576134
$319$	10.0250	0.561292
$320$	1.06478	0.0595231
$321$	−8.42666	−0.470330
$322$	−3.41160	−0.190121
$323$	−39.3488	−2.18943
$324$	7.08885	0.393825
$325$	15.6285	0.866911
$326$	13.1192	0.726607
$327$	0.600212	0.0331918
$328$	−3.87954	−0.214212
$329$	−8.58131	−0.473103
$330$	0.770099	0.0423926
$331$	11.0466	0.607177	0.303589	−	0.952803i	$-0.401815\pi$
$331$	11.0466	0.607177	0.303589	+	0.952803i	$0.401815\pi$
$332$	−13.6361	−0.748377
$333$	26.4570	1.44983
$334$	6.43423	0.352065
$335$	6.81441	0.372311
$336$	−0.955318	−0.0521169
$337$	−4.22461	−0.230129	−0.115065	−	0.993358i	$-0.536708\pi$
$337$	−4.22461	−0.230129	−0.115065	+	0.993358i	$0.536708\pi$
$338$	3.34008	0.181676
$339$	1.62191	0.0880900
$340$	−7.01499	−0.380441
$341$	−14.4144	−0.780584
$342$	−16.6181	−0.898606
$343$	−20.0819	−1.08432
$344$	−8.89119	−0.479381
$345$	0.827472	0.0445496
$346$	20.0889	1.07998
$347$	−26.1594	−1.40431	−0.702155	−	0.712024i	$-0.747779\pi$
$347$	−26.1594	−1.40431	−0.702155	+	0.712024i	$0.747779\pi$
$348$	3.01634	0.161693
$349$	−15.9888	−0.855862	−0.427931	−	0.903812i	$-0.640757\pi$
$349$	−15.9888	−0.855862	−0.427931	+	0.903812i	$0.640757\pi$
$350$	−7.91764	−0.423216
$351$	−10.9037	−0.581999
$352$	−1.55040	−0.0826368
$353$	25.9904	1.38333	0.691666	−	0.722218i	$-0.256877\pi$
$353$	25.9904	1.38333	0.691666	+	0.722218i	$0.256877\pi$
$354$	1.07139	0.0569437
$355$	2.62272	0.139200
$356$	−0.000578536 0	−3.06623e−5 0
$357$	6.29383	0.333105
$358$	−17.7008	−0.935516
$359$	4.21368	0.222390	0.111195	−	0.993799i	$-0.464532\pi$
$359$	4.21368	0.222390	0.111195	+	0.993799i	$0.464532\pi$
$360$	−2.96263	−0.156144
$361$	16.6722	0.877483
$362$	−0.426354	−0.0224087
$363$	4.01005	0.210473
$364$	−8.27816	−0.433893
$365$	3.19213	0.167084
$366$	−3.30583	−0.172798
$367$	−30.2015	−1.57651	−0.788253	−	0.615351i	$-0.789014\pi$
$367$	−30.2015	−1.57651	−0.788253	+	0.615351i	$0.789014\pi$
$368$	−1.66591	−0.0868416
$369$	10.7944	0.561934
$370$	−10.1247	−0.526359
$371$	4.51027	0.234161
$372$	−4.33704	−0.224865
$373$	29.8267	1.54437	0.772183	−	0.635400i	$-0.219165\pi$
$373$	29.8267	1.54437	0.772183	+	0.635400i	$0.219165\pi$
$374$	10.2144	0.528173
$375$	4.40394	0.227418
$376$	−4.19032	−0.216099
$377$	26.1376	1.34616
$378$	5.52402	0.284125
$379$	35.2337	1.80983	0.904916	−	0.425590i	$-0.139933\pi$
$379$	35.2337	1.80983	0.904916	+	0.425590i	$0.139933\pi$
$380$	6.35953	0.326237
$381$	−6.33474	−0.324539
$382$	−26.1783	−1.33940
$383$	13.5199	0.690834	0.345417	−	0.938449i	$-0.387737\pi$
$383$	13.5199	0.690834	0.345417	+	0.938449i	$0.387737\pi$
$384$	−0.466489	−0.0238054
$385$	−3.38074	−0.172299
$386$	14.6292	0.744609
$387$	24.7387	1.25754
$388$	−5.18607	−0.263283
$389$	7.00882	0.355361	0.177681	−	0.984088i	$-0.443141\pi$
$389$	7.00882	0.355361	0.177681	+	0.984088i	$0.443141\pi$
$390$	2.00784	0.101671
$391$	10.9754	0.555047
$392$	−2.80615	−0.141732
$393$	−9.60352	−0.484433
$394$	−7.13353	−0.359382
$395$	−12.2570	−0.616718
$396$	4.31383	0.216778
$397$	−30.7612	−1.54386	−0.771930	−	0.635708i	$-0.780708\pi$
$397$	−30.7612	−1.54386	−0.771930	+	0.635708i	$0.780708\pi$
$398$	−5.26144	−0.263732
$399$	−5.70575	−0.285645
$400$	−3.86624	−0.193312
$401$	9.31409	0.465124	0.232562	−	0.972582i	$-0.425289\pi$
$401$	9.31409	0.465124	0.232562	+	0.972582i	$0.425289\pi$
$402$	−2.98545	−0.148901
$403$	−37.5819	−1.87209
$404$	−18.1712	−0.904052
$405$	7.54807	0.375066
$406$	−13.2418	−0.657177
$407$	14.7424	0.730753
$408$	3.07332	0.152152
$409$	−37.5145	−1.85497	−0.927486	−	0.373858i	$-0.878035\pi$
$409$	−37.5145	−1.85497	−0.927486	+	0.373858i	$0.878035\pi$
$410$	−4.13086	−0.204009
$411$	7.04121	0.347317
$412$	18.3055	0.901850
$413$	−4.70341	−0.231439
$414$	4.63521	0.227808
$415$	−14.5194	−0.712731
$416$	−4.04229	−0.198189
$417$	0.397086	0.0194454
$418$	−9.25998	−0.452920
$419$	−30.4002	−1.48515	−0.742574	−	0.669764i	$-0.766395\pi$
$419$	−30.4002	−1.48515	−0.742574	+	0.669764i	$0.766395\pi$
$420$	−1.01720	−0.0496345
$421$	−27.1973	−1.32552	−0.662758	−	0.748833i	$-0.730614\pi$
$421$	−27.1973	−1.32552	−0.662758	+	0.748833i	$0.730614\pi$
$422$	1.70465	0.0829809
$423$	11.6591	0.566884
$424$	2.20240	0.106958
$425$	25.4716	1.23555
$426$	−1.14904	−0.0556710
$427$	14.5126	0.702314
$428$	18.0640	0.873156
$429$	−2.92357	−0.141151
$430$	−9.46716	−0.456547
$431$	17.3646	0.836423	0.418212	−	0.908350i	$-0.362657\pi$
$431$	17.3646	0.836423	0.418212	+	0.908350i	$0.362657\pi$
$432$	2.69742	0.129780
$433$	−15.3082	−0.735666	−0.367833	−	0.929892i	$-0.619900\pi$
$433$	−15.3082	−0.735666	−0.367833	+	0.929892i	$0.619900\pi$
$434$	19.0396	0.913931
$435$	3.21174	0.153991
$436$	−1.28666	−0.0616197
$437$	−9.94984	−0.475966
$438$	−1.39850	−0.0668228
$439$	23.6445	1.12849	0.564245	−	0.825607i	$-0.309167\pi$
$439$	23.6445	1.12849	0.564245	+	0.825607i	$0.309167\pi$
$440$	−1.65084	−0.0787008
$441$	7.80779	0.371799
$442$	26.6314	1.26673
$443$	21.6992	1.03096	0.515479	−	0.856902i	$-0.327614\pi$
$443$	21.6992	1.03096	0.515479	+	0.856902i	$0.327614\pi$
$444$	4.43572	0.210510
$445$	−0.000616014 0	−2.92018e−5 0
$446$	8.74841	0.414249
$447$	6.06463	0.286847
$448$	2.04789	0.0967537
$449$	33.3703	1.57484	0.787421	−	0.616416i	$-0.211416\pi$
$449$	33.3703	1.57484	0.787421	+	0.616416i	$0.211416\pi$
$450$	10.7574	0.507108
$451$	6.01486	0.283229
$452$	−3.47684	−0.163537
$453$	2.58438	0.121425
$454$	−3.50226	−0.164369
$455$	−8.81442	−0.413227
$456$	−2.78616	−0.130474
$457$	−11.3206	−0.529557	−0.264778	−	0.964309i	$-0.585299\pi$
$457$	−11.3206	−0.529557	−0.264778	+	0.964309i	$0.585299\pi$
$458$	7.62737	0.356404
$459$	−17.7712	−0.829487
$460$	−1.77383	−0.0827052
$461$	33.2619	1.54916	0.774581	−	0.632474i	$-0.217961\pi$
$461$	33.2619	1.54916	0.774581	+	0.632474i	$0.217961\pi$
$462$	1.48113	0.0689084
$463$	0.931641	0.0432970	0.0216485	−	0.999766i	$-0.493109\pi$
$463$	0.931641	0.0432970	0.0216485	+	0.999766i	$0.493109\pi$
$464$	−6.46605	−0.300179
$465$	−4.61799	−0.214154
$466$	12.0808	0.559632
$467$	9.92622	0.459331	0.229665	−	0.973270i	$-0.426237\pi$
$467$	9.92622	0.459331	0.229665	+	0.973270i	$0.426237\pi$
$468$	11.2472	0.519902
$469$	13.1061	0.605185
$470$	−4.46177	−0.205806
$471$	−3.80676	−0.175406
$472$	−2.29671	−0.105715
$473$	13.7849	0.633832
$474$	5.36990	0.246648
$475$	−23.0916	−1.05951
$476$	−13.4919	−0.618401
$477$	−6.12792	−0.280578
$478$	−7.10961	−0.325186
$479$	10.2867	0.470012	0.235006	−	0.971994i	$-0.424489\pi$
$479$	10.2867	0.470012	0.235006	+	0.971994i	$0.424489\pi$
$480$	−0.496709	−0.0226715
$481$	38.4370	1.75258
$482$	−7.13001	−0.324763
$483$	1.59147	0.0724146
$484$	−8.59625	−0.390738
$485$	−5.52203	−0.250742
$486$	−11.3991	−0.517075
$487$	8.44811	0.382820	0.191410	−	0.981510i	$-0.438694\pi$
$487$	8.44811	0.382820	0.191410	+	0.981510i	$0.438694\pi$
$488$	7.08661	0.320796
$489$	−6.11998	−0.276755
$490$	−2.98793	−0.134981
$491$	19.5608	0.882766	0.441383	−	0.897319i	$-0.354488\pi$
$491$	19.5608	0.882766	0.441383	+	0.897319i	$0.354488\pi$
$492$	1.80976	0.0815905
$493$	42.5996	1.91859
$494$	−24.1430	−1.08625
$495$	4.59328	0.206453
$496$	9.29719	0.417456
$497$	5.04428	0.226267
$498$	6.36108	0.285047
$499$	8.63851	0.386713	0.193356	−	0.981129i	$-0.438063\pi$
$499$	8.63851	0.386713	0.193356	+	0.981129i	$0.438063\pi$
$500$	−9.44060	−0.422197
$501$	−3.00150	−0.134097
$502$	−27.3738	−1.22175
$503$	−5.02623	−0.224108	−0.112054	−	0.993702i	$-0.535743\pi$
$503$	−5.02623	−0.224108	−0.112054	+	0.993702i	$0.535743\pi$
$504$	−5.69802	−0.253810
$505$	−19.3484	−0.860991
$506$	2.58283	0.114821
$507$	−1.55811	−0.0691981
$508$	13.5796	0.602498
$509$	−28.3581	−1.25695	−0.628475	−	0.777830i	$-0.716321\pi$
$509$	−28.3581	−1.25695	−0.628475	+	0.777830i	$0.716321\pi$
$510$	3.27242	0.144905
$511$	6.13941	0.271592
$512$	1.00000	0.0441942
$513$	16.1107	0.711303
$514$	27.3506	1.20638
$515$	19.4914	0.858894
$516$	4.14764	0.182590
$517$	6.49669	0.285724
$518$	−19.4728	−0.855587
$519$	−9.37124	−0.411352
$520$	−4.30415	−0.188749
$521$	0.770698	0.0337649	0.0168824	−	0.999857i	$-0.494626\pi$
$521$	0.770698	0.0337649	0.0168824	+	0.999857i	$0.494626\pi$
$522$	17.9911	0.787447
$523$	23.5687	1.03059	0.515295	−	0.857013i	$-0.327683\pi$
$523$	23.5687	1.03059	0.515295	+	0.857013i	$0.327683\pi$
$524$	20.5868	0.899339
$525$	3.69349	0.161197
$526$	−4.68244	−0.204164
$527$	−61.2518	−2.66817
$528$	0.723247	0.0314753
$529$	−20.2247	−0.879337
$530$	2.34507	0.101863
$531$	6.39033	0.277317
$532$	12.2313	0.530293
$533$	15.6822	0.679273
$534$	0.000269881 0	1.16789e−5 0
$535$	19.2342	0.831567
$536$	6.39982	0.276430
$537$	8.25722	0.356325
$538$	11.6457	0.502083
$539$	4.35066	0.187396
$540$	2.87216	0.123598
$541$	−1.22706	−0.0527555	−0.0263778	−	0.999652i	$-0.508397\pi$
$541$	−1.22706	−0.0527555	−0.0263778	+	0.999652i	$0.508397\pi$
$542$	−10.6490	−0.457416
$543$	0.198889	0.00853516
$544$	−6.58820	−0.282467
$545$	−1.37001	−0.0586847
$546$	3.86167	0.165264
$547$	9.09087	0.388697	0.194349	−	0.980932i	$-0.437741\pi$
$547$	9.09087	0.388697	0.194349	+	0.980932i	$0.437741\pi$
$548$	−15.0941	−0.644786
$549$	−19.7177	−0.841531
$550$	5.99424	0.255595
$551$	−38.6192	−1.64523
$552$	0.777129	0.0330768
$553$	−23.5739	−1.00246
$554$	−24.0027	−1.01978
$555$	4.72307	0.200483
$556$	−0.851224	−0.0360999
$557$	18.7417	0.794110	0.397055	−	0.917795i	$-0.370032\pi$
$557$	18.7417	0.794110	0.397055	+	0.917795i	$0.370032\pi$
$558$	−25.8684	−1.09510
$559$	35.9407	1.52013
$560$	2.18055	0.0921452
$561$	−4.76490	−0.201174
$562$	−26.8933	−1.13442
$563$	18.1834	0.766339	0.383170	−	0.923678i	$-0.374832\pi$
$563$	18.1834	0.766339	0.383170	+	0.923678i	$0.374832\pi$
$564$	1.95474	0.0823093
$565$	−3.70208	−0.155747
$566$	1.68514	0.0708318
$567$	14.5172	0.609664
$568$	2.46316	0.103352
$569$	10.4661	0.438760	0.219380	−	0.975639i	$-0.429596\pi$
$569$	10.4661	0.438760	0.219380	+	0.975639i	$0.429596\pi$
$570$	−2.96665	−0.124259
$571$	30.0964	1.25949	0.629747	−	0.776801i	$-0.283159\pi$
$571$	30.0964	1.25949	0.629747	+	0.776801i	$0.283159\pi$
$572$	6.26718	0.262044
$573$	12.2119	0.510159
$574$	−7.94488	−0.331613
$575$	6.44081	0.268600
$576$	−2.78239	−0.115933
$577$	−23.2320	−0.967159	−0.483580	−	0.875300i	$-0.660664\pi$
$577$	−23.2320	−0.967159	−0.483580	+	0.875300i	$0.660664\pi$
$578$	26.4044	1.09828
$579$	−6.82438	−0.283612
$580$	−6.88492	−0.285881
$581$	−27.9252	−1.15853
$582$	2.41924	0.100281
$583$	−3.41461	−0.141419
$584$	2.99792	0.124055
$585$	11.9758	0.495139
$586$	4.76186	0.196710
$587$	−26.9824	−1.11368	−0.556841	−	0.830619i	$-0.687987\pi$
$587$	−26.9824	−1.11368	−0.556841	+	0.830619i	$0.687987\pi$
$588$	1.30904	0.0539838
$589$	55.5286	2.28801
$590$	−2.44549	−0.100679
$591$	3.32771	0.136884
$592$	−9.50873	−0.390806
$593$	1.52517	0.0626312	0.0313156	−	0.999510i	$-0.490030\pi$
$593$	1.52517	0.0626312	0.0313156	+	0.999510i	$0.490030\pi$
$594$	−4.18209	−0.171593
$595$	−14.3659	−0.588946
$596$	−13.0006	−0.532525
$597$	2.45440	0.100452
$598$	6.73408	0.275377
$599$	−22.4580	−0.917609	−0.458804	−	0.888537i	$-0.651722\pi$
$599$	−22.4580	−0.917609	−0.458804	+	0.888537i	$0.651722\pi$
$600$	1.80356	0.0736300
$601$	36.1327	1.47388	0.736942	−	0.675956i	$-0.236269\pi$
$601$	36.1327	1.47388	0.736942	+	0.675956i	$0.236269\pi$
$602$	−18.2082	−0.742110
$603$	−17.8068	−0.725149
$604$	−5.54007	−0.225422
$605$	−9.15312	−0.372127
$606$	8.47667	0.344341
$607$	−2.81203	−0.114137	−0.0570685	−	0.998370i	$-0.518175\pi$
$607$	−2.81203	−0.114137	−0.0570685	+	0.998370i	$0.518175\pi$
$608$	5.97262	0.242222
$609$	6.17713	0.250310
$610$	7.54569	0.305516
$611$	16.9385	0.685257
$612$	18.3309	0.740984
$613$	4.28815	0.173197	0.0865984	−	0.996243i	$-0.472400\pi$
$613$	4.28815	0.173197	0.0865984	+	0.996243i	$0.472400\pi$
$614$	2.06903	0.0834992
$615$	1.92700	0.0777042
$616$	−3.17506	−0.127927
$617$	19.0002	0.764919	0.382459	−	0.923972i	$-0.375077\pi$
$617$	19.0002	0.764919	0.382459	+	0.923972i	$0.375077\pi$
$618$	−8.53934	−0.343503
$619$	−33.6745	−1.35349	−0.676746	−	0.736217i	$-0.736610\pi$
$619$	−33.6745	−1.35349	−0.676746	+	0.736217i	$0.736610\pi$
$620$	9.89947	0.397572
$621$	−4.49366	−0.180324
$622$	9.75720	0.391228
$623$	−0.00118478	−4.74671e−5 0
$624$	1.88568	0.0754877
$625$	9.27904	0.371161
$626$	−16.9023	−0.675554
$627$	4.31968	0.172511
$628$	8.16046	0.325638
$629$	62.6454	2.49784
$630$	−6.06715	−0.241721
$631$	−2.38545	−0.0949634	−0.0474817	−	0.998872i	$-0.515120\pi$
$631$	−2.38545	−0.0949634	−0.0474817	+	0.998872i	$0.515120\pi$
$632$	−11.5113	−0.457896
$633$	−0.795199	−0.0316063
$634$	−17.4359	−0.692466
$635$	14.4593	0.573801
$636$	−1.02739	−0.0407388
$637$	11.3432	0.449436
$638$	10.0250	0.396893
$639$	−6.85346	−0.271119
$640$	1.06478	0.0420892
$641$	26.5482	1.04859	0.524295	−	0.851537i	$-0.324329\pi$
$641$	26.5482	1.04859	0.524295	+	0.851537i	$0.324329\pi$
$642$	−8.42666	−0.332574
$643$	−4.12870	−0.162820	−0.0814100	−	0.996681i	$-0.525942\pi$
$643$	−4.12870	−0.162820	−0.0814100	+	0.996681i	$0.525942\pi$
$644$	−3.41160	−0.134436
$645$	4.41633	0.173893
$646$	−39.3488	−1.54816
$647$	47.4018	1.86356	0.931778	−	0.363029i	$-0.118257\pi$
$647$	47.4018	1.86356	0.931778	+	0.363029i	$0.118257\pi$
$648$	7.08885	0.278476
$649$	3.56083	0.139775
$650$	15.6285	0.612999
$651$	−8.88177	−0.348104
$652$	13.1192	0.513789
$653$	−1.56147	−0.0611050	−0.0305525	−	0.999533i	$-0.509727\pi$
$653$	−1.56147	−0.0611050	−0.0305525	+	0.999533i	$0.509727\pi$
$654$	0.600212	0.0234701
$655$	21.9204	0.856502
$656$	−3.87954	−0.151471
$657$	−8.34138	−0.325428
$658$	−8.58131	−0.334534
$659$	−26.7941	−1.04375	−0.521874	−	0.853023i	$-0.674767\pi$
$659$	−26.7941	−1.04375	−0.521874	+	0.853023i	$0.674767\pi$
$660$	0.770099	0.0299761
$661$	8.06648	0.313750	0.156875	−	0.987618i	$-0.449858\pi$
$661$	8.06648	0.313750	0.156875	+	0.987618i	$0.449858\pi$
$662$	11.0466	0.429339
$663$	−12.4233	−0.482479
$664$	−13.6361	−0.529182
$665$	13.0236	0.505034
$666$	26.4570	1.02519
$667$	10.7719	0.417088
$668$	6.43423	0.248948
$669$	−4.08104	−0.157782
$670$	6.81441	0.263264
$671$	−10.9871	−0.424153
$672$	−0.955318	−0.0368522
$673$	28.6658	1.10499	0.552493	−	0.833518i	$-0.313677\pi$
$673$	28.6658	1.10499	0.552493	+	0.833518i	$0.313677\pi$
$674$	−4.22461	−0.162726
$675$	−10.4289	−0.401408
$676$	3.34008	0.128465
$677$	−31.2676	−1.20171	−0.600856	−	0.799357i	$-0.705174\pi$
$677$	−31.2676	−1.20171	−0.600856	+	0.799357i	$0.705174\pi$
$678$	1.62191	0.0622891
$679$	−10.6205	−0.407577
$680$	−7.01499	−0.269013
$681$	1.63377	0.0626060
$682$	−14.4144	−0.551956
$683$	−7.37225	−0.282091	−0.141046	−	0.990003i	$-0.545046\pi$
$683$	−7.37225	−0.282091	−0.141046	+	0.990003i	$0.545046\pi$
$684$	−16.6181	−0.635411
$685$	−16.0719	−0.614074
$686$	−20.0819	−0.766731
$687$	−3.55809	−0.135749
$688$	−8.89119	−0.338973
$689$	−8.90272	−0.339166
$690$	0.827472	0.0315013
$691$	46.2498	1.75942	0.879712	−	0.475507i	$-0.157736\pi$
$691$	46.2498	1.75942	0.879712	+	0.475507i	$0.157736\pi$
$692$	20.0889	0.763665
$693$	8.83424	0.335585
$694$	−26.1594	−0.992997
$695$	−0.906367	−0.0343804
$696$	3.01634	0.114334
$697$	25.5592	0.968124
$698$	−15.9888	−0.605185
$699$	−5.63555	−0.213156
$700$	−7.91764	−0.299259
$701$	7.00232	0.264474	0.132237	−	0.991218i	$-0.457784\pi$
$701$	7.00232	0.264474	0.132237	+	0.991218i	$0.457784\pi$
$702$	−10.9037	−0.411535
$703$	−56.7920	−2.14195
$704$	−1.55040	−0.0584331
$705$	2.08137	0.0783888
$706$	25.9904	0.978163
$707$	−37.2127	−1.39953
$708$	1.07139	0.0402653
$709$	−38.5981	−1.44958	−0.724791	−	0.688969i	$-0.758064\pi$
$709$	−38.5981	−1.44958	−0.724791	+	0.688969i	$0.758064\pi$
$710$	2.62272	0.0984291
$711$	32.0290	1.20118
$712$	−0.000578536 0	−2.16815e−5 0
$713$	−15.4883	−0.580041
$714$	6.29383	0.235541
$715$	6.67317	0.249562
$716$	−17.7008	−0.661509
$717$	3.31656	0.123859
$718$	4.21368	0.157253
$719$	−48.6986	−1.81615	−0.908076	−	0.418805i	$-0.862449\pi$
$719$	−48.6986	−1.81615	−0.908076	+	0.418805i	$0.862449\pi$
$720$	−2.96263	−0.110411
$721$	37.4878	1.39612
$722$	16.6722	0.620474
$723$	3.32607	0.123698
$724$	−0.426354	−0.0158453
$725$	24.9993	0.928451
$726$	4.01005	0.148827
$727$	36.4413	1.35153	0.675767	−	0.737116i	$-0.263813\pi$
$727$	36.4413	1.35153	0.675767	+	0.737116i	$0.263813\pi$
$728$	−8.27816	−0.306809
$729$	−15.9490	−0.590703
$730$	3.19213	0.118146
$731$	58.5769	2.16655
$732$	−3.30583	−0.122187
$733$	11.3347	0.418657	0.209328	−	0.977845i	$-0.432872\pi$
$733$	11.3347	0.418657	0.209328	+	0.977845i	$0.432872\pi$
$734$	−30.2015	−1.11476
$735$	1.39384	0.0514124
$736$	−1.66591	−0.0614063
$737$	−9.92232	−0.365493
$738$	10.7944	0.397347
$739$	9.18301	0.337802	0.168901	−	0.985633i	$-0.445978\pi$
$739$	9.18301	0.337802	0.168901	+	0.985633i	$0.445978\pi$
$740$	−10.1247	−0.372192
$741$	11.2625	0.413737
$742$	4.51027	0.165577
$743$	40.9137	1.50098	0.750489	−	0.660883i	$-0.229818\pi$
$743$	40.9137	1.50098	0.750489	+	0.660883i	$0.229818\pi$
$744$	−4.33704	−0.159003
$745$	−13.8428	−0.507160
$746$	29.8267	1.09203
$747$	37.9408	1.38818
$748$	10.2144	0.373475
$749$	36.9931	1.35170
$750$	4.40394	0.160809
$751$	−10.8275	−0.395102	−0.197551	−	0.980293i	$-0.563299\pi$
$751$	−10.8275	−0.395102	−0.197551	+	0.980293i	$0.563299\pi$
$752$	−4.19032	−0.152805
$753$	12.7696	0.465349
$754$	26.1376	0.951876
$755$	−5.89896	−0.214685
$756$	5.52402	0.200907
$757$	24.5029	0.890575	0.445287	−	0.895388i	$-0.353102\pi$
$757$	24.5029	0.890575	0.445287	+	0.895388i	$0.353102\pi$
$758$	35.2337	1.27974
$759$	−1.20486	−0.0437338
$760$	6.35953	0.230684
$761$	12.8662	0.466398	0.233199	−	0.972429i	$-0.425081\pi$
$761$	12.8662	0.466398	0.233199	+	0.972429i	$0.425081\pi$
$762$	−6.33474	−0.229484
$763$	−2.63493	−0.0953910
$764$	−26.1783	−0.947098
$765$	19.5184	0.705690
$766$	13.5199	0.488493
$767$	9.28395	0.335224
$768$	−0.466489	−0.0168330
$769$	13.1591	0.474529	0.237265	−	0.971445i	$-0.423749\pi$
$769$	13.1591	0.474529	0.237265	+	0.971445i	$0.423749\pi$
$770$	−3.38074	−0.121833
$771$	−12.7587	−0.459495
$772$	14.6292	0.526518
$773$	22.1001	0.794887	0.397443	−	0.917627i	$-0.369897\pi$
$773$	22.1001	0.794887	0.397443	+	0.917627i	$0.369897\pi$
$774$	24.7387	0.889215
$775$	−35.9452	−1.29119
$776$	−5.18607	−0.186169
$777$	9.08386	0.325882
$778$	7.00882	0.251278
$779$	−23.1710	−0.830188
$780$	2.00784	0.0718921
$781$	−3.81889	−0.136651
$782$	10.9754	0.392478
$783$	−17.4416	−0.623314
$784$	−2.80615	−0.100220
$785$	8.68910	0.310127
$786$	−9.60352	−0.342546
$787$	31.0701	1.10753	0.553765	−	0.832673i	$-0.313191\pi$
$787$	31.0701	1.10753	0.553765	+	0.832673i	$0.313191\pi$
$788$	−7.13353	−0.254121
$789$	2.18430	0.0777633
$790$	−12.2570	−0.436086
$791$	−7.12019	−0.253165
$792$	4.31383	0.153285
$793$	−28.6461	−1.01725
$794$	−30.7612	−1.09167
$795$	−1.09395	−0.0387984
$796$	−5.26144	−0.186487
$797$	−43.9698	−1.55749	−0.778745	−	0.627341i	$-0.784143\pi$
$797$	−43.9698	−1.55749	−0.778745	+	0.627341i	$0.784143\pi$
$798$	−5.70575	−0.201981
$799$	27.6067	0.976653
$800$	−3.86624	−0.136692
$801$	0.00160971	5.68763e−5 0
$802$	9.31409	0.328892
$803$	−4.64799	−0.164024
$804$	−2.98545	−0.105289
$805$	−3.63261	−0.128033
$806$	−37.5819	−1.32377
$807$	−5.43261	−0.191237
$808$	−18.1712	−0.639261
$809$	−7.27889	−0.255912	−0.127956	−	0.991780i	$-0.540842\pi$
$809$	−7.27889	−0.255912	−0.127956	+	0.991780i	$0.540842\pi$
$810$	7.54807	0.265212
$811$	26.6403	0.935468	0.467734	−	0.883869i	$-0.345070\pi$
$811$	26.6403	0.935468	0.467734	+	0.883869i	$0.345070\pi$
$812$	−13.2418	−0.464694
$813$	4.96766	0.174224
$814$	14.7424	0.516720
$815$	13.9691	0.489316
$816$	3.07332	0.107588
$817$	−53.1037	−1.85786
$818$	−37.5145	−1.31166
$819$	23.0330	0.804840
$820$	−4.13086	−0.144256
$821$	−45.5302	−1.58901	−0.794507	−	0.607255i	$-0.792271\pi$
$821$	−45.5302	−1.58901	−0.794507	+	0.607255i	$0.792271\pi$
$822$	7.04121	0.245590
$823$	56.3022	1.96257	0.981286	−	0.192555i	$-0.0616773\pi$
$823$	56.3022	1.96257	0.981286	+	0.192555i	$0.0616773\pi$
$824$	18.3055	0.637704
$825$	−2.79625	−0.0973528
$826$	−4.70341	−0.163652
$827$	−20.7864	−0.722813	−0.361407	−	0.932408i	$-0.617703\pi$
$827$	−20.7864	−0.722813	−0.361407	+	0.932408i	$0.617703\pi$
$828$	4.63521	0.161085
$829$	−10.3869	−0.360751	−0.180375	−	0.983598i	$-0.557731\pi$
$829$	−10.3869	−0.360751	−0.180375	+	0.983598i	$0.557731\pi$
$830$	−14.5194	−0.503977
$831$	11.1970	0.388420
$832$	−4.04229	−0.140141
$833$	18.4875	0.640553
$834$	0.397086	0.0137500
$835$	6.85104	0.237090
$836$	−9.25998	−0.320263
$837$	25.0784	0.866837
$838$	−30.4002	−1.05016
$839$	48.4100	1.67130	0.835649	−	0.549264i	$-0.185092\pi$
$839$	48.4100	1.67130	0.835649	+	0.549264i	$0.185092\pi$
$840$	−1.01720	−0.0350969
$841$	12.8098	0.441716
$842$	−27.1973	−0.937282
$843$	12.5454	0.432087
$844$	1.70465	0.0586763
$845$	3.55645	0.122346
$846$	11.6591	0.400848
$847$	−17.6042	−0.604886
$848$	2.20240	0.0756306
$849$	−0.786101	−0.0269789
$850$	25.4716	0.873668
$851$	15.8407	0.543012
$852$	−1.14904	−0.0393653
$853$	−13.0229	−0.445894	−0.222947	−	0.974831i	$-0.571568\pi$
$853$	−13.0229	−0.445894	−0.222947	+	0.974831i	$0.571568\pi$
$854$	14.5126	0.496611
$855$	−17.6947	−0.605145
$856$	18.0640	0.617415
$857$	16.9762	0.579894	0.289947	−	0.957043i	$-0.406362\pi$
$857$	16.9762	0.579894	0.289947	+	0.957043i	$0.406362\pi$
$858$	−2.92357	−0.0998091
$859$	4.84653	0.165361	0.0826807	−	0.996576i	$-0.473652\pi$
$859$	4.84653	0.165361	0.0826807	+	0.996576i	$0.473652\pi$
$860$	−9.46716	−0.322828
$861$	3.70620	0.126307
$862$	17.3646	0.591441
$863$	−22.9441	−0.781027	−0.390514	−	0.920597i	$-0.627703\pi$
$863$	−22.9441	−0.781027	−0.390514	+	0.920597i	$0.627703\pi$
$864$	2.69742	0.0917681
$865$	21.3902	0.727290
$866$	−15.3082	−0.520194
$867$	−12.3174	−0.418320
$868$	19.0396	0.646247
$869$	17.8472	0.605425
$870$	3.21174	0.108888
$871$	−25.8699	−0.876569
$872$	−1.28666	−0.0435717
$873$	14.4297	0.488370
$874$	−9.94984	−0.336559
$875$	−19.3333	−0.653585
$876$	−1.39850	−0.0472509
$877$	−6.40659	−0.216335	−0.108168	−	0.994133i	$-0.534498\pi$
$877$	−6.40659	−0.216335	−0.108168	+	0.994133i	$0.534498\pi$
$878$	23.6445	0.797963
$879$	−2.22135	−0.0749244
$880$	−1.65084	−0.0556498
$881$	−51.5568	−1.73699	−0.868496	−	0.495696i	$-0.834913\pi$
$881$	−51.5568	−1.73699	−0.868496	+	0.495696i	$0.834913\pi$
$882$	7.80779	0.262902
$883$	19.8995	0.669673	0.334837	−	0.942276i	$-0.391319\pi$
$883$	19.8995	0.669673	0.334837	+	0.942276i	$0.391319\pi$
$884$	26.6314	0.895711
$885$	1.14079	0.0383474
$886$	21.6992	0.728997
$887$	10.7631	0.361389	0.180694	−	0.983539i	$-0.442165\pi$
$887$	10.7631	0.361389	0.180694	+	0.983539i	$0.442165\pi$
$888$	4.43572	0.148853
$889$	27.8096	0.932703
$890$	−0.000616014 0	−2.06488e−5 0
$891$	−10.9906	−0.368198
$892$	8.74841	0.292918
$893$	−25.0272	−0.837502
$894$	6.06463	0.202832
$895$	−18.8475	−0.630001
$896$	2.04789	0.0684152
$897$	−3.14138	−0.104888
$898$	33.3703	1.11358
$899$	−60.1161	−2.00498
$900$	10.7574	0.358580
$901$	−14.5098	−0.483393
$902$	6.01486	0.200273
$903$	8.49391	0.282660
$904$	−3.47684	−0.115638
$905$	−0.453973	−0.0150906
$906$	2.58438	0.0858604
$907$	36.2673	1.20423	0.602117	−	0.798408i	$-0.294324\pi$
$907$	36.2673	1.20423	0.602117	+	0.798408i	$0.294324\pi$
$908$	−3.50226	−0.116227
$909$	50.5594	1.67695
$910$	−8.81442	−0.292195
$911$	−52.6988	−1.74599	−0.872995	−	0.487730i	$-0.837825\pi$
$911$	−52.6988	−1.74599	−0.872995	+	0.487730i	$0.837825\pi$
$912$	−2.78616	−0.0922590
$913$	21.1414	0.699679
$914$	−11.3206	−0.374453
$915$	−3.51998	−0.116367
$916$	7.62737	0.252016
$917$	42.1595	1.39223
$918$	−17.7712	−0.586536
$919$	−56.9812	−1.87964	−0.939818	−	0.341676i	$-0.889005\pi$
$919$	−56.9812	−1.87964	−0.939818	+	0.341676i	$0.889005\pi$
$920$	−1.77383	−0.0584814
$921$	−0.965179	−0.0318037
$922$	33.2619	1.09542
$923$	−9.95679	−0.327732
$924$	1.48113	0.0487256
$925$	36.7631	1.20876
$926$	0.931641	0.0306156
$927$	−50.9331	−1.67286
$928$	−6.46605	−0.212258
$929$	−18.4795	−0.606294	−0.303147	−	0.952944i	$-0.598037\pi$
$929$	−18.4795	−0.606294	−0.303147	+	0.952944i	$0.598037\pi$
$930$	−4.61799	−0.151430
$931$	−16.7600	−0.549288
$932$	12.0808	0.395719
$933$	−4.55163	−0.149014
$934$	9.92622	0.324796
$935$	10.8761	0.355686
$936$	11.2472	0.367627
$937$	−17.8542	−0.583272	−0.291636	−	0.956529i	$-0.594200\pi$
$937$	−17.8542	−0.583272	−0.291636	+	0.956529i	$0.594200\pi$
$938$	13.1061	0.427931
$939$	7.88476	0.257309
$940$	−4.46177	−0.145527
$941$	24.2165	0.789436	0.394718	−	0.918802i	$-0.370842\pi$
$941$	24.2165	0.789436	0.394718	+	0.918802i	$0.370842\pi$
$942$	−3.80676	−0.124031
$943$	6.46297	0.210463
$944$	−2.29671	−0.0747515
$945$	5.88187	0.191337
$946$	13.7849	0.448187
$947$	19.3477	0.628717	0.314359	−	0.949304i	$-0.398211\pi$
$947$	19.3477	0.628717	0.314359	+	0.949304i	$0.398211\pi$
$948$	5.36990	0.174406
$949$	−12.1185	−0.393382
$950$	−23.0916	−0.749190
$951$	8.13363	0.263751
$952$	−13.4919	−0.437276
$953$	0.401723	0.0130131	0.00650654	−	0.999979i	$-0.497929\pi$
$953$	0.401723	0.0130131	0.00650654	+	0.999979i	$0.497929\pi$
$954$	−6.12792	−0.198399
$955$	−27.8742	−0.901987
$956$	−7.10961	−0.229941
$957$	−4.67655	−0.151171
$958$	10.2867	0.332349
$959$	−30.9110	−0.998167
$960$	−0.496709	−0.0160312
$961$	55.4377	1.78831
$962$	38.4370	1.23926
$963$	−50.2610	−1.61964
$964$	−7.13001	−0.229642
$965$	15.5769	0.501439
$966$	1.59147	0.0512048
$967$	−19.0424	−0.612363	−0.306181	−	0.951973i	$-0.599051\pi$
$967$	−19.0424	−0.612363	−0.306181	+	0.951973i	$0.599051\pi$
$968$	−8.59625	−0.276294
$969$	18.3558	0.589673
$970$	−5.52203	−0.177302
$971$	−61.8529	−1.98495	−0.992477	−	0.122428i	$-0.960932\pi$
$971$	−61.8529	−1.98495	−0.992477	+	0.122428i	$0.960932\pi$
$972$	−11.3991	−0.365627
$973$	−1.74321	−0.0558848
$974$	8.44811	0.270695
$975$	−7.29050	−0.233483
$976$	7.08661	0.226837
$977$	−29.9345	−0.957688	−0.478844	−	0.877900i	$-0.658944\pi$
$977$	−29.9345	−0.957688	−0.478844	+	0.877900i	$0.658944\pi$
$978$	−6.11998	−0.195695
$979$	0.000896964 0	2.86671e−5 0
$980$	−2.98793	−0.0954459
$981$	3.57998	0.114300
$982$	19.5608	0.624209
$983$	13.4943	0.430401	0.215200	−	0.976570i	$-0.430960\pi$
$983$	13.4943	0.430401	0.215200	+	0.976570i	$0.430960\pi$
$984$	1.80976	0.0576932
$985$	−7.59564	−0.242017
$986$	42.5996	1.35665
$987$	4.00309	0.127420
$988$	−24.1430	−0.768092
$989$	14.8119	0.470992
$990$	4.59328	0.145984
$991$	22.8497	0.725846	0.362923	−	0.931819i	$-0.381779\pi$
$991$	22.8497	0.725846	0.362923	+	0.931819i	$0.381779\pi$
$992$	9.29719	0.295186
$993$	−5.15313	−0.163530
$994$	5.04428	0.159995
$995$	−5.60228	−0.177604
$996$	6.36108	0.201558
$997$	−11.2957	−0.357739	−0.178869	−	0.983873i	$-0.557244\pi$
$997$	−11.2957	−0.357739	−0.178869	+	0.983873i	$0.557244\pi$
$998$	8.63851	0.273447
$999$	−25.6490	−0.811500

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.f.1.16	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.f.1.16	✓	31	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4006 = 2 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.466489 q^{3} +1.00000 q^{4} +1.06478 q^{5} -0.466489 q^{6} +2.04789 q^{7} +1.00000 q^{8} -2.78239 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.466489 q^{3} +1.00000 q^{4} +1.06478 q^{5} -0.466489 q^{6} +2.04789 q^{7} +1.00000 q^{8} -2.78239 q^{9} +1.06478 q^{10} -1.55040 q^{11} -0.466489 q^{12} -4.04229 q^{13} +2.04789 q^{14} -0.496709 q^{15} +1.00000 q^{16} -6.58820 q^{17} -2.78239 q^{18} +5.97262 q^{19} +1.06478 q^{20} -0.955318 q^{21} -1.55040 q^{22} -1.66591 q^{23} -0.466489 q^{24} -3.86624 q^{25} -4.04229 q^{26} +2.69742 q^{27} +2.04789 q^{28} -6.46605 q^{29} -0.496709 q^{30} +9.29719 q^{31} +1.00000 q^{32} +0.723247 q^{33} -6.58820 q^{34} +2.18055 q^{35} -2.78239 q^{36} -9.50873 q^{37} +5.97262 q^{38} +1.88568 q^{39} +1.06478 q^{40} -3.87954 q^{41} -0.955318 q^{42} -8.89119 q^{43} -1.55040 q^{44} -2.96263 q^{45} -1.66591 q^{46} -4.19032 q^{47} -0.466489 q^{48} -2.80615 q^{49} -3.86624 q^{50} +3.07332 q^{51} -4.04229 q^{52} +2.20240 q^{53} +2.69742 q^{54} -1.65084 q^{55} +2.04789 q^{56} -2.78616 q^{57} -6.46605 q^{58} -2.29671 q^{59} -0.496709 q^{60} +7.08661 q^{61} +9.29719 q^{62} -5.69802 q^{63} +1.00000 q^{64} -4.30415 q^{65} +0.723247 q^{66} +6.39982 q^{67} -6.58820 q^{68} +0.777129 q^{69} +2.18055 q^{70} +2.46316 q^{71} -2.78239 q^{72} +2.99792 q^{73} -9.50873 q^{74} +1.80356 q^{75} +5.97262 q^{76} -3.17506 q^{77} +1.88568 q^{78} -11.5113 q^{79} +1.06478 q^{80} +7.08885 q^{81} -3.87954 q^{82} -13.6361 q^{83} -0.955318 q^{84} -7.01499 q^{85} -8.89119 q^{86} +3.01634 q^{87} -1.55040 q^{88} -0.000578536 q^{89} -2.96263 q^{90} -8.27816 q^{91} -1.66591 q^{92} -4.33704 q^{93} -4.19032 q^{94} +6.35953 q^{95} -0.466489 q^{96} -5.18607 q^{97} -2.80615 q^{98} +4.31383 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9}+O(q^{10}) \) 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9	\( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9} - 23 q^{10} - 32 q^{11} - 13 q^{12} - 8 q^{13} - 18 q^{14} - 14 q^{15} + 31 q^{16} - 30 q^{17} + 20 q^{18} - 38 q^{19} - 23 q^{20} - 16 q^{21} - 32 q^{22} - 24 q^{23} - 13 q^{24} + 40 q^{25} - 8 q^{26} - 28 q^{27} - 18 q^{28} - 7 q^{29} - 14 q^{30} - 32 q^{31} + 31 q^{32} - 9 q^{33} - 30 q^{34} - 14 q^{35} + 20 q^{36} + 11 q^{37} - 38 q^{38} - 9 q^{39} - 23 q^{40} - 76 q^{41} - 16 q^{42} - 33 q^{43} - 32 q^{44} - 40 q^{45} - 24 q^{46} - 96 q^{47} - 13 q^{48} + 15 q^{49} + 40 q^{50} - 55 q^{51} - 8 q^{52} - 28 q^{53} - 28 q^{54} - 52 q^{55} - 18 q^{56} - 21 q^{57} - 7 q^{58} - 72 q^{59} - 14 q^{60} - 9 q^{61} - 32 q^{62} - 54 q^{63} + 31 q^{64} - 38 q^{65} - 9 q^{66} - 4 q^{67} - 30 q^{68} - 17 q^{69} - 14 q^{70} - 61 q^{71} + 20 q^{72} - 62 q^{73} + 11 q^{74} - 63 q^{75} - 38 q^{76} - 9 q^{77} - 9 q^{78} - 30 q^{79} - 23 q^{80} - 13 q^{81} - 76 q^{82} - 90 q^{83} - 16 q^{84} + 26 q^{85} - 33 q^{86} - 34 q^{87} - 32 q^{88} - 99 q^{89} - 40 q^{90} - 47 q^{91} - 24 q^{92} - 6 q^{93} - 96 q^{94} - 24 q^{95} - 13 q^{96} - 46 q^{97} + 15 q^{98} - 48 q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9 - 23 * q^10 - 32 * q^11 - 13 * q^12 - 8 * q^13 - 18 * q^14 - 14 * q^15 + 31 * q^16 - 30 * q^17 + 20 * q^18 - 38 * q^19 - 23 * q^20 - 16 * q^21 - 32 * q^22 - 24 * q^23 - 13 * q^24 + 40 * q^25 - 8 * q^26 - 28 * q^27 - 18 * q^28 - 7 * q^29 - 14 * q^30 - 32 * q^31 + 31 * q^32 - 9 * q^33 - 30 * q^34 - 14 * q^35 + 20 * q^36 + 11 * q^37 - 38 * q^38 - 9 * q^39 - 23 * q^40 - 76 * q^41 - 16 * q^42 - 33 * q^43 - 32 * q^44 - 40 * q^45 - 24 * q^46 - 96 * q^47 - 13 * q^48 + 15 * q^49 + 40 * q^50 - 55 * q^51 - 8 * q^52 - 28 * q^53 - 28 * q^54 - 52 * q^55 - 18 * q^56 - 21 * q^57 - 7 * q^58 - 72 * q^59 - 14 * q^60 - 9 * q^61 - 32 * q^62 - 54 * q^63 + 31 * q^64 - 38 * q^65 - 9 * q^66 - 4 * q^67 - 30 * q^68 - 17 * q^69 - 14 * q^70 - 61 * q^71 + 20 * q^72 - 62 * q^73 + 11 * q^74 - 63 * q^75 - 38 * q^76 - 9 * q^77 - 9 * q^78 - 30 * q^79 - 23 * q^80 - 13 * q^81 - 76 * q^82 - 90 * q^83 - 16 * q^84 + 26 * q^85 - 33 * q^86 - 34 * q^87 - 32 * q^88 - 99 * q^89 - 40 * q^90 - 47 * q^91 - 24 * q^92 - 6 * q^93 - 96 * q^94 - 24 * q^95 - 13 * q^96 - 46 * q^97 + 15 * q^98 - 48 * q^99

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