LMFDB - Embedded newform 4003.2.a.b.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4003,2,Mod(1,4003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4003, 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("4003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4003.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:	$31.9641159291$
Analytic rank:	$1$
Dimension:	$152$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	4003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64207 q^{2} +1.02501 q^{3} +4.98054 q^{4} +2.32673 q^{5} -2.70814 q^{6} -0.493904 q^{7} -7.87479 q^{8} -1.94936 q^{9} +O(q^{10})$	\(q-2.64207 q^{2} +1.02501 q^{3} +4.98054 q^{4} +2.32673 q^{5} -2.70814 q^{6} -0.493904 q^{7} -7.87479 q^{8} -1.94936 q^{9} -6.14739 q^{10} +2.96142 q^{11} +5.10508 q^{12} +0.776601 q^{13} +1.30493 q^{14} +2.38491 q^{15} +10.8447 q^{16} +6.61446 q^{17} +5.15036 q^{18} -7.31111 q^{19} +11.5884 q^{20} -0.506254 q^{21} -7.82428 q^{22} -8.89506 q^{23} -8.07170 q^{24} +0.413674 q^{25} -2.05183 q^{26} -5.07312 q^{27} -2.45991 q^{28} -2.08930 q^{29} -6.30110 q^{30} +0.878413 q^{31} -12.9028 q^{32} +3.03547 q^{33} -17.4759 q^{34} -1.14918 q^{35} -9.70888 q^{36} +4.25838 q^{37} +19.3165 q^{38} +0.796020 q^{39} -18.3225 q^{40} -11.1405 q^{41} +1.33756 q^{42} -7.88750 q^{43} +14.7495 q^{44} -4.53564 q^{45} +23.5014 q^{46} +10.9205 q^{47} +11.1158 q^{48} -6.75606 q^{49} -1.09296 q^{50} +6.77986 q^{51} +3.86789 q^{52} -10.8022 q^{53} +13.4036 q^{54} +6.89043 q^{55} +3.88939 q^{56} -7.49393 q^{57} +5.52008 q^{58} +2.76488 q^{59} +11.8781 q^{60} +8.06297 q^{61} -2.32083 q^{62} +0.962798 q^{63} +12.4008 q^{64} +1.80694 q^{65} -8.01993 q^{66} -0.138871 q^{67} +32.9436 q^{68} -9.11748 q^{69} +3.03622 q^{70} -10.2433 q^{71} +15.3508 q^{72} -11.2894 q^{73} -11.2509 q^{74} +0.424018 q^{75} -36.4132 q^{76} -1.46266 q^{77} -2.10314 q^{78} +12.0755 q^{79} +25.2326 q^{80} +0.648113 q^{81} +29.4339 q^{82} -5.70804 q^{83} -2.52142 q^{84} +15.3901 q^{85} +20.8393 q^{86} -2.14155 q^{87} -23.3206 q^{88} -11.7510 q^{89} +11.9835 q^{90} -0.383566 q^{91} -44.3022 q^{92} +0.900378 q^{93} -28.8528 q^{94} -17.0110 q^{95} -13.2254 q^{96} -6.74086 q^{97} +17.8500 q^{98} -5.77289 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9}+O(q^{10}) $ 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9} - 15 q^{10} - 40 q^{11} - 53 q^{12} - 59 q^{13} - 36 q^{14} - 40 q^{15} + 118 q^{16} - 93 q^{17} - 59 q^{18} - 16 q^{19} - 108 q^{20} - 62 q^{21} - 37 q^{22} - 107 q^{23} - 31 q^{24} + 101 q^{25} - 64 q^{26} - 63 q^{27} - 53 q^{28} - 124 q^{29} - 68 q^{30} - 15 q^{31} - 129 q^{32} - 49 q^{33} - 76 q^{35} + 45 q^{36} - 98 q^{37} - 125 q^{38} - 47 q^{39} - 7 q^{40} - 56 q^{41} - 84 q^{42} - 62 q^{43} - 114 q^{44} - 142 q^{45} - 3 q^{46} - 111 q^{47} - 92 q^{48} + 117 q^{49} - 64 q^{50} - 21 q^{51} - 85 q^{52} - 347 q^{53} + 3 q^{54} - 16 q^{55} - 73 q^{56} - 115 q^{57} - 29 q^{58} - 50 q^{59} - 54 q^{60} - 62 q^{61} - 55 q^{62} - 70 q^{63} + 64 q^{64} - 147 q^{65} + 34 q^{66} - 86 q^{67} - 174 q^{68} - 104 q^{69} - 7 q^{70} - 86 q^{71} - 139 q^{72} - 27 q^{73} - 52 q^{74} - 49 q^{75} - 11 q^{76} - 346 q^{77} - 59 q^{78} - 17 q^{79} - 149 q^{80} - 8 q^{81} - 31 q^{82} - 106 q^{83} - 51 q^{84} - 69 q^{85} - 85 q^{86} - 32 q^{87} - 113 q^{88} - 59 q^{89} + 10 q^{90} - 9 q^{91} - 314 q^{92} - 230 q^{93} + 7 q^{94} - 74 q^{95} - 54 q^{96} - 60 q^{97} - 77 q^{98} - 96 q^{99}+O(q^{100}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9 - 15 * q^10 - 40 * q^11 - 53 * q^12 - 59 * q^13 - 36 * q^14 - 40 * q^15 + 118 * q^16 - 93 * q^17 - 59 * q^18 - 16 * q^19 - 108 * q^20 - 62 * q^21 - 37 * q^22 - 107 * q^23 - 31 * q^24 + 101 * q^25 - 64 * q^26 - 63 * q^27 - 53 * q^28 - 124 * q^29 - 68 * q^30 - 15 * q^31 - 129 * q^32 - 49 * q^33 - 76 * q^35 + 45 * q^36 - 98 * q^37 - 125 * q^38 - 47 * q^39 - 7 * q^40 - 56 * q^41 - 84 * q^42 - 62 * q^43 - 114 * q^44 - 142 * q^45 - 3 * q^46 - 111 * q^47 - 92 * q^48 + 117 * q^49 - 64 * q^50 - 21 * q^51 - 85 * q^52 - 347 * q^53 + 3 * q^54 - 16 * q^55 - 73 * q^56 - 115 * q^57 - 29 * q^58 - 50 * q^59 - 54 * q^60 - 62 * q^61 - 55 * q^62 - 70 * q^63 + 64 * q^64 - 147 * q^65 + 34 * q^66 - 86 * q^67 - 174 * q^68 - 104 * q^69 - 7 * q^70 - 86 * q^71 - 139 * q^72 - 27 * q^73 - 52 * q^74 - 49 * q^75 - 11 * q^76 - 346 * q^77 - 59 * q^78 - 17 * q^79 - 149 * q^80 - 8 * q^81 - 31 * q^82 - 106 * q^83 - 51 * q^84 - 69 * q^85 - 85 * q^86 - 32 * q^87 - 113 * q^88 - 59 * q^89 + 10 * q^90 - 9 * q^91 - 314 * q^92 - 230 * q^93 + 7 * q^94 - 74 * q^95 - 54 * q^96 - 60 * q^97 - 77 * q^98 - 96 * 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 $

$2$	−2.64207	−1.86823	−0.934113	−	0.356978i	$-0.883807\pi$
$2$	−2.64207	−1.86823	−0.934113	+	0.356978i	$0.883807\pi$
$3$	1.02501	0.591787	0.295894	−	0.955221i	$-0.404383\pi$
$3$	1.02501	0.591787	0.295894	+	0.955221i	$0.404383\pi$
$4$	4.98054	2.49027
$5$	2.32673	1.04055	0.520273	−	0.854000i	$-0.325830\pi$
$5$	2.32673	1.04055	0.520273	+	0.854000i	$0.325830\pi$
$6$	−2.70814	−1.10559
$7$	−0.493904	−0.186678	−0.0933391	−	0.995634i	$-0.529754\pi$
$7$	−0.493904	−0.186678	−0.0933391	+	0.995634i	$0.529754\pi$
$8$	−7.87479	−2.78416
$9$	−1.94936	−0.649788
$10$	−6.14739	−1.94397
$11$	2.96142	0.892902	0.446451	−	0.894808i	$-0.352688\pi$
$11$	2.96142	0.892902	0.446451	+	0.894808i	$0.352688\pi$
$12$	5.10508	1.47371
$13$	0.776601	0.215390	0.107695	−	0.994184i	$-0.465653\pi$
$13$	0.776601	0.215390	0.107695	+	0.994184i	$0.465653\pi$
$14$	1.30493	0.348757
$15$	2.38491	0.615781
$16$	10.8447	2.71117
$17$	6.61446	1.60424	0.802122	−	0.597161i	$-0.203705\pi$
$17$	6.61446	1.60424	0.802122	+	0.597161i	$0.203705\pi$
$18$	5.15036	1.21395
$19$	−7.31111	−1.67728	−0.838642	−	0.544683i	$-0.816650\pi$
$19$	−7.31111	−1.67728	−0.838642	+	0.544683i	$0.816650\pi$
$20$	11.5884	2.59124
$21$	−0.506254	−0.110474
$22$	−7.82428	−1.66814
$23$	−8.89506	−1.85475	−0.927374	−	0.374135i	$-0.877940\pi$
$23$	−8.89506	−1.85475	−0.927374	+	0.374135i	$0.877940\pi$
$24$	−8.07170	−1.64763
$25$	0.413674	0.0827348
$26$	−2.05183	−0.402398
$27$	−5.07312	−0.976323
$28$	−2.45991	−0.464879
$29$	−2.08930	−0.387974	−0.193987	−	0.981004i	$-0.562142\pi$
$29$	−2.08930	−0.387974	−0.193987	+	0.981004i	$0.562142\pi$
$30$	−6.30110	−1.15042
$31$	0.878413	0.157768	0.0788838	−	0.996884i	$-0.474864\pi$
$31$	0.878413	0.157768	0.0788838	+	0.996884i	$0.474864\pi$
$32$	−12.9028	−2.28091
$33$	3.03547	0.528408
$34$	−17.4759	−2.99709
$35$	−1.14918	−0.194247
$36$	−9.70888	−1.61815
$37$	4.25838	0.700074	0.350037	−	0.936736i	$-0.386169\pi$
$37$	4.25838	0.700074	0.350037	+	0.936736i	$0.386169\pi$
$38$	19.3165	3.13354
$39$	0.796020	0.127465
$40$	−18.3225	−2.89704
$41$	−11.1405	−1.73985	−0.869926	−	0.493183i	$-0.835833\pi$
$41$	−11.1405	−1.73985	−0.869926	+	0.493183i	$0.835833\pi$
$42$	1.33756	0.206390
$43$	−7.88750	−1.20283	−0.601416	−	0.798936i	$-0.705397\pi$
$43$	−7.88750	−1.20283	−0.601416	+	0.798936i	$0.705397\pi$
$44$	14.7495	2.22357
$45$	−4.53564	−0.676134
$46$	23.5014	3.46509
$47$	10.9205	1.59292	0.796460	−	0.604691i	$-0.206704\pi$
$47$	10.9205	1.59292	0.796460	+	0.604691i	$0.206704\pi$
$48$	11.1158	1.60443
$49$	−6.75606	−0.965151
$50$	−1.09296	−0.154567
$51$	6.77986	0.949370
$52$	3.86789	0.536380
$53$	−10.8022	−1.48380	−0.741898	−	0.670513i	$-0.766074\pi$
$53$	−10.8022	−1.48380	−0.741898	+	0.670513i	$0.766074\pi$
$54$	13.4036	1.82399
$55$	6.89043	0.929105
$56$	3.88939	0.519741
$57$	−7.49393	−0.992595
$58$	5.52008	0.724823
$59$	2.76488	0.359957	0.179979	−	0.983671i	$-0.442397\pi$
$59$	2.76488	0.359957	0.179979	+	0.983671i	$0.442397\pi$
$60$	11.8781	1.53346
$61$	8.06297	1.03236	0.516179	−	0.856481i	$-0.327354\pi$
$61$	8.06297	1.03236	0.516179	+	0.856481i	$0.327354\pi$
$62$	−2.32083	−0.294746
$63$	0.962798	0.121301
$64$	12.4008	1.55010
$65$	1.80694	0.224123
$66$	−8.01993	−0.987185
$67$	−0.138871	−0.0169657	−0.00848287	−	0.999964i	$-0.502700\pi$
$67$	−0.138871	−0.0169657	−0.00848287	+	0.999964i	$0.502700\pi$
$68$	32.9436	3.99500
$69$	−9.11748	−1.09762
$70$	3.03622	0.362897
$71$	−10.2433	−1.21566	−0.607829	−	0.794068i	$-0.707960\pi$
$71$	−10.2433	−1.21566	−0.607829	+	0.794068i	$0.707960\pi$
$72$	15.3508	1.80911
$73$	−11.2894	−1.32132	−0.660662	−	0.750684i	$-0.729724\pi$
$73$	−11.2894	−1.32132	−0.660662	+	0.750684i	$0.729724\pi$
$74$	−11.2509	−1.30790
$75$	0.424018	0.0489614
$76$	−36.4132	−4.17689
$77$	−1.46266	−0.166685
$78$	−2.10314	−0.238134
$79$	12.0755	1.35860	0.679299	−	0.733862i	$-0.262284\pi$
$79$	12.0755	1.35860	0.679299	+	0.733862i	$0.262284\pi$
$80$	25.2326	2.82109
$81$	0.648113	0.0720126
$82$	29.4339	3.25044
$83$	−5.70804	−0.626539	−0.313269	−	0.949664i	$-0.601424\pi$
$83$	−5.70804	−0.626539	−0.313269	+	0.949664i	$0.601424\pi$
$84$	−2.52142	−0.275109
$85$	15.3901	1.66929
$86$	20.8393	2.24716
$87$	−2.14155	−0.229598
$88$	−23.3206	−2.48598
$89$	−11.7510	−1.24560	−0.622801	−	0.782381i	$-0.714005\pi$
$89$	−11.7510	−1.24560	−0.622801	+	0.782381i	$0.714005\pi$
$90$	11.9835	1.26317
$91$	−0.383566	−0.0402087
$92$	−44.3022	−4.61882
$93$	0.900378	0.0933649
$94$	−28.8528	−2.97593
$95$	−17.0110	−1.74529
$96$	−13.2254	−1.34982
$97$	−6.74086	−0.684431	−0.342215	−	0.939622i	$-0.611177\pi$
$97$	−6.74086	−0.684431	−0.342215	+	0.939622i	$0.611177\pi$
$98$	17.8500	1.80312
$99$	−5.77289	−0.580197
$100$	2.06032	0.206032
$101$	−1.13676	−0.113112	−0.0565561	−	0.998399i	$-0.518012\pi$
$101$	−1.13676	−0.113112	−0.0565561	+	0.998399i	$0.518012\pi$
$102$	−17.9129	−1.77364
$103$	−6.15488	−0.606458	−0.303229	−	0.952918i	$-0.598065\pi$
$103$	−6.15488	−0.606458	−0.303229	+	0.952918i	$0.598065\pi$
$104$	−6.11556	−0.599680
$105$	−1.17792	−0.114953
$106$	28.5402	2.77207
$107$	11.1794	1.08075	0.540374	−	0.841425i	$-0.318283\pi$
$107$	11.1794	1.08075	0.540374	+	0.841425i	$0.318283\pi$
$108$	−25.2669	−2.43131
$109$	13.4404	1.28736	0.643679	−	0.765295i	$-0.277407\pi$
$109$	13.4404	1.28736	0.643679	+	0.765295i	$0.277407\pi$
$110$	−18.2050	−1.73578
$111$	4.36486	0.414294
$112$	−5.35622	−0.506115
$113$	−13.2241	−1.24402	−0.622011	−	0.783009i	$-0.713684\pi$
$113$	−13.2241	−1.24402	−0.622011	+	0.783009i	$0.713684\pi$
$114$	19.7995	1.85439
$115$	−20.6964	−1.92995
$116$	−10.4058	−0.966158
$117$	−1.51388	−0.139958
$118$	−7.30501	−0.672481
$119$	−3.26691	−0.299477
$120$	−18.7807	−1.71443
$121$	−2.22999	−0.202726
$122$	−21.3029	−1.92868
$123$	−11.4191	−1.02962
$124$	4.37497	0.392884
$125$	−10.6711	−0.954456
$126$	−2.54378	−0.226618
$127$	0.798224	0.0708310	0.0354155	−	0.999373i	$-0.488725\pi$
$127$	0.798224	0.0708310	0.0354155	+	0.999373i	$0.488725\pi$
$128$	−6.95811	−0.615016
$129$	−8.08473	−0.711820
$130$	−4.77406	−0.418713
$131$	−15.7436	−1.37552	−0.687760	−	0.725938i	$-0.741406\pi$
$131$	−15.7436	−1.37552	−0.687760	+	0.725938i	$0.741406\pi$
$132$	15.1183	1.31588
$133$	3.61099	0.313112
$134$	0.366906	0.0316958
$135$	−11.8038	−1.01591
$136$	−52.0875	−4.46646
$137$	−18.6132	−1.59023	−0.795115	−	0.606459i	$-0.792590\pi$
$137$	−18.6132	−1.59023	−0.795115	+	0.606459i	$0.792590\pi$
$138$	24.0890	2.05059
$139$	10.9726	0.930680	0.465340	−	0.885132i	$-0.345932\pi$
$139$	10.9726	0.930680	0.465340	+	0.885132i	$0.345932\pi$
$140$	−5.72354	−0.483727
$141$	11.1936	0.942669
$142$	27.0636	2.27113
$143$	2.29984	0.192322
$144$	−21.1402	−1.76168
$145$	−4.86124	−0.403704
$146$	29.8274	2.46853
$147$	−6.92500	−0.571164
$148$	21.2090	1.74337
$149$	1.99896	0.163761	0.0818807	−	0.996642i	$-0.473907\pi$
$149$	1.99896	0.163761	0.0818807	+	0.996642i	$0.473907\pi$
$150$	−1.12029	−0.0914710
$151$	−4.34393	−0.353504	−0.176752	−	0.984255i	$-0.556559\pi$
$151$	−4.34393	−0.353504	−0.176752	+	0.984255i	$0.556559\pi$
$152$	57.5734	4.66982
$153$	−12.8940	−1.04242
$154$	3.86444	0.311406
$155$	2.04383	0.164164
$156$	3.96461	0.317422
$157$	8.39098	0.669673	0.334837	−	0.942276i	$-0.391319\pi$
$157$	8.39098	0.669673	0.334837	+	0.942276i	$0.391319\pi$
$158$	−31.9043	−2.53817
$159$	−11.0723	−0.878091
$160$	−30.0213	−2.37339
$161$	4.39330	0.346241
$162$	−1.71236	−0.134536
$163$	−4.93579	−0.386601	−0.193300	−	0.981140i	$-0.561919\pi$
$163$	−4.93579	−0.386601	−0.193300	+	0.981140i	$0.561919\pi$
$164$	−55.4856	−4.33270
$165$	7.06272	0.549832
$166$	15.0810	1.17052
$167$	7.80826	0.604221	0.302110	−	0.953273i	$-0.402309\pi$
$167$	7.80826	0.604221	0.302110	+	0.953273i	$0.402309\pi$
$168$	3.98664	0.307576
$169$	−12.3969	−0.953607
$170$	−40.6617	−3.11861
$171$	14.2520	1.08988
$172$	−39.2840	−2.99537
$173$	15.1926	1.15507	0.577535	−	0.816366i	$-0.304015\pi$
$173$	15.1926	1.15507	0.577535	+	0.816366i	$0.304015\pi$
$174$	5.65811	0.428941
$175$	−0.204315	−0.0154448
$176$	32.1156	2.42081
$177$	2.83402	0.213018
$178$	31.0469	2.32706
$179$	11.6776	0.872828	0.436414	−	0.899746i	$-0.356248\pi$
$179$	11.6776	0.872828	0.436414	+	0.899746i	$0.356248\pi$
$180$	−22.5899	−1.68375
$181$	22.1145	1.64376	0.821878	−	0.569663i	$-0.192926\pi$
$181$	22.1145	1.64376	0.821878	+	0.569663i	$0.192926\pi$
$182$	1.01341	0.0751188
$183$	8.26459	0.610936
$184$	70.0467	5.16391
$185$	9.90811	0.728458
$186$	−2.37886	−0.174427
$187$	19.5882	1.43243
$188$	54.3900	3.96680
$189$	2.50564	0.182258
$190$	44.9442	3.26060
$191$	3.56942	0.258274	0.129137	−	0.991627i	$-0.458779\pi$
$191$	3.56942	0.258274	0.129137	+	0.991627i	$0.458779\pi$
$192$	12.7109	0.917327
$193$	14.1884	1.02131	0.510653	−	0.859787i	$-0.329404\pi$
$193$	14.1884	1.02131	0.510653	+	0.859787i	$0.329404\pi$
$194$	17.8098	1.27867
$195$	1.85212	0.132633
$196$	−33.6488	−2.40349
$197$	−18.4248	−1.31271	−0.656356	−	0.754452i	$-0.727903\pi$
$197$	−18.4248	−1.31271	−0.656356	+	0.754452i	$0.727903\pi$
$198$	15.2524	1.08394
$199$	15.9680	1.13194	0.565971	−	0.824425i	$-0.308502\pi$
$199$	15.9680	1.13194	0.565971	+	0.824425i	$0.308502\pi$
$200$	−3.25760	−0.230347
$201$	−0.142343	−0.0100401
$202$	3.00341	0.211319
$203$	1.03191	0.0724262
$204$	33.7673	2.36419
$205$	−25.9209	−1.81039
$206$	16.2616	1.13300
$207$	17.3397	1.20519
$208$	8.42198	0.583959
$209$	−21.6513	−1.49765
$210$	3.11214	0.214758
$211$	8.51245	0.586021	0.293010	−	0.956109i	$-0.405343\pi$
$211$	8.51245	0.586021	0.293010	+	0.956109i	$0.405343\pi$
$212$	−53.8007	−3.69505
$213$	−10.4995	−0.719411
$214$	−29.5366	−2.01908
$215$	−18.3521	−1.25160
$216$	39.9498	2.71824
$217$	−0.433852	−0.0294518
$218$	−35.5105	−2.40508
$219$	−11.5717	−0.781942
$220$	34.3180	2.31372
$221$	5.13680	0.345538
$222$	−11.5323	−0.773996
$223$	13.2510	0.887353	0.443676	−	0.896187i	$-0.353674\pi$
$223$	13.2510	0.887353	0.443676	+	0.896187i	$0.353674\pi$
$224$	6.37274	0.425797
$225$	−0.806402	−0.0537601
$226$	34.9391	2.32411
$227$	−6.83741	−0.453815	−0.226908	−	0.973916i	$-0.572862\pi$
$227$	−6.83741	−0.453815	−0.226908	+	0.973916i	$0.572862\pi$
$228$	−37.3238	−2.47183
$229$	−19.9059	−1.31542	−0.657709	−	0.753273i	$-0.728474\pi$
$229$	−19.9059	−1.31542	−0.657709	+	0.753273i	$0.728474\pi$
$230$	54.6814	3.60558
$231$	−1.49923	−0.0986422
$232$	16.4528	1.08018
$233$	−13.6658	−0.895276	−0.447638	−	0.894215i	$-0.647735\pi$
$233$	−13.6658	−0.895276	−0.447638	+	0.894215i	$0.647735\pi$
$234$	3.99977	0.261473
$235$	25.4091	1.65751
$236$	13.7706	0.896389
$237$	12.3774	0.804001
$238$	8.63140	0.559491
$239$	15.2687	0.987648	0.493824	−	0.869562i	$-0.335599\pi$
$239$	15.2687	0.987648	0.493824	+	0.869562i	$0.335599\pi$
$240$	25.8636	1.66949
$241$	4.11200	0.264877	0.132439	−	0.991191i	$-0.457719\pi$
$241$	4.11200	0.264877	0.132439	+	0.991191i	$0.457719\pi$
$242$	5.89178	0.378738
$243$	15.8837	1.01894
$244$	40.1579	2.57085
$245$	−15.7195	−1.00428
$246$	30.1700	1.92357
$247$	−5.67781	−0.361271
$248$	−6.91732	−0.439250
$249$	−5.85077	−0.370777
$250$	28.1939	1.78314
$251$	19.8514	1.25301	0.626504	−	0.779418i	$-0.284485\pi$
$251$	19.8514	1.25301	0.626504	+	0.779418i	$0.284485\pi$
$252$	4.79525	0.302073
$253$	−26.3420	−1.65611
$254$	−2.10896	−0.132328
$255$	15.7749	0.987863
$256$	−6.41772	−0.401107
$257$	−23.0628	−1.43862	−0.719309	−	0.694690i	$-0.755542\pi$
$257$	−23.0628	−1.43862	−0.719309	+	0.694690i	$0.755542\pi$
$258$	21.3604	1.32984
$259$	−2.10323	−0.130688
$260$	8.99953	0.558127
$261$	4.07281	0.252101
$262$	41.5956	2.56978
$263$	−5.93944	−0.366242	−0.183121	−	0.983090i	$-0.558620\pi$
$263$	−5.93944	−0.366242	−0.183121	+	0.983090i	$0.558620\pi$
$264$	−23.9037	−1.47117
$265$	−25.1338	−1.54396
$266$	−9.54048	−0.584964
$267$	−12.0448	−0.737131
$268$	−0.691650	−0.0422493
$269$	−18.0932	−1.10316	−0.551582	−	0.834120i	$-0.685976\pi$
$269$	−18.0932	−1.10316	−0.551582	+	0.834120i	$0.685976\pi$
$270$	31.1865	1.89795
$271$	18.0167	1.09444	0.547220	−	0.836989i	$-0.315686\pi$
$271$	18.0167	1.09444	0.547220	+	0.836989i	$0.315686\pi$
$272$	71.7316	4.34937
$273$	−0.393157	−0.0237950
$274$	49.1773	2.97091
$275$	1.22506	0.0738741
$276$	−45.4100	−2.73336
$277$	−29.7144	−1.78537	−0.892683	−	0.450686i	$-0.851179\pi$
$277$	−29.7144	−1.78537	−0.892683	+	0.450686i	$0.851179\pi$
$278$	−28.9903	−1.73872
$279$	−1.71235	−0.102516
$280$	9.04955	0.540814
$281$	1.14465	0.0682843	0.0341421	−	0.999417i	$-0.489130\pi$
$281$	1.14465	0.0682843	0.0341421	+	0.999417i	$0.489130\pi$
$282$	−29.5742	−1.76112
$283$	19.4627	1.15694	0.578470	−	0.815704i	$-0.303650\pi$
$283$	19.4627	1.15694	0.578470	+	0.815704i	$0.303650\pi$
$284$	−51.0172	−3.02732
$285$	−17.4363	−1.03284
$286$	−6.07634	−0.359302
$287$	5.50233	0.324792
$288$	25.1523	1.48211
$289$	26.7511	1.57360
$290$	12.8437	0.754211
$291$	−6.90942	−0.405037
$292$	−56.2272	−3.29045
$293$	−2.31783	−0.135409	−0.0677046	−	0.997705i	$-0.521568\pi$
$293$	−2.31783	−0.135409	−0.0677046	+	0.997705i	$0.521568\pi$
$294$	18.2963	1.06706
$295$	6.43314	0.374552
$296$	−33.5338	−1.94911
$297$	−15.0237	−0.871761
$298$	−5.28140	−0.305943
$299$	−6.90791	−0.399495
$300$	2.11184	0.121927
$301$	3.89566	0.224542
$302$	11.4770	0.660425
$303$	−1.16519	−0.0669384
$304$	−79.2865	−4.54740
$305$	18.7604	1.07422
$306$	34.0669	1.94747
$307$	−13.0843	−0.746759	−0.373380	−	0.927679i	$-0.621801\pi$
$307$	−13.0843	−0.746759	−0.373380	+	0.927679i	$0.621801\pi$
$308$	−7.28482	−0.415091
$309$	−6.30878	−0.358894
$310$	−5.39994	−0.306696
$311$	15.8629	0.899504	0.449752	−	0.893153i	$-0.351512\pi$
$311$	15.8629	0.899504	0.449752	+	0.893153i	$0.351512\pi$
$312$	−6.26849	−0.354883
$313$	−6.77606	−0.383006	−0.191503	−	0.981492i	$-0.561336\pi$
$313$	−6.77606	−0.383006	−0.191503	+	0.981492i	$0.561336\pi$
$314$	−22.1696	−1.25110
$315$	2.24017	0.126219
$316$	60.1424	3.38327
$317$	17.5246	0.984278	0.492139	−	0.870517i	$-0.336215\pi$
$317$	17.5246	0.984278	0.492139	+	0.870517i	$0.336215\pi$
$318$	29.2538	1.64047
$319$	−6.18730	−0.346422
$320$	28.8533	1.61295
$321$	11.4589	0.639573
$322$	−11.6074	−0.646856
$323$	−48.3591	−2.69077
$324$	3.22795	0.179331
$325$	0.321260	0.0178203
$326$	13.0407	0.722257
$327$	13.7765	0.761842
$328$	87.7289	4.84402
$329$	−5.39368	−0.297363
$330$	−18.6602	−1.02721
$331$	−12.0475	−0.662188	−0.331094	−	0.943598i	$-0.607418\pi$
$331$	−12.0475	−0.662188	−0.331094	+	0.943598i	$0.607418\pi$
$332$	−28.4291	−1.56025
$333$	−8.30114	−0.454899
$334$	−20.6300	−1.12882
$335$	−0.323114	−0.0176536
$336$	−5.49016	−0.299513
$337$	18.8516	1.02691	0.513457	−	0.858115i	$-0.328365\pi$
$337$	18.8516	1.02691	0.513457	+	0.858115i	$0.328365\pi$
$338$	32.7535	1.78155
$339$	−13.5548	−0.736196
$340$	76.6508	4.15697
$341$	2.60135	0.140871
$342$	−37.6548	−2.03614
$343$	6.79417	0.366851
$344$	62.1123	3.34887
$345$	−21.2139	−1.14212
$346$	−40.1399	−2.15793
$347$	−26.1619	−1.40445	−0.702223	−	0.711957i	$-0.747809\pi$
$347$	−26.1619	−1.40445	−0.702223	+	0.711957i	$0.747809\pi$
$348$	−10.6660	−0.571760
$349$	12.5396	0.671230	0.335615	−	0.941999i	$-0.391056\pi$
$349$	12.5396	0.671230	0.335615	+	0.941999i	$0.391056\pi$
$350$	0.539815	0.0288543
$351$	−3.93979	−0.210291
$352$	−38.2106	−2.03663
$353$	11.1603	0.594001	0.297001	−	0.954877i	$-0.404014\pi$
$353$	11.1603	0.594001	0.297001	+	0.954877i	$0.404014\pi$
$354$	−7.48768	−0.397966
$355$	−23.8335	−1.26495
$356$	−58.5262	−3.10188
$357$	−3.34860	−0.177227
$358$	−30.8531	−1.63064
$359$	32.8588	1.73422	0.867110	−	0.498117i	$-0.165975\pi$
$359$	32.8588	1.73422	0.867110	+	0.498117i	$0.165975\pi$
$360$	35.7172	1.88246
$361$	34.4523	1.81328
$362$	−58.4280	−3.07091
$363$	−2.28575	−0.119971
$364$	−1.91036	−0.100130
$365$	−26.2674	−1.37490
$366$	−21.8356	−1.14137
$367$	−13.3290	−0.695768	−0.347884	−	0.937538i	$-0.613100\pi$
$367$	−13.3290	−0.695768	−0.347884	+	0.937538i	$0.613100\pi$
$368$	−96.4640	−5.02853
$369$	21.7169	1.13053
$370$	−26.1779	−1.36092
$371$	5.33525	0.276992
$372$	4.48437	0.232504
$373$	8.88503	0.460050	0.230025	−	0.973185i	$-0.426119\pi$
$373$	8.88503	0.460050	0.230025	+	0.973185i	$0.426119\pi$
$374$	−51.7534	−2.67611
$375$	−10.9380	−0.564835
$376$	−85.9967	−4.43494
$377$	−1.62255	−0.0835658
$378$	−6.62007	−0.340499
$379$	22.6120	1.16150	0.580750	−	0.814082i	$-0.302759\pi$
$379$	22.6120	1.16150	0.580750	+	0.814082i	$0.302759\pi$
$380$	−84.7238	−4.34624
$381$	0.818184	0.0419168
$382$	−9.43065	−0.482514
$383$	−0.491019	−0.0250899	−0.0125450	−	0.999921i	$-0.503993\pi$
$383$	−0.491019	−0.0250899	−0.0125450	+	0.999921i	$0.503993\pi$
$384$	−7.13210	−0.363959
$385$	−3.40321	−0.173444
$386$	−37.4868	−1.90803
$387$	15.3756	0.781586
$388$	−33.5731	−1.70442
$389$	−33.0524	−1.67583	−0.837913	−	0.545804i	$-0.816224\pi$
$389$	−33.0524	−1.67583	−0.837913	+	0.545804i	$0.816224\pi$
$390$	−4.89344	−0.247789
$391$	−58.8361	−2.97547
$392$	53.2025	2.68713
$393$	−16.1372	−0.814016
$394$	48.6796	2.45244
$395$	28.0964	1.41368
$396$	−28.7521	−1.44485
$397$	−24.9472	−1.25206	−0.626031	−	0.779798i	$-0.715322\pi$
$397$	−24.9472	−1.25206	−0.626031	+	0.779798i	$0.715322\pi$
$398$	−42.1886	−2.11472
$399$	3.70128	0.185296
$400$	4.48616	0.224308
$401$	4.24375	0.211923	0.105961	−	0.994370i	$-0.466208\pi$
$401$	4.24375	0.211923	0.105961	+	0.994370i	$0.466208\pi$
$402$	0.376081	0.0187572
$403$	0.682176	0.0339816
$404$	−5.66169	−0.281680
$405$	1.50798	0.0749324
$406$	−2.72639	−0.135308
$407$	12.6109	0.625097
$408$	−53.3899	−2.64320
$409$	−9.07084	−0.448524	−0.224262	−	0.974529i	$-0.571997\pi$
$409$	−9.07084	−0.448524	−0.224262	+	0.974529i	$0.571997\pi$
$410$	68.4849	3.38223
$411$	−19.0786	−0.941078
$412$	−30.6546	−1.51024
$413$	−1.36559	−0.0671961
$414$	−45.8127	−2.25157
$415$	−13.2811	−0.651942
$416$	−10.0203	−0.491287
$417$	11.2469	0.550765
$418$	57.2042	2.79795
$419$	22.3057	1.08970	0.544852	−	0.838532i	$-0.316586\pi$
$419$	22.3057	1.08970	0.544852	+	0.838532i	$0.316586\pi$
$420$	−5.86666	−0.286264
$421$	−19.7844	−0.964234	−0.482117	−	0.876107i	$-0.660132\pi$
$421$	−19.7844	−0.964234	−0.482117	+	0.876107i	$0.660132\pi$
$422$	−22.4905	−1.09482
$423$	−21.2880	−1.03506
$424$	85.0650	4.13112
$425$	2.73623	0.132727
$426$	27.7403	1.34402
$427$	−3.98233	−0.192719
$428$	55.6792	2.69135
$429$	2.35735	0.113814
$430$	48.4875	2.33827
$431$	−13.1267	−0.632289	−0.316145	−	0.948711i	$-0.602388\pi$
$431$	−13.1267	−0.632289	−0.316145	+	0.948711i	$0.602388\pi$
$432$	−55.0163	−2.64697
$433$	23.5459	1.13154	0.565772	−	0.824562i	$-0.308578\pi$
$433$	23.5459	1.13154	0.565772	+	0.824562i	$0.308578\pi$
$434$	1.14627	0.0550226
$435$	−4.98280	−0.238907
$436$	66.9405	3.20587
$437$	65.0328	3.11094
$438$	30.5732	1.46084
$439$	−5.08283	−0.242590	−0.121295	−	0.992616i	$-0.538705\pi$
$439$	−5.08283	−0.242590	−0.121295	+	0.992616i	$0.538705\pi$
$440$	−54.2606	−2.58677
$441$	13.1700	0.627144
$442$	−13.5718	−0.645544
$443$	−27.4957	−1.30636	−0.653180	−	0.757202i	$-0.726566\pi$
$443$	−27.4957	−1.30636	−0.653180	+	0.757202i	$0.726566\pi$
$444$	21.7394	1.03170
$445$	−27.3414	−1.29610
$446$	−35.0101	−1.65778
$447$	2.04895	0.0969119
$448$	−6.12479	−0.289369
$449$	−20.1737	−0.952054	−0.476027	−	0.879431i	$-0.657924\pi$
$449$	−20.1737	−0.952054	−0.476027	+	0.879431i	$0.657924\pi$
$450$	2.13057	0.100436
$451$	−32.9917	−1.55352
$452$	−65.8632	−3.09795
$453$	−4.45255	−0.209199
$454$	18.0649	0.847829
$455$	−0.892455	−0.0418389
$456$	59.0131	2.76354
$457$	36.5943	1.71181	0.855905	−	0.517133i	$-0.173001\pi$
$457$	36.5943	1.71181	0.855905	+	0.517133i	$0.173001\pi$
$458$	52.5927	2.45750
$459$	−33.5560	−1.56626
$460$	−103.079	−4.80609
$461$	−19.9762	−0.930385	−0.465193	−	0.885209i	$-0.654015\pi$
$461$	−19.9762	−0.930385	−0.465193	+	0.885209i	$0.654015\pi$
$462$	3.96107	0.184286
$463$	−26.6366	−1.23791	−0.618955	−	0.785427i	$-0.712444\pi$
$463$	−26.6366	−1.23791	−0.618955	+	0.785427i	$0.712444\pi$
$464$	−22.6578	−1.05186
$465$	2.09494	0.0971504
$466$	36.1060	1.67258
$467$	−21.8308	−1.01021	−0.505103	−	0.863059i	$-0.668546\pi$
$467$	−21.8308	−1.01021	−0.505103	+	0.863059i	$0.668546\pi$
$468$	−7.53992	−0.348533
$469$	0.0685887	0.00316713
$470$	−67.1326	−3.09659
$471$	8.60080	0.396304
$472$	−21.7729	−1.00218
$473$	−23.3582	−1.07401
$474$	−32.7021	−1.50205
$475$	−3.02442	−0.138770
$476$	−16.2710	−0.745778
$477$	21.0574	0.964153
$478$	−40.3409	−1.84515
$479$	9.28019	0.424023	0.212011	−	0.977267i	$-0.431999\pi$
$479$	9.28019	0.424023	0.212011	+	0.977267i	$0.431999\pi$
$480$	−30.7720	−1.40454
$481$	3.30706	0.150789
$482$	−10.8642	−0.494851
$483$	4.50316	0.204901
$484$	−11.1065	−0.504842
$485$	−15.6842	−0.712181
$486$	−41.9658	−1.90361
$487$	28.8945	1.30934	0.654668	−	0.755917i	$-0.272809\pi$
$487$	28.8945	1.30934	0.654668	+	0.755917i	$0.272809\pi$
$488$	−63.4942	−2.87425
$489$	−5.05921	−0.228785
$490$	41.5321	1.87623
$491$	22.0320	0.994290	0.497145	−	0.867667i	$-0.334382\pi$
$491$	22.0320	0.994290	0.497145	+	0.867667i	$0.334382\pi$
$492$	−56.8730	−2.56403
$493$	−13.8196	−0.622404
$494$	15.0012	0.674935
$495$	−13.4320	−0.603721
$496$	9.52610	0.427734
$497$	5.05922	0.226937
$498$	15.4581	0.692696
$499$	−13.3085	−0.595769	−0.297884	−	0.954602i	$-0.596281\pi$
$499$	−13.3085	−0.595769	−0.297884	+	0.954602i	$0.596281\pi$
$500$	−53.1480	−2.37685
$501$	8.00350	0.357570
$502$	−52.4487	−2.34090
$503$	−38.8551	−1.73246	−0.866231	−	0.499643i	$-0.833464\pi$
$503$	−38.8551	−1.73246	−0.866231	+	0.499643i	$0.833464\pi$
$504$	−7.58183	−0.337722
$505$	−2.64494	−0.117698
$506$	69.5975	3.09398
$507$	−12.7069	−0.564332
$508$	3.97559	0.176388
$509$	11.9069	0.527765	0.263883	−	0.964555i	$-0.414997\pi$
$509$	11.9069	0.527765	0.263883	+	0.964555i	$0.414997\pi$
$510$	−41.6784	−1.84555
$511$	5.57587	0.246662
$512$	30.8723	1.36438
$513$	37.0902	1.63757
$514$	60.9336	2.68766
$515$	−14.3207	−0.631047
$516$	−40.2663	−1.77262
$517$	32.3402	1.42232
$518$	5.55688	0.244155
$519$	15.5725	0.683556
$520$	−14.2293	−0.623995
$521$	13.1056	0.574167	0.287083	−	0.957906i	$-0.407314\pi$
$521$	13.1056	0.574167	0.287083	+	0.957906i	$0.407314\pi$
$522$	−10.7607	−0.470981
$523$	9.88203	0.432111	0.216056	−	0.976381i	$-0.430681\pi$
$523$	9.88203	0.432111	0.216056	+	0.976381i	$0.430681\pi$
$524$	−78.4113	−3.42542
$525$	−0.209424	−0.00914002
$526$	15.6924	0.684222
$527$	5.81023	0.253098
$528$	32.9187	1.43260
$529$	56.1221	2.44009
$530$	66.4053	2.88446
$531$	−5.38976	−0.233896
$532$	17.9846	0.779733
$533$	−8.65171	−0.374747
$534$	31.8232	1.37713
$535$	26.0113	1.12457
$536$	1.09358	0.0472353
$537$	11.9696	0.516528
$538$	47.8036	2.06096
$539$	−20.0075	−0.861785
$540$	−58.7892	−2.52988
$541$	15.8552	0.681669	0.340835	−	0.940123i	$-0.389290\pi$
$541$	15.8552	0.681669	0.340835	+	0.940123i	$0.389290\pi$
$542$	−47.6015	−2.04466
$543$	22.6675	0.972754
$544$	−85.3451	−3.65914
$545$	31.2722	1.33955
$546$	1.03875	0.0444544
$547$	5.31702	0.227339	0.113670	−	0.993519i	$-0.463739\pi$
$547$	5.31702	0.227339	0.113670	+	0.993519i	$0.463739\pi$
$548$	−92.7035	−3.96010
$549$	−15.7177	−0.670814
$550$	−3.23670	−0.138014
$551$	15.2751	0.650742
$552$	71.7982	3.05594
$553$	−5.96413	−0.253621
$554$	78.5075	3.33547
$555$	10.1559	0.431092
$556$	54.6492	2.31764
$557$	−36.5698	−1.54951	−0.774755	−	0.632261i	$-0.782127\pi$
$557$	−36.5698	−1.54951	−0.774755	+	0.632261i	$0.782127\pi$
$558$	4.52414	0.191522
$559$	−6.12544	−0.259078
$560$	−12.4625	−0.526636
$561$	20.0780	0.847695
$562$	−3.02425	−0.127570
$563$	27.3771	1.15381	0.576904	−	0.816812i	$-0.304261\pi$
$563$	27.3771	1.15381	0.576904	+	0.816812i	$0.304261\pi$
$564$	55.7500	2.34750
$565$	−30.7690	−1.29446
$566$	−51.4219	−2.16142
$567$	−0.320106	−0.0134432
$568$	80.6640	3.38459
$569$	−12.6810	−0.531617	−0.265808	−	0.964026i	$-0.585639\pi$
$569$	−12.6810	−0.531617	−0.265808	+	0.964026i	$0.585639\pi$
$570$	46.0681	1.92958
$571$	13.2561	0.554751	0.277376	−	0.960762i	$-0.410535\pi$
$571$	13.2561	0.554751	0.277376	+	0.960762i	$0.410535\pi$
$572$	11.4544	0.478934
$573$	3.65867	0.152843
$574$	−14.5375	−0.606785
$575$	−3.67966	−0.153452
$576$	−24.1736	−1.00723
$577$	−41.1712	−1.71398	−0.856990	−	0.515333i	$-0.827668\pi$
$577$	−41.1712	−1.71398	−0.856990	+	0.515333i	$0.827668\pi$
$578$	−70.6784	−2.93983
$579$	14.5432	0.604396
$580$	−24.2116	−1.00533
$581$	2.81922	0.116961
$582$	18.2552	0.756701
$583$	−31.9898	−1.32488
$584$	88.9015	3.67877
$585$	−3.52239	−0.145633
$586$	6.12387	0.252975
$587$	5.17284	0.213506	0.106753	−	0.994286i	$-0.465955\pi$
$587$	5.17284	0.213506	0.106753	+	0.994286i	$0.465955\pi$
$588$	−34.4902	−1.42235
$589$	−6.42218	−0.264621
$590$	−16.9968	−0.699747
$591$	−18.8855	−0.776846
$592$	46.1807	1.89802
$593$	−34.4971	−1.41663	−0.708313	−	0.705899i	$-0.750543\pi$
$593$	−34.4971	−1.41663	−0.708313	+	0.705899i	$0.750543\pi$
$594$	39.6936	1.62865
$595$	−7.60122	−0.311619
$596$	9.95590	0.407810
$597$	16.3673	0.669868
$598$	18.2512	0.746346
$599$	22.3063	0.911410	0.455705	−	0.890131i	$-0.349387\pi$
$599$	22.3063	0.911410	0.455705	+	0.890131i	$0.349387\pi$
$600$	−3.33905	−0.136316
$601$	−6.22133	−0.253773	−0.126887	−	0.991917i	$-0.540498\pi$
$601$	−6.22133	−0.253773	−0.126887	+	0.991917i	$0.540498\pi$
$602$	−10.2926	−0.419496
$603$	0.270709	0.0110241
$604$	−21.6351	−0.880319
$605$	−5.18858	−0.210946
$606$	3.07851	0.125056
$607$	−41.2831	−1.67563	−0.837814	−	0.545955i	$-0.816167\pi$
$607$	−41.2831	−1.67563	−0.837814	+	0.545955i	$0.816167\pi$
$608$	94.3338	3.82574
$609$	1.05772	0.0428609
$610$	−49.5662	−2.00688
$611$	8.48087	0.343099
$612$	−64.2190	−2.59590
$613$	25.8987	1.04604	0.523018	−	0.852321i	$-0.324806\pi$
$613$	25.8987	1.04604	0.523018	+	0.852321i	$0.324806\pi$
$614$	34.5696	1.39511
$615$	−26.5691	−1.07137
$616$	11.5181	0.464078
$617$	38.4627	1.54845	0.774225	−	0.632910i	$-0.218140\pi$
$617$	38.4627	1.54845	0.774225	+	0.632910i	$0.218140\pi$
$618$	16.6682	0.670495
$619$	−25.8680	−1.03972	−0.519860	−	0.854251i	$-0.674016\pi$
$619$	−25.8680	−1.03972	−0.519860	+	0.854251i	$0.674016\pi$
$620$	10.1794	0.408813
$621$	45.1257	1.81083
$622$	−41.9110	−1.68048
$623$	5.80385	0.232526
$624$	8.63257	0.345579
$625$	−26.8972	−1.07589
$626$	17.9028	0.715541
$627$	−22.1927	−0.886290
$628$	41.7916	1.66767
$629$	28.1669	1.12309
$630$	−5.91869	−0.235806
$631$	−25.1744	−1.00218	−0.501088	−	0.865396i	$-0.667067\pi$
$631$	−25.1744	−1.00218	−0.501088	+	0.865396i	$0.667067\pi$
$632$	−95.0918	−3.78255
$633$	8.72530	0.346800
$634$	−46.3011	−1.83885
$635$	1.85725	0.0737028
$636$	−55.1460	−2.18668
$637$	−5.24676	−0.207884
$638$	16.3473	0.647195
$639$	19.9680	0.789921
$640$	−16.1897	−0.639952
$641$	0.360829	0.0142519	0.00712594	−	0.999975i	$-0.497732\pi$
$641$	0.360829	0.0142519	0.00712594	+	0.999975i	$0.497732\pi$
$642$	−30.2752	−1.19487
$643$	3.78669	0.149332	0.0746662	−	0.997209i	$-0.476211\pi$
$643$	3.78669	0.149332	0.0746662	+	0.997209i	$0.476211\pi$
$644$	21.8810	0.862233
$645$	−18.8110	−0.740681
$646$	127.768	5.02697
$647$	−11.5272	−0.453179	−0.226590	−	0.973990i	$-0.572758\pi$
$647$	−11.5272	−0.453179	−0.226590	+	0.973990i	$0.572758\pi$
$648$	−5.10375	−0.200494
$649$	8.18798	0.321406
$650$	−0.848791	−0.0332923
$651$	−0.444700	−0.0174292
$652$	−24.5829	−0.962739
$653$	−35.2744	−1.38039	−0.690197	−	0.723622i	$-0.742476\pi$
$653$	−35.2744	−1.38039	−0.690197	+	0.723622i	$0.742476\pi$
$654$	−36.3985	−1.42329
$655$	−36.6310	−1.43129
$656$	−120.815	−4.71703
$657$	22.0071	0.858580
$658$	14.2505	0.555542
$659$	−6.14081	−0.239212	−0.119606	−	0.992821i	$-0.538163\pi$
$659$	−6.14081	−0.239212	−0.119606	+	0.992821i	$0.538163\pi$
$660$	35.1762	1.36923
$661$	0.711705	0.0276821	0.0138411	−	0.999904i	$-0.495594\pi$
$661$	0.711705	0.0276821	0.0138411	+	0.999904i	$0.495594\pi$
$662$	31.8302	1.23712
$663$	5.26524	0.204485
$664$	44.9496	1.74438
$665$	8.40179	0.325807
$666$	21.9322	0.849855
$667$	18.5845	0.719594
$668$	38.8893	1.50467
$669$	13.5823	0.525124
$670$	0.853691	0.0329810
$671$	23.8779	0.921794
$672$	6.53209	0.251981
$673$	−44.1131	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$673$	−44.1131	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$674$	−49.8074	−1.91851
$675$	−2.09862	−0.0807760
$676$	−61.7432	−2.37474
$677$	−9.04562	−0.347652	−0.173826	−	0.984776i	$-0.555613\pi$
$677$	−9.04562	−0.347652	−0.173826	+	0.984776i	$0.555613\pi$
$678$	35.8127	1.37538
$679$	3.32934	0.127768
$680$	−121.194	−4.64756
$681$	−7.00839	−0.268562
$682$	−6.87295	−0.263179
$683$	−9.59421	−0.367112	−0.183556	−	0.983009i	$-0.558761\pi$
$683$	−9.59421	−0.367112	−0.183556	+	0.983009i	$0.558761\pi$
$684$	70.9827	2.71409
$685$	−43.3078	−1.65471
$686$	−17.9507	−0.685360
$687$	−20.4036	−0.778447
$688$	−85.5373	−3.26108
$689$	−8.38899	−0.319595
$690$	56.0487	2.13374
$691$	8.70008	0.330967	0.165483	−	0.986213i	$-0.447082\pi$
$691$	8.70008	0.330967	0.165483	+	0.986213i	$0.447082\pi$
$692$	75.6672	2.87644
$693$	2.85125	0.108310
$694$	69.1217	2.62382
$695$	25.5302	0.968415
$696$	16.8642	0.639236
$697$	−73.6883	−2.79114
$698$	−33.1305	−1.25401
$699$	−14.0075	−0.529813
$700$	−1.01760	−0.0384617
$701$	−30.4119	−1.14864	−0.574321	−	0.818630i	$-0.694734\pi$
$701$	−30.4119	−1.14864	−0.574321	+	0.818630i	$0.694734\pi$
$702$	10.4092	0.392870
$703$	−31.1335	−1.17422
$704$	36.7239	1.38408
$705$	26.0444	0.980890
$706$	−29.4862	−1.10973
$707$	0.561452	0.0211156
$708$	14.1149	0.530472
$709$	44.2104	1.66036	0.830179	−	0.557497i	$-0.188238\pi$
$709$	44.2104	1.66036	0.830179	+	0.557497i	$0.188238\pi$
$710$	62.9697	2.36321
$711$	−23.5395	−0.882801
$712$	92.5364	3.46795
$713$	−7.81354	−0.292619
$714$	8.84723	0.331099
$715$	5.35111	0.200120
$716$	58.1609	2.17357
$717$	15.6505	0.584477
$718$	−86.8152	−3.23991
$719$	−37.5002	−1.39852	−0.699261	−	0.714866i	$-0.746488\pi$
$719$	−37.5002	−1.39852	−0.699261	+	0.714866i	$0.746488\pi$
$720$	−49.1876	−1.83311
$721$	3.03992	0.113212
$722$	−91.0255	−3.38762
$723$	4.21482	0.156751
$724$	110.142	4.09339
$725$	−0.864291	−0.0320989
$726$	6.03911	0.224132
$727$	−50.2627	−1.86414	−0.932071	−	0.362276i	$-0.882000\pi$
$727$	−50.2627	−1.86414	−0.932071	+	0.362276i	$0.882000\pi$
$728$	3.02050	0.111947
$729$	14.3365	0.530983
$730$	69.4002	2.56862
$731$	−52.1716	−1.92963
$732$	41.1621	1.52139
$733$	42.4330	1.56730	0.783648	−	0.621205i	$-0.213356\pi$
$733$	42.4330	1.56730	0.783648	+	0.621205i	$0.213356\pi$
$734$	35.2162	1.29985
$735$	−16.1126	−0.594322
$736$	114.771	4.23052
$737$	−0.411254	−0.0151487
$738$	−57.3775	−2.11209
$739$	4.92702	0.181243	0.0906217	−	0.995885i	$-0.471115\pi$
$739$	4.92702	0.181243	0.0906217	+	0.995885i	$0.471115\pi$
$740$	49.3477	1.81406
$741$	−5.81979	−0.213795
$742$	−14.0961	−0.517484
$743$	−44.3385	−1.62662	−0.813310	−	0.581830i	$-0.802337\pi$
$743$	−44.3385	−1.62662	−0.813310	+	0.581830i	$0.802337\pi$
$744$	−7.09029	−0.259942
$745$	4.65105	0.170401
$746$	−23.4749	−0.859476
$747$	11.1270	0.407117
$748$	97.5598	3.56714
$749$	−5.52153	−0.201752
$750$	28.8989	1.05524
$751$	−7.54396	−0.275283	−0.137642	−	0.990482i	$-0.543952\pi$
$751$	−7.54396	−0.275283	−0.137642	+	0.990482i	$0.543952\pi$
$752$	118.429	4.31867
$753$	20.3478	0.741514
$754$	4.28690	0.156120
$755$	−10.1071	−0.367837
$756$	12.4794	0.453872
$757$	−3.41994	−0.124300	−0.0621499	−	0.998067i	$-0.519796\pi$
$757$	−3.41994	−0.124300	−0.0621499	+	0.998067i	$0.519796\pi$
$758$	−59.7425	−2.16995
$759$	−27.0007	−0.980064
$760$	133.958	4.85916
$761$	0.188752	0.00684227	0.00342113	−	0.999994i	$-0.498911\pi$
$761$	0.188752	0.00684227	0.00342113	+	0.999994i	$0.498911\pi$
$762$	−2.16170	−0.0783101
$763$	−6.63827	−0.240322
$764$	17.7776	0.643171
$765$	−30.0009	−1.08468
$766$	1.29731	0.0468736
$767$	2.14721	0.0775312
$768$	−6.57819	−0.237370
$769$	−3.54424	−0.127809	−0.0639043	−	0.997956i	$-0.520355\pi$
$769$	−3.54424	−0.127809	−0.0639043	+	0.997956i	$0.520355\pi$
$770$	8.99152	0.324032
$771$	−23.6395	−0.851356
$772$	70.6660	2.54333
$773$	46.2757	1.66442	0.832211	−	0.554459i	$-0.187075\pi$
$773$	46.2757	1.66442	0.832211	+	0.554459i	$0.187075\pi$
$774$	−40.6234	−1.46018
$775$	0.363377	0.0130529
$776$	53.0828	1.90556
$777$	−2.15582	−0.0773397
$778$	87.3269	3.13082
$779$	81.4493	2.91822
$780$	9.22457	0.330293
$781$	−30.3348	−1.08546
$782$	155.449	5.55884
$783$	10.5993	0.378788
$784$	−73.2672	−2.61669
$785$	19.5236	0.696826
$786$	42.6357	1.52076
$787$	−32.9728	−1.17535	−0.587677	−	0.809095i	$-0.699958\pi$
$787$	−32.9728	−1.17535	−0.587677	+	0.809095i	$0.699958\pi$
$788$	−91.7653	−3.26900
$789$	−6.08796	−0.216737
$790$	−74.2326	−2.64108
$791$	6.53145	0.232232
$792$	45.4603	1.61536
$793$	6.26171	0.222360
$794$	65.9122	2.33914
$795$	−25.7623	−0.913694
$796$	79.5292	2.81884
$797$	−22.2197	−0.787062	−0.393531	−	0.919311i	$-0.628747\pi$
$797$	−22.2197	−0.787062	−0.393531	+	0.919311i	$0.628747\pi$
$798$	−9.77904	−0.346174
$799$	72.2333	2.55543
$800$	−5.33756	−0.188711
$801$	22.9069	0.809377
$802$	−11.2123	−0.395919
$803$	−33.4326	−1.17981
$804$	−0.708945	−0.0250026
$805$	10.2220	0.360279
$806$	−1.80236	−0.0634854
$807$	−18.5457	−0.652839
$808$	8.95177	0.314922
$809$	−2.63715	−0.0927172	−0.0463586	−	0.998925i	$-0.514762\pi$
$809$	−2.63715	−0.0927172	−0.0463586	+	0.998925i	$0.514762\pi$
$810$	−3.98420	−0.139991
$811$	−0.783721	−0.0275201	−0.0137601	−	0.999905i	$-0.504380\pi$
$811$	−0.783721	−0.0275201	−0.0137601	+	0.999905i	$0.504380\pi$
$812$	5.13949	0.180361
$813$	18.4673	0.647675
$814$	−33.3188	−1.16782
$815$	−11.4842	−0.402275
$816$	73.5253	2.57390
$817$	57.6664	2.01749
$818$	23.9658	0.837945
$819$	0.747710	0.0261271
$820$	−129.100	−4.50837
$821$	−2.12851	−0.0742856	−0.0371428	−	0.999310i	$-0.511826\pi$
$821$	−2.12851	−0.0742856	−0.0371428	+	0.999310i	$0.511826\pi$
$822$	50.4070	1.75815
$823$	−31.8659	−1.11077	−0.555387	−	0.831592i	$-0.687430\pi$
$823$	−31.8659	−1.11077	−0.555387	+	0.831592i	$0.687430\pi$
$824$	48.4683	1.68847
$825$	1.25570	0.0437177
$826$	3.60797	0.125537
$827$	32.4426	1.12814	0.564070	−	0.825727i	$-0.309235\pi$
$827$	32.4426	1.12814	0.564070	+	0.825727i	$0.309235\pi$
$828$	86.3611	3.00125
$829$	38.8086	1.34788	0.673939	−	0.738787i	$-0.264601\pi$
$829$	38.8086	1.34788	0.673939	+	0.738787i	$0.264601\pi$
$830$	35.0895	1.21797
$831$	−30.4574	−1.05656
$832$	9.63045	0.333876
$833$	−44.6877	−1.54834
$834$	−29.7152	−1.02895
$835$	18.1677	0.628719
$836$	−107.835	−3.72955
$837$	−4.45630	−0.154032
$838$	−58.9332	−2.03581
$839$	24.5690	0.848217	0.424108	−	0.905611i	$-0.360588\pi$
$839$	24.5690	0.848217	0.424108	+	0.905611i	$0.360588\pi$
$840$	9.27584	0.320047
$841$	−24.6348	−0.849476
$842$	52.2719	1.80141
$843$	1.17328	0.0404098
$844$	42.3965	1.45935
$845$	−28.8442	−0.992271
$846$	56.2445	1.93373
$847$	1.10140	0.0378445
$848$	−117.146	−4.02282
$849$	19.9494	0.684662
$850$	−7.22932	−0.247964
$851$	−37.8786	−1.29846
$852$	−52.2929	−1.79153
$853$	3.84572	0.131675	0.0658374	−	0.997830i	$-0.479028\pi$
$853$	3.84572	0.131675	0.0658374	+	0.997830i	$0.479028\pi$
$854$	10.5216	0.360042
$855$	33.1606	1.13407
$856$	−88.0350	−3.00897
$857$	−3.82526	−0.130668	−0.0653342	−	0.997863i	$-0.520811\pi$
$857$	−3.82526	−0.130668	−0.0653342	+	0.997863i	$0.520811\pi$
$858$	−6.22828	−0.212630
$859$	−8.00146	−0.273006	−0.136503	−	0.990640i	$-0.543586\pi$
$859$	−8.00146	−0.273006	−0.136503	+	0.990640i	$0.543586\pi$
$860$	−91.4032	−3.11682
$861$	5.63992	0.192208
$862$	34.6816	1.18126
$863$	25.1596	0.856443	0.428221	−	0.903674i	$-0.359140\pi$
$863$	25.1596	0.856443	0.428221	+	0.903674i	$0.359140\pi$
$864$	65.4575	2.22691
$865$	35.3491	1.20190
$866$	−62.2100	−2.11398
$867$	27.4200	0.931234
$868$	−2.16081	−0.0733428
$869$	35.7606	1.21309
$870$	13.1649	0.446332
$871$	−0.107847	−0.00365426
$872$	−105.840	−3.58421
$873$	13.1404	0.444735
$874$	−171.821	−5.81194
$875$	5.27052	0.178176
$876$	−57.6332	−1.94725
$877$	47.0932	1.59023	0.795113	−	0.606462i	$-0.207412\pi$
$877$	47.0932	1.59023	0.795113	+	0.606462i	$0.207412\pi$
$878$	13.4292	0.453213
$879$	−2.37579	−0.0801334
$880$	74.7244	2.51896
$881$	−32.7410	−1.10307	−0.551537	−	0.834150i	$-0.685958\pi$
$881$	−32.7410	−1.10307	−0.551537	+	0.834150i	$0.685958\pi$
$882$	−34.7961	−1.17165
$883$	9.65046	0.324764	0.162382	−	0.986728i	$-0.448082\pi$
$883$	9.65046	0.324764	0.162382	+	0.986728i	$0.448082\pi$
$884$	25.5840	0.860483
$885$	6.59400	0.221655
$886$	72.6456	2.44058
$887$	−26.5622	−0.891871	−0.445935	−	0.895065i	$-0.647129\pi$
$887$	−26.5622	−0.891871	−0.445935	+	0.895065i	$0.647129\pi$
$888$	−34.3724	−1.15346
$889$	−0.394246	−0.0132226
$890$	72.2378	2.42142
$891$	1.91934	0.0643002
$892$	65.9971	2.20975
$893$	−79.8410	−2.67178
$894$	−5.41346	−0.181053
$895$	27.1707	0.908217
$896$	3.43664	0.114810
$897$	−7.08064	−0.236416
$898$	53.3002	1.77865
$899$	−1.83527	−0.0612097
$900$	−4.01631	−0.133877
$901$	−71.4507	−2.38037
$902$	87.1663	2.90232
$903$	3.99308	0.132881
$904$	104.137	3.46355
$905$	51.4544	1.71040
$906$	11.7639	0.390831
$907$	25.7393	0.854660	0.427330	−	0.904096i	$-0.359454\pi$
$907$	25.7393	0.854660	0.427330	+	0.904096i	$0.359454\pi$
$908$	−34.0540	−1.13012
$909$	2.21597	0.0734990
$910$	2.35793	0.0781646
$911$	−25.9725	−0.860507	−0.430253	−	0.902708i	$-0.641576\pi$
$911$	−25.9725	−0.860507	−0.430253	+	0.902708i	$0.641576\pi$
$912$	−81.2691	−2.69109
$913$	−16.9039	−0.559438
$914$	−96.6848	−3.19805
$915$	19.2295	0.635707
$916$	−99.1419	−3.27574
$917$	7.77580	0.256780
$918$	88.6573	2.92613
$919$	14.8970	0.491408	0.245704	−	0.969345i	$-0.420981\pi$
$919$	14.8970	0.491408	0.245704	+	0.969345i	$0.420981\pi$
$920$	162.980	5.37328
$921$	−13.4115	−0.441922
$922$	52.7786	1.73817
$923$	−7.95497	−0.261841
$924$	−7.46698	−0.245645
$925$	1.76158	0.0579205
$926$	70.3759	2.31269
$927$	11.9981	0.394069
$928$	26.9578	0.884935
$929$	−23.8673	−0.783062	−0.391531	−	0.920165i	$-0.628054\pi$
$929$	−23.8673	−0.783062	−0.391531	+	0.920165i	$0.628054\pi$
$930$	−5.53497	−0.181499
$931$	49.3943	1.61883
$932$	−68.0630	−2.22948
$933$	16.2596	0.532315
$934$	57.6784	1.88729
$935$	45.5765	1.49051
$936$	11.9215	0.389665
$937$	27.8986	0.911407	0.455704	−	0.890132i	$-0.349388\pi$
$937$	27.8986	0.911407	0.455704	+	0.890132i	$0.349388\pi$
$938$	−0.181216	−0.00591692
$939$	−6.94550	−0.226658
$940$	126.551	4.12763
$941$	56.3426	1.83672	0.918359	−	0.395749i	$-0.129515\pi$
$941$	56.3426	1.83672	0.918359	+	0.395749i	$0.129515\pi$
$942$	−22.7239	−0.740386
$943$	99.0953	3.22699
$944$	29.9842	0.975903
$945$	5.82994	0.189648
$946$	61.7140	2.00650
$947$	30.6416	0.995717	0.497859	−	0.867258i	$-0.334120\pi$
$947$	30.6416	0.995717	0.497859	+	0.867258i	$0.334120\pi$
$948$	61.6463	2.00218
$949$	−8.76735	−0.284600
$950$	7.99072	0.259253
$951$	17.9628	0.582483
$952$	25.7262	0.833791
$953$	−16.6997	−0.540958	−0.270479	−	0.962726i	$-0.587182\pi$
$953$	−16.6997	−0.540958	−0.270479	+	0.962726i	$0.587182\pi$
$954$	−55.6352	−1.80125
$955$	8.30507	0.268746
$956$	76.0461	2.45951
$957$	−6.34202	−0.205008
$958$	−24.5189	−0.792170
$959$	9.19311	0.296861
$960$	29.5747	0.954520
$961$	−30.2284	−0.975109
$962$	−8.73749	−0.281708
$963$	−21.7926	−0.702258
$964$	20.4800	0.659616
$965$	33.0127	1.06272
$966$	−11.8977	−0.382801
$967$	−50.3165	−1.61807	−0.809034	−	0.587762i	$-0.800009\pi$
$967$	−50.3165	−1.61807	−0.809034	+	0.587762i	$0.800009\pi$
$968$	17.5607	0.564421
$969$	−49.5683	−1.59236
$970$	41.4387	1.33052
$971$	−53.8266	−1.72738	−0.863689	−	0.504026i	$-0.831852\pi$
$971$	−53.8266	−1.72738	−0.863689	+	0.504026i	$0.831852\pi$
$972$	79.1093	2.53743
$973$	−5.41939	−0.173738
$974$	−76.3414	−2.44613
$975$	0.329293	0.0105458
$976$	87.4402	2.79889
$977$	−51.7565	−1.65584	−0.827918	−	0.560849i	$-0.810475\pi$
$977$	−51.7565	−1.65584	−0.827918	+	0.560849i	$0.810475\pi$
$978$	13.3668	0.427422
$979$	−34.7996	−1.11220
$980$	−78.2917	−2.50094
$981$	−26.2003	−0.836510
$982$	−58.2101	−1.85756
$983$	−15.5608	−0.496312	−0.248156	−	0.968720i	$-0.579825\pi$
$983$	−15.5608	−0.496312	−0.248156	+	0.968720i	$0.579825\pi$
$984$	89.9226	2.86663
$985$	−42.8695	−1.36594
$986$	36.5124	1.16279
$987$	−5.52855	−0.175976
$988$	−28.2786	−0.899661
$989$	70.1598	2.23095
$990$	35.4882	1.12789
$991$	−9.59922	−0.304929	−0.152465	−	0.988309i	$-0.548721\pi$
$991$	−9.59922	−0.304929	−0.152465	+	0.988309i	$0.548721\pi$
$992$	−11.3340	−0.359855
$993$	−12.3487	−0.391874
$994$	−13.3668	−0.423969
$995$	37.1532	1.17784
$996$	−29.1400	−0.923335
$997$	−41.8528	−1.32549	−0.662745	−	0.748845i	$-0.730609\pi$
$997$	−41.8528	−1.32549	−0.662745	+	0.748845i	$0.730609\pi$
$998$	35.1619	1.11303
$999$	−21.6033	−0.683498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4003.2.a.b.1.8	✓	152

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4003.2.a.b.1.8	✓	152	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:	\( N \)	\(=\)	\( 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64207 q^{2} +1.02501 q^{3} +4.98054 q^{4} +2.32673 q^{5} -2.70814 q^{6} -0.493904 q^{7} -7.87479 q^{8} -1.94936 q^{9} +O(q^{10})\)	\(q-2.64207 q^{2} +1.02501 q^{3} +4.98054 q^{4} +2.32673 q^{5} -2.70814 q^{6} -0.493904 q^{7} -7.87479 q^{8} -1.94936 q^{9} -6.14739 q^{10} +2.96142 q^{11} +5.10508 q^{12} +0.776601 q^{13} +1.30493 q^{14} +2.38491 q^{15} +10.8447 q^{16} +6.61446 q^{17} +5.15036 q^{18} -7.31111 q^{19} +11.5884 q^{20} -0.506254 q^{21} -7.82428 q^{22} -8.89506 q^{23} -8.07170 q^{24} +0.413674 q^{25} -2.05183 q^{26} -5.07312 q^{27} -2.45991 q^{28} -2.08930 q^{29} -6.30110 q^{30} +0.878413 q^{31} -12.9028 q^{32} +3.03547 q^{33} -17.4759 q^{34} -1.14918 q^{35} -9.70888 q^{36} +4.25838 q^{37} +19.3165 q^{38} +0.796020 q^{39} -18.3225 q^{40} -11.1405 q^{41} +1.33756 q^{42} -7.88750 q^{43} +14.7495 q^{44} -4.53564 q^{45} +23.5014 q^{46} +10.9205 q^{47} +11.1158 q^{48} -6.75606 q^{49} -1.09296 q^{50} +6.77986 q^{51} +3.86789 q^{52} -10.8022 q^{53} +13.4036 q^{54} +6.89043 q^{55} +3.88939 q^{56} -7.49393 q^{57} +5.52008 q^{58} +2.76488 q^{59} +11.8781 q^{60} +8.06297 q^{61} -2.32083 q^{62} +0.962798 q^{63} +12.4008 q^{64} +1.80694 q^{65} -8.01993 q^{66} -0.138871 q^{67} +32.9436 q^{68} -9.11748 q^{69} +3.03622 q^{70} -10.2433 q^{71} +15.3508 q^{72} -11.2894 q^{73} -11.2509 q^{74} +0.424018 q^{75} -36.4132 q^{76} -1.46266 q^{77} -2.10314 q^{78} +12.0755 q^{79} +25.2326 q^{80} +0.648113 q^{81} +29.4339 q^{82} -5.70804 q^{83} -2.52142 q^{84} +15.3901 q^{85} +20.8393 q^{86} -2.14155 q^{87} -23.3206 q^{88} -11.7510 q^{89} +11.9835 q^{90} -0.383566 q^{91} -44.3022 q^{92} +0.900378 q^{93} -28.8528 q^{94} -17.0110 q^{95} -13.2254 q^{96} -6.74086 q^{97} +17.8500 q^{98} -5.77289 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9}+O(q^{10}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9} - 15 q^{10} - 40 q^{11} - 53 q^{12} - 59 q^{13} - 36 q^{14} - 40 q^{15} + 118 q^{16} - 93 q^{17} - 59 q^{18} - 16 q^{19} - 108 q^{20} - 62 q^{21} - 37 q^{22} - 107 q^{23} - 31 q^{24} + 101 q^{25} - 64 q^{26} - 63 q^{27} - 53 q^{28} - 124 q^{29} - 68 q^{30} - 15 q^{31} - 129 q^{32} - 49 q^{33} - 76 q^{35} + 45 q^{36} - 98 q^{37} - 125 q^{38} - 47 q^{39} - 7 q^{40} - 56 q^{41} - 84 q^{42} - 62 q^{43} - 114 q^{44} - 142 q^{45} - 3 q^{46} - 111 q^{47} - 92 q^{48} + 117 q^{49} - 64 q^{50} - 21 q^{51} - 85 q^{52} - 347 q^{53} + 3 q^{54} - 16 q^{55} - 73 q^{56} - 115 q^{57} - 29 q^{58} - 50 q^{59} - 54 q^{60} - 62 q^{61} - 55 q^{62} - 70 q^{63} + 64 q^{64} - 147 q^{65} + 34 q^{66} - 86 q^{67} - 174 q^{68} - 104 q^{69} - 7 q^{70} - 86 q^{71} - 139 q^{72} - 27 q^{73} - 52 q^{74} - 49 q^{75} - 11 q^{76} - 346 q^{77} - 59 q^{78} - 17 q^{79} - 149 q^{80} - 8 q^{81} - 31 q^{82} - 106 q^{83} - 51 q^{84} - 69 q^{85} - 85 q^{86} - 32 q^{87} - 113 q^{88} - 59 q^{89} + 10 q^{90} - 9 q^{91} - 314 q^{92} - 230 q^{93} + 7 q^{94} - 74 q^{95} - 54 q^{96} - 60 q^{97} - 77 q^{98} - 96 q^{99}+O(q^{100}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9 - 15 * q^10 - 40 * q^11 - 53 * q^12 - 59 * q^13 - 36 * q^14 - 40 * q^15 + 118 * q^16 - 93 * q^17 - 59 * q^18 - 16 * q^19 - 108 * q^20 - 62 * q^21 - 37 * q^22 - 107 * q^23 - 31 * q^24 + 101 * q^25 - 64 * q^26 - 63 * q^27 - 53 * q^28 - 124 * q^29 - 68 * q^30 - 15 * q^31 - 129 * q^32 - 49 * q^33 - 76 * q^35 + 45 * q^36 - 98 * q^37 - 125 * q^38 - 47 * q^39 - 7 * q^40 - 56 * q^41 - 84 * q^42 - 62 * q^43 - 114 * q^44 - 142 * q^45 - 3 * q^46 - 111 * q^47 - 92 * q^48 + 117 * q^49 - 64 * q^50 - 21 * q^51 - 85 * q^52 - 347 * q^53 + 3 * q^54 - 16 * q^55 - 73 * q^56 - 115 * q^57 - 29 * q^58 - 50 * q^59 - 54 * q^60 - 62 * q^61 - 55 * q^62 - 70 * q^63 + 64 * q^64 - 147 * q^65 + 34 * q^66 - 86 * q^67 - 174 * q^68 - 104 * q^69 - 7 * q^70 - 86 * q^71 - 139 * q^72 - 27 * q^73 - 52 * q^74 - 49 * q^75 - 11 * q^76 - 346 * q^77 - 59 * q^78 - 17 * q^79 - 149 * q^80 - 8 * q^81 - 31 * q^82 - 106 * q^83 - 51 * q^84 - 69 * q^85 - 85 * q^86 - 32 * q^87 - 113 * q^88 - 59 * q^89 + 10 * q^90 - 9 * q^91 - 314 * q^92 - 230 * q^93 + 7 * q^94 - 74 * q^95 - 54 * q^96 - 60 * q^97 - 77 * q^98 - 96 * q^99

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