LMFDB - Embedded newform 2005.2.a.f.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,2,Mod(1,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2005.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:	$16.0100056053$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0750780 q^{2} +3.37448 q^{3} -1.99436 q^{4} -1.00000 q^{5} +0.253350 q^{6} +1.31500 q^{7} -0.299889 q^{8} +8.38713 q^{9} +O(q^{10})$	\(q+0.0750780 q^{2} +3.37448 q^{3} -1.99436 q^{4} -1.00000 q^{5} +0.253350 q^{6} +1.31500 q^{7} -0.299889 q^{8} +8.38713 q^{9} -0.0750780 q^{10} -0.660677 q^{11} -6.72994 q^{12} +3.48964 q^{13} +0.0987277 q^{14} -3.37448 q^{15} +3.96621 q^{16} +3.95506 q^{17} +0.629689 q^{18} -3.86927 q^{19} +1.99436 q^{20} +4.43745 q^{21} -0.0496023 q^{22} -2.37568 q^{23} -1.01197 q^{24} +1.00000 q^{25} +0.261995 q^{26} +18.1788 q^{27} -2.62259 q^{28} -5.70320 q^{29} -0.253350 q^{30} -2.14115 q^{31} +0.897553 q^{32} -2.22944 q^{33} +0.296938 q^{34} -1.31500 q^{35} -16.7270 q^{36} +5.63704 q^{37} -0.290497 q^{38} +11.7757 q^{39} +0.299889 q^{40} +12.1365 q^{41} +0.333155 q^{42} +0.449823 q^{43} +1.31763 q^{44} -8.38713 q^{45} -0.178361 q^{46} -8.97365 q^{47} +13.3839 q^{48} -5.27077 q^{49} +0.0750780 q^{50} +13.3463 q^{51} -6.95960 q^{52} +9.21713 q^{53} +1.36483 q^{54} +0.660677 q^{55} -0.394354 q^{56} -13.0568 q^{57} -0.428185 q^{58} +13.2681 q^{59} +6.72994 q^{60} +3.38617 q^{61} -0.160754 q^{62} +11.0291 q^{63} -7.86504 q^{64} -3.48964 q^{65} -0.167382 q^{66} -5.41739 q^{67} -7.88783 q^{68} -8.01668 q^{69} -0.0987277 q^{70} +0.493335 q^{71} -2.51521 q^{72} -1.48480 q^{73} +0.423218 q^{74} +3.37448 q^{75} +7.71673 q^{76} -0.868791 q^{77} +0.884098 q^{78} -1.31133 q^{79} -3.96621 q^{80} +36.1826 q^{81} +0.911181 q^{82} +14.1598 q^{83} -8.84989 q^{84} -3.95506 q^{85} +0.0337718 q^{86} -19.2453 q^{87} +0.198130 q^{88} -10.5141 q^{89} -0.629689 q^{90} +4.58888 q^{91} +4.73796 q^{92} -7.22529 q^{93} -0.673724 q^{94} +3.86927 q^{95} +3.02878 q^{96} +7.11241 q^{97} -0.395719 q^{98} -5.54118 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9}+O(q^{10}) $ 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9} - 7 q^{10} + 42 q^{11} - 13 q^{13} + 14 q^{14} - 3 q^{15} + 63 q^{16} + 18 q^{17} + 22 q^{18} + 22 q^{19} - 43 q^{20} + 16 q^{21} - 10 q^{22} + 23 q^{23} + 23 q^{24} + 37 q^{25} + 21 q^{26} + 3 q^{27} - 18 q^{28} + 33 q^{29} - 8 q^{30} + 11 q^{31} + 54 q^{32} + 2 q^{33} + 8 q^{34} + 16 q^{35} + 91 q^{36} - 11 q^{37} + 29 q^{38} + 25 q^{39} - 21 q^{40} + 24 q^{41} + 4 q^{42} + 25 q^{43} + 84 q^{44} - 54 q^{45} + 31 q^{46} + 7 q^{47} + 4 q^{48} + 45 q^{49} + 7 q^{50} + 94 q^{51} - 43 q^{52} + 49 q^{53} + 38 q^{54} - 42 q^{55} + 46 q^{56} + 6 q^{57} + 15 q^{58} + 69 q^{59} + 9 q^{61} + 17 q^{62} - 38 q^{63} + 107 q^{64} + 13 q^{65} + 74 q^{66} + 13 q^{67} + 86 q^{68} - 14 q^{70} + 51 q^{71} + 81 q^{72} - 47 q^{73} + 79 q^{74} + 3 q^{75} + 59 q^{76} + 2 q^{77} + 20 q^{78} + 67 q^{79} - 63 q^{80} + 125 q^{81} - 24 q^{82} + 80 q^{83} + 50 q^{84} - 18 q^{85} + 69 q^{86} - 32 q^{87} - 12 q^{88} + 34 q^{89} - 22 q^{90} + 39 q^{91} + 85 q^{92} + q^{93} + 12 q^{94} - 22 q^{95} + 77 q^{96} - 14 q^{97} + 40 q^{98} + 133 q^{99}+O(q^{100}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9 - 7 * q^10 + 42 * q^11 - 13 * q^13 + 14 * q^14 - 3 * q^15 + 63 * q^16 + 18 * q^17 + 22 * q^18 + 22 * q^19 - 43 * q^20 + 16 * q^21 - 10 * q^22 + 23 * q^23 + 23 * q^24 + 37 * q^25 + 21 * q^26 + 3 * q^27 - 18 * q^28 + 33 * q^29 - 8 * q^30 + 11 * q^31 + 54 * q^32 + 2 * q^33 + 8 * q^34 + 16 * q^35 + 91 * q^36 - 11 * q^37 + 29 * q^38 + 25 * q^39 - 21 * q^40 + 24 * q^41 + 4 * q^42 + 25 * q^43 + 84 * q^44 - 54 * q^45 + 31 * q^46 + 7 * q^47 + 4 * q^48 + 45 * q^49 + 7 * q^50 + 94 * q^51 - 43 * q^52 + 49 * q^53 + 38 * q^54 - 42 * q^55 + 46 * q^56 + 6 * q^57 + 15 * q^58 + 69 * q^59 + 9 * q^61 + 17 * q^62 - 38 * q^63 + 107 * q^64 + 13 * q^65 + 74 * q^66 + 13 * q^67 + 86 * q^68 - 14 * q^70 + 51 * q^71 + 81 * q^72 - 47 * q^73 + 79 * q^74 + 3 * q^75 + 59 * q^76 + 2 * q^77 + 20 * q^78 + 67 * q^79 - 63 * q^80 + 125 * q^81 - 24 * q^82 + 80 * q^83 + 50 * q^84 - 18 * q^85 + 69 * q^86 - 32 * q^87 - 12 * q^88 + 34 * q^89 - 22 * q^90 + 39 * q^91 + 85 * q^92 + q^93 + 12 * q^94 - 22 * q^95 + 77 * q^96 - 14 * q^97 + 40 * q^98 + 133 * 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 $

$2$	0.0750780	0.0530882	0.0265441	−	0.999648i	$-0.491550\pi$
$2$	0.0750780	0.0530882	0.0265441	+	0.999648i	$0.491550\pi$
$3$	3.37448	1.94826	0.974129	−	0.225992i	$-0.0725622\pi$
$3$	3.37448	1.94826	0.974129	+	0.225992i	$0.0725622\pi$
$4$	−1.99436	−0.997182
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.253350	0.103430
$7$	1.31500	0.497024	0.248512	−	0.968629i	$-0.420058\pi$
$7$	1.31500	0.497024	0.248512	+	0.968629i	$0.420058\pi$
$8$	−0.299889	−0.106027
$9$	8.38713	2.79571
$10$	−0.0750780	−0.0237418
$11$	−0.660677	−0.199202	−0.0996008	−	0.995027i	$-0.531757\pi$
$11$	−0.660677	−0.199202	−0.0996008	+	0.995027i	$0.531757\pi$
$12$	−6.72994	−1.94277
$13$	3.48964	0.967851	0.483925	−	0.875109i	$-0.339211\pi$
$13$	3.48964	0.967851	0.483925	+	0.875109i	$0.339211\pi$
$14$	0.0987277	0.0263861
$15$	−3.37448	−0.871288
$16$	3.96621	0.991553
$17$	3.95506	0.959243	0.479622	−	0.877475i	$-0.340774\pi$
$17$	3.95506	0.959243	0.479622	+	0.877475i	$0.340774\pi$
$18$	0.629689	0.148419
$19$	−3.86927	−0.887671	−0.443836	−	0.896108i	$-0.646383\pi$
$19$	−3.86927	−0.887671	−0.443836	+	0.896108i	$0.646383\pi$
$20$	1.99436	0.445953
$21$	4.43745	0.968331
$22$	−0.0496023	−0.0105753
$23$	−2.37568	−0.495363	−0.247682	−	0.968842i	$-0.579669\pi$
$23$	−2.37568	−0.495363	−0.247682	+	0.968842i	$0.579669\pi$
$24$	−1.01197	−0.206568
$25$	1.00000	0.200000
$26$	0.261995	0.0513815
$27$	18.1788	3.49851
$28$	−2.62259	−0.495623
$29$	−5.70320	−1.05906	−0.529528	−	0.848292i	$-0.677631\pi$
$29$	−5.70320	−1.05906	−0.529528	+	0.848292i	$0.677631\pi$
$30$	−0.253350	−0.0462551
$31$	−2.14115	−0.384563	−0.192281	−	0.981340i	$-0.561589\pi$
$31$	−2.14115	−0.384563	−0.192281	+	0.981340i	$0.561589\pi$
$32$	0.897553	0.158666
$33$	−2.22944	−0.388096
$34$	0.296938	0.0509245
$35$	−1.31500	−0.222276
$36$	−16.7270	−2.78783
$37$	5.63704	0.926724	0.463362	−	0.886169i	$-0.346643\pi$
$37$	5.63704	0.926724	0.463362	+	0.886169i	$0.346643\pi$
$38$	−0.290497	−0.0471249
$39$	11.7757	1.88562
$40$	0.299889	0.0474166
$41$	12.1365	1.89540	0.947698	−	0.319168i	$-0.103404\pi$
$41$	12.1365	1.89540	0.947698	+	0.319168i	$0.103404\pi$
$42$	0.333155	0.0514069
$43$	0.449823	0.0685973	0.0342987	−	0.999412i	$-0.489080\pi$
$43$	0.449823	0.0685973	0.0342987	+	0.999412i	$0.489080\pi$
$44$	1.31763	0.198640
$45$	−8.38713	−1.25028
$46$	−0.178361	−0.0262979
$47$	−8.97365	−1.30894	−0.654470	−	0.756088i	$-0.727108\pi$
$47$	−8.97365	−1.30894	−0.654470	+	0.756088i	$0.727108\pi$
$48$	13.3839	1.93180
$49$	−5.27077	−0.752967
$50$	0.0750780	0.0106176
$51$	13.3463	1.86885
$52$	−6.95960	−0.965123
$53$	9.21713	1.26607	0.633035	−	0.774123i	$-0.281809\pi$
$53$	9.21713	1.26607	0.633035	+	0.774123i	$0.281809\pi$
$54$	1.36483	0.185730
$55$	0.660677	0.0890857
$56$	−0.394354	−0.0526978
$57$	−13.0568	−1.72941
$58$	−0.428185	−0.0562234
$59$	13.2681	1.72737	0.863683	−	0.504035i	$-0.168152\pi$
$59$	13.2681	1.72737	0.863683	+	0.504035i	$0.168152\pi$
$60$	6.72994	0.868832
$61$	3.38617	0.433554	0.216777	−	0.976221i	$-0.430446\pi$
$61$	3.38617	0.433554	0.216777	+	0.976221i	$0.430446\pi$
$62$	−0.160754	−0.0204157
$63$	11.0291	1.38954
$64$	−7.86504	−0.983130
$65$	−3.48964	−0.432836
$66$	−0.167382	−0.0206033
$67$	−5.41739	−0.661839	−0.330920	−	0.943659i	$-0.607359\pi$
$67$	−5.41739	−0.661839	−0.330920	+	0.943659i	$0.607359\pi$
$68$	−7.88783	−0.956540
$69$	−8.01668	−0.965095
$70$	−0.0987277	−0.0118002
$71$	0.493335	0.0585481	0.0292741	−	0.999571i	$-0.490680\pi$
$71$	0.493335	0.0585481	0.0292741	+	0.999571i	$0.490680\pi$
$72$	−2.51521	−0.296420
$73$	−1.48480	−0.173783	−0.0868914	−	0.996218i	$-0.527693\pi$
$73$	−1.48480	−0.173783	−0.0868914	+	0.996218i	$0.527693\pi$
$74$	0.423218	0.0491981
$75$	3.37448	0.389652
$76$	7.71673	0.885169
$77$	−0.868791	−0.0990079
$78$	0.884098	0.100104
$79$	−1.31133	−0.147536	−0.0737678	−	0.997275i	$-0.523502\pi$
$79$	−1.31133	−0.147536	−0.0737678	+	0.997275i	$0.523502\pi$
$80$	−3.96621	−0.443436
$81$	36.1826	4.02029
$82$	0.911181	0.100623
$83$	14.1598	1.55424	0.777119	−	0.629354i	$-0.216680\pi$
$83$	14.1598	1.55424	0.777119	+	0.629354i	$0.216680\pi$
$84$	−8.84989	−0.965602
$85$	−3.95506	−0.428987
$86$	0.0337718	0.00364171
$87$	−19.2453	−2.06332
$88$	0.198130	0.0211207
$89$	−10.5141	−1.11449	−0.557246	−	0.830348i	$-0.688142\pi$
$89$	−10.5141	−1.11449	−0.557246	+	0.830348i	$0.688142\pi$
$90$	−0.629689	−0.0663751
$91$	4.58888	0.481045
$92$	4.73796	0.493967
$93$	−7.22529	−0.749227
$94$	−0.673724	−0.0694893
$95$	3.86927	0.396979
$96$	3.02878	0.309123
$97$	7.11241	0.722155	0.361078	−	0.932536i	$-0.382409\pi$
$97$	7.11241	0.722155	0.361078	+	0.932536i	$0.382409\pi$
$98$	−0.395719	−0.0399737
$99$	−5.54118	−0.556910
$100$	−1.99436	−0.199436
$101$	−5.58071	−0.555301	−0.277651	−	0.960682i	$-0.589556\pi$
$101$	−5.58071	−0.555301	−0.277651	+	0.960682i	$0.589556\pi$
$102$	1.00201	0.0992140
$103$	8.86733	0.873724	0.436862	−	0.899529i	$-0.356090\pi$
$103$	8.86733	0.873724	0.436862	+	0.899529i	$0.356090\pi$
$104$	−1.04650	−0.102618
$105$	−4.43745	−0.433051
$106$	0.692004	0.0672134
$107$	2.62086	0.253368	0.126684	−	0.991943i	$-0.459567\pi$
$107$	2.62086	0.253368	0.126684	+	0.991943i	$0.459567\pi$
$108$	−36.2551	−3.48865
$109$	−19.1495	−1.83419	−0.917096	−	0.398667i	$-0.869473\pi$
$109$	−19.1495	−1.83419	−0.917096	+	0.398667i	$0.869473\pi$
$110$	0.0496023	0.00472940
$111$	19.0221	1.80550
$112$	5.21557	0.492825
$113$	19.0654	1.79352	0.896762	−	0.442513i	$-0.145913\pi$
$113$	19.0654	1.79352	0.896762	+	0.442513i	$0.145913\pi$
$114$	−0.980277	−0.0918114
$115$	2.37568	0.221533
$116$	11.3742	1.05607
$117$	29.2680	2.70583
$118$	0.996147	0.0917027
$119$	5.20091	0.476767
$120$	1.01197	0.0923798
$121$	−10.5635	−0.960319
$122$	0.254227	0.0230166
$123$	40.9543	3.69272
$124$	4.27024	0.383479
$125$	−1.00000	−0.0894427
$126$	0.828043	0.0737679
$127$	20.8044	1.84609	0.923044	−	0.384695i	$-0.125693\pi$
$127$	20.8044	1.84609	0.923044	+	0.384695i	$0.125693\pi$
$128$	−2.38560	−0.210859
$129$	1.51792	0.133645
$130$	−0.261995	−0.0229785
$131$	1.17534	0.102690	0.0513451	−	0.998681i	$-0.483649\pi$
$131$	1.17534	0.102690	0.0513451	+	0.998681i	$0.483649\pi$
$132$	4.44632	0.387002
$133$	−5.08809	−0.441194
$134$	−0.406727	−0.0351359
$135$	−18.1788	−1.56458
$136$	−1.18608	−0.101705
$137$	−17.9544	−1.53395	−0.766974	−	0.641678i	$-0.778239\pi$
$137$	−17.9544	−1.53395	−0.766974	+	0.641678i	$0.778239\pi$
$138$	−0.601877	−0.0512352
$139$	−1.25046	−0.106063	−0.0530314	−	0.998593i	$-0.516888\pi$
$139$	−1.25046	−0.106063	−0.0530314	+	0.998593i	$0.516888\pi$
$140$	2.62259	0.221649
$141$	−30.2814	−2.55016
$142$	0.0370386	0.00310821
$143$	−2.30552	−0.192797
$144$	33.2651	2.77210
$145$	5.70320	0.473625
$146$	−0.111476	−0.00922581
$147$	−17.7861	−1.46697
$148$	−11.2423	−0.924112
$149$	−7.69797	−0.630643	−0.315321	−	0.948985i	$-0.602112\pi$
$149$	−7.69797	−0.630643	−0.315321	+	0.948985i	$0.602112\pi$
$150$	0.253350	0.0206859
$151$	−15.0094	−1.22144	−0.610722	−	0.791845i	$-0.709121\pi$
$151$	−15.0094	−1.22144	−0.610722	+	0.791845i	$0.709121\pi$
$152$	1.16035	0.0941169
$153$	33.1716	2.68177
$154$	−0.0652271	−0.00525615
$155$	2.14115	0.171982
$156$	−23.4851	−1.88031
$157$	−6.23756	−0.497812	−0.248906	−	0.968528i	$-0.580071\pi$
$157$	−6.23756	−0.497812	−0.248906	+	0.968528i	$0.580071\pi$
$158$	−0.0984518	−0.00783240
$159$	31.1030	2.46663
$160$	−0.897553	−0.0709578
$161$	−3.12402	−0.246207
$162$	2.71652	0.213430
$163$	10.2884	0.805852	0.402926	−	0.915233i	$-0.367993\pi$
$163$	10.2884	0.805852	0.402926	+	0.915233i	$0.367993\pi$
$164$	−24.2045	−1.89005
$165$	2.22944	0.173562
$166$	1.06309	0.0825117
$167$	6.03284	0.466835	0.233418	−	0.972377i	$-0.425009\pi$
$167$	6.03284	0.466835	0.233418	+	0.972377i	$0.425009\pi$
$168$	−1.33074	−0.102669
$169$	−0.822439	−0.0632645
$170$	−0.296938	−0.0227741
$171$	−32.4521	−2.48167
$172$	−0.897110	−0.0684040
$173$	−3.80696	−0.289438	−0.144719	−	0.989473i	$-0.546228\pi$
$173$	−3.80696	−0.289438	−0.144719	+	0.989473i	$0.546228\pi$
$174$	−1.44490	−0.109538
$175$	1.31500	0.0994048
$176$	−2.62038	−0.197519
$177$	44.7731	3.36536
$178$	−0.789378	−0.0591664
$179$	−17.8366	−1.33317	−0.666584	−	0.745430i	$-0.732244\pi$
$179$	−17.8366	−1.33317	−0.666584	+	0.745430i	$0.732244\pi$
$180$	16.7270	1.24676
$181$	−23.8965	−1.77621	−0.888105	−	0.459640i	$-0.847978\pi$
$181$	−23.8965	−1.77621	−0.888105	+	0.459640i	$0.847978\pi$
$182$	0.344524	0.0255378
$183$	11.4266	0.844676
$184$	0.712439	0.0525217
$185$	−5.63704	−0.414444
$186$	−0.542460	−0.0397751
$187$	−2.61302	−0.191083
$188$	17.8967	1.30525
$189$	23.9051	1.73884
$190$	0.290497	0.0210749
$191$	−8.23241	−0.595676	−0.297838	−	0.954616i	$-0.596266\pi$
$191$	−8.23241	−0.595676	−0.297838	+	0.954616i	$0.596266\pi$
$192$	−26.5404	−1.91539
$193$	−24.0341	−1.73002	−0.865008	−	0.501758i	$-0.832687\pi$
$193$	−24.0341	−1.73002	−0.865008	+	0.501758i	$0.832687\pi$
$194$	0.533985	0.0383379
$195$	−11.7757	−0.843277
$196$	10.5118	0.750845
$197$	−13.0548	−0.930115	−0.465058	−	0.885280i	$-0.653966\pi$
$197$	−13.0548	−0.930115	−0.465058	+	0.885280i	$0.653966\pi$
$198$	−0.416021	−0.0295653
$199$	0.444844	0.0315342	0.0157671	−	0.999876i	$-0.494981\pi$
$199$	0.444844	0.0315342	0.0157671	+	0.999876i	$0.494981\pi$
$200$	−0.299889	−0.0212054
$201$	−18.2809	−1.28943
$202$	−0.418989	−0.0294799
$203$	−7.49971	−0.526377
$204$	−26.6173	−1.86359
$205$	−12.1365	−0.847647
$206$	0.665741	0.0463844
$207$	−19.9251	−1.38489
$208$	13.8406	0.959675
$209$	2.55634	0.176826
$210$	−0.333155	−0.0229899
$211$	−8.03968	−0.553475	−0.276737	−	0.960946i	$-0.589253\pi$
$211$	−8.03968	−0.553475	−0.276737	+	0.960946i	$0.589253\pi$
$212$	−18.3823	−1.26250
$213$	1.66475	0.114067
$214$	0.196769	0.0134509
$215$	−0.449823	−0.0306777
$216$	−5.45162	−0.370936
$217$	−2.81562	−0.191137
$218$	−1.43771	−0.0973739
$219$	−5.01043	−0.338574
$220$	−1.31763	−0.0888346
$221$	13.8017	0.928404
$222$	1.42814	0.0958506
$223$	−2.24769	−0.150517	−0.0752583	−	0.997164i	$-0.523978\pi$
$223$	−2.24769	−0.150517	−0.0752583	+	0.997164i	$0.523978\pi$
$224$	1.18028	0.0788610
$225$	8.38713	0.559142
$226$	1.43139	0.0952149
$227$	12.4429	0.825865	0.412933	−	0.910762i	$-0.364504\pi$
$227$	12.4429	0.825865	0.412933	+	0.910762i	$0.364504\pi$
$228$	26.0400	1.72454
$229$	−14.3392	−0.947562	−0.473781	−	0.880643i	$-0.657111\pi$
$229$	−14.3392	−0.947562	−0.473781	+	0.880643i	$0.657111\pi$
$230$	0.178361	0.0117608
$231$	−2.93172	−0.192893
$232$	1.71033	0.112288
$233$	−22.0483	−1.44443	−0.722215	−	0.691669i	$-0.756876\pi$
$233$	−22.0483	−1.44443	−0.722215	+	0.691669i	$0.756876\pi$
$234$	2.19739	0.143648
$235$	8.97365	0.585376
$236$	−26.4615	−1.72250
$237$	−4.42505	−0.287438
$238$	0.390474	0.0253107
$239$	−5.09258	−0.329412	−0.164706	−	0.986343i	$-0.552668\pi$
$239$	−5.09258	−0.329412	−0.164706	+	0.986343i	$0.552668\pi$
$240$	−13.3839	−0.863928
$241$	−16.5724	−1.06752	−0.533761	−	0.845636i	$-0.679222\pi$
$241$	−16.5724	−1.06752	−0.533761	+	0.845636i	$0.679222\pi$
$242$	−0.793087	−0.0509816
$243$	67.5612	4.33405
$244$	−6.75325	−0.432332
$245$	5.27077	0.336737
$246$	3.07477	0.196040
$247$	−13.5023	−0.859133
$248$	0.642108	0.0407739
$249$	47.7819	3.02806
$250$	−0.0750780	−0.00474835
$251$	1.03866	0.0655597	0.0327798	−	0.999463i	$-0.489564\pi$
$251$	1.03866	0.0655597	0.0327798	+	0.999463i	$0.489564\pi$
$252$	−21.9960	−1.38562
$253$	1.56956	0.0986771
$254$	1.56195	0.0980054
$255$	−13.3463	−0.835777
$256$	15.5510	0.971935
$257$	−24.1130	−1.50413	−0.752063	−	0.659091i	$-0.770941\pi$
$257$	−24.1130	−1.50413	−0.752063	+	0.659091i	$0.770941\pi$
$258$	0.113962	0.00709499
$259$	7.41272	0.460604
$260$	6.95960	0.431616
$261$	−47.8335	−2.96082
$262$	0.0882424	0.00545164
$263$	3.47517	0.214288	0.107144	−	0.994243i	$-0.465829\pi$
$263$	3.47517	0.214288	0.107144	+	0.994243i	$0.465829\pi$
$264$	0.668585	0.0411486
$265$	−9.21713	−0.566204
$266$	−0.382004	−0.0234222
$267$	−35.4796	−2.17132
$268$	10.8042	0.659974
$269$	−8.31820	−0.507170	−0.253585	−	0.967313i	$-0.581610\pi$
$269$	−8.31820	−0.507170	−0.253585	+	0.967313i	$0.581610\pi$
$270$	−1.36483	−0.0830608
$271$	4.90054	0.297687	0.148843	−	0.988861i	$-0.452445\pi$
$271$	4.90054	0.297687	0.148843	+	0.988861i	$0.452445\pi$
$272$	15.6866	0.951140
$273$	15.4851	0.937200
$274$	−1.34798	−0.0814345
$275$	−0.660677	−0.0398403
$276$	15.9882	0.962375
$277$	−13.3264	−0.800706	−0.400353	−	0.916361i	$-0.631113\pi$
$277$	−13.3264	−0.800706	−0.400353	+	0.916361i	$0.631113\pi$
$278$	−0.0938823	−0.00563069
$279$	−17.9581	−1.07513
$280$	0.394354	0.0235672
$281$	−2.20268	−0.131401	−0.0657003	−	0.997839i	$-0.520928\pi$
$281$	−2.20268	−0.131401	−0.0657003	+	0.997839i	$0.520928\pi$
$282$	−2.27347	−0.135383
$283$	−21.0428	−1.25086	−0.625432	−	0.780279i	$-0.715077\pi$
$283$	−21.0428	−1.25086	−0.625432	+	0.780279i	$0.715077\pi$
$284$	−0.983890	−0.0583831
$285$	13.0568	0.773417
$286$	−0.173094	−0.0102353
$287$	15.9595	0.942057
$288$	7.52790	0.443586
$289$	−1.35749	−0.0798526
$290$	0.428185	0.0251439
$291$	24.0007	1.40695
$292$	2.96123	0.173293
$293$	13.7314	0.802199	0.401100	−	0.916034i	$-0.368628\pi$
$293$	13.7314	0.802199	0.401100	+	0.916034i	$0.368628\pi$
$294$	−1.33535	−0.0778790
$295$	−13.2681	−0.772502
$296$	−1.69049	−0.0982575
$297$	−12.0103	−0.696909
$298$	−0.577949	−0.0334797
$299$	−8.29025	−0.479438
$300$	−6.72994	−0.388554
$301$	0.591518	0.0340945
$302$	−1.12687	−0.0648443
$303$	−18.8320	−1.08187
$304$	−15.3463	−0.880173
$305$	−3.38617	−0.193891
$306$	2.49046	0.142370
$307$	−15.8246	−0.903159	−0.451580	−	0.892231i	$-0.649139\pi$
$307$	−15.8246	−0.903159	−0.451580	+	0.892231i	$0.649139\pi$
$308$	1.73269	0.0987289
$309$	29.9226	1.70224
$310$	0.160754	0.00913019
$311$	30.6955	1.74058	0.870290	−	0.492539i	$-0.163931\pi$
$311$	30.6955	1.74058	0.870290	+	0.492539i	$0.163931\pi$
$312$	−3.53141	−0.199927
$313$	24.5576	1.38808	0.694040	−	0.719937i	$-0.255829\pi$
$313$	24.5576	1.38808	0.694040	+	0.719937i	$0.255829\pi$
$314$	−0.468304	−0.0264279
$315$	−11.0291	−0.621419
$316$	2.61526	0.147120
$317$	7.33419	0.411929	0.205965	−	0.978559i	$-0.433967\pi$
$317$	7.33419	0.411929	0.205965	+	0.978559i	$0.433967\pi$
$318$	2.33515	0.130949
$319$	3.76797	0.210966
$320$	7.86504	0.439669
$321$	8.84405	0.493627
$322$	−0.234545	−0.0130707
$323$	−15.3032	−0.851492
$324$	−72.1612	−4.00896
$325$	3.48964	0.193570
$326$	0.772435	0.0427812
$327$	−64.6197	−3.57348
$328$	−3.63959	−0.200963
$329$	−11.8004	−0.650575
$330$	0.167382	0.00921409
$331$	19.3787	1.06515	0.532574	−	0.846384i	$-0.321225\pi$
$331$	19.3787	1.06515	0.532574	+	0.846384i	$0.321225\pi$
$332$	−28.2397	−1.54986
$333$	47.2786	2.59085
$334$	0.452934	0.0247834
$335$	5.41739	0.295984
$336$	17.5999	0.960151
$337$	5.09573	0.277582	0.138791	−	0.990322i	$-0.455678\pi$
$337$	5.09573	0.277582	0.138791	+	0.990322i	$0.455678\pi$
$338$	−0.0617471	−0.00335860
$339$	64.3359	3.49425
$340$	7.88783	0.427778
$341$	1.41461	0.0766055
$342$	−2.43644	−0.131747
$343$	−16.1361	−0.871267
$344$	−0.134897	−0.00727315
$345$	8.01668	0.431604
$346$	−0.285819	−0.0153657
$347$	13.5447	0.727118	0.363559	−	0.931571i	$-0.381561\pi$
$347$	13.5447	0.727118	0.363559	+	0.931571i	$0.381561\pi$
$348$	38.3822	2.05750
$349$	10.5623	0.565388	0.282694	−	0.959210i	$-0.408772\pi$
$349$	10.5623	0.565388	0.282694	+	0.959210i	$0.408772\pi$
$350$	0.0987277	0.00527722
$351$	63.4373	3.38604
$352$	−0.592993	−0.0316066
$353$	32.3012	1.71922	0.859610	−	0.510950i	$-0.170706\pi$
$353$	32.3012	1.71922	0.859610	+	0.510950i	$0.170706\pi$
$354$	3.36148	0.178661
$355$	−0.493335	−0.0261835
$356$	20.9689	1.11135
$357$	17.5504	0.928865
$358$	−1.33913	−0.0707755
$359$	28.6999	1.51472	0.757361	−	0.652996i	$-0.226488\pi$
$359$	28.6999	1.51472	0.757361	+	0.652996i	$0.226488\pi$
$360$	2.51521	0.132563
$361$	−4.02876	−0.212040
$362$	−1.79410	−0.0942958
$363$	−35.6464	−1.87095
$364$	−9.15189	−0.479689
$365$	1.48480	0.0777180
$366$	0.857884	0.0448423
$367$	20.9142	1.09171	0.545857	−	0.837878i	$-0.316204\pi$
$367$	20.9142	1.09171	0.545857	+	0.837878i	$0.316204\pi$
$368$	−9.42244	−0.491179
$369$	101.790	5.29898
$370$	−0.423218	−0.0220021
$371$	12.1205	0.629267
$372$	14.4098	0.747116
$373$	10.6228	0.550030	0.275015	−	0.961440i	$-0.411317\pi$
$373$	10.6228	0.550030	0.275015	+	0.961440i	$0.411317\pi$
$374$	−0.196180	−0.0101442
$375$	−3.37448	−0.174258
$376$	2.69110	0.138783
$377$	−19.9021	−1.02501
$378$	1.79475	0.0923120
$379$	−33.5262	−1.72212	−0.861062	−	0.508500i	$-0.830200\pi$
$379$	−33.5262	−1.72212	−0.861062	+	0.508500i	$0.830200\pi$
$380$	−7.71673	−0.395860
$381$	70.2039	3.59666
$382$	−0.618073	−0.0316234
$383$	−20.7428	−1.05991	−0.529953	−	0.848027i	$-0.677790\pi$
$383$	−20.7428	−1.05991	−0.529953	+	0.848027i	$0.677790\pi$
$384$	−8.05016	−0.410808
$385$	0.868791	0.0442777
$386$	−1.80444	−0.0918434
$387$	3.77272	0.191778
$388$	−14.1847	−0.720120
$389$	−19.4484	−0.986072	−0.493036	−	0.870009i	$-0.664113\pi$
$389$	−19.4484	−0.986072	−0.493036	+	0.870009i	$0.664113\pi$
$390$	−0.884098	−0.0447680
$391$	−9.39595	−0.475174
$392$	1.58065	0.0798347
$393$	3.96617	0.200067
$394$	−0.980128	−0.0493781
$395$	1.31133	0.0659800
$396$	11.0511	0.555340
$397$	15.7938	0.792668	0.396334	−	0.918106i	$-0.370282\pi$
$397$	15.7938	0.792668	0.396334	+	0.918106i	$0.370282\pi$
$398$	0.0333980	0.00167409
$399$	−17.1697	−0.859559
$400$	3.96621	0.198311
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	−1.37249	−0.0684537
$403$	−7.47185	−0.372199
$404$	11.1300	0.553736
$405$	−36.1826	−1.79793
$406$	−0.563064	−0.0279444
$407$	−3.72426	−0.184605
$408$	−4.00240	−0.198148
$409$	−4.90950	−0.242759	−0.121379	−	0.992606i	$-0.538732\pi$
$409$	−4.90950	−0.242759	−0.121379	+	0.992606i	$0.538732\pi$
$410$	−0.911181	−0.0450000
$411$	−60.5868	−2.98853
$412$	−17.6847	−0.871261
$413$	17.4476	0.858542
$414$	−1.49594	−0.0735214
$415$	−14.1598	−0.695076
$416$	3.13213	0.153566
$417$	−4.21967	−0.206638
$418$	0.191925	0.00938735
$419$	7.05094	0.344461	0.172231	−	0.985057i	$-0.444903\pi$
$419$	7.05094	0.344461	0.172231	+	0.985057i	$0.444903\pi$
$420$	8.84989	0.431830
$421$	−4.97555	−0.242494	−0.121247	−	0.992622i	$-0.538689\pi$
$421$	−4.97555	−0.242494	−0.121247	+	0.992622i	$0.538689\pi$
$422$	−0.603604	−0.0293830
$423$	−75.2632	−3.65942
$424$	−2.76411	−0.134237
$425$	3.95506	0.191849
$426$	0.124986	0.00605560
$427$	4.45282	0.215487
$428$	−5.22695	−0.252654
$429$	−7.77994	−0.375619
$430$	−0.0337718	−0.00162862
$431$	−14.0621	−0.677347	−0.338674	−	0.940904i	$-0.609978\pi$
$431$	−14.0621	−0.677347	−0.338674	+	0.940904i	$0.609978\pi$
$432$	72.1009	3.46896
$433$	26.6923	1.28275	0.641375	−	0.767227i	$-0.278364\pi$
$433$	26.6923	1.28275	0.641375	+	0.767227i	$0.278364\pi$
$434$	−0.211391	−0.0101471
$435$	19.2453	0.922743
$436$	38.1911	1.82902
$437$	9.19214	0.439720
$438$	−0.376173	−0.0179743
$439$	5.87957	0.280616	0.140308	−	0.990108i	$-0.455191\pi$
$439$	5.87957	0.280616	0.140308	+	0.990108i	$0.455191\pi$
$440$	−0.198130	−0.00944546
$441$	−44.2067	−2.10508
$442$	1.03621	0.0492873
$443$	12.1211	0.575891	0.287945	−	0.957647i	$-0.407028\pi$
$443$	12.1211	0.575891	0.287945	+	0.957647i	$0.407028\pi$
$444$	−37.9370	−1.80041
$445$	10.5141	0.498416
$446$	−0.168752	−0.00799065
$447$	−25.9767	−1.22865
$448$	−10.3425	−0.488639
$449$	−20.4264	−0.963979	−0.481990	−	0.876177i	$-0.660086\pi$
$449$	−20.4264	−0.963979	−0.481990	+	0.876177i	$0.660086\pi$
$450$	0.629689	0.0296838
$451$	−8.01828	−0.377566
$452$	−38.0234	−1.78847
$453$	−50.6488	−2.37969
$454$	0.934190	0.0438437
$455$	−4.58888	−0.215130
$456$	3.91558	0.183364
$457$	31.1548	1.45736	0.728680	−	0.684854i	$-0.240134\pi$
$457$	31.1548	1.45736	0.728680	+	0.684854i	$0.240134\pi$
$458$	−1.07656	−0.0503043
$459$	71.8982	3.35592
$460$	−4.73796	−0.220909
$461$	−22.7307	−1.05867	−0.529337	−	0.848412i	$-0.677559\pi$
$461$	−22.7307	−1.05867	−0.529337	+	0.848412i	$0.677559\pi$
$462$	−0.220108	−0.0102403
$463$	−8.74901	−0.406601	−0.203301	−	0.979116i	$-0.565167\pi$
$463$	−8.74901	−0.406601	−0.203301	+	0.979116i	$0.565167\pi$
$464$	−22.6201	−1.05011
$465$	7.22529	0.335065
$466$	−1.65534	−0.0766821
$467$	2.01565	0.0932733	0.0466367	−	0.998912i	$-0.485150\pi$
$467$	2.01565	0.0932733	0.0466367	+	0.998912i	$0.485150\pi$
$468$	−58.3711	−2.69821
$469$	−7.12388	−0.328950
$470$	0.673724	0.0310766
$471$	−21.0486	−0.969866
$472$	−3.97897	−0.183147
$473$	−0.297188	−0.0136647
$474$	−0.332224	−0.0152595
$475$	−3.86927	−0.177534
$476$	−10.3725	−0.475423
$477$	77.3053	3.53957
$478$	−0.382341	−0.0174879
$479$	1.91080	0.0873066	0.0436533	−	0.999047i	$-0.486100\pi$
$479$	1.91080	0.0873066	0.0436533	+	0.999047i	$0.486100\pi$
$480$	−3.02878	−0.138244
$481$	19.6712	0.896931
$482$	−1.24422	−0.0566728
$483$	−10.5420	−0.479675
$484$	21.0675	0.957612
$485$	−7.11241	−0.322958
$486$	5.07236	0.230087
$487$	−10.6425	−0.482259	−0.241129	−	0.970493i	$-0.577518\pi$
$487$	−10.6425	−0.482259	−0.241129	+	0.970493i	$0.577518\pi$
$488$	−1.01547	−0.0459684
$489$	34.7181	1.57001
$490$	0.395719	0.0178768
$491$	−4.42579	−0.199733	−0.0998665	−	0.995001i	$-0.531842\pi$
$491$	−4.42579	−0.199733	−0.0998665	+	0.995001i	$0.531842\pi$
$492$	−81.6777	−3.68231
$493$	−22.5565	−1.01589
$494$	−1.01373	−0.0456098
$495$	5.54118	0.249058
$496$	−8.49227	−0.381314
$497$	0.648737	0.0290998
$498$	3.58737	0.160754
$499$	−18.1145	−0.810915	−0.405458	−	0.914114i	$-0.632888\pi$
$499$	−18.1145	−0.810915	−0.405458	+	0.914114i	$0.632888\pi$
$500$	1.99436	0.0891906
$501$	20.3577	0.909516
$502$	0.0779806	0.00348044
$503$	−24.7208	−1.10225	−0.551123	−	0.834424i	$-0.685800\pi$
$503$	−24.7208	−1.10225	−0.551123	+	0.834424i	$0.685800\pi$
$504$	−3.30750	−0.147328
$505$	5.58071	0.248338
$506$	0.117839	0.00523859
$507$	−2.77531	−0.123256
$508$	−41.4914	−1.84088
$509$	−35.9106	−1.59171	−0.795855	−	0.605488i	$-0.792978\pi$
$509$	−35.9106	−1.59171	−0.795855	+	0.605488i	$0.792978\pi$
$510$	−1.00201	−0.0443699
$511$	−1.95251	−0.0863742
$512$	5.93873	0.262457
$513$	−70.3386	−3.10553
$514$	−1.81036	−0.0798514
$515$	−8.86733	−0.390741
$516$	−3.02728	−0.133269
$517$	5.92868	0.260743
$518$	0.556532	0.0244526
$519$	−12.8465	−0.563899
$520$	1.04650	0.0458922
$521$	−19.0832	−0.836051	−0.418025	−	0.908435i	$-0.637278\pi$
$521$	−19.0832	−0.836051	−0.418025	+	0.908435i	$0.637278\pi$
$522$	−3.59124	−0.157184
$523$	−2.65416	−0.116058	−0.0580292	−	0.998315i	$-0.518482\pi$
$523$	−2.65416	−0.116058	−0.0580292	+	0.998315i	$0.518482\pi$
$524$	−2.34406	−0.102401
$525$	4.43745	0.193666
$526$	0.260909	0.0113762
$527$	−8.46839	−0.368889
$528$	−8.84244	−0.384818
$529$	−17.3562	−0.754615
$530$	−0.692004	−0.0300587
$531$	111.282	4.82922
$532$	10.1475	0.439950
$533$	42.3518	1.83446
$534$	−2.66374	−0.115271
$535$	−2.62086	−0.113310
$536$	1.62462	0.0701727
$537$	−60.1892	−2.59736
$538$	−0.624514	−0.0269247
$539$	3.48228	0.149992
$540$	36.2551	1.56017
$541$	20.8331	0.895685	0.447843	−	0.894112i	$-0.352192\pi$
$541$	20.8331	0.895685	0.447843	+	0.894112i	$0.352192\pi$
$542$	0.367923	0.0158036
$543$	−80.6382	−3.46052
$544$	3.54988	0.152200
$545$	19.1495	0.820275
$546$	1.16259	0.0497542
$547$	26.3486	1.12658	0.563292	−	0.826258i	$-0.309535\pi$
$547$	26.3486	1.12658	0.563292	+	0.826258i	$0.309535\pi$
$548$	35.8076	1.52963
$549$	28.4002	1.21209
$550$	−0.0496023	−0.00211505
$551$	22.0672	0.940094
$552$	2.40411	0.102326
$553$	−1.72440	−0.0733288
$554$	−1.00052	−0.0425080
$555$	−19.0221	−0.807443
$556$	2.49388	0.105764
$557$	−34.1922	−1.44877	−0.724385	−	0.689396i	$-0.757876\pi$
$557$	−34.1922	−1.44877	−0.724385	+	0.689396i	$0.757876\pi$
$558$	−1.34826	−0.0570765
$559$	1.56972	0.0663920
$560$	−5.21557	−0.220398
$561$	−8.81758	−0.372279
$562$	−0.165373	−0.00697582
$563$	39.7964	1.67722	0.838609	−	0.544734i	$-0.183369\pi$
$563$	39.7964	1.67722	0.838609	+	0.544734i	$0.183369\pi$
$564$	60.3921	2.54297
$565$	−19.0654	−0.802088
$566$	−1.57985	−0.0664061
$567$	47.5802	1.99818
$568$	−0.147946	−0.00620767
$569$	33.9810	1.42456	0.712279	−	0.701897i	$-0.247663\pi$
$569$	33.9810	1.42456	0.712279	+	0.701897i	$0.247663\pi$
$570$	0.980277	0.0410593
$571$	12.6258	0.528374	0.264187	−	0.964471i	$-0.414896\pi$
$571$	12.6258	0.528374	0.264187	+	0.964471i	$0.414896\pi$
$572$	4.59805	0.192254
$573$	−27.7801	−1.16053
$574$	1.19820	0.0500121
$575$	−2.37568	−0.0990726
$576$	−65.9651	−2.74855
$577$	−7.13070	−0.296855	−0.148427	−	0.988923i	$-0.547421\pi$
$577$	−7.13070	−0.296855	−0.148427	+	0.988923i	$0.547421\pi$
$578$	−0.101918	−0.00423923
$579$	−81.1028	−3.37052
$580$	−11.3742	−0.472290
$581$	18.6201	0.772493
$582$	1.80192	0.0746922
$583$	−6.08954	−0.252203
$584$	0.445275	0.0184256
$585$	−29.2680	−1.21008
$586$	1.03093	0.0425873
$587$	−27.2035	−1.12281	−0.561404	−	0.827542i	$-0.689739\pi$
$587$	−27.2035	−1.12281	−0.561404	+	0.827542i	$0.689739\pi$
$588$	35.4720	1.46284
$589$	8.28470	0.341365
$590$	−0.996147	−0.0410107
$591$	−44.0532	−1.81211
$592$	22.3577	0.918896
$593$	28.6199	1.17528	0.587638	−	0.809124i	$-0.300058\pi$
$593$	28.6199	1.17528	0.587638	+	0.809124i	$0.300058\pi$
$594$	−0.901710	−0.0369976
$595$	−5.20091	−0.213217
$596$	15.3526	0.628865
$597$	1.50112	0.0614367
$598$	−0.622416	−0.0254525
$599$	29.6563	1.21172	0.605862	−	0.795570i	$-0.292828\pi$
$599$	29.6563	1.21172	0.605862	+	0.795570i	$0.292828\pi$
$600$	−1.01197	−0.0413135
$601$	−7.50272	−0.306043	−0.153021	−	0.988223i	$-0.548900\pi$
$601$	−7.50272	−0.306043	−0.153021	+	0.988223i	$0.548900\pi$
$602$	0.0444100	0.00181002
$603$	−45.4364	−1.85031
$604$	29.9341	1.21800
$605$	10.5635	0.429468
$606$	−1.41387	−0.0574345
$607$	−17.8317	−0.723766	−0.361883	−	0.932223i	$-0.617866\pi$
$607$	−17.8317	−0.723766	−0.361883	+	0.932223i	$0.617866\pi$
$608$	−3.47288	−0.140844
$609$	−25.3076	−1.02552
$610$	−0.254227	−0.0102933
$611$	−31.3148	−1.26686
$612$	−66.1563	−2.67421
$613$	−3.26694	−0.131950	−0.0659752	−	0.997821i	$-0.521016\pi$
$613$	−3.26694	−0.131950	−0.0659752	+	0.997821i	$0.521016\pi$
$614$	−1.18808	−0.0479471
$615$	−40.9543	−1.65144
$616$	0.260541	0.0104975
$617$	0.721330	0.0290397	0.0145198	−	0.999895i	$-0.495378\pi$
$617$	0.721330	0.0290397	0.0145198	+	0.999895i	$0.495378\pi$
$618$	2.24653	0.0903688
$619$	26.3948	1.06090	0.530448	−	0.847717i	$-0.322024\pi$
$619$	26.3948	1.06090	0.530448	+	0.847717i	$0.322024\pi$
$620$	−4.27024	−0.171497
$621$	−43.1869	−1.73303
$622$	2.30456	0.0924043
$623$	−13.8261	−0.553929
$624$	46.7050	1.86970
$625$	1.00000	0.0400000
$626$	1.84374	0.0736906
$627$	8.62631	0.344502
$628$	12.4400	0.496409
$629$	22.2948	0.888953
$630$	−0.828043	−0.0329900
$631$	23.3748	0.930537	0.465268	−	0.885170i	$-0.345958\pi$
$631$	23.3748	0.930537	0.465268	+	0.885170i	$0.345958\pi$
$632$	0.393252	0.0156427
$633$	−27.1298	−1.07831
$634$	0.550637	0.0218686
$635$	−20.8044	−0.825596
$636$	−62.0308	−2.45968
$637$	−18.3931	−0.728760
$638$	0.282892	0.0111998
$639$	4.13767	0.163684
$640$	2.38560	0.0942990
$641$	−5.25472	−0.207549	−0.103774	−	0.994601i	$-0.533092\pi$
$641$	−5.25472	−0.207549	−0.103774	+	0.994601i	$0.533092\pi$
$642$	0.663994	0.0262058
$643$	−30.9857	−1.22196	−0.610978	−	0.791647i	$-0.709224\pi$
$643$	−30.9857	−1.22196	−0.610978	+	0.791647i	$0.709224\pi$
$644$	6.23043	0.245513
$645$	−1.51792	−0.0597680
$646$	−1.14893	−0.0452042
$647$	−46.2919	−1.81992	−0.909961	−	0.414694i	$-0.863889\pi$
$647$	−46.2919	−1.81992	−0.909961	+	0.414694i	$0.863889\pi$
$648$	−10.8508	−0.426258
$649$	−8.76596	−0.344094
$650$	0.261995	0.0102763
$651$	−9.50126	−0.372384
$652$	−20.5189	−0.803581
$653$	32.3371	1.26545	0.632725	−	0.774377i	$-0.281937\pi$
$653$	32.3371	1.26545	0.632725	+	0.774377i	$0.281937\pi$
$654$	−4.85152	−0.189710
$655$	−1.17534	−0.0459244
$656$	48.1358	1.87939
$657$	−12.4532	−0.485846
$658$	−0.885948	−0.0345378
$659$	44.4615	1.73197	0.865987	−	0.500067i	$-0.166691\pi$
$659$	44.4615	1.73197	0.865987	+	0.500067i	$0.166691\pi$
$660$	−4.44632	−0.173073
$661$	−26.3859	−1.02629	−0.513147	−	0.858301i	$-0.671520\pi$
$661$	−26.3859	−1.02629	−0.513147	+	0.858301i	$0.671520\pi$
$662$	1.45491	0.0565467
$663$	46.5737	1.80877
$664$	−4.24636	−0.164791
$665$	5.08809	0.197308
$666$	3.54959	0.137544
$667$	13.5490	0.524618
$668$	−12.0317	−0.465520
$669$	−7.58480	−0.293245
$670$	0.406727	0.0157132
$671$	−2.23716	−0.0863647
$672$	3.98285	0.153642
$673$	3.06950	0.118320	0.0591602	−	0.998249i	$-0.481158\pi$
$673$	3.06950	0.118320	0.0591602	+	0.998249i	$0.481158\pi$
$674$	0.382577	0.0147363
$675$	18.1788	0.699702
$676$	1.64024	0.0630862
$677$	37.2622	1.43210	0.716051	−	0.698048i	$-0.245948\pi$
$677$	37.2622	1.43210	0.716051	+	0.698048i	$0.245948\pi$
$678$	4.83022	0.185503
$679$	9.35282	0.358928
$680$	1.18608	0.0454841
$681$	41.9884	1.60900
$682$	0.106206	0.00406685
$683$	−3.41297	−0.130594	−0.0652969	−	0.997866i	$-0.520799\pi$
$683$	−3.41297	−0.130594	−0.0652969	+	0.997866i	$0.520799\pi$
$684$	64.7212	2.47468
$685$	17.9544	0.686003
$686$	−1.21147	−0.0462540
$687$	−48.3874	−1.84609
$688$	1.78409	0.0680179
$689$	32.1644	1.22537
$690$	0.601877	0.0229131
$691$	29.9798	1.14048	0.570242	−	0.821477i	$-0.306849\pi$
$691$	29.9798	1.14048	0.570242	+	0.821477i	$0.306849\pi$
$692$	7.59245	0.288622
$693$	−7.28667	−0.276798
$694$	1.01691	0.0386014
$695$	1.25046	0.0474328
$696$	5.77146	0.218767
$697$	48.0004	1.81815
$698$	0.792998	0.0300154
$699$	−74.4014	−2.81412
$700$	−2.62259	−0.0991246
$701$	26.1968	0.989439	0.494719	−	0.869053i	$-0.335271\pi$
$701$	26.1968	0.989439	0.494719	+	0.869053i	$0.335271\pi$
$702$	4.76275	0.179758
$703$	−21.8112	−0.822626
$704$	5.19625	0.195841
$705$	30.2814	1.14046
$706$	2.42511	0.0912703
$707$	−7.33864	−0.275998
$708$	−89.2939	−3.35587
$709$	−20.5891	−0.773241	−0.386621	−	0.922239i	$-0.626358\pi$
$709$	−20.5891	−0.773241	−0.386621	+	0.922239i	$0.626358\pi$
$710$	−0.0370386	−0.00139004
$711$	−10.9983	−0.412467
$712$	3.15306	0.118166
$713$	5.08669	0.190498
$714$	1.31765	0.0493117
$715$	2.30552	0.0862216
$716$	35.5726	1.32941
$717$	−17.1848	−0.641779
$718$	2.15473	0.0804139
$719$	34.6411	1.29189	0.645947	−	0.763383i	$-0.276463\pi$
$719$	34.6411	1.29189	0.645947	+	0.763383i	$0.276463\pi$
$720$	−33.2651	−1.23972
$721$	11.6605	0.434262
$722$	−0.302471	−0.0112568
$723$	−55.9232	−2.07981
$724$	47.6582	1.77120
$725$	−5.70320	−0.211811
$726$	−2.67626	−0.0993253
$727$	−32.8528	−1.21844	−0.609222	−	0.793000i	$-0.708518\pi$
$727$	−32.8528	−1.21844	−0.609222	+	0.793000i	$0.708518\pi$
$728$	−1.37615	−0.0510036
$729$	119.436	4.42357
$730$	0.111476	0.00412591
$731$	1.77908	0.0658015
$732$	−22.7887	−0.842295
$733$	−48.3584	−1.78616	−0.893079	−	0.449901i	$-0.851459\pi$
$733$	−48.3584	−1.78616	−0.893079	+	0.449901i	$0.851459\pi$
$734$	1.57020	0.0579571
$735$	17.7861	0.656051
$736$	−2.13230	−0.0785975
$737$	3.57914	0.131839
$738$	7.64220	0.281313
$739$	−8.53062	−0.313804	−0.156902	−	0.987614i	$-0.550151\pi$
$739$	−8.53062	−0.313804	−0.156902	+	0.987614i	$0.550151\pi$
$740$	11.2423	0.413275
$741$	−45.5634	−1.67381
$742$	0.909986	0.0334066
$743$	−21.3555	−0.783457	−0.391729	−	0.920081i	$-0.628123\pi$
$743$	−21.3555	−0.783457	−0.391729	+	0.920081i	$0.628123\pi$
$744$	2.16678	0.0794382
$745$	7.69797	0.282032
$746$	0.797542	0.0292001
$747$	118.760	4.34520
$748$	5.21131	0.190544
$749$	3.44644	0.125930
$750$	−0.253350	−0.00925102
$751$	−15.2422	−0.556195	−0.278097	−	0.960553i	$-0.589704\pi$
$751$	−15.2422	−0.556195	−0.278097	+	0.960553i	$0.589704\pi$
$752$	−35.5914	−1.29788
$753$	3.50494	0.127727
$754$	−1.49421	−0.0544159
$755$	15.0094	0.546246
$756$	−47.6755	−1.73394
$757$	−40.5967	−1.47551	−0.737757	−	0.675066i	$-0.764115\pi$
$757$	−40.5967	−1.47551	−0.737757	+	0.675066i	$0.764115\pi$
$758$	−2.51708	−0.0914244
$759$	5.29644	0.192248
$760$	−1.16035	−0.0420904
$761$	−13.6897	−0.496253	−0.248126	−	0.968728i	$-0.579815\pi$
$761$	−13.6897	−0.496253	−0.248126	+	0.968728i	$0.579815\pi$
$762$	5.27077	0.190940
$763$	−25.1816	−0.911637
$764$	16.4184	0.593997
$765$	−33.1716	−1.19932
$766$	−1.55732	−0.0562684
$767$	46.3010	1.67183
$768$	52.4765	1.89358
$769$	−30.8769	−1.11345	−0.556726	−	0.830696i	$-0.687943\pi$
$769$	−30.8769	−1.11345	−0.556726	+	0.830696i	$0.687943\pi$
$770$	0.0652271	0.00235062
$771$	−81.3689	−2.93043
$772$	47.9328	1.72514
$773$	−22.7443	−0.818056	−0.409028	−	0.912522i	$-0.634132\pi$
$773$	−22.7443	−0.818056	−0.409028	+	0.912522i	$0.634132\pi$
$774$	0.283249	0.0101812
$775$	−2.14115	−0.0769125
$776$	−2.13293	−0.0765678
$777$	25.0141	0.897375
$778$	−1.46015	−0.0523488
$779$	−46.9592	−1.68249
$780$	23.4851	0.840900
$781$	−0.325935	−0.0116629
$782$	−0.705429	−0.0252261
$783$	−103.677	−3.70512
$784$	−20.9050	−0.746607
$785$	6.23756	0.222628
$786$	0.297773	0.0106212
$787$	12.9018	0.459899	0.229950	−	0.973203i	$-0.426144\pi$
$787$	12.9018	0.459899	0.229950	+	0.973203i	$0.426144\pi$
$788$	26.0360	0.927494
$789$	11.7269	0.417489
$790$	0.0984518	0.00350276
$791$	25.0711	0.891424
$792$	1.66174	0.0590474
$793$	11.8165	0.419616
$794$	1.18577	0.0420813
$795$	−31.1030	−1.10311
$796$	−0.887181	−0.0314453
$797$	−7.33725	−0.259899	−0.129949	−	0.991521i	$-0.541481\pi$
$797$	−7.33725	−0.259899	−0.129949	+	0.991521i	$0.541481\pi$
$798$	−1.28907	−0.0456325
$799$	−35.4913	−1.25559
$800$	0.897553	0.0317333
$801$	−88.1831	−3.11580
$802$	0.0750780	0.00265110
$803$	0.980973	0.0346178
$804$	36.4587	1.28580
$805$	3.12402	0.110107
$806$	−0.560972	−0.0197594
$807$	−28.0696	−0.988097
$808$	1.67359	0.0588768
$809$	17.1584	0.603259	0.301629	−	0.953425i	$-0.402469\pi$
$809$	17.1584	0.603259	0.301629	+	0.953425i	$0.402469\pi$
$810$	−2.71652	−0.0954487
$811$	28.8541	1.01320	0.506602	−	0.862180i	$-0.330902\pi$
$811$	28.8541	1.01320	0.506602	+	0.862180i	$0.330902\pi$
$812$	14.9571	0.524893
$813$	16.5368	0.579970
$814$	−0.279610	−0.00980034
$815$	−10.2884	−0.360388
$816$	52.9342	1.85307
$817$	−1.74049	−0.0608919
$818$	−0.368595	−0.0128876
$819$	38.4875	1.34486
$820$	24.2045	0.845258
$821$	−10.8745	−0.379524	−0.189762	−	0.981830i	$-0.560772\pi$
$821$	−10.8745	−0.379524	−0.189762	+	0.981830i	$0.560772\pi$
$822$	−4.54874	−0.158656
$823$	−15.5332	−0.541455	−0.270727	−	0.962656i	$-0.587264\pi$
$823$	−15.5332	−0.541455	−0.270727	+	0.962656i	$0.587264\pi$
$824$	−2.65921	−0.0926381
$825$	−2.22944	−0.0776192
$826$	1.30993	0.0455784
$827$	56.9437	1.98013	0.990063	−	0.140627i	$-0.0449119\pi$
$827$	56.9437	1.98013	0.990063	+	0.140627i	$0.0449119\pi$
$828$	39.7379	1.38099
$829$	−25.8445	−0.897617	−0.448808	−	0.893628i	$-0.648151\pi$
$829$	−25.8445	−0.897617	−0.448808	+	0.893628i	$0.648151\pi$
$830$	−1.06309	−0.0369003
$831$	−44.9697	−1.55998
$832$	−27.4461	−0.951523
$833$	−20.8462	−0.722279
$834$	−0.316804	−0.0109700
$835$	−6.03284	−0.208775
$836$	−5.09826	−0.176327
$837$	−38.9236	−1.34540
$838$	0.529371	0.0182868
$839$	8.82629	0.304717	0.152359	−	0.988325i	$-0.451313\pi$
$839$	8.82629	0.304717	0.152359	+	0.988325i	$0.451313\pi$
$840$	1.33074	0.0459150
$841$	3.52645	0.121602
$842$	−0.373555	−0.0128735
$843$	−7.43289	−0.256002
$844$	16.0341	0.551915
$845$	0.822439	0.0282928
$846$	−5.65061	−0.194272
$847$	−13.8910	−0.477301
$848$	36.5571	1.25538
$849$	−71.0085	−2.43701
$850$	0.296938	0.0101849
$851$	−13.3918	−0.459065
$852$	−3.32012	−0.113745
$853$	31.1146	1.06535	0.532673	−	0.846321i	$-0.321188\pi$
$853$	31.1146	1.06535	0.532673	+	0.846321i	$0.321188\pi$
$854$	0.334309	0.0114398
$855$	32.4521	1.10984
$856$	−0.785968	−0.0268638
$857$	−2.36523	−0.0807948	−0.0403974	−	0.999184i	$-0.512862\pi$
$857$	−2.36523	−0.0807948	−0.0403974	+	0.999184i	$0.512862\pi$
$858$	−0.584103	−0.0199409
$859$	15.1802	0.517943	0.258972	−	0.965885i	$-0.416616\pi$
$859$	15.1802	0.517943	0.258972	+	0.965885i	$0.416616\pi$
$860$	0.897110	0.0305912
$861$	53.8549	1.83537
$862$	−1.05575	−0.0359591
$863$	−3.09289	−0.105283	−0.0526415	−	0.998613i	$-0.516764\pi$
$863$	−3.09289	−0.105283	−0.0526415	+	0.998613i	$0.516764\pi$
$864$	16.3164	0.555096
$865$	3.80696	0.129440
$866$	2.00401	0.0680989
$867$	−4.58084	−0.155574
$868$	5.61537	0.190598
$869$	0.866363	0.0293893
$870$	1.44490	0.0489868
$871$	−18.9047	−0.640562
$872$	5.74273	0.194473
$873$	59.6527	2.01894
$874$	0.690128	0.0233439
$875$	−1.31500	−0.0444552
$876$	9.99262	0.337619
$877$	−28.4945	−0.962192	−0.481096	−	0.876668i	$-0.659761\pi$
$877$	−28.4945	−0.962192	−0.481096	+	0.876668i	$0.659761\pi$
$878$	0.441426	0.0148974
$879$	46.3365	1.56289
$880$	2.62038	0.0883331
$881$	29.4672	0.992777	0.496388	−	0.868101i	$-0.334659\pi$
$881$	29.4672	0.992777	0.496388	+	0.868101i	$0.334659\pi$
$882$	−3.31895	−0.111755
$883$	14.3947	0.484421	0.242211	−	0.970224i	$-0.422127\pi$
$883$	14.3947	0.484421	0.242211	+	0.970224i	$0.422127\pi$
$884$	−27.5256	−0.925788
$885$	−44.7731	−1.50503
$886$	0.910028	0.0305730
$887$	−12.5539	−0.421519	−0.210759	−	0.977538i	$-0.567594\pi$
$887$	−12.5539	−0.421519	−0.210759	+	0.977538i	$0.567594\pi$
$888$	−5.70452	−0.191431
$889$	27.3578	0.917550
$890$	0.789378	0.0264600
$891$	−23.9050	−0.800848
$892$	4.48272	0.150092
$893$	34.7215	1.16191
$894$	−1.95028	−0.0652271
$895$	17.8366	0.596211
$896$	−3.13706	−0.104802
$897$	−27.9753	−0.934068
$898$	−1.53357	−0.0511759
$899$	12.2114	0.407274
$900$	−16.7270	−0.557566
$901$	36.4543	1.21447
$902$	−0.601996	−0.0200443
$903$	1.99607	0.0664249
$904$	−5.71751	−0.190162
$905$	23.8965	0.794345
$906$	−3.80261	−0.126333
$907$	51.5605	1.71204	0.856019	−	0.516944i	$-0.172931\pi$
$907$	51.5605	1.71204	0.856019	+	0.516944i	$0.172931\pi$
$908$	−24.8157	−0.823538
$909$	−46.8061	−1.55246
$910$	−0.344524	−0.0114209
$911$	29.1677	0.966369	0.483185	−	0.875518i	$-0.339480\pi$
$911$	29.1677	0.966369	0.483185	+	0.875518i	$0.339480\pi$
$912$	−51.7860	−1.71480
$913$	−9.35504	−0.309607
$914$	2.33904	0.0773687
$915$	−11.4266	−0.377751
$916$	28.5976	0.944891
$917$	1.54558	0.0510395
$918$	5.39798	0.178160
$919$	−55.8557	−1.84251	−0.921254	−	0.388961i	$-0.872834\pi$
$919$	−55.8557	−1.84251	−0.921254	+	0.388961i	$0.872834\pi$
$920$	−0.712439	−0.0234884
$921$	−53.3999	−1.75959
$922$	−1.70658	−0.0562031
$923$	1.72156	0.0566659
$924$	5.84692	0.192349
$925$	5.63704	0.185345
$926$	−0.656859	−0.0215857
$927$	74.3714	2.44268
$928$	−5.11892	−0.168037
$929$	50.3171	1.65085	0.825425	−	0.564512i	$-0.190935\pi$
$929$	50.3171	1.65085	0.825425	+	0.564512i	$0.190935\pi$
$930$	0.542460	0.0177880
$931$	20.3940	0.668387
$932$	43.9722	1.44036
$933$	103.581	3.39110
$934$	0.151331	0.00495171
$935$	2.61302	0.0854548
$936$	−8.77716	−0.286891
$937$	13.2660	0.433381	0.216690	−	0.976240i	$-0.430474\pi$
$937$	13.2660	0.433381	0.216690	+	0.976240i	$0.430474\pi$
$938$	−0.534847	−0.0174634
$939$	82.8693	2.70434
$940$	−17.8967	−0.583726
$941$	46.7272	1.52326	0.761632	−	0.648010i	$-0.224398\pi$
$941$	46.7272	1.52326	0.761632	+	0.648010i	$0.224398\pi$
$942$	−1.58028	−0.0514884
$943$	−28.8323	−0.938909
$944$	52.6243	1.71277
$945$	−23.9051	−0.777634
$946$	−0.0223123	−0.000725434 0
$947$	17.2847	0.561677	0.280839	−	0.959755i	$-0.409387\pi$
$947$	17.2847	0.561677	0.280839	+	0.959755i	$0.409387\pi$
$948$	8.82515	0.286628
$949$	−5.18141	−0.168196
$950$	−0.290497	−0.00942497
$951$	24.7491	0.802545
$952$	−1.55970	−0.0505500
$953$	−20.8440	−0.675205	−0.337602	−	0.941289i	$-0.609616\pi$
$953$	−20.8440	−0.675205	−0.337602	+	0.941289i	$0.609616\pi$
$954$	5.80393	0.187909
$955$	8.23241	0.266394
$956$	10.1565	0.328483
$957$	12.7149	0.411016
$958$	0.143459	0.00463495
$959$	−23.6101	−0.762409
$960$	26.5404	0.856589
$961$	−26.4155	−0.852112
$962$	1.47688	0.0476164
$963$	21.9815	0.708345
$964$	33.0514	1.06451
$965$	24.0341	0.773687
$966$	−0.791469	−0.0254651
$967$	−39.8351	−1.28101	−0.640505	−	0.767954i	$-0.721275\pi$
$967$	−39.8351	−1.28101	−0.640505	+	0.767954i	$0.721275\pi$
$968$	3.16788	0.101819
$969$	−51.6404	−1.65893
$970$	−0.533985	−0.0171452
$971$	21.4739	0.689131	0.344566	−	0.938762i	$-0.388026\pi$
$971$	21.4739	0.689131	0.344566	+	0.938762i	$0.388026\pi$
$972$	−134.742	−4.32184
$973$	−1.64436	−0.0527158
$974$	−0.799020	−0.0256022
$975$	11.7757	0.377125
$976$	13.4303	0.429892
$977$	−8.47283	−0.271070	−0.135535	−	0.990773i	$-0.543275\pi$
$977$	−8.47283	−0.271070	−0.135535	+	0.990773i	$0.543275\pi$
$978$	2.60657	0.0833488
$979$	6.94642	0.222009
$980$	−10.5118	−0.335788
$981$	−160.610	−5.12787
$982$	−0.332280	−0.0106035
$983$	27.3091	0.871026	0.435513	−	0.900183i	$-0.356567\pi$
$983$	27.3091	0.871026	0.435513	+	0.900183i	$0.356567\pi$
$984$	−12.2817	−0.391527
$985$	13.0548	0.415960
$986$	−1.69350	−0.0539319
$987$	−39.8201	−1.26749
$988$	26.9286	0.856712
$989$	−1.06863	−0.0339806
$990$	0.416021	0.0132220
$991$	−23.1409	−0.735093	−0.367547	−	0.930005i	$-0.619802\pi$
$991$	−23.1409	−0.735093	−0.367547	+	0.930005i	$0.619802\pi$
$992$	−1.92180	−0.0610172
$993$	65.3929	2.07518
$994$	0.0487059	0.00154486
$995$	−0.444844	−0.0141025
$996$	−95.2945	−3.01952
$997$	17.4921	0.553981	0.276991	−	0.960873i	$-0.410663\pi$
$997$	17.4921	0.553981	0.276991	+	0.960873i	$0.410663\pi$
$998$	−1.36000	−0.0430500
$999$	102.475	3.24215

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.f.1.18	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.f.1.18	✓	37	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:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2005.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0750780 q^{2} +3.37448 q^{3} -1.99436 q^{4} -1.00000 q^{5} +0.253350 q^{6} +1.31500 q^{7} -0.299889 q^{8} +8.38713 q^{9} +O(q^{10})\)	\(q+0.0750780 q^{2} +3.37448 q^{3} -1.99436 q^{4} -1.00000 q^{5} +0.253350 q^{6} +1.31500 q^{7} -0.299889 q^{8} +8.38713 q^{9} -0.0750780 q^{10} -0.660677 q^{11} -6.72994 q^{12} +3.48964 q^{13} +0.0987277 q^{14} -3.37448 q^{15} +3.96621 q^{16} +3.95506 q^{17} +0.629689 q^{18} -3.86927 q^{19} +1.99436 q^{20} +4.43745 q^{21} -0.0496023 q^{22} -2.37568 q^{23} -1.01197 q^{24} +1.00000 q^{25} +0.261995 q^{26} +18.1788 q^{27} -2.62259 q^{28} -5.70320 q^{29} -0.253350 q^{30} -2.14115 q^{31} +0.897553 q^{32} -2.22944 q^{33} +0.296938 q^{34} -1.31500 q^{35} -16.7270 q^{36} +5.63704 q^{37} -0.290497 q^{38} +11.7757 q^{39} +0.299889 q^{40} +12.1365 q^{41} +0.333155 q^{42} +0.449823 q^{43} +1.31763 q^{44} -8.38713 q^{45} -0.178361 q^{46} -8.97365 q^{47} +13.3839 q^{48} -5.27077 q^{49} +0.0750780 q^{50} +13.3463 q^{51} -6.95960 q^{52} +9.21713 q^{53} +1.36483 q^{54} +0.660677 q^{55} -0.394354 q^{56} -13.0568 q^{57} -0.428185 q^{58} +13.2681 q^{59} +6.72994 q^{60} +3.38617 q^{61} -0.160754 q^{62} +11.0291 q^{63} -7.86504 q^{64} -3.48964 q^{65} -0.167382 q^{66} -5.41739 q^{67} -7.88783 q^{68} -8.01668 q^{69} -0.0987277 q^{70} +0.493335 q^{71} -2.51521 q^{72} -1.48480 q^{73} +0.423218 q^{74} +3.37448 q^{75} +7.71673 q^{76} -0.868791 q^{77} +0.884098 q^{78} -1.31133 q^{79} -3.96621 q^{80} +36.1826 q^{81} +0.911181 q^{82} +14.1598 q^{83} -8.84989 q^{84} -3.95506 q^{85} +0.0337718 q^{86} -19.2453 q^{87} +0.198130 q^{88} -10.5141 q^{89} -0.629689 q^{90} +4.58888 q^{91} +4.73796 q^{92} -7.22529 q^{93} -0.673724 q^{94} +3.86927 q^{95} +3.02878 q^{96} +7.11241 q^{97} -0.395719 q^{98} -5.54118 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9}+O(q^{10}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9} - 7 q^{10} + 42 q^{11} - 13 q^{13} + 14 q^{14} - 3 q^{15} + 63 q^{16} + 18 q^{17} + 22 q^{18} + 22 q^{19} - 43 q^{20} + 16 q^{21} - 10 q^{22} + 23 q^{23} + 23 q^{24} + 37 q^{25} + 21 q^{26} + 3 q^{27} - 18 q^{28} + 33 q^{29} - 8 q^{30} + 11 q^{31} + 54 q^{32} + 2 q^{33} + 8 q^{34} + 16 q^{35} + 91 q^{36} - 11 q^{37} + 29 q^{38} + 25 q^{39} - 21 q^{40} + 24 q^{41} + 4 q^{42} + 25 q^{43} + 84 q^{44} - 54 q^{45} + 31 q^{46} + 7 q^{47} + 4 q^{48} + 45 q^{49} + 7 q^{50} + 94 q^{51} - 43 q^{52} + 49 q^{53} + 38 q^{54} - 42 q^{55} + 46 q^{56} + 6 q^{57} + 15 q^{58} + 69 q^{59} + 9 q^{61} + 17 q^{62} - 38 q^{63} + 107 q^{64} + 13 q^{65} + 74 q^{66} + 13 q^{67} + 86 q^{68} - 14 q^{70} + 51 q^{71} + 81 q^{72} - 47 q^{73} + 79 q^{74} + 3 q^{75} + 59 q^{76} + 2 q^{77} + 20 q^{78} + 67 q^{79} - 63 q^{80} + 125 q^{81} - 24 q^{82} + 80 q^{83} + 50 q^{84} - 18 q^{85} + 69 q^{86} - 32 q^{87} - 12 q^{88} + 34 q^{89} - 22 q^{90} + 39 q^{91} + 85 q^{92} + q^{93} + 12 q^{94} - 22 q^{95} + 77 q^{96} - 14 q^{97} + 40 q^{98} + 133 q^{99}+O(q^{100}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9 - 7 * q^10 + 42 * q^11 - 13 * q^13 + 14 * q^14 - 3 * q^15 + 63 * q^16 + 18 * q^17 + 22 * q^18 + 22 * q^19 - 43 * q^20 + 16 * q^21 - 10 * q^22 + 23 * q^23 + 23 * q^24 + 37 * q^25 + 21 * q^26 + 3 * q^27 - 18 * q^28 + 33 * q^29 - 8 * q^30 + 11 * q^31 + 54 * q^32 + 2 * q^33 + 8 * q^34 + 16 * q^35 + 91 * q^36 - 11 * q^37 + 29 * q^38 + 25 * q^39 - 21 * q^40 + 24 * q^41 + 4 * q^42 + 25 * q^43 + 84 * q^44 - 54 * q^45 + 31 * q^46 + 7 * q^47 + 4 * q^48 + 45 * q^49 + 7 * q^50 + 94 * q^51 - 43 * q^52 + 49 * q^53 + 38 * q^54 - 42 * q^55 + 46 * q^56 + 6 * q^57 + 15 * q^58 + 69 * q^59 + 9 * q^61 + 17 * q^62 - 38 * q^63 + 107 * q^64 + 13 * q^65 + 74 * q^66 + 13 * q^67 + 86 * q^68 - 14 * q^70 + 51 * q^71 + 81 * q^72 - 47 * q^73 + 79 * q^74 + 3 * q^75 + 59 * q^76 + 2 * q^77 + 20 * q^78 + 67 * q^79 - 63 * q^80 + 125 * q^81 - 24 * q^82 + 80 * q^83 + 50 * q^84 - 18 * q^85 + 69 * q^86 - 32 * q^87 - 12 * q^88 + 34 * q^89 - 22 * q^90 + 39 * q^91 + 85 * q^92 + q^93 + 12 * q^94 - 22 * q^95 + 77 * q^96 - 14 * q^97 + 40 * q^98 + 133 * q^99

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