LMFDB - Embedded newform 4031.2.a.c.1.33

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, 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("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$32.1876970548$
Analytic rank:	$1$
Dimension:	$61$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.33
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.138106 q^{2} -0.467503 q^{3} -1.98093 q^{4} -3.97741 q^{5} -0.0645649 q^{6} +0.0696388 q^{7} -0.549789 q^{8} -2.78144 q^{9} +O(q^{10})$	\(q+0.138106 q^{2} -0.467503 q^{3} -1.98093 q^{4} -3.97741 q^{5} -0.0645649 q^{6} +0.0696388 q^{7} -0.549789 q^{8} -2.78144 q^{9} -0.549304 q^{10} +1.78259 q^{11} +0.926089 q^{12} -4.67947 q^{13} +0.00961753 q^{14} +1.85945 q^{15} +3.88592 q^{16} -2.31017 q^{17} -0.384133 q^{18} +8.18821 q^{19} +7.87896 q^{20} -0.0325564 q^{21} +0.246186 q^{22} +5.67889 q^{23} +0.257028 q^{24} +10.8198 q^{25} -0.646262 q^{26} +2.70284 q^{27} -0.137949 q^{28} -1.00000 q^{29} +0.256801 q^{30} -0.155832 q^{31} +1.63625 q^{32} -0.833365 q^{33} -0.319047 q^{34} -0.276982 q^{35} +5.50983 q^{36} +3.98674 q^{37} +1.13084 q^{38} +2.18767 q^{39} +2.18674 q^{40} -4.18466 q^{41} -0.00449622 q^{42} +0.605145 q^{43} -3.53118 q^{44} +11.0629 q^{45} +0.784287 q^{46} +5.16981 q^{47} -1.81668 q^{48} -6.99515 q^{49} +1.49428 q^{50} +1.08001 q^{51} +9.26969 q^{52} -1.19869 q^{53} +0.373278 q^{54} -7.09009 q^{55} -0.0382867 q^{56} -3.82801 q^{57} -0.138106 q^{58} -1.10372 q^{59} -3.68344 q^{60} +5.63615 q^{61} -0.0215212 q^{62} -0.193696 q^{63} -7.54587 q^{64} +18.6122 q^{65} -0.115093 q^{66} -10.9328 q^{67} +4.57627 q^{68} -2.65490 q^{69} -0.0382529 q^{70} +13.1365 q^{71} +1.52921 q^{72} -9.97887 q^{73} +0.550591 q^{74} -5.05829 q^{75} -16.2202 q^{76} +0.124137 q^{77} +0.302129 q^{78} -8.85133 q^{79} -15.4559 q^{80} +7.08074 q^{81} -0.577926 q^{82} -8.17936 q^{83} +0.0644918 q^{84} +9.18848 q^{85} +0.0835740 q^{86} +0.467503 q^{87} -0.980048 q^{88} +3.01533 q^{89} +1.52786 q^{90} -0.325873 q^{91} -11.2495 q^{92} +0.0728517 q^{93} +0.713981 q^{94} -32.5679 q^{95} -0.764950 q^{96} +11.2107 q^{97} -0.966071 q^{98} -4.95817 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) $ 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * 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 $

$2$	0.138106	0.0976555	0.0488278	−	0.998807i	$-0.484451\pi$
$2$	0.138106	0.0976555	0.0488278	+	0.998807i	$0.484451\pi$
$3$	−0.467503	−0.269913	−0.134956	−	0.990852i	$-0.543089\pi$
$3$	−0.467503	−0.269913	−0.134956	+	0.990852i	$0.543089\pi$
$4$	−1.98093	−0.990463
$5$	−3.97741	−1.77875	−0.889376	−	0.457176i	$-0.848861\pi$
$5$	−3.97741	−1.77875	−0.889376	+	0.457176i	$0.848861\pi$
$6$	−0.0645649	−0.0263585
$7$	0.0696388	0.0263210	0.0131605	−	0.999913i	$-0.495811\pi$
$7$	0.0696388	0.0263210	0.0131605	+	0.999913i	$0.495811\pi$
$8$	−0.549789	−0.194380
$9$	−2.78144	−0.927147
$10$	−0.549304	−0.173705
$11$	1.78259	0.537471	0.268735	−	0.963214i	$-0.413394\pi$
$11$	1.78259	0.537471	0.268735	+	0.963214i	$0.413394\pi$
$12$	0.926089	0.267339
$13$	−4.67947	−1.29785	−0.648926	−	0.760852i	$-0.724781\pi$
$13$	−4.67947	−1.29785	−0.648926	+	0.760852i	$0.724781\pi$
$14$	0.00961753	0.00257039
$15$	1.85945	0.480108
$16$	3.88592	0.971481
$17$	−2.31017	−0.560298	−0.280149	−	0.959957i	$-0.590384\pi$
$17$	−2.31017	−0.560298	−0.280149	+	0.959957i	$0.590384\pi$
$18$	−0.384133	−0.0905410
$19$	8.18821	1.87850	0.939252	−	0.343229i	$-0.111521\pi$
$19$	8.18821	1.87850	0.939252	+	0.343229i	$0.111521\pi$
$20$	7.87896	1.76179
$21$	−0.0325564	−0.00710438
$22$	0.246186	0.0524870
$23$	5.67889	1.18413	0.592065	−	0.805890i	$-0.298313\pi$
$23$	5.67889	1.18413	0.592065	+	0.805890i	$0.298313\pi$
$24$	0.257028	0.0524656
$25$	10.8198	2.16396
$26$	−0.646262	−0.126742
$27$	2.70284	0.520162
$28$	−0.137949	−0.0260700
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0.256801	0.0468852
$31$	−0.155832	−0.0279882	−0.0139941	−	0.999902i	$-0.504455\pi$
$31$	−0.155832	−0.0279882	−0.0139941	+	0.999902i	$0.504455\pi$
$32$	1.63625	0.289250
$33$	−0.833365	−0.145070
$34$	−0.319047	−0.0547162
$35$	−0.276982	−0.0468186
$36$	5.50983	0.918305
$37$	3.98674	0.655415	0.327708	−	0.944779i	$-0.393724\pi$
$37$	3.98674	0.655415	0.327708	+	0.944779i	$0.393724\pi$
$38$	1.13084	0.183446
$39$	2.18767	0.350307
$40$	2.18674	0.345754
$41$	−4.18466	−0.653534	−0.326767	−	0.945105i	$-0.605959\pi$
$41$	−4.18466	−0.653534	−0.326767	+	0.945105i	$0.605959\pi$
$42$	−0.00449622	−0.000693782 0
$43$	0.605145	0.0922838	0.0461419	−	0.998935i	$-0.485307\pi$
$43$	0.605145	0.0922838	0.0461419	+	0.998935i	$0.485307\pi$
$44$	−3.53118	−0.532345
$45$	11.0629	1.64917
$46$	0.784287	0.115637
$47$	5.16981	0.754095	0.377047	−	0.926194i	$-0.376939\pi$
$47$	5.16981	0.754095	0.377047	+	0.926194i	$0.376939\pi$
$48$	−1.81668	−0.262215
$49$	−6.99515	−0.999307
$50$	1.49428	0.211323
$51$	1.08001	0.151232
$52$	9.26969	1.28547
$53$	−1.19869	−0.164652	−0.0823261	−	0.996605i	$-0.526235\pi$
$53$	−1.19869	−0.164652	−0.0823261	+	0.996605i	$0.526235\pi$
$54$	0.373278	0.0507967
$55$	−7.09009	−0.956028
$56$	−0.0382867	−0.00511627
$57$	−3.82801	−0.507032
$58$	−0.138106	−0.0181342
$59$	−1.10372	−0.143693	−0.0718463	−	0.997416i	$-0.522889\pi$
$59$	−1.10372	−0.143693	−0.0718463	+	0.997416i	$0.522889\pi$
$60$	−3.68344	−0.475530
$61$	5.63615	0.721635	0.360818	−	0.932636i	$-0.382498\pi$
$61$	5.63615	0.721635	0.360818	+	0.932636i	$0.382498\pi$
$62$	−0.0215212	−0.00273320
$63$	−0.193696	−0.0244034
$64$	−7.54587	−0.943234
$65$	18.6122	2.30856
$66$	−0.115093	−0.0141669
$67$	−10.9328	−1.33566	−0.667829	−	0.744314i	$-0.732776\pi$
$67$	−10.9328	−1.33566	−0.667829	+	0.744314i	$0.732776\pi$
$68$	4.57627	0.554954
$69$	−2.65490	−0.319612
$70$	−0.0382529	−0.00457209
$71$	13.1365	1.55901	0.779506	−	0.626395i	$-0.215470\pi$
$71$	13.1365	1.55901	0.779506	+	0.626395i	$0.215470\pi$
$72$	1.52921	0.180219
$73$	−9.97887	−1.16794	−0.583969	−	0.811776i	$-0.698501\pi$
$73$	−9.97887	−1.16794	−0.583969	+	0.811776i	$0.698501\pi$
$74$	0.550591	0.0640049
$75$	−5.05829	−0.584081
$76$	−16.2202	−1.86059
$77$	0.124137	0.0141468
$78$	0.302129	0.0342094
$79$	−8.85133	−0.995852	−0.497926	−	0.867219i	$-0.665905\pi$
$79$	−8.85133	−0.995852	−0.497926	+	0.867219i	$0.665905\pi$
$80$	−15.4559	−1.72802
$81$	7.08074	0.786749
$82$	−0.577926	−0.0638212
$83$	−8.17936	−0.897801	−0.448900	−	0.893582i	$-0.648184\pi$
$83$	−8.17936	−0.897801	−0.448900	+	0.893582i	$0.648184\pi$
$84$	0.0644918	0.00703663
$85$	9.18848	0.996631
$86$	0.0835740	0.00901202
$87$	0.467503	0.0501216
$88$	−0.980048	−0.104473
$89$	3.01533	0.319624	0.159812	−	0.987147i	$-0.448911\pi$
$89$	3.01533	0.319624	0.159812	+	0.987147i	$0.448911\pi$
$90$	1.52786	0.161050
$91$	−0.325873	−0.0341608
$92$	−11.2495	−1.17284
$93$	0.0728517	0.00755437
$94$	0.713981	0.0736415
$95$	−32.5679	−3.34139
$96$	−0.764950	−0.0780724
$97$	11.2107	1.13828	0.569139	−	0.822242i	$-0.307277\pi$
$97$	11.2107	1.13828	0.569139	+	0.822242i	$0.307277\pi$
$98$	−0.966071	−0.0975879
$99$	−4.95817	−0.498314
$100$	−21.4332	−2.14332
$101$	18.4039	1.83126	0.915628	−	0.402026i	$-0.131694\pi$
$101$	18.4039	1.83126	0.915628	+	0.402026i	$0.131694\pi$
$102$	0.149156	0.0147686
$103$	−1.66062	−0.163626	−0.0818130	−	0.996648i	$-0.526071\pi$
$103$	−1.66062	−0.163626	−0.0818130	+	0.996648i	$0.526071\pi$
$104$	2.57272	0.252276
$105$	0.129490	0.0126369
$106$	−0.165545	−0.0160792
$107$	−9.00293	−0.870346	−0.435173	−	0.900347i	$-0.643313\pi$
$107$	−9.00293	−0.870346	−0.435173	+	0.900347i	$0.643313\pi$
$108$	−5.35413	−0.515201
$109$	−15.8211	−1.51538	−0.757692	−	0.652612i	$-0.773673\pi$
$109$	−15.8211	−1.51538	−0.757692	+	0.652612i	$0.773673\pi$
$110$	−0.979183	−0.0933614
$111$	−1.86381	−0.176905
$112$	0.270611	0.0255704
$113$	8.84813	0.832362	0.416181	−	0.909282i	$-0.363368\pi$
$113$	8.84813	0.832362	0.416181	+	0.909282i	$0.363368\pi$
$114$	−0.528670	−0.0495145
$115$	−22.5873	−2.10627
$116$	1.98093	0.183924
$117$	13.0157	1.20330
$118$	−0.152431	−0.0140324
$119$	−0.160877	−0.0147476
$120$	−1.02231	−0.0933234
$121$	−7.82238	−0.711125
$122$	0.778385	0.0704717
$123$	1.95634	0.176397
$124$	0.308691	0.0277213
$125$	−23.1478	−2.07040
$126$	−0.0267506	−0.00238313
$127$	−2.56849	−0.227917	−0.113958	−	0.993486i	$-0.536353\pi$
$127$	−2.56849	−0.227917	−0.113958	+	0.993486i	$0.536353\pi$
$128$	−4.31462	−0.381362
$129$	−0.282907	−0.0249086
$130$	2.57045	0.225443
$131$	9.98345	0.872258	0.436129	−	0.899884i	$-0.356349\pi$
$131$	9.98345	0.872258	0.436129	+	0.899884i	$0.356349\pi$
$132$	1.65084	0.143687
$133$	0.570217	0.0494441
$134$	−1.50989	−0.130434
$135$	−10.7503	−0.925239
$136$	1.27010	0.108911
$137$	7.62258	0.651241	0.325620	−	0.945501i	$-0.394427\pi$
$137$	7.62258	0.651241	0.325620	+	0.945501i	$0.394427\pi$
$138$	−0.366656	−0.0312119
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	0.548682	0.0463721
$141$	−2.41690	−0.203540
$142$	1.81422	0.152246
$143$	−8.34157	−0.697557
$144$	−10.8085	−0.900706
$145$	3.97741	0.330306
$146$	−1.37814	−0.114056
$147$	3.27025	0.269726
$148$	−7.89743	−0.649165
$149$	−10.0802	−0.825798	−0.412899	−	0.910777i	$-0.635484\pi$
$149$	−10.0802	−0.825798	−0.412899	+	0.910777i	$0.635484\pi$
$150$	−0.698579	−0.0570388
$151$	−1.62328	−0.132100	−0.0660501	−	0.997816i	$-0.521040\pi$
$151$	−1.62328	−0.132100	−0.0660501	+	0.997816i	$0.521040\pi$
$152$	−4.50179	−0.365143
$153$	6.42559	0.519478
$154$	0.0171441	0.00138151
$155$	0.619806	0.0497840
$156$	−4.33361	−0.346966
$157$	11.1375	0.888867	0.444434	−	0.895812i	$-0.353405\pi$
$157$	11.1375	0.888867	0.444434	+	0.895812i	$0.353405\pi$
$158$	−1.22242	−0.0972505
$159$	0.560389	0.0444417
$160$	−6.50803	−0.514505
$161$	0.395471	0.0311675
$162$	0.977891	0.0768304
$163$	3.05645	0.239399	0.119700	−	0.992810i	$-0.461807\pi$
$163$	3.05645	0.239399	0.119700	+	0.992810i	$0.461807\pi$
$164$	8.28950	0.647301
$165$	3.31464	0.258044
$166$	−1.12962	−0.0876752
$167$	4.96735	0.384385	0.192193	−	0.981357i	$-0.438440\pi$
$167$	4.96735	0.384385	0.192193	+	0.981357i	$0.438440\pi$
$168$	0.0178991	0.00138095
$169$	8.89744	0.684419
$170$	1.26898	0.0973265
$171$	−22.7750	−1.74165
$172$	−1.19875	−0.0914037
$173$	17.8106	1.35411	0.677057	−	0.735931i	$-0.263255\pi$
$173$	17.8106	1.35411	0.677057	+	0.735931i	$0.263255\pi$
$174$	0.0645649	0.00489465
$175$	0.753479	0.0569576
$176$	6.92700	0.522143
$177$	0.515994	0.0387845
$178$	0.416434	0.0312131
$179$	14.8464	1.10967	0.554836	−	0.831960i	$-0.312781\pi$
$179$	14.8464	1.10967	0.554836	+	0.831960i	$0.312781\pi$
$180$	−21.9149	−1.63344
$181$	−5.91151	−0.439399	−0.219699	−	0.975568i	$-0.570508\pi$
$181$	−5.91151	−0.439399	−0.219699	+	0.975568i	$0.570508\pi$
$182$	−0.0450049	−0.00333599
$183$	−2.63492	−0.194779
$184$	−3.12219	−0.230171
$185$	−15.8569	−1.16582
$186$	0.0100612	0.000737726 0
$187$	−4.11808	−0.301144
$188$	−10.2410	−0.746903
$189$	0.188223	0.0136912
$190$	−4.49781	−0.326306
$191$	−19.1850	−1.38818	−0.694089	−	0.719889i	$-0.744193\pi$
$191$	−19.1850	−1.38818	−0.694089	+	0.719889i	$0.744193\pi$
$192$	3.52772	0.254591
$193$	4.12867	0.297188	0.148594	−	0.988898i	$-0.452525\pi$
$193$	4.12867	0.297188	0.148594	+	0.988898i	$0.452525\pi$
$194$	1.54827	0.111159
$195$	−8.70125	−0.623109
$196$	13.8569	0.989777
$197$	14.6732	1.04542	0.522709	−	0.852511i	$-0.324921\pi$
$197$	14.6732	1.04542	0.522709	+	0.852511i	$0.324921\pi$
$198$	−0.684751	−0.0486632
$199$	−3.96117	−0.280800	−0.140400	−	0.990095i	$-0.544839\pi$
$199$	−3.96117	−0.280800	−0.140400	+	0.990095i	$0.544839\pi$
$200$	−5.94861	−0.420630
$201$	5.11113	0.360512
$202$	2.54168	0.178832
$203$	−0.0696388	−0.00488769
$204$	−2.13942	−0.149789
$205$	16.6441	1.16248
$206$	−0.229341	−0.0159790
$207$	−15.7955	−1.09786
$208$	−18.1841	−1.26084
$209$	14.5962	1.00964
$210$	0.0178833	0.00123407
$211$	0.0156904	0.00108017	0.000540087	−	1.00000i	$-0.499828\pi$
$211$	0.0156904	0.00108017	0.000540087		1.00000i	$0.499828\pi$
$212$	2.37451	0.163082
$213$	−6.14134	−0.420797
$214$	−1.24336	−0.0849941
$215$	−2.40691	−0.164150
$216$	−1.48599	−0.101109
$217$	−0.0108519	−0.000736677 0
$218$	−2.18498	−0.147986
$219$	4.66515	0.315242
$220$	14.0449	0.946910
$221$	10.8104	0.727183
$222$	−0.257403	−0.0172758
$223$	19.4969	1.30561	0.652805	−	0.757526i	$-0.273592\pi$
$223$	19.4969	1.30561	0.652805	+	0.757526i	$0.273592\pi$
$224$	0.113946	0.00761336
$225$	−30.0947	−2.00631
$226$	1.22198	0.0812848
$227$	−9.69966	−0.643789	−0.321894	−	0.946776i	$-0.604320\pi$
$227$	−9.69966	−0.643789	−0.321894	+	0.946776i	$0.604320\pi$
$228$	7.58301	0.502197
$229$	7.76305	0.512997	0.256499	−	0.966545i	$-0.417431\pi$
$229$	7.76305	0.512997	0.256499	+	0.966545i	$0.417431\pi$
$230$	−3.11943	−0.205689
$231$	−0.0580346	−0.00381840
$232$	0.549789	0.0360954
$233$	4.97330	0.325812	0.162906	−	0.986642i	$-0.447913\pi$
$233$	4.97330	0.325812	0.162906	+	0.986642i	$0.447913\pi$
$234$	1.79754	0.117509
$235$	−20.5625	−1.34135
$236$	2.18640	0.142322
$237$	4.13802	0.268793
$238$	−0.0222181	−0.00144018
$239$	−17.1631	−1.11019	−0.555096	−	0.831786i	$-0.687318\pi$
$239$	−17.1631	−1.11019	−0.555096	+	0.831786i	$0.687318\pi$
$240$	7.22569	0.466416
$241$	−18.6857	−1.20365	−0.601827	−	0.798626i	$-0.705560\pi$
$241$	−18.6857	−1.20365	−0.601827	+	0.798626i	$0.705560\pi$
$242$	−1.08032	−0.0694453
$243$	−11.4188	−0.732515
$244$	−11.1648	−0.714753
$245$	27.8226	1.77752
$246$	0.270182	0.0172262
$247$	−38.3165	−2.43802
$248$	0.0856745	0.00544033
$249$	3.82387	0.242328
$250$	−3.19684	−0.202186
$251$	−18.5491	−1.17081	−0.585405	−	0.810741i	$-0.699065\pi$
$251$	−18.5491	−1.17081	−0.585405	+	0.810741i	$0.699065\pi$
$252$	0.383698	0.0241707
$253$	10.1231	0.636435
$254$	−0.354724	−0.0222573
$255$	−4.29564	−0.269004
$256$	14.4959	0.905992
$257$	−2.41744	−0.150796	−0.0753978	−	0.997154i	$-0.524023\pi$
$257$	−2.41744	−0.150796	−0.0753978	+	0.997154i	$0.524023\pi$
$258$	−0.0390711	−0.00243246
$259$	0.277632	0.0172512
$260$	−36.8694	−2.28654
$261$	2.78144	0.172167
$262$	1.37877	0.0851809
$263$	5.10341	0.314690	0.157345	−	0.987544i	$-0.449707\pi$
$263$	5.10341	0.314690	0.157345	+	0.987544i	$0.449707\pi$
$264$	0.458175	0.0281987
$265$	4.76767	0.292876
$266$	0.0787503	0.00482849
$267$	−1.40967	−0.0862707
$268$	21.6571	1.32292
$269$	−11.8125	−0.720220	−0.360110	−	0.932910i	$-0.617261\pi$
$269$	−11.8125	−0.720220	−0.360110	+	0.932910i	$0.617261\pi$
$270$	−1.48468	−0.0903548
$271$	−1.56588	−0.0951207	−0.0475603	−	0.998868i	$-0.515145\pi$
$271$	−1.56588	−0.0951207	−0.0475603	+	0.998868i	$0.515145\pi$
$272$	−8.97713	−0.544319
$273$	0.152347	0.00922043
$274$	1.05272	0.0635973
$275$	19.2873	1.16307
$276$	5.25915	0.316564
$277$	−32.1779	−1.93338	−0.966692	−	0.255941i	$-0.917615\pi$
$277$	−32.1779	−1.93338	−0.966692	+	0.255941i	$0.917615\pi$
$278$	−0.138106	−0.00828304
$279$	0.433436	0.0259492
$280$	0.152282	0.00910058
$281$	−22.9230	−1.36747	−0.683736	−	0.729730i	$-0.739646\pi$
$281$	−22.9230	−1.36747	−0.683736	+	0.729730i	$0.739646\pi$
$282$	−0.333788	−0.0198768
$283$	4.19342	0.249273	0.124637	−	0.992202i	$-0.460224\pi$
$283$	4.19342	0.249273	0.124637	+	0.992202i	$0.460224\pi$
$284$	−26.0224	−1.54414
$285$	15.2256	0.901885
$286$	−1.15202	−0.0681203
$287$	−0.291415	−0.0172017
$288$	−4.55112	−0.268178
$289$	−11.6631	−0.686067
$290$	0.549304	0.0322562
$291$	−5.24105	−0.307236
$292$	19.7674	1.15680
$293$	−11.0427	−0.645122	−0.322561	−	0.946549i	$-0.604544\pi$
$293$	−11.0427	−0.645122	−0.322561	+	0.946549i	$0.604544\pi$
$294$	0.451641	0.0263402
$295$	4.38997	0.255594
$296$	−2.19186	−0.127399
$297$	4.81805	0.279572
$298$	−1.39213	−0.0806438
$299$	−26.5742	−1.53682
$300$	10.0201	0.578511
$301$	0.0421416	0.00242900
$302$	−0.224184	−0.0129003
$303$	−8.60388	−0.494280
$304$	31.8188	1.82493
$305$	−22.4173	−1.28361
$306$	0.887411	0.0507299
$307$	28.2741	1.61369	0.806844	−	0.590764i	$-0.201174\pi$
$307$	28.2741	1.61369	0.806844	+	0.590764i	$0.201174\pi$
$308$	−0.245907	−0.0140119
$309$	0.776345	0.0441647
$310$	0.0855988	0.00486169
$311$	−16.6361	−0.943344	−0.471672	−	0.881774i	$-0.656349\pi$
$311$	−16.6361	−0.943344	−0.471672	+	0.881774i	$0.656349\pi$
$312$	−1.20275	−0.0680926
$313$	−0.792239	−0.0447800	−0.0223900	−	0.999749i	$-0.507128\pi$
$313$	−0.792239	−0.0447800	−0.0223900	+	0.999749i	$0.507128\pi$
$314$	1.53815	0.0868028
$315$	0.770410	0.0434077
$316$	17.5338	0.986355
$317$	−24.7159	−1.38818	−0.694091	−	0.719888i	$-0.744193\pi$
$317$	−24.7159	−1.38818	−0.694091	+	0.719888i	$0.744193\pi$
$318$	0.0773930	0.00433998
$319$	−1.78259	−0.0998058
$320$	30.0131	1.67778
$321$	4.20890	0.234918
$322$	0.0546168	0.00304368
$323$	−18.9161	−1.05252
$324$	−14.0264	−0.779246
$325$	−50.6310	−2.80850
$326$	0.422113	0.0233787
$327$	7.39640	0.409022
$328$	2.30068	0.127034
$329$	0.360020	0.0198485
$330$	0.457771	0.0251994
$331$	5.23899	0.287961	0.143980	−	0.989581i	$-0.454010\pi$
$331$	5.23899	0.287961	0.143980	+	0.989581i	$0.454010\pi$
$332$	16.2027	0.889239
$333$	−11.0889	−0.607666
$334$	0.686020	0.0375373
$335$	43.4844	2.37581
$336$	−0.126512	−0.00690177
$337$	−31.5697	−1.71971	−0.859856	−	0.510537i	$-0.829447\pi$
$337$	−31.5697	−1.71971	−0.859856	+	0.510537i	$0.829447\pi$
$338$	1.22879	0.0668373
$339$	−4.13653	−0.224665
$340$	−18.2017	−0.987126
$341$	−0.277784	−0.0150428
$342$	−3.14536	−0.170082
$343$	−0.974606	−0.0526238
$344$	−0.332702	−0.0179381
$345$	10.5596	0.568510
$346$	2.45975	0.132237
$347$	0.591559	0.0317566	0.0158783	−	0.999874i	$-0.494946\pi$
$347$	0.591559	0.0317566	0.0158783	+	0.999874i	$0.494946\pi$
$348$	−0.926089	−0.0496436
$349$	31.9631	1.71095	0.855473	−	0.517847i	$-0.173266\pi$
$349$	31.9631	1.71095	0.855473	+	0.517847i	$0.173266\pi$
$350$	0.104060	0.00556223
$351$	−12.6479	−0.675093
$352$	2.91676	0.155464
$353$	27.5685	1.46732	0.733662	−	0.679515i	$-0.237810\pi$
$353$	27.5685	1.46732	0.733662	+	0.679515i	$0.237810\pi$
$354$	0.0712618	0.00378752
$355$	−52.2491	−2.77310
$356$	−5.97314	−0.316576
$357$	0.0752106	0.00398057
$358$	2.05037	0.108366
$359$	−33.7725	−1.78244	−0.891222	−	0.453568i	$-0.850151\pi$
$359$	−33.7725	−1.78244	−0.891222	+	0.453568i	$0.850151\pi$
$360$	−6.08228	−0.320564
$361$	48.0467	2.52878
$362$	−0.816414	−0.0429097
$363$	3.65698	0.191942
$364$	0.645530	0.0338350
$365$	39.6901	2.07747
$366$	−0.363897	−0.0190212
$367$	11.8385	0.617963	0.308981	−	0.951068i	$-0.400012\pi$
$367$	11.8385	0.617963	0.308981	+	0.951068i	$0.400012\pi$
$368$	22.0677	1.15036
$369$	11.6394	0.605922
$370$	−2.18993	−0.113849
$371$	−0.0834751	−0.00433381
$372$	−0.144314	−0.00748233
$373$	−5.78370	−0.299469	−0.149734	−	0.988726i	$-0.547842\pi$
$373$	−5.78370	−0.299469	−0.149734	+	0.988726i	$0.547842\pi$
$374$	−0.568730	−0.0294083
$375$	10.8217	0.558828
$376$	−2.84231	−0.146581
$377$	4.67947	0.241005
$378$	0.0259946	0.00133702
$379$	15.3397	0.787946	0.393973	−	0.919122i	$-0.371100\pi$
$379$	15.3397	0.787946	0.393973	+	0.919122i	$0.371100\pi$
$380$	64.5146	3.30953
$381$	1.20078	0.0615177
$382$	−2.64956	−0.135563
$383$	10.6902	0.546242	0.273121	−	0.961980i	$-0.411944\pi$
$383$	10.6902	0.546242	0.273121	+	0.961980i	$0.411944\pi$
$384$	2.01710	0.102935
$385$	−0.493746	−0.0251636
$386$	0.570194	0.0290221
$387$	−1.68318	−0.0855606
$388$	−22.2076	−1.12742
$389$	−0.256564	−0.0130083	−0.00650416	−	0.999979i	$-0.502070\pi$
$389$	−0.256564	−0.0130083	−0.00650416	+	0.999979i	$0.502070\pi$
$390$	−1.20169	−0.0608501
$391$	−13.1192	−0.663465
$392$	3.84586	0.194245
$393$	−4.66729	−0.235434
$394$	2.02645	0.102091
$395$	35.2054	1.77138
$396$	9.82176	0.493562
$397$	11.3223	0.568252	0.284126	−	0.958787i	$-0.408297\pi$
$397$	11.3223	0.568252	0.284126	+	0.958787i	$0.408297\pi$
$398$	−0.547061	−0.0274217
$399$	−0.266578	−0.0133456
$400$	42.0450	2.10225
$401$	36.3292	1.81419	0.907097	−	0.420923i	$-0.138294\pi$
$401$	36.3292	1.81419	0.907097	+	0.420923i	$0.138294\pi$
$402$	0.705877	0.0352059
$403$	0.729209	0.0363245
$404$	−36.4568	−1.81379
$405$	−28.1630	−1.39943
$406$	−0.00961753	−0.000477310 0
$407$	7.10671	0.352267
$408$	−0.593777	−0.0293964
$409$	−23.2692	−1.15059	−0.575295	−	0.817946i	$-0.695113\pi$
$409$	−23.2692	−1.15059	−0.575295	+	0.817946i	$0.695113\pi$
$410$	2.29865	0.113522
$411$	−3.56358	−0.175778
$412$	3.28957	0.162065
$413$	−0.0768621	−0.00378213
$414$	−2.18145	−0.107212
$415$	32.5327	1.59697
$416$	−7.65677	−0.375404
$417$	0.467503	0.0228937
$418$	2.01582	0.0985970
$419$	12.5980	0.615453	0.307726	−	0.951475i	$-0.400432\pi$
$419$	12.5980	0.615453	0.307726	+	0.951475i	$0.400432\pi$
$420$	−0.256510	−0.0125164
$421$	−27.4038	−1.33558	−0.667789	−	0.744351i	$-0.732759\pi$
$421$	−27.4038	−1.33558	−0.667789	+	0.744351i	$0.732759\pi$
$422$	0.00216694	0.000105485 0
$423$	−14.3795	−0.699157
$424$	0.659024	0.0320051
$425$	−24.9956	−1.21246
$426$	−0.848154	−0.0410932
$427$	0.392495	0.0189942
$428$	17.8341	0.862046
$429$	3.89971	0.188280
$430$	−0.332408	−0.0160302
$431$	−23.8399	−1.14833	−0.574163	−	0.818741i	$-0.694673\pi$
$431$	−23.8399	−1.14833	−0.574163	+	0.818741i	$0.694673\pi$
$432$	10.5030	0.505327
$433$	−19.4512	−0.934763	−0.467381	−	0.884056i	$-0.654802\pi$
$433$	−19.4512	−0.934763	−0.467381	+	0.884056i	$0.654802\pi$
$434$	−0.00149871	−7.19406e−5 0
$435$	−1.85945	−0.0891539
$436$	31.3404	1.50093
$437$	46.4999	2.22439
$438$	0.644285	0.0307851
$439$	−0.577376	−0.0275566	−0.0137783	−	0.999905i	$-0.504386\pi$
$439$	−0.577376	−0.0275566	−0.0137783	+	0.999905i	$0.504386\pi$
$440$	3.89805	0.185832
$441$	19.4566	0.926505
$442$	1.49297	0.0710135
$443$	15.0579	0.715420	0.357710	−	0.933833i	$-0.383558\pi$
$443$	15.0579	0.715420	0.357710	+	0.933833i	$0.383558\pi$
$444$	3.69207	0.175218
$445$	−11.9932	−0.568532
$446$	2.69264	0.127500
$447$	4.71250	0.222894
$448$	−0.525486	−0.0248269
$449$	−36.6915	−1.73158	−0.865789	−	0.500409i	$-0.833183\pi$
$449$	−36.6915	−1.73158	−0.865789	+	0.500409i	$0.833183\pi$
$450$	−4.15625	−0.195927
$451$	−7.45953	−0.351255
$452$	−17.5275	−0.824424
$453$	0.758886	0.0356556
$454$	−1.33958	−0.0628696
$455$	1.29613	0.0607635
$456$	2.10460	0.0985568
$457$	−5.80418	−0.271508	−0.135754	−	0.990743i	$-0.543346\pi$
$457$	−5.80418	−0.271508	−0.135754	+	0.990743i	$0.543346\pi$
$458$	1.07212	0.0500970
$459$	−6.24401	−0.291445
$460$	44.7437	2.08619
$461$	16.5235	0.769575	0.384788	−	0.923005i	$-0.374275\pi$
$461$	16.5235	0.769575	0.384788	+	0.923005i	$0.374275\pi$
$462$	−0.00801491	−0.000372888 0
$463$	−9.13202	−0.424401	−0.212201	−	0.977226i	$-0.568063\pi$
$463$	−9.13202	−0.424401	−0.212201	+	0.977226i	$0.568063\pi$
$464$	−3.88592	−0.180400
$465$	−0.289761	−0.0134374
$466$	0.686842	0.0318173
$467$	−0.527300	−0.0244006	−0.0122003	−	0.999926i	$-0.503884\pi$
$467$	−0.527300	−0.0244006	−0.0122003	+	0.999926i	$0.503884\pi$
$468$	−25.7831	−1.19182
$469$	−0.761350	−0.0351559
$470$	−2.83980	−0.130990
$471$	−5.20680	−0.239917
$472$	0.606815	0.0279309
$473$	1.07872	0.0495998
$474$	0.571485	0.0262492
$475$	88.5948	4.06501
$476$	0.318686	0.0146070
$477$	3.33407	0.152657
$478$	−2.37033	−0.108416
$479$	20.7297	0.947166	0.473583	−	0.880749i	$-0.342960\pi$
$479$	20.7297	0.947166	0.473583	+	0.880749i	$0.342960\pi$
$480$	3.04252	0.138871
$481$	−18.6558	−0.850632
$482$	−2.58061	−0.117544
$483$	−0.184884	−0.00841251
$484$	15.4956	0.704344
$485$	−44.5897	−2.02471
$486$	−1.57700	−0.0715342
$487$	4.24222	0.192233	0.0961167	−	0.995370i	$-0.469358\pi$
$487$	4.24222	0.192233	0.0961167	+	0.995370i	$0.469358\pi$
$488$	−3.09869	−0.140271
$489$	−1.42890	−0.0646170
$490$	3.84246	0.173585
$491$	8.33185	0.376011	0.188006	−	0.982168i	$-0.439798\pi$
$491$	8.33185	0.376011	0.188006	+	0.982168i	$0.439798\pi$
$492$	−3.87537	−0.174715
$493$	2.31017	0.104045
$494$	−5.29173	−0.238086
$495$	19.7207	0.886378
$496$	−0.605550	−0.0271900
$497$	0.914808	0.0410348
$498$	0.528099	0.0236647
$499$	−26.4804	−1.18543	−0.592714	−	0.805413i	$-0.701943\pi$
$499$	−26.4804	−1.18543	−0.592714	+	0.805413i	$0.701943\pi$
$500$	45.8540	2.05066
$501$	−2.32225	−0.103751
$502$	−2.56174	−0.114336
$503$	−3.67649	−0.163927	−0.0819634	−	0.996635i	$-0.526119\pi$
$503$	−3.67649	−0.163927	−0.0819634	+	0.996635i	$0.526119\pi$
$504$	0.106492	0.00474354
$505$	−73.1999	−3.25735
$506$	1.39806	0.0621514
$507$	−4.15958	−0.184733
$508$	5.08799	0.225743
$509$	27.5177	1.21970	0.609851	−	0.792516i	$-0.291229\pi$
$509$	27.5177	1.21970	0.609851	+	0.792516i	$0.291229\pi$
$510$	−0.593253	−0.0262697
$511$	−0.694917	−0.0307413
$512$	10.6312	0.469838
$513$	22.1314	0.977126
$514$	−0.333862	−0.0147260
$515$	6.60498	0.291050
$516$	0.560418	0.0246710
$517$	9.21565	0.405304
$518$	0.0383425	0.00168467
$519$	−8.32650	−0.365493
$520$	−10.2328	−0.448737
$521$	13.9805	0.612499	0.306249	−	0.951951i	$-0.400926\pi$
$521$	13.9805	0.612499	0.306249	+	0.951951i	$0.400926\pi$
$522$	0.384133	0.0168131
$523$	−13.4596	−0.588547	−0.294273	−	0.955721i	$-0.595078\pi$
$523$	−13.4596	−0.588547	−0.294273	+	0.955721i	$0.595078\pi$
$524$	−19.7765	−0.863940
$525$	−0.352254	−0.0153736
$526$	0.704810	0.0307312
$527$	0.359997	0.0156817
$528$	−3.23839	−0.140933
$529$	9.24974	0.402163
$530$	0.658443	0.0286009
$531$	3.06994	0.133224
$532$	−1.12956	−0.0489726
$533$	19.5820	0.848190
$534$	−0.194684	−0.00842481
$535$	35.8084	1.54813
$536$	6.01075	0.259625
$537$	−6.94074	−0.299515
$538$	−1.63137	−0.0703335
$539$	−12.4695	−0.537098
$540$	21.2956	0.916416
$541$	−27.6511	−1.18881	−0.594406	−	0.804165i	$-0.702613\pi$
$541$	−27.6511	−1.18881	−0.594406	+	0.804165i	$0.702613\pi$
$542$	−0.216258	−0.00928906
$543$	2.76365	0.118599
$544$	−3.78000	−0.162066
$545$	62.9269	2.69549
$546$	0.0210399	0.000900426 0
$547$	9.54660	0.408183	0.204091	−	0.978952i	$-0.434576\pi$
$547$	9.54660	0.408183	0.204091	+	0.978952i	$0.434576\pi$
$548$	−15.0998	−0.645030
$549$	−15.6766	−0.669062
$550$	2.66368	0.113580
$551$	−8.18821	−0.348829
$552$	1.45963	0.0621261
$553$	−0.616396	−0.0262118
$554$	−4.44396	−0.188806
$555$	7.41314	0.314670
$556$	1.98093	0.0840100
$557$	30.8697	1.30799	0.653996	−	0.756498i	$-0.273091\pi$
$557$	30.8697	1.30799	0.653996	+	0.756498i	$0.273091\pi$
$558$	0.0598601	0.00253408
$559$	−2.83176	−0.119771
$560$	−1.07633	−0.0454834
$561$	1.92521	0.0812825
$562$	−3.16580	−0.133541
$563$	−47.4033	−1.99781	−0.998905	−	0.0467824i	$-0.985103\pi$
$563$	−47.4033	−1.99781	−0.998905	+	0.0467824i	$0.985103\pi$
$564$	4.78771	0.201599
$565$	−35.1927	−1.48057
$566$	0.579136	0.0243429
$567$	0.493094	0.0207080
$568$	−7.22229	−0.303040
$569$	3.87346	0.162384	0.0811919	−	0.996698i	$-0.474127\pi$
$569$	3.87346	0.162384	0.0811919	+	0.996698i	$0.474127\pi$
$570$	2.10274	0.0880741
$571$	−23.7397	−0.993475	−0.496738	−	0.867901i	$-0.665469\pi$
$571$	−23.7397	−0.993475	−0.496738	+	0.867901i	$0.665469\pi$
$572$	16.5240	0.690905
$573$	8.96904	0.374687
$574$	−0.0402461	−0.00167984
$575$	61.4444	2.56241
$576$	20.9884	0.874517
$577$	37.6464	1.56724	0.783620	−	0.621240i	$-0.213371\pi$
$577$	37.6464	1.56724	0.783620	+	0.621240i	$0.213371\pi$
$578$	−1.61075	−0.0669982
$579$	−1.93017	−0.0802150
$580$	−7.87896	−0.327156
$581$	−0.569601	−0.0236310
$582$	−0.723819	−0.0300033
$583$	−2.13676	−0.0884957
$584$	5.48628	0.227024
$585$	−51.7687	−2.14037
$586$	−1.52506	−0.0629997
$587$	8.69532	0.358894	0.179447	−	0.983768i	$-0.442569\pi$
$587$	8.69532	0.358894	0.179447	+	0.983768i	$0.442569\pi$
$588$	−6.47813	−0.267154
$589$	−1.27598	−0.0525759
$590$	0.606280	0.0249601
$591$	−6.85974	−0.282172
$592$	15.4922	0.636724
$593$	−24.6528	−1.01237	−0.506186	−	0.862425i	$-0.668945\pi$
$593$	−24.6528	−1.01237	−0.506186	+	0.862425i	$0.668945\pi$
$594$	0.665401	0.0273017
$595$	0.639875	0.0262323
$596$	19.9680	0.817923
$597$	1.85186	0.0757916
$598$	−3.67005	−0.150079
$599$	10.9872	0.448926	0.224463	−	0.974483i	$-0.427937\pi$
$599$	10.9872	0.448926	0.224463	+	0.974483i	$0.427937\pi$
$600$	2.78099	0.113534
$601$	−11.9473	−0.487342	−0.243671	−	0.969858i	$-0.578352\pi$
$601$	−11.9473	−0.487342	−0.243671	+	0.969858i	$0.578352\pi$
$602$	0.00582000	0.000237205 0
$603$	30.4090	1.23835
$604$	3.21559	0.130841
$605$	31.1128	1.26492
$606$	−1.18825	−0.0482692
$607$	−25.4719	−1.03387	−0.516937	−	0.856024i	$-0.672928\pi$
$607$	−25.4719	−1.03387	−0.516937	+	0.856024i	$0.672928\pi$
$608$	13.3979	0.543358
$609$	0.0325564	0.00131925
$610$	−3.09596	−0.125352
$611$	−24.1920	−0.978703
$612$	−12.7286	−0.514524
$613$	−11.4873	−0.463967	−0.231984	−	0.972720i	$-0.574522\pi$
$613$	−11.4873	−0.463967	−0.231984	+	0.972720i	$0.574522\pi$
$614$	3.90482	0.157586
$615$	−7.78117	−0.313767
$616$	−0.0682494	−0.00274985
$617$	10.2139	0.411194	0.205597	−	0.978637i	$-0.434086\pi$
$617$	10.2139	0.411194	0.205597	+	0.978637i	$0.434086\pi$
$618$	0.107218	0.00431293
$619$	−25.9027	−1.04112	−0.520558	−	0.853826i	$-0.674276\pi$
$619$	−25.9027	−1.04112	−0.520558	+	0.853826i	$0.674276\pi$
$620$	−1.22779	−0.0493093
$621$	15.3491	0.615939
$622$	−2.29754	−0.0921228
$623$	0.209984	0.00841283
$624$	8.50110	0.340317
$625$	37.9692	1.51877
$626$	−0.109413	−0.00437302
$627$	−6.82377	−0.272515
$628$	−22.0625	−0.880391
$629$	−9.21002	−0.367228
$630$	0.106398	0.00423900
$631$	−49.3350	−1.96400	−0.981998	−	0.188891i	$-0.939511\pi$
$631$	−49.3350	−1.96400	−0.981998	+	0.188891i	$0.939511\pi$
$632$	4.86636	0.193574
$633$	−0.00733533	−0.000291553 0
$634$	−3.41340	−0.135564
$635$	10.2159	0.405408
$636$	−1.11009	−0.0440179
$637$	32.7336	1.29695
$638$	−0.246186	−0.00974659
$639$	−36.5383	−1.44543
$640$	17.1610	0.678349
$641$	35.4238	1.39915	0.699577	−	0.714557i	$-0.253372\pi$
$641$	35.4238	1.39915	0.699577	+	0.714557i	$0.253372\pi$
$642$	0.581273	0.0229410
$643$	−39.6862	−1.56507	−0.782535	−	0.622607i	$-0.786074\pi$
$643$	−39.6862	−1.56507	−0.782535	+	0.622607i	$0.786074\pi$
$644$	−0.783399	−0.0308702
$645$	1.12524	0.0443062
$646$	−2.61243	−0.102785
$647$	39.1928	1.54083	0.770415	−	0.637543i	$-0.220049\pi$
$647$	39.1928	1.54083	0.770415	+	0.637543i	$0.220049\pi$
$648$	−3.89291	−0.152928
$649$	−1.96749	−0.0772306
$650$	−6.99243	−0.274266
$651$	0.00507331	0.000198839 0
$652$	−6.05459	−0.237116
$653$	0.616022	0.0241068	0.0120534	−	0.999927i	$-0.496163\pi$
$653$	0.616022	0.0241068	0.0120534	+	0.999927i	$0.496163\pi$
$654$	1.02149	0.0399432
$655$	−39.7083	−1.55153
$656$	−16.2613	−0.634896
$657$	27.7556	1.08285
$658$	0.0497208	0.00193832
$659$	−37.5862	−1.46415	−0.732074	−	0.681225i	$-0.761448\pi$
$659$	−37.5862	−1.46415	−0.732074	+	0.681225i	$0.761448\pi$
$660$	−6.56605	−0.255583
$661$	43.4166	1.68871	0.844356	−	0.535783i	$-0.179984\pi$
$661$	43.4166	1.68871	0.844356	+	0.535783i	$0.179984\pi$
$662$	0.723535	0.0281210
$663$	−5.05387	−0.196276
$664$	4.49692	0.174514
$665$	−2.26799	−0.0879488
$666$	−1.53144	−0.0593420
$667$	−5.67889	−0.219887
$668$	−9.83996	−0.380719
$669$	−9.11486	−0.352401
$670$	6.00545	0.232011
$671$	10.0469	0.387858
$672$	−0.0532702	−0.00205494
$673$	−17.0369	−0.656723	−0.328362	−	0.944552i	$-0.606496\pi$
$673$	−17.0369	−0.656723	−0.328362	+	0.944552i	$0.606496\pi$
$674$	−4.35996	−0.167939
$675$	29.2442	1.12561
$676$	−17.6252	−0.677892
$677$	6.77866	0.260525	0.130262	−	0.991480i	$-0.458418\pi$
$677$	6.77866	0.260525	0.130262	+	0.991480i	$0.458418\pi$
$678$	−0.571278	−0.0219398
$679$	0.780702	0.0299606
$680$	−5.05173	−0.193725
$681$	4.53462	0.173767
$682$	−0.0383635	−0.00146902
$683$	43.9406	1.68134	0.840671	−	0.541547i	$-0.182161\pi$
$683$	43.9406	1.68134	0.840671	+	0.541547i	$0.182161\pi$
$684$	45.1156	1.72504
$685$	−30.3181	−1.15840
$686$	−0.134599	−0.00513900
$687$	−3.62925	−0.138465
$688$	2.35155	0.0896519
$689$	5.60922	0.213694
$690$	1.45834	0.0555182
$691$	−31.1972	−1.18680	−0.593399	−	0.804909i	$-0.702214\pi$
$691$	−31.1972	−1.18680	−0.593399	+	0.804909i	$0.702214\pi$
$692$	−35.2815	−1.34120
$693$	−0.345281	−0.0131161
$694$	0.0816978	0.00310120
$695$	3.97741	0.150872
$696$	−0.257028	−0.00974262
$697$	9.66726	0.366174
$698$	4.41429	0.167083
$699$	−2.32503	−0.0879408
$700$	−1.49259	−0.0564145
$701$	−20.2081	−0.763250	−0.381625	−	0.924317i	$-0.624635\pi$
$701$	−20.2081	−0.763250	−0.381625	+	0.924317i	$0.624635\pi$
$702$	−1.74674	−0.0659266
$703$	32.6442	1.23120
$704$	−13.4512	−0.506961
$705$	9.61302	0.362047
$706$	3.80737	0.143292
$707$	1.28163	0.0482005
$708$	−1.02215	−0.0384146
$709$	43.2740	1.62519	0.812595	−	0.582828i	$-0.198054\pi$
$709$	43.2740	1.62519	0.812595	+	0.582828i	$0.198054\pi$
$710$	−7.21591	−0.270808
$711$	24.6195	0.923302
$712$	−1.65779	−0.0621285
$713$	−0.884950	−0.0331416
$714$	0.0103870	0.000388724 0
$715$	33.1779	1.24078
$716$	−29.4096	−1.09909
$717$	8.02382	0.299655
$718$	−4.66417	−0.174065
$719$	−1.39103	−0.0518768	−0.0259384	−	0.999664i	$-0.508257\pi$
$719$	−1.39103	−0.0518768	−0.0259384	+	0.999664i	$0.508257\pi$
$720$	42.9897	1.60213
$721$	−0.115644	−0.00430680
$722$	6.63553	0.246949
$723$	8.73564	0.324882
$724$	11.7103	0.435209
$725$	−10.8198	−0.401838
$726$	0.505051	0.0187442
$727$	1.11377	0.0413074	0.0206537	−	0.999787i	$-0.493425\pi$
$727$	1.11377	0.0413074	0.0206537	+	0.999787i	$0.493425\pi$
$728$	0.179161	0.00664016
$729$	−15.9039	−0.589033
$730$	5.48143	0.202877
$731$	−1.39799	−0.0517064
$732$	5.21958	0.192921
$733$	−0.605330	−0.0223584	−0.0111792	−	0.999938i	$-0.503559\pi$
$733$	−0.605330	−0.0223584	−0.0111792	+	0.999938i	$0.503559\pi$
$734$	1.63496	0.0603475
$735$	−13.0071	−0.479776
$736$	9.29206	0.342510
$737$	−19.4888	−0.717877
$738$	1.60747	0.0591717
$739$	−6.12699	−0.225385	−0.112693	−	0.993630i	$-0.535948\pi$
$739$	−6.12699	−0.225385	−0.112693	+	0.993630i	$0.535948\pi$
$740$	31.4113	1.15470
$741$	17.9131	0.658053
$742$	−0.0115284	−0.000423221 0
$743$	−7.25572	−0.266187	−0.133093	−	0.991104i	$-0.542491\pi$
$743$	−7.25572	−0.266187	−0.133093	+	0.991104i	$0.542491\pi$
$744$	−0.0400531	−0.00146842
$745$	40.0929	1.46889
$746$	−0.798762	−0.0292448
$747$	22.7504	0.832393
$748$	8.15761	0.298272
$749$	−0.626954	−0.0229084
$750$	1.49453	0.0545726
$751$	22.0942	0.806228	0.403114	−	0.915150i	$-0.367928\pi$
$751$	22.0942	0.806228	0.403114	+	0.915150i	$0.367928\pi$
$752$	20.0895	0.732589
$753$	8.67177	0.316017
$754$	0.646262	0.0235355
$755$	6.45644	0.234974
$756$	−0.372855	−0.0135606
$757$	−3.83404	−0.139350	−0.0696752	−	0.997570i	$-0.522196\pi$
$757$	−3.83404	−0.139350	−0.0696752	+	0.997570i	$0.522196\pi$
$758$	2.11850	0.0769473
$759$	−4.73259	−0.171782
$760$	17.9055	0.649499
$761$	−5.86170	−0.212486	−0.106243	−	0.994340i	$-0.533882\pi$
$761$	−5.86170	−0.212486	−0.106243	+	0.994340i	$0.533882\pi$
$762$	0.165834	0.00600754
$763$	−1.10176	−0.0398864
$764$	38.0041	1.37494
$765$	−25.5572	−0.924023
$766$	1.47637	0.0533436
$767$	5.16484	0.186492
$768$	−6.77686	−0.244539
$769$	−23.0862	−0.832511	−0.416256	−	0.909248i	$-0.636658\pi$
$769$	−23.0862	−0.832511	−0.416256	+	0.909248i	$0.636658\pi$
$770$	−0.0681891	−0.00245737
$771$	1.13016	0.0407017
$772$	−8.17860	−0.294354
$773$	−14.5117	−0.521949	−0.260974	−	0.965346i	$-0.584044\pi$
$773$	−14.5117	−0.521949	−0.260974	+	0.965346i	$0.584044\pi$
$774$	−0.232456	−0.00835547
$775$	−1.68607	−0.0605653
$776$	−6.16354	−0.221258
$777$	−0.129794	−0.00465632
$778$	−0.0354330	−0.00127033
$779$	−34.2649	−1.22767
$780$	17.2365	0.617167
$781$	23.4169	0.837923
$782$	−1.81183	−0.0647910
$783$	−2.70284	−0.0965916
$784$	−27.1826	−0.970808
$785$	−44.2983	−1.58108
$786$	−0.644580	−0.0229914
$787$	−12.7586	−0.454796	−0.227398	−	0.973802i	$-0.573022\pi$
$787$	−12.7586	−0.454796	−0.227398	+	0.973802i	$0.573022\pi$
$788$	−29.0664	−1.03545
$789$	−2.38586	−0.0849388
$790$	4.86207	0.172985
$791$	0.616173	0.0219086
$792$	2.72595	0.0968622
$793$	−26.3742	−0.936575
$794$	1.56368	0.0554930
$795$	−2.22890	−0.0790509
$796$	7.84679	0.278122
$797$	−26.7186	−0.946422	−0.473211	−	0.880949i	$-0.656905\pi$
$797$	−26.7186	−0.946422	−0.473211	+	0.880949i	$0.656905\pi$
$798$	−0.0368160	−0.00130327
$799$	−11.9431	−0.422517
$800$	17.7039	0.625927
$801$	−8.38695	−0.296338
$802$	5.01727	0.177166
$803$	−17.7882	−0.627733
$804$	−10.1248	−0.357073
$805$	−1.57295	−0.0554392
$806$	0.100708	0.00354729
$807$	5.52237	0.194397
$808$	−10.1183	−0.355959
$809$	−32.3204	−1.13633	−0.568163	−	0.822916i	$-0.692346\pi$
$809$	−32.3204	−1.13633	−0.568163	+	0.822916i	$0.692346\pi$
$810$	−3.88948	−0.136662
$811$	8.47682	0.297661	0.148831	−	0.988863i	$-0.452449\pi$
$811$	8.47682	0.297661	0.148831	+	0.988863i	$0.452449\pi$
$812$	0.137949	0.00484108
$813$	0.732055	0.0256743
$814$	0.981478	0.0344008
$815$	−12.1567	−0.425832
$816$	4.19684	0.146919
$817$	4.95505	0.173355
$818$	−3.21362	−0.112361
$819$	0.906396	0.0316720
$820$	−32.9708	−1.15139
$821$	2.00056	0.0698201	0.0349101	−	0.999390i	$-0.488886\pi$
$821$	2.00056	0.0698201	0.0349101	+	0.999390i	$0.488886\pi$
$822$	−0.492151	−0.0171657
$823$	−3.21207	−0.111966	−0.0559828	−	0.998432i	$-0.517829\pi$
$823$	−3.21207	−0.111966	−0.0559828	+	0.998432i	$0.517829\pi$
$824$	0.912992	0.0318056
$825$	−9.01685	−0.313927
$826$	−0.0106151	−0.000369346 0
$827$	−17.2582	−0.600126	−0.300063	−	0.953919i	$-0.597008\pi$
$827$	−17.2582	−0.600126	−0.300063	+	0.953919i	$0.597008\pi$
$828$	31.2897	1.08739
$829$	−6.72042	−0.233410	−0.116705	−	0.993167i	$-0.537233\pi$
$829$	−6.72042	−0.233410	−0.116705	+	0.993167i	$0.537233\pi$
$830$	4.49295	0.155953
$831$	15.0433	0.521846
$832$	35.3107	1.22418
$833$	16.1600	0.559909
$834$	0.0645649	0.00223570
$835$	−19.7572	−0.683726
$836$	−28.9140	−1.00001
$837$	−0.421188	−0.0145584
$838$	1.73986	0.0601024
$839$	4.88350	0.168597	0.0842986	−	0.996441i	$-0.473135\pi$
$839$	4.88350	0.168597	0.0842986	+	0.996441i	$0.473135\pi$
$840$	−0.0711922	−0.00245636
$841$	1.00000	0.0344828
$842$	−3.78462	−0.130427
$843$	10.7166	0.369098
$844$	−0.0310816	−0.00106987
$845$	−35.3888	−1.21741
$846$	−1.98590	−0.0682765
$847$	−0.544741	−0.0187175
$848$	−4.65800	−0.159956
$849$	−1.96044	−0.0672820
$850$	−3.45203	−0.118404
$851$	22.6402	0.776097
$852$	12.1655	0.416784
$853$	−39.8691	−1.36509	−0.682545	−	0.730843i	$-0.739127\pi$
$853$	−39.8691	−1.36509	−0.682545	+	0.730843i	$0.739127\pi$
$854$	0.0542058	0.00185489
$855$	90.5856	3.09796
$856$	4.94971	0.169178
$857$	−48.3829	−1.65273	−0.826364	−	0.563136i	$-0.809595\pi$
$857$	−48.3829	−1.65273	−0.826364	+	0.563136i	$0.809595\pi$
$858$	0.538572	0.0183866
$859$	−3.05356	−0.104186	−0.0520930	−	0.998642i	$-0.516589\pi$
$859$	−3.05356	−0.104186	−0.0520930	+	0.998642i	$0.516589\pi$
$860$	4.76791	0.162585
$861$	0.136237	0.00464295
$862$	−3.29243	−0.112140
$863$	27.5411	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$863$	27.5411	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$864$	4.42251	0.150457
$865$	−70.8400	−2.40863
$866$	−2.68632	−0.0912848
$867$	5.45255	0.185178
$868$	0.0214969	0.000729651 0
$869$	−15.7783	−0.535241
$870$	−0.256801	−0.00870637
$871$	51.1599	1.73349
$872$	8.69825	0.294560
$873$	−31.1820	−1.05535
$874$	6.42190	0.217224
$875$	−1.61198	−0.0544950
$876$	−9.24132	−0.312235
$877$	−28.7840	−0.971966	−0.485983	−	0.873968i	$-0.661538\pi$
$877$	−28.7840	−0.971966	−0.485983	+	0.873968i	$0.661538\pi$
$878$	−0.0797390	−0.00269106
$879$	5.16250	0.174127
$880$	−27.5516	−0.928763
$881$	25.3781	0.855009	0.427504	−	0.904013i	$-0.359393\pi$
$881$	25.3781	0.855009	0.427504	+	0.904013i	$0.359393\pi$
$882$	2.68707	0.0904783
$883$	14.1918	0.477591	0.238795	−	0.971070i	$-0.423247\pi$
$883$	14.1918	0.477591	0.238795	+	0.971070i	$0.423247\pi$
$884$	−21.4145	−0.720248
$885$	−2.05232	−0.0689880
$886$	2.07958	0.0698648
$887$	17.2068	0.577746	0.288873	−	0.957367i	$-0.406719\pi$
$887$	17.2068	0.577746	0.288873	+	0.957367i	$0.406719\pi$
$888$	1.02470	0.0343868
$889$	−0.178867	−0.00599900
$890$	−1.65633	−0.0555203
$891$	12.6220	0.422854
$892$	−38.6219	−1.29316
$893$	42.3315	1.41657
$894$	0.650824	0.0217668
$895$	−59.0503	−1.97383
$896$	−0.300465	−0.0100378
$897$	12.4235	0.414809
$898$	−5.06731	−0.169098
$899$	0.155832	0.00519727
$900$	59.6153	1.98718
$901$	2.76916	0.0922542
$902$	−1.03020	−0.0343020
$903$	−0.0197013	−0.000655619 0
$904$	−4.86460	−0.161794
$905$	23.5125	0.781582
$906$	0.104807	0.00348196
$907$	39.5747	1.31406	0.657029	−	0.753865i	$-0.271813\pi$
$907$	39.5747	1.31406	0.657029	+	0.753865i	$0.271813\pi$
$908$	19.2143	0.637649
$909$	−51.1894	−1.69784
$910$	0.179003	0.00593390
$911$	14.8936	0.493446	0.246723	−	0.969086i	$-0.420646\pi$
$911$	14.8936	0.493446	0.246723	+	0.969086i	$0.420646\pi$
$912$	−14.8754	−0.492572
$913$	−14.5804	−0.482542
$914$	−0.801591	−0.0265143
$915$	10.4802	0.346463
$916$	−15.3780	−0.508105
$917$	0.695236	0.0229587
$918$	−0.862334	−0.0284613
$919$	49.7858	1.64228	0.821141	−	0.570725i	$-0.193338\pi$
$919$	49.7858	1.64228	0.821141	+	0.570725i	$0.193338\pi$
$920$	12.4182	0.409417
$921$	−13.2182	−0.435555
$922$	2.28199	0.0751533
$923$	−61.4717	−2.02337
$924$	0.114962	0.00378198
$925$	43.1357	1.41829
$926$	−1.26119	−0.0414451
$927$	4.61892	0.151705
$928$	−1.63625	−0.0537124
$929$	−18.0145	−0.591037	−0.295519	−	0.955337i	$-0.595492\pi$
$929$	−18.0145	−0.591037	−0.295519	+	0.955337i	$0.595492\pi$
$930$	−0.0400177	−0.00131223
$931$	−57.2777	−1.87720
$932$	−9.85174	−0.322705
$933$	7.77741	0.254621
$934$	−0.0728232	−0.00238285
$935$	16.3793	0.535660
$936$	−7.15587	−0.233897
$937$	−1.75707	−0.0574008	−0.0287004	−	0.999588i	$-0.509137\pi$
$937$	−1.75707	−0.0574008	−0.0287004	+	0.999588i	$0.509137\pi$
$938$	−0.105147	−0.00343317
$939$	0.370374	0.0120867
$940$	40.7328	1.32856
$941$	59.0981	1.92654	0.963271	−	0.268529i	$-0.0865377\pi$
$941$	59.0981	1.92654	0.963271	+	0.268529i	$0.0865377\pi$
$942$	−0.719090	−0.0234292
$943$	−23.7642	−0.773869
$944$	−4.28899	−0.139595
$945$	−0.748639	−0.0243532
$946$	0.148978	0.00484370
$947$	−17.7707	−0.577469	−0.288735	−	0.957409i	$-0.593235\pi$
$947$	−17.7707	−0.577469	−0.288735	+	0.957409i	$0.593235\pi$
$948$	−8.19712	−0.266230
$949$	46.6958	1.51581
$950$	12.2355	0.396971
$951$	11.5547	0.374688
$952$	0.0884486	0.00286663
$953$	−47.2170	−1.52951	−0.764754	−	0.644323i	$-0.777139\pi$
$953$	−47.2170	−1.52951	−0.764754	+	0.644323i	$0.777139\pi$
$954$	0.460455	0.0149078
$955$	76.3066	2.46923
$956$	33.9989	1.09960
$957$	0.833365	0.0269389
$958$	2.86290	0.0924960
$959$	0.530828	0.0171413
$960$	−14.0312	−0.452855
$961$	−30.9757	−0.999217
$962$	−2.57648	−0.0830689
$963$	25.0411	0.806939
$964$	37.0151	1.19218
$965$	−16.4214	−0.528625
$966$	−0.0255335	−0.000821528 0
$967$	−14.9803	−0.481735	−0.240867	−	0.970558i	$-0.577432\pi$
$967$	−14.9803	−0.481735	−0.240867	+	0.970558i	$0.577432\pi$
$968$	4.30066	0.138228
$969$	8.84334	0.284089
$970$	−6.15810	−0.197725
$971$	−28.5503	−0.916223	−0.458111	−	0.888895i	$-0.651474\pi$
$971$	−28.5503	−0.916223	−0.458111	+	0.888895i	$0.651474\pi$
$972$	22.6198	0.725530
$973$	−0.0696388	−0.00223252
$974$	0.585875	0.0187727
$975$	23.6701	0.758051
$976$	21.9017	0.701055
$977$	6.66918	0.213366	0.106683	−	0.994293i	$-0.465977\pi$
$977$	6.66918	0.213366	0.106683	+	0.994293i	$0.465977\pi$
$978$	−0.197339	−0.00631020
$979$	5.37509	0.171789
$980$	−55.1145	−1.76057
$981$	44.0054	1.40498
$982$	1.15068	0.0367196
$983$	−48.1948	−1.53718	−0.768588	−	0.639744i	$-0.779040\pi$
$983$	−48.1948	−1.53718	−0.768588	+	0.639744i	$0.779040\pi$
$984$	−1.07557	−0.0342881
$985$	−58.3612	−1.85954
$986$	0.319047	0.0101605
$987$	−0.168310	−0.00535738
$988$	75.9021	2.41477
$989$	3.43655	0.109276
$990$	2.72354	0.0865597
$991$	6.60497	0.209814	0.104907	−	0.994482i	$-0.466546\pi$
$991$	6.60497	0.209814	0.104907	+	0.994482i	$0.466546\pi$
$992$	−0.254979	−0.00809559
$993$	−2.44924	−0.0777244
$994$	0.126340	0.00400727
$995$	15.7552	0.499474
$996$	−7.57481	−0.240017
$997$	35.6959	1.13050	0.565250	−	0.824920i	$-0.308780\pi$
$997$	35.6959	1.13050	0.565250	+	0.824920i	$0.308780\pi$
$998$	−3.65710	−0.115764
$999$	10.7755	0.340922

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.33	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.33	✓	61	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q+0.138106 q^{2} -0.467503 q^{3} -1.98093 q^{4} -3.97741 q^{5} -0.0645649 q^{6} +0.0696388 q^{7} -0.549789 q^{8} -2.78144 q^{9} +O(q^{10})\)	\(q+0.138106 q^{2} -0.467503 q^{3} -1.98093 q^{4} -3.97741 q^{5} -0.0645649 q^{6} +0.0696388 q^{7} -0.549789 q^{8} -2.78144 q^{9} -0.549304 q^{10} +1.78259 q^{11} +0.926089 q^{12} -4.67947 q^{13} +0.00961753 q^{14} +1.85945 q^{15} +3.88592 q^{16} -2.31017 q^{17} -0.384133 q^{18} +8.18821 q^{19} +7.87896 q^{20} -0.0325564 q^{21} +0.246186 q^{22} +5.67889 q^{23} +0.257028 q^{24} +10.8198 q^{25} -0.646262 q^{26} +2.70284 q^{27} -0.137949 q^{28} -1.00000 q^{29} +0.256801 q^{30} -0.155832 q^{31} +1.63625 q^{32} -0.833365 q^{33} -0.319047 q^{34} -0.276982 q^{35} +5.50983 q^{36} +3.98674 q^{37} +1.13084 q^{38} +2.18767 q^{39} +2.18674 q^{40} -4.18466 q^{41} -0.00449622 q^{42} +0.605145 q^{43} -3.53118 q^{44} +11.0629 q^{45} +0.784287 q^{46} +5.16981 q^{47} -1.81668 q^{48} -6.99515 q^{49} +1.49428 q^{50} +1.08001 q^{51} +9.26969 q^{52} -1.19869 q^{53} +0.373278 q^{54} -7.09009 q^{55} -0.0382867 q^{56} -3.82801 q^{57} -0.138106 q^{58} -1.10372 q^{59} -3.68344 q^{60} +5.63615 q^{61} -0.0215212 q^{62} -0.193696 q^{63} -7.54587 q^{64} +18.6122 q^{65} -0.115093 q^{66} -10.9328 q^{67} +4.57627 q^{68} -2.65490 q^{69} -0.0382529 q^{70} +13.1365 q^{71} +1.52921 q^{72} -9.97887 q^{73} +0.550591 q^{74} -5.05829 q^{75} -16.2202 q^{76} +0.124137 q^{77} +0.302129 q^{78} -8.85133 q^{79} -15.4559 q^{80} +7.08074 q^{81} -0.577926 q^{82} -8.17936 q^{83} +0.0644918 q^{84} +9.18848 q^{85} +0.0835740 q^{86} +0.467503 q^{87} -0.980048 q^{88} +3.01533 q^{89} +1.52786 q^{90} -0.325873 q^{91} -11.2495 q^{92} +0.0728517 q^{93} +0.713981 q^{94} -32.5679 q^{95} -0.764950 q^{96} +11.2107 q^{97} -0.966071 q^{98} -4.95817 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

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