LMFDB - Embedded newform 4031.2.a.c.1.34

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			1.34
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)

$f(q)$	$=$	$q+0.148593 q^{2} -0.406255 q^{3} -1.97792 q^{4} +2.45728 q^{5} -0.0603664 q^{6} +1.94332 q^{7} -0.591089 q^{8} -2.83496 q^{9} +O(q^{10})$	\(q+0.148593 q^{2} -0.406255 q^{3} -1.97792 q^{4} +2.45728 q^{5} -0.0603664 q^{6} +1.94332 q^{7} -0.591089 q^{8} -2.83496 q^{9} +0.365134 q^{10} -3.61318 q^{11} +0.803539 q^{12} +2.14245 q^{13} +0.288763 q^{14} -0.998281 q^{15} +3.86801 q^{16} +1.64189 q^{17} -0.421254 q^{18} -3.09547 q^{19} -4.86030 q^{20} -0.789484 q^{21} -0.536892 q^{22} +1.14788 q^{23} +0.240133 q^{24} +1.03823 q^{25} +0.318352 q^{26} +2.37048 q^{27} -3.84374 q^{28} -1.00000 q^{29} -0.148337 q^{30} -4.10731 q^{31} +1.75694 q^{32} +1.46787 q^{33} +0.243973 q^{34} +4.77529 q^{35} +5.60732 q^{36} +4.64405 q^{37} -0.459964 q^{38} -0.870379 q^{39} -1.45247 q^{40} -6.43978 q^{41} -0.117311 q^{42} +5.49462 q^{43} +7.14658 q^{44} -6.96629 q^{45} +0.170566 q^{46} +6.53587 q^{47} -1.57140 q^{48} -3.22350 q^{49} +0.154273 q^{50} -0.667026 q^{51} -4.23759 q^{52} -8.55973 q^{53} +0.352235 q^{54} -8.87860 q^{55} -1.14868 q^{56} +1.25755 q^{57} -0.148593 q^{58} -9.77078 q^{59} +1.97452 q^{60} +5.39585 q^{61} -0.610316 q^{62} -5.50924 q^{63} -7.47495 q^{64} +5.26459 q^{65} +0.218115 q^{66} -13.1007 q^{67} -3.24753 q^{68} -0.466330 q^{69} +0.709573 q^{70} -4.73466 q^{71} +1.67571 q^{72} -11.4728 q^{73} +0.690072 q^{74} -0.421785 q^{75} +6.12260 q^{76} -7.02158 q^{77} -0.129332 q^{78} +5.60125 q^{79} +9.50478 q^{80} +7.54185 q^{81} -0.956903 q^{82} +12.2758 q^{83} +1.56154 q^{84} +4.03459 q^{85} +0.816460 q^{86} +0.406255 q^{87} +2.13571 q^{88} -7.23495 q^{89} -1.03514 q^{90} +4.16347 q^{91} -2.27041 q^{92} +1.66861 q^{93} +0.971183 q^{94} -7.60644 q^{95} -0.713763 q^{96} -9.07685 q^{97} -0.478988 q^{98} +10.2432 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.148593	0.105071	0.0525354	−	0.998619i	$-0.483270\pi$
$2$	0.148593	0.105071	0.0525354	+	0.998619i	$0.483270\pi$
$3$	−0.406255	−0.234551	−0.117276	−	0.993099i	$-0.537416\pi$
$3$	−0.406255	−0.234551	−0.117276	+	0.993099i	$0.537416\pi$
$4$	−1.97792	−0.988960
$5$	2.45728	1.09893	0.549465	−	0.835517i	$-0.314832\pi$
$5$	2.45728	1.09893	0.549465	+	0.835517i	$0.314832\pi$
$6$	−0.0603664	−0.0246445
$7$	1.94332	0.734507	0.367254	−	0.930121i	$-0.380298\pi$
$7$	1.94332	0.734507	0.367254	+	0.930121i	$0.380298\pi$
$8$	−0.591089	−0.208982
$9$	−2.83496	−0.944986
$10$	0.365134	0.115465
$11$	−3.61318	−1.08941	−0.544707	−	0.838626i	$-0.683359\pi$
$11$	−3.61318	−1.08941	−0.544707	+	0.838626i	$0.683359\pi$
$12$	0.803539	0.231962
$13$	2.14245	0.594208	0.297104	−	0.954845i	$-0.403979\pi$
$13$	2.14245	0.594208	0.297104	+	0.954845i	$0.403979\pi$
$14$	0.288763	0.0771753
$15$	−0.998281	−0.257755
$16$	3.86801	0.967002
$17$	1.64189	0.398217	0.199109	−	0.979977i	$-0.436195\pi$
$17$	1.64189	0.398217	0.199109	+	0.979977i	$0.436195\pi$
$18$	−0.421254	−0.0992904
$19$	−3.09547	−0.710150	−0.355075	−	0.934838i	$-0.615545\pi$
$19$	−3.09547	−0.710150	−0.355075	+	0.934838i	$0.615545\pi$
$20$	−4.86030	−1.08680
$21$	−0.789484	−0.172279
$22$	−0.536892	−0.114466
$23$	1.14788	0.239349	0.119674	−	0.992813i	$-0.461815\pi$
$23$	1.14788	0.239349	0.119674	+	0.992813i	$0.461815\pi$
$24$	0.240133	0.0490169
$25$	1.03823	0.207646
$26$	0.318352	0.0624339
$27$	2.37048	0.456199
$28$	−3.84374	−0.726398
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−0.148337	−0.0270825
$31$	−4.10731	−0.737695	−0.368848	−	0.929490i	$-0.620248\pi$
$31$	−4.10731	−0.737695	−0.368848	+	0.929490i	$0.620248\pi$
$32$	1.75694	0.310585
$33$	1.46787	0.255524
$34$	0.243973	0.0418410
$35$	4.77529	0.807171
$36$	5.60732	0.934553
$37$	4.64405	0.763477	0.381739	−	0.924270i	$-0.375325\pi$
$37$	4.64405	0.763477	0.381739	+	0.924270i	$0.375325\pi$
$38$	−0.459964	−0.0746161
$39$	−0.870379	−0.139372
$40$	−1.45247	−0.229656
$41$	−6.43978	−1.00572	−0.502862	−	0.864367i	$-0.667720\pi$
$41$	−6.43978	−1.00572	−0.502862	+	0.864367i	$0.667720\pi$
$42$	−0.117311	−0.0181015
$43$	5.49462	0.837922	0.418961	−	0.908004i	$-0.362394\pi$
$43$	5.49462	0.837922	0.418961	+	0.908004i	$0.362394\pi$
$44$	7.14658	1.07739
$45$	−6.96629	−1.03847
$46$	0.170566	0.0251486
$47$	6.53587	0.953355	0.476678	−	0.879078i	$-0.341841\pi$
$47$	6.53587	0.953355	0.476678	+	0.879078i	$0.341841\pi$
$48$	−1.57140	−0.226811
$49$	−3.22350	−0.460499
$50$	0.154273	0.0218175
$51$	−0.667026	−0.0934023
$52$	−4.23759	−0.587648
$53$	−8.55973	−1.17577	−0.587885	−	0.808945i	$-0.700039\pi$
$53$	−8.55973	−1.17577	−0.587885	+	0.808945i	$0.700039\pi$
$54$	0.352235	0.0479332
$55$	−8.87860	−1.19719
$56$	−1.14868	−0.153499
$57$	1.25755	0.166567
$58$	−0.148593	−0.0195112
$59$	−9.77078	−1.27205	−0.636024	−	0.771670i	$-0.719422\pi$
$59$	−9.77078	−1.27205	−0.636024	+	0.771670i	$0.719422\pi$
$60$	1.97452	0.254910
$61$	5.39585	0.690867	0.345434	−	0.938443i	$-0.387732\pi$
$61$	5.39585	0.690867	0.345434	+	0.938443i	$0.387732\pi$
$62$	−0.610316	−0.0775103
$63$	−5.50924	−0.694099
$64$	−7.47495	−0.934369
$65$	5.26459	0.652992
$66$	0.218115	0.0268481
$67$	−13.1007	−1.60050	−0.800251	−	0.599665i	$-0.795301\pi$
$67$	−13.1007	−1.60050	−0.800251	+	0.599665i	$0.795301\pi$
$68$	−3.24753	−0.393821
$69$	−0.466330	−0.0561396
$70$	0.709573	0.0848102
$71$	−4.73466	−0.561900	−0.280950	−	0.959722i	$-0.590650\pi$
$71$	−4.73466	−0.561900	−0.280950	+	0.959722i	$0.590650\pi$
$72$	1.67571	0.197485
$73$	−11.4728	−1.34279	−0.671393	−	0.741102i	$-0.734304\pi$
$73$	−11.4728	−1.34279	−0.671393	+	0.741102i	$0.734304\pi$
$74$	0.690072	0.0802192
$75$	−0.421785	−0.0487035
$76$	6.12260	0.702310
$77$	−7.02158	−0.800183
$78$	−0.129332	−0.0146439
$79$	5.60125	0.630190	0.315095	−	0.949060i	$-0.397964\pi$
$79$	5.60125	0.630190	0.315095	+	0.949060i	$0.397964\pi$
$80$	9.50478	1.06267
$81$	7.54185	0.837984
$82$	−0.956903	−0.105672
$83$	12.2758	1.34744	0.673719	−	0.738987i	$-0.264696\pi$
$83$	12.2758	1.34744	0.673719	+	0.738987i	$0.264696\pi$
$84$	1.56154	0.170378
$85$	4.03459	0.437612
$86$	0.816460	0.0880411
$87$	0.406255	0.0435551
$88$	2.13571	0.227668
$89$	−7.23495	−0.766903	−0.383452	−	0.923561i	$-0.625265\pi$
$89$	−7.23495	−0.766903	−0.383452	+	0.923561i	$0.625265\pi$
$90$	−1.03514	−0.109113
$91$	4.16347	0.436450
$92$	−2.27041	−0.236707
$93$	1.66861	0.173027
$94$	0.971183	0.100170
$95$	−7.60644	−0.780405
$96$	−0.713763	−0.0728482
$97$	−9.07685	−0.921615	−0.460807	−	0.887500i	$-0.652440\pi$
$97$	−9.07685	−0.921615	−0.460807	+	0.887500i	$0.652440\pi$
$98$	−0.478988	−0.0483851
$99$	10.2432	1.02948
$100$	−2.05353	−0.205353
$101$	9.68517	0.963711	0.481855	−	0.876251i	$-0.339963\pi$
$101$	9.68517	0.963711	0.481855	+	0.876251i	$0.339963\pi$
$102$	−0.0991151	−0.00981385
$103$	14.6611	1.44460	0.722302	−	0.691577i	$-0.243084\pi$
$103$	14.6611	1.44460	0.722302	+	0.691577i	$0.243084\pi$
$104$	−1.26638	−0.124179
$105$	−1.93998	−0.189323
$106$	−1.27191	−0.123539
$107$	−1.89728	−0.183417	−0.0917083	−	0.995786i	$-0.529233\pi$
$107$	−1.89728	−0.183417	−0.0917083	+	0.995786i	$0.529233\pi$
$108$	−4.68862	−0.451162
$109$	−1.98286	−0.189923	−0.0949616	−	0.995481i	$-0.530273\pi$
$109$	−1.98286	−0.189923	−0.0949616	+	0.995481i	$0.530273\pi$
$110$	−1.31929	−0.125790
$111$	−1.88667	−0.179074
$112$	7.51679	0.710270
$113$	−11.3367	−1.06647	−0.533236	−	0.845967i	$-0.679024\pi$
$113$	−11.3367	−1.06647	−0.533236	+	0.845967i	$0.679024\pi$
$114$	0.186863	0.0175013
$115$	2.82066	0.263028
$116$	1.97792	0.183645
$117$	−6.07375	−0.561518
$118$	−1.45187	−0.133655
$119$	3.19072	0.292493
$120$	0.590074	0.0538661
$121$	2.05507	0.186825
$122$	0.801783	0.0725900
$123$	2.61619	0.235894
$124$	8.12394	0.729551
$125$	−9.73519	−0.870741
$126$	−0.818632	−0.0729295
$127$	−11.2300	−0.996498	−0.498249	−	0.867034i	$-0.666023\pi$
$127$	−11.2300	−0.996498	−0.498249	+	0.867034i	$0.666023\pi$
$128$	−4.62459	−0.408760
$129$	−2.23221	−0.196536
$130$	0.782279	0.0686104
$131$	8.90874	0.778360	0.389180	−	0.921162i	$-0.372758\pi$
$131$	8.90874	0.778360	0.389180	+	0.921162i	$0.372758\pi$
$132$	−2.90333	−0.252703
$133$	−6.01550	−0.521610
$134$	−1.94666	−0.168166
$135$	5.82493	0.501330
$136$	−0.970504	−0.0832201
$137$	16.0983	1.37537	0.687687	−	0.726007i	$-0.258626\pi$
$137$	16.0983	1.37537	0.687687	+	0.726007i	$0.258626\pi$
$138$	−0.0692932	−0.00589863
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	−9.44514	−0.798260
$141$	−2.65523	−0.223611
$142$	−0.703535	−0.0590393
$143$	−7.74105	−0.647339
$144$	−10.9656	−0.913803
$145$	−2.45728	−0.204066
$146$	−1.70477	−0.141088
$147$	1.30956	0.108011
$148$	−9.18556	−0.755049
$149$	−12.7090	−1.04116	−0.520580	−	0.853813i	$-0.674284\pi$
$149$	−12.7090	−1.04116	−0.520580	+	0.853813i	$0.674284\pi$
$150$	−0.0626741	−0.00511732
$151$	−7.99423	−0.650561	−0.325280	−	0.945618i	$-0.605459\pi$
$151$	−7.99423	−0.650561	−0.325280	+	0.945618i	$0.605459\pi$
$152$	1.82970	0.148408
$153$	−4.65469	−0.376309
$154$	−1.04335	−0.0840759
$155$	−10.0928	−0.810675
$156$	1.72154	0.137833
$157$	−19.7609	−1.57710	−0.788548	−	0.614974i	$-0.789167\pi$
$157$	−19.7609	−1.57710	−0.788548	+	0.614974i	$0.789167\pi$
$158$	0.832305	0.0662146
$159$	3.47743	0.275778
$160$	4.31729	0.341311
$161$	2.23070	0.175803
$162$	1.12066	0.0880477
$163$	−24.5400	−1.92212	−0.961060	−	0.276340i	$-0.910878\pi$
$163$	−24.5400	−1.92212	−0.961060	+	0.276340i	$0.910878\pi$
$164$	12.7374	0.994621
$165$	3.60697	0.280802
$166$	1.82409	0.141577
$167$	5.83808	0.451764	0.225882	−	0.974155i	$-0.427474\pi$
$167$	5.83808	0.451764	0.225882	+	0.974155i	$0.427474\pi$
$168$	0.466655	0.0360033
$169$	−8.40992	−0.646917
$170$	0.599510	0.0459803
$171$	8.77553	0.671082
$172$	−10.8679	−0.828671
$173$	22.4683	1.70823	0.854117	−	0.520080i	$-0.174098\pi$
$173$	22.4683	1.70823	0.854117	+	0.520080i	$0.174098\pi$
$174$	0.0603664	0.00457637
$175$	2.01761	0.152517
$176$	−13.9758	−1.05347
$177$	3.96942	0.298360
$178$	−1.07506	−0.0805792
$179$	−4.93537	−0.368887	−0.184444	−	0.982843i	$-0.559048\pi$
$179$	−4.93537	−0.368887	−0.184444	+	0.982843i	$0.559048\pi$
$180$	13.7788	1.02701
$181$	−3.61730	−0.268872	−0.134436	−	0.990922i	$-0.542922\pi$
$181$	−3.61730	−0.268872	−0.134436	+	0.990922i	$0.542922\pi$
$182$	0.618660	0.0458581
$183$	−2.19209	−0.162044
$184$	−0.678498	−0.0500195
$185$	11.4117	0.839008
$186$	0.247944	0.0181801
$187$	−5.93245	−0.433824
$188$	−12.9274	−0.942830
$189$	4.60660	0.335081
$190$	−1.13026	−0.0819978
$191$	11.7117	0.847427	0.423714	−	0.905796i	$-0.360726\pi$
$191$	11.7117	0.847427	0.423714	+	0.905796i	$0.360726\pi$
$192$	3.03673	0.219157
$193$	−0.502777	−0.0361907	−0.0180953	−	0.999836i	$-0.505760\pi$
$193$	−0.502777	−0.0361907	−0.0180953	+	0.999836i	$0.505760\pi$
$194$	−1.34875	−0.0968348
$195$	−2.13876	−0.153160
$196$	6.37582	0.455416
$197$	−13.2591	−0.944669	−0.472335	−	0.881419i	$-0.656589\pi$
$197$	−13.2591	−0.944669	−0.472335	+	0.881419i	$0.656589\pi$
$198$	1.52207	0.108168
$199$	6.86233	0.486458	0.243229	−	0.969969i	$-0.421793\pi$
$199$	6.86233	0.486458	0.243229	+	0.969969i	$0.421793\pi$
$200$	−0.613686	−0.0433941
$201$	5.32221	0.375400
$202$	1.43915	0.101258
$203$	−1.94332	−0.136395
$204$	1.31932	0.0923711
$205$	−15.8243	−1.10522
$206$	2.17854	0.151786
$207$	−3.25418	−0.226181
$208$	8.28700	0.574600
$209$	11.1845	0.773648
$210$	−0.288267	−0.0198923
$211$	−10.1628	−0.699633	−0.349817	−	0.936818i	$-0.613756\pi$
$211$	−10.1628	−0.699633	−0.349817	+	0.936818i	$0.613756\pi$
$212$	16.9305	1.16279
$213$	1.92348	0.131794
$214$	−0.281921	−0.0192717
$215$	13.5018	0.920817
$216$	−1.40116	−0.0953372
$217$	−7.98184	−0.541842
$218$	−0.294638	−0.0199554
$219$	4.66086	0.314952
$220$	17.5612	1.18397
$221$	3.51766	0.236624
$222$	−0.280345	−0.0188155
$223$	−4.19293	−0.280779	−0.140390	−	0.990096i	$-0.544836\pi$
$223$	−4.19293	−0.280779	−0.140390	+	0.990096i	$0.544836\pi$
$224$	3.41429	0.228127
$225$	−2.94333	−0.196222
$226$	−1.68456	−0.112055
$227$	−19.2945	−1.28062	−0.640309	−	0.768117i	$-0.721194\pi$
$227$	−19.2945	−1.28062	−0.640309	+	0.768117i	$0.721194\pi$
$228$	−2.48733	−0.164728
$229$	−26.5171	−1.75230	−0.876148	−	0.482042i	$-0.839895\pi$
$229$	−26.5171	−1.75230	−0.876148	+	0.482042i	$0.839895\pi$
$230$	0.419129	0.0276365
$231$	2.85255	0.187684
$232$	0.591089	0.0388069
$233$	−24.0920	−1.57832	−0.789159	−	0.614189i	$-0.789483\pi$
$233$	−24.0920	−1.57832	−0.789159	+	0.614189i	$0.789483\pi$
$234$	−0.902514	−0.0589992
$235$	16.0605	1.04767
$236$	19.3258	1.25800
$237$	−2.27553	−0.147812
$238$	0.474118	0.0307325
$239$	−1.68028	−0.108689	−0.0543443	−	0.998522i	$-0.517307\pi$
$239$	−1.68028	−0.108689	−0.0543443	+	0.998522i	$0.517307\pi$
$240$	−3.86136	−0.249250
$241$	14.1597	0.912107	0.456053	−	0.889952i	$-0.349263\pi$
$241$	14.1597	0.912107	0.456053	+	0.889952i	$0.349263\pi$
$242$	0.305369	0.0196298
$243$	−10.1753	−0.652749
$244$	−10.6726	−0.683240
$245$	−7.92103	−0.506056
$246$	0.388746	0.0247856
$247$	−6.63189	−0.421977
$248$	2.42779	0.154165
$249$	−4.98708	−0.316043
$250$	−1.44658	−0.0914895
$251$	−3.25261	−0.205303	−0.102652	−	0.994717i	$-0.532733\pi$
$251$	−3.25261	−0.205303	−0.102652	+	0.994717i	$0.532733\pi$
$252$	10.8968	0.686436
$253$	−4.14749	−0.260750
$254$	−1.66869	−0.104703
$255$	−1.63907	−0.102642
$256$	14.2627	0.891420
$257$	−17.4032	−1.08558	−0.542790	−	0.839869i	$-0.682632\pi$
$257$	−17.4032	−1.08558	−0.542790	+	0.839869i	$0.682632\pi$
$258$	−0.331691	−0.0206502
$259$	9.02489	0.560779
$260$	−10.4129	−0.645783
$261$	2.83496	0.175479
$262$	1.32377	0.0817829
$263$	−0.746499	−0.0460311	−0.0230155	−	0.999735i	$-0.507327\pi$
$263$	−0.746499	−0.0460311	−0.0230155	+	0.999735i	$0.507327\pi$
$264$	−0.867643	−0.0533997
$265$	−21.0337	−1.29209
$266$	−0.893859	−0.0548060
$267$	2.93923	0.179878
$268$	25.9121	1.58283
$269$	−12.3435	−0.752595	−0.376298	−	0.926499i	$-0.622803\pi$
$269$	−12.3435	−0.752595	−0.376298	+	0.926499i	$0.622803\pi$
$270$	0.865541	0.0526752
$271$	17.7184	1.07631	0.538157	−	0.842844i	$-0.319121\pi$
$271$	17.7184	1.07631	0.538157	+	0.842844i	$0.319121\pi$
$272$	6.35085	0.385077
$273$	−1.69143	−0.102370
$274$	2.39209	0.144512
$275$	−3.75130	−0.226212
$276$	0.922364	0.0555198
$277$	−4.93444	−0.296482	−0.148241	−	0.988951i	$-0.547361\pi$
$277$	−4.93444	−0.296482	−0.148241	+	0.988951i	$0.547361\pi$
$278$	−0.148593	−0.00891199
$279$	11.6441	0.697112
$280$	−2.82262	−0.168684
$281$	−0.0903245	−0.00538831	−0.00269415	−	0.999996i	$-0.500858\pi$
$281$	−0.0903245	−0.00538831	−0.00269415	+	0.999996i	$0.500858\pi$
$282$	−0.394547	−0.0234949
$283$	1.94947	0.115884	0.0579421	−	0.998320i	$-0.481546\pi$
$283$	1.94947	0.115884	0.0579421	+	0.998320i	$0.481546\pi$
$284$	9.36478	0.555697
$285$	3.09015	0.183045
$286$	−1.15026	−0.0680164
$287$	−12.5146	−0.738712
$288$	−4.98084	−0.293499
$289$	−14.3042	−0.841423
$290$	−0.365134	−0.0214414
$291$	3.68751	0.216166
$292$	22.6922	1.32796
$293$	−9.14092	−0.534018	−0.267009	−	0.963694i	$-0.586035\pi$
$293$	−9.14092	−0.534018	−0.267009	+	0.963694i	$0.586035\pi$
$294$	0.194591	0.0113488
$295$	−24.0095	−1.39789
$296$	−2.74505	−0.159553
$297$	−8.56496	−0.496990
$298$	−1.88846	−0.109396
$299$	2.45927	0.142223
$300$	0.834257	0.0481658
$301$	10.6778	0.615459
$302$	−1.18788	−0.0683550
$303$	−3.93465	−0.226039
$304$	−11.9733	−0.686717
$305$	13.2591	0.759214
$306$	−0.691653	−0.0395391
$307$	10.1112	0.577077	0.288538	−	0.957468i	$-0.406831\pi$
$307$	10.1112	0.577077	0.288538	+	0.957468i	$0.406831\pi$
$308$	13.8881	0.791349
$309$	−5.95615	−0.338834
$310$	−1.49972	−0.0851783
$311$	11.0309	0.625503	0.312752	−	0.949835i	$-0.398749\pi$
$311$	11.0309	0.625503	0.312752	+	0.949835i	$0.398749\pi$
$312$	0.514472	0.0291262
$313$	1.56694	0.0885689	0.0442844	−	0.999019i	$-0.485899\pi$
$313$	1.56694	0.0885689	0.0442844	+	0.999019i	$0.485899\pi$
$314$	−2.93633	−0.165707
$315$	−13.5377	−0.762765
$316$	−11.0788	−0.623233
$317$	−0.165147	−0.00927556	−0.00463778	−	0.999989i	$-0.501476\pi$
$317$	−0.165147	−0.00927556	−0.00463778	+	0.999989i	$0.501476\pi$
$318$	0.516720	0.0289762
$319$	3.61318	0.202299
$320$	−18.3680	−1.02681
$321$	0.770777	0.0430206
$322$	0.331465	0.0184718
$323$	−5.08243	−0.282794
$324$	−14.9172	−0.828733
$325$	2.22435	0.123385
$326$	−3.64646	−0.201959
$327$	0.805544	0.0445467
$328$	3.80648	0.210178
$329$	12.7013	0.700246
$330$	0.535969	0.0295041
$331$	−0.128818	−0.00708047	−0.00354024	−	0.999994i	$-0.501127\pi$
$331$	−0.128818	−0.00708047	−0.00354024	+	0.999994i	$0.501127\pi$
$332$	−24.2805	−1.33256
$333$	−13.1657	−0.721475
$334$	0.867495	0.0474672
$335$	−32.1920	−1.75884
$336$	−3.05373	−0.166595
$337$	31.9154	1.73854	0.869270	−	0.494338i	$-0.164589\pi$
$337$	31.9154	1.73854	0.869270	+	0.494338i	$0.164589\pi$
$338$	−1.24965	−0.0679721
$339$	4.60560	0.250142
$340$	−7.98009	−0.432781
$341$	14.8405	0.803656
$342$	1.30398	0.0705111
$343$	−19.8676	−1.07275
$344$	−3.24781	−0.175110
$345$	−1.14590	−0.0616934
$346$	3.33863	0.179486
$347$	−16.6711	−0.894951	−0.447476	−	0.894296i	$-0.647677\pi$
$347$	−16.6711	−0.894951	−0.447476	+	0.894296i	$0.647677\pi$
$348$	−0.803539	−0.0430742
$349$	5.42984	0.290652	0.145326	−	0.989384i	$-0.453577\pi$
$349$	5.42984	0.290652	0.145326	+	0.989384i	$0.453577\pi$
$350$	0.299802	0.0160251
$351$	5.07862	0.271077
$352$	−6.34813	−0.338356
$353$	−34.3300	−1.82720	−0.913600	−	0.406614i	$-0.866709\pi$
$353$	−34.3300	−1.82720	−0.913600	+	0.406614i	$0.866709\pi$
$354$	0.589827	0.0313489
$355$	−11.6344	−0.617489
$356$	14.3102	0.758437
$357$	−1.29625	−0.0686046
$358$	−0.733360	−0.0387593
$359$	7.48370	0.394974	0.197487	−	0.980305i	$-0.436722\pi$
$359$	7.48370	0.394974	0.197487	+	0.980305i	$0.436722\pi$
$360$	4.11770	0.217022
$361$	−9.41805	−0.495687
$362$	−0.537505	−0.0282506
$363$	−0.834883	−0.0438200
$364$	−8.23500	−0.431631
$365$	−28.1918	−1.47563
$366$	−0.325728	−0.0170261
$367$	−9.12790	−0.476473	−0.238236	−	0.971207i	$-0.576569\pi$
$367$	−9.12790	−0.476473	−0.238236	+	0.971207i	$0.576569\pi$
$368$	4.44000	0.231451
$369$	18.2565	0.950395
$370$	1.69570	0.0881552
$371$	−16.6343	−0.863611
$372$	−3.30039	−0.171117
$373$	27.2530	1.41111	0.705554	−	0.708656i	$-0.250698\pi$
$373$	27.2530	1.41111	0.705554	+	0.708656i	$0.250698\pi$
$374$	−0.881518	−0.0455822
$375$	3.95496	0.204233
$376$	−3.86329	−0.199234
$377$	−2.14245	−0.110342
$378$	0.684507	0.0352072
$379$	24.3335	1.24993	0.624963	−	0.780654i	$-0.285114\pi$
$379$	24.3335	1.24993	0.624963	+	0.780654i	$0.285114\pi$
$380$	15.0449	0.771789
$381$	4.56222	0.233730
$382$	1.74027	0.0890399
$383$	3.17130	0.162046	0.0810229	−	0.996712i	$-0.474181\pi$
$383$	3.17130	0.162046	0.0810229	+	0.996712i	$0.474181\pi$
$384$	1.87876	0.0958752
$385$	−17.2540	−0.879344
$386$	−0.0747089	−0.00380258
$387$	−15.5770	−0.791824
$388$	17.9533	0.911440
$389$	6.26583	0.317690	0.158845	−	0.987304i	$-0.449223\pi$
$389$	6.26583	0.317690	0.158845	+	0.987304i	$0.449223\pi$
$390$	−0.317805	−0.0160927
$391$	1.88469	0.0953128
$392$	1.90537	0.0962359
$393$	−3.61921	−0.182565
$394$	−1.97020	−0.0992572
$395$	13.7639	0.692535
$396$	−20.2603	−1.01812
$397$	−17.7780	−0.892253	−0.446126	−	0.894970i	$-0.647197\pi$
$397$	−17.7780	−0.892253	−0.446126	+	0.894970i	$0.647197\pi$
$398$	1.01969	0.0511125
$399$	2.44383	0.122344
$400$	4.01587	0.200794
$401$	−10.7198	−0.535319	−0.267660	−	0.963514i	$-0.586250\pi$
$401$	−10.7198	−0.535319	−0.267660	+	0.963514i	$0.586250\pi$
$402$	0.790841	0.0394436
$403$	−8.79970	−0.438344
$404$	−19.1565	−0.953072
$405$	18.5325	0.920885
$406$	−0.288763	−0.0143311
$407$	−16.7798	−0.831744
$408$	0.394272	0.0195194
$409$	−2.82829	−0.139850	−0.0699251	−	0.997552i	$-0.522276\pi$
$409$	−2.82829	−0.139850	−0.0699251	+	0.997552i	$0.522276\pi$
$410$	−2.35138	−0.116126
$411$	−6.54002	−0.322596
$412$	−28.9986	−1.42866
$413$	−18.9878	−0.934328
$414$	−0.483547	−0.0237651
$415$	30.1650	1.48074
$416$	3.76414	0.184552
$417$	0.406255	0.0198944
$418$	1.66193	0.0812878
$419$	−11.9306	−0.582846	−0.291423	−	0.956594i	$-0.594129\pi$
$419$	−11.9306	−0.582846	−0.291423	+	0.956594i	$0.594129\pi$
$420$	3.83713	0.187233
$421$	−32.5505	−1.58641	−0.793207	−	0.608952i	$-0.791590\pi$
$421$	−32.5505	−1.58641	−0.793207	+	0.608952i	$0.791590\pi$
$422$	−1.51011	−0.0735111
$423$	−18.5289	−0.900907
$424$	5.05957	0.245714
$425$	1.70466	0.0826880
$426$	0.285814	0.0138477
$427$	10.4859	0.507447
$428$	3.75266	0.181392
$429$	3.14484	0.151834
$430$	2.00627	0.0967510
$431$	4.59726	0.221442	0.110721	−	0.993852i	$-0.464684\pi$
$431$	4.59726	0.221442	0.110721	+	0.993852i	$0.464684\pi$
$432$	9.16903	0.441145
$433$	−0.439356	−0.0211141	−0.0105571	−	0.999944i	$-0.503360\pi$
$433$	−0.439356	−0.0211141	−0.0105571	+	0.999944i	$0.503360\pi$
$434$	−1.18604	−0.0569318
$435$	0.998281	0.0478639
$436$	3.92193	0.187826
$437$	−3.55322	−0.169974
$438$	0.692570	0.0330923
$439$	−6.76965	−0.323098	−0.161549	−	0.986865i	$-0.551649\pi$
$439$	−6.76965	−0.323098	−0.161549	+	0.986865i	$0.551649\pi$
$440$	5.24805	0.250191
$441$	9.13847	0.435165
$442$	0.522699	0.0248622
$443$	20.4420	0.971229	0.485614	−	0.874173i	$-0.338596\pi$
$443$	20.4420	0.971229	0.485614	+	0.874173i	$0.338596\pi$
$444$	3.73168	0.177098
$445$	−17.7783	−0.842773
$446$	−0.623038	−0.0295017
$447$	5.16308	0.244205
$448$	−14.5262	−0.686300
$449$	40.7667	1.92390	0.961949	−	0.273228i	$-0.0880913\pi$
$449$	40.7667	1.92390	0.961949	+	0.273228i	$0.0880913\pi$
$450$	−0.437357	−0.0206172
$451$	23.2681	1.09565
$452$	22.4232	1.05470
$453$	3.24769	0.152590
$454$	−2.86701	−0.134556
$455$	10.2308	0.479627
$456$	−0.743324	−0.0348094
$457$	−5.26240	−0.246165	−0.123082	−	0.992396i	$-0.539278\pi$
$457$	−5.26240	−0.246165	−0.123082	+	0.992396i	$0.539278\pi$
$458$	−3.94024	−0.184115
$459$	3.89207	0.181666
$460$	−5.57903	−0.260124
$461$	−4.99238	−0.232518	−0.116259	−	0.993219i	$-0.537090\pi$
$461$	−4.99238	−0.232518	−0.116259	+	0.993219i	$0.537090\pi$
$462$	0.423867	0.0197201
$463$	−2.81692	−0.130913	−0.0654567	−	0.997855i	$-0.520850\pi$
$463$	−2.81692	−0.130913	−0.0654567	+	0.997855i	$0.520850\pi$
$464$	−3.86801	−0.179568
$465$	4.10025	0.190145
$466$	−3.57989	−0.165835
$467$	−1.11958	−0.0518081	−0.0259041	−	0.999664i	$-0.508246\pi$
$467$	−1.11958	−0.0518081	−0.0259041	+	0.999664i	$0.508246\pi$
$468$	12.0134	0.555319
$469$	−25.4588	−1.17558
$470$	2.38647	0.110080
$471$	8.02797	0.369909
$472$	5.77541	0.265835
$473$	−19.8531	−0.912845
$474$	−0.338128	−0.0155307
$475$	−3.21381	−0.147460
$476$	−6.31100	−0.289264
$477$	24.2665	1.11109
$478$	−0.249678	−0.0114200
$479$	−25.5151	−1.16581	−0.582907	−	0.812539i	$-0.698085\pi$
$479$	−25.5151	−1.16581	−0.582907	+	0.812539i	$0.698085\pi$
$480$	−1.75392	−0.0800550
$481$	9.94963	0.453664
$482$	2.10403	0.0958358
$483$	−0.906230	−0.0412349
$484$	−4.06477	−0.184762
$485$	−22.3044	−1.01279
$486$	−1.51198	−0.0685848
$487$	38.4583	1.74271	0.871356	−	0.490651i	$-0.163241\pi$
$487$	38.4583	1.74271	0.871356	+	0.490651i	$0.163241\pi$
$488$	−3.18943	−0.144379
$489$	9.96948	0.450835
$490$	−1.17701	−0.0531718
$491$	18.4039	0.830558	0.415279	−	0.909694i	$-0.363684\pi$
$491$	18.4039	0.830558	0.415279	+	0.909694i	$0.363684\pi$
$492$	−5.17461	−0.233290
$493$	−1.64189	−0.0739471
$494$	−0.985449	−0.0443374
$495$	25.1704	1.13133
$496$	−15.8871	−0.713353
$497$	−9.20097	−0.412720
$498$	−0.741043	−0.0332069
$499$	−31.0066	−1.38805	−0.694023	−	0.719953i	$-0.744164\pi$
$499$	−31.0066	−1.38805	−0.694023	+	0.719953i	$0.744164\pi$
$500$	19.2554	0.861129
$501$	−2.37175	−0.105962
$502$	−0.483314	−0.0215714
$503$	−21.5946	−0.962855	−0.481427	−	0.876486i	$-0.659882\pi$
$503$	−21.5946	−0.962855	−0.481427	+	0.876486i	$0.659882\pi$
$504$	3.25645	0.145054
$505$	23.7992	1.05905
$506$	−0.616286	−0.0273973
$507$	3.41657	0.151735
$508$	22.2120	0.985497
$509$	14.9994	0.664837	0.332419	−	0.943132i	$-0.392135\pi$
$509$	14.9994	0.664837	0.332419	+	0.943132i	$0.392135\pi$
$510$	−0.243554	−0.0107847
$511$	−22.2953	−0.986286
$512$	11.3685	0.502423
$513$	−7.33775	−0.323970
$514$	−2.58598	−0.114063
$515$	36.0265	1.58752
$516$	4.41514	0.194366
$517$	−23.6153	−1.03860
$518$	1.34103	0.0589216
$519$	−9.12786	−0.400668
$520$	−3.11185	−0.136463
$521$	−33.1567	−1.45262	−0.726310	−	0.687367i	$-0.758766\pi$
$521$	−33.1567	−1.45262	−0.726310	+	0.687367i	$0.758766\pi$
$522$	0.421254	0.0184378
$523$	19.1259	0.836318	0.418159	−	0.908374i	$-0.362675\pi$
$523$	19.1259	0.836318	0.418159	+	0.908374i	$0.362675\pi$
$524$	−17.6208	−0.769767
$525$	−0.819664	−0.0357731
$526$	−0.110924	−0.00483652
$527$	−6.74376	−0.293763
$528$	5.67774	0.247092
$529$	−21.6824	−0.942712
$530$	−3.12545	−0.135761
$531$	27.6997	1.20207
$532$	11.8982	0.515852
$533$	−13.7969	−0.597609
$534$	0.436748	0.0188999
$535$	−4.66214	−0.201562
$536$	7.74367	0.334476
$537$	2.00502	0.0865229
$538$	−1.83415	−0.0790758
$539$	11.6471	0.501675
$540$	−11.5212	−0.495795
$541$	31.9252	1.37257	0.686286	−	0.727332i	$-0.259240\pi$
$541$	31.9252	1.37257	0.686286	+	0.727332i	$0.259240\pi$
$542$	2.63282	0.113089
$543$	1.46955	0.0630643
$544$	2.88470	0.123680
$545$	−4.87243	−0.208712
$546$	−0.251333	−0.0107561
$547$	28.5716	1.22164	0.610818	−	0.791771i	$-0.290841\pi$
$547$	28.5716	1.22164	0.610818	+	0.791771i	$0.290841\pi$
$548$	−31.8412	−1.36019
$549$	−15.2970	−0.652860
$550$	−0.557416	−0.0237683
$551$	3.09547	0.131872
$552$	0.275643	0.0117321
$553$	10.8850	0.462879
$554$	−0.733221	−0.0311516
$555$	−4.63607	−0.196790
$556$	1.97792	0.0838825
$557$	17.6925	0.749656	0.374828	−	0.927094i	$-0.377702\pi$
$557$	17.6925	0.749656	0.374828	+	0.927094i	$0.377702\pi$
$558$	1.73022	0.0732461
$559$	11.7719	0.497900
$560$	18.4709	0.780536
$561$	2.41008	0.101754
$562$	−0.0134216	−0.000566154 0
$563$	−13.8617	−0.584200	−0.292100	−	0.956388i	$-0.594354\pi$
$563$	−13.8617	−0.584200	−0.292100	+	0.956388i	$0.594354\pi$
$564$	5.25183	0.221142
$565$	−27.8576	−1.17198
$566$	0.289677	0.0121761
$567$	14.6563	0.615505
$568$	2.79861	0.117427
$569$	40.1240	1.68208	0.841042	−	0.540970i	$-0.181943\pi$
$569$	40.1240	1.68208	0.841042	+	0.540970i	$0.181943\pi$
$570$	0.459174	0.0192327
$571$	40.4541	1.69295	0.846476	−	0.532427i	$-0.178720\pi$
$571$	40.4541	1.69295	0.846476	+	0.532427i	$0.178720\pi$
$572$	15.3112	0.640192
$573$	−4.75792	−0.198765
$574$	−1.85957	−0.0776170
$575$	1.19176	0.0496997
$576$	21.1912	0.882965
$577$	25.2784	1.05235	0.526177	−	0.850375i	$-0.323625\pi$
$577$	25.2784	1.05235	0.526177	+	0.850375i	$0.323625\pi$
$578$	−2.12550	−0.0884090
$579$	0.204255	0.00848856
$580$	4.86030	0.201813
$581$	23.8558	0.989703
$582$	0.547937	0.0227127
$583$	30.9278	1.28090
$584$	6.78143	0.280618
$585$	−14.9249	−0.617069
$586$	−1.35827	−0.0561097
$587$	15.3280	0.632655	0.316327	−	0.948650i	$-0.397550\pi$
$587$	15.3280	0.632655	0.316327	+	0.948650i	$0.397550\pi$
$588$	−2.59020	−0.106818
$589$	12.7141	0.523874
$590$	−3.56764	−0.146877
$591$	5.38655	0.221573
$592$	17.9632	0.738284
$593$	−37.6517	−1.54617	−0.773084	−	0.634304i	$-0.781287\pi$
$593$	−37.6517	−1.54617	−0.773084	+	0.634304i	$0.781287\pi$
$594$	−1.27269	−0.0522191
$595$	7.84051	0.321429
$596$	25.1374	1.02967
$597$	−2.78785	−0.114099
$598$	0.365429	0.0149435
$599$	8.40067	0.343242	0.171621	−	0.985163i	$-0.445100\pi$
$599$	8.40067	0.343242	0.171621	+	0.985163i	$0.445100\pi$
$600$	0.249313	0.0101781
$601$	19.3777	0.790435	0.395217	−	0.918588i	$-0.370669\pi$
$601$	19.3777	0.790435	0.395217	+	0.918588i	$0.370669\pi$
$602$	1.58665	0.0646668
$603$	37.1399	1.51245
$604$	15.8119	0.643379
$605$	5.04989	0.205307
$606$	−0.584659	−0.0237502
$607$	16.0842	0.652837	0.326419	−	0.945225i	$-0.394158\pi$
$607$	16.0842	0.652837	0.326419	+	0.945225i	$0.394158\pi$
$608$	−5.43855	−0.220562
$609$	0.789484	0.0319915
$610$	1.97021	0.0797713
$611$	14.0028	0.566491
$612$	9.20661	0.372155
$613$	−38.0263	−1.53587	−0.767934	−	0.640529i	$-0.778715\pi$
$613$	−38.0263	−1.53587	−0.767934	+	0.640529i	$0.778715\pi$
$614$	1.50245	0.0606340
$615$	6.42871	0.259231
$616$	4.15038	0.167224
$617$	38.2789	1.54105	0.770525	−	0.637410i	$-0.219994\pi$
$617$	38.2789	1.54105	0.770525	+	0.637410i	$0.219994\pi$
$618$	−0.885040	−0.0356015
$619$	12.8251	0.515485	0.257743	−	0.966214i	$-0.417021\pi$
$619$	12.8251	0.515485	0.257743	+	0.966214i	$0.417021\pi$
$620$	19.9628	0.801725
$621$	2.72102	0.109191
$622$	1.63911	0.0657221
$623$	−14.0598	−0.563296
$624$	−3.36663	−0.134773
$625$	−29.1132	−1.16453
$626$	0.232836	0.00930600
$627$	−4.54375	−0.181460
$628$	39.0856	1.55968
$629$	7.62503	0.304030
$630$	−2.01161	−0.0801444
$631$	16.3265	0.649948	0.324974	−	0.945723i	$-0.394645\pi$
$631$	16.3265	0.649948	0.324974	+	0.945723i	$0.394645\pi$
$632$	−3.31084	−0.131698
$633$	4.12867	0.164100
$634$	−0.0245396	−0.000974591 0
$635$	−27.5952	−1.09508
$636$	−6.87808	−0.272733
$637$	−6.90617	−0.273632
$638$	0.536892	0.0212558
$639$	13.4226	0.530988
$640$	−11.3639	−0.449199
$641$	−17.1372	−0.676878	−0.338439	−	0.940988i	$-0.609899\pi$
$641$	−17.1372	−0.676878	−0.338439	+	0.940988i	$0.609899\pi$
$642$	0.114532	0.00452021
$643$	4.49404	0.177228	0.0886139	−	0.996066i	$-0.471756\pi$
$643$	4.49404	0.177228	0.0886139	+	0.996066i	$0.471756\pi$
$644$	−4.41214	−0.173863
$645$	−5.48518	−0.215979
$646$	−0.755211	−0.0297134
$647$	−22.1955	−0.872597	−0.436299	−	0.899802i	$-0.643711\pi$
$647$	−22.1955	−0.872597	−0.436299	+	0.899802i	$0.643711\pi$
$648$	−4.45791	−0.175123
$649$	35.3036	1.38579
$650$	0.330522	0.0129641
$651$	3.24266	0.127090
$652$	48.5381	1.90090
$653$	5.12445	0.200535	0.100268	−	0.994961i	$-0.468030\pi$
$653$	5.12445	0.200535	0.100268	+	0.994961i	$0.468030\pi$
$654$	0.119698	0.00468056
$655$	21.8913	0.855362
$656$	−24.9091	−0.972538
$657$	32.5248	1.26891
$658$	1.88732	0.0735754
$659$	33.6252	1.30985	0.654926	−	0.755693i	$-0.272700\pi$
$659$	33.6252	1.30985	0.654926	+	0.755693i	$0.272700\pi$
$660$	−7.13430	−0.277702
$661$	0.635983	0.0247369	0.0123684	−	0.999924i	$-0.496063\pi$
$661$	0.635983	0.0247369	0.0123684	+	0.999924i	$0.496063\pi$
$662$	−0.0191414	−0.000743951 0
$663$	−1.42907	−0.0555004
$664$	−7.25607	−0.281590
$665$	−14.7818	−0.573213
$666$	−1.95632	−0.0758060
$667$	−1.14788	−0.0444460
$668$	−11.5473	−0.446777
$669$	1.70340	0.0658571
$670$	−4.78350	−0.184803
$671$	−19.4962	−0.752641
$672$	−1.38707	−0.0535075
$673$	−20.4400	−0.787903	−0.393951	−	0.919131i	$-0.628892\pi$
$673$	−20.4400	−0.787903	−0.393951	+	0.919131i	$0.628892\pi$
$674$	4.74238	0.182670
$675$	2.46110	0.0947276
$676$	16.6342	0.639775
$677$	33.9032	1.30301	0.651503	−	0.758646i	$-0.274139\pi$
$677$	33.9032	1.30301	0.651503	+	0.758646i	$0.274139\pi$
$678$	0.684359	0.0262826
$679$	−17.6393	−0.676933
$680$	−2.38480	−0.0914530
$681$	7.83846	0.300371
$682$	2.20518	0.0844408
$683$	0.510069	0.0195172	0.00975862	−	0.999952i	$-0.496894\pi$
$683$	0.510069	0.0195172	0.00975862	+	0.999952i	$0.496894\pi$
$684$	−17.3573	−0.663673
$685$	39.5581	1.51144
$686$	−2.95217	−0.112714
$687$	10.7727	0.411003
$688$	21.2532	0.810272
$689$	−18.3388	−0.698651
$690$	−0.170273	−0.00648218
$691$	−51.1849	−1.94717	−0.973583	−	0.228335i	$-0.926672\pi$
$691$	−51.1849	−1.94717	−0.973583	+	0.228335i	$0.926672\pi$
$692$	−44.4405	−1.68938
$693$	19.9059	0.756161
$694$	−2.47720	−0.0940333
$695$	−2.45728	−0.0932100
$696$	−0.240133	−0.00910221
$697$	−10.5734	−0.400497
$698$	0.806833	0.0305391
$699$	9.78747	0.370196
$700$	−3.99068	−0.150833
$701$	9.38860	0.354603	0.177301	−	0.984157i	$-0.443263\pi$
$701$	9.38860	0.354603	0.177301	+	0.984157i	$0.443263\pi$
$702$	0.754646	0.0284823
$703$	−14.3755	−0.542184
$704$	27.0083	1.01792
$705$	−6.52464	−0.245732
$706$	−5.10118	−0.191985
$707$	18.8214	0.707852
$708$	−7.85120	−0.295066
$709$	11.8796	0.446146	0.223073	−	0.974802i	$-0.428391\pi$
$709$	11.8796	0.446146	0.223073	+	0.974802i	$0.428391\pi$
$710$	−1.72878	−0.0648801
$711$	−15.8793	−0.595521
$712$	4.27650	0.160269
$713$	−4.71469	−0.176567
$714$	−0.192613	−0.00720834
$715$	−19.0219	−0.711380
$716$	9.76177	0.364815
$717$	0.682623	0.0254930
$718$	1.11202	0.0415003
$719$	20.5281	0.765570	0.382785	−	0.923838i	$-0.374965\pi$
$719$	20.5281	0.765570	0.382785	+	0.923838i	$0.374965\pi$
$720$	−26.9457	−1.00421
$721$	28.4913	1.06107
$722$	−1.39945	−0.0520822
$723$	−5.75245	−0.213936
$724$	7.15474	0.265904
$725$	−1.03823	−0.0385588
$726$	−0.124057	−0.00460420
$727$	7.22165	0.267836	0.133918	−	0.990992i	$-0.457244\pi$
$727$	7.22165	0.267836	0.133918	+	0.990992i	$0.457244\pi$
$728$	−2.46098	−0.0912100
$729$	−18.4918	−0.684881
$730$	−4.18909	−0.155045
$731$	9.02157	0.333675
$732$	4.33577	0.160255
$733$	45.1869	1.66902	0.834508	−	0.550996i	$-0.185752\pi$
$733$	45.1869	1.66902	0.834508	+	0.550996i	$0.185752\pi$
$734$	−1.35634	−0.0500634
$735$	3.21796	0.118696
$736$	2.01675	0.0743383
$737$	47.3351	1.74361
$738$	2.71278	0.0998588
$739$	53.4101	1.96472	0.982361	−	0.186997i	$-0.0598753\pi$
$739$	53.4101	1.96472	0.982361	+	0.186997i	$0.0598753\pi$
$740$	−22.5715	−0.829745
$741$	2.69423	0.0989751
$742$	−2.47174	−0.0907403
$743$	30.2727	1.11060	0.555298	−	0.831651i	$-0.312604\pi$
$743$	30.2727	1.11060	0.555298	+	0.831651i	$0.312604\pi$
$744$	−0.986301	−0.0361595
$745$	−31.2295	−1.14416
$746$	4.04960	0.148266
$747$	−34.8012	−1.27331
$748$	11.7339	0.429034
$749$	−3.68702	−0.134721
$750$	0.587678	0.0214590
$751$	−38.2954	−1.39742	−0.698709	−	0.715406i	$-0.746242\pi$
$751$	−38.2954	−1.39742	−0.698709	+	0.715406i	$0.746242\pi$
$752$	25.2808	0.921897
$753$	1.32139	0.0481541
$754$	−0.318352	−0.0115937
$755$	−19.6441	−0.714920
$756$	−9.11149	−0.331382
$757$	12.8079	0.465510	0.232755	−	0.972535i	$-0.425226\pi$
$757$	12.8079	0.465510	0.232755	+	0.972535i	$0.425226\pi$
$758$	3.61577	0.131331
$759$	1.68494	0.0611593
$760$	4.49609	0.163090
$761$	41.8266	1.51621	0.758106	−	0.652132i	$-0.226125\pi$
$761$	41.8266	1.51621	0.758106	+	0.652132i	$0.226125\pi$
$762$	0.677913	0.0245582
$763$	−3.85333	−0.139500
$764$	−23.1648	−0.838072
$765$	−11.4379	−0.413537
$766$	0.471232	0.0170263
$767$	−20.9334	−0.755860
$768$	−5.79429	−0.209084
$769$	5.76961	0.208058	0.104029	−	0.994574i	$-0.466827\pi$
$769$	5.76961	0.208058	0.104029	+	0.994574i	$0.466827\pi$
$770$	−2.56381	−0.0923934
$771$	7.07011	0.254624
$772$	0.994453	0.0357911
$773$	7.70761	0.277223	0.138612	−	0.990347i	$-0.455736\pi$
$773$	7.70761	0.277223	0.138612	+	0.990347i	$0.455736\pi$
$774$	−2.31463	−0.0831976
$775$	−4.26433	−0.153179
$776$	5.36523	0.192601
$777$	−3.66640	−0.131531
$778$	0.931056	0.0333800
$779$	19.9342	0.714215
$780$	4.23031	0.151469
$781$	17.1072	0.612143
$782$	0.280051	0.0100146
$783$	−2.37048	−0.0847140
$784$	−12.4685	−0.445304
$785$	−48.5582	−1.73312
$786$	−0.537788	−0.0191823
$787$	−2.64808	−0.0943938	−0.0471969	−	0.998886i	$-0.515029\pi$
$787$	−2.64808	−0.0943938	−0.0471969	+	0.998886i	$0.515029\pi$
$788$	26.2254	0.934240
$789$	0.303269	0.0107966
$790$	2.04521	0.0727652
$791$	−22.0310	−0.783331
$792$	−6.05466	−0.215143
$793$	11.5603	0.410519
$794$	−2.64168	−0.0937497
$795$	8.54502	0.303061
$796$	−13.5731	−0.481087
$797$	34.4037	1.21864	0.609320	−	0.792924i	$-0.291442\pi$
$797$	34.4037	1.21864	0.609320	+	0.792924i	$0.291442\pi$
$798$	0.363134	0.0128548
$799$	10.7312	0.379642
$800$	1.82410	0.0644917
$801$	20.5108	0.724713
$802$	−1.59288	−0.0562465
$803$	41.4532	1.46285
$804$	−10.5269	−0.371255
$805$	5.48145	0.193196
$806$	−1.30757	−0.0460572
$807$	5.01459	0.176522
$808$	−5.72480	−0.201398
$809$	−18.4660	−0.649231	−0.324615	−	0.945846i	$-0.605235\pi$
$809$	−18.4660	−0.649231	−0.324615	+	0.945846i	$0.605235\pi$
$810$	2.75379	0.0967581
$811$	8.76007	0.307608	0.153804	−	0.988101i	$-0.450848\pi$
$811$	8.76007	0.307608	0.153804	+	0.988101i	$0.450848\pi$
$812$	3.84374	0.134889
$813$	−7.19817	−0.252451
$814$	−2.49335	−0.0873920
$815$	−60.3016	−2.11227
$816$	−2.58006	−0.0903202
$817$	−17.0084	−0.595050
$818$	−0.420263	−0.0146942
$819$	−11.8032	−0.412439
$820$	31.2993	1.09302
$821$	51.7160	1.80490	0.902451	−	0.430792i	$-0.141766\pi$
$821$	51.7160	1.80490	0.902451	+	0.430792i	$0.141766\pi$
$822$	−0.971799	−0.0338954
$823$	−32.7640	−1.14208	−0.571040	−	0.820922i	$-0.693460\pi$
$823$	−32.7640	−1.14208	−0.571040	+	0.820922i	$0.693460\pi$
$824$	−8.66604	−0.301896
$825$	1.52398	0.0530583
$826$	−2.82144	−0.0981706
$827$	−7.67179	−0.266774	−0.133387	−	0.991064i	$-0.542585\pi$
$827$	−7.67179	−0.266774	−0.133387	+	0.991064i	$0.542585\pi$
$828$	6.43651	0.223684
$829$	−21.8159	−0.757696	−0.378848	−	0.925459i	$-0.623680\pi$
$829$	−21.8159	−0.757696	−0.378848	+	0.925459i	$0.623680\pi$
$830$	4.48229	0.155583
$831$	2.00464	0.0695401
$832$	−16.0147	−0.555209
$833$	−5.29263	−0.183379
$834$	0.0603664	0.00209032
$835$	14.3458	0.496457
$836$	−22.1221	−0.765107
$837$	−9.73630	−0.336536
$838$	−1.77279	−0.0612401
$839$	−46.8058	−1.61592	−0.807958	−	0.589240i	$-0.799427\pi$
$839$	−46.8058	−1.61592	−0.807958	+	0.589240i	$0.799427\pi$
$840$	1.14670	0.0395650
$841$	1.00000	0.0344828
$842$	−4.83676	−0.166686
$843$	0.0366947	0.00126383
$844$	20.1011	0.691909
$845$	−20.6655	−0.710916
$846$	−2.75326	−0.0946591
$847$	3.99367	0.137224
$848$	−33.1091	−1.13697
$849$	−0.791983	−0.0271808
$850$	0.253299	0.00868810
$851$	5.33080	0.182737
$852$	−3.80448	−0.130339
$853$	−13.5077	−0.462494	−0.231247	−	0.972895i	$-0.574281\pi$
$853$	−13.5077	−0.462494	−0.231247	+	0.972895i	$0.574281\pi$
$854$	1.55812	0.0533179
$855$	21.5639	0.737471
$856$	1.12146	0.0383307
$857$	38.1051	1.30165	0.650823	−	0.759229i	$-0.274424\pi$
$857$	38.1051	1.30165	0.650823	+	0.759229i	$0.274424\pi$
$858$	0.467299	0.0159533
$859$	−26.0657	−0.889350	−0.444675	−	0.895692i	$-0.646681\pi$
$859$	−26.0657	−0.889350	−0.444675	+	0.895692i	$0.646681\pi$
$860$	−26.7055	−0.910651
$861$	5.08410	0.173266
$862$	0.683119	0.0232671
$863$	−8.64907	−0.294418	−0.147209	−	0.989105i	$-0.547029\pi$
$863$	−8.64907	−0.294418	−0.147209	+	0.989105i	$0.547029\pi$
$864$	4.16478	0.141689
$865$	55.2110	1.87723
$866$	−0.0652851	−0.00221848
$867$	5.81114	0.197357
$868$	15.7874	0.535861
$869$	−20.2383	−0.686539
$870$	0.148337	0.00502910
$871$	−28.0675	−0.951031
$872$	1.17204	0.0396904
$873$	25.7325	0.870913
$874$	−0.527982	−0.0178593
$875$	−18.9186	−0.639566
$876$	−9.21882	−0.311475
$877$	−37.7025	−1.27312	−0.636561	−	0.771226i	$-0.719644\pi$
$877$	−37.7025	−1.27312	−0.636561	+	0.771226i	$0.719644\pi$
$878$	−1.00592	−0.0339481
$879$	3.71354	0.125255
$880$	−34.3425	−1.15769
$881$	−38.6661	−1.30269	−0.651347	−	0.758780i	$-0.725796\pi$
$881$	−38.6661	−1.30269	−0.651347	+	0.758780i	$0.725796\pi$
$882$	1.35791	0.0457232
$883$	−33.7540	−1.13591	−0.567957	−	0.823058i	$-0.692266\pi$
$883$	−33.7540	−1.13591	−0.567957	+	0.823058i	$0.692266\pi$
$884$	−6.95766	−0.234011
$885$	9.75399	0.327877
$886$	3.03753	0.102048
$887$	−41.1066	−1.38022	−0.690112	−	0.723703i	$-0.742439\pi$
$887$	−41.1066	−1.38022	−0.690112	+	0.723703i	$0.742439\pi$
$888$	1.11519	0.0374233
$889$	−21.8234	−0.731935
$890$	−2.64172	−0.0885508
$891$	−27.2501	−0.912912
$892$	8.29328	0.277679
$893$	−20.2316	−0.677025
$894$	0.767196	0.0256589
$895$	−12.1276	−0.405381
$896$	−8.98708	−0.300237
$897$	−0.999088	−0.0333586
$898$	6.05763	0.202146
$899$	4.10731	0.136987
$900$	5.82168	0.194056
$901$	−14.0541	−0.468211
$902$	3.45746	0.115121
$903$	−4.33791	−0.144357
$904$	6.70103	0.222873
$905$	−8.88873	−0.295471
$906$	0.482583	0.0160327
$907$	10.9236	0.362713	0.181356	−	0.983417i	$-0.441951\pi$
$907$	10.9236	0.362713	0.181356	+	0.983417i	$0.441951\pi$
$908$	38.1629	1.26648
$909$	−27.4571	−0.910693
$910$	1.52022	0.0503949
$911$	37.9871	1.25857	0.629285	−	0.777175i	$-0.283348\pi$
$911$	37.9871	1.25857	0.629285	+	0.777175i	$0.283348\pi$
$912$	4.86421	0.161070
$913$	−44.3545	−1.46792
$914$	−0.781954	−0.0258648
$915$	−5.38657	−0.178075
$916$	52.4486	1.73295
$917$	17.3126	0.571711
$918$	0.578332	0.0190878
$919$	7.74181	0.255379	0.127689	−	0.991814i	$-0.459244\pi$
$919$	7.74181	0.255379	0.127689	+	0.991814i	$0.459244\pi$
$920$	−1.66726	−0.0549679
$921$	−4.10772	−0.135354
$922$	−0.741831	−0.0244309
$923$	−10.1438	−0.333886
$924$	−5.64211	−0.185612
$925$	4.82158	0.158533
$926$	−0.418574	−0.0137552
$927$	−41.5637	−1.36513
$928$	−1.75694	−0.0576743
$929$	52.6893	1.72868	0.864341	−	0.502907i	$-0.167736\pi$
$929$	52.6893	1.72868	0.864341	+	0.502907i	$0.167736\pi$
$930$	0.609267	0.0199787
$931$	9.97824	0.327024
$932$	47.6520	1.56089
$933$	−4.48134	−0.146712
$934$	−0.166362	−0.00544352
$935$	−14.5777	−0.476741
$936$	3.59013	0.117347
$937$	21.0163	0.686574	0.343287	−	0.939231i	$-0.388460\pi$
$937$	21.0163	0.686574	0.343287	+	0.939231i	$0.388460\pi$
$938$	−3.78300	−0.123519
$939$	−0.636578	−0.0207739
$940$	−31.7663	−1.03610
$941$	43.4322	1.41585	0.707925	−	0.706288i	$-0.249632\pi$
$941$	43.4322	1.41585	0.707925	+	0.706288i	$0.249632\pi$
$942$	1.19290	0.0388667
$943$	−7.39207	−0.240719
$944$	−37.7935	−1.23007
$945$	11.3197	0.368230
$946$	−2.95002	−0.0959133
$947$	−31.5630	−1.02566	−0.512831	−	0.858490i	$-0.671403\pi$
$947$	−31.5630	−1.02566	−0.512831	+	0.858490i	$0.671403\pi$
$948$	4.50083	0.146180
$949$	−24.5798	−0.797894
$950$	−0.477548	−0.0154937
$951$	0.0670916	0.00217559
$952$	−1.88600	−0.0611257
$953$	−6.97812	−0.226043	−0.113022	−	0.993593i	$-0.536053\pi$
$953$	−6.97812	−0.226043	−0.113022	+	0.993593i	$0.536053\pi$
$954$	3.60582	0.116743
$955$	28.7789	0.931262
$956$	3.32347	0.107489
$957$	−1.46787	−0.0474495
$958$	−3.79135	−0.122493
$959$	31.2843	1.01022
$960$	7.46210	0.240838
$961$	−14.1300	−0.455806
$962$	1.47844	0.0476669
$963$	5.37870	0.173326
$964$	−28.0068	−0.902037
$965$	−1.23546	−0.0397710
$966$	−0.134659	−0.00433259
$967$	−8.74687	−0.281280	−0.140640	−	0.990061i	$-0.544916\pi$
$967$	−8.74687	−0.281280	−0.140640	+	0.990061i	$0.544916\pi$
$968$	−1.21473	−0.0390430
$969$	2.06476	0.0663296
$970$	−3.31427	−0.106415
$971$	21.7661	0.698507	0.349253	−	0.937028i	$-0.386435\pi$
$971$	21.7661	0.698507	0.349253	+	0.937028i	$0.386435\pi$
$972$	20.1260	0.645542
$973$	−1.94332	−0.0623001
$974$	5.71462	0.183108
$975$	−0.903651	−0.0289400
$976$	20.8712	0.668070
$977$	36.0043	1.15188	0.575939	−	0.817492i	$-0.304636\pi$
$977$	36.0043	1.15188	0.575939	+	0.817492i	$0.304636\pi$
$978$	1.48139	0.0473697
$979$	26.1412	0.835476
$980$	15.6672	0.500470
$981$	5.62131	0.179475
$982$	2.73469	0.0872674
$983$	49.2700	1.57147	0.785734	−	0.618564i	$-0.212285\pi$
$983$	49.2700	1.57147	0.785734	+	0.618564i	$0.212285\pi$
$984$	−1.54640	−0.0492975
$985$	−32.5812	−1.03812
$986$	−0.243973	−0.00776968
$987$	−5.15997	−0.164244
$988$	13.1173	0.417318
$989$	6.30715	0.200556
$990$	3.74014	0.118870
$991$	45.2440	1.43722	0.718612	−	0.695411i	$-0.244778\pi$
$991$	45.2440	1.43722	0.718612	+	0.695411i	$0.244778\pi$
$992$	−7.21629	−0.229117
$993$	0.0523329	0.00166073
$994$	−1.36720	−0.0433648
$995$	16.8627	0.534583
$996$	9.86405	0.312554
$997$	−60.6567	−1.92102	−0.960508	−	0.278253i	$-0.910245\pi$
$997$	−60.6567	−1.92102	−0.960508	+	0.278253i	$0.910245\pi$
$998$	−4.60735	−0.145843
$999$	11.0086	0.348297

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.34	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.34	✓	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)

\(f(q)\)	\(=\)	\(q+0.148593 q^{2} -0.406255 q^{3} -1.97792 q^{4} +2.45728 q^{5} -0.0603664 q^{6} +1.94332 q^{7} -0.591089 q^{8} -2.83496 q^{9} +O(q^{10})\)	\(q+0.148593 q^{2} -0.406255 q^{3} -1.97792 q^{4} +2.45728 q^{5} -0.0603664 q^{6} +1.94332 q^{7} -0.591089 q^{8} -2.83496 q^{9} +0.365134 q^{10} -3.61318 q^{11} +0.803539 q^{12} +2.14245 q^{13} +0.288763 q^{14} -0.998281 q^{15} +3.86801 q^{16} +1.64189 q^{17} -0.421254 q^{18} -3.09547 q^{19} -4.86030 q^{20} -0.789484 q^{21} -0.536892 q^{22} +1.14788 q^{23} +0.240133 q^{24} +1.03823 q^{25} +0.318352 q^{26} +2.37048 q^{27} -3.84374 q^{28} -1.00000 q^{29} -0.148337 q^{30} -4.10731 q^{31} +1.75694 q^{32} +1.46787 q^{33} +0.243973 q^{34} +4.77529 q^{35} +5.60732 q^{36} +4.64405 q^{37} -0.459964 q^{38} -0.870379 q^{39} -1.45247 q^{40} -6.43978 q^{41} -0.117311 q^{42} +5.49462 q^{43} +7.14658 q^{44} -6.96629 q^{45} +0.170566 q^{46} +6.53587 q^{47} -1.57140 q^{48} -3.22350 q^{49} +0.154273 q^{50} -0.667026 q^{51} -4.23759 q^{52} -8.55973 q^{53} +0.352235 q^{54} -8.87860 q^{55} -1.14868 q^{56} +1.25755 q^{57} -0.148593 q^{58} -9.77078 q^{59} +1.97452 q^{60} +5.39585 q^{61} -0.610316 q^{62} -5.50924 q^{63} -7.47495 q^{64} +5.26459 q^{65} +0.218115 q^{66} -13.1007 q^{67} -3.24753 q^{68} -0.466330 q^{69} +0.709573 q^{70} -4.73466 q^{71} +1.67571 q^{72} -11.4728 q^{73} +0.690072 q^{74} -0.421785 q^{75} +6.12260 q^{76} -7.02158 q^{77} -0.129332 q^{78} +5.60125 q^{79} +9.50478 q^{80} +7.54185 q^{81} -0.956903 q^{82} +12.2758 q^{83} +1.56154 q^{84} +4.03459 q^{85} +0.816460 q^{86} +0.406255 q^{87} +2.13571 q^{88} -7.23495 q^{89} -1.03514 q^{90} +4.16347 q^{91} -2.27041 q^{92} +1.66861 q^{93} +0.971183 q^{94} -7.60644 q^{95} -0.713763 q^{96} -9.07685 q^{97} -0.478988 q^{98} +10.2432 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 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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