LMFDB - Embedded newform 4031.2.a.b.1.8

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:	$59$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-2.19361 q^{2} -1.71235 q^{3} +2.81195 q^{4} -2.23633 q^{5} +3.75624 q^{6} -3.04720 q^{7} -1.78110 q^{8} -0.0678453 q^{9} +O(q^{10})$	\(q-2.19361 q^{2} -1.71235 q^{3} +2.81195 q^{4} -2.23633 q^{5} +3.75624 q^{6} -3.04720 q^{7} -1.78110 q^{8} -0.0678453 q^{9} +4.90564 q^{10} -1.23615 q^{11} -4.81505 q^{12} +4.62416 q^{13} +6.68439 q^{14} +3.82938 q^{15} -1.71685 q^{16} -2.82011 q^{17} +0.148826 q^{18} -2.42496 q^{19} -6.28843 q^{20} +5.21789 q^{21} +2.71163 q^{22} -1.68438 q^{23} +3.04987 q^{24} +0.00115310 q^{25} -10.1436 q^{26} +5.25324 q^{27} -8.56857 q^{28} +1.00000 q^{29} -8.40019 q^{30} -3.83960 q^{31} +7.32830 q^{32} +2.11672 q^{33} +6.18624 q^{34} +6.81454 q^{35} -0.190777 q^{36} -11.7579 q^{37} +5.31943 q^{38} -7.91820 q^{39} +3.98311 q^{40} +5.64161 q^{41} -11.4460 q^{42} +1.28438 q^{43} -3.47598 q^{44} +0.151724 q^{45} +3.69487 q^{46} +2.94154 q^{47} +2.93986 q^{48} +2.28545 q^{49} -0.00252946 q^{50} +4.82903 q^{51} +13.0029 q^{52} +6.00738 q^{53} -11.5236 q^{54} +2.76442 q^{55} +5.42737 q^{56} +4.15239 q^{57} -2.19361 q^{58} -2.00477 q^{59} +10.7680 q^{60} -0.627562 q^{61} +8.42261 q^{62} +0.206738 q^{63} -12.6418 q^{64} -10.3411 q^{65} -4.64327 q^{66} -0.175788 q^{67} -7.93000 q^{68} +2.88425 q^{69} -14.9485 q^{70} +10.5450 q^{71} +0.120839 q^{72} +8.86291 q^{73} +25.7922 q^{74} -0.00197452 q^{75} -6.81886 q^{76} +3.76679 q^{77} +17.3695 q^{78} +12.9873 q^{79} +3.83944 q^{80} -8.79186 q^{81} -12.3755 q^{82} +6.51740 q^{83} +14.6724 q^{84} +6.30668 q^{85} -2.81744 q^{86} -1.71235 q^{87} +2.20170 q^{88} +6.91860 q^{89} -0.332824 q^{90} -14.0908 q^{91} -4.73637 q^{92} +6.57475 q^{93} -6.45260 q^{94} +5.42300 q^{95} -12.5486 q^{96} -6.03515 q^{97} -5.01339 q^{98} +0.0838666 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) $ 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.19361	−1.55112	−0.775560	−	0.631274i	$-0.782532\pi$
$2$	−2.19361	−1.55112	−0.775560	+	0.631274i	$0.782532\pi$
$3$	−1.71235	−0.988628	−0.494314	−	0.869283i	$-0.664581\pi$
$3$	−1.71235	−0.988628	−0.494314	+	0.869283i	$0.664581\pi$
$4$	2.81195	1.40597
$5$	−2.23633	−1.00012	−0.500058	−	0.865992i	$-0.666688\pi$
$5$	−2.23633	−1.00012	−0.500058	+	0.865992i	$0.666688\pi$
$6$	3.75624	1.53348
$7$	−3.04720	−1.15173	−0.575867	−	0.817543i	$-0.695336\pi$
$7$	−3.04720	−1.15173	−0.575867	+	0.817543i	$0.695336\pi$
$8$	−1.78110	−0.629713
$9$	−0.0678453	−0.0226151
$10$	4.90564	1.55130
$11$	−1.23615	−0.372712	−0.186356	−	0.982482i	$-0.559668\pi$
$11$	−1.23615	−0.372712	−0.186356	+	0.982482i	$0.559668\pi$
$12$	−4.81505	−1.38998
$13$	4.62416	1.28251	0.641256	−	0.767327i	$-0.278414\pi$
$13$	4.62416	1.28251	0.641256	+	0.767327i	$0.278414\pi$
$14$	6.68439	1.78648
$15$	3.82938	0.988742
$16$	−1.71685	−0.429213
$17$	−2.82011	−0.683977	−0.341989	−	0.939704i	$-0.611100\pi$
$17$	−2.82011	−0.683977	−0.341989	+	0.939704i	$0.611100\pi$
$18$	0.148826	0.0350787
$19$	−2.42496	−0.556324	−0.278162	−	0.960534i	$-0.589725\pi$
$19$	−2.42496	−0.556324	−0.278162	+	0.960534i	$0.589725\pi$
$20$	−6.28843	−1.40614
$21$	5.21789	1.13864
$22$	2.71163	0.578121
$23$	−1.68438	−0.351217	−0.175608	−	0.984460i	$-0.556189\pi$
$23$	−1.68438	−0.351217	−0.175608	+	0.984460i	$0.556189\pi$
$24$	3.04987	0.622552
$25$	0.00115310	0.000230620 0
$26$	−10.1436	−1.98933
$27$	5.25324	1.01099
$28$	−8.56857	−1.61931
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−8.40019	−1.53366
$31$	−3.83960	−0.689613	−0.344806	−	0.938674i	$-0.612055\pi$
$31$	−3.83960	−0.689613	−0.344806	+	0.938674i	$0.612055\pi$
$32$	7.32830	1.29547
$33$	2.11672	0.368473
$34$	6.18624	1.06093
$35$	6.81454	1.15187
$36$	−0.190777	−0.0317962
$37$	−11.7579	−1.93298	−0.966491	−	0.256700i	$-0.917365\pi$
$37$	−11.7579	−1.93298	−0.966491	+	0.256700i	$0.917365\pi$
$38$	5.31943	0.862925
$39$	−7.91820	−1.26793
$40$	3.98311	0.629786
$41$	5.64161	0.881071	0.440536	−	0.897735i	$-0.354789\pi$
$41$	5.64161	0.881071	0.440536	+	0.897735i	$0.354789\pi$
$42$	−11.4460	−1.76616
$43$	1.28438	0.195867	0.0979334	−	0.995193i	$-0.468777\pi$
$43$	1.28438	0.195867	0.0979334	+	0.995193i	$0.468777\pi$
$44$	−3.47598	−0.524023
$45$	0.151724	0.0226177
$46$	3.69487	0.544779
$47$	2.94154	0.429067	0.214534	−	0.976717i	$-0.431177\pi$
$47$	2.94154	0.429067	0.214534	+	0.976717i	$0.431177\pi$
$48$	2.93986	0.424332
$49$	2.28545	0.326493
$50$	−0.00252946	−0.000357720 0
$51$	4.82903	0.676199
$52$	13.0029	1.80318
$53$	6.00738	0.825177	0.412589	−	0.910917i	$-0.364625\pi$
$53$	6.00738	0.825177	0.412589	+	0.910917i	$0.364625\pi$
$54$	−11.5236	−1.56816
$55$	2.76442	0.372755
$56$	5.42737	0.725262
$57$	4.15239	0.549997
$58$	−2.19361	−0.288036
$59$	−2.00477	−0.260999	−0.130500	−	0.991448i	$-0.541658\pi$
$59$	−2.00477	−0.260999	−0.130500	+	0.991448i	$0.541658\pi$
$60$	10.7680	1.39014
$61$	−0.627562	−0.0803511	−0.0401756	−	0.999193i	$-0.512792\pi$
$61$	−0.627562	−0.0803511	−0.0401756	+	0.999193i	$0.512792\pi$
$62$	8.42261	1.06967
$63$	0.206738	0.0260466
$64$	−12.6418	−1.58022
$65$	−10.3411	−1.28266
$66$	−4.64327	−0.571546
$67$	−0.175788	−0.0214759	−0.0107380	−	0.999942i	$-0.503418\pi$
$67$	−0.175788	−0.0214759	−0.0107380	+	0.999942i	$0.503418\pi$
$68$	−7.93000	−0.961654
$69$	2.88425	0.347222
$70$	−14.9485	−1.78668
$71$	10.5450	1.25146	0.625728	−	0.780041i	$-0.284802\pi$
$71$	10.5450	1.25146	0.625728	+	0.780041i	$0.284802\pi$
$72$	0.120839	0.0142410
$73$	8.86291	1.03733	0.518663	−	0.854979i	$-0.326430\pi$
$73$	8.86291	1.03733	0.518663	+	0.854979i	$0.326430\pi$
$74$	25.7922	2.99829
$75$	−0.00197452	−0.000227998 0
$76$	−6.81886	−0.782177
$77$	3.76679	0.429265
$78$	17.3695	1.96671
$79$	12.9873	1.46119	0.730593	−	0.682813i	$-0.239244\pi$
$79$	12.9873	1.46119	0.730593	+	0.682813i	$0.239244\pi$
$80$	3.83944	0.429262
$81$	−8.79186	−0.976873
$82$	−12.3755	−1.36665
$83$	6.51740	0.715377	0.357689	−	0.933841i	$-0.383565\pi$
$83$	6.51740	0.715377	0.357689	+	0.933841i	$0.383565\pi$
$84$	14.6724	1.60089
$85$	6.30668	0.684056
$86$	−2.81744	−0.303813
$87$	−1.71235	−0.183584
$88$	2.20170	0.234702
$89$	6.91860	0.733370	0.366685	−	0.930345i	$-0.380493\pi$
$89$	6.91860	0.733370	0.366685	+	0.930345i	$0.380493\pi$
$90$	−0.332824	−0.0350828
$91$	−14.0908	−1.47711
$92$	−4.73637	−0.493801
$93$	6.57475	0.681770
$94$	−6.45260	−0.665535
$95$	5.42300	0.556388
$96$	−12.5486	−1.28074
$97$	−6.03515	−0.612776	−0.306388	−	0.951907i	$-0.599121\pi$
$97$	−6.03515	−0.612776	−0.306388	+	0.951907i	$0.599121\pi$
$98$	−5.01339	−0.506429
$99$	0.0838666	0.00842892
$100$	0.00324246	0.000324246 0
$101$	0.315225	0.0313660	0.0156830	−	0.999877i	$-0.495008\pi$
$101$	0.315225	0.0313660	0.0156830	+	0.999877i	$0.495008\pi$
$102$	−10.5930	−1.04887
$103$	−1.62823	−0.160434	−0.0802170	−	0.996777i	$-0.525561\pi$
$103$	−1.62823	−0.160434	−0.0802170	+	0.996777i	$0.525561\pi$
$104$	−8.23608	−0.807614
$105$	−11.6689	−1.13877
$106$	−13.1779	−1.27995
$107$	16.3813	1.58364	0.791818	−	0.610757i	$-0.209135\pi$
$107$	16.3813	1.58364	0.791818	+	0.610757i	$0.209135\pi$
$108$	14.7718	1.42142
$109$	0.606072	0.0580511	0.0290256	−	0.999579i	$-0.490760\pi$
$109$	0.606072	0.0580511	0.0290256	+	0.999579i	$0.490760\pi$
$110$	−6.06408	−0.578188
$111$	20.1336	1.91100
$112$	5.23159	0.494339
$113$	−8.91723	−0.838862	−0.419431	−	0.907787i	$-0.637770\pi$
$113$	−8.91723	−0.838862	−0.419431	+	0.907787i	$0.637770\pi$
$114$	−9.10874	−0.853112
$115$	3.76681	0.351257
$116$	2.81195	0.261083
$117$	−0.313727	−0.0290041
$118$	4.39770	0.404841
$119$	8.59345	0.787760
$120$	−6.82050	−0.622624
$121$	−9.47194	−0.861086
$122$	1.37663	0.124634
$123$	−9.66043	−0.871052
$124$	−10.7968	−0.969577
$125$	11.1791	0.999885
$126$	−0.453504	−0.0404014
$127$	10.0274	0.889789	0.444894	−	0.895583i	$-0.353241\pi$
$127$	10.0274	0.889789	0.444894	+	0.895583i	$0.353241\pi$
$128$	13.0746	1.15564
$129$	−2.19932	−0.193639
$130$	22.6845	1.98956
$131$	−6.75402	−0.590102	−0.295051	−	0.955482i	$-0.595337\pi$
$131$	−6.75402	−0.590102	−0.295051	+	0.955482i	$0.595337\pi$
$132$	5.95210	0.518064
$133$	7.38935	0.640738
$134$	0.385611	0.0333117
$135$	−11.7479	−1.01110
$136$	5.02289	0.430709
$137$	1.75918	0.150297	0.0751486	−	0.997172i	$-0.476057\pi$
$137$	1.75918	0.150297	0.0751486	+	0.997172i	$0.476057\pi$
$138$	−6.32693	−0.538584
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	19.1621	1.61949
$141$	−5.03695	−0.424188
$142$	−23.1316	−1.94116
$143$	−5.71614	−0.478007
$144$	0.116480	0.00970668
$145$	−2.23633	−0.185717
$146$	−19.4418	−1.60902
$147$	−3.91349	−0.322780
$148$	−33.0625	−2.71772
$149$	−10.9341	−0.895760	−0.447880	−	0.894094i	$-0.647821\pi$
$149$	−10.9341	−0.895760	−0.447880	+	0.894094i	$0.647821\pi$
$150$	0.00433133	0.000353652 0
$151$	7.42367	0.604129	0.302065	−	0.953287i	$-0.402324\pi$
$151$	7.42367	0.604129	0.302065	+	0.953287i	$0.402324\pi$
$152$	4.31909	0.350324
$153$	0.191331	0.0154682
$154$	−8.26288	−0.665842
$155$	8.58660	0.689692
$156$	−22.2655	−1.78267
$157$	−0.334951	−0.0267320	−0.0133660	−	0.999911i	$-0.504255\pi$
$157$	−0.334951	−0.0267320	−0.0133660	+	0.999911i	$0.504255\pi$
$158$	−28.4891	−2.26647
$159$	−10.2868	−0.815793
$160$	−16.3885	−1.29562
$161$	5.13263	0.404508
$162$	19.2860	1.51525
$163$	10.5105	0.823244	0.411622	−	0.911355i	$-0.364962\pi$
$163$	10.5105	0.823244	0.411622	+	0.911355i	$0.364962\pi$
$164$	15.8639	1.23876
$165$	−4.73367	−0.368516
$166$	−14.2967	−1.10964
$167$	8.16987	0.632203	0.316102	−	0.948725i	$-0.397626\pi$
$167$	8.16987	0.632203	0.316102	+	0.948725i	$0.397626\pi$
$168$	−9.29357	−0.717014
$169$	8.38286	0.644835
$170$	−13.8344	−1.06105
$171$	0.164522	0.0125813
$172$	3.61162	0.275383
$173$	−1.36012	−0.103408	−0.0517040	−	0.998662i	$-0.516465\pi$
$173$	−1.36012	−0.103408	−0.0517040	+	0.998662i	$0.516465\pi$
$174$	3.75624	0.284760
$175$	−0.00351373	−0.000265613 0
$176$	2.12228	0.159973
$177$	3.43288	0.258031
$178$	−15.1767	−1.13754
$179$	14.6215	1.09286	0.546431	−	0.837504i	$-0.315986\pi$
$179$	14.6215	1.09286	0.546431	+	0.837504i	$0.315986\pi$
$180$	0.426640	0.0317999
$181$	−20.3589	−1.51327	−0.756634	−	0.653838i	$-0.773158\pi$
$181$	−20.3589	−1.51327	−0.756634	+	0.653838i	$0.773158\pi$
$182$	30.9097	2.29118
$183$	1.07461	0.0794373
$184$	3.00004	0.221166
$185$	26.2944	1.93321
$186$	−14.4225	−1.05751
$187$	3.48607	0.254926
$188$	8.27145	0.603257
$189$	−16.0077	−1.16439
$190$	−11.8960	−0.863025
$191$	4.13976	0.299542	0.149771	−	0.988721i	$-0.452146\pi$
$191$	4.13976	0.299542	0.149771	+	0.988721i	$0.452146\pi$
$192$	21.6472	1.56225
$193$	13.4746	0.969922	0.484961	−	0.874536i	$-0.338834\pi$
$193$	13.4746	0.969922	0.484961	+	0.874536i	$0.338834\pi$
$194$	13.2388	0.950490
$195$	17.7077	1.26807
$196$	6.42656	0.459040
$197$	9.49257	0.676318	0.338159	−	0.941089i	$-0.390196\pi$
$197$	9.49257	0.676318	0.338159	+	0.941089i	$0.390196\pi$
$198$	−0.183971	−0.0130743
$199$	−18.3412	−1.30017	−0.650085	−	0.759861i	$-0.725267\pi$
$199$	−18.3412	−1.30017	−0.650085	+	0.759861i	$0.725267\pi$
$200$	−0.00205379	−0.000145225 0
$201$	0.301011	0.0212317
$202$	−0.691482	−0.0486525
$203$	−3.04720	−0.213872
$204$	13.5790	0.950717
$205$	−12.6165	−0.881173
$206$	3.57170	0.248852
$207$	0.114277	0.00794279
$208$	−7.93899	−0.550470
$209$	2.99760	0.207349
$210$	25.5971	1.76637
$211$	0.812851	0.0559590	0.0279795	−	0.999608i	$-0.491093\pi$
$211$	0.812851	0.0559590	0.0279795	+	0.999608i	$0.491093\pi$
$212$	16.8924	1.16018
$213$	−18.0567	−1.23722
$214$	−35.9342	−2.45641
$215$	−2.87230	−0.195889
$216$	−9.35653	−0.636631
$217$	11.7000	0.794251
$218$	−1.32949	−0.0900443
$219$	−15.1764	−1.02553
$220$	7.77341	0.524083
$221$	−13.0406	−0.877208
$222$	−44.1654	−2.96419
$223$	28.8515	1.93204	0.966021	−	0.258464i	$-0.0832162\pi$
$223$	28.8515	1.93204	0.966021	+	0.258464i	$0.0832162\pi$
$224$	−22.3308	−1.49204
$225$	−7.82325e−5 0	−5.21550e−6 0
$226$	19.5610	1.30118
$227$	7.40465	0.491464	0.245732	−	0.969338i	$-0.420972\pi$
$227$	7.40465	0.491464	0.245732	+	0.969338i	$0.420972\pi$
$228$	11.6763	0.773281
$229$	−7.08862	−0.468429	−0.234215	−	0.972185i	$-0.575252\pi$
$229$	−7.08862	−0.468429	−0.234215	+	0.972185i	$0.575252\pi$
$230$	−8.26294	−0.544842
$231$	−6.45007	−0.424384
$232$	−1.78110	−0.116935
$233$	27.9754	1.83273	0.916363	−	0.400348i	$-0.131111\pi$
$233$	27.9754	1.83273	0.916363	+	0.400348i	$0.131111\pi$
$234$	0.688197	0.0449889
$235$	−6.57824	−0.429117
$236$	−5.63731	−0.366958
$237$	−22.2389	−1.44457
$238$	−18.8507	−1.22191
$239$	−17.3927	−1.12504	−0.562519	−	0.826784i	$-0.690168\pi$
$239$	−17.3927	−1.12504	−0.562519	+	0.826784i	$0.690168\pi$
$240$	−6.57447	−0.424380
$241$	−0.628817	−0.0405056	−0.0202528	−	0.999795i	$-0.506447\pi$
$241$	−0.628817	−0.0405056	−0.0202528	+	0.999795i	$0.506447\pi$
$242$	20.7778	1.33565
$243$	−0.704933	−0.0452214
$244$	−1.76467	−0.112972
$245$	−5.11101	−0.326530
$246$	21.1913	1.35111
$247$	−11.2134	−0.713492
$248$	6.83870	0.434258
$249$	−11.1601	−0.707242
$250$	−24.5225	−1.55094
$251$	−6.75212	−0.426190	−0.213095	−	0.977031i	$-0.568354\pi$
$251$	−6.75212	−0.426190	−0.213095	+	0.977031i	$0.568354\pi$
$252$	0.581337	0.0366208
$253$	2.08213	0.130903
$254$	−21.9963	−1.38017
$255$	−10.7993	−0.676277
$256$	−3.39704	−0.212315
$257$	−10.3738	−0.647101	−0.323550	−	0.946211i	$-0.604877\pi$
$257$	−10.3738	−0.647101	−0.323550	+	0.946211i	$0.604877\pi$
$258$	4.82446	0.300358
$259$	35.8286	2.22628
$260$	−29.0787	−1.80338
$261$	−0.0678453	−0.00419952
$262$	14.8157	0.915319
$263$	22.7679	1.40393	0.701964	−	0.712212i	$-0.252307\pi$
$263$	22.7679	1.40393	0.701964	+	0.712212i	$0.252307\pi$
$264$	−3.77008	−0.232033
$265$	−13.4345	−0.825272
$266$	−16.2094	−0.993861
$267$	−11.8471	−0.725030
$268$	−0.494307	−0.0301946
$269$	20.4883	1.24919	0.624596	−	0.780948i	$-0.285264\pi$
$269$	20.4883	1.24919	0.624596	+	0.780948i	$0.285264\pi$
$270$	25.7705	1.56834
$271$	−6.49749	−0.394694	−0.197347	−	0.980334i	$-0.563233\pi$
$271$	−6.49749	−0.394694	−0.197347	+	0.980334i	$0.563233\pi$
$272$	4.84171	0.293572
$273$	24.1284	1.46031
$274$	−3.85897	−0.233129
$275$	−0.00142540	−8.59549e−5 0
$276$	8.11035	0.488185
$277$	−12.2209	−0.734283	−0.367142	−	0.930165i	$-0.619664\pi$
$277$	−12.2209	−0.734283	−0.367142	+	0.930165i	$0.619664\pi$
$278$	−2.19361	−0.131564
$279$	0.260499	0.0155957
$280$	−12.1374	−0.725346
$281$	7.61619	0.454344	0.227172	−	0.973855i	$-0.427052\pi$
$281$	7.61619	0.454344	0.227172	+	0.973855i	$0.427052\pi$
$282$	11.0491	0.657966
$283$	14.9057	0.886052	0.443026	−	0.896509i	$-0.353905\pi$
$283$	14.9057	0.886052	0.443026	+	0.896509i	$0.353905\pi$
$284$	29.6518	1.75951
$285$	−9.28609	−0.550061
$286$	12.5390	0.741447
$287$	−17.1911	−1.01476
$288$	−0.497191	−0.0292973
$289$	−9.04698	−0.532175
$290$	4.90564	0.288069
$291$	10.3343	0.605808
$292$	24.9220	1.45845
$293$	−6.27915	−0.366832	−0.183416	−	0.983035i	$-0.558716\pi$
$293$	−6.27915	−0.366832	−0.183416	+	0.983035i	$0.558716\pi$
$294$	8.58470	0.500670
$295$	4.48332	0.261029
$296$	20.9419	1.21722
$297$	−6.49376	−0.376806
$298$	23.9853	1.38943
$299$	−7.78882	−0.450439
$300$	−0.00555224	−0.000320558 0
$301$	−3.91378	−0.225587
$302$	−16.2847	−0.937077
$303$	−0.539776	−0.0310093
$304$	4.16329	0.238781
$305$	1.40343	0.0803604
$306$	−0.419707	−0.0239930
$307$	−21.7635	−1.24211	−0.621055	−	0.783767i	$-0.713296\pi$
$307$	−21.7635	−1.24211	−0.621055	+	0.783767i	$0.713296\pi$
$308$	10.5920	0.603535
$309$	2.78810	0.158609
$310$	−18.8357	−1.06980
$311$	−25.6032	−1.45182	−0.725912	−	0.687787i	$-0.758582\pi$
$311$	−25.6032	−1.45182	−0.725912	+	0.687787i	$0.758582\pi$
$312$	14.1031	0.798430
$313$	−3.84534	−0.217351	−0.108676	−	0.994077i	$-0.534661\pi$
$313$	−3.84534	−0.217351	−0.108676	+	0.994077i	$0.534661\pi$
$314$	0.734753	0.0414645
$315$	−0.462334	−0.0260496
$316$	36.5196	2.05439
$317$	−15.4804	−0.869464	−0.434732	−	0.900560i	$-0.643157\pi$
$317$	−15.4804	−0.869464	−0.434732	+	0.900560i	$0.643157\pi$
$318$	22.5652	1.26539
$319$	−1.23615	−0.0692109
$320$	28.2711	1.58040
$321$	−28.0505	−1.56563
$322$	−11.2590	−0.627441
$323$	6.83865	0.380513
$324$	−24.7222	−1.37346
$325$	0.00533212	0.000295773 0
$326$	−23.0559	−1.27695
$327$	−1.03781	−0.0573910
$328$	−10.0483	−0.554822
$329$	−8.96346	−0.494172
$330$	10.3839	0.571612
$331$	−29.2569	−1.60810	−0.804052	−	0.594559i	$-0.797327\pi$
$331$	−29.2569	−1.60810	−0.804052	+	0.594559i	$0.797327\pi$
$332$	18.3266	1.00580
$333$	0.797716	0.0437146
$334$	−17.9215	−0.980623
$335$	0.393119	0.0214784
$336$	−8.95834	−0.488717
$337$	−5.85699	−0.319050	−0.159525	−	0.987194i	$-0.550996\pi$
$337$	−5.85699	−0.319050	−0.159525	+	0.987194i	$0.550996\pi$
$338$	−18.3888	−1.00022
$339$	15.2694	0.829322
$340$	17.7341	0.961764
$341$	4.74631	0.257027
$342$	−0.360898	−0.0195151
$343$	14.3662	0.775702
$344$	−2.28761	−0.123340
$345$	−6.45011	−0.347262
$346$	2.98358	0.160398
$347$	−6.63929	−0.356416	−0.178208	−	0.983993i	$-0.557030\pi$
$347$	−6.63929	−0.356416	−0.178208	+	0.983993i	$0.557030\pi$
$348$	−4.81505	−0.258114
$349$	24.0057	1.28499	0.642497	−	0.766288i	$-0.277899\pi$
$349$	24.0057	1.28499	0.642497	+	0.766288i	$0.277899\pi$
$350$	0.00770778	0.000411998 0
$351$	24.2918	1.29660
$352$	−9.05885	−0.482838
$353$	−22.1705	−1.18001	−0.590007	−	0.807398i	$-0.700875\pi$
$353$	−22.1705	−1.18001	−0.590007	+	0.807398i	$0.700875\pi$
$354$	−7.53041	−0.400237
$355$	−23.5819	−1.25160
$356$	19.4547	1.03110
$357$	−14.7150	−0.778802
$358$	−32.0739	−1.69516
$359$	−6.91733	−0.365083	−0.182541	−	0.983198i	$-0.558432\pi$
$359$	−6.91733	−0.365083	−0.182541	+	0.983198i	$0.558432\pi$
$360$	−0.270236	−0.0142427
$361$	−13.1196	−0.690504
$362$	44.6597	2.34726
$363$	16.2193	0.851293
$364$	−39.6224	−2.07678
$365$	−19.8204	−1.03744
$366$	−2.35728	−0.123217
$367$	−4.41163	−0.230285	−0.115143	−	0.993349i	$-0.536733\pi$
$367$	−4.41163	−0.230285	−0.115143	+	0.993349i	$0.536733\pi$
$368$	2.89182	0.150747
$369$	−0.382757	−0.0199255
$370$	−57.6799	−2.99863
$371$	−18.3057	−0.950385
$372$	18.4879	0.958551
$373$	−23.0975	−1.19594	−0.597972	−	0.801517i	$-0.704027\pi$
$373$	−23.0975	−1.19594	−0.597972	+	0.801517i	$0.704027\pi$
$374$	−7.64709	−0.395422
$375$	−19.1425	−0.988514
$376$	−5.23917	−0.270189
$377$	4.62416	0.238156
$378$	35.1147	1.80610
$379$	3.94939	0.202866	0.101433	−	0.994842i	$-0.467657\pi$
$379$	3.94939	0.202866	0.101433	+	0.994842i	$0.467657\pi$
$380$	15.2492	0.782267
$381$	−17.1705	−0.879670
$382$	−9.08103	−0.464626
$383$	−14.7771	−0.755076	−0.377538	−	0.925994i	$-0.623229\pi$
$383$	−14.7771	−0.755076	−0.377538	+	0.925994i	$0.623229\pi$
$384$	−22.3883	−1.14250
$385$	−8.42376	−0.429315
$386$	−29.5580	−1.50446
$387$	−0.0871394	−0.00442954
$388$	−16.9705	−0.861547
$389$	15.5718	0.789522	0.394761	−	0.918784i	$-0.370827\pi$
$389$	15.5718	0.789522	0.394761	+	0.918784i	$0.370827\pi$
$390$	−38.8438	−1.96693
$391$	4.75012	0.240224
$392$	−4.07061	−0.205597
$393$	11.5653	0.583391
$394$	−20.8230	−1.04905
$395$	−29.0438	−1.46135
$396$	0.235829	0.0118508
$397$	0.931099	0.0467305	0.0233653	−	0.999727i	$-0.492562\pi$
$397$	0.931099	0.0467305	0.0233653	+	0.999727i	$0.492562\pi$
$398$	40.2334	2.01672
$399$	−12.6532	−0.633451
$400$	−0.00197970	−9.89851e−5 0
$401$	3.06120	0.152869	0.0764346	−	0.997075i	$-0.475646\pi$
$401$	3.06120	0.152869	0.0764346	+	0.997075i	$0.475646\pi$
$402$	−0.660303	−0.0329329
$403$	−17.7549	−0.884436
$404$	0.886395	0.0440998
$405$	19.6615	0.976986
$406$	6.68439	0.331741
$407$	14.5344	0.720446
$408$	−8.60097	−0.425811
$409$	−16.0618	−0.794207	−0.397103	−	0.917774i	$-0.629985\pi$
$409$	−16.0618	−0.794207	−0.397103	+	0.917774i	$0.629985\pi$
$410$	27.6757	1.36680
$411$	−3.01234	−0.148588
$412$	−4.57849	−0.225566
$413$	6.10895	0.300602
$414$	−0.250680	−0.0123202
$415$	−14.5750	−0.715460
$416$	33.8873	1.66146
$417$	−1.71235	−0.0838543
$418$	−6.57559	−0.321623
$419$	−10.4654	−0.511271	−0.255635	−	0.966773i	$-0.582285\pi$
$419$	−10.4654	−0.511271	−0.255635	+	0.966773i	$0.582285\pi$
$420$	−32.8123	−1.60108
$421$	−36.7832	−1.79271	−0.896353	−	0.443341i	$-0.853793\pi$
$421$	−36.7832	−1.79271	−0.896353	+	0.443341i	$0.853793\pi$
$422$	−1.78308	−0.0867991
$423$	−0.199569	−0.00970340
$424$	−10.6997	−0.519625
$425$	−0.00325187	−0.000157739 0
$426$	39.6094	1.91908
$427$	1.91231	0.0925432
$428$	46.0632	2.22655
$429$	9.78804	0.472571
$430$	6.30072	0.303848
$431$	−11.8286	−0.569761	−0.284881	−	0.958563i	$-0.591954\pi$
$431$	−11.8286	−0.569761	−0.284881	+	0.958563i	$0.591954\pi$
$432$	−9.01902	−0.433928
$433$	−14.7847	−0.710506	−0.355253	−	0.934770i	$-0.615605\pi$
$433$	−14.7847	−0.710506	−0.355253	+	0.934770i	$0.615605\pi$
$434$	−25.6654	−1.23198
$435$	3.82938	0.183605
$436$	1.70424	0.0816183
$437$	4.08454	0.195390
$438$	33.2913	1.59072
$439$	−5.68147	−0.271162	−0.135581	−	0.990766i	$-0.543290\pi$
$439$	−5.68147	−0.271162	−0.135581	+	0.990766i	$0.543290\pi$
$440$	−4.92371	−0.234729
$441$	−0.155057	−0.00738366
$442$	28.6061	1.36066
$443$	−28.8245	−1.36950	−0.684748	−	0.728780i	$-0.740088\pi$
$443$	−28.8245	−1.36950	−0.684748	+	0.728780i	$0.740088\pi$
$444$	56.6147	2.68681
$445$	−15.4722	−0.733455
$446$	−63.2892	−2.99683
$447$	18.7231	0.885573
$448$	38.5221	1.82000
$449$	−6.84205	−0.322897	−0.161448	−	0.986881i	$-0.551616\pi$
$449$	−6.84205	−0.322897	−0.161448	+	0.986881i	$0.551616\pi$
$450$	0.000171612 0	8.08986e−6 0
$451$	−6.97385	−0.328386
$452$	−25.0748	−1.17942
$453$	−12.7119	−0.597259
$454$	−16.2430	−0.762320
$455$	31.5115	1.47728
$456$	−7.39581	−0.346341
$457$	35.6615	1.66818	0.834088	−	0.551631i	$-0.185994\pi$
$457$	35.6615	1.66818	0.834088	+	0.551631i	$0.185994\pi$
$458$	15.5497	0.726590
$459$	−14.8147	−0.691491
$460$	10.5921	0.493858
$461$	−26.9967	−1.25736	−0.628681	−	0.777663i	$-0.716405\pi$
$461$	−26.9967	−1.25736	−0.628681	+	0.777663i	$0.716405\pi$
$462$	14.1490	0.658270
$463$	16.1665	0.751321	0.375660	−	0.926757i	$-0.377416\pi$
$463$	16.1665	0.751321	0.375660	+	0.926757i	$0.377416\pi$
$464$	−1.71685	−0.0797028
$465$	−14.7033	−0.681849
$466$	−61.3672	−2.84278
$467$	0.339338	0.0157027	0.00785135	−	0.999969i	$-0.497501\pi$
$467$	0.339338	0.0157027	0.00785135	+	0.999969i	$0.497501\pi$
$468$	−0.882185	−0.0407790
$469$	0.535662	0.0247346
$470$	14.4301	0.665612
$471$	0.573554	0.0264280
$472$	3.57069	0.164355
$473$	−1.58769	−0.0730019
$474$	48.7835	2.24070
$475$	−0.00279622	−0.000128300 0
$476$	24.1643	1.10757
$477$	−0.407572	−0.0186615
$478$	38.1528	1.74507
$479$	12.9319	0.590876	0.295438	−	0.955362i	$-0.404534\pi$
$479$	12.9319	0.590876	0.295438	+	0.955362i	$0.404534\pi$
$480$	28.0629	1.28089
$481$	−54.3703	−2.47907
$482$	1.37938	0.0628291
$483$	−8.78888	−0.399908
$484$	−26.6346	−1.21066
$485$	13.4966	0.612847
$486$	1.54635	0.0701439
$487$	−5.19086	−0.235220	−0.117610	−	0.993060i	$-0.537523\pi$
$487$	−5.19086	−0.235220	−0.117610	+	0.993060i	$0.537523\pi$
$488$	1.11775	0.0505981
$489$	−17.9977	−0.813882
$490$	11.2116	0.506488
$491$	−26.8894	−1.21350	−0.606752	−	0.794891i	$-0.707528\pi$
$491$	−26.8894	−1.21350	−0.606752	+	0.794891i	$0.707528\pi$
$492$	−27.1646	−1.22468
$493$	−2.82011	−0.127011
$494$	24.5979	1.10671
$495$	−0.187553	−0.00842989
$496$	6.59202	0.295991
$497$	−32.1326	−1.44134
$498$	24.4809	1.09702
$499$	−5.81465	−0.260299	−0.130150	−	0.991494i	$-0.541546\pi$
$499$	−5.81465	−0.260299	−0.130150	+	0.991494i	$0.541546\pi$
$500$	31.4349	1.40581
$501$	−13.9897	−0.625014
$502$	14.8116	0.661072
$503$	14.1721	0.631902	0.315951	−	0.948775i	$-0.397676\pi$
$503$	14.1721	0.631902	0.315951	+	0.948775i	$0.397676\pi$
$504$	−0.368221	−0.0164019
$505$	−0.704946	−0.0313697
$506$	−4.56740	−0.203046
$507$	−14.3544	−0.637502
$508$	28.1965	1.25102
$509$	−27.1766	−1.20458	−0.602292	−	0.798276i	$-0.705746\pi$
$509$	−27.1766	−1.20458	−0.602292	+	0.798276i	$0.705746\pi$
$510$	23.6894	1.04899
$511$	−27.0071	−1.19472
$512$	−18.6974	−0.826314
$513$	−12.7389	−0.562436
$514$	22.7562	1.00373
$515$	3.64125	0.160452
$516$	−6.18437	−0.272252
$517$	−3.63617	−0.159919
$518$	−78.5942	−3.45323
$519$	2.32900	0.102232
$520$	18.4186	0.807707
$521$	−36.5787	−1.60254	−0.801271	−	0.598301i	$-0.795843\pi$
$521$	−36.5787	−1.60254	−0.801271	+	0.598301i	$0.795843\pi$
$522$	0.148826	0.00651396
$523$	16.1921	0.708031	0.354016	−	0.935240i	$-0.384816\pi$
$523$	16.1921	0.708031	0.354016	+	0.935240i	$0.384816\pi$
$524$	−18.9920	−0.829667
$525$	0.00601675	0.000262593 0
$526$	−49.9440	−2.17766
$527$	10.8281	0.471679
$528$	−3.63409	−0.158153
$529$	−20.1629	−0.876647
$530$	29.4700	1.28010
$531$	0.136014	0.00590252
$532$	20.7784	0.900860
$533$	26.0877	1.12998
$534$	25.9879	1.12461
$535$	−36.6338	−1.58382
$536$	0.313096	0.0135237
$537$	−25.0372	−1.08043
$538$	−44.9434	−1.93765
$539$	−2.82515	−0.121688
$540$	−33.0346	−1.42158
$541$	12.8529	0.552588	0.276294	−	0.961073i	$-0.410894\pi$
$541$	12.8529	0.552588	0.276294	+	0.961073i	$0.410894\pi$
$542$	14.2530	0.612218
$543$	34.8617	1.49606
$544$	−20.6666	−0.886074
$545$	−1.35537	−0.0580578
$546$	−52.9283	−2.26512
$547$	−2.73862	−0.117095	−0.0585476	−	0.998285i	$-0.518647\pi$
$547$	−2.73862	−0.117095	−0.0585476	+	0.998285i	$0.518647\pi$
$548$	4.94673	0.211314
$549$	0.0425771	0.00181715
$550$	0.00312678	0.000133326 0
$551$	−2.42496	−0.103307
$552$	−5.13712	−0.218651
$553$	−39.5750	−1.68290
$554$	26.8080	1.13896
$555$	−45.0254	−1.91122
$556$	2.81195	0.119253
$557$	17.8413	0.755959	0.377979	−	0.925814i	$-0.376619\pi$
$557$	17.8413	0.755959	0.377979	+	0.925814i	$0.376619\pi$
$558$	−0.571434	−0.0241907
$559$	5.93920	0.251201
$560$	−11.6995	−0.494396
$561$	−5.96938	−0.252027
$562$	−16.7070	−0.704741
$563$	−7.04099	−0.296742	−0.148371	−	0.988932i	$-0.547403\pi$
$563$	−7.04099	−0.296742	−0.148371	+	0.988932i	$0.547403\pi$
$564$	−14.1636	−0.596397
$565$	19.9418	0.838959
$566$	−32.6974	−1.37437
$567$	26.7906	1.12510
$568$	−18.7816	−0.788058
$569$	15.8440	0.664213	0.332107	−	0.943242i	$-0.392241\pi$
$569$	15.8440	0.664213	0.332107	+	0.943242i	$0.392241\pi$
$570$	20.3701	0.853210
$571$	46.5722	1.94899	0.974494	−	0.224415i	$-0.0720470\pi$
$571$	46.5722	1.94899	0.974494	+	0.224415i	$0.0720470\pi$
$572$	−16.0735	−0.672065
$573$	−7.08873	−0.296136
$574$	37.7107	1.57401
$575$	−0.00194226	−8.09976e−5 0
$576$	0.857685	0.0357369
$577$	−11.0696	−0.460834	−0.230417	−	0.973092i	$-0.574009\pi$
$577$	−11.0696	−0.460834	−0.230417	+	0.973092i	$0.574009\pi$
$578$	19.8456	0.825468
$579$	−23.0732	−0.958891
$580$	−6.28843	−0.261113
$581$	−19.8598	−0.823925
$582$	−22.6695	−0.939680
$583$	−7.42600	−0.307553
$584$	−15.7857	−0.653217
$585$	0.701597	0.0290075
$586$	13.7740	0.569000
$587$	36.0609	1.48839	0.744195	−	0.667962i	$-0.232833\pi$
$587$	36.0609	1.48839	0.744195	+	0.667962i	$0.232833\pi$
$588$	−11.0045	−0.453819
$589$	9.31088	0.383648
$590$	−9.83469	−0.404888
$591$	−16.2546	−0.668626
$592$	20.1865	0.829661
$593$	14.7547	0.605905	0.302952	−	0.953006i	$-0.402028\pi$
$593$	14.7547	0.605905	0.302952	+	0.953006i	$0.402028\pi$
$594$	14.2448	0.584472
$595$	−19.2178	−0.787851
$596$	−30.7462	−1.25941
$597$	31.4066	1.28538
$598$	17.0857	0.698685
$599$	−0.544719	−0.0222566	−0.0111283	−	0.999938i	$-0.503542\pi$
$599$	−0.544719	−0.0222566	−0.0111283	+	0.999938i	$0.503542\pi$
$600$	0.00351681	0.000143573 0
$601$	15.8966	0.648437	0.324219	−	0.945982i	$-0.394899\pi$
$601$	15.8966	0.648437	0.324219	+	0.945982i	$0.394899\pi$
$602$	8.58533	0.349912
$603$	0.0119264	0.000485680 0
$604$	20.8750	0.849390
$605$	21.1824	0.861185
$606$	1.18406	0.0480992
$607$	−6.41339	−0.260312	−0.130156	−	0.991494i	$-0.541548\pi$
$607$	−6.41339	−0.260312	−0.130156	+	0.991494i	$0.541548\pi$
$608$	−17.7708	−0.720703
$609$	5.21789	0.211440
$610$	−3.07859	−0.124649
$611$	13.6021	0.550284
$612$	0.538013	0.0217479
$613$	18.1907	0.734715	0.367357	−	0.930080i	$-0.380263\pi$
$613$	18.1907	0.734715	0.367357	+	0.930080i	$0.380263\pi$
$614$	47.7408	1.92666
$615$	21.6039	0.871152
$616$	−6.70902	−0.270314
$617$	27.4921	1.10679	0.553395	−	0.832919i	$-0.313332\pi$
$617$	27.4921	1.10679	0.553395	+	0.832919i	$0.313332\pi$
$618$	−6.11602	−0.246022
$619$	1.55256	0.0624026	0.0312013	−	0.999513i	$-0.490067\pi$
$619$	1.55256	0.0624026	0.0312013	+	0.999513i	$0.490067\pi$
$620$	24.1451	0.969689
$621$	−8.84842	−0.355075
$622$	56.1636	2.25195
$623$	−21.0824	−0.844648
$624$	13.5944	0.544210
$625$	−25.0058	−1.00023
$626$	8.43519	0.337138
$627$	−5.13296	−0.204991
$628$	−0.941863	−0.0375844
$629$	33.1585	1.32212
$630$	1.01418	0.0404060
$631$	2.38646	0.0950035	0.0475017	−	0.998871i	$-0.484874\pi$
$631$	2.38646	0.0950035	0.0475017	+	0.998871i	$0.484874\pi$
$632$	−23.1317	−0.920128
$633$	−1.39189	−0.0553226
$634$	33.9579	1.34864
$635$	−22.4246	−0.889891
$636$	−28.9258	−1.14698
$637$	10.5683	0.418730
$638$	2.71163	0.107354
$639$	−0.715425	−0.0283018
$640$	−29.2390	−1.15577
$641$	−12.4619	−0.492217	−0.246108	−	0.969242i	$-0.579152\pi$
$641$	−12.4619	−0.492217	−0.246108	+	0.969242i	$0.579152\pi$
$642$	61.5320	2.42847
$643$	30.3287	1.19605	0.598024	−	0.801478i	$-0.295953\pi$
$643$	30.3287	1.19605	0.598024	+	0.801478i	$0.295953\pi$
$644$	14.4327	0.568728
$645$	4.91840	0.193662
$646$	−15.0014	−0.590221
$647$	1.67157	0.0657163	0.0328582	−	0.999460i	$-0.489539\pi$
$647$	1.67157	0.0657163	0.0328582	+	0.999460i	$0.489539\pi$
$648$	15.6592	0.615150
$649$	2.47819	0.0972775
$650$	−0.0116966	−0.000458779 0
$651$	−20.0346	−0.785218
$652$	29.5549	1.15746
$653$	1.68651	0.0659982	0.0329991	−	0.999455i	$-0.489494\pi$
$653$	1.68651	0.0659982	0.0329991	+	0.999455i	$0.489494\pi$
$654$	2.27655	0.0890203
$655$	15.1042	0.590170
$656$	−9.68580	−0.378167
$657$	−0.601307	−0.0234592
$658$	19.6624	0.766520
$659$	−7.66163	−0.298455	−0.149227	−	0.988803i	$-0.547679\pi$
$659$	−7.66163	−0.298455	−0.149227	+	0.988803i	$0.547679\pi$
$660$	−13.3108	−0.518123
$661$	−24.5560	−0.955117	−0.477559	−	0.878600i	$-0.658478\pi$
$661$	−24.5560	−0.955117	−0.477559	+	0.878600i	$0.658478\pi$
$662$	64.1784	2.49436
$663$	22.3302	0.867233
$664$	−11.6081	−0.450482
$665$	−16.5250	−0.640811
$666$	−1.74988	−0.0678066
$667$	−1.68438	−0.0652193
$668$	22.9732	0.888861
$669$	−49.4040	−1.91007
$670$	−0.862353	−0.0333156
$671$	0.775758	0.0299478
$672$	38.2383	1.47507
$673$	−14.1036	−0.543654	−0.271827	−	0.962346i	$-0.587628\pi$
$673$	−14.1036	−0.543654	−0.271827	+	0.962346i	$0.587628\pi$
$674$	12.8480	0.494886
$675$	0.00605751	0.000233154 0
$676$	23.5721	0.906621
$677$	35.5685	1.36701	0.683504	−	0.729947i	$-0.260455\pi$
$677$	35.5685	1.36701	0.683504	+	0.729947i	$0.260455\pi$
$678$	−33.4953	−1.28638
$679$	18.3903	0.705756
$680$	−11.2328	−0.430759
$681$	−12.6794	−0.485875
$682$	−10.4116	−0.398680
$683$	10.4193	0.398682	0.199341	−	0.979930i	$-0.436120\pi$
$683$	10.4193	0.398682	0.199341	+	0.979930i	$0.436120\pi$
$684$	0.462627	0.0176890
$685$	−3.93411	−0.150315
$686$	−31.5139	−1.20321
$687$	12.1382	0.463102
$688$	−2.20510	−0.0840685
$689$	27.7791	1.05830
$690$	14.1491	0.538646
$691$	37.1399	1.41287	0.706435	−	0.707778i	$-0.250302\pi$
$691$	37.1399	1.41287	0.706435	+	0.707778i	$0.250302\pi$
$692$	−3.82458	−0.145389
$693$	−0.255559	−0.00970787
$694$	14.5640	0.552843
$695$	−2.23633	−0.0848287
$696$	3.04987	0.115605
$697$	−15.9100	−0.602633
$698$	−52.6592	−1.99318
$699$	−47.9037	−1.81188
$700$	−0.00988043	−0.000373445 0
$701$	−22.8356	−0.862490	−0.431245	−	0.902235i	$-0.641926\pi$
$701$	−22.8356	−0.862490	−0.431245	+	0.902235i	$0.641926\pi$
$702$	−53.2869	−2.01118
$703$	28.5124	1.07536
$704$	15.6271	0.588968
$705$	11.2643	0.424237
$706$	48.6335	1.83034
$707$	−0.960554	−0.0361254
$708$	9.65307	0.362785
$709$	45.5934	1.71230	0.856149	−	0.516730i	$-0.172851\pi$
$709$	45.5934	1.71230	0.856149	+	0.516730i	$0.172851\pi$
$710$	51.7297	1.94138
$711$	−0.881127	−0.0330448
$712$	−12.3227	−0.461813
$713$	6.46733	0.242203
$714$	32.2791	1.20801
$715$	12.7831	0.478062
$716$	41.1148	1.53653
$717$	29.7824	1.11224
$718$	15.1740	0.566287
$719$	22.2633	0.830280	0.415140	−	0.909758i	$-0.363733\pi$
$719$	22.2633	0.830280	0.415140	+	0.909758i	$0.363733\pi$
$720$	−0.260488	−0.00970780
$721$	4.96154	0.184777
$722$	28.7793	1.07105
$723$	1.07676	0.0400450
$724$	−57.2483	−2.12761
$725$	0.00115310	4.28251e−5 0
$726$	−35.5789	−1.32046
$727$	−37.4973	−1.39070	−0.695350	−	0.718672i	$-0.744750\pi$
$727$	−37.4973	−1.39070	−0.695350	+	0.718672i	$0.744750\pi$
$728$	25.0970	0.930157
$729$	27.5827	1.02158
$730$	43.4782	1.60920
$731$	−3.62210	−0.133968
$732$	3.02174	0.111687
$733$	−10.8492	−0.400725	−0.200363	−	0.979722i	$-0.564212\pi$
$733$	−10.8492	−0.400725	−0.200363	+	0.979722i	$0.564212\pi$
$734$	9.67743	0.357200
$735$	8.75185	0.322817
$736$	−12.3436	−0.454992
$737$	0.217300	0.00800434
$738$	0.839620	0.0309069
$739$	0.762676	0.0280555	0.0140277	−	0.999902i	$-0.495535\pi$
$739$	0.762676	0.0280555	0.0140277	+	0.999902i	$0.495535\pi$
$740$	73.9385	2.71803
$741$	19.2013	0.705378
$742$	40.1557	1.47416
$743$	−8.51803	−0.312496	−0.156248	−	0.987718i	$-0.549940\pi$
$743$	−8.51803	−0.312496	−0.156248	+	0.987718i	$0.549940\pi$
$744$	−11.7103	−0.429320
$745$	24.4523	0.895863
$746$	50.6670	1.85505
$747$	−0.442175	−0.0161783
$748$	9.80263	0.358420
$749$	−49.9170	−1.82393
$750$	41.9912	1.53330
$751$	30.0749	1.09745	0.548724	−	0.836003i	$-0.315114\pi$
$751$	30.0749	1.09745	0.548724	+	0.836003i	$0.315114\pi$
$752$	−5.05018	−0.184161
$753$	11.5620	0.421344
$754$	−10.1436	−0.369409
$755$	−16.6017	−0.604199
$756$	−45.0127	−1.63710
$757$	−38.6729	−1.40559	−0.702796	−	0.711392i	$-0.748065\pi$
$757$	−38.6729	−1.40559	−0.702796	+	0.711392i	$0.748065\pi$
$758$	−8.66343	−0.314670
$759$	−3.56535	−0.129414
$760$	−9.65889	−0.350365
$761$	−13.9600	−0.506049	−0.253025	−	0.967460i	$-0.581425\pi$
$761$	−13.9600	−0.506049	−0.253025	+	0.967460i	$0.581425\pi$
$762$	37.6654	1.36447
$763$	−1.84682	−0.0668595
$764$	11.6408	0.421148
$765$	−0.427879	−0.0154700
$766$	32.4153	1.17121
$767$	−9.27039	−0.334734
$768$	5.81694	0.209901
$769$	−43.0406	−1.55208	−0.776041	−	0.630682i	$-0.782775\pi$
$769$	−43.0406	−1.55208	−0.776041	+	0.630682i	$0.782775\pi$
$770$	18.4785	0.665919
$771$	17.7636	0.639742
$772$	37.8898	1.36368
$773$	9.81779	0.353121	0.176561	−	0.984290i	$-0.443503\pi$
$773$	9.81779	0.353121	0.176561	+	0.984290i	$0.443503\pi$
$774$	0.191150	0.00687076
$775$	−0.00442745	−0.000159039 0
$776$	10.7492	0.385873
$777$	−61.3513	−2.20097
$778$	−34.1585	−1.22464
$779$	−13.6807	−0.490161
$780$	49.7930	1.78288
$781$	−13.0351	−0.466432
$782$	−10.4199	−0.372616
$783$	5.25324	0.187735
$784$	−3.92377	−0.140135
$785$	0.749059	0.0267351
$786$	−25.3698	−0.904909
$787$	−18.1905	−0.648421	−0.324210	−	0.945985i	$-0.605099\pi$
$787$	−18.1905	−0.648421	−0.324210	+	0.945985i	$0.605099\pi$
$788$	26.6926	0.950884
$789$	−38.9867	−1.38796
$790$	63.7110	2.26674
$791$	27.1726	0.966146
$792$	−0.149375	−0.00530780
$793$	−2.90195	−0.103051
$794$	−2.04247	−0.0724846
$795$	23.0045	0.815887
$796$	−51.5744	−1.82800
$797$	23.9899	0.849767	0.424884	−	0.905248i	$-0.360315\pi$
$797$	23.9899	0.849767	0.424884	+	0.905248i	$0.360315\pi$
$798$	27.7562	0.982558
$799$	−8.29546	−0.293472
$800$	0.00845028	0.000298762 0
$801$	−0.469394	−0.0165852
$802$	−6.71510	−0.237118
$803$	−10.9558	−0.386624
$804$	0.846428	0.0298512
$805$	−11.4782	−0.404555
$806$	38.9475	1.37187
$807$	−35.0832	−1.23499
$808$	−0.561446	−0.0197516
$809$	−21.2688	−0.747771	−0.373886	−	0.927475i	$-0.621975\pi$
$809$	−21.2688	−0.747771	−0.373886	+	0.927475i	$0.621975\pi$
$810$	−43.1297	−1.51542
$811$	24.1280	0.847250	0.423625	−	0.905838i	$-0.360757\pi$
$811$	24.1280	0.847250	0.423625	+	0.905838i	$0.360757\pi$
$812$	−8.56857	−0.300698
$813$	11.1260	0.390206
$814$	−31.8830	−1.11750
$815$	−23.5049	−0.823339
$816$	−8.29072	−0.290233
$817$	−3.11458	−0.108965
$818$	35.2335	1.23191
$819$	0.955991	0.0334050
$820$	−35.4769	−1.23891
$821$	46.8952	1.63665	0.818326	−	0.574754i	$-0.194902\pi$
$821$	46.8952	1.63665	0.818326	+	0.574754i	$0.194902\pi$
$822$	6.60792	0.230478
$823$	−20.4049	−0.711272	−0.355636	−	0.934625i	$-0.615736\pi$
$823$	−20.4049	−0.711272	−0.355636	+	0.934625i	$0.615736\pi$
$824$	2.90003	0.101027
$825$	0.00244079	8.49774e−5 0
$826$	−13.4007	−0.466269
$827$	8.64849	0.300737	0.150369	−	0.988630i	$-0.451954\pi$
$827$	8.64849	0.300737	0.150369	+	0.988630i	$0.451954\pi$
$828$	0.321341	0.0111674
$829$	16.5277	0.574030	0.287015	−	0.957926i	$-0.407337\pi$
$829$	16.5277	0.574030	0.287015	+	0.957926i	$0.407337\pi$
$830$	31.9720	1.10976
$831$	20.9265	0.725933
$832$	−58.4576	−2.02665
$833$	−6.44521	−0.223313
$834$	3.75624	0.130068
$835$	−18.2705	−0.632276
$836$	8.42910	0.291527
$837$	−20.1703	−0.697189
$838$	22.9572	0.793042
$839$	−6.84408	−0.236284	−0.118142	−	0.992997i	$-0.537694\pi$
$839$	−6.84408	−0.236284	−0.118142	+	0.992997i	$0.537694\pi$
$840$	20.7835	0.717097
$841$	1.00000	0.0344828
$842$	80.6883	2.78070
$843$	−13.0416	−0.449177
$844$	2.28569	0.0786768
$845$	−18.7468	−0.644909
$846$	0.437779	0.0150511
$847$	28.8629	0.991742
$848$	−10.3138	−0.354176
$849$	−25.5238	−0.875976
$850$	0.00713336	0.000244672 0
$851$	19.8047	0.678895
$852$	−50.7744	−1.73950
$853$	−18.3716	−0.629033	−0.314517	−	0.949252i	$-0.601842\pi$
$853$	−18.3716	−0.629033	−0.314517	+	0.949252i	$0.601842\pi$
$854$	−4.19487	−0.143546
$855$	−0.367925	−0.0125828
$856$	−29.1766	−0.997236
$857$	12.4559	0.425487	0.212743	−	0.977108i	$-0.431760\pi$
$857$	12.4559	0.425487	0.212743	+	0.977108i	$0.431760\pi$
$858$	−21.4712	−0.733015
$859$	−30.2817	−1.03320	−0.516600	−	0.856227i	$-0.672803\pi$
$859$	−30.2817	−1.03320	−0.516600	+	0.856227i	$0.672803\pi$
$860$	−8.07676	−0.275415
$861$	29.4373	1.00322
$862$	25.9473	0.883768
$863$	16.3159	0.555399	0.277699	−	0.960668i	$-0.410428\pi$
$863$	16.3159	0.555399	0.277699	+	0.960668i	$0.410428\pi$
$864$	38.4973	1.30971
$865$	3.04167	0.103420
$866$	32.4319	1.10208
$867$	15.4916	0.526123
$868$	32.8999	1.11670
$869$	−16.0542	−0.544601
$870$	−8.40019	−0.284793
$871$	−0.812872	−0.0275431
$872$	−1.07947	−0.0365556
$873$	0.409456	0.0138580
$874$	−8.95991	−0.303074
$875$	−34.0648	−1.15160
$876$	−42.6753	−1.44187
$877$	23.7024	0.800374	0.400187	−	0.916433i	$-0.368945\pi$
$877$	23.7024	0.800374	0.400187	+	0.916433i	$0.368945\pi$
$878$	12.4630	0.420605
$879$	10.7521	0.362660
$880$	−4.74610	−0.159991
$881$	18.9059	0.636955	0.318477	−	0.947930i	$-0.396828\pi$
$881$	18.9059	0.636955	0.318477	+	0.947930i	$0.396828\pi$
$882$	0.340135	0.0114529
$883$	23.7780	0.800193	0.400097	−	0.916473i	$-0.368977\pi$
$883$	23.7780	0.800193	0.400097	+	0.916473i	$0.368977\pi$
$884$	−36.6696	−1.23333
$885$	−7.67704	−0.258061
$886$	63.2299	2.12425
$887$	16.1515	0.542315	0.271157	−	0.962535i	$-0.412594\pi$
$887$	16.1515	0.542315	0.271157	+	0.962535i	$0.412594\pi$
$888$	−35.8600	−1.20338
$889$	−30.5556	−1.02480
$890$	33.9401	1.13768
$891$	10.8680	0.364092
$892$	81.1290	2.71640
$893$	−7.13311	−0.238700
$894$	−41.0713	−1.37363
$895$	−32.6984	−1.09299
$896$	−39.8409	−1.33099
$897$	13.3372	0.445317
$898$	15.0088	0.500851
$899$	−3.83960	−0.128058
$900$	−0.000219985 0	−7.33285e−6 0
$901$	−16.9415	−0.564402
$902$	15.2979	0.509366
$903$	6.70177	0.223021
$904$	15.8824	0.528242
$905$	45.5292	1.51344
$906$	27.8851	0.926421
$907$	−34.7712	−1.15456	−0.577280	−	0.816546i	$-0.695886\pi$
$907$	−34.7712	−1.15456	−0.577280	+	0.816546i	$0.695886\pi$
$908$	20.8215	0.690985
$909$	−0.0213865	−0.000709346 0
$910$	−69.1241	−2.29144
$911$	−52.5178	−1.73999	−0.869996	−	0.493059i	$-0.835879\pi$
$911$	−52.5178	−1.73999	−0.869996	+	0.493059i	$0.835879\pi$
$912$	−7.12903	−0.236066
$913$	−8.05645	−0.266630
$914$	−78.2277	−2.58754
$915$	−2.40317	−0.0794465
$916$	−19.9328	−0.658599
$917$	20.5809	0.679641
$918$	32.4978	1.07259
$919$	−19.6793	−0.649159	−0.324579	−	0.945858i	$-0.605223\pi$
$919$	−19.6793	−0.649159	−0.324579	+	0.945858i	$0.605223\pi$
$920$	−6.70906	−0.221191
$921$	37.2668	1.22798
$922$	59.2204	1.95032
$923$	48.7615	1.60501
$924$	−18.1373	−0.596672
$925$	−0.0135580	−0.000445785 0
$926$	−35.4631	−1.16539
$927$	0.110467	0.00362823
$928$	7.32830	0.240563
$929$	52.0527	1.70780	0.853898	−	0.520441i	$-0.174232\pi$
$929$	52.0527	1.70780	0.853898	+	0.520441i	$0.174232\pi$
$930$	32.2534	1.05763
$931$	−5.54212	−0.181636
$932$	78.6652	2.57676
$933$	43.8417	1.43531
$934$	−0.744378	−0.0243568
$935$	−7.79598	−0.254956
$936$	0.558779	0.0182643
$937$	−24.0301	−0.785028	−0.392514	−	0.919746i	$-0.628395\pi$
$937$	−24.0301	−0.785028	−0.392514	+	0.919746i	$0.628395\pi$
$938$	−1.17504	−0.0383663
$939$	6.58458	0.214880
$940$	−18.4976	−0.603327
$941$	−48.0339	−1.56586	−0.782930	−	0.622110i	$-0.786276\pi$
$941$	−48.0339	−1.56586	−0.782930	+	0.622110i	$0.786276\pi$
$942$	−1.25816	−0.0409930
$943$	−9.50259	−0.309447
$944$	3.44189	0.112024
$945$	35.7984	1.16452
$946$	3.48277	0.113235
$947$	−2.76166	−0.0897419	−0.0448710	−	0.998993i	$-0.514288\pi$
$947$	−2.76166	−0.0897419	−0.0448710	+	0.998993i	$0.514288\pi$
$948$	−62.5345	−2.03102
$949$	40.9835	1.33038
$950$	0.00613384	0.000199008 0
$951$	26.5078	0.859576
$952$	−15.3058	−0.496063
$953$	−12.5120	−0.405305	−0.202652	−	0.979251i	$-0.564956\pi$
$953$	−12.5120	−0.405305	−0.202652	+	0.979251i	$0.564956\pi$
$954$	0.894057	0.0289462
$955$	−9.25784	−0.299577
$956$	−48.9072	−1.58177
$957$	2.11672	0.0684238
$958$	−28.3677	−0.916519
$959$	−5.36059	−0.173102
$960$	−48.4102	−1.56243
$961$	−16.2575	−0.524434
$962$	119.267	3.84534
$963$	−1.11139	−0.0358141
$964$	−1.76820	−0.0569498
$965$	−30.1336	−0.970033
$966$	19.2794	0.620305
$967$	−18.6051	−0.598299	−0.299150	−	0.954206i	$-0.596703\pi$
$967$	−18.6051	−0.598299	−0.299150	+	0.954206i	$0.596703\pi$
$968$	16.8705	0.542237
$969$	−11.7102	−0.376186
$970$	−29.6062	−0.950599
$971$	41.4220	1.32930	0.664648	−	0.747157i	$-0.268582\pi$
$971$	41.4220	1.32930	0.664648	+	0.747157i	$0.268582\pi$
$972$	−1.98223	−0.0635801
$973$	−3.04720	−0.0976889
$974$	11.3867	0.364855
$975$	−0.00913048	−0.000292409 0
$976$	1.07743	0.0344877
$977$	−47.5754	−1.52207	−0.761036	−	0.648709i	$-0.775309\pi$
$977$	−47.5754	−1.52207	−0.761036	+	0.648709i	$0.775309\pi$
$978$	39.4799	1.26243
$979$	−8.55240	−0.273336
$980$	−14.3719	−0.459093
$981$	−0.0411191	−0.00131283
$982$	58.9851	1.88229
$983$	31.3046	0.998461	0.499231	−	0.866469i	$-0.333616\pi$
$983$	31.3046	0.998461	0.499231	+	0.866469i	$0.333616\pi$
$984$	17.2062	0.548513
$985$	−21.2285	−0.676396
$986$	6.18624	0.197010
$987$	15.3486	0.488552
$988$	−31.5315	−1.00315
$989$	−2.16339	−0.0687916
$990$	0.411419	0.0130758
$991$	−36.6547	−1.16437	−0.582187	−	0.813055i	$-0.697803\pi$
$991$	−36.6547	−1.16437	−0.582187	+	0.813055i	$0.697803\pi$
$992$	−28.1378	−0.893375
$993$	50.0982	1.58982
$994$	70.4866	2.23570
$995$	41.0168	1.30032
$996$	−31.3816	−0.994363
$997$	11.6378	0.368574	0.184287	−	0.982872i	$-0.441002\pi$
$997$	11.6378	0.368574	0.184287	+	0.982872i	$0.441002\pi$
$998$	12.7551	0.403755
$999$	−61.7669	−1.95422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.b.1.8	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.b.1.8	✓	59	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)

\(f(q)\)	\(=\)	\(q-2.19361 q^{2} -1.71235 q^{3} +2.81195 q^{4} -2.23633 q^{5} +3.75624 q^{6} -3.04720 q^{7} -1.78110 q^{8} -0.0678453 q^{9} +O(q^{10})\)	\(q-2.19361 q^{2} -1.71235 q^{3} +2.81195 q^{4} -2.23633 q^{5} +3.75624 q^{6} -3.04720 q^{7} -1.78110 q^{8} -0.0678453 q^{9} +4.90564 q^{10} -1.23615 q^{11} -4.81505 q^{12} +4.62416 q^{13} +6.68439 q^{14} +3.82938 q^{15} -1.71685 q^{16} -2.82011 q^{17} +0.148826 q^{18} -2.42496 q^{19} -6.28843 q^{20} +5.21789 q^{21} +2.71163 q^{22} -1.68438 q^{23} +3.04987 q^{24} +0.00115310 q^{25} -10.1436 q^{26} +5.25324 q^{27} -8.56857 q^{28} +1.00000 q^{29} -8.40019 q^{30} -3.83960 q^{31} +7.32830 q^{32} +2.11672 q^{33} +6.18624 q^{34} +6.81454 q^{35} -0.190777 q^{36} -11.7579 q^{37} +5.31943 q^{38} -7.91820 q^{39} +3.98311 q^{40} +5.64161 q^{41} -11.4460 q^{42} +1.28438 q^{43} -3.47598 q^{44} +0.151724 q^{45} +3.69487 q^{46} +2.94154 q^{47} +2.93986 q^{48} +2.28545 q^{49} -0.00252946 q^{50} +4.82903 q^{51} +13.0029 q^{52} +6.00738 q^{53} -11.5236 q^{54} +2.76442 q^{55} +5.42737 q^{56} +4.15239 q^{57} -2.19361 q^{58} -2.00477 q^{59} +10.7680 q^{60} -0.627562 q^{61} +8.42261 q^{62} +0.206738 q^{63} -12.6418 q^{64} -10.3411 q^{65} -4.64327 q^{66} -0.175788 q^{67} -7.93000 q^{68} +2.88425 q^{69} -14.9485 q^{70} +10.5450 q^{71} +0.120839 q^{72} +8.86291 q^{73} +25.7922 q^{74} -0.00197452 q^{75} -6.81886 q^{76} +3.76679 q^{77} +17.3695 q^{78} +12.9873 q^{79} +3.83944 q^{80} -8.79186 q^{81} -12.3755 q^{82} +6.51740 q^{83} +14.6724 q^{84} +6.30668 q^{85} -2.81744 q^{86} -1.71235 q^{87} +2.20170 q^{88} +6.91860 q^{89} -0.332824 q^{90} -14.0908 q^{91} -4.73637 q^{92} +6.57475 q^{93} -6.45260 q^{94} +5.42300 q^{95} -12.5486 q^{96} -6.03515 q^{97} -5.01339 q^{98} +0.0838666 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * q^99

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