LMFDB - Embedded newform 2015.2.a.j.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2015,2,Mod(1,2015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2015, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2015 = 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2015.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$16.0898560073$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 5 x^{19} - 20 x^{18} + 130 x^{17} + 119 x^{16} - 1374 x^{15} + 129 x^{14} + 7595 x^{13} + \cdots - 80 $ x^20 - 5x^19 - 20x^18 + 130x^17 + 119x^16 - 1374x^15 + 129x^14 + 7595x^13 - 4447x^12 - 23555x^11 + 20653x^10 + 40907x^9 - 44507x^8 - 37338x^7 + 49026x^6 + 14851x^5 - 26733x^4 - 224x^3 + 5841x^2 - 968*x - 80
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$2.54458$ of defining polynomial
Character	$\chi$	$=$	2015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.54458 q^{2} +1.69062 q^{3} +4.47491 q^{4} -1.00000 q^{5} +4.30194 q^{6} +2.93829 q^{7} +6.29761 q^{8} -0.141787 q^{9} +O(q^{10})$	\(q+2.54458 q^{2} +1.69062 q^{3} +4.47491 q^{4} -1.00000 q^{5} +4.30194 q^{6} +2.93829 q^{7} +6.29761 q^{8} -0.141787 q^{9} -2.54458 q^{10} +4.38815 q^{11} +7.56539 q^{12} +1.00000 q^{13} +7.47672 q^{14} -1.69062 q^{15} +7.07498 q^{16} -2.45982 q^{17} -0.360789 q^{18} -7.95103 q^{19} -4.47491 q^{20} +4.96754 q^{21} +11.1660 q^{22} -8.52209 q^{23} +10.6469 q^{24} +1.00000 q^{25} +2.54458 q^{26} -5.31158 q^{27} +13.1486 q^{28} +6.61504 q^{29} -4.30194 q^{30} -1.00000 q^{31} +5.40766 q^{32} +7.41871 q^{33} -6.25921 q^{34} -2.93829 q^{35} -0.634484 q^{36} +8.09954 q^{37} -20.2321 q^{38} +1.69062 q^{39} -6.29761 q^{40} -8.81257 q^{41} +12.6403 q^{42} +2.27968 q^{43} +19.6366 q^{44} +0.141787 q^{45} -21.6852 q^{46} +7.00292 q^{47} +11.9611 q^{48} +1.63353 q^{49} +2.54458 q^{50} -4.15863 q^{51} +4.47491 q^{52} -6.46149 q^{53} -13.5158 q^{54} -4.38815 q^{55} +18.5042 q^{56} -13.4422 q^{57} +16.8325 q^{58} -4.24895 q^{59} -7.56539 q^{60} +12.7294 q^{61} -2.54458 q^{62} -0.416611 q^{63} -0.389712 q^{64} -1.00000 q^{65} +18.8775 q^{66} -5.15855 q^{67} -11.0074 q^{68} -14.4077 q^{69} -7.47672 q^{70} -4.40773 q^{71} -0.892920 q^{72} +11.5095 q^{73} +20.6100 q^{74} +1.69062 q^{75} -35.5801 q^{76} +12.8936 q^{77} +4.30194 q^{78} +5.70631 q^{79} -7.07498 q^{80} -8.55453 q^{81} -22.4243 q^{82} -12.8844 q^{83} +22.2293 q^{84} +2.45982 q^{85} +5.80084 q^{86} +11.1836 q^{87} +27.6348 q^{88} -14.4016 q^{89} +0.360789 q^{90} +2.93829 q^{91} -38.1356 q^{92} -1.69062 q^{93} +17.8195 q^{94} +7.95103 q^{95} +9.14233 q^{96} +5.21382 q^{97} +4.15666 q^{98} -0.622183 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{2} + q^{3} + 25 q^{4} - 20 q^{5} + 9 q^{6} + 11 q^{7} + 15 q^{8} + 29 q^{9}+O(q^{10}) $ 20 * q + 5 * q^2 + q^3 + 25 * q^4 - 20 * q^5 + 9 * q^6 + 11 * q^7 + 15 * q^8 + 29 * q^9	\( 20 q + 5 q^{2} + q^{3} + 25 q^{4} - 20 q^{5} + 9 q^{6} + 11 q^{7} + 15 q^{8} + 29 q^{9} - 5 q^{10} + 5 q^{11} + 20 q^{13} - 7 q^{14} - q^{15} + 39 q^{16} + 3 q^{17} + 30 q^{18} + 15 q^{19} - 25 q^{20} + 11 q^{22} - 5 q^{23} + 8 q^{24} + 20 q^{25} + 5 q^{26} - 8 q^{27} + 46 q^{28} - 2 q^{29} - 9 q^{30} - 20 q^{31} + 50 q^{32} + 43 q^{33} - q^{34} - 11 q^{35} + 10 q^{36} + 46 q^{37} + 7 q^{38} + q^{39} - 15 q^{40} - 12 q^{41} + 39 q^{42} + 13 q^{43} + 8 q^{44} - 29 q^{45} - 14 q^{46} + 21 q^{47} + 27 q^{48} + 55 q^{49} + 5 q^{50} - 9 q^{51} + 25 q^{52} + 7 q^{53} - 32 q^{54} - 5 q^{55} - 15 q^{56} + 70 q^{57} + 10 q^{58} + 5 q^{59} - 21 q^{61} - 5 q^{62} + 78 q^{63} + 51 q^{64} - 20 q^{65} - 25 q^{66} + 57 q^{67} + 23 q^{68} - 28 q^{69} + 7 q^{70} - 15 q^{71} + 16 q^{72} + 67 q^{73} + 18 q^{74} + q^{75} + 61 q^{76} - 22 q^{77} + 9 q^{78} + 12 q^{79} - 39 q^{80} + 68 q^{81} - 6 q^{82} + 21 q^{83} - 43 q^{84} - 3 q^{85} + 12 q^{86} + 20 q^{87} + 79 q^{88} - 30 q^{90} + 11 q^{91} + 16 q^{92} - q^{93} + 25 q^{94} - 15 q^{95} + 86 q^{96} + 105 q^{97} + 4 q^{98} - 11 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 + q^3 + 25 * q^4 - 20 * q^5 + 9 * q^6 + 11 * q^7 + 15 * q^8 + 29 * q^9 - 5 * q^10 + 5 * q^11 + 20 * q^13 - 7 * q^14 - q^15 + 39 * q^16 + 3 * q^17 + 30 * q^18 + 15 * q^19 - 25 * q^20 + 11 * q^22 - 5 * q^23 + 8 * q^24 + 20 * q^25 + 5 * q^26 - 8 * q^27 + 46 * q^28 - 2 * q^29 - 9 * q^30 - 20 * q^31 + 50 * q^32 + 43 * q^33 - q^34 - 11 * q^35 + 10 * q^36 + 46 * q^37 + 7 * q^38 + q^39 - 15 * q^40 - 12 * q^41 + 39 * q^42 + 13 * q^43 + 8 * q^44 - 29 * q^45 - 14 * q^46 + 21 * q^47 + 27 * q^48 + 55 * q^49 + 5 * q^50 - 9 * q^51 + 25 * q^52 + 7 * q^53 - 32 * q^54 - 5 * q^55 - 15 * q^56 + 70 * q^57 + 10 * q^58 + 5 * q^59 - 21 * q^61 - 5 * q^62 + 78 * q^63 + 51 * q^64 - 20 * q^65 - 25 * q^66 + 57 * q^67 + 23 * q^68 - 28 * q^69 + 7 * q^70 - 15 * q^71 + 16 * q^72 + 67 * q^73 + 18 * q^74 + q^75 + 61 * q^76 - 22 * q^77 + 9 * q^78 + 12 * q^79 - 39 * q^80 + 68 * q^81 - 6 * q^82 + 21 * q^83 - 43 * q^84 - 3 * q^85 + 12 * q^86 + 20 * q^87 + 79 * q^88 - 30 * q^90 + 11 * q^91 + 16 * q^92 - q^93 + 25 * q^94 - 15 * q^95 + 86 * q^96 + 105 * q^97 + 4 * q^98 - 11 * 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 $

$2$	2.54458	1.79929	0.899646	−	0.436620i	$-0.143824\pi$
$2$	2.54458	1.79929	0.899646	+	0.436620i	$0.143824\pi$
$3$	1.69062	0.976083	0.488041	−	0.872820i	$-0.337712\pi$
$3$	1.69062	0.976083	0.488041	+	0.872820i	$0.337712\pi$
$4$	4.47491	2.23745
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	4.30194	1.75626
$7$	2.93829	1.11057	0.555284	−	0.831661i	$-0.312610\pi$
$7$	2.93829	1.11057	0.555284	+	0.831661i	$0.312610\pi$
$8$	6.29761	2.22654
$9$	−0.141787	−0.0472624
$10$	−2.54458	−0.804668
$11$	4.38815	1.32308	0.661538	−	0.749912i	$-0.269904\pi$
$11$	4.38815	1.32308	0.661538	+	0.749912i	$0.269904\pi$
$12$	7.56539	2.18394
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	7.47672	1.99824
$15$	−1.69062	−0.436517
$16$	7.07498	1.76874
$17$	−2.45982	−0.596593	−0.298296	−	0.954473i	$-0.596418\pi$
$17$	−2.45982	−0.596593	−0.298296	+	0.954473i	$0.596418\pi$
$18$	−0.360789	−0.0850388
$19$	−7.95103	−1.82409	−0.912046	−	0.410088i	$-0.865498\pi$
$19$	−7.95103	−1.82409	−0.912046	+	0.410088i	$0.865498\pi$
$20$	−4.47491	−1.00062
$21$	4.96754	1.08401
$22$	11.1660	2.38060
$23$	−8.52209	−1.77698	−0.888489	−	0.458898i	$-0.848245\pi$
$23$	−8.52209	−1.77698	−0.888489	+	0.458898i	$0.848245\pi$
$24$	10.6469	2.17329
$25$	1.00000	0.200000
$26$	2.54458	0.499034
$27$	−5.31158	−1.02221
$28$	13.1486	2.48484
$29$	6.61504	1.22838	0.614191	−	0.789157i	$-0.289482\pi$
$29$	6.61504	1.22838	0.614191	+	0.789157i	$0.289482\pi$
$30$	−4.30194	−0.785423
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	5.40766	0.955948
$33$	7.41871	1.29143
$34$	−6.25921	−1.07344
$35$	−2.93829	−0.496661
$36$	−0.634484	−0.105747
$37$	8.09954	1.33156	0.665778	−	0.746150i	$-0.268100\pi$
$37$	8.09954	1.33156	0.665778	+	0.746150i	$0.268100\pi$
$38$	−20.2321	−3.28208
$39$	1.69062	0.270717
$40$	−6.29761	−0.995739
$41$	−8.81257	−1.37629	−0.688146	−	0.725572i	$-0.741575\pi$
$41$	−8.81257	−1.37629	−0.688146	+	0.725572i	$0.741575\pi$
$42$	12.6403	1.95044
$43$	2.27968	0.347648	0.173824	−	0.984777i	$-0.444388\pi$
$43$	2.27968	0.347648	0.173824	+	0.984777i	$0.444388\pi$
$44$	19.6366	2.96032
$45$	0.141787	0.0211364
$46$	−21.6852	−3.19730
$47$	7.00292	1.02148	0.510741	−	0.859735i	$-0.329371\pi$
$47$	7.00292	1.02148	0.510741	+	0.859735i	$0.329371\pi$
$48$	11.9611	1.72644
$49$	1.63353	0.233362
$50$	2.54458	0.359859
$51$	−4.15863	−0.582324
$52$	4.47491	0.620558
$53$	−6.46149	−0.887554	−0.443777	−	0.896137i	$-0.646362\pi$
$53$	−6.46149	−0.887554	−0.443777	+	0.896137i	$0.646362\pi$
$54$	−13.5158	−1.83926
$55$	−4.38815	−0.591698
$56$	18.5042	2.47273
$57$	−13.4422	−1.78046
$58$	16.8325	2.21022
$59$	−4.24895	−0.553166	−0.276583	−	0.960990i	$-0.589202\pi$
$59$	−4.24895	−0.553166	−0.276583	+	0.960990i	$0.589202\pi$
$60$	−7.56539	−0.976688
$61$	12.7294	1.62984	0.814919	−	0.579574i	$-0.196781\pi$
$61$	12.7294	1.62984	0.814919	+	0.579574i	$0.196781\pi$
$62$	−2.54458	−0.323162
$63$	−0.416611	−0.0524881
$64$	−0.389712	−0.0487140
$65$	−1.00000	−0.124035
$66$	18.8775	2.32366
$67$	−5.15855	−0.630217	−0.315108	−	0.949056i	$-0.602041\pi$
$67$	−5.15855	−0.630217	−0.315108	+	0.949056i	$0.602041\pi$
$68$	−11.0074	−1.33485
$69$	−14.4077	−1.73448
$70$	−7.47672	−0.893639
$71$	−4.40773	−0.523101	−0.261551	−	0.965190i	$-0.584234\pi$
$71$	−4.40773	−0.523101	−0.261551	+	0.965190i	$0.584234\pi$
$72$	−0.892920	−0.105232
$73$	11.5095	1.34708	0.673540	−	0.739151i	$-0.264773\pi$
$73$	11.5095	1.34708	0.673540	+	0.739151i	$0.264773\pi$
$74$	20.6100	2.39586
$75$	1.69062	0.195217
$76$	−35.5801	−4.08132
$77$	12.8936	1.46937
$78$	4.30194	0.487098
$79$	5.70631	0.642011	0.321005	−	0.947077i	$-0.395979\pi$
$79$	5.70631	0.642011	0.321005	+	0.947077i	$0.395979\pi$
$80$	−7.07498	−0.791007
$81$	−8.55453	−0.950504
$82$	−22.4243	−2.47635
$83$	−12.8844	−1.41425	−0.707126	−	0.707088i	$-0.750009\pi$
$83$	−12.8844	−1.41425	−0.707126	+	0.707088i	$0.750009\pi$
$84$	22.2293	2.42541
$85$	2.45982	0.266804
$86$	5.80084	0.625520
$87$	11.1836	1.19900
$88$	27.6348	2.94588
$89$	−14.4016	−1.52657	−0.763286	−	0.646061i	$-0.776415\pi$
$89$	−14.4016	−1.52657	−0.763286	+	0.646061i	$0.776415\pi$
$90$	0.360789	0.0380305
$91$	2.93829	0.308016
$92$	−38.1356	−3.97591
$93$	−1.69062	−0.175310
$94$	17.8195	1.83794
$95$	7.95103	0.815759
$96$	9.14233	0.933085
$97$	5.21382	0.529384	0.264692	−	0.964333i	$-0.414730\pi$
$97$	5.21382	0.529384	0.264692	+	0.964333i	$0.414730\pi$
$98$	4.15666	0.419886
$99$	−0.622183	−0.0625317
$100$	4.47491	0.447491
$101$	14.7471	1.46739	0.733695	−	0.679478i	$-0.237794\pi$
$101$	14.7471	1.46739	0.733695	+	0.679478i	$0.237794\pi$
$102$	−10.5820	−1.04777
$103$	−2.57242	−0.253468	−0.126734	−	0.991937i	$-0.540449\pi$
$103$	−2.57242	−0.253468	−0.126734	+	0.991937i	$0.540449\pi$
$104$	6.29761	0.617531
$105$	−4.96754	−0.484782
$106$	−16.4418	−1.59697
$107$	14.6432	1.41561	0.707804	−	0.706409i	$-0.249686\pi$
$107$	14.6432	1.41561	0.707804	+	0.706409i	$0.249686\pi$
$108$	−23.7688	−2.28716
$109$	10.5986	1.01516	0.507579	−	0.861605i	$-0.330541\pi$
$109$	10.5986	1.01516	0.507579	+	0.861605i	$0.330541\pi$
$110$	−11.1660	−1.06464
$111$	13.6933	1.29971
$112$	20.7883	1.96431
$113$	18.9412	1.78184	0.890920	−	0.454159i	$-0.150060\pi$
$113$	18.9412	1.78184	0.890920	+	0.454159i	$0.150060\pi$
$114$	−34.2048	−3.20358
$115$	8.52209	0.794689
$116$	29.6017	2.74845
$117$	−0.141787	−0.0131082
$118$	−10.8118	−0.995308
$119$	−7.22764	−0.662557
$120$	−10.6469	−0.971924
$121$	8.25584	0.750530
$122$	32.3911	2.93256
$123$	−14.8987	−1.34337
$124$	−4.47491	−0.401859
$125$	−1.00000	−0.0894427
$126$	−1.06010	−0.0944414
$127$	2.43196	0.215802	0.107901	−	0.994162i	$-0.465587\pi$
$127$	2.43196	0.215802	0.107901	+	0.994162i	$0.465587\pi$
$128$	−11.8070	−1.04360
$129$	3.85408	0.339333
$130$	−2.54458	−0.223175
$131$	11.3245	0.989424	0.494712	−	0.869057i	$-0.335273\pi$
$131$	11.3245	0.989424	0.494712	+	0.869057i	$0.335273\pi$
$132$	33.1980	2.88952
$133$	−23.3624	−2.02578
$134$	−13.1264	−1.13394
$135$	5.31158	0.457148
$136$	−15.4910	−1.32834
$137$	−18.2329	−1.55774	−0.778869	−	0.627186i	$-0.784207\pi$
$137$	−18.2329	−1.55774	−0.778869	+	0.627186i	$0.784207\pi$
$138$	−36.6615	−3.12083
$139$	−18.0331	−1.52954	−0.764772	−	0.644301i	$-0.777148\pi$
$139$	−18.0331	−1.52954	−0.764772	+	0.644301i	$0.777148\pi$
$140$	−13.1486	−1.11126
$141$	11.8393	0.997050
$142$	−11.2158	−0.941212
$143$	4.38815	0.366955
$144$	−1.00314	−0.0835951
$145$	−6.61504	−0.549349
$146$	29.2868	2.42379
$147$	2.76169	0.227780
$148$	36.2447	2.97929
$149$	8.92446	0.731121	0.365560	−	0.930788i	$-0.380877\pi$
$149$	8.92446	0.731121	0.365560	+	0.930788i	$0.380877\pi$
$150$	4.30194	0.351252
$151$	−3.71077	−0.301978	−0.150989	−	0.988535i	$-0.548246\pi$
$151$	−3.71077	−0.301978	−0.150989	+	0.988535i	$0.548246\pi$
$152$	−50.0725	−4.06142
$153$	0.348770	0.0281964
$154$	32.8089	2.64382
$155$	1.00000	0.0803219
$156$	7.56539	0.605716
$157$	−19.6713	−1.56994	−0.784972	−	0.619531i	$-0.787323\pi$
$157$	−19.6713	−1.56994	−0.784972	+	0.619531i	$0.787323\pi$
$158$	14.5202	1.15516
$159$	−10.9240	−0.866327
$160$	−5.40766	−0.427513
$161$	−25.0403	−1.97346
$162$	−21.7677	−1.71023
$163$	−13.4772	−1.05562	−0.527808	−	0.849364i	$-0.676986\pi$
$163$	−13.4772	−1.05562	−0.527808	+	0.849364i	$0.676986\pi$
$164$	−39.4354	−3.07939
$165$	−7.41871	−0.577546
$166$	−32.7855	−2.54465
$167$	−9.06044	−0.701118	−0.350559	−	0.936541i	$-0.614008\pi$
$167$	−9.06044	−0.701118	−0.350559	+	0.936541i	$0.614008\pi$
$168$	31.2836	2.41358
$169$	1.00000	0.0769231
$170$	6.25921	0.480059
$171$	1.12735	0.0862109
$172$	10.2014	0.777846
$173$	−10.5659	−0.803311	−0.401656	−	0.915791i	$-0.631565\pi$
$173$	−10.5659	−0.803311	−0.401656	+	0.915791i	$0.631565\pi$
$174$	28.4575	2.15736
$175$	2.93829	0.222114
$176$	31.0460	2.34018
$177$	−7.18338	−0.539936
$178$	−36.6462	−2.74675
$179$	10.0337	0.749952	0.374976	−	0.927034i	$-0.377651\pi$
$179$	10.0337	0.749952	0.374976	+	0.927034i	$0.377651\pi$
$180$	0.634484	0.0472917
$181$	−2.15062	−0.159854	−0.0799271	−	0.996801i	$-0.525469\pi$
$181$	−2.15062	−0.159854	−0.0799271	+	0.996801i	$0.525469\pi$
$182$	7.47672	0.554211
$183$	21.5207	1.59086
$184$	−53.6688	−3.95652
$185$	−8.09954	−0.595490
$186$	−4.30194	−0.315433
$187$	−10.7940	−0.789338
$188$	31.3374	2.28552
$189$	−15.6070	−1.13524
$190$	20.2321	1.46779
$191$	2.06286	0.149263	0.0746316	−	0.997211i	$-0.476222\pi$
$191$	2.06286	0.149263	0.0746316	+	0.997211i	$0.476222\pi$
$192$	−0.658857	−0.0475489
$193$	18.4173	1.32571	0.662854	−	0.748748i	$-0.269345\pi$
$193$	18.4173	1.32571	0.662854	+	0.748748i	$0.269345\pi$
$194$	13.2670	0.952516
$195$	−1.69062	−0.121068
$196$	7.30990	0.522136
$197$	4.32257	0.307971	0.153985	−	0.988073i	$-0.450789\pi$
$197$	4.32257	0.307971	0.153985	+	0.988073i	$0.450789\pi$
$198$	−1.58320	−0.112513
$199$	−21.9969	−1.55932	−0.779658	−	0.626206i	$-0.784607\pi$
$199$	−21.9969	−1.55932	−0.779658	+	0.626206i	$0.784607\pi$
$200$	6.29761	0.445308
$201$	−8.72117	−0.615144
$202$	37.5252	2.64027
$203$	19.4369	1.36420
$204$	−18.6095	−1.30292
$205$	8.81257	0.615496
$206$	−6.54573	−0.456063
$207$	1.20832	0.0839842
$208$	7.07498	0.490562
$209$	−34.8903	−2.41341
$210$	−12.6403	−0.872265
$211$	−3.41833	−0.235327	−0.117664	−	0.993054i	$-0.537540\pi$
$211$	−3.41833	−0.235327	−0.117664	+	0.993054i	$0.537540\pi$
$212$	−28.9146	−1.98586
$213$	−7.45182	−0.510590
$214$	37.2608	2.54709
$215$	−2.27968	−0.155473
$216$	−33.4503	−2.27600
$217$	−2.93829	−0.199464
$218$	26.9689	1.82657
$219$	19.4582	1.31486
$220$	−19.6366	−1.32390
$221$	−2.45982	−0.165465
$222$	34.8437	2.33856
$223$	−9.06614	−0.607114	−0.303557	−	0.952813i	$-0.598174\pi$
$223$	−9.06614	−0.607114	−0.303557	+	0.952813i	$0.598174\pi$
$224$	15.8893	1.06165
$225$	−0.141787	−0.00945248
$226$	48.1975	3.20605
$227$	27.3726	1.81679	0.908393	−	0.418117i	$-0.137310\pi$
$227$	27.3726	1.81679	0.908393	+	0.418117i	$0.137310\pi$
$228$	−60.1527	−3.98371
$229$	8.03259	0.530808	0.265404	−	0.964137i	$-0.414495\pi$
$229$	8.03259	0.530808	0.265404	+	0.964137i	$0.414495\pi$
$230$	21.6852	1.42988
$231$	21.7983	1.43422
$232$	41.6589	2.73504
$233$	6.14019	0.402257	0.201129	−	0.979565i	$-0.435539\pi$
$233$	6.14019	0.402257	0.201129	+	0.979565i	$0.435539\pi$
$234$	−0.360789	−0.0235855
$235$	−7.00292	−0.456820
$236$	−19.0137	−1.23768
$237$	9.64724	0.626655
$238$	−18.3913	−1.19213
$239$	13.9789	0.904218	0.452109	−	0.891963i	$-0.350672\pi$
$239$	13.9789	0.904218	0.452109	+	0.891963i	$0.350672\pi$
$240$	−11.9611	−0.772088
$241$	−16.4685	−1.06083	−0.530415	−	0.847738i	$-0.677964\pi$
$241$	−16.4685	−1.06083	−0.530415	+	0.847738i	$0.677964\pi$
$242$	21.0077	1.35042
$243$	1.47224	0.0944443
$244$	56.9631	3.64669
$245$	−1.63353	−0.104363
$246$	−37.9111	−2.41712
$247$	−7.95103	−0.505912
$248$	−6.29761	−0.399899
$249$	−21.7828	−1.38043
$250$	−2.54458	−0.160934
$251$	11.9721	0.755674	0.377837	−	0.925872i	$-0.376668\pi$
$251$	11.9721	0.755674	0.377837	+	0.925872i	$0.376668\pi$
$252$	−1.86430	−0.117440
$253$	−37.3962	−2.35108
$254$	6.18833	0.388290
$255$	4.15863	0.260423
$256$	−29.2644	−1.82903
$257$	1.89197	0.118018	0.0590088	−	0.998257i	$-0.481206\pi$
$257$	1.89197	0.118018	0.0590088	+	0.998257i	$0.481206\pi$
$258$	9.80704	0.610560
$259$	23.7988	1.47878
$260$	−4.47491	−0.277522
$261$	−0.937928	−0.0580563
$262$	28.8161	1.78026
$263$	−20.2626	−1.24944	−0.624721	−	0.780848i	$-0.714787\pi$
$263$	−20.2626	−1.24944	−0.624721	+	0.780848i	$0.714787\pi$
$264$	46.7201	2.87543
$265$	6.46149	0.396926
$266$	−59.4476	−3.64497
$267$	−24.3478	−1.49006
$268$	−23.0840	−1.41008
$269$	2.22970	0.135947	0.0679735	−	0.997687i	$-0.478347\pi$
$269$	2.22970	0.135947	0.0679735	+	0.997687i	$0.478347\pi$
$270$	13.5158	0.822544
$271$	30.5792	1.85756	0.928778	−	0.370636i	$-0.120860\pi$
$271$	30.5792	1.85756	0.928778	+	0.370636i	$0.120860\pi$
$272$	−17.4031	−1.05522
$273$	4.96754	0.300649
$274$	−46.3951	−2.80283
$275$	4.38815	0.264615
$276$	−64.4729	−3.88081
$277$	−16.9133	−1.01622	−0.508110	−	0.861292i	$-0.669656\pi$
$277$	−16.9133	−1.01622	−0.508110	+	0.861292i	$0.669656\pi$
$278$	−45.8866	−2.75210
$279$	0.141787	0.00848857
$280$	−18.5042	−1.10584
$281$	−8.26494	−0.493045	−0.246523	−	0.969137i	$-0.579288\pi$
$281$	−8.26494	−0.493045	−0.246523	+	0.969137i	$0.579288\pi$
$282$	30.1261	1.79399
$283$	2.98274	0.177305	0.0886526	−	0.996063i	$-0.471744\pi$
$283$	2.98274	0.177305	0.0886526	+	0.996063i	$0.471744\pi$
$284$	−19.7242	−1.17041
$285$	13.4422	0.796248
$286$	11.1660	0.660260
$287$	−25.8939	−1.52847
$288$	−0.766737	−0.0451804
$289$	−10.9493	−0.644077
$290$	−16.8325	−0.988440
$291$	8.81462	0.516722
$292$	51.5038	3.01403
$293$	−2.73645	−0.159865	−0.0799324	−	0.996800i	$-0.525470\pi$
$293$	−2.73645	−0.159865	−0.0799324	+	0.996800i	$0.525470\pi$
$294$	7.02735	0.409844
$295$	4.24895	0.247383
$296$	51.0077	2.96476
$297$	−23.3080	−1.35247
$298$	22.7090	1.31550
$299$	−8.52209	−0.492845
$300$	7.56539	0.436788
$301$	6.69835	0.386087
$302$	−9.44236	−0.543347
$303$	24.9318	1.43230
$304$	−56.2534	−3.22635
$305$	−12.7294	−0.728886
$306$	0.887475	0.0507336
$307$	14.7066	0.839351	0.419675	−	0.907674i	$-0.362144\pi$
$307$	14.7066	0.839351	0.419675	+	0.907674i	$0.362144\pi$
$308$	57.6978	3.28764
$309$	−4.34899	−0.247406
$310$	2.54458	0.144523
$311$	−32.7121	−1.85493	−0.927465	−	0.373909i	$-0.878017\pi$
$311$	−32.7121	−1.85493	−0.927465	+	0.373909i	$0.878017\pi$
$312$	10.6469	0.602762
$313$	9.00939	0.509241	0.254621	−	0.967041i	$-0.418049\pi$
$313$	9.00939	0.509241	0.254621	+	0.967041i	$0.418049\pi$
$314$	−50.0554	−2.82479
$315$	0.416611	0.0234734
$316$	25.5352	1.43647
$317$	5.76695	0.323904	0.161952	−	0.986799i	$-0.448221\pi$
$317$	5.76695	0.323904	0.161952	+	0.986799i	$0.448221\pi$
$318$	−27.7969	−1.55877
$319$	29.0278	1.62524
$320$	0.389712	0.0217856
$321$	24.7561	1.38175
$322$	−63.7173	−3.55082
$323$	19.5581	1.08824
$324$	−38.2807	−2.12671
$325$	1.00000	0.0554700
$326$	−34.2939	−1.89936
$327$	17.9182	0.990878
$328$	−55.4981	−3.06437
$329$	20.5766	1.13442
$330$	−18.8775	−1.03917
$331$	16.0981	0.884831	0.442416	−	0.896810i	$-0.354122\pi$
$331$	16.0981	0.884831	0.442416	+	0.896810i	$0.354122\pi$
$332$	−57.6567	−3.16432
$333$	−1.14841	−0.0629325
$334$	−23.0550	−1.26152
$335$	5.15855	0.281842
$336$	35.1453	1.91733
$337$	6.52042	0.355190	0.177595	−	0.984104i	$-0.443168\pi$
$337$	6.52042	0.355190	0.177595	+	0.984104i	$0.443168\pi$
$338$	2.54458	0.138407
$339$	32.0225	1.73922
$340$	11.0074	0.596962
$341$	−4.38815	−0.237631
$342$	2.86865	0.155119
$343$	−15.7682	−0.851404
$344$	14.3565	0.774052
$345$	14.4077	0.775682
$346$	−26.8858	−1.44539
$347$	27.4115	1.47152	0.735762	−	0.677240i	$-0.236824\pi$
$347$	27.4115	1.47152	0.735762	+	0.677240i	$0.236824\pi$
$348$	50.0454	2.68271
$349$	25.9259	1.38778	0.693892	−	0.720080i	$-0.255895\pi$
$349$	25.9259	1.38778	0.693892	+	0.720080i	$0.255895\pi$
$350$	7.47672	0.399647
$351$	−5.31158	−0.283511
$352$	23.7296	1.26479
$353$	−29.3196	−1.56052	−0.780262	−	0.625453i	$-0.784914\pi$
$353$	−29.3196	−1.56052	−0.780262	+	0.625453i	$0.784914\pi$
$354$	−18.2787	−0.971503
$355$	4.40773	0.233938
$356$	−64.4460	−3.41563
$357$	−12.2192	−0.646710
$358$	25.5315	1.34938
$359$	−11.8270	−0.624204	−0.312102	−	0.950049i	$-0.601033\pi$
$359$	−11.8270	−0.624204	−0.312102	+	0.950049i	$0.601033\pi$
$360$	0.892920	0.0470610
$361$	44.2189	2.32731
$362$	−5.47243	−0.287625
$363$	13.9575	0.732580
$364$	13.1486	0.689172
$365$	−11.5095	−0.602433
$366$	54.7613	2.86242
$367$	26.9295	1.40571	0.702854	−	0.711334i	$-0.251909\pi$
$367$	26.9295	1.40571	0.702854	+	0.711334i	$0.251909\pi$
$368$	−60.2936	−3.14302
$369$	1.24951	0.0650468
$370$	−20.6100	−1.07146
$371$	−18.9857	−0.985690
$372$	−7.56539	−0.392247
$373$	−23.4032	−1.21177	−0.605885	−	0.795552i	$-0.707181\pi$
$373$	−23.4032	−1.21177	−0.605885	+	0.795552i	$0.707181\pi$
$374$	−27.4663	−1.42025
$375$	−1.69062	−0.0873035
$376$	44.1017	2.27437
$377$	6.61504	0.340692
$378$	−39.7132	−2.04263
$379$	16.5495	0.850093	0.425046	−	0.905172i	$-0.360258\pi$
$379$	16.5495	0.850093	0.425046	+	0.905172i	$0.360258\pi$
$380$	35.5801	1.82522
$381$	4.11153	0.210640
$382$	5.24911	0.268568
$383$	−17.0709	−0.872281	−0.436140	−	0.899879i	$-0.643655\pi$
$383$	−17.0709	−0.872281	−0.436140	+	0.899879i	$0.643655\pi$
$384$	−19.9612	−1.01864
$385$	−12.8936	−0.657121
$386$	46.8644	2.38534
$387$	−0.323229	−0.0164307
$388$	23.3314	1.18447
$389$	−32.2314	−1.63420	−0.817099	−	0.576497i	$-0.804419\pi$
$389$	−32.2314	−1.63420	−0.817099	+	0.576497i	$0.804419\pi$
$390$	−4.30194	−0.217837
$391$	20.9628	1.06013
$392$	10.2873	0.519589
$393$	19.1454	0.965760
$394$	10.9991	0.554129
$395$	−5.70631	−0.287116
$396$	−2.78421	−0.139912
$397$	19.9299	1.00025	0.500127	−	0.865952i	$-0.333287\pi$
$397$	19.9299	1.00025	0.500127	+	0.865952i	$0.333287\pi$
$398$	−55.9729	−2.80567
$399$	−39.4971	−1.97733
$400$	7.07498	0.353749
$401$	11.0533	0.551978	0.275989	−	0.961161i	$-0.410995\pi$
$401$	11.0533	0.551978	0.275989	+	0.961161i	$0.410995\pi$
$402$	−22.1917	−1.10682
$403$	−1.00000	−0.0498135
$404$	65.9919	3.28322
$405$	8.55453	0.425078
$406$	49.4588	2.45460
$407$	35.5420	1.76175
$408$	−26.1894	−1.29657
$409$	−14.9261	−0.738046	−0.369023	−	0.929420i	$-0.620308\pi$
$409$	−14.9261	−0.738046	−0.369023	+	0.929420i	$0.620308\pi$
$410$	22.4243	1.10746
$411$	−30.8249	−1.52048
$412$	−11.5113	−0.567123
$413$	−12.4846	−0.614329
$414$	3.07468	0.151112
$415$	12.8844	0.632472
$416$	5.40766	0.265132
$417$	−30.4871	−1.49296
$418$	−88.7813	−4.34244
$419$	17.8852	0.873747	0.436874	−	0.899523i	$-0.356086\pi$
$419$	17.8852	0.873747	0.436874	+	0.899523i	$0.356086\pi$
$420$	−22.2293	−1.08468
$421$	−10.8817	−0.530343	−0.265172	−	0.964201i	$-0.585429\pi$
$421$	−10.8817	−0.530343	−0.265172	+	0.964201i	$0.585429\pi$
$422$	−8.69822	−0.423423
$423$	−0.992924	−0.0482776
$424$	−40.6920	−1.97618
$425$	−2.45982	−0.119319
$426$	−18.9618	−0.918701
$427$	37.4028	1.81005
$428$	65.5268	3.16736
$429$	7.41871	0.358179
$430$	−5.80084	−0.279741
$431$	34.4024	1.65711	0.828553	−	0.559911i	$-0.189165\pi$
$431$	34.4024	1.65711	0.828553	+	0.559911i	$0.189165\pi$
$432$	−37.5793	−1.80804
$433$	4.44400	0.213565	0.106783	−	0.994282i	$-0.465945\pi$
$433$	4.44400	0.213565	0.106783	+	0.994282i	$0.465945\pi$
$434$	−7.47672	−0.358894
$435$	−11.1836	−0.536210
$436$	47.4276	2.27137
$437$	67.7594	3.24137
$438$	49.5130	2.36582
$439$	20.4206	0.974620	0.487310	−	0.873229i	$-0.337978\pi$
$439$	20.4206	0.974620	0.487310	+	0.873229i	$0.337978\pi$
$440$	−27.6348	−1.31744
$441$	−0.231614	−0.0110292
$442$	−6.25921	−0.297720
$443$	−3.40133	−0.161602	−0.0808010	−	0.996730i	$-0.525748\pi$
$443$	−3.40133	−0.161602	−0.0808010	+	0.996730i	$0.525748\pi$
$444$	61.2762	2.90804
$445$	14.4016	0.682704
$446$	−23.0696	−1.09238
$447$	15.0879	0.713634
$448$	−1.14509	−0.0541002
$449$	−2.72788	−0.128737	−0.0643684	−	0.997926i	$-0.520503\pi$
$449$	−2.72788	−0.128737	−0.0643684	+	0.997926i	$0.520503\pi$
$450$	−0.360789	−0.0170078
$451$	−38.6708	−1.82094
$452$	84.7602	3.98679
$453$	−6.27352	−0.294756
$454$	69.6520	3.26893
$455$	−2.93829	−0.137749
$456$	−84.6538	−3.96428
$457$	16.7169	0.781984	0.390992	−	0.920394i	$-0.372132\pi$
$457$	16.7169	0.781984	0.390992	+	0.920394i	$0.372132\pi$
$458$	20.4396	0.955080
$459$	13.0655	0.609846
$460$	38.1356	1.77808
$461$	3.14496	0.146475	0.0732377	−	0.997315i	$-0.476667\pi$
$461$	3.14496	0.146475	0.0732377	+	0.997315i	$0.476667\pi$
$462$	55.4676	2.58059
$463$	−18.5241	−0.860888	−0.430444	−	0.902617i	$-0.641643\pi$
$463$	−18.5241	−0.860888	−0.430444	+	0.902617i	$0.641643\pi$
$464$	46.8013	2.17269
$465$	1.69062	0.0784009
$466$	15.6242	0.723778
$467$	−14.0758	−0.651352	−0.325676	−	0.945482i	$-0.605592\pi$
$467$	−14.0758	−0.651352	−0.325676	+	0.945482i	$0.605592\pi$
$468$	−0.634484	−0.0293290
$469$	−15.1573	−0.699899
$470$	−17.8195	−0.821953
$471$	−33.2569	−1.53240
$472$	−26.7582	−1.23165
$473$	10.0036	0.459965
$474$	24.5482	1.12754
$475$	−7.95103	−0.364818
$476$	−32.3430	−1.48244
$477$	0.916157	0.0419479
$478$	35.5704	1.62695
$479$	10.7632	0.491782	0.245891	−	0.969297i	$-0.420919\pi$
$479$	10.7632	0.491782	0.245891	+	0.969297i	$0.420919\pi$
$480$	−9.14233	−0.417288
$481$	8.09954	0.369307
$482$	−41.9055	−1.90874
$483$	−42.3338	−1.92626
$484$	36.9441	1.67928
$485$	−5.21382	−0.236748
$486$	3.74624	0.169933
$487$	−22.6744	−1.02748	−0.513738	−	0.857947i	$-0.671740\pi$
$487$	−22.6744	−1.02748	−0.513738	+	0.857947i	$0.671740\pi$
$488$	80.1651	3.62890
$489$	−22.7849	−1.03037
$490$	−4.15666	−0.187779
$491$	−0.00860548	−0.000388360 0	−0.000194180	−	1.00000i	$-0.500062\pi$
$491$	−0.00860548	−0.000388360 0	−0.000194180		1.00000i	$0.500062\pi$
$492$	−66.6705	−3.00574
$493$	−16.2718	−0.732844
$494$	−20.2321	−0.910284
$495$	0.622183	0.0279650
$496$	−7.07498	−0.317676
$497$	−12.9512	−0.580940
$498$	−55.4281	−2.48379
$499$	35.0143	1.56745	0.783727	−	0.621106i	$-0.213316\pi$
$499$	35.0143	1.56745	0.783727	+	0.621106i	$0.213316\pi$
$500$	−4.47491	−0.200124
$501$	−15.3178	−0.684349
$502$	30.4641	1.35968
$503$	−7.02741	−0.313337	−0.156668	−	0.987651i	$-0.550075\pi$
$503$	−7.02741	−0.313337	−0.156668	+	0.987651i	$0.550075\pi$
$504$	−2.62366	−0.116867
$505$	−14.7471	−0.656237
$506$	−95.1577	−4.23028
$507$	1.69062	0.0750833
$508$	10.8828	0.482846
$509$	25.3816	1.12502	0.562511	−	0.826790i	$-0.309835\pi$
$509$	25.3816	1.12502	0.562511	+	0.826790i	$0.309835\pi$
$510$	10.5820	0.468578
$511$	33.8181	1.49602
$512$	−50.8518	−2.24735
$513$	42.2326	1.86461
$514$	4.81427	0.212348
$515$	2.57242	0.113354
$516$	17.2467	0.759242
$517$	30.7299	1.35150
$518$	60.5580	2.66076
$519$	−17.8630	−0.784098
$520$	−6.29761	−0.276168
$521$	−11.1515	−0.488556	−0.244278	−	0.969705i	$-0.578551\pi$
$521$	−11.1515	−0.488556	−0.244278	+	0.969705i	$0.578551\pi$
$522$	−2.38664	−0.104460
$523$	−20.5830	−0.900031	−0.450016	−	0.893021i	$-0.648582\pi$
$523$	−20.5830	−0.900031	−0.450016	+	0.893021i	$0.648582\pi$
$524$	50.6760	2.21379
$525$	4.96754	0.216801
$526$	−51.5598	−2.24811
$527$	2.45982	0.107151
$528$	52.4872	2.28421
$529$	49.6260	2.15765
$530$	16.4418	0.714187
$531$	0.602446	0.0261440
$532$	−104.545	−4.53259
$533$	−8.81257	−0.381715
$534$	−61.9550	−2.68105
$535$	−14.6432	−0.633079
$536$	−32.4865	−1.40320
$537$	16.9632	0.732015
$538$	5.67365	0.244608
$539$	7.16818	0.308755
$540$	23.7688	1.02285
$541$	11.6458	0.500691	0.250346	−	0.968157i	$-0.419456\pi$
$541$	11.6458	0.500691	0.250346	+	0.968157i	$0.419456\pi$
$542$	77.8114	3.34229
$543$	−3.63589	−0.156031
$544$	−13.3018	−0.570312
$545$	−10.5986	−0.453992
$546$	12.6403	0.540956
$547$	9.59466	0.410238	0.205119	−	0.978737i	$-0.434242\pi$
$547$	9.59466	0.410238	0.205119	+	0.978737i	$0.434242\pi$
$548$	−81.5904	−3.48537
$549$	−1.80487	−0.0770301
$550$	11.1660	0.476120
$551$	−52.5964	−2.24068
$552$	−90.7338	−3.86189
$553$	16.7668	0.712996
$554$	−43.0373	−1.82848
$555$	−13.6933	−0.581248
$556$	−80.6963	−3.42228
$557$	−25.6121	−1.08522	−0.542610	−	0.839984i	$-0.682564\pi$
$557$	−25.6121	−1.08522	−0.542610	+	0.839984i	$0.682564\pi$
$558$	0.360789	0.0152734
$559$	2.27968	0.0964202
$560$	−20.7883	−0.878467
$561$	−18.2487	−0.770459
$562$	−21.0308	−0.887133
$563$	−11.8480	−0.499332	−0.249666	−	0.968332i	$-0.580321\pi$
$563$	−11.8480	−0.499332	−0.249666	+	0.968332i	$0.580321\pi$
$564$	52.9798	2.23085
$565$	−18.9412	−0.796863
$566$	7.58982	0.319024
$567$	−25.1357	−1.05560
$568$	−27.7582	−1.16471
$569$	−18.2523	−0.765176	−0.382588	−	0.923919i	$-0.624967\pi$
$569$	−18.2523	−0.765176	−0.382588	+	0.923919i	$0.624967\pi$
$570$	34.2048	1.43268
$571$	14.5164	0.607494	0.303747	−	0.952753i	$-0.401762\pi$
$571$	14.5164	0.607494	0.303747	+	0.952753i	$0.401762\pi$
$572$	19.6366	0.821045
$573$	3.48752	0.145693
$574$	−65.8891	−2.75016
$575$	−8.52209	−0.355396
$576$	0.0552562	0.00230234
$577$	26.3337	1.09629	0.548143	−	0.836385i	$-0.315335\pi$
$577$	26.3337	1.09629	0.548143	+	0.836385i	$0.315335\pi$
$578$	−27.8614	−1.15888
$579$	31.1368	1.29400
$580$	−29.6017	−1.22914
$581$	−37.8582	−1.57062
$582$	22.4295	0.929735
$583$	−28.3540	−1.17430
$584$	72.4821	2.99933
$585$	0.141787	0.00586218
$586$	−6.96312	−0.287644
$587$	−8.57478	−0.353919	−0.176960	−	0.984218i	$-0.556626\pi$
$587$	−8.57478	−0.353919	−0.176960	+	0.984218i	$0.556626\pi$
$588$	12.3583	0.509648
$589$	7.95103	0.327617
$590$	10.8118	0.445115
$591$	7.30785	0.300605
$592$	57.3041	2.35518
$593$	39.0021	1.60163	0.800813	−	0.598915i	$-0.204401\pi$
$593$	39.0021	1.60163	0.800813	+	0.598915i	$0.204401\pi$
$594$	−59.3092	−2.43349
$595$	7.22764	0.296305
$596$	39.9361	1.63585
$597$	−37.1884	−1.52202
$598$	−21.6852	−0.886773
$599$	−18.6872	−0.763539	−0.381769	−	0.924258i	$-0.624685\pi$
$599$	−18.6872	−0.763539	−0.381769	+	0.924258i	$0.624685\pi$
$600$	10.6469	0.434658
$601$	47.2857	1.92882	0.964412	−	0.264404i	$-0.0851751\pi$
$601$	47.2857	1.92882	0.964412	+	0.264404i	$0.0851751\pi$
$602$	17.0445	0.694683
$603$	0.731416	0.0297855
$604$	−16.6053	−0.675662
$605$	−8.25584	−0.335647
$606$	63.4411	2.57712
$607$	21.2664	0.863177	0.431589	−	0.902071i	$-0.357953\pi$
$607$	21.2664	0.863177	0.431589	+	0.902071i	$0.357953\pi$
$608$	−42.9965	−1.74374
$609$	32.8605	1.33157
$610$	−32.3911	−1.31148
$611$	7.00292	0.283308
$612$	1.56071	0.0630881
$613$	13.5124	0.545762	0.272881	−	0.962048i	$-0.412023\pi$
$613$	13.5124	0.545762	0.272881	+	0.962048i	$0.412023\pi$
$614$	37.4222	1.51024
$615$	14.8987	0.600775
$616$	81.1991	3.27160
$617$	21.0225	0.846335	0.423167	−	0.906052i	$-0.360918\pi$
$617$	21.0225	0.846335	0.423167	+	0.906052i	$0.360918\pi$
$618$	−11.0664	−0.445155
$619$	26.8737	1.08014	0.540072	−	0.841619i	$-0.318397\pi$
$619$	26.8737	1.08014	0.540072	+	0.841619i	$0.318397\pi$
$620$	4.47491	0.179717
$621$	45.2658	1.81645
$622$	−83.2386	−3.33756
$623$	−42.3162	−1.69536
$624$	11.9611	0.478829
$625$	1.00000	0.0400000
$626$	22.9252	0.916274
$627$	−58.9864	−2.35569
$628$	−88.0274	−3.51268
$629$	−19.9234	−0.794397
$630$	1.06010	0.0422355
$631$	−4.59639	−0.182980	−0.0914898	−	0.995806i	$-0.529163\pi$
$631$	−4.59639	−0.182980	−0.0914898	+	0.995806i	$0.529163\pi$
$632$	35.9361	1.42946
$633$	−5.77911	−0.229699
$634$	14.6745	0.582798
$635$	−2.43196	−0.0965094
$636$	−48.8837	−1.93837
$637$	1.63353	0.0647229
$638$	73.8636	2.92429
$639$	0.624959	0.0247230
$640$	11.8070	0.466712
$641$	−28.9236	−1.14241	−0.571207	−	0.820806i	$-0.693525\pi$
$641$	−28.9236	−1.14241	−0.571207	+	0.820806i	$0.693525\pi$
$642$	62.9940	2.48617
$643$	26.4273	1.04219	0.521095	−	0.853499i	$-0.325524\pi$
$643$	26.4273	1.04219	0.521095	+	0.853499i	$0.325524\pi$
$644$	−112.053	−4.41552
$645$	−3.85408	−0.151754
$646$	49.7672	1.95806
$647$	−11.7897	−0.463500	−0.231750	−	0.972775i	$-0.574445\pi$
$647$	−11.7897	−0.463500	−0.231750	+	0.972775i	$0.574445\pi$
$648$	−53.8731	−2.11634
$649$	−18.6450	−0.731881
$650$	2.54458	0.0998068
$651$	−4.96754	−0.194693
$652$	−60.3092	−2.36189
$653$	47.5403	1.86040	0.930198	−	0.367059i	$-0.119635\pi$
$653$	47.5403	1.86040	0.930198	+	0.367059i	$0.119635\pi$
$654$	45.5943	1.78288
$655$	−11.3245	−0.442484
$656$	−62.3487	−2.43431
$657$	−1.63189	−0.0636662
$658$	52.3589	2.04116
$659$	−29.0843	−1.13296	−0.566481	−	0.824075i	$-0.691696\pi$
$659$	−29.0843	−1.13296	−0.566481	+	0.824075i	$0.691696\pi$
$660$	−33.1980	−1.29223
$661$	19.3953	0.754389	0.377195	−	0.926134i	$-0.376889\pi$
$661$	19.3953	0.754389	0.377195	+	0.926134i	$0.376889\pi$
$662$	40.9630	1.59207
$663$	−4.15863	−0.161508
$664$	−81.1412	−3.14889
$665$	23.3624	0.905956
$666$	−2.92223	−0.113234
$667$	−56.3740	−2.18281
$668$	−40.5446	−1.56872
$669$	−15.3275	−0.592593
$670$	13.1264	0.507115
$671$	55.8587	2.15640
$672$	26.8628	1.03625
$673$	−3.30923	−0.127561	−0.0637807	−	0.997964i	$-0.520316\pi$
$673$	−3.30923	−0.127561	−0.0637807	+	0.997964i	$0.520316\pi$
$674$	16.5917	0.639090
$675$	−5.31158	−0.204443
$676$	4.47491	0.172112
$677$	−14.2579	−0.547975	−0.273987	−	0.961733i	$-0.588343\pi$
$677$	−14.2579	−0.547975	−0.273987	+	0.961733i	$0.588343\pi$
$678$	81.4840	3.12937
$679$	15.3197	0.587917
$680$	15.4910	0.594051
$681$	46.2769	1.77333
$682$	−11.1660	−0.427569
$683$	−50.1882	−1.92040	−0.960199	−	0.279316i	$-0.909892\pi$
$683$	−50.1882	−1.92040	−0.960199	+	0.279316i	$0.909892\pi$
$684$	5.04481	0.192893
$685$	18.2329	0.696642
$686$	−40.1236	−1.53192
$687$	13.5801	0.518113
$688$	16.1287	0.614900
$689$	−6.46149	−0.246163
$690$	36.6615	1.39568
$691$	−3.87996	−0.147600	−0.0738002	−	0.997273i	$-0.523513\pi$
$691$	−3.87996	−0.147600	−0.0738002	+	0.997273i	$0.523513\pi$
$692$	−47.2815	−1.79737
$693$	−1.82815	−0.0694457
$694$	69.7508	2.64770
$695$	18.0331	0.684033
$696$	70.4296	2.66963
$697$	21.6773	0.821086
$698$	65.9707	2.49703
$699$	10.3808	0.392636
$700$	13.1486	0.496969
$701$	−14.3232	−0.540979	−0.270489	−	0.962723i	$-0.587185\pi$
$701$	−14.3232	−0.540979	−0.270489	+	0.962723i	$0.587185\pi$
$702$	−13.5158	−0.510120
$703$	−64.3997	−2.42888
$704$	−1.71011	−0.0644524
$705$	−11.8393	−0.445894
$706$	−74.6061	−2.80784
$707$	43.3312	1.62964
$708$	−32.1450	−1.20808
$709$	9.48057	0.356050	0.178025	−	0.984026i	$-0.443029\pi$
$709$	9.48057	0.356050	0.178025	+	0.984026i	$0.443029\pi$
$710$	11.2158	0.420923
$711$	−0.809082	−0.0303429
$712$	−90.6959	−3.39897
$713$	8.52209	0.319155
$714$	−31.0929	−1.16362
$715$	−4.38815	−0.164107
$716$	44.8998	1.67798
$717$	23.6330	0.882592
$718$	−30.0947	−1.12313
$719$	−7.84748	−0.292662	−0.146331	−	0.989236i	$-0.546746\pi$
$719$	−7.84748	−0.292662	−0.146331	+	0.989236i	$0.546746\pi$
$720$	1.00314	0.0373849
$721$	−7.55850	−0.281493
$722$	112.519	4.18751
$723$	−27.8421	−1.03546
$724$	−9.62382	−0.357666
$725$	6.61504	0.245676
$726$	35.5161	1.31813
$727$	−4.06991	−0.150944	−0.0754722	−	0.997148i	$-0.524046\pi$
$727$	−4.06991	−0.150944	−0.0754722	+	0.997148i	$0.524046\pi$
$728$	18.5042	0.685811
$729$	28.1526	1.04269
$730$	−29.2868	−1.08395
$731$	−5.60759	−0.207404
$732$	96.3032	3.55947
$733$	0.176375	0.00651458	0.00325729	−	0.999995i	$-0.498963\pi$
$733$	0.176375	0.00651458	0.00325729	+	0.999995i	$0.498963\pi$
$734$	68.5244	2.52928
$735$	−2.76169	−0.101866
$736$	−46.0846	−1.69870
$737$	−22.6365	−0.833825
$738$	3.17948	0.117038
$739$	−41.3707	−1.52185	−0.760923	−	0.648843i	$-0.775253\pi$
$739$	−41.3707	−1.52185	−0.760923	+	0.648843i	$0.775253\pi$
$740$	−36.2447	−1.33238
$741$	−13.4422	−0.493812
$742$	−48.3108	−1.77354
$743$	6.32185	0.231926	0.115963	−	0.993254i	$-0.463005\pi$
$743$	6.32185	0.231926	0.115963	+	0.993254i	$0.463005\pi$
$744$	−10.6469	−0.390334
$745$	−8.92446	−0.326967
$746$	−59.5513	−2.18033
$747$	1.82685	0.0668409
$748$	−48.3023	−1.76611
$749$	43.0258	1.57213
$750$	−4.30194	−0.157085
$751$	−37.9383	−1.38439	−0.692194	−	0.721712i	$-0.743356\pi$
$751$	−37.9383	−1.38439	−0.692194	+	0.721712i	$0.743356\pi$
$752$	49.5455	1.80674
$753$	20.2404	0.737600
$754$	16.8325	0.613004
$755$	3.71077	0.135049
$756$	−69.8397	−2.54005
$757$	23.4562	0.852530	0.426265	−	0.904598i	$-0.359829\pi$
$757$	23.4562	0.852530	0.426265	+	0.904598i	$0.359829\pi$
$758$	42.1117	1.52957
$759$	−63.2229	−2.29485
$760$	50.0725	1.81632
$761$	−18.0823	−0.655482	−0.327741	−	0.944768i	$-0.606287\pi$
$761$	−18.0823	−0.655482	−0.327741	+	0.944768i	$0.606287\pi$
$762$	10.4621	0.379003
$763$	31.1416	1.12740
$764$	9.23109	0.333969
$765$	−0.348770	−0.0126098
$766$	−43.4383	−1.56949
$767$	−4.24895	−0.153421
$768$	−49.4752	−1.78528
$769$	29.1094	1.04971	0.524857	−	0.851191i	$-0.324119\pi$
$769$	29.1094	1.04971	0.524857	+	0.851191i	$0.324119\pi$
$770$	−32.8089	−1.18235
$771$	3.19861	0.115195
$772$	82.4158	2.96621
$773$	16.4846	0.592910	0.296455	−	0.955047i	$-0.404195\pi$
$773$	16.4846	0.592910	0.296455	+	0.955047i	$0.404195\pi$
$774$	−0.822484	−0.0295636
$775$	−1.00000	−0.0359211
$776$	32.8346	1.17869
$777$	40.2348	1.44342
$778$	−82.0156	−2.94040
$779$	70.0690	2.51048
$780$	−7.56539	−0.270884
$781$	−19.3418	−0.692103
$782$	53.3415	1.90749
$783$	−35.1363	−1.25567
$784$	11.5572	0.412757
$785$	19.6713	0.702100
$786$	48.7172	1.73769
$787$	16.2720	0.580034	0.290017	−	0.957022i	$-0.406339\pi$
$787$	16.2720	0.580034	0.290017	+	0.957022i	$0.406339\pi$
$788$	19.3431	0.689070
$789$	−34.2564	−1.21956
$790$	−14.5202	−0.516605
$791$	55.6548	1.97886
$792$	−3.91826	−0.139229
$793$	12.7294	0.452036
$794$	50.7134	1.79975
$795$	10.9240	0.387433
$796$	−98.4339	−3.48890
$797$	50.4119	1.78568	0.892841	−	0.450371i	$-0.148708\pi$
$797$	50.4119	1.78568	0.892841	+	0.450371i	$0.148708\pi$
$798$	−100.504	−3.55779
$799$	−17.2259	−0.609408
$800$	5.40766	0.191190
$801$	2.04197	0.0721494
$802$	28.1262	0.993169
$803$	50.5052	1.78229
$804$	−39.0264	−1.37636
$805$	25.0403	0.882556
$806$	−2.54458	−0.0896291
$807$	3.76958	0.132696
$808$	92.8714	3.26721
$809$	12.4516	0.437776	0.218888	−	0.975750i	$-0.429757\pi$
$809$	12.4516	0.437776	0.218888	+	0.975750i	$0.429757\pi$
$810$	21.7677	0.764840
$811$	−42.6719	−1.49841	−0.749206	−	0.662337i	$-0.769565\pi$
$811$	−42.6719	−1.49841	−0.749206	+	0.662337i	$0.769565\pi$
$812$	86.9783	3.05234
$813$	51.6980	1.81313
$814$	90.4395	3.16990
$815$	13.4772	0.472086
$816$	−29.4222	−1.02998
$817$	−18.1258	−0.634142
$818$	−37.9806	−1.32796
$819$	−0.416611	−0.0145576
$820$	39.4354	1.37714
$821$	10.5849	0.369416	0.184708	−	0.982793i	$-0.440866\pi$
$821$	10.5849	0.369416	0.184708	+	0.982793i	$0.440866\pi$
$822$	−78.4366	−2.73579
$823$	6.59657	0.229942	0.114971	−	0.993369i	$-0.463323\pi$
$823$	6.59657	0.229942	0.114971	+	0.993369i	$0.463323\pi$
$824$	−16.2001	−0.564357
$825$	7.41871	0.258286
$826$	−31.7682	−1.10536
$827$	−1.08941	−0.0378826	−0.0189413	−	0.999821i	$-0.506030\pi$
$827$	−1.08941	−0.0378826	−0.0189413	+	0.999821i	$0.506030\pi$
$828$	5.40713	0.187911
$829$	−42.8301	−1.48755	−0.743775	−	0.668430i	$-0.766967\pi$
$829$	−42.8301	−1.48755	−0.743775	+	0.668430i	$0.766967\pi$
$830$	32.7855	1.13800
$831$	−28.5940	−0.991915
$832$	−0.389712	−0.0135108
$833$	−4.01819	−0.139222
$834$	−77.5771	−2.68627
$835$	9.06044	0.313549
$836$	−156.131	−5.39990
$837$	5.31158	0.183595
$838$	45.5103	1.57213
$839$	12.1502	0.419472	0.209736	−	0.977758i	$-0.432740\pi$
$839$	12.1502	0.419472	0.209736	+	0.977758i	$0.432740\pi$
$840$	−31.2836	−1.07939
$841$	14.7588	0.508922
$842$	−27.6895	−0.954243
$843$	−13.9729	−0.481253
$844$	−15.2967	−0.526534
$845$	−1.00000	−0.0344010
$846$	−2.52658	−0.0868656
$847$	24.2580	0.833515
$848$	−45.7149	−1.56986
$849$	5.04269	0.173065
$850$	−6.25921	−0.214689
$851$	−69.0250	−2.36615
$852$	−33.3462	−1.14242
$853$	41.1754	1.40982	0.704909	−	0.709298i	$-0.250988\pi$
$853$	41.1754	1.40982	0.704909	+	0.709298i	$0.250988\pi$
$854$	95.1745	3.25680
$855$	−1.12735	−0.0385547
$856$	92.2169	3.15191
$857$	−46.5704	−1.59082	−0.795408	−	0.606074i	$-0.792743\pi$
$857$	−46.5704	−1.59082	−0.795408	+	0.606074i	$0.792743\pi$
$858$	18.8775	0.644468
$859$	−43.7827	−1.49385	−0.746923	−	0.664911i	$-0.768470\pi$
$859$	−43.7827	−1.49385	−0.746923	+	0.664911i	$0.768470\pi$
$860$	−10.2014	−0.347863
$861$	−43.7768	−1.49191
$862$	87.5398	2.98162
$863$	−43.1659	−1.46939	−0.734693	−	0.678400i	$-0.762674\pi$
$863$	−43.1659	−1.46939	−0.734693	+	0.678400i	$0.762674\pi$
$864$	−28.7232	−0.977185
$865$	10.5659	0.359252
$866$	11.3081	0.384266
$867$	−18.5112	−0.628672
$868$	−13.1486	−0.446291
$869$	25.0401	0.849429
$870$	−28.4575	−0.964799
$871$	−5.15855	−0.174791
$872$	66.7456	2.26029
$873$	−0.739253	−0.0250199
$874$	172.420	5.83218
$875$	−2.93829	−0.0993322
$876$	87.0735	2.94194
$877$	21.2364	0.717102	0.358551	−	0.933510i	$-0.383271\pi$
$877$	21.2364	0.717102	0.358551	+	0.933510i	$0.383271\pi$
$878$	51.9618	1.75363
$879$	−4.62630	−0.156041
$880$	−31.0460	−1.04656
$881$	−51.9949	−1.75175	−0.875876	−	0.482536i	$-0.839716\pi$
$881$	−51.9949	−1.75175	−0.875876	+	0.482536i	$0.839716\pi$
$882$	−0.589361	−0.0198448
$883$	44.7280	1.50522	0.752609	−	0.658467i	$-0.228795\pi$
$883$	44.7280	1.50522	0.752609	+	0.658467i	$0.228795\pi$
$884$	−11.0074	−0.370220
$885$	7.18338	0.241467
$886$	−8.65497	−0.290769
$887$	−8.97102	−0.301217	−0.150609	−	0.988593i	$-0.548123\pi$
$887$	−8.97102	−0.301217	−0.150609	+	0.988593i	$0.548123\pi$
$888$	86.2349	2.89386
$889$	7.14580	0.239662
$890$	36.6462	1.22838
$891$	−37.5386	−1.25759
$892$	−40.5702	−1.35839
$893$	−55.6805	−1.86328
$894$	38.3925	1.28404
$895$	−10.0337	−0.335389
$896$	−34.6923	−1.15899
$897$	−14.4077	−0.481058
$898$	−6.94133	−0.231635
$899$	−6.61504	−0.220624
$900$	−0.634484	−0.0211495
$901$	15.8941	0.529509
$902$	−98.4012	−3.27640
$903$	11.3244	0.376853
$904$	119.284	3.96734
$905$	2.15062	0.0714890
$906$	−15.9635	−0.530352
$907$	−7.69045	−0.255357	−0.127679	−	0.991816i	$-0.540753\pi$
$907$	−7.69045	−0.255357	−0.127679	+	0.991816i	$0.540753\pi$
$908$	122.490	4.06497
$909$	−2.09095	−0.0693524
$910$	−7.47672	−0.247851
$911$	−5.32873	−0.176549	−0.0882743	−	0.996096i	$-0.528135\pi$
$911$	−5.32873	−0.176549	−0.0882743	+	0.996096i	$0.528135\pi$
$912$	−95.1034	−3.14919
$913$	−56.5388	−1.87116
$914$	42.5376	1.40702
$915$	−21.5207	−0.711453
$916$	35.9451	1.18766
$917$	33.2746	1.09882
$918$	33.2463	1.09729
$919$	−4.99594	−0.164801	−0.0824005	−	0.996599i	$-0.526259\pi$
$919$	−4.99594	−0.164801	−0.0824005	+	0.996599i	$0.526259\pi$
$920$	53.6688	1.76941
$921$	24.8634	0.819276
$922$	8.00262	0.263552
$923$	−4.40773	−0.145082
$924$	97.5454	3.20901
$925$	8.09954	0.266311
$926$	−47.1361	−1.54899
$927$	0.364736	0.0119795
$928$	35.7719	1.17427
$929$	−26.6894	−0.875650	−0.437825	−	0.899060i	$-0.644251\pi$
$929$	−26.6894	−0.875650	−0.437825	+	0.899060i	$0.644251\pi$
$930$	4.30194	0.141066
$931$	−12.9883	−0.425673
$932$	27.4768	0.900032
$933$	−55.3038	−1.81057
$934$	−35.8171	−1.17197
$935$	10.7940	0.353003
$936$	−0.892920	−0.0291860
$937$	−15.3585	−0.501741	−0.250870	−	0.968021i	$-0.580717\pi$
$937$	−15.3585	−0.501741	−0.250870	+	0.968021i	$0.580717\pi$
$938$	−38.5690	−1.25932
$939$	15.2315	0.497061
$940$	−31.3374	−1.02211
$941$	−47.5468	−1.54998	−0.774991	−	0.631972i	$-0.782246\pi$
$941$	−47.5468	−1.54998	−0.774991	+	0.631972i	$0.782246\pi$
$942$	−84.6249	−2.75723
$943$	75.1015	2.44564
$944$	−30.0612	−0.978410
$945$	15.6070	0.507694
$946$	25.4549	0.827611
$947$	−9.30093	−0.302240	−0.151120	−	0.988515i	$-0.548288\pi$
$947$	−9.30093	−0.302240	−0.151120	+	0.988515i	$0.548288\pi$
$948$	43.1705	1.40211
$949$	11.5095	0.373613
$950$	−20.2321	−0.656415
$951$	9.74975	0.316157
$952$	−45.5169	−1.47521
$953$	25.4120	0.823174	0.411587	−	0.911370i	$-0.364975\pi$
$953$	25.4120	0.823174	0.411587	+	0.911370i	$0.364975\pi$
$954$	2.33124	0.0754766
$955$	−2.06286	−0.0667525
$956$	62.5542	2.02315
$957$	49.0751	1.58637
$958$	27.3878	0.884860
$959$	−53.5734	−1.72998
$960$	0.658857	0.0212645
$961$	1.00000	0.0322581
$962$	20.6100	0.664492
$963$	−2.07621	−0.0669050
$964$	−73.6950	−2.37356
$965$	−18.4173	−0.592875
$966$	−107.722	−3.46590
$967$	30.0897	0.967618	0.483809	−	0.875174i	$-0.339253\pi$
$967$	30.0897	0.967618	0.483809	+	0.875174i	$0.339253\pi$
$968$	51.9920	1.67109
$969$	33.0654	1.06221
$970$	−13.2670	−0.425978
$971$	11.3534	0.364348	0.182174	−	0.983266i	$-0.441687\pi$
$971$	11.3534	0.364348	0.182174	+	0.983266i	$0.441687\pi$
$972$	6.58814	0.211315
$973$	−52.9863	−1.69866
$974$	−57.6970	−1.84873
$975$	1.69062	0.0541433
$976$	90.0606	2.88277
$977$	30.7523	0.983855	0.491927	−	0.870636i	$-0.336293\pi$
$977$	30.7523	0.983855	0.491927	+	0.870636i	$0.336293\pi$
$978$	−57.9781	−1.85393
$979$	−63.1965	−2.01977
$980$	−7.30990	−0.233506
$981$	−1.50274	−0.0479788
$982$	−0.0218974	−0.000698773 0
$983$	−29.8754	−0.952877	−0.476439	−	0.879208i	$-0.658073\pi$
$983$	−29.8754	−0.952877	−0.476439	+	0.879208i	$0.658073\pi$
$984$	−93.8265	−2.99108
$985$	−4.32257	−0.137729
$986$	−41.4049	−1.31860
$987$	34.7873	1.10729
$988$	−35.5801	−1.13195
$989$	−19.4276	−0.617763
$990$	1.58320	0.0503173
$991$	−3.35258	−0.106498	−0.0532492	−	0.998581i	$-0.516958\pi$
$991$	−3.35258	−0.106498	−0.0532492	+	0.998581i	$0.516958\pi$
$992$	−5.40766	−0.171693
$993$	27.2158	0.863669
$994$	−32.9554	−1.04528
$995$	21.9969	0.697347
$996$	−97.4758	−3.08864
$997$	−25.3931	−0.804208	−0.402104	−	0.915594i	$-0.631721\pi$
$997$	−25.3931	−0.804208	−0.402104	+	0.915594i	$0.631721\pi$
$998$	89.0967	2.82031
$999$	−43.0214	−1.36114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2015.2.a.j.1.18	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2015.2.a.j.1.18	✓	20	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:	\( N \)	\(=\)	\( 2015 = 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2015.a (trivial)

\(f(q)\)	\(=\)	\(q+2.54458 q^{2} +1.69062 q^{3} +4.47491 q^{4} -1.00000 q^{5} +4.30194 q^{6} +2.93829 q^{7} +6.29761 q^{8} -0.141787 q^{9} +O(q^{10})\)	\(q+2.54458 q^{2} +1.69062 q^{3} +4.47491 q^{4} -1.00000 q^{5} +4.30194 q^{6} +2.93829 q^{7} +6.29761 q^{8} -0.141787 q^{9} -2.54458 q^{10} +4.38815 q^{11} +7.56539 q^{12} +1.00000 q^{13} +7.47672 q^{14} -1.69062 q^{15} +7.07498 q^{16} -2.45982 q^{17} -0.360789 q^{18} -7.95103 q^{19} -4.47491 q^{20} +4.96754 q^{21} +11.1660 q^{22} -8.52209 q^{23} +10.6469 q^{24} +1.00000 q^{25} +2.54458 q^{26} -5.31158 q^{27} +13.1486 q^{28} +6.61504 q^{29} -4.30194 q^{30} -1.00000 q^{31} +5.40766 q^{32} +7.41871 q^{33} -6.25921 q^{34} -2.93829 q^{35} -0.634484 q^{36} +8.09954 q^{37} -20.2321 q^{38} +1.69062 q^{39} -6.29761 q^{40} -8.81257 q^{41} +12.6403 q^{42} +2.27968 q^{43} +19.6366 q^{44} +0.141787 q^{45} -21.6852 q^{46} +7.00292 q^{47} +11.9611 q^{48} +1.63353 q^{49} +2.54458 q^{50} -4.15863 q^{51} +4.47491 q^{52} -6.46149 q^{53} -13.5158 q^{54} -4.38815 q^{55} +18.5042 q^{56} -13.4422 q^{57} +16.8325 q^{58} -4.24895 q^{59} -7.56539 q^{60} +12.7294 q^{61} -2.54458 q^{62} -0.416611 q^{63} -0.389712 q^{64} -1.00000 q^{65} +18.8775 q^{66} -5.15855 q^{67} -11.0074 q^{68} -14.4077 q^{69} -7.47672 q^{70} -4.40773 q^{71} -0.892920 q^{72} +11.5095 q^{73} +20.6100 q^{74} +1.69062 q^{75} -35.5801 q^{76} +12.8936 q^{77} +4.30194 q^{78} +5.70631 q^{79} -7.07498 q^{80} -8.55453 q^{81} -22.4243 q^{82} -12.8844 q^{83} +22.2293 q^{84} +2.45982 q^{85} +5.80084 q^{86} +11.1836 q^{87} +27.6348 q^{88} -14.4016 q^{89} +0.360789 q^{90} +2.93829 q^{91} -38.1356 q^{92} -1.69062 q^{93} +17.8195 q^{94} +7.95103 q^{95} +9.14233 q^{96} +5.21382 q^{97} +4.15666 q^{98} -0.622183 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} + q^{3} + 25 q^{4} - 20 q^{5} + 9 q^{6} + 11 q^{7} + 15 q^{8} + 29 q^{9}+O(q^{10}) \) 20 * q + 5 * q^2 + q^3 + 25 * q^4 - 20 * q^5 + 9 * q^6 + 11 * q^7 + 15 * q^8 + 29 * q^9	\( 20 q + 5 q^{2} + q^{3} + 25 q^{4} - 20 q^{5} + 9 q^{6} + 11 q^{7} + 15 q^{8} + 29 q^{9} - 5 q^{10} + 5 q^{11} + 20 q^{13} - 7 q^{14} - q^{15} + 39 q^{16} + 3 q^{17} + 30 q^{18} + 15 q^{19} - 25 q^{20} + 11 q^{22} - 5 q^{23} + 8 q^{24} + 20 q^{25} + 5 q^{26} - 8 q^{27} + 46 q^{28} - 2 q^{29} - 9 q^{30} - 20 q^{31} + 50 q^{32} + 43 q^{33} - q^{34} - 11 q^{35} + 10 q^{36} + 46 q^{37} + 7 q^{38} + q^{39} - 15 q^{40} - 12 q^{41} + 39 q^{42} + 13 q^{43} + 8 q^{44} - 29 q^{45} - 14 q^{46} + 21 q^{47} + 27 q^{48} + 55 q^{49} + 5 q^{50} - 9 q^{51} + 25 q^{52} + 7 q^{53} - 32 q^{54} - 5 q^{55} - 15 q^{56} + 70 q^{57} + 10 q^{58} + 5 q^{59} - 21 q^{61} - 5 q^{62} + 78 q^{63} + 51 q^{64} - 20 q^{65} - 25 q^{66} + 57 q^{67} + 23 q^{68} - 28 q^{69} + 7 q^{70} - 15 q^{71} + 16 q^{72} + 67 q^{73} + 18 q^{74} + q^{75} + 61 q^{76} - 22 q^{77} + 9 q^{78} + 12 q^{79} - 39 q^{80} + 68 q^{81} - 6 q^{82} + 21 q^{83} - 43 q^{84} - 3 q^{85} + 12 q^{86} + 20 q^{87} + 79 q^{88} - 30 q^{90} + 11 q^{91} + 16 q^{92} - q^{93} + 25 q^{94} - 15 q^{95} + 86 q^{96} + 105 q^{97} + 4 q^{98} - 11 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 + q^3 + 25 * q^4 - 20 * q^5 + 9 * q^6 + 11 * q^7 + 15 * q^8 + 29 * q^9 - 5 * q^10 + 5 * q^11 + 20 * q^13 - 7 * q^14 - q^15 + 39 * q^16 + 3 * q^17 + 30 * q^18 + 15 * q^19 - 25 * q^20 + 11 * q^22 - 5 * q^23 + 8 * q^24 + 20 * q^25 + 5 * q^26 - 8 * q^27 + 46 * q^28 - 2 * q^29 - 9 * q^30 - 20 * q^31 + 50 * q^32 + 43 * q^33 - q^34 - 11 * q^35 + 10 * q^36 + 46 * q^37 + 7 * q^38 + q^39 - 15 * q^40 - 12 * q^41 + 39 * q^42 + 13 * q^43 + 8 * q^44 - 29 * q^45 - 14 * q^46 + 21 * q^47 + 27 * q^48 + 55 * q^49 + 5 * q^50 - 9 * q^51 + 25 * q^52 + 7 * q^53 - 32 * q^54 - 5 * q^55 - 15 * q^56 + 70 * q^57 + 10 * q^58 + 5 * q^59 - 21 * q^61 - 5 * q^62 + 78 * q^63 + 51 * q^64 - 20 * q^65 - 25 * q^66 + 57 * q^67 + 23 * q^68 - 28 * q^69 + 7 * q^70 - 15 * q^71 + 16 * q^72 + 67 * q^73 + 18 * q^74 + q^75 + 61 * q^76 - 22 * q^77 + 9 * q^78 + 12 * q^79 - 39 * q^80 + 68 * q^81 - 6 * q^82 + 21 * q^83 - 43 * q^84 - 3 * q^85 + 12 * q^86 + 20 * q^87 + 79 * q^88 - 30 * q^90 + 11 * q^91 + 16 * q^92 - q^93 + 25 * q^94 - 15 * q^95 + 86 * q^96 + 105 * q^97 + 4 * q^98 - 11 * q^99

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