LMFDB - Embedded newform 1690.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1690,2,Mod(1,1690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1690, 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("1690.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.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:	$13.4947179416$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	1690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.445042 q^{6} -0.890084 q^{7} -1.00000 q^{8} -2.80194 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.445042 q^{6} -0.890084 q^{7} -1.00000 q^{8} -2.80194 q^{9} -1.00000 q^{10} +2.75302 q^{11} -0.445042 q^{12} +0.890084 q^{14} -0.445042 q^{15} +1.00000 q^{16} -2.13706 q^{17} +2.80194 q^{18} -0.692021 q^{19} +1.00000 q^{20} +0.396125 q^{21} -2.75302 q^{22} -0.396125 q^{23} +0.445042 q^{24} +1.00000 q^{25} +2.58211 q^{27} -0.890084 q^{28} +0.493959 q^{29} +0.445042 q^{30} +7.87800 q^{31} -1.00000 q^{32} -1.22521 q^{33} +2.13706 q^{34} -0.890084 q^{35} -2.80194 q^{36} +10.8116 q^{37} +0.692021 q^{38} -1.00000 q^{40} +4.41789 q^{41} -0.396125 q^{42} -11.9976 q^{43} +2.75302 q^{44} -2.80194 q^{45} +0.396125 q^{46} +0.176292 q^{47} -0.445042 q^{48} -6.20775 q^{49} -1.00000 q^{50} +0.951083 q^{51} +9.30559 q^{53} -2.58211 q^{54} +2.75302 q^{55} +0.890084 q^{56} +0.307979 q^{57} -0.493959 q^{58} +6.98254 q^{59} -0.445042 q^{60} +9.30559 q^{61} -7.87800 q^{62} +2.49396 q^{63} +1.00000 q^{64} +1.22521 q^{66} -4.98254 q^{67} -2.13706 q^{68} +0.176292 q^{69} +0.890084 q^{70} +4.67025 q^{71} +2.80194 q^{72} -10.6407 q^{73} -10.8116 q^{74} -0.445042 q^{75} -0.692021 q^{76} -2.45042 q^{77} +9.70171 q^{79} +1.00000 q^{80} +7.25667 q^{81} -4.41789 q^{82} +2.96615 q^{83} +0.396125 q^{84} -2.13706 q^{85} +11.9976 q^{86} -0.219833 q^{87} -2.75302 q^{88} +13.4330 q^{89} +2.80194 q^{90} -0.396125 q^{92} -3.50604 q^{93} -0.176292 q^{94} -0.692021 q^{95} +0.445042 q^{96} +8.35690 q^{97} +6.20775 q^{98} -7.71379 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 2 q^{7} - 3 q^{8} - 4 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + 3 * q^5 + q^6 - 2 * q^7 - 3 * q^8 - 4 * q^9	$ 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 2 q^{7} - 3 q^{8} - 4 q^{9} - 3 q^{10} + 13 q^{11} - q^{12} + 2 q^{14} - q^{15} + 3 q^{16} - q^{17} + 4 q^{18} + 3 q^{19} + 3 q^{20} + 10 q^{21} - 13 q^{22} - 10 q^{23} + q^{24} + 3 q^{25} + 2 q^{27} - 2 q^{28} - 8 q^{29} + q^{30} + 4 q^{31} - 3 q^{32} - 2 q^{33} + q^{34} - 2 q^{35} - 4 q^{36} + 6 q^{37} - 3 q^{38} - 3 q^{40} + 19 q^{41} - 10 q^{42} + 5 q^{43} + 13 q^{44} - 4 q^{45} + 10 q^{46} + 8 q^{47} - q^{48} - q^{49} - 3 q^{50} + 12 q^{51} - 8 q^{53} - 2 q^{54} + 13 q^{55} + 2 q^{56} + 6 q^{57} + 8 q^{58} + 5 q^{59} - q^{60} - 8 q^{61} - 4 q^{62} - 2 q^{63} + 3 q^{64} + 2 q^{66} + q^{67} - q^{68} + 8 q^{69} + 2 q^{70} + 12 q^{71} + 4 q^{72} + 5 q^{73} - 6 q^{74} - q^{75} + 3 q^{76} - 4 q^{77} + 2 q^{79} + 3 q^{80} - 5 q^{81} - 19 q^{82} - 7 q^{83} + 10 q^{84} - q^{85} - 5 q^{86} - 2 q^{87} - 13 q^{88} + 21 q^{89} + 4 q^{90} - 10 q^{92} - 20 q^{93} - 8 q^{94} + 3 q^{95} + q^{96} + 21 q^{97} + q^{98} - 15 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + 3 * q^5 + q^6 - 2 * q^7 - 3 * q^8 - 4 * q^9 - 3 * q^10 + 13 * q^11 - q^12 + 2 * q^14 - q^15 + 3 * q^16 - q^17 + 4 * q^18 + 3 * q^19 + 3 * q^20 + 10 * q^21 - 13 * q^22 - 10 * q^23 + q^24 + 3 * q^25 + 2 * q^27 - 2 * q^28 - 8 * q^29 + q^30 + 4 * q^31 - 3 * q^32 - 2 * q^33 + q^34 - 2 * q^35 - 4 * q^36 + 6 * q^37 - 3 * q^38 - 3 * q^40 + 19 * q^41 - 10 * q^42 + 5 * q^43 + 13 * q^44 - 4 * q^45 + 10 * q^46 + 8 * q^47 - q^48 - q^49 - 3 * q^50 + 12 * q^51 - 8 * q^53 - 2 * q^54 + 13 * q^55 + 2 * q^56 + 6 * q^57 + 8 * q^58 + 5 * q^59 - q^60 - 8 * q^61 - 4 * q^62 - 2 * q^63 + 3 * q^64 + 2 * q^66 + q^67 - q^68 + 8 * q^69 + 2 * q^70 + 12 * q^71 + 4 * q^72 + 5 * q^73 - 6 * q^74 - q^75 + 3 * q^76 - 4 * q^77 + 2 * q^79 + 3 * q^80 - 5 * q^81 - 19 * q^82 - 7 * q^83 + 10 * q^84 - q^85 - 5 * q^86 - 2 * q^87 - 13 * q^88 + 21 * q^89 + 4 * q^90 - 10 * q^92 - 20 * q^93 - 8 * q^94 + 3 * q^95 + q^96 + 21 * q^97 + q^98 - 15 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.445042	−0.256945	−0.128473	−	0.991713i	$-0.541007\pi$
$3$	−0.445042	−0.256945	−0.128473	+	0.991713i	$0.541007\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.445042	0.181688
$7$	−0.890084	−0.336420	−0.168210	−	0.985751i	$-0.553799\pi$
$7$	−0.890084	−0.336420	−0.168210	+	0.985751i	$0.553799\pi$
$8$	−1.00000	−0.353553
$9$	−2.80194	−0.933979
$10$	−1.00000	−0.316228
$11$	2.75302	0.830067	0.415033	−	0.909806i	$-0.363770\pi$
$11$	2.75302	0.830067	0.415033	+	0.909806i	$0.363770\pi$
$12$	−0.445042	−0.128473
$13$	0	0
$13$	0	0
$14$	0.890084	0.237885
$15$	−0.445042	−0.114909
$16$	1.00000	0.250000
$17$	−2.13706	−0.518314	−0.259157	−	0.965835i	$-0.583445\pi$
$17$	−2.13706	−0.518314	−0.259157	+	0.965835i	$0.583445\pi$
$18$	2.80194	0.660423
$19$	−0.692021	−0.158761	−0.0793803	−	0.996844i	$-0.525294\pi$
$19$	−0.692021	−0.158761	−0.0793803	+	0.996844i	$0.525294\pi$
$20$	1.00000	0.223607
$21$	0.396125	0.0864415
$22$	−2.75302	−0.586946
$23$	−0.396125	−0.0825977	−0.0412988	−	0.999147i	$-0.513150\pi$
$23$	−0.396125	−0.0825977	−0.0412988	+	0.999147i	$0.513150\pi$
$24$	0.445042	0.0908438
$25$	1.00000	0.200000
$26$	0	0
$27$	2.58211	0.496926
$28$	−0.890084	−0.168210
$29$	0.493959	0.0917259	0.0458630	−	0.998948i	$-0.485396\pi$
$29$	0.493959	0.0917259	0.0458630	+	0.998948i	$0.485396\pi$
$30$	0.445042	0.0812532
$31$	7.87800	1.41493	0.707465	−	0.706748i	$-0.249838\pi$
$31$	7.87800	1.41493	0.707465	+	0.706748i	$0.249838\pi$
$32$	−1.00000	−0.176777
$33$	−1.22521	−0.213282
$34$	2.13706	0.366503
$35$	−0.890084	−0.150452
$36$	−2.80194	−0.466990
$37$	10.8116	1.77742	0.888710	−	0.458469i	$-0.151602\pi$
$37$	10.8116	1.77742	0.888710	+	0.458469i	$0.151602\pi$
$38$	0.692021	0.112261
$39$	0	0
$40$	−1.00000	−0.158114
$41$	4.41789	0.689959	0.344980	−	0.938610i	$-0.387886\pi$
$41$	4.41789	0.689959	0.344980	+	0.938610i	$0.387886\pi$
$42$	−0.396125	−0.0611233
$43$	−11.9976	−1.82962	−0.914809	−	0.403887i	$-0.867659\pi$
$43$	−11.9976	−1.82962	−0.914809	+	0.403887i	$0.867659\pi$
$44$	2.75302	0.415033
$45$	−2.80194	−0.417688
$46$	0.396125	0.0584054
$47$	0.176292	0.0257148	0.0128574	−	0.999917i	$-0.495907\pi$
$47$	0.176292	0.0257148	0.0128574	+	0.999917i	$0.495907\pi$
$48$	−0.445042	−0.0642363
$49$	−6.20775	−0.886822
$50$	−1.00000	−0.141421
$51$	0.951083	0.133178
$52$	0	0
$53$	9.30559	1.27822	0.639110	−	0.769115i	$-0.279303\pi$
$53$	9.30559	1.27822	0.639110	+	0.769115i	$0.279303\pi$
$54$	−2.58211	−0.351380
$55$	2.75302	0.371217
$56$	0.890084	0.118942
$57$	0.307979	0.0407928
$58$	−0.493959	−0.0648600
$59$	6.98254	0.909049	0.454525	−	0.890734i	$-0.349809\pi$
$59$	6.98254	0.909049	0.454525	+	0.890734i	$0.349809\pi$
$60$	−0.445042	−0.0574547
$61$	9.30559	1.19146	0.595729	−	0.803185i	$-0.296863\pi$
$61$	9.30559	1.19146	0.595729	+	0.803185i	$0.296863\pi$
$62$	−7.87800	−1.00051
$63$	2.49396	0.314209
$64$	1.00000	0.125000
$65$	0	0
$66$	1.22521	0.150813
$67$	−4.98254	−0.608714	−0.304357	−	0.952558i	$-0.598442\pi$
$67$	−4.98254	−0.608714	−0.304357	+	0.952558i	$0.598442\pi$
$68$	−2.13706	−0.259157
$69$	0.176292	0.0212231
$70$	0.890084	0.106385
$71$	4.67025	0.554257	0.277128	−	0.960833i	$-0.410617\pi$
$71$	4.67025	0.554257	0.277128	+	0.960833i	$0.410617\pi$
$72$	2.80194	0.330212
$73$	−10.6407	−1.24540	−0.622701	−	0.782460i	$-0.713965\pi$
$73$	−10.6407	−1.24540	−0.622701	+	0.782460i	$0.713965\pi$
$74$	−10.8116	−1.25683
$75$	−0.445042	−0.0513890
$76$	−0.692021	−0.0793803
$77$	−2.45042	−0.279251
$78$	0	0
$79$	9.70171	1.09153	0.545764	−	0.837939i	$-0.316240\pi$
$79$	9.70171	1.09153	0.545764	+	0.837939i	$0.316240\pi$
$80$	1.00000	0.111803
$81$	7.25667	0.806296
$82$	−4.41789	−0.487875
$83$	2.96615	0.325577	0.162789	−	0.986661i	$-0.447951\pi$
$83$	2.96615	0.325577	0.162789	+	0.986661i	$0.447951\pi$
$84$	0.396125	0.0432207
$85$	−2.13706	−0.231797
$86$	11.9976	1.29374
$87$	−0.219833	−0.0235685
$88$	−2.75302	−0.293473
$89$	13.4330	1.42389	0.711945	−	0.702235i	$-0.247814\pi$
$89$	13.4330	1.42389	0.711945	+	0.702235i	$0.247814\pi$
$90$	2.80194	0.295350
$91$	0	0
$92$	−0.396125	−0.0412988
$93$	−3.50604	−0.363559
$94$	−0.176292	−0.0181831
$95$	−0.692021	−0.0709999
$96$	0.445042	0.0454219
$97$	8.35690	0.848514	0.424257	−	0.905542i	$-0.360535\pi$
$97$	8.35690	0.848514	0.424257	+	0.905542i	$0.360535\pi$
$98$	6.20775	0.627078
$99$	−7.71379	−0.775265
$100$	1.00000	0.100000
$101$	−16.7875	−1.67042	−0.835208	−	0.549935i	$-0.814653\pi$
$101$	−16.7875	−1.67042	−0.835208	+	0.549935i	$0.814653\pi$
$102$	−0.951083	−0.0941712
$103$	11.8780	1.17037	0.585187	−	0.810898i	$-0.301021\pi$
$103$	11.8780	1.17037	0.585187	+	0.810898i	$0.301021\pi$
$104$	0	0
$105$	0.396125	0.0386578
$106$	−9.30559	−0.903838
$107$	−13.3817	−1.29365	−0.646826	−	0.762637i	$-0.723904\pi$
$107$	−13.3817	−1.29365	−0.646826	+	0.762637i	$0.723904\pi$
$108$	2.58211	0.248463
$109$	−7.32975	−0.702063	−0.351031	−	0.936364i	$-0.614169\pi$
$109$	−7.32975	−0.702063	−0.351031	+	0.936364i	$0.614169\pi$
$110$	−2.75302	−0.262490
$111$	−4.81163	−0.456699
$112$	−0.890084	−0.0841050
$113$	0.225209	0.0211859	0.0105930	−	0.999944i	$-0.496628\pi$
$113$	0.225209	0.0211859	0.0105930	+	0.999944i	$0.496628\pi$
$114$	−0.307979	−0.0288448
$115$	−0.396125	−0.0369388
$116$	0.493959	0.0458630
$117$	0	0
$118$	−6.98254	−0.642795
$119$	1.90217	0.174371
$120$	0.445042	0.0406266
$121$	−3.42088	−0.310989
$122$	−9.30559	−0.842488
$123$	−1.96615	−0.177282
$124$	7.87800	0.707465
$125$	1.00000	0.0894427
$126$	−2.49396	−0.222180
$127$	10.6896	0.948551	0.474276	−	0.880376i	$-0.342710\pi$
$127$	10.6896	0.948551	0.474276	+	0.880376i	$0.342710\pi$
$128$	−1.00000	−0.0883883
$129$	5.33944	0.470111
$130$	0	0
$131$	10.5157	0.918764	0.459382	−	0.888239i	$-0.348071\pi$
$131$	10.5157	0.918764	0.459382	+	0.888239i	$0.348071\pi$
$132$	−1.22521	−0.106641
$133$	0.615957	0.0534103
$134$	4.98254	0.430426
$135$	2.58211	0.222232
$136$	2.13706	0.183252
$137$	6.44504	0.550637	0.275319	−	0.961353i	$-0.411217\pi$
$137$	6.44504	0.550637	0.275319	+	0.961353i	$0.411217\pi$
$138$	−0.176292	−0.0150070
$139$	13.5211	1.14685	0.573423	−	0.819260i	$-0.305615\pi$
$139$	13.5211	1.14685	0.573423	+	0.819260i	$0.305615\pi$
$140$	−0.890084	−0.0752258
$141$	−0.0784573	−0.00660730
$142$	−4.67025	−0.391919
$143$	0	0
$144$	−2.80194	−0.233495
$145$	0.493959	0.0410211
$146$	10.6407	0.880632
$147$	2.76271	0.227864
$148$	10.8116	0.888710
$149$	2.51812	0.206293	0.103146	−	0.994666i	$-0.467109\pi$
$149$	2.51812	0.206293	0.103146	+	0.994666i	$0.467109\pi$
$150$	0.445042	0.0363375
$151$	22.9095	1.86435	0.932173	−	0.362014i	$-0.117911\pi$
$151$	22.9095	1.86435	0.932173	+	0.362014i	$0.117911\pi$
$152$	0.692021	0.0561304
$153$	5.98792	0.484095
$154$	2.45042	0.197460
$155$	7.87800	0.632776
$156$	0	0
$157$	−10.7681	−0.859387	−0.429693	−	0.902975i	$-0.641378\pi$
$157$	−10.7681	−0.859387	−0.429693	+	0.902975i	$0.641378\pi$
$158$	−9.70171	−0.771827
$159$	−4.14138	−0.328432
$160$	−1.00000	−0.0790569
$161$	0.352584	0.0277875
$162$	−7.25667	−0.570138
$163$	2.41119	0.188859	0.0944295	−	0.995532i	$-0.469897\pi$
$163$	2.41119	0.188859	0.0944295	+	0.995532i	$0.469897\pi$
$164$	4.41789	0.344980
$165$	−1.22521	−0.0953824
$166$	−2.96615	−0.230218
$167$	21.0315	1.62746	0.813732	−	0.581241i	$-0.197433\pi$
$167$	21.0315	1.62746	0.813732	+	0.581241i	$0.197433\pi$
$168$	−0.396125	−0.0305617
$169$	0	0
$170$	2.13706	0.163905
$171$	1.93900	0.148279
$172$	−11.9976	−0.914809
$173$	−6.04354	−0.459482	−0.229741	−	0.973252i	$-0.573788\pi$
$173$	−6.04354	−0.459482	−0.229741	+	0.973252i	$0.573788\pi$
$174$	0.219833	0.0166655
$175$	−0.890084	−0.0672840
$176$	2.75302	0.207517
$177$	−3.10752	−0.233576
$178$	−13.4330	−1.00684
$179$	16.0151	1.19702	0.598511	−	0.801115i	$-0.295759\pi$
$179$	16.0151	1.19702	0.598511	+	0.801115i	$0.295759\pi$
$180$	−2.80194	−0.208844
$181$	9.83446	0.730990	0.365495	−	0.930813i	$-0.380900\pi$
$181$	9.83446	0.730990	0.365495	+	0.930813i	$0.380900\pi$
$182$	0	0
$183$	−4.14138	−0.306139
$184$	0.396125	0.0292027
$185$	10.8116	0.794887
$186$	3.50604	0.257075
$187$	−5.88338	−0.430235
$188$	0.176292	0.0128574
$189$	−2.29829	−0.167176
$190$	0.692021	0.0502045
$191$	24.9879	1.80806	0.904031	−	0.427467i	$-0.140594\pi$
$191$	24.9879	1.80806	0.904031	+	0.427467i	$0.140594\pi$
$192$	−0.445042	−0.0321181
$193$	−16.1631	−1.16345	−0.581724	−	0.813386i	$-0.697622\pi$
$193$	−16.1631	−1.16345	−0.581724	+	0.813386i	$0.697622\pi$
$194$	−8.35690	−0.599990
$195$	0	0
$196$	−6.20775	−0.443411
$197$	11.5603	0.823640	0.411820	−	0.911265i	$-0.364893\pi$
$197$	11.5603	0.823640	0.411820	+	0.911265i	$0.364893\pi$
$198$	7.71379	0.548195
$199$	−3.92154	−0.277991	−0.138995	−	0.990293i	$-0.544387\pi$
$199$	−3.92154	−0.277991	−0.138995	+	0.990293i	$0.544387\pi$
$200$	−1.00000	−0.0707107
$201$	2.21744	0.156406
$202$	16.7875	1.18116
$203$	−0.439665	−0.0308584
$204$	0.951083	0.0665891
$205$	4.41789	0.308559
$206$	−11.8780	−0.827580
$207$	1.10992	0.0771445
$208$	0	0
$209$	−1.90515	−0.131782
$210$	−0.396125	−0.0273352
$211$	17.1685	1.18193	0.590965	−	0.806697i	$-0.298747\pi$
$211$	17.1685	1.18193	0.590965	+	0.806697i	$0.298747\pi$
$212$	9.30559	0.639110
$213$	−2.07846	−0.142414
$214$	13.3817	0.914751
$215$	−11.9976	−0.818230
$216$	−2.58211	−0.175690
$217$	−7.01208	−0.476011
$218$	7.32975	0.496433
$219$	4.73556	0.320000
$220$	2.75302	0.185609
$221$	0	0
$222$	4.81163	0.322935
$223$	−8.59179	−0.575349	−0.287675	−	0.957728i	$-0.592882\pi$
$223$	−8.59179	−0.575349	−0.287675	+	0.957728i	$0.592882\pi$
$224$	0.890084	0.0594712
$225$	−2.80194	−0.186796
$226$	−0.225209	−0.0149807
$227$	19.8485	1.31739	0.658694	−	0.752411i	$-0.271109\pi$
$227$	19.8485	1.31739	0.658694	+	0.752411i	$0.271109\pi$
$228$	0.307979	0.0203964
$229$	−21.6582	−1.43121	−0.715606	−	0.698504i	$-0.753849\pi$
$229$	−21.6582	−1.43121	−0.715606	+	0.698504i	$0.753849\pi$
$230$	0.396125	0.0261197
$231$	1.09054	0.0717522
$232$	−0.493959	−0.0324300
$233$	−1.75063	−0.114687	−0.0573437	−	0.998354i	$-0.518263\pi$
$233$	−1.75063	−0.114687	−0.0573437	+	0.998354i	$0.518263\pi$
$234$	0	0
$235$	0.176292	0.0115000
$236$	6.98254	0.454525
$237$	−4.31767	−0.280463
$238$	−1.90217	−0.123299
$239$	−12.8224	−0.829411	−0.414705	−	0.909956i	$-0.636115\pi$
$239$	−12.8224	−0.829411	−0.414705	+	0.909956i	$0.636115\pi$
$240$	−0.445042	−0.0287273
$241$	−20.6112	−1.32768	−0.663841	−	0.747874i	$-0.731075\pi$
$241$	−20.6112	−1.32768	−0.663841	+	0.747874i	$0.731075\pi$
$242$	3.42088	0.219902
$243$	−10.9758	−0.704100
$244$	9.30559	0.595729
$245$	−6.20775	−0.396599
$246$	1.96615	0.125357
$247$	0	0
$248$	−7.87800	−0.500254
$249$	−1.32006	−0.0836554
$250$	−1.00000	−0.0632456
$251$	28.4547	1.79605	0.898023	−	0.439948i	$-0.145003\pi$
$251$	28.4547	1.79605	0.898023	+	0.439948i	$0.145003\pi$
$252$	2.49396	0.157105
$253$	−1.09054	−0.0685616
$254$	−10.6896	−0.670727
$255$	0.951083	0.0595591
$256$	1.00000	0.0625000
$257$	10.4480	0.651730	0.325865	−	0.945416i	$-0.394345\pi$
$257$	10.4480	0.651730	0.325865	+	0.945416i	$0.394345\pi$
$258$	−5.33944	−0.332419
$259$	−9.62325	−0.597960
$260$	0	0
$261$	−1.38404	−0.0856701
$262$	−10.5157	−0.649664
$263$	−6.43967	−0.397087	−0.198543	−	0.980092i	$-0.563621\pi$
$263$	−6.43967	−0.397087	−0.198543	+	0.980092i	$0.563621\pi$
$264$	1.22521	0.0754064
$265$	9.30559	0.571638
$266$	−0.615957	−0.0377668
$267$	−5.97823	−0.365862
$268$	−4.98254	−0.304357
$269$	10.2935	0.627606	0.313803	−	0.949488i	$-0.398397\pi$
$269$	10.2935	0.627606	0.313803	+	0.949488i	$0.398397\pi$
$270$	−2.58211	−0.157142
$271$	−31.6340	−1.92163	−0.960815	−	0.277192i	$-0.910596\pi$
$271$	−31.6340	−1.92163	−0.960815	+	0.277192i	$0.910596\pi$
$272$	−2.13706	−0.129578
$273$	0	0
$274$	−6.44504	−0.389359
$275$	2.75302	0.166013
$276$	0.176292	0.0106115
$277$	−21.1099	−1.26837	−0.634186	−	0.773181i	$-0.718665\pi$
$277$	−21.1099	−1.26837	−0.634186	+	0.773181i	$0.718665\pi$
$278$	−13.5211	−0.810942
$279$	−22.0737	−1.32152
$280$	0.890084	0.0531927
$281$	17.6082	1.05042	0.525208	−	0.850974i	$-0.323987\pi$
$281$	17.6082	1.05042	0.525208	+	0.850974i	$0.323987\pi$
$282$	0.0784573	0.00467207
$283$	18.4131	1.09455	0.547273	−	0.836954i	$-0.315666\pi$
$283$	18.4131	1.09455	0.547273	+	0.836954i	$0.315666\pi$
$284$	4.67025	0.277128
$285$	0.307979	0.0182431
$286$	0	0
$287$	−3.93230	−0.232116
$288$	2.80194	0.165106
$289$	−12.4330	−0.731351
$290$	−0.493959	−0.0290063
$291$	−3.71917	−0.218022
$292$	−10.6407	−0.622701
$293$	−0.0241632	−0.00141163	−0.000705814	−	1.00000i	$-0.500225\pi$
$293$	−0.0241632	−0.00141163	−0.000705814		1.00000i	$0.500225\pi$
$294$	−2.76271	−0.161124
$295$	6.98254	0.406539
$296$	−10.8116	−0.628413
$297$	7.10859	0.412482
$298$	−2.51812	−0.145871
$299$	0	0
$300$	−0.445042	−0.0256945
$301$	10.6789	0.615520
$302$	−22.9095	−1.31829
$303$	7.47112	0.429205
$304$	−0.692021	−0.0396902
$305$	9.30559	0.532836
$306$	−5.98792	−0.342307
$307$	−14.1371	−0.806845	−0.403422	−	0.915014i	$-0.632179\pi$
$307$	−14.1371	−0.806845	−0.403422	+	0.915014i	$0.632179\pi$
$308$	−2.45042	−0.139626
$309$	−5.28621	−0.300722
$310$	−7.87800	−0.447440
$311$	−4.09783	−0.232367	−0.116183	−	0.993228i	$-0.537066\pi$
$311$	−4.09783	−0.232367	−0.116183	+	0.993228i	$0.537066\pi$
$312$	0	0
$313$	−16.9051	−0.955536	−0.477768	−	0.878486i	$-0.658554\pi$
$313$	−16.9051	−0.955536	−0.477768	+	0.878486i	$0.658554\pi$
$314$	10.7681	0.607678
$315$	2.49396	0.140519
$316$	9.70171	0.545764
$317$	−27.1594	−1.52543	−0.762713	−	0.646738i	$-0.776133\pi$
$317$	−27.1594	−1.52543	−0.762713	+	0.646738i	$0.776133\pi$
$318$	4.14138	0.232237
$319$	1.35988	0.0761387
$320$	1.00000	0.0559017
$321$	5.95539	0.332398
$322$	−0.352584	−0.0196487
$323$	1.47889	0.0822878
$324$	7.25667	0.403148
$325$	0	0
$326$	−2.41119	−0.133543
$327$	3.26205	0.180392
$328$	−4.41789	−0.243937
$329$	−0.156915	−0.00865098
$330$	1.22521	0.0674456
$331$	−13.8672	−0.762213	−0.381106	−	0.924531i	$-0.624457\pi$
$331$	−13.8672	−0.762213	−0.381106	+	0.924531i	$0.624457\pi$
$332$	2.96615	0.162789
$333$	−30.2935	−1.66007
$334$	−21.0315	−1.15079
$335$	−4.98254	−0.272225
$336$	0.396125	0.0216104
$337$	−33.9433	−1.84901	−0.924505	−	0.381170i	$-0.875521\pi$
$337$	−33.9433	−1.84901	−0.924505	+	0.381170i	$0.875521\pi$
$338$	0	0
$339$	−0.100228	−0.00544362
$340$	−2.13706	−0.115899
$341$	21.6883	1.17449
$342$	−1.93900	−0.104849
$343$	11.7560	0.634765
$344$	11.9976	0.646868
$345$	0.176292	0.00949124
$346$	6.04354	0.324903
$347$	−24.2828	−1.30357	−0.651783	−	0.758405i	$-0.725979\pi$
$347$	−24.2828	−1.30357	−0.651783	+	0.758405i	$0.725979\pi$
$348$	−0.219833	−0.0117843
$349$	−31.9517	−1.71033	−0.855167	−	0.518353i	$-0.826545\pi$
$349$	−31.9517	−1.71033	−0.855167	+	0.518353i	$0.826545\pi$
$350$	0.890084	0.0475770
$351$	0	0
$352$	−2.75302	−0.146736
$353$	30.8025	1.63945	0.819727	−	0.572755i	$-0.194125\pi$
$353$	30.8025	1.63945	0.819727	+	0.572755i	$0.194125\pi$
$354$	3.10752	0.165163
$355$	4.67025	0.247871
$356$	13.4330	0.711945
$357$	−0.846543	−0.0448038
$358$	−16.0151	−0.846423
$359$	21.9517	1.15857	0.579283	−	0.815127i	$-0.303333\pi$
$359$	21.9517	1.15857	0.579283	+	0.815127i	$0.303333\pi$
$360$	2.80194	0.147675
$361$	−18.5211	−0.974795
$362$	−9.83446	−0.516888
$363$	1.52243	0.0799071
$364$	0	0
$365$	−10.6407	−0.556960
$366$	4.14138	0.216473
$367$	−8.61596	−0.449749	−0.224875	−	0.974388i	$-0.572197\pi$
$367$	−8.61596	−0.449749	−0.224875	+	0.974388i	$0.572197\pi$
$368$	−0.396125	−0.0206494
$369$	−12.3787	−0.644408
$370$	−10.8116	−0.562070
$371$	−8.28275	−0.430019
$372$	−3.50604	−0.181780
$373$	6.93362	0.359009	0.179505	−	0.983757i	$-0.442550\pi$
$373$	6.93362	0.359009	0.179505	+	0.983757i	$0.442550\pi$
$374$	5.88338	0.304222
$375$	−0.445042	−0.0229819
$376$	−0.176292	−0.00909157
$377$	0	0
$378$	2.29829	0.118211
$379$	−10.2494	−0.526475	−0.263237	−	0.964731i	$-0.584790\pi$
$379$	−10.2494	−0.526475	−0.263237	+	0.964731i	$0.584790\pi$
$380$	−0.692021	−0.0355000
$381$	−4.75733	−0.243726
$382$	−24.9879	−1.27849
$383$	−27.2814	−1.39402	−0.697008	−	0.717064i	$-0.745486\pi$
$383$	−27.2814	−1.39402	−0.697008	+	0.717064i	$0.745486\pi$
$384$	0.445042	0.0227109
$385$	−2.45042	−0.124885
$386$	16.1631	0.822682
$387$	33.6165	1.70883
$388$	8.35690	0.424257
$389$	−32.5133	−1.64849	−0.824246	−	0.566232i	$-0.808401\pi$
$389$	−32.5133	−1.64849	−0.824246	+	0.566232i	$0.808401\pi$
$390$	0	0
$391$	0.846543	0.0428115
$392$	6.20775	0.313539
$393$	−4.67994	−0.236072
$394$	−11.5603	−0.582401
$395$	9.70171	0.488146
$396$	−7.71379	−0.387633
$397$	3.86725	0.194092	0.0970458	−	0.995280i	$-0.469061\pi$
$397$	3.86725	0.194092	0.0970458	+	0.995280i	$0.469061\pi$
$398$	3.92154	0.196569
$399$	−0.274127	−0.0137235
$400$	1.00000	0.0500000
$401$	−13.7168	−0.684983	−0.342492	−	0.939521i	$-0.611271\pi$
$401$	−13.7168	−0.684983	−0.342492	+	0.939521i	$0.611271\pi$
$402$	−2.21744	−0.110596
$403$	0	0
$404$	−16.7875	−0.835208
$405$	7.25667	0.360587
$406$	0.439665	0.0218202
$407$	29.7646	1.47538
$408$	−0.951083	−0.0470856
$409$	−1.67696	−0.0829201	−0.0414601	−	0.999140i	$-0.513201\pi$
$409$	−1.67696	−0.0829201	−0.0414601	+	0.999140i	$0.513201\pi$
$410$	−4.41789	−0.218184
$411$	−2.86831	−0.141483
$412$	11.8780	0.585187
$413$	−6.21505	−0.305822
$414$	−1.10992	−0.0545494
$415$	2.96615	0.145602
$416$	0	0
$417$	−6.01746	−0.294676
$418$	1.90515	0.0931839
$419$	−26.6112	−1.30004	−0.650020	−	0.759917i	$-0.725240\pi$
$419$	−26.6112	−1.30004	−0.650020	+	0.759917i	$0.725240\pi$
$420$	0.396125	0.0193289
$421$	24.8418	1.21071	0.605357	−	0.795954i	$-0.293031\pi$
$421$	24.8418	1.21071	0.605357	+	0.795954i	$0.293031\pi$
$422$	−17.1685	−0.835751
$423$	−0.493959	−0.0240171
$424$	−9.30559	−0.451919
$425$	−2.13706	−0.103663
$426$	2.07846	0.100702
$427$	−8.28275	−0.400830
$428$	−13.3817	−0.646826
$429$	0	0
$430$	11.9976	0.578576
$431$	12.4155	0.598034	0.299017	−	0.954248i	$-0.403341\pi$
$431$	12.4155	0.598034	0.299017	+	0.954248i	$0.403341\pi$
$432$	2.58211	0.124232
$433$	5.31096	0.255229	0.127614	−	0.991824i	$-0.459268\pi$
$433$	5.31096	0.255229	0.127614	+	0.991824i	$0.459268\pi$
$434$	7.01208	0.336591
$435$	−0.219833	−0.0105402
$436$	−7.32975	−0.351031
$437$	0.274127	0.0131133
$438$	−4.73556	−0.226274
$439$	7.15346	0.341416	0.170708	−	0.985322i	$-0.445395\pi$
$439$	7.15346	0.341416	0.170708	+	0.985322i	$0.445395\pi$
$440$	−2.75302	−0.131245
$441$	17.3937	0.828273
$442$	0	0
$443$	25.8146	1.22649	0.613245	−	0.789893i	$-0.289864\pi$
$443$	25.8146	1.22649	0.613245	+	0.789893i	$0.289864\pi$
$444$	−4.81163	−0.228350
$445$	13.4330	0.636783
$446$	8.59179	0.406833
$447$	−1.12067	−0.0530059
$448$	−0.890084	−0.0420525
$449$	11.9782	0.565288	0.282644	−	0.959225i	$-0.408789\pi$
$449$	11.9782	0.565288	0.282644	+	0.959225i	$0.408789\pi$
$450$	2.80194	0.132085
$451$	12.1626	0.572712
$452$	0.225209	0.0105930
$453$	−10.1957	−0.479034
$454$	−19.8485	−0.931534
$455$	0	0
$456$	−0.307979	−0.0144224
$457$	8.32842	0.389587	0.194794	−	0.980844i	$-0.437596\pi$
$457$	8.32842	0.389587	0.194794	+	0.980844i	$0.437596\pi$
$458$	21.6582	1.01202
$459$	−5.51812	−0.257564
$460$	−0.396125	−0.0184694
$461$	−3.94571	−0.183770	−0.0918849	−	0.995770i	$-0.529289\pi$
$461$	−3.94571	−0.183770	−0.0918849	+	0.995770i	$0.529289\pi$
$462$	−1.09054	−0.0507365
$463$	16.5918	0.771086	0.385543	−	0.922690i	$-0.374014\pi$
$463$	16.5918	0.771086	0.385543	+	0.922690i	$0.374014\pi$
$464$	0.493959	0.0229315
$465$	−3.50604	−0.162589
$466$	1.75063	0.0810963
$467$	−34.5187	−1.59734	−0.798668	−	0.601772i	$-0.794462\pi$
$467$	−34.5187	−1.59734	−0.798668	+	0.601772i	$0.794462\pi$
$468$	0	0
$469$	4.43488	0.204784
$470$	−0.176292	−0.00813174
$471$	4.79225	0.220815
$472$	−6.98254	−0.321398
$473$	−33.0297	−1.51871
$474$	4.31767	0.198317
$475$	−0.692021	−0.0317521
$476$	1.90217	0.0871856
$477$	−26.0737	−1.19383
$478$	12.8224	0.586482
$479$	3.96508	0.181169	0.0905846	−	0.995889i	$-0.471126\pi$
$479$	3.96508	0.181169	0.0905846	+	0.995889i	$0.471126\pi$
$480$	0.445042	0.0203133
$481$	0	0
$482$	20.6112	0.938813
$483$	−0.156915	−0.00713986
$484$	−3.42088	−0.155494
$485$	8.35690	0.379467
$486$	10.9758	0.497874
$487$	7.18837	0.325736	0.162868	−	0.986648i	$-0.447925\pi$
$487$	7.18837	0.325736	0.162868	+	0.986648i	$0.447925\pi$
$488$	−9.30559	−0.421244
$489$	−1.07308	−0.0485264
$490$	6.20775	0.280438
$491$	0.0295400	0.00133312	0.000666560	−	1.00000i	$-0.499788\pi$
$491$	0.0295400	0.00133312	0.000666560		1.00000i	$0.499788\pi$
$492$	−1.96615	−0.0886408
$493$	−1.05562	−0.0475428
$494$	0	0
$495$	−7.71379	−0.346709
$496$	7.87800	0.353733
$497$	−4.15691	−0.186463
$498$	1.32006	0.0591533
$499$	9.88231	0.442393	0.221197	−	0.975229i	$-0.429004\pi$
$499$	9.88231	0.442393	0.221197	+	0.975229i	$0.429004\pi$
$500$	1.00000	0.0447214
$501$	−9.35988	−0.418169
$502$	−28.4547	−1.27000
$503$	13.8345	0.616848	0.308424	−	0.951249i	$-0.400198\pi$
$503$	13.8345	0.616848	0.308424	+	0.951249i	$0.400198\pi$
$504$	−2.49396	−0.111090
$505$	−16.7875	−0.747032
$506$	1.09054	0.0484804
$507$	0	0
$508$	10.6896	0.474276
$509$	−3.34913	−0.148447	−0.0742237	−	0.997242i	$-0.523648\pi$
$509$	−3.34913	−0.148447	−0.0742237	+	0.997242i	$0.523648\pi$
$510$	−0.951083	−0.0421146
$511$	9.47112	0.418978
$512$	−1.00000	−0.0441942
$513$	−1.78687	−0.0788923
$514$	−10.4480	−0.460843
$515$	11.8780	0.523407
$516$	5.33944	0.235056
$517$	0.485335	0.0213450
$518$	9.62325	0.422821
$519$	2.68963	0.118062
$520$	0	0
$521$	−7.75063	−0.339561	−0.169781	−	0.985482i	$-0.554306\pi$
$521$	−7.75063	−0.339561	−0.169781	+	0.985482i	$0.554306\pi$
$522$	1.38404	0.0605779
$523$	−19.3207	−0.844833	−0.422417	−	0.906402i	$-0.638818\pi$
$523$	−19.3207	−0.844833	−0.422417	+	0.906402i	$0.638818\pi$
$524$	10.5157	0.459382
$525$	0.396125	0.0172883
$526$	6.43967	0.280783
$527$	−16.8358	−0.733379
$528$	−1.22521	−0.0533204
$529$	−22.8431	−0.993178
$530$	−9.30559	−0.404209
$531$	−19.5646	−0.849033
$532$	0.615957	0.0267051
$533$	0	0
$534$	5.97823	0.258703
$535$	−13.3817	−0.578539
$536$	4.98254	0.215213
$537$	−7.12737	−0.307569
$538$	−10.2935	−0.443784
$539$	−17.0901	−0.736121
$540$	2.58211	0.111116
$541$	9.83446	0.422817	0.211408	−	0.977398i	$-0.432195\pi$
$541$	9.83446	0.422817	0.211408	+	0.977398i	$0.432195\pi$
$542$	31.6340	1.35880
$543$	−4.37675	−0.187824
$544$	2.13706	0.0916258
$545$	−7.32975	−0.313972
$546$	0	0
$547$	−11.3980	−0.487345	−0.243673	−	0.969858i	$-0.578352\pi$
$547$	−11.3980	−0.487345	−0.243673	+	0.969858i	$0.578352\pi$
$548$	6.44504	0.275319
$549$	−26.0737	−1.11280
$550$	−2.75302	−0.117389
$551$	−0.341830	−0.0145625
$552$	−0.176292	−0.00750349
$553$	−8.63533	−0.367212
$554$	21.1099	0.896874
$555$	−4.81163	−0.204242
$556$	13.5211	0.573423
$557$	38.6762	1.63876	0.819382	−	0.573248i	$-0.194317\pi$
$557$	38.6762	1.63876	0.819382	+	0.573248i	$0.194317\pi$
$558$	22.0737	0.934453
$559$	0	0
$560$	−0.890084	−0.0376129
$561$	2.61835	0.110547
$562$	−17.6082	−0.742757
$563$	19.4034	0.817757	0.408878	−	0.912589i	$-0.365920\pi$
$563$	19.4034	0.817757	0.408878	+	0.912589i	$0.365920\pi$
$564$	−0.0784573	−0.00330365
$565$	0.225209	0.00947463
$566$	−18.4131	−0.773961
$567$	−6.45904	−0.271254
$568$	−4.67025	−0.195959
$569$	−30.0790	−1.26098	−0.630490	−	0.776198i	$-0.717146\pi$
$569$	−30.0790	−1.26098	−0.630490	+	0.776198i	$0.717146\pi$
$570$	−0.307979	−0.0128998
$571$	9.56571	0.400313	0.200156	−	0.979764i	$-0.435855\pi$
$571$	9.56571	0.400313	0.200156	+	0.979764i	$0.435855\pi$
$572$	0	0
$573$	−11.1207	−0.464573
$574$	3.93230	0.164131
$575$	−0.396125	−0.0165195
$576$	−2.80194	−0.116747
$577$	19.6752	0.819087	0.409544	−	0.912291i	$-0.365688\pi$
$577$	19.6752	0.819087	0.409544	+	0.912291i	$0.365688\pi$
$578$	12.4330	0.517143
$579$	7.19328	0.298942
$580$	0.493959	0.0205105
$581$	−2.64012	−0.109531
$582$	3.71917	0.154164
$583$	25.6185	1.06101
$584$	10.6407	0.440316
$585$	0	0
$586$	0.0241632	0.000998171 0
$587$	−40.6980	−1.67979	−0.839893	−	0.542752i	$-0.817382\pi$
$587$	−40.6980	−1.67979	−0.839893	+	0.542752i	$0.817382\pi$
$588$	2.76271	0.113932
$589$	−5.45175	−0.224635
$590$	−6.98254	−0.287467
$591$	−5.14483	−0.211630
$592$	10.8116	0.444355
$593$	12.4741	0.512250	0.256125	−	0.966644i	$-0.417554\pi$
$593$	12.4741	0.512250	0.256125	+	0.966644i	$0.417554\pi$
$594$	−7.10859	−0.291669
$595$	1.90217	0.0779812
$596$	2.51812	0.103146
$597$	1.74525	0.0714284
$598$	0	0
$599$	−12.8358	−0.524456	−0.262228	−	0.965006i	$-0.584457\pi$
$599$	−12.8358	−0.524456	−0.262228	+	0.965006i	$0.584457\pi$
$600$	0.445042	0.0181688
$601$	20.8092	0.848826	0.424413	−	0.905469i	$-0.360480\pi$
$601$	20.8092	0.848826	0.424413	+	0.905469i	$0.360480\pi$
$602$	−10.6789	−0.435238
$603$	13.9608	0.568527
$604$	22.9095	0.932173
$605$	−3.42088	−0.139078
$606$	−7.47112	−0.303494
$607$	−28.4805	−1.15599	−0.577995	−	0.816040i	$-0.696165\pi$
$607$	−28.4805	−1.15599	−0.577995	+	0.816040i	$0.696165\pi$
$608$	0.692021	0.0280652
$609$	0.195669	0.00792892
$610$	−9.30559	−0.376772
$611$	0	0
$612$	5.98792	0.242047
$613$	14.1220	0.570382	0.285191	−	0.958471i	$-0.407943\pi$
$613$	14.1220	0.570382	0.285191	+	0.958471i	$0.407943\pi$
$614$	14.1371	0.570525
$615$	−1.96615	−0.0792827
$616$	2.45042	0.0987302
$617$	47.0911	1.89582	0.947909	−	0.318542i	$-0.103193\pi$
$617$	47.0911	1.89582	0.947909	+	0.318542i	$0.103193\pi$
$618$	5.28621	0.212642
$619$	−22.8068	−0.916684	−0.458342	−	0.888776i	$-0.651557\pi$
$619$	−22.8068	−0.916684	−0.458342	+	0.888776i	$0.651557\pi$
$620$	7.87800	0.316388
$621$	−1.02284	−0.0410450
$622$	4.09783	0.164308
$623$	−11.9565	−0.479025
$624$	0	0
$625$	1.00000	0.0400000
$626$	16.9051	0.675666
$627$	0.847871	0.0338607
$628$	−10.7681	−0.429693
$629$	−23.1051	−0.921262
$630$	−2.49396	−0.0993617
$631$	−22.3370	−0.889224	−0.444612	−	0.895723i	$-0.646658\pi$
$631$	−22.3370	−0.889224	−0.444612	+	0.895723i	$0.646658\pi$
$632$	−9.70171	−0.385913
$633$	−7.64071	−0.303691
$634$	27.1594	1.07864
$635$	10.6896	0.424205
$636$	−4.14138	−0.164216
$637$	0	0
$638$	−1.35988	−0.0538382
$639$	−13.0858	−0.517664
$640$	−1.00000	−0.0395285
$641$	−11.0291	−0.435622	−0.217811	−	0.975991i	$-0.569892\pi$
$641$	−11.0291	−0.435622	−0.217811	+	0.975991i	$0.569892\pi$
$642$	−5.95539	−0.235041
$643$	−5.42519	−0.213949	−0.106974	−	0.994262i	$-0.534116\pi$
$643$	−5.42519	−0.213949	−0.106974	+	0.994262i	$0.534116\pi$
$644$	0.352584	0.0138938
$645$	5.33944	0.210240
$646$	−1.47889	−0.0581863
$647$	4.42626	0.174014	0.0870070	−	0.996208i	$-0.472270\pi$
$647$	4.42626	0.174014	0.0870070	+	0.996208i	$0.472270\pi$
$648$	−7.25667	−0.285069
$649$	19.2231	0.754572
$650$	0	0
$651$	3.12067	0.122309
$652$	2.41119	0.0944295
$653$	−2.78150	−0.108848	−0.0544242	−	0.998518i	$-0.517332\pi$
$653$	−2.78150	−0.108848	−0.0544242	+	0.998518i	$0.517332\pi$
$654$	−3.26205	−0.127556
$655$	10.5157	0.410884
$656$	4.41789	0.172490
$657$	29.8146	1.16318
$658$	0.156915	0.00611717
$659$	7.97046	0.310485	0.155243	−	0.987876i	$-0.450384\pi$
$659$	7.97046	0.310485	0.155243	+	0.987876i	$0.450384\pi$
$660$	−1.22521	−0.0476912
$661$	−7.34913	−0.285848	−0.142924	−	0.989734i	$-0.545650\pi$
$661$	−7.34913	−0.285848	−0.142924	+	0.989734i	$0.545650\pi$
$662$	13.8672	0.538966
$663$	0	0
$664$	−2.96615	−0.115109
$665$	0.615957	0.0238858
$666$	30.2935	1.17385
$667$	−0.195669	−0.00757635
$668$	21.0315	0.813732
$669$	3.82371	0.147833
$670$	4.98254	0.192492
$671$	25.6185	0.988990
$672$	−0.396125	−0.0152808
$673$	−33.4088	−1.28781	−0.643907	−	0.765104i	$-0.722688\pi$
$673$	−33.4088	−1.28781	−0.643907	+	0.765104i	$0.722688\pi$
$674$	33.9433	1.30745
$675$	2.58211	0.0993853
$676$	0	0
$677$	−9.25129	−0.355556	−0.177778	−	0.984071i	$-0.556891\pi$
$677$	−9.25129	−0.355556	−0.177778	+	0.984071i	$0.556891\pi$
$678$	0.100228	0.00384922
$679$	−7.43834	−0.285457
$680$	2.13706	0.0819526
$681$	−8.83340	−0.338496
$682$	−21.6883	−0.830488
$683$	46.1041	1.76412	0.882062	−	0.471134i	$-0.156155\pi$
$683$	46.1041	1.76412	0.882062	+	0.471134i	$0.156155\pi$
$684$	1.93900	0.0741396
$685$	6.44504	0.246252
$686$	−11.7560	−0.448846
$687$	9.63879	0.367743
$688$	−11.9976	−0.457404
$689$	0	0
$690$	−0.176292	−0.00671132
$691$	−34.1631	−1.29963	−0.649814	−	0.760094i	$-0.725153\pi$
$691$	−34.1631	−1.29963	−0.649814	+	0.760094i	$0.725153\pi$
$692$	−6.04354	−0.229741
$693$	6.86592	0.260815
$694$	24.2828	0.921761
$695$	13.5211	0.512885
$696$	0.219833	0.00833273
$697$	−9.44132	−0.357616
$698$	31.9517	1.20939
$699$	0.779103	0.0294684
$700$	−0.890084	−0.0336420
$701$	23.9409	0.904236	0.452118	−	0.891958i	$-0.350669\pi$
$701$	23.9409	0.904236	0.452118	+	0.891958i	$0.350669\pi$
$702$	0	0
$703$	−7.48188	−0.282184
$704$	2.75302	0.103758
$705$	−0.0784573	−0.00295487
$706$	−30.8025	−1.15927
$707$	14.9422	0.561961
$708$	−3.10752	−0.116788
$709$	24.0435	0.902974	0.451487	−	0.892278i	$-0.350894\pi$
$709$	24.0435	0.902974	0.451487	+	0.892278i	$0.350894\pi$
$710$	−4.67025	−0.175271
$711$	−27.1836	−1.01946
$712$	−13.4330	−0.503421
$713$	−3.12067	−0.116870
$714$	0.846543	0.0316811
$715$	0	0
$716$	16.0151	0.598511
$717$	5.70650	0.213113
$718$	−21.9517	−0.819229
$719$	−11.3840	−0.424553	−0.212277	−	0.977210i	$-0.568088\pi$
$719$	−11.3840	−0.424553	−0.212277	+	0.977210i	$0.568088\pi$
$720$	−2.80194	−0.104422
$721$	−10.5724	−0.393737
$722$	18.5211	0.689284
$723$	9.17283	0.341141
$724$	9.83446	0.365495
$725$	0.493959	0.0183452
$726$	−1.52243	−0.0565028
$727$	22.3478	0.828834	0.414417	−	0.910087i	$-0.363986\pi$
$727$	22.3478	0.828834	0.414417	+	0.910087i	$0.363986\pi$
$728$	0	0
$729$	−16.8853	−0.625381
$730$	10.6407	0.393830
$731$	25.6396	0.948317
$732$	−4.14138	−0.153070
$733$	−22.0978	−0.816202	−0.408101	−	0.912937i	$-0.633809\pi$
$733$	−22.0978	−0.816202	−0.408101	+	0.912937i	$0.633809\pi$
$734$	8.61596	0.318021
$735$	2.76271	0.101904
$736$	0.396125	0.0146013
$737$	−13.7170	−0.505274
$738$	12.3787	0.455665
$739$	−22.4838	−0.827080	−0.413540	−	0.910486i	$-0.635708\pi$
$739$	−22.4838	−0.827080	−0.413540	+	0.910486i	$0.635708\pi$
$740$	10.8116	0.397443
$741$	0	0
$742$	8.28275	0.304069
$743$	−47.2331	−1.73281	−0.866407	−	0.499338i	$-0.833577\pi$
$743$	−47.2331	−1.73281	−0.866407	+	0.499338i	$0.833577\pi$
$744$	3.50604	0.128538
$745$	2.51812	0.0922569
$746$	−6.93362	−0.253858
$747$	−8.31096	−0.304082
$748$	−5.88338	−0.215118
$749$	11.9108	0.435211
$750$	0.445042	0.0162506
$751$	12.8552	0.469092	0.234546	−	0.972105i	$-0.424640\pi$
$751$	12.8552	0.469092	0.234546	+	0.972105i	$0.424640\pi$
$752$	0.176292	0.00642871
$753$	−12.6635	−0.461485
$754$	0	0
$755$	22.9095	0.833761
$756$	−2.29829	−0.0835880
$757$	37.6668	1.36902	0.684511	−	0.729002i	$-0.260016\pi$
$757$	37.6668	1.36902	0.684511	+	0.729002i	$0.260016\pi$
$758$	10.2494	0.372274
$759$	0.485335	0.0176166
$760$	0.692021	0.0251023
$761$	53.9211	1.95464	0.977319	−	0.211774i	$-0.0679242\pi$
$761$	53.9211	1.95464	0.977319	+	0.211774i	$0.0679242\pi$
$762$	4.75733	0.172340
$763$	6.52409	0.236188
$764$	24.9879	0.904031
$765$	5.98792	0.216494
$766$	27.2814	0.985718
$767$	0	0
$768$	−0.445042	−0.0160591
$769$	5.52973	0.199407	0.0997036	−	0.995017i	$-0.468211\pi$
$769$	5.52973	0.199407	0.0997036	+	0.995017i	$0.468211\pi$
$770$	2.45042	0.0883070
$771$	−4.64981	−0.167459
$772$	−16.1631	−0.581724
$773$	34.1473	1.22819	0.614097	−	0.789230i	$-0.289520\pi$
$773$	34.1473	1.22819	0.614097	+	0.789230i	$0.289520\pi$
$774$	−33.6165	−1.20832
$775$	7.87800	0.282986
$776$	−8.35690	−0.299995
$777$	4.28275	0.153643
$778$	32.5133	1.16566
$779$	−3.05728	−0.109538
$780$	0	0
$781$	12.8573	0.460070
$782$	−0.846543	−0.0302723
$783$	1.27545	0.0455810
$784$	−6.20775	−0.221705
$785$	−10.7681	−0.384329
$786$	4.67994	0.166928
$787$	6.57673	0.234435	0.117218	−	0.993106i	$-0.462603\pi$
$787$	6.57673	0.234435	0.117218	+	0.993106i	$0.462603\pi$
$788$	11.5603	0.411820
$789$	2.86592	0.102029
$790$	−9.70171	−0.345171
$791$	−0.200455	−0.00712737
$792$	7.71379	0.274098
$793$	0	0
$794$	−3.86725	−0.137244
$795$	−4.14138	−0.146879
$796$	−3.92154	−0.138995
$797$	30.8224	1.09178	0.545892	−	0.837855i	$-0.316191\pi$
$797$	30.8224	1.09178	0.545892	+	0.837855i	$0.316191\pi$
$798$	0.274127	0.00970398
$799$	−0.376747	−0.0133284
$800$	−1.00000	−0.0353553
$801$	−37.6383	−1.32988
$802$	13.7168	0.484356
$803$	−29.2941	−1.03377
$804$	2.21744	0.0782031
$805$	0.352584	0.0124270
$806$	0	0
$807$	−4.58104	−0.161260
$808$	16.7875	0.590581
$809$	8.32603	0.292728	0.146364	−	0.989231i	$-0.453243\pi$
$809$	8.32603	0.292728	0.146364	+	0.989231i	$0.453243\pi$
$810$	−7.25667	−0.254973
$811$	41.7469	1.46593	0.732966	−	0.680265i	$-0.238135\pi$
$811$	41.7469	1.46593	0.732966	+	0.680265i	$0.238135\pi$
$812$	−0.439665	−0.0154292
$813$	14.0785	0.493753
$814$	−29.7646	−1.04325
$815$	2.41119	0.0844603
$816$	0.951083	0.0332946
$817$	8.30260	0.290471
$818$	1.67696	0.0586334
$819$	0	0
$820$	4.41789	0.154280
$821$	−26.5327	−0.925998	−0.462999	−	0.886359i	$-0.653227\pi$
$821$	−26.5327	−0.925998	−0.462999	+	0.886359i	$0.653227\pi$
$822$	2.86831	0.100044
$823$	−34.7827	−1.21245	−0.606224	−	0.795294i	$-0.707316\pi$
$823$	−34.7827	−1.21245	−0.606224	+	0.795294i	$0.707316\pi$
$824$	−11.8780	−0.413790
$825$	−1.22521	−0.0426563
$826$	6.21505	0.216249
$827$	−4.19029	−0.145711	−0.0728554	−	0.997343i	$-0.523211\pi$
$827$	−4.19029	−0.145711	−0.0728554	+	0.997343i	$0.523211\pi$
$828$	1.10992	0.0385723
$829$	−38.5676	−1.33951	−0.669755	−	0.742583i	$-0.733601\pi$
$829$	−38.5676	−1.33951	−0.669755	+	0.742583i	$0.733601\pi$
$830$	−2.96615	−0.102957
$831$	9.39480	0.325902
$832$	0	0
$833$	13.2664	0.459652
$834$	6.01746	0.208368
$835$	21.0315	0.727824
$836$	−1.90515	−0.0658910
$837$	20.3418	0.703117
$838$	26.6112	0.919268
$839$	16.7439	0.578064	0.289032	−	0.957319i	$-0.406667\pi$
$839$	16.7439	0.578064	0.289032	+	0.957319i	$0.406667\pi$
$840$	−0.396125	−0.0136676
$841$	−28.7560	−0.991586
$842$	−24.8418	−0.856103
$843$	−7.83638	−0.269899
$844$	17.1685	0.590965
$845$	0	0
$846$	0.493959	0.0169827
$847$	3.04487	0.104623
$848$	9.30559	0.319555
$849$	−8.19460	−0.281238
$850$	2.13706	0.0733007
$851$	−4.28275	−0.146811
$852$	−2.07846	−0.0712068
$853$	27.0422	0.925908	0.462954	−	0.886382i	$-0.346790\pi$
$853$	27.0422	0.925908	0.462954	+	0.886382i	$0.346790\pi$
$854$	8.28275	0.283430
$855$	1.93900	0.0663124
$856$	13.3817	0.457375
$857$	28.2301	0.964322	0.482161	−	0.876083i	$-0.339852\pi$
$857$	28.2301	0.964322	0.482161	+	0.876083i	$0.339852\pi$
$858$	0	0
$859$	53.1648	1.81396	0.906980	−	0.421174i	$-0.138382\pi$
$859$	53.1648	1.81396	0.906980	+	0.421174i	$0.138382\pi$
$860$	−11.9976	−0.409115
$861$	1.75004	0.0596411
$862$	−12.4155	−0.422874
$863$	−17.9711	−0.611742	−0.305871	−	0.952073i	$-0.598948\pi$
$863$	−17.9711	−0.611742	−0.305871	+	0.952073i	$0.598948\pi$
$864$	−2.58211	−0.0878450
$865$	−6.04354	−0.205487
$866$	−5.31096	−0.180474
$867$	5.53319	0.187917
$868$	−7.01208	−0.238006
$869$	26.7090	0.906041
$870$	0.219833	0.00745302
$871$	0	0
$872$	7.32975	0.248217
$873$	−23.4155	−0.792495
$874$	−0.274127	−0.00927247
$875$	−0.890084	−0.0300903
$876$	4.73556	0.160000
$877$	17.1185	0.578052	0.289026	−	0.957321i	$-0.406669\pi$
$877$	17.1185	0.578052	0.289026	+	0.957321i	$0.406669\pi$
$878$	−7.15346	−0.241417
$879$	0.0107536	0.000362711 0
$880$	2.75302	0.0928043
$881$	−46.8133	−1.57718	−0.788590	−	0.614920i	$-0.789188\pi$
$881$	−46.8133	−1.57718	−0.788590	+	0.614920i	$0.789188\pi$
$882$	−17.3937	−0.585677
$883$	40.0344	1.34727	0.673633	−	0.739066i	$-0.264733\pi$
$883$	40.0344	1.34727	0.673633	+	0.739066i	$0.264733\pi$
$884$	0	0
$885$	−3.10752	−0.104458
$886$	−25.8146	−0.867259
$887$	−41.3793	−1.38938	−0.694690	−	0.719310i	$-0.744458\pi$
$887$	−41.3793	−1.38938	−0.694690	+	0.719310i	$0.744458\pi$
$888$	4.81163	0.161468
$889$	−9.51466	−0.319112
$890$	−13.4330	−0.450274
$891$	19.9778	0.669280
$892$	−8.59179	−0.287675
$893$	−0.121998	−0.00408250
$894$	1.12067	0.0374808
$895$	16.0151	0.535325
$896$	0.890084	0.0297356
$897$	0	0
$898$	−11.9782	−0.399719
$899$	3.89141	0.129786
$900$	−2.80194	−0.0933979
$901$	−19.8866	−0.662520
$902$	−12.1626	−0.404969
$903$	−4.75255	−0.158155
$904$	−0.225209	−0.00749035
$905$	9.83446	0.326909
$906$	10.1957	0.338728
$907$	30.2532	1.00454	0.502271	−	0.864710i	$-0.332498\pi$
$907$	30.2532	1.00454	0.502271	+	0.864710i	$0.332498\pi$
$908$	19.8485	0.658694
$909$	47.0374	1.56013
$910$	0	0
$911$	−35.8538	−1.18789	−0.593945	−	0.804505i	$-0.702430\pi$
$911$	−35.8538	−1.18789	−0.593945	+	0.804505i	$0.702430\pi$
$912$	0.307979	0.0101982
$913$	8.16587	0.270251
$914$	−8.32842	−0.275480
$915$	−4.14138	−0.136910
$916$	−21.6582	−0.715606
$917$	−9.35988	−0.309090
$918$	5.51812	0.182125
$919$	10.6025	0.349746	0.174873	−	0.984591i	$-0.444049\pi$
$919$	10.6025	0.349746	0.174873	+	0.984591i	$0.444049\pi$
$920$	0.396125	0.0130598
$921$	6.29159	0.207315
$922$	3.94571	0.129945
$923$	0	0
$924$	1.09054	0.0358761
$925$	10.8116	0.355484
$926$	−16.5918	−0.545240
$927$	−33.2814	−1.09311
$928$	−0.493959	−0.0162150
$929$	−35.7294	−1.17225	−0.586123	−	0.810222i	$-0.699346\pi$
$929$	−35.7294	−1.17225	−0.586123	+	0.810222i	$0.699346\pi$
$930$	3.50604	0.114968
$931$	4.29590	0.140792
$932$	−1.75063	−0.0573437
$933$	1.82371	0.0597055
$934$	34.5187	1.12949
$935$	−5.88338	−0.192407
$936$	0	0
$937$	−5.39506	−0.176249	−0.0881245	−	0.996109i	$-0.528087\pi$
$937$	−5.39506	−0.176249	−0.0881245	+	0.996109i	$0.528087\pi$
$938$	−4.43488	−0.144804
$939$	7.52350	0.245520
$940$	0.176292	0.00575001
$941$	5.19700	0.169417	0.0847086	−	0.996406i	$-0.473004\pi$
$941$	5.19700	0.169417	0.0847086	+	0.996406i	$0.473004\pi$
$942$	−4.79225	−0.156140
$943$	−1.75004	−0.0569890
$944$	6.98254	0.227262
$945$	−2.29829	−0.0747634
$946$	33.0297	1.07389
$947$	−5.58211	−0.181394	−0.0906970	−	0.995879i	$-0.528909\pi$
$947$	−5.58211	−0.181394	−0.0906970	+	0.995879i	$0.528909\pi$
$948$	−4.31767	−0.140231
$949$	0	0
$950$	0.692021	0.0224521
$951$	12.0871	0.391950
$952$	−1.90217	−0.0616495
$953$	18.4916	0.599001	0.299500	−	0.954096i	$-0.403180\pi$
$953$	18.4916	0.599001	0.299500	+	0.954096i	$0.403180\pi$
$954$	26.0737	0.844166
$955$	24.9879	0.808590
$956$	−12.8224	−0.414705
$957$	−0.605203	−0.0195634
$958$	−3.96508	−0.128106
$959$	−5.73663	−0.185245
$960$	−0.445042	−0.0143637
$961$	31.0629	1.00203
$962$	0	0
$963$	37.4946	1.20824
$964$	−20.6112	−0.663841
$965$	−16.1631	−0.520310
$966$	0.156915	0.00504865
$967$	41.4819	1.33397	0.666984	−	0.745072i	$-0.267585\pi$
$967$	41.4819	1.33397	0.666984	+	0.745072i	$0.267585\pi$
$968$	3.42088	0.109951
$969$	−0.658170	−0.0211435
$970$	−8.35690	−0.268324
$971$	20.4916	0.657606	0.328803	−	0.944399i	$-0.393355\pi$
$971$	20.4916	0.657606	0.328803	+	0.944399i	$0.393355\pi$
$972$	−10.9758	−0.352050
$973$	−12.0349	−0.385822
$974$	−7.18837	−0.230330
$975$	0	0
$976$	9.30559	0.297865
$977$	−40.3588	−1.29119	−0.645596	−	0.763679i	$-0.723391\pi$
$977$	−40.3588	−1.29119	−0.645596	+	0.763679i	$0.723391\pi$
$978$	1.07308	0.0343133
$979$	36.9812	1.18192
$980$	−6.20775	−0.198299
$981$	20.5375	0.655712
$982$	−0.0295400	−0.000942659 0
$983$	31.0422	0.990093	0.495046	−	0.868867i	$-0.335151\pi$
$983$	31.0422	0.990093	0.495046	+	0.868867i	$0.335151\pi$
$984$	1.96615	0.0626785
$985$	11.5603	0.368343
$986$	1.05562	0.0336179
$987$	0.0698336	0.00222283
$988$	0	0
$989$	4.75255	0.151122
$990$	7.71379	0.245160
$991$	−57.5206	−1.82720	−0.913602	−	0.406611i	$-0.866711\pi$
$991$	−57.5206	−1.82720	−0.913602	+	0.406611i	$0.866711\pi$
$992$	−7.87800	−0.250127
$993$	6.17151	0.195847
$994$	4.15691	0.131849
$995$	−3.92154	−0.124321
$996$	−1.32006	−0.0418277
$997$	−33.2486	−1.05299	−0.526497	−	0.850177i	$-0.676495\pi$
$997$	−33.2486	−1.05299	−0.526497	+	0.850177i	$0.676495\pi$
$998$	−9.88231	−0.312819
$999$	27.9168	0.883247

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.a.p.1.2	✓	3
5.4	even	2		8450.2.a.cg.1.2		3
13.2	odd	12		1690.2.l.m.1161.2		12
13.3	even	3		1690.2.e.r.191.2		6
13.4	even	6		1690.2.e.p.991.2		6
13.5	odd	4		1690.2.d.i.1351.5		6
13.6	odd	12		1690.2.l.m.361.5		12
13.7	odd	12		1690.2.l.m.361.2		12
13.8	odd	4		1690.2.d.i.1351.2		6
13.9	even	3		1690.2.e.r.991.2		6
13.10	even	6		1690.2.e.p.191.2		6
13.11	odd	12		1690.2.l.m.1161.5		12
13.12	even	2		1690.2.a.r.1.2	yes	3
65.64	even	2		8450.2.a.bv.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.a.p.1.2	✓	3	1.1	even	1	trivial
1690.2.a.r.1.2	yes	3	13.12	even	2
1690.2.d.i.1351.2		6	13.8	odd	4
1690.2.d.i.1351.5		6	13.5	odd	4
1690.2.e.p.191.2		6	13.10	even	6
1690.2.e.p.991.2		6	13.4	even	6
1690.2.e.r.191.2		6	13.3	even	3
1690.2.e.r.991.2		6	13.9	even	3
1690.2.l.m.361.2		12	13.7	odd	12
1690.2.l.m.361.5		12	13.6	odd	12
1690.2.l.m.1161.2		12	13.2	odd	12
1690.2.l.m.1161.5		12	13.11	odd	12
8450.2.a.bv.1.2		3	65.64	even	2
8450.2.a.cg.1.2		3	5.4	even	2

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:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.445042 q^{6} -0.890084 q^{7} -1.00000 q^{8} -2.80194 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.445042 q^{6} -0.890084 q^{7} -1.00000 q^{8} -2.80194 q^{9} -1.00000 q^{10} +2.75302 q^{11} -0.445042 q^{12} +0.890084 q^{14} -0.445042 q^{15} +1.00000 q^{16} -2.13706 q^{17} +2.80194 q^{18} -0.692021 q^{19} +1.00000 q^{20} +0.396125 q^{21} -2.75302 q^{22} -0.396125 q^{23} +0.445042 q^{24} +1.00000 q^{25} +2.58211 q^{27} -0.890084 q^{28} +0.493959 q^{29} +0.445042 q^{30} +7.87800 q^{31} -1.00000 q^{32} -1.22521 q^{33} +2.13706 q^{34} -0.890084 q^{35} -2.80194 q^{36} +10.8116 q^{37} +0.692021 q^{38} -1.00000 q^{40} +4.41789 q^{41} -0.396125 q^{42} -11.9976 q^{43} +2.75302 q^{44} -2.80194 q^{45} +0.396125 q^{46} +0.176292 q^{47} -0.445042 q^{48} -6.20775 q^{49} -1.00000 q^{50} +0.951083 q^{51} +9.30559 q^{53} -2.58211 q^{54} +2.75302 q^{55} +0.890084 q^{56} +0.307979 q^{57} -0.493959 q^{58} +6.98254 q^{59} -0.445042 q^{60} +9.30559 q^{61} -7.87800 q^{62} +2.49396 q^{63} +1.00000 q^{64} +1.22521 q^{66} -4.98254 q^{67} -2.13706 q^{68} +0.176292 q^{69} +0.890084 q^{70} +4.67025 q^{71} +2.80194 q^{72} -10.6407 q^{73} -10.8116 q^{74} -0.445042 q^{75} -0.692021 q^{76} -2.45042 q^{77} +9.70171 q^{79} +1.00000 q^{80} +7.25667 q^{81} -4.41789 q^{82} +2.96615 q^{83} +0.396125 q^{84} -2.13706 q^{85} +11.9976 q^{86} -0.219833 q^{87} -2.75302 q^{88} +13.4330 q^{89} +2.80194 q^{90} -0.396125 q^{92} -3.50604 q^{93} -0.176292 q^{94} -0.692021 q^{95} +0.445042 q^{96} +8.35690 q^{97} +6.20775 q^{98} -7.71379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 2 q^{7} - 3 q^{8} - 4 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - q^3 + 3 * q^4 + 3 * q^5 + q^6 - 2 * q^7 - 3 * q^8 - 4 * q^9	\( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 2 q^{7} - 3 q^{8} - 4 q^{9} - 3 q^{10} + 13 q^{11} - q^{12} + 2 q^{14} - q^{15} + 3 q^{16} - q^{17} + 4 q^{18} + 3 q^{19} + 3 q^{20} + 10 q^{21} - 13 q^{22} - 10 q^{23} + q^{24} + 3 q^{25} + 2 q^{27} - 2 q^{28} - 8 q^{29} + q^{30} + 4 q^{31} - 3 q^{32} - 2 q^{33} + q^{34} - 2 q^{35} - 4 q^{36} + 6 q^{37} - 3 q^{38} - 3 q^{40} + 19 q^{41} - 10 q^{42} + 5 q^{43} + 13 q^{44} - 4 q^{45} + 10 q^{46} + 8 q^{47} - q^{48} - q^{49} - 3 q^{50} + 12 q^{51} - 8 q^{53} - 2 q^{54} + 13 q^{55} + 2 q^{56} + 6 q^{57} + 8 q^{58} + 5 q^{59} - q^{60} - 8 q^{61} - 4 q^{62} - 2 q^{63} + 3 q^{64} + 2 q^{66} + q^{67} - q^{68} + 8 q^{69} + 2 q^{70} + 12 q^{71} + 4 q^{72} + 5 q^{73} - 6 q^{74} - q^{75} + 3 q^{76} - 4 q^{77} + 2 q^{79} + 3 q^{80} - 5 q^{81} - 19 q^{82} - 7 q^{83} + 10 q^{84} - q^{85} - 5 q^{86} - 2 q^{87} - 13 q^{88} + 21 q^{89} + 4 q^{90} - 10 q^{92} - 20 q^{93} - 8 q^{94} + 3 q^{95} + q^{96} + 21 q^{97} + q^{98} - 15 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - q^3 + 3 * q^4 + 3 * q^5 + q^6 - 2 * q^7 - 3 * q^8 - 4 * q^9 - 3 * q^10 + 13 * q^11 - q^12 + 2 * q^14 - q^15 + 3 * q^16 - q^17 + 4 * q^18 + 3 * q^19 + 3 * q^20 + 10 * q^21 - 13 * q^22 - 10 * q^23 + q^24 + 3 * q^25 + 2 * q^27 - 2 * q^28 - 8 * q^29 + q^30 + 4 * q^31 - 3 * q^32 - 2 * q^33 + q^34 - 2 * q^35 - 4 * q^36 + 6 * q^37 - 3 * q^38 - 3 * q^40 + 19 * q^41 - 10 * q^42 + 5 * q^43 + 13 * q^44 - 4 * q^45 + 10 * q^46 + 8 * q^47 - q^48 - q^49 - 3 * q^50 + 12 * q^51 - 8 * q^53 - 2 * q^54 + 13 * q^55 + 2 * q^56 + 6 * q^57 + 8 * q^58 + 5 * q^59 - q^60 - 8 * q^61 - 4 * q^62 - 2 * q^63 + 3 * q^64 + 2 * q^66 + q^67 - q^68 + 8 * q^69 + 2 * q^70 + 12 * q^71 + 4 * q^72 + 5 * q^73 - 6 * q^74 - q^75 + 3 * q^76 - 4 * q^77 + 2 * q^79 + 3 * q^80 - 5 * q^81 - 19 * q^82 - 7 * q^83 + 10 * q^84 - q^85 - 5 * q^86 - 2 * q^87 - 13 * q^88 + 21 * q^89 + 4 * q^90 - 10 * q^92 - 20 * q^93 - 8 * q^94 + 3 * q^95 + q^96 + 21 * q^97 + q^98 - 15 * q^99

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