LMFDB - Embedded newform 169.2.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,2,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	169.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:	$1.34947179416$
Analytic rank:	$1$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.801938 q^{2} -2.24698 q^{3} -1.35690 q^{4} +0.246980 q^{5} -1.80194 q^{6} -2.35690 q^{7} -2.69202 q^{8} +2.04892 q^{9} +O(q^{10})$	\(q+0.801938 q^{2} -2.24698 q^{3} -1.35690 q^{4} +0.246980 q^{5} -1.80194 q^{6} -2.35690 q^{7} -2.69202 q^{8} +2.04892 q^{9} +0.198062 q^{10} -4.24698 q^{11} +3.04892 q^{12} -1.89008 q^{14} -0.554958 q^{15} +0.554958 q^{16} +2.15883 q^{17} +1.64310 q^{18} -0.0881460 q^{19} -0.335126 q^{20} +5.29590 q^{21} -3.40581 q^{22} +1.49396 q^{23} +6.04892 q^{24} -4.93900 q^{25} +2.13706 q^{27} +3.19806 q^{28} +4.63102 q^{29} -0.445042 q^{30} -6.63102 q^{31} +5.82908 q^{32} +9.54288 q^{33} +1.73125 q^{34} -0.582105 q^{35} -2.78017 q^{36} +5.69202 q^{37} -0.0706876 q^{38} -0.664874 q^{40} -11.5918 q^{41} +4.24698 q^{42} -0.295897 q^{43} +5.76271 q^{44} +0.506041 q^{45} +1.19806 q^{46} -7.35690 q^{47} -1.24698 q^{48} -1.44504 q^{49} -3.96077 q^{50} -4.85086 q^{51} -10.3937 q^{53} +1.71379 q^{54} -1.04892 q^{55} +6.34481 q^{56} +0.198062 q^{57} +3.71379 q^{58} -6.78017 q^{59} +0.753020 q^{60} +3.47219 q^{61} -5.31767 q^{62} -4.82908 q^{63} +3.56465 q^{64} +7.65279 q^{66} +7.67994 q^{67} -2.92931 q^{68} -3.35690 q^{69} -0.466812 q^{70} -8.66487 q^{71} -5.51573 q^{72} +6.73556 q^{73} +4.56465 q^{74} +11.0978 q^{75} +0.119605 q^{76} +10.0097 q^{77} +9.97046 q^{79} +0.137063 q^{80} -10.9487 q^{81} -9.29590 q^{82} +1.60925 q^{83} -7.18598 q^{84} +0.533188 q^{85} -0.237291 q^{86} -10.4058 q^{87} +11.4330 q^{88} -2.88471 q^{89} +0.405813 q^{90} -2.02715 q^{92} +14.8998 q^{93} -5.89977 q^{94} -0.0217703 q^{95} -13.0978 q^{96} -8.05861 q^{97} -1.15883 q^{98} -8.70171 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^2 - 2 * q^3 - 4 * q^5 - q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9	$ 3 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9} + 5 q^{10} - 8 q^{11} - 5 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{17} + 9 q^{18} - 4 q^{19} + 2 q^{21} + 3 q^{22} - 5 q^{23} + 9 q^{24} - 5 q^{25} + q^{27} + 14 q^{28} - q^{29} - q^{30} - 5 q^{31} + 7 q^{32} + 10 q^{33} + 13 q^{34} + 4 q^{35} - 7 q^{36} + 12 q^{37} + 12 q^{38} - 3 q^{40} - 7 q^{41} + 8 q^{42} + 13 q^{43} + 11 q^{45} + 8 q^{46} - 18 q^{47} + q^{48} - 4 q^{49} + q^{50} - q^{51} + q^{53} - 3 q^{54} + 6 q^{55} - 4 q^{56} + 5 q^{57} + 3 q^{58} - 19 q^{59} + 7 q^{60} + 4 q^{61} + q^{62} - 4 q^{63} - 11 q^{64} + 5 q^{66} - q^{67} - 21 q^{68} - 6 q^{69} + 2 q^{70} - 27 q^{71} - 4 q^{72} + 9 q^{73} - 8 q^{74} + 15 q^{75} - 21 q^{76} + 8 q^{77} - 5 q^{79} - 5 q^{80} - q^{81} - 14 q^{82} - 7 q^{83} - 7 q^{84} + 5 q^{85} - 18 q^{86} - 18 q^{87} + 15 q^{88} - 11 q^{89} - 12 q^{90} + 22 q^{93} + 5 q^{94} + 3 q^{95} - 21 q^{96} + 7 q^{97} + 5 q^{98} + q^{99}+O(q^{100}) $ 3 * q - 2 * q^2 - 2 * q^3 - 4 * q^5 - q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9 + 5 * q^10 - 8 * q^11 - 5 * q^14 - 2 * q^15 + 2 * q^16 - 2 * q^17 + 9 * q^18 - 4 * q^19 + 2 * q^21 + 3 * q^22 - 5 * q^23 + 9 * q^24 - 5 * q^25 + q^27 + 14 * q^28 - q^29 - q^30 - 5 * q^31 + 7 * q^32 + 10 * q^33 + 13 * q^34 + 4 * q^35 - 7 * q^36 + 12 * q^37 + 12 * q^38 - 3 * q^40 - 7 * q^41 + 8 * q^42 + 13 * q^43 + 11 * q^45 + 8 * q^46 - 18 * q^47 + q^48 - 4 * q^49 + q^50 - q^51 + q^53 - 3 * q^54 + 6 * q^55 - 4 * q^56 + 5 * q^57 + 3 * q^58 - 19 * q^59 + 7 * q^60 + 4 * q^61 + q^62 - 4 * q^63 - 11 * q^64 + 5 * q^66 - q^67 - 21 * q^68 - 6 * q^69 + 2 * q^70 - 27 * q^71 - 4 * q^72 + 9 * q^73 - 8 * q^74 + 15 * q^75 - 21 * q^76 + 8 * q^77 - 5 * q^79 - 5 * q^80 - q^81 - 14 * q^82 - 7 * q^83 - 7 * q^84 + 5 * q^85 - 18 * q^86 - 18 * q^87 + 15 * q^88 - 11 * q^89 - 12 * q^90 + 22 * q^93 + 5 * q^94 + 3 * q^95 - 21 * q^96 + 7 * q^97 + 5 * q^98 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.801938	0.567056	0.283528	−	0.958964i	$-0.408495\pi$
$2$	0.801938	0.567056	0.283528	+	0.958964i	$0.408495\pi$
$3$	−2.24698	−1.29729	−0.648647	−	0.761089i	$-0.724665\pi$
$3$	−2.24698	−1.29729	−0.648647	+	0.761089i	$0.724665\pi$
$4$	−1.35690	−0.678448
$5$	0.246980	0.110453	0.0552263	−	0.998474i	$-0.482412\pi$
$5$	0.246980	0.110453	0.0552263	+	0.998474i	$0.482412\pi$
$6$	−1.80194	−0.735638
$7$	−2.35690	−0.890823	−0.445411	−	0.895326i	$-0.646943\pi$
$7$	−2.35690	−0.890823	−0.445411	+	0.895326i	$0.646943\pi$
$8$	−2.69202	−0.951773
$9$	2.04892	0.682972
$10$	0.198062	0.0626328
$11$	−4.24698	−1.28051	−0.640256	−	0.768161i	$-0.721172\pi$
$11$	−4.24698	−1.28051	−0.640256	+	0.768161i	$0.721172\pi$
$12$	3.04892	0.880147
$13$	0	0
$13$	0	0
$14$	−1.89008	−0.505146
$15$	−0.554958	−0.143290
$16$	0.554958	0.138740
$17$	2.15883	0.523594	0.261797	−	0.965123i	$-0.415685\pi$
$17$	2.15883	0.523594	0.261797	+	0.965123i	$0.415685\pi$
$18$	1.64310	0.387283
$19$	−0.0881460	−0.0202221	−0.0101110	−	0.999949i	$-0.503218\pi$
$19$	−0.0881460	−0.0202221	−0.0101110	+	0.999949i	$0.503218\pi$
$20$	−0.335126	−0.0749364
$21$	5.29590	1.15566
$22$	−3.40581	−0.726122
$23$	1.49396	0.311512	0.155756	−	0.987796i	$-0.450219\pi$
$23$	1.49396	0.311512	0.155756	+	0.987796i	$0.450219\pi$
$24$	6.04892	1.23473
$25$	−4.93900	−0.987800
$26$	0	0
$27$	2.13706	0.411278
$28$	3.19806	0.604377
$29$	4.63102	0.859959	0.429980	−	0.902839i	$-0.358521\pi$
$29$	4.63102	0.859959	0.429980	+	0.902839i	$0.358521\pi$
$30$	−0.445042	−0.0812532
$31$	−6.63102	−1.19097	−0.595483	−	0.803368i	$-0.703039\pi$
$31$	−6.63102	−1.19097	−0.595483	+	0.803368i	$0.703039\pi$
$32$	5.82908	1.03045
$33$	9.54288	1.66120
$34$	1.73125	0.296907
$35$	−0.582105	−0.0983937
$36$	−2.78017	−0.463361
$37$	5.69202	0.935763	0.467881	−	0.883791i	$-0.345017\pi$
$37$	5.69202	0.935763	0.467881	+	0.883791i	$0.345017\pi$
$38$	−0.0706876	−0.0114670
$39$	0	0
$40$	−0.664874	−0.105126
$41$	−11.5918	−1.81033	−0.905167	−	0.425056i	$-0.860254\pi$
$41$	−11.5918	−1.81033	−0.905167	+	0.425056i	$0.860254\pi$
$42$	4.24698	0.655323
$43$	−0.295897	−0.0451239	−0.0225619	−	0.999745i	$-0.507182\pi$
$43$	−0.295897	−0.0451239	−0.0225619	+	0.999745i	$0.507182\pi$
$44$	5.76271	0.868761
$45$	0.506041	0.0754361
$46$	1.19806	0.176645
$47$	−7.35690	−1.07311	−0.536557	−	0.843864i	$-0.680275\pi$
$47$	−7.35690	−1.07311	−0.536557	+	0.843864i	$0.680275\pi$
$48$	−1.24698	−0.179986
$49$	−1.44504	−0.206435
$50$	−3.96077	−0.560138
$51$	−4.85086	−0.679256
$52$	0	0
$53$	−10.3937	−1.42769	−0.713844	−	0.700304i	$-0.753048\pi$
$53$	−10.3937	−1.42769	−0.713844	+	0.700304i	$0.753048\pi$
$54$	1.71379	0.233218
$55$	−1.04892	−0.141436
$56$	6.34481	0.847861
$57$	0.198062	0.0262340
$58$	3.71379	0.487645
$59$	−6.78017	−0.882703	−0.441351	−	0.897334i	$-0.645501\pi$
$59$	−6.78017	−0.882703	−0.441351	+	0.897334i	$0.645501\pi$
$60$	0.753020	0.0972145
$61$	3.47219	0.444568	0.222284	−	0.974982i	$-0.428649\pi$
$61$	3.47219	0.444568	0.222284	+	0.974982i	$0.428649\pi$
$62$	−5.31767	−0.675344
$63$	−4.82908	−0.608407
$64$	3.56465	0.445581
$65$	0	0
$66$	7.65279	0.941994
$67$	7.67994	0.938254	0.469127	−	0.883131i	$-0.344569\pi$
$67$	7.67994	0.938254	0.469127	+	0.883131i	$0.344569\pi$
$68$	−2.92931	−0.355231
$69$	−3.35690	−0.404123
$70$	−0.466812	−0.0557947
$71$	−8.66487	−1.02833	−0.514166	−	0.857691i	$-0.671898\pi$
$71$	−8.66487	−1.02833	−0.514166	+	0.857691i	$0.671898\pi$
$72$	−5.51573	−0.650035
$73$	6.73556	0.788338	0.394169	−	0.919038i	$-0.371032\pi$
$73$	6.73556	0.788338	0.394169	+	0.919038i	$0.371032\pi$
$74$	4.56465	0.530629
$75$	11.0978	1.28147
$76$	0.119605	0.0137196
$77$	10.0097	1.14071
$78$	0	0
$79$	9.97046	1.12176	0.560882	−	0.827896i	$-0.310462\pi$
$79$	9.97046	1.12176	0.560882	+	0.827896i	$0.310462\pi$
$80$	0.137063	0.0153241
$81$	−10.9487	−1.21652
$82$	−9.29590	−1.02656
$83$	1.60925	0.176638	0.0883192	−	0.996092i	$-0.471850\pi$
$83$	1.60925	0.176638	0.0883192	+	0.996092i	$0.471850\pi$
$84$	−7.18598	−0.784055
$85$	0.533188	0.0578323
$86$	−0.237291	−0.0255877
$87$	−10.4058	−1.11562
$88$	11.4330	1.21876
$89$	−2.88471	−0.305778	−0.152889	−	0.988243i	$-0.548858\pi$
$89$	−2.88471	−0.305778	−0.152889	+	0.988243i	$0.548858\pi$
$90$	0.405813	0.0427765
$91$	0	0
$92$	−2.02715	−0.211345
$93$	14.8998	1.54503
$94$	−5.89977	−0.608515
$95$	−0.0217703	−0.00223358
$96$	−13.0978	−1.33679
$97$	−8.05861	−0.818227	−0.409114	−	0.912483i	$-0.634162\pi$
$97$	−8.05861	−0.818227	−0.409114	+	0.912483i	$0.634162\pi$
$98$	−1.15883	−0.117060
$99$	−8.70171	−0.874555
$100$	6.70171	0.670171
$101$	−13.3545	−1.32882	−0.664411	−	0.747367i	$-0.731318\pi$
$101$	−13.3545	−1.32882	−0.664411	+	0.747367i	$0.731318\pi$
$102$	−3.89008	−0.385176
$103$	1.36227	0.134229	0.0671144	−	0.997745i	$-0.478621\pi$
$103$	1.36227	0.134229	0.0671144	+	0.997745i	$0.478621\pi$
$104$	0	0
$105$	1.30798	0.127646
$106$	−8.33513	−0.809579
$107$	3.26875	0.316002	0.158001	−	0.987439i	$-0.449495\pi$
$107$	3.26875	0.316002	0.158001	+	0.987439i	$0.449495\pi$
$108$	−2.89977	−0.279031
$109$	15.7017	1.50395	0.751976	−	0.659191i	$-0.229101\pi$
$109$	15.7017	1.50395	0.751976	+	0.659191i	$0.229101\pi$
$110$	−0.841166	−0.0802021
$111$	−12.7899	−1.21396
$112$	−1.30798	−0.123592
$113$	12.0489	1.13347	0.566733	−	0.823901i	$-0.308207\pi$
$113$	12.0489	1.13347	0.566733	+	0.823901i	$0.308207\pi$
$114$	0.158834	0.0148761
$115$	0.368977	0.0344073
$116$	−6.28382	−0.583438
$117$	0	0
$118$	−5.43727	−0.500541
$119$	−5.08815	−0.466430
$120$	1.49396	0.136379
$121$	7.03684	0.639712
$122$	2.78448	0.252095
$123$	26.0465	2.34854
$124$	8.99761	0.808009
$125$	−2.45473	−0.219558
$126$	−3.87263	−0.345001
$127$	−9.80731	−0.870258	−0.435129	−	0.900368i	$-0.643297\pi$
$127$	−9.80731	−0.870258	−0.435129	+	0.900368i	$0.643297\pi$
$128$	−8.79954	−0.777777
$129$	0.664874	0.0585389
$130$	0	0
$131$	−6.57673	−0.574611	−0.287306	−	0.957839i	$-0.592760\pi$
$131$	−6.57673	−0.574611	−0.287306	+	0.957839i	$0.592760\pi$
$132$	−12.9487	−1.12704
$133$	0.207751	0.0180143
$134$	6.15883	0.532042
$135$	0.527811	0.0454267
$136$	−5.81163	−0.498343
$137$	6.21983	0.531396	0.265698	−	0.964056i	$-0.414398\pi$
$137$	6.21983	0.531396	0.265698	+	0.964056i	$0.414398\pi$
$138$	−2.69202	−0.229160
$139$	−14.7071	−1.24744	−0.623719	−	0.781648i	$-0.714379\pi$
$139$	−14.7071	−1.24744	−0.623719	+	0.781648i	$0.714379\pi$
$140$	0.789856	0.0667550
$141$	16.5308	1.39214
$142$	−6.94869	−0.583121
$143$	0	0
$144$	1.13706	0.0947553
$145$	1.14377	0.0949848
$146$	5.40150	0.447031
$147$	3.24698	0.267806
$148$	−7.72348	−0.634866
$149$	4.33513	0.355147	0.177574	−	0.984108i	$-0.443175\pi$
$149$	4.33513	0.355147	0.177574	+	0.984108i	$0.443175\pi$
$150$	8.89977	0.726663
$151$	3.94438	0.320989	0.160494	−	0.987037i	$-0.448691\pi$
$151$	3.94438	0.320989	0.160494	+	0.987037i	$0.448691\pi$
$152$	0.237291	0.0192468
$153$	4.42327	0.357600
$154$	8.02715	0.646846
$155$	−1.63773	−0.131545
$156$	0	0
$157$	4.45473	0.355526	0.177763	−	0.984073i	$-0.443114\pi$
$157$	4.45473	0.355526	0.177763	+	0.984073i	$0.443114\pi$
$158$	7.99569	0.636103
$159$	23.3545	1.85213
$160$	1.43967	0.113816
$161$	−3.52111	−0.277502
$162$	−8.78017	−0.689835
$163$	16.1588	1.26566	0.632829	−	0.774292i	$-0.281894\pi$
$163$	16.1588	1.26566	0.632829	+	0.774292i	$0.281894\pi$
$164$	15.7289	1.22822
$165$	2.35690	0.183484
$166$	1.29052	0.100164
$167$	16.1172	1.24719	0.623594	−	0.781749i	$-0.285672\pi$
$167$	16.1172	1.24719	0.623594	+	0.781749i	$0.285672\pi$
$168$	−14.2567	−1.09993
$169$	0	0
$170$	0.427583	0.0327942
$171$	−0.180604	−0.0138111
$172$	0.401501	0.0306142
$173$	−21.5362	−1.63736	−0.818682	−	0.574247i	$-0.805295\pi$
$173$	−21.5362	−1.63736	−0.818682	+	0.574247i	$0.805295\pi$
$174$	−8.34481	−0.632619
$175$	11.6407	0.879955
$176$	−2.35690	−0.177658
$177$	15.2349	1.14513
$178$	−2.31336	−0.173393
$179$	11.4330	0.854540	0.427270	−	0.904124i	$-0.359475\pi$
$179$	11.4330	0.854540	0.427270	+	0.904124i	$0.359475\pi$
$180$	−0.686645	−0.0511795
$181$	20.9705	1.55872	0.779361	−	0.626575i	$-0.215544\pi$
$181$	20.9705	1.55872	0.779361	+	0.626575i	$0.215544\pi$
$182$	0	0
$183$	−7.80194	−0.576736
$184$	−4.02177	−0.296489
$185$	1.40581	0.103357
$186$	11.9487	0.876120
$187$	−9.16852	−0.670469
$188$	9.98254	0.728052
$189$	−5.03684	−0.366376
$190$	−0.0174584	−0.00126657
$191$	−14.4373	−1.04464	−0.522322	−	0.852748i	$-0.674934\pi$
$191$	−14.4373	−1.04464	−0.522322	+	0.852748i	$0.674934\pi$
$192$	−8.00969	−0.578049
$193$	−13.5797	−0.977489	−0.488745	−	0.872427i	$-0.662545\pi$
$193$	−13.5797	−0.977489	−0.488745	+	0.872427i	$0.662545\pi$
$194$	−6.46250	−0.463980
$195$	0	0
$196$	1.96077	0.140055
$197$	−0.560335	−0.0399222	−0.0199611	−	0.999801i	$-0.506354\pi$
$197$	−0.560335	−0.0399222	−0.0199611	+	0.999801i	$0.506354\pi$
$198$	−6.97823	−0.495921
$199$	11.4916	0.814616	0.407308	−	0.913291i	$-0.366468\pi$
$199$	11.4916	0.814616	0.407308	+	0.913291i	$0.366468\pi$
$200$	13.2959	0.940162
$201$	−17.2567	−1.21719
$202$	−10.7095	−0.753516
$203$	−10.9148	−0.766071
$204$	6.58211	0.460840
$205$	−2.86294	−0.199956
$206$	1.09246	0.0761151
$207$	3.06100	0.212754
$208$	0	0
$209$	0.374354	0.0258946
$210$	1.04892	0.0723822
$211$	8.78448	0.604748	0.302374	−	0.953189i	$-0.402221\pi$
$211$	8.78448	0.604748	0.302374	+	0.953189i	$0.402221\pi$
$212$	14.1032	0.968613
$213$	19.4698	1.33405
$214$	2.62133	0.179191
$215$	−0.0730805	−0.00498405
$216$	−5.75302	−0.391443
$217$	15.6286	1.06094
$218$	12.5918	0.852824
$219$	−15.1347	−1.02271
$220$	1.42327	0.0959570
$221$	0	0
$222$	−10.2567	−0.688383
$223$	−2.25906	−0.151278	−0.0756390	−	0.997135i	$-0.524100\pi$
$223$	−2.25906	−0.151278	−0.0756390	+	0.997135i	$0.524100\pi$
$224$	−13.7385	−0.917945
$225$	−10.1196	−0.674640
$226$	9.66248	0.642739
$227$	−6.96615	−0.462359	−0.231180	−	0.972911i	$-0.574259\pi$
$227$	−6.96615	−0.462359	−0.231180	+	0.972911i	$0.574259\pi$
$228$	−0.268750	−0.0177984
$229$	−24.1739	−1.59746	−0.798728	−	0.601692i	$-0.794493\pi$
$229$	−24.1739	−1.59746	−0.798728	+	0.601692i	$0.794493\pi$
$230$	0.295897	0.0195109
$231$	−22.4916	−1.47984
$232$	−12.4668	−0.818486
$233$	−3.06100	−0.200533	−0.100266	−	0.994961i	$-0.531969\pi$
$233$	−3.06100	−0.200533	−0.100266	+	0.994961i	$0.531969\pi$
$234$	0	0
$235$	−1.81700	−0.118528
$236$	9.19998	0.598868
$237$	−22.4034	−1.45526
$238$	−4.08038	−0.264492
$239$	−25.1468	−1.62661	−0.813304	−	0.581839i	$-0.802333\pi$
$239$	−25.1468	−1.62661	−0.813304	+	0.581839i	$0.802333\pi$
$240$	−0.307979	−0.0198799
$241$	−20.2664	−1.30547	−0.652735	−	0.757586i	$-0.726379\pi$
$241$	−20.2664	−1.30547	−0.652735	+	0.757586i	$0.726379\pi$
$242$	5.64310	0.362752
$243$	18.1903	1.16691
$244$	−4.71140	−0.301616
$245$	−0.356896	−0.0228012
$246$	20.8877	1.33175
$247$	0	0
$248$	17.8509	1.13353
$249$	−3.61596	−0.229152
$250$	−1.96854	−0.124501
$251$	−23.7211	−1.49726	−0.748631	−	0.662987i	$-0.769288\pi$
$251$	−23.7211	−1.49726	−0.748631	+	0.662987i	$0.769288\pi$
$252$	6.55257	0.412773
$253$	−6.34481	−0.398895
$254$	−7.86486	−0.493485
$255$	−1.19806	−0.0750256
$256$	−14.1860	−0.886624
$257$	14.2241	0.887278	0.443639	−	0.896206i	$-0.353687\pi$
$257$	14.2241	0.887278	0.443639	+	0.896206i	$0.353687\pi$
$258$	0.533188	0.0331948
$259$	−13.4155	−0.833599
$260$	0	0
$261$	9.48858	0.587329
$262$	−5.27413	−0.325837
$263$	−17.0954	−1.05415	−0.527075	−	0.849819i	$-0.676711\pi$
$263$	−17.0954	−1.05415	−0.527075	+	0.849819i	$0.676711\pi$
$264$	−25.6896	−1.58109
$265$	−2.56704	−0.157692
$266$	0.166603	0.0102151
$267$	6.48188	0.396684
$268$	−10.4209	−0.636556
$269$	−6.46681	−0.394288	−0.197144	−	0.980374i	$-0.563167\pi$
$269$	−6.46681	−0.394288	−0.197144	+	0.980374i	$0.563167\pi$
$270$	0.423272	0.0257595
$271$	6.44803	0.391690	0.195845	−	0.980635i	$-0.437255\pi$
$271$	6.44803	0.391690	0.195845	+	0.980635i	$0.437255\pi$
$272$	1.19806	0.0726432
$273$	0	0
$274$	4.98792	0.301331
$275$	20.9758	1.26489
$276$	4.55496	0.274176
$277$	13.4601	0.808739	0.404370	−	0.914596i	$-0.367491\pi$
$277$	13.4601	0.808739	0.404370	+	0.914596i	$0.367491\pi$
$278$	−11.7942	−0.707367
$279$	−13.5864	−0.813398
$280$	1.56704	0.0936485
$281$	5.03684	0.300472	0.150236	−	0.988650i	$-0.451997\pi$
$281$	5.03684	0.300472	0.150236	+	0.988650i	$0.451997\pi$
$282$	13.2567	0.789423
$283$	22.1280	1.31537	0.657686	−	0.753293i	$-0.271536\pi$
$283$	22.1280	1.31537	0.657686	+	0.753293i	$0.271536\pi$
$284$	11.7573	0.697669
$285$	0.0489173	0.00289761
$286$	0	0
$287$	27.3207	1.61269
$288$	11.9433	0.703766
$289$	−12.3394	−0.725849
$290$	0.917231	0.0538616
$291$	18.1075	1.06148
$292$	−9.13946	−0.534846
$293$	14.9463	0.873172	0.436586	−	0.899663i	$-0.356187\pi$
$293$	14.9463	0.873172	0.436586	+	0.899663i	$0.356187\pi$
$294$	2.60388	0.151861
$295$	−1.67456	−0.0974968
$296$	−15.3230	−0.890634
$297$	−9.07606	−0.526647
$298$	3.47650	0.201388
$299$	0	0
$300$	−15.0586	−0.869409
$301$	0.697398	0.0401974
$302$	3.16315	0.182019
$303$	30.0073	1.72387
$304$	−0.0489173	−0.00280560
$305$	0.857560	0.0491037
$306$	3.54719	0.202779
$307$	19.1293	1.09177	0.545883	−	0.837861i	$-0.316194\pi$
$307$	19.1293	1.09177	0.545883	+	0.837861i	$0.316194\pi$
$308$	−13.5821	−0.773912
$309$	−3.06100	−0.174134
$310$	−1.31336	−0.0745936
$311$	−0.269815	−0.0152998	−0.00764990	−	0.999971i	$-0.502435\pi$
$311$	−0.269815	−0.0152998	−0.00764990	+	0.999971i	$0.502435\pi$
$312$	0	0
$313$	−23.3937	−1.32229	−0.661146	−	0.750257i	$-0.729930\pi$
$313$	−23.3937	−1.32229	−0.661146	+	0.750257i	$0.729930\pi$
$314$	3.57242	0.201603
$315$	−1.19269	−0.0672002
$316$	−13.5289	−0.761059
$317$	−13.9952	−0.786050	−0.393025	−	0.919528i	$-0.628571\pi$
$317$	−13.9952	−0.786050	−0.393025	+	0.919528i	$0.628571\pi$
$318$	18.7289	1.05026
$319$	−19.6679	−1.10119
$320$	0.880395	0.0492156
$321$	−7.34481	−0.409948
$322$	−2.82371	−0.157359
$323$	−0.190293	−0.0105882
$324$	14.8562	0.825346
$325$	0	0
$326$	12.9584	0.717698
$327$	−35.2814	−1.95107
$328$	31.2054	1.72303
$329$	17.3394	0.955954
$330$	1.89008	0.104046
$331$	17.8213	0.979548	0.489774	−	0.871849i	$-0.337079\pi$
$331$	17.8213	0.979548	0.489774	+	0.871849i	$0.337079\pi$
$332$	−2.18359	−0.119840
$333$	11.6625	0.639100
$334$	12.9250	0.707225
$335$	1.89679	0.103633
$336$	2.93900	0.160336
$337$	−27.8485	−1.51700	−0.758501	−	0.651672i	$-0.774068\pi$
$337$	−27.8485	−1.51700	−0.758501	+	0.651672i	$0.774068\pi$
$338$	0	0
$339$	−27.0737	−1.47044
$340$	−0.723480	−0.0392362
$341$	28.1618	1.52505
$342$	−0.144833	−0.00783167
$343$	19.9041	1.07472
$344$	0.796561	0.0429477
$345$	−0.829085	−0.0446364
$346$	−17.2707	−0.928477
$347$	1.50365	0.0807200	0.0403600	−	0.999185i	$-0.487150\pi$
$347$	1.50365	0.0807200	0.0403600	+	0.999185i	$0.487150\pi$
$348$	14.1196	0.756890
$349$	−14.1860	−0.759358	−0.379679	−	0.925118i	$-0.623966\pi$
$349$	−14.1860	−0.759358	−0.379679	+	0.925118i	$0.623966\pi$
$350$	9.33513	0.498983
$351$	0	0
$352$	−24.7560	−1.31950
$353$	−7.16852	−0.381542	−0.190771	−	0.981635i	$-0.561099\pi$
$353$	−7.16852	−0.381542	−0.190771	+	0.981635i	$0.561099\pi$
$354$	12.2174	0.649350
$355$	−2.14005	−0.113582
$356$	3.91425	0.207455
$357$	11.4330	0.605096
$358$	9.16852	0.484571
$359$	19.8853	1.04951	0.524753	−	0.851255i	$-0.324158\pi$
$359$	19.8853	1.04951	0.524753	+	0.851255i	$0.324158\pi$
$360$	−1.36227	−0.0717981
$361$	−18.9922	−0.999591
$362$	16.8170	0.883882
$363$	−15.8116	−0.829895
$364$	0	0
$365$	1.66355	0.0870740
$366$	−6.25667	−0.327041
$367$	1.08383	0.0565757	0.0282878	−	0.999600i	$-0.490994\pi$
$367$	1.08383	0.0565757	0.0282878	+	0.999600i	$0.490994\pi$
$368$	0.829085	0.0432190
$369$	−23.7506	−1.23641
$370$	1.12737	0.0586094
$371$	24.4969	1.27182
$372$	−20.2174	−1.04823
$373$	−6.13036	−0.317418	−0.158709	−	0.987325i	$-0.550733\pi$
$373$	−6.13036	−0.317418	−0.158709	+	0.987325i	$0.550733\pi$
$374$	−7.35258	−0.380193
$375$	5.51573	0.284831
$376$	19.8049	1.02136
$377$	0	0
$378$	−4.03923	−0.207756
$379$	−2.40880	−0.123732	−0.0618658	−	0.998084i	$-0.519705\pi$
$379$	−2.40880	−0.123732	−0.0618658	+	0.998084i	$0.519705\pi$
$380$	0.0295400	0.00151537
$381$	22.0368	1.12898
$382$	−11.5778	−0.592371
$383$	−30.3913	−1.55292	−0.776462	−	0.630164i	$-0.782988\pi$
$383$	−30.3913	−1.55292	−0.776462	+	0.630164i	$0.782988\pi$
$384$	19.7724	1.00901
$385$	2.47219	0.125994
$386$	−10.8901	−0.554291
$387$	−0.606268	−0.0308184
$388$	10.9347	0.555125
$389$	−15.9409	−0.808237	−0.404118	−	0.914707i	$-0.632422\pi$
$389$	−15.9409	−0.808237	−0.404118	+	0.914707i	$0.632422\pi$
$390$	0	0
$391$	3.22521	0.163106
$392$	3.89008	0.196479
$393$	14.7778	0.745440
$394$	−0.449354	−0.0226381
$395$	2.46250	0.123902
$396$	11.8073	0.593340
$397$	16.9148	0.848931	0.424466	−	0.905444i	$-0.360462\pi$
$397$	16.9148	0.848931	0.424466	+	0.905444i	$0.360462\pi$
$398$	9.21552	0.461932
$399$	−0.466812	−0.0233698
$400$	−2.74094	−0.137047
$401$	26.6625	1.33146	0.665730	−	0.746192i	$-0.268120\pi$
$401$	26.6625	1.33146	0.665730	+	0.746192i	$0.268120\pi$
$402$	−13.8388	−0.690215
$403$	0	0
$404$	18.1207	0.901537
$405$	−2.70410	−0.134368
$406$	−8.75302	−0.434405
$407$	−24.1739	−1.19826
$408$	13.0586	0.646497
$409$	28.5163	1.41004	0.705021	−	0.709187i	$-0.250938\pi$
$409$	28.5163	1.41004	0.705021	+	0.709187i	$0.250938\pi$
$410$	−2.29590	−0.113386
$411$	−13.9758	−0.689377
$412$	−1.84846	−0.0910672
$413$	15.9801	0.786332
$414$	2.45473	0.120643
$415$	0.397452	0.0195102
$416$	0	0
$417$	33.0465	1.61830
$418$	0.300209	0.0146837
$419$	−29.6093	−1.44651	−0.723253	−	0.690583i	$-0.757354\pi$
$419$	−29.6093	−1.44651	−0.723253	+	0.690583i	$0.757354\pi$
$420$	−1.77479	−0.0866009
$421$	−11.6606	−0.568301	−0.284151	−	0.958780i	$-0.591712\pi$
$421$	−11.6606	−0.568301	−0.284151	+	0.958780i	$0.591712\pi$
$422$	7.04461	0.342926
$423$	−15.0737	−0.732907
$424$	27.9801	1.35884
$425$	−10.6625	−0.517206
$426$	15.6136	0.756480
$427$	−8.18359	−0.396032
$428$	−4.43535	−0.214391
$429$	0	0
$430$	−0.0586060	−0.00282623
$431$	4.34913	0.209490	0.104745	−	0.994499i	$-0.466597\pi$
$431$	4.34913	0.209490	0.104745	+	0.994499i	$0.466597\pi$
$432$	1.18598	0.0570605
$433$	−14.3884	−0.691460	−0.345730	−	0.938334i	$-0.612369\pi$
$433$	−14.3884	−0.691460	−0.345730	+	0.938334i	$0.612369\pi$
$434$	12.5332	0.601612
$435$	−2.57002	−0.123223
$436$	−21.3056	−1.02035
$437$	−0.131687	−0.00629942
$438$	−12.1371	−0.579931
$439$	−20.2325	−0.965645	−0.482822	−	0.875718i	$-0.660388\pi$
$439$	−20.2325	−0.965645	−0.482822	+	0.875718i	$0.660388\pi$
$440$	2.82371	0.134615
$441$	−2.96077	−0.140989
$442$	0	0
$443$	8.12200	0.385888	0.192944	−	0.981210i	$-0.438196\pi$
$443$	8.12200	0.385888	0.192944	+	0.981210i	$0.438196\pi$
$444$	17.3545	0.823608
$445$	−0.712464	−0.0337740
$446$	−1.81163	−0.0857830
$447$	−9.74094	−0.460731
$448$	−8.40150	−0.396934
$449$	12.4916	0.589513	0.294757	−	0.955572i	$-0.404761\pi$
$449$	12.4916	0.589513	0.294757	+	0.955572i	$0.404761\pi$
$450$	−8.11529	−0.382559
$451$	49.2301	2.31816
$452$	−16.3491	−0.768998
$453$	−8.86294	−0.416417
$454$	−5.58642	−0.262184
$455$	0	0
$456$	−0.533188	−0.0249688
$457$	5.98121	0.279789	0.139895	−	0.990166i	$-0.455324\pi$
$457$	5.98121	0.279789	0.139895	+	0.990166i	$0.455324\pi$
$458$	−19.3860	−0.905847
$459$	4.61356	0.215343
$460$	−0.500664	−0.0233436
$461$	−2.05669	−0.0957895	−0.0478947	−	0.998852i	$-0.515251\pi$
$461$	−2.05669	−0.0957895	−0.0478947	+	0.998852i	$0.515251\pi$
$462$	−18.0368	−0.839150
$463$	−8.44935	−0.392675	−0.196337	−	0.980536i	$-0.562905\pi$
$463$	−8.44935	−0.392675	−0.196337	+	0.980536i	$0.562905\pi$
$464$	2.57002	0.119310
$465$	3.67994	0.170653
$466$	−2.45473	−0.113713
$467$	33.5139	1.55084	0.775420	−	0.631446i	$-0.217538\pi$
$467$	33.5139	1.55084	0.775420	+	0.631446i	$0.217538\pi$
$468$	0	0
$469$	−18.1008	−0.835818
$470$	−1.45712	−0.0672121
$471$	−10.0097	−0.461222
$472$	18.2524	0.840133
$473$	1.25667	0.0577817
$474$	−17.9661	−0.825213
$475$	0.435353	0.0199754
$476$	6.90408	0.316448
$477$	−21.2959	−0.975072
$478$	−20.1661	−0.922377
$479$	−24.7313	−1.13000	−0.565000	−	0.825091i	$-0.691124\pi$
$479$	−24.7313	−1.13000	−0.565000	+	0.825091i	$0.691124\pi$
$480$	−3.23490	−0.147652
$481$	0	0
$482$	−16.2524	−0.740275
$483$	7.91185	0.360002
$484$	−9.54825	−0.434012
$485$	−1.99031	−0.0903754
$486$	14.5875	0.661702
$487$	−37.7555	−1.71087	−0.855433	−	0.517913i	$-0.826709\pi$
$487$	−37.7555	−1.71087	−0.855433	+	0.517913i	$0.826709\pi$
$488$	−9.34721	−0.423128
$489$	−36.3086	−1.64193
$490$	−0.286208	−0.0129296
$491$	31.3110	1.41304	0.706522	−	0.707691i	$-0.250263\pi$
$491$	31.3110	1.41304	0.706522	+	0.707691i	$0.250263\pi$
$492$	−35.3424	−1.59336
$493$	9.99761	0.450270
$494$	0	0
$495$	−2.14914	−0.0965969
$496$	−3.67994	−0.165234
$497$	20.4222	0.916061
$498$	−2.89977	−0.129942
$499$	21.4873	0.961902	0.480951	−	0.876748i	$-0.340292\pi$
$499$	21.4873	0.961902	0.480951	+	0.876748i	$0.340292\pi$
$500$	3.33081	0.148959
$501$	−36.2150	−1.61797
$502$	−19.0228	−0.849031
$503$	37.5924	1.67616	0.838081	−	0.545546i	$-0.183678\pi$
$503$	37.5924	1.67616	0.838081	+	0.545546i	$0.183678\pi$
$504$	13.0000	0.579066
$505$	−3.29829	−0.146772
$506$	−5.08815	−0.226196
$507$	0	0
$508$	13.3075	0.590425
$509$	−17.1075	−0.758278	−0.379139	−	0.925340i	$-0.623780\pi$
$509$	−17.1075	−0.758278	−0.379139	+	0.925340i	$0.623780\pi$
$510$	−0.960771	−0.0425437
$511$	−15.8750	−0.702269
$512$	6.22282	0.275012
$513$	−0.188374	−0.00831690
$514$	11.4069	0.503136
$515$	0.336454	0.0148259
$516$	−0.902165	−0.0397156
$517$	31.2446	1.37414
$518$	−10.7584	−0.472697
$519$	48.3913	2.12414
$520$	0	0
$521$	−19.8465	−0.869493	−0.434746	−	0.900553i	$-0.643162\pi$
$521$	−19.8465	−0.869493	−0.434746	+	0.900553i	$0.643162\pi$
$522$	7.60925	0.333048
$523$	−11.4300	−0.499798	−0.249899	−	0.968272i	$-0.580397\pi$
$523$	−11.4300	−0.499798	−0.249899	+	0.968272i	$0.580397\pi$
$524$	8.92394	0.389844
$525$	−26.1564	−1.14156
$526$	−13.7095	−0.597762
$527$	−14.3153	−0.623583
$528$	5.29590	0.230474
$529$	−20.7681	−0.902960
$530$	−2.05861	−0.0894201
$531$	−13.8920	−0.602862
$532$	−0.281896	−0.0122218
$533$	0	0
$534$	5.19806	0.224942
$535$	0.807315	0.0349033
$536$	−20.6746	−0.893005
$537$	−25.6896	−1.10859
$538$	−5.18598	−0.223584
$539$	6.13706	0.264342
$540$	−0.716185	−0.0308197
$541$	16.1884	0.695993	0.347996	−	0.937496i	$-0.386862\pi$
$541$	16.1884	0.695993	0.347996	+	0.937496i	$0.386862\pi$
$542$	5.17092	0.222110
$543$	−47.1202	−2.02212
$544$	12.5840	0.539536
$545$	3.87800	0.166115
$546$	0	0
$547$	5.33081	0.227929	0.113965	−	0.993485i	$-0.463645\pi$
$547$	5.33081	0.227929	0.113965	+	0.993485i	$0.463645\pi$
$548$	−8.43967	−0.360525
$549$	7.11423	0.303628
$550$	16.8213	0.717263
$551$	−0.408206	−0.0173902
$552$	9.03684	0.384633
$553$	−23.4993	−0.999293
$554$	10.7942	0.458600
$555$	−3.15883	−0.134085
$556$	19.9560	0.846322
$557$	7.39075	0.313156	0.156578	−	0.987666i	$-0.449954\pi$
$557$	7.39075	0.313156	0.156578	+	0.987666i	$0.449954\pi$
$558$	−10.8955	−0.461242
$559$	0	0
$560$	−0.323044	−0.0136511
$561$	20.6015	0.869795
$562$	4.03923	0.170385
$563$	−9.47889	−0.399488	−0.199744	−	0.979848i	$-0.564011\pi$
$563$	−9.47889	−0.399488	−0.199744	+	0.979848i	$0.564011\pi$
$564$	−22.4306	−0.944497
$565$	2.97584	0.125194
$566$	17.7453	0.745889
$567$	25.8049	1.08370
$568$	23.3260	0.978738
$569$	−10.1438	−0.425249	−0.212624	−	0.977134i	$-0.568201\pi$
$569$	−10.1438	−0.425249	−0.212624	+	0.977134i	$0.568201\pi$
$570$	0.0392287	0.00164311
$571$	−14.0925	−0.589751	−0.294876	−	0.955536i	$-0.595278\pi$
$571$	−14.0925	−0.589751	−0.294876	+	0.955536i	$0.595278\pi$
$572$	0	0
$573$	32.4403	1.35521
$574$	21.9095	0.914483
$575$	−7.37867	−0.307712
$576$	7.30367	0.304319
$577$	25.1545	1.04720	0.523598	−	0.851965i	$-0.324589\pi$
$577$	25.1545	1.04720	0.523598	+	0.851965i	$0.324589\pi$
$578$	−9.89546	−0.411597
$579$	30.5133	1.26809
$580$	−1.55197	−0.0644422
$581$	−3.79284	−0.157354
$582$	14.5211	0.601919
$583$	44.1420	1.82817
$584$	−18.1323	−0.750319
$585$	0	0
$586$	11.9860	0.495137
$587$	43.8353	1.80928	0.904639	−	0.426180i	$-0.140141\pi$
$587$	43.8353	1.80928	0.904639	+	0.426180i	$0.140141\pi$
$588$	−4.40581	−0.181693
$589$	0.584498	0.0240838
$590$	−1.34290	−0.0552861
$591$	1.25906	0.0517909
$592$	3.15883	0.129827
$593$	−24.9965	−1.02648	−0.513242	−	0.858244i	$-0.671556\pi$
$593$	−24.9965	−1.02648	−0.513242	+	0.858244i	$0.671556\pi$
$594$	−7.27844	−0.298638
$595$	−1.25667	−0.0515184
$596$	−5.88231	−0.240949
$597$	−25.8213	−1.05680
$598$	0	0
$599$	−6.24027	−0.254971	−0.127485	−	0.991840i	$-0.540691\pi$
$599$	−6.24027	−0.254971	−0.127485	+	0.991840i	$0.540691\pi$
$600$	−29.8756	−1.21967
$601$	6.32975	0.258196	0.129098	−	0.991632i	$-0.458792\pi$
$601$	6.32975	0.258196	0.129098	+	0.991632i	$0.458792\pi$
$602$	0.559270	0.0227941
$603$	15.7356	0.640802
$604$	−5.35211	−0.217774
$605$	1.73795	0.0706579
$606$	24.0640	0.977532
$607$	−43.6480	−1.77162	−0.885809	−	0.464050i	$-0.846396\pi$
$607$	−43.6480	−1.77162	−0.885809	+	0.464050i	$0.846396\pi$
$608$	−0.513811	−0.0208378
$609$	24.5254	0.993820
$610$	0.687710	0.0278445
$611$	0	0
$612$	−6.00192	−0.242613
$613$	−25.9541	−1.04827	−0.524137	−	0.851634i	$-0.675612\pi$
$613$	−25.9541	−1.04827	−0.524137	+	0.851634i	$0.675612\pi$
$614$	15.3405	0.619092
$615$	6.43296	0.259402
$616$	−26.9463	−1.08570
$617$	45.9396	1.84946	0.924729	−	0.380626i	$-0.124291\pi$
$617$	45.9396	1.84946	0.924729	+	0.380626i	$0.124291\pi$
$618$	−2.45473	−0.0987437
$619$	6.73556	0.270725	0.135363	−	0.990796i	$-0.456780\pi$
$619$	6.73556	0.270725	0.135363	+	0.990796i	$0.456780\pi$
$620$	2.22223	0.0892467
$621$	3.19269	0.128118
$622$	−0.216375	−0.00867583
$623$	6.79895	0.272394
$624$	0	0
$625$	24.0887	0.963549
$626$	−18.7603	−0.749813
$627$	−0.841166	−0.0335930
$628$	−6.04461	−0.241206
$629$	12.2881	0.489960
$630$	−0.956459	−0.0381063
$631$	45.0998	1.79539	0.897696	−	0.440614i	$-0.145239\pi$
$631$	45.0998	1.79539	0.897696	+	0.440614i	$0.145239\pi$
$632$	−26.8407	−1.06767
$633$	−19.7385	−0.784537
$634$	−11.2233	−0.445734
$635$	−2.42221	−0.0961223
$636$	−31.6896	−1.25658
$637$	0	0
$638$	−15.7724	−0.624435
$639$	−17.7536	−0.702322
$640$	−2.17331	−0.0859075
$641$	32.5821	1.28692	0.643458	−	0.765482i	$-0.277499\pi$
$641$	32.5821	1.28692	0.643458	+	0.765482i	$0.277499\pi$
$642$	−5.89008	−0.232463
$643$	25.5754	1.00860	0.504298	−	0.863530i	$-0.331751\pi$
$643$	25.5754	1.00860	0.504298	+	0.863530i	$0.331751\pi$
$644$	4.77777	0.188271
$645$	0.164210	0.00646578
$646$	−0.152603	−0.00600408
$647$	−30.1715	−1.18616	−0.593082	−	0.805142i	$-0.702089\pi$
$647$	−30.1715	−1.18616	−0.593082	+	0.805142i	$0.702089\pi$
$648$	29.4741	1.15785
$649$	28.7952	1.13031
$650$	0	0
$651$	−35.1172	−1.37635
$652$	−21.9259	−0.858683
$653$	36.9028	1.44412	0.722058	−	0.691832i	$-0.243196\pi$
$653$	36.9028	1.44412	0.722058	+	0.691832i	$0.243196\pi$
$654$	−28.2935	−1.10636
$655$	−1.62432	−0.0634673
$656$	−6.43296	−0.251165
$657$	13.8006	0.538413
$658$	13.9051	0.542079
$659$	23.6866	0.922701	0.461350	−	0.887218i	$-0.347365\pi$
$659$	23.6866	0.922701	0.461350	+	0.887218i	$0.347365\pi$
$660$	−3.19806	−0.124484
$661$	−31.7590	−1.23528	−0.617641	−	0.786460i	$-0.711911\pi$
$661$	−31.7590	−1.23528	−0.617641	+	0.786460i	$0.711911\pi$
$662$	14.2916	0.555458
$663$	0	0
$664$	−4.33214	−0.168120
$665$	0.0513102	0.00198973
$666$	9.35258	0.362405
$667$	6.91856	0.267888
$668$	−21.8694	−0.846152
$669$	5.07606	0.196252
$670$	1.52111	0.0587655
$671$	−14.7463	−0.569275
$672$	30.8702	1.19085
$673$	−7.50232	−0.289193	−0.144597	−	0.989491i	$-0.546188\pi$
$673$	−7.50232	−0.289193	−0.144597	+	0.989491i	$0.546188\pi$
$674$	−22.3327	−0.860225
$675$	−10.5550	−0.406261
$676$	0	0
$677$	−35.0315	−1.34637	−0.673184	−	0.739475i	$-0.735074\pi$
$677$	−35.0315	−1.34637	−0.673184	+	0.739475i	$0.735074\pi$
$678$	−21.7114	−0.833821
$679$	18.9933	0.728896
$680$	−1.43535	−0.0550433
$681$	15.6528	0.599816
$682$	22.5840	0.864787
$683$	−24.0834	−0.921524	−0.460762	−	0.887524i	$-0.652424\pi$
$683$	−24.0834	−0.921524	−0.460762	+	0.887524i	$0.652424\pi$
$684$	0.245061	0.00937013
$685$	1.53617	0.0586941
$686$	15.9618	0.609426
$687$	54.3183	2.07237
$688$	−0.164210	−0.00626046
$689$	0	0
$690$	−0.664874	−0.0253113
$691$	2.01447	0.0766342	0.0383171	−	0.999266i	$-0.487800\pi$
$691$	2.01447	0.0766342	0.0383171	+	0.999266i	$0.487800\pi$
$692$	29.2223	1.11087
$693$	20.5090	0.779073
$694$	1.20583	0.0457728
$695$	−3.63235	−0.137783
$696$	28.0127	1.06182
$697$	−25.0248	−0.947880
$698$	−11.3763	−0.430598
$699$	6.87800	0.260150
$700$	−15.7952	−0.597004
$701$	−48.8189	−1.84387	−0.921933	−	0.387350i	$-0.873390\pi$
$701$	−48.8189	−1.84387	−0.921933	+	0.387350i	$0.873390\pi$
$702$	0	0
$703$	−0.501729	−0.0189231
$704$	−15.1390	−0.570572
$705$	4.08277	0.153766
$706$	−5.74871	−0.216355
$707$	31.4752	1.18375
$708$	−20.6722	−0.776908
$709$	−20.8060	−0.781385	−0.390693	−	0.920521i	$-0.627764\pi$
$709$	−20.8060	−0.781385	−0.390693	+	0.920521i	$0.627764\pi$
$710$	−1.71618	−0.0644073
$711$	20.4286	0.766134
$712$	7.76569	0.291032
$713$	−9.90648	−0.371000
$714$	9.16852	0.343123
$715$	0	0
$716$	−15.5133	−0.579761
$717$	56.5042	2.11019
$718$	15.9468	0.595128
$719$	21.4306	0.799225	0.399613	−	0.916684i	$-0.369145\pi$
$719$	21.4306	0.799225	0.399613	+	0.916684i	$0.369145\pi$
$720$	0.280831	0.0104660
$721$	−3.21073	−0.119574
$722$	−15.2306	−0.566824
$723$	45.5381	1.69358
$724$	−28.4547	−1.05751
$725$	−22.8726	−0.849468
$726$	−12.6799	−0.470597
$727$	13.4862	0.500175	0.250088	−	0.968223i	$-0.419541\pi$
$727$	13.4862	0.500175	0.250088	+	0.968223i	$0.419541\pi$
$728$	0	0
$729$	−8.02715	−0.297302
$730$	1.33406	0.0493758
$731$	−0.638792	−0.0236266
$732$	10.5864	0.391285
$733$	−43.5424	−1.60828	−0.804138	−	0.594443i	$-0.797373\pi$
$733$	−43.5424	−1.60828	−0.804138	+	0.594443i	$0.797373\pi$
$734$	0.869167	0.0320816
$735$	0.801938	0.0295799
$736$	8.70841	0.320996
$737$	−32.6165	−1.20145
$738$	−19.0465	−0.701112
$739$	20.0543	0.737709	0.368855	−	0.929487i	$-0.379750\pi$
$739$	20.0543	0.737709	0.368855	+	0.929487i	$0.379750\pi$
$740$	−1.90754	−0.0701226
$741$	0	0
$742$	19.6450	0.721191
$743$	−33.1685	−1.21684	−0.608418	−	0.793617i	$-0.708195\pi$
$743$	−33.1685	−1.21684	−0.608418	+	0.793617i	$0.708195\pi$
$744$	−40.1105	−1.47052
$745$	1.07069	0.0392270
$746$	−4.91617	−0.179994
$747$	3.29722	0.120639
$748$	12.4407	0.454878
$749$	−7.70410	−0.281502
$750$	4.42327	0.161515
$751$	39.2814	1.43340	0.716700	−	0.697382i	$-0.245652\pi$
$751$	39.2814	1.43340	0.716700	+	0.697382i	$0.245652\pi$
$752$	−4.08277	−0.148883
$753$	53.3008	1.94239
$754$	0	0
$755$	0.974181	0.0354541
$756$	6.83446	0.248567
$757$	−46.6426	−1.69526	−0.847628	−	0.530592i	$-0.821970\pi$
$757$	−46.6426	−1.69526	−0.847628	+	0.530592i	$0.821970\pi$
$758$	−1.93171	−0.0701627
$759$	14.2567	0.517484
$760$	0.0586060	0.00212586
$761$	−21.8984	−0.793818	−0.396909	−	0.917858i	$-0.629917\pi$
$761$	−21.8984	−0.793818	−0.396909	+	0.917858i	$0.629917\pi$
$762$	17.6722	0.640195
$763$	−37.0073	−1.33975
$764$	19.5899	0.708737
$765$	1.09246	0.0394979
$766$	−24.3720	−0.880595
$767$	0	0
$768$	31.8756	1.15021
$769$	46.7096	1.68439	0.842196	−	0.539172i	$-0.181263\pi$
$769$	46.7096	1.68439	0.842196	+	0.539172i	$0.181263\pi$
$770$	1.98254	0.0714458
$771$	−31.9614	−1.15106
$772$	18.4263	0.663175
$773$	−30.2416	−1.08771	−0.543857	−	0.839178i	$-0.683037\pi$
$773$	−30.2416	−1.08771	−0.543857	+	0.839178i	$0.683037\pi$
$774$	−0.486189	−0.0174757
$775$	32.7506	1.17644
$776$	21.6939	0.778767
$777$	30.1444	1.08142
$778$	−12.7836	−0.458315
$779$	1.02177	0.0366087
$780$	0	0
$781$	36.7995	1.31679
$782$	2.58642	0.0924901
$783$	9.89679	0.353682
$784$	−0.801938	−0.0286406
$785$	1.10023	0.0392688
$786$	11.8509	0.422706
$787$	28.7023	1.02313	0.511563	−	0.859246i	$-0.329067\pi$
$787$	28.7023	1.02313	0.511563	+	0.859246i	$0.329067\pi$
$788$	0.760316	0.0270851
$789$	38.4131	1.36754
$790$	1.97477	0.0702592
$791$	−28.3980	−1.00972
$792$	23.4252	0.832378
$793$	0	0
$794$	13.5646	0.481391
$795$	5.76809	0.204573
$796$	−15.5929	−0.552674
$797$	−18.5418	−0.656785	−0.328392	−	0.944541i	$-0.606507\pi$
$797$	−18.5418	−0.656785	−0.328392	+	0.944541i	$0.606507\pi$
$798$	−0.374354	−0.0132520
$799$	−15.8823	−0.561876
$800$	−28.7899	−1.01788
$801$	−5.91053	−0.208838
$802$	21.3817	0.755012
$803$	−28.6058	−1.00948
$804$	23.4155	0.825801
$805$	−0.869641	−0.0306508
$806$	0	0
$807$	14.5308	0.511508
$808$	35.9506	1.26474
$809$	−10.0677	−0.353962	−0.176981	−	0.984214i	$-0.556633\pi$
$809$	−10.0677	−0.353962	−0.176981	+	0.984214i	$0.556633\pi$
$810$	−2.16852	−0.0761941
$811$	−10.0285	−0.352147	−0.176074	−	0.984377i	$-0.556340\pi$
$811$	−10.0285	−0.352147	−0.176074	+	0.984377i	$0.556340\pi$
$812$	14.8103	0.519740
$813$	−14.4886	−0.508137
$814$	−19.3860	−0.679478
$815$	3.99090	0.139795
$816$	−2.69202	−0.0942396
$817$	0.0260821	0.000912498 0
$818$	22.8683	0.799572
$819$	0	0
$820$	3.88471	0.135660
$821$	−26.1704	−0.913355	−0.456677	−	0.889632i	$-0.650961\pi$
$821$	−26.1704	−0.913355	−0.456677	+	0.889632i	$0.650961\pi$
$822$	−11.2078	−0.390915
$823$	1.82238	0.0635242	0.0317621	−	0.999495i	$-0.489888\pi$
$823$	1.82238	0.0635242	0.0317621	+	0.999495i	$0.489888\pi$
$824$	−3.66727	−0.127755
$825$	−47.1323	−1.64094
$826$	12.8151	0.445894
$827$	−32.2941	−1.12298	−0.561488	−	0.827485i	$-0.689771\pi$
$827$	−32.2941	−1.12298	−0.561488	+	0.827485i	$0.689771\pi$
$828$	−4.15346	−0.144343
$829$	15.1002	0.524453	0.262226	−	0.965006i	$-0.415543\pi$
$829$	15.1002	0.524453	0.262226	+	0.965006i	$0.415543\pi$
$830$	0.318732	0.0110634
$831$	−30.2446	−1.04917
$832$	0	0
$833$	−3.11960	−0.108088
$834$	26.5013	0.917663
$835$	3.98062	0.137755
$836$	−0.507960	−0.0175682
$837$	−14.1709	−0.489818
$838$	−23.7448	−0.820250
$839$	−32.9965	−1.13917	−0.569584	−	0.821933i	$-0.692896\pi$
$839$	−32.9965	−1.13917	−0.569584	+	0.821933i	$0.692896\pi$
$840$	−3.52111	−0.121490
$841$	−7.55363	−0.260470
$842$	−9.35105	−0.322258
$843$	−11.3177	−0.389801
$844$	−11.9196	−0.410290
$845$	0	0
$846$	−12.0881	−0.415599
$847$	−16.5851	−0.569870
$848$	−5.76809	−0.198077
$849$	−49.7211	−1.70642
$850$	−8.55065	−0.293285
$851$	8.50365	0.291501
$852$	−26.4185	−0.905082
$853$	−37.7802	−1.29357	−0.646784	−	0.762673i	$-0.723887\pi$
$853$	−37.7802	−1.29357	−0.646784	+	0.762673i	$0.723887\pi$
$854$	−6.56273	−0.224572
$855$	−0.0446055	−0.00152547
$856$	−8.79954	−0.300762
$857$	27.3623	0.934677	0.467339	−	0.884078i	$-0.345213\pi$
$857$	27.3623	0.934677	0.467339	+	0.884078i	$0.345213\pi$
$858$	0	0
$859$	−20.0629	−0.684538	−0.342269	−	0.939602i	$-0.611195\pi$
$859$	−20.0629	−0.684538	−0.342269	+	0.939602i	$0.611195\pi$
$860$	0.0991626	0.00338142
$861$	−61.3889	−2.09213
$862$	3.48773	0.118792
$863$	−6.14483	−0.209173	−0.104586	−	0.994516i	$-0.533352\pi$
$863$	−6.14483	−0.209173	−0.104586	+	0.994516i	$0.533352\pi$
$864$	12.4571	0.423800
$865$	−5.31900	−0.180851
$866$	−11.5386	−0.392096
$867$	27.7265	0.941640
$868$	−21.2064	−0.719793
$869$	−42.3443	−1.43643
$870$	−2.06100	−0.0698744
$871$	0	0
$872$	−42.2693	−1.43142
$873$	−16.5114	−0.558827
$874$	−0.105604	−0.00357212
$875$	5.78554	0.195587
$876$	20.5362	0.693853
$877$	−13.5077	−0.456123	−0.228061	−	0.973647i	$-0.573239\pi$
$877$	−13.5077	−0.456123	−0.228061	+	0.973647i	$0.573239\pi$
$878$	−16.2252	−0.547574
$879$	−33.5840	−1.13276
$880$	−0.582105	−0.0196228
$881$	−5.23431	−0.176348	−0.0881741	−	0.996105i	$-0.528103\pi$
$881$	−5.23431	−0.176348	−0.0881741	+	0.996105i	$0.528103\pi$
$882$	−2.37435	−0.0799487
$883$	−4.57301	−0.153894	−0.0769470	−	0.997035i	$-0.524517\pi$
$883$	−4.57301	−0.153894	−0.0769470	+	0.997035i	$0.524517\pi$
$884$	0	0
$885$	3.76271	0.126482
$886$	6.51334	0.218820
$887$	−1.64071	−0.0550897	−0.0275448	−	0.999621i	$-0.508769\pi$
$887$	−1.64071	−0.0550897	−0.0275448	+	0.999621i	$0.508769\pi$
$888$	34.4306	1.15541
$889$	23.1148	0.775246
$890$	−0.571352	−0.0191517
$891$	46.4989	1.55777
$892$	3.06531	0.102634
$893$	0.648481	0.0217006
$894$	−7.81163	−0.261260
$895$	2.82371	0.0943861
$896$	20.7396	0.692862
$897$	0	0
$898$	10.0175	0.334287
$899$	−30.7084	−1.02418
$900$	13.7313	0.457708
$901$	−22.4383	−0.747529
$902$	39.4795	1.31452
$903$	−1.56704	−0.0521478
$904$	−32.4359	−1.07880
$905$	5.17928	0.172165
$906$	−7.10752	−0.236132
$907$	8.10215	0.269027	0.134514	−	0.990912i	$-0.457053\pi$
$907$	8.10215	0.269027	0.134514	+	0.990912i	$0.457053\pi$
$908$	9.45234	0.313687
$909$	−27.3623	−0.907549
$910$	0	0
$911$	−9.18119	−0.304187	−0.152093	−	0.988366i	$-0.548601\pi$
$911$	−9.18119	−0.304187	−0.152093	+	0.988366i	$0.548601\pi$
$912$	0.109916	0.00363969
$913$	−6.83446	−0.226188
$914$	4.79656	0.158656
$915$	−1.92692	−0.0637020
$916$	32.8015	1.08379
$917$	15.5007	0.511877
$918$	3.69979	0.122111
$919$	27.5036	0.907262	0.453631	−	0.891190i	$-0.350128\pi$
$919$	27.5036	0.907262	0.453631	+	0.891190i	$0.350128\pi$
$920$	−0.993295	−0.0327480
$921$	−42.9831	−1.41634
$922$	−1.64933	−0.0543180
$923$	0	0
$924$	30.5187	1.00399
$925$	−28.1129	−0.924346
$926$	−6.77586	−0.222668
$927$	2.79118	0.0916745
$928$	26.9946	0.886142
$929$	24.2131	0.794407	0.397203	−	0.917731i	$-0.369981\pi$
$929$	24.2131	0.794407	0.397203	+	0.917731i	$0.369981\pi$
$930$	2.95108	0.0967698
$931$	0.127375	0.00417454
$932$	4.15346	0.136051
$933$	0.606268	0.0198483
$934$	26.8761	0.879412
$935$	−2.26444	−0.0740550
$936$	0	0
$937$	11.1830	0.365333	0.182666	−	0.983175i	$-0.441527\pi$
$937$	11.1830	0.365333	0.182666	+	0.983175i	$0.441527\pi$
$938$	−14.5157	−0.473955
$939$	52.5652	1.71540
$940$	2.46548	0.0804152
$941$	−15.9638	−0.520404	−0.260202	−	0.965554i	$-0.583789\pi$
$941$	−15.9638	−0.520404	−0.260202	+	0.965554i	$0.583789\pi$
$942$	−8.02715	−0.261539
$943$	−17.3177	−0.563941
$944$	−3.76271	−0.122466
$945$	−1.24400	−0.0404672
$946$	1.00777	0.0327654
$947$	6.51466	0.211698	0.105849	−	0.994382i	$-0.466244\pi$
$947$	6.51466	0.211698	0.105849	+	0.994382i	$0.466244\pi$
$948$	30.3991	0.987317
$949$	0	0
$950$	0.349126	0.0113271
$951$	31.4470	1.01974
$952$	13.6974	0.443935
$953$	47.6469	1.54344	0.771718	−	0.635965i	$-0.219398\pi$
$953$	47.6469	1.54344	0.771718	+	0.635965i	$0.219398\pi$
$954$	−17.0780	−0.552920
$955$	−3.56571	−0.115384
$956$	34.1215	1.10357
$957$	44.1933	1.42857
$958$	−19.8329	−0.640773
$959$	−14.6595	−0.473380
$960$	−1.97823	−0.0638471
$961$	12.9705	0.418402
$962$	0	0
$963$	6.69740	0.215821
$964$	27.4993	0.885694
$965$	−3.35391	−0.107966
$966$	6.34481	0.204141
$967$	−43.8122	−1.40891	−0.704453	−	0.709751i	$-0.748808\pi$
$967$	−43.8122	−1.40891	−0.704453	+	0.709751i	$0.748808\pi$
$968$	−18.9433	−0.608861
$969$	0.427583	0.0137360
$970$	−1.59611	−0.0512479
$971$	4.29483	0.137828	0.0689139	−	0.997623i	$-0.478047\pi$
$971$	4.29483	0.137828	0.0689139	+	0.997623i	$0.478047\pi$
$972$	−24.6823	−0.791686
$973$	34.6631	1.11125
$974$	−30.2776	−0.970156
$975$	0	0
$976$	1.92692	0.0616792
$977$	26.8019	0.857470	0.428735	−	0.903430i	$-0.358959\pi$
$977$	26.8019	0.857470	0.428735	+	0.903430i	$0.358959\pi$
$978$	−29.1172	−0.931066
$979$	12.2513	0.391553
$980$	0.484271	0.0154695
$981$	32.1715	1.02716
$982$	25.1094	0.801275
$983$	27.2495	0.869124	0.434562	−	0.900642i	$-0.356903\pi$
$983$	27.2495	0.869124	0.434562	+	0.900642i	$0.356903\pi$
$984$	−70.1178	−2.23527
$985$	−0.138391	−0.00440951
$986$	8.01746	0.255328
$987$	−38.9614	−1.24015
$988$	0	0
$989$	−0.442058	−0.0140566
$990$	−1.72348	−0.0547758
$991$	24.3889	0.774740	0.387370	−	0.921924i	$-0.373384\pi$
$991$	24.3889	0.774740	0.387370	+	0.921924i	$0.373384\pi$
$992$	−38.6528	−1.22723
$993$	−40.0441	−1.27076
$994$	16.3773	0.519458
$995$	2.83818	0.0899764
$996$	4.90648	0.155468
$997$	31.3207	0.991935	0.495967	−	0.868341i	$-0.334814\pi$
$997$	31.3207	0.991935	0.495967	+	0.868341i	$0.334814\pi$
$998$	17.2314	0.545452
$999$	12.1642	0.384859

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.a.b.1.3	✓	3
3.2	odd	2		1521.2.a.r.1.1		3
4.3	odd	2		2704.2.a.z.1.3		3
5.4	even	2		4225.2.a.bg.1.1		3
7.6	odd	2		8281.2.a.bf.1.3		3
13.2	odd	12		169.2.e.b.147.5		12
13.3	even	3		169.2.c.c.22.1		6
13.4	even	6		169.2.c.b.146.3		6
13.5	odd	4		169.2.b.b.168.2		6
13.6	odd	12		169.2.e.b.23.2		12
13.7	odd	12		169.2.e.b.23.5		12
13.8	odd	4		169.2.b.b.168.5		6
13.9	even	3		169.2.c.c.146.1		6
13.10	even	6		169.2.c.b.22.3		6
13.11	odd	12		169.2.e.b.147.2		12
13.12	even	2		169.2.a.c.1.1	yes	3
39.5	even	4		1521.2.b.l.1351.5		6
39.8	even	4		1521.2.b.l.1351.2		6
39.38	odd	2		1521.2.a.o.1.3		3
52.31	even	4		2704.2.f.o.337.5		6
52.47	even	4		2704.2.f.o.337.6		6
52.51	odd	2		2704.2.a.ba.1.3		3
65.64	even	2		4225.2.a.bb.1.3		3
91.90	odd	2		8281.2.a.bj.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.3	✓	3	1.1	even	1	trivial
169.2.a.c.1.1	yes	3	13.12	even	2
169.2.b.b.168.2		6	13.5	odd	4
169.2.b.b.168.5		6	13.8	odd	4
169.2.c.b.22.3		6	13.10	even	6
169.2.c.b.146.3		6	13.4	even	6
169.2.c.c.22.1		6	13.3	even	3
169.2.c.c.146.1		6	13.9	even	3
169.2.e.b.23.2		12	13.6	odd	12
169.2.e.b.23.5		12	13.7	odd	12
169.2.e.b.147.2		12	13.11	odd	12
169.2.e.b.147.5		12	13.2	odd	12
1521.2.a.o.1.3		3	39.38	odd	2
1521.2.a.r.1.1		3	3.2	odd	2
1521.2.b.l.1351.2		6	39.8	even	4
1521.2.b.l.1351.5		6	39.5	even	4
2704.2.a.z.1.3		3	4.3	odd	2
2704.2.a.ba.1.3		3	52.51	odd	2
2704.2.f.o.337.5		6	52.31	even	4
2704.2.f.o.337.6		6	52.47	even	4
4225.2.a.bb.1.3		3	65.64	even	2
4225.2.a.bg.1.1		3	5.4	even	2
8281.2.a.bf.1.3		3	7.6	odd	2
8281.2.a.bj.1.1		3	91.90	odd	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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q+0.801938 q^{2} -2.24698 q^{3} -1.35690 q^{4} +0.246980 q^{5} -1.80194 q^{6} -2.35690 q^{7} -2.69202 q^{8} +2.04892 q^{9} +O(q^{10})\)	\(q+0.801938 q^{2} -2.24698 q^{3} -1.35690 q^{4} +0.246980 q^{5} -1.80194 q^{6} -2.35690 q^{7} -2.69202 q^{8} +2.04892 q^{9} +0.198062 q^{10} -4.24698 q^{11} +3.04892 q^{12} -1.89008 q^{14} -0.554958 q^{15} +0.554958 q^{16} +2.15883 q^{17} +1.64310 q^{18} -0.0881460 q^{19} -0.335126 q^{20} +5.29590 q^{21} -3.40581 q^{22} +1.49396 q^{23} +6.04892 q^{24} -4.93900 q^{25} +2.13706 q^{27} +3.19806 q^{28} +4.63102 q^{29} -0.445042 q^{30} -6.63102 q^{31} +5.82908 q^{32} +9.54288 q^{33} +1.73125 q^{34} -0.582105 q^{35} -2.78017 q^{36} +5.69202 q^{37} -0.0706876 q^{38} -0.664874 q^{40} -11.5918 q^{41} +4.24698 q^{42} -0.295897 q^{43} +5.76271 q^{44} +0.506041 q^{45} +1.19806 q^{46} -7.35690 q^{47} -1.24698 q^{48} -1.44504 q^{49} -3.96077 q^{50} -4.85086 q^{51} -10.3937 q^{53} +1.71379 q^{54} -1.04892 q^{55} +6.34481 q^{56} +0.198062 q^{57} +3.71379 q^{58} -6.78017 q^{59} +0.753020 q^{60} +3.47219 q^{61} -5.31767 q^{62} -4.82908 q^{63} +3.56465 q^{64} +7.65279 q^{66} +7.67994 q^{67} -2.92931 q^{68} -3.35690 q^{69} -0.466812 q^{70} -8.66487 q^{71} -5.51573 q^{72} +6.73556 q^{73} +4.56465 q^{74} +11.0978 q^{75} +0.119605 q^{76} +10.0097 q^{77} +9.97046 q^{79} +0.137063 q^{80} -10.9487 q^{81} -9.29590 q^{82} +1.60925 q^{83} -7.18598 q^{84} +0.533188 q^{85} -0.237291 q^{86} -10.4058 q^{87} +11.4330 q^{88} -2.88471 q^{89} +0.405813 q^{90} -2.02715 q^{92} +14.8998 q^{93} -5.89977 q^{94} -0.0217703 q^{95} -13.0978 q^{96} -8.05861 q^{97} -1.15883 q^{98} -8.70171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^2 - 2 * q^3 - 4 * q^5 - q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9	\( 3 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9} + 5 q^{10} - 8 q^{11} - 5 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{17} + 9 q^{18} - 4 q^{19} + 2 q^{21} + 3 q^{22} - 5 q^{23} + 9 q^{24} - 5 q^{25} + q^{27} + 14 q^{28} - q^{29} - q^{30} - 5 q^{31} + 7 q^{32} + 10 q^{33} + 13 q^{34} + 4 q^{35} - 7 q^{36} + 12 q^{37} + 12 q^{38} - 3 q^{40} - 7 q^{41} + 8 q^{42} + 13 q^{43} + 11 q^{45} + 8 q^{46} - 18 q^{47} + q^{48} - 4 q^{49} + q^{50} - q^{51} + q^{53} - 3 q^{54} + 6 q^{55} - 4 q^{56} + 5 q^{57} + 3 q^{58} - 19 q^{59} + 7 q^{60} + 4 q^{61} + q^{62} - 4 q^{63} - 11 q^{64} + 5 q^{66} - q^{67} - 21 q^{68} - 6 q^{69} + 2 q^{70} - 27 q^{71} - 4 q^{72} + 9 q^{73} - 8 q^{74} + 15 q^{75} - 21 q^{76} + 8 q^{77} - 5 q^{79} - 5 q^{80} - q^{81} - 14 q^{82} - 7 q^{83} - 7 q^{84} + 5 q^{85} - 18 q^{86} - 18 q^{87} + 15 q^{88} - 11 q^{89} - 12 q^{90} + 22 q^{93} + 5 q^{94} + 3 q^{95} - 21 q^{96} + 7 q^{97} + 5 q^{98} + q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 - 2 * q^3 - 4 * q^5 - q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9 + 5 * q^10 - 8 * q^11 - 5 * q^14 - 2 * q^15 + 2 * q^16 - 2 * q^17 + 9 * q^18 - 4 * q^19 + 2 * q^21 + 3 * q^22 - 5 * q^23 + 9 * q^24 - 5 * q^25 + q^27 + 14 * q^28 - q^29 - q^30 - 5 * q^31 + 7 * q^32 + 10 * q^33 + 13 * q^34 + 4 * q^35 - 7 * q^36 + 12 * q^37 + 12 * q^38 - 3 * q^40 - 7 * q^41 + 8 * q^42 + 13 * q^43 + 11 * q^45 + 8 * q^46 - 18 * q^47 + q^48 - 4 * q^49 + q^50 - q^51 + q^53 - 3 * q^54 + 6 * q^55 - 4 * q^56 + 5 * q^57 + 3 * q^58 - 19 * q^59 + 7 * q^60 + 4 * q^61 + q^62 - 4 * q^63 - 11 * q^64 + 5 * q^66 - q^67 - 21 * q^68 - 6 * q^69 + 2 * q^70 - 27 * q^71 - 4 * q^72 + 9 * q^73 - 8 * q^74 + 15 * q^75 - 21 * q^76 + 8 * q^77 - 5 * q^79 - 5 * q^80 - q^81 - 14 * q^82 - 7 * q^83 - 7 * q^84 + 5 * q^85 - 18 * q^86 - 18 * q^87 + 15 * q^88 - 11 * q^89 - 12 * q^90 + 22 * q^93 + 5 * q^94 + 3 * q^95 - 21 * q^96 + 7 * q^97 + 5 * q^98 + q^99

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