LMFDB - Embedded newform 1690.2.a.p.1.3

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			1.3
Root			$-1.24698$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.24698 q^{6} +2.49396 q^{7} -1.00000 q^{8} -1.44504 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.24698 q^{6} +2.49396 q^{7} -1.00000 q^{8} -1.44504 q^{9} -1.00000 q^{10} +5.80194 q^{11} +1.24698 q^{12} -2.49396 q^{14} +1.24698 q^{15} +1.00000 q^{16} +4.29590 q^{17} +1.44504 q^{18} +4.04892 q^{19} +1.00000 q^{20} +3.10992 q^{21} -5.80194 q^{22} -3.10992 q^{23} -1.24698 q^{24} +1.00000 q^{25} -5.54288 q^{27} +2.49396 q^{28} -5.60388 q^{29} -1.24698 q^{30} -7.70171 q^{31} -1.00000 q^{32} +7.23490 q^{33} -4.29590 q^{34} +2.49396 q^{35} -1.44504 q^{36} +2.67025 q^{37} -4.04892 q^{38} -1.00000 q^{40} +12.5429 q^{41} -3.10992 q^{42} +6.98254 q^{43} +5.80194 q^{44} -1.44504 q^{45} +3.10992 q^{46} -3.87800 q^{47} +1.24698 q^{48} -0.780167 q^{49} -1.00000 q^{50} +5.35690 q^{51} -4.93362 q^{53} +5.54288 q^{54} +5.80194 q^{55} -2.49396 q^{56} +5.04892 q^{57} +5.60388 q^{58} +10.0151 q^{59} +1.24698 q^{60} -4.93362 q^{61} +7.70171 q^{62} -3.60388 q^{63} +1.00000 q^{64} -7.23490 q^{66} -8.01507 q^{67} +4.29590 q^{68} -3.87800 q^{69} -2.49396 q^{70} -5.48188 q^{71} +1.44504 q^{72} +8.67456 q^{73} -2.67025 q^{74} +1.24698 q^{75} +4.04892 q^{76} +14.4698 q^{77} -1.82371 q^{79} +1.00000 q^{80} -2.57673 q^{81} -12.5429 q^{82} -14.6407 q^{83} +3.10992 q^{84} +4.29590 q^{85} -6.98254 q^{86} -6.98792 q^{87} -5.80194 q^{88} -0.454731 q^{89} +1.44504 q^{90} -3.10992 q^{92} -9.60388 q^{93} +3.87800 q^{94} +4.04892 q^{95} -1.24698 q^{96} +8.69202 q^{97} +0.780167 q^{98} -8.38404 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.24698	0.719944	0.359972	−	0.932963i	$-0.382786\pi$
$3$	1.24698	0.719944	0.359972	+	0.932963i	$0.382786\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.24698	−0.509077
$7$	2.49396	0.942628	0.471314	−	0.881965i	$-0.343780\pi$
$7$	2.49396	0.942628	0.471314	+	0.881965i	$0.343780\pi$
$8$	−1.00000	−0.353553
$9$	−1.44504	−0.481681
$10$	−1.00000	−0.316228
$11$	5.80194	1.74935	0.874675	−	0.484710i	$-0.161075\pi$
$11$	5.80194	1.74935	0.874675	+	0.484710i	$0.161075\pi$
$12$	1.24698	0.359972
$13$	0	0
$13$	0	0
$14$	−2.49396	−0.666539
$15$	1.24698	0.321969
$16$	1.00000	0.250000
$17$	4.29590	1.04191	0.520954	−	0.853585i	$-0.325576\pi$
$17$	4.29590	1.04191	0.520954	+	0.853585i	$0.325576\pi$
$18$	1.44504	0.340600
$19$	4.04892	0.928885	0.464443	−	0.885603i	$-0.346255\pi$
$19$	4.04892	0.928885	0.464443	+	0.885603i	$0.346255\pi$
$20$	1.00000	0.223607
$21$	3.10992	0.678639
$22$	−5.80194	−1.23698
$23$	−3.10992	−0.648462	−0.324231	−	0.945978i	$-0.605106\pi$
$23$	−3.10992	−0.648462	−0.324231	+	0.945978i	$0.605106\pi$
$24$	−1.24698	−0.254539
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.54288	−1.06673
$28$	2.49396	0.471314
$29$	−5.60388	−1.04061	−0.520307	−	0.853979i	$-0.674182\pi$
$29$	−5.60388	−1.04061	−0.520307	+	0.853979i	$0.674182\pi$
$30$	−1.24698	−0.227666
$31$	−7.70171	−1.38327	−0.691634	−	0.722248i	$-0.743109\pi$
$31$	−7.70171	−1.38327	−0.691634	+	0.722248i	$0.743109\pi$
$32$	−1.00000	−0.176777
$33$	7.23490	1.25943
$34$	−4.29590	−0.736740
$35$	2.49396	0.421556
$36$	−1.44504	−0.240840
$37$	2.67025	0.438987	0.219493	−	0.975614i	$-0.429560\pi$
$37$	2.67025	0.438987	0.219493	+	0.975614i	$0.429560\pi$
$38$	−4.04892	−0.656821
$39$	0	0
$40$	−1.00000	−0.158114
$41$	12.5429	1.95887	0.979434	−	0.201764i	$-0.0646675\pi$
$41$	12.5429	1.95887	0.979434	+	0.201764i	$0.0646675\pi$
$42$	−3.10992	−0.479870
$43$	6.98254	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$43$	6.98254	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$44$	5.80194	0.874675
$45$	−1.44504	−0.215414
$46$	3.10992	0.458532
$47$	−3.87800	−0.565665	−0.282832	−	0.959169i	$-0.591274\pi$
$47$	−3.87800	−0.565665	−0.282832	+	0.959169i	$0.591274\pi$
$48$	1.24698	0.179986
$49$	−0.780167	−0.111452
$50$	−1.00000	−0.141421
$51$	5.35690	0.750115
$52$	0	0
$53$	−4.93362	−0.677685	−0.338843	−	0.940843i	$-0.610035\pi$
$53$	−4.93362	−0.677685	−0.338843	+	0.940843i	$0.610035\pi$
$54$	5.54288	0.754290
$55$	5.80194	0.782333
$56$	−2.49396	−0.333269
$57$	5.04892	0.668745
$58$	5.60388	0.735825
$59$	10.0151	1.30385	0.651925	−	0.758283i	$-0.273962\pi$
$59$	10.0151	1.30385	0.651925	+	0.758283i	$0.273962\pi$
$60$	1.24698	0.160984
$61$	−4.93362	−0.631686	−0.315843	−	0.948811i	$-0.602287\pi$
$61$	−4.93362	−0.631686	−0.315843	+	0.948811i	$0.602287\pi$
$62$	7.70171	0.978118
$63$	−3.60388	−0.454046
$64$	1.00000	0.125000
$65$	0	0
$66$	−7.23490	−0.890554
$67$	−8.01507	−0.979196	−0.489598	−	0.871948i	$-0.662856\pi$
$67$	−8.01507	−0.979196	−0.489598	+	0.871948i	$0.662856\pi$
$68$	4.29590	0.520954
$69$	−3.87800	−0.466857
$70$	−2.49396	−0.298085
$71$	−5.48188	−0.650579	−0.325290	−	0.945614i	$-0.605462\pi$
$71$	−5.48188	−0.650579	−0.325290	+	0.945614i	$0.605462\pi$
$72$	1.44504	0.170300
$73$	8.67456	1.01528	0.507640	−	0.861569i	$-0.330518\pi$
$73$	8.67456	1.01528	0.507640	+	0.861569i	$0.330518\pi$
$74$	−2.67025	−0.310410
$75$	1.24698	0.143989
$76$	4.04892	0.464443
$77$	14.4698	1.64899
$78$	0	0
$79$	−1.82371	−0.205183	−0.102592	−	0.994724i	$-0.532713\pi$
$79$	−1.82371	−0.205183	−0.102592	+	0.994724i	$0.532713\pi$
$80$	1.00000	0.111803
$81$	−2.57673	−0.286303
$82$	−12.5429	−1.38513
$83$	−14.6407	−1.60703	−0.803513	−	0.595287i	$-0.797039\pi$
$83$	−14.6407	−1.60703	−0.803513	+	0.595287i	$0.797039\pi$
$84$	3.10992	0.339320
$85$	4.29590	0.465955
$86$	−6.98254	−0.752947
$87$	−6.98792	−0.749183
$88$	−5.80194	−0.618489
$89$	−0.454731	−0.0482013	−0.0241007	−	0.999710i	$-0.507672\pi$
$89$	−0.454731	−0.0482013	−0.0241007	+	0.999710i	$0.507672\pi$
$90$	1.44504	0.152321
$91$	0	0
$92$	−3.10992	−0.324231
$93$	−9.60388	−0.995875
$94$	3.87800	0.399985
$95$	4.04892	0.415410
$96$	−1.24698	−0.127269
$97$	8.69202	0.882541	0.441271	−	0.897374i	$-0.354528\pi$
$97$	8.69202	0.882541	0.441271	+	0.897374i	$0.354528\pi$
$98$	0.780167	0.0788088
$99$	−8.38404	−0.842628
$100$	1.00000	0.100000
$101$	15.7453	1.56671	0.783355	−	0.621574i	$-0.213506\pi$
$101$	15.7453	1.56671	0.783355	+	0.621574i	$0.213506\pi$
$102$	−5.35690	−0.530412
$103$	−3.70171	−0.364740	−0.182370	−	0.983230i	$-0.558377\pi$
$103$	−3.70171	−0.364740	−0.182370	+	0.983230i	$0.558377\pi$
$104$	0	0
$105$	3.10992	0.303497
$106$	4.93362	0.479196
$107$	15.0804	1.45788	0.728938	−	0.684580i	$-0.240014\pi$
$107$	15.0804	1.45788	0.728938	+	0.684580i	$0.240014\pi$
$108$	−5.54288	−0.533364
$109$	−17.4819	−1.67446	−0.837230	−	0.546851i	$-0.815827\pi$
$109$	−17.4819	−1.67446	−0.837230	+	0.546851i	$0.815827\pi$
$110$	−5.80194	−0.553193
$111$	3.32975	0.316046
$112$	2.49396	0.235657
$113$	−8.23490	−0.774674	−0.387337	−	0.921938i	$-0.626605\pi$
$113$	−8.23490	−0.774674	−0.387337	+	0.921938i	$0.626605\pi$
$114$	−5.04892	−0.472874
$115$	−3.10992	−0.290001
$116$	−5.60388	−0.520307
$117$	0	0
$118$	−10.0151	−0.921962
$119$	10.7138	0.982132
$120$	−1.24698	−0.113833
$121$	22.6625	2.06023
$122$	4.93362	0.446669
$123$	15.6407	1.41028
$124$	−7.70171	−0.691634
$125$	1.00000	0.0894427
$126$	3.60388	0.321059
$127$	−13.0315	−1.15635	−0.578177	−	0.815911i	$-0.696236\pi$
$127$	−13.0315	−1.15635	−0.578177	+	0.815911i	$0.696236\pi$
$128$	−1.00000	−0.0883883
$129$	8.70709	0.766616
$130$	0	0
$131$	9.82908	0.858771	0.429386	−	0.903121i	$-0.358730\pi$
$131$	9.82908	0.858771	0.429386	+	0.903121i	$0.358730\pi$
$132$	7.23490	0.629717
$133$	10.0978	0.875593
$134$	8.01507	0.692396
$135$	−5.54288	−0.477055
$136$	−4.29590	−0.368370
$137$	4.75302	0.406078	0.203039	−	0.979171i	$-0.434918\pi$
$137$	4.75302	0.406078	0.203039	+	0.979171i	$0.434918\pi$
$138$	3.87800	0.330117
$139$	−2.39373	−0.203034	−0.101517	−	0.994834i	$-0.532370\pi$
$139$	−2.39373	−0.203034	−0.101517	+	0.994834i	$0.532370\pi$
$140$	2.49396	0.210778
$141$	−4.83579	−0.407247
$142$	5.48188	0.460029
$143$	0	0
$144$	−1.44504	−0.120420
$145$	−5.60388	−0.465377
$146$	−8.67456	−0.717912
$147$	−0.972853	−0.0802396
$148$	2.67025	0.219493
$149$	20.8116	1.70495	0.852477	−	0.522764i	$-0.175099\pi$
$149$	20.8116	1.70495	0.852477	+	0.522764i	$0.175099\pi$
$150$	−1.24698	−0.101815
$151$	5.95646	0.484730	0.242365	−	0.970185i	$-0.422077\pi$
$151$	5.95646	0.484730	0.242365	+	0.970185i	$0.422077\pi$
$152$	−4.04892	−0.328411
$153$	−6.20775	−0.501867
$154$	−14.4698	−1.16601
$155$	−7.70171	−0.618616
$156$	0	0
$157$	8.19567	0.654086	0.327043	−	0.945010i	$-0.393948\pi$
$157$	8.19567	0.654086	0.327043	+	0.945010i	$0.393948\pi$
$158$	1.82371	0.145086
$159$	−6.15213	−0.487896
$160$	−1.00000	−0.0790569
$161$	−7.75600	−0.611259
$162$	2.57673	0.202447
$163$	−16.8877	−1.32275	−0.661373	−	0.750057i	$-0.730026\pi$
$163$	−16.8877	−1.32275	−0.661373	+	0.750057i	$0.730026\pi$
$164$	12.5429	0.979434
$165$	7.23490	0.563236
$166$	14.6407	1.13634
$167$	19.6582	1.52119	0.760597	−	0.649224i	$-0.224906\pi$
$167$	19.6582	1.52119	0.760597	+	0.649224i	$0.224906\pi$
$168$	−3.10992	−0.239935
$169$	0	0
$170$	−4.29590	−0.329480
$171$	−5.85086	−0.447426
$172$	6.98254	0.532414
$173$	−16.8659	−1.28229	−0.641146	−	0.767419i	$-0.721541\pi$
$173$	−16.8659	−1.28229	−0.641146	+	0.767419i	$0.721541\pi$
$174$	6.98792	0.529753
$175$	2.49396	0.188526
$176$	5.80194	0.437338
$177$	12.4886	0.938699
$178$	0.454731	0.0340835
$179$	−5.99761	−0.448282	−0.224141	−	0.974557i	$-0.571958\pi$
$179$	−5.99761	−0.448282	−0.224141	+	0.974557i	$0.571958\pi$
$180$	−1.44504	−0.107707
$181$	−16.5676	−1.23146	−0.615731	−	0.787956i	$-0.711139\pi$
$181$	−16.5676	−1.23146	−0.615731	+	0.787956i	$0.711139\pi$
$182$	0	0
$183$	−6.15213	−0.454778
$184$	3.10992	0.229266
$185$	2.67025	0.196321
$186$	9.60388	0.704190
$187$	24.9245	1.82266
$188$	−3.87800	−0.282832
$189$	−13.8237	−1.00553
$190$	−4.04892	−0.293739
$191$	12.7922	0.925615	0.462807	−	0.886459i	$-0.346842\pi$
$191$	12.7922	0.925615	0.462807	+	0.886459i	$0.346842\pi$
$192$	1.24698	0.0899930
$193$	−23.5851	−1.69769	−0.848846	−	0.528640i	$-0.822702\pi$
$193$	−23.5851	−1.69769	−0.848846	+	0.528640i	$0.822702\pi$
$194$	−8.69202	−0.624051
$195$	0	0
$196$	−0.780167	−0.0557262
$197$	−1.97584	−0.140773	−0.0703863	−	0.997520i	$-0.522423\pi$
$197$	−1.97584	−0.140773	−0.0703863	+	0.997520i	$0.522423\pi$
$198$	8.38404	0.595828
$199$	0.835790	0.0592476	0.0296238	−	0.999561i	$-0.490569\pi$
$199$	0.835790	0.0592476	0.0296238	+	0.999561i	$0.490569\pi$
$200$	−1.00000	−0.0707107
$201$	−9.99462	−0.704966
$202$	−15.7453	−1.10783
$203$	−13.9758	−0.980911
$204$	5.35690	0.375058
$205$	12.5429	0.876032
$206$	3.70171	0.257910
$207$	4.49396	0.312352
$208$	0	0
$209$	23.4916	1.62495
$210$	−3.10992	−0.214605
$211$	9.36227	0.644525	0.322263	−	0.946650i	$-0.395557\pi$
$211$	9.36227	0.644525	0.322263	+	0.946650i	$0.395557\pi$
$212$	−4.93362	−0.338843
$213$	−6.83579	−0.468381
$214$	−15.0804	−1.03087
$215$	6.98254	0.476205
$216$	5.54288	0.377145
$217$	−19.2078	−1.30391
$218$	17.4819	1.18402
$219$	10.8170	0.730945
$220$	5.80194	0.391167
$221$	0	0
$222$	−3.32975	−0.223478
$223$	6.31767	0.423062	0.211531	−	0.977371i	$-0.432155\pi$
$223$	6.31767	0.423062	0.211531	+	0.977371i	$0.432155\pi$
$224$	−2.49396	−0.166635
$225$	−1.44504	−0.0963361
$226$	8.23490	0.547777
$227$	−4.89440	−0.324852	−0.162426	−	0.986721i	$-0.551932\pi$
$227$	−4.89440	−0.324852	−0.162426	+	0.986721i	$0.551932\pi$
$228$	5.04892	0.334373
$229$	0.689629	0.0455719	0.0227860	−	0.999740i	$-0.492746\pi$
$229$	0.689629	0.0455719	0.0227860	+	0.999740i	$0.492746\pi$
$230$	3.10992	0.205062
$231$	18.0435	1.18718
$232$	5.60388	0.367912
$233$	14.1806	0.929002	0.464501	−	0.885573i	$-0.346234\pi$
$233$	14.1806	0.929002	0.464501	+	0.885573i	$0.346234\pi$
$234$	0	0
$235$	−3.87800	−0.252973
$236$	10.0151	0.651925
$237$	−2.27413	−0.147720
$238$	−10.7138	−0.694472
$239$	25.7754	1.66727	0.833635	−	0.552315i	$-0.186255\pi$
$239$	25.7754	1.66727	0.833635	+	0.552315i	$0.186255\pi$
$240$	1.24698	0.0804922
$241$	7.86725	0.506774	0.253387	−	0.967365i	$-0.418455\pi$
$241$	7.86725	0.506774	0.253387	+	0.967365i	$0.418455\pi$
$242$	−22.6625	−1.45680
$243$	13.4155	0.860605
$244$	−4.93362	−0.315843
$245$	−0.780167	−0.0498431
$246$	−15.6407	−0.997215
$247$	0	0
$248$	7.70171	0.489059
$249$	−18.2567	−1.15697
$250$	−1.00000	−0.0632456
$251$	19.9782	1.26101	0.630507	−	0.776183i	$-0.282847\pi$
$251$	19.9782	1.26101	0.630507	+	0.776183i	$0.282847\pi$
$252$	−3.60388	−0.227023
$253$	−18.0435	−1.13439
$254$	13.0315	0.817666
$255$	5.35690	0.335462
$256$	1.00000	0.0625000
$257$	−25.4523	−1.58767	−0.793837	−	0.608131i	$-0.791919\pi$
$257$	−25.4523	−1.58767	−0.793837	+	0.608131i	$0.791919\pi$
$258$	−8.70709	−0.542080
$259$	6.65950	0.413801
$260$	0	0
$261$	8.09783	0.501243
$262$	−9.82908	−0.607243
$263$	−19.9758	−1.23176	−0.615881	−	0.787839i	$-0.711200\pi$
$263$	−19.9758	−1.23176	−0.615881	+	0.787839i	$0.711200\pi$
$264$	−7.23490	−0.445277
$265$	−4.93362	−0.303070
$266$	−10.0978	−0.619138
$267$	−0.567040	−0.0347023
$268$	−8.01507	−0.489598
$269$	−16.1414	−0.984157	−0.492079	−	0.870551i	$-0.663763\pi$
$269$	−16.1414	−0.984157	−0.492079	+	0.870551i	$0.663763\pi$
$270$	5.54288	0.337329
$271$	15.1051	0.917571	0.458786	−	0.888547i	$-0.348285\pi$
$271$	15.1051	0.917571	0.458786	+	0.888547i	$0.348285\pi$
$272$	4.29590	0.260477
$273$	0	0
$274$	−4.75302	−0.287140
$275$	5.80194	0.349870
$276$	−3.87800	−0.233428
$277$	−24.4940	−1.47170	−0.735850	−	0.677145i	$-0.763217\pi$
$277$	−24.4940	−1.47170	−0.735850	+	0.677145i	$0.763217\pi$
$278$	2.39373	0.143566
$279$	11.1293	0.666293
$280$	−2.49396	−0.149043
$281$	23.3381	1.39223	0.696117	−	0.717928i	$-0.254909\pi$
$281$	23.3381	1.39223	0.696117	+	0.717928i	$0.254909\pi$
$282$	4.83579	0.287967
$283$	−11.4222	−0.678980	−0.339490	−	0.940610i	$-0.610254\pi$
$283$	−11.4222	−0.678980	−0.339490	+	0.940610i	$0.610254\pi$
$284$	−5.48188	−0.325290
$285$	5.04892	0.299072
$286$	0	0
$287$	31.2814	1.84648
$288$	1.44504	0.0851499
$289$	1.45473	0.0855724
$290$	5.60388	0.329071
$291$	10.8388	0.635380
$292$	8.67456	0.507640
$293$	−24.4155	−1.42637	−0.713184	−	0.700976i	$-0.752748\pi$
$293$	−24.4155	−1.42637	−0.713184	+	0.700976i	$0.752748\pi$
$294$	0.972853	0.0567379
$295$	10.0151	0.583100
$296$	−2.67025	−0.155205
$297$	−32.1594	−1.86608
$298$	−20.8116	−1.20559
$299$	0	0
$300$	1.24698	0.0719944
$301$	17.4142	1.00374
$302$	−5.95646	−0.342756
$303$	19.6340	1.12794
$304$	4.04892	0.232221
$305$	−4.93362	−0.282499
$306$	6.20775	0.354874
$307$	−7.70410	−0.439696	−0.219848	−	0.975534i	$-0.570556\pi$
$307$	−7.70410	−0.439696	−0.219848	+	0.975534i	$0.570556\pi$
$308$	14.4698	0.824493
$309$	−4.61596	−0.262593
$310$	7.70171	0.437428
$311$	4.71379	0.267295	0.133647	−	0.991029i	$-0.457331\pi$
$311$	4.71379	0.267295	0.133647	+	0.991029i	$0.457331\pi$
$312$	0	0
$313$	8.49157	0.479972	0.239986	−	0.970776i	$-0.422857\pi$
$313$	8.49157	0.479972	0.239986	+	0.970776i	$0.422857\pi$
$314$	−8.19567	−0.462508
$315$	−3.60388	−0.203055
$316$	−1.82371	−0.102592
$317$	27.0508	1.51933	0.759663	−	0.650317i	$-0.225364\pi$
$317$	27.0508	1.51933	0.759663	+	0.650317i	$0.225364\pi$
$318$	6.15213	0.344994
$319$	−32.5133	−1.82040
$320$	1.00000	0.0559017
$321$	18.8049	1.04959
$322$	7.75600	0.432225
$323$	17.3937	0.967813
$324$	−2.57673	−0.143152
$325$	0	0
$326$	16.8877	0.935323
$327$	−21.7995	−1.20552
$328$	−12.5429	−0.692564
$329$	−9.67158	−0.533211
$330$	−7.23490	−0.398268
$331$	−28.7439	−1.57991	−0.789954	−	0.613166i	$-0.789896\pi$
$331$	−28.7439	−1.57991	−0.789954	+	0.613166i	$0.789896\pi$
$332$	−14.6407	−0.803513
$333$	−3.85862	−0.211451
$334$	−19.6582	−1.07565
$335$	−8.01507	−0.437910
$336$	3.10992	0.169660
$337$	−34.5972	−1.88463	−0.942314	−	0.334730i	$-0.891355\pi$
$337$	−34.5972	−1.88463	−0.942314	+	0.334730i	$0.891355\pi$
$338$	0	0
$339$	−10.2687	−0.557722
$340$	4.29590	0.232978
$341$	−44.6848	−2.41982
$342$	5.85086	0.316378
$343$	−19.4034	−1.04769
$344$	−6.98254	−0.376473
$345$	−3.87800	−0.208785
$346$	16.8659	0.906718
$347$	−28.3043	−1.51945	−0.759726	−	0.650243i	$-0.774667\pi$
$347$	−28.3043	−1.51945	−0.759726	+	0.650243i	$0.774667\pi$
$348$	−6.98792	−0.374592
$349$	16.8310	0.900943	0.450471	−	0.892791i	$-0.351256\pi$
$349$	16.8310	0.900943	0.450471	+	0.892791i	$0.351256\pi$
$350$	−2.49396	−0.133308
$351$	0	0
$352$	−5.80194	−0.309244
$353$	−23.7429	−1.26370	−0.631852	−	0.775089i	$-0.717705\pi$
$353$	−23.7429	−1.26370	−0.631852	+	0.775089i	$0.717705\pi$
$354$	−12.4886	−0.663761
$355$	−5.48188	−0.290948
$356$	−0.454731	−0.0241007
$357$	13.3599	0.707080
$358$	5.99761	0.316983
$359$	−26.8310	−1.41609	−0.708043	−	0.706169i	$-0.750422\pi$
$359$	−26.8310	−1.41609	−0.708043	+	0.706169i	$0.750422\pi$
$360$	1.44504	0.0761604
$361$	−2.60627	−0.137172
$362$	16.5676	0.870775
$363$	28.2597	1.48325
$364$	0	0
$365$	8.67456	0.454047
$366$	6.15213	0.321577
$367$	−18.0978	−0.944699	−0.472350	−	0.881411i	$-0.656594\pi$
$367$	−18.0978	−0.944699	−0.472350	+	0.881411i	$0.656594\pi$
$368$	−3.10992	−0.162116
$369$	−18.1250	−0.943549
$370$	−2.67025	−0.138820
$371$	−12.3043	−0.638805
$372$	−9.60388	−0.497938
$373$	14.3720	0.744152	0.372076	−	0.928202i	$-0.378646\pi$
$373$	14.3720	0.744152	0.372076	+	0.928202i	$0.378646\pi$
$374$	−24.9245	−1.28882
$375$	1.24698	0.0643937
$376$	3.87800	0.199993
$377$	0	0
$378$	13.8237	0.711015
$379$	−26.1806	−1.34481	−0.672404	−	0.740185i	$-0.734738\pi$
$379$	−26.1806	−1.34481	−0.672404	+	0.740185i	$0.734738\pi$
$380$	4.04892	0.207705
$381$	−16.2500	−0.832511
$382$	−12.7922	−0.654508
$383$	11.3491	0.579913	0.289957	−	0.957040i	$-0.406359\pi$
$383$	11.3491	0.579913	0.289957	+	0.957040i	$0.406359\pi$
$384$	−1.24698	−0.0636347
$385$	14.4698	0.737449
$386$	23.5851	1.20045
$387$	−10.0901	−0.512907
$388$	8.69202	0.441271
$389$	−12.8465	−0.651346	−0.325673	−	0.945483i	$-0.605591\pi$
$389$	−12.8465	−0.651346	−0.325673	+	0.945483i	$0.605591\pi$
$390$	0	0
$391$	−13.3599	−0.675638
$392$	0.780167	0.0394044
$393$	12.2567	0.618267
$394$	1.97584	0.0995412
$395$	−1.82371	−0.0917607
$396$	−8.38404	−0.421314
$397$	18.7439	0.940731	0.470365	−	0.882472i	$-0.344122\pi$
$397$	18.7439	0.940731	0.470365	+	0.882472i	$0.344122\pi$
$398$	−0.835790	−0.0418943
$399$	12.5918	0.630378
$400$	1.00000	0.0500000
$401$	19.8213	0.989829	0.494915	−	0.868942i	$-0.335199\pi$
$401$	19.8213	0.989829	0.494915	+	0.868942i	$0.335199\pi$
$402$	9.99462	0.498486
$403$	0	0
$404$	15.7453	0.783355
$405$	−2.57673	−0.128039
$406$	13.9758	0.693609
$407$	15.4926	0.767941
$408$	−5.35690	−0.265206
$409$	−18.9487	−0.936952	−0.468476	−	0.883476i	$-0.655197\pi$
$409$	−18.9487	−0.936952	−0.468476	+	0.883476i	$0.655197\pi$
$410$	−12.5429	−0.619449
$411$	5.92692	0.292353
$412$	−3.70171	−0.182370
$413$	24.9772	1.22905
$414$	−4.49396	−0.220866
$415$	−14.6407	−0.718684
$416$	0	0
$417$	−2.98493	−0.146173
$418$	−23.4916	−1.14901
$419$	1.86725	0.0912211	0.0456105	−	0.998959i	$-0.485477\pi$
$419$	1.86725	0.0912211	0.0456105	+	0.998959i	$0.485477\pi$
$420$	3.10992	0.151748
$421$	−27.3250	−1.33174	−0.665869	−	0.746069i	$-0.731939\pi$
$421$	−27.3250	−1.33174	−0.665869	+	0.746069i	$0.731939\pi$
$422$	−9.36227	−0.455748
$423$	5.60388	0.272470
$424$	4.93362	0.239598
$425$	4.29590	0.208382
$426$	6.83579	0.331195
$427$	−12.3043	−0.595445
$428$	15.0804	0.728938
$429$	0	0
$430$	−6.98254	−0.336728
$431$	1.56033	0.0751587	0.0375793	−	0.999294i	$-0.488035\pi$
$431$	1.56033	0.0751587	0.0375793	+	0.999294i	$0.488035\pi$
$432$	−5.54288	−0.266682
$433$	−24.1564	−1.16088	−0.580442	−	0.814301i	$-0.697120\pi$
$433$	−24.1564	−1.16088	−0.580442	+	0.814301i	$0.697120\pi$
$434$	19.2078	0.922002
$435$	−6.98792	−0.335045
$436$	−17.4819	−0.837230
$437$	−12.5918	−0.602347
$438$	−10.8170	−0.516856
$439$	21.3599	1.01945	0.509726	−	0.860337i	$-0.329747\pi$
$439$	21.3599	1.01945	0.509726	+	0.860337i	$0.329747\pi$
$440$	−5.80194	−0.276597
$441$	1.12737	0.0536845
$442$	0	0
$443$	−16.5351	−0.785607	−0.392803	−	0.919623i	$-0.628495\pi$
$443$	−16.5351	−0.785607	−0.392803	+	0.919623i	$0.628495\pi$
$444$	3.32975	0.158023
$445$	−0.454731	−0.0215563
$446$	−6.31767	−0.299150
$447$	25.9517	1.22747
$448$	2.49396	0.117828
$449$	6.56704	0.309918	0.154959	−	0.987921i	$-0.450475\pi$
$449$	6.56704	0.309918	0.154959	+	0.987921i	$0.450475\pi$
$450$	1.44504	0.0681199
$451$	72.7730	3.42675
$452$	−8.23490	−0.387337
$453$	7.42758	0.348978
$454$	4.89440	0.229705
$455$	0	0
$456$	−5.04892	−0.236437
$457$	−24.1715	−1.13070	−0.565348	−	0.824853i	$-0.691258\pi$
$457$	−24.1715	−1.13070	−0.565348	+	0.824853i	$0.691258\pi$
$458$	−0.689629	−0.0322242
$459$	−23.8116	−1.11143
$460$	−3.10992	−0.145001
$461$	−23.5797	−1.09822	−0.549108	−	0.835751i	$-0.685033\pi$
$461$	−23.5797	−1.09822	−0.549108	+	0.835751i	$0.685033\pi$
$462$	−18.0435	−0.839461
$463$	1.68233	0.0781846	0.0390923	−	0.999236i	$-0.487553\pi$
$463$	1.68233	0.0781846	0.0390923	+	0.999236i	$0.487553\pi$
$464$	−5.60388	−0.260153
$465$	−9.60388	−0.445369
$466$	−14.1806	−0.656904
$467$	0.376273	0.0174118	0.00870592	−	0.999962i	$-0.497229\pi$
$467$	0.376273	0.0174118	0.00870592	+	0.999962i	$0.497229\pi$
$468$	0	0
$469$	−19.9892	−0.923018
$470$	3.87800	0.178879
$471$	10.2198	0.470905
$472$	−10.0151	−0.460981
$473$	40.5123	1.86276
$474$	2.27413	0.104454
$475$	4.04892	0.185777
$476$	10.7138	0.491066
$477$	7.12929	0.326428
$478$	−25.7754	−1.17894
$479$	10.0301	0.458288	0.229144	−	0.973392i	$-0.426407\pi$
$479$	10.0301	0.458288	0.229144	+	0.973392i	$0.426407\pi$
$480$	−1.24698	−0.0569166
$481$	0	0
$482$	−7.86725	−0.358343
$483$	−9.67158	−0.440072
$484$	22.6625	1.03011
$485$	8.69202	0.394684
$486$	−13.4155	−0.608540
$487$	15.3297	0.694657	0.347329	−	0.937743i	$-0.387089\pi$
$487$	15.3297	0.694657	0.347329	+	0.937743i	$0.387089\pi$
$488$	4.93362	0.223335
$489$	−21.0586	−0.952303
$490$	0.780167	0.0352444
$491$	9.19269	0.414860	0.207430	−	0.978250i	$-0.433490\pi$
$491$	9.19269	0.414860	0.207430	+	0.978250i	$0.433490\pi$
$492$	15.6407	0.705138
$493$	−24.0737	−1.08422
$494$	0	0
$495$	−8.38404	−0.376835
$496$	−7.70171	−0.345817
$497$	−13.6716	−0.613254
$498$	18.2567	0.818101
$499$	2.74632	0.122942	0.0614710	−	0.998109i	$-0.480421\pi$
$499$	2.74632	0.122942	0.0614710	+	0.998109i	$0.480421\pi$
$500$	1.00000	0.0447214
$501$	24.5133	1.09518
$502$	−19.9782	−0.891672
$503$	−12.5676	−0.560363	−0.280181	−	0.959947i	$-0.590395\pi$
$503$	−12.5676	−0.560363	−0.280181	+	0.959947i	$0.590395\pi$
$504$	3.60388	0.160529
$505$	15.7453	0.700654
$506$	18.0435	0.802133
$507$	0	0
$508$	−13.0315	−0.578177
$509$	0.0677037	0.00300091	0.00150046	−	0.999999i	$-0.499522\pi$
$509$	0.0677037	0.00300091	0.00150046	+	0.999999i	$0.499522\pi$
$510$	−5.35690	−0.237207
$511$	21.6340	0.957032
$512$	−1.00000	−0.0441942
$513$	−22.4426	−0.990867
$514$	25.4523	1.12265
$515$	−3.70171	−0.163117
$516$	8.70709	0.383308
$517$	−22.4999	−0.989546
$518$	−6.65950	−0.292602
$519$	−21.0315	−0.923179
$520$	0	0
$521$	8.18060	0.358399	0.179199	−	0.983813i	$-0.442649\pi$
$521$	8.18060	0.358399	0.179199	+	0.983813i	$0.442649\pi$
$522$	−8.09783	−0.354433
$523$	16.9312	0.740351	0.370176	−	0.928962i	$-0.379298\pi$
$523$	16.9312	0.740351	0.370176	+	0.928962i	$0.379298\pi$
$524$	9.82908	0.429386
$525$	3.10992	0.135728
$526$	19.9758	0.870988
$527$	−33.0858	−1.44124
$528$	7.23490	0.314859
$529$	−13.3284	−0.579497
$530$	4.93362	0.214303
$531$	−14.4722	−0.628040
$532$	10.0978	0.437797
$533$	0	0
$534$	0.567040	0.0245382
$535$	15.0804	0.651982
$536$	8.01507	0.346198
$537$	−7.47889	−0.322738
$538$	16.1414	0.695904
$539$	−4.52648	−0.194969
$540$	−5.54288	−0.238527
$541$	−16.5676	−0.712298	−0.356149	−	0.934429i	$-0.615910\pi$
$541$	−16.5676	−0.712298	−0.356149	+	0.934429i	$0.615910\pi$
$542$	−15.1051	−0.648821
$543$	−20.6595	−0.886584
$544$	−4.29590	−0.184185
$545$	−17.4819	−0.748841
$546$	0	0
$547$	−3.57540	−0.152873	−0.0764365	−	0.997074i	$-0.524354\pi$
$547$	−3.57540	−0.152873	−0.0764365	+	0.997074i	$0.524354\pi$
$548$	4.75302	0.203039
$549$	7.12929	0.304271
$550$	−5.80194	−0.247395
$551$	−22.6896	−0.966611
$552$	3.87800	0.165059
$553$	−4.54825	−0.193411
$554$	24.4940	1.04065
$555$	3.32975	0.141340
$556$	−2.39373	−0.101517
$557$	−39.8926	−1.69030	−0.845152	−	0.534526i	$-0.820490\pi$
$557$	−39.8926	−1.69030	−0.845152	+	0.534526i	$0.820490\pi$
$558$	−11.1293	−0.471141
$559$	0	0
$560$	2.49396	0.105389
$561$	31.0804	1.31221
$562$	−23.3381	−0.984459
$563$	−3.64742	−0.153720	−0.0768601	−	0.997042i	$-0.524489\pi$
$563$	−3.64742	−0.153720	−0.0768601	+	0.997042i	$0.524489\pi$
$564$	−4.83579	−0.203623
$565$	−8.23490	−0.346445
$566$	11.4222	0.480111
$567$	−6.42626	−0.269877
$568$	5.48188	0.230014
$569$	18.3521	0.769360	0.384680	−	0.923050i	$-0.374312\pi$
$569$	18.3521	0.769360	0.384680	+	0.923050i	$0.374312\pi$
$570$	−5.04892	−0.211476
$571$	−19.1987	−0.803439	−0.401719	−	0.915763i	$-0.631587\pi$
$571$	−19.1987	−0.803439	−0.401719	+	0.915763i	$0.631587\pi$
$572$	0	0
$573$	15.9517	0.666391
$574$	−31.2814	−1.30566
$575$	−3.10992	−0.129692
$576$	−1.44504	−0.0602101
$577$	−35.2218	−1.46630	−0.733150	−	0.680067i	$-0.761951\pi$
$577$	−35.2218	−1.46630	−0.733150	+	0.680067i	$0.761951\pi$
$578$	−1.45473	−0.0605088
$579$	−29.4101	−1.22224
$580$	−5.60388	−0.232688
$581$	−36.5133	−1.51483
$582$	−10.8388	−0.449282
$583$	−28.6246	−1.18551
$584$	−8.67456	−0.358956
$585$	0	0
$586$	24.4155	1.00860
$587$	32.4596	1.33975	0.669876	−	0.742473i	$-0.266347\pi$
$587$	32.4596	1.33975	0.669876	+	0.742473i	$0.266347\pi$
$588$	−0.972853	−0.0401198
$589$	−31.1836	−1.28490
$590$	−10.0151	−0.412314
$591$	−2.46383	−0.101348
$592$	2.67025	0.109747
$593$	−9.57135	−0.393048	−0.196524	−	0.980499i	$-0.562965\pi$
$593$	−9.57135	−0.393048	−0.196524	+	0.980499i	$0.562965\pi$
$594$	32.1594	1.31952
$595$	10.7138	0.439223
$596$	20.8116	0.852477
$597$	1.04221	0.0426549
$598$	0	0
$599$	−29.0858	−1.18841	−0.594206	−	0.804313i	$-0.702534\pi$
$599$	−29.0858	−1.18841	−0.594206	+	0.804313i	$0.702534\pi$
$600$	−1.24698	−0.0509077
$601$	−6.31229	−0.257484	−0.128742	−	0.991678i	$-0.541094\pi$
$601$	−6.31229	−0.257484	−0.128742	+	0.991678i	$0.541094\pi$
$602$	−17.4142	−0.709749
$603$	11.5821	0.471660
$604$	5.95646	0.242365
$605$	22.6625	0.921361
$606$	−19.6340	−0.797577
$607$	32.4650	1.31771	0.658857	−	0.752268i	$-0.271040\pi$
$607$	32.4650	1.31771	0.658857	+	0.752268i	$0.271040\pi$
$608$	−4.04892	−0.164205
$609$	−17.4276	−0.706201
$610$	4.93362	0.199757
$611$	0	0
$612$	−6.20775	−0.250933
$613$	29.7017	1.19964	0.599820	−	0.800135i	$-0.295239\pi$
$613$	29.7017	1.19964	0.599820	+	0.800135i	$0.295239\pi$
$614$	7.70410	0.310912
$615$	15.6407	0.630694
$616$	−14.4698	−0.583005
$617$	10.8556	0.437032	0.218516	−	0.975833i	$-0.429878\pi$
$617$	10.8556	0.437032	0.218516	+	0.975833i	$0.429878\pi$
$618$	4.61596	0.185681
$619$	23.2948	0.936298	0.468149	−	0.883649i	$-0.344921\pi$
$619$	23.2948	0.936298	0.468149	+	0.883649i	$0.344921\pi$
$620$	−7.70171	−0.309308
$621$	17.2379	0.691732
$622$	−4.71379	−0.189006
$623$	−1.13408	−0.0454359
$624$	0	0
$625$	1.00000	0.0400000
$626$	−8.49157	−0.339391
$627$	29.2935	1.16987
$628$	8.19567	0.327043
$629$	11.4711	0.457384
$630$	3.60388	0.143582
$631$	−6.72455	−0.267700	−0.133850	−	0.991002i	$-0.542734\pi$
$631$	−6.72455	−0.267700	−0.133850	+	0.991002i	$0.542734\pi$
$632$	1.82371	0.0725432
$633$	11.6746	0.464022
$634$	−27.0508	−1.07433
$635$	−13.0315	−0.517138
$636$	−6.15213	−0.243948
$637$	0	0
$638$	32.5133	1.28722
$639$	7.92154	0.313371
$640$	−1.00000	−0.0395285
$641$	9.32437	0.368291	0.184145	−	0.982899i	$-0.441048\pi$
$641$	9.32437	0.368291	0.184145	+	0.982899i	$0.441048\pi$
$642$	−18.8049	−0.742171
$643$	12.2145	0.481691	0.240846	−	0.970563i	$-0.422575\pi$
$643$	12.2145	0.481691	0.240846	+	0.970563i	$0.422575\pi$
$644$	−7.75600	−0.305629
$645$	8.70709	0.342841
$646$	−17.3937	−0.684347
$647$	−36.8853	−1.45011	−0.725055	−	0.688691i	$-0.758186\pi$
$647$	−36.8853	−1.45011	−0.725055	+	0.688691i	$0.758186\pi$
$648$	2.57673	0.101223
$649$	58.1068	2.28089
$650$	0	0
$651$	−23.9517	−0.938740
$652$	−16.8877	−0.661373
$653$	−38.6655	−1.51310	−0.756548	−	0.653938i	$-0.773116\pi$
$653$	−38.6655	−1.51310	−0.756548	+	0.653938i	$0.773116\pi$
$654$	21.7995	0.852430
$655$	9.82908	0.384054
$656$	12.5429	0.489717
$657$	−12.5351	−0.489041
$658$	9.67158	0.377037
$659$	−1.19269	−0.0464604	−0.0232302	−	0.999730i	$-0.507395\pi$
$659$	−1.19269	−0.0464604	−0.0232302	+	0.999730i	$0.507395\pi$
$660$	7.23490	0.281618
$661$	−3.93230	−0.152949	−0.0764743	−	0.997072i	$-0.524366\pi$
$661$	−3.93230	−0.152949	−0.0764743	+	0.997072i	$0.524366\pi$
$662$	28.7439	1.11716
$663$	0	0
$664$	14.6407	0.568170
$665$	10.0978	0.391577
$666$	3.85862	0.149519
$667$	17.4276	0.674799
$668$	19.6582	0.760597
$669$	7.87800	0.304581
$670$	8.01507	0.309649
$671$	−28.6246	−1.10504
$672$	−3.10992	−0.119968
$673$	4.87023	0.187734	0.0938668	−	0.995585i	$-0.470077\pi$
$673$	4.87023	0.187734	0.0938668	+	0.995585i	$0.470077\pi$
$674$	34.5972	1.33263
$675$	−5.54288	−0.213345
$676$	0	0
$677$	−14.6461	−0.562895	−0.281447	−	0.959577i	$-0.590815\pi$
$677$	−14.6461	−0.562895	−0.281447	+	0.959577i	$0.590815\pi$
$678$	10.2687	0.394369
$679$	21.6775	0.831908
$680$	−4.29590	−0.164740
$681$	−6.10321	−0.233876
$682$	44.6848	1.71107
$683$	35.1997	1.34688	0.673440	−	0.739242i	$-0.264816\pi$
$683$	35.1997	1.34688	0.673440	+	0.739242i	$0.264816\pi$
$684$	−5.85086	−0.223713
$685$	4.75302	0.181604
$686$	19.4034	0.740826
$687$	0.859953	0.0328092
$688$	6.98254	0.266207
$689$	0	0
$690$	3.87800	0.147633
$691$	−41.5851	−1.58197	−0.790986	−	0.611835i	$-0.790432\pi$
$691$	−41.5851	−1.58197	−0.790986	+	0.611835i	$0.790432\pi$
$692$	−16.8659	−0.641146
$693$	−20.9095	−0.794285
$694$	28.3043	1.07441
$695$	−2.39373	−0.0907994
$696$	6.98792	0.264876
$697$	53.8829	2.04096
$698$	−16.8310	−0.637063
$699$	17.6829	0.668830
$700$	2.49396	0.0942628
$701$	5.61463	0.212062	0.106031	−	0.994363i	$-0.466186\pi$
$701$	5.61463	0.212062	0.106031	+	0.994363i	$0.466186\pi$
$702$	0	0
$703$	10.8116	0.407768
$704$	5.80194	0.218669
$705$	−4.83579	−0.182126
$706$	23.7429	0.893574
$707$	39.2680	1.47683
$708$	12.4886	0.469350
$709$	34.8659	1.30942	0.654709	−	0.755881i	$-0.272791\pi$
$709$	34.8659	1.30942	0.654709	+	0.755881i	$0.272791\pi$
$710$	5.48188	0.205731
$711$	2.63533	0.0988328
$712$	0.454731	0.0170417
$713$	23.9517	0.896997
$714$	−13.3599	−0.499981
$715$	0	0
$716$	−5.99761	−0.224141
$717$	32.1414	1.20034
$718$	26.8310	1.00132
$719$	−1.90217	−0.0709388	−0.0354694	−	0.999371i	$-0.511293\pi$
$719$	−1.90217	−0.0709388	−0.0354694	+	0.999371i	$0.511293\pi$
$720$	−1.44504	−0.0538535
$721$	−9.23191	−0.343814
$722$	2.60627	0.0969953
$723$	9.81030	0.364849
$724$	−16.5676	−0.615731
$725$	−5.60388	−0.208123
$726$	−28.2597	−1.04881
$727$	−23.7211	−0.879766	−0.439883	−	0.898055i	$-0.644980\pi$
$727$	−23.7211	−0.879766	−0.439883	+	0.898055i	$0.644980\pi$
$728$	0	0
$729$	24.4590	0.905890
$730$	−8.67456	−0.321060
$731$	29.9963	1.10945
$732$	−6.15213	−0.227389
$733$	−13.2862	−0.490737	−0.245369	−	0.969430i	$-0.578909\pi$
$733$	−13.2862	−0.490737	−0.245369	+	0.969430i	$0.578909\pi$
$734$	18.0978	0.668003
$735$	−0.972853	−0.0358842
$736$	3.10992	0.114633
$737$	−46.5029	−1.71296
$738$	18.1250	0.667190
$739$	6.34614	0.233447	0.116723	−	0.993164i	$-0.462761\pi$
$739$	6.34614	0.233447	0.116723	+	0.993164i	$0.462761\pi$
$740$	2.67025	0.0981604
$741$	0	0
$742$	12.3043	0.451704
$743$	40.1801	1.47407	0.737033	−	0.675857i	$-0.236226\pi$
$743$	40.1801	1.47407	0.737033	+	0.675857i	$0.236226\pi$
$744$	9.60388	0.352095
$745$	20.8116	0.762479
$746$	−14.3720	−0.526195
$747$	21.1564	0.774074
$748$	24.9245	0.911331
$749$	37.6098	1.37423
$750$	−1.24698	−0.0455333
$751$	15.5362	0.566923	0.283461	−	0.958984i	$-0.408517\pi$
$751$	15.5362	0.566923	0.283461	+	0.958984i	$0.408517\pi$
$752$	−3.87800	−0.141416
$753$	24.9124	0.907860
$754$	0	0
$755$	5.95646	0.216778
$756$	−13.8237	−0.502763
$757$	32.2064	1.17056	0.585281	−	0.810830i	$-0.300984\pi$
$757$	32.2064	1.17056	0.585281	+	0.810830i	$0.300984\pi$
$758$	26.1806	0.950922
$759$	−22.4999	−0.816696
$760$	−4.04892	−0.146870
$761$	19.6472	0.712209	0.356104	−	0.934446i	$-0.384105\pi$
$761$	19.6472	0.712209	0.356104	+	0.934446i	$0.384105\pi$
$762$	16.2500	0.588674
$763$	−43.5991	−1.57839
$764$	12.7922	0.462807
$765$	−6.20775	−0.224442
$766$	−11.3491	−0.410061
$767$	0	0
$768$	1.24698	0.0449965
$769$	6.50232	0.234480	0.117240	−	0.993104i	$-0.462595\pi$
$769$	6.50232	0.234480	0.117240	+	0.993104i	$0.462595\pi$
$770$	−14.4698	−0.521455
$771$	−31.7385	−1.14304
$772$	−23.5851	−0.848846
$773$	−32.2586	−1.16026	−0.580130	−	0.814524i	$-0.696998\pi$
$773$	−32.2586	−1.16026	−0.580130	+	0.814524i	$0.696998\pi$
$774$	10.0901	0.362680
$775$	−7.70171	−0.276654
$776$	−8.69202	−0.312025
$777$	8.30426	0.297914
$778$	12.8465	0.460571
$779$	50.7851	1.81956
$780$	0	0
$781$	−31.8055	−1.13809
$782$	13.3599	0.477748
$783$	31.0616	1.11005
$784$	−0.780167	−0.0278631
$785$	8.19567	0.292516
$786$	−12.2567	−0.437181
$787$	13.6799	0.487637	0.243819	−	0.969821i	$-0.421600\pi$
$787$	13.6799	0.487637	0.243819	+	0.969821i	$0.421600\pi$
$788$	−1.97584	−0.0703863
$789$	−24.9095	−0.886800
$790$	1.82371	0.0648846
$791$	−20.5375	−0.730229
$792$	8.38404	0.297914
$793$	0	0
$794$	−18.7439	−0.665197
$795$	−6.15213	−0.218194
$796$	0.835790	0.0296238
$797$	−7.77538	−0.275418	−0.137709	−	0.990473i	$-0.543974\pi$
$797$	−7.77538	−0.275418	−0.137709	+	0.990473i	$0.543974\pi$
$798$	−12.5918	−0.445745
$799$	−16.6595	−0.589371
$800$	−1.00000	−0.0353553
$801$	0.657105	0.0232177
$802$	−19.8213	−0.699915
$803$	50.3293	1.77608
$804$	−9.99462	−0.352483
$805$	−7.75600	−0.273363
$806$	0	0
$807$	−20.1280	−0.708538
$808$	−15.7453	−0.553916
$809$	−43.1540	−1.51722	−0.758608	−	0.651548i	$-0.774120\pi$
$809$	−43.1540	−1.51722	−0.758608	+	0.651548i	$0.774120\pi$
$810$	2.57673	0.0905370
$811$	−35.8165	−1.25769	−0.628844	−	0.777531i	$-0.716472\pi$
$811$	−35.8165	−1.25769	−0.628844	+	0.777531i	$0.716472\pi$
$812$	−13.9758	−0.490456
$813$	18.8358	0.660600
$814$	−15.4926	−0.543016
$815$	−16.8877	−0.591550
$816$	5.35690	0.187529
$817$	28.2717	0.989103
$818$	18.9487	0.662525
$819$	0	0
$820$	12.5429	0.438016
$821$	6.70304	0.233938	0.116969	−	0.993136i	$-0.462682\pi$
$821$	6.70304	0.233938	0.116969	+	0.993136i	$0.462682\pi$
$822$	−5.92692	−0.206725
$823$	35.7103	1.24478	0.622392	−	0.782706i	$-0.286161\pi$
$823$	35.7103	1.24478	0.622392	+	0.782706i	$0.286161\pi$
$824$	3.70171	0.128955
$825$	7.23490	0.251887
$826$	−24.9772	−0.869067
$827$	−1.79523	−0.0624264	−0.0312132	−	0.999513i	$-0.509937\pi$
$827$	−1.79523	−0.0624264	−0.0312132	+	0.999513i	$0.509937\pi$
$828$	4.49396	0.156176
$829$	0.733169	0.0254640	0.0127320	−	0.999919i	$-0.495947\pi$
$829$	0.733169	0.0254640	0.0127320	+	0.999919i	$0.495947\pi$
$830$	14.6407	0.508187
$831$	−30.5435	−1.05954
$832$	0	0
$833$	−3.35152	−0.116123
$834$	2.98493	0.103360
$835$	19.6582	0.680299
$836$	23.4916	0.812473
$837$	42.6896	1.47557
$838$	−1.86725	−0.0645030
$839$	−26.6112	−0.918720	−0.459360	−	0.888250i	$-0.651921\pi$
$839$	−26.6112	−0.918720	−0.459360	+	0.888250i	$0.651921\pi$
$840$	−3.10992	−0.107302
$841$	2.40342	0.0828766
$842$	27.3250	0.941680
$843$	29.1021	1.00233
$844$	9.36227	0.322263
$845$	0	0
$846$	−5.60388	−0.192665
$847$	56.5193	1.94203
$848$	−4.93362	−0.169421
$849$	−14.2433	−0.488827
$850$	−4.29590	−0.147348
$851$	−8.30426	−0.284666
$852$	−6.83579	−0.234190
$853$	−4.78746	−0.163920	−0.0819598	−	0.996636i	$-0.526118\pi$
$853$	−4.78746	−0.163920	−0.0819598	+	0.996636i	$0.526118\pi$
$854$	12.3043	0.421043
$855$	−5.85086	−0.200095
$856$	−15.0804	−0.515437
$857$	−24.9748	−0.853122	−0.426561	−	0.904459i	$-0.640275\pi$
$857$	−24.9748	−0.853122	−0.426561	+	0.904459i	$0.640275\pi$
$858$	0	0
$859$	−16.2737	−0.555250	−0.277625	−	0.960690i	$-0.589547\pi$
$859$	−16.2737	−0.555250	−0.277625	+	0.960690i	$0.589547\pi$
$860$	6.98254	0.238103
$861$	39.0073	1.32937
$862$	−1.56033	−0.0531452
$863$	44.3806	1.51073	0.755366	−	0.655303i	$-0.227459\pi$
$863$	44.3806	1.51073	0.755366	+	0.655303i	$0.227459\pi$
$864$	5.54288	0.188572
$865$	−16.8659	−0.573459
$866$	24.1564	0.820869
$867$	1.81402	0.0616073
$868$	−19.2078	−0.651954
$869$	−10.5810	−0.358937
$870$	6.98792	0.236913
$871$	0	0
$872$	17.4819	0.592011
$873$	−12.5603	−0.425103
$874$	12.5918	0.425924
$875$	2.49396	0.0843112
$876$	10.8170	0.365473
$877$	37.3900	1.26257	0.631285	−	0.775551i	$-0.282528\pi$
$877$	37.3900	1.26257	0.631285	+	0.775551i	$0.282528\pi$
$878$	−21.3599	−0.720861
$879$	−30.4456	−1.02691
$880$	5.80194	0.195583
$881$	38.1885	1.28660	0.643301	−	0.765613i	$-0.277564\pi$
$881$	38.1885	1.28660	0.643301	+	0.765613i	$0.277564\pi$
$882$	−1.12737	−0.0379607
$883$	4.45281	0.149849	0.0749245	−	0.997189i	$-0.476128\pi$
$883$	4.45281	0.149849	0.0749245	+	0.997189i	$0.476128\pi$
$884$	0	0
$885$	12.4886	0.419799
$886$	16.5351	0.555508
$887$	6.06292	0.203573	0.101786	−	0.994806i	$-0.467544\pi$
$887$	6.06292	0.203573	0.101786	+	0.994806i	$0.467544\pi$
$888$	−3.32975	−0.111739
$889$	−32.4999	−1.09001
$890$	0.454731	0.0152426
$891$	−14.9500	−0.500844
$892$	6.31767	0.211531
$893$	−15.7017	−0.525438
$894$	−25.9517	−0.867954
$895$	−5.99761	−0.200478
$896$	−2.49396	−0.0833173
$897$	0	0
$898$	−6.56704	−0.219145
$899$	43.1594	1.43945
$900$	−1.44504	−0.0481681
$901$	−21.1943	−0.706086
$902$	−72.7730	−2.42308
$903$	21.7151	0.722634
$904$	8.23490	0.273889
$905$	−16.5676	−0.550727
$906$	−7.42758	−0.246765
$907$	25.1116	0.833816	0.416908	−	0.908949i	$-0.363114\pi$
$907$	25.1116	0.833816	0.416908	+	0.908949i	$0.363114\pi$
$908$	−4.89440	−0.162426
$909$	−22.7525	−0.754654
$910$	0	0
$911$	4.11721	0.136409	0.0682047	−	0.997671i	$-0.478273\pi$
$911$	4.11721	0.136409	0.0682047	+	0.997671i	$0.478273\pi$
$912$	5.04892	0.167186
$913$	−84.9445	−2.81125
$914$	24.1715	0.799522
$915$	−6.15213	−0.203383
$916$	0.689629	0.0227860
$917$	24.5133	0.809502
$918$	23.8116	0.785901
$919$	−34.7633	−1.14674	−0.573368	−	0.819298i	$-0.694363\pi$
$919$	−34.7633	−1.14674	−0.573368	+	0.819298i	$0.694363\pi$
$920$	3.10992	0.102531
$921$	−9.60686	−0.316557
$922$	23.5797	0.776556
$923$	0	0
$924$	18.0435	0.593589
$925$	2.67025	0.0877973
$926$	−1.68233	−0.0552849
$927$	5.34913	0.175688
$928$	5.60388	0.183956
$929$	38.8015	1.27303	0.636517	−	0.771262i	$-0.280374\pi$
$929$	38.8015	1.27303	0.636517	+	0.771262i	$0.280374\pi$
$930$	9.60388	0.314923
$931$	−3.15883	−0.103527
$932$	14.1806	0.464501
$933$	5.87800	0.192437
$934$	−0.376273	−0.0123120
$935$	24.9245	0.815119
$936$	0	0
$937$	−31.7808	−1.03823	−0.519116	−	0.854704i	$-0.673739\pi$
$937$	−31.7808	−1.03823	−0.519116	+	0.854704i	$0.673739\pi$
$938$	19.9892	0.652672
$939$	10.5888	0.345553
$940$	−3.87800	−0.126486
$941$	30.2258	0.985333	0.492666	−	0.870218i	$-0.336022\pi$
$941$	30.2258	0.985333	0.492666	+	0.870218i	$0.336022\pi$
$942$	−10.2198	−0.332980
$943$	−39.0073	−1.27025
$944$	10.0151	0.325963
$945$	−13.8237	−0.449685
$946$	−40.5123	−1.31717
$947$	2.54288	0.0826324	0.0413162	−	0.999146i	$-0.486845\pi$
$947$	2.54288	0.0826324	0.0413162	+	0.999146i	$0.486845\pi$
$948$	−2.27413	−0.0738602
$949$	0	0
$950$	−4.04892	−0.131364
$951$	33.7318	1.09383
$952$	−10.7138	−0.347236
$953$	−6.58642	−0.213355	−0.106677	−	0.994294i	$-0.534021\pi$
$953$	−6.58642	−0.213355	−0.106677	+	0.994294i	$0.534021\pi$
$954$	−7.12929	−0.230819
$955$	12.7922	0.413947
$956$	25.7754	0.833635
$957$	−40.5435	−1.31058
$958$	−10.0301	−0.324059
$959$	11.8538	0.382780
$960$	1.24698	0.0402461
$961$	28.3163	0.913430
$962$	0	0
$963$	−21.7918	−0.702230
$964$	7.86725	0.253387
$965$	−23.5851	−0.759231
$966$	9.67158	0.311178
$967$	23.1884	0.745688	0.372844	−	0.927894i	$-0.378383\pi$
$967$	23.1884	0.745688	0.372844	+	0.927894i	$0.378383\pi$
$968$	−22.6625	−0.728400
$969$	21.6896	0.696771
$970$	−8.69202	−0.279084
$971$	−4.58642	−0.147185	−0.0735926	−	0.997288i	$-0.523446\pi$
$971$	−4.58642	−0.147185	−0.0735926	+	0.997288i	$0.523446\pi$
$972$	13.4155	0.430302
$973$	−5.96987	−0.191385
$974$	−15.3297	−0.491197
$975$	0	0
$976$	−4.93362	−0.157921
$977$	−30.1575	−0.964824	−0.482412	−	0.875944i	$-0.660239\pi$
$977$	−30.1575	−0.964824	−0.482412	+	0.875944i	$0.660239\pi$
$978$	21.0586	0.673380
$979$	−2.63832	−0.0843210
$980$	−0.780167	−0.0249215
$981$	25.2620	0.806555
$982$	−9.19269	−0.293350
$983$	−0.787463	−0.0251162	−0.0125581	−	0.999921i	$-0.503997\pi$
$983$	−0.787463	−0.0251162	−0.0125581	+	0.999921i	$0.503997\pi$
$984$	−15.6407	−0.498608
$985$	−1.97584	−0.0629554
$986$	24.0737	0.766662
$987$	−12.0603	−0.383882
$988$	0	0
$989$	−21.7151	−0.690501
$990$	8.38404	0.266462
$991$	−12.0892	−0.384026	−0.192013	−	0.981392i	$-0.561502\pi$
$991$	−12.0892	−0.384026	−0.192013	+	0.981392i	$0.561502\pi$
$992$	7.70171	0.244530
$993$	−35.8431	−1.13745
$994$	13.6716	0.433636
$995$	0.835790	0.0264963
$996$	−18.2567	−0.578485
$997$	46.6607	1.47776	0.738879	−	0.673838i	$-0.235355\pi$
$997$	46.6607	1.47776	0.738879	+	0.673838i	$0.235355\pi$
$998$	−2.74632	−0.0869331
$999$	−14.8009	−0.468279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.a.p.1.3	✓	3
5.4	even	2		8450.2.a.cg.1.1		3
13.2	odd	12		1690.2.l.m.1161.1		12
13.3	even	3		1690.2.e.r.191.1		6
13.4	even	6		1690.2.e.p.991.1		6
13.5	odd	4		1690.2.d.i.1351.6		6
13.6	odd	12		1690.2.l.m.361.4		12
13.7	odd	12		1690.2.l.m.361.1		12
13.8	odd	4		1690.2.d.i.1351.3		6
13.9	even	3		1690.2.e.r.991.1		6
13.10	even	6		1690.2.e.p.191.1		6
13.11	odd	12		1690.2.l.m.1161.4		12
13.12	even	2		1690.2.a.r.1.3	yes	3
65.64	even	2		8450.2.a.bv.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.a.p.1.3	✓	3	1.1	even	1	trivial
1690.2.a.r.1.3	yes	3	13.12	even	2
1690.2.d.i.1351.3		6	13.8	odd	4
1690.2.d.i.1351.6		6	13.5	odd	4
1690.2.e.p.191.1		6	13.10	even	6
1690.2.e.p.991.1		6	13.4	even	6
1690.2.e.r.191.1		6	13.3	even	3
1690.2.e.r.991.1		6	13.9	even	3
1690.2.l.m.361.1		12	13.7	odd	12
1690.2.l.m.361.4		12	13.6	odd	12
1690.2.l.m.1161.1		12	13.2	odd	12
1690.2.l.m.1161.4		12	13.11	odd	12
8450.2.a.bv.1.1		3	65.64	even	2
8450.2.a.cg.1.1		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.24698 q^{6} +2.49396 q^{7} -1.00000 q^{8} -1.44504 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.24698 q^{6} +2.49396 q^{7} -1.00000 q^{8} -1.44504 q^{9} -1.00000 q^{10} +5.80194 q^{11} +1.24698 q^{12} -2.49396 q^{14} +1.24698 q^{15} +1.00000 q^{16} +4.29590 q^{17} +1.44504 q^{18} +4.04892 q^{19} +1.00000 q^{20} +3.10992 q^{21} -5.80194 q^{22} -3.10992 q^{23} -1.24698 q^{24} +1.00000 q^{25} -5.54288 q^{27} +2.49396 q^{28} -5.60388 q^{29} -1.24698 q^{30} -7.70171 q^{31} -1.00000 q^{32} +7.23490 q^{33} -4.29590 q^{34} +2.49396 q^{35} -1.44504 q^{36} +2.67025 q^{37} -4.04892 q^{38} -1.00000 q^{40} +12.5429 q^{41} -3.10992 q^{42} +6.98254 q^{43} +5.80194 q^{44} -1.44504 q^{45} +3.10992 q^{46} -3.87800 q^{47} +1.24698 q^{48} -0.780167 q^{49} -1.00000 q^{50} +5.35690 q^{51} -4.93362 q^{53} +5.54288 q^{54} +5.80194 q^{55} -2.49396 q^{56} +5.04892 q^{57} +5.60388 q^{58} +10.0151 q^{59} +1.24698 q^{60} -4.93362 q^{61} +7.70171 q^{62} -3.60388 q^{63} +1.00000 q^{64} -7.23490 q^{66} -8.01507 q^{67} +4.29590 q^{68} -3.87800 q^{69} -2.49396 q^{70} -5.48188 q^{71} +1.44504 q^{72} +8.67456 q^{73} -2.67025 q^{74} +1.24698 q^{75} +4.04892 q^{76} +14.4698 q^{77} -1.82371 q^{79} +1.00000 q^{80} -2.57673 q^{81} -12.5429 q^{82} -14.6407 q^{83} +3.10992 q^{84} +4.29590 q^{85} -6.98254 q^{86} -6.98792 q^{87} -5.80194 q^{88} -0.454731 q^{89} +1.44504 q^{90} -3.10992 q^{92} -9.60388 q^{93} +3.87800 q^{94} +4.04892 q^{95} -1.24698 q^{96} +8.69202 q^{97} +0.780167 q^{98} -8.38404 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 2 q^{7} - 3 q^{8} - 4 q^{9}+O(q^{10}) \) 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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