LMFDB - Embedded newform 1690.2.a.v.1.4

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:	$1$
Dimension:	$6$
Coefficient field:	6.6.20439713.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 15x^{4} + 22x^{3} + 70x^{2} - 56x - 91 $ x^6 - 2x^5 - 15x^4 + 22x^3 + 70x^2 - 56*x - 91
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.988450$ of defining polynomial
Character	$\chi$	$=$	1690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.988450 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.988450 q^{6} +3.47315 q^{7} -1.00000 q^{8} -2.02297 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.988450 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.988450 q^{6} +3.47315 q^{7} -1.00000 q^{8} -2.02297 q^{9} +1.00000 q^{10} -6.04251 q^{11} +0.988450 q^{12} -3.47315 q^{14} -0.988450 q^{15} +1.00000 q^{16} +2.05533 q^{17} +2.02297 q^{18} -3.11752 q^{19} -1.00000 q^{20} +3.43303 q^{21} +6.04251 q^{22} +7.97448 q^{23} -0.988450 q^{24} +1.00000 q^{25} -4.96495 q^{27} +3.47315 q^{28} -6.73035 q^{29} +0.988450 q^{30} -8.96482 q^{31} -1.00000 q^{32} -5.97272 q^{33} -2.05533 q^{34} -3.47315 q^{35} -2.02297 q^{36} -2.69527 q^{37} +3.11752 q^{38} +1.00000 q^{40} -3.23543 q^{41} -3.43303 q^{42} -2.03505 q^{43} -6.04251 q^{44} +2.02297 q^{45} -7.97448 q^{46} -9.03057 q^{47} +0.988450 q^{48} +5.06275 q^{49} -1.00000 q^{50} +2.03159 q^{51} -11.1847 q^{53} +4.96495 q^{54} +6.04251 q^{55} -3.47315 q^{56} -3.08152 q^{57} +6.73035 q^{58} +6.40187 q^{59} -0.988450 q^{60} +13.0482 q^{61} +8.96482 q^{62} -7.02606 q^{63} +1.00000 q^{64} +5.97272 q^{66} +8.49642 q^{67} +2.05533 q^{68} +7.88237 q^{69} +3.47315 q^{70} -0.121998 q^{71} +2.02297 q^{72} +0.138022 q^{73} +2.69527 q^{74} +0.988450 q^{75} -3.11752 q^{76} -20.9865 q^{77} -6.71618 q^{79} -1.00000 q^{80} +1.16129 q^{81} +3.23543 q^{82} +3.03849 q^{83} +3.43303 q^{84} -2.05533 q^{85} +2.03505 q^{86} -6.65262 q^{87} +6.04251 q^{88} -3.38412 q^{89} -2.02297 q^{90} +7.97448 q^{92} -8.86128 q^{93} +9.03057 q^{94} +3.11752 q^{95} -0.988450 q^{96} +5.82521 q^{97} -5.06275 q^{98} +12.2238 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 6 q^{5} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 16 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 6 * q^5 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 16 * q^9	$ 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 6 q^{5} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 16 q^{9} + 6 q^{10} - 15 q^{11} - 2 q^{12} + 3 q^{14} + 2 q^{15} + 6 q^{16} - 3 q^{17} - 16 q^{18} - q^{19} - 6 q^{20} + 2 q^{21} + 15 q^{22} - 3 q^{23} + 2 q^{24} + 6 q^{25} - 20 q^{27} - 3 q^{28} + 7 q^{29} - 2 q^{30} - 6 q^{32} - 4 q^{33} + 3 q^{34} + 3 q^{35} + 16 q^{36} - 6 q^{37} + q^{38} + 6 q^{40} - 2 q^{41} - 2 q^{42} - 22 q^{43} - 15 q^{44} - 16 q^{45} + 3 q^{46} - 7 q^{47} - 2 q^{48} + 31 q^{49} - 6 q^{50} - 22 q^{51} - 16 q^{53} + 20 q^{54} + 15 q^{55} + 3 q^{56} - 2 q^{57} - 7 q^{58} - 15 q^{59} + 2 q^{60} + 33 q^{61} - 25 q^{63} + 6 q^{64} + 4 q^{66} + 8 q^{67} - 3 q^{68} - 6 q^{69} - 3 q^{70} - 40 q^{71} - 16 q^{72} - 21 q^{73} + 6 q^{74} - 2 q^{75} - q^{76} - 34 q^{77} + 20 q^{79} - 6 q^{80} - 2 q^{81} + 2 q^{82} - 22 q^{83} + 2 q^{84} + 3 q^{85} + 22 q^{86} - 39 q^{87} + 15 q^{88} - 20 q^{89} + 16 q^{90} - 3 q^{92} - 48 q^{93} + 7 q^{94} + q^{95} + 2 q^{96} + 7 q^{97} - 31 q^{98} - 55 q^{99}+O(q^{100}) $ 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 6 * q^5 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 16 * q^9 + 6 * q^10 - 15 * q^11 - 2 * q^12 + 3 * q^14 + 2 * q^15 + 6 * q^16 - 3 * q^17 - 16 * q^18 - q^19 - 6 * q^20 + 2 * q^21 + 15 * q^22 - 3 * q^23 + 2 * q^24 + 6 * q^25 - 20 * q^27 - 3 * q^28 + 7 * q^29 - 2 * q^30 - 6 * q^32 - 4 * q^33 + 3 * q^34 + 3 * q^35 + 16 * q^36 - 6 * q^37 + q^38 + 6 * q^40 - 2 * q^41 - 2 * q^42 - 22 * q^43 - 15 * q^44 - 16 * q^45 + 3 * q^46 - 7 * q^47 - 2 * q^48 + 31 * q^49 - 6 * q^50 - 22 * q^51 - 16 * q^53 + 20 * q^54 + 15 * q^55 + 3 * q^56 - 2 * q^57 - 7 * q^58 - 15 * q^59 + 2 * q^60 + 33 * q^61 - 25 * q^63 + 6 * q^64 + 4 * q^66 + 8 * q^67 - 3 * q^68 - 6 * q^69 - 3 * q^70 - 40 * q^71 - 16 * q^72 - 21 * q^73 + 6 * q^74 - 2 * q^75 - q^76 - 34 * q^77 + 20 * q^79 - 6 * q^80 - 2 * q^81 + 2 * q^82 - 22 * q^83 + 2 * q^84 + 3 * q^85 + 22 * q^86 - 39 * q^87 + 15 * q^88 - 20 * q^89 + 16 * q^90 - 3 * q^92 - 48 * q^93 + 7 * q^94 + q^95 + 2 * q^96 + 7 * q^97 - 31 * q^98 - 55 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.988450	0.570682	0.285341	−	0.958426i	$-0.407893\pi$
$3$	0.988450	0.570682	0.285341	+	0.958426i	$0.407893\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.988450	−0.403533
$7$	3.47315	1.31273	0.656363	−	0.754445i	$-0.272094\pi$
$7$	3.47315	1.31273	0.656363	+	0.754445i	$0.272094\pi$
$8$	−1.00000	−0.353553
$9$	−2.02297	−0.674322
$10$	1.00000	0.316228
$11$	−6.04251	−1.82188	−0.910942	−	0.412534i	$-0.864644\pi$
$11$	−6.04251	−1.82188	−0.910942	+	0.412534i	$0.864644\pi$
$12$	0.988450	0.285341
$13$	0	0
$13$	0	0
$14$	−3.47315	−0.928238
$15$	−0.988450	−0.255217
$16$	1.00000	0.250000
$17$	2.05533	0.498490	0.249245	−	0.968440i	$-0.419818\pi$
$17$	2.05533	0.498490	0.249245	+	0.968440i	$0.419818\pi$
$18$	2.02297	0.476818
$19$	−3.11752	−0.715208	−0.357604	−	0.933873i	$-0.616406\pi$
$19$	−3.11752	−0.715208	−0.357604	+	0.933873i	$0.616406\pi$
$20$	−1.00000	−0.223607
$21$	3.43303	0.749150
$22$	6.04251	1.28827
$23$	7.97448	1.66279	0.831396	−	0.555680i	$-0.187542\pi$
$23$	7.97448	1.66279	0.831396	+	0.555680i	$0.187542\pi$
$24$	−0.988450	−0.201767
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.96495	−0.955506
$28$	3.47315	0.656363
$29$	−6.73035	−1.24979	−0.624897	−	0.780707i	$-0.714859\pi$
$29$	−6.73035	−1.24979	−0.624897	+	0.780707i	$0.714859\pi$
$30$	0.988450	0.180466
$31$	−8.96482	−1.61013	−0.805065	−	0.593187i	$-0.797869\pi$
$31$	−8.96482	−1.61013	−0.805065	+	0.593187i	$0.797869\pi$
$32$	−1.00000	−0.176777
$33$	−5.97272	−1.03972
$34$	−2.05533	−0.352486
$35$	−3.47315	−0.587069
$36$	−2.02297	−0.337161
$37$	−2.69527	−0.443099	−0.221550	−	0.975149i	$-0.571111\pi$
$37$	−2.69527	−0.443099	−0.221550	+	0.975149i	$0.571111\pi$
$38$	3.11752	0.505729
$39$	0	0
$40$	1.00000	0.158114
$41$	−3.23543	−0.505289	−0.252645	−	0.967559i	$-0.581300\pi$
$41$	−3.23543	−0.505289	−0.252645	+	0.967559i	$0.581300\pi$
$42$	−3.43303	−0.529729
$43$	−2.03505	−0.310342	−0.155171	−	0.987888i	$-0.549593\pi$
$43$	−2.03505	−0.310342	−0.155171	+	0.987888i	$0.549593\pi$
$44$	−6.04251	−0.910942
$45$	2.02297	0.301566
$46$	−7.97448	−1.17577
$47$	−9.03057	−1.31724	−0.658622	−	0.752474i	$-0.728860\pi$
$47$	−9.03057	−1.31724	−0.658622	+	0.752474i	$0.728860\pi$
$48$	0.988450	0.142671
$49$	5.06275	0.723251
$50$	−1.00000	−0.141421
$51$	2.03159	0.284479
$52$	0	0
$53$	−11.1847	−1.53633	−0.768165	−	0.640252i	$-0.778830\pi$
$53$	−11.1847	−1.53633	−0.768165	+	0.640252i	$0.778830\pi$
$54$	4.96495	0.675644
$55$	6.04251	0.814772
$56$	−3.47315	−0.464119
$57$	−3.08152	−0.408157
$58$	6.73035	0.883738
$59$	6.40187	0.833452	0.416726	−	0.909032i	$-0.363177\pi$
$59$	6.40187	0.833452	0.416726	+	0.909032i	$0.363177\pi$
$60$	−0.988450	−0.127608
$61$	13.0482	1.67065	0.835327	−	0.549754i	$-0.185279\pi$
$61$	13.0482	1.67065	0.835327	+	0.549754i	$0.185279\pi$
$62$	8.96482	1.13853
$63$	−7.02606	−0.885200
$64$	1.00000	0.125000
$65$	0	0
$66$	5.97272	0.735191
$67$	8.49642	1.03800	0.519002	−	0.854773i	$-0.326304\pi$
$67$	8.49642	1.03800	0.519002	+	0.854773i	$0.326304\pi$
$68$	2.05533	0.249245
$69$	7.88237	0.948926
$70$	3.47315	0.415121
$71$	−0.121998	−0.0144785	−0.00723924	−	0.999974i	$-0.502304\pi$
$71$	−0.121998	−0.0144785	−0.00723924	+	0.999974i	$0.502304\pi$
$72$	2.02297	0.238409
$73$	0.138022	0.0161543	0.00807713	−	0.999967i	$-0.497429\pi$
$73$	0.138022	0.0161543	0.00807713	+	0.999967i	$0.497429\pi$
$74$	2.69527	0.313319
$75$	0.988450	0.114136
$76$	−3.11752	−0.357604
$77$	−20.9865	−2.39164
$78$	0	0
$79$	−6.71618	−0.755630	−0.377815	−	0.925881i	$-0.623324\pi$
$79$	−6.71618	−0.755630	−0.377815	+	0.925881i	$0.623324\pi$
$80$	−1.00000	−0.111803
$81$	1.16129	0.129032
$82$	3.23543	0.357293
$83$	3.03849	0.333518	0.166759	−	0.985998i	$-0.446670\pi$
$83$	3.03849	0.333518	0.166759	+	0.985998i	$0.446670\pi$
$84$	3.43303	0.374575
$85$	−2.05533	−0.222931
$86$	2.03505	0.219445
$87$	−6.65262	−0.713235
$88$	6.04251	0.644133
$89$	−3.38412	−0.358716	−0.179358	−	0.983784i	$-0.557402\pi$
$89$	−3.38412	−0.358716	−0.179358	+	0.983784i	$0.557402\pi$
$90$	−2.02297	−0.213239
$91$	0	0
$92$	7.97448	0.831396
$93$	−8.86128	−0.918872
$94$	9.03057	0.931432
$95$	3.11752	0.319851
$96$	−0.988450	−0.100883
$97$	5.82521	0.591461	0.295730	−	0.955271i	$-0.404437\pi$
$97$	5.82521	0.591461	0.295730	+	0.955271i	$0.404437\pi$
$98$	−5.06275	−0.511415
$99$	12.2238	1.22854
$100$	1.00000	0.100000
$101$	−8.47871	−0.843664	−0.421832	−	0.906674i	$-0.638613\pi$
$101$	−8.47871	−0.843664	−0.421832	+	0.906674i	$0.638613\pi$
$102$	−2.03159	−0.201157
$103$	−4.96891	−0.489601	−0.244801	−	0.969573i	$-0.578722\pi$
$103$	−4.96891	−0.489601	−0.244801	+	0.969573i	$0.578722\pi$
$104$	0	0
$105$	−3.43303	−0.335030
$106$	11.1847	1.08635
$107$	−5.16942	−0.499747	−0.249874	−	0.968278i	$-0.580389\pi$
$107$	−5.16942	−0.499747	−0.249874	+	0.968278i	$0.580389\pi$
$108$	−4.96495	−0.477753
$109$	−8.09832	−0.775678	−0.387839	−	0.921727i	$-0.626778\pi$
$109$	−8.09832	−0.775678	−0.387839	+	0.921727i	$0.626778\pi$
$110$	−6.04251	−0.576131
$111$	−2.66414	−0.252869
$112$	3.47315	0.328182
$113$	−18.2568	−1.71745	−0.858726	−	0.512435i	$-0.828744\pi$
$113$	−18.2568	−1.71745	−0.858726	+	0.512435i	$0.828744\pi$
$114$	3.08152	0.288610
$115$	−7.97448	−0.743624
$116$	−6.73035	−0.624897
$117$	0	0
$118$	−6.40187	−0.589340
$119$	7.13845	0.654381
$120$	0.988450	0.0902328
$121$	25.5119	2.31926
$122$	−13.0482	−1.18133
$123$	−3.19806	−0.288360
$124$	−8.96482	−0.805065
$125$	−1.00000	−0.0894427
$126$	7.02606	0.625931
$127$	0.289626	0.0257001	0.0128501	−	0.999917i	$-0.495910\pi$
$127$	0.289626	0.0257001	0.0128501	+	0.999917i	$0.495910\pi$
$128$	−1.00000	−0.0883883
$129$	−2.01154	−0.177107
$130$	0	0
$131$	−19.4914	−1.70297	−0.851486	−	0.524378i	$-0.824298\pi$
$131$	−19.4914	−1.70297	−0.851486	+	0.524378i	$0.824298\pi$
$132$	−5.97272	−0.519859
$133$	−10.8276	−0.938873
$134$	−8.49642	−0.733979
$135$	4.96495	0.427315
$136$	−2.05533	−0.176243
$137$	3.70598	0.316623	0.158312	−	0.987389i	$-0.449395\pi$
$137$	3.70598	0.316623	0.158312	+	0.987389i	$0.449395\pi$
$138$	−7.88237	−0.670992
$139$	−10.3833	−0.880698	−0.440349	−	0.897827i	$-0.645145\pi$
$139$	−10.3833	−0.880698	−0.440349	+	0.897827i	$0.645145\pi$
$140$	−3.47315	−0.293535
$141$	−8.92627	−0.751728
$142$	0.121998	0.0102378
$143$	0	0
$144$	−2.02297	−0.168580
$145$	6.73035	0.558925
$146$	−0.138022	−0.0114228
$147$	5.00428	0.412746
$148$	−2.69527	−0.221550
$149$	−20.7318	−1.69842	−0.849209	−	0.528058i	$-0.822921\pi$
$149$	−20.7318	−1.69842	−0.849209	+	0.528058i	$0.822921\pi$
$150$	−0.988450	−0.0807066
$151$	20.5005	1.66831	0.834154	−	0.551531i	$-0.185956\pi$
$151$	20.5005	1.66831	0.834154	+	0.551531i	$0.185956\pi$
$152$	3.11752	0.252864
$153$	−4.15786	−0.336143
$154$	20.9865	1.69114
$155$	8.96482	0.720072
$156$	0	0
$157$	11.6082	0.926434	0.463217	−	0.886245i	$-0.346695\pi$
$157$	11.6082	0.926434	0.463217	+	0.886245i	$0.346695\pi$
$158$	6.71618	0.534311
$159$	−11.0555	−0.876756
$160$	1.00000	0.0790569
$161$	27.6965	2.18279
$162$	−1.16129	−0.0912394
$163$	8.77539	0.687342	0.343671	−	0.939090i	$-0.388330\pi$
$163$	8.77539	0.687342	0.343671	+	0.939090i	$0.388330\pi$
$164$	−3.23543	−0.252645
$165$	5.97272	0.464976
$166$	−3.03849	−0.235833
$167$	9.66796	0.748129	0.374065	−	0.927403i	$-0.377964\pi$
$167$	9.66796	0.748129	0.374065	+	0.927403i	$0.377964\pi$
$168$	−3.43303	−0.264864
$169$	0	0
$170$	2.05533	0.157636
$171$	6.30664	0.482281
$172$	−2.03505	−0.155171
$173$	−9.00074	−0.684313	−0.342157	−	0.939643i	$-0.611157\pi$
$173$	−9.00074	−0.684313	−0.342157	+	0.939643i	$0.611157\pi$
$174$	6.65262	0.504333
$175$	3.47315	0.262545
$176$	−6.04251	−0.455471
$177$	6.32793	0.475636
$178$	3.38412	0.253650
$179$	−11.1345	−0.832233	−0.416117	−	0.909311i	$-0.636609\pi$
$179$	−11.1345	−0.832233	−0.416117	+	0.909311i	$0.636609\pi$
$180$	2.02297	0.150783
$181$	1.63511	0.121536	0.0607682	−	0.998152i	$-0.480645\pi$
$181$	1.63511	0.121536	0.0607682	+	0.998152i	$0.480645\pi$
$182$	0	0
$183$	12.8975	0.953412
$184$	−7.97448	−0.587886
$185$	2.69527	0.198160
$186$	8.86128	0.649741
$187$	−12.4193	−0.908191
$188$	−9.03057	−0.658622
$189$	−17.2440	−1.25432
$190$	−3.11752	−0.226169
$191$	9.23847	0.668472	0.334236	−	0.942489i	$-0.391522\pi$
$191$	9.23847	0.668472	0.334236	+	0.942489i	$0.391522\pi$
$192$	0.988450	0.0713353
$193$	−7.28291	−0.524235	−0.262118	−	0.965036i	$-0.584421\pi$
$193$	−7.28291	−0.524235	−0.262118	+	0.965036i	$0.584421\pi$
$194$	−5.82521	−0.418226
$195$	0	0
$196$	5.06275	0.361625
$197$	−20.5744	−1.46587	−0.732933	−	0.680301i	$-0.761849\pi$
$197$	−20.5744	−1.46587	−0.732933	+	0.680301i	$0.761849\pi$
$198$	−12.2238	−0.868707
$199$	−2.77107	−0.196436	−0.0982179	−	0.995165i	$-0.531314\pi$
$199$	−2.77107	−0.196436	−0.0982179	+	0.995165i	$0.531314\pi$
$200$	−1.00000	−0.0707107
$201$	8.39829	0.592370
$202$	8.47871	0.596560
$203$	−23.3755	−1.64064
$204$	2.03159	0.142240
$205$	3.23543	0.225972
$206$	4.96891	0.346200
$207$	−16.1321	−1.12126
$208$	0	0
$209$	18.8376	1.30303
$210$	3.43303	0.236902
$211$	21.2755	1.46466	0.732332	−	0.680947i	$-0.238432\pi$
$211$	21.2755	1.46466	0.732332	+	0.680947i	$0.238432\pi$
$212$	−11.1847	−0.768165
$213$	−0.120589	−0.00826261
$214$	5.16942	0.353375
$215$	2.03505	0.138789
$216$	4.96495	0.337822
$217$	−31.1361	−2.11366
$218$	8.09832	0.548487
$219$	0.136428	0.00921895
$220$	6.04251	0.407386
$221$	0	0
$222$	2.66414	0.178805
$223$	−7.35898	−0.492793	−0.246397	−	0.969169i	$-0.579247\pi$
$223$	−7.35898	−0.492793	−0.246397	+	0.969169i	$0.579247\pi$
$224$	−3.47315	−0.232059
$225$	−2.02297	−0.134864
$226$	18.2568	1.21442
$227$	21.3232	1.41527	0.707637	−	0.706576i	$-0.249761\pi$
$227$	21.3232	1.41527	0.707637	+	0.706576i	$0.249761\pi$
$228$	−3.08152	−0.204078
$229$	−15.7929	−1.04363	−0.521813	−	0.853060i	$-0.674744\pi$
$229$	−15.7929	−1.04363	−0.521813	+	0.853060i	$0.674744\pi$
$230$	7.97448	0.525821
$231$	−20.7441	−1.36486
$232$	6.73035	0.441869
$233$	8.13040	0.532640	0.266320	−	0.963885i	$-0.414192\pi$
$233$	8.13040	0.532640	0.266320	+	0.963885i	$0.414192\pi$
$234$	0	0
$235$	9.03057	0.589089
$236$	6.40187	0.416726
$237$	−6.63861	−0.431224
$238$	−7.13845	−0.462717
$239$	−12.5247	−0.810154	−0.405077	−	0.914283i	$-0.632755\pi$
$239$	−12.5247	−0.810154	−0.405077	+	0.914283i	$0.632755\pi$
$240$	−0.988450	−0.0638042
$241$	3.58778	0.231109	0.115555	−	0.993301i	$-0.463135\pi$
$241$	3.58778	0.231109	0.115555	+	0.993301i	$0.463135\pi$
$242$	−25.5119	−1.63997
$243$	16.0427	1.02914
$244$	13.0482	0.835327
$245$	−5.06275	−0.323448
$246$	3.19806	0.203901
$247$	0	0
$248$	8.96482	0.569267
$249$	3.00340	0.190333
$250$	1.00000	0.0632456
$251$	−15.0726	−0.951372	−0.475686	−	0.879615i	$-0.657800\pi$
$251$	−15.0726	−0.951372	−0.475686	+	0.879615i	$0.657800\pi$
$252$	−7.02606	−0.442600
$253$	−48.1858	−3.02942
$254$	−0.289626	−0.0181727
$255$	−2.03159	−0.127223
$256$	1.00000	0.0625000
$257$	22.3804	1.39605	0.698026	−	0.716072i	$-0.254062\pi$
$257$	22.3804	1.39605	0.698026	+	0.716072i	$0.254062\pi$
$258$	2.01154	0.125233
$259$	−9.36106	−0.581668
$260$	0	0
$261$	13.6153	0.842764
$262$	19.4914	1.20418
$263$	18.7301	1.15495	0.577475	−	0.816408i	$-0.304038\pi$
$263$	18.7301	1.15495	0.577475	+	0.816408i	$0.304038\pi$
$264$	5.97272	0.367595
$265$	11.1847	0.687068
$266$	10.8276	0.663883
$267$	−3.34503	−0.204713
$268$	8.49642	0.519002
$269$	6.53932	0.398709	0.199355	−	0.979927i	$-0.436115\pi$
$269$	6.53932	0.398709	0.199355	+	0.979927i	$0.436115\pi$
$270$	−4.96495	−0.302157
$271$	−2.86445	−0.174003	−0.0870013	−	0.996208i	$-0.527728\pi$
$271$	−2.86445	−0.174003	−0.0870013	+	0.996208i	$0.527728\pi$
$272$	2.05533	0.124622
$273$	0	0
$274$	−3.70598	−0.223886
$275$	−6.04251	−0.364377
$276$	7.88237	0.474463
$277$	−2.80684	−0.168647	−0.0843233	−	0.996438i	$-0.526873\pi$
$277$	−2.80684	−0.168647	−0.0843233	+	0.996438i	$0.526873\pi$
$278$	10.3833	0.622747
$279$	18.1355	1.08575
$280$	3.47315	0.207560
$281$	25.8974	1.54491	0.772453	−	0.635071i	$-0.219029\pi$
$281$	25.8974	1.54491	0.772453	+	0.635071i	$0.219029\pi$
$282$	8.92627	0.531552
$283$	−28.8774	−1.71658	−0.858292	−	0.513162i	$-0.828474\pi$
$283$	−28.8774	−1.71658	−0.858292	+	0.513162i	$0.828474\pi$
$284$	−0.121998	−0.00723924
$285$	3.08152	0.182533
$286$	0	0
$287$	−11.2371	−0.663307
$288$	2.02297	0.119204
$289$	−12.7756	−0.751508
$290$	−6.73035	−0.395220
$291$	5.75794	0.337536
$292$	0.138022	0.00807713
$293$	11.0820	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$293$	11.0820	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$294$	−5.00428	−0.291856
$295$	−6.40187	−0.372731
$296$	2.69527	0.156659
$297$	30.0008	1.74082
$298$	20.7318	1.20096
$299$	0	0
$300$	0.988450	0.0570682
$301$	−7.06802	−0.407394
$302$	−20.5005	−1.17967
$303$	−8.38079	−0.481464
$304$	−3.11752	−0.178802
$305$	−13.0482	−0.747139
$306$	4.15786	0.237689
$307$	6.38045	0.364152	0.182076	−	0.983284i	$-0.441718\pi$
$307$	6.38045	0.364152	0.182076	+	0.983284i	$0.441718\pi$
$308$	−20.9865	−1.19582
$309$	−4.91152	−0.279407
$310$	−8.96482	−0.509168
$311$	18.9443	1.07423	0.537115	−	0.843509i	$-0.319514\pi$
$311$	18.9443	1.07423	0.537115	+	0.843509i	$0.319514\pi$
$312$	0	0
$313$	−17.5007	−0.989196	−0.494598	−	0.869122i	$-0.664685\pi$
$313$	−17.5007	−0.989196	−0.494598	+	0.869122i	$0.664685\pi$
$314$	−11.6082	−0.655088
$315$	7.02606	0.395874
$316$	−6.71618	−0.377815
$317$	34.2450	1.92339	0.961695	−	0.274121i	$-0.0883869\pi$
$317$	34.2450	1.92339	0.961695	+	0.274121i	$0.0883869\pi$
$318$	11.0555	0.619960
$319$	40.6682	2.27698
$320$	−1.00000	−0.0559017
$321$	−5.10972	−0.285197
$322$	−27.6965	−1.54347
$323$	−6.40753	−0.356524
$324$	1.16129	0.0645160
$325$	0	0
$326$	−8.77539	−0.486024
$327$	−8.00479	−0.442666
$328$	3.23543	0.178647
$329$	−31.3645	−1.72918
$330$	−5.97272	−0.328787
$331$	−8.24786	−0.453343	−0.226672	−	0.973971i	$-0.572784\pi$
$331$	−8.24786	−0.453343	−0.226672	+	0.973971i	$0.572784\pi$
$332$	3.03849	0.166759
$333$	5.45243	0.298792
$334$	−9.66796	−0.529007
$335$	−8.49642	−0.464209
$336$	3.43303	0.187287
$337$	12.1509	0.661902	0.330951	−	0.943648i	$-0.392631\pi$
$337$	12.1509	0.661902	0.330951	+	0.943648i	$0.392631\pi$
$338$	0	0
$339$	−18.0459	−0.980120
$340$	−2.05533	−0.111466
$341$	54.1700	2.93347
$342$	−6.30664	−0.341024
$343$	−6.72834	−0.363296
$344$	2.03505	0.109722
$345$	−7.88237	−0.424373
$346$	9.00074	0.483883
$347$	−14.7075	−0.789541	−0.394770	−	0.918780i	$-0.629176\pi$
$347$	−14.7075	−0.789541	−0.394770	+	0.918780i	$0.629176\pi$
$348$	−6.65262	−0.356618
$349$	27.6452	1.47982	0.739908	−	0.672708i	$-0.234869\pi$
$349$	27.6452	1.47982	0.739908	+	0.672708i	$0.234869\pi$
$350$	−3.47315	−0.185648
$351$	0	0
$352$	6.04251	0.322067
$353$	−15.0902	−0.803172	−0.401586	−	0.915821i	$-0.631541\pi$
$353$	−15.0902	−0.803172	−0.401586	+	0.915821i	$0.631541\pi$
$354$	−6.32793	−0.336326
$355$	0.121998	0.00647497
$356$	−3.38412	−0.179358
$357$	7.05601	0.373444
$358$	11.1345	0.588478
$359$	−10.0380	−0.529786	−0.264893	−	0.964278i	$-0.585337\pi$
$359$	−10.0380	−0.529786	−0.264893	+	0.964278i	$0.585337\pi$
$360$	−2.02297	−0.106620
$361$	−9.28106	−0.488477
$362$	−1.63511	−0.0859392
$363$	25.2172	1.32356
$364$	0	0
$365$	−0.138022	−0.00722440
$366$	−12.8975	−0.674164
$367$	−24.5353	−1.28073	−0.640365	−	0.768071i	$-0.721217\pi$
$367$	−24.5353	−1.28073	−0.640365	+	0.768071i	$0.721217\pi$
$368$	7.97448	0.415698
$369$	6.54516	0.340728
$370$	−2.69527	−0.140120
$371$	−38.8459	−2.01678
$372$	−8.86128	−0.459436
$373$	−1.24367	−0.0643947	−0.0321974	−	0.999482i	$-0.510251\pi$
$373$	−1.24367	−0.0643947	−0.0321974	+	0.999482i	$0.510251\pi$
$374$	12.4193	0.642188
$375$	−0.988450	−0.0510434
$376$	9.03057	0.465716
$377$	0	0
$378$	17.2440	0.886936
$379$	−18.4312	−0.946747	−0.473373	−	0.880862i	$-0.656964\pi$
$379$	−18.4312	−0.946747	−0.473373	+	0.880862i	$0.656964\pi$
$380$	3.11752	0.159925
$381$	0.286281	0.0146666
$382$	−9.23847	−0.472681
$383$	25.8315	1.31993	0.659963	−	0.751298i	$-0.270572\pi$
$383$	25.8315	1.31993	0.659963	+	0.751298i	$0.270572\pi$
$384$	−0.988450	−0.0504417
$385$	20.9865	1.06957
$386$	7.28291	0.370690
$387$	4.11683	0.209270
$388$	5.82521	0.295730
$389$	31.2329	1.58357	0.791784	−	0.610801i	$-0.209152\pi$
$389$	31.2329	1.58357	0.791784	+	0.610801i	$0.209152\pi$
$390$	0	0
$391$	16.3902	0.828886
$392$	−5.06275	−0.255708
$393$	−19.2663	−0.971855
$394$	20.5744	1.03652
$395$	6.71618	0.337928
$396$	12.2238	0.614268
$397$	28.1161	1.41111	0.705553	−	0.708657i	$-0.250699\pi$
$397$	28.1161	1.41111	0.705553	+	0.708657i	$0.250699\pi$
$398$	2.77107	0.138901
$399$	−10.7026	−0.535798
$400$	1.00000	0.0500000
$401$	−6.15725	−0.307478	−0.153739	−	0.988111i	$-0.549132\pi$
$401$	−6.15725	−0.307478	−0.153739	+	0.988111i	$0.549132\pi$
$402$	−8.39829	−0.418869
$403$	0	0
$404$	−8.47871	−0.421832
$405$	−1.16129	−0.0577048
$406$	23.3755	1.16011
$407$	16.2862	0.807276
$408$	−2.03159	−0.100579
$409$	26.4807	1.30939	0.654693	−	0.755895i	$-0.272798\pi$
$409$	26.4807	1.30939	0.654693	+	0.755895i	$0.272798\pi$
$410$	−3.23543	−0.159787
$411$	3.66318	0.180691
$412$	−4.96891	−0.244801
$413$	22.2346	1.09409
$414$	16.1321	0.792849
$415$	−3.03849	−0.149154
$416$	0	0
$417$	−10.2634	−0.502599
$418$	−18.8376	−0.921379
$419$	−11.8540	−0.579105	−0.289553	−	0.957162i	$-0.593507\pi$
$419$	−11.8540	−0.579105	−0.289553	+	0.957162i	$0.593507\pi$
$420$	−3.43303	−0.167515
$421$	−9.48437	−0.462240	−0.231120	−	0.972925i	$-0.574239\pi$
$421$	−9.48437	−0.462240	−0.231120	+	0.972925i	$0.574239\pi$
$422$	−21.2755	−1.03567
$423$	18.2685	0.888247
$424$	11.1847	0.543175
$425$	2.05533	0.0996980
$426$	0.120589	0.00584255
$427$	45.3184	2.19311
$428$	−5.16942	−0.249874
$429$	0	0
$430$	−2.03505	−0.0981387
$431$	14.1175	0.680016	0.340008	−	0.940423i	$-0.389570\pi$
$431$	14.1175	0.680016	0.340008	+	0.940423i	$0.389570\pi$
$432$	−4.96495	−0.238876
$433$	26.3194	1.26483	0.632414	−	0.774631i	$-0.282064\pi$
$433$	26.3194	1.26483	0.632414	+	0.774631i	$0.282064\pi$
$434$	31.1361	1.49458
$435$	6.65262	0.318968
$436$	−8.09832	−0.387839
$437$	−24.8606	−1.18924
$438$	−0.136428	−0.00651878
$439$	−26.5970	−1.26940	−0.634702	−	0.772757i	$-0.718877\pi$
$439$	−26.5970	−1.26940	−0.634702	+	0.772757i	$0.718877\pi$
$440$	−6.04251	−0.288065
$441$	−10.2418	−0.487704
$442$	0	0
$443$	15.9166	0.756222	0.378111	−	0.925760i	$-0.376574\pi$
$443$	15.9166	0.756222	0.378111	+	0.925760i	$0.376574\pi$
$444$	−2.66414	−0.126434
$445$	3.38412	0.160423
$446$	7.35898	0.348458
$447$	−20.4924	−0.969256
$448$	3.47315	0.164091
$449$	38.7349	1.82801	0.914007	−	0.405698i	$-0.132971\pi$
$449$	38.7349	1.82801	0.914007	+	0.405698i	$0.132971\pi$
$450$	2.02297	0.0953635
$451$	19.5501	0.920579
$452$	−18.2568	−0.858726
$453$	20.2637	0.952074
$454$	−21.3232	−1.00075
$455$	0	0
$456$	3.08152	0.144305
$457$	14.0340	0.656484	0.328242	−	0.944594i	$-0.393544\pi$
$457$	14.0340	0.656484	0.328242	+	0.944594i	$0.393544\pi$
$458$	15.7929	0.737955
$459$	−10.2046	−0.476310
$460$	−7.97448	−0.371812
$461$	−9.00148	−0.419241	−0.209620	−	0.977783i	$-0.567223\pi$
$461$	−9.00148	−0.419241	−0.209620	+	0.977783i	$0.567223\pi$
$462$	20.7441	0.965105
$463$	−34.7780	−1.61627	−0.808135	−	0.588997i	$-0.799523\pi$
$463$	−34.7780	−1.61627	−0.808135	+	0.588997i	$0.799523\pi$
$464$	−6.73035	−0.312449
$465$	8.86128	0.410932
$466$	−8.13040	−0.376633
$467$	18.1864	0.841568	0.420784	−	0.907161i	$-0.361755\pi$
$467$	18.1864	0.841568	0.420784	+	0.907161i	$0.361755\pi$
$468$	0	0
$469$	29.5093	1.36261
$470$	−9.03057	−0.416549
$471$	11.4741	0.528699
$472$	−6.40187	−0.294670
$473$	12.2968	0.565407
$474$	6.63861	0.304922
$475$	−3.11752	−0.143042
$476$	7.13845	0.327190
$477$	22.6262	1.03598
$478$	12.5247	0.572866
$479$	−32.0954	−1.46648	−0.733238	−	0.679973i	$-0.761992\pi$
$479$	−32.0954	−1.46648	−0.733238	+	0.679973i	$0.761992\pi$
$480$	0.988450	0.0451164
$481$	0	0
$482$	−3.58778	−0.163419
$483$	27.3766	1.24568
$484$	25.5119	1.15963
$485$	−5.82521	−0.264509
$486$	−16.0427	−0.727713
$487$	31.3280	1.41961	0.709804	−	0.704399i	$-0.248783\pi$
$487$	31.3280	1.41961	0.709804	+	0.704399i	$0.248783\pi$
$488$	−13.0482	−0.590665
$489$	8.67404	0.392254
$490$	5.06275	0.228712
$491$	−7.27229	−0.328194	−0.164097	−	0.986444i	$-0.552471\pi$
$491$	−7.27229	−0.328194	−0.164097	+	0.986444i	$0.552471\pi$
$492$	−3.19806	−0.144180
$493$	−13.8331	−0.623010
$494$	0	0
$495$	−12.2238	−0.549418
$496$	−8.96482	−0.402532
$497$	−0.423717	−0.0190063
$498$	−3.00340	−0.134586
$499$	−9.57876	−0.428804	−0.214402	−	0.976745i	$-0.568780\pi$
$499$	−9.57876	−0.428804	−0.214402	+	0.976745i	$0.568780\pi$
$500$	−1.00000	−0.0447214
$501$	9.55630	0.426944
$502$	15.0726	0.672722
$503$	6.45424	0.287780	0.143890	−	0.989594i	$-0.454039\pi$
$503$	6.45424	0.287780	0.143890	+	0.989594i	$0.454039\pi$
$504$	7.02606	0.312966
$505$	8.47871	0.377298
$506$	48.1858	2.14212
$507$	0	0
$508$	0.289626	0.0128501
$509$	−3.11126	−0.137904	−0.0689521	−	0.997620i	$-0.521966\pi$
$509$	−3.11126	−0.137904	−0.0689521	+	0.997620i	$0.521966\pi$
$510$	2.03159	0.0899603
$511$	0.479371	0.0212061
$512$	−1.00000	−0.0441942
$513$	15.4783	0.683386
$514$	−22.3804	−0.987158
$515$	4.96891	0.218956
$516$	−2.01154	−0.0885533
$517$	54.5673	2.39987
$518$	9.36106	0.411301
$519$	−8.89678	−0.390525
$520$	0	0
$521$	42.6003	1.86635	0.933177	−	0.359416i	$-0.117024\pi$
$521$	42.6003	1.86635	0.933177	+	0.359416i	$0.117024\pi$
$522$	−13.6153	−0.595924
$523$	−1.34603	−0.0588576	−0.0294288	−	0.999567i	$-0.509369\pi$
$523$	−1.34603	−0.0588576	−0.0294288	+	0.999567i	$0.509369\pi$
$524$	−19.4914	−0.851486
$525$	3.43303	0.149830
$526$	−18.7301	−0.816673
$527$	−18.4256	−0.802633
$528$	−5.97272	−0.259929
$529$	40.5923	1.76488
$530$	−11.1847	−0.485830
$531$	−12.9508	−0.562015
$532$	−10.8276	−0.469436
$533$	0	0
$534$	3.34503	0.144754
$535$	5.16942	0.223494
$536$	−8.49642	−0.366990
$537$	−11.0059	−0.474941
$538$	−6.53932	−0.281930
$539$	−30.5917	−1.31768
$540$	4.96495	0.213658
$541$	21.9468	0.943567	0.471784	−	0.881714i	$-0.343610\pi$
$541$	21.9468	0.943567	0.471784	+	0.881714i	$0.343610\pi$
$542$	2.86445	0.123038
$543$	1.61622	0.0693587
$544$	−2.05533	−0.0881214
$545$	8.09832	0.346894
$546$	0	0
$547$	−26.7344	−1.14308	−0.571540	−	0.820574i	$-0.693654\pi$
$547$	−26.7344	−1.14308	−0.571540	+	0.820574i	$0.693654\pi$
$548$	3.70598	0.158312
$549$	−26.3961	−1.12656
$550$	6.04251	0.257653
$551$	20.9820	0.893863
$552$	−7.88237	−0.335496
$553$	−23.3263	−0.991935
$554$	2.80684	0.119251
$555$	2.66414	0.113086
$556$	−10.3833	−0.440349
$557$	17.2868	0.732465	0.366233	−	0.930523i	$-0.380647\pi$
$557$	17.2868	0.732465	0.366233	+	0.930523i	$0.380647\pi$
$558$	−18.1355	−0.767738
$559$	0	0
$560$	−3.47315	−0.146767
$561$	−12.2759	−0.518289
$562$	−25.8974	−1.09241
$563$	−36.2325	−1.52702	−0.763508	−	0.645798i	$-0.776525\pi$
$563$	−36.2325	−1.52702	−0.763508	+	0.645798i	$0.776525\pi$
$564$	−8.92627	−0.375864
$565$	18.2568	0.768068
$566$	28.8774	1.21381
$567$	4.03332	0.169384
$568$	0.121998	0.00511892
$569$	21.4862	0.900749	0.450375	−	0.892840i	$-0.351290\pi$
$569$	21.4862	0.900749	0.450375	+	0.892840i	$0.351290\pi$
$570$	−3.08152	−0.129070
$571$	16.8212	0.703944	0.351972	−	0.936011i	$-0.385511\pi$
$571$	16.8212	0.703944	0.351972	+	0.936011i	$0.385511\pi$
$572$	0	0
$573$	9.13177	0.381485
$574$	11.2371	0.469029
$575$	7.97448	0.332559
$576$	−2.02297	−0.0842902
$577$	−33.5996	−1.39877	−0.699384	−	0.714746i	$-0.746542\pi$
$577$	−33.5996	−1.39877	−0.699384	+	0.714746i	$0.746542\pi$
$578$	12.7756	0.531396
$579$	−7.19880	−0.299172
$580$	6.73035	0.279462
$581$	10.5531	0.437818
$582$	−5.75794	−0.238674
$583$	67.5833	2.79902
$584$	−0.138022	−0.00571139
$585$	0	0
$586$	−11.0820	−0.457792
$587$	−26.4983	−1.09370	−0.546850	−	0.837230i	$-0.684173\pi$
$587$	−26.4983	−1.09370	−0.546850	+	0.837230i	$0.684173\pi$
$588$	5.00428	0.206373
$589$	27.9480	1.15158
$590$	6.40187	0.263561
$591$	−20.3368	−0.836543
$592$	−2.69527	−0.110775
$593$	33.8602	1.39047	0.695235	−	0.718783i	$-0.255300\pi$
$593$	33.8602	1.39047	0.695235	+	0.718783i	$0.255300\pi$
$594$	−30.0008	−1.23095
$595$	−7.13845	−0.292648
$596$	−20.7318	−0.849209
$597$	−2.73906	−0.112102
$598$	0	0
$599$	15.5745	0.636358	0.318179	−	0.948031i	$-0.396929\pi$
$599$	15.5745	0.636358	0.318179	+	0.948031i	$0.396929\pi$
$600$	−0.988450	−0.0403533
$601$	−33.7603	−1.37711	−0.688555	−	0.725184i	$-0.741755\pi$
$601$	−33.7603	−1.37711	−0.688555	+	0.725184i	$0.741755\pi$
$602$	7.06802	0.288071
$603$	−17.1880	−0.699948
$604$	20.5005	0.834154
$605$	−25.5119	−1.03721
$606$	8.38079	0.340446
$607$	−7.10319	−0.288310	−0.144155	−	0.989555i	$-0.546046\pi$
$607$	−7.10319	−0.288310	−0.144155	+	0.989555i	$0.546046\pi$
$608$	3.11752	0.126432
$609$	−23.1055	−0.936283
$610$	13.0482	0.528307
$611$	0	0
$612$	−4.15786	−0.168071
$613$	17.1320	0.691953	0.345976	−	0.938243i	$-0.387548\pi$
$613$	17.1320	0.691953	0.345976	+	0.938243i	$0.387548\pi$
$614$	−6.38045	−0.257494
$615$	3.19806	0.128958
$616$	20.9865	0.845571
$617$	−21.2367	−0.854959	−0.427479	−	0.904025i	$-0.640598\pi$
$617$	−21.2367	−0.854959	−0.427479	+	0.904025i	$0.640598\pi$
$618$	4.91152	0.197570
$619$	−23.8169	−0.957284	−0.478642	−	0.878010i	$-0.658871\pi$
$619$	−23.8169	−0.957284	−0.478642	+	0.878010i	$0.658871\pi$
$620$	8.96482	0.360036
$621$	−39.5929	−1.58881
$622$	−18.9443	−0.759596
$623$	−11.7535	−0.470896
$624$	0	0
$625$	1.00000	0.0400000
$626$	17.5007	0.699467
$627$	18.6201	0.743614
$628$	11.6082	0.463217
$629$	−5.53966	−0.220881
$630$	−7.02606	−0.279925
$631$	−11.1351	−0.443280	−0.221640	−	0.975129i	$-0.571141\pi$
$631$	−11.1351	−0.443280	−0.221640	+	0.975129i	$0.571141\pi$
$632$	6.71618	0.267155
$633$	21.0298	0.835858
$634$	−34.2450	−1.36004
$635$	−0.289626	−0.0114934
$636$	−11.0555	−0.438378
$637$	0	0
$638$	−40.6682	−1.61007
$639$	0.246797	0.00976316
$640$	1.00000	0.0395285
$641$	−1.04349	−0.0412152	−0.0206076	−	0.999788i	$-0.506560\pi$
$641$	−1.04349	−0.0412152	−0.0206076	+	0.999788i	$0.506560\pi$
$642$	5.10972	0.201665
$643$	6.78818	0.267700	0.133850	−	0.991002i	$-0.457266\pi$
$643$	6.78818	0.267700	0.133850	+	0.991002i	$0.457266\pi$
$644$	27.6965	1.09140
$645$	2.01154	0.0792044
$646$	6.40753	0.252101
$647$	−11.0037	−0.432601	−0.216300	−	0.976327i	$-0.569399\pi$
$647$	−11.0037	−0.432601	−0.216300	+	0.976327i	$0.569399\pi$
$648$	−1.16129	−0.0456197
$649$	−38.6833	−1.51845
$650$	0	0
$651$	−30.7765	−1.20623
$652$	8.77539	0.343671
$653$	−3.35145	−0.131152	−0.0655762	−	0.997848i	$-0.520889\pi$
$653$	−3.35145	−0.131152	−0.0655762	+	0.997848i	$0.520889\pi$
$654$	8.00479	0.313012
$655$	19.4914	0.761592
$656$	−3.23543	−0.126322
$657$	−0.279214	−0.0108932
$658$	31.3645	1.22272
$659$	−4.46293	−0.173851	−0.0869256	−	0.996215i	$-0.527704\pi$
$659$	−4.46293	−0.173851	−0.0869256	+	0.996215i	$0.527704\pi$
$660$	5.97272	0.232488
$661$	10.6936	0.415933	0.207967	−	0.978136i	$-0.433315\pi$
$661$	10.6936	0.415933	0.207967	+	0.978136i	$0.433315\pi$
$662$	8.24786	0.320562
$663$	0	0
$664$	−3.03849	−0.117916
$665$	10.8276	0.419877
$666$	−5.45243	−0.211278
$667$	−53.6710	−2.07815
$668$	9.66796	0.374065
$669$	−7.27398	−0.281228
$670$	8.49642	0.328246
$671$	−78.8440	−3.04374
$672$	−3.43303	−0.132432
$673$	8.33410	0.321256	0.160628	−	0.987015i	$-0.448648\pi$
$673$	8.33410	0.321256	0.160628	+	0.987015i	$0.448648\pi$
$674$	−12.1509	−0.468035
$675$	−4.96495	−0.191101
$676$	0	0
$677$	−10.4945	−0.403338	−0.201669	−	0.979454i	$-0.564637\pi$
$677$	−10.4945	−0.403338	−0.201669	+	0.979454i	$0.564637\pi$
$678$	18.0459	0.693049
$679$	20.2318	0.776426
$680$	2.05533	0.0788182
$681$	21.0770	0.807671
$682$	−54.1700	−2.07428
$683$	4.29065	0.164177	0.0820886	−	0.996625i	$-0.473841\pi$
$683$	4.29065	0.164177	0.0820886	+	0.996625i	$0.473841\pi$
$684$	6.30664	0.241140
$685$	−3.70598	−0.141598
$686$	6.72834	0.256889
$687$	−15.6105	−0.595579
$688$	−2.03505	−0.0775855
$689$	0	0
$690$	7.88237	0.300077
$691$	30.6269	1.16510	0.582550	−	0.812795i	$-0.302055\pi$
$691$	30.6269	1.16510	0.582550	+	0.812795i	$0.302055\pi$
$692$	−9.00074	−0.342157
$693$	42.4550	1.61273
$694$	14.7075	0.558290
$695$	10.3833	0.393860
$696$	6.65262	0.252167
$697$	−6.64987	−0.251882
$698$	−27.6452	−1.04639
$699$	8.03650	0.303968
$700$	3.47315	0.131273
$701$	8.16045	0.308216	0.154108	−	0.988054i	$-0.450750\pi$
$701$	8.16045	0.308216	0.154108	+	0.988054i	$0.450750\pi$
$702$	0	0
$703$	8.40255	0.316908
$704$	−6.04251	−0.227736
$705$	8.92627	0.336183
$706$	15.0902	0.567929
$707$	−29.4478	−1.10750
$708$	6.32793	0.237818
$709$	−32.1295	−1.20665	−0.603325	−	0.797495i	$-0.706158\pi$
$709$	−32.1295	−1.20665	−0.603325	+	0.797495i	$0.706158\pi$
$710$	−0.121998	−0.00457850
$711$	13.5866	0.509538
$712$	3.38412	0.126825
$713$	−71.4897	−2.67731
$714$	−7.05601	−0.264064
$715$	0	0
$716$	−11.1345	−0.416117
$717$	−12.3800	−0.462341
$718$	10.0380	0.374616
$719$	0.829030	0.0309176	0.0154588	−	0.999881i	$-0.495079\pi$
$719$	0.829030	0.0309176	0.0154588	+	0.999881i	$0.495079\pi$
$720$	2.02297	0.0753915
$721$	−17.2578	−0.642712
$722$	9.28106	0.345405
$723$	3.54634	0.131890
$724$	1.63511	0.0607682
$725$	−6.73035	−0.249959
$726$	−25.2172	−0.935900
$727$	−42.6451	−1.58162	−0.790810	−	0.612061i	$-0.790341\pi$
$727$	−42.6451	−1.58162	−0.790810	+	0.612061i	$0.790341\pi$
$728$	0	0
$729$	12.3736	0.458281
$730$	0.138022	0.00510842
$731$	−4.18269	−0.154702
$732$	12.8975	0.476706
$733$	−30.2477	−1.11722	−0.558612	−	0.829429i	$-0.688666\pi$
$733$	−30.2477	−1.11722	−0.558612	+	0.829429i	$0.688666\pi$
$734$	24.5353	0.905613
$735$	−5.00428	−0.184586
$736$	−7.97448	−0.293943
$737$	−51.3397	−1.89112
$738$	−6.54516	−0.240931
$739$	−15.8951	−0.584712	−0.292356	−	0.956309i	$-0.594439\pi$
$739$	−15.8951	−0.584712	−0.292356	+	0.956309i	$0.594439\pi$
$740$	2.69527	0.0990800
$741$	0	0
$742$	38.8459	1.42608
$743$	−7.65278	−0.280753	−0.140377	−	0.990098i	$-0.544831\pi$
$743$	−7.65278	−0.280753	−0.140377	+	0.990098i	$0.544831\pi$
$744$	8.86128	0.324870
$745$	20.7318	0.759555
$746$	1.24367	0.0455340
$747$	−6.14677	−0.224898
$748$	−12.4193	−0.454096
$749$	−17.9542	−0.656031
$750$	0.988450	0.0360931
$751$	30.3260	1.10661	0.553305	−	0.832979i	$-0.313366\pi$
$751$	30.3260	1.10661	0.553305	+	0.832979i	$0.313366\pi$
$752$	−9.03057	−0.329311
$753$	−14.8985	−0.542931
$754$	0	0
$755$	−20.5005	−0.746090
$756$	−17.2440	−0.627159
$757$	−20.9308	−0.760743	−0.380372	−	0.924834i	$-0.624204\pi$
$757$	−20.9308	−0.760743	−0.380372	+	0.924834i	$0.624204\pi$
$758$	18.4312	0.669451
$759$	−47.6293	−1.72883
$760$	−3.11752	−0.113084
$761$	−15.2635	−0.553301	−0.276651	−	0.960971i	$-0.589225\pi$
$761$	−15.2635	−0.553301	−0.276651	+	0.960971i	$0.589225\pi$
$762$	−0.286281	−0.0103709
$763$	−28.1267	−1.01825
$764$	9.23847	0.334236
$765$	4.15786	0.150328
$766$	−25.8315	−0.933329
$767$	0	0
$768$	0.988450	0.0356676
$769$	−16.3758	−0.590527	−0.295263	−	0.955416i	$-0.595407\pi$
$769$	−16.3758	−0.590527	−0.295263	+	0.955416i	$0.595407\pi$
$770$	−20.9865	−0.756302
$771$	22.1219	0.796702
$772$	−7.28291	−0.262118
$773$	5.38202	0.193578	0.0967890	−	0.995305i	$-0.469143\pi$
$773$	5.38202	0.193578	0.0967890	+	0.995305i	$0.469143\pi$
$774$	−4.11683	−0.147976
$775$	−8.96482	−0.322026
$776$	−5.82521	−0.209113
$777$	−9.25295	−0.331948
$778$	−31.2329	−1.11975
$779$	10.0865	0.361387
$780$	0	0
$781$	0.737173	0.0263781
$782$	−16.3902	−0.586111
$783$	33.4159	1.19419
$784$	5.06275	0.180813
$785$	−11.6082	−0.414314
$786$	19.2663	0.687206
$787$	−26.3189	−0.938167	−0.469084	−	0.883154i	$-0.655416\pi$
$787$	−26.3189	−0.938167	−0.469084	+	0.883154i	$0.655416\pi$
$788$	−20.5744	−0.732933
$789$	18.5138	0.659109
$790$	−6.71618	−0.238951
$791$	−63.4085	−2.25455
$792$	−12.2238	−0.434353
$793$	0	0
$794$	−28.1161	−0.997802
$795$	11.0555	0.392097
$796$	−2.77107	−0.0982179
$797$	−22.8724	−0.810182	−0.405091	−	0.914276i	$-0.632760\pi$
$797$	−22.8724	−0.810182	−0.405091	+	0.914276i	$0.632760\pi$
$798$	10.7026	0.378866
$799$	−18.5608	−0.656633
$800$	−1.00000	−0.0353553
$801$	6.84595	0.241890
$802$	6.15725	0.217420
$803$	−0.833999	−0.0294312
$804$	8.39829	0.296185
$805$	−27.6965	−0.976174
$806$	0	0
$807$	6.46379	0.227536
$808$	8.47871	0.298280
$809$	−19.1225	−0.672310	−0.336155	−	0.941807i	$-0.609127\pi$
$809$	−19.1225	−0.672310	−0.336155	+	0.941807i	$0.609127\pi$
$810$	1.16129	0.0408035
$811$	13.2144	0.464019	0.232009	−	0.972714i	$-0.425470\pi$
$811$	13.2144	0.464019	0.232009	+	0.972714i	$0.425470\pi$
$812$	−23.3755	−0.820319
$813$	−2.83136	−0.0993002
$814$	−16.2862	−0.570830
$815$	−8.77539	−0.307389
$816$	2.03159	0.0711198
$817$	6.34430	0.221959
$818$	−26.4807	−0.925876
$819$	0	0
$820$	3.23543	0.112986
$821$	9.35719	0.326568	0.163284	−	0.986579i	$-0.447791\pi$
$821$	9.35719	0.326568	0.163284	+	0.986579i	$0.447791\pi$
$822$	−3.66318	−0.127768
$823$	10.5591	0.368065	0.184033	−	0.982920i	$-0.441085\pi$
$823$	10.5591	0.368065	0.184033	+	0.982920i	$0.441085\pi$
$824$	4.96891	0.173100
$825$	−5.97272	−0.207943
$826$	−22.2346	−0.773642
$827$	−12.6564	−0.440107	−0.220054	−	0.975488i	$-0.570623\pi$
$827$	−12.6564	−0.440107	−0.220054	+	0.975488i	$0.570623\pi$
$828$	−16.1321	−0.560629
$829$	54.5753	1.89548	0.947740	−	0.319044i	$-0.103362\pi$
$829$	54.5753	1.89548	0.947740	+	0.319044i	$0.103362\pi$
$830$	3.03849	0.105468
$831$	−2.77442	−0.0962436
$832$	0	0
$833$	10.4056	0.360533
$834$	10.2634	0.355391
$835$	−9.66796	−0.334574
$836$	18.8376	0.651514
$837$	44.5099	1.53849
$838$	11.8540	0.409489
$839$	−12.0853	−0.417231	−0.208616	−	0.977998i	$-0.566896\pi$
$839$	−12.0853	−0.417231	−0.208616	+	0.977998i	$0.566896\pi$
$840$	3.43303	0.118451
$841$	16.2976	0.561986
$842$	9.48437	0.326853
$843$	25.5983	0.881651
$844$	21.2755	0.732332
$845$	0	0
$846$	−18.2685	−0.628085
$847$	88.6066	3.04456
$848$	−11.1847	−0.384083
$849$	−28.5439	−0.979624
$850$	−2.05533	−0.0704971
$851$	−21.4933	−0.736782
$852$	−0.120589	−0.00413131
$853$	16.2371	0.555947	0.277973	−	0.960589i	$-0.410337\pi$
$853$	16.2371	0.555947	0.277973	+	0.960589i	$0.410337\pi$
$854$	−45.3184	−1.55076
$855$	−6.30664	−0.215682
$856$	5.16942	0.176687
$857$	30.2096	1.03194	0.515970	−	0.856607i	$-0.327432\pi$
$857$	30.2096	1.03194	0.515970	+	0.856607i	$0.327432\pi$
$858$	0	0
$859$	18.9280	0.645814	0.322907	−	0.946431i	$-0.395340\pi$
$859$	18.9280	0.645814	0.322907	+	0.946431i	$0.395340\pi$
$860$	2.03505	0.0693945
$861$	−11.1073	−0.378537
$862$	−14.1175	−0.480844
$863$	9.02786	0.307312	0.153656	−	0.988124i	$-0.450895\pi$
$863$	9.02786	0.307312	0.153656	+	0.988124i	$0.450895\pi$
$864$	4.96495	0.168911
$865$	9.00074	0.306034
$866$	−26.3194	−0.894368
$867$	−12.6281	−0.428872
$868$	−31.1361	−1.05683
$869$	40.5826	1.37667
$870$	−6.65262	−0.225545
$871$	0	0
$872$	8.09832	0.274244
$873$	−11.7842	−0.398835
$874$	24.8606	0.840922
$875$	−3.47315	−0.117414
$876$	0.136428	0.00460947
$877$	42.5655	1.43733	0.718667	−	0.695354i	$-0.244753\pi$
$877$	42.5655	1.43733	0.718667	+	0.695354i	$0.244753\pi$
$878$	26.5970	0.897604
$879$	10.9540	0.369468
$880$	6.04251	0.203693
$881$	39.9325	1.34536	0.672680	−	0.739933i	$-0.265143\pi$
$881$	39.9325	1.34536	0.672680	+	0.739933i	$0.265143\pi$
$882$	10.2418	0.344859
$883$	−26.1186	−0.878962	−0.439481	−	0.898252i	$-0.644838\pi$
$883$	−26.1186	−0.878962	−0.439481	+	0.898252i	$0.644838\pi$
$884$	0	0
$885$	−6.32793	−0.212711
$886$	−15.9166	−0.534729
$887$	23.2009	0.779011	0.389506	−	0.921024i	$-0.372646\pi$
$887$	23.2009	0.779011	0.389506	+	0.921024i	$0.372646\pi$
$888$	2.66414	0.0894026
$889$	1.00591	0.0337372
$890$	−3.38412	−0.113436
$891$	−7.01709	−0.235081
$892$	−7.35898	−0.246397
$893$	28.1530	0.942104
$894$	20.4924	0.685368
$895$	11.1345	0.372186
$896$	−3.47315	−0.116030
$897$	0	0
$898$	−38.7349	−1.29260
$899$	60.3364	2.01233
$900$	−2.02297	−0.0674322
$901$	−22.9881	−0.765845
$902$	−19.5501	−0.650948
$903$	−6.98639	−0.232492
$904$	18.2568	0.607211
$905$	−1.63511	−0.0543527
$906$	−20.2637	−0.673218
$907$	22.1574	0.735724	0.367862	−	0.929880i	$-0.380090\pi$
$907$	22.1574	0.735724	0.367862	+	0.929880i	$0.380090\pi$
$908$	21.3232	0.707637
$909$	17.1521	0.568901
$910$	0	0
$911$	−10.5539	−0.349666	−0.174833	−	0.984598i	$-0.555939\pi$
$911$	−10.5539	−0.349666	−0.174833	+	0.984598i	$0.555939\pi$
$912$	−3.08152	−0.102039
$913$	−18.3601	−0.607631
$914$	−14.0340	−0.464204
$915$	−12.8975	−0.426379
$916$	−15.7929	−0.521813
$917$	−67.6965	−2.23554
$918$	10.2046	0.336802
$919$	29.8994	0.986291	0.493146	−	0.869947i	$-0.335847\pi$
$919$	29.8994	0.986291	0.493146	+	0.869947i	$0.335847\pi$
$920$	7.97448	0.262911
$921$	6.30676	0.207815
$922$	9.00148	0.296448
$923$	0	0
$924$	−20.7441	−0.682432
$925$	−2.69527	−0.0886199
$926$	34.7780	1.14288
$927$	10.0519	0.330149
$928$	6.73035	0.220934
$929$	−56.6561	−1.85883	−0.929413	−	0.369042i	$-0.879686\pi$
$929$	−56.6561	−1.85883	−0.929413	+	0.369042i	$0.879686\pi$
$930$	−8.86128	−0.290573
$931$	−15.7832	−0.517275
$932$	8.13040	0.266320
$933$	18.7255	0.613044
$934$	−18.1864	−0.595078
$935$	12.4193	0.406155
$936$	0	0
$937$	−37.4212	−1.22250	−0.611249	−	0.791439i	$-0.709332\pi$
$937$	−37.4212	−1.22250	−0.611249	+	0.791439i	$0.709332\pi$
$938$	−29.5093	−0.963514
$939$	−17.2985	−0.564517
$940$	9.03057	0.294545
$941$	−0.100449	−0.00327454	−0.00163727	−	0.999999i	$-0.500521\pi$
$941$	−0.100449	−0.00327454	−0.00163727	+	0.999999i	$0.500521\pi$
$942$	−11.4741	−0.373847
$943$	−25.8009	−0.840191
$944$	6.40187	0.208363
$945$	17.2440	0.560948
$946$	−12.2968	−0.399803
$947$	−27.9119	−0.907014	−0.453507	−	0.891253i	$-0.649827\pi$
$947$	−27.9119	−0.907014	−0.453507	+	0.891253i	$0.649827\pi$
$948$	−6.63861	−0.215612
$949$	0	0
$950$	3.11752	0.101146
$951$	33.8495	1.09764
$952$	−7.13845	−0.231359
$953$	4.28506	0.138807	0.0694033	−	0.997589i	$-0.477890\pi$
$953$	4.28506	0.138807	0.0694033	+	0.997589i	$0.477890\pi$
$954$	−22.6262	−0.732549
$955$	−9.23847	−0.298950
$956$	−12.5247	−0.405077
$957$	40.1985	1.29943
$958$	32.0954	1.03695
$959$	12.8714	0.415640
$960$	−0.988450	−0.0319021
$961$	49.3680	1.59252
$962$	0	0
$963$	10.4576	0.336990
$964$	3.58778	0.115555
$965$	7.28291	0.234445
$966$	−27.3766	−0.880829
$967$	28.0511	0.902062	0.451031	−	0.892508i	$-0.351056\pi$
$967$	28.0511	0.902062	0.451031	+	0.892508i	$0.351056\pi$
$968$	−25.5119	−0.819984
$969$	−6.33352	−0.203462
$970$	5.82521	0.187036
$971$	−29.3042	−0.940418	−0.470209	−	0.882555i	$-0.655821\pi$
$971$	−29.3042	−0.940418	−0.470209	+	0.882555i	$0.655821\pi$
$972$	16.0427	0.514571
$973$	−36.0626	−1.15612
$974$	−31.3280	−1.00382
$975$	0	0
$976$	13.0482	0.417663
$977$	21.1328	0.676099	0.338049	−	0.941128i	$-0.390233\pi$
$977$	21.1328	0.676099	0.338049	+	0.941128i	$0.390233\pi$
$978$	−8.67404	−0.277365
$979$	20.4486	0.653539
$980$	−5.06275	−0.161724
$981$	16.3826	0.523057
$982$	7.27229	0.232068
$983$	30.8904	0.985250	0.492625	−	0.870242i	$-0.336037\pi$
$983$	30.8904	0.985250	0.492625	+	0.870242i	$0.336037\pi$
$984$	3.19806	0.101951
$985$	20.5744	0.655555
$986$	13.8331	0.440535
$987$	−31.0023	−0.986813
$988$	0	0
$989$	−16.2284	−0.516034
$990$	12.2238	0.388497
$991$	19.9662	0.634247	0.317124	−	0.948384i	$-0.397283\pi$
$991$	19.9662	0.634247	0.317124	+	0.948384i	$0.397283\pi$
$992$	8.96482	0.284633
$993$	−8.15260	−0.258715
$994$	0.423717	0.0134395
$995$	2.77107	0.0878487
$996$	3.00340	0.0951664
$997$	1.55915	0.0493788	0.0246894	−	0.999695i	$-0.492140\pi$
$997$	1.55915	0.0493788	0.0246894	+	0.999695i	$0.492140\pi$
$998$	9.57876	0.303210
$999$	13.3819	0.423384

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.a.v.1.4	✓	6
5.4	even	2		8450.2.a.cq.1.3		6
13.2	odd	12		1690.2.l.n.1161.3		24
13.3	even	3		1690.2.e.v.191.3		12
13.4	even	6		1690.2.e.u.991.3		12
13.5	odd	4		1690.2.d.l.1351.10		12
13.6	odd	12		1690.2.l.n.361.8		24
13.7	odd	12		1690.2.l.n.361.3		24
13.8	odd	4		1690.2.d.l.1351.4		12
13.9	even	3		1690.2.e.v.991.3		12
13.10	even	6		1690.2.e.u.191.3		12
13.11	odd	12		1690.2.l.n.1161.8		24
13.12	even	2		1690.2.a.w.1.4	yes	6
65.64	even	2		8450.2.a.cp.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.a.v.1.4	✓	6	1.1	even	1	trivial
1690.2.a.w.1.4	yes	6	13.12	even	2
1690.2.d.l.1351.4		12	13.8	odd	4
1690.2.d.l.1351.10		12	13.5	odd	4
1690.2.e.u.191.3		12	13.10	even	6
1690.2.e.u.991.3		12	13.4	even	6
1690.2.e.v.191.3		12	13.3	even	3
1690.2.e.v.991.3		12	13.9	even	3
1690.2.l.n.361.3		24	13.7	odd	12
1690.2.l.n.361.8		24	13.6	odd	12
1690.2.l.n.1161.3		24	13.2	odd	12
1690.2.l.n.1161.8		24	13.11	odd	12
8450.2.a.cp.1.3		6	65.64	even	2
8450.2.a.cq.1.3		6	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.988450 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.988450 q^{6} +3.47315 q^{7} -1.00000 q^{8} -2.02297 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.988450 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.988450 q^{6} +3.47315 q^{7} -1.00000 q^{8} -2.02297 q^{9} +1.00000 q^{10} -6.04251 q^{11} +0.988450 q^{12} -3.47315 q^{14} -0.988450 q^{15} +1.00000 q^{16} +2.05533 q^{17} +2.02297 q^{18} -3.11752 q^{19} -1.00000 q^{20} +3.43303 q^{21} +6.04251 q^{22} +7.97448 q^{23} -0.988450 q^{24} +1.00000 q^{25} -4.96495 q^{27} +3.47315 q^{28} -6.73035 q^{29} +0.988450 q^{30} -8.96482 q^{31} -1.00000 q^{32} -5.97272 q^{33} -2.05533 q^{34} -3.47315 q^{35} -2.02297 q^{36} -2.69527 q^{37} +3.11752 q^{38} +1.00000 q^{40} -3.23543 q^{41} -3.43303 q^{42} -2.03505 q^{43} -6.04251 q^{44} +2.02297 q^{45} -7.97448 q^{46} -9.03057 q^{47} +0.988450 q^{48} +5.06275 q^{49} -1.00000 q^{50} +2.03159 q^{51} -11.1847 q^{53} +4.96495 q^{54} +6.04251 q^{55} -3.47315 q^{56} -3.08152 q^{57} +6.73035 q^{58} +6.40187 q^{59} -0.988450 q^{60} +13.0482 q^{61} +8.96482 q^{62} -7.02606 q^{63} +1.00000 q^{64} +5.97272 q^{66} +8.49642 q^{67} +2.05533 q^{68} +7.88237 q^{69} +3.47315 q^{70} -0.121998 q^{71} +2.02297 q^{72} +0.138022 q^{73} +2.69527 q^{74} +0.988450 q^{75} -3.11752 q^{76} -20.9865 q^{77} -6.71618 q^{79} -1.00000 q^{80} +1.16129 q^{81} +3.23543 q^{82} +3.03849 q^{83} +3.43303 q^{84} -2.05533 q^{85} +2.03505 q^{86} -6.65262 q^{87} +6.04251 q^{88} -3.38412 q^{89} -2.02297 q^{90} +7.97448 q^{92} -8.86128 q^{93} +9.03057 q^{94} +3.11752 q^{95} -0.988450 q^{96} +5.82521 q^{97} -5.06275 q^{98} +12.2238 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 6 q^{5} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 6 * q^5 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 16 * q^9	\( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 6 q^{5} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 16 q^{9} + 6 q^{10} - 15 q^{11} - 2 q^{12} + 3 q^{14} + 2 q^{15} + 6 q^{16} - 3 q^{17} - 16 q^{18} - q^{19} - 6 q^{20} + 2 q^{21} + 15 q^{22} - 3 q^{23} + 2 q^{24} + 6 q^{25} - 20 q^{27} - 3 q^{28} + 7 q^{29} - 2 q^{30} - 6 q^{32} - 4 q^{33} + 3 q^{34} + 3 q^{35} + 16 q^{36} - 6 q^{37} + q^{38} + 6 q^{40} - 2 q^{41} - 2 q^{42} - 22 q^{43} - 15 q^{44} - 16 q^{45} + 3 q^{46} - 7 q^{47} - 2 q^{48} + 31 q^{49} - 6 q^{50} - 22 q^{51} - 16 q^{53} + 20 q^{54} + 15 q^{55} + 3 q^{56} - 2 q^{57} - 7 q^{58} - 15 q^{59} + 2 q^{60} + 33 q^{61} - 25 q^{63} + 6 q^{64} + 4 q^{66} + 8 q^{67} - 3 q^{68} - 6 q^{69} - 3 q^{70} - 40 q^{71} - 16 q^{72} - 21 q^{73} + 6 q^{74} - 2 q^{75} - q^{76} - 34 q^{77} + 20 q^{79} - 6 q^{80} - 2 q^{81} + 2 q^{82} - 22 q^{83} + 2 q^{84} + 3 q^{85} + 22 q^{86} - 39 q^{87} + 15 q^{88} - 20 q^{89} + 16 q^{90} - 3 q^{92} - 48 q^{93} + 7 q^{94} + q^{95} + 2 q^{96} + 7 q^{97} - 31 q^{98} - 55 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 6 * q^5 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 16 * q^9 + 6 * q^10 - 15 * q^11 - 2 * q^12 + 3 * q^14 + 2 * q^15 + 6 * q^16 - 3 * q^17 - 16 * q^18 - q^19 - 6 * q^20 + 2 * q^21 + 15 * q^22 - 3 * q^23 + 2 * q^24 + 6 * q^25 - 20 * q^27 - 3 * q^28 + 7 * q^29 - 2 * q^30 - 6 * q^32 - 4 * q^33 + 3 * q^34 + 3 * q^35 + 16 * q^36 - 6 * q^37 + q^38 + 6 * q^40 - 2 * q^41 - 2 * q^42 - 22 * q^43 - 15 * q^44 - 16 * q^45 + 3 * q^46 - 7 * q^47 - 2 * q^48 + 31 * q^49 - 6 * q^50 - 22 * q^51 - 16 * q^53 + 20 * q^54 + 15 * q^55 + 3 * q^56 - 2 * q^57 - 7 * q^58 - 15 * q^59 + 2 * q^60 + 33 * q^61 - 25 * q^63 + 6 * q^64 + 4 * q^66 + 8 * q^67 - 3 * q^68 - 6 * q^69 - 3 * q^70 - 40 * q^71 - 16 * q^72 - 21 * q^73 + 6 * q^74 - 2 * q^75 - q^76 - 34 * q^77 + 20 * q^79 - 6 * q^80 - 2 * q^81 + 2 * q^82 - 22 * q^83 + 2 * q^84 + 3 * q^85 + 22 * q^86 - 39 * q^87 + 15 * q^88 - 20 * q^89 + 16 * q^90 - 3 * q^92 - 48 * q^93 + 7 * q^94 + q^95 + 2 * q^96 + 7 * q^97 - 31 * q^98 - 55 * q^99

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