LMFDB - Embedded newform 418.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [418,2,Mod(1,418)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(418, 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("418.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 418 = 2 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	418.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:	$3.33774680449$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	418.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} -2.30278 q^{5} -0.302776 q^{6} +1.30278 q^{7} -1.00000 q^{8} -2.90833 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} -2.30278 q^{5} -0.302776 q^{6} +1.30278 q^{7} -1.00000 q^{8} -2.90833 q^{9} +2.30278 q^{10} +1.00000 q^{11} +0.302776 q^{12} -3.30278 q^{13} -1.30278 q^{14} -0.697224 q^{15} +1.00000 q^{16} -4.60555 q^{17} +2.90833 q^{18} +1.00000 q^{19} -2.30278 q^{20} +0.394449 q^{21} -1.00000 q^{22} -1.39445 q^{23} -0.302776 q^{24} +0.302776 q^{25} +3.30278 q^{26} -1.78890 q^{27} +1.30278 q^{28} -8.30278 q^{29} +0.697224 q^{30} -3.30278 q^{31} -1.00000 q^{32} +0.302776 q^{33} +4.60555 q^{34} -3.00000 q^{35} -2.90833 q^{36} +5.21110 q^{37} -1.00000 q^{38} -1.00000 q^{39} +2.30278 q^{40} +3.90833 q^{41} -0.394449 q^{42} +1.09167 q^{43} +1.00000 q^{44} +6.69722 q^{45} +1.39445 q^{46} -6.00000 q^{47} +0.302776 q^{48} -5.30278 q^{49} -0.302776 q^{50} -1.39445 q^{51} -3.30278 q^{52} -10.6056 q^{53} +1.78890 q^{54} -2.30278 q^{55} -1.30278 q^{56} +0.302776 q^{57} +8.30278 q^{58} -0.697224 q^{60} +11.2111 q^{61} +3.30278 q^{62} -3.78890 q^{63} +1.00000 q^{64} +7.60555 q^{65} -0.302776 q^{66} -6.51388 q^{67} -4.60555 q^{68} -0.422205 q^{69} +3.00000 q^{70} -9.69722 q^{71} +2.90833 q^{72} -2.60555 q^{73} -5.21110 q^{74} +0.0916731 q^{75} +1.00000 q^{76} +1.30278 q^{77} +1.00000 q^{78} +15.8167 q^{79} -2.30278 q^{80} +8.18335 q^{81} -3.90833 q^{82} +14.3028 q^{83} +0.394449 q^{84} +10.6056 q^{85} -1.09167 q^{86} -2.51388 q^{87} -1.00000 q^{88} +4.60555 q^{89} -6.69722 q^{90} -4.30278 q^{91} -1.39445 q^{92} -1.00000 q^{93} +6.00000 q^{94} -2.30278 q^{95} -0.302776 q^{96} +8.00000 q^{97} +5.30278 q^{98} -2.90833 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{5} + 3 q^{6} - q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 3 * q^3 + 2 * q^4 - q^5 + 3 * q^6 - q^7 - 2 * q^8 + 5 * q^9	\( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{5} + 3 q^{6} - q^{7} - 2 q^{8} + 5 q^{9} + q^{10} + 2 q^{11} - 3 q^{12} - 3 q^{13} + q^{14} - 5 q^{15} + 2 q^{16} - 2 q^{17} - 5 q^{18} + 2 q^{19} - q^{20} + 8 q^{21} - 2 q^{22} - 10 q^{23} + 3 q^{24} - 3 q^{25} + 3 q^{26} - 18 q^{27} - q^{28} - 13 q^{29} + 5 q^{30} - 3 q^{31} - 2 q^{32} - 3 q^{33} + 2 q^{34} - 6 q^{35} + 5 q^{36} - 4 q^{37} - 2 q^{38} - 2 q^{39} + q^{40} - 3 q^{41} - 8 q^{42} + 13 q^{43} + 2 q^{44} + 17 q^{45} + 10 q^{46} - 12 q^{47} - 3 q^{48} - 7 q^{49} + 3 q^{50} - 10 q^{51} - 3 q^{52} - 14 q^{53} + 18 q^{54} - q^{55} + q^{56} - 3 q^{57} + 13 q^{58} - 5 q^{60} + 8 q^{61} + 3 q^{62} - 22 q^{63} + 2 q^{64} + 8 q^{65} + 3 q^{66} + 5 q^{67} - 2 q^{68} + 28 q^{69} + 6 q^{70} - 23 q^{71} - 5 q^{72} + 2 q^{73} + 4 q^{74} + 11 q^{75} + 2 q^{76} - q^{77} + 2 q^{78} + 10 q^{79} - q^{80} + 38 q^{81} + 3 q^{82} + 25 q^{83} + 8 q^{84} + 14 q^{85} - 13 q^{86} + 13 q^{87} - 2 q^{88} + 2 q^{89} - 17 q^{90} - 5 q^{91} - 10 q^{92} - 2 q^{93} + 12 q^{94} - q^{95} + 3 q^{96} + 16 q^{97} + 7 q^{98} + 5 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 3 * q^3 + 2 * q^4 - q^5 + 3 * q^6 - q^7 - 2 * q^8 + 5 * q^9 + q^10 + 2 * q^11 - 3 * q^12 - 3 * q^13 + q^14 - 5 * q^15 + 2 * q^16 - 2 * q^17 - 5 * q^18 + 2 * q^19 - q^20 + 8 * q^21 - 2 * q^22 - 10 * q^23 + 3 * q^24 - 3 * q^25 + 3 * q^26 - 18 * q^27 - q^28 - 13 * q^29 + 5 * q^30 - 3 * q^31 - 2 * q^32 - 3 * q^33 + 2 * q^34 - 6 * q^35 + 5 * q^36 - 4 * q^37 - 2 * q^38 - 2 * q^39 + q^40 - 3 * q^41 - 8 * q^42 + 13 * q^43 + 2 * q^44 + 17 * q^45 + 10 * q^46 - 12 * q^47 - 3 * q^48 - 7 * q^49 + 3 * q^50 - 10 * q^51 - 3 * q^52 - 14 * q^53 + 18 * q^54 - q^55 + q^56 - 3 * q^57 + 13 * q^58 - 5 * q^60 + 8 * q^61 + 3 * q^62 - 22 * q^63 + 2 * q^64 + 8 * q^65 + 3 * q^66 + 5 * q^67 - 2 * q^68 + 28 * q^69 + 6 * q^70 - 23 * q^71 - 5 * q^72 + 2 * q^73 + 4 * q^74 + 11 * q^75 + 2 * q^76 - q^77 + 2 * q^78 + 10 * q^79 - q^80 + 38 * q^81 + 3 * q^82 + 25 * q^83 + 8 * q^84 + 14 * q^85 - 13 * q^86 + 13 * q^87 - 2 * q^88 + 2 * q^89 - 17 * q^90 - 5 * q^91 - 10 * q^92 - 2 * q^93 + 12 * q^94 - q^95 + 3 * q^96 + 16 * q^97 + 7 * q^98 + 5 * 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
$3$	0.302776	0.174808	0.0874038	−	0.996173i	$-0.472143\pi$
$3$	0.302776	0.174808	0.0874038	+	0.996173i	$0.472143\pi$
$4$	1.00000	0.500000
$5$	−2.30278	−1.02983	−0.514916	−	0.857240i	$-0.672177\pi$
$5$	−2.30278	−1.02983	−0.514916	+	0.857240i	$0.672177\pi$
$6$	−0.302776	−0.123608
$7$	1.30278	0.492403	0.246201	−	0.969219i	$-0.420818\pi$
$7$	1.30278	0.492403	0.246201	+	0.969219i	$0.420818\pi$
$8$	−1.00000	−0.353553
$9$	−2.90833	−0.969442
$10$	2.30278	0.728202
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.302776	0.0874038
$13$	−3.30278	−0.916025	−0.458013	−	0.888946i	$-0.651439\pi$
$13$	−3.30278	−0.916025	−0.458013	+	0.888946i	$0.651439\pi$
$14$	−1.30278	−0.348181
$15$	−0.697224	−0.180023
$16$	1.00000	0.250000
$17$	−4.60555	−1.11701	−0.558505	−	0.829501i	$-0.688625\pi$
$17$	−4.60555	−1.11701	−0.558505	+	0.829501i	$0.688625\pi$
$18$	2.90833	0.685499
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.30278	−0.514916
$21$	0.394449	0.0860758
$22$	−1.00000	−0.213201
$23$	−1.39445	−0.290763	−0.145381	−	0.989376i	$-0.546441\pi$
$23$	−1.39445	−0.290763	−0.145381	+	0.989376i	$0.546441\pi$
$24$	−0.302776	−0.0618038
$25$	0.302776	0.0605551
$26$	3.30278	0.647728
$27$	−1.78890	−0.344273
$28$	1.30278	0.246201
$29$	−8.30278	−1.54179	−0.770893	−	0.636964i	$-0.780190\pi$
$29$	−8.30278	−1.54179	−0.770893	+	0.636964i	$0.780190\pi$
$30$	0.697224	0.127295
$31$	−3.30278	−0.593196	−0.296598	−	0.955002i	$-0.595852\pi$
$31$	−3.30278	−0.593196	−0.296598	+	0.955002i	$0.595852\pi$
$32$	−1.00000	−0.176777
$33$	0.302776	0.0527065
$34$	4.60555	0.789846
$35$	−3.00000	−0.507093
$36$	−2.90833	−0.484721
$37$	5.21110	0.856700	0.428350	−	0.903613i	$-0.359095\pi$
$37$	5.21110	0.856700	0.428350	+	0.903613i	$0.359095\pi$
$38$	−1.00000	−0.162221
$39$	−1.00000	−0.160128
$40$	2.30278	0.364101
$41$	3.90833	0.610378	0.305189	−	0.952292i	$-0.401280\pi$
$41$	3.90833	0.610378	0.305189	+	0.952292i	$0.401280\pi$
$42$	−0.394449	−0.0608648
$43$	1.09167	0.166479	0.0832393	−	0.996530i	$-0.473473\pi$
$43$	1.09167	0.166479	0.0832393	+	0.996530i	$0.473473\pi$
$44$	1.00000	0.150756
$45$	6.69722	0.998363
$46$	1.39445	0.205600
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0.302776	0.0437019
$49$	−5.30278	−0.757539
$50$	−0.302776	−0.0428189
$51$	−1.39445	−0.195262
$52$	−3.30278	−0.458013
$53$	−10.6056	−1.45678	−0.728392	−	0.685160i	$-0.759732\pi$
$53$	−10.6056	−1.45678	−0.728392	+	0.685160i	$0.759732\pi$
$54$	1.78890	0.243438
$55$	−2.30278	−0.310506
$56$	−1.30278	−0.174091
$57$	0.302776	0.0401036
$58$	8.30278	1.09021
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−0.697224	−0.0900113
$61$	11.2111	1.43543	0.717717	−	0.696335i	$-0.245187\pi$
$61$	11.2111	1.43543	0.717717	+	0.696335i	$0.245187\pi$
$62$	3.30278	0.419453
$63$	−3.78890	−0.477356
$64$	1.00000	0.125000
$65$	7.60555	0.943353
$66$	−0.302776	−0.0372691
$67$	−6.51388	−0.795797	−0.397898	−	0.917429i	$-0.630260\pi$
$67$	−6.51388	−0.795797	−0.397898	+	0.917429i	$0.630260\pi$
$68$	−4.60555	−0.558505
$69$	−0.422205	−0.0508275
$70$	3.00000	0.358569
$71$	−9.69722	−1.15085	−0.575424	−	0.817855i	$-0.695163\pi$
$71$	−9.69722	−1.15085	−0.575424	+	0.817855i	$0.695163\pi$
$72$	2.90833	0.342750
$73$	−2.60555	−0.304957	−0.152478	−	0.988307i	$-0.548725\pi$
$73$	−2.60555	−0.304957	−0.152478	+	0.988307i	$0.548725\pi$
$74$	−5.21110	−0.605778
$75$	0.0916731	0.0105855
$76$	1.00000	0.114708
$77$	1.30278	0.148465
$78$	1.00000	0.113228
$79$	15.8167	1.77951	0.889756	−	0.456436i	$-0.150874\pi$
$79$	15.8167	1.77951	0.889756	+	0.456436i	$0.150874\pi$
$80$	−2.30278	−0.257458
$81$	8.18335	0.909261
$82$	−3.90833	−0.431603
$83$	14.3028	1.56993	0.784967	−	0.619538i	$-0.212680\pi$
$83$	14.3028	1.56993	0.784967	+	0.619538i	$0.212680\pi$
$84$	0.394449	0.0430379
$85$	10.6056	1.15033
$86$	−1.09167	−0.117718
$87$	−2.51388	−0.269516
$88$	−1.00000	−0.106600
$89$	4.60555	0.488187	0.244094	−	0.969752i	$-0.421510\pi$
$89$	4.60555	0.488187	0.244094	+	0.969752i	$0.421510\pi$
$90$	−6.69722	−0.705949
$91$	−4.30278	−0.451053
$92$	−1.39445	−0.145381
$93$	−1.00000	−0.103695
$94$	6.00000	0.618853
$95$	−2.30278	−0.236260
$96$	−0.302776	−0.0309019
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	5.30278	0.535661
$99$	−2.90833	−0.292298
$100$	0.302776	0.0302776
$101$	7.81665	0.777786	0.388893	−	0.921283i	$-0.372858\pi$
$101$	7.81665	0.777786	0.388893	+	0.921283i	$0.372858\pi$
$102$	1.39445	0.138071
$103$	−1.69722	−0.167232	−0.0836162	−	0.996498i	$-0.526647\pi$
$103$	−1.69722	−0.167232	−0.0836162	+	0.996498i	$0.526647\pi$
$104$	3.30278	0.323864
$105$	−0.908327	−0.0886436
$106$	10.6056	1.03010
$107$	3.21110	0.310429	0.155215	−	0.987881i	$-0.450393\pi$
$107$	3.21110	0.310429	0.155215	+	0.987881i	$0.450393\pi$
$108$	−1.78890	−0.172137
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	2.30278	0.219561
$111$	1.57779	0.149758
$112$	1.30278	0.123101
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	−0.302776	−0.0283575
$115$	3.21110	0.299437
$116$	−8.30278	−0.770893
$117$	9.60555	0.888034
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0.697224	0.0636476
$121$	1.00000	0.0909091
$122$	−11.2111	−1.01501
$123$	1.18335	0.106699
$124$	−3.30278	−0.296598
$125$	10.8167	0.967471
$126$	3.78890	0.337542
$127$	−5.39445	−0.478680	−0.239340	−	0.970936i	$-0.576931\pi$
$127$	−5.39445	−0.478680	−0.239340	+	0.970936i	$0.576931\pi$
$128$	−1.00000	−0.0883883
$129$	0.330532	0.0291017
$130$	−7.60555	−0.667051
$131$	19.1194	1.67047	0.835236	−	0.549891i	$-0.185331\pi$
$131$	19.1194	1.67047	0.835236	+	0.549891i	$0.185331\pi$
$132$	0.302776	0.0263532
$133$	1.30278	0.112965
$134$	6.51388	0.562713
$135$	4.11943	0.354544
$136$	4.60555	0.394923
$137$	−3.48612	−0.297839	−0.148920	−	0.988849i	$-0.547580\pi$
$137$	−3.48612	−0.297839	−0.148920	+	0.988849i	$0.547580\pi$
$138$	0.422205	0.0359405
$139$	−6.30278	−0.534594	−0.267297	−	0.963614i	$-0.586131\pi$
$139$	−6.30278	−0.534594	−0.267297	+	0.963614i	$0.586131\pi$
$140$	−3.00000	−0.253546
$141$	−1.81665	−0.152990
$142$	9.69722	0.813773
$143$	−3.30278	−0.276192
$144$	−2.90833	−0.242361
$145$	19.1194	1.58778
$146$	2.60555	0.215637
$147$	−1.60555	−0.132424
$148$	5.21110	0.428350
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	−0.0916731	−0.00748508
$151$	−16.4222	−1.33642	−0.668210	−	0.743973i	$-0.732939\pi$
$151$	−16.4222	−1.33642	−0.668210	+	0.743973i	$0.732939\pi$
$152$	−1.00000	−0.0811107
$153$	13.3944	1.08288
$154$	−1.30278	−0.104981
$155$	7.60555	0.610893
$156$	−1.00000	−0.0800641
$157$	1.09167	0.0871250	0.0435625	−	0.999051i	$-0.486129\pi$
$157$	1.09167	0.0871250	0.0435625	+	0.999051i	$0.486129\pi$
$158$	−15.8167	−1.25831
$159$	−3.21110	−0.254657
$160$	2.30278	0.182050
$161$	−1.81665	−0.143172
$162$	−8.18335	−0.642944
$163$	−11.3944	−0.892482	−0.446241	−	0.894913i	$-0.647238\pi$
$163$	−11.3944	−0.892482	−0.446241	+	0.894913i	$0.647238\pi$
$164$	3.90833	0.305189
$165$	−0.697224	−0.0542788
$166$	−14.3028	−1.11011
$167$	−12.4222	−0.961259	−0.480630	−	0.876924i	$-0.659592\pi$
$167$	−12.4222	−0.961259	−0.480630	+	0.876924i	$0.659592\pi$
$168$	−0.394449	−0.0304324
$169$	−2.09167	−0.160898
$170$	−10.6056	−0.813409
$171$	−2.90833	−0.222405
$172$	1.09167	0.0832393
$173$	−19.3305	−1.46967	−0.734837	−	0.678244i	$-0.762741\pi$
$173$	−19.3305	−1.46967	−0.734837	+	0.678244i	$0.762741\pi$
$174$	2.51388	0.190577
$175$	0.394449	0.0298175
$176$	1.00000	0.0753778
$177$	0	0
$178$	−4.60555	−0.345201
$179$	−13.1194	−0.980592	−0.490296	−	0.871556i	$-0.663111\pi$
$179$	−13.1194	−0.980592	−0.490296	+	0.871556i	$0.663111\pi$
$180$	6.69722	0.499182
$181$	−15.0278	−1.11700	−0.558502	−	0.829503i	$-0.688624\pi$
$181$	−15.0278	−1.11700	−0.558502	+	0.829503i	$0.688624\pi$
$182$	4.30278	0.318943
$183$	3.39445	0.250925
$184$	1.39445	0.102800
$185$	−12.0000	−0.882258
$186$	1.00000	0.0733236
$187$	−4.60555	−0.336791
$188$	−6.00000	−0.437595
$189$	−2.33053	−0.169521
$190$	2.30278	0.167061
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0.302776	0.0218509
$193$	−24.7250	−1.77974	−0.889872	−	0.456211i	$-0.849206\pi$
$193$	−24.7250	−1.77974	−0.889872	+	0.456211i	$0.849206\pi$
$194$	−8.00000	−0.574367
$195$	2.30278	0.164905
$196$	−5.30278	−0.378770
$197$	21.6333	1.54131	0.770655	−	0.637253i	$-0.219929\pi$
$197$	21.6333	1.54131	0.770655	+	0.637253i	$0.219929\pi$
$198$	2.90833	0.206686
$199$	−2.18335	−0.154773	−0.0773867	−	0.997001i	$-0.524658\pi$
$199$	−2.18335	−0.154773	−0.0773867	+	0.997001i	$0.524658\pi$
$200$	−0.302776	−0.0214095
$201$	−1.97224	−0.139111
$202$	−7.81665	−0.549978
$203$	−10.8167	−0.759180
$204$	−1.39445	−0.0976309
$205$	−9.00000	−0.628587
$206$	1.69722	0.118251
$207$	4.05551	0.281878
$208$	−3.30278	−0.229006
$209$	1.00000	0.0691714
$210$	0.908327	0.0626805
$211$	24.6056	1.69392	0.846958	−	0.531660i	$-0.178431\pi$
$211$	24.6056	1.69392	0.846958	+	0.531660i	$0.178431\pi$
$212$	−10.6056	−0.728392
$213$	−2.93608	−0.201177
$214$	−3.21110	−0.219506
$215$	−2.51388	−0.171445
$216$	1.78890	0.121719
$217$	−4.30278	−0.292091
$218$	−2.00000	−0.135457
$219$	−0.788897	−0.0533087
$220$	−2.30278	−0.155253
$221$	15.2111	1.02321
$222$	−1.57779	−0.105895
$223$	−25.2111	−1.68826	−0.844130	−	0.536138i	$-0.819883\pi$
$223$	−25.2111	−1.68826	−0.844130	+	0.536138i	$0.819883\pi$
$224$	−1.30278	−0.0870454
$225$	−0.880571	−0.0587047
$226$	0	0
$227$	1.81665	0.120576	0.0602878	−	0.998181i	$-0.480798\pi$
$227$	1.81665	0.120576	0.0602878	+	0.998181i	$0.480798\pi$
$228$	0.302776	0.0200518
$229$	11.9083	0.786924	0.393462	−	0.919341i	$-0.371277\pi$
$229$	11.9083	0.786924	0.393462	+	0.919341i	$0.371277\pi$
$230$	−3.21110	−0.211734
$231$	0.394449	0.0259528
$232$	8.30278	0.545104
$233$	9.21110	0.603439	0.301720	−	0.953397i	$-0.402439\pi$
$233$	9.21110	0.603439	0.301720	+	0.953397i	$0.402439\pi$
$234$	−9.60555	−0.627935
$235$	13.8167	0.901299
$236$	0	0
$237$	4.78890	0.311072
$238$	6.00000	0.388922
$239$	−23.9361	−1.54830	−0.774148	−	0.633004i	$-0.781822\pi$
$239$	−23.9361	−1.54830	−0.774148	+	0.633004i	$0.781822\pi$
$240$	−0.697224	−0.0450056
$241$	13.0917	0.843309	0.421654	−	0.906757i	$-0.361450\pi$
$241$	13.0917	0.843309	0.421654	+	0.906757i	$0.361450\pi$
$242$	−1.00000	−0.0642824
$243$	7.84441	0.503219
$244$	11.2111	0.717717
$245$	12.2111	0.780139
$246$	−1.18335	−0.0754474
$247$	−3.30278	−0.210151
$248$	3.30278	0.209726
$249$	4.33053	0.274436
$250$	−10.8167	−0.684105
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	−3.78890	−0.238678
$253$	−1.39445	−0.0876682
$254$	5.39445	0.338478
$255$	3.21110	0.201087
$256$	1.00000	0.0625000
$257$	−4.60555	−0.287286	−0.143643	−	0.989630i	$-0.545882\pi$
$257$	−4.60555	−0.287286	−0.143643	+	0.989630i	$0.545882\pi$
$258$	−0.330532	−0.0205780
$259$	6.78890	0.421842
$260$	7.60555	0.471676
$261$	24.1472	1.49467
$262$	−19.1194	−1.18120
$263$	−20.3028	−1.25192	−0.625961	−	0.779854i	$-0.715293\pi$
$263$	−20.3028	−1.25192	−0.625961	+	0.779854i	$0.715293\pi$
$264$	−0.302776	−0.0186346
$265$	24.4222	1.50024
$266$	−1.30278	−0.0798783
$267$	1.39445	0.0853389
$268$	−6.51388	−0.397898
$269$	−19.8167	−1.20824	−0.604121	−	0.796892i	$-0.706476\pi$
$269$	−19.8167	−1.20824	−0.604121	+	0.796892i	$0.706476\pi$
$270$	−4.11943	−0.250700
$271$	−6.51388	−0.395690	−0.197845	−	0.980233i	$-0.563394\pi$
$271$	−6.51388	−0.395690	−0.197845	+	0.980233i	$0.563394\pi$
$272$	−4.60555	−0.279253
$273$	−1.30278	−0.0788476
$274$	3.48612	0.210604
$275$	0.302776	0.0182581
$276$	−0.422205	−0.0254138
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	6.30278	0.378015
$279$	9.60555	0.575069
$280$	3.00000	0.179284
$281$	−0.486122	−0.0289996	−0.0144998	−	0.999895i	$-0.504616\pi$
$281$	−0.486122	−0.0289996	−0.0144998	+	0.999895i	$0.504616\pi$
$282$	1.81665	0.108180
$283$	−18.9361	−1.12563	−0.562817	−	0.826582i	$-0.690282\pi$
$283$	−18.9361	−1.12563	−0.562817	+	0.826582i	$0.690282\pi$
$284$	−9.69722	−0.575424
$285$	−0.697224	−0.0413000
$286$	3.30278	0.195297
$287$	5.09167	0.300552
$288$	2.90833	0.171375
$289$	4.21110	0.247712
$290$	−19.1194	−1.12273
$291$	2.42221	0.141992
$292$	−2.60555	−0.152478
$293$	−25.5416	−1.49216	−0.746079	−	0.665857i	$-0.768066\pi$
$293$	−25.5416	−1.49216	−0.746079	+	0.665857i	$0.768066\pi$
$294$	1.60555	0.0936377
$295$	0	0
$296$	−5.21110	−0.302889
$297$	−1.78890	−0.103802
$298$	0	0
$299$	4.60555	0.266346
$300$	0.0916731	0.00529275
$301$	1.42221	0.0819745
$302$	16.4222	0.944992
$303$	2.36669	0.135963
$304$	1.00000	0.0573539
$305$	−25.8167	−1.47826
$306$	−13.3944	−0.765710
$307$	−2.18335	−0.124610	−0.0623051	−	0.998057i	$-0.519845\pi$
$307$	−2.18335	−0.124610	−0.0623051	+	0.998057i	$0.519845\pi$
$308$	1.30278	0.0742325
$309$	−0.513878	−0.0292335
$310$	−7.60555	−0.431966
$311$	25.8167	1.46393	0.731964	−	0.681343i	$-0.238604\pi$
$311$	25.8167	1.46393	0.731964	+	0.681343i	$0.238604\pi$
$312$	1.00000	0.0566139
$313$	18.1194	1.02417	0.512085	−	0.858935i	$-0.328873\pi$
$313$	18.1194	1.02417	0.512085	+	0.858935i	$0.328873\pi$
$314$	−1.09167	−0.0616067
$315$	8.72498	0.491597
$316$	15.8167	0.889756
$317$	−13.8167	−0.776021	−0.388010	−	0.921655i	$-0.626838\pi$
$317$	−13.8167	−0.776021	−0.388010	+	0.921655i	$0.626838\pi$
$318$	3.21110	0.180070
$319$	−8.30278	−0.464866
$320$	−2.30278	−0.128729
$321$	0.972244	0.0542653
$322$	1.81665	0.101238
$323$	−4.60555	−0.256260
$324$	8.18335	0.454630
$325$	−1.00000	−0.0554700
$326$	11.3944	0.631080
$327$	0.605551	0.0334871
$328$	−3.90833	−0.215801
$329$	−7.81665	−0.430946
$330$	0.697224	0.0383809
$331$	22.7250	1.24908	0.624539	−	0.780994i	$-0.285287\pi$
$331$	22.7250	1.24908	0.624539	+	0.780994i	$0.285287\pi$
$332$	14.3028	0.784967
$333$	−15.1556	−0.830521
$334$	12.4222	0.679713
$335$	15.0000	0.819538
$336$	0.394449	0.0215189
$337$	−9.30278	−0.506754	−0.253377	−	0.967368i	$-0.581541\pi$
$337$	−9.30278	−0.506754	−0.253377	+	0.967368i	$0.581541\pi$
$338$	2.09167	0.113772
$339$	0	0
$340$	10.6056	0.575167
$341$	−3.30278	−0.178855
$342$	2.90833	0.157264
$343$	−16.0278	−0.865417
$344$	−1.09167	−0.0588591
$345$	0.972244	0.0523438
$346$	19.3305	1.03922
$347$	2.78890	0.149716	0.0748579	−	0.997194i	$-0.476150\pi$
$347$	2.78890	0.149716	0.0748579	+	0.997194i	$0.476150\pi$
$348$	−2.51388	−0.134758
$349$	32.4222	1.73552	0.867760	−	0.496983i	$-0.165559\pi$
$349$	32.4222	1.73552	0.867760	+	0.496983i	$0.165559\pi$
$350$	−0.394449	−0.0210842
$351$	5.90833	0.315363
$352$	−1.00000	−0.0533002
$353$	−15.2111	−0.809605	−0.404803	−	0.914404i	$-0.632660\pi$
$353$	−15.2111	−0.809605	−0.404803	+	0.914404i	$0.632660\pi$
$354$	0	0
$355$	22.3305	1.18518
$356$	4.60555	0.244094
$357$	−1.81665	−0.0961475
$358$	13.1194	0.693383
$359$	−9.48612	−0.500658	−0.250329	−	0.968161i	$-0.580539\pi$
$359$	−9.48612	−0.500658	−0.250329	+	0.968161i	$0.580539\pi$
$360$	−6.69722	−0.352975
$361$	1.00000	0.0526316
$362$	15.0278	0.789841
$363$	0.302776	0.0158916
$364$	−4.30278	−0.225527
$365$	6.00000	0.314054
$366$	−3.39445	−0.177431
$367$	−7.21110	−0.376416	−0.188208	−	0.982129i	$-0.560268\pi$
$367$	−7.21110	−0.376416	−0.188208	+	0.982129i	$0.560268\pi$
$368$	−1.39445	−0.0726907
$369$	−11.3667	−0.591726
$370$	12.0000	0.623850
$371$	−13.8167	−0.717325
$372$	−1.00000	−0.0518476
$373$	7.09167	0.367193	0.183596	−	0.983002i	$-0.441226\pi$
$373$	7.09167	0.367193	0.183596	+	0.983002i	$0.441226\pi$
$374$	4.60555	0.238147
$375$	3.27502	0.169121
$376$	6.00000	0.309426
$377$	27.4222	1.41232
$378$	2.33053	0.119870
$379$	−13.4861	−0.692736	−0.346368	−	0.938099i	$-0.612585\pi$
$379$	−13.4861	−0.692736	−0.346368	+	0.938099i	$0.612585\pi$
$380$	−2.30278	−0.118130
$381$	−1.63331	−0.0836769
$382$	6.00000	0.306987
$383$	3.48612	0.178133	0.0890663	−	0.996026i	$-0.471612\pi$
$383$	3.48612	0.178133	0.0890663	+	0.996026i	$0.471612\pi$
$384$	−0.302776	−0.0154510
$385$	−3.00000	−0.152894
$386$	24.7250	1.25847
$387$	−3.17494	−0.161391
$388$	8.00000	0.406138
$389$	34.5416	1.75133	0.875665	−	0.482919i	$-0.160423\pi$
$389$	34.5416	1.75133	0.875665	+	0.482919i	$0.160423\pi$
$390$	−2.30278	−0.116606
$391$	6.42221	0.324785
$392$	5.30278	0.267831
$393$	5.78890	0.292011
$394$	−21.6333	−1.08987
$395$	−36.4222	−1.83260
$396$	−2.90833	−0.146149
$397$	−32.7527	−1.64381	−0.821906	−	0.569623i	$-0.807089\pi$
$397$	−32.7527	−1.64381	−0.821906	+	0.569623i	$0.807089\pi$
$398$	2.18335	0.109441
$399$	0.394449	0.0197471
$400$	0.302776	0.0151388
$401$	22.6056	1.12887	0.564434	−	0.825478i	$-0.309095\pi$
$401$	22.6056	1.12887	0.564434	+	0.825478i	$0.309095\pi$
$402$	1.97224	0.0983666
$403$	10.9083	0.543382
$404$	7.81665	0.388893
$405$	−18.8444	−0.936386
$406$	10.8167	0.536822
$407$	5.21110	0.258305
$408$	1.39445	0.0690355
$409$	23.1472	1.14455	0.572277	−	0.820060i	$-0.306060\pi$
$409$	23.1472	1.14455	0.572277	+	0.820060i	$0.306060\pi$
$410$	9.00000	0.444478
$411$	−1.05551	−0.0520646
$412$	−1.69722	−0.0836162
$413$	0	0
$414$	−4.05551	−0.199318
$415$	−32.9361	−1.61677
$416$	3.30278	0.161932
$417$	−1.90833	−0.0934512
$418$	−1.00000	−0.0489116
$419$	8.78890	0.429366	0.214683	−	0.976684i	$-0.431128\pi$
$419$	8.78890	0.429366	0.214683	+	0.976684i	$0.431128\pi$
$420$	−0.908327	−0.0443218
$421$	−30.2389	−1.47375	−0.736876	−	0.676028i	$-0.763700\pi$
$421$	−30.2389	−1.47375	−0.736876	+	0.676028i	$0.763700\pi$
$422$	−24.6056	−1.19778
$423$	17.4500	0.848446
$424$	10.6056	0.515051
$425$	−1.39445	−0.0676407
$426$	2.93608	0.142254
$427$	14.6056	0.706812
$428$	3.21110	0.155215
$429$	−1.00000	−0.0482805
$430$	2.51388	0.121230
$431$	2.78890	0.134336	0.0671682	−	0.997742i	$-0.478604\pi$
$431$	2.78890	0.134336	0.0671682	+	0.997742i	$0.478604\pi$
$432$	−1.78890	−0.0860684
$433$	−24.2389	−1.16485	−0.582423	−	0.812886i	$-0.697895\pi$
$433$	−24.2389	−1.16485	−0.582423	+	0.812886i	$0.697895\pi$
$434$	4.30278	0.206540
$435$	5.78890	0.277556
$436$	2.00000	0.0957826
$437$	−1.39445	−0.0667055
$438$	0.788897	0.0376950
$439$	−25.2111	−1.20326	−0.601630	−	0.798775i	$-0.705482\pi$
$439$	−25.2111	−1.20326	−0.601630	+	0.798775i	$0.705482\pi$
$440$	2.30278	0.109781
$441$	15.4222	0.734391
$442$	−15.2111	−0.723518
$443$	39.6333	1.88304	0.941518	−	0.336964i	$-0.109400\pi$
$443$	39.6333	1.88304	0.941518	+	0.336964i	$0.109400\pi$
$444$	1.57779	0.0748788
$445$	−10.6056	−0.502751
$446$	25.2111	1.19378
$447$	0	0
$448$	1.30278	0.0615504
$449$	−31.8167	−1.50152	−0.750760	−	0.660575i	$-0.770313\pi$
$449$	−31.8167	−1.50152	−0.750760	+	0.660575i	$0.770313\pi$
$450$	0.880571	0.0415105
$451$	3.90833	0.184036
$452$	0	0
$453$	−4.97224	−0.233616
$454$	−1.81665	−0.0852598
$455$	9.90833	0.464510
$456$	−0.302776	−0.0141788
$457$	21.8167	1.02054	0.510270	−	0.860014i	$-0.329545\pi$
$457$	21.8167	1.02054	0.510270	+	0.860014i	$0.329545\pi$
$458$	−11.9083	−0.556440
$459$	8.23886	0.384557
$460$	3.21110	0.149718
$461$	6.42221	0.299112	0.149556	−	0.988753i	$-0.452216\pi$
$461$	6.42221	0.299112	0.149556	+	0.988753i	$0.452216\pi$
$462$	−0.394449	−0.0183514
$463$	−1.21110	−0.0562847	−0.0281424	−	0.999604i	$-0.508959\pi$
$463$	−1.21110	−0.0562847	−0.0281424	+	0.999604i	$0.508959\pi$
$464$	−8.30278	−0.385447
$465$	2.30278	0.106789
$466$	−9.21110	−0.426696
$467$	−22.6056	−1.04606	−0.523030	−	0.852314i	$-0.675198\pi$
$467$	−22.6056	−1.04606	−0.523030	+	0.852314i	$0.675198\pi$
$468$	9.60555	0.444017
$469$	−8.48612	−0.391853
$470$	−13.8167	−0.637315
$471$	0.330532	0.0152301
$472$	0	0
$473$	1.09167	0.0501952
$474$	−4.78890	−0.219961
$475$	0.302776	0.0138923
$476$	−6.00000	−0.275010
$477$	30.8444	1.41227
$478$	23.9361	1.09481
$479$	35.7250	1.63232	0.816158	−	0.577829i	$-0.196100\pi$
$479$	35.7250	1.63232	0.816158	+	0.577829i	$0.196100\pi$
$480$	0.697224	0.0318238
$481$	−17.2111	−0.784759
$482$	−13.0917	−0.596309
$483$	−0.550039	−0.0250276
$484$	1.00000	0.0454545
$485$	−18.4222	−0.836509
$486$	−7.84441	−0.355830
$487$	−29.3305	−1.32909	−0.664547	−	0.747247i	$-0.731375\pi$
$487$	−29.3305	−1.32909	−0.664547	+	0.747247i	$0.731375\pi$
$488$	−11.2111	−0.507503
$489$	−3.44996	−0.156013
$490$	−12.2111	−0.551641
$491$	−14.9361	−0.674056	−0.337028	−	0.941495i	$-0.609422\pi$
$491$	−14.9361	−0.674056	−0.337028	+	0.941495i	$0.609422\pi$
$492$	1.18335	0.0533494
$493$	38.2389	1.72219
$494$	3.30278	0.148599
$495$	6.69722	0.301018
$496$	−3.30278	−0.148299
$497$	−12.6333	−0.566681
$498$	−4.33053	−0.194056
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	10.8167	0.483735
$501$	−3.76114	−0.168035
$502$	−24.0000	−1.07117
$503$	22.1194	0.986257	0.493128	−	0.869957i	$-0.335853\pi$
$503$	22.1194	0.986257	0.493128	+	0.869957i	$0.335853\pi$
$504$	3.78890	0.168771
$505$	−18.0000	−0.800989
$506$	1.39445	0.0619908
$507$	−0.633308	−0.0281262
$508$	−5.39445	−0.239340
$509$	−18.4222	−0.816550	−0.408275	−	0.912859i	$-0.633870\pi$
$509$	−18.4222	−0.816550	−0.408275	+	0.912859i	$0.633870\pi$
$510$	−3.21110	−0.142190
$511$	−3.39445	−0.150162
$512$	−1.00000	−0.0441942
$513$	−1.78890	−0.0789818
$514$	4.60555	0.203142
$515$	3.90833	0.172221
$516$	0.330532	0.0145509
$517$	−6.00000	−0.263880
$518$	−6.78890	−0.298287
$519$	−5.85281	−0.256910
$520$	−7.60555	−0.333525
$521$	36.4222	1.59569	0.797843	−	0.602865i	$-0.205974\pi$
$521$	36.4222	1.59569	0.797843	+	0.602865i	$0.205974\pi$
$522$	−24.1472	−1.05689
$523$	19.0278	0.832026	0.416013	−	0.909359i	$-0.363427\pi$
$523$	19.0278	0.832026	0.416013	+	0.909359i	$0.363427\pi$
$524$	19.1194	0.835236
$525$	0.119429	0.00521233
$526$	20.3028	0.885243
$527$	15.2111	0.662606
$528$	0.302776	0.0131766
$529$	−21.0555	−0.915457
$530$	−24.4222	−1.06083
$531$	0	0
$532$	1.30278	0.0564825
$533$	−12.9083	−0.559122
$534$	−1.39445	−0.0603437
$535$	−7.39445	−0.319690
$536$	6.51388	0.281357
$537$	−3.97224	−0.171415
$538$	19.8167	0.854357
$539$	−5.30278	−0.228407
$540$	4.11943	0.177272
$541$	3.81665	0.164091	0.0820454	−	0.996629i	$-0.473855\pi$
$541$	3.81665	0.164091	0.0820454	+	0.996629i	$0.473855\pi$
$542$	6.51388	0.279795
$543$	−4.55004	−0.195261
$544$	4.60555	0.197461
$545$	−4.60555	−0.197280
$546$	1.30278	0.0557537
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−3.48612	−0.148920
$549$	−32.6056	−1.39157
$550$	−0.302776	−0.0129104
$551$	−8.30278	−0.353710
$552$	0.422205	0.0179702
$553$	20.6056	0.876237
$554$	10.0000	0.424859
$555$	−3.63331	−0.154225
$556$	−6.30278	−0.267297
$557$	−27.6333	−1.17086	−0.585430	−	0.810723i	$-0.699074\pi$
$557$	−27.6333	−1.17086	−0.585430	+	0.810723i	$0.699074\pi$
$558$	−9.60555	−0.406635
$559$	−3.60555	−0.152499
$560$	−3.00000	−0.126773
$561$	−1.39445	−0.0588737
$562$	0.486122	0.0205058
$563$	21.6333	0.911735	0.455868	−	0.890048i	$-0.349329\pi$
$563$	21.6333	0.911735	0.455868	+	0.890048i	$0.349329\pi$
$564$	−1.81665	−0.0764949
$565$	0	0
$566$	18.9361	0.795943
$567$	10.6611	0.447723
$568$	9.69722	0.406886
$569$	12.2750	0.514596	0.257298	−	0.966332i	$-0.417168\pi$
$569$	12.2750	0.514596	0.257298	+	0.966332i	$0.417168\pi$
$570$	0.697224	0.0292035
$571$	−6.93608	−0.290266	−0.145133	−	0.989412i	$-0.546361\pi$
$571$	−6.93608	−0.290266	−0.145133	+	0.989412i	$0.546361\pi$
$572$	−3.30278	−0.138096
$573$	−1.81665	−0.0758918
$574$	−5.09167	−0.212522
$575$	−0.422205	−0.0176072
$576$	−2.90833	−0.121180
$577$	−18.0917	−0.753166	−0.376583	−	0.926383i	$-0.622901\pi$
$577$	−18.0917	−0.753166	−0.376583	+	0.926383i	$0.622901\pi$
$578$	−4.21110	−0.175159
$579$	−7.48612	−0.311113
$580$	19.1194	0.793891
$581$	18.6333	0.773040
$582$	−2.42221	−0.100404
$583$	−10.6056	−0.439237
$584$	2.60555	0.107818
$585$	−22.1194	−0.914526
$586$	25.5416	1.05512
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	−1.60555	−0.0662118
$589$	−3.30278	−0.136088
$590$	0	0
$591$	6.55004	0.269433
$592$	5.21110	0.214175
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	1.78890	0.0733994
$595$	13.8167	0.566428
$596$	0	0
$597$	−0.661064	−0.0270555
$598$	−4.60555	−0.188335
$599$	−19.8806	−0.812298	−0.406149	−	0.913807i	$-0.633129\pi$
$599$	−19.8806	−0.812298	−0.406149	+	0.913807i	$0.633129\pi$
$600$	−0.0916731	−0.00374254
$601$	4.09167	0.166903	0.0834514	−	0.996512i	$-0.473406\pi$
$601$	4.09167	0.166903	0.0834514	+	0.996512i	$0.473406\pi$
$602$	−1.42221	−0.0579648
$603$	18.9445	0.771479
$604$	−16.4222	−0.668210
$605$	−2.30278	−0.0936211
$606$	−2.36669	−0.0961403
$607$	3.81665	0.154913	0.0774566	−	0.996996i	$-0.475320\pi$
$607$	3.81665	0.154913	0.0774566	+	0.996996i	$0.475320\pi$
$608$	−1.00000	−0.0405554
$609$	−3.27502	−0.132710
$610$	25.8167	1.04529
$611$	19.8167	0.801696
$612$	13.3944	0.541438
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	2.18335	0.0881127
$615$	−2.72498	−0.109882
$616$	−1.30278	−0.0524903
$617$	2.09167	0.0842076	0.0421038	−	0.999113i	$-0.486594\pi$
$617$	2.09167	0.0842076	0.0421038	+	0.999113i	$0.486594\pi$
$618$	0.513878	0.0206712
$619$	−34.8444	−1.40052	−0.700258	−	0.713890i	$-0.746932\pi$
$619$	−34.8444	−1.40052	−0.700258	+	0.713890i	$0.746932\pi$
$620$	7.60555	0.305446
$621$	2.49453	0.100102
$622$	−25.8167	−1.03515
$623$	6.00000	0.240385
$624$	−1.00000	−0.0400320
$625$	−26.4222	−1.05689
$626$	−18.1194	−0.724198
$627$	0.302776	0.0120917
$628$	1.09167	0.0435625
$629$	−24.0000	−0.956943
$630$	−8.72498	−0.347612
$631$	−45.0278	−1.79253	−0.896263	−	0.443522i	$-0.853729\pi$
$631$	−45.0278	−1.79253	−0.896263	+	0.443522i	$0.853729\pi$
$632$	−15.8167	−0.629153
$633$	7.44996	0.296109
$634$	13.8167	0.548729
$635$	12.4222	0.492960
$636$	−3.21110	−0.127328
$637$	17.5139	0.693925
$638$	8.30278	0.328710
$639$	28.2027	1.11568
$640$	2.30278	0.0910252
$641$	20.2389	0.799387	0.399693	−	0.916649i	$-0.369117\pi$
$641$	20.2389	0.799387	0.399693	+	0.916649i	$0.369117\pi$
$642$	−0.972244	−0.0383714
$643$	26.4222	1.04199	0.520995	−	0.853560i	$-0.325561\pi$
$643$	26.4222	1.04199	0.520995	+	0.853560i	$0.325561\pi$
$644$	−1.81665	−0.0715862
$645$	−0.761141	−0.0299699
$646$	4.60555	0.181203
$647$	25.8167	1.01496	0.507479	−	0.861664i	$-0.330578\pi$
$647$	25.8167	1.01496	0.507479	+	0.861664i	$0.330578\pi$
$648$	−8.18335	−0.321472
$649$	0	0
$650$	1.00000	0.0392232
$651$	−1.30278	−0.0510598
$652$	−11.3944	−0.446241
$653$	22.3305	0.873861	0.436931	−	0.899495i	$-0.356066\pi$
$653$	22.3305	0.873861	0.436931	+	0.899495i	$0.356066\pi$
$654$	−0.605551	−0.0236789
$655$	−44.0278	−1.72031
$656$	3.90833	0.152595
$657$	7.57779	0.295638
$658$	7.81665	0.304725
$659$	29.0278	1.13076	0.565380	−	0.824830i	$-0.308729\pi$
$659$	29.0278	1.13076	0.565380	+	0.824830i	$0.308729\pi$
$660$	−0.697224	−0.0271394
$661$	23.6333	0.919229	0.459615	−	0.888118i	$-0.347988\pi$
$661$	23.6333	0.919229	0.459615	+	0.888118i	$0.347988\pi$
$662$	−22.7250	−0.883231
$663$	4.60555	0.178865
$664$	−14.3028	−0.555055
$665$	−3.00000	−0.116335
$666$	15.1556	0.587267
$667$	11.5778	0.448294
$668$	−12.4222	−0.480630
$669$	−7.63331	−0.295121
$670$	−15.0000	−0.579501
$671$	11.2111	0.432800
$672$	−0.394449	−0.0152162
$673$	1.30278	0.0502183	0.0251092	−	0.999685i	$-0.492007\pi$
$673$	1.30278	0.0502183	0.0251092	+	0.999685i	$0.492007\pi$
$674$	9.30278	0.358330
$675$	−0.541635	−0.0208475
$676$	−2.09167	−0.0804490
$677$	−8.30278	−0.319102	−0.159551	−	0.987190i	$-0.551005\pi$
$677$	−8.30278	−0.319102	−0.159551	+	0.987190i	$0.551005\pi$
$678$	0	0
$679$	10.4222	0.399968
$680$	−10.6056	−0.406704
$681$	0.550039	0.0210775
$682$	3.30278	0.126470
$683$	−17.5778	−0.672596	−0.336298	−	0.941756i	$-0.609175\pi$
$683$	−17.5778	−0.672596	−0.336298	+	0.941756i	$0.609175\pi$
$684$	−2.90833	−0.111203
$685$	8.02776	0.306725
$686$	16.0278	0.611943
$687$	3.60555	0.137560
$688$	1.09167	0.0416196
$689$	35.0278	1.33445
$690$	−0.972244	−0.0370127
$691$	32.4222	1.23340	0.616699	−	0.787199i	$-0.288469\pi$
$691$	32.4222	1.23340	0.616699	+	0.787199i	$0.288469\pi$
$692$	−19.3305	−0.734837
$693$	−3.78890	−0.143928
$694$	−2.78890	−0.105865
$695$	14.5139	0.550543
$696$	2.51388	0.0952883
$697$	−18.0000	−0.681799
$698$	−32.4222	−1.22720
$699$	2.78890	0.105486
$700$	0.394449	0.0149088
$701$	−27.6333	−1.04370	−0.521848	−	0.853039i	$-0.674757\pi$
$701$	−27.6333	−1.04370	−0.521848	+	0.853039i	$0.674757\pi$
$702$	−5.90833	−0.222995
$703$	5.21110	0.196540
$704$	1.00000	0.0376889
$705$	4.18335	0.157554
$706$	15.2111	0.572477
$707$	10.1833	0.382984
$708$	0	0
$709$	14.6972	0.551966	0.275983	−	0.961163i	$-0.410997\pi$
$709$	14.6972	0.551966	0.275983	+	0.961163i	$0.410997\pi$
$710$	−22.3305	−0.838050
$711$	−46.0000	−1.72513
$712$	−4.60555	−0.172600
$713$	4.60555	0.172479
$714$	1.81665	0.0679866
$715$	7.60555	0.284431
$716$	−13.1194	−0.490296
$717$	−7.24726	−0.270654
$718$	9.48612	0.354019
$719$	−25.8167	−0.962799	−0.481399	−	0.876501i	$-0.659871\pi$
$719$	−25.8167	−0.962799	−0.481399	+	0.876501i	$0.659871\pi$
$720$	6.69722	0.249591
$721$	−2.21110	−0.0823458
$722$	−1.00000	−0.0372161
$723$	3.96384	0.147417
$724$	−15.0278	−0.558502
$725$	−2.51388	−0.0933631
$726$	−0.302776	−0.0112371
$727$	−0.366692	−0.0135999	−0.00679993	−	0.999977i	$-0.502165\pi$
$727$	−0.366692	−0.0135999	−0.00679993	+	0.999977i	$0.502165\pi$
$728$	4.30278	0.159471
$729$	−22.1749	−0.821294
$730$	−6.00000	−0.222070
$731$	−5.02776	−0.185958
$732$	3.39445	0.125462
$733$	−7.21110	−0.266348	−0.133174	−	0.991093i	$-0.542517\pi$
$733$	−7.21110	−0.266348	−0.133174	+	0.991093i	$0.542517\pi$
$734$	7.21110	0.266167
$735$	3.69722	0.136374
$736$	1.39445	0.0514001
$737$	−6.51388	−0.239942
$738$	11.3667	0.418414
$739$	−40.3583	−1.48460	−0.742302	−	0.670066i	$-0.766266\pi$
$739$	−40.3583	−1.48460	−0.742302	+	0.670066i	$0.766266\pi$
$740$	−12.0000	−0.441129
$741$	−1.00000	−0.0367359
$742$	13.8167	0.507225
$743$	2.36669	0.0868255	0.0434128	−	0.999057i	$-0.486177\pi$
$743$	2.36669	0.0868255	0.0434128	+	0.999057i	$0.486177\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	−7.09167	−0.259645
$747$	−41.5971	−1.52196
$748$	−4.60555	−0.168396
$749$	4.18335	0.152856
$750$	−3.27502	−0.119587
$751$	−40.8444	−1.49043	−0.745217	−	0.666822i	$-0.767654\pi$
$751$	−40.8444	−1.49043	−0.745217	+	0.666822i	$0.767654\pi$
$752$	−6.00000	−0.218797
$753$	7.26662	0.264810
$754$	−27.4222	−0.998658
$755$	37.8167	1.37629
$756$	−2.33053	−0.0847606
$757$	30.5416	1.11005	0.555027	−	0.831832i	$-0.312708\pi$
$757$	30.5416	1.11005	0.555027	+	0.831832i	$0.312708\pi$
$758$	13.4861	0.489838
$759$	−0.422205	−0.0153251
$760$	2.30278	0.0835305
$761$	20.2389	0.733658	0.366829	−	0.930288i	$-0.380443\pi$
$761$	20.2389	0.733658	0.366829	+	0.930288i	$0.380443\pi$
$762$	1.63331	0.0591685
$763$	2.60555	0.0943273
$764$	−6.00000	−0.217072
$765$	−30.8444	−1.11518
$766$	−3.48612	−0.125959
$767$	0	0
$768$	0.302776	0.0109255
$769$	−37.6333	−1.35709	−0.678546	−	0.734558i	$-0.737390\pi$
$769$	−37.6333	−1.35709	−0.678546	+	0.734558i	$0.737390\pi$
$770$	3.00000	0.108112
$771$	−1.39445	−0.0502198
$772$	−24.7250	−0.889872
$773$	48.4222	1.74163	0.870813	−	0.491615i	$-0.163593\pi$
$773$	48.4222	1.74163	0.870813	+	0.491615i	$0.163593\pi$
$774$	3.17494	0.114121
$775$	−1.00000	−0.0359211
$776$	−8.00000	−0.287183
$777$	2.05551	0.0737411
$778$	−34.5416	−1.23838
$779$	3.90833	0.140030
$780$	2.30278	0.0824526
$781$	−9.69722	−0.346994
$782$	−6.42221	−0.229658
$783$	14.8528	0.530796
$784$	−5.30278	−0.189385
$785$	−2.51388	−0.0897242
$786$	−5.78890	−0.206483
$787$	−15.0278	−0.535682	−0.267841	−	0.963463i	$-0.586310\pi$
$787$	−15.0278	−0.535682	−0.267841	+	0.963463i	$0.586310\pi$
$788$	21.6333	0.770655
$789$	−6.14719	−0.218846
$790$	36.4222	1.29584
$791$	0	0
$792$	2.90833	0.103343
$793$	−37.0278	−1.31489
$794$	32.7527	1.16235
$795$	7.39445	0.262254
$796$	−2.18335	−0.0773867
$797$	16.6056	0.588199	0.294099	−	0.955775i	$-0.404980\pi$
$797$	16.6056	0.588199	0.294099	+	0.955775i	$0.404980\pi$
$798$	−0.394449	−0.0139633
$799$	27.6333	0.977596
$800$	−0.302776	−0.0107047
$801$	−13.3944	−0.473270
$802$	−22.6056	−0.798230
$803$	−2.60555	−0.0919479
$804$	−1.97224	−0.0695557
$805$	4.18335	0.147444
$806$	−10.9083	−0.384229
$807$	−6.00000	−0.211210
$808$	−7.81665	−0.274989
$809$	33.2111	1.16764	0.583820	−	0.811883i	$-0.301557\pi$
$809$	33.2111	1.16764	0.583820	+	0.811883i	$0.301557\pi$
$810$	18.8444	0.662125
$811$	29.6333	1.04057	0.520283	−	0.853994i	$-0.325826\pi$
$811$	29.6333	1.04057	0.520283	+	0.853994i	$0.325826\pi$
$812$	−10.8167	−0.379590
$813$	−1.97224	−0.0691696
$814$	−5.21110	−0.182649
$815$	26.2389	0.919107
$816$	−1.39445	−0.0488155
$817$	1.09167	0.0381928
$818$	−23.1472	−0.809322
$819$	12.5139	0.437270
$820$	−9.00000	−0.314294
$821$	−34.6056	−1.20774	−0.603871	−	0.797082i	$-0.706376\pi$
$821$	−34.6056	−1.20774	−0.603871	+	0.797082i	$0.706376\pi$
$822$	1.05551	0.0368152
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	1.69722	0.0591256
$825$	0.0916731	0.00319165
$826$	0	0
$827$	22.0555	0.766945	0.383473	−	0.923552i	$-0.374728\pi$
$827$	22.0555	0.766945	0.383473	+	0.923552i	$0.374728\pi$
$828$	4.05551	0.140939
$829$	−27.5778	−0.957816	−0.478908	−	0.877865i	$-0.658967\pi$
$829$	−27.5778	−0.957816	−0.478908	+	0.877865i	$0.658967\pi$
$830$	32.9361	1.14323
$831$	−3.02776	−0.105032
$832$	−3.30278	−0.114503
$833$	24.4222	0.846179
$834$	1.90833	0.0660800
$835$	28.6056	0.989936
$836$	1.00000	0.0345857
$837$	5.90833	0.204222
$838$	−8.78890	−0.303607
$839$	19.8806	0.686354	0.343177	−	0.939271i	$-0.388497\pi$
$839$	19.8806	0.686354	0.343177	+	0.939271i	$0.388497\pi$
$840$	0.908327	0.0313403
$841$	39.9361	1.37711
$842$	30.2389	1.04210
$843$	−0.147186	−0.00506935
$844$	24.6056	0.846958
$845$	4.81665	0.165698
$846$	−17.4500	−0.599942
$847$	1.30278	0.0447639
$848$	−10.6056	−0.364196
$849$	−5.73338	−0.196769
$850$	1.39445	0.0478292
$851$	−7.26662	−0.249096
$852$	−2.93608	−0.100589
$853$	−46.4222	−1.58947	−0.794733	−	0.606959i	$-0.792389\pi$
$853$	−46.4222	−1.58947	−0.794733	+	0.606959i	$0.792389\pi$
$854$	−14.6056	−0.499792
$855$	6.69722	0.229040
$856$	−3.21110	−0.109753
$857$	−43.5416	−1.48735	−0.743677	−	0.668539i	$-0.766920\pi$
$857$	−43.5416	−1.48735	−0.743677	+	0.668539i	$0.766920\pi$
$858$	1.00000	0.0341394
$859$	8.84441	0.301767	0.150884	−	0.988552i	$-0.451788\pi$
$859$	8.84441	0.301767	0.150884	+	0.988552i	$0.451788\pi$
$860$	−2.51388	−0.0857225
$861$	1.54163	0.0525388
$862$	−2.78890	−0.0949902
$863$	−17.7250	−0.603365	−0.301683	−	0.953408i	$-0.597548\pi$
$863$	−17.7250	−0.603365	−0.301683	+	0.953408i	$0.597548\pi$
$864$	1.78890	0.0608595
$865$	44.5139	1.51352
$866$	24.2389	0.823670
$867$	1.27502	0.0433019
$868$	−4.30278	−0.146046
$869$	15.8167	0.536543
$870$	−5.78890	−0.196262
$871$	21.5139	0.728970
$872$	−2.00000	−0.0677285
$873$	−23.2666	−0.787456
$874$	1.39445	0.0471679
$875$	14.0917	0.476385
$876$	−0.788897	−0.0266544
$877$	47.5694	1.60630	0.803152	−	0.595774i	$-0.203155\pi$
$877$	47.5694	1.60630	0.803152	+	0.595774i	$0.203155\pi$
$878$	25.2111	0.850833
$879$	−7.73338	−0.260841
$880$	−2.30278	−0.0776266
$881$	−33.9083	−1.14240	−0.571200	−	0.820811i	$-0.693522\pi$
$881$	−33.9083	−1.14240	−0.571200	+	0.820811i	$0.693522\pi$
$882$	−15.4222	−0.519293
$883$	−39.0278	−1.31339	−0.656694	−	0.754157i	$-0.728046\pi$
$883$	−39.0278	−1.31339	−0.656694	+	0.754157i	$0.728046\pi$
$884$	15.2111	0.511605
$885$	0	0
$886$	−39.6333	−1.33151
$887$	24.8444	0.834194	0.417097	−	0.908862i	$-0.363048\pi$
$887$	24.8444	0.834194	0.417097	+	0.908862i	$0.363048\pi$
$888$	−1.57779	−0.0529473
$889$	−7.02776	−0.235703
$890$	10.6056	0.355499
$891$	8.18335	0.274152
$892$	−25.2111	−0.844130
$893$	−6.00000	−0.200782
$894$	0	0
$895$	30.2111	1.00985
$896$	−1.30278	−0.0435227
$897$	1.39445	0.0465593
$898$	31.8167	1.06174
$899$	27.4222	0.914582
$900$	−0.880571	−0.0293524
$901$	48.8444	1.62724
$902$	−3.90833	−0.130133
$903$	0.430609	0.0143298
$904$	0	0
$905$	34.6056	1.15033
$906$	4.97224	0.165192
$907$	−4.84441	−0.160856	−0.0804280	−	0.996760i	$-0.525629\pi$
$907$	−4.84441	−0.160856	−0.0804280	+	0.996760i	$0.525629\pi$
$908$	1.81665	0.0602878
$909$	−22.7334	−0.754019
$910$	−9.90833	−0.328458
$911$	27.6333	0.915532	0.457766	−	0.889073i	$-0.348650\pi$
$911$	27.6333	0.915532	0.457766	+	0.889073i	$0.348650\pi$
$912$	0.302776	0.0100259
$913$	14.3028	0.473353
$914$	−21.8167	−0.721631
$915$	−7.81665	−0.258411
$916$	11.9083	0.393462
$917$	24.9083	0.822545
$918$	−8.23886	−0.271923
$919$	6.11943	0.201861	0.100931	−	0.994893i	$-0.467818\pi$
$919$	6.11943	0.201861	0.100931	+	0.994893i	$0.467818\pi$
$920$	−3.21110	−0.105867
$921$	−0.661064	−0.0217828
$922$	−6.42221	−0.211504
$923$	32.0278	1.05421
$924$	0.394449	0.0129764
$925$	1.57779	0.0518776
$926$	1.21110	0.0397993
$927$	4.93608	0.162122
$928$	8.30278	0.272552
$929$	−8.51388	−0.279331	−0.139666	−	0.990199i	$-0.544603\pi$
$929$	−8.51388	−0.279331	−0.139666	+	0.990199i	$0.544603\pi$
$930$	−2.30278	−0.0755110
$931$	−5.30278	−0.173791
$932$	9.21110	0.301720
$933$	7.81665	0.255906
$934$	22.6056	0.739676
$935$	10.6056	0.346839
$936$	−9.60555	−0.313967
$937$	1.15559	0.0377515	0.0188757	−	0.999822i	$-0.493991\pi$
$937$	1.15559	0.0377515	0.0188757	+	0.999822i	$0.493991\pi$
$938$	8.48612	0.277082
$939$	5.48612	0.179033
$940$	13.8167	0.450650
$941$	−57.6333	−1.87879	−0.939396	−	0.342834i	$-0.888613\pi$
$941$	−57.6333	−1.87879	−0.939396	+	0.342834i	$0.888613\pi$
$942$	−0.330532	−0.0107693
$943$	−5.44996	−0.177475
$944$	0	0
$945$	5.36669	0.174579
$946$	−1.09167	−0.0354934
$947$	25.8167	0.838929	0.419464	−	0.907772i	$-0.362218\pi$
$947$	25.8167	0.838929	0.419464	+	0.907772i	$0.362218\pi$
$948$	4.78890	0.155536
$949$	8.60555	0.279348
$950$	−0.302776	−0.00982334
$951$	−4.18335	−0.135654
$952$	6.00000	0.194461
$953$	−21.6333	−0.700772	−0.350386	−	0.936605i	$-0.613950\pi$
$953$	−21.6333	−0.700772	−0.350386	+	0.936605i	$0.613950\pi$
$954$	−30.8444	−0.998625
$955$	13.8167	0.447096
$956$	−23.9361	−0.774148
$957$	−2.51388	−0.0812621
$958$	−35.7250	−1.15422
$959$	−4.54163	−0.146657
$960$	−0.697224	−0.0225028
$961$	−20.0917	−0.648118
$962$	17.2111	0.554908
$963$	−9.33894	−0.300943
$964$	13.0917	0.421654
$965$	56.9361	1.83284
$966$	0.550039	0.0176972
$967$	23.6333	0.759996	0.379998	−	0.924987i	$-0.375925\pi$
$967$	23.6333	0.759996	0.379998	+	0.924987i	$0.375925\pi$
$968$	−1.00000	−0.0321412
$969$	−1.39445	−0.0447961
$970$	18.4222	0.591501
$971$	−23.9361	−0.768145	−0.384073	−	0.923303i	$-0.625479\pi$
$971$	−23.9361	−0.768145	−0.384073	+	0.923303i	$0.625479\pi$
$972$	7.84441	0.251610
$973$	−8.21110	−0.263236
$974$	29.3305	0.939811
$975$	−0.302776	−0.00969658
$976$	11.2111	0.358859
$977$	27.6333	0.884068	0.442034	−	0.896998i	$-0.354257\pi$
$977$	27.6333	0.884068	0.442034	+	0.896998i	$0.354257\pi$
$978$	3.44996	0.110318
$979$	4.60555	0.147194
$980$	12.2111	0.390069
$981$	−5.81665	−0.185711
$982$	14.9361	0.476630
$983$	23.3028	0.743243	0.371622	−	0.928384i	$-0.378802\pi$
$983$	23.3028	0.743243	0.371622	+	0.928384i	$0.378802\pi$
$984$	−1.18335	−0.0377237
$985$	−49.8167	−1.58729
$986$	−38.2389	−1.21777
$987$	−2.36669	−0.0753326
$988$	−3.30278	−0.105075
$989$	−1.52228	−0.0484058
$990$	−6.69722	−0.212852
$991$	−25.9083	−0.823005	−0.411503	−	0.911409i	$-0.634996\pi$
$991$	−25.9083	−0.823005	−0.411503	+	0.911409i	$0.634996\pi$
$992$	3.30278	0.104863
$993$	6.88057	0.218348
$994$	12.6333	0.400704
$995$	5.02776	0.159391
$996$	4.33053	0.137218
$997$	0.183346	0.00580663	0.00290332	−	0.999996i	$-0.499076\pi$
$997$	0.183346	0.00580663	0.00290332	+	0.999996i	$0.499076\pi$
$998$	34.0000	1.07625
$999$	−9.32213	−0.294939

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	418.2.a.d.1.2	✓	2
3.2	odd	2		3762.2.a.bb.1.2		2
4.3	odd	2		3344.2.a.o.1.1		2
11.10	odd	2		4598.2.a.bc.1.2		2
19.18	odd	2		7942.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.d.1.2	✓	2	1.1	even	1	trivial
3344.2.a.o.1.1		2	4.3	odd	2
3762.2.a.bb.1.2		2	3.2	odd	2
4598.2.a.bc.1.2		2	11.10	odd	2
7942.2.a.bb.1.1		2	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 418 = 2 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	418.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} -2.30278 q^{5} -0.302776 q^{6} +1.30278 q^{7} -1.00000 q^{8} -2.90833 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} -2.30278 q^{5} -0.302776 q^{6} +1.30278 q^{7} -1.00000 q^{8} -2.90833 q^{9} +2.30278 q^{10} +1.00000 q^{11} +0.302776 q^{12} -3.30278 q^{13} -1.30278 q^{14} -0.697224 q^{15} +1.00000 q^{16} -4.60555 q^{17} +2.90833 q^{18} +1.00000 q^{19} -2.30278 q^{20} +0.394449 q^{21} -1.00000 q^{22} -1.39445 q^{23} -0.302776 q^{24} +0.302776 q^{25} +3.30278 q^{26} -1.78890 q^{27} +1.30278 q^{28} -8.30278 q^{29} +0.697224 q^{30} -3.30278 q^{31} -1.00000 q^{32} +0.302776 q^{33} +4.60555 q^{34} -3.00000 q^{35} -2.90833 q^{36} +5.21110 q^{37} -1.00000 q^{38} -1.00000 q^{39} +2.30278 q^{40} +3.90833 q^{41} -0.394449 q^{42} +1.09167 q^{43} +1.00000 q^{44} +6.69722 q^{45} +1.39445 q^{46} -6.00000 q^{47} +0.302776 q^{48} -5.30278 q^{49} -0.302776 q^{50} -1.39445 q^{51} -3.30278 q^{52} -10.6056 q^{53} +1.78890 q^{54} -2.30278 q^{55} -1.30278 q^{56} +0.302776 q^{57} +8.30278 q^{58} -0.697224 q^{60} +11.2111 q^{61} +3.30278 q^{62} -3.78890 q^{63} +1.00000 q^{64} +7.60555 q^{65} -0.302776 q^{66} -6.51388 q^{67} -4.60555 q^{68} -0.422205 q^{69} +3.00000 q^{70} -9.69722 q^{71} +2.90833 q^{72} -2.60555 q^{73} -5.21110 q^{74} +0.0916731 q^{75} +1.00000 q^{76} +1.30278 q^{77} +1.00000 q^{78} +15.8167 q^{79} -2.30278 q^{80} +8.18335 q^{81} -3.90833 q^{82} +14.3028 q^{83} +0.394449 q^{84} +10.6056 q^{85} -1.09167 q^{86} -2.51388 q^{87} -1.00000 q^{88} +4.60555 q^{89} -6.69722 q^{90} -4.30278 q^{91} -1.39445 q^{92} -1.00000 q^{93} +6.00000 q^{94} -2.30278 q^{95} -0.302776 q^{96} +8.00000 q^{97} +5.30278 q^{98} -2.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{5} + 3 q^{6} - q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 3 * q^3 + 2 * q^4 - q^5 + 3 * q^6 - q^7 - 2 * q^8 + 5 * q^9	\( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{5} + 3 q^{6} - q^{7} - 2 q^{8} + 5 q^{9} + q^{10} + 2 q^{11} - 3 q^{12} - 3 q^{13} + q^{14} - 5 q^{15} + 2 q^{16} - 2 q^{17} - 5 q^{18} + 2 q^{19} - q^{20} + 8 q^{21} - 2 q^{22} - 10 q^{23} + 3 q^{24} - 3 q^{25} + 3 q^{26} - 18 q^{27} - q^{28} - 13 q^{29} + 5 q^{30} - 3 q^{31} - 2 q^{32} - 3 q^{33} + 2 q^{34} - 6 q^{35} + 5 q^{36} - 4 q^{37} - 2 q^{38} - 2 q^{39} + q^{40} - 3 q^{41} - 8 q^{42} + 13 q^{43} + 2 q^{44} + 17 q^{45} + 10 q^{46} - 12 q^{47} - 3 q^{48} - 7 q^{49} + 3 q^{50} - 10 q^{51} - 3 q^{52} - 14 q^{53} + 18 q^{54} - q^{55} + q^{56} - 3 q^{57} + 13 q^{58} - 5 q^{60} + 8 q^{61} + 3 q^{62} - 22 q^{63} + 2 q^{64} + 8 q^{65} + 3 q^{66} + 5 q^{67} - 2 q^{68} + 28 q^{69} + 6 q^{70} - 23 q^{71} - 5 q^{72} + 2 q^{73} + 4 q^{74} + 11 q^{75} + 2 q^{76} - q^{77} + 2 q^{78} + 10 q^{79} - q^{80} + 38 q^{81} + 3 q^{82} + 25 q^{83} + 8 q^{84} + 14 q^{85} - 13 q^{86} + 13 q^{87} - 2 q^{88} + 2 q^{89} - 17 q^{90} - 5 q^{91} - 10 q^{92} - 2 q^{93} + 12 q^{94} - q^{95} + 3 q^{96} + 16 q^{97} + 7 q^{98} + 5 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 3 * q^3 + 2 * q^4 - q^5 + 3 * q^6 - q^7 - 2 * q^8 + 5 * q^9 + q^10 + 2 * q^11 - 3 * q^12 - 3 * q^13 + q^14 - 5 * q^15 + 2 * q^16 - 2 * q^17 - 5 * q^18 + 2 * q^19 - q^20 + 8 * q^21 - 2 * q^22 - 10 * q^23 + 3 * q^24 - 3 * q^25 + 3 * q^26 - 18 * q^27 - q^28 - 13 * q^29 + 5 * q^30 - 3 * q^31 - 2 * q^32 - 3 * q^33 + 2 * q^34 - 6 * q^35 + 5 * q^36 - 4 * q^37 - 2 * q^38 - 2 * q^39 + q^40 - 3 * q^41 - 8 * q^42 + 13 * q^43 + 2 * q^44 + 17 * q^45 + 10 * q^46 - 12 * q^47 - 3 * q^48 - 7 * q^49 + 3 * q^50 - 10 * q^51 - 3 * q^52 - 14 * q^53 + 18 * q^54 - q^55 + q^56 - 3 * q^57 + 13 * q^58 - 5 * q^60 + 8 * q^61 + 3 * q^62 - 22 * q^63 + 2 * q^64 + 8 * q^65 + 3 * q^66 + 5 * q^67 - 2 * q^68 + 28 * q^69 + 6 * q^70 - 23 * q^71 - 5 * q^72 + 2 * q^73 + 4 * q^74 + 11 * q^75 + 2 * q^76 - q^77 + 2 * q^78 + 10 * q^79 - q^80 + 38 * q^81 + 3 * q^82 + 25 * q^83 + 8 * q^84 + 14 * q^85 - 13 * q^86 + 13 * q^87 - 2 * q^88 + 2 * q^89 - 17 * q^90 - 5 * q^91 - 10 * q^92 - 2 * q^93 + 12 * q^94 - q^95 + 3 * q^96 + 16 * q^97 + 7 * q^98 + 5 * q^99

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