LMFDB - Embedded newform 123.2.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [123,2,Mod(1,123)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(123, 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("123.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 123 = 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	123.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:	$0.982159944862$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	123.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.34292 q^{2} -1.00000 q^{3} +3.48929 q^{4} +0.853635 q^{5} -2.34292 q^{6} -3.83221 q^{7} +3.48929 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.34292 q^{2} -1.00000 q^{3} +3.48929 q^{4} +0.853635 q^{5} -2.34292 q^{6} -3.83221 q^{7} +3.48929 q^{8} +1.00000 q^{9} +2.00000 q^{10} -3.34292 q^{11} -3.48929 q^{12} +3.14637 q^{13} -8.97858 q^{14} -0.853635 q^{15} +1.19656 q^{16} +3.63565 q^{17} +2.34292 q^{18} +1.14637 q^{19} +2.97858 q^{20} +3.83221 q^{21} -7.83221 q^{22} -2.85363 q^{23} -3.48929 q^{24} -4.27131 q^{25} +7.37169 q^{26} -1.00000 q^{27} -13.3717 q^{28} -8.02877 q^{29} -2.00000 q^{30} +9.86098 q^{31} -4.17513 q^{32} +3.34292 q^{33} +8.51806 q^{34} -3.27131 q^{35} +3.48929 q^{36} +8.19656 q^{37} +2.68585 q^{38} -3.14637 q^{39} +2.97858 q^{40} +1.00000 q^{41} +8.97858 q^{42} +11.7606 q^{43} -11.6644 q^{44} +0.853635 q^{45} -6.68585 q^{46} +8.32150 q^{47} -1.19656 q^{48} +7.68585 q^{49} -10.0073 q^{50} -3.63565 q^{51} +10.9786 q^{52} +1.60688 q^{53} -2.34292 q^{54} -2.85363 q^{55} -13.3717 q^{56} -1.14637 q^{57} -18.8108 q^{58} -11.6644 q^{59} -2.97858 q^{60} -4.19656 q^{61} +23.1035 q^{62} -3.83221 q^{63} -12.1751 q^{64} +2.68585 q^{65} +7.83221 q^{66} +6.10038 q^{67} +12.6858 q^{68} +2.85363 q^{69} -7.66442 q^{70} -8.65708 q^{71} +3.48929 q^{72} -10.1537 q^{73} +19.2039 q^{74} +4.27131 q^{75} +4.00000 q^{76} +12.8108 q^{77} -7.37169 q^{78} -3.60688 q^{79} +1.02142 q^{80} +1.00000 q^{81} +2.34292 q^{82} -10.1249 q^{83} +13.3717 q^{84} +3.10352 q^{85} +27.5542 q^{86} +8.02877 q^{87} -11.6644 q^{88} -3.37169 q^{89} +2.00000 q^{90} -12.0575 q^{91} -9.95715 q^{92} -9.86098 q^{93} +19.4966 q^{94} +0.978577 q^{95} +4.17513 q^{96} -7.53948 q^{97} +18.0073 q^{98} -3.34292 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 3 q^{3} + 3 q^{4} + 4 q^{5} - q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + q^2 - 3 * q^3 + 3 * q^4 + 4 * q^5 - q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} - 3 q^{3} + 3 q^{4} + 4 q^{5} - q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} + 6 q^{10} - 4 q^{11} - 3 q^{12} + 8 q^{13} - 12 q^{14} - 4 q^{15} - q^{16} + 2 q^{17} + q^{18} + 2 q^{19} - 6 q^{20} - 2 q^{21} - 10 q^{22} - 10 q^{23} - 3 q^{24} + 5 q^{25} - 2 q^{26} - 3 q^{27} - 16 q^{28} - 6 q^{29} - 6 q^{30} - 2 q^{31} + 7 q^{32} + 4 q^{33} + 8 q^{35} + 3 q^{36} + 20 q^{37} - 4 q^{38} - 8 q^{39} - 6 q^{40} + 3 q^{41} + 12 q^{42} + 10 q^{43} - 8 q^{44} + 4 q^{45} - 8 q^{46} + 4 q^{47} + q^{48} + 11 q^{49} + 3 q^{50} - 2 q^{51} + 18 q^{52} + 14 q^{53} - q^{54} - 10 q^{55} - 16 q^{56} - 2 q^{57} - 28 q^{58} - 8 q^{59} + 6 q^{60} - 8 q^{61} + 38 q^{62} + 2 q^{63} - 17 q^{64} - 4 q^{65} + 10 q^{66} + 12 q^{67} + 26 q^{68} + 10 q^{69} + 4 q^{70} - 32 q^{71} + 3 q^{72} + 4 q^{73} + 20 q^{74} - 5 q^{75} + 12 q^{76} + 10 q^{77} + 2 q^{78} - 20 q^{79} + 18 q^{80} + 3 q^{81} + q^{82} - 14 q^{83} + 16 q^{84} - 22 q^{85} + 6 q^{86} + 6 q^{87} - 8 q^{88} + 14 q^{89} + 6 q^{90} + 2 q^{93} + 18 q^{94} - 12 q^{95} - 7 q^{96} - 12 q^{97} + 21 q^{98} - 4 q^{99}+O(q^{100}) \) 3 * q + q^2 - 3 * q^3 + 3 * q^4 + 4 * q^5 - q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 + 6 * q^10 - 4 * q^11 - 3 * q^12 + 8 * q^13 - 12 * q^14 - 4 * q^15 - q^16 + 2 * q^17 + q^18 + 2 * q^19 - 6 * q^20 - 2 * q^21 - 10 * q^22 - 10 * q^23 - 3 * q^24 + 5 * q^25 - 2 * q^26 - 3 * q^27 - 16 * q^28 - 6 * q^29 - 6 * q^30 - 2 * q^31 + 7 * q^32 + 4 * q^33 + 8 * q^35 + 3 * q^36 + 20 * q^37 - 4 * q^38 - 8 * q^39 - 6 * q^40 + 3 * q^41 + 12 * q^42 + 10 * q^43 - 8 * q^44 + 4 * q^45 - 8 * q^46 + 4 * q^47 + q^48 + 11 * q^49 + 3 * q^50 - 2 * q^51 + 18 * q^52 + 14 * q^53 - q^54 - 10 * q^55 - 16 * q^56 - 2 * q^57 - 28 * q^58 - 8 * q^59 + 6 * q^60 - 8 * q^61 + 38 * q^62 + 2 * q^63 - 17 * q^64 - 4 * q^65 + 10 * q^66 + 12 * q^67 + 26 * q^68 + 10 * q^69 + 4 * q^70 - 32 * q^71 + 3 * q^72 + 4 * q^73 + 20 * q^74 - 5 * q^75 + 12 * q^76 + 10 * q^77 + 2 * q^78 - 20 * q^79 + 18 * q^80 + 3 * q^81 + q^82 - 14 * q^83 + 16 * q^84 - 22 * q^85 + 6 * q^86 + 6 * q^87 - 8 * q^88 + 14 * q^89 + 6 * q^90 + 2 * q^93 + 18 * q^94 - 12 * q^95 - 7 * q^96 - 12 * q^97 + 21 * q^98 - 4 * 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 $

$2$	2.34292	1.65670	0.828348	−	0.560213i	$-0.189281\pi$
$2$	2.34292	1.65670	0.828348	+	0.560213i	$0.189281\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.48929	1.74464
$5$	0.853635	0.381757	0.190878	−	0.981614i	$-0.438866\pi$
$5$	0.853635	0.381757	0.190878	+	0.981614i	$0.438866\pi$
$6$	−2.34292	−0.956494
$7$	−3.83221	−1.44844	−0.724220	−	0.689569i	$-0.757800\pi$
$7$	−3.83221	−1.44844	−0.724220	+	0.689569i	$0.757800\pi$
$8$	3.48929	1.23365
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−3.34292	−1.00793	−0.503965	−	0.863724i	$-0.668126\pi$
$11$	−3.34292	−1.00793	−0.503965	+	0.863724i	$0.668126\pi$
$12$	−3.48929	−1.00727
$13$	3.14637	0.872645	0.436322	−	0.899790i	$-0.356281\pi$
$13$	3.14637	0.872645	0.436322	+	0.899790i	$0.356281\pi$
$14$	−8.97858	−2.39963
$15$	−0.853635	−0.220407
$16$	1.19656	0.299139
$17$	3.63565	0.881776	0.440888	−	0.897562i	$-0.354664\pi$
$17$	3.63565	0.881776	0.440888	+	0.897562i	$0.354664\pi$
$18$	2.34292	0.552232
$19$	1.14637	0.262994	0.131497	−	0.991317i	$-0.458022\pi$
$19$	1.14637	0.262994	0.131497	+	0.991317i	$0.458022\pi$
$20$	2.97858	0.666030
$21$	3.83221	0.836257
$22$	−7.83221	−1.66983
$23$	−2.85363	−0.595024	−0.297512	−	0.954718i	$-0.596157\pi$
$23$	−2.85363	−0.595024	−0.297512	+	0.954718i	$0.596157\pi$
$24$	−3.48929	−0.712248
$25$	−4.27131	−0.854262
$26$	7.37169	1.44571
$27$	−1.00000	−0.192450
$28$	−13.3717	−2.52701
$29$	−8.02877	−1.49091	−0.745453	−	0.666559i	$-0.767767\pi$
$29$	−8.02877	−1.49091	−0.745453	+	0.666559i	$0.767767\pi$
$30$	−2.00000	−0.365148
$31$	9.86098	1.77108	0.885542	−	0.464559i	$-0.153787\pi$
$31$	9.86098	1.77108	0.885542	+	0.464559i	$0.153787\pi$
$32$	−4.17513	−0.738067
$33$	3.34292	0.581928
$34$	8.51806	1.46083
$35$	−3.27131	−0.552952
$36$	3.48929	0.581548
$37$	8.19656	1.34751	0.673753	−	0.738957i	$-0.264681\pi$
$37$	8.19656	1.34751	0.673753	+	0.738957i	$0.264681\pi$
$38$	2.68585	0.435702
$39$	−3.14637	−0.503822
$40$	2.97858	0.470954
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	8.97858	1.38542
$43$	11.7606	1.79347	0.896737	−	0.442564i	$-0.145931\pi$
$43$	11.7606	1.79347	0.896737	+	0.442564i	$0.145931\pi$
$44$	−11.6644	−1.75848
$45$	0.853635	0.127252
$46$	−6.68585	−0.985774
$47$	8.32150	1.21382	0.606908	−	0.794772i	$-0.292410\pi$
$47$	8.32150	1.21382	0.606908	+	0.794772i	$0.292410\pi$
$48$	−1.19656	−0.172708
$49$	7.68585	1.09798
$50$	−10.0073	−1.41525
$51$	−3.63565	−0.509093
$52$	10.9786	1.52245
$53$	1.60688	0.220723	0.110361	−	0.993892i	$-0.464799\pi$
$53$	1.60688	0.220723	0.110361	+	0.993892i	$0.464799\pi$
$54$	−2.34292	−0.318831
$55$	−2.85363	−0.384784
$56$	−13.3717	−1.78687
$57$	−1.14637	−0.151840
$58$	−18.8108	−2.46998
$59$	−11.6644	−1.51858	−0.759289	−	0.650753i	$-0.774453\pi$
$59$	−11.6644	−1.51858	−0.759289	+	0.650753i	$0.774453\pi$
$60$	−2.97858	−0.384533
$61$	−4.19656	−0.537314	−0.268657	−	0.963236i	$-0.586580\pi$
$61$	−4.19656	−0.537314	−0.268657	+	0.963236i	$0.586580\pi$
$62$	23.1035	2.93415
$63$	−3.83221	−0.482813
$64$	−12.1751	−1.52189
$65$	2.68585	0.333138
$66$	7.83221	0.964079
$67$	6.10038	0.745281	0.372640	−	0.927976i	$-0.378453\pi$
$67$	6.10038	0.745281	0.372640	+	0.927976i	$0.378453\pi$
$68$	12.6858	1.53838
$69$	2.85363	0.343537
$70$	−7.66442	−0.916074
$71$	−8.65708	−1.02741	−0.513703	−	0.857968i	$-0.671727\pi$
$71$	−8.65708	−1.02741	−0.513703	+	0.857968i	$0.671727\pi$
$72$	3.48929	0.411217
$73$	−10.1537	−1.18840	−0.594201	−	0.804317i	$-0.702532\pi$
$73$	−10.1537	−1.18840	−0.594201	+	0.804317i	$0.702532\pi$
$74$	19.2039	2.23241
$75$	4.27131	0.493208
$76$	4.00000	0.458831
$77$	12.8108	1.45992
$78$	−7.37169	−0.834680
$79$	−3.60688	−0.405806	−0.202903	−	0.979199i	$-0.565038\pi$
$79$	−3.60688	−0.405806	−0.202903	+	0.979199i	$0.565038\pi$
$80$	1.02142	0.114199
$81$	1.00000	0.111111
$82$	2.34292	0.258733
$83$	−10.1249	−1.11136	−0.555678	−	0.831397i	$-0.687541\pi$
$83$	−10.1249	−1.11136	−0.555678	+	0.831397i	$0.687541\pi$
$84$	13.3717	1.45897
$85$	3.10352	0.336624
$86$	27.5542	2.97124
$87$	8.02877	0.860774
$88$	−11.6644	−1.24343
$89$	−3.37169	−0.357399	−0.178699	−	0.983904i	$-0.557189\pi$
$89$	−3.37169	−0.357399	−0.178699	+	0.983904i	$0.557189\pi$
$90$	2.00000	0.210819
$91$	−12.0575	−1.26397
$92$	−9.95715	−1.03811
$93$	−9.86098	−1.02254
$94$	19.4966	2.01092
$95$	0.978577	0.100400
$96$	4.17513	0.426123
$97$	−7.53948	−0.765518	−0.382759	−	0.923848i	$-0.625026\pi$
$97$	−7.53948	−0.765518	−0.382759	+	0.923848i	$0.625026\pi$
$98$	18.0073	1.81902
$99$	−3.34292	−0.335976
$100$	−14.9038	−1.49038
$101$	5.00735	0.498250	0.249125	−	0.968471i	$-0.419857\pi$
$101$	5.00735	0.498250	0.249125	+	0.968471i	$0.419857\pi$
$102$	−8.51806	−0.843413
$103$	1.21798	0.120011	0.0600056	−	0.998198i	$-0.480888\pi$
$103$	1.21798	0.120011	0.0600056	+	0.998198i	$0.480888\pi$
$104$	10.9786	1.07654
$105$	3.27131	0.319247
$106$	3.76481	0.365670
$107$	13.7648	1.33069	0.665347	−	0.746534i	$-0.268284\pi$
$107$	13.7648	1.33069	0.665347	+	0.746534i	$0.268284\pi$
$108$	−3.48929	−0.335757
$109$	3.14637	0.301367	0.150684	−	0.988582i	$-0.451853\pi$
$109$	3.14637	0.301367	0.150684	+	0.988582i	$0.451853\pi$
$110$	−6.68585	−0.637470
$111$	−8.19656	−0.777983
$112$	−4.58546	−0.433285
$113$	−7.73183	−0.727349	−0.363675	−	0.931526i	$-0.618478\pi$
$113$	−7.73183	−0.727349	−0.363675	+	0.931526i	$0.618478\pi$
$114$	−2.68585	−0.251553
$115$	−2.43596	−0.227155
$116$	−28.0147	−2.60110
$117$	3.14637	0.290882
$118$	−27.3288	−2.51582
$119$	−13.9326	−1.27720
$120$	−2.97858	−0.271906
$121$	0.175135	0.0159213
$122$	−9.83221	−0.890167
$123$	−1.00000	−0.0901670
$124$	34.4078	3.08991
$125$	−7.91431	−0.707877
$126$	−8.97858	−0.799875
$127$	−12.0575	−1.06993	−0.534967	−	0.844873i	$-0.679676\pi$
$127$	−12.0575	−1.06993	−0.534967	+	0.844873i	$0.679676\pi$
$128$	−20.1751	−1.78325
$129$	−11.7606	−1.03546
$130$	6.29273	0.551909
$131$	12.2253	1.06813	0.534066	−	0.845443i	$-0.320663\pi$
$131$	12.2253	1.06813	0.534066	+	0.845443i	$0.320663\pi$
$132$	11.6644	1.01526
$133$	−4.39312	−0.380931
$134$	14.2927	1.23470
$135$	−0.853635	−0.0734692
$136$	12.6858	1.08780
$137$	7.69319	0.657274	0.328637	−	0.944456i	$-0.393411\pi$
$137$	7.69319	0.657274	0.328637	+	0.944456i	$0.393411\pi$
$138$	6.68585	0.569137
$139$	−18.0147	−1.52799	−0.763993	−	0.645224i	$-0.776764\pi$
$139$	−18.0147	−1.52799	−0.763993	+	0.645224i	$0.776764\pi$
$140$	−11.4145	−0.964705
$141$	−8.32150	−0.700797
$142$	−20.2829	−1.70210
$143$	−10.5181	−0.879564
$144$	1.19656	0.0997131
$145$	−6.85363	−0.569163
$146$	−23.7894	−1.96882
$147$	−7.68585	−0.633918
$148$	28.6002	2.35092
$149$	12.5426	1.02753	0.513766	−	0.857931i	$-0.328250\pi$
$149$	12.5426	1.02753	0.513766	+	0.857931i	$0.328250\pi$
$150$	10.0073	0.817096
$151$	2.62831	0.213889	0.106944	−	0.994265i	$-0.465893\pi$
$151$	2.62831	0.213889	0.106944	+	0.994265i	$0.465893\pi$
$152$	4.00000	0.324443
$153$	3.63565	0.293925
$154$	30.0147	2.41865
$155$	8.41767	0.676124
$156$	−10.9786	−0.878990
$157$	20.1579	1.60878	0.804389	−	0.594103i	$-0.202493\pi$
$157$	20.1579	1.60878	0.804389	+	0.594103i	$0.202493\pi$
$158$	−8.45065	−0.672298
$159$	−1.60688	−0.127434
$160$	−3.56404	−0.281762
$161$	10.9357	0.861856
$162$	2.34292	0.184077
$163$	18.2541	1.42977	0.714886	−	0.699241i	$-0.246479\pi$
$163$	18.2541	1.42977	0.714886	+	0.699241i	$0.246479\pi$
$164$	3.48929	0.272468
$165$	2.85363	0.222155
$166$	−23.7220	−1.84118
$167$	1.31415	0.101692	0.0508461	−	0.998706i	$-0.483808\pi$
$167$	1.31415	0.101692	0.0508461	+	0.998706i	$0.483808\pi$
$168$	13.3717	1.03165
$169$	−3.10038	−0.238491
$170$	7.27131	0.557684
$171$	1.14637	0.0876648
$172$	41.0361	3.12897
$173$	1.18921	0.0904141	0.0452070	−	0.998978i	$-0.485605\pi$
$173$	1.18921	0.0904141	0.0452070	+	0.998978i	$0.485605\pi$
$174$	18.8108	1.42604
$175$	16.3686	1.23735
$176$	−4.00000	−0.301511
$177$	11.6644	0.876752
$178$	−7.89962	−0.592101
$179$	−19.0073	−1.42068	−0.710338	−	0.703861i	$-0.751458\pi$
$179$	−19.0073	−1.42068	−0.710338	+	0.703861i	$0.751458\pi$
$180$	2.97858	0.222010
$181$	10.3931	0.772514	0.386257	−	0.922391i	$-0.373768\pi$
$181$	10.3931	0.772514	0.386257	+	0.922391i	$0.373768\pi$
$182$	−28.2499	−2.09402
$183$	4.19656	0.310218
$184$	−9.95715	−0.734051
$185$	6.99686	0.514420
$186$	−23.1035	−1.69403
$187$	−12.1537	−0.888767
$188$	29.0361	2.11768
$189$	3.83221	0.278752
$190$	2.29273	0.166332
$191$	−6.87819	−0.497689	−0.248844	−	0.968544i	$-0.580051\pi$
$191$	−6.87819	−0.497689	−0.248844	+	0.968544i	$0.580051\pi$
$192$	12.1751	0.878665
$193$	−19.5970	−1.41062	−0.705312	−	0.708897i	$-0.749193\pi$
$193$	−19.5970	−1.41062	−0.705312	+	0.708897i	$0.749193\pi$
$194$	−17.6644	−1.26823
$195$	−2.68585	−0.192337
$196$	26.8181	1.91558
$197$	−24.7679	−1.76464	−0.882321	−	0.470647i	$-0.844020\pi$
$197$	−24.7679	−1.76464	−0.882321	+	0.470647i	$0.844020\pi$
$198$	−7.83221	−0.556611
$199$	12.8108	0.908133	0.454066	−	0.890968i	$-0.349973\pi$
$199$	12.8108	0.908133	0.454066	+	0.890968i	$0.349973\pi$
$200$	−14.9038	−1.05386
$201$	−6.10038	−0.430288
$202$	11.7318	0.825448
$203$	30.7679	2.15949
$204$	−12.6858	−0.888187
$205$	0.853635	0.0596204
$206$	2.85363	0.198822
$207$	−2.85363	−0.198341
$208$	3.76481	0.261042
$209$	−3.83221	−0.265080
$210$	7.66442	0.528895
$211$	3.66442	0.252269	0.126135	−	0.992013i	$-0.459743\pi$
$211$	3.66442	0.252269	0.126135	+	0.992013i	$0.459743\pi$
$212$	5.60688	0.385082
$213$	8.65708	0.593173
$214$	32.2499	2.20456
$215$	10.0393	0.684671
$216$	−3.48929	−0.237416
$217$	−37.7894	−2.56531
$218$	7.37169	0.499274
$219$	10.1537	0.686124
$220$	−9.95715	−0.671311
$221$	11.4391	0.769477
$222$	−19.2039	−1.28888
$223$	9.17092	0.614130	0.307065	−	0.951688i	$-0.400653\pi$
$223$	9.17092	0.614130	0.307065	+	0.951688i	$0.400653\pi$
$224$	16.0000	1.06904
$225$	−4.27131	−0.284754
$226$	−18.1151	−1.20500
$227$	0.263962	0.0175198	0.00875988	−	0.999962i	$-0.497212\pi$
$227$	0.263962	0.0175198	0.00875988	+	0.999962i	$0.497212\pi$
$228$	−4.00000	−0.264906
$229$	7.20390	0.476047	0.238024	−	0.971259i	$-0.423500\pi$
$229$	7.20390	0.476047	0.238024	+	0.971259i	$0.423500\pi$
$230$	−5.70727	−0.376326
$231$	−12.8108	−0.842888
$232$	−28.0147	−1.83925
$233$	16.1579	1.05854	0.529270	−	0.848453i	$-0.322466\pi$
$233$	16.1579	1.05854	0.529270	+	0.848453i	$0.322466\pi$
$234$	7.37169	0.481903
$235$	7.10352	0.463383
$236$	−40.7005	−2.64938
$237$	3.60688	0.234292
$238$	−32.6430	−2.11593
$239$	−15.6644	−1.01325	−0.506624	−	0.862167i	$-0.669107\pi$
$239$	−15.6644	−1.01325	−0.506624	+	0.862167i	$0.669107\pi$
$240$	−1.02142	−0.0659326
$241$	10.5468	0.679381	0.339690	−	0.940537i	$-0.389678\pi$
$241$	10.5468	0.679381	0.339690	+	0.940537i	$0.389678\pi$
$242$	0.410327	0.0263768
$243$	−1.00000	−0.0641500
$244$	−14.6430	−0.937422
$245$	6.56090	0.419161
$246$	−2.34292	−0.149379
$247$	3.60688	0.229501
$248$	34.4078	2.18490
$249$	10.1249	0.641642
$250$	−18.5426	−1.17274
$251$	21.4292	1.35260	0.676301	−	0.736626i	$-0.263582\pi$
$251$	21.4292	1.35260	0.676301	+	0.736626i	$0.263582\pi$
$252$	−13.3717	−0.842337
$253$	9.53948	0.599742
$254$	−28.2499	−1.77256
$255$	−3.10352	−0.194350
$256$	−22.9185	−1.43241
$257$	−7.10773	−0.443368	−0.221684	−	0.975119i	$-0.571155\pi$
$257$	−7.10773	−0.443368	−0.221684	+	0.975119i	$0.571155\pi$
$258$	−27.5542	−1.71545
$259$	−31.4109	−1.95178
$260$	9.37169	0.581208
$261$	−8.02877	−0.496968
$262$	28.6430	1.76957
$263$	−9.97123	−0.614852	−0.307426	−	0.951572i	$-0.599468\pi$
$263$	−9.97123	−0.614852	−0.307426	+	0.951572i	$0.599468\pi$
$264$	11.6644	0.717896
$265$	1.37169	0.0842624
$266$	−10.2927	−0.631088
$267$	3.37169	0.206344
$268$	21.2860	1.30025
$269$	−0.685846	−0.0418168	−0.0209084	−	0.999781i	$-0.506656\pi$
$269$	−0.685846	−0.0418168	−0.0209084	+	0.999781i	$0.506656\pi$
$270$	−2.00000	−0.121716
$271$	13.8034	0.838499	0.419250	−	0.907871i	$-0.362293\pi$
$271$	13.8034	0.838499	0.419250	+	0.907871i	$0.362293\pi$
$272$	4.35027	0.263774
$273$	12.0575	0.729755
$274$	18.0246	1.08890
$275$	14.2787	0.861035
$276$	9.95715	0.599350
$277$	22.2113	1.33454	0.667272	−	0.744814i	$-0.267462\pi$
$277$	22.2113	1.33454	0.667272	+	0.744814i	$0.267462\pi$
$278$	−42.2070	−2.53141
$279$	9.86098	0.590361
$280$	−11.4145	−0.682149
$281$	4.61423	0.275262	0.137631	−	0.990484i	$-0.456051\pi$
$281$	4.61423	0.275262	0.137631	+	0.990484i	$0.456051\pi$
$282$	−19.4966	−1.16101
$283$	9.13229	0.542858	0.271429	−	0.962458i	$-0.412504\pi$
$283$	9.13229	0.542858	0.271429	+	0.962458i	$0.412504\pi$
$284$	−30.2070	−1.79246
$285$	−0.978577	−0.0579659
$286$	−24.6430	−1.45717
$287$	−3.83221	−0.226208
$288$	−4.17513	−0.246022
$289$	−3.78202	−0.222472
$290$	−16.0575	−0.942931
$291$	7.53948	0.441972
$292$	−35.4292	−2.07334
$293$	23.4433	1.36957	0.684786	−	0.728744i	$-0.259896\pi$
$293$	23.4433	1.36957	0.684786	+	0.728744i	$0.259896\pi$
$294$	−18.0073	−1.05021
$295$	−9.95715	−0.579728
$296$	28.6002	1.66235
$297$	3.34292	0.193976
$298$	29.3864	1.70231
$299$	−8.97858	−0.519245
$300$	14.9038	0.860473
$301$	−45.0691	−2.59774
$302$	6.15792	0.354349
$303$	−5.00735	−0.287665
$304$	1.37169	0.0786720
$305$	−3.58233	−0.205123
$306$	8.51806	0.486945
$307$	18.2541	1.04182	0.520908	−	0.853613i	$-0.325593\pi$
$307$	18.2541	1.04182	0.520908	+	0.853613i	$0.325593\pi$
$308$	44.7005	2.54705
$309$	−1.21798	−0.0692885
$310$	19.7220	1.12013
$311$	14.8782	0.843665	0.421832	−	0.906674i	$-0.361387\pi$
$311$	14.8782	0.843665	0.421832	+	0.906674i	$0.361387\pi$
$312$	−10.9786	−0.621540
$313$	−26.4752	−1.49647	−0.748234	−	0.663435i	$-0.769098\pi$
$313$	−26.4752	−1.49647	−0.748234	+	0.663435i	$0.769098\pi$
$314$	47.2285	2.66526
$315$	−3.27131	−0.184317
$316$	−12.5855	−0.707988
$317$	29.6503	1.66533	0.832665	−	0.553777i	$-0.186814\pi$
$317$	29.6503	1.66533	0.832665	+	0.553777i	$0.186814\pi$
$318$	−3.76481	−0.211120
$319$	26.8396	1.50273
$320$	−10.3931	−0.580993
$321$	−13.7648	−0.768277
$322$	25.6216	1.42783
$323$	4.16779	0.231902
$324$	3.48929	0.193849
$325$	−13.4391	−0.745467
$326$	42.7679	2.36870
$327$	−3.14637	−0.173994
$328$	3.48929	0.192664
$329$	−31.8898	−1.75814
$330$	6.68585	0.368044
$331$	2.60375	0.143115	0.0715575	−	0.997436i	$-0.477203\pi$
$331$	2.60375	0.143115	0.0715575	+	0.997436i	$0.477203\pi$
$332$	−35.3288	−1.93892
$333$	8.19656	0.449169
$334$	3.07896	0.168473
$335$	5.20750	0.284516
$336$	4.58546	0.250157
$337$	−1.51071	−0.0822937	−0.0411468	−	0.999153i	$-0.513101\pi$
$337$	−1.51071	−0.0822937	−0.0411468	+	0.999153i	$0.513101\pi$
$338$	−7.26396	−0.395107
$339$	7.73183	0.419935
$340$	10.8291	0.587289
$341$	−32.9645	−1.78513
$342$	2.68585	0.145234
$343$	−2.62831	−0.141915
$344$	41.0361	2.21252
$345$	2.43596	0.131148
$346$	2.78623	0.149789
$347$	−14.2787	−0.766518	−0.383259	−	0.923641i	$-0.625198\pi$
$347$	−14.2787	−0.766518	−0.383259	+	0.923641i	$0.625198\pi$
$348$	28.0147	1.50175
$349$	−16.2541	−0.870062	−0.435031	−	0.900416i	$-0.643263\pi$
$349$	−16.2541	−0.870062	−0.435031	+	0.900416i	$0.643263\pi$
$350$	38.3503	2.04991
$351$	−3.14637	−0.167941
$352$	13.9572	0.743919
$353$	11.1709	0.594568	0.297284	−	0.954789i	$-0.403919\pi$
$353$	11.1709	0.594568	0.297284	+	0.954789i	$0.403919\pi$
$354$	27.3288	1.45251
$355$	−7.38998	−0.392219
$356$	−11.7648	−0.623534
$357$	13.9326	0.737391
$358$	−44.5328	−2.35363
$359$	−16.2253	−0.856340	−0.428170	−	0.903698i	$-0.640842\pi$
$359$	−16.2253	−0.856340	−0.428170	+	0.903698i	$0.640842\pi$
$360$	2.97858	0.156985
$361$	−17.6858	−0.930834
$362$	24.3503	1.27982
$363$	−0.175135	−0.00919219
$364$	−42.0722	−2.20518
$365$	−8.66756	−0.453681
$366$	9.83221	0.513938
$367$	0.431750	0.0225372	0.0112686	−	0.999937i	$-0.496413\pi$
$367$	0.431750	0.0225372	0.0112686	+	0.999937i	$0.496413\pi$
$368$	−3.41454	−0.177995
$369$	1.00000	0.0520579
$370$	16.3931	0.852237
$371$	−6.15792	−0.319703
$372$	−34.4078	−1.78396
$373$	−3.66863	−0.189955	−0.0949773	−	0.995479i	$-0.530278\pi$
$373$	−3.66863	−0.189955	−0.0949773	+	0.995479i	$0.530278\pi$
$374$	−28.4752	−1.47242
$375$	7.91431	0.408693
$376$	29.0361	1.49742
$377$	−25.2614	−1.30103
$378$	8.97858	0.461808
$379$	−9.89962	−0.508509	−0.254255	−	0.967137i	$-0.581830\pi$
$379$	−9.89962	−0.508509	−0.254255	+	0.967137i	$0.581830\pi$
$380$	3.41454	0.175162
$381$	12.0575	0.617726
$382$	−16.1151	−0.824519
$383$	−31.3148	−1.60011	−0.800055	−	0.599927i	$-0.795196\pi$
$383$	−31.3148	−1.60011	−0.800055	+	0.599927i	$0.795196\pi$
$384$	20.1751	1.02956
$385$	10.9357	0.557336
$386$	−45.9143	−2.33698
$387$	11.7606	0.597825
$388$	−26.3074	−1.33556
$389$	7.39625	0.375005	0.187502	−	0.982264i	$-0.439961\pi$
$389$	7.39625	0.375005	0.187502	+	0.982264i	$0.439961\pi$
$390$	−6.29273	−0.318645
$391$	−10.3748	−0.524678
$392$	26.8181	1.35452
$393$	−12.2253	−0.616686
$394$	−58.0294	−2.92348
$395$	−3.07896	−0.154919
$396$	−11.6644	−0.586159
$397$	11.9326	0.598880	0.299440	−	0.954115i	$-0.403200\pi$
$397$	11.9326	0.598880	0.299440	+	0.954115i	$0.403200\pi$
$398$	30.0147	1.50450
$399$	4.39312	0.219931
$400$	−5.11087	−0.255543
$401$	−15.7648	−0.787257	−0.393628	−	0.919270i	$-0.628780\pi$
$401$	−15.7648	−0.787257	−0.393628	+	0.919270i	$0.628780\pi$
$402$	−14.2927	−0.712857
$403$	31.0263	1.54553
$404$	17.4721	0.869268
$405$	0.853635	0.0424174
$406$	72.0869	3.57761
$407$	−27.4005	−1.35819
$408$	−12.6858	−0.628043
$409$	−7.55356	−0.373499	−0.186750	−	0.982408i	$-0.559795\pi$
$409$	−7.55356	−0.373499	−0.186750	+	0.982408i	$0.559795\pi$
$410$	2.00000	0.0987730
$411$	−7.69319	−0.379477
$412$	4.24989	0.209377
$413$	44.7005	2.19957
$414$	−6.68585	−0.328591
$415$	−8.64300	−0.424268
$416$	−13.1365	−0.644070
$417$	18.0147	0.882183
$418$	−8.97858	−0.439157
$419$	9.59702	0.468845	0.234423	−	0.972135i	$-0.424680\pi$
$419$	9.59702	0.468845	0.234423	+	0.972135i	$0.424680\pi$
$420$	11.4145	0.556972
$421$	12.3503	0.601915	0.300958	−	0.953638i	$-0.402694\pi$
$421$	12.3503	0.601915	0.300958	+	0.953638i	$0.402694\pi$
$422$	8.58546	0.417934
$423$	8.32150	0.404605
$424$	5.60688	0.272294
$425$	−15.5290	−0.753267
$426$	20.2829	0.982708
$427$	16.0821	0.778267
$428$	48.0294	2.32159
$429$	10.5181	0.507817
$430$	23.5212	1.13429
$431$	5.14637	0.247892	0.123946	−	0.992289i	$-0.460445\pi$
$431$	5.14637	0.247892	0.123946	+	0.992289i	$0.460445\pi$
$432$	−1.19656	−0.0575694
$433$	17.0894	0.821266	0.410633	−	0.911801i	$-0.365308\pi$
$433$	17.0894	0.821266	0.410633	+	0.911801i	$0.365308\pi$
$434$	−88.5376	−4.24994
$435$	6.85363	0.328607
$436$	10.9786	0.525778
$437$	−3.27131	−0.156488
$438$	23.7894	1.13670
$439$	−22.3503	−1.06672	−0.533360	−	0.845888i	$-0.679071\pi$
$439$	−22.3503	−1.06672	−0.533360	+	0.845888i	$0.679071\pi$
$440$	−9.95715	−0.474689
$441$	7.68585	0.365993
$442$	26.8009	1.27479
$443$	23.4145	1.11246	0.556229	−	0.831029i	$-0.312248\pi$
$443$	23.4145	1.11246	0.556229	+	0.831029i	$0.312248\pi$
$444$	−28.6002	−1.35730
$445$	−2.87819	−0.136439
$446$	21.4868	1.01743
$447$	−12.5426	−0.593245
$448$	46.6577	2.20437
$449$	−33.2713	−1.57017	−0.785085	−	0.619388i	$-0.787381\pi$
$449$	−33.2713	−1.57017	−0.785085	+	0.619388i	$0.787381\pi$
$450$	−10.0073	−0.471751
$451$	−3.34292	−0.157412
$452$	−26.9786	−1.26897
$453$	−2.62831	−0.123489
$454$	0.618442	0.0290249
$455$	−10.2927	−0.482531
$456$	−4.00000	−0.187317
$457$	−26.3931	−1.23462	−0.617309	−	0.786721i	$-0.711777\pi$
$457$	−26.3931	−1.23462	−0.617309	+	0.786721i	$0.711777\pi$
$458$	16.8782	0.788666
$459$	−3.63565	−0.169698
$460$	−8.49977	−0.396304
$461$	2.20077	0.102500	0.0512500	−	0.998686i	$-0.483679\pi$
$461$	2.20077	0.102500	0.0512500	+	0.998686i	$0.483679\pi$
$462$	−30.0147	−1.39641
$463$	24.5328	1.14013	0.570067	−	0.821598i	$-0.306917\pi$
$463$	24.5328	1.14013	0.570067	+	0.821598i	$0.306917\pi$
$464$	−9.60688	−0.445988
$465$	−8.41767	−0.390360
$466$	37.8568	1.75368
$467$	−22.9357	−1.06134	−0.530670	−	0.847579i	$-0.678059\pi$
$467$	−22.9357	−1.06134	−0.530670	+	0.847579i	$0.678059\pi$
$468$	10.9786	0.507485
$469$	−23.3780	−1.07949
$470$	16.6430	0.767684
$471$	−20.1579	−0.928828
$472$	−40.7005	−1.87339
$473$	−39.3148	−1.80770
$474$	8.45065	0.388151
$475$	−4.89648	−0.224666
$476$	−48.6148	−2.22826
$477$	1.60688	0.0735742
$478$	−36.7005	−1.67864
$479$	41.6075	1.90110	0.950548	−	0.310579i	$-0.100523\pi$
$479$	41.6075	1.90110	0.950548	+	0.310579i	$0.100523\pi$
$480$	3.56404	0.162675
$481$	25.7894	1.17589
$482$	24.7104	1.12553
$483$	−10.9357	−0.497593
$484$	0.611096	0.0277771
$485$	−6.43596	−0.292242
$486$	−2.34292	−0.106277
$487$	40.4036	1.83086	0.915431	−	0.402475i	$-0.131850\pi$
$487$	40.4036	1.83086	0.915431	+	0.402475i	$0.131850\pi$
$488$	−14.6430	−0.662857
$489$	−18.2541	−0.825479
$490$	15.3717	0.694422
$491$	−16.6184	−0.749980	−0.374990	−	0.927029i	$-0.622354\pi$
$491$	−16.6184	−0.749980	−0.374990	+	0.927029i	$0.622354\pi$
$492$	−3.48929	−0.157309
$493$	−29.1898	−1.31464
$494$	8.45065	0.380213
$495$	−2.85363	−0.128261
$496$	11.7992	0.529801
$497$	33.1758	1.48814
$498$	23.7220	1.06301
$499$	−0.921039	−0.0412314	−0.0206157	−	0.999787i	$-0.506563\pi$
$499$	−0.921039	−0.0412314	−0.0206157	+	0.999787i	$0.506563\pi$
$500$	−27.6153	−1.23499
$501$	−1.31415	−0.0587121
$502$	50.2070	2.24085
$503$	18.9217	0.843675	0.421837	−	0.906671i	$-0.361385\pi$
$503$	18.9217	0.843675	0.421837	+	0.906671i	$0.361385\pi$
$504$	−13.3717	−0.595623
$505$	4.27444	0.190210
$506$	22.3503	0.993591
$507$	3.10038	0.137693
$508$	−42.0722	−1.86665
$509$	27.8855	1.23600	0.618002	−	0.786176i	$-0.287942\pi$
$509$	27.8855	1.23600	0.618002	+	0.786176i	$0.287942\pi$
$510$	−7.27131	−0.321979
$511$	38.9112	1.72133
$512$	−13.3461	−0.589818
$513$	−1.14637	−0.0506133
$514$	−16.6529	−0.734526
$515$	1.03971	0.0458151
$516$	−41.0361	−1.80651
$517$	−27.8181	−1.22344
$518$	−73.5934	−3.23351
$519$	−1.18921	−0.0522006
$520$	9.37169	0.410976
$521$	9.93681	0.435339	0.217670	−	0.976022i	$-0.430154\pi$
$521$	9.93681	0.435339	0.217670	+	0.976022i	$0.430154\pi$
$522$	−18.8108	−0.823326
$523$	−35.9718	−1.57294	−0.786470	−	0.617629i	$-0.788093\pi$
$523$	−35.9718	−1.57294	−0.786470	+	0.617629i	$0.788093\pi$
$524$	42.6577	1.86351
$525$	−16.3686	−0.714382
$526$	−23.3618	−1.01862
$527$	35.8511	1.56170
$528$	4.00000	0.174078
$529$	−14.8568	−0.645947
$530$	3.21377	0.139597
$531$	−11.6644	−0.506193
$532$	−15.3288	−0.664590
$533$	3.14637	0.136284
$534$	7.89962	0.341850
$535$	11.7501	0.508002
$536$	21.2860	0.919415
$537$	19.0073	0.820228
$538$	−1.60688	−0.0692777
$539$	−25.6932	−1.10668
$540$	−2.97858	−0.128178
$541$	9.79923	0.421302	0.210651	−	0.977561i	$-0.432442\pi$
$541$	9.79923	0.421302	0.210651	+	0.977561i	$0.432442\pi$
$542$	32.3404	1.38914
$543$	−10.3931	−0.446011
$544$	−15.1793	−0.650809
$545$	2.68585	0.115049
$546$	28.2499	1.20898
$547$	−33.8223	−1.44614	−0.723070	−	0.690775i	$-0.757269\pi$
$547$	−33.8223	−1.44614	−0.723070	+	0.690775i	$0.757269\pi$
$548$	26.8438	1.14671
$549$	−4.19656	−0.179105
$550$	33.4538	1.42647
$551$	−9.20390	−0.392099
$552$	9.95715	0.423805
$553$	13.8223	0.587786
$554$	52.0393	2.21094
$555$	−6.99686	−0.297000
$556$	−62.8585	−2.66579
$557$	−20.9217	−0.886479	−0.443239	−	0.896403i	$-0.646171\pi$
$557$	−20.9217	−0.886479	−0.443239	+	0.896403i	$0.646171\pi$
$558$	23.1035	0.978050
$559$	37.0031	1.56507
$560$	−3.91431	−0.165410
$561$	12.1537	0.513130
$562$	10.8108	0.456026
$563$	−28.6289	−1.20657	−0.603283	−	0.797527i	$-0.706141\pi$
$563$	−28.6289	−1.20657	−0.603283	+	0.797527i	$0.706141\pi$
$564$	−29.0361	−1.22264
$565$	−6.60015	−0.277671
$566$	21.3963	0.899352
$567$	−3.83221	−0.160938
$568$	−30.2070	−1.26746
$569$	−20.9603	−0.878701	−0.439351	−	0.898316i	$-0.644791\pi$
$569$	−20.9603	−0.878701	−0.439351	+	0.898316i	$0.644791\pi$
$570$	−2.29273	−0.0960319
$571$	−13.2860	−0.556002	−0.278001	−	0.960581i	$-0.589672\pi$
$571$	−13.2860	−0.556002	−0.278001	+	0.960581i	$0.589672\pi$
$572$	−36.7005	−1.53453
$573$	6.87819	0.287341
$574$	−8.97858	−0.374759
$575$	12.1888	0.508306
$576$	−12.1751	−0.507297
$577$	1.35700	0.0564926	0.0282463	−	0.999601i	$-0.491008\pi$
$577$	1.35700	0.0564926	0.0282463	+	0.999601i	$0.491008\pi$
$578$	−8.86098	−0.368568
$579$	19.5970	0.814424
$580$	−23.9143	−0.992988
$581$	38.8009	1.60973
$582$	17.6644	0.732214
$583$	−5.37169	−0.222473
$584$	−35.4292	−1.46607
$585$	2.68585	0.111046
$586$	54.9259	2.26897
$587$	29.9431	1.23588	0.617942	−	0.786224i	$-0.287967\pi$
$587$	29.9431	1.23588	0.617942	+	0.786224i	$0.287967\pi$
$588$	−26.8181	−1.10596
$589$	11.3043	0.465785
$590$	−23.3288	−0.960433
$591$	24.7679	1.01882
$592$	9.80765	0.403092
$593$	−6.01408	−0.246969	−0.123484	−	0.992347i	$-0.539407\pi$
$593$	−6.01408	−0.246969	−0.123484	+	0.992347i	$0.539407\pi$
$594$	7.83221	0.321360
$595$	−11.8933	−0.487580
$596$	43.7648	1.79268
$597$	−12.8108	−0.524311
$598$	−21.0361	−0.860231
$599$	30.0477	1.22771	0.613857	−	0.789417i	$-0.289617\pi$
$599$	30.0477	1.22771	0.613857	+	0.789417i	$0.289617\pi$
$600$	14.9038	0.608446
$601$	−38.1642	−1.55675	−0.778375	−	0.627800i	$-0.783956\pi$
$601$	−38.1642	−1.55675	−0.778375	+	0.627800i	$0.783956\pi$
$602$	−105.593	−4.30367
$603$	6.10038	0.248427
$604$	9.17092	0.373160
$605$	0.149501	0.00607808
$606$	−11.7318	−0.476573
$607$	35.7220	1.44991	0.724955	−	0.688796i	$-0.241861\pi$
$607$	35.7220	1.44991	0.724955	+	0.688796i	$0.241861\pi$
$608$	−4.78623	−0.194107
$609$	−30.7679	−1.24678
$610$	−8.39312	−0.339827
$611$	26.1825	1.05923
$612$	12.6858	0.512795
$613$	−8.68585	−0.350818	−0.175409	−	0.984496i	$-0.556125\pi$
$613$	−8.68585	−0.350818	−0.175409	+	0.984496i	$0.556125\pi$
$614$	42.7679	1.72597
$615$	−0.853635	−0.0344219
$616$	44.7005	1.80104
$617$	25.4391	1.02414	0.512070	−	0.858944i	$-0.328879\pi$
$617$	25.4391	1.02414	0.512070	+	0.858944i	$0.328879\pi$
$618$	−2.85363	−0.114790
$619$	4.28852	0.172370	0.0861851	−	0.996279i	$-0.472532\pi$
$619$	4.28852	0.172370	0.0861851	+	0.996279i	$0.472532\pi$
$620$	29.3717	1.17960
$621$	2.85363	0.114512
$622$	34.8585	1.39770
$623$	12.9210	0.517670
$624$	−3.76481	−0.150713
$625$	14.6006	0.584025
$626$	−62.0294	−2.47919
$627$	3.83221	0.153044
$628$	70.3368	2.80674
$629$	29.7998	1.18820
$630$	−7.66442	−0.305358
$631$	−3.17513	−0.126400	−0.0632001	−	0.998001i	$-0.520131\pi$
$631$	−3.17513	−0.126400	−0.0632001	+	0.998001i	$0.520131\pi$
$632$	−12.5855	−0.500623
$633$	−3.66442	−0.145648
$634$	69.4685	2.75895
$635$	−10.2927	−0.408455
$636$	−5.60688	−0.222327
$637$	24.1825	0.958145
$638$	62.8830	2.48956
$639$	−8.65708	−0.342469
$640$	−17.2222	−0.680767
$641$	−21.0565	−0.831680	−0.415840	−	0.909438i	$-0.636512\pi$
$641$	−21.0565	−0.831680	−0.415840	+	0.909438i	$0.636512\pi$
$642$	−32.2499	−1.27280
$643$	−18.2070	−0.718016	−0.359008	−	0.933335i	$-0.616885\pi$
$643$	−18.2070	−0.718016	−0.359008	+	0.933335i	$0.616885\pi$
$644$	38.1579	1.50363
$645$	−10.0393	−0.395295
$646$	9.76481	0.384191
$647$	−19.6890	−0.774054	−0.387027	−	0.922068i	$-0.626498\pi$
$647$	−19.6890	−0.774054	−0.387027	+	0.922068i	$0.626498\pi$
$648$	3.48929	0.137072
$649$	38.9933	1.53062
$650$	−31.4868	−1.23501
$651$	37.7894	1.48108
$652$	63.6938	2.49444
$653$	−2.80031	−0.109584	−0.0547922	−	0.998498i	$-0.517450\pi$
$653$	−2.80031	−0.109584	−0.0547922	+	0.998498i	$0.517450\pi$
$654$	−7.37169	−0.288256
$655$	10.4360	0.407767
$656$	1.19656	0.0467177
$657$	−10.1537	−0.396134
$658$	−74.7152	−2.91270
$659$	−7.07896	−0.275757	−0.137879	−	0.990449i	$-0.544028\pi$
$659$	−7.07896	−0.275757	−0.137879	+	0.990449i	$0.544028\pi$
$660$	9.95715	0.387582
$661$	−15.6791	−0.609847	−0.304923	−	0.952377i	$-0.598631\pi$
$661$	−15.6791	−0.609847	−0.304923	+	0.952377i	$0.598631\pi$
$662$	6.10038	0.237098
$663$	−11.4391	−0.444258
$664$	−35.3288	−1.37103
$665$	−3.75011	−0.145423
$666$	19.2039	0.744136
$667$	22.9112	0.887124
$668$	4.58546	0.177417
$669$	−9.17092	−0.354568
$670$	12.2008	0.471357
$671$	14.0288	0.541575
$672$	−16.0000	−0.617213
$673$	−12.2070	−0.470547	−0.235273	−	0.971929i	$-0.575599\pi$
$673$	−12.2070	−0.470547	−0.235273	+	0.971929i	$0.575599\pi$
$674$	−3.53948	−0.136336
$675$	4.27131	0.164403
$676$	−10.8181	−0.416082
$677$	−39.7121	−1.52626	−0.763130	−	0.646245i	$-0.776338\pi$
$677$	−39.7121	−1.52626	−0.763130	+	0.646245i	$0.776338\pi$
$678$	18.1151	0.695705
$679$	28.8929	1.10881
$680$	10.8291	0.415276
$681$	−0.263962	−0.0101150
$682$	−77.2333	−2.95742
$683$	9.22846	0.353117	0.176559	−	0.984290i	$-0.443503\pi$
$683$	9.22846	0.353117	0.176559	+	0.984290i	$0.443503\pi$
$684$	4.00000	0.152944
$685$	6.56717	0.250919
$686$	−6.15792	−0.235111
$687$	−7.20390	−0.274846
$688$	14.0722	0.536499
$689$	5.05585	0.192612
$690$	5.70727	0.217272
$691$	−4.75325	−0.180822	−0.0904111	−	0.995905i	$-0.528818\pi$
$691$	−4.75325	−0.180822	−0.0904111	+	0.995905i	$0.528818\pi$
$692$	4.14950	0.157740
$693$	12.8108	0.486642
$694$	−33.4538	−1.26989
$695$	−15.3780	−0.583319
$696$	28.0147	1.06189
$697$	3.63565	0.137710
$698$	−38.0821	−1.44143
$699$	−16.1579	−0.611149
$700$	57.1146	2.15873
$701$	−21.0790	−0.796141	−0.398071	−	0.917355i	$-0.630320\pi$
$701$	−21.0790	−0.796141	−0.398071	+	0.917355i	$0.630320\pi$
$702$	−7.37169	−0.278227
$703$	9.39625	0.354386
$704$	40.7005	1.53396
$705$	−7.10352	−0.267534
$706$	26.1726	0.985019
$707$	−19.1892	−0.721685
$708$	40.7005	1.52962
$709$	42.7497	1.60550	0.802749	−	0.596318i	$-0.203370\pi$
$709$	42.7497	1.60550	0.802749	+	0.596318i	$0.203370\pi$
$710$	−17.3142	−0.649789
$711$	−3.60688	−0.135269
$712$	−11.7648	−0.440905
$713$	−28.1396	−1.05384
$714$	32.6430	1.22163
$715$	−8.97858	−0.335780
$716$	−66.3221	−2.47857
$717$	15.6644	0.584999
$718$	−38.0147	−1.41870
$719$	−7.00735	−0.261330	−0.130665	−	0.991427i	$-0.541711\pi$
$719$	−7.00735	−0.261330	−0.130665	+	0.991427i	$0.541711\pi$
$720$	1.02142	0.0380662
$721$	−4.66756	−0.173829
$722$	−41.4366	−1.54211
$723$	−10.5468	−0.392241
$724$	36.2646	1.34776
$725$	34.2933	1.27362
$726$	−0.410327	−0.0152287
$727$	−33.3963	−1.23860	−0.619299	−	0.785155i	$-0.712583\pi$
$727$	−33.3963	−1.23860	−0.619299	+	0.785155i	$0.712583\pi$
$728$	−42.0722	−1.55930
$729$	1.00000	0.0370370
$730$	−20.3074	−0.751611
$731$	42.7575	1.58144
$732$	14.6430	0.541221
$733$	13.1176	0.484509	0.242255	−	0.970213i	$-0.422113\pi$
$733$	13.1176	0.484509	0.242255	+	0.970213i	$0.422113\pi$
$734$	1.01156	0.0373373
$735$	−6.56090	−0.242003
$736$	11.9143	0.439167
$737$	−20.3931	−0.751190
$738$	2.34292	0.0862442
$739$	−47.2902	−1.73960	−0.869799	−	0.493406i	$-0.835752\pi$
$739$	−47.2902	−1.73960	−0.869799	+	0.493406i	$0.835752\pi$
$740$	24.4141	0.897479
$741$	−3.60688	−0.132502
$742$	−14.4275	−0.529652
$743$	−18.5510	−0.680572	−0.340286	−	0.940322i	$-0.610524\pi$
$743$	−18.5510	−0.680572	−0.340286	+	0.940322i	$0.610524\pi$
$744$	−34.4078	−1.26145
$745$	10.7068	0.392267
$746$	−8.59533	−0.314697
$747$	−10.1249	−0.370452
$748$	−42.4078	−1.55058
$749$	−52.7497	−1.92743
$750$	18.5426	0.677081
$751$	−13.6791	−0.499158	−0.249579	−	0.968354i	$-0.580292\pi$
$751$	−13.6791	−0.499158	−0.249579	+	0.968354i	$0.580292\pi$
$752$	9.95715	0.363100
$753$	−21.4292	−0.780925
$754$	−59.1856	−2.15541
$755$	2.24361	0.0816535
$756$	13.3717	0.486324
$757$	−23.0031	−0.836063	−0.418032	−	0.908432i	$-0.637280\pi$
$757$	−23.0031	−0.836063	−0.418032	+	0.908432i	$0.637280\pi$
$758$	−23.1940	−0.842445
$759$	−9.53948	−0.346261
$760$	3.41454	0.123858
$761$	20.4851	0.742583	0.371292	−	0.928516i	$-0.378915\pi$
$761$	20.4851	0.742583	0.371292	+	0.928516i	$0.378915\pi$
$762$	28.2499	1.02339
$763$	−12.0575	−0.436512
$764$	−24.0000	−0.868290
$765$	3.10352	0.112208
$766$	−73.3681	−2.65090
$767$	−36.7005	−1.32518
$768$	22.9185	0.827001
$769$	−17.5107	−0.631452	−0.315726	−	0.948850i	$-0.602248\pi$
$769$	−17.5107	−0.631452	−0.315726	+	0.948850i	$0.602248\pi$
$770$	25.6216	0.923338
$771$	7.10773	0.255979
$772$	−68.3797	−2.46104
$773$	−31.8083	−1.14406	−0.572032	−	0.820231i	$-0.693845\pi$
$773$	−31.8083	−1.14406	−0.572032	+	0.820231i	$0.693845\pi$
$774$	27.5542	0.990414
$775$	−42.1193	−1.51297
$776$	−26.3074	−0.944381
$777$	31.4109	1.12686
$778$	17.3288	0.621269
$779$	1.14637	0.0410728
$780$	−9.37169	−0.335560
$781$	28.9399	1.03555
$782$	−24.3074	−0.869232
$783$	8.02877	0.286925
$784$	9.19656	0.328448
$785$	17.2075	0.614162
$786$	−28.6430	−1.02166
$787$	14.5040	0.517011	0.258506	−	0.966010i	$-0.416770\pi$
$787$	14.5040	0.517011	0.258506	+	0.966010i	$0.416770\pi$
$788$	−86.4225	−3.07867
$789$	9.97123	0.354985
$790$	−7.21377	−0.256654
$791$	29.6300	1.05352
$792$	−11.6644	−0.414477
$793$	−13.2039	−0.468884
$794$	27.9572	0.992162
$795$	−1.37169	−0.0486489
$796$	44.7005	1.58437
$797$	50.3650	1.78402	0.892009	−	0.452017i	$-0.149295\pi$
$797$	50.3650	1.78402	0.892009	+	0.452017i	$0.149295\pi$
$798$	10.2927	0.364359
$799$	30.2541	1.07031
$800$	17.8333	0.630502
$801$	−3.37169	−0.119133
$802$	−36.9357	−1.30425
$803$	33.9431	1.19783
$804$	−21.2860	−0.750699
$805$	9.33512	0.329020
$806$	72.6921	2.56047
$807$	0.685846	0.0241429
$808$	17.4721	0.614666
$809$	35.2369	1.23886	0.619431	−	0.785051i	$-0.287363\pi$
$809$	35.2369	1.23886	0.619431	+	0.785051i	$0.287363\pi$
$810$	2.00000	0.0702728
$811$	−13.8610	−0.486725	−0.243362	−	0.969935i	$-0.578250\pi$
$811$	−13.8610	−0.486725	−0.243362	+	0.969935i	$0.578250\pi$
$812$	107.358	3.76754
$813$	−13.8034	−0.484108
$814$	−64.1972	−2.25011
$815$	15.5823	0.545825
$816$	−4.35027	−0.152290
$817$	13.4819	0.471673
$818$	−17.6974	−0.618775
$819$	−12.0575	−0.421324
$820$	2.97858	0.104016
$821$	7.20390	0.251418	0.125709	−	0.992067i	$-0.459879\pi$
$821$	7.20390	0.251418	0.125709	+	0.992067i	$0.459879\pi$
$822$	−18.0246	−0.628679
$823$	−9.81392	−0.342092	−0.171046	−	0.985263i	$-0.554715\pi$
$823$	−9.81392	−0.342092	−0.171046	+	0.985263i	$0.554715\pi$
$824$	4.24989	0.148052
$825$	−14.2787	−0.497119
$826$	104.730	3.64402
$827$	24.8788	0.865121	0.432560	−	0.901605i	$-0.357610\pi$
$827$	24.8788	0.865121	0.432560	+	0.901605i	$0.357610\pi$
$828$	−9.95715	−0.346035
$829$	13.3759	0.464564	0.232282	−	0.972648i	$-0.425381\pi$
$829$	13.3759	0.464564	0.232282	+	0.972648i	$0.425381\pi$
$830$	−20.2499	−0.702884
$831$	−22.2113	−0.770500
$832$	−38.3074	−1.32807
$833$	27.9431	0.968170
$834$	42.2070	1.46151
$835$	1.12181	0.0388217
$836$	−13.3717	−0.462470
$837$	−9.86098	−0.340845
$838$	22.4851	0.776734
$839$	−12.9210	−0.446084	−0.223042	−	0.974809i	$-0.571599\pi$
$839$	−12.9210	−0.446084	−0.223042	+	0.974809i	$0.571599\pi$
$840$	11.4145	0.393839
$841$	35.4611	1.22280
$842$	28.9357	0.997191
$843$	−4.61423	−0.158923
$844$	12.7862	0.440120
$845$	−2.64659	−0.0910456
$846$	19.4966	0.670308
$847$	−0.671153	−0.0230611
$848$	1.92273	0.0660268
$849$	−9.13229	−0.313419
$850$	−36.3832	−1.24794
$851$	−23.3900	−0.801798
$852$	30.2070	1.03488
$853$	16.4851	0.564438	0.282219	−	0.959350i	$-0.408929\pi$
$853$	16.4851	0.564438	0.282219	+	0.959350i	$0.408929\pi$
$854$	37.6791	1.28935
$855$	0.978577	0.0334666
$856$	48.0294	1.64161
$857$	−11.5149	−0.393342	−0.196671	−	0.980470i	$-0.563013\pi$
$857$	−11.5149	−0.393342	−0.196671	+	0.980470i	$0.563013\pi$
$858$	24.6430	0.841298
$859$	−28.7392	−0.980568	−0.490284	−	0.871563i	$-0.663107\pi$
$859$	−28.7392	−0.980568	−0.490284	+	0.871563i	$0.663107\pi$
$860$	35.0298	1.19451
$861$	3.83221	0.130601
$862$	12.0575	0.410681
$863$	51.1611	1.74154	0.870771	−	0.491688i	$-0.163620\pi$
$863$	51.1611	1.74154	0.870771	+	0.491688i	$0.163620\pi$
$864$	4.17513	0.142041
$865$	1.01515	0.0345162
$866$	40.0393	1.36059
$867$	3.78202	0.128444
$868$	−131.858	−4.47555
$869$	12.0575	0.409024
$870$	16.0575	0.544402
$871$	19.1940	0.650365
$872$	10.9786	0.371782
$873$	−7.53948	−0.255173
$874$	−7.66442	−0.259253
$875$	30.3293	1.02532
$876$	35.4292	1.19704
$877$	34.3179	1.15883	0.579417	−	0.815031i	$-0.303280\pi$
$877$	34.3179	1.15883	0.579417	+	0.815031i	$0.303280\pi$
$878$	−52.3650	−1.76723
$879$	−23.4433	−0.790723
$880$	−3.41454	−0.115104
$881$	−38.9786	−1.31322	−0.656611	−	0.754230i	$-0.728011\pi$
$881$	−38.9786	−1.31322	−0.656611	+	0.754230i	$0.728011\pi$
$882$	18.0073	0.606339
$883$	−28.2253	−0.949858	−0.474929	−	0.880024i	$-0.657526\pi$
$883$	−28.2253	−0.949858	−0.474929	+	0.880024i	$0.657526\pi$
$884$	39.9143	1.34246
$885$	9.95715	0.334706
$886$	54.8585	1.84301
$887$	−30.1438	−1.01213	−0.506066	−	0.862495i	$-0.668901\pi$
$887$	−30.1438	−1.01213	−0.506066	+	0.862495i	$0.668901\pi$
$888$	−28.6002	−0.959758
$889$	46.2070	1.54973
$890$	−6.74338	−0.226039
$891$	−3.34292	−0.111992
$892$	32.0000	1.07144
$893$	9.53948	0.319227
$894$	−29.3864	−0.982828
$895$	−16.2253	−0.542353
$896$	77.3154	2.58293
$897$	8.97858	0.299786
$898$	−77.9521	−2.60130
$899$	−79.1715	−2.64052
$900$	−14.9038	−0.496794
$901$	5.84208	0.194628
$902$	−7.83221	−0.260784
$903$	45.0691	1.49981
$904$	−26.9786	−0.897294
$905$	8.87192	0.294913
$906$	−6.15792	−0.204583
$907$	41.3435	1.37279	0.686395	−	0.727229i	$-0.259192\pi$
$907$	41.3435	1.37279	0.686395	+	0.727229i	$0.259192\pi$
$908$	0.921039	0.0305657
$909$	5.00735	0.166083
$910$	−24.1151	−0.799407
$911$	−1.23206	−0.0408199	−0.0204099	−	0.999792i	$-0.506497\pi$
$911$	−1.23206	−0.0408199	−0.0204099	+	0.999792i	$0.506497\pi$
$912$	−1.37169	−0.0454213
$913$	33.8469	1.12017
$914$	−61.8370	−2.04539
$915$	3.58233	0.118428
$916$	25.1365	0.830533
$917$	−46.8500	−1.54712
$918$	−8.51806	−0.281138
$919$	−21.4047	−0.706075	−0.353037	−	0.935609i	$-0.614851\pi$
$919$	−21.4047	−0.706075	−0.353037	+	0.935609i	$0.614851\pi$
$920$	−8.49977	−0.280229
$921$	−18.2541	−0.601493
$922$	5.15623	0.169811
$923$	−27.2383	−0.896560
$924$	−44.7005	−1.47054
$925$	−35.0100	−1.15112
$926$	57.4783	1.88886
$927$	1.21798	0.0400037
$928$	33.5212	1.10039
$929$	32.8150	1.07663	0.538313	−	0.842745i	$-0.319062\pi$
$929$	32.8150	1.07663	0.538313	+	0.842745i	$0.319062\pi$
$930$	−19.7220	−0.646709
$931$	8.81079	0.288762
$932$	56.3797	1.84678
$933$	−14.8782	−0.487090
$934$	−53.7367	−1.75832
$935$	−10.3748	−0.339293
$936$	10.9786	0.358846
$937$	27.4783	0.897678	0.448839	−	0.893613i	$-0.351838\pi$
$937$	27.4783	0.897678	0.448839	+	0.893613i	$0.351838\pi$
$938$	−54.7728	−1.78839
$939$	26.4752	0.863986
$940$	24.7862	0.808438
$941$	38.1151	1.24252	0.621258	−	0.783606i	$-0.286622\pi$
$941$	38.1151	1.24252	0.621258	+	0.783606i	$0.286622\pi$
$942$	−47.2285	−1.53879
$943$	−2.85363	−0.0929271
$944$	−13.9572	−0.454267
$945$	3.27131	0.106416
$946$	−92.1115	−2.99480
$947$	−9.85050	−0.320098	−0.160049	−	0.987109i	$-0.551165\pi$
$947$	−9.85050	−0.320098	−0.160049	+	0.987109i	$0.551165\pi$
$948$	12.5855	0.408757
$949$	−31.9473	−1.03705
$950$	−11.4721	−0.372203
$951$	−29.6503	−0.961478
$952$	−48.6148	−1.57562
$953$	3.92417	0.127116	0.0635582	−	0.997978i	$-0.479755\pi$
$953$	3.92417	0.127116	0.0635582	+	0.997978i	$0.479755\pi$
$954$	3.76481	0.121890
$955$	−5.87146	−0.189996
$956$	−54.6577	−1.76776
$957$	−26.8396	−0.867600
$958$	97.4832	3.14954
$959$	−29.4819	−0.952022
$960$	10.3931	0.335436
$961$	66.2389	2.13674
$962$	60.4225	1.94810
$963$	13.7648	0.443565
$964$	36.8009	1.18528
$965$	−16.7287	−0.538516
$966$	−25.6216	−0.824361
$967$	−40.3074	−1.29620	−0.648100	−	0.761556i	$-0.724436\pi$
$967$	−40.3074	−1.29620	−0.648100	+	0.761556i	$0.724436\pi$
$968$	0.611096	0.0196414
$969$	−4.16779	−0.133889
$970$	−15.0790	−0.484156
$971$	37.8280	1.21396	0.606979	−	0.794718i	$-0.292381\pi$
$971$	37.8280	1.21396	0.606979	+	0.794718i	$0.292381\pi$
$972$	−3.48929	−0.111919
$973$	69.0361	2.21320
$974$	94.6625	3.03318
$975$	13.4391	0.430396
$976$	−5.02142	−0.160732
$977$	8.89227	0.284489	0.142244	−	0.989832i	$-0.454568\pi$
$977$	8.89227	0.284489	0.142244	+	0.989832i	$0.454568\pi$
$978$	−42.7679	−1.36757
$979$	11.2713	0.360233
$980$	22.8929	0.731286
$981$	3.14637	0.100456
$982$	−38.9357	−1.24249
$983$	−30.6247	−0.976777	−0.488388	−	0.872626i	$-0.662415\pi$
$983$	−30.6247	−0.976777	−0.488388	+	0.872626i	$0.662415\pi$
$984$	−3.48929	−0.111234
$985$	−21.1428	−0.673665
$986$	−68.3895	−2.17797
$987$	31.8898	1.01506
$988$	12.5855	0.400397
$989$	−33.5604	−1.06716
$990$	−6.68585	−0.212490
$991$	−53.1512	−1.68840	−0.844202	−	0.536026i	$-0.819925\pi$
$991$	−53.1512	−1.68840	−0.844202	+	0.536026i	$0.819925\pi$
$992$	−41.1709	−1.30718
$993$	−2.60375	−0.0826275
$994$	77.7282	2.46539
$995$	10.9357	0.346686
$996$	35.3288	1.11944
$997$	−44.5181	−1.40990	−0.704951	−	0.709256i	$-0.749031\pi$
$997$	−44.5181	−1.40990	−0.704951	+	0.709256i	$0.749031\pi$
$998$	−2.15792	−0.0683078
$999$	−8.19656	−0.259328

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.2.a.d.1.3	✓	3
3.2	odd	2		369.2.a.e.1.1		3
4.3	odd	2		1968.2.a.w.1.2		3
5.4	even	2		3075.2.a.t.1.1		3
7.6	odd	2		6027.2.a.s.1.3		3
8.3	odd	2		7872.2.a.bs.1.2		3
8.5	even	2		7872.2.a.bx.1.2		3
12.11	even	2		5904.2.a.bd.1.2		3
15.14	odd	2		9225.2.a.bx.1.3		3
41.40	even	2		5043.2.a.n.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.a.d.1.3	✓	3	1.1	even	1	trivial
369.2.a.e.1.1		3	3.2	odd	2
1968.2.a.w.1.2		3	4.3	odd	2
3075.2.a.t.1.1		3	5.4	even	2
5043.2.a.n.1.3		3	41.40	even	2
5904.2.a.bd.1.2		3	12.11	even	2
6027.2.a.s.1.3		3	7.6	odd	2
7872.2.a.bs.1.2		3	8.3	odd	2
7872.2.a.bx.1.2		3	8.5	even	2
9225.2.a.bx.1.3		3	15.14	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:	\( N \)	\(=\)	\( 123 = 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	123.a (trivial)

\(f(q)\)	\(=\)	\(q+2.34292 q^{2} -1.00000 q^{3} +3.48929 q^{4} +0.853635 q^{5} -2.34292 q^{6} -3.83221 q^{7} +3.48929 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.34292 q^{2} -1.00000 q^{3} +3.48929 q^{4} +0.853635 q^{5} -2.34292 q^{6} -3.83221 q^{7} +3.48929 q^{8} +1.00000 q^{9} +2.00000 q^{10} -3.34292 q^{11} -3.48929 q^{12} +3.14637 q^{13} -8.97858 q^{14} -0.853635 q^{15} +1.19656 q^{16} +3.63565 q^{17} +2.34292 q^{18} +1.14637 q^{19} +2.97858 q^{20} +3.83221 q^{21} -7.83221 q^{22} -2.85363 q^{23} -3.48929 q^{24} -4.27131 q^{25} +7.37169 q^{26} -1.00000 q^{27} -13.3717 q^{28} -8.02877 q^{29} -2.00000 q^{30} +9.86098 q^{31} -4.17513 q^{32} +3.34292 q^{33} +8.51806 q^{34} -3.27131 q^{35} +3.48929 q^{36} +8.19656 q^{37} +2.68585 q^{38} -3.14637 q^{39} +2.97858 q^{40} +1.00000 q^{41} +8.97858 q^{42} +11.7606 q^{43} -11.6644 q^{44} +0.853635 q^{45} -6.68585 q^{46} +8.32150 q^{47} -1.19656 q^{48} +7.68585 q^{49} -10.0073 q^{50} -3.63565 q^{51} +10.9786 q^{52} +1.60688 q^{53} -2.34292 q^{54} -2.85363 q^{55} -13.3717 q^{56} -1.14637 q^{57} -18.8108 q^{58} -11.6644 q^{59} -2.97858 q^{60} -4.19656 q^{61} +23.1035 q^{62} -3.83221 q^{63} -12.1751 q^{64} +2.68585 q^{65} +7.83221 q^{66} +6.10038 q^{67} +12.6858 q^{68} +2.85363 q^{69} -7.66442 q^{70} -8.65708 q^{71} +3.48929 q^{72} -10.1537 q^{73} +19.2039 q^{74} +4.27131 q^{75} +4.00000 q^{76} +12.8108 q^{77} -7.37169 q^{78} -3.60688 q^{79} +1.02142 q^{80} +1.00000 q^{81} +2.34292 q^{82} -10.1249 q^{83} +13.3717 q^{84} +3.10352 q^{85} +27.5542 q^{86} +8.02877 q^{87} -11.6644 q^{88} -3.37169 q^{89} +2.00000 q^{90} -12.0575 q^{91} -9.95715 q^{92} -9.86098 q^{93} +19.4966 q^{94} +0.978577 q^{95} +4.17513 q^{96} -7.53948 q^{97} +18.0073 q^{98} -3.34292 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 3 q^{3} + 3 q^{4} + 4 q^{5} - q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + q^2 - 3 * q^3 + 3 * q^4 + 4 * q^5 - q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} - 3 q^{3} + 3 q^{4} + 4 q^{5} - q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} + 6 q^{10} - 4 q^{11} - 3 q^{12} + 8 q^{13} - 12 q^{14} - 4 q^{15} - q^{16} + 2 q^{17} + q^{18} + 2 q^{19} - 6 q^{20} - 2 q^{21} - 10 q^{22} - 10 q^{23} - 3 q^{24} + 5 q^{25} - 2 q^{26} - 3 q^{27} - 16 q^{28} - 6 q^{29} - 6 q^{30} - 2 q^{31} + 7 q^{32} + 4 q^{33} + 8 q^{35} + 3 q^{36} + 20 q^{37} - 4 q^{38} - 8 q^{39} - 6 q^{40} + 3 q^{41} + 12 q^{42} + 10 q^{43} - 8 q^{44} + 4 q^{45} - 8 q^{46} + 4 q^{47} + q^{48} + 11 q^{49} + 3 q^{50} - 2 q^{51} + 18 q^{52} + 14 q^{53} - q^{54} - 10 q^{55} - 16 q^{56} - 2 q^{57} - 28 q^{58} - 8 q^{59} + 6 q^{60} - 8 q^{61} + 38 q^{62} + 2 q^{63} - 17 q^{64} - 4 q^{65} + 10 q^{66} + 12 q^{67} + 26 q^{68} + 10 q^{69} + 4 q^{70} - 32 q^{71} + 3 q^{72} + 4 q^{73} + 20 q^{74} - 5 q^{75} + 12 q^{76} + 10 q^{77} + 2 q^{78} - 20 q^{79} + 18 q^{80} + 3 q^{81} + q^{82} - 14 q^{83} + 16 q^{84} - 22 q^{85} + 6 q^{86} + 6 q^{87} - 8 q^{88} + 14 q^{89} + 6 q^{90} + 2 q^{93} + 18 q^{94} - 12 q^{95} - 7 q^{96} - 12 q^{97} + 21 q^{98} - 4 q^{99}+O(q^{100}) \) 3 * q + q^2 - 3 * q^3 + 3 * q^4 + 4 * q^5 - q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 + 6 * q^10 - 4 * q^11 - 3 * q^12 + 8 * q^13 - 12 * q^14 - 4 * q^15 - q^16 + 2 * q^17 + q^18 + 2 * q^19 - 6 * q^20 - 2 * q^21 - 10 * q^22 - 10 * q^23 - 3 * q^24 + 5 * q^25 - 2 * q^26 - 3 * q^27 - 16 * q^28 - 6 * q^29 - 6 * q^30 - 2 * q^31 + 7 * q^32 + 4 * q^33 + 8 * q^35 + 3 * q^36 + 20 * q^37 - 4 * q^38 - 8 * q^39 - 6 * q^40 + 3 * q^41 + 12 * q^42 + 10 * q^43 - 8 * q^44 + 4 * q^45 - 8 * q^46 + 4 * q^47 + q^48 + 11 * q^49 + 3 * q^50 - 2 * q^51 + 18 * q^52 + 14 * q^53 - q^54 - 10 * q^55 - 16 * q^56 - 2 * q^57 - 28 * q^58 - 8 * q^59 + 6 * q^60 - 8 * q^61 + 38 * q^62 + 2 * q^63 - 17 * q^64 - 4 * q^65 + 10 * q^66 + 12 * q^67 + 26 * q^68 + 10 * q^69 + 4 * q^70 - 32 * q^71 + 3 * q^72 + 4 * q^73 + 20 * q^74 - 5 * q^75 + 12 * q^76 + 10 * q^77 + 2 * q^78 - 20 * q^79 + 18 * q^80 + 3 * q^81 + q^82 - 14 * q^83 + 16 * q^84 - 22 * q^85 + 6 * q^86 + 6 * q^87 - 8 * q^88 + 14 * q^89 + 6 * q^90 + 2 * q^93 + 18 * q^94 - 12 * q^95 - 7 * q^96 - 12 * q^97 + 21 * q^98 - 4 * q^99

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