LMFDB - Embedded newform 1045.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 28x^{4} - 17x^{3} - 28x^{2} + 6x + 8 $ x^8 - 2x^7 - 9x^6 + 12x^5 + 28x^4 - 17x^3 - 28x^2 + 6*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.649219$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.64922 q^{2} -2.98027 q^{3} +0.719922 q^{4} +1.00000 q^{5} +4.91512 q^{6} -5.09280 q^{7} +2.11113 q^{8} +5.88203 q^{9} +O(q^{10})$	\(q-1.64922 q^{2} -2.98027 q^{3} +0.719922 q^{4} +1.00000 q^{5} +4.91512 q^{6} -5.09280 q^{7} +2.11113 q^{8} +5.88203 q^{9} -1.64922 q^{10} +1.00000 q^{11} -2.14556 q^{12} -6.30960 q^{13} +8.39913 q^{14} -2.98027 q^{15} -4.92156 q^{16} +6.73010 q^{17} -9.70075 q^{18} -1.00000 q^{19} +0.719922 q^{20} +15.1779 q^{21} -1.64922 q^{22} +4.03742 q^{23} -6.29174 q^{24} +1.00000 q^{25} +10.4059 q^{26} -8.58923 q^{27} -3.66641 q^{28} +5.50320 q^{29} +4.91512 q^{30} +4.01900 q^{31} +3.89446 q^{32} -2.98027 q^{33} -11.0994 q^{34} -5.09280 q^{35} +4.23460 q^{36} +3.62306 q^{37} +1.64922 q^{38} +18.8043 q^{39} +2.11113 q^{40} -3.31685 q^{41} -25.0317 q^{42} -2.86498 q^{43} +0.719922 q^{44} +5.88203 q^{45} -6.65859 q^{46} +6.37629 q^{47} +14.6676 q^{48} +18.9366 q^{49} -1.64922 q^{50} -20.0575 q^{51} -4.54242 q^{52} +3.60804 q^{53} +14.1655 q^{54} +1.00000 q^{55} -10.7515 q^{56} +2.98027 q^{57} -9.07597 q^{58} -13.9464 q^{59} -2.14556 q^{60} -11.7640 q^{61} -6.62822 q^{62} -29.9560 q^{63} +3.42029 q^{64} -6.30960 q^{65} +4.91512 q^{66} -3.05805 q^{67} +4.84515 q^{68} -12.0326 q^{69} +8.39913 q^{70} +4.13233 q^{71} +12.4177 q^{72} +0.708095 q^{73} -5.97522 q^{74} -2.98027 q^{75} -0.719922 q^{76} -5.09280 q^{77} -31.0124 q^{78} -10.0436 q^{79} -4.92156 q^{80} +7.95216 q^{81} +5.47022 q^{82} -1.16974 q^{83} +10.9269 q^{84} +6.73010 q^{85} +4.72497 q^{86} -16.4010 q^{87} +2.11113 q^{88} -0.0957258 q^{89} -9.70075 q^{90} +32.1335 q^{91} +2.90663 q^{92} -11.9777 q^{93} -10.5159 q^{94} -1.00000 q^{95} -11.6066 q^{96} +3.15397 q^{97} -31.2306 q^{98} +5.88203 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) $ 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 17 q^{13} + 12 q^{14} - 7 q^{15} + 18 q^{16} - 9 q^{17} - 2 q^{18} - 8 q^{19} + 10 q^{20} + q^{21} - 6 q^{22} - 8 q^{23} + q^{24} + 8 q^{25} + 10 q^{26} - 34 q^{27} - 22 q^{28} - 3 q^{29} - q^{31} - 37 q^{32} - 7 q^{33} - 8 q^{34} - 11 q^{35} + 30 q^{36} - 17 q^{37} + 6 q^{38} + 14 q^{39} - 18 q^{40} - 5 q^{41} + 15 q^{42} - 21 q^{43} + 10 q^{44} + 11 q^{45} - 2 q^{46} - 8 q^{47} + 10 q^{48} + 19 q^{49} - 6 q^{50} - 16 q^{51} + 9 q^{52} - 19 q^{53} - 3 q^{54} + 8 q^{55} + 24 q^{56} + 7 q^{57} + 37 q^{58} - 33 q^{59} - 7 q^{60} - q^{61} - 42 q^{62} - 20 q^{63} + 48 q^{64} - 17 q^{65} - 18 q^{67} - 37 q^{68} + 16 q^{69} + 12 q^{70} - 18 q^{71} + 13 q^{72} - 18 q^{73} + 15 q^{74} - 7 q^{75} - 10 q^{76} - 11 q^{77} - 51 q^{78} - 5 q^{79} + 18 q^{80} + 32 q^{81} + 12 q^{82} - 33 q^{83} - 51 q^{84} - 9 q^{85} - 16 q^{86} - 26 q^{87} - 18 q^{88} - 20 q^{89} - 2 q^{90} + 6 q^{91} - 3 q^{92} + 18 q^{93} + 30 q^{94} - 8 q^{95} + 21 q^{96} - 69 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 17 * q^13 + 12 * q^14 - 7 * q^15 + 18 * q^16 - 9 * q^17 - 2 * q^18 - 8 * q^19 + 10 * q^20 + q^21 - 6 * q^22 - 8 * q^23 + q^24 + 8 * q^25 + 10 * q^26 - 34 * q^27 - 22 * q^28 - 3 * q^29 - q^31 - 37 * q^32 - 7 * q^33 - 8 * q^34 - 11 * q^35 + 30 * q^36 - 17 * q^37 + 6 * q^38 + 14 * q^39 - 18 * q^40 - 5 * q^41 + 15 * q^42 - 21 * q^43 + 10 * q^44 + 11 * q^45 - 2 * q^46 - 8 * q^47 + 10 * q^48 + 19 * q^49 - 6 * q^50 - 16 * q^51 + 9 * q^52 - 19 * q^53 - 3 * q^54 + 8 * q^55 + 24 * q^56 + 7 * q^57 + 37 * q^58 - 33 * q^59 - 7 * q^60 - q^61 - 42 * q^62 - 20 * q^63 + 48 * q^64 - 17 * q^65 - 18 * q^67 - 37 * q^68 + 16 * q^69 + 12 * q^70 - 18 * q^71 + 13 * q^72 - 18 * q^73 + 15 * q^74 - 7 * q^75 - 10 * q^76 - 11 * q^77 - 51 * q^78 - 5 * q^79 + 18 * q^80 + 32 * q^81 + 12 * q^82 - 33 * q^83 - 51 * q^84 - 9 * q^85 - 16 * q^86 - 26 * q^87 - 18 * q^88 - 20 * q^89 - 2 * q^90 + 6 * q^91 - 3 * q^92 + 18 * q^93 + 30 * q^94 - 8 * q^95 + 21 * q^96 - 69 * q^98 + 11 * 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 $

$2$	−1.64922	−1.16617	−0.583087	−	0.812410i	$-0.698155\pi$
$2$	−1.64922	−1.16617	−0.583087	+	0.812410i	$0.698155\pi$
$3$	−2.98027	−1.72066	−0.860331	−	0.509736i	$-0.829743\pi$
$3$	−2.98027	−1.72066	−0.860331	+	0.509736i	$0.829743\pi$
$4$	0.719922	0.359961
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	4.91512	2.00659
$7$	−5.09280	−1.92490	−0.962448	−	0.271466i	$-0.912492\pi$
$7$	−5.09280	−1.92490	−0.962448	+	0.271466i	$0.912492\pi$
$8$	2.11113	0.746397
$9$	5.88203	1.96068
$10$	−1.64922	−0.521529
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.14556	−0.619371
$13$	−6.30960	−1.74997	−0.874984	−	0.484152i	$-0.839128\pi$
$13$	−6.30960	−1.74997	−0.874984	+	0.484152i	$0.839128\pi$
$14$	8.39913	2.24476
$15$	−2.98027	−0.769503
$16$	−4.92156	−1.23039
$17$	6.73010	1.63229	0.816145	−	0.577847i	$-0.196107\pi$
$17$	6.73010	1.63229	0.816145	+	0.577847i	$0.196107\pi$
$18$	−9.70075	−2.28649
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.719922	0.160979
$21$	15.1779	3.31209
$22$	−1.64922	−0.351615
$23$	4.03742	0.841860	0.420930	−	0.907093i	$-0.361704\pi$
$23$	4.03742	0.841860	0.420930	+	0.907093i	$0.361704\pi$
$24$	−6.29174	−1.28430
$25$	1.00000	0.200000
$26$	10.4059	2.04077
$27$	−8.58923	−1.65300
$28$	−3.66641	−0.692887
$29$	5.50320	1.02192	0.510959	−	0.859605i	$-0.329290\pi$
$29$	5.50320	1.02192	0.510959	+	0.859605i	$0.329290\pi$
$30$	4.91512	0.897374
$31$	4.01900	0.721834	0.360917	−	0.932598i	$-0.382464\pi$
$31$	4.01900	0.721834	0.360917	+	0.932598i	$0.382464\pi$
$32$	3.89446	0.688450
$33$	−2.98027	−0.518799
$34$	−11.0994	−1.90353
$35$	−5.09280	−0.860840
$36$	4.23460	0.705766
$37$	3.62306	0.595628	0.297814	−	0.954624i	$-0.403743\pi$
$37$	3.62306	0.595628	0.297814	+	0.954624i	$0.403743\pi$
$38$	1.64922	0.267539
$39$	18.8043	3.01110
$40$	2.11113	0.333799
$41$	−3.31685	−0.518006	−0.259003	−	0.965877i	$-0.583394\pi$
$41$	−3.31685	−0.518006	−0.259003	+	0.965877i	$0.583394\pi$
$42$	−25.0317	−3.86248
$43$	−2.86498	−0.436905	−0.218453	−	0.975848i	$-0.570101\pi$
$43$	−2.86498	−0.436905	−0.218453	+	0.975848i	$0.570101\pi$
$44$	0.719922	0.108532
$45$	5.88203	0.876841
$46$	−6.65859	−0.981755
$47$	6.37629	0.930078	0.465039	−	0.885290i	$-0.346040\pi$
$47$	6.37629	0.930078	0.465039	+	0.885290i	$0.346040\pi$
$48$	14.6676	2.11708
$49$	18.9366	2.70523
$50$	−1.64922	−0.233235
$51$	−20.0575	−2.80862
$52$	−4.54242	−0.629920
$53$	3.60804	0.495602	0.247801	−	0.968811i	$-0.420292\pi$
$53$	3.60804	0.495602	0.247801	+	0.968811i	$0.420292\pi$
$54$	14.1655	1.92768
$55$	1.00000	0.134840
$56$	−10.7515	−1.43674
$57$	2.98027	0.394747
$58$	−9.07597	−1.19173
$59$	−13.9464	−1.81566	−0.907831	−	0.419337i	$-0.862263\pi$
$59$	−13.9464	−1.81566	−0.907831	+	0.419337i	$0.862263\pi$
$60$	−2.14556	−0.276991
$61$	−11.7640	−1.50623	−0.753114	−	0.657890i	$-0.771449\pi$
$61$	−11.7640	−1.50623	−0.753114	+	0.657890i	$0.771449\pi$
$62$	−6.62822	−0.841784
$63$	−29.9560	−3.77410
$64$	3.42029	0.427536
$65$	−6.30960	−0.782609
$66$	4.91512	0.605010
$67$	−3.05805	−0.373600	−0.186800	−	0.982398i	$-0.559812\pi$
$67$	−3.05805	−0.373600	−0.186800	+	0.982398i	$0.559812\pi$
$68$	4.84515	0.587560
$69$	−12.0326	−1.44856
$70$	8.39913	1.00389
$71$	4.13233	0.490417	0.245208	−	0.969470i	$-0.421144\pi$
$71$	4.13233	0.490417	0.245208	+	0.969470i	$0.421144\pi$
$72$	12.4177	1.46344
$73$	0.708095	0.0828763	0.0414381	−	0.999141i	$-0.486806\pi$
$73$	0.708095	0.0828763	0.0414381	+	0.999141i	$0.486806\pi$
$74$	−5.97522	−0.694605
$75$	−2.98027	−0.344132
$76$	−0.719922	−0.0825807
$77$	−5.09280	−0.580378
$78$	−31.0124	−3.51147
$79$	−10.0436	−1.12999	−0.564995	−	0.825095i	$-0.691122\pi$
$79$	−10.0436	−1.12999	−0.564995	+	0.825095i	$0.691122\pi$
$80$	−4.92156	−0.550247
$81$	7.95216	0.883573
$82$	5.47022	0.604084
$83$	−1.16974	−0.128395	−0.0641976	−	0.997937i	$-0.520449\pi$
$83$	−1.16974	−0.128395	−0.0641976	+	0.997937i	$0.520449\pi$
$84$	10.9269	1.19222
$85$	6.73010	0.729982
$86$	4.72497	0.509507
$87$	−16.4010	−1.75837
$88$	2.11113	0.225047
$89$	−0.0957258	−0.0101469	−0.00507346	−	0.999987i	$-0.501615\pi$
$89$	−0.0957258	−0.0101469	−0.00507346	+	0.999987i	$0.501615\pi$
$90$	−9.70075	−1.02255
$91$	32.1335	3.36851
$92$	2.90663	0.303037
$93$	−11.9777	−1.24203
$94$	−10.5159	−1.08463
$95$	−1.00000	−0.102598
$96$	−11.6066	−1.18459
$97$	3.15397	0.320237	0.160119	−	0.987098i	$-0.448812\pi$
$97$	3.15397	0.320237	0.160119	+	0.987098i	$0.448812\pi$
$98$	−31.2306	−3.15476
$99$	5.88203	0.591166
$100$	0.719922	0.0719922
$101$	8.31492	0.827365	0.413683	−	0.910421i	$-0.364242\pi$
$101$	8.31492	0.827365	0.413683	+	0.910421i	$0.364242\pi$
$102$	33.0793	3.27534
$103$	−18.7440	−1.84690	−0.923452	−	0.383714i	$-0.874645\pi$
$103$	−18.7440	−1.84690	−0.923452	+	0.383714i	$0.874645\pi$
$104$	−13.3204	−1.30617
$105$	15.1779	1.48121
$106$	−5.95044	−0.577958
$107$	5.98720	0.578804	0.289402	−	0.957208i	$-0.406543\pi$
$107$	5.98720	0.578804	0.289402	+	0.957208i	$0.406543\pi$
$108$	−6.18357	−0.595014
$109$	−6.50866	−0.623417	−0.311708	−	0.950178i	$-0.600901\pi$
$109$	−6.50866	−0.623417	−0.311708	+	0.950178i	$0.600901\pi$
$110$	−1.64922	−0.157247
$111$	−10.7977	−1.02487
$112$	25.0645	2.36837
$113$	−16.7997	−1.58038	−0.790189	−	0.612863i	$-0.790018\pi$
$113$	−16.7997	−1.58038	−0.790189	+	0.612863i	$0.790018\pi$
$114$	−4.91512	−0.460343
$115$	4.03742	0.376491
$116$	3.96187	0.367850
$117$	−37.1132	−3.43112
$118$	23.0006	2.11738
$119$	−34.2750	−3.14199
$120$	−6.29174	−0.574355
$121$	1.00000	0.0909091
$122$	19.4014	1.75652
$123$	9.88513	0.891312
$124$	2.89337	0.259832
$125$	1.00000	0.0894427
$126$	49.4039	4.40125
$127$	5.78468	0.513307	0.256654	−	0.966503i	$-0.417380\pi$
$127$	5.78468	0.513307	0.256654	+	0.966503i	$0.417380\pi$
$128$	−13.4297	−1.18703
$129$	8.53842	0.751766
$130$	10.4059	0.912658
$131$	−8.66140	−0.756750	−0.378375	−	0.925652i	$-0.623517\pi$
$131$	−8.66140	−0.756750	−0.378375	+	0.925652i	$0.623517\pi$
$132$	−2.14556	−0.186747
$133$	5.09280	0.441601
$134$	5.04340	0.435683
$135$	−8.58923	−0.739243
$136$	14.2081	1.21834
$137$	4.95755	0.423552	0.211776	−	0.977318i	$-0.432075\pi$
$137$	4.95755	0.423552	0.211776	+	0.977318i	$0.432075\pi$
$138$	19.8444	1.68927
$139$	−6.76849	−0.574096	−0.287048	−	0.957916i	$-0.592674\pi$
$139$	−6.76849	−0.574096	−0.287048	+	0.957916i	$0.592674\pi$
$140$	−3.66641	−0.309869
$141$	−19.0031	−1.60035
$142$	−6.81511	−0.571911
$143$	−6.30960	−0.527635
$144$	−28.9487	−2.41239
$145$	5.50320	0.457015
$146$	−1.16780	−0.0966481
$147$	−56.4362	−4.65478
$148$	2.60832	0.214403
$149$	3.61286	0.295977	0.147988	−	0.988989i	$-0.452720\pi$
$149$	3.61286	0.295977	0.147988	+	0.988989i	$0.452720\pi$
$150$	4.91512	0.401318
$151$	−3.59897	−0.292880	−0.146440	−	0.989220i	$-0.546782\pi$
$151$	−3.59897	−0.292880	−0.146440	+	0.989220i	$0.546782\pi$
$152$	−2.11113	−0.171235
$153$	39.5867	3.20039
$154$	8.39913	0.676822
$155$	4.01900	0.322814
$156$	13.5376	1.08388
$157$	2.82711	0.225628	0.112814	−	0.993616i	$-0.464014\pi$
$157$	2.82711	0.225628	0.112814	+	0.993616i	$0.464014\pi$
$158$	16.5640	1.31776
$159$	−10.7529	−0.852763
$160$	3.89446	0.307884
$161$	−20.5618	−1.62049
$162$	−13.1148	−1.03040
$163$	−16.3983	−1.28441	−0.642207	−	0.766531i	$-0.721981\pi$
$163$	−16.3983	−1.28441	−0.642207	+	0.766531i	$0.721981\pi$
$164$	−2.38788	−0.186462
$165$	−2.98027	−0.232014
$166$	1.92915	0.149731
$167$	15.4420	1.19494	0.597471	−	0.801891i	$-0.296172\pi$
$167$	15.4420	1.19494	0.597471	+	0.801891i	$0.296172\pi$
$168$	32.0426	2.47214
$169$	26.8110	2.06239
$170$	−11.0994	−0.851286
$171$	−5.88203	−0.449810
$172$	−2.06256	−0.157269
$173$	−3.15651	−0.239985	−0.119992	−	0.992775i	$-0.538287\pi$
$173$	−3.15651	−0.239985	−0.119992	+	0.992775i	$0.538287\pi$
$174$	27.0489	2.05057
$175$	−5.09280	−0.384979
$176$	−4.92156	−0.370976
$177$	41.5640	3.12414
$178$	0.157873	0.0118331
$179$	−0.0274136	−0.00204899	−0.00102450	−	0.999999i	$-0.500326\pi$
$179$	−0.0274136	−0.00204899	−0.00102450	+	0.999999i	$0.500326\pi$
$180$	4.23460	0.315628
$181$	−5.54723	−0.412322	−0.206161	−	0.978518i	$-0.566097\pi$
$181$	−5.54723	−0.412322	−0.206161	+	0.978518i	$0.566097\pi$
$182$	−52.9952	−3.92826
$183$	35.0600	2.59171
$184$	8.52352	0.628362
$185$	3.62306	0.266373
$186$	19.7539	1.44843
$187$	6.73010	0.492154
$188$	4.59043	0.334792
$189$	43.7432	3.18185
$190$	1.64922	0.119647
$191$	16.0183	1.15905	0.579523	−	0.814956i	$-0.303239\pi$
$191$	16.0183	1.15905	0.579523	+	0.814956i	$0.303239\pi$
$192$	−10.1934	−0.735645
$193$	−18.7111	−1.34685	−0.673426	−	0.739255i	$-0.735178\pi$
$193$	−18.7111	−1.34685	−0.673426	+	0.739255i	$0.735178\pi$
$194$	−5.20159	−0.373452
$195$	18.8043	1.34661
$196$	13.6329	0.973775
$197$	19.2044	1.36826	0.684130	−	0.729360i	$-0.260182\pi$
$197$	19.2044	1.36826	0.684130	+	0.729360i	$0.260182\pi$
$198$	−9.70075	−0.689402
$199$	2.62898	0.186364	0.0931818	−	0.995649i	$-0.470296\pi$
$199$	2.62898	0.186364	0.0931818	+	0.995649i	$0.470296\pi$
$200$	2.11113	0.149279
$201$	9.11383	0.642840
$202$	−13.7131	−0.964851
$203$	−28.0267	−1.96709
$204$	−14.4399	−1.01099
$205$	−3.31685	−0.231659
$206$	30.9130	2.15381
$207$	23.7482	1.65062
$208$	31.0530	2.15314
$209$	−1.00000	−0.0691714
$210$	−25.0317	−1.72735
$211$	8.35469	0.575160	0.287580	−	0.957757i	$-0.407149\pi$
$211$	8.35469	0.575160	0.287580	+	0.957757i	$0.407149\pi$
$212$	2.59750	0.178397
$213$	−12.3155	−0.843841
$214$	−9.87420	−0.674986
$215$	−2.86498	−0.195390
$216$	−18.1330	−1.23379
$217$	−20.4680	−1.38946
$218$	10.7342	0.727012
$219$	−2.11032	−0.142602
$220$	0.719922	0.0485371
$221$	−42.4643	−2.85645
$222$	17.8078	1.19518
$223$	11.3812	0.762141	0.381070	−	0.924546i	$-0.375556\pi$
$223$	11.3812	0.762141	0.381070	+	0.924546i	$0.375556\pi$
$224$	−19.8337	−1.32520
$225$	5.88203	0.392135
$226$	27.7063	1.84300
$227$	−19.3330	−1.28318	−0.641589	−	0.767049i	$-0.721724\pi$
$227$	−19.3330	−1.28318	−0.641589	+	0.767049i	$0.721724\pi$
$228$	2.14556	0.142093
$229$	−25.0452	−1.65503	−0.827517	−	0.561441i	$-0.810247\pi$
$229$	−25.0452	−1.65503	−0.827517	+	0.561441i	$0.810247\pi$
$230$	−6.65859	−0.439054
$231$	15.1779	0.998634
$232$	11.6180	0.762756
$233$	4.81142	0.315206	0.157603	−	0.987503i	$-0.449623\pi$
$233$	4.81142	0.315206	0.157603	+	0.987503i	$0.449623\pi$
$234$	61.2078	4.00128
$235$	6.37629	0.415943
$236$	−10.0403	−0.653567
$237$	29.9326	1.94433
$238$	56.5270	3.66410
$239$	22.0809	1.42830	0.714149	−	0.699994i	$-0.246814\pi$
$239$	22.0809	1.42830	0.714149	+	0.699994i	$0.246814\pi$
$240$	14.6676	0.946788
$241$	−16.6213	−1.07067	−0.535336	−	0.844639i	$-0.679815\pi$
$241$	−16.6213	−1.07067	−0.535336	+	0.844639i	$0.679815\pi$
$242$	−1.64922	−0.106016
$243$	2.06808	0.132667
$244$	−8.46917	−0.542183
$245$	18.9366	1.20981
$246$	−16.3027	−1.03942
$247$	6.30960	0.401470
$248$	8.48463	0.538775
$249$	3.48613	0.220925
$250$	−1.64922	−0.104306
$251$	23.6165	1.49066	0.745330	−	0.666696i	$-0.232292\pi$
$251$	23.6165	1.49066	0.745330	+	0.666696i	$0.232292\pi$
$252$	−21.5659	−1.35853
$253$	4.03742	0.253830
$254$	−9.54020	−0.598605
$255$	−20.0575	−1.25605
$256$	15.3080	0.956749
$257$	14.0882	0.878801	0.439400	−	0.898291i	$-0.355191\pi$
$257$	14.0882	0.878801	0.439400	+	0.898291i	$0.355191\pi$
$258$	−14.0817	−0.876689
$259$	−18.4515	−1.14652
$260$	−4.54242	−0.281709
$261$	32.3699	2.00365
$262$	14.2845	0.882502
$263$	−19.3505	−1.19320	−0.596602	−	0.802537i	$-0.703483\pi$
$263$	−19.3505	−1.19320	−0.596602	+	0.802537i	$0.703483\pi$
$264$	−6.29174	−0.387230
$265$	3.60804	0.221640
$266$	−8.39913	−0.514984
$267$	0.285289	0.0174594
$268$	−2.20156	−0.134482
$269$	−9.03734	−0.551016	−0.275508	−	0.961299i	$-0.588846\pi$
$269$	−9.03734	−0.551016	−0.275508	+	0.961299i	$0.588846\pi$
$270$	14.1655	0.862086
$271$	−18.6248	−1.13138	−0.565688	−	0.824619i	$-0.691389\pi$
$271$	−18.6248	−1.13138	−0.565688	+	0.824619i	$0.691389\pi$
$272$	−33.1226	−2.00835
$273$	−95.7666	−5.79606
$274$	−8.17608	−0.493935
$275$	1.00000	0.0603023
$276$	−8.66254	−0.521424
$277$	4.55182	0.273492	0.136746	−	0.990606i	$-0.456336\pi$
$277$	4.55182	0.273492	0.136746	+	0.990606i	$0.456336\pi$
$278$	11.1627	0.669495
$279$	23.6399	1.41528
$280$	−10.7515	−0.642528
$281$	−13.2430	−0.790011	−0.395006	−	0.918679i	$-0.629257\pi$
$281$	−13.2430	−0.790011	−0.395006	+	0.918679i	$0.629257\pi$
$282$	31.3402	1.86628
$283$	12.2945	0.730831	0.365416	−	0.930844i	$-0.380927\pi$
$283$	12.2945	0.730831	0.365416	+	0.930844i	$0.380927\pi$
$284$	2.97495	0.176531
$285$	2.98027	0.176536
$286$	10.4059	0.615314
$287$	16.8921	0.997107
$288$	22.9073	1.34983
$289$	28.2943	1.66437
$290$	−9.07597	−0.532959
$291$	−9.39969	−0.551020
$292$	0.509773	0.0298322
$293$	0.782299	0.0457024	0.0228512	−	0.999739i	$-0.492726\pi$
$293$	0.782299	0.0457024	0.0228512	+	0.999739i	$0.492726\pi$
$294$	93.0756	5.42828
$295$	−13.9464	−0.811988
$296$	7.64875	0.444574
$297$	−8.58923	−0.498397
$298$	−5.95839	−0.345160
$299$	−25.4745	−1.47323
$300$	−2.14556	−0.123874
$301$	14.5907	0.840997
$302$	5.93549	0.341549
$303$	−24.7807	−1.42362
$304$	4.92156	0.282271
$305$	−11.7640	−0.673605
$306$	−65.2870	−3.73221
$307$	−27.0921	−1.54623	−0.773114	−	0.634267i	$-0.781302\pi$
$307$	−27.0921	−1.54623	−0.773114	+	0.634267i	$0.781302\pi$
$308$	−3.66641	−0.208913
$309$	55.8623	3.17790
$310$	−6.62822	−0.376457
$311$	−16.1121	−0.913631	−0.456815	−	0.889562i	$-0.651010\pi$
$311$	−16.1121	−0.913631	−0.456815	+	0.889562i	$0.651010\pi$
$312$	39.6984	2.24748
$313$	−20.9961	−1.18677	−0.593386	−	0.804918i	$-0.702209\pi$
$313$	−20.9961	−1.18677	−0.593386	+	0.804918i	$0.702209\pi$
$314$	−4.66253	−0.263122
$315$	−29.9560	−1.68783
$316$	−7.23058	−0.406752
$317$	13.7082	0.769930	0.384965	−	0.922931i	$-0.374214\pi$
$317$	13.7082	0.769930	0.384965	+	0.922931i	$0.374214\pi$
$318$	17.7339	0.994470
$319$	5.50320	0.308120
$320$	3.42029	0.191200
$321$	−17.8435	−0.995926
$322$	33.9108	1.88978
$323$	−6.73010	−0.374473
$324$	5.72493	0.318052
$325$	−6.30960	−0.349994
$326$	27.0444	1.49785
$327$	19.3976	1.07269
$328$	−7.00231	−0.386638
$329$	−32.4732	−1.79030
$330$	4.91512	0.270569
$331$	−27.2520	−1.49790	−0.748952	−	0.662624i	$-0.769443\pi$
$331$	−27.2520	−1.49790	−0.748952	+	0.662624i	$0.769443\pi$
$332$	−0.842118	−0.0462173
$333$	21.3109	1.16783
$334$	−25.4673	−1.39351
$335$	−3.05805	−0.167079
$336$	−74.6990	−4.07516
$337$	−22.8071	−1.24238	−0.621191	−	0.783659i	$-0.713351\pi$
$337$	−22.8071	−1.24238	−0.621191	+	0.783659i	$0.713351\pi$
$338$	−44.2173	−2.40510
$339$	50.0676	2.71930
$340$	4.84515	0.262765
$341$	4.01900	0.217641
$342$	9.70075	0.524556
$343$	−60.7906	−3.28238
$344$	−6.04834	−0.326104
$345$	−12.0326	−0.647814
$346$	5.20577	0.279864
$347$	−28.2667	−1.51744	−0.758719	−	0.651418i	$-0.774174\pi$
$347$	−28.2667	−1.51744	−0.758719	+	0.651418i	$0.774174\pi$
$348$	−11.8075	−0.632946
$349$	12.7377	0.681835	0.340918	−	0.940093i	$-0.389262\pi$
$349$	12.7377	0.681835	0.340918	+	0.940093i	$0.389262\pi$
$350$	8.39913	0.448953
$351$	54.1946	2.89269
$352$	3.89446	0.207576
$353$	−6.26998	−0.333717	−0.166859	−	0.985981i	$-0.553362\pi$
$353$	−6.26998	−0.333717	−0.166859	+	0.985981i	$0.553362\pi$
$354$	−68.5481	−3.64329
$355$	4.13233	0.219321
$356$	−0.0689151	−0.00365249
$357$	102.149	5.40630
$358$	0.0452111	0.00238948
$359$	−26.4647	−1.39675	−0.698377	−	0.715730i	$-0.746094\pi$
$359$	−26.4647	−1.39675	−0.698377	+	0.715730i	$0.746094\pi$
$360$	12.4177	0.654471
$361$	1.00000	0.0526316
$362$	9.14859	0.480839
$363$	−2.98027	−0.156424
$364$	23.1336	1.21253
$365$	0.708095	0.0370634
$366$	−57.8216	−3.02238
$367$	1.29974	0.0678458	0.0339229	−	0.999424i	$-0.489200\pi$
$367$	1.29974	0.0678458	0.0339229	+	0.999424i	$0.489200\pi$
$368$	−19.8704	−1.03582
$369$	−19.5098	−1.01564
$370$	−5.97522	−0.310637
$371$	−18.3750	−0.953982
$372$	−8.62303	−0.447083
$373$	−5.61834	−0.290907	−0.145453	−	0.989365i	$-0.546464\pi$
$373$	−5.61834	−0.290907	−0.145453	+	0.989365i	$0.546464\pi$
$374$	−11.0994	−0.573937
$375$	−2.98027	−0.153901
$376$	13.4612	0.694207
$377$	−34.7230	−1.78832
$378$	−72.1421	−3.71059
$379$	17.4494	0.896317	0.448158	−	0.893954i	$-0.352080\pi$
$379$	17.4494	0.896317	0.448158	+	0.893954i	$0.352080\pi$
$380$	−0.719922	−0.0369312
$381$	−17.2399	−0.883228
$382$	−26.4177	−1.35165
$383$	23.0893	1.17981	0.589903	−	0.807474i	$-0.299166\pi$
$383$	23.0893	1.17981	0.589903	+	0.807474i	$0.299166\pi$
$384$	40.0243	2.04248
$385$	−5.09280	−0.259553
$386$	30.8586	1.57066
$387$	−16.8519	−0.856629
$388$	2.27061	0.115273
$389$	−0.0574602	−0.00291335	−0.00145667	−	0.999999i	$-0.500464\pi$
$389$	−0.0574602	−0.00291335	−0.00145667	+	0.999999i	$0.500464\pi$
$390$	−31.0124	−1.57038
$391$	27.1723	1.37416
$392$	39.9776	2.01917
$393$	25.8133	1.30211
$394$	−31.6723	−1.59563
$395$	−10.0436	−0.505347
$396$	4.23460	0.212797
$397$	3.54710	0.178024	0.0890120	−	0.996031i	$-0.471629\pi$
$397$	3.54710	0.178024	0.0890120	+	0.996031i	$0.471629\pi$
$398$	−4.33577	−0.217332
$399$	−15.1779	−0.759847
$400$	−4.92156	−0.246078
$401$	−9.80786	−0.489781	−0.244891	−	0.969551i	$-0.578752\pi$
$401$	−9.80786	−0.489781	−0.244891	+	0.969551i	$0.578752\pi$
$402$	−15.0307	−0.749663
$403$	−25.3583	−1.26319
$404$	5.98609	0.297819
$405$	7.95216	0.395146
$406$	46.2221	2.29396
$407$	3.62306	0.179588
$408$	−42.3441	−2.09634
$409$	12.3825	0.612275	0.306137	−	0.951987i	$-0.400963\pi$
$409$	12.3825	0.612275	0.306137	+	0.951987i	$0.400963\pi$
$410$	5.47022	0.270155
$411$	−14.7748	−0.728790
$412$	−13.4942	−0.664813
$413$	71.0260	3.49496
$414$	−39.1660	−1.92490
$415$	−1.16974	−0.0574201
$416$	−24.5725	−1.20477
$417$	20.1719	0.987824
$418$	1.64922	0.0806659
$419$	−11.8235	−0.577615	−0.288807	−	0.957387i	$-0.593259\pi$
$419$	−11.8235	−0.577615	−0.288807	+	0.957387i	$0.593259\pi$
$420$	10.9269	0.533179
$421$	18.5511	0.904123	0.452061	−	0.891987i	$-0.350689\pi$
$421$	18.5511	0.904123	0.452061	+	0.891987i	$0.350689\pi$
$422$	−13.7787	−0.670737
$423$	37.5055	1.82358
$424$	7.61703	0.369916
$425$	6.73010	0.326458
$426$	20.3109	0.984066
$427$	59.9117	2.89933
$428$	4.31031	0.208347
$429$	18.8043	0.907881
$430$	4.72497	0.227859
$431$	40.0750	1.93035	0.965173	−	0.261613i	$-0.0842543\pi$
$431$	40.0750	1.93035	0.965173	+	0.261613i	$0.0842543\pi$
$432$	42.2724	2.03383
$433$	−22.0830	−1.06124	−0.530621	−	0.847609i	$-0.678041\pi$
$433$	−22.0830	−1.06124	−0.530621	+	0.847609i	$0.678041\pi$
$434$	33.7562	1.62035
$435$	−16.4010	−0.786369
$436$	−4.68573	−0.224406
$437$	−4.03742	−0.193136
$438$	3.48037	0.166299
$439$	7.50445	0.358168	0.179084	−	0.983834i	$-0.442687\pi$
$439$	7.50445	0.358168	0.179084	+	0.983834i	$0.442687\pi$
$440$	2.11113	0.100644
$441$	111.385	5.30407
$442$	70.0328	3.33112
$443$	17.4855	0.830762	0.415381	−	0.909647i	$-0.363648\pi$
$443$	17.4855	0.830762	0.415381	+	0.909647i	$0.363648\pi$
$444$	−7.77351	−0.368914
$445$	−0.0957258	−0.00453784
$446$	−18.7701	−0.888788
$447$	−10.7673	−0.509276
$448$	−17.4188	−0.822963
$449$	−13.8130	−0.651878	−0.325939	−	0.945391i	$-0.605680\pi$
$449$	−13.8130	−0.651878	−0.325939	+	0.945391i	$0.605680\pi$
$450$	−9.70075	−0.457298
$451$	−3.31685	−0.156185
$452$	−12.0944	−0.568874
$453$	10.7259	0.503947
$454$	31.8844	1.49641
$455$	32.1335	1.50644
$456$	6.29174	0.294638
$457$	−27.0921	−1.26732	−0.633658	−	0.773614i	$-0.718447\pi$
$457$	−27.0921	−1.26732	−0.633658	+	0.773614i	$0.718447\pi$
$458$	41.3050	1.93006
$459$	−57.8064	−2.69817
$460$	2.90663	0.135522
$461$	−22.2365	−1.03566	−0.517829	−	0.855484i	$-0.673260\pi$
$461$	−22.2365	−1.03566	−0.517829	+	0.855484i	$0.673260\pi$
$462$	−25.0317	−1.16458
$463$	13.5690	0.630605	0.315302	−	0.948991i	$-0.397894\pi$
$463$	13.5690	0.630605	0.315302	+	0.948991i	$0.397894\pi$
$464$	−27.0843	−1.25736
$465$	−11.9777	−0.555454
$466$	−7.93508	−0.367585
$467$	16.2179	0.750476	0.375238	−	0.926929i	$-0.377561\pi$
$467$	16.2179	0.750476	0.375238	+	0.926929i	$0.377561\pi$
$468$	−26.7186	−1.23507
$469$	15.5740	0.719142
$470$	−10.5159	−0.485062
$471$	−8.42557	−0.388230
$472$	−29.4426	−1.35520
$473$	−2.86498	−0.131732
$474$	−49.3653	−2.26743
$475$	−1.00000	−0.0458831
$476$	−24.6754	−1.13099
$477$	21.2226	0.971714
$478$	−36.4163	−1.66564
$479$	1.34758	0.0615725	0.0307862	−	0.999526i	$-0.490199\pi$
$479$	1.34758	0.0615725	0.0307862	+	0.999526i	$0.490199\pi$
$480$	−11.6066	−0.529765
$481$	−22.8601	−1.04233
$482$	27.4122	1.24859
$483$	61.2797	2.78832
$484$	0.719922	0.0327237
$485$	3.15397	0.143214
$486$	−3.41071	−0.154713
$487$	−24.4018	−1.10575	−0.552876	−	0.833263i	$-0.686470\pi$
$487$	−24.4018	−1.10575	−0.552876	+	0.833263i	$0.686470\pi$
$488$	−24.8353	−1.12424
$489$	48.8714	2.21004
$490$	−31.2306	−1.41085
$491$	−17.8525	−0.805671	−0.402835	−	0.915272i	$-0.631975\pi$
$491$	−17.8525	−0.805671	−0.402835	+	0.915272i	$0.631975\pi$
$492$	7.11652	0.320838
$493$	37.0371	1.66807
$494$	−10.4059	−0.468184
$495$	5.88203	0.264377
$496$	−19.7798	−0.888137
$497$	−21.0451	−0.944002
$498$	−5.74939	−0.257637
$499$	−25.7709	−1.15366	−0.576832	−	0.816863i	$-0.695711\pi$
$499$	−25.7709	−1.15366	−0.576832	+	0.816863i	$0.695711\pi$
$500$	0.719922	0.0321959
$501$	−46.0215	−2.05609
$502$	−38.9488	−1.73837
$503$	−4.38299	−0.195428	−0.0977140	−	0.995215i	$-0.531153\pi$
$503$	−4.38299	−0.195428	−0.0977140	+	0.995215i	$0.531153\pi$
$504$	−63.2409	−2.81697
$505$	8.31492	0.370009
$506$	−6.65859	−0.296010
$507$	−79.9042	−3.54867
$508$	4.16451	0.184770
$509$	−32.0167	−1.41912	−0.709558	−	0.704647i	$-0.751105\pi$
$509$	−32.0167	−1.41912	−0.709558	+	0.704647i	$0.751105\pi$
$510$	33.0793	1.46478
$511$	−3.60619	−0.159528
$512$	1.61326	0.0712965
$513$	8.58923	0.379224
$514$	−23.2346	−1.02483
$515$	−18.7440	−0.825961
$516$	6.14699	0.270606
$517$	6.37629	0.280429
$518$	30.4306	1.33704
$519$	9.40725	0.412932
$520$	−13.3204	−0.584137
$521$	16.1802	0.708869	0.354434	−	0.935081i	$-0.384673\pi$
$521$	16.1802	0.708869	0.354434	+	0.935081i	$0.384673\pi$
$522$	−53.3851	−2.33660
$523$	6.06040	0.265003	0.132501	−	0.991183i	$-0.457699\pi$
$523$	6.06040	0.265003	0.132501	+	0.991183i	$0.457699\pi$
$524$	−6.23553	−0.272400
$525$	15.1779	0.662419
$526$	31.9133	1.39148
$527$	27.0483	1.17824
$528$	14.6676	0.638325
$529$	−6.69923	−0.291271
$530$	−5.95044	−0.258471
$531$	−82.0329	−3.55992
$532$	3.66641	0.158959
$533$	20.9280	0.906493
$534$	−0.470504	−0.0203607
$535$	5.98720	0.258849
$536$	−6.45594	−0.278854
$537$	0.0817002	0.00352562
$538$	14.9045	0.642581
$539$	18.9366	0.815656
$540$	−6.18357	−0.266098
$541$	−11.5841	−0.498039	−0.249019	−	0.968498i	$-0.580108\pi$
$541$	−11.5841	−0.498039	−0.249019	+	0.968498i	$0.580108\pi$
$542$	30.7164	1.31938
$543$	16.5322	0.709467
$544$	26.2101	1.12375
$545$	−6.50866	−0.278800
$546$	157.940	6.75921
$547$	32.4033	1.38547	0.692733	−	0.721194i	$-0.256406\pi$
$547$	32.4033	1.38547	0.692733	+	0.721194i	$0.256406\pi$
$548$	3.56905	0.152462
$549$	−69.1962	−2.95322
$550$	−1.64922	−0.0703229
$551$	−5.50320	−0.234444
$552$	−25.4024	−1.08120
$553$	51.1498	2.17511
$554$	−7.50695	−0.318940
$555$	−10.7977	−0.458337
$556$	−4.87278	−0.206652
$557$	−17.9768	−0.761703	−0.380851	−	0.924636i	$-0.624369\pi$
$557$	−17.9768	−0.761703	−0.380851	+	0.924636i	$0.624369\pi$
$558$	−38.9873	−1.65047
$559$	18.0769	0.764570
$560$	25.0645	1.05917
$561$	−20.0575	−0.846830
$562$	21.8406	0.921290
$563$	−5.47706	−0.230831	−0.115415	−	0.993317i	$-0.536820\pi$
$563$	−5.47706	−0.230831	−0.115415	+	0.993317i	$0.536820\pi$
$564$	−13.6807	−0.576063
$565$	−16.7997	−0.706767
$566$	−20.2763	−0.852276
$567$	−40.4987	−1.70079
$568$	8.72387	0.366046
$569$	−30.3126	−1.27077	−0.635385	−	0.772196i	$-0.719158\pi$
$569$	−30.3126	−1.27077	−0.635385	+	0.772196i	$0.719158\pi$
$570$	−4.91512	−0.205872
$571$	20.0927	0.840855	0.420428	−	0.907326i	$-0.361880\pi$
$571$	20.0927	0.840855	0.420428	+	0.907326i	$0.361880\pi$
$572$	−4.54242	−0.189928
$573$	−47.7390	−1.99432
$574$	−27.8587	−1.16280
$575$	4.03742	0.168372
$576$	20.1182	0.838260
$577$	1.87788	0.0781770	0.0390885	−	0.999236i	$-0.487555\pi$
$577$	1.87788	0.0781770	0.0390885	+	0.999236i	$0.487555\pi$
$578$	−46.6635	−1.94094
$579$	55.7640	2.31747
$580$	3.96187	0.164508
$581$	5.95723	0.247147
$582$	15.5021	0.642585
$583$	3.60804	0.149430
$584$	1.49488	0.0618586
$585$	−37.1132	−1.53444
$586$	−1.29018	−0.0532969
$587$	−43.7126	−1.80421	−0.902106	−	0.431515i	$-0.857979\pi$
$587$	−43.7126	−1.80421	−0.902106	+	0.431515i	$0.857979\pi$
$588$	−40.6296	−1.67554
$589$	−4.01900	−0.165600
$590$	23.0006	0.946920
$591$	−57.2345	−2.35431
$592$	−17.8311	−0.732854
$593$	−24.9360	−1.02400	−0.511999	−	0.858986i	$-0.671095\pi$
$593$	−24.9360	−1.02400	−0.511999	+	0.858986i	$0.671095\pi$
$594$	14.1655	0.581218
$595$	−34.2750	−1.40514
$596$	2.60097	0.106540
$597$	−7.83509	−0.320669
$598$	42.0130	1.71804
$599$	7.93735	0.324311	0.162156	−	0.986765i	$-0.448155\pi$
$599$	7.93735	0.324311	0.162156	+	0.986765i	$0.448155\pi$
$600$	−6.29174	−0.256859
$601$	−21.7410	−0.886833	−0.443417	−	0.896316i	$-0.646234\pi$
$601$	−21.7410	−0.886833	−0.443417	+	0.896316i	$0.646234\pi$
$602$	−24.0633	−0.980748
$603$	−17.9875	−0.732509
$604$	−2.59098	−0.105425
$605$	1.00000	0.0406558
$606$	40.8688	1.66018
$607$	16.9344	0.687345	0.343673	−	0.939090i	$-0.388329\pi$
$607$	16.9344	0.687345	0.343673	+	0.939090i	$0.388329\pi$
$608$	−3.89446	−0.157941
$609$	83.5271	3.38469
$610$	19.4014	0.785541
$611$	−40.2318	−1.62761
$612$	28.4993	1.15202
$613$	−41.6796	−1.68342	−0.841712	−	0.539926i	$-0.818452\pi$
$613$	−41.6796	−1.68342	−0.841712	+	0.539926i	$0.818452\pi$
$614$	44.6808	1.80317
$615$	9.88513	0.398607
$616$	−10.7515	−0.433192
$617$	29.9607	1.20617	0.603086	−	0.797676i	$-0.293937\pi$
$617$	29.9607	1.20617	0.603086	+	0.797676i	$0.293937\pi$
$618$	−92.1292	−3.70598
$619$	39.1040	1.57172	0.785861	−	0.618403i	$-0.212220\pi$
$619$	39.1040	1.57172	0.785861	+	0.618403i	$0.212220\pi$
$620$	2.89337	0.116200
$621$	−34.6783	−1.39159
$622$	26.5723	1.06545
$623$	0.487512	0.0195318
$624$	−92.5465	−3.70483
$625$	1.00000	0.0400000
$626$	34.6272	1.38398
$627$	2.98027	0.119021
$628$	2.03530	0.0812174
$629$	24.3836	0.972237
$630$	49.4039	1.96830
$631$	−16.6480	−0.662745	−0.331373	−	0.943500i	$-0.607512\pi$
$631$	−16.6480	−0.662745	−0.331373	+	0.943500i	$0.607512\pi$
$632$	−21.2033	−0.843420
$633$	−24.8992	−0.989656
$634$	−22.6078	−0.897872
$635$	5.78468	0.229558
$636$	−7.74127	−0.306961
$637$	−119.482	−4.73406
$638$	−9.07597	−0.359321
$639$	24.3065	0.961548
$640$	−13.4297	−0.530857
$641$	−17.8252	−0.704054	−0.352027	−	0.935990i	$-0.614507\pi$
$641$	−17.8252	−0.704054	−0.352027	+	0.935990i	$0.614507\pi$
$642$	29.4278	1.16142
$643$	15.3843	0.606698	0.303349	−	0.952880i	$-0.401895\pi$
$643$	15.3843	0.606698	0.303349	+	0.952880i	$0.401895\pi$
$644$	−14.8029	−0.583314
$645$	8.53842	0.336200
$646$	11.0994	0.436701
$647$	2.78815	0.109613	0.0548067	−	0.998497i	$-0.482546\pi$
$647$	2.78815	0.109613	0.0548067	+	0.998497i	$0.482546\pi$
$648$	16.7880	0.659496
$649$	−13.9464	−0.547443
$650$	10.4059	0.408153
$651$	61.0001	2.39078
$652$	−11.8055	−0.462339
$653$	−2.01987	−0.0790437	−0.0395219	−	0.999219i	$-0.512583\pi$
$653$	−2.01987	−0.0790437	−0.0395219	+	0.999219i	$0.512583\pi$
$654$	−31.9909	−1.25094
$655$	−8.66140	−0.338429
$656$	16.3241	0.637348
$657$	4.16504	0.162494
$658$	53.5553	2.08780
$659$	40.8023	1.58943	0.794716	−	0.606981i	$-0.207620\pi$
$659$	40.8023	1.58943	0.794716	+	0.606981i	$0.207620\pi$
$660$	−2.14556	−0.0835159
$661$	3.61956	0.140785	0.0703924	−	0.997519i	$-0.477575\pi$
$661$	3.61956	0.140785	0.0703924	+	0.997519i	$0.477575\pi$
$662$	44.9445	1.74682
$663$	126.555	4.91499
$664$	−2.46946	−0.0958338
$665$	5.09280	0.197490
$666$	−35.1464	−1.36190
$667$	22.2187	0.860312
$668$	11.1171	0.430132
$669$	−33.9190	−1.31139
$670$	5.04340	0.194843
$671$	−11.7640	−0.454145
$672$	59.1099	2.28021
$673$	36.4403	1.40467	0.702335	−	0.711847i	$-0.252141\pi$
$673$	36.4403	1.40467	0.702335	+	0.711847i	$0.252141\pi$
$674$	37.6139	1.44883
$675$	−8.58923	−0.330599
$676$	19.3018	0.742379
$677$	−18.7228	−0.719575	−0.359787	−	0.933034i	$-0.617151\pi$
$677$	−18.7228	−0.719575	−0.359787	+	0.933034i	$0.617151\pi$
$678$	−82.5723	−3.17117
$679$	−16.0625	−0.616423
$680$	14.2081	0.544856
$681$	57.6177	2.20791
$682$	−6.62822	−0.253807
$683$	32.6452	1.24913	0.624566	−	0.780972i	$-0.285276\pi$
$683$	32.6452	1.24913	0.624566	+	0.780972i	$0.285276\pi$
$684$	−4.23460	−0.161914
$685$	4.95755	0.189418
$686$	100.257	3.82783
$687$	74.6415	2.84775
$688$	14.1001	0.537563
$689$	−22.7653	−0.867287
$690$	19.8444	0.755464
$691$	−9.55831	−0.363615	−0.181808	−	0.983334i	$-0.558195\pi$
$691$	−9.55831	−0.363615	−0.181808	+	0.983334i	$0.558195\pi$
$692$	−2.27244	−0.0863851
$693$	−29.9560	−1.13793
$694$	46.6180	1.76960
$695$	−6.76849	−0.256743
$696$	−34.6247	−1.31244
$697$	−22.3228	−0.845535
$698$	−21.0073	−0.795138
$699$	−14.3393	−0.542363
$700$	−3.66641	−0.138577
$701$	19.1270	0.722415	0.361208	−	0.932485i	$-0.382365\pi$
$701$	19.1270	0.722415	0.361208	+	0.932485i	$0.382365\pi$
$702$	−89.3787	−3.37338
$703$	−3.62306	−0.136646
$704$	3.42029	0.128907
$705$	−19.0031	−0.715698
$706$	10.3406	0.389172
$707$	−42.3462	−1.59259
$708$	29.9228	1.12457
$709$	21.5003	0.807462	0.403731	−	0.914878i	$-0.367713\pi$
$709$	21.5003	0.807462	0.403731	+	0.914878i	$0.367713\pi$
$710$	−6.81511	−0.255766
$711$	−59.0765	−2.21554
$712$	−0.202090	−0.00757363
$713$	16.2264	0.607684
$714$	−168.466	−6.30468
$715$	−6.30960	−0.235966
$716$	−0.0197357	−0.000737557 0
$717$	−65.8073	−2.45762
$718$	43.6461	1.62886
$719$	−8.80422	−0.328342	−0.164171	−	0.986432i	$-0.552495\pi$
$719$	−8.80422	−0.328342	−0.164171	+	0.986432i	$0.552495\pi$
$720$	−28.9487	−1.07886
$721$	95.4595	3.55510
$722$	−1.64922	−0.0613776
$723$	49.5360	1.84226
$724$	−3.99357	−0.148420
$725$	5.50320	0.204384
$726$	4.91512	0.182417
$727$	39.8800	1.47907	0.739533	−	0.673120i	$-0.235046\pi$
$727$	39.8800	1.47907	0.739533	+	0.673120i	$0.235046\pi$
$728$	67.8380	2.51424
$729$	−30.0199	−1.11185
$730$	−1.16780	−0.0432224
$731$	−19.2816	−0.713156
$732$	25.2404	0.932913
$733$	−36.9625	−1.36524	−0.682621	−	0.730772i	$-0.739160\pi$
$733$	−36.9625	−1.36524	−0.682621	+	0.730772i	$0.739160\pi$
$734$	−2.14355	−0.0791200
$735$	−56.4362	−2.08168
$736$	15.7236	0.579579
$737$	−3.05805	−0.112645
$738$	32.1760	1.18441
$739$	−38.3106	−1.40928	−0.704640	−	0.709565i	$-0.748891\pi$
$739$	−38.3106	−1.40928	−0.704640	+	0.709565i	$0.748891\pi$
$740$	2.60832	0.0958838
$741$	−18.8043	−0.690794
$742$	30.3044	1.11251
$743$	39.3406	1.44327	0.721633	−	0.692276i	$-0.243392\pi$
$743$	39.3406	1.44327	0.721633	+	0.692276i	$0.243392\pi$
$744$	−25.2865	−0.927049
$745$	3.61286	0.132365
$746$	9.26587	0.339248
$747$	−6.88042	−0.251741
$748$	4.84515	0.177156
$749$	−30.4916	−1.11414
$750$	4.91512	0.179475
$751$	−9.56270	−0.348948	−0.174474	−	0.984662i	$-0.555822\pi$
$751$	−9.56270	−0.348948	−0.174474	+	0.984662i	$0.555822\pi$
$752$	−31.3813	−1.14436
$753$	−70.3836	−2.56492
$754$	57.2657	2.08550
$755$	−3.59897	−0.130980
$756$	31.4917	1.14534
$757$	11.5414	0.419481	0.209740	−	0.977757i	$-0.432738\pi$
$757$	11.5414	0.419481	0.209740	+	0.977757i	$0.432738\pi$
$758$	−28.7779	−1.04526
$759$	−12.0326	−0.436756
$760$	−2.11113	−0.0765787
$761$	−46.3862	−1.68150	−0.840749	−	0.541425i	$-0.817885\pi$
$761$	−46.3862	−1.68150	−0.840749	+	0.541425i	$0.817885\pi$
$762$	28.4324	1.03000
$763$	33.1473	1.20001
$764$	11.5319	0.417211
$765$	39.5867	1.43126
$766$	−38.0792	−1.37586
$767$	87.9959	3.17735
$768$	−45.6220	−1.64624
$769$	−7.14097	−0.257510	−0.128755	−	0.991676i	$-0.541098\pi$
$769$	−7.14097	−0.257510	−0.128755	+	0.991676i	$0.541098\pi$
$770$	8.39913	0.302684
$771$	−41.9868	−1.51212
$772$	−13.4705	−0.484814
$773$	−12.8253	−0.461296	−0.230648	−	0.973037i	$-0.574085\pi$
$773$	−12.8253	−0.461296	−0.230648	+	0.973037i	$0.574085\pi$
$774$	27.7924	0.998978
$775$	4.01900	0.144367
$776$	6.65844	0.239024
$777$	54.9905	1.97277
$778$	0.0947643	0.00339747
$779$	3.31685	0.118839
$780$	13.5376	0.484725
$781$	4.13233	0.147866
$782$	−44.8130	−1.60251
$783$	−47.2682	−1.68923
$784$	−93.1974	−3.32848
$785$	2.82711	0.100904
$786$	−42.5718	−1.51849
$787$	−3.04208	−0.108438	−0.0542192	−	0.998529i	$-0.517267\pi$
$787$	−3.04208	−0.108438	−0.0542192	+	0.998529i	$0.517267\pi$
$788$	13.8257	0.492520
$789$	57.6699	2.05310
$790$	16.5640	0.589322
$791$	85.5572	3.04206
$792$	12.4177	0.441244
$793$	74.2262	2.63585
$794$	−5.84995	−0.207607
$795$	−10.7529	−0.381367
$796$	1.89266	0.0670836
$797$	−5.41140	−0.191681	−0.0958407	−	0.995397i	$-0.530554\pi$
$797$	−5.41140	−0.191681	−0.0958407	+	0.995397i	$0.530554\pi$
$798$	25.0317	0.886113
$799$	42.9131	1.51816
$800$	3.89446	0.137690
$801$	−0.563062	−0.0198948
$802$	16.1753	0.571170
$803$	0.708095	0.0249881
$804$	6.56124	0.231397
$805$	−20.5618	−0.724707
$806$	41.8214	1.47310
$807$	26.9337	0.948112
$808$	17.5539	0.617543
$809$	−21.4487	−0.754098	−0.377049	−	0.926193i	$-0.623061\pi$
$809$	−21.4487	−0.754098	−0.377049	+	0.926193i	$0.623061\pi$
$810$	−13.1148	−0.460809
$811$	−43.4139	−1.52447	−0.762235	−	0.647301i	$-0.775898\pi$
$811$	−43.4139	−1.52447	−0.762235	+	0.647301i	$0.775898\pi$
$812$	−20.1770	−0.708074
$813$	55.5070	1.94671
$814$	−5.97522	−0.209431
$815$	−16.3983	−0.574407
$816$	98.7143	3.45569
$817$	2.86498	0.100233
$818$	−20.4214	−0.714019
$819$	189.010	6.60455
$820$	−2.38788	−0.0833882
$821$	−0.607722	−0.0212097	−0.0106048	−	0.999944i	$-0.503376\pi$
$821$	−0.607722	−0.0212097	−0.0106048	+	0.999944i	$0.503376\pi$
$822$	24.3670	0.849895
$823$	4.20077	0.146430	0.0732149	−	0.997316i	$-0.476674\pi$
$823$	4.20077	0.146430	0.0732149	+	0.997316i	$0.476674\pi$
$824$	−39.5711	−1.37852
$825$	−2.98027	−0.103760
$826$	−117.137	−4.07573
$827$	32.7788	1.13983	0.569915	−	0.821704i	$-0.306976\pi$
$827$	32.7788	1.13983	0.569915	+	0.821704i	$0.306976\pi$
$828$	17.0969	0.594157
$829$	24.7062	0.858081	0.429041	−	0.903285i	$-0.358852\pi$
$829$	24.7062	0.858081	0.429041	+	0.903285i	$0.358852\pi$
$830$	1.92915	0.0669618
$831$	−13.5657	−0.470588
$832$	−21.5807	−0.748175
$833$	127.445	4.41571
$834$	−33.2679	−1.15197
$835$	15.4420	0.534394
$836$	−0.719922	−0.0248990
$837$	−34.5201	−1.19319
$838$	19.4995	0.673599
$839$	−39.7983	−1.37399	−0.686994	−	0.726663i	$-0.741070\pi$
$839$	−39.7983	−1.37399	−0.686994	+	0.726663i	$0.741070\pi$
$840$	32.0426	1.10557
$841$	1.28516	0.0443157
$842$	−30.5947	−1.05436
$843$	39.4678	1.35934
$844$	6.01472	0.207035
$845$	26.8110	0.922328
$846$	−61.8548	−2.12661
$847$	−5.09280	−0.174991
$848$	−17.7571	−0.609783
$849$	−36.6409	−1.25751
$850$	−11.0994	−0.380707
$851$	14.6278	0.501435
$852$	−8.86617	−0.303750
$853$	−18.1553	−0.621627	−0.310813	−	0.950471i	$-0.600601\pi$
$853$	−18.1553	−0.621627	−0.310813	+	0.950471i	$0.600601\pi$
$854$	−98.8075	−3.38112
$855$	−5.88203	−0.201161
$856$	12.6397	0.432018
$857$	1.67609	0.0572543	0.0286271	−	0.999590i	$-0.490886\pi$
$857$	1.67609	0.0572543	0.0286271	+	0.999590i	$0.490886\pi$
$858$	−31.0124	−1.05875
$859$	−41.6306	−1.42042	−0.710209	−	0.703991i	$-0.751400\pi$
$859$	−41.6306	−1.42042	−0.710209	+	0.703991i	$0.751400\pi$
$860$	−2.06256	−0.0703327
$861$	−50.3430	−1.71568
$862$	−66.0925	−2.25112
$863$	−6.17721	−0.210275	−0.105137	−	0.994458i	$-0.533528\pi$
$863$	−6.17721	−0.210275	−0.105137	+	0.994458i	$0.533528\pi$
$864$	−33.4504	−1.13801
$865$	−3.15651	−0.107324
$866$	36.4197	1.23759
$867$	−84.3247	−2.86382
$868$	−14.7353	−0.500150
$869$	−10.0436	−0.340705
$870$	27.0489	0.917043
$871$	19.2951	0.653789
$872$	−13.7406	−0.465316
$873$	18.5517	0.627881
$874$	6.65859	0.225230
$875$	−5.09280	−0.172168
$876$	−1.51926	−0.0513312
$877$	−19.6413	−0.663238	−0.331619	−	0.943413i	$-0.607595\pi$
$877$	−19.6413	−0.663238	−0.331619	+	0.943413i	$0.607595\pi$
$878$	−12.3765	−0.417686
$879$	−2.33146	−0.0786384
$880$	−4.92156	−0.165906
$881$	56.0786	1.88934	0.944669	−	0.328026i	$-0.106384\pi$
$881$	56.0786	1.88934	0.944669	+	0.328026i	$0.106384\pi$
$882$	−183.699	−6.18547
$883$	7.77161	0.261536	0.130768	−	0.991413i	$-0.458256\pi$
$883$	7.77161	0.261536	0.130768	+	0.991413i	$0.458256\pi$
$884$	−30.5709	−1.02821
$885$	41.5640	1.39716
$886$	−28.8374	−0.968813
$887$	−26.4601	−0.888445	−0.444222	−	0.895917i	$-0.646520\pi$
$887$	−26.4601	−0.888445	−0.444222	+	0.895917i	$0.646520\pi$
$888$	−22.7954	−0.764962
$889$	−29.4602	−0.988063
$890$	0.157873	0.00529191
$891$	7.95216	0.266407
$892$	8.19356	0.274341
$893$	−6.37629	−0.213374
$894$	17.7576	0.593904
$895$	−0.0274136	−0.000916337 0
$896$	68.3949	2.28491
$897$	75.9210	2.53493
$898$	22.7807	0.760203
$899$	22.1174	0.737655
$900$	4.23460	0.141153
$901$	24.2825	0.808966
$902$	5.47022	0.182138
$903$	−43.4844	−1.44707
$904$	−35.4662	−1.17959
$905$	−5.54723	−0.184396
$906$	−17.6894	−0.587690
$907$	−41.4323	−1.37574	−0.687869	−	0.725835i	$-0.741454\pi$
$907$	−41.4323	−1.37574	−0.687869	+	0.725835i	$0.741454\pi$
$908$	−13.9183	−0.461894
$909$	48.9086	1.62219
$910$	−52.9952	−1.75677
$911$	−10.1150	−0.335127	−0.167563	−	0.985861i	$-0.553590\pi$
$911$	−10.1150	−0.335127	−0.167563	+	0.985861i	$0.553590\pi$
$912$	−14.6676	−0.485692
$913$	−1.16974	−0.0387126
$914$	44.6808	1.47791
$915$	35.0600	1.15905
$916$	−18.0306	−0.595747
$917$	44.1108	1.45667
$918$	95.3354	3.14654
$919$	49.3816	1.62895	0.814474	−	0.580200i	$-0.197026\pi$
$919$	49.3816	1.62895	0.814474	+	0.580200i	$0.197026\pi$
$920$	8.52352	0.281012
$921$	80.7419	2.66054
$922$	36.6729	1.20776
$923$	−26.0733	−0.858214
$924$	10.9269	0.359469
$925$	3.62306	0.119126
$926$	−22.3782	−0.735395
$927$	−110.253	−3.62118
$928$	21.4320	0.703540
$929$	−6.51051	−0.213603	−0.106801	−	0.994280i	$-0.534061\pi$
$929$	−6.51051	−0.213603	−0.106801	+	0.994280i	$0.534061\pi$
$930$	19.7539	0.647756
$931$	−18.9366	−0.620621
$932$	3.46384	0.113462
$933$	48.0183	1.57205
$934$	−26.7469	−0.875185
$935$	6.73010	0.220098
$936$	−78.3508	−2.56098
$937$	−31.1621	−1.01802	−0.509010	−	0.860761i	$-0.669988\pi$
$937$	−31.1621	−1.01802	−0.509010	+	0.860761i	$0.669988\pi$
$938$	−25.6850	−0.838645
$939$	62.5742	2.04203
$940$	4.59043	0.149723
$941$	26.3879	0.860221	0.430111	−	0.902776i	$-0.358475\pi$
$941$	26.3879	0.860221	0.430111	+	0.902776i	$0.358475\pi$
$942$	13.8956	0.452744
$943$	−13.3915	−0.436088
$944$	68.6378	2.23397
$945$	43.7432	1.42297
$946$	4.72497	0.153622
$947$	−27.8703	−0.905663	−0.452832	−	0.891596i	$-0.649586\pi$
$947$	−27.8703	−0.905663	−0.452832	+	0.891596i	$0.649586\pi$
$948$	21.5491	0.699882
$949$	−4.46780	−0.145031
$950$	1.64922	0.0535077
$951$	−40.8542	−1.32479
$952$	−72.3590	−2.34517
$953$	−7.86299	−0.254707	−0.127354	−	0.991857i	$-0.540648\pi$
$953$	−7.86299	−0.254707	−0.127354	+	0.991857i	$0.540648\pi$
$954$	−35.0006	−1.13319
$955$	16.0183	0.518341
$956$	15.8966	0.514131
$957$	−16.4010	−0.530170
$958$	−2.22245	−0.0718042
$959$	−25.2478	−0.815294
$960$	−10.1934	−0.328991
$961$	−14.8476	−0.478955
$962$	37.7012	1.21554
$963$	35.2169	1.13485
$964$	−11.9660	−0.385400
$965$	−18.7111	−0.602330
$966$	−101.064	−3.25167
$967$	20.9490	0.673675	0.336838	−	0.941563i	$-0.390643\pi$
$967$	20.9490	0.673675	0.336838	+	0.941563i	$0.390643\pi$
$968$	2.11113	0.0678542
$969$	20.0575	0.644341
$970$	−5.20159	−0.167013
$971$	−8.59455	−0.275812	−0.137906	−	0.990445i	$-0.544037\pi$
$971$	−8.59455	−0.275812	−0.137906	+	0.990445i	$0.544037\pi$
$972$	1.48885	0.0477550
$973$	34.4705	1.10507
$974$	40.2439	1.28950
$975$	18.8043	0.602220
$976$	57.8973	1.85325
$977$	20.4552	0.654419	0.327209	−	0.944952i	$-0.393892\pi$
$977$	20.4552	0.654419	0.327209	+	0.944952i	$0.393892\pi$
$978$	−80.5996	−2.57729
$979$	−0.0957258	−0.00305941
$980$	13.6329	0.435485
$981$	−38.2841	−1.22232
$982$	29.4426	0.939552
$983$	31.6450	1.00932	0.504660	−	0.863318i	$-0.331618\pi$
$983$	31.6450	1.00932	0.504660	+	0.863318i	$0.331618\pi$
$984$	20.8688	0.665273
$985$	19.2044	0.611904
$986$	−61.0822	−1.94525
$987$	96.7789	3.08051
$988$	4.54242	0.144514
$989$	−11.5671	−0.367813
$990$	−9.70075	−0.308310
$991$	24.2097	0.769047	0.384524	−	0.923115i	$-0.374366\pi$
$991$	24.2097	0.769047	0.384524	+	0.923115i	$0.374366\pi$
$992$	15.6519	0.496947
$993$	81.2183	2.57739
$994$	34.7080	1.10087
$995$	2.62898	0.0833444
$996$	2.50974	0.0795242
$997$	−58.6751	−1.85826	−0.929129	−	0.369755i	$-0.879441\pi$
$997$	−58.6751	−1.85826	−0.929129	+	0.369755i	$0.879441\pi$
$998$	42.5018	1.34537
$999$	−31.1193	−0.984571

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.i.1.4	✓	8
3.2	odd	2		9405.2.a.bf.1.5		8
5.4	even	2		5225.2.a.o.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.i.1.4	✓	8	1.1	even	1	trivial
5225.2.a.o.1.5		8	5.4	even	2
9405.2.a.bf.1.5		8	3.2	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-1.64922 q^{2} -2.98027 q^{3} +0.719922 q^{4} +1.00000 q^{5} +4.91512 q^{6} -5.09280 q^{7} +2.11113 q^{8} +5.88203 q^{9} +O(q^{10})\)	\(q-1.64922 q^{2} -2.98027 q^{3} +0.719922 q^{4} +1.00000 q^{5} +4.91512 q^{6} -5.09280 q^{7} +2.11113 q^{8} +5.88203 q^{9} -1.64922 q^{10} +1.00000 q^{11} -2.14556 q^{12} -6.30960 q^{13} +8.39913 q^{14} -2.98027 q^{15} -4.92156 q^{16} +6.73010 q^{17} -9.70075 q^{18} -1.00000 q^{19} +0.719922 q^{20} +15.1779 q^{21} -1.64922 q^{22} +4.03742 q^{23} -6.29174 q^{24} +1.00000 q^{25} +10.4059 q^{26} -8.58923 q^{27} -3.66641 q^{28} +5.50320 q^{29} +4.91512 q^{30} +4.01900 q^{31} +3.89446 q^{32} -2.98027 q^{33} -11.0994 q^{34} -5.09280 q^{35} +4.23460 q^{36} +3.62306 q^{37} +1.64922 q^{38} +18.8043 q^{39} +2.11113 q^{40} -3.31685 q^{41} -25.0317 q^{42} -2.86498 q^{43} +0.719922 q^{44} +5.88203 q^{45} -6.65859 q^{46} +6.37629 q^{47} +14.6676 q^{48} +18.9366 q^{49} -1.64922 q^{50} -20.0575 q^{51} -4.54242 q^{52} +3.60804 q^{53} +14.1655 q^{54} +1.00000 q^{55} -10.7515 q^{56} +2.98027 q^{57} -9.07597 q^{58} -13.9464 q^{59} -2.14556 q^{60} -11.7640 q^{61} -6.62822 q^{62} -29.9560 q^{63} +3.42029 q^{64} -6.30960 q^{65} +4.91512 q^{66} -3.05805 q^{67} +4.84515 q^{68} -12.0326 q^{69} +8.39913 q^{70} +4.13233 q^{71} +12.4177 q^{72} +0.708095 q^{73} -5.97522 q^{74} -2.98027 q^{75} -0.719922 q^{76} -5.09280 q^{77} -31.0124 q^{78} -10.0436 q^{79} -4.92156 q^{80} +7.95216 q^{81} +5.47022 q^{82} -1.16974 q^{83} +10.9269 q^{84} +6.73010 q^{85} +4.72497 q^{86} -16.4010 q^{87} +2.11113 q^{88} -0.0957258 q^{89} -9.70075 q^{90} +32.1335 q^{91} +2.90663 q^{92} -11.9777 q^{93} -10.5159 q^{94} -1.00000 q^{95} -11.6066 q^{96} +3.15397 q^{97} -31.2306 q^{98} +5.88203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 17 q^{13} + 12 q^{14} - 7 q^{15} + 18 q^{16} - 9 q^{17} - 2 q^{18} - 8 q^{19} + 10 q^{20} + q^{21} - 6 q^{22} - 8 q^{23} + q^{24} + 8 q^{25} + 10 q^{26} - 34 q^{27} - 22 q^{28} - 3 q^{29} - q^{31} - 37 q^{32} - 7 q^{33} - 8 q^{34} - 11 q^{35} + 30 q^{36} - 17 q^{37} + 6 q^{38} + 14 q^{39} - 18 q^{40} - 5 q^{41} + 15 q^{42} - 21 q^{43} + 10 q^{44} + 11 q^{45} - 2 q^{46} - 8 q^{47} + 10 q^{48} + 19 q^{49} - 6 q^{50} - 16 q^{51} + 9 q^{52} - 19 q^{53} - 3 q^{54} + 8 q^{55} + 24 q^{56} + 7 q^{57} + 37 q^{58} - 33 q^{59} - 7 q^{60} - q^{61} - 42 q^{62} - 20 q^{63} + 48 q^{64} - 17 q^{65} - 18 q^{67} - 37 q^{68} + 16 q^{69} + 12 q^{70} - 18 q^{71} + 13 q^{72} - 18 q^{73} + 15 q^{74} - 7 q^{75} - 10 q^{76} - 11 q^{77} - 51 q^{78} - 5 q^{79} + 18 q^{80} + 32 q^{81} + 12 q^{82} - 33 q^{83} - 51 q^{84} - 9 q^{85} - 16 q^{86} - 26 q^{87} - 18 q^{88} - 20 q^{89} - 2 q^{90} + 6 q^{91} - 3 q^{92} + 18 q^{93} + 30 q^{94} - 8 q^{95} + 21 q^{96} - 69 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 17 * q^13 + 12 * q^14 - 7 * q^15 + 18 * q^16 - 9 * q^17 - 2 * q^18 - 8 * q^19 + 10 * q^20 + q^21 - 6 * q^22 - 8 * q^23 + q^24 + 8 * q^25 + 10 * q^26 - 34 * q^27 - 22 * q^28 - 3 * q^29 - q^31 - 37 * q^32 - 7 * q^33 - 8 * q^34 - 11 * q^35 + 30 * q^36 - 17 * q^37 + 6 * q^38 + 14 * q^39 - 18 * q^40 - 5 * q^41 + 15 * q^42 - 21 * q^43 + 10 * q^44 + 11 * q^45 - 2 * q^46 - 8 * q^47 + 10 * q^48 + 19 * q^49 - 6 * q^50 - 16 * q^51 + 9 * q^52 - 19 * q^53 - 3 * q^54 + 8 * q^55 + 24 * q^56 + 7 * q^57 + 37 * q^58 - 33 * q^59 - 7 * q^60 - q^61 - 42 * q^62 - 20 * q^63 + 48 * q^64 - 17 * q^65 - 18 * q^67 - 37 * q^68 + 16 * q^69 + 12 * q^70 - 18 * q^71 + 13 * q^72 - 18 * q^73 + 15 * q^74 - 7 * q^75 - 10 * q^76 - 11 * q^77 - 51 * q^78 - 5 * q^79 + 18 * q^80 + 32 * q^81 + 12 * q^82 - 33 * q^83 - 51 * q^84 - 9 * q^85 - 16 * q^86 - 26 * q^87 - 18 * q^88 - 20 * q^89 - 2 * q^90 + 6 * q^91 - 3 * q^92 + 18 * q^93 + 30 * q^94 - 8 * q^95 + 21 * q^96 - 69 * q^98 + 11 * q^99

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