LMFDB - Embedded newform 867.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,2,Mod(1,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, 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("867.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	867.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:	$6.92302985525$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.765367$ of defining polynomial
Character	$\chi$	$=$	867.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.765367 q^{2} +1.00000 q^{3} -1.41421 q^{4} +0.0823922 q^{5} +0.765367 q^{6} -2.76537 q^{7} -2.61313 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.765367 q^{2} +1.00000 q^{3} -1.41421 q^{4} +0.0823922 q^{5} +0.765367 q^{6} -2.76537 q^{7} -2.61313 q^{8} +1.00000 q^{9} +0.0630603 q^{10} -0.668179 q^{11} -1.41421 q^{12} -3.28130 q^{13} -2.11652 q^{14} +0.0823922 q^{15} +0.828427 q^{16} +0.765367 q^{18} +3.64047 q^{19} -0.116520 q^{20} -2.76537 q^{21} -0.511402 q^{22} -9.30864 q^{23} -2.61313 q^{24} -4.99321 q^{25} -2.51140 q^{26} +1.00000 q^{27} +3.91082 q^{28} -6.24264 q^{29} +0.0630603 q^{30} -5.04373 q^{31} +5.86030 q^{32} -0.668179 q^{33} -0.227845 q^{35} -1.41421 q^{36} -2.40621 q^{37} +2.78629 q^{38} -3.28130 q^{39} -0.215301 q^{40} +0.480217 q^{41} -2.11652 q^{42} +8.27452 q^{43} +0.944947 q^{44} +0.0823922 q^{45} -7.12453 q^{46} -8.88311 q^{47} +0.828427 q^{48} +0.647254 q^{49} -3.82164 q^{50} +4.64047 q^{52} +11.3524 q^{53} +0.765367 q^{54} -0.0550527 q^{55} +7.22625 q^{56} +3.64047 q^{57} -4.77791 q^{58} +9.59379 q^{59} -0.116520 q^{60} +2.58701 q^{61} -3.86030 q^{62} -2.76537 q^{63} +2.82843 q^{64} -0.270354 q^{65} -0.511402 q^{66} -0.944947 q^{67} -9.30864 q^{69} -0.174385 q^{70} +3.28809 q^{71} -2.61313 q^{72} -15.6819 q^{73} -1.84163 q^{74} -4.99321 q^{75} -5.14840 q^{76} +1.84776 q^{77} -2.51140 q^{78} +8.37170 q^{79} +0.0682559 q^{80} +1.00000 q^{81} +0.367542 q^{82} +0.899869 q^{83} +3.91082 q^{84} +6.33304 q^{86} -6.24264 q^{87} +1.74603 q^{88} +5.64431 q^{89} +0.0630603 q^{90} +9.07401 q^{91} +13.1644 q^{92} -5.04373 q^{93} -6.79884 q^{94} +0.299946 q^{95} +5.86030 q^{96} +10.9723 q^{97} +0.495387 q^{98} -0.668179 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{5} - 8 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^5 - 8 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{5} - 8 q^{7} + 4 q^{9} - 8 q^{10} - 4 q^{11} - 4 q^{13} - 8 q^{14} - 4 q^{15} - 8 q^{16} - 12 q^{19} - 8 q^{21} + 8 q^{22} - 12 q^{23} + 4 q^{27} - 8 q^{29} - 8 q^{30} - 8 q^{31} - 4 q^{33} + 16 q^{35} - 24 q^{37} + 16 q^{38} - 4 q^{39} - 12 q^{41} - 8 q^{42} + 4 q^{43} - 8 q^{44} - 4 q^{45} - 8 q^{46} + 8 q^{47} - 8 q^{48} - 4 q^{49} + 16 q^{50} - 8 q^{52} + 8 q^{53} - 12 q^{55} + 8 q^{56} - 12 q^{57} + 24 q^{59} - 24 q^{61} + 8 q^{62} - 8 q^{63} - 12 q^{65} + 8 q^{66} + 8 q^{67} - 12 q^{69} + 24 q^{70} + 24 q^{71} - 8 q^{73} + 8 q^{74} - 8 q^{76} + 8 q^{80} + 4 q^{81} + 8 q^{82} + 8 q^{83} - 16 q^{86} - 8 q^{87} + 16 q^{89} - 8 q^{90} + 8 q^{91} - 8 q^{93} - 16 q^{94} + 12 q^{95} + 16 q^{97} + 32 q^{98} - 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^5 - 8 * q^7 + 4 * q^9 - 8 * q^10 - 4 * q^11 - 4 * q^13 - 8 * q^14 - 4 * q^15 - 8 * q^16 - 12 * q^19 - 8 * q^21 + 8 * q^22 - 12 * q^23 + 4 * q^27 - 8 * q^29 - 8 * q^30 - 8 * q^31 - 4 * q^33 + 16 * q^35 - 24 * q^37 + 16 * q^38 - 4 * q^39 - 12 * q^41 - 8 * q^42 + 4 * q^43 - 8 * q^44 - 4 * q^45 - 8 * q^46 + 8 * q^47 - 8 * q^48 - 4 * q^49 + 16 * q^50 - 8 * q^52 + 8 * q^53 - 12 * q^55 + 8 * q^56 - 12 * q^57 + 24 * q^59 - 24 * q^61 + 8 * q^62 - 8 * q^63 - 12 * q^65 + 8 * q^66 + 8 * q^67 - 12 * q^69 + 24 * q^70 + 24 * q^71 - 8 * q^73 + 8 * q^74 - 8 * q^76 + 8 * q^80 + 4 * q^81 + 8 * q^82 + 8 * q^83 - 16 * q^86 - 8 * q^87 + 16 * q^89 - 8 * q^90 + 8 * q^91 - 8 * q^93 - 16 * q^94 + 12 * q^95 + 16 * q^97 + 32 * 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 $

$2$	0.765367	0.541196	0.270598	−	0.962692i	$-0.412779\pi$
$2$	0.765367	0.541196	0.270598	+	0.962692i	$0.412779\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.41421	−0.707107
$5$	0.0823922	0.0368469	0.0184235	−	0.999830i	$-0.494135\pi$
$5$	0.0823922	0.0368469	0.0184235	+	0.999830i	$0.494135\pi$
$6$	0.765367	0.312460
$7$	−2.76537	−1.04521	−0.522605	−	0.852575i	$-0.675040\pi$
$7$	−2.76537	−1.04521	−0.522605	+	0.852575i	$0.675040\pi$
$8$	−2.61313	−0.923880
$9$	1.00000	0.333333
$10$	0.0630603	0.0199414
$11$	−0.668179	−0.201463	−0.100732	−	0.994914i	$-0.532118\pi$
$11$	−0.668179	−0.201463	−0.100732	+	0.994914i	$0.532118\pi$
$12$	−1.41421	−0.408248
$13$	−3.28130	−0.910070	−0.455035	−	0.890474i	$-0.650373\pi$
$13$	−3.28130	−0.910070	−0.455035	+	0.890474i	$0.650373\pi$
$14$	−2.11652	−0.565664
$15$	0.0823922	0.0212736
$16$	0.828427	0.207107
$17$	0	0
$17$	0	0
$18$	0.765367	0.180399
$19$	3.64047	0.835180	0.417590	−	0.908636i	$-0.362875\pi$
$19$	3.64047	0.835180	0.417590	+	0.908636i	$0.362875\pi$
$20$	−0.116520	−0.0260547
$21$	−2.76537	−0.603453
$22$	−0.511402	−0.109031
$23$	−9.30864	−1.94099	−0.970493	−	0.241128i	$-0.922483\pi$
$23$	−9.30864	−1.94099	−0.970493	+	0.241128i	$0.922483\pi$
$24$	−2.61313	−0.533402
$25$	−4.99321	−0.998642
$26$	−2.51140	−0.492526
$27$	1.00000	0.192450
$28$	3.91082	0.739075
$29$	−6.24264	−1.15923	−0.579615	−	0.814891i	$-0.696797\pi$
$29$	−6.24264	−1.15923	−0.579615	+	0.814891i	$0.696797\pi$
$30$	0.0630603	0.0115132
$31$	−5.04373	−0.905880	−0.452940	−	0.891541i	$-0.649625\pi$
$31$	−5.04373	−0.905880	−0.452940	+	0.891541i	$0.649625\pi$
$32$	5.86030	1.03596
$33$	−0.668179	−0.116315
$34$	0	0
$35$	−0.227845	−0.0385128
$36$	−1.41421	−0.235702
$37$	−2.40621	−0.395578	−0.197789	−	0.980245i	$-0.563376\pi$
$37$	−2.40621	−0.395578	−0.197789	+	0.980245i	$0.563376\pi$
$38$	2.78629	0.451996
$39$	−3.28130	−0.525429
$40$	−0.215301	−0.0340421
$41$	0.480217	0.0749973	0.0374986	−	0.999297i	$-0.488061\pi$
$41$	0.480217	0.0749973	0.0374986	+	0.999297i	$0.488061\pi$
$42$	−2.11652	−0.326586
$43$	8.27452	1.26185	0.630926	−	0.775843i	$-0.282675\pi$
$43$	8.27452	1.26185	0.630926	+	0.775843i	$0.282675\pi$
$44$	0.944947	0.142456
$45$	0.0823922	0.0122823
$46$	−7.12453	−1.05045
$47$	−8.88311	−1.29573	−0.647867	−	0.761753i	$-0.724339\pi$
$47$	−8.88311	−1.29573	−0.647867	+	0.761753i	$0.724339\pi$
$48$	0.828427	0.119573
$49$	0.647254	0.0924648
$50$	−3.82164	−0.540461
$51$	0	0
$52$	4.64047	0.643517
$53$	11.3524	1.55937	0.779684	−	0.626173i	$-0.215380\pi$
$53$	11.3524	1.55937	0.779684	+	0.626173i	$0.215380\pi$
$54$	0.765367	0.104153
$55$	−0.0550527	−0.00742331
$56$	7.22625	0.965649
$57$	3.64047	0.482191
$58$	−4.77791	−0.627370
$59$	9.59379	1.24901	0.624503	−	0.781023i	$-0.285302\pi$
$59$	9.59379	1.24901	0.624503	+	0.781023i	$0.285302\pi$
$60$	−0.116520	−0.0150427
$61$	2.58701	0.331232	0.165616	−	0.986190i	$-0.447039\pi$
$61$	2.58701	0.331232	0.165616	+	0.986190i	$0.447039\pi$
$62$	−3.86030	−0.490259
$63$	−2.76537	−0.348403
$64$	2.82843	0.353553
$65$	−0.270354	−0.0335333
$66$	−0.511402	−0.0629492
$67$	−0.944947	−0.115444	−0.0577218	−	0.998333i	$-0.518384\pi$
$67$	−0.944947	−0.115444	−0.0577218	+	0.998333i	$0.518384\pi$
$68$	0	0
$69$	−9.30864	−1.12063
$70$	−0.174385	−0.0208430
$71$	3.28809	0.390225	0.195112	−	0.980781i	$-0.437493\pi$
$71$	3.28809	0.390225	0.195112	+	0.980781i	$0.437493\pi$
$72$	−2.61313	−0.307960
$73$	−15.6819	−1.83543	−0.917716	−	0.397237i	$-0.869969\pi$
$73$	−15.6819	−1.83543	−0.917716	+	0.397237i	$0.869969\pi$
$74$	−1.84163	−0.214085
$75$	−4.99321	−0.576566
$76$	−5.14840	−0.590561
$77$	1.84776	0.210572
$78$	−2.51140	−0.284360
$79$	8.37170	0.941890	0.470945	−	0.882162i	$-0.343913\pi$
$79$	8.37170	0.941890	0.470945	+	0.882162i	$0.343913\pi$
$80$	0.0682559	0.00763125
$81$	1.00000	0.111111
$82$	0.367542	0.0405882
$83$	0.899869	0.0987734	0.0493867	−	0.998780i	$-0.484273\pi$
$83$	0.899869	0.0987734	0.0493867	+	0.998780i	$0.484273\pi$
$84$	3.91082	0.426705
$85$	0	0
$86$	6.33304	0.682909
$87$	−6.24264	−0.669281
$88$	1.74603	0.186128
$89$	5.64431	0.598296	0.299148	−	0.954207i	$-0.403298\pi$
$89$	5.64431	0.598296	0.299148	+	0.954207i	$0.403298\pi$
$90$	0.0630603	0.00664714
$91$	9.07401	0.951215
$92$	13.1644	1.37248
$93$	−5.04373	−0.523010
$94$	−6.79884	−0.701246
$95$	0.299946	0.0307738
$96$	5.86030	0.598115
$97$	10.9723	1.11407	0.557033	−	0.830490i	$-0.311940\pi$
$97$	10.9723	1.11407	0.557033	+	0.830490i	$0.311940\pi$
$98$	0.495387	0.0500416
$99$	−0.668179	−0.0671545
$100$	7.06147	0.706147
$101$	−3.27677	−0.326051	−0.163025	−	0.986622i	$-0.552125\pi$
$101$	−3.27677	−0.326051	−0.163025	+	0.986622i	$0.552125\pi$
$102$	0	0
$103$	−12.0228	−1.18464	−0.592321	−	0.805702i	$-0.701788\pi$
$103$	−12.0228	−1.18464	−0.592321	+	0.805702i	$0.701788\pi$
$104$	8.57446	0.840795
$105$	−0.227845	−0.0222354
$106$	8.68873	0.843924
$107$	8.72927	0.843891	0.421945	−	0.906621i	$-0.361347\pi$
$107$	8.72927	0.843891	0.421945	+	0.906621i	$0.361347\pi$
$108$	−1.41421	−0.136083
$109$	4.26998	0.408990	0.204495	−	0.978868i	$-0.434445\pi$
$109$	4.26998	0.408990	0.204495	+	0.978868i	$0.434445\pi$
$110$	−0.0421355	−0.00401746
$111$	−2.40621	−0.228387
$112$	−2.29090	−0.216470
$113$	−13.8558	−1.30344	−0.651720	−	0.758459i	$-0.725952\pi$
$113$	−13.8558	−1.30344	−0.651720	+	0.758459i	$0.725952\pi$
$114$	2.78629	0.260960
$115$	−0.766960	−0.0715194
$116$	8.82843	0.819699
$117$	−3.28130	−0.303357
$118$	7.34277	0.675957
$119$	0	0
$120$	−0.215301	−0.0196542
$121$	−10.5535	−0.959412
$122$	1.98001	0.179262
$123$	0.480217	0.0432997
$124$	7.13291	0.640554
$125$	−0.823363	−0.0736438
$126$	−2.11652	−0.188555
$127$	0.633677	0.0562297	0.0281149	−	0.999605i	$-0.491050\pi$
$127$	0.633677	0.0562297	0.0281149	+	0.999605i	$0.491050\pi$
$128$	−9.55582	−0.844623
$129$	8.27452	0.728531
$130$	−0.206920	−0.0181481
$131$	15.2081	1.32874	0.664371	−	0.747403i	$-0.268700\pi$
$131$	15.2081	1.32874	0.664371	+	0.747403i	$0.268700\pi$
$132$	0.944947	0.0822471
$133$	−10.0672	−0.872939
$134$	−0.723231	−0.0624777
$135$	0.0823922	0.00709119
$136$	0	0
$137$	2.38847	0.204060	0.102030	−	0.994781i	$-0.467466\pi$
$137$	2.38847	0.204060	0.102030	+	0.994781i	$0.467466\pi$
$138$	−7.12453	−0.606480
$139$	−1.61472	−0.136959	−0.0684793	−	0.997653i	$-0.521815\pi$
$139$	−1.61472	−0.136959	−0.0684793	+	0.997653i	$0.521815\pi$
$140$	0.322221	0.0272326
$141$	−8.88311	−0.748092
$142$	2.51660	0.211188
$143$	2.19250	0.183346
$144$	0.828427	0.0690356
$145$	−0.514345	−0.0427140
$146$	−12.0024	−0.993329
$147$	0.647254	0.0533846
$148$	3.40289	0.279716
$149$	10.8082	0.885442	0.442721	−	0.896660i	$-0.354013\pi$
$149$	10.8082	0.885442	0.442721	+	0.896660i	$0.354013\pi$
$150$	−3.82164	−0.312035
$151$	−21.7566	−1.77053	−0.885264	−	0.465089i	$-0.846022\pi$
$151$	−21.7566	−1.77053	−0.885264	+	0.465089i	$0.846022\pi$
$152$	−9.51299	−0.771606
$153$	0	0
$154$	1.41421	0.113961
$155$	−0.415564	−0.0333789
$156$	4.64047	0.371535
$157$	−19.8535	−1.58448	−0.792241	−	0.610208i	$-0.791086\pi$
$157$	−19.8535	−1.58448	−0.792241	+	0.610208i	$0.791086\pi$
$158$	6.40743	0.509747
$159$	11.3524	0.900302
$160$	0.482843	0.0381721
$161$	25.7418	2.02874
$162$	0.765367	0.0601329
$163$	1.75511	0.137471	0.0687353	−	0.997635i	$-0.478104\pi$
$163$	1.75511	0.137471	0.0687353	+	0.997635i	$0.478104\pi$
$164$	−0.679129	−0.0530311
$165$	−0.0550527	−0.00428585
$166$	0.688730	0.0534558
$167$	8.05921	0.623641	0.311820	−	0.950141i	$-0.399061\pi$
$167$	8.05921	0.623641	0.311820	+	0.950141i	$0.399061\pi$
$168$	7.22625	0.557517
$169$	−2.23304	−0.171772
$170$	0	0
$171$	3.64047	0.278393
$172$	−11.7019	−0.892264
$173$	−5.75282	−0.437379	−0.218690	−	0.975794i	$-0.570178\pi$
$173$	−5.75282	−0.437379	−0.218690	+	0.975794i	$0.570178\pi$
$174$	−4.77791	−0.362212
$175$	13.8081	1.04379
$176$	−0.553537	−0.0417244
$177$	9.59379	0.721114
$178$	4.31997	0.323795
$179$	14.6173	1.09255	0.546274	−	0.837607i	$-0.316046\pi$
$179$	14.6173	1.09255	0.546274	+	0.837607i	$0.316046\pi$
$180$	−0.116520	−0.00868490
$181$	−14.1000	−1.04804	−0.524022	−	0.851704i	$-0.675569\pi$
$181$	−14.1000	−1.04804	−0.524022	+	0.851704i	$0.675569\pi$
$182$	6.94495	0.514794
$183$	2.58701	0.191237
$184$	24.3247	1.79324
$185$	−0.198253	−0.0145758
$186$	−3.86030	−0.283051
$187$	0	0
$188$	12.5626	0.916222
$189$	−2.76537	−0.201151
$190$	0.229569	0.0166547
$191$	16.0167	1.15893	0.579464	−	0.814998i	$-0.303262\pi$
$191$	16.0167	1.15893	0.579464	+	0.814998i	$0.303262\pi$
$192$	2.82843	0.204124
$193$	4.59539	0.330783	0.165392	−	0.986228i	$-0.447111\pi$
$193$	4.59539	0.330783	0.165392	+	0.986228i	$0.447111\pi$
$194$	8.39782	0.602929
$195$	−0.270354	−0.0193604
$196$	−0.915355	−0.0653825
$197$	−21.1689	−1.50822	−0.754112	−	0.656745i	$-0.771933\pi$
$197$	−21.1689	−1.50822	−0.754112	+	0.656745i	$0.771933\pi$
$198$	−0.511402	−0.0363437
$199$	2.32541	0.164844	0.0824219	−	0.996598i	$-0.473735\pi$
$199$	2.32541	0.164844	0.0824219	+	0.996598i	$0.473735\pi$
$200$	13.0479	0.922625
$201$	−0.944947	−0.0666514
$202$	−2.50793	−0.176457
$203$	17.2632	1.21164
$204$	0	0
$205$	0.0395661	0.00276342
$206$	−9.20186	−0.641124
$207$	−9.30864	−0.646995
$208$	−2.71832	−0.188482
$209$	−2.43248	−0.168258
$210$	−0.174385	−0.0120337
$211$	−7.58541	−0.522201	−0.261101	−	0.965312i	$-0.584085\pi$
$211$	−7.58541	−0.522201	−0.261101	+	0.965312i	$0.584085\pi$
$212$	−16.0547	−1.10264
$213$	3.28809	0.225296
$214$	6.68110	0.456710
$215$	0.681756	0.0464953
$216$	−2.61313	−0.177801
$217$	13.9478	0.946836
$218$	3.26810	0.221344
$219$	−15.6819	−1.05969
$220$	0.0778563	0.00524907
$221$	0	0
$222$	−1.84163	−0.123602
$223$	−15.6624	−1.04883	−0.524415	−	0.851463i	$-0.675716\pi$
$223$	−15.6624	−1.04883	−0.524415	+	0.851463i	$0.675716\pi$
$224$	−16.2059	−1.08280
$225$	−4.99321	−0.332881
$226$	−10.6047	−0.705417
$227$	−14.6314	−0.971122	−0.485561	−	0.874203i	$-0.661385\pi$
$227$	−14.6314	−0.971122	−0.485561	+	0.874203i	$0.661385\pi$
$228$	−5.14840	−0.340961
$229$	16.3451	1.08011	0.540056	−	0.841629i	$-0.318403\pi$
$229$	16.3451	1.08011	0.540056	+	0.841629i	$0.318403\pi$
$230$	−0.587006	−0.0387060
$231$	1.84776	0.121574
$232$	16.3128	1.07099
$233$	0.357811	0.0234409	0.0117205	−	0.999931i	$-0.496269\pi$
$233$	0.357811	0.0234409	0.0117205	+	0.999931i	$0.496269\pi$
$234$	−2.51140	−0.164175
$235$	−0.731899	−0.0477438
$236$	−13.5677	−0.883180
$237$	8.37170	0.543801
$238$	0	0
$239$	−9.71153	−0.628187	−0.314093	−	0.949392i	$-0.601701\pi$
$239$	−9.71153	−0.628187	−0.314093	+	0.949392i	$0.601701\pi$
$240$	0.0682559	0.00440590
$241$	−9.36173	−0.603042	−0.301521	−	0.953460i	$-0.597494\pi$
$241$	−9.36173	−0.603042	−0.301521	+	0.953460i	$0.597494\pi$
$242$	−8.07733	−0.519230
$243$	1.00000	0.0641500
$244$	−3.65858	−0.234216
$245$	0.0533287	0.00340704
$246$	0.367542	0.0234336
$247$	−11.9455	−0.760072
$248$	13.1799	0.836924
$249$	0.899869	0.0570269
$250$	−0.630175	−0.0398557
$251$	3.28196	0.207156	0.103578	−	0.994621i	$-0.466971\pi$
$251$	3.28196	0.207156	0.103578	+	0.994621i	$0.466971\pi$
$252$	3.91082	0.246358
$253$	6.21984	0.391038
$254$	0.484995	0.0304313
$255$	0	0
$256$	−12.9706	−0.810660
$257$	3.54500	0.221131	0.110566	−	0.993869i	$-0.464734\pi$
$257$	3.54500	0.221131	0.110566	+	0.993869i	$0.464734\pi$
$258$	6.33304	0.394278
$259$	6.65404	0.413462
$260$	0.382338	0.0237116
$261$	−6.24264	−0.386410
$262$	11.6398	0.719110
$263$	13.5133	0.833265	0.416632	−	0.909075i	$-0.363210\pi$
$263$	13.5133	0.833265	0.416632	+	0.909075i	$0.363210\pi$
$264$	1.74603	0.107461
$265$	0.935347	0.0574579
$266$	−7.70512	−0.472431
$267$	5.64431	0.345426
$268$	1.33636	0.0816310
$269$	−5.16895	−0.315156	−0.157578	−	0.987507i	$-0.550369\pi$
$269$	−5.16895	−0.315156	−0.157578	+	0.987507i	$0.550369\pi$
$270$	0.0630603	0.00383773
$271$	−6.68592	−0.406141	−0.203070	−	0.979164i	$-0.565092\pi$
$271$	−6.68592	−0.406141	−0.203070	+	0.979164i	$0.565092\pi$
$272$	0	0
$273$	9.07401	0.549184
$274$	1.82805	0.110437
$275$	3.33636	0.201190
$276$	13.1644	0.792404
$277$	−2.90159	−0.174340	−0.0871699	−	0.996193i	$-0.527782\pi$
$277$	−2.90159	−0.174340	−0.0871699	+	0.996193i	$0.527782\pi$
$278$	−1.23585	−0.0741215
$279$	−5.04373	−0.301960
$280$	0.595387	0.0355812
$281$	−22.9207	−1.36734	−0.683668	−	0.729793i	$-0.739616\pi$
$281$	−22.9207	−1.36734	−0.683668	+	0.729793i	$0.739616\pi$
$282$	−6.79884	−0.404865
$283$	−27.7209	−1.64784	−0.823918	−	0.566709i	$-0.808216\pi$
$283$	−27.7209	−1.64784	−0.823918	+	0.566709i	$0.808216\pi$
$284$	−4.65007	−0.275931
$285$	0.299946	0.0177673
$286$	1.67807	0.0992261
$287$	−1.32798	−0.0783879
$288$	5.86030	0.345322
$289$	0	0
$290$	−0.393663	−0.0231167
$291$	10.9723	0.643207
$292$	22.1776	1.29785
$293$	11.4677	0.669949	0.334974	−	0.942227i	$-0.391272\pi$
$293$	11.4677	0.669949	0.334974	+	0.942227i	$0.391272\pi$
$294$	0.495387	0.0288915
$295$	0.790454	0.0460220
$296$	6.28772	0.365466
$297$	−0.668179	−0.0387717
$298$	8.27223	0.479198
$299$	30.5445	1.76643
$300$	7.06147	0.407694
$301$	−22.8821	−1.31890
$302$	−16.6518	−0.958203
$303$	−3.27677	−0.188245
$304$	3.01586	0.172971
$305$	0.213149	0.0122049
$306$	0	0
$307$	10.5245	0.600663	0.300332	−	0.953835i	$-0.402903\pi$
$307$	10.5245	0.600663	0.300332	+	0.953835i	$0.402903\pi$
$308$	−2.61313	−0.148897
$309$	−12.0228	−0.683953
$310$	−0.318059	−0.0180645
$311$	5.41421	0.307012	0.153506	−	0.988148i	$-0.450944\pi$
$311$	5.41421	0.307012	0.153506	+	0.988148i	$0.450944\pi$
$312$	8.57446	0.485433
$313$	0.951362	0.0537742	0.0268871	−	0.999638i	$-0.491441\pi$
$313$	0.951362	0.0537742	0.0268871	+	0.999638i	$0.491441\pi$
$314$	−15.1952	−0.857516
$315$	−0.227845	−0.0128376
$316$	−11.8394	−0.666017
$317$	21.5647	1.21119	0.605596	−	0.795772i	$-0.292935\pi$
$317$	21.5647	1.21119	0.605596	+	0.795772i	$0.292935\pi$
$318$	8.68873	0.487240
$319$	4.17120	0.233542
$320$	0.233040	0.0130274
$321$	8.72927	0.487220
$322$	19.7019	1.09795
$323$	0	0
$324$	−1.41421	−0.0785674
$325$	16.3842	0.908835
$326$	1.34330	0.0743985
$327$	4.26998	0.236130
$328$	−1.25487	−0.0692885
$329$	24.5650	1.35431
$330$	−0.0421355	−0.00231948
$331$	−20.7366	−1.13979	−0.569894	−	0.821718i	$-0.693016\pi$
$331$	−20.7366	−1.13979	−0.569894	+	0.821718i	$0.693016\pi$
$332$	−1.27261	−0.0698434
$333$	−2.40621	−0.131859
$334$	6.16826	0.337512
$335$	−0.0778563	−0.00425374
$336$	−2.29090	−0.124979
$337$	13.9275	0.758681	0.379340	−	0.925257i	$-0.376151\pi$
$337$	13.9275	0.758681	0.379340	+	0.925257i	$0.376151\pi$
$338$	−1.70910	−0.0929625
$339$	−13.8558	−0.752542
$340$	0	0
$341$	3.37011	0.182502
$342$	2.78629	0.150665
$343$	17.5677	0.948565
$344$	−21.6224	−1.16580
$345$	−0.766960	−0.0412917
$346$	−4.40302	−0.236708
$347$	−16.0894	−0.863722	−0.431861	−	0.901940i	$-0.642143\pi$
$347$	−16.0894	−0.863722	−0.431861	+	0.901940i	$0.642143\pi$
$348$	8.82843	0.473253
$349$	9.30051	0.497845	0.248922	−	0.968523i	$-0.419924\pi$
$349$	9.30051	0.497845	0.248922	+	0.968523i	$0.419924\pi$
$350$	10.5682	0.564896
$351$	−3.28130	−0.175143
$352$	−3.91573	−0.208709
$353$	−27.8142	−1.48040	−0.740201	−	0.672385i	$-0.765270\pi$
$353$	−27.8142	−1.48040	−0.740201	+	0.672385i	$0.765270\pi$
$354$	7.34277	0.390264
$355$	0.270913	0.0143786
$356$	−7.98226	−0.423059
$357$	0	0
$358$	11.1876	0.591282
$359$	4.64022	0.244902	0.122451	−	0.992475i	$-0.460925\pi$
$359$	4.64022	0.244902	0.122451	+	0.992475i	$0.460925\pi$
$360$	−0.215301	−0.0113474
$361$	−5.74701	−0.302474
$362$	−10.7917	−0.567198
$363$	−10.5535	−0.553917
$364$	−12.8326	−0.672610
$365$	−1.29207	−0.0676300
$366$	1.98001	0.103497
$367$	−4.32119	−0.225564	−0.112782	−	0.993620i	$-0.535976\pi$
$367$	−4.32119	−0.225564	−0.112782	+	0.993620i	$0.535976\pi$
$368$	−7.71153	−0.401991
$369$	0.480217	0.0249991
$370$	−0.151736	−0.00788838
$371$	−31.3935	−1.62987
$372$	7.13291	0.369824
$373$	−8.44020	−0.437017	−0.218509	−	0.975835i	$-0.570119\pi$
$373$	−8.44020	−0.437017	−0.218509	+	0.975835i	$0.570119\pi$
$374$	0	0
$375$	−0.823363	−0.0425183
$376$	23.2127	1.19710
$377$	20.4840	1.05498
$378$	−2.11652	−0.108862
$379$	7.65616	0.393271	0.196635	−	0.980477i	$-0.436998\pi$
$379$	7.65616	0.393271	0.196635	+	0.980477i	$0.436998\pi$
$380$	−0.424188	−0.0217604
$381$	0.633677	0.0324643
$382$	12.2587	0.627207
$383$	18.5264	0.946654	0.473327	−	0.880887i	$-0.343053\pi$
$383$	18.5264	0.946654	0.473327	+	0.880887i	$0.343053\pi$
$384$	−9.55582	−0.487643
$385$	0.152241	0.00775892
$386$	3.51716	0.179019
$387$	8.27452	0.420617
$388$	−15.5172	−0.787764
$389$	26.8124	1.35944	0.679720	−	0.733472i	$-0.262101\pi$
$389$	26.8124	1.35944	0.679720	+	0.733472i	$0.262101\pi$
$390$	−0.206920	−0.0104778
$391$	0	0
$392$	−1.69136	−0.0854264
$393$	15.2081	0.767149
$394$	−16.2020	−0.816245
$395$	0.689763	0.0347058
$396$	0.944947	0.0474854
$397$	−3.14026	−0.157605	−0.0788025	−	0.996890i	$-0.525110\pi$
$397$	−3.14026	−0.157605	−0.0788025	+	0.996890i	$0.525110\pi$
$398$	1.77979	0.0892128
$399$	−10.0672	−0.503992
$400$	−4.13651	−0.206826
$401$	−26.4666	−1.32168	−0.660840	−	0.750526i	$-0.729800\pi$
$401$	−26.4666	−1.32168	−0.660840	+	0.750526i	$0.729800\pi$
$402$	−0.723231	−0.0360715
$403$	16.5500	0.824415
$404$	4.63405	0.230553
$405$	0.0823922	0.00409410
$406$	13.2127	0.655734
$407$	1.60778	0.0796945
$408$	0	0
$409$	−31.0443	−1.53504	−0.767521	−	0.641024i	$-0.778510\pi$
$409$	−31.0443	−1.53504	−0.767521	+	0.641024i	$0.778510\pi$
$410$	0.0302826	0.00149555
$411$	2.38847	0.117814
$412$	17.0028	0.837668
$413$	−26.5304	−1.30547
$414$	−7.12453	−0.350151
$415$	0.0741422	0.00363950
$416$	−19.2294	−0.942801
$417$	−1.61472	−0.0790731
$418$	−1.86174	−0.0910607
$419$	38.7394	1.89255	0.946273	−	0.323370i	$-0.104816\pi$
$419$	38.7394	1.89255	0.946273	+	0.323370i	$0.104816\pi$
$420$	0.322221	0.0157228
$421$	−19.0379	−0.927851	−0.463926	−	0.885874i	$-0.653560\pi$
$421$	−19.0379	−0.927851	−0.463926	+	0.885874i	$0.653560\pi$
$422$	−5.80562	−0.282613
$423$	−8.88311	−0.431911
$424$	−29.6652	−1.44067
$425$	0	0
$426$	2.51660	0.121930
$427$	−7.15402	−0.346207
$428$	−12.3451	−0.596721
$429$	2.19250	0.105855
$430$	0.521793	0.0251631
$431$	11.9054	0.573462	0.286731	−	0.958011i	$-0.407431\pi$
$431$	11.9054	0.573462	0.286731	+	0.958011i	$0.407431\pi$
$432$	0.828427	0.0398577
$433$	−21.3675	−1.02686	−0.513428	−	0.858133i	$-0.671625\pi$
$433$	−21.3675	−1.02686	−0.513428	+	0.858133i	$0.671625\pi$
$434$	10.6752	0.512424
$435$	−0.514345	−0.0246610
$436$	−6.03866	−0.289200
$437$	−33.8878	−1.62107
$438$	−12.0024	−0.573499
$439$	−21.1872	−1.01121	−0.505606	−	0.862764i	$-0.668731\pi$
$439$	−21.1872	−1.01121	−0.505606	+	0.862764i	$0.668731\pi$
$440$	0.143860	0.00685824
$441$	0.647254	0.0308216
$442$	0	0
$443$	−21.5467	−1.02372	−0.511858	−	0.859070i	$-0.671043\pi$
$443$	−21.5467	−1.02372	−0.511858	+	0.859070i	$0.671043\pi$
$444$	3.40289	0.161494
$445$	0.465047	0.0220454
$446$	−11.9875	−0.567623
$447$	10.8082	0.511210
$448$	−7.82164	−0.369538
$449$	−36.1769	−1.70729	−0.853646	−	0.520854i	$-0.825614\pi$
$449$	−36.1769	−1.70729	−0.853646	+	0.520854i	$0.825614\pi$
$450$	−3.82164	−0.180154
$451$	−0.320871	−0.0151092
$452$	19.5950	0.921672
$453$	−21.7566	−1.02221
$454$	−11.1984	−0.525567
$455$	0.747628	0.0350493
$456$	−9.51299	−0.445487
$457$	28.2159	1.31988	0.659942	−	0.751316i	$-0.270581\pi$
$457$	28.2159	1.31988	0.659942	+	0.751316i	$0.270581\pi$
$458$	12.5100	0.584552
$459$	0	0
$460$	1.08464	0.0505718
$461$	−28.4543	−1.32525	−0.662623	−	0.748953i	$-0.730557\pi$
$461$	−28.4543	−1.32525	−0.662623	+	0.748953i	$0.730557\pi$
$462$	1.41421	0.0657952
$463$	−6.05277	−0.281296	−0.140648	−	0.990060i	$-0.544919\pi$
$463$	−6.05277	−0.281296	−0.140648	+	0.990060i	$0.544919\pi$
$464$	−5.17157	−0.240084
$465$	−0.415564	−0.0192713
$466$	0.273856	0.0126861
$467$	−32.1428	−1.48739	−0.743696	−	0.668518i	$-0.766929\pi$
$467$	−32.1428	−1.48739	−0.743696	+	0.668518i	$0.766929\pi$
$468$	4.64047	0.214506
$469$	2.61313	0.120663
$470$	−0.560171	−0.0258388
$471$	−19.8535	−0.914802
$472$	−25.0698	−1.15393
$473$	−5.52885	−0.254217
$474$	6.40743	0.294303
$475$	−18.1776	−0.834046
$476$	0	0
$477$	11.3524	0.519789
$478$	−7.43289	−0.339972
$479$	15.5513	0.710556	0.355278	−	0.934761i	$-0.384386\pi$
$479$	15.5513	0.710556	0.355278	+	0.934761i	$0.384386\pi$
$480$	0.482843	0.0220387
$481$	7.89549	0.360004
$482$	−7.16516	−0.326364
$483$	25.7418	1.17129
$484$	14.9250	0.678407
$485$	0.904031	0.0410499
$486$	0.765367	0.0347177
$487$	20.0717	0.909537	0.454768	−	0.890610i	$-0.349722\pi$
$487$	20.0717	0.909537	0.454768	+	0.890610i	$0.349722\pi$
$488$	−6.76017	−0.306019
$489$	1.75511	0.0793687
$490$	0.0408160	0.00184388
$491$	5.37086	0.242383	0.121192	−	0.992629i	$-0.461328\pi$
$491$	5.37086	0.242383	0.121192	+	0.992629i	$0.461328\pi$
$492$	−0.679129	−0.0306175
$493$	0	0
$494$	−9.14267	−0.411348
$495$	−0.0550527	−0.00247444
$496$	−4.17836	−0.187614
$497$	−9.09278	−0.407867
$498$	0.688730	0.0308627
$499$	−28.0658	−1.25640	−0.628199	−	0.778053i	$-0.716207\pi$
$499$	−28.0658	−1.25640	−0.628199	+	0.778053i	$0.716207\pi$
$500$	1.16441	0.0520740
$501$	8.05921	0.360059
$502$	2.51191	0.112112
$503$	−8.76921	−0.391000	−0.195500	−	0.980704i	$-0.562633\pi$
$503$	−8.76921	−0.391000	−0.195500	+	0.980704i	$0.562633\pi$
$504$	7.22625	0.321883
$505$	−0.269980	−0.0120140
$506$	4.76046	0.211628
$507$	−2.23304	−0.0991728
$508$	−0.896155	−0.0397604
$509$	−13.5248	−0.599475	−0.299737	−	0.954022i	$-0.596899\pi$
$509$	−13.5248	−0.599475	−0.299737	+	0.954022i	$0.596899\pi$
$510$	0	0
$511$	43.3663	1.91841
$512$	9.18440	0.405897
$513$	3.64047	0.160730
$514$	2.71323	0.119675
$515$	−0.990585	−0.0436504
$516$	−11.7019	−0.515149
$517$	5.93550	0.261043
$518$	5.09278	0.223764
$519$	−5.75282	−0.252521
$520$	0.706469	0.0309807
$521$	9.32360	0.408474	0.204237	−	0.978921i	$-0.434529\pi$
$521$	9.32360	0.408474	0.204237	+	0.978921i	$0.434529\pi$
$522$	−4.77791	−0.209123
$523$	−17.0634	−0.746129	−0.373065	−	0.927805i	$-0.621693\pi$
$523$	−17.0634	−0.746129	−0.373065	+	0.927805i	$0.621693\pi$
$524$	−21.5076	−0.939562
$525$	13.8081	0.602633
$526$	10.3426	0.450960
$527$	0	0
$528$	−0.553537	−0.0240896
$529$	63.6509	2.76743
$530$	0.715884	0.0310960
$531$	9.59379	0.416335
$532$	14.2372	0.617261
$533$	−1.57574	−0.0682528
$534$	4.31997	0.186943
$535$	0.719224	0.0310948
$536$	2.46927	0.106656
$537$	14.6173	0.630783
$538$	−3.95614	−0.170561
$539$	−0.432481	−0.0186283
$540$	−0.116520	−0.00501423
$541$	−3.81082	−0.163840	−0.0819200	−	0.996639i	$-0.526105\pi$
$541$	−3.81082	−0.163840	−0.0819200	+	0.996639i	$0.526105\pi$
$542$	−5.11718	−0.219802
$543$	−14.1000	−0.605089
$544$	0	0
$545$	0.351813	0.0150700
$546$	6.94495	0.297216
$547$	−26.0859	−1.11535	−0.557677	−	0.830058i	$-0.688307\pi$
$547$	−26.0859	−1.11535	−0.557677	+	0.830058i	$0.688307\pi$
$548$	−3.37780	−0.144293
$549$	2.58701	0.110411
$550$	2.55354	0.108883
$551$	−22.7261	−0.968165
$552$	24.3247	1.03533
$553$	−23.1508	−0.984474
$554$	−2.22078	−0.0943520
$555$	−0.198253	−0.00841535
$556$	2.28356	0.0968444
$557$	2.00763	0.0850662	0.0425331	−	0.999095i	$-0.486457\pi$
$557$	2.00763	0.0850662	0.0425331	+	0.999095i	$0.486457\pi$
$558$	−3.86030	−0.163420
$559$	−27.1512	−1.14837
$560$	−0.188753	−0.00797626
$561$	0	0
$562$	−17.5428	−0.739997
$563$	−27.9031	−1.17598	−0.587988	−	0.808869i	$-0.700080\pi$
$563$	−27.9031	−1.17598	−0.587988	+	0.808869i	$0.700080\pi$
$564$	12.5626	0.528981
$565$	−1.14161	−0.0480278
$566$	−21.2167	−0.891802
$567$	−2.76537	−0.116134
$568$	−8.59220	−0.360521
$569$	−22.8126	−0.956356	−0.478178	−	0.878263i	$-0.658703\pi$
$569$	−22.8126	−0.956356	−0.478178	+	0.878263i	$0.658703\pi$
$570$	0.229569	0.00961557
$571$	15.7098	0.657435	0.328718	−	0.944428i	$-0.393384\pi$
$571$	15.7098	0.657435	0.328718	+	0.944428i	$0.393384\pi$
$572$	−3.10066	−0.129645
$573$	16.0167	0.669107
$574$	−1.01639	−0.0424233
$575$	46.4800	1.93835
$576$	2.82843	0.117851
$577$	29.1062	1.21171	0.605853	−	0.795577i	$-0.292832\pi$
$577$	29.1062	1.21171	0.605853	+	0.795577i	$0.292832\pi$
$578$	0	0
$579$	4.59539	0.190978
$580$	0.727394	0.0302034
$581$	−2.48847	−0.103239
$582$	8.39782	0.348101
$583$	−7.58541	−0.314156
$584$	40.9789	1.69572
$585$	−0.270354	−0.0111778
$586$	8.77698	0.362574
$587$	19.0186	0.784983	0.392492	−	0.919756i	$-0.371613\pi$
$587$	19.0186	0.784983	0.392492	+	0.919756i	$0.371613\pi$
$588$	−0.915355	−0.0377486
$589$	−18.3615	−0.756573
$590$	0.604987	0.0249069
$591$	−21.1689	−0.870774
$592$	−1.99337	−0.0819269
$593$	−10.4425	−0.428820	−0.214410	−	0.976744i	$-0.568783\pi$
$593$	−10.4425	−0.428820	−0.214410	+	0.976744i	$0.568783\pi$
$594$	−0.511402	−0.0209831
$595$	0	0
$596$	−15.2851	−0.626102
$597$	2.32541	0.0951726
$598$	23.3777	0.955987
$599$	−22.1338	−0.904361	−0.452180	−	0.891927i	$-0.649354\pi$
$599$	−22.1338	−0.904361	−0.452180	+	0.891927i	$0.649354\pi$
$600$	13.0479	0.532678
$601$	−2.96951	−0.121129	−0.0605643	−	0.998164i	$-0.519290\pi$
$601$	−2.96951	−0.121129	−0.0605643	+	0.998164i	$0.519290\pi$
$602$	−17.5132	−0.713784
$603$	−0.944947	−0.0384812
$604$	30.7685	1.25195
$605$	−0.869529	−0.0353514
$606$	−2.50793	−0.101878
$607$	−29.6467	−1.20332	−0.601661	−	0.798752i	$-0.705494\pi$
$607$	−29.6467	−1.20332	−0.601661	+	0.798752i	$0.705494\pi$
$608$	21.3342	0.865217
$609$	17.2632	0.699540
$610$	0.163137	0.00660523
$611$	29.1482	1.17921
$612$	0	0
$613$	30.3538	1.22598	0.612988	−	0.790092i	$-0.289967\pi$
$613$	30.3538	1.22598	0.612988	+	0.790092i	$0.289967\pi$
$614$	8.05508	0.325077
$615$	0.0395661	0.00159546
$616$	−4.82843	−0.194543
$617$	−35.4772	−1.42826	−0.714129	−	0.700014i	$-0.753177\pi$
$617$	−35.4772	−1.42826	−0.714129	+	0.700014i	$0.753177\pi$
$618$	−9.20186	−0.370153
$619$	26.2024	1.05316	0.526582	−	0.850124i	$-0.323473\pi$
$619$	26.2024	1.05316	0.526582	+	0.850124i	$0.323473\pi$
$620$	0.587696	0.0236024
$621$	−9.30864	−0.373543
$622$	4.14386	0.166154
$623$	−15.6086	−0.625345
$624$	−2.71832	−0.108820
$625$	24.8982	0.995929
$626$	0.728141	0.0291024
$627$	−2.43248	−0.0971439
$628$	28.0771	1.12040
$629$	0	0
$630$	−0.174385	−0.00694765
$631$	−0.438772	−0.0174672	−0.00873362	−	0.999962i	$-0.502780\pi$
$631$	−0.438772	−0.0174672	−0.00873362	+	0.999962i	$0.502780\pi$
$632$	−21.8763	−0.870193
$633$	−7.58541	−0.301493
$634$	16.5049	0.655493
$635$	0.0522100	0.00207189
$636$	−16.0547	−0.636609
$637$	−2.12384	−0.0841495
$638$	3.19250	0.126392
$639$	3.28809	0.130075
$640$	−0.787325	−0.0311218
$641$	−4.78793	−0.189112	−0.0945559	−	0.995520i	$-0.530143\pi$
$641$	−4.78793	−0.189112	−0.0945559	+	0.995520i	$0.530143\pi$
$642$	6.68110	0.263682
$643$	20.5340	0.809782	0.404891	−	0.914365i	$-0.367310\pi$
$643$	20.5340	0.809782	0.404891	+	0.914365i	$0.367310\pi$
$644$	−36.4044	−1.43454
$645$	0.681756	0.0268441
$646$	0	0
$647$	18.7594	0.737508	0.368754	−	0.929527i	$-0.379784\pi$
$647$	18.7594	0.737508	0.368754	+	0.929527i	$0.379784\pi$
$648$	−2.61313	−0.102653
$649$	−6.41037	−0.251629
$650$	12.5400	0.491858
$651$	13.9478	0.546656
$652$	−2.48210	−0.0972064
$653$	−14.2176	−0.556379	−0.278189	−	0.960526i	$-0.589734\pi$
$653$	−14.2176	−0.556379	−0.278189	+	0.960526i	$0.589734\pi$
$654$	3.26810	0.127793
$655$	1.25303	0.0489600
$656$	0.397825	0.0155324
$657$	−15.6819	−0.611811
$658$	18.8013	0.732950
$659$	34.5061	1.34417	0.672084	−	0.740475i	$-0.265399\pi$
$659$	34.5061	1.34417	0.672084	+	0.740475i	$0.265399\pi$
$660$	0.0778563	0.00303055
$661$	12.7758	0.496922	0.248461	−	0.968642i	$-0.420075\pi$
$661$	12.7758	0.496922	0.248461	+	0.968642i	$0.420075\pi$
$662$	−15.8711	−0.616849
$663$	0	0
$664$	−2.35147	−0.0912547
$665$	−0.829461	−0.0321651
$666$	−1.84163	−0.0713617
$667$	58.1105	2.25005
$668$	−11.3975	−0.440981
$669$	−15.6624	−0.605542
$670$	−0.0595886	−0.00230211
$671$	−1.72858	−0.0667312
$672$	−16.2059	−0.625156
$673$	−33.8641	−1.30536	−0.652682	−	0.757632i	$-0.726356\pi$
$673$	−33.8641	−1.30536	−0.652682	+	0.757632i	$0.726356\pi$
$674$	10.6597	0.410595
$675$	−4.99321	−0.192189
$676$	3.15800	0.121461
$677$	35.6921	1.37176	0.685880	−	0.727714i	$-0.259417\pi$
$677$	35.6921	1.37176	0.685880	+	0.727714i	$0.259417\pi$
$678$	−10.6047	−0.407273
$679$	−30.3424	−1.16443
$680$	0	0
$681$	−14.6314	−0.560677
$682$	2.57937	0.0987692
$683$	22.2664	0.851999	0.426000	−	0.904723i	$-0.359922\pi$
$683$	22.2664	0.851999	0.426000	+	0.904723i	$0.359922\pi$
$684$	−5.14840	−0.196854
$685$	0.196791	0.00751900
$686$	13.4457	0.513360
$687$	16.3451	0.623603
$688$	6.85483	0.261338
$689$	−37.2506	−1.41913
$690$	−0.587006	−0.0223469
$691$	29.0193	1.10395	0.551973	−	0.833862i	$-0.313875\pi$
$691$	29.0193	1.10395	0.551973	+	0.833862i	$0.313875\pi$
$692$	8.13572	0.309274
$693$	1.84776	0.0701906
$694$	−12.3143	−0.467443
$695$	−0.133040	−0.00504650
$696$	16.3128	0.618335
$697$	0	0
$698$	7.11830	0.269432
$699$	0.357811	0.0135336
$700$	−19.5275	−0.738072
$701$	−33.5632	−1.26766	−0.633832	−	0.773471i	$-0.718519\pi$
$701$	−33.5632	−1.26766	−0.633832	+	0.773471i	$0.718519\pi$
$702$	−2.51140	−0.0947868
$703$	−8.75971	−0.330379
$704$	−1.88989	−0.0712281
$705$	−0.731899	−0.0275649
$706$	−21.2881	−0.801188
$707$	9.06147	0.340792
$708$	−13.5677	−0.509904
$709$	41.7240	1.56698	0.783489	−	0.621405i	$-0.213438\pi$
$709$	41.7240	1.56698	0.783489	+	0.621405i	$0.213438\pi$
$710$	0.207348	0.00778163
$711$	8.37170	0.313963
$712$	−14.7493	−0.552753
$713$	46.9503	1.75830
$714$	0	0
$715$	0.180645	0.00675573
$716$	−20.6720	−0.772548
$717$	−9.71153	−0.362684
$718$	3.55147	0.132540
$719$	50.6576	1.88921	0.944604	−	0.328212i	$-0.106446\pi$
$719$	50.6576	1.88921	0.944604	+	0.328212i	$0.106446\pi$
$720$	0.0682559	0.00254375
$721$	33.2475	1.23820
$722$	−4.39857	−0.163698
$723$	−9.36173	−0.348166
$724$	19.9404	0.741080
$725$	31.1708	1.15766
$726$	−8.07733	−0.299778
$727$	11.4948	0.426317	0.213159	−	0.977018i	$-0.431625\pi$
$727$	11.4948	0.426317	0.213159	+	0.977018i	$0.431625\pi$
$728$	−23.7115	−0.878808
$729$	1.00000	0.0370370
$730$	−0.988907	−0.0366011
$731$	0	0
$732$	−3.65858	−0.135225
$733$	−48.4680	−1.79021	−0.895103	−	0.445859i	$-0.852898\pi$
$733$	−48.4680	−1.79021	−0.895103	+	0.445859i	$0.852898\pi$
$734$	−3.30729	−0.122074
$735$	0.0533287	0.00196706
$736$	−54.5515	−2.01079
$737$	0.631394	0.0232577
$738$	0.367542	0.0135294
$739$	2.50512	0.0921523	0.0460761	−	0.998938i	$-0.485328\pi$
$739$	2.50512	0.0921523	0.0460761	+	0.998938i	$0.485328\pi$
$740$	0.280372	0.0103067
$741$	−11.9455	−0.438828
$742$	−24.0275	−0.882078
$743$	14.1003	0.517289	0.258645	−	0.965973i	$-0.416724\pi$
$743$	14.1003	0.517289	0.258645	+	0.965973i	$0.416724\pi$
$744$	13.1799	0.483198
$745$	0.890511	0.0326258
$746$	−6.45985	−0.236512
$747$	0.899869	0.0329245
$748$	0	0
$749$	−24.1396	−0.882043
$750$	−0.630175	−0.0230107
$751$	22.3611	0.815969	0.407984	−	0.912989i	$-0.366232\pi$
$751$	22.3611	0.815969	0.407984	+	0.912989i	$0.366232\pi$
$752$	−7.35901	−0.268355
$753$	3.28196	0.119601
$754$	15.6778	0.570951
$755$	−1.79258	−0.0652385
$756$	3.91082	0.142235
$757$	−9.15121	−0.332606	−0.166303	−	0.986075i	$-0.553183\pi$
$757$	−9.15121	−0.332606	−0.166303	+	0.986075i	$0.553183\pi$
$758$	5.85977	0.212837
$759$	6.21984	0.225766
$760$	−0.783797	−0.0284313
$761$	−29.3561	−1.06416	−0.532079	−	0.846694i	$-0.678589\pi$
$761$	−29.3561	−1.06416	−0.532079	+	0.846694i	$0.678589\pi$
$762$	0.484995	0.0175695
$763$	−11.8081	−0.427481
$764$	−22.6510	−0.819486
$765$	0	0
$766$	14.1795	0.512325
$767$	−31.4802	−1.13668
$768$	−12.9706	−0.468035
$769$	−17.3301	−0.624939	−0.312470	−	0.949928i	$-0.601156\pi$
$769$	−17.3301	−0.624939	−0.312470	+	0.949928i	$0.601156\pi$
$770$	0.116520	0.00419910
$771$	3.54500	0.127670
$772$	−6.49886	−0.233899
$773$	45.5488	1.63828	0.819139	−	0.573595i	$-0.194452\pi$
$773$	45.5488	1.63828	0.819139	+	0.573595i	$0.194452\pi$
$774$	6.33304	0.227636
$775$	25.1844	0.904650
$776$	−28.6720	−1.02926
$777$	6.65404	0.238712
$778$	20.5213	0.735724
$779$	1.74821	0.0626362
$780$	0.382338	0.0136899
$781$	−2.19703	−0.0786160
$782$	0	0
$783$	−6.24264	−0.223094
$784$	0.536203	0.0191501
$785$	−1.63577	−0.0583833
$786$	11.6398	0.415178
$787$	33.4248	1.19147	0.595733	−	0.803183i	$-0.296862\pi$
$787$	33.4248	1.19147	0.595733	+	0.803183i	$0.296862\pi$
$788$	29.9374	1.06648
$789$	13.5133	0.481086
$790$	0.527922	0.0187826
$791$	38.3163	1.36237
$792$	1.74603	0.0620426
$793$	−8.48875	−0.301444
$794$	−2.40345	−0.0852952
$795$	0.935347	0.0331733
$796$	−3.28862	−0.116562
$797$	−9.90666	−0.350912	−0.175456	−	0.984487i	$-0.556140\pi$
$797$	−9.90666	−0.350912	−0.175456	+	0.984487i	$0.556140\pi$
$798$	−7.70512	−0.272758
$799$	0	0
$800$	−29.2617	−1.03456
$801$	5.64431	0.199432
$802$	−20.2567	−0.715289
$803$	10.4783	0.369773
$804$	1.33636	0.0471297
$805$	2.12092	0.0747528
$806$	12.6668	0.446170
$807$	−5.16895	−0.181956
$808$	8.56261	0.301232
$809$	−4.21493	−0.148189	−0.0740945	−	0.997251i	$-0.523607\pi$
$809$	−4.21493	−0.148189	−0.0740945	+	0.997251i	$0.523607\pi$
$810$	0.0630603	0.00221571
$811$	25.0270	0.878816	0.439408	−	0.898288i	$-0.355188\pi$
$811$	25.0270	0.878816	0.439408	+	0.898288i	$0.355188\pi$
$812$	−24.4138	−0.856758
$813$	−6.68592	−0.234485
$814$	1.23054	0.0431303
$815$	0.144607	0.00506537
$816$	0	0
$817$	30.1231	1.05387
$818$	−23.7603	−0.830758
$819$	9.07401	0.317072
$820$	−0.0559550	−0.00195403
$821$	21.1135	0.736867	0.368433	−	0.929654i	$-0.379894\pi$
$821$	21.1135	0.736867	0.368433	+	0.929654i	$0.379894\pi$
$822$	1.82805	0.0637607
$823$	25.2967	0.881786	0.440893	−	0.897560i	$-0.354662\pi$
$823$	25.2967	0.881786	0.440893	+	0.897560i	$0.354662\pi$
$824$	31.4171	1.09447
$825$	3.33636	0.116157
$826$	−20.3055	−0.706517
$827$	−17.8491	−0.620675	−0.310338	−	0.950626i	$-0.600442\pi$
$827$	−17.8491	−0.620675	−0.310338	+	0.950626i	$0.600442\pi$
$828$	13.1644	0.457495
$829$	8.81114	0.306023	0.153012	−	0.988224i	$-0.451103\pi$
$829$	8.81114	0.306023	0.153012	+	0.988224i	$0.451103\pi$
$830$	0.0567460	0.00196968
$831$	−2.90159	−0.100655
$832$	−9.28093	−0.321758
$833$	0	0
$834$	−1.23585	−0.0427941
$835$	0.664016	0.0229792
$836$	3.44005	0.118977
$837$	−5.04373	−0.174337
$838$	29.6499	1.02424
$839$	−9.83630	−0.339587	−0.169793	−	0.985480i	$-0.554310\pi$
$839$	−9.83630	−0.339587	−0.169793	+	0.985480i	$0.554310\pi$
$840$	0.595387	0.0205428
$841$	9.97056	0.343813
$842$	−14.5710	−0.502149
$843$	−22.9207	−0.789432
$844$	10.7274	0.369252
$845$	−0.183985	−0.00632928
$846$	−6.79884	−0.233749
$847$	29.1844	1.00279
$848$	9.40461	0.322956
$849$	−27.7209	−0.951379
$850$	0	0
$851$	22.3985	0.767811
$852$	−4.65007	−0.159309
$853$	−52.8949	−1.81109	−0.905543	−	0.424253i	$-0.860537\pi$
$853$	−52.8949	−1.81109	−0.905543	+	0.424253i	$0.860537\pi$
$854$	−5.47545	−0.187366
$855$	0.299946	0.0102579
$856$	−22.8107	−0.779653
$857$	48.0500	1.64136	0.820678	−	0.571392i	$-0.193596\pi$
$857$	48.0500	1.64136	0.820678	+	0.571392i	$0.193596\pi$
$858$	1.67807	0.0572882
$859$	18.0931	0.617329	0.308665	−	0.951171i	$-0.400118\pi$
$859$	18.0931	0.617329	0.308665	+	0.951171i	$0.400118\pi$
$860$	−0.964148	−0.0328772
$861$	−1.32798	−0.0452573
$862$	9.11198	0.310355
$863$	29.4477	1.00241	0.501206	−	0.865328i	$-0.332890\pi$
$863$	29.4477	1.00241	0.501206	+	0.865328i	$0.332890\pi$
$864$	5.86030	0.199372
$865$	−0.473988	−0.0161161
$866$	−16.3540	−0.555730
$867$	0	0
$868$	−19.7251	−0.669514
$869$	−5.59379	−0.189756
$870$	−0.393663	−0.0133464
$871$	3.10066	0.105062
$872$	−11.1580	−0.377857
$873$	10.9723	0.371356
$874$	−25.9366	−0.877318
$875$	2.27690	0.0769733
$876$	22.1776	0.749312
$877$	−35.7165	−1.20606	−0.603030	−	0.797719i	$-0.706040\pi$
$877$	−35.7165	−1.20606	−0.603030	+	0.797719i	$0.706040\pi$
$878$	−16.2160	−0.547264
$879$	11.4677	0.386795
$880$	−0.0456072	−0.00153742
$881$	8.14664	0.274467	0.137234	−	0.990539i	$-0.456179\pi$
$881$	8.14664	0.274467	0.137234	+	0.990539i	$0.456179\pi$
$882$	0.495387	0.0166805
$883$	28.3255	0.953228	0.476614	−	0.879113i	$-0.341864\pi$
$883$	28.3255	0.953228	0.476614	+	0.879113i	$0.341864\pi$
$884$	0	0
$885$	0.790454	0.0265708
$886$	−16.4912	−0.554032
$887$	40.2883	1.35275	0.676375	−	0.736558i	$-0.263550\pi$
$887$	40.2883	1.35275	0.676375	+	0.736558i	$0.263550\pi$
$888$	6.28772	0.211002
$889$	−1.75235	−0.0587719
$890$	0.355932	0.0119309
$891$	−0.668179	−0.0223848
$892$	22.1499	0.741635
$893$	−32.3386	−1.08217
$894$	8.27223	0.276665
$895$	1.20435	0.0402570
$896$	26.4253	0.882809
$897$	30.5445	1.01985
$898$	−27.6886	−0.923980
$899$	31.4862	1.05012
$900$	7.06147	0.235382
$901$	0	0
$902$	−0.245584	−0.00817705
$903$	−22.8821	−0.761468
$904$	36.2069	1.20422
$905$	−1.16173	−0.0386172
$906$	−16.6518	−0.553219
$907$	43.4425	1.44248	0.721242	−	0.692683i	$-0.243572\pi$
$907$	43.4425	1.44248	0.721242	+	0.692683i	$0.243572\pi$
$908$	20.6920	0.686687
$909$	−3.27677	−0.108684
$910$	0.572209	0.0189686
$911$	2.22475	0.0737091	0.0368546	−	0.999321i	$-0.488266\pi$
$911$	2.22475	0.0737091	0.0368546	+	0.999321i	$0.488266\pi$
$912$	3.01586	0.0998651
$913$	−0.601273	−0.0198992
$914$	21.5955	0.714316
$915$	0.213149	0.00704649
$916$	−23.1154	−0.763754
$917$	−42.0561	−1.38881
$918$	0	0
$919$	−47.4595	−1.56554	−0.782772	−	0.622308i	$-0.786195\pi$
$919$	−47.4595	−1.56554	−0.782772	+	0.622308i	$0.786195\pi$
$920$	2.00416	0.0660753
$921$	10.5245	0.346793
$922$	−21.7779	−0.717218
$923$	−10.7892	−0.355132
$924$	−2.61313	−0.0859655
$925$	12.0147	0.395041
$926$	−4.63259	−0.152236
$927$	−12.0228	−0.394881
$928$	−36.5838	−1.20092
$929$	32.0598	1.05185	0.525924	−	0.850532i	$-0.323720\pi$
$929$	32.0598	1.05185	0.525924	+	0.850532i	$0.323720\pi$
$930$	−0.318059	−0.0104296
$931$	2.35631	0.0772248
$932$	−0.506021	−0.0165753
$933$	5.41421	0.177253
$934$	−24.6011	−0.804971
$935$	0	0
$936$	8.57446	0.280265
$937$	16.9740	0.554517	0.277258	−	0.960795i	$-0.410574\pi$
$937$	16.9740	0.554517	0.277258	+	0.960795i	$0.410574\pi$
$938$	2.00000	0.0653023
$939$	0.951362	0.0310465
$940$	1.03506	0.0337600
$941$	14.2357	0.464070	0.232035	−	0.972707i	$-0.425462\pi$
$941$	14.2357	0.464070	0.232035	+	0.972707i	$0.425462\pi$
$942$	−15.1952	−0.495087
$943$	−4.47017	−0.145569
$944$	7.94776	0.258678
$945$	−0.227845	−0.00741179
$946$	−4.23160	−0.137581
$947$	38.5283	1.25200	0.626001	−	0.779823i	$-0.284691\pi$
$947$	38.5283	1.25200	0.626001	+	0.779823i	$0.284691\pi$
$948$	−11.8394	−0.384525
$949$	51.4572	1.67037
$950$	−13.9125	−0.451383
$951$	21.5647	0.699282
$952$	0	0
$953$	17.8637	0.578663	0.289331	−	0.957229i	$-0.406567\pi$
$953$	17.8637	0.578663	0.289331	+	0.957229i	$0.406567\pi$
$954$	8.68873	0.281308
$955$	1.31965	0.0427029
$956$	13.7342	0.444195
$957$	4.17120	0.134836
$958$	11.9024	0.384550
$959$	−6.60499	−0.213286
$960$	0.233040	0.00752134
$961$	−5.56080	−0.179381
$962$	6.04295	0.194833
$963$	8.72927	0.281297
$964$	13.2395	0.426415
$965$	0.378624	0.0121883
$966$	19.7019	0.633899
$967$	22.2654	0.716008	0.358004	−	0.933720i	$-0.383457\pi$
$967$	22.2654	0.716008	0.358004	+	0.933720i	$0.383457\pi$
$968$	27.5777	0.886382
$969$	0	0
$970$	0.691915	0.0222161
$971$	−4.38252	−0.140642	−0.0703209	−	0.997524i	$-0.522402\pi$
$971$	−4.38252	−0.140642	−0.0703209	+	0.997524i	$0.522402\pi$
$972$	−1.41421	−0.0453609
$973$	4.46529	0.143151
$974$	15.3622	0.492238
$975$	16.3842	0.524716
$976$	2.14315	0.0686004
$977$	−27.5043	−0.879940	−0.439970	−	0.898013i	$-0.645011\pi$
$977$	−27.5043	−0.879940	−0.439970	+	0.898013i	$0.645011\pi$
$978$	1.34330	0.0429540
$979$	−3.77141	−0.120535
$980$	−0.0754181	−0.00240914
$981$	4.26998	0.136330
$982$	4.11068	0.131177
$983$	−31.1372	−0.993123	−0.496562	−	0.868001i	$-0.665404\pi$
$983$	−31.1372	−0.993123	−0.496562	+	0.868001i	$0.665404\pi$
$984$	−1.25487	−0.0400037
$985$	−1.74416	−0.0555734
$986$	0	0
$987$	24.5650	0.781914
$988$	16.8935	0.537452
$989$	−77.0245	−2.44924
$990$	−0.0421355	−0.00133915
$991$	0.764676	0.0242907	0.0121454	−	0.999926i	$-0.496134\pi$
$991$	0.764676	0.0242907	0.0121454	+	0.999926i	$0.496134\pi$
$992$	−29.5578	−0.938460
$993$	−20.7366	−0.658057
$994$	−6.95932	−0.220736
$995$	0.191595	0.00607398
$996$	−1.27261	−0.0403241
$997$	1.19938	0.0379849	0.0189924	−	0.999820i	$-0.493954\pi$
$997$	1.19938	0.0379849	0.0189924	+	0.999820i	$0.493954\pi$
$998$	−21.4806	−0.679957
$999$	−2.40621	−0.0761290

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.2.a.n.1.3		4
3.2	odd	2		2601.2.a.bd.1.2		4
17.2	even	8		867.2.e.h.616.3		8
17.3	odd	16		867.2.h.f.757.2		8
17.4	even	4		867.2.d.e.577.3		8
17.5	odd	16		51.2.h.a.25.1	✓	8
17.6	odd	16		867.2.h.f.733.2		8
17.7	odd	16		51.2.h.a.49.1	yes	8
17.8	even	8		867.2.e.i.829.2		8
17.9	even	8		867.2.e.h.829.2		8
17.10	odd	16		867.2.h.g.712.1		8
17.11	odd	16		867.2.h.b.733.2		8
17.12	odd	16		867.2.h.g.688.1		8
17.13	even	4		867.2.d.e.577.4		8
17.14	odd	16		867.2.h.b.757.2		8
17.15	even	8		867.2.e.i.616.3		8
17.16	even	2		867.2.a.m.1.3		4
51.5	even	16		153.2.l.e.127.2		8
51.41	even	16		153.2.l.e.100.2		8
51.50	odd	2		2601.2.a.bc.1.2		4
68.7	even	16		816.2.bq.a.49.2		8
68.39	even	16		816.2.bq.a.433.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.h.a.25.1	✓	8	17.5	odd	16
51.2.h.a.49.1	yes	8	17.7	odd	16
153.2.l.e.100.2		8	51.41	even	16
153.2.l.e.127.2		8	51.5	even	16
816.2.bq.a.49.2		8	68.7	even	16
816.2.bq.a.433.2		8	68.39	even	16
867.2.a.m.1.3		4	17.16	even	2
867.2.a.n.1.3		4	1.1	even	1	trivial
867.2.d.e.577.3		8	17.4	even	4
867.2.d.e.577.4		8	17.13	even	4
867.2.e.h.616.3		8	17.2	even	8
867.2.e.h.829.2		8	17.9	even	8
867.2.e.i.616.3		8	17.15	even	8
867.2.e.i.829.2		8	17.8	even	8
867.2.h.b.733.2		8	17.11	odd	16
867.2.h.b.757.2		8	17.14	odd	16
867.2.h.f.733.2		8	17.6	odd	16
867.2.h.f.757.2		8	17.3	odd	16
867.2.h.g.688.1		8	17.12	odd	16
867.2.h.g.712.1		8	17.10	odd	16
2601.2.a.bc.1.2		4	51.50	odd	2
2601.2.a.bd.1.2		4	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:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.a (trivial)

\(f(q)\)	\(=\)	\(q+0.765367 q^{2} +1.00000 q^{3} -1.41421 q^{4} +0.0823922 q^{5} +0.765367 q^{6} -2.76537 q^{7} -2.61313 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.765367 q^{2} +1.00000 q^{3} -1.41421 q^{4} +0.0823922 q^{5} +0.765367 q^{6} -2.76537 q^{7} -2.61313 q^{8} +1.00000 q^{9} +0.0630603 q^{10} -0.668179 q^{11} -1.41421 q^{12} -3.28130 q^{13} -2.11652 q^{14} +0.0823922 q^{15} +0.828427 q^{16} +0.765367 q^{18} +3.64047 q^{19} -0.116520 q^{20} -2.76537 q^{21} -0.511402 q^{22} -9.30864 q^{23} -2.61313 q^{24} -4.99321 q^{25} -2.51140 q^{26} +1.00000 q^{27} +3.91082 q^{28} -6.24264 q^{29} +0.0630603 q^{30} -5.04373 q^{31} +5.86030 q^{32} -0.668179 q^{33} -0.227845 q^{35} -1.41421 q^{36} -2.40621 q^{37} +2.78629 q^{38} -3.28130 q^{39} -0.215301 q^{40} +0.480217 q^{41} -2.11652 q^{42} +8.27452 q^{43} +0.944947 q^{44} +0.0823922 q^{45} -7.12453 q^{46} -8.88311 q^{47} +0.828427 q^{48} +0.647254 q^{49} -3.82164 q^{50} +4.64047 q^{52} +11.3524 q^{53} +0.765367 q^{54} -0.0550527 q^{55} +7.22625 q^{56} +3.64047 q^{57} -4.77791 q^{58} +9.59379 q^{59} -0.116520 q^{60} +2.58701 q^{61} -3.86030 q^{62} -2.76537 q^{63} +2.82843 q^{64} -0.270354 q^{65} -0.511402 q^{66} -0.944947 q^{67} -9.30864 q^{69} -0.174385 q^{70} +3.28809 q^{71} -2.61313 q^{72} -15.6819 q^{73} -1.84163 q^{74} -4.99321 q^{75} -5.14840 q^{76} +1.84776 q^{77} -2.51140 q^{78} +8.37170 q^{79} +0.0682559 q^{80} +1.00000 q^{81} +0.367542 q^{82} +0.899869 q^{83} +3.91082 q^{84} +6.33304 q^{86} -6.24264 q^{87} +1.74603 q^{88} +5.64431 q^{89} +0.0630603 q^{90} +9.07401 q^{91} +13.1644 q^{92} -5.04373 q^{93} -6.79884 q^{94} +0.299946 q^{95} +5.86030 q^{96} +10.9723 q^{97} +0.495387 q^{98} -0.668179 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{5} - 8 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^5 - 8 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{5} - 8 q^{7} + 4 q^{9} - 8 q^{10} - 4 q^{11} - 4 q^{13} - 8 q^{14} - 4 q^{15} - 8 q^{16} - 12 q^{19} - 8 q^{21} + 8 q^{22} - 12 q^{23} + 4 q^{27} - 8 q^{29} - 8 q^{30} - 8 q^{31} - 4 q^{33} + 16 q^{35} - 24 q^{37} + 16 q^{38} - 4 q^{39} - 12 q^{41} - 8 q^{42} + 4 q^{43} - 8 q^{44} - 4 q^{45} - 8 q^{46} + 8 q^{47} - 8 q^{48} - 4 q^{49} + 16 q^{50} - 8 q^{52} + 8 q^{53} - 12 q^{55} + 8 q^{56} - 12 q^{57} + 24 q^{59} - 24 q^{61} + 8 q^{62} - 8 q^{63} - 12 q^{65} + 8 q^{66} + 8 q^{67} - 12 q^{69} + 24 q^{70} + 24 q^{71} - 8 q^{73} + 8 q^{74} - 8 q^{76} + 8 q^{80} + 4 q^{81} + 8 q^{82} + 8 q^{83} - 16 q^{86} - 8 q^{87} + 16 q^{89} - 8 q^{90} + 8 q^{91} - 8 q^{93} - 16 q^{94} + 12 q^{95} + 16 q^{97} + 32 q^{98} - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^5 - 8 * q^7 + 4 * q^9 - 8 * q^10 - 4 * q^11 - 4 * q^13 - 8 * q^14 - 4 * q^15 - 8 * q^16 - 12 * q^19 - 8 * q^21 + 8 * q^22 - 12 * q^23 + 4 * q^27 - 8 * q^29 - 8 * q^30 - 8 * q^31 - 4 * q^33 + 16 * q^35 - 24 * q^37 + 16 * q^38 - 4 * q^39 - 12 * q^41 - 8 * q^42 + 4 * q^43 - 8 * q^44 - 4 * q^45 - 8 * q^46 + 8 * q^47 - 8 * q^48 - 4 * q^49 + 16 * q^50 - 8 * q^52 + 8 * q^53 - 12 * q^55 + 8 * q^56 - 12 * q^57 + 24 * q^59 - 24 * q^61 + 8 * q^62 - 8 * q^63 - 12 * q^65 + 8 * q^66 + 8 * q^67 - 12 * q^69 + 24 * q^70 + 24 * q^71 - 8 * q^73 + 8 * q^74 - 8 * q^76 + 8 * q^80 + 4 * q^81 + 8 * q^82 + 8 * q^83 - 16 * q^86 - 8 * q^87 + 16 * q^89 - 8 * q^90 + 8 * q^91 - 8 * q^93 - 16 * q^94 + 12 * q^95 + 16 * q^97 + 32 * q^98 - 4 * q^99

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