LMFDB - Embedded newform 4002.2.a.be.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2389280.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 11x^{3} + 7x^{2} + 26x - 18 $ x^5 - x^4 - 11x^3 + 7x^2 + 26*x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.35912$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.26670 q^{5} +1.00000 q^{6} -3.10583 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.26670 q^{5} +1.00000 q^{6} -3.10583 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.26670 q^{10} -4.62582 q^{11} -1.00000 q^{12} +0.359125 q^{13} +3.10583 q^{14} -3.26670 q^{15} +1.00000 q^{16} +3.61242 q^{17} -1.00000 q^{18} -3.61242 q^{19} +3.26670 q^{20} +3.10583 q^{21} +4.62582 q^{22} -1.00000 q^{23} +1.00000 q^{24} +5.67130 q^{25} -0.359125 q^{26} -1.00000 q^{27} -3.10583 q^{28} -1.00000 q^{29} +3.26670 q^{30} +7.25499 q^{31} -1.00000 q^{32} +4.62582 q^{33} -3.61242 q^{34} -10.1458 q^{35} +1.00000 q^{36} +7.37615 q^{37} +3.61242 q^{38} -0.359125 q^{39} -3.26670 q^{40} +3.60880 q^{41} -3.10583 q^{42} +1.96792 q^{43} -4.62582 q^{44} +3.26670 q^{45} +1.00000 q^{46} +7.50235 q^{47} -1.00000 q^{48} +2.64619 q^{49} -5.67130 q^{50} -3.61242 q^{51} +0.359125 q^{52} -12.4762 q^{53} +1.00000 q^{54} -15.1112 q^{55} +3.10583 q^{56} +3.61242 q^{57} +1.00000 q^{58} +6.85616 q^{59} -3.26670 q^{60} -10.8557 q^{61} -7.25499 q^{62} -3.10583 q^{63} +1.00000 q^{64} +1.17315 q^{65} -4.62582 q^{66} +5.78195 q^{67} +3.61242 q^{68} +1.00000 q^{69} +10.1458 q^{70} -12.8925 q^{71} -1.00000 q^{72} -14.7979 q^{73} -7.37615 q^{74} -5.67130 q^{75} -3.61242 q^{76} +14.3670 q^{77} +0.359125 q^{78} +1.92460 q^{79} +3.26670 q^{80} +1.00000 q^{81} -3.60880 q^{82} -17.6984 q^{83} +3.10583 q^{84} +11.8007 q^{85} -1.96792 q^{86} +1.00000 q^{87} +4.62582 q^{88} -7.10780 q^{89} -3.26670 q^{90} -1.11538 q^{91} -1.00000 q^{92} -7.25499 q^{93} -7.50235 q^{94} -11.8007 q^{95} +1.00000 q^{96} +8.47620 q^{97} -2.64619 q^{98} -4.62582 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 - 4 * q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9} - q^{10} + 5 q^{11} - 5 q^{12} - 11 q^{13} + 4 q^{14} - q^{15} + 5 q^{16} + 4 q^{17} - 5 q^{18} - 4 q^{19} + q^{20} + 4 q^{21} - 5 q^{22} - 5 q^{23} + 5 q^{24} + 12 q^{25} + 11 q^{26} - 5 q^{27} - 4 q^{28} - 5 q^{29} + q^{30} + 5 q^{31} - 5 q^{32} - 5 q^{33} - 4 q^{34} - 6 q^{35} + 5 q^{36} + 9 q^{37} + 4 q^{38} + 11 q^{39} - q^{40} + 5 q^{41} - 4 q^{42} - 16 q^{43} + 5 q^{44} + q^{45} + 5 q^{46} + 8 q^{47} - 5 q^{48} - 5 q^{49} - 12 q^{50} - 4 q^{51} - 11 q^{52} - 4 q^{53} + 5 q^{54} - 33 q^{55} + 4 q^{56} + 4 q^{57} + 5 q^{58} + 23 q^{59} - q^{60} + 7 q^{61} - 5 q^{62} - 4 q^{63} + 5 q^{64} - 5 q^{65} + 5 q^{66} + 5 q^{67} + 4 q^{68} + 5 q^{69} + 6 q^{70} - 21 q^{71} - 5 q^{72} - 8 q^{73} - 9 q^{74} - 12 q^{75} - 4 q^{76} - 11 q^{78} - 8 q^{79} + q^{80} + 5 q^{81} - 5 q^{82} - 18 q^{83} + 4 q^{84} - 10 q^{85} + 16 q^{86} + 5 q^{87} - 5 q^{88} + 32 q^{89} - q^{90} + 10 q^{91} - 5 q^{92} - 5 q^{93} - 8 q^{94} + 10 q^{95} + 5 q^{96} - 16 q^{97} + 5 q^{98} + 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 - 4 * q^7 - 5 * q^8 + 5 * q^9 - q^10 + 5 * q^11 - 5 * q^12 - 11 * q^13 + 4 * q^14 - q^15 + 5 * q^16 + 4 * q^17 - 5 * q^18 - 4 * q^19 + q^20 + 4 * q^21 - 5 * q^22 - 5 * q^23 + 5 * q^24 + 12 * q^25 + 11 * q^26 - 5 * q^27 - 4 * q^28 - 5 * q^29 + q^30 + 5 * q^31 - 5 * q^32 - 5 * q^33 - 4 * q^34 - 6 * q^35 + 5 * q^36 + 9 * q^37 + 4 * q^38 + 11 * q^39 - q^40 + 5 * q^41 - 4 * q^42 - 16 * q^43 + 5 * q^44 + q^45 + 5 * q^46 + 8 * q^47 - 5 * q^48 - 5 * q^49 - 12 * q^50 - 4 * q^51 - 11 * q^52 - 4 * q^53 + 5 * q^54 - 33 * q^55 + 4 * q^56 + 4 * q^57 + 5 * q^58 + 23 * q^59 - q^60 + 7 * q^61 - 5 * q^62 - 4 * q^63 + 5 * q^64 - 5 * q^65 + 5 * q^66 + 5 * q^67 + 4 * q^68 + 5 * q^69 + 6 * q^70 - 21 * q^71 - 5 * q^72 - 8 * q^73 - 9 * q^74 - 12 * q^75 - 4 * q^76 - 11 * q^78 - 8 * q^79 + q^80 + 5 * q^81 - 5 * q^82 - 18 * q^83 + 4 * q^84 - 10 * q^85 + 16 * q^86 + 5 * q^87 - 5 * q^88 + 32 * q^89 - q^90 + 10 * q^91 - 5 * q^92 - 5 * q^93 - 8 * q^94 + 10 * q^95 + 5 * q^96 - 16 * q^97 + 5 * q^98 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	3.26670	1.46091	0.730455	−	0.682960i	$-0.239308\pi$
$5$	3.26670	1.46091	0.730455	+	0.682960i	$0.239308\pi$
$6$	1.00000	0.408248
$7$	−3.10583	−1.17389	−0.586947	−	0.809625i	$-0.699670\pi$
$7$	−3.10583	−1.17389	−0.586947	+	0.809625i	$0.699670\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−3.26670	−1.03302
$11$	−4.62582	−1.39474	−0.697369	−	0.716713i	$-0.745646\pi$
$11$	−4.62582	−1.39474	−0.697369	+	0.716713i	$0.745646\pi$
$12$	−1.00000	−0.288675
$13$	0.359125	0.0996033	0.0498017	−	0.998759i	$-0.484141\pi$
$13$	0.359125	0.0996033	0.0498017	+	0.998759i	$0.484141\pi$
$14$	3.10583	0.830069
$15$	−3.26670	−0.843457
$16$	1.00000	0.250000
$17$	3.61242	0.876140	0.438070	−	0.898941i	$-0.355662\pi$
$17$	3.61242	0.876140	0.438070	+	0.898941i	$0.355662\pi$
$18$	−1.00000	−0.235702
$19$	−3.61242	−0.828745	−0.414373	−	0.910107i	$-0.635999\pi$
$19$	−3.61242	−0.828745	−0.414373	+	0.910107i	$0.635999\pi$
$20$	3.26670	0.730455
$21$	3.10583	0.677748
$22$	4.62582	0.986228
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	5.67130	1.13426
$26$	−0.359125	−0.0704302
$27$	−1.00000	−0.192450
$28$	−3.10583	−0.586947
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	3.26670	0.596414
$31$	7.25499	1.30303	0.651517	−	0.758634i	$-0.274133\pi$
$31$	7.25499	1.30303	0.651517	+	0.758634i	$0.274133\pi$
$32$	−1.00000	−0.176777
$33$	4.62582	0.805252
$34$	−3.61242	−0.619525
$35$	−10.1458	−1.71495
$36$	1.00000	0.166667
$37$	7.37615	1.21263	0.606316	−	0.795224i	$-0.292647\pi$
$37$	7.37615	1.21263	0.606316	+	0.795224i	$0.292647\pi$
$38$	3.61242	0.586012
$39$	−0.359125	−0.0575060
$40$	−3.26670	−0.516510
$41$	3.60880	0.563599	0.281800	−	0.959473i	$-0.409069\pi$
$41$	3.60880	0.563599	0.281800	+	0.959473i	$0.409069\pi$
$42$	−3.10583	−0.479240
$43$	1.96792	0.300105	0.150053	−	0.988678i	$-0.452056\pi$
$43$	1.96792	0.300105	0.150053	+	0.988678i	$0.452056\pi$
$44$	−4.62582	−0.697369
$45$	3.26670	0.486970
$46$	1.00000	0.147442
$47$	7.50235	1.09433	0.547165	−	0.837024i	$-0.315707\pi$
$47$	7.50235	1.09433	0.547165	+	0.837024i	$0.315707\pi$
$48$	−1.00000	−0.144338
$49$	2.64619	0.378028
$50$	−5.67130	−0.802043
$51$	−3.61242	−0.505840
$52$	0.359125	0.0498017
$53$	−12.4762	−1.71374	−0.856869	−	0.515534i	$-0.827594\pi$
$53$	−12.4762	−1.71374	−0.856869	+	0.515534i	$0.827594\pi$
$54$	1.00000	0.136083
$55$	−15.1112	−2.03759
$56$	3.10583	0.415034
$57$	3.61242	0.478476
$58$	1.00000	0.131306
$59$	6.85616	0.892596	0.446298	−	0.894884i	$-0.352742\pi$
$59$	6.85616	0.892596	0.446298	+	0.894884i	$0.352742\pi$
$60$	−3.26670	−0.421729
$61$	−10.8557	−1.38993	−0.694965	−	0.719043i	$-0.744580\pi$
$61$	−10.8557	−1.38993	−0.694965	+	0.719043i	$0.744580\pi$
$62$	−7.25499	−0.921385
$63$	−3.10583	−0.391298
$64$	1.00000	0.125000
$65$	1.17315	0.145512
$66$	−4.62582	−0.569399
$67$	5.78195	0.706377	0.353189	−	0.935552i	$-0.385097\pi$
$67$	5.78195	0.706377	0.353189	+	0.935552i	$0.385097\pi$
$68$	3.61242	0.438070
$69$	1.00000	0.120386
$70$	10.1458	1.21266
$71$	−12.8925	−1.53006	−0.765030	−	0.643994i	$-0.777276\pi$
$71$	−12.8925	−1.53006	−0.765030	+	0.643994i	$0.777276\pi$
$72$	−1.00000	−0.117851
$73$	−14.7979	−1.73197	−0.865983	−	0.500073i	$-0.833307\pi$
$73$	−14.7979	−1.73197	−0.865983	+	0.500073i	$0.833307\pi$
$74$	−7.37615	−0.857460
$75$	−5.67130	−0.654866
$76$	−3.61242	−0.414373
$77$	14.3670	1.63727
$78$	0.359125	0.0406629
$79$	1.92460	0.216534	0.108267	−	0.994122i	$-0.465470\pi$
$79$	1.92460	0.216534	0.108267	+	0.994122i	$0.465470\pi$
$80$	3.26670	0.365228
$81$	1.00000	0.111111
$82$	−3.60880	−0.398525
$83$	−17.6984	−1.94265	−0.971325	−	0.237756i	$-0.923588\pi$
$83$	−17.6984	−1.94265	−0.971325	+	0.237756i	$0.923588\pi$
$84$	3.10583	0.338874
$85$	11.8007	1.27996
$86$	−1.96792	−0.212206
$87$	1.00000	0.107211
$88$	4.62582	0.493114
$89$	−7.10780	−0.753426	−0.376713	−	0.926330i	$-0.622946\pi$
$89$	−7.10780	−0.753426	−0.376713	+	0.926330i	$0.622946\pi$
$90$	−3.26670	−0.344340
$91$	−1.11538	−0.116924
$92$	−1.00000	−0.104257
$93$	−7.25499	−0.752307
$94$	−7.50235	−0.773809
$95$	−11.8007	−1.21072
$96$	1.00000	0.102062
$97$	8.47620	0.860628	0.430314	−	0.902679i	$-0.358403\pi$
$97$	8.47620	0.860628	0.430314	+	0.902679i	$0.358403\pi$
$98$	−2.64619	−0.267306
$99$	−4.62582	−0.464913
$100$	5.67130	0.567130
$101$	13.5423	1.34751	0.673756	−	0.738954i	$-0.264680\pi$
$101$	13.5423	1.34751	0.673756	+	0.738954i	$0.264680\pi$
$102$	3.61242	0.357683
$103$	−18.5253	−1.82535	−0.912676	−	0.408683i	$-0.865988\pi$
$103$	−18.5253	−1.82535	−0.912676	+	0.408683i	$0.865988\pi$
$104$	−0.359125	−0.0352151
$105$	10.1458	0.990130
$106$	12.4762	1.21180
$107$	−0.667682	−0.0645472	−0.0322736	−	0.999479i	$-0.510275\pi$
$107$	−0.667682	−0.0645472	−0.0322736	+	0.999479i	$0.510275\pi$
$108$	−1.00000	−0.0962250
$109$	−10.8433	−1.03860	−0.519298	−	0.854593i	$-0.673807\pi$
$109$	−10.8433	−1.03860	−0.519298	+	0.854593i	$0.673807\pi$
$110$	15.1112	1.44079
$111$	−7.37615	−0.700113
$112$	−3.10583	−0.293474
$113$	−0.303588	−0.0285592	−0.0142796	−	0.999898i	$-0.504545\pi$
$113$	−0.303588	−0.0285592	−0.0142796	+	0.999898i	$0.504545\pi$
$114$	−3.61242	−0.338334
$115$	−3.26670	−0.304621
$116$	−1.00000	−0.0928477
$117$	0.359125	0.0332011
$118$	−6.85616	−0.631161
$119$	−11.2196	−1.02850
$120$	3.26670	0.298207
$121$	10.3982	0.945293
$122$	10.8557	0.982829
$123$	−3.60880	−0.325394
$124$	7.25499	0.651517
$125$	2.19294	0.196143
$126$	3.10583	0.276690
$127$	−10.1925	−0.904437	−0.452218	−	0.891907i	$-0.649367\pi$
$127$	−10.1925	−0.904437	−0.452218	+	0.891907i	$0.649367\pi$
$128$	−1.00000	−0.0883883
$129$	−1.96792	−0.173266
$130$	−1.17315	−0.102892
$131$	1.47786	0.129121	0.0645604	−	0.997914i	$-0.479435\pi$
$131$	1.47786	0.129121	0.0645604	+	0.997914i	$0.479435\pi$
$132$	4.62582	0.402626
$133$	11.2196	0.972860
$134$	−5.78195	−0.499484
$135$	−3.26670	−0.281152
$136$	−3.61242	−0.309762
$137$	−8.57441	−0.732561	−0.366281	−	0.930504i	$-0.619369\pi$
$137$	−8.57441	−0.732561	−0.366281	+	0.930504i	$0.619369\pi$
$138$	−1.00000	−0.0851257
$139$	−0.506312	−0.0429449	−0.0214724	−	0.999769i	$-0.506835\pi$
$139$	−0.506312	−0.0429449	−0.0214724	+	0.999769i	$0.506835\pi$
$140$	−10.1458	−0.857477
$141$	−7.50235	−0.631812
$142$	12.8925	1.08192
$143$	−1.66125	−0.138921
$144$	1.00000	0.0833333
$145$	−3.26670	−0.271284
$146$	14.7979	1.22469
$147$	−2.64619	−0.218254
$148$	7.37615	0.606316
$149$	8.45051	0.692293	0.346146	−	0.938180i	$-0.387490\pi$
$149$	8.45051	0.692293	0.346146	+	0.938180i	$0.387490\pi$
$150$	5.67130	0.463060
$151$	−9.89949	−0.805609	−0.402804	−	0.915286i	$-0.631965\pi$
$151$	−9.89949	−0.805609	−0.402804	+	0.915286i	$0.631965\pi$
$152$	3.61242	0.293006
$153$	3.61242	0.292047
$154$	−14.3670	−1.15773
$155$	23.6998	1.90362
$156$	−0.359125	−0.0287530
$157$	5.89683	0.470618	0.235309	−	0.971921i	$-0.424390\pi$
$157$	5.89683	0.470618	0.235309	+	0.971921i	$0.424390\pi$
$158$	−1.92460	−0.153113
$159$	12.4762	0.989427
$160$	−3.26670	−0.258255
$161$	3.10583	0.244774
$162$	−1.00000	−0.0785674
$163$	−19.5668	−1.53259	−0.766296	−	0.642487i	$-0.777903\pi$
$163$	−19.5668	−1.53259	−0.766296	+	0.642487i	$0.777903\pi$
$164$	3.60880	0.281800
$165$	15.1112	1.17640
$166$	17.6984	1.37366
$167$	−13.2735	−1.02713	−0.513566	−	0.858050i	$-0.671676\pi$
$167$	−13.2735	−1.02713	−0.513566	+	0.858050i	$0.671676\pi$
$168$	−3.10583	−0.239620
$169$	−12.8710	−0.990079
$170$	−11.8007	−0.905070
$171$	−3.61242	−0.276248
$172$	1.96792	0.150053
$173$	15.9792	1.21487	0.607437	−	0.794368i	$-0.292198\pi$
$173$	15.9792	1.21487	0.607437	+	0.794368i	$0.292198\pi$
$174$	−1.00000	−0.0758098
$175$	−17.6141	−1.33150
$176$	−4.62582	−0.348684
$177$	−6.85616	−0.515341
$178$	7.10780	0.532752
$179$	20.3635	1.52204	0.761019	−	0.648729i	$-0.224699\pi$
$179$	20.3635	1.52204	0.761019	+	0.648729i	$0.224699\pi$
$180$	3.26670	0.243485
$181$	−9.32677	−0.693254	−0.346627	−	0.938003i	$-0.612673\pi$
$181$	−9.32677	−0.693254	−0.346627	+	0.938003i	$0.612673\pi$
$182$	1.11538	0.0826776
$183$	10.8557	0.802477
$184$	1.00000	0.0737210
$185$	24.0956	1.77155
$186$	7.25499	0.531962
$187$	−16.7104	−1.22199
$188$	7.50235	0.547165
$189$	3.10583	0.225916
$190$	11.8007	0.856111
$191$	−10.8017	−0.781586	−0.390793	−	0.920479i	$-0.627799\pi$
$191$	−10.8017	−0.781586	−0.390793	+	0.920479i	$0.627799\pi$
$192$	−1.00000	−0.0721688
$193$	8.32540	0.599275	0.299638	−	0.954053i	$-0.403134\pi$
$193$	8.32540	0.599275	0.299638	+	0.954053i	$0.403134\pi$
$194$	−8.47620	−0.608556
$195$	−1.17315	−0.0840112
$196$	2.64619	0.189014
$197$	3.38859	0.241427	0.120713	−	0.992687i	$-0.461482\pi$
$197$	3.38859	0.241427	0.120713	+	0.992687i	$0.461482\pi$
$198$	4.62582	0.328743
$199$	20.9764	1.48698	0.743489	−	0.668748i	$-0.233170\pi$
$199$	20.9764	1.48698	0.743489	+	0.668748i	$0.233170\pi$
$200$	−5.67130	−0.401022
$201$	−5.78195	−0.407827
$202$	−13.5423	−0.952835
$203$	3.10583	0.217987
$204$	−3.61242	−0.252920
$205$	11.7888	0.823368
$206$	18.5253	1.29072
$207$	−1.00000	−0.0695048
$208$	0.359125	0.0249008
$209$	16.7104	1.15588
$210$	−10.1458	−0.700127
$211$	−11.6237	−0.800212	−0.400106	−	0.916469i	$-0.631027\pi$
$211$	−11.6237	−0.800212	−0.400106	+	0.916469i	$0.631027\pi$
$212$	−12.4762	−0.856869
$213$	12.8925	0.883381
$214$	0.667682	0.0456418
$215$	6.42860	0.438427
$216$	1.00000	0.0680414
$217$	−22.5328	−1.52962
$218$	10.8433	0.734398
$219$	14.7979	0.999952
$220$	−15.1112	−1.01879
$221$	1.29731	0.0872665
$222$	7.37615	0.495055
$223$	−19.2179	−1.28692	−0.643462	−	0.765478i	$-0.722502\pi$
$223$	−19.2179	−1.28692	−0.643462	+	0.765478i	$0.722502\pi$
$224$	3.10583	0.207517
$225$	5.67130	0.378087
$226$	0.303588	0.0201944
$227$	−15.3287	−1.01740	−0.508702	−	0.860943i	$-0.669875\pi$
$227$	−15.3287	−1.01740	−0.508702	+	0.860943i	$0.669875\pi$
$228$	3.61242	0.239238
$229$	0.895633	0.0591851	0.0295925	−	0.999562i	$-0.490579\pi$
$229$	0.895633	0.0591851	0.0295925	+	0.999562i	$0.490579\pi$
$230$	3.26670	0.215400
$231$	−14.3670	−0.945281
$232$	1.00000	0.0656532
$233$	21.3532	1.39890	0.699449	−	0.714683i	$-0.253429\pi$
$233$	21.3532	1.39890	0.699449	+	0.714683i	$0.253429\pi$
$234$	−0.359125	−0.0234767
$235$	24.5079	1.59872
$236$	6.85616	0.446298
$237$	−1.92460	−0.125016
$238$	11.2196	0.727256
$239$	0.850151	0.0549917	0.0274959	−	0.999622i	$-0.491247\pi$
$239$	0.850151	0.0549917	0.0274959	+	0.999622i	$0.491247\pi$
$240$	−3.26670	−0.210864
$241$	−20.4259	−1.31575	−0.657874	−	0.753128i	$-0.728544\pi$
$241$	−20.4259	−1.31575	−0.657874	+	0.753128i	$0.728544\pi$
$242$	−10.3982	−0.668423
$243$	−1.00000	−0.0641500
$244$	−10.8557	−0.694965
$245$	8.64431	0.552265
$246$	3.60880	0.230088
$247$	−1.29731	−0.0825458
$248$	−7.25499	−0.460692
$249$	17.6984	1.12159
$250$	−2.19294	−0.138694
$251$	19.7082	1.24397	0.621984	−	0.783030i	$-0.286327\pi$
$251$	19.7082	1.24397	0.621984	+	0.783030i	$0.286327\pi$
$252$	−3.10583	−0.195649
$253$	4.62582	0.290823
$254$	10.1925	0.639533
$255$	−11.8007	−0.738987
$256$	1.00000	0.0625000
$257$	−24.0750	−1.50176	−0.750880	−	0.660439i	$-0.770370\pi$
$257$	−24.0750	−1.50176	−0.750880	+	0.660439i	$0.770370\pi$
$258$	1.96792	0.122517
$259$	−22.9091	−1.42350
$260$	1.17315	0.0727558
$261$	−1.00000	−0.0618984
$262$	−1.47786	−0.0913022
$263$	2.87935	0.177548	0.0887740	−	0.996052i	$-0.471705\pi$
$263$	2.87935	0.177548	0.0887740	+	0.996052i	$0.471705\pi$
$264$	−4.62582	−0.284700
$265$	−40.7560	−2.50362
$266$	−11.2196	−0.687916
$267$	7.10780	0.434990
$268$	5.78195	0.353189
$269$	−4.29231	−0.261707	−0.130853	−	0.991402i	$-0.541772\pi$
$269$	−4.29231	−0.261707	−0.130853	+	0.991402i	$0.541772\pi$
$270$	3.26670	0.198805
$271$	14.7073	0.893406	0.446703	−	0.894682i	$-0.352598\pi$
$271$	14.7073	0.893406	0.446703	+	0.894682i	$0.352598\pi$
$272$	3.61242	0.219035
$273$	1.11538	0.0675060
$274$	8.57441	0.517999
$275$	−26.2344	−1.58200
$276$	1.00000	0.0601929
$277$	−18.4692	−1.10971	−0.554853	−	0.831948i	$-0.687225\pi$
$277$	−18.4692	−1.10971	−0.554853	+	0.831948i	$0.687225\pi$
$278$	0.506312	0.0303666
$279$	7.25499	0.434345
$280$	10.1458	0.606328
$281$	−22.5510	−1.34528	−0.672640	−	0.739970i	$-0.734840\pi$
$281$	−22.5510	−1.34528	−0.672640	+	0.739970i	$0.734840\pi$
$282$	7.50235	0.446759
$283$	−28.8834	−1.71694	−0.858470	−	0.512864i	$-0.828584\pi$
$283$	−28.8834	−1.71694	−0.858470	+	0.512864i	$0.828584\pi$
$284$	−12.8925	−0.765030
$285$	11.8007	0.699011
$286$	1.66125	0.0982316
$287$	−11.2083	−0.661606
$288$	−1.00000	−0.0589256
$289$	−3.95044	−0.232379
$290$	3.26670	0.191827
$291$	−8.47620	−0.496884
$292$	−14.7979	−0.865983
$293$	26.4574	1.54566	0.772828	−	0.634615i	$-0.218841\pi$
$293$	26.4574	1.54566	0.772828	+	0.634615i	$0.218841\pi$
$294$	2.64619	0.154329
$295$	22.3970	1.30400
$296$	−7.37615	−0.428730
$297$	4.62582	0.268417
$298$	−8.45051	−0.489525
$299$	−0.359125	−0.0207687
$300$	−5.67130	−0.327433
$301$	−6.11203	−0.352292
$302$	9.89949	0.569651
$303$	−13.5423	−0.777987
$304$	−3.61242	−0.207186
$305$	−35.4623	−2.03056
$306$	−3.61242	−0.206508
$307$	−3.62266	−0.206756	−0.103378	−	0.994642i	$-0.532965\pi$
$307$	−3.62266	−0.206756	−0.103378	+	0.994642i	$0.532965\pi$
$308$	14.3670	0.818637
$309$	18.5253	1.05387
$310$	−23.6998	−1.34606
$311$	14.9177	0.845904	0.422952	−	0.906152i	$-0.360994\pi$
$311$	14.9177	0.845904	0.422952	+	0.906152i	$0.360994\pi$
$312$	0.359125	0.0203314
$313$	−7.17427	−0.405514	−0.202757	−	0.979229i	$-0.564990\pi$
$313$	−7.17427	−0.405514	−0.202757	+	0.979229i	$0.564990\pi$
$314$	−5.89683	−0.332777
$315$	−10.1458	−0.571652
$316$	1.92460	0.108267
$317$	−7.43985	−0.417864	−0.208932	−	0.977930i	$-0.566999\pi$
$317$	−7.43985	−0.417864	−0.208932	+	0.977930i	$0.566999\pi$
$318$	−12.4762	−0.699631
$319$	4.62582	0.258996
$320$	3.26670	0.182614
$321$	0.667682	0.0372664
$322$	−3.10583	−0.173081
$323$	−13.0496	−0.726097
$324$	1.00000	0.0555556
$325$	2.03671	0.112976
$326$	19.5668	1.08371
$327$	10.8433	0.599634
$328$	−3.60880	−0.199262
$329$	−23.3011	−1.28463
$330$	−15.1112	−0.831841
$331$	19.3628	1.06428	0.532138	−	0.846658i	$-0.321389\pi$
$331$	19.3628	1.06428	0.532138	+	0.846658i	$0.321389\pi$
$332$	−17.6984	−0.971325
$333$	7.37615	0.404211
$334$	13.2735	0.726293
$335$	18.8879	1.03195
$336$	3.10583	0.169437
$337$	−8.87065	−0.483215	−0.241608	−	0.970374i	$-0.577675\pi$
$337$	−8.87065	−0.483215	−0.241608	+	0.970374i	$0.577675\pi$
$338$	12.8710	0.700092
$339$	0.303588	0.0164886
$340$	11.8007	0.639981
$341$	−33.5603	−1.81739
$342$	3.61242	0.195337
$343$	13.5222	0.730130
$344$	−1.96792	−0.106103
$345$	3.26670	0.175873
$346$	−15.9792	−0.859045
$347$	1.24967	0.0670859	0.0335429	−	0.999437i	$-0.489321\pi$
$347$	1.24967	0.0670859	0.0335429	+	0.999437i	$0.489321\pi$
$348$	1.00000	0.0536056
$349$	22.2709	1.19214	0.596068	−	0.802934i	$-0.296729\pi$
$349$	22.2709	1.19214	0.596068	+	0.802934i	$0.296729\pi$
$350$	17.6141	0.941514
$351$	−0.359125	−0.0191687
$352$	4.62582	0.246557
$353$	−4.18752	−0.222879	−0.111439	−	0.993771i	$-0.535546\pi$
$353$	−4.18752	−0.222879	−0.111439	+	0.993771i	$0.535546\pi$
$354$	6.85616	0.364401
$355$	−42.1159	−2.23528
$356$	−7.10780	−0.376713
$357$	11.2196	0.593802
$358$	−20.3635	−1.07624
$359$	−19.8518	−1.04774	−0.523868	−	0.851799i	$-0.675512\pi$
$359$	−19.8518	−1.04774	−0.523868	+	0.851799i	$0.675512\pi$
$360$	−3.26670	−0.172170
$361$	−5.95044	−0.313181
$362$	9.32677	0.490204
$363$	−10.3982	−0.545765
$364$	−1.11538	−0.0584619
$365$	−48.3403	−2.53025
$366$	−10.8557	−0.567437
$367$	−12.3913	−0.646822	−0.323411	−	0.946259i	$-0.604830\pi$
$367$	−12.3913	−0.646822	−0.323411	+	0.946259i	$0.604830\pi$
$368$	−1.00000	−0.0521286
$369$	3.60880	0.187866
$370$	−24.0956	−1.25267
$371$	38.7490	2.01175
$372$	−7.25499	−0.376154
$373$	−14.0367	−0.726793	−0.363397	−	0.931635i	$-0.618383\pi$
$373$	−14.0367	−0.726793	−0.363397	+	0.931635i	$0.618383\pi$
$374$	16.7104	0.864074
$375$	−2.19294	−0.113243
$376$	−7.50235	−0.386904
$377$	−0.359125	−0.0184959
$378$	−3.10583	−0.159747
$379$	−23.3581	−1.19983	−0.599913	−	0.800065i	$-0.704798\pi$
$379$	−23.3581	−1.19983	−0.599913	+	0.800065i	$0.704798\pi$
$380$	−11.8007	−0.605362
$381$	10.1925	0.522177
$382$	10.8017	0.552665
$383$	−16.6697	−0.851779	−0.425890	−	0.904775i	$-0.640039\pi$
$383$	−16.6697	−0.851779	−0.425890	+	0.904775i	$0.640039\pi$
$384$	1.00000	0.0510310
$385$	46.9327	2.39191
$386$	−8.32540	−0.423751
$387$	1.96792	0.100035
$388$	8.47620	0.430314
$389$	−3.90931	−0.198210	−0.0991049	−	0.995077i	$-0.531598\pi$
$389$	−3.90931	−0.198210	−0.0991049	+	0.995077i	$0.531598\pi$
$390$	1.17315	0.0594049
$391$	−3.61242	−0.182688
$392$	−2.64619	−0.133653
$393$	−1.47786	−0.0745479
$394$	−3.38859	−0.170715
$395$	6.28707	0.316337
$396$	−4.62582	−0.232456
$397$	−4.31415	−0.216521	−0.108260	−	0.994123i	$-0.534528\pi$
$397$	−4.31415	−0.216521	−0.108260	+	0.994123i	$0.534528\pi$
$398$	−20.9764	−1.05145
$399$	−11.2196	−0.561681
$400$	5.67130	0.283565
$401$	37.0780	1.85159	0.925794	−	0.378029i	$-0.123398\pi$
$401$	37.0780	1.85159	0.925794	+	0.378029i	$0.123398\pi$
$402$	5.78195	0.288377
$403$	2.60545	0.129787
$404$	13.5423	0.673756
$405$	3.26670	0.162323
$406$	−3.10583	−0.154140
$407$	−34.1207	−1.69130
$408$	3.61242	0.178841
$409$	−16.1746	−0.799784	−0.399892	−	0.916562i	$-0.630952\pi$
$409$	−16.1746	−0.799784	−0.399892	+	0.916562i	$0.630952\pi$
$410$	−11.7888	−0.582209
$411$	8.57441	0.422944
$412$	−18.5253	−0.912676
$413$	−21.2941	−1.04781
$414$	1.00000	0.0491473
$415$	−57.8152	−2.83804
$416$	−0.359125	−0.0176076
$417$	0.506312	0.0247942
$418$	−16.7104	−0.817332
$419$	27.9302	1.36448	0.682241	−	0.731128i	$-0.261006\pi$
$419$	27.9302	1.36448	0.682241	+	0.731128i	$0.261006\pi$
$420$	10.1458	0.495065
$421$	0.972735	0.0474082	0.0237041	−	0.999719i	$-0.492454\pi$
$421$	0.972735	0.0474082	0.0237041	+	0.999719i	$0.492454\pi$
$422$	11.6237	0.565835
$423$	7.50235	0.364777
$424$	12.4762	0.605898
$425$	20.4871	0.993771
$426$	−12.8925	−0.624644
$427$	33.7160	1.63163
$428$	−0.667682	−0.0322736
$429$	1.66125	0.0802058
$430$	−6.42860	−0.310015
$431$	−9.74698	−0.469496	−0.234748	−	0.972056i	$-0.575426\pi$
$431$	−9.74698	−0.469496	−0.234748	+	0.972056i	$0.575426\pi$
$432$	−1.00000	−0.0481125
$433$	−21.5728	−1.03672	−0.518360	−	0.855162i	$-0.673457\pi$
$433$	−21.5728	−1.03672	−0.518360	+	0.855162i	$0.673457\pi$
$434$	22.5328	1.08161
$435$	3.26670	0.156626
$436$	−10.8433	−0.519298
$437$	3.61242	0.172805
$438$	−14.7979	−0.707073
$439$	−19.7678	−0.943465	−0.471732	−	0.881742i	$-0.656371\pi$
$439$	−19.7678	−0.943465	−0.471732	+	0.881742i	$0.656371\pi$
$440$	15.1112	0.720396
$441$	2.64619	0.126009
$442$	−1.29731	−0.0617067
$443$	1.19445	0.0567502	0.0283751	−	0.999597i	$-0.490967\pi$
$443$	1.19445	0.0567502	0.0283751	+	0.999597i	$0.490967\pi$
$444$	−7.37615	−0.350057
$445$	−23.2190	−1.10069
$446$	19.2179	0.909992
$447$	−8.45051	−0.399696
$448$	−3.10583	−0.146737
$449$	31.0697	1.46627	0.733135	−	0.680083i	$-0.238056\pi$
$449$	31.0697	1.46627	0.733135	+	0.680083i	$0.238056\pi$
$450$	−5.67130	−0.267348
$451$	−16.6936	−0.786073
$452$	−0.303588	−0.0142796
$453$	9.89949	0.465118
$454$	15.3287	0.719414
$455$	−3.64361	−0.170815
$456$	−3.61242	−0.169167
$457$	−7.97981	−0.373280	−0.186640	−	0.982428i	$-0.559760\pi$
$457$	−7.97981	−0.373280	−0.186640	+	0.982428i	$0.559760\pi$
$458$	−0.895633	−0.0418502
$459$	−3.61242	−0.168613
$460$	−3.26670	−0.152310
$461$	−25.8318	−1.20311	−0.601553	−	0.798833i	$-0.705451\pi$
$461$	−25.8318	−1.20311	−0.601553	+	0.798833i	$0.705451\pi$
$462$	14.3670	0.668414
$463$	−7.28202	−0.338424	−0.169212	−	0.985580i	$-0.554122\pi$
$463$	−7.28202	−0.338424	−0.169212	+	0.985580i	$0.554122\pi$
$464$	−1.00000	−0.0464238
$465$	−23.6998	−1.09905
$466$	−21.3532	−0.989170
$467$	−24.4559	−1.13168	−0.565842	−	0.824513i	$-0.691449\pi$
$467$	−24.4559	−1.13168	−0.565842	+	0.824513i	$0.691449\pi$
$468$	0.359125	0.0166006
$469$	−17.9578	−0.829212
$470$	−24.5079	−1.13047
$471$	−5.89683	−0.271712
$472$	−6.85616	−0.315580
$473$	−9.10325	−0.418568
$474$	1.92460	0.0883996
$475$	−20.4871	−0.940013
$476$	−11.2196	−0.514248
$477$	−12.4762	−0.571246
$478$	−0.850151	−0.0388850
$479$	−28.2731	−1.29183	−0.645916	−	0.763409i	$-0.723524\pi$
$479$	−28.2731	−1.29183	−0.645916	+	0.763409i	$0.723524\pi$
$480$	3.26670	0.149104
$481$	2.64896	0.120782
$482$	20.4259	0.930375
$483$	−3.10583	−0.141320
$484$	10.3982	0.472646
$485$	27.6892	1.25730
$486$	1.00000	0.0453609
$487$	26.8492	1.21666	0.608328	−	0.793686i	$-0.291841\pi$
$487$	26.8492	1.21666	0.608328	+	0.793686i	$0.291841\pi$
$488$	10.8557	0.491415
$489$	19.5668	0.884843
$490$	−8.64431	−0.390510
$491$	28.0890	1.26764	0.633819	−	0.773482i	$-0.281486\pi$
$491$	28.0890	1.26764	0.633819	+	0.773482i	$0.281486\pi$
$492$	−3.60880	−0.162697
$493$	−3.61242	−0.162695
$494$	1.29731	0.0583687
$495$	−15.1112	−0.679196
$496$	7.25499	0.325759
$497$	40.0420	1.79613
$498$	−17.6984	−0.793083
$499$	−17.3727	−0.777708	−0.388854	−	0.921299i	$-0.627129\pi$
$499$	−17.3727	−0.777708	−0.388854	+	0.921299i	$0.627129\pi$
$500$	2.19294	0.0980714
$501$	13.2735	0.593015
$502$	−19.7082	−0.879618
$503$	−0.577635	−0.0257555	−0.0128777	−	0.999917i	$-0.504099\pi$
$503$	−0.577635	−0.0257555	−0.0128777	+	0.999917i	$0.504099\pi$
$504$	3.10583	0.138345
$505$	44.2387	1.96860
$506$	−4.62582	−0.205643
$507$	12.8710	0.571622
$508$	−10.1925	−0.452218
$509$	33.1739	1.47041	0.735204	−	0.677846i	$-0.237087\pi$
$509$	33.1739	1.47041	0.735204	+	0.677846i	$0.237087\pi$
$510$	11.8007	0.522542
$511$	45.9599	2.03315
$512$	−1.00000	−0.0441942
$513$	3.61242	0.159492
$514$	24.0750	1.06190
$515$	−60.5165	−2.66668
$516$	−1.96792	−0.0866329
$517$	−34.7045	−1.52630
$518$	22.9091	1.00657
$519$	−15.9792	−0.701408
$520$	−1.17315	−0.0514461
$521$	−35.4092	−1.55130	−0.775652	−	0.631161i	$-0.782579\pi$
$521$	−35.4092	−1.55130	−0.775652	+	0.631161i	$0.782579\pi$
$522$	1.00000	0.0437688
$523$	−37.5135	−1.64035	−0.820175	−	0.572113i	$-0.806124\pi$
$523$	−37.5135	−1.64035	−0.820175	+	0.572113i	$0.806124\pi$
$524$	1.47786	0.0645604
$525$	17.6141	0.768743
$526$	−2.87935	−0.125545
$527$	26.2081	1.14164
$528$	4.62582	0.201313
$529$	1.00000	0.0434783
$530$	40.7560	1.77033
$531$	6.85616	0.297532
$532$	11.2196	0.486430
$533$	1.29601	0.0561364
$534$	−7.10780	−0.307585
$535$	−2.18111	−0.0942977
$536$	−5.78195	−0.249742
$537$	−20.3635	−0.878749
$538$	4.29231	0.185055
$539$	−12.2408	−0.527249
$540$	−3.26670	−0.140576
$541$	−15.9464	−0.685591	−0.342795	−	0.939410i	$-0.611374\pi$
$541$	−15.9464	−0.685591	−0.342795	+	0.939410i	$0.611374\pi$
$542$	−14.7073	−0.631734
$543$	9.32677	0.400250
$544$	−3.61242	−0.154881
$545$	−35.4216	−1.51730
$546$	−1.11538	−0.0477339
$547$	26.4702	1.13179	0.565893	−	0.824479i	$-0.308532\pi$
$547$	26.4702	1.13179	0.565893	+	0.824479i	$0.308532\pi$
$548$	−8.57441	−0.366281
$549$	−10.8557	−0.463310
$550$	26.2344	1.11864
$551$	3.61242	0.153894
$552$	−1.00000	−0.0425628
$553$	−5.97747	−0.254188
$554$	18.4692	0.784681
$555$	−24.0956	−1.02280
$556$	−0.506312	−0.0214724
$557$	4.65521	0.197247	0.0986237	−	0.995125i	$-0.468556\pi$
$557$	4.65521	0.197247	0.0986237	+	0.995125i	$0.468556\pi$
$558$	−7.25499	−0.307128
$559$	0.706730	0.0298915
$560$	−10.1458	−0.428739
$561$	16.7104	0.705514
$562$	22.5510	0.951256
$563$	−9.14072	−0.385236	−0.192618	−	0.981274i	$-0.561698\pi$
$563$	−9.14072	−0.385236	−0.192618	+	0.981274i	$0.561698\pi$
$564$	−7.50235	−0.315906
$565$	−0.991730	−0.0417224
$566$	28.8834	1.21406
$567$	−3.10583	−0.130433
$568$	12.8925	0.540958
$569$	36.9167	1.54763	0.773813	−	0.633414i	$-0.218347\pi$
$569$	36.9167	1.54763	0.773813	+	0.633414i	$0.218347\pi$
$570$	−11.8007	−0.494276
$571$	26.6720	1.11619	0.558094	−	0.829778i	$-0.311533\pi$
$571$	26.6720	1.11619	0.558094	+	0.829778i	$0.311533\pi$
$572$	−1.66125	−0.0694603
$573$	10.8017	0.451249
$574$	11.2083	0.467826
$575$	−5.67130	−0.236510
$576$	1.00000	0.0416667
$577$	−15.1091	−0.629002	−0.314501	−	0.949257i	$-0.601837\pi$
$577$	−15.1091	−0.629002	−0.314501	+	0.949257i	$0.601837\pi$
$578$	3.95044	0.164317
$579$	−8.32540	−0.345992
$580$	−3.26670	−0.135642
$581$	54.9682	2.28047
$582$	8.47620	0.351350
$583$	57.7127	2.39022
$584$	14.7979	0.612343
$585$	1.17315	0.0485039
$586$	−26.4574	−1.09294
$587$	7.71136	0.318282	0.159141	−	0.987256i	$-0.449128\pi$
$587$	7.71136	0.318282	0.159141	+	0.987256i	$0.449128\pi$
$588$	−2.64619	−0.109127
$589$	−26.2081	−1.07988
$590$	−22.3970	−0.922070
$591$	−3.38859	−0.139388
$592$	7.37615	0.303158
$593$	14.0782	0.578122	0.289061	−	0.957311i	$-0.406657\pi$
$593$	14.0782	0.578122	0.289061	+	0.957311i	$0.406657\pi$
$594$	−4.62582	−0.189800
$595$	−36.6509	−1.50254
$596$	8.45051	0.346146
$597$	−20.9764	−0.858507
$598$	0.359125	0.0146857
$599$	−10.8162	−0.441937	−0.220969	−	0.975281i	$-0.570922\pi$
$599$	−10.8162	−0.441937	−0.220969	+	0.975281i	$0.570922\pi$
$600$	5.67130	0.231530
$601$	20.9080	0.852855	0.426427	−	0.904522i	$-0.359772\pi$
$601$	20.9080	0.852855	0.426427	+	0.904522i	$0.359772\pi$
$602$	6.11203	0.249108
$603$	5.78195	0.235459
$604$	−9.89949	−0.402804
$605$	33.9678	1.38099
$606$	13.5423	0.550120
$607$	−13.7361	−0.557530	−0.278765	−	0.960359i	$-0.589925\pi$
$607$	−13.7361	−0.557530	−0.278765	+	0.960359i	$0.589925\pi$
$608$	3.61242	0.146503
$609$	−3.10583	−0.125855
$610$	35.4623	1.43583
$611$	2.69428	0.108999
$612$	3.61242	0.146023
$613$	3.97366	0.160495	0.0802473	−	0.996775i	$-0.474429\pi$
$613$	3.97366	0.160495	0.0802473	+	0.996775i	$0.474429\pi$
$614$	3.62266	0.146199
$615$	−11.7888	−0.475372
$616$	−14.3670	−0.578864
$617$	−39.2761	−1.58120	−0.790598	−	0.612335i	$-0.790230\pi$
$617$	−39.2761	−1.58120	−0.790598	+	0.612335i	$0.790230\pi$
$618$	−18.5253	−0.745197
$619$	−5.08297	−0.204302	−0.102151	−	0.994769i	$-0.532572\pi$
$619$	−5.08297	−0.204302	−0.102151	+	0.994769i	$0.532572\pi$
$620$	23.6998	0.951809
$621$	1.00000	0.0401286
$622$	−14.9177	−0.598144
$623$	22.0756	0.884442
$624$	−0.359125	−0.0143765
$625$	−21.1928	−0.847714
$626$	7.17427	0.286741
$627$	−16.7104	−0.667349
$628$	5.89683	0.235309
$629$	26.6457	1.06243
$630$	10.1458	0.404219
$631$	36.8546	1.46716	0.733580	−	0.679603i	$-0.237848\pi$
$631$	36.8546	1.46716	0.733580	+	0.679603i	$0.237848\pi$
$632$	−1.92460	−0.0765563
$633$	11.6237	0.462002
$634$	7.43985	0.295474
$635$	−33.2957	−1.32130
$636$	12.4762	0.494714
$637$	0.950314	0.0376528
$638$	−4.62582	−0.183138
$639$	−12.8925	−0.510020
$640$	−3.26670	−0.129127
$641$	30.4929	1.20440	0.602198	−	0.798347i	$-0.294292\pi$
$641$	30.4929	1.20440	0.602198	+	0.798347i	$0.294292\pi$
$642$	−0.667682	−0.0263513
$643$	10.9114	0.430305	0.215152	−	0.976581i	$-0.430975\pi$
$643$	10.9114	0.430305	0.215152	+	0.976581i	$0.430975\pi$
$644$	3.10583	0.122387
$645$	−6.42860	−0.253126
$646$	13.0496	0.513428
$647$	31.3323	1.23180	0.615900	−	0.787825i	$-0.288793\pi$
$647$	31.3323	1.23180	0.615900	+	0.787825i	$0.288793\pi$
$648$	−1.00000	−0.0392837
$649$	−31.7154	−1.24494
$650$	−2.03671	−0.0798862
$651$	22.5328	0.883129
$652$	−19.5668	−0.766296
$653$	30.0794	1.17710	0.588549	−	0.808462i	$-0.299699\pi$
$653$	30.0794	1.17710	0.588549	+	0.808462i	$0.299699\pi$
$654$	−10.8433	−0.424005
$655$	4.82770	0.188634
$656$	3.60880	0.140900
$657$	−14.7979	−0.577322
$658$	23.3011	0.908370
$659$	49.2474	1.91841	0.959203	−	0.282718i	$-0.0912361\pi$
$659$	49.2474	1.91841	0.959203	+	0.282718i	$0.0912361\pi$
$660$	15.1112	0.588201
$661$	−3.65440	−0.142140	−0.0710698	−	0.997471i	$-0.522641\pi$
$661$	−3.65440	−0.142140	−0.0710698	+	0.997471i	$0.522641\pi$
$662$	−19.3628	−0.752556
$663$	−1.29731	−0.0503833
$664$	17.6984	0.686830
$665$	36.6509	1.42126
$666$	−7.37615	−0.285820
$667$	1.00000	0.0387202
$668$	−13.2735	−0.513566
$669$	19.2179	0.743006
$670$	−18.8879	−0.729702
$671$	50.2165	1.93859
$672$	−3.10583	−0.119810
$673$	−42.8060	−1.65005	−0.825024	−	0.565097i	$-0.808839\pi$
$673$	−42.8060	−1.65005	−0.825024	+	0.565097i	$0.808839\pi$
$674$	8.87065	0.341685
$675$	−5.67130	−0.218289
$676$	−12.8710	−0.495040
$677$	−3.54241	−0.136146	−0.0680730	−	0.997680i	$-0.521685\pi$
$677$	−3.54241	−0.136146	−0.0680730	+	0.997680i	$0.521685\pi$
$678$	−0.303588	−0.0116592
$679$	−26.3257	−1.01029
$680$	−11.8007	−0.452535
$681$	15.3287	0.587399
$682$	33.5603	1.28509
$683$	30.7284	1.17579	0.587895	−	0.808938i	$-0.299957\pi$
$683$	30.7284	1.17579	0.587895	+	0.808938i	$0.299957\pi$
$684$	−3.61242	−0.138124
$685$	−28.0100	−1.07021
$686$	−13.5222	−0.516280
$687$	−0.895633	−0.0341705
$688$	1.96792	0.0750263
$689$	−4.48052	−0.170694
$690$	−3.26670	−0.124361
$691$	2.84653	0.108287	0.0541435	−	0.998533i	$-0.482757\pi$
$691$	2.84653	0.108287	0.0541435	+	0.998533i	$0.482757\pi$
$692$	15.9792	0.607437
$693$	14.3670	0.545758
$694$	−1.24967	−0.0474369
$695$	−1.65397	−0.0627386
$696$	−1.00000	−0.0379049
$697$	13.0365	0.493792
$698$	−22.2709	−0.842968
$699$	−21.3532	−0.807654
$700$	−17.6141	−0.665751
$701$	−49.6380	−1.87480	−0.937401	−	0.348253i	$-0.886775\pi$
$701$	−49.6380	−1.87480	−0.937401	+	0.348253i	$0.886775\pi$
$702$	0.359125	0.0135543
$703$	−26.6457	−1.00496
$704$	−4.62582	−0.174342
$705$	−24.5079	−0.923021
$706$	4.18752	0.157599
$707$	−42.0602	−1.58184
$708$	−6.85616	−0.257670
$709$	16.7414	0.628735	0.314367	−	0.949301i	$-0.398208\pi$
$709$	16.7414	0.628735	0.314367	+	0.949301i	$0.398208\pi$
$710$	42.1159	1.58058
$711$	1.92460	0.0721780
$712$	7.10780	0.266376
$713$	−7.25499	−0.271702
$714$	−11.2196	−0.419882
$715$	−5.42679	−0.202951
$716$	20.3635	0.761019
$717$	−0.850151	−0.0317495
$718$	19.8518	0.740862
$719$	34.6691	1.29294	0.646469	−	0.762940i	$-0.276245\pi$
$719$	34.6691	1.29294	0.646469	+	0.762940i	$0.276245\pi$
$720$	3.26670	0.121743
$721$	57.5365	2.14277
$722$	5.95044	0.221452
$723$	20.4259	0.759648
$724$	−9.32677	−0.346627
$725$	−5.67130	−0.210627
$726$	10.3982	0.385914
$727$	40.2472	1.49269	0.746343	−	0.665562i	$-0.231808\pi$
$727$	40.2472	1.49269	0.746343	+	0.665562i	$0.231808\pi$
$728$	1.11538	0.0413388
$729$	1.00000	0.0370370
$730$	48.3403	1.78916
$731$	7.10895	0.262934
$732$	10.8557	0.401238
$733$	−36.3150	−1.34132	−0.670662	−	0.741763i	$-0.733990\pi$
$733$	−36.3150	−1.34132	−0.670662	+	0.741763i	$0.733990\pi$
$734$	12.3913	0.457372
$735$	−8.64431	−0.318850
$736$	1.00000	0.0368605
$737$	−26.7463	−0.985211
$738$	−3.60880	−0.132842
$739$	27.1465	0.998602	0.499301	−	0.866429i	$-0.333590\pi$
$739$	27.1465	0.998602	0.499301	+	0.866429i	$0.333590\pi$
$740$	24.0956	0.885773
$741$	1.29731	0.0476579
$742$	−38.7490	−1.42252
$743$	46.2863	1.69808	0.849040	−	0.528328i	$-0.177181\pi$
$743$	46.2863	1.69808	0.849040	+	0.528328i	$0.177181\pi$
$744$	7.25499	0.265981
$745$	27.6053	1.01138
$746$	14.0367	0.513920
$747$	−17.6984	−0.647550
$748$	−16.7104	−0.610993
$749$	2.07371	0.0757716
$750$	2.19294	0.0800749
$751$	−5.08531	−0.185566	−0.0927828	−	0.995686i	$-0.529576\pi$
$751$	−5.08531	−0.185566	−0.0927828	+	0.995686i	$0.529576\pi$
$752$	7.50235	0.273583
$753$	−19.7082	−0.718205
$754$	0.359125	0.0130786
$755$	−32.3386	−1.17692
$756$	3.10583	0.112958
$757$	52.1553	1.89561	0.947807	−	0.318843i	$-0.103294\pi$
$757$	52.1553	1.89561	0.947807	+	0.318843i	$0.103294\pi$
$758$	23.3581	0.848406
$759$	−4.62582	−0.167907
$760$	11.8007	0.428055
$761$	35.4073	1.28351	0.641756	−	0.766909i	$-0.278206\pi$
$761$	35.4073	1.28351	0.641756	+	0.766909i	$0.278206\pi$
$762$	−10.1925	−0.369235
$763$	33.6774	1.21920
$764$	−10.8017	−0.390793
$765$	11.8007	0.426654
$766$	16.6697	0.602299
$767$	2.46222	0.0889056
$768$	−1.00000	−0.0360844
$769$	−15.9026	−0.573463	−0.286731	−	0.958011i	$-0.592569\pi$
$769$	−15.9026	−0.573463	−0.286731	+	0.958011i	$0.592569\pi$
$770$	−46.9327	−1.69134
$771$	24.0750	0.867041
$772$	8.32540	0.299638
$773$	23.4226	0.842452	0.421226	−	0.906956i	$-0.361600\pi$
$773$	23.4226	0.842452	0.421226	+	0.906956i	$0.361600\pi$
$774$	−1.96792	−0.0707355
$775$	41.1452	1.47798
$776$	−8.47620	−0.304278
$777$	22.9091	0.821859
$778$	3.90931	0.140156
$779$	−13.0365	−0.467080
$780$	−1.17315	−0.0420056
$781$	59.6385	2.13403
$782$	3.61242	0.129180
$783$	1.00000	0.0357371
$784$	2.64619	0.0945069
$785$	19.2631	0.687531
$786$	1.47786	0.0527133
$787$	−48.8891	−1.74271	−0.871355	−	0.490653i	$-0.836758\pi$
$787$	−48.8891	−1.74271	−0.871355	+	0.490653i	$0.836758\pi$
$788$	3.38859	0.120713
$789$	−2.87935	−0.102507
$790$	−6.28707	−0.223684
$791$	0.942894	0.0335254
$792$	4.62582	0.164371
$793$	−3.89855	−0.138442
$794$	4.31415	0.153103
$795$	40.7560	1.44547
$796$	20.9764	0.743489
$797$	−10.6985	−0.378962	−0.189481	−	0.981884i	$-0.560681\pi$
$797$	−10.6985	−0.378962	−0.189481	+	0.981884i	$0.560681\pi$
$798$	11.2196	0.397168
$799$	27.1016	0.958787
$800$	−5.67130	−0.200511
$801$	−7.10780	−0.251142
$802$	−37.0780	−1.30927
$803$	68.4526	2.41564
$804$	−5.78195	−0.203914
$805$	10.1458	0.357593
$806$	−2.60545	−0.0917730
$807$	4.29231	0.151096
$808$	−13.5423	−0.476418
$809$	−11.1844	−0.393224	−0.196612	−	0.980481i	$-0.562994\pi$
$809$	−11.1844	−0.393224	−0.196612	+	0.980481i	$0.562994\pi$
$810$	−3.26670	−0.114780
$811$	−2.50631	−0.0880085	−0.0440043	−	0.999031i	$-0.514012\pi$
$811$	−2.50631	−0.0880085	−0.0440043	+	0.999031i	$0.514012\pi$
$812$	3.10583	0.108993
$813$	−14.7073	−0.515808
$814$	34.1207	1.19593
$815$	−63.9189	−2.23898
$816$	−3.61242	−0.126460
$817$	−7.10895	−0.248711
$818$	16.1746	0.565532
$819$	−1.11538	−0.0389746
$820$	11.7888	0.411684
$821$	−35.7863	−1.24895	−0.624476	−	0.781044i	$-0.714687\pi$
$821$	−35.7863	−1.24895	−0.624476	+	0.781044i	$0.714687\pi$
$822$	−8.57441	−0.299067
$823$	52.1949	1.81940	0.909700	−	0.415266i	$-0.136312\pi$
$823$	52.1949	1.81940	0.909700	+	0.415266i	$0.136312\pi$
$824$	18.5253	0.645360
$825$	26.2344	0.913366
$826$	21.2941	0.740916
$827$	−32.4142	−1.12715	−0.563576	−	0.826064i	$-0.690575\pi$
$827$	−32.4142	−1.12715	−0.563576	+	0.826064i	$0.690575\pi$
$828$	−1.00000	−0.0347524
$829$	9.63060	0.334485	0.167242	−	0.985916i	$-0.446514\pi$
$829$	9.63060	0.334485	0.167242	+	0.985916i	$0.446514\pi$
$830$	57.8152	2.00680
$831$	18.4692	0.640689
$832$	0.359125	0.0124504
$833$	9.55916	0.331205
$834$	−0.506312	−0.0175322
$835$	−43.3604	−1.50055
$836$	16.7104	0.577941
$837$	−7.25499	−0.250769
$838$	−27.9302	−0.964834
$839$	−20.2383	−0.698705	−0.349352	−	0.936991i	$-0.613598\pi$
$839$	−20.2383	−0.698705	−0.349352	+	0.936991i	$0.613598\pi$
$840$	−10.1458	−0.350064
$841$	1.00000	0.0344828
$842$	−0.972735	−0.0335227
$843$	22.5510	0.776698
$844$	−11.6237	−0.400106
$845$	−42.0457	−1.44642
$846$	−7.50235	−0.257936
$847$	−32.2951	−1.10967
$848$	−12.4762	−0.428435
$849$	28.8834	0.991275
$850$	−20.4871	−0.702702
$851$	−7.37615	−0.252851
$852$	12.8925	0.441690
$853$	−6.49196	−0.222280	−0.111140	−	0.993805i	$-0.535450\pi$
$853$	−6.49196	−0.222280	−0.111140	+	0.993805i	$0.535450\pi$
$854$	−33.7160	−1.15374
$855$	−11.8007	−0.403574
$856$	0.667682	0.0228209
$857$	−43.6971	−1.49267	−0.746333	−	0.665573i	$-0.768187\pi$
$857$	−43.6971	−1.49267	−0.746333	+	0.665573i	$0.768187\pi$
$858$	−1.66125	−0.0567141
$859$	12.0156	0.409966	0.204983	−	0.978766i	$-0.434286\pi$
$859$	12.0156	0.409966	0.204983	+	0.978766i	$0.434286\pi$
$860$	6.42860	0.219213
$861$	11.2083	0.381978
$862$	9.74698	0.331983
$863$	8.39556	0.285788	0.142894	−	0.989738i	$-0.454359\pi$
$863$	8.39556	0.285788	0.142894	+	0.989738i	$0.454359\pi$
$864$	1.00000	0.0340207
$865$	52.1991	1.77482
$866$	21.5728	0.733072
$867$	3.95044	0.134164
$868$	−22.5328	−0.764812
$869$	−8.90283	−0.302008
$870$	−3.26670	−0.110751
$871$	2.07644	0.0703576
$872$	10.8433	0.367199
$873$	8.47620	0.286876
$874$	−3.61242	−0.122192
$875$	−6.81091	−0.230251
$876$	14.7979	0.499976
$877$	−33.3264	−1.12535	−0.562676	−	0.826677i	$-0.690228\pi$
$877$	−33.3264	−1.12535	−0.562676	+	0.826677i	$0.690228\pi$
$878$	19.7678	0.667130
$879$	−26.4574	−0.892385
$880$	−15.1112	−0.509397
$881$	−29.9013	−1.00740	−0.503700	−	0.863878i	$-0.668028\pi$
$881$	−29.9013	−1.00740	−0.503700	+	0.863878i	$0.668028\pi$
$882$	−2.64619	−0.0891020
$883$	−51.0240	−1.71710	−0.858548	−	0.512733i	$-0.828633\pi$
$883$	−51.0240	−1.71710	−0.858548	+	0.512733i	$0.828633\pi$
$884$	1.29731	0.0436332
$885$	−22.3970	−0.752867
$886$	−1.19445	−0.0401285
$887$	−46.7438	−1.56950	−0.784751	−	0.619811i	$-0.787209\pi$
$887$	−46.7438	−1.56950	−0.784751	+	0.619811i	$0.787209\pi$
$888$	7.37615	0.247527
$889$	31.6561	1.06171
$890$	23.2190	0.778304
$891$	−4.62582	−0.154971
$892$	−19.2179	−0.643462
$893$	−27.1016	−0.906922
$894$	8.45051	0.282627
$895$	66.5213	2.22356
$896$	3.10583	0.103759
$897$	0.359125	0.0119908
$898$	−31.0697	−1.03681
$899$	−7.25499	−0.241967
$900$	5.67130	0.189043
$901$	−45.0693	−1.50147
$902$	16.6936	0.555838
$903$	6.11203	0.203396
$904$	0.303588	0.0100972
$905$	−30.4677	−1.01278
$906$	−9.89949	−0.328888
$907$	52.7849	1.75269	0.876347	−	0.481681i	$-0.159974\pi$
$907$	52.7849	1.75269	0.876347	+	0.481681i	$0.159974\pi$
$908$	−15.3287	−0.508702
$909$	13.5423	0.449171
$910$	3.64361	0.120785
$911$	26.2357	0.869226	0.434613	−	0.900617i	$-0.356885\pi$
$911$	26.2357	0.869226	0.434613	+	0.900617i	$0.356885\pi$
$912$	3.61242	0.119619
$913$	81.8695	2.70949
$914$	7.97981	0.263949
$915$	35.4623	1.17235
$916$	0.895633	0.0295925
$917$	−4.58997	−0.151574
$918$	3.61242	0.119228
$919$	−48.6753	−1.60565	−0.802824	−	0.596216i	$-0.796670\pi$
$919$	−48.6753	−1.60565	−0.802824	+	0.596216i	$0.796670\pi$
$920$	3.26670	0.107700
$921$	3.62266	0.119371
$922$	25.8318	0.850725
$923$	−4.63002	−0.152399
$924$	−14.3670	−0.472640
$925$	41.8324	1.37544
$926$	7.28202	0.239302
$927$	−18.5253	−0.608451
$928$	1.00000	0.0328266
$929$	1.81147	0.0594325	0.0297163	−	0.999558i	$-0.490540\pi$
$929$	1.81147	0.0594325	0.0297163	+	0.999558i	$0.490540\pi$
$930$	23.6998	0.777149
$931$	−9.55916	−0.313289
$932$	21.3532	0.699449
$933$	−14.9177	−0.488383
$934$	24.4559	0.800222
$935$	−54.5878	−1.78521
$936$	−0.359125	−0.0117384
$937$	43.8218	1.43160	0.715798	−	0.698307i	$-0.246063\pi$
$937$	43.8218	1.43160	0.715798	+	0.698307i	$0.246063\pi$
$938$	17.9578	0.586342
$939$	7.17427	0.234123
$940$	24.5079	0.799360
$941$	−6.93576	−0.226099	−0.113050	−	0.993589i	$-0.536062\pi$
$941$	−6.93576	−0.226099	−0.113050	+	0.993589i	$0.536062\pi$
$942$	5.89683	0.192129
$943$	−3.60880	−0.117519
$944$	6.85616	0.223149
$945$	10.1458	0.330043
$946$	9.10325	0.295972
$947$	−19.8868	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$947$	−19.8868	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$948$	−1.92460	−0.0625080
$949$	−5.31431	−0.172510
$950$	20.4871	0.664690
$951$	7.43985	0.241254
$952$	11.2196	0.363628
$953$	25.2710	0.818608	0.409304	−	0.912398i	$-0.365771\pi$
$953$	25.2710	0.818608	0.409304	+	0.912398i	$0.365771\pi$
$954$	12.4762	0.403932
$955$	−35.2860	−1.14183
$956$	0.850151	0.0274959
$957$	−4.62582	−0.149532
$958$	28.2731	0.913462
$959$	26.6307	0.859950
$960$	−3.26670	−0.105432
$961$	21.6349	0.697899
$962$	−2.64896	−0.0854059
$963$	−0.667682	−0.0215157
$964$	−20.4259	−0.657874
$965$	27.1965	0.875487
$966$	3.10583	0.0999285
$967$	−8.93318	−0.287272	−0.143636	−	0.989631i	$-0.545879\pi$
$967$	−8.93318	−0.287272	−0.143636	+	0.989631i	$0.545879\pi$
$968$	−10.3982	−0.334211
$969$	13.0496	0.419212
$970$	−27.6892	−0.889046
$971$	−24.7511	−0.794301	−0.397150	−	0.917754i	$-0.630001\pi$
$971$	−24.7511	−0.794301	−0.397150	+	0.917754i	$0.630001\pi$
$972$	−1.00000	−0.0320750
$973$	1.57252	0.0504127
$974$	−26.8492	−0.860305
$975$	−2.03671	−0.0652268
$976$	−10.8557	−0.347483
$977$	18.7878	0.601076	0.300538	−	0.953770i	$-0.402834\pi$
$977$	18.7878	0.601076	0.300538	+	0.953770i	$0.402834\pi$
$978$	−19.5668	−0.625678
$979$	32.8794	1.05083
$980$	8.64431	0.276132
$981$	−10.8433	−0.346199
$982$	−28.0890	−0.896355
$983$	35.2480	1.12424	0.562119	−	0.827057i	$-0.309986\pi$
$983$	35.2480	1.12424	0.562119	+	0.827057i	$0.309986\pi$
$984$	3.60880	0.115044
$985$	11.0695	0.352703
$986$	3.61242	0.115043
$987$	23.3011	0.741681
$988$	−1.29731	−0.0412729
$989$	−1.96792	−0.0625763
$990$	15.1112	0.480264
$991$	14.3289	0.455172	0.227586	−	0.973758i	$-0.426917\pi$
$991$	14.3289	0.455172	0.227586	+	0.973758i	$0.426917\pi$
$992$	−7.25499	−0.230346
$993$	−19.3628	−0.614460
$994$	−40.0420	−1.27005
$995$	68.5235	2.17234
$996$	17.6984	0.560795
$997$	−28.0773	−0.889218	−0.444609	−	0.895725i	$-0.646657\pi$
$997$	−28.0773	−0.889218	−0.444609	+	0.895725i	$0.646657\pi$
$998$	17.3727	0.549923
$999$	−7.37615	−0.233371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.be.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.be.1.4	✓	5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.26670 q^{5} +1.00000 q^{6} -3.10583 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.26670 q^{5} +1.00000 q^{6} -3.10583 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.26670 q^{10} -4.62582 q^{11} -1.00000 q^{12} +0.359125 q^{13} +3.10583 q^{14} -3.26670 q^{15} +1.00000 q^{16} +3.61242 q^{17} -1.00000 q^{18} -3.61242 q^{19} +3.26670 q^{20} +3.10583 q^{21} +4.62582 q^{22} -1.00000 q^{23} +1.00000 q^{24} +5.67130 q^{25} -0.359125 q^{26} -1.00000 q^{27} -3.10583 q^{28} -1.00000 q^{29} +3.26670 q^{30} +7.25499 q^{31} -1.00000 q^{32} +4.62582 q^{33} -3.61242 q^{34} -10.1458 q^{35} +1.00000 q^{36} +7.37615 q^{37} +3.61242 q^{38} -0.359125 q^{39} -3.26670 q^{40} +3.60880 q^{41} -3.10583 q^{42} +1.96792 q^{43} -4.62582 q^{44} +3.26670 q^{45} +1.00000 q^{46} +7.50235 q^{47} -1.00000 q^{48} +2.64619 q^{49} -5.67130 q^{50} -3.61242 q^{51} +0.359125 q^{52} -12.4762 q^{53} +1.00000 q^{54} -15.1112 q^{55} +3.10583 q^{56} +3.61242 q^{57} +1.00000 q^{58} +6.85616 q^{59} -3.26670 q^{60} -10.8557 q^{61} -7.25499 q^{62} -3.10583 q^{63} +1.00000 q^{64} +1.17315 q^{65} -4.62582 q^{66} +5.78195 q^{67} +3.61242 q^{68} +1.00000 q^{69} +10.1458 q^{70} -12.8925 q^{71} -1.00000 q^{72} -14.7979 q^{73} -7.37615 q^{74} -5.67130 q^{75} -3.61242 q^{76} +14.3670 q^{77} +0.359125 q^{78} +1.92460 q^{79} +3.26670 q^{80} +1.00000 q^{81} -3.60880 q^{82} -17.6984 q^{83} +3.10583 q^{84} +11.8007 q^{85} -1.96792 q^{86} +1.00000 q^{87} +4.62582 q^{88} -7.10780 q^{89} -3.26670 q^{90} -1.11538 q^{91} -1.00000 q^{92} -7.25499 q^{93} -7.50235 q^{94} -11.8007 q^{95} +1.00000 q^{96} +8.47620 q^{97} -2.64619 q^{98} -4.62582 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 - 4 * q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9} - q^{10} + 5 q^{11} - 5 q^{12} - 11 q^{13} + 4 q^{14} - q^{15} + 5 q^{16} + 4 q^{17} - 5 q^{18} - 4 q^{19} + q^{20} + 4 q^{21} - 5 q^{22} - 5 q^{23} + 5 q^{24} + 12 q^{25} + 11 q^{26} - 5 q^{27} - 4 q^{28} - 5 q^{29} + q^{30} + 5 q^{31} - 5 q^{32} - 5 q^{33} - 4 q^{34} - 6 q^{35} + 5 q^{36} + 9 q^{37} + 4 q^{38} + 11 q^{39} - q^{40} + 5 q^{41} - 4 q^{42} - 16 q^{43} + 5 q^{44} + q^{45} + 5 q^{46} + 8 q^{47} - 5 q^{48} - 5 q^{49} - 12 q^{50} - 4 q^{51} - 11 q^{52} - 4 q^{53} + 5 q^{54} - 33 q^{55} + 4 q^{56} + 4 q^{57} + 5 q^{58} + 23 q^{59} - q^{60} + 7 q^{61} - 5 q^{62} - 4 q^{63} + 5 q^{64} - 5 q^{65} + 5 q^{66} + 5 q^{67} + 4 q^{68} + 5 q^{69} + 6 q^{70} - 21 q^{71} - 5 q^{72} - 8 q^{73} - 9 q^{74} - 12 q^{75} - 4 q^{76} - 11 q^{78} - 8 q^{79} + q^{80} + 5 q^{81} - 5 q^{82} - 18 q^{83} + 4 q^{84} - 10 q^{85} + 16 q^{86} + 5 q^{87} - 5 q^{88} + 32 q^{89} - q^{90} + 10 q^{91} - 5 q^{92} - 5 q^{93} - 8 q^{94} + 10 q^{95} + 5 q^{96} - 16 q^{97} + 5 q^{98} + 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 - 4 * q^7 - 5 * q^8 + 5 * q^9 - q^10 + 5 * q^11 - 5 * q^12 - 11 * q^13 + 4 * q^14 - q^15 + 5 * q^16 + 4 * q^17 - 5 * q^18 - 4 * q^19 + q^20 + 4 * q^21 - 5 * q^22 - 5 * q^23 + 5 * q^24 + 12 * q^25 + 11 * q^26 - 5 * q^27 - 4 * q^28 - 5 * q^29 + q^30 + 5 * q^31 - 5 * q^32 - 5 * q^33 - 4 * q^34 - 6 * q^35 + 5 * q^36 + 9 * q^37 + 4 * q^38 + 11 * q^39 - q^40 + 5 * q^41 - 4 * q^42 - 16 * q^43 + 5 * q^44 + q^45 + 5 * q^46 + 8 * q^47 - 5 * q^48 - 5 * q^49 - 12 * q^50 - 4 * q^51 - 11 * q^52 - 4 * q^53 + 5 * q^54 - 33 * q^55 + 4 * q^56 + 4 * q^57 + 5 * q^58 + 23 * q^59 - q^60 + 7 * q^61 - 5 * q^62 - 4 * q^63 + 5 * q^64 - 5 * q^65 + 5 * q^66 + 5 * q^67 + 4 * q^68 + 5 * q^69 + 6 * q^70 - 21 * q^71 - 5 * q^72 - 8 * q^73 - 9 * q^74 - 12 * q^75 - 4 * q^76 - 11 * q^78 - 8 * q^79 + q^80 + 5 * q^81 - 5 * q^82 - 18 * q^83 + 4 * q^84 - 10 * q^85 + 16 * q^86 + 5 * q^87 - 5 * q^88 + 32 * q^89 - q^90 + 10 * q^91 - 5 * q^92 - 5 * q^93 - 8 * q^94 + 10 * q^95 + 5 * q^96 - 16 * q^97 + 5 * q^98 + 5 * q^99

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