LMFDB - Embedded newform 4002.2.a.bd.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.19796.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + x + 8 $ x^4 - x^3 - 7*x^2 + x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.15927$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.76696 q^{5} +1.00000 q^{6} -0.607688 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.76696 q^{5} +1.00000 q^{6} -0.607688 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.76696 q^{10} +2.84073 q^{11} +1.00000 q^{12} +6.97463 q^{13} -0.607688 q^{14} +1.76696 q^{15} +1.00000 q^{16} -0.607688 q^{17} +1.00000 q^{18} +0.607688 q^{19} +1.76696 q^{20} -0.607688 q^{21} +2.84073 q^{22} -1.00000 q^{23} +1.00000 q^{24} -1.87784 q^{25} +6.97463 q^{26} +1.00000 q^{27} -0.607688 q^{28} +1.00000 q^{29} +1.76696 q^{30} -1.44071 q^{31} +1.00000 q^{32} +2.84073 q^{33} -0.607688 q^{34} -1.07376 q^{35} +1.00000 q^{36} -0.937515 q^{37} +0.607688 q^{38} +6.97463 q^{39} +1.76696 q^{40} +1.12216 q^{41} -0.607688 q^{42} -5.94926 q^{43} +2.84073 q^{44} +1.76696 q^{45} -1.00000 q^{46} +2.82945 q^{47} +1.00000 q^{48} -6.63071 q^{49} -1.87784 q^{50} -0.607688 q^{51} +6.97463 q^{52} -10.1877 q^{53} +1.00000 q^{54} +5.01946 q^{55} -0.607688 q^{56} +0.607688 q^{57} +1.00000 q^{58} -3.19639 q^{59} +1.76696 q^{60} +10.8868 q^{61} -1.44071 q^{62} -0.607688 q^{63} +1.00000 q^{64} +12.3239 q^{65} +2.84073 q^{66} +6.76058 q^{67} -0.607688 q^{68} -1.00000 q^{69} -1.07376 q^{70} -7.87146 q^{71} +1.00000 q^{72} -12.6537 q^{73} -0.937515 q^{74} -1.87784 q^{75} +0.607688 q^{76} -1.72628 q^{77} +6.97463 q^{78} -12.4602 q^{79} +1.76696 q^{80} +1.00000 q^{81} +1.12216 q^{82} +9.26143 q^{83} -0.607688 q^{84} -1.07376 q^{85} -5.94926 q^{86} +1.00000 q^{87} +2.84073 q^{88} +14.1877 q^{89} +1.76696 q^{90} -4.23840 q^{91} -1.00000 q^{92} -1.44071 q^{93} +2.82945 q^{94} +1.07376 q^{95} +1.00000 q^{96} -7.56333 q^{97} -6.63071 q^{98} +2.84073 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 4 * q^6 + 2 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9} - q^{10} + 15 q^{11} + 4 q^{12} + 11 q^{13} + 2 q^{14} - q^{15} + 4 q^{16} + 2 q^{17} + 4 q^{18} - 2 q^{19} - q^{20} + 2 q^{21} + 15 q^{22} - 4 q^{23} + 4 q^{24} - q^{25} + 11 q^{26} + 4 q^{27} + 2 q^{28} + 4 q^{29} - q^{30} - 5 q^{31} + 4 q^{32} + 15 q^{33} + 2 q^{34} - 16 q^{35} + 4 q^{36} + 3 q^{37} - 2 q^{38} + 11 q^{39} - q^{40} + 11 q^{41} + 2 q^{42} + 10 q^{43} + 15 q^{44} - q^{45} - 4 q^{46} + 10 q^{47} + 4 q^{48} - q^{50} + 2 q^{51} + 11 q^{52} + 24 q^{53} + 4 q^{54} - 7 q^{55} + 2 q^{56} - 2 q^{57} + 4 q^{58} + q^{59} - q^{60} + 3 q^{61} - 5 q^{62} + 2 q^{63} + 4 q^{64} + 3 q^{65} + 15 q^{66} + 7 q^{67} + 2 q^{68} - 4 q^{69} - 16 q^{70} - 13 q^{71} + 4 q^{72} - 2 q^{73} + 3 q^{74} - q^{75} - 2 q^{76} - 4 q^{77} + 11 q^{78} - 22 q^{79} - q^{80} + 4 q^{81} + 11 q^{82} - 16 q^{83} + 2 q^{84} - 16 q^{85} + 10 q^{86} + 4 q^{87} + 15 q^{88} - 8 q^{89} - q^{90} + 14 q^{91} - 4 q^{92} - 5 q^{93} + 10 q^{94} + 16 q^{95} + 4 q^{96} - 4 q^{97} + 15 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 4 * q^6 + 2 * q^7 + 4 * q^8 + 4 * q^9 - q^10 + 15 * q^11 + 4 * q^12 + 11 * q^13 + 2 * q^14 - q^15 + 4 * q^16 + 2 * q^17 + 4 * q^18 - 2 * q^19 - q^20 + 2 * q^21 + 15 * q^22 - 4 * q^23 + 4 * q^24 - q^25 + 11 * q^26 + 4 * q^27 + 2 * q^28 + 4 * q^29 - q^30 - 5 * q^31 + 4 * q^32 + 15 * q^33 + 2 * q^34 - 16 * q^35 + 4 * q^36 + 3 * q^37 - 2 * q^38 + 11 * q^39 - q^40 + 11 * q^41 + 2 * q^42 + 10 * q^43 + 15 * q^44 - q^45 - 4 * q^46 + 10 * q^47 + 4 * q^48 - q^50 + 2 * q^51 + 11 * q^52 + 24 * q^53 + 4 * q^54 - 7 * q^55 + 2 * q^56 - 2 * q^57 + 4 * q^58 + q^59 - q^60 + 3 * q^61 - 5 * q^62 + 2 * q^63 + 4 * q^64 + 3 * q^65 + 15 * q^66 + 7 * q^67 + 2 * q^68 - 4 * q^69 - 16 * q^70 - 13 * q^71 + 4 * q^72 - 2 * q^73 + 3 * q^74 - q^75 - 2 * q^76 - 4 * q^77 + 11 * q^78 - 22 * q^79 - q^80 + 4 * q^81 + 11 * q^82 - 16 * q^83 + 2 * q^84 - 16 * q^85 + 10 * q^86 + 4 * q^87 + 15 * q^88 - 8 * q^89 - q^90 + 14 * q^91 - 4 * q^92 - 5 * q^93 + 10 * q^94 + 16 * q^95 + 4 * q^96 - 4 * q^97 + 15 * 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
$5$	1.76696	0.790210	0.395105	−	0.918636i	$-0.370708\pi$
$5$	1.76696	0.790210	0.395105	+	0.918636i	$0.370708\pi$
$6$	1.00000	0.408248
$7$	−0.607688	−0.229685	−0.114842	−	0.993384i	$-0.536636\pi$
$7$	−0.607688	−0.229685	−0.114842	+	0.993384i	$0.536636\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.76696	0.558763
$11$	2.84073	0.856511	0.428256	−	0.903658i	$-0.359128\pi$
$11$	2.84073	0.856511	0.428256	+	0.903658i	$0.359128\pi$
$12$	1.00000	0.288675
$13$	6.97463	1.93441	0.967207	−	0.253988i	$-0.0817423\pi$
$13$	6.97463	1.93441	0.967207	+	0.253988i	$0.0817423\pi$
$14$	−0.607688	−0.162412
$15$	1.76696	0.456228
$16$	1.00000	0.250000
$17$	−0.607688	−0.147386	−0.0736931	−	0.997281i	$-0.523479\pi$
$17$	−0.607688	−0.147386	−0.0736931	+	0.997281i	$0.523479\pi$
$18$	1.00000	0.235702
$19$	0.607688	0.139413	0.0697066	−	0.997568i	$-0.477794\pi$
$19$	0.607688	0.139413	0.0697066	+	0.997568i	$0.477794\pi$
$20$	1.76696	0.395105
$21$	−0.607688	−0.132608
$22$	2.84073	0.605645
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	−1.87784	−0.375569
$26$	6.97463	1.36784
$27$	1.00000	0.192450
$28$	−0.607688	−0.114842
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	1.76696	0.322602
$31$	−1.44071	−0.258758	−0.129379	−	0.991595i	$-0.541298\pi$
$31$	−1.44071	−0.258758	−0.129379	+	0.991595i	$0.541298\pi$
$32$	1.00000	0.176777
$33$	2.84073	0.494507
$34$	−0.607688	−0.104218
$35$	−1.07376	−0.181499
$36$	1.00000	0.166667
$37$	−0.937515	−0.154126	−0.0770632	−	0.997026i	$-0.524554\pi$
$37$	−0.937515	−0.154126	−0.0770632	+	0.997026i	$0.524554\pi$
$38$	0.607688	0.0985801
$39$	6.97463	1.11683
$40$	1.76696	0.279381
$41$	1.12216	0.175252	0.0876258	−	0.996153i	$-0.472072\pi$
$41$	1.12216	0.175252	0.0876258	+	0.996153i	$0.472072\pi$
$42$	−0.607688	−0.0937684
$43$	−5.94926	−0.907254	−0.453627	−	0.891192i	$-0.649870\pi$
$43$	−5.94926	−0.907254	−0.453627	+	0.891192i	$0.649870\pi$
$44$	2.84073	0.428256
$45$	1.76696	0.263403
$46$	−1.00000	−0.147442
$47$	2.82945	0.412717	0.206359	−	0.978476i	$-0.433839\pi$
$47$	2.82945	0.412717	0.206359	+	0.978476i	$0.433839\pi$
$48$	1.00000	0.144338
$49$	−6.63071	−0.947245
$50$	−1.87784	−0.265567
$51$	−0.607688	−0.0850934
$52$	6.97463	0.967207
$53$	−10.1877	−1.39938	−0.699692	−	0.714445i	$-0.746679\pi$
$53$	−10.1877	−1.39938	−0.699692	+	0.714445i	$0.746679\pi$
$54$	1.00000	0.136083
$55$	5.01946	0.676823
$56$	−0.607688	−0.0812058
$57$	0.607688	0.0804903
$58$	1.00000	0.131306
$59$	−3.19639	−0.416135	−0.208067	−	0.978115i	$-0.566717\pi$
$59$	−3.19639	−0.416135	−0.208067	+	0.978115i	$0.566717\pi$
$60$	1.76696	0.228114
$61$	10.8868	1.39391	0.696955	−	0.717115i	$-0.254538\pi$
$61$	10.8868	1.39391	0.696955	+	0.717115i	$0.254538\pi$
$62$	−1.44071	−0.182970
$63$	−0.607688	−0.0765615
$64$	1.00000	0.125000
$65$	12.3239	1.52859
$66$	2.84073	0.349669
$67$	6.76058	0.825936	0.412968	−	0.910745i	$-0.364492\pi$
$67$	6.76058	0.825936	0.412968	+	0.910745i	$0.364492\pi$
$68$	−0.607688	−0.0736931
$69$	−1.00000	−0.120386
$70$	−1.07376	−0.128339
$71$	−7.87146	−0.934170	−0.467085	−	0.884212i	$-0.654696\pi$
$71$	−7.87146	−0.934170	−0.467085	+	0.884212i	$0.654696\pi$
$72$	1.00000	0.117851
$73$	−12.6537	−1.48101	−0.740504	−	0.672052i	$-0.765413\pi$
$73$	−12.6537	−1.48101	−0.740504	+	0.672052i	$0.765413\pi$
$74$	−0.937515	−0.108984
$75$	−1.87784	−0.216835
$76$	0.607688	0.0697066
$77$	−1.72628	−0.196727
$78$	6.97463	0.789722
$79$	−12.4602	−1.40188	−0.700939	−	0.713221i	$-0.747235\pi$
$79$	−12.4602	−1.40188	−0.700939	+	0.713221i	$0.747235\pi$
$80$	1.76696	0.197552
$81$	1.00000	0.111111
$82$	1.12216	0.123922
$83$	9.26143	1.01657	0.508287	−	0.861188i	$-0.330279\pi$
$83$	9.26143	1.01657	0.508287	+	0.861188i	$0.330279\pi$
$84$	−0.607688	−0.0663042
$85$	−1.07376	−0.116466
$86$	−5.94926	−0.641526
$87$	1.00000	0.107211
$88$	2.84073	0.302822
$89$	14.1877	1.50389	0.751945	−	0.659226i	$-0.229116\pi$
$89$	14.1877	1.50389	0.751945	+	0.659226i	$0.229116\pi$
$90$	1.76696	0.186254
$91$	−4.23840	−0.444305
$92$	−1.00000	−0.104257
$93$	−1.44071	−0.149394
$94$	2.82945	0.291835
$95$	1.07376	0.110166
$96$	1.00000	0.102062
$97$	−7.56333	−0.767940	−0.383970	−	0.923346i	$-0.625443\pi$
$97$	−7.56333	−0.767940	−0.383970	+	0.923346i	$0.625443\pi$
$98$	−6.63071	−0.669803
$99$	2.84073	0.285504
$100$	−1.87784	−0.187784
$101$	4.65608	0.463298	0.231649	−	0.972799i	$-0.425588\pi$
$101$	4.65608	0.463298	0.231649	+	0.972799i	$0.425588\pi$
$102$	−0.607688	−0.0601701
$103$	17.0578	1.68076	0.840378	−	0.542001i	$-0.182333\pi$
$103$	17.0578	1.68076	0.840378	+	0.542001i	$0.182333\pi$
$104$	6.97463	0.683919
$105$	−1.07376	−0.104789
$106$	−10.1877	−0.989514
$107$	−13.1940	−1.27552	−0.637758	−	0.770236i	$-0.720138\pi$
$107$	−13.1940	−1.27552	−0.637758	+	0.770236i	$0.720138\pi$
$108$	1.00000	0.0962250
$109$	13.9493	1.33610	0.668049	−	0.744118i	$-0.267130\pi$
$109$	13.9493	1.33610	0.668049	+	0.744118i	$0.267130\pi$
$110$	5.01946	0.478586
$111$	−0.937515	−0.0889850
$112$	−0.607688	−0.0574212
$113$	14.5121	1.36519	0.682593	−	0.730799i	$-0.260852\pi$
$113$	14.5121	1.36519	0.682593	+	0.730799i	$0.260852\pi$
$114$	0.607688	0.0569152
$115$	−1.76696	−0.164770
$116$	1.00000	0.0928477
$117$	6.97463	0.644805
$118$	−3.19639	−0.294252
$119$	0.369285	0.0338523
$120$	1.76696	0.161301
$121$	−2.93028	−0.266389
$122$	10.8868	0.985643
$123$	1.12216	0.101182
$124$	−1.44071	−0.129379
$125$	−12.1529	−1.08699
$126$	−0.607688	−0.0541372
$127$	−2.07780	−0.184375	−0.0921876	−	0.995742i	$-0.529386\pi$
$127$	−2.07780	−0.184375	−0.0921876	+	0.995742i	$0.529386\pi$
$128$	1.00000	0.0883883
$129$	−5.94926	−0.523803
$130$	12.3239	1.08088
$131$	−7.69810	−0.672586	−0.336293	−	0.941757i	$-0.609173\pi$
$131$	−7.69810	−0.672586	−0.336293	+	0.941757i	$0.609173\pi$
$132$	2.84073	0.247253
$133$	−0.369285	−0.0320211
$134$	6.76058	0.584025
$135$	1.76696	0.152076
$136$	−0.607688	−0.0521089
$137$	−3.03844	−0.259592	−0.129796	−	0.991541i	$-0.541432\pi$
$137$	−3.03844	−0.259592	−0.129796	+	0.991541i	$0.541432\pi$
$138$	−1.00000	−0.0851257
$139$	−6.01664	−0.510325	−0.255163	−	0.966898i	$-0.582129\pi$
$139$	−6.01664	−0.510325	−0.255163	+	0.966898i	$0.582129\pi$
$140$	−1.07376	−0.0907495
$141$	2.82945	0.238283
$142$	−7.87146	−0.660558
$143$	19.8130	1.65685
$144$	1.00000	0.0833333
$145$	1.76696	0.146738
$146$	−12.6537	−1.04723
$147$	−6.63071	−0.546892
$148$	−0.937515	−0.0770632
$149$	−5.93113	−0.485897	−0.242949	−	0.970039i	$-0.578115\pi$
$149$	−5.93113	−0.485897	−0.242949	+	0.970039i	$0.578115\pi$
$150$	−1.87784	−0.153325
$151$	−1.27419	−0.103692	−0.0518462	−	0.998655i	$-0.516511\pi$
$151$	−1.27419	−0.103692	−0.0518462	+	0.998655i	$0.516511\pi$
$152$	0.607688	0.0492900
$153$	−0.607688	−0.0491287
$154$	−1.72628	−0.139107
$155$	−2.54567	−0.204473
$156$	6.97463	0.558417
$157$	13.3063	1.06195	0.530977	−	0.847386i	$-0.321825\pi$
$157$	13.3063	1.06195	0.530977	+	0.847386i	$0.321825\pi$
$158$	−12.4602	−0.991277
$159$	−10.1877	−0.807935
$160$	1.76696	0.139691
$161$	0.607688	0.0478926
$162$	1.00000	0.0785674
$163$	9.24244	0.723924	0.361962	−	0.932193i	$-0.382107\pi$
$163$	9.24244	0.723924	0.361962	+	0.932193i	$0.382107\pi$
$164$	1.12216	0.0876258
$165$	5.01946	0.390764
$166$	9.26143	0.718826
$167$	6.14518	0.475529	0.237764	−	0.971323i	$-0.423585\pi$
$167$	6.14518	0.475529	0.237764	+	0.971323i	$0.423585\pi$
$168$	−0.607688	−0.0468842
$169$	35.6455	2.74196
$170$	−1.07376	−0.0823539
$171$	0.607688	0.0464711
$172$	−5.94926	−0.453627
$173$	7.91985	0.602135	0.301068	−	0.953603i	$-0.402657\pi$
$173$	7.91985	0.602135	0.301068	+	0.953603i	$0.402657\pi$
$174$	1.00000	0.0758098
$175$	1.14114	0.0862623
$176$	2.84073	0.214128
$177$	−3.19639	−0.240255
$178$	14.1877	1.06341
$179$	−21.0397	−1.57258	−0.786289	−	0.617858i	$-0.788000\pi$
$179$	−21.0397	−1.57258	−0.786289	+	0.617858i	$0.788000\pi$
$180$	1.76696	0.131702
$181$	24.4543	1.81767	0.908835	−	0.417156i	$-0.136973\pi$
$181$	24.4543	1.81767	0.908835	+	0.417156i	$0.136973\pi$
$182$	−4.23840	−0.314171
$183$	10.8868	0.804774
$184$	−1.00000	−0.0737210
$185$	−1.65655	−0.121792
$186$	−1.44071	−0.105638
$187$	−1.72628	−0.126238
$188$	2.82945	0.206359
$189$	−0.607688	−0.0442028
$190$	1.07376	0.0778989
$191$	7.75155	0.560882	0.280441	−	0.959871i	$-0.409519\pi$
$191$	7.75155	0.560882	0.280441	+	0.959871i	$0.409519\pi$
$192$	1.00000	0.0721688
$193$	−26.1369	−1.88138	−0.940689	−	0.339271i	$-0.889820\pi$
$193$	−26.1369	−1.88138	−0.940689	+	0.339271i	$0.889820\pi$
$194$	−7.56333	−0.543016
$195$	12.3239	0.882534
$196$	−6.63071	−0.473622
$197$	−2.84609	−0.202776	−0.101388	−	0.994847i	$-0.532328\pi$
$197$	−2.84609	−0.202776	−0.101388	+	0.994847i	$0.532328\pi$
$198$	2.84073	0.201882
$199$	−8.24968	−0.584804	−0.292402	−	0.956295i	$-0.594455\pi$
$199$	−8.24968	−0.584804	−0.292402	+	0.956295i	$0.594455\pi$
$200$	−1.87784	−0.132784
$201$	6.76058	0.476855
$202$	4.65608	0.327601
$203$	−0.607688	−0.0426514
$204$	−0.607688	−0.0425467
$205$	1.98281	0.138485
$206$	17.0578	1.18847
$207$	−1.00000	−0.0695048
$208$	6.97463	0.483604
$209$	1.72628	0.119409
$210$	−1.07376	−0.0740967
$211$	19.1329	1.31716	0.658581	−	0.752510i	$-0.271157\pi$
$211$	19.1329	1.31716	0.658581	+	0.752510i	$0.271157\pi$
$212$	−10.1877	−0.699692
$213$	−7.87146	−0.539344
$214$	−13.1940	−0.901927
$215$	−10.5121	−0.716921
$216$	1.00000	0.0680414
$217$	0.875500	0.0594328
$218$	13.9493	0.944763
$219$	−12.6537	−0.855061
$220$	5.01946	0.338412
$221$	−4.23840	−0.285106
$222$	−0.937515	−0.0629219
$223$	−9.72628	−0.651320	−0.325660	−	0.945487i	$-0.605586\pi$
$223$	−9.72628	−0.651320	−0.325660	+	0.945487i	$0.605586\pi$
$224$	−0.607688	−0.0406029
$225$	−1.87784	−0.125190
$226$	14.5121	0.965332
$227$	11.9872	0.795621	0.397810	−	0.917468i	$-0.369770\pi$
$227$	11.9872	0.795621	0.397810	+	0.917468i	$0.369770\pi$
$228$	0.607688	0.0402452
$229$	−23.1026	−1.52666	−0.763332	−	0.646006i	$-0.776438\pi$
$229$	−23.1026	−1.52666	−0.763332	+	0.646006i	$0.776438\pi$
$230$	−1.76696	−0.116510
$231$	−1.72628	−0.113581
$232$	1.00000	0.0656532
$233$	17.1646	1.12449	0.562246	−	0.826970i	$-0.309937\pi$
$233$	17.1646	1.12449	0.562246	+	0.826970i	$0.309937\pi$
$234$	6.97463	0.455946
$235$	4.99953	0.326133
$236$	−3.19639	−0.208067
$237$	−12.4602	−0.809375
$238$	0.369285	0.0239372
$239$	−22.7116	−1.46909	−0.734547	−	0.678558i	$-0.762605\pi$
$239$	−22.7116	−1.46909	−0.734547	+	0.678558i	$0.762605\pi$
$240$	1.76696	0.114057
$241$	−1.69810	−0.109384	−0.0546920	−	0.998503i	$-0.517418\pi$
$241$	−1.69810	−0.109384	−0.0546920	+	0.998503i	$0.517418\pi$
$242$	−2.93028	−0.188365
$243$	1.00000	0.0641500
$244$	10.8868	0.696955
$245$	−11.7162	−0.748522
$246$	1.12216	0.0715461
$247$	4.23840	0.269683
$248$	−1.44071	−0.0914849
$249$	9.26143	0.586919
$250$	−12.1529	−0.768616
$251$	−12.8348	−0.810126	−0.405063	−	0.914289i	$-0.632750\pi$
$251$	−12.8348	−0.810126	−0.405063	+	0.914289i	$0.632750\pi$
$252$	−0.607688	−0.0382808
$253$	−2.84073	−0.178595
$254$	−2.07780	−0.130373
$255$	−1.07376	−0.0672416
$256$	1.00000	0.0625000
$257$	28.4709	1.77597	0.887983	−	0.459877i	$-0.152106\pi$
$257$	28.4709	1.77597	0.887983	+	0.459877i	$0.152106\pi$
$258$	−5.94926	−0.370385
$259$	0.569717	0.0354005
$260$	12.3239	0.764297
$261$	1.00000	0.0618984
$262$	−7.69810	−0.475590
$263$	−31.2401	−1.92635	−0.963174	−	0.268880i	$-0.913346\pi$
$263$	−31.2401	−1.92635	−0.963174	+	0.268880i	$0.913346\pi$
$264$	2.84073	0.174835
$265$	−18.0012	−1.10581
$266$	−0.369285	−0.0226423
$267$	14.1877	0.868271
$268$	6.76058	0.412968
$269$	−9.61050	−0.585963	−0.292981	−	0.956118i	$-0.594647\pi$
$269$	−9.61050	−0.585963	−0.292981	+	0.956118i	$0.594647\pi$
$270$	1.76696	0.107534
$271$	8.49314	0.515922	0.257961	−	0.966155i	$-0.416949\pi$
$271$	8.49314	0.515922	0.257961	+	0.966155i	$0.416949\pi$
$272$	−0.607688	−0.0368465
$273$	−4.23840	−0.256520
$274$	−3.03844	−0.183559
$275$	−5.33444	−0.321679
$276$	−1.00000	−0.0601929
$277$	−14.8036	−0.889463	−0.444731	−	0.895664i	$-0.646701\pi$
$277$	−14.8036	−0.889463	−0.444731	+	0.895664i	$0.646701\pi$
$278$	−6.01664	−0.360854
$279$	−1.44071	−0.0862528
$280$	−1.07376	−0.0641696
$281$	15.2602	0.910347	0.455174	−	0.890403i	$-0.349577\pi$
$281$	15.2602	0.910347	0.455174	+	0.890403i	$0.349577\pi$
$282$	2.82945	0.168491
$283$	−13.3918	−0.796058	−0.398029	−	0.917373i	$-0.630306\pi$
$283$	−13.3918	−0.796058	−0.398029	+	0.917373i	$0.630306\pi$
$284$	−7.87146	−0.467085
$285$	1.07376	0.0636042
$286$	19.8130	1.17157
$287$	−0.681922	−0.0402526
$288$	1.00000	0.0589256
$289$	−16.6307	−0.978277
$290$	1.76696	0.103760
$291$	−7.56333	−0.443371
$292$	−12.6537	−0.740504
$293$	1.63540	0.0955410	0.0477705	−	0.998858i	$-0.484788\pi$
$293$	1.63540	0.0955410	0.0477705	+	0.998858i	$0.484788\pi$
$294$	−6.63071	−0.386711
$295$	−5.64790	−0.328834
$296$	−0.937515	−0.0544919
$297$	2.84073	0.164836
$298$	−5.93113	−0.343581
$299$	−6.97463	−0.403353
$300$	−1.87784	−0.108417
$301$	3.61530	0.208382
$302$	−1.27419	−0.0733217
$303$	4.65608	0.267485
$304$	0.607688	0.0348533
$305$	19.2365	1.10148
$306$	−0.607688	−0.0347392
$307$	−11.7004	−0.667779	−0.333890	−	0.942612i	$-0.608361\pi$
$307$	−11.7004	−0.667779	−0.333890	+	0.942612i	$0.608361\pi$
$308$	−1.72628	−0.0983637
$309$	17.0578	0.970384
$310$	−2.54567	−0.144585
$311$	−0.693746	−0.0393387	−0.0196694	−	0.999807i	$-0.506261\pi$
$311$	−0.693746	−0.0393387	−0.0196694	+	0.999807i	$0.506261\pi$
$312$	6.97463	0.394861
$313$	5.33349	0.301467	0.150733	−	0.988574i	$-0.451836\pi$
$313$	5.33349	0.301467	0.150733	+	0.988574i	$0.451836\pi$
$314$	13.3063	0.750915
$315$	−1.07376	−0.0604997
$316$	−12.4602	−0.700939
$317$	9.78540	0.549603	0.274801	−	0.961501i	$-0.411388\pi$
$317$	9.78540	0.549603	0.274801	+	0.961501i	$0.411388\pi$
$318$	−10.1877	−0.571296
$319$	2.84073	0.159050
$320$	1.76696	0.0987762
$321$	−13.1940	−0.736420
$322$	0.607688	0.0338652
$323$	−0.369285	−0.0205476
$324$	1.00000	0.0555556
$325$	−13.0973	−0.726505
$326$	9.24244	0.511892
$327$	13.9493	0.771396
$328$	1.12216	0.0619608
$329$	−1.71942	−0.0947949
$330$	5.01946	0.276312
$331$	−24.2678	−1.33388	−0.666940	−	0.745112i	$-0.732396\pi$
$331$	−24.2678	−1.33388	−0.666940	+	0.745112i	$0.732396\pi$
$332$	9.26143	0.508287
$333$	−0.937515	−0.0513755
$334$	6.14518	0.336249
$335$	11.9457	0.652663
$336$	−0.607688	−0.0331521
$337$	−12.4100	−0.676014	−0.338007	−	0.941144i	$-0.609753\pi$
$337$	−12.4100	−0.676014	−0.338007	+	0.941144i	$0.609753\pi$
$338$	35.6455	1.93886
$339$	14.5121	0.788191
$340$	−1.07376	−0.0582330
$341$	−4.09265	−0.221629
$342$	0.607688	0.0328600
$343$	8.28323	0.447252
$344$	−5.94926	−0.320763
$345$	−1.76696	−0.0951301
$346$	7.91985	0.425774
$347$	−11.3629	−0.609993	−0.304996	−	0.952354i	$-0.598655\pi$
$347$	−11.3629	−0.609993	−0.304996	+	0.952354i	$0.598655\pi$
$348$	1.00000	0.0536056
$349$	−20.6267	−1.10412	−0.552060	−	0.833804i	$-0.686158\pi$
$349$	−20.6267	−1.10412	−0.552060	+	0.833804i	$0.686158\pi$
$350$	1.14114	0.0609967
$351$	6.97463	0.372278
$352$	2.84073	0.151411
$353$	−8.55695	−0.455441	−0.227720	−	0.973727i	$-0.573127\pi$
$353$	−8.55695	−0.455441	−0.227720	+	0.973727i	$0.573127\pi$
$354$	−3.19639	−0.169886
$355$	−13.9086	−0.738191
$356$	14.1877	0.751945
$357$	0.369285	0.0195446
$358$	−21.0397	−1.11198
$359$	−4.86742	−0.256893	−0.128446	−	0.991716i	$-0.540999\pi$
$359$	−4.86742	−0.256893	−0.128446	+	0.991716i	$0.540999\pi$
$360$	1.76696	0.0931271
$361$	−18.6307	−0.980564
$362$	24.4543	1.28529
$363$	−2.93028	−0.153800
$364$	−4.23840	−0.222153
$365$	−22.3587	−1.17031
$366$	10.8868	0.569061
$367$	−6.77186	−0.353488	−0.176744	−	0.984257i	$-0.556556\pi$
$367$	−6.77186	−0.353488	−0.176744	+	0.984257i	$0.556556\pi$
$368$	−1.00000	−0.0521286
$369$	1.12216	0.0584172
$370$	−1.65655	−0.0861201
$371$	6.19093	0.321417
$372$	−1.44071	−0.0746971
$373$	−22.3967	−1.15966	−0.579828	−	0.814739i	$-0.696880\pi$
$373$	−22.3967	−1.15966	−0.579828	+	0.814739i	$0.696880\pi$
$374$	−1.72628	−0.0892636
$375$	−12.1529	−0.627573
$376$	2.82945	0.145918
$377$	6.97463	0.359212
$378$	−0.607688	−0.0312561
$379$	0.999529	0.0513424	0.0256712	−	0.999670i	$-0.491828\pi$
$379$	0.999529	0.0513424	0.0256712	+	0.999670i	$0.491828\pi$
$380$	1.07376	0.0550829
$381$	−2.07780	−0.106449
$382$	7.75155	0.396604
$383$	−12.5288	−0.640190	−0.320095	−	0.947385i	$-0.603715\pi$
$383$	−12.5288	−0.640190	−0.320095	+	0.947385i	$0.603715\pi$
$384$	1.00000	0.0510310
$385$	−3.05027	−0.155456
$386$	−26.1369	−1.33033
$387$	−5.94926	−0.302418
$388$	−7.56333	−0.383970
$389$	19.1659	0.971748	0.485874	−	0.874029i	$-0.338501\pi$
$389$	19.1659	0.971748	0.485874	+	0.874029i	$0.338501\pi$
$390$	12.3239	0.624046
$391$	0.607688	0.0307321
$392$	−6.63071	−0.334902
$393$	−7.69810	−0.388318
$394$	−2.84609	−0.143384
$395$	−22.0166	−1.10778
$396$	2.84073	0.142752
$397$	4.89214	0.245530	0.122765	−	0.992436i	$-0.460824\pi$
$397$	4.89214	0.245530	0.122765	+	0.992436i	$0.460824\pi$
$398$	−8.24968	−0.413519
$399$	−0.369285	−0.0184874
$400$	−1.87784	−0.0938921
$401$	17.2207	0.859963	0.429981	−	0.902838i	$-0.358520\pi$
$401$	17.2207	0.859963	0.429981	+	0.902838i	$0.358520\pi$
$402$	6.76058	0.337187
$403$	−10.0484	−0.500546
$404$	4.65608	0.231649
$405$	1.76696	0.0878011
$406$	−0.607688	−0.0301591
$407$	−2.66322	−0.132011
$408$	−0.607688	−0.0300851
$409$	12.3325	0.609805	0.304902	−	0.952384i	$-0.401376\pi$
$409$	12.3325	0.609805	0.304902	+	0.952384i	$0.401376\pi$
$410$	1.98281	0.0979240
$411$	−3.03844	−0.149875
$412$	17.0578	0.840378
$413$	1.94241	0.0955797
$414$	−1.00000	−0.0491473
$415$	16.3646	0.803307
$416$	6.97463	0.341959
$417$	−6.01664	−0.294636
$418$	1.72628	0.0844349
$419$	−27.6179	−1.34923	−0.674613	−	0.738172i	$-0.735689\pi$
$419$	−27.6179	−1.34923	−0.674613	+	0.738172i	$0.735689\pi$
$420$	−1.07376	−0.0523943
$421$	−6.26122	−0.305153	−0.152577	−	0.988292i	$-0.548757\pi$
$421$	−6.26122	−0.305153	−0.152577	+	0.988292i	$0.548757\pi$
$422$	19.1329	0.931374
$423$	2.82945	0.137572
$424$	−10.1877	−0.494757
$425$	1.14114	0.0553536
$426$	−7.87146	−0.381373
$427$	−6.61577	−0.320160
$428$	−13.1940	−0.637758
$429$	19.8130	0.956581
$430$	−10.5121	−0.506940
$431$	11.1506	0.537108	0.268554	−	0.963265i	$-0.413454\pi$
$431$	11.1506	0.537108	0.268554	+	0.963265i	$0.413454\pi$
$432$	1.00000	0.0481125
$433$	−17.6954	−0.850389	−0.425194	−	0.905102i	$-0.639794\pi$
$433$	−17.6954	−0.850389	−0.425194	+	0.905102i	$0.639794\pi$
$434$	0.875500	0.0420254
$435$	1.76696	0.0847194
$436$	13.9493	0.668049
$437$	−0.607688	−0.0290697
$438$	−12.6537	−0.604619
$439$	24.4721	1.16799	0.583995	−	0.811757i	$-0.301489\pi$
$439$	24.4721	1.16799	0.583995	+	0.811757i	$0.301489\pi$
$440$	5.01946	0.239293
$441$	−6.63071	−0.315748
$442$	−4.23840	−0.201600
$443$	28.0022	1.33042	0.665212	−	0.746655i	$-0.268341\pi$
$443$	28.0022	1.33042	0.665212	+	0.746655i	$0.268341\pi$
$444$	−0.937515	−0.0444925
$445$	25.0691	1.18839
$446$	−9.72628	−0.460553
$447$	−5.93113	−0.280533
$448$	−0.607688	−0.0287106
$449$	−13.9149	−0.656683	−0.328341	−	0.944559i	$-0.606490\pi$
$449$	−13.9149	−0.656683	−0.328341	+	0.944559i	$0.606490\pi$
$450$	−1.87784	−0.0885223
$451$	3.18774	0.150105
$452$	14.5121	0.682593
$453$	−1.27419	−0.0598669
$454$	11.9872	0.562589
$455$	−7.48910	−0.351094
$456$	0.607688	0.0284576
$457$	35.0913	1.64150	0.820752	−	0.571285i	$-0.193555\pi$
$457$	35.0913	1.64150	0.820752	+	0.571285i	$0.193555\pi$
$458$	−23.1026	−1.07951
$459$	−0.607688	−0.0283645
$460$	−1.76696	−0.0823851
$461$	19.2285	0.895558	0.447779	−	0.894144i	$-0.352215\pi$
$461$	19.2285	0.895558	0.447779	+	0.894144i	$0.352215\pi$
$462$	−1.72628	−0.0803136
$463$	−15.8620	−0.737169	−0.368584	−	0.929594i	$-0.620157\pi$
$463$	−15.8620	−0.737169	−0.368584	+	0.929594i	$0.620157\pi$
$464$	1.00000	0.0464238
$465$	−2.54567	−0.118053
$466$	17.1646	0.795137
$467$	24.8868	1.15162	0.575811	−	0.817583i	$-0.304686\pi$
$467$	24.8868	1.15162	0.575811	+	0.817583i	$0.304686\pi$
$468$	6.97463	0.322402
$469$	−4.10833	−0.189705
$470$	4.99953	0.230611
$471$	13.3063	0.613120
$472$	−3.19639	−0.147126
$473$	−16.9002	−0.777073
$474$	−12.4602	−0.572314
$475$	−1.14114	−0.0523592
$476$	0.369285	0.0169262
$477$	−10.1877	−0.466461
$478$	−22.7116	−1.03881
$479$	39.5453	1.80687	0.903436	−	0.428722i	$-0.141036\pi$
$479$	39.5453	1.80687	0.903436	+	0.428722i	$0.141036\pi$
$480$	1.76696	0.0806504
$481$	−6.53882	−0.298145
$482$	−1.69810	−0.0773461
$483$	0.607688	0.0276508
$484$	−2.93028	−0.133194
$485$	−13.3641	−0.606834
$486$	1.00000	0.0453609
$487$	39.0516	1.76960	0.884799	−	0.465973i	$-0.154295\pi$
$487$	39.0516	1.76960	0.884799	+	0.465973i	$0.154295\pi$
$488$	10.8868	0.492821
$489$	9.24244	0.417958
$490$	−11.7162	−0.529285
$491$	11.8845	0.536342	0.268171	−	0.963371i	$-0.413581\pi$
$491$	11.8845	0.536342	0.268171	+	0.963371i	$0.413581\pi$
$492$	1.12216	0.0505908
$493$	−0.607688	−0.0273689
$494$	4.23840	0.190695
$495$	5.01946	0.225608
$496$	−1.44071	−0.0646896
$497$	4.78340	0.214565
$498$	9.26143	0.415015
$499$	−12.4815	−0.558748	−0.279374	−	0.960182i	$-0.590127\pi$
$499$	−12.4815	−0.558748	−0.279374	+	0.960182i	$0.590127\pi$
$500$	−12.1529	−0.543494
$501$	6.14518	0.274547
$502$	−12.8348	−0.572846
$503$	−10.5109	−0.468658	−0.234329	−	0.972157i	$-0.575289\pi$
$503$	−10.5109	−0.468658	−0.234329	+	0.972157i	$0.575289\pi$
$504$	−0.607688	−0.0270686
$505$	8.22713	0.366102
$506$	−2.84073	−0.126286
$507$	35.6455	1.58307
$508$	−2.07780	−0.0921876
$509$	13.0610	0.578919	0.289459	−	0.957190i	$-0.406525\pi$
$509$	13.0610	0.578919	0.289459	+	0.957190i	$0.406525\pi$
$510$	−1.07376	−0.0475470
$511$	7.68953	0.340165
$512$	1.00000	0.0441942
$513$	0.607688	0.0268301
$514$	28.4709	1.25580
$515$	30.1405	1.32815
$516$	−5.94926	−0.261902
$517$	8.03769	0.353497
$518$	0.569717	0.0250319
$519$	7.91985	0.347643
$520$	12.3239	0.540439
$521$	15.0691	0.660188	0.330094	−	0.943948i	$-0.392919\pi$
$521$	15.0691	0.660188	0.330094	+	0.943948i	$0.392919\pi$
$522$	1.00000	0.0437688
$523$	−32.5187	−1.42194	−0.710972	−	0.703220i	$-0.751745\pi$
$523$	−32.5187	−1.42194	−0.710972	+	0.703220i	$0.751745\pi$
$524$	−7.69810	−0.336293
$525$	1.14114	0.0498036
$526$	−31.2401	−1.36213
$527$	0.875500	0.0381374
$528$	2.84073	0.123627
$529$	1.00000	0.0434783
$530$	−18.0012	−0.781923
$531$	−3.19639	−0.138712
$532$	−0.369285	−0.0160105
$533$	7.82664	0.339009
$534$	14.1877	0.613960
$535$	−23.3134	−1.00793
$536$	6.76058	0.292013
$537$	−21.0397	−0.907929
$538$	−9.61050	−0.414338
$539$	−18.8360	−0.811326
$540$	1.76696	0.0760380
$541$	−9.27685	−0.398843	−0.199421	−	0.979914i	$-0.563906\pi$
$541$	−9.27685	−0.398843	−0.199421	+	0.979914i	$0.563906\pi$
$542$	8.49314	0.364812
$543$	24.4543	1.04943
$544$	−0.607688	−0.0260544
$545$	24.6478	1.05580
$546$	−4.23840	−0.181387
$547$	−38.0930	−1.62874	−0.814371	−	0.580345i	$-0.802918\pi$
$547$	−38.0930	−1.62874	−0.814371	+	0.580345i	$0.802918\pi$
$548$	−3.03844	−0.129796
$549$	10.8868	0.464636
$550$	−5.33444	−0.227461
$551$	0.607688	0.0258884
$552$	−1.00000	−0.0425628
$553$	7.57190	0.321990
$554$	−14.8036	−0.628945
$555$	−1.65655	−0.0703168
$556$	−6.01664	−0.255163
$557$	13.8792	0.588079	0.294040	−	0.955793i	$-0.405000\pi$
$557$	13.8792	0.588079	0.294040	+	0.955793i	$0.405000\pi$
$558$	−1.44071	−0.0609900
$559$	−41.4939	−1.75501
$560$	−1.07376	−0.0453748
$561$	−1.72628	−0.0728834
$562$	15.2602	0.643713
$563$	−7.09454	−0.298999	−0.149500	−	0.988762i	$-0.547766\pi$
$563$	−7.09454	−0.298999	−0.149500	+	0.988762i	$0.547766\pi$
$564$	2.82945	0.119141
$565$	25.6424	1.07878
$566$	−13.3918	−0.562898
$567$	−0.607688	−0.0255205
$568$	−7.87146	−0.330279
$569$	8.25382	0.346018	0.173009	−	0.984920i	$-0.444651\pi$
$569$	8.25382	0.346018	0.173009	+	0.984920i	$0.444651\pi$
$570$	1.07376	0.0449750
$571$	−37.1487	−1.55462	−0.777312	−	0.629115i	$-0.783417\pi$
$571$	−37.1487	−1.55462	−0.777312	+	0.629115i	$0.783417\pi$
$572$	19.8130	0.828424
$573$	7.75155	0.323826
$574$	−0.681922	−0.0284629
$575$	1.87784	0.0783114
$576$	1.00000	0.0416667
$577$	−46.2112	−1.92380	−0.961898	−	0.273409i	$-0.911849\pi$
$577$	−46.2112	−1.92380	−0.961898	+	0.273409i	$0.911849\pi$
$578$	−16.6307	−0.691747
$579$	−26.1369	−1.08621
$580$	1.76696	0.0733691
$581$	−5.62806	−0.233491
$582$	−7.56333	−0.313510
$583$	−28.9404	−1.19859
$584$	−12.6537	−0.523616
$585$	12.3239	0.509531
$586$	1.63540	0.0675577
$587$	−32.4768	−1.34046	−0.670231	−	0.742153i	$-0.733805\pi$
$587$	−32.4768	−1.34046	−0.670231	+	0.742153i	$0.733805\pi$
$588$	−6.63071	−0.273446
$589$	−0.875500	−0.0360744
$590$	−5.64790	−0.232521
$591$	−2.84609	−0.117073
$592$	−0.937515	−0.0385316
$593$	5.35822	0.220036	0.110018	−	0.993930i	$-0.464909\pi$
$593$	5.35822	0.220036	0.110018	+	0.993930i	$0.464909\pi$
$594$	2.84073	0.116556
$595$	0.652513	0.0267504
$596$	−5.93113	−0.242949
$597$	−8.24968	−0.337637
$598$	−6.97463	−0.285214
$599$	−6.28914	−0.256967	−0.128484	−	0.991712i	$-0.541011\pi$
$599$	−6.28914	−0.256967	−0.128484	+	0.991712i	$0.541011\pi$
$600$	−1.87784	−0.0766626
$601$	31.1540	1.27080	0.635400	−	0.772183i	$-0.280835\pi$
$601$	31.1540	1.27080	0.635400	+	0.772183i	$0.280835\pi$
$602$	3.61530	0.147349
$603$	6.76058	0.275312
$604$	−1.27419	−0.0518462
$605$	−5.17769	−0.210503
$606$	4.65608	0.189140
$607$	20.7755	0.843250	0.421625	−	0.906770i	$-0.361460\pi$
$607$	20.7755	0.843250	0.421625	+	0.906770i	$0.361460\pi$
$608$	0.607688	0.0246450
$609$	−0.607688	−0.0246248
$610$	19.2365	0.778864
$611$	19.7344	0.798367
$612$	−0.607688	−0.0245644
$613$	5.89045	0.237913	0.118956	−	0.992899i	$-0.462045\pi$
$613$	5.89045	0.237913	0.118956	+	0.992899i	$0.462045\pi$
$614$	−11.7004	−0.472191
$615$	1.98281	0.0799546
$616$	−1.72628	−0.0695537
$617$	−2.43789	−0.0981459	−0.0490729	−	0.998795i	$-0.515627\pi$
$617$	−2.43789	−0.0981459	−0.0490729	+	0.998795i	$0.515627\pi$
$618$	17.0578	0.686165
$619$	8.88018	0.356925	0.178462	−	0.983947i	$-0.442888\pi$
$619$	8.88018	0.356925	0.178462	+	0.983947i	$0.442888\pi$
$620$	−2.54567	−0.102237
$621$	−1.00000	−0.0401286
$622$	−0.693746	−0.0278167
$623$	−8.62168	−0.345420
$624$	6.97463	0.279209
$625$	−12.0845	−0.483380
$626$	5.33349	0.213169
$627$	1.72628	0.0689408
$628$	13.3063	0.530977
$629$	0.569717	0.0227161
$630$	−1.07376	−0.0427797
$631$	−20.3052	−0.808339	−0.404169	−	0.914684i	$-0.632439\pi$
$631$	−20.3052	−0.808339	−0.404169	+	0.914684i	$0.632439\pi$
$632$	−12.4602	−0.495639
$633$	19.1329	0.760464
$634$	9.78540	0.388628
$635$	−3.67140	−0.145695
$636$	−10.1877	−0.403967
$637$	−46.2468	−1.83236
$638$	2.84073	0.112465
$639$	−7.87146	−0.311390
$640$	1.76696	0.0698453
$641$	36.0742	1.42485	0.712423	−	0.701750i	$-0.247598\pi$
$641$	36.0742	1.42485	0.712423	+	0.701750i	$0.247598\pi$
$642$	−13.1940	−0.520728
$643$	19.5684	0.771704	0.385852	−	0.922561i	$-0.373908\pi$
$643$	19.5684	0.771704	0.385852	+	0.922561i	$0.373908\pi$
$644$	0.607688	0.0239463
$645$	−10.5121	−0.413915
$646$	−0.369285	−0.0145293
$647$	32.4852	1.27712	0.638562	−	0.769570i	$-0.279529\pi$
$647$	32.4852	1.27712	0.638562	+	0.769570i	$0.279529\pi$
$648$	1.00000	0.0392837
$649$	−9.08007	−0.356424
$650$	−13.0973	−0.513717
$651$	0.875500	0.0343136
$652$	9.24244	0.361962
$653$	−16.7271	−0.654580	−0.327290	−	0.944924i	$-0.606135\pi$
$653$	−16.7271	−0.654580	−0.327290	+	0.944924i	$0.606135\pi$
$654$	13.9493	0.545459
$655$	−13.6022	−0.531484
$656$	1.12216	0.0438129
$657$	−12.6537	−0.493669
$658$	−1.71942	−0.0670301
$659$	20.2717	0.789673	0.394836	−	0.918751i	$-0.370801\pi$
$659$	20.2717	0.789673	0.394836	+	0.918751i	$0.370801\pi$
$660$	5.01946	0.195382
$661$	−0.492222	−0.0191452	−0.00957262	−	0.999954i	$-0.503047\pi$
$661$	−0.492222	−0.0191452	−0.00957262	+	0.999954i	$0.503047\pi$
$662$	−24.2678	−0.943195
$663$	−4.23840	−0.164606
$664$	9.26143	0.359413
$665$	−0.652513	−0.0253034
$666$	−0.937515	−0.0363280
$667$	−1.00000	−0.0387202
$668$	6.14518	0.237764
$669$	−9.72628	−0.376040
$670$	11.9457	0.461502
$671$	30.9264	1.19390
$672$	−0.607688	−0.0234421
$673$	8.71398	0.335899	0.167950	−	0.985796i	$-0.446285\pi$
$673$	8.71398	0.335899	0.167950	+	0.985796i	$0.446285\pi$
$674$	−12.4100	−0.478014
$675$	−1.87784	−0.0722782
$676$	35.6455	1.37098
$677$	11.8145	0.454068	0.227034	−	0.973887i	$-0.427097\pi$
$677$	11.8145	0.454068	0.227034	+	0.973887i	$0.427097\pi$
$678$	14.5121	0.557335
$679$	4.59615	0.176384
$680$	−1.07376	−0.0411769
$681$	11.9872	0.459352
$682$	−4.09265	−0.156716
$683$	32.3483	1.23777	0.618886	−	0.785481i	$-0.287584\pi$
$683$	32.3483	1.23777	0.618886	+	0.785481i	$0.287584\pi$
$684$	0.607688	0.0232355
$685$	−5.36881	−0.205132
$686$	8.28323	0.316255
$687$	−23.1026	−0.881420
$688$	−5.94926	−0.226814
$689$	−71.0552	−2.70699
$690$	−1.76696	−0.0672671
$691$	−37.1365	−1.41274	−0.706369	−	0.707844i	$-0.749668\pi$
$691$	−37.1365	−1.41274	−0.706369	+	0.707844i	$0.749668\pi$
$692$	7.91985	0.301068
$693$	−1.72628	−0.0655758
$694$	−11.3629	−0.431330
$695$	−10.6312	−0.403264
$696$	1.00000	0.0379049
$697$	−0.681922	−0.0258296
$698$	−20.6267	−0.780731
$699$	17.1646	0.649226
$700$	1.14114	0.0431312
$701$	−11.9439	−0.451115	−0.225557	−	0.974230i	$-0.572420\pi$
$701$	−11.9439	−0.451115	−0.225557	+	0.974230i	$0.572420\pi$
$702$	6.97463	0.263241
$703$	−0.569717	−0.0214873
$704$	2.84073	0.107064
$705$	4.99953	0.188293
$706$	−8.55695	−0.322045
$707$	−2.82945	−0.106412
$708$	−3.19639	−0.120128
$709$	−48.4356	−1.81904	−0.909518	−	0.415664i	$-0.863549\pi$
$709$	−48.4356	−1.81904	−0.909518	+	0.415664i	$0.863549\pi$
$710$	−13.9086	−0.521980
$711$	−12.4602	−0.467293
$712$	14.1877	0.531705
$713$	1.44071	0.0539549
$714$	0.369285	0.0138202
$715$	35.0089	1.30926
$716$	−21.0397	−0.786289
$717$	−22.7116	−0.848182
$718$	−4.86742	−0.181650
$719$	−39.1870	−1.46143	−0.730715	−	0.682683i	$-0.760813\pi$
$719$	−39.1870	−1.46143	−0.730715	+	0.682683i	$0.760813\pi$
$720$	1.76696	0.0658508
$721$	−10.3658	−0.386044
$722$	−18.6307	−0.693363
$723$	−1.69810	−0.0631528
$724$	24.4543	0.908835
$725$	−1.87784	−0.0697413
$726$	−2.93028	−0.108753
$727$	11.9267	0.442337	0.221169	−	0.975236i	$-0.429013\pi$
$727$	11.9267	0.442337	0.221169	+	0.975236i	$0.429013\pi$
$728$	−4.23840	−0.157086
$729$	1.00000	0.0370370
$730$	−22.3587	−0.827532
$731$	3.61530	0.133717
$732$	10.8868	0.402387
$733$	5.00490	0.184860	0.0924300	−	0.995719i	$-0.470537\pi$
$733$	5.00490	0.184860	0.0924300	+	0.995719i	$0.470537\pi$
$734$	−6.77186	−0.249954
$735$	−11.7162	−0.432160
$736$	−1.00000	−0.0368605
$737$	19.2050	0.707424
$738$	1.12216	0.0413072
$739$	−43.8327	−1.61241	−0.806206	−	0.591635i	$-0.798483\pi$
$739$	−43.8327	−1.61241	−0.806206	+	0.591635i	$0.798483\pi$
$740$	−1.65655	−0.0608961
$741$	4.23840	0.155702
$742$	6.19093	0.227276
$743$	1.73878	0.0637897	0.0318949	−	0.999491i	$-0.489846\pi$
$743$	1.73878	0.0637897	0.0318949	+	0.999491i	$0.489846\pi$
$744$	−1.44071	−0.0528188
$745$	−10.4801	−0.383961
$746$	−22.3967	−0.820000
$747$	9.26143	0.338858
$748$	−1.72628	−0.0631189
$749$	8.01787	0.292967
$750$	−12.1529	−0.443761
$751$	28.3694	1.03521	0.517607	−	0.855618i	$-0.326823\pi$
$751$	28.3694	1.03521	0.517607	+	0.855618i	$0.326823\pi$
$752$	2.82945	0.103179
$753$	−12.8348	−0.467727
$754$	6.97463	0.254001
$755$	−2.25145	−0.0819388
$756$	−0.607688	−0.0221014
$757$	−3.45792	−0.125680	−0.0628401	−	0.998024i	$-0.520016\pi$
$757$	−3.45792	−0.125680	−0.0628401	+	0.998024i	$0.520016\pi$
$758$	0.999529	0.0363045
$759$	−2.84073	−0.103112
$760$	1.07376	0.0389495
$761$	42.0649	1.52485	0.762425	−	0.647076i	$-0.224009\pi$
$761$	42.0649	1.52485	0.762425	+	0.647076i	$0.224009\pi$
$762$	−2.07780	−0.0752709
$763$	−8.47681	−0.306881
$764$	7.75155	0.280441
$765$	−1.07376	−0.0388220
$766$	−12.5288	−0.452683
$767$	−22.2937	−0.804977
$768$	1.00000	0.0360844
$769$	15.7388	0.567555	0.283777	−	0.958890i	$-0.408412\pi$
$769$	15.7388	0.567555	0.283777	+	0.958890i	$0.408412\pi$
$770$	−3.05027	−0.109924
$771$	28.4709	1.02535
$772$	−26.1369	−0.940689
$773$	−35.1673	−1.26488	−0.632440	−	0.774610i	$-0.717946\pi$
$773$	−35.1673	−1.26488	−0.632440	+	0.774610i	$0.717946\pi$
$774$	−5.94926	−0.213842
$775$	2.70542	0.0971815
$776$	−7.56333	−0.271508
$777$	0.569717	0.0204385
$778$	19.1659	0.687130
$779$	0.681922	0.0244324
$780$	12.3239	0.441267
$781$	−22.3607	−0.800127
$782$	0.607688	0.0217309
$783$	1.00000	0.0357371
$784$	−6.63071	−0.236811
$785$	23.5117	0.839167
$786$	−7.69810	−0.274582
$787$	4.00863	0.142892	0.0714461	−	0.997444i	$-0.477239\pi$
$787$	4.00863	0.142892	0.0714461	+	0.997444i	$0.477239\pi$
$788$	−2.84609	−0.101388
$789$	−31.2401	−1.11218
$790$	−22.0166	−0.783317
$791$	−8.81885	−0.313562
$792$	2.84073	0.100941
$793$	75.9313	2.69640
$794$	4.89214	0.173616
$795$	−18.0012	−0.638438
$796$	−8.24968	−0.292402
$797$	−39.0727	−1.38403	−0.692013	−	0.721886i	$-0.743276\pi$
$797$	−39.0727	−1.38403	−0.692013	+	0.721886i	$0.743276\pi$
$798$	−0.369285	−0.0130726
$799$	−1.71942	−0.0608288
$800$	−1.87784	−0.0663918
$801$	14.1877	0.501297
$802$	17.2207	0.608086
$803$	−35.9458	−1.26850
$804$	6.76058	0.238427
$805$	1.07376	0.0378452
$806$	−10.0484	−0.353940
$807$	−9.61050	−0.338306
$808$	4.65608	0.163800
$809$	−10.2138	−0.359098	−0.179549	−	0.983749i	$-0.557464\pi$
$809$	−10.2138	−0.359098	−0.179549	+	0.983749i	$0.557464\pi$
$810$	1.76696	0.0620847
$811$	−10.1390	−0.356027	−0.178014	−	0.984028i	$-0.556967\pi$
$811$	−10.1390	−0.356027	−0.178014	+	0.984028i	$0.556967\pi$
$812$	−0.607688	−0.0213257
$813$	8.49314	0.297867
$814$	−2.66322	−0.0933459
$815$	16.3311	0.572052
$816$	−0.607688	−0.0212734
$817$	−3.61530	−0.126483
$818$	12.3325	0.431197
$819$	−4.23840	−0.148102
$820$	1.98281	0.0692427
$821$	32.6971	1.14114	0.570569	−	0.821250i	$-0.306723\pi$
$821$	32.6971	1.14114	0.570569	+	0.821250i	$0.306723\pi$
$822$	−3.03844	−0.105978
$823$	24.8058	0.864675	0.432338	−	0.901712i	$-0.357689\pi$
$823$	24.8058	0.864675	0.432338	+	0.901712i	$0.357689\pi$
$824$	17.0578	0.594237
$825$	−5.33444	−0.185721
$826$	1.94241	0.0675851
$827$	−42.5223	−1.47865	−0.739323	−	0.673351i	$-0.764854\pi$
$827$	−42.5223	−1.47865	−0.739323	+	0.673351i	$0.764854\pi$
$828$	−1.00000	−0.0347524
$829$	34.8328	1.20979	0.604897	−	0.796303i	$-0.293214\pi$
$829$	34.8328	1.20979	0.604897	+	0.796303i	$0.293214\pi$
$830$	16.3646	0.568024
$831$	−14.8036	−0.513532
$832$	6.97463	0.241802
$833$	4.02941	0.139611
$834$	−6.01664	−0.208339
$835$	10.8583	0.375767
$836$	1.72628	0.0597045
$837$	−1.44071	−0.0497981
$838$	−27.6179	−0.954046
$839$	−40.7206	−1.40583	−0.702915	−	0.711274i	$-0.748119\pi$
$839$	−40.7206	−1.40583	−0.702915	+	0.711274i	$0.748119\pi$
$840$	−1.07376	−0.0370483
$841$	1.00000	0.0344828
$842$	−6.26122	−0.215776
$843$	15.2602	0.525589
$844$	19.1329	0.658581
$845$	62.9842	2.16672
$846$	2.82945	0.0972784
$847$	1.78070	0.0611854
$848$	−10.1877	−0.349846
$849$	−13.3918	−0.459604
$850$	1.14114	0.0391409
$851$	0.937515	0.0321376
$852$	−7.87146	−0.269672
$853$	−50.7468	−1.73754	−0.868768	−	0.495219i	$-0.835088\pi$
$853$	−50.7468	−1.73754	−0.868768	+	0.495219i	$0.835088\pi$
$854$	−6.61577	−0.226387
$855$	1.07376	0.0367219
$856$	−13.1940	−0.450963
$857$	−34.4496	−1.17677	−0.588387	−	0.808579i	$-0.700237\pi$
$857$	−34.4496	−1.17677	−0.588387	+	0.808579i	$0.700237\pi$
$858$	19.8130	0.676405
$859$	6.31386	0.215426	0.107713	−	0.994182i	$-0.465647\pi$
$859$	6.31386	0.215426	0.107713	+	0.994182i	$0.465647\pi$
$860$	−10.5121	−0.358461
$861$	−0.681922	−0.0232398
$862$	11.1506	0.379793
$863$	−34.1900	−1.16384	−0.581921	−	0.813246i	$-0.697699\pi$
$863$	−34.1900	−1.16384	−0.581921	+	0.813246i	$0.697699\pi$
$864$	1.00000	0.0340207
$865$	13.9941	0.475813
$866$	−17.6954	−0.601316
$867$	−16.6307	−0.564809
$868$	0.875500	0.0297164
$869$	−35.3959	−1.20072
$870$	1.76696	0.0599056
$871$	47.1526	1.59770
$872$	13.9493	0.472382
$873$	−7.56333	−0.255980
$874$	−0.607688	−0.0205554
$875$	7.38517	0.249664
$876$	−12.6537	−0.427530
$877$	−56.4214	−1.90521	−0.952607	−	0.304202i	$-0.901610\pi$
$877$	−56.4214	−1.90521	−0.952607	+	0.304202i	$0.901610\pi$
$878$	24.4721	0.825894
$879$	1.63540	0.0551606
$880$	5.01946	0.169206
$881$	22.0213	0.741917	0.370959	−	0.928649i	$-0.379029\pi$
$881$	22.0213	0.741917	0.370959	+	0.928649i	$0.379029\pi$
$882$	−6.63071	−0.223268
$883$	9.60939	0.323382	0.161691	−	0.986841i	$-0.448305\pi$
$883$	9.60939	0.323382	0.161691	+	0.986841i	$0.448305\pi$
$884$	−4.23840	−0.142553
$885$	−5.64790	−0.189852
$886$	28.0022	0.940751
$887$	41.2214	1.38408	0.692040	−	0.721859i	$-0.256712\pi$
$887$	41.2214	1.38408	0.692040	+	0.721859i	$0.256712\pi$
$888$	−0.937515	−0.0314609
$889$	1.26266	0.0423482
$890$	25.0691	0.840317
$891$	2.84073	0.0951679
$892$	−9.72628	−0.325660
$893$	1.71942	0.0575383
$894$	−5.93113	−0.198367
$895$	−37.1763	−1.24267
$896$	−0.607688	−0.0203014
$897$	−6.97463	−0.232876
$898$	−13.9149	−0.464345
$899$	−1.44071	−0.0480502
$900$	−1.87784	−0.0625948
$901$	6.19093	0.206250
$902$	3.18774	0.106140
$903$	3.61530	0.120310
$904$	14.5121	0.482666
$905$	43.2098	1.43634
$906$	−1.27419	−0.0423323
$907$	15.5339	0.515796	0.257898	−	0.966172i	$-0.416970\pi$
$907$	15.5339	0.515796	0.257898	+	0.966172i	$0.416970\pi$
$908$	11.9872	0.397810
$909$	4.65608	0.154433
$910$	−7.48910	−0.248261
$911$	−8.12939	−0.269339	−0.134669	−	0.990891i	$-0.542997\pi$
$911$	−8.12939	−0.269339	−0.134669	+	0.990891i	$0.542997\pi$
$912$	0.607688	0.0201226
$913$	26.3092	0.870707
$914$	35.0913	1.16072
$915$	19.2365	0.635940
$916$	−23.1026	−0.763332
$917$	4.67804	0.154483
$918$	−0.607688	−0.0200567
$919$	−56.6093	−1.86737	−0.933684	−	0.358097i	$-0.883426\pi$
$919$	−56.6093	−1.86737	−0.933684	+	0.358097i	$0.883426\pi$
$920$	−1.76696	−0.0582550
$921$	−11.7004	−0.385543
$922$	19.2285	0.633255
$923$	−54.9005	−1.80707
$924$	−1.72628	−0.0567903
$925$	1.76051	0.0578851
$926$	−15.8620	−0.521257
$927$	17.0578	0.560252
$928$	1.00000	0.0328266
$929$	−24.9595	−0.818895	−0.409448	−	0.912334i	$-0.634279\pi$
$929$	−24.9595	−0.818895	−0.409448	+	0.912334i	$0.634279\pi$
$930$	−2.54567	−0.0834759
$931$	−4.02941	−0.132059
$932$	17.1646	0.562246
$933$	−0.693746	−0.0227122
$934$	24.8868	0.814320
$935$	−3.05027	−0.0997544
$936$	6.97463	0.227973
$937$	−3.18842	−0.104161	−0.0520806	−	0.998643i	$-0.516585\pi$
$937$	−3.18842	−0.104161	−0.0520806	+	0.998643i	$0.516585\pi$
$938$	−4.10833	−0.134142
$939$	5.33349	0.174052
$940$	4.99953	0.163067
$941$	−31.8980	−1.03984	−0.519922	−	0.854214i	$-0.674039\pi$
$941$	−31.8980	−1.03984	−0.519922	+	0.854214i	$0.674039\pi$
$942$	13.3063	0.433541
$943$	−1.12216	−0.0365425
$944$	−3.19639	−0.104034
$945$	−1.07376	−0.0349295
$946$	−16.9002	−0.549474
$947$	−8.40943	−0.273270	−0.136635	−	0.990621i	$-0.543629\pi$
$947$	−8.40943	−0.273270	−0.136635	+	0.990621i	$0.543629\pi$
$948$	−12.4602	−0.404687
$949$	−88.2552	−2.86488
$950$	−1.14114	−0.0370236
$951$	9.78540	0.317313
$952$	0.369285	0.0119686
$953$	47.9188	1.55224	0.776120	−	0.630585i	$-0.217185\pi$
$953$	47.9188	1.55224	0.776120	+	0.630585i	$0.217185\pi$
$954$	−10.1877	−0.329838
$955$	13.6967	0.443215
$956$	−22.7116	−0.734547
$957$	2.84073	0.0918276
$958$	39.5453	1.27765
$959$	1.84643	0.0596242
$960$	1.76696	0.0570285
$961$	−28.9244	−0.933044
$962$	−6.53882	−0.210820
$963$	−13.1940	−0.425172
$964$	−1.69810	−0.0546920
$965$	−46.1830	−1.48668
$966$	0.607688	0.0195521
$967$	−38.3472	−1.23316	−0.616581	−	0.787292i	$-0.711483\pi$
$967$	−38.3472	−1.23316	−0.616581	+	0.787292i	$0.711483\pi$
$968$	−2.93028	−0.0941827
$969$	−0.369285	−0.0118632
$970$	−13.3641	−0.429096
$971$	44.7179	1.43507	0.717533	−	0.696524i	$-0.245271\pi$
$971$	44.7179	1.43507	0.717533	+	0.696524i	$0.245271\pi$
$972$	1.00000	0.0320750
$973$	3.65624	0.117214
$974$	39.0516	1.25129
$975$	−13.0973	−0.419448
$976$	10.8868	0.348477
$977$	53.7757	1.72044	0.860218	−	0.509927i	$-0.170327\pi$
$977$	53.7757	1.72044	0.860218	+	0.509927i	$0.170327\pi$
$978$	9.24244	0.295541
$979$	40.3033	1.28810
$980$	−11.7162	−0.374261
$981$	13.9493	0.445366
$982$	11.8845	0.379251
$983$	2.80153	0.0893548	0.0446774	−	0.999001i	$-0.485774\pi$
$983$	2.80153	0.0893548	0.0446774	+	0.999001i	$0.485774\pi$
$984$	1.12216	0.0357731
$985$	−5.02894	−0.160235
$986$	−0.607688	−0.0193527
$987$	−1.71942	−0.0547298
$988$	4.23840	0.134842
$989$	5.94926	0.189176
$990$	5.01946	0.159529
$991$	−24.8367	−0.788965	−0.394482	−	0.918904i	$-0.629076\pi$
$991$	−24.8367	−0.788965	−0.394482	+	0.918904i	$0.629076\pi$
$992$	−1.44071	−0.0457425
$993$	−24.2678	−0.770116
$994$	4.78340	0.151720
$995$	−14.5769	−0.462118
$996$	9.26143	0.293460
$997$	−18.8529	−0.597079	−0.298539	−	0.954397i	$-0.596499\pi$
$997$	−18.8529	−0.597079	−0.298539	+	0.954397i	$0.596499\pi$
$998$	−12.4815	−0.395095
$999$	−0.937515	−0.0296617

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bd.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bd.1.4	✓	4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.76696 q^{5} +1.00000 q^{6} -0.607688 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.76696 q^{5} +1.00000 q^{6} -0.607688 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.76696 q^{10} +2.84073 q^{11} +1.00000 q^{12} +6.97463 q^{13} -0.607688 q^{14} +1.76696 q^{15} +1.00000 q^{16} -0.607688 q^{17} +1.00000 q^{18} +0.607688 q^{19} +1.76696 q^{20} -0.607688 q^{21} +2.84073 q^{22} -1.00000 q^{23} +1.00000 q^{24} -1.87784 q^{25} +6.97463 q^{26} +1.00000 q^{27} -0.607688 q^{28} +1.00000 q^{29} +1.76696 q^{30} -1.44071 q^{31} +1.00000 q^{32} +2.84073 q^{33} -0.607688 q^{34} -1.07376 q^{35} +1.00000 q^{36} -0.937515 q^{37} +0.607688 q^{38} +6.97463 q^{39} +1.76696 q^{40} +1.12216 q^{41} -0.607688 q^{42} -5.94926 q^{43} +2.84073 q^{44} +1.76696 q^{45} -1.00000 q^{46} +2.82945 q^{47} +1.00000 q^{48} -6.63071 q^{49} -1.87784 q^{50} -0.607688 q^{51} +6.97463 q^{52} -10.1877 q^{53} +1.00000 q^{54} +5.01946 q^{55} -0.607688 q^{56} +0.607688 q^{57} +1.00000 q^{58} -3.19639 q^{59} +1.76696 q^{60} +10.8868 q^{61} -1.44071 q^{62} -0.607688 q^{63} +1.00000 q^{64} +12.3239 q^{65} +2.84073 q^{66} +6.76058 q^{67} -0.607688 q^{68} -1.00000 q^{69} -1.07376 q^{70} -7.87146 q^{71} +1.00000 q^{72} -12.6537 q^{73} -0.937515 q^{74} -1.87784 q^{75} +0.607688 q^{76} -1.72628 q^{77} +6.97463 q^{78} -12.4602 q^{79} +1.76696 q^{80} +1.00000 q^{81} +1.12216 q^{82} +9.26143 q^{83} -0.607688 q^{84} -1.07376 q^{85} -5.94926 q^{86} +1.00000 q^{87} +2.84073 q^{88} +14.1877 q^{89} +1.76696 q^{90} -4.23840 q^{91} -1.00000 q^{92} -1.44071 q^{93} +2.82945 q^{94} +1.07376 q^{95} +1.00000 q^{96} -7.56333 q^{97} -6.63071 q^{98} +2.84073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 4 * q^6 + 2 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9} - q^{10} + 15 q^{11} + 4 q^{12} + 11 q^{13} + 2 q^{14} - q^{15} + 4 q^{16} + 2 q^{17} + 4 q^{18} - 2 q^{19} - q^{20} + 2 q^{21} + 15 q^{22} - 4 q^{23} + 4 q^{24} - q^{25} + 11 q^{26} + 4 q^{27} + 2 q^{28} + 4 q^{29} - q^{30} - 5 q^{31} + 4 q^{32} + 15 q^{33} + 2 q^{34} - 16 q^{35} + 4 q^{36} + 3 q^{37} - 2 q^{38} + 11 q^{39} - q^{40} + 11 q^{41} + 2 q^{42} + 10 q^{43} + 15 q^{44} - q^{45} - 4 q^{46} + 10 q^{47} + 4 q^{48} - q^{50} + 2 q^{51} + 11 q^{52} + 24 q^{53} + 4 q^{54} - 7 q^{55} + 2 q^{56} - 2 q^{57} + 4 q^{58} + q^{59} - q^{60} + 3 q^{61} - 5 q^{62} + 2 q^{63} + 4 q^{64} + 3 q^{65} + 15 q^{66} + 7 q^{67} + 2 q^{68} - 4 q^{69} - 16 q^{70} - 13 q^{71} + 4 q^{72} - 2 q^{73} + 3 q^{74} - q^{75} - 2 q^{76} - 4 q^{77} + 11 q^{78} - 22 q^{79} - q^{80} + 4 q^{81} + 11 q^{82} - 16 q^{83} + 2 q^{84} - 16 q^{85} + 10 q^{86} + 4 q^{87} + 15 q^{88} - 8 q^{89} - q^{90} + 14 q^{91} - 4 q^{92} - 5 q^{93} + 10 q^{94} + 16 q^{95} + 4 q^{96} - 4 q^{97} + 15 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 4 * q^6 + 2 * q^7 + 4 * q^8 + 4 * q^9 - q^10 + 15 * q^11 + 4 * q^12 + 11 * q^13 + 2 * q^14 - q^15 + 4 * q^16 + 2 * q^17 + 4 * q^18 - 2 * q^19 - q^20 + 2 * q^21 + 15 * q^22 - 4 * q^23 + 4 * q^24 - q^25 + 11 * q^26 + 4 * q^27 + 2 * q^28 + 4 * q^29 - q^30 - 5 * q^31 + 4 * q^32 + 15 * q^33 + 2 * q^34 - 16 * q^35 + 4 * q^36 + 3 * q^37 - 2 * q^38 + 11 * q^39 - q^40 + 11 * q^41 + 2 * q^42 + 10 * q^43 + 15 * q^44 - q^45 - 4 * q^46 + 10 * q^47 + 4 * q^48 - q^50 + 2 * q^51 + 11 * q^52 + 24 * q^53 + 4 * q^54 - 7 * q^55 + 2 * q^56 - 2 * q^57 + 4 * q^58 + q^59 - q^60 + 3 * q^61 - 5 * q^62 + 2 * q^63 + 4 * q^64 + 3 * q^65 + 15 * q^66 + 7 * q^67 + 2 * q^68 - 4 * q^69 - 16 * q^70 - 13 * q^71 + 4 * q^72 - 2 * q^73 + 3 * q^74 - q^75 - 2 * q^76 - 4 * q^77 + 11 * q^78 - 22 * q^79 - q^80 + 4 * q^81 + 11 * q^82 - 16 * q^83 + 2 * q^84 - 16 * q^85 + 10 * q^86 + 4 * q^87 + 15 * q^88 - 8 * q^89 - q^90 + 14 * q^91 - 4 * q^92 - 5 * q^93 + 10 * q^94 + 16 * q^95 + 4 * q^96 - 4 * q^97 + 15 * q^99

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