LMFDB - Embedded newform 6038.2.a.d.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$0$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.40625 q^{3} +1.00000 q^{4} -0.733460 q^{5} +2.40625 q^{6} -2.00296 q^{7} -1.00000 q^{8} +2.79002 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.40625 q^{3} +1.00000 q^{4} -0.733460 q^{5} +2.40625 q^{6} -2.00296 q^{7} -1.00000 q^{8} +2.79002 q^{9} +0.733460 q^{10} +2.85569 q^{11} -2.40625 q^{12} -4.47786 q^{13} +2.00296 q^{14} +1.76488 q^{15} +1.00000 q^{16} +6.62251 q^{17} -2.79002 q^{18} -1.71031 q^{19} -0.733460 q^{20} +4.81961 q^{21} -2.85569 q^{22} +4.93978 q^{23} +2.40625 q^{24} -4.46204 q^{25} +4.47786 q^{26} +0.505268 q^{27} -2.00296 q^{28} +4.08107 q^{29} -1.76488 q^{30} -6.07305 q^{31} -1.00000 q^{32} -6.87149 q^{33} -6.62251 q^{34} +1.46909 q^{35} +2.79002 q^{36} -0.150299 q^{37} +1.71031 q^{38} +10.7748 q^{39} +0.733460 q^{40} +6.64393 q^{41} -4.81961 q^{42} +0.0163745 q^{43} +2.85569 q^{44} -2.04637 q^{45} -4.93978 q^{46} +6.50946 q^{47} -2.40625 q^{48} -2.98816 q^{49} +4.46204 q^{50} -15.9354 q^{51} -4.47786 q^{52} -0.866226 q^{53} -0.505268 q^{54} -2.09453 q^{55} +2.00296 q^{56} +4.11543 q^{57} -4.08107 q^{58} -12.0323 q^{59} +1.76488 q^{60} -7.44459 q^{61} +6.07305 q^{62} -5.58829 q^{63} +1.00000 q^{64} +3.28433 q^{65} +6.87149 q^{66} -0.214268 q^{67} +6.62251 q^{68} -11.8863 q^{69} -1.46909 q^{70} -7.17094 q^{71} -2.79002 q^{72} +14.0020 q^{73} +0.150299 q^{74} +10.7368 q^{75} -1.71031 q^{76} -5.71982 q^{77} -10.7748 q^{78} -7.32816 q^{79} -0.733460 q^{80} -9.58585 q^{81} -6.64393 q^{82} +12.7315 q^{83} +4.81961 q^{84} -4.85734 q^{85} -0.0163745 q^{86} -9.82006 q^{87} -2.85569 q^{88} -10.3414 q^{89} +2.04637 q^{90} +8.96896 q^{91} +4.93978 q^{92} +14.6133 q^{93} -6.50946 q^{94} +1.25444 q^{95} +2.40625 q^{96} -11.3416 q^{97} +2.98816 q^{98} +7.96742 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) $ 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * 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
$3$	−2.40625	−1.38925	−0.694623	−	0.719374i	$-0.744429\pi$
$3$	−2.40625	−1.38925	−0.694623	+	0.719374i	$0.744429\pi$
$4$	1.00000	0.500000
$5$	−0.733460	−0.328013	−0.164007	−	0.986459i	$-0.552442\pi$
$5$	−0.733460	−0.328013	−0.164007	+	0.986459i	$0.552442\pi$
$6$	2.40625	0.982346
$7$	−2.00296	−0.757047	−0.378523	−	0.925592i	$-0.623568\pi$
$7$	−2.00296	−0.757047	−0.378523	+	0.925592i	$0.623568\pi$
$8$	−1.00000	−0.353553
$9$	2.79002	0.930006
$10$	0.733460	0.231940
$11$	2.85569	0.861023	0.430511	−	0.902585i	$-0.358333\pi$
$11$	2.85569	0.861023	0.430511	+	0.902585i	$0.358333\pi$
$12$	−2.40625	−0.694623
$13$	−4.47786	−1.24193	−0.620967	−	0.783837i	$-0.713260\pi$
$13$	−4.47786	−1.24193	−0.620967	+	0.783837i	$0.713260\pi$
$14$	2.00296	0.535313
$15$	1.76488	0.455691
$16$	1.00000	0.250000
$17$	6.62251	1.60619	0.803097	−	0.595848i	$-0.203184\pi$
$17$	6.62251	1.60619	0.803097	+	0.595848i	$0.203184\pi$
$18$	−2.79002	−0.657614
$19$	−1.71031	−0.392372	−0.196186	−	0.980567i	$-0.562856\pi$
$19$	−1.71031	−0.392372	−0.196186	+	0.980567i	$0.562856\pi$
$20$	−0.733460	−0.164007
$21$	4.81961	1.05172
$22$	−2.85569	−0.608835
$23$	4.93978	1.03002	0.515008	−	0.857186i	$-0.327789\pi$
$23$	4.93978	1.03002	0.515008	+	0.857186i	$0.327789\pi$
$24$	2.40625	0.491173
$25$	−4.46204	−0.892407
$26$	4.47786	0.878180
$27$	0.505268	0.0972390
$28$	−2.00296	−0.378523
$29$	4.08107	0.757836	0.378918	−	0.925430i	$-0.376296\pi$
$29$	4.08107	0.757836	0.378918	+	0.925430i	$0.376296\pi$
$30$	−1.76488	−0.322222
$31$	−6.07305	−1.09075	−0.545376	−	0.838191i	$-0.683613\pi$
$31$	−6.07305	−1.09075	−0.545376	+	0.838191i	$0.683613\pi$
$32$	−1.00000	−0.176777
$33$	−6.87149	−1.19617
$34$	−6.62251	−1.13575
$35$	1.46909	0.248321
$36$	2.79002	0.465003
$37$	−0.150299	−0.0247091	−0.0123545	−	0.999924i	$-0.503933\pi$
$37$	−0.150299	−0.0247091	−0.0123545	+	0.999924i	$0.503933\pi$
$38$	1.71031	0.277449
$39$	10.7748	1.72535
$40$	0.733460	0.115970
$41$	6.64393	1.03761	0.518803	−	0.854894i	$-0.326378\pi$
$41$	6.64393	1.03761	0.518803	+	0.854894i	$0.326378\pi$
$42$	−4.81961	−0.743682
$43$	0.0163745	0.00249709	0.00124854	−	0.999999i	$-0.499603\pi$
$43$	0.0163745	0.00249709	0.00124854	+	0.999999i	$0.499603\pi$
$44$	2.85569	0.430511
$45$	−2.04637	−0.305054
$46$	−4.93978	−0.728331
$47$	6.50946	0.949503	0.474751	−	0.880120i	$-0.342538\pi$
$47$	6.50946	0.949503	0.474751	+	0.880120i	$0.342538\pi$
$48$	−2.40625	−0.347312
$49$	−2.98816	−0.426880
$50$	4.46204	0.631027
$51$	−15.9354	−2.23140
$52$	−4.47786	−0.620967
$53$	−0.866226	−0.118985	−0.0594926	−	0.998229i	$-0.518948\pi$
$53$	−0.866226	−0.118985	−0.0594926	+	0.998229i	$0.518948\pi$
$54$	−0.505268	−0.0687583
$55$	−2.09453	−0.282427
$56$	2.00296	0.267656
$57$	4.11543	0.545102
$58$	−4.08107	−0.535871
$59$	−12.0323	−1.56647	−0.783236	−	0.621724i	$-0.786432\pi$
$59$	−12.0323	−1.56647	−0.783236	+	0.621724i	$0.786432\pi$
$60$	1.76488	0.227846
$61$	−7.44459	−0.953183	−0.476591	−	0.879125i	$-0.658128\pi$
$61$	−7.44459	−0.953183	−0.476591	+	0.879125i	$0.658128\pi$
$62$	6.07305	0.771279
$63$	−5.58829	−0.704058
$64$	1.00000	0.125000
$65$	3.28433	0.407371
$66$	6.87149	0.845822
$67$	−0.214268	−0.0261770	−0.0130885	−	0.999914i	$-0.504166\pi$
$67$	−0.214268	−0.0261770	−0.0130885	+	0.999914i	$0.504166\pi$
$68$	6.62251	0.803097
$69$	−11.8863	−1.43095
$70$	−1.46909	−0.175590
$71$	−7.17094	−0.851034	−0.425517	−	0.904951i	$-0.639908\pi$
$71$	−7.17094	−0.851034	−0.425517	+	0.904951i	$0.639908\pi$
$72$	−2.79002	−0.328807
$73$	14.0020	1.63881	0.819407	−	0.573213i	$-0.194303\pi$
$73$	14.0020	1.63881	0.819407	+	0.573213i	$0.194303\pi$
$74$	0.150299	0.0174720
$75$	10.7368	1.23977
$76$	−1.71031	−0.196186
$77$	−5.71982	−0.651835
$78$	−10.7748	−1.22001
$79$	−7.32816	−0.824483	−0.412241	−	0.911075i	$-0.635254\pi$
$79$	−7.32816	−0.824483	−0.412241	+	0.911075i	$0.635254\pi$
$80$	−0.733460	−0.0820033
$81$	−9.58585	−1.06509
$82$	−6.64393	−0.733699
$83$	12.7315	1.39747	0.698734	−	0.715382i	$-0.253747\pi$
$83$	12.7315	1.39747	0.698734	+	0.715382i	$0.253747\pi$
$84$	4.81961	0.525862
$85$	−4.85734	−0.526853
$86$	−0.0163745	−0.00176571
$87$	−9.82006	−1.05282
$88$	−2.85569	−0.304417
$89$	−10.3414	−1.09618	−0.548092	−	0.836418i	$-0.684645\pi$
$89$	−10.3414	−1.09618	−0.548092	+	0.836418i	$0.684645\pi$
$90$	2.04637	0.215706
$91$	8.96896	0.940203
$92$	4.93978	0.515008
$93$	14.6133	1.51532
$94$	−6.50946	−0.671400
$95$	1.25444	0.128703
$96$	2.40625	0.245586
$97$	−11.3416	−1.15157	−0.575783	−	0.817603i	$-0.695303\pi$
$97$	−11.3416	−1.15157	−0.575783	+	0.817603i	$0.695303\pi$
$98$	2.98816	0.301850
$99$	7.96742	0.800756
$100$	−4.46204	−0.446204
$101$	19.2656	1.91700	0.958498	−	0.285098i	$-0.0920260\pi$
$101$	19.2656	1.91700	0.958498	+	0.285098i	$0.0920260\pi$
$102$	15.9354	1.57784
$103$	7.90522	0.778925	0.389462	−	0.921042i	$-0.372661\pi$
$103$	7.90522	0.778925	0.389462	+	0.921042i	$0.372661\pi$
$104$	4.47786	0.439090
$105$	−3.53499	−0.344980
$106$	0.866226	0.0841353
$107$	0.985850	0.0953057	0.0476529	−	0.998864i	$-0.484826\pi$
$107$	0.985850	0.0953057	0.0476529	+	0.998864i	$0.484826\pi$
$108$	0.505268	0.0486195
$109$	11.1537	1.06833	0.534164	−	0.845381i	$-0.320627\pi$
$109$	11.1537	1.06833	0.534164	+	0.845381i	$0.320627\pi$
$110$	2.09453	0.199706
$111$	0.361657	0.0343270
$112$	−2.00296	−0.189262
$113$	−14.7077	−1.38359	−0.691794	−	0.722095i	$-0.743179\pi$
$113$	−14.7077	−1.38359	−0.691794	+	0.722095i	$0.743179\pi$
$114$	−4.11543	−0.385445
$115$	−3.62313	−0.337859
$116$	4.08107	0.378918
$117$	−12.4933	−1.15501
$118$	12.0323	1.10766
$119$	−13.2646	−1.21596
$120$	−1.76488	−0.161111
$121$	−2.84504	−0.258640
$122$	7.44459	0.674002
$123$	−15.9869	−1.44149
$124$	−6.07305	−0.545376
$125$	6.94002	0.620735
$126$	5.58829	0.497844
$127$	−15.8291	−1.40461	−0.702304	−	0.711877i	$-0.747845\pi$
$127$	−15.8291	−1.40461	−0.702304	+	0.711877i	$0.747845\pi$
$128$	−1.00000	−0.0883883
$129$	−0.0394010	−0.00346907
$130$	−3.28433	−0.288055
$131$	−15.3326	−1.33962	−0.669809	−	0.742534i	$-0.733624\pi$
$131$	−15.3326	−1.33962	−0.669809	+	0.742534i	$0.733624\pi$
$132$	−6.87149	−0.598086
$133$	3.42568	0.297044
$134$	0.214268	0.0185100
$135$	−0.370594	−0.0318957
$136$	−6.62251	−0.567876
$137$	0.467215	0.0399169	0.0199584	−	0.999801i	$-0.493647\pi$
$137$	0.467215	0.0399169	0.0199584	+	0.999801i	$0.493647\pi$
$138$	11.8863	1.01183
$139$	1.97075	0.167157	0.0835784	−	0.996501i	$-0.473365\pi$
$139$	1.97075	0.167157	0.0835784	+	0.996501i	$0.473365\pi$
$140$	1.46909	0.124161
$141$	−15.6634	−1.31909
$142$	7.17094	0.601772
$143$	−12.7874	−1.06933
$144$	2.79002	0.232501
$145$	−2.99330	−0.248580
$146$	−14.0020	−1.15882
$147$	7.19025	0.593042
$148$	−0.150299	−0.0123545
$149$	−5.66896	−0.464419	−0.232210	−	0.972666i	$-0.574596\pi$
$149$	−5.66896	−0.464419	−0.232210	+	0.972666i	$0.574596\pi$
$150$	−10.7368	−0.876653
$151$	−17.1231	−1.39346	−0.696730	−	0.717334i	$-0.745362\pi$
$151$	−17.1231	−1.39346	−0.696730	+	0.717334i	$0.745362\pi$
$152$	1.71031	0.138724
$153$	18.4769	1.49377
$154$	5.71982	0.460917
$155$	4.45434	0.357781
$156$	10.7748	0.862677
$157$	6.36867	0.508275	0.254137	−	0.967168i	$-0.418208\pi$
$157$	6.36867	0.508275	0.254137	+	0.967168i	$0.418208\pi$
$158$	7.32816	0.582997
$159$	2.08435	0.165300
$160$	0.733460	0.0579851
$161$	−9.89417	−0.779770
$162$	9.58585	0.753136
$163$	−20.3688	−1.59541	−0.797703	−	0.603050i	$-0.793952\pi$
$163$	−20.3688	−1.59541	−0.797703	+	0.603050i	$0.793952\pi$
$164$	6.64393	0.518803
$165$	5.03996	0.392360
$166$	−12.7315	−0.988159
$167$	15.8227	1.22439	0.612197	−	0.790705i	$-0.290286\pi$
$167$	15.8227	1.22439	0.612197	+	0.790705i	$0.290286\pi$
$168$	−4.81961	−0.371841
$169$	7.05121	0.542401
$170$	4.85734	0.372541
$171$	−4.77180	−0.364908
$172$	0.0163745	0.00124854
$173$	23.4288	1.78126	0.890628	−	0.454732i	$-0.150265\pi$
$173$	23.4288	1.78126	0.890628	+	0.454732i	$0.150265\pi$
$174$	9.82006	0.744457
$175$	8.93727	0.675594
$176$	2.85569	0.215256
$177$	28.9527	2.17622
$178$	10.3414	0.775119
$179$	12.1028	0.904605	0.452302	−	0.891865i	$-0.350603\pi$
$179$	12.1028	0.904605	0.452302	+	0.891865i	$0.350603\pi$
$180$	−2.04637	−0.152527
$181$	−8.08655	−0.601069	−0.300534	−	0.953771i	$-0.597165\pi$
$181$	−8.08655	−0.601069	−0.300534	+	0.953771i	$0.597165\pi$
$182$	−8.96896	−0.664824
$183$	17.9135	1.32421
$184$	−4.93978	−0.364165
$185$	0.110239	0.00810490
$186$	−14.6133	−1.07150
$187$	18.9118	1.38297
$188$	6.50946	0.474751
$189$	−1.01203	−0.0736145
$190$	−1.25444	−0.0910069
$191$	5.37718	0.389079	0.194539	−	0.980895i	$-0.437679\pi$
$191$	5.37718	0.389079	0.194539	+	0.980895i	$0.437679\pi$
$192$	−2.40625	−0.173656
$193$	0.223394	0.0160803	0.00804013	−	0.999968i	$-0.497441\pi$
$193$	0.223394	0.0160803	0.00804013	+	0.999968i	$0.497441\pi$
$194$	11.3416	0.814280
$195$	−7.90290	−0.565939
$196$	−2.98816	−0.213440
$197$	10.1552	0.723528	0.361764	−	0.932270i	$-0.382175\pi$
$197$	10.1552	0.723528	0.361764	+	0.932270i	$0.382175\pi$
$198$	−7.96742	−0.566220
$199$	−16.7437	−1.18693	−0.593466	−	0.804859i	$-0.702241\pi$
$199$	−16.7437	−1.18693	−0.593466	+	0.804859i	$0.702241\pi$
$200$	4.46204	0.315514
$201$	0.515582	0.0363663
$202$	−19.2656	−1.35552
$203$	−8.17421	−0.573717
$204$	−15.9354	−1.11570
$205$	−4.87305	−0.340349
$206$	−7.90522	−0.550783
$207$	13.7821	0.957921
$208$	−4.47786	−0.310484
$209$	−4.88411	−0.337841
$210$	3.53499	0.243937
$211$	24.7334	1.70272	0.851359	−	0.524584i	$-0.175779\pi$
$211$	24.7334	1.70272	0.851359	+	0.524584i	$0.175779\pi$
$212$	−0.866226	−0.0594926
$213$	17.2550	1.18230
$214$	−0.985850	−0.0673913
$215$	−0.0120100	−0.000819077 0
$216$	−0.505268	−0.0343792
$217$	12.1641	0.825751
$218$	−11.1537	−0.755421
$219$	−33.6923	−2.27672
$220$	−2.09453	−0.141213
$221$	−29.6547	−1.99479
$222$	−0.361657	−0.0242728
$223$	−20.3986	−1.36599	−0.682997	−	0.730421i	$-0.739324\pi$
$223$	−20.3986	−1.36599	−0.682997	+	0.730421i	$0.739324\pi$
$224$	2.00296	0.133828
$225$	−12.4492	−0.829944
$226$	14.7077	0.978344
$227$	−14.4031	−0.955965	−0.477982	−	0.878369i	$-0.658632\pi$
$227$	−14.4031	−0.955965	−0.477982	+	0.878369i	$0.658632\pi$
$228$	4.11543	0.272551
$229$	15.3323	1.01319	0.506594	−	0.862185i	$-0.330904\pi$
$229$	15.3323	1.01319	0.506594	+	0.862185i	$0.330904\pi$
$230$	3.62313	0.238902
$231$	13.7633	0.905559
$232$	−4.08107	−0.267935
$233$	−2.72756	−0.178688	−0.0893442	−	0.996001i	$-0.528477\pi$
$233$	−2.72756	−0.178688	−0.0893442	+	0.996001i	$0.528477\pi$
$234$	12.4933	0.816713
$235$	−4.77443	−0.311449
$236$	−12.0323	−0.783236
$237$	17.6334	1.14541
$238$	13.2646	0.859817
$239$	−2.02696	−0.131113	−0.0655566	−	0.997849i	$-0.520882\pi$
$239$	−2.02696	−0.131113	−0.0655566	+	0.997849i	$0.520882\pi$
$240$	1.76488	0.113923
$241$	−7.18086	−0.462560	−0.231280	−	0.972887i	$-0.574291\pi$
$241$	−7.18086	−0.462560	−0.231280	+	0.972887i	$0.574291\pi$
$242$	2.84504	0.182886
$243$	21.5501	1.38244
$244$	−7.44459	−0.476591
$245$	2.19169	0.140022
$246$	15.9869	1.01929
$247$	7.65853	0.487300
$248$	6.07305	0.385639
$249$	−30.6352	−1.94143
$250$	−6.94002	−0.438926
$251$	−13.9840	−0.882662	−0.441331	−	0.897344i	$-0.645493\pi$
$251$	−13.9840	−0.882662	−0.441331	+	0.897344i	$0.645493\pi$
$252$	−5.58829	−0.352029
$253$	14.1065	0.886867
$254$	15.8291	0.993208
$255$	11.6880	0.731929
$256$	1.00000	0.0625000
$257$	10.2999	0.642490	0.321245	−	0.946996i	$-0.395899\pi$
$257$	10.2999	0.642490	0.321245	+	0.946996i	$0.395899\pi$
$258$	0.0394010	0.00245300
$259$	0.301043	0.0187059
$260$	3.28433	0.203685
$261$	11.3863	0.704792
$262$	15.3326	0.947253
$263$	−1.06475	−0.0656553	−0.0328276	−	0.999461i	$-0.510451\pi$
$263$	−1.06475	−0.0656553	−0.0328276	+	0.999461i	$0.510451\pi$
$264$	6.87149	0.422911
$265$	0.635342	0.0390287
$266$	−3.42568	−0.210042
$267$	24.8839	1.52287
$268$	−0.214268	−0.0130885
$269$	−20.8510	−1.27131	−0.635653	−	0.771975i	$-0.719269\pi$
$269$	−20.8510	−1.27131	−0.635653	+	0.771975i	$0.719269\pi$
$270$	0.370594	0.0225536
$271$	14.3967	0.874534	0.437267	−	0.899332i	$-0.355946\pi$
$271$	14.3967	0.874534	0.437267	+	0.899332i	$0.355946\pi$
$272$	6.62251	0.401549
$273$	−21.5815	−1.30617
$274$	−0.467215	−0.0282255
$275$	−12.7422	−0.768383
$276$	−11.8863	−0.715473
$277$	21.6722	1.30216	0.651079	−	0.759010i	$-0.274317\pi$
$277$	21.6722	1.30216	0.651079	+	0.759010i	$0.274317\pi$
$278$	−1.97075	−0.118198
$279$	−16.9439	−1.01441
$280$	−1.46909	−0.0877949
$281$	−1.92923	−0.115088	−0.0575441	−	0.998343i	$-0.518327\pi$
$281$	−1.92923	−0.115088	−0.0575441	+	0.998343i	$0.518327\pi$
$282$	15.6634	0.932740
$283$	−14.8195	−0.880925	−0.440462	−	0.897771i	$-0.645186\pi$
$283$	−14.8195	−0.880925	−0.440462	+	0.897771i	$0.645186\pi$
$284$	−7.17094	−0.425517
$285$	−3.01850	−0.178800
$286$	12.7874	0.756133
$287$	−13.3075	−0.785517
$288$	−2.79002	−0.164403
$289$	26.8576	1.57986
$290$	2.99330	0.175773
$291$	27.2907	1.59981
$292$	14.0020	0.819407
$293$	−8.42583	−0.492242	−0.246121	−	0.969239i	$-0.579156\pi$
$293$	−8.42583	−0.492242	−0.246121	+	0.969239i	$0.579156\pi$
$294$	−7.19025	−0.419344
$295$	8.82521	0.513824
$296$	0.150299	0.00873598
$297$	1.44289	0.0837249
$298$	5.66896	0.328394
$299$	−22.1196	−1.27921
$300$	10.7368	0.619887
$301$	−0.0327974	−0.00189041
$302$	17.1231	0.985325
$303$	−46.3577	−2.66318
$304$	−1.71031	−0.0980930
$305$	5.46031	0.312656
$306$	−18.4769	−1.05626
$307$	32.3254	1.84491	0.922453	−	0.386109i	$-0.126181\pi$
$307$	32.3254	1.84491	0.922453	+	0.386109i	$0.126181\pi$
$308$	−5.71982	−0.325917
$309$	−19.0219	−1.08212
$310$	−4.45434	−0.252990
$311$	−12.4298	−0.704830	−0.352415	−	0.935844i	$-0.614639\pi$
$311$	−12.4298	−0.704830	−0.352415	+	0.935844i	$0.614639\pi$
$312$	−10.7748	−0.610004
$313$	−5.19391	−0.293577	−0.146789	−	0.989168i	$-0.546894\pi$
$313$	−5.19391	−0.293577	−0.146789	+	0.989168i	$0.546894\pi$
$314$	−6.36867	−0.359405
$315$	4.09878	0.230940
$316$	−7.32816	−0.412241
$317$	19.0861	1.07198	0.535992	−	0.844223i	$-0.319938\pi$
$317$	19.0861	1.07198	0.535992	+	0.844223i	$0.319938\pi$
$318$	−2.08435	−0.116885
$319$	11.6543	0.652514
$320$	−0.733460	−0.0410016
$321$	−2.37220	−0.132403
$322$	9.89417	0.551381
$323$	−11.3265	−0.630226
$324$	−9.58585	−0.532547
$325$	19.9804	1.10831
$326$	20.3688	1.12812
$327$	−26.8385	−1.48417
$328$	−6.64393	−0.366849
$329$	−13.0382	−0.718818
$330$	−5.03996	−0.277441
$331$	20.0677	1.10302	0.551510	−	0.834168i	$-0.314052\pi$
$331$	20.0677	1.10302	0.551510	+	0.834168i	$0.314052\pi$
$332$	12.7315	0.698734
$333$	−0.419338	−0.0229796
$334$	−15.8227	−0.865777
$335$	0.157157	0.00858641
$336$	4.81961	0.262931
$337$	20.5678	1.12040	0.560199	−	0.828358i	$-0.310725\pi$
$337$	20.5678	1.12040	0.560199	+	0.828358i	$0.310725\pi$
$338$	−7.05121	−0.383536
$339$	35.3904	1.92214
$340$	−4.85734	−0.263426
$341$	−17.3428	−0.939163
$342$	4.77180	0.258029
$343$	20.0059	1.08022
$344$	−0.0163745	−0.000882853 0
$345$	8.71814	0.469369
$346$	−23.4288	−1.25954
$347$	−7.32686	−0.393326	−0.196663	−	0.980471i	$-0.563011\pi$
$347$	−7.32686	−0.393326	−0.196663	+	0.980471i	$0.563011\pi$
$348$	−9.82006	−0.526411
$349$	11.5412	0.617785	0.308892	−	0.951097i	$-0.400042\pi$
$349$	11.5412	0.617785	0.308892	+	0.951097i	$0.400042\pi$
$350$	−8.93727	−0.477717
$351$	−2.26252	−0.120764
$352$	−2.85569	−0.152209
$353$	12.7664	0.679486	0.339743	−	0.940518i	$-0.389660\pi$
$353$	12.7664	0.679486	0.339743	+	0.940518i	$0.389660\pi$
$354$	−28.9527	−1.53882
$355$	5.25959	0.279150
$356$	−10.3414	−0.548092
$357$	31.9179	1.68927
$358$	−12.1028	−0.639652
$359$	34.8371	1.83863	0.919317	−	0.393518i	$-0.128742\pi$
$359$	34.8371	1.83863	0.919317	+	0.393518i	$0.128742\pi$
$360$	2.04637	0.107853
$361$	−16.0748	−0.846044
$362$	8.08655	0.425020
$363$	6.84587	0.359315
$364$	8.96896	0.470101
$365$	−10.2699	−0.537552
$366$	−17.9135	−0.936355
$367$	0.802797	0.0419057	0.0209528	−	0.999780i	$-0.493330\pi$
$367$	0.802797	0.0419057	0.0209528	+	0.999780i	$0.493330\pi$
$368$	4.93978	0.257504
$369$	18.5367	0.964981
$370$	−0.110239	−0.00573103
$371$	1.73501	0.0900774
$372$	14.6133	0.757662
$373$	−27.4441	−1.42100	−0.710500	−	0.703697i	$-0.751531\pi$
$373$	−27.4441	−1.42100	−0.710500	+	0.703697i	$0.751531\pi$
$374$	−18.9118	−0.977907
$375$	−16.6994	−0.862353
$376$	−6.50946	−0.335700
$377$	−18.2745	−0.941183
$378$	1.01203	0.0520533
$379$	−31.1543	−1.60029	−0.800146	−	0.599806i	$-0.795244\pi$
$379$	−31.1543	−1.60029	−0.800146	+	0.599806i	$0.795244\pi$
$380$	1.25444	0.0643516
$381$	38.0888	1.95135
$382$	−5.37718	−0.275120
$383$	−22.3977	−1.14447	−0.572234	−	0.820090i	$-0.693923\pi$
$383$	−22.3977	−1.14447	−0.572234	+	0.820090i	$0.693923\pi$
$384$	2.40625	0.122793
$385$	4.19526	0.213810
$386$	−0.223394	−0.0113705
$387$	0.0456851	0.00232231
$388$	−11.3416	−0.575783
$389$	0.635852	0.0322390	0.0161195	−	0.999870i	$-0.494869\pi$
$389$	0.635852	0.0322390	0.0161195	+	0.999870i	$0.494869\pi$
$390$	7.90290	0.400179
$391$	32.7137	1.65441
$392$	2.98816	0.150925
$393$	36.8941	1.86106
$394$	−10.1552	−0.511612
$395$	5.37491	0.270441
$396$	7.96742	0.400378
$397$	27.1997	1.36512	0.682558	−	0.730831i	$-0.260867\pi$
$397$	27.1997	1.36512	0.682558	+	0.730831i	$0.260867\pi$
$398$	16.7437	0.839287
$399$	−8.24303	−0.412667
$400$	−4.46204	−0.223102
$401$	8.72385	0.435648	0.217824	−	0.975988i	$-0.430104\pi$
$401$	8.72385	0.435648	0.217824	+	0.975988i	$0.430104\pi$
$402$	−0.515582	−0.0257149
$403$	27.1943	1.35464
$404$	19.2656	0.958498
$405$	7.03084	0.349365
$406$	8.17421	0.405679
$407$	−0.429208	−0.0212751
$408$	15.9354	0.788919
$409$	2.53177	0.125188	0.0625939	−	0.998039i	$-0.480063\pi$
$409$	2.53177	0.125188	0.0625939	+	0.998039i	$0.480063\pi$
$410$	4.87305	0.240663
$411$	−1.12423	−0.0554544
$412$	7.90522	0.389462
$413$	24.1002	1.18589
$414$	−13.7821	−0.677352
$415$	−9.33807	−0.458388
$416$	4.47786	0.219545
$417$	−4.74211	−0.232222
$418$	4.88411	0.238890
$419$	9.80052	0.478787	0.239393	−	0.970923i	$-0.423051\pi$
$419$	9.80052	0.478787	0.239393	+	0.970923i	$0.423051\pi$
$420$	−3.53499	−0.172490
$421$	18.2810	0.890960	0.445480	−	0.895292i	$-0.353033\pi$
$421$	18.2810	0.890960	0.445480	+	0.895292i	$0.353033\pi$
$422$	−24.7334	−1.20400
$423$	18.1615	0.883043
$424$	0.866226	0.0420676
$425$	−29.5499	−1.43338
$426$	−17.2550	−0.836009
$427$	14.9112	0.721604
$428$	0.985850	0.0476529
$429$	30.7696	1.48557
$430$	0.0120100	0.000579175 0
$431$	−3.10782	−0.149698	−0.0748492	−	0.997195i	$-0.523848\pi$
$431$	−3.10782	−0.149698	−0.0748492	+	0.997195i	$0.523848\pi$
$432$	0.505268	0.0243097
$433$	37.0148	1.77882	0.889410	−	0.457111i	$-0.151116\pi$
$433$	37.0148	1.77882	0.889410	+	0.457111i	$0.151116\pi$
$434$	−12.1641	−0.583894
$435$	7.20262	0.345339
$436$	11.1537	0.534164
$437$	−8.44856	−0.404149
$438$	33.6923	1.60988
$439$	−24.8220	−1.18469	−0.592345	−	0.805684i	$-0.701798\pi$
$439$	−24.8220	−1.18469	−0.592345	+	0.805684i	$0.701798\pi$
$440$	2.09453	0.0998529
$441$	−8.33702	−0.397001
$442$	29.6547	1.41053
$443$	11.2870	0.536262	0.268131	−	0.963382i	$-0.413594\pi$
$443$	11.2870	0.536262	0.268131	+	0.963382i	$0.413594\pi$
$444$	0.361657	0.0171635
$445$	7.58498	0.359563
$446$	20.3986	0.965904
$447$	13.6409	0.645193
$448$	−2.00296	−0.0946309
$449$	9.08416	0.428708	0.214354	−	0.976756i	$-0.431235\pi$
$449$	9.08416	0.428708	0.214354	+	0.976756i	$0.431235\pi$
$450$	12.4492	0.586859
$451$	18.9730	0.893403
$452$	−14.7077	−0.691794
$453$	41.2024	1.93586
$454$	14.4031	0.675969
$455$	−6.57837	−0.308399
$456$	−4.11543	−0.192723
$457$	18.0217	0.843020	0.421510	−	0.906824i	$-0.361500\pi$
$457$	18.0217	0.843020	0.421510	+	0.906824i	$0.361500\pi$
$458$	−15.3323	−0.716432
$459$	3.34615	0.156185
$460$	−3.62313	−0.168929
$461$	34.1247	1.58935	0.794673	−	0.607037i	$-0.207642\pi$
$461$	34.1247	1.58935	0.794673	+	0.607037i	$0.207642\pi$
$462$	−13.7633	−0.640327
$463$	−3.51932	−0.163557	−0.0817783	−	0.996651i	$-0.526060\pi$
$463$	−3.51932	−0.163557	−0.0817783	+	0.996651i	$0.526060\pi$
$464$	4.08107	0.189459
$465$	−10.7182	−0.497046
$466$	2.72756	0.126352
$467$	−7.75729	−0.358964	−0.179482	−	0.983761i	$-0.557442\pi$
$467$	−7.75729	−0.358964	−0.179482	+	0.983761i	$0.557442\pi$
$468$	−12.4933	−0.577503
$469$	0.429170	0.0198172
$470$	4.77443	0.220228
$471$	−15.3246	−0.706119
$472$	12.0323	0.553832
$473$	0.0467605	0.00215005
$474$	−17.6334	−0.809927
$475$	7.63147	0.350156
$476$	−13.2646	−0.607982
$477$	−2.41678	−0.110657
$478$	2.02696	0.0927110
$479$	37.4969	1.71328	0.856639	−	0.515917i	$-0.172549\pi$
$479$	37.4969	1.71328	0.856639	+	0.515917i	$0.172549\pi$
$480$	−1.76488	−0.0805556
$481$	0.673019	0.0306870
$482$	7.18086	0.327079
$483$	23.8078	1.08329
$484$	−2.84504	−0.129320
$485$	8.31861	0.377729
$486$	−21.5501	−0.977533
$487$	7.69191	0.348554	0.174277	−	0.984697i	$-0.444241\pi$
$487$	7.69191	0.348554	0.174277	+	0.984697i	$0.444241\pi$
$488$	7.44459	0.337001
$489$	49.0123	2.21641
$490$	−2.19169	−0.0990107
$491$	26.6291	1.20175	0.600877	−	0.799341i	$-0.294818\pi$
$491$	26.6291	1.20175	0.600877	+	0.799341i	$0.294818\pi$
$492$	−15.9869	−0.720746
$493$	27.0269	1.21723
$494$	−7.65853	−0.344573
$495$	−5.84378	−0.262659
$496$	−6.07305	−0.272688
$497$	14.3631	0.644272
$498$	30.6352	1.37280
$499$	−19.5968	−0.877273	−0.438637	−	0.898665i	$-0.644538\pi$
$499$	−19.5968	−0.877273	−0.438637	+	0.898665i	$0.644538\pi$
$500$	6.94002	0.310367
$501$	−38.0732	−1.70099
$502$	13.9840	0.624136
$503$	3.29623	0.146971	0.0734857	−	0.997296i	$-0.476588\pi$
$503$	3.29623	0.146971	0.0734857	+	0.997296i	$0.476588\pi$
$504$	5.58829	0.248922
$505$	−14.1305	−0.628800
$506$	−14.1065	−0.627109
$507$	−16.9670	−0.753529
$508$	−15.8291	−0.702304
$509$	21.1660	0.938166	0.469083	−	0.883154i	$-0.344585\pi$
$509$	21.1660	0.938166	0.469083	+	0.883154i	$0.344585\pi$
$510$	−11.6880	−0.517552
$511$	−28.0455	−1.24066
$512$	−1.00000	−0.0441942
$513$	−0.864166	−0.0381539
$514$	−10.2999	−0.454309
$515$	−5.79816	−0.255498
$516$	−0.0394010	−0.00173453
$517$	18.5890	0.817543
$518$	−0.301043	−0.0132271
$519$	−56.3754	−2.47460
$520$	−3.28433	−0.144027
$521$	22.4432	0.983253	0.491627	−	0.870806i	$-0.336402\pi$
$521$	22.4432	0.983253	0.491627	+	0.870806i	$0.336402\pi$
$522$	−11.3863	−0.498363
$523$	12.5399	0.548330	0.274165	−	0.961683i	$-0.411599\pi$
$523$	12.5399	0.548330	0.274165	+	0.961683i	$0.411599\pi$
$524$	−15.3326	−0.669809
$525$	−21.5053	−0.938567
$526$	1.06475	0.0464253
$527$	−40.2189	−1.75196
$528$	−6.87149	−0.299043
$529$	1.40143	0.0609318
$530$	−0.635342	−0.0275975
$531$	−33.5703	−1.45683
$532$	3.42568	0.148522
$533$	−29.7506	−1.28864
$534$	−24.8839	−1.07683
$535$	−0.723081	−0.0312615
$536$	0.214268	0.00925498
$537$	−29.1223	−1.25672
$538$	20.8510	0.898949
$539$	−8.53326	−0.367553
$540$	−0.370594	−0.0159478
$541$	30.5296	1.31257	0.656284	−	0.754514i	$-0.272127\pi$
$541$	30.5296	1.31257	0.656284	+	0.754514i	$0.272127\pi$
$542$	−14.3967	−0.618389
$543$	19.4582	0.835033
$544$	−6.62251	−0.283938
$545$	−8.18076	−0.350425
$546$	21.5815	0.923604
$547$	−3.89256	−0.166434	−0.0832170	−	0.996531i	$-0.526519\pi$
$547$	−3.89256	−0.166434	−0.0832170	+	0.996531i	$0.526519\pi$
$548$	0.467215	0.0199584
$549$	−20.7706	−0.886466
$550$	12.7422	0.543329
$551$	−6.97990	−0.297354
$552$	11.8863	0.505916
$553$	14.6780	0.624172
$554$	−21.6722	−0.920765
$555$	−0.265261	−0.0112597
$556$	1.97075	0.0835784
$557$	31.2164	1.32268	0.661342	−	0.750085i	$-0.269987\pi$
$557$	31.2164	1.32268	0.661342	+	0.750085i	$0.269987\pi$
$558$	16.9439	0.717294
$559$	−0.0733227	−0.00310122
$560$	1.46909	0.0620803
$561$	−45.5065	−1.92129
$562$	1.92923	0.0813797
$563$	13.9717	0.588838	0.294419	−	0.955676i	$-0.404874\pi$
$563$	13.9717	0.588838	0.294419	+	0.955676i	$0.404874\pi$
$564$	−15.6634	−0.659547
$565$	10.7875	0.453835
$566$	14.8195	0.622908
$567$	19.2001	0.806327
$568$	7.17094	0.300886
$569$	18.4068	0.771651	0.385826	−	0.922572i	$-0.373917\pi$
$569$	18.4068	0.771651	0.385826	+	0.922572i	$0.373917\pi$
$570$	3.01850	0.126431
$571$	−12.3032	−0.514871	−0.257436	−	0.966295i	$-0.582878\pi$
$571$	−12.3032	−0.514871	−0.257436	+	0.966295i	$0.582878\pi$
$572$	−12.7874	−0.534667
$573$	−12.9388	−0.540526
$574$	13.3075	0.555444
$575$	−22.0415	−0.919193
$576$	2.79002	0.116251
$577$	10.6627	0.443893	0.221947	−	0.975059i	$-0.428759\pi$
$577$	10.6627	0.443893	0.221947	+	0.975059i	$0.428759\pi$
$578$	−26.8576	−1.11713
$579$	−0.537541	−0.0223394
$580$	−2.99330	−0.124290
$581$	−25.5007	−1.05795
$582$	−27.2907	−1.13124
$583$	−2.47367	−0.102449
$584$	−14.0020	−0.579408
$585$	9.16334	0.378857
$586$	8.42583	0.348068
$587$	2.96754	0.122483	0.0612416	−	0.998123i	$-0.480494\pi$
$587$	2.96754	0.122483	0.0612416	+	0.998123i	$0.480494\pi$
$588$	7.19025	0.296521
$589$	10.3868	0.427981
$590$	−8.82521	−0.363328
$591$	−24.4359	−1.00516
$592$	−0.150299	−0.00617727
$593$	2.20210	0.0904293	0.0452147	−	0.998977i	$-0.485603\pi$
$593$	2.20210	0.0904293	0.0452147	+	0.998977i	$0.485603\pi$
$594$	−1.44289	−0.0592025
$595$	9.72906	0.398852
$596$	−5.66896	−0.232210
$597$	40.2895	1.64894
$598$	22.1196	0.904539
$599$	−6.74790	−0.275712	−0.137856	−	0.990452i	$-0.544021\pi$
$599$	−6.74790	−0.275712	−0.137856	+	0.990452i	$0.544021\pi$
$600$	−10.7368	−0.438326
$601$	9.41750	0.384148	0.192074	−	0.981380i	$-0.438479\pi$
$601$	9.41750	0.384148	0.192074	+	0.981380i	$0.438479\pi$
$602$	0.0327974	0.00133672
$603$	−0.597812	−0.0243448
$604$	−17.1231	−0.696730
$605$	2.08672	0.0848373
$606$	46.3577	1.88315
$607$	38.8020	1.57493	0.787463	−	0.616362i	$-0.211394\pi$
$607$	38.8020	1.57493	0.787463	+	0.616362i	$0.211394\pi$
$608$	1.71031	0.0693622
$609$	19.6692	0.797035
$610$	−5.46031	−0.221081
$611$	−29.1485	−1.17922
$612$	18.4769	0.746885
$613$	−26.2943	−1.06202	−0.531008	−	0.847367i	$-0.678186\pi$
$613$	−26.2943	−1.06202	−0.531008	+	0.847367i	$0.678186\pi$
$614$	−32.3254	−1.30455
$615$	11.7258	0.472828
$616$	5.71982	0.230458
$617$	−9.47259	−0.381352	−0.190676	−	0.981653i	$-0.561068\pi$
$617$	−9.47259	−0.381352	−0.190676	+	0.981653i	$0.561068\pi$
$618$	19.0219	0.765173
$619$	5.50016	0.221070	0.110535	−	0.993872i	$-0.464744\pi$
$619$	5.50016	0.221070	0.110535	+	0.993872i	$0.464744\pi$
$620$	4.45434	0.178891
$621$	2.49592	0.100158
$622$	12.4298	0.498390
$623$	20.7133	0.829862
$624$	10.7748	0.431338
$625$	17.2200	0.688798
$626$	5.19391	0.207590
$627$	11.7524	0.469345
$628$	6.36867	0.254137
$629$	−0.995359	−0.0396876
$630$	−4.09878	−0.163299
$631$	−32.4501	−1.29182	−0.645909	−	0.763415i	$-0.723521\pi$
$631$	−32.4501	−1.29182	−0.645909	+	0.763415i	$0.723521\pi$
$632$	7.32816	0.291499
$633$	−59.5146	−2.36549
$634$	−19.0861	−0.758007
$635$	11.6100	0.460730
$636$	2.08435	0.0826499
$637$	13.3806	0.530157
$638$	−11.6543	−0.461397
$639$	−20.0070	−0.791466
$640$	0.733460	0.0289925
$641$	−9.67000	−0.381942	−0.190971	−	0.981596i	$-0.561164\pi$
$641$	−9.67000	−0.381942	−0.190971	+	0.981596i	$0.561164\pi$
$642$	2.37220	0.0936232
$643$	26.6969	1.05282	0.526412	−	0.850229i	$-0.323537\pi$
$643$	26.6969	1.05282	0.526412	+	0.850229i	$0.323537\pi$
$644$	−9.89417	−0.389885
$645$	0.0288991	0.00113790
$646$	11.3265	0.445637
$647$	34.7881	1.36766	0.683830	−	0.729641i	$-0.260313\pi$
$647$	34.7881	1.36766	0.683830	+	0.729641i	$0.260313\pi$
$648$	9.58585	0.376568
$649$	−34.3605	−1.34877
$650$	−19.9804	−0.783695
$651$	−29.2697	−1.14717
$652$	−20.3688	−0.797703
$653$	−30.8327	−1.20658	−0.603289	−	0.797523i	$-0.706143\pi$
$653$	−30.8327	−1.20658	−0.603289	+	0.797523i	$0.706143\pi$
$654$	26.8385	1.04947
$655$	11.2459	0.439412
$656$	6.64393	0.259402
$657$	39.0659	1.52411
$658$	13.0382	0.508281
$659$	19.3245	0.752775	0.376388	−	0.926462i	$-0.377166\pi$
$659$	19.3245	0.752775	0.376388	+	0.926462i	$0.377166\pi$
$660$	5.03996	0.196180
$661$	7.03659	0.273692	0.136846	−	0.990592i	$-0.456304\pi$
$661$	7.03659	0.273692	0.136846	+	0.990592i	$0.456304\pi$
$662$	−20.0677	−0.779954
$663$	71.3564	2.77125
$664$	−12.7315	−0.494079
$665$	−2.51260	−0.0974344
$666$	0.419338	0.0162490
$667$	20.1596	0.780583
$668$	15.8227	0.612197
$669$	49.0842	1.89770
$670$	−0.157157	−0.00607151
$671$	−21.2594	−0.820712
$672$	−4.81961	−0.185920
$673$	−1.63110	−0.0628743	−0.0314371	−	0.999506i	$-0.510008\pi$
$673$	−1.63110	−0.0628743	−0.0314371	+	0.999506i	$0.510008\pi$
$674$	−20.5678	−0.792241
$675$	−2.25453	−0.0867768
$676$	7.05121	0.271201
$677$	32.6826	1.25609	0.628046	−	0.778176i	$-0.283855\pi$
$677$	32.6826	1.25609	0.628046	+	0.778176i	$0.283855\pi$
$678$	−35.3904	−1.35916
$679$	22.7168	0.871789
$680$	4.85734	0.186271
$681$	34.6573	1.32807
$682$	17.3428	0.664088
$683$	−41.5020	−1.58803	−0.794015	−	0.607899i	$-0.792013\pi$
$683$	−41.5020	−1.58803	−0.794015	+	0.607899i	$0.792013\pi$
$684$	−4.77180	−0.182454
$685$	−0.342683	−0.0130933
$686$	−20.0059	−0.763827
$687$	−36.8933	−1.40757
$688$	0.0163745	0.000624272 0
$689$	3.87884	0.147772
$690$	−8.71814	−0.331894
$691$	−12.8803	−0.489990	−0.244995	−	0.969524i	$-0.578786\pi$
$691$	−12.8803	−0.489990	−0.244995	+	0.969524i	$0.578786\pi$
$692$	23.4288	0.890628
$693$	−15.9584	−0.606210
$694$	7.32686	0.278124
$695$	−1.44546	−0.0548296
$696$	9.82006	0.372228
$697$	43.9995	1.66660
$698$	−11.5412	−0.436840
$699$	6.56318	0.248242
$700$	8.93727	0.337797
$701$	−4.46131	−0.168501	−0.0842506	−	0.996445i	$-0.526850\pi$
$701$	−4.46131	−0.168501	−0.0842506	+	0.996445i	$0.526850\pi$
$702$	2.26252	0.0853933
$703$	0.257059	0.00969515
$704$	2.85569	0.107628
$705$	11.4884	0.432680
$706$	−12.7664	−0.480469
$707$	−38.5881	−1.45126
$708$	28.9527	1.08811
$709$	15.4755	0.581195	0.290597	−	0.956845i	$-0.406146\pi$
$709$	15.4755	0.581195	0.290597	+	0.956845i	$0.406146\pi$
$710$	−5.25959	−0.197389
$711$	−20.4457	−0.766774
$712$	10.3414	0.387559
$713$	−29.9996	−1.12349
$714$	−31.9179	−1.19450
$715$	9.37902	0.350755
$716$	12.1028	0.452302
$717$	4.87736	0.182148
$718$	−34.8371	−1.30011
$719$	20.5093	0.764867	0.382433	−	0.923983i	$-0.375086\pi$
$719$	20.5093	0.764867	0.382433	+	0.923983i	$0.375086\pi$
$720$	−2.04637	−0.0762635
$721$	−15.8338	−0.589683
$722$	16.0748	0.598244
$723$	17.2789	0.642610
$724$	−8.08655	−0.300534
$725$	−18.2099	−0.676298
$726$	−6.84587	−0.254074
$727$	7.05681	0.261723	0.130861	−	0.991401i	$-0.458226\pi$
$727$	7.05681	0.261723	0.130861	+	0.991401i	$0.458226\pi$
$728$	−8.96896	−0.332412
$729$	−23.0973	−0.855456
$730$	10.2699	0.380107
$731$	0.108440	0.00401081
$732$	17.9135	0.662103
$733$	−4.66363	−0.172255	−0.0861275	−	0.996284i	$-0.527449\pi$
$733$	−4.66363	−0.172255	−0.0861275	+	0.996284i	$0.527449\pi$
$734$	−0.802797	−0.0296318
$735$	−5.27376	−0.194525
$736$	−4.93978	−0.182083
$737$	−0.611883	−0.0225390
$738$	−18.5367	−0.682344
$739$	20.8160	0.765730	0.382865	−	0.923804i	$-0.374937\pi$
$739$	20.8160	0.765730	0.382865	+	0.923804i	$0.374937\pi$
$740$	0.110239	0.00405245
$741$	−18.4283	−0.676980
$742$	−1.73501	−0.0636943
$743$	−2.78716	−0.102251	−0.0511255	−	0.998692i	$-0.516281\pi$
$743$	−2.78716	−0.102251	−0.0511255	+	0.998692i	$0.516281\pi$
$744$	−14.6133	−0.535748
$745$	4.15795	0.152336
$746$	27.4441	1.00480
$747$	35.5212	1.29965
$748$	18.9118	0.691485
$749$	−1.97462	−0.0721509
$750$	16.6994	0.609776
$751$	−51.2413	−1.86982	−0.934911	−	0.354883i	$-0.884521\pi$
$751$	−51.2413	−1.86982	−0.934911	+	0.354883i	$0.884521\pi$
$752$	6.50946	0.237376
$753$	33.6489	1.22623
$754$	18.2745	0.665517
$755$	12.5591	0.457073
$756$	−1.01203	−0.0368072
$757$	2.56964	0.0933953	0.0466976	−	0.998909i	$-0.485130\pi$
$757$	2.56964	0.0933953	0.0466976	+	0.998909i	$0.485130\pi$
$758$	31.1543	1.13158
$759$	−33.9436	−1.23208
$760$	−1.25444	−0.0455035
$761$	15.5501	0.563692	0.281846	−	0.959460i	$-0.409053\pi$
$761$	15.5501	0.563692	0.281846	+	0.959460i	$0.409053\pi$
$762$	−38.0888	−1.37981
$763$	−22.3403	−0.808774
$764$	5.37718	0.194539
$765$	−13.5521	−0.489976
$766$	22.3977	0.809262
$767$	53.8789	1.94546
$768$	−2.40625	−0.0868279
$769$	−37.8631	−1.36538	−0.682690	−	0.730708i	$-0.739190\pi$
$769$	−37.8631	−1.36538	−0.682690	+	0.730708i	$0.739190\pi$
$770$	−4.19526	−0.151187
$771$	−24.7841	−0.892577
$772$	0.223394	0.00804013
$773$	28.1795	1.01355	0.506774	−	0.862079i	$-0.330838\pi$
$773$	28.1795	1.01355	0.506774	+	0.862079i	$0.330838\pi$
$774$	−0.0456851	−0.00164212
$775$	27.0982	0.973396
$776$	11.3416	0.407140
$777$	−0.724384	−0.0259871
$778$	−0.635852	−0.0227964
$779$	−11.3632	−0.407128
$780$	−7.90290	−0.282969
$781$	−20.4780	−0.732759
$782$	−32.7137	−1.16984
$783$	2.06204	0.0736912
$784$	−2.98816	−0.106720
$785$	−4.67116	−0.166721
$786$	−36.8941	−1.31597
$787$	13.6456	0.486411	0.243206	−	0.969975i	$-0.421801\pi$
$787$	13.6456	0.486411	0.243206	+	0.969975i	$0.421801\pi$
$788$	10.1552	0.361764
$789$	2.56205	0.0912114
$790$	−5.37491	−0.191231
$791$	29.4590	1.04744
$792$	−7.96742	−0.283110
$793$	33.3358	1.18379
$794$	−27.1997	−0.965283
$795$	−1.52879	−0.0542205
$796$	−16.7437	−0.593466
$797$	48.5718	1.72050	0.860251	−	0.509871i	$-0.170307\pi$
$797$	48.5718	1.72050	0.860251	+	0.509871i	$0.170307\pi$
$798$	8.24303	0.291800
$799$	43.1090	1.52509
$800$	4.46204	0.157757
$801$	−28.8526	−1.01946
$802$	−8.72385	−0.308050
$803$	39.9854	1.41106
$804$	0.515582	0.0181832
$805$	7.25698	0.255775
$806$	−27.1943	−0.957878
$807$	50.1725	1.76616
$808$	−19.2656	−0.677761
$809$	−5.53467	−0.194589	−0.0972943	−	0.995256i	$-0.531019\pi$
$809$	−5.53467	−0.194589	−0.0972943	+	0.995256i	$0.531019\pi$
$810$	−7.03084	−0.247038
$811$	−19.3413	−0.679166	−0.339583	−	0.940576i	$-0.610286\pi$
$811$	−19.3413	−0.679166	−0.339583	+	0.940576i	$0.610286\pi$
$812$	−8.17421	−0.286859
$813$	−34.6419	−1.21494
$814$	0.429208	0.0150437
$815$	14.9397	0.523314
$816$	−15.9354	−0.557850
$817$	−0.0280055	−0.000979787 0
$818$	−2.53177	−0.0885212
$819$	25.0236	0.874394
$820$	−4.87305	−0.170174
$821$	32.1533	1.12216	0.561080	−	0.827762i	$-0.310386\pi$
$821$	32.1533	1.12216	0.561080	+	0.827762i	$0.310386\pi$
$822$	1.12423	0.0392122
$823$	18.9498	0.660547	0.330273	−	0.943885i	$-0.392859\pi$
$823$	18.9498	0.660547	0.330273	+	0.943885i	$0.392859\pi$
$824$	−7.90522	−0.275392
$825$	30.6608	1.06747
$826$	−24.1002	−0.838553
$827$	−14.9027	−0.518219	−0.259110	−	0.965848i	$-0.583429\pi$
$827$	−14.9027	−0.518219	−0.259110	+	0.965848i	$0.583429\pi$
$828$	13.7821	0.478960
$829$	2.83704	0.0985343	0.0492672	−	0.998786i	$-0.484311\pi$
$829$	2.83704	0.0985343	0.0492672	+	0.998786i	$0.484311\pi$
$830$	9.33807	0.324129
$831$	−52.1487	−1.80902
$832$	−4.47786	−0.155242
$833$	−19.7891	−0.685652
$834$	4.74211	0.164206
$835$	−11.6053	−0.401617
$836$	−4.88411	−0.168921
$837$	−3.06852	−0.106064
$838$	−9.80052	−0.338553
$839$	20.0069	0.690716	0.345358	−	0.938471i	$-0.387758\pi$
$839$	20.0069	0.690716	0.345358	+	0.938471i	$0.387758\pi$
$840$	3.53499	0.121969
$841$	−12.3449	−0.425685
$842$	−18.2810	−0.630004
$843$	4.64220	0.159886
$844$	24.7334	0.851359
$845$	−5.17178	−0.177915
$846$	−18.1615	−0.624406
$847$	5.69850	0.195803
$848$	−0.866226	−0.0297463
$849$	35.6592	1.22382
$850$	29.5499	1.01355
$851$	−0.742446	−0.0254507
$852$	17.2550	0.591148
$853$	−19.8367	−0.679195	−0.339598	−	0.940571i	$-0.610291\pi$
$853$	−19.8367	−0.679195	−0.339598	+	0.940571i	$0.610291\pi$
$854$	−14.9112	−0.510251
$855$	3.49992	0.119695
$856$	−0.985850	−0.0336957
$857$	−48.5645	−1.65893	−0.829466	−	0.558558i	$-0.811355\pi$
$857$	−48.5645	−1.65893	−0.829466	+	0.558558i	$0.811355\pi$
$858$	−30.7696	−1.05046
$859$	17.2958	0.590124	0.295062	−	0.955478i	$-0.404660\pi$
$859$	17.2958	0.590124	0.295062	+	0.955478i	$0.404660\pi$
$860$	−0.0120100	−0.000409539 0
$861$	32.0211	1.09128
$862$	3.10782	0.105853
$863$	−12.0364	−0.409725	−0.204863	−	0.978791i	$-0.565675\pi$
$863$	−12.0364	−0.409725	−0.204863	+	0.978791i	$0.565675\pi$
$864$	−0.505268	−0.0171896
$865$	−17.1841	−0.584276
$866$	−37.0148	−1.25782
$867$	−64.6261	−2.19482
$868$	12.1641	0.412875
$869$	−20.9270	−0.709898
$870$	−7.20262	−0.244192
$871$	0.959463	0.0325101
$872$	−11.1537	−0.377711
$873$	−31.6433	−1.07096
$874$	8.44856	0.285777
$875$	−13.9006	−0.469925
$876$	−33.6923	−1.13836
$877$	−5.39148	−0.182057	−0.0910287	−	0.995848i	$-0.529015\pi$
$877$	−5.39148	−0.182057	−0.0910287	+	0.995848i	$0.529015\pi$
$878$	24.8220	0.837703
$879$	20.2746	0.683846
$880$	−2.09453	−0.0706067
$881$	40.0699	1.34999	0.674995	−	0.737823i	$-0.264146\pi$
$881$	40.0699	1.34999	0.674995	+	0.737823i	$0.264146\pi$
$882$	8.33702	0.280722
$883$	−47.4404	−1.59650	−0.798248	−	0.602329i	$-0.794240\pi$
$883$	−47.4404	−1.59650	−0.798248	+	0.602329i	$0.794240\pi$
$884$	−29.6547	−0.997394
$885$	−21.2356	−0.713828
$886$	−11.2870	−0.379195
$887$	−26.8737	−0.902331	−0.451165	−	0.892440i	$-0.648992\pi$
$887$	−26.8737	−0.902331	−0.451165	+	0.892440i	$0.648992\pi$
$888$	−0.361657	−0.0121364
$889$	31.7051	1.06335
$890$	−7.58498	−0.254249
$891$	−27.3742	−0.917071
$892$	−20.3986	−0.682997
$893$	−11.1332	−0.372558
$894$	−13.6409	−0.456220
$895$	−8.87691	−0.296722
$896$	2.00296	0.0669141
$897$	53.2253	1.77714
$898$	−9.08416	−0.303142
$899$	−24.7846	−0.826612
$900$	−12.4492	−0.414972
$901$	−5.73659	−0.191113
$902$	−18.9730	−0.631731
$903$	0.0789186	0.00262625
$904$	14.7077	0.489172
$905$	5.93116	0.197158
$906$	−41.2024	−1.36886
$907$	28.7177	0.953556	0.476778	−	0.879024i	$-0.341805\pi$
$907$	28.7177	0.953556	0.476778	+	0.879024i	$0.341805\pi$
$908$	−14.4031	−0.477982
$909$	53.7513	1.78282
$910$	6.57837	0.218071
$911$	42.9966	1.42454	0.712270	−	0.701906i	$-0.247667\pi$
$911$	42.9966	1.42454	0.712270	+	0.701906i	$0.247667\pi$
$912$	4.11543	0.136275
$913$	36.3573	1.20325
$914$	−18.0217	−0.596105
$915$	−13.1388	−0.434357
$916$	15.3323	0.506594
$917$	30.7106	1.01415
$918$	−3.34615	−0.110439
$919$	19.5484	0.644842	0.322421	−	0.946596i	$-0.395503\pi$
$919$	19.5484	0.644842	0.322421	+	0.946596i	$0.395503\pi$
$920$	3.62313	0.119451
$921$	−77.7828	−2.56303
$922$	−34.1247	−1.12384
$923$	32.1104	1.05693
$924$	13.7633	0.452779
$925$	0.670641	0.0220506
$926$	3.51932	0.115652
$927$	22.0557	0.724405
$928$	−4.08107	−0.133968
$929$	−44.0495	−1.44522	−0.722608	−	0.691258i	$-0.757057\pi$
$929$	−44.0495	−1.44522	−0.722608	+	0.691258i	$0.757057\pi$
$930$	10.7182	0.351465
$931$	5.11068	0.167496
$932$	−2.72756	−0.0893442
$933$	29.9092	0.979183
$934$	7.75729	0.253826
$935$	−13.8711	−0.453632
$936$	12.4933	0.408356
$937$	20.7686	0.678480	0.339240	−	0.940700i	$-0.389830\pi$
$937$	20.7686	0.678480	0.339240	+	0.940700i	$0.389830\pi$
$938$	−0.429170	−0.0140129
$939$	12.4978	0.407851
$940$	−4.77443	−0.155725
$941$	−1.75525	−0.0572194	−0.0286097	−	0.999591i	$-0.509108\pi$
$941$	−1.75525	−0.0572194	−0.0286097	+	0.999591i	$0.509108\pi$
$942$	15.3246	0.499302
$943$	32.8195	1.06875
$944$	−12.0323	−0.391618
$945$	0.742284	0.0241465
$946$	−0.0467605	−0.00152031
$947$	−58.9176	−1.91457	−0.957283	−	0.289152i	$-0.906627\pi$
$947$	−58.9176	−1.91457	−0.957283	+	0.289152i	$0.906627\pi$
$948$	17.6334	0.572705
$949$	−62.6991	−2.03530
$950$	−7.63147	−0.247598
$951$	−45.9259	−1.48925
$952$	13.2646	0.429908
$953$	33.2334	1.07653	0.538267	−	0.842774i	$-0.319079\pi$
$953$	33.2334	1.07653	0.538267	+	0.842774i	$0.319079\pi$
$954$	2.41678	0.0782463
$955$	−3.94394	−0.127623
$956$	−2.02696	−0.0655566
$957$	−28.0430	−0.906503
$958$	−37.4969	−1.21147
$959$	−0.935812	−0.0302189
$960$	1.76488	0.0569614
$961$	5.88199	0.189741
$962$	−0.673019	−0.0216990
$963$	2.75054	0.0886349
$964$	−7.18086	−0.231280
$965$	−0.163851	−0.00527454
$966$	−23.8078	−0.766004
$967$	−2.49118	−0.0801110	−0.0400555	−	0.999197i	$-0.512753\pi$
$967$	−2.49118	−0.0801110	−0.0400555	+	0.999197i	$0.512753\pi$
$968$	2.84504	0.0914431
$969$	27.2545	0.875539
$970$	−8.31861	−0.267095
$971$	−45.9289	−1.47393	−0.736965	−	0.675931i	$-0.763742\pi$
$971$	−45.9289	−1.47393	−0.736965	+	0.675931i	$0.763742\pi$
$972$	21.5501	0.691220
$973$	−3.94733	−0.126545
$974$	−7.69191	−0.246465
$975$	−48.0777	−1.53972
$976$	−7.44459	−0.238296
$977$	−47.7697	−1.52829	−0.764145	−	0.645045i	$-0.776839\pi$
$977$	−47.7697	−1.52829	−0.764145	+	0.645045i	$0.776839\pi$
$978$	−49.0123	−1.56724
$979$	−29.5317	−0.943839
$980$	2.19169	0.0700111
$981$	31.1189	0.993551
$982$	−26.6291	−0.849768
$983$	51.1291	1.63077	0.815383	−	0.578922i	$-0.196526\pi$
$983$	51.1291	1.63077	0.815383	+	0.578922i	$0.196526\pi$
$984$	15.9869	0.509644
$985$	−7.44843	−0.237327
$986$	−27.0269	−0.860713
$987$	31.3731	0.998616
$988$	7.65853	0.243650
$989$	0.0808864	0.00257204
$990$	5.84378	0.185728
$991$	−46.6079	−1.48055	−0.740274	−	0.672305i	$-0.765304\pi$
$991$	−46.6079	−1.48055	−0.740274	+	0.672305i	$0.765304\pi$
$992$	6.07305	0.192820
$993$	−48.2878	−1.53237
$994$	−14.3631	−0.455569
$995$	12.2809	0.389329
$996$	−30.6352	−0.970714
$997$	35.4931	1.12408	0.562038	−	0.827111i	$-0.310017\pi$
$997$	35.4931	1.12408	0.562038	+	0.827111i	$0.310017\pi$
$998$	19.5968	0.620326
$999$	−0.0759415	−0.00240268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.d.1.9	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.d.1.9	✓	69	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.40625 q^{3} +1.00000 q^{4} -0.733460 q^{5} +2.40625 q^{6} -2.00296 q^{7} -1.00000 q^{8} +2.79002 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.40625 q^{3} +1.00000 q^{4} -0.733460 q^{5} +2.40625 q^{6} -2.00296 q^{7} -1.00000 q^{8} +2.79002 q^{9} +0.733460 q^{10} +2.85569 q^{11} -2.40625 q^{12} -4.47786 q^{13} +2.00296 q^{14} +1.76488 q^{15} +1.00000 q^{16} +6.62251 q^{17} -2.79002 q^{18} -1.71031 q^{19} -0.733460 q^{20} +4.81961 q^{21} -2.85569 q^{22} +4.93978 q^{23} +2.40625 q^{24} -4.46204 q^{25} +4.47786 q^{26} +0.505268 q^{27} -2.00296 q^{28} +4.08107 q^{29} -1.76488 q^{30} -6.07305 q^{31} -1.00000 q^{32} -6.87149 q^{33} -6.62251 q^{34} +1.46909 q^{35} +2.79002 q^{36} -0.150299 q^{37} +1.71031 q^{38} +10.7748 q^{39} +0.733460 q^{40} +6.64393 q^{41} -4.81961 q^{42} +0.0163745 q^{43} +2.85569 q^{44} -2.04637 q^{45} -4.93978 q^{46} +6.50946 q^{47} -2.40625 q^{48} -2.98816 q^{49} +4.46204 q^{50} -15.9354 q^{51} -4.47786 q^{52} -0.866226 q^{53} -0.505268 q^{54} -2.09453 q^{55} +2.00296 q^{56} +4.11543 q^{57} -4.08107 q^{58} -12.0323 q^{59} +1.76488 q^{60} -7.44459 q^{61} +6.07305 q^{62} -5.58829 q^{63} +1.00000 q^{64} +3.28433 q^{65} +6.87149 q^{66} -0.214268 q^{67} +6.62251 q^{68} -11.8863 q^{69} -1.46909 q^{70} -7.17094 q^{71} -2.79002 q^{72} +14.0020 q^{73} +0.150299 q^{74} +10.7368 q^{75} -1.71031 q^{76} -5.71982 q^{77} -10.7748 q^{78} -7.32816 q^{79} -0.733460 q^{80} -9.58585 q^{81} -6.64393 q^{82} +12.7315 q^{83} +4.81961 q^{84} -4.85734 q^{85} -0.0163745 q^{86} -9.82006 q^{87} -2.85569 q^{88} -10.3414 q^{89} +2.04637 q^{90} +8.96896 q^{91} +4.93978 q^{92} +14.6133 q^{93} -6.50946 q^{94} +1.25444 q^{95} +2.40625 q^{96} -11.3416 q^{97} +2.98816 q^{98} +7.96742 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * q^99

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