LMFDB - Embedded newform 6046.2.a.f.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6046,2,Mod(1,6046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6046, 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("6046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6046 = 2 \cdot 3023 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6046.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:	$48.2775530621$
Analytic rank:	$0$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	6046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.41094 q^{3} +1.00000 q^{4} +0.848657 q^{5} -2.41094 q^{6} -1.16640 q^{7} +1.00000 q^{8} +2.81262 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.41094 q^{3} +1.00000 q^{4} +0.848657 q^{5} -2.41094 q^{6} -1.16640 q^{7} +1.00000 q^{8} +2.81262 q^{9} +0.848657 q^{10} +6.18894 q^{11} -2.41094 q^{12} +4.65645 q^{13} -1.16640 q^{14} -2.04606 q^{15} +1.00000 q^{16} +7.47114 q^{17} +2.81262 q^{18} +6.33992 q^{19} +0.848657 q^{20} +2.81211 q^{21} +6.18894 q^{22} -3.85585 q^{23} -2.41094 q^{24} -4.27978 q^{25} +4.65645 q^{26} +0.451765 q^{27} -1.16640 q^{28} -6.00987 q^{29} -2.04606 q^{30} +8.23626 q^{31} +1.00000 q^{32} -14.9212 q^{33} +7.47114 q^{34} -0.989871 q^{35} +2.81262 q^{36} +1.03372 q^{37} +6.33992 q^{38} -11.2264 q^{39} +0.848657 q^{40} +7.46899 q^{41} +2.81211 q^{42} -2.30049 q^{43} +6.18894 q^{44} +2.38695 q^{45} -3.85585 q^{46} +8.82762 q^{47} -2.41094 q^{48} -5.63952 q^{49} -4.27978 q^{50} -18.0125 q^{51} +4.65645 q^{52} +1.37934 q^{53} +0.451765 q^{54} +5.25229 q^{55} -1.16640 q^{56} -15.2851 q^{57} -6.00987 q^{58} +3.48597 q^{59} -2.04606 q^{60} -0.648486 q^{61} +8.23626 q^{62} -3.28063 q^{63} +1.00000 q^{64} +3.95173 q^{65} -14.9212 q^{66} -11.8327 q^{67} +7.47114 q^{68} +9.29622 q^{69} -0.989871 q^{70} -11.1178 q^{71} +2.81262 q^{72} -15.6374 q^{73} +1.03372 q^{74} +10.3183 q^{75} +6.33992 q^{76} -7.21876 q^{77} -11.2264 q^{78} +11.3574 q^{79} +0.848657 q^{80} -9.52703 q^{81} +7.46899 q^{82} -2.61838 q^{83} +2.81211 q^{84} +6.34044 q^{85} -2.30049 q^{86} +14.4894 q^{87} +6.18894 q^{88} -13.4885 q^{89} +2.38695 q^{90} -5.43126 q^{91} -3.85585 q^{92} -19.8571 q^{93} +8.82762 q^{94} +5.38042 q^{95} -2.41094 q^{96} +14.8873 q^{97} -5.63952 q^{98} +17.4071 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9 + 21 * q^10 + 56 * q^11 + 21 * q^12 + 33 * q^13 + 38 * q^14 + 25 * q^15 + 67 * q^16 + 30 * q^17 + 90 * q^18 + 36 * q^19 + 21 * q^20 + 20 * q^21 + 56 * q^22 + 65 * q^23 + 21 * q^24 + 72 * q^25 + 33 * q^26 + 57 * q^27 + 38 * q^28 + 84 * q^29 + 25 * q^30 + 52 * q^31 + 67 * q^32 - 9 * q^33 + 30 * q^34 + 30 * q^35 + 90 * q^36 + 52 * q^37 + 36 * q^38 + 41 * q^39 + 21 * q^40 + 46 * q^41 + 20 * q^42 + 61 * q^43 + 56 * q^44 + 23 * q^45 + 65 * q^46 + 51 * q^47 + 21 * q^48 + 81 * q^49 + 72 * q^50 + 33 * q^51 + 33 * q^52 + 72 * q^53 + 57 * q^54 + 14 * q^55 + 38 * q^56 - 26 * q^57 + 84 * q^58 + 71 * q^59 + 25 * q^60 + 42 * q^61 + 52 * q^62 + 63 * q^63 + 67 * q^64 - 2 * q^65 - 9 * q^66 + 70 * q^67 + 30 * q^68 + 21 * q^69 + 30 * q^70 + 104 * q^71 + 90 * q^72 - 31 * q^73 + 52 * q^74 + 69 * q^75 + 36 * q^76 + 48 * q^77 + 41 * q^78 + 79 * q^79 + 21 * q^80 + 123 * q^81 + 46 * q^82 + 41 * q^83 + 20 * q^84 + 6 * q^85 + 61 * q^86 + 19 * q^87 + 56 * q^88 + 58 * q^89 + 23 * q^90 + 31 * q^91 + 65 * q^92 + 13 * q^93 + 51 * q^94 + 77 * q^95 + 21 * q^96 - 8 * q^97 + 81 * q^98 + 129 * 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
$3$	−2.41094	−1.39196	−0.695978	−	0.718063i	$-0.745029\pi$
$3$	−2.41094	−1.39196	−0.695978	+	0.718063i	$0.745029\pi$
$4$	1.00000	0.500000
$5$	0.848657	0.379531	0.189766	−	0.981829i	$-0.439227\pi$
$5$	0.848657	0.379531	0.189766	+	0.981829i	$0.439227\pi$
$6$	−2.41094	−0.984261
$7$	−1.16640	−0.440856	−0.220428	−	0.975403i	$-0.570745\pi$
$7$	−1.16640	−0.440856	−0.220428	+	0.975403i	$0.570745\pi$
$8$	1.00000	0.353553
$9$	2.81262	0.937540
$10$	0.848657	0.268369
$11$	6.18894	1.86604	0.933019	−	0.359828i	$-0.117165\pi$
$11$	6.18894	1.86604	0.933019	+	0.359828i	$0.117165\pi$
$12$	−2.41094	−0.695978
$13$	4.65645	1.29147	0.645733	−	0.763563i	$-0.276552\pi$
$13$	4.65645	1.29147	0.645733	+	0.763563i	$0.276552\pi$
$14$	−1.16640	−0.311733
$15$	−2.04606	−0.528290
$16$	1.00000	0.250000
$17$	7.47114	1.81202	0.906009	−	0.423258i	$-0.139114\pi$
$17$	7.47114	1.81202	0.906009	+	0.423258i	$0.139114\pi$
$18$	2.81262	0.662941
$19$	6.33992	1.45448	0.727238	−	0.686385i	$-0.240804\pi$
$19$	6.33992	1.45448	0.727238	+	0.686385i	$0.240804\pi$
$20$	0.848657	0.189766
$21$	2.81211	0.613652
$22$	6.18894	1.31949
$23$	−3.85585	−0.804001	−0.402001	−	0.915639i	$-0.631685\pi$
$23$	−3.85585	−0.804001	−0.402001	+	0.915639i	$0.631685\pi$
$24$	−2.41094	−0.492131
$25$	−4.27978	−0.855956
$26$	4.65645	0.913205
$27$	0.451765	0.0869421
$28$	−1.16640	−0.220428
$29$	−6.00987	−1.11601	−0.558003	−	0.829839i	$-0.688432\pi$
$29$	−6.00987	−1.11601	−0.558003	+	0.829839i	$0.688432\pi$
$30$	−2.04606	−0.373558
$31$	8.23626	1.47928	0.739638	−	0.673005i	$-0.234997\pi$
$31$	8.23626	1.47928	0.739638	+	0.673005i	$0.234997\pi$
$32$	1.00000	0.176777
$33$	−14.9212	−2.59744
$34$	7.47114	1.28129
$35$	−0.989871	−0.167319
$36$	2.81262	0.468770
$37$	1.03372	0.169942	0.0849711	−	0.996383i	$-0.472920\pi$
$37$	1.03372	0.169942	0.0849711	+	0.996383i	$0.472920\pi$
$38$	6.33992	1.02847
$39$	−11.2264	−1.79766
$40$	0.848657	0.134184
$41$	7.46899	1.16646	0.583230	−	0.812307i	$-0.301788\pi$
$41$	7.46899	1.16646	0.583230	+	0.812307i	$0.301788\pi$
$42$	2.81211	0.433918
$43$	−2.30049	−0.350821	−0.175411	−	0.984495i	$-0.556125\pi$
$43$	−2.30049	−0.350821	−0.175411	+	0.984495i	$0.556125\pi$
$44$	6.18894	0.933019
$45$	2.38695	0.355825
$46$	−3.85585	−0.568515
$47$	8.82762	1.28764	0.643820	−	0.765177i	$-0.277348\pi$
$47$	8.82762	1.28764	0.643820	+	0.765177i	$0.277348\pi$
$48$	−2.41094	−0.347989
$49$	−5.63952	−0.805646
$50$	−4.27978	−0.605252
$51$	−18.0125	−2.52225
$52$	4.65645	0.645733
$53$	1.37934	0.189467	0.0947333	−	0.995503i	$-0.469800\pi$
$53$	1.37934	0.189467	0.0947333	+	0.995503i	$0.469800\pi$
$54$	0.451765	0.0614774
$55$	5.25229	0.708219
$56$	−1.16640	−0.155866
$57$	−15.2851	−2.02457
$58$	−6.00987	−0.789135
$59$	3.48597	0.453835	0.226917	−	0.973914i	$-0.427135\pi$
$59$	3.48597	0.453835	0.226917	+	0.973914i	$0.427135\pi$
$60$	−2.04606	−0.264145
$61$	−0.648486	−0.0830301	−0.0415151	−	0.999138i	$-0.513218\pi$
$61$	−0.648486	−0.0830301	−0.0415151	+	0.999138i	$0.513218\pi$
$62$	8.23626	1.04601
$63$	−3.28063	−0.413320
$64$	1.00000	0.125000
$65$	3.95173	0.490152
$66$	−14.9212	−1.83667
$67$	−11.8327	−1.44560	−0.722798	−	0.691059i	$-0.757144\pi$
$67$	−11.8327	−1.44560	−0.722798	+	0.691059i	$0.757144\pi$
$68$	7.47114	0.906009
$69$	9.29622	1.11913
$70$	−0.989871	−0.118312
$71$	−11.1178	−1.31944	−0.659722	−	0.751509i	$-0.729326\pi$
$71$	−11.1178	−1.31944	−0.659722	+	0.751509i	$0.729326\pi$
$72$	2.81262	0.331470
$73$	−15.6374	−1.83022	−0.915109	−	0.403206i	$-0.867896\pi$
$73$	−15.6374	−1.83022	−0.915109	+	0.403206i	$0.867896\pi$
$74$	1.03372	0.120167
$75$	10.3183	1.19145
$76$	6.33992	0.727238
$77$	−7.21876	−0.822654
$78$	−11.2264	−1.27114
$79$	11.3574	1.27781	0.638904	−	0.769286i	$-0.279388\pi$
$79$	11.3574	1.27781	0.638904	+	0.769286i	$0.279388\pi$
$80$	0.848657	0.0948828
$81$	−9.52703	−1.05856
$82$	7.46899	0.824812
$83$	−2.61838	−0.287404	−0.143702	−	0.989621i	$-0.545901\pi$
$83$	−2.61838	−0.287404	−0.143702	+	0.989621i	$0.545901\pi$
$84$	2.81211	0.306826
$85$	6.34044	0.687717
$86$	−2.30049	−0.248068
$87$	14.4894	1.55343
$88$	6.18894	0.659744
$89$	−13.4885	−1.42978	−0.714889	−	0.699238i	$-0.753523\pi$
$89$	−13.4885	−1.42978	−0.714889	+	0.699238i	$0.753523\pi$
$90$	2.38695	0.251607
$91$	−5.43126	−0.569351
$92$	−3.85585	−0.402001
$93$	−19.8571	−2.05909
$94$	8.82762	0.910499
$95$	5.38042	0.552019
$96$	−2.41094	−0.246065
$97$	14.8873	1.51158	0.755789	−	0.654815i	$-0.227253\pi$
$97$	14.8873	1.51158	0.755789	+	0.654815i	$0.227253\pi$
$98$	−5.63952	−0.569678
$99$	17.4071	1.74948
$100$	−4.27978	−0.427978
$101$	−0.979701	−0.0974839	−0.0487420	−	0.998811i	$-0.515521\pi$
$101$	−0.979701	−0.0974839	−0.0487420	+	0.998811i	$0.515521\pi$
$102$	−18.0125	−1.78350
$103$	−11.9547	−1.17793	−0.588965	−	0.808159i	$-0.700464\pi$
$103$	−11.9547	−1.17793	−0.588965	+	0.808159i	$0.700464\pi$
$104$	4.65645	0.456602
$105$	2.38652	0.232900
$106$	1.37934	0.133973
$107$	−5.76300	−0.557130	−0.278565	−	0.960417i	$-0.589859\pi$
$107$	−5.76300	−0.557130	−0.278565	+	0.960417i	$0.589859\pi$
$108$	0.451765	0.0434711
$109$	11.3105	1.08335	0.541674	−	0.840589i	$-0.317791\pi$
$109$	11.3105	1.08335	0.541674	+	0.840589i	$0.317791\pi$
$110$	5.25229	0.500786
$111$	−2.49223	−0.236552
$112$	−1.16640	−0.110214
$113$	−2.12662	−0.200056	−0.100028	−	0.994985i	$-0.531893\pi$
$113$	−2.12662	−0.200056	−0.100028	+	0.994985i	$0.531893\pi$
$114$	−15.2851	−1.43158
$115$	−3.27230	−0.305143
$116$	−6.00987	−0.558003
$117$	13.0968	1.21080
$118$	3.48597	0.320909
$119$	−8.71431	−0.798840
$120$	−2.04606	−0.186779
$121$	27.3030	2.48209
$122$	−0.648486	−0.0587112
$123$	−18.0073	−1.62366
$124$	8.23626	0.739638
$125$	−7.87535	−0.704393
$126$	−3.28063	−0.292262
$127$	13.1106	1.16338	0.581690	−	0.813411i	$-0.302392\pi$
$127$	13.1106	1.16338	0.581690	+	0.813411i	$0.302392\pi$
$128$	1.00000	0.0883883
$129$	5.54633	0.488327
$130$	3.95173	0.346589
$131$	−2.05321	−0.179390	−0.0896949	−	0.995969i	$-0.528589\pi$
$131$	−2.05321	−0.179390	−0.0896949	+	0.995969i	$0.528589\pi$
$132$	−14.9212	−1.29872
$133$	−7.39485	−0.641215
$134$	−11.8327	−1.02219
$135$	0.383393	0.0329972
$136$	7.47114	0.640645
$137$	19.9906	1.70791	0.853956	−	0.520345i	$-0.174197\pi$
$137$	19.9906	1.70791	0.853956	+	0.520345i	$0.174197\pi$
$138$	9.29622	0.791347
$139$	−0.592949	−0.0502933	−0.0251466	−	0.999684i	$-0.508005\pi$
$139$	−0.592949	−0.0502933	−0.0251466	+	0.999684i	$0.508005\pi$
$140$	−0.989871	−0.0836593
$141$	−21.2828	−1.79234
$142$	−11.1178	−0.932988
$143$	28.8185	2.40992
$144$	2.81262	0.234385
$145$	−5.10032	−0.423559
$146$	−15.6374	−1.29416
$147$	13.5965	1.12142
$148$	1.03372	0.0849711
$149$	−6.29929	−0.516058	−0.258029	−	0.966137i	$-0.583073\pi$
$149$	−6.29929	−0.516058	−0.258029	+	0.966137i	$0.583073\pi$
$150$	10.3183	0.842484
$151$	−15.3599	−1.24997	−0.624985	−	0.780637i	$-0.714895\pi$
$151$	−15.3599	−1.24997	−0.624985	+	0.780637i	$0.714895\pi$
$152$	6.33992	0.514235
$153$	21.0135	1.69884
$154$	−7.21876	−0.581704
$155$	6.98976	0.561431
$156$	−11.2264	−0.898832
$157$	−4.04883	−0.323132	−0.161566	−	0.986862i	$-0.551654\pi$
$157$	−4.04883	−0.323132	−0.161566	+	0.986862i	$0.551654\pi$
$158$	11.3574	0.903547
$159$	−3.32550	−0.263729
$160$	0.848657	0.0670922
$161$	4.49745	0.354449
$162$	−9.52703	−0.748514
$163$	15.9812	1.25175	0.625874	−	0.779924i	$-0.284743\pi$
$163$	15.9812	1.25175	0.625874	+	0.779924i	$0.284743\pi$
$164$	7.46899	0.583230
$165$	−12.6629	−0.985809
$166$	−2.61838	−0.203225
$167$	−8.57680	−0.663692	−0.331846	−	0.943333i	$-0.607672\pi$
$167$	−8.57680	−0.663692	−0.331846	+	0.943333i	$0.607672\pi$
$168$	2.81211	0.216959
$169$	8.68251	0.667885
$170$	6.34044	0.486290
$171$	17.8318	1.36363
$172$	−2.30049	−0.175411
$173$	−11.4463	−0.870243	−0.435122	−	0.900372i	$-0.643295\pi$
$173$	−11.4463	−0.870243	−0.435122	+	0.900372i	$0.643295\pi$
$174$	14.4894	1.09844
$175$	4.99192	0.377354
$176$	6.18894	0.466509
$177$	−8.40445	−0.631717
$178$	−13.4885	−1.01101
$179$	11.3748	0.850192	0.425096	−	0.905148i	$-0.360240\pi$
$179$	11.3748	0.850192	0.425096	+	0.905148i	$0.360240\pi$
$180$	2.38695	0.177913
$181$	−8.68464	−0.645525	−0.322762	−	0.946480i	$-0.604611\pi$
$181$	−8.68464	−0.645525	−0.322762	+	0.946480i	$0.604611\pi$
$182$	−5.43126	−0.402592
$183$	1.56346	0.115574
$184$	−3.85585	−0.284257
$185$	0.877272	0.0644983
$186$	−19.8571	−1.45599
$187$	46.2385	3.38129
$188$	8.82762	0.643820
$189$	−0.526936	−0.0383290
$190$	5.38042	0.390336
$191$	19.0546	1.37874	0.689372	−	0.724408i	$-0.257887\pi$
$191$	19.0546	1.37874	0.689372	+	0.724408i	$0.257887\pi$
$192$	−2.41094	−0.173994
$193$	−2.06479	−0.148627	−0.0743133	−	0.997235i	$-0.523676\pi$
$193$	−2.06479	−0.148627	−0.0743133	+	0.997235i	$0.523676\pi$
$194$	14.8873	1.06885
$195$	−9.52737	−0.682269
$196$	−5.63952	−0.402823
$197$	−20.3330	−1.44867	−0.724333	−	0.689451i	$-0.757852\pi$
$197$	−20.3330	−1.44867	−0.724333	+	0.689451i	$0.757852\pi$
$198$	17.4071	1.23707
$199$	14.0311	0.994642	0.497321	−	0.867567i	$-0.334317\pi$
$199$	14.0311	0.994642	0.497321	+	0.867567i	$0.334317\pi$
$200$	−4.27978	−0.302626
$201$	28.5279	2.01220
$202$	−0.979701	−0.0689316
$203$	7.00989	0.491998
$204$	−18.0125	−1.26112
$205$	6.33862	0.442708
$206$	−11.9547	−0.832922
$207$	−10.8450	−0.753783
$208$	4.65645	0.322867
$209$	39.2374	2.71411
$210$	2.38652	0.164685
$211$	−18.6381	−1.28310	−0.641551	−	0.767081i	$-0.721709\pi$
$211$	−18.6381	−1.28310	−0.641551	+	0.767081i	$0.721709\pi$
$212$	1.37934	0.0947333
$213$	26.8044	1.83661
$214$	−5.76300	−0.393950
$215$	−1.95233	−0.133148
$216$	0.451765	0.0307387
$217$	−9.60675	−0.652148
$218$	11.3105	0.766043
$219$	37.7008	2.54758
$220$	5.25229	0.354110
$221$	34.7890	2.34016
$222$	−2.49223	−0.167267
$223$	6.24146	0.417959	0.208979	−	0.977920i	$-0.432986\pi$
$223$	6.24146	0.417959	0.208979	+	0.977920i	$0.432986\pi$
$224$	−1.16640	−0.0779331
$225$	−12.0374	−0.802493
$226$	−2.12662	−0.141461
$227$	19.1127	1.26855	0.634277	−	0.773106i	$-0.281298\pi$
$227$	19.1127	1.26855	0.634277	+	0.773106i	$0.281298\pi$
$228$	−15.2851	−1.01228
$229$	−2.14540	−0.141772	−0.0708860	−	0.997484i	$-0.522583\pi$
$229$	−2.14540	−0.141772	−0.0708860	+	0.997484i	$0.522583\pi$
$230$	−3.27230	−0.215769
$231$	17.4040	1.14510
$232$	−6.00987	−0.394568
$233$	−11.4355	−0.749164	−0.374582	−	0.927194i	$-0.622214\pi$
$233$	−11.4355	−0.749164	−0.374582	+	0.927194i	$0.622214\pi$
$234$	13.0968	0.856165
$235$	7.49162	0.488700
$236$	3.48597	0.226917
$237$	−27.3820	−1.77865
$238$	−8.71431	−0.564865
$239$	6.95062	0.449598	0.224799	−	0.974405i	$-0.427827\pi$
$239$	6.95062	0.449598	0.224799	+	0.974405i	$0.427827\pi$
$240$	−2.04606	−0.132073
$241$	−12.2637	−0.789977	−0.394989	−	0.918686i	$-0.629251\pi$
$241$	−12.2637	−0.789977	−0.394989	+	0.918686i	$0.629251\pi$
$242$	27.3030	1.75511
$243$	21.6138	1.38652
$244$	−0.648486	−0.0415151
$245$	−4.78602	−0.305768
$246$	−18.0073	−1.14810
$247$	29.5215	1.87841
$248$	8.23626	0.523003
$249$	6.31274	0.400054
$250$	−7.87535	−0.498081
$251$	22.0689	1.39298	0.696490	−	0.717567i	$-0.254744\pi$
$251$	22.0689	1.39298	0.696490	+	0.717567i	$0.254744\pi$
$252$	−3.28063	−0.206660
$253$	−23.8637	−1.50030
$254$	13.1106	0.822634
$255$	−15.2864	−0.957272
$256$	1.00000	0.0625000
$257$	−24.0264	−1.49873	−0.749363	−	0.662160i	$-0.769640\pi$
$257$	−24.0264	−1.49873	−0.749363	+	0.662160i	$0.769640\pi$
$258$	5.54633	0.345300
$259$	−1.20572	−0.0749201
$260$	3.95173	0.245076
$261$	−16.9035	−1.04630
$262$	−2.05321	−0.126848
$263$	−21.8953	−1.35012	−0.675061	−	0.737762i	$-0.735883\pi$
$263$	−21.8953	−1.35012	−0.675061	+	0.737762i	$0.735883\pi$
$264$	−14.9212	−0.918334
$265$	1.17058	0.0719084
$266$	−7.39485	−0.453408
$267$	32.5199	1.99019
$268$	−11.8327	−0.722798
$269$	1.08143	0.0659358	0.0329679	−	0.999456i	$-0.489504\pi$
$269$	1.08143	0.0659358	0.0329679	+	0.999456i	$0.489504\pi$
$270$	0.383393	0.0233326
$271$	−1.54996	−0.0941534	−0.0470767	−	0.998891i	$-0.514991\pi$
$271$	−1.54996	−0.0941534	−0.0470767	+	0.998891i	$0.514991\pi$
$272$	7.47114	0.453005
$273$	13.0944	0.792511
$274$	19.9906	1.20768
$275$	−26.4873	−1.59725
$276$	9.29622	0.559567
$277$	10.5069	0.631300	0.315650	−	0.948876i	$-0.397778\pi$
$277$	10.5069	0.631300	0.315650	+	0.948876i	$0.397778\pi$
$278$	−0.592949	−0.0355627
$279$	23.1655	1.38688
$280$	−0.989871	−0.0591561
$281$	16.8033	1.00240	0.501200	−	0.865332i	$-0.332892\pi$
$281$	16.8033	1.00240	0.501200	+	0.865332i	$0.332892\pi$
$282$	−21.2828	−1.26737
$283$	4.18494	0.248769	0.124384	−	0.992234i	$-0.460304\pi$
$283$	4.18494	0.248769	0.124384	+	0.992234i	$0.460304\pi$
$284$	−11.1178	−0.659722
$285$	−12.9718	−0.768386
$286$	28.8185	1.70407
$287$	−8.71181	−0.514242
$288$	2.81262	0.165735
$289$	38.8180	2.28341
$290$	−5.10032	−0.299501
$291$	−35.8924	−2.10405
$292$	−15.6374	−0.915109
$293$	−15.3173	−0.894849	−0.447425	−	0.894322i	$-0.647659\pi$
$293$	−15.3173	−0.894849	−0.447425	+	0.894322i	$0.647659\pi$
$294$	13.5965	0.792966
$295$	2.95839	0.172244
$296$	1.03372	0.0600836
$297$	2.79595	0.162237
$298$	−6.29929	−0.364908
$299$	−17.9546	−1.03834
$300$	10.3183	0.595726
$301$	2.68328	0.154662
$302$	−15.3599	−0.883862
$303$	2.36200	0.135693
$304$	6.33992	0.363619
$305$	−0.550342	−0.0315125
$306$	21.0135	1.20126
$307$	−11.3461	−0.647557	−0.323779	−	0.946133i	$-0.604953\pi$
$307$	−11.3461	−0.647557	−0.323779	+	0.946133i	$0.604953\pi$
$308$	−7.21876	−0.411327
$309$	28.8220	1.63963
$310$	6.98976	0.396992
$311$	23.3510	1.32412	0.662058	−	0.749453i	$-0.269683\pi$
$311$	23.3510	1.32412	0.662058	+	0.749453i	$0.269683\pi$
$312$	−11.2264	−0.635570
$313$	−27.6344	−1.56199	−0.780994	−	0.624539i	$-0.785287\pi$
$313$	−27.6344	−1.56199	−0.780994	+	0.624539i	$0.785287\pi$
$314$	−4.04883	−0.228489
$315$	−2.78413	−0.156868
$316$	11.3574	0.638904
$317$	−18.1025	−1.01674	−0.508370	−	0.861139i	$-0.669752\pi$
$317$	−18.1025	−1.01674	−0.508370	+	0.861139i	$0.669752\pi$
$318$	−3.32550	−0.186485
$319$	−37.1948	−2.08251
$320$	0.848657	0.0474414
$321$	13.8942	0.775500
$322$	4.49745	0.250633
$323$	47.3664	2.63554
$324$	−9.52703	−0.529280
$325$	−19.9286	−1.10544
$326$	15.9812	0.885119
$327$	−27.2689	−1.50797
$328$	7.46899	0.412406
$329$	−10.2965	−0.567665
$330$	−12.6629	−0.697072
$331$	−32.7554	−1.80040	−0.900200	−	0.435478i	$-0.856579\pi$
$331$	−32.7554	−1.80040	−0.900200	+	0.435478i	$0.856579\pi$
$332$	−2.61838	−0.143702
$333$	2.90745	0.159327
$334$	−8.57680	−0.469301
$335$	−10.0419	−0.548649
$336$	2.81211	0.153413
$337$	−31.5378	−1.71797	−0.858986	−	0.512000i	$-0.828905\pi$
$337$	−31.5378	−1.71797	−0.858986	+	0.512000i	$0.828905\pi$
$338$	8.68251	0.472266
$339$	5.12715	0.278469
$340$	6.34044	0.343859
$341$	50.9738	2.76038
$342$	17.8318	0.964231
$343$	14.7427	0.796030
$344$	−2.30049	−0.124034
$345$	7.88931	0.424746
$346$	−11.4463	−0.615355
$347$	−1.80014	−0.0966365	−0.0483182	−	0.998832i	$-0.515386\pi$
$347$	−1.80014	−0.0966365	−0.0483182	+	0.998832i	$0.515386\pi$
$348$	14.4894	0.776715
$349$	−29.2851	−1.56760	−0.783798	−	0.621016i	$-0.786720\pi$
$349$	−29.2851	−1.56760	−0.783798	+	0.621016i	$0.786720\pi$
$350$	4.99192	0.266829
$351$	2.10362	0.112283
$352$	6.18894	0.329872
$353$	−5.99786	−0.319234	−0.159617	−	0.987179i	$-0.551026\pi$
$353$	−5.99786	−0.319234	−0.159617	+	0.987179i	$0.551026\pi$
$354$	−8.40445	−0.446692
$355$	−9.43524	−0.500770
$356$	−13.4885	−0.714889
$357$	21.0097	1.11195
$358$	11.3748	0.601177
$359$	33.3789	1.76167	0.880836	−	0.473422i	$-0.156982\pi$
$359$	33.3789	1.76167	0.880836	+	0.473422i	$0.156982\pi$
$360$	2.38695	0.125803
$361$	21.1945	1.11550
$362$	−8.68464	−0.456455
$363$	−65.8259	−3.45496
$364$	−5.43126	−0.284676
$365$	−13.2708	−0.694625
$366$	1.56346	0.0817233
$367$	24.9929	1.30462	0.652308	−	0.757954i	$-0.273801\pi$
$367$	24.9929	1.30462	0.652308	+	0.757954i	$0.273801\pi$
$368$	−3.85585	−0.201000
$369$	21.0074	1.09360
$370$	0.877272	0.0456072
$371$	−1.60885	−0.0835275
$372$	−19.8571	−1.02954
$373$	5.17930	0.268174	0.134087	−	0.990970i	$-0.457190\pi$
$373$	5.17930	0.268174	0.134087	+	0.990970i	$0.457190\pi$
$374$	46.2385	2.39094
$375$	18.9870	0.980484
$376$	8.82762	0.455250
$377$	−27.9847	−1.44128
$378$	−0.526936	−0.0271027
$379$	19.2315	0.987857	0.493929	−	0.869502i	$-0.335560\pi$
$379$	19.2315	0.987857	0.493929	+	0.869502i	$0.335560\pi$
$380$	5.38042	0.276009
$381$	−31.6089	−1.61937
$382$	19.0546	0.974919
$383$	21.6689	1.10723	0.553614	−	0.832773i	$-0.313248\pi$
$383$	21.6689	1.10723	0.553614	+	0.832773i	$0.313248\pi$
$384$	−2.41094	−0.123033
$385$	−6.12626	−0.312223
$386$	−2.06479	−0.105095
$387$	−6.47039	−0.328909
$388$	14.8873	0.755789
$389$	7.90806	0.400955	0.200477	−	0.979698i	$-0.435751\pi$
$389$	7.90806	0.400955	0.200477	+	0.979698i	$0.435751\pi$
$390$	−9.52737	−0.482437
$391$	−28.8076	−1.45686
$392$	−5.63952	−0.284839
$393$	4.95016	0.249703
$394$	−20.3330	−1.02436
$395$	9.63855	0.484968
$396$	17.4071	0.874742
$397$	15.2470	0.765227	0.382613	−	0.923909i	$-0.375024\pi$
$397$	15.2470	0.765227	0.382613	+	0.923909i	$0.375024\pi$
$398$	14.0311	0.703318
$399$	17.8285	0.892543
$400$	−4.27978	−0.213989
$401$	−5.35451	−0.267392	−0.133696	−	0.991022i	$-0.542685\pi$
$401$	−5.35451	−0.267392	−0.133696	+	0.991022i	$0.542685\pi$
$402$	28.5279	1.42284
$403$	38.3517	1.91044
$404$	−0.979701	−0.0487420
$405$	−8.08518	−0.401756
$406$	7.00989	0.347895
$407$	6.39762	0.317118
$408$	−18.0125	−0.891750
$409$	6.77021	0.334765	0.167383	−	0.985892i	$-0.446468\pi$
$409$	6.77021	0.334765	0.167383	+	0.985892i	$0.446468\pi$
$410$	6.33862	0.313042
$411$	−48.1961	−2.37734
$412$	−11.9547	−0.588965
$413$	−4.06602	−0.200076
$414$	−10.8450	−0.533005
$415$	−2.22210	−0.109079
$416$	4.65645	0.228301
$417$	1.42956	0.0700060
$418$	39.2374	1.91916
$419$	−26.6921	−1.30399	−0.651997	−	0.758222i	$-0.726068\pi$
$419$	−26.6921	−1.30399	−0.651997	+	0.758222i	$0.726068\pi$
$420$	2.38652	0.116450
$421$	5.60340	0.273093	0.136547	−	0.990634i	$-0.456400\pi$
$421$	5.60340	0.273093	0.136547	+	0.990634i	$0.456400\pi$
$422$	−18.6381	−0.907290
$423$	24.8287	1.20721
$424$	1.37934	0.0669865
$425$	−31.9749	−1.55101
$426$	26.8044	1.29868
$427$	0.756392	0.0366044
$428$	−5.76300	−0.278565
$429$	−69.4796	−3.35451
$430$	−1.95233	−0.0941495
$431$	0.163554	0.00787813	0.00393907	−	0.999992i	$-0.498746\pi$
$431$	0.163554	0.00787813	0.00393907	+	0.999992i	$0.498746\pi$
$432$	0.451765	0.0217355
$433$	32.8894	1.58057	0.790283	−	0.612742i	$-0.209934\pi$
$433$	32.8894	1.58057	0.790283	+	0.612742i	$0.209934\pi$
$434$	−9.60675	−0.461139
$435$	12.2966	0.589575
$436$	11.3105	0.541674
$437$	−24.4458	−1.16940
$438$	37.7008	1.80141
$439$	30.2264	1.44263	0.721313	−	0.692609i	$-0.243539\pi$
$439$	30.2264	1.44263	0.721313	+	0.692609i	$0.243539\pi$
$440$	5.25229	0.250393
$441$	−15.8618	−0.755325
$442$	34.7890	1.65474
$443$	−24.5001	−1.16403	−0.582017	−	0.813177i	$-0.697736\pi$
$443$	−24.5001	−1.16403	−0.582017	+	0.813177i	$0.697736\pi$
$444$	−2.49223	−0.118276
$445$	−11.4471	−0.542645
$446$	6.24146	0.295542
$447$	15.1872	0.718330
$448$	−1.16640	−0.0551070
$449$	−28.4725	−1.34370	−0.671849	−	0.740688i	$-0.734500\pi$
$449$	−28.4725	−1.34370	−0.671849	+	0.740688i	$0.734500\pi$
$450$	−12.0374	−0.567448
$451$	46.2252	2.17666
$452$	−2.12662	−0.100028
$453$	37.0317	1.73990
$454$	19.1127	0.897004
$455$	−4.60928	−0.216086
$456$	−15.2851	−0.715792
$457$	29.8534	1.39648	0.698242	−	0.715861i	$-0.253966\pi$
$457$	29.8534	1.39648	0.698242	+	0.715861i	$0.253966\pi$
$458$	−2.14540	−0.100248
$459$	3.37520	0.157541
$460$	−3.27230	−0.152572
$461$	21.4879	1.00079	0.500397	−	0.865796i	$-0.333188\pi$
$461$	21.4879	1.00079	0.500397	+	0.865796i	$0.333188\pi$
$462$	17.4040	0.809707
$463$	14.8051	0.688052	0.344026	−	0.938960i	$-0.388209\pi$
$463$	14.8051	0.688052	0.344026	+	0.938960i	$0.388209\pi$
$464$	−6.00987	−0.279001
$465$	−16.8519	−0.781487
$466$	−11.4355	−0.529739
$467$	−28.7389	−1.32988	−0.664938	−	0.746899i	$-0.731542\pi$
$467$	−28.7389	−1.32988	−0.664938	+	0.746899i	$0.731542\pi$
$468$	13.0968	0.605400
$469$	13.8016	0.637300
$470$	7.49162	0.345563
$471$	9.76147	0.449785
$472$	3.48597	0.160455
$473$	−14.2376	−0.654645
$474$	−27.3820	−1.25770
$475$	−27.1335	−1.24497
$476$	−8.71431	−0.399420
$477$	3.87955	0.177632
$478$	6.95062	0.317914
$479$	6.66037	0.304320	0.152160	−	0.988356i	$-0.451377\pi$
$479$	6.66037	0.304320	0.152160	+	0.988356i	$0.451377\pi$
$480$	−2.04606	−0.0933894
$481$	4.81345	0.219475
$482$	−12.2637	−0.558598
$483$	−10.8431	−0.493377
$484$	27.3030	1.24105
$485$	12.6342	0.573691
$486$	21.6138	0.980421
$487$	29.4864	1.33616	0.668079	−	0.744091i	$-0.267117\pi$
$487$	29.4864	1.33616	0.668079	+	0.744091i	$0.267117\pi$
$488$	−0.648486	−0.0293556
$489$	−38.5298	−1.74238
$490$	−4.78602	−0.216210
$491$	−26.9185	−1.21481	−0.607407	−	0.794391i	$-0.707790\pi$
$491$	−26.9185	−1.21481	−0.607407	+	0.794391i	$0.707790\pi$
$492$	−18.0073	−0.811831
$493$	−44.9006	−2.02222
$494$	29.5215	1.32823
$495$	14.7727	0.663983
$496$	8.23626	0.369819
$497$	12.9678	0.581686
$498$	6.31274	0.282881
$499$	23.3067	1.04335	0.521675	−	0.853144i	$-0.325307\pi$
$499$	23.3067	1.04335	0.521675	+	0.853144i	$0.325307\pi$
$500$	−7.87535	−0.352197
$501$	20.6781	0.923830
$502$	22.0689	0.984985
$503$	−26.4955	−1.18138	−0.590688	−	0.806900i	$-0.701144\pi$
$503$	−26.4955	−1.18138	−0.590688	+	0.806900i	$0.701144\pi$
$504$	−3.28063	−0.146131
$505$	−0.831431	−0.0369982
$506$	−23.8637	−1.06087
$507$	−20.9330	−0.929666
$508$	13.1106	0.581690
$509$	−5.30904	−0.235319	−0.117659	−	0.993054i	$-0.537539\pi$
$509$	−5.30904	−0.235319	−0.117659	+	0.993054i	$0.537539\pi$
$510$	−15.2864	−0.676893
$511$	18.2394	0.806864
$512$	1.00000	0.0441942
$513$	2.86415	0.126455
$514$	−24.0264	−1.05976
$515$	−10.1454	−0.447061
$516$	5.54633	0.244164
$517$	54.6336	2.40278
$518$	−1.20572	−0.0529765
$519$	27.5962	1.21134
$520$	3.95173	0.173295
$521$	−29.7505	−1.30339	−0.651697	−	0.758480i	$-0.725942\pi$
$521$	−29.7505	−1.30339	−0.651697	+	0.758480i	$0.725942\pi$
$522$	−16.9035	−0.739845
$523$	22.5516	0.986112	0.493056	−	0.869998i	$-0.335880\pi$
$523$	22.5516	0.986112	0.493056	+	0.869998i	$0.335880\pi$
$524$	−2.05321	−0.0896949
$525$	−12.0352	−0.525260
$526$	−21.8953	−0.954680
$527$	61.5343	2.68048
$528$	−14.9212	−0.649360
$529$	−8.13239	−0.353582
$530$	1.17058	0.0508469
$531$	9.80470	0.425488
$532$	−7.39485	−0.320608
$533$	34.7790	1.50644
$534$	32.5199	1.40727
$535$	−4.89081	−0.211448
$536$	−11.8327	−0.511095
$537$	−27.4239	−1.18343
$538$	1.08143	0.0466236
$539$	−34.9027	−1.50336
$540$	0.383393	0.0164986
$541$	−7.90997	−0.340076	−0.170038	−	0.985438i	$-0.554389\pi$
$541$	−7.90997	−0.340076	−0.170038	+	0.985438i	$0.554389\pi$
$542$	−1.54996	−0.0665765
$543$	20.9381	0.898541
$544$	7.47114	0.320323
$545$	9.59873	0.411164
$546$	13.0944	0.560390
$547$	−46.4603	−1.98650	−0.993249	−	0.116005i	$-0.962991\pi$
$547$	−46.4603	−1.98650	−0.993249	+	0.116005i	$0.962991\pi$
$548$	19.9906	0.853956
$549$	−1.82394	−0.0778440
$550$	−26.4873	−1.12942
$551$	−38.1021	−1.62320
$552$	9.29622	0.395673
$553$	−13.2472	−0.563330
$554$	10.5069	0.446396
$555$	−2.11505	−0.0897788
$556$	−0.592949	−0.0251466
$557$	−31.4650	−1.33321	−0.666606	−	0.745410i	$-0.732254\pi$
$557$	−31.4650	−1.33321	−0.666606	+	0.745410i	$0.732254\pi$
$558$	23.1655	0.980672
$559$	−10.7121	−0.453074
$560$	−0.989871	−0.0418297
$561$	−111.478	−4.70661
$562$	16.8033	0.708803
$563$	13.7100	0.577809	0.288905	−	0.957358i	$-0.406709\pi$
$563$	13.7100	0.577809	0.288905	+	0.957358i	$0.406709\pi$
$564$	−21.2828	−0.896169
$565$	−1.80477	−0.0759274
$566$	4.18494	0.175906
$567$	11.1123	0.466673
$568$	−11.1178	−0.466494
$569$	4.16928	0.174785	0.0873927	−	0.996174i	$-0.472147\pi$
$569$	4.16928	0.174785	0.0873927	+	0.996174i	$0.472147\pi$
$570$	−12.9718	−0.543331
$571$	29.5528	1.23675	0.618373	−	0.785885i	$-0.287792\pi$
$571$	29.5528	1.23675	0.618373	+	0.785885i	$0.287792\pi$
$572$	28.8185	1.20496
$573$	−45.9395	−1.91915
$574$	−8.71181	−0.363624
$575$	16.5022	0.688190
$576$	2.81262	0.117192
$577$	−38.2545	−1.59256	−0.796279	−	0.604930i	$-0.793201\pi$
$577$	−38.2545	−1.59256	−0.796279	+	0.604930i	$0.793201\pi$
$578$	38.8180	1.61462
$579$	4.97807	0.206882
$580$	−5.10032	−0.211779
$581$	3.05406	0.126704
$582$	−35.8924	−1.48779
$583$	8.53664	0.353552
$584$	−15.6374	−0.647080
$585$	11.1147	0.459536
$586$	−15.3173	−0.632754
$587$	15.0070	0.619404	0.309702	−	0.950834i	$-0.399771\pi$
$587$	15.0070	0.619404	0.309702	+	0.950834i	$0.399771\pi$
$588$	13.5965	0.560711
$589$	52.2172	2.15157
$590$	2.95839	0.121795
$591$	49.0215	2.01648
$592$	1.03372	0.0424855
$593$	−41.8572	−1.71887	−0.859435	−	0.511246i	$-0.829184\pi$
$593$	−41.8572	−1.71887	−0.859435	+	0.511246i	$0.829184\pi$
$594$	2.79595	0.114719
$595$	−7.39547	−0.303185
$596$	−6.29929	−0.258029
$597$	−33.8282	−1.38450
$598$	−17.9546	−0.734217
$599$	26.3658	1.07728	0.538639	−	0.842537i	$-0.318939\pi$
$599$	26.3658	1.07728	0.538639	+	0.842537i	$0.318939\pi$
$600$	10.3183	0.421242
$601$	31.8156	1.29779	0.648893	−	0.760880i	$-0.275232\pi$
$601$	31.8156	1.29779	0.648893	+	0.760880i	$0.275232\pi$
$602$	2.68328	0.109362
$603$	−33.2809	−1.35530
$604$	−15.3599	−0.624985
$605$	23.1709	0.942032
$606$	2.36200	0.0959496
$607$	−47.2208	−1.91663	−0.958317	−	0.285707i	$-0.907772\pi$
$607$	−47.2208	−1.91663	−0.958317	+	0.285707i	$0.907772\pi$
$608$	6.33992	0.257118
$609$	−16.9004	−0.684839
$610$	−0.550342	−0.0222827
$611$	41.1053	1.66294
$612$	21.0135	0.849420
$613$	−46.6742	−1.88515	−0.942577	−	0.333989i	$-0.891605\pi$
$613$	−46.6742	−1.88515	−0.942577	+	0.333989i	$0.891605\pi$
$614$	−11.3461	−0.457892
$615$	−15.2820	−0.616230
$616$	−7.21876	−0.290852
$617$	−40.5285	−1.63162	−0.815809	−	0.578322i	$-0.803708\pi$
$617$	−40.5285	−1.63162	−0.815809	+	0.578322i	$0.803708\pi$
$618$	28.8220	1.15939
$619$	−22.0488	−0.886215	−0.443107	−	0.896469i	$-0.646124\pi$
$619$	−22.0488	−0.886215	−0.443107	+	0.896469i	$0.646124\pi$
$620$	6.98976	0.280716
$621$	−1.74194	−0.0699016
$622$	23.3510	0.936291
$623$	15.7329	0.630327
$624$	−11.2264	−0.449416
$625$	14.7154	0.588617
$626$	−27.6344	−1.10449
$627$	−94.5989	−3.77792
$628$	−4.04883	−0.161566
$629$	7.72305	0.307938
$630$	−2.78413	−0.110922
$631$	−37.5567	−1.49511	−0.747554	−	0.664201i	$-0.768772\pi$
$631$	−37.5567	−1.49511	−0.747554	+	0.664201i	$0.768772\pi$
$632$	11.3574	0.451773
$633$	44.9354	1.78602
$634$	−18.1025	−0.718943
$635$	11.1264	0.441539
$636$	−3.32550	−0.131864
$637$	−26.2601	−1.04046
$638$	−37.1948	−1.47256
$639$	−31.2702	−1.23703
$640$	0.848657	0.0335461
$641$	17.4612	0.689677	0.344838	−	0.938662i	$-0.387934\pi$
$641$	17.4612	0.689677	0.344838	+	0.938662i	$0.387934\pi$
$642$	13.8942	0.548361
$643$	28.0391	1.10575	0.552877	−	0.833263i	$-0.313530\pi$
$643$	28.0391	1.10575	0.552877	+	0.833263i	$0.313530\pi$
$644$	4.49745	0.177224
$645$	4.70693	0.185335
$646$	47.3664	1.86361
$647$	−5.62869	−0.221286	−0.110643	−	0.993860i	$-0.535291\pi$
$647$	−5.62869	−0.221286	−0.110643	+	0.993860i	$0.535291\pi$
$648$	−9.52703	−0.374257
$649$	21.5745	0.846872
$650$	−19.9286	−0.781663
$651$	23.1613	0.907761
$652$	15.9812	0.625874
$653$	36.7069	1.43645	0.718227	−	0.695809i	$-0.244954\pi$
$653$	36.7069	1.43645	0.718227	+	0.695809i	$0.244954\pi$
$654$	−27.2689	−1.06630
$655$	−1.74247	−0.0680840
$656$	7.46899	0.291615
$657$	−43.9820	−1.71590
$658$	−10.2965	−0.401399
$659$	1.15584	0.0450250	0.0225125	−	0.999747i	$-0.492833\pi$
$659$	1.15584	0.0450250	0.0225125	+	0.999747i	$0.492833\pi$
$660$	−12.6629	−0.492905
$661$	31.0338	1.20708	0.603538	−	0.797334i	$-0.293757\pi$
$661$	31.0338	1.20708	0.603538	+	0.797334i	$0.293757\pi$
$662$	−32.7554	−1.27307
$663$	−83.8741	−3.25740
$664$	−2.61838	−0.101613
$665$	−6.27570	−0.243361
$666$	2.90745	0.112662
$667$	23.1732	0.897270
$668$	−8.57680	−0.331846
$669$	−15.0478	−0.581780
$670$	−10.0419	−0.387953
$671$	−4.01345	−0.154937
$672$	2.81211	0.108479
$673$	−8.86134	−0.341580	−0.170790	−	0.985307i	$-0.554632\pi$
$673$	−8.86134	−0.341580	−0.170790	+	0.985307i	$0.554632\pi$
$674$	−31.5378	−1.21479
$675$	−1.93345	−0.0744186
$676$	8.68251	0.333943
$677$	23.4135	0.899854	0.449927	−	0.893065i	$-0.351450\pi$
$677$	23.4135	0.899854	0.449927	+	0.893065i	$0.351450\pi$
$678$	5.12715	0.196907
$679$	−17.3645	−0.666389
$680$	6.34044	0.243145
$681$	−46.0795	−1.76577
$682$	50.9738	1.95189
$683$	19.9227	0.762323	0.381161	−	0.924509i	$-0.375524\pi$
$683$	19.9227	0.762323	0.381161	+	0.924509i	$0.375524\pi$
$684$	17.8318	0.681815
$685$	16.9652	0.648206
$686$	14.7427	0.562878
$687$	5.17243	0.197340
$688$	−2.30049	−0.0877053
$689$	6.42281	0.244690
$690$	7.88931	0.300341
$691$	−25.5639	−0.972498	−0.486249	−	0.873820i	$-0.661635\pi$
$691$	−25.5639	−0.972498	−0.486249	+	0.873820i	$0.661635\pi$
$692$	−11.4463	−0.435122
$693$	−20.3036	−0.771271
$694$	−1.80014	−0.0683323
$695$	−0.503210	−0.0190879
$696$	14.4894	0.549220
$697$	55.8019	2.11365
$698$	−29.2851	−1.10846
$699$	27.5703	1.04280
$700$	4.99192	0.188677
$701$	12.6145	0.476444	0.238222	−	0.971211i	$-0.423435\pi$
$701$	12.6145	0.476444	0.238222	+	0.971211i	$0.423435\pi$
$702$	2.10362	0.0793959
$703$	6.55368	0.247177
$704$	6.18894	0.233255
$705$	−18.0618	−0.680248
$706$	−5.99786	−0.225732
$707$	1.14272	0.0429764
$708$	−8.40445	−0.315859
$709$	13.5311	0.508172	0.254086	−	0.967182i	$-0.418225\pi$
$709$	13.5311	0.508172	0.254086	+	0.967182i	$0.418225\pi$
$710$	−9.43524	−0.354098
$711$	31.9441	1.19800
$712$	−13.4885	−0.505503
$713$	−31.7578	−1.18934
$714$	21.0097	0.786267
$715$	24.4570	0.914641
$716$	11.3748	0.425096
$717$	−16.7575	−0.625821
$718$	33.3789	1.24569
$719$	−6.92260	−0.258170	−0.129085	−	0.991634i	$-0.541204\pi$
$719$	−6.92260	−0.258170	−0.129085	+	0.991634i	$0.541204\pi$
$720$	2.38695	0.0889563
$721$	13.9439	0.519298
$722$	21.1945	0.788779
$723$	29.5671	1.09961
$724$	−8.68464	−0.322762
$725$	25.7209	0.955252
$726$	−65.8259	−2.44303
$727$	6.46204	0.239664	0.119832	−	0.992794i	$-0.461764\pi$
$727$	6.46204	0.239664	0.119832	+	0.992794i	$0.461764\pi$
$728$	−5.43126	−0.201296
$729$	−23.5284	−0.871422
$730$	−13.2708	−0.491174
$731$	−17.1873	−0.635694
$732$	1.56346	0.0577871
$733$	16.0483	0.592758	0.296379	−	0.955070i	$-0.404221\pi$
$733$	16.0483	0.592758	0.296379	+	0.955070i	$0.404221\pi$
$734$	24.9929	0.922503
$735$	11.5388	0.425615
$736$	−3.85585	−0.142129
$737$	−73.2320	−2.69754
$738$	21.0074	0.773294
$739$	−26.8604	−0.988076	−0.494038	−	0.869440i	$-0.664480\pi$
$739$	−26.8604	−0.988076	−0.494038	+	0.869440i	$0.664480\pi$
$740$	0.877272	0.0322492
$741$	−71.1745	−2.61466
$742$	−1.60885	−0.0590629
$743$	13.1819	0.483596	0.241798	−	0.970327i	$-0.422263\pi$
$743$	13.1819	0.483596	0.241798	+	0.970327i	$0.422263\pi$
$744$	−19.8571	−0.727997
$745$	−5.34594	−0.195860
$746$	5.17930	0.189628
$747$	−7.36449	−0.269453
$748$	46.2385	1.69065
$749$	6.72194	0.245614
$750$	18.9870	0.693307
$751$	−38.8269	−1.41681	−0.708406	−	0.705805i	$-0.750586\pi$
$751$	−38.8269	−1.41681	−0.708406	+	0.705805i	$0.750586\pi$
$752$	8.82762	0.321910
$753$	−53.2068	−1.93896
$754$	−27.9847	−1.01914
$755$	−13.0353	−0.474403
$756$	−0.526936	−0.0191645
$757$	12.8765	0.468004	0.234002	−	0.972236i	$-0.424818\pi$
$757$	12.8765	0.468004	0.234002	+	0.972236i	$0.424818\pi$
$758$	19.2315	0.698521
$759$	57.5338	2.08834
$760$	5.38042	0.195168
$761$	25.3379	0.918499	0.459250	−	0.888307i	$-0.348118\pi$
$761$	25.3379	0.918499	0.459250	+	0.888307i	$0.348118\pi$
$762$	−31.6089	−1.14507
$763$	−13.1925	−0.477601
$764$	19.0546	0.689372
$765$	17.8332	0.644762
$766$	21.6689	0.782929
$767$	16.2322	0.586112
$768$	−2.41094	−0.0869972
$769$	−3.21674	−0.115999	−0.0579993	−	0.998317i	$-0.518472\pi$
$769$	−3.21674	−0.115999	−0.0579993	+	0.998317i	$0.518472\pi$
$770$	−6.12626	−0.220775
$771$	57.9261	2.08616
$772$	−2.06479	−0.0743133
$773$	−29.7685	−1.07070	−0.535349	−	0.844631i	$-0.679820\pi$
$773$	−29.7685	−1.07070	−0.535349	+	0.844631i	$0.679820\pi$
$774$	−6.47039	−0.232574
$775$	−35.2494	−1.26620
$776$	14.8873	0.534424
$777$	2.90693	0.104285
$778$	7.90806	0.283518
$779$	47.3528	1.69659
$780$	−9.52737	−0.341135
$781$	−68.8077	−2.46213
$782$	−28.8076	−1.03016
$783$	−2.71505	−0.0970279
$784$	−5.63952	−0.201411
$785$	−3.43607	−0.122638
$786$	4.95016	0.176566
$787$	−11.2307	−0.400330	−0.200165	−	0.979762i	$-0.564148\pi$
$787$	−11.2307	−0.400330	−0.200165	+	0.979762i	$0.564148\pi$
$788$	−20.3330	−0.724333
$789$	52.7882	1.87931
$790$	9.63855	0.342924
$791$	2.48048	0.0881958
$792$	17.4071	0.618536
$793$	−3.01964	−0.107231
$794$	15.2470	0.541097
$795$	−2.82221	−0.100093
$796$	14.0311	0.497321
$797$	32.9078	1.16565	0.582827	−	0.812596i	$-0.301947\pi$
$797$	32.9078	1.16565	0.582827	+	0.812596i	$0.301947\pi$
$798$	17.8285	0.631123
$799$	65.9524	2.33323
$800$	−4.27978	−0.151313
$801$	−37.9380	−1.34047
$802$	−5.35451	−0.189074
$803$	−96.7790	−3.41526
$804$	28.5279	1.00610
$805$	3.81680	0.134524
$806$	38.3517	1.35088
$807$	−2.60725	−0.0917796
$808$	−0.979701	−0.0344658
$809$	−6.69766	−0.235477	−0.117739	−	0.993045i	$-0.537564\pi$
$809$	−6.69766	−0.235477	−0.117739	+	0.993045i	$0.537564\pi$
$810$	−8.08518	−0.284084
$811$	19.6892	0.691382	0.345691	−	0.938348i	$-0.387645\pi$
$811$	19.6892	0.691382	0.345691	+	0.938348i	$0.387645\pi$
$812$	7.00989	0.245999
$813$	3.73686	0.131057
$814$	6.39762	0.224236
$815$	13.5626	0.475077
$816$	−18.0125	−0.630562
$817$	−14.5849	−0.510261
$818$	6.77021	0.236715
$819$	−15.2761	−0.533789
$820$	6.33862	0.221354
$821$	26.2495	0.916114	0.458057	−	0.888923i	$-0.348546\pi$
$821$	26.2495	0.916114	0.458057	+	0.888923i	$0.348546\pi$
$822$	−48.1961	−1.68103
$823$	−40.2737	−1.40385	−0.701927	−	0.712249i	$-0.747677\pi$
$823$	−40.2737	−1.40385	−0.701927	+	0.712249i	$0.747677\pi$
$824$	−11.9547	−0.416461
$825$	63.8593	2.22329
$826$	−4.06602	−0.141475
$827$	−16.9808	−0.590479	−0.295240	−	0.955423i	$-0.595399\pi$
$827$	−16.9808	−0.590479	−0.295240	+	0.955423i	$0.595399\pi$
$828$	−10.8450	−0.376891
$829$	−0.709057	−0.0246266	−0.0123133	−	0.999924i	$-0.503920\pi$
$829$	−0.709057	−0.0246266	−0.0123133	+	0.999924i	$0.503920\pi$
$830$	−2.22210	−0.0771303
$831$	−25.3315	−0.878741
$832$	4.65645	0.161433
$833$	−42.1337	−1.45984
$834$	1.42956	0.0495017
$835$	−7.27876	−0.251892
$836$	39.2374	1.35705
$837$	3.72085	0.128611
$838$	−26.6921	−0.922062
$839$	33.8497	1.16862	0.584311	−	0.811530i	$-0.301365\pi$
$839$	33.8497	1.16862	0.584311	+	0.811530i	$0.301365\pi$
$840$	2.38652	0.0823426
$841$	7.11858	0.245468
$842$	5.60340	0.193106
$843$	−40.5116	−1.39530
$844$	−18.6381	−0.641551
$845$	7.36847	0.253483
$846$	24.8287	0.853629
$847$	−31.8462	−1.09425
$848$	1.37934	0.0473666
$849$	−10.0896	−0.346275
$850$	−31.9749	−1.09673
$851$	−3.98586	−0.136634
$852$	26.8044	0.918304
$853$	44.3232	1.51760	0.758799	−	0.651325i	$-0.225786\pi$
$853$	44.3232	1.51760	0.758799	+	0.651325i	$0.225786\pi$
$854$	0.756392	0.0258832
$855$	15.1331	0.517540
$856$	−5.76300	−0.196975
$857$	26.7084	0.912341	0.456171	−	0.889892i	$-0.349221\pi$
$857$	26.7084	0.912341	0.456171	+	0.889892i	$0.349221\pi$
$858$	−69.4796	−2.37199
$859$	10.2803	0.350759	0.175379	−	0.984501i	$-0.443885\pi$
$859$	10.2803	0.350759	0.175379	+	0.984501i	$0.443885\pi$
$860$	−1.95233	−0.0665738
$861$	21.0036	0.715802
$862$	0.163554	0.00557068
$863$	−52.8231	−1.79812	−0.899059	−	0.437827i	$-0.855748\pi$
$863$	−52.8231	−1.79812	−0.899059	+	0.437827i	$0.855748\pi$
$864$	0.451765	0.0153693
$865$	−9.71395	−0.330284
$866$	32.8894	1.11763
$867$	−93.5877	−3.17841
$868$	−9.60675	−0.326074
$869$	70.2904	2.38444
$870$	12.2966	0.416892
$871$	−55.0984	−1.86694
$872$	11.3105	0.383021
$873$	41.8724	1.41716
$874$	−24.4458	−0.826891
$875$	9.18578	0.310536
$876$	37.7008	1.27379
$877$	48.4301	1.63537	0.817685	−	0.575666i	$-0.195257\pi$
$877$	48.4301	1.63537	0.817685	+	0.575666i	$0.195257\pi$
$878$	30.2264	1.02009
$879$	36.9292	1.24559
$880$	5.25229	0.177055
$881$	−13.5545	−0.456661	−0.228331	−	0.973584i	$-0.573327\pi$
$881$	−13.5545	−0.456661	−0.228331	+	0.973584i	$0.573327\pi$
$882$	−15.8618	−0.534095
$883$	−43.7974	−1.47390	−0.736950	−	0.675947i	$-0.763735\pi$
$883$	−43.7974	−1.47390	−0.736950	+	0.675947i	$0.763735\pi$
$884$	34.7890	1.17008
$885$	−7.13250	−0.239756
$886$	−24.5001	−0.823096
$887$	24.4796	0.821943	0.410972	−	0.911648i	$-0.365190\pi$
$887$	24.4796	0.821943	0.410972	+	0.911648i	$0.365190\pi$
$888$	−2.49223	−0.0836337
$889$	−15.2922	−0.512883
$890$	−11.4471	−0.383708
$891$	−58.9623	−1.97531
$892$	6.24146	0.208979
$893$	55.9664	1.87284
$894$	15.1872	0.507936
$895$	9.65330	0.322674
$896$	−1.16640	−0.0389666
$897$	43.2874	1.44532
$898$	−28.4725	−0.950139
$899$	−49.4989	−1.65088
$900$	−12.0374	−0.401246
$901$	10.3052	0.343317
$902$	46.2252	1.53913
$903$	−6.46922	−0.215282
$904$	−2.12662	−0.0707304
$905$	−7.37029	−0.244997
$906$	37.0317	1.23030
$907$	23.9348	0.794743	0.397371	−	0.917658i	$-0.369922\pi$
$907$	23.9348	0.794743	0.397371	+	0.917658i	$0.369922\pi$
$908$	19.1127	0.634277
$909$	−2.75553	−0.0913950
$910$	−4.60928	−0.152796
$911$	25.2169	0.835472	0.417736	−	0.908569i	$-0.362824\pi$
$911$	25.2169	0.835472	0.417736	+	0.908569i	$0.362824\pi$
$912$	−15.2851	−0.506142
$913$	−16.2050	−0.536307
$914$	29.8534	0.987464
$915$	1.32684	0.0438640
$916$	−2.14540	−0.0708860
$917$	2.39486	0.0790852
$918$	3.37520	0.111398
$919$	−44.1444	−1.45619	−0.728094	−	0.685477i	$-0.759594\pi$
$919$	−44.1444	−1.45619	−0.728094	+	0.685477i	$0.759594\pi$
$920$	−3.27230	−0.107884
$921$	27.3548	0.901371
$922$	21.4879	0.707668
$923$	−51.7696	−1.70402
$924$	17.4040	0.572549
$925$	−4.42408	−0.145463
$926$	14.8051	0.486526
$927$	−33.6240	−1.10436
$928$	−6.00987	−0.197284
$929$	8.80476	0.288875	0.144437	−	0.989514i	$-0.453863\pi$
$929$	8.80476	0.288875	0.144437	+	0.989514i	$0.453863\pi$
$930$	−16.8519	−0.552595
$931$	−35.7541	−1.17179
$932$	−11.4355	−0.374582
$933$	−56.2979	−1.84311
$934$	−28.7389	−0.940364
$935$	39.2406	1.28331
$936$	13.0968	0.428083
$937$	39.7302	1.29793	0.648965	−	0.760819i	$-0.275202\pi$
$937$	39.7302	1.29793	0.648965	+	0.760819i	$0.275202\pi$
$938$	13.8016	0.450639
$939$	66.6248	2.17422
$940$	7.49162	0.244350
$941$	−9.67169	−0.315288	−0.157644	−	0.987496i	$-0.550390\pi$
$941$	−9.67169	−0.315288	−0.157644	+	0.987496i	$0.550390\pi$
$942$	9.76147	0.318046
$943$	−28.7993	−0.937836
$944$	3.48597	0.113459
$945$	−0.447188	−0.0145470
$946$	−14.2376	−0.462904
$947$	−27.4473	−0.891917	−0.445958	−	0.895054i	$-0.647137\pi$
$947$	−27.4473	−0.891917	−0.445958	+	0.895054i	$0.647137\pi$
$948$	−27.3820	−0.889326
$949$	−72.8147	−2.36367
$950$	−27.1335	−0.880325
$951$	43.6441	1.41526
$952$	−8.71431	−0.282433
$953$	6.40350	0.207430	0.103715	−	0.994607i	$-0.466927\pi$
$953$	6.40350	0.207430	0.103715	+	0.994607i	$0.466927\pi$
$954$	3.87955	0.125605
$955$	16.1708	0.523276
$956$	6.95062	0.224799
$957$	89.6743	2.89876
$958$	6.66037	0.215187
$959$	−23.3170	−0.752944
$960$	−2.04606	−0.0660363
$961$	36.8360	1.18826
$962$	4.81345	0.155192
$963$	−16.2091	−0.522331
$964$	−12.2637	−0.394989
$965$	−1.75230	−0.0564084
$966$	−10.8431	−0.348870
$967$	14.7954	0.475788	0.237894	−	0.971291i	$-0.423543\pi$
$967$	14.7954	0.475788	0.237894	+	0.971291i	$0.423543\pi$
$968$	27.3030	0.877553
$969$	−114.197	−3.66855
$970$	12.6342	0.405661
$971$	40.2728	1.29242	0.646208	−	0.763161i	$-0.276354\pi$
$971$	40.2728	1.29242	0.646208	+	0.763161i	$0.276354\pi$
$972$	21.6138	0.693262
$973$	0.691613	0.0221721
$974$	29.4864	0.944806
$975$	48.0465	1.53872
$976$	−0.648486	−0.0207575
$977$	1.79903	0.0575560	0.0287780	−	0.999586i	$-0.490838\pi$
$977$	1.79903	0.0575560	0.0287780	+	0.999586i	$0.490838\pi$
$978$	−38.5298	−1.23205
$979$	−83.4796	−2.66802
$980$	−4.78602	−0.152884
$981$	31.8121	1.01568
$982$	−26.9185	−0.859004
$983$	−10.1606	−0.324074	−0.162037	−	0.986785i	$-0.551806\pi$
$983$	−10.1606	−0.324074	−0.162037	+	0.986785i	$0.551806\pi$
$984$	−18.0073	−0.574051
$985$	−17.2557	−0.549813
$986$	−44.9006	−1.42993
$987$	24.8242	0.790164
$988$	29.5215	0.939204
$989$	8.87034	0.282061
$990$	14.7727	0.469507
$991$	47.3795	1.50506	0.752530	−	0.658559i	$-0.228833\pi$
$991$	47.3795	1.50506	0.752530	+	0.658559i	$0.228833\pi$
$992$	8.23626	0.261502
$993$	78.9712	2.50608
$994$	12.9678	0.411314
$995$	11.9076	0.377497
$996$	6.31274	0.200027
$997$	16.0875	0.509496	0.254748	−	0.967007i	$-0.418007\pi$
$997$	16.0875	0.509496	0.254748	+	0.967007i	$0.418007\pi$
$998$	23.3067	0.737761
$999$	0.466997	0.0147751

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.f.1.9	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.f.1.9	✓	67	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:	\( N \)	\(=\)	\( 6046 = 2 \cdot 3023 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.41094 q^{3} +1.00000 q^{4} +0.848657 q^{5} -2.41094 q^{6} -1.16640 q^{7} +1.00000 q^{8} +2.81262 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.41094 q^{3} +1.00000 q^{4} +0.848657 q^{5} -2.41094 q^{6} -1.16640 q^{7} +1.00000 q^{8} +2.81262 q^{9} +0.848657 q^{10} +6.18894 q^{11} -2.41094 q^{12} +4.65645 q^{13} -1.16640 q^{14} -2.04606 q^{15} +1.00000 q^{16} +7.47114 q^{17} +2.81262 q^{18} +6.33992 q^{19} +0.848657 q^{20} +2.81211 q^{21} +6.18894 q^{22} -3.85585 q^{23} -2.41094 q^{24} -4.27978 q^{25} +4.65645 q^{26} +0.451765 q^{27} -1.16640 q^{28} -6.00987 q^{29} -2.04606 q^{30} +8.23626 q^{31} +1.00000 q^{32} -14.9212 q^{33} +7.47114 q^{34} -0.989871 q^{35} +2.81262 q^{36} +1.03372 q^{37} +6.33992 q^{38} -11.2264 q^{39} +0.848657 q^{40} +7.46899 q^{41} +2.81211 q^{42} -2.30049 q^{43} +6.18894 q^{44} +2.38695 q^{45} -3.85585 q^{46} +8.82762 q^{47} -2.41094 q^{48} -5.63952 q^{49} -4.27978 q^{50} -18.0125 q^{51} +4.65645 q^{52} +1.37934 q^{53} +0.451765 q^{54} +5.25229 q^{55} -1.16640 q^{56} -15.2851 q^{57} -6.00987 q^{58} +3.48597 q^{59} -2.04606 q^{60} -0.648486 q^{61} +8.23626 q^{62} -3.28063 q^{63} +1.00000 q^{64} +3.95173 q^{65} -14.9212 q^{66} -11.8327 q^{67} +7.47114 q^{68} +9.29622 q^{69} -0.989871 q^{70} -11.1178 q^{71} +2.81262 q^{72} -15.6374 q^{73} +1.03372 q^{74} +10.3183 q^{75} +6.33992 q^{76} -7.21876 q^{77} -11.2264 q^{78} +11.3574 q^{79} +0.848657 q^{80} -9.52703 q^{81} +7.46899 q^{82} -2.61838 q^{83} +2.81211 q^{84} +6.34044 q^{85} -2.30049 q^{86} +14.4894 q^{87} +6.18894 q^{88} -13.4885 q^{89} +2.38695 q^{90} -5.43126 q^{91} -3.85585 q^{92} -19.8571 q^{93} +8.82762 q^{94} +5.38042 q^{95} -2.41094 q^{96} +14.8873 q^{97} -5.63952 q^{98} +17.4071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9 + 21 * q^10 + 56 * q^11 + 21 * q^12 + 33 * q^13 + 38 * q^14 + 25 * q^15 + 67 * q^16 + 30 * q^17 + 90 * q^18 + 36 * q^19 + 21 * q^20 + 20 * q^21 + 56 * q^22 + 65 * q^23 + 21 * q^24 + 72 * q^25 + 33 * q^26 + 57 * q^27 + 38 * q^28 + 84 * q^29 + 25 * q^30 + 52 * q^31 + 67 * q^32 - 9 * q^33 + 30 * q^34 + 30 * q^35 + 90 * q^36 + 52 * q^37 + 36 * q^38 + 41 * q^39 + 21 * q^40 + 46 * q^41 + 20 * q^42 + 61 * q^43 + 56 * q^44 + 23 * q^45 + 65 * q^46 + 51 * q^47 + 21 * q^48 + 81 * q^49 + 72 * q^50 + 33 * q^51 + 33 * q^52 + 72 * q^53 + 57 * q^54 + 14 * q^55 + 38 * q^56 - 26 * q^57 + 84 * q^58 + 71 * q^59 + 25 * q^60 + 42 * q^61 + 52 * q^62 + 63 * q^63 + 67 * q^64 - 2 * q^65 - 9 * q^66 + 70 * q^67 + 30 * q^68 + 21 * q^69 + 30 * q^70 + 104 * q^71 + 90 * q^72 - 31 * q^73 + 52 * q^74 + 69 * q^75 + 36 * q^76 + 48 * q^77 + 41 * q^78 + 79 * q^79 + 21 * q^80 + 123 * q^81 + 46 * q^82 + 41 * q^83 + 20 * q^84 + 6 * q^85 + 61 * q^86 + 19 * q^87 + 56 * q^88 + 58 * q^89 + 23 * q^90 + 31 * q^91 + 65 * q^92 + 13 * q^93 + 51 * q^94 + 77 * q^95 + 21 * q^96 - 8 * q^97 + 81 * q^98 + 129 * q^99

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