LMFDB - Embedded newform 6046.2.a.g.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:	$69$
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.73247 q^{3} +1.00000 q^{4} -4.08673 q^{5} +2.73247 q^{6} -1.35525 q^{7} -1.00000 q^{8} +4.46639 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.73247 q^{3} +1.00000 q^{4} -4.08673 q^{5} +2.73247 q^{6} -1.35525 q^{7} -1.00000 q^{8} +4.46639 q^{9} +4.08673 q^{10} +4.30075 q^{11} -2.73247 q^{12} -5.03138 q^{13} +1.35525 q^{14} +11.1669 q^{15} +1.00000 q^{16} -4.20743 q^{17} -4.46639 q^{18} +6.88446 q^{19} -4.08673 q^{20} +3.70317 q^{21} -4.30075 q^{22} -0.530712 q^{23} +2.73247 q^{24} +11.7013 q^{25} +5.03138 q^{26} -4.00686 q^{27} -1.35525 q^{28} +4.32835 q^{29} -11.1669 q^{30} -0.860776 q^{31} -1.00000 q^{32} -11.7517 q^{33} +4.20743 q^{34} +5.53853 q^{35} +4.46639 q^{36} -2.72088 q^{37} -6.88446 q^{38} +13.7481 q^{39} +4.08673 q^{40} +5.26479 q^{41} -3.70317 q^{42} -3.64472 q^{43} +4.30075 q^{44} -18.2529 q^{45} +0.530712 q^{46} +13.5096 q^{47} -2.73247 q^{48} -5.16330 q^{49} -11.7013 q^{50} +11.4967 q^{51} -5.03138 q^{52} -11.0833 q^{53} +4.00686 q^{54} -17.5760 q^{55} +1.35525 q^{56} -18.8116 q^{57} -4.32835 q^{58} -2.93144 q^{59} +11.1669 q^{60} -14.4856 q^{61} +0.860776 q^{62} -6.05306 q^{63} +1.00000 q^{64} +20.5619 q^{65} +11.7517 q^{66} +6.99569 q^{67} -4.20743 q^{68} +1.45015 q^{69} -5.53853 q^{70} -8.33336 q^{71} -4.46639 q^{72} -14.6670 q^{73} +2.72088 q^{74} -31.9735 q^{75} +6.88446 q^{76} -5.82858 q^{77} -13.7481 q^{78} +16.6767 q^{79} -4.08673 q^{80} -2.45054 q^{81} -5.26479 q^{82} +11.2314 q^{83} +3.70317 q^{84} +17.1946 q^{85} +3.64472 q^{86} -11.8271 q^{87} -4.30075 q^{88} +4.51767 q^{89} +18.2529 q^{90} +6.81876 q^{91} -0.530712 q^{92} +2.35204 q^{93} -13.5096 q^{94} -28.1349 q^{95} +2.73247 q^{96} +16.1322 q^{97} +5.16330 q^{98} +19.2088 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9}+O(q^{10}) $ 69 * q - 69 * q^2 + 69 * q^4 + 13 * q^5 - 27 * q^7 - 69 * q^8 + 99 * q^9	\( 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9} - 13 q^{10} + 42 q^{11} - 5 q^{13} + 27 q^{14} + 18 q^{15} + 69 q^{16} + 24 q^{17} - 99 q^{18} + q^{19} + 13 q^{20} + 7 q^{21} - 42 q^{22} + 25 q^{23} + 100 q^{25} + 5 q^{26} + 15 q^{27} - 27 q^{28} + 87 q^{29} - 18 q^{30} + 5 q^{31} - 69 q^{32} + 28 q^{33} - 24 q^{34} + 33 q^{35} + 99 q^{36} - 5 q^{37} - q^{38} + 22 q^{39} - 13 q^{40} + 47 q^{41} - 7 q^{42} - 23 q^{43} + 42 q^{44} + 14 q^{45} - 25 q^{46} + 13 q^{47} + 106 q^{49} - 100 q^{50} + 2 q^{51} - 5 q^{52} + 51 q^{53} - 15 q^{54} - 11 q^{55} + 27 q^{56} + 52 q^{57} - 87 q^{58} + 73 q^{59} + 18 q^{60} + 4 q^{61} - 5 q^{62} - 86 q^{63} + 69 q^{64} + 70 q^{65} - 28 q^{66} - 24 q^{67} + 24 q^{68} + 56 q^{69} - 33 q^{70} + 84 q^{71} - 99 q^{72} + 27 q^{73} + 5 q^{74} + 27 q^{75} + q^{76} + 45 q^{77} - 22 q^{78} + 42 q^{79} + 13 q^{80} + 205 q^{81} - 47 q^{82} + q^{83} + 7 q^{84} - 18 q^{85} + 23 q^{86} - q^{87} - 42 q^{88} + 94 q^{89} - 14 q^{90} + 6 q^{91} + 25 q^{92} - 13 q^{93} - 13 q^{94} + 86 q^{95} + 35 q^{97} - 106 q^{98} + 83 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 69 * q^4 + 13 * q^5 - 27 * q^7 - 69 * q^8 + 99 * q^9 - 13 * q^10 + 42 * q^11 - 5 * q^13 + 27 * q^14 + 18 * q^15 + 69 * q^16 + 24 * q^17 - 99 * q^18 + q^19 + 13 * q^20 + 7 * q^21 - 42 * q^22 + 25 * q^23 + 100 * q^25 + 5 * q^26 + 15 * q^27 - 27 * q^28 + 87 * q^29 - 18 * q^30 + 5 * q^31 - 69 * q^32 + 28 * q^33 - 24 * q^34 + 33 * q^35 + 99 * q^36 - 5 * q^37 - q^38 + 22 * q^39 - 13 * q^40 + 47 * q^41 - 7 * q^42 - 23 * q^43 + 42 * q^44 + 14 * q^45 - 25 * q^46 + 13 * q^47 + 106 * q^49 - 100 * q^50 + 2 * q^51 - 5 * q^52 + 51 * q^53 - 15 * q^54 - 11 * q^55 + 27 * q^56 + 52 * q^57 - 87 * q^58 + 73 * q^59 + 18 * q^60 + 4 * q^61 - 5 * q^62 - 86 * q^63 + 69 * q^64 + 70 * q^65 - 28 * q^66 - 24 * q^67 + 24 * q^68 + 56 * q^69 - 33 * q^70 + 84 * q^71 - 99 * q^72 + 27 * q^73 + 5 * q^74 + 27 * q^75 + q^76 + 45 * q^77 - 22 * q^78 + 42 * q^79 + 13 * q^80 + 205 * q^81 - 47 * q^82 + q^83 + 7 * q^84 - 18 * q^85 + 23 * q^86 - q^87 - 42 * q^88 + 94 * q^89 - 14 * q^90 + 6 * q^91 + 25 * q^92 - 13 * q^93 - 13 * q^94 + 86 * q^95 + 35 * q^97 - 106 * q^98 + 83 * 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
$3$	−2.73247	−1.57759	−0.788796	−	0.614655i	$-0.789295\pi$
$3$	−2.73247	−1.57759	−0.788796	+	0.614655i	$0.789295\pi$
$4$	1.00000	0.500000
$5$	−4.08673	−1.82764	−0.913820	−	0.406120i	$-0.866882\pi$
$5$	−4.08673	−1.82764	−0.913820	+	0.406120i	$0.866882\pi$
$6$	2.73247	1.11553
$7$	−1.35525	−0.512235	−0.256118	−	0.966646i	$-0.582443\pi$
$7$	−1.35525	−0.512235	−0.256118	+	0.966646i	$0.582443\pi$
$8$	−1.00000	−0.353553
$9$	4.46639	1.48880
$10$	4.08673	1.29234
$11$	4.30075	1.29672	0.648362	−	0.761332i	$-0.275454\pi$
$11$	4.30075	1.29672	0.648362	+	0.761332i	$0.275454\pi$
$12$	−2.73247	−0.788796
$13$	−5.03138	−1.39545	−0.697726	−	0.716365i	$-0.745805\pi$
$13$	−5.03138	−1.39545	−0.697726	+	0.716365i	$0.745805\pi$
$14$	1.35525	0.362205
$15$	11.1669	2.88327
$16$	1.00000	0.250000
$17$	−4.20743	−1.02045	−0.510226	−	0.860040i	$-0.670438\pi$
$17$	−4.20743	−1.02045	−0.510226	+	0.860040i	$0.670438\pi$
$18$	−4.46639	−1.05274
$19$	6.88446	1.57940	0.789701	−	0.613492i	$-0.210236\pi$
$19$	6.88446	1.57940	0.789701	+	0.613492i	$0.210236\pi$
$20$	−4.08673	−0.913820
$21$	3.70317	0.808098
$22$	−4.30075	−0.916922
$23$	−0.530712	−0.110661	−0.0553305	−	0.998468i	$-0.517621\pi$
$23$	−0.530712	−0.110661	−0.0553305	+	0.998468i	$0.517621\pi$
$24$	2.73247	0.557763
$25$	11.7013	2.34027
$26$	5.03138	0.986734
$27$	−4.00686	−0.771121
$28$	−1.35525	−0.256118
$29$	4.32835	0.803754	0.401877	−	0.915694i	$-0.368358\pi$
$29$	4.32835	0.803754	0.401877	+	0.915694i	$0.368358\pi$
$30$	−11.1669	−2.03878
$31$	−0.860776	−0.154600	−0.0772999	−	0.997008i	$-0.524630\pi$
$31$	−0.860776	−0.154600	−0.0772999	+	0.997008i	$0.524630\pi$
$32$	−1.00000	−0.176777
$33$	−11.7517	−2.04570
$34$	4.20743	0.721569
$35$	5.53853	0.936182
$36$	4.46639	0.744398
$37$	−2.72088	−0.447310	−0.223655	−	0.974668i	$-0.571799\pi$
$37$	−2.72088	−0.447310	−0.223655	+	0.974668i	$0.571799\pi$
$38$	−6.88446	−1.11681
$39$	13.7481	2.20145
$40$	4.08673	0.646168
$41$	5.26479	0.822222	0.411111	−	0.911585i	$-0.365141\pi$
$41$	5.26479	0.822222	0.411111	+	0.911585i	$0.365141\pi$
$42$	−3.70317	−0.571412
$43$	−3.64472	−0.555814	−0.277907	−	0.960608i	$-0.589641\pi$
$43$	−3.64472	−0.555814	−0.277907	+	0.960608i	$0.589641\pi$
$44$	4.30075	0.648362
$45$	−18.2529	−2.72098
$46$	0.530712	0.0782492
$47$	13.5096	1.97058	0.985290	−	0.170889i	$-0.0546638\pi$
$47$	13.5096	1.97058	0.985290	+	0.170889i	$0.0546638\pi$
$48$	−2.73247	−0.394398
$49$	−5.16330	−0.737615
$50$	−11.7013	−1.65482
$51$	11.4967	1.60986
$52$	−5.03138	−0.697726
$53$	−11.0833	−1.52240	−0.761202	−	0.648515i	$-0.775390\pi$
$53$	−11.0833	−1.52240	−0.761202	+	0.648515i	$0.775390\pi$
$54$	4.00686	0.545265
$55$	−17.5760	−2.36994
$56$	1.35525	0.181103
$57$	−18.8116	−2.49165
$58$	−4.32835	−0.568340
$59$	−2.93144	−0.381641	−0.190821	−	0.981625i	$-0.561115\pi$
$59$	−2.93144	−0.381641	−0.190821	+	0.981625i	$0.561115\pi$
$60$	11.1669	1.44163
$61$	−14.4856	−1.85469	−0.927347	−	0.374202i	$-0.877917\pi$
$61$	−14.4856	−1.85469	−0.927347	+	0.374202i	$0.877917\pi$
$62$	0.860776	0.109319
$63$	−6.05306	−0.762614
$64$	1.00000	0.125000
$65$	20.5619	2.55038
$66$	11.7517	1.44653
$67$	6.99569	0.854660	0.427330	−	0.904096i	$-0.359454\pi$
$67$	6.99569	0.854660	0.427330	+	0.904096i	$0.359454\pi$
$68$	−4.20743	−0.510226
$69$	1.45015	0.174578
$70$	−5.53853	−0.661980
$71$	−8.33336	−0.988988	−0.494494	−	0.869181i	$-0.664647\pi$
$71$	−8.33336	−0.988988	−0.494494	+	0.869181i	$0.664647\pi$
$72$	−4.46639	−0.526369
$73$	−14.6670	−1.71664	−0.858320	−	0.513115i	$-0.828491\pi$
$73$	−14.6670	−1.71664	−0.858320	+	0.513115i	$0.828491\pi$
$74$	2.72088	0.316296
$75$	−31.9735	−3.69199
$76$	6.88446	0.789701
$77$	−5.82858	−0.664228
$78$	−13.7481	−1.55666
$79$	16.6767	1.87628	0.938138	−	0.346260i	$-0.112549\pi$
$79$	16.6767	1.87628	0.938138	+	0.346260i	$0.112549\pi$
$80$	−4.08673	−0.456910
$81$	−2.45054	−0.272282
$82$	−5.26479	−0.581399
$83$	11.2314	1.23281	0.616406	−	0.787429i	$-0.288588\pi$
$83$	11.2314	1.23281	0.616406	+	0.787429i	$0.288588\pi$
$84$	3.70317	0.404049
$85$	17.1946	1.86502
$86$	3.64472	0.393020
$87$	−11.8271	−1.26800
$88$	−4.30075	−0.458461
$89$	4.51767	0.478872	0.239436	−	0.970912i	$-0.423038\pi$
$89$	4.51767	0.478872	0.239436	+	0.970912i	$0.423038\pi$
$90$	18.2529	1.92403
$91$	6.81876	0.714800
$92$	−0.530712	−0.0553305
$93$	2.35204	0.243896
$94$	−13.5096	−1.39341
$95$	−28.1349	−2.88658
$96$	2.73247	0.278881
$97$	16.1322	1.63798	0.818988	−	0.573810i	$-0.194535\pi$
$97$	16.1322	1.63798	0.818988	+	0.573810i	$0.194535\pi$
$98$	5.16330	0.521572
$99$	19.2088	1.93056
$100$	11.7013	1.17013
$101$	−4.09717	−0.407684	−0.203842	−	0.979004i	$-0.565343\pi$
$101$	−4.09717	−0.407684	−0.203842	+	0.979004i	$0.565343\pi$
$102$	−11.4967	−1.13834
$103$	−14.2313	−1.40225	−0.701124	−	0.713039i	$-0.747318\pi$
$103$	−14.2313	−1.40225	−0.701124	+	0.713039i	$0.747318\pi$
$104$	5.03138	0.493367
$105$	−15.1339	−1.47691
$106$	11.0833	1.07650
$107$	7.64197	0.738777	0.369388	−	0.929275i	$-0.379567\pi$
$107$	7.64197	0.738777	0.369388	+	0.929275i	$0.379567\pi$
$108$	−4.00686	−0.385561
$109$	−2.39965	−0.229845	−0.114922	−	0.993374i	$-0.536662\pi$
$109$	−2.39965	−0.229845	−0.114922	+	0.993374i	$0.536662\pi$
$110$	17.5760	1.67580
$111$	7.43472	0.705672
$112$	−1.35525	−0.128059
$113$	−5.26389	−0.495185	−0.247592	−	0.968864i	$-0.579639\pi$
$113$	−5.26389	−0.495185	−0.247592	+	0.968864i	$0.579639\pi$
$114$	18.8116	1.76186
$115$	2.16887	0.202248
$116$	4.32835	0.401877
$117$	−22.4721	−2.07754
$118$	2.93144	0.269861
$119$	5.70211	0.522712
$120$	−11.1669	−1.01939
$121$	7.49643	0.681494
$122$	14.4856	1.31147
$123$	−14.3859	−1.29713
$124$	−0.860776	−0.0772999
$125$	−27.3865	−2.44952
$126$	6.05306	0.539250
$127$	−17.4334	−1.54697	−0.773484	−	0.633816i	$-0.781488\pi$
$127$	−17.4334	−1.54697	−0.773484	+	0.633816i	$0.781488\pi$
$128$	−1.00000	−0.0883883
$129$	9.95908	0.876848
$130$	−20.5619	−1.80339
$131$	−22.0489	−1.92642	−0.963211	−	0.268746i	$-0.913391\pi$
$131$	−22.0489	−1.92642	−0.963211	+	0.268746i	$0.913391\pi$
$132$	−11.7517	−1.02285
$133$	−9.33014	−0.809026
$134$	−6.99569	−0.604336
$135$	16.3750	1.40933
$136$	4.20743	0.360784
$137$	8.77608	0.749791	0.374896	−	0.927067i	$-0.377679\pi$
$137$	8.77608	0.749791	0.374896	+	0.927067i	$0.377679\pi$
$138$	−1.45015	−0.123445
$139$	−16.3355	−1.38556	−0.692781	−	0.721148i	$-0.743615\pi$
$139$	−16.3355	−1.38556	−0.692781	+	0.721148i	$0.743615\pi$
$140$	5.53853	0.468091
$141$	−36.9146	−3.10877
$142$	8.33336	0.699320
$143$	−21.6387	−1.80952
$144$	4.46639	0.372199
$145$	−17.6888	−1.46897
$146$	14.6670	1.21385
$147$	14.1086	1.16366
$148$	−2.72088	−0.223655
$149$	−17.6163	−1.44318	−0.721592	−	0.692319i	$-0.756589\pi$
$149$	−17.6163	−1.44318	−0.721592	+	0.692319i	$0.756589\pi$
$150$	31.9735	2.61063
$151$	−9.04873	−0.736375	−0.368187	−	0.929752i	$-0.620022\pi$
$151$	−9.04873	−0.736375	−0.368187	+	0.929752i	$0.620022\pi$
$152$	−6.88446	−0.558403
$153$	−18.7920	−1.51925
$154$	5.82858	0.469680
$155$	3.51776	0.282553
$156$	13.7481	1.10073
$157$	−15.8574	−1.26556	−0.632781	−	0.774331i	$-0.718087\pi$
$157$	−15.8574	−1.26556	−0.632781	+	0.774331i	$0.718087\pi$
$158$	−16.6767	−1.32673
$159$	30.2847	2.40173
$160$	4.08673	0.323084
$161$	0.719246	0.0566845
$162$	2.45054	0.192532
$163$	10.3982	0.814453	0.407227	−	0.913327i	$-0.366496\pi$
$163$	10.3982	0.814453	0.407227	+	0.913327i	$0.366496\pi$
$164$	5.26479	0.411111
$165$	48.0258	3.73881
$166$	−11.2314	−0.871729
$167$	2.66790	0.206448	0.103224	−	0.994658i	$-0.467084\pi$
$167$	2.66790	0.206448	0.103224	+	0.994658i	$0.467084\pi$
$168$	−3.70317	−0.285706
$169$	12.3147	0.947287
$170$	−17.1946	−1.31877
$171$	30.7487	2.35141
$172$	−3.64472	−0.277907
$173$	−14.7665	−1.12268	−0.561338	−	0.827586i	$-0.689713\pi$
$173$	−14.7665	−1.12268	−0.561338	+	0.827586i	$0.689713\pi$
$174$	11.8271	0.896609
$175$	−15.8582	−1.19877
$176$	4.30075	0.324181
$177$	8.01007	0.602074
$178$	−4.51767	−0.338613
$179$	9.75730	0.729295	0.364647	−	0.931146i	$-0.381190\pi$
$179$	9.75730	0.729295	0.364647	+	0.931146i	$0.381190\pi$
$180$	−18.2529	−1.36049
$181$	−7.03006	−0.522540	−0.261270	−	0.965266i	$-0.584141\pi$
$181$	−7.03006	−0.522540	−0.261270	+	0.965266i	$0.584141\pi$
$182$	−6.81876	−0.505440
$183$	39.5815	2.92595
$184$	0.530712	0.0391246
$185$	11.1195	0.817521
$186$	−2.35204	−0.172460
$187$	−18.0951	−1.32325
$188$	13.5096	0.985290
$189$	5.43029	0.394996
$190$	28.1349	2.04112
$191$	20.0088	1.44779	0.723893	−	0.689913i	$-0.242351\pi$
$191$	20.0088	1.44779	0.723893	+	0.689913i	$0.242351\pi$
$192$	−2.73247	−0.197199
$193$	5.20957	0.374993	0.187497	−	0.982265i	$-0.439963\pi$
$193$	5.20957	0.374993	0.187497	+	0.982265i	$0.439963\pi$
$194$	−16.1322	−1.15822
$195$	−56.1846	−4.02347
$196$	−5.16330	−0.368807
$197$	24.1840	1.72304	0.861518	−	0.507726i	$-0.169514\pi$
$197$	24.1840	1.72304	0.861518	+	0.507726i	$0.169514\pi$
$198$	−19.2088	−1.36511
$199$	−10.4690	−0.742130	−0.371065	−	0.928607i	$-0.621007\pi$
$199$	−10.4690	−0.742130	−0.371065	+	0.928607i	$0.621007\pi$
$200$	−11.7013	−0.827409
$201$	−19.1155	−1.34830
$202$	4.09717	0.288276
$203$	−5.86599	−0.411711
$204$	11.4967	0.804929
$205$	−21.5157	−1.50273
$206$	14.2313	0.991540
$207$	−2.37036	−0.164752
$208$	−5.03138	−0.348863
$209$	29.6083	2.04805
$210$	15.1339	1.04434
$211$	−8.32219	−0.572923	−0.286462	−	0.958092i	$-0.592479\pi$
$211$	−8.32219	−0.572923	−0.286462	+	0.958092i	$0.592479\pi$
$212$	−11.0833	−0.761202
$213$	22.7707	1.56022
$214$	−7.64197	−0.522394
$215$	14.8950	1.01583
$216$	4.00686	0.272632
$217$	1.16656	0.0791915
$218$	2.39965	0.162525
$219$	40.0771	2.70816
$220$	−17.5760	−1.18497
$221$	21.1692	1.42399
$222$	−7.43472	−0.498986
$223$	−17.8789	−1.19726	−0.598629	−	0.801026i	$-0.704288\pi$
$223$	−17.8789	−1.19726	−0.598629	+	0.801026i	$0.704288\pi$
$224$	1.35525	0.0905513
$225$	52.2627	3.48418
$226$	5.26389	0.350148
$227$	25.6469	1.70224	0.851121	−	0.524969i	$-0.175923\pi$
$227$	25.6469	1.70224	0.851121	+	0.524969i	$0.175923\pi$
$228$	−18.8116	−1.24583
$229$	9.41763	0.622335	0.311167	−	0.950355i	$-0.399280\pi$
$229$	9.41763	0.622335	0.311167	+	0.950355i	$0.399280\pi$
$230$	−2.16887	−0.143011
$231$	15.9264	1.04788
$232$	−4.32835	−0.284170
$233$	−11.9230	−0.781105	−0.390552	−	0.920581i	$-0.627716\pi$
$233$	−11.9230	−0.781105	−0.390552	+	0.920581i	$0.627716\pi$
$234$	22.4721	1.46905
$235$	−55.2101	−3.60151
$236$	−2.93144	−0.190821
$237$	−45.5686	−2.96000
$238$	−5.70211	−0.369613
$239$	−11.4096	−0.738023	−0.369011	−	0.929425i	$-0.620304\pi$
$239$	−11.4096	−0.738023	−0.369011	+	0.929425i	$0.620304\pi$
$240$	11.1669	0.720817
$241$	13.6935	0.882075	0.441038	−	0.897489i	$-0.354611\pi$
$241$	13.6935	0.882075	0.441038	+	0.897489i	$0.354611\pi$
$242$	−7.49643	−0.481889
$243$	18.7166	1.20067
$244$	−14.4856	−0.927347
$245$	21.1010	1.34809
$246$	14.3859	0.917210
$247$	−34.6383	−2.20398
$248$	0.860776	0.0546593
$249$	−30.6896	−1.94487
$250$	27.3865	1.73208
$251$	6.72690	0.424598	0.212299	−	0.977205i	$-0.431905\pi$
$251$	6.72690	0.424598	0.212299	+	0.977205i	$0.431905\pi$
$252$	−6.05306	−0.381307
$253$	−2.28246	−0.143497
$254$	17.4334	1.09387
$255$	−46.9838	−2.94224
$256$	1.00000	0.0625000
$257$	−29.3739	−1.83230	−0.916148	−	0.400840i	$-0.868718\pi$
$257$	−29.3739	−1.83230	−0.916148	+	0.400840i	$0.868718\pi$
$258$	−9.95908	−0.620025
$259$	3.68746	0.229128
$260$	20.5619	1.27519
$261$	19.3321	1.19663
$262$	22.0489	1.36219
$263$	31.2786	1.92872	0.964359	−	0.264596i	$-0.0852388\pi$
$263$	31.2786	1.92872	0.964359	+	0.264596i	$0.0852388\pi$
$264$	11.7517	0.723265
$265$	45.2943	2.78240
$266$	9.33014	0.572068
$267$	−12.3444	−0.755464
$268$	6.99569	0.427330
$269$	10.1829	0.620864	0.310432	−	0.950596i	$-0.399526\pi$
$269$	10.1829	0.620864	0.310432	+	0.950596i	$0.399526\pi$
$270$	−16.3750	−0.996548
$271$	25.3241	1.53833	0.769164	−	0.639051i	$-0.220673\pi$
$271$	25.3241	1.53833	0.769164	+	0.639051i	$0.220673\pi$
$272$	−4.20743	−0.255113
$273$	−18.6320	−1.12766
$274$	−8.77608	−0.530182
$275$	50.3245	3.03468
$276$	1.45015	0.0872890
$277$	−7.14994	−0.429598	−0.214799	−	0.976658i	$-0.568910\pi$
$277$	−7.14994	−0.429598	−0.214799	+	0.976658i	$0.568910\pi$
$278$	16.3355	0.979741
$279$	−3.84456	−0.230168
$280$	−5.53853	−0.330990
$281$	3.56118	0.212442	0.106221	−	0.994343i	$-0.466125\pi$
$281$	3.56118	0.212442	0.106221	+	0.994343i	$0.466125\pi$
$282$	36.9146	2.19823
$283$	−1.38023	−0.0820461	−0.0410230	−	0.999158i	$-0.513062\pi$
$283$	−1.38023	−0.0820461	−0.0410230	+	0.999158i	$0.513062\pi$
$284$	−8.33336	−0.494494
$285$	76.8777	4.55384
$286$	21.6387	1.27952
$287$	−7.13509	−0.421171
$288$	−4.46639	−0.263184
$289$	0.702490	0.0413229
$290$	17.6888	1.03872
$291$	−44.0807	−2.58406
$292$	−14.6670	−0.858320
$293$	−23.9333	−1.39820	−0.699101	−	0.715023i	$-0.746416\pi$
$293$	−23.9333	−1.39820	−0.699101	+	0.715023i	$0.746416\pi$
$294$	−14.1086	−0.822829
$295$	11.9800	0.697502
$296$	2.72088	0.158148
$297$	−17.2325	−0.999931
$298$	17.6163	1.02049
$299$	2.67021	0.154422
$300$	−31.9735	−1.84599
$301$	4.93950	0.284708
$302$	9.04873	0.520696
$303$	11.1954	0.643159
$304$	6.88446	0.394851
$305$	59.1988	3.38971
$306$	18.7920	1.07427
$307$	−19.4761	−1.11156	−0.555779	−	0.831330i	$-0.687580\pi$
$307$	−19.4761	−1.11156	−0.555779	+	0.831330i	$0.687580\pi$
$308$	−5.82858	−0.332114
$309$	38.8865	2.21218
$310$	−3.51776	−0.199795
$311$	1.15629	0.0655673	0.0327837	−	0.999462i	$-0.489563\pi$
$311$	1.15629	0.0655673	0.0327837	+	0.999462i	$0.489563\pi$
$312$	−13.7481	−0.778332
$313$	−5.48561	−0.310065	−0.155033	−	0.987909i	$-0.549548\pi$
$313$	−5.48561	−0.310065	−0.155033	+	0.987909i	$0.549548\pi$
$314$	15.8574	0.894887
$315$	24.7372	1.39378
$316$	16.6767	0.938138
$317$	24.3406	1.36710	0.683552	−	0.729902i	$-0.260434\pi$
$317$	24.3406	1.36710	0.683552	+	0.729902i	$0.260434\pi$
$318$	−30.2847	−1.69828
$319$	18.6151	1.04225
$320$	−4.08673	−0.228455
$321$	−20.8814	−1.16549
$322$	−0.719246	−0.0400820
$323$	−28.9659	−1.61170
$324$	−2.45054	−0.136141
$325$	−58.8738	−3.26573
$326$	−10.3982	−0.575905
$327$	6.55697	0.362601
$328$	−5.26479	−0.290699
$329$	−18.3089	−1.00940
$330$	−48.0258	−2.64373
$331$	−2.84215	−0.156219	−0.0781094	−	0.996945i	$-0.524888\pi$
$331$	−2.84215	−0.156219	−0.0781094	+	0.996945i	$0.524888\pi$
$332$	11.2314	0.616406
$333$	−12.1525	−0.665953
$334$	−2.66790	−0.145981
$335$	−28.5895	−1.56201
$336$	3.70317	0.202025
$337$	7.51623	0.409435	0.204718	−	0.978821i	$-0.434372\pi$
$337$	7.51623	0.409435	0.204718	+	0.978821i	$0.434372\pi$
$338$	−12.3147	−0.669833
$339$	14.3834	0.781199
$340$	17.1946	0.932510
$341$	−3.70198	−0.200473
$342$	−30.7487	−1.66270
$343$	16.4843	0.890068
$344$	3.64472	0.196510
$345$	−5.92638	−0.319066
$346$	14.7665	0.793852
$347$	−19.2586	−1.03386	−0.516928	−	0.856029i	$-0.672925\pi$
$347$	−19.2586	−1.03386	−0.516928	+	0.856029i	$0.672925\pi$
$348$	−11.8271	−0.633998
$349$	0.352320	0.0188593	0.00942964	−	0.999956i	$-0.496998\pi$
$349$	0.352320	0.0188593	0.00942964	+	0.999956i	$0.496998\pi$
$350$	15.8582	0.847657
$351$	20.1600	1.07606
$352$	−4.30075	−0.229231
$353$	31.9361	1.69979	0.849894	−	0.526954i	$-0.176666\pi$
$353$	31.9361	1.69979	0.849894	+	0.526954i	$0.176666\pi$
$354$	−8.01007	−0.425730
$355$	34.0562	1.80751
$356$	4.51767	0.239436
$357$	−15.5808	−0.824626
$358$	−9.75730	−0.515689
$359$	−24.0711	−1.27043	−0.635213	−	0.772337i	$-0.719088\pi$
$359$	−24.0711	−1.27043	−0.635213	+	0.772337i	$0.719088\pi$
$360$	18.2529	0.962013
$361$	28.3957	1.49451
$362$	7.03006	0.369492
$363$	−20.4838	−1.07512
$364$	6.81876	0.357400
$365$	59.9399	3.13740
$366$	−39.5815	−2.06896
$367$	23.1662	1.20927	0.604633	−	0.796504i	$-0.293320\pi$
$367$	23.1662	1.20927	0.604633	+	0.796504i	$0.293320\pi$
$368$	−0.530712	−0.0276653
$369$	23.5146	1.22412
$370$	−11.1195	−0.578075
$371$	15.0206	0.779829
$372$	2.35204	0.121948
$373$	19.5954	1.01461	0.507305	−	0.861766i	$-0.330642\pi$
$373$	19.5954	1.01461	0.507305	+	0.861766i	$0.330642\pi$
$374$	18.0951	0.935676
$375$	74.8328	3.86435
$376$	−13.5096	−0.696706
$377$	−21.7776	−1.12160
$378$	−5.43029	−0.279304
$379$	−14.1326	−0.725944	−0.362972	−	0.931800i	$-0.618238\pi$
$379$	−14.1326	−0.725944	−0.362972	+	0.931800i	$0.618238\pi$
$380$	−28.1349	−1.44329
$381$	47.6363	2.44048
$382$	−20.0088	−1.02374
$383$	−29.2470	−1.49445	−0.747225	−	0.664571i	$-0.768614\pi$
$383$	−29.2470	−1.49445	−0.747225	+	0.664571i	$0.768614\pi$
$384$	2.73247	0.139441
$385$	23.8198	1.21397
$386$	−5.20957	−0.265160
$387$	−16.2787	−0.827495
$388$	16.1322	0.818988
$389$	−4.16116	−0.210979	−0.105490	−	0.994420i	$-0.533641\pi$
$389$	−4.16116	−0.210979	−0.105490	+	0.994420i	$0.533641\pi$
$390$	56.1846	2.84502
$391$	2.23293	0.112924
$392$	5.16330	0.260786
$393$	60.2480	3.03911
$394$	−24.1840	−1.21837
$395$	−68.1532	−3.42916
$396$	19.2088	0.965279
$397$	3.82312	0.191877	0.0959385	−	0.995387i	$-0.469415\pi$
$397$	3.82312	0.191877	0.0959385	+	0.995387i	$0.469415\pi$
$398$	10.4690	0.524765
$399$	25.4943	1.27631
$400$	11.7013	0.585067
$401$	8.05470	0.402232	0.201116	−	0.979567i	$-0.435543\pi$
$401$	8.05470	0.402232	0.201116	+	0.979567i	$0.435543\pi$
$402$	19.1155	0.953395
$403$	4.33089	0.215737
$404$	−4.09717	−0.203842
$405$	10.0147	0.497633
$406$	5.86599	0.291124
$407$	−11.7018	−0.580037
$408$	−11.4967	−0.569171
$409$	−14.8360	−0.733594	−0.366797	−	0.930301i	$-0.619546\pi$
$409$	−14.8360	−0.733594	−0.366797	+	0.930301i	$0.619546\pi$
$410$	21.5157	1.06259
$411$	−23.9804	−1.18286
$412$	−14.2313	−0.701124
$413$	3.97283	0.195490
$414$	2.37036	0.116497
$415$	−45.8998	−2.25313
$416$	5.03138	0.246683
$417$	44.6364	2.18585
$418$	−29.6083	−1.44819
$419$	35.0106	1.71038	0.855189	−	0.518316i	$-0.173441\pi$
$419$	35.0106	1.71038	0.855189	+	0.518316i	$0.173441\pi$
$420$	−15.1339	−0.738456
$421$	−14.9502	−0.728627	−0.364313	−	0.931276i	$-0.618696\pi$
$421$	−14.9502	−0.728627	−0.364313	+	0.931276i	$0.618696\pi$
$422$	8.32219	0.405118
$423$	60.3392	2.93379
$424$	11.0833	0.538251
$425$	−49.2326	−2.38813
$426$	−22.7707	−1.10324
$427$	19.6316	0.950040
$428$	7.64197	0.369388
$429$	59.1270	2.85468
$430$	−14.8950	−0.718299
$431$	14.8212	0.713914	0.356957	−	0.934121i	$-0.383814\pi$
$431$	14.8212	0.713914	0.356957	+	0.934121i	$0.383814\pi$
$432$	−4.00686	−0.192780
$433$	−15.2126	−0.731069	−0.365535	−	0.930798i	$-0.619114\pi$
$433$	−15.2126	−0.731069	−0.365535	+	0.930798i	$0.619114\pi$
$434$	−1.16656	−0.0559969
$435$	48.3341	2.31744
$436$	−2.39965	−0.114922
$437$	−3.65366	−0.174778
$438$	−40.0771	−1.91496
$439$	−33.6882	−1.60785	−0.803925	−	0.594731i	$-0.797259\pi$
$439$	−33.6882	−1.60785	−0.803925	+	0.594731i	$0.797259\pi$
$440$	17.5760	0.837902
$441$	−23.0613	−1.09816
$442$	−21.1692	−1.00691
$443$	−21.6006	−1.02628	−0.513138	−	0.858306i	$-0.671517\pi$
$443$	−21.6006	−1.02628	−0.513138	+	0.858306i	$0.671517\pi$
$444$	7.43472	0.352836
$445$	−18.4625	−0.875205
$446$	17.8789	0.846590
$447$	48.1360	2.27676
$448$	−1.35525	−0.0640294
$449$	15.4668	0.729923	0.364962	−	0.931023i	$-0.381082\pi$
$449$	15.4668	0.729923	0.364962	+	0.931023i	$0.381082\pi$
$450$	−52.2627	−2.46369
$451$	22.6425	1.06619
$452$	−5.26389	−0.247592
$453$	24.7254	1.16170
$454$	−25.6469	−1.20367
$455$	−27.8664	−1.30640
$456$	18.8116	0.880932
$457$	31.6315	1.47966	0.739830	−	0.672794i	$-0.234906\pi$
$457$	31.6315	1.47966	0.739830	+	0.672794i	$0.234906\pi$
$458$	−9.41763	−0.440057
$459$	16.8586	0.786892
$460$	2.16887	0.101124
$461$	−31.2176	−1.45395	−0.726974	−	0.686665i	$-0.759074\pi$
$461$	−31.2176	−1.45395	−0.726974	+	0.686665i	$0.759074\pi$
$462$	−15.9264	−0.740964
$463$	6.66590	0.309791	0.154895	−	0.987931i	$-0.450496\pi$
$463$	6.66590	0.309791	0.154895	+	0.987931i	$0.450496\pi$
$464$	4.32835	0.200939
$465$	−9.61216	−0.445753
$466$	11.9230	0.552324
$467$	24.2057	1.12011	0.560053	−	0.828457i	$-0.310781\pi$
$467$	24.2057	1.12011	0.560053	+	0.828457i	$0.310781\pi$
$468$	−22.4721	−1.03877
$469$	−9.48090	−0.437787
$470$	55.2101	2.54665
$471$	43.3300	1.99654
$472$	2.93144	0.134930
$473$	−15.6750	−0.720738
$474$	45.5686	2.09304
$475$	80.5573	3.69622
$476$	5.70211	0.261356
$477$	−49.5022	−2.26655
$478$	11.4096	0.521861
$479$	22.7448	1.03924	0.519618	−	0.854399i	$-0.326074\pi$
$479$	22.7448	1.03924	0.519618	+	0.854399i	$0.326074\pi$
$480$	−11.1669	−0.509695
$481$	13.6898	0.624199
$482$	−13.6935	−0.623721
$483$	−1.96532	−0.0894250
$484$	7.49643	0.340747
$485$	−65.9279	−2.99363
$486$	−18.7166	−0.849002
$487$	43.4025	1.96675	0.983377	−	0.181577i	$-0.0581201\pi$
$487$	43.4025	1.96675	0.983377	+	0.181577i	$0.0581201\pi$
$488$	14.4856	0.655734
$489$	−28.4129	−1.28487
$490$	−21.1010	−0.953247
$491$	15.8142	0.713683	0.356842	−	0.934165i	$-0.383854\pi$
$491$	15.8142	0.713683	0.356842	+	0.934165i	$0.383854\pi$
$492$	−14.3859	−0.648565
$493$	−18.2112	−0.820193
$494$	34.6383	1.55845
$495$	−78.5012	−3.52836
$496$	−0.860776	−0.0386500
$497$	11.2938	0.506595
$498$	30.6896	1.37523
$499$	−7.38346	−0.330529	−0.165265	−	0.986249i	$-0.552848\pi$
$499$	−7.38346	−0.330529	−0.165265	+	0.986249i	$0.552848\pi$
$500$	−27.3865	−1.22476
$501$	−7.28995	−0.325691
$502$	−6.72690	−0.300236
$503$	31.6908	1.41302	0.706512	−	0.707701i	$-0.250268\pi$
$503$	31.6908	1.41302	0.706512	+	0.707701i	$0.250268\pi$
$504$	6.05306	0.269625
$505$	16.7440	0.745099
$506$	2.28246	0.101468
$507$	−33.6496	−1.49443
$508$	−17.4334	−0.773484
$509$	−29.8553	−1.32331	−0.661657	−	0.749806i	$-0.730147\pi$
$509$	−29.8553	−1.32331	−0.661657	+	0.749806i	$0.730147\pi$
$510$	46.9838	2.08048
$511$	19.8774	0.879323
$512$	−1.00000	−0.0441942
$513$	−27.5851	−1.21791
$514$	29.3739	1.29563
$515$	58.1593	2.56281
$516$	9.95908	0.438424
$517$	58.1015	2.55530
$518$	−3.68746	−0.162018
$519$	40.3490	1.77113
$520$	−20.5619	−0.901697
$521$	42.5519	1.86423	0.932116	−	0.362159i	$-0.117960\pi$
$521$	42.5519	1.86423	0.932116	+	0.362159i	$0.117960\pi$
$522$	−19.3321	−0.846143
$523$	1.33999	0.0585936	0.0292968	−	0.999571i	$-0.490673\pi$
$523$	1.33999	0.0585936	0.0292968	+	0.999571i	$0.490673\pi$
$524$	−22.0489	−0.963211
$525$	43.3321	1.89117
$526$	−31.2786	−1.36381
$527$	3.62166	0.157762
$528$	−11.7517	−0.511425
$529$	−22.7183	−0.987754
$530$	−45.2943	−1.96746
$531$	−13.0930	−0.568186
$532$	−9.33014	−0.404513
$533$	−26.4891	−1.14737
$534$	12.3444	0.534194
$535$	−31.2306	−1.35022
$536$	−6.99569	−0.302168
$537$	−26.6615	−1.15053
$538$	−10.1829	−0.439017
$539$	−22.2061	−0.956483
$540$	16.3750	0.704666
$541$	−6.79469	−0.292127	−0.146063	−	0.989275i	$-0.546660\pi$
$541$	−6.79469	−0.292127	−0.146063	+	0.989275i	$0.546660\pi$
$542$	−25.3241	−1.08776
$543$	19.2094	0.824355
$544$	4.20743	0.180392
$545$	9.80671	0.420073
$546$	18.6320	0.797378
$547$	5.43049	0.232191	0.116096	−	0.993238i	$-0.462962\pi$
$547$	5.43049	0.232191	0.116096	+	0.993238i	$0.462962\pi$
$548$	8.77608	0.374896
$549$	−64.6985	−2.76126
$550$	−50.3245	−2.14584
$551$	29.7983	1.26945
$552$	−1.45015	−0.0617226
$553$	−22.6011	−0.961095
$554$	7.14994	0.303772
$555$	−30.3837	−1.28971
$556$	−16.3355	−0.692781
$557$	9.28644	0.393479	0.196740	−	0.980456i	$-0.436965\pi$
$557$	9.28644	0.393479	0.196740	+	0.980456i	$0.436965\pi$
$558$	3.84456	0.162753
$559$	18.3379	0.775613
$560$	5.53853	0.234045
$561$	49.4443	2.08754
$562$	−3.56118	−0.150219
$563$	−23.9763	−1.01048	−0.505239	−	0.862979i	$-0.668596\pi$
$563$	−23.9763	−1.01048	−0.505239	+	0.862979i	$0.668596\pi$
$564$	−36.9146	−1.55439
$565$	21.5121	0.905019
$566$	1.38023	0.0580153
$567$	3.32108	0.139472
$568$	8.33336	0.349660
$569$	7.01610	0.294130	0.147065	−	0.989127i	$-0.453017\pi$
$569$	7.01610	0.294130	0.147065	+	0.989127i	$0.453017\pi$
$570$	−76.8777	−3.22005
$571$	−28.3412	−1.18604	−0.593020	−	0.805187i	$-0.702065\pi$
$571$	−28.3412	−1.18604	−0.593020	+	0.805187i	$0.702065\pi$
$572$	−21.6387	−0.904758
$573$	−54.6734	−2.28401
$574$	7.13509	0.297813
$575$	−6.21003	−0.258976
$576$	4.46639	0.186100
$577$	−15.3417	−0.638684	−0.319342	−	0.947640i	$-0.603462\pi$
$577$	−15.3417	−0.638684	−0.319342	+	0.947640i	$0.603462\pi$
$578$	−0.702490	−0.0292197
$579$	−14.2350	−0.591586
$580$	−17.6888	−0.734487
$581$	−15.2214	−0.631490
$582$	44.0807	1.82721
$583$	−47.6663	−1.97414
$584$	14.6670	0.606924
$585$	91.8372	3.79700
$586$	23.9333	0.988678
$587$	−24.4406	−1.00877	−0.504386	−	0.863478i	$-0.668281\pi$
$587$	−24.4406	−1.00877	−0.504386	+	0.863478i	$0.668281\pi$
$588$	14.1086	0.581828
$589$	−5.92597	−0.244175
$590$	−11.9800	−0.493209
$591$	−66.0820	−2.71825
$592$	−2.72088	−0.111827
$593$	23.5283	0.966193	0.483097	−	0.875567i	$-0.339512\pi$
$593$	23.5283	0.966193	0.483097	+	0.875567i	$0.339512\pi$
$594$	17.2325	0.707058
$595$	−23.3030	−0.955329
$596$	−17.6163	−0.721592
$597$	28.6063	1.17078
$598$	−2.67021	−0.109193
$599$	−12.4485	−0.508632	−0.254316	−	0.967121i	$-0.581850\pi$
$599$	−12.4485	−0.508632	−0.254316	+	0.967121i	$0.581850\pi$
$600$	31.9735	1.30531
$601$	−33.6740	−1.37359	−0.686795	−	0.726851i	$-0.740983\pi$
$601$	−33.6740	−1.37359	−0.686795	+	0.726851i	$0.740983\pi$
$602$	−4.93950	−0.201319
$603$	31.2455	1.27241
$604$	−9.04873	−0.368187
$605$	−30.6359	−1.24552
$606$	−11.1954	−0.454782
$607$	28.0003	1.13650	0.568248	−	0.822857i	$-0.307621\pi$
$607$	28.0003	1.13650	0.568248	+	0.822857i	$0.307621\pi$
$608$	−6.88446	−0.279202
$609$	16.0286	0.649513
$610$	−59.1988	−2.39689
$611$	−67.9720	−2.74985
$612$	−18.7920	−0.759623
$613$	34.1042	1.37746	0.688729	−	0.725019i	$-0.258169\pi$
$613$	34.1042	1.37746	0.688729	+	0.725019i	$0.258169\pi$
$614$	19.4761	0.785990
$615$	58.7911	2.37069
$616$	5.82858	0.234840
$617$	32.6245	1.31341	0.656706	−	0.754147i	$-0.271949\pi$
$617$	32.6245	1.31341	0.656706	+	0.754147i	$0.271949\pi$
$618$	−38.8865	−1.56424
$619$	−21.7147	−0.872789	−0.436395	−	0.899755i	$-0.643745\pi$
$619$	−21.7147	−0.872789	−0.436395	+	0.899755i	$0.643745\pi$
$620$	3.51776	0.141276
$621$	2.12649	0.0853330
$622$	−1.15629	−0.0463631
$623$	−6.12256	−0.245295
$624$	13.7481	0.550364
$625$	53.4145	2.13658
$626$	5.48561	0.219249
$627$	−80.9038	−3.23099
$628$	−15.8574	−0.632781
$629$	11.4479	0.456458
$630$	−24.7372	−0.985554
$631$	4.50262	0.179246	0.0896232	−	0.995976i	$-0.471434\pi$
$631$	4.50262	0.179246	0.0896232	+	0.995976i	$0.471434\pi$
$632$	−16.6767	−0.663364
$633$	22.7401	0.903839
$634$	−24.3406	−0.966689
$635$	71.2457	2.82730
$636$	30.2847	1.20087
$637$	25.9785	1.02931
$638$	−18.6151	−0.736981
$639$	−37.2200	−1.47240
$640$	4.08673	0.161542
$641$	−23.0199	−0.909231	−0.454616	−	0.890688i	$-0.650223\pi$
$641$	−23.0199	−0.909231	−0.454616	+	0.890688i	$0.650223\pi$
$642$	20.8814	0.824125
$643$	21.7780	0.858841	0.429421	−	0.903105i	$-0.358718\pi$
$643$	21.7780	0.858841	0.429421	+	0.903105i	$0.358718\pi$
$644$	0.719246	0.0283422
$645$	−40.7001	−1.60256
$646$	28.9659	1.13965
$647$	−14.6821	−0.577215	−0.288607	−	0.957448i	$-0.593192\pi$
$647$	−14.6821	−0.577215	−0.288607	+	0.957448i	$0.593192\pi$
$648$	2.45054	0.0962662
$649$	−12.6074	−0.494883
$650$	58.8738	2.30922
$651$	−3.18760	−0.124932
$652$	10.3982	0.407227
$653$	10.4544	0.409114	0.204557	−	0.978855i	$-0.434425\pi$
$653$	10.4544	0.409114	0.204557	+	0.978855i	$0.434425\pi$
$654$	−6.55697	−0.256398
$655$	90.1079	3.52081
$656$	5.26479	0.205555
$657$	−65.5084	−2.55573
$658$	18.3089	0.713754
$659$	−35.4732	−1.38184	−0.690920	−	0.722931i	$-0.742794\pi$
$659$	−35.4732	−1.38184	−0.690920	+	0.722931i	$0.742794\pi$
$660$	48.0258	1.86940
$661$	22.5736	0.878010	0.439005	−	0.898485i	$-0.355331\pi$
$661$	22.5736	0.878010	0.439005	+	0.898485i	$0.355331\pi$
$662$	2.84215	0.110463
$663$	−57.8441	−2.24648
$664$	−11.2314	−0.435865
$665$	38.1297	1.47861
$666$	12.1525	0.470900
$667$	−2.29711	−0.0889443
$668$	2.66790	0.103224
$669$	48.8535	1.88879
$670$	28.5895	1.10451
$671$	−62.2990	−2.40503
$672$	−3.70317	−0.142853
$673$	5.79258	0.223288	0.111644	−	0.993748i	$-0.464388\pi$
$673$	5.79258	0.223288	0.111644	+	0.993748i	$0.464388\pi$
$674$	−7.51623	−0.289514
$675$	−46.8856	−1.80463
$676$	12.3147	0.473644
$677$	−38.7919	−1.49089	−0.745446	−	0.666566i	$-0.767764\pi$
$677$	−38.7919	−1.49089	−0.745446	+	0.666566i	$0.767764\pi$
$678$	−14.3834	−0.552391
$679$	−21.8631	−0.839030
$680$	−17.1946	−0.659384
$681$	−70.0793	−2.68544
$682$	3.70198	0.141756
$683$	18.3544	0.702310	0.351155	−	0.936317i	$-0.385789\pi$
$683$	18.3544	0.702310	0.351155	+	0.936317i	$0.385789\pi$
$684$	30.7487	1.17570
$685$	−35.8654	−1.37035
$686$	−16.4843	−0.629373
$687$	−25.7334	−0.981790
$688$	−3.64472	−0.138954
$689$	55.7641	2.12444
$690$	5.92638	0.225613
$691$	−11.5724	−0.440234	−0.220117	−	0.975473i	$-0.570644\pi$
$691$	−11.5724	−0.440234	−0.220117	+	0.975473i	$0.570644\pi$
$692$	−14.7665	−0.561338
$693$	−26.0327	−0.988900
$694$	19.2586	0.731047
$695$	66.7589	2.53231
$696$	11.8271	0.448304
$697$	−22.1512	−0.839038
$698$	−0.352320	−0.0133355
$699$	32.5794	1.23226
$700$	−15.8582	−0.599384
$701$	3.15134	0.119025	0.0595123	−	0.998228i	$-0.481045\pi$
$701$	3.15134	0.119025	0.0595123	+	0.998228i	$0.481045\pi$
$702$	−20.1600	−0.760891
$703$	−18.7318	−0.706482
$704$	4.30075	0.162091
$705$	150.860	5.68172
$706$	−31.9361	−1.20193
$707$	5.55268	0.208830
$708$	8.01007	0.301037
$709$	−9.92621	−0.372787	−0.186393	−	0.982475i	$-0.559680\pi$
$709$	−9.92621	−0.372787	−0.186393	+	0.982475i	$0.559680\pi$
$710$	−34.0562	−1.27811
$711$	74.4847	2.79339
$712$	−4.51767	−0.169307
$713$	0.456824	0.0171082
$714$	15.5808	0.583099
$715$	88.4313	3.30714
$716$	9.75730	0.364647
$717$	31.1763	1.16430
$718$	24.0711	0.898327
$719$	37.4728	1.39750	0.698749	−	0.715367i	$-0.253740\pi$
$719$	37.4728	1.39750	0.698749	+	0.715367i	$0.253740\pi$
$720$	−18.2529	−0.680246
$721$	19.2869	0.718281
$722$	−28.3957	−1.05678
$723$	−37.4170	−1.39155
$724$	−7.03006	−0.261270
$725$	50.6475	1.88100
$726$	20.4838	0.760224
$727$	23.6589	0.877458	0.438729	−	0.898619i	$-0.355429\pi$
$727$	23.6589	0.877458	0.438729	+	0.898619i	$0.355429\pi$
$728$	−6.81876	−0.252720
$729$	−43.7909	−1.62189
$730$	−59.9399	−2.21848
$731$	15.3349	0.567182
$732$	39.5815	1.46298
$733$	5.58107	0.206142	0.103071	−	0.994674i	$-0.467133\pi$
$733$	5.58107	0.206142	0.103071	+	0.994674i	$0.467133\pi$
$734$	−23.1662	−0.855081
$735$	−57.6579	−2.12674
$736$	0.530712	0.0195623
$737$	30.0867	1.10826
$738$	−23.5146	−0.865584
$739$	−29.1816	−1.07346	−0.536731	−	0.843753i	$-0.680341\pi$
$739$	−29.1816	−1.07346	−0.536731	+	0.843753i	$0.680341\pi$
$740$	11.1195	0.408761
$741$	94.6480	3.47698
$742$	−15.0206	−0.551422
$743$	41.7940	1.53327	0.766637	−	0.642081i	$-0.221929\pi$
$743$	41.7940	1.53327	0.766637	+	0.642081i	$0.221929\pi$
$744$	−2.35204	−0.0862301
$745$	71.9930	2.63762
$746$	−19.5954	−0.717438
$747$	50.1640	1.83540
$748$	−18.0951	−0.661623
$749$	−10.3568	−0.378428
$750$	−74.8328	−2.73251
$751$	−33.0255	−1.20512	−0.602558	−	0.798075i	$-0.705852\pi$
$751$	−33.0255	−1.20512	−0.602558	+	0.798075i	$0.705852\pi$
$752$	13.5096	0.492645
$753$	−18.3810	−0.669843
$754$	21.7776	0.793092
$755$	36.9797	1.34583
$756$	5.43029	0.197498
$757$	−17.4126	−0.632870	−0.316435	−	0.948614i	$-0.602486\pi$
$757$	−17.4126	−0.632870	−0.316435	+	0.948614i	$0.602486\pi$
$758$	14.1326	0.513320
$759$	6.23674	0.226379
$760$	28.1349	1.02056
$761$	−5.06596	−0.183641	−0.0918205	−	0.995776i	$-0.529269\pi$
$761$	−5.06596	−0.183641	−0.0918205	+	0.995776i	$0.529269\pi$
$762$	−47.6363	−1.72568
$763$	3.25212	0.117735
$764$	20.0088	0.723893
$765$	76.7979	2.77663
$766$	29.2470	1.05674
$767$	14.7492	0.532562
$768$	−2.73247	−0.0985995
$769$	30.3624	1.09490	0.547448	−	0.836840i	$-0.315599\pi$
$769$	30.3624	1.09490	0.547448	+	0.836840i	$0.315599\pi$
$770$	−23.8198	−0.858406
$771$	80.2634	2.89062
$772$	5.20957	0.187497
$773$	45.1782	1.62495	0.812473	−	0.582999i	$-0.198121\pi$
$773$	45.1782	1.62495	0.812473	+	0.582999i	$0.198121\pi$
$774$	16.2787	0.585127
$775$	−10.0722	−0.361805
$776$	−16.1322	−0.579112
$777$	−10.0759	−0.361470
$778$	4.16116	0.149185
$779$	36.2452	1.29862
$780$	−56.1846	−2.01173
$781$	−35.8397	−1.28244
$782$	−2.23293	−0.0798495
$783$	−17.3431	−0.619792
$784$	−5.16330	−0.184404
$785$	64.8050	2.31299
$786$	−60.2480	−2.14897
$787$	−8.13352	−0.289929	−0.144964	−	0.989437i	$-0.546307\pi$
$787$	−8.13352	−0.289929	−0.144964	+	0.989437i	$0.546307\pi$
$788$	24.1840	0.861518
$789$	−85.4677	−3.04273
$790$	68.1532	2.42478
$791$	7.13387	0.253651
$792$	−19.2088	−0.682555
$793$	72.8826	2.58814
$794$	−3.82312	−0.135678
$795$	−123.765	−4.38950
$796$	−10.4690	−0.371065
$797$	−16.3018	−0.577440	−0.288720	−	0.957414i	$-0.593230\pi$
$797$	−16.3018	−0.577440	−0.288720	+	0.957414i	$0.593230\pi$
$798$	−25.4943	−0.902489
$799$	−56.8408	−2.01088
$800$	−11.7013	−0.413705
$801$	20.1777	0.712942
$802$	−8.05470	−0.284421
$803$	−63.0789	−2.22601
$804$	−19.1155	−0.674152
$805$	−2.93936	−0.103599
$806$	−4.33089	−0.152549
$807$	−27.8245	−0.979470
$808$	4.09717	0.144138
$809$	20.7436	0.729305	0.364653	−	0.931144i	$-0.381188\pi$
$809$	20.7436	0.729305	0.364653	+	0.931144i	$0.381188\pi$
$810$	−10.0147	−0.351880
$811$	−7.10759	−0.249581	−0.124791	−	0.992183i	$-0.539826\pi$
$811$	−7.10759	−0.249581	−0.124791	+	0.992183i	$0.539826\pi$
$812$	−5.86599	−0.205856
$813$	−69.1973	−2.42685
$814$	11.7018	0.410148
$815$	−42.4948	−1.48853
$816$	11.4967	0.402464
$817$	−25.0919	−0.877855
$818$	14.8360	0.518729
$819$	30.4552	1.06419
$820$	−21.5157	−0.751363
$821$	3.17227	0.110713	0.0553566	−	0.998467i	$-0.482370\pi$
$821$	3.17227	0.110713	0.0553566	+	0.998467i	$0.482370\pi$
$822$	23.9804	0.836412
$823$	−13.8923	−0.484254	−0.242127	−	0.970245i	$-0.577845\pi$
$823$	−13.8923	−0.484254	−0.242127	+	0.970245i	$0.577845\pi$
$824$	14.2313	0.495770
$825$	−137.510	−4.78749
$826$	−3.97283	−0.138232
$827$	−46.6230	−1.62124	−0.810620	−	0.585572i	$-0.800870\pi$
$827$	−46.6230	−1.62124	−0.810620	+	0.585572i	$0.800870\pi$
$828$	−2.37036	−0.0823759
$829$	−5.99542	−0.208230	−0.104115	−	0.994565i	$-0.533201\pi$
$829$	−5.99542	−0.208230	−0.104115	+	0.994565i	$0.533201\pi$
$830$	45.8998	1.59321
$831$	19.5370	0.677730
$832$	−5.03138	−0.174432
$833$	21.7243	0.752701
$834$	−44.6364	−1.54563
$835$	−10.9030	−0.377313
$836$	29.6083	1.02402
$837$	3.44901	0.119215
$838$	−35.0106	−1.20942
$839$	−24.9133	−0.860104	−0.430052	−	0.902804i	$-0.641505\pi$
$839$	−24.9133	−0.860104	−0.430052	+	0.902804i	$0.641505\pi$
$840$	15.1339	0.522168
$841$	−10.2654	−0.353979
$842$	14.9502	0.515217
$843$	−9.73081	−0.335147
$844$	−8.32219	−0.286462
$845$	−50.3270	−1.73130
$846$	−60.3392	−2.07451
$847$	−10.1595	−0.349085
$848$	−11.0833	−0.380601
$849$	3.77143	0.129435
$850$	49.2326	1.68866
$851$	1.44400	0.0494998
$852$	22.7707	0.780110
$853$	−6.28403	−0.215161	−0.107581	−	0.994196i	$-0.534310\pi$
$853$	−6.28403	−0.215161	−0.107581	+	0.994196i	$0.534310\pi$
$854$	−19.6316	−0.671780
$855$	−125.661	−4.29753
$856$	−7.64197	−0.261197
$857$	−7.54397	−0.257697	−0.128848	−	0.991664i	$-0.541128\pi$
$857$	−7.54397	−0.257697	−0.128848	+	0.991664i	$0.541128\pi$
$858$	−59.1270	−2.01856
$859$	−41.0522	−1.40068	−0.700341	−	0.713808i	$-0.746969\pi$
$859$	−41.0522	−1.40068	−0.700341	+	0.713808i	$0.746969\pi$
$860$	14.8950	0.507914
$861$	19.4964	0.664436
$862$	−14.8212	−0.504813
$863$	11.2545	0.383109	0.191554	−	0.981482i	$-0.438647\pi$
$863$	11.2545	0.383109	0.191554	+	0.981482i	$0.438647\pi$
$864$	4.00686	0.136316
$865$	60.3467	2.05185
$866$	15.2126	0.516944
$867$	−1.91953	−0.0651907
$868$	1.16656	0.0395958
$869$	71.7223	2.43301
$870$	−48.3341	−1.63868
$871$	−35.1980	−1.19264
$872$	2.39965	0.0812624
$873$	72.0527	2.43861
$874$	3.65366	0.123587
$875$	37.1155	1.25473
$876$	40.0771	1.35408
$877$	−22.1861	−0.749171	−0.374585	−	0.927192i	$-0.622215\pi$
$877$	−22.1861	−0.749171	−0.374585	+	0.927192i	$0.622215\pi$
$878$	33.6882	1.13692
$879$	65.3971	2.20579
$880$	−17.5760	−0.592486
$881$	−18.2818	−0.615928	−0.307964	−	0.951398i	$-0.599648\pi$
$881$	−18.2818	−0.615928	−0.307964	+	0.951398i	$0.599648\pi$
$882$	23.0613	0.776515
$883$	−7.06550	−0.237773	−0.118887	−	0.992908i	$-0.537932\pi$
$883$	−7.06550	−0.237773	−0.118887	+	0.992908i	$0.537932\pi$
$884$	21.1692	0.711996
$885$	−32.7350	−1.10037
$886$	21.6006	0.725686
$887$	15.0990	0.506976	0.253488	−	0.967339i	$-0.418422\pi$
$887$	15.0990	0.506976	0.253488	+	0.967339i	$0.418422\pi$
$888$	−7.43472	−0.249493
$889$	23.6266	0.792411
$890$	18.4625	0.618863
$891$	−10.5391	−0.353075
$892$	−17.8789	−0.598629
$893$	93.0064	3.11234
$894$	−48.1360	−1.60991
$895$	−39.8754	−1.33289
$896$	1.35525	0.0452756
$897$	−7.29626	−0.243615
$898$	−15.4668	−0.516134
$899$	−3.72574	−0.124260
$900$	52.2627	1.74209
$901$	46.6321	1.55354
$902$	−22.6425	−0.753914
$903$	−13.4970	−0.449153
$904$	5.26389	0.175074
$905$	28.7299	0.955015
$906$	−24.7254	−0.821445
$907$	−21.0072	−0.697534	−0.348767	−	0.937210i	$-0.613399\pi$
$907$	−21.0072	−0.697534	−0.348767	+	0.937210i	$0.613399\pi$
$908$	25.6469	0.851121
$909$	−18.2996	−0.606958
$910$	27.8664	0.923762
$911$	44.3939	1.47084	0.735418	−	0.677613i	$-0.236986\pi$
$911$	44.3939	1.47084	0.735418	+	0.677613i	$0.236986\pi$
$912$	−18.8116	−0.622913
$913$	48.3036	1.59862
$914$	−31.6315	−1.04628
$915$	−161.759	−5.34758
$916$	9.41763	0.311167
$917$	29.8817	0.986782
$918$	−16.8586	−0.556417
$919$	−27.4570	−0.905724	−0.452862	−	0.891581i	$-0.649597\pi$
$919$	−27.4570	−0.905724	−0.452862	+	0.891581i	$0.649597\pi$
$920$	−2.16887	−0.0715056
$921$	53.2178	1.75359
$922$	31.2176	1.02810
$923$	41.9283	1.38009
$924$	15.9264	0.523940
$925$	−31.8379	−1.04682
$926$	−6.66590	−0.219055
$927$	−63.5624	−2.08766
$928$	−4.32835	−0.142085
$929$	10.1079	0.331628	0.165814	−	0.986157i	$-0.446975\pi$
$929$	10.1079	0.331628	0.165814	+	0.986157i	$0.446975\pi$
$930$	9.61216	0.315195
$931$	−35.5465	−1.16499
$932$	−11.9230	−0.390552
$933$	−3.15953	−0.103438
$934$	−24.2057	−0.792035
$935$	73.9497	2.41842
$936$	22.4721	0.734523
$937$	1.02721	0.0335575	0.0167787	−	0.999859i	$-0.494659\pi$
$937$	1.02721	0.0335575	0.0167787	+	0.999859i	$0.494659\pi$
$938$	9.48090	0.309562
$939$	14.9893	0.489156
$940$	−55.2101	−1.80076
$941$	14.4376	0.470651	0.235325	−	0.971917i	$-0.424384\pi$
$941$	14.4376	0.470651	0.235325	+	0.971917i	$0.424384\pi$
$942$	−43.3300	−1.41177
$943$	−2.79408	−0.0909879
$944$	−2.93144	−0.0954103
$945$	−22.1921	−0.721909
$946$	15.6750	0.509639
$947$	8.51934	0.276841	0.138421	−	0.990374i	$-0.455797\pi$
$947$	8.51934	0.276841	0.138421	+	0.990374i	$0.455797\pi$
$948$	−45.5686	−1.48000
$949$	73.7950	2.39549
$950$	−80.5573	−2.61362
$951$	−66.5100	−2.15673
$952$	−5.70211	−0.184807
$953$	−35.1353	−1.13814	−0.569072	−	0.822288i	$-0.692697\pi$
$953$	−35.1353	−1.13814	−0.569072	+	0.822288i	$0.692697\pi$
$954$	49.5022	1.60269
$955$	−81.7705	−2.64603
$956$	−11.4096	−0.369011
$957$	−50.8653	−1.64424
$958$	−22.7448	−0.734851
$959$	−11.8938	−0.384070
$960$	11.1669	0.360409
$961$	−30.2591	−0.976099
$962$	−13.6898	−0.441376
$963$	34.1320	1.09989
$964$	13.6935	0.441038
$965$	−21.2901	−0.685353
$966$	1.96532	0.0632330
$967$	−29.6366	−0.953050	−0.476525	−	0.879161i	$-0.658104\pi$
$967$	−29.6366	−0.953050	−0.476525	+	0.879161i	$0.658104\pi$
$968$	−7.49643	−0.240944
$969$	79.1484	2.54261
$970$	65.9279	2.11682
$971$	6.51679	0.209134	0.104567	−	0.994518i	$-0.466654\pi$
$971$	6.51679	0.209134	0.104567	+	0.994518i	$0.466654\pi$
$972$	18.7166	0.600335
$973$	22.1387	0.709734
$974$	−43.4025	−1.39070
$975$	160.871	5.15199
$976$	−14.4856	−0.463674
$977$	14.3363	0.458658	0.229329	−	0.973349i	$-0.426347\pi$
$977$	14.3363	0.458658	0.229329	+	0.973349i	$0.426347\pi$
$978$	28.4129	0.908543
$979$	19.4293	0.620964
$980$	21.1010	0.674047
$981$	−10.7178	−0.342192
$982$	−15.8142	−0.504650
$983$	−51.8644	−1.65422	−0.827108	−	0.562042i	$-0.810016\pi$
$983$	−51.8644	−1.65422	−0.827108	+	0.562042i	$0.810016\pi$
$984$	14.3859	0.458605
$985$	−98.8333	−3.14909
$986$	18.2112	0.579964
$987$	50.0285	1.59242
$988$	−34.6383	−1.10199
$989$	1.93429	0.0615070
$990$	78.5012	2.49493
$991$	−7.85193	−0.249425	−0.124712	−	0.992193i	$-0.539801\pi$
$991$	−7.85193	−0.249425	−0.124712	+	0.992193i	$0.539801\pi$
$992$	0.860776	0.0273297
$993$	7.76609	0.246449
$994$	−11.2938	−0.358217
$995$	42.7841	1.35635
$996$	−30.6896	−0.972437
$997$	52.0003	1.64687	0.823433	−	0.567413i	$-0.192056\pi$
$997$	52.0003	1.64687	0.823433	+	0.567413i	$0.192056\pi$
$998$	7.38346	0.233719
$999$	10.9022	0.344930

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.g.1.9	✓	69

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.73247 q^{3} +1.00000 q^{4} -4.08673 q^{5} +2.73247 q^{6} -1.35525 q^{7} -1.00000 q^{8} +4.46639 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.73247 q^{3} +1.00000 q^{4} -4.08673 q^{5} +2.73247 q^{6} -1.35525 q^{7} -1.00000 q^{8} +4.46639 q^{9} +4.08673 q^{10} +4.30075 q^{11} -2.73247 q^{12} -5.03138 q^{13} +1.35525 q^{14} +11.1669 q^{15} +1.00000 q^{16} -4.20743 q^{17} -4.46639 q^{18} +6.88446 q^{19} -4.08673 q^{20} +3.70317 q^{21} -4.30075 q^{22} -0.530712 q^{23} +2.73247 q^{24} +11.7013 q^{25} +5.03138 q^{26} -4.00686 q^{27} -1.35525 q^{28} +4.32835 q^{29} -11.1669 q^{30} -0.860776 q^{31} -1.00000 q^{32} -11.7517 q^{33} +4.20743 q^{34} +5.53853 q^{35} +4.46639 q^{36} -2.72088 q^{37} -6.88446 q^{38} +13.7481 q^{39} +4.08673 q^{40} +5.26479 q^{41} -3.70317 q^{42} -3.64472 q^{43} +4.30075 q^{44} -18.2529 q^{45} +0.530712 q^{46} +13.5096 q^{47} -2.73247 q^{48} -5.16330 q^{49} -11.7013 q^{50} +11.4967 q^{51} -5.03138 q^{52} -11.0833 q^{53} +4.00686 q^{54} -17.5760 q^{55} +1.35525 q^{56} -18.8116 q^{57} -4.32835 q^{58} -2.93144 q^{59} +11.1669 q^{60} -14.4856 q^{61} +0.860776 q^{62} -6.05306 q^{63} +1.00000 q^{64} +20.5619 q^{65} +11.7517 q^{66} +6.99569 q^{67} -4.20743 q^{68} +1.45015 q^{69} -5.53853 q^{70} -8.33336 q^{71} -4.46639 q^{72} -14.6670 q^{73} +2.72088 q^{74} -31.9735 q^{75} +6.88446 q^{76} -5.82858 q^{77} -13.7481 q^{78} +16.6767 q^{79} -4.08673 q^{80} -2.45054 q^{81} -5.26479 q^{82} +11.2314 q^{83} +3.70317 q^{84} +17.1946 q^{85} +3.64472 q^{86} -11.8271 q^{87} -4.30075 q^{88} +4.51767 q^{89} +18.2529 q^{90} +6.81876 q^{91} -0.530712 q^{92} +2.35204 q^{93} -13.5096 q^{94} -28.1349 q^{95} +2.73247 q^{96} +16.1322 q^{97} +5.16330 q^{98} +19.2088 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9}+O(q^{10}) \) 69 * q - 69 * q^2 + 69 * q^4 + 13 * q^5 - 27 * q^7 - 69 * q^8 + 99 * q^9	\( 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9} - 13 q^{10} + 42 q^{11} - 5 q^{13} + 27 q^{14} + 18 q^{15} + 69 q^{16} + 24 q^{17} - 99 q^{18} + q^{19} + 13 q^{20} + 7 q^{21} - 42 q^{22} + 25 q^{23} + 100 q^{25} + 5 q^{26} + 15 q^{27} - 27 q^{28} + 87 q^{29} - 18 q^{30} + 5 q^{31} - 69 q^{32} + 28 q^{33} - 24 q^{34} + 33 q^{35} + 99 q^{36} - 5 q^{37} - q^{38} + 22 q^{39} - 13 q^{40} + 47 q^{41} - 7 q^{42} - 23 q^{43} + 42 q^{44} + 14 q^{45} - 25 q^{46} + 13 q^{47} + 106 q^{49} - 100 q^{50} + 2 q^{51} - 5 q^{52} + 51 q^{53} - 15 q^{54} - 11 q^{55} + 27 q^{56} + 52 q^{57} - 87 q^{58} + 73 q^{59} + 18 q^{60} + 4 q^{61} - 5 q^{62} - 86 q^{63} + 69 q^{64} + 70 q^{65} - 28 q^{66} - 24 q^{67} + 24 q^{68} + 56 q^{69} - 33 q^{70} + 84 q^{71} - 99 q^{72} + 27 q^{73} + 5 q^{74} + 27 q^{75} + q^{76} + 45 q^{77} - 22 q^{78} + 42 q^{79} + 13 q^{80} + 205 q^{81} - 47 q^{82} + q^{83} + 7 q^{84} - 18 q^{85} + 23 q^{86} - q^{87} - 42 q^{88} + 94 q^{89} - 14 q^{90} + 6 q^{91} + 25 q^{92} - 13 q^{93} - 13 q^{94} + 86 q^{95} + 35 q^{97} - 106 q^{98} + 83 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 69 * q^4 + 13 * q^5 - 27 * q^7 - 69 * q^8 + 99 * q^9 - 13 * q^10 + 42 * q^11 - 5 * q^13 + 27 * q^14 + 18 * q^15 + 69 * q^16 + 24 * q^17 - 99 * q^18 + q^19 + 13 * q^20 + 7 * q^21 - 42 * q^22 + 25 * q^23 + 100 * q^25 + 5 * q^26 + 15 * q^27 - 27 * q^28 + 87 * q^29 - 18 * q^30 + 5 * q^31 - 69 * q^32 + 28 * q^33 - 24 * q^34 + 33 * q^35 + 99 * q^36 - 5 * q^37 - q^38 + 22 * q^39 - 13 * q^40 + 47 * q^41 - 7 * q^42 - 23 * q^43 + 42 * q^44 + 14 * q^45 - 25 * q^46 + 13 * q^47 + 106 * q^49 - 100 * q^50 + 2 * q^51 - 5 * q^52 + 51 * q^53 - 15 * q^54 - 11 * q^55 + 27 * q^56 + 52 * q^57 - 87 * q^58 + 73 * q^59 + 18 * q^60 + 4 * q^61 - 5 * q^62 - 86 * q^63 + 69 * q^64 + 70 * q^65 - 28 * q^66 - 24 * q^67 + 24 * q^68 + 56 * q^69 - 33 * q^70 + 84 * q^71 - 99 * q^72 + 27 * q^73 + 5 * q^74 + 27 * q^75 + q^76 + 45 * q^77 - 22 * q^78 + 42 * q^79 + 13 * q^80 + 205 * q^81 - 47 * q^82 + q^83 + 7 * q^84 - 18 * q^85 + 23 * q^86 - q^87 - 42 * q^88 + 94 * q^89 - 14 * q^90 + 6 * q^91 + 25 * q^92 - 13 * q^93 - 13 * q^94 + 86 * q^95 + 35 * q^97 - 106 * q^98 + 83 * q^99

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