LMFDB - Embedded newform 6046.2.a.g.1.4

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			1.4
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)

$f(q)$	$=$	$q-1.00000 q^{2} -3.27820 q^{3} +1.00000 q^{4} +0.761195 q^{5} +3.27820 q^{6} +3.33846 q^{7} -1.00000 q^{8} +7.74657 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.27820 q^{3} +1.00000 q^{4} +0.761195 q^{5} +3.27820 q^{6} +3.33846 q^{7} -1.00000 q^{8} +7.74657 q^{9} -0.761195 q^{10} -2.95576 q^{11} -3.27820 q^{12} +5.24380 q^{13} -3.33846 q^{14} -2.49535 q^{15} +1.00000 q^{16} +6.37841 q^{17} -7.74657 q^{18} -1.65365 q^{19} +0.761195 q^{20} -10.9441 q^{21} +2.95576 q^{22} +4.38368 q^{23} +3.27820 q^{24} -4.42058 q^{25} -5.24380 q^{26} -15.5602 q^{27} +3.33846 q^{28} +4.88660 q^{29} +2.49535 q^{30} +4.51244 q^{31} -1.00000 q^{32} +9.68956 q^{33} -6.37841 q^{34} +2.54122 q^{35} +7.74657 q^{36} +8.03426 q^{37} +1.65365 q^{38} -17.1902 q^{39} -0.761195 q^{40} +8.92900 q^{41} +10.9441 q^{42} +4.20984 q^{43} -2.95576 q^{44} +5.89665 q^{45} -4.38368 q^{46} +5.45343 q^{47} -3.27820 q^{48} +4.14533 q^{49} +4.42058 q^{50} -20.9097 q^{51} +5.24380 q^{52} +2.15408 q^{53} +15.5602 q^{54} -2.24991 q^{55} -3.33846 q^{56} +5.42099 q^{57} -4.88660 q^{58} +4.40280 q^{59} -2.49535 q^{60} -0.893734 q^{61} -4.51244 q^{62} +25.8616 q^{63} +1.00000 q^{64} +3.99155 q^{65} -9.68956 q^{66} -6.32230 q^{67} +6.37841 q^{68} -14.3705 q^{69} -2.54122 q^{70} +7.10128 q^{71} -7.74657 q^{72} +4.16901 q^{73} -8.03426 q^{74} +14.4915 q^{75} -1.65365 q^{76} -9.86769 q^{77} +17.1902 q^{78} -5.92315 q^{79} +0.761195 q^{80} +27.7697 q^{81} -8.92900 q^{82} +12.8468 q^{83} -10.9441 q^{84} +4.85522 q^{85} -4.20984 q^{86} -16.0193 q^{87} +2.95576 q^{88} +9.19216 q^{89} -5.89665 q^{90} +17.5062 q^{91} +4.38368 q^{92} -14.7927 q^{93} -5.45343 q^{94} -1.25875 q^{95} +3.27820 q^{96} +14.5367 q^{97} -4.14533 q^{98} -22.8970 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.27820	−1.89267	−0.946334	−	0.323191i	$-0.895244\pi$
$3$	−3.27820	−1.89267	−0.946334	+	0.323191i	$0.895244\pi$
$4$	1.00000	0.500000
$5$	0.761195	0.340417	0.170208	−	0.985408i	$-0.445556\pi$
$5$	0.761195	0.340417	0.170208	+	0.985408i	$0.445556\pi$
$6$	3.27820	1.33832
$7$	3.33846	1.26182	0.630910	−	0.775856i	$-0.282682\pi$
$7$	3.33846	1.26182	0.630910	+	0.775856i	$0.282682\pi$
$8$	−1.00000	−0.353553
$9$	7.74657	2.58219
$10$	−0.761195	−0.240711
$11$	−2.95576	−0.891195	−0.445598	−	0.895233i	$-0.647009\pi$
$11$	−2.95576	−0.891195	−0.445598	+	0.895233i	$0.647009\pi$
$12$	−3.27820	−0.946334
$13$	5.24380	1.45437	0.727184	−	0.686443i	$-0.240829\pi$
$13$	5.24380	1.45437	0.727184	+	0.686443i	$0.240829\pi$
$14$	−3.33846	−0.892241
$15$	−2.49535	−0.644296
$16$	1.00000	0.250000
$17$	6.37841	1.54699	0.773496	−	0.633801i	$-0.218506\pi$
$17$	6.37841	1.54699	0.773496	+	0.633801i	$0.218506\pi$
$18$	−7.74657	−1.82588
$19$	−1.65365	−0.379374	−0.189687	−	0.981845i	$-0.560747\pi$
$19$	−1.65365	−0.379374	−0.189687	+	0.981845i	$0.560747\pi$
$20$	0.761195	0.170208
$21$	−10.9441	−2.38821
$22$	2.95576	0.630170
$23$	4.38368	0.914059	0.457030	−	0.889451i	$-0.348913\pi$
$23$	4.38368	0.914059	0.457030	+	0.889451i	$0.348913\pi$
$24$	3.27820	0.669159
$25$	−4.42058	−0.884116
$26$	−5.24380	−1.02839
$27$	−15.5602	−2.99456
$28$	3.33846	0.630910
$29$	4.88660	0.907420	0.453710	−	0.891149i	$-0.350100\pi$
$29$	4.88660	0.907420	0.453710	+	0.891149i	$0.350100\pi$
$30$	2.49535	0.455586
$31$	4.51244	0.810458	0.405229	−	0.914215i	$-0.367192\pi$
$31$	4.51244	0.810458	0.405229	+	0.914215i	$0.367192\pi$
$32$	−1.00000	−0.176777
$33$	9.68956	1.68674
$34$	−6.37841	−1.09389
$35$	2.54122	0.429545
$36$	7.74657	1.29110
$37$	8.03426	1.32082	0.660412	−	0.750904i	$-0.270382\pi$
$37$	8.03426	1.32082	0.660412	+	0.750904i	$0.270382\pi$
$38$	1.65365	0.268258
$39$	−17.1902	−2.75263
$40$	−0.761195	−0.120356
$41$	8.92900	1.39448	0.697238	−	0.716840i	$-0.254412\pi$
$41$	8.92900	1.39448	0.697238	+	0.716840i	$0.254412\pi$
$42$	10.9441	1.68872
$43$	4.20984	0.641994	0.320997	−	0.947080i	$-0.395982\pi$
$43$	4.20984	0.641994	0.320997	+	0.947080i	$0.395982\pi$
$44$	−2.95576	−0.445598
$45$	5.89665	0.879021
$46$	−4.38368	−0.646338
$47$	5.45343	0.795464	0.397732	−	0.917502i	$-0.369797\pi$
$47$	5.45343	0.795464	0.397732	+	0.917502i	$0.369797\pi$
$48$	−3.27820	−0.473167
$49$	4.14533	0.592189
$50$	4.42058	0.625165
$51$	−20.9097	−2.92794
$52$	5.24380	0.727184
$53$	2.15408	0.295885	0.147943	−	0.988996i	$-0.452735\pi$
$53$	2.15408	0.295885	0.147943	+	0.988996i	$0.452735\pi$
$54$	15.5602	2.11747
$55$	−2.24991	−0.303378
$56$	−3.33846	−0.446121
$57$	5.42099	0.718028
$58$	−4.88660	−0.641643
$59$	4.40280	0.573195	0.286598	−	0.958051i	$-0.407476\pi$
$59$	4.40280	0.573195	0.286598	+	0.958051i	$0.407476\pi$
$60$	−2.49535	−0.322148
$61$	−0.893734	−0.114431	−0.0572155	−	0.998362i	$-0.518222\pi$
$61$	−0.893734	−0.114431	−0.0572155	+	0.998362i	$0.518222\pi$
$62$	−4.51244	−0.573080
$63$	25.8616	3.25826
$64$	1.00000	0.125000
$65$	3.99155	0.495091
$66$	−9.68956	−1.19270
$67$	−6.32230	−0.772392	−0.386196	−	0.922417i	$-0.626211\pi$
$67$	−6.32230	−0.772392	−0.386196	+	0.922417i	$0.626211\pi$
$68$	6.37841	0.773496
$69$	−14.3705	−1.73001
$70$	−2.54122	−0.303734
$71$	7.10128	0.842766	0.421383	−	0.906883i	$-0.361545\pi$
$71$	7.10128	0.842766	0.421383	+	0.906883i	$0.361545\pi$
$72$	−7.74657	−0.912942
$73$	4.16901	0.487946	0.243973	−	0.969782i	$-0.421549\pi$
$73$	4.16901	0.487946	0.243973	+	0.969782i	$0.421549\pi$
$74$	−8.03426	−0.933963
$75$	14.4915	1.67334
$76$	−1.65365	−0.189687
$77$	−9.86769	−1.12453
$78$	17.1902	1.94641
$79$	−5.92315	−0.666406	−0.333203	−	0.942855i	$-0.608130\pi$
$79$	−5.92315	−0.666406	−0.333203	+	0.942855i	$0.608130\pi$
$80$	0.761195	0.0851042
$81$	27.7697	3.08552
$82$	−8.92900	−0.986043
$83$	12.8468	1.41012	0.705059	−	0.709149i	$-0.250921\pi$
$83$	12.8468	1.41012	0.705059	+	0.709149i	$0.250921\pi$
$84$	−10.9441	−1.19410
$85$	4.85522	0.526622
$86$	−4.20984	−0.453958
$87$	−16.0193	−1.71744
$88$	2.95576	0.315085
$89$	9.19216	0.974367	0.487184	−	0.873300i	$-0.338024\pi$
$89$	9.19216	0.974367	0.487184	+	0.873300i	$0.338024\pi$
$90$	−5.89665	−0.621562
$91$	17.5062	1.83515
$92$	4.38368	0.457030
$93$	−14.7927	−1.53393
$94$	−5.45343	−0.562478
$95$	−1.25875	−0.129145
$96$	3.27820	0.334580
$97$	14.5367	1.47598	0.737990	−	0.674812i	$-0.235775\pi$
$97$	14.5367	1.47598	0.737990	+	0.674812i	$0.235775\pi$
$98$	−4.14533	−0.418741
$99$	−22.8970	−2.30124
$100$	−4.42058	−0.442058
$101$	−5.08908	−0.506382	−0.253191	−	0.967416i	$-0.581480\pi$
$101$	−5.08908	−0.506382	−0.253191	+	0.967416i	$0.581480\pi$
$102$	20.9097	2.07037
$103$	−4.73884	−0.466932	−0.233466	−	0.972365i	$-0.575007\pi$
$103$	−4.73884	−0.466932	−0.233466	+	0.972365i	$0.575007\pi$
$104$	−5.24380	−0.514196
$105$	−8.33062	−0.812985
$106$	−2.15408	−0.209222
$107$	2.30489	0.222822	0.111411	−	0.993774i	$-0.464463\pi$
$107$	2.30489	0.222822	0.111411	+	0.993774i	$0.464463\pi$
$108$	−15.5602	−1.49728
$109$	2.22969	0.213566	0.106783	−	0.994282i	$-0.465945\pi$
$109$	2.22969	0.213566	0.106783	+	0.994282i	$0.465945\pi$
$110$	2.24991	0.214520
$111$	−26.3379	−2.49988
$112$	3.33846	0.315455
$113$	−4.90926	−0.461824	−0.230912	−	0.972975i	$-0.574171\pi$
$113$	−4.90926	−0.461824	−0.230912	+	0.972975i	$0.574171\pi$
$114$	−5.42099	−0.507723
$115$	3.33683	0.311161
$116$	4.88660	0.453710
$117$	40.6214	3.75545
$118$	−4.40280	−0.405310
$119$	21.2941	1.95203
$120$	2.49535	0.227793
$121$	−2.26348	−0.205771
$122$	0.893734	0.0809149
$123$	−29.2710	−2.63928
$124$	4.51244	0.405229
$125$	−7.17090	−0.641385
$126$	−25.8616	−2.30394
$127$	−21.0834	−1.87085	−0.935426	−	0.353522i	$-0.884984\pi$
$127$	−21.0834	−1.87085	−0.935426	+	0.353522i	$0.884984\pi$
$128$	−1.00000	−0.0883883
$129$	−13.8007	−1.21508
$130$	−3.99155	−0.350082
$131$	16.1376	1.40995	0.704973	−	0.709234i	$-0.250959\pi$
$131$	16.1376	1.40995	0.704973	+	0.709234i	$0.250959\pi$
$132$	9.68956	0.843368
$133$	−5.52065	−0.478701
$134$	6.32230	0.546163
$135$	−11.8443	−1.01940
$136$	−6.37841	−0.546944
$137$	−5.44709	−0.465376	−0.232688	−	0.972551i	$-0.574752\pi$
$137$	−5.44709	−0.465376	−0.232688	+	0.972551i	$0.574752\pi$
$138$	14.3705	1.22330
$139$	−19.5995	−1.66240	−0.831202	−	0.555971i	$-0.812347\pi$
$139$	−19.5995	−1.66240	−0.831202	+	0.555971i	$0.812347\pi$
$140$	2.54122	0.214772
$141$	−17.8774	−1.50555
$142$	−7.10128	−0.595926
$143$	−15.4994	−1.29612
$144$	7.74657	0.645548
$145$	3.71966	0.308901
$146$	−4.16901	−0.345030
$147$	−13.5892	−1.12082
$148$	8.03426	0.660412
$149$	−17.8342	−1.46103	−0.730516	−	0.682896i	$-0.760720\pi$
$149$	−17.8342	−1.46103	−0.730516	+	0.682896i	$0.760720\pi$
$150$	−14.4915	−1.18323
$151$	4.32343	0.351836	0.175918	−	0.984405i	$-0.443711\pi$
$151$	4.32343	0.351836	0.175918	+	0.984405i	$0.443711\pi$
$152$	1.65365	0.134129
$153$	49.4108	3.99463
$154$	9.86769	0.795161
$155$	3.43484	0.275893
$156$	−17.1902	−1.37632
$157$	4.05820	0.323880	0.161940	−	0.986801i	$-0.448225\pi$
$157$	4.05820	0.323880	0.161940	+	0.986801i	$0.448225\pi$
$158$	5.92315	0.471220
$159$	−7.06148	−0.560012
$160$	−0.761195	−0.0601778
$161$	14.6347	1.15338
$162$	−27.7697	−2.18179
$163$	−23.4617	−1.83766	−0.918829	−	0.394655i	$-0.870864\pi$
$163$	−23.4617	−1.83766	−0.918829	+	0.394655i	$0.870864\pi$
$164$	8.92900	0.697238
$165$	7.37565	0.574193
$166$	−12.8468	−0.997104
$167$	−17.5162	−1.35545	−0.677723	−	0.735317i	$-0.737033\pi$
$167$	−17.5162	−1.35545	−0.677723	+	0.735317i	$0.737033\pi$
$168$	10.9441	0.844358
$169$	14.4974	1.11518
$170$	−4.85522	−0.372378
$171$	−12.8101	−0.979615
$172$	4.20984	0.320997
$173$	−19.0447	−1.44794	−0.723972	−	0.689829i	$-0.757686\pi$
$173$	−19.0447	−1.44794	−0.723972	+	0.689829i	$0.757686\pi$
$174$	16.0193	1.21442
$175$	−14.7579	−1.11560
$176$	−2.95576	−0.222799
$177$	−14.4332	−1.08487
$178$	−9.19216	−0.688982
$179$	−8.56264	−0.640002	−0.320001	−	0.947417i	$-0.603683\pi$
$179$	−8.56264	−0.640002	−0.320001	+	0.947417i	$0.603683\pi$
$180$	5.89665	0.439511
$181$	−21.7761	−1.61860	−0.809301	−	0.587394i	$-0.800154\pi$
$181$	−21.7761	−1.61860	−0.809301	+	0.587394i	$0.800154\pi$
$182$	−17.5062	−1.29765
$183$	2.92984	0.216580
$184$	−4.38368	−0.323169
$185$	6.11564	0.449631
$186$	14.7927	1.08465
$187$	−18.8531	−1.37867
$188$	5.45343	0.397732
$189$	−51.9471	−3.77860
$190$	1.25875	0.0913194
$191$	24.5732	1.77805	0.889026	−	0.457857i	$-0.151383\pi$
$191$	24.5732	1.77805	0.889026	+	0.457857i	$0.151383\pi$
$192$	−3.27820	−0.236583
$193$	9.13136	0.657290	0.328645	−	0.944454i	$-0.393408\pi$
$193$	9.13136	0.657290	0.328645	+	0.944454i	$0.393408\pi$
$194$	−14.5367	−1.04368
$195$	−13.0851	−0.937043
$196$	4.14533	0.296095
$197$	16.9560	1.20806	0.604032	−	0.796960i	$-0.293560\pi$
$197$	16.9560	1.20806	0.604032	+	0.796960i	$0.293560\pi$
$198$	22.8970	1.62722
$199$	−14.3152	−1.01478	−0.507389	−	0.861717i	$-0.669389\pi$
$199$	−14.3152	−1.01478	−0.507389	+	0.861717i	$0.669389\pi$
$200$	4.42058	0.312582
$201$	20.7257	1.46188
$202$	5.08908	0.358066
$203$	16.3137	1.14500
$204$	−20.9097	−1.46397
$205$	6.79671	0.474703
$206$	4.73884	0.330171
$207$	33.9585	2.36028
$208$	5.24380	0.363592
$209$	4.88780	0.338096
$210$	8.33062	0.574867
$211$	2.90045	0.199675	0.0998377	−	0.995004i	$-0.468168\pi$
$211$	2.90045	0.199675	0.0998377	+	0.995004i	$0.468168\pi$
$212$	2.15408	0.147943
$213$	−23.2794	−1.59508
$214$	−2.30489	−0.157559
$215$	3.20451	0.218546
$216$	15.5602	1.05874
$217$	15.0646	1.02265
$218$	−2.22969	−0.151014
$219$	−13.6668	−0.923520
$220$	−2.24991	−0.151689
$221$	33.4471	2.24989
$222$	26.3379	1.76768
$223$	−6.87674	−0.460501	−0.230250	−	0.973131i	$-0.573955\pi$
$223$	−6.87674	−0.460501	−0.230250	+	0.973131i	$0.573955\pi$
$224$	−3.33846	−0.223060
$225$	−34.2444	−2.28296
$226$	4.90926	0.326559
$227$	3.08506	0.204763	0.102381	−	0.994745i	$-0.467354\pi$
$227$	3.08506	0.204763	0.102381	+	0.994745i	$0.467354\pi$
$228$	5.42099	0.359014
$229$	21.0567	1.39147	0.695734	−	0.718300i	$-0.255079\pi$
$229$	21.0567	1.39147	0.695734	+	0.718300i	$0.255079\pi$
$230$	−3.33683	−0.220024
$231$	32.3482	2.12836
$232$	−4.88660	−0.320821
$233$	−7.82937	−0.512919	−0.256459	−	0.966555i	$-0.582556\pi$
$233$	−7.82937	−0.512919	−0.256459	+	0.966555i	$0.582556\pi$
$234$	−40.6214	−2.65551
$235$	4.15112	0.270789
$236$	4.40280	0.286598
$237$	19.4172	1.26129
$238$	−21.2941	−1.38029
$239$	−8.31411	−0.537795	−0.268898	−	0.963169i	$-0.586659\pi$
$239$	−8.31411	−0.537795	−0.268898	+	0.963169i	$0.586659\pi$
$240$	−2.49535	−0.161074
$241$	−14.3510	−0.924430	−0.462215	−	0.886768i	$-0.652945\pi$
$241$	−14.3510	−0.924430	−0.462215	+	0.886768i	$0.652945\pi$
$242$	2.26348	0.145502
$243$	−44.3538	−2.84530
$244$	−0.893734	−0.0572155
$245$	3.15540	0.201591
$246$	29.2710	1.86625
$247$	−8.67141	−0.551749
$248$	−4.51244	−0.286540
$249$	−42.1143	−2.66888
$250$	7.17090	0.453528
$251$	−19.7668	−1.24767	−0.623835	−	0.781556i	$-0.714426\pi$
$251$	−19.7668	−1.24767	−0.623835	+	0.781556i	$0.714426\pi$
$252$	25.8616	1.62913
$253$	−12.9571	−0.814605
$254$	21.0834	1.32289
$255$	−15.9164	−0.996721
$256$	1.00000	0.0625000
$257$	19.1481	1.19442	0.597212	−	0.802083i	$-0.296275\pi$
$257$	19.1481	1.19442	0.597212	+	0.802083i	$0.296275\pi$
$258$	13.8007	0.859192
$259$	26.8221	1.66664
$260$	3.99155	0.247546
$261$	37.8544	2.34313
$262$	−16.1376	−0.996982
$263$	12.4430	0.767270	0.383635	−	0.923485i	$-0.374672\pi$
$263$	12.4430	0.767270	0.383635	+	0.923485i	$0.374672\pi$
$264$	−9.68956	−0.596351
$265$	1.63967	0.100724
$266$	5.52065	0.338493
$267$	−30.1337	−1.84415
$268$	−6.32230	−0.386196
$269$	11.3223	0.690332	0.345166	−	0.938542i	$-0.387823\pi$
$269$	11.3223	0.690332	0.345166	+	0.938542i	$0.387823\pi$
$270$	11.8443	0.720824
$271$	20.2924	1.23268	0.616338	−	0.787482i	$-0.288616\pi$
$271$	20.2924	1.23268	0.616338	+	0.787482i	$0.288616\pi$
$272$	6.37841	0.386748
$273$	−57.3888	−3.47333
$274$	5.44709	0.329071
$275$	13.0662	0.787920
$276$	−14.3705	−0.865005
$277$	−16.5513	−0.994472	−0.497236	−	0.867615i	$-0.665652\pi$
$277$	−16.5513	−0.994472	−0.497236	+	0.867615i	$0.665652\pi$
$278$	19.5995	1.17550
$279$	34.9559	2.09276
$280$	−2.54122	−0.151867
$281$	−4.35137	−0.259581	−0.129790	−	0.991541i	$-0.541430\pi$
$281$	−4.35137	−0.259581	−0.129790	+	0.991541i	$0.541430\pi$
$282$	17.8774	1.06458
$283$	−21.5293	−1.27979	−0.639893	−	0.768464i	$-0.721021\pi$
$283$	−21.5293	−1.27979	−0.639893	+	0.768464i	$0.721021\pi$
$284$	7.10128	0.421383
$285$	4.12643	0.244429
$286$	15.4994	0.916499
$287$	29.8091	1.75958
$288$	−7.74657	−0.456471
$289$	23.6841	1.39318
$290$	−3.71966	−0.218426
$291$	−47.6542	−2.79354
$292$	4.16901	0.243973
$293$	−15.3030	−0.894010	−0.447005	−	0.894531i	$-0.647509\pi$
$293$	−15.3030	−0.894010	−0.447005	+	0.894531i	$0.647509\pi$
$294$	13.5892	0.792538
$295$	3.35139	0.195125
$296$	−8.03426	−0.466982
$297$	45.9922	2.66874
$298$	17.8342	1.03311
$299$	22.9871	1.32938
$300$	14.4915	0.836669
$301$	14.0544	0.810081
$302$	−4.32343	−0.248785
$303$	16.6830	0.958414
$304$	−1.65365	−0.0948434
$305$	−0.680306	−0.0389542
$306$	−49.4108	−2.82463
$307$	30.1516	1.72084	0.860422	−	0.509583i	$-0.170200\pi$
$307$	30.1516	1.72084	0.860422	+	0.509583i	$0.170200\pi$
$308$	−9.86769	−0.562264
$309$	15.5349	0.883747
$310$	−3.43484	−0.195086
$311$	−11.8466	−0.671759	−0.335880	−	0.941905i	$-0.609034\pi$
$311$	−11.8466	−0.671759	−0.335880	+	0.941905i	$0.609034\pi$
$312$	17.1902	0.973203
$313$	−18.1584	−1.02637	−0.513186	−	0.858278i	$-0.671535\pi$
$313$	−18.1584	−1.02637	−0.513186	+	0.858278i	$0.671535\pi$
$314$	−4.05820	−0.229018
$315$	19.6857	1.10917
$316$	−5.92315	−0.333203
$317$	−11.6801	−0.656022	−0.328011	−	0.944674i	$-0.606378\pi$
$317$	−11.6801	−0.656022	−0.328011	+	0.944674i	$0.606378\pi$
$318$	7.06148	0.395988
$319$	−14.4436	−0.808688
$320$	0.761195	0.0425521
$321$	−7.55587	−0.421727
$322$	−14.6347	−0.815562
$323$	−10.5477	−0.586888
$324$	27.7697	1.54276
$325$	−23.1806	−1.28583
$326$	23.4617	1.29942
$327$	−7.30937	−0.404209
$328$	−8.92900	−0.493021
$329$	18.2061	1.00373
$330$	−7.37565	−0.406016
$331$	−13.2810	−0.729989	−0.364994	−	0.931010i	$-0.618929\pi$
$331$	−13.2810	−0.729989	−0.364994	+	0.931010i	$0.618929\pi$
$332$	12.8468	0.705059
$333$	62.2379	3.41062
$334$	17.5162	0.958446
$335$	−4.81250	−0.262935
$336$	−10.9441	−0.597051
$337$	−17.1843	−0.936088	−0.468044	−	0.883705i	$-0.655041\pi$
$337$	−17.1843	−0.936088	−0.468044	+	0.883705i	$0.655041\pi$
$338$	−14.4974	−0.788554
$339$	16.0935	0.874080
$340$	4.85522	0.263311
$341$	−13.3377	−0.722276
$342$	12.8101	0.692692
$343$	−9.53022	−0.514584
$344$	−4.20984	−0.226979
$345$	−10.9388	−0.588925
$346$	19.0447	1.02385
$347$	−6.17418	−0.331447	−0.165724	−	0.986172i	$-0.552996\pi$
$347$	−6.17418	−0.331447	−0.165724	+	0.986172i	$0.552996\pi$
$348$	−16.0193	−0.858722
$349$	−18.8072	−1.00673	−0.503363	−	0.864075i	$-0.667904\pi$
$349$	−18.8072	−1.00673	−0.503363	+	0.864075i	$0.667904\pi$
$350$	14.7579	0.788845
$351$	−81.5945	−4.35519
$352$	2.95576	0.157543
$353$	−0.937462	−0.0498960	−0.0249480	−	0.999689i	$-0.507942\pi$
$353$	−0.937462	−0.0498960	−0.0249480	+	0.999689i	$0.507942\pi$
$354$	14.4332	0.767117
$355$	5.40546	0.286892
$356$	9.19216	0.487184
$357$	−69.8062	−3.69453
$358$	8.56264	0.452550
$359$	34.2828	1.80938	0.904689	−	0.426072i	$-0.140103\pi$
$359$	34.2828	1.80938	0.904689	+	0.426072i	$0.140103\pi$
$360$	−5.89665	−0.310781
$361$	−16.2654	−0.856076
$362$	21.7761	1.14452
$363$	7.42015	0.389457
$364$	17.5062	0.917575
$365$	3.17343	0.166105
$366$	−2.92984	−0.153145
$367$	−18.2924	−0.954855	−0.477427	−	0.878671i	$-0.658431\pi$
$367$	−18.2924	−0.954855	−0.477427	+	0.878671i	$0.658431\pi$
$368$	4.38368	0.228515
$369$	69.1691	3.60080
$370$	−6.11564	−0.317937
$371$	7.19130	0.373354
$372$	−14.7927	−0.766963
$373$	−23.6304	−1.22354	−0.611769	−	0.791037i	$-0.709542\pi$
$373$	−23.6304	−1.22354	−0.611769	+	0.791037i	$0.709542\pi$
$374$	18.8531	0.974868
$375$	23.5076	1.21393
$376$	−5.45343	−0.281239
$377$	25.6244	1.31972
$378$	51.9471	2.67187
$379$	22.9131	1.17697	0.588484	−	0.808509i	$-0.299725\pi$
$379$	22.9131	1.17697	0.588484	+	0.808509i	$0.299725\pi$
$380$	−1.25875	−0.0645726
$381$	69.1156	3.54090
$382$	−24.5732	−1.25727
$383$	33.5645	1.71507	0.857534	−	0.514428i	$-0.171996\pi$
$383$	33.5645	1.71507	0.857534	+	0.514428i	$0.171996\pi$
$384$	3.27820	0.167290
$385$	−7.51124	−0.382808
$386$	−9.13136	−0.464774
$387$	32.6118	1.65775
$388$	14.5367	0.737990
$389$	−19.1177	−0.969305	−0.484652	−	0.874707i	$-0.661054\pi$
$389$	−19.1177	−0.969305	−0.484652	+	0.874707i	$0.661054\pi$
$390$	13.0851	0.662589
$391$	27.9609	1.41404
$392$	−4.14533	−0.209371
$393$	−52.9021	−2.66856
$394$	−16.9560	−0.854230
$395$	−4.50867	−0.226856
$396$	−22.8970	−1.15062
$397$	10.2435	0.514108	0.257054	−	0.966397i	$-0.417248\pi$
$397$	10.2435	0.514108	0.257054	+	0.966397i	$0.417248\pi$
$398$	14.3152	0.717557
$399$	18.0978	0.906022
$400$	−4.42058	−0.221029
$401$	−33.7507	−1.68543	−0.842715	−	0.538359i	$-0.819044\pi$
$401$	−33.7507	−1.68543	−0.842715	+	0.538359i	$0.819044\pi$
$402$	−20.7257	−1.03371
$403$	23.6623	1.17870
$404$	−5.08908	−0.253191
$405$	21.1381	1.05036
$406$	−16.3137	−0.809637
$407$	−23.7473	−1.17711
$408$	20.9097	1.03518
$409$	20.0616	0.991981	0.495990	−	0.868328i	$-0.334805\pi$
$409$	20.0616	0.991981	0.495990	+	0.868328i	$0.334805\pi$
$410$	−6.79671	−0.335666
$411$	17.8566	0.880802
$412$	−4.73884	−0.233466
$413$	14.6986	0.723269
$414$	−33.9585	−1.66897
$415$	9.77891	0.480028
$416$	−5.24380	−0.257098
$417$	64.2508	3.14638
$418$	−4.88780	−0.239070
$419$	14.4360	0.705245	0.352622	−	0.935766i	$-0.385290\pi$
$419$	14.4360	0.705245	0.352622	+	0.935766i	$0.385290\pi$
$420$	−8.33062	−0.406493
$421$	−0.501621	−0.0244475	−0.0122238	−	0.999925i	$-0.503891\pi$
$421$	−0.501621	−0.0244475	−0.0122238	+	0.999925i	$0.503891\pi$
$422$	−2.90045	−0.141192
$423$	42.2454	2.05404
$424$	−2.15408	−0.104611
$425$	−28.1963	−1.36772
$426$	23.2794	1.12789
$427$	−2.98370	−0.144391
$428$	2.30489	0.111411
$429$	50.8101	2.45313
$430$	−3.20451	−0.154535
$431$	18.6947	0.900493	0.450246	−	0.892904i	$-0.351336\pi$
$431$	18.6947	0.900493	0.450246	+	0.892904i	$0.351336\pi$
$432$	−15.5602	−0.748640
$433$	−6.55103	−0.314822	−0.157411	−	0.987533i	$-0.550315\pi$
$433$	−6.55103	−0.314822	−0.157411	+	0.987533i	$0.550315\pi$
$434$	−15.0646	−0.723124
$435$	−12.1938	−0.584647
$436$	2.22969	0.106783
$437$	−7.24907	−0.346770
$438$	13.6668	0.653027
$439$	−7.05395	−0.336666	−0.168333	−	0.985730i	$-0.553838\pi$
$439$	−7.05395	−0.336666	−0.168333	+	0.985730i	$0.553838\pi$
$440$	2.24991	0.107260
$441$	32.1121	1.52915
$442$	−33.4471	−1.59092
$443$	4.63593	0.220260	0.110130	−	0.993917i	$-0.464873\pi$
$443$	4.63593	0.220260	0.110130	+	0.993917i	$0.464873\pi$
$444$	−26.3379	−1.24994
$445$	6.99703	0.331691
$446$	6.87674	0.325623
$447$	58.4639	2.76525
$448$	3.33846	0.157727
$449$	41.4052	1.95403	0.977015	−	0.213171i	$-0.0683791\pi$
$449$	41.4052	1.95403	0.977015	+	0.213171i	$0.0683791\pi$
$450$	34.2444	1.61429
$451$	−26.3920	−1.24275
$452$	−4.90926	−0.230912
$453$	−14.1731	−0.665908
$454$	−3.08506	−0.144789
$455$	13.3256	0.624716
$456$	−5.42099	−0.253861
$457$	−7.75944	−0.362971	−0.181486	−	0.983394i	$-0.558091\pi$
$457$	−7.75944	−0.362971	−0.181486	+	0.983394i	$0.558091\pi$
$458$	−21.0567	−0.983916
$459$	−99.2493	−4.63256
$460$	3.33683	0.155581
$461$	4.29248	0.199921	0.0999604	−	0.994991i	$-0.468128\pi$
$461$	4.29248	0.199921	0.0999604	+	0.994991i	$0.468128\pi$
$462$	−32.3482	−1.50498
$463$	−0.948591	−0.0440848	−0.0220424	−	0.999757i	$-0.507017\pi$
$463$	−0.948591	−0.0440848	−0.0220424	+	0.999757i	$0.507017\pi$
$464$	4.88660	0.226855
$465$	−11.2601	−0.522174
$466$	7.82937	0.362688
$467$	−33.6381	−1.55659	−0.778293	−	0.627902i	$-0.783914\pi$
$467$	−33.6381	−1.55659	−0.778293	+	0.627902i	$0.783914\pi$
$468$	40.6214	1.87773
$469$	−21.1067	−0.974619
$470$	−4.15112	−0.191477
$471$	−13.3036	−0.612997
$472$	−4.40280	−0.202655
$473$	−12.4433	−0.572142
$474$	−19.4172	−0.891864
$475$	7.31010	0.335410
$476$	21.2941	0.976013
$477$	16.6867	0.764032
$478$	8.31411	0.380278
$479$	−0.721571	−0.0329694	−0.0164847	−	0.999864i	$-0.505247\pi$
$479$	−0.721571	−0.0329694	−0.0164847	+	0.999864i	$0.505247\pi$
$480$	2.49535	0.113896
$481$	42.1300	1.92096
$482$	14.3510	0.653671
$483$	−47.9755	−2.18296
$484$	−2.26348	−0.102886
$485$	11.0653	0.502448
$486$	44.3538	2.01193
$487$	30.3156	1.37373	0.686866	−	0.726784i	$-0.258986\pi$
$487$	30.3156	1.37373	0.686866	+	0.726784i	$0.258986\pi$
$488$	0.893734	0.0404575
$489$	76.9119	3.47808
$490$	−3.15540	−0.142547
$491$	−23.1038	−1.04266	−0.521331	−	0.853355i	$-0.674564\pi$
$491$	−23.1038	−1.04266	−0.521331	+	0.853355i	$0.674564\pi$
$492$	−29.2710	−1.31964
$493$	31.1688	1.40377
$494$	8.67141	0.390145
$495$	−17.4291	−0.783379
$496$	4.51244	0.202614
$497$	23.7073	1.06342
$498$	42.1143	1.88719
$499$	23.6414	1.05833	0.529167	−	0.848518i	$-0.322504\pi$
$499$	23.6414	1.05833	0.529167	+	0.848518i	$0.322504\pi$
$500$	−7.17090	−0.320692
$501$	57.4216	2.56541
$502$	19.7668	0.882236
$503$	−19.5138	−0.870078	−0.435039	−	0.900412i	$-0.643265\pi$
$503$	−19.5138	−0.870078	−0.435039	+	0.900412i	$0.643265\pi$
$504$	−25.8616	−1.15197
$505$	−3.87378	−0.172381
$506$	12.9571	0.576013
$507$	−47.5253	−2.11067
$508$	−21.0834	−0.935426
$509$	26.8009	1.18793	0.593965	−	0.804491i	$-0.297562\pi$
$509$	26.8009	1.18793	0.593965	+	0.804491i	$0.297562\pi$
$510$	15.9164	0.704788
$511$	13.9181	0.615700
$512$	−1.00000	−0.0441942
$513$	25.7311	1.13606
$514$	−19.1481	−0.844585
$515$	−3.60718	−0.158952
$516$	−13.8007	−0.607541
$517$	−16.1190	−0.708914
$518$	−26.8221	−1.17849
$519$	62.4324	2.74048
$520$	−3.99155	−0.175041
$521$	23.8395	1.04443	0.522214	−	0.852814i	$-0.325106\pi$
$521$	23.8395	1.04443	0.522214	+	0.852814i	$0.325106\pi$
$522$	−37.8544	−1.65684
$523$	24.7214	1.08099	0.540496	−	0.841347i	$-0.318237\pi$
$523$	24.7214	1.08099	0.540496	+	0.841347i	$0.318237\pi$
$524$	16.1376	0.704973
$525$	48.3794	2.11145
$526$	−12.4430	−0.542542
$527$	28.7822	1.25377
$528$	9.68956	0.421684
$529$	−3.78339	−0.164495
$530$	−1.63967	−0.0712228
$531$	34.1066	1.48010
$532$	−5.52065	−0.239351
$533$	46.8218	2.02808
$534$	30.1337	1.30401
$535$	1.75447	0.0758523
$536$	6.32230	0.273082
$537$	28.0700	1.21131
$538$	−11.3223	−0.488138
$539$	−12.2526	−0.527756
$540$	−11.8443	−0.509699
$541$	21.7122	0.933482	0.466741	−	0.884394i	$-0.345428\pi$
$541$	21.7122	0.933482	0.466741	+	0.884394i	$0.345428\pi$
$542$	−20.2924	−0.871633
$543$	71.3862	3.06348
$544$	−6.37841	−0.273472
$545$	1.69723	0.0727014
$546$	57.3888	2.45601
$547$	19.5639	0.836493	0.418246	−	0.908334i	$-0.362645\pi$
$547$	19.5639	0.836493	0.418246	+	0.908334i	$0.362645\pi$
$548$	−5.44709	−0.232688
$549$	−6.92338	−0.295483
$550$	−13.0662	−0.557144
$551$	−8.08074	−0.344251
$552$	14.3705	0.611651
$553$	−19.7742	−0.840885
$554$	16.5513	0.703198
$555$	−20.0483	−0.851001
$556$	−19.5995	−0.831202
$557$	−13.1269	−0.556204	−0.278102	−	0.960552i	$-0.589705\pi$
$557$	−13.1269	−0.556204	−0.278102	+	0.960552i	$0.589705\pi$
$558$	−34.9559	−1.47980
$559$	22.0755	0.933695
$560$	2.54122	0.107386
$561$	61.8040	2.60937
$562$	4.35137	0.183551
$563$	−21.2696	−0.896406	−0.448203	−	0.893932i	$-0.647936\pi$
$563$	−21.2696	−0.896406	−0.448203	+	0.893932i	$0.647936\pi$
$564$	−17.8774	−0.752774
$565$	−3.73690	−0.157213
$566$	21.5293	0.904945
$567$	92.7079	3.89337
$568$	−7.10128	−0.297963
$569$	−0.209625	−0.00878792	−0.00439396	−	0.999990i	$-0.501399\pi$
$569$	−0.209625	−0.00878792	−0.00439396	+	0.999990i	$0.501399\pi$
$570$	−4.12643	−0.172837
$571$	−47.2895	−1.97900	−0.989502	−	0.144520i	$-0.953836\pi$
$571$	−47.2895	−1.97900	−0.989502	+	0.144520i	$0.953836\pi$
$572$	−15.4994	−0.648062
$573$	−80.5557	−3.36526
$574$	−29.8091	−1.24421
$575$	−19.3784	−0.808135
$576$	7.74657	0.322774
$577$	−26.0373	−1.08395	−0.541974	−	0.840395i	$-0.682323\pi$
$577$	−26.0373	−1.08395	−0.541974	+	0.840395i	$0.682323\pi$
$578$	−23.6841	−0.985130
$579$	−29.9344	−1.24403
$580$	3.71966	0.154450
$581$	42.8885	1.77931
$582$	47.6542	1.97533
$583$	−6.36693	−0.263691
$584$	−4.16901	−0.172515
$585$	30.9208	1.27842
$586$	15.3030	0.632161
$587$	23.6968	0.978073	0.489037	−	0.872263i	$-0.337348\pi$
$587$	23.6968	0.978073	0.489037	+	0.872263i	$0.337348\pi$
$588$	−13.5892	−0.560409
$589$	−7.46200	−0.307466
$590$	−3.35139	−0.137974
$591$	−55.5850	−2.28646
$592$	8.03426	0.330206
$593$	−6.22823	−0.255763	−0.127881	−	0.991789i	$-0.540818\pi$
$593$	−6.22823	−0.255763	−0.127881	+	0.991789i	$0.540818\pi$
$594$	−45.9922	−1.88708
$595$	16.2090	0.664502
$596$	−17.8342	−0.730516
$597$	46.9281	1.92064
$598$	−22.9871	−0.940012
$599$	7.23171	0.295480	0.147740	−	0.989026i	$-0.452800\pi$
$599$	7.23171	0.295480	0.147740	+	0.989026i	$0.452800\pi$
$600$	−14.4915	−0.591614
$601$	17.9110	0.730605	0.365303	−	0.930889i	$-0.380965\pi$
$601$	17.9110	0.730605	0.365303	+	0.930889i	$0.380965\pi$
$602$	−14.0544	−0.572814
$603$	−48.9761	−1.99446
$604$	4.32343	0.175918
$605$	−1.72295	−0.0700480
$606$	−16.6830	−0.677701
$607$	22.0995	0.896991	0.448495	−	0.893785i	$-0.351960\pi$
$607$	22.0995	0.896991	0.448495	+	0.893785i	$0.351960\pi$
$608$	1.65365	0.0670644
$609$	−53.4797	−2.16710
$610$	0.680306	0.0275448
$611$	28.5967	1.15690
$612$	49.4108	1.99731
$613$	−40.9859	−1.65540	−0.827702	−	0.561168i	$-0.810352\pi$
$613$	−40.9859	−1.65540	−0.827702	+	0.561168i	$0.810352\pi$
$614$	−30.1516	−1.21682
$615$	−22.2810	−0.898455
$616$	9.86769	0.397581
$617$	24.2725	0.977173	0.488587	−	0.872515i	$-0.337513\pi$
$617$	24.2725	0.977173	0.488587	+	0.872515i	$0.337513\pi$
$618$	−15.5349	−0.624904
$619$	−26.9673	−1.08391	−0.541954	−	0.840408i	$-0.682315\pi$
$619$	−26.9673	−1.08391	−0.541954	+	0.840408i	$0.682315\pi$
$620$	3.43484	0.137947
$621$	−68.2108	−2.73721
$622$	11.8466	0.475005
$623$	30.6877	1.22948
$624$	−17.1902	−0.688158
$625$	16.6445	0.665778
$626$	18.1584	0.725754
$627$	−16.0232	−0.639903
$628$	4.05820	0.161940
$629$	51.2458	2.04330
$630$	−19.6857	−0.784299
$631$	33.2048	1.32186	0.660932	−	0.750446i	$-0.270161\pi$
$631$	33.2048	1.32186	0.660932	+	0.750446i	$0.270161\pi$
$632$	5.92315	0.235610
$633$	−9.50826	−0.377919
$634$	11.6801	0.463878
$635$	−16.0486	−0.636870
$636$	−7.06148	−0.280006
$637$	21.7372	0.861261
$638$	14.4436	0.571829
$639$	55.0105	2.17618
$640$	−0.761195	−0.0300889
$641$	24.2671	0.958491	0.479245	−	0.877681i	$-0.340910\pi$
$641$	24.2671	0.958491	0.479245	+	0.877681i	$0.340910\pi$
$642$	7.55587	0.298206
$643$	17.0810	0.673611	0.336805	−	0.941574i	$-0.390654\pi$
$643$	17.0810	0.673611	0.336805	+	0.941574i	$0.390654\pi$
$644$	14.6347	0.576689
$645$	−10.5050	−0.413634
$646$	10.5477	0.414992
$647$	−44.6692	−1.75613	−0.878064	−	0.478543i	$-0.841165\pi$
$647$	−44.6692	−1.75613	−0.878064	+	0.478543i	$0.841165\pi$
$648$	−27.7697	−1.09090
$649$	−13.0136	−0.510829
$650$	23.1806	0.909219
$651$	−49.3847	−1.93554
$652$	−23.4617	−0.918829
$653$	−3.08397	−0.120685	−0.0603425	−	0.998178i	$-0.519219\pi$
$653$	−3.08397	−0.120685	−0.0603425	+	0.998178i	$0.519219\pi$
$654$	7.30937	0.285819
$655$	12.2838	0.479969
$656$	8.92900	0.348619
$657$	32.2956	1.25997
$658$	−18.2061	−0.709746
$659$	−45.4095	−1.76890	−0.884452	−	0.466631i	$-0.845468\pi$
$659$	−45.4095	−1.76890	−0.884452	+	0.466631i	$0.845468\pi$
$660$	7.37565	0.287097
$661$	35.3946	1.37669	0.688345	−	0.725383i	$-0.258337\pi$
$661$	35.3946	1.37669	0.688345	+	0.725383i	$0.258337\pi$
$662$	13.2810	0.516180
$663$	−109.646	−4.25830
$664$	−12.8468	−0.498552
$665$	−4.20229	−0.162958
$666$	−62.2379	−2.41167
$667$	21.4213	0.829436
$668$	−17.5162	−0.677723
$669$	22.5433	0.871575
$670$	4.81250	0.185923
$671$	2.64166	0.101980
$672$	10.9441	0.422179
$673$	−38.7665	−1.49434	−0.747169	−	0.664634i	$-0.768587\pi$
$673$	−38.7665	−1.49434	−0.747169	+	0.664634i	$0.768587\pi$
$674$	17.1843	0.661914
$675$	68.7851	2.64754
$676$	14.4974	0.557592
$677$	−42.5199	−1.63417	−0.817087	−	0.576514i	$-0.804412\pi$
$677$	−42.5199	−1.63417	−0.817087	+	0.576514i	$0.804412\pi$
$678$	−16.0935	−0.618068
$679$	48.5303	1.86242
$680$	−4.85522	−0.186189
$681$	−10.1134	−0.387548
$682$	13.3377	0.510726
$683$	−7.10806	−0.271982	−0.135991	−	0.990710i	$-0.543422\pi$
$683$	−7.10806	−0.271982	−0.135991	+	0.990710i	$0.543422\pi$
$684$	−12.8101	−0.489808
$685$	−4.14630	−0.158422
$686$	9.53022	0.363866
$687$	−69.0281	−2.63359
$688$	4.20984	0.160498
$689$	11.2955	0.430326
$690$	10.9388	0.416433
$691$	−23.5955	−0.897613	−0.448807	−	0.893629i	$-0.648151\pi$
$691$	−23.5955	−0.897613	−0.448807	+	0.893629i	$0.648151\pi$
$692$	−19.0447	−0.723972
$693$	−76.4408	−2.90374
$694$	6.17418	0.234368
$695$	−14.9190	−0.565910
$696$	16.0193	0.607208
$697$	56.9528	2.15724
$698$	18.8072	0.711862
$699$	25.6662	0.970785
$700$	−14.7579	−0.557798
$701$	−33.0020	−1.24647	−0.623234	−	0.782036i	$-0.714181\pi$
$701$	−33.0020	−1.24647	−0.623234	+	0.782036i	$0.714181\pi$
$702$	81.5945	3.07959
$703$	−13.2859	−0.501086
$704$	−2.95576	−0.111399
$705$	−13.6082	−0.512514
$706$	0.937462	0.0352818
$707$	−16.9897	−0.638963
$708$	−14.4332	−0.542434
$709$	30.2767	1.13707	0.568533	−	0.822661i	$-0.307511\pi$
$709$	30.2767	1.13707	0.568533	+	0.822661i	$0.307511\pi$
$710$	−5.40546	−0.202863
$711$	−45.8841	−1.72079
$712$	−9.19216	−0.344491
$713$	19.7811	0.740806
$714$	69.8062	2.61243
$715$	−11.7981	−0.441223
$716$	−8.56264	−0.320001
$717$	27.2553	1.01787
$718$	−34.2828	−1.27942
$719$	−18.0402	−0.672785	−0.336392	−	0.941722i	$-0.609207\pi$
$719$	−18.0402	−0.672785	−0.336392	+	0.941722i	$0.609207\pi$
$720$	5.89665	0.219755
$721$	−15.8204	−0.589184
$722$	16.2654	0.605337
$723$	47.0454	1.74964
$724$	−21.7761	−0.809301
$725$	−21.6016	−0.802265
$726$	−7.42015	−0.275388
$727$	−52.8817	−1.96127	−0.980636	−	0.195839i	$-0.937257\pi$
$727$	−52.8817	−1.96127	−0.980636	+	0.195839i	$0.937257\pi$
$728$	−17.5062	−0.648823
$729$	62.0915	2.29969
$730$	−3.17343	−0.117454
$731$	26.8521	0.993159
$732$	2.92984	0.108290
$733$	29.0935	1.07459	0.537296	−	0.843394i	$-0.319446\pi$
$733$	29.0935	1.07459	0.537296	+	0.843394i	$0.319446\pi$
$734$	18.2924	0.675184
$735$	−10.3440	−0.381545
$736$	−4.38368	−0.161584
$737$	18.6872	0.688352
$738$	−69.1691	−2.54615
$739$	9.24349	0.340027	0.170014	−	0.985442i	$-0.445619\pi$
$739$	9.24349	0.340027	0.170014	+	0.985442i	$0.445619\pi$
$740$	6.11564	0.224815
$741$	28.4266	1.04428
$742$	−7.19130	−0.264001
$743$	−33.4012	−1.22537	−0.612685	−	0.790327i	$-0.709911\pi$
$743$	−33.4012	−1.22537	−0.612685	+	0.790327i	$0.709911\pi$
$744$	14.7927	0.542325
$745$	−13.5753	−0.497360
$746$	23.6304	0.865171
$747$	99.5185	3.64119
$748$	−18.8531	−0.689336
$749$	7.69477	0.281161
$750$	−23.5076	−0.858377
$751$	−13.1405	−0.479503	−0.239752	−	0.970834i	$-0.577066\pi$
$751$	−13.1405	−0.479503	−0.239752	+	0.970834i	$0.577066\pi$
$752$	5.45343	0.198866
$753$	64.7995	2.36143
$754$	−25.6244	−0.933184
$755$	3.29097	0.119771
$756$	−51.9471	−1.88930
$757$	−2.65158	−0.0963735	−0.0481867	−	0.998838i	$-0.515344\pi$
$757$	−2.65158	−0.0963735	−0.0481867	+	0.998838i	$0.515344\pi$
$758$	−22.9131	−0.832242
$759$	42.4759	1.54178
$760$	1.25875	0.0456597
$761$	37.8570	1.37232	0.686158	−	0.727452i	$-0.259296\pi$
$761$	37.8570	1.37232	0.686158	+	0.727452i	$0.259296\pi$
$762$	−69.1156	−2.50380
$763$	7.44375	0.269482
$764$	24.5732	0.889026
$765$	37.6113	1.35984
$766$	−33.5645	−1.21274
$767$	23.0874	0.833636
$768$	−3.27820	−0.118292
$769$	12.2828	0.442928	0.221464	−	0.975169i	$-0.428916\pi$
$769$	12.2828	0.442928	0.221464	+	0.975169i	$0.428916\pi$
$770$	7.51124	0.270686
$771$	−62.7712	−2.26065
$772$	9.13136	0.328645
$773$	−25.4590	−0.915698	−0.457849	−	0.889030i	$-0.651380\pi$
$773$	−25.4590	−0.915698	−0.457849	+	0.889030i	$0.651380\pi$
$774$	−32.6118	−1.17221
$775$	−19.9476	−0.716539
$776$	−14.5367	−0.521838
$777$	−87.9280	−3.15440
$778$	19.1177	0.685402
$779$	−14.7655	−0.529027
$780$	−13.0851	−0.468521
$781$	−20.9897	−0.751069
$782$	−27.9609	−0.999879
$783$	−76.0365	−2.71732
$784$	4.14533	0.148047
$785$	3.08908	0.110254
$786$	52.9021	1.88696
$787$	−32.5873	−1.16161	−0.580806	−	0.814042i	$-0.697263\pi$
$787$	−32.5873	−1.16161	−0.580806	+	0.814042i	$0.697263\pi$
$788$	16.9560	0.604032
$789$	−40.7907	−1.45219
$790$	4.50867	0.160411
$791$	−16.3894	−0.582739
$792$	22.8970	0.813610
$793$	−4.68656	−0.166425
$794$	−10.2435	−0.363530
$795$	−5.37517	−0.190638
$796$	−14.3152	−0.507389
$797$	−30.3962	−1.07669	−0.538344	−	0.842725i	$-0.680950\pi$
$797$	−30.3962	−1.07669	−0.538344	+	0.842725i	$0.680950\pi$
$798$	−18.0978	−0.640655
$799$	34.7842	1.23058
$800$	4.42058	0.156291
$801$	71.2077	2.51600
$802$	33.7507	1.19178
$803$	−12.3226	−0.434855
$804$	20.7257	0.730940
$805$	11.1399	0.392629
$806$	−23.6623	−0.833469
$807$	−37.1167	−1.30657
$808$	5.08908	0.179033
$809$	−25.6437	−0.901584	−0.450792	−	0.892629i	$-0.648858\pi$
$809$	−25.6437	−0.901584	−0.450792	+	0.892629i	$0.648858\pi$
$810$	−21.1381	−0.742718
$811$	15.7315	0.552408	0.276204	−	0.961099i	$-0.410924\pi$
$811$	15.7315	0.552408	0.276204	+	0.961099i	$0.410924\pi$
$812$	16.3137	0.572500
$813$	−66.5225	−2.33305
$814$	23.7473	0.832343
$815$	−17.8589	−0.625570
$816$	−20.9097	−0.731985
$817$	−6.96160	−0.243556
$818$	−20.0616	−0.701436
$819$	135.613	4.73871
$820$	6.79671	0.237351
$821$	9.01503	0.314627	0.157313	−	0.987549i	$-0.449717\pi$
$821$	9.01503	0.314627	0.157313	+	0.987549i	$0.449717\pi$
$822$	−17.8566	−0.622821
$823$	−29.8518	−1.04057	−0.520284	−	0.853993i	$-0.674174\pi$
$823$	−29.8518	−1.04057	−0.520284	+	0.853993i	$0.674174\pi$
$824$	4.73884	0.165085
$825$	−42.8335	−1.49127
$826$	−14.6986	−0.511428
$827$	40.4232	1.40565	0.702826	−	0.711362i	$-0.251921\pi$
$827$	40.4232	1.40565	0.702826	+	0.711362i	$0.251921\pi$
$828$	33.9585	1.18014
$829$	40.2885	1.39928	0.699639	−	0.714497i	$-0.253344\pi$
$829$	40.2885	1.39928	0.699639	+	0.714497i	$0.253344\pi$
$830$	−9.77891	−0.339431
$831$	54.2585	1.88221
$832$	5.24380	0.181796
$833$	26.4406	0.916112
$834$	−64.2508	−2.22482
$835$	−13.3333	−0.461417
$836$	4.88780	0.169048
$837$	−70.2144	−2.42696
$838$	−14.4360	−0.498683
$839$	−46.4070	−1.60215	−0.801073	−	0.598567i	$-0.795737\pi$
$839$	−46.4070	−1.60215	−0.801073	+	0.598567i	$0.795737\pi$
$840$	8.33062	0.287434
$841$	−5.12109	−0.176589
$842$	0.501621	0.0172870
$843$	14.2646	0.491300
$844$	2.90045	0.0998377
$845$	11.0353	0.379628
$846$	−42.2454	−1.45243
$847$	−7.55656	−0.259646
$848$	2.15408	0.0739713
$849$	70.5773	2.42221
$850$	28.1963	0.967125
$851$	35.2196	1.20731
$852$	−23.2794	−0.797538
$853$	12.3727	0.423635	0.211817	−	0.977309i	$-0.432062\pi$
$853$	12.3727	0.423635	0.211817	+	0.977309i	$0.432062\pi$
$854$	2.98370	0.102100
$855$	−9.75101	−0.333477
$856$	−2.30489	−0.0787794
$857$	11.9095	0.406822	0.203411	−	0.979093i	$-0.434797\pi$
$857$	11.9095	0.406822	0.203411	+	0.979093i	$0.434797\pi$
$858$	−50.8101	−1.73463
$859$	10.8432	0.369965	0.184982	−	0.982742i	$-0.440777\pi$
$859$	10.8432	0.369965	0.184982	+	0.982742i	$0.440777\pi$
$860$	3.20451	0.109273
$861$	−97.7201	−3.33029
$862$	−18.6947	−0.636745
$863$	18.6454	0.634697	0.317348	−	0.948309i	$-0.397208\pi$
$863$	18.6454	0.634697	0.317348	+	0.948309i	$0.397208\pi$
$864$	15.5602	0.529369
$865$	−14.4968	−0.492905
$866$	6.55103	0.222613
$867$	−77.6412	−2.63683
$868$	15.0646	0.511326
$869$	17.5074	0.593898
$870$	12.1938	0.413408
$871$	−33.1528	−1.12334
$872$	−2.22969	−0.0755070
$873$	112.610	3.81126
$874$	7.24907	0.245203
$875$	−23.9398	−0.809312
$876$	−13.6668	−0.461760
$877$	18.2893	0.617584	0.308792	−	0.951130i	$-0.400075\pi$
$877$	18.2893	0.617584	0.308792	+	0.951130i	$0.400075\pi$
$878$	7.05395	0.238059
$879$	50.1662	1.69206
$880$	−2.24991	−0.0758445
$881$	−23.8852	−0.804713	−0.402357	−	0.915483i	$-0.631809\pi$
$881$	−23.8852	−0.804713	−0.402357	+	0.915483i	$0.631809\pi$
$882$	−32.1121	−1.08127
$883$	−28.1485	−0.947271	−0.473636	−	0.880721i	$-0.657059\pi$
$883$	−28.1485	−0.947271	−0.473636	+	0.880721i	$0.657059\pi$
$884$	33.4471	1.12495
$885$	−10.9865	−0.369307
$886$	−4.63593	−0.155747
$887$	46.4221	1.55870	0.779351	−	0.626588i	$-0.215549\pi$
$887$	46.4221	1.55870	0.779351	+	0.626588i	$0.215549\pi$
$888$	26.3379	0.883841
$889$	−70.3862	−2.36068
$890$	−6.99703	−0.234541
$891$	−82.0804	−2.74980
$892$	−6.87674	−0.230250
$893$	−9.01807	−0.301778
$894$	−58.4639	−1.95533
$895$	−6.51784	−0.217867
$896$	−3.33846	−0.111530
$897$	−75.3562	−2.51607
$898$	−41.4052	−1.38171
$899$	22.0505	0.735425
$900$	−34.2444	−1.14148
$901$	13.7396	0.457732
$902$	26.3920	0.878757
$903$	−46.0730	−1.53321
$904$	4.90926	0.163280
$905$	−16.5758	−0.551000
$906$	14.1731	0.470868
$907$	40.8114	1.35512	0.677560	−	0.735468i	$-0.263038\pi$
$907$	40.8114	1.35512	0.677560	+	0.735468i	$0.263038\pi$
$908$	3.08506	0.102381
$909$	−39.4229	−1.30758
$910$	−13.3256	−0.441741
$911$	31.1970	1.03360	0.516802	−	0.856105i	$-0.327122\pi$
$911$	31.1970	1.03360	0.516802	+	0.856105i	$0.327122\pi$
$912$	5.42099	0.179507
$913$	−37.9720	−1.25669
$914$	7.75944	0.256660
$915$	2.23018	0.0737274
$916$	21.0567	0.695734
$917$	53.8746	1.77910
$918$	99.2493	3.27572
$919$	39.9149	1.31667	0.658336	−	0.752724i	$-0.271261\pi$
$919$	39.9149	1.31667	0.658336	+	0.752724i	$0.271261\pi$
$920$	−3.33683	−0.110012
$921$	−98.8429	−3.25698
$922$	−4.29248	−0.141365
$923$	37.2376	1.22569
$924$	32.3482	1.06418
$925$	−35.5161	−1.16776
$926$	0.948591	0.0311726
$927$	−36.7098	−1.20571
$928$	−4.88660	−0.160411
$929$	−29.2769	−0.960545	−0.480272	−	0.877119i	$-0.659462\pi$
$929$	−29.2769	−0.960545	−0.480272	+	0.877119i	$0.659462\pi$
$930$	11.2601	0.369233
$931$	−6.85492	−0.224661
$932$	−7.82937	−0.256459
$933$	38.8355	1.27142
$934$	33.6381	1.10067
$935$	−14.3509	−0.469323
$936$	−40.6214	−1.32775
$937$	−48.6823	−1.59038	−0.795192	−	0.606358i	$-0.792630\pi$
$937$	−48.6823	−1.59038	−0.795192	+	0.606358i	$0.792630\pi$
$938$	21.1067	0.689160
$939$	59.5267	1.94258
$940$	4.15112	0.135395
$941$	31.1318	1.01487	0.507434	−	0.861690i	$-0.330594\pi$
$941$	31.1318	1.01487	0.507434	+	0.861690i	$0.330594\pi$
$942$	13.3036	0.433454
$943$	39.1418	1.27463
$944$	4.40280	0.143299
$945$	−39.5419	−1.28630
$946$	12.4433	0.404565
$947$	−46.4354	−1.50895	−0.754474	−	0.656330i	$-0.772108\pi$
$947$	−46.4354	−1.50895	−0.754474	+	0.656330i	$0.772108\pi$
$948$	19.4172	0.630643
$949$	21.8615	0.709653
$950$	−7.31010	−0.237171
$951$	38.2898	1.24163
$952$	−21.2941	−0.690145
$953$	57.5645	1.86470	0.932348	−	0.361561i	$-0.117756\pi$
$953$	57.5645	1.86470	0.932348	+	0.361561i	$0.117756\pi$
$954$	−16.6867	−0.540252
$955$	18.7050	0.605279
$956$	−8.31411	−0.268898
$957$	47.3491	1.53058
$958$	0.721571	0.0233129
$959$	−18.1849	−0.587221
$960$	−2.49535	−0.0805370
$961$	−10.6379	−0.343159
$962$	−42.1300	−1.35833
$963$	17.8550	0.575368
$964$	−14.3510	−0.462215
$965$	6.95075	0.223752
$966$	47.9755	1.54359
$967$	−14.2182	−0.457228	−0.228614	−	0.973517i	$-0.573419\pi$
$967$	−14.2182	−0.457228	−0.228614	+	0.973517i	$0.573419\pi$
$968$	2.26348	0.0727512
$969$	34.5773	1.11078
$970$	−11.0653	−0.355285
$971$	40.3296	1.29424	0.647119	−	0.762389i	$-0.275974\pi$
$971$	40.3296	1.29424	0.647119	+	0.762389i	$0.275974\pi$
$972$	−44.3538	−1.42265
$973$	−65.4320	−2.09765
$974$	−30.3156	−0.971375
$975$	75.9907	2.43365
$976$	−0.893734	−0.0286077
$977$	−13.8194	−0.442123	−0.221061	−	0.975260i	$-0.570952\pi$
$977$	−13.8194	−0.442123	−0.221061	+	0.975260i	$0.570952\pi$
$978$	−76.9119	−2.45937
$979$	−27.1698	−0.868351
$980$	3.15540	0.100796
$981$	17.2725	0.551468
$982$	23.1038	0.737273
$983$	−30.5896	−0.975657	−0.487828	−	0.872940i	$-0.662211\pi$
$983$	−30.5896	−0.975657	−0.487828	+	0.872940i	$0.662211\pi$
$984$	29.2710	0.933126
$985$	12.9068	0.411245
$986$	−31.1688	−0.992616
$987$	−59.6830	−1.89973
$988$	−8.67141	−0.275874
$989$	18.4546	0.586821
$990$	17.4291	0.553933
$991$	14.7208	0.467621	0.233810	−	0.972282i	$-0.424880\pi$
$991$	14.7208	0.467621	0.233810	+	0.972282i	$0.424880\pi$
$992$	−4.51244	−0.143270
$993$	43.5377	1.38163
$994$	−23.7073	−0.751951
$995$	−10.8967	−0.345448
$996$	−42.1143	−1.33444
$997$	−47.5849	−1.50703	−0.753514	−	0.657431i	$-0.771643\pi$
$997$	−47.5849	−1.50703	−0.753514	+	0.657431i	$0.771643\pi$
$998$	−23.6414	−0.748355
$999$	−125.015	−3.95529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.g.1.4	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.g.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.27820 q^{3} +1.00000 q^{4} +0.761195 q^{5} +3.27820 q^{6} +3.33846 q^{7} -1.00000 q^{8} +7.74657 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.27820 q^{3} +1.00000 q^{4} +0.761195 q^{5} +3.27820 q^{6} +3.33846 q^{7} -1.00000 q^{8} +7.74657 q^{9} -0.761195 q^{10} -2.95576 q^{11} -3.27820 q^{12} +5.24380 q^{13} -3.33846 q^{14} -2.49535 q^{15} +1.00000 q^{16} +6.37841 q^{17} -7.74657 q^{18} -1.65365 q^{19} +0.761195 q^{20} -10.9441 q^{21} +2.95576 q^{22} +4.38368 q^{23} +3.27820 q^{24} -4.42058 q^{25} -5.24380 q^{26} -15.5602 q^{27} +3.33846 q^{28} +4.88660 q^{29} +2.49535 q^{30} +4.51244 q^{31} -1.00000 q^{32} +9.68956 q^{33} -6.37841 q^{34} +2.54122 q^{35} +7.74657 q^{36} +8.03426 q^{37} +1.65365 q^{38} -17.1902 q^{39} -0.761195 q^{40} +8.92900 q^{41} +10.9441 q^{42} +4.20984 q^{43} -2.95576 q^{44} +5.89665 q^{45} -4.38368 q^{46} +5.45343 q^{47} -3.27820 q^{48} +4.14533 q^{49} +4.42058 q^{50} -20.9097 q^{51} +5.24380 q^{52} +2.15408 q^{53} +15.5602 q^{54} -2.24991 q^{55} -3.33846 q^{56} +5.42099 q^{57} -4.88660 q^{58} +4.40280 q^{59} -2.49535 q^{60} -0.893734 q^{61} -4.51244 q^{62} +25.8616 q^{63} +1.00000 q^{64} +3.99155 q^{65} -9.68956 q^{66} -6.32230 q^{67} +6.37841 q^{68} -14.3705 q^{69} -2.54122 q^{70} +7.10128 q^{71} -7.74657 q^{72} +4.16901 q^{73} -8.03426 q^{74} +14.4915 q^{75} -1.65365 q^{76} -9.86769 q^{77} +17.1902 q^{78} -5.92315 q^{79} +0.761195 q^{80} +27.7697 q^{81} -8.92900 q^{82} +12.8468 q^{83} -10.9441 q^{84} +4.85522 q^{85} -4.20984 q^{86} -16.0193 q^{87} +2.95576 q^{88} +9.19216 q^{89} -5.89665 q^{90} +17.5062 q^{91} +4.38368 q^{92} -14.7927 q^{93} -5.45343 q^{94} -1.25875 q^{95} +3.27820 q^{96} +14.5367 q^{97} -4.14533 q^{98} -22.8970 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9}+O(q^{10}) \) 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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