LMFDB - Embedded newform 8047.2.a.e.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$0$
Dimension:	$168$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43547 q^{2} -1.20585 q^{3} +3.93153 q^{4} +2.83456 q^{5} +2.93681 q^{6} -2.65946 q^{7} -4.70420 q^{8} -1.54594 q^{9} +O(q^{10})$	\(q-2.43547 q^{2} -1.20585 q^{3} +3.93153 q^{4} +2.83456 q^{5} +2.93681 q^{6} -2.65946 q^{7} -4.70420 q^{8} -1.54594 q^{9} -6.90350 q^{10} +5.01339 q^{11} -4.74082 q^{12} +1.00000 q^{13} +6.47706 q^{14} -3.41804 q^{15} +3.59389 q^{16} -4.13573 q^{17} +3.76509 q^{18} -3.13256 q^{19} +11.1442 q^{20} +3.20690 q^{21} -12.2100 q^{22} +1.23317 q^{23} +5.67254 q^{24} +3.03474 q^{25} -2.43547 q^{26} +5.48170 q^{27} -10.4558 q^{28} -7.71676 q^{29} +8.32456 q^{30} -0.678022 q^{31} +0.655581 q^{32} -6.04538 q^{33} +10.0725 q^{34} -7.53842 q^{35} -6.07790 q^{36} +11.2780 q^{37} +7.62927 q^{38} -1.20585 q^{39} -13.3343 q^{40} -6.92832 q^{41} -7.81033 q^{42} -7.99706 q^{43} +19.7103 q^{44} -4.38205 q^{45} -3.00336 q^{46} +4.93419 q^{47} -4.33367 q^{48} +0.0727519 q^{49} -7.39102 q^{50} +4.98706 q^{51} +3.93153 q^{52} +0.663461 q^{53} -13.3505 q^{54} +14.2108 q^{55} +12.5106 q^{56} +3.77739 q^{57} +18.7940 q^{58} -3.01717 q^{59} -13.4382 q^{60} -6.52869 q^{61} +1.65130 q^{62} +4.11136 q^{63} -8.78442 q^{64} +2.83456 q^{65} +14.7234 q^{66} +8.35556 q^{67} -16.2598 q^{68} -1.48702 q^{69} +18.3596 q^{70} -8.35489 q^{71} +7.27239 q^{72} +5.18595 q^{73} -27.4673 q^{74} -3.65943 q^{75} -12.3158 q^{76} -13.3329 q^{77} +2.93681 q^{78} +10.8042 q^{79} +10.1871 q^{80} -1.97228 q^{81} +16.8737 q^{82} -12.6009 q^{83} +12.6081 q^{84} -11.7230 q^{85} +19.4766 q^{86} +9.30522 q^{87} -23.5840 q^{88} +12.8564 q^{89} +10.6724 q^{90} -2.65946 q^{91} +4.84826 q^{92} +0.817590 q^{93} -12.0171 q^{94} -8.87943 q^{95} -0.790530 q^{96} -3.72103 q^{97} -0.177185 q^{98} -7.75038 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) $ 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * 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 $

$2$	−2.43547	−1.72214	−0.861070	−	0.508486i	$-0.830205\pi$
$2$	−2.43547	−1.72214	−0.861070	+	0.508486i	$0.830205\pi$
$3$	−1.20585	−0.696195	−0.348098	−	0.937458i	$-0.613172\pi$
$3$	−1.20585	−0.696195	−0.348098	+	0.937458i	$0.613172\pi$
$4$	3.93153	1.96577
$5$	2.83456	1.26765	0.633827	−	0.773475i	$-0.281483\pi$
$5$	2.83456	1.26765	0.633827	+	0.773475i	$0.281483\pi$
$6$	2.93681	1.19895
$7$	−2.65946	−1.00518	−0.502592	−	0.864524i	$-0.667620\pi$
$7$	−2.65946	−1.00518	−0.502592	+	0.864524i	$0.667620\pi$
$8$	−4.70420	−1.66319
$9$	−1.54594	−0.515312
$10$	−6.90350	−2.18308
$11$	5.01339	1.51159	0.755797	−	0.654806i	$-0.227250\pi$
$11$	5.01339	1.51159	0.755797	+	0.654806i	$0.227250\pi$
$12$	−4.74082	−1.36856
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	6.47706	1.73107
$15$	−3.41804	−0.882535
$16$	3.59389	0.898471
$17$	−4.13573	−1.00306	−0.501531	−	0.865139i	$-0.667230\pi$
$17$	−4.13573	−1.00306	−0.501531	+	0.865139i	$0.667230\pi$
$18$	3.76509	0.887439
$19$	−3.13256	−0.718659	−0.359329	−	0.933211i	$-0.616995\pi$
$19$	−3.13256	−0.718659	−0.359329	+	0.933211i	$0.616995\pi$
$20$	11.1442	2.49191
$21$	3.20690	0.699804
$22$	−12.2100	−2.60318
$23$	1.23317	0.257135	0.128567	−	0.991701i	$-0.458962\pi$
$23$	1.23317	0.257135	0.128567	+	0.991701i	$0.458962\pi$
$24$	5.67254	1.15790
$25$	3.03474	0.606947
$26$	−2.43547	−0.477636
$27$	5.48170	1.05495
$28$	−10.4558	−1.97596
$29$	−7.71676	−1.43297	−0.716483	−	0.697605i	$-0.754249\pi$
$29$	−7.71676	−1.43297	−0.716483	+	0.697605i	$0.754249\pi$
$30$	8.32456	1.51985
$31$	−0.678022	−0.121776	−0.0608881	−	0.998145i	$-0.519393\pi$
$31$	−0.678022	−0.121776	−0.0608881	+	0.998145i	$0.519393\pi$
$32$	0.655581	0.115891
$33$	−6.04538	−1.05237
$34$	10.0725	1.72741
$35$	−7.53842	−1.27422
$36$	−6.07790	−1.01298
$37$	11.2780	1.85409	0.927047	−	0.374946i	$-0.122339\pi$
$37$	11.2780	1.85409	0.927047	+	0.374946i	$0.122339\pi$
$38$	7.62927	1.23763
$39$	−1.20585	−0.193090
$40$	−13.3343	−2.10834
$41$	−6.92832	−1.08202	−0.541011	−	0.841015i	$-0.681958\pi$
$41$	−6.92832	−1.08202	−0.541011	+	0.841015i	$0.681958\pi$
$42$	−7.81033	−1.20516
$43$	−7.99706	−1.21954	−0.609770	−	0.792578i	$-0.708738\pi$
$43$	−7.99706	−1.21954	−0.609770	+	0.792578i	$0.708738\pi$
$44$	19.7103	2.97144
$45$	−4.38205	−0.653237
$46$	−3.00336	−0.442822
$47$	4.93419	0.719725	0.359863	−	0.933005i	$-0.382824\pi$
$47$	4.93419	0.719725	0.359863	+	0.933005i	$0.382824\pi$
$48$	−4.33367	−0.625512
$49$	0.0727519	0.0103931
$50$	−7.39102	−1.04525
$51$	4.98706	0.698328
$52$	3.93153	0.545206
$53$	0.663461	0.0911333	0.0455667	−	0.998961i	$-0.485491\pi$
$53$	0.663461	0.0911333	0.0455667	+	0.998961i	$0.485491\pi$
$54$	−13.3505	−1.81678
$55$	14.2108	1.91618
$56$	12.5106	1.67181
$57$	3.77739	0.500327
$58$	18.7940	2.46777
$59$	−3.01717	−0.392802	−0.196401	−	0.980524i	$-0.562925\pi$
$59$	−3.01717	−0.392802	−0.196401	+	0.980524i	$0.562925\pi$
$60$	−13.4382	−1.73486
$61$	−6.52869	−0.835913	−0.417956	−	0.908467i	$-0.637254\pi$
$61$	−6.52869	−0.835913	−0.417956	+	0.908467i	$0.637254\pi$
$62$	1.65130	0.209716
$63$	4.11136	0.517983
$64$	−8.78442	−1.09805
$65$	2.83456	0.351584
$66$	14.7234	1.81232
$67$	8.35556	1.02079	0.510397	−	0.859939i	$-0.329498\pi$
$67$	8.35556	1.02079	0.510397	+	0.859939i	$0.329498\pi$
$68$	−16.2598	−1.97179
$69$	−1.48702	−0.179016
$70$	18.3596	2.19439
$71$	−8.35489	−0.991543	−0.495772	−	0.868453i	$-0.665115\pi$
$71$	−8.35489	−0.991543	−0.495772	+	0.868453i	$0.665115\pi$
$72$	7.27239	0.857059
$73$	5.18595	0.606970	0.303485	−	0.952836i	$-0.401850\pi$
$73$	5.18595	0.606970	0.303485	+	0.952836i	$0.401850\pi$
$74$	−27.4673	−3.19301
$75$	−3.65943	−0.422554
$76$	−12.3158	−1.41272
$77$	−13.3329	−1.51943
$78$	2.93681	0.332528
$79$	10.8042	1.21557	0.607786	−	0.794101i	$-0.292058\pi$
$79$	10.8042	1.21557	0.607786	+	0.794101i	$0.292058\pi$
$80$	10.1871	1.13895
$81$	−1.97228	−0.219142
$82$	16.8737	1.86339
$83$	−12.6009	−1.38313	−0.691567	−	0.722313i	$-0.743079\pi$
$83$	−12.6009	−1.38313	−0.691567	+	0.722313i	$0.743079\pi$
$84$	12.6081	1.37565
$85$	−11.7230	−1.27154
$86$	19.4766	2.10022
$87$	9.30522	0.997624
$88$	−23.5840	−2.51406
$89$	12.8564	1.36277	0.681385	−	0.731925i	$-0.261378\pi$
$89$	12.8564	1.36277	0.681385	+	0.731925i	$0.261378\pi$
$90$	10.6724	1.12497
$91$	−2.65946	−0.278788
$92$	4.84826	0.505466
$93$	0.817590	0.0847801
$94$	−12.0171	−1.23947
$95$	−8.87943	−0.911011
$96$	−0.790530	−0.0806831
$97$	−3.72103	−0.377814	−0.188907	−	0.981995i	$-0.560494\pi$
$97$	−3.72103	−0.377814	−0.188907	+	0.981995i	$0.560494\pi$
$98$	−0.177185	−0.0178984
$99$	−7.75038	−0.778943
$100$	11.9312	1.19312
$101$	15.0825	1.50076	0.750382	−	0.661005i	$-0.229870\pi$
$101$	15.0825	1.50076	0.750382	+	0.661005i	$0.229870\pi$
$102$	−12.1458	−1.20262
$103$	1.35017	0.133036	0.0665181	−	0.997785i	$-0.478811\pi$
$103$	1.35017	0.133036	0.0665181	+	0.997785i	$0.478811\pi$
$104$	−4.70420	−0.461285
$105$	9.09017	0.887109
$106$	−1.61584	−0.156944
$107$	−7.03070	−0.679684	−0.339842	−	0.940483i	$-0.610374\pi$
$107$	−7.03070	−0.679684	−0.339842	+	0.940483i	$0.610374\pi$
$108$	21.5515	2.07379
$109$	−0.738234	−0.0707100	−0.0353550	−	0.999375i	$-0.511256\pi$
$109$	−0.738234	−0.0707100	−0.0353550	+	0.999375i	$0.511256\pi$
$110$	−34.6099	−3.29993
$111$	−13.5995	−1.29081
$112$	−9.55781	−0.903128
$113$	−11.3369	−1.06648	−0.533241	−	0.845963i	$-0.679026\pi$
$113$	−11.3369	−1.06648	−0.533241	+	0.845963i	$0.679026\pi$
$114$	−9.19972	−0.861633
$115$	3.49551	0.325958
$116$	−30.3387	−2.81688
$117$	−1.54594	−0.142922
$118$	7.34824	0.676460
$119$	10.9988	1.00826
$120$	16.0792	1.46782
$121$	14.1341	1.28492
$122$	15.9004	1.43956
$123$	8.35449	0.753299
$124$	−2.66566	−0.239384
$125$	−5.57066	−0.498255
$126$	−10.0131	−0.892039
$127$	−3.36471	−0.298569	−0.149285	−	0.988794i	$-0.547697\pi$
$127$	−3.36471	−0.298569	−0.149285	+	0.988794i	$0.547697\pi$
$128$	20.0831	1.77511
$129$	9.64322	0.849038
$130$	−6.90350	−0.605477
$131$	−17.7988	−1.55508	−0.777542	−	0.628830i	$-0.783534\pi$
$131$	−17.7988	−1.55508	−0.777542	+	0.628830i	$0.783534\pi$
$132$	−23.7676	−2.06870
$133$	8.33093	0.722384
$134$	−20.3497	−1.75795
$135$	15.5382	1.33732
$136$	19.4553	1.66828
$137$	14.9284	1.27542	0.637711	−	0.770276i	$-0.279881\pi$
$137$	14.9284	1.27542	0.637711	+	0.770276i	$0.279881\pi$
$138$	3.62159	0.308290
$139$	−8.53581	−0.723998	−0.361999	−	0.932179i	$-0.617906\pi$
$139$	−8.53581	−0.723998	−0.361999	+	0.932179i	$0.617906\pi$
$140$	−29.6375	−2.50483
$141$	−5.94987	−0.501070
$142$	20.3481	1.70758
$143$	5.01339	0.419241
$144$	−5.55592	−0.462993
$145$	−21.8736	−1.81651
$146$	−12.6303	−1.04529
$147$	−0.0877275	−0.00723565
$148$	44.3399	3.64471
$149$	8.10850	0.664274	0.332137	−	0.943231i	$-0.392230\pi$
$149$	8.10850	0.664274	0.332137	+	0.943231i	$0.392230\pi$
$150$	8.91244	0.727697
$151$	21.9496	1.78623	0.893115	−	0.449828i	$-0.148515\pi$
$151$	21.9496	1.78623	0.893115	+	0.449828i	$0.148515\pi$
$152$	14.7362	1.19526
$153$	6.39358	0.516890
$154$	32.4720	2.61667
$155$	−1.92189	−0.154370
$156$	−4.74082	−0.379570
$157$	−6.93961	−0.553841	−0.276921	−	0.960893i	$-0.589314\pi$
$157$	−6.93961	−0.553841	−0.276921	+	0.960893i	$0.589314\pi$
$158$	−26.3134	−2.09338
$159$	−0.800031	−0.0634466
$160$	1.85828	0.146910
$161$	−3.27958	−0.258467
$162$	4.80343	0.377393
$163$	25.1217	1.96768	0.983842	−	0.179038i	$-0.0572984\pi$
$163$	25.1217	1.96768	0.983842	+	0.179038i	$0.0572984\pi$
$164$	−27.2389	−2.12700
$165$	−17.1360	−1.33404
$166$	30.6893	2.38195
$167$	18.3235	1.41792	0.708959	−	0.705250i	$-0.249165\pi$
$167$	18.3235	1.41792	0.708959	+	0.705250i	$0.249165\pi$
$168$	−15.0859	−1.16390
$169$	1.00000	0.0769231
$170$	28.5510	2.18976
$171$	4.84274	0.370333
$172$	−31.4407	−2.39733
$173$	20.7816	1.57999	0.789997	−	0.613110i	$-0.210082\pi$
$173$	20.7816	1.57999	0.789997	+	0.613110i	$0.210082\pi$
$174$	−22.6626	−1.71805
$175$	−8.07078	−0.610093
$176$	18.0176	1.35812
$177$	3.63824	0.273467
$178$	−31.3113	−2.34688
$179$	6.41274	0.479310	0.239655	−	0.970858i	$-0.422966\pi$
$179$	6.41274	0.479310	0.239655	+	0.970858i	$0.422966\pi$
$180$	−17.2282	−1.28411
$181$	−1.50824	−0.112106	−0.0560531	−	0.998428i	$-0.517852\pi$
$181$	−1.50824	−0.112106	−0.0560531	+	0.998428i	$0.517852\pi$
$182$	6.47706	0.480111
$183$	7.87259	0.581959
$184$	−5.80109	−0.427662
$185$	31.9682	2.35035
$186$	−1.99122	−0.146003
$187$	−20.7341	−1.51622
$188$	19.3989	1.41481
$189$	−14.5784	−1.06042
$190$	21.6256	1.56889
$191$	9.99803	0.723432	0.361716	−	0.932288i	$-0.382191\pi$
$191$	9.99803	0.723432	0.361716	+	0.932288i	$0.382191\pi$
$192$	10.5927	0.764459
$193$	−0.707623	−0.0509358	−0.0254679	−	0.999676i	$-0.508108\pi$
$193$	−0.707623	−0.0509358	−0.0254679	+	0.999676i	$0.508108\pi$
$194$	9.06248	0.650648
$195$	−3.41804	−0.244771
$196$	0.286026	0.0204305
$197$	9.79006	0.697513	0.348756	−	0.937213i	$-0.386604\pi$
$197$	9.79006	0.697513	0.348756	+	0.937213i	$0.386604\pi$
$198$	18.8758	1.34145
$199$	10.3365	0.732738	0.366369	−	0.930470i	$-0.380601\pi$
$199$	10.3365	0.732738	0.366369	+	0.930470i	$0.380601\pi$
$200$	−14.2760	−1.00947
$201$	−10.0755	−0.710672
$202$	−36.7330	−2.58453
$203$	20.5224	1.44039
$204$	19.6068	1.37275
$205$	−19.6388	−1.37163
$206$	−3.28830	−0.229107
$207$	−1.90641	−0.132504
$208$	3.59389	0.249191
$209$	−15.7048	−1.08632
$210$	−22.1389	−1.52773
$211$	−22.0521	−1.51813	−0.759065	−	0.651015i	$-0.774344\pi$
$211$	−22.0521	−1.51813	−0.759065	+	0.651015i	$0.774344\pi$
$212$	2.60842	0.179147
$213$	10.0747	0.690308
$214$	17.1231	1.17051
$215$	−22.6681	−1.54595
$216$	−25.7870	−1.75458
$217$	1.80317	0.122407
$218$	1.79795	0.121773
$219$	−6.25346	−0.422570
$220$	55.8701	3.76676
$221$	−4.13573	−0.278200
$222$	33.1213	2.22296
$223$	20.3248	1.36105	0.680526	−	0.732724i	$-0.261751\pi$
$223$	20.3248	1.36105	0.680526	+	0.732724i	$0.261751\pi$
$224$	−1.74349	−0.116492
$225$	−4.69151	−0.312767
$226$	27.6106	1.83663
$227$	−6.10944	−0.405498	−0.202749	−	0.979231i	$-0.564987\pi$
$227$	−6.10944	−0.405498	−0.202749	+	0.979231i	$0.564987\pi$
$228$	14.8509	0.983526
$229$	7.76958	0.513428	0.256714	−	0.966487i	$-0.417360\pi$
$229$	7.76958	0.513428	0.256714	+	0.966487i	$0.417360\pi$
$230$	−8.51322	−0.561345
$231$	16.0775	1.05782
$232$	36.3012	2.38329
$233$	−13.9333	−0.912798	−0.456399	−	0.889775i	$-0.650861\pi$
$233$	−13.9333	−0.912798	−0.456399	+	0.889775i	$0.650861\pi$
$234$	3.76509	0.246131
$235$	13.9863	0.912363
$236$	−11.8621	−0.772157
$237$	−13.0282	−0.846275
$238$	−26.7874	−1.73637
$239$	6.56350	0.424557	0.212279	−	0.977209i	$-0.431912\pi$
$239$	6.56350	0.424557	0.212279	+	0.977209i	$0.431912\pi$
$240$	−12.2841	−0.792933
$241$	2.52260	0.162495	0.0812476	−	0.996694i	$-0.474110\pi$
$241$	2.52260	0.162495	0.0812476	+	0.996694i	$0.474110\pi$
$242$	−34.4232	−2.21281
$243$	−14.0668	−0.902388
$244$	−25.6678	−1.64321
$245$	0.206220	0.0131749
$246$	−20.3471	−1.29729
$247$	−3.13256	−0.199320
$248$	3.18955	0.202537
$249$	15.1948	0.962931
$250$	13.5672	0.858064
$251$	−5.27198	−0.332764	−0.166382	−	0.986061i	$-0.553209\pi$
$251$	−5.27198	−0.332764	−0.166382	+	0.986061i	$0.553209\pi$
$252$	16.1640	1.01823
$253$	6.18238	0.388683
$254$	8.19465	0.514178
$255$	14.1361	0.885238
$256$	−31.3429	−1.95893
$257$	23.8348	1.48677	0.743387	−	0.668861i	$-0.233218\pi$
$257$	23.8348	1.48677	0.743387	+	0.668861i	$0.233218\pi$
$258$	−23.4858	−1.46216
$259$	−29.9935	−1.86370
$260$	11.1442	0.691132
$261$	11.9296	0.738424
$262$	43.3484	2.67807
$263$	−29.2837	−1.80571	−0.902857	−	0.429942i	$-0.858534\pi$
$263$	−29.2837	−1.80571	−0.902857	+	0.429942i	$0.858534\pi$
$264$	28.4387	1.75028
$265$	1.88062	0.115526
$266$	−20.2898	−1.24405
$267$	−15.5028	−0.948755
$268$	32.8502	2.00664
$269$	18.3194	1.11695	0.558476	−	0.829521i	$-0.311386\pi$
$269$	18.3194	1.11695	0.558476	+	0.829521i	$0.311386\pi$
$270$	−37.8429	−2.30305
$271$	−11.3125	−0.687186	−0.343593	−	0.939119i	$-0.611644\pi$
$271$	−11.3125	−0.687186	−0.343593	+	0.939119i	$0.611644\pi$
$272$	−14.8634	−0.901223
$273$	3.20690	0.194091
$274$	−36.3578	−2.19646
$275$	15.2143	0.917458
$276$	−5.84626	−0.351903
$277$	30.1493	1.81150	0.905749	−	0.423814i	$-0.139309\pi$
$277$	30.1493	1.81150	0.905749	+	0.423814i	$0.139309\pi$
$278$	20.7887	1.24683
$279$	1.04818	0.0627528
$280$	35.4622	2.11927
$281$	−4.61973	−0.275590	−0.137795	−	0.990461i	$-0.544002\pi$
$281$	−4.61973	−0.275590	−0.137795	+	0.990461i	$0.544002\pi$
$282$	14.4908	0.862912
$283$	1.85629	0.110345	0.0551724	−	0.998477i	$-0.482429\pi$
$283$	1.85629	0.110345	0.0551724	+	0.998477i	$0.482429\pi$
$284$	−32.8475	−1.94914
$285$	10.7072	0.634242
$286$	−12.2100	−0.721991
$287$	18.4256	1.08763
$288$	−1.01349	−0.0597203
$289$	0.104290	0.00613471
$290$	53.2726	3.12828
$291$	4.48699	0.263032
$292$	20.3888	1.19316
$293$	15.0896	0.881545	0.440772	−	0.897619i	$-0.354705\pi$
$293$	15.0896	0.881545	0.440772	+	0.897619i	$0.354705\pi$
$294$	0.213658	0.0124608
$295$	−8.55235	−0.497937
$296$	−53.0540	−3.08370
$297$	27.4819	1.59466
$298$	−19.7480	−1.14397
$299$	1.23317	0.0713163
$300$	−14.3872	−0.830643
$301$	21.2679	1.22586
$302$	−53.4576	−3.07614
$303$	−18.1872	−1.04482
$304$	−11.2581	−0.645694
$305$	−18.5060	−1.05965
$306$	−15.5714	−0.890157
$307$	−1.39525	−0.0796312	−0.0398156	−	0.999207i	$-0.512677\pi$
$307$	−1.39525	−0.0796312	−0.0398156	+	0.999207i	$0.512677\pi$
$308$	−52.4189	−2.98684
$309$	−1.62810	−0.0926192
$310$	4.68072	0.265847
$311$	−9.70885	−0.550538	−0.275269	−	0.961367i	$-0.588767\pi$
$311$	−9.70885	−0.550538	−0.275269	+	0.961367i	$0.588767\pi$
$312$	5.67254	0.321144
$313$	1.21312	0.0685694	0.0342847	−	0.999412i	$-0.489085\pi$
$313$	1.21312	0.0685694	0.0342847	+	0.999412i	$0.489085\pi$
$314$	16.9012	0.953792
$315$	11.6539	0.656623
$316$	42.4772	2.38953
$317$	−31.7628	−1.78398	−0.891988	−	0.452058i	$-0.850690\pi$
$317$	−31.7628	−1.78398	−0.891988	+	0.452058i	$0.850690\pi$
$318$	1.94845	0.109264
$319$	−38.6871	−2.16606
$320$	−24.9000	−1.39195
$321$	8.47794	0.473193
$322$	7.98734	0.445117
$323$	12.9554	0.720860
$324$	−7.75407	−0.430782
$325$	3.03474	0.168337
$326$	−61.1833	−3.38863
$327$	0.890197	0.0492280
$328$	32.5922	1.79960
$329$	−13.1223	−0.723456
$330$	41.7343	2.29740
$331$	−12.6839	−0.697170	−0.348585	−	0.937277i	$-0.613338\pi$
$331$	−12.6839	−0.697170	−0.348585	+	0.937277i	$0.613338\pi$
$332$	−49.5410	−2.71892
$333$	−17.4351	−0.955436
$334$	−44.6265	−2.44185
$335$	23.6843	1.29401
$336$	11.5252	0.628754
$337$	−4.06048	−0.221188	−0.110594	−	0.993866i	$-0.535275\pi$
$337$	−4.06048	−0.221188	−0.110594	+	0.993866i	$0.535275\pi$
$338$	−2.43547	−0.132472
$339$	13.6705	0.742480
$340$	−46.0893	−2.49954
$341$	−3.39919	−0.184076
$342$	−11.7944	−0.637766
$343$	18.4228	0.994736
$344$	37.6197	2.02832
$345$	−4.21504	−0.226930
$346$	−50.6130	−2.72097
$347$	3.72134	0.199772	0.0998859	−	0.994999i	$-0.468152\pi$
$347$	3.72134	0.199772	0.0998859	+	0.994999i	$0.468152\pi$
$348$	36.5838	1.96110
$349$	−17.4152	−0.932212	−0.466106	−	0.884729i	$-0.654343\pi$
$349$	−17.4152	−0.932212	−0.466106	+	0.884729i	$0.654343\pi$
$350$	19.6562	1.05067
$351$	5.48170	0.292591
$352$	3.28668	0.175181
$353$	8.07489	0.429783	0.214891	−	0.976638i	$-0.431060\pi$
$353$	8.07489	0.429783	0.214891	+	0.976638i	$0.431060\pi$
$354$	−8.86084	−0.470948
$355$	−23.6825	−1.25693
$356$	50.5452	2.67889
$357$	−13.2629	−0.701947
$358$	−15.6181	−0.825440
$359$	−1.34292	−0.0708764	−0.0354382	−	0.999372i	$-0.511283\pi$
$359$	−1.34292	−0.0708764	−0.0354382	+	0.999372i	$0.511283\pi$
$360$	20.6140	1.08645
$361$	−9.18707	−0.483530
$362$	3.67327	0.193063
$363$	−17.0435	−0.894554
$364$	−10.4558	−0.548031
$365$	14.6999	0.769428
$366$	−19.1735	−1.00221
$367$	4.55534	0.237787	0.118894	−	0.992907i	$-0.462065\pi$
$367$	4.55534	0.237787	0.118894	+	0.992907i	$0.462065\pi$
$368$	4.43189	0.231028
$369$	10.7107	0.557579
$370$	−77.8577	−4.04763
$371$	−1.76445	−0.0916057
$372$	3.21438	0.166658
$373$	−8.54649	−0.442521	−0.221260	−	0.975215i	$-0.571017\pi$
$373$	−8.54649	−0.442521	−0.221260	+	0.975215i	$0.571017\pi$
$374$	50.4972	2.61115
$375$	6.71735	0.346883
$376$	−23.2114	−1.19704
$377$	−7.71676	−0.397433
$378$	35.5053	1.82619
$379$	−12.4021	−0.637053	−0.318526	−	0.947914i	$-0.603188\pi$
$379$	−12.4021	−0.637053	−0.318526	+	0.947914i	$0.603188\pi$
$380$	−34.9098	−1.79083
$381$	4.05732	0.207863
$382$	−24.3499	−1.24585
$383$	−8.66598	−0.442811	−0.221405	−	0.975182i	$-0.571064\pi$
$383$	−8.66598	−0.442811	−0.221405	+	0.975182i	$0.571064\pi$
$384$	−24.2171	−1.23582
$385$	−37.7930	−1.92611
$386$	1.72340	0.0877186
$387$	12.3629	0.628443
$388$	−14.6294	−0.742693
$389$	6.03794	0.306135	0.153068	−	0.988216i	$-0.451085\pi$
$389$	6.03794	0.306135	0.153068	+	0.988216i	$0.451085\pi$
$390$	8.32456	0.421530
$391$	−5.10008	−0.257922
$392$	−0.342239	−0.0172857
$393$	21.4626	1.08264
$394$	−23.8434	−1.20121
$395$	30.6253	1.54092
$396$	−30.4709	−1.53122
$397$	−8.12600	−0.407832	−0.203916	−	0.978988i	$-0.565367\pi$
$397$	−8.12600	−0.407832	−0.203916	+	0.978988i	$0.565367\pi$
$398$	−25.1744	−1.26188
$399$	−10.0458	−0.502920
$400$	10.9065	0.545325
$401$	9.39153	0.468990	0.234495	−	0.972117i	$-0.424656\pi$
$401$	9.39153	0.468990	0.234495	+	0.972117i	$0.424656\pi$
$402$	24.5387	1.22388
$403$	−0.678022	−0.0337747
$404$	59.2973	2.95015
$405$	−5.59054	−0.277796
$406$	−49.9819	−2.48056
$407$	56.5411	2.80264
$408$	−23.4601	−1.16145
$409$	−26.3330	−1.30208	−0.651040	−	0.759043i	$-0.725667\pi$
$409$	−26.3330	−1.30208	−0.651040	+	0.759043i	$0.725667\pi$
$410$	47.8297	2.36214
$411$	−18.0014	−0.887943
$412$	5.30824	0.261518
$413$	8.02405	0.394838
$414$	4.64301	0.228191
$415$	−35.7182	−1.75334
$416$	0.655581	0.0321425
$417$	10.2929	0.504044
$418$	38.2485	1.87080
$419$	32.1528	1.57077	0.785383	−	0.619010i	$-0.212466\pi$
$419$	32.1528	1.57077	0.785383	+	0.619010i	$0.212466\pi$
$420$	35.7383	1.74385
$421$	−27.1511	−1.32327	−0.661633	−	0.749828i	$-0.730136\pi$
$421$	−27.1511	−1.32327	−0.661633	+	0.749828i	$0.730136\pi$
$422$	53.7073	2.61443
$423$	−7.62794	−0.370883
$424$	−3.12105	−0.151572
$425$	−12.5509	−0.608806
$426$	−24.5367	−1.18881
$427$	17.3628	0.840246
$428$	−27.6414	−1.33610
$429$	−6.04538	−0.291874
$430$	55.2077	2.66235
$431$	−4.12993	−0.198932	−0.0994660	−	0.995041i	$-0.531713\pi$
$431$	−4.12993	−0.198932	−0.0994660	+	0.995041i	$0.531713\pi$
$432$	19.7006	0.947845
$433$	34.2604	1.64645	0.823225	−	0.567715i	$-0.192173\pi$
$433$	34.2604	1.64645	0.823225	+	0.567715i	$0.192173\pi$
$434$	−4.39158	−0.210803
$435$	26.3762	1.26464
$436$	−2.90239	−0.138999
$437$	−3.86299	−0.184792
$438$	15.2301	0.727724
$439$	−10.9670	−0.523428	−0.261714	−	0.965146i	$-0.584288\pi$
$439$	−10.9670	−0.523428	−0.261714	+	0.965146i	$0.584288\pi$
$440$	−66.8503	−3.18696
$441$	−0.112470	−0.00535570
$442$	10.0725	0.479099
$443$	4.80071	0.228089	0.114044	−	0.993476i	$-0.463619\pi$
$443$	4.80071	0.228089	0.114044	+	0.993476i	$0.463619\pi$
$444$	−53.4670	−2.53743
$445$	36.4421	1.72752
$446$	−49.5006	−2.34392
$447$	−9.77760	−0.462465
$448$	23.3619	1.10374
$449$	15.5260	0.732717	0.366358	−	0.930474i	$-0.380604\pi$
$449$	15.5260	0.732717	0.366358	+	0.930474i	$0.380604\pi$
$450$	11.4260	0.538629
$451$	−34.7344	−1.63558
$452$	−44.5712	−2.09645
$453$	−26.4678	−1.24357
$454$	14.8794	0.698324
$455$	−7.53842	−0.353406
$456$	−17.7696	−0.832136
$457$	−8.26741	−0.386733	−0.193367	−	0.981127i	$-0.561941\pi$
$457$	−8.26741	−0.386733	−0.193367	+	0.981127i	$0.561941\pi$
$458$	−18.9226	−0.884196
$459$	−22.6708	−1.05818
$460$	13.7427	0.640757
$461$	2.86624	0.133494	0.0667469	−	0.997770i	$-0.478738\pi$
$461$	2.86624	0.133494	0.0667469	+	0.997770i	$0.478738\pi$
$462$	−39.1563	−1.82171
$463$	−22.1784	−1.03072	−0.515359	−	0.856974i	$-0.672341\pi$
$463$	−22.1784	−1.03072	−0.515359	+	0.856974i	$0.672341\pi$
$464$	−27.7331	−1.28748
$465$	2.31751	0.107472
$466$	33.9341	1.57197
$467$	2.21612	0.102550	0.0512749	−	0.998685i	$-0.483672\pi$
$467$	2.21612	0.102550	0.0512749	+	0.998685i	$0.483672\pi$
$468$	−6.07790	−0.280951
$469$	−22.2213	−1.02609
$470$	−34.0632	−1.57122
$471$	8.36810	0.385582
$472$	14.1934	0.653302
$473$	−40.0924	−1.84345
$474$	31.7299	1.45740
$475$	−9.50650	−0.436188
$476$	43.2423	1.98201
$477$	−1.02567	−0.0469621
$478$	−15.9852	−0.731147
$479$	12.5842	0.574986	0.287493	−	0.957783i	$-0.407178\pi$
$479$	12.5842	0.574986	0.287493	+	0.957783i	$0.407178\pi$
$480$	−2.24081	−0.102278
$481$	11.2780	0.514233
$482$	−6.14374	−0.279840
$483$	3.95467	0.179944
$484$	55.5687	2.52585
$485$	−10.5475	−0.478937
$486$	34.2594	1.55404
$487$	8.77703	0.397725	0.198862	−	0.980027i	$-0.436275\pi$
$487$	8.77703	0.397725	0.198862	+	0.980027i	$0.436275\pi$
$488$	30.7122	1.39028
$489$	−30.2929	−1.36989
$490$	−0.502242	−0.0226890
$491$	−7.36802	−0.332514	−0.166257	−	0.986082i	$-0.553168\pi$
$491$	−7.36802	−0.332514	−0.166257	+	0.986082i	$0.553168\pi$
$492$	32.8459	1.48081
$493$	31.9145	1.43735
$494$	7.62927	0.343257
$495$	−21.9689	−0.987430
$496$	−2.43673	−0.109413
$497$	22.2195	0.996683
$498$	−37.0065	−1.65830
$499$	−40.5141	−1.81366	−0.906830	−	0.421497i	$-0.861505\pi$
$499$	−40.5141	−1.81366	−0.906830	+	0.421497i	$0.861505\pi$
$500$	−21.9012	−0.979452
$501$	−22.0954	−0.987147
$502$	12.8398	0.573067
$503$	10.0359	0.447478	0.223739	−	0.974649i	$-0.428174\pi$
$503$	10.0359	0.447478	0.223739	+	0.974649i	$0.428174\pi$
$504$	−19.3407	−0.861501
$505$	42.7522	1.90245
$506$	−15.0570	−0.669367
$507$	−1.20585	−0.0535535
$508$	−13.2284	−0.586918
$509$	−6.96617	−0.308770	−0.154385	−	0.988011i	$-0.549340\pi$
$509$	−6.96617	−0.308770	−0.154385	+	0.988011i	$0.549340\pi$
$510$	−34.4281	−1.52450
$511$	−13.7919	−0.610116
$512$	36.1688	1.59845
$513$	−17.1717	−0.758151
$514$	−58.0491	−2.56043
$515$	3.82714	0.168644
$516$	37.9126	1.66901
$517$	24.7370	1.08793
$518$	73.0483	3.20956
$519$	−25.0594	−1.09999
$520$	−13.3343	−0.584749
$521$	24.3081	1.06496	0.532478	−	0.846444i	$-0.321261\pi$
$521$	24.3081	1.06496	0.532478	+	0.846444i	$0.321261\pi$
$522$	−29.0543	−1.27167
$523$	23.3599	1.02146	0.510728	−	0.859743i	$-0.329376\pi$
$523$	23.3599	1.02146	0.510728	+	0.859743i	$0.329376\pi$
$524$	−69.9764	−3.05693
$525$	9.73211	0.424744
$526$	71.3198	3.10969
$527$	2.80412	0.122149
$528$	−21.7264	−0.945520
$529$	−21.4793	−0.933882
$530$	−4.58020	−0.198951
$531$	4.66435	0.202416
$532$	32.7533	1.42004
$533$	−6.92832	−0.300099
$534$	37.7566	1.63389
$535$	−19.9290	−0.861604
$536$	−39.3062	−1.69777
$537$	−7.73277	−0.333694
$538$	−44.6164	−1.92355
$539$	0.364734	0.0157102
$540$	61.0890	2.62885
$541$	17.6456	0.758645	0.379322	−	0.925265i	$-0.376157\pi$
$541$	17.6456	0.758645	0.379322	+	0.925265i	$0.376157\pi$
$542$	27.5513	1.18343
$543$	1.81870	0.0780479
$544$	−2.71131	−0.116246
$545$	−2.09257	−0.0896358
$546$	−7.81033	−0.334251
$547$	7.99519	0.341850	0.170925	−	0.985284i	$-0.445324\pi$
$547$	7.99519	0.341850	0.170925	+	0.985284i	$0.445324\pi$
$548$	58.6916	2.50718
$549$	10.0929	0.430756
$550$	−37.0541	−1.57999
$551$	24.1732	1.02981
$552$	6.99523	0.297737
$553$	−28.7335	−1.22187
$554$	−73.4279	−3.11965
$555$	−38.5487	−1.63630
$556$	−33.5588	−1.42321
$557$	44.9838	1.90603	0.953013	−	0.302931i	$-0.0979651\pi$
$557$	44.9838	1.90603	0.953013	+	0.302931i	$0.0979651\pi$
$558$	−2.55281	−0.108069
$559$	−7.99706	−0.338239
$560$	−27.0922	−1.14485
$561$	25.0021	1.05559
$562$	11.2512	0.474605
$563$	−38.6622	−1.62942	−0.814708	−	0.579872i	$-0.803103\pi$
$563$	−38.6622	−1.62942	−0.814708	+	0.579872i	$0.803103\pi$
$564$	−23.3921	−0.984986
$565$	−32.1350	−1.35193
$566$	−4.52093	−0.190029
$567$	5.24520	0.220278
$568$	39.3031	1.64912
$569$	−36.9248	−1.54797	−0.773985	−	0.633204i	$-0.781739\pi$
$569$	−36.9248	−1.54797	−0.773985	+	0.633204i	$0.781739\pi$
$570$	−26.0772	−1.09225
$571$	−3.92687	−0.164334	−0.0821671	−	0.996619i	$-0.526184\pi$
$571$	−3.92687	−0.164334	−0.0821671	+	0.996619i	$0.526184\pi$
$572$	19.7103	0.824130
$573$	−12.0561	−0.503650
$574$	−44.8751	−1.87305
$575$	3.74236	0.156067
$576$	13.5802	0.565840
$577$	4.16919	0.173566	0.0867828	−	0.996227i	$-0.472341\pi$
$577$	4.16919	0.173566	0.0867828	+	0.996227i	$0.472341\pi$
$578$	−0.253996	−0.0105648
$579$	0.853284	0.0354613
$580$	−85.9969	−3.57083
$581$	33.5118	1.39030
$582$	−10.9279	−0.452978
$583$	3.32619	0.137757
$584$	−24.3958	−1.00950
$585$	−4.38205	−0.181175
$586$	−36.7504	−1.51814
$587$	−4.78396	−0.197455	−0.0987275	−	0.995115i	$-0.531477\pi$
$587$	−4.78396	−0.197455	−0.0987275	+	0.995115i	$0.531477\pi$
$588$	−0.344904	−0.0142236
$589$	2.12394	0.0875156
$590$	20.8290	0.857517
$591$	−11.8053	−0.485605
$592$	40.5319	1.66585
$593$	10.9570	0.449949	0.224974	−	0.974365i	$-0.427770\pi$
$593$	10.9570	0.449949	0.224974	+	0.974365i	$0.427770\pi$
$594$	−66.9314	−2.74623
$595$	31.1769	1.27813
$596$	31.8788	1.30581
$597$	−12.4643	−0.510129
$598$	−3.00336	−0.122817
$599$	34.5043	1.40981	0.704904	−	0.709302i	$-0.250990\pi$
$599$	34.5043	1.40981	0.704904	+	0.709302i	$0.250990\pi$
$600$	17.2147	0.702786
$601$	−19.0624	−0.777570	−0.388785	−	0.921329i	$-0.627105\pi$
$601$	−19.0624	−0.777570	−0.388785	+	0.921329i	$0.627105\pi$
$602$	−51.7974	−2.11110
$603$	−12.9172	−0.526027
$604$	86.2955	3.51131
$605$	40.0640	1.62883
$606$	44.2943	1.79933
$607$	−19.5419	−0.793181	−0.396591	−	0.917996i	$-0.629807\pi$
$607$	−19.5419	−0.793181	−0.396591	+	0.917996i	$0.629807\pi$
$608$	−2.05365	−0.0832864
$609$	−24.7469	−1.00280
$610$	45.0708	1.82486
$611$	4.93419	0.199616
$612$	25.1366	1.01609
$613$	45.9886	1.85746	0.928731	−	0.370756i	$-0.120901\pi$
$613$	45.9886	1.85746	0.928731	+	0.370756i	$0.120901\pi$
$614$	3.39810	0.137136
$615$	23.6813	0.954923
$616$	62.7208	2.52709
$617$	48.2733	1.94341	0.971706	−	0.236195i	$-0.0759005\pi$
$617$	48.2733	1.94341	0.971706	+	0.236195i	$0.0759005\pi$
$618$	3.96519	0.159503
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−7.55599	−0.303456
$621$	6.75989	0.271265
$622$	23.6457	0.948104
$623$	−34.1910	−1.36983
$624$	−4.33367	−0.173486
$625$	−30.9641	−1.23856
$626$	−2.95451	−0.118086
$627$	18.9375	0.756291
$628$	−27.2833	−1.08872
$629$	−46.6428	−1.85977
$630$	−28.3828	−1.13080
$631$	18.4267	0.733555	0.366778	−	0.930309i	$-0.380461\pi$
$631$	18.4267	0.733555	0.366778	+	0.930309i	$0.380461\pi$
$632$	−50.8253	−2.02172
$633$	26.5915	1.05692
$634$	77.3575	3.07226
$635$	−9.53746	−0.378483
$636$	−3.14535	−0.124721
$637$	0.0727519	0.00288253
$638$	94.2215	3.73026
$639$	12.9161	0.510954
$640$	56.9267	2.25023
$641$	13.8590	0.547397	0.273698	−	0.961816i	$-0.411753\pi$
$641$	13.8590	0.547397	0.273698	+	0.961816i	$0.411753\pi$
$642$	−20.6478	−0.814904
$643$	−7.43069	−0.293038	−0.146519	−	0.989208i	$-0.546807\pi$
$643$	−7.43069	−0.293038	−0.146519	+	0.989208i	$0.546807\pi$
$644$	−12.8938	−0.508086
$645$	27.3343	1.07629
$646$	−31.5526	−1.24142
$647$	−3.87656	−0.152403	−0.0762017	−	0.997092i	$-0.524279\pi$
$647$	−3.87656	−0.152403	−0.0762017	+	0.997092i	$0.524279\pi$
$648$	9.27798	0.364473
$649$	−15.1262	−0.593757
$650$	−7.39102	−0.289900
$651$	−2.17435	−0.0852195
$652$	98.7668	3.86801
$653$	46.2296	1.80910	0.904551	−	0.426365i	$-0.140206\pi$
$653$	46.2296	1.80910	0.904551	+	0.426365i	$0.140206\pi$
$654$	−2.16805	−0.0847775
$655$	−50.4517	−1.97131
$656$	−24.8996	−0.972166
$657$	−8.01715	−0.312779
$658$	31.9590	1.24589
$659$	24.5772	0.957392	0.478696	−	0.877981i	$-0.341110\pi$
$659$	24.5772	0.957392	0.478696	+	0.877981i	$0.341110\pi$
$660$	−67.3707	−2.62240
$661$	20.3698	0.792292	0.396146	−	0.918188i	$-0.370347\pi$
$661$	20.3698	0.792292	0.396146	+	0.918188i	$0.370347\pi$
$662$	30.8913	1.20062
$663$	4.98706	0.193681
$664$	59.2774	2.30041
$665$	23.6145	0.915733
$666$	42.4627	1.64540
$667$	−9.51610	−0.368465
$668$	72.0396	2.78729
$669$	−24.5086	−0.947558
$670$	−57.6826	−2.22847
$671$	−32.7309	−1.26356
$672$	2.10239	0.0811013
$673$	−11.9595	−0.461005	−0.230503	−	0.973072i	$-0.574037\pi$
$673$	−11.9595	−0.461005	−0.230503	+	0.973072i	$0.574037\pi$
$674$	9.88920	0.380918
$675$	16.6355	0.640301
$676$	3.93153	0.151213
$677$	48.9955	1.88305	0.941525	−	0.336943i	$-0.109393\pi$
$677$	48.9955	1.88305	0.941525	+	0.336943i	$0.109393\pi$
$678$	−33.2942	−1.27865
$679$	9.89595	0.379772
$680$	55.1473	2.11480
$681$	7.36704	0.282306
$682$	8.27863	0.317005
$683$	−37.3227	−1.42811	−0.714057	−	0.700087i	$-0.753144\pi$
$683$	−37.3227	−1.42811	−0.714057	+	0.700087i	$0.753144\pi$
$684$	19.0394	0.727989
$685$	42.3156	1.61679
$686$	−44.8682	−1.71307
$687$	−9.36892	−0.357447
$688$	−28.7405	−1.09572
$689$	0.663461	0.0252758
$690$	10.2656	0.390806
$691$	−14.4603	−0.550095	−0.275048	−	0.961431i	$-0.588694\pi$
$691$	−14.4603	−0.550095	−0.275048	+	0.961431i	$0.588694\pi$
$692$	81.7035	3.10590
$693$	20.6119	0.782980
$694$	−9.06322	−0.344035
$695$	−24.1953	−0.917779
$696$	−43.7736	−1.65923
$697$	28.6537	1.08534
$698$	42.4142	1.60540
$699$	16.8014	0.635486
$700$	−31.7305	−1.19930
$701$	28.4227	1.07351	0.536755	−	0.843738i	$-0.319650\pi$
$701$	28.4227	1.07351	0.536755	+	0.843738i	$0.319650\pi$
$702$	−13.3505	−0.503883
$703$	−35.3290	−1.33246
$704$	−44.0397	−1.65981
$705$	−16.8653	−0.635183
$706$	−19.6662	−0.740146
$707$	−40.1113	−1.50854
$708$	14.3039	0.537572
$709$	32.3371	1.21445	0.607223	−	0.794532i	$-0.292284\pi$
$709$	32.3371	1.21445	0.607223	+	0.794532i	$0.292284\pi$
$710$	57.6780	2.16462
$711$	−16.7026	−0.626398
$712$	−60.4788	−2.26654
$713$	−0.836118	−0.0313129
$714$	32.3015	1.20885
$715$	14.2108	0.531452
$716$	25.2119	0.942213
$717$	−7.91457	−0.295575
$718$	3.27064	0.122059
$719$	7.11029	0.265169	0.132585	−	0.991172i	$-0.457672\pi$
$719$	7.11029	0.265169	0.132585	+	0.991172i	$0.457672\pi$
$720$	−15.7486	−0.586915
$721$	−3.59073	−0.133726
$722$	22.3749	0.832706
$723$	−3.04187	−0.113128
$724$	−5.92968	−0.220375
$725$	−23.4183	−0.869735
$726$	41.5091	1.54055
$727$	30.6763	1.13772	0.568860	−	0.822434i	$-0.307385\pi$
$727$	30.6763	1.13772	0.568860	+	0.822434i	$0.307385\pi$
$728$	12.5106	0.463675
$729$	22.8793	0.847380
$730$	−35.8012	−1.32506
$731$	33.0737	1.22327
$732$	30.9514	1.14399
$733$	44.1296	1.62996	0.814981	−	0.579487i	$-0.196747\pi$
$733$	44.1296	1.62996	0.814981	+	0.579487i	$0.196747\pi$
$734$	−11.0944	−0.409503
$735$	−0.248669	−0.00917230
$736$	0.808445	0.0297997
$737$	41.8897	1.54303
$738$	−26.0857	−0.960229
$739$	−43.4782	−1.59937	−0.799686	−	0.600419i	$-0.795001\pi$
$739$	−43.4782	−1.59937	−0.799686	+	0.600419i	$0.795001\pi$
$740$	125.684	4.62024
$741$	3.77739	0.138766
$742$	4.29727	0.157758
$743$	23.8625	0.875430	0.437715	−	0.899114i	$-0.355788\pi$
$743$	23.8625	0.875430	0.437715	+	0.899114i	$0.355788\pi$
$744$	−3.84610	−0.141005
$745$	22.9840	0.842070
$746$	20.8148	0.762083
$747$	19.4803	0.712745
$748$	−81.5166	−2.98054
$749$	18.6979	0.683207
$750$	−16.3599	−0.597381
$751$	−13.7053	−0.500114	−0.250057	−	0.968231i	$-0.580449\pi$
$751$	−13.7053	−0.500114	−0.250057	+	0.968231i	$0.580449\pi$
$752$	17.7329	0.646653
$753$	6.35719	0.231669
$754$	18.7940	0.684436
$755$	62.2174	2.26432
$756$	−57.3154	−2.08454
$757$	15.8161	0.574845	0.287423	−	0.957804i	$-0.407202\pi$
$757$	15.8161	0.574845	0.287423	+	0.957804i	$0.407202\pi$
$758$	30.2050	1.09709
$759$	−7.45500	−0.270599
$760$	41.7706	1.51518
$761$	15.1767	0.550155	0.275078	−	0.961422i	$-0.411296\pi$
$761$	15.1767	0.550155	0.275078	+	0.961422i	$0.411296\pi$
$762$	−9.88149	−0.357968
$763$	1.96331	0.0710765
$764$	39.3076	1.42210
$765$	18.1230	0.655238
$766$	21.1058	0.762582
$767$	−3.01717	−0.108944
$768$	37.7948	1.36380
$769$	45.3957	1.63701	0.818505	−	0.574499i	$-0.194803\pi$
$769$	45.3957	1.63701	0.818505	+	0.574499i	$0.194803\pi$
$770$	92.0439	3.31703
$771$	−28.7411	−1.03509
$772$	−2.78204	−0.100128
$773$	−43.3930	−1.56074	−0.780369	−	0.625319i	$-0.784969\pi$
$773$	−43.3930	−1.56074	−0.780369	+	0.625319i	$0.784969\pi$
$774$	−30.1096	−1.08227
$775$	−2.05762	−0.0739118
$776$	17.5045	0.628374
$777$	36.1675	1.29750
$778$	−14.7052	−0.527208
$779$	21.7034	0.777605
$780$	−13.4382	−0.481163
$781$	−41.8863	−1.49881
$782$	12.4211	0.444178
$783$	−42.3009	−1.51171
$784$	0.261462	0.00933793
$785$	−19.6707	−0.702079
$786$	−52.2715	−1.86446
$787$	14.3415	0.511218	0.255609	−	0.966780i	$-0.417724\pi$
$787$	14.3415	0.511218	0.255609	+	0.966780i	$0.417724\pi$
$788$	38.4899	1.37115
$789$	35.3117	1.25713
$790$	−74.5870	−2.65369
$791$	30.1500	1.07201
$792$	36.4593	1.29553
$793$	−6.52869	−0.231841
$794$	19.7906	0.702344
$795$	−2.26774	−0.0804284
$796$	40.6385	1.44039
$797$	−19.0078	−0.673292	−0.336646	−	0.941631i	$-0.609293\pi$
$797$	−19.0078	−0.673292	−0.336646	+	0.941631i	$0.609293\pi$
$798$	24.4663	0.866099
$799$	−20.4065	−0.721930
$800$	1.98952	0.0703400
$801$	−19.8751	−0.702252
$802$	−22.8728	−0.807667
$803$	25.9992	0.917493
$804$	−39.6122	−1.39702
$805$	−9.29618	−0.327647
$806$	1.65130	0.0581647
$807$	−22.0903	−0.777617
$808$	−70.9510	−2.49605
$809$	−10.8444	−0.381269	−0.190634	−	0.981661i	$-0.561055\pi$
$809$	−10.8444	−0.381269	−0.190634	+	0.981661i	$0.561055\pi$
$810$	13.6156	0.478404
$811$	28.7055	1.00799	0.503994	−	0.863707i	$-0.331864\pi$
$811$	28.7055	1.00799	0.503994	+	0.863707i	$0.331864\pi$
$812$	80.6847	2.83148
$813$	13.6411	0.478416
$814$	−137.704	−4.82653
$815$	71.2090	2.49434
$816$	17.9229	0.627428
$817$	25.0513	0.876433
$818$	64.1332	2.24237
$819$	4.11136	0.143663
$820$	−77.2104	−2.69630
$821$	41.5124	1.44879	0.724396	−	0.689384i	$-0.242119\pi$
$821$	41.5124	1.44879	0.724396	+	0.689384i	$0.242119\pi$
$822$	43.8419	1.52916
$823$	9.92533	0.345975	0.172988	−	0.984924i	$-0.444658\pi$
$823$	9.92533	0.345975	0.172988	+	0.984924i	$0.444658\pi$
$824$	−6.35147	−0.221264
$825$	−18.3461	−0.638730
$826$	−19.5424	−0.679966
$827$	16.4518	0.572085	0.286043	−	0.958217i	$-0.407660\pi$
$827$	16.4518	0.572085	0.286043	+	0.958217i	$0.407660\pi$
$828$	−7.49510	−0.260473
$829$	−45.3946	−1.57662	−0.788310	−	0.615278i	$-0.789044\pi$
$829$	−45.3946	−1.57662	−0.788310	+	0.615278i	$0.789044\pi$
$830$	86.9906	3.01949
$831$	−36.3555	−1.26116
$832$	−8.78442	−0.304545
$833$	−0.300882	−0.0104250
$834$	−25.0680	−0.868034
$835$	51.9392	1.79743
$836$	−61.7438	−2.13545
$837$	−3.71671	−0.128468
$838$	−78.3073	−2.70508
$839$	40.3681	1.39366	0.696831	−	0.717236i	$-0.254593\pi$
$839$	40.3681	1.39366	0.696831	+	0.717236i	$0.254593\pi$
$840$	−42.7619	−1.47543
$841$	30.5483	1.05339
$842$	66.1259	2.27885
$843$	5.57069	0.191865
$844$	−86.6986	−2.98429
$845$	2.83456	0.0975119
$846$	18.5776	0.638713
$847$	−37.5891	−1.29158
$848$	2.38440	0.0818807
$849$	−2.23839	−0.0768215
$850$	30.5673	1.04845
$851$	13.9077	0.476751
$852$	39.6091	1.35698
$853$	15.0725	0.516072	0.258036	−	0.966135i	$-0.416925\pi$
$853$	15.0725	0.516072	0.258036	+	0.966135i	$0.416925\pi$
$854$	−42.2867	−1.44702
$855$	13.7270	0.469455
$856$	33.0738	1.13044
$857$	18.0090	0.615174	0.307587	−	0.951520i	$-0.400478\pi$
$857$	18.0090	0.615174	0.307587	+	0.951520i	$0.400478\pi$
$858$	14.7234	0.502647
$859$	25.6202	0.874151	0.437076	−	0.899425i	$-0.356014\pi$
$859$	25.6202	0.874151	0.437076	+	0.899425i	$0.356014\pi$
$860$	−89.1206	−3.03899
$861$	−22.2185	−0.757203
$862$	10.0583	0.342589
$863$	−14.6731	−0.499477	−0.249738	−	0.968313i	$-0.580345\pi$
$863$	−14.6731	−0.499477	−0.249738	+	0.968313i	$0.580345\pi$
$864$	3.59370	0.122260
$865$	58.9067	2.00289
$866$	−83.4403	−2.83542
$867$	−0.125758	−0.00427096
$868$	7.08924	0.240624
$869$	54.1658	1.83745
$870$	−64.2386	−2.17789
$871$	8.35556	0.283117
$872$	3.47280	0.117604
$873$	5.75248	0.194692
$874$	9.40821	0.318238
$875$	14.8150	0.500837
$876$	−24.5857	−0.830674
$877$	6.33714	0.213990	0.106995	−	0.994260i	$-0.465877\pi$
$877$	6.33714	0.213990	0.106995	+	0.994260i	$0.465877\pi$
$878$	26.7099	0.901416
$879$	−18.1958	−0.613728
$880$	51.0719	1.72163
$881$	44.7255	1.50684	0.753421	−	0.657539i	$-0.228402\pi$
$881$	44.7255	1.50684	0.753421	+	0.657539i	$0.228402\pi$
$882$	0.273917	0.00922327
$883$	−2.48539	−0.0836401	−0.0418200	−	0.999125i	$-0.513316\pi$
$883$	−2.48539	−0.0836401	−0.0418200	+	0.999125i	$0.513316\pi$
$884$	−16.2598	−0.546875
$885$	10.3128	0.346662
$886$	−11.6920	−0.392801
$887$	38.7282	1.30037	0.650183	−	0.759777i	$-0.274692\pi$
$887$	38.7282	1.30037	0.650183	+	0.759777i	$0.274692\pi$
$888$	63.9749	2.14686
$889$	8.94831	0.300117
$890$	−88.7538	−2.97504
$891$	−9.88779	−0.331253
$892$	79.9078	2.67551
$893$	−15.4566	−0.517237
$894$	23.8131	0.796429
$895$	18.1773	0.607600
$896$	−53.4102	−1.78431
$897$	−1.48702	−0.0496501
$898$	−37.8131	−1.26184
$899$	5.23213	0.174501
$900$	−18.4448	−0.614827
$901$	−2.74390	−0.0914124
$902$	84.5947	2.81670
$903$	−25.6458	−0.853439
$904$	53.3308	1.77376
$905$	−4.27519	−0.142112
$906$	64.4616	2.14159
$907$	12.3146	0.408899	0.204449	−	0.978877i	$-0.434460\pi$
$907$	12.3146	0.408899	0.204449	+	0.978877i	$0.434460\pi$
$908$	−24.0195	−0.797114
$909$	−23.3166	−0.773361
$910$	18.3596	0.608615
$911$	3.81734	0.126474	0.0632370	−	0.997999i	$-0.479858\pi$
$911$	3.81734	0.126474	0.0632370	+	0.997999i	$0.479858\pi$
$912$	13.5755	0.449529
$913$	−63.1735	−2.09074
$914$	20.1351	0.666009
$915$	22.3153	0.737722
$916$	30.5464	1.00928
$917$	47.3352	1.56315
$918$	55.2142	1.82234
$919$	16.8994	0.557461	0.278731	−	0.960369i	$-0.410086\pi$
$919$	16.8994	0.557461	0.278731	+	0.960369i	$0.410086\pi$
$920$	−16.4436	−0.542128
$921$	1.68246	0.0554389
$922$	−6.98064	−0.229895
$923$	−8.35489	−0.275005
$924$	63.2091	2.07943
$925$	34.2258	1.12534
$926$	54.0150	1.77504
$927$	−2.08728	−0.0685551
$928$	−5.05896	−0.166069
$929$	−43.5974	−1.43038	−0.715192	−	0.698928i	$-0.753661\pi$
$929$	−43.5974	−1.43038	−0.715192	+	0.698928i	$0.753661\pi$
$930$	−5.64423	−0.185082
$931$	−0.227900	−0.00746911
$932$	−54.7791	−1.79435
$933$	11.7074	0.383282
$934$	−5.39730	−0.176605
$935$	−58.7719	−1.92205
$936$	7.27239	0.237705
$937$	40.4406	1.32114	0.660568	−	0.750766i	$-0.270316\pi$
$937$	40.4406	1.32114	0.660568	+	0.750766i	$0.270316\pi$
$938$	54.1194	1.76706
$939$	−1.46283	−0.0477377
$940$	54.9875	1.79349
$941$	23.7352	0.773745	0.386873	−	0.922133i	$-0.373555\pi$
$941$	23.7352	0.773745	0.386873	+	0.922133i	$0.373555\pi$
$942$	−20.3803	−0.664026
$943$	−8.54382	−0.278225
$944$	−10.8434	−0.352921
$945$	−41.3233	−1.34425
$946$	97.6439	3.17468
$947$	−28.6271	−0.930256	−0.465128	−	0.885243i	$-0.653992\pi$
$947$	−28.6271	−0.930256	−0.465128	+	0.885243i	$0.653992\pi$
$948$	−51.2210	−1.66358
$949$	5.18595	0.168343
$950$	23.1528	0.751177
$951$	38.3010	1.24200
$952$	−51.7407	−1.67693
$953$	−37.7655	−1.22334	−0.611672	−	0.791111i	$-0.709503\pi$
$953$	−37.7655	−1.22334	−0.611672	+	0.791111i	$0.709503\pi$
$954$	2.49799	0.0808753
$955$	28.3400	0.917062
$956$	25.8046	0.834581
$957$	46.6507	1.50800
$958$	−30.6484	−0.990206
$959$	−39.7016	−1.28203
$960$	30.0255	0.969070
$961$	−30.5403	−0.985171
$962$	−27.4673	−0.885581
$963$	10.8690	0.350249
$964$	9.91770	0.319428
$965$	−2.00580	−0.0645690
$966$	−9.63150	−0.309888
$967$	48.2737	1.55238	0.776189	−	0.630501i	$-0.217150\pi$
$967$	48.2737	1.55238	0.776189	+	0.630501i	$0.217150\pi$
$968$	−66.4896	−2.13706
$969$	−15.6223	−0.501859
$970$	25.6881	0.824797
$971$	20.1189	0.645646	0.322823	−	0.946459i	$-0.395368\pi$
$971$	20.1189	0.645646	0.322823	+	0.946459i	$0.395368\pi$
$972$	−55.3042	−1.77388
$973$	22.7007	0.727750
$974$	−21.3762	−0.684938
$975$	−3.65943	−0.117195
$976$	−23.4634	−0.751044
$977$	58.2140	1.86243	0.931215	−	0.364470i	$-0.118750\pi$
$977$	58.2140	1.86243	0.931215	+	0.364470i	$0.118750\pi$
$978$	73.7776	2.35915
$979$	64.4539	2.05996
$980$	0.810759	0.0258988
$981$	1.14126	0.0364377
$982$	17.9446	0.572636
$983$	−30.0889	−0.959687	−0.479844	−	0.877354i	$-0.659307\pi$
$983$	−30.0889	−0.959687	−0.479844	+	0.877354i	$0.659307\pi$
$984$	−39.3012	−1.25288
$985$	27.7505	0.884205
$986$	−77.7268	−2.47533
$987$	15.8235	0.503667
$988$	−12.3158	−0.391817
$989$	−9.86176	−0.313586
$990$	53.5048	1.70049
$991$	6.18942	0.196613	0.0983067	−	0.995156i	$-0.468657\pi$
$991$	6.18942	0.196613	0.0983067	+	0.995156i	$0.468657\pi$
$992$	−0.444498	−0.0141128
$993$	15.2948	0.485367
$994$	−54.1151	−1.71643
$995$	29.2996	0.928859
$996$	59.7389	1.89290
$997$	7.85178	0.248668	0.124334	−	0.992240i	$-0.460321\pi$
$997$	7.85178	0.248668	0.124334	+	0.992240i	$0.460321\pi$
$998$	98.6710	3.12338
$999$	61.8226	1.95598

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.e.1.16	✓	168

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.e.1.16	✓	168	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.43547 q^{2} -1.20585 q^{3} +3.93153 q^{4} +2.83456 q^{5} +2.93681 q^{6} -2.65946 q^{7} -4.70420 q^{8} -1.54594 q^{9} +O(q^{10})\)	\(q-2.43547 q^{2} -1.20585 q^{3} +3.93153 q^{4} +2.83456 q^{5} +2.93681 q^{6} -2.65946 q^{7} -4.70420 q^{8} -1.54594 q^{9} -6.90350 q^{10} +5.01339 q^{11} -4.74082 q^{12} +1.00000 q^{13} +6.47706 q^{14} -3.41804 q^{15} +3.59389 q^{16} -4.13573 q^{17} +3.76509 q^{18} -3.13256 q^{19} +11.1442 q^{20} +3.20690 q^{21} -12.2100 q^{22} +1.23317 q^{23} +5.67254 q^{24} +3.03474 q^{25} -2.43547 q^{26} +5.48170 q^{27} -10.4558 q^{28} -7.71676 q^{29} +8.32456 q^{30} -0.678022 q^{31} +0.655581 q^{32} -6.04538 q^{33} +10.0725 q^{34} -7.53842 q^{35} -6.07790 q^{36} +11.2780 q^{37} +7.62927 q^{38} -1.20585 q^{39} -13.3343 q^{40} -6.92832 q^{41} -7.81033 q^{42} -7.99706 q^{43} +19.7103 q^{44} -4.38205 q^{45} -3.00336 q^{46} +4.93419 q^{47} -4.33367 q^{48} +0.0727519 q^{49} -7.39102 q^{50} +4.98706 q^{51} +3.93153 q^{52} +0.663461 q^{53} -13.3505 q^{54} +14.2108 q^{55} +12.5106 q^{56} +3.77739 q^{57} +18.7940 q^{58} -3.01717 q^{59} -13.4382 q^{60} -6.52869 q^{61} +1.65130 q^{62} +4.11136 q^{63} -8.78442 q^{64} +2.83456 q^{65} +14.7234 q^{66} +8.35556 q^{67} -16.2598 q^{68} -1.48702 q^{69} +18.3596 q^{70} -8.35489 q^{71} +7.27239 q^{72} +5.18595 q^{73} -27.4673 q^{74} -3.65943 q^{75} -12.3158 q^{76} -13.3329 q^{77} +2.93681 q^{78} +10.8042 q^{79} +10.1871 q^{80} -1.97228 q^{81} +16.8737 q^{82} -12.6009 q^{83} +12.6081 q^{84} -11.7230 q^{85} +19.4766 q^{86} +9.30522 q^{87} -23.5840 q^{88} +12.8564 q^{89} +10.6724 q^{90} -2.65946 q^{91} +4.84826 q^{92} +0.817590 q^{93} -12.0171 q^{94} -8.87943 q^{95} -0.790530 q^{96} -3.72103 q^{97} -0.177185 q^{98} -7.75038 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

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