LMFDB - Embedded newform 8047.2.a.d.1.1

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:	$156$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-2.78003 q^{2} +2.35788 q^{3} +5.72856 q^{4} +3.19176 q^{5} -6.55498 q^{6} -2.60821 q^{7} -10.3655 q^{8} +2.55961 q^{9} +O(q^{10})$	\(q-2.78003 q^{2} +2.35788 q^{3} +5.72856 q^{4} +3.19176 q^{5} -6.55498 q^{6} -2.60821 q^{7} -10.3655 q^{8} +2.55961 q^{9} -8.87319 q^{10} +4.15318 q^{11} +13.5073 q^{12} -1.00000 q^{13} +7.25089 q^{14} +7.52580 q^{15} +17.3593 q^{16} +2.99376 q^{17} -7.11579 q^{18} +7.00799 q^{19} +18.2842 q^{20} -6.14984 q^{21} -11.5460 q^{22} +8.82835 q^{23} -24.4407 q^{24} +5.18735 q^{25} +2.78003 q^{26} -1.03839 q^{27} -14.9413 q^{28} -8.41598 q^{29} -20.9220 q^{30} +1.30536 q^{31} -27.5284 q^{32} +9.79271 q^{33} -8.32274 q^{34} -8.32477 q^{35} +14.6629 q^{36} +9.66612 q^{37} -19.4824 q^{38} -2.35788 q^{39} -33.0843 q^{40} +4.57527 q^{41} +17.0967 q^{42} +9.67756 q^{43} +23.7917 q^{44} +8.16967 q^{45} -24.5431 q^{46} +10.7544 q^{47} +40.9312 q^{48} -0.197263 q^{49} -14.4210 q^{50} +7.05893 q^{51} -5.72856 q^{52} -10.3092 q^{53} +2.88674 q^{54} +13.2560 q^{55} +27.0354 q^{56} +16.5240 q^{57} +23.3967 q^{58} -3.05416 q^{59} +43.1120 q^{60} -14.3630 q^{61} -3.62894 q^{62} -6.67599 q^{63} +41.8110 q^{64} -3.19176 q^{65} -27.2240 q^{66} +9.09616 q^{67} +17.1499 q^{68} +20.8162 q^{69} +23.1431 q^{70} +0.542927 q^{71} -26.5317 q^{72} +1.18119 q^{73} -26.8721 q^{74} +12.2312 q^{75} +40.1457 q^{76} -10.8323 q^{77} +6.55498 q^{78} +1.48202 q^{79} +55.4068 q^{80} -10.1272 q^{81} -12.7194 q^{82} +9.96792 q^{83} -35.2298 q^{84} +9.55537 q^{85} -26.9039 q^{86} -19.8439 q^{87} -43.0498 q^{88} -6.18819 q^{89} -22.7119 q^{90} +2.60821 q^{91} +50.5738 q^{92} +3.07788 q^{93} -29.8976 q^{94} +22.3679 q^{95} -64.9086 q^{96} -16.4821 q^{97} +0.548398 q^{98} +10.6305 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) $ 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * 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 $

$2$	−2.78003	−1.96578	−0.982889	−	0.184200i	$-0.941031\pi$
$2$	−2.78003	−1.96578	−0.982889	+	0.184200i	$0.941031\pi$
$3$	2.35788	1.36132	0.680662	−	0.732597i	$-0.261692\pi$
$3$	2.35788	1.36132	0.680662	+	0.732597i	$0.261692\pi$
$4$	5.72856	2.86428
$5$	3.19176	1.42740	0.713700	−	0.700452i	$-0.247018\pi$
$5$	3.19176	1.42740	0.713700	+	0.700452i	$0.247018\pi$
$6$	−6.55498	−2.67606
$7$	−2.60821	−0.985809	−0.492905	−	0.870083i	$-0.664065\pi$
$7$	−2.60821	−0.985809	−0.492905	+	0.870083i	$0.664065\pi$
$8$	−10.3655	−3.66476
$9$	2.55961	0.853204
$10$	−8.87319	−2.80595
$11$	4.15318	1.25223	0.626115	−	0.779731i	$-0.284644\pi$
$11$	4.15318	1.25223	0.626115	+	0.779731i	$0.284644\pi$
$12$	13.5073	3.89922
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	7.25089	1.93788
$15$	7.52580	1.94315
$16$	17.3593	4.33983
$17$	2.99376	0.726093	0.363047	−	0.931771i	$-0.381737\pi$
$17$	2.99376	0.726093	0.363047	+	0.931771i	$0.381737\pi$
$18$	−7.11579	−1.67721
$19$	7.00799	1.60774	0.803872	−	0.594802i	$-0.202770\pi$
$19$	7.00799	1.60774	0.803872	+	0.594802i	$0.202770\pi$
$20$	18.2842	4.08847
$21$	−6.14984	−1.34201
$22$	−11.5460	−2.46161
$23$	8.82835	1.84084	0.920419	−	0.390933i	$-0.127847\pi$
$23$	8.82835	1.84084	0.920419	+	0.390933i	$0.127847\pi$
$24$	−24.4407	−4.98893
$25$	5.18735	1.03747
$26$	2.78003	0.545209
$27$	−1.03839	−0.199837
$28$	−14.9413	−2.82363
$29$	−8.41598	−1.56281	−0.781404	−	0.624026i	$-0.785496\pi$
$29$	−8.41598	−1.56281	−0.781404	+	0.624026i	$0.785496\pi$
$30$	−20.9220	−3.81981
$31$	1.30536	0.234449	0.117225	−	0.993105i	$-0.462600\pi$
$31$	1.30536	0.234449	0.117225	+	0.993105i	$0.462600\pi$
$32$	−27.5284	−4.86637
$33$	9.79271	1.70469
$34$	−8.32274	−1.42734
$35$	−8.32477	−1.40714
$36$	14.6629	2.44382
$37$	9.66612	1.58910	0.794551	−	0.607198i	$-0.207707\pi$
$37$	9.66612	1.58910	0.794551	+	0.607198i	$0.207707\pi$
$38$	−19.4824	−3.16047
$39$	−2.35788	−0.377563
$40$	−33.0843	−5.23108
$41$	4.57527	0.714537	0.357268	−	0.934002i	$-0.383708\pi$
$41$	4.57527	0.714537	0.357268	+	0.934002i	$0.383708\pi$
$42$	17.0967	2.63808
$43$	9.67756	1.47581	0.737907	−	0.674902i	$-0.235814\pi$
$43$	9.67756	1.47581	0.737907	+	0.674902i	$0.235814\pi$
$44$	23.7917	3.58674
$45$	8.16967	1.21786
$46$	−24.5431	−3.61868
$47$	10.7544	1.56869	0.784346	−	0.620323i	$-0.212998\pi$
$47$	10.7544	1.56869	0.784346	+	0.620323i	$0.212998\pi$
$48$	40.9312	5.90791
$49$	−0.197263	−0.0281805
$50$	−14.4210	−2.03943
$51$	7.05893	0.988449
$52$	−5.72856	−0.794409
$53$	−10.3092	−1.41607	−0.708036	−	0.706177i	$-0.750418\pi$
$53$	−10.3092	−1.41607	−0.708036	+	0.706177i	$0.750418\pi$
$54$	2.88674	0.392836
$55$	13.2560	1.78743
$56$	27.0354	3.61276
$57$	16.5240	2.18866
$58$	23.3967	3.07213
$59$	−3.05416	−0.397618	−0.198809	−	0.980038i	$-0.563707\pi$
$59$	−3.05416	−0.397618	−0.198809	+	0.980038i	$0.563707\pi$
$60$	43.1120	5.56574
$61$	−14.3630	−1.83900	−0.919498	−	0.393095i	$-0.871404\pi$
$61$	−14.3630	−1.83900	−0.919498	+	0.393095i	$0.871404\pi$
$62$	−3.62894	−0.460875
$63$	−6.67599	−0.841096
$64$	41.8110	5.22638
$65$	−3.19176	−0.395889
$66$	−27.2240	−3.35104
$67$	9.09616	1.11127	0.555636	−	0.831425i	$-0.312475\pi$
$67$	9.09616	1.11127	0.555636	+	0.831425i	$0.312475\pi$
$68$	17.1499	2.07974
$69$	20.8162	2.50598
$70$	23.1431	2.76613
$71$	0.542927	0.0644336	0.0322168	−	0.999481i	$-0.489743\pi$
$71$	0.542927	0.0644336	0.0322168	+	0.999481i	$0.489743\pi$
$72$	−26.5317	−3.12679
$73$	1.18119	0.138248	0.0691240	−	0.997608i	$-0.477980\pi$
$73$	1.18119	0.138248	0.0691240	+	0.997608i	$0.477980\pi$
$74$	−26.8721	−3.12382
$75$	12.2312	1.41233
$76$	40.1457	4.60503
$77$	−10.8323	−1.23446
$78$	6.55498	0.742206
$79$	1.48202	0.166741	0.0833704	−	0.996519i	$-0.473432\pi$
$79$	1.48202	0.166741	0.0833704	+	0.996519i	$0.473432\pi$
$80$	55.4068	6.19467
$81$	−10.1272	−1.12525
$82$	−12.7194	−1.40462
$83$	9.96792	1.09412	0.547061	−	0.837093i	$-0.315747\pi$
$83$	9.96792	1.09412	0.547061	+	0.837093i	$0.315747\pi$
$84$	−35.2298	−3.84388
$85$	9.55537	1.03643
$86$	−26.9039	−2.90112
$87$	−19.8439	−2.12749
$88$	−43.0498	−4.58913
$89$	−6.18819	−0.655947	−0.327973	−	0.944687i	$-0.606366\pi$
$89$	−6.18819	−0.655947	−0.327973	+	0.944687i	$0.606366\pi$
$90$	−22.7119	−2.39405
$91$	2.60821	0.273414
$92$	50.5738	5.27268
$93$	3.07788	0.319162
$94$	−29.8976	−3.08370
$95$	22.3679	2.29489
$96$	−64.9086	−6.62471
$97$	−16.4821	−1.67350	−0.836751	−	0.547584i	$-0.815548\pi$
$97$	−16.4821	−1.67350	−0.836751	+	0.547584i	$0.815548\pi$
$98$	0.548398	0.0553966
$99$	10.6305	1.06841
$100$	29.7161	2.97161
$101$	−9.10791	−0.906271	−0.453136	−	0.891442i	$-0.649695\pi$
$101$	−9.10791	−0.906271	−0.453136	+	0.891442i	$0.649695\pi$
$102$	−19.6240	−1.94307
$103$	−8.39834	−0.827513	−0.413756	−	0.910388i	$-0.635783\pi$
$103$	−8.39834	−0.827513	−0.413756	+	0.910388i	$0.635783\pi$
$104$	10.3655	1.01642
$105$	−19.6288	−1.91558
$106$	28.6598	2.78368
$107$	3.64463	0.352339	0.176170	−	0.984360i	$-0.443629\pi$
$107$	3.64463	0.352339	0.176170	+	0.984360i	$0.443629\pi$
$108$	−5.94846	−0.572391
$109$	0.273138	0.0261619	0.0130809	−	0.999914i	$-0.495836\pi$
$109$	0.273138	0.0261619	0.0130809	+	0.999914i	$0.495836\pi$
$110$	−36.8519	−3.51370
$111$	22.7916	2.16328
$112$	−45.2766	−4.27824
$113$	−16.8969	−1.58953	−0.794765	−	0.606918i	$-0.792406\pi$
$113$	−16.8969	−1.58953	−0.794765	+	0.606918i	$0.792406\pi$
$114$	−45.9373	−4.30242
$115$	28.1780	2.62761
$116$	−48.2115	−4.47632
$117$	−2.55961	−0.236636
$118$	8.49066	0.781628
$119$	−7.80834	−0.715789
$120$	−78.0088	−7.12120
$121$	6.24888	0.568080
$122$	39.9296	3.61506
$123$	10.7879	0.972716
$124$	7.47783	0.671529
$125$	0.597973	0.0534843
$126$	18.5595	1.65341
$127$	−4.23787	−0.376050	−0.188025	−	0.982164i	$-0.560209\pi$
$127$	−4.23787	−0.376050	−0.188025	+	0.982164i	$0.560209\pi$
$128$	−61.1791	−5.40752
$129$	22.8186	2.00906
$130$	8.87319	0.778231
$131$	9.64545	0.842727	0.421364	−	0.906892i	$-0.361552\pi$
$131$	9.64545	0.842727	0.421364	+	0.906892i	$0.361552\pi$
$132$	56.0981	4.88272
$133$	−18.2783	−1.58493
$134$	−25.2876	−2.18451
$135$	−3.31428	−0.285248
$136$	−31.0319	−2.66096
$137$	2.71820	0.232231	0.116116	−	0.993236i	$-0.462956\pi$
$137$	2.71820	0.232231	0.116116	+	0.993236i	$0.462956\pi$
$138$	−57.8697	−4.92620
$139$	−8.50444	−0.721337	−0.360669	−	0.932694i	$-0.617451\pi$
$139$	−8.50444	−0.721337	−0.360669	+	0.932694i	$0.617451\pi$
$140$	−47.6890	−4.03046
$141$	25.3577	2.13550
$142$	−1.50935	−0.126662
$143$	−4.15318	−0.347306
$144$	44.4331	3.70276
$145$	−26.8618	−2.23075
$146$	−3.28374	−0.271765
$147$	−0.465124	−0.0383628
$148$	55.3730	4.55163
$149$	−15.5631	−1.27498	−0.637490	−	0.770459i	$-0.720027\pi$
$149$	−15.5631	−1.27498	−0.637490	+	0.770459i	$0.720027\pi$
$150$	−34.0030	−2.77633
$151$	−7.90041	−0.642926	−0.321463	−	0.946922i	$-0.604175\pi$
$151$	−7.90041	−0.642926	−0.321463	+	0.946922i	$0.604175\pi$
$152$	−72.6415	−5.89200
$153$	7.66286	0.619506
$154$	30.1142	2.42667
$155$	4.16640	0.334653
$156$	−13.5073	−1.08145
$157$	7.98745	0.637468	0.318734	−	0.947844i	$-0.396742\pi$
$157$	7.98745	0.637468	0.318734	+	0.947844i	$0.396742\pi$
$158$	−4.12007	−0.327775
$159$	−24.3078	−1.92773
$160$	−87.8640	−6.94626
$161$	−23.0262	−1.81472
$162$	28.1540	2.21199
$163$	2.73624	0.214318	0.107159	−	0.994242i	$-0.465825\pi$
$163$	2.73624	0.214318	0.107159	+	0.994242i	$0.465825\pi$
$164$	26.2097	2.04663
$165$	31.2560	2.43328
$166$	−27.7111	−2.15080
$167$	−14.4645	−1.11929	−0.559647	−	0.828731i	$-0.689063\pi$
$167$	−14.4645	−1.11929	−0.559647	+	0.828731i	$0.689063\pi$
$168$	63.7463	4.91813
$169$	1.00000	0.0769231
$170$	−26.5642	−2.03738
$171$	17.9377	1.37173
$172$	55.4385	4.22715
$173$	−12.3683	−0.940344	−0.470172	−	0.882575i	$-0.655808\pi$
$173$	−12.3683	−0.940344	−0.470172	+	0.882575i	$0.655808\pi$
$174$	55.1666	4.18217
$175$	−13.5297	−1.02275
$176$	72.0963	5.43446
$177$	−7.20135	−0.541287
$178$	17.2033	1.28945
$179$	3.36344	0.251396	0.125698	−	0.992069i	$-0.459883\pi$
$179$	3.36344	0.251396	0.125698	+	0.992069i	$0.459883\pi$
$180$	46.8005	3.48830
$181$	−17.7070	−1.31615	−0.658074	−	0.752953i	$-0.728629\pi$
$181$	−17.7070	−1.31615	−0.658074	+	0.752953i	$0.728629\pi$
$182$	−7.25089	−0.537472
$183$	−33.8663	−2.50347
$184$	−91.5104	−6.74624
$185$	30.8520	2.26828
$186$	−8.55661	−0.627401
$187$	12.4336	0.909236
$188$	61.6074	4.49318
$189$	2.70832	0.197002
$190$	−62.1833	−4.51125
$191$	8.59542	0.621943	0.310971	−	0.950419i	$-0.399346\pi$
$191$	8.59542	0.621943	0.310971	+	0.950419i	$0.399346\pi$
$192$	98.5855	7.11479
$193$	9.94061	0.715541	0.357770	−	0.933810i	$-0.383537\pi$
$193$	9.94061	0.715541	0.357770	+	0.933810i	$0.383537\pi$
$194$	45.8207	3.28973
$195$	−7.52580	−0.538934
$196$	−1.13004	−0.0807168
$197$	18.5884	1.32437	0.662185	−	0.749340i	$-0.269629\pi$
$197$	18.5884	1.32437	0.662185	+	0.749340i	$0.269629\pi$
$198$	−29.5532	−2.10025
$199$	21.7281	1.54026	0.770131	−	0.637886i	$-0.220191\pi$
$199$	21.7281	1.54026	0.770131	+	0.637886i	$0.220191\pi$
$200$	−53.7695	−3.80208
$201$	21.4477	1.51280
$202$	25.3203	1.78153
$203$	21.9506	1.54063
$204$	40.4375	2.83119
$205$	14.6032	1.01993
$206$	23.3476	1.62671
$207$	22.5971	1.57061
$208$	−17.3593	−1.20365
$209$	29.1054	2.01327
$210$	54.5687	3.76560
$211$	−11.1820	−0.769798	−0.384899	−	0.922959i	$-0.625764\pi$
$211$	−11.1820	−0.769798	−0.384899	+	0.922959i	$0.625764\pi$
$212$	−59.0567	−4.05603
$213$	1.28016	0.0877150
$214$	−10.1322	−0.692621
$215$	30.8885	2.10658
$216$	10.7634	0.732357
$217$	−3.40465	−0.231122
$218$	−0.759331	−0.0514284
$219$	2.78511	0.188200
$220$	75.9376	5.11971
$221$	−2.99376	−0.201382
$222$	−63.3613	−4.25253
$223$	−6.83437	−0.457663	−0.228832	−	0.973466i	$-0.573490\pi$
$223$	−6.83437	−0.457663	−0.228832	+	0.973466i	$0.573490\pi$
$224$	71.7996	4.79731
$225$	13.2776	0.885173
$226$	46.9740	3.12466
$227$	6.29871	0.418060	0.209030	−	0.977909i	$-0.432969\pi$
$227$	6.29871	0.418060	0.209030	+	0.977909i	$0.432969\pi$
$228$	94.6589	6.26894
$229$	−8.08374	−0.534188	−0.267094	−	0.963670i	$-0.586063\pi$
$229$	−8.08374	−0.534188	−0.267094	+	0.963670i	$0.586063\pi$
$230$	−78.3357	−5.16530
$231$	−25.5414	−1.68050
$232$	87.2359	5.72732
$233$	4.96049	0.324973	0.162486	−	0.986711i	$-0.448049\pi$
$233$	4.96049	0.324973	0.162486	+	0.986711i	$0.448049\pi$
$234$	7.11579	0.465174
$235$	34.3255	2.23915
$236$	−17.4960	−1.13889
$237$	3.49444	0.226988
$238$	21.7074	1.40708
$239$	−13.4251	−0.868397	−0.434198	−	0.900817i	$-0.642968\pi$
$239$	−13.4251	−0.868397	−0.434198	+	0.900817i	$0.642968\pi$
$240$	130.643	8.43295
$241$	−7.71305	−0.496841	−0.248421	−	0.968652i	$-0.579912\pi$
$241$	−7.71305	−0.496841	−0.248421	+	0.968652i	$0.579912\pi$
$242$	−17.3721	−1.11672
$243$	−20.7637	−1.33199
$244$	−82.2794	−5.26740
$245$	−0.629618	−0.0402248
$246$	−29.9908	−1.91214
$247$	−7.00799	−0.445908
$248$	−13.5307	−0.859202
$249$	23.5032	1.48945
$250$	−1.66238	−0.105138
$251$	8.60234	0.542975	0.271488	−	0.962442i	$-0.412484\pi$
$251$	8.60234	0.542975	0.271488	+	0.962442i	$0.412484\pi$
$252$	−38.2438	−2.40914
$253$	36.6657	2.30515
$254$	11.7814	0.739230
$255$	22.5304	1.41091
$256$	86.4578	5.40361
$257$	11.8584	0.739705	0.369852	−	0.929091i	$-0.379408\pi$
$257$	11.8584	0.739705	0.369852	+	0.929091i	$0.379408\pi$
$258$	−63.4363	−3.94937
$259$	−25.2112	−1.56655
$260$	−18.2842	−1.13394
$261$	−21.5416	−1.33339
$262$	−26.8146	−1.65661
$263$	24.8627	1.53310	0.766549	−	0.642186i	$-0.221972\pi$
$263$	24.8627	1.53310	0.766549	+	0.642186i	$0.221972\pi$
$264$	−101.506	−6.24729
$265$	−32.9044	−2.02130
$266$	50.8142	3.11562
$267$	−14.5910	−0.892956
$268$	52.1079	3.18300
$269$	18.5739	1.13247	0.566235	−	0.824244i	$-0.308399\pi$
$269$	18.5739	1.13247	0.566235	+	0.824244i	$0.308399\pi$
$270$	9.21380	0.560734
$271$	−10.7194	−0.651158	−0.325579	−	0.945515i	$-0.605559\pi$
$271$	−10.7194	−0.651158	−0.325579	+	0.945515i	$0.605559\pi$
$272$	51.9696	3.15112
$273$	6.14984	0.372205
$274$	−7.55666	−0.456515
$275$	21.5440	1.29915
$276$	119.247	7.17783
$277$	−15.0263	−0.902842	−0.451421	−	0.892311i	$-0.649083\pi$
$277$	−15.0263	−0.902842	−0.451421	+	0.892311i	$0.649083\pi$
$278$	23.6426	1.41799
$279$	3.34121	0.200033
$280$	86.2906	5.15685
$281$	11.4341	0.682102	0.341051	−	0.940045i	$-0.389217\pi$
$281$	11.4341	0.682102	0.341051	+	0.940045i	$0.389217\pi$
$282$	−70.4950	−4.19792
$283$	7.65996	0.455338	0.227669	−	0.973739i	$-0.426890\pi$
$283$	7.65996	0.455338	0.227669	+	0.973739i	$0.426890\pi$
$284$	3.11019	0.184556
$285$	52.7408	3.12409
$286$	11.5460	0.682727
$287$	−11.9332	−0.704397
$288$	−70.4619	−4.15201
$289$	−8.03740	−0.472788
$290$	74.6766	4.38516
$291$	−38.8628	−2.27818
$292$	6.76652	0.395981
$293$	9.19083	0.536934	0.268467	−	0.963289i	$-0.413483\pi$
$293$	9.19083	0.536934	0.268467	+	0.963289i	$0.413483\pi$
$294$	1.29306	0.0754127
$295$	−9.74816	−0.567560
$296$	−100.194	−5.82368
$297$	−4.31260	−0.250242
$298$	43.2659	2.50633
$299$	−8.82835	−0.510557
$300$	70.0670	4.04532
$301$	−25.2411	−1.45487
$302$	21.9634	1.26385
$303$	−21.4754	−1.23373
$304$	121.654	6.97733
$305$	−45.8433	−2.62498
$306$	−21.3030	−1.21781
$307$	−29.6702	−1.69337	−0.846684	−	0.532096i	$-0.821404\pi$
$307$	−29.6702	−1.69337	−0.846684	+	0.532096i	$0.821404\pi$
$308$	−62.0537	−3.53584
$309$	−19.8023	−1.12651
$310$	−11.5827	−0.657853
$311$	−11.2469	−0.637754	−0.318877	−	0.947796i	$-0.603306\pi$
$311$	−11.2469	−0.637754	−0.318877	+	0.947796i	$0.603306\pi$
$312$	24.4407	1.38368
$313$	7.61921	0.430663	0.215332	−	0.976541i	$-0.430917\pi$
$313$	7.61921	0.430663	0.215332	+	0.976541i	$0.430917\pi$
$314$	−22.2053	−1.25312
$315$	−21.3082	−1.20058
$316$	8.48987	0.477593
$317$	3.53754	0.198688	0.0993441	−	0.995053i	$-0.468326\pi$
$317$	3.53754	0.198688	0.0993441	+	0.995053i	$0.468326\pi$
$318$	67.5764	3.78949
$319$	−34.9530	−1.95699
$320$	133.451	7.46013
$321$	8.59360	0.479648
$322$	64.0134	3.56733
$323$	20.9803	1.16737
$324$	−58.0144	−3.22302
$325$	−5.18735	−0.287742
$326$	−7.60681	−0.421302
$327$	0.644027	0.0356148
$328$	−47.4250	−2.61861
$329$	−28.0497	−1.54643
$330$	−86.8926	−4.78328
$331$	23.2170	1.27612	0.638062	−	0.769985i	$-0.279736\pi$
$331$	23.2170	1.27612	0.638062	+	0.769985i	$0.279736\pi$
$332$	57.1018	3.13387
$333$	24.7415	1.35583
$334$	40.2116	2.20028
$335$	29.0328	1.58623
$336$	−106.757	−5.82407
$337$	−10.2197	−0.556704	−0.278352	−	0.960479i	$-0.589788\pi$
$337$	−10.2197	−0.556704	−0.278352	+	0.960479i	$0.589788\pi$
$338$	−2.78003	−0.151214
$339$	−39.8410	−2.16387
$340$	54.7385	2.96861
$341$	5.42139	0.293585
$342$	−49.8674	−2.69652
$343$	18.7719	1.01359
$344$	−100.313	−5.40851
$345$	66.4404	3.57703
$346$	34.3842	1.84851
$347$	11.6392	0.624824	0.312412	−	0.949947i	$-0.398863\pi$
$347$	11.6392	0.624824	0.312412	+	0.949947i	$0.398863\pi$
$348$	−113.677	−6.09372
$349$	2.06019	0.110279	0.0551397	−	0.998479i	$-0.482440\pi$
$349$	2.06019	0.110279	0.0551397	+	0.998479i	$0.482440\pi$
$350$	37.6129	2.01049
$351$	1.03839	0.0554249
$352$	−114.330	−6.09382
$353$	25.3745	1.35055	0.675275	−	0.737566i	$-0.264025\pi$
$353$	25.3745	1.35055	0.675275	+	0.737566i	$0.264025\pi$
$354$	20.0200	1.06405
$355$	1.73289	0.0919725
$356$	−35.4494	−1.87882
$357$	−18.4112	−0.974421
$358$	−9.35047	−0.494188
$359$	−6.78018	−0.357844	−0.178922	−	0.983863i	$-0.557261\pi$
$359$	−6.78018	−0.357844	−0.178922	+	0.983863i	$0.557261\pi$
$360$	−84.6828	−4.46318
$361$	30.1120	1.58484
$362$	49.2259	2.58725
$363$	14.7341	0.773342
$364$	14.9413	0.783135
$365$	3.77008	0.197335
$366$	94.1493	4.92126
$367$	−17.4157	−0.909091	−0.454545	−	0.890724i	$-0.650198\pi$
$367$	−17.4157	−0.909091	−0.454545	+	0.890724i	$0.650198\pi$
$368$	153.254	7.98892
$369$	11.7109	0.609645
$370$	−85.7694	−4.45894
$371$	26.8884	1.39598
$372$	17.6319	0.914169
$373$	−20.1082	−1.04116	−0.520581	−	0.853812i	$-0.674285\pi$
$373$	−20.1082	−1.04116	−0.520581	+	0.853812i	$0.674285\pi$
$374$	−34.5658	−1.78736
$375$	1.40995	0.0728095
$376$	−111.475	−5.74889
$377$	8.41598	0.433445
$378$	−7.52922	−0.387261
$379$	2.03762	0.104665	0.0523327	−	0.998630i	$-0.483334\pi$
$379$	2.03762	0.104665	0.0523327	+	0.998630i	$0.483334\pi$
$380$	128.136	6.57322
$381$	−9.99239	−0.511926
$382$	−23.8955	−1.22260
$383$	−0.386169	−0.0197323	−0.00986614	−	0.999951i	$-0.503141\pi$
$383$	−0.386169	−0.0197323	−0.00986614	+	0.999951i	$0.503141\pi$
$384$	−144.253	−7.36139
$385$	−34.5743	−1.76207
$386$	−27.6352	−1.40659
$387$	24.7708	1.25917
$388$	−94.4187	−4.79338
$389$	14.6655	0.743569	0.371785	−	0.928319i	$-0.378746\pi$
$389$	14.6655	0.743569	0.371785	+	0.928319i	$0.378746\pi$
$390$	20.9220	1.05942
$391$	26.4300	1.33662
$392$	2.04474	0.103275
$393$	22.7428	1.14722
$394$	−51.6764	−2.60342
$395$	4.73027	0.238006
$396$	60.8976	3.06022
$397$	24.0919	1.20914	0.604568	−	0.796553i	$-0.293346\pi$
$397$	24.0919	1.20914	0.604568	+	0.796553i	$0.293346\pi$
$398$	−60.4047	−3.02781
$399$	−43.0981	−2.15760
$400$	90.0488	4.50244
$401$	5.92327	0.295794	0.147897	−	0.989003i	$-0.452750\pi$
$401$	5.92327	0.295794	0.147897	+	0.989003i	$0.452750\pi$
$402$	−59.6252	−2.97383
$403$	−1.30536	−0.0650246
$404$	−52.1752	−2.59582
$405$	−32.3237	−1.60618
$406$	−61.0233	−3.02854
$407$	40.1451	1.98992
$408$	−73.1695	−3.62243
$409$	−18.2236	−0.901101	−0.450550	−	0.892751i	$-0.648772\pi$
$409$	−18.2236	−0.901101	−0.450550	+	0.892751i	$0.648772\pi$
$410$	−40.5972	−2.00495
$411$	6.40919	0.316142
$412$	−48.1104	−2.37023
$413$	7.96588	0.391975
$414$	−62.8207	−3.08747
$415$	31.8152	1.56175
$416$	27.5284	1.34969
$417$	−20.0525	−0.981974
$418$	−80.9140	−3.95763
$419$	19.8924	0.971805	0.485903	−	0.874013i	$-0.338491\pi$
$419$	19.8924	0.971805	0.485903	+	0.874013i	$0.338491\pi$
$420$	−112.445	−5.48676
$421$	1.88453	0.0918463	0.0459232	−	0.998945i	$-0.485377\pi$
$421$	1.88453	0.0918463	0.0459232	+	0.998945i	$0.485377\pi$
$422$	31.0862	1.51325
$423$	27.5271	1.33841
$424$	106.860	5.18957
$425$	15.5297	0.753300
$426$	−3.55888	−0.172428
$427$	37.4617	1.81290
$428$	20.8785	1.00920
$429$	−9.79271	−0.472796
$430$	−85.8709	−4.14106
$431$	13.2686	0.639124	0.319562	−	0.947565i	$-0.396464\pi$
$431$	13.2686	0.639124	0.319562	+	0.947565i	$0.396464\pi$
$432$	−18.0257	−0.867260
$433$	23.6248	1.13534	0.567668	−	0.823258i	$-0.307846\pi$
$433$	23.6248	1.13534	0.567668	+	0.823258i	$0.307846\pi$
$434$	9.46501	0.454335
$435$	−63.3370	−3.03678
$436$	1.56469	0.0749349
$437$	61.8690	2.95960
$438$	−7.74268	−0.369960
$439$	18.5675	0.886177	0.443088	−	0.896478i	$-0.353883\pi$
$439$	18.5675	0.886177	0.443088	+	0.896478i	$0.353883\pi$
$440$	−137.405	−6.55052
$441$	−0.504917	−0.0240437
$442$	8.32274	0.395872
$443$	−15.0799	−0.716468	−0.358234	−	0.933632i	$-0.616621\pi$
$443$	−15.0799	−0.716468	−0.358234	+	0.933632i	$0.616621\pi$
$444$	130.563	6.19625
$445$	−19.7512	−0.936298
$446$	18.9997	0.899664
$447$	−36.6960	−1.73566
$448$	−109.052	−5.15221
$449$	33.6060	1.58596	0.792982	−	0.609246i	$-0.208528\pi$
$449$	33.6060	1.58596	0.792982	+	0.609246i	$0.208528\pi$
$450$	−36.9121	−1.74005
$451$	19.0019	0.894764
$452$	−96.7951	−4.55286
$453$	−18.6282	−0.875231
$454$	−17.5106	−0.821813
$455$	8.32477	0.390271
$456$	−171.280	−8.02092
$457$	−18.4676	−0.863879	−0.431940	−	0.901903i	$-0.642171\pi$
$457$	−18.4676	−0.863879	−0.431940	+	0.901903i	$0.642171\pi$
$458$	22.4730	1.05010
$459$	−3.10868	−0.145101
$460$	161.419	7.52622
$461$	19.8335	0.923739	0.461870	−	0.886948i	$-0.347179\pi$
$461$	19.8335	0.923739	0.461870	+	0.886948i	$0.347179\pi$
$462$	71.0058	3.30349
$463$	33.5828	1.56072	0.780362	−	0.625328i	$-0.215035\pi$
$463$	33.5828	1.56072	0.780362	+	0.625328i	$0.215035\pi$
$464$	−146.096	−6.78231
$465$	9.82388	0.455571
$466$	−13.7903	−0.638824
$467$	−7.57650	−0.350599	−0.175299	−	0.984515i	$-0.556089\pi$
$467$	−7.57650	−0.350599	−0.175299	+	0.984515i	$0.556089\pi$
$468$	−14.6629	−0.677792
$469$	−23.7247	−1.09550
$470$	−95.4260	−4.40167
$471$	18.8335	0.867800
$472$	31.6579	1.45718
$473$	40.1926	1.84806
$474$	−9.71465	−0.446208
$475$	36.3529	1.66799
$476$	−44.7306	−2.05022
$477$	−26.3874	−1.20820
$478$	37.3221	1.70707
$479$	−33.3521	−1.52390	−0.761948	−	0.647638i	$-0.775757\pi$
$479$	−33.3521	−1.52390	−0.761948	+	0.647638i	$0.775757\pi$
$480$	−207.173	−9.45611
$481$	−9.66612	−0.440737
$482$	21.4425	0.976679
$483$	−54.2930	−2.47042
$484$	35.7971	1.62714
$485$	−52.6069	−2.38876
$486$	57.7236	2.61839
$487$	20.4176	0.925212	0.462606	−	0.886564i	$-0.346915\pi$
$487$	20.4176	0.925212	0.462606	+	0.886564i	$0.346915\pi$
$488$	148.880	6.73948
$489$	6.45172	0.291757
$490$	1.75036	0.0790730
$491$	11.0508	0.498716	0.249358	−	0.968411i	$-0.419780\pi$
$491$	11.0508	0.498716	0.249358	+	0.968411i	$0.419780\pi$
$492$	61.7994	2.78613
$493$	−25.1954	−1.13474
$494$	19.4824	0.876556
$495$	33.9301	1.52504
$496$	22.6601	1.01747
$497$	−1.41607	−0.0635192
$498$	−65.3395	−2.92793
$499$	−37.6726	−1.68646	−0.843229	−	0.537554i	$-0.819348\pi$
$499$	−37.6726	−1.68646	−0.843229	+	0.537554i	$0.819348\pi$
$500$	3.42552	0.153194
$501$	−34.1055	−1.52372
$502$	−23.9148	−1.06737
$503$	28.9031	1.28873	0.644363	−	0.764720i	$-0.277123\pi$
$503$	28.9031	1.28873	0.644363	+	0.764720i	$0.277123\pi$
$504$	69.2001	3.08242
$505$	−29.0703	−1.29361
$506$	−101.932	−4.53142
$507$	2.35788	0.104717
$508$	−24.2769	−1.07711
$509$	2.66320	0.118044	0.0590220	−	0.998257i	$-0.481202\pi$
$509$	2.66320	0.118044	0.0590220	+	0.998257i	$0.481202\pi$
$510$	−62.6353	−2.77354
$511$	−3.08079	−0.136286
$512$	−117.997	−5.21478
$513$	−7.27700	−0.321287
$514$	−32.9666	−1.45410
$515$	−26.8055	−1.18119
$516$	130.718	5.75452
$517$	44.6650	1.96436
$518$	70.0880	3.07949
$519$	−29.1630	−1.28011
$520$	33.0843	1.45084
$521$	−26.9469	−1.18056	−0.590282	−	0.807197i	$-0.700984\pi$
$521$	−26.9469	−1.18056	−0.590282	+	0.807197i	$0.700984\pi$
$522$	59.8864	2.62115
$523$	32.9340	1.44010	0.720051	−	0.693921i	$-0.244118\pi$
$523$	32.9340	1.44010	0.720051	+	0.693921i	$0.244118\pi$
$524$	55.2546	2.41381
$525$	−31.9014	−1.39229
$526$	−69.1190	−3.01373
$527$	3.90793	0.170232
$528$	169.995	7.39806
$529$	54.9398	2.38869
$530$	91.4751	3.97343
$531$	−7.81746	−0.339249
$532$	−104.708	−4.53968
$533$	−4.57527	−0.198177
$534$	40.5635	1.75535
$535$	11.6328	0.502929
$536$	−94.2864	−4.07255
$537$	7.93061	0.342231
$538$	−51.6360	−2.22618
$539$	−0.819270	−0.0352884
$540$	−18.9861	−0.817030
$541$	−9.68796	−0.416518	−0.208259	−	0.978074i	$-0.566780\pi$
$541$	−9.68796	−0.416518	−0.208259	+	0.978074i	$0.566780\pi$
$542$	29.8003	1.28003
$543$	−41.7509	−1.79170
$544$	−82.4133	−3.53344
$545$	0.871791	0.0373434
$546$	−17.0967	−0.731673
$547$	−29.0144	−1.24056	−0.620282	−	0.784379i	$-0.712982\pi$
$547$	−29.0144	−1.24056	−0.620282	+	0.784379i	$0.712982\pi$
$548$	15.5714	0.665175
$549$	−36.7637	−1.56904
$550$	−59.8929	−2.55384
$551$	−58.9791	−2.51259
$552$	−215.771	−9.18382
$553$	−3.86542	−0.164375
$554$	41.7735	1.77479
$555$	72.7453	3.08787
$556$	−48.7182	−2.06611
$557$	4.25474	0.180279	0.0901396	−	0.995929i	$-0.471269\pi$
$557$	4.25474	0.180279	0.0901396	+	0.995929i	$0.471269\pi$
$558$	−9.28867	−0.393221
$559$	−9.67756	−0.409317
$560$	−144.512	−6.10676
$561$	29.3170	1.23777
$562$	−31.7872	−1.34086
$563$	23.0694	0.972259	0.486129	−	0.873887i	$-0.338408\pi$
$563$	23.0694	0.972259	0.486129	+	0.873887i	$0.338408\pi$
$564$	145.263	6.11667
$565$	−53.9310	−2.26889
$566$	−21.2949	−0.895092
$567$	26.4139	1.10928
$568$	−5.62772	−0.236134
$569$	−1.56892	−0.0657725	−0.0328862	−	0.999459i	$-0.510470\pi$
$569$	−1.56892	−0.0657725	−0.0328862	+	0.999459i	$0.510470\pi$
$570$	−146.621	−6.14127
$571$	37.4844	1.56867	0.784337	−	0.620335i	$-0.213003\pi$
$571$	37.4844	1.56867	0.784337	+	0.620335i	$0.213003\pi$
$572$	−23.7917	−0.994783
$573$	20.2670	0.846666
$574$	33.1747	1.38469
$575$	45.7957	1.90981
$576$	107.020	4.45916
$577$	−7.45823	−0.310490	−0.155245	−	0.987876i	$-0.549617\pi$
$577$	−7.45823	−0.310490	−0.155245	+	0.987876i	$0.549617\pi$
$578$	22.3442	0.929397
$579$	23.4388	0.974083
$580$	−153.880	−6.38950
$581$	−25.9984	−1.07859
$582$	108.040	4.47839
$583$	−42.8158	−1.77325
$584$	−12.2436	−0.506646
$585$	−8.16967	−0.337774
$586$	−25.5508	−1.05549
$587$	−33.6984	−1.39088	−0.695440	−	0.718584i	$-0.744791\pi$
$587$	−33.6984	−1.39088	−0.695440	+	0.718584i	$0.744791\pi$
$588$	−2.66449	−0.109882
$589$	9.14795	0.376935
$590$	27.1002	1.11570
$591$	43.8293	1.80290
$592$	167.797	6.89642
$593$	46.0606	1.89148	0.945742	−	0.324920i	$-0.105337\pi$
$593$	46.0606	1.89148	0.945742	+	0.324920i	$0.105337\pi$
$594$	11.9892	0.491921
$595$	−24.9224	−1.02172
$596$	−89.1542	−3.65190
$597$	51.2322	2.09680
$598$	24.5431	1.00364
$599$	−22.4589	−0.917647	−0.458823	−	0.888527i	$-0.651729\pi$
$599$	−22.4589	−0.917647	−0.458823	+	0.888527i	$0.651729\pi$
$600$	−126.782	−5.17586
$601$	−17.5590	−0.716245	−0.358122	−	0.933675i	$-0.616583\pi$
$601$	−17.5590	−0.716245	−0.358122	+	0.933675i	$0.616583\pi$
$602$	70.1709	2.85995
$603$	23.2826	0.948142
$604$	−45.2580	−1.84152
$605$	19.9450	0.810878
$606$	59.7022	2.42524
$607$	−4.84461	−0.196637	−0.0983183	−	0.995155i	$-0.531346\pi$
$607$	−4.84461	−0.196637	−0.0983183	+	0.995155i	$0.531346\pi$
$608$	−192.919	−7.82388
$609$	51.7569	2.09730
$610$	127.446	5.16013
$611$	−10.7544	−0.435077
$612$	43.8972	1.77444
$613$	−43.9209	−1.77395	−0.886975	−	0.461817i	$-0.847198\pi$
$613$	−43.9209	−1.77395	−0.886975	+	0.461817i	$0.847198\pi$
$614$	82.4840	3.32878
$615$	34.4326	1.38845
$616$	112.283	4.52400
$617$	0.810564	0.0326321	0.0163160	−	0.999867i	$-0.494806\pi$
$617$	0.810564	0.0326321	0.0163160	+	0.999867i	$0.494806\pi$
$618$	55.0510	2.21447
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	23.8675	0.958541
$621$	−9.16723	−0.367868
$622$	31.2668	1.25368
$623$	16.1401	0.646638
$624$	−40.9312	−1.63856
$625$	−24.0282	−0.961126
$626$	−21.1816	−0.846589
$627$	68.6272	2.74071
$628$	45.7566	1.82589
$629$	28.9381	1.15384
$630$	59.2374	2.36007
$631$	16.9345	0.674150	0.337075	−	0.941478i	$-0.390562\pi$
$631$	16.9345	0.674150	0.337075	+	0.941478i	$0.390562\pi$
$632$	−15.3619	−0.611065
$633$	−26.3657	−1.04794
$634$	−9.83447	−0.390577
$635$	−13.5263	−0.536773
$636$	−139.249	−5.52157
$637$	0.197263	0.00781586
$638$	97.1705	3.84702
$639$	1.38968	0.0549750
$640$	−195.269	−7.71870
$641$	−26.7137	−1.05513	−0.527564	−	0.849515i	$-0.676894\pi$
$641$	−26.7137	−1.05513	−0.527564	+	0.849515i	$0.676894\pi$
$642$	−23.8905	−0.942881
$643$	40.7618	1.60749	0.803744	−	0.594975i	$-0.202838\pi$
$643$	40.7618	1.60749	0.803744	+	0.594975i	$0.202838\pi$
$644$	−131.907	−5.19786
$645$	72.8314	2.86773
$646$	−58.3257	−2.29479
$647$	9.67460	0.380348	0.190174	−	0.981750i	$-0.439095\pi$
$647$	9.67460	0.380348	0.190174	+	0.981750i	$0.439095\pi$
$648$	104.974	4.12376
$649$	−12.6845	−0.497909
$650$	14.4210	0.565637
$651$	−8.02775	−0.314633
$652$	15.6747	0.613868
$653$	−31.2149	−1.22153	−0.610767	−	0.791811i	$-0.709139\pi$
$653$	−31.2149	−1.22153	−0.610767	+	0.791811i	$0.709139\pi$
$654$	−1.79041	−0.0700107
$655$	30.7860	1.20291
$656$	79.4235	3.10097
$657$	3.02339	0.117954
$658$	77.9791	3.03994
$659$	−9.34238	−0.363928	−0.181964	−	0.983305i	$-0.558245\pi$
$659$	−9.34238	−0.363928	−0.181964	+	0.983305i	$0.558245\pi$
$660$	179.052	6.96959
$661$	−10.9754	−0.426895	−0.213448	−	0.976955i	$-0.568469\pi$
$661$	−10.9754	−0.426895	−0.213448	+	0.976955i	$0.568469\pi$
$662$	−64.5440	−2.50858
$663$	−7.05893	−0.274146
$664$	−103.323	−4.00969
$665$	−58.3400	−2.26233
$666$	−68.7821	−2.66525
$667$	−74.2992	−2.87688
$668$	−82.8605	−3.20597
$669$	−16.1146	−0.623028
$670$	−80.7120	−3.11818
$671$	−59.6522	−2.30285
$672$	169.295	6.53070
$673$	30.7587	1.18566	0.592830	−	0.805328i	$-0.298011\pi$
$673$	30.7587	1.18566	0.592830	+	0.805328i	$0.298011\pi$
$674$	28.4111	1.09436
$675$	−5.38647	−0.207325
$676$	5.72856	0.220329
$677$	50.2064	1.92959	0.964794	−	0.263008i	$-0.0847144\pi$
$677$	50.2064	1.92959	0.964794	+	0.263008i	$0.0847144\pi$
$678$	110.759	4.25368
$679$	42.9887	1.64975
$680$	−99.0463	−3.79825
$681$	14.8516	0.569115
$682$	−15.0716	−0.577122
$683$	−18.4432	−0.705711	−0.352856	−	0.935678i	$-0.614789\pi$
$683$	−18.4432	−0.705711	−0.352856	+	0.935678i	$0.614789\pi$
$684$	102.757	3.92903
$685$	8.67584	0.331487
$686$	−52.1866	−1.99249
$687$	−19.0605	−0.727204
$688$	167.996	6.40478
$689$	10.3092	0.392748
$690$	−184.706	−7.03165
$691$	−23.8925	−0.908912	−0.454456	−	0.890769i	$-0.650166\pi$
$691$	−23.8925	−0.908912	−0.454456	+	0.890769i	$0.650166\pi$
$692$	−70.8525	−2.69341
$693$	−27.7266	−1.05325
$694$	−32.3573	−1.22827
$695$	−27.1442	−1.02964
$696$	205.692	7.79674
$697$	13.6972	0.518820
$698$	−5.72739	−0.216785
$699$	11.6963	0.442393
$700$	−77.5056	−2.92944
$701$	1.17674	0.0444449	0.0222224	−	0.999753i	$-0.492926\pi$
$701$	1.17674	0.0444449	0.0222224	+	0.999753i	$0.492926\pi$
$702$	−2.88674	−0.108953
$703$	67.7401	2.55487
$704$	173.649	6.54463
$705$	80.9356	3.04821
$706$	−70.5420	−2.65488
$707$	23.7553	0.893410
$708$	−41.2534	−1.55040
$709$	−30.0427	−1.12828	−0.564139	−	0.825680i	$-0.690792\pi$
$709$	−30.0427	−1.12828	−0.564139	+	0.825680i	$0.690792\pi$
$710$	−4.81750	−0.180798
$711$	3.79341	0.142264
$712$	64.1438	2.40389
$713$	11.5242	0.431584
$714$	51.1835	1.91550
$715$	−13.2560	−0.495745
$716$	19.2677	0.720068
$717$	−31.6548	−1.18217
$718$	18.8491	0.703442
$719$	−18.4011	−0.686246	−0.343123	−	0.939291i	$-0.611485\pi$
$719$	−18.4011	−0.686246	−0.343123	+	0.939291i	$0.611485\pi$
$720$	141.820	5.28531
$721$	21.9046	0.815770
$722$	−83.7122	−3.11545
$723$	−18.1865	−0.676362
$724$	−101.435	−3.76982
$725$	−43.6566	−1.62137
$726$	−40.9613	−1.52022
$727$	−29.1584	−1.08143	−0.540713	−	0.841207i	$-0.681846\pi$
$727$	−29.1584	−1.08143	−0.540713	+	0.841207i	$0.681846\pi$
$728$	−27.0354	−1.00200
$729$	−18.5766	−0.688021
$730$	−10.4809	−0.387917
$731$	28.9723	1.07158
$732$	−194.005	−7.17064
$733$	−50.1068	−1.85074	−0.925369	−	0.379067i	$-0.876245\pi$
$733$	−50.1068	−1.85074	−0.925369	+	0.379067i	$0.876245\pi$
$734$	48.4161	1.78707
$735$	−1.48456	−0.0547590
$736$	−243.030	−8.95820
$737$	37.7780	1.39157
$738$	−32.5567	−1.19843
$739$	−16.1901	−0.595564	−0.297782	−	0.954634i	$-0.596247\pi$
$739$	−16.1901	−0.595564	−0.297782	+	0.954634i	$0.596247\pi$
$740$	176.737	6.49700
$741$	−16.5240	−0.607025
$742$	−74.7505	−2.74418
$743$	16.3058	0.598203	0.299101	−	0.954221i	$-0.403313\pi$
$743$	16.3058	0.598203	0.299101	+	0.954221i	$0.403313\pi$
$744$	−31.9039	−1.16965
$745$	−49.6737	−1.81991
$746$	55.9013	2.04669
$747$	25.5140	0.933508
$748$	71.2267	2.60431
$749$	−9.50593	−0.347339
$750$	−3.91970	−0.143127
$751$	−33.1320	−1.20900	−0.604502	−	0.796604i	$-0.706628\pi$
$751$	−33.1320	−1.20900	−0.604502	+	0.796604i	$0.706628\pi$
$752$	186.689	6.80786
$753$	20.2833	0.739165
$754$	−23.3967	−0.852056
$755$	−25.2162	−0.917712
$756$	15.5148	0.564268
$757$	−5.39106	−0.195942	−0.0979708	−	0.995189i	$-0.531235\pi$
$757$	−5.39106	−0.195942	−0.0979708	+	0.995189i	$0.531235\pi$
$758$	−5.66464	−0.205749
$759$	86.4534	3.13806
$760$	−231.854	−8.41024
$761$	−47.9153	−1.73693	−0.868465	−	0.495751i	$-0.834893\pi$
$761$	−47.9153	−1.73693	−0.868465	+	0.495751i	$0.834893\pi$
$762$	27.7791	1.00633
$763$	−0.712399	−0.0257906
$764$	49.2394	1.78142
$765$	24.4580	0.884282
$766$	1.07356	0.0387893
$767$	3.05416	0.110279
$768$	203.857	7.35607
$769$	−30.8086	−1.11099	−0.555494	−	0.831521i	$-0.687471\pi$
$769$	−30.8086	−1.11099	−0.555494	+	0.831521i	$0.687471\pi$
$770$	96.1175	3.46383
$771$	27.9606	1.00698
$772$	56.9454	2.04951
$773$	36.9854	1.33027	0.665136	−	0.746722i	$-0.268373\pi$
$773$	36.9854	1.33027	0.665136	+	0.746722i	$0.268373\pi$
$774$	−68.8635	−2.47525
$775$	6.77135	0.243234
$776$	170.845	6.13299
$777$	−59.4451	−2.13258
$778$	−40.7705	−1.46169
$779$	32.0634	1.14879
$780$	−43.1120	−1.54366
$781$	2.25487	0.0806857
$782$	−73.4761	−2.62750
$783$	8.73903	0.312307
$784$	−3.42436	−0.122298
$785$	25.4940	0.909921
$786$	−63.2258	−2.25519
$787$	0.955000	0.0340421	0.0170210	−	0.999855i	$-0.494582\pi$
$787$	0.955000	0.0340421	0.0170210	+	0.999855i	$0.494582\pi$
$788$	106.485	3.79337
$789$	58.6233	2.08704
$790$	−13.1503	−0.467866
$791$	44.0707	1.56697
$792$	−110.191	−3.91546
$793$	14.3630	0.510046
$794$	−66.9761	−2.37689
$795$	−77.5847	−2.75164
$796$	124.471	4.41174
$797$	−42.1612	−1.49342	−0.746712	−	0.665147i	$-0.768369\pi$
$797$	−42.1612	−1.49342	−0.746712	+	0.665147i	$0.768369\pi$
$798$	119.814	4.24137
$799$	32.1961	1.13902
$800$	−142.799	−5.04871
$801$	−15.8394	−0.559656
$802$	−16.4669	−0.581465
$803$	4.90569	0.173118
$804$	122.864	4.33309
$805$	−73.4940	−2.59032
$806$	3.62894	0.127824
$807$	43.7951	1.54166
$808$	94.4082	3.32127
$809$	29.8006	1.04773	0.523867	−	0.851800i	$-0.324489\pi$
$809$	29.8006	1.04773	0.523867	+	0.851800i	$0.324489\pi$
$810$	89.8608	3.15739
$811$	−20.9640	−0.736145	−0.368073	−	0.929797i	$-0.619982\pi$
$811$	−20.9640	−0.736145	−0.368073	+	0.929797i	$0.619982\pi$
$812$	125.745	4.41280
$813$	−25.2751	−0.886437
$814$	−111.605	−3.91174
$815$	8.73341	0.305918
$816$	122.538	4.28970
$817$	67.8203	2.37273
$818$	50.6622	1.77136
$819$	6.67599	0.233278
$820$	83.6552	2.92136
$821$	35.7823	1.24881	0.624406	−	0.781100i	$-0.285341\pi$
$821$	35.7823	1.24881	0.624406	+	0.781100i	$0.285341\pi$
$822$	−17.8177	−0.621465
$823$	−17.4331	−0.607679	−0.303840	−	0.952723i	$-0.598269\pi$
$823$	−17.4331	−0.607679	−0.303840	+	0.952723i	$0.598269\pi$
$824$	87.0531	3.03264
$825$	50.7982	1.76857
$826$	−22.1454	−0.770536
$827$	5.04515	0.175437	0.0877185	−	0.996145i	$-0.472042\pi$
$827$	5.04515	0.175437	0.0877185	+	0.996145i	$0.472042\pi$
$828$	129.449	4.49867
$829$	14.8133	0.514488	0.257244	−	0.966347i	$-0.417186\pi$
$829$	14.8133	0.514488	0.257244	+	0.966347i	$0.417186\pi$
$830$	−88.4473	−3.07005
$831$	−35.4302	−1.22906
$832$	−41.8110	−1.44954
$833$	−0.590559	−0.0204617
$834$	55.7465	1.93034
$835$	−46.1671	−1.59768
$836$	166.732	5.76656
$837$	−1.35547	−0.0468518
$838$	−55.3013	−1.91035
$839$	34.1655	1.17952	0.589761	−	0.807578i	$-0.299222\pi$
$839$	34.1655	1.17952	0.589761	+	0.807578i	$0.299222\pi$
$840$	203.463	7.02014
$841$	41.8287	1.44237
$842$	−5.23905	−0.180549
$843$	26.9603	0.928562
$844$	−64.0565	−2.20492
$845$	3.19176	0.109800
$846$	−76.5262	−2.63103
$847$	−16.2984	−0.560019
$848$	−178.960	−6.14551
$849$	18.0613	0.619862
$850$	−43.1730	−1.48082
$851$	85.3359	2.92528
$852$	7.33347	0.251241
$853$	−53.2244	−1.82237	−0.911185	−	0.411998i	$-0.864831\pi$
$853$	−53.2244	−1.82237	−0.911185	+	0.411998i	$0.864831\pi$
$854$	−104.145	−3.56376
$855$	57.2530	1.95801
$856$	−37.7784	−1.29124
$857$	32.0421	1.09454	0.547269	−	0.836957i	$-0.315668\pi$
$857$	32.0421	1.09454	0.547269	+	0.836957i	$0.315668\pi$
$858$	27.2240	0.929412
$859$	−39.2217	−1.33823	−0.669113	−	0.743161i	$-0.733326\pi$
$859$	−39.2217	−1.33823	−0.669113	+	0.743161i	$0.733326\pi$
$860$	176.947	6.03383
$861$	−28.1372	−0.958912
$862$	−36.8870	−1.25638
$863$	46.8830	1.59592	0.797959	−	0.602712i	$-0.205913\pi$
$863$	46.8830	1.59592	0.797959	+	0.602712i	$0.205913\pi$
$864$	28.5851	0.972483
$865$	−39.4767	−1.34225
$866$	−65.6776	−2.23182
$867$	−18.9513	−0.643618
$868$	−19.5037	−0.662000
$869$	6.15511	0.208798
$870$	176.079	5.96963
$871$	−9.09616	−0.308212
$872$	−2.83121	−0.0958770
$873$	−42.1877	−1.42784
$874$	−171.998	−5.81791
$875$	−1.55964	−0.0527253
$876$	15.9547	0.539058
$877$	36.1238	1.21981	0.609906	−	0.792473i	$-0.291207\pi$
$877$	36.1238	1.21981	0.609906	+	0.792473i	$0.291207\pi$
$878$	−51.6181	−1.74203
$879$	21.6709	0.730941
$880$	230.114	7.75715
$881$	−20.5073	−0.690907	−0.345454	−	0.938436i	$-0.612275\pi$
$881$	−20.5073	−0.690907	−0.345454	+	0.938436i	$0.612275\pi$
$882$	1.40369	0.0472645
$883$	−13.7895	−0.464053	−0.232027	−	0.972709i	$-0.574536\pi$
$883$	−13.7895	−0.464053	−0.232027	+	0.972709i	$0.574536\pi$
$884$	−17.1499	−0.576815
$885$	−22.9850	−0.772633
$886$	41.9226	1.40842
$887$	23.2027	0.779070	0.389535	−	0.921012i	$-0.372636\pi$
$887$	23.2027	0.779070	0.389535	+	0.921012i	$0.372636\pi$
$888$	−236.247	−7.92791
$889$	11.0532	0.370713
$890$	54.9090	1.84055
$891$	−42.0602	−1.40907
$892$	−39.1511	−1.31088
$893$	75.3669	2.52206
$894$	102.016	3.41192
$895$	10.7353	0.358842
$896$	159.568	5.33079
$897$	−20.8162	−0.695033
$898$	−93.4256	−3.11765
$899$	−10.9859	−0.366399
$900$	76.0615	2.53538
$901$	−30.8631	−1.02820
$902$	−52.8258	−1.75891
$903$	−59.5155	−1.98055
$904$	175.145	5.82525
$905$	−56.5164	−1.87867
$906$	51.7870	1.72051
$907$	−50.9648	−1.69226	−0.846129	−	0.532978i	$-0.821073\pi$
$907$	−50.9648	−1.69226	−0.846129	+	0.532978i	$0.821073\pi$
$908$	36.0825	1.19744
$909$	−23.3127	−0.773234
$910$	−23.1431	−0.767187
$911$	−44.6956	−1.48083	−0.740415	−	0.672149i	$-0.765371\pi$
$911$	−44.6956	−1.48083	−0.740415	+	0.672149i	$0.765371\pi$
$912$	286.846	9.49841
$913$	41.3985	1.37009
$914$	51.3406	1.69819
$915$	−108.093	−3.57345
$916$	−46.3082	−1.53007
$917$	−25.1573	−0.830768
$918$	8.64221	0.285236
$919$	16.7227	0.551630	0.275815	−	0.961211i	$-0.411052\pi$
$919$	16.7227	0.551630	0.275815	+	0.961211i	$0.411052\pi$
$920$	−292.079	−9.62958
$921$	−69.9589	−2.30522
$922$	−55.1378	−1.81587
$923$	−0.542927	−0.0178707
$924$	−146.315	−4.81343
$925$	50.1416	1.64864
$926$	−93.3611	−3.06804
$927$	−21.4965	−0.706037
$928$	231.678	7.60520
$929$	28.7085	0.941897	0.470948	−	0.882161i	$-0.343912\pi$
$929$	28.7085	0.941897	0.470948	+	0.882161i	$0.343912\pi$
$930$	−27.3107	−0.895552
$931$	−1.38242	−0.0453070
$932$	28.4165	0.930813
$933$	−26.5189	−0.868191
$934$	21.0629	0.689199
$935$	39.6851	1.29784
$936$	26.5317	0.867215
$937$	55.3982	1.80978	0.904890	−	0.425646i	$-0.139953\pi$
$937$	55.3982	1.80978	0.904890	+	0.425646i	$0.139953\pi$
$938$	65.9552	2.15351
$939$	17.9652	0.586273
$940$	196.636	6.41356
$941$	−40.2350	−1.31162	−0.655811	−	0.754925i	$-0.727673\pi$
$941$	−40.2350	−1.31162	−0.655811	+	0.754925i	$0.727673\pi$
$942$	−52.3576	−1.70590
$943$	40.3921	1.31535
$944$	−53.0181	−1.72559
$945$	8.64433	0.281200
$946$	−111.737	−3.63287
$947$	−13.5152	−0.439184	−0.219592	−	0.975592i	$-0.570473\pi$
$947$	−13.5152	−0.439184	−0.219592	+	0.975592i	$0.570473\pi$
$948$	20.0181	0.650158
$949$	−1.18119	−0.0383431
$950$	−101.062	−3.27889
$951$	8.34111	0.270479
$952$	80.9375	2.62320
$953$	27.6858	0.896832	0.448416	−	0.893825i	$-0.351988\pi$
$953$	27.6858	0.896832	0.448416	+	0.893825i	$0.351988\pi$
$954$	73.3578	2.37505
$955$	27.4345	0.887761
$956$	−76.9064	−2.48733
$957$	−82.4152	−2.66410
$958$	92.7198	2.99564
$959$	−7.08961	−0.228936
$960$	314.661	10.1557
$961$	−29.2960	−0.945033
$962$	26.8721	0.866392
$963$	9.32883	0.300617
$964$	−44.1847	−1.42309
$965$	31.7281	1.02136
$966$	150.936	4.85629
$967$	27.6657	0.889668	0.444834	−	0.895613i	$-0.353263\pi$
$967$	27.6657	0.889668	0.444834	+	0.895613i	$0.353263\pi$
$968$	−64.7729	−2.08188
$969$	49.4690	1.58917
$970$	146.249	4.69576
$971$	−7.45891	−0.239368	−0.119684	−	0.992812i	$-0.538188\pi$
$971$	−7.45891	−0.239368	−0.119684	+	0.992812i	$0.538188\pi$
$972$	−118.946	−3.81519
$973$	22.1813	0.711101
$974$	−56.7617	−1.81876
$975$	−12.2312	−0.391711
$976$	−249.332	−7.98092
$977$	10.8269	0.346382	0.173191	−	0.984888i	$-0.444592\pi$
$977$	10.8269	0.346382	0.173191	+	0.984888i	$0.444592\pi$
$978$	−17.9360	−0.573529
$979$	−25.7007	−0.821396
$980$	−3.60681	−0.115215
$981$	0.699126	0.0223214
$982$	−30.7216	−0.980365
$983$	−22.1738	−0.707234	−0.353617	−	0.935390i	$-0.615048\pi$
$983$	−22.1738	−0.707234	−0.353617	+	0.935390i	$0.615048\pi$
$984$	−111.823	−3.56477
$985$	59.3299	1.89041
$986$	70.0440	2.23065
$987$	−66.1380	−2.10519
$988$	−40.1457	−1.27721
$989$	85.4369	2.71674
$990$	−94.3267	−2.99790
$991$	−2.58771	−0.0822012	−0.0411006	−	0.999155i	$-0.513086\pi$
$991$	−2.58771	−0.0822012	−0.0411006	+	0.999155i	$0.513086\pi$
$992$	−35.9344	−1.14092
$993$	54.7430	1.73722
$994$	3.93670	0.124865
$995$	69.3508	2.19857
$996$	134.639	4.26621
$997$	−47.6160	−1.50802	−0.754008	−	0.656866i	$-0.771882\pi$
$997$	−47.6160	−1.50802	−0.754008	+	0.656866i	$0.771882\pi$
$998$	104.731	3.31520
$999$	−10.0372	−0.317562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.d.1.1	✓	156

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.d.1.1	✓	156	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)

\(f(q)\)	\(=\)	\(q-2.78003 q^{2} +2.35788 q^{3} +5.72856 q^{4} +3.19176 q^{5} -6.55498 q^{6} -2.60821 q^{7} -10.3655 q^{8} +2.55961 q^{9} +O(q^{10})\)	\(q-2.78003 q^{2} +2.35788 q^{3} +5.72856 q^{4} +3.19176 q^{5} -6.55498 q^{6} -2.60821 q^{7} -10.3655 q^{8} +2.55961 q^{9} -8.87319 q^{10} +4.15318 q^{11} +13.5073 q^{12} -1.00000 q^{13} +7.25089 q^{14} +7.52580 q^{15} +17.3593 q^{16} +2.99376 q^{17} -7.11579 q^{18} +7.00799 q^{19} +18.2842 q^{20} -6.14984 q^{21} -11.5460 q^{22} +8.82835 q^{23} -24.4407 q^{24} +5.18735 q^{25} +2.78003 q^{26} -1.03839 q^{27} -14.9413 q^{28} -8.41598 q^{29} -20.9220 q^{30} +1.30536 q^{31} -27.5284 q^{32} +9.79271 q^{33} -8.32274 q^{34} -8.32477 q^{35} +14.6629 q^{36} +9.66612 q^{37} -19.4824 q^{38} -2.35788 q^{39} -33.0843 q^{40} +4.57527 q^{41} +17.0967 q^{42} +9.67756 q^{43} +23.7917 q^{44} +8.16967 q^{45} -24.5431 q^{46} +10.7544 q^{47} +40.9312 q^{48} -0.197263 q^{49} -14.4210 q^{50} +7.05893 q^{51} -5.72856 q^{52} -10.3092 q^{53} +2.88674 q^{54} +13.2560 q^{55} +27.0354 q^{56} +16.5240 q^{57} +23.3967 q^{58} -3.05416 q^{59} +43.1120 q^{60} -14.3630 q^{61} -3.62894 q^{62} -6.67599 q^{63} +41.8110 q^{64} -3.19176 q^{65} -27.2240 q^{66} +9.09616 q^{67} +17.1499 q^{68} +20.8162 q^{69} +23.1431 q^{70} +0.542927 q^{71} -26.5317 q^{72} +1.18119 q^{73} -26.8721 q^{74} +12.2312 q^{75} +40.1457 q^{76} -10.8323 q^{77} +6.55498 q^{78} +1.48202 q^{79} +55.4068 q^{80} -10.1272 q^{81} -12.7194 q^{82} +9.96792 q^{83} -35.2298 q^{84} +9.55537 q^{85} -26.9039 q^{86} -19.8439 q^{87} -43.0498 q^{88} -6.18819 q^{89} -22.7119 q^{90} +2.60821 q^{91} +50.5738 q^{92} +3.07788 q^{93} -29.8976 q^{94} +22.3679 q^{95} -64.9086 q^{96} -16.4821 q^{97} +0.548398 q^{98} +10.6305 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * q^99

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