LMFDB - Embedded newform 8009.2.a.a.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8009,2,Mod(1,8009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8009, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8009 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8009.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:	$63.9521869788$
Analytic rank:	$1$
Dimension:	$306$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.60787 q^{2} +1.96510 q^{3} +4.80101 q^{4} +3.32724 q^{5} -5.12474 q^{6} +0.944759 q^{7} -7.30469 q^{8} +0.861625 q^{9} +O(q^{10})$	\(q-2.60787 q^{2} +1.96510 q^{3} +4.80101 q^{4} +3.32724 q^{5} -5.12474 q^{6} +0.944759 q^{7} -7.30469 q^{8} +0.861625 q^{9} -8.67702 q^{10} -1.90952 q^{11} +9.43448 q^{12} -3.86745 q^{13} -2.46381 q^{14} +6.53836 q^{15} +9.44769 q^{16} -4.12359 q^{17} -2.24701 q^{18} +5.70365 q^{19} +15.9741 q^{20} +1.85655 q^{21} +4.97979 q^{22} +2.08903 q^{23} -14.3545 q^{24} +6.07050 q^{25} +10.0858 q^{26} -4.20212 q^{27} +4.53580 q^{28} +0.744721 q^{29} -17.0512 q^{30} -3.49976 q^{31} -10.0290 q^{32} -3.75240 q^{33} +10.7538 q^{34} +3.14344 q^{35} +4.13667 q^{36} -10.9937 q^{37} -14.8744 q^{38} -7.59993 q^{39} -24.3044 q^{40} -8.58131 q^{41} -4.84164 q^{42} +0.429043 q^{43} -9.16763 q^{44} +2.86683 q^{45} -5.44793 q^{46} +1.99756 q^{47} +18.5657 q^{48} -6.10743 q^{49} -15.8311 q^{50} -8.10328 q^{51} -18.5677 q^{52} -5.40615 q^{53} +10.9586 q^{54} -6.35343 q^{55} -6.90117 q^{56} +11.2082 q^{57} -1.94214 q^{58} +7.07409 q^{59} +31.3907 q^{60} -6.17781 q^{61} +9.12694 q^{62} +0.814028 q^{63} +7.25904 q^{64} -12.8679 q^{65} +9.78580 q^{66} +1.41848 q^{67} -19.7974 q^{68} +4.10516 q^{69} -8.19769 q^{70} -9.38250 q^{71} -6.29390 q^{72} -10.5558 q^{73} +28.6701 q^{74} +11.9292 q^{75} +27.3833 q^{76} -1.80404 q^{77} +19.8197 q^{78} -8.39725 q^{79} +31.4347 q^{80} -10.8425 q^{81} +22.3790 q^{82} +3.74904 q^{83} +8.91330 q^{84} -13.7202 q^{85} -1.11889 q^{86} +1.46345 q^{87} +13.9485 q^{88} +13.0631 q^{89} -7.47634 q^{90} -3.65381 q^{91} +10.0295 q^{92} -6.87739 q^{93} -5.20939 q^{94} +18.9774 q^{95} -19.7080 q^{96} +2.03327 q^{97} +15.9274 q^{98} -1.64529 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) $ 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9} - 61 q^{10} - 43 q^{11} - 50 q^{12} - 89 q^{13} - 40 q^{14} - 61 q^{15} + 151 q^{16} - 52 q^{17} - 57 q^{18} - 185 q^{19} - 66 q^{20} - 63 q^{21} - 55 q^{22} - 62 q^{23} - 131 q^{24} + 209 q^{25} - 57 q^{26} - 88 q^{27} - 182 q^{28} - 67 q^{29} - 68 q^{30} - 240 q^{31} - 64 q^{32} - 52 q^{33} - 128 q^{34} - 99 q^{35} + 106 q^{36} - 49 q^{37} - 45 q^{38} - 190 q^{39} - 158 q^{40} - 72 q^{41} - 36 q^{42} - 141 q^{43} - 80 q^{44} - 100 q^{45} - 91 q^{46} - 105 q^{47} - 85 q^{48} + 116 q^{49} - 51 q^{50} - 145 q^{51} - 237 q^{52} - 48 q^{53} - 156 q^{54} - 420 q^{55} - 116 q^{56} - 35 q^{57} - 43 q^{58} - 139 q^{59} - 73 q^{60} - 233 q^{61} - 58 q^{62} - 252 q^{63} - 3 q^{64} - 45 q^{65} - 127 q^{66} - 108 q^{67} - 85 q^{68} - 164 q^{69} - 56 q^{70} - 131 q^{71} - 117 q^{72} - 118 q^{73} - 47 q^{74} - 112 q^{75} - 389 q^{76} - 36 q^{77} + 9 q^{78} - 382 q^{79} - 119 q^{80} + 102 q^{81} - 131 q^{82} - 59 q^{83} - 144 q^{84} - 140 q^{85} - 38 q^{86} - 301 q^{87} - 131 q^{88} - 98 q^{89} - 138 q^{90} - 176 q^{91} - 97 q^{92} - 60 q^{93} - 342 q^{94} - 154 q^{95} - 243 q^{96} - 109 q^{97} - 21 q^{98} - 173 q^{99}+O(q^{100}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9 - 61 * q^10 - 43 * q^11 - 50 * q^12 - 89 * q^13 - 40 * q^14 - 61 * q^15 + 151 * q^16 - 52 * q^17 - 57 * q^18 - 185 * q^19 - 66 * q^20 - 63 * q^21 - 55 * q^22 - 62 * q^23 - 131 * q^24 + 209 * q^25 - 57 * q^26 - 88 * q^27 - 182 * q^28 - 67 * q^29 - 68 * q^30 - 240 * q^31 - 64 * q^32 - 52 * q^33 - 128 * q^34 - 99 * q^35 + 106 * q^36 - 49 * q^37 - 45 * q^38 - 190 * q^39 - 158 * q^40 - 72 * q^41 - 36 * q^42 - 141 * q^43 - 80 * q^44 - 100 * q^45 - 91 * q^46 - 105 * q^47 - 85 * q^48 + 116 * q^49 - 51 * q^50 - 145 * q^51 - 237 * q^52 - 48 * q^53 - 156 * q^54 - 420 * q^55 - 116 * q^56 - 35 * q^57 - 43 * q^58 - 139 * q^59 - 73 * q^60 - 233 * q^61 - 58 * q^62 - 252 * q^63 - 3 * q^64 - 45 * q^65 - 127 * q^66 - 108 * q^67 - 85 * q^68 - 164 * q^69 - 56 * q^70 - 131 * q^71 - 117 * q^72 - 118 * q^73 - 47 * q^74 - 112 * q^75 - 389 * q^76 - 36 * q^77 + 9 * q^78 - 382 * q^79 - 119 * q^80 + 102 * q^81 - 131 * q^82 - 59 * q^83 - 144 * q^84 - 140 * q^85 - 38 * q^86 - 301 * q^87 - 131 * q^88 - 98 * q^89 - 138 * q^90 - 176 * q^91 - 97 * q^92 - 60 * q^93 - 342 * q^94 - 154 * q^95 - 243 * q^96 - 109 * q^97 - 21 * q^98 - 173 * 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 $

$2$	−2.60787	−1.84405	−0.922023	−	0.387135i	$-0.873465\pi$
$2$	−2.60787	−1.84405	−0.922023	+	0.387135i	$0.873465\pi$
$3$	1.96510	1.13455	0.567276	−	0.823528i	$-0.307997\pi$
$3$	1.96510	1.13455	0.567276	+	0.823528i	$0.307997\pi$
$4$	4.80101	2.40051
$5$	3.32724	1.48799	0.743993	−	0.668188i	$-0.232930\pi$
$5$	3.32724	1.48799	0.743993	+	0.668188i	$0.232930\pi$
$6$	−5.12474	−2.09217
$7$	0.944759	0.357085	0.178543	−	0.983932i	$-0.442862\pi$
$7$	0.944759	0.357085	0.178543	+	0.983932i	$0.442862\pi$
$8$	−7.30469	−2.58260
$9$	0.861625	0.287208
$10$	−8.67702	−2.74391
$11$	−1.90952	−0.575742	−0.287871	−	0.957669i	$-0.592947\pi$
$11$	−1.90952	−0.575742	−0.287871	+	0.957669i	$0.592947\pi$
$12$	9.43448	2.72350
$13$	−3.86745	−1.07264	−0.536319	−	0.844016i	$-0.680185\pi$
$13$	−3.86745	−1.07264	−0.536319	+	0.844016i	$0.680185\pi$
$14$	−2.46381	−0.658482
$15$	6.53836	1.68820
$16$	9.44769	2.36192
$17$	−4.12359	−1.00012	−0.500059	−	0.865991i	$-0.666688\pi$
$17$	−4.12359	−1.00012	−0.500059	+	0.865991i	$0.666688\pi$
$18$	−2.24701	−0.529626
$19$	5.70365	1.30851	0.654253	−	0.756276i	$-0.272983\pi$
$19$	5.70365	1.30851	0.654253	+	0.756276i	$0.272983\pi$
$20$	15.9741	3.57192
$21$	1.85655	0.405132
$22$	4.97979	1.06170
$23$	2.08903	0.435593	0.217797	−	0.975994i	$-0.430113\pi$
$23$	2.08903	0.435593	0.217797	+	0.975994i	$0.430113\pi$
$24$	−14.3545	−2.93009
$25$	6.07050	1.21410
$26$	10.0858	1.97799
$27$	−4.20212	−0.808699
$28$	4.53580	0.857185
$29$	0.744721	0.138291	0.0691456	−	0.997607i	$-0.477973\pi$
$29$	0.744721	0.138291	0.0691456	+	0.997607i	$0.477973\pi$
$30$	−17.0512	−3.11311
$31$	−3.49976	−0.628576	−0.314288	−	0.949328i	$-0.601766\pi$
$31$	−3.49976	−0.628576	−0.314288	+	0.949328i	$0.601766\pi$
$32$	−10.0290	−1.77290
$33$	−3.75240	−0.653209
$34$	10.7538	1.84426
$35$	3.14344	0.531338
$36$	4.13667	0.689445
$37$	−10.9937	−1.80734	−0.903672	−	0.428224i	$-0.859139\pi$
$37$	−10.9937	−1.80734	−0.903672	+	0.428224i	$0.859139\pi$
$38$	−14.8744	−2.41295
$39$	−7.59993	−1.21696
$40$	−24.3044	−3.84287
$41$	−8.58131	−1.34018	−0.670088	−	0.742282i	$-0.733744\pi$
$41$	−8.58131	−1.34018	−0.670088	+	0.742282i	$0.733744\pi$
$42$	−4.84164	−0.747082
$43$	0.429043	0.0654285	0.0327143	−	0.999465i	$-0.489585\pi$
$43$	0.429043	0.0654285	0.0327143	+	0.999465i	$0.489585\pi$
$44$	−9.16763	−1.38207
$45$	2.86683	0.427362
$46$	−5.44793	−0.803254
$47$	1.99756	0.291374	0.145687	−	0.989331i	$-0.453461\pi$
$47$	1.99756	0.291374	0.145687	+	0.989331i	$0.453461\pi$
$48$	18.5657	2.67972
$49$	−6.10743	−0.872490
$50$	−15.8311	−2.23886
$51$	−8.10328	−1.13469
$52$	−18.5677	−2.57487
$53$	−5.40615	−0.742592	−0.371296	−	0.928515i	$-0.621087\pi$
$53$	−5.40615	−0.742592	−0.371296	+	0.928515i	$0.621087\pi$
$54$	10.9586	1.49128
$55$	−6.35343	−0.856696
$56$	−6.90117	−0.922207
$57$	11.2082	1.48457
$58$	−1.94214	−0.255015
$59$	7.07409	0.920968	0.460484	−	0.887668i	$-0.347676\pi$
$59$	7.07409	0.920968	0.460484	+	0.887668i	$0.347676\pi$
$60$	31.3907	4.05253
$61$	−6.17781	−0.790988	−0.395494	−	0.918469i	$-0.629427\pi$
$61$	−6.17781	−0.790988	−0.395494	+	0.918469i	$0.629427\pi$
$62$	9.12694	1.15912
$63$	0.814028	0.102558
$64$	7.25904	0.907380
$65$	−12.8679	−1.59607
$66$	9.78580	1.20455
$67$	1.41848	0.173294	0.0866472	−	0.996239i	$-0.472385\pi$
$67$	1.41848	0.173294	0.0866472	+	0.996239i	$0.472385\pi$
$68$	−19.7974	−2.40079
$69$	4.10516	0.494203
$70$	−8.19769	−0.979811
$71$	−9.38250	−1.11350	−0.556749	−	0.830681i	$-0.687952\pi$
$71$	−9.38250	−1.11350	−0.556749	+	0.830681i	$0.687952\pi$
$72$	−6.29390	−0.741744
$73$	−10.5558	−1.23546	−0.617729	−	0.786391i	$-0.711947\pi$
$73$	−10.5558	−1.23546	−0.617729	+	0.786391i	$0.711947\pi$
$74$	28.6701	3.33283
$75$	11.9292	1.37746
$76$	27.3833	3.14108
$77$	−1.80404	−0.205589
$78$	19.8197	2.24414
$79$	−8.39725	−0.944764	−0.472382	−	0.881394i	$-0.656606\pi$
$79$	−8.39725	−0.944764	−0.472382	+	0.881394i	$0.656606\pi$
$80$	31.4347	3.51451
$81$	−10.8425	−1.20472
$82$	22.3790	2.47135
$83$	3.74904	0.411511	0.205756	−	0.978603i	$-0.434035\pi$
$83$	3.74904	0.411511	0.205756	+	0.978603i	$0.434035\pi$
$84$	8.91330	0.972521
$85$	−13.7202	−1.48816
$86$	−1.11889	−0.120653
$87$	1.46345	0.156899
$88$	13.9485	1.48691
$89$	13.0631	1.38469	0.692345	−	0.721567i	$-0.256578\pi$
$89$	13.0631	1.38469	0.692345	+	0.721567i	$0.256578\pi$
$90$	−7.47634	−0.788075
$91$	−3.65381	−0.383023
$92$	10.0295	1.04564
$93$	−6.87739	−0.713152
$94$	−5.20939	−0.537307
$95$	18.9774	1.94704
$96$	−19.7080	−2.01144
$97$	2.03327	0.206447	0.103223	−	0.994658i	$-0.467084\pi$
$97$	2.03327	0.206447	0.103223	+	0.994658i	$0.467084\pi$
$98$	15.9274	1.60891
$99$	−1.64529	−0.165358
$100$	29.1446	2.91446
$101$	−13.1224	−1.30572	−0.652862	−	0.757477i	$-0.726432\pi$
$101$	−13.1224	−1.30572	−0.652862	+	0.757477i	$0.726432\pi$
$102$	21.1323	2.09241
$103$	10.5901	1.04347	0.521736	−	0.853107i	$-0.325285\pi$
$103$	10.5901	1.04347	0.521736	+	0.853107i	$0.325285\pi$
$104$	28.2505	2.77019
$105$	6.17717	0.602830
$106$	14.0986	1.36937
$107$	0.961493	0.0929511	0.0464755	−	0.998919i	$-0.485201\pi$
$107$	0.961493	0.0929511	0.0464755	+	0.998919i	$0.485201\pi$
$108$	−20.1744	−1.94129
$109$	−12.4896	−1.19629	−0.598144	−	0.801389i	$-0.704095\pi$
$109$	−12.4896	−1.19629	−0.598144	+	0.801389i	$0.704095\pi$
$110$	16.5689	1.57979
$111$	−21.6036	−2.05053
$112$	8.92579	0.843408
$113$	19.7940	1.86207	0.931033	−	0.364934i	$-0.118908\pi$
$113$	19.7940	1.86207	0.931033	+	0.364934i	$0.118908\pi$
$114$	−29.2297	−2.73761
$115$	6.95070	0.648156
$116$	3.57541	0.331969
$117$	−3.33229	−0.308070
$118$	−18.4483	−1.69831
$119$	−3.89580	−0.357127
$120$	−47.7607	−4.35993
$121$	−7.35373	−0.668521
$122$	16.1110	1.45862
$123$	−16.8631	−1.52050
$124$	−16.8024	−1.50890
$125$	3.56181	0.318578
$126$	−2.12288	−0.189121
$127$	3.02451	0.268382	0.134191	−	0.990956i	$-0.457156\pi$
$127$	3.02451	0.268382	0.134191	+	0.990956i	$0.457156\pi$
$128$	1.12736	0.0996454
$129$	0.843114	0.0742320
$130$	33.5579	2.94322
$131$	6.93965	0.606320	0.303160	−	0.952940i	$-0.401958\pi$
$131$	6.93965	0.606320	0.303160	+	0.952940i	$0.401958\pi$
$132$	−18.0153	−1.56803
$133$	5.38857	0.467248
$134$	−3.69921	−0.319563
$135$	−13.9815	−1.20333
$136$	30.1215	2.58290
$137$	−0.767340	−0.0655583	−0.0327791	−	0.999463i	$-0.510436\pi$
$137$	−0.767340	−0.0655583	−0.0327791	+	0.999463i	$0.510436\pi$
$138$	−10.7057	−0.911334
$139$	8.43813	0.715713	0.357857	−	0.933777i	$-0.383508\pi$
$139$	8.43813	0.715713	0.357857	+	0.933777i	$0.383508\pi$
$140$	15.0917	1.27548
$141$	3.92541	0.330579
$142$	24.4684	2.05334
$143$	7.38498	0.617563
$144$	8.14037	0.678364
$145$	2.47786	0.205775
$146$	27.5281	2.27824
$147$	−12.0017	−0.989886
$148$	−52.7806	−4.33854
$149$	−18.9250	−1.55040	−0.775199	−	0.631717i	$-0.782350\pi$
$149$	−18.9250	−1.55040	−0.775199	+	0.631717i	$0.782350\pi$
$150$	−31.1097	−2.54010
$151$	−13.3076	−1.08296	−0.541478	−	0.840715i	$-0.682135\pi$
$151$	−13.3076	−1.08296	−0.541478	+	0.840715i	$0.682135\pi$
$152$	−41.6634	−3.37934
$153$	−3.55299	−0.287242
$154$	4.70470	0.379116
$155$	−11.6445	−0.935311
$156$	−36.4874	−2.92133
$157$	−14.3346	−1.14403	−0.572014	−	0.820244i	$-0.693838\pi$
$157$	−14.3346	−1.14403	−0.572014	+	0.820244i	$0.693838\pi$
$158$	21.8990	1.74219
$159$	−10.6236	−0.842510
$160$	−33.3689	−2.63804
$161$	1.97363	0.155544
$162$	28.2758	2.22156
$163$	9.74044	0.762930	0.381465	−	0.924383i	$-0.375420\pi$
$163$	9.74044	0.762930	0.381465	+	0.924383i	$0.375420\pi$
$164$	−41.1990	−3.21710
$165$	−12.4851	−0.971966
$166$	−9.77704	−0.758846
$167$	−15.6128	−1.20815	−0.604077	−	0.796926i	$-0.706458\pi$
$167$	−15.6128	−1.20815	−0.604077	+	0.796926i	$0.706458\pi$
$168$	−13.5615	−1.04629
$169$	1.95716	0.150551
$170$	35.7805	2.74424
$171$	4.91441	0.375814
$172$	2.05984	0.157062
$173$	−11.2742	−0.857163	−0.428582	−	0.903503i	$-0.640987\pi$
$173$	−11.2742	−0.857163	−0.428582	+	0.903503i	$0.640987\pi$
$174$	−3.81650	−0.289328
$175$	5.73516	0.433537
$176$	−18.0406	−1.35986
$177$	13.9013	1.04489
$178$	−34.0670	−2.55343
$179$	26.3581	1.97010	0.985048	−	0.172282i	$-0.0551141\pi$
$179$	26.3581	1.97010	0.985048	+	0.172282i	$0.0551141\pi$
$180$	13.7637	1.02588
$181$	−12.2287	−0.908955	−0.454478	−	0.890758i	$-0.650174\pi$
$181$	−12.2287	−0.908955	−0.454478	+	0.890758i	$0.650174\pi$
$182$	9.52867	0.706312
$183$	−12.1400	−0.897417
$184$	−15.2597	−1.12496
$185$	−36.5785	−2.68930
$186$	17.9354	1.31508
$187$	7.87408	0.575810
$188$	9.59031	0.699445
$189$	−3.96999	−0.288775
$190$	−49.4906	−3.59043
$191$	8.31312	0.601516	0.300758	−	0.953700i	$-0.402760\pi$
$191$	8.31312	0.601516	0.300758	+	0.953700i	$0.402760\pi$
$192$	14.2648	1.02947
$193$	18.7876	1.35236	0.676180	−	0.736736i	$-0.263634\pi$
$193$	18.7876	1.35236	0.676180	+	0.736736i	$0.263634\pi$
$194$	−5.30250	−0.380698
$195$	−25.2868	−1.81082
$196$	−29.3218	−2.09442
$197$	10.4207	0.742443	0.371221	−	0.928544i	$-0.378939\pi$
$197$	10.4207	0.742443	0.371221	+	0.928544i	$0.378939\pi$
$198$	4.29071	0.304928
$199$	2.23604	0.158509	0.0792545	−	0.996854i	$-0.474746\pi$
$199$	2.23604	0.158509	0.0792545	+	0.996854i	$0.474746\pi$
$200$	−44.3431	−3.13553
$201$	2.78745	0.196612
$202$	34.2215	2.40781
$203$	0.703582	0.0493818
$204$	−38.9039	−2.72382
$205$	−28.5520	−1.99416
$206$	−27.6176	−1.92421
$207$	1.79996	0.125106
$208$	−36.5385	−2.53349
$209$	−10.8912	−0.753362
$210$	−16.1093	−1.11165
$211$	6.76775	0.465911	0.232956	−	0.972487i	$-0.425160\pi$
$211$	6.76775	0.465911	0.232956	+	0.972487i	$0.425160\pi$
$212$	−25.9550	−1.78260
$213$	−18.4376	−1.26332
$214$	−2.50745	−0.171406
$215$	1.42753	0.0973567
$216$	30.6952	2.08854
$217$	−3.30643	−0.224455
$218$	32.5713	2.20601
$219$	−20.7431	−1.40169
$220$	−30.5029	−2.05650
$221$	15.9478	1.07276
$222$	56.3396	3.78127
$223$	10.5504	0.706507	0.353254	−	0.935528i	$-0.385075\pi$
$223$	10.5504	0.706507	0.353254	+	0.935528i	$0.385075\pi$
$224$	−9.47500	−0.633075
$225$	5.23050	0.348700
$226$	−51.6204	−3.43374
$227$	18.0481	1.19789	0.598947	−	0.800789i	$-0.295586\pi$
$227$	18.0481	1.19789	0.598947	+	0.800789i	$0.295586\pi$
$228$	53.8109	3.56372
$229$	28.0116	1.85106	0.925530	−	0.378674i	$-0.123620\pi$
$229$	28.0116	1.85106	0.925530	+	0.378674i	$0.123620\pi$
$230$	−18.1266	−1.19523
$231$	−3.54512	−0.233251
$232$	−5.43996	−0.357151
$233$	−9.05336	−0.593105	−0.296553	−	0.955017i	$-0.595837\pi$
$233$	−9.05336	−0.593105	−0.296553	+	0.955017i	$0.595837\pi$
$234$	8.69020	0.568096
$235$	6.64635	0.433560
$236$	33.9628	2.21079
$237$	−16.5014	−1.07188
$238$	10.1598	0.658559
$239$	−12.4797	−0.807245	−0.403623	−	0.914926i	$-0.632249\pi$
$239$	−12.4797	−0.807245	−0.403623	+	0.914926i	$0.632249\pi$
$240$	61.7724	3.98739
$241$	−6.34996	−0.409037	−0.204518	−	0.978863i	$-0.565563\pi$
$241$	−6.34996	−0.409037	−0.204518	+	0.978863i	$0.565563\pi$
$242$	19.1776	1.23278
$243$	−8.70020	−0.558118
$244$	−29.6597	−1.89877
$245$	−20.3209	−1.29825
$246$	43.9770	2.80387
$247$	−22.0586	−1.40355
$248$	25.5647	1.62336
$249$	7.36726	0.466881
$250$	−9.28876	−0.587473
$251$	−30.7498	−1.94091	−0.970454	−	0.241287i	$-0.922431\pi$
$251$	−30.7498	−1.94091	−0.970454	+	0.241287i	$0.922431\pi$
$252$	3.90816	0.246191
$253$	−3.98905	−0.250789
$254$	−7.88754	−0.494908
$255$	−26.9615	−1.68840
$256$	−17.4581	−1.09113
$257$	−22.2156	−1.38577	−0.692886	−	0.721047i	$-0.743661\pi$
$257$	−22.2156	−1.38577	−0.692886	+	0.721047i	$0.743661\pi$
$258$	−2.19874	−0.136887
$259$	−10.3863	−0.645376
$260$	−61.7790	−3.83137
$261$	0.641670	0.0397184
$262$	−18.0977	−1.11808
$263$	5.79734	0.357479	0.178740	−	0.983896i	$-0.442798\pi$
$263$	5.79734	0.357479	0.178740	+	0.983896i	$0.442798\pi$
$264$	27.4101	1.68698
$265$	−17.9875	−1.10497
$266$	−14.0527	−0.861627
$267$	25.6704	1.57100
$268$	6.81012	0.415994
$269$	10.3778	0.632748	0.316374	−	0.948635i	$-0.397535\pi$
$269$	10.3778	0.632748	0.316374	+	0.948635i	$0.397535\pi$
$270$	36.4619	2.21900
$271$	−26.4106	−1.60433	−0.802166	−	0.597101i	$-0.796319\pi$
$271$	−26.4106	−1.60433	−0.802166	+	0.597101i	$0.796319\pi$
$272$	−38.9584	−2.36220
$273$	−7.18010	−0.434560
$274$	2.00113	0.120892
$275$	−11.5917	−0.699009
$276$	19.7089	1.18634
$277$	3.99029	0.239753	0.119877	−	0.992789i	$-0.461750\pi$
$277$	3.99029	0.239753	0.119877	+	0.992789i	$0.461750\pi$
$278$	−22.0056	−1.31981
$279$	−3.01548	−0.180532
$280$	−22.9618	−1.37223
$281$	−17.3912	−1.03747	−0.518737	−	0.854934i	$-0.673598\pi$
$281$	−17.3912	−1.03747	−0.518737	+	0.854934i	$0.673598\pi$
$282$	−10.2370	−0.609603
$283$	30.7930	1.83045	0.915226	−	0.402940i	$-0.132012\pi$
$283$	30.7930	1.83045	0.915226	+	0.402940i	$0.132012\pi$
$284$	−45.0455	−2.67296
$285$	37.2925	2.20902
$286$	−19.2591	−1.13881
$287$	−8.10727	−0.478557
$288$	−8.64125	−0.509191
$289$	0.00400521	0.000235600 0
$290$	−6.46196	−0.379459
$291$	3.99558	0.234225
$292$	−50.6783	−2.96573
$293$	26.4198	1.54346	0.771730	−	0.635950i	$-0.219392\pi$
$293$	26.4198	1.54346	0.771730	+	0.635950i	$0.219392\pi$
$294$	31.2990	1.82539
$295$	23.5372	1.37039
$296$	80.3052	4.66764
$297$	8.02404	0.465602
$298$	49.3541	2.85900
$299$	−8.07922	−0.467234
$300$	57.2720	3.30660
$301$	0.405342	0.0233636
$302$	34.7045	1.99702
$303$	−25.7868	−1.48141
$304$	53.8863	3.09059
$305$	−20.5550	−1.17698
$306$	9.26575	0.529688
$307$	−24.5520	−1.40126	−0.700630	−	0.713525i	$-0.747097\pi$
$307$	−24.5520	−1.40126	−0.700630	+	0.713525i	$0.747097\pi$
$308$	−8.66120	−0.493518
$309$	20.8106	1.18387
$310$	30.3675	1.72476
$311$	31.5962	1.79165	0.895827	−	0.444402i	$-0.146584\pi$
$311$	31.5962	1.79165	0.895827	+	0.444402i	$0.146584\pi$
$312$	55.5151	3.14293
$313$	−0.633574	−0.0358117	−0.0179059	−	0.999840i	$-0.505700\pi$
$313$	−0.633574	−0.0358117	−0.0179059	+	0.999840i	$0.505700\pi$
$314$	37.3829	2.10964
$315$	2.70846	0.152605
$316$	−40.3153	−2.26791
$317$	−0.536936	−0.0301573	−0.0150787	−	0.999886i	$-0.504800\pi$
$317$	−0.536936	−0.0301573	−0.0150787	+	0.999886i	$0.504800\pi$
$318$	27.7051	1.55363
$319$	−1.42206	−0.0796201
$320$	24.1525	1.35017
$321$	1.88943	0.105458
$322$	−5.14698	−0.286830
$323$	−23.5195	−1.30866
$324$	−52.0549	−2.89194
$325$	−23.4774	−1.30229
$326$	−25.4018	−1.40688
$327$	−24.5434	−1.35725
$328$	62.6838	3.46113
$329$	1.88721	0.104045
$330$	32.5597	1.79235
$331$	−13.7948	−0.758228	−0.379114	−	0.925350i	$-0.623771\pi$
$331$	−13.7948	−0.758228	−0.379114	+	0.925350i	$0.623771\pi$
$332$	17.9992	0.987835
$333$	−9.47241	−0.519085
$334$	40.7162	2.22789
$335$	4.71961	0.257860
$336$	17.5401	0.956890
$337$	8.51552	0.463870	0.231935	−	0.972731i	$-0.425494\pi$
$337$	8.51552	0.463870	0.231935	+	0.972731i	$0.425494\pi$
$338$	−5.10404	−0.277623
$339$	38.8973	2.11261
$340$	−65.8707	−3.57234
$341$	6.68287	0.361898
$342$	−12.8162	−0.693018
$343$	−12.3834	−0.668639
$344$	−3.13403	−0.168975
$345$	13.6588	0.735367
$346$	29.4018	1.58065
$347$	2.02832	0.108886	0.0544429	−	0.998517i	$-0.482662\pi$
$347$	2.02832	0.108886	0.0544429	+	0.998517i	$0.482662\pi$
$348$	7.02605	0.376636
$349$	−20.7939	−1.11307	−0.556536	−	0.830824i	$-0.687870\pi$
$349$	−20.7939	−1.11307	−0.556536	+	0.830824i	$0.687870\pi$
$350$	−14.9566	−0.799463
$351$	16.2515	0.867441
$352$	19.1506	1.02073
$353$	14.5046	0.772001	0.386000	−	0.922499i	$-0.373856\pi$
$353$	14.5046	0.772001	0.386000	+	0.922499i	$0.373856\pi$
$354$	−36.2529	−1.92682
$355$	−31.2178	−1.65687
$356$	62.7163	3.32396
$357$	−7.65564	−0.405179
$358$	−68.7386	−3.63295
$359$	−17.0089	−0.897693	−0.448847	−	0.893609i	$-0.648165\pi$
$359$	−17.0089	−0.897693	−0.448847	+	0.893609i	$0.648165\pi$
$360$	−20.9413	−1.10370
$361$	13.5316	0.712189
$362$	31.8910	1.67616
$363$	−14.4508	−0.758472
$364$	−17.5420	−0.919449
$365$	−35.1215	−1.83834
$366$	31.6597	1.65488
$367$	18.9385	0.988581	0.494291	−	0.869297i	$-0.335428\pi$
$367$	18.9385	0.988581	0.494291	+	0.869297i	$0.335428\pi$
$368$	19.7365	1.02884
$369$	−7.39387	−0.384910
$370$	95.3921	4.95920
$371$	−5.10751	−0.265169
$372$	−33.0184	−1.71193
$373$	14.0456	0.727254	0.363627	−	0.931545i	$-0.381538\pi$
$373$	14.0456	0.727254	0.363627	+	0.931545i	$0.381538\pi$
$374$	−20.5346	−1.06182
$375$	6.99932	0.361444
$376$	−14.5916	−0.752502
$377$	−2.88017	−0.148336
$378$	10.3532	0.532514
$379$	17.3921	0.893370	0.446685	−	0.894691i	$-0.352604\pi$
$379$	17.3921	0.893370	0.446685	+	0.894691i	$0.352604\pi$
$380$	91.1106	4.67388
$381$	5.94347	0.304493
$382$	−21.6796	−1.10922
$383$	31.5466	1.61196	0.805979	−	0.591944i	$-0.201639\pi$
$383$	31.5466	1.61196	0.805979	+	0.591944i	$0.201639\pi$
$384$	2.21538	0.113053
$385$	−6.00246	−0.305913
$386$	−48.9957	−2.49382
$387$	0.369675	0.0187916
$388$	9.76174	0.495577
$389$	−10.3208	−0.523284	−0.261642	−	0.965165i	$-0.584264\pi$
$389$	−10.3208	−0.523284	−0.261642	+	0.965165i	$0.584264\pi$
$390$	65.9447	3.33924
$391$	−8.61431	−0.435645
$392$	44.6129	2.25329
$393$	13.6371	0.687902
$394$	−27.1758	−1.36910
$395$	−27.9396	−1.40579
$396$	−7.89906	−0.396943
$397$	−38.4373	−1.92911	−0.964556	−	0.263878i	$-0.914998\pi$
$397$	−38.4373	−1.92911	−0.964556	+	0.263878i	$0.914998\pi$
$398$	−5.83132	−0.292298
$399$	10.5891	0.530117
$400$	57.3522	2.86761
$401$	−3.58204	−0.178878	−0.0894392	−	0.995992i	$-0.528507\pi$
$401$	−3.58204	−0.178878	−0.0894392	+	0.995992i	$0.528507\pi$
$402$	−7.26932	−0.362561
$403$	13.5351	0.674234
$404$	−63.0006	−3.13440
$405$	−36.0755	−1.79261
$406$	−1.83485	−0.0910622
$407$	20.9926	1.04056
$408$	59.1919	2.93044
$409$	2.47823	0.122541	0.0612703	−	0.998121i	$-0.480485\pi$
$409$	2.47823	0.122541	0.0612703	+	0.998121i	$0.480485\pi$
$410$	74.4602	3.67733
$411$	−1.50790	−0.0743793
$412$	50.8431	2.50486
$413$	6.68331	0.328864
$414$	−4.69408	−0.230701
$415$	12.4740	0.612323
$416$	38.7867	1.90168
$417$	16.5818	0.812014
$418$	28.4030	1.38923
$419$	5.75145	0.280977	0.140488	−	0.990082i	$-0.455133\pi$
$419$	5.75145	0.280977	0.140488	+	0.990082i	$0.455133\pi$
$420$	29.6567	1.44710
$421$	−38.2793	−1.86562	−0.932810	−	0.360369i	$-0.882651\pi$
$421$	−38.2793	−1.86562	−0.932810	+	0.360369i	$0.882651\pi$
$422$	−17.6495	−0.859162
$423$	1.72115	0.0836851
$424$	39.4903	1.91782
$425$	−25.0323	−1.21424
$426$	48.0829	2.32962
$427$	−5.83654	−0.282450
$428$	4.61614	0.223130
$429$	14.5122	0.700657
$430$	−3.72282	−0.179530
$431$	−7.83970	−0.377625	−0.188812	−	0.982013i	$-0.560464\pi$
$431$	−7.83970	−0.377625	−0.188812	+	0.982013i	$0.560464\pi$
$432$	−39.7004	−1.91008
$433$	−6.70863	−0.322396	−0.161198	−	0.986922i	$-0.551536\pi$
$433$	−6.70863	−0.322396	−0.161198	+	0.986922i	$0.551536\pi$
$434$	8.62275	0.413905
$435$	4.86925	0.233463
$436$	−59.9628	−2.87170
$437$	11.9151	0.569977
$438$	54.0955	2.58478
$439$	−11.1996	−0.534530	−0.267265	−	0.963623i	$-0.586120\pi$
$439$	−11.1996	−0.534530	−0.267265	+	0.963623i	$0.586120\pi$
$440$	46.4098	2.21250
$441$	−5.26232	−0.250587
$442$	−41.5898	−1.97823
$443$	−28.0047	−1.33055	−0.665273	−	0.746600i	$-0.731685\pi$
$443$	−28.0047	−1.33055	−0.665273	+	0.746600i	$0.731685\pi$
$444$	−103.719	−4.92230
$445$	43.4642	2.06040
$446$	−27.5141	−1.30283
$447$	−37.1896	−1.75901
$448$	6.85804	0.324012
$449$	20.8960	0.986142	0.493071	−	0.869989i	$-0.335874\pi$
$449$	20.8960	0.986142	0.493071	+	0.869989i	$0.335874\pi$
$450$	−13.6405	−0.643019
$451$	16.3862	0.771595
$452$	95.0314	4.46990
$453$	−26.1508	−1.22867
$454$	−47.0672	−2.20897
$455$	−12.1571	−0.569933
$456$	−81.8728	−3.83404
$457$	−14.7519	−0.690067	−0.345034	−	0.938590i	$-0.612132\pi$
$457$	−14.7519	−0.690067	−0.345034	+	0.938590i	$0.612132\pi$
$458$	−73.0508	−3.41344
$459$	17.3278	0.808794
$460$	33.3704	1.55590
$461$	10.4103	0.484857	0.242428	−	0.970169i	$-0.422056\pi$
$461$	10.4103	0.484857	0.242428	+	0.970169i	$0.422056\pi$
$462$	9.24522	0.430126
$463$	16.0522	0.746009	0.373004	−	0.927830i	$-0.378328\pi$
$463$	16.0522	0.746009	0.373004	+	0.927830i	$0.378328\pi$
$464$	7.03589	0.326633
$465$	−22.8827	−1.06116
$466$	23.6100	1.09371
$467$	0.329090	0.0152285	0.00761424	−	0.999971i	$-0.497576\pi$
$467$	0.329090	0.0152285	0.00761424	+	0.999971i	$0.497576\pi$
$468$	−15.9984	−0.739525
$469$	1.34012	0.0618809
$470$	−17.3329	−0.799505
$471$	−28.1690	−1.29796
$472$	−51.6740	−2.37849
$473$	−0.819267	−0.0376700
$474$	43.0337	1.97660
$475$	34.6240	1.58866
$476$	−18.7038	−0.857286
$477$	−4.65808	−0.213279
$478$	32.5455	1.48860
$479$	30.3024	1.38455	0.692276	−	0.721633i	$-0.256608\pi$
$479$	30.3024	1.38455	0.692276	+	0.721633i	$0.256608\pi$
$480$	−65.5733	−2.99300
$481$	42.5174	1.93863
$482$	16.5599	0.754283
$483$	3.87839	0.176473
$484$	−35.3053	−1.60479
$485$	6.76516	0.307190
$486$	22.6890	1.02920
$487$	−1.06308	−0.0481728	−0.0240864	−	0.999710i	$-0.507668\pi$
$487$	−1.06308	−0.0481728	−0.0240864	+	0.999710i	$0.507668\pi$
$488$	45.1270	2.04280
$489$	19.1409	0.865584
$490$	52.9943	2.39404
$491$	−37.3865	−1.68723	−0.843614	−	0.536951i	$-0.819576\pi$
$491$	−37.3865	−1.68723	−0.843614	+	0.536951i	$0.819576\pi$
$492$	−80.9602	−3.64997
$493$	−3.07093	−0.138308
$494$	57.5260	2.58822
$495$	−5.47427	−0.246050
$496$	−33.0647	−1.48465
$497$	−8.86420	−0.397614
$498$	−19.2129	−0.860950
$499$	4.01973	0.179948	0.0899738	−	0.995944i	$-0.471322\pi$
$499$	4.01973	0.179948	0.0899738	+	0.995944i	$0.471322\pi$
$500$	17.1003	0.764749
$501$	−30.6807	−1.37071
$502$	80.1915	3.57912
$503$	−5.98439	−0.266831	−0.133415	−	0.991060i	$-0.542594\pi$
$503$	−5.98439	−0.266831	−0.133415	+	0.991060i	$0.542594\pi$
$504$	−5.94622	−0.264866
$505$	−43.6612	−1.94290
$506$	10.4029	0.462467
$507$	3.84603	0.170808
$508$	14.5207	0.644252
$509$	27.3699	1.21315	0.606575	−	0.795026i	$-0.292543\pi$
$509$	27.3699	1.21315	0.606575	+	0.795026i	$0.292543\pi$
$510$	70.3123	3.11348
$511$	−9.97265	−0.441164
$512$	43.2738	1.91245
$513$	−23.9674	−1.05819
$514$	57.9356	2.55543
$515$	35.2357	1.55267
$516$	4.04780	0.178194
$517$	−3.81438	−0.167756
$518$	27.0863	1.19010
$519$	−22.1550	−0.972497
$520$	93.9961	4.12200
$521$	9.50482	0.416414	0.208207	−	0.978085i	$-0.433237\pi$
$521$	9.50482	0.416414	0.208207	+	0.978085i	$0.433237\pi$
$522$	−1.67340	−0.0732426
$523$	26.7089	1.16790	0.583950	−	0.811790i	$-0.301506\pi$
$523$	26.7089	1.16790	0.583950	+	0.811790i	$0.301506\pi$
$524$	33.3173	1.45548
$525$	11.2702	0.491871
$526$	−15.1187	−0.659209
$527$	14.4316	0.628650
$528$	−35.4515	−1.54283
$529$	−18.6359	−0.810259
$530$	46.9093	2.03761
$531$	6.09521	0.264510
$532$	25.8706	1.12163
$533$	33.1878	1.43752
$534$	−66.9452	−2.89700
$535$	3.19912	0.138310
$536$	−10.3615	−0.447550
$537$	51.7963	2.23518
$538$	−27.0641	−1.16682
$539$	11.6623	0.502329
$540$	−67.1252	−2.88861
$541$	9.50496	0.408650	0.204325	−	0.978903i	$-0.434500\pi$
$541$	9.50496	0.408650	0.204325	+	0.978903i	$0.434500\pi$
$542$	68.8756	2.95846
$543$	−24.0307	−1.03126
$544$	41.3556	1.77311
$545$	−41.5559	−1.78006
$546$	18.7248	0.801348
$547$	−6.18292	−0.264363	−0.132181	−	0.991226i	$-0.542198\pi$
$547$	−6.18292	−0.264363	−0.132181	+	0.991226i	$0.542198\pi$
$548$	−3.68401	−0.157373
$549$	−5.32296	−0.227178
$550$	30.2298	1.28900
$551$	4.24763	0.180955
$552$	−29.9869	−1.27633
$553$	−7.93337	−0.337361
$554$	−10.4062	−0.442116
$555$	−71.8804	−3.05115
$556$	40.5116	1.71807
$557$	−19.0472	−0.807055	−0.403528	−	0.914967i	$-0.632216\pi$
$557$	−19.0472	−0.807055	−0.403528	+	0.914967i	$0.632216\pi$
$558$	7.86400	0.332910
$559$	−1.65930	−0.0701811
$560$	29.6982	1.25498
$561$	15.4734	0.653286
$562$	45.3541	1.91315
$563$	6.08319	0.256376	0.128188	−	0.991750i	$-0.459084\pi$
$563$	6.08319	0.256376	0.128188	+	0.991750i	$0.459084\pi$
$564$	18.8459	0.793557
$565$	65.8595	2.77073
$566$	−80.3042	−3.37544
$567$	−10.2435	−0.430188
$568$	68.5363	2.87572
$569$	30.6971	1.28689	0.643444	−	0.765493i	$-0.277505\pi$
$569$	30.6971	1.28689	0.643444	+	0.765493i	$0.277505\pi$
$570$	−97.2541	−4.07353
$571$	−3.04483	−0.127422	−0.0637112	−	0.997968i	$-0.520294\pi$
$571$	−3.04483	−0.127422	−0.0637112	+	0.997968i	$0.520294\pi$
$572$	35.4554	1.48246
$573$	16.3361	0.682451
$574$	21.1427	0.882481
$575$	12.6815	0.528854
$576$	6.25457	0.260607
$577$	15.5304	0.646537	0.323269	−	0.946307i	$-0.395218\pi$
$577$	15.5304	0.646537	0.323269	+	0.946307i	$0.395218\pi$
$578$	−0.0104451	−0.000434458 0
$579$	36.9195	1.53432
$580$	11.8962	0.493965
$581$	3.54194	0.146945
$582$	−10.4200	−0.431921
$583$	10.3232	0.427542
$584$	77.1065	3.19069
$585$	−11.0873	−0.458404
$586$	−68.8995	−2.84621
$587$	8.36802	0.345385	0.172693	−	0.984976i	$-0.444753\pi$
$587$	8.36802	0.345385	0.172693	+	0.984976i	$0.444753\pi$
$588$	−57.6204	−2.37623
$589$	−19.9614	−0.822495
$590$	−61.3820	−2.52706
$591$	20.4777	0.842340
$592$	−103.865	−4.26881
$593$	32.3086	1.32675	0.663377	−	0.748285i	$-0.269123\pi$
$593$	32.3086	1.32675	0.663377	+	0.748285i	$0.269123\pi$
$594$	−20.9257	−0.858592
$595$	−12.9622	−0.531400
$596$	−90.8592	−3.72174
$597$	4.39405	0.179837
$598$	21.0696	0.861600
$599$	6.53388	0.266967	0.133483	−	0.991051i	$-0.457384\pi$
$599$	6.53388	0.266967	0.133483	+	0.991051i	$0.457384\pi$
$600$	−87.1388	−3.55742
$601$	19.2504	0.785239	0.392620	−	0.919701i	$-0.371569\pi$
$601$	19.2504	0.785239	0.392620	+	0.919701i	$0.371569\pi$
$602$	−1.05708	−0.0430835
$603$	1.22219	0.0497716
$604$	−63.8899	−2.59964
$605$	−24.4676	−0.994749
$606$	67.2487	2.73179
$607$	−33.2854	−1.35101	−0.675506	−	0.737355i	$-0.736075\pi$
$607$	−33.2854	−1.35101	−0.675506	+	0.737355i	$0.736075\pi$
$608$	−57.2020	−2.31985
$609$	1.38261	0.0560262
$610$	53.6050	2.17040
$611$	−7.72546	−0.312539
$612$	−17.0579	−0.689527
$613$	−26.5972	−1.07425	−0.537125	−	0.843503i	$-0.680489\pi$
$613$	−26.5972	−1.07425	−0.537125	+	0.843503i	$0.680489\pi$
$614$	64.0287	2.58399
$615$	−56.1077	−2.26248
$616$	13.1779	0.530954
$617$	−40.2072	−1.61868	−0.809341	−	0.587339i	$-0.800176\pi$
$617$	−40.2072	−1.61868	−0.809341	+	0.587339i	$0.800176\pi$
$618$	−54.2714	−2.18312
$619$	34.0122	1.36707	0.683533	−	0.729919i	$-0.260442\pi$
$619$	34.0122	1.36707	0.683533	+	0.729919i	$0.260442\pi$
$620$	−55.9055	−2.24522
$621$	−8.77837	−0.352264
$622$	−82.3989	−3.30389
$623$	12.3415	0.494452
$624$	−71.8018	−2.87437
$625$	−18.5015	−0.740061
$626$	1.65228	0.0660385
$627$	−21.4024	−0.854729
$628$	−68.8207	−2.74624
$629$	45.3333	1.80756
$630$	−7.06333	−0.281410
$631$	−27.1034	−1.07897	−0.539485	−	0.841995i	$-0.681381\pi$
$631$	−27.1034	−1.07897	−0.539485	+	0.841995i	$0.681381\pi$
$632$	61.3393	2.43994
$633$	13.2993	0.528601
$634$	1.40026	0.0556115
$635$	10.0633	0.399348
$636$	−51.0042	−2.02245
$637$	23.6202	0.935866
$638$	3.70856	0.146823
$639$	−8.08420	−0.319806
$640$	3.75099	0.148271
$641$	5.15698	0.203688	0.101844	−	0.994800i	$-0.467526\pi$
$641$	5.15698	0.203688	0.101844	+	0.994800i	$0.467526\pi$
$642$	−4.92740	−0.194469
$643$	−16.5782	−0.653782	−0.326891	−	0.945062i	$-0.606001\pi$
$643$	−16.5782	−0.653782	−0.326891	+	0.945062i	$0.606001\pi$
$644$	9.47543	0.373384
$645$	2.80524	0.110456
$646$	61.3359	2.41323
$647$	42.0632	1.65367	0.826837	−	0.562441i	$-0.190138\pi$
$647$	42.0632	1.65367	0.826837	+	0.562441i	$0.190138\pi$
$648$	79.2009	3.11131
$649$	−13.5081	−0.530240
$650$	61.2260	2.40148
$651$	−6.49747	−0.254656
$652$	46.7639	1.83142
$653$	−38.4746	−1.50563	−0.752813	−	0.658234i	$-0.771304\pi$
$653$	−38.4746	−1.50563	−0.752813	+	0.658234i	$0.771304\pi$
$654$	64.0060	2.50283
$655$	23.0899	0.902196
$656$	−81.0735	−3.16539
$657$	−9.09511	−0.354834
$658$	−4.92161	−0.191864
$659$	35.3101	1.37549	0.687744	−	0.725953i	$-0.258601\pi$
$659$	35.3101	1.37549	0.687744	+	0.725953i	$0.258601\pi$
$660$	−59.9413	−2.33321
$661$	−7.07623	−0.275233	−0.137617	−	0.990486i	$-0.543944\pi$
$661$	−7.07623	−0.275233	−0.137617	+	0.990486i	$0.543944\pi$
$662$	35.9750	1.39821
$663$	31.3390	1.21711
$664$	−27.3856	−1.06277
$665$	17.9290	0.695259
$666$	24.7028	0.957216
$667$	1.55575	0.0602387
$668$	−74.9572	−2.90018
$669$	20.7326	0.801569
$670$	−12.3081	−0.475505
$671$	11.7967	0.455405
$672$	−18.6193	−0.718257
$673$	8.97654	0.346020	0.173010	−	0.984920i	$-0.444651\pi$
$673$	8.97654	0.346020	0.173010	+	0.984920i	$0.444651\pi$
$674$	−22.2074	−0.855397
$675$	−25.5090	−0.981842
$676$	9.39637	0.361399
$677$	25.6286	0.984988	0.492494	−	0.870316i	$-0.336085\pi$
$677$	25.6286	0.984988	0.492494	+	0.870316i	$0.336085\pi$
$678$	−101.439	−3.89575
$679$	1.92095	0.0737191
$680$	100.222	3.84332
$681$	35.4664	1.35907
$682$	−17.4281	−0.667356
$683$	27.7938	1.06350	0.531751	−	0.846901i	$-0.321534\pi$
$683$	27.7938	1.06350	0.531751	+	0.846901i	$0.321534\pi$
$684$	23.5941	0.902144
$685$	−2.55312	−0.0975498
$686$	32.2943	1.23300
$687$	55.0457	2.10012
$688$	4.05347	0.154537
$689$	20.9080	0.796532
$690$	−35.6205	−1.35605
$691$	−4.67794	−0.177957	−0.0889785	−	0.996034i	$-0.528360\pi$
$691$	−4.67794	−0.177957	−0.0889785	+	0.996034i	$0.528360\pi$
$692$	−54.1277	−2.05763
$693$	−1.55440	−0.0590469
$694$	−5.28960	−0.200791
$695$	28.0757	1.06497
$696$	−10.6901	−0.405206
$697$	35.3858	1.34033
$698$	54.2279	2.05256
$699$	−17.7908	−0.672909
$700$	27.5346	1.04071
$701$	−7.22156	−0.272755	−0.136377	−	0.990657i	$-0.543546\pi$
$701$	−7.22156	−0.272755	−0.136377	+	0.990657i	$0.543546\pi$
$702$	−42.3819	−1.59960
$703$	−62.7039	−2.36492
$704$	−13.8613	−0.522417
$705$	13.0608	0.491897
$706$	−37.8261	−1.42361
$707$	−12.3975	−0.466254
$708$	66.7403	2.50825
$709$	−1.63805	−0.0615183	−0.0307591	−	0.999527i	$-0.509792\pi$
$709$	−1.63805	−0.0615183	−0.0307591	+	0.999527i	$0.509792\pi$
$710$	81.4121	3.05534
$711$	−7.23528	−0.271344
$712$	−95.4222	−3.57610
$713$	−7.31111	−0.273803
$714$	19.9650	0.747170
$715$	24.5716	0.918924
$716$	126.545	4.72923
$717$	−24.5239	−0.915862
$718$	44.3570	1.65539
$719$	2.56606	0.0956978	0.0478489	−	0.998855i	$-0.484763\pi$
$719$	2.56606	0.0956978	0.0478489	+	0.998855i	$0.484763\pi$
$720$	27.0849	1.00940
$721$	10.0051	0.372608
$722$	−35.2887	−1.31331
$723$	−12.4783	−0.464074
$724$	−58.7103	−2.18195
$725$	4.52083	0.167899
$726$	37.6860	1.39866
$727$	−27.4060	−1.01643	−0.508216	−	0.861230i	$-0.669695\pi$
$727$	−27.4060	−1.01643	−0.508216	+	0.861230i	$0.669695\pi$
$728$	26.6899	0.989194
$729$	15.4307	0.571506
$730$	91.5925	3.38999
$731$	−1.76920	−0.0654362
$732$	−58.2844	−2.15425
$733$	−5.69086	−0.210197	−0.105098	−	0.994462i	$-0.533516\pi$
$733$	−5.69086	−0.210197	−0.105098	+	0.994462i	$0.533516\pi$
$734$	−49.3892	−1.82299
$735$	−39.9326	−1.47294
$736$	−20.9509	−0.772262
$737$	−2.70861	−0.0997729
$738$	19.2823	0.709791
$739$	−4.40936	−0.162201	−0.0811005	−	0.996706i	$-0.525843\pi$
$739$	−4.40936	−0.162201	−0.0811005	+	0.996706i	$0.525843\pi$
$740$	−175.614	−6.45569
$741$	−43.3473	−1.59240
$742$	13.3197	0.488983
$743$	13.1360	0.481914	0.240957	−	0.970536i	$-0.422539\pi$
$743$	13.1360	0.481914	0.240957	+	0.970536i	$0.422539\pi$
$744$	50.2372	1.84178
$745$	−62.9680	−2.30697
$746$	−36.6292	−1.34109
$747$	3.23027	0.118189
$748$	37.8036	1.38224
$749$	0.908379	0.0331915
$750$	−18.2534	−0.666519
$751$	34.6668	1.26501	0.632505	−	0.774556i	$-0.282027\pi$
$751$	34.6668	1.26501	0.632505	+	0.774556i	$0.282027\pi$
$752$	18.8723	0.688203
$753$	−60.4264	−2.20206
$754$	7.51113	0.273539
$755$	−44.2775	−1.61142
$756$	−19.0600	−0.693205
$757$	−28.8824	−1.04975	−0.524874	−	0.851180i	$-0.675888\pi$
$757$	−28.8824	−1.04975	−0.524874	+	0.851180i	$0.675888\pi$
$758$	−45.3563	−1.64742
$759$	−7.83889	−0.284534
$760$	−138.624	−5.02842
$761$	−1.35095	−0.0489718	−0.0244859	−	0.999700i	$-0.507795\pi$
$761$	−1.35095	−0.0489718	−0.0244859	+	0.999700i	$0.507795\pi$
$762$	−15.4998	−0.561499
$763$	−11.7997	−0.427177
$764$	39.9114	1.44394
$765$	−11.8216	−0.427412
$766$	−82.2697	−2.97253
$767$	−27.3587	−0.987864
$768$	−34.3069	−1.23794
$769$	19.5255	0.704108	0.352054	−	0.935980i	$-0.385483\pi$
$769$	19.5255	0.704108	0.352054	+	0.935980i	$0.385483\pi$
$770$	15.6537	0.564118
$771$	−43.6560	−1.57223
$772$	90.1995	3.24635
$773$	25.8923	0.931280	0.465640	−	0.884974i	$-0.345824\pi$
$773$	25.8923	0.931280	0.465640	+	0.884974i	$0.345824\pi$
$774$	−0.964065	−0.0346526
$775$	−21.2453	−0.763154
$776$	−14.8524	−0.533169
$777$	−20.4102	−0.732213
$778$	26.9153	0.964960
$779$	−48.9448	−1.75363
$780$	−121.402	−4.34689
$781$	17.9161	0.641088
$782$	22.4651	0.803349
$783$	−3.12941	−0.111836
$784$	−57.7011	−2.06075
$785$	−47.6947	−1.70230
$786$	−35.5639	−1.26852
$787$	−24.0951	−0.858896	−0.429448	−	0.903092i	$-0.641292\pi$
$787$	−24.0951	−0.858896	−0.429448	+	0.903092i	$0.641292\pi$
$788$	50.0298	1.78224
$789$	11.3924	0.405579
$790$	72.8630	2.59235
$791$	18.7006	0.664916
$792$	12.0183	0.427053
$793$	23.8924	0.848443
$794$	100.240	3.55737
$795$	−35.3474	−1.25364
$796$	10.7353	0.380502
$797$	52.5938	1.86297	0.931483	−	0.363784i	$-0.118515\pi$
$797$	52.5938	1.86297	0.931483	+	0.363784i	$0.118515\pi$
$798$	−27.6150	−0.977561
$799$	−8.23712	−0.291408
$800$	−60.8812	−2.15247
$801$	11.2555	0.397695
$802$	9.34150	0.329860
$803$	20.1564	0.711306
$804$	13.3826	0.471967
$805$	6.56674	0.231447
$806$	−35.2980	−1.24332
$807$	20.3935	0.717886
$808$	95.8547	3.37216
$809$	−39.7432	−1.39730	−0.698648	−	0.715465i	$-0.746215\pi$
$809$	−39.7432	−1.39730	−0.698648	+	0.715465i	$0.746215\pi$
$810$	94.0804	3.30565
$811$	29.7734	1.04548	0.522742	−	0.852491i	$-0.324909\pi$
$811$	29.7734	1.04548	0.522742	+	0.852491i	$0.324909\pi$
$812$	3.37790	0.118541
$813$	−51.8996	−1.82020
$814$	−54.7461	−1.91885
$815$	32.4087	1.13523
$816$	−76.5572	−2.68004
$817$	2.44711	0.0856136
$818$	−6.46292	−0.225971
$819$	−3.14821	−0.110007
$820$	−137.079	−4.78700
$821$	−24.4837	−0.854489	−0.427244	−	0.904136i	$-0.640516\pi$
$821$	−24.4837	−0.854489	−0.427244	+	0.904136i	$0.640516\pi$
$822$	3.93242	0.137159
$823$	−12.9075	−0.449928	−0.224964	−	0.974367i	$-0.572226\pi$
$823$	−12.9075	−0.449928	−0.224964	+	0.974367i	$0.572226\pi$
$824$	−77.3572	−2.69487
$825$	−22.7790	−0.793062
$826$	−17.4292	−0.606440
$827$	−27.0404	−0.940288	−0.470144	−	0.882590i	$-0.655798\pi$
$827$	−27.0404	−0.940288	−0.470144	+	0.882590i	$0.655798\pi$
$828$	8.64164	0.300318
$829$	−38.7573	−1.34610	−0.673048	−	0.739599i	$-0.735015\pi$
$829$	−38.7573	−1.34610	−0.673048	+	0.739599i	$0.735015\pi$
$830$	−32.5305	−1.12915
$831$	7.84132	0.272012
$832$	−28.0740	−0.973290
$833$	25.1845	0.872593
$834$	−43.2432	−1.49739
$835$	−51.9474	−1.79772
$836$	−52.2889	−1.80845
$837$	14.7064	0.508329
$838$	−14.9991	−0.518134
$839$	29.7721	1.02785	0.513924	−	0.857835i	$-0.328191\pi$
$839$	29.7721	1.02785	0.513924	+	0.857835i	$0.328191\pi$
$840$	−45.1223	−1.55687
$841$	−28.4454	−0.980876
$842$	99.8277	3.44029
$843$	−34.1755	−1.17707
$844$	32.4921	1.11842
$845$	6.51195	0.224018
$846$	−4.48854	−0.154319
$847$	−6.94750	−0.238719
$848$	−51.0757	−1.75395
$849$	60.5113	2.07674
$850$	65.2810	2.23912
$851$	−22.9661	−0.787267
$852$	−88.5190	−3.03261
$853$	−15.9292	−0.545406	−0.272703	−	0.962098i	$-0.587918\pi$
$853$	−15.9292	−0.545406	−0.272703	+	0.962098i	$0.587918\pi$
$854$	15.2210	0.520851
$855$	16.3514	0.559206
$856$	−7.02341	−0.240055
$857$	−4.46336	−0.152466	−0.0762328	−	0.997090i	$-0.524289\pi$
$857$	−4.46336	−0.152466	−0.0762328	+	0.997090i	$0.524289\pi$
$858$	−37.8461	−1.29204
$859$	−34.0145	−1.16056	−0.580280	−	0.814417i	$-0.697057\pi$
$859$	−34.0145	−1.16056	−0.580280	+	0.814417i	$0.697057\pi$
$860$	6.85358	0.233705
$861$	−15.9316	−0.542948
$862$	20.4450	0.696358
$863$	42.5445	1.44823	0.724116	−	0.689679i	$-0.242248\pi$
$863$	42.5445	1.44823	0.724116	+	0.689679i	$0.242248\pi$
$864$	42.1432	1.43374
$865$	−37.5120	−1.27545
$866$	17.4953	0.594514
$867$	0.00787064	0.000267301 0
$868$	−15.8742	−0.538806
$869$	16.0347	0.543940
$870$	−12.6984	−0.430516
$871$	−5.48589	−0.185882
$872$	91.2327	3.08953
$873$	1.75191	0.0592933
$874$	−31.0731	−1.05106
$875$	3.36505	0.113760
$876$	−99.5881	−3.36477
$877$	−52.8129	−1.78336	−0.891682	−	0.452663i	$-0.850474\pi$
$877$	−52.8129	−1.78336	−0.891682	+	0.452663i	$0.850474\pi$
$878$	29.2073	0.985698
$879$	51.9176	1.75114
$880$	−60.0252	−2.02345
$881$	−7.92977	−0.267161	−0.133580	−	0.991038i	$-0.542647\pi$
$881$	−7.92977	−0.267161	−0.133580	+	0.991038i	$0.542647\pi$
$882$	13.7235	0.462093
$883$	29.1613	0.981357	0.490679	−	0.871341i	$-0.336749\pi$
$883$	29.1613	0.981357	0.490679	+	0.871341i	$0.336749\pi$
$884$	76.5655	2.57518
$885$	46.2529	1.55477
$886$	73.0329	2.45359
$887$	−4.92911	−0.165503	−0.0827517	−	0.996570i	$-0.526371\pi$
$887$	−4.92911	−0.165503	−0.0827517	+	0.996570i	$0.526371\pi$
$888$	157.808	5.29569
$889$	2.85743	0.0958351
$890$	−113.349	−3.79947
$891$	20.7039	0.693608
$892$	50.6526	1.69597
$893$	11.3934	0.381265
$894$	96.9858	3.24369
$895$	87.6996	2.93147
$896$	1.06508	0.0355819
$897$	−15.8765	−0.530101
$898$	−54.4941	−1.81849
$899$	−2.60635	−0.0869265
$900$	25.1117	0.837056
$901$	22.2928	0.742680
$902$	−42.7331	−1.42286
$903$	0.796539	0.0265072
$904$	−144.589	−4.80897
$905$	−40.6879	−1.35251
$906$	68.1979	2.26572
$907$	12.9808	0.431020	0.215510	−	0.976502i	$-0.430859\pi$
$907$	12.9808	0.431020	0.215510	+	0.976502i	$0.430859\pi$
$908$	86.6491	2.87555
$909$	−11.3066	−0.375015
$910$	31.7041	1.05098
$911$	−13.5872	−0.450165	−0.225082	−	0.974340i	$-0.572265\pi$
$911$	−13.5872	−0.450165	−0.225082	+	0.974340i	$0.572265\pi$
$912$	105.892	3.50644
$913$	−7.15888	−0.236924
$914$	38.4712	1.27252
$915$	−40.3927	−1.33534
$916$	134.484	4.44348
$917$	6.55630	0.216508
$918$	−45.1888	−1.49145
$919$	−16.4173	−0.541557	−0.270779	−	0.962642i	$-0.587281\pi$
$919$	−16.4173	−0.541557	−0.270779	+	0.962642i	$0.587281\pi$
$920$	−50.7727	−1.67393
$921$	−48.2473	−1.58980
$922$	−27.1488	−0.894099
$923$	36.2864	1.19438
$924$	−17.0201	−0.559921
$925$	−66.7370	−2.19430
$926$	−41.8621	−1.37567
$927$	9.12468	0.299694
$928$	−7.46882	−0.245176
$929$	−28.3907	−0.931467	−0.465734	−	0.884925i	$-0.654209\pi$
$929$	−28.3907	−0.931467	−0.465734	+	0.884925i	$0.654209\pi$
$930$	59.6752	1.95683
$931$	−34.8346	−1.14166
$932$	−43.4653	−1.42375
$933$	62.0897	2.03273
$934$	−0.858226	−0.0280820
$935$	26.1989	0.856797
$936$	24.3414	0.795622
$937$	−47.4681	−1.55072	−0.775358	−	0.631522i	$-0.782431\pi$
$937$	−47.4681	−1.55072	−0.775358	+	0.631522i	$0.782431\pi$
$938$	−3.49486	−0.114111
$939$	−1.24504	−0.0406303
$940$	31.9092	1.04076
$941$	−6.00762	−0.195843	−0.0979213	−	0.995194i	$-0.531219\pi$
$941$	−6.00762	−0.195843	−0.0979213	+	0.995194i	$0.531219\pi$
$942$	73.4612	2.39350
$943$	−17.9266	−0.583771
$944$	66.8338	2.17525
$945$	−13.2091	−0.429692
$946$	2.13655	0.0694651
$947$	22.8080	0.741160	0.370580	−	0.928800i	$-0.379159\pi$
$947$	22.8080	0.741160	0.370580	+	0.928800i	$0.379159\pi$
$948$	−79.2236	−2.57306
$949$	40.8239	1.32520
$950$	−90.2951	−2.92956
$951$	−1.05513	−0.0342150
$952$	28.4576	0.922316
$953$	19.6776	0.637421	0.318710	−	0.947852i	$-0.396750\pi$
$953$	19.6776	0.637421	0.318710	+	0.947852i	$0.396750\pi$
$954$	12.1477	0.393296
$955$	27.6597	0.895047
$956$	−59.9152	−1.93780
$957$	−2.79449	−0.0903331
$958$	−79.0249	−2.55318
$959$	−0.724951	−0.0234099
$960$	47.4622	1.53184
$961$	−18.7517	−0.604893
$962$	−110.880	−3.57492
$963$	0.828447	0.0266963
$964$	−30.4862	−0.981895
$965$	62.5108	2.01229
$966$	−10.1143	−0.325424
$967$	54.6184	1.75641	0.878205	−	0.478285i	$-0.158742\pi$
$967$	54.6184	1.75641	0.878205	+	0.478285i	$0.158742\pi$
$968$	53.7167	1.72652
$969$	−46.2182	−1.48474
$970$	−17.6427	−0.566473
$971$	−53.2363	−1.70844	−0.854218	−	0.519916i	$-0.825963\pi$
$971$	−53.2363	−1.70844	−0.854218	+	0.519916i	$0.825963\pi$
$972$	−41.7698	−1.33977
$973$	7.97200	0.255571
$974$	2.77238	0.0888328
$975$	−46.1354	−1.47752
$976$	−58.3660	−1.86825
$977$	−20.0191	−0.640466	−0.320233	−	0.947339i	$-0.603761\pi$
$977$	−20.0191	−0.640466	−0.320233	+	0.947339i	$0.603761\pi$
$978$	−49.9172	−1.59618
$979$	−24.9443	−0.797224
$980$	−97.5607	−3.11646
$981$	−10.7614	−0.343584
$982$	97.4992	3.11132
$983$	−38.6488	−1.23270	−0.616352	−	0.787471i	$-0.711390\pi$
$983$	−38.6488	−1.23270	−0.616352	+	0.787471i	$0.711390\pi$
$984$	123.180	3.92684
$985$	34.6721	1.10474
$986$	8.00859	0.255045
$987$	3.70856	0.118045
$988$	−105.903	−3.36924
$989$	0.896285	0.0285002
$990$	14.2762	0.453728
$991$	3.54542	0.112624	0.0563119	−	0.998413i	$-0.482066\pi$
$991$	3.54542	0.112624	0.0563119	+	0.998413i	$0.482066\pi$
$992$	35.0992	1.11440
$993$	−27.1081	−0.860249
$994$	23.1167	0.733218
$995$	7.43985	0.235859
$996$	35.3703	1.12075
$997$	−13.9553	−0.441970	−0.220985	−	0.975277i	$-0.570927\pi$
$997$	−13.9553	−0.441970	−0.220985	+	0.975277i	$0.570927\pi$
$998$	−10.4829	−0.331832
$999$	46.1967	1.46160

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.a.1.11	✓	306

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.a.1.11	✓	306	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:	\( N \)	\(=\)	\( 8009 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.60787 q^{2} +1.96510 q^{3} +4.80101 q^{4} +3.32724 q^{5} -5.12474 q^{6} +0.944759 q^{7} -7.30469 q^{8} +0.861625 q^{9} +O(q^{10})\)	\(q-2.60787 q^{2} +1.96510 q^{3} +4.80101 q^{4} +3.32724 q^{5} -5.12474 q^{6} +0.944759 q^{7} -7.30469 q^{8} +0.861625 q^{9} -8.67702 q^{10} -1.90952 q^{11} +9.43448 q^{12} -3.86745 q^{13} -2.46381 q^{14} +6.53836 q^{15} +9.44769 q^{16} -4.12359 q^{17} -2.24701 q^{18} +5.70365 q^{19} +15.9741 q^{20} +1.85655 q^{21} +4.97979 q^{22} +2.08903 q^{23} -14.3545 q^{24} +6.07050 q^{25} +10.0858 q^{26} -4.20212 q^{27} +4.53580 q^{28} +0.744721 q^{29} -17.0512 q^{30} -3.49976 q^{31} -10.0290 q^{32} -3.75240 q^{33} +10.7538 q^{34} +3.14344 q^{35} +4.13667 q^{36} -10.9937 q^{37} -14.8744 q^{38} -7.59993 q^{39} -24.3044 q^{40} -8.58131 q^{41} -4.84164 q^{42} +0.429043 q^{43} -9.16763 q^{44} +2.86683 q^{45} -5.44793 q^{46} +1.99756 q^{47} +18.5657 q^{48} -6.10743 q^{49} -15.8311 q^{50} -8.10328 q^{51} -18.5677 q^{52} -5.40615 q^{53} +10.9586 q^{54} -6.35343 q^{55} -6.90117 q^{56} +11.2082 q^{57} -1.94214 q^{58} +7.07409 q^{59} +31.3907 q^{60} -6.17781 q^{61} +9.12694 q^{62} +0.814028 q^{63} +7.25904 q^{64} -12.8679 q^{65} +9.78580 q^{66} +1.41848 q^{67} -19.7974 q^{68} +4.10516 q^{69} -8.19769 q^{70} -9.38250 q^{71} -6.29390 q^{72} -10.5558 q^{73} +28.6701 q^{74} +11.9292 q^{75} +27.3833 q^{76} -1.80404 q^{77} +19.8197 q^{78} -8.39725 q^{79} +31.4347 q^{80} -10.8425 q^{81} +22.3790 q^{82} +3.74904 q^{83} +8.91330 q^{84} -13.7202 q^{85} -1.11889 q^{86} +1.46345 q^{87} +13.9485 q^{88} +13.0631 q^{89} -7.47634 q^{90} -3.65381 q^{91} +10.0295 q^{92} -6.87739 q^{93} -5.20939 q^{94} +18.9774 q^{95} -19.7080 q^{96} +2.03327 q^{97} +15.9274 q^{98} -1.64529 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9} - 61 q^{10} - 43 q^{11} - 50 q^{12} - 89 q^{13} - 40 q^{14} - 61 q^{15} + 151 q^{16} - 52 q^{17} - 57 q^{18} - 185 q^{19} - 66 q^{20} - 63 q^{21} - 55 q^{22} - 62 q^{23} - 131 q^{24} + 209 q^{25} - 57 q^{26} - 88 q^{27} - 182 q^{28} - 67 q^{29} - 68 q^{30} - 240 q^{31} - 64 q^{32} - 52 q^{33} - 128 q^{34} - 99 q^{35} + 106 q^{36} - 49 q^{37} - 45 q^{38} - 190 q^{39} - 158 q^{40} - 72 q^{41} - 36 q^{42} - 141 q^{43} - 80 q^{44} - 100 q^{45} - 91 q^{46} - 105 q^{47} - 85 q^{48} + 116 q^{49} - 51 q^{50} - 145 q^{51} - 237 q^{52} - 48 q^{53} - 156 q^{54} - 420 q^{55} - 116 q^{56} - 35 q^{57} - 43 q^{58} - 139 q^{59} - 73 q^{60} - 233 q^{61} - 58 q^{62} - 252 q^{63} - 3 q^{64} - 45 q^{65} - 127 q^{66} - 108 q^{67} - 85 q^{68} - 164 q^{69} - 56 q^{70} - 131 q^{71} - 117 q^{72} - 118 q^{73} - 47 q^{74} - 112 q^{75} - 389 q^{76} - 36 q^{77} + 9 q^{78} - 382 q^{79} - 119 q^{80} + 102 q^{81} - 131 q^{82} - 59 q^{83} - 144 q^{84} - 140 q^{85} - 38 q^{86} - 301 q^{87} - 131 q^{88} - 98 q^{89} - 138 q^{90} - 176 q^{91} - 97 q^{92} - 60 q^{93} - 342 q^{94} - 154 q^{95} - 243 q^{96} - 109 q^{97} - 21 q^{98} - 173 q^{99}+O(q^{100}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9 - 61 * q^10 - 43 * q^11 - 50 * q^12 - 89 * q^13 - 40 * q^14 - 61 * q^15 + 151 * q^16 - 52 * q^17 - 57 * q^18 - 185 * q^19 - 66 * q^20 - 63 * q^21 - 55 * q^22 - 62 * q^23 - 131 * q^24 + 209 * q^25 - 57 * q^26 - 88 * q^27 - 182 * q^28 - 67 * q^29 - 68 * q^30 - 240 * q^31 - 64 * q^32 - 52 * q^33 - 128 * q^34 - 99 * q^35 + 106 * q^36 - 49 * q^37 - 45 * q^38 - 190 * q^39 - 158 * q^40 - 72 * q^41 - 36 * q^42 - 141 * q^43 - 80 * q^44 - 100 * q^45 - 91 * q^46 - 105 * q^47 - 85 * q^48 + 116 * q^49 - 51 * q^50 - 145 * q^51 - 237 * q^52 - 48 * q^53 - 156 * q^54 - 420 * q^55 - 116 * q^56 - 35 * q^57 - 43 * q^58 - 139 * q^59 - 73 * q^60 - 233 * q^61 - 58 * q^62 - 252 * q^63 - 3 * q^64 - 45 * q^65 - 127 * q^66 - 108 * q^67 - 85 * q^68 - 164 * q^69 - 56 * q^70 - 131 * q^71 - 117 * q^72 - 118 * q^73 - 47 * q^74 - 112 * q^75 - 389 * q^76 - 36 * q^77 + 9 * q^78 - 382 * q^79 - 119 * q^80 + 102 * q^81 - 131 * q^82 - 59 * q^83 - 144 * q^84 - 140 * q^85 - 38 * q^86 - 301 * q^87 - 131 * q^88 - 98 * q^89 - 138 * q^90 - 176 * q^91 - 97 * q^92 - 60 * q^93 - 342 * q^94 - 154 * q^95 - 243 * q^96 - 109 * q^97 - 21 * q^98 - 173 * q^99

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