LMFDB - Embedded newform 4009.2.a.c.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.74061 q^{2} -3.34117 q^{3} +1.02972 q^{4} -2.66559 q^{5} +5.81567 q^{6} +0.0455688 q^{7} +1.68888 q^{8} +8.16342 q^{9} +O(q^{10})$	\(q-1.74061 q^{2} -3.34117 q^{3} +1.02972 q^{4} -2.66559 q^{5} +5.81567 q^{6} +0.0455688 q^{7} +1.68888 q^{8} +8.16342 q^{9} +4.63975 q^{10} -5.77378 q^{11} -3.44046 q^{12} +5.44313 q^{13} -0.0793174 q^{14} +8.90619 q^{15} -4.99912 q^{16} -2.61660 q^{17} -14.2093 q^{18} +1.00000 q^{19} -2.74481 q^{20} -0.152253 q^{21} +10.0499 q^{22} -5.93212 q^{23} -5.64284 q^{24} +2.10537 q^{25} -9.47436 q^{26} -17.2519 q^{27} +0.0469230 q^{28} -1.01835 q^{29} -15.5022 q^{30} +5.87824 q^{31} +5.32374 q^{32} +19.2912 q^{33} +4.55448 q^{34} -0.121468 q^{35} +8.40602 q^{36} -5.35484 q^{37} -1.74061 q^{38} -18.1864 q^{39} -4.50187 q^{40} -6.75452 q^{41} +0.265013 q^{42} +8.74809 q^{43} -5.94537 q^{44} -21.7603 q^{45} +10.3255 q^{46} -9.44683 q^{47} +16.7029 q^{48} -6.99792 q^{49} -3.66463 q^{50} +8.74252 q^{51} +5.60489 q^{52} +0.930586 q^{53} +30.0287 q^{54} +15.3905 q^{55} +0.0769603 q^{56} -3.34117 q^{57} +1.77254 q^{58} +6.84591 q^{59} +9.17087 q^{60} +3.42660 q^{61} -10.2317 q^{62} +0.371997 q^{63} +0.731682 q^{64} -14.5092 q^{65} -33.5784 q^{66} +12.3645 q^{67} -2.69436 q^{68} +19.8202 q^{69} +0.211428 q^{70} +1.90288 q^{71} +13.7870 q^{72} +0.651226 q^{73} +9.32067 q^{74} -7.03441 q^{75} +1.02972 q^{76} -0.263104 q^{77} +31.6554 q^{78} -8.80260 q^{79} +13.3256 q^{80} +33.1512 q^{81} +11.7570 q^{82} -2.19419 q^{83} -0.156778 q^{84} +6.97480 q^{85} -15.2270 q^{86} +3.40247 q^{87} -9.75123 q^{88} -5.58740 q^{89} +37.8762 q^{90} +0.248037 q^{91} -6.10841 q^{92} -19.6402 q^{93} +16.4432 q^{94} -2.66559 q^{95} -17.7875 q^{96} +15.4317 q^{97} +12.1806 q^{98} -47.1338 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.74061	−1.23080	−0.615398	−	0.788217i	$-0.711005\pi$
$2$	−1.74061	−1.23080	−0.615398	+	0.788217i	$0.711005\pi$
$3$	−3.34117	−1.92903	−0.964513	−	0.264036i	$-0.914946\pi$
$3$	−3.34117	−1.92903	−0.964513	+	0.264036i	$0.914946\pi$
$4$	1.02972	0.514859
$5$	−2.66559	−1.19209	−0.596044	−	0.802952i	$-0.703262\pi$
$5$	−2.66559	−1.19209	−0.596044	+	0.802952i	$0.703262\pi$
$6$	5.81567	2.37424
$7$	0.0455688	0.0172234	0.00861169	−	0.999963i	$-0.497259\pi$
$7$	0.0455688	0.0172234	0.00861169	+	0.999963i	$0.497259\pi$
$8$	1.68888	0.597110
$9$	8.16342	2.72114
$10$	4.63975	1.46722
$11$	−5.77378	−1.74086	−0.870430	−	0.492292i	$-0.836159\pi$
$11$	−5.77378	−1.74086	−0.870430	+	0.492292i	$0.836159\pi$
$12$	−3.44046	−0.993176
$13$	5.44313	1.50965	0.754826	−	0.655925i	$-0.227721\pi$
$13$	5.44313	1.50965	0.754826	+	0.655925i	$0.227721\pi$
$14$	−0.0793174	−0.0211985
$15$	8.90619	2.29957
$16$	−4.99912	−1.24978
$17$	−2.61660	−0.634620	−0.317310	−	0.948322i	$-0.602780\pi$
$17$	−2.61660	−0.634620	−0.317310	+	0.948322i	$0.602780\pi$
$18$	−14.2093	−3.34917
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.74481	−0.613757
$21$	−0.152253	−0.0332244
$22$	10.0499	2.14264
$23$	−5.93212	−1.23693	−0.618466	−	0.785812i	$-0.712246\pi$
$23$	−5.93212	−1.23693	−0.618466	+	0.785812i	$0.712246\pi$
$24$	−5.64284	−1.15184
$25$	2.10537	0.421075
$26$	−9.47436	−1.85807
$27$	−17.2519	−3.32012
$28$	0.0469230	0.00886761
$29$	−1.01835	−0.189102	−0.0945510	−	0.995520i	$-0.530142\pi$
$29$	−1.01835	−0.189102	−0.0945510	+	0.995520i	$0.530142\pi$
$30$	−15.5022	−2.83030
$31$	5.87824	1.05576	0.527881	−	0.849318i	$-0.322987\pi$
$31$	5.87824	1.05576	0.527881	+	0.849318i	$0.322987\pi$
$32$	5.32374	0.941114
$33$	19.2912	3.35816
$34$	4.55448	0.781087
$35$	−0.121468	−0.0205318
$36$	8.40602	1.40100
$37$	−5.35484	−0.880330	−0.440165	−	0.897917i	$-0.645080\pi$
$37$	−5.35484	−0.880330	−0.440165	+	0.897917i	$0.645080\pi$
$38$	−1.74061	−0.282364
$39$	−18.1864	−2.91216
$40$	−4.50187	−0.711808
$41$	−6.75452	−1.05488	−0.527439	−	0.849593i	$-0.676848\pi$
$41$	−6.75452	−1.05488	−0.527439	+	0.849593i	$0.676848\pi$
$42$	0.265013	0.0408924
$43$	8.74809	1.33407	0.667036	−	0.745026i	$-0.267563\pi$
$43$	8.74809	1.33407	0.667036	+	0.745026i	$0.267563\pi$
$44$	−5.94537	−0.896298
$45$	−21.7603	−3.24384
$46$	10.3255	1.52241
$47$	−9.44683	−1.37796	−0.688981	−	0.724779i	$-0.741942\pi$
$47$	−9.44683	−1.37796	−0.688981	+	0.724779i	$0.741942\pi$
$48$	16.7029	2.41086
$49$	−6.99792	−0.999703
$50$	−3.66463	−0.518257
$51$	8.74252	1.22420
$52$	5.60489	0.777258
$53$	0.930586	0.127826	0.0639129	−	0.997955i	$-0.479642\pi$
$53$	0.930586	0.127826	0.0639129	+	0.997955i	$0.479642\pi$
$54$	30.0287	4.08639
$55$	15.3905	2.07526
$56$	0.0769603	0.0102843
$57$	−3.34117	−0.442549
$58$	1.77254	0.232746
$59$	6.84591	0.891261	0.445631	−	0.895217i	$-0.352979\pi$
$59$	6.84591	0.891261	0.445631	+	0.895217i	$0.352979\pi$
$60$	9.17087	1.18395
$61$	3.42660	0.438731	0.219366	−	0.975643i	$-0.429601\pi$
$61$	3.42660	0.438731	0.219366	+	0.975643i	$0.429601\pi$
$62$	−10.2317	−1.29943
$63$	0.371997	0.0468672
$64$	0.731682	0.0914603
$65$	−14.5092	−1.79964
$66$	−33.5784	−4.13322
$67$	12.3645	1.51057	0.755285	−	0.655397i	$-0.227499\pi$
$67$	12.3645	1.51057	0.755285	+	0.655397i	$0.227499\pi$
$68$	−2.69436	−0.326740
$69$	19.8202	2.38607
$70$	0.211428	0.0252705
$71$	1.90288	0.225831	0.112915	−	0.993605i	$-0.463981\pi$
$71$	1.90288	0.225831	0.112915	+	0.993605i	$0.463981\pi$
$72$	13.7870	1.62482
$73$	0.651226	0.0762203	0.0381101	−	0.999274i	$-0.487866\pi$
$73$	0.651226	0.0762203	0.0381101	+	0.999274i	$0.487866\pi$
$74$	9.32067	1.08351
$75$	−7.03441	−0.812264
$76$	1.02972	0.118117
$77$	−0.263104	−0.0299835
$78$	31.6554	3.58427
$79$	−8.80260	−0.990370	−0.495185	−	0.868788i	$-0.664900\pi$
$79$	−8.80260	−0.990370	−0.495185	+	0.868788i	$0.664900\pi$
$80$	13.3256	1.48985
$81$	33.1512	3.68346
$82$	11.7570	1.29834
$83$	−2.19419	−0.240843	−0.120422	−	0.992723i	$-0.538425\pi$
$83$	−2.19419	−0.240843	−0.120422	+	0.992723i	$0.538425\pi$
$84$	−0.156778	−0.0171059
$85$	6.97480	0.756523
$86$	−15.2270	−1.64197
$87$	3.40247	0.364783
$88$	−9.75123	−1.03948
$89$	−5.58740	−0.592264	−0.296132	−	0.955147i	$-0.595697\pi$
$89$	−5.58740	−0.592264	−0.296132	+	0.955147i	$0.595697\pi$
$90$	37.8762	3.99250
$91$	0.248037	0.0260013
$92$	−6.10841	−0.636846
$93$	−19.6402	−2.03659
$94$	16.4432	1.69599
$95$	−2.66559	−0.273484
$96$	−17.7875	−1.81543
$97$	15.4317	1.56685	0.783427	−	0.621484i	$-0.213470\pi$
$97$	15.4317	1.56685	0.783427	+	0.621484i	$0.213470\pi$
$98$	12.1806	1.23043
$99$	−47.1338	−4.73713
$100$	2.16794	0.216794
$101$	16.2442	1.61636	0.808181	−	0.588934i	$-0.200452\pi$
$101$	16.2442	1.61636	0.808181	+	0.588934i	$0.200452\pi$
$102$	−15.2173	−1.50674
$103$	−19.4403	−1.91551	−0.957753	−	0.287592i	$-0.907145\pi$
$103$	−19.4403	−1.91551	−0.957753	+	0.287592i	$0.907145\pi$
$104$	9.19280	0.901428
$105$	0.405845	0.0396064
$106$	−1.61979	−0.157327
$107$	19.9334	1.92704	0.963519	−	0.267641i	$-0.0862442\pi$
$107$	19.9334	1.92704	0.963519	+	0.267641i	$0.0862442\pi$
$108$	−17.7646	−1.70939
$109$	3.64550	0.349176	0.174588	−	0.984642i	$-0.444141\pi$
$109$	3.64550	0.349176	0.174588	+	0.984642i	$0.444141\pi$
$110$	−26.7889	−2.55422
$111$	17.8914	1.69818
$112$	−0.227804	−0.0215254
$113$	16.2367	1.52742	0.763710	−	0.645559i	$-0.223376\pi$
$113$	16.2367	1.52742	0.763710	+	0.645559i	$0.223376\pi$
$114$	5.81567	0.544687
$115$	15.8126	1.47453
$116$	−1.04861	−0.0973609
$117$	44.4345	4.10798
$118$	−11.9160	−1.09696
$119$	−0.119236	−0.0109303
$120$	15.0415	1.37310
$121$	22.3366	2.03060
$122$	−5.96437	−0.539989
$123$	22.5680	2.03489
$124$	6.05293	0.543569
$125$	7.71589	0.690130
$126$	−0.647502	−0.0576840
$127$	−9.77740	−0.867604	−0.433802	−	0.901008i	$-0.642828\pi$
$127$	−9.77740	−0.867604	−0.433802	+	0.901008i	$0.642828\pi$
$128$	−11.9211	−1.05368
$129$	−29.2289	−2.57346
$130$	25.2548	2.21499
$131$	−3.36722	−0.294196	−0.147098	−	0.989122i	$-0.546993\pi$
$131$	−3.36722	−0.294196	−0.147098	+	0.989122i	$0.546993\pi$
$132$	19.8645	1.72898
$133$	0.0455688	0.00395132
$134$	−21.5218	−1.85920
$135$	45.9864	3.95788
$136$	−4.41913	−0.378938
$137$	12.3070	1.05145	0.525727	−	0.850653i	$-0.323793\pi$
$137$	12.3070	1.05145	0.525727	+	0.850653i	$0.323793\pi$
$138$	−34.4992	−2.93677
$139$	−5.54031	−0.469923	−0.234962	−	0.972005i	$-0.575496\pi$
$139$	−5.54031	−0.469923	−0.234962	+	0.972005i	$0.575496\pi$
$140$	−0.125078	−0.0105710
$141$	31.5635	2.65812
$142$	−3.31217	−0.277951
$143$	−31.4274	−2.62809
$144$	−40.8099	−3.40082
$145$	2.71449	0.225426
$146$	−1.13353	−0.0938116
$147$	23.3813	1.92845
$148$	−5.51397	−0.453246
$149$	4.46211	0.365551	0.182775	−	0.983155i	$-0.441492\pi$
$149$	4.46211	0.365551	0.182775	+	0.983155i	$0.441492\pi$
$150$	12.2442	0.999732
$151$	−16.9640	−1.38051	−0.690257	−	0.723564i	$-0.742503\pi$
$151$	−16.9640	−1.38051	−0.690257	+	0.723564i	$0.742503\pi$
$152$	1.68888	0.136986
$153$	−21.3604	−1.72689
$154$	0.457962	0.0369036
$155$	−15.6690	−1.25856
$156$	−18.7269	−1.49935
$157$	6.02503	0.480849	0.240425	−	0.970668i	$-0.422713\pi$
$157$	6.02503	0.480849	0.240425	+	0.970668i	$0.422713\pi$
$158$	15.3219	1.21894
$159$	−3.10925	−0.246579
$160$	−14.1909	−1.12189
$161$	−0.270319	−0.0213042
$162$	−57.7032	−4.53359
$163$	0.00745152	0.000583648 0	0.000291824	−	1.00000i	$-0.499907\pi$
$163$	0.00745152	0.000583648 0	0.000291824		1.00000i	$0.499907\pi$
$164$	−6.95525	−0.543114
$165$	−51.4224	−4.00323
$166$	3.81922	0.296429
$167$	19.1424	1.48128	0.740642	−	0.671900i	$-0.234521\pi$
$167$	19.1424	1.48128	0.740642	+	0.671900i	$0.234521\pi$
$168$	−0.257137	−0.0198386
$169$	16.6277	1.27905
$170$	−12.1404	−0.931125
$171$	8.16342	0.624272
$172$	9.00807	0.686859
$173$	−0.0597609	−0.00454354	−0.00227177	−	0.999997i	$-0.500723\pi$
$173$	−0.0597609	−0.00454354	−0.00227177	+	0.999997i	$0.500723\pi$
$174$	−5.92236	−0.448973
$175$	0.0959394	0.00725234
$176$	28.8638	2.17569
$177$	−22.8733	−1.71927
$178$	9.72548	0.728956
$179$	−13.8722	−1.03685	−0.518427	−	0.855122i	$-0.673482\pi$
$179$	−13.8722	−1.03685	−0.518427	+	0.855122i	$0.673482\pi$
$180$	−22.4070	−1.67012
$181$	3.34790	0.248848	0.124424	−	0.992229i	$-0.460292\pi$
$181$	3.34790	0.248848	0.124424	+	0.992229i	$0.460292\pi$
$182$	−0.431735	−0.0320023
$183$	−11.4489	−0.846324
$184$	−10.0186	−0.738584
$185$	14.2738	1.04943
$186$	34.1859	2.50663
$187$	15.1077	1.10478
$188$	−9.72757	−0.709456
$189$	−0.786147	−0.0571838
$190$	4.63975	0.336603
$191$	21.5384	1.55846	0.779231	−	0.626736i	$-0.215610\pi$
$191$	21.5384	1.55846	0.779231	+	0.626736i	$0.215610\pi$
$192$	−2.44467	−0.176429
$193$	1.73663	0.125005	0.0625026	−	0.998045i	$-0.480092\pi$
$193$	1.73663	0.125005	0.0625026	+	0.998045i	$0.480092\pi$
$194$	−26.8606	−1.92848
$195$	48.4776	3.47155
$196$	−7.20589	−0.514706
$197$	−22.6717	−1.61529	−0.807644	−	0.589670i	$-0.799258\pi$
$197$	−22.6717	−1.61529	−0.807644	+	0.589670i	$0.799258\pi$
$198$	82.0415	5.83044
$199$	−14.8533	−1.05292	−0.526460	−	0.850200i	$-0.676481\pi$
$199$	−14.8533	−1.05292	−0.526460	+	0.850200i	$0.676481\pi$
$200$	3.55573	0.251428
$201$	−41.3120	−2.91393
$202$	−28.2749	−1.98941
$203$	−0.0464048	−0.00325698
$204$	9.00233	0.630289
$205$	18.0048	1.25751
$206$	33.8379	2.35760
$207$	−48.4264	−3.36587
$208$	−27.2108	−1.88673
$209$	−5.77378	−0.399381
$210$	−0.706416	−0.0487474
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	0.958241	0.0658123
$213$	−6.35785	−0.435633
$214$	−34.6963	−2.37179
$215$	−23.3188	−1.59033
$216$	−29.1364	−1.98248
$217$	0.267864	0.0181838
$218$	−6.34539	−0.429764
$219$	−2.17586	−0.147031
$220$	15.8479	1.06847
$221$	−14.2425	−0.958055
$222$	−31.1420	−2.09011
$223$	7.03060	0.470804	0.235402	−	0.971898i	$-0.424359\pi$
$223$	7.03060	0.470804	0.235402	+	0.971898i	$0.424359\pi$
$224$	0.242597	0.0162092
$225$	17.1871	1.14580
$226$	−28.2617	−1.87994
$227$	−20.5078	−1.36115	−0.680576	−	0.732677i	$-0.738271\pi$
$227$	−20.5078	−1.36115	−0.680576	+	0.732677i	$0.738271\pi$
$228$	−3.44046	−0.227850
$229$	18.4436	1.21879	0.609395	−	0.792867i	$-0.291413\pi$
$229$	18.4436	1.21879	0.609395	+	0.792867i	$0.291413\pi$
$230$	−27.5235	−1.81485
$231$	0.879076	0.0578390
$232$	−1.71986	−0.112915
$233$	23.4767	1.53801	0.769004	−	0.639244i	$-0.220753\pi$
$233$	23.4767	1.53801	0.769004	+	0.639244i	$0.220753\pi$
$234$	−77.3431	−5.05608
$235$	25.1814	1.64265
$236$	7.04935	0.458874
$237$	29.4110	1.91045
$238$	0.207542	0.0134530
$239$	−9.84627	−0.636902	−0.318451	−	0.947939i	$-0.603163\pi$
$239$	−9.84627	−0.636902	−0.318451	+	0.947939i	$0.603163\pi$
$240$	−44.5231	−2.87395
$241$	−1.95165	−0.125717	−0.0628584	−	0.998022i	$-0.520022\pi$
$241$	−1.95165	−0.125717	−0.0628584	+	0.998022i	$0.520022\pi$
$242$	−38.8792	−2.49925
$243$	−59.0081	−3.78537
$244$	3.52843	0.225885
$245$	18.6536	1.19173
$246$	−39.2821	−2.50453
$247$	5.44313	0.346338
$248$	9.92765	0.630406
$249$	7.33115	0.464593
$250$	−13.4303	−0.849409
$251$	−0.0590057	−0.00372441	−0.00186220	−	0.999998i	$-0.500593\pi$
$251$	−0.0590057	−0.00372441	−0.00186220	+	0.999998i	$0.500593\pi$
$252$	0.383052	0.0241300
$253$	34.2508	2.15333
$254$	17.0186	1.06784
$255$	−23.3040	−1.45935
$256$	19.2865	1.20541
$257$	13.9070	0.867495	0.433747	−	0.901035i	$-0.357191\pi$
$257$	13.9070	0.867495	0.433747	+	0.901035i	$0.357191\pi$
$258$	50.8760	3.16740
$259$	−0.244013	−0.0151623
$260$	−14.9403	−0.926560
$261$	−8.31318	−0.514573
$262$	5.86102	0.362095
$263$	21.7293	1.33988	0.669942	−	0.742413i	$-0.266319\pi$
$263$	21.7293	1.33988	0.669942	+	0.742413i	$0.266319\pi$
$264$	32.5805	2.00519
$265$	−2.48056	−0.152380
$266$	−0.0793174	−0.00486326
$267$	18.6685	1.14249
$268$	12.7320	0.777730
$269$	−24.6374	−1.50217	−0.751084	−	0.660207i	$-0.770468\pi$
$269$	−24.6374	−1.50217	−0.751084	+	0.660207i	$0.770468\pi$
$270$	−80.0443	−4.87134
$271$	−19.3262	−1.17399	−0.586993	−	0.809592i	$-0.699688\pi$
$271$	−19.3262	−1.17399	−0.586993	+	0.809592i	$0.699688\pi$
$272$	13.0807	0.793134
$273$	−0.828733	−0.0501572
$274$	−21.4216	−1.29413
$275$	−12.1560	−0.733033
$276$	20.4092	1.22849
$277$	−5.80451	−0.348759	−0.174380	−	0.984679i	$-0.555792\pi$
$277$	−5.80451	−0.348759	−0.174380	+	0.984679i	$0.555792\pi$
$278$	9.64351	0.578380
$279$	47.9865	2.87288
$280$	−0.205145	−0.0122597
$281$	2.12615	0.126836	0.0634178	−	0.997987i	$-0.479800\pi$
$281$	2.12615	0.126836	0.0634178	+	0.997987i	$0.479800\pi$
$282$	−54.9397	−3.27161
$283$	−2.58168	−0.153465	−0.0767326	−	0.997052i	$-0.524449\pi$
$283$	−2.58168	−0.153465	−0.0767326	+	0.997052i	$0.524449\pi$
$284$	1.95943	0.116271
$285$	8.90619	0.527557
$286$	54.7029	3.23465
$287$	−0.307795	−0.0181686
$288$	43.4599	2.56090
$289$	−10.1534	−0.597258
$290$	−4.72487	−0.277454
$291$	−51.5600	−3.02250
$292$	0.670580	0.0392427
$293$	21.6572	1.26523	0.632614	−	0.774468i	$-0.281982\pi$
$293$	21.6572	1.26523	0.632614	+	0.774468i	$0.281982\pi$
$294$	−40.6976	−2.37353
$295$	−18.2484	−1.06246
$296$	−9.04368	−0.525653
$297$	99.6085	5.77987
$298$	−7.76679	−0.449918
$299$	−32.2893	−1.86734
$300$	−7.24346	−0.418201
$301$	0.398640	0.0229772
$302$	29.5278	1.69913
$303$	−54.2748	−3.11800
$304$	−4.99912	−0.286719
$305$	−9.13392	−0.523007
$306$	37.1802	2.12545
$307$	27.9345	1.59430	0.797152	−	0.603779i	$-0.206339\pi$
$307$	27.9345	1.59430	0.797152	+	0.603779i	$0.206339\pi$
$308$	−0.270923	−0.0154373
$309$	64.9532	3.69506
$310$	27.2736	1.54903
$311$	11.4609	0.649886	0.324943	−	0.945734i	$-0.394655\pi$
$311$	11.4609	0.649886	0.324943	+	0.945734i	$0.394655\pi$
$312$	−30.7147	−1.73888
$313$	−5.97637	−0.337804	−0.168902	−	0.985633i	$-0.554022\pi$
$313$	−5.97637	−0.337804	−0.168902	+	0.985633i	$0.554022\pi$
$314$	−10.4872	−0.591828
$315$	−0.991592	−0.0558699
$316$	−9.06419	−0.509901
$317$	12.0415	0.676320	0.338160	−	0.941089i	$-0.390195\pi$
$317$	12.0415	0.676320	0.338160	+	0.941089i	$0.390195\pi$
$318$	5.41198	0.303489
$319$	5.87970	0.329200
$320$	−1.95036	−0.109029
$321$	−66.6010	−3.71730
$322$	0.470520	0.0262211
$323$	−2.61660	−0.145592
$324$	34.1363	1.89646
$325$	11.4598	0.635677
$326$	−0.0129702	−0.000718351 0
$327$	−12.1802	−0.673569
$328$	−11.4076	−0.629878
$329$	−0.430481	−0.0237332
$330$	89.5063	4.92716
$331$	−29.6154	−1.62781	−0.813904	−	0.581000i	$-0.802662\pi$
$331$	−29.6154	−1.62781	−0.813904	+	0.581000i	$0.802662\pi$
$332$	−2.25939	−0.124000
$333$	−43.7138	−2.39550
$334$	−33.3194	−1.82316
$335$	−32.9588	−1.80073
$336$	0.761131	0.0415231
$337$	17.7117	0.964817	0.482409	−	0.875946i	$-0.339762\pi$
$337$	17.7117	0.964817	0.482409	+	0.875946i	$0.339762\pi$
$338$	−28.9422	−1.57425
$339$	−54.2496	−2.94643
$340$	7.18207	0.389503
$341$	−33.9397	−1.83794
$342$	−14.2093	−0.768352
$343$	−0.637869	−0.0344417
$344$	14.7745	0.796587
$345$	−52.8326	−2.84441
$346$	0.104020	0.00559217
$347$	13.0704	0.701658	0.350829	−	0.936440i	$-0.385900\pi$
$347$	13.0704	0.701658	0.350829	+	0.936440i	$0.385900\pi$
$348$	3.50358	0.187812
$349$	0.495731	0.0265359	0.0132679	−	0.999912i	$-0.495777\pi$
$349$	0.495731	0.0265359	0.0132679	+	0.999912i	$0.495777\pi$
$350$	−0.166993	−0.00892615
$351$	−93.9041	−5.01223
$352$	−30.7381	−1.63835
$353$	21.6028	1.14980	0.574901	−	0.818223i	$-0.305040\pi$
$353$	21.6028	1.14980	0.574901	+	0.818223i	$0.305040\pi$
$354$	39.8135	2.11607
$355$	−5.07231	−0.269210
$356$	−5.75345	−0.304932
$357$	0.398386	0.0210848
$358$	24.1460	1.27616
$359$	−29.3675	−1.54995	−0.774977	−	0.631989i	$-0.782239\pi$
$359$	−29.3675	−1.54995	−0.774977	+	0.631989i	$0.782239\pi$
$360$	−36.7506	−1.93693
$361$	1.00000	0.0526316
$362$	−5.82739	−0.306281
$363$	−74.6302	−3.91707
$364$	0.255408	0.0133870
$365$	−1.73590	−0.0908613
$366$	19.9280	1.04165
$367$	1.84372	0.0962414	0.0481207	−	0.998842i	$-0.484677\pi$
$367$	1.84372	0.0962414	0.0481207	+	0.998842i	$0.484677\pi$
$368$	29.6553	1.54589
$369$	−55.1400	−2.87047
$370$	−24.8451	−1.29164
$371$	0.0424057	0.00220159
$372$	−20.2239	−1.04856
$373$	−5.04481	−0.261210	−0.130605	−	0.991434i	$-0.541692\pi$
$373$	−5.04481	−0.261210	−0.130605	+	0.991434i	$0.541692\pi$
$374$	−26.2966	−1.35976
$375$	−25.7801	−1.33128
$376$	−15.9546	−0.822795
$377$	−5.54299	−0.285478
$378$	1.36837	0.0703815
$379$	13.9046	0.714233	0.357117	−	0.934060i	$-0.383760\pi$
$379$	13.9046	0.714233	0.357117	+	0.934060i	$0.383760\pi$
$380$	−2.74481	−0.140806
$381$	32.6680	1.67363
$382$	−37.4899	−1.91815
$383$	−36.4675	−1.86340	−0.931701	−	0.363226i	$-0.881675\pi$
$383$	−36.4675	−1.86340	−0.931701	+	0.363226i	$0.881675\pi$
$384$	39.8303	2.03258
$385$	0.701328	0.0357430
$386$	−3.02279	−0.153856
$387$	71.4143	3.63020
$388$	15.8903	0.806708
$389$	−27.3415	−1.38627	−0.693134	−	0.720808i	$-0.743771\pi$
$389$	−27.3415	−1.38627	−0.693134	+	0.720808i	$0.743771\pi$
$390$	−84.3805	−4.27277
$391$	15.5220	0.784981
$392$	−11.8187	−0.596933
$393$	11.2505	0.567511
$394$	39.4625	1.98809
$395$	23.4641	1.18061
$396$	−48.5345	−2.43895
$397$	6.47117	0.324779	0.162389	−	0.986727i	$-0.448080\pi$
$397$	6.47117	0.324779	0.162389	+	0.986727i	$0.448080\pi$
$398$	25.8537	1.29593
$399$	−0.152253	−0.00762219
$400$	−10.5250	−0.526251
$401$	−22.4117	−1.11919	−0.559594	−	0.828767i	$-0.689043\pi$
$401$	−22.4117	−1.11919	−0.559594	+	0.828767i	$0.689043\pi$
$402$	71.9081	3.58645
$403$	31.9960	1.59383
$404$	16.7270	0.832198
$405$	−88.3674	−4.39101
$406$	0.0807725	0.00400867
$407$	30.9177	1.53253
$408$	14.7651	0.730980
$409$	36.2751	1.79369	0.896845	−	0.442345i	$-0.145853\pi$
$409$	36.2751	1.79369	0.896845	+	0.442345i	$0.145853\pi$
$410$	−31.3393	−1.54774
$411$	−41.1196	−2.02828
$412$	−20.0180	−0.986216
$413$	0.311960	0.0153505
$414$	84.2913	4.14269
$415$	5.84880	0.287106
$416$	28.9778	1.42075
$417$	18.5111	0.906494
$418$	10.0499	0.491556
$419$	−15.5144	−0.757928	−0.378964	−	0.925411i	$-0.623720\pi$
$419$	−15.5144	−0.757928	−0.378964	+	0.925411i	$0.623720\pi$
$420$	0.417905	0.0203917
$421$	4.98991	0.243193	0.121597	−	0.992580i	$-0.461199\pi$
$421$	4.98991	0.243193	0.121597	+	0.992580i	$0.461199\pi$
$422$	−1.74061	−0.0847315
$423$	−77.1185	−3.74963
$424$	1.57165	0.0763260
$425$	−5.50893	−0.267222
$426$	11.0665	0.536175
$427$	0.156146	0.00755644
$428$	20.5258	0.992152
$429$	105.004	5.06966
$430$	40.5890	1.95737
$431$	−0.927561	−0.0446791	−0.0223395	−	0.999750i	$-0.507111\pi$
$431$	−0.927561	−0.0446791	−0.0223395	+	0.999750i	$0.507111\pi$
$432$	86.2441	4.14942
$433$	4.49916	0.216216	0.108108	−	0.994139i	$-0.465521\pi$
$433$	4.49916	0.216216	0.108108	+	0.994139i	$0.465521\pi$
$434$	−0.466247	−0.0223806
$435$	−9.06958	−0.434853
$436$	3.75384	0.179776
$437$	−5.93212	−0.283772
$438$	3.78732	0.180965
$439$	−20.3729	−0.972345	−0.486172	−	0.873863i	$-0.661607\pi$
$439$	−20.3729	−0.972345	−0.486172	+	0.873863i	$0.661607\pi$
$440$	25.9928	1.23916
$441$	−57.1270	−2.72033
$442$	24.7906	1.17917
$443$	39.2967	1.86704	0.933522	−	0.358521i	$-0.116719\pi$
$443$	39.2967	1.86704	0.933522	+	0.358521i	$0.116719\pi$
$444$	18.4231	0.874322
$445$	14.8937	0.706031
$446$	−12.2375	−0.579464
$447$	−14.9087	−0.705156
$448$	0.0333419	0.00157526
$449$	15.8914	0.749964	0.374982	−	0.927032i	$-0.377649\pi$
$449$	15.8914	0.749964	0.374982	+	0.927032i	$0.377649\pi$
$450$	−29.9159	−1.41025
$451$	38.9991	1.83640
$452$	16.7192	0.786406
$453$	56.6798	2.66305
$454$	35.6961	1.67530
$455$	−0.661165	−0.0309959
$456$	−5.64284	−0.264250
$457$	−14.8024	−0.692426	−0.346213	−	0.938156i	$-0.612533\pi$
$457$	−14.8024	−0.692426	−0.346213	+	0.938156i	$0.612533\pi$
$458$	−32.1031	−1.50008
$459$	45.1413	2.10702
$460$	16.2825	0.759176
$461$	31.9799	1.48945	0.744726	−	0.667371i	$-0.232580\pi$
$461$	31.9799	1.48945	0.744726	+	0.667371i	$0.232580\pi$
$462$	−1.53013	−0.0711880
$463$	9.86644	0.458532	0.229266	−	0.973364i	$-0.426367\pi$
$463$	9.86644	0.458532	0.229266	+	0.973364i	$0.426367\pi$
$464$	5.09083	0.236336
$465$	52.3527	2.42780
$466$	−40.8637	−1.89297
$467$	28.9588	1.34005	0.670026	−	0.742337i	$-0.266283\pi$
$467$	28.9588	1.34005	0.670026	+	0.742337i	$0.266283\pi$
$468$	45.7550	2.11503
$469$	0.563437	0.0260171
$470$	−43.8310	−2.02177
$471$	−20.1306	−0.927571
$472$	11.5619	0.532181
$473$	−50.5096	−2.32243
$474$	−51.1930	−2.35137
$475$	2.10537	0.0966012
$476$	−0.122779	−0.00562756
$477$	7.59676	0.347832
$478$	17.1385	0.783897
$479$	−12.6325	−0.577193	−0.288597	−	0.957451i	$-0.593189\pi$
$479$	−12.6325	−0.577193	−0.288597	+	0.957451i	$0.593189\pi$
$480$	47.4143	2.16416
$481$	−29.1471	−1.32899
$482$	3.39706	0.154732
$483$	0.903183	0.0410963
$484$	23.0004	1.04547
$485$	−41.1346	−1.86783
$486$	102.710	4.65902
$487$	−23.9161	−1.08374	−0.541870	−	0.840462i	$-0.682283\pi$
$487$	−23.9161	−1.08374	−0.541870	+	0.840462i	$0.682283\pi$
$488$	5.78712	0.261971
$489$	−0.0248968	−0.00112587
$490$	−32.4686	−1.46678
$491$	1.45598	0.0657076	0.0328538	−	0.999460i	$-0.489540\pi$
$491$	1.45598	0.0657076	0.0328538	+	0.999460i	$0.489540\pi$
$492$	23.2387	1.04768
$493$	2.66461	0.120008
$494$	−9.47436	−0.426271
$495$	125.639	5.64707
$496$	−29.3860	−1.31947
$497$	0.0867121	0.00388957
$498$	−12.7607	−0.571819
$499$	−15.2554	−0.682925	−0.341463	−	0.939895i	$-0.610922\pi$
$499$	−15.2554	−0.682925	−0.341463	+	0.939895i	$0.610922\pi$
$500$	7.94519	0.355320
$501$	−63.9581	−2.85744
$502$	0.102706	0.00458399
$503$	28.9694	1.29168	0.645842	−	0.763471i	$-0.276507\pi$
$503$	28.9694	1.29168	0.645842	+	0.763471i	$0.276507\pi$
$504$	0.628259	0.0279849
$505$	−43.3005	−1.92685
$506$	−59.6172	−2.65031
$507$	−55.5558	−2.46732
$508$	−10.0680	−0.446694
$509$	4.96622	0.220124	0.110062	−	0.993925i	$-0.464895\pi$
$509$	4.96622	0.220124	0.110062	+	0.993925i	$0.464895\pi$
$510$	40.5631	1.79616
$511$	0.0296756	0.00131277
$512$	−9.72818	−0.429929
$513$	−17.2519	−0.761688
$514$	−24.2066	−1.06771
$515$	51.8198	2.28345
$516$	−30.0975	−1.32497
$517$	54.5440	2.39884
$518$	0.424732	0.0186616
$519$	0.199671	0.00876460
$520$	−24.5042	−1.07458
$521$	14.7790	0.647481	0.323741	−	0.946146i	$-0.395059\pi$
$521$	14.7790	0.647481	0.323741	+	0.946146i	$0.395059\pi$
$522$	14.4700	0.633334
$523$	−4.11833	−0.180082	−0.0900410	−	0.995938i	$-0.528700\pi$
$523$	−4.11833	−0.180082	−0.0900410	+	0.995938i	$0.528700\pi$
$524$	−3.46729	−0.151469
$525$	−0.320550	−0.0139899
$526$	−37.8222	−1.64912
$527$	−15.3810	−0.670008
$528$	−96.4389	−4.19696
$529$	12.1900	0.530001
$530$	4.31769	0.187548
$531$	55.8860	2.42525
$532$	0.0469230	0.00203437
$533$	−36.7657	−1.59250
$534$	−32.4945	−1.40617
$535$	−53.1344	−2.29720
$536$	20.8822	0.901975
$537$	46.3492	2.00012
$538$	42.8840	1.84886
$539$	40.4045	1.74034
$540$	47.3530	2.03775
$541$	−12.5585	−0.539933	−0.269967	−	0.962870i	$-0.587013\pi$
$541$	−12.5585	−0.539933	−0.269967	+	0.962870i	$0.587013\pi$
$542$	33.6394	1.44494
$543$	−11.1859	−0.480033
$544$	−13.9301	−0.597249
$545$	−9.71741	−0.416248
$546$	1.44250	0.0617333
$547$	−4.37116	−0.186897	−0.0934486	−	0.995624i	$-0.529789\pi$
$547$	−4.37116	−0.186897	−0.0934486	+	0.995624i	$0.529789\pi$
$548$	12.6727	0.541351
$549$	27.9728	1.19385
$550$	21.1588	0.902214
$551$	−1.01835	−0.0433830
$552$	33.4740	1.42475
$553$	−0.401124	−0.0170575
$554$	10.1034	0.429251
$555$	−47.6912	−2.02438
$556$	−5.70496	−0.241944
$557$	−26.0003	−1.10167	−0.550835	−	0.834614i	$-0.685690\pi$
$557$	−26.0003	−1.10167	−0.550835	+	0.834614i	$0.685690\pi$
$558$	−83.5257	−3.53593
$559$	47.6170	2.01398
$560$	0.607232	0.0256602
$561$	−50.4774	−2.13116
$562$	−3.70080	−0.156109
$563$	−16.9417	−0.714007	−0.357004	−	0.934103i	$-0.616202\pi$
$563$	−16.9417	−0.714007	−0.357004	+	0.934103i	$0.616202\pi$
$564$	32.5015	1.36856
$565$	−43.2804	−1.82082
$566$	4.49370	0.188884
$567$	1.51066	0.0634417
$568$	3.21374	0.134846
$569$	−39.1407	−1.64086	−0.820431	−	0.571745i	$-0.806267\pi$
$569$	−39.1407	−1.64086	−0.820431	+	0.571745i	$0.806267\pi$
$570$	−15.5022	−0.649316
$571$	−14.0289	−0.587090	−0.293545	−	0.955945i	$-0.594835\pi$
$571$	−14.0289	−0.587090	−0.293545	+	0.955945i	$0.594835\pi$
$572$	−32.3614	−1.35310
$573$	−71.9634	−3.00631
$574$	0.535751	0.0223618
$575$	−12.4893	−0.520841
$576$	5.97303	0.248876
$577$	−37.9296	−1.57903	−0.789515	−	0.613731i	$-0.789668\pi$
$577$	−37.9296	−1.57903	−0.789515	+	0.613731i	$0.789668\pi$
$578$	17.6731	0.735103
$579$	−5.80237	−0.241138
$580$	2.79516	0.116063
$581$	−0.0999864	−0.00414814
$582$	89.7458	3.72008
$583$	−5.37300	−0.222527
$584$	1.09984	0.0455119
$585$	−118.444	−4.89707
$586$	−37.6967	−1.55724
$587$	−1.21754	−0.0502534	−0.0251267	−	0.999684i	$-0.507999\pi$
$587$	−1.21754	−0.0502534	−0.0251267	+	0.999684i	$0.507999\pi$
$588$	24.0761	0.992881
$589$	5.87824	0.242209
$590$	31.7633	1.30767
$591$	75.7499	3.11593
$592$	26.7695	1.10022
$593$	38.6941	1.58898	0.794488	−	0.607279i	$-0.207739\pi$
$593$	38.6941	1.58898	0.794488	+	0.607279i	$0.207739\pi$
$594$	−173.379	−7.11384
$595$	0.317833	0.0130299
$596$	4.59472	0.188207
$597$	49.6273	2.03111
$598$	56.2030	2.29831
$599$	16.5459	0.676048	0.338024	−	0.941137i	$-0.390242\pi$
$599$	16.5459	0.676048	0.338024	+	0.941137i	$0.390242\pi$
$600$	−11.8803	−0.485011
$601$	−32.3580	−1.31991	−0.659954	−	0.751306i	$-0.729424\pi$
$601$	−32.3580	−1.31991	−0.659954	+	0.751306i	$0.729424\pi$
$602$	−0.693876	−0.0282803
$603$	100.937	4.11047
$604$	−17.4682	−0.710770
$605$	−59.5401	−2.42065
$606$	94.4711	3.83763
$607$	34.4933	1.40004	0.700019	−	0.714124i	$-0.253175\pi$
$607$	34.4933	1.40004	0.700019	+	0.714124i	$0.253175\pi$
$608$	5.32374	0.215906
$609$	0.155046	0.00628279
$610$	15.8986	0.643714
$611$	−51.4203	−2.08024
$612$	−21.9952	−0.889104
$613$	−1.13192	−0.0457177	−0.0228589	−	0.999739i	$-0.507277\pi$
$613$	−1.13192	−0.0457177	−0.0228589	+	0.999739i	$0.507277\pi$
$614$	−48.6230	−1.96226
$615$	−60.1570	−2.42577
$616$	−0.444352	−0.0179034
$617$	9.00443	0.362505	0.181252	−	0.983437i	$-0.441985\pi$
$617$	9.00443	0.362505	0.181252	+	0.983437i	$0.441985\pi$
$618$	−113.058	−4.54787
$619$	20.1918	0.811579	0.405789	−	0.913967i	$-0.366997\pi$
$619$	20.1918	0.811579	0.405789	+	0.913967i	$0.366997\pi$
$620$	−16.1346	−0.647982
$621$	102.340	4.10677
$622$	−19.9489	−0.799877
$623$	−0.254611	−0.0102008
$624$	90.9160	3.63955
$625$	−31.0943	−1.24377
$626$	10.4025	0.415768
$627$	19.2912	0.770416
$628$	6.20408	0.247570
$629$	14.0115	0.558674
$630$	1.72597	0.0687645
$631$	−32.4333	−1.29115	−0.645575	−	0.763696i	$-0.723382\pi$
$631$	−32.4333	−1.29115	−0.645575	+	0.763696i	$0.723382\pi$
$632$	−14.8665	−0.591359
$633$	−3.34117	−0.132800
$634$	−20.9596	−0.832413
$635$	26.0626	1.03426
$636$	−3.20165	−0.126954
$637$	−38.0906	−1.50920
$638$	−10.2343	−0.405178
$639$	15.5340	0.614517
$640$	31.7767	1.25608
$641$	11.0757	0.437462	0.218731	−	0.975785i	$-0.429808\pi$
$641$	11.0757	0.437462	0.218731	+	0.975785i	$0.429808\pi$
$642$	115.926	4.57524
$643$	10.7261	0.422995	0.211498	−	0.977378i	$-0.432166\pi$
$643$	10.7261	0.422995	0.211498	+	0.977378i	$0.432166\pi$
$644$	−0.278353	−0.0109686
$645$	77.9122	3.06779
$646$	4.55448	0.179194
$647$	−7.37626	−0.289991	−0.144995	−	0.989432i	$-0.546317\pi$
$647$	−7.37626	−0.289991	−0.144995	+	0.989432i	$0.546317\pi$
$648$	55.9884	2.19943
$649$	−39.5268	−1.55156
$650$	−19.9471	−0.782388
$651$	−0.894980	−0.0350770
$652$	0.00767296	0.000300496 0
$653$	21.5605	0.843726	0.421863	−	0.906660i	$-0.361376\pi$
$653$	21.5605	0.843726	0.421863	+	0.906660i	$0.361376\pi$
$654$	21.2010	0.829026
$655$	8.97564	0.350707
$656$	33.7666	1.31837
$657$	5.31623	0.207406
$658$	0.749299	0.0292107
$659$	−13.1817	−0.513488	−0.256744	−	0.966480i	$-0.582650\pi$
$659$	−13.1817	−0.513488	−0.256744	+	0.966480i	$0.582650\pi$
$660$	−52.9506	−2.06110
$661$	40.8506	1.58890	0.794452	−	0.607326i	$-0.207758\pi$
$661$	40.8506	1.58890	0.794452	+	0.607326i	$0.207758\pi$
$662$	51.5487	2.00350
$663$	47.5867	1.84811
$664$	−3.70572	−0.143810
$665$	−0.121468	−0.00471032
$666$	76.0886	2.94837
$667$	6.04094	0.233906
$668$	19.7113	0.762652
$669$	−23.4904	−0.908193
$670$	57.3684	2.21633
$671$	−19.7844	−0.763770
$672$	−0.810556	−0.0312679
$673$	−12.9417	−0.498865	−0.249432	−	0.968392i	$-0.580244\pi$
$673$	−12.9417	−0.498865	−0.249432	+	0.968392i	$0.580244\pi$
$674$	−30.8291	−1.18749
$675$	−36.3216	−1.39802
$676$	17.1218	0.658531
$677$	−42.5408	−1.63498	−0.817488	−	0.575946i	$-0.804634\pi$
$677$	−42.5408	−1.63498	−0.817488	+	0.575946i	$0.804634\pi$
$678$	94.4273	3.62646
$679$	0.703205	0.0269865
$680$	11.7796	0.451727
$681$	68.5202	2.62570
$682$	59.0757	2.26212
$683$	−22.0731	−0.844603	−0.422301	−	0.906455i	$-0.638778\pi$
$683$	−22.0731	−0.844603	−0.422301	+	0.906455i	$0.638778\pi$
$684$	8.40602	0.321412
$685$	−32.8053	−1.25343
$686$	1.11028	0.0423907
$687$	−61.6233	−2.35108
$688$	−43.7327	−1.66729
$689$	5.06530	0.192973
$690$	91.9608	3.50089
$691$	−18.0550	−0.686845	−0.343422	−	0.939181i	$-0.611586\pi$
$691$	−18.0550	−0.686845	−0.343422	+	0.939181i	$0.611586\pi$
$692$	−0.0615369	−0.00233928
$693$	−2.14783	−0.0815893
$694$	−22.7505	−0.863598
$695$	14.7682	0.560190
$696$	5.74636	0.217815
$697$	17.6739	0.669447
$698$	−0.862874	−0.0326603
$699$	−78.4395	−2.96686
$700$	0.0987905	0.00373393
$701$	6.75406	0.255097	0.127549	−	0.991832i	$-0.459289\pi$
$701$	6.75406	0.255097	0.127549	+	0.991832i	$0.459289\pi$
$702$	163.450	6.16903
$703$	−5.35484	−0.201961
$704$	−4.22457	−0.159220
$705$	−84.1353	−3.16872
$706$	−37.6021	−1.41517
$707$	0.740230	0.0278392
$708$	−23.5531	−0.885179
$709$	−20.9893	−0.788268	−0.394134	−	0.919053i	$-0.628955\pi$
$709$	−20.9893	−0.788268	−0.394134	+	0.919053i	$0.628955\pi$
$710$	8.82890	0.331343
$711$	−71.8593	−2.69493
$712$	−9.43646	−0.353646
$713$	−34.8704	−1.30591
$714$	−0.693434	−0.0259511
$715$	83.7727	3.13292
$716$	−14.2844	−0.533833
$717$	32.8981	1.22860
$718$	51.1172	1.90768
$719$	22.2685	0.830476	0.415238	−	0.909713i	$-0.363698\pi$
$719$	22.2685	0.830476	0.415238	+	0.909713i	$0.363698\pi$
$720$	108.782	4.05408
$721$	−0.885870	−0.0329915
$722$	−1.74061	−0.0647787
$723$	6.52079	0.242511
$724$	3.44739	0.128121
$725$	−2.14400	−0.0796261
$726$	129.902	4.82112
$727$	−35.3705	−1.31182	−0.655910	−	0.754839i	$-0.727715\pi$
$727$	−35.3705	−1.31182	−0.655910	+	0.754839i	$0.727715\pi$
$728$	0.418905	0.0155256
$729$	97.7026	3.61861
$730$	3.02153	0.111832
$731$	−22.8903	−0.846628
$732$	−11.7891	−0.435737
$733$	44.8057	1.65494	0.827469	−	0.561511i	$-0.189780\pi$
$733$	44.8057	1.65494	0.827469	+	0.561511i	$0.189780\pi$
$734$	−3.20919	−0.118454
$735$	−62.3249	−2.29889
$736$	−31.5811	−1.16409
$737$	−71.3902	−2.62969
$738$	95.9771	3.53297
$739$	11.3703	0.418263	0.209132	−	0.977887i	$-0.432936\pi$
$739$	11.3703	0.418263	0.209132	+	0.977887i	$0.432936\pi$
$740$	14.6980	0.540309
$741$	−18.1864	−0.668095
$742$	−0.0738117	−0.00270971
$743$	−48.6553	−1.78499	−0.892496	−	0.451056i	$-0.851047\pi$
$743$	−48.6553	−1.78499	−0.892496	+	0.451056i	$0.851047\pi$
$744$	−33.1700	−1.21607
$745$	−11.8942	−0.435769
$746$	8.78103	0.321496
$747$	−17.9121	−0.655368
$748$	15.5567	0.568808
$749$	0.908342	0.0331901
$750$	44.8731	1.63853
$751$	22.6187	0.825369	0.412685	−	0.910874i	$-0.364591\pi$
$751$	22.6187	0.825369	0.412685	+	0.910874i	$0.364591\pi$
$752$	47.2258	1.72215
$753$	0.197148	0.00718448
$754$	9.64817	0.351366
$755$	45.2192	1.64570
$756$	−0.809509	−0.0294416
$757$	−40.2314	−1.46223	−0.731117	−	0.682252i	$-0.761001\pi$
$757$	−40.2314	−1.46223	−0.731117	+	0.682252i	$0.761001\pi$
$758$	−24.2025	−0.879075
$759$	−114.438	−4.15382
$760$	−4.50187	−0.163300
$761$	10.9158	0.395696	0.197848	−	0.980233i	$-0.436605\pi$
$761$	10.9158	0.395696	0.197848	+	0.980233i	$0.436605\pi$
$762$	−56.8621	−2.05990
$763$	0.166121	0.00601399
$764$	22.1785	0.802388
$765$	56.9382	2.05860
$766$	63.4756	2.29347
$767$	37.2632	1.34549
$768$	−64.4396	−2.32526
$769$	−40.0451	−1.44406	−0.722031	−	0.691861i	$-0.756791\pi$
$769$	−40.0451	−1.44406	−0.722031	+	0.691861i	$0.756791\pi$
$770$	−1.22074	−0.0439924
$771$	−46.4657	−1.67342
$772$	1.78824	0.0643600
$773$	15.4759	0.556630	0.278315	−	0.960490i	$-0.410224\pi$
$773$	15.4759	0.556630	0.278315	+	0.960490i	$0.410224\pi$
$774$	−124.304	−4.46803
$775$	12.3759	0.444555
$776$	26.0623	0.935583
$777$	0.815291	0.0292484
$778$	47.5908	1.70621
$779$	−6.75452	−0.242006
$780$	49.9182	1.78736
$781$	−10.9868	−0.393140
$782$	−27.0177	−0.966152
$783$	17.5684	0.627842
$784$	34.9834	1.24941
$785$	−16.0603	−0.573215
$786$	−19.5827	−0.698490
$787$	25.3333	0.903036	0.451518	−	0.892262i	$-0.350883\pi$
$787$	25.3333	0.903036	0.451518	+	0.892262i	$0.350883\pi$
$788$	−23.3454	−0.831646
$789$	−72.6012	−2.58467
$790$	−40.8418	−1.45309
$791$	0.739887	0.0263074
$792$	−79.6034	−2.82858
$793$	18.6514	0.662332
$794$	−11.2638	−0.399736
$795$	8.28798	0.293944
$796$	−15.2947	−0.542105
$797$	−15.9386	−0.564573	−0.282286	−	0.959330i	$-0.591093\pi$
$797$	−15.9386	−0.564573	−0.282286	+	0.959330i	$0.591093\pi$
$798$	0.265013	0.00938136
$799$	24.7186	0.874482
$800$	11.2085	0.396279
$801$	−45.6123	−1.61163
$802$	39.0100	1.37749
$803$	−3.76004	−0.132689
$804$	−42.5397	−1.50026
$805$	0.720561	0.0253964
$806$	−55.6925	−1.96169
$807$	82.3177	2.89772
$808$	27.4346	0.965145
$809$	21.5734	0.758481	0.379240	−	0.925298i	$-0.376185\pi$
$809$	21.5734	0.758481	0.379240	+	0.925298i	$0.376185\pi$
$810$	153.813	5.40444
$811$	−28.2124	−0.990673	−0.495336	−	0.868701i	$-0.664955\pi$
$811$	−28.2124	−0.990673	−0.495336	+	0.868701i	$0.664955\pi$
$812$	−0.0477838	−0.00167688
$813$	64.5723	2.26465
$814$	−53.8155	−1.88623
$815$	−0.0198627	−0.000695760 0
$816$	−43.7049	−1.52998
$817$	8.74809	0.306057
$818$	−63.1408	−2.20767
$819$	2.02483	0.0707532
$820$	18.5398	0.647440
$821$	14.2937	0.498853	0.249426	−	0.968394i	$-0.419758\pi$
$821$	14.2937	0.498853	0.249426	+	0.968394i	$0.419758\pi$
$822$	71.5732	2.49640
$823$	8.02738	0.279817	0.139908	−	0.990164i	$-0.455319\pi$
$823$	8.02738	0.279817	0.139908	+	0.990164i	$0.455319\pi$
$824$	−32.8323	−1.14377
$825$	40.6152	1.41404
$826$	−0.543000	−0.0188934
$827$	35.4635	1.23319	0.616594	−	0.787281i	$-0.288512\pi$
$827$	35.4635	1.23319	0.616594	+	0.787281i	$0.288512\pi$
$828$	−49.8655	−1.73295
$829$	1.85139	0.0643013	0.0321507	−	0.999483i	$-0.489764\pi$
$829$	1.85139	0.0643013	0.0321507	+	0.999483i	$0.489764\pi$
$830$	−10.1805	−0.353369
$831$	19.3939	0.672765
$832$	3.98264	0.138073
$833$	18.3108	0.634431
$834$	−32.2206	−1.11571
$835$	−51.0258	−1.76582
$836$	−5.94537	−0.205625
$837$	−101.411	−3.50526
$838$	27.0045	0.932855
$839$	−15.8569	−0.547440	−0.273720	−	0.961809i	$-0.588254\pi$
$839$	−15.8569	−0.547440	−0.273720	+	0.961809i	$0.588254\pi$
$840$	0.685423	0.0236493
$841$	−27.9630	−0.964240
$842$	−8.68548	−0.299321
$843$	−7.10383	−0.244669
$844$	1.02972	0.0354444
$845$	−44.3225	−1.52474
$846$	134.233	4.61503
$847$	1.01785	0.0349737
$848$	−4.65211	−0.159754
$849$	8.62584	0.296038
$850$	9.58889	0.328896
$851$	31.7655	1.08891
$852$	−6.54680	−0.224290
$853$	37.3340	1.27829	0.639146	−	0.769085i	$-0.279288\pi$
$853$	37.3340	1.27829	0.639146	+	0.769085i	$0.279288\pi$
$854$	−0.271789	−0.00930044
$855$	−21.7603	−0.744188
$856$	33.6652	1.15065
$857$	−29.6198	−1.01179	−0.505896	−	0.862594i	$-0.668838\pi$
$857$	−29.6198	−1.01179	−0.505896	+	0.862594i	$0.668838\pi$
$858$	−182.772	−6.23972
$859$	42.0851	1.43593	0.717963	−	0.696082i	$-0.245075\pi$
$859$	42.0851	1.43593	0.717963	+	0.696082i	$0.245075\pi$
$860$	−24.0118	−0.818796
$861$	1.02840	0.0350477
$862$	1.61452	0.0549908
$863$	5.08070	0.172949	0.0864745	−	0.996254i	$-0.472440\pi$
$863$	5.08070	0.172949	0.0864745	+	0.996254i	$0.472440\pi$
$864$	−91.8445	−3.12461
$865$	0.159298	0.00541630
$866$	−7.83127	−0.266117
$867$	33.9242	1.15213
$868$	0.275825	0.00936210
$869$	50.8243	1.72410
$870$	15.7866	0.535215
$871$	67.3018	2.28043
$872$	6.15682	0.208496
$873$	125.976	4.26363
$874$	10.3255	0.349265
$875$	0.351604	0.0118864
$876$	−2.24052	−0.0757002
$877$	31.7332	1.07155	0.535777	−	0.844359i	$-0.320019\pi$
$877$	31.7332	1.07155	0.535777	+	0.844359i	$0.320019\pi$
$878$	35.4612	1.19676
$879$	−72.3604	−2.44066
$880$	−76.9391	−2.59362
$881$	−17.1482	−0.577737	−0.288868	−	0.957369i	$-0.593279\pi$
$881$	−17.1482	−0.577737	−0.288868	+	0.957369i	$0.593279\pi$
$882$	99.4357	3.34817
$883$	−29.1692	−0.981622	−0.490811	−	0.871266i	$-0.663300\pi$
$883$	−29.1692	−0.981622	−0.490811	+	0.871266i	$0.663300\pi$
$884$	−14.6658	−0.493263
$885$	60.9710	2.04952
$886$	−68.4002	−2.29795
$887$	8.50695	0.285635	0.142818	−	0.989749i	$-0.454384\pi$
$887$	8.50695	0.285635	0.142818	+	0.989749i	$0.454384\pi$
$888$	30.2165	1.01400
$889$	−0.445544	−0.0149431
$890$	−25.9242	−0.868980
$891$	−191.408	−6.41239
$892$	7.23954	0.242398
$893$	−9.44683	−0.316126
$894$	25.9502	0.867904
$895$	36.9775	1.23602
$896$	−0.543228	−0.0181480
$897$	107.884	3.60214
$898$	−27.6608	−0.923052
$899$	−5.98608	−0.199647
$900$	17.6978	0.589927
$901$	−2.43497	−0.0811208
$902$	−67.8822	−2.26023
$903$	−1.33192	−0.0443237
$904$	27.4219	0.912038
$905$	−8.92414	−0.296648
$906$	−98.6573	−3.27767
$907$	−33.1020	−1.09913	−0.549566	−	0.835450i	$-0.685207\pi$
$907$	−33.1020	−1.09913	−0.549566	+	0.835450i	$0.685207\pi$
$908$	−21.1173	−0.700802
$909$	132.609	4.39835
$910$	1.15083	0.0381496
$911$	13.7116	0.454285	0.227143	−	0.973861i	$-0.427062\pi$
$911$	13.7116	0.454285	0.227143	+	0.973861i	$0.427062\pi$
$912$	16.7029	0.553088
$913$	12.6688	0.419274
$914$	25.7651	0.852235
$915$	30.5180	1.00889
$916$	18.9917	0.627504
$917$	−0.153440	−0.00506705
$918$	−78.5733	−2.59331
$919$	−22.0703	−0.728031	−0.364016	−	0.931393i	$-0.618594\pi$
$919$	−22.0703	−0.728031	−0.364016	+	0.931393i	$0.618594\pi$
$920$	26.7056	0.880458
$921$	−93.3338	−3.07545
$922$	−55.6644	−1.83321
$923$	10.3576	0.340926
$924$	0.905201	0.0297789
$925$	−11.2739	−0.370685
$926$	−17.1736	−0.564360
$927$	−158.699	−5.21236
$928$	−5.42141	−0.177966
$929$	−20.5367	−0.673788	−0.336894	−	0.941543i	$-0.609376\pi$
$929$	−20.5367	−0.673788	−0.336894	+	0.941543i	$0.609376\pi$
$930$	−91.1256	−2.98813
$931$	−6.99792	−0.229348
$932$	24.1743	0.791857
$933$	−38.2927	−1.25365
$934$	−50.4059	−1.64933
$935$	−40.2709	−1.31700
$936$	75.0447	2.45291
$937$	−39.4019	−1.28720	−0.643602	−	0.765360i	$-0.722561\pi$
$937$	−39.4019	−1.28720	−0.643602	+	0.765360i	$0.722561\pi$
$938$	−0.980724	−0.0320218
$939$	19.9681	0.651633
$940$	25.9297	0.845735
$941$	−28.9711	−0.944432	−0.472216	−	0.881483i	$-0.656546\pi$
$941$	−28.9711	−0.944432	−0.472216	+	0.881483i	$0.656546\pi$
$942$	35.0396	1.14165
$943$	40.0686	1.30481
$944$	−34.2235	−1.11388
$945$	2.09555	0.0681681
$946$	87.9174	2.85844
$947$	8.24181	0.267823	0.133911	−	0.990993i	$-0.457246\pi$
$947$	8.24181	0.267823	0.133911	+	0.990993i	$0.457246\pi$
$948$	30.2850	0.983611
$949$	3.54471	0.115066
$950$	−3.66463	−0.118896
$951$	−40.2328	−1.30464
$952$	−0.201375	−0.00652659
$953$	−28.2162	−0.914012	−0.457006	−	0.889463i	$-0.651078\pi$
$953$	−28.2162	−0.914012	−0.457006	+	0.889463i	$0.651078\pi$
$954$	−13.2230	−0.428110
$955$	−57.4125	−1.85783
$956$	−10.1389	−0.327915
$957$	−19.6451	−0.635036
$958$	21.9882	0.710407
$959$	0.560813	0.0181096
$960$	6.51650	0.210319
$961$	3.55368	0.114635
$962$	50.7336	1.63572
$963$	162.725	5.24374
$964$	−2.00965	−0.0647264
$965$	−4.62914	−0.149017
$966$	−1.57209	−0.0505811
$967$	−24.0414	−0.773120	−0.386560	−	0.922264i	$-0.626337\pi$
$967$	−24.0414	−0.773120	−0.386560	+	0.922264i	$0.626337\pi$
$968$	37.7238	1.21249
$969$	8.74252	0.280850
$970$	71.5993	2.29892
$971$	47.5150	1.52483	0.762415	−	0.647089i	$-0.224014\pi$
$971$	47.5150	1.52483	0.762415	+	0.647089i	$0.224014\pi$
$972$	−60.7617	−1.94893
$973$	−0.252465	−0.00809367
$974$	41.6285	1.33386
$975$	−38.2892	−1.22624
$976$	−17.1300	−0.548317
$977$	−50.4242	−1.61321	−0.806607	−	0.591088i	$-0.798698\pi$
$977$	−50.4242	−1.61321	−0.806607	+	0.591088i	$0.798698\pi$
$978$	0.0433356	0.00138572
$979$	32.2604	1.03105
$980$	19.2079	0.613575
$981$	29.7598	0.950156
$982$	−2.53430	−0.0808726
$983$	−37.6059	−1.19944	−0.599722	−	0.800209i	$-0.704722\pi$
$983$	−37.6059	−1.19944	−0.599722	+	0.800209i	$0.704722\pi$
$984$	38.1147	1.21505
$985$	60.4334	1.92557
$986$	−4.63804	−0.147705
$987$	1.43831	0.0457819
$988$	5.60489	0.178315
$989$	−51.8947	−1.65016
$990$	−218.689	−6.95039
$991$	−24.7989	−0.787763	−0.393882	−	0.919161i	$-0.628868\pi$
$991$	−24.7989	−0.787763	−0.393882	+	0.919161i	$0.628868\pi$
$992$	31.2942	0.993593
$993$	98.9499	3.14008
$994$	−0.150932	−0.00478727
$995$	39.5927	1.25517
$996$	7.54901	0.239200
$997$	31.1879	0.987732	0.493866	−	0.869538i	$-0.335583\pi$
$997$	31.1879	0.987732	0.493866	+	0.869538i	$0.335583\pi$
$998$	26.5537	0.840542
$999$	92.3809	2.92280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.19	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.19	✓	71	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.74061 q^{2} -3.34117 q^{3} +1.02972 q^{4} -2.66559 q^{5} +5.81567 q^{6} +0.0455688 q^{7} +1.68888 q^{8} +8.16342 q^{9} +O(q^{10})\)	\(q-1.74061 q^{2} -3.34117 q^{3} +1.02972 q^{4} -2.66559 q^{5} +5.81567 q^{6} +0.0455688 q^{7} +1.68888 q^{8} +8.16342 q^{9} +4.63975 q^{10} -5.77378 q^{11} -3.44046 q^{12} +5.44313 q^{13} -0.0793174 q^{14} +8.90619 q^{15} -4.99912 q^{16} -2.61660 q^{17} -14.2093 q^{18} +1.00000 q^{19} -2.74481 q^{20} -0.152253 q^{21} +10.0499 q^{22} -5.93212 q^{23} -5.64284 q^{24} +2.10537 q^{25} -9.47436 q^{26} -17.2519 q^{27} +0.0469230 q^{28} -1.01835 q^{29} -15.5022 q^{30} +5.87824 q^{31} +5.32374 q^{32} +19.2912 q^{33} +4.55448 q^{34} -0.121468 q^{35} +8.40602 q^{36} -5.35484 q^{37} -1.74061 q^{38} -18.1864 q^{39} -4.50187 q^{40} -6.75452 q^{41} +0.265013 q^{42} +8.74809 q^{43} -5.94537 q^{44} -21.7603 q^{45} +10.3255 q^{46} -9.44683 q^{47} +16.7029 q^{48} -6.99792 q^{49} -3.66463 q^{50} +8.74252 q^{51} +5.60489 q^{52} +0.930586 q^{53} +30.0287 q^{54} +15.3905 q^{55} +0.0769603 q^{56} -3.34117 q^{57} +1.77254 q^{58} +6.84591 q^{59} +9.17087 q^{60} +3.42660 q^{61} -10.2317 q^{62} +0.371997 q^{63} +0.731682 q^{64} -14.5092 q^{65} -33.5784 q^{66} +12.3645 q^{67} -2.69436 q^{68} +19.8202 q^{69} +0.211428 q^{70} +1.90288 q^{71} +13.7870 q^{72} +0.651226 q^{73} +9.32067 q^{74} -7.03441 q^{75} +1.02972 q^{76} -0.263104 q^{77} +31.6554 q^{78} -8.80260 q^{79} +13.3256 q^{80} +33.1512 q^{81} +11.7570 q^{82} -2.19419 q^{83} -0.156778 q^{84} +6.97480 q^{85} -15.2270 q^{86} +3.40247 q^{87} -9.75123 q^{88} -5.58740 q^{89} +37.8762 q^{90} +0.248037 q^{91} -6.10841 q^{92} -19.6402 q^{93} +16.4432 q^{94} -2.66559 q^{95} -17.7875 q^{96} +15.4317 q^{97} +12.1806 q^{98} -47.1338 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

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