LMFDB - Embedded newform 8047.2.a.e.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2556185065$
Analytic rank:	$0$
Dimension:	$168$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43925 q^{2} -2.47662 q^{3} +3.94994 q^{4} +4.18680 q^{5} +6.04110 q^{6} +3.85857 q^{7} -4.75640 q^{8} +3.13366 q^{9} +O(q^{10})$	\(q-2.43925 q^{2} -2.47662 q^{3} +3.94994 q^{4} +4.18680 q^{5} +6.04110 q^{6} +3.85857 q^{7} -4.75640 q^{8} +3.13366 q^{9} -10.2127 q^{10} +3.54092 q^{11} -9.78252 q^{12} +1.00000 q^{13} -9.41202 q^{14} -10.3691 q^{15} +3.70216 q^{16} +1.46406 q^{17} -7.64378 q^{18} +7.43716 q^{19} +16.5376 q^{20} -9.55622 q^{21} -8.63720 q^{22} +3.58276 q^{23} +11.7798 q^{24} +12.5293 q^{25} -2.43925 q^{26} -0.331028 q^{27} +15.2411 q^{28} -3.84691 q^{29} +25.2929 q^{30} -9.75626 q^{31} +0.482302 q^{32} -8.76953 q^{33} -3.57122 q^{34} +16.1551 q^{35} +12.3778 q^{36} -6.76186 q^{37} -18.1411 q^{38} -2.47662 q^{39} -19.9141 q^{40} +5.89377 q^{41} +23.3100 q^{42} +4.42445 q^{43} +13.9864 q^{44} +13.1200 q^{45} -8.73925 q^{46} -7.00332 q^{47} -9.16885 q^{48} +7.88856 q^{49} -30.5621 q^{50} -3.62593 q^{51} +3.94994 q^{52} -9.39244 q^{53} +0.807459 q^{54} +14.8251 q^{55} -18.3529 q^{56} -18.4190 q^{57} +9.38357 q^{58} -13.9416 q^{59} -40.9574 q^{60} -13.5957 q^{61} +23.7980 q^{62} +12.0915 q^{63} -8.58077 q^{64} +4.18680 q^{65} +21.3911 q^{66} -2.54988 q^{67} +5.78296 q^{68} -8.87314 q^{69} -39.4062 q^{70} +1.13878 q^{71} -14.9049 q^{72} +1.96202 q^{73} +16.4939 q^{74} -31.0303 q^{75} +29.3763 q^{76} +13.6629 q^{77} +6.04110 q^{78} +9.60173 q^{79} +15.5002 q^{80} -8.58115 q^{81} -14.3764 q^{82} +2.32718 q^{83} -37.7465 q^{84} +6.12974 q^{85} -10.7923 q^{86} +9.52734 q^{87} -16.8420 q^{88} +9.45610 q^{89} -32.0030 q^{90} +3.85857 q^{91} +14.1517 q^{92} +24.1626 q^{93} +17.0829 q^{94} +31.1379 q^{95} -1.19448 q^{96} +15.3532 q^{97} -19.2422 q^{98} +11.0961 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) $ 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.43925	−1.72481	−0.862405	−	0.506219i	$-0.831043\pi$
$2$	−2.43925	−1.72481	−0.862405	+	0.506219i	$0.831043\pi$
$3$	−2.47662	−1.42988	−0.714939	−	0.699186i	$-0.753546\pi$
$3$	−2.47662	−1.42988	−0.714939	+	0.699186i	$0.753546\pi$
$4$	3.94994	1.97497
$5$	4.18680	1.87239	0.936197	−	0.351476i	$-0.114320\pi$
$5$	4.18680	1.87239	0.936197	+	0.351476i	$0.114320\pi$
$6$	6.04110	2.46627
$7$	3.85857	1.45840	0.729201	−	0.684299i	$-0.239892\pi$
$7$	3.85857	1.45840	0.729201	+	0.684299i	$0.239892\pi$
$8$	−4.75640	−1.68164
$9$	3.13366	1.04455
$10$	−10.2127	−3.22952
$11$	3.54092	1.06763	0.533814	−	0.845602i	$-0.320758\pi$
$11$	3.54092	1.06763	0.533814	+	0.845602i	$0.320758\pi$
$12$	−9.78252	−2.82397
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−9.41202	−2.51547
$15$	−10.3691	−2.67730
$16$	3.70216	0.925540
$17$	1.46406	0.355087	0.177544	−	0.984113i	$-0.443185\pi$
$17$	1.46406	0.355087	0.177544	+	0.984113i	$0.443185\pi$
$18$	−7.64378	−1.80166
$19$	7.43716	1.70620	0.853100	−	0.521747i	$-0.174719\pi$
$19$	7.43716	1.70620	0.853100	+	0.521747i	$0.174719\pi$
$20$	16.5376	3.69792
$21$	−9.55622	−2.08534
$22$	−8.63720	−1.84146
$23$	3.58276	0.747057	0.373528	−	0.927619i	$-0.378148\pi$
$23$	3.58276	0.747057	0.373528	+	0.927619i	$0.378148\pi$
$24$	11.7798	2.40454
$25$	12.5293	2.50586
$26$	−2.43925	−0.478376
$27$	−0.331028	−0.0637063
$28$	15.2411	2.88030
$29$	−3.84691	−0.714353	−0.357176	−	0.934037i	$-0.616261\pi$
$29$	−3.84691	−0.714353	−0.357176	+	0.934037i	$0.616261\pi$
$30$	25.2929	4.61783
$31$	−9.75626	−1.75228	−0.876138	−	0.482060i	$-0.839889\pi$
$31$	−9.75626	−1.75228	−0.876138	+	0.482060i	$0.839889\pi$
$32$	0.482302	0.0852597
$33$	−8.76953	−1.52658
$34$	−3.57122	−0.612458
$35$	16.1551	2.73070
$36$	12.3778	2.06296
$37$	−6.76186	−1.11164	−0.555822	−	0.831302i	$-0.687596\pi$
$37$	−6.76186	−1.11164	−0.555822	+	0.831302i	$0.687596\pi$
$38$	−18.1411	−2.94287
$39$	−2.47662	−0.396577
$40$	−19.9141	−3.14869
$41$	5.89377	0.920452	0.460226	−	0.887802i	$-0.347768\pi$
$41$	5.89377	0.920452	0.460226	+	0.887802i	$0.347768\pi$
$42$	23.3100	3.59681
$43$	4.42445	0.674722	0.337361	−	0.941375i	$-0.390466\pi$
$43$	4.42445	0.674722	0.337361	+	0.941375i	$0.390466\pi$
$44$	13.9864	2.10854
$45$	13.1200	1.95582
$46$	−8.73925	−1.28853
$47$	−7.00332	−1.02154	−0.510770	−	0.859717i	$-0.670640\pi$
$47$	−7.00332	−1.02154	−0.510770	+	0.859717i	$0.670640\pi$
$48$	−9.16885	−1.32341
$49$	7.88856	1.12694
$50$	−30.5621	−4.32213
$51$	−3.62593	−0.507732
$52$	3.94994	0.547758
$53$	−9.39244	−1.29015	−0.645075	−	0.764119i	$-0.723174\pi$
$53$	−9.39244	−1.29015	−0.645075	+	0.764119i	$0.723174\pi$
$54$	0.807459	0.109881
$55$	14.8251	1.99902
$56$	−18.3529	−2.45251
$57$	−18.4190	−2.43966
$58$	9.38357	1.23212
$59$	−13.9416	−1.81504	−0.907522	−	0.420004i	$-0.862029\pi$
$59$	−13.9416	−1.81504	−0.907522	+	0.420004i	$0.862029\pi$
$60$	−40.9574	−5.28758
$61$	−13.5957	−1.74075	−0.870377	−	0.492386i	$-0.836125\pi$
$61$	−13.5957	−1.74075	−0.870377	+	0.492386i	$0.836125\pi$
$62$	23.7980	3.02234
$63$	12.0915	1.52338
$64$	−8.58077	−1.07260
$65$	4.18680	0.519309
$66$	21.3911	2.63306
$67$	−2.54988	−0.311517	−0.155758	−	0.987795i	$-0.549782\pi$
$67$	−2.54988	−0.311517	−0.155758	+	0.987795i	$0.549782\pi$
$68$	5.78296	0.701287
$69$	−8.87314	−1.06820
$70$	−39.4062	−4.70995
$71$	1.13878	0.135148	0.0675742	−	0.997714i	$-0.478474\pi$
$71$	1.13878	0.135148	0.0675742	+	0.997714i	$0.478474\pi$
$72$	−14.9049	−1.75656
$73$	1.96202	0.229637	0.114819	−	0.993386i	$-0.463371\pi$
$73$	1.96202	0.229637	0.114819	+	0.993386i	$0.463371\pi$
$74$	16.4939	1.91737
$75$	−31.0303	−3.58308
$76$	29.3763	3.36970
$77$	13.6629	1.55703
$78$	6.04110	0.684020
$79$	9.60173	1.08028	0.540139	−	0.841576i	$-0.318371\pi$
$79$	9.60173	1.08028	0.540139	+	0.841576i	$0.318371\pi$
$80$	15.5002	1.73298
$81$	−8.58115	−0.953461
$82$	−14.3764	−1.58761
$83$	2.32718	0.255442	0.127721	−	0.991810i	$-0.459234\pi$
$83$	2.32718	0.255442	0.127721	+	0.991810i	$0.459234\pi$
$84$	−37.7465	−4.11848
$85$	6.12974	0.664864
$86$	−10.7923	−1.16377
$87$	9.52734	1.02144
$88$	−16.8420	−1.79537
$89$	9.45610	1.00234	0.501172	−	0.865347i	$-0.332902\pi$
$89$	9.45610	1.00234	0.501172	+	0.865347i	$0.332902\pi$
$90$	−32.0030	−3.37341
$91$	3.85857	0.404488
$92$	14.1517	1.47542
$93$	24.1626	2.50554
$94$	17.0829	1.76196
$95$	31.1379	3.19468
$96$	−1.19448	−0.121911
$97$	15.3532	1.55888	0.779441	−	0.626475i	$-0.215503\pi$
$97$	15.3532	1.55888	0.779441	+	0.626475i	$0.215503\pi$
$98$	−19.2422	−1.94375
$99$	11.0961	1.11520
$100$	49.4900	4.94900
$101$	6.21121	0.618039	0.309019	−	0.951056i	$-0.399999\pi$
$101$	6.21121	0.618039	0.309019	+	0.951056i	$0.399999\pi$
$102$	8.84456	0.875741
$103$	−10.1621	−1.00130	−0.500649	−	0.865650i	$-0.666905\pi$
$103$	−10.1621	−1.00130	−0.500649	+	0.865650i	$0.666905\pi$
$104$	−4.75640	−0.466403
$105$	−40.0100	−3.90458
$106$	22.9105	2.22527
$107$	6.57430	0.635561	0.317781	−	0.948164i	$-0.397062\pi$
$107$	6.57430	0.635561	0.317781	+	0.948164i	$0.397062\pi$
$108$	−1.30754	−0.125818
$109$	3.86401	0.370105	0.185053	−	0.982729i	$-0.440754\pi$
$109$	3.86401	0.370105	0.185053	+	0.982729i	$0.440754\pi$
$110$	−36.1622	−3.44793
$111$	16.7466	1.58952
$112$	14.2850	1.34981
$113$	14.1576	1.33184	0.665918	−	0.746025i	$-0.268040\pi$
$113$	14.1576	1.33184	0.665918	+	0.746025i	$0.268040\pi$
$114$	44.9286	4.20795
$115$	15.0003	1.39878
$116$	−15.1951	−1.41083
$117$	3.13366	0.289707
$118$	34.0071	3.13061
$119$	5.64919	0.517860
$120$	49.3197	4.50225
$121$	1.53813	0.139830
$122$	33.1634	3.00247
$123$	−14.5966	−1.31614
$124$	−38.5367	−3.46070
$125$	31.5237	2.81956
$126$	−29.4941	−2.62754
$127$	−12.2239	−1.08470	−0.542348	−	0.840154i	$-0.682465\pi$
$127$	−12.2239	−1.08470	−0.542348	+	0.840154i	$0.682465\pi$
$128$	19.9660	1.76477
$129$	−10.9577	−0.964771
$130$	−10.2127	−0.895709
$131$	7.52485	0.657450	0.328725	−	0.944426i	$-0.393381\pi$
$131$	7.52485	0.657450	0.328725	+	0.944426i	$0.393381\pi$
$132$	−34.6391	−3.01495
$133$	28.6968	2.48833
$134$	6.21979	0.537308
$135$	−1.38595	−0.119283
$136$	−6.96366	−0.597129
$137$	13.6924	1.16982	0.584912	−	0.811097i	$-0.301129\pi$
$137$	13.6924	1.16982	0.584912	+	0.811097i	$0.301129\pi$
$138$	21.6438	1.84244
$139$	3.21793	0.272941	0.136471	−	0.990644i	$-0.456424\pi$
$139$	3.21793	0.272941	0.136471	+	0.990644i	$0.456424\pi$
$140$	63.8116	5.39306
$141$	17.3446	1.46068
$142$	−2.77777	−0.233105
$143$	3.54092	0.296107
$144$	11.6013	0.966776
$145$	−16.1062	−1.33755
$146$	−4.78587	−0.396081
$147$	−19.5370	−1.61138
$148$	−26.7090	−2.19546
$149$	−16.8142	−1.37748	−0.688738	−	0.725010i	$-0.741835\pi$
$149$	−16.8142	−1.37748	−0.688738	+	0.725010i	$0.741835\pi$
$150$	75.6908	6.18013
$151$	0.613201	0.0499016	0.0249508	−	0.999689i	$-0.492057\pi$
$151$	0.613201	0.0499016	0.0249508	+	0.999689i	$0.492057\pi$
$152$	−35.3741	−2.86922
$153$	4.58788	0.370908
$154$	−33.3272	−2.68558
$155$	−40.8475	−3.28095
$156$	−9.78252	−0.783228
$157$	1.88882	0.150744	0.0753720	−	0.997155i	$-0.475986\pi$
$157$	1.88882	0.150744	0.0753720	+	0.997155i	$0.475986\pi$
$158$	−23.4210	−1.86328
$159$	23.2615	1.84476
$160$	2.01930	0.159640
$161$	13.8243	1.08951
$162$	20.9316	1.64454
$163$	−3.72076	−0.291433	−0.145716	−	0.989326i	$-0.546549\pi$
$163$	−3.72076	−0.291433	−0.145716	+	0.989326i	$0.546549\pi$
$164$	23.2800	1.81787
$165$	−36.7163	−2.85836
$166$	−5.67658	−0.440588
$167$	2.35430	0.182181	0.0910907	−	0.995843i	$-0.470965\pi$
$167$	2.35430	0.182181	0.0910907	+	0.995843i	$0.470965\pi$
$168$	45.4532	3.50679
$169$	1.00000	0.0769231
$170$	−14.9520	−1.14676
$171$	23.3055	1.78222
$172$	17.4763	1.33256
$173$	13.2416	1.00674	0.503369	−	0.864072i	$-0.332094\pi$
$173$	13.2416	1.00674	0.503369	+	0.864072i	$0.332094\pi$
$174$	−23.2396	−1.76179
$175$	48.3452	3.65455
$176$	13.1091	0.988132
$177$	34.5281	2.59529
$178$	−23.0658	−1.72885
$179$	−2.90782	−0.217340	−0.108670	−	0.994078i	$-0.534659\pi$
$179$	−2.90782	−0.217340	−0.108670	+	0.994078i	$0.534659\pi$
$180$	51.8233	3.86268
$181$	−20.2992	−1.50883	−0.754416	−	0.656397i	$-0.772080\pi$
$181$	−20.2992	−1.50883	−0.754416	+	0.656397i	$0.772080\pi$
$182$	−9.41202	−0.697665
$183$	33.6715	2.48907
$184$	−17.0410	−1.25628
$185$	−28.3106	−2.08143
$186$	−58.9386	−4.32159
$187$	5.18413	0.379101
$188$	−27.6627	−2.01751
$189$	−1.27729	−0.0929094
$190$	−75.9531	−5.51022
$191$	−20.9736	−1.51759	−0.758797	−	0.651327i	$-0.774213\pi$
$191$	−20.9736	−1.51759	−0.758797	+	0.651327i	$0.774213\pi$
$192$	21.2513	1.53368
$193$	26.2852	1.89205	0.946026	−	0.324089i	$-0.105058\pi$
$193$	26.2852	1.89205	0.946026	+	0.324089i	$0.105058\pi$
$194$	−37.4503	−2.68878
$195$	−10.3691	−0.742548
$196$	31.1594	2.22567
$197$	0.197104	0.0140431	0.00702154	−	0.999975i	$-0.497765\pi$
$197$	0.197104	0.0140431	0.00702154	+	0.999975i	$0.497765\pi$
$198$	−27.0660	−1.92350
$199$	9.96990	0.706747	0.353374	−	0.935482i	$-0.385034\pi$
$199$	9.96990	0.706747	0.353374	+	0.935482i	$0.385034\pi$
$200$	−59.5943	−4.21395
$201$	6.31508	0.445431
$202$	−15.1507	−1.06600
$203$	−14.8436	−1.04181
$204$	−14.3222	−1.00276
$205$	24.6760	1.72345
$206$	24.7878	1.72705
$207$	11.2272	0.780341
$208$	3.70216	0.256699
$209$	26.3344	1.82159
$210$	97.5944	6.73465
$211$	13.2796	0.914204	0.457102	−	0.889414i	$-0.348888\pi$
$211$	13.2796	0.914204	0.457102	+	0.889414i	$0.348888\pi$
$212$	−37.0996	−2.54801
$213$	−2.82033	−0.193246
$214$	−16.0364	−1.09622
$215$	18.5243	1.26335
$216$	1.57450	0.107131
$217$	−37.6452	−2.55552
$218$	−9.42530	−0.638362
$219$	−4.85919	−0.328354
$220$	58.5584	3.94801
$221$	1.46406	0.0984835
$222$	−40.8491	−2.74161
$223$	21.9654	1.47091	0.735455	−	0.677574i	$-0.236969\pi$
$223$	21.9654	1.47091	0.735455	+	0.677574i	$0.236969\pi$
$224$	1.86100	0.124343
$225$	39.2626	2.61750
$226$	−34.5340	−2.29716
$227$	19.4316	1.28972	0.644861	−	0.764300i	$-0.276915\pi$
$227$	19.4316	1.28972	0.644861	+	0.764300i	$0.276915\pi$
$228$	−72.7541	−4.81826
$229$	21.6419	1.43014	0.715069	−	0.699054i	$-0.246395\pi$
$229$	21.6419	1.43014	0.715069	+	0.699054i	$0.246395\pi$
$230$	−36.5895	−2.41264
$231$	−33.8378	−2.22637
$232$	18.2974	1.20128
$233$	−3.56442	−0.233513	−0.116757	−	0.993161i	$-0.537250\pi$
$233$	−3.56442	−0.233513	−0.116757	+	0.993161i	$0.537250\pi$
$234$	−7.64378	−0.499690
$235$	−29.3215	−1.91273
$236$	−55.0686	−3.58466
$237$	−23.7799	−1.54467
$238$	−13.7798	−0.893211
$239$	14.6533	0.947846	0.473923	−	0.880566i	$-0.342838\pi$
$239$	14.6533	0.947846	0.473923	+	0.880566i	$0.342838\pi$
$240$	−38.3882	−2.47794
$241$	−2.80131	−0.180448	−0.0902240	−	0.995921i	$-0.528758\pi$
$241$	−2.80131	−0.180448	−0.0902240	+	0.995921i	$0.528758\pi$
$242$	−3.75189	−0.241181
$243$	22.2454	1.42704
$244$	−53.7023	−3.43794
$245$	33.0278	2.11007
$246$	35.6049	2.27008
$247$	7.43716	0.473215
$248$	46.4047	2.94670
$249$	−5.76355	−0.365250
$250$	−76.8941	−4.86321
$251$	−0.930060	−0.0587048	−0.0293524	−	0.999569i	$-0.509345\pi$
$251$	−0.930060	−0.0587048	−0.0293524	+	0.999569i	$0.509345\pi$
$252$	47.7605	3.00863
$253$	12.6863	0.797579
$254$	29.8172	1.87090
$255$	−15.1811	−0.950674
$256$	−31.5406	−1.97129
$257$	−20.8470	−1.30040	−0.650199	−	0.759764i	$-0.725315\pi$
$257$	−20.8470	−1.30040	−0.650199	+	0.759764i	$0.725315\pi$
$258$	26.7286	1.66405
$259$	−26.0911	−1.62122
$260$	16.5376	1.02562
$261$	−12.0549	−0.746180
$262$	−18.3550	−1.13398
$263$	0.920833	0.0567810	0.0283905	−	0.999597i	$-0.490962\pi$
$263$	0.920833	0.0567810	0.0283905	+	0.999597i	$0.490962\pi$
$264$	41.7114	2.56716
$265$	−39.3243	−2.41567
$266$	−69.9987	−4.29189
$267$	−23.4192	−1.43323
$268$	−10.0719	−0.615237
$269$	29.7328	1.81284	0.906420	−	0.422377i	$-0.138804\pi$
$269$	29.7328	1.81284	0.906420	+	0.422377i	$0.138804\pi$
$270$	3.38067	0.205741
$271$	12.3241	0.748635	0.374317	−	0.927301i	$-0.377877\pi$
$271$	12.3241	0.748635	0.374317	+	0.927301i	$0.377877\pi$
$272$	5.42019	0.328648
$273$	−9.55622	−0.578369
$274$	−33.3993	−2.01772
$275$	44.3653	2.67533
$276$	−35.0484	−2.10967
$277$	22.6664	1.36189	0.680945	−	0.732335i	$-0.261569\pi$
$277$	22.6664	1.36189	0.680945	+	0.732335i	$0.261569\pi$
$278$	−7.84934	−0.470772
$279$	−30.5728	−1.83035
$280$	−76.8399	−4.59206
$281$	27.4321	1.63646	0.818232	−	0.574889i	$-0.194955\pi$
$281$	27.4321	1.63646	0.818232	+	0.574889i	$0.194955\pi$
$282$	−42.3078	−2.51939
$283$	−14.3012	−0.850121	−0.425061	−	0.905165i	$-0.639747\pi$
$283$	−14.3012	−0.850121	−0.425061	+	0.905165i	$0.639747\pi$
$284$	4.49812	0.266914
$285$	−77.1168	−4.56801
$286$	−8.63720	−0.510728
$287$	22.7415	1.34239
$288$	1.51137	0.0890583
$289$	−14.8565	−0.873913
$290$	39.2871	2.30702
$291$	−38.0241	−2.22901
$292$	7.74988	0.453527
$293$	−18.5938	−1.08626	−0.543130	−	0.839648i	$-0.682761\pi$
$293$	−18.5938	−1.08626	−0.543130	+	0.839648i	$0.682761\pi$
$294$	47.6556	2.77933
$295$	−58.3708	−3.39848
$296$	32.1621	1.86938
$297$	−1.17214	−0.0680147
$298$	41.0142	2.37589
$299$	3.58276	0.207196
$300$	−122.568	−7.07647
$301$	17.0720	0.984017
$302$	−1.49575	−0.0860708
$303$	−15.3828	−0.883721
$304$	27.5335	1.57916
$305$	−56.9226	−3.25938
$306$	−11.1910	−0.639746
$307$	−11.3519	−0.647889	−0.323944	−	0.946076i	$-0.605009\pi$
$307$	−11.3519	−0.647889	−0.323944	+	0.946076i	$0.605009\pi$
$308$	53.9677	3.07509
$309$	25.1676	1.43174
$310$	99.6373	5.65902
$311$	−29.8924	−1.69504	−0.847521	−	0.530762i	$-0.821906\pi$
$311$	−29.8924	−1.69504	−0.847521	+	0.530762i	$0.821906\pi$
$312$	11.7798	0.666900
$313$	−9.42035	−0.532470	−0.266235	−	0.963908i	$-0.585780\pi$
$313$	−9.42035	−0.532470	−0.266235	+	0.963908i	$0.585780\pi$
$314$	−4.60730	−0.260005
$315$	50.6245	2.85237
$316$	37.9263	2.13352
$317$	22.6758	1.27360	0.636799	−	0.771030i	$-0.280258\pi$
$317$	22.6758	1.27360	0.636799	+	0.771030i	$0.280258\pi$
$318$	−56.7407	−3.18186
$319$	−13.6216	−0.762663
$320$	−35.9260	−2.00832
$321$	−16.2821	−0.908776
$322$	−33.7210	−1.87920
$323$	10.8885	0.605850
$324$	−33.8951	−1.88306
$325$	12.5293	0.695000
$326$	9.07588	0.502666
$327$	−9.56971	−0.529206
$328$	−28.0331	−1.54787
$329$	−27.0228	−1.48982
$330$	89.5602	4.93013
$331$	−20.5096	−1.12731	−0.563654	−	0.826011i	$-0.690605\pi$
$331$	−20.5096	−1.12731	−0.563654	+	0.826011i	$0.690605\pi$
$332$	9.19224	0.504490
$333$	−21.1894	−1.16117
$334$	−5.74273	−0.314228
$335$	−10.6758	−0.583282
$336$	−35.3787	−1.93006
$337$	5.57475	0.303676	0.151838	−	0.988405i	$-0.451481\pi$
$337$	5.57475	0.303676	0.151838	+	0.988405i	$0.451481\pi$
$338$	−2.43925	−0.132678
$339$	−35.0631	−1.90436
$340$	24.2121	1.31309
$341$	−34.5462	−1.87078
$342$	−56.8480	−3.07399
$343$	3.42859	0.185126
$344$	−21.0444	−1.13464
$345$	−37.1501	−2.00009
$346$	−32.2995	−1.73643
$347$	28.9189	1.55245	0.776224	−	0.630458i	$-0.217133\pi$
$347$	28.9189	1.55245	0.776224	+	0.630458i	$0.217133\pi$
$348$	37.6324	2.01731
$349$	−20.0806	−1.07489	−0.537446	−	0.843298i	$-0.680611\pi$
$349$	−20.0806	−1.07489	−0.537446	+	0.843298i	$0.680611\pi$
$350$	−117.926	−6.30341
$351$	−0.331028	−0.0176690
$352$	1.70779	0.0910257
$353$	28.5408	1.51907	0.759537	−	0.650465i	$-0.225426\pi$
$353$	28.5408	1.51907	0.759537	+	0.650465i	$0.225426\pi$
$354$	−84.2228	−4.47639
$355$	4.76785	0.253051
$356$	37.3511	1.97960
$357$	−13.9909	−0.740478
$358$	7.09289	0.374871
$359$	30.6246	1.61630	0.808152	−	0.588975i	$-0.200468\pi$
$359$	30.6246	1.61630	0.808152	+	0.588975i	$0.200468\pi$
$360$	−62.4040	−3.28898
$361$	36.3113	1.91112
$362$	49.5149	2.60245
$363$	−3.80938	−0.199940
$364$	15.2411	0.798852
$365$	8.21460	0.429972
$366$	−82.1332	−4.29317
$367$	16.1847	0.844836	0.422418	−	0.906401i	$-0.361181\pi$
$367$	16.1847	0.844836	0.422418	+	0.906401i	$0.361181\pi$
$368$	13.2639	0.691431
$369$	18.4691	0.961462
$370$	69.0566	3.59008
$371$	−36.2414	−1.88156
$372$	95.4408	4.94838
$373$	−5.11565	−0.264878	−0.132439	−	0.991191i	$-0.542281\pi$
$373$	−5.11565	−0.264878	−0.132439	+	0.991191i	$0.542281\pi$
$374$	−12.6454	−0.653878
$375$	−78.0722	−4.03163
$376$	33.3106	1.71786
$377$	−3.84691	−0.198126
$378$	3.11564	0.160251
$379$	5.72248	0.293944	0.146972	−	0.989141i	$-0.453047\pi$
$379$	5.72248	0.293944	0.146972	+	0.989141i	$0.453047\pi$
$380$	122.993	6.30940
$381$	30.2740	1.55098
$382$	51.1598	2.61756
$383$	28.6420	1.46354	0.731768	−	0.681554i	$-0.238695\pi$
$383$	28.6420	1.46354	0.731768	+	0.681554i	$0.238695\pi$
$384$	−49.4484	−2.52340
$385$	57.2038	2.91538
$386$	−64.1163	−3.26343
$387$	13.8647	0.704784
$388$	60.6443	3.07875
$389$	9.49504	0.481418	0.240709	−	0.970597i	$-0.422620\pi$
$389$	9.49504	0.481418	0.240709	+	0.970597i	$0.422620\pi$
$390$	25.2929	1.28076
$391$	5.24538	0.265271
$392$	−37.5211	−1.89510
$393$	−18.6362	−0.940073
$394$	−0.480786	−0.0242216
$395$	40.2005	2.02271
$396$	43.8288	2.20248
$397$	−18.9418	−0.950664	−0.475332	−	0.879806i	$-0.657672\pi$
$397$	−18.9418	−0.950664	−0.475332	+	0.879806i	$0.657672\pi$
$398$	−24.3191	−1.21900
$399$	−71.0711	−3.55801
$400$	46.3854	2.31927
$401$	−35.3413	−1.76486	−0.882429	−	0.470445i	$-0.844093\pi$
$401$	−35.3413	−1.76486	−0.882429	+	0.470445i	$0.844093\pi$
$402$	−15.4041	−0.768285
$403$	−9.75626	−0.485994
$404$	24.5339	1.22061
$405$	−35.9276	−1.78526
$406$	36.2072	1.79693
$407$	−23.9432	−1.18682
$408$	17.2464	0.853823
$409$	7.20236	0.356134	0.178067	−	0.984018i	$-0.443016\pi$
$409$	7.20236	0.356134	0.178067	+	0.984018i	$0.443016\pi$
$410$	−60.1910	−2.97262
$411$	−33.9110	−1.67271
$412$	−40.1396	−1.97754
$413$	−53.7947	−2.64706
$414$	−27.3858	−1.34594
$415$	9.74345	0.478287
$416$	0.482302	0.0236468
$417$	−7.96960	−0.390273
$418$	−64.2362	−3.14189
$419$	−21.8829	−1.06905	−0.534526	−	0.845152i	$-0.679510\pi$
$419$	−21.8829	−1.06905	−0.534526	+	0.845152i	$0.679510\pi$
$420$	−158.037	−7.71142
$421$	−2.63008	−0.128182	−0.0640910	−	0.997944i	$-0.520415\pi$
$421$	−2.63008	−0.128182	−0.0640910	+	0.997944i	$0.520415\pi$
$422$	−32.3922	−1.57683
$423$	−21.9460	−1.06705
$424$	44.6742	2.16957
$425$	18.3437	0.889799
$426$	6.87949	0.333312
$427$	−52.4600	−2.53872
$428$	25.9681	1.25522
$429$	−8.76953	−0.423397
$430$	−45.1854	−2.17903
$431$	−12.9648	−0.624492	−0.312246	−	0.950001i	$-0.601081\pi$
$431$	−12.9648	−0.624492	−0.312246	+	0.950001i	$0.601081\pi$
$432$	−1.22552	−0.0589627
$433$	14.6502	0.704042	0.352021	−	0.935992i	$-0.385495\pi$
$433$	14.6502	0.704042	0.352021	+	0.935992i	$0.385495\pi$
$434$	91.8261	4.40780
$435$	39.8891	1.91253
$436$	15.2626	0.730948
$437$	26.6455	1.27463
$438$	11.8528	0.566348
$439$	−24.9169	−1.18922	−0.594609	−	0.804015i	$-0.702693\pi$
$439$	−24.9169	−1.18922	−0.594609	+	0.804015i	$0.702693\pi$
$440$	−70.5142	−3.36163
$441$	24.7201	1.17715
$442$	−3.57122	−0.169865
$443$	−7.55945	−0.359160	−0.179580	−	0.983743i	$-0.557474\pi$
$443$	−7.55945	−0.359160	−0.179580	+	0.983743i	$0.557474\pi$
$444$	66.1480	3.13925
$445$	39.5908	1.87678
$446$	−53.5790	−2.53704
$447$	41.6426	1.96963
$448$	−33.1095	−1.56428
$449$	−24.3539	−1.14933	−0.574666	−	0.818388i	$-0.694868\pi$
$449$	−24.3539	−1.14933	−0.574666	+	0.818388i	$0.694868\pi$
$450$	−95.7712	−4.51470
$451$	20.8694	0.982701
$452$	55.9217	2.63034
$453$	−1.51867	−0.0713533
$454$	−47.3986	−2.22453
$455$	16.1551	0.757361
$456$	87.6082	4.10263
$457$	16.5161	0.772590	0.386295	−	0.922375i	$-0.373755\pi$
$457$	16.5161	0.772590	0.386295	+	0.922375i	$0.373755\pi$
$458$	−52.7900	−2.46672
$459$	−0.484645	−0.0226213
$460$	59.2503	2.76256
$461$	19.3095	0.899331	0.449666	−	0.893197i	$-0.351543\pi$
$461$	19.3095	0.899331	0.449666	+	0.893197i	$0.351543\pi$
$462$	82.5390	3.84006
$463$	−28.9305	−1.34451	−0.672257	−	0.740318i	$-0.734675\pi$
$463$	−28.9305	−1.34451	−0.672257	+	0.740318i	$0.734675\pi$
$464$	−14.2419	−0.661162
$465$	101.164	4.69136
$466$	8.69452	0.402766
$467$	−16.2014	−0.749713	−0.374856	−	0.927083i	$-0.622308\pi$
$467$	−16.2014	−0.749713	−0.374856	+	0.927083i	$0.622308\pi$
$468$	12.3778	0.572163
$469$	−9.83887	−0.454317
$470$	71.5225	3.29909
$471$	−4.67789	−0.215546
$472$	66.3119	3.05225
$473$	15.6666	0.720353
$474$	58.0050	2.66426
$475$	93.1823	4.27550
$476$	22.3140	1.02276
$477$	−29.4327	−1.34763
$478$	−35.7432	−1.63485
$479$	11.1649	0.510138	0.255069	−	0.966923i	$-0.417902\pi$
$479$	11.1649	0.510138	0.255069	+	0.966923i	$0.417902\pi$
$480$	−5.00105	−0.228266
$481$	−6.76186	−0.308314
$482$	6.83309	0.311239
$483$	−34.2376	−1.55787
$484$	6.07554	0.276161
$485$	64.2808	2.91884
$486$	−54.2620	−2.46137
$487$	−32.7521	−1.48414	−0.742071	−	0.670322i	$-0.766156\pi$
$487$	−32.7521	−1.48414	−0.742071	+	0.670322i	$0.766156\pi$
$488$	64.6666	2.92732
$489$	9.21493	0.416714
$490$	−80.5632	−3.63947
$491$	10.0710	0.454497	0.227249	−	0.973837i	$-0.427027\pi$
$491$	10.0710	0.454497	0.227249	+	0.973837i	$0.427027\pi$
$492$	−57.6559	−2.59933
$493$	−5.63211	−0.253658
$494$	−18.1411	−0.816206
$495$	46.4569	2.08808
$496$	−36.1192	−1.62180
$497$	4.39406	0.197101
$498$	14.0588	0.629988
$499$	15.5052	0.694106	0.347053	−	0.937845i	$-0.387182\pi$
$499$	15.5052	0.694106	0.347053	+	0.937845i	$0.387182\pi$
$500$	124.517	5.56855
$501$	−5.83072	−0.260497
$502$	2.26865	0.101255
$503$	9.46831	0.422171	0.211086	−	0.977468i	$-0.432300\pi$
$503$	9.46831	0.422171	0.211086	+	0.977468i	$0.432300\pi$
$504$	−57.5117	−2.56178
$505$	26.0051	1.15721
$506$	−30.9450	−1.37567
$507$	−2.47662	−0.109991
$508$	−48.2837	−2.14224
$509$	−18.6336	−0.825918	−0.412959	−	0.910750i	$-0.635505\pi$
$509$	−18.6336	−0.825918	−0.412959	+	0.910750i	$0.635505\pi$
$510$	37.0304	1.63973
$511$	7.57061	0.334904
$512$	37.0034	1.63534
$513$	−2.46191	−0.108696
$514$	50.8510	2.24294
$515$	−42.5466	−1.87483
$516$	−43.2823	−1.90539
$517$	−24.7982	−1.09062
$518$	63.6428	2.79630
$519$	−32.7944	−1.43951
$520$	−19.9141	−0.873290
$521$	30.5975	1.34050	0.670250	−	0.742136i	$-0.266187\pi$
$521$	30.5975	1.34050	0.670250	+	0.742136i	$0.266187\pi$
$522$	29.4049	1.28702
$523$	3.34452	0.146246	0.0731229	−	0.997323i	$-0.476703\pi$
$523$	3.34452	0.146246	0.0731229	+	0.997323i	$0.476703\pi$
$524$	29.7227	1.29844
$525$	−119.733	−5.22557
$526$	−2.24614	−0.0979364
$527$	−14.2838	−0.622211
$528$	−32.4662	−1.41291
$529$	−10.1638	−0.441906
$530$	95.9217	4.16657
$531$	−43.6883	−1.89591
$532$	113.351	4.91437
$533$	5.89377	0.255288
$534$	57.1253	2.47205
$535$	27.5253	1.19002
$536$	12.1282	0.523859
$537$	7.20157	0.310771
$538$	−72.5257	−3.12681
$539$	27.9328	1.20315
$540$	−5.47441	−0.235581
$541$	−11.3383	−0.487470	−0.243735	−	0.969842i	$-0.578373\pi$
$541$	−11.3383	−0.487470	−0.243735	+	0.969842i	$0.578373\pi$
$542$	−30.0615	−1.29125
$543$	50.2736	2.15745
$544$	0.706120	0.0302746
$545$	16.1779	0.692983
$546$	23.3100	0.997577
$547$	−1.75048	−0.0748450	−0.0374225	−	0.999300i	$-0.511915\pi$
$547$	−1.75048	−0.0748450	−0.0374225	+	0.999300i	$0.511915\pi$
$548$	54.0843	2.31037
$549$	−42.6044	−1.81831
$550$	−108.218	−4.61443
$551$	−28.6101	−1.21883
$552$	42.2042	1.79633
$553$	37.0489	1.57548
$554$	−55.2889	−2.34900
$555$	70.1146	2.97620
$556$	12.7106	0.539051
$557$	25.4528	1.07847	0.539235	−	0.842156i	$-0.318714\pi$
$557$	25.4528	1.07847	0.539235	+	0.842156i	$0.318714\pi$
$558$	74.5748	3.15700
$559$	4.42445	0.187134
$560$	59.8086	2.52738
$561$	−12.8391	−0.542069
$562$	−66.9138	−2.82259
$563$	35.5933	1.50008	0.750038	−	0.661394i	$-0.230035\pi$
$563$	35.5933	1.50008	0.750038	+	0.661394i	$0.230035\pi$
$564$	68.5101	2.88480
$565$	59.2751	2.49372
$566$	34.8843	1.46630
$567$	−33.1110	−1.39053
$568$	−5.41649	−0.227271
$569$	24.0092	1.00652	0.503259	−	0.864136i	$-0.332134\pi$
$569$	24.0092	1.00652	0.503259	+	0.864136i	$0.332134\pi$
$570$	188.107	7.87894
$571$	−41.6224	−1.74184	−0.870921	−	0.491424i	$-0.836477\pi$
$571$	−41.6224	−1.74184	−0.870921	+	0.491424i	$0.836477\pi$
$572$	13.9864	0.584802
$573$	51.9436	2.16998
$574$	−55.4723	−2.31537
$575$	44.8895	1.87202
$576$	−26.8892	−1.12038
$577$	6.08861	0.253472	0.126736	−	0.991936i	$-0.459550\pi$
$577$	6.08861	0.253472	0.126736	+	0.991936i	$0.459550\pi$
$578$	36.2388	1.50733
$579$	−65.0986	−2.70541
$580$	−63.6187	−2.64162
$581$	8.97960	0.372537
$582$	92.7503	3.84462
$583$	−33.2579	−1.37740
$584$	−9.33216	−0.386168
$585$	13.1200	0.542446
$586$	45.3549	1.87359
$587$	7.35900	0.303738	0.151869	−	0.988401i	$-0.451471\pi$
$587$	7.35900	0.303738	0.151869	+	0.988401i	$0.451471\pi$
$588$	−77.1700	−3.18244
$589$	−72.5589	−2.98974
$590$	142.381	5.86173
$591$	−0.488152	−0.0200799
$592$	−25.0335	−1.02887
$593$	29.1214	1.19587	0.597936	−	0.801544i	$-0.295988\pi$
$593$	29.1214	1.19587	0.597936	+	0.801544i	$0.295988\pi$
$594$	2.85915	0.117312
$595$	23.6520	0.969639
$596$	−66.4153	−2.72048
$597$	−24.6917	−1.01056
$598$	−8.73925	−0.357374
$599$	−26.3421	−1.07631	−0.538155	−	0.842846i	$-0.680879\pi$
$599$	−26.3421	−1.07631	−0.538155	+	0.842846i	$0.680879\pi$
$600$	147.593	6.02544
$601$	−7.97714	−0.325395	−0.162697	−	0.986676i	$-0.552019\pi$
$601$	−7.97714	−0.325395	−0.162697	+	0.986676i	$0.552019\pi$
$602$	−41.6430	−1.69724
$603$	−7.99045	−0.325396
$604$	2.42211	0.0985543
$605$	6.43986	0.261817
$606$	37.5226	1.52425
$607$	7.70564	0.312762	0.156381	−	0.987697i	$-0.450017\pi$
$607$	7.70564	0.312762	0.156381	+	0.987697i	$0.450017\pi$
$608$	3.58695	0.145470
$609$	36.7619	1.48967
$610$	138.848	5.62181
$611$	−7.00332	−0.283324
$612$	18.1218	0.732532
$613$	28.9286	1.16841	0.584207	−	0.811604i	$-0.301405\pi$
$613$	28.9286	1.16841	0.584207	+	0.811604i	$0.301405\pi$
$614$	27.6902	1.11749
$615$	−61.1132	−2.46432
$616$	−64.9862	−2.61837
$617$	−25.6440	−1.03239	−0.516195	−	0.856471i	$-0.672652\pi$
$617$	−25.6440	−1.03239	−0.516195	+	0.856471i	$0.672652\pi$
$618$	−61.3901	−2.46947
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−161.345	−6.47979
$621$	−1.18599	−0.0475922
$622$	72.9150	2.92363
$623$	36.4870	1.46182
$624$	−9.16885	−0.367048
$625$	69.3368	2.77347
$626$	22.9786	0.918409
$627$	−65.2204	−2.60465
$628$	7.46072	0.297715
$629$	−9.89979	−0.394731
$630$	−123.486	−4.91979
$631$	−16.4311	−0.654111	−0.327056	−	0.945005i	$-0.606056\pi$
$631$	−16.4311	−0.654111	−0.327056	+	0.945005i	$0.606056\pi$
$632$	−45.6696	−1.81664
$633$	−32.8885	−1.30720
$634$	−55.3118	−2.19671
$635$	−51.1791	−2.03098
$636$	91.8817	3.64335
$637$	7.88856	0.312556
$638$	33.2265	1.31545
$639$	3.56855	0.141170
$640$	83.5939	3.30434
$641$	25.7745	1.01803	0.509016	−	0.860757i	$-0.330009\pi$
$641$	25.7745	1.01803	0.509016	+	0.860757i	$0.330009\pi$
$642$	39.7160	1.56747
$643$	−46.1732	−1.82089	−0.910447	−	0.413626i	$-0.864262\pi$
$643$	−46.1732	−1.82089	−0.910447	+	0.413626i	$0.864262\pi$
$644$	54.6053	2.15175
$645$	−45.8777	−1.80643
$646$	−26.5597	−1.04498
$647$	8.29842	0.326244	0.163122	−	0.986606i	$-0.447844\pi$
$647$	8.29842	0.326244	0.163122	+	0.986606i	$0.447844\pi$
$648$	40.8154	1.60338
$649$	−49.3662	−1.93779
$650$	−30.5621	−1.19874
$651$	93.2330	3.65409
$652$	−14.6968	−0.575571
$653$	5.23112	0.204710	0.102355	−	0.994748i	$-0.467362\pi$
$653$	5.23112	0.204710	0.102355	+	0.994748i	$0.467362\pi$
$654$	23.3429	0.912780
$655$	31.5051	1.23100
$656$	21.8197	0.851915
$657$	6.14832	0.239869
$658$	65.9154	2.56965
$659$	−25.0007	−0.973888	−0.486944	−	0.873433i	$-0.661888\pi$
$659$	−25.0007	−0.973888	−0.486944	+	0.873433i	$0.661888\pi$
$660$	−145.027	−5.64517
$661$	−26.5683	−1.03339	−0.516693	−	0.856171i	$-0.672837\pi$
$661$	−26.5683	−1.03339	−0.516693	+	0.856171i	$0.672837\pi$
$662$	50.0279	1.94439
$663$	−3.62593	−0.140820
$664$	−11.0690	−0.429561
$665$	120.148	4.65913
$666$	51.6862	2.00280
$667$	−13.7825	−0.533662
$668$	9.29936	0.359803
$669$	−54.3999	−2.10322
$670$	26.0410	1.00605
$671$	−48.1414	−1.85848
$672$	−4.60898	−0.177795
$673$	−27.0323	−1.04202	−0.521010	−	0.853551i	$-0.674444\pi$
$673$	−27.0323	−1.04202	−0.521010	+	0.853551i	$0.674444\pi$
$674$	−13.5982	−0.523783
$675$	−4.14754	−0.159639
$676$	3.94994	0.151921
$677$	17.2648	0.663541	0.331770	−	0.943360i	$-0.392354\pi$
$677$	17.2648	0.663541	0.331770	+	0.943360i	$0.392354\pi$
$678$	85.5276	3.28467
$679$	59.2414	2.27348
$680$	−29.1555	−1.11806
$681$	−48.1248	−1.84415
$682$	84.2668	3.22674
$683$	30.6619	1.17324	0.586622	−	0.809861i	$-0.300458\pi$
$683$	30.6619	1.17324	0.586622	+	0.809861i	$0.300458\pi$
$684$	92.0555	3.51983
$685$	57.3275	2.19037
$686$	−8.36318	−0.319308
$687$	−53.5988	−2.04492
$688$	16.3800	0.624482
$689$	−9.39244	−0.357823
$690$	90.6183	3.44978
$691$	4.77071	0.181486	0.0907431	−	0.995874i	$-0.471076\pi$
$691$	4.77071	0.181486	0.0907431	+	0.995874i	$0.471076\pi$
$692$	52.3034	1.98828
$693$	42.8149	1.62640
$694$	−70.5404	−2.67768
$695$	13.4728	0.511054
$696$	−45.3158	−1.71769
$697$	8.62885	0.326841
$698$	48.9817	1.85398
$699$	8.82774	0.333896
$700$	190.961	7.21763
$701$	−28.6990	−1.08395	−0.541973	−	0.840396i	$-0.682322\pi$
$701$	−28.6990	−1.08395	−0.541973	+	0.840396i	$0.682322\pi$
$702$	0.807459	0.0304756
$703$	−50.2890	−1.89669
$704$	−30.3839	−1.14513
$705$	72.6183	2.73497
$706$	−69.6181	−2.62011
$707$	23.9664	0.901349
$708$	136.384	5.12563
$709$	30.5563	1.14756	0.573782	−	0.819008i	$-0.305476\pi$
$709$	30.5563	1.14756	0.573782	+	0.819008i	$0.305476\pi$
$710$	−11.6300	−0.436465
$711$	30.0886	1.12841
$712$	−44.9770	−1.68558
$713$	−34.9543	−1.30905
$714$	34.1273	1.27718
$715$	14.8251	0.554429
$716$	−11.4857	−0.429241
$717$	−36.2908	−1.35530
$718$	−74.7010	−2.78782
$719$	−50.8141	−1.89505	−0.947523	−	0.319687i	$-0.896422\pi$
$719$	−50.8141	−1.89505	−0.947523	+	0.319687i	$0.896422\pi$
$720$	48.5724	1.81019
$721$	−39.2111	−1.46030
$722$	−88.5724	−3.29632
$723$	6.93778	0.258019
$724$	−80.1809	−2.97990
$725$	−48.1990	−1.79007
$726$	9.29202	0.344859
$727$	−41.2403	−1.52952	−0.764759	−	0.644316i	$-0.777142\pi$
$727$	−41.2403	−1.52952	−0.764759	+	0.644316i	$0.777142\pi$
$728$	−18.3529	−0.680203
$729$	−29.3499	−1.08703
$730$	−20.0375	−0.741620
$731$	6.47767	0.239585
$732$	133.000	4.91584
$733$	0.850985	0.0314319	0.0157159	−	0.999876i	$-0.494997\pi$
$733$	0.850985	0.0314319	0.0157159	+	0.999876i	$0.494997\pi$
$734$	−39.4786	−1.45718
$735$	−81.7975	−3.01715
$736$	1.72797	0.0636939
$737$	−9.02891	−0.332584
$738$	−45.0507	−1.65834
$739$	−2.03555	−0.0748789	−0.0374395	−	0.999299i	$-0.511920\pi$
$739$	−2.03555	−0.0748789	−0.0374395	+	0.999299i	$0.511920\pi$
$740$	−111.825	−4.11077
$741$	−18.4190	−0.676640
$742$	88.4018	3.24533
$743$	−11.0549	−0.405567	−0.202783	−	0.979224i	$-0.564999\pi$
$743$	−11.0549	−0.405567	−0.202783	+	0.979224i	$0.564999\pi$
$744$	−114.927	−4.21342
$745$	−70.3979	−2.57918
$746$	12.4783	0.456864
$747$	7.29260	0.266822
$748$	20.4770	0.748714
$749$	25.3674	0.926904
$750$	190.438	6.95380
$751$	32.0410	1.16919	0.584596	−	0.811325i	$-0.301253\pi$
$751$	32.0410	1.16919	0.584596	+	0.811325i	$0.301253\pi$
$752$	−25.9274	−0.945476
$753$	2.30341	0.0839408
$754$	9.38357	0.341730
$755$	2.56735	0.0934355
$756$	−5.04524	−0.183493
$757$	14.7671	0.536721	0.268361	−	0.963319i	$-0.413518\pi$
$757$	14.7671	0.536721	0.268361	+	0.963319i	$0.413518\pi$
$758$	−13.9586	−0.506998
$759$	−31.4191	−1.14044
$760$	−148.104	−5.37230
$761$	33.3577	1.20922	0.604608	−	0.796523i	$-0.293330\pi$
$761$	33.3577	1.20922	0.604608	+	0.796523i	$0.293330\pi$
$762$	−73.8459	−2.67515
$763$	14.9096	0.539763
$764$	−82.8444	−2.99720
$765$	19.2085	0.694486
$766$	−69.8649	−2.52432
$767$	−13.9416	−0.503403
$768$	78.1143	2.81871
$769$	20.8136	0.750558	0.375279	−	0.926912i	$-0.377547\pi$
$769$	20.8136	0.750558	0.375279	+	0.926912i	$0.377547\pi$
$770$	−139.534	−5.02847
$771$	51.6301	1.85941
$772$	103.825	3.73675
$773$	12.0299	0.432687	0.216344	−	0.976317i	$-0.430587\pi$
$773$	12.0299	0.432687	0.216344	+	0.976317i	$0.430587\pi$
$774$	−33.8195	−1.21562
$775$	−122.239	−4.39096
$776$	−73.0260	−2.62148
$777$	64.6179	2.31815
$778$	−23.1608	−0.830354
$779$	43.8329	1.57048
$780$	−40.9574	−1.46651
$781$	4.03233	0.144288
$782$	−12.7948	−0.457541
$783$	1.27343	0.0455088
$784$	29.2047	1.04303
$785$	7.90810	0.282252
$786$	45.4584	1.62145
$787$	38.5759	1.37508	0.687540	−	0.726146i	$-0.258690\pi$
$787$	38.5759	1.37508	0.687540	+	0.726146i	$0.258690\pi$
$788$	0.778549	0.0277347
$789$	−2.28055	−0.0811899
$790$	−98.0591	−3.48879
$791$	54.6281	1.94235
$792$	−52.7772	−1.87536
$793$	−13.5957	−0.482798
$794$	46.2039	1.63972
$795$	97.3914	3.45412
$796$	39.3805	1.39581
$797$	−10.2328	−0.362463	−0.181231	−	0.983440i	$-0.558008\pi$
$797$	−10.2328	−0.362463	−0.181231	+	0.983440i	$0.558008\pi$
$798$	173.360	6.13689
$799$	−10.2533	−0.362736
$800$	6.04290	0.213649
$801$	29.6322	1.04700
$802$	86.2062	3.04405
$803$	6.94737	0.245167
$804$	24.9442	0.879714
$805$	57.8797	2.03999
$806$	23.7980	0.838248
$807$	−73.6369	−2.59214
$808$	−29.5430	−1.03932
$809$	26.4563	0.930156	0.465078	−	0.885270i	$-0.346026\pi$
$809$	26.4563	0.930156	0.465078	+	0.885270i	$0.346026\pi$
$810$	87.6363	3.07923
$811$	−23.8351	−0.836962	−0.418481	−	0.908225i	$-0.637437\pi$
$811$	−23.8351	−0.836962	−0.418481	+	0.908225i	$0.637437\pi$
$812$	−58.6312	−2.05755
$813$	−30.5221	−1.07046
$814$	58.4035	2.04704
$815$	−15.5781	−0.545677
$816$	−13.4238	−0.469926
$817$	32.9053	1.15121
$818$	−17.5684	−0.614263
$819$	12.0915	0.422509
$820$	97.4689	3.40376
$821$	13.8129	0.482075	0.241037	−	0.970516i	$-0.422512\pi$
$821$	13.8129	0.482075	0.241037	+	0.970516i	$0.422512\pi$
$822$	82.7174	2.88510
$823$	−17.7833	−0.619888	−0.309944	−	0.950755i	$-0.600310\pi$
$823$	−17.7833	−0.619888	−0.309944	+	0.950755i	$0.600310\pi$
$824$	48.3349	1.68382
$825$	−109.876	−3.82539
$826$	131.219	4.56569
$827$	47.8449	1.66373	0.831866	−	0.554977i	$-0.187273\pi$
$827$	47.8449	1.66373	0.831866	+	0.554977i	$0.187273\pi$
$828$	44.3466	1.54115
$829$	−2.66635	−0.0926062	−0.0463031	−	0.998927i	$-0.514744\pi$
$829$	−2.66635	−0.0926062	−0.0463031	+	0.998927i	$0.514744\pi$
$830$	−23.7667	−0.824955
$831$	−56.1360	−1.94734
$832$	−8.58077	−0.297485
$833$	11.5494	0.400161
$834$	19.4398	0.673147
$835$	9.85699	0.341115
$836$	104.019	3.59758
$837$	3.22959	0.111631
$838$	53.3780	1.84391
$839$	11.6931	0.403691	0.201846	−	0.979417i	$-0.435306\pi$
$839$	11.6931	0.403691	0.201846	+	0.979417i	$0.435306\pi$
$840$	190.303	6.56609
$841$	−14.2013	−0.489700
$842$	6.41541	0.221090
$843$	−67.9390	−2.33994
$844$	52.4535	1.80553
$845$	4.18680	0.144030
$846$	53.5319	1.84046
$847$	5.93500	0.203929
$848$	−34.7723	−1.19409
$849$	35.4188	1.21557
$850$	−44.7448	−1.53473
$851$	−24.2261	−0.830461
$852$	−11.1401	−0.381655
$853$	4.61643	0.158063	0.0790317	−	0.996872i	$-0.474817\pi$
$853$	4.61643	0.158063	0.0790317	+	0.996872i	$0.474817\pi$
$854$	127.963	4.37881
$855$	97.5756	3.33701
$856$	−31.2700	−1.06879
$857$	−4.61237	−0.157555	−0.0787777	−	0.996892i	$-0.525102\pi$
$857$	−4.61237	−0.157555	−0.0787777	+	0.996892i	$0.525102\pi$
$858$	21.3911	0.730279
$859$	−46.4122	−1.58356	−0.791782	−	0.610804i	$-0.790846\pi$
$859$	−46.4122	−1.58356	−0.791782	+	0.610804i	$0.790846\pi$
$860$	73.1699	2.49507
$861$	−56.3222	−1.91945
$862$	31.6244	1.07713
$863$	9.93149	0.338072	0.169036	−	0.985610i	$-0.445935\pi$
$863$	9.93149	0.338072	0.169036	+	0.985610i	$0.445935\pi$
$864$	−0.159655	−0.00543158
$865$	55.4398	1.88501
$866$	−35.7354	−1.21434
$867$	36.7940	1.24959
$868$	−148.696	−5.04709
$869$	33.9990	1.15334
$870$	−97.2994	−3.29876
$871$	−2.54988	−0.0863992
$872$	−18.3788	−0.622384
$873$	48.1118	1.62834
$874$	−64.9952	−2.19849
$875$	121.636	4.11206
$876$	−19.1935	−0.648489
$877$	−10.4338	−0.352325	−0.176162	−	0.984361i	$-0.556368\pi$
$877$	−10.4338	−0.352325	−0.176162	+	0.984361i	$0.556368\pi$
$878$	60.7785	2.05117
$879$	46.0498	1.55322
$880$	54.8850	1.85017
$881$	−29.2855	−0.986653	−0.493326	−	0.869844i	$-0.664219\pi$
$881$	−29.2855	−0.986653	−0.493326	+	0.869844i	$0.664219\pi$
$882$	−60.2985	−2.03036
$883$	33.8637	1.13961	0.569803	−	0.821782i	$-0.307020\pi$
$883$	33.8637	1.13961	0.569803	+	0.821782i	$0.307020\pi$
$884$	5.78296	0.194502
$885$	144.562	4.85941
$886$	18.4394	0.619483
$887$	−34.3366	−1.15291	−0.576455	−	0.817129i	$-0.695564\pi$
$887$	−34.3366	−1.15291	−0.576455	+	0.817129i	$0.695564\pi$
$888$	−79.6534	−2.67299
$889$	−47.1668	−1.58192
$890$	−96.5719	−3.23710
$891$	−30.3852	−1.01794
$892$	86.7619	2.90500
$893$	−52.0848	−1.74295
$894$	−101.577	−3.39723
$895$	−12.1745	−0.406947
$896$	77.0404	2.57374
$897$	−8.87314	−0.296266
$898$	59.4053	1.98238
$899$	37.5314	1.25174
$900$	155.085	5.16950
$901$	−13.7511	−0.458116
$902$	−50.9056	−1.69497
$903$	−42.2810	−1.40702
$904$	−67.3392	−2.23967
$905$	−84.9889	−2.82513
$906$	3.70441	0.123071
$907$	−23.3873	−0.776563	−0.388282	−	0.921541i	$-0.626931\pi$
$907$	−23.3873	−0.776563	−0.388282	+	0.921541i	$0.626931\pi$
$908$	76.7538	2.54716
$909$	19.4638	0.645575
$910$	−39.4062	−1.30630
$911$	−5.80238	−0.192241	−0.0961207	−	0.995370i	$-0.530643\pi$
$911$	−5.80238	−0.192241	−0.0961207	+	0.995370i	$0.530643\pi$
$912$	−68.1902	−2.25800
$913$	8.24037	0.272717
$914$	−40.2869	−1.33257
$915$	140.976	4.66051
$916$	85.4843	2.82448
$917$	29.0352	0.958826
$918$	1.18217	0.0390175
$919$	−47.7760	−1.57598	−0.787992	−	0.615686i	$-0.788879\pi$
$919$	−47.7760	−1.57598	−0.787992	+	0.615686i	$0.788879\pi$
$920$	−71.3474	−2.35225
$921$	28.1144	0.926403
$922$	−47.1006	−1.55118
$923$	1.13878	0.0374834
$924$	−133.658	−4.39701
$925$	−84.7214	−2.78562
$926$	70.5687	2.31903
$927$	−31.8445	−1.04591
$928$	−1.85537	−0.0609055
$929$	33.8477	1.11051	0.555253	−	0.831681i	$-0.312621\pi$
$929$	33.8477	1.11051	0.555253	+	0.831681i	$0.312621\pi$
$930$	−246.764	−8.09171
$931$	58.6685	1.92278
$932$	−14.0793	−0.461182
$933$	74.0322	2.42371
$934$	39.5193	1.29311
$935$	21.7049	0.709827
$936$	−14.9049	−0.487183
$937$	47.0626	1.53747	0.768734	−	0.639569i	$-0.220887\pi$
$937$	47.0626	1.53747	0.768734	+	0.639569i	$0.220887\pi$
$938$	23.9995	0.783611
$939$	23.3307	0.761367
$940$	−115.818	−3.77758
$941$	55.5265	1.81011	0.905057	−	0.425291i	$-0.139828\pi$
$941$	55.5265	1.81011	0.905057	+	0.425291i	$0.139828\pi$
$942$	11.4105	0.371775
$943$	21.1160	0.687630
$944$	−51.6141	−1.67990
$945$	−5.34777	−0.173963
$946$	−38.2148	−1.24247
$947$	−15.2912	−0.496899	−0.248449	−	0.968645i	$-0.579921\pi$
$947$	−15.2912	−0.496899	−0.248449	+	0.968645i	$0.579921\pi$
$948$	−93.9291	−3.05067
$949$	1.96202	0.0636900
$950$	−227.295	−7.37443
$951$	−56.1593	−1.82109
$952$	−26.8698	−0.870855
$953$	25.3022	0.819620	0.409810	−	0.912171i	$-0.365595\pi$
$953$	25.3022	0.819620	0.409810	+	0.912171i	$0.365595\pi$
$954$	71.7938	2.32441
$955$	−87.8122	−2.84153
$956$	57.8798	1.87197
$957$	33.7356	1.09052
$958$	−27.2340	−0.879891
$959$	52.8332	1.70607
$960$	88.9751	2.87166
$961$	64.1847	2.07047
$962$	16.4939	0.531784
$963$	20.6016	0.663878
$964$	−11.0650	−0.356380
$965$	110.051	3.54267
$966$	83.5142	2.68703
$967$	−57.0787	−1.83553	−0.917764	−	0.397126i	$-0.870008\pi$
$967$	−57.0787	−1.83553	−0.917764	+	0.397126i	$0.870008\pi$
$968$	−7.31597	−0.235144
$969$	−26.9666	−0.866293
$970$	−156.797	−5.03445
$971$	−26.0342	−0.835478	−0.417739	−	0.908567i	$-0.637177\pi$
$971$	−26.0342	−0.835478	−0.417739	+	0.908567i	$0.637177\pi$
$972$	87.8679	2.81836
$973$	12.4166	0.398058
$974$	79.8907	2.55986
$975$	−31.0303	−0.993766
$976$	−50.3335	−1.61114
$977$	−24.8886	−0.796256	−0.398128	−	0.917330i	$-0.630340\pi$
$977$	−24.8886	−0.796256	−0.398128	+	0.917330i	$0.630340\pi$
$978$	−22.4775	−0.718752
$979$	33.4833	1.07013
$980$	130.458	4.16733
$981$	12.1085	0.386595
$982$	−24.5656	−0.783921
$983$	−24.4837	−0.780909	−0.390454	−	0.920622i	$-0.627682\pi$
$983$	−24.4837	−0.780909	−0.390454	+	0.920622i	$0.627682\pi$
$984$	69.4274	2.21327
$985$	0.825235	0.0262942
$986$	13.7381	0.437511
$987$	66.9253	2.13026
$988$	29.3763	0.934586
$989$	15.8517	0.504056
$990$	−113.320	−3.60155
$991$	11.5972	0.368398	0.184199	−	0.982889i	$-0.441031\pi$
$991$	11.5972	0.368398	0.184199	+	0.982889i	$0.441031\pi$
$992$	−4.70546	−0.149399
$993$	50.7944	1.61191
$994$	−10.7182	−0.339961
$995$	41.7420	1.32331
$996$	−22.7657	−0.721359
$997$	7.82189	0.247722	0.123861	−	0.992300i	$-0.460472\pi$
$997$	7.82189	0.247722	0.123861	+	0.992300i	$0.460472\pi$
$998$	−37.8210	−1.19720
$999$	2.23836	0.0708187

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.e.1.15	✓	168

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.e.1.15	✓	168	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.43925 q^{2} -2.47662 q^{3} +3.94994 q^{4} +4.18680 q^{5} +6.04110 q^{6} +3.85857 q^{7} -4.75640 q^{8} +3.13366 q^{9} +O(q^{10})\)	\(q-2.43925 q^{2} -2.47662 q^{3} +3.94994 q^{4} +4.18680 q^{5} +6.04110 q^{6} +3.85857 q^{7} -4.75640 q^{8} +3.13366 q^{9} -10.2127 q^{10} +3.54092 q^{11} -9.78252 q^{12} +1.00000 q^{13} -9.41202 q^{14} -10.3691 q^{15} +3.70216 q^{16} +1.46406 q^{17} -7.64378 q^{18} +7.43716 q^{19} +16.5376 q^{20} -9.55622 q^{21} -8.63720 q^{22} +3.58276 q^{23} +11.7798 q^{24} +12.5293 q^{25} -2.43925 q^{26} -0.331028 q^{27} +15.2411 q^{28} -3.84691 q^{29} +25.2929 q^{30} -9.75626 q^{31} +0.482302 q^{32} -8.76953 q^{33} -3.57122 q^{34} +16.1551 q^{35} +12.3778 q^{36} -6.76186 q^{37} -18.1411 q^{38} -2.47662 q^{39} -19.9141 q^{40} +5.89377 q^{41} +23.3100 q^{42} +4.42445 q^{43} +13.9864 q^{44} +13.1200 q^{45} -8.73925 q^{46} -7.00332 q^{47} -9.16885 q^{48} +7.88856 q^{49} -30.5621 q^{50} -3.62593 q^{51} +3.94994 q^{52} -9.39244 q^{53} +0.807459 q^{54} +14.8251 q^{55} -18.3529 q^{56} -18.4190 q^{57} +9.38357 q^{58} -13.9416 q^{59} -40.9574 q^{60} -13.5957 q^{61} +23.7980 q^{62} +12.0915 q^{63} -8.58077 q^{64} +4.18680 q^{65} +21.3911 q^{66} -2.54988 q^{67} +5.78296 q^{68} -8.87314 q^{69} -39.4062 q^{70} +1.13878 q^{71} -14.9049 q^{72} +1.96202 q^{73} +16.4939 q^{74} -31.0303 q^{75} +29.3763 q^{76} +13.6629 q^{77} +6.04110 q^{78} +9.60173 q^{79} +15.5002 q^{80} -8.58115 q^{81} -14.3764 q^{82} +2.32718 q^{83} -37.7465 q^{84} +6.12974 q^{85} -10.7923 q^{86} +9.52734 q^{87} -16.8420 q^{88} +9.45610 q^{89} -32.0030 q^{90} +3.85857 q^{91} +14.1517 q^{92} +24.1626 q^{93} +17.0829 q^{94} +31.1379 q^{95} -1.19448 q^{96} +15.3532 q^{97} -19.2422 q^{98} +11.0961 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

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