LMFDB - Embedded newform 8047.2.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2556185065$
Analytic rank:	$0$
Dimension:	$156$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-2.65345 q^{2} +0.515937 q^{3} +5.04077 q^{4} -0.00500829 q^{5} -1.36901 q^{6} +0.475978 q^{7} -8.06853 q^{8} -2.73381 q^{9} +O(q^{10})$	\(q-2.65345 q^{2} +0.515937 q^{3} +5.04077 q^{4} -0.00500829 q^{5} -1.36901 q^{6} +0.475978 q^{7} -8.06853 q^{8} -2.73381 q^{9} +0.0132892 q^{10} +1.65099 q^{11} +2.60072 q^{12} -1.00000 q^{13} -1.26298 q^{14} -0.00258396 q^{15} +11.3279 q^{16} +3.63802 q^{17} +7.25401 q^{18} -5.84505 q^{19} -0.0252457 q^{20} +0.245575 q^{21} -4.38082 q^{22} -7.42216 q^{23} -4.16285 q^{24} -4.99997 q^{25} +2.65345 q^{26} -2.95828 q^{27} +2.39930 q^{28} -0.0556594 q^{29} +0.00685641 q^{30} +2.36468 q^{31} -13.9208 q^{32} +0.851808 q^{33} -9.65330 q^{34} -0.00238384 q^{35} -13.7805 q^{36} -6.53383 q^{37} +15.5095 q^{38} -0.515937 q^{39} +0.0404096 q^{40} -8.86805 q^{41} -0.651619 q^{42} +5.29287 q^{43} +8.32228 q^{44} +0.0136917 q^{45} +19.6943 q^{46} +7.95646 q^{47} +5.84446 q^{48} -6.77344 q^{49} +13.2672 q^{50} +1.87699 q^{51} -5.04077 q^{52} -8.94334 q^{53} +7.84965 q^{54} -0.00826865 q^{55} -3.84044 q^{56} -3.01568 q^{57} +0.147689 q^{58} +2.92756 q^{59} -0.0130252 q^{60} +2.66939 q^{61} -6.27456 q^{62} -1.30123 q^{63} +14.2824 q^{64} +0.00500829 q^{65} -2.26023 q^{66} +3.28724 q^{67} +18.3385 q^{68} -3.82937 q^{69} +0.00632538 q^{70} +2.08943 q^{71} +22.0578 q^{72} -2.96308 q^{73} +17.3372 q^{74} -2.57967 q^{75} -29.4636 q^{76} +0.785836 q^{77} +1.36901 q^{78} +10.7511 q^{79} -0.0567332 q^{80} +6.67514 q^{81} +23.5309 q^{82} +15.5905 q^{83} +1.23789 q^{84} -0.0182203 q^{85} -14.0444 q^{86} -0.0287167 q^{87} -13.3211 q^{88} +0.666900 q^{89} -0.0363302 q^{90} -0.475978 q^{91} -37.4134 q^{92} +1.22003 q^{93} -21.1120 q^{94} +0.0292737 q^{95} -7.18226 q^{96} -8.96468 q^{97} +17.9730 q^{98} -4.51350 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) $ 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.65345	−1.87627	−0.938135	−	0.346270i	$-0.887448\pi$
$2$	−2.65345	−1.87627	−0.938135	+	0.346270i	$0.887448\pi$
$3$	0.515937	0.297876	0.148938	−	0.988847i	$-0.452414\pi$
$3$	0.515937	0.297876	0.148938	+	0.988847i	$0.452414\pi$
$4$	5.04077	2.52039
$5$	−0.00500829	−0.00223978	−0.00111989	−	0.999999i	$-0.500356\pi$
$5$	−0.00500829	−0.00223978	−0.00111989	+	0.999999i	$0.500356\pi$
$6$	−1.36901	−0.558896
$7$	0.475978	0.179903	0.0899514	−	0.995946i	$-0.471329\pi$
$7$	0.475978	0.179903	0.0899514	+	0.995946i	$0.471329\pi$
$8$	−8.06853	−2.85266
$9$	−2.73381	−0.911270
$10$	0.0132892	0.00420242
$11$	1.65099	0.497793	0.248896	−	0.968530i	$-0.419932\pi$
$11$	1.65099	0.497793	0.248896	+	0.968530i	$0.419932\pi$
$12$	2.60072	0.750764
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.26298	−0.337546
$15$	−0.00258396	−0.000667176 0
$16$	11.3279	2.83197
$17$	3.63802	0.882350	0.441175	−	0.897421i	$-0.354562\pi$
$17$	3.63802	0.882350	0.441175	+	0.897421i	$0.354562\pi$
$18$	7.25401	1.70979
$19$	−5.84505	−1.34095	−0.670473	−	0.741934i	$-0.733909\pi$
$19$	−5.84505	−1.34095	−0.670473	+	0.741934i	$0.733909\pi$
$20$	−0.0252457	−0.00564510
$21$	0.245575	0.0535888
$22$	−4.38082	−0.933993
$23$	−7.42216	−1.54763	−0.773814	−	0.633413i	$-0.781653\pi$
$23$	−7.42216	−1.54763	−0.773814	+	0.633413i	$0.781653\pi$
$24$	−4.16285	−0.849739
$25$	−4.99997	−0.999995
$26$	2.65345	0.520384
$27$	−2.95828	−0.569322
$28$	2.39930	0.453425
$29$	−0.0556594	−0.0103357	−0.00516784	−	0.999987i	$-0.501645\pi$
$29$	−0.0556594	−0.0103357	−0.00516784	+	0.999987i	$0.501645\pi$
$30$	0.00685641	0.00125180
$31$	2.36468	0.424710	0.212355	−	0.977193i	$-0.431887\pi$
$31$	2.36468	0.424710	0.212355	+	0.977193i	$0.431887\pi$
$32$	−13.9208	−2.46087
$33$	0.851808	0.148281
$34$	−9.65330	−1.65553
$35$	−0.00238384	−0.000402942 0
$36$	−13.7805	−2.29675
$37$	−6.53383	−1.07416	−0.537078	−	0.843533i	$-0.680472\pi$
$37$	−6.53383	−1.07416	−0.537078	+	0.843533i	$0.680472\pi$
$38$	15.5095	2.51598
$39$	−0.515937	−0.0826161
$40$	0.0404096	0.00638931
$41$	−8.86805	−1.38496	−0.692478	−	0.721439i	$-0.743481\pi$
$41$	−8.86805	−1.38496	−0.692478	+	0.721439i	$0.743481\pi$
$42$	−0.651619	−0.100547
$43$	5.29287	0.807156	0.403578	−	0.914945i	$-0.367766\pi$
$43$	5.29287	0.807156	0.403578	+	0.914945i	$0.367766\pi$
$44$	8.32228	1.25463
$45$	0.0136917	0.00204104
$46$	19.6943	2.90377
$47$	7.95646	1.16057	0.580284	−	0.814414i	$-0.302941\pi$
$47$	7.95646	1.16057	0.580284	+	0.814414i	$0.302941\pi$
$48$	5.84446	0.843576
$49$	−6.77344	−0.967635
$50$	13.2672	1.87626
$51$	1.87699	0.262831
$52$	−5.04077	−0.699030
$53$	−8.94334	−1.22846	−0.614231	−	0.789126i	$-0.710534\pi$
$53$	−8.94334	−1.22846	−0.614231	+	0.789126i	$0.710534\pi$
$54$	7.84965	1.06820
$55$	−0.00826865	−0.00111494
$56$	−3.84044	−0.513201
$57$	−3.01568	−0.399436
$58$	0.147689	0.0193925
$59$	2.92756	0.381136	0.190568	−	0.981674i	$-0.438967\pi$
$59$	2.92756	0.381136	0.190568	+	0.981674i	$0.438967\pi$
$60$	−0.0130252	−0.00168154
$61$	2.66939	0.341780	0.170890	−	0.985290i	$-0.445336\pi$
$61$	2.66939	0.341780	0.170890	+	0.985290i	$0.445336\pi$
$62$	−6.27456	−0.796870
$63$	−1.30123	−0.163940
$64$	14.2824	1.78530
$65$	0.00500829	0.000621202 0
$66$	−2.26023	−0.278215
$67$	3.28724	0.401601	0.200800	−	0.979632i	$-0.435646\pi$
$67$	3.28724	0.401601	0.200800	+	0.979632i	$0.435646\pi$
$68$	18.3385	2.22386
$69$	−3.82937	−0.461002
$70$	0.00632538	0.000756028 0
$71$	2.08943	0.247969	0.123985	−	0.992284i	$-0.460433\pi$
$71$	2.08943	0.247969	0.123985	+	0.992284i	$0.460433\pi$
$72$	22.0578	2.59954
$73$	−2.96308	−0.346802	−0.173401	−	0.984851i	$-0.555476\pi$
$73$	−2.96308	−0.346802	−0.173401	+	0.984851i	$0.555476\pi$
$74$	17.3372	2.01541
$75$	−2.57967	−0.297875
$76$	−29.4636	−3.37970
$77$	0.785836	0.0895543
$78$	1.36901	0.155010
$79$	10.7511	1.20959	0.604797	−	0.796379i	$-0.293254\pi$
$79$	10.7511	1.20959	0.604797	+	0.796379i	$0.293254\pi$
$80$	−0.0567332	−0.00634297
$81$	6.67514	0.741682
$82$	23.5309	2.59855
$83$	15.5905	1.71128	0.855639	−	0.517574i	$-0.173165\pi$
$83$	15.5905	1.71128	0.855639	+	0.517574i	$0.173165\pi$
$84$	1.23789	0.135065
$85$	−0.0182203	−0.00197627
$86$	−14.0444	−1.51444
$87$	−0.0287167	−0.00307876
$88$	−13.3211	−1.42003
$89$	0.666900	0.0706913	0.0353457	−	0.999375i	$-0.488747\pi$
$89$	0.666900	0.0706913	0.0353457	+	0.999375i	$0.488747\pi$
$90$	−0.0363302	−0.00382954
$91$	−0.475978	−0.0498961
$92$	−37.4134	−3.90062
$93$	1.22003	0.126511
$94$	−21.1120	−2.17754
$95$	0.0292737	0.00300342
$96$	−7.18226	−0.733036
$97$	−8.96468	−0.910225	−0.455113	−	0.890434i	$-0.650401\pi$
$97$	−8.96468	−0.910225	−0.455113	+	0.890434i	$0.650401\pi$
$98$	17.9730	1.81554
$99$	−4.51350	−0.453623
$100$	−25.2037	−2.52037
$101$	3.30741	0.329100	0.164550	−	0.986369i	$-0.447383\pi$
$101$	3.30741	0.329100	0.164550	+	0.986369i	$0.447383\pi$
$102$	−4.98050	−0.493143
$103$	−12.8857	−1.26967	−0.634833	−	0.772650i	$-0.718931\pi$
$103$	−12.8857	−1.26967	−0.634833	+	0.772650i	$0.718931\pi$
$104$	8.06853	0.791185
$105$	−0.00122991	−0.000120027 0
$106$	23.7307	2.30493
$107$	10.4861	1.01373	0.506866	−	0.862025i	$-0.330804\pi$
$107$	10.4861	1.01373	0.506866	+	0.862025i	$0.330804\pi$
$108$	−14.9120	−1.43491
$109$	7.93761	0.760285	0.380143	−	0.924928i	$-0.375875\pi$
$109$	7.93761	0.760285	0.380143	+	0.924928i	$0.375875\pi$
$110$	0.0219404	0.00209194
$111$	−3.37105	−0.319966
$112$	5.39181	0.509479
$113$	−8.11503	−0.763398	−0.381699	−	0.924287i	$-0.624661\pi$
$113$	−8.11503	−0.763398	−0.381699	+	0.924287i	$0.624661\pi$
$114$	8.00194	0.749450
$115$	0.0371723	0.00346634
$116$	−0.280566	−0.0260499
$117$	2.73381	0.252741
$118$	−7.76813	−0.715114
$119$	1.73162	0.158737
$120$	0.0208488	0.00190323
$121$	−8.27423	−0.752202
$122$	−7.08308	−0.641271
$123$	−4.57536	−0.412546
$124$	11.9198	1.07043
$125$	0.0500828	0.00447954
$126$	3.45275	0.307596
$127$	7.74521	0.687276	0.343638	−	0.939102i	$-0.388341\pi$
$127$	7.74521	0.687276	0.343638	+	0.939102i	$0.388341\pi$
$128$	−10.0559	−0.888825
$129$	2.73079	0.240433
$130$	−0.0132892	−0.00116554
$131$	−5.88653	−0.514308	−0.257154	−	0.966370i	$-0.582785\pi$
$131$	−5.88653	−0.514308	−0.257154	+	0.966370i	$0.582785\pi$
$132$	4.29377	0.373725
$133$	−2.78211	−0.241240
$134$	−8.72253	−0.753511
$135$	0.0148160	0.00127515
$136$	−29.3535	−2.51704
$137$	13.4566	1.14968	0.574839	−	0.818267i	$-0.305065\pi$
$137$	13.4566	1.14968	0.574839	+	0.818267i	$0.305065\pi$
$138$	10.1610	0.864963
$139$	−17.2788	−1.46557	−0.732786	−	0.680459i	$-0.761780\pi$
$139$	−17.2788	−1.46557	−0.732786	+	0.680459i	$0.761780\pi$
$140$	−0.0120164	−0.00101557
$141$	4.10503	0.345706
$142$	−5.54418	−0.465258
$143$	−1.65099	−0.138063
$144$	−30.9682	−2.58068
$145$	0.000278758 0	2.31496e−5 0
$146$	7.86237	0.650695
$147$	−3.49467	−0.288236
$148$	−32.9356	−2.70729
$149$	12.5580	1.02879	0.514395	−	0.857554i	$-0.328017\pi$
$149$	12.5580	1.02879	0.514395	+	0.857554i	$0.328017\pi$
$150$	6.84502	0.558894
$151$	15.7683	1.28320	0.641601	−	0.767038i	$-0.278270\pi$
$151$	15.7683	1.28320	0.641601	+	0.767038i	$0.278270\pi$
$152$	47.1609	3.82526
$153$	−9.94566	−0.804059
$154$	−2.08517	−0.168028
$155$	−0.0118430	−0.000951254 0
$156$	−2.60072	−0.208224
$157$	23.4257	1.86957	0.934786	−	0.355212i	$-0.115591\pi$
$157$	23.4257	1.86957	0.934786	+	0.355212i	$0.115591\pi$
$158$	−28.5275	−2.26953
$159$	−4.61420	−0.365930
$160$	0.0697194	0.00551181
$161$	−3.53279	−0.278423
$162$	−17.7121	−1.39160
$163$	−4.10226	−0.321314	−0.160657	−	0.987010i	$-0.551361\pi$
$163$	−4.10226	−0.321314	−0.160657	+	0.987010i	$0.551361\pi$
$164$	−44.7018	−3.49063
$165$	−0.00426610	−0.000332116 0
$166$	−41.3685	−3.21082
$167$	−3.44344	−0.266462	−0.133231	−	0.991085i	$-0.542535\pi$
$167$	−3.44344	−0.266462	−0.133231	+	0.991085i	$0.542535\pi$
$168$	−1.98143	−0.152870
$169$	1.00000	0.0769231
$170$	0.0483465	0.00370801
$171$	15.9792	1.22196
$172$	26.6802	2.03435
$173$	11.5721	0.879811	0.439906	−	0.898044i	$-0.355012\pi$
$173$	11.5721	0.879811	0.439906	+	0.898044i	$0.355012\pi$
$174$	0.0761983	0.00577658
$175$	−2.37988	−0.179902
$176$	18.7022	1.40973
$177$	1.51044	0.113532
$178$	−1.76958	−0.132636
$179$	8.58201	0.641449	0.320725	−	0.947172i	$-0.396074\pi$
$179$	8.58201	0.641449	0.320725	+	0.947172i	$0.396074\pi$
$180$	0.0690168	0.00514421
$181$	−8.74813	−0.650243	−0.325122	−	0.945672i	$-0.605405\pi$
$181$	−8.74813	−0.650243	−0.325122	+	0.945672i	$0.605405\pi$
$182$	1.26298	0.0936185
$183$	1.37724	0.101808
$184$	59.8859	4.41485
$185$	0.0327233	0.00240587
$186$	−3.23728	−0.237369
$187$	6.00635	0.439228
$188$	40.1067	2.92508
$189$	−1.40808	−0.102423
$190$	−0.0776762	−0.00563522
$191$	17.5140	1.26727	0.633634	−	0.773633i	$-0.281562\pi$
$191$	17.5140	1.26727	0.633634	+	0.773633i	$0.281562\pi$
$192$	7.36881	0.531798
$193$	3.43729	0.247421	0.123711	−	0.992318i	$-0.460521\pi$
$193$	3.43729	0.247421	0.123711	+	0.992318i	$0.460521\pi$
$194$	23.7873	1.70783
$195$	0.00258396	0.000185041 0
$196$	−34.1434	−2.43882
$197$	−2.59164	−0.184647	−0.0923234	−	0.995729i	$-0.529429\pi$
$197$	−2.59164	−0.184647	−0.0923234	+	0.995729i	$0.529429\pi$
$198$	11.9763	0.851120
$199$	−7.46291	−0.529032	−0.264516	−	0.964381i	$-0.585212\pi$
$199$	−7.46291	−0.529032	−0.264516	+	0.964381i	$0.585212\pi$
$200$	40.3425	2.85264
$201$	1.69601	0.119627
$202$	−8.77604	−0.617480
$203$	−0.0264927	−0.00185942
$204$	9.46149	0.662437
$205$	0.0444138	0.00310199
$206$	34.1915	2.38223
$207$	20.2908	1.41031
$208$	−11.3279	−0.785446
$209$	−9.65012	−0.667513
$210$	0.00326350	0.000225203 0
$211$	−4.59355	−0.316233	−0.158117	−	0.987420i	$-0.550542\pi$
$211$	−4.59355	−0.316233	−0.158117	+	0.987420i	$0.550542\pi$
$212$	−45.0814	−3.09620
$213$	1.07801	0.0738643
$214$	−27.8243	−1.90203
$215$	−0.0265083	−0.00180785
$216$	23.8690	1.62408
$217$	1.12554	0.0764064
$218$	−21.0620	−1.42650
$219$	−1.52876	−0.103304
$220$	−0.0416804	−0.00281009
$221$	−3.63802	−0.244720
$222$	8.94489	0.600342
$223$	22.7650	1.52446	0.762228	−	0.647308i	$-0.224105\pi$
$223$	22.7650	1.52446	0.762228	+	0.647308i	$0.224105\pi$
$224$	−6.62600	−0.442718
$225$	13.6690	0.911265
$226$	21.5328	1.43234
$227$	14.7090	0.976267	0.488134	−	0.872769i	$-0.337678\pi$
$227$	14.7090	0.976267	0.488134	+	0.872769i	$0.337678\pi$
$228$	−15.2013	−1.00673
$229$	10.5556	0.697531	0.348766	−	0.937210i	$-0.386601\pi$
$229$	10.5556	0.697531	0.348766	+	0.937210i	$0.386601\pi$
$230$	−0.0986348	−0.00650379
$231$	0.405442	0.0266761
$232$	0.449090	0.0294842
$233$	−3.19984	−0.209629	−0.104814	−	0.994492i	$-0.533425\pi$
$233$	−3.19984	−0.209629	−0.104814	+	0.994492i	$0.533425\pi$
$234$	−7.25401	−0.474210
$235$	−0.0398483	−0.00259941
$236$	14.7572	0.960611
$237$	5.54690	0.360310
$238$	−4.59476	−0.297834
$239$	23.3898	1.51296	0.756482	−	0.654015i	$-0.226917\pi$
$239$	23.3898	1.51296	0.756482	+	0.654015i	$0.226917\pi$
$240$	−0.0292708	−0.00188942
$241$	−17.0415	−1.09774	−0.548871	−	0.835907i	$-0.684942\pi$
$241$	−17.0415	−1.09774	−0.548871	+	0.835907i	$0.684942\pi$
$242$	21.9552	1.41133
$243$	12.3188	0.790252
$244$	13.4558	0.861418
$245$	0.0339234	0.00216729
$246$	12.1405	0.774048
$247$	5.84505	0.371911
$248$	−19.0795	−1.21155
$249$	8.04371	0.509749
$250$	−0.132892	−0.00840483
$251$	12.3446	0.779185	0.389592	−	0.920987i	$-0.372616\pi$
$251$	12.3446	0.779185	0.389592	+	0.920987i	$0.372616\pi$
$252$	−6.55922	−0.413192
$253$	−12.2539	−0.770398
$254$	−20.5515	−1.28951
$255$	−0.00940052	−0.000588683 0
$256$	−1.88195	−0.117622
$257$	2.39685	0.149511	0.0747556	−	0.997202i	$-0.476182\pi$
$257$	2.39685	0.149511	0.0747556	+	0.997202i	$0.476182\pi$
$258$	−7.24600	−0.451117
$259$	−3.10996	−0.193244
$260$	0.0252457	0.00156567
$261$	0.152162	0.00941860
$262$	15.6196	0.964981
$263$	−12.6291	−0.778745	−0.389372	−	0.921080i	$-0.627308\pi$
$263$	−12.6291	−0.778745	−0.389372	+	0.921080i	$0.627308\pi$
$264$	−6.87284	−0.422994
$265$	0.0447909	0.00275148
$266$	7.38219	0.452631
$267$	0.344079	0.0210573
$268$	16.5703	1.01219
$269$	6.53901	0.398690	0.199345	−	0.979929i	$-0.436119\pi$
$269$	6.53901	0.398690	0.199345	+	0.979929i	$0.436119\pi$
$270$	−0.0393133	−0.00239253
$271$	3.69104	0.224215	0.112107	−	0.993696i	$-0.464240\pi$
$271$	3.69104	0.224215	0.112107	+	0.993696i	$0.464240\pi$
$272$	41.2110	2.49879
$273$	−0.245575	−0.0148629
$274$	−35.7064	−2.15710
$275$	−8.25492	−0.497790
$276$	−19.3030	−1.16190
$277$	−10.5010	−0.630947	−0.315473	−	0.948934i	$-0.602163\pi$
$277$	−10.5010	−0.630947	−0.315473	+	0.948934i	$0.602163\pi$
$278$	45.8485	2.74981
$279$	−6.46459	−0.387025
$280$	0.0192341	0.00114946
$281$	12.3150	0.734654	0.367327	−	0.930092i	$-0.380273\pi$
$281$	12.3150	0.734654	0.367327	+	0.930092i	$0.380273\pi$
$282$	−10.8925	−0.648638
$283$	6.98483	0.415205	0.207603	−	0.978213i	$-0.433434\pi$
$283$	6.98483	0.415205	0.207603	+	0.978213i	$0.433434\pi$
$284$	10.5323	0.624979
$285$	0.0151034	0.000894647 0
$286$	4.38082	0.259043
$287$	−4.22100	−0.249158
$288$	38.0568	2.24252
$289$	−3.76478	−0.221458
$290$	−0.000739670 0	−4.34349e−5 0
$291$	−4.62521	−0.271135
$292$	−14.9362	−0.874076
$293$	0.0137949	0.000805909 0	0.000402954	−	1.00000i	$-0.499872\pi$
$293$	0.0137949	0.000805909 0	0.000402954		1.00000i	$0.499872\pi$
$294$	9.27292	0.540808
$295$	−0.0146621	−0.000853660 0
$296$	52.7184	3.06420
$297$	−4.88410	−0.283404
$298$	−33.3219	−1.93029
$299$	7.42216	0.429235
$300$	−13.0035	−0.750760
$301$	2.51929	0.145210
$302$	−41.8402	−2.40763
$303$	1.70642	0.0980311
$304$	−66.2119	−3.79751
$305$	−0.0133691	−0.000765511 0
$306$	26.3903	1.50863
$307$	12.7245	0.726225	0.363113	−	0.931745i	$-0.381714\pi$
$307$	12.7245	0.726225	0.363113	+	0.931745i	$0.381714\pi$
$308$	3.96122	0.225712
$309$	−6.64821	−0.378203
$310$	0.0314248	0.00178481
$311$	22.8436	1.29534	0.647670	−	0.761921i	$-0.275743\pi$
$311$	22.8436	1.29534	0.647670	+	0.761921i	$0.275743\pi$
$312$	4.16285	0.235675
$313$	6.11890	0.345860	0.172930	−	0.984934i	$-0.444676\pi$
$313$	6.11890	0.345860	0.172930	+	0.984934i	$0.444676\pi$
$314$	−62.1587	−3.50782
$315$	0.00651695	0.000367189 0
$316$	54.1939	3.04865
$317$	10.8002	0.606601	0.303300	−	0.952895i	$-0.401911\pi$
$317$	10.8002	0.606601	0.303300	+	0.952895i	$0.401911\pi$
$318$	12.2435	0.686584
$319$	−0.0918932	−0.00514503
$320$	−0.0715303	−0.00399867
$321$	5.41018	0.301967
$322$	9.37406	0.522396
$323$	−21.2644	−1.18318
$324$	33.6479	1.86933
$325$	4.99997	0.277349
$326$	10.8851	0.602871
$327$	4.09531	0.226471
$328$	71.5522	3.95081
$329$	3.78710	0.208790
$330$	0.0113199	0.000623138 0
$331$	9.59881	0.527599	0.263799	−	0.964578i	$-0.415024\pi$
$331$	9.59881	0.527599	0.263799	+	0.964578i	$0.415024\pi$
$332$	78.5881	4.31308
$333$	17.8623	0.978845
$334$	9.13699	0.499954
$335$	−0.0164635	−0.000899496 0
$336$	2.78184	0.151762
$337$	2.52387	0.137484	0.0687420	−	0.997634i	$-0.478101\pi$
$337$	2.52387	0.137484	0.0687420	+	0.997634i	$0.478101\pi$
$338$	−2.65345	−0.144328
$339$	−4.18684	−0.227398
$340$	−0.0918443	−0.00498096
$341$	3.90407	0.211417
$342$	−42.4001	−2.29273
$343$	−6.55586	−0.353983
$344$	−42.7057	−2.30254
$345$	0.0191786	0.00103254
$346$	−30.7060	−1.65076
$347$	12.6819	0.680801	0.340401	−	0.940280i	$-0.389437\pi$
$347$	12.6819	0.680801	0.340401	+	0.940280i	$0.389437\pi$
$348$	−0.144755	−0.00775966
$349$	0.762243	0.0408020	0.0204010	−	0.999792i	$-0.493506\pi$
$349$	0.762243	0.0408020	0.0204010	+	0.999792i	$0.493506\pi$
$350$	6.31488	0.337544
$351$	2.95828	0.157902
$352$	−22.9831	−1.22501
$353$	35.0468	1.86535	0.932675	−	0.360717i	$-0.117468\pi$
$353$	35.0468	1.86535	0.932675	+	0.360717i	$0.117468\pi$
$354$	−4.00787	−0.213016
$355$	−0.0104645	−0.000555396 0
$356$	3.36170	0.178169
$357$	0.893407	0.0472841
$358$	−22.7719	−1.20353
$359$	−28.3780	−1.49774	−0.748868	−	0.662720i	$-0.769402\pi$
$359$	−28.3780	−1.49774	−0.748868	+	0.662720i	$0.769402\pi$
$360$	−0.110472	−0.00582239
$361$	15.1646	0.798136
$362$	23.2127	1.22003
$363$	−4.26898	−0.224063
$364$	−2.39930	−0.125757
$365$	0.0148400	0.000776759 0
$366$	−3.65442	−0.191020
$367$	−22.7527	−1.18768	−0.593840	−	0.804583i	$-0.702389\pi$
$367$	−22.7527	−1.18768	−0.593840	+	0.804583i	$0.702389\pi$
$368$	−84.0772	−4.38283
$369$	24.2436	1.26207
$370$	−0.0868296	−0.00451406
$371$	−4.25684	−0.221004
$372$	6.14988	0.318857
$373$	−31.8029	−1.64669	−0.823346	−	0.567540i	$-0.807895\pi$
$373$	−31.8029	−1.64669	−0.823346	+	0.567540i	$0.807895\pi$
$374$	−15.9375	−0.824109
$375$	0.0258396	0.00133435
$376$	−64.1969	−3.31070
$377$	0.0556594	0.00286660
$378$	3.73626	0.192173
$379$	−27.8757	−1.43188	−0.715940	−	0.698162i	$-0.754001\pi$
$379$	−27.8757	−1.43188	−0.715940	+	0.698162i	$0.754001\pi$
$380$	0.147562	0.00756978
$381$	3.99604	0.204723
$382$	−46.4725	−2.37774
$383$	3.33238	0.170277	0.0851384	−	0.996369i	$-0.472867\pi$
$383$	3.33238	0.170277	0.0851384	+	0.996369i	$0.472867\pi$
$384$	−5.18822	−0.264760
$385$	−0.00393569	−0.000200582 0
$386$	−9.12065	−0.464229
$387$	−14.4697	−0.735537
$388$	−45.1889	−2.29412
$389$	−21.2115	−1.07547	−0.537734	−	0.843115i	$-0.680719\pi$
$389$	−21.2115	−1.07547	−0.537734	+	0.843115i	$0.680719\pi$
$390$	−0.00685641	−0.000347188 0
$391$	−27.0020	−1.36555
$392$	54.6518	2.76033
$393$	−3.03708	−0.153200
$394$	6.87678	0.346447
$395$	−0.0538447	−0.00270922
$396$	−22.7515	−1.14331
$397$	4.20883	0.211235	0.105618	−	0.994407i	$-0.466318\pi$
$397$	4.20883	0.211235	0.105618	+	0.994407i	$0.466318\pi$
$398$	19.8024	0.992607
$399$	−1.43540	−0.0718597
$400$	−56.6390	−2.83195
$401$	−33.8744	−1.69161	−0.845803	−	0.533496i	$-0.820878\pi$
$401$	−33.8744	−1.69161	−0.845803	+	0.533496i	$0.820878\pi$
$402$	−4.50027	−0.224453
$403$	−2.36468	−0.117793
$404$	16.6719	0.829460
$405$	−0.0334310	−0.00166120
$406$	0.0702968	0.00348877
$407$	−10.7873	−0.534707
$408$	−15.1446	−0.749768
$409$	−34.8372	−1.72259	−0.861293	−	0.508108i	$-0.830345\pi$
$409$	−34.8372	−1.72259	−0.861293	+	0.508108i	$0.830345\pi$
$410$	−0.117850	−0.00582017
$411$	6.94277	0.342462
$412$	−64.9539	−3.20005
$413$	1.39346	0.0685675
$414$	−53.8405	−2.64611
$415$	−0.0780816	−0.00383288
$416$	13.9208	0.682524
$417$	−8.91479	−0.436559
$418$	25.6061	1.25243
$419$	−30.4052	−1.48539	−0.742697	−	0.669628i	$-0.766454\pi$
$419$	−30.4052	−1.48539	−0.742697	+	0.669628i	$0.766454\pi$
$420$	−0.00619970	−0.000302514 0
$421$	9.98329	0.486556	0.243278	−	0.969957i	$-0.421777\pi$
$421$	9.98329	0.486556	0.243278	+	0.969957i	$0.421777\pi$
$422$	12.1887	0.593338
$423$	−21.7514	−1.05759
$424$	72.1597	3.50438
$425$	−18.1900	−0.882346
$426$	−2.86045	−0.138589
$427$	1.27057	0.0614872
$428$	52.8581	2.55499
$429$	−0.851808	−0.0411257
$430$	0.0703382	0.00339201
$431$	−4.65355	−0.224153	−0.112077	−	0.993700i	$-0.535750\pi$
$431$	−4.65355	−0.224153	−0.112077	+	0.993700i	$0.535750\pi$
$432$	−33.5110	−1.61230
$433$	−12.7533	−0.612884	−0.306442	−	0.951889i	$-0.599139\pi$
$433$	−12.7533	−0.612884	−0.306442	+	0.951889i	$0.599139\pi$
$434$	−2.98655	−0.143359
$435$	0.000143822 0	6.89573e−6 0
$436$	40.0117	1.91621
$437$	43.3829	2.07528
$438$	4.05649	0.193827
$439$	9.54567	0.455590	0.227795	−	0.973709i	$-0.426848\pi$
$439$	9.54567	0.455590	0.227795	+	0.973709i	$0.426848\pi$
$440$	0.0667158	0.00318055
$441$	18.5173	0.881776
$442$	9.65330	0.459161
$443$	38.2436	1.81701	0.908504	−	0.417877i	$-0.137226\pi$
$443$	38.2436	1.81701	0.908504	+	0.417877i	$0.137226\pi$
$444$	−16.9927	−0.806437
$445$	−0.00334003	−0.000158333 0
$446$	−60.4057	−2.86029
$447$	6.47913	0.306452
$448$	6.79810	0.321180
$449$	−9.31882	−0.439783	−0.219891	−	0.975524i	$-0.570570\pi$
$449$	−9.31882	−0.439783	−0.219891	+	0.975524i	$0.570570\pi$
$450$	−36.2699	−1.70978
$451$	−14.6411	−0.689421
$452$	−40.9060	−1.92406
$453$	8.13543	0.382236
$454$	−39.0294	−1.83174
$455$	0.00238384	0.000111756 0
$456$	24.3321	1.13945
$457$	−2.09500	−0.0980000	−0.0490000	−	0.998799i	$-0.515603\pi$
$457$	−2.09500	−0.0980000	−0.0490000	+	0.998799i	$0.515603\pi$
$458$	−28.0086	−1.30876
$459$	−10.7623	−0.502342
$460$	0.187377	0.00873652
$461$	7.03041	0.327439	0.163719	−	0.986507i	$-0.447651\pi$
$461$	7.03041	0.327439	0.163719	+	0.986507i	$0.447651\pi$
$462$	−1.07582	−0.0500516
$463$	−41.7166	−1.93873	−0.969367	−	0.245618i	$-0.921009\pi$
$463$	−41.7166	−1.93873	−0.969367	+	0.245618i	$0.921009\pi$
$464$	−0.630502	−0.0292703
$465$	−0.00611025	−0.000283356 0
$466$	8.49060	0.393320
$467$	−10.0452	−0.464836	−0.232418	−	0.972616i	$-0.574664\pi$
$467$	−10.0452	−0.464836	−0.232418	+	0.972616i	$0.574664\pi$
$468$	13.7805	0.637005
$469$	1.56466	0.0722491
$470$	0.105735	0.00487720
$471$	12.0862	0.556901
$472$	−23.6211	−1.08725
$473$	8.73849	0.401796
$474$	−14.7184	−0.676038
$475$	29.2251	1.34094
$476$	8.72871	0.400080
$477$	24.4494	1.11946
$478$	−62.0637	−2.83873
$479$	−39.0021	−1.78205	−0.891026	−	0.453952i	$-0.850014\pi$
$479$	−39.0021	−1.78205	−0.891026	+	0.453952i	$0.850014\pi$
$480$	0.0359708	0.00164184
$481$	6.53383	0.297917
$482$	45.2188	2.05966
$483$	−1.82270	−0.0829355
$484$	−41.7085	−1.89584
$485$	0.0448977	0.00203870
$486$	−32.6873	−1.48273
$487$	6.16441	0.279336	0.139668	−	0.990198i	$-0.455396\pi$
$487$	6.16441	0.279336	0.139668	+	0.990198i	$0.455396\pi$
$488$	−21.5380	−0.974981
$489$	−2.11651	−0.0957117
$490$	−0.0900139	−0.00406641
$491$	−17.4046	−0.785458	−0.392729	−	0.919654i	$-0.628469\pi$
$491$	−17.4046	−0.785458	−0.392729	+	0.919654i	$0.628469\pi$
$492$	−23.0633	−1.03978
$493$	−0.202490	−0.00911970
$494$	−15.5095	−0.697806
$495$	0.0226049	0.00101601
$496$	26.7868	1.20276
$497$	0.994522	0.0446104
$498$	−21.3435	−0.956427
$499$	33.9519	1.51990	0.759948	−	0.649983i	$-0.225224\pi$
$499$	33.9519	1.51990	0.759948	+	0.649983i	$0.225224\pi$
$500$	0.252456	0.0112902
$501$	−1.77660	−0.0793726
$502$	−32.7557	−1.46196
$503$	12.1637	0.542351	0.271176	−	0.962530i	$-0.412588\pi$
$503$	12.1637	0.542351	0.271176	+	0.962530i	$0.412588\pi$
$504$	10.4990	0.467664
$505$	−0.0165645	−0.000737110 0
$506$	32.5151	1.44547
$507$	0.515937	0.0229136
$508$	39.0418	1.73220
$509$	33.1871	1.47099	0.735496	−	0.677529i	$-0.236949\pi$
$509$	33.1871	1.47099	0.735496	+	0.677529i	$0.236949\pi$
$510$	0.0249438	0.00110453
$511$	−1.41036	−0.0623907
$512$	25.1055	1.10952
$513$	17.2913	0.763430
$514$	−6.35990	−0.280523
$515$	0.0645353	0.00284377
$516$	13.7653	0.605983
$517$	13.1360	0.577723
$518$	8.25211	0.362577
$519$	5.97048	0.262075
$520$	−0.0404096	−0.00177208
$521$	12.2627	0.537238	0.268619	−	0.963246i	$-0.413433\pi$
$521$	12.2627	0.537238	0.268619	+	0.963246i	$0.413433\pi$
$522$	−0.403754	−0.0176718
$523$	10.6567	0.465986	0.232993	−	0.972478i	$-0.425148\pi$
$523$	10.6567	0.465986	0.232993	+	0.972478i	$0.425148\pi$
$524$	−29.6727	−1.29626
$525$	−1.22787	−0.0535885
$526$	33.5107	1.46114
$527$	8.60277	0.374743
$528$	9.64916	0.419926
$529$	32.0885	1.39515
$530$	−0.118850	−0.00516252
$531$	−8.00340	−0.347318
$532$	−14.0240	−0.608018
$533$	8.86805	0.384118
$534$	−0.912994	−0.0395091
$535$	−0.0525175	−0.00227053
$536$	−26.5232	−1.14563
$537$	4.42778	0.191073
$538$	−17.3509	−0.748050
$539$	−11.1829	−0.481682
$540$	0.0746839	0.00321388
$541$	12.0032	0.516058	0.258029	−	0.966137i	$-0.416927\pi$
$541$	12.0032	0.516058	0.258029	+	0.966137i	$0.416927\pi$
$542$	−9.79398	−0.420687
$543$	−4.51348	−0.193692
$544$	−50.6442	−2.17135
$545$	−0.0397539	−0.00170287
$546$	0.651619	0.0278867
$547$	−27.8620	−1.19129	−0.595647	−	0.803246i	$-0.703104\pi$
$547$	−27.8620	−1.19129	−0.595647	+	0.803246i	$0.703104\pi$
$548$	67.8318	2.89763
$549$	−7.29760	−0.311454
$550$	21.9040	0.933989
$551$	0.325332	0.0138596
$552$	30.8974	1.31508
$553$	5.11729	0.217609
$554$	27.8640	1.18383
$555$	0.0168832	0.000716651 0
$556$	−87.0987	−3.69381
$557$	14.7421	0.624644	0.312322	−	0.949976i	$-0.398893\pi$
$557$	14.7421	0.624644	0.312322	+	0.949976i	$0.398893\pi$
$558$	17.1534	0.726163
$559$	−5.29287	−0.223865
$560$	−0.0270038	−0.00114112
$561$	3.09890	0.130836
$562$	−32.6773	−1.37841
$563$	−3.37440	−0.142214	−0.0711069	−	0.997469i	$-0.522653\pi$
$563$	−3.37440	−0.142214	−0.0711069	+	0.997469i	$0.522653\pi$
$564$	20.6925	0.871313
$565$	0.0406424	0.00170984
$566$	−18.5339	−0.779037
$567$	3.17722	0.133431
$568$	−16.8586	−0.707372
$569$	−8.57801	−0.359609	−0.179804	−	0.983702i	$-0.557546\pi$
$569$	−8.57801	−0.359609	−0.179804	+	0.983702i	$0.557546\pi$
$570$	−0.0400760	−0.00167860
$571$	−34.5580	−1.44621	−0.723104	−	0.690739i	$-0.757285\pi$
$571$	−34.5580	−1.44621	−0.723104	+	0.690739i	$0.757285\pi$
$572$	−8.32228	−0.347972
$573$	9.03612	0.377489
$574$	11.2002	0.467487
$575$	37.1106	1.54762
$576$	−39.0453	−1.62689
$577$	41.5208	1.72853	0.864267	−	0.503033i	$-0.167783\pi$
$577$	41.5208	1.72853	0.864267	+	0.503033i	$0.167783\pi$
$578$	9.98964	0.415514
$579$	1.77342	0.0737010
$580$	0.00140516	5.83460e−5 0
$581$	7.42073	0.307864
$582$	12.2727	0.508722
$583$	−14.7654	−0.611520
$584$	23.9077	0.989308
$585$	−0.0136917	−0.000566083 0
$586$	−0.0366041	−0.00151210
$587$	41.0504	1.69433	0.847166	−	0.531328i	$-0.178307\pi$
$587$	41.0504	1.69433	0.847166	+	0.531328i	$0.178307\pi$
$588$	−17.6159	−0.726466
$589$	−13.8217	−0.569512
$590$	0.0389051	0.00160170
$591$	−1.33712	−0.0550019
$592$	−74.0144	−3.04197
$593$	6.70002	0.275137	0.137568	−	0.990492i	$-0.456071\pi$
$593$	6.70002	0.275137	0.137568	+	0.990492i	$0.456071\pi$
$594$	12.9597	0.531743
$595$	−0.00867246	−0.000355536 0
$596$	63.3019	2.59295
$597$	−3.85039	−0.157586
$598$	−19.6943	−0.805360
$599$	−31.5675	−1.28981	−0.644906	−	0.764262i	$-0.723103\pi$
$599$	−31.5675	−1.28981	−0.644906	+	0.764262i	$0.723103\pi$
$600$	20.8142	0.849735
$601$	8.44281	0.344389	0.172195	−	0.985063i	$-0.444914\pi$
$601$	8.44281	0.344389	0.172195	+	0.985063i	$0.444914\pi$
$602$	−6.68481	−0.272452
$603$	−8.98670	−0.365967
$604$	79.4842	3.23417
$605$	0.0414397	0.00168476
$606$	−4.52789	−0.183933
$607$	−27.4718	−1.11505	−0.557524	−	0.830161i	$-0.688249\pi$
$607$	−27.4718	−1.11505	−0.557524	+	0.830161i	$0.688249\pi$
$608$	81.3678	3.29990
$609$	−0.0136685	−0.000553877 0
$610$	0.0354741	0.00143630
$611$	−7.95646	−0.321884
$612$	−50.1338	−2.02654
$613$	−34.8804	−1.40881	−0.704403	−	0.709800i	$-0.748785\pi$
$613$	−34.8804	−1.40881	−0.704403	+	0.709800i	$0.748785\pi$
$614$	−33.7638	−1.36259
$615$	0.0229147	0.000924011 0
$616$	−6.34054	−0.255468
$617$	32.4039	1.30453	0.652266	−	0.757990i	$-0.273818\pi$
$617$	32.4039	1.30453	0.652266	+	0.757990i	$0.273818\pi$
$618$	17.6407	0.709611
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	−0.0596980	−0.00239753
$621$	21.9569	0.881099
$622$	−60.6142	−2.43041
$623$	0.317430	0.0127176
$624$	−5.84446	−0.233966
$625$	24.9996	0.999985
$626$	−16.2362	−0.648927
$627$	−4.97886	−0.198836
$628$	118.084	4.71204
$629$	−23.7702	−0.947782
$630$	−0.0172924	−0.000688945 0
$631$	1.94293	0.0773467	0.0386734	−	0.999252i	$-0.487687\pi$
$631$	1.94293	0.0773467	0.0386734	+	0.999252i	$0.487687\pi$
$632$	−86.7457	−3.45056
$633$	−2.36998	−0.0941984
$634$	−28.6578	−1.13815
$635$	−0.0387902	−0.00153934
$636$	−23.2592	−0.922286
$637$	6.77344	0.268374
$638$	0.243834	0.00965347
$639$	−5.71210	−0.225967
$640$	0.0503629	0.00199077
$641$	6.59171	0.260357	0.130178	−	0.991491i	$-0.458445\pi$
$641$	6.59171	0.260357	0.130178	+	0.991491i	$0.458445\pi$
$642$	−14.3556	−0.566571
$643$	31.1716	1.22929	0.614643	−	0.788805i	$-0.289300\pi$
$643$	31.1716	1.22929	0.614643	+	0.788805i	$0.289300\pi$
$644$	−17.8080	−0.701733
$645$	−0.0136766	−0.000538515 0
$646$	56.4240	2.21997
$647$	42.8632	1.68513	0.842563	−	0.538597i	$-0.181045\pi$
$647$	42.8632	1.68513	0.842563	+	0.538597i	$0.181045\pi$
$648$	−53.8586	−2.11576
$649$	4.83338	0.189727
$650$	−13.2672	−0.520381
$651$	0.580706	0.0227597
$652$	−20.6786	−0.809835
$653$	−4.52802	−0.177195	−0.0885975	−	0.996068i	$-0.528238\pi$
$653$	−4.52802	−0.177195	−0.0885975	+	0.996068i	$0.528238\pi$
$654$	−10.8667	−0.424921
$655$	0.0294814	0.00115194
$656$	−100.456	−3.92215
$657$	8.10050	0.316030
$658$	−10.0489	−0.391746
$659$	6.97938	0.271878	0.135939	−	0.990717i	$-0.456595\pi$
$659$	6.97938	0.271878	0.135939	+	0.990717i	$0.456595\pi$
$660$	−0.0215045	−0.000837060 0
$661$	−16.7494	−0.651475	−0.325737	−	0.945460i	$-0.605612\pi$
$661$	−16.7494	−0.651475	−0.325737	+	0.945460i	$0.605612\pi$
$662$	−25.4699	−0.989917
$663$	−1.87699	−0.0728963
$664$	−125.792	−4.88169
$665$	0.0139336	0.000540323 0
$666$	−47.3965	−1.83658
$667$	0.413113	0.0159958
$668$	−17.3576	−0.671587
$669$	11.7453	0.454100
$670$	0.0436849	0.00168770
$671$	4.40714	0.170136
$672$	−3.41860	−0.131875
$673$	38.4954	1.48389	0.741944	−	0.670462i	$-0.233904\pi$
$673$	38.4954	1.48389	0.741944	+	0.670462i	$0.233904\pi$
$674$	−6.69696	−0.257957
$675$	14.7913	0.569319
$676$	5.04077	0.193876
$677$	−18.4536	−0.709229	−0.354615	−	0.935013i	$-0.615388\pi$
$677$	−18.4536	−0.709229	−0.354615	+	0.935013i	$0.615388\pi$
$678$	11.1096	0.426660
$679$	−4.26699	−0.163752
$680$	0.147011	0.00563761
$681$	7.58889	0.290807
$682$	−10.3592	−0.396676
$683$	−7.28490	−0.278749	−0.139375	−	0.990240i	$-0.544509\pi$
$683$	−7.28490	−0.278749	−0.139375	+	0.990240i	$0.544509\pi$
$684$	80.5478	3.07982
$685$	−0.0673947	−0.00257502
$686$	17.3956	0.664168
$687$	5.44600	0.207778
$688$	59.9570	2.28584
$689$	8.94334	0.340714
$690$	−0.0508893	−0.00193732
$691$	21.5639	0.820330	0.410165	−	0.912011i	$-0.365471\pi$
$691$	21.5639	0.820330	0.410165	+	0.912011i	$0.365471\pi$
$692$	58.3324	2.21747
$693$	−2.14833	−0.0816081
$694$	−33.6508	−1.27737
$695$	0.0865374	0.00328255
$696$	0.231702	0.00878264
$697$	−32.2622	−1.22202
$698$	−2.02257	−0.0765555
$699$	−1.65092	−0.0624434
$700$	−11.9964	−0.453423
$701$	−26.4193	−0.997842	−0.498921	−	0.866648i	$-0.666270\pi$
$701$	−26.4193	−0.997842	−0.498921	+	0.866648i	$0.666270\pi$
$702$	−7.84965	−0.296266
$703$	38.1906	1.44038
$704$	23.5801	0.888708
$705$	−0.0205592	−0.000774304 0
$706$	−92.9947	−3.49990
$707$	1.57426	0.0592060
$708$	7.61378	0.286143
$709$	41.6078	1.56261	0.781306	−	0.624148i	$-0.214554\pi$
$709$	41.6078	1.56261	0.781306	+	0.624148i	$0.214554\pi$
$710$	0.0277669	0.00104207
$711$	−29.3915	−1.10227
$712$	−5.38091	−0.201658
$713$	−17.5511	−0.657292
$714$	−2.37061	−0.0887177
$715$	0.00826865	0.000309230 0
$716$	43.2600	1.61670
$717$	12.0677	0.450676
$718$	75.2996	2.81016
$719$	30.9501	1.15424	0.577122	−	0.816658i	$-0.304176\pi$
$719$	30.9501	1.15424	0.577122	+	0.816658i	$0.304176\pi$
$720$	0.155098	0.00578015
$721$	−6.13331	−0.228416
$722$	−40.2384	−1.49752
$723$	−8.79236	−0.326992
$724$	−44.0974	−1.63887
$725$	0.278296	0.0103356
$726$	11.3275	0.420403
$727$	−12.1071	−0.449028	−0.224514	−	0.974471i	$-0.572079\pi$
$727$	−12.1071	−0.449028	−0.224514	+	0.974471i	$0.572079\pi$
$728$	3.84044	0.142336
$729$	−13.6697	−0.506285
$730$	−0.0393771	−0.00145741
$731$	19.2556	0.712194
$732$	6.94234	0.256596
$733$	−1.86559	−0.0689070	−0.0344535	−	0.999406i	$-0.510969\pi$
$733$	−1.86559	−0.0689070	−0.0344535	+	0.999406i	$0.510969\pi$
$734$	60.3730	2.22841
$735$	0.0175023	0.000645583 0
$736$	103.322	3.80852
$737$	5.42721	0.199914
$738$	−64.3290	−2.36798
$739$	7.80076	0.286956	0.143478	−	0.989654i	$-0.454171\pi$
$739$	7.80076	0.286956	0.143478	+	0.989654i	$0.454171\pi$
$740$	0.164951	0.00606372
$741$	3.01568	0.110784
$742$	11.2953	0.414663
$743$	12.6571	0.464343	0.232172	−	0.972675i	$-0.425417\pi$
$743$	12.6571	0.464343	0.232172	+	0.972675i	$0.425417\pi$
$744$	−9.84383	−0.360892
$745$	−0.0628940	−0.00230426
$746$	84.3872	3.08964
$747$	−42.6214	−1.55944
$748$	30.2766	1.10702
$749$	4.99116	0.182373
$750$	−0.0685639	−0.00250360
$751$	−17.7165	−0.646486	−0.323243	−	0.946316i	$-0.604773\pi$
$751$	−17.7165	−0.646486	−0.323243	+	0.946316i	$0.604773\pi$
$752$	90.1297	3.28669
$753$	6.36904	0.232101
$754$	−0.147689	−0.00537852
$755$	−0.0789720	−0.00287409
$756$	−7.09781	−0.258145
$757$	2.58644	0.0940059	0.0470029	−	0.998895i	$-0.485033\pi$
$757$	2.58644	0.0940059	0.0470029	+	0.998895i	$0.485033\pi$
$758$	73.9667	2.68659
$759$	−6.32225	−0.229483
$760$	−0.236196	−0.00856772
$761$	17.0680	0.618713	0.309357	−	0.950946i	$-0.399886\pi$
$761$	17.0680	0.618713	0.309357	+	0.950946i	$0.399886\pi$
$762$	−10.6033	−0.384116
$763$	3.77813	0.136777
$764$	88.2841	3.19401
$765$	0.0498108	0.00180091
$766$	−8.84230	−0.319485
$767$	−2.92756	−0.105708
$768$	−0.970967	−0.0350368
$769$	−7.51879	−0.271134	−0.135567	−	0.990768i	$-0.543286\pi$
$769$	−7.51879	−0.271134	−0.135567	+	0.990768i	$0.543286\pi$
$770$	0.0104432	0.000376345 0
$771$	1.23662	0.0445358
$772$	17.3266	0.623598
$773$	7.85500	0.282525	0.141262	−	0.989972i	$-0.454884\pi$
$773$	7.85500	0.282525	0.141262	+	0.989972i	$0.454884\pi$
$774$	38.3946	1.38006
$775$	−11.8234	−0.424707
$776$	72.3318	2.59656
$777$	−1.60454	−0.0575627
$778$	56.2836	2.01787
$779$	51.8342	1.85715
$780$	0.0130252	0.000466376 0
$781$	3.44963	0.123437
$782$	71.6483	2.56214
$783$	0.164656	0.00588434
$784$	−76.7286	−2.74031
$785$	−0.117323	−0.00418742
$786$	8.05872	0.287445
$787$	39.3063	1.40112	0.700560	−	0.713594i	$-0.252934\pi$
$787$	39.3063	1.40112	0.700560	+	0.713594i	$0.252934\pi$
$788$	−13.0639	−0.465381
$789$	−6.51583	−0.231970
$790$	0.142874	0.00508323
$791$	−3.86258	−0.137337
$792$	36.4173	1.29403
$793$	−2.66939	−0.0947927
$794$	−11.1679	−0.396334
$795$	0.0231093	0.000819601 0
$796$	−37.6189	−1.33337
$797$	13.3509	0.472913	0.236457	−	0.971642i	$-0.424014\pi$
$797$	13.3509	0.472913	0.236457	+	0.971642i	$0.424014\pi$
$798$	3.80875	0.134828
$799$	28.9458	1.02403
$800$	69.6037	2.46086
$801$	−1.82318	−0.0644188
$802$	89.8838	3.17391
$803$	−4.89202	−0.172636
$804$	8.54921	0.301507
$805$	0.0176932	0.000623604 0
$806$	6.27456	0.221012
$807$	3.37372	0.118760
$808$	−26.6860	−0.938809
$809$	−1.11585	−0.0392311	−0.0196156	−	0.999808i	$-0.506244\pi$
$809$	−1.11585	−0.0392311	−0.0196156	+	0.999808i	$0.506244\pi$
$810$	0.0887074	0.00311686
$811$	−18.3014	−0.642648	−0.321324	−	0.946969i	$-0.604128\pi$
$811$	−18.3014	−0.642648	−0.321324	+	0.946969i	$0.604128\pi$
$812$	−0.133543	−0.00468646
$813$	1.90434	0.0667883
$814$	28.6235	1.00325
$815$	0.0205453	0.000719670 0
$816$	21.2623	0.744329
$817$	−30.9371	−1.08235
$818$	92.4385	3.23204
$819$	1.30123	0.0454688
$820$	0.223880	0.00781822
$821$	−9.42950	−0.329092	−0.164546	−	0.986369i	$-0.552616\pi$
$821$	−9.42950	−0.329092	−0.164546	+	0.986369i	$0.552616\pi$
$822$	−18.4223	−0.642551
$823$	19.3028	0.672854	0.336427	−	0.941710i	$-0.390781\pi$
$823$	19.3028	0.672854	0.336427	+	0.941710i	$0.390781\pi$
$824$	103.969	3.62192
$825$	−4.25902	−0.148280
$826$	−3.69746	−0.128651
$827$	−12.1323	−0.421881	−0.210941	−	0.977499i	$-0.567653\pi$
$827$	−12.1323	−0.421881	−0.210941	+	0.977499i	$0.567653\pi$
$828$	102.281	3.55452
$829$	−23.7879	−0.826188	−0.413094	−	0.910688i	$-0.635552\pi$
$829$	−23.7879	−0.826188	−0.413094	+	0.910688i	$0.635552\pi$
$830$	0.207185	0.00719151
$831$	−5.41788	−0.187944
$832$	−14.2824	−0.495152
$833$	−24.6420	−0.853793
$834$	23.6549	0.819103
$835$	0.0172458	0.000596814 0
$836$	−48.6441	−1.68239
$837$	−6.99540	−0.241797
$838$	80.6787	2.78700
$839$	24.3103	0.839285	0.419643	−	0.907689i	$-0.362155\pi$
$839$	24.3103	0.839285	0.419643	+	0.907689i	$0.362155\pi$
$840$	0.00992357	0.000342396 0
$841$	−28.9969	−0.999893
$842$	−26.4901	−0.912910
$843$	6.35379	0.218836
$844$	−23.1551	−0.797030
$845$	−0.00500829	−0.000172290 0
$846$	57.7163	1.98433
$847$	−3.93835	−0.135323
$848$	−101.309	−3.47896
$849$	3.60373	0.123680
$850$	48.2663	1.65552
$851$	48.4952	1.66239
$852$	5.43402	0.186167
$853$	−20.3283	−0.696028	−0.348014	−	0.937489i	$-0.613144\pi$
$853$	−20.3283	−0.696028	−0.348014	+	0.937489i	$0.613144\pi$
$854$	−3.37139	−0.115367
$855$	−0.0800287	−0.00273692
$856$	−84.6076	−2.89183
$857$	−5.81877	−0.198766	−0.0993828	−	0.995049i	$-0.531687\pi$
$857$	−5.81877	−0.198766	−0.0993828	+	0.995049i	$0.531687\pi$
$858$	2.26023	0.0771628
$859$	17.2590	0.588871	0.294435	−	0.955671i	$-0.404868\pi$
$859$	17.2590	0.588871	0.294435	+	0.955671i	$0.404868\pi$
$860$	−0.133622	−0.00455648
$861$	−2.17777	−0.0742182
$862$	12.3479	0.420572
$863$	−39.7257	−1.35228	−0.676139	−	0.736774i	$-0.736348\pi$
$863$	−39.7257	−1.35228	−0.676139	+	0.736774i	$0.736348\pi$
$864$	41.1817	1.40103
$865$	−0.0579565	−0.00197058
$866$	33.8402	1.14994
$867$	−1.94239	−0.0659670
$868$	5.67358	0.192574
$869$	17.7500	0.602127
$870$	−0.000381623 0	−1.29382e−5 0
$871$	−3.28724	−0.111384
$872$	−64.0449	−2.16883
$873$	24.5077	0.829461
$874$	−115.114	−3.89379
$875$	0.0238383	0.000805882 0
$876$	−7.70615	−0.260367
$877$	13.6176	0.459835	0.229917	−	0.973210i	$-0.426154\pi$
$877$	13.6176	0.459835	0.229917	+	0.973210i	$0.426154\pi$
$878$	−25.3289	−0.854810
$879$	0.00711732	0.000240061 0
$880$	−0.0936661	−0.00315748
$881$	−30.9788	−1.04370	−0.521851	−	0.853036i	$-0.674758\pi$
$881$	−30.9788	−1.04370	−0.521851	+	0.853036i	$0.674758\pi$
$882$	−49.1347	−1.65445
$883$	46.0102	1.54837	0.774184	−	0.632960i	$-0.218160\pi$
$883$	46.0102	1.54837	0.774184	+	0.632960i	$0.218160\pi$
$884$	−18.3385	−0.616789
$885$	−0.00756472	−0.000254285 0
$886$	−101.477	−3.40920
$887$	−11.4962	−0.386005	−0.193002	−	0.981198i	$-0.561823\pi$
$887$	−11.4962	−0.386005	−0.193002	+	0.981198i	$0.561823\pi$
$888$	27.1994	0.912752
$889$	3.68655	0.123643
$890$	0.00886259	0.000297075 0
$891$	11.0206	0.369204
$892$	114.753	3.84222
$893$	−46.5059	−1.55626
$894$	−17.1920	−0.574987
$895$	−0.0429812	−0.00143670
$896$	−4.78639	−0.159902
$897$	3.82937	0.127859
$898$	24.7270	0.825151
$899$	−0.131617	−0.00438967
$900$	68.9022	2.29674
$901$	−32.5361	−1.08393
$902$	38.8493	1.29354
$903$	1.29980	0.0432545
$904$	65.4764	2.17771
$905$	0.0438132	0.00145640
$906$	−21.5869	−0.717177
$907$	2.07117	0.0687722	0.0343861	−	0.999409i	$-0.489052\pi$
$907$	2.07117	0.0687722	0.0343861	+	0.999409i	$0.489052\pi$
$908$	74.1445	2.46057
$909$	−9.04184	−0.299899
$910$	−0.00632538	−0.000209684 0
$911$	26.3507	0.873038	0.436519	−	0.899695i	$-0.356211\pi$
$911$	26.3507	0.873038	0.436519	+	0.899695i	$0.356211\pi$
$912$	−34.1612	−1.13119
$913$	25.7398	0.851861
$914$	5.55897	0.183874
$915$	−0.00689760	−0.000228028 0
$916$	53.2082	1.75805
$917$	−2.80186	−0.0925255
$918$	28.5572	0.942528
$919$	7.01806	0.231504	0.115752	−	0.993278i	$-0.463072\pi$
$919$	7.01806	0.231504	0.115752	+	0.993278i	$0.463072\pi$
$920$	−0.299926	−0.00988827
$921$	6.56504	0.216325
$922$	−18.6548	−0.614364
$923$	−2.08943	−0.0687744
$924$	2.04374	0.0672342
$925$	32.6690	1.07415
$926$	110.693	3.63759
$927$	35.2270	1.15701
$928$	0.774824	0.0254348
$929$	33.7765	1.10817	0.554086	−	0.832459i	$-0.313068\pi$
$929$	33.7765	1.10817	0.554086	+	0.832459i	$0.313068\pi$
$930$	0.0162132	0.000531653 0
$931$	39.5911	1.29755
$932$	−16.1297	−0.528345
$933$	11.7858	0.385851
$934$	26.6544	0.872158
$935$	−0.0300815	−0.000983771 0
$936$	−22.0578	−0.720983
$937$	−41.2222	−1.34667	−0.673335	−	0.739338i	$-0.735139\pi$
$937$	−41.2222	−1.34667	−0.673335	+	0.739338i	$0.735139\pi$
$938$	−4.15173	−0.135559
$939$	3.15697	0.103024
$940$	−0.200866	−0.00655153
$941$	46.8750	1.52808	0.764040	−	0.645169i	$-0.223213\pi$
$941$	46.8750	1.52808	0.764040	+	0.645169i	$0.223213\pi$
$942$	−32.0700	−1.04490
$943$	65.8201	2.14340
$944$	33.1630	1.07936
$945$	0.00705207	0.000229404 0
$946$	−23.1871	−0.753878
$947$	−55.3568	−1.79885	−0.899426	−	0.437072i	$-0.856015\pi$
$947$	−55.3568	−1.79885	−0.899426	+	0.437072i	$0.856015\pi$
$948$	27.9607	0.908120
$949$	2.96308	0.0961857
$950$	−77.5472	−2.51596
$951$	5.57223	0.180692
$952$	−13.9716	−0.452823
$953$	35.7661	1.15858	0.579289	−	0.815122i	$-0.303330\pi$
$953$	35.7661	1.15858	0.579289	+	0.815122i	$0.303330\pi$
$954$	−64.8751	−2.10041
$955$	−0.0877152	−0.00283840
$956$	117.903	3.81325
$957$	−0.0474111	−0.00153258
$958$	103.490	3.34361
$959$	6.40506	0.206830
$960$	−0.0369051	−0.00119111
$961$	−25.4083	−0.819622
$962$	−17.3372	−0.558973
$963$	−28.6670	−0.923782
$964$	−85.9026	−2.76674
$965$	−0.0172149	−0.000554168 0
$966$	4.83642	0.155609
$967$	6.61843	0.212835	0.106417	−	0.994322i	$-0.466062\pi$
$967$	6.61843	0.212835	0.106417	+	0.994322i	$0.466062\pi$
$968$	66.7609	2.14578
$969$	−10.9711	−0.352443
$970$	−0.119134	−0.00382515
$971$	26.7164	0.857371	0.428686	−	0.903454i	$-0.358977\pi$
$971$	26.7164	0.857371	0.428686	+	0.903454i	$0.358977\pi$
$972$	62.0963	1.99174
$973$	−8.22435	−0.263660
$974$	−16.3569	−0.524110
$975$	2.57967	0.0826156
$976$	30.2385	0.967909
$977$	30.1264	0.963828	0.481914	−	0.876219i	$-0.339942\pi$
$977$	30.1264	0.963828	0.481914	+	0.876219i	$0.339942\pi$
$978$	5.61604	0.179581
$979$	1.10105	0.0351896
$980$	0.171000	0.00546240
$981$	−21.6999	−0.692825
$982$	46.1821	1.47373
$983$	−6.58163	−0.209921	−0.104961	−	0.994476i	$-0.533472\pi$
$983$	−6.58163	−0.209921	−0.104961	+	0.994476i	$0.533472\pi$
$984$	36.9164	1.17685
$985$	0.0129797	0.000413567 0
$986$	0.537297	0.0171110
$987$	1.95391	0.0621935
$988$	29.4636	0.937361
$989$	−39.2846	−1.24918
$990$	−0.0599809	−0.00190632
$991$	59.2791	1.88306	0.941531	−	0.336928i	$-0.109388\pi$
$991$	59.2791	1.88306	0.941531	+	0.336928i	$0.109388\pi$
$992$	−32.9183	−1.04516
$993$	4.95238	0.157159
$994$	−2.63891	−0.0837011
$995$	0.0373764	0.00118491
$996$	40.5465	1.28477
$997$	36.8420	1.16680	0.583399	−	0.812186i	$-0.301723\pi$
$997$	36.8420	1.16680	0.583399	+	0.812186i	$0.301723\pi$
$998$	−90.0896	−2.85174
$999$	19.3289	0.611541

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.d.1.5	✓	156

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.d.1.5	✓	156	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.65345 q^{2} +0.515937 q^{3} +5.04077 q^{4} -0.00500829 q^{5} -1.36901 q^{6} +0.475978 q^{7} -8.06853 q^{8} -2.73381 q^{9} +O(q^{10})\)	\(q-2.65345 q^{2} +0.515937 q^{3} +5.04077 q^{4} -0.00500829 q^{5} -1.36901 q^{6} +0.475978 q^{7} -8.06853 q^{8} -2.73381 q^{9} +0.0132892 q^{10} +1.65099 q^{11} +2.60072 q^{12} -1.00000 q^{13} -1.26298 q^{14} -0.00258396 q^{15} +11.3279 q^{16} +3.63802 q^{17} +7.25401 q^{18} -5.84505 q^{19} -0.0252457 q^{20} +0.245575 q^{21} -4.38082 q^{22} -7.42216 q^{23} -4.16285 q^{24} -4.99997 q^{25} +2.65345 q^{26} -2.95828 q^{27} +2.39930 q^{28} -0.0556594 q^{29} +0.00685641 q^{30} +2.36468 q^{31} -13.9208 q^{32} +0.851808 q^{33} -9.65330 q^{34} -0.00238384 q^{35} -13.7805 q^{36} -6.53383 q^{37} +15.5095 q^{38} -0.515937 q^{39} +0.0404096 q^{40} -8.86805 q^{41} -0.651619 q^{42} +5.29287 q^{43} +8.32228 q^{44} +0.0136917 q^{45} +19.6943 q^{46} +7.95646 q^{47} +5.84446 q^{48} -6.77344 q^{49} +13.2672 q^{50} +1.87699 q^{51} -5.04077 q^{52} -8.94334 q^{53} +7.84965 q^{54} -0.00826865 q^{55} -3.84044 q^{56} -3.01568 q^{57} +0.147689 q^{58} +2.92756 q^{59} -0.0130252 q^{60} +2.66939 q^{61} -6.27456 q^{62} -1.30123 q^{63} +14.2824 q^{64} +0.00500829 q^{65} -2.26023 q^{66} +3.28724 q^{67} +18.3385 q^{68} -3.82937 q^{69} +0.00632538 q^{70} +2.08943 q^{71} +22.0578 q^{72} -2.96308 q^{73} +17.3372 q^{74} -2.57967 q^{75} -29.4636 q^{76} +0.785836 q^{77} +1.36901 q^{78} +10.7511 q^{79} -0.0567332 q^{80} +6.67514 q^{81} +23.5309 q^{82} +15.5905 q^{83} +1.23789 q^{84} -0.0182203 q^{85} -14.0444 q^{86} -0.0287167 q^{87} -13.3211 q^{88} +0.666900 q^{89} -0.0363302 q^{90} -0.475978 q^{91} -37.4134 q^{92} +1.22003 q^{93} -21.1120 q^{94} +0.0292737 q^{95} -7.18226 q^{96} -8.96468 q^{97} +17.9730 q^{98} -4.51350 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * q^99

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