LMFDB - Embedded newform 8002.2.a.d.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.62997 q^{3} +1.00000 q^{4} -0.133179 q^{5} -2.62997 q^{6} -2.98290 q^{7} +1.00000 q^{8} +3.91674 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.62997 q^{3} +1.00000 q^{4} -0.133179 q^{5} -2.62997 q^{6} -2.98290 q^{7} +1.00000 q^{8} +3.91674 q^{9} -0.133179 q^{10} -3.21116 q^{11} -2.62997 q^{12} +6.39878 q^{13} -2.98290 q^{14} +0.350258 q^{15} +1.00000 q^{16} -3.30489 q^{17} +3.91674 q^{18} +4.41196 q^{19} -0.133179 q^{20} +7.84493 q^{21} -3.21116 q^{22} -1.70409 q^{23} -2.62997 q^{24} -4.98226 q^{25} +6.39878 q^{26} -2.41100 q^{27} -2.98290 q^{28} -4.05713 q^{29} +0.350258 q^{30} +9.35394 q^{31} +1.00000 q^{32} +8.44527 q^{33} -3.30489 q^{34} +0.397260 q^{35} +3.91674 q^{36} -11.1712 q^{37} +4.41196 q^{38} -16.8286 q^{39} -0.133179 q^{40} -9.01963 q^{41} +7.84493 q^{42} +7.51380 q^{43} -3.21116 q^{44} -0.521629 q^{45} -1.70409 q^{46} +7.66372 q^{47} -2.62997 q^{48} +1.89767 q^{49} -4.98226 q^{50} +8.69175 q^{51} +6.39878 q^{52} -3.41893 q^{53} -2.41100 q^{54} +0.427661 q^{55} -2.98290 q^{56} -11.6033 q^{57} -4.05713 q^{58} +6.87216 q^{59} +0.350258 q^{60} +12.8840 q^{61} +9.35394 q^{62} -11.6832 q^{63} +1.00000 q^{64} -0.852185 q^{65} +8.44527 q^{66} +6.96556 q^{67} -3.30489 q^{68} +4.48170 q^{69} +0.397260 q^{70} -3.36425 q^{71} +3.91674 q^{72} +8.97955 q^{73} -11.1712 q^{74} +13.1032 q^{75} +4.41196 q^{76} +9.57857 q^{77} -16.8286 q^{78} -6.45762 q^{79} -0.133179 q^{80} -5.40936 q^{81} -9.01963 q^{82} -2.50616 q^{83} +7.84493 q^{84} +0.440142 q^{85} +7.51380 q^{86} +10.6701 q^{87} -3.21116 q^{88} +6.26390 q^{89} -0.521629 q^{90} -19.0869 q^{91} -1.70409 q^{92} -24.6006 q^{93} +7.66372 q^{94} -0.587583 q^{95} -2.62997 q^{96} +4.35742 q^{97} +1.89767 q^{98} -12.5773 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.62997	−1.51841	−0.759207	−	0.650849i	$-0.774413\pi$
$3$	−2.62997	−1.51841	−0.759207	+	0.650849i	$0.774413\pi$
$4$	1.00000	0.500000
$5$	−0.133179	−0.0595596	−0.0297798	−	0.999556i	$-0.509481\pi$
$5$	−0.133179	−0.0595596	−0.0297798	+	0.999556i	$0.509481\pi$
$6$	−2.62997	−1.07368
$7$	−2.98290	−1.12743	−0.563714	−	0.825970i	$-0.690628\pi$
$7$	−2.98290	−1.12743	−0.563714	+	0.825970i	$0.690628\pi$
$8$	1.00000	0.353553
$9$	3.91674	1.30558
$10$	−0.133179	−0.0421150
$11$	−3.21116	−0.968203	−0.484101	−	0.875012i	$-0.660853\pi$
$11$	−3.21116	−0.968203	−0.484101	+	0.875012i	$0.660853\pi$
$12$	−2.62997	−0.759207
$13$	6.39878	1.77470	0.887351	−	0.461095i	$-0.152543\pi$
$13$	6.39878	1.77470	0.887351	+	0.461095i	$0.152543\pi$
$14$	−2.98290	−0.797212
$15$	0.350258	0.0904361
$16$	1.00000	0.250000
$17$	−3.30489	−0.801553	−0.400776	−	0.916176i	$-0.631260\pi$
$17$	−3.30489	−0.801553	−0.400776	+	0.916176i	$0.631260\pi$
$18$	3.91674	0.923185
$19$	4.41196	1.01217	0.506087	−	0.862482i	$-0.331091\pi$
$19$	4.41196	1.01217	0.506087	+	0.862482i	$0.331091\pi$
$20$	−0.133179	−0.0297798
$21$	7.84493	1.71190
$22$	−3.21116	−0.684623
$23$	−1.70409	−0.355327	−0.177663	−	0.984091i	$-0.556854\pi$
$23$	−1.70409	−0.355327	−0.177663	+	0.984091i	$0.556854\pi$
$24$	−2.62997	−0.536840
$25$	−4.98226	−0.996453
$26$	6.39878	1.25490
$27$	−2.41100	−0.463998
$28$	−2.98290	−0.563714
$29$	−4.05713	−0.753390	−0.376695	−	0.926337i	$-0.622939\pi$
$29$	−4.05713	−0.753390	−0.376695	+	0.926337i	$0.622939\pi$
$30$	0.350258	0.0639480
$31$	9.35394	1.68002	0.840009	−	0.542573i	$-0.182550\pi$
$31$	9.35394	1.68002	0.840009	+	0.542573i	$0.182550\pi$
$32$	1.00000	0.176777
$33$	8.44527	1.47013
$34$	−3.30489	−0.566783
$35$	0.397260	0.0671492
$36$	3.91674	0.652790
$37$	−11.1712	−1.83653	−0.918267	−	0.395961i	$-0.870411\pi$
$37$	−11.1712	−1.83653	−0.918267	+	0.395961i	$0.870411\pi$
$38$	4.41196	0.715715
$39$	−16.8286	−2.69473
$40$	−0.133179	−0.0210575
$41$	−9.01963	−1.40863	−0.704315	−	0.709888i	$-0.748746\pi$
$41$	−9.01963	−1.40863	−0.704315	+	0.709888i	$0.748746\pi$
$42$	7.84493	1.21050
$43$	7.51380	1.14584	0.572922	−	0.819610i	$-0.305810\pi$
$43$	7.51380	1.14584	0.572922	+	0.819610i	$0.305810\pi$
$44$	−3.21116	−0.484101
$45$	−0.521629	−0.0777599
$46$	−1.70409	−0.251254
$47$	7.66372	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$47$	7.66372	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$48$	−2.62997	−0.379603
$49$	1.89767	0.271095
$50$	−4.98226	−0.704598
$51$	8.69175	1.21709
$52$	6.39878	0.887351
$53$	−3.41893	−0.469626	−0.234813	−	0.972041i	$-0.575448\pi$
$53$	−3.41893	−0.469626	−0.234813	+	0.972041i	$0.575448\pi$
$54$	−2.41100	−0.328096
$55$	0.427661	0.0576658
$56$	−2.98290	−0.398606
$57$	−11.6033	−1.53690
$58$	−4.05713	−0.532727
$59$	6.87216	0.894679	0.447339	−	0.894364i	$-0.352372\pi$
$59$	6.87216	0.894679	0.447339	+	0.894364i	$0.352372\pi$
$60$	0.350258	0.0452181
$61$	12.8840	1.64963	0.824813	−	0.565405i	$-0.191280\pi$
$61$	12.8840	1.64963	0.824813	+	0.565405i	$0.191280\pi$
$62$	9.35394	1.18795
$63$	−11.6832	−1.47195
$64$	1.00000	0.125000
$65$	−0.852185	−0.105701
$66$	8.44527	1.03954
$67$	6.96556	0.850978	0.425489	−	0.904964i	$-0.360102\pi$
$67$	6.96556	0.850978	0.425489	+	0.904964i	$0.360102\pi$
$68$	−3.30489	−0.400776
$69$	4.48170	0.539533
$70$	0.397260	0.0474817
$71$	−3.36425	−0.399263	−0.199632	−	0.979871i	$-0.563975\pi$
$71$	−3.36425	−0.399263	−0.199632	+	0.979871i	$0.563975\pi$
$72$	3.91674	0.461592
$73$	8.97955	1.05098	0.525488	−	0.850801i	$-0.323883\pi$
$73$	8.97955	1.05098	0.525488	+	0.850801i	$0.323883\pi$
$74$	−11.1712	−1.29863
$75$	13.1032	1.51303
$76$	4.41196	0.506087
$77$	9.57857	1.09158
$78$	−16.8286	−1.90546
$79$	−6.45762	−0.726539	−0.363269	−	0.931684i	$-0.618340\pi$
$79$	−6.45762	−0.726539	−0.363269	+	0.931684i	$0.618340\pi$
$80$	−0.133179	−0.0148899
$81$	−5.40936	−0.601040
$82$	−9.01963	−0.996051
$83$	−2.50616	−0.275087	−0.137543	−	0.990496i	$-0.543921\pi$
$83$	−2.50616	−0.275087	−0.137543	+	0.990496i	$0.543921\pi$
$84$	7.84493	0.855952
$85$	0.440142	0.0477402
$86$	7.51380	0.810234
$87$	10.6701	1.14396
$88$	−3.21116	−0.342311
$89$	6.26390	0.663972	0.331986	−	0.943284i	$-0.392281\pi$
$89$	6.26390	0.663972	0.331986	+	0.943284i	$0.392281\pi$
$90$	−0.521629	−0.0549845
$91$	−19.0869	−2.00085
$92$	−1.70409	−0.177663
$93$	−24.6006	−2.55096
$94$	7.66372	0.790452
$95$	−0.587583	−0.0602847
$96$	−2.62997	−0.268420
$97$	4.35742	0.442429	0.221215	−	0.975225i	$-0.428998\pi$
$97$	4.35742	0.442429	0.221215	+	0.975225i	$0.428998\pi$
$98$	1.89767	0.191693
$99$	−12.5773	−1.26407
$100$	−4.98226	−0.498226
$101$	−5.87049	−0.584135	−0.292068	−	0.956398i	$-0.594343\pi$
$101$	−5.87049	−0.584135	−0.292068	+	0.956398i	$0.594343\pi$
$102$	8.69175	0.860612
$103$	2.77219	0.273152	0.136576	−	0.990630i	$-0.456390\pi$
$103$	2.77219	0.273152	0.136576	+	0.990630i	$0.456390\pi$
$104$	6.39878	0.627452
$105$	−1.04478	−0.101960
$106$	−3.41893	−0.332076
$107$	4.07490	0.393936	0.196968	−	0.980410i	$-0.436891\pi$
$107$	4.07490	0.393936	0.196968	+	0.980410i	$0.436891\pi$
$108$	−2.41100	−0.231999
$109$	12.3408	1.18203	0.591015	−	0.806661i	$-0.298727\pi$
$109$	12.3408	1.18203	0.591015	+	0.806661i	$0.298727\pi$
$110$	0.427661	0.0407759
$111$	29.3799	2.78862
$112$	−2.98290	−0.281857
$113$	−10.3693	−0.975458	−0.487729	−	0.872995i	$-0.662175\pi$
$113$	−10.3693	−0.975458	−0.487729	+	0.872995i	$0.662175\pi$
$114$	−11.6033	−1.08675
$115$	0.226949	0.0211631
$116$	−4.05713	−0.376695
$117$	25.0624	2.31702
$118$	6.87216	0.632633
$119$	9.85813	0.903693
$120$	0.350258	0.0319740
$121$	−0.688423	−0.0625839
$122$	12.8840	1.16646
$123$	23.7214	2.13888
$124$	9.35394	0.840009
$125$	1.32943	0.118908
$126$	−11.6832	−1.04083
$127$	4.89366	0.434242	0.217121	−	0.976145i	$-0.430333\pi$
$127$	4.89366	0.434242	0.217121	+	0.976145i	$0.430333\pi$
$128$	1.00000	0.0883883
$129$	−19.7611	−1.73987
$130$	−0.852185	−0.0747416
$131$	−15.7955	−1.38006	−0.690029	−	0.723781i	$-0.742402\pi$
$131$	−15.7955	−1.38006	−0.690029	+	0.723781i	$0.742402\pi$
$132$	8.44527	0.735066
$133$	−13.1604	−1.14115
$134$	6.96556	0.601732
$135$	0.321096	0.0276355
$136$	−3.30489	−0.283392
$137$	−10.6305	−0.908226	−0.454113	−	0.890944i	$-0.650044\pi$
$137$	−10.6305	−0.908226	−0.454113	+	0.890944i	$0.650044\pi$
$138$	4.48170	0.381508
$139$	7.14185	0.605764	0.302882	−	0.953028i	$-0.402051\pi$
$139$	7.14185	0.605764	0.302882	+	0.953028i	$0.402051\pi$
$140$	0.397260	0.0335746
$141$	−20.1553	−1.69739
$142$	−3.36425	−0.282322
$143$	−20.5475	−1.71827
$144$	3.91674	0.326395
$145$	0.540326	0.0448716
$146$	8.97955	0.743153
$147$	−4.99080	−0.411635
$148$	−11.1712	−0.918267
$149$	3.29780	0.270166	0.135083	−	0.990834i	$-0.456870\pi$
$149$	3.29780	0.270166	0.135083	+	0.990834i	$0.456870\pi$
$150$	13.1032	1.06987
$151$	−4.60869	−0.375050	−0.187525	−	0.982260i	$-0.560047\pi$
$151$	−4.60869	−0.375050	−0.187525	+	0.982260i	$0.560047\pi$
$152$	4.41196	0.357858
$153$	−12.9444	−1.04649
$154$	9.57857	0.771863
$155$	−1.24575	−0.100061
$156$	−16.8286	−1.34737
$157$	−19.9945	−1.59573	−0.797867	−	0.602833i	$-0.794038\pi$
$157$	−19.9945	−1.59573	−0.797867	+	0.602833i	$0.794038\pi$
$158$	−6.45762	−0.513740
$159$	8.99168	0.713087
$160$	−0.133179	−0.0105288
$161$	5.08312	0.400606
$162$	−5.40936	−0.424999
$163$	8.64548	0.677166	0.338583	−	0.940936i	$-0.390052\pi$
$163$	8.64548	0.677166	0.338583	+	0.940936i	$0.390052\pi$
$164$	−9.01963	−0.704315
$165$	−1.12473	−0.0875605
$166$	−2.50616	−0.194516
$167$	−1.48837	−0.115173	−0.0575866	−	0.998341i	$-0.518341\pi$
$167$	−1.48837	−0.115173	−0.0575866	+	0.998341i	$0.518341\pi$
$168$	7.84493	0.605249
$169$	27.9444	2.14957
$170$	0.440142	0.0337574
$171$	17.2805	1.32147
$172$	7.51380	0.572922
$173$	−22.2013	−1.68793	−0.843965	−	0.536398i	$-0.819785\pi$
$173$	−22.2013	−1.68793	−0.843965	+	0.536398i	$0.819785\pi$
$174$	10.6701	0.808900
$175$	14.8616	1.12343
$176$	−3.21116	−0.242051
$177$	−18.0736	−1.35849
$178$	6.26390	0.469499
$179$	15.6212	1.16758	0.583792	−	0.811903i	$-0.301568\pi$
$179$	15.6212	1.16758	0.583792	+	0.811903i	$0.301568\pi$
$180$	−0.521629	−0.0388799
$181$	20.7058	1.53905	0.769526	−	0.638615i	$-0.220492\pi$
$181$	20.7058	1.53905	0.769526	+	0.638615i	$0.220492\pi$
$182$	−19.0869	−1.41481
$183$	−33.8845	−2.50482
$184$	−1.70409	−0.125627
$185$	1.48777	0.109383
$186$	−24.6006	−1.80380
$187$	10.6125	0.776065
$188$	7.66372	0.558934
$189$	7.19177	0.523125
$190$	−0.587583	−0.0426277
$191$	−10.5596	−0.764067	−0.382034	−	0.924148i	$-0.624776\pi$
$191$	−10.5596	−0.764067	−0.382034	+	0.924148i	$0.624776\pi$
$192$	−2.62997	−0.189802
$193$	−24.0218	−1.72912	−0.864562	−	0.502526i	$-0.832404\pi$
$193$	−24.0218	−1.72912	−0.864562	+	0.502526i	$0.832404\pi$
$194$	4.35742	0.312845
$195$	2.24122	0.160497
$196$	1.89767	0.135548
$197$	−9.67603	−0.689388	−0.344694	−	0.938715i	$-0.612017\pi$
$197$	−9.67603	−0.689388	−0.344694	+	0.938715i	$0.612017\pi$
$198$	−12.5773	−0.893830
$199$	−19.0908	−1.35331	−0.676656	−	0.736300i	$-0.736571\pi$
$199$	−19.0908	−1.35331	−0.676656	+	0.736300i	$0.736571\pi$
$200$	−4.98226	−0.352299
$201$	−18.3192	−1.29214
$202$	−5.87049	−0.413046
$203$	12.1020	0.849393
$204$	8.69175	0.608544
$205$	1.20123	0.0838974
$206$	2.77219	0.193147
$207$	−6.67447	−0.463908
$208$	6.39878	0.443675
$209$	−14.1675	−0.979989
$210$	−1.04478	−0.0720968
$211$	−6.74610	−0.464421	−0.232210	−	0.972666i	$-0.574596\pi$
$211$	−6.74610	−0.464421	−0.232210	+	0.972666i	$0.574596\pi$
$212$	−3.41893	−0.234813
$213$	8.84788	0.606247
$214$	4.07490	0.278555
$215$	−1.00068	−0.0682460
$216$	−2.41100	−0.164048
$217$	−27.9018	−1.89410
$218$	12.3408	0.835821
$219$	−23.6159	−1.59582
$220$	0.427661	0.0288329
$221$	−21.1472	−1.42252
$222$	29.3799	1.97185
$223$	−9.99394	−0.669244	−0.334622	−	0.942352i	$-0.608609\pi$
$223$	−9.99394	−0.669244	−0.334622	+	0.942352i	$0.608609\pi$
$224$	−2.98290	−0.199303
$225$	−19.5142	−1.30095
$226$	−10.3693	−0.689753
$227$	−26.2775	−1.74410	−0.872051	−	0.489416i	$-0.837210\pi$
$227$	−26.2775	−1.74410	−0.872051	+	0.489416i	$0.837210\pi$
$228$	−11.6033	−0.768450
$229$	−8.88024	−0.586823	−0.293412	−	0.955986i	$-0.594791\pi$
$229$	−8.88024	−0.586823	−0.293412	+	0.955986i	$0.594791\pi$
$230$	0.226949	0.0149646
$231$	−25.1913	−1.65747
$232$	−4.05713	−0.266363
$233$	−14.2979	−0.936687	−0.468343	−	0.883547i	$-0.655149\pi$
$233$	−14.2979	−0.936687	−0.468343	+	0.883547i	$0.655149\pi$
$234$	25.0624	1.63838
$235$	−1.02065	−0.0665798
$236$	6.87216	0.447339
$237$	16.9833	1.10319
$238$	9.85813	0.639008
$239$	0.234563	0.0151726	0.00758631	−	0.999971i	$-0.497585\pi$
$239$	0.234563	0.0151726	0.00758631	+	0.999971i	$0.497585\pi$
$240$	0.350258	0.0226090
$241$	−18.6942	−1.20420	−0.602101	−	0.798420i	$-0.705670\pi$
$241$	−18.6942	−1.20420	−0.602101	+	0.798420i	$0.705670\pi$
$242$	−0.688423	−0.0442535
$243$	21.4595	1.37663
$244$	12.8840	0.824813
$245$	−0.252730	−0.0161463
$246$	23.7214	1.51242
$247$	28.2312	1.79631
$248$	9.35394	0.593976
$249$	6.59112	0.417695
$250$	1.32943	0.0840806
$251$	−16.5827	−1.04669	−0.523344	−	0.852122i	$-0.675316\pi$
$251$	−16.5827	−1.04669	−0.523344	+	0.852122i	$0.675316\pi$
$252$	−11.6832	−0.735974
$253$	5.47211	0.344028
$254$	4.89366	0.307056
$255$	−1.15756	−0.0724893
$256$	1.00000	0.0625000
$257$	−0.577994	−0.0360543	−0.0180271	−	0.999837i	$-0.505739\pi$
$257$	−0.577994	−0.0360543	−0.0180271	+	0.999837i	$0.505739\pi$
$258$	−19.7611	−1.23027
$259$	33.3225	2.07056
$260$	−0.852185	−0.0528503
$261$	−15.8907	−0.983611
$262$	−15.7955	−0.975849
$263$	−9.09976	−0.561115	−0.280558	−	0.959837i	$-0.590519\pi$
$263$	−9.09976	−0.561115	−0.280558	+	0.959837i	$0.590519\pi$
$264$	8.44527	0.519770
$265$	0.455331	0.0279707
$266$	−13.1604	−0.806918
$267$	−16.4739	−1.00818
$268$	6.96556	0.425489
$269$	6.92422	0.422177	0.211089	−	0.977467i	$-0.432299\pi$
$269$	6.92422	0.422177	0.211089	+	0.977467i	$0.432299\pi$
$270$	0.321096	0.0195413
$271$	3.32939	0.202246	0.101123	−	0.994874i	$-0.467756\pi$
$271$	3.32939	0.202246	0.101123	+	0.994874i	$0.467756\pi$
$272$	−3.30489	−0.200388
$273$	50.1979	3.03812
$274$	−10.6305	−0.642212
$275$	15.9989	0.964768
$276$	4.48170	0.269767
$277$	0.190648	0.0114549	0.00572747	−	0.999984i	$-0.498177\pi$
$277$	0.190648	0.0114549	0.00572747	+	0.999984i	$0.498177\pi$
$278$	7.14185	0.428340
$279$	36.6370	2.19340
$280$	0.397260	0.0237408
$281$	−8.34937	−0.498081	−0.249041	−	0.968493i	$-0.580115\pi$
$281$	−8.34937	−0.498081	−0.249041	+	0.968493i	$0.580115\pi$
$282$	−20.1553	−1.20023
$283$	−12.0914	−0.718758	−0.359379	−	0.933192i	$-0.617011\pi$
$283$	−12.0914	−0.718758	−0.359379	+	0.933192i	$0.617011\pi$
$284$	−3.36425	−0.199632
$285$	1.54532	0.0915371
$286$	−20.5475	−1.21500
$287$	26.9046	1.58813
$288$	3.91674	0.230796
$289$	−6.07773	−0.357514
$290$	0.540326	0.0317290
$291$	−11.4599	−0.671791
$292$	8.97955	0.525488
$293$	22.0633	1.28895	0.644477	−	0.764624i	$-0.277075\pi$
$293$	22.0633	1.28895	0.644477	+	0.764624i	$0.277075\pi$
$294$	−4.99080	−0.291070
$295$	−0.915230	−0.0532867
$296$	−11.1712	−0.649313
$297$	7.74213	0.449244
$298$	3.29780	0.191036
$299$	−10.9041	−0.630599
$300$	13.1032	0.756514
$301$	−22.4129	−1.29186
$302$	−4.60869	−0.265200
$303$	15.4392	0.886959
$304$	4.41196	0.253044
$305$	−1.71588	−0.0982511
$306$	−12.9444	−0.739981
$307$	5.65068	0.322501	0.161251	−	0.986913i	$-0.448447\pi$
$307$	5.65068	0.322501	0.161251	+	0.986913i	$0.448447\pi$
$308$	9.57857	0.545790
$309$	−7.29077	−0.414757
$310$	−1.24575	−0.0707539
$311$	−17.5752	−0.996597	−0.498299	−	0.867006i	$-0.666042\pi$
$311$	−17.5752	−0.996597	−0.498299	+	0.867006i	$0.666042\pi$
$312$	−16.8286	−0.952732
$313$	−10.6930	−0.604403	−0.302201	−	0.953244i	$-0.597722\pi$
$313$	−10.6930	−0.604403	−0.302201	+	0.953244i	$0.597722\pi$
$314$	−19.9945	−1.12835
$315$	1.55597	0.0876687
$316$	−6.45762	−0.363269
$317$	7.51322	0.421985	0.210992	−	0.977488i	$-0.432331\pi$
$317$	7.51322	0.421985	0.210992	+	0.977488i	$0.432331\pi$
$318$	8.99168	0.504228
$319$	13.0281	0.729434
$320$	−0.133179	−0.00744495
$321$	−10.7169	−0.598158
$322$	5.08312	0.283271
$323$	−14.5810	−0.811311
$324$	−5.40936	−0.300520
$325$	−31.8804	−1.76841
$326$	8.64548	0.478829
$327$	−32.4558	−1.79481
$328$	−9.01963	−0.498026
$329$	−22.8601	−1.26032
$330$	−1.12473	−0.0619146
$331$	−1.19105	−0.0654659	−0.0327330	−	0.999464i	$-0.510421\pi$
$331$	−1.19105	−0.0654659	−0.0327330	+	0.999464i	$0.510421\pi$
$332$	−2.50616	−0.137543
$333$	−43.7547	−2.39774
$334$	−1.48837	−0.0814398
$335$	−0.927668	−0.0506839
$336$	7.84493	0.427976
$337$	−15.7956	−0.860441	−0.430220	−	0.902724i	$-0.641564\pi$
$337$	−15.7956	−0.860441	−0.430220	+	0.902724i	$0.641564\pi$
$338$	27.9444	1.51997
$339$	27.2709	1.48115
$340$	0.440142	0.0238701
$341$	−30.0370	−1.62660
$342$	17.2805	0.934424
$343$	15.2197	0.821788
$344$	7.51380	0.405117
$345$	−0.596870	−0.0321344
$346$	−22.2013	−1.19355
$347$	−29.3425	−1.57519	−0.787593	−	0.616196i	$-0.788673\pi$
$347$	−29.3425	−1.57519	−0.787593	+	0.616196i	$0.788673\pi$
$348$	10.6701	0.571979
$349$	−11.1330	−0.595936	−0.297968	−	0.954576i	$-0.596309\pi$
$349$	−11.1330	−0.595936	−0.297968	+	0.954576i	$0.596309\pi$
$350$	14.8616	0.794384
$351$	−15.4275	−0.823458
$352$	−3.21116	−0.171156
$353$	−11.1144	−0.591559	−0.295780	−	0.955256i	$-0.595579\pi$
$353$	−11.1144	−0.591559	−0.295780	+	0.955256i	$0.595579\pi$
$354$	−18.0736	−0.960599
$355$	0.448049	0.0237800
$356$	6.26390	0.331986
$357$	−25.9266	−1.37218
$358$	15.6212	0.825606
$359$	26.5411	1.40078	0.700392	−	0.713758i	$-0.253009\pi$
$359$	26.5411	1.40078	0.700392	+	0.713758i	$0.253009\pi$
$360$	−0.521629	−0.0274923
$361$	0.465431	0.0244964
$362$	20.7058	1.08827
$363$	1.81053	0.0950283
$364$	−19.0869	−1.00042
$365$	−1.19589	−0.0625958
$366$	−33.8845	−1.77117
$367$	16.7371	0.873668	0.436834	−	0.899542i	$-0.356100\pi$
$367$	16.7371	0.873668	0.436834	+	0.899542i	$0.356100\pi$
$368$	−1.70409	−0.0888317
$369$	−35.3276	−1.83908
$370$	1.48777	0.0773457
$371$	10.1983	0.529470
$372$	−24.6006	−1.27548
$373$	−20.5677	−1.06496	−0.532478	−	0.846444i	$-0.678739\pi$
$373$	−20.5677	−1.06496	−0.532478	+	0.846444i	$0.678739\pi$
$374$	10.6125	0.548761
$375$	−3.49636	−0.180551
$376$	7.66372	0.395226
$377$	−25.9607	−1.33704
$378$	7.19177	0.369905
$379$	6.19803	0.318371	0.159186	−	0.987249i	$-0.449113\pi$
$379$	6.19803	0.318371	0.159186	+	0.987249i	$0.449113\pi$
$380$	−0.587583	−0.0301423
$381$	−12.8702	−0.659359
$382$	−10.5596	−0.540277
$383$	−0.542879	−0.0277398	−0.0138699	−	0.999904i	$-0.504415\pi$
$383$	−0.542879	−0.0277398	−0.0138699	+	0.999904i	$0.504415\pi$
$384$	−2.62997	−0.134210
$385$	−1.27567	−0.0650140
$386$	−24.0218	−1.22268
$387$	29.4296	1.49599
$388$	4.35742	0.221215
$389$	−21.7112	−1.10080	−0.550401	−	0.834901i	$-0.685525\pi$
$389$	−21.7112	−1.10080	−0.550401	+	0.834901i	$0.685525\pi$
$390$	2.24122	0.113489
$391$	5.63182	0.284813
$392$	1.89767	0.0958466
$393$	41.5417	2.09550
$394$	−9.67603	−0.487471
$395$	0.860021	0.0432724
$396$	−12.5773	−0.632033
$397$	32.2232	1.61724	0.808618	−	0.588334i	$-0.200216\pi$
$397$	32.2232	1.61724	0.808618	+	0.588334i	$0.200216\pi$
$398$	−19.0908	−0.956935
$399$	34.6115	1.73274
$400$	−4.98226	−0.249113
$401$	34.0790	1.70182	0.850912	−	0.525308i	$-0.176050\pi$
$401$	34.0790	1.70182	0.850912	+	0.525308i	$0.176050\pi$
$402$	−18.3192	−0.913679
$403$	59.8538	2.98153
$404$	−5.87049	−0.292068
$405$	0.720415	0.0357977
$406$	12.1020	0.600612
$407$	35.8726	1.77814
$408$	8.69175	0.430306
$409$	−4.18983	−0.207174	−0.103587	−	0.994620i	$-0.533032\pi$
$409$	−4.18983	−0.207174	−0.103587	+	0.994620i	$0.533032\pi$
$410$	1.20123	0.0593244
$411$	27.9579	1.37906
$412$	2.77219	0.136576
$413$	−20.4989	−1.00869
$414$	−6.67447	−0.328032
$415$	0.333768	0.0163841
$416$	6.39878	0.313726
$417$	−18.7828	−0.919800
$418$	−14.1675	−0.692957
$419$	17.5670	0.858203	0.429101	−	0.903256i	$-0.358830\pi$
$419$	17.5670	0.858203	0.429101	+	0.903256i	$0.358830\pi$
$420$	−1.04478	−0.0509801
$421$	−25.4373	−1.23974	−0.619869	−	0.784705i	$-0.712814\pi$
$421$	−25.4373	−1.23974	−0.619869	+	0.784705i	$0.712814\pi$
$422$	−6.74610	−0.328395
$423$	30.0168	1.45947
$424$	−3.41893	−0.166038
$425$	16.4658	0.798709
$426$	8.84788	0.428681
$427$	−38.4316	−1.85984
$428$	4.07490	0.196968
$429$	54.0394	2.60905
$430$	−1.00068	−0.0482572
$431$	−7.70426	−0.371101	−0.185551	−	0.982635i	$-0.559407\pi$
$431$	−7.70426	−0.371101	−0.185551	+	0.982635i	$0.559407\pi$
$432$	−2.41100	−0.115999
$433$	33.2194	1.59642	0.798211	−	0.602378i	$-0.205780\pi$
$433$	33.2194	1.59642	0.798211	+	0.602378i	$0.205780\pi$
$434$	−27.9018	−1.33933
$435$	−1.42104	−0.0681337
$436$	12.3408	0.591015
$437$	−7.51838	−0.359653
$438$	−23.6159	−1.12841
$439$	−40.5548	−1.93558	−0.967788	−	0.251767i	$-0.918988\pi$
$439$	−40.5548	−1.93558	−0.967788	+	0.251767i	$0.918988\pi$
$440$	0.427661	0.0203879
$441$	7.43267	0.353937
$442$	−21.1472	−1.00587
$443$	−20.2385	−0.961558	−0.480779	−	0.876842i	$-0.659646\pi$
$443$	−20.2385	−0.961558	−0.480779	+	0.876842i	$0.659646\pi$
$444$	29.3799	1.39431
$445$	−0.834221	−0.0395459
$446$	−9.99394	−0.473227
$447$	−8.67311	−0.410224
$448$	−2.98290	−0.140929
$449$	7.57351	0.357416	0.178708	−	0.983902i	$-0.442808\pi$
$449$	7.57351	0.357416	0.178708	+	0.983902i	$0.442808\pi$
$450$	−19.5142	−0.919910
$451$	28.9635	1.36384
$452$	−10.3693	−0.487729
$453$	12.1207	0.569481
$454$	−26.2775	−1.23327
$455$	2.54198	0.119170
$456$	−11.6033	−0.543376
$457$	−0.569802	−0.0266542	−0.0133271	−	0.999911i	$-0.504242\pi$
$457$	−0.569802	−0.0266542	−0.0133271	+	0.999911i	$0.504242\pi$
$458$	−8.88024	−0.414947
$459$	7.96809	0.371919
$460$	0.226949	0.0105816
$461$	3.89924	0.181606	0.0908028	−	0.995869i	$-0.471057\pi$
$461$	3.89924	0.181606	0.0908028	+	0.995869i	$0.471057\pi$
$462$	−25.1913	−1.17201
$463$	30.2977	1.40806	0.704028	−	0.710173i	$-0.251383\pi$
$463$	30.2977	1.40806	0.704028	+	0.710173i	$0.251383\pi$
$464$	−4.05713	−0.188347
$465$	3.27629	0.151934
$466$	−14.2979	−0.662338
$467$	−18.0191	−0.833825	−0.416913	−	0.908947i	$-0.636888\pi$
$467$	−18.0191	−0.833825	−0.416913	+	0.908947i	$0.636888\pi$
$468$	25.0624	1.15851
$469$	−20.7775	−0.959417
$470$	−1.02065	−0.0470790
$471$	52.5849	2.42299
$472$	6.87216	0.316317
$473$	−24.1280	−1.10941
$474$	16.9833	0.780071
$475$	−21.9816	−1.00858
$476$	9.85813	0.451847
$477$	−13.3911	−0.613135
$478$	0.234563	0.0107287
$479$	−7.99029	−0.365086	−0.182543	−	0.983198i	$-0.558433\pi$
$479$	−7.99029	−0.365086	−0.182543	+	0.983198i	$0.558433\pi$
$480$	0.350258	0.0159870
$481$	−71.4821	−3.25930
$482$	−18.6942	−0.851500
$483$	−13.3684	−0.608285
$484$	−0.688423	−0.0312920
$485$	−0.580319	−0.0263509
$486$	21.4595	0.973421
$487$	21.1513	0.958455	0.479228	−	0.877691i	$-0.340917\pi$
$487$	21.1513	0.958455	0.479228	+	0.877691i	$0.340917\pi$
$488$	12.8840	0.583231
$489$	−22.7374	−1.02822
$490$	−0.252730	−0.0114172
$491$	35.1515	1.58637	0.793183	−	0.608983i	$-0.208422\pi$
$491$	35.1515	1.58637	0.793183	+	0.608983i	$0.208422\pi$
$492$	23.7214	1.06944
$493$	13.4083	0.603881
$494$	28.2312	1.27018
$495$	1.67504	0.0752873
$496$	9.35394	0.420004
$497$	10.0352	0.450141
$498$	6.59112	0.295355
$499$	11.9801	0.536303	0.268152	−	0.963377i	$-0.413587\pi$
$499$	11.9801	0.536303	0.268152	+	0.963377i	$0.413587\pi$
$500$	1.32943	0.0594540
$501$	3.91436	0.174881
$502$	−16.5827	−0.740120
$503$	−34.3504	−1.53161	−0.765805	−	0.643073i	$-0.777659\pi$
$503$	−34.3504	−1.53161	−0.765805	+	0.643073i	$0.777659\pi$
$504$	−11.6832	−0.520413
$505$	0.781828	0.0347909
$506$	5.47211	0.243265
$507$	−73.4929	−3.26393
$508$	4.89366	0.217121
$509$	5.95234	0.263833	0.131916	−	0.991261i	$-0.457887\pi$
$509$	5.95234	0.263833	0.131916	+	0.991261i	$0.457887\pi$
$510$	−1.15756	−0.0512577
$511$	−26.7851	−1.18490
$512$	1.00000	0.0441942
$513$	−10.6373	−0.469647
$514$	−0.577994	−0.0254942
$515$	−0.369198	−0.0162688
$516$	−19.7611	−0.869933
$517$	−24.6095	−1.08232
$518$	33.3225	1.46411
$519$	58.3886	2.56298
$520$	−0.852185	−0.0373708
$521$	−4.23233	−0.185422	−0.0927109	−	0.995693i	$-0.529553\pi$
$521$	−4.23233	−0.185422	−0.0927109	+	0.995693i	$0.529553\pi$
$522$	−15.8907	−0.695518
$523$	25.0692	1.09620	0.548100	−	0.836413i	$-0.315351\pi$
$523$	25.0692	1.09620	0.548100	+	0.836413i	$0.315351\pi$
$524$	−15.7955	−0.690029
$525$	−39.0855	−1.70583
$526$	−9.09976	−0.396768
$527$	−30.9137	−1.34662
$528$	8.44527	0.367533
$529$	−20.0961	−0.873743
$530$	0.455331	0.0197783
$531$	26.9165	1.16808
$532$	−13.1604	−0.570577
$533$	−57.7146	−2.49990
$534$	−16.4739	−0.712894
$535$	−0.542693	−0.0234627
$536$	6.96556	0.300866
$537$	−41.0833	−1.77288
$538$	6.92422	0.298524
$539$	−6.09372	−0.262475
$540$	0.321096	0.0138178
$541$	44.6416	1.91929	0.959646	−	0.281212i	$-0.0907364\pi$
$541$	44.6416	1.91929	0.959646	+	0.281212i	$0.0907364\pi$
$542$	3.32939	0.143010
$543$	−54.4557	−2.33692
$544$	−3.30489	−0.141696
$545$	−1.64353	−0.0704012
$546$	50.1979	2.14827
$547$	−19.2707	−0.823957	−0.411978	−	0.911194i	$-0.635162\pi$
$547$	−19.2707	−0.823957	−0.411978	+	0.911194i	$0.635162\pi$
$548$	−10.6305	−0.454113
$549$	50.4633	2.15372
$550$	15.9989	0.682194
$551$	−17.8999	−0.762562
$552$	4.48170	0.190754
$553$	19.2624	0.819120
$554$	0.190648	0.00809986
$555$	−3.91280	−0.166089
$556$	7.14185	0.302882
$557$	−17.2624	−0.731430	−0.365715	−	0.930727i	$-0.619175\pi$
$557$	−17.2624	−0.731430	−0.365715	+	0.930727i	$0.619175\pi$
$558$	36.6370	1.55097
$559$	48.0791	2.03353
$560$	0.397260	0.0167873
$561$	−27.9106	−1.17839
$562$	−8.34937	−0.352197
$563$	−3.14065	−0.132363	−0.0661813	−	0.997808i	$-0.521082\pi$
$563$	−3.14065	−0.132363	−0.0661813	+	0.997808i	$0.521082\pi$
$564$	−20.1553	−0.848693
$565$	1.38097	0.0580979
$566$	−12.0914	−0.508238
$567$	16.1355	0.677629
$568$	−3.36425	−0.141161
$569$	10.3020	0.431881	0.215941	−	0.976406i	$-0.430718\pi$
$569$	10.3020	0.431881	0.215941	+	0.976406i	$0.430718\pi$
$570$	1.54532	0.0647265
$571$	−35.0932	−1.46860	−0.734301	−	0.678824i	$-0.762490\pi$
$571$	−35.0932	−1.46860	−0.734301	+	0.678824i	$0.762490\pi$
$572$	−20.5475	−0.859135
$573$	27.7715	1.16017
$574$	26.9046	1.12298
$575$	8.49022	0.354066
$576$	3.91674	0.163198
$577$	15.4046	0.641303	0.320651	−	0.947197i	$-0.396098\pi$
$577$	15.4046	0.641303	0.320651	+	0.947197i	$0.396098\pi$
$578$	−6.07773	−0.252800
$579$	63.1765	2.62553
$580$	0.540326	0.0224358
$581$	7.47561	0.310140
$582$	−11.4599	−0.475028
$583$	10.9787	0.454693
$584$	8.97955	0.371576
$585$	−3.33779	−0.138001
$586$	22.0633	0.911428
$587$	−45.9000	−1.89450	−0.947248	−	0.320502i	$-0.896148\pi$
$587$	−45.9000	−1.89450	−0.947248	+	0.320502i	$0.896148\pi$
$588$	−4.99080	−0.205817
$589$	41.2693	1.70047
$590$	−0.915230	−0.0376794
$591$	25.4477	1.04678
$592$	−11.1712	−0.459134
$593$	3.55411	0.145950	0.0729749	−	0.997334i	$-0.476751\pi$
$593$	3.55411	0.145950	0.0729749	+	0.997334i	$0.476751\pi$
$594$	7.74213	0.317663
$595$	−1.31290	−0.0538236
$596$	3.29780	0.135083
$597$	50.2082	2.05489
$598$	−10.9041	−0.445901
$599$	−47.5119	−1.94128	−0.970642	−	0.240529i	$-0.922679\pi$
$599$	−47.5119	−1.94128	−0.970642	+	0.240529i	$0.922679\pi$
$600$	13.1032	0.534936
$601$	−42.2330	−1.72272	−0.861360	−	0.507995i	$-0.830387\pi$
$601$	−42.2330	−1.72272	−0.861360	+	0.507995i	$0.830387\pi$
$602$	−22.4129	−0.913481
$603$	27.2823	1.11102
$604$	−4.60869	−0.187525
$605$	0.0916837	0.00372747
$606$	15.4392	0.627175
$607$	−43.1107	−1.74981	−0.874904	−	0.484296i	$-0.839076\pi$
$607$	−43.1107	−1.74981	−0.874904	+	0.484296i	$0.839076\pi$
$608$	4.41196	0.178929
$609$	−31.8279	−1.28973
$610$	−1.71588	−0.0694740
$611$	49.0384	1.98388
$612$	−12.9444	−0.523246
$613$	−20.2861	−0.819348	−0.409674	−	0.912232i	$-0.634358\pi$
$613$	−20.2861	−0.819348	−0.409674	+	0.912232i	$0.634358\pi$
$614$	5.65068	0.228043
$615$	−3.15919	−0.127391
$616$	9.57857	0.385932
$617$	22.0173	0.886384	0.443192	−	0.896427i	$-0.353846\pi$
$617$	22.0173	0.886384	0.443192	+	0.896427i	$0.353846\pi$
$618$	−7.29077	−0.293278
$619$	26.6371	1.07063	0.535317	−	0.844651i	$-0.320192\pi$
$619$	26.6371	1.07063	0.535317	+	0.844651i	$0.320192\pi$
$620$	−1.24575	−0.0500306
$621$	4.10856	0.164871
$622$	−17.5752	−0.704701
$623$	−18.6845	−0.748581
$624$	−16.8286	−0.673683
$625$	24.7343	0.989371
$626$	−10.6930	−0.427377
$627$	37.2602	1.48803
$628$	−19.9945	−0.797867
$629$	36.9196	1.47208
$630$	1.55597	0.0619911
$631$	−23.0720	−0.918482	−0.459241	−	0.888312i	$-0.651879\pi$
$631$	−23.0720	−0.918482	−0.459241	+	0.888312i	$0.651879\pi$
$632$	−6.45762	−0.256870
$633$	17.7421	0.705183
$634$	7.51322	0.298388
$635$	−0.651734	−0.0258633
$636$	8.99168	0.356543
$637$	12.1427	0.481113
$638$	13.0281	0.515788
$639$	−13.1769	−0.521270
$640$	−0.133179	−0.00526438
$641$	−4.86034	−0.191972	−0.0959860	−	0.995383i	$-0.530600\pi$
$641$	−4.86034	−0.191972	−0.0959860	+	0.995383i	$0.530600\pi$
$642$	−10.7169	−0.422961
$643$	−29.0047	−1.14384	−0.571918	−	0.820311i	$-0.693800\pi$
$643$	−29.0047	−1.14384	−0.571918	+	0.820311i	$0.693800\pi$
$644$	5.08312	0.200303
$645$	2.63177	0.103626
$646$	−14.5810	−0.573683
$647$	−43.7061	−1.71826	−0.859132	−	0.511754i	$-0.828996\pi$
$647$	−43.7061	−1.71826	−0.859132	+	0.511754i	$0.828996\pi$
$648$	−5.40936	−0.212500
$649$	−22.0676	−0.866230
$650$	−31.8804	−1.25045
$651$	73.3810	2.87603
$652$	8.64548	0.338583
$653$	−23.7803	−0.930593	−0.465297	−	0.885155i	$-0.654052\pi$
$653$	−23.7803	−0.930593	−0.465297	+	0.885155i	$0.654052\pi$
$654$	−32.4558	−1.26912
$655$	2.10363	0.0821958
$656$	−9.01963	−0.352157
$657$	35.1706	1.37213
$658$	−22.8601	−0.891178
$659$	29.1985	1.13741	0.568705	−	0.822541i	$-0.307444\pi$
$659$	29.1985	1.13741	0.568705	+	0.822541i	$0.307444\pi$
$660$	−1.12473	−0.0437802
$661$	−32.6219	−1.26884	−0.634422	−	0.772987i	$-0.718762\pi$
$661$	−32.6219	−1.26884	−0.634422	+	0.772987i	$0.718762\pi$
$662$	−1.19105	−0.0462914
$663$	55.6166	2.15997
$664$	−2.50616	−0.0972578
$665$	1.75270	0.0679667
$666$	−43.7547	−1.69546
$667$	6.91370	0.267700
$668$	−1.48837	−0.0575866
$669$	26.2838	1.01619
$670$	−0.927668	−0.0358389
$671$	−41.3726	−1.59717
$672$	7.84493	0.302625
$673$	−37.4977	−1.44543	−0.722715	−	0.691146i	$-0.757106\pi$
$673$	−37.4977	−1.44543	−0.722715	+	0.691146i	$0.757106\pi$
$674$	−15.7956	−0.608423
$675$	12.0123	0.462352
$676$	27.9444	1.07478
$677$	−22.8796	−0.879333	−0.439667	−	0.898161i	$-0.644903\pi$
$677$	−22.8796	−0.879333	−0.439667	+	0.898161i	$0.644903\pi$
$678$	27.2709	1.04733
$679$	−12.9977	−0.498807
$680$	0.440142	0.0168787
$681$	69.1091	2.64827
$682$	−30.0370	−1.15018
$683$	2.77870	0.106324	0.0531621	−	0.998586i	$-0.483070\pi$
$683$	2.77870	0.106324	0.0531621	+	0.998586i	$0.483070\pi$
$684$	17.2805	0.660737
$685$	1.41576	0.0540936
$686$	15.2197	0.581092
$687$	23.3548	0.891040
$688$	7.51380	0.286461
$689$	−21.8770	−0.833446
$690$	−0.596870	−0.0227224
$691$	−7.68775	−0.292456	−0.146228	−	0.989251i	$-0.546713\pi$
$691$	−7.68775	−0.292456	−0.146228	+	0.989251i	$0.546713\pi$
$692$	−22.2013	−0.843965
$693$	37.5168	1.42514
$694$	−29.3425	−1.11382
$695$	−0.951147	−0.0360791
$696$	10.6701	0.404450
$697$	29.8088	1.12909
$698$	−11.1330	−0.421391
$699$	37.6031	1.42228
$700$	14.8616	0.561715
$701$	−29.7679	−1.12432	−0.562159	−	0.827029i	$-0.690029\pi$
$701$	−29.7679	−1.12432	−0.562159	+	0.827029i	$0.690029\pi$
$702$	−15.4275	−0.582273
$703$	−49.2870	−1.85889
$704$	−3.21116	−0.121025
$705$	2.68428	0.101096
$706$	−11.1144	−0.418296
$707$	17.5111	0.658571
$708$	−18.0736	−0.679246
$709$	3.74967	0.140822	0.0704109	−	0.997518i	$-0.477569\pi$
$709$	3.74967	0.140822	0.0704109	+	0.997518i	$0.477569\pi$
$710$	0.448049	0.0168150
$711$	−25.2928	−0.948555
$712$	6.26390	0.234749
$713$	−15.9399	−0.596955
$714$	−25.9266	−0.970278
$715$	2.73651	0.102340
$716$	15.6212	0.583792
$717$	−0.616894	−0.0230383
$718$	26.5411	0.990504
$719$	38.2410	1.42615	0.713074	−	0.701089i	$-0.247302\pi$
$719$	38.2410	1.42615	0.713074	+	0.701089i	$0.247302\pi$
$720$	−0.521629	−0.0194400
$721$	−8.26915	−0.307959
$722$	0.465431	0.0173216
$723$	49.1653	1.82848
$724$	20.7058	0.769526
$725$	20.2137	0.750717
$726$	1.81053	0.0671951
$727$	21.0432	0.780450	0.390225	−	0.920720i	$-0.372397\pi$
$727$	21.0432	0.780450	0.390225	+	0.920720i	$0.372397\pi$
$728$	−19.0869	−0.707407
$729$	−40.2097	−1.48925
$730$	−1.19589	−0.0442619
$731$	−24.8323	−0.918454
$732$	−33.8845	−1.25241
$733$	51.3367	1.89616	0.948082	−	0.318025i	$-0.103020\pi$
$733$	51.3367	1.89616	0.948082	+	0.318025i	$0.103020\pi$
$734$	16.7371	0.617777
$735$	0.664672	0.0245168
$736$	−1.70409	−0.0628135
$737$	−22.3675	−0.823919
$738$	−35.3276	−1.30043
$739$	−41.4344	−1.52419	−0.762094	−	0.647467i	$-0.775829\pi$
$739$	−41.4344	−1.52419	−0.762094	+	0.647467i	$0.775829\pi$
$740$	1.48777	0.0546916
$741$	−74.2472	−2.72754
$742$	10.1983	0.374392
$743$	−4.65789	−0.170881	−0.0854406	−	0.996343i	$-0.527230\pi$
$743$	−4.65789	−0.170881	−0.0854406	+	0.996343i	$0.527230\pi$
$744$	−24.6006	−0.901901
$745$	−0.439198	−0.0160910
$746$	−20.5677	−0.753038
$747$	−9.81598	−0.359148
$748$	10.6125	0.388033
$749$	−12.1550	−0.444134
$750$	−3.49636	−0.127669
$751$	44.7500	1.63295	0.816476	−	0.577380i	$-0.195925\pi$
$751$	44.7500	1.63295	0.816476	+	0.577380i	$0.195925\pi$
$752$	7.66372	0.279467
$753$	43.6119	1.58931
$754$	−25.9607	−0.945432
$755$	0.613783	0.0223378
$756$	7.19177	0.261562
$757$	−18.1450	−0.659490	−0.329745	−	0.944070i	$-0.606963\pi$
$757$	−18.1450	−0.659490	−0.329745	+	0.944070i	$0.606963\pi$
$758$	6.19803	0.225122
$759$	−14.3915	−0.522378
$760$	−0.587583	−0.0213139
$761$	3.29321	0.119379	0.0596894	−	0.998217i	$-0.480989\pi$
$761$	3.29321	0.119379	0.0596894	+	0.998217i	$0.480989\pi$
$762$	−12.8702	−0.466237
$763$	−36.8112	−1.33265
$764$	−10.5596	−0.382034
$765$	1.72392	0.0623286
$766$	−0.542879	−0.0196150
$767$	43.9734	1.58779
$768$	−2.62997	−0.0949009
$769$	31.6709	1.14208	0.571042	−	0.820921i	$-0.306540\pi$
$769$	31.6709	1.14208	0.571042	+	0.820921i	$0.306540\pi$
$770$	−1.27567	−0.0459719
$771$	1.52011	0.0547453
$772$	−24.0218	−0.864562
$773$	25.2491	0.908147	0.454074	−	0.890964i	$-0.349970\pi$
$773$	25.2491	0.908147	0.454074	+	0.890964i	$0.349970\pi$
$774$	29.4296	1.05783
$775$	−46.6038	−1.67406
$776$	4.35742	0.156422
$777$	−87.6373	−3.14397
$778$	−21.7112	−0.778384
$779$	−39.7943	−1.42578
$780$	2.24122	0.0802486
$781$	10.8032	0.386567
$782$	5.63182	0.201393
$783$	9.78175	0.349571
$784$	1.89767	0.0677738
$785$	2.66285	0.0950413
$786$	41.5417	1.48174
$787$	19.7923	0.705521	0.352760	−	0.935714i	$-0.385243\pi$
$787$	19.7923	0.705521	0.352760	+	0.935714i	$0.385243\pi$
$788$	−9.67603	−0.344694
$789$	23.9321	0.852005
$790$	0.860021	0.0305982
$791$	30.9304	1.09976
$792$	−12.5773	−0.446915
$793$	82.4418	2.92760
$794$	32.2232	1.14356
$795$	−1.19751	−0.0424712
$796$	−19.0908	−0.676656
$797$	15.6704	0.555075	0.277538	−	0.960715i	$-0.410482\pi$
$797$	15.6704	0.555075	0.277538	+	0.960715i	$0.410482\pi$
$798$	34.6115	1.22524
$799$	−25.3277	−0.896030
$800$	−4.98226	−0.176150
$801$	24.5341	0.866869
$802$	34.0790	1.20337
$803$	−28.8348	−1.01756
$804$	−18.3192	−0.646069
$805$	−0.676966	−0.0238599
$806$	59.8538	2.10826
$807$	−18.2105	−0.641039
$808$	−5.87049	−0.206523
$809$	33.4793	1.17707	0.588534	−	0.808472i	$-0.299705\pi$
$809$	33.4793	1.17707	0.588534	+	0.808472i	$0.299705\pi$
$810$	0.720415	0.0253128
$811$	30.9366	1.08633	0.543166	−	0.839626i	$-0.317226\pi$
$811$	30.9366	1.08633	0.543166	+	0.839626i	$0.317226\pi$
$812$	12.1020	0.424697
$813$	−8.75620	−0.307093
$814$	35.8726	1.25733
$815$	−1.15140	−0.0403318
$816$	8.69175	0.304272
$817$	33.1506	1.15979
$818$	−4.18983	−0.146494
$819$	−74.7584	−2.61227
$820$	1.20123	0.0419487
$821$	−6.18660	−0.215914	−0.107957	−	0.994156i	$-0.534431\pi$
$821$	−6.18660	−0.215914	−0.107957	+	0.994156i	$0.534431\pi$
$822$	27.9579	0.975144
$823$	−12.3996	−0.432224	−0.216112	−	0.976369i	$-0.569338\pi$
$823$	−12.3996	−0.432224	−0.216112	+	0.976369i	$0.569338\pi$
$824$	2.77219	0.0965737
$825$	−42.0765	−1.46492
$826$	−20.4989	−0.713249
$827$	34.1205	1.18649	0.593243	−	0.805023i	$-0.297847\pi$
$827$	34.1205	1.18649	0.593243	+	0.805023i	$0.297847\pi$
$828$	−6.67447	−0.231954
$829$	−11.9221	−0.414070	−0.207035	−	0.978334i	$-0.566381\pi$
$829$	−11.9221	−0.414070	−0.207035	+	0.978334i	$0.566381\pi$
$830$	0.333768	0.0115853
$831$	−0.501399	−0.0173933
$832$	6.39878	0.221838
$833$	−6.27157	−0.217297
$834$	−18.7828	−0.650397
$835$	0.198220	0.00685967
$836$	−14.1675	−0.489995
$837$	−22.5524	−0.779525
$838$	17.5670	0.606841
$839$	31.2257	1.07803	0.539016	−	0.842296i	$-0.318796\pi$
$839$	31.2257	1.07803	0.539016	+	0.842296i	$0.318796\pi$
$840$	−1.04478	−0.0360484
$841$	−12.5397	−0.432404
$842$	−25.4373	−0.876628
$843$	21.9586	0.756294
$844$	−6.74610	−0.232210
$845$	−3.72161	−0.128027
$846$	30.0168	1.03200
$847$	2.05349	0.0705589
$848$	−3.41893	−0.117407
$849$	31.7999	1.09137
$850$	16.4658	0.564773
$851$	19.0367	0.652570
$852$	8.84788	0.303123
$853$	−31.4513	−1.07687	−0.538437	−	0.842666i	$-0.680985\pi$
$853$	−31.4513	−1.07687	−0.538437	+	0.842666i	$0.680985\pi$
$854$	−38.4316	−1.31510
$855$	−2.30141	−0.0787065
$856$	4.07490	0.139277
$857$	42.9199	1.46612	0.733059	−	0.680165i	$-0.238092\pi$
$857$	42.9199	1.46612	0.733059	+	0.680165i	$0.238092\pi$
$858$	54.0394	1.84487
$859$	−35.7822	−1.22087	−0.610437	−	0.792065i	$-0.709006\pi$
$859$	−35.7822	−1.22087	−0.610437	+	0.792065i	$0.709006\pi$
$860$	−1.00068	−0.0341230
$861$	−70.7583	−2.41144
$862$	−7.70426	−0.262408
$863$	−10.4621	−0.356136	−0.178068	−	0.984018i	$-0.556985\pi$
$863$	−10.4621	−0.356136	−0.178068	+	0.984018i	$0.556985\pi$
$864$	−2.41100	−0.0820240
$865$	2.95675	0.100532
$866$	33.2194	1.12884
$867$	15.9842	0.542853
$868$	−27.9018	−0.947050
$869$	20.7365	0.703437
$870$	−1.42104	−0.0481778
$871$	44.5711	1.51023
$872$	12.3408	0.417911
$873$	17.0669	0.577627
$874$	−7.51838	−0.254313
$875$	−3.96555	−0.134060
$876$	−23.6159	−0.797909
$877$	−36.6331	−1.23701	−0.618506	−	0.785780i	$-0.712262\pi$
$877$	−36.6331	−1.23701	−0.618506	+	0.785780i	$0.712262\pi$
$878$	−40.5548	−1.36866
$879$	−58.0259	−1.95717
$880$	0.427661	0.0144164
$881$	−6.11592	−0.206050	−0.103025	−	0.994679i	$-0.532852\pi$
$881$	−6.11592	−0.206050	−0.103025	+	0.994679i	$0.532852\pi$
$882$	7.43267	0.250271
$883$	−12.4748	−0.419812	−0.209906	−	0.977722i	$-0.567316\pi$
$883$	−12.4748	−0.419812	−0.209906	+	0.977722i	$0.567316\pi$
$884$	−21.1472	−0.711258
$885$	2.40703	0.0809113
$886$	−20.2385	−0.679924
$887$	−2.96844	−0.0996705	−0.0498353	−	0.998757i	$-0.515870\pi$
$887$	−2.96844	−0.0996705	−0.0498353	+	0.998757i	$0.515870\pi$
$888$	29.3799	0.985926
$889$	−14.5973	−0.489577
$890$	−0.834221	−0.0279632
$891$	17.3703	0.581928
$892$	−9.99394	−0.334622
$893$	33.8120	1.13148
$894$	−8.67311	−0.290072
$895$	−2.08042	−0.0695408
$896$	−2.98290	−0.0996515
$897$	28.6774	0.957511
$898$	7.57351	0.252731
$899$	−37.9501	−1.26571
$900$	−19.5142	−0.650475
$901$	11.2992	0.376430
$902$	28.9635	0.964379
$903$	58.9452	1.96157
$904$	−10.3693	−0.344877
$905$	−2.75759	−0.0916654
$906$	12.1207	0.402684
$907$	−31.0898	−1.03232	−0.516160	−	0.856492i	$-0.672639\pi$
$907$	−31.0898	−1.03232	−0.516160	+	0.856492i	$0.672639\pi$
$908$	−26.2775	−0.872051
$909$	−22.9932	−0.762636
$910$	2.54198	0.0842658
$911$	40.8917	1.35480	0.677401	−	0.735614i	$-0.263106\pi$
$911$	40.8917	1.35480	0.677401	+	0.735614i	$0.263106\pi$
$912$	−11.6033	−0.384225
$913$	8.04769	0.266340
$914$	−0.569802	−0.0188474
$915$	4.51272	0.149186
$916$	−8.88024	−0.293412
$917$	47.1163	1.55592
$918$	7.96809	0.262986
$919$	−3.20026	−0.105567	−0.0527834	−	0.998606i	$-0.516809\pi$
$919$	−3.20026	−0.105567	−0.0527834	+	0.998606i	$0.516809\pi$
$920$	0.226949	0.00748230
$921$	−14.8611	−0.489690
$922$	3.89924	0.128415
$923$	−21.5271	−0.708573
$924$	−25.1913	−0.828734
$925$	55.6579	1.83002
$926$	30.2977	0.995646
$927$	10.8579	0.356622
$928$	−4.05713	−0.133182
$929$	34.4417	1.12999	0.564997	−	0.825093i	$-0.308877\pi$
$929$	34.4417	1.12999	0.564997	+	0.825093i	$0.308877\pi$
$930$	3.27629	0.107434
$931$	8.37244	0.274395
$932$	−14.2979	−0.468343
$933$	46.2222	1.51325
$934$	−18.0191	−0.589604
$935$	−1.41337	−0.0462221
$936$	25.0624	0.819189
$937$	2.20212	0.0719402	0.0359701	−	0.999353i	$-0.488548\pi$
$937$	2.20212	0.0719402	0.0359701	+	0.999353i	$0.488548\pi$
$938$	−20.7775	−0.678410
$939$	28.1222	0.917733
$940$	−1.02065	−0.0332899
$941$	−1.34998	−0.0440082	−0.0220041	−	0.999758i	$-0.507005\pi$
$941$	−1.34998	−0.0440082	−0.0220041	+	0.999758i	$0.507005\pi$
$942$	52.5849	1.71331
$943$	15.3702	0.500524
$944$	6.87216	0.223670
$945$	−0.957796	−0.0311571
$946$	−24.1280	−0.784471
$947$	5.09719	0.165637	0.0828183	−	0.996565i	$-0.473608\pi$
$947$	5.09719	0.165637	0.0828183	+	0.996565i	$0.473608\pi$
$948$	16.9833	0.551593
$949$	57.4581	1.86517
$950$	−21.9816	−0.713176
$951$	−19.7595	−0.640747
$952$	9.85813	0.319504
$953$	−36.0854	−1.16892	−0.584460	−	0.811422i	$-0.698694\pi$
$953$	−36.0854	−1.16892	−0.584460	+	0.811422i	$0.698694\pi$
$954$	−13.3911	−0.433552
$955$	1.40632	0.0455075
$956$	0.234563	0.00758631
$957$	−34.2635	−1.10758
$958$	−7.99029	−0.258155
$959$	31.7097	1.02396
$960$	0.350258	0.0113045
$961$	56.4962	1.82246
$962$	−71.4821	−2.30467
$963$	15.9603	0.514315
$964$	−18.6942	−0.602101
$965$	3.19920	0.102986
$966$	−13.3684	−0.430123
$967$	−35.7635	−1.15008	−0.575039	−	0.818126i	$-0.695013\pi$
$967$	−35.7635	−1.15008	−0.575039	+	0.818126i	$0.695013\pi$
$968$	−0.688423	−0.0221268
$969$	38.3477	1.23191
$970$	−0.580319	−0.0186329
$971$	26.4057	0.847400	0.423700	−	0.905802i	$-0.360731\pi$
$971$	26.4057	0.847400	0.423700	+	0.905802i	$0.360731\pi$
$972$	21.4595	0.688313
$973$	−21.3034	−0.682955
$974$	21.1513	0.677730
$975$	83.8445	2.68517
$976$	12.8840	0.412407
$977$	11.2346	0.359428	0.179714	−	0.983719i	$-0.442483\pi$
$977$	11.2346	0.359428	0.179714	+	0.983719i	$0.442483\pi$
$978$	−22.7374	−0.727061
$979$	−20.1144	−0.642859
$980$	−0.252730	−0.00807316
$981$	48.3355	1.54324
$982$	35.1515	1.12173
$983$	−40.6174	−1.29549	−0.647746	−	0.761856i	$-0.724288\pi$
$983$	−40.6174	−1.29549	−0.647746	+	0.761856i	$0.724288\pi$
$984$	23.7214	0.756209
$985$	1.28865	0.0410597
$986$	13.4083	0.427009
$987$	60.1213	1.91368
$988$	28.2312	0.898154
$989$	−12.8042	−0.407149
$990$	1.67504	0.0532362
$991$	−10.5683	−0.335712	−0.167856	−	0.985811i	$-0.553684\pi$
$991$	−10.5683	−0.335712	−0.167856	+	0.985811i	$0.553684\pi$
$992$	9.35394	0.296988
$993$	3.13242	0.0994044
$994$	10.0352	0.318297
$995$	2.54250	0.0806027
$996$	6.59112	0.208848
$997$	10.1443	0.321274	0.160637	−	0.987014i	$-0.448645\pi$
$997$	10.1443	0.321274	0.160637	+	0.987014i	$0.448645\pi$
$998$	11.9801	0.379224
$999$	26.9338	0.852148

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.12	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.12	✓	69	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.62997 q^{3} +1.00000 q^{4} -0.133179 q^{5} -2.62997 q^{6} -2.98290 q^{7} +1.00000 q^{8} +3.91674 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.62997 q^{3} +1.00000 q^{4} -0.133179 q^{5} -2.62997 q^{6} -2.98290 q^{7} +1.00000 q^{8} +3.91674 q^{9} -0.133179 q^{10} -3.21116 q^{11} -2.62997 q^{12} +6.39878 q^{13} -2.98290 q^{14} +0.350258 q^{15} +1.00000 q^{16} -3.30489 q^{17} +3.91674 q^{18} +4.41196 q^{19} -0.133179 q^{20} +7.84493 q^{21} -3.21116 q^{22} -1.70409 q^{23} -2.62997 q^{24} -4.98226 q^{25} +6.39878 q^{26} -2.41100 q^{27} -2.98290 q^{28} -4.05713 q^{29} +0.350258 q^{30} +9.35394 q^{31} +1.00000 q^{32} +8.44527 q^{33} -3.30489 q^{34} +0.397260 q^{35} +3.91674 q^{36} -11.1712 q^{37} +4.41196 q^{38} -16.8286 q^{39} -0.133179 q^{40} -9.01963 q^{41} +7.84493 q^{42} +7.51380 q^{43} -3.21116 q^{44} -0.521629 q^{45} -1.70409 q^{46} +7.66372 q^{47} -2.62997 q^{48} +1.89767 q^{49} -4.98226 q^{50} +8.69175 q^{51} +6.39878 q^{52} -3.41893 q^{53} -2.41100 q^{54} +0.427661 q^{55} -2.98290 q^{56} -11.6033 q^{57} -4.05713 q^{58} +6.87216 q^{59} +0.350258 q^{60} +12.8840 q^{61} +9.35394 q^{62} -11.6832 q^{63} +1.00000 q^{64} -0.852185 q^{65} +8.44527 q^{66} +6.96556 q^{67} -3.30489 q^{68} +4.48170 q^{69} +0.397260 q^{70} -3.36425 q^{71} +3.91674 q^{72} +8.97955 q^{73} -11.1712 q^{74} +13.1032 q^{75} +4.41196 q^{76} +9.57857 q^{77} -16.8286 q^{78} -6.45762 q^{79} -0.133179 q^{80} -5.40936 q^{81} -9.01963 q^{82} -2.50616 q^{83} +7.84493 q^{84} +0.440142 q^{85} +7.51380 q^{86} +10.6701 q^{87} -3.21116 q^{88} +6.26390 q^{89} -0.521629 q^{90} -19.0869 q^{91} -1.70409 q^{92} -24.6006 q^{93} +7.66372 q^{94} -0.587583 q^{95} -2.62997 q^{96} +4.35742 q^{97} +1.89767 q^{98} -12.5773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

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