LMFDB - Embedded newform 8002.2.a.d.1.47

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			1.47
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)

$f(q)$	$=$	$q+1.00000 q^{2} +0.839268 q^{3} +1.00000 q^{4} -4.35272 q^{5} +0.839268 q^{6} +1.00705 q^{7} +1.00000 q^{8} -2.29563 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.839268 q^{3} +1.00000 q^{4} -4.35272 q^{5} +0.839268 q^{6} +1.00705 q^{7} +1.00000 q^{8} -2.29563 q^{9} -4.35272 q^{10} -1.96804 q^{11} +0.839268 q^{12} +5.57768 q^{13} +1.00705 q^{14} -3.65309 q^{15} +1.00000 q^{16} -2.49817 q^{17} -2.29563 q^{18} +4.77825 q^{19} -4.35272 q^{20} +0.845183 q^{21} -1.96804 q^{22} -4.66373 q^{23} +0.839268 q^{24} +13.9461 q^{25} +5.57768 q^{26} -4.44445 q^{27} +1.00705 q^{28} +5.68625 q^{29} -3.65309 q^{30} +2.33391 q^{31} +1.00000 q^{32} -1.65172 q^{33} -2.49817 q^{34} -4.38340 q^{35} -2.29563 q^{36} -3.12092 q^{37} +4.77825 q^{38} +4.68117 q^{39} -4.35272 q^{40} -7.26367 q^{41} +0.845183 q^{42} +2.54913 q^{43} -1.96804 q^{44} +9.99222 q^{45} -4.66373 q^{46} -4.14408 q^{47} +0.839268 q^{48} -5.98585 q^{49} +13.9461 q^{50} -2.09663 q^{51} +5.57768 q^{52} +0.781697 q^{53} -4.44445 q^{54} +8.56634 q^{55} +1.00705 q^{56} +4.01023 q^{57} +5.68625 q^{58} -6.13762 q^{59} -3.65309 q^{60} -5.77896 q^{61} +2.33391 q^{62} -2.31181 q^{63} +1.00000 q^{64} -24.2781 q^{65} -1.65172 q^{66} -2.65942 q^{67} -2.49817 q^{68} -3.91412 q^{69} -4.38340 q^{70} +5.71342 q^{71} -2.29563 q^{72} +5.86712 q^{73} -3.12092 q^{74} +11.7045 q^{75} +4.77825 q^{76} -1.98192 q^{77} +4.68117 q^{78} -7.46831 q^{79} -4.35272 q^{80} +3.15681 q^{81} -7.26367 q^{82} +12.8391 q^{83} +0.845183 q^{84} +10.8738 q^{85} +2.54913 q^{86} +4.77228 q^{87} -1.96804 q^{88} -4.21114 q^{89} +9.99222 q^{90} +5.61700 q^{91} -4.66373 q^{92} +1.95877 q^{93} -4.14408 q^{94} -20.7984 q^{95} +0.839268 q^{96} +1.11834 q^{97} -5.98585 q^{98} +4.51790 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.839268	0.484551	0.242276	−	0.970207i	$-0.422106\pi$
$3$	0.839268	0.484551	0.242276	+	0.970207i	$0.422106\pi$
$4$	1.00000	0.500000
$5$	−4.35272	−1.94659	−0.973297	−	0.229550i	$-0.926275\pi$
$5$	−4.35272	−1.94659	−0.973297	+	0.229550i	$0.926275\pi$
$6$	0.839268	0.342630
$7$	1.00705	0.380629	0.190314	−	0.981723i	$-0.439049\pi$
$7$	1.00705	0.380629	0.190314	+	0.981723i	$0.439049\pi$
$8$	1.00000	0.353553
$9$	−2.29563	−0.765210
$10$	−4.35272	−1.37645
$11$	−1.96804	−0.593388	−0.296694	−	0.954973i	$-0.595884\pi$
$11$	−1.96804	−0.593388	−0.296694	+	0.954973i	$0.595884\pi$
$12$	0.839268	0.242276
$13$	5.57768	1.54697	0.773485	−	0.633815i	$-0.218512\pi$
$13$	5.57768	1.54697	0.773485	+	0.633815i	$0.218512\pi$
$14$	1.00705	0.269145
$15$	−3.65309	−0.943225
$16$	1.00000	0.250000
$17$	−2.49817	−0.605894	−0.302947	−	0.953007i	$-0.597971\pi$
$17$	−2.49817	−0.605894	−0.302947	+	0.953007i	$0.597971\pi$
$18$	−2.29563	−0.541085
$19$	4.77825	1.09621	0.548103	−	0.836411i	$-0.315350\pi$
$19$	4.77825	1.09621	0.548103	+	0.836411i	$0.315350\pi$
$20$	−4.35272	−0.973297
$21$	0.845183	0.184434
$22$	−1.96804	−0.419588
$23$	−4.66373	−0.972455	−0.486228	−	0.873832i	$-0.661627\pi$
$23$	−4.66373	−0.972455	−0.486228	+	0.873832i	$0.661627\pi$
$24$	0.839268	0.171315
$25$	13.9461	2.78923
$26$	5.57768	1.09387
$27$	−4.44445	−0.855335
$28$	1.00705	0.190314
$29$	5.68625	1.05591	0.527955	−	0.849272i	$-0.322959\pi$
$29$	5.68625	1.05591	0.527955	+	0.849272i	$0.322959\pi$
$30$	−3.65309	−0.666961
$31$	2.33391	0.419182	0.209591	−	0.977789i	$-0.432787\pi$
$31$	2.33391	0.419182	0.209591	+	0.977789i	$0.432787\pi$
$32$	1.00000	0.176777
$33$	−1.65172	−0.287527
$34$	−2.49817	−0.428432
$35$	−4.38340	−0.740929
$36$	−2.29563	−0.382605
$37$	−3.12092	−0.513077	−0.256538	−	0.966534i	$-0.582582\pi$
$37$	−3.12092	−0.513077	−0.256538	+	0.966534i	$0.582582\pi$
$38$	4.77825	0.775135
$39$	4.68117	0.749587
$40$	−4.35272	−0.688225
$41$	−7.26367	−1.13440	−0.567198	−	0.823582i	$-0.691972\pi$
$41$	−7.26367	−1.13440	−0.567198	+	0.823582i	$0.691972\pi$
$42$	0.845183	0.130415
$43$	2.54913	0.388738	0.194369	−	0.980928i	$-0.437734\pi$
$43$	2.54913	0.388738	0.194369	+	0.980928i	$0.437734\pi$
$44$	−1.96804	−0.296694
$45$	9.99222	1.48955
$46$	−4.66373	−0.687630
$47$	−4.14408	−0.604476	−0.302238	−	0.953232i	$-0.597734\pi$
$47$	−4.14408	−0.604476	−0.302238	+	0.953232i	$0.597734\pi$
$48$	0.839268	0.121138
$49$	−5.98585	−0.855122
$50$	13.9461	1.97228
$51$	−2.09663	−0.293587
$52$	5.57768	0.773485
$53$	0.781697	0.107374	0.0536872	−	0.998558i	$-0.482903\pi$
$53$	0.781697	0.107374	0.0536872	+	0.998558i	$0.482903\pi$
$54$	−4.44445	−0.604813
$55$	8.56634	1.15508
$56$	1.00705	0.134573
$57$	4.01023	0.531168
$58$	5.68625	0.746641
$59$	−6.13762	−0.799050	−0.399525	−	0.916722i	$-0.630825\pi$
$59$	−6.13762	−0.799050	−0.399525	+	0.916722i	$0.630825\pi$
$60$	−3.65309	−0.471612
$61$	−5.77896	−0.739920	−0.369960	−	0.929048i	$-0.620629\pi$
$61$	−5.77896	−0.739920	−0.369960	+	0.929048i	$0.620629\pi$
$62$	2.33391	0.296407
$63$	−2.31181	−0.291261
$64$	1.00000	0.125000
$65$	−24.2781	−3.01132
$66$	−1.65172	−0.203312
$67$	−2.65942	−0.324900	−0.162450	−	0.986717i	$-0.551940\pi$
$67$	−2.65942	−0.324900	−0.162450	+	0.986717i	$0.551940\pi$
$68$	−2.49817	−0.302947
$69$	−3.91412	−0.471205
$70$	−4.38340	−0.523916
$71$	5.71342	0.678058	0.339029	−	0.940776i	$-0.389902\pi$
$71$	5.71342	0.678058	0.339029	+	0.940776i	$0.389902\pi$
$72$	−2.29563	−0.270543
$73$	5.86712	0.686695	0.343347	−	0.939208i	$-0.388439\pi$
$73$	5.86712	0.686695	0.343347	+	0.939208i	$0.388439\pi$
$74$	−3.12092	−0.362800
$75$	11.7045	1.35152
$76$	4.77825	0.548103
$77$	−1.98192	−0.225860
$78$	4.68117	0.530038
$79$	−7.46831	−0.840251	−0.420125	−	0.907466i	$-0.638014\pi$
$79$	−7.46831	−0.840251	−0.420125	+	0.907466i	$0.638014\pi$
$80$	−4.35272	−0.486648
$81$	3.15681	0.350756
$82$	−7.26367	−0.802139
$83$	12.8391	1.40928	0.704638	−	0.709566i	$-0.251109\pi$
$83$	12.8391	1.40928	0.704638	+	0.709566i	$0.251109\pi$
$84$	0.845183	0.0922171
$85$	10.8738	1.17943
$86$	2.54913	0.274880
$87$	4.77228	0.511643
$88$	−1.96804	−0.209794
$89$	−4.21114	−0.446380	−0.223190	−	0.974775i	$-0.571647\pi$
$89$	−4.21114	−0.446380	−0.223190	+	0.974775i	$0.571647\pi$
$90$	9.99222	1.05327
$91$	5.61700	0.588821
$92$	−4.66373	−0.486228
$93$	1.95877	0.203115
$94$	−4.14408	−0.427429
$95$	−20.7984	−2.13387
$96$	0.839268	0.0856574
$97$	1.11834	0.113551	0.0567753	−	0.998387i	$-0.481918\pi$
$97$	1.11834	0.113551	0.0567753	+	0.998387i	$0.481918\pi$
$98$	−5.98585	−0.604662
$99$	4.51790	0.454066
$100$	13.9461	1.39461
$101$	−6.66266	−0.662959	−0.331480	−	0.943462i	$-0.607548\pi$
$101$	−6.66266	−0.662959	−0.331480	+	0.943462i	$0.607548\pi$
$102$	−2.09663	−0.207597
$103$	−9.91091	−0.976551	−0.488276	−	0.872689i	$-0.662374\pi$
$103$	−9.91091	−0.976551	−0.488276	+	0.872689i	$0.662374\pi$
$104$	5.57768	0.546937
$105$	−3.67884	−0.359018
$106$	0.781697	0.0759252
$107$	7.98313	0.771759	0.385879	−	0.922549i	$-0.373898\pi$
$107$	7.98313	0.771759	0.385879	+	0.922549i	$0.373898\pi$
$108$	−4.44445	−0.427667
$109$	−9.43374	−0.903588	−0.451794	−	0.892122i	$-0.649216\pi$
$109$	−9.43374	−0.903588	−0.451794	+	0.892122i	$0.649216\pi$
$110$	8.56634	0.816768
$111$	−2.61929	−0.248612
$112$	1.00705	0.0951572
$113$	6.34267	0.596668	0.298334	−	0.954461i	$-0.403569\pi$
$113$	6.34267	0.596668	0.298334	+	0.954461i	$0.403569\pi$
$114$	4.01023	0.375593
$115$	20.2999	1.89298
$116$	5.68625	0.527955
$117$	−12.8043	−1.18376
$118$	−6.13762	−0.565013
$119$	−2.51577	−0.230621
$120$	−3.65309	−0.333480
$121$	−7.12680	−0.647891
$122$	−5.77896	−0.523203
$123$	−6.09617	−0.549673
$124$	2.33391	0.209591
$125$	−38.9400	−3.48290
$126$	−2.31181	−0.205952
$127$	−16.0386	−1.42320	−0.711598	−	0.702587i	$-0.752028\pi$
$127$	−16.0386	−1.42320	−0.711598	+	0.702587i	$0.752028\pi$
$128$	1.00000	0.0883883
$129$	2.13940	0.188364
$130$	−24.2781	−2.12933
$131$	−16.7778	−1.46588	−0.732942	−	0.680291i	$-0.761853\pi$
$131$	−16.7778	−1.46588	−0.732942	+	0.680291i	$0.761853\pi$
$132$	−1.65172	−0.143763
$133$	4.81193	0.417247
$134$	−2.65942	−0.229739
$135$	19.3454	1.66499
$136$	−2.49817	−0.214216
$137$	−15.8368	−1.35303	−0.676516	−	0.736428i	$-0.736511\pi$
$137$	−15.8368	−1.35303	−0.676516	+	0.736428i	$0.736511\pi$
$138$	−3.91412	−0.333192
$139$	14.1296	1.19846	0.599229	−	0.800578i	$-0.295474\pi$
$139$	14.1296	1.19846	0.599229	+	0.800578i	$0.295474\pi$
$140$	−4.38340	−0.370465
$141$	−3.47799	−0.292900
$142$	5.71342	0.479459
$143$	−10.9771	−0.917953
$144$	−2.29563	−0.191302
$145$	−24.7506	−2.05543
$146$	5.86712	0.485567
$147$	−5.02373	−0.414351
$148$	−3.12092	−0.256538
$149$	13.4482	1.10172	0.550858	−	0.834599i	$-0.314301\pi$
$149$	13.4482	1.10172	0.550858	+	0.834599i	$0.314301\pi$
$150$	11.7045	0.955672
$151$	1.13317	0.0922160	0.0461080	−	0.998936i	$-0.485318\pi$
$151$	1.13317	0.0922160	0.0461080	+	0.998936i	$0.485318\pi$
$152$	4.77825	0.387567
$153$	5.73486	0.463636
$154$	−1.98192	−0.159707
$155$	−10.1588	−0.815977
$156$	4.68117	0.374793
$157$	6.78956	0.541866	0.270933	−	0.962598i	$-0.412668\pi$
$157$	6.78956	0.541866	0.270933	+	0.962598i	$0.412668\pi$
$158$	−7.46831	−0.594147
$159$	0.656053	0.0520284
$160$	−4.35272	−0.344112
$161$	−4.69660	−0.370144
$162$	3.15681	0.248022
$163$	−8.36463	−0.655168	−0.327584	−	0.944822i	$-0.606235\pi$
$163$	−8.36463	−0.655168	−0.327584	+	0.944822i	$0.606235\pi$
$164$	−7.26367	−0.567198
$165$	7.18945	0.559698
$166$	12.8391	0.996509
$167$	−21.4223	−1.65770	−0.828852	−	0.559468i	$-0.811006\pi$
$167$	−21.4223	−1.65770	−0.828852	+	0.559468i	$0.811006\pi$
$168$	0.845183	0.0652073
$169$	18.1105	1.39312
$170$	10.8738	0.833983
$171$	−10.9691	−0.838828
$172$	2.54913	0.194369
$173$	−17.3683	−1.32049	−0.660246	−	0.751050i	$-0.729548\pi$
$173$	−17.3683	−1.32049	−0.660246	+	0.751050i	$0.729548\pi$
$174$	4.77228	0.361786
$175$	14.0444	1.06166
$176$	−1.96804	−0.148347
$177$	−5.15110	−0.387181
$178$	−4.21114	−0.315638
$179$	4.26442	0.318737	0.159369	−	0.987219i	$-0.449054\pi$
$179$	4.26442	0.318737	0.159369	+	0.987219i	$0.449054\pi$
$180$	9.99222	0.744776
$181$	−11.5621	−0.859401	−0.429700	−	0.902972i	$-0.641381\pi$
$181$	−11.5621	−0.859401	−0.429700	+	0.902972i	$0.641381\pi$
$182$	5.61700	0.416359
$183$	−4.85010	−0.358529
$184$	−4.66373	−0.343815
$185$	13.5845	0.998752
$186$	1.95877	0.143624
$187$	4.91650	0.359530
$188$	−4.14408	−0.302238
$189$	−4.47578	−0.325565
$190$	−20.7984	−1.50887
$191$	6.96180	0.503738	0.251869	−	0.967761i	$-0.418955\pi$
$191$	6.96180	0.503738	0.251869	+	0.967761i	$0.418955\pi$
$192$	0.839268	0.0605689
$193$	−26.0231	−1.87318	−0.936590	−	0.350427i	$-0.886037\pi$
$193$	−26.0231	−1.87318	−0.936590	+	0.350427i	$0.886037\pi$
$194$	1.11834	0.0802924
$195$	−20.3758	−1.45914
$196$	−5.98585	−0.427561
$197$	8.34990	0.594906	0.297453	−	0.954736i	$-0.403863\pi$
$197$	8.34990	0.594906	0.297453	+	0.954736i	$0.403863\pi$
$198$	4.51790	0.321073
$199$	2.58777	0.183442	0.0917210	−	0.995785i	$-0.470763\pi$
$199$	2.58777	0.183442	0.0917210	+	0.995785i	$0.470763\pi$
$200$	13.9461	0.986141
$201$	−2.23196	−0.157431
$202$	−6.66266	−0.468783
$203$	5.72633	0.401909
$204$	−2.09663	−0.146793
$205$	31.6167	2.20821
$206$	−9.91091	−0.690526
$207$	10.7062	0.744132
$208$	5.57768	0.386743
$209$	−9.40381	−0.650475
$210$	−3.67884	−0.253864
$211$	−15.6623	−1.07824	−0.539119	−	0.842230i	$-0.681243\pi$
$211$	−15.6623	−1.07824	−0.539119	+	0.842230i	$0.681243\pi$
$212$	0.781697	0.0536872
$213$	4.79509	0.328554
$214$	7.98313	0.545716
$215$	−11.0956	−0.756716
$216$	−4.44445	−0.302407
$217$	2.35036	0.159553
$218$	−9.43374	−0.638933
$219$	4.92409	0.332739
$220$	8.56634	0.577542
$221$	−13.9340	−0.937300
$222$	−2.61929	−0.175795
$223$	−29.5562	−1.97923	−0.989615	−	0.143744i	$-0.954086\pi$
$223$	−29.5562	−1.97923	−0.989615	+	0.143744i	$0.954086\pi$
$224$	1.00705	0.0672863
$225$	−32.0152	−2.13434
$226$	6.34267	0.421908
$227$	−13.4756	−0.894406	−0.447203	−	0.894432i	$-0.647580\pi$
$227$	−13.4756	−0.894406	−0.447203	+	0.894432i	$0.647580\pi$
$228$	4.01023	0.265584
$229$	0.115554	0.00763600	0.00381800	−	0.999993i	$-0.498785\pi$
$229$	0.115554	0.00763600	0.00381800	+	0.999993i	$0.498785\pi$
$230$	20.2999	1.33854
$231$	−1.66336	−0.109441
$232$	5.68625	0.373320
$233$	−17.8326	−1.16825	−0.584126	−	0.811663i	$-0.698563\pi$
$233$	−17.8326	−1.16825	−0.584126	+	0.811663i	$0.698563\pi$
$234$	−12.8043	−0.837043
$235$	18.0380	1.17667
$236$	−6.13762	−0.399525
$237$	−6.26791	−0.407145
$238$	−2.51577	−0.163073
$239$	2.00799	0.129886	0.0649430	−	0.997889i	$-0.479313\pi$
$239$	2.00799	0.129886	0.0649430	+	0.997889i	$0.479313\pi$
$240$	−3.65309	−0.235806
$241$	−11.5416	−0.743462	−0.371731	−	0.928340i	$-0.621236\pi$
$241$	−11.5416	−0.743462	−0.371731	+	0.928340i	$0.621236\pi$
$242$	−7.12680	−0.458128
$243$	15.9828	1.02529
$244$	−5.77896	−0.369960
$245$	26.0547	1.66457
$246$	−6.09617	−0.388677
$247$	26.6516	1.69580
$248$	2.33391	0.148203
$249$	10.7755	0.682867
$250$	−38.9400	−2.46278
$251$	−23.3277	−1.47243	−0.736217	−	0.676745i	$-0.763390\pi$
$251$	−23.3277	−1.47243	−0.736217	+	0.676745i	$0.763390\pi$
$252$	−2.31181	−0.145630
$253$	9.17843	0.577043
$254$	−16.0386	−1.00635
$255$	9.12603	0.571494
$256$	1.00000	0.0625000
$257$	−28.3175	−1.76640	−0.883200	−	0.468997i	$-0.844616\pi$
$257$	−28.3175	−1.76640	−0.883200	+	0.468997i	$0.844616\pi$
$258$	2.13940	0.133193
$259$	−3.14292	−0.195292
$260$	−24.2781	−1.50566
$261$	−13.0535	−0.807993
$262$	−16.7778	−1.03654
$263$	26.3494	1.62477	0.812386	−	0.583120i	$-0.198168\pi$
$263$	26.3494	1.62477	0.812386	+	0.583120i	$0.198168\pi$
$264$	−1.65172	−0.101656
$265$	−3.40251	−0.209014
$266$	4.81193	0.295039
$267$	−3.53427	−0.216294
$268$	−2.65942	−0.162450
$269$	−4.99057	−0.304280	−0.152140	−	0.988359i	$-0.548616\pi$
$269$	−4.99057	−0.304280	−0.152140	+	0.988359i	$0.548616\pi$
$270$	19.3454	1.17733
$271$	17.7391	1.07758	0.538788	−	0.842442i	$-0.318882\pi$
$271$	17.7391	1.07758	0.538788	+	0.842442i	$0.318882\pi$
$272$	−2.49817	−0.151474
$273$	4.71416	0.285314
$274$	−15.8368	−0.956738
$275$	−27.4466	−1.65509
$276$	−3.91412	−0.235602
$277$	−12.2862	−0.738206	−0.369103	−	0.929388i	$-0.620335\pi$
$277$	−12.2862	−0.738206	−0.369103	+	0.929388i	$0.620335\pi$
$278$	14.1296	0.847437
$279$	−5.35779	−0.320762
$280$	−4.38340	−0.261958
$281$	1.12583	0.0671614	0.0335807	−	0.999436i	$-0.489309\pi$
$281$	1.12583	0.0671614	0.0335807	+	0.999436i	$0.489309\pi$
$282$	−3.47799	−0.207111
$283$	31.0247	1.84423	0.922113	−	0.386921i	$-0.126461\pi$
$283$	31.0247	1.84423	0.922113	+	0.386921i	$0.126461\pi$
$284$	5.71342	0.339029
$285$	−17.4554	−1.03397
$286$	−10.9771	−0.649091
$287$	−7.31487	−0.431783
$288$	−2.29563	−0.135271
$289$	−10.7592	−0.632892
$290$	−24.7506	−1.45341
$291$	0.938590	0.0550211
$292$	5.86712	0.343347
$293$	26.3634	1.54017	0.770084	−	0.637943i	$-0.220214\pi$
$293$	26.3634	1.54017	0.770084	+	0.637943i	$0.220214\pi$
$294$	−5.02373	−0.292990
$295$	26.7153	1.55543
$296$	−3.12092	−0.181400
$297$	8.74687	0.507545
$298$	13.4482	0.779031
$299$	−26.0128	−1.50436
$300$	11.7045	0.675762
$301$	2.56710	0.147965
$302$	1.13317	0.0652066
$303$	−5.59175	−0.321238
$304$	4.77825	0.274052
$305$	25.1542	1.44032
$306$	5.73486	0.327840
$307$	23.8079	1.35879	0.679395	−	0.733773i	$-0.262242\pi$
$307$	23.8079	1.35879	0.679395	+	0.733773i	$0.262242\pi$
$308$	−1.98192	−0.112930
$309$	−8.31791	−0.473189
$310$	−10.1588	−0.576983
$311$	−1.55153	−0.0879793	−0.0439896	−	0.999032i	$-0.514007\pi$
$311$	−1.55153	−0.0879793	−0.0439896	+	0.999032i	$0.514007\pi$
$312$	4.68117	0.265019
$313$	27.9268	1.57851	0.789257	−	0.614063i	$-0.210466\pi$
$313$	27.9268	1.57851	0.789257	+	0.614063i	$0.210466\pi$
$314$	6.78956	0.383157
$315$	10.0627	0.566966
$316$	−7.46831	−0.420125
$317$	−24.6664	−1.38540	−0.692702	−	0.721224i	$-0.743580\pi$
$317$	−24.6664	−1.38540	−0.692702	+	0.721224i	$0.743580\pi$
$318$	0.656053	0.0367896
$319$	−11.1908	−0.626564
$320$	−4.35272	−0.243324
$321$	6.69999	0.373957
$322$	−4.69660	−0.261732
$323$	−11.9369	−0.664185
$324$	3.15681	0.175378
$325$	77.7871	4.31485
$326$	−8.36463	−0.463274
$327$	−7.91743	−0.437835
$328$	−7.26367	−0.401069
$329$	−4.17329	−0.230081
$330$	7.18945	0.395766
$331$	1.77668	0.0976551	0.0488276	−	0.998807i	$-0.484452\pi$
$331$	1.77668	0.0976551	0.0488276	+	0.998807i	$0.484452\pi$
$332$	12.8391	0.704638
$333$	7.16449	0.392611
$334$	−21.4223	−1.17217
$335$	11.5757	0.632448
$336$	0.845183	0.0461085
$337$	−21.5374	−1.17322	−0.586608	−	0.809871i	$-0.699537\pi$
$337$	−21.5374	−1.17322	−0.586608	+	0.809871i	$0.699537\pi$
$338$	18.1105	0.985082
$339$	5.32320	0.289116
$340$	10.8738	0.589715
$341$	−4.59323	−0.248738
$342$	−10.9691	−0.593141
$343$	−13.0774	−0.706112
$344$	2.54913	0.137440
$345$	17.0371	0.917244
$346$	−17.3683	−0.933728
$347$	−11.4700	−0.615744	−0.307872	−	0.951428i	$-0.599617\pi$
$347$	−11.4700	−0.615744	−0.307872	+	0.951428i	$0.599617\pi$
$348$	4.77228	0.255821
$349$	26.0108	1.39232	0.696162	−	0.717885i	$-0.254889\pi$
$349$	26.0108	1.39232	0.696162	+	0.717885i	$0.254889\pi$
$350$	14.0444	0.750707
$351$	−24.7897	−1.32318
$352$	−1.96804	−0.104897
$353$	18.6762	0.994032	0.497016	−	0.867741i	$-0.334429\pi$
$353$	18.6762	0.994032	0.497016	+	0.867741i	$0.334429\pi$
$354$	−5.15110	−0.273778
$355$	−24.8689	−1.31990
$356$	−4.21114	−0.223190
$357$	−2.11141	−0.111748
$358$	4.26442	0.225381
$359$	−26.7936	−1.41411	−0.707057	−	0.707156i	$-0.749978\pi$
$359$	−26.7936	−1.41411	−0.707057	+	0.707156i	$0.749978\pi$
$360$	9.99222	0.526636
$361$	3.83169	0.201668
$362$	−11.5621	−0.607688
$363$	−5.98130	−0.313937
$364$	5.61700	0.294411
$365$	−25.5379	−1.33672
$366$	−4.85010	−0.253519
$367$	−5.74757	−0.300021	−0.150010	−	0.988684i	$-0.547931\pi$
$367$	−5.74757	−0.300021	−0.150010	+	0.988684i	$0.547931\pi$
$368$	−4.66373	−0.243114
$369$	16.6747	0.868050
$370$	13.5845	0.706224
$371$	0.787207	0.0408698
$372$	1.95877	0.101558
$373$	−14.1397	−0.732126	−0.366063	−	0.930590i	$-0.619294\pi$
$373$	−14.1397	−0.732126	−0.366063	+	0.930590i	$0.619294\pi$
$374$	4.91650	0.254226
$375$	−32.6811	−1.68764
$376$	−4.14408	−0.213715
$377$	31.7161	1.63346
$378$	−4.47578	−0.230209
$379$	−32.4209	−1.66535	−0.832675	−	0.553763i	$-0.813191\pi$
$379$	−32.4209	−1.66535	−0.832675	+	0.553763i	$0.813191\pi$
$380$	−20.7984	−1.06693
$381$	−13.4607	−0.689611
$382$	6.96180	0.356197
$383$	10.9227	0.558125	0.279063	−	0.960273i	$-0.409976\pi$
$383$	10.9227	0.558125	0.279063	+	0.960273i	$0.409976\pi$
$384$	0.839268	0.0428287
$385$	8.62672	0.439658
$386$	−26.0231	−1.32454
$387$	−5.85186	−0.297467
$388$	1.11834	0.0567753
$389$	11.1305	0.564338	0.282169	−	0.959365i	$-0.408946\pi$
$389$	11.1305	0.564338	0.282169	+	0.959365i	$0.408946\pi$
$390$	−20.3758	−1.03177
$391$	11.6508	0.589205
$392$	−5.98585	−0.302331
$393$	−14.0811	−0.710296
$394$	8.34990	0.420662
$395$	32.5074	1.63563
$396$	4.51790	0.227033
$397$	32.9613	1.65428	0.827139	−	0.561997i	$-0.189967\pi$
$397$	32.9613	1.65428	0.827139	+	0.561997i	$0.189967\pi$
$398$	2.58777	0.129713
$399$	4.03850	0.202178
$400$	13.9461	0.697307
$401$	33.8059	1.68819	0.844093	−	0.536197i	$-0.180139\pi$
$401$	33.8059	1.68819	0.844093	+	0.536197i	$0.180139\pi$
$402$	−2.23196	−0.111320
$403$	13.0178	0.648462
$404$	−6.66266	−0.331480
$405$	−13.7407	−0.682780
$406$	5.72633	0.284193
$407$	6.14212	0.304453
$408$	−2.09663	−0.103799
$409$	25.4403	1.25794	0.628970	−	0.777430i	$-0.283477\pi$
$409$	25.4403	1.25794	0.628970	+	0.777430i	$0.283477\pi$
$410$	31.6167	1.56144
$411$	−13.2913	−0.655614
$412$	−9.91091	−0.488276
$413$	−6.18088	−0.304141
$414$	10.7062	0.526181
$415$	−55.8851	−2.74329
$416$	5.57768	0.273468
$417$	11.8585	0.580714
$418$	−9.40381	−0.459955
$419$	−22.7743	−1.11260	−0.556298	−	0.830983i	$-0.687779\pi$
$419$	−22.7743	−1.11260	−0.556298	+	0.830983i	$0.687779\pi$
$420$	−3.67884	−0.179509
$421$	25.6387	1.24955	0.624777	−	0.780803i	$-0.285190\pi$
$421$	25.6387	1.24955	0.624777	+	0.780803i	$0.285190\pi$
$422$	−15.6623	−0.762429
$423$	9.51327	0.462551
$424$	0.781697	0.0379626
$425$	−34.8398	−1.68998
$426$	4.79509	0.232323
$427$	−5.81970	−0.281635
$428$	7.98313	0.385879
$429$	−9.21274	−0.444795
$430$	−11.0956	−0.535079
$431$	−14.0308	−0.675840	−0.337920	−	0.941175i	$-0.609723\pi$
$431$	−14.0308	−0.675840	−0.337920	+	0.941175i	$0.609723\pi$
$432$	−4.44445	−0.213834
$433$	−22.3591	−1.07451	−0.537254	−	0.843421i	$-0.680538\pi$
$433$	−22.3591	−1.07451	−0.537254	+	0.843421i	$0.680538\pi$
$434$	2.35036	0.112821
$435$	−20.7724	−0.995960
$436$	−9.43374	−0.451794
$437$	−22.2845	−1.06601
$438$	4.92409	0.235282
$439$	−1.21479	−0.0579789	−0.0289895	−	0.999580i	$-0.509229\pi$
$439$	−1.21479	−0.0579789	−0.0289895	+	0.999580i	$0.509229\pi$
$440$	8.56634	0.408384
$441$	13.7413	0.654348
$442$	−13.9340	−0.662771
$443$	−24.0379	−1.14207	−0.571037	−	0.820924i	$-0.693459\pi$
$443$	−24.0379	−1.14207	−0.571037	+	0.820924i	$0.693459\pi$
$444$	−2.61929	−0.124306
$445$	18.3299	0.868920
$446$	−29.5562	−1.39953
$447$	11.2866	0.533838
$448$	1.00705	0.0475786
$449$	14.6670	0.692177	0.346089	−	0.938202i	$-0.387510\pi$
$449$	14.6670	0.692177	0.346089	+	0.938202i	$0.387510\pi$
$450$	−32.0152	−1.50921
$451$	14.2952	0.673136
$452$	6.34267	0.298334
$453$	0.951033	0.0446834
$454$	−13.4756	−0.632441
$455$	−24.4492	−1.14620
$456$	4.01023	0.187796
$457$	12.3474	0.577587	0.288794	−	0.957391i	$-0.406746\pi$
$457$	12.3474	0.577587	0.288794	+	0.957391i	$0.406746\pi$
$458$	0.115554	0.00539947
$459$	11.1030	0.518242
$460$	20.2999	0.946488
$461$	−15.2867	−0.711970	−0.355985	−	0.934492i	$-0.615855\pi$
$461$	−15.2867	−0.711970	−0.355985	+	0.934492i	$0.615855\pi$
$462$	−1.66336	−0.0773864
$463$	21.7773	1.01208	0.506039	−	0.862511i	$-0.331109\pi$
$463$	21.7773	1.01208	0.506039	+	0.862511i	$0.331109\pi$
$464$	5.68625	0.263977
$465$	−8.52598	−0.395383
$466$	−17.8326	−0.826079
$467$	−10.8803	−0.503482	−0.251741	−	0.967795i	$-0.581003\pi$
$467$	−10.8803	−0.503482	−0.251741	+	0.967795i	$0.581003\pi$
$468$	−12.8043	−0.591878
$469$	−2.67816	−0.123666
$470$	18.0380	0.832031
$471$	5.69826	0.262562
$472$	−6.13762	−0.282507
$473$	−5.01680	−0.230673
$474$	−6.26791	−0.287895
$475$	66.6382	3.05757
$476$	−2.51577	−0.115310
$477$	−1.79449	−0.0821639
$478$	2.00799	0.0918433
$479$	−11.2921	−0.515949	−0.257975	−	0.966152i	$-0.583055\pi$
$479$	−11.2921	−0.515949	−0.257975	+	0.966152i	$0.583055\pi$
$480$	−3.65309	−0.166740
$481$	−17.4075	−0.793715
$482$	−11.5416	−0.525707
$483$	−3.94171	−0.179354
$484$	−7.12680	−0.323946
$485$	−4.86784	−0.221037
$486$	15.9828	0.724993
$487$	−25.2296	−1.14326	−0.571632	−	0.820510i	$-0.693689\pi$
$487$	−25.2296	−1.14326	−0.571632	+	0.820510i	$0.693689\pi$
$488$	−5.77896	−0.261601
$489$	−7.02016	−0.317463
$490$	26.0547	1.17703
$491$	−10.0858	−0.455167	−0.227584	−	0.973759i	$-0.573082\pi$
$491$	−10.0858	−0.455167	−0.227584	+	0.973759i	$0.573082\pi$
$492$	−6.09617	−0.274836
$493$	−14.2052	−0.639770
$494$	26.6516	1.19911
$495$	−19.6651	−0.883882
$496$	2.33391	0.104796
$497$	5.75369	0.258088
$498$	10.7755	0.482860
$499$	25.5181	1.14235	0.571173	−	0.820830i	$-0.306489\pi$
$499$	25.5181	1.14235	0.571173	+	0.820830i	$0.306489\pi$
$500$	−38.9400	−1.74145
$501$	−17.9790	−0.803243
$502$	−23.3277	−1.04117
$503$	19.3543	0.862964	0.431482	−	0.902122i	$-0.357991\pi$
$503$	19.3543	0.862964	0.431482	+	0.902122i	$0.357991\pi$
$504$	−2.31181	−0.102976
$505$	29.0007	1.29051
$506$	9.17843	0.408031
$507$	15.1996	0.675037
$508$	−16.0386	−0.711598
$509$	−24.5356	−1.08752	−0.543761	−	0.839240i	$-0.683000\pi$
$509$	−24.5356	−1.08752	−0.543761	+	0.839240i	$0.683000\pi$
$510$	9.12603	0.404108
$511$	5.90848	0.261376
$512$	1.00000	0.0441942
$513$	−21.2367	−0.937624
$514$	−28.3175	−1.24903
$515$	43.1394	1.90095
$516$	2.13940	0.0941819
$517$	8.15573	0.358689
$518$	−3.14292	−0.138092
$519$	−14.5767	−0.639846
$520$	−24.2781	−1.06466
$521$	35.7744	1.56731	0.783653	−	0.621198i	$-0.213354\pi$
$521$	35.7744	1.56731	0.783653	+	0.621198i	$0.213354\pi$
$522$	−13.0535	−0.571337
$523$	−17.2910	−0.756083	−0.378041	−	0.925789i	$-0.623402\pi$
$523$	−17.2910	−0.756083	−0.378041	+	0.925789i	$0.623402\pi$
$524$	−16.7778	−0.732942
$525$	11.7870	0.514429
$526$	26.3494	1.14889
$527$	−5.83049	−0.253980
$528$	−1.65172	−0.0718817
$529$	−1.24960	−0.0543306
$530$	−3.40251	−0.147795
$531$	14.0897	0.611441
$532$	4.81193	0.208624
$533$	−40.5144	−1.75488
$534$	−3.53427	−0.152943
$535$	−34.7483	−1.50230
$536$	−2.65942	−0.114869
$537$	3.57899	0.154445
$538$	−4.99057	−0.215159
$539$	11.7804	0.507419
$540$	19.3454	0.832495
$541$	−31.9277	−1.37268	−0.686340	−	0.727281i	$-0.740784\pi$
$541$	−31.9277	−1.37268	−0.686340	+	0.727281i	$0.740784\pi$
$542$	17.7391	0.761961
$543$	−9.70366	−0.416424
$544$	−2.49817	−0.107108
$545$	41.0624	1.75892
$546$	4.71416	0.201748
$547$	19.8460	0.848555	0.424277	−	0.905532i	$-0.360528\pi$
$547$	19.8460	0.848555	0.424277	+	0.905532i	$0.360528\pi$
$548$	−15.8368	−0.676516
$549$	13.2664	0.566194
$550$	−27.4466	−1.17033
$551$	27.1703	1.15749
$552$	−3.91412	−0.166596
$553$	−7.52095	−0.319823
$554$	−12.2862	−0.521991
$555$	11.4010	0.483947
$556$	14.1296	0.599229
$557$	6.99118	0.296226	0.148113	−	0.988970i	$-0.452680\pi$
$557$	6.99118	0.296226	0.148113	+	0.988970i	$0.452680\pi$
$558$	−5.35779	−0.226813
$559$	14.2182	0.601367
$560$	−4.38340	−0.185232
$561$	4.12626	0.174211
$562$	1.12583	0.0474903
$563$	−13.0053	−0.548109	−0.274055	−	0.961714i	$-0.588365\pi$
$563$	−13.0053	−0.548109	−0.274055	+	0.961714i	$0.588365\pi$
$564$	−3.47799	−0.146450
$565$	−27.6078	−1.16147
$566$	31.0247	1.30406
$567$	3.17906	0.133508
$568$	5.71342	0.239730
$569$	29.5659	1.23947	0.619734	−	0.784812i	$-0.287240\pi$
$569$	29.5659	1.23947	0.619734	+	0.784812i	$0.287240\pi$
$570$	−17.4554	−0.731126
$571$	19.5020	0.816133	0.408067	−	0.912952i	$-0.366203\pi$
$571$	19.5020	0.816133	0.408067	+	0.912952i	$0.366203\pi$
$572$	−10.9771	−0.458976
$573$	5.84281	0.244087
$574$	−7.31487	−0.305317
$575$	−65.0410	−2.71240
$576$	−2.29563	−0.0956512
$577$	10.8440	0.451443	0.225722	−	0.974192i	$-0.427526\pi$
$577$	10.8440	0.451443	0.225722	+	0.974192i	$0.427526\pi$
$578$	−10.7592	−0.447522
$579$	−21.8403	−0.907652
$580$	−24.7506	−1.02771
$581$	12.9296	0.536411
$582$	0.938590	0.0389058
$583$	−1.53841	−0.0637146
$584$	5.86712	0.242783
$585$	55.7334	2.30429
$586$	26.3634	1.08906
$587$	23.1147	0.954045	0.477022	−	0.878891i	$-0.341716\pi$
$587$	23.1147	0.954045	0.477022	+	0.878891i	$0.341716\pi$
$588$	−5.02373	−0.207175
$589$	11.1520	0.459510
$590$	26.7153	1.09985
$591$	7.00780	0.288263
$592$	−3.12092	−0.128269
$593$	0.478952	0.0196682	0.00983411	−	0.999952i	$-0.496870\pi$
$593$	0.478952	0.0196682	0.00983411	+	0.999952i	$0.496870\pi$
$594$	8.74687	0.358889
$595$	10.9505	0.448925
$596$	13.4482	0.550858
$597$	2.17183	0.0888871
$598$	−26.0128	−1.06374
$599$	23.3060	0.952259	0.476129	−	0.879375i	$-0.342039\pi$
$599$	23.3060	0.952259	0.476129	+	0.879375i	$0.342039\pi$
$600$	11.7045	0.477836
$601$	25.2623	1.03047	0.515235	−	0.857049i	$-0.327705\pi$
$601$	25.2623	1.03047	0.515235	+	0.857049i	$0.327705\pi$
$602$	2.56710	0.104627
$603$	6.10504	0.248616
$604$	1.13317	0.0461080
$605$	31.0210	1.26118
$606$	−5.59175	−0.227149
$607$	−6.56382	−0.266417	−0.133209	−	0.991088i	$-0.542528\pi$
$607$	−6.56382	−0.266417	−0.133209	+	0.991088i	$0.542528\pi$
$608$	4.77825	0.193784
$609$	4.80592	0.194746
$610$	25.1542	1.01846
$611$	−23.1143	−0.935106
$612$	5.73486	0.231818
$613$	24.7193	0.998402	0.499201	−	0.866486i	$-0.333627\pi$
$613$	24.7193	0.998402	0.499201	+	0.866486i	$0.333627\pi$
$614$	23.8079	0.960809
$615$	26.5349	1.06999
$616$	−1.98192	−0.0798537
$617$	−0.434665	−0.0174989	−0.00874947	−	0.999962i	$-0.502785\pi$
$617$	−0.434665	−0.0174989	−0.00874947	+	0.999962i	$0.502785\pi$
$618$	−8.31791	−0.334595
$619$	−5.45297	−0.219173	−0.109587	−	0.993977i	$-0.534953\pi$
$619$	−5.45297	−0.219173	−0.109587	+	0.993977i	$0.534953\pi$
$620$	−10.1588	−0.407989
$621$	20.7277	0.831775
$622$	−1.55153	−0.0622107
$623$	−4.24082	−0.169905
$624$	4.68117	0.187397
$625$	99.7640	3.99056
$626$	27.9268	1.11618
$627$	−7.89231	−0.315189
$628$	6.78956	0.270933
$629$	7.79659	0.310870
$630$	10.0627	0.400906
$631$	26.5021	1.05503	0.527517	−	0.849545i	$-0.323123\pi$
$631$	26.5021	1.05503	0.527517	+	0.849545i	$0.323123\pi$
$632$	−7.46831	−0.297073
$633$	−13.1449	−0.522462
$634$	−24.6664	−0.979628
$635$	69.8115	2.77038
$636$	0.656053	0.0260142
$637$	−33.3872	−1.32285
$638$	−11.1908	−0.443047
$639$	−13.1159	−0.518857
$640$	−4.35272	−0.172056
$641$	30.9131	1.22099	0.610497	−	0.792018i	$-0.290970\pi$
$641$	30.9131	1.22099	0.610497	+	0.792018i	$0.290970\pi$
$642$	6.69999	0.264427
$643$	−14.6444	−0.577519	−0.288760	−	0.957402i	$-0.593243\pi$
$643$	−14.6444	−0.577519	−0.288760	+	0.957402i	$0.593243\pi$
$644$	−4.69660	−0.185072
$645$	−9.31221	−0.366668
$646$	−11.9369	−0.469650
$647$	−18.4153	−0.723979	−0.361990	−	0.932182i	$-0.617902\pi$
$647$	−18.4153	−0.723979	−0.361990	+	0.932182i	$0.617902\pi$
$648$	3.15681	0.124011
$649$	12.0791	0.474146
$650$	77.7871	3.05106
$651$	1.97258	0.0773115
$652$	−8.36463	−0.327584
$653$	28.2543	1.10567	0.552837	−	0.833289i	$-0.313545\pi$
$653$	28.2543	1.10567	0.552837	+	0.833289i	$0.313545\pi$
$654$	−7.91743	−0.309596
$655$	73.0290	2.85348
$656$	−7.26367	−0.283599
$657$	−13.4687	−0.525466
$658$	−4.17329	−0.162692
$659$	−24.8566	−0.968277	−0.484139	−	0.874991i	$-0.660867\pi$
$659$	−24.8566	−0.968277	−0.484139	+	0.874991i	$0.660867\pi$
$660$	7.18945	0.279849
$661$	−13.5300	−0.526254	−0.263127	−	0.964761i	$-0.584754\pi$
$661$	−13.5300	−0.526254	−0.263127	+	0.964761i	$0.584754\pi$
$662$	1.77668	0.0690526
$663$	−11.6943	−0.454170
$664$	12.8391	0.498255
$665$	−20.9450	−0.812211
$666$	7.16449	0.277618
$667$	−26.5191	−1.02683
$668$	−21.4223	−0.828852
$669$	−24.8056	−0.959039
$670$	11.5757	0.447208
$671$	11.3733	0.439060
$672$	0.845183	0.0326037
$673$	6.76830	0.260899	0.130449	−	0.991455i	$-0.458358\pi$
$673$	6.76830	0.260899	0.130449	+	0.991455i	$0.458358\pi$
$674$	−21.5374	−0.829589
$675$	−61.9829	−2.38572
$676$	18.1105	0.696558
$677$	6.33535	0.243487	0.121744	−	0.992562i	$-0.461151\pi$
$677$	6.33535	0.243487	0.121744	+	0.992562i	$0.461151\pi$
$678$	5.32320	0.204436
$679$	1.12623	0.0432206
$680$	10.8738	0.416991
$681$	−11.3096	−0.433386
$682$	−4.59323	−0.175884
$683$	−6.34967	−0.242963	−0.121482	−	0.992594i	$-0.538765\pi$
$683$	−6.34967	−0.242963	−0.121482	+	0.992594i	$0.538765\pi$
$684$	−10.9691	−0.419414
$685$	68.9333	2.63380
$686$	−13.0774	−0.499297
$687$	0.0969805	0.00370004
$688$	2.54913	0.0971846
$689$	4.36006	0.166105
$690$	17.0371	0.648589
$691$	5.89707	0.224335	0.112168	−	0.993689i	$-0.464221\pi$
$691$	5.89707	0.224335	0.112168	+	0.993689i	$0.464221\pi$
$692$	−17.3683	−0.660246
$693$	4.54975	0.172831
$694$	−11.4700	−0.435397
$695$	−61.5021	−2.33291
$696$	4.77228	0.180893
$697$	18.1459	0.687323
$698$	26.0108	0.984522
$699$	−14.9663	−0.566078
$700$	14.0444	0.530830
$701$	30.9198	1.16783	0.583913	−	0.811816i	$-0.301521\pi$
$701$	30.9198	1.16783	0.583913	+	0.811816i	$0.301521\pi$
$702$	−24.7897	−0.935628
$703$	−14.9126	−0.562438
$704$	−1.96804	−0.0741734
$705$	15.1387	0.570157
$706$	18.6762	0.702887
$707$	−6.70962	−0.252341
$708$	−5.15110	−0.193590
$709$	−9.13979	−0.343252	−0.171626	−	0.985162i	$-0.554902\pi$
$709$	−9.13979	−0.343252	−0.171626	+	0.985162i	$0.554902\pi$
$710$	−24.8689	−0.933313
$711$	17.1445	0.642968
$712$	−4.21114	−0.157819
$713$	−10.8847	−0.407636
$714$	−2.11141	−0.0790175
$715$	47.7803	1.78688
$716$	4.26442	0.159369
$717$	1.68524	0.0629364
$718$	−26.7936	−0.999930
$719$	−5.40012	−0.201391	−0.100695	−	0.994917i	$-0.532107\pi$
$719$	−5.40012	−0.201391	−0.100695	+	0.994917i	$0.532107\pi$
$720$	9.99222	0.372388
$721$	−9.98077	−0.371703
$722$	3.83169	0.142601
$723$	−9.68653	−0.360246
$724$	−11.5621	−0.429700
$725$	79.3012	2.94517
$726$	−5.98130	−0.221987
$727$	0.103810	0.00385011	0.00192505	−	0.999998i	$-0.499387\pi$
$727$	0.103810	0.00385011	0.00192505	+	0.999998i	$0.499387\pi$
$728$	5.61700	0.208180
$729$	3.94340	0.146052
$730$	−25.5379	−0.945201
$731$	−6.36815	−0.235534
$732$	−4.85010	−0.179265
$733$	−24.2234	−0.894713	−0.447357	−	0.894356i	$-0.647635\pi$
$733$	−24.2234	−0.894713	−0.447357	+	0.894356i	$0.647635\pi$
$734$	−5.74757	−0.212147
$735$	21.8669	0.806572
$736$	−4.66373	−0.171907
$737$	5.23385	0.192791
$738$	16.6747	0.613804
$739$	−32.3876	−1.19140	−0.595698	−	0.803209i	$-0.703124\pi$
$739$	−32.3876	−1.19140	−0.595698	+	0.803209i	$0.703124\pi$
$740$	13.5845	0.499376
$741$	22.3678	0.821701
$742$	0.787207	0.0288993
$743$	−15.9536	−0.585282	−0.292641	−	0.956222i	$-0.594534\pi$
$743$	−15.9536	−0.585282	−0.292641	+	0.956222i	$0.594534\pi$
$744$	1.95877	0.0718121
$745$	−58.5360	−2.14459
$746$	−14.1397	−0.517691
$747$	−29.4739	−1.07839
$748$	4.91650	0.179765
$749$	8.03940	0.293753
$750$	−32.6811	−1.19334
$751$	−39.1811	−1.42974	−0.714869	−	0.699259i	$-0.753514\pi$
$751$	−39.1811	−1.42974	−0.714869	+	0.699259i	$0.753514\pi$
$752$	−4.14408	−0.151119
$753$	−19.5782	−0.713470
$754$	31.7161	1.15503
$755$	−4.93237	−0.179507
$756$	−4.47578	−0.162782
$757$	14.1708	0.515046	0.257523	−	0.966272i	$-0.417094\pi$
$757$	14.1708	0.515046	0.257523	+	0.966272i	$0.417094\pi$
$758$	−32.4209	−1.17758
$759$	7.70316	0.279607
$760$	−20.7984	−0.754436
$761$	−53.8981	−1.95380	−0.976902	−	0.213686i	$-0.931453\pi$
$761$	−53.8981	−1.95380	−0.976902	+	0.213686i	$0.931453\pi$
$762$	−13.4607	−0.487629
$763$	−9.50023	−0.343932
$764$	6.96180	0.251869
$765$	−24.9622	−0.902511
$766$	10.9227	0.394654
$767$	−34.2337	−1.23611
$768$	0.839268	0.0302845
$769$	13.9020	0.501320	0.250660	−	0.968075i	$-0.419352\pi$
$769$	13.9020	0.501320	0.250660	+	0.968075i	$0.419352\pi$
$770$	8.62672	0.310885
$771$	−23.7660	−0.855911
$772$	−26.0231	−0.936590
$773$	20.4226	0.734551	0.367276	−	0.930112i	$-0.380291\pi$
$773$	20.4226	0.734551	0.367276	+	0.930112i	$0.380291\pi$
$774$	−5.85186	−0.210341
$775$	32.5490	1.16919
$776$	1.11834	0.0401462
$777$	−2.63775	−0.0946289
$778$	11.1305	0.399047
$779$	−34.7077	−1.24353
$780$	−20.3758	−0.729570
$781$	−11.2443	−0.402351
$782$	11.6508	0.416631
$783$	−25.2722	−0.903157
$784$	−5.98585	−0.213780
$785$	−29.5530	−1.05479
$786$	−14.0811	−0.502255
$787$	−44.3522	−1.58098	−0.790492	−	0.612472i	$-0.790175\pi$
$787$	−44.3522	−1.58098	−0.790492	+	0.612472i	$0.790175\pi$
$788$	8.34990	0.297453
$789$	22.1142	0.787286
$790$	32.5074	1.15656
$791$	6.38738	0.227109
$792$	4.51790	0.160537
$793$	−32.2332	−1.14463
$794$	32.9613	1.16975
$795$	−2.85561	−0.101278
$796$	2.58777	0.0917210
$797$	41.2986	1.46287	0.731435	−	0.681911i	$-0.238851\pi$
$797$	41.2986	1.46287	0.731435	+	0.681911i	$0.238851\pi$
$798$	4.03850	0.142961
$799$	10.3526	0.366249
$800$	13.9461	0.493070
$801$	9.66722	0.341574
$802$	33.8059	1.19373
$803$	−11.5468	−0.407476
$804$	−2.23196	−0.0787153
$805$	20.4430	0.720521
$806$	13.0178	0.458532
$807$	−4.18842	−0.147439
$808$	−6.66266	−0.234391
$809$	−22.1880	−0.780089	−0.390044	−	0.920796i	$-0.627540\pi$
$809$	−22.1880	−0.780089	−0.390044	+	0.920796i	$0.627540\pi$
$810$	−13.7407	−0.482798
$811$	−28.0191	−0.983882	−0.491941	−	0.870628i	$-0.663712\pi$
$811$	−28.0191	−0.983882	−0.491941	+	0.870628i	$0.663712\pi$
$812$	5.72633	0.200955
$813$	14.8879	0.522141
$814$	6.14212	0.215281
$815$	36.4089	1.27535
$816$	−2.09663	−0.0733967
$817$	12.1804	0.426138
$818$	25.4403	0.889498
$819$	−12.8945	−0.450572
$820$	31.6167	1.10410
$821$	39.5289	1.37957	0.689783	−	0.724016i	$-0.257706\pi$
$821$	39.5289	1.37957	0.689783	+	0.724016i	$0.257706\pi$
$822$	−13.2913	−0.463589
$823$	50.5327	1.76146	0.880730	−	0.473618i	$-0.157052\pi$
$823$	50.5327	1.76146	0.880730	+	0.473618i	$0.157052\pi$
$824$	−9.91091	−0.345263
$825$	−23.0351	−0.801978
$826$	−6.18088	−0.215060
$827$	−26.7165	−0.929023	−0.464512	−	0.885567i	$-0.653770\pi$
$827$	−26.7165	−0.929023	−0.464512	+	0.885567i	$0.653770\pi$
$828$	10.7062	0.372066
$829$	14.3424	0.498133	0.249066	−	0.968486i	$-0.419876\pi$
$829$	14.3424	0.498133	0.249066	+	0.968486i	$0.419876\pi$
$830$	−55.8851	−1.93980
$831$	−10.3114	−0.357699
$832$	5.57768	0.193371
$833$	14.9537	0.518113
$834$	11.8585	0.410627
$835$	93.2450	3.22688
$836$	−9.40381	−0.325238
$837$	−10.3729	−0.358541
$838$	−22.7743	−0.786724
$839$	10.9503	0.378045	0.189023	−	0.981973i	$-0.439468\pi$
$839$	10.9503	0.378045	0.189023	+	0.981973i	$0.439468\pi$
$840$	−3.67884	−0.126932
$841$	3.33342	0.114945
$842$	25.6387	0.883569
$843$	0.944873	0.0325432
$844$	−15.6623	−0.539119
$845$	−78.8299	−2.71183
$846$	9.51327	0.327073
$847$	−7.17704	−0.246606
$848$	0.781697	0.0268436
$849$	26.0380	0.893622
$850$	−34.8398	−1.19499
$851$	14.5552	0.498944
$852$	4.79509	0.164277
$853$	−24.3430	−0.833489	−0.416745	−	0.909024i	$-0.636829\pi$
$853$	−24.3430	−0.833489	−0.416745	+	0.909024i	$0.636829\pi$
$854$	−5.81970	−0.199146
$855$	47.7454	1.63286
$856$	7.98313	0.272858
$857$	−1.28208	−0.0437952	−0.0218976	−	0.999760i	$-0.506971\pi$
$857$	−1.28208	−0.0437952	−0.0218976	+	0.999760i	$0.506971\pi$
$858$	−9.21274	−0.314518
$859$	24.6510	0.841082	0.420541	−	0.907273i	$-0.361840\pi$
$859$	24.6510	0.841082	0.420541	+	0.907273i	$0.361840\pi$
$860$	−11.0956	−0.378358
$861$	−6.13914	−0.209221
$862$	−14.0308	−0.477891
$863$	30.6700	1.04402	0.522010	−	0.852940i	$-0.325182\pi$
$863$	30.6700	1.04402	0.522010	+	0.852940i	$0.325182\pi$
$864$	−4.44445	−0.151203
$865$	75.5995	2.57046
$866$	−22.3591	−0.759792
$867$	−9.02982	−0.306669
$868$	2.35036	0.0797764
$869$	14.6980	0.498594
$870$	−20.7724	−0.704250
$871$	−14.8334	−0.502610
$872$	−9.43374	−0.319467
$873$	−2.56730	−0.0868901
$874$	−22.2845	−0.753784
$875$	−39.2145	−1.32569
$876$	4.92409	0.166369
$877$	−3.95663	−0.133606	−0.0668029	−	0.997766i	$-0.521280\pi$
$877$	−3.95663	−0.133606	−0.0668029	+	0.997766i	$0.521280\pi$
$878$	−1.21479	−0.0409973
$879$	22.1260	0.746290
$880$	8.56634	0.288771
$881$	31.1523	1.04955	0.524774	−	0.851242i	$-0.324150\pi$
$881$	31.1523	1.04955	0.524774	+	0.851242i	$0.324150\pi$
$882$	13.7413	0.462694
$883$	24.7196	0.831882	0.415941	−	0.909392i	$-0.363452\pi$
$883$	24.7196	0.831882	0.415941	+	0.909392i	$0.363452\pi$
$884$	−13.9340	−0.468650
$885$	22.4213	0.753683
$886$	−24.0379	−0.807569
$887$	49.2267	1.65287	0.826435	−	0.563032i	$-0.190365\pi$
$887$	49.2267	1.65287	0.826435	+	0.563032i	$0.190365\pi$
$888$	−2.61929	−0.0878977
$889$	−16.1516	−0.541709
$890$	18.3299	0.614419
$891$	−6.21273	−0.208134
$892$	−29.5562	−0.989615
$893$	−19.8015	−0.662630
$894$	11.2866	0.377480
$895$	−18.5618	−0.620452
$896$	1.00705	0.0336431
$897$	−21.8317	−0.728939
$898$	14.6670	0.489443
$899$	13.2712	0.442619
$900$	−32.0152	−1.06717
$901$	−1.95281	−0.0650575
$902$	14.2952	0.475979
$903$	2.15448	0.0716966
$904$	6.34267	0.210954
$905$	50.3263	1.67290
$906$	0.951033	0.0315959
$907$	48.7385	1.61834	0.809168	−	0.587578i	$-0.199918\pi$
$907$	48.7385	1.61834	0.809168	+	0.587578i	$0.199918\pi$
$908$	−13.4756	−0.447203
$909$	15.2950	0.507303
$910$	−24.4492	−0.810483
$911$	−34.0554	−1.12831	−0.564153	−	0.825671i	$-0.690797\pi$
$911$	−34.0554	−1.12831	−0.564153	+	0.825671i	$0.690797\pi$
$912$	4.01023	0.132792
$913$	−25.2680	−0.836247
$914$	12.3474	0.408416
$915$	21.1111	0.697911
$916$	0.115554	0.00381800
$917$	−16.8961	−0.557957
$918$	11.1030	0.366453
$919$	−20.2260	−0.667195	−0.333597	−	0.942716i	$-0.608263\pi$
$919$	−20.2260	−0.667195	−0.333597	+	0.942716i	$0.608263\pi$
$920$	20.2999	0.669268
$921$	19.9812	0.658403
$922$	−15.2867	−0.503439
$923$	31.8676	1.04894
$924$	−1.66336	−0.0547205
$925$	−43.5248	−1.43109
$926$	21.7773	0.715647
$927$	22.7518	0.747267
$928$	5.68625	0.186660
$929$	9.03590	0.296458	0.148229	−	0.988953i	$-0.452643\pi$
$929$	9.03590	0.296458	0.148229	+	0.988953i	$0.452643\pi$
$930$	−8.52598	−0.279578
$931$	−28.6019	−0.937390
$932$	−17.8326	−0.584126
$933$	−1.30215	−0.0426305
$934$	−10.8803	−0.356015
$935$	−21.4001	−0.699859
$936$	−12.8043	−0.418521
$937$	−36.2882	−1.18549	−0.592743	−	0.805392i	$-0.701955\pi$
$937$	−36.2882	−1.18549	−0.592743	+	0.805392i	$0.701955\pi$
$938$	−2.67816	−0.0874452
$939$	23.4380	0.764872
$940$	18.0380	0.588335
$941$	−42.3219	−1.37966	−0.689828	−	0.723974i	$-0.742314\pi$
$941$	−42.3219	−1.37966	−0.689828	+	0.723974i	$0.742314\pi$
$942$	5.69826	0.185659
$943$	33.8758	1.10315
$944$	−6.13762	−0.199762
$945$	19.4818	0.633743
$946$	−5.01680	−0.163110
$947$	−21.0982	−0.685599	−0.342800	−	0.939409i	$-0.611375\pi$
$947$	−21.0982	−0.685599	−0.342800	+	0.939409i	$0.611375\pi$
$948$	−6.26791	−0.203572
$949$	32.7249	1.06230
$950$	66.6382	2.16203
$951$	−20.7017	−0.671299
$952$	−2.51577	−0.0815367
$953$	10.8476	0.351389	0.175695	−	0.984445i	$-0.443783\pi$
$953$	10.8476	0.351389	0.175695	+	0.984445i	$0.443783\pi$
$954$	−1.79449	−0.0580987
$955$	−30.3027	−0.980573
$956$	2.00799	0.0649430
$957$	−9.39206	−0.303602
$958$	−11.2921	−0.364831
$959$	−15.9485	−0.515003
$960$	−3.65309	−0.117903
$961$	−25.5529	−0.824286
$962$	−17.4075	−0.561241
$963$	−18.3263	−0.590557
$964$	−11.5416	−0.371731
$965$	113.271	3.64632
$966$	−3.94171	−0.126822
$967$	−12.6410	−0.406507	−0.203253	−	0.979126i	$-0.565152\pi$
$967$	−12.6410	−0.406507	−0.203253	+	0.979126i	$0.565152\pi$
$968$	−7.12680	−0.229064
$969$	−10.0182	−0.321832
$970$	−4.86784	−0.156297
$971$	48.1762	1.54605	0.773023	−	0.634378i	$-0.218744\pi$
$971$	48.1762	1.54605	0.773023	+	0.634378i	$0.218744\pi$
$972$	15.9828	0.512647
$973$	14.2292	0.456167
$974$	−25.2296	−0.808409
$975$	65.2842	2.09077
$976$	−5.77896	−0.184980
$977$	26.4154	0.845102	0.422551	−	0.906339i	$-0.361135\pi$
$977$	26.4154	0.845102	0.422551	+	0.906339i	$0.361135\pi$
$978$	−7.02016	−0.224480
$979$	8.28771	0.264876
$980$	26.0547	0.832287
$981$	21.6564	0.691435
$982$	−10.0858	−0.321852
$983$	15.6933	0.500538	0.250269	−	0.968176i	$-0.419481\pi$
$983$	15.6933	0.500538	0.250269	+	0.968176i	$0.419481\pi$
$984$	−6.09617	−0.194339
$985$	−36.3448	−1.15804
$986$	−14.2052	−0.452385
$987$	−3.50251	−0.111486
$988$	26.6516	0.847899
$989$	−11.8885	−0.378031
$990$	−19.6651	−0.624999
$991$	12.4090	0.394186	0.197093	−	0.980385i	$-0.436850\pi$
$991$	12.4090	0.394186	0.197093	+	0.980385i	$0.436850\pi$
$992$	2.33391	0.0741016
$993$	1.49111	0.0473189
$994$	5.75369	0.182496
$995$	−11.2638	−0.357087
$996$	10.7755	0.341434
$997$	−53.2182	−1.68544	−0.842719	−	0.538354i	$-0.819046\pi$
$997$	−53.2182	−1.68544	−0.842719	+	0.538354i	$0.819046\pi$
$998$	25.5181	0.807760
$999$	13.8708	0.438853

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.47	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.47	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.839268 q^{3} +1.00000 q^{4} -4.35272 q^{5} +0.839268 q^{6} +1.00705 q^{7} +1.00000 q^{8} -2.29563 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.839268 q^{3} +1.00000 q^{4} -4.35272 q^{5} +0.839268 q^{6} +1.00705 q^{7} +1.00000 q^{8} -2.29563 q^{9} -4.35272 q^{10} -1.96804 q^{11} +0.839268 q^{12} +5.57768 q^{13} +1.00705 q^{14} -3.65309 q^{15} +1.00000 q^{16} -2.49817 q^{17} -2.29563 q^{18} +4.77825 q^{19} -4.35272 q^{20} +0.845183 q^{21} -1.96804 q^{22} -4.66373 q^{23} +0.839268 q^{24} +13.9461 q^{25} +5.57768 q^{26} -4.44445 q^{27} +1.00705 q^{28} +5.68625 q^{29} -3.65309 q^{30} +2.33391 q^{31} +1.00000 q^{32} -1.65172 q^{33} -2.49817 q^{34} -4.38340 q^{35} -2.29563 q^{36} -3.12092 q^{37} +4.77825 q^{38} +4.68117 q^{39} -4.35272 q^{40} -7.26367 q^{41} +0.845183 q^{42} +2.54913 q^{43} -1.96804 q^{44} +9.99222 q^{45} -4.66373 q^{46} -4.14408 q^{47} +0.839268 q^{48} -5.98585 q^{49} +13.9461 q^{50} -2.09663 q^{51} +5.57768 q^{52} +0.781697 q^{53} -4.44445 q^{54} +8.56634 q^{55} +1.00705 q^{56} +4.01023 q^{57} +5.68625 q^{58} -6.13762 q^{59} -3.65309 q^{60} -5.77896 q^{61} +2.33391 q^{62} -2.31181 q^{63} +1.00000 q^{64} -24.2781 q^{65} -1.65172 q^{66} -2.65942 q^{67} -2.49817 q^{68} -3.91412 q^{69} -4.38340 q^{70} +5.71342 q^{71} -2.29563 q^{72} +5.86712 q^{73} -3.12092 q^{74} +11.7045 q^{75} +4.77825 q^{76} -1.98192 q^{77} +4.68117 q^{78} -7.46831 q^{79} -4.35272 q^{80} +3.15681 q^{81} -7.26367 q^{82} +12.8391 q^{83} +0.845183 q^{84} +10.8738 q^{85} +2.54913 q^{86} +4.77228 q^{87} -1.96804 q^{88} -4.21114 q^{89} +9.99222 q^{90} +5.61700 q^{91} -4.66373 q^{92} +1.95877 q^{93} -4.14408 q^{94} -20.7984 q^{95} +0.839268 q^{96} +1.11834 q^{97} -5.98585 q^{98} +4.51790 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 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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