LMFDB - Embedded newform 8002.2.a.d.1.31

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			1.31
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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.781162 q^{3} +1.00000 q^{4} +3.65427 q^{5} -0.781162 q^{6} -1.77653 q^{7} +1.00000 q^{8} -2.38979 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.781162 q^{3} +1.00000 q^{4} +3.65427 q^{5} -0.781162 q^{6} -1.77653 q^{7} +1.00000 q^{8} -2.38979 q^{9} +3.65427 q^{10} +0.0295158 q^{11} -0.781162 q^{12} -5.14709 q^{13} -1.77653 q^{14} -2.85457 q^{15} +1.00000 q^{16} +1.42942 q^{17} -2.38979 q^{18} -5.54548 q^{19} +3.65427 q^{20} +1.38776 q^{21} +0.0295158 q^{22} +4.75025 q^{23} -0.781162 q^{24} +8.35366 q^{25} -5.14709 q^{26} +4.21030 q^{27} -1.77653 q^{28} -7.72500 q^{29} -2.85457 q^{30} +1.19995 q^{31} +1.00000 q^{32} -0.0230567 q^{33} +1.42942 q^{34} -6.49192 q^{35} -2.38979 q^{36} +9.02149 q^{37} -5.54548 q^{38} +4.02071 q^{39} +3.65427 q^{40} -3.96596 q^{41} +1.38776 q^{42} -2.00786 q^{43} +0.0295158 q^{44} -8.73292 q^{45} +4.75025 q^{46} -0.629273 q^{47} -0.781162 q^{48} -3.84393 q^{49} +8.35366 q^{50} -1.11661 q^{51} -5.14709 q^{52} +7.35976 q^{53} +4.21030 q^{54} +0.107859 q^{55} -1.77653 q^{56} +4.33192 q^{57} -7.72500 q^{58} -1.87935 q^{59} -2.85457 q^{60} -0.913567 q^{61} +1.19995 q^{62} +4.24553 q^{63} +1.00000 q^{64} -18.8088 q^{65} -0.0230567 q^{66} +7.41328 q^{67} +1.42942 q^{68} -3.71071 q^{69} -6.49192 q^{70} +1.73099 q^{71} -2.38979 q^{72} +0.281521 q^{73} +9.02149 q^{74} -6.52556 q^{75} -5.54548 q^{76} -0.0524359 q^{77} +4.02071 q^{78} -12.2558 q^{79} +3.65427 q^{80} +3.88044 q^{81} -3.96596 q^{82} -9.43188 q^{83} +1.38776 q^{84} +5.22349 q^{85} -2.00786 q^{86} +6.03448 q^{87} +0.0295158 q^{88} -8.60631 q^{89} -8.73292 q^{90} +9.14397 q^{91} +4.75025 q^{92} -0.937359 q^{93} -0.629273 q^{94} -20.2647 q^{95} -0.781162 q^{96} -14.8521 q^{97} -3.84393 q^{98} -0.0705366 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.781162	−0.451004	−0.225502	−	0.974243i	$-0.572402\pi$
$3$	−0.781162	−0.451004	−0.225502	+	0.974243i	$0.572402\pi$
$4$	1.00000	0.500000
$5$	3.65427	1.63424	0.817119	−	0.576469i	$-0.195570\pi$
$5$	3.65427	1.63424	0.817119	+	0.576469i	$0.195570\pi$
$6$	−0.781162	−0.318908
$7$	−1.77653	−0.671466	−0.335733	−	0.941957i	$-0.608984\pi$
$7$	−1.77653	−0.671466	−0.335733	+	0.941957i	$0.608984\pi$
$8$	1.00000	0.353553
$9$	−2.38979	−0.796595
$10$	3.65427	1.15558
$11$	0.0295158	0.00889936	0.00444968	−	0.999990i	$-0.498584\pi$
$11$	0.0295158	0.00889936	0.00444968	+	0.999990i	$0.498584\pi$
$12$	−0.781162	−0.225502
$13$	−5.14709	−1.42754	−0.713772	−	0.700378i	$-0.753015\pi$
$13$	−5.14709	−1.42754	−0.713772	+	0.700378i	$0.753015\pi$
$14$	−1.77653	−0.474798
$15$	−2.85457	−0.737048
$16$	1.00000	0.250000
$17$	1.42942	0.346686	0.173343	−	0.984862i	$-0.444543\pi$
$17$	1.42942	0.346686	0.173343	+	0.984862i	$0.444543\pi$
$18$	−2.38979	−0.563278
$19$	−5.54548	−1.27222	−0.636110	−	0.771598i	$-0.719458\pi$
$19$	−5.54548	−1.27222	−0.636110	+	0.771598i	$0.719458\pi$
$20$	3.65427	0.817119
$21$	1.38776	0.302834
$22$	0.0295158	0.00629280
$23$	4.75025	0.990495	0.495248	−	0.868752i	$-0.335077\pi$
$23$	4.75025	0.990495	0.495248	+	0.868752i	$0.335077\pi$
$24$	−0.781162	−0.159454
$25$	8.35366	1.67073
$26$	−5.14709	−1.00943
$27$	4.21030	0.810272
$28$	−1.77653	−0.335733
$29$	−7.72500	−1.43450	−0.717249	−	0.696817i	$-0.754599\pi$
$29$	−7.72500	−1.43450	−0.717249	+	0.696817i	$0.754599\pi$
$30$	−2.85457	−0.521171
$31$	1.19995	0.215518	0.107759	−	0.994177i	$-0.465632\pi$
$31$	1.19995	0.215518	0.107759	+	0.994177i	$0.465632\pi$
$32$	1.00000	0.176777
$33$	−0.0230567	−0.00401365
$34$	1.42942	0.245144
$35$	−6.49192	−1.09734
$36$	−2.38979	−0.398298
$37$	9.02149	1.48312	0.741562	−	0.670885i	$-0.234085\pi$
$37$	9.02149	1.48312	0.741562	+	0.670885i	$0.234085\pi$
$38$	−5.54548	−0.899595
$39$	4.02071	0.643829
$40$	3.65427	0.577790
$41$	−3.96596	−0.619379	−0.309689	−	0.950838i	$-0.600225\pi$
$41$	−3.96596	−0.619379	−0.309689	+	0.950838i	$0.600225\pi$
$42$	1.38776	0.214136
$43$	−2.00786	−0.306196	−0.153098	−	0.988211i	$-0.548925\pi$
$43$	−2.00786	−0.306196	−0.153098	+	0.988211i	$0.548925\pi$
$44$	0.0295158	0.00444968
$45$	−8.73292	−1.30183
$46$	4.75025	0.700386
$47$	−0.629273	−0.0917889	−0.0458944	−	0.998946i	$-0.514614\pi$
$47$	−0.629273	−0.0917889	−0.0458944	+	0.998946i	$0.514614\pi$
$48$	−0.781162	−0.112751
$49$	−3.84393	−0.549133
$50$	8.35366	1.18139
$51$	−1.11661	−0.156357
$52$	−5.14709	−0.713772
$53$	7.35976	1.01094	0.505471	−	0.862844i	$-0.331319\pi$
$53$	7.35976	1.01094	0.505471	+	0.862844i	$0.331319\pi$
$54$	4.21030	0.572949
$55$	0.107859	0.0145437
$56$	−1.77653	−0.237399
$57$	4.33192	0.573776
$58$	−7.72500	−1.01434
$59$	−1.87935	−0.244671	−0.122335	−	0.992489i	$-0.539038\pi$
$59$	−1.87935	−0.244671	−0.122335	+	0.992489i	$0.539038\pi$
$60$	−2.85457	−0.368524
$61$	−0.913567	−0.116970	−0.0584851	−	0.998288i	$-0.518627\pi$
$61$	−0.913567	−0.116970	−0.0584851	+	0.998288i	$0.518627\pi$
$62$	1.19995	0.152394
$63$	4.24553	0.534887
$64$	1.00000	0.125000
$65$	−18.8088	−2.33295
$66$	−0.0230567	−0.00283808
$67$	7.41328	0.905676	0.452838	−	0.891593i	$-0.350412\pi$
$67$	7.41328	0.905676	0.452838	+	0.891593i	$0.350412\pi$
$68$	1.42942	0.173343
$69$	−3.71071	−0.446717
$70$	−6.49192	−0.775933
$71$	1.73099	0.205431	0.102715	−	0.994711i	$-0.467247\pi$
$71$	1.73099	0.205431	0.102715	+	0.994711i	$0.467247\pi$
$72$	−2.38979	−0.281639
$73$	0.281521	0.0329496	0.0164748	−	0.999864i	$-0.494756\pi$
$73$	0.281521	0.0329496	0.0164748	+	0.999864i	$0.494756\pi$
$74$	9.02149	1.04873
$75$	−6.52556	−0.753507
$76$	−5.54548	−0.636110
$77$	−0.0524359	−0.00597562
$78$	4.02071	0.455256
$79$	−12.2558	−1.37888	−0.689440	−	0.724343i	$-0.742143\pi$
$79$	−12.2558	−1.37888	−0.689440	+	0.724343i	$0.742143\pi$
$80$	3.65427	0.408559
$81$	3.88044	0.431160
$82$	−3.96596	−0.437967
$83$	−9.43188	−1.03528	−0.517642	−	0.855597i	$-0.673190\pi$
$83$	−9.43188	−1.03528	−0.517642	+	0.855597i	$0.673190\pi$
$84$	1.38776	0.151417
$85$	5.22349	0.566568
$86$	−2.00786	−0.216513
$87$	6.03448	0.646964
$88$	0.0295158	0.00314640
$89$	−8.60631	−0.912267	−0.456133	−	0.889911i	$-0.650766\pi$
$89$	−8.60631	−0.912267	−0.456133	+	0.889911i	$0.650766\pi$
$90$	−8.73292	−0.920530
$91$	9.14397	0.958548
$92$	4.75025	0.495248
$93$	−0.937359	−0.0971996
$94$	−0.629273	−0.0649045
$95$	−20.2647	−2.07911
$96$	−0.781162	−0.0797270
$97$	−14.8521	−1.50800	−0.754000	−	0.656874i	$-0.771878\pi$
$97$	−14.8521	−1.50800	−0.754000	+	0.656874i	$0.771878\pi$
$98$	−3.84393	−0.388296
$99$	−0.0705366	−0.00708919
$100$	8.35366	0.835366
$101$	−18.6214	−1.85290	−0.926451	−	0.376414i	$-0.877157\pi$
$101$	−18.6214	−1.85290	−0.926451	+	0.376414i	$0.877157\pi$
$102$	−1.11661	−0.110561
$103$	−10.3886	−1.02362	−0.511809	−	0.859099i	$-0.671025\pi$
$103$	−10.3886	−1.02362	−0.511809	+	0.859099i	$0.671025\pi$
$104$	−5.14709	−0.504713
$105$	5.07124	0.494903
$106$	7.35976	0.714843
$107$	7.94048	0.767636	0.383818	−	0.923409i	$-0.374609\pi$
$107$	7.94048	0.767636	0.383818	+	0.923409i	$0.374609\pi$
$108$	4.21030	0.405136
$109$	−16.2432	−1.55581	−0.777906	−	0.628380i	$-0.783718\pi$
$109$	−16.2432	−1.55581	−0.777906	+	0.628380i	$0.783718\pi$
$110$	0.107859	0.0102839
$111$	−7.04724	−0.668895
$112$	−1.77653	−0.167867
$113$	−13.6945	−1.28827	−0.644134	−	0.764912i	$-0.722782\pi$
$113$	−13.6945	−1.28827	−0.644134	+	0.764912i	$0.722782\pi$
$114$	4.33192	0.405721
$115$	17.3587	1.61870
$116$	−7.72500	−0.717249
$117$	12.3004	1.13718
$118$	−1.87935	−0.173008
$119$	−2.53942	−0.232788
$120$	−2.85457	−0.260586
$121$	−10.9991	−0.999921
$122$	−0.913567	−0.0827105
$123$	3.09806	0.279342
$124$	1.19995	0.107759
$125$	12.2552	1.09614
$126$	4.24553	0.378222
$127$	14.7212	1.30630	0.653148	−	0.757231i	$-0.273448\pi$
$127$	14.7212	1.30630	0.653148	+	0.757231i	$0.273448\pi$
$128$	1.00000	0.0883883
$129$	1.56846	0.138095
$130$	−18.8088	−1.64964
$131$	−12.3898	−1.08250	−0.541252	−	0.840860i	$-0.682050\pi$
$131$	−12.3898	−1.08250	−0.541252	+	0.840860i	$0.682050\pi$
$132$	−0.0230567	−0.00200682
$133$	9.85172	0.854253
$134$	7.41328	0.640410
$135$	15.3855	1.32418
$136$	1.42942	0.122572
$137$	12.6668	1.08220	0.541100	−	0.840958i	$-0.318008\pi$
$137$	12.6668	1.08220	0.541100	+	0.840958i	$0.318008\pi$
$138$	−3.71071	−0.315877
$139$	−15.5451	−1.31852	−0.659258	−	0.751917i	$-0.729130\pi$
$139$	−15.5451	−1.31852	−0.659258	+	0.751917i	$0.729130\pi$
$140$	−6.49192	−0.548668
$141$	0.491564	0.0413971
$142$	1.73099	0.145262
$143$	−0.151921	−0.0127042
$144$	−2.38979	−0.199149
$145$	−28.2292	−2.34431
$146$	0.281521	0.0232989
$147$	3.00273	0.247661
$148$	9.02149	0.741562
$149$	−6.03452	−0.494368	−0.247184	−	0.968969i	$-0.579505\pi$
$149$	−6.03452	−0.494368	−0.247184	+	0.968969i	$0.579505\pi$
$150$	−6.52556	−0.532810
$151$	−11.1295	−0.905708	−0.452854	−	0.891585i	$-0.649594\pi$
$151$	−11.1295	−0.905708	−0.452854	+	0.891585i	$0.649594\pi$
$152$	−5.54548	−0.449798
$153$	−3.41602	−0.276169
$154$	−0.0524359	−0.00422540
$155$	4.38496	0.352208
$156$	4.02071	0.321914
$157$	−1.99723	−0.159396	−0.0796980	−	0.996819i	$-0.525396\pi$
$157$	−1.99723	−0.159396	−0.0796980	+	0.996819i	$0.525396\pi$
$158$	−12.2558	−0.975015
$159$	−5.74917	−0.455939
$160$	3.65427	0.288895
$161$	−8.43897	−0.665084
$162$	3.88044	0.304876
$163$	−10.4653	−0.819706	−0.409853	−	0.912151i	$-0.634420\pi$
$163$	−10.4653	−0.819706	−0.409853	+	0.912151i	$0.634420\pi$
$164$	−3.96596	−0.309689
$165$	−0.0842552	−0.00655925
$166$	−9.43188	−0.732056
$167$	−6.55201	−0.507010	−0.253505	−	0.967334i	$-0.581583\pi$
$167$	−6.55201	−0.507010	−0.253505	+	0.967334i	$0.581583\pi$
$168$	1.38776	0.107068
$169$	13.4925	1.03788
$170$	5.22349	0.400624
$171$	13.2525	1.01344
$172$	−2.00786	−0.153098
$173$	−12.4545	−0.946902	−0.473451	−	0.880820i	$-0.656992\pi$
$173$	−12.4545	−0.946902	−0.473451	+	0.880820i	$0.656992\pi$
$174$	6.03448	0.457473
$175$	−14.8406	−1.12184
$176$	0.0295158	0.00222484
$177$	1.46808	0.110347
$178$	−8.60631	−0.645070
$179$	1.22439	0.0915149	0.0457574	−	0.998953i	$-0.485430\pi$
$179$	1.22439	0.0915149	0.0457574	+	0.998953i	$0.485430\pi$
$180$	−8.73292	−0.650913
$181$	−12.6863	−0.942967	−0.471484	−	0.881875i	$-0.656281\pi$
$181$	−12.6863	−0.942967	−0.471484	+	0.881875i	$0.656281\pi$
$182$	9.14397	0.677796
$183$	0.713644	0.0527540
$184$	4.75025	0.350193
$185$	32.9669	2.42378
$186$	−0.937359	−0.0687305
$187$	0.0421906	0.00308529
$188$	−0.629273	−0.0458944
$189$	−7.47973	−0.544070
$190$	−20.2647	−1.47015
$191$	9.36365	0.677530	0.338765	−	0.940871i	$-0.389991\pi$
$191$	9.36365	0.677530	0.338765	+	0.940871i	$0.389991\pi$
$192$	−0.781162	−0.0563755
$193$	13.5319	0.974050	0.487025	−	0.873388i	$-0.338082\pi$
$193$	13.5319	0.974050	0.487025	+	0.873388i	$0.338082\pi$
$194$	−14.8521	−1.06632
$195$	14.6927	1.05217
$196$	−3.84393	−0.274567
$197$	−1.17913	−0.0840098	−0.0420049	−	0.999117i	$-0.513375\pi$
$197$	−1.17913	−0.0840098	−0.0420049	+	0.999117i	$0.513375\pi$
$198$	−0.0705366	−0.00501281
$199$	−17.3093	−1.22703	−0.613513	−	0.789685i	$-0.710244\pi$
$199$	−17.3093	−1.22703	−0.613513	+	0.789685i	$0.710244\pi$
$200$	8.35366	0.590693
$201$	−5.79097	−0.408464
$202$	−18.6214	−1.31020
$203$	13.7237	0.963216
$204$	−1.11661	−0.0781784
$205$	−14.4927	−1.01221
$206$	−10.3886	−0.723807
$207$	−11.3521	−0.789024
$208$	−5.14709	−0.356886
$209$	−0.163679	−0.0113219
$210$	5.07124	0.349949
$211$	16.1299	1.11043	0.555215	−	0.831707i	$-0.312636\pi$
$211$	16.1299	1.11043	0.555215	+	0.831707i	$0.312636\pi$
$212$	7.35976	0.505471
$213$	−1.35218	−0.0926501
$214$	7.94048	0.542800
$215$	−7.33725	−0.500397
$216$	4.21030	0.286474
$217$	−2.13176	−0.144713
$218$	−16.2432	−1.10013
$219$	−0.219914	−0.0148604
$220$	0.107859	0.00727184
$221$	−7.35737	−0.494910
$222$	−7.04724	−0.472980
$223$	16.8915	1.13114	0.565570	−	0.824700i	$-0.308656\pi$
$223$	16.8915	1.13114	0.565570	+	0.824700i	$0.308656\pi$
$224$	−1.77653	−0.118700
$225$	−19.9635	−1.33090
$226$	−13.6945	−0.910944
$227$	17.4080	1.15541	0.577704	−	0.816246i	$-0.303949\pi$
$227$	17.4080	1.15541	0.577704	+	0.816246i	$0.303949\pi$
$228$	4.33192	0.286888
$229$	−8.91162	−0.588896	−0.294448	−	0.955667i	$-0.595136\pi$
$229$	−8.91162	−0.588896	−0.294448	+	0.955667i	$0.595136\pi$
$230$	17.3587	1.14460
$231$	0.0409609	0.00269503
$232$	−7.72500	−0.507171
$233$	−16.7363	−1.09643	−0.548215	−	0.836337i	$-0.684693\pi$
$233$	−16.7363	−1.09643	−0.548215	+	0.836337i	$0.684693\pi$
$234$	12.3004	0.804105
$235$	−2.29953	−0.150005
$236$	−1.87935	−0.122335
$237$	9.57373	0.621880
$238$	−2.53942	−0.164606
$239$	9.15914	0.592455	0.296228	−	0.955117i	$-0.404271\pi$
$239$	9.15914	0.592455	0.296228	+	0.955117i	$0.404271\pi$
$240$	−2.85457	−0.184262
$241$	−13.9673	−0.899711	−0.449855	−	0.893101i	$-0.648524\pi$
$241$	−13.9673	−0.899711	−0.449855	+	0.893101i	$0.648524\pi$
$242$	−10.9991	−0.707051
$243$	−15.6621	−1.00473
$244$	−0.913567	−0.0584851
$245$	−14.0468	−0.897414
$246$	3.09806	0.197525
$247$	28.5431	1.81615
$248$	1.19995	0.0761972
$249$	7.36783	0.466917
$250$	12.2552	0.775086
$251$	14.3801	0.907665	0.453832	−	0.891087i	$-0.350057\pi$
$251$	14.3801	0.907665	0.453832	+	0.891087i	$0.350057\pi$
$252$	4.24553	0.267443
$253$	0.140208	0.00881478
$254$	14.7212	0.923690
$255$	−4.08040	−0.255524
$256$	1.00000	0.0625000
$257$	3.01301	0.187946	0.0939732	−	0.995575i	$-0.470043\pi$
$257$	3.01301	0.187946	0.0939732	+	0.995575i	$0.470043\pi$
$258$	1.56846	0.0976483
$259$	−16.0270	−0.995867
$260$	−18.8088	−1.16647
$261$	18.4611	1.14271
$262$	−12.3898	−0.765446
$263$	16.5344	1.01956	0.509779	−	0.860306i	$-0.329727\pi$
$263$	16.5344	1.01956	0.509779	+	0.860306i	$0.329727\pi$
$264$	−0.0230567	−0.00141904
$265$	26.8945	1.65212
$266$	9.85172	0.604048
$267$	6.72292	0.411436
$268$	7.41328	0.452838
$269$	19.1614	1.16829	0.584145	−	0.811649i	$-0.301430\pi$
$269$	19.1614	1.16829	0.584145	+	0.811649i	$0.301430\pi$
$270$	15.3855	0.936334
$271$	12.9344	0.785706	0.392853	−	0.919601i	$-0.371488\pi$
$271$	12.9344	0.785706	0.392853	+	0.919601i	$0.371488\pi$
$272$	1.42942	0.0866715
$273$	−7.14292	−0.432309
$274$	12.6668	0.765231
$275$	0.246565	0.0148685
$276$	−3.71071	−0.223359
$277$	−7.48598	−0.449789	−0.224894	−	0.974383i	$-0.572204\pi$
$277$	−7.48598	−0.449789	−0.224894	+	0.974383i	$0.572204\pi$
$278$	−15.5451	−0.932332
$279$	−2.86764	−0.171681
$280$	−6.49192	−0.387967
$281$	15.6566	0.933996	0.466998	−	0.884258i	$-0.345335\pi$
$281$	15.6566	0.933996	0.466998	+	0.884258i	$0.345335\pi$
$282$	0.491564	0.0292722
$283$	−16.1260	−0.958588	−0.479294	−	0.877654i	$-0.659107\pi$
$283$	−16.1260	−0.958588	−0.479294	+	0.877654i	$0.659107\pi$
$284$	1.73099	0.102715
$285$	15.8300	0.937687
$286$	−0.151921	−0.00898325
$287$	7.04566	0.415892
$288$	−2.38979	−0.140819
$289$	−14.9567	−0.879809
$290$	−28.2292	−1.65768
$291$	11.6019	0.680114
$292$	0.281521	0.0164748
$293$	1.09082	0.0637262	0.0318631	−	0.999492i	$-0.489856\pi$
$293$	1.09082	0.0637262	0.0318631	+	0.999492i	$0.489856\pi$
$294$	3.00273	0.175123
$295$	−6.86765	−0.399850
$296$	9.02149	0.524363
$297$	0.124270	0.00721090
$298$	−6.03452	−0.349571
$299$	−24.4499	−1.41398
$300$	−6.52556	−0.376754
$301$	3.56703	0.205600
$302$	−11.1295	−0.640432
$303$	14.5464	0.835667
$304$	−5.54548	−0.318055
$305$	−3.33842	−0.191157
$306$	−3.41602	−0.195281
$307$	−6.73632	−0.384462	−0.192231	−	0.981350i	$-0.561572\pi$
$307$	−6.73632	−0.384462	−0.192231	+	0.981350i	$0.561572\pi$
$308$	−0.0524359	−0.00298781
$309$	8.11517	0.461656
$310$	4.38496	0.249049
$311$	−3.89996	−0.221146	−0.110573	−	0.993868i	$-0.535269\pi$
$311$	−3.89996	−0.221146	−0.110573	+	0.993868i	$0.535269\pi$
$312$	4.02071	0.227628
$313$	12.9421	0.731529	0.365765	−	0.930707i	$-0.380808\pi$
$313$	12.9421	0.731529	0.365765	+	0.930707i	$0.380808\pi$
$314$	−1.99723	−0.112710
$315$	15.5143	0.874132
$316$	−12.2558	−0.689440
$317$	−17.1481	−0.963131	−0.481566	−	0.876410i	$-0.659932\pi$
$317$	−17.1481	−0.963131	−0.481566	+	0.876410i	$0.659932\pi$
$318$	−5.74917	−0.322397
$319$	−0.228010	−0.0127661
$320$	3.65427	0.204280
$321$	−6.20280	−0.346207
$322$	−8.43897	−0.470285
$323$	−7.92684	−0.441061
$324$	3.88044	0.215580
$325$	−42.9970	−2.38505
$326$	−10.4653	−0.579620
$327$	12.6885	0.701678
$328$	−3.96596	−0.218983
$329$	1.11792	0.0616331
$330$	−0.0842552	−0.00463809
$331$	2.07597	0.114105	0.0570527	−	0.998371i	$-0.481830\pi$
$331$	2.07597	0.114105	0.0570527	+	0.998371i	$0.481830\pi$
$332$	−9.43188	−0.517642
$333$	−21.5594	−1.18145
$334$	−6.55201	−0.358510
$335$	27.0901	1.48009
$336$	1.38776	0.0757085
$337$	8.20823	0.447131	0.223565	−	0.974689i	$-0.428230\pi$
$337$	8.20823	0.447131	0.223565	+	0.974689i	$0.428230\pi$
$338$	13.4925	0.733895
$339$	10.6976	0.581014
$340$	5.22349	0.283284
$341$	0.0354177	0.00191798
$342$	13.2525	0.716614
$343$	19.2646	1.04019
$344$	−2.00786	−0.108257
$345$	−13.5599	−0.730042
$346$	−12.4545	−0.669561
$347$	26.0956	1.40088	0.700442	−	0.713710i	$-0.252986\pi$
$347$	26.0956	1.40088	0.700442	+	0.713710i	$0.252986\pi$
$348$	6.03448	0.323482
$349$	12.3285	0.659928	0.329964	−	0.943994i	$-0.392963\pi$
$349$	12.3285	0.659928	0.329964	+	0.943994i	$0.392963\pi$
$350$	−14.8406	−0.793261
$351$	−21.6708	−1.15670
$352$	0.0295158	0.00157320
$353$	35.4516	1.88690	0.943448	−	0.331520i	$-0.107561\pi$
$353$	35.4516	1.88690	0.943448	+	0.331520i	$0.107561\pi$
$354$	1.46808	0.0780275
$355$	6.32550	0.335723
$356$	−8.60631	−0.456133
$357$	1.98370	0.104988
$358$	1.22439	0.0647108
$359$	27.2635	1.43891	0.719456	−	0.694538i	$-0.244391\pi$
$359$	27.2635	1.43891	0.719456	+	0.694538i	$0.244391\pi$
$360$	−8.73292	−0.460265
$361$	11.7523	0.618544
$362$	−12.6863	−0.666779
$363$	8.59210	0.450968
$364$	9.14397	0.479274
$365$	1.02875	0.0538474
$366$	0.713644	0.0373027
$367$	6.61632	0.345369	0.172685	−	0.984977i	$-0.444756\pi$
$367$	6.61632	0.345369	0.172685	+	0.984977i	$0.444756\pi$
$368$	4.75025	0.247624
$369$	9.47779	0.493394
$370$	32.9669	1.71387
$371$	−13.0749	−0.678813
$372$	−0.937359	−0.0485998
$373$	6.32926	0.327717	0.163858	−	0.986484i	$-0.447606\pi$
$373$	6.32926	0.327717	0.163858	+	0.986484i	$0.447606\pi$
$374$	0.0421906	0.00218163
$375$	−9.57328	−0.494362
$376$	−0.629273	−0.0324523
$377$	39.7613	2.04781
$378$	−7.47973	−0.384716
$379$	26.2938	1.35062	0.675311	−	0.737533i	$-0.264010\pi$
$379$	26.2938	1.35062	0.675311	+	0.737533i	$0.264010\pi$
$380$	−20.2647	−1.03955
$381$	−11.4996	−0.589144
$382$	9.36365	0.479086
$383$	0.311482	0.0159160	0.00795799	−	0.999968i	$-0.497467\pi$
$383$	0.311482	0.0159160	0.00795799	+	0.999968i	$0.497467\pi$
$384$	−0.781162	−0.0398635
$385$	−0.191615	−0.00976558
$386$	13.5319	0.688757
$387$	4.79835	0.243914
$388$	−14.8521	−0.754000
$389$	15.6920	0.795617	0.397808	−	0.917469i	$-0.369771\pi$
$389$	15.6920	0.795617	0.397808	+	0.917469i	$0.369771\pi$
$390$	14.6927	0.743996
$391$	6.79012	0.343391
$392$	−3.84393	−0.194148
$393$	9.67846	0.488214
$394$	−1.17913	−0.0594039
$395$	−44.7858	−2.25342
$396$	−0.0705366	−0.00354460
$397$	−8.85195	−0.444267	−0.222133	−	0.975016i	$-0.571302\pi$
$397$	−8.85195	−0.444267	−0.222133	+	0.975016i	$0.571302\pi$
$398$	−17.3093	−0.867639
$399$	−7.69579	−0.385271
$400$	8.35366	0.417683
$401$	27.2693	1.36177	0.680883	−	0.732392i	$-0.261596\pi$
$401$	27.2693	1.36177	0.680883	+	0.732392i	$0.261596\pi$
$402$	−5.79097	−0.288827
$403$	−6.17627	−0.307662
$404$	−18.6214	−0.926451
$405$	14.1801	0.704617
$406$	13.7237	0.681097
$407$	0.266277	0.0131989
$408$	−1.11661	−0.0552805
$409$	12.3589	0.611108	0.305554	−	0.952175i	$-0.401158\pi$
$409$	12.3589	0.611108	0.305554	+	0.952175i	$0.401158\pi$
$410$	−14.4927	−0.715742
$411$	−9.89484	−0.488077
$412$	−10.3886	−0.511809
$413$	3.33873	0.164288
$414$	−11.3521	−0.557924
$415$	−34.4666	−1.69190
$416$	−5.14709	−0.252357
$417$	12.1432	0.594656
$418$	−0.163679	−0.00800582
$419$	−5.94846	−0.290601	−0.145301	−	0.989388i	$-0.546415\pi$
$419$	−5.94846	−0.290601	−0.145301	+	0.989388i	$0.546415\pi$
$420$	5.07124	0.247451
$421$	−38.4741	−1.87511	−0.937557	−	0.347831i	$-0.886918\pi$
$421$	−38.4741	−1.87511	−0.937557	+	0.347831i	$0.886918\pi$
$422$	16.1299	0.785193
$423$	1.50383	0.0731186
$424$	7.35976	0.357422
$425$	11.9409	0.579220
$426$	−1.35218	−0.0655135
$427$	1.62298	0.0785416
$428$	7.94048	0.383818
$429$	0.118675	0.00572966
$430$	−7.33725	−0.353834
$431$	19.1214	0.921046	0.460523	−	0.887648i	$-0.347662\pi$
$431$	19.1214	0.921046	0.460523	+	0.887648i	$0.347662\pi$
$432$	4.21030	0.202568
$433$	−11.0669	−0.531841	−0.265920	−	0.963995i	$-0.585676\pi$
$433$	−11.0669	−0.531841	−0.265920	+	0.963995i	$0.585676\pi$
$434$	−2.13176	−0.102328
$435$	22.0516	1.05729
$436$	−16.2432	−0.777906
$437$	−26.3424	−1.26013
$438$	−0.219914	−0.0105079
$439$	6.47554	0.309061	0.154530	−	0.987988i	$-0.450614\pi$
$439$	6.47554	0.309061	0.154530	+	0.987988i	$0.450614\pi$
$440$	0.107859	0.00514196
$441$	9.18618	0.437437
$442$	−7.35737	−0.349954
$443$	−21.2290	−1.00862	−0.504309	−	0.863523i	$-0.668253\pi$
$443$	−21.2290	−1.00862	−0.504309	+	0.863523i	$0.668253\pi$
$444$	−7.04724	−0.334447
$445$	−31.4497	−1.49086
$446$	16.8915	0.799837
$447$	4.71394	0.222962
$448$	−1.77653	−0.0839333
$449$	−0.740595	−0.0349508	−0.0174754	−	0.999847i	$-0.505563\pi$
$449$	−0.740595	−0.0349508	−0.0174754	+	0.999847i	$0.505563\pi$
$450$	−19.9635	−0.941087
$451$	−0.117059	−0.00551208
$452$	−13.6945	−0.644134
$453$	8.69396	0.408478
$454$	17.4080	0.816997
$455$	33.4145	1.56650
$456$	4.33192	0.202861
$457$	−19.0483	−0.891040	−0.445520	−	0.895272i	$-0.646981\pi$
$457$	−19.0483	−0.891040	−0.445520	+	0.895272i	$0.646981\pi$
$458$	−8.91162	−0.416413
$459$	6.01830	0.280910
$460$	17.3587	0.809352
$461$	35.3130	1.64469	0.822344	−	0.568990i	$-0.192666\pi$
$461$	35.3130	1.64469	0.822344	+	0.568990i	$0.192666\pi$
$462$	0.0409609	0.00190567
$463$	−29.8924	−1.38922	−0.694610	−	0.719387i	$-0.744423\pi$
$463$	−29.8924	−1.38922	−0.694610	+	0.719387i	$0.744423\pi$
$464$	−7.72500	−0.358624
$465$	−3.42536	−0.158847
$466$	−16.7363	−0.775294
$467$	29.7723	1.37770	0.688848	−	0.724906i	$-0.258117\pi$
$467$	29.7723	1.37770	0.688848	+	0.724906i	$0.258117\pi$
$468$	12.3004	0.568588
$469$	−13.1699	−0.608131
$470$	−2.29953	−0.106069
$471$	1.56016	0.0718882
$472$	−1.87935	−0.0865042
$473$	−0.0592637	−0.00272495
$474$	9.57373	0.439736
$475$	−46.3251	−2.12554
$476$	−2.53942	−0.116394
$477$	−17.5883	−0.805311
$478$	9.15914	0.418929
$479$	−30.7545	−1.40521	−0.702603	−	0.711582i	$-0.747979\pi$
$479$	−30.7545	−1.40521	−0.702603	+	0.711582i	$0.747979\pi$
$480$	−2.85457	−0.130293
$481$	−46.4344	−2.11723
$482$	−13.9673	−0.636192
$483$	6.59220	0.299956
$484$	−10.9991	−0.499960
$485$	−54.2735	−2.46443
$486$	−15.6621	−0.710449
$487$	16.0368	0.726696	0.363348	−	0.931653i	$-0.381634\pi$
$487$	16.0368	0.726696	0.363348	+	0.931653i	$0.381634\pi$
$488$	−0.913567	−0.0413552
$489$	8.17510	0.369691
$490$	−14.0468	−0.634568
$491$	23.4201	1.05693	0.528466	−	0.848954i	$-0.322767\pi$
$491$	23.4201	1.05693	0.528466	+	0.848954i	$0.322767\pi$
$492$	3.09806	0.139671
$493$	−11.0423	−0.497320
$494$	28.5431	1.28421
$495$	−0.257759	−0.0115854
$496$	1.19995	0.0538796
$497$	−3.07516	−0.137940
$498$	7.36783	0.330160
$499$	23.4838	1.05128	0.525640	−	0.850707i	$-0.323826\pi$
$499$	23.4838	1.05128	0.525640	+	0.850707i	$0.323826\pi$
$500$	12.2552	0.548068
$501$	5.11818	0.228663
$502$	14.3801	0.641816
$503$	−22.6235	−1.00873	−0.504366	−	0.863490i	$-0.668274\pi$
$503$	−22.6235	−1.00873	−0.504366	+	0.863490i	$0.668274\pi$
$504$	4.24553	0.189111
$505$	−68.0477	−3.02808
$506$	0.140208	0.00623299
$507$	−10.5398	−0.468090
$508$	14.7212	0.653148
$509$	0.967341	0.0428766	0.0214383	−	0.999770i	$-0.493175\pi$
$509$	0.967341	0.0428766	0.0214383	+	0.999770i	$0.493175\pi$
$510$	−4.08040	−0.180683
$511$	−0.500131	−0.0221245
$512$	1.00000	0.0441942
$513$	−23.3481	−1.03084
$514$	3.01301	0.132898
$515$	−37.9627	−1.67284
$516$	1.56846	0.0690477
$517$	−0.0185735	−0.000816862 0
$518$	−16.0270	−0.704185
$519$	9.72902	0.427056
$520$	−18.8088	−0.824822
$521$	−4.47250	−0.195944	−0.0979719	−	0.995189i	$-0.531236\pi$
$521$	−4.47250	−0.195944	−0.0979719	+	0.995189i	$0.531236\pi$
$522$	18.4611	0.808021
$523$	−27.6909	−1.21084	−0.605420	−	0.795907i	$-0.706995\pi$
$523$	−27.6909	−1.21084	−0.605420	+	0.795907i	$0.706995\pi$
$524$	−12.3898	−0.541252
$525$	11.5929	0.505955
$526$	16.5344	0.720936
$527$	1.71524	0.0747172
$528$	−0.0230567	−0.00100341
$529$	−0.435134	−0.0189189
$530$	26.8945	1.16822
$531$	4.49125	0.194904
$532$	9.85172	0.427126
$533$	20.4131	0.884191
$534$	6.72292	0.290929
$535$	29.0166	1.25450
$536$	7.41328	0.320205
$537$	−0.956443	−0.0412736
$538$	19.1614	0.826106
$539$	−0.113457	−0.00488694
$540$	15.3855	0.662088
$541$	−7.92338	−0.340653	−0.170326	−	0.985388i	$-0.554482\pi$
$541$	−7.92338	−0.340653	−0.170326	+	0.985388i	$0.554482\pi$
$542$	12.9344	0.555578
$543$	9.91008	0.425282
$544$	1.42942	0.0612860
$545$	−59.3568	−2.54257
$546$	−7.14292	−0.305689
$547$	−8.44443	−0.361058	−0.180529	−	0.983570i	$-0.557781\pi$
$547$	−8.44443	−0.361058	−0.180529	+	0.983570i	$0.557781\pi$
$548$	12.6668	0.541100
$549$	2.18323	0.0931780
$550$	0.246565	0.0105136
$551$	42.8388	1.82500
$552$	−3.71071	−0.157938
$553$	21.7727	0.925871
$554$	−7.48598	−0.318049
$555$	−25.7525	−1.09313
$556$	−15.5451	−0.659258
$557$	5.30332	0.224709	0.112355	−	0.993668i	$-0.464161\pi$
$557$	5.30332	0.224709	0.112355	+	0.993668i	$0.464161\pi$
$558$	−2.86764	−0.121397
$559$	10.3346	0.437108
$560$	−6.49192	−0.274334
$561$	−0.0329577	−0.00139148
$562$	15.6566	0.660435
$563$	3.07329	0.129524	0.0647619	−	0.997901i	$-0.479371\pi$
$563$	3.07329	0.129524	0.0647619	+	0.997901i	$0.479371\pi$
$564$	0.491564	0.0206986
$565$	−50.0433	−2.10534
$566$	−16.1260	−0.677824
$567$	−6.89372	−0.289509
$568$	1.73099	0.0726308
$569$	10.3227	0.432750	0.216375	−	0.976310i	$-0.430577\pi$
$569$	10.3227	0.432750	0.216375	+	0.976310i	$0.430577\pi$
$570$	15.8300	0.663045
$571$	−23.7796	−0.995144	−0.497572	−	0.867423i	$-0.665775\pi$
$571$	−23.7796	−0.995144	−0.497572	+	0.867423i	$0.665775\pi$
$572$	−0.151921	−0.00635212
$573$	−7.31453	−0.305569
$574$	7.04566	0.294080
$575$	39.6820	1.65485
$576$	−2.38979	−0.0995744
$577$	−24.4826	−1.01922	−0.509612	−	0.860405i	$-0.670211\pi$
$577$	−24.4826	−1.01922	−0.509612	+	0.860405i	$0.670211\pi$
$578$	−14.9567	−0.622119
$579$	−10.5706	−0.439300
$580$	−28.2292	−1.17215
$581$	16.7560	0.695158
$582$	11.6019	0.480913
$583$	0.217230	0.00899673
$584$	0.281521	0.0116494
$585$	44.9491	1.85842
$586$	1.09082	0.0450612
$587$	−5.29284	−0.218459	−0.109229	−	0.994017i	$-0.534838\pi$
$587$	−5.29284	−0.218459	−0.109229	+	0.994017i	$0.534838\pi$
$588$	3.00273	0.123831
$589$	−6.65432	−0.274187
$590$	−6.86765	−0.282737
$591$	0.921095	0.0378888
$592$	9.02149	0.370781
$593$	5.78221	0.237447	0.118724	−	0.992927i	$-0.462120\pi$
$593$	5.78221	0.237447	0.118724	+	0.992927i	$0.462120\pi$
$594$	0.124270	0.00509888
$595$	−9.27971	−0.380431
$596$	−6.03452	−0.247184
$597$	13.5214	0.553394
$598$	−24.4499	−0.999833
$599$	−11.2470	−0.459542	−0.229771	−	0.973245i	$-0.573798\pi$
$599$	−11.2470	−0.459542	−0.229771	+	0.973245i	$0.573798\pi$
$600$	−6.52556	−0.266405
$601$	−15.7728	−0.643385	−0.321692	−	0.946844i	$-0.604252\pi$
$601$	−15.7728	−0.643385	−0.321692	+	0.946844i	$0.604252\pi$
$602$	3.56703	0.145381
$603$	−17.7162	−0.721458
$604$	−11.1295	−0.452854
$605$	−40.1937	−1.63411
$606$	14.5464	0.590906
$607$	14.9806	0.608042	0.304021	−	0.952665i	$-0.401671\pi$
$607$	14.9806	0.608042	0.304021	+	0.952665i	$0.401671\pi$
$608$	−5.54548	−0.224899
$609$	−10.7204	−0.434414
$610$	−3.33842	−0.135169
$611$	3.23892	0.131033
$612$	−3.41602	−0.138084
$613$	20.8925	0.843840	0.421920	−	0.906633i	$-0.361356\pi$
$613$	20.8925	0.843840	0.421920	+	0.906633i	$0.361356\pi$
$614$	−6.73632	−0.271856
$615$	11.3211	0.456512
$616$	−0.0524359	−0.00211270
$617$	47.0950	1.89597	0.947987	−	0.318308i	$-0.103115\pi$
$617$	47.0950	1.89597	0.947987	+	0.318308i	$0.103115\pi$
$618$	8.11517	0.326440
$619$	22.6093	0.908745	0.454373	−	0.890812i	$-0.349864\pi$
$619$	22.6093	0.908745	0.454373	+	0.890812i	$0.349864\pi$
$620$	4.38496	0.176104
$621$	20.0000	0.802570
$622$	−3.89996	−0.156374
$623$	15.2894	0.612556
$624$	4.02071	0.160957
$625$	3.01538	0.120615
$626$	12.9421	0.517269
$627$	0.127860	0.00510624
$628$	−1.99723	−0.0796980
$629$	12.8955	0.514178
$630$	15.5143	0.618105
$631$	−43.5514	−1.73376	−0.866878	−	0.498521i	$-0.833877\pi$
$631$	−43.5514	−1.73376	−0.866878	+	0.498521i	$0.833877\pi$
$632$	−12.2558	−0.487508
$633$	−12.6001	−0.500808
$634$	−17.1481	−0.681037
$635$	53.7952	2.13480
$636$	−5.74917	−0.227969
$637$	19.7851	0.783912
$638$	−0.228010	−0.00902700
$639$	−4.13670	−0.163645
$640$	3.65427	0.144448
$641$	8.87124	0.350393	0.175196	−	0.984534i	$-0.443944\pi$
$641$	8.87124	0.350393	0.175196	+	0.984534i	$0.443944\pi$
$642$	−6.20280	−0.244805
$643$	5.82993	0.229910	0.114955	−	0.993371i	$-0.463328\pi$
$643$	5.82993	0.229910	0.114955	+	0.993371i	$0.463328\pi$
$644$	−8.43897	−0.332542
$645$	5.73158	0.225681
$646$	−7.92684	−0.311877
$647$	−19.1993	−0.754802	−0.377401	−	0.926050i	$-0.623182\pi$
$647$	−19.1993	−0.754802	−0.377401	+	0.926050i	$0.623182\pi$
$648$	3.88044	0.152438
$649$	−0.0554706	−0.00217741
$650$	−42.9970	−1.68648
$651$	1.66525	0.0652662
$652$	−10.4653	−0.409853
$653$	−18.5491	−0.725884	−0.362942	−	0.931812i	$-0.618228\pi$
$653$	−18.5491	−0.725884	−0.362942	+	0.931812i	$0.618228\pi$
$654$	12.6885	0.496161
$655$	−45.2757	−1.76907
$656$	−3.96596	−0.154845
$657$	−0.672775	−0.0262475
$658$	1.11792	0.0435812
$659$	−24.9495	−0.971894	−0.485947	−	0.873988i	$-0.661525\pi$
$659$	−24.9495	−0.971894	−0.485947	+	0.873988i	$0.661525\pi$
$660$	−0.0842552	−0.00327963
$661$	4.24958	0.165289	0.0826447	−	0.996579i	$-0.473663\pi$
$661$	4.24958	0.165289	0.0826447	+	0.996579i	$0.473663\pi$
$662$	2.07597	0.0806847
$663$	5.74729	0.223206
$664$	−9.43188	−0.366028
$665$	36.0008	1.39605
$666$	−21.5594	−0.835411
$667$	−36.6957	−1.42086
$668$	−6.55201	−0.253505
$669$	−13.1950	−0.510149
$670$	27.0901	1.04658
$671$	−0.0269647	−0.00104096
$672$	1.38776	0.0535340
$673$	−50.6388	−1.95198	−0.975992	−	0.217807i	$-0.930110\pi$
$673$	−50.6388	−1.95198	−0.975992	+	0.217807i	$0.930110\pi$
$674$	8.20823	0.316169
$675$	35.1714	1.35375
$676$	13.4925	0.518942
$677$	6.22068	0.239080	0.119540	−	0.992829i	$-0.461858\pi$
$677$	6.22068	0.239080	0.119540	+	0.992829i	$0.461858\pi$
$678$	10.6976	0.410839
$679$	26.3852	1.01257
$680$	5.22349	0.200312
$681$	−13.5984	−0.521094
$682$	0.0354177	0.00135621
$683$	46.4271	1.77648	0.888242	−	0.459376i	$-0.151927\pi$
$683$	46.4271	1.77648	0.888242	+	0.459376i	$0.151927\pi$
$684$	13.2525	0.506722
$685$	46.2880	1.76857
$686$	19.2646	0.735526
$687$	6.96142	0.265595
$688$	−2.00786	−0.0765489
$689$	−37.8813	−1.44316
$690$	−13.5599	−0.516218
$691$	−42.1366	−1.60295	−0.801476	−	0.598027i	$-0.795951\pi$
$691$	−42.1366	−1.60295	−0.801476	+	0.598027i	$0.795951\pi$
$692$	−12.4545	−0.473451
$693$	0.125310	0.00476015
$694$	26.0956	0.990574
$695$	−56.8059	−2.15477
$696$	6.03448	0.228736
$697$	−5.66904	−0.214730
$698$	12.3285	0.466639
$699$	13.0738	0.494495
$700$	−14.8406	−0.560920
$701$	−18.7603	−0.708566	−0.354283	−	0.935138i	$-0.615275\pi$
$701$	−18.7603	−0.708566	−0.354283	+	0.935138i	$0.615275\pi$
$702$	−21.6708	−0.817910
$703$	−50.0285	−1.88686
$704$	0.0295158	0.00111242
$705$	1.79631	0.0676528
$706$	35.4516	1.33424
$707$	33.0816	1.24416
$708$	1.46808	0.0551737
$709$	−30.3910	−1.14136	−0.570679	−	0.821173i	$-0.693320\pi$
$709$	−30.3910	−1.14136	−0.570679	+	0.821173i	$0.693320\pi$
$710$	6.32550	0.237392
$711$	29.2886	1.09841
$712$	−8.60631	−0.322535
$713$	5.70008	0.213470
$714$	1.98370	0.0742380
$715$	−0.555158	−0.0207617
$716$	1.22439	0.0457574
$717$	−7.15477	−0.267200
$718$	27.2635	1.01746
$719$	10.2806	0.383402	0.191701	−	0.981453i	$-0.438600\pi$
$719$	10.2806	0.383402	0.191701	+	0.981453i	$0.438600\pi$
$720$	−8.73292	−0.325457
$721$	18.4557	0.687325
$722$	11.7523	0.437376
$723$	10.9107	0.405773
$724$	−12.6863	−0.471484
$725$	−64.5321	−2.39666
$726$	8.59210	0.318883
$727$	39.8581	1.47826	0.739128	−	0.673565i	$-0.235238\pi$
$727$	39.8581	1.47826	0.739128	+	0.673565i	$0.235238\pi$
$728$	9.14397	0.338898
$729$	0.593355	0.0219761
$730$	1.02875	0.0380759
$731$	−2.87008	−0.106154
$732$	0.713644	0.0263770
$733$	6.65657	0.245866	0.122933	−	0.992415i	$-0.460770\pi$
$733$	6.65657	0.245866	0.122933	+	0.992415i	$0.460770\pi$
$734$	6.61632	0.244213
$735$	10.9728	0.404737
$736$	4.75025	0.175097
$737$	0.218809	0.00805994
$738$	9.47779	0.348882
$739$	−7.70081	−0.283279	−0.141639	−	0.989918i	$-0.545237\pi$
$739$	−7.70081	−0.283279	−0.141639	+	0.989918i	$0.545237\pi$
$740$	32.9669	1.21189
$741$	−22.2967	−0.819092
$742$	−13.0749	−0.479993
$743$	30.3385	1.11301	0.556506	−	0.830844i	$-0.312142\pi$
$743$	30.3385	1.11301	0.556506	+	0.830844i	$0.312142\pi$
$744$	−0.937359	−0.0343653
$745$	−22.0518	−0.807914
$746$	6.32926	0.231731
$747$	22.5402	0.824702
$748$	0.0421906	0.00154264
$749$	−14.1065	−0.515441
$750$	−9.57328	−0.349567
$751$	31.9020	1.16412	0.582061	−	0.813145i	$-0.302247\pi$
$751$	31.9020	1.16412	0.582061	+	0.813145i	$0.302247\pi$
$752$	−0.629273	−0.0229472
$753$	−11.2332	−0.409360
$754$	39.7613	1.44802
$755$	−40.6703	−1.48014
$756$	−7.47973	−0.272035
$757$	13.9225	0.506023	0.253011	−	0.967463i	$-0.418579\pi$
$757$	13.9225	0.506023	0.253011	+	0.967463i	$0.418579\pi$
$758$	26.2938	0.955034
$759$	−0.109525	−0.00397550
$760$	−20.2647	−0.735076
$761$	−2.58147	−0.0935783	−0.0467891	−	0.998905i	$-0.514899\pi$
$761$	−2.58147	−0.0935783	−0.0467891	+	0.998905i	$0.514899\pi$
$762$	−11.4996	−0.416588
$763$	28.8565	1.04468
$764$	9.36365	0.338765
$765$	−12.4830	−0.451325
$766$	0.311482	0.0112543
$767$	9.67318	0.349278
$768$	−0.781162	−0.0281878
$769$	30.6593	1.10560	0.552801	−	0.833313i	$-0.313559\pi$
$769$	30.6593	1.10560	0.552801	+	0.833313i	$0.313559\pi$
$770$	−0.191615	−0.00690531
$771$	−2.35365	−0.0847646
$772$	13.5319	0.487025
$773$	−27.3439	−0.983493	−0.491746	−	0.870738i	$-0.663641\pi$
$773$	−27.3439	−0.983493	−0.491746	+	0.870738i	$0.663641\pi$
$774$	4.79835	0.172473
$775$	10.0240	0.360073
$776$	−14.8521	−0.533159
$777$	12.5197	0.449140
$778$	15.6920	0.562586
$779$	21.9931	0.787986
$780$	14.6927	0.526084
$781$	0.0510917	0.00182820
$782$	6.79012	0.242814
$783$	−32.5246	−1.16233
$784$	−3.84393	−0.137283
$785$	−7.29840	−0.260491
$786$	9.67846	0.345219
$787$	16.7244	0.596161	0.298080	−	0.954541i	$-0.403654\pi$
$787$	16.7244	0.596161	0.298080	+	0.954541i	$0.403654\pi$
$788$	−1.17913	−0.0420049
$789$	−12.9161	−0.459825
$790$	−44.7858	−1.59341
$791$	24.3287	0.865029
$792$	−0.0705366	−0.00250641
$793$	4.70221	0.166980
$794$	−8.85195	−0.314144
$795$	−21.0090	−0.745112
$796$	−17.3093	−0.613513
$797$	23.9519	0.848419	0.424209	−	0.905564i	$-0.360552\pi$
$797$	23.9519	0.848419	0.424209	+	0.905564i	$0.360552\pi$
$798$	−7.69579	−0.272428
$799$	−0.899497	−0.0318219
$800$	8.35366	0.295347
$801$	20.5672	0.726708
$802$	27.2693	0.962914
$803$	0.00830933	0.000293230 0
$804$	−5.79097	−0.204232
$805$	−30.8383	−1.08691
$806$	−6.17627	−0.217550
$807$	−14.9682	−0.526904
$808$	−18.6214	−0.655100
$809$	29.4380	1.03499	0.517493	−	0.855687i	$-0.326865\pi$
$809$	29.4380	1.03499	0.517493	+	0.855687i	$0.326865\pi$
$810$	14.1801	0.498240
$811$	−17.2149	−0.604497	−0.302248	−	0.953229i	$-0.597737\pi$
$811$	−17.2149	−0.604497	−0.302248	+	0.953229i	$0.597737\pi$
$812$	13.7237	0.481608
$813$	−10.1038	−0.354357
$814$	0.266277	0.00933300
$815$	−38.2430	−1.33960
$816$	−1.11661	−0.0390892
$817$	11.1345	0.389548
$818$	12.3589	0.432119
$819$	−21.8521	−0.763575
$820$	−14.4927	−0.506106
$821$	2.16570	0.0755834	0.0377917	−	0.999286i	$-0.487968\pi$
$821$	2.16570	0.0755834	0.0377917	+	0.999286i	$0.487968\pi$
$822$	−9.89484	−0.345122
$823$	39.9175	1.39144	0.695719	−	0.718314i	$-0.255086\pi$
$823$	39.9175	1.39144	0.695719	+	0.718314i	$0.255086\pi$
$824$	−10.3886	−0.361904
$825$	−0.192608	−0.00670573
$826$	3.33873	0.116169
$827$	48.2770	1.67875	0.839377	−	0.543549i	$-0.182920\pi$
$827$	48.2770	1.67875	0.839377	+	0.543549i	$0.182920\pi$
$828$	−11.3521	−0.394512
$829$	3.17508	0.110275	0.0551375	−	0.998479i	$-0.482440\pi$
$829$	3.17508	0.110275	0.0551375	+	0.998479i	$0.482440\pi$
$830$	−34.4666	−1.19635
$831$	5.84776	0.202857
$832$	−5.14709	−0.178443
$833$	−5.49461	−0.190377
$834$	12.1432	0.420485
$835$	−23.9428	−0.828574
$836$	−0.163679	−0.00566097
$837$	5.05216	0.174628
$838$	−5.94846	−0.205486
$839$	9.39172	0.324238	0.162119	−	0.986771i	$-0.448167\pi$
$839$	9.39172	0.324238	0.162119	+	0.986771i	$0.448167\pi$
$840$	5.07124	0.174974
$841$	30.6757	1.05778
$842$	−38.4741	−1.32591
$843$	−12.2304	−0.421236
$844$	16.1299	0.555215
$845$	49.3052	1.69615
$846$	1.50383	0.0517026
$847$	19.5403	0.671413
$848$	7.35976	0.252735
$849$	12.5970	0.432327
$850$	11.9409	0.409570
$851$	42.8543	1.46903
$852$	−1.35218	−0.0463251
$853$	12.5458	0.429562	0.214781	−	0.976662i	$-0.431096\pi$
$853$	12.5458	0.429562	0.214781	+	0.976662i	$0.431096\pi$
$854$	1.62298	0.0555373
$855$	48.4282	1.65621
$856$	7.94048	0.271400
$857$	−19.5710	−0.668532	−0.334266	−	0.942479i	$-0.608488\pi$
$857$	−19.5710	−0.668532	−0.334266	+	0.942479i	$0.608488\pi$
$858$	0.118675	0.00405148
$859$	−2.72103	−0.0928403	−0.0464202	−	0.998922i	$-0.514781\pi$
$859$	−2.72103	−0.0928403	−0.0464202	+	0.998922i	$0.514781\pi$
$860$	−7.33725	−0.250198
$861$	−5.50380	−0.187569
$862$	19.1214	0.651278
$863$	−18.4914	−0.629456	−0.314728	−	0.949182i	$-0.601913\pi$
$863$	−18.4914	−0.629456	−0.314728	+	0.949182i	$0.601913\pi$
$864$	4.21030	0.143237
$865$	−45.5122	−1.54746
$866$	−11.0669	−0.376068
$867$	11.6836	0.396797
$868$	−2.13176	−0.0723566
$869$	−0.361739	−0.0122712
$870$	22.0516	0.747619
$871$	−38.1568	−1.29289
$872$	−16.2432	−0.550063
$873$	35.4933	1.20127
$874$	−26.3424	−0.891045
$875$	−21.7717	−0.736019
$876$	−0.219914	−0.00743019
$877$	18.5682	0.627005	0.313503	−	0.949587i	$-0.398498\pi$
$877$	18.5682	0.627005	0.313503	+	0.949587i	$0.398498\pi$
$878$	6.47554	0.218539
$879$	−0.852105	−0.0287408
$880$	0.107859	0.00363592
$881$	−1.88917	−0.0636476	−0.0318238	−	0.999493i	$-0.510132\pi$
$881$	−1.88917	−0.0636476	−0.0318238	+	0.999493i	$0.510132\pi$
$882$	9.18618	0.309315
$883$	24.7543	0.833049	0.416525	−	0.909124i	$-0.363248\pi$
$883$	24.7543	0.833049	0.416525	+	0.909124i	$0.363248\pi$
$884$	−7.35737	−0.247455
$885$	5.36475	0.180334
$886$	−21.2290	−0.713201
$887$	−52.8199	−1.77352	−0.886759	−	0.462232i	$-0.847049\pi$
$887$	−52.8199	−1.77352	−0.886759	+	0.462232i	$0.847049\pi$
$888$	−7.04724	−0.236490
$889$	−26.1527	−0.877133
$890$	−31.4497	−1.05420
$891$	0.114534	0.00383704
$892$	16.8915	0.565570
$893$	3.48962	0.116776
$894$	4.71394	0.157658
$895$	4.47423	0.149557
$896$	−1.77653	−0.0593498
$897$	19.0994	0.637709
$898$	−0.740595	−0.0247140
$899$	−9.26966	−0.309160
$900$	−19.9635	−0.665449
$901$	10.5202	0.350479
$902$	−0.117059	−0.00389763
$903$	−2.78643	−0.0927265
$904$	−13.6945	−0.455472
$905$	−46.3592	−1.54103
$906$	8.69396	0.288838
$907$	6.93136	0.230152	0.115076	−	0.993357i	$-0.463289\pi$
$907$	6.93136	0.230152	0.115076	+	0.993357i	$0.463289\pi$
$908$	17.4080	0.577704
$909$	44.5013	1.47601
$910$	33.4145	1.10768
$911$	2.97913	0.0987029	0.0493514	−	0.998781i	$-0.484285\pi$
$911$	2.97913	0.0987029	0.0493514	+	0.998781i	$0.484285\pi$
$912$	4.33192	0.143444
$913$	−0.278390	−0.00921336
$914$	−19.0483	−0.630060
$915$	2.60784	0.0862127
$916$	−8.91162	−0.294448
$917$	22.0109	0.726865
$918$	6.01830	0.198633
$919$	16.7252	0.551713	0.275857	−	0.961199i	$-0.411039\pi$
$919$	16.7252	0.551713	0.275857	+	0.961199i	$0.411039\pi$
$920$	17.3587	0.572299
$921$	5.26216	0.173394
$922$	35.3130	1.16297
$923$	−8.90956	−0.293262
$924$	0.0409609	0.00134751
$925$	75.3625	2.47790
$926$	−29.8924	−0.982326
$927$	24.8265	0.815410
$928$	−7.72500	−0.253586
$929$	5.07782	0.166598	0.0832990	−	0.996525i	$-0.473454\pi$
$929$	5.07782	0.166598	0.0832990	+	0.996525i	$0.473454\pi$
$930$	−3.42536	−0.112322
$931$	21.3164	0.698618
$932$	−16.7363	−0.548215
$933$	3.04650	0.0997379
$934$	29.7723	0.974178
$935$	0.154176	0.00504209
$936$	12.3004	0.402052
$937$	−18.1423	−0.592683	−0.296342	−	0.955082i	$-0.595767\pi$
$937$	−18.1423	−0.592683	−0.296342	+	0.955082i	$0.595767\pi$
$938$	−13.1699	−0.430014
$939$	−10.1099	−0.329923
$940$	−2.29953	−0.0750024
$941$	37.4488	1.22079	0.610397	−	0.792095i	$-0.291010\pi$
$941$	37.4488	1.22079	0.610397	+	0.792095i	$0.291010\pi$
$942$	1.56016	0.0508326
$943$	−18.8393	−0.613492
$944$	−1.87935	−0.0611677
$945$	−27.3329	−0.889140
$946$	−0.0592637	−0.00192683
$947$	−52.9324	−1.72007	−0.860035	−	0.510235i	$-0.829559\pi$
$947$	−52.9324	−1.72007	−0.860035	+	0.510235i	$0.829559\pi$
$948$	9.57373	0.310940
$949$	−1.44901	−0.0470370
$950$	−46.3251	−1.50298
$951$	13.3954	0.434376
$952$	−2.53942	−0.0823030
$953$	9.78625	0.317008	0.158504	−	0.987358i	$-0.449333\pi$
$953$	9.78625	0.317008	0.158504	+	0.987358i	$0.449333\pi$
$954$	−17.5883	−0.569441
$955$	34.2173	1.10725
$956$	9.15914	0.296228
$957$	0.178113	0.00575757
$958$	−30.7545	−0.993631
$959$	−22.5030	−0.726661
$960$	−2.85457	−0.0921310
$961$	−29.5601	−0.953552
$962$	−46.4344	−1.49710
$963$	−18.9761	−0.611495
$964$	−13.9673	−0.449855
$965$	49.4493	1.59183
$966$	6.59220	0.212101
$967$	29.5916	0.951603	0.475802	−	0.879553i	$-0.342158\pi$
$967$	29.5916	0.951603	0.475802	+	0.879553i	$0.342158\pi$
$968$	−10.9991	−0.353525
$969$	6.19214	0.198920
$970$	−54.2735	−1.74262
$971$	−31.9228	−1.02445	−0.512225	−	0.858851i	$-0.671179\pi$
$971$	−31.9228	−1.02445	−0.512225	+	0.858851i	$0.671179\pi$
$972$	−15.6621	−0.502363
$973$	27.6163	0.885339
$974$	16.0368	0.513852
$975$	33.5876	1.07567
$976$	−0.913567	−0.0292426
$977$	−51.4234	−1.64518	−0.822590	−	0.568635i	$-0.807472\pi$
$977$	−51.4234	−1.64518	−0.822590	+	0.568635i	$0.807472\pi$
$978$	8.17510	0.261411
$979$	−0.254022	−0.00811859
$980$	−14.0468	−0.448707
$981$	38.8177	1.23935
$982$	23.4201	0.747364
$983$	35.7450	1.14009	0.570044	−	0.821614i	$-0.306926\pi$
$983$	35.7450	1.14009	0.570044	+	0.821614i	$0.306926\pi$
$984$	3.09806	0.0987624
$985$	−4.30887	−0.137292
$986$	−11.0423	−0.351659
$987$	−0.873279	−0.0277968
$988$	28.5431	0.908076
$989$	−9.53783	−0.303285
$990$	−0.257759	−0.00819213
$991$	−32.3604	−1.02796	−0.513981	−	0.857802i	$-0.671830\pi$
$991$	−32.3604	−1.02796	−0.513981	+	0.857802i	$0.671830\pi$
$992$	1.19995	0.0380986
$993$	−1.62167	−0.0514620
$994$	−3.07516	−0.0975382
$995$	−63.2529	−2.00525
$996$	7.36783	0.233459
$997$	−38.3224	−1.21368	−0.606842	−	0.794823i	$-0.707564\pi$
$997$	−38.3224	−1.21368	−0.606842	+	0.794823i	$0.707564\pi$
$998$	23.4838	0.743368
$999$	37.9831	1.20173

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.31	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.31	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.781162 q^{3} +1.00000 q^{4} +3.65427 q^{5} -0.781162 q^{6} -1.77653 q^{7} +1.00000 q^{8} -2.38979 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.781162 q^{3} +1.00000 q^{4} +3.65427 q^{5} -0.781162 q^{6} -1.77653 q^{7} +1.00000 q^{8} -2.38979 q^{9} +3.65427 q^{10} +0.0295158 q^{11} -0.781162 q^{12} -5.14709 q^{13} -1.77653 q^{14} -2.85457 q^{15} +1.00000 q^{16} +1.42942 q^{17} -2.38979 q^{18} -5.54548 q^{19} +3.65427 q^{20} +1.38776 q^{21} +0.0295158 q^{22} +4.75025 q^{23} -0.781162 q^{24} +8.35366 q^{25} -5.14709 q^{26} +4.21030 q^{27} -1.77653 q^{28} -7.72500 q^{29} -2.85457 q^{30} +1.19995 q^{31} +1.00000 q^{32} -0.0230567 q^{33} +1.42942 q^{34} -6.49192 q^{35} -2.38979 q^{36} +9.02149 q^{37} -5.54548 q^{38} +4.02071 q^{39} +3.65427 q^{40} -3.96596 q^{41} +1.38776 q^{42} -2.00786 q^{43} +0.0295158 q^{44} -8.73292 q^{45} +4.75025 q^{46} -0.629273 q^{47} -0.781162 q^{48} -3.84393 q^{49} +8.35366 q^{50} -1.11661 q^{51} -5.14709 q^{52} +7.35976 q^{53} +4.21030 q^{54} +0.107859 q^{55} -1.77653 q^{56} +4.33192 q^{57} -7.72500 q^{58} -1.87935 q^{59} -2.85457 q^{60} -0.913567 q^{61} +1.19995 q^{62} +4.24553 q^{63} +1.00000 q^{64} -18.8088 q^{65} -0.0230567 q^{66} +7.41328 q^{67} +1.42942 q^{68} -3.71071 q^{69} -6.49192 q^{70} +1.73099 q^{71} -2.38979 q^{72} +0.281521 q^{73} +9.02149 q^{74} -6.52556 q^{75} -5.54548 q^{76} -0.0524359 q^{77} +4.02071 q^{78} -12.2558 q^{79} +3.65427 q^{80} +3.88044 q^{81} -3.96596 q^{82} -9.43188 q^{83} +1.38776 q^{84} +5.22349 q^{85} -2.00786 q^{86} +6.03448 q^{87} +0.0295158 q^{88} -8.60631 q^{89} -8.73292 q^{90} +9.14397 q^{91} +4.75025 q^{92} -0.937359 q^{93} -0.629273 q^{94} -20.2647 q^{95} -0.781162 q^{96} -14.8521 q^{97} -3.84393 q^{98} -0.0705366 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 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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