LMFDB - Embedded newform 8002.2.a.d.1.15

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			1.15
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.15867 q^{3} +1.00000 q^{4} +0.496482 q^{5} -2.15867 q^{6} -0.313465 q^{7} +1.00000 q^{8} +1.65985 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.15867 q^{3} +1.00000 q^{4} +0.496482 q^{5} -2.15867 q^{6} -0.313465 q^{7} +1.00000 q^{8} +1.65985 q^{9} +0.496482 q^{10} -0.947161 q^{11} -2.15867 q^{12} +0.165631 q^{13} -0.313465 q^{14} -1.07174 q^{15} +1.00000 q^{16} -6.79807 q^{17} +1.65985 q^{18} +1.01270 q^{19} +0.496482 q^{20} +0.676667 q^{21} -0.947161 q^{22} +4.85504 q^{23} -2.15867 q^{24} -4.75351 q^{25} +0.165631 q^{26} +2.89293 q^{27} -0.313465 q^{28} +8.41712 q^{29} -1.07174 q^{30} -0.965191 q^{31} +1.00000 q^{32} +2.04461 q^{33} -6.79807 q^{34} -0.155629 q^{35} +1.65985 q^{36} +10.9954 q^{37} +1.01270 q^{38} -0.357542 q^{39} +0.496482 q^{40} -1.43868 q^{41} +0.676667 q^{42} -7.10074 q^{43} -0.947161 q^{44} +0.824087 q^{45} +4.85504 q^{46} -6.38432 q^{47} -2.15867 q^{48} -6.90174 q^{49} -4.75351 q^{50} +14.6748 q^{51} +0.165631 q^{52} +7.98758 q^{53} +2.89293 q^{54} -0.470248 q^{55} -0.313465 q^{56} -2.18608 q^{57} +8.41712 q^{58} +0.213356 q^{59} -1.07174 q^{60} +4.95459 q^{61} -0.965191 q^{62} -0.520306 q^{63} +1.00000 q^{64} +0.0822325 q^{65} +2.04461 q^{66} +8.95755 q^{67} -6.79807 q^{68} -10.4804 q^{69} -0.155629 q^{70} +0.235117 q^{71} +1.65985 q^{72} -13.4876 q^{73} +10.9954 q^{74} +10.2612 q^{75} +1.01270 q^{76} +0.296902 q^{77} -0.357542 q^{78} -12.3178 q^{79} +0.496482 q^{80} -11.2244 q^{81} -1.43868 q^{82} -3.57389 q^{83} +0.676667 q^{84} -3.37511 q^{85} -7.10074 q^{86} -18.1698 q^{87} -0.947161 q^{88} -2.22111 q^{89} +0.824087 q^{90} -0.0519193 q^{91} +4.85504 q^{92} +2.08353 q^{93} -6.38432 q^{94} +0.502786 q^{95} -2.15867 q^{96} -14.0019 q^{97} -6.90174 q^{98} -1.57215 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.15867	−1.24631	−0.623154	−	0.782099i	$-0.714149\pi$
$3$	−2.15867	−1.24631	−0.623154	+	0.782099i	$0.714149\pi$
$4$	1.00000	0.500000
$5$	0.496482	0.222033	0.111017	−	0.993819i	$-0.464589\pi$
$5$	0.496482	0.222033	0.111017	+	0.993819i	$0.464589\pi$
$6$	−2.15867	−0.881273
$7$	−0.313465	−0.118479	−0.0592393	−	0.998244i	$-0.518867\pi$
$7$	−0.313465	−0.118479	−0.0592393	+	0.998244i	$0.518867\pi$
$8$	1.00000	0.353553
$9$	1.65985	0.553285
$10$	0.496482	0.157001
$11$	−0.947161	−0.285580	−0.142790	−	0.989753i	$-0.545607\pi$
$11$	−0.947161	−0.285580	−0.142790	+	0.989753i	$0.545607\pi$
$12$	−2.15867	−0.623154
$13$	0.165631	0.0459377	0.0229688	−	0.999736i	$-0.492688\pi$
$13$	0.165631	0.0459377	0.0229688	+	0.999736i	$0.492688\pi$
$14$	−0.313465	−0.0837770
$15$	−1.07174	−0.276722
$16$	1.00000	0.250000
$17$	−6.79807	−1.64877	−0.824387	−	0.566027i	$-0.808480\pi$
$17$	−6.79807	−1.64877	−0.824387	+	0.566027i	$0.808480\pi$
$18$	1.65985	0.391231
$19$	1.01270	0.232329	0.116164	−	0.993230i	$-0.462940\pi$
$19$	1.01270	0.232329	0.116164	+	0.993230i	$0.462940\pi$
$20$	0.496482	0.111017
$21$	0.676667	0.147661
$22$	−0.947161	−0.201935
$23$	4.85504	1.01235	0.506173	−	0.862432i	$-0.331060\pi$
$23$	4.85504	1.01235	0.506173	+	0.862432i	$0.331060\pi$
$24$	−2.15867	−0.440637
$25$	−4.75351	−0.950701
$26$	0.165631	0.0324828
$27$	2.89293	0.556745
$28$	−0.313465	−0.0592393
$29$	8.41712	1.56302	0.781510	−	0.623892i	$-0.214450\pi$
$29$	8.41712	1.56302	0.781510	+	0.623892i	$0.214450\pi$
$30$	−1.07174	−0.195672
$31$	−0.965191	−0.173353	−0.0866767	−	0.996236i	$-0.527625\pi$
$31$	−0.965191	−0.173353	−0.0866767	+	0.996236i	$0.527625\pi$
$32$	1.00000	0.176777
$33$	2.04461	0.355920
$34$	−6.79807	−1.16586
$35$	−0.155629	−0.0263062
$36$	1.65985	0.276642
$37$	10.9954	1.80764	0.903820	−	0.427913i	$-0.140751\pi$
$37$	10.9954	1.80764	0.903820	+	0.427913i	$0.140751\pi$
$38$	1.01270	0.164281
$39$	−0.357542	−0.0572525
$40$	0.496482	0.0785006
$41$	−1.43868	−0.224683	−0.112342	−	0.993670i	$-0.535835\pi$
$41$	−1.43868	−0.224683	−0.112342	+	0.993670i	$0.535835\pi$
$42$	0.676667	0.104412
$43$	−7.10074	−1.08285	−0.541426	−	0.840748i	$-0.682115\pi$
$43$	−7.10074	−1.08285	−0.541426	+	0.840748i	$0.682115\pi$
$44$	−0.947161	−0.142790
$45$	0.824087	0.122848
$46$	4.85504	0.715837
$47$	−6.38432	−0.931249	−0.465624	−	0.884983i	$-0.654170\pi$
$47$	−6.38432	−0.931249	−0.465624	+	0.884983i	$0.654170\pi$
$48$	−2.15867	−0.311577
$49$	−6.90174	−0.985963
$50$	−4.75351	−0.672247
$51$	14.6748	2.05488
$52$	0.165631	0.0229688
$53$	7.98758	1.09718	0.548590	−	0.836092i	$-0.315165\pi$
$53$	7.98758	1.09718	0.548590	+	0.836092i	$0.315165\pi$
$54$	2.89293	0.393678
$55$	−0.470248	−0.0634082
$56$	−0.313465	−0.0418885
$57$	−2.18608	−0.289553
$58$	8.41712	1.10522
$59$	0.213356	0.0277766	0.0138883	−	0.999904i	$-0.495579\pi$
$59$	0.213356	0.0277766	0.0138883	+	0.999904i	$0.495579\pi$
$60$	−1.07174	−0.138361
$61$	4.95459	0.634370	0.317185	−	0.948364i	$-0.397262\pi$
$61$	4.95459	0.634370	0.317185	+	0.948364i	$0.397262\pi$
$62$	−0.965191	−0.122579
$63$	−0.520306	−0.0655523
$64$	1.00000	0.125000
$65$	0.0822325	0.0101997
$66$	2.04461	0.251674
$67$	8.95755	1.09434	0.547169	−	0.837022i	$-0.315705\pi$
$67$	8.95755	1.09434	0.547169	+	0.837022i	$0.315705\pi$
$68$	−6.79807	−0.824387
$69$	−10.4804	−1.26170
$70$	−0.155629	−0.0186013
$71$	0.235117	0.0279033	0.0139516	−	0.999903i	$-0.495559\pi$
$71$	0.235117	0.0279033	0.0139516	+	0.999903i	$0.495559\pi$
$72$	1.65985	0.195616
$73$	−13.4876	−1.57861	−0.789305	−	0.614002i	$-0.789559\pi$
$73$	−13.4876	−1.57861	−0.789305	+	0.614002i	$0.789559\pi$
$74$	10.9954	1.27819
$75$	10.2612	1.18487
$76$	1.01270	0.116164
$77$	0.296902	0.0338351
$78$	−0.357542	−0.0404836
$79$	−12.3178	−1.38586	−0.692932	−	0.721003i	$-0.743682\pi$
$79$	−12.3178	−1.38586	−0.692932	+	0.721003i	$0.743682\pi$
$80$	0.496482	0.0555083
$81$	−11.2244	−1.24716
$82$	−1.43868	−0.158875
$83$	−3.57389	−0.392285	−0.196143	−	0.980575i	$-0.562842\pi$
$83$	−3.57389	−0.392285	−0.196143	+	0.980575i	$0.562842\pi$
$84$	0.676667	0.0738304
$85$	−3.37511	−0.366083
$86$	−7.10074	−0.765692
$87$	−18.1698	−1.94801
$88$	−0.947161	−0.100968
$89$	−2.22111	−0.235438	−0.117719	−	0.993047i	$-0.537558\pi$
$89$	−2.22111	−0.235438	−0.117719	+	0.993047i	$0.537558\pi$
$90$	0.824087	0.0868664
$91$	−0.0519193	−0.00544263
$92$	4.85504	0.506173
$93$	2.08353	0.216052
$94$	−6.38432	−0.658492
$95$	0.502786	0.0515847
$96$	−2.15867	−0.220318
$97$	−14.0019	−1.42168	−0.710841	−	0.703353i	$-0.751685\pi$
$97$	−14.0019	−1.42168	−0.710841	+	0.703353i	$0.751685\pi$
$98$	−6.90174	−0.697181
$99$	−1.57215	−0.158007
$100$	−4.75351	−0.475351
$101$	−10.5233	−1.04711	−0.523554	−	0.851992i	$-0.675394\pi$
$101$	−10.5233	−1.04711	−0.523554	+	0.851992i	$0.675394\pi$
$102$	14.6748	1.45302
$103$	7.98345	0.786632	0.393316	−	0.919403i	$-0.371328\pi$
$103$	7.98345	0.786632	0.393316	+	0.919403i	$0.371328\pi$
$104$	0.165631	0.0162414
$105$	0.335953	0.0327856
$106$	7.98758	0.775823
$107$	−12.2727	−1.18644	−0.593222	−	0.805039i	$-0.702144\pi$
$107$	−12.2727	−1.18644	−0.593222	+	0.805039i	$0.702144\pi$
$108$	2.89293	0.278373
$109$	6.20572	0.594400	0.297200	−	0.954815i	$-0.403947\pi$
$109$	6.20572	0.594400	0.297200	+	0.954815i	$0.403947\pi$
$110$	−0.470248	−0.0448364
$111$	−23.7355	−2.25288
$112$	−0.313465	−0.0296196
$113$	16.5787	1.55960	0.779798	−	0.626031i	$-0.215322\pi$
$113$	16.5787	1.55960	0.779798	+	0.626031i	$0.215322\pi$
$114$	−2.18608	−0.204745
$115$	2.41044	0.224775
$116$	8.41712	0.781510
$117$	0.274922	0.0254166
$118$	0.213356	0.0196410
$119$	2.13095	0.195344
$120$	−1.07174	−0.0978360
$121$	−10.1029	−0.918444
$122$	4.95459	0.448567
$123$	3.10563	0.280025
$124$	−0.965191	−0.0866767
$125$	−4.84244	−0.433121
$126$	−0.520306	−0.0463525
$127$	0.215287	0.0191037	0.00955183	−	0.999954i	$-0.496960\pi$
$127$	0.215287	0.0191037	0.00955183	+	0.999954i	$0.496960\pi$
$128$	1.00000	0.0883883
$129$	15.3282	1.34957
$130$	0.0822325	0.00721227
$131$	3.83192	0.334796	0.167398	−	0.985889i	$-0.446463\pi$
$131$	3.83192	0.334796	0.167398	+	0.985889i	$0.446463\pi$
$132$	2.04461	0.177960
$133$	−0.317445	−0.0275260
$134$	8.95755	0.773814
$135$	1.43629	0.123616
$136$	−6.79807	−0.582929
$137$	−11.8464	−1.01211	−0.506054	−	0.862502i	$-0.668896\pi$
$137$	−11.8464	−1.01211	−0.506054	+	0.862502i	$0.668896\pi$
$138$	−10.4804	−0.892153
$139$	23.2156	1.96912	0.984562	−	0.175039i	$-0.0560050\pi$
$139$	23.2156	1.96912	0.984562	+	0.175039i	$0.0560050\pi$
$140$	−0.155629	−0.0131531
$141$	13.7816	1.16062
$142$	0.235117	0.0197306
$143$	−0.156879	−0.0131189
$144$	1.65985	0.138321
$145$	4.17895	0.347043
$146$	−13.4876	−1.11625
$147$	14.8986	1.22881
$148$	10.9954	0.903820
$149$	2.97121	0.243411	0.121705	−	0.992566i	$-0.461164\pi$
$149$	2.97121	0.243411	0.121705	+	0.992566i	$0.461164\pi$
$150$	10.2612	0.837827
$151$	−9.49867	−0.772991	−0.386495	−	0.922291i	$-0.626314\pi$
$151$	−9.49867	−0.772991	−0.386495	+	0.922291i	$0.626314\pi$
$152$	1.01270	0.0821406
$153$	−11.2838	−0.912241
$154$	0.296902	0.0239250
$155$	−0.479200	−0.0384902
$156$	−0.357542	−0.0286262
$157$	−13.5598	−1.08219	−0.541097	−	0.840960i	$-0.681991\pi$
$157$	−13.5598	−1.08219	−0.541097	+	0.840960i	$0.681991\pi$
$158$	−12.3178	−0.979954
$159$	−17.2426	−1.36742
$160$	0.496482	0.0392503
$161$	−1.52188	−0.119941
$162$	−11.2244	−0.881876
$163$	3.12522	0.244786	0.122393	−	0.992482i	$-0.460943\pi$
$163$	3.12522	0.244786	0.122393	+	0.992482i	$0.460943\pi$
$164$	−1.43868	−0.112342
$165$	1.01511	0.0790262
$166$	−3.57389	−0.277387
$167$	21.3478	1.65194	0.825971	−	0.563713i	$-0.190627\pi$
$167$	21.3478	1.65194	0.825971	+	0.563713i	$0.190627\pi$
$168$	0.676667	0.0522060
$169$	−12.9726	−0.997890
$170$	−3.37511	−0.258859
$171$	1.68093	0.128544
$172$	−7.10074	−0.541426
$173$	−24.8639	−1.89036	−0.945182	−	0.326544i	$-0.894116\pi$
$173$	−24.8639	−1.89036	−0.945182	+	0.326544i	$0.894116\pi$
$174$	−18.1698	−1.37745
$175$	1.49006	0.112638
$176$	−0.947161	−0.0713949
$177$	−0.460566	−0.0346182
$178$	−2.22111	−0.166480
$179$	−3.43967	−0.257093	−0.128547	−	0.991703i	$-0.541031\pi$
$179$	−3.43967	−0.257093	−0.128547	+	0.991703i	$0.541031\pi$
$180$	0.824087	0.0614238
$181$	8.01064	0.595426	0.297713	−	0.954655i	$-0.403776\pi$
$181$	8.01064	0.595426	0.297713	+	0.954655i	$0.403776\pi$
$182$	−0.0519193	−0.00384852
$183$	−10.6953	−0.790621
$184$	4.85504	0.357918
$185$	5.45904	0.401356
$186$	2.08353	0.152772
$187$	6.43886	0.470856
$188$	−6.38432	−0.465624
$189$	−0.906833	−0.0659624
$190$	0.502786	0.0364759
$191$	−3.09608	−0.224025	−0.112012	−	0.993707i	$-0.535730\pi$
$191$	−3.09608	−0.224025	−0.112012	+	0.993707i	$0.535730\pi$
$192$	−2.15867	−0.155789
$193$	12.4919	0.899186	0.449593	−	0.893233i	$-0.351569\pi$
$193$	12.4919	0.899186	0.449593	+	0.893233i	$0.351569\pi$
$194$	−14.0019	−1.00528
$195$	−0.177513	−0.0127120
$196$	−6.90174	−0.492981
$197$	13.3977	0.954546	0.477273	−	0.878755i	$-0.341625\pi$
$197$	13.3977	0.954546	0.477273	+	0.878755i	$0.341625\pi$
$198$	−1.57215	−0.111728
$199$	−12.6721	−0.898304	−0.449152	−	0.893455i	$-0.648274\pi$
$199$	−12.6721	−0.898304	−0.449152	+	0.893455i	$0.648274\pi$
$200$	−4.75351	−0.336124
$201$	−19.3364	−1.36388
$202$	−10.5233	−0.740417
$203$	−2.63847	−0.185184
$204$	14.6748	1.02744
$205$	−0.714276	−0.0498872
$206$	7.98345	0.556233
$207$	8.05866	0.560115
$208$	0.165631	0.0114844
$209$	−0.959188	−0.0663484
$210$	0.335953	0.0231829
$211$	−12.7324	−0.876537	−0.438269	−	0.898844i	$-0.644408\pi$
$211$	−12.7324	−0.876537	−0.438269	+	0.898844i	$0.644408\pi$
$212$	7.98758	0.548590
$213$	−0.507540	−0.0347761
$214$	−12.2727	−0.838942
$215$	−3.52539	−0.240429
$216$	2.89293	0.196839
$217$	0.302553	0.0205387
$218$	6.20572	0.420304
$219$	29.1154	1.96743
$220$	−0.470248	−0.0317041
$221$	−1.12597	−0.0757408
$222$	−23.7355	−1.59302
$223$	−29.4755	−1.97382	−0.986911	−	0.161265i	$-0.948443\pi$
$223$	−29.4755	−1.97382	−0.986911	+	0.161265i	$0.948443\pi$
$224$	−0.313465	−0.0209442
$225$	−7.89012	−0.526008
$226$	16.5787	1.10280
$227$	−12.6452	−0.839294	−0.419647	−	0.907687i	$-0.637846\pi$
$227$	−12.6452	−0.839294	−0.419647	+	0.907687i	$0.637846\pi$
$228$	−2.18608	−0.144777
$229$	10.5157	0.694894	0.347447	−	0.937700i	$-0.387049\pi$
$229$	10.5157	0.694894	0.347447	+	0.937700i	$0.387049\pi$
$230$	2.41044	0.158940
$231$	−0.640912	−0.0421689
$232$	8.41712	0.552611
$233$	9.57748	0.627442	0.313721	−	0.949515i	$-0.398424\pi$
$233$	9.57748	0.627442	0.313721	+	0.949515i	$0.398424\pi$
$234$	0.274922	0.0179722
$235$	−3.16970	−0.206768
$236$	0.213356	0.0138883
$237$	26.5901	1.72721
$238$	2.13095	0.138129
$239$	8.94706	0.578737	0.289369	−	0.957218i	$-0.406555\pi$
$239$	8.94706	0.578737	0.289369	+	0.957218i	$0.406555\pi$
$240$	−1.07174	−0.0691805
$241$	−12.3296	−0.794219	−0.397109	−	0.917771i	$-0.629987\pi$
$241$	−12.3296	−0.794219	−0.397109	+	0.917771i	$0.629987\pi$
$242$	−10.1029	−0.649438
$243$	15.5511	0.997602
$244$	4.95459	0.317185
$245$	−3.42659	−0.218917
$246$	3.10563	0.198007
$247$	0.167734	0.0106726
$248$	−0.965191	−0.0612897
$249$	7.71484	0.488908
$250$	−4.84244	−0.306263
$251$	13.3033	0.839696	0.419848	−	0.907594i	$-0.362083\pi$
$251$	13.3033	0.839696	0.419848	+	0.907594i	$0.362083\pi$
$252$	−0.520306	−0.0327762
$253$	−4.59850	−0.289106
$254$	0.215287	0.0135083
$255$	7.28576	0.456252
$256$	1.00000	0.0625000
$257$	−25.7708	−1.60754	−0.803769	−	0.594942i	$-0.797175\pi$
$257$	−25.7708	−1.60754	−0.803769	+	0.594942i	$0.797175\pi$
$258$	15.3282	0.954289
$259$	−3.44668	−0.214167
$260$	0.0822325	0.00509984
$261$	13.9712	0.864795
$262$	3.83192	0.236737
$263$	8.07327	0.497819	0.248910	−	0.968527i	$-0.419928\pi$
$263$	8.07327	0.497819	0.248910	+	0.968527i	$0.419928\pi$
$264$	2.04461	0.125837
$265$	3.96569	0.243610
$266$	−0.317445	−0.0194638
$267$	4.79465	0.293428
$268$	8.95755	0.547169
$269$	−23.8405	−1.45358	−0.726791	−	0.686859i	$-0.758989\pi$
$269$	−23.8405	−1.45358	−0.726791	+	0.686859i	$0.758989\pi$
$270$	1.43629	0.0874097
$271$	2.13690	0.129807	0.0649036	−	0.997892i	$-0.479326\pi$
$271$	2.13690	0.129807	0.0649036	+	0.997892i	$0.479326\pi$
$272$	−6.79807	−0.412193
$273$	0.112077	0.00678319
$274$	−11.8464	−0.715668
$275$	4.50233	0.271501
$276$	−10.4804	−0.630848
$277$	−18.1933	−1.09313	−0.546564	−	0.837417i	$-0.684064\pi$
$277$	−18.1933	−1.09313	−0.546564	+	0.837417i	$0.684064\pi$
$278$	23.2156	1.39238
$279$	−1.60208	−0.0959138
$280$	−0.155629	−0.00930064
$281$	−17.3775	−1.03665	−0.518327	−	0.855182i	$-0.673445\pi$
$281$	−17.3775	−1.03665	−0.518327	+	0.855182i	$0.673445\pi$
$282$	13.7816	0.820684
$283$	−6.93786	−0.412413	−0.206206	−	0.978509i	$-0.566112\pi$
$283$	−6.93786	−0.412413	−0.206206	+	0.978509i	$0.566112\pi$
$284$	0.235117	0.0139516
$285$	−1.08535	−0.0642905
$286$	−0.156879	−0.00927644
$287$	0.450974	0.0266202
$288$	1.65985	0.0978078
$289$	29.2137	1.71845
$290$	4.17895	0.245396
$291$	30.2256	1.77185
$292$	−13.4876	−0.789305
$293$	−33.5837	−1.96198	−0.980990	−	0.194061i	$-0.937834\pi$
$293$	−33.5837	−1.96198	−0.980990	+	0.194061i	$0.937834\pi$
$294$	14.8986	0.868903
$295$	0.105927	0.00616733
$296$	10.9954	0.639097
$297$	−2.74007	−0.158995
$298$	2.97121	0.172118
$299$	0.804143	0.0465048
$300$	10.2612	0.592433
$301$	2.22583	0.128295
$302$	−9.49867	−0.546587
$303$	22.7163	1.30502
$304$	1.01270	0.0580822
$305$	2.45986	0.140851
$306$	−11.2838	−0.645052
$307$	1.47970	0.0844507	0.0422254	−	0.999108i	$-0.486555\pi$
$307$	1.47970	0.0844507	0.0422254	+	0.999108i	$0.486555\pi$
$308$	0.296902	0.0169175
$309$	−17.2336	−0.980386
$310$	−0.479200	−0.0272167
$311$	−31.9898	−1.81397	−0.906987	−	0.421158i	$-0.861624\pi$
$311$	−31.9898	−1.81397	−0.906987	+	0.421158i	$0.861624\pi$
$312$	−0.357542	−0.0202418
$313$	−7.83489	−0.442854	−0.221427	−	0.975177i	$-0.571072\pi$
$313$	−7.83489	−0.442854	−0.221427	+	0.975177i	$0.571072\pi$
$314$	−13.5598	−0.765226
$315$	−0.258322	−0.0145548
$316$	−12.3178	−0.692932
$317$	−13.0466	−0.732772	−0.366386	−	0.930463i	$-0.619405\pi$
$317$	−13.0466	−0.732772	−0.366386	+	0.930463i	$0.619405\pi$
$318$	−17.2426	−0.966914
$319$	−7.97237	−0.446367
$320$	0.496482	0.0277542
$321$	26.4926	1.47867
$322$	−1.52188	−0.0848113
$323$	−6.88439	−0.383058
$324$	−11.2244	−0.623580
$325$	−0.787326	−0.0436730
$326$	3.12522	0.173090
$327$	−13.3961	−0.740806
$328$	−1.43868	−0.0794376
$329$	2.00126	0.110333
$330$	1.01511	0.0558799
$331$	10.7342	0.590005	0.295003	−	0.955496i	$-0.404679\pi$
$331$	10.7342	0.590005	0.295003	+	0.955496i	$0.404679\pi$
$332$	−3.57389	−0.196143
$333$	18.2508	1.00014
$334$	21.3478	1.16810
$335$	4.44726	0.242980
$336$	0.676667	0.0369152
$337$	12.1708	0.662984	0.331492	−	0.943458i	$-0.392448\pi$
$337$	12.1708	0.662984	0.331492	+	0.943458i	$0.392448\pi$
$338$	−12.9726	−0.705615
$339$	−35.7880	−1.94374
$340$	−3.37511	−0.183041
$341$	0.914191	0.0495062
$342$	1.68093	0.0908943
$343$	4.35771	0.235294
$344$	−7.10074	−0.382846
$345$	−5.20334	−0.280138
$346$	−24.8639	−1.33669
$347$	1.17081	0.0628524	0.0314262	−	0.999506i	$-0.489995\pi$
$347$	1.17081	0.0628524	0.0314262	+	0.999506i	$0.489995\pi$
$348$	−18.1698	−0.974003
$349$	27.3594	1.46451	0.732257	−	0.681028i	$-0.238467\pi$
$349$	27.3594	1.46451	0.732257	+	0.681028i	$0.238467\pi$
$350$	1.49006	0.0796469
$351$	0.479158	0.0255756
$352$	−0.947161	−0.0504838
$353$	23.1708	1.23326	0.616629	−	0.787254i	$-0.288498\pi$
$353$	23.1708	1.23326	0.616629	+	0.787254i	$0.288498\pi$
$354$	−0.460566	−0.0244788
$355$	0.116731	0.00619546
$356$	−2.22111	−0.117719
$357$	−4.60003	−0.243459
$358$	−3.43967	−0.181792
$359$	9.50037	0.501410	0.250705	−	0.968064i	$-0.419338\pi$
$359$	9.50037	0.501410	0.250705	+	0.968064i	$0.419338\pi$
$360$	0.824087	0.0434332
$361$	−17.9744	−0.946023
$362$	8.01064	0.421030
$363$	21.8088	1.14466
$364$	−0.0519193	−0.00272131
$365$	−6.69637	−0.350504
$366$	−10.6953	−0.559053
$367$	−26.2152	−1.36842	−0.684212	−	0.729283i	$-0.739854\pi$
$367$	−26.2152	−1.36842	−0.684212	+	0.729283i	$0.739854\pi$
$368$	4.85504	0.253087
$369$	−2.38799	−0.124314
$370$	5.45904	0.283802
$371$	−2.50383	−0.129992
$372$	2.08353	0.108026
$373$	−31.0937	−1.60997	−0.804986	−	0.593293i	$-0.797827\pi$
$373$	−31.0937	−1.60997	−0.804986	+	0.593293i	$0.797827\pi$
$374$	6.43886	0.332946
$375$	10.4532	0.539802
$376$	−6.38432	−0.329246
$377$	1.39413	0.0718015
$378$	−0.906833	−0.0466424
$379$	19.5885	1.00619	0.503097	−	0.864230i	$-0.332194\pi$
$379$	19.5885	1.00619	0.503097	+	0.864230i	$0.332194\pi$
$380$	0.502786	0.0257924
$381$	−0.464734	−0.0238091
$382$	−3.09608	−0.158409
$383$	−19.6501	−1.00407	−0.502036	−	0.864847i	$-0.667415\pi$
$383$	−19.6501	−1.00407	−0.502036	+	0.864847i	$0.667415\pi$
$384$	−2.15867	−0.110159
$385$	0.147406	0.00751251
$386$	12.4919	0.635821
$387$	−11.7862	−0.599126
$388$	−14.0019	−0.710841
$389$	−21.4746	−1.08881	−0.544403	−	0.838824i	$-0.683244\pi$
$389$	−21.4746	−1.08881	−0.544403	+	0.838824i	$0.683244\pi$
$390$	−0.177513	−0.00898871
$391$	−33.0049	−1.66913
$392$	−6.90174	−0.348591
$393$	−8.27185	−0.417259
$394$	13.3977	0.674966
$395$	−6.11558	−0.307708
$396$	−1.57215	−0.0790034
$397$	22.5403	1.13127	0.565633	−	0.824657i	$-0.308632\pi$
$397$	22.5403	1.13127	0.565633	+	0.824657i	$0.308632\pi$
$398$	−12.6721	−0.635197
$399$	0.685259	0.0343058
$400$	−4.75351	−0.237675
$401$	−11.4310	−0.570837	−0.285419	−	0.958403i	$-0.592133\pi$
$401$	−11.4310	−0.570837	−0.285419	+	0.958403i	$0.592133\pi$
$402$	−19.3364	−0.964411
$403$	−0.159865	−0.00796345
$404$	−10.5233	−0.523554
$405$	−5.57273	−0.276911
$406$	−2.63847	−0.130945
$407$	−10.4145	−0.516225
$408$	14.6748	0.726510
$409$	10.8461	0.536307	0.268154	−	0.963376i	$-0.413587\pi$
$409$	10.8461	0.536307	0.268154	+	0.963376i	$0.413587\pi$
$410$	−0.714276	−0.0352756
$411$	25.5725	1.26140
$412$	7.98345	0.393316
$413$	−0.0668796	−0.00329093
$414$	8.05866	0.396061
$415$	−1.77437	−0.0871003
$416$	0.165631	0.00812071
$417$	−50.1148	−2.45413
$418$	−0.959188	−0.0469154
$419$	17.0512	0.833006	0.416503	−	0.909134i	$-0.363256\pi$
$419$	17.0512	0.833006	0.416503	+	0.909134i	$0.363256\pi$
$420$	0.335953	0.0163928
$421$	−26.6281	−1.29777	−0.648887	−	0.760885i	$-0.724765\pi$
$421$	−26.6281	−1.29777	−0.648887	+	0.760885i	$0.724765\pi$
$422$	−12.7324	−0.619806
$423$	−10.5970	−0.515245
$424$	7.98758	0.387911
$425$	32.3147	1.56749
$426$	−0.507540	−0.0245904
$427$	−1.55309	−0.0751592
$428$	−12.2727	−0.593222
$429$	0.338649	0.0163501
$430$	−3.52539	−0.170009
$431$	−2.97008	−0.143064	−0.0715319	−	0.997438i	$-0.522789\pi$
$431$	−2.97008	−0.143064	−0.0715319	+	0.997438i	$0.522789\pi$
$432$	2.89293	0.139186
$433$	−5.02363	−0.241420	−0.120710	−	0.992688i	$-0.538517\pi$
$433$	−5.02363	−0.241420	−0.120710	+	0.992688i	$0.538517\pi$
$434$	0.302553	0.0145230
$435$	−9.02096	−0.432522
$436$	6.20572	0.297200
$437$	4.91669	0.235197
$438$	29.1154	1.39119
$439$	9.22859	0.440457	0.220228	−	0.975448i	$-0.429320\pi$
$439$	9.22859	0.440457	0.220228	+	0.975448i	$0.429320\pi$
$440$	−0.470248	−0.0224182
$441$	−11.4559	−0.545518
$442$	−1.12597	−0.0535568
$443$	−30.2716	−1.43825	−0.719123	−	0.694883i	$-0.755456\pi$
$443$	−30.2716	−1.43825	−0.719123	+	0.694883i	$0.755456\pi$
$444$	−23.7355	−1.12644
$445$	−1.10274	−0.0522750
$446$	−29.4755	−1.39570
$447$	−6.41386	−0.303365
$448$	−0.313465	−0.0148098
$449$	−16.0401	−0.756980	−0.378490	−	0.925605i	$-0.623557\pi$
$449$	−16.0401	−0.756980	−0.378490	+	0.925605i	$0.623557\pi$
$450$	−7.89012	−0.371944
$451$	1.36266	0.0641650
$452$	16.5787	0.779798
$453$	20.5045	0.963385
$454$	−12.6452	−0.593470
$455$	−0.0257770	−0.00120844
$456$	−2.18608	−0.102373
$457$	9.06796	0.424181	0.212091	−	0.977250i	$-0.431973\pi$
$457$	9.06796	0.424181	0.212091	+	0.977250i	$0.431973\pi$
$458$	10.5157	0.491365
$459$	−19.6664	−0.917947
$460$	2.41044	0.112387
$461$	−13.0767	−0.609044	−0.304522	−	0.952505i	$-0.598497\pi$
$461$	−13.0767	−0.609044	−0.304522	+	0.952505i	$0.598497\pi$
$462$	−0.640912	−0.0298179
$463$	−9.00584	−0.418537	−0.209268	−	0.977858i	$-0.567108\pi$
$463$	−9.00584	−0.418537	−0.209268	+	0.977858i	$0.567108\pi$
$464$	8.41712	0.390755
$465$	1.03443	0.0479707
$466$	9.57748	0.443668
$467$	7.81614	0.361688	0.180844	−	0.983512i	$-0.442117\pi$
$467$	7.81614	0.361688	0.180844	+	0.983512i	$0.442117\pi$
$468$	0.274922	0.0127083
$469$	−2.80787	−0.129656
$470$	−3.16970	−0.146207
$471$	29.2712	1.34875
$472$	0.213356	0.00982052
$473$	6.72554	0.309241
$474$	26.5901	1.22133
$475$	−4.81386	−0.220875
$476$	2.13095	0.0976721
$477$	13.2582	0.607052
$478$	8.94706	0.409229
$479$	37.7812	1.72627	0.863134	−	0.504974i	$-0.168498\pi$
$479$	37.7812	1.72627	0.863134	+	0.504974i	$0.168498\pi$
$480$	−1.07174	−0.0489180
$481$	1.82118	0.0830387
$482$	−12.3296	−0.561597
$483$	3.28524	0.149484
$484$	−10.1029	−0.459222
$485$	−6.95170	−0.315661
$486$	15.5511	0.705411
$487$	−15.7752	−0.714841	−0.357420	−	0.933944i	$-0.616344\pi$
$487$	−15.7752	−0.714841	−0.357420	+	0.933944i	$0.616344\pi$
$488$	4.95459	0.224284
$489$	−6.74632	−0.305079
$490$	−3.42659	−0.154797
$491$	−7.37765	−0.332949	−0.166474	−	0.986046i	$-0.553238\pi$
$491$	−7.37765	−0.332949	−0.166474	+	0.986046i	$0.553238\pi$
$492$	3.10563	0.140012
$493$	−57.2202	−2.57707
$494$	0.167734	0.00754670
$495$	−0.780543	−0.0350828
$496$	−0.965191	−0.0433384
$497$	−0.0737010	−0.00330594
$498$	7.71484	0.345710
$499$	−13.4031	−0.600007	−0.300003	−	0.953938i	$-0.596988\pi$
$499$	−13.4031	−0.600007	−0.300003	+	0.953938i	$0.596988\pi$
$500$	−4.84244	−0.216560
$501$	−46.0828	−2.05883
$502$	13.3033	0.593755
$503$	−31.6257	−1.41012	−0.705060	−	0.709147i	$-0.749080\pi$
$503$	−31.6257	−1.41012	−0.705060	+	0.709147i	$0.749080\pi$
$504$	−0.520306	−0.0231763
$505$	−5.22463	−0.232493
$506$	−4.59850	−0.204428
$507$	28.0035	1.24368
$508$	0.215287	0.00955183
$509$	−16.1894	−0.717582	−0.358791	−	0.933418i	$-0.616811\pi$
$509$	−16.1894	−0.717582	−0.358791	+	0.933418i	$0.616811\pi$
$510$	7.28576	0.322619
$511$	4.22790	0.187031
$512$	1.00000	0.0441942
$513$	2.92967	0.129348
$514$	−25.7708	−1.13670
$515$	3.96363	0.174659
$516$	15.3282	0.674784
$517$	6.04698	0.265946
$518$	−3.44668	−0.151439
$519$	53.6728	2.35598
$520$	0.0822325	0.00360613
$521$	5.48548	0.240323	0.120162	−	0.992754i	$-0.461659\pi$
$521$	5.48548	0.240323	0.120162	+	0.992754i	$0.461659\pi$
$522$	13.9712	0.611502
$523$	43.1354	1.88618	0.943090	−	0.332538i	$-0.107905\pi$
$523$	43.1354	1.88618	0.943090	+	0.332538i	$0.107905\pi$
$524$	3.83192	0.167398
$525$	−3.21654	−0.140381
$526$	8.07327	0.352011
$527$	6.56143	0.285821
$528$	2.04461	0.0889801
$529$	0.571423	0.0248445
$530$	3.96569	0.172258
$531$	0.354140	0.0153684
$532$	−0.317445	−0.0137630
$533$	−0.238289	−0.0103214
$534$	4.79465	0.207485
$535$	−6.09315	−0.263430
$536$	8.95755	0.386907
$537$	7.42512	0.320418
$538$	−23.8405	−1.02784
$539$	6.53706	0.281571
$540$	1.43629	0.0618080
$541$	14.5612	0.626034	0.313017	−	0.949748i	$-0.398660\pi$
$541$	14.5612	0.626034	0.313017	+	0.949748i	$0.398660\pi$
$542$	2.13690	0.0917875
$543$	−17.2923	−0.742084
$544$	−6.79807	−0.291465
$545$	3.08102	0.131977
$546$	0.112077	0.00479644
$547$	−24.2392	−1.03639	−0.518197	−	0.855261i	$-0.673397\pi$
$547$	−24.2392	−1.03639	−0.518197	+	0.855261i	$0.673397\pi$
$548$	−11.8464	−0.506054
$549$	8.22389	0.350987
$550$	4.50233	0.191980
$551$	8.52400	0.363135
$552$	−10.4804	−0.446077
$553$	3.86121	0.164195
$554$	−18.1933	−0.772958
$555$	−11.7843	−0.500214
$556$	23.2156	0.984562
$557$	−24.3627	−1.03228	−0.516141	−	0.856504i	$-0.672632\pi$
$557$	−24.3627	−1.03228	−0.516141	+	0.856504i	$0.672632\pi$
$558$	−1.60208	−0.0678213
$559$	−1.17610	−0.0497437
$560$	−0.155629	−0.00657654
$561$	−13.8994	−0.586832
$562$	−17.3775	−0.733026
$563$	−20.1340	−0.848548	−0.424274	−	0.905534i	$-0.639471\pi$
$563$	−20.1340	−0.848548	−0.424274	+	0.905534i	$0.639471\pi$
$564$	13.7816	0.580311
$565$	8.23104	0.346282
$566$	−6.93786	−0.291620
$567$	3.51847	0.147762
$568$	0.235117	0.00986530
$569$	−22.5303	−0.944519	−0.472260	−	0.881460i	$-0.656562\pi$
$569$	−22.5303	−0.944519	−0.472260	+	0.881460i	$0.656562\pi$
$570$	−1.08535	−0.0454602
$571$	33.3064	1.39383	0.696914	−	0.717155i	$-0.254556\pi$
$571$	33.3064	1.39383	0.696914	+	0.717155i	$0.254556\pi$
$572$	−0.156879	−0.00655943
$573$	6.68342	0.279204
$574$	0.450974	0.0188233
$575$	−23.0785	−0.962439
$576$	1.65985	0.0691606
$577$	−27.7946	−1.15711	−0.578553	−	0.815645i	$-0.696382\pi$
$577$	−27.7946	−1.15711	−0.578553	+	0.815645i	$0.696382\pi$
$578$	29.2137	1.21513
$579$	−26.9659	−1.12066
$580$	4.17895	0.173521
$581$	1.12029	0.0464774
$582$	30.2256	1.25289
$583$	−7.56553	−0.313332
$584$	−13.4876	−0.558123
$585$	0.136494	0.00564333
$586$	−33.5837	−1.38733
$587$	−0.857469	−0.0353915	−0.0176958	−	0.999843i	$-0.505633\pi$
$587$	−0.857469	−0.0353915	−0.0176958	+	0.999843i	$0.505633\pi$
$588$	14.8986	0.614407
$589$	−0.977447	−0.0402750
$590$	0.105927	0.00436096
$591$	−28.9212	−1.18966
$592$	10.9954	0.451910
$593$	17.7236	0.727822	0.363911	−	0.931434i	$-0.381441\pi$
$593$	17.7236	0.727822	0.363911	+	0.931434i	$0.381441\pi$
$594$	−2.74007	−0.112427
$595$	1.05798	0.0433729
$596$	2.97121	0.121705
$597$	27.3549	1.11956
$598$	0.804143	0.0328839
$599$	−36.4639	−1.48987	−0.744937	−	0.667134i	$-0.767521\pi$
$599$	−36.4639	−1.48987	−0.744937	+	0.667134i	$0.767521\pi$
$600$	10.2612	0.418914
$601$	−14.5317	−0.592761	−0.296381	−	0.955070i	$-0.595780\pi$
$601$	−14.5317	−0.592761	−0.296381	+	0.955070i	$0.595780\pi$
$602$	2.22583	0.0907181
$603$	14.8682	0.605481
$604$	−9.49867	−0.386495
$605$	−5.01590	−0.203925
$606$	22.7163	0.922788
$607$	19.3202	0.784182	0.392091	−	0.919926i	$-0.371752\pi$
$607$	19.3202	0.784182	0.392091	+	0.919926i	$0.371752\pi$
$608$	1.01270	0.0410703
$609$	5.69559	0.230797
$610$	2.45986	0.0995969
$611$	−1.05744	−0.0427794
$612$	−11.2838	−0.456120
$613$	−12.8765	−0.520077	−0.260039	−	0.965598i	$-0.583735\pi$
$613$	−12.8765	−0.520077	−0.260039	+	0.965598i	$0.583735\pi$
$614$	1.47970	0.0597157
$615$	1.54189	0.0621748
$616$	0.296902	0.0119625
$617$	−17.9693	−0.723416	−0.361708	−	0.932291i	$-0.617806\pi$
$617$	−17.9693	−0.723416	−0.361708	+	0.932291i	$0.617806\pi$
$618$	−17.2336	−0.693238
$619$	−31.5851	−1.26951	−0.634756	−	0.772712i	$-0.718899\pi$
$619$	−31.5851	−1.26951	−0.634756	+	0.772712i	$0.718899\pi$
$620$	−0.479200	−0.0192451
$621$	14.0453	0.563619
$622$	−31.9898	−1.28267
$623$	0.696241	0.0278943
$624$	−0.357542	−0.0143131
$625$	21.3634	0.854534
$626$	−7.83489	−0.313145
$627$	2.07057	0.0826906
$628$	−13.5598	−0.541097
$629$	−74.7478	−2.98039
$630$	−0.258322	−0.0102918
$631$	37.2956	1.48471	0.742356	−	0.670005i	$-0.233708\pi$
$631$	37.2956	1.48471	0.742356	+	0.670005i	$0.233708\pi$
$632$	−12.3178	−0.489977
$633$	27.4851	1.09244
$634$	−13.0466	−0.518148
$635$	0.106886	0.00424165
$636$	−17.2426	−0.683712
$637$	−1.14314	−0.0452928
$638$	−7.97237	−0.315629
$639$	0.390260	0.0154385
$640$	0.496482	0.0196252
$641$	30.8298	1.21770	0.608851	−	0.793284i	$-0.291631\pi$
$641$	30.8298	1.21770	0.608851	+	0.793284i	$0.291631\pi$
$642$	26.4926	1.04558
$643$	21.5872	0.851317	0.425658	−	0.904884i	$-0.360042\pi$
$643$	21.5872	0.851317	0.425658	+	0.904884i	$0.360042\pi$
$644$	−1.52188	−0.0599706
$645$	7.61014	0.299649
$646$	−6.88439	−0.270863
$647$	14.2129	0.558767	0.279383	−	0.960180i	$-0.409870\pi$
$647$	14.2129	0.558767	0.279383	+	0.960180i	$0.409870\pi$
$648$	−11.2244	−0.440938
$649$	−0.202083	−0.00793244
$650$	−0.787326	−0.0308815
$651$	−0.653113	−0.0255975
$652$	3.12522	0.122393
$653$	−25.2831	−0.989402	−0.494701	−	0.869063i	$-0.664723\pi$
$653$	−25.2831	−0.989402	−0.494701	+	0.869063i	$0.664723\pi$
$654$	−13.3961	−0.523829
$655$	1.90248	0.0743359
$656$	−1.43868	−0.0561709
$657$	−22.3875	−0.873420
$658$	2.00126	0.0780172
$659$	16.0532	0.625345	0.312672	−	0.949861i	$-0.398776\pi$
$659$	16.0532	0.625345	0.312672	+	0.949861i	$0.398776\pi$
$660$	1.01511	0.0395131
$661$	26.0691	1.01397	0.506985	−	0.861955i	$-0.330760\pi$
$661$	26.0691	1.01397	0.506985	+	0.861955i	$0.330760\pi$
$662$	10.7342	0.417197
$663$	2.43059	0.0943964
$664$	−3.57389	−0.138694
$665$	−0.157606	−0.00611168
$666$	18.2508	0.707205
$667$	40.8655	1.58232
$668$	21.3478	0.825971
$669$	63.6278	2.45999
$670$	4.44726	0.171813
$671$	−4.69279	−0.181163
$672$	0.676667	0.0261030
$673$	2.58440	0.0996212	0.0498106	−	0.998759i	$-0.484138\pi$
$673$	2.58440	0.0996212	0.0498106	+	0.998759i	$0.484138\pi$
$674$	12.1708	0.468801
$675$	−13.7516	−0.529298
$676$	−12.9726	−0.498945
$677$	−24.2058	−0.930303	−0.465152	−	0.885231i	$-0.654000\pi$
$677$	−24.2058	−0.930303	−0.465152	+	0.885231i	$0.654000\pi$
$678$	−35.7880	−1.37443
$679$	4.38911	0.168439
$680$	−3.37511	−0.129430
$681$	27.2969	1.04602
$682$	0.914191	0.0350062
$683$	17.5887	0.673011	0.336506	−	0.941681i	$-0.390755\pi$
$683$	17.5887	0.673011	0.336506	+	0.941681i	$0.390755\pi$
$684$	1.68093	0.0642720
$685$	−5.88153	−0.224722
$686$	4.35771	0.166378
$687$	−22.6998	−0.866053
$688$	−7.10074	−0.270713
$689$	1.32299	0.0504018
$690$	−5.20334	−0.198088
$691$	42.1265	1.60257	0.801284	−	0.598284i	$-0.204151\pi$
$691$	42.1265	1.60257	0.801284	+	0.598284i	$0.204151\pi$
$692$	−24.8639	−0.945182
$693$	0.492813	0.0187204
$694$	1.17081	0.0444434
$695$	11.5261	0.437211
$696$	−18.1698	−0.688724
$697$	9.78022	0.370452
$698$	27.3594	1.03557
$699$	−20.6746	−0.781986
$700$	1.49006	0.0563188
$701$	−16.4316	−0.620611	−0.310306	−	0.950637i	$-0.600431\pi$
$701$	−16.4316	−0.620611	−0.310306	+	0.950637i	$0.600431\pi$
$702$	0.479158	0.0180847
$703$	11.1351	0.419967
$704$	−0.947161	−0.0356975
$705$	6.84233	0.257697
$706$	23.1708	0.872045
$707$	3.29869	0.124060
$708$	−0.460566	−0.0173091
$709$	9.81023	0.368431	0.184215	−	0.982886i	$-0.441026\pi$
$709$	9.81023	0.368431	0.184215	+	0.982886i	$0.441026\pi$
$710$	0.116731	0.00438085
$711$	−20.4458	−0.766777
$712$	−2.22111	−0.0832398
$713$	−4.68604	−0.175494
$714$	−4.60003	−0.172152
$715$	−0.0778874	−0.00291282
$716$	−3.43967	−0.128547
$717$	−19.3138	−0.721285
$718$	9.50037	0.354550
$719$	−3.25745	−0.121482	−0.0607412	−	0.998154i	$-0.519346\pi$
$719$	−3.25745	−0.121482	−0.0607412	+	0.998154i	$0.519346\pi$
$720$	0.824087	0.0307119
$721$	−2.50253	−0.0931990
$722$	−17.9744	−0.668940
$723$	26.6155	0.989842
$724$	8.01064	0.297713
$725$	−40.0108	−1.48597
$726$	21.8088	0.809400
$727$	−42.4730	−1.57524	−0.787618	−	0.616164i	$-0.788686\pi$
$727$	−42.4730	−1.57524	−0.787618	+	0.616164i	$0.788686\pi$
$728$	−0.0519193	−0.00192426
$729$	0.103721	0.00384153
$730$	−6.69637	−0.247844
$731$	48.2713	1.78538
$732$	−10.6953	−0.395310
$733$	−23.4279	−0.865331	−0.432666	−	0.901555i	$-0.642427\pi$
$733$	−23.4279	−0.865331	−0.432666	+	0.901555i	$0.642427\pi$
$734$	−26.2152	−0.967622
$735$	7.39687	0.272838
$736$	4.85504	0.178959
$737$	−8.48424	−0.312521
$738$	−2.38799	−0.0879032
$739$	−36.9772	−1.36023	−0.680114	−	0.733106i	$-0.738070\pi$
$739$	−36.9772	−1.36023	−0.680114	+	0.733106i	$0.738070\pi$
$740$	5.45904	0.200678
$741$	−0.362082	−0.0133014
$742$	−2.50383	−0.0919183
$743$	29.5559	1.08430	0.542150	−	0.840282i	$-0.317610\pi$
$743$	29.5559	1.08430	0.542150	+	0.840282i	$0.317610\pi$
$744$	2.08353	0.0763859
$745$	1.47515	0.0540453
$746$	−31.0937	−1.13842
$747$	−5.93213	−0.217045
$748$	6.43886	0.235428
$749$	3.84705	0.140568
$750$	10.4532	0.381698
$751$	10.5651	0.385527	0.192764	−	0.981245i	$-0.438255\pi$
$751$	10.5651	0.385527	0.192764	+	0.981245i	$0.438255\pi$
$752$	−6.38432	−0.232812
$753$	−28.7174	−1.04652
$754$	1.39413	0.0507713
$755$	−4.71591	−0.171630
$756$	−0.906833	−0.0329812
$757$	43.1175	1.56713	0.783566	−	0.621309i	$-0.213399\pi$
$757$	43.1175	1.56713	0.783566	+	0.621309i	$0.213399\pi$
$758$	19.5885	0.711487
$759$	9.92665	0.360315
$760$	0.502786	0.0182380
$761$	32.2537	1.16920	0.584598	−	0.811323i	$-0.301252\pi$
$761$	32.2537	1.16920	0.584598	+	0.811323i	$0.301252\pi$
$762$	−0.464734	−0.0168355
$763$	−1.94527	−0.0704236
$764$	−3.09608	−0.112012
$765$	−5.60220	−0.202548
$766$	−19.6501	−0.709986
$767$	0.0353383	0.00127599
$768$	−2.15867	−0.0778943
$769$	−5.08801	−0.183478	−0.0917391	−	0.995783i	$-0.529243\pi$
$769$	−5.08801	−0.183478	−0.0917391	+	0.995783i	$0.529243\pi$
$770$	0.147406	0.00531215
$771$	55.6306	2.00349
$772$	12.4919	0.449593
$773$	−38.8251	−1.39644	−0.698222	−	0.715882i	$-0.746025\pi$
$773$	−38.8251	−1.39644	−0.698222	+	0.715882i	$0.746025\pi$
$774$	−11.7862	−0.423646
$775$	4.58804	0.164807
$776$	−14.0019	−0.502640
$777$	7.44025	0.266918
$778$	−21.4746	−0.769902
$779$	−1.45694	−0.0522004
$780$	−0.177513	−0.00635598
$781$	−0.222694	−0.00796861
$782$	−33.0049	−1.18025
$783$	24.3502	0.870204
$784$	−6.90174	−0.246491
$785$	−6.73221	−0.240283
$786$	−8.27185	−0.295047
$787$	−20.7356	−0.739143	−0.369571	−	0.929202i	$-0.620496\pi$
$787$	−20.7356	−0.739143	−0.369571	+	0.929202i	$0.620496\pi$
$788$	13.3977	0.477273
$789$	−17.4275	−0.620436
$790$	−6.11558	−0.217582
$791$	−5.19685	−0.184779
$792$	−1.57215	−0.0558639
$793$	0.820631	0.0291415
$794$	22.5403	0.799926
$795$	−8.56061	−0.303614
$796$	−12.6721	−0.449152
$797$	−47.2442	−1.67348	−0.836738	−	0.547603i	$-0.815540\pi$
$797$	−47.2442	−1.67348	−0.836738	+	0.547603i	$0.815540\pi$
$798$	0.685259	0.0242579
$799$	43.4010	1.53542
$800$	−4.75351	−0.168062
$801$	−3.68672	−0.130264
$802$	−11.4310	−0.403643
$803$	12.7750	0.450819
$804$	−19.3364	−0.681942
$805$	−0.755587	−0.0266310
$806$	−0.159865	−0.00563101
$807$	51.4638	1.81161
$808$	−10.5233	−0.370209
$809$	−14.5940	−0.513098	−0.256549	−	0.966531i	$-0.582585\pi$
$809$	−14.5940	−0.513098	−0.256549	+	0.966531i	$0.582585\pi$
$810$	−5.57273	−0.195806
$811$	25.2081	0.885177	0.442589	−	0.896725i	$-0.354060\pi$
$811$	25.2081	0.885177	0.442589	+	0.896725i	$0.354060\pi$
$812$	−2.63847	−0.0925922
$813$	−4.61285	−0.161780
$814$	−10.4145	−0.365026
$815$	1.55161	0.0543507
$816$	14.6748	0.513720
$817$	−7.19090	−0.251578
$818$	10.8461	0.379226
$819$	−0.0861785	−0.00301132
$820$	−0.714276	−0.0249436
$821$	10.3738	0.362048	0.181024	−	0.983479i	$-0.442059\pi$
$821$	10.3738	0.362048	0.181024	+	0.983479i	$0.442059\pi$
$822$	25.5725	0.891943
$823$	34.1058	1.18885	0.594427	−	0.804149i	$-0.297379\pi$
$823$	34.1058	1.18885	0.594427	+	0.804149i	$0.297379\pi$
$824$	7.98345	0.278117
$825$	−9.71905	−0.338374
$826$	−0.0668796	−0.00232704
$827$	36.0318	1.25295	0.626475	−	0.779442i	$-0.284497\pi$
$827$	36.0318	1.25295	0.626475	+	0.779442i	$0.284497\pi$
$828$	8.05866	0.280058
$829$	10.9138	0.379054	0.189527	−	0.981876i	$-0.439305\pi$
$829$	10.9138	0.379054	0.189527	+	0.981876i	$0.439305\pi$
$830$	−1.77437	−0.0615892
$831$	39.2733	1.36237
$832$	0.165631	0.00574221
$833$	46.9185	1.62563
$834$	−50.1148	−1.73534
$835$	10.5988	0.366786
$836$	−0.959188	−0.0331742
$837$	−2.79223	−0.0965137
$838$	17.0512	0.589024
$839$	−12.3669	−0.426952	−0.213476	−	0.976948i	$-0.568479\pi$
$839$	−12.3669	−0.426952	−0.213476	+	0.976948i	$0.568479\pi$
$840$	0.335953	0.0115915
$841$	41.8480	1.44303
$842$	−26.6281	−0.917665
$843$	37.5123	1.29199
$844$	−12.7324	−0.438269
$845$	−6.44064	−0.221565
$846$	−10.5970	−0.364334
$847$	3.16690	0.108816
$848$	7.98758	0.274295
$849$	14.9765	0.513994
$850$	32.3147	1.10838
$851$	53.3833	1.82996
$852$	−0.507540	−0.0173880
$853$	17.2970	0.592239	0.296119	−	0.955151i	$-0.404307\pi$
$853$	17.2970	0.592239	0.296119	+	0.955151i	$0.404307\pi$
$854$	−1.55309	−0.0531456
$855$	0.834551	0.0285410
$856$	−12.2727	−0.419471
$857$	−36.9891	−1.26352	−0.631762	−	0.775162i	$-0.717668\pi$
$857$	−36.9891	−1.26352	−0.631762	+	0.775162i	$0.717668\pi$
$858$	0.338649	0.0115613
$859$	38.6362	1.31825	0.659126	−	0.752033i	$-0.270927\pi$
$859$	38.6362	1.31825	0.659126	+	0.752033i	$0.270927\pi$
$860$	−3.52539	−0.120215
$861$	−0.973504	−0.0331769
$862$	−2.97008	−0.101161
$863$	11.7063	0.398488	0.199244	−	0.979950i	$-0.436151\pi$
$863$	11.7063	0.398488	0.199244	+	0.979950i	$0.436151\pi$
$864$	2.89293	0.0984196
$865$	−12.3444	−0.419724
$866$	−5.02363	−0.170710
$867$	−63.0628	−2.14172
$868$	0.302553	0.0102693
$869$	11.6670	0.395775
$870$	−9.02096	−0.305839
$871$	1.48364	0.0502713
$872$	6.20572	0.210152
$873$	−23.2412	−0.786594
$874$	4.91669	0.166309
$875$	1.51793	0.0513155
$876$	29.1154	0.983717
$877$	−16.6424	−0.561973	−0.280986	−	0.959712i	$-0.590662\pi$
$877$	−16.6424	−0.561973	−0.280986	+	0.959712i	$0.590662\pi$
$878$	9.22859	0.311450
$879$	72.4960	2.44523
$880$	−0.470248	−0.0158521
$881$	−2.49252	−0.0839750	−0.0419875	−	0.999118i	$-0.513369\pi$
$881$	−2.49252	−0.0839750	−0.0419875	+	0.999118i	$0.513369\pi$
$882$	−11.4559	−0.385739
$883$	35.5553	1.19653	0.598266	−	0.801298i	$-0.295857\pi$
$883$	35.5553	1.19653	0.598266	+	0.801298i	$0.295857\pi$
$884$	−1.12597	−0.0378704
$885$	−0.228662	−0.00768640
$886$	−30.2716	−1.01699
$887$	55.1552	1.85193	0.925966	−	0.377608i	$-0.123253\pi$
$887$	55.1552	1.85193	0.925966	+	0.377608i	$0.123253\pi$
$888$	−23.7355	−0.796512
$889$	−0.0674850	−0.00226337
$890$	−1.10274	−0.0369640
$891$	10.6314	0.356164
$892$	−29.4755	−0.986911
$893$	−6.46538	−0.216356
$894$	−6.41386	−0.214511
$895$	−1.70773	−0.0570833
$896$	−0.313465	−0.0104721
$897$	−1.73588	−0.0579593
$898$	−16.0401	−0.535266
$899$	−8.12413	−0.270955
$900$	−7.89012	−0.263004
$901$	−54.3001	−1.80900
$902$	1.36266	0.0453715
$903$	−4.80483	−0.159895
$904$	16.5787	0.551401
$905$	3.97713	0.132204
$906$	20.5045	0.681216
$907$	20.2270	0.671627	0.335813	−	0.941929i	$-0.390989\pi$
$907$	20.2270	0.671627	0.335813	+	0.941929i	$0.390989\pi$
$908$	−12.6452	−0.419647
$909$	−17.4671	−0.579349
$910$	−0.0257770	−0.000854499 0
$911$	−2.66666	−0.0883503	−0.0441752	−	0.999024i	$-0.514066\pi$
$911$	−2.66666	−0.0883503	−0.0441752	+	0.999024i	$0.514066\pi$
$912$	−2.18608	−0.0723883
$913$	3.38505	0.112029
$914$	9.06796	0.299941
$915$	−5.31003	−0.175544
$916$	10.5157	0.347447
$917$	−1.20117	−0.0396662
$918$	−19.6664	−0.649086
$919$	28.9288	0.954274	0.477137	−	0.878829i	$-0.341675\pi$
$919$	28.9288	0.954274	0.477137	+	0.878829i	$0.341675\pi$
$920$	2.41044	0.0794698
$921$	−3.19417	−0.105252
$922$	−13.0767	−0.430659
$923$	0.0389426	0.00128181
$924$	−0.640912	−0.0210845
$925$	−52.2669	−1.71853
$926$	−9.00584	−0.295950
$927$	13.2514	0.435231
$928$	8.41712	0.276306
$929$	−3.62246	−0.118849	−0.0594246	−	0.998233i	$-0.518927\pi$
$929$	−3.62246	−0.118849	−0.0594246	+	0.998233i	$0.518927\pi$
$930$	1.03443	0.0339204
$931$	−6.98938	−0.229068
$932$	9.57748	0.313721
$933$	69.0554	2.26077
$934$	7.81614	0.255752
$935$	3.19678	0.104546
$936$	0.274922	0.00898612
$937$	0.0206771	0.000675490 0	0.000337745	−	1.00000i	$-0.499892\pi$
$937$	0.0206771	0.000675490 0	0.000337745		1.00000i	$0.499892\pi$
$938$	−2.80787	−0.0916804
$939$	16.9129	0.551933
$940$	−3.16970	−0.103384
$941$	19.9670	0.650906	0.325453	−	0.945558i	$-0.394483\pi$
$941$	19.9670	0.650906	0.325453	+	0.945558i	$0.394483\pi$
$942$	29.2712	0.953708
$943$	−6.98483	−0.227457
$944$	0.213356	0.00694415
$945$	−0.450226	−0.0146458
$946$	6.72554	0.218666
$947$	−10.4608	−0.339929	−0.169965	−	0.985450i	$-0.554365\pi$
$947$	−10.4608	−0.339929	−0.169965	+	0.985450i	$0.554365\pi$
$948$	26.5901	0.863607
$949$	−2.23397	−0.0725176
$950$	−4.81386	−0.156182
$951$	28.1634	0.913260
$952$	2.13095	0.0690646
$953$	16.9418	0.548800	0.274400	−	0.961616i	$-0.411521\pi$
$953$	16.9418	0.548800	0.274400	+	0.961616i	$0.411521\pi$
$954$	13.2582	0.429251
$955$	−1.53715	−0.0497410
$956$	8.94706	0.289369
$957$	17.2097	0.556311
$958$	37.7812	1.22066
$959$	3.71343	0.119913
$960$	−1.07174	−0.0345902
$961$	−30.0684	−0.969949
$962$	1.82118	0.0587173
$963$	−20.3708	−0.656441
$964$	−12.3296	−0.397109
$965$	6.20199	0.199649
$966$	3.28524	0.105701
$967$	26.9773	0.867532	0.433766	−	0.901026i	$-0.357184\pi$
$967$	26.9773	0.867532	0.433766	+	0.901026i	$0.357184\pi$
$968$	−10.1029	−0.324719
$969$	14.8611	0.477408
$970$	−6.95170	−0.223206
$971$	40.7916	1.30906	0.654532	−	0.756034i	$-0.272866\pi$
$971$	40.7916	1.30906	0.654532	+	0.756034i	$0.272866\pi$
$972$	15.5511	0.498801
$973$	−7.27728	−0.233299
$974$	−15.7752	−0.505469
$975$	1.69958	0.0544300
$976$	4.95459	0.158592
$977$	38.1376	1.22013	0.610065	−	0.792351i	$-0.291143\pi$
$977$	38.1376	1.22013	0.610065	+	0.792351i	$0.291143\pi$
$978$	−6.74632	−0.215723
$979$	2.10375	0.0672362
$980$	−3.42659	−0.109458
$981$	10.3006	0.328872
$982$	−7.37765	−0.235430
$983$	56.8439	1.81304	0.906520	−	0.422163i	$-0.138729\pi$
$983$	56.8439	1.81304	0.906520	+	0.422163i	$0.138729\pi$
$984$	3.10563	0.0990037
$985$	6.65171	0.211941
$986$	−57.2202	−1.82226
$987$	−4.32006	−0.137509
$988$	0.167734	0.00533632
$989$	−34.4744	−1.09622
$990$	−0.780543	−0.0248073
$991$	23.4411	0.744632	0.372316	−	0.928106i	$-0.378564\pi$
$991$	23.4411	0.744632	0.372316	+	0.928106i	$0.378564\pi$
$992$	−0.965191	−0.0306448
$993$	−23.1716	−0.735329
$994$	−0.0737010	−0.00233765
$995$	−6.29148	−0.199453
$996$	7.71484	0.244454
$997$	−43.4111	−1.37484	−0.687421	−	0.726259i	$-0.741257\pi$
$997$	−43.4111	−1.37484	−0.687421	+	0.726259i	$0.741257\pi$
$998$	−13.4031	−0.424269
$999$	31.8091	1.00640

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.15	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.15	✓	69	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.15867 q^{3} +1.00000 q^{4} +0.496482 q^{5} -2.15867 q^{6} -0.313465 q^{7} +1.00000 q^{8} +1.65985 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.15867 q^{3} +1.00000 q^{4} +0.496482 q^{5} -2.15867 q^{6} -0.313465 q^{7} +1.00000 q^{8} +1.65985 q^{9} +0.496482 q^{10} -0.947161 q^{11} -2.15867 q^{12} +0.165631 q^{13} -0.313465 q^{14} -1.07174 q^{15} +1.00000 q^{16} -6.79807 q^{17} +1.65985 q^{18} +1.01270 q^{19} +0.496482 q^{20} +0.676667 q^{21} -0.947161 q^{22} +4.85504 q^{23} -2.15867 q^{24} -4.75351 q^{25} +0.165631 q^{26} +2.89293 q^{27} -0.313465 q^{28} +8.41712 q^{29} -1.07174 q^{30} -0.965191 q^{31} +1.00000 q^{32} +2.04461 q^{33} -6.79807 q^{34} -0.155629 q^{35} +1.65985 q^{36} +10.9954 q^{37} +1.01270 q^{38} -0.357542 q^{39} +0.496482 q^{40} -1.43868 q^{41} +0.676667 q^{42} -7.10074 q^{43} -0.947161 q^{44} +0.824087 q^{45} +4.85504 q^{46} -6.38432 q^{47} -2.15867 q^{48} -6.90174 q^{49} -4.75351 q^{50} +14.6748 q^{51} +0.165631 q^{52} +7.98758 q^{53} +2.89293 q^{54} -0.470248 q^{55} -0.313465 q^{56} -2.18608 q^{57} +8.41712 q^{58} +0.213356 q^{59} -1.07174 q^{60} +4.95459 q^{61} -0.965191 q^{62} -0.520306 q^{63} +1.00000 q^{64} +0.0822325 q^{65} +2.04461 q^{66} +8.95755 q^{67} -6.79807 q^{68} -10.4804 q^{69} -0.155629 q^{70} +0.235117 q^{71} +1.65985 q^{72} -13.4876 q^{73} +10.9954 q^{74} +10.2612 q^{75} +1.01270 q^{76} +0.296902 q^{77} -0.357542 q^{78} -12.3178 q^{79} +0.496482 q^{80} -11.2244 q^{81} -1.43868 q^{82} -3.57389 q^{83} +0.676667 q^{84} -3.37511 q^{85} -7.10074 q^{86} -18.1698 q^{87} -0.947161 q^{88} -2.22111 q^{89} +0.824087 q^{90} -0.0519193 q^{91} +4.85504 q^{92} +2.08353 q^{93} -6.38432 q^{94} +0.502786 q^{95} -2.15867 q^{96} -14.0019 q^{97} -6.90174 q^{98} -1.57215 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 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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