LMFDB - Embedded newform 8002.2.a.d.1.56

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			1.56
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.69344 q^{3} +1.00000 q^{4} +0.696607 q^{5} +1.69344 q^{6} +3.85329 q^{7} +1.00000 q^{8} -0.132273 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.69344 q^{3} +1.00000 q^{4} +0.696607 q^{5} +1.69344 q^{6} +3.85329 q^{7} +1.00000 q^{8} -0.132273 q^{9} +0.696607 q^{10} -4.77751 q^{11} +1.69344 q^{12} -6.50068 q^{13} +3.85329 q^{14} +1.17966 q^{15} +1.00000 q^{16} -7.20090 q^{17} -0.132273 q^{18} -2.46527 q^{19} +0.696607 q^{20} +6.52530 q^{21} -4.77751 q^{22} -5.15618 q^{23} +1.69344 q^{24} -4.51474 q^{25} -6.50068 q^{26} -5.30431 q^{27} +3.85329 q^{28} -1.51726 q^{29} +1.17966 q^{30} +8.39045 q^{31} +1.00000 q^{32} -8.09040 q^{33} -7.20090 q^{34} +2.68423 q^{35} -0.132273 q^{36} +6.98155 q^{37} -2.46527 q^{38} -11.0085 q^{39} +0.696607 q^{40} -6.26883 q^{41} +6.52530 q^{42} -0.489424 q^{43} -4.77751 q^{44} -0.0921426 q^{45} -5.15618 q^{46} -5.34729 q^{47} +1.69344 q^{48} +7.84783 q^{49} -4.51474 q^{50} -12.1943 q^{51} -6.50068 q^{52} +12.7475 q^{53} -5.30431 q^{54} -3.32805 q^{55} +3.85329 q^{56} -4.17477 q^{57} -1.51726 q^{58} -1.77586 q^{59} +1.17966 q^{60} -1.44871 q^{61} +8.39045 q^{62} -0.509687 q^{63} +1.00000 q^{64} -4.52842 q^{65} -8.09040 q^{66} +8.42545 q^{67} -7.20090 q^{68} -8.73167 q^{69} +2.68423 q^{70} +0.00331013 q^{71} -0.132273 q^{72} +4.15202 q^{73} +6.98155 q^{74} -7.64542 q^{75} -2.46527 q^{76} -18.4091 q^{77} -11.0085 q^{78} -7.32396 q^{79} +0.696607 q^{80} -8.58568 q^{81} -6.26883 q^{82} +0.983760 q^{83} +6.52530 q^{84} -5.01620 q^{85} -0.489424 q^{86} -2.56938 q^{87} -4.77751 q^{88} -13.3109 q^{89} -0.0921426 q^{90} -25.0490 q^{91} -5.15618 q^{92} +14.2087 q^{93} -5.34729 q^{94} -1.71732 q^{95} +1.69344 q^{96} +0.339392 q^{97} +7.84783 q^{98} +0.631936 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.69344	0.977706	0.488853	−	0.872366i	$-0.337415\pi$
$3$	1.69344	0.977706	0.488853	+	0.872366i	$0.337415\pi$
$4$	1.00000	0.500000
$5$	0.696607	0.311532	0.155766	−	0.987794i	$-0.450215\pi$
$5$	0.696607	0.311532	0.155766	+	0.987794i	$0.450215\pi$
$6$	1.69344	0.691342
$7$	3.85329	1.45641	0.728203	−	0.685362i	$-0.240356\pi$
$7$	3.85329	1.45641	0.728203	+	0.685362i	$0.240356\pi$
$8$	1.00000	0.353553
$9$	−0.132273	−0.0440911
$10$	0.696607	0.220287
$11$	−4.77751	−1.44047	−0.720236	−	0.693729i	$-0.755967\pi$
$11$	−4.77751	−1.44047	−0.720236	+	0.693729i	$0.755967\pi$
$12$	1.69344	0.488853
$13$	−6.50068	−1.80297	−0.901483	−	0.432816i	$-0.857520\pi$
$13$	−6.50068	−1.80297	−0.901483	+	0.432816i	$0.857520\pi$
$14$	3.85329	1.02983
$15$	1.17966	0.304587
$16$	1.00000	0.250000
$17$	−7.20090	−1.74647	−0.873237	−	0.487296i	$-0.837983\pi$
$17$	−7.20090	−1.74647	−0.873237	+	0.487296i	$0.837983\pi$
$18$	−0.132273	−0.0311771
$19$	−2.46527	−0.565571	−0.282785	−	0.959183i	$-0.591258\pi$
$19$	−2.46527	−0.565571	−0.282785	+	0.959183i	$0.591258\pi$
$20$	0.696607	0.155766
$21$	6.52530	1.42394
$22$	−4.77751	−1.01857
$23$	−5.15618	−1.07514	−0.537569	−	0.843220i	$-0.680657\pi$
$23$	−5.15618	−1.07514	−0.537569	+	0.843220i	$0.680657\pi$
$24$	1.69344	0.345671
$25$	−4.51474	−0.902948
$26$	−6.50068	−1.27489
$27$	−5.30431	−1.02081
$28$	3.85329	0.728203
$29$	−1.51726	−0.281748	−0.140874	−	0.990028i	$-0.544991\pi$
$29$	−1.51726	−0.281748	−0.140874	+	0.990028i	$0.544991\pi$
$30$	1.17966	0.215376
$31$	8.39045	1.50697	0.753484	−	0.657466i	$-0.228372\pi$
$31$	8.39045	1.50697	0.753484	+	0.657466i	$0.228372\pi$
$32$	1.00000	0.176777
$33$	−8.09040	−1.40836
$34$	−7.20090	−1.23494
$35$	2.68423	0.453717
$36$	−0.132273	−0.0220456
$37$	6.98155	1.14776	0.573880	−	0.818939i	$-0.305438\pi$
$37$	6.98155	1.14776	0.573880	+	0.818939i	$0.305438\pi$
$38$	−2.46527	−0.399919
$39$	−11.0085	−1.76277
$40$	0.696607	0.110143
$41$	−6.26883	−0.979027	−0.489513	−	0.871996i	$-0.662826\pi$
$41$	−6.26883	−0.979027	−0.489513	+	0.871996i	$0.662826\pi$
$42$	6.52530	1.00688
$43$	−0.489424	−0.0746365	−0.0373182	−	0.999303i	$-0.511882\pi$
$43$	−0.489424	−0.0746365	−0.0373182	+	0.999303i	$0.511882\pi$
$44$	−4.77751	−0.720236
$45$	−0.0921426	−0.0137358
$46$	−5.15618	−0.760237
$47$	−5.34729	−0.779983	−0.389991	−	0.920819i	$-0.627522\pi$
$47$	−5.34729	−0.779983	−0.389991	+	0.920819i	$0.627522\pi$
$48$	1.69344	0.244426
$49$	7.84783	1.12112
$50$	−4.51474	−0.638480
$51$	−12.1943	−1.70754
$52$	−6.50068	−0.901483
$53$	12.7475	1.75100	0.875501	−	0.483217i	$-0.160532\pi$
$53$	12.7475	1.75100	0.875501	+	0.483217i	$0.160532\pi$
$54$	−5.30431	−0.721825
$55$	−3.32805	−0.448754
$56$	3.85329	0.514917
$57$	−4.17477	−0.552962
$58$	−1.51726	−0.199226
$59$	−1.77586	−0.231198	−0.115599	−	0.993296i	$-0.536879\pi$
$59$	−1.77586	−0.231198	−0.115599	+	0.993296i	$0.536879\pi$
$60$	1.17966	0.152293
$61$	−1.44871	−0.185488	−0.0927442	−	0.995690i	$-0.529564\pi$
$61$	−1.44871	−0.185488	−0.0927442	+	0.995690i	$0.529564\pi$
$62$	8.39045	1.06559
$63$	−0.509687	−0.0642146
$64$	1.00000	0.125000
$65$	−4.52842	−0.561682
$66$	−8.09040	−0.995860
$67$	8.42545	1.02933	0.514666	−	0.857390i	$-0.327916\pi$
$67$	8.42545	1.02933	0.514666	+	0.857390i	$0.327916\pi$
$68$	−7.20090	−0.873237
$69$	−8.73167	−1.05117
$70$	2.68423	0.320827
$71$	0.00331013	0.000392840 0	0.000196420	−	1.00000i	$-0.499937\pi$
$71$	0.00331013	0.000392840 0	0.000196420		1.00000i	$0.499937\pi$
$72$	−0.132273	−0.0155886
$73$	4.15202	0.485958	0.242979	−	0.970032i	$-0.421875\pi$
$73$	4.15202	0.485958	0.242979	+	0.970032i	$0.421875\pi$
$74$	6.98155	0.811589
$75$	−7.64542	−0.882817
$76$	−2.46527	−0.282785
$77$	−18.4091	−2.09791
$78$	−11.0085	−1.24647
$79$	−7.32396	−0.824010	−0.412005	−	0.911182i	$-0.635171\pi$
$79$	−7.32396	−0.824010	−0.412005	+	0.911182i	$0.635171\pi$
$80$	0.696607	0.0778831
$81$	−8.58568	−0.953965
$82$	−6.26883	−0.692276
$83$	0.983760	0.107982	0.0539908	−	0.998541i	$-0.482806\pi$
$83$	0.983760	0.107982	0.0539908	+	0.998541i	$0.482806\pi$
$84$	6.52530	0.711968
$85$	−5.01620	−0.544083
$86$	−0.489424	−0.0527759
$87$	−2.56938	−0.275467
$88$	−4.77751	−0.509284
$89$	−13.3109	−1.41095	−0.705476	−	0.708733i	$-0.749267\pi$
$89$	−13.3109	−1.41095	−0.705476	+	0.708733i	$0.749267\pi$
$90$	−0.0921426	−0.00971268
$91$	−25.0490	−2.62585
$92$	−5.15618	−0.537569
$93$	14.2087	1.47337
$94$	−5.34729	−0.551531
$95$	−1.71732	−0.176194
$96$	1.69344	0.172836
$97$	0.339392	0.0344601	0.0172300	−	0.999852i	$-0.494515\pi$
$97$	0.339392	0.0344601	0.0172300	+	0.999852i	$0.494515\pi$
$98$	7.84783	0.792750
$99$	0.631936	0.0635120
$100$	−4.51474	−0.451474
$101$	−19.8605	−1.97619	−0.988095	−	0.153842i	$-0.950835\pi$
$101$	−19.8605	−1.97619	−0.988095	+	0.153842i	$0.950835\pi$
$102$	−12.1943	−1.20741
$103$	6.20849	0.611741	0.305870	−	0.952073i	$-0.401053\pi$
$103$	6.20849	0.611741	0.305870	+	0.952073i	$0.401053\pi$
$104$	−6.50068	−0.637444
$105$	4.54557	0.443602
$106$	12.7475	1.23815
$107$	16.5325	1.59826	0.799128	−	0.601161i	$-0.205295\pi$
$107$	16.5325	1.59826	0.799128	+	0.601161i	$0.205295\pi$
$108$	−5.30431	−0.510407
$109$	−12.9219	−1.23769	−0.618845	−	0.785513i	$-0.712399\pi$
$109$	−12.9219	−1.23769	−0.618845	+	0.785513i	$0.712399\pi$
$110$	−3.32805	−0.317317
$111$	11.8228	1.12217
$112$	3.85329	0.364102
$113$	18.1040	1.70308	0.851539	−	0.524291i	$-0.175670\pi$
$113$	18.1040	1.70308	0.851539	+	0.524291i	$0.175670\pi$
$114$	−4.17477	−0.391003
$115$	−3.59183	−0.334940
$116$	−1.51726	−0.140874
$117$	0.859867	0.0794947
$118$	−1.77586	−0.163482
$119$	−27.7471	−2.54358
$120$	1.17966	0.107688
$121$	11.8246	1.07496
$122$	−1.44871	−0.131160
$123$	−10.6159	−0.957200
$124$	8.39045	0.753484
$125$	−6.62804	−0.592830
$126$	−0.509687	−0.0454065
$127$	8.97301	0.796226	0.398113	−	0.917336i	$-0.369665\pi$
$127$	8.97301	0.796226	0.398113	+	0.917336i	$0.369665\pi$
$128$	1.00000	0.0883883
$129$	−0.828808	−0.0729725
$130$	−4.52842	−0.397169
$131$	−15.1069	−1.31989	−0.659947	−	0.751312i	$-0.729421\pi$
$131$	−15.1069	−1.31989	−0.659947	+	0.751312i	$0.729421\pi$
$132$	−8.09040	−0.704179
$133$	−9.49938	−0.823701
$134$	8.42545	0.727848
$135$	−3.69502	−0.318017
$136$	−7.20090	−0.617472
$137$	−9.38913	−0.802167	−0.401084	−	0.916041i	$-0.631366\pi$
$137$	−9.38913	−0.802167	−0.401084	+	0.916041i	$0.631366\pi$
$138$	−8.73167	−0.743289
$139$	2.21022	0.187469	0.0937343	−	0.995597i	$-0.470120\pi$
$139$	2.21022	0.187469	0.0937343	+	0.995597i	$0.470120\pi$
$140$	2.68423	0.226859
$141$	−9.05530	−0.762594
$142$	0.00331013	0.000277780 0
$143$	31.0570	2.59712
$144$	−0.132273	−0.0110228
$145$	−1.05693	−0.0877736
$146$	4.15202	0.343624
$147$	13.2898	1.09612
$148$	6.98155	0.573880
$149$	21.8214	1.78768	0.893838	−	0.448390i	$-0.148003\pi$
$149$	21.8214	1.78768	0.893838	+	0.448390i	$0.148003\pi$
$150$	−7.64542	−0.624246
$151$	6.90392	0.561833	0.280917	−	0.959732i	$-0.409362\pi$
$151$	6.90392	0.561833	0.280917	+	0.959732i	$0.409362\pi$
$152$	−2.46527	−0.199959
$153$	0.952486	0.0770040
$154$	−18.4091	−1.48345
$155$	5.84485	0.469469
$156$	−11.0085	−0.881385
$157$	3.57101	0.284998	0.142499	−	0.989795i	$-0.454486\pi$
$157$	3.57101	0.284998	0.142499	+	0.989795i	$0.454486\pi$
$158$	−7.32396	−0.582663
$159$	21.5871	1.71196
$160$	0.696607	0.0550716
$161$	−19.8683	−1.56584
$162$	−8.58568	−0.674555
$163$	23.7379	1.85930	0.929648	−	0.368448i	$-0.120111\pi$
$163$	23.7379	1.85930	0.929648	+	0.368448i	$0.120111\pi$
$164$	−6.26883	−0.489513
$165$	−5.63583	−0.438749
$166$	0.983760	0.0763546
$167$	−10.8619	−0.840517	−0.420258	−	0.907405i	$-0.638061\pi$
$167$	−10.8619	−0.840517	−0.420258	+	0.907405i	$0.638061\pi$
$168$	6.52530	0.503438
$169$	29.2589	2.25068
$170$	−5.01620	−0.384725
$171$	0.326089	0.0249366
$172$	−0.489424	−0.0373182
$173$	3.57776	0.272012	0.136006	−	0.990708i	$-0.456573\pi$
$173$	3.57776	0.272012	0.136006	+	0.990708i	$0.456573\pi$
$174$	−2.56938	−0.194784
$175$	−17.3966	−1.31506
$176$	−4.77751	−0.360118
$177$	−3.00731	−0.226044
$178$	−13.3109	−0.997694
$179$	−15.3840	−1.14985	−0.574926	−	0.818205i	$-0.694969\pi$
$179$	−15.3840	−1.14985	−0.574926	+	0.818205i	$0.694969\pi$
$180$	−0.0921426	−0.00686790
$181$	−19.4743	−1.44751	−0.723757	−	0.690055i	$-0.757586\pi$
$181$	−19.4743	−1.44751	−0.723757	+	0.690055i	$0.757586\pi$
$182$	−25.0490	−1.85676
$183$	−2.45330	−0.181353
$184$	−5.15618	−0.380119
$185$	4.86340	0.357564
$186$	14.2087	1.04183
$187$	34.4023	2.51575
$188$	−5.34729	−0.389991
$189$	−20.4390	−1.48672
$190$	−1.71732	−0.124588
$191$	−17.1405	−1.24024	−0.620121	−	0.784506i	$-0.712916\pi$
$191$	−17.1405	−1.24024	−0.620121	+	0.784506i	$0.712916\pi$
$192$	1.69344	0.122213
$193$	10.5647	0.760465	0.380233	−	0.924891i	$-0.375844\pi$
$193$	10.5647	0.760465	0.380233	+	0.924891i	$0.375844\pi$
$194$	0.339392	0.0243669
$195$	−7.66860	−0.549160
$196$	7.84783	0.560559
$197$	3.73193	0.265889	0.132945	−	0.991123i	$-0.457557\pi$
$197$	3.73193	0.265889	0.132945	+	0.991123i	$0.457557\pi$
$198$	0.631936	0.0449098
$199$	−5.30337	−0.375946	−0.187973	−	0.982174i	$-0.560192\pi$
$199$	−5.30337	−0.375946	−0.187973	+	0.982174i	$0.560192\pi$
$200$	−4.51474	−0.319240
$201$	14.2680	1.00638
$202$	−19.8605	−1.39738
$203$	−5.84644	−0.410339
$204$	−12.1943	−0.853769
$205$	−4.36691	−0.304998
$206$	6.20849	0.432566
$207$	0.682025	0.0474040
$208$	−6.50068	−0.450741
$209$	11.7778	0.814689
$210$	4.54557	0.313674
$211$	−3.37342	−0.232236	−0.116118	−	0.993235i	$-0.537045\pi$
$211$	−3.37342	−0.232236	−0.116118	+	0.993235i	$0.537045\pi$
$212$	12.7475	0.875501
$213$	0.00560550	0.000384082 0
$214$	16.5325	1.13014
$215$	−0.340936	−0.0232517
$216$	−5.30431	−0.360912
$217$	32.3308	2.19476
$218$	−12.9219	−0.875178
$219$	7.03119	0.475124
$220$	−3.32805	−0.224377
$221$	46.8107	3.14883
$222$	11.8228	0.793495
$223$	−18.1602	−1.21610	−0.608049	−	0.793900i	$-0.708047\pi$
$223$	−18.1602	−1.21610	−0.608049	+	0.793900i	$0.708047\pi$
$224$	3.85329	0.257459
$225$	0.597179	0.0398120
$226$	18.1040	1.20426
$227$	−2.01088	−0.133467	−0.0667335	−	0.997771i	$-0.521258\pi$
$227$	−2.01088	−0.133467	−0.0667335	+	0.997771i	$0.521258\pi$
$228$	−4.17477	−0.276481
$229$	−21.5315	−1.42284	−0.711420	−	0.702767i	$-0.751948\pi$
$229$	−21.5315	−1.42284	−0.711420	+	0.702767i	$0.751948\pi$
$230$	−3.59183	−0.236839
$231$	−31.1746	−2.05114
$232$	−1.51726	−0.0996130
$233$	9.51368	0.623262	0.311631	−	0.950203i	$-0.399125\pi$
$233$	9.51368	0.623262	0.311631	+	0.950203i	$0.399125\pi$
$234$	0.859867	0.0562113
$235$	−3.72496	−0.242990
$236$	−1.77586	−0.115599
$237$	−12.4027	−0.805640
$238$	−27.7471	−1.79858
$239$	−15.3278	−0.991475	−0.495738	−	0.868472i	$-0.665102\pi$
$239$	−15.3278	−0.991475	−0.495738	+	0.868472i	$0.665102\pi$
$240$	1.17966	0.0761467
$241$	16.9717	1.09324	0.546622	−	0.837379i	$-0.315913\pi$
$241$	16.9717	1.09324	0.546622	+	0.837379i	$0.315913\pi$
$242$	11.8246	0.760111
$243$	1.37361	0.0881170
$244$	−1.44871	−0.0927442
$245$	5.46686	0.349265
$246$	−10.6159	−0.676843
$247$	16.0259	1.01970
$248$	8.39045	0.532794
$249$	1.66593	0.105574
$250$	−6.62804	−0.419194
$251$	3.54328	0.223650	0.111825	−	0.993728i	$-0.464330\pi$
$251$	3.54328	0.223650	0.111825	+	0.993728i	$0.464330\pi$
$252$	−0.509687	−0.0321073
$253$	24.6337	1.54871
$254$	8.97301	0.563017
$255$	−8.49461	−0.531953
$256$	1.00000	0.0625000
$257$	17.1332	1.06874	0.534371	−	0.845250i	$-0.320549\pi$
$257$	17.1332	1.06874	0.534371	+	0.845250i	$0.320549\pi$
$258$	−0.828808	−0.0515994
$259$	26.9019	1.67160
$260$	−4.52842	−0.280841
$261$	0.200693	0.0124226
$262$	−15.1069	−0.933306
$263$	−25.6335	−1.58063	−0.790316	−	0.612700i	$-0.790084\pi$
$263$	−25.6335	−1.58063	−0.790316	+	0.612700i	$0.790084\pi$
$264$	−8.09040	−0.497930
$265$	8.87999	0.545494
$266$	−9.49938	−0.582444
$267$	−22.5412	−1.37950
$268$	8.42545	0.514666
$269$	14.1918	0.865290	0.432645	−	0.901564i	$-0.357580\pi$
$269$	14.1918	0.865290	0.432645	+	0.901564i	$0.357580\pi$
$270$	−3.69502	−0.224872
$271$	0.991546	0.0602321	0.0301161	−	0.999546i	$-0.490412\pi$
$271$	0.991546	0.0602321	0.0301161	+	0.999546i	$0.490412\pi$
$272$	−7.20090	−0.436618
$273$	−42.4189	−2.56731
$274$	−9.38913	−0.567218
$275$	21.5692	1.30067
$276$	−8.73167	−0.525584
$277$	30.5165	1.83356	0.916781	−	0.399391i	$-0.130778\pi$
$277$	30.5165	1.83356	0.916781	+	0.399391i	$0.130778\pi$
$278$	2.21022	0.132560
$279$	−1.10983	−0.0664439
$280$	2.68423	0.160413
$281$	8.84850	0.527857	0.263929	−	0.964542i	$-0.414982\pi$
$281$	8.84850	0.527857	0.263929	+	0.964542i	$0.414982\pi$
$282$	−9.05530	−0.539235
$283$	−19.3888	−1.15254	−0.576272	−	0.817258i	$-0.695493\pi$
$283$	−19.3888	−1.15254	−0.576272	+	0.817258i	$0.695493\pi$
$284$	0.00331013	0.000196420 0
$285$	−2.90818	−0.172265
$286$	31.0570	1.83644
$287$	−24.1556	−1.42586
$288$	−0.132273	−0.00779428
$289$	34.8529	2.05017
$290$	−1.05693	−0.0620653
$291$	0.574739	0.0336918
$292$	4.15202	0.242979
$293$	14.4705	0.845378	0.422689	−	0.906275i	$-0.361086\pi$
$293$	14.4705	0.845378	0.422689	+	0.906275i	$0.361086\pi$
$294$	13.2898	0.775077
$295$	−1.23708	−0.0720256
$296$	6.98155	0.405794
$297$	25.3413	1.47045
$298$	21.8214	1.26408
$299$	33.5187	1.93844
$300$	−7.64542	−0.441409
$301$	−1.88589	−0.108701
$302$	6.90392	0.397276
$303$	−33.6324	−1.93213
$304$	−2.46527	−0.141393
$305$	−1.00918	−0.0577856
$306$	0.952486	0.0544500
$307$	−5.26760	−0.300638	−0.150319	−	0.988638i	$-0.548030\pi$
$307$	−5.26760	−0.300638	−0.150319	+	0.988638i	$0.548030\pi$
$308$	−18.4091	−1.04896
$309$	10.5137	0.598103
$310$	5.84485	0.331965
$311$	0.500328	0.0283710	0.0141855	−	0.999899i	$-0.495484\pi$
$311$	0.500328	0.0283710	0.0141855	+	0.999899i	$0.495484\pi$
$312$	−11.0085	−0.623233
$313$	−5.00347	−0.282813	−0.141406	−	0.989952i	$-0.545162\pi$
$313$	−5.00347	−0.282813	−0.141406	+	0.989952i	$0.545162\pi$
$314$	3.57101	0.201524
$315$	−0.355052	−0.0200049
$316$	−7.32396	−0.412005
$317$	−4.79286	−0.269194	−0.134597	−	0.990900i	$-0.542974\pi$
$317$	−4.79286	−0.269194	−0.134597	+	0.990900i	$0.542974\pi$
$318$	21.5871	1.21054
$319$	7.24871	0.405850
$320$	0.696607	0.0389415
$321$	27.9967	1.56262
$322$	−19.8683	−1.10721
$323$	17.7521	0.987755
$324$	−8.58568	−0.476982
$325$	29.3489	1.62798
$326$	23.7379	1.31472
$327$	−21.8823	−1.21010
$328$	−6.26883	−0.346138
$329$	−20.6047	−1.13597
$330$	−5.63583	−0.310242
$331$	−9.97439	−0.548242	−0.274121	−	0.961695i	$-0.588387\pi$
$331$	−9.97439	−0.548242	−0.274121	+	0.961695i	$0.588387\pi$
$332$	0.983760	0.0539908
$333$	−0.923473	−0.0506060
$334$	−10.8619	−0.594335
$335$	5.86923	0.320670
$336$	6.52530	0.355984
$337$	27.7668	1.51255	0.756276	−	0.654253i	$-0.227017\pi$
$337$	27.7668	1.51255	0.756276	+	0.654253i	$0.227017\pi$
$338$	29.2589	1.59147
$339$	30.6579	1.66511
$340$	−5.01620	−0.272042
$341$	−40.0854	−2.17075
$342$	0.326089	0.0176329
$343$	3.26693	0.176398
$344$	−0.489424	−0.0263880
$345$	−6.08254	−0.327473
$346$	3.57776	0.192342
$347$	18.3122	0.983053	0.491526	−	0.870863i	$-0.336439\pi$
$347$	18.3122	0.983053	0.491526	+	0.870863i	$0.336439\pi$
$348$	−2.56938	−0.137733
$349$	−17.3355	−0.927947	−0.463973	−	0.885849i	$-0.653577\pi$
$349$	−17.3355	−0.927947	−0.463973	+	0.885849i	$0.653577\pi$
$350$	−17.3966	−0.929887
$351$	34.4816	1.84049
$352$	−4.77751	−0.254642
$353$	−26.0117	−1.38446	−0.692231	−	0.721676i	$-0.743372\pi$
$353$	−26.0117	−1.38446	−0.692231	+	0.721676i	$0.743372\pi$
$354$	−3.00731	−0.159837
$355$	0.00230586	0.000122382 0
$356$	−13.3109	−0.705476
$357$	−46.9880	−2.48687
$358$	−15.3840	−0.813068
$359$	−34.7012	−1.83146	−0.915728	−	0.401798i	$-0.868385\pi$
$359$	−34.7012	−1.83146	−0.915728	+	0.401798i	$0.868385\pi$
$360$	−0.0921426	−0.00485634
$361$	−12.9225	−0.680130
$362$	−19.4743	−1.02355
$363$	20.0241	1.05099
$364$	−25.0490	−1.31292
$365$	2.89233	0.151391
$366$	−2.45330	−0.128236
$367$	0.628499	0.0328074	0.0164037	−	0.999865i	$-0.494778\pi$
$367$	0.628499	0.0328074	0.0164037	+	0.999865i	$0.494778\pi$
$368$	−5.15618	−0.268785
$369$	0.829199	0.0431664
$370$	4.86340	0.252836
$371$	49.1197	2.55017
$372$	14.2087	0.736686
$373$	−19.1796	−0.993083	−0.496542	−	0.868013i	$-0.665397\pi$
$373$	−19.1796	−0.993083	−0.496542	+	0.868013i	$0.665397\pi$
$374$	34.4023	1.77890
$375$	−11.2242	−0.579613
$376$	−5.34729	−0.275766
$377$	9.86322	0.507982
$378$	−20.4390	−1.05127
$379$	−5.28616	−0.271532	−0.135766	−	0.990741i	$-0.543350\pi$
$379$	−5.28616	−0.271532	−0.135766	+	0.990741i	$0.543350\pi$
$380$	−1.71732	−0.0880968
$381$	15.1952	0.778475
$382$	−17.1405	−0.876983
$383$	22.8583	1.16800	0.584002	−	0.811753i	$-0.301486\pi$
$383$	22.8583	1.16800	0.584002	+	0.811753i	$0.301486\pi$
$384$	1.69344	0.0864178
$385$	−12.8239	−0.653567
$386$	10.5647	0.537730
$387$	0.0647377	0.00329080
$388$	0.339392	0.0172300
$389$	−28.3865	−1.43925	−0.719626	−	0.694361i	$-0.755687\pi$
$389$	−28.3865	−1.43925	−0.719626	+	0.694361i	$0.755687\pi$
$390$	−7.66860	−0.388315
$391$	37.1291	1.87770
$392$	7.84783	0.396375
$393$	−25.5825	−1.29047
$394$	3.73193	0.188012
$395$	−5.10193	−0.256706
$396$	0.631936	0.0317560
$397$	10.3486	0.519379	0.259690	−	0.965692i	$-0.416380\pi$
$397$	10.3486	0.519379	0.259690	+	0.965692i	$0.416380\pi$
$398$	−5.30337	−0.265834
$399$	−16.0866	−0.805337
$400$	−4.51474	−0.225737
$401$	−27.8515	−1.39084	−0.695420	−	0.718604i	$-0.744782\pi$
$401$	−27.8515	−1.39084	−0.695420	+	0.718604i	$0.744782\pi$
$402$	14.2680	0.711622
$403$	−54.5436	−2.71701
$404$	−19.8605	−0.988095
$405$	−5.98085	−0.297191
$406$	−5.84644	−0.290154
$407$	−33.3544	−1.65332
$408$	−12.1943	−0.603706
$409$	−15.2472	−0.753927	−0.376964	−	0.926228i	$-0.623032\pi$
$409$	−15.2472	−0.753927	−0.376964	+	0.926228i	$0.623032\pi$
$410$	−4.36691	−0.215666
$411$	−15.8999	−0.784284
$412$	6.20849	0.305870
$413$	−6.84292	−0.336718
$414$	0.682025	0.0335197
$415$	0.685294	0.0336398
$416$	−6.50068	−0.318722
$417$	3.74287	0.183289
$418$	11.7778	0.576072
$419$	34.7462	1.69746	0.848731	−	0.528826i	$-0.177367\pi$
$419$	34.7462	1.69746	0.848731	+	0.528826i	$0.177367\pi$
$420$	4.54557	0.221801
$421$	7.25244	0.353463	0.176731	−	0.984259i	$-0.443448\pi$
$421$	7.25244	0.353463	0.176731	+	0.984259i	$0.443448\pi$
$422$	−3.37342	−0.164215
$423$	0.707304	0.0343903
$424$	12.7475	0.619073
$425$	32.5102	1.57697
$426$	0.00560550	0.000271587 0
$427$	−5.58230	−0.270146
$428$	16.5325	0.799128
$429$	52.5931	2.53922
$430$	−0.340936	−0.0164414
$431$	37.0642	1.78532	0.892660	−	0.450730i	$-0.148836\pi$
$431$	37.0642	1.78532	0.892660	+	0.450730i	$0.148836\pi$
$432$	−5.30431	−0.255204
$433$	−33.2937	−1.59999	−0.799997	−	0.600004i	$-0.795166\pi$
$433$	−33.2937	−1.59999	−0.799997	+	0.600004i	$0.795166\pi$
$434$	32.3308	1.55193
$435$	−1.78985	−0.0858168
$436$	−12.9219	−0.618845
$437$	12.7114	0.608067
$438$	7.03119	0.335963
$439$	−32.3964	−1.54620	−0.773098	−	0.634287i	$-0.781294\pi$
$439$	−32.3964	−1.54620	−0.773098	+	0.634287i	$0.781294\pi$
$440$	−3.32805	−0.158658
$441$	−1.03806	−0.0494314
$442$	46.8107	2.22656
$443$	−17.2854	−0.821253	−0.410627	−	0.911804i	$-0.634690\pi$
$443$	−17.2854	−0.821253	−0.410627	+	0.911804i	$0.634690\pi$
$444$	11.8228	0.561086
$445$	−9.27247	−0.439557
$446$	−18.1602	−0.859910
$447$	36.9531	1.74782
$448$	3.85329	0.182051
$449$	19.6498	0.927331	0.463665	−	0.886010i	$-0.346534\pi$
$449$	19.6498	0.927331	0.463665	+	0.886010i	$0.346534\pi$
$450$	0.597179	0.0281513
$451$	29.9494	1.41026
$452$	18.1040	0.851539
$453$	11.6914	0.549308
$454$	−2.01088	−0.0943754
$455$	−17.4493	−0.818037
$456$	−4.17477	−0.195502
$457$	29.5235	1.38105	0.690525	−	0.723308i	$-0.257379\pi$
$457$	29.5235	1.38105	0.690525	+	0.723308i	$0.257379\pi$
$458$	−21.5315	−1.00610
$459$	38.1958	1.78283
$460$	−3.59183	−0.167470
$461$	−2.17064	−0.101097	−0.0505484	−	0.998722i	$-0.516097\pi$
$461$	−2.17064	−0.101097	−0.0505484	+	0.998722i	$0.516097\pi$
$462$	−31.1746	−1.45038
$463$	38.9689	1.81104	0.905520	−	0.424304i	$-0.139481\pi$
$463$	38.9689	1.81104	0.905520	+	0.424304i	$0.139481\pi$
$464$	−1.51726	−0.0704370
$465$	9.89787	0.459003
$466$	9.51368	0.440713
$467$	−18.8293	−0.871316	−0.435658	−	0.900112i	$-0.643484\pi$
$467$	−18.8293	−0.871316	−0.435658	+	0.900112i	$0.643484\pi$
$468$	0.859867	0.0397474
$469$	32.4657	1.49913
$470$	−3.72496	−0.171820
$471$	6.04728	0.278644
$472$	−1.77586	−0.0817408
$473$	2.33823	0.107512
$474$	−12.4027	−0.569673
$475$	11.1300	0.510681
$476$	−27.7471	−1.27179
$477$	−1.68615	−0.0772036
$478$	−15.3278	−0.701079
$479$	−16.9560	−0.774739	−0.387369	−	0.921925i	$-0.626616\pi$
$479$	−16.9560	−0.774739	−0.387369	+	0.921925i	$0.626616\pi$
$480$	1.17966	0.0538439
$481$	−45.3848	−2.06937
$482$	16.9717	0.773041
$483$	−33.6456	−1.53093
$484$	11.8246	0.537480
$485$	0.236423	0.0107354
$486$	1.37361	0.0623081
$487$	−0.690603	−0.0312942	−0.0156471	−	0.999878i	$-0.504981\pi$
$487$	−0.690603	−0.0312942	−0.0156471	+	0.999878i	$0.504981\pi$
$488$	−1.44871	−0.0655800
$489$	40.1986	1.81785
$490$	5.46686	0.246967
$491$	−11.5863	−0.522880	−0.261440	−	0.965220i	$-0.584197\pi$
$491$	−11.5863	−0.522880	−0.261440	+	0.965220i	$0.584197\pi$
$492$	−10.6159	−0.478600
$493$	10.9256	0.492065
$494$	16.0259	0.721040
$495$	0.440212	0.0197860
$496$	8.39045	0.376742
$497$	0.0127549	0.000572135 0
$498$	1.66593	0.0746523
$499$	−36.5994	−1.63842	−0.819208	−	0.573496i	$-0.805587\pi$
$499$	−36.5994	−1.63842	−0.819208	+	0.573496i	$0.805587\pi$
$500$	−6.62804	−0.296415
$501$	−18.3939	−0.821778
$502$	3.54328	0.158145
$503$	−35.0124	−1.56113	−0.780563	−	0.625077i	$-0.785068\pi$
$503$	−35.0124	−1.56113	−0.780563	+	0.625077i	$0.785068\pi$
$504$	−0.509687	−0.0227033
$505$	−13.8350	−0.615647
$506$	24.6337	1.09510
$507$	49.5480	2.20051
$508$	8.97301	0.398113
$509$	8.07791	0.358047	0.179024	−	0.983845i	$-0.442706\pi$
$509$	8.07791	0.358047	0.179024	+	0.983845i	$0.442706\pi$
$510$	−8.49461	−0.376148
$511$	15.9989	0.707752
$512$	1.00000	0.0441942
$513$	13.0765	0.577343
$514$	17.1332	0.755714
$515$	4.32488	0.190577
$516$	−0.828808	−0.0364863
$517$	25.5467	1.12354
$518$	26.9019	1.18200
$519$	6.05871	0.265948
$520$	−4.52842	−0.198585
$521$	26.5511	1.16323	0.581613	−	0.813465i	$-0.302422\pi$
$521$	26.5511	1.16323	0.581613	+	0.813465i	$0.302422\pi$
$522$	0.200693	0.00878409
$523$	−14.2212	−0.621848	−0.310924	−	0.950435i	$-0.600638\pi$
$523$	−14.2212	−0.621848	−0.310924	+	0.950435i	$0.600638\pi$
$524$	−15.1069	−0.659947
$525$	−29.4600	−1.28574
$526$	−25.6335	−1.11768
$527$	−60.4187	−2.63188
$528$	−8.09040	−0.352090
$529$	3.58621	0.155922
$530$	8.87999	0.385722
$531$	0.234900	0.0101938
$532$	−9.49938	−0.411850
$533$	40.7517	1.76515
$534$	−22.5412	−0.975452
$535$	11.5166	0.497908
$536$	8.42545	0.363924
$537$	−26.0518	−1.12422
$538$	14.1918	0.611852
$539$	−37.4930	−1.61494
$540$	−3.69502	−0.159008
$541$	−2.87083	−0.123427	−0.0617134	−	0.998094i	$-0.519656\pi$
$541$	−2.87083	−0.123427	−0.0617134	+	0.998094i	$0.519656\pi$
$542$	0.991546	0.0425906
$543$	−32.9785	−1.41524
$544$	−7.20090	−0.308736
$545$	−9.00146	−0.385580
$546$	−42.4189	−1.81536
$547$	−22.8554	−0.977227	−0.488614	−	0.872500i	$-0.662497\pi$
$547$	−22.8554	−0.977227	−0.488614	+	0.872500i	$0.662497\pi$
$548$	−9.38913	−0.401084
$549$	0.191626	0.00817839
$550$	21.5692	0.919713
$551$	3.74045	0.159348
$552$	−8.73167	−0.371644
$553$	−28.2213	−1.20009
$554$	30.5165	1.29652
$555$	8.23586	0.349593
$556$	2.21022	0.0937343
$557$	7.61741	0.322760	0.161380	−	0.986892i	$-0.448406\pi$
$557$	7.61741	0.322760	0.161380	+	0.986892i	$0.448406\pi$
$558$	−1.10983	−0.0469829
$559$	3.18159	0.134567
$560$	2.68423	0.113429
$561$	58.2581	2.45966
$562$	8.84850	0.373251
$563$	11.6860	0.492505	0.246253	−	0.969206i	$-0.420801\pi$
$563$	11.6860	0.492505	0.246253	+	0.969206i	$0.420801\pi$
$564$	−9.05530	−0.381297
$565$	12.6114	0.530564
$566$	−19.3888	−0.814971
$567$	−33.0831	−1.38936
$568$	0.00331013	0.000138890 0
$569$	−34.6434	−1.45233	−0.726163	−	0.687523i	$-0.758698\pi$
$569$	−34.6434	−1.45233	−0.726163	+	0.687523i	$0.758698\pi$
$570$	−2.90818	−0.121810
$571$	−34.4915	−1.44342	−0.721712	−	0.692193i	$-0.756645\pi$
$571$	−34.4915	−1.44342	−0.721712	+	0.692193i	$0.756645\pi$
$572$	31.0570	1.29856
$573$	−29.0263	−1.21259
$574$	−24.1556	−1.00824
$575$	23.2788	0.970793
$576$	−0.132273	−0.00551139
$577$	7.48815	0.311736	0.155868	−	0.987778i	$-0.450183\pi$
$577$	7.48815	0.311736	0.155868	+	0.987778i	$0.450183\pi$
$578$	34.8529	1.44969
$579$	17.8907	0.743511
$580$	−1.05693	−0.0438868
$581$	3.79071	0.157265
$582$	0.574739	0.0238237
$583$	−60.9012	−2.52227
$584$	4.15202	0.171812
$585$	0.598990	0.0247652
$586$	14.4705	0.597773
$587$	−12.7810	−0.527529	−0.263764	−	0.964587i	$-0.584964\pi$
$587$	−12.7810	−0.527529	−0.263764	+	0.964587i	$0.584964\pi$
$588$	13.2898	0.548062
$589$	−20.6847	−0.852297
$590$	−1.23708	−0.0509298
$591$	6.31979	0.259961
$592$	6.98155	0.286940
$593$	−12.2112	−0.501454	−0.250727	−	0.968058i	$-0.580670\pi$
$593$	−12.2112	−0.501454	−0.250727	+	0.968058i	$0.580670\pi$
$594$	25.3413	1.03977
$595$	−19.3289	−0.792406
$596$	21.8214	0.893838
$597$	−8.98092	−0.367565
$598$	33.5187	1.37068
$599$	5.95225	0.243202	0.121601	−	0.992579i	$-0.461197\pi$
$599$	5.95225	0.243202	0.121601	+	0.992579i	$0.461197\pi$
$600$	−7.64542	−0.312123
$601$	−37.7437	−1.53960	−0.769799	−	0.638286i	$-0.779644\pi$
$601$	−37.7437	−1.53960	−0.769799	+	0.638286i	$0.779644\pi$
$602$	−1.88589	−0.0768632
$603$	−1.11446	−0.0453844
$604$	6.90392	0.280917
$605$	8.23707	0.334885
$606$	−33.6324	−1.36622
$607$	1.78387	0.0724050	0.0362025	−	0.999344i	$-0.488474\pi$
$607$	1.78387	0.0724050	0.0362025	+	0.999344i	$0.488474\pi$
$608$	−2.46527	−0.0999797
$609$	−9.90057	−0.401191
$610$	−1.00918	−0.0408606
$611$	34.7611	1.40628
$612$	0.952486	0.0385020
$613$	−21.1928	−0.855970	−0.427985	−	0.903786i	$-0.640776\pi$
$613$	−21.1928	−0.855970	−0.427985	+	0.903786i	$0.640776\pi$
$614$	−5.26760	−0.212583
$615$	−7.39509	−0.298199
$616$	−18.4091	−0.741724
$617$	13.6741	0.550497	0.275248	−	0.961373i	$-0.411240\pi$
$617$	13.6741	0.550497	0.275248	+	0.961373i	$0.411240\pi$
$618$	10.5137	0.422923
$619$	20.3270	0.817013	0.408506	−	0.912755i	$-0.366050\pi$
$619$	20.3270	0.817013	0.408506	+	0.912755i	$0.366050\pi$
$620$	5.84485	0.234735
$621$	27.3500	1.09752
$622$	0.500328	0.0200613
$623$	−51.2907	−2.05492
$624$	−11.0085	−0.440692
$625$	17.9566	0.718262
$626$	−5.00347	−0.199979
$627$	19.9450	0.796526
$628$	3.57101	0.142499
$629$	−50.2734	−2.00453
$630$	−0.355052	−0.0141456
$631$	22.5266	0.896771	0.448386	−	0.893840i	$-0.351999\pi$
$631$	22.5266	0.896771	0.448386	+	0.893840i	$0.351999\pi$
$632$	−7.32396	−0.291332
$633$	−5.71267	−0.227058
$634$	−4.79286	−0.190349
$635$	6.25066	0.248050
$636$	21.5871	0.855982
$637$	−51.0162	−2.02134
$638$	7.24871	0.286979
$639$	−0.000437842 0	−1.73208e−5 0
$640$	0.696607	0.0275358
$641$	−35.4569	−1.40046	−0.700231	−	0.713916i	$-0.746920\pi$
$641$	−35.4569	−1.40046	−0.700231	+	0.713916i	$0.746920\pi$
$642$	27.9967	1.10494
$643$	30.5459	1.20461	0.602306	−	0.798265i	$-0.294249\pi$
$643$	30.5459	1.20461	0.602306	+	0.798265i	$0.294249\pi$
$644$	−19.8683	−0.782919
$645$	−0.577354	−0.0227333
$646$	17.7521	0.698448
$647$	−5.05297	−0.198653	−0.0993264	−	0.995055i	$-0.531669\pi$
$647$	−5.05297	−0.198653	−0.0993264	+	0.995055i	$0.531669\pi$
$648$	−8.58568	−0.337278
$649$	8.48420	0.333034
$650$	29.3489	1.15116
$651$	54.7502	2.14583
$652$	23.7379	0.929648
$653$	39.6759	1.55264	0.776318	−	0.630341i	$-0.217085\pi$
$653$	39.6759	1.55264	0.776318	+	0.630341i	$0.217085\pi$
$654$	−21.8823	−0.855667
$655$	−10.5236	−0.411190
$656$	−6.26883	−0.244757
$657$	−0.549202	−0.0214264
$658$	−20.6047	−0.803253
$659$	−36.5564	−1.42404	−0.712018	−	0.702162i	$-0.752218\pi$
$659$	−36.5564	−1.42404	−0.712018	+	0.702162i	$0.752218\pi$
$660$	−5.63583	−0.219375
$661$	−17.5869	−0.684053	−0.342026	−	0.939690i	$-0.611113\pi$
$661$	−17.5869	−0.684053	−0.342026	+	0.939690i	$0.611113\pi$
$662$	−9.97439	−0.387666
$663$	79.2710	3.07863
$664$	0.983760	0.0381773
$665$	−6.61734	−0.256609
$666$	−0.923473	−0.0357838
$667$	7.82326	0.302918
$668$	−10.8619	−0.420258
$669$	−30.7531	−1.18899
$670$	5.86923	0.226748
$671$	6.92122	0.267191
$672$	6.52530	0.251719
$673$	−13.1881	−0.508363	−0.254181	−	0.967157i	$-0.581806\pi$
$673$	−13.1881	−0.508363	−0.254181	+	0.967157i	$0.581806\pi$
$674$	27.7668	1.06954
$675$	23.9476	0.921742
$676$	29.2589	1.12534
$677$	−25.0451	−0.962561	−0.481280	−	0.876567i	$-0.659828\pi$
$677$	−25.0451	−0.962561	−0.481280	+	0.876567i	$0.659828\pi$
$678$	30.6579	1.17741
$679$	1.30778	0.0501878
$680$	−5.01620	−0.192362
$681$	−3.40530	−0.130491
$682$	−40.0854	−1.53495
$683$	11.8417	0.453112	0.226556	−	0.973998i	$-0.427253\pi$
$683$	11.8417	0.453112	0.226556	+	0.973998i	$0.427253\pi$
$684$	0.326089	0.0124683
$685$	−6.54054	−0.249901
$686$	3.26693	0.124732
$687$	−36.4622	−1.39112
$688$	−0.489424	−0.0186591
$689$	−82.8674	−3.15699
$690$	−6.08254	−0.231558
$691$	−0.672436	−0.0255807	−0.0127903	−	0.999918i	$-0.504071\pi$
$691$	−0.672436	−0.0255807	−0.0127903	+	0.999918i	$0.504071\pi$
$692$	3.57776	0.136006
$693$	2.43503	0.0924993
$694$	18.3122	0.695123
$695$	1.53966	0.0584025
$696$	−2.56938	−0.0973922
$697$	45.1412	1.70984
$698$	−17.3355	−0.656158
$699$	16.1108	0.609367
$700$	−17.3966	−0.657529
$701$	2.56088	0.0967231	0.0483615	−	0.998830i	$-0.484600\pi$
$701$	2.56088	0.0967231	0.0483615	+	0.998830i	$0.484600\pi$
$702$	34.4816	1.30142
$703$	−17.2114	−0.649139
$704$	−4.77751	−0.180059
$705$	−6.30799	−0.237573
$706$	−26.0117	−0.978963
$707$	−76.5281	−2.87814
$708$	−3.00731	−0.113022
$709$	−1.69832	−0.0637819	−0.0318910	−	0.999491i	$-0.510153\pi$
$709$	−1.69832	−0.0637819	−0.0318910	+	0.999491i	$0.510153\pi$
$710$	0.00230586	8.65375e−5 0
$711$	0.968765	0.0363315
$712$	−13.3109	−0.498847
$713$	−43.2627	−1.62020
$714$	−46.9880	−1.75848
$715$	21.6346	0.809087
$716$	−15.3840	−0.574926
$717$	−25.9567	−0.969371
$718$	−34.7012	−1.29504
$719$	−48.4494	−1.80686	−0.903430	−	0.428736i	$-0.858959\pi$
$719$	−48.4494	−1.80686	−0.903430	+	0.428736i	$0.858959\pi$
$720$	−0.0921426	−0.00343395
$721$	23.9231	0.890943
$722$	−12.9225	−0.480924
$723$	28.7405	1.06887
$724$	−19.4743	−0.723757
$725$	6.85003	0.254404
$726$	20.0241	0.743165
$727$	−13.1150	−0.486407	−0.243203	−	0.969975i	$-0.578198\pi$
$727$	−13.1150	−0.486407	−0.243203	+	0.969975i	$0.578198\pi$
$728$	−25.0490	−0.928378
$729$	28.0832	1.04012
$730$	2.89233	0.107050
$731$	3.52429	0.130351
$732$	−2.45330	−0.0906765
$733$	12.5308	0.462834	0.231417	−	0.972855i	$-0.425664\pi$
$733$	12.5308	0.462834	0.231417	+	0.972855i	$0.425664\pi$
$734$	0.628499	0.0231983
$735$	9.25777	0.341478
$736$	−5.15618	−0.190059
$737$	−40.2526	−1.48273
$738$	0.829199	0.0305232
$739$	−19.1273	−0.703608	−0.351804	−	0.936074i	$-0.614432\pi$
$739$	−19.1273	−0.703608	−0.351804	+	0.936074i	$0.614432\pi$
$740$	4.86340	0.178782
$741$	27.1389	0.996971
$742$	49.1197	1.80324
$743$	−8.94213	−0.328055	−0.164028	−	0.986456i	$-0.552449\pi$
$743$	−8.94213	−0.328055	−0.164028	+	0.986456i	$0.552449\pi$
$744$	14.2087	0.520916
$745$	15.2009	0.556919
$746$	−19.1796	−0.702216
$747$	−0.130125	−0.00476103
$748$	34.4023	1.25787
$749$	63.7044	2.32771
$750$	−11.2242	−0.409848
$751$	12.7547	0.465426	0.232713	−	0.972545i	$-0.425240\pi$
$751$	12.7547	0.465426	0.232713	+	0.972545i	$0.425240\pi$
$752$	−5.34729	−0.194996
$753$	6.00033	0.218664
$754$	9.86322	0.359197
$755$	4.80932	0.175029
$756$	−20.4390	−0.743360
$757$	8.55775	0.311037	0.155518	−	0.987833i	$-0.450295\pi$
$757$	8.55775	0.311037	0.155518	+	0.987833i	$0.450295\pi$
$758$	−5.28616	−0.192002
$759$	41.7156	1.51418
$760$	−1.71732	−0.0622938
$761$	33.7258	1.22256	0.611279	−	0.791415i	$-0.290655\pi$
$761$	33.7258	1.22256	0.611279	+	0.791415i	$0.290655\pi$
$762$	15.1952	0.550465
$763$	−49.7916	−1.80258
$764$	−17.1405	−0.620121
$765$	0.663509	0.0239892
$766$	22.8583	0.825903
$767$	11.5443	0.416842
$768$	1.69344	0.0611066
$769$	3.90982	0.140992	0.0704959	−	0.997512i	$-0.477542\pi$
$769$	3.90982	0.140992	0.0704959	+	0.997512i	$0.477542\pi$
$770$	−12.8239	−0.462142
$771$	29.0140	1.04492
$772$	10.5647	0.380233
$773$	−4.71821	−0.169702	−0.0848511	−	0.996394i	$-0.527041\pi$
$773$	−4.71821	−0.169702	−0.0848511	+	0.996394i	$0.527041\pi$
$774$	0.0647377	0.00232695
$775$	−37.8807	−1.36071
$776$	0.339392	0.0121835
$777$	45.5567	1.63434
$778$	−28.3865	−1.01771
$779$	15.4543	0.553709
$780$	−7.66860	−0.274580
$781$	−0.0158142	−0.000565876 0
$782$	37.1291	1.32773
$783$	8.04801	0.287612
$784$	7.84783	0.280280
$785$	2.48759	0.0887859
$786$	−25.5825	−0.912499
$787$	−37.6889	−1.34347	−0.671733	−	0.740793i	$-0.734450\pi$
$787$	−37.6889	−1.34347	−0.671733	+	0.740793i	$0.734450\pi$
$788$	3.73193	0.132945
$789$	−43.4088	−1.54539
$790$	−5.10193	−0.181518
$791$	69.7598	2.48037
$792$	0.631936	0.0224549
$793$	9.41761	0.334429
$794$	10.3486	0.367257
$795$	15.0377	0.533332
$796$	−5.30337	−0.187973
$797$	−7.53381	−0.266861	−0.133431	−	0.991058i	$-0.542599\pi$
$797$	−7.53381	−0.266861	−0.133431	+	0.991058i	$0.542599\pi$
$798$	−16.0866	−0.569459
$799$	38.5053	1.36222
$800$	−4.51474	−0.159620
$801$	1.76068	0.0622105
$802$	−27.8515	−0.983472
$803$	−19.8363	−0.700008
$804$	14.2680	0.503192
$805$	−13.8404	−0.487809
$806$	−54.5436	−1.92122
$807$	24.0329	0.845999
$808$	−19.8605	−0.698689
$809$	9.02052	0.317145	0.158572	−	0.987347i	$-0.449311\pi$
$809$	9.02052	0.317145	0.158572	+	0.987347i	$0.449311\pi$
$810$	−5.98085	−0.210146
$811$	−45.4970	−1.59762	−0.798808	−	0.601587i	$-0.794535\pi$
$811$	−45.4970	−1.59762	−0.798808	+	0.601587i	$0.794535\pi$
$812$	−5.84644	−0.205170
$813$	1.67912	0.0588893
$814$	−33.3544	−1.16907
$815$	16.5360	0.579231
$816$	−12.1943	−0.426884
$817$	1.20656	0.0422122
$818$	−15.2472	−0.533107
$819$	3.31331	0.115777
$820$	−4.36691	−0.152499
$821$	−39.5616	−1.38071	−0.690355	−	0.723471i	$-0.742545\pi$
$821$	−39.5616	−1.38071	−0.690355	+	0.723471i	$0.742545\pi$
$822$	−15.8999	−0.554572
$823$	26.8180	0.934815	0.467408	−	0.884042i	$-0.345188\pi$
$823$	26.8180	0.934815	0.467408	+	0.884042i	$0.345188\pi$
$824$	6.20849	0.216283
$825$	36.5260	1.27167
$826$	−6.84292	−0.238096
$827$	−25.0821	−0.872191	−0.436096	−	0.899900i	$-0.643639\pi$
$827$	−25.0821	−0.872191	−0.436096	+	0.899900i	$0.643639\pi$
$828$	0.682025	0.0237020
$829$	6.31690	0.219395	0.109697	−	0.993965i	$-0.465012\pi$
$829$	6.31690	0.219395	0.109697	+	0.993965i	$0.465012\pi$
$830$	0.685294	0.0237869
$831$	51.6778	1.79268
$832$	−6.50068	−0.225371
$833$	−56.5114	−1.95800
$834$	3.74287	0.129605
$835$	−7.56646	−0.261848
$836$	11.7778	0.407344
$837$	−44.5055	−1.53833
$838$	34.7462	1.20029
$839$	34.7921	1.20116	0.600578	−	0.799566i	$-0.294937\pi$
$839$	34.7921	1.20116	0.600578	+	0.799566i	$0.294937\pi$
$840$	4.54557	0.156837
$841$	−26.6979	−0.920618
$842$	7.25244	0.249936
$843$	14.9844	0.516089
$844$	−3.37342	−0.116118
$845$	20.3820	0.701160
$846$	0.707304	0.0243176
$847$	45.5634	1.56558
$848$	12.7475	0.437750
$849$	−32.8337	−1.12685
$850$	32.5102	1.11509
$851$	−35.9981	−1.23400
$852$	0.00560550	0.000192041 0
$853$	16.5138	0.565423	0.282712	−	0.959205i	$-0.408766\pi$
$853$	16.5138	0.565423	0.282712	+	0.959205i	$0.408766\pi$
$854$	−5.58230	−0.191022
$855$	0.227156	0.00776857
$856$	16.5325	0.565069
$857$	−2.75578	−0.0941355	−0.0470677	−	0.998892i	$-0.514988\pi$
$857$	−2.75578	−0.0941355	−0.0470677	+	0.998892i	$0.514988\pi$
$858$	52.5931	1.79550
$859$	49.5623	1.69105	0.845523	−	0.533939i	$-0.179289\pi$
$859$	49.5623	1.69105	0.845523	+	0.533939i	$0.179289\pi$
$860$	−0.340936	−0.0116258
$861$	−40.9060	−1.39407
$862$	37.0642	1.26241
$863$	10.7341	0.365393	0.182696	−	0.983169i	$-0.441518\pi$
$863$	10.7341	0.365393	0.182696	+	0.983169i	$0.441518\pi$
$864$	−5.30431	−0.180456
$865$	2.49229	0.0847405
$866$	−33.2937	−1.13137
$867$	59.0212	2.00446
$868$	32.3308	1.09738
$869$	34.9903	1.18696
$870$	−1.78985	−0.0606816
$871$	−54.7712	−1.85585
$872$	−12.9219	−0.437589
$873$	−0.0448925	−0.00151938
$874$	12.7114	0.429968
$875$	−25.5397	−0.863401
$876$	7.03119	0.237562
$877$	31.6244	1.06788	0.533940	−	0.845522i	$-0.320711\pi$
$877$	31.6244	1.06788	0.533940	+	0.845522i	$0.320711\pi$
$878$	−32.3964	−1.09333
$879$	24.5049	0.826531
$880$	−3.32805	−0.112188
$881$	11.6683	0.393114	0.196557	−	0.980492i	$-0.437024\pi$
$881$	11.6683	0.393114	0.196557	+	0.980492i	$0.437024\pi$
$882$	−1.03806	−0.0349532
$883$	−9.68621	−0.325967	−0.162983	−	0.986629i	$-0.552112\pi$
$883$	−9.68621	−0.325967	−0.162983	+	0.986629i	$0.552112\pi$
$884$	46.8107	1.57442
$885$	−2.09492	−0.0704199
$886$	−17.2854	−0.580714
$887$	−41.5835	−1.39624	−0.698119	−	0.715982i	$-0.745979\pi$
$887$	−41.5835	−1.39624	−0.698119	+	0.715982i	$0.745979\pi$
$888$	11.8228	0.396748
$889$	34.5756	1.15963
$890$	−9.27247	−0.310814
$891$	41.0182	1.37416
$892$	−18.1602	−0.608049
$893$	13.1825	0.441135
$894$	36.9531	1.23590
$895$	−10.7166	−0.358216
$896$	3.85329	0.128729
$897$	56.7618	1.89522
$898$	19.6498	0.655722
$899$	−12.7305	−0.424585
$900$	0.597179	0.0199060
$901$	−91.7933	−3.05808
$902$	29.9494	0.997205
$903$	−3.19364	−0.106278
$904$	18.1040	0.602129
$905$	−13.5659	−0.450947
$906$	11.6914	0.388419
$907$	−56.7576	−1.88461	−0.942303	−	0.334762i	$-0.891344\pi$
$907$	−56.7576	−1.88461	−0.942303	+	0.334762i	$0.891344\pi$
$908$	−2.01088	−0.0667335
$909$	2.62701	0.0871324
$910$	−17.4493	−0.578439
$911$	28.7813	0.953567	0.476783	−	0.879021i	$-0.341803\pi$
$911$	28.7813	0.953567	0.476783	+	0.879021i	$0.341803\pi$
$912$	−4.17477	−0.138240
$913$	−4.69992	−0.155545
$914$	29.5235	0.976550
$915$	−1.70899	−0.0564973
$916$	−21.5315	−0.711420
$917$	−58.2112	−1.92230
$918$	38.1958	1.26065
$919$	13.8685	0.457481	0.228740	−	0.973487i	$-0.426539\pi$
$919$	13.8685	0.457481	0.228740	+	0.973487i	$0.426539\pi$
$920$	−3.59183	−0.118419
$921$	−8.92035	−0.293935
$922$	−2.17064	−0.0714862
$923$	−0.0215181	−0.000708278 0
$924$	−31.1746	−1.02557
$925$	−31.5199	−1.03637
$926$	38.9689	1.28060
$927$	−0.821218	−0.0269723
$928$	−1.51726	−0.0498065
$929$	51.0993	1.67651	0.838257	−	0.545275i	$-0.183575\pi$
$929$	51.0993	1.67651	0.838257	+	0.545275i	$0.183575\pi$
$930$	9.89787	0.324564
$931$	−19.3470	−0.634072
$932$	9.51368	0.311631
$933$	0.847274	0.0277385
$934$	−18.8293	−0.616113
$935$	23.9649	0.783736
$936$	0.859867	0.0281056
$937$	45.0736	1.47249	0.736245	−	0.676715i	$-0.236597\pi$
$937$	45.0736	1.47249	0.736245	+	0.676715i	$0.236597\pi$
$938$	32.4657	1.06004
$939$	−8.47305	−0.276508
$940$	−3.72496	−0.121495
$941$	−49.7699	−1.62245	−0.811227	−	0.584732i	$-0.801200\pi$
$941$	−49.7699	−1.62245	−0.811227	+	0.584732i	$0.801200\pi$
$942$	6.04728	0.197031
$943$	32.3232	1.05259
$944$	−1.77586	−0.0577995
$945$	−14.2380	−0.463161
$946$	2.33823	0.0760223
$947$	52.7573	1.71438	0.857191	−	0.514998i	$-0.172207\pi$
$947$	52.7573	1.71438	0.857191	+	0.514998i	$0.172207\pi$
$948$	−12.4027	−0.402820
$949$	−26.9910	−0.876165
$950$	11.1300	0.361106
$951$	−8.11641	−0.263193
$952$	−27.7471	−0.899290
$953$	10.2971	0.333555	0.166777	−	0.985995i	$-0.446664\pi$
$953$	10.2971	0.333555	0.166777	+	0.985995i	$0.446664\pi$
$954$	−1.68615	−0.0545912
$955$	−11.9402	−0.386375
$956$	−15.3278	−0.495738
$957$	12.2752	0.396802
$958$	−16.9560	−0.547823
$959$	−36.1790	−1.16828
$960$	1.17966	0.0380734
$961$	39.3996	1.27095
$962$	−45.3848	−1.46327
$963$	−2.18681	−0.0704688
$964$	16.9717	0.546622
$965$	7.35946	0.236909
$966$	−33.6456	−1.08253
$967$	−12.1864	−0.391889	−0.195945	−	0.980615i	$-0.562777\pi$
$967$	−12.1864	−0.391889	−0.195945	+	0.980615i	$0.562777\pi$
$968$	11.8246	0.380056
$969$	30.0621	0.965733
$970$	0.236423	0.00759109
$971$	−8.21398	−0.263599	−0.131800	−	0.991276i	$-0.542076\pi$
$971$	−8.21398	−0.263599	−0.131800	+	0.991276i	$0.542076\pi$
$972$	1.37361	0.0440585
$973$	8.51662	0.273030
$974$	−0.690603	−0.0221283
$975$	49.7005	1.59169
$976$	−1.44871	−0.0463721
$977$	42.3887	1.35613	0.678067	−	0.735000i	$-0.262818\pi$
$977$	42.3887	1.35613	0.678067	+	0.735000i	$0.262818\pi$
$978$	40.1986	1.28541
$979$	63.5929	2.03244
$980$	5.46686	0.174632
$981$	1.70922	0.0545711
$982$	−11.5863	−0.369732
$983$	−13.7846	−0.439661	−0.219831	−	0.975538i	$-0.570550\pi$
$983$	−13.7846	−0.439661	−0.219831	+	0.975538i	$0.570550\pi$
$984$	−10.6159	−0.338421
$985$	2.59969	0.0828331
$986$	10.9256	0.347943
$987$	−34.8927	−1.11065
$988$	16.0259	0.509852
$989$	2.52356	0.0802445
$990$	0.440212	0.0139908
$991$	−1.42984	−0.0454204	−0.0227102	−	0.999742i	$-0.507230\pi$
$991$	−1.42984	−0.0454204	−0.0227102	+	0.999742i	$0.507230\pi$
$992$	8.39045	0.266397
$993$	−16.8910	−0.536019
$994$	0.0127549	0.000404561 0
$995$	−3.69437	−0.117119
$996$	1.66593	0.0527872
$997$	−42.8618	−1.35745	−0.678723	−	0.734395i	$-0.737466\pi$
$997$	−42.8618	−1.35745	−0.678723	+	0.734395i	$0.737466\pi$
$998$	−36.5994	−1.15854
$999$	−37.0323	−1.17165

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.56	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.56	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.69344 q^{3} +1.00000 q^{4} +0.696607 q^{5} +1.69344 q^{6} +3.85329 q^{7} +1.00000 q^{8} -0.132273 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.69344 q^{3} +1.00000 q^{4} +0.696607 q^{5} +1.69344 q^{6} +3.85329 q^{7} +1.00000 q^{8} -0.132273 q^{9} +0.696607 q^{10} -4.77751 q^{11} +1.69344 q^{12} -6.50068 q^{13} +3.85329 q^{14} +1.17966 q^{15} +1.00000 q^{16} -7.20090 q^{17} -0.132273 q^{18} -2.46527 q^{19} +0.696607 q^{20} +6.52530 q^{21} -4.77751 q^{22} -5.15618 q^{23} +1.69344 q^{24} -4.51474 q^{25} -6.50068 q^{26} -5.30431 q^{27} +3.85329 q^{28} -1.51726 q^{29} +1.17966 q^{30} +8.39045 q^{31} +1.00000 q^{32} -8.09040 q^{33} -7.20090 q^{34} +2.68423 q^{35} -0.132273 q^{36} +6.98155 q^{37} -2.46527 q^{38} -11.0085 q^{39} +0.696607 q^{40} -6.26883 q^{41} +6.52530 q^{42} -0.489424 q^{43} -4.77751 q^{44} -0.0921426 q^{45} -5.15618 q^{46} -5.34729 q^{47} +1.69344 q^{48} +7.84783 q^{49} -4.51474 q^{50} -12.1943 q^{51} -6.50068 q^{52} +12.7475 q^{53} -5.30431 q^{54} -3.32805 q^{55} +3.85329 q^{56} -4.17477 q^{57} -1.51726 q^{58} -1.77586 q^{59} +1.17966 q^{60} -1.44871 q^{61} +8.39045 q^{62} -0.509687 q^{63} +1.00000 q^{64} -4.52842 q^{65} -8.09040 q^{66} +8.42545 q^{67} -7.20090 q^{68} -8.73167 q^{69} +2.68423 q^{70} +0.00331013 q^{71} -0.132273 q^{72} +4.15202 q^{73} +6.98155 q^{74} -7.64542 q^{75} -2.46527 q^{76} -18.4091 q^{77} -11.0085 q^{78} -7.32396 q^{79} +0.696607 q^{80} -8.58568 q^{81} -6.26883 q^{82} +0.983760 q^{83} +6.52530 q^{84} -5.01620 q^{85} -0.489424 q^{86} -2.56938 q^{87} -4.77751 q^{88} -13.3109 q^{89} -0.0921426 q^{90} -25.0490 q^{91} -5.15618 q^{92} +14.2087 q^{93} -5.34729 q^{94} -1.71732 q^{95} +1.69344 q^{96} +0.339392 q^{97} +7.84783 q^{98} +0.631936 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 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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