LMFDB - Embedded newform 6042.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.17609.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + 10x - 1 $ x^4 - x^3 - 7x^2 + 10x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.21368$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.434198 q^{5} -1.00000 q^{6} +1.90038 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.434198 q^{5} -1.00000 q^{6} +1.90038 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.434198 q^{10} -4.67987 q^{11} -1.00000 q^{12} +6.14605 q^{13} +1.90038 q^{14} -0.434198 q^{15} +1.00000 q^{16} +7.97563 q^{17} +1.00000 q^{18} -1.00000 q^{19} +0.434198 q^{20} -1.90038 q^{21} -4.67987 q^{22} +6.11407 q^{23} -1.00000 q^{24} -4.81147 q^{25} +6.14605 q^{26} -1.00000 q^{27} +1.90038 q^{28} +1.86156 q^{29} -0.434198 q^{30} +2.77948 q^{31} +1.00000 q^{32} +4.67987 q^{33} +7.97563 q^{34} +0.825143 q^{35} +1.00000 q^{36} +9.45935 q^{37} -1.00000 q^{38} -6.14605 q^{39} +0.434198 q^{40} -5.54143 q^{41} -1.90038 q^{42} -10.6054 q^{43} -4.67987 q^{44} +0.434198 q^{45} +6.11407 q^{46} -1.94364 q^{47} -1.00000 q^{48} -3.38854 q^{49} -4.81147 q^{50} -7.97563 q^{51} +6.14605 q^{52} -1.00000 q^{53} -1.00000 q^{54} -2.03199 q^{55} +1.90038 q^{56} +1.00000 q^{57} +1.86156 q^{58} -1.78632 q^{59} -0.434198 q^{60} -8.62978 q^{61} +2.77948 q^{62} +1.90038 q^{63} +1.00000 q^{64} +2.66860 q^{65} +4.67987 q^{66} +3.41291 q^{67} +7.97563 q^{68} -6.11407 q^{69} +0.825143 q^{70} +6.22813 q^{71} +1.00000 q^{72} -2.32775 q^{73} +9.45935 q^{74} +4.81147 q^{75} -1.00000 q^{76} -8.89355 q^{77} -6.14605 q^{78} +2.98246 q^{79} +0.434198 q^{80} +1.00000 q^{81} -5.54143 q^{82} +0.665417 q^{83} -1.90038 q^{84} +3.46300 q^{85} -10.6054 q^{86} -1.86156 q^{87} -4.67987 q^{88} +3.09962 q^{89} +0.434198 q^{90} +11.6799 q^{91} +6.11407 q^{92} -2.77948 q^{93} -1.94364 q^{94} -0.434198 q^{95} -1.00000 q^{96} +4.47745 q^{97} -3.38854 q^{98} -4.67987 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} + 3 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 + 3 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} + 3 q^{7} + 4 q^{8} + 4 q^{9} + 6 q^{10} - 2 q^{11} - 4 q^{12} - q^{13} + 3 q^{14} - 6 q^{15} + 4 q^{16} + 8 q^{17} + 4 q^{18} - 4 q^{19} + 6 q^{20} - 3 q^{21} - 2 q^{22} + 12 q^{23} - 4 q^{24} + 6 q^{25} - q^{26} - 4 q^{27} + 3 q^{28} - 4 q^{29} - 6 q^{30} - q^{31} + 4 q^{32} + 2 q^{33} + 8 q^{34} + 18 q^{35} + 4 q^{36} + 9 q^{37} - 4 q^{38} + q^{39} + 6 q^{40} + 6 q^{41} - 3 q^{42} + 12 q^{43} - 2 q^{44} + 6 q^{45} + 12 q^{46} + 3 q^{47} - 4 q^{48} + 9 q^{49} + 6 q^{50} - 8 q^{51} - q^{52} - 4 q^{53} - 4 q^{54} + 5 q^{55} + 3 q^{56} + 4 q^{57} - 4 q^{58} - 15 q^{59} - 6 q^{60} - 4 q^{61} - q^{62} + 3 q^{63} + 4 q^{64} - 13 q^{65} + 2 q^{66} + 15 q^{67} + 8 q^{68} - 12 q^{69} + 18 q^{70} + 4 q^{72} + 11 q^{73} + 9 q^{74} - 6 q^{75} - 4 q^{76} - 11 q^{77} + q^{78} + 8 q^{79} + 6 q^{80} + 4 q^{81} + 6 q^{82} + 3 q^{83} - 3 q^{84} + 29 q^{85} + 12 q^{86} + 4 q^{87} - 2 q^{88} + 17 q^{89} + 6 q^{90} + 30 q^{91} + 12 q^{92} + q^{93} + 3 q^{94} - 6 q^{95} - 4 q^{96} + 16 q^{97} + 9 q^{98} - 2 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 + 3 * q^7 + 4 * q^8 + 4 * q^9 + 6 * q^10 - 2 * q^11 - 4 * q^12 - q^13 + 3 * q^14 - 6 * q^15 + 4 * q^16 + 8 * q^17 + 4 * q^18 - 4 * q^19 + 6 * q^20 - 3 * q^21 - 2 * q^22 + 12 * q^23 - 4 * q^24 + 6 * q^25 - q^26 - 4 * q^27 + 3 * q^28 - 4 * q^29 - 6 * q^30 - q^31 + 4 * q^32 + 2 * q^33 + 8 * q^34 + 18 * q^35 + 4 * q^36 + 9 * q^37 - 4 * q^38 + q^39 + 6 * q^40 + 6 * q^41 - 3 * q^42 + 12 * q^43 - 2 * q^44 + 6 * q^45 + 12 * q^46 + 3 * q^47 - 4 * q^48 + 9 * q^49 + 6 * q^50 - 8 * q^51 - q^52 - 4 * q^53 - 4 * q^54 + 5 * q^55 + 3 * q^56 + 4 * q^57 - 4 * q^58 - 15 * q^59 - 6 * q^60 - 4 * q^61 - q^62 + 3 * q^63 + 4 * q^64 - 13 * q^65 + 2 * q^66 + 15 * q^67 + 8 * q^68 - 12 * q^69 + 18 * q^70 + 4 * q^72 + 11 * q^73 + 9 * q^74 - 6 * q^75 - 4 * q^76 - 11 * q^77 + q^78 + 8 * q^79 + 6 * q^80 + 4 * q^81 + 6 * q^82 + 3 * q^83 - 3 * q^84 + 29 * q^85 + 12 * q^86 + 4 * q^87 - 2 * q^88 + 17 * q^89 + 6 * q^90 + 30 * q^91 + 12 * q^92 + q^93 + 3 * q^94 - 6 * q^95 - 4 * q^96 + 16 * q^97 + 9 * q^98 - 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0.434198	0.194179	0.0970896	−	0.995276i	$-0.469047\pi$
$5$	0.434198	0.194179	0.0970896	+	0.995276i	$0.469047\pi$
$6$	−1.00000	−0.408248
$7$	1.90038	0.718278	0.359139	−	0.933284i	$-0.383070\pi$
$7$	1.90038	0.718278	0.359139	+	0.933284i	$0.383070\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0.434198	0.137305
$11$	−4.67987	−1.41103	−0.705517	−	0.708693i	$-0.749285\pi$
$11$	−4.67987	−1.41103	−0.705517	+	0.708693i	$0.749285\pi$
$12$	−1.00000	−0.288675
$13$	6.14605	1.70461	0.852304	−	0.523046i	$-0.175204\pi$
$13$	6.14605	1.70461	0.852304	+	0.523046i	$0.175204\pi$
$14$	1.90038	0.507899
$15$	−0.434198	−0.112109
$16$	1.00000	0.250000
$17$	7.97563	1.93437	0.967187	−	0.254067i	$-0.0817682\pi$
$17$	7.97563	1.93437	0.967187	+	0.254067i	$0.0817682\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.434198	0.0970896
$21$	−1.90038	−0.414698
$22$	−4.67987	−0.997751
$23$	6.11407	1.27487	0.637435	−	0.770504i	$-0.279995\pi$
$23$	6.11407	1.27487	0.637435	+	0.770504i	$0.279995\pi$
$24$	−1.00000	−0.204124
$25$	−4.81147	−0.962294
$26$	6.14605	1.20534
$27$	−1.00000	−0.192450
$28$	1.90038	0.359139
$29$	1.86156	0.345683	0.172842	−	0.984950i	$-0.444705\pi$
$29$	1.86156	0.345683	0.172842	+	0.984950i	$0.444705\pi$
$30$	−0.434198	−0.0792733
$31$	2.77948	0.499210	0.249605	−	0.968348i	$-0.419699\pi$
$31$	2.77948	0.499210	0.249605	+	0.968348i	$0.419699\pi$
$32$	1.00000	0.176777
$33$	4.67987	0.814660
$34$	7.97563	1.36781
$35$	0.825143	0.139475
$36$	1.00000	0.166667
$37$	9.45935	1.55511	0.777554	−	0.628816i	$-0.216460\pi$
$37$	9.45935	1.55511	0.777554	+	0.628816i	$0.216460\pi$
$38$	−1.00000	−0.162221
$39$	−6.14605	−0.984156
$40$	0.434198	0.0686527
$41$	−5.54143	−0.865426	−0.432713	−	0.901532i	$-0.642444\pi$
$41$	−5.54143	−0.865426	−0.432713	+	0.901532i	$0.642444\pi$
$42$	−1.90038	−0.293236
$43$	−10.6054	−1.61731	−0.808655	−	0.588284i	$-0.799804\pi$
$43$	−10.6054	−1.61731	−0.808655	+	0.588284i	$0.799804\pi$
$44$	−4.67987	−0.705517
$45$	0.434198	0.0647264
$46$	6.11407	0.901470
$47$	−1.94364	−0.283509	−0.141754	−	0.989902i	$-0.545274\pi$
$47$	−1.94364	−0.283509	−0.141754	+	0.989902i	$0.545274\pi$
$48$	−1.00000	−0.144338
$49$	−3.38854	−0.484077
$50$	−4.81147	−0.680445
$51$	−7.97563	−1.11681
$52$	6.14605	0.852304
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	−1.00000	−0.136083
$55$	−2.03199	−0.273993
$56$	1.90038	0.253950
$57$	1.00000	0.132453
$58$	1.86156	0.244435
$59$	−1.78632	−0.232559	−0.116279	−	0.993217i	$-0.537097\pi$
$59$	−1.78632	−0.232559	−0.116279	+	0.993217i	$0.537097\pi$
$60$	−0.434198	−0.0560547
$61$	−8.62978	−1.10493	−0.552465	−	0.833536i	$-0.686313\pi$
$61$	−8.62978	−1.10493	−0.552465	+	0.833536i	$0.686313\pi$
$62$	2.77948	0.352995
$63$	1.90038	0.239426
$64$	1.00000	0.125000
$65$	2.66860	0.331000
$66$	4.67987	0.576052
$67$	3.41291	0.416954	0.208477	−	0.978027i	$-0.433149\pi$
$67$	3.41291	0.416954	0.208477	+	0.978027i	$0.433149\pi$
$68$	7.97563	0.967187
$69$	−6.11407	−0.736047
$70$	0.825143	0.0986235
$71$	6.22813	0.739143	0.369572	−	0.929202i	$-0.379504\pi$
$71$	6.22813	0.739143	0.369572	+	0.929202i	$0.379504\pi$
$72$	1.00000	0.117851
$73$	−2.32775	−0.272442	−0.136221	−	0.990678i	$-0.543496\pi$
$73$	−2.32775	−0.272442	−0.136221	+	0.990678i	$0.543496\pi$
$74$	9.45935	1.09963
$75$	4.81147	0.555581
$76$	−1.00000	−0.114708
$77$	−8.89355	−1.01351
$78$	−6.14605	−0.695904
$79$	2.98246	0.335553	0.167777	−	0.985825i	$-0.446341\pi$
$79$	2.98246	0.335553	0.167777	+	0.985825i	$0.446341\pi$
$80$	0.434198	0.0485448
$81$	1.00000	0.111111
$82$	−5.54143	−0.611948
$83$	0.665417	0.0730391	0.0365195	−	0.999333i	$-0.488373\pi$
$83$	0.665417	0.0730391	0.0365195	+	0.999333i	$0.488373\pi$
$84$	−1.90038	−0.207349
$85$	3.46300	0.375615
$86$	−10.6054	−1.14361
$87$	−1.86156	−0.199580
$88$	−4.67987	−0.498876
$89$	3.09962	0.328559	0.164279	−	0.986414i	$-0.447470\pi$
$89$	3.09962	0.328559	0.164279	+	0.986414i	$0.447470\pi$
$90$	0.434198	0.0457685
$91$	11.6799	1.22438
$92$	6.11407	0.637435
$93$	−2.77948	−0.288219
$94$	−1.94364	−0.200471
$95$	−0.434198	−0.0445478
$96$	−1.00000	−0.102062
$97$	4.47745	0.454616	0.227308	−	0.973823i	$-0.427008\pi$
$97$	4.47745	0.454616	0.227308	+	0.973823i	$0.427008\pi$
$98$	−3.38854	−0.342294
$99$	−4.67987	−0.470344
$100$	−4.81147	−0.481147
$101$	17.1461	1.70610	0.853048	−	0.521832i	$-0.174751\pi$
$101$	17.1461	1.70610	0.853048	+	0.521832i	$0.174751\pi$
$102$	−7.97563	−0.789705
$103$	5.37409	0.529525	0.264762	−	0.964314i	$-0.414707\pi$
$103$	5.37409	0.529525	0.264762	+	0.964314i	$0.414707\pi$
$104$	6.14605	0.602670
$105$	−0.825143	−0.0805257
$106$	−1.00000	−0.0971286
$107$	−8.14605	−0.787509	−0.393754	−	0.919216i	$-0.628824\pi$
$107$	−8.14605	−0.787509	−0.393754	+	0.919216i	$0.628824\pi$
$108$	−1.00000	−0.0962250
$109$	7.30646	0.699832	0.349916	−	0.936781i	$-0.386210\pi$
$109$	7.30646	0.699832	0.349916	+	0.936781i	$0.386210\pi$
$110$	−2.03199	−0.193743
$111$	−9.45935	−0.897842
$112$	1.90038	0.179569
$113$	17.4913	1.64545	0.822723	−	0.568442i	$-0.192454\pi$
$113$	17.4913	1.64545	0.822723	+	0.568442i	$0.192454\pi$
$114$	1.00000	0.0936586
$115$	2.65471	0.247553
$116$	1.86156	0.172842
$117$	6.14605	0.568203
$118$	−1.78632	−0.164444
$119$	15.1568	1.38942
$120$	−0.434198	−0.0396367
$121$	10.9012	0.991015
$122$	−8.62978	−0.781304
$123$	5.54143	0.499654
$124$	2.77948	0.249605
$125$	−4.26012	−0.381037
$126$	1.90038	0.169300
$127$	−15.5270	−1.37780	−0.688898	−	0.724858i	$-0.741905\pi$
$127$	−15.5270	−1.37780	−0.688898	+	0.724858i	$0.741905\pi$
$128$	1.00000	0.0883883
$129$	10.6054	0.933754
$130$	2.66860	0.234052
$131$	12.1893	1.06498	0.532492	−	0.846435i	$-0.321256\pi$
$131$	12.1893	1.06498	0.532492	+	0.846435i	$0.321256\pi$
$132$	4.67987	0.407330
$133$	−1.90038	−0.164784
$134$	3.41291	0.294831
$135$	−0.434198	−0.0373698
$136$	7.97563	0.683904
$137$	16.1361	1.37860	0.689301	−	0.724475i	$-0.257918\pi$
$137$	16.1361	1.37860	0.689301	+	0.724475i	$0.257918\pi$
$138$	−6.11407	−0.520464
$139$	−8.76878	−0.743758	−0.371879	−	0.928281i	$-0.621286\pi$
$139$	−8.76878	−0.743758	−0.371879	+	0.928281i	$0.621286\pi$
$140$	0.825143	0.0697373
$141$	1.94364	0.163684
$142$	6.22813	0.522653
$143$	−28.7627	−2.40526
$144$	1.00000	0.0833333
$145$	0.808286	0.0671245
$146$	−2.32775	−0.192646
$147$	3.38854	0.279482
$148$	9.45935	0.777554
$149$	10.7119	0.877550	0.438775	−	0.898597i	$-0.355413\pi$
$149$	10.7119	0.877550	0.438775	+	0.898597i	$0.355413\pi$
$150$	4.81147	0.392855
$151$	−2.79075	−0.227108	−0.113554	−	0.993532i	$-0.536223\pi$
$151$	−2.79075	−0.227108	−0.113554	+	0.993532i	$0.536223\pi$
$152$	−1.00000	−0.0811107
$153$	7.97563	0.644791
$154$	−8.89355	−0.716663
$155$	1.20685	0.0969362
$156$	−6.14605	−0.492078
$157$	−5.65107	−0.451004	−0.225502	−	0.974243i	$-0.572402\pi$
$157$	−5.65107	−0.451004	−0.225502	+	0.974243i	$0.572402\pi$
$158$	2.98246	0.237272
$159$	1.00000	0.0793052
$160$	0.434198	0.0343264
$161$	11.6191	0.915712
$162$	1.00000	0.0785674
$163$	−17.8760	−1.40016	−0.700079	−	0.714066i	$-0.746852\pi$
$163$	−17.8760	−1.40016	−0.700079	+	0.714066i	$0.746852\pi$
$164$	−5.54143	−0.432713
$165$	2.03199	0.158190
$166$	0.665417	0.0516464
$167$	5.30578	0.410574	0.205287	−	0.978702i	$-0.434187\pi$
$167$	5.30578	0.410574	0.205287	+	0.978702i	$0.434187\pi$
$168$	−1.90038	−0.146618
$169$	24.7740	1.90569
$170$	3.46300	0.265600
$171$	−1.00000	−0.0764719
$172$	−10.6054	−0.808655
$173$	−2.88285	−0.219179	−0.109589	−	0.993977i	$-0.534954\pi$
$173$	−2.88285	−0.219179	−0.109589	+	0.993977i	$0.534954\pi$
$174$	−1.86156	−0.141125
$175$	−9.14365	−0.691195
$176$	−4.67987	−0.352758
$177$	1.78632	0.134268
$178$	3.09962	0.232326
$179$	12.2570	0.916134	0.458067	−	0.888918i	$-0.348542\pi$
$179$	12.2570	0.916134	0.458067	+	0.888918i	$0.348542\pi$
$180$	0.434198	0.0323632
$181$	−8.84037	−0.657100	−0.328550	−	0.944487i	$-0.606560\pi$
$181$	−8.84037	−0.657100	−0.328550	+	0.944487i	$0.606560\pi$
$182$	11.6799	0.865769
$183$	8.62978	0.637932
$184$	6.11407	0.450735
$185$	4.10723	0.301970
$186$	−2.77948	−0.203802
$187$	−37.3249	−2.72947
$188$	−1.94364	−0.141754
$189$	−1.90038	−0.138233
$190$	−0.434198	−0.0315000
$191$	−0.736792	−0.0533124	−0.0266562	−	0.999645i	$-0.508486\pi$
$191$	−0.736792	−0.0533124	−0.0266562	+	0.999645i	$0.508486\pi$
$192$	−1.00000	−0.0721688
$193$	−5.01888	−0.361267	−0.180633	−	0.983550i	$-0.557815\pi$
$193$	−5.01888	−0.361267	−0.180633	+	0.983550i	$0.557815\pi$
$194$	4.47745	0.321462
$195$	−2.66860	−0.191103
$196$	−3.38854	−0.242038
$197$	−13.0396	−0.929033	−0.464517	−	0.885564i	$-0.653772\pi$
$197$	−13.0396	−0.929033	−0.464517	+	0.885564i	$0.653772\pi$
$198$	−4.67987	−0.332584
$199$	−19.3542	−1.37199	−0.685993	−	0.727608i	$-0.740632\pi$
$199$	−19.3542	−1.37199	−0.685993	+	0.727608i	$0.740632\pi$
$200$	−4.81147	−0.340222
$201$	−3.41291	−0.240728
$202$	17.1461	1.20639
$203$	3.53768	0.248297
$204$	−7.97563	−0.558406
$205$	−2.40608	−0.168048
$206$	5.37409	0.374430
$207$	6.11407	0.424957
$208$	6.14605	0.426152
$209$	4.67987	0.323713
$210$	−0.825143	−0.0569403
$211$	9.85416	0.678389	0.339194	−	0.940716i	$-0.389846\pi$
$211$	9.85416	0.678389	0.339194	+	0.940716i	$0.389846\pi$
$212$	−1.00000	−0.0686803
$213$	−6.22813	−0.426745
$214$	−8.14605	−0.556853
$215$	−4.60485	−0.314048
$216$	−1.00000	−0.0680414
$217$	5.28209	0.358571
$218$	7.30646	0.494856
$219$	2.32775	0.157295
$220$	−2.03199	−0.136997
$221$	49.0186	3.29735
$222$	−9.45935	−0.634870
$223$	−4.13294	−0.276762	−0.138381	−	0.990379i	$-0.544190\pi$
$223$	−4.13294	−0.276762	−0.138381	+	0.990379i	$0.544190\pi$
$224$	1.90038	0.126975
$225$	−4.81147	−0.320765
$226$	17.4913	1.16351
$227$	−9.76513	−0.648135	−0.324067	−	0.946034i	$-0.605050\pi$
$227$	−9.76513	−0.648135	−0.324067	+	0.946034i	$0.605050\pi$
$228$	1.00000	0.0662266
$229$	−5.40243	−0.357002	−0.178501	−	0.983940i	$-0.557125\pi$
$229$	−5.40243	−0.357002	−0.178501	+	0.983940i	$0.557125\pi$
$230$	2.65471	0.175047
$231$	8.89355	0.585153
$232$	1.86156	0.122217
$233$	−2.26696	−0.148513	−0.0742566	−	0.997239i	$-0.523658\pi$
$233$	−2.26696	−0.148513	−0.0742566	+	0.997239i	$0.523658\pi$
$234$	6.14605	0.401780
$235$	−0.843923	−0.0550515
$236$	−1.78632	−0.116279
$237$	−2.98246	−0.193732
$238$	15.1568	0.982467
$239$	24.9895	1.61644	0.808219	−	0.588882i	$-0.200432\pi$
$239$	24.9895	1.61644	0.808219	+	0.588882i	$0.200432\pi$
$240$	−0.434198	−0.0280274
$241$	0.447989	0.0288575	0.0144287	−	0.999896i	$-0.495407\pi$
$241$	0.447989	0.0288575	0.0144287	+	0.999896i	$0.495407\pi$
$242$	10.9012	0.700753
$243$	−1.00000	−0.0641500
$244$	−8.62978	−0.552465
$245$	−1.47130	−0.0939977
$246$	5.54143	0.353309
$247$	−6.14605	−0.391064
$248$	2.77948	0.176497
$249$	−0.665417	−0.0421691
$250$	−4.26012	−0.269434
$251$	−7.61589	−0.480711	−0.240355	−	0.970685i	$-0.577264\pi$
$251$	−7.61589	−0.480711	−0.240355	+	0.970685i	$0.577264\pi$
$252$	1.90038	0.119713
$253$	−28.6130	−1.79889
$254$	−15.5270	−0.974249
$255$	−3.46300	−0.216861
$256$	1.00000	0.0625000
$257$	14.5658	0.908590	0.454295	−	0.890851i	$-0.349891\pi$
$257$	14.5658	0.908590	0.454295	+	0.890851i	$0.349891\pi$
$258$	10.6054	0.660264
$259$	17.9764	1.11700
$260$	2.66860	0.165500
$261$	1.86156	0.115228
$262$	12.1893	0.753058
$263$	11.6599	0.718982	0.359491	−	0.933149i	$-0.382950\pi$
$263$	11.6599	0.718982	0.359491	+	0.933149i	$0.382950\pi$
$264$	4.67987	0.288026
$265$	−0.434198	−0.0266726
$266$	−1.90038	−0.116520
$267$	−3.09962	−0.189693
$268$	3.41291	0.208477
$269$	9.67225	0.589728	0.294864	−	0.955539i	$-0.404726\pi$
$269$	9.67225	0.589728	0.294864	+	0.955539i	$0.404726\pi$
$270$	−0.434198	−0.0264244
$271$	13.5201	0.821290	0.410645	−	0.911795i	$-0.365304\pi$
$271$	13.5201	0.821290	0.410645	+	0.911795i	$0.365304\pi$
$272$	7.97563	0.483593
$273$	−11.6799	−0.706898
$274$	16.1361	0.974819
$275$	22.5171	1.35783
$276$	−6.11407	−0.368024
$277$	25.8574	1.55362	0.776809	−	0.629737i	$-0.216837\pi$
$277$	25.8574	1.55362	0.776809	+	0.629737i	$0.216837\pi$
$278$	−8.76878	−0.525917
$279$	2.77948	0.166403
$280$	0.825143	0.0493117
$281$	−8.40618	−0.501470	−0.250735	−	0.968056i	$-0.580672\pi$
$281$	−8.40618	−0.501470	−0.250735	+	0.968056i	$0.580672\pi$
$282$	1.94364	0.115742
$283$	25.8981	1.53948	0.769740	−	0.638357i	$-0.220386\pi$
$283$	25.8981	1.53948	0.769740	+	0.638357i	$0.220386\pi$
$284$	6.22813	0.369572
$285$	0.434198	0.0257197
$286$	−28.7627	−1.70078
$287$	−10.5308	−0.621616
$288$	1.00000	0.0589256
$289$	46.6106	2.74180
$290$	0.808286	0.0474642
$291$	−4.47745	−0.262473
$292$	−2.32775	−0.136221
$293$	11.1491	0.651340	0.325670	−	0.945484i	$-0.394410\pi$
$293$	11.1491	0.651340	0.325670	+	0.945484i	$0.394410\pi$
$294$	3.38854	0.197624
$295$	−0.775616	−0.0451581
$296$	9.45935	0.549814
$297$	4.67987	0.271553
$298$	10.7119	0.620521
$299$	37.5774	2.17316
$300$	4.81147	0.277790
$301$	−20.1544	−1.16168
$302$	−2.79075	−0.160589
$303$	−17.1461	−0.985015
$304$	−1.00000	−0.0573539
$305$	−3.74703	−0.214554
$306$	7.97563	0.455936
$307$	−8.40530	−0.479716	−0.239858	−	0.970808i	$-0.577101\pi$
$307$	−8.40530	−0.479716	−0.239858	+	0.970808i	$0.577101\pi$
$308$	−8.89355	−0.506757
$309$	−5.37409	−0.305721
$310$	1.20685	0.0685442
$311$	−9.51946	−0.539799	−0.269900	−	0.962888i	$-0.586991\pi$
$311$	−9.51946	−0.539799	−0.269900	+	0.962888i	$0.586991\pi$
$312$	−6.14605	−0.347952
$313$	−11.5797	−0.654523	−0.327261	−	0.944934i	$-0.606126\pi$
$313$	−11.5797	−0.654523	−0.327261	+	0.944934i	$0.606126\pi$
$314$	−5.65107	−0.318908
$315$	0.825143	0.0464915
$316$	2.98246	0.167777
$317$	−31.2578	−1.75561	−0.877807	−	0.479015i	$-0.840994\pi$
$317$	−31.2578	−1.75561	−0.877807	+	0.479015i	$0.840994\pi$
$318$	1.00000	0.0560772
$319$	−8.71186	−0.487770
$320$	0.434198	0.0242724
$321$	8.14605	0.454668
$322$	11.6191	0.647506
$323$	−7.97563	−0.443776
$324$	1.00000	0.0555556
$325$	−29.5716	−1.64034
$326$	−17.8760	−0.990061
$327$	−7.30646	−0.404048
$328$	−5.54143	−0.305974
$329$	−3.69366	−0.203638
$330$	2.03199	0.111857
$331$	21.5779	1.18603	0.593016	−	0.805191i	$-0.297937\pi$
$331$	21.5779	1.18603	0.593016	+	0.805191i	$0.297937\pi$
$332$	0.665417	0.0365195
$333$	9.45935	0.518369
$334$	5.30578	0.290319
$335$	1.48188	0.0809637
$336$	−1.90038	−0.103674
$337$	20.9120	1.13915	0.569574	−	0.821940i	$-0.307108\pi$
$337$	20.9120	1.13915	0.569574	+	0.821940i	$0.307108\pi$
$338$	24.7740	1.34753
$339$	−17.4913	−0.949999
$340$	3.46300	0.187808
$341$	−13.0076	−0.704402
$342$	−1.00000	−0.0540738
$343$	−19.7422	−1.06598
$344$	−10.6054	−0.571805
$345$	−2.65471	−0.142925
$346$	−2.88285	−0.154983
$347$	10.2695	0.551295	0.275647	−	0.961259i	$-0.411108\pi$
$347$	10.2695	0.551295	0.275647	+	0.961259i	$0.411108\pi$
$348$	−1.86156	−0.0997901
$349$	11.6259	0.622321	0.311160	−	0.950357i	$-0.399282\pi$
$349$	11.6259	0.622321	0.311160	+	0.950357i	$0.399282\pi$
$350$	−9.14365	−0.488749
$351$	−6.14605	−0.328052
$352$	−4.67987	−0.249438
$353$	−27.2929	−1.45265	−0.726327	−	0.687349i	$-0.758774\pi$
$353$	−27.2929	−1.45265	−0.726327	+	0.687349i	$0.758774\pi$
$354$	1.78632	0.0949418
$355$	2.70424	0.143526
$356$	3.09962	0.164279
$357$	−15.1568	−0.802181
$358$	12.2570	0.647804
$359$	−8.52333	−0.449844	−0.224922	−	0.974377i	$-0.572213\pi$
$359$	−8.52333	−0.449844	−0.224922	+	0.974377i	$0.572213\pi$
$360$	0.434198	0.0228842
$361$	1.00000	0.0526316
$362$	−8.84037	−0.464640
$363$	−10.9012	−0.572163
$364$	11.6799	0.612191
$365$	−1.01070	−0.0529026
$366$	8.62978	0.451086
$367$	18.1117	0.945421	0.472710	−	0.881218i	$-0.343276\pi$
$367$	18.1117	0.945421	0.472710	+	0.881218i	$0.343276\pi$
$368$	6.11407	0.318718
$369$	−5.54143	−0.288475
$370$	4.10723	0.213525
$371$	−1.90038	−0.0986631
$372$	−2.77948	−0.144109
$373$	−22.5972	−1.17004	−0.585020	−	0.811019i	$-0.698913\pi$
$373$	−22.5972	−1.17004	−0.585020	+	0.811019i	$0.698913\pi$
$374$	−37.3249	−1.93002
$375$	4.26012	0.219992
$376$	−1.94364	−0.100235
$377$	11.4413	0.589254
$378$	−1.90038	−0.0977452
$379$	17.7187	0.910148	0.455074	−	0.890454i	$-0.349613\pi$
$379$	17.7187	0.910148	0.455074	+	0.890454i	$0.349613\pi$
$380$	−0.434198	−0.0222739
$381$	15.5270	0.795471
$382$	−0.736792	−0.0376975
$383$	−20.4838	−1.04667	−0.523337	−	0.852126i	$-0.675313\pi$
$383$	−20.4838	−1.04667	−0.523337	+	0.852126i	$0.675313\pi$
$384$	−1.00000	−0.0510310
$385$	−3.86156	−0.196803
$386$	−5.01888	−0.255454
$387$	−10.6054	−0.539103
$388$	4.47745	0.227308
$389$	16.9413	0.858959	0.429480	−	0.903076i	$-0.358697\pi$
$389$	16.9413	0.858959	0.429480	+	0.903076i	$0.358697\pi$
$390$	−2.66860	−0.135130
$391$	48.7635	2.46608
$392$	−3.38854	−0.171147
$393$	−12.1893	−0.614869
$394$	−13.0396	−0.656926
$395$	1.29498	0.0651574
$396$	−4.67987	−0.235172
$397$	−12.8254	−0.643686	−0.321843	−	0.946793i	$-0.604302\pi$
$397$	−12.8254	−0.643686	−0.321843	+	0.946793i	$0.604302\pi$
$398$	−19.3542	−0.970140
$399$	1.90038	0.0951382
$400$	−4.81147	−0.240574
$401$	20.1881	1.00814	0.504072	−	0.863662i	$-0.331835\pi$
$401$	20.1881	1.00814	0.504072	+	0.863662i	$0.331835\pi$
$402$	−3.41291	−0.170221
$403$	17.0829	0.850958
$404$	17.1461	0.853048
$405$	0.434198	0.0215755
$406$	3.53768	0.175572
$407$	−44.2685	−2.19431
$408$	−7.97563	−0.394852
$409$	6.43420	0.318151	0.159075	−	0.987266i	$-0.449149\pi$
$409$	6.43420	0.318151	0.159075	+	0.987266i	$0.449149\pi$
$410$	−2.40608	−0.118828
$411$	−16.1361	−0.795937
$412$	5.37409	0.264762
$413$	−3.39469	−0.167042
$414$	6.11407	0.300490
$415$	0.288923	0.0141827
$416$	6.14605	0.301335
$417$	8.76878	0.429409
$418$	4.67987	0.228900
$419$	9.52708	0.465428	0.232714	−	0.972545i	$-0.425239\pi$
$419$	9.52708	0.465428	0.232714	+	0.972545i	$0.425239\pi$
$420$	−0.825143	−0.0402629
$421$	−31.0503	−1.51330	−0.756650	−	0.653821i	$-0.773165\pi$
$421$	−31.0503	−1.51330	−0.756650	+	0.653821i	$0.773165\pi$
$422$	9.85416	0.479693
$423$	−1.94364	−0.0945029
$424$	−1.00000	−0.0485643
$425$	−38.3745	−1.86144
$426$	−6.22813	−0.301754
$427$	−16.3999	−0.793647
$428$	−8.14605	−0.393754
$429$	28.7627	1.38868
$430$	−4.60485	−0.222065
$431$	−18.2381	−0.878496	−0.439248	−	0.898366i	$-0.644755\pi$
$431$	−18.2381	−0.878496	−0.439248	+	0.898366i	$0.644755\pi$
$432$	−1.00000	−0.0481125
$433$	12.9279	0.621277	0.310639	−	0.950528i	$-0.399457\pi$
$433$	12.9279	0.621277	0.310639	+	0.950528i	$0.399457\pi$
$434$	5.28209	0.253548
$435$	−0.808286	−0.0387543
$436$	7.30646	0.349916
$437$	−6.11407	−0.292475
$438$	2.32775	0.111224
$439$	−17.3516	−0.828145	−0.414072	−	0.910244i	$-0.635894\pi$
$439$	−17.3516	−0.828145	−0.414072	+	0.910244i	$0.635894\pi$
$440$	−2.03199	−0.0968713
$441$	−3.38854	−0.161359
$442$	49.0186	2.33158
$443$	−28.6030	−1.35897	−0.679485	−	0.733690i	$-0.737797\pi$
$443$	−28.6030	−1.35897	−0.679485	+	0.733690i	$0.737797\pi$
$444$	−9.45935	−0.448921
$445$	1.34585	0.0637992
$446$	−4.13294	−0.195701
$447$	−10.7119	−0.506653
$448$	1.90038	0.0897847
$449$	−28.0097	−1.32186	−0.660931	−	0.750447i	$-0.729838\pi$
$449$	−28.0097	−1.32186	−0.660931	+	0.750447i	$0.729838\pi$
$450$	−4.81147	−0.226815
$451$	25.9332	1.22114
$452$	17.4913	0.822723
$453$	2.79075	0.131121
$454$	−9.76513	−0.458300
$455$	5.07137	0.237750
$456$	1.00000	0.0468293
$457$	5.38720	0.252002	0.126001	−	0.992030i	$-0.459786\pi$
$457$	5.38720	0.252002	0.126001	+	0.992030i	$0.459786\pi$
$458$	−5.40243	−0.252439
$459$	−7.97563	−0.372270
$460$	2.65471	0.123777
$461$	−2.93680	−0.136781	−0.0683903	−	0.997659i	$-0.521786\pi$
$461$	−2.93680	−0.136781	−0.0683903	+	0.997659i	$0.521786\pi$
$462$	8.89355	0.413765
$463$	−4.31205	−0.200398	−0.100199	−	0.994967i	$-0.531948\pi$
$463$	−4.31205	−0.200398	−0.100199	+	0.994967i	$0.531948\pi$
$464$	1.86156	0.0864208
$465$	−1.20685	−0.0559661
$466$	−2.26696	−0.105015
$467$	−13.1284	−0.607511	−0.303755	−	0.952750i	$-0.598241\pi$
$467$	−13.1284	−0.607511	−0.303755	+	0.952750i	$0.598241\pi$
$468$	6.14605	0.284101
$469$	6.48585	0.299489
$470$	−0.843923	−0.0389273
$471$	5.65107	0.260387
$472$	−1.78632	−0.0822220
$473$	49.6319	2.28208
$474$	−2.98246	−0.136989
$475$	4.81147	0.220765
$476$	15.1568	0.694709
$477$	−1.00000	−0.0457869
$478$	24.9895	1.14299
$479$	12.3854	0.565901	0.282951	−	0.959134i	$-0.408687\pi$
$479$	12.3854	0.565901	0.282951	+	0.959134i	$0.408687\pi$
$480$	−0.434198	−0.0198183
$481$	58.1377	2.65085
$482$	0.447989	0.0204053
$483$	−11.6191	−0.528686
$484$	10.9012	0.495507
$485$	1.94410	0.0882770
$486$	−1.00000	−0.0453609
$487$	40.0197	1.81346	0.906732	−	0.421708i	$-0.138569\pi$
$487$	40.0197	1.81346	0.906732	+	0.421708i	$0.138569\pi$
$488$	−8.62978	−0.390652
$489$	17.8760	0.808381
$490$	−1.47130	−0.0664664
$491$	25.4783	1.14982	0.574910	−	0.818216i	$-0.305037\pi$
$491$	25.4783	1.14982	0.574910	+	0.818216i	$0.305037\pi$
$492$	5.54143	0.249827
$493$	14.8471	0.668680
$494$	−6.14605	−0.276524
$495$	−2.03199	−0.0913311
$496$	2.77948	0.124802
$497$	11.8358	0.530910
$498$	−0.665417	−0.0298181
$499$	−17.0852	−0.764837	−0.382419	−	0.923989i	$-0.624909\pi$
$499$	−17.0852	−0.764837	−0.382419	+	0.923989i	$0.624909\pi$
$500$	−4.26012	−0.190518
$501$	−5.30578	−0.237045
$502$	−7.61589	−0.339914
$503$	24.6987	1.10126	0.550631	−	0.834749i	$-0.314387\pi$
$503$	24.6987	1.10126	0.550631	+	0.834749i	$0.314387\pi$
$504$	1.90038	0.0846499
$505$	7.44478	0.331288
$506$	−28.6130	−1.27200
$507$	−24.7740	−1.10025
$508$	−15.5270	−0.688898
$509$	−14.0114	−0.621043	−0.310521	−	0.950566i	$-0.600504\pi$
$509$	−14.0114	−0.621043	−0.310521	+	0.950566i	$0.600504\pi$
$510$	−3.46300	−0.153344
$511$	−4.42361	−0.195689
$512$	1.00000	0.0441942
$513$	1.00000	0.0441511
$514$	14.5658	0.642470
$515$	2.33342	0.102823
$516$	10.6054	0.466877
$517$	9.09597	0.400040
$518$	17.9764	0.789838
$519$	2.88285	0.126543
$520$	2.66860	0.117026
$521$	−6.49180	−0.284411	−0.142206	−	0.989837i	$-0.545419\pi$
$521$	−6.49180	−0.284411	−0.142206	+	0.989837i	$0.545419\pi$
$522$	1.86156	0.0814783
$523$	−37.1285	−1.62352	−0.811758	−	0.583994i	$-0.801489\pi$
$523$	−37.1285	−1.62352	−0.811758	+	0.583994i	$0.801489\pi$
$524$	12.1893	0.532492
$525$	9.14365	0.399062
$526$	11.6599	0.508397
$527$	22.1681	0.965658
$528$	4.67987	0.203665
$529$	14.3818	0.625296
$530$	−0.434198	−0.0188604
$531$	−1.78632	−0.0775196
$532$	−1.90038	−0.0823921
$533$	−34.0579	−1.47521
$534$	−3.09962	−0.134133
$535$	−3.53700	−0.152918
$536$	3.41291	0.147415
$537$	−12.2570	−0.528930
$538$	9.67225	0.417000
$539$	15.8579	0.683049
$540$	−0.434198	−0.0186849
$541$	−45.5857	−1.95988	−0.979941	−	0.199289i	$-0.936137\pi$
$541$	−45.5857	−1.95988	−0.979941	+	0.199289i	$0.936137\pi$
$542$	13.5201	0.580740
$543$	8.84037	0.379377
$544$	7.97563	0.341952
$545$	3.17245	0.135893
$546$	−11.6799	−0.499852
$547$	16.7207	0.714926	0.357463	−	0.933927i	$-0.383642\pi$
$547$	16.7207	0.714926	0.357463	+	0.933927i	$0.383642\pi$
$548$	16.1361	0.689301
$549$	−8.62978	−0.368310
$550$	22.5171	0.960130
$551$	−1.86156	−0.0793051
$552$	−6.11407	−0.260232
$553$	5.66782	0.241020
$554$	25.8574	1.09857
$555$	−4.10723	−0.174342
$556$	−8.76878	−0.371879
$557$	3.62535	0.153611	0.0768055	−	0.997046i	$-0.475528\pi$
$557$	3.62535	0.153611	0.0768055	+	0.997046i	$0.475528\pi$
$558$	2.77948	0.117665
$559$	−65.1814	−2.75688
$560$	0.825143	0.0348687
$561$	37.3249	1.57586
$562$	−8.40618	−0.354593
$563$	−20.4256	−0.860837	−0.430419	−	0.902629i	$-0.641634\pi$
$563$	−20.4256	−0.860837	−0.430419	+	0.902629i	$0.641634\pi$
$564$	1.94364	0.0818419
$565$	7.59470	0.319512
$566$	25.8981	1.08858
$567$	1.90038	0.0798087
$568$	6.22813	0.261327
$569$	−8.05405	−0.337644	−0.168822	−	0.985647i	$-0.553996\pi$
$569$	−8.05405	−0.337644	−0.168822	+	0.985647i	$0.553996\pi$
$570$	0.434198	0.0181865
$571$	−23.8654	−0.998737	−0.499369	−	0.866390i	$-0.666435\pi$
$571$	−23.8654	−0.998737	−0.499369	+	0.866390i	$0.666435\pi$
$572$	−28.7627	−1.20263
$573$	0.736792	0.0307799
$574$	−10.5308	−0.439549
$575$	−29.4177	−1.22680
$576$	1.00000	0.0416667
$577$	35.6914	1.48585	0.742927	−	0.669372i	$-0.233437\pi$
$577$	35.6914	1.48585	0.742927	+	0.669372i	$0.233437\pi$
$578$	46.6106	1.93875
$579$	5.01888	0.208577
$580$	0.808286	0.0335622
$581$	1.26455	0.0524623
$582$	−4.47745	−0.185596
$583$	4.67987	0.193820
$584$	−2.32775	−0.0963229
$585$	2.66860	0.110333
$586$	11.1491	0.460567
$587$	37.3256	1.54059	0.770295	−	0.637687i	$-0.220109\pi$
$587$	37.3256	1.54059	0.770295	+	0.637687i	$0.220109\pi$
$588$	3.38854	0.139741
$589$	−2.77948	−0.114527
$590$	−0.775616	−0.0319316
$591$	13.0396	0.536378
$592$	9.45935	0.388777
$593$	10.7413	0.441093	0.220547	−	0.975376i	$-0.429216\pi$
$593$	10.7413	0.441093	0.220547	+	0.975376i	$0.429216\pi$
$594$	4.67987	0.192017
$595$	6.58103	0.269796
$596$	10.7119	0.438775
$597$	19.3542	0.792116
$598$	37.5774	1.53665
$599$	18.7752	0.767132	0.383566	−	0.923513i	$-0.374696\pi$
$599$	18.7752	0.767132	0.383566	+	0.923513i	$0.374696\pi$
$600$	4.81147	0.196428
$601$	−15.9831	−0.651966	−0.325983	−	0.945376i	$-0.605695\pi$
$601$	−15.9831	−0.651966	−0.325983	+	0.945376i	$0.605695\pi$
$602$	−20.1544	−0.821430
$603$	3.41291	0.138985
$604$	−2.79075	−0.113554
$605$	4.73326	0.192434
$606$	−17.1461	−0.696511
$607$	−41.2911	−1.67596	−0.837978	−	0.545704i	$-0.816262\pi$
$607$	−41.2911	−1.67596	−0.837978	+	0.545704i	$0.816262\pi$
$608$	−1.00000	−0.0405554
$609$	−3.53768	−0.143354
$610$	−3.74703	−0.151713
$611$	−11.9457	−0.483271
$612$	7.97563	0.322396
$613$	21.0799	0.851409	0.425704	−	0.904862i	$-0.360026\pi$
$613$	21.0799	0.851409	0.425704	+	0.904862i	$0.360026\pi$
$614$	−8.40530	−0.339210
$615$	2.40608	0.0970224
$616$	−8.89355	−0.358331
$617$	7.05646	0.284082	0.142041	−	0.989861i	$-0.454633\pi$
$617$	7.05646	0.284082	0.142041	+	0.989861i	$0.454633\pi$
$618$	−5.37409	−0.216178
$619$	0.479956	0.0192911	0.00964554	−	0.999953i	$-0.496930\pi$
$619$	0.479956	0.0192911	0.00964554	+	0.999953i	$0.496930\pi$
$620$	1.20685	0.0484681
$621$	−6.11407	−0.245349
$622$	−9.51946	−0.381696
$623$	5.89046	0.235996
$624$	−6.14605	−0.246039
$625$	22.2076	0.888305
$626$	−11.5797	−0.462818
$627$	−4.67987	−0.186896
$628$	−5.65107	−0.225502
$629$	75.4443	3.00816
$630$	0.825143	0.0328745
$631$	−10.1140	−0.402631	−0.201315	−	0.979526i	$-0.564522\pi$
$631$	−10.1140	−0.402631	−0.201315	+	0.979526i	$0.564522\pi$
$632$	2.98246	0.118636
$633$	−9.85416	−0.391668
$634$	−31.2578	−1.24141
$635$	−6.74178	−0.267539
$636$	1.00000	0.0396526
$637$	−20.8261	−0.825162
$638$	−8.71186	−0.344906
$639$	6.22813	0.246381
$640$	0.434198	0.0171632
$641$	−15.8742	−0.626992	−0.313496	−	0.949590i	$-0.601500\pi$
$641$	−15.8742	−0.626992	−0.313496	+	0.949590i	$0.601500\pi$
$642$	8.14605	0.321499
$643$	−21.8517	−0.861748	−0.430874	−	0.902412i	$-0.641795\pi$
$643$	−21.8517	−0.861748	−0.430874	+	0.902412i	$0.641795\pi$
$644$	11.6191	0.457856
$645$	4.60485	0.181316
$646$	−7.97563	−0.313797
$647$	17.6831	0.695193	0.347596	−	0.937644i	$-0.386998\pi$
$647$	17.6831	0.695193	0.347596	+	0.937644i	$0.386998\pi$
$648$	1.00000	0.0392837
$649$	8.35974	0.328148
$650$	−29.5716	−1.15989
$651$	−5.28209	−0.207021
$652$	−17.8760	−0.700079
$653$	−22.7732	−0.891185	−0.445592	−	0.895236i	$-0.647007\pi$
$653$	−22.7732	−0.891185	−0.445592	+	0.895236i	$0.647007\pi$
$654$	−7.30646	−0.285705
$655$	5.29257	0.206798
$656$	−5.54143	−0.216356
$657$	−2.32775	−0.0908141
$658$	−3.69366	−0.143994
$659$	40.0547	1.56031	0.780155	−	0.625586i	$-0.215140\pi$
$659$	40.0547	1.56031	0.780155	+	0.625586i	$0.215140\pi$
$660$	2.03199	0.0790951
$661$	−46.9697	−1.82691	−0.913455	−	0.406941i	$-0.866596\pi$
$661$	−46.9697	−1.82691	−0.913455	+	0.406941i	$0.866596\pi$
$662$	21.5779	0.838651
$663$	−49.0186	−1.90373
$664$	0.665417	0.0258232
$665$	−0.825143	−0.0319977
$666$	9.45935	0.366542
$667$	11.3817	0.440701
$668$	5.30578	0.205287
$669$	4.13294	0.159789
$670$	1.48188	0.0572500
$671$	40.3862	1.55909
$672$	−1.90038	−0.0733089
$673$	−23.9620	−0.923666	−0.461833	−	0.886967i	$-0.652808\pi$
$673$	−23.9620	−0.923666	−0.461833	+	0.886967i	$0.652808\pi$
$674$	20.9120	0.805499
$675$	4.81147	0.185194
$676$	24.7740	0.952846
$677$	40.7403	1.56578	0.782889	−	0.622162i	$-0.213745\pi$
$677$	40.7403	1.56578	0.782889	+	0.622162i	$0.213745\pi$
$678$	−17.4913	−0.671751
$679$	8.50888	0.326541
$680$	3.46300	0.132800
$681$	9.76513	0.374201
$682$	−13.0076	−0.498087
$683$	−26.3361	−1.00772	−0.503862	−	0.863784i	$-0.668088\pi$
$683$	−26.3361	−1.00772	−0.503862	+	0.863784i	$0.668088\pi$
$684$	−1.00000	−0.0382360
$685$	7.00627	0.267696
$686$	−19.7422	−0.753761
$687$	5.40243	0.206115
$688$	−10.6054	−0.404327
$689$	−6.14605	−0.234146
$690$	−2.65471	−0.101063
$691$	−30.2320	−1.15008	−0.575040	−	0.818125i	$-0.695013\pi$
$691$	−30.2320	−1.15008	−0.575040	+	0.818125i	$0.695013\pi$
$692$	−2.88285	−0.109589
$693$	−8.89355	−0.337838
$694$	10.2695	0.389824
$695$	−3.80739	−0.144422
$696$	−1.86156	−0.0705623
$697$	−44.1964	−1.67406
$698$	11.6259	0.440047
$699$	2.26696	0.0857442
$700$	−9.14365	−0.345597
$701$	−19.7552	−0.746144	−0.373072	−	0.927802i	$-0.621696\pi$
$701$	−19.7552	−0.746144	−0.373072	+	0.927802i	$0.621696\pi$
$702$	−6.14605	−0.231968
$703$	−9.45935	−0.356766
$704$	−4.67987	−0.176379
$705$	0.843923	0.0317840
$706$	−27.2929	−1.02718
$707$	32.5841	1.22545
$708$	1.78632	0.0671340
$709$	−37.1226	−1.39417	−0.697084	−	0.716990i	$-0.745519\pi$
$709$	−37.1226	−1.39417	−0.697084	+	0.716990i	$0.745519\pi$
$710$	2.70424	0.101488
$711$	2.98246	0.111851
$712$	3.09962	0.116163
$713$	16.9939	0.636428
$714$	−15.1568	−0.567227
$715$	−12.4887	−0.467051
$716$	12.2570	0.458067
$717$	−24.9895	−0.933251
$718$	−8.52333	−0.318088
$719$	17.9977	0.671201	0.335600	−	0.942004i	$-0.391061\pi$
$719$	17.9977	0.671201	0.335600	+	0.942004i	$0.391061\pi$
$720$	0.434198	0.0161816
$721$	10.2128	0.380346
$722$	1.00000	0.0372161
$723$	−0.447989	−0.0166609
$724$	−8.84037	−0.328550
$725$	−8.95685	−0.332649
$726$	−10.9012	−0.404580
$727$	28.3530	1.05156	0.525778	−	0.850622i	$-0.323774\pi$
$727$	28.3530	1.05156	0.525778	+	0.850622i	$0.323774\pi$
$728$	11.6799	0.432885
$729$	1.00000	0.0370370
$730$	−1.01070	−0.0374078
$731$	−84.5848	−3.12848
$732$	8.62978	0.318966
$733$	14.4745	0.534627	0.267314	−	0.963610i	$-0.413864\pi$
$733$	14.4745	0.534627	0.267314	+	0.963610i	$0.413864\pi$
$734$	18.1117	0.668514
$735$	1.47130	0.0542696
$736$	6.11407	0.225367
$737$	−15.9720	−0.588335
$738$	−5.54143	−0.203983
$739$	−37.1063	−1.36498	−0.682488	−	0.730897i	$-0.739102\pi$
$739$	−37.1063	−1.36498	−0.682488	+	0.730897i	$0.739102\pi$
$740$	4.10723	0.150985
$741$	6.14605	0.225781
$742$	−1.90038	−0.0697653
$743$	−54.1766	−1.98755	−0.993774	−	0.111415i	$-0.964462\pi$
$743$	−54.1766	−1.98755	−0.993774	+	0.111415i	$0.964462\pi$
$744$	−2.77948	−0.101901
$745$	4.65107	0.170402
$746$	−22.5972	−0.827343
$747$	0.665417	0.0243464
$748$	−37.3249	−1.36473
$749$	−15.4806	−0.565650
$750$	4.26012	0.155558
$751$	−19.2481	−0.702372	−0.351186	−	0.936306i	$-0.614222\pi$
$751$	−19.2481	−0.702372	−0.351186	+	0.936306i	$0.614222\pi$
$752$	−1.94364	−0.0708772
$753$	7.61589	0.277538
$754$	11.4413	0.416666
$755$	−1.21174	−0.0440996
$756$	−1.90038	−0.0691163
$757$	−1.43296	−0.0520817	−0.0260408	−	0.999661i	$-0.508290\pi$
$757$	−1.43296	−0.0520817	−0.0260408	+	0.999661i	$0.508290\pi$
$758$	17.7187	0.643572
$759$	28.6130	1.03859
$760$	−0.434198	−0.0157500
$761$	4.80897	0.174325	0.0871625	−	0.996194i	$-0.472220\pi$
$761$	4.80897	0.174325	0.0871625	+	0.996194i	$0.472220\pi$
$762$	15.5270	0.562483
$763$	13.8851	0.502674
$764$	−0.736792	−0.0266562
$765$	3.46300	0.125205
$766$	−20.4838	−0.740111
$767$	−10.9788	−0.396422
$768$	−1.00000	−0.0360844
$769$	46.9892	1.69447	0.847237	−	0.531214i	$-0.178264\pi$
$769$	46.9892	1.69447	0.847237	+	0.531214i	$0.178264\pi$
$770$	−3.86156	−0.139161
$771$	−14.5658	−0.524575
$772$	−5.01888	−0.180633
$773$	−33.4076	−1.20159	−0.600795	−	0.799403i	$-0.705149\pi$
$773$	−33.4076	−1.20159	−0.600795	+	0.799403i	$0.705149\pi$
$774$	−10.6054	−0.381203
$775$	−13.3734	−0.480387
$776$	4.47745	0.160731
$777$	−17.9764	−0.644900
$778$	16.9413	0.607376
$779$	5.54143	0.198542
$780$	−2.66860	−0.0955514
$781$	−29.1468	−1.04296
$782$	48.7635	1.74378
$783$	−1.86156	−0.0665267
$784$	−3.38854	−0.121019
$785$	−2.45368	−0.0875756
$786$	−12.1893	−0.434778
$787$	−40.6565	−1.44925	−0.724624	−	0.689145i	$-0.757986\pi$
$787$	−40.6565	−1.44925	−0.724624	+	0.689145i	$0.757986\pi$
$788$	−13.0396	−0.464517
$789$	−11.6599	−0.415104
$790$	1.29498	0.0460733
$791$	33.2403	1.18189
$792$	−4.67987	−0.166292
$793$	−53.0391	−1.88347
$794$	−12.8254	−0.455155
$795$	0.434198	0.0153994
$796$	−19.3542	−0.685993
$797$	−39.5059	−1.39937	−0.699685	−	0.714451i	$-0.746676\pi$
$797$	−39.5059	−1.39937	−0.699685	+	0.714451i	$0.746676\pi$
$798$	1.90038	0.0672729
$799$	−15.5017	−0.548412
$800$	−4.81147	−0.170111
$801$	3.09962	0.109520
$802$	20.1881	0.712865
$803$	10.8935	0.384425
$804$	−3.41291	−0.120364
$805$	5.04498	0.177812
$806$	17.0829	0.601718
$807$	−9.67225	−0.340479
$808$	17.1461	0.603196
$809$	−27.4991	−0.966815	−0.483408	−	0.875395i	$-0.660601\pi$
$809$	−27.4991	−0.966815	−0.483408	+	0.875395i	$0.660601\pi$
$810$	0.434198	0.0152562
$811$	−4.28814	−0.150577	−0.0752885	−	0.997162i	$-0.523988\pi$
$811$	−4.28814	−0.150577	−0.0752885	+	0.997162i	$0.523988\pi$
$812$	3.53768	0.124148
$813$	−13.5201	−0.474172
$814$	−44.2685	−1.55161
$815$	−7.76173	−0.271881
$816$	−7.97563	−0.279203
$817$	10.6054	0.371036
$818$	6.43420	0.224966
$819$	11.6799	0.408128
$820$	−2.40608	−0.0840238
$821$	42.2464	1.47441	0.737206	−	0.675668i	$-0.236145\pi$
$821$	42.2464	1.47441	0.737206	+	0.675668i	$0.236145\pi$
$822$	−16.1361	−0.562812
$823$	12.5915	0.438913	0.219456	−	0.975622i	$-0.429572\pi$
$823$	12.5915	0.438913	0.219456	+	0.975622i	$0.429572\pi$
$824$	5.37409	0.187215
$825$	−22.5171	−0.783943
$826$	−3.39469	−0.118116
$827$	14.6232	0.508497	0.254249	−	0.967139i	$-0.418172\pi$
$827$	14.6232	0.508497	0.254249	+	0.967139i	$0.418172\pi$
$828$	6.11407	0.212478
$829$	19.7725	0.686728	0.343364	−	0.939202i	$-0.388434\pi$
$829$	19.7725	0.686728	0.343364	+	0.939202i	$0.388434\pi$
$830$	0.288923	0.0100287
$831$	−25.8574	−0.896982
$832$	6.14605	0.213076
$833$	−27.0257	−0.936386
$834$	8.76878	0.303638
$835$	2.30376	0.0797249
$836$	4.67987	0.161857
$837$	−2.77948	−0.0960730
$838$	9.52708	0.329107
$839$	−12.2934	−0.424414	−0.212207	−	0.977225i	$-0.568065\pi$
$839$	−12.2934	−0.424414	−0.212207	+	0.977225i	$0.568065\pi$
$840$	−0.825143	−0.0284701
$841$	−25.5346	−0.880503
$842$	−31.0503	−1.07006
$843$	8.40618	0.289524
$844$	9.85416	0.339194
$845$	10.7568	0.370046
$846$	−1.94364	−0.0668236
$847$	20.7164	0.711824
$848$	−1.00000	−0.0343401
$849$	−25.8981	−0.888820
$850$	−38.3745	−1.31623
$851$	57.8351	1.98256
$852$	−6.22813	−0.213372
$853$	13.5877	0.465233	0.232616	−	0.972569i	$-0.425271\pi$
$853$	13.5877	0.465233	0.232616	+	0.972569i	$0.425271\pi$
$854$	−16.3999	−0.561193
$855$	−0.434198	−0.0148493
$856$	−8.14605	−0.278426
$857$	26.9619	0.921002	0.460501	−	0.887659i	$-0.347670\pi$
$857$	26.9619	0.921002	0.460501	+	0.887659i	$0.347670\pi$
$858$	28.7627	0.981943
$859$	26.4609	0.902835	0.451417	−	0.892313i	$-0.350919\pi$
$859$	26.4609	0.902835	0.451417	+	0.892313i	$0.350919\pi$
$860$	−4.60485	−0.157024
$861$	10.5308	0.358890
$862$	−18.2381	−0.621191
$863$	13.8045	0.469911	0.234956	−	0.972006i	$-0.424506\pi$
$863$	13.8045	0.469911	0.234956	+	0.972006i	$0.424506\pi$
$864$	−1.00000	−0.0340207
$865$	−1.25173	−0.0425600
$866$	12.9279	0.439309
$867$	−46.6106	−1.58298
$868$	5.28209	0.179286
$869$	−13.9575	−0.473477
$870$	−0.808286	−0.0274034
$871$	20.9759	0.710743
$872$	7.30646	0.247428
$873$	4.47745	0.151539
$874$	−6.11407	−0.206811
$875$	−8.09587	−0.273690
$876$	2.32775	0.0786473
$877$	38.8974	1.31347	0.656736	−	0.754120i	$-0.271936\pi$
$877$	38.8974	1.31347	0.656736	+	0.754120i	$0.271936\pi$
$878$	−17.3516	−0.585587
$879$	−11.1491	−0.376051
$880$	−2.03199	−0.0684983
$881$	33.5648	1.13083	0.565413	−	0.824808i	$-0.308717\pi$
$881$	33.5648	1.13083	0.565413	+	0.824808i	$0.308717\pi$
$882$	−3.38854	−0.114098
$883$	−51.0887	−1.71927	−0.859636	−	0.510907i	$-0.829310\pi$
$883$	−51.0887	−1.71927	−0.859636	+	0.510907i	$0.829310\pi$
$884$	49.0186	1.64868
$885$	0.775616	0.0260720
$886$	−28.6030	−0.960937
$887$	−24.3603	−0.817939	−0.408969	−	0.912548i	$-0.634112\pi$
$887$	−24.3603	−0.817939	−0.408969	+	0.912548i	$0.634112\pi$
$888$	−9.45935	−0.317435
$889$	−29.5072	−0.989641
$890$	1.34585	0.0451129
$891$	−4.67987	−0.156781
$892$	−4.13294	−0.138381
$893$	1.94364	0.0650413
$894$	−10.7119	−0.358258
$895$	5.32198	0.177894
$896$	1.90038	0.0634874
$897$	−37.5774	−1.25467
$898$	−28.0097	−0.934697
$899$	5.17418	0.172568
$900$	−4.81147	−0.160382
$901$	−7.97563	−0.265707
$902$	25.9332	0.863480
$903$	20.1544	0.670695
$904$	17.4913	0.581753
$905$	−3.83847	−0.127595
$906$	2.79075	0.0927164
$907$	11.8808	0.394494	0.197247	−	0.980354i	$-0.436800\pi$
$907$	11.8808	0.394494	0.197247	+	0.980354i	$0.436800\pi$
$908$	−9.76513	−0.324067
$909$	17.1461	0.568699
$910$	5.07137	0.168114
$911$	39.4413	1.30675	0.653373	−	0.757036i	$-0.273353\pi$
$911$	39.4413	1.30675	0.653373	+	0.757036i	$0.273353\pi$
$912$	1.00000	0.0331133
$913$	−3.11407	−0.103061
$914$	5.38720	0.178193
$915$	3.74703	0.123873
$916$	−5.40243	−0.178501
$917$	23.1644	0.764955
$918$	−7.97563	−0.263235
$919$	43.4015	1.43168	0.715841	−	0.698263i	$-0.246044\pi$
$919$	43.4015	1.43168	0.715841	+	0.698263i	$0.246044\pi$
$920$	2.65471	0.0875233
$921$	8.40530	0.276964
$922$	−2.93680	−0.0967184
$923$	38.2784	1.25995
$924$	8.89355	0.292576
$925$	−45.5134	−1.49647
$926$	−4.31205	−0.141703
$927$	5.37409	0.176508
$928$	1.86156	0.0611087
$929$	−50.9440	−1.67142	−0.835708	−	0.549173i	$-0.814943\pi$
$929$	−50.9440	−1.67142	−0.835708	+	0.549173i	$0.814943\pi$
$930$	−1.20685	−0.0395740
$931$	3.38854	0.111055
$932$	−2.26696	−0.0742566
$933$	9.51946	0.311653
$934$	−13.1284	−0.429575
$935$	−16.2064	−0.530005
$936$	6.14605	0.200890
$937$	−2.46927	−0.0806677	−0.0403338	−	0.999186i	$-0.512842\pi$
$937$	−2.46927	−0.0806677	−0.0403338	+	0.999186i	$0.512842\pi$
$938$	6.48585	0.211770
$939$	11.5797	0.377889
$940$	−0.843923	−0.0275257
$941$	−20.5176	−0.668855	−0.334428	−	0.942421i	$-0.608543\pi$
$941$	−20.5176	−0.668855	−0.334428	+	0.942421i	$0.608543\pi$
$942$	5.65107	0.184122
$943$	−33.8807	−1.10331
$944$	−1.78632	−0.0581397
$945$	−0.825143	−0.0268419
$946$	49.6319	1.61367
$947$	−32.4747	−1.05529	−0.527643	−	0.849466i	$-0.676924\pi$
$947$	−32.4747	−1.05529	−0.527643	+	0.849466i	$0.676924\pi$
$948$	−2.98246	−0.0968659
$949$	−14.3065	−0.464407
$950$	4.81147	0.156105
$951$	31.2578	1.01360
$952$	15.1568	0.491233
$953$	−42.4233	−1.37423	−0.687113	−	0.726551i	$-0.741122\pi$
$953$	−42.4233	−1.37423	−0.687113	+	0.726551i	$0.741122\pi$
$954$	−1.00000	−0.0323762
$955$	−0.319913	−0.0103522
$956$	24.9895	0.808219
$957$	8.71186	0.281614
$958$	12.3854	0.400153
$959$	30.6649	0.990220
$960$	−0.434198	−0.0140137
$961$	−23.2745	−0.750789
$962$	58.1377	1.87443
$963$	−8.14605	−0.262503
$964$	0.447989	0.0144287
$965$	−2.17919	−0.0701505
$966$	−11.6191	−0.373838
$967$	−21.2945	−0.684785	−0.342393	−	0.939557i	$-0.611237\pi$
$967$	−21.2945	−0.684785	−0.342393	+	0.939557i	$0.611237\pi$
$968$	10.9012	0.350377
$969$	7.97563	0.256214
$970$	1.94410	0.0624213
$971$	−15.3841	−0.493700	−0.246850	−	0.969054i	$-0.579395\pi$
$971$	−15.3841	−0.493700	−0.246850	+	0.969054i	$0.579395\pi$
$972$	−1.00000	−0.0320750
$973$	−16.6641	−0.534225
$974$	40.0197	1.28231
$975$	29.5716	0.947048
$976$	−8.62978	−0.276233
$977$	−39.8209	−1.27398	−0.636992	−	0.770870i	$-0.719822\pi$
$977$	−39.8209	−1.27398	−0.636992	+	0.770870i	$0.719822\pi$
$978$	17.8760	0.571612
$979$	−14.5058	−0.463607
$980$	−1.47130	−0.0469988
$981$	7.30646	0.233277
$982$	25.4783	0.813046
$983$	32.9444	1.05076	0.525382	−	0.850867i	$-0.323922\pi$
$983$	32.9444	1.05076	0.525382	+	0.850867i	$0.323922\pi$
$984$	5.54143	0.176654
$985$	−5.66177	−0.180399
$986$	14.8471	0.472828
$987$	3.69366	0.117570
$988$	−6.14605	−0.195532
$989$	−64.8422	−2.06186
$990$	−2.03199	−0.0645808
$991$	−49.7590	−1.58065	−0.790323	−	0.612691i	$-0.790087\pi$
$991$	−49.7590	−1.58065	−0.790323	+	0.612691i	$0.790087\pi$
$992$	2.77948	0.0882487
$993$	−21.5779	−0.684755
$994$	11.8358	0.375410
$995$	−8.40357	−0.266411
$996$	−0.665417	−0.0210846
$997$	−55.7923	−1.76696	−0.883480	−	0.468469i	$-0.844806\pi$
$997$	−55.7923	−1.76696	−0.883480	+	0.468469i	$0.844806\pi$
$998$	−17.0852	−0.540821
$999$	−9.45935	−0.299281

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.t.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.t.1.2	✓	4	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:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.434198 q^{5} -1.00000 q^{6} +1.90038 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.434198 q^{5} -1.00000 q^{6} +1.90038 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.434198 q^{10} -4.67987 q^{11} -1.00000 q^{12} +6.14605 q^{13} +1.90038 q^{14} -0.434198 q^{15} +1.00000 q^{16} +7.97563 q^{17} +1.00000 q^{18} -1.00000 q^{19} +0.434198 q^{20} -1.90038 q^{21} -4.67987 q^{22} +6.11407 q^{23} -1.00000 q^{24} -4.81147 q^{25} +6.14605 q^{26} -1.00000 q^{27} +1.90038 q^{28} +1.86156 q^{29} -0.434198 q^{30} +2.77948 q^{31} +1.00000 q^{32} +4.67987 q^{33} +7.97563 q^{34} +0.825143 q^{35} +1.00000 q^{36} +9.45935 q^{37} -1.00000 q^{38} -6.14605 q^{39} +0.434198 q^{40} -5.54143 q^{41} -1.90038 q^{42} -10.6054 q^{43} -4.67987 q^{44} +0.434198 q^{45} +6.11407 q^{46} -1.94364 q^{47} -1.00000 q^{48} -3.38854 q^{49} -4.81147 q^{50} -7.97563 q^{51} +6.14605 q^{52} -1.00000 q^{53} -1.00000 q^{54} -2.03199 q^{55} +1.90038 q^{56} +1.00000 q^{57} +1.86156 q^{58} -1.78632 q^{59} -0.434198 q^{60} -8.62978 q^{61} +2.77948 q^{62} +1.90038 q^{63} +1.00000 q^{64} +2.66860 q^{65} +4.67987 q^{66} +3.41291 q^{67} +7.97563 q^{68} -6.11407 q^{69} +0.825143 q^{70} +6.22813 q^{71} +1.00000 q^{72} -2.32775 q^{73} +9.45935 q^{74} +4.81147 q^{75} -1.00000 q^{76} -8.89355 q^{77} -6.14605 q^{78} +2.98246 q^{79} +0.434198 q^{80} +1.00000 q^{81} -5.54143 q^{82} +0.665417 q^{83} -1.90038 q^{84} +3.46300 q^{85} -10.6054 q^{86} -1.86156 q^{87} -4.67987 q^{88} +3.09962 q^{89} +0.434198 q^{90} +11.6799 q^{91} +6.11407 q^{92} -2.77948 q^{93} -1.94364 q^{94} -0.434198 q^{95} -1.00000 q^{96} +4.47745 q^{97} -3.38854 q^{98} -4.67987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} + 3 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 + 3 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} + 3 q^{7} + 4 q^{8} + 4 q^{9} + 6 q^{10} - 2 q^{11} - 4 q^{12} - q^{13} + 3 q^{14} - 6 q^{15} + 4 q^{16} + 8 q^{17} + 4 q^{18} - 4 q^{19} + 6 q^{20} - 3 q^{21} - 2 q^{22} + 12 q^{23} - 4 q^{24} + 6 q^{25} - q^{26} - 4 q^{27} + 3 q^{28} - 4 q^{29} - 6 q^{30} - q^{31} + 4 q^{32} + 2 q^{33} + 8 q^{34} + 18 q^{35} + 4 q^{36} + 9 q^{37} - 4 q^{38} + q^{39} + 6 q^{40} + 6 q^{41} - 3 q^{42} + 12 q^{43} - 2 q^{44} + 6 q^{45} + 12 q^{46} + 3 q^{47} - 4 q^{48} + 9 q^{49} + 6 q^{50} - 8 q^{51} - q^{52} - 4 q^{53} - 4 q^{54} + 5 q^{55} + 3 q^{56} + 4 q^{57} - 4 q^{58} - 15 q^{59} - 6 q^{60} - 4 q^{61} - q^{62} + 3 q^{63} + 4 q^{64} - 13 q^{65} + 2 q^{66} + 15 q^{67} + 8 q^{68} - 12 q^{69} + 18 q^{70} + 4 q^{72} + 11 q^{73} + 9 q^{74} - 6 q^{75} - 4 q^{76} - 11 q^{77} + q^{78} + 8 q^{79} + 6 q^{80} + 4 q^{81} + 6 q^{82} + 3 q^{83} - 3 q^{84} + 29 q^{85} + 12 q^{86} + 4 q^{87} - 2 q^{88} + 17 q^{89} + 6 q^{90} + 30 q^{91} + 12 q^{92} + q^{93} + 3 q^{94} - 6 q^{95} - 4 q^{96} + 16 q^{97} + 9 q^{98} - 2 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 + 3 * q^7 + 4 * q^8 + 4 * q^9 + 6 * q^10 - 2 * q^11 - 4 * q^12 - q^13 + 3 * q^14 - 6 * q^15 + 4 * q^16 + 8 * q^17 + 4 * q^18 - 4 * q^19 + 6 * q^20 - 3 * q^21 - 2 * q^22 + 12 * q^23 - 4 * q^24 + 6 * q^25 - q^26 - 4 * q^27 + 3 * q^28 - 4 * q^29 - 6 * q^30 - q^31 + 4 * q^32 + 2 * q^33 + 8 * q^34 + 18 * q^35 + 4 * q^36 + 9 * q^37 - 4 * q^38 + q^39 + 6 * q^40 + 6 * q^41 - 3 * q^42 + 12 * q^43 - 2 * q^44 + 6 * q^45 + 12 * q^46 + 3 * q^47 - 4 * q^48 + 9 * q^49 + 6 * q^50 - 8 * q^51 - q^52 - 4 * q^53 - 4 * q^54 + 5 * q^55 + 3 * q^56 + 4 * q^57 - 4 * q^58 - 15 * q^59 - 6 * q^60 - 4 * q^61 - q^62 + 3 * q^63 + 4 * q^64 - 13 * q^65 + 2 * q^66 + 15 * q^67 + 8 * q^68 - 12 * q^69 + 18 * q^70 + 4 * q^72 + 11 * q^73 + 9 * q^74 - 6 * q^75 - 4 * q^76 - 11 * q^77 + q^78 + 8 * q^79 + 6 * q^80 + 4 * q^81 + 6 * q^82 + 3 * q^83 - 3 * q^84 + 29 * q^85 + 12 * q^86 + 4 * q^87 - 2 * q^88 + 17 * q^89 + 6 * q^90 + 30 * q^91 + 12 * q^92 + q^93 + 3 * q^94 - 6 * q^95 - 4 * q^96 + 16 * q^97 + 9 * q^98 - 2 * q^99

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