LMFDB - Embedded newform 6038.2.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, 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("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$0$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.91924 q^{3} +1.00000 q^{4} -2.24533 q^{5} +2.91924 q^{6} -3.88903 q^{7} -1.00000 q^{8} +5.52197 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.91924 q^{3} +1.00000 q^{4} -2.24533 q^{5} +2.91924 q^{6} -3.88903 q^{7} -1.00000 q^{8} +5.52197 q^{9} +2.24533 q^{10} +0.797741 q^{11} -2.91924 q^{12} +5.48386 q^{13} +3.88903 q^{14} +6.55465 q^{15} +1.00000 q^{16} +5.85624 q^{17} -5.52197 q^{18} -2.04598 q^{19} -2.24533 q^{20} +11.3530 q^{21} -0.797741 q^{22} +1.32664 q^{23} +2.91924 q^{24} +0.0414955 q^{25} -5.48386 q^{26} -7.36225 q^{27} -3.88903 q^{28} -2.99583 q^{29} -6.55465 q^{30} +8.09746 q^{31} -1.00000 q^{32} -2.32880 q^{33} -5.85624 q^{34} +8.73214 q^{35} +5.52197 q^{36} +5.36957 q^{37} +2.04598 q^{38} -16.0087 q^{39} +2.24533 q^{40} +5.25796 q^{41} -11.3530 q^{42} +11.5220 q^{43} +0.797741 q^{44} -12.3986 q^{45} -1.32664 q^{46} +2.12851 q^{47} -2.91924 q^{48} +8.12454 q^{49} -0.0414955 q^{50} -17.0958 q^{51} +5.48386 q^{52} +6.17073 q^{53} +7.36225 q^{54} -1.79119 q^{55} +3.88903 q^{56} +5.97271 q^{57} +2.99583 q^{58} -0.546527 q^{59} +6.55465 q^{60} +3.30978 q^{61} -8.09746 q^{62} -21.4751 q^{63} +1.00000 q^{64} -12.3131 q^{65} +2.32880 q^{66} +2.52447 q^{67} +5.85624 q^{68} -3.87278 q^{69} -8.73214 q^{70} -9.00291 q^{71} -5.52197 q^{72} -6.37453 q^{73} -5.36957 q^{74} -0.121135 q^{75} -2.04598 q^{76} -3.10244 q^{77} +16.0087 q^{78} +4.19591 q^{79} -2.24533 q^{80} +4.92626 q^{81} -5.25796 q^{82} -6.83296 q^{83} +11.3530 q^{84} -13.1492 q^{85} -11.5220 q^{86} +8.74554 q^{87} -0.797741 q^{88} -2.75079 q^{89} +12.3986 q^{90} -21.3269 q^{91} +1.32664 q^{92} -23.6384 q^{93} -2.12851 q^{94} +4.59390 q^{95} +2.91924 q^{96} +11.9841 q^{97} -8.12454 q^{98} +4.40510 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) $ 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.91924	−1.68542	−0.842712	−	0.538364i	$-0.819043\pi$
$3$	−2.91924	−1.68542	−0.842712	+	0.538364i	$0.819043\pi$
$4$	1.00000	0.500000
$5$	−2.24533	−1.00414	−0.502070	−	0.864827i	$-0.667428\pi$
$5$	−2.24533	−1.00414	−0.502070	+	0.864827i	$0.667428\pi$
$6$	2.91924	1.19178
$7$	−3.88903	−1.46991	−0.734957	−	0.678114i	$-0.762798\pi$
$7$	−3.88903	−1.46991	−0.734957	+	0.678114i	$0.762798\pi$
$8$	−1.00000	−0.353553
$9$	5.52197	1.84066
$10$	2.24533	0.710035
$11$	0.797741	0.240528	0.120264	−	0.992742i	$-0.461626\pi$
$11$	0.797741	0.240528	0.120264	+	0.992742i	$0.461626\pi$
$12$	−2.91924	−0.842712
$13$	5.48386	1.52095	0.760475	−	0.649368i	$-0.224966\pi$
$13$	5.48386	1.52095	0.760475	+	0.649368i	$0.224966\pi$
$14$	3.88903	1.03939
$15$	6.55465	1.69240
$16$	1.00000	0.250000
$17$	5.85624	1.42035	0.710173	−	0.704027i	$-0.248617\pi$
$17$	5.85624	1.42035	0.710173	+	0.704027i	$0.248617\pi$
$18$	−5.52197	−1.30154
$19$	−2.04598	−0.469380	−0.234690	−	0.972070i	$-0.575408\pi$
$19$	−2.04598	−0.469380	−0.234690	+	0.972070i	$0.575408\pi$
$20$	−2.24533	−0.502070
$21$	11.3530	2.47743
$22$	−0.797741	−0.170079
$23$	1.32664	0.276624	0.138312	−	0.990389i	$-0.455832\pi$
$23$	1.32664	0.276624	0.138312	+	0.990389i	$0.455832\pi$
$24$	2.91924	0.595888
$25$	0.0414955	0.00829909
$26$	−5.48386	−1.07547
$27$	−7.36225	−1.41686
$28$	−3.88903	−0.734957
$29$	−2.99583	−0.556311	−0.278156	−	0.960536i	$-0.589723\pi$
$29$	−2.99583	−0.556311	−0.278156	+	0.960536i	$0.589723\pi$
$30$	−6.55465	−1.19671
$31$	8.09746	1.45435	0.727173	−	0.686454i	$-0.240834\pi$
$31$	8.09746	1.45435	0.727173	+	0.686454i	$0.240834\pi$
$32$	−1.00000	−0.176777
$33$	−2.32880	−0.405392
$34$	−5.85624	−1.00434
$35$	8.73214	1.47600
$36$	5.52197	0.920329
$37$	5.36957	0.882752	0.441376	−	0.897322i	$-0.354490\pi$
$37$	5.36957	0.882752	0.441376	+	0.897322i	$0.354490\pi$
$38$	2.04598	0.331902
$39$	−16.0087	−2.56345
$40$	2.24533	0.355017
$41$	5.25796	0.821156	0.410578	−	0.911825i	$-0.365327\pi$
$41$	5.25796	0.821156	0.410578	+	0.911825i	$0.365327\pi$
$42$	−11.3530	−1.75181
$43$	11.5220	1.75710	0.878548	−	0.477655i	$-0.158513\pi$
$43$	11.5220	1.75710	0.878548	+	0.477655i	$0.158513\pi$
$44$	0.797741	0.120264
$45$	−12.3986	−1.84828
$46$	−1.32664	−0.195602
$47$	2.12851	0.310475	0.155238	−	0.987877i	$-0.450386\pi$
$47$	2.12851	0.310475	0.155238	+	0.987877i	$0.450386\pi$
$48$	−2.91924	−0.421356
$49$	8.12454	1.16065
$50$	−0.0414955	−0.00586834
$51$	−17.0958	−2.39389
$52$	5.48386	0.760475
$53$	6.17073	0.847615	0.423807	−	0.905752i	$-0.360693\pi$
$53$	6.17073	0.847615	0.423807	+	0.905752i	$0.360693\pi$
$54$	7.36225	1.00187
$55$	−1.79119	−0.241524
$56$	3.88903	0.519693
$57$	5.97271	0.791105
$58$	2.99583	0.393371
$59$	−0.546527	−0.0711518	−0.0355759	−	0.999367i	$-0.511327\pi$
$59$	−0.546527	−0.0711518	−0.0355759	+	0.999367i	$0.511327\pi$
$60$	6.55465	0.846202
$61$	3.30978	0.423773	0.211887	−	0.977294i	$-0.432039\pi$
$61$	3.30978	0.423773	0.211887	+	0.977294i	$0.432039\pi$
$62$	−8.09746	−1.02838
$63$	−21.4751	−2.70561
$64$	1.00000	0.125000
$65$	−12.3131	−1.52725
$66$	2.32880	0.286655
$67$	2.52447	0.308413	0.154206	−	0.988039i	$-0.450718\pi$
$67$	2.52447	0.308413	0.154206	+	0.988039i	$0.450718\pi$
$68$	5.85624	0.710173
$69$	−3.87278	−0.466228
$70$	−8.73214	−1.04369
$71$	−9.00291	−1.06845	−0.534224	−	0.845343i	$-0.679396\pi$
$71$	−9.00291	−1.06845	−0.534224	+	0.845343i	$0.679396\pi$
$72$	−5.52197	−0.650771
$73$	−6.37453	−0.746083	−0.373041	−	0.927815i	$-0.621685\pi$
$73$	−6.37453	−0.746083	−0.373041	+	0.927815i	$0.621685\pi$
$74$	−5.36957	−0.624200
$75$	−0.121135	−0.0139875
$76$	−2.04598	−0.234690
$77$	−3.10244	−0.353555
$78$	16.0087	1.81263
$79$	4.19591	0.472077	0.236038	−	0.971744i	$-0.424151\pi$
$79$	4.19591	0.472077	0.236038	+	0.971744i	$0.424151\pi$
$80$	−2.24533	−0.251035
$81$	4.92626	0.547362
$82$	−5.25796	−0.580645
$83$	−6.83296	−0.750015	−0.375008	−	0.927022i	$-0.622360\pi$
$83$	−6.83296	−0.750015	−0.375008	+	0.927022i	$0.622360\pi$
$84$	11.3530	1.23872
$85$	−13.1492	−1.42623
$86$	−11.5220	−1.24245
$87$	8.74554	0.937621
$88$	−0.797741	−0.0850394
$89$	−2.75079	−0.291583	−0.145791	−	0.989315i	$-0.546573\pi$
$89$	−2.75079	−0.291583	−0.145791	+	0.989315i	$0.546573\pi$
$90$	12.3986	1.30693
$91$	−21.3269	−2.23567
$92$	1.32664	0.138312
$93$	−23.6384	−2.45119
$94$	−2.12851	−0.219539
$95$	4.59390	0.471324
$96$	2.91924	0.297944
$97$	11.9841	1.21680	0.608401	−	0.793630i	$-0.291811\pi$
$97$	11.9841	1.21680	0.608401	+	0.793630i	$0.291811\pi$
$98$	−8.12454	−0.820702
$99$	4.40510	0.442729
$100$	0.0414955	0.00414955
$101$	−13.3126	−1.32465	−0.662324	−	0.749217i	$-0.730430\pi$
$101$	−13.3126	−1.32465	−0.662324	+	0.749217i	$0.730430\pi$
$102$	17.0958	1.69273
$103$	−16.0283	−1.57931	−0.789656	−	0.613550i	$-0.789741\pi$
$103$	−16.0283	−1.57931	−0.789656	+	0.613550i	$0.789741\pi$
$104$	−5.48386	−0.537737
$105$	−25.4912	−2.48769
$106$	−6.17073	−0.599354
$107$	3.01997	0.291951	0.145976	−	0.989288i	$-0.453368\pi$
$107$	3.01997	0.291951	0.145976	+	0.989288i	$0.453368\pi$
$108$	−7.36225	−0.708432
$109$	3.69840	0.354242	0.177121	−	0.984189i	$-0.443322\pi$
$109$	3.69840	0.354242	0.177121	+	0.984189i	$0.443322\pi$
$110$	1.79119	0.170783
$111$	−15.6751	−1.48781
$112$	−3.88903	−0.367479
$113$	1.50403	0.141488	0.0707438	−	0.997495i	$-0.477463\pi$
$113$	1.50403	0.141488	0.0707438	+	0.997495i	$0.477463\pi$
$114$	−5.97271	−0.559396
$115$	−2.97874	−0.277769
$116$	−2.99583	−0.278156
$117$	30.2817	2.79955
$118$	0.546527	0.0503119
$119$	−22.7751	−2.08779
$120$	−6.55465	−0.598355
$121$	−10.3636	−0.942146
$122$	−3.30978	−0.299653
$123$	−15.3493	−1.38400
$124$	8.09746	0.727173
$125$	11.1335	0.995808
$126$	21.4751	1.91315
$127$	6.64186	0.589370	0.294685	−	0.955594i	$-0.404785\pi$
$127$	6.64186	0.589370	0.294685	+	0.955594i	$0.404785\pi$
$128$	−1.00000	−0.0883883
$129$	−33.6356	−2.96145
$130$	12.3131	1.07993
$131$	5.20929	0.455138	0.227569	−	0.973762i	$-0.426922\pi$
$131$	5.20929	0.455138	0.227569	+	0.973762i	$0.426922\pi$
$132$	−2.32880	−0.202696
$133$	7.95688	0.689949
$134$	−2.52447	−0.218081
$135$	16.5307	1.42273
$136$	−5.85624	−0.502168
$137$	15.6297	1.33534	0.667668	−	0.744460i	$-0.267293\pi$
$137$	15.6297	1.33534	0.667668	+	0.744460i	$0.267293\pi$
$138$	3.87278	0.329673
$139$	−13.3566	−1.13289	−0.566446	−	0.824099i	$-0.691682\pi$
$139$	−13.3566	−1.13289	−0.566446	+	0.824099i	$0.691682\pi$
$140$	8.73214	0.738001
$141$	−6.21363	−0.523282
$142$	9.00291	0.755507
$143$	4.37470	0.365831
$144$	5.52197	0.460164
$145$	6.72661	0.558615
$146$	6.37453	0.527560
$147$	−23.7175	−1.95618
$148$	5.36957	0.441376
$149$	18.6970	1.53172	0.765859	−	0.643008i	$-0.222314\pi$
$149$	18.6970	1.53172	0.765859	+	0.643008i	$0.222314\pi$
$150$	0.121135	0.00989065
$151$	−12.3293	−1.00335	−0.501674	−	0.865057i	$-0.667282\pi$
$151$	−12.3293	−1.00335	−0.501674	+	0.865057i	$0.667282\pi$
$152$	2.04598	0.165951
$153$	32.3380	2.61437
$154$	3.10244	0.250001
$155$	−18.1814	−1.46037
$156$	−16.0087	−1.28172
$157$	−7.08039	−0.565076	−0.282538	−	0.959256i	$-0.591176\pi$
$157$	−7.08039	−0.565076	−0.282538	+	0.959256i	$0.591176\pi$
$158$	−4.19591	−0.333809
$159$	−18.0138	−1.42859
$160$	2.24533	0.177509
$161$	−5.15934	−0.406613
$162$	−4.92626	−0.387043
$163$	−9.97115	−0.781000	−0.390500	−	0.920603i	$-0.627698\pi$
$163$	−9.97115	−0.781000	−0.390500	+	0.920603i	$0.627698\pi$
$164$	5.25796	0.410578
$165$	5.22891	0.407070
$166$	6.83296	0.530341
$167$	−3.01205	−0.233080	−0.116540	−	0.993186i	$-0.537180\pi$
$167$	−3.01205	−0.233080	−0.116540	+	0.993186i	$0.537180\pi$
$168$	−11.3530	−0.875904
$169$	17.0727	1.31329
$170$	13.1492	1.00850
$171$	−11.2979	−0.863968
$172$	11.5220	0.878548
$173$	20.6595	1.57071	0.785355	−	0.619046i	$-0.212481\pi$
$173$	20.6595	1.57071	0.785355	+	0.619046i	$0.212481\pi$
$174$	−8.74554	−0.662998
$175$	−0.161377	−0.0121990
$176$	0.797741	0.0601320
$177$	1.59545	0.119921
$178$	2.75079	0.206180
$179$	−5.67504	−0.424173	−0.212086	−	0.977251i	$-0.568026\pi$
$179$	−5.67504	−0.424173	−0.212086	+	0.977251i	$0.568026\pi$
$180$	−12.3986	−0.924140
$181$	2.58426	0.192087	0.0960434	−	0.995377i	$-0.469381\pi$
$181$	2.58426	0.192087	0.0960434	+	0.995377i	$0.469381\pi$
$182$	21.3269	1.58085
$183$	−9.66203	−0.714238
$184$	−1.32664	−0.0978012
$185$	−12.0564	−0.886408
$186$	23.6384	1.73325
$187$	4.67176	0.341633
$188$	2.12851	0.155238
$189$	28.6320	2.08267
$190$	−4.59390	−0.333276
$191$	−11.4693	−0.829893	−0.414946	−	0.909846i	$-0.636200\pi$
$191$	−11.4693	−0.829893	−0.414946	+	0.909846i	$0.636200\pi$
$192$	−2.91924	−0.210678
$193$	12.6615	0.911395	0.455697	−	0.890135i	$-0.349390\pi$
$193$	12.6615	0.911395	0.455697	+	0.890135i	$0.349390\pi$
$194$	−11.9841	−0.860409
$195$	35.9448	2.57406
$196$	8.12454	0.580324
$197$	−4.38989	−0.312766	−0.156383	−	0.987696i	$-0.549983\pi$
$197$	−4.38989	−0.312766	−0.156383	+	0.987696i	$0.549983\pi$
$198$	−4.40510	−0.313057
$199$	4.38731	0.311008	0.155504	−	0.987835i	$-0.450300\pi$
$199$	4.38731	0.311008	0.155504	+	0.987835i	$0.450300\pi$
$200$	−0.0414955	−0.00293417
$201$	−7.36953	−0.519806
$202$	13.3126	0.936668
$203$	11.6509	0.817730
$204$	−17.0958	−1.19694
$205$	−11.8059	−0.824556
$206$	16.0283	1.11674
$207$	7.32567	0.509169
$208$	5.48386	0.380237
$209$	−1.63216	−0.112899
$210$	25.4912	1.75906
$211$	8.25701	0.568436	0.284218	−	0.958760i	$-0.408266\pi$
$211$	8.25701	0.568436	0.284218	+	0.958760i	$0.408266\pi$
$212$	6.17073	0.423807
$213$	26.2817	1.80079
$214$	−3.01997	−0.206441
$215$	−25.8708	−1.76437
$216$	7.36225	0.500937
$217$	−31.4912	−2.13776
$218$	−3.69840	−0.250487
$219$	18.6088	1.25747
$220$	−1.79119	−0.120762
$221$	32.1148	2.16028
$222$	15.6751	1.05204
$223$	24.4281	1.63582	0.817912	−	0.575343i	$-0.195132\pi$
$223$	24.4281	1.63582	0.817912	+	0.575343i	$0.195132\pi$
$224$	3.88903	0.259847
$225$	0.229137	0.0152758
$226$	−1.50403	−0.100047
$227$	−7.87359	−0.522589	−0.261294	−	0.965259i	$-0.584149\pi$
$227$	−7.87359	−0.522589	−0.261294	+	0.965259i	$0.584149\pi$
$228$	5.97271	0.395553
$229$	−1.20220	−0.0794439	−0.0397219	−	0.999211i	$-0.512647\pi$
$229$	−1.20220	−0.0794439	−0.0397219	+	0.999211i	$0.512647\pi$
$230$	2.97874	0.196412
$231$	9.05676	0.595891
$232$	2.99583	0.196686
$233$	14.0487	0.920359	0.460180	−	0.887826i	$-0.347785\pi$
$233$	14.0487	0.920359	0.460180	+	0.887826i	$0.347785\pi$
$234$	−30.2817	−1.97958
$235$	−4.77920	−0.311761
$236$	−0.546527	−0.0355759
$237$	−12.2489	−0.795650
$238$	22.7751	1.47629
$239$	0.784400	0.0507386	0.0253693	−	0.999678i	$-0.491924\pi$
$239$	0.784400	0.0507386	0.0253693	+	0.999678i	$0.491924\pi$
$240$	6.55465	0.423101
$241$	7.67889	0.494641	0.247320	−	0.968934i	$-0.420450\pi$
$241$	7.67889	0.494641	0.247320	+	0.968934i	$0.420450\pi$
$242$	10.3636	0.666198
$243$	7.70580	0.494327
$244$	3.30978	0.211887
$245$	−18.2422	−1.16545
$246$	15.3493	0.978634
$247$	−11.2199	−0.713904
$248$	−8.09746	−0.514189
$249$	19.9471	1.26409
$250$	−11.1335	−0.704142
$251$	−1.60802	−0.101497	−0.0507487	−	0.998711i	$-0.516161\pi$
$251$	−1.60802	−0.101497	−0.0507487	+	0.998711i	$0.516161\pi$
$252$	−21.4751	−1.35280
$253$	1.05832	0.0665357
$254$	−6.64186	−0.416748
$255$	38.3856	2.40380
$256$	1.00000	0.0625000
$257$	−18.6047	−1.16053	−0.580265	−	0.814427i	$-0.697051\pi$
$257$	−18.6047	−1.16053	−0.580265	+	0.814427i	$0.697051\pi$
$258$	33.6356	2.09406
$259$	−20.8824	−1.29757
$260$	−12.3131	−0.763624
$261$	−16.5429	−1.02398
$262$	−5.20929	−0.321831
$263$	−14.2627	−0.879478	−0.439739	−	0.898126i	$-0.644929\pi$
$263$	−14.2627	−0.879478	−0.439739	+	0.898126i	$0.644929\pi$
$264$	2.32880	0.143328
$265$	−13.8553	−0.851125
$266$	−7.95688	−0.487868
$267$	8.03021	0.491441
$268$	2.52447	0.154206
$269$	6.37324	0.388583	0.194292	−	0.980944i	$-0.437759\pi$
$269$	6.37324	0.388583	0.194292	+	0.980944i	$0.437759\pi$
$270$	−16.5307	−1.00602
$271$	−7.13259	−0.433274	−0.216637	−	0.976252i	$-0.569509\pi$
$271$	−7.13259	−0.433274	−0.216637	+	0.976252i	$0.569509\pi$
$272$	5.85624	0.355087
$273$	62.2583	3.76805
$274$	−15.6297	−0.944225
$275$	0.0331026	0.00199616
$276$	−3.87278	−0.233114
$277$	10.2143	0.613720	0.306860	−	0.951755i	$-0.400722\pi$
$277$	10.2143	0.613720	0.306860	+	0.951755i	$0.400722\pi$
$278$	13.3566	0.801076
$279$	44.7139	2.67695
$280$	−8.73214	−0.521845
$281$	−28.9420	−1.72654	−0.863269	−	0.504745i	$-0.831587\pi$
$281$	−28.9420	−1.72654	−0.863269	+	0.504745i	$0.831587\pi$
$282$	6.21363	0.370017
$283$	4.60549	0.273768	0.136884	−	0.990587i	$-0.456291\pi$
$283$	4.60549	0.273768	0.136884	+	0.990587i	$0.456291\pi$
$284$	−9.00291	−0.534224
$285$	−13.4107	−0.794381
$286$	−4.37470	−0.258681
$287$	−20.4484	−1.20703
$288$	−5.52197	−0.325385
$289$	17.2955	1.01738
$290$	−6.72661	−0.395000
$291$	−34.9845	−2.05083
$292$	−6.37453	−0.373041
$293$	2.43538	0.142276	0.0711382	−	0.997466i	$-0.477337\pi$
$293$	2.43538	0.142276	0.0711382	+	0.997466i	$0.477337\pi$
$294$	23.7175	1.38323
$295$	1.22713	0.0714465
$296$	−5.36957	−0.312100
$297$	−5.87316	−0.340795
$298$	−18.6970	−1.08309
$299$	7.27511	0.420731
$300$	−0.121135	−0.00699375
$301$	−44.8095	−2.58278
$302$	12.3293	0.709474
$303$	38.8626	2.23260
$304$	−2.04598	−0.117345
$305$	−7.43153	−0.425528
$306$	−32.3380	−1.84864
$307$	14.4906	0.827023	0.413512	−	0.910499i	$-0.364302\pi$
$307$	14.4906	0.827023	0.413512	+	0.910499i	$0.364302\pi$
$308$	−3.10244	−0.176778
$309$	46.7904	2.66181
$310$	18.1814	1.03264
$311$	18.6608	1.05815	0.529077	−	0.848574i	$-0.322538\pi$
$311$	18.6608	1.05815	0.529077	+	0.848574i	$0.322538\pi$
$312$	16.0087	0.906315
$313$	−25.4577	−1.43895	−0.719476	−	0.694517i	$-0.755618\pi$
$313$	−25.4577	−1.43895	−0.719476	+	0.694517i	$0.755618\pi$
$314$	7.08039	0.399569
$315$	48.2186	2.71681
$316$	4.19591	0.236038
$317$	18.9879	1.06647	0.533233	−	0.845969i	$-0.320977\pi$
$317$	18.9879	1.06647	0.533233	+	0.845969i	$0.320977\pi$
$318$	18.0138	1.01017
$319$	−2.38989	−0.133808
$320$	−2.24533	−0.125518
$321$	−8.81602	−0.492062
$322$	5.15934	0.287519
$323$	−11.9818	−0.666683
$324$	4.92626	0.273681
$325$	0.227555	0.0126225
$326$	9.97115	0.552251
$327$	−10.7965	−0.597049
$328$	−5.25796	−0.290323
$329$	−8.27783	−0.456372
$330$	−5.22891	−0.287842
$331$	−20.4321	−1.12305	−0.561525	−	0.827460i	$-0.689785\pi$
$331$	−20.4321	−1.12305	−0.561525	+	0.827460i	$0.689785\pi$
$332$	−6.83296	−0.375008
$333$	29.6506	1.62484
$334$	3.01205	0.164812
$335$	−5.66825	−0.309690
$336$	11.3530	0.619358
$337$	23.4523	1.27753	0.638765	−	0.769402i	$-0.279446\pi$
$337$	23.4523	1.27753	0.638765	+	0.769402i	$0.279446\pi$
$338$	−17.0727	−0.928634
$339$	−4.39064	−0.238467
$340$	−13.1492	−0.713114
$341$	6.45967	0.349811
$342$	11.2979	0.610918
$343$	−4.37335	−0.236139
$344$	−11.5220	−0.621227
$345$	8.69567	0.468159
$346$	−20.6595	−1.11066
$347$	−2.86379	−0.153736	−0.0768681	−	0.997041i	$-0.524492\pi$
$347$	−2.86379	−0.153736	−0.0768681	+	0.997041i	$0.524492\pi$
$348$	8.74554	0.468810
$349$	27.9999	1.49880	0.749400	−	0.662118i	$-0.230342\pi$
$349$	27.9999	1.49880	0.749400	+	0.662118i	$0.230342\pi$
$350$	0.161377	0.00862596
$351$	−40.3735	−2.15498
$352$	−0.797741	−0.0425197
$353$	2.08668	0.111063	0.0555314	−	0.998457i	$-0.482315\pi$
$353$	2.08668	0.111063	0.0555314	+	0.998457i	$0.482315\pi$
$354$	−1.59545	−0.0847970
$355$	20.2145	1.07287
$356$	−2.75079	−0.145791
$357$	66.4859	3.51881
$358$	5.67504	0.299935
$359$	−3.87901	−0.204726	−0.102363	−	0.994747i	$-0.532640\pi$
$359$	−3.87901	−0.204726	−0.102363	+	0.994747i	$0.532640\pi$
$360$	12.3986	0.653465
$361$	−14.8140	−0.779682
$362$	−2.58426	−0.135826
$363$	30.2539	1.58792
$364$	−21.3269	−1.11783
$365$	14.3129	0.749172
$366$	9.66203	0.505043
$367$	22.3753	1.16798	0.583991	−	0.811760i	$-0.301490\pi$
$367$	22.3753	1.16798	0.583991	+	0.811760i	$0.301490\pi$
$368$	1.32664	0.0691559
$369$	29.0343	1.51147
$370$	12.0564	0.626785
$371$	−23.9981	−1.24592
$372$	−23.6384	−1.22560
$373$	−1.12720	−0.0583643	−0.0291822	−	0.999574i	$-0.509290\pi$
$373$	−1.12720	−0.0583643	−0.0291822	+	0.999574i	$0.509290\pi$
$374$	−4.67176	−0.241571
$375$	−32.5013	−1.67836
$376$	−2.12851	−0.109770
$377$	−16.4287	−0.846121
$378$	−28.6320	−1.47267
$379$	−3.03883	−0.156094	−0.0780471	−	0.996950i	$-0.524868\pi$
$379$	−3.03883	−0.156094	−0.0780471	+	0.996950i	$0.524868\pi$
$380$	4.59390	0.235662
$381$	−19.3892	−0.993339
$382$	11.4693	0.586823
$383$	−8.09999	−0.413890	−0.206945	−	0.978353i	$-0.566352\pi$
$383$	−8.09999	−0.413890	−0.206945	+	0.978353i	$0.566352\pi$
$384$	2.91924	0.148972
$385$	6.96598	0.355019
$386$	−12.6615	−0.644453
$387$	63.6244	3.23421
$388$	11.9841	0.608401
$389$	−37.1316	−1.88264	−0.941322	−	0.337509i	$-0.890416\pi$
$389$	−37.1316	−1.88264	−0.941322	+	0.337509i	$0.890416\pi$
$390$	−35.9448	−1.82014
$391$	7.76913	0.392902
$392$	−8.12454	−0.410351
$393$	−15.2072	−0.767100
$394$	4.38989	0.221159
$395$	−9.42119	−0.474031
$396$	4.40510	0.221365
$397$	11.2927	0.566765	0.283382	−	0.959007i	$-0.408543\pi$
$397$	11.2927	0.566765	0.283382	+	0.959007i	$0.408543\pi$
$398$	−4.38731	−0.219916
$399$	−23.2281	−1.16286
$400$	0.0414955	0.00207477
$401$	−31.6669	−1.58137	−0.790686	−	0.612222i	$-0.790276\pi$
$401$	−31.6669	−1.58137	−0.790686	+	0.612222i	$0.790276\pi$
$402$	7.36953	0.367559
$403$	44.4053	2.21199
$404$	−13.3126	−0.662324
$405$	−11.0611	−0.549629
$406$	−11.6509	−0.578222
$407$	4.28353	0.212326
$408$	17.0958	0.846367
$409$	18.6776	0.923547	0.461774	−	0.886998i	$-0.347213\pi$
$409$	18.6776	0.923547	0.461774	+	0.886998i	$0.347213\pi$
$410$	11.8059	0.583049
$411$	−45.6269	−2.25061
$412$	−16.0283	−0.789656
$413$	2.12546	0.104587
$414$	−7.32567	−0.360037
$415$	15.3422	0.753121
$416$	−5.48386	−0.268868
$417$	38.9912	1.90941
$418$	1.63216	0.0798317
$419$	40.1396	1.96095	0.980473	−	0.196652i	$-0.0630068\pi$
$419$	40.1396	1.96095	0.980473	+	0.196652i	$0.0630068\pi$
$420$	−25.4912	−1.24384
$421$	7.62559	0.371649	0.185824	−	0.982583i	$-0.440504\pi$
$421$	7.62559	0.371649	0.185824	+	0.982583i	$0.440504\pi$
$422$	−8.25701	−0.401945
$423$	11.7536	0.571478
$424$	−6.17073	−0.299677
$425$	0.243007	0.0117876
$426$	−26.2817	−1.27335
$427$	−12.8718	−0.622910
$428$	3.01997	0.145976
$429$	−12.7708	−0.616580
$430$	25.8708	1.24760
$431$	−16.4802	−0.793823	−0.396912	−	0.917857i	$-0.629918\pi$
$431$	−16.4802	−0.793823	−0.396912	+	0.917857i	$0.629918\pi$
$432$	−7.36225	−0.354216
$433$	6.20884	0.298378	0.149189	−	0.988809i	$-0.452334\pi$
$433$	6.20884	0.298378	0.149189	+	0.988809i	$0.452334\pi$
$434$	31.4912	1.51163
$435$	−19.6366	−0.941503
$436$	3.69840	0.177121
$437$	−2.71428	−0.129842
$438$	−18.6088	−0.889163
$439$	5.23978	0.250081	0.125040	−	0.992152i	$-0.460094\pi$
$439$	5.23978	0.250081	0.125040	+	0.992152i	$0.460094\pi$
$440$	1.79119	0.0853916
$441$	44.8635	2.13636
$442$	−32.1148	−1.52755
$443$	13.3091	0.632336	0.316168	−	0.948703i	$-0.397604\pi$
$443$	13.3091	0.632336	0.316168	+	0.948703i	$0.397604\pi$
$444$	−15.6751	−0.743906
$445$	6.17642	0.292790
$446$	−24.4281	−1.15670
$447$	−54.5811	−2.58160
$448$	−3.88903	−0.183739
$449$	24.7210	1.16665	0.583327	−	0.812237i	$-0.301750\pi$
$449$	24.7210	1.16665	0.583327	+	0.812237i	$0.301750\pi$
$450$	−0.229137	−0.0108016
$451$	4.19449	0.197511
$452$	1.50403	0.0707438
$453$	35.9923	1.69107
$454$	7.87359	0.369526
$455$	47.8858	2.24492
$456$	−5.97271	−0.279698
$457$	3.73799	0.174856	0.0874278	−	0.996171i	$-0.472135\pi$
$457$	3.73799	0.174856	0.0874278	+	0.996171i	$0.472135\pi$
$458$	1.20220	0.0561753
$459$	−43.1151	−2.01244
$460$	−2.97874	−0.138885
$461$	40.4148	1.88231	0.941153	−	0.337980i	$-0.109744\pi$
$461$	40.4148	1.88231	0.941153	+	0.337980i	$0.109744\pi$
$462$	−9.05676	−0.421359
$463$	14.7347	0.684779	0.342389	−	0.939558i	$-0.388764\pi$
$463$	14.7347	0.684779	0.342389	+	0.939558i	$0.388764\pi$
$464$	−2.99583	−0.139078
$465$	53.0760	2.46134
$466$	−14.0487	−0.650792
$467$	−31.1594	−1.44188	−0.720942	−	0.692995i	$-0.756291\pi$
$467$	−31.1594	−1.44188	−0.720942	+	0.692995i	$0.756291\pi$
$468$	30.2817	1.39977
$469$	−9.81772	−0.453340
$470$	4.77920	0.220448
$471$	20.6694	0.952394
$472$	0.546527	0.0251560
$473$	9.19160	0.422630
$474$	12.2489	0.562609
$475$	−0.0848989	−0.00389543
$476$	−22.7751	−1.04389
$477$	34.0746	1.56017
$478$	−0.784400	−0.0358776
$479$	−32.4461	−1.48250	−0.741250	−	0.671229i	$-0.765767\pi$
$479$	−32.4461	−1.48250	−0.741250	+	0.671229i	$0.765767\pi$
$480$	−6.55465	−0.299178
$481$	29.4460	1.34262
$482$	−7.67889	−0.349764
$483$	15.0614	0.685316
$484$	−10.3636	−0.471073
$485$	−26.9082	−1.22184
$486$	−7.70580	−0.349542
$487$	30.1774	1.36747	0.683733	−	0.729732i	$-0.260355\pi$
$487$	30.1774	1.36747	0.683733	+	0.729732i	$0.260355\pi$
$488$	−3.30978	−0.149826
$489$	29.1082	1.31632
$490$	18.2422	0.824101
$491$	−24.2527	−1.09451	−0.547254	−	0.836967i	$-0.684327\pi$
$491$	−24.2527	−1.09451	−0.547254	+	0.836967i	$0.684327\pi$
$492$	−15.3493	−0.691998
$493$	−17.5443	−0.790155
$494$	11.2199	0.504806
$495$	−9.89090	−0.444563
$496$	8.09746	0.363587
$497$	35.0126	1.57053
$498$	−19.9471	−0.893850
$499$	39.6935	1.77693	0.888464	−	0.458947i	$-0.151773\pi$
$499$	39.6935	1.77693	0.888464	+	0.458947i	$0.151773\pi$
$500$	11.1335	0.497904
$501$	8.79291	0.392838
$502$	1.60802	0.0717696
$503$	−22.1624	−0.988174	−0.494087	−	0.869412i	$-0.664498\pi$
$503$	−22.1624	−0.988174	−0.494087	+	0.869412i	$0.664498\pi$
$504$	21.4751	0.956577
$505$	29.8910	1.33013
$506$	−1.05832	−0.0470478
$507$	−49.8394	−2.21345
$508$	6.64186	0.294685
$509$	−5.14189	−0.227910	−0.113955	−	0.993486i	$-0.536352\pi$
$509$	−5.14189	−0.227910	−0.113955	+	0.993486i	$0.536352\pi$
$510$	−38.3856	−1.69974
$511$	24.7907	1.09668
$512$	−1.00000	−0.0441942
$513$	15.0630	0.665049
$514$	18.6047	0.820619
$515$	35.9887	1.58585
$516$	−33.6356	−1.48073
$517$	1.69800	0.0746779
$518$	20.8824	0.917520
$519$	−60.3099	−2.64731
$520$	12.3131	0.539964
$521$	25.6297	1.12286	0.561428	−	0.827525i	$-0.310252\pi$
$521$	25.6297	1.12286	0.561428	+	0.827525i	$0.310252\pi$
$522$	16.5429	0.724062
$523$	−26.7352	−1.16905	−0.584524	−	0.811377i	$-0.698719\pi$
$523$	−26.7352	−1.16905	−0.584524	+	0.811377i	$0.698719\pi$
$524$	5.20929	0.227569
$525$	0.471098	0.0205604
$526$	14.2627	0.621885
$527$	47.4206	2.06568
$528$	−2.32880	−0.101348
$529$	−21.2400	−0.923479
$530$	13.8553	0.601836
$531$	−3.01791	−0.130966
$532$	7.95688	0.344974
$533$	28.8339	1.24894
$534$	−8.03021	−0.347501
$535$	−6.78082	−0.293160
$536$	−2.52447	−0.109040
$537$	16.5668	0.714911
$538$	−6.37324	−0.274770
$539$	6.48127	0.279168
$540$	16.5307	0.711366
$541$	−7.05769	−0.303434	−0.151717	−	0.988424i	$-0.548480\pi$
$541$	−7.05769	−0.303434	−0.151717	+	0.988424i	$0.548480\pi$
$542$	7.13259	0.306371
$543$	−7.54409	−0.323748
$544$	−5.85624	−0.251084
$545$	−8.30411	−0.355709
$546$	−62.2583	−2.66441
$547$	8.74293	0.373821	0.186910	−	0.982377i	$-0.440153\pi$
$547$	8.74293	0.373821	0.186910	+	0.982377i	$0.440153\pi$
$548$	15.6297	0.667668
$549$	18.2765	0.780021
$550$	−0.0331026	−0.00141150
$551$	6.12941	0.261122
$552$	3.87278	0.164837
$553$	−16.3180	−0.693912
$554$	−10.2143	−0.433966
$555$	35.1957	1.49397
$556$	−13.3566	−0.566446
$557$	−9.29828	−0.393981	−0.196990	−	0.980405i	$-0.563117\pi$
$557$	−9.29828	−0.393981	−0.196990	+	0.980405i	$0.563117\pi$
$558$	−44.7139	−1.89289
$559$	63.1853	2.67245
$560$	8.73214	0.369000
$561$	−13.6380	−0.575797
$562$	28.9420	1.22085
$563$	28.2016	1.18856	0.594278	−	0.804259i	$-0.297438\pi$
$563$	28.2016	1.18856	0.594278	+	0.804259i	$0.297438\pi$
$564$	−6.21363	−0.261641
$565$	−3.37705	−0.142074
$566$	−4.60549	−0.193583
$567$	−19.1584	−0.804575
$568$	9.00291	0.377754
$569$	−24.2899	−1.01828	−0.509142	−	0.860683i	$-0.670037\pi$
$569$	−24.2899	−1.01828	−0.509142	+	0.860683i	$0.670037\pi$
$570$	13.4107	0.561712
$571$	2.90481	0.121563	0.0607813	−	0.998151i	$-0.480641\pi$
$571$	2.90481	0.121563	0.0607813	+	0.998151i	$0.480641\pi$
$572$	4.37470	0.182915
$573$	33.4818	1.39872
$574$	20.4484	0.853498
$575$	0.0550496	0.00229573
$576$	5.52197	0.230082
$577$	12.8732	0.535919	0.267959	−	0.963430i	$-0.413651\pi$
$577$	12.8732	0.535919	0.267959	+	0.963430i	$0.413651\pi$
$578$	−17.2955	−0.719400
$579$	−36.9620	−1.53609
$580$	6.72661	0.279307
$581$	26.5736	1.10246
$582$	34.9845	1.45015
$583$	4.92264	0.203875
$584$	6.37453	0.263780
$585$	−67.9924	−2.81114
$586$	−2.43538	−0.100605
$587$	−40.2789	−1.66249	−0.831244	−	0.555908i	$-0.812371\pi$
$587$	−40.2789	−1.66249	−0.831244	+	0.555908i	$0.812371\pi$
$588$	−23.7175	−0.978092
$589$	−16.5672	−0.682642
$590$	−1.22713	−0.0505203
$591$	12.8151	0.527144
$592$	5.36957	0.220688
$593$	−25.4870	−1.04663	−0.523313	−	0.852140i	$-0.675304\pi$
$593$	−25.4870	−1.04663	−0.523313	+	0.852140i	$0.675304\pi$
$594$	5.87316	0.240979
$595$	51.1375	2.09643
$596$	18.6970	0.765859
$597$	−12.8076	−0.524181
$598$	−7.27511	−0.297502
$599$	−5.35661	−0.218865	−0.109432	−	0.993994i	$-0.534903\pi$
$599$	−5.35661	−0.218865	−0.109432	+	0.993994i	$0.534903\pi$
$600$	0.121135	0.00494533
$601$	31.2686	1.27547	0.637737	−	0.770255i	$-0.279871\pi$
$601$	31.2686	1.27547	0.637737	+	0.770255i	$0.279871\pi$
$602$	44.8095	1.82630
$603$	13.9400	0.567682
$604$	−12.3293	−0.501674
$605$	23.2697	0.946048
$606$	−38.8626	−1.57868
$607$	9.02107	0.366154	0.183077	−	0.983099i	$-0.441394\pi$
$607$	9.02107	0.366154	0.183077	+	0.983099i	$0.441394\pi$
$608$	2.04598	0.0829755
$609$	−34.0117	−1.37822
$610$	7.43153	0.300894
$611$	11.6725	0.472217
$612$	32.3380	1.30719
$613$	−31.2458	−1.26200	−0.631002	−	0.775781i	$-0.717356\pi$
$613$	−31.2458	−1.26200	−0.631002	+	0.775781i	$0.717356\pi$
$614$	−14.4906	−0.584794
$615$	34.4641	1.38973
$616$	3.10244	0.125001
$617$	−21.3178	−0.858224	−0.429112	−	0.903251i	$-0.641173\pi$
$617$	−21.3178	−0.858224	−0.429112	+	0.903251i	$0.641173\pi$
$618$	−46.7904	−1.88218
$619$	41.8428	1.68181	0.840903	−	0.541186i	$-0.182025\pi$
$619$	41.8428	1.68181	0.840903	+	0.541186i	$0.182025\pi$
$620$	−18.1814	−0.730184
$621$	−9.76705	−0.391938
$622$	−18.6608	−0.748228
$623$	10.6979	0.428602
$624$	−16.0087	−0.640862
$625$	−25.2058	−1.00823
$626$	25.4577	1.01749
$627$	4.76468	0.190283
$628$	−7.08039	−0.282538
$629$	31.4455	1.25381
$630$	−48.2186	−1.92108
$631$	−19.0654	−0.758980	−0.379490	−	0.925196i	$-0.623901\pi$
$631$	−19.0654	−0.758980	−0.379490	+	0.925196i	$0.623901\pi$
$632$	−4.19591	−0.166904
$633$	−24.1042	−0.958056
$634$	−18.9879	−0.754105
$635$	−14.9132	−0.591811
$636$	−18.0138	−0.714296
$637$	44.5538	1.76529
$638$	2.38989	0.0946168
$639$	−49.7138	−1.96665
$640$	2.24533	0.0887544
$641$	−9.31990	−0.368114	−0.184057	−	0.982916i	$-0.558923\pi$
$641$	−9.31990	−0.368114	−0.184057	+	0.982916i	$0.558923\pi$
$642$	8.81602	0.347940
$643$	49.0624	1.93483	0.967417	−	0.253188i	$-0.0814793\pi$
$643$	49.0624	1.93483	0.967417	+	0.253188i	$0.0814793\pi$
$644$	−5.15934	−0.203307
$645$	75.5230	2.97372
$646$	11.9818	0.471416
$647$	−46.1867	−1.81579	−0.907894	−	0.419201i	$-0.862310\pi$
$647$	−46.1867	−1.81579	−0.907894	+	0.419201i	$0.862310\pi$
$648$	−4.92626	−0.193522
$649$	−0.435987	−0.0171140
$650$	−0.227555	−0.00892545
$651$	91.9305	3.60304
$652$	−9.97115	−0.390500
$653$	−17.9082	−0.700804	−0.350402	−	0.936599i	$-0.613955\pi$
$653$	−17.9082	−0.700804	−0.350402	+	0.936599i	$0.613955\pi$
$654$	10.7965	0.422177
$655$	−11.6966	−0.457022
$656$	5.25796	0.205289
$657$	−35.2000	−1.37328
$658$	8.27783	0.322704
$659$	26.8042	1.04414	0.522072	−	0.852901i	$-0.325159\pi$
$659$	26.8042	1.04414	0.522072	+	0.852901i	$0.325159\pi$
$660$	5.22891	0.203535
$661$	24.1214	0.938214	0.469107	−	0.883141i	$-0.344576\pi$
$661$	24.1214	0.938214	0.469107	+	0.883141i	$0.344576\pi$
$662$	20.4321	0.794116
$663$	−93.7509	−3.64098
$664$	6.83296	0.265170
$665$	−17.8658	−0.692806
$666$	−29.6506	−1.14894
$667$	−3.97439	−0.153889
$668$	−3.01205	−0.116540
$669$	−71.3114	−2.75706
$670$	5.66825	0.218984
$671$	2.64034	0.101929
$672$	−11.3530	−0.437952
$673$	−11.3151	−0.436167	−0.218083	−	0.975930i	$-0.569980\pi$
$673$	−11.3151	−0.436167	−0.218083	+	0.975930i	$0.569980\pi$
$674$	−23.4523	−0.903350
$675$	−0.305500	−0.0117587
$676$	17.0727	0.656644
$677$	−51.0682	−1.96271	−0.981356	−	0.192198i	$-0.938438\pi$
$677$	−51.0682	−1.96271	−0.981356	+	0.192198i	$0.938438\pi$
$678$	4.39064	0.168622
$679$	−46.6065	−1.78859
$680$	13.1492	0.504248
$681$	22.9849	0.880784
$682$	−6.45967	−0.247354
$683$	−38.0101	−1.45442	−0.727208	−	0.686417i	$-0.759182\pi$
$683$	−38.0101	−1.45442	−0.727208	+	0.686417i	$0.759182\pi$
$684$	−11.2979	−0.431984
$685$	−35.0938	−1.34086
$686$	4.37335	0.166975
$687$	3.50952	0.133897
$688$	11.5220	0.439274
$689$	33.8394	1.28918
$690$	−8.69567	−0.331039
$691$	−6.01995	−0.229010	−0.114505	−	0.993423i	$-0.536528\pi$
$691$	−6.01995	−0.229010	−0.114505	+	0.993423i	$0.536528\pi$
$692$	20.6595	0.785355
$693$	−17.1316	−0.650774
$694$	2.86379	0.108708
$695$	29.9900	1.13758
$696$	−8.74554	−0.331499
$697$	30.7919	1.16633
$698$	−27.9999	−1.05981
$699$	−41.0115	−1.55120
$700$	−0.161377	−0.00609948
$701$	−44.8111	−1.69249	−0.846247	−	0.532792i	$-0.821143\pi$
$701$	−44.8111	−1.69249	−0.846247	+	0.532792i	$0.821143\pi$
$702$	40.3735	1.52380
$703$	−10.9860	−0.414347
$704$	0.797741	0.0300660
$705$	13.9516	0.525449
$706$	−2.08668	−0.0785332
$707$	51.7729	1.94712
$708$	1.59545	0.0599605
$709$	5.08781	0.191077	0.0955383	−	0.995426i	$-0.469543\pi$
$709$	5.08781	0.191077	0.0955383	+	0.995426i	$0.469543\pi$
$710$	−20.2145	−0.758636
$711$	23.1697	0.868931
$712$	2.75079	0.103090
$713$	10.7424	0.402307
$714$	−66.4859	−2.48817
$715$	−9.82263	−0.367346
$716$	−5.67504	−0.212086
$717$	−2.28985	−0.0855162
$718$	3.87901	0.144763
$719$	−48.6215	−1.81328	−0.906639	−	0.421908i	$-0.861361\pi$
$719$	−48.6215	−1.81328	−0.906639	+	0.421908i	$0.861361\pi$
$720$	−12.3986	−0.462070
$721$	62.3343	2.32145
$722$	14.8140	0.551318
$723$	−22.4165	−0.833680
$724$	2.58426	0.0960434
$725$	−0.124313	−0.00461688
$726$	−30.2539	−1.12283
$727$	38.9242	1.44362	0.721810	−	0.692091i	$-0.243310\pi$
$727$	38.9242	1.44362	0.721810	+	0.692091i	$0.243310\pi$
$728$	21.3269	0.790427
$729$	−37.2739	−1.38051
$730$	−14.3129	−0.529745
$731$	67.4758	2.49568
$732$	−9.66203	−0.357119
$733$	47.6338	1.75939	0.879696	−	0.475536i	$-0.157746\pi$
$733$	47.6338	1.75939	0.879696	+	0.475536i	$0.157746\pi$
$734$	−22.3753	−0.825888
$735$	53.2535	1.96429
$736$	−1.32664	−0.0489006
$737$	2.01387	0.0741818
$738$	−29.0343	−1.06877
$739$	27.1573	0.998997	0.499498	−	0.866315i	$-0.333518\pi$
$739$	27.1573	0.998997	0.499498	+	0.866315i	$0.333518\pi$
$740$	−12.0564	−0.443204
$741$	32.7535	1.20323
$742$	23.9981	0.880999
$743$	5.19144	0.190456	0.0952278	−	0.995456i	$-0.469642\pi$
$743$	5.19144	0.190456	0.0952278	+	0.995456i	$0.469642\pi$
$744$	23.6384	0.866627
$745$	−41.9809	−1.53806
$746$	1.12720	0.0412698
$747$	−37.7314	−1.38052
$748$	4.67176	0.170816
$749$	−11.7447	−0.429144
$750$	32.5013	1.18678
$751$	21.5463	0.786236	0.393118	−	0.919488i	$-0.371396\pi$
$751$	21.5463	0.786236	0.393118	+	0.919488i	$0.371396\pi$
$752$	2.12851	0.0776188
$753$	4.69421	0.171066
$754$	16.4287	0.598298
$755$	27.6834	1.00750
$756$	28.6320	1.04133
$757$	27.8188	1.01109	0.505546	−	0.862800i	$-0.331291\pi$
$757$	27.8188	1.01109	0.505546	+	0.862800i	$0.331291\pi$
$758$	3.03883	0.110375
$759$	−3.08948	−0.112141
$760$	−4.59390	−0.166638
$761$	−9.56539	−0.346745	−0.173373	−	0.984856i	$-0.555466\pi$
$761$	−9.56539	−0.346745	−0.173373	+	0.984856i	$0.555466\pi$
$762$	19.3892	0.702397
$763$	−14.3832	−0.520706
$764$	−11.4693	−0.414946
$765$	−72.6094	−2.62520
$766$	8.09999	0.292664
$767$	−2.99708	−0.108218
$768$	−2.91924	−0.105339
$769$	−32.4744	−1.17106	−0.585528	−	0.810652i	$-0.699113\pi$
$769$	−32.4744	−1.17106	−0.585528	+	0.810652i	$0.699113\pi$
$770$	−6.96598	−0.251037
$771$	54.3117	1.95599
$772$	12.6615	0.455697
$773$	39.4890	1.42032	0.710160	−	0.704041i	$-0.248623\pi$
$773$	39.4890	1.42032	0.710160	+	0.704041i	$0.248623\pi$
$774$	−63.6244	−2.28693
$775$	0.336008	0.0120698
$776$	−11.9841	−0.430204
$777$	60.9608	2.18696
$778$	37.1316	1.33123
$779$	−10.7577	−0.385435
$780$	35.9448	1.28703
$781$	−7.18199	−0.256992
$782$	−7.76913	−0.277823
$783$	22.0560	0.788218
$784$	8.12454	0.290162
$785$	15.8978	0.567416
$786$	15.2072	0.542422
$787$	−25.2540	−0.900207	−0.450103	−	0.892976i	$-0.648613\pi$
$787$	−25.2540	−0.900207	−0.450103	+	0.892976i	$0.648613\pi$
$788$	−4.38989	−0.156383
$789$	41.6364	1.48229
$790$	9.42119	0.335191
$791$	−5.84923	−0.207975
$792$	−4.40510	−0.156528
$793$	18.1503	0.644538
$794$	−11.2927	−0.400763
$795$	40.4470	1.43451
$796$	4.38731	0.155504
$797$	−38.9678	−1.38031	−0.690155	−	0.723662i	$-0.742458\pi$
$797$	−38.9678	−1.38031	−0.690155	+	0.723662i	$0.742458\pi$
$798$	23.2281	0.822264
$799$	12.4651	0.440982
$800$	−0.0414955	−0.00146709
$801$	−15.1898	−0.536704
$802$	31.6669	1.11820
$803$	−5.08523	−0.179454
$804$	−7.36953	−0.259903
$805$	11.5844	0.408297
$806$	−44.4053	−1.56411
$807$	−18.6050	−0.654928
$808$	13.3126	0.468334
$809$	28.6228	1.00632	0.503162	−	0.864192i	$-0.332170\pi$
$809$	28.6228	1.00632	0.503162	+	0.864192i	$0.332170\pi$
$810$	11.0611	0.388646
$811$	−29.4622	−1.03456	−0.517278	−	0.855817i	$-0.673055\pi$
$811$	−29.4622	−1.03456	−0.517278	+	0.855817i	$0.673055\pi$
$812$	11.6509	0.408865
$813$	20.8217	0.730250
$814$	−4.28353	−0.150137
$815$	22.3885	0.784234
$816$	−17.0958	−0.598472
$817$	−23.5739	−0.824746
$818$	−18.6776	−0.653047
$819$	−117.766	−4.11509
$820$	−11.8059	−0.412278
$821$	−4.41766	−0.154178	−0.0770888	−	0.997024i	$-0.524562\pi$
$821$	−4.41766	−0.154178	−0.0770888	+	0.997024i	$0.524562\pi$
$822$	45.6269	1.59142
$823$	13.2773	0.462816	0.231408	−	0.972857i	$-0.425667\pi$
$823$	13.2773	0.462816	0.231408	+	0.972857i	$0.425667\pi$
$824$	16.0283	0.558371
$825$	−0.0966345	−0.00336438
$826$	−2.12546	−0.0739542
$827$	35.5976	1.23785	0.618924	−	0.785451i	$-0.287569\pi$
$827$	35.5976	1.23785	0.618924	+	0.785451i	$0.287569\pi$
$828$	7.32567	0.254585
$829$	−33.5589	−1.16555	−0.582774	−	0.812634i	$-0.698033\pi$
$829$	−33.5589	−1.16555	−0.582774	+	0.812634i	$0.698033\pi$
$830$	−15.3422	−0.532537
$831$	−29.8181	−1.03438
$832$	5.48386	0.190119
$833$	47.5792	1.64852
$834$	−38.9912	−1.35015
$835$	6.76304	0.234045
$836$	−1.63216	−0.0564495
$837$	−59.6155	−2.06061
$838$	−40.1396	−1.38660
$839$	7.57357	0.261469	0.130734	−	0.991417i	$-0.458267\pi$
$839$	7.57357	0.261469	0.130734	+	0.991417i	$0.458267\pi$
$840$	25.4912	0.879531
$841$	−20.0250	−0.690518
$842$	−7.62559	−0.262795
$843$	84.4888	2.90995
$844$	8.25701	0.284218
$845$	−38.3339	−1.31873
$846$	−11.7536	−0.404096
$847$	40.3044	1.38487
$848$	6.17073	0.211904
$849$	−13.4445	−0.461415
$850$	−0.243007	−0.00833508
$851$	7.12349	0.244190
$852$	26.2817	0.900395
$853$	35.8805	1.22853	0.614263	−	0.789101i	$-0.289453\pi$
$853$	35.8805	1.22853	0.614263	+	0.789101i	$0.289453\pi$
$854$	12.8718	0.440464
$855$	25.3674	0.867546
$856$	−3.01997	−0.103220
$857$	33.8103	1.15494	0.577469	−	0.816413i	$-0.304040\pi$
$857$	33.8103	1.15494	0.577469	+	0.816413i	$0.304040\pi$
$858$	12.7708	0.435988
$859$	32.7952	1.11896	0.559479	−	0.828845i	$-0.311001\pi$
$859$	32.7952	1.11896	0.559479	+	0.828845i	$0.311001\pi$
$860$	−25.8708	−0.882186
$861$	59.6937	2.03436
$862$	16.4802	0.561318
$863$	13.5469	0.461140	0.230570	−	0.973056i	$-0.425941\pi$
$863$	13.5469	0.461140	0.230570	+	0.973056i	$0.425941\pi$
$864$	7.36225	0.250469
$865$	−46.3872	−1.57721
$866$	−6.20884	−0.210985
$867$	−50.4899	−1.71473
$868$	−31.4912	−1.06888
$869$	3.34725	0.113548
$870$	19.6366	0.665743
$871$	13.8438	0.469080
$872$	−3.69840	−0.125244
$873$	66.1759	2.23971
$874$	2.71428	0.0918120
$875$	−43.2984	−1.46375
$876$	18.6088	0.628733
$877$	−29.5763	−0.998722	−0.499361	−	0.866394i	$-0.666432\pi$
$877$	−29.5763	−0.998722	−0.499361	+	0.866394i	$0.666432\pi$
$878$	−5.23978	−0.176834
$879$	−7.10946	−0.239796
$880$	−1.79119	−0.0603810
$881$	11.7516	0.395921	0.197960	−	0.980210i	$-0.436568\pi$
$881$	11.7516	0.395921	0.197960	+	0.980210i	$0.436568\pi$
$882$	−44.8635	−1.51063
$883$	−34.0811	−1.14692	−0.573461	−	0.819233i	$-0.694399\pi$
$883$	−34.0811	−1.14692	−0.573461	+	0.819233i	$0.694399\pi$
$884$	32.1148	1.08014
$885$	−3.58230	−0.120418
$886$	−13.3091	−0.447129
$887$	33.2233	1.11553	0.557765	−	0.829999i	$-0.311659\pi$
$887$	33.2233	1.11553	0.557765	+	0.829999i	$0.311659\pi$
$888$	15.6751	0.526021
$889$	−25.8304	−0.866324
$890$	−6.17642	−0.207034
$891$	3.92988	0.131656
$892$	24.4281	0.817912
$893$	−4.35489	−0.145731
$894$	54.5811	1.82546
$895$	12.7423	0.425929
$896$	3.88903	0.129923
$897$	−21.2378	−0.709110
$898$	−24.7210	−0.824950
$899$	−24.2586	−0.809069
$900$	0.229137	0.00763789
$901$	36.1373	1.20391
$902$	−4.19449	−0.139661
$903$	130.810	4.35308
$904$	−1.50403	−0.0500235
$905$	−5.80252	−0.192882
$906$	−35.9923	−1.19577
$907$	−26.2121	−0.870358	−0.435179	−	0.900344i	$-0.643315\pi$
$907$	−26.2121	−0.870358	−0.435179	+	0.900344i	$0.643315\pi$
$908$	−7.87359	−0.261294
$909$	−73.5116	−2.43822
$910$	−47.8858	−1.58740
$911$	−44.1166	−1.46165	−0.730824	−	0.682566i	$-0.760864\pi$
$911$	−44.1166	−1.46165	−0.730824	+	0.682566i	$0.760864\pi$
$912$	5.97271	0.197776
$913$	−5.45093	−0.180400
$914$	−3.73799	−0.123642
$915$	21.6944	0.717196
$916$	−1.20220	−0.0397219
$917$	−20.2591	−0.669013
$918$	43.1151	1.42301
$919$	−37.8134	−1.24735	−0.623674	−	0.781685i	$-0.714361\pi$
$919$	−37.8134	−1.24735	−0.623674	+	0.781685i	$0.714361\pi$
$920$	2.97874	0.0982062
$921$	−42.3016	−1.39389
$922$	−40.4148	−1.33099
$923$	−49.3707	−1.62506
$924$	9.05676	0.297945
$925$	0.222813	0.00732604
$926$	−14.7347	−0.484212
$927$	−88.5076	−2.90697
$928$	2.99583	0.0983429
$929$	41.5380	1.36282	0.681409	−	0.731903i	$-0.261368\pi$
$929$	41.5380	1.36282	0.681409	+	0.731903i	$0.261368\pi$
$930$	−53.0760	−1.74043
$931$	−16.6227	−0.544785
$932$	14.0487	0.460180
$933$	−54.4752	−1.78344
$934$	31.1594	1.01957
$935$	−10.4896	−0.343048
$936$	−30.2817	−0.989789
$937$	−40.4067	−1.32003	−0.660015	−	0.751252i	$-0.729450\pi$
$937$	−40.4067	−1.32003	−0.660015	+	0.751252i	$0.729450\pi$
$938$	9.81772	0.320560
$939$	74.3171	2.42525
$940$	−4.77920	−0.155880
$941$	39.4097	1.28472	0.642360	−	0.766403i	$-0.277955\pi$
$941$	39.4097	1.28472	0.642360	+	0.766403i	$0.277955\pi$
$942$	−20.6694	−0.673444
$943$	6.97543	0.227151
$944$	−0.546527	−0.0177880
$945$	−64.2882	−2.09129
$946$	−9.19160	−0.298845
$947$	51.3746	1.66945	0.834724	−	0.550668i	$-0.185627\pi$
$947$	51.3746	1.66945	0.834724	+	0.550668i	$0.185627\pi$
$948$	−12.2489	−0.397825
$949$	−34.9571	−1.13475
$950$	0.0848989	0.00275449
$951$	−55.4302	−1.79745
$952$	22.7751	0.738144
$953$	−53.5387	−1.73429	−0.867144	−	0.498058i	$-0.834047\pi$
$953$	−53.5387	−1.73429	−0.867144	+	0.498058i	$0.834047\pi$
$954$	−34.0746	−1.10321
$955$	25.7524	0.833329
$956$	0.784400	0.0253693
$957$	6.97668	0.225524
$958$	32.4461	1.04829
$959$	−60.7843	−1.96283
$960$	6.55465	0.211551
$961$	34.5688	1.11512
$962$	−29.4460	−0.949377
$963$	16.6762	0.537382
$964$	7.67889	0.247320
$965$	−28.4292	−0.915169
$966$	−15.0614	−0.484592
$967$	−38.0404	−1.22330	−0.611649	−	0.791130i	$-0.709493\pi$
$967$	−38.0404	−1.22330	−0.611649	+	0.791130i	$0.709493\pi$
$968$	10.3636	0.333099
$969$	34.9776	1.12364
$970$	26.9082	0.863972
$971$	−32.5244	−1.04376	−0.521879	−	0.853020i	$-0.674769\pi$
$971$	−32.5244	−1.04376	−0.521879	+	0.853020i	$0.674769\pi$
$972$	7.70580	0.247164
$973$	51.9442	1.66525
$974$	−30.1774	−0.966945
$975$	−0.664289	−0.0212743
$976$	3.30978	0.105943
$977$	21.3607	0.683389	0.341695	−	0.939811i	$-0.388999\pi$
$977$	21.3607	0.683389	0.341695	+	0.939811i	$0.388999\pi$
$978$	−29.1082	−0.930777
$979$	−2.19441	−0.0701338
$980$	−18.2422	−0.582727
$981$	20.4224	0.652039
$982$	24.2527	0.773934
$983$	−11.0402	−0.352128	−0.176064	−	0.984379i	$-0.556337\pi$
$983$	−11.0402	−0.352128	−0.176064	+	0.984379i	$0.556337\pi$
$984$	15.3493	0.489317
$985$	9.85673	0.314062
$986$	17.5443	0.558724
$987$	24.1650	0.769180
$988$	−11.2199	−0.356952
$989$	15.2856	0.486054
$990$	9.89090	0.314353
$991$	10.4645	0.332416	0.166208	−	0.986091i	$-0.446848\pi$
$991$	10.4645	0.332416	0.166208	+	0.986091i	$0.446848\pi$
$992$	−8.09746	−0.257095
$993$	59.6463	1.89282
$994$	−35.0126	−1.11053
$995$	−9.85094	−0.312296
$996$	19.9471	0.632047
$997$	−15.8793	−0.502902	−0.251451	−	0.967870i	$-0.580908\pi$
$997$	−15.8793	−0.502902	−0.251451	+	0.967870i	$0.580908\pi$
$998$	−39.6935	−1.25648
$999$	−39.5321	−1.25074

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.d.1.5	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.d.1.5	✓	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.91924 q^{3} +1.00000 q^{4} -2.24533 q^{5} +2.91924 q^{6} -3.88903 q^{7} -1.00000 q^{8} +5.52197 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.91924 q^{3} +1.00000 q^{4} -2.24533 q^{5} +2.91924 q^{6} -3.88903 q^{7} -1.00000 q^{8} +5.52197 q^{9} +2.24533 q^{10} +0.797741 q^{11} -2.91924 q^{12} +5.48386 q^{13} +3.88903 q^{14} +6.55465 q^{15} +1.00000 q^{16} +5.85624 q^{17} -5.52197 q^{18} -2.04598 q^{19} -2.24533 q^{20} +11.3530 q^{21} -0.797741 q^{22} +1.32664 q^{23} +2.91924 q^{24} +0.0414955 q^{25} -5.48386 q^{26} -7.36225 q^{27} -3.88903 q^{28} -2.99583 q^{29} -6.55465 q^{30} +8.09746 q^{31} -1.00000 q^{32} -2.32880 q^{33} -5.85624 q^{34} +8.73214 q^{35} +5.52197 q^{36} +5.36957 q^{37} +2.04598 q^{38} -16.0087 q^{39} +2.24533 q^{40} +5.25796 q^{41} -11.3530 q^{42} +11.5220 q^{43} +0.797741 q^{44} -12.3986 q^{45} -1.32664 q^{46} +2.12851 q^{47} -2.91924 q^{48} +8.12454 q^{49} -0.0414955 q^{50} -17.0958 q^{51} +5.48386 q^{52} +6.17073 q^{53} +7.36225 q^{54} -1.79119 q^{55} +3.88903 q^{56} +5.97271 q^{57} +2.99583 q^{58} -0.546527 q^{59} +6.55465 q^{60} +3.30978 q^{61} -8.09746 q^{62} -21.4751 q^{63} +1.00000 q^{64} -12.3131 q^{65} +2.32880 q^{66} +2.52447 q^{67} +5.85624 q^{68} -3.87278 q^{69} -8.73214 q^{70} -9.00291 q^{71} -5.52197 q^{72} -6.37453 q^{73} -5.36957 q^{74} -0.121135 q^{75} -2.04598 q^{76} -3.10244 q^{77} +16.0087 q^{78} +4.19591 q^{79} -2.24533 q^{80} +4.92626 q^{81} -5.25796 q^{82} -6.83296 q^{83} +11.3530 q^{84} -13.1492 q^{85} -11.5220 q^{86} +8.74554 q^{87} -0.797741 q^{88} -2.75079 q^{89} +12.3986 q^{90} -21.3269 q^{91} +1.32664 q^{92} -23.6384 q^{93} -2.12851 q^{94} +4.59390 q^{95} +2.91924 q^{96} +11.9841 q^{97} -8.12454 q^{98} +4.40510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * q^99

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