LMFDB - Embedded newform 6046.2.a.f.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6046,2,Mod(1,6046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6046, 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("6046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6046 = 2 \cdot 3023 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6046.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:	$48.2775530621$
Analytic rank:	$0$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	6046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.73291 q^{3} +1.00000 q^{4} -0.132829 q^{5} -2.73291 q^{6} +3.97016 q^{7} +1.00000 q^{8} +4.46882 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.73291 q^{3} +1.00000 q^{4} -0.132829 q^{5} -2.73291 q^{6} +3.97016 q^{7} +1.00000 q^{8} +4.46882 q^{9} -0.132829 q^{10} -3.95767 q^{11} -2.73291 q^{12} -4.52290 q^{13} +3.97016 q^{14} +0.363011 q^{15} +1.00000 q^{16} -7.47960 q^{17} +4.46882 q^{18} -3.36283 q^{19} -0.132829 q^{20} -10.8501 q^{21} -3.95767 q^{22} -0.752780 q^{23} -2.73291 q^{24} -4.98236 q^{25} -4.52290 q^{26} -4.01416 q^{27} +3.97016 q^{28} +10.4771 q^{29} +0.363011 q^{30} -3.99154 q^{31} +1.00000 q^{32} +10.8160 q^{33} -7.47960 q^{34} -0.527354 q^{35} +4.46882 q^{36} -0.0616936 q^{37} -3.36283 q^{38} +12.3607 q^{39} -0.132829 q^{40} +9.24746 q^{41} -10.8501 q^{42} +9.63765 q^{43} -3.95767 q^{44} -0.593590 q^{45} -0.752780 q^{46} -7.24049 q^{47} -2.73291 q^{48} +8.76219 q^{49} -4.98236 q^{50} +20.4411 q^{51} -4.52290 q^{52} +5.25725 q^{53} -4.01416 q^{54} +0.525695 q^{55} +3.97016 q^{56} +9.19033 q^{57} +10.4771 q^{58} +5.05595 q^{59} +0.363011 q^{60} +4.94315 q^{61} -3.99154 q^{62} +17.7419 q^{63} +1.00000 q^{64} +0.600773 q^{65} +10.8160 q^{66} +3.99877 q^{67} -7.47960 q^{68} +2.05728 q^{69} -0.527354 q^{70} +2.27818 q^{71} +4.46882 q^{72} -0.804029 q^{73} -0.0616936 q^{74} +13.6164 q^{75} -3.36283 q^{76} -15.7126 q^{77} +12.3607 q^{78} +3.37500 q^{79} -0.132829 q^{80} -2.43611 q^{81} +9.24746 q^{82} +7.28297 q^{83} -10.8501 q^{84} +0.993510 q^{85} +9.63765 q^{86} -28.6330 q^{87} -3.95767 q^{88} -3.26298 q^{89} -0.593590 q^{90} -17.9567 q^{91} -0.752780 q^{92} +10.9085 q^{93} -7.24049 q^{94} +0.446682 q^{95} -2.73291 q^{96} -15.6101 q^{97} +8.76219 q^{98} -17.6861 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9 + 21 * q^10 + 56 * q^11 + 21 * q^12 + 33 * q^13 + 38 * q^14 + 25 * q^15 + 67 * q^16 + 30 * q^17 + 90 * q^18 + 36 * q^19 + 21 * q^20 + 20 * q^21 + 56 * q^22 + 65 * q^23 + 21 * q^24 + 72 * q^25 + 33 * q^26 + 57 * q^27 + 38 * q^28 + 84 * q^29 + 25 * q^30 + 52 * q^31 + 67 * q^32 - 9 * q^33 + 30 * q^34 + 30 * q^35 + 90 * q^36 + 52 * q^37 + 36 * q^38 + 41 * q^39 + 21 * q^40 + 46 * q^41 + 20 * q^42 + 61 * q^43 + 56 * q^44 + 23 * q^45 + 65 * q^46 + 51 * q^47 + 21 * q^48 + 81 * q^49 + 72 * q^50 + 33 * q^51 + 33 * q^52 + 72 * q^53 + 57 * q^54 + 14 * q^55 + 38 * q^56 - 26 * q^57 + 84 * q^58 + 71 * q^59 + 25 * q^60 + 42 * q^61 + 52 * q^62 + 63 * q^63 + 67 * q^64 - 2 * q^65 - 9 * q^66 + 70 * q^67 + 30 * q^68 + 21 * q^69 + 30 * q^70 + 104 * q^71 + 90 * q^72 - 31 * q^73 + 52 * q^74 + 69 * q^75 + 36 * q^76 + 48 * q^77 + 41 * q^78 + 79 * q^79 + 21 * q^80 + 123 * q^81 + 46 * q^82 + 41 * q^83 + 20 * q^84 + 6 * q^85 + 61 * q^86 + 19 * q^87 + 56 * q^88 + 58 * q^89 + 23 * q^90 + 31 * q^91 + 65 * q^92 + 13 * q^93 + 51 * q^94 + 77 * q^95 + 21 * q^96 - 8 * q^97 + 81 * q^98 + 129 * 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
$3$	−2.73291	−1.57785	−0.788924	−	0.614490i	$-0.789362\pi$
$3$	−2.73291	−1.57785	−0.788924	+	0.614490i	$0.789362\pi$
$4$	1.00000	0.500000
$5$	−0.132829	−0.0594030	−0.0297015	−	0.999559i	$-0.509456\pi$
$5$	−0.132829	−0.0594030	−0.0297015	+	0.999559i	$0.509456\pi$
$6$	−2.73291	−1.11571
$7$	3.97016	1.50058	0.750290	−	0.661109i	$-0.229914\pi$
$7$	3.97016	1.50058	0.750290	+	0.661109i	$0.229914\pi$
$8$	1.00000	0.353553
$9$	4.46882	1.48961
$10$	−0.132829	−0.0420043
$11$	−3.95767	−1.19328	−0.596642	−	0.802508i	$-0.703499\pi$
$11$	−3.95767	−1.19328	−0.596642	+	0.802508i	$0.703499\pi$
$12$	−2.73291	−0.788924
$13$	−4.52290	−1.25443	−0.627214	−	0.778847i	$-0.715805\pi$
$13$	−4.52290	−1.25443	−0.627214	+	0.778847i	$0.715805\pi$
$14$	3.97016	1.06107
$15$	0.363011	0.0937290
$16$	1.00000	0.250000
$17$	−7.47960	−1.81407	−0.907035	−	0.421055i	$-0.861660\pi$
$17$	−7.47960	−1.81407	−0.907035	+	0.421055i	$0.861660\pi$
$18$	4.46882	1.05331
$19$	−3.36283	−0.771487	−0.385743	−	0.922606i	$-0.626055\pi$
$19$	−3.36283	−0.771487	−0.385743	+	0.922606i	$0.626055\pi$
$20$	−0.132829	−0.0297015
$21$	−10.8501	−2.36769
$22$	−3.95767	−0.843779
$23$	−0.752780	−0.156965	−0.0784827	−	0.996915i	$-0.525008\pi$
$23$	−0.752780	−0.156965	−0.0784827	+	0.996915i	$0.525008\pi$
$24$	−2.73291	−0.557854
$25$	−4.98236	−0.996471
$26$	−4.52290	−0.887014
$27$	−4.01416	−0.772525
$28$	3.97016	0.750290
$29$	10.4771	1.94555	0.972776	−	0.231749i	$-0.0744450\pi$
$29$	10.4771	1.94555	0.972776	+	0.231749i	$0.0744450\pi$
$30$	0.363011	0.0662764
$31$	−3.99154	−0.716902	−0.358451	−	0.933549i	$-0.616695\pi$
$31$	−3.99154	−0.716902	−0.358451	+	0.933549i	$0.616695\pi$
$32$	1.00000	0.176777
$33$	10.8160	1.88282
$34$	−7.47960	−1.28274
$35$	−0.527354	−0.0891390
$36$	4.46882	0.744803
$37$	−0.0616936	−0.0101424	−0.00507119	−	0.999987i	$-0.501614\pi$
$37$	−0.0616936	−0.0101424	−0.00507119	+	0.999987i	$0.501614\pi$
$38$	−3.36283	−0.545524
$39$	12.3607	1.97930
$40$	−0.132829	−0.0210021
$41$	9.24746	1.44421	0.722106	−	0.691783i	$-0.243174\pi$
$41$	9.24746	1.44421	0.722106	+	0.691783i	$0.243174\pi$
$42$	−10.8501	−1.67421
$43$	9.63765	1.46973	0.734864	−	0.678215i	$-0.237246\pi$
$43$	9.63765	1.46973	0.734864	+	0.678215i	$0.237246\pi$
$44$	−3.95767	−0.596642
$45$	−0.593590	−0.0884871
$46$	−0.752780	−0.110991
$47$	−7.24049	−1.05613	−0.528067	−	0.849203i	$-0.677083\pi$
$47$	−7.24049	−1.05613	−0.528067	+	0.849203i	$0.677083\pi$
$48$	−2.73291	−0.394462
$49$	8.76219	1.25174
$50$	−4.98236	−0.704612
$51$	20.4411	2.86233
$52$	−4.52290	−0.627214
$53$	5.25725	0.722139	0.361070	−	0.932539i	$-0.382412\pi$
$53$	5.25725	0.722139	0.361070	+	0.932539i	$0.382412\pi$
$54$	−4.01416	−0.546257
$55$	0.525695	0.0708846
$56$	3.97016	0.530535
$57$	9.19033	1.21729
$58$	10.4771	1.37571
$59$	5.05595	0.658228	0.329114	−	0.944290i	$-0.393250\pi$
$59$	5.05595	0.658228	0.329114	+	0.944290i	$0.393250\pi$
$60$	0.363011	0.0468645
$61$	4.94315	0.632906	0.316453	−	0.948608i	$-0.397508\pi$
$61$	4.94315	0.632906	0.316453	+	0.948608i	$0.397508\pi$
$62$	−3.99154	−0.506926
$63$	17.7419	2.23527
$64$	1.00000	0.125000
$65$	0.600773	0.0745168
$66$	10.8160	1.33136
$67$	3.99877	0.488528	0.244264	−	0.969709i	$-0.421454\pi$
$67$	3.99877	0.488528	0.244264	+	0.969709i	$0.421454\pi$
$68$	−7.47960	−0.907035
$69$	2.05728	0.247668
$70$	−0.527354	−0.0630308
$71$	2.27818	0.270371	0.135185	−	0.990820i	$-0.456837\pi$
$71$	2.27818	0.270371	0.135185	+	0.990820i	$0.456837\pi$
$72$	4.46882	0.526655
$73$	−0.804029	−0.0941044	−0.0470522	−	0.998892i	$-0.514983\pi$
$73$	−0.804029	−0.0941044	−0.0470522	+	0.998892i	$0.514983\pi$
$74$	−0.0616936	−0.00717174
$75$	13.6164	1.57228
$76$	−3.36283	−0.385743
$77$	−15.7126	−1.79062
$78$	12.3607	1.39957
$79$	3.37500	0.379717	0.189858	−	0.981811i	$-0.439197\pi$
$79$	3.37500	0.379717	0.189858	+	0.981811i	$0.439197\pi$
$80$	−0.132829	−0.0148508
$81$	−2.43611	−0.270679
$82$	9.24746	1.02121
$83$	7.28297	0.799410	0.399705	−	0.916644i	$-0.369113\pi$
$83$	7.28297	0.799410	0.399705	+	0.916644i	$0.369113\pi$
$84$	−10.8501	−1.18384
$85$	0.993510	0.107761
$86$	9.63765	1.03925
$87$	−28.6330	−3.06979
$88$	−3.95767	−0.421889
$89$	−3.26298	−0.345875	−0.172938	−	0.984933i	$-0.555326\pi$
$89$	−3.26298	−0.345875	−0.172938	+	0.984933i	$0.555326\pi$
$90$	−0.593590	−0.0625698
$91$	−17.9567	−1.88237
$92$	−0.752780	−0.0784827
$93$	10.9085	1.13116
$94$	−7.24049	−0.746800
$95$	0.446682	0.0458287
$96$	−2.73291	−0.278927
$97$	−15.6101	−1.58497	−0.792485	−	0.609891i	$-0.791213\pi$
$97$	−15.6101	−1.58497	−0.792485	+	0.609891i	$0.791213\pi$
$98$	8.76219	0.885115
$99$	−17.6861	−1.77752
$100$	−4.98236	−0.498236
$101$	12.0752	1.20153	0.600765	−	0.799426i	$-0.294863\pi$
$101$	12.0752	1.20153	0.600765	+	0.799426i	$0.294863\pi$
$102$	20.4411	2.02397
$103$	−9.98174	−0.983530	−0.491765	−	0.870728i	$-0.663648\pi$
$103$	−9.98174	−0.983530	−0.491765	+	0.870728i	$0.663648\pi$
$104$	−4.52290	−0.443507
$105$	1.44121	0.140648
$106$	5.25725	0.510630
$107$	2.89844	0.280202	0.140101	−	0.990137i	$-0.455257\pi$
$107$	2.89844	0.280202	0.140101	+	0.990137i	$0.455257\pi$
$108$	−4.01416	−0.386262
$109$	4.70627	0.450779	0.225390	−	0.974269i	$-0.427635\pi$
$109$	4.70627	0.450779	0.225390	+	0.974269i	$0.427635\pi$
$110$	0.525695	0.0501230
$111$	0.168603	0.0160031
$112$	3.97016	0.375145
$113$	−1.33778	−0.125848	−0.0629241	−	0.998018i	$-0.520043\pi$
$113$	−1.33778	−0.125848	−0.0629241	+	0.998018i	$0.520043\pi$
$114$	9.19033	0.860754
$115$	0.0999911	0.00932422
$116$	10.4771	0.972776
$117$	−20.2120	−1.86860
$118$	5.05595	0.465437
$119$	−29.6952	−2.72216
$120$	0.363011	0.0331382
$121$	4.66318	0.423925
$122$	4.94315	0.447532
$123$	−25.2725	−2.27875
$124$	−3.99154	−0.358451
$125$	1.32595	0.118596
$126$	17.7419	1.58058
$127$	15.8527	1.40670	0.703351	−	0.710843i	$-0.251686\pi$
$127$	15.8527	1.40670	0.703351	+	0.710843i	$0.251686\pi$
$128$	1.00000	0.0883883
$129$	−26.3389	−2.31901
$130$	0.600773	0.0526913
$131$	1.61217	0.140856	0.0704278	−	0.997517i	$-0.477564\pi$
$131$	1.61217	0.140856	0.0704278	+	0.997517i	$0.477564\pi$
$132$	10.8160	0.941410
$133$	−13.3510	−1.15768
$134$	3.99877	0.345441
$135$	0.533197	0.0458903
$136$	−7.47960	−0.641371
$137$	16.9308	1.44650	0.723248	−	0.690588i	$-0.242648\pi$
$137$	16.9308	1.44650	0.723248	+	0.690588i	$0.242648\pi$
$138$	2.05728	0.175127
$139$	4.36198	0.369978	0.184989	−	0.982741i	$-0.440775\pi$
$139$	4.36198	0.369978	0.184989	+	0.982741i	$0.440775\pi$
$140$	−0.527354	−0.0445695
$141$	19.7876	1.66642
$142$	2.27818	0.191181
$143$	17.9002	1.49689
$144$	4.46882	0.372402
$145$	−1.39167	−0.115572
$146$	−0.804029	−0.0665419
$147$	−23.9463	−1.97506
$148$	−0.0616936	−0.00507119
$149$	−4.08149	−0.334369	−0.167184	−	0.985926i	$-0.553468\pi$
$149$	−4.08149	−0.334369	−0.167184	+	0.985926i	$0.553468\pi$
$150$	13.6164	1.11177
$151$	14.9130	1.21360	0.606801	−	0.794854i	$-0.292452\pi$
$151$	14.9130	1.21360	0.606801	+	0.794854i	$0.292452\pi$
$152$	−3.36283	−0.272762
$153$	−33.4250	−2.70225
$154$	−15.7126	−1.26616
$155$	0.530193	0.0425861
$156$	12.3607	0.989648
$157$	22.4100	1.78851	0.894257	−	0.447553i	$-0.147705\pi$
$157$	22.4100	1.78851	0.894257	+	0.447553i	$0.147705\pi$
$158$	3.37500	0.268500
$159$	−14.3676	−1.13943
$160$	−0.132829	−0.0105011
$161$	−2.98866	−0.235539
$162$	−2.43611	−0.191399
$163$	−0.0243186	−0.00190478	−0.000952389	−	1.00000i	$-0.500303\pi$
$163$	−0.0243186	−0.00190478	−0.000952389		1.00000i	$0.500303\pi$
$164$	9.24746	0.722106
$165$	−1.43668	−0.111845
$166$	7.28297	0.565268
$167$	18.9794	1.46867	0.734334	−	0.678789i	$-0.237495\pi$
$167$	18.9794	1.46867	0.734334	+	0.678789i	$0.237495\pi$
$168$	−10.8501	−0.837104
$169$	7.45664	0.573588
$170$	0.993510	0.0761987
$171$	−15.0279	−1.14921
$172$	9.63765	0.734864
$173$	−24.6486	−1.87400	−0.937000	−	0.349330i	$-0.886409\pi$
$173$	−24.6486	−1.87400	−0.937000	+	0.349330i	$0.886409\pi$
$174$	−28.6330	−2.17067
$175$	−19.7808	−1.49529
$176$	−3.95767	−0.298321
$177$	−13.8175	−1.03858
$178$	−3.26298	−0.244571
$179$	−10.5036	−0.785079	−0.392540	−	0.919735i	$-0.628403\pi$
$179$	−10.5036	−0.785079	−0.392540	+	0.919735i	$0.628403\pi$
$180$	−0.593590	−0.0442436
$181$	10.5727	0.785864	0.392932	−	0.919568i	$-0.371461\pi$
$181$	10.5727	0.785864	0.392932	+	0.919568i	$0.371461\pi$
$182$	−17.9567	−1.33104
$183$	−13.5092	−0.998630
$184$	−0.752780	−0.0554956
$185$	0.00819472	0.000602488 0
$186$	10.9085	0.799853
$187$	29.6018	2.16470
$188$	−7.24049	−0.528067
$189$	−15.9369	−1.15924
$190$	0.446682	0.0324058
$191$	13.4771	0.975167	0.487584	−	0.873076i	$-0.337878\pi$
$191$	13.4771	0.975167	0.487584	+	0.873076i	$0.337878\pi$
$192$	−2.73291	−0.197231
$193$	−11.7274	−0.844159	−0.422080	−	0.906559i	$-0.638700\pi$
$193$	−11.7274	−0.844159	−0.422080	+	0.906559i	$0.638700\pi$
$194$	−15.6101	−1.12074
$195$	−1.64186	−0.117576
$196$	8.76219	0.625871
$197$	17.3538	1.23640	0.618202	−	0.786019i	$-0.287861\pi$
$197$	17.3538	1.23640	0.618202	+	0.786019i	$0.287861\pi$
$198$	−17.6861	−1.25690
$199$	−8.01019	−0.567827	−0.283914	−	0.958850i	$-0.591633\pi$
$199$	−8.01019	−0.567827	−0.283914	+	0.958850i	$0.591633\pi$
$200$	−4.98236	−0.352306
$201$	−10.9283	−0.770823
$202$	12.0752	0.849609
$203$	41.5958	2.91946
$204$	20.4411	1.43116
$205$	−1.22833	−0.0857905
$206$	−9.98174	−0.695461
$207$	−3.36404	−0.233817
$208$	−4.52290	−0.313607
$209$	13.3090	0.920603
$210$	1.44121	0.0994531
$211$	13.6803	0.941789	0.470894	−	0.882190i	$-0.343931\pi$
$211$	13.6803	0.941789	0.470894	+	0.882190i	$0.343931\pi$
$212$	5.25725	0.361070
$213$	−6.22608	−0.426604
$214$	2.89844	0.198133
$215$	−1.28016	−0.0873062
$216$	−4.01416	−0.273129
$217$	−15.8471	−1.07577
$218$	4.70627	0.318749
$219$	2.19734	0.148483
$220$	0.525695	0.0354423
$221$	33.8295	2.27562
$222$	0.168603	0.0113159
$223$	9.43511	0.631822	0.315911	−	0.948789i	$-0.397690\pi$
$223$	9.43511	0.631822	0.315911	+	0.948789i	$0.397690\pi$
$224$	3.97016	0.265268
$225$	−22.2652	−1.48435
$226$	−1.33778	−0.0889881
$227$	−3.51015	−0.232977	−0.116489	−	0.993192i	$-0.537164\pi$
$227$	−3.51015	−0.232977	−0.116489	+	0.993192i	$0.537164\pi$
$228$	9.19033	0.608645
$229$	20.0647	1.32591	0.662957	−	0.748658i	$-0.269301\pi$
$229$	20.0647	1.32591	0.662957	+	0.748658i	$0.269301\pi$
$230$	0.0999911	0.00659322
$231$	42.9412	2.82532
$232$	10.4771	0.687856
$233$	−2.75079	−0.180210	−0.0901050	−	0.995932i	$-0.528720\pi$
$233$	−2.75079	−0.180210	−0.0901050	+	0.995932i	$0.528720\pi$
$234$	−20.2120	−1.32130
$235$	0.961749	0.0627376
$236$	5.05595	0.329114
$237$	−9.22358	−0.599136
$238$	−29.6952	−1.92486
$239$	−26.7416	−1.72977	−0.864887	−	0.501967i	$-0.832610\pi$
$239$	−26.7416	−1.72977	−0.864887	+	0.501967i	$0.832610\pi$
$240$	0.363011	0.0234322
$241$	−3.46681	−0.223317	−0.111658	−	0.993747i	$-0.535616\pi$
$241$	−3.46681	−0.223317	−0.111658	+	0.993747i	$0.535616\pi$
$242$	4.66318	0.299761
$243$	18.7002	1.19962
$244$	4.94315	0.316453
$245$	−1.16387	−0.0743572
$246$	−25.2725	−1.61132
$247$	15.2098	0.967774
$248$	−3.99154	−0.253463
$249$	−19.9037	−1.26135
$250$	1.32595	0.0838603
$251$	19.4897	1.23018	0.615090	−	0.788457i	$-0.289120\pi$
$251$	19.4897	1.23018	0.615090	+	0.788457i	$0.289120\pi$
$252$	17.7419	1.11764
$253$	2.97926	0.187304
$254$	15.8527	0.994689
$255$	−2.71518	−0.170031
$256$	1.00000	0.0625000
$257$	−28.3294	−1.76714	−0.883570	−	0.468300i	$-0.844867\pi$
$257$	−28.3294	−1.76714	−0.883570	+	0.468300i	$0.844867\pi$
$258$	−26.3389	−1.63979
$259$	−0.244934	−0.0152194
$260$	0.600773	0.0372584
$261$	46.8203	2.89811
$262$	1.61217	0.0996000
$263$	13.7570	0.848291	0.424145	−	0.905594i	$-0.360575\pi$
$263$	13.7570	0.848291	0.424145	+	0.905594i	$0.360575\pi$
$264$	10.8160	0.665678
$265$	−0.698317	−0.0428973
$266$	−13.3510	−0.818602
$267$	8.91744	0.545739
$268$	3.99877	0.244264
$269$	−14.1023	−0.859831	−0.429916	−	0.902869i	$-0.641457\pi$
$269$	−14.1023	−0.859831	−0.429916	+	0.902869i	$0.641457\pi$
$270$	0.533197	0.0324493
$271$	15.7703	0.957977	0.478988	−	0.877821i	$-0.341004\pi$
$271$	15.7703	0.957977	0.478988	+	0.877821i	$0.341004\pi$
$272$	−7.47960	−0.453518
$273$	49.0740	2.97009
$274$	16.9308	1.02283
$275$	19.7185	1.18907
$276$	2.05728	0.123834
$277$	−4.77610	−0.286968	−0.143484	−	0.989653i	$-0.545831\pi$
$277$	−4.77610	−0.286968	−0.143484	+	0.989653i	$0.545831\pi$
$278$	4.36198	0.261614
$279$	−17.8375	−1.06790
$280$	−0.527354	−0.0315154
$281$	−0.674478	−0.0402360	−0.0201180	−	0.999798i	$-0.506404\pi$
$281$	−0.674478	−0.0402360	−0.0201180	+	0.999798i	$0.506404\pi$
$282$	19.7876	1.17834
$283$	−25.3505	−1.50693	−0.753464	−	0.657489i	$-0.771619\pi$
$283$	−25.3505	−1.50693	−0.753464	+	0.657489i	$0.771619\pi$
$284$	2.27818	0.135185
$285$	−1.22074	−0.0723107
$286$	17.9002	1.05846
$287$	36.7139	2.16715
$288$	4.46882	0.263328
$289$	38.9445	2.29085
$290$	−1.39167	−0.0817215
$291$	42.6612	2.50084
$292$	−0.804029	−0.0470522
$293$	−15.8651	−0.926847	−0.463424	−	0.886137i	$-0.653379\pi$
$293$	−15.8651	−0.926847	−0.463424	+	0.886137i	$0.653379\pi$
$294$	−23.9463	−1.39658
$295$	−0.671577	−0.0391007
$296$	−0.0616936	−0.00358587
$297$	15.8867	0.921841
$298$	−4.08149	−0.236435
$299$	3.40475	0.196902
$300$	13.6164	0.786140
$301$	38.2630	2.20544
$302$	14.9130	0.858147
$303$	−33.0005	−1.89583
$304$	−3.36283	−0.192872
$305$	−0.656595	−0.0375965
$306$	−33.4250	−1.91078
$307$	10.4216	0.594792	0.297396	−	0.954754i	$-0.403882\pi$
$307$	10.4216	0.594792	0.297396	+	0.954754i	$0.403882\pi$
$308$	−15.7126	−0.895309
$309$	27.2792	1.55186
$310$	0.530193	0.0301129
$311$	26.4103	1.49759	0.748795	−	0.662802i	$-0.230633\pi$
$311$	26.4103	1.49759	0.748795	+	0.662802i	$0.230633\pi$
$312$	12.3607	0.699787
$313$	31.7056	1.79211	0.896054	−	0.443945i	$-0.146421\pi$
$313$	31.7056	1.79211	0.896054	+	0.443945i	$0.146421\pi$
$314$	22.4100	1.26467
$315$	−2.35665	−0.132782
$316$	3.37500	0.189858
$317$	9.79155	0.549948	0.274974	−	0.961452i	$-0.411331\pi$
$317$	9.79155	0.549948	0.274974	+	0.961452i	$0.411331\pi$
$318$	−14.3676	−0.805696
$319$	−41.4650	−2.32159
$320$	−0.132829	−0.00742538
$321$	−7.92118	−0.442117
$322$	−2.98866	−0.166551
$323$	25.1527	1.39953
$324$	−2.43611	−0.135340
$325$	22.5347	1.25000
$326$	−0.0243186	−0.00134688
$327$	−12.8618	−0.711261
$328$	9.24746	0.510606
$329$	−28.7459	−1.58481
$330$	−1.43668	−0.0790865
$331$	−35.6726	−1.96074	−0.980371	−	0.197161i	$-0.936828\pi$
$331$	−35.6726	−1.96074	−0.980371	+	0.197161i	$0.936828\pi$
$332$	7.28297	0.399705
$333$	−0.275698	−0.0151081
$334$	18.9794	1.03850
$335$	−0.531154	−0.0290200
$336$	−10.8501	−0.591922
$337$	−34.6385	−1.88688	−0.943441	−	0.331541i	$-0.892431\pi$
$337$	−34.6385	−1.88688	−0.943441	+	0.331541i	$0.892431\pi$
$338$	7.45664	0.405588
$339$	3.65605	0.198569
$340$	0.993510	0.0538806
$341$	15.7972	0.855467
$342$	−15.0279	−0.812615
$343$	6.99619	0.377759
$344$	9.63765	0.519627
$345$	−0.273267	−0.0147122
$346$	−24.6486	−1.32512
$347$	−23.6916	−1.27183	−0.635915	−	0.771759i	$-0.719377\pi$
$347$	−23.6916	−1.27183	−0.635915	+	0.771759i	$0.719377\pi$
$348$	−28.6330	−1.53489
$349$	−12.3943	−0.663452	−0.331726	−	0.943376i	$-0.607631\pi$
$349$	−12.3943	−0.663452	−0.331726	+	0.943376i	$0.607631\pi$
$350$	−19.7808	−1.05733
$351$	18.1556	0.969076
$352$	−3.95767	−0.210945
$353$	6.94306	0.369542	0.184771	−	0.982782i	$-0.440846\pi$
$353$	6.94306	0.369542	0.184771	+	0.982782i	$0.440846\pi$
$354$	−13.8175	−0.734390
$355$	−0.302609	−0.0160608
$356$	−3.26298	−0.172938
$357$	81.1545	4.29515
$358$	−10.5036	−0.555135
$359$	2.50004	0.131947	0.0659734	−	0.997821i	$-0.478985\pi$
$359$	2.50004	0.131947	0.0659734	+	0.997821i	$0.478985\pi$
$360$	−0.593590	−0.0312849
$361$	−7.69135	−0.404808
$362$	10.5727	0.555690
$363$	−12.7441	−0.668890
$364$	−17.9567	−0.941184
$365$	0.106798	0.00559009
$366$	−13.5092	−0.706138
$367$	−33.5719	−1.75244	−0.876218	−	0.481914i	$-0.839942\pi$
$367$	−33.5719	−1.75244	−0.876218	+	0.481914i	$0.839942\pi$
$368$	−0.752780	−0.0392413
$369$	41.3252	2.15131
$370$	0.00819472	0.000426023 0
$371$	20.8722	1.08363
$372$	10.9085	0.565581
$373$	−14.2093	−0.735728	−0.367864	−	0.929880i	$-0.619911\pi$
$373$	−14.2093	−0.735728	−0.367864	+	0.929880i	$0.619911\pi$
$374$	29.6018	1.53067
$375$	−3.62370	−0.187127
$376$	−7.24049	−0.373400
$377$	−47.3870	−2.44055
$378$	−15.9369	−0.819703
$379$	12.7063	0.652681	0.326341	−	0.945252i	$-0.394184\pi$
$379$	12.7063	0.652681	0.326341	+	0.945252i	$0.394184\pi$
$380$	0.446682	0.0229143
$381$	−43.3242	−2.21956
$382$	13.4771	0.689547
$383$	16.1351	0.824463	0.412231	−	0.911079i	$-0.364750\pi$
$383$	16.1351	0.824463	0.412231	+	0.911079i	$0.364750\pi$
$384$	−2.73291	−0.139463
$385$	2.08709	0.106368
$386$	−11.7274	−0.596911
$387$	43.0689	2.18931
$388$	−15.6101	−0.792485
$389$	3.04018	0.154143	0.0770717	−	0.997026i	$-0.475443\pi$
$389$	3.04018	0.154143	0.0770717	+	0.997026i	$0.475443\pi$
$390$	−1.64186	−0.0831389
$391$	5.63049	0.284746
$392$	8.76219	0.442558
$393$	−4.40591	−0.222249
$394$	17.3538	0.874270
$395$	−0.448298	−0.0225563
$396$	−17.6861	−0.888761
$397$	−11.7677	−0.590606	−0.295303	−	0.955404i	$-0.595421\pi$
$397$	−11.7677	−0.590606	−0.295303	+	0.955404i	$0.595421\pi$
$398$	−8.01019	−0.401514
$399$	36.4871	1.82664
$400$	−4.98236	−0.249118
$401$	19.9351	0.995510	0.497755	−	0.867318i	$-0.334158\pi$
$401$	19.9351	0.995510	0.497755	+	0.867318i	$0.334158\pi$
$402$	−10.9283	−0.545054
$403$	18.0533	0.899301
$404$	12.0752	0.600765
$405$	0.323587	0.0160792
$406$	41.5958	2.06437
$407$	0.244163	0.0121027
$408$	20.4411	1.01199
$409$	12.5124	0.618699	0.309349	−	0.950948i	$-0.399889\pi$
$409$	12.5124	0.618699	0.309349	+	0.950948i	$0.399889\pi$
$410$	−1.22833	−0.0606630
$411$	−46.2705	−2.28235
$412$	−9.98174	−0.491765
$413$	20.0729	0.987724
$414$	−3.36404	−0.165333
$415$	−0.967391	−0.0474873
$416$	−4.52290	−0.221753
$417$	−11.9209	−0.583769
$418$	13.3090	0.650964
$419$	−25.0141	−1.22202	−0.611010	−	0.791623i	$-0.709237\pi$
$419$	−25.0141	−1.22202	−0.611010	+	0.791623i	$0.709237\pi$
$420$	1.44121	0.0703239
$421$	−15.2787	−0.744637	−0.372318	−	0.928105i	$-0.621437\pi$
$421$	−15.2787	−0.744637	−0.372318	+	0.928105i	$0.621437\pi$
$422$	13.6803	0.665945
$423$	−32.3565	−1.57322
$424$	5.25725	0.255315
$425$	37.2660	1.80767
$426$	−6.22608	−0.301655
$427$	19.6251	0.949726
$428$	2.89844	0.140101
$429$	−48.9196	−2.36186
$430$	−1.28016	−0.0617348
$431$	23.6955	1.14137	0.570685	−	0.821169i	$-0.306678\pi$
$431$	23.6955	1.14137	0.570685	+	0.821169i	$0.306678\pi$
$432$	−4.01416	−0.193131
$433$	−10.7561	−0.516906	−0.258453	−	0.966024i	$-0.583213\pi$
$433$	−10.7561	−0.516906	−0.258453	+	0.966024i	$0.583213\pi$
$434$	−15.8471	−0.760684
$435$	3.80331	0.182355
$436$	4.70627	0.225390
$437$	2.53147	0.121097
$438$	2.19734	0.104993
$439$	0.942755	0.0449953	0.0224976	−	0.999747i	$-0.492838\pi$
$439$	0.942755	0.0449953	0.0224976	+	0.999747i	$0.492838\pi$
$440$	0.525695	0.0250615
$441$	39.1567	1.86460
$442$	33.8295	1.60911
$443$	13.4068	0.636975	0.318487	−	0.947927i	$-0.396825\pi$
$443$	13.4068	0.636975	0.318487	+	0.947927i	$0.396825\pi$
$444$	0.168603	0.00800156
$445$	0.433419	0.0205460
$446$	9.43511	0.446766
$447$	11.1544	0.527584
$448$	3.97016	0.187573
$449$	27.7810	1.31107	0.655534	−	0.755166i	$-0.272444\pi$
$449$	27.7810	1.31107	0.655534	+	0.755166i	$0.272444\pi$
$450$	−22.2652	−1.04959
$451$	−36.5984	−1.72335
$452$	−1.33778	−0.0629241
$453$	−40.7559	−1.91488
$454$	−3.51015	−0.164740
$455$	2.38517	0.111818
$456$	9.19033	0.430377
$457$	−7.77092	−0.363508	−0.181754	−	0.983344i	$-0.558177\pi$
$457$	−7.77092	−0.363508	−0.181754	+	0.983344i	$0.558177\pi$
$458$	20.0647	0.937563
$459$	30.0243	1.40141
$460$	0.0999911	0.00466211
$461$	24.3430	1.13377	0.566883	−	0.823799i	$-0.308149\pi$
$461$	24.3430	1.13377	0.566883	+	0.823799i	$0.308149\pi$
$462$	42.9412	1.99781
$463$	−30.6059	−1.42238	−0.711189	−	0.703001i	$-0.751843\pi$
$463$	−30.6059	−1.42238	−0.711189	+	0.703001i	$0.751843\pi$
$464$	10.4771	0.486388
$465$	−1.44897	−0.0671945
$466$	−2.75079	−0.127428
$467$	31.0267	1.43574	0.717872	−	0.696175i	$-0.245116\pi$
$467$	31.0267	1.43574	0.717872	+	0.696175i	$0.245116\pi$
$468$	−20.2120	−0.934301
$469$	15.8758	0.733075
$470$	0.961749	0.0443622
$471$	−61.2447	−2.82201
$472$	5.05595	0.232719
$473$	−38.1427	−1.75380
$474$	−9.22358	−0.423653
$475$	16.7548	0.768765
$476$	−29.6952	−1.36108
$477$	23.4937	1.07570
$478$	−26.7416	−1.22313
$479$	9.35231	0.427318	0.213659	−	0.976908i	$-0.431462\pi$
$479$	9.35231	0.427318	0.213659	+	0.976908i	$0.431462\pi$
$480$	0.363011	0.0165691
$481$	0.279034	0.0127229
$482$	−3.46681	−0.157909
$483$	8.16774	0.371645
$484$	4.66318	0.211963
$485$	2.07348	0.0941520
$486$	18.7002	0.848256
$487$	30.4826	1.38130	0.690649	−	0.723190i	$-0.257325\pi$
$487$	30.4826	1.38130	0.690649	+	0.723190i	$0.257325\pi$
$488$	4.94315	0.223766
$489$	0.0664606	0.00300545
$490$	−1.16387	−0.0525785
$491$	24.7080	1.11506	0.557529	−	0.830158i	$-0.311750\pi$
$491$	24.7080	1.11506	0.557529	+	0.830158i	$0.311750\pi$
$492$	−25.2725	−1.13937
$493$	−78.3646	−3.52937
$494$	15.2098	0.684320
$495$	2.34923	0.105590
$496$	−3.99154	−0.179225
$497$	9.04476	0.405713
$498$	−19.9037	−0.891907
$499$	20.6830	0.925898	0.462949	−	0.886385i	$-0.346791\pi$
$499$	20.6830	0.925898	0.462949	+	0.886385i	$0.346791\pi$
$500$	1.32595	0.0592982
$501$	−51.8690	−2.31734
$502$	19.4897	0.869869
$503$	42.3889	1.89003	0.945015	−	0.327028i	$-0.106047\pi$
$503$	42.3889	1.89003	0.945015	+	0.327028i	$0.106047\pi$
$504$	17.7419	0.790289
$505$	−1.60394	−0.0713745
$506$	2.97926	0.132444
$507$	−20.3784	−0.905034
$508$	15.8527	0.703351
$509$	2.25292	0.0998591	0.0499296	−	0.998753i	$-0.484100\pi$
$509$	2.25292	0.0998591	0.0499296	+	0.998753i	$0.484100\pi$
$510$	−2.71518	−0.120230
$511$	−3.19213	−0.141211
$512$	1.00000	0.0441942
$513$	13.4989	0.595993
$514$	−28.3294	−1.24956
$515$	1.32587	0.0584247
$516$	−26.3389	−1.15950
$517$	28.6555	1.26027
$518$	−0.244934	−0.0107618
$519$	67.3626	2.95689
$520$	0.600773	0.0263457
$521$	−8.98734	−0.393743	−0.196871	−	0.980429i	$-0.563078\pi$
$521$	−8.98734	−0.393743	−0.196871	+	0.980429i	$0.563078\pi$
$522$	46.8203	2.04927
$523$	−25.0241	−1.09423	−0.547114	−	0.837058i	$-0.684274\pi$
$523$	−25.0241	−1.09423	−0.547114	+	0.837058i	$0.684274\pi$
$524$	1.61217	0.0704278
$525$	54.0591	2.35933
$526$	13.7570	0.599832
$527$	29.8551	1.30051
$528$	10.8160	0.470705
$529$	−22.4333	−0.975362
$530$	−0.698317	−0.0303329
$531$	22.5941	0.980501
$532$	−13.3510	−0.578839
$533$	−41.8254	−1.81166
$534$	8.91744	0.385896
$535$	−0.384997	−0.0166449
$536$	3.99877	0.172721
$537$	28.7056	1.23874
$538$	−14.1023	−0.607993
$539$	−34.6779	−1.49368
$540$	0.533197	0.0229452
$541$	−11.0017	−0.473002	−0.236501	−	0.971631i	$-0.576001\pi$
$541$	−11.0017	−0.473002	−0.236501	+	0.971631i	$0.576001\pi$
$542$	15.7703	0.677392
$543$	−28.8943	−1.23997
$544$	−7.47960	−0.320685
$545$	−0.625131	−0.0267776
$546$	49.0740	2.10017
$547$	0.421365	0.0180163	0.00900814	−	0.999959i	$-0.497133\pi$
$547$	0.421365	0.0180163	0.00900814	+	0.999959i	$0.497133\pi$
$548$	16.9308	0.723248
$549$	22.0901	0.942781
$550$	19.7185	0.840801
$551$	−35.2328	−1.50097
$552$	2.05728	0.0875637
$553$	13.3993	0.569796
$554$	−4.77610	−0.202917
$555$	−0.0223955	−0.000950634 0
$556$	4.36198	0.184989
$557$	18.3993	0.779604	0.389802	−	0.920899i	$-0.372543\pi$
$557$	18.3993	0.779604	0.389802	+	0.920899i	$0.372543\pi$
$558$	−17.8375	−0.755120
$559$	−43.5901	−1.84367
$560$	−0.527354	−0.0222848
$561$	−80.8992	−3.41557
$562$	−0.674478	−0.0284512
$563$	25.1282	1.05903	0.529513	−	0.848302i	$-0.322375\pi$
$563$	25.1282	1.05903	0.529513	+	0.848302i	$0.322375\pi$
$564$	19.7876	0.833210
$565$	0.177697	0.00747576
$566$	−25.3505	−1.06556
$567$	−9.67177	−0.406176
$568$	2.27818	0.0955905
$569$	16.5309	0.693012	0.346506	−	0.938048i	$-0.387368\pi$
$569$	16.5309	0.693012	0.346506	+	0.938048i	$0.387368\pi$
$570$	−1.22074	−0.0511314
$571$	16.8060	0.703311	0.351655	−	0.936129i	$-0.385619\pi$
$571$	16.8060	0.703311	0.351655	+	0.936129i	$0.385619\pi$
$572$	17.9002	0.748444
$573$	−36.8317	−1.53867
$574$	36.7139	1.53241
$575$	3.75062	0.156411
$576$	4.46882	0.186201
$577$	−44.0552	−1.83404	−0.917021	−	0.398839i	$-0.869413\pi$
$577$	−44.0552	−1.83404	−0.917021	+	0.398839i	$0.869413\pi$
$578$	38.9445	1.61988
$579$	32.0501	1.33196
$580$	−1.39167	−0.0577858
$581$	28.9146	1.19958
$582$	42.6612	1.76836
$583$	−20.8065	−0.861717
$584$	−0.804029	−0.0332709
$585$	2.68475	0.111001
$586$	−15.8651	−0.655380
$587$	13.7567	0.567801	0.283900	−	0.958854i	$-0.408371\pi$
$587$	13.7567	0.567801	0.283900	+	0.958854i	$0.408371\pi$
$588$	−23.9463	−0.987529
$589$	13.4229	0.553080
$590$	−0.671577	−0.0276484
$591$	−47.4263	−1.95086
$592$	−0.0616936	−0.00253559
$593$	45.5471	1.87039	0.935197	−	0.354129i	$-0.115222\pi$
$593$	45.5471	1.87039	0.935197	+	0.354129i	$0.115222\pi$
$594$	15.8867	0.651840
$595$	3.94440	0.161704
$596$	−4.08149	−0.167184
$597$	21.8912	0.895945
$598$	3.40475	0.139230
$599$	6.36416	0.260033	0.130016	−	0.991512i	$-0.458497\pi$
$599$	6.36416	0.260033	0.130016	+	0.991512i	$0.458497\pi$
$600$	13.6164	0.555885
$601$	−3.23336	−0.131892	−0.0659458	−	0.997823i	$-0.521006\pi$
$601$	−3.23336	−0.131892	−0.0659458	+	0.997823i	$0.521006\pi$
$602$	38.2630	1.55948
$603$	17.8698	0.727714
$604$	14.9130	0.606801
$605$	−0.619406	−0.0251825
$606$	−33.0005	−1.34055
$607$	3.19127	0.129530	0.0647648	−	0.997901i	$-0.479370\pi$
$607$	3.19127	0.129530	0.0647648	+	0.997901i	$0.479370\pi$
$608$	−3.36283	−0.136381
$609$	−113.678	−4.60646
$610$	−0.656595	−0.0265848
$611$	32.7480	1.32484
$612$	−33.4250	−1.35113
$613$	1.49818	0.0605109	0.0302555	−	0.999542i	$-0.490368\pi$
$613$	1.49818	0.0605109	0.0302555	+	0.999542i	$0.490368\pi$
$614$	10.4216	0.420582
$615$	3.35693	0.135364
$616$	−15.7126	−0.633079
$617$	−36.3697	−1.46419	−0.732095	−	0.681203i	$-0.761457\pi$
$617$	−36.3697	−1.46419	−0.732095	+	0.681203i	$0.761457\pi$
$618$	27.2792	1.09733
$619$	−1.61419	−0.0648797	−0.0324398	−	0.999474i	$-0.510328\pi$
$619$	−1.61419	−0.0648797	−0.0324398	+	0.999474i	$0.510328\pi$
$620$	0.530193	0.0212931
$621$	3.02177	0.121260
$622$	26.4103	1.05896
$623$	−12.9546	−0.519014
$624$	12.3607	0.494824
$625$	24.7357	0.989426
$626$	31.7056	1.26721
$627$	−36.3723	−1.45257
$628$	22.4100	0.894257
$629$	0.461444	0.0183990
$630$	−2.35665	−0.0938911
$631$	−11.6634	−0.464315	−0.232157	−	0.972678i	$-0.574578\pi$
$631$	−11.6634	−0.464315	−0.232157	+	0.972678i	$0.574578\pi$
$632$	3.37500	0.134250
$633$	−37.3870	−1.48600
$634$	9.79155	0.388872
$635$	−2.10571	−0.0835624
$636$	−14.3676	−0.569713
$637$	−39.6305	−1.57022
$638$	−41.4650	−1.64161
$639$	10.1808	0.402746
$640$	−0.132829	−0.00525054
$641$	−43.6807	−1.72529	−0.862643	−	0.505814i	$-0.831192\pi$
$641$	−43.6807	−1.72529	−0.862643	+	0.505814i	$0.831192\pi$
$642$	−7.92118	−0.312624
$643$	44.8249	1.76772	0.883861	−	0.467750i	$-0.154935\pi$
$643$	44.8249	1.76772	0.883861	+	0.467750i	$0.154935\pi$
$644$	−2.98866	−0.117770
$645$	3.49857	0.137756
$646$	25.1527	0.989618
$647$	2.41041	0.0947631	0.0473815	−	0.998877i	$-0.484912\pi$
$647$	2.41041	0.0947631	0.0473815	+	0.998877i	$0.484912\pi$
$648$	−2.43611	−0.0956996
$649$	−20.0098	−0.785453
$650$	22.5347	0.883884
$651$	43.3087	1.69740
$652$	−0.0243186	−0.000952389 0
$653$	−22.3324	−0.873933	−0.436967	−	0.899478i	$-0.643947\pi$
$653$	−22.3324	−0.873933	−0.436967	+	0.899478i	$0.643947\pi$
$654$	−12.8618	−0.502938
$655$	−0.214143	−0.00836725
$656$	9.24746	0.361053
$657$	−3.59306	−0.140179
$658$	−28.7459	−1.12063
$659$	28.5705	1.11295	0.556475	−	0.830864i	$-0.312153\pi$
$659$	28.5705	1.11295	0.556475	+	0.830864i	$0.312153\pi$
$660$	−1.43668	−0.0559226
$661$	−22.2364	−0.864896	−0.432448	−	0.901659i	$-0.642350\pi$
$661$	−22.2364	−0.864896	−0.432448	+	0.901659i	$0.642350\pi$
$662$	−35.6726	−1.38645
$663$	−92.4531	−3.59058
$664$	7.28297	0.282634
$665$	1.77340	0.0687696
$666$	−0.275698	−0.0106831
$667$	−7.88696	−0.305384
$668$	18.9794	0.734334
$669$	−25.7854	−0.996919
$670$	−0.531154	−0.0205203
$671$	−19.5634	−0.755236
$672$	−10.8501	−0.418552
$673$	36.7504	1.41662	0.708312	−	0.705899i	$-0.249457\pi$
$673$	36.7504	1.41662	0.708312	+	0.705899i	$0.249457\pi$
$674$	−34.6385	−1.33423
$675$	20.0000	0.769799
$676$	7.45664	0.286794
$677$	−12.7977	−0.491856	−0.245928	−	0.969288i	$-0.579093\pi$
$677$	−12.7977	−0.491856	−0.245928	+	0.969288i	$0.579093\pi$
$678$	3.65605	0.140410
$679$	−61.9748	−2.37837
$680$	0.993510	0.0380994
$681$	9.59295	0.367602
$682$	15.7972	0.604907
$683$	50.3558	1.92681	0.963405	−	0.268050i	$-0.0863794\pi$
$683$	50.3558	1.92681	0.963405	+	0.268050i	$0.0863794\pi$
$684$	−15.0279	−0.574606
$685$	−2.24891	−0.0859263
$686$	6.99619	0.267116
$687$	−54.8351	−2.09209
$688$	9.63765	0.367432
$689$	−23.7780	−0.905871
$690$	−0.273267	−0.0104031
$691$	43.0771	1.63873	0.819365	−	0.573272i	$-0.194326\pi$
$691$	43.0771	1.63873	0.819365	+	0.573272i	$0.194326\pi$
$692$	−24.6486	−0.937000
$693$	−70.2168	−2.66732
$694$	−23.6916	−0.899320
$695$	−0.579398	−0.0219778
$696$	−28.6330	−1.08533
$697$	−69.1674	−2.61990
$698$	−12.3943	−0.469131
$699$	7.51766	0.284344
$700$	−19.7808	−0.747643
$701$	5.66200	0.213851	0.106925	−	0.994267i	$-0.465899\pi$
$701$	5.66200	0.213851	0.106925	+	0.994267i	$0.465899\pi$
$702$	18.1556	0.685240
$703$	0.207465	0.00782471
$704$	−3.95767	−0.149160
$705$	−2.62838	−0.0989904
$706$	6.94306	0.261306
$707$	47.9406	1.80299
$708$	−13.8175	−0.519292
$709$	2.09013	0.0784966	0.0392483	−	0.999229i	$-0.487504\pi$
$709$	2.09013	0.0784966	0.0392483	+	0.999229i	$0.487504\pi$
$710$	−0.302609	−0.0113567
$711$	15.0823	0.565629
$712$	−3.26298	−0.122285
$713$	3.00475	0.112529
$714$	81.1545	3.03713
$715$	−2.37767	−0.0889196
$716$	−10.5036	−0.392540
$717$	73.0826	2.72932
$718$	2.50004	0.0933005
$719$	−3.67508	−0.137057	−0.0685286	−	0.997649i	$-0.521830\pi$
$719$	−3.67508	−0.137057	−0.0685286	+	0.997649i	$0.521830\pi$
$720$	−0.593590	−0.0221218
$721$	−39.6291	−1.47587
$722$	−7.69135	−0.286242
$723$	9.47450	0.352360
$724$	10.5727	0.392932
$725$	−52.2007	−1.93869
$726$	−12.7441	−0.472977
$727$	25.6900	0.952788	0.476394	−	0.879232i	$-0.341944\pi$
$727$	25.6900	0.952788	0.476394	+	0.879232i	$0.341944\pi$
$728$	−17.9567	−0.665518
$729$	−43.7976	−1.62213
$730$	0.106798	0.00395279
$731$	−72.0858	−2.66619
$732$	−13.5092	−0.499315
$733$	−11.0052	−0.406488	−0.203244	−	0.979128i	$-0.565148\pi$
$733$	−11.0052	−0.406488	−0.203244	+	0.979128i	$0.565148\pi$
$734$	−33.5719	−1.23916
$735$	3.18077	0.117324
$736$	−0.752780	−0.0277478
$737$	−15.8258	−0.582952
$738$	41.3252	1.52120
$739$	16.7511	0.616198	0.308099	−	0.951354i	$-0.400307\pi$
$739$	16.7511	0.616198	0.308099	+	0.951354i	$0.400307\pi$
$740$	0.00819472	0.000301244 0
$741$	−41.5670	−1.52700
$742$	20.8722	0.766241
$743$	−45.4949	−1.66905	−0.834524	−	0.550972i	$-0.814257\pi$
$743$	−45.4949	−1.66905	−0.834524	+	0.550972i	$0.814257\pi$
$744$	10.9085	0.399926
$745$	0.542142	0.0198625
$746$	−14.2093	−0.520238
$747$	32.5463	1.19081
$748$	29.6018	1.08235
$749$	11.5073	0.420466
$750$	−3.62370	−0.132319
$751$	−34.6128	−1.26304	−0.631519	−	0.775360i	$-0.717568\pi$
$751$	−34.6128	−1.26304	−0.631519	+	0.775360i	$0.717568\pi$
$752$	−7.24049	−0.264034
$753$	−53.2637	−1.94104
$754$	−47.3870	−1.72573
$755$	−1.98088	−0.0720917
$756$	−15.9369	−0.579618
$757$	−16.4295	−0.597139	−0.298569	−	0.954388i	$-0.596509\pi$
$757$	−16.4295	−0.597139	−0.298569	+	0.954388i	$0.596509\pi$
$758$	12.7063	0.461515
$759$	−8.14205	−0.295538
$760$	0.446682	0.0162029
$761$	8.01736	0.290629	0.145314	−	0.989386i	$-0.453581\pi$
$761$	8.01736	0.290629	0.145314	+	0.989386i	$0.453581\pi$
$762$	−43.3242	−1.56947
$763$	18.6847	0.676430
$764$	13.4771	0.487584
$765$	4.43981	0.160522
$766$	16.1351	0.582983
$767$	−22.8675	−0.825699
$768$	−2.73291	−0.0986155
$769$	33.2568	1.19927	0.599636	−	0.800273i	$-0.295312\pi$
$769$	33.2568	1.19927	0.599636	+	0.800273i	$0.295312\pi$
$770$	2.08709	0.0752136
$771$	77.4218	2.78828
$772$	−11.7274	−0.422080
$773$	32.5104	1.16932	0.584658	−	0.811280i	$-0.301229\pi$
$773$	32.5104	1.16932	0.584658	+	0.811280i	$0.301229\pi$
$774$	43.0689	1.54808
$775$	19.8873	0.714372
$776$	−15.6101	−0.560371
$777$	0.669383	0.0240140
$778$	3.04018	0.108996
$779$	−31.0977	−1.11419
$780$	−1.64186	−0.0587881
$781$	−9.01631	−0.322629
$782$	5.63049	0.201346
$783$	−42.0568	−1.50299
$784$	8.76219	0.312935
$785$	−2.97671	−0.106243
$786$	−4.40591	−0.157154
$787$	51.1698	1.82401	0.912003	−	0.410183i	$-0.134535\pi$
$787$	51.1698	1.82401	0.912003	+	0.410183i	$0.134535\pi$
$788$	17.3538	0.618202
$789$	−37.5966	−1.33847
$790$	−0.448298	−0.0159497
$791$	−5.31122	−0.188845
$792$	−17.6861	−0.628449
$793$	−22.3574	−0.793934
$794$	−11.7677	−0.417621
$795$	1.90844	0.0676854
$796$	−8.01019	−0.283914
$797$	5.37367	0.190345	0.0951725	−	0.995461i	$-0.469660\pi$
$797$	5.37367	0.190345	0.0951725	+	0.995461i	$0.469660\pi$
$798$	36.4871	1.29163
$799$	54.1560	1.91590
$800$	−4.98236	−0.176153
$801$	−14.5817	−0.515218
$802$	19.9351	0.703932
$803$	3.18208	0.112293
$804$	−10.9283	−0.385412
$805$	0.396981	0.0139917
$806$	18.0533	0.635902
$807$	38.5403	1.35668
$808$	12.0752	0.424805
$809$	−2.30729	−0.0811201	−0.0405600	−	0.999177i	$-0.512914\pi$
$809$	−2.30729	−0.0811201	−0.0405600	+	0.999177i	$0.512914\pi$
$810$	0.323587	0.0113697
$811$	−36.3572	−1.27667	−0.638337	−	0.769757i	$-0.720377\pi$
$811$	−36.3572	−1.27667	−0.638337	+	0.769757i	$0.720377\pi$
$812$	41.5958	1.45973
$813$	−43.0988	−1.51154
$814$	0.244163	0.00855792
$815$	0.00323022	0.000113150 0
$816$	20.4411	0.715582
$817$	−32.4098	−1.13388
$818$	12.5124	0.437486
$819$	−80.2450	−2.80399
$820$	−1.22833	−0.0428953
$821$	−51.6000	−1.80085	−0.900426	−	0.435008i	$-0.856745\pi$
$821$	−51.6000	−1.80085	−0.900426	+	0.435008i	$0.856745\pi$
$822$	−46.2705	−1.61387
$823$	16.1150	0.561733	0.280866	−	0.959747i	$-0.409378\pi$
$823$	16.1150	0.561733	0.280866	+	0.959747i	$0.409378\pi$
$824$	−9.98174	−0.347730
$825$	−53.8891	−1.87618
$826$	20.0729	0.698426
$827$	40.9207	1.42295	0.711476	−	0.702710i	$-0.248027\pi$
$827$	40.9207	1.42295	0.711476	+	0.702710i	$0.248027\pi$
$828$	−3.36404	−0.116908
$829$	−39.4661	−1.37071	−0.685357	−	0.728207i	$-0.740354\pi$
$829$	−39.4661	−1.37071	−0.685357	+	0.728207i	$0.740354\pi$
$830$	−0.967391	−0.0335786
$831$	13.0527	0.452792
$832$	−4.52290	−0.156803
$833$	−65.5377	−2.27075
$834$	−11.9209	−0.412787
$835$	−2.52101	−0.0872433
$836$	13.3090	0.460301
$837$	16.0227	0.553824
$838$	−25.0141	−0.864099
$839$	−29.4491	−1.01670	−0.508348	−	0.861152i	$-0.669744\pi$
$839$	−29.4491	−1.01670	−0.508348	+	0.861152i	$0.669744\pi$
$840$	1.44121	0.0497265
$841$	80.7699	2.78517
$842$	−15.2787	−0.526538
$843$	1.84329	0.0634863
$844$	13.6803	0.470894
$845$	−0.990459	−0.0340728
$846$	−32.3565	−1.11244
$847$	18.5136	0.636134
$848$	5.25725	0.180535
$849$	69.2806	2.37770
$850$	37.2660	1.27821
$851$	0.0464417	0.00159200
$852$	−6.22608	−0.213302
$853$	5.80409	0.198728	0.0993641	−	0.995051i	$-0.468319\pi$
$853$	5.80409	0.198728	0.0993641	+	0.995051i	$0.468319\pi$
$854$	19.6251	0.671558
$855$	1.99614	0.0682667
$856$	2.89844	0.0990665
$857$	19.1140	0.652923	0.326462	−	0.945210i	$-0.394144\pi$
$857$	19.1140	0.652923	0.326462	+	0.945210i	$0.394144\pi$
$858$	−48.9196	−1.67009
$859$	−11.4999	−0.392372	−0.196186	−	0.980567i	$-0.562856\pi$
$859$	−11.4999	−0.392372	−0.196186	+	0.980567i	$0.562856\pi$
$860$	−1.28016	−0.0436531
$861$	−100.336	−3.41944
$862$	23.6955	0.807070
$863$	−2.04632	−0.0696576	−0.0348288	−	0.999393i	$-0.511089\pi$
$863$	−2.04632	−0.0696576	−0.0348288	+	0.999393i	$0.511089\pi$
$864$	−4.01416	−0.136564
$865$	3.27406	0.111321
$866$	−10.7561	−0.365508
$867$	−106.432	−3.61462
$868$	−15.8471	−0.537884
$869$	−13.3571	−0.453110
$870$	3.80331	0.128944
$871$	−18.0861	−0.612823
$872$	4.70627	0.159375
$873$	−69.7589	−2.36098
$874$	2.53147	0.0856283
$875$	5.26423	0.177963
$876$	2.19734	0.0742413
$877$	−52.4411	−1.77081	−0.885405	−	0.464821i	$-0.846119\pi$
$877$	−52.4411	−1.77081	−0.885405	+	0.464821i	$0.846119\pi$
$878$	0.942755	0.0318165
$879$	43.3579	1.46242
$880$	0.525695	0.0177212
$881$	−18.4891	−0.622915	−0.311458	−	0.950260i	$-0.600817\pi$
$881$	−18.4891	−0.622915	−0.311458	+	0.950260i	$0.600817\pi$
$882$	39.1567	1.31847
$883$	−0.00458912	−0.000154436 0	−7.72181e−5	−	1.00000i	$-0.500025\pi$
$883$	−0.00458912	−0.000154436 0	−7.72181e−5		1.00000i	$0.500025\pi$
$884$	33.8295	1.13781
$885$	1.83536	0.0616950
$886$	13.4068	0.450409
$887$	46.7159	1.56857	0.784283	−	0.620404i	$-0.213031\pi$
$887$	46.7159	1.56857	0.784283	+	0.620404i	$0.213031\pi$
$888$	0.168603	0.00565796
$889$	62.9379	2.11087
$890$	0.433419	0.0145282
$891$	9.64134	0.322997
$892$	9.43511	0.315911
$893$	24.3486	0.814794
$894$	11.1544	0.373058
$895$	1.39519	0.0466361
$896$	3.97016	0.132634
$897$	−9.30488	−0.310681
$898$	27.7810	0.927065
$899$	−41.8198	−1.39477
$900$	−22.2652	−0.742175
$901$	−39.3222	−1.31001
$902$	−36.5984	−1.21859
$903$	−104.570	−3.47986
$904$	−1.33778	−0.0444940
$905$	−1.40437	−0.0466827
$906$	−40.7559	−1.35403
$907$	24.6567	0.818714	0.409357	−	0.912374i	$-0.365753\pi$
$907$	24.6567	0.818714	0.409357	+	0.912374i	$0.365753\pi$
$908$	−3.51015	−0.116489
$909$	53.9620	1.78981
$910$	2.38517	0.0790676
$911$	13.9677	0.462771	0.231385	−	0.972862i	$-0.425674\pi$
$911$	13.9677	0.462771	0.231385	+	0.972862i	$0.425674\pi$
$912$	9.19033	0.304322
$913$	−28.8236	−0.953922
$914$	−7.77092	−0.257039
$915$	1.79442	0.0593216
$916$	20.0647	0.662957
$917$	6.40056	0.211365
$918$	30.0243	0.990949
$919$	−27.0921	−0.893687	−0.446843	−	0.894612i	$-0.647452\pi$
$919$	−27.0921	−0.893687	−0.446843	+	0.894612i	$0.647452\pi$
$920$	0.0999911	0.00329661
$921$	−28.4813	−0.938492
$922$	24.3430	0.801693
$923$	−10.3040	−0.339160
$924$	42.9412	1.41266
$925$	0.307380	0.0101066
$926$	−30.6059	−1.00577
$927$	−44.6066	−1.46507
$928$	10.4771	0.343928
$929$	−51.6961	−1.69609	−0.848047	−	0.529921i	$-0.822222\pi$
$929$	−51.6961	−1.69609	−0.848047	+	0.529921i	$0.822222\pi$
$930$	−1.44897	−0.0475137
$931$	−29.4658	−0.965702
$932$	−2.75079	−0.0901050
$933$	−72.1770	−2.36297
$934$	31.0267	1.01522
$935$	−3.93199	−0.128590
$936$	−20.2120	−0.660651
$937$	−5.71962	−0.186852	−0.0934259	−	0.995626i	$-0.529782\pi$
$937$	−5.71962	−0.186852	−0.0934259	+	0.995626i	$0.529782\pi$
$938$	15.8758	0.518363
$939$	−86.6487	−2.82768
$940$	0.961749	0.0313688
$941$	6.07128	0.197918	0.0989590	−	0.995092i	$-0.468449\pi$
$941$	6.07128	0.197918	0.0989590	+	0.995092i	$0.468449\pi$
$942$	−61.2447	−1.99546
$943$	−6.96130	−0.226691
$944$	5.05595	0.164557
$945$	2.11688	0.0688621
$946$	−38.1427	−1.24012
$947$	−8.02150	−0.260664	−0.130332	−	0.991470i	$-0.541604\pi$
$947$	−8.02150	−0.260664	−0.130332	+	0.991470i	$0.541604\pi$
$948$	−9.22358	−0.299568
$949$	3.63654	0.118047
$950$	16.7548	0.543599
$951$	−26.7595	−0.867735
$952$	−29.6952	−0.962428
$953$	40.4783	1.31122	0.655610	−	0.755099i	$-0.272411\pi$
$953$	40.4783	1.31122	0.655610	+	0.755099i	$0.272411\pi$
$954$	23.4937	0.760637
$955$	−1.79015	−0.0579279
$956$	−26.7416	−0.864887
$957$	113.320	3.66312
$958$	9.35231	0.302159
$959$	67.2181	2.17059
$960$	0.363011	0.0117161
$961$	−15.0676	−0.486052
$962$	0.279034	0.00899643
$963$	12.9526	0.417391
$964$	−3.46681	−0.111658
$965$	1.55775	0.0501456
$966$	8.16774	0.262793
$967$	−25.9249	−0.833689	−0.416845	−	0.908978i	$-0.636864\pi$
$967$	−25.9249	−0.833689	−0.416845	+	0.908978i	$0.636864\pi$
$968$	4.66318	0.149880
$969$	−68.7401	−2.20825
$970$	2.07348	0.0665755
$971$	40.1035	1.28698	0.643491	−	0.765454i	$-0.277485\pi$
$971$	40.1035	1.28698	0.643491	+	0.765454i	$0.277485\pi$
$972$	18.7002	0.599808
$973$	17.3178	0.555182
$974$	30.4826	0.976725
$975$	−61.5854	−1.97231
$976$	4.94315	0.158226
$977$	40.2414	1.28744	0.643718	−	0.765262i	$-0.277391\pi$
$977$	40.2414	1.28744	0.643718	+	0.765262i	$0.277391\pi$
$978$	0.0664606	0.00212518
$979$	12.9138	0.412727
$980$	−1.16387	−0.0371786
$981$	21.0315	0.671484
$982$	24.7080	0.788465
$983$	45.6713	1.45669	0.728345	−	0.685211i	$-0.240290\pi$
$983$	45.6713	1.45669	0.728345	+	0.685211i	$0.240290\pi$
$984$	−25.2725	−0.805659
$985$	−2.30509	−0.0734462
$986$	−78.3646	−2.49564
$987$	78.5602	2.50060
$988$	15.2098	0.483887
$989$	−7.25502	−0.230696
$990$	2.34923	0.0746636
$991$	38.9381	1.23691	0.618454	−	0.785821i	$-0.287759\pi$
$991$	38.9381	1.23691	0.618454	+	0.785821i	$0.287759\pi$
$992$	−3.99154	−0.126732
$993$	97.4901	3.09375
$994$	9.04476	0.286882
$995$	1.06399	0.0337307
$996$	−19.9037	−0.630674
$997$	4.27248	0.135311	0.0676554	−	0.997709i	$-0.478448\pi$
$997$	4.27248	0.135311	0.0676554	+	0.997709i	$0.478448\pi$
$998$	20.6830	0.654708
$999$	0.247648	0.00783523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.f.1.7	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.f.1.7	✓	67	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:	\( N \)	\(=\)	\( 6046 = 2 \cdot 3023 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.73291 q^{3} +1.00000 q^{4} -0.132829 q^{5} -2.73291 q^{6} +3.97016 q^{7} +1.00000 q^{8} +4.46882 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.73291 q^{3} +1.00000 q^{4} -0.132829 q^{5} -2.73291 q^{6} +3.97016 q^{7} +1.00000 q^{8} +4.46882 q^{9} -0.132829 q^{10} -3.95767 q^{11} -2.73291 q^{12} -4.52290 q^{13} +3.97016 q^{14} +0.363011 q^{15} +1.00000 q^{16} -7.47960 q^{17} +4.46882 q^{18} -3.36283 q^{19} -0.132829 q^{20} -10.8501 q^{21} -3.95767 q^{22} -0.752780 q^{23} -2.73291 q^{24} -4.98236 q^{25} -4.52290 q^{26} -4.01416 q^{27} +3.97016 q^{28} +10.4771 q^{29} +0.363011 q^{30} -3.99154 q^{31} +1.00000 q^{32} +10.8160 q^{33} -7.47960 q^{34} -0.527354 q^{35} +4.46882 q^{36} -0.0616936 q^{37} -3.36283 q^{38} +12.3607 q^{39} -0.132829 q^{40} +9.24746 q^{41} -10.8501 q^{42} +9.63765 q^{43} -3.95767 q^{44} -0.593590 q^{45} -0.752780 q^{46} -7.24049 q^{47} -2.73291 q^{48} +8.76219 q^{49} -4.98236 q^{50} +20.4411 q^{51} -4.52290 q^{52} +5.25725 q^{53} -4.01416 q^{54} +0.525695 q^{55} +3.97016 q^{56} +9.19033 q^{57} +10.4771 q^{58} +5.05595 q^{59} +0.363011 q^{60} +4.94315 q^{61} -3.99154 q^{62} +17.7419 q^{63} +1.00000 q^{64} +0.600773 q^{65} +10.8160 q^{66} +3.99877 q^{67} -7.47960 q^{68} +2.05728 q^{69} -0.527354 q^{70} +2.27818 q^{71} +4.46882 q^{72} -0.804029 q^{73} -0.0616936 q^{74} +13.6164 q^{75} -3.36283 q^{76} -15.7126 q^{77} +12.3607 q^{78} +3.37500 q^{79} -0.132829 q^{80} -2.43611 q^{81} +9.24746 q^{82} +7.28297 q^{83} -10.8501 q^{84} +0.993510 q^{85} +9.63765 q^{86} -28.6330 q^{87} -3.95767 q^{88} -3.26298 q^{89} -0.593590 q^{90} -17.9567 q^{91} -0.752780 q^{92} +10.9085 q^{93} -7.24049 q^{94} +0.446682 q^{95} -2.73291 q^{96} -15.6101 q^{97} +8.76219 q^{98} -17.6861 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9 + 21 * q^10 + 56 * q^11 + 21 * q^12 + 33 * q^13 + 38 * q^14 + 25 * q^15 + 67 * q^16 + 30 * q^17 + 90 * q^18 + 36 * q^19 + 21 * q^20 + 20 * q^21 + 56 * q^22 + 65 * q^23 + 21 * q^24 + 72 * q^25 + 33 * q^26 + 57 * q^27 + 38 * q^28 + 84 * q^29 + 25 * q^30 + 52 * q^31 + 67 * q^32 - 9 * q^33 + 30 * q^34 + 30 * q^35 + 90 * q^36 + 52 * q^37 + 36 * q^38 + 41 * q^39 + 21 * q^40 + 46 * q^41 + 20 * q^42 + 61 * q^43 + 56 * q^44 + 23 * q^45 + 65 * q^46 + 51 * q^47 + 21 * q^48 + 81 * q^49 + 72 * q^50 + 33 * q^51 + 33 * q^52 + 72 * q^53 + 57 * q^54 + 14 * q^55 + 38 * q^56 - 26 * q^57 + 84 * q^58 + 71 * q^59 + 25 * q^60 + 42 * q^61 + 52 * q^62 + 63 * q^63 + 67 * q^64 - 2 * q^65 - 9 * q^66 + 70 * q^67 + 30 * q^68 + 21 * q^69 + 30 * q^70 + 104 * q^71 + 90 * q^72 - 31 * q^73 + 52 * q^74 + 69 * q^75 + 36 * q^76 + 48 * q^77 + 41 * q^78 + 79 * q^79 + 21 * q^80 + 123 * q^81 + 46 * q^82 + 41 * q^83 + 20 * q^84 + 6 * q^85 + 61 * q^86 + 19 * q^87 + 56 * q^88 + 58 * q^89 + 23 * q^90 + 31 * q^91 + 65 * q^92 + 13 * q^93 + 51 * q^94 + 77 * q^95 + 21 * q^96 - 8 * q^97 + 81 * q^98 + 129 * q^99

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