LMFDB - Embedded newform 4015.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4015,2,Mod(1,4015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4015, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4015 = 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4015.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:	$32.0599364115$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.76981 q^{2} +1.77605 q^{3} +5.67182 q^{4} +1.00000 q^{5} -4.91932 q^{6} -0.953960 q^{7} -10.1702 q^{8} +0.154367 q^{9} +O(q^{10})$	\(q-2.76981 q^{2} +1.77605 q^{3} +5.67182 q^{4} +1.00000 q^{5} -4.91932 q^{6} -0.953960 q^{7} -10.1702 q^{8} +0.154367 q^{9} -2.76981 q^{10} -1.00000 q^{11} +10.0735 q^{12} -6.96950 q^{13} +2.64228 q^{14} +1.77605 q^{15} +16.8259 q^{16} -1.81899 q^{17} -0.427568 q^{18} -0.165649 q^{19} +5.67182 q^{20} -1.69428 q^{21} +2.76981 q^{22} +1.66553 q^{23} -18.0629 q^{24} +1.00000 q^{25} +19.3042 q^{26} -5.05400 q^{27} -5.41069 q^{28} +7.43543 q^{29} -4.91932 q^{30} +5.38941 q^{31} -26.2640 q^{32} -1.77605 q^{33} +5.03824 q^{34} -0.953960 q^{35} +0.875544 q^{36} -1.22357 q^{37} +0.458816 q^{38} -12.3782 q^{39} -10.1702 q^{40} +9.23166 q^{41} +4.69284 q^{42} +7.53958 q^{43} -5.67182 q^{44} +0.154367 q^{45} -4.61318 q^{46} +3.29987 q^{47} +29.8837 q^{48} -6.08996 q^{49} -2.76981 q^{50} -3.23062 q^{51} -39.5297 q^{52} -2.58351 q^{53} +13.9986 q^{54} -1.00000 q^{55} +9.70198 q^{56} -0.294202 q^{57} -20.5947 q^{58} +2.86589 q^{59} +10.0735 q^{60} -12.3747 q^{61} -14.9276 q^{62} -0.147260 q^{63} +39.0944 q^{64} -6.96950 q^{65} +4.91932 q^{66} +5.28055 q^{67} -10.3170 q^{68} +2.95807 q^{69} +2.64228 q^{70} -2.20096 q^{71} -1.56995 q^{72} -1.00000 q^{73} +3.38905 q^{74} +1.77605 q^{75} -0.939532 q^{76} +0.953960 q^{77} +34.2852 q^{78} -7.82311 q^{79} +16.8259 q^{80} -9.43927 q^{81} -25.5699 q^{82} +11.7340 q^{83} -9.60967 q^{84} -1.81899 q^{85} -20.8832 q^{86} +13.2057 q^{87} +10.1702 q^{88} +9.62676 q^{89} -0.427568 q^{90} +6.64862 q^{91} +9.44657 q^{92} +9.57188 q^{93} -9.14000 q^{94} -0.165649 q^{95} -46.6463 q^{96} -3.04958 q^{97} +16.8680 q^{98} -0.154367 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9}+O(q^{10}) $ 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9	\( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9} + 4 q^{10} - 38 q^{11} + 12 q^{12} - q^{13} + 23 q^{14} + 5 q^{15} + 74 q^{16} + 26 q^{17} + 16 q^{18} - 10 q^{19} + 50 q^{20} + 21 q^{21} - 4 q^{22} + 10 q^{23} + 41 q^{24} + 38 q^{25} + 25 q^{26} + 5 q^{27} + 2 q^{28} + 28 q^{29} + 11 q^{30} + 24 q^{31} + 39 q^{32} - 5 q^{33} + 38 q^{34} + 111 q^{36} + 12 q^{37} + 19 q^{38} - 18 q^{39} + 15 q^{40} + 62 q^{41} - 17 q^{42} - 32 q^{43} - 50 q^{44} + 63 q^{45} - 9 q^{46} + 31 q^{47} + 53 q^{48} + 88 q^{49} + 4 q^{50} - 3 q^{51} - 21 q^{52} + 30 q^{53} + 49 q^{54} - 38 q^{55} + 32 q^{56} + 49 q^{57} + 12 q^{58} + 31 q^{59} + 12 q^{60} + 25 q^{61} + 12 q^{62} + 15 q^{63} + 137 q^{64} - q^{65} - 11 q^{66} + 20 q^{67} + 75 q^{68} + 92 q^{69} + 23 q^{70} + 32 q^{71} + 6 q^{72} - 38 q^{73} + 55 q^{74} + 5 q^{75} - 57 q^{76} - 17 q^{78} - 2 q^{79} + 74 q^{80} + 118 q^{81} + 14 q^{82} + 4 q^{83} + 22 q^{84} + 26 q^{85} + 5 q^{86} + 24 q^{87} - 15 q^{88} + 143 q^{89} + 16 q^{90} + 66 q^{91} + 29 q^{92} - 8 q^{93} - 7 q^{94} - 10 q^{95} + 59 q^{96} + 41 q^{97} - 10 q^{98} - 63 q^{99}+O(q^{100}) \) 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9 + 4 * q^10 - 38 * q^11 + 12 * q^12 - q^13 + 23 * q^14 + 5 * q^15 + 74 * q^16 + 26 * q^17 + 16 * q^18 - 10 * q^19 + 50 * q^20 + 21 * q^21 - 4 * q^22 + 10 * q^23 + 41 * q^24 + 38 * q^25 + 25 * q^26 + 5 * q^27 + 2 * q^28 + 28 * q^29 + 11 * q^30 + 24 * q^31 + 39 * q^32 - 5 * q^33 + 38 * q^34 + 111 * q^36 + 12 * q^37 + 19 * q^38 - 18 * q^39 + 15 * q^40 + 62 * q^41 - 17 * q^42 - 32 * q^43 - 50 * q^44 + 63 * q^45 - 9 * q^46 + 31 * q^47 + 53 * q^48 + 88 * q^49 + 4 * q^50 - 3 * q^51 - 21 * q^52 + 30 * q^53 + 49 * q^54 - 38 * q^55 + 32 * q^56 + 49 * q^57 + 12 * q^58 + 31 * q^59 + 12 * q^60 + 25 * q^61 + 12 * q^62 + 15 * q^63 + 137 * q^64 - q^65 - 11 * q^66 + 20 * q^67 + 75 * q^68 + 92 * q^69 + 23 * q^70 + 32 * q^71 + 6 * q^72 - 38 * q^73 + 55 * q^74 + 5 * q^75 - 57 * q^76 - 17 * q^78 - 2 * q^79 + 74 * q^80 + 118 * q^81 + 14 * q^82 + 4 * q^83 + 22 * q^84 + 26 * q^85 + 5 * q^86 + 24 * q^87 - 15 * q^88 + 143 * q^89 + 16 * q^90 + 66 * q^91 + 29 * q^92 - 8 * q^93 - 7 * q^94 - 10 * q^95 + 59 * q^96 + 41 * q^97 - 10 * q^98 - 63 * 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 $

$2$	−2.76981	−1.95855	−0.979274	−	0.202540i	$-0.935080\pi$
$2$	−2.76981	−1.95855	−0.979274	+	0.202540i	$0.935080\pi$
$3$	1.77605	1.02541	0.512703	−	0.858566i	$-0.328644\pi$
$3$	1.77605	1.02541	0.512703	+	0.858566i	$0.328644\pi$
$4$	5.67182	2.83591
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−4.91932	−2.00831
$7$	−0.953960	−0.360563	−0.180281	−	0.983615i	$-0.557701\pi$
$7$	−0.953960	−0.360563	−0.180281	+	0.983615i	$0.557701\pi$
$8$	−10.1702	−3.59572
$9$	0.154367	0.0514558
$10$	−2.76981	−0.875889
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	10.0735	2.90796
$13$	−6.96950	−1.93299	−0.966496	−	0.256683i	$-0.917370\pi$
$13$	−6.96950	−1.93299	−0.966496	+	0.256683i	$0.917370\pi$
$14$	2.64228	0.706180
$15$	1.77605	0.458575
$16$	16.8259	4.20648
$17$	−1.81899	−0.441169	−0.220584	−	0.975368i	$-0.570796\pi$
$17$	−1.81899	−0.441169	−0.220584	+	0.975368i	$0.570796\pi$
$18$	−0.427568	−0.100779
$19$	−0.165649	−0.0380025	−0.0190013	−	0.999819i	$-0.506049\pi$
$19$	−0.165649	−0.0380025	−0.0190013	+	0.999819i	$0.506049\pi$
$20$	5.67182	1.26826
$21$	−1.69428	−0.369723
$22$	2.76981	0.590524
$23$	1.66553	0.347286	0.173643	−	0.984809i	$-0.444446\pi$
$23$	1.66553	0.347286	0.173643	+	0.984809i	$0.444446\pi$
$24$	−18.0629	−3.68707
$25$	1.00000	0.200000
$26$	19.3042	3.78586
$27$	−5.05400	−0.972642
$28$	−5.41069	−1.02252
$29$	7.43543	1.38072	0.690362	−	0.723464i	$-0.257451\pi$
$29$	7.43543	1.38072	0.690362	+	0.723464i	$0.257451\pi$
$30$	−4.91932	−0.898141
$31$	5.38941	0.967967	0.483983	−	0.875077i	$-0.339189\pi$
$31$	5.38941	0.967967	0.483983	+	0.875077i	$0.339189\pi$
$32$	−26.2640	−4.64287
$33$	−1.77605	−0.309171
$34$	5.03824	0.864050
$35$	−0.953960	−0.161249
$36$	0.875544	0.145924
$37$	−1.22357	−0.201154	−0.100577	−	0.994929i	$-0.532069\pi$
$37$	−1.22357	−0.201154	−0.100577	+	0.994929i	$0.532069\pi$
$38$	0.458816	0.0744298
$39$	−12.3782	−1.98210
$40$	−10.1702	−1.60805
$41$	9.23166	1.44174	0.720872	−	0.693068i	$-0.243742\pi$
$41$	9.23166	1.44174	0.720872	+	0.693068i	$0.243742\pi$
$42$	4.69284	0.724120
$43$	7.53958	1.14977	0.574887	−	0.818233i	$-0.305046\pi$
$43$	7.53958	1.14977	0.574887	+	0.818233i	$0.305046\pi$
$44$	−5.67182	−0.855059
$45$	0.154367	0.0230117
$46$	−4.61318	−0.680177
$47$	3.29987	0.481335	0.240668	−	0.970608i	$-0.422634\pi$
$47$	3.29987	0.481335	0.240668	+	0.970608i	$0.422634\pi$
$48$	29.8837	4.31334
$49$	−6.08996	−0.869994
$50$	−2.76981	−0.391710
$51$	−3.23062	−0.452377
$52$	−39.5297	−5.48179
$53$	−2.58351	−0.354872	−0.177436	−	0.984132i	$-0.556780\pi$
$53$	−2.58351	−0.354872	−0.177436	+	0.984132i	$0.556780\pi$
$54$	13.9986	1.90497
$55$	−1.00000	−0.134840
$56$	9.70198	1.29648
$57$	−0.294202	−0.0389680
$58$	−20.5947	−2.70421
$59$	2.86589	0.373107	0.186554	−	0.982445i	$-0.440268\pi$
$59$	2.86589	0.373107	0.186554	+	0.982445i	$0.440268\pi$
$60$	10.0735	1.30048
$61$	−12.3747	−1.58442	−0.792211	−	0.610247i	$-0.791070\pi$
$61$	−12.3747	−1.58442	−0.792211	+	0.610247i	$0.791070\pi$
$62$	−14.9276	−1.89581
$63$	−0.147260	−0.0185531
$64$	39.0944	4.88680
$65$	−6.96950	−0.864460
$66$	4.91932	0.605527
$67$	5.28055	0.645122	0.322561	−	0.946549i	$-0.395456\pi$
$67$	5.28055	0.645122	0.322561	+	0.946549i	$0.395456\pi$
$68$	−10.3170	−1.25112
$69$	2.95807	0.356109
$70$	2.64228	0.315813
$71$	−2.20096	−0.261206	−0.130603	−	0.991435i	$-0.541691\pi$
$71$	−2.20096	−0.261206	−0.130603	+	0.991435i	$0.541691\pi$
$72$	−1.56995	−0.185021
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	3.38905	0.393970
$75$	1.77605	0.205081
$76$	−0.939532	−0.107772
$77$	0.953960	0.108714
$78$	34.2852	3.88204
$79$	−7.82311	−0.880169	−0.440085	−	0.897956i	$-0.645052\pi$
$79$	−7.82311	−0.880169	−0.440085	+	0.897956i	$0.645052\pi$
$80$	16.8259	1.88119
$81$	−9.43927	−1.04881
$82$	−25.5699	−2.82372
$83$	11.7340	1.28797	0.643985	−	0.765038i	$-0.277280\pi$
$83$	11.7340	1.28797	0.643985	+	0.765038i	$0.277280\pi$
$84$	−9.60967	−1.04850
$85$	−1.81899	−0.197297
$86$	−20.8832	−2.25189
$87$	13.2057	1.41580
$88$	10.1702	1.08415
$89$	9.62676	1.02043	0.510217	−	0.860045i	$-0.329565\pi$
$89$	9.62676	1.02043	0.510217	+	0.860045i	$0.329565\pi$
$90$	−0.427568	−0.0450696
$91$	6.64862	0.696965
$92$	9.44657	0.984873
$93$	9.57188	0.992558
$94$	−9.14000	−0.942719
$95$	−0.165649	−0.0169952
$96$	−46.6463	−4.76082
$97$	−3.04958	−0.309637	−0.154819	−	0.987943i	$-0.549479\pi$
$97$	−3.04958	−0.309637	−0.154819	+	0.987943i	$0.549479\pi$
$98$	16.8680	1.70393
$99$	−0.154367	−0.0155145
$100$	5.67182	0.567182
$101$	9.14251	0.909713	0.454857	−	0.890565i	$-0.349690\pi$
$101$	9.14251	0.909713	0.454857	+	0.890565i	$0.349690\pi$
$102$	8.94818	0.886002
$103$	14.9200	1.47012	0.735058	−	0.678004i	$-0.237155\pi$
$103$	14.9200	1.47012	0.735058	+	0.678004i	$0.237155\pi$
$104$	70.8814	6.95049
$105$	−1.69428	−0.165345
$106$	7.15582	0.695034
$107$	11.6337	1.12467	0.562336	−	0.826909i	$-0.309903\pi$
$107$	11.6337	1.12467	0.562336	+	0.826909i	$0.309903\pi$
$108$	−28.6654	−2.75833
$109$	17.8557	1.71026	0.855132	−	0.518411i	$-0.173476\pi$
$109$	17.8557	1.71026	0.855132	+	0.518411i	$0.173476\pi$
$110$	2.76981	0.264091
$111$	−2.17313	−0.206264
$112$	−16.0512	−1.51670
$113$	−8.33295	−0.783898	−0.391949	−	0.919987i	$-0.628199\pi$
$113$	−8.33295	−0.783898	−0.391949	+	0.919987i	$0.628199\pi$
$114$	0.814882	0.0763207
$115$	1.66553	0.155311
$116$	42.1724	3.91561
$117$	−1.07586	−0.0994636
$118$	−7.93796	−0.730748
$119$	1.73524	0.159069
$120$	−18.0629	−1.64891
$121$	1.00000	0.0909091
$122$	34.2756	3.10317
$123$	16.3959	1.47837
$124$	30.5678	2.74507
$125$	1.00000	0.0894427
$126$	0.407882	0.0363370
$127$	−4.24449	−0.376637	−0.188319	−	0.982108i	$-0.560304\pi$
$127$	−4.24449	−0.376637	−0.188319	+	0.982108i	$0.560304\pi$
$128$	−55.7558	−4.92817
$129$	13.3907	1.17898
$130$	19.3042	1.69309
$131$	−9.33467	−0.815574	−0.407787	−	0.913077i	$-0.633699\pi$
$131$	−9.33467	−0.815574	−0.407787	+	0.913077i	$0.633699\pi$
$132$	−10.0735	−0.876782
$133$	0.158023	0.0137023
$134$	−14.6261	−1.26350
$135$	−5.05400	−0.434979
$136$	18.4995	1.58632
$137$	18.8825	1.61324	0.806622	−	0.591068i	$-0.201293\pi$
$137$	18.8825	1.61324	0.806622	+	0.591068i	$0.201293\pi$
$138$	−8.19327	−0.697457
$139$	5.13835	0.435829	0.217915	−	0.975968i	$-0.430075\pi$
$139$	5.13835	0.435829	0.217915	+	0.975968i	$0.430075\pi$
$140$	−5.41069	−0.457287
$141$	5.86075	0.493564
$142$	6.09624	0.511585
$143$	6.96950	0.582819
$144$	2.59737	0.216448
$145$	7.43543	0.617478
$146$	2.76981	0.229231
$147$	−10.8161	−0.892097
$148$	−6.93988	−0.570454
$149$	1.45256	0.118999	0.0594993	−	0.998228i	$-0.481050\pi$
$149$	1.45256	0.118999	0.0594993	+	0.998228i	$0.481050\pi$
$150$	−4.91932	−0.401661
$151$	−1.83795	−0.149571	−0.0747853	−	0.997200i	$-0.523827\pi$
$151$	−1.83795	−0.149571	−0.0747853	+	0.997200i	$0.523827\pi$
$152$	1.68469	0.136646
$153$	−0.280792	−0.0227007
$154$	−2.64228	−0.212921
$155$	5.38941	0.432888
$156$	−70.2070	−5.62105
$157$	−7.92583	−0.632550	−0.316275	−	0.948668i	$-0.602432\pi$
$157$	−7.92583	−0.632550	−0.316275	+	0.948668i	$0.602432\pi$
$158$	21.6685	1.72385
$159$	−4.58845	−0.363888
$160$	−26.2640	−2.07635
$161$	−1.58885	−0.125219
$162$	26.1449	2.05414
$163$	−1.20536	−0.0944112	−0.0472056	−	0.998885i	$-0.515032\pi$
$163$	−1.20536	−0.0944112	−0.0472056	+	0.998885i	$0.515032\pi$
$164$	52.3603	4.08865
$165$	−1.77605	−0.138266
$166$	−32.5008	−2.52255
$167$	13.5710	1.05016	0.525078	−	0.851054i	$-0.324036\pi$
$167$	13.5710	1.05016	0.525078	+	0.851054i	$0.324036\pi$
$168$	17.2312	1.32942
$169$	35.5739	2.73646
$170$	5.03824	0.386415
$171$	−0.0255708	−0.00195545
$172$	42.7631	3.26066
$173$	13.7258	1.04355	0.521775	−	0.853083i	$-0.325270\pi$
$173$	13.7258	1.04355	0.521775	+	0.853083i	$0.325270\pi$
$174$	−36.5773	−2.77291
$175$	−0.953960	−0.0721126
$176$	−16.8259	−1.26830
$177$	5.08998	0.382586
$178$	−26.6643	−1.99857
$179$	15.5325	1.16095	0.580477	−	0.814277i	$-0.302866\pi$
$179$	15.5325	1.16095	0.580477	+	0.814277i	$0.302866\pi$
$180$	0.875544	0.0652592
$181$	−13.9041	−1.03349	−0.516743	−	0.856140i	$-0.672856\pi$
$181$	−13.9041	−1.03349	−0.516743	+	0.856140i	$0.672856\pi$
$182$	−18.4154	−1.36504
$183$	−21.9782	−1.62467
$184$	−16.9388	−1.24874
$185$	−1.22357	−0.0899588
$186$	−26.5123	−1.94397
$187$	1.81899	0.133017
$188$	18.7163	1.36502
$189$	4.82131	0.350699
$190$	0.458816	0.0332860
$191$	2.14835	0.155449	0.0777246	−	0.996975i	$-0.475235\pi$
$191$	2.14835	0.155449	0.0777246	+	0.996975i	$0.475235\pi$
$192$	69.4338	5.01095
$193$	−6.82172	−0.491038	−0.245519	−	0.969392i	$-0.578958\pi$
$193$	−6.82172	−0.491038	−0.245519	+	0.969392i	$0.578958\pi$
$194$	8.44673	0.606440
$195$	−12.3782	−0.886422
$196$	−34.5412	−2.46723
$197$	−18.2202	−1.29813	−0.649067	−	0.760732i	$-0.724840\pi$
$197$	−18.2202	−1.29813	−0.649067	+	0.760732i	$0.724840\pi$
$198$	0.427568	0.0303859
$199$	7.03098	0.498413	0.249207	−	0.968450i	$-0.419830\pi$
$199$	7.03098	0.498413	0.249207	+	0.968450i	$0.419830\pi$
$200$	−10.1702	−0.719143
$201$	9.37855	0.661512
$202$	−25.3230	−1.78172
$203$	−7.09310	−0.497838
$204$	−18.3235	−1.28290
$205$	9.23166	0.644767
$206$	−41.3256	−2.87929
$207$	0.257103	0.0178699
$208$	−117.268	−8.13108
$209$	0.165649	0.0114582
$210$	4.69284	0.323836
$211$	7.87317	0.542011	0.271006	−	0.962578i	$-0.412644\pi$
$211$	7.87317	0.542011	0.271006	+	0.962578i	$0.412644\pi$
$212$	−14.6532	−1.00639
$213$	−3.90903	−0.267842
$214$	−32.2230	−2.20272
$215$	7.53958	0.514195
$216$	51.4003	3.49735
$217$	−5.14128	−0.349013
$218$	−49.4567	−3.34963
$219$	−1.77605	−0.120015
$220$	−5.67182	−0.382394
$221$	12.6774	0.852775
$222$	6.01914	0.403979
$223$	−22.6290	−1.51535	−0.757674	−	0.652634i	$-0.773664\pi$
$223$	−22.6290	−1.51535	−0.757674	+	0.652634i	$0.773664\pi$
$224$	25.0548	1.67404
$225$	0.154367	0.0102912
$226$	23.0806	1.53530
$227$	16.3088	1.08246	0.541228	−	0.840876i	$-0.317960\pi$
$227$	16.3088	1.08246	0.541228	+	0.840876i	$0.317960\pi$
$228$	−1.66866	−0.110510
$229$	−1.50147	−0.0992200	−0.0496100	−	0.998769i	$-0.515798\pi$
$229$	−1.50147	−0.0992200	−0.0496100	+	0.998769i	$0.515798\pi$
$230$	−4.61318	−0.304184
$231$	1.69428	0.111476
$232$	−75.6199	−4.96469
$233$	9.52075	0.623725	0.311862	−	0.950127i	$-0.399047\pi$
$233$	9.52075	0.623725	0.311862	+	0.950127i	$0.399047\pi$
$234$	2.97993	0.194804
$235$	3.29987	0.215260
$236$	16.2548	1.05810
$237$	−13.8943	−0.902530
$238$	−4.80627	−0.311544
$239$	8.33709	0.539281	0.269641	−	0.962961i	$-0.413095\pi$
$239$	8.33709	0.539281	0.269641	+	0.962961i	$0.413095\pi$
$240$	29.8837	1.92898
$241$	20.7267	1.33513	0.667563	−	0.744553i	$-0.267337\pi$
$241$	20.7267	1.33513	0.667563	+	0.744553i	$0.267337\pi$
$242$	−2.76981	−0.178050
$243$	−1.60267	−0.102811
$244$	−70.1872	−4.49328
$245$	−6.08996	−0.389073
$246$	−45.4135	−2.89546
$247$	1.15449	0.0734585
$248$	−54.8115	−3.48054
$249$	20.8402	1.32069
$250$	−2.76981	−0.175178
$251$	20.5145	1.29486	0.647431	−	0.762124i	$-0.275843\pi$
$251$	20.5145	1.29486	0.647431	+	0.762124i	$0.275843\pi$
$252$	−0.835234	−0.0526148
$253$	−1.66553	−0.104711
$254$	11.7564	0.737662
$255$	−3.23062	−0.202309
$256$	76.2440	4.76525
$257$	−26.8600	−1.67548	−0.837740	−	0.546069i	$-0.816123\pi$
$257$	−26.8600	−1.67548	−0.837740	+	0.546069i	$0.816123\pi$
$258$	−37.0896	−2.30910
$259$	1.16724	0.0725286
$260$	−39.5297	−2.45153
$261$	1.14779	0.0710463
$262$	25.8552	1.59734
$263$	−6.88182	−0.424351	−0.212176	−	0.977232i	$-0.568055\pi$
$263$	−6.88182	−0.424351	−0.212176	+	0.977232i	$0.568055\pi$
$264$	18.0629	1.11169
$265$	−2.58351	−0.158704
$266$	−0.437692	−0.0268366
$267$	17.0976	1.04636
$268$	29.9504	1.82951
$269$	18.0149	1.09839	0.549193	−	0.835695i	$-0.314935\pi$
$269$	18.0149	1.09839	0.549193	+	0.835695i	$0.314935\pi$
$270$	13.9986	0.851927
$271$	10.1009	0.613588	0.306794	−	0.951776i	$-0.400744\pi$
$271$	10.1009	0.613588	0.306794	+	0.951776i	$0.400744\pi$
$272$	−30.6061	−1.85577
$273$	11.8083	0.714671
$274$	−52.3009	−3.15961
$275$	−1.00000	−0.0603023
$276$	16.7776	1.00989
$277$	32.5431	1.95533	0.977663	−	0.210179i	$-0.0674048\pi$
$277$	32.5431	1.95533	0.977663	+	0.210179i	$0.0674048\pi$
$278$	−14.2322	−0.853593
$279$	0.831950	0.0498075
$280$	9.70198	0.579804
$281$	−30.3835	−1.81253	−0.906265	−	0.422711i	$-0.861079\pi$
$281$	−30.3835	−1.81253	−0.906265	+	0.422711i	$0.861079\pi$
$282$	−16.2331	−0.966669
$283$	−7.98040	−0.474386	−0.237193	−	0.971463i	$-0.576227\pi$
$283$	−7.98040	−0.474386	−0.237193	+	0.971463i	$0.576227\pi$
$284$	−12.4835	−0.740758
$285$	−0.294202	−0.0174270
$286$	−19.3042	−1.14148
$287$	−8.80663	−0.519839
$288$	−4.05431	−0.238902
$289$	−13.6913	−0.805370
$290$	−20.5947	−1.20936
$291$	−5.41621	−0.317504
$292$	−5.67182	−0.331918
$293$	2.14074	0.125063	0.0625315	−	0.998043i	$-0.480083\pi$
$293$	2.14074	0.125063	0.0625315	+	0.998043i	$0.480083\pi$
$294$	29.9585	1.74721
$295$	2.86589	0.166859
$296$	12.4440	0.723293
$297$	5.05400	0.293263
$298$	−4.02332	−0.233064
$299$	−11.6079	−0.671302
$300$	10.0735	0.581591
$301$	−7.19245	−0.414566
$302$	5.09077	0.292941
$303$	16.2376	0.932825
$304$	−2.78720	−0.159857
$305$	−12.3747	−0.708575
$306$	0.777740	0.0444604
$307$	−2.84047	−0.162114	−0.0810572	−	0.996709i	$-0.525830\pi$
$307$	−2.84047	−0.162114	−0.0810572	+	0.996709i	$0.525830\pi$
$308$	5.41069	0.308303
$309$	26.4988	1.50746
$310$	−14.9276	−0.847832
$311$	−2.93589	−0.166479	−0.0832394	−	0.996530i	$-0.526527\pi$
$311$	−2.93589	−0.166479	−0.0832394	+	0.996530i	$0.526527\pi$
$312$	125.889	7.12707
$313$	1.88084	0.106311	0.0531556	−	0.998586i	$-0.483072\pi$
$313$	1.88084	0.106311	0.0531556	+	0.998586i	$0.483072\pi$
$314$	21.9530	1.23888
$315$	−0.147260	−0.00829718
$316$	−44.3713	−2.49608
$317$	−19.3838	−1.08870	−0.544351	−	0.838857i	$-0.683224\pi$
$317$	−19.3838	−1.08870	−0.544351	+	0.838857i	$0.683224\pi$
$318$	12.7091	0.712692
$319$	−7.43543	−0.416304
$320$	39.0944	2.18544
$321$	20.6621	1.15324
$322$	4.40079	0.245247
$323$	0.301313	0.0167655
$324$	−53.5379	−2.97433
$325$	−6.96950	−0.386598
$326$	3.33862	0.184909
$327$	31.7126	1.75371
$328$	−93.8881	−5.18410
$329$	−3.14794	−0.173552
$330$	4.91932	0.270800
$331$	−24.4717	−1.34509	−0.672543	−	0.740058i	$-0.734798\pi$
$331$	−24.4717	−1.34509	−0.672543	+	0.740058i	$0.734798\pi$
$332$	66.5530	3.65257
$333$	−0.188880	−0.0103505
$334$	−37.5890	−2.05678
$335$	5.28055	0.288508
$336$	−28.5079	−1.55523
$337$	−18.9485	−1.03219	−0.516096	−	0.856531i	$-0.672615\pi$
$337$	−18.9485	−1.03219	−0.516096	+	0.856531i	$0.672615\pi$
$338$	−98.5328	−5.35948
$339$	−14.7998	−0.803813
$340$	−10.3170	−0.559516
$341$	−5.38941	−0.291853
$342$	0.0708262	0.00382984
$343$	12.4873	0.674251
$344$	−76.6792	−4.13426
$345$	2.95807	0.159257
$346$	−38.0177	−2.04384
$347$	31.5903	1.69585	0.847927	−	0.530113i	$-0.177851\pi$
$347$	31.5903	1.69585	0.847927	+	0.530113i	$0.177851\pi$
$348$	74.9004	4.01509
$349$	7.31864	0.391758	0.195879	−	0.980628i	$-0.437244\pi$
$349$	7.31864	0.391758	0.195879	+	0.980628i	$0.437244\pi$
$350$	2.64228	0.141236
$351$	35.2238	1.88011
$352$	26.2640	1.39988
$353$	7.32179	0.389699	0.194850	−	0.980833i	$-0.437578\pi$
$353$	7.32179	0.389699	0.194850	+	0.980833i	$0.437578\pi$
$354$	−14.0982	−0.749313
$355$	−2.20096	−0.116815
$356$	54.6013	2.89386
$357$	3.08188	0.163110
$358$	−43.0220	−2.27378
$359$	18.1638	0.958650	0.479325	−	0.877637i	$-0.340881\pi$
$359$	18.1638	0.958650	0.479325	+	0.877637i	$0.340881\pi$
$360$	−1.56995	−0.0827437
$361$	−18.9726	−0.998556
$362$	38.5118	2.02413
$363$	1.77605	0.0932187
$364$	37.7098	1.97653
$365$	−1.00000	−0.0523424
$366$	60.8753	3.18200
$367$	−19.2436	−1.00451	−0.502255	−	0.864720i	$-0.667496\pi$
$367$	−19.2436	−1.00451	−0.502255	+	0.864720i	$0.667496\pi$
$368$	28.0240	1.46085
$369$	1.42507	0.0741861
$370$	3.38905	0.176189
$371$	2.46456	0.127954
$372$	54.2900	2.81481
$373$	−17.5709	−0.909785	−0.454893	−	0.890546i	$-0.650322\pi$
$373$	−17.5709	−0.909785	−0.454893	+	0.890546i	$0.650322\pi$
$374$	−5.03824	−0.260521
$375$	1.77605	0.0917150
$376$	−33.5604	−1.73075
$377$	−51.8212	−2.66893
$378$	−13.3541	−0.686860
$379$	12.6295	0.648732	0.324366	−	0.945932i	$-0.394849\pi$
$379$	12.6295	0.648732	0.324366	+	0.945932i	$0.394849\pi$
$380$	−0.939532	−0.0481970
$381$	−7.53844	−0.386206
$382$	−5.95051	−0.304455
$383$	32.0514	1.63775	0.818875	−	0.573972i	$-0.194598\pi$
$383$	32.0514	1.63775	0.818875	+	0.573972i	$0.194598\pi$
$384$	−99.0253	−5.05337
$385$	0.953960	0.0486183
$386$	18.8948	0.961722
$387$	1.16387	0.0591626
$388$	−17.2966	−0.878104
$389$	0.938915	0.0476049	0.0238024	−	0.999717i	$-0.492423\pi$
$389$	0.938915	0.0476049	0.0238024	+	0.999717i	$0.492423\pi$
$390$	34.2852	1.73610
$391$	−3.02957	−0.153212
$392$	61.9363	3.12825
$393$	−16.5789	−0.836294
$394$	50.4663	2.54246
$395$	−7.82311	−0.393624
$396$	−0.875544	−0.0439978
$397$	25.5429	1.28196	0.640981	−	0.767557i	$-0.278528\pi$
$397$	25.5429	1.28196	0.640981	+	0.767557i	$0.278528\pi$
$398$	−19.4745	−0.976166
$399$	0.280657	0.0140504
$400$	16.8259	0.841295
$401$	13.7192	0.685103	0.342551	−	0.939499i	$-0.388709\pi$
$401$	13.7192	0.685103	0.342551	+	0.939499i	$0.388709\pi$
$402$	−25.9768	−1.29560
$403$	−37.5615	−1.87107
$404$	51.8546	2.57986
$405$	−9.43927	−0.469041
$406$	19.6465	0.975039
$407$	1.22357	0.0606502
$408$	32.8561	1.62662
$409$	−11.3245	−0.559961	−0.279981	−	0.960006i	$-0.590328\pi$
$409$	−11.3245	−0.559961	−0.279981	+	0.960006i	$0.590328\pi$
$410$	−25.5699	−1.26281
$411$	33.5364	1.65423
$412$	84.6238	4.16912
$413$	−2.73394	−0.134529
$414$	−0.712126	−0.0349991
$415$	11.7340	0.575998
$416$	183.047	8.97462
$417$	9.12599	0.446902
$418$	−0.458816	−0.0224414
$419$	−37.1730	−1.81602	−0.908010	−	0.418948i	$-0.862399\pi$
$419$	−37.1730	−1.81602	−0.908010	+	0.418948i	$0.862399\pi$
$420$	−9.60967	−0.468904
$421$	36.3409	1.77115	0.885574	−	0.464498i	$-0.153765\pi$
$421$	36.3409	1.77115	0.885574	+	0.464498i	$0.153765\pi$
$422$	−21.8071	−1.06155
$423$	0.509392	0.0247675
$424$	26.2749	1.27602
$425$	−1.81899	−0.0882338
$426$	10.8273	0.524582
$427$	11.8050	0.571284
$428$	65.9842	3.18947
$429$	12.3782	0.597625
$430$	−20.8832	−1.00708
$431$	0.593741	0.0285995	0.0142998	−	0.999898i	$-0.495448\pi$
$431$	0.593741	0.0285995	0.0142998	+	0.999898i	$0.495448\pi$
$432$	−85.0380	−4.09139
$433$	8.33654	0.400628	0.200314	−	0.979732i	$-0.435804\pi$
$433$	8.33654	0.400628	0.200314	+	0.979732i	$0.435804\pi$
$434$	14.2403	0.683558
$435$	13.2057	0.633166
$436$	101.274	4.85015
$437$	−0.275893	−0.0131978
$438$	4.91932	0.235054
$439$	−2.16774	−0.103460	−0.0517302	−	0.998661i	$-0.516474\pi$
$439$	−2.16774	−0.103460	−0.0517302	+	0.998661i	$0.516474\pi$
$440$	10.1702	0.484846
$441$	−0.940092	−0.0447663
$442$	−35.1140	−1.67020
$443$	−22.0261	−1.04649	−0.523246	−	0.852182i	$-0.675279\pi$
$443$	−22.0261	−1.04649	−0.523246	+	0.852182i	$0.675279\pi$
$444$	−12.3256	−0.584947
$445$	9.62676	0.456352
$446$	62.6778	2.96788
$447$	2.57983	0.122022
$448$	−37.2945	−1.76200
$449$	40.4560	1.90924	0.954618	−	0.297834i	$-0.0962643\pi$
$449$	40.4560	1.90924	0.954618	+	0.297834i	$0.0962643\pi$
$450$	−0.427568	−0.0201557
$451$	−9.23166	−0.434702
$452$	−47.2630	−2.22306
$453$	−3.26431	−0.153370
$454$	−45.1723	−2.12004
$455$	6.64862	0.311692
$456$	2.99210	0.140118
$457$	12.8281	0.600071	0.300036	−	0.953928i	$-0.403001\pi$
$457$	12.8281	0.600071	0.300036	+	0.953928i	$0.403001\pi$
$458$	4.15878	0.194327
$459$	9.19315	0.429099
$460$	9.44657	0.440449
$461$	−36.5189	−1.70085	−0.850427	−	0.526093i	$-0.823656\pi$
$461$	−36.5189	−1.70085	−0.850427	+	0.526093i	$0.823656\pi$
$462$	−4.69284	−0.218330
$463$	29.0389	1.34955	0.674775	−	0.738023i	$-0.264240\pi$
$463$	29.0389	1.34955	0.674775	+	0.738023i	$0.264240\pi$
$464$	125.108	5.80798
$465$	9.57188	0.443886
$466$	−26.3706	−1.22159
$467$	5.45507	0.252430	0.126215	−	0.992003i	$-0.459717\pi$
$467$	5.45507	0.252430	0.126215	+	0.992003i	$0.459717\pi$
$468$	−6.10211	−0.282070
$469$	−5.03744	−0.232607
$470$	−9.14000	−0.421597
$471$	−14.0767	−0.648620
$472$	−29.1467	−1.34159
$473$	−7.53958	−0.346670
$474$	38.4844	1.76765
$475$	−0.165649	−0.00760050
$476$	9.84196	0.451106
$477$	−0.398810	−0.0182602
$478$	−23.0921	−1.05621
$479$	−9.75885	−0.445893	−0.222947	−	0.974831i	$-0.571568\pi$
$479$	−9.75885	−0.445893	−0.222947	+	0.974831i	$0.571568\pi$
$480$	−46.6463	−2.12910
$481$	8.52768	0.388829
$482$	−57.4090	−2.61491
$483$	−2.82188	−0.128400
$484$	5.67182	0.257810
$485$	−3.04958	−0.138474
$486$	4.43908	0.201361
$487$	14.4693	0.655667	0.327833	−	0.944736i	$-0.393682\pi$
$487$	14.4693	0.655667	0.327833	+	0.944736i	$0.393682\pi$
$488$	125.854	5.69713
$489$	−2.14079	−0.0968097
$490$	16.8680	0.762019
$491$	40.4375	1.82492	0.912459	−	0.409168i	$-0.134181\pi$
$491$	40.4375	1.82492	0.912459	+	0.409168i	$0.134181\pi$
$492$	92.9948	4.19253
$493$	−13.5249	−0.609132
$494$	−3.19772	−0.143872
$495$	−0.154367	−0.00693830
$496$	90.6817	4.07173
$497$	2.09963	0.0941813
$498$	−57.7232	−2.58664
$499$	−2.35698	−0.105513	−0.0527565	−	0.998607i	$-0.516801\pi$
$499$	−2.35698	−0.105513	−0.0527565	+	0.998607i	$0.516801\pi$
$500$	5.67182	0.253651
$501$	24.1028	1.07683
$502$	−56.8211	−2.53605
$503$	15.7284	0.701293	0.350646	−	0.936508i	$-0.385962\pi$
$503$	15.7284	0.701293	0.350646	+	0.936508i	$0.385962\pi$
$504$	1.49767	0.0667115
$505$	9.14251	0.406836
$506$	4.61318	0.205081
$507$	63.1812	2.80598
$508$	−24.0740	−1.06811
$509$	9.95434	0.441218	0.220609	−	0.975362i	$-0.429195\pi$
$509$	9.95434	0.441218	0.220609	+	0.975362i	$0.429195\pi$
$510$	8.94818	0.396232
$511$	0.953960	0.0422007
$512$	−99.6693	−4.40480
$513$	0.837190	0.0369629
$514$	74.3969	3.28151
$515$	14.9200	0.657456
$516$	75.9496	3.34350
$517$	−3.29987	−0.145128
$518$	−3.23302	−0.142051
$519$	24.3777	1.07006
$520$	70.8814	3.10835
$521$	−22.5434	−0.987645	−0.493822	−	0.869563i	$-0.664401\pi$
$521$	−22.5434	−0.987645	−0.493822	+	0.869563i	$0.664401\pi$
$522$	−3.17915	−0.139148
$523$	5.77953	0.252721	0.126361	−	0.991984i	$-0.459670\pi$
$523$	5.77953	0.252721	0.126361	+	0.991984i	$0.459670\pi$
$524$	−52.9446	−2.31289
$525$	−1.69428	−0.0739446
$526$	19.0613	0.831112
$527$	−9.80326	−0.427037
$528$	−29.8837	−1.30052
$529$	−20.2260	−0.879392
$530$	7.15582	0.310829
$531$	0.442400	0.0191985
$532$	0.896276	0.0388585
$533$	−64.3401	−2.78688
$534$	−47.3572	−2.04934
$535$	11.6337	0.502968
$536$	−53.7044	−2.31968
$537$	27.5866	1.19045
$538$	−49.8977	−2.15124
$539$	6.08996	0.262313
$540$	−28.6654	−1.23356
$541$	0.905873	0.0389465	0.0194733	−	0.999810i	$-0.493801\pi$
$541$	0.905873	0.0389465	0.0194733	+	0.999810i	$0.493801\pi$
$542$	−27.9776	−1.20174
$543$	−24.6945	−1.05974
$544$	47.7739	2.04829
$545$	17.8557	0.764853
$546$	−32.7067	−1.39972
$547$	−5.08035	−0.217220	−0.108610	−	0.994084i	$-0.534640\pi$
$547$	−5.08035	−0.217220	−0.108610	+	0.994084i	$0.534640\pi$
$548$	107.098	4.57501
$549$	−1.91026	−0.0815277
$550$	2.76981	0.118105
$551$	−1.23167	−0.0524710
$552$	−30.0842	−1.28047
$553$	7.46294	0.317356
$554$	−90.1381	−3.82960
$555$	−2.17313	−0.0922442
$556$	29.1438	1.23597
$557$	−7.79717	−0.330377	−0.165188	−	0.986262i	$-0.552823\pi$
$557$	−7.79717	−0.330377	−0.165188	+	0.986262i	$0.552823\pi$
$558$	−2.30434	−0.0975504
$559$	−52.5471	−2.22250
$560$	−16.0512	−0.678288
$561$	3.23062	0.136397
$562$	84.1564	3.54993
$563$	−38.9853	−1.64303	−0.821517	−	0.570185i	$-0.806872\pi$
$563$	−38.9853	−1.64303	−0.821517	+	0.570185i	$0.806872\pi$
$564$	33.2411	1.39970
$565$	−8.33295	−0.350570
$566$	22.1042	0.929107
$567$	9.00469	0.378161
$568$	22.3843	0.939224
$569$	−2.65245	−0.111196	−0.0555982	−	0.998453i	$-0.517707\pi$
$569$	−2.65245	−0.111196	−0.0555982	+	0.998453i	$0.517707\pi$
$570$	0.814882	0.0341316
$571$	−7.50977	−0.314274	−0.157137	−	0.987577i	$-0.550226\pi$
$571$	−7.50977	−0.314274	−0.157137	+	0.987577i	$0.550226\pi$
$572$	39.5297	1.65282
$573$	3.81559	0.159398
$574$	24.3927	1.01813
$575$	1.66553	0.0694573
$576$	6.03490	0.251454
$577$	32.9784	1.37291	0.686455	−	0.727172i	$-0.259166\pi$
$577$	32.9784	1.37291	0.686455	+	0.727172i	$0.259166\pi$
$578$	37.9222	1.57736
$579$	−12.1157	−0.503513
$580$	42.1724	1.75111
$581$	−11.1937	−0.464394
$582$	15.0018	0.621847
$583$	2.58351	0.106998
$584$	10.1702	0.420847
$585$	−1.07586	−0.0444815
$586$	−5.92942	−0.244942
$587$	22.2961	0.920257	0.460128	−	0.887852i	$-0.347803\pi$
$587$	22.2961	0.920257	0.460128	+	0.887852i	$0.347803\pi$
$588$	−61.3470	−2.52991
$589$	−0.892751	−0.0367852
$590$	−7.93796	−0.326801
$591$	−32.3600	−1.33111
$592$	−20.5877	−0.846149
$593$	43.3936	1.78196	0.890981	−	0.454040i	$-0.150018\pi$
$593$	43.3936	1.78196	0.890981	+	0.454040i	$0.150018\pi$
$594$	−13.9986	−0.574369
$595$	1.73524	0.0711379
$596$	8.23868	0.337469
$597$	12.4874	0.511076
$598$	32.1516	1.31478
$599$	42.5052	1.73671	0.868357	−	0.495940i	$-0.165176\pi$
$599$	42.5052	1.73671	0.868357	+	0.495940i	$0.165176\pi$
$600$	−18.0629	−0.737413
$601$	−21.9341	−0.894710	−0.447355	−	0.894356i	$-0.647634\pi$
$601$	−21.9341	−0.894710	−0.447355	+	0.894356i	$0.647634\pi$
$602$	19.9217	0.811947
$603$	0.815146	0.0331953
$604$	−10.4245	−0.424169
$605$	1.00000	0.0406558
$606$	−44.9749	−1.82698
$607$	−23.2102	−0.942071	−0.471036	−	0.882114i	$-0.656120\pi$
$607$	−23.2102	−0.942071	−0.471036	+	0.882114i	$0.656120\pi$
$608$	4.35061	0.176441
$609$	−12.5977	−0.510485
$610$	34.2756	1.38778
$611$	−22.9984	−0.930417
$612$	−1.59260	−0.0643771
$613$	−25.7894	−1.04162	−0.520812	−	0.853672i	$-0.674371\pi$
$613$	−25.7894	−1.04162	−0.520812	+	0.853672i	$0.674371\pi$
$614$	7.86755	0.317509
$615$	16.3959	0.661148
$616$	−9.70198	−0.390904
$617$	16.2962	0.656060	0.328030	−	0.944667i	$-0.393615\pi$
$617$	16.2962	0.656060	0.328030	+	0.944667i	$0.393615\pi$
$618$	−73.3965	−2.95244
$619$	−11.8460	−0.476130	−0.238065	−	0.971249i	$-0.576513\pi$
$619$	−11.8460	−0.476130	−0.238065	+	0.971249i	$0.576513\pi$
$620$	30.5678	1.22763
$621$	−8.41757	−0.337785
$622$	8.13183	0.326057
$623$	−9.18354	−0.367931
$624$	−208.274	−8.33765
$625$	1.00000	0.0400000
$626$	−5.20955	−0.208216
$627$	0.294202	0.0117493
$628$	−44.9539	−1.79386
$629$	2.22566	0.0887429
$630$	0.407882	0.0162504
$631$	6.41918	0.255544	0.127772	−	0.991804i	$-0.459217\pi$
$631$	6.41918	0.255544	0.127772	+	0.991804i	$0.459217\pi$
$632$	79.5628	3.16484
$633$	13.9832	0.555781
$634$	53.6893	2.13228
$635$	−4.24449	−0.168437
$636$	−26.0249	−1.03195
$637$	42.4440	1.68169
$638$	20.5947	0.815351
$639$	−0.339757	−0.0134406
$640$	−55.7558	−2.20394
$641$	9.26792	0.366061	0.183030	−	0.983107i	$-0.441409\pi$
$641$	9.26792	0.366061	0.183030	+	0.983107i	$0.441409\pi$
$642$	−57.2299	−2.25868
$643$	−28.3637	−1.11856	−0.559278	−	0.828980i	$-0.688921\pi$
$643$	−28.3637	−1.11856	−0.559278	+	0.828980i	$0.688921\pi$
$644$	−9.01165	−0.355109
$645$	13.3907	0.527258
$646$	−0.834580	−0.0328361
$647$	−30.2270	−1.18835	−0.594174	−	0.804337i	$-0.702521\pi$
$647$	−30.2270	−1.18835	−0.594174	+	0.804337i	$0.702521\pi$
$648$	95.9995	3.77122
$649$	−2.86589	−0.112496
$650$	19.3042	0.757171
$651$	−9.13119	−0.357880
$652$	−6.83659	−0.267742
$653$	−42.8144	−1.67546	−0.837729	−	0.546087i	$-0.816117\pi$
$653$	−42.8144	−1.67546	−0.837729	+	0.546087i	$0.816117\pi$
$654$	−87.8378	−3.43473
$655$	−9.33467	−0.364736
$656$	155.331	6.06466
$657$	−0.154367	−0.00602245
$658$	8.71919	0.339909
$659$	41.4404	1.61429	0.807144	−	0.590355i	$-0.201012\pi$
$659$	41.4404	1.61429	0.807144	+	0.590355i	$0.201012\pi$
$660$	−10.0735	−0.392109
$661$	31.8924	1.24047	0.620236	−	0.784415i	$-0.287037\pi$
$661$	31.8924	1.24047	0.620236	+	0.784415i	$0.287037\pi$
$662$	67.7818	2.63442
$663$	22.5158	0.874440
$664$	−119.337	−4.63118
$665$	0.158023	0.00612785
$666$	0.523160	0.0202720
$667$	12.3839	0.479507
$668$	76.9723	2.97815
$669$	−40.1902	−1.55384
$670$	−14.6261	−0.565056
$671$	12.3747	0.477721
$672$	44.4987	1.71657
$673$	−17.0052	−0.655504	−0.327752	−	0.944764i	$-0.606291\pi$
$673$	−17.0052	−0.655504	−0.327752	+	0.944764i	$0.606291\pi$
$674$	52.4837	2.02160
$675$	−5.05400	−0.194528
$676$	201.769	7.76034
$677$	13.3257	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$677$	13.3257	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$678$	40.9925	1.57431
$679$	2.90917	0.111644
$680$	18.4995	0.709423
$681$	28.9654	1.10996
$682$	14.9276	0.571608
$683$	−22.0655	−0.844314	−0.422157	−	0.906523i	$-0.638727\pi$
$683$	−22.0655	−0.844314	−0.422157	+	0.906523i	$0.638727\pi$
$684$	−0.145033	−0.00554548
$685$	18.8825	0.721464
$686$	−34.5874	−1.32055
$687$	−2.66669	−0.101741
$688$	126.860	4.83650
$689$	18.0058	0.685965
$690$	−8.19327	−0.311912
$691$	−32.1862	−1.22442	−0.612210	−	0.790695i	$-0.709719\pi$
$691$	−32.1862	−1.22442	−0.612210	+	0.790695i	$0.709719\pi$
$692$	77.8500	2.95941
$693$	0.147260	0.00559396
$694$	−87.4988	−3.32141
$695$	5.13835	0.194909
$696$	−134.305	−5.09082
$697$	−16.7923	−0.636052
$698$	−20.2712	−0.767276
$699$	16.9094	0.639571
$700$	−5.41069	−0.204505
$701$	−44.5572	−1.68290	−0.841451	−	0.540334i	$-0.818298\pi$
$701$	−44.5572	−1.68290	−0.841451	+	0.540334i	$0.818298\pi$
$702$	−97.5631	−3.68228
$703$	0.202684	0.00764436
$704$	−39.0944	−1.47343
$705$	5.86075	0.220728
$706$	−20.2799	−0.763245
$707$	−8.72158	−0.328009
$708$	28.8694	1.08498
$709$	−11.9405	−0.448434	−0.224217	−	0.974539i	$-0.571983\pi$
$709$	−11.9405	−0.448434	−0.224217	+	0.974539i	$0.571983\pi$
$710$	6.09624	0.228788
$711$	−1.20763	−0.0452898
$712$	−97.9063	−3.66919
$713$	8.97621	0.336162
$714$	−8.53620	−0.319459
$715$	6.96950	0.260644
$716$	88.0976	3.29236
$717$	14.8071	0.552982
$718$	−50.3103	−1.87756
$719$	−41.8443	−1.56053	−0.780264	−	0.625450i	$-0.784916\pi$
$719$	−41.8443	−1.56053	−0.780264	+	0.625450i	$0.784916\pi$
$720$	2.59737	0.0967983
$721$	−14.2331	−0.530069
$722$	52.5503	1.95572
$723$	36.8118	1.36905
$724$	−78.8618	−2.93088
$725$	7.43543	0.276145
$726$	−4.91932	−0.182573
$727$	−15.0599	−0.558542	−0.279271	−	0.960212i	$-0.590093\pi$
$727$	−15.0599	−0.558542	−0.279271	+	0.960212i	$0.590093\pi$
$728$	−67.6180	−2.50609
$729$	25.4714	0.943385
$730$	2.76981	0.102515
$731$	−13.7144	−0.507245
$732$	−124.656	−4.60743
$733$	20.7532	0.766538	0.383269	−	0.923637i	$-0.374798\pi$
$733$	20.7532	0.766538	0.383269	+	0.923637i	$0.374798\pi$
$734$	53.3011	1.96738
$735$	−10.8161	−0.398958
$736$	−43.7434	−1.61240
$737$	−5.28055	−0.194512
$738$	−3.94716	−0.145297
$739$	−29.8468	−1.09793	−0.548966	−	0.835845i	$-0.684978\pi$
$739$	−29.8468	−1.09793	−0.548966	+	0.835845i	$0.684978\pi$
$740$	−6.93988	−0.255115
$741$	2.05044	0.0753248
$742$	−6.82636	−0.250604
$743$	−24.1066	−0.884386	−0.442193	−	0.896920i	$-0.645799\pi$
$743$	−24.1066	−0.884386	−0.442193	+	0.896920i	$0.645799\pi$
$744$	−97.3482	−3.56896
$745$	1.45256	0.0532178
$746$	48.6679	1.78186
$747$	1.81134	0.0662736
$748$	10.3170	0.377225
$749$	−11.0981	−0.405515
$750$	−4.91932	−0.179628
$751$	−39.2560	−1.43247	−0.716236	−	0.697858i	$-0.754137\pi$
$751$	−39.2560	−1.43247	−0.716236	+	0.697858i	$0.754137\pi$
$752$	55.5233	2.02473
$753$	36.4348	1.32776
$754$	143.535	5.22722
$755$	−1.83795	−0.0668900
$756$	27.3456	0.994550
$757$	2.41397	0.0877372	0.0438686	−	0.999037i	$-0.486032\pi$
$757$	2.41397	0.0877372	0.0438686	+	0.999037i	$0.486032\pi$
$758$	−34.9812	−1.27057
$759$	−2.95807	−0.107371
$760$	1.68469	0.0611101
$761$	7.03785	0.255122	0.127561	−	0.991831i	$-0.459285\pi$
$761$	7.03785	0.255122	0.127561	+	0.991831i	$0.459285\pi$
$762$	20.8800	0.756403
$763$	−17.0336	−0.616658
$764$	12.1851	0.440840
$765$	−0.280792	−0.0101521
$766$	−88.7761	−3.20761
$767$	−19.9738	−0.721213
$768$	135.413	4.88631
$769$	−19.4061	−0.699803	−0.349902	−	0.936786i	$-0.613785\pi$
$769$	−19.4061	−0.699803	−0.349902	+	0.936786i	$0.613785\pi$
$770$	−2.64228	−0.0952212
$771$	−47.7048	−1.71805
$772$	−38.6916	−1.39254
$773$	−38.3340	−1.37878	−0.689389	−	0.724391i	$-0.742121\pi$
$773$	−38.3340	−1.37878	−0.689389	+	0.724391i	$0.742121\pi$
$774$	−3.22368	−0.115873
$775$	5.38941	0.193593
$776$	31.0149	1.11337
$777$	2.07308	0.0743712
$778$	−2.60061	−0.0932364
$779$	−1.52922	−0.0547899
$780$	−70.2070	−2.51381
$781$	2.20096	0.0787567
$782$	8.39132	0.300073
$783$	−37.5786	−1.34295
$784$	−102.469	−3.65961
$785$	−7.92583	−0.282885
$786$	45.9202	1.63792
$787$	35.2259	1.25567	0.627833	−	0.778348i	$-0.283942\pi$
$787$	35.2259	1.25567	0.627833	+	0.778348i	$0.283942\pi$
$788$	−103.341	−3.68139
$789$	−12.2225	−0.435132
$790$	21.6685	0.770931
$791$	7.94930	0.282645
$792$	1.56995	0.0557858
$793$	86.2457	3.06267
$794$	−70.7488	−2.51078
$795$	−4.58845	−0.162736
$796$	39.8785	1.41346
$797$	−9.47583	−0.335651	−0.167826	−	0.985817i	$-0.553675\pi$
$797$	−9.47583	−0.335651	−0.167826	+	0.985817i	$0.553675\pi$
$798$	−0.777364	−0.0275184
$799$	−6.00242	−0.212350
$800$	−26.2640	−0.928573
$801$	1.48606	0.0525073
$802$	−37.9994	−1.34181
$803$	1.00000	0.0352892
$804$	53.1934	1.87599
$805$	−1.58885	−0.0559994
$806$	104.038	3.66458
$807$	31.9954	1.12629
$808$	−92.9813	−3.27107
$809$	46.4509	1.63313	0.816564	−	0.577255i	$-0.195876\pi$
$809$	46.4509	1.63313	0.816564	+	0.577255i	$0.195876\pi$
$810$	26.1449	0.918640
$811$	−20.8152	−0.730921	−0.365460	−	0.930827i	$-0.619088\pi$
$811$	−20.8152	−0.730921	−0.365460	+	0.930827i	$0.619088\pi$
$812$	−40.2308	−1.41182
$813$	17.9398	0.629176
$814$	−3.38905	−0.118786
$815$	−1.20536	−0.0422220
$816$	−54.3580	−1.90291
$817$	−1.24892	−0.0436943
$818$	31.3667	1.09671
$819$	1.02633	0.0358629
$820$	52.3603	1.82850
$821$	−50.3496	−1.75721	−0.878607	−	0.477546i	$-0.841526\pi$
$821$	−50.3496	−1.75721	−0.878607	+	0.477546i	$0.841526\pi$
$822$	−92.8893	−3.23989
$823$	43.7357	1.52453	0.762265	−	0.647265i	$-0.224087\pi$
$823$	43.7357	1.52453	0.762265	+	0.647265i	$0.224087\pi$
$824$	−151.740	−5.28612
$825$	−1.77605	−0.0618343
$826$	7.57249	0.263481
$827$	48.3865	1.68256	0.841281	−	0.540597i	$-0.181802\pi$
$827$	48.3865	1.68256	0.841281	+	0.540597i	$0.181802\pi$
$828$	1.45824	0.0506774
$829$	16.8389	0.584839	0.292419	−	0.956290i	$-0.405540\pi$
$829$	16.8389	0.584839	0.292419	+	0.956290i	$0.405540\pi$
$830$	−32.5008	−1.12812
$831$	57.7983	2.00500
$832$	−272.468	−9.44614
$833$	11.0776	0.383814
$834$	−25.2772	−0.875279
$835$	13.5710	0.469644
$836$	0.939532	0.0324944
$837$	−27.2381	−0.941485
$838$	102.962	3.55676
$839$	−40.5520	−1.40001	−0.700006	−	0.714137i	$-0.746819\pi$
$839$	−40.5520	−1.40001	−0.700006	+	0.714137i	$0.746819\pi$
$840$	17.2312	0.594535
$841$	26.2855	0.906398
$842$	−100.657	−3.46888
$843$	−53.9628	−1.85858
$844$	44.6552	1.53709
$845$	35.5739	1.22378
$846$	−1.41092	−0.0485084
$847$	−0.953960	−0.0327784
$848$	−43.4699	−1.49276
$849$	−14.1736	−0.486437
$850$	5.03824	0.172810
$851$	−2.03789	−0.0698580
$852$	−22.1713	−0.759577
$853$	35.6658	1.22117	0.610587	−	0.791949i	$-0.290934\pi$
$853$	35.6658	1.22117	0.610587	+	0.791949i	$0.290934\pi$
$854$	−32.6975	−1.11889
$855$	−0.0255708	−0.000874504 0
$856$	−118.317	−4.04400
$857$	39.2273	1.33998	0.669990	−	0.742370i	$-0.266298\pi$
$857$	39.2273	1.33998	0.669990	+	0.742370i	$0.266298\pi$
$858$	−34.2852	−1.17048
$859$	35.9344	1.22607	0.613033	−	0.790057i	$-0.289949\pi$
$859$	35.9344	1.22607	0.613033	+	0.790057i	$0.289949\pi$
$860$	42.7631	1.45821
$861$	−15.6411	−0.533046
$862$	−1.64455	−0.0560135
$863$	−14.8460	−0.505364	−0.252682	−	0.967549i	$-0.581313\pi$
$863$	−14.8460	−0.505364	−0.252682	+	0.967549i	$0.581313\pi$
$864$	132.738	4.51585
$865$	13.7258	0.466690
$866$	−23.0906	−0.784650
$867$	−24.3165	−0.825831
$868$	−29.1604	−0.989769
$869$	7.82311	0.265381
$870$	−36.5773	−1.24009
$871$	−36.8028	−1.24702
$872$	−181.596	−6.14962
$873$	−0.470755	−0.0159326
$874$	0.764170	0.0258484
$875$	−0.953960	−0.0322497
$876$	−10.0735	−0.340351
$877$	−23.1514	−0.781766	−0.390883	−	0.920440i	$-0.627830\pi$
$877$	−23.1514	−0.781766	−0.390883	+	0.920440i	$0.627830\pi$
$878$	6.00421	0.202632
$879$	3.80206	0.128240
$880$	−16.8259	−0.567201
$881$	−15.6457	−0.527117	−0.263558	−	0.964643i	$-0.584896\pi$
$881$	−15.6457	−0.527117	−0.263558	+	0.964643i	$0.584896\pi$
$882$	2.60387	0.0876769
$883$	−42.6443	−1.43510	−0.717548	−	0.696509i	$-0.754736\pi$
$883$	−42.6443	−1.43510	−0.717548	+	0.696509i	$0.754736\pi$
$884$	71.9040	2.41839
$885$	5.08998	0.171098
$886$	61.0080	2.04961
$887$	29.7915	1.00030	0.500150	−	0.865939i	$-0.333278\pi$
$887$	29.7915	1.00030	0.500150	+	0.865939i	$0.333278\pi$
$888$	22.1012	0.741668
$889$	4.04907	0.135801
$890$	−26.6643	−0.893788
$891$	9.43927	0.316228
$892$	−128.347	−4.29739
$893$	−0.546621	−0.0182920
$894$	−7.14563	−0.238986
$895$	15.5325	0.519194
$896$	53.1888	1.77691
$897$	−20.6162	−0.688356
$898$	−112.055	−3.73933
$899$	40.0726	1.33649
$900$	0.875544	0.0291848
$901$	4.69937	0.156559
$902$	25.5699	0.851385
$903$	−12.7742	−0.425098
$904$	84.7480	2.81868
$905$	−13.9041	−0.462189
$906$	9.04149	0.300383
$907$	−20.9131	−0.694409	−0.347204	−	0.937790i	$-0.612869\pi$
$907$	−20.9131	−0.694409	−0.347204	+	0.937790i	$0.612869\pi$
$908$	92.5008	3.06975
$909$	1.41131	0.0468100
$910$	−18.4154	−0.610464
$911$	26.6537	0.883077	0.441539	−	0.897242i	$-0.354433\pi$
$911$	26.6537	0.883077	0.441539	+	0.897242i	$0.354433\pi$
$912$	−4.95021	−0.163918
$913$	−11.7340	−0.388338
$914$	−35.5312	−1.17527
$915$	−21.9782	−0.726576
$916$	−8.51607	−0.281379
$917$	8.90490	0.294066
$918$	−25.4632	−0.840412
$919$	53.0666	1.75051	0.875254	−	0.483664i	$-0.160694\pi$
$919$	53.0666	1.75051	0.875254	+	0.483664i	$0.160694\pi$
$920$	−16.9388	−0.558455
$921$	−5.04483	−0.166233
$922$	101.150	3.33120
$923$	15.3396	0.504910
$924$	9.60967	0.316135
$925$	−1.22357	−0.0402308
$926$	−80.4320	−2.64316
$927$	2.30317	0.0756460
$928$	−195.284	−6.41051
$929$	43.5299	1.42817	0.714085	−	0.700059i	$-0.246843\pi$
$929$	43.5299	1.42817	0.714085	+	0.700059i	$0.246843\pi$
$930$	−26.5123	−0.869371
$931$	1.00880	0.0330620
$932$	54.0000	1.76883
$933$	−5.21429	−0.170708
$934$	−15.1095	−0.494397
$935$	1.81899	0.0594872
$936$	10.9418	0.357643
$937$	10.3210	0.337172	0.168586	−	0.985687i	$-0.446080\pi$
$937$	10.3210	0.337172	0.168586	+	0.985687i	$0.446080\pi$
$938$	13.9527	0.455572
$939$	3.34047	0.109012
$940$	18.7163	0.610457
$941$	−37.6713	−1.22805	−0.614024	−	0.789287i	$-0.710450\pi$
$941$	−37.6713	−1.22805	−0.614024	+	0.789287i	$0.710450\pi$
$942$	38.9897	1.27035
$943$	15.3756	0.500698
$944$	48.2212	1.56947
$945$	4.82131	0.156837
$946$	20.8832	0.678970
$947$	−2.39138	−0.0777093	−0.0388547	−	0.999245i	$-0.512371\pi$
$947$	−2.39138	−0.0777093	−0.0388547	+	0.999245i	$0.512371\pi$
$948$	−78.8058	−2.55949
$949$	6.96950	0.226240
$950$	0.458816	0.0148860
$951$	−34.4267	−1.11636
$952$	−17.6478	−0.571968
$953$	−34.2780	−1.11037	−0.555187	−	0.831726i	$-0.687353\pi$
$953$	−34.2780	−1.11037	−0.555187	+	0.831726i	$0.687353\pi$
$954$	1.10463	0.0357636
$955$	2.14835	0.0695190
$956$	47.2865	1.52935
$957$	−13.2057	−0.426880
$958$	27.0301	0.873303
$959$	−18.0132	−0.581676
$960$	69.4338	2.24096
$961$	−1.95425	−0.0630402
$962$	−23.6200	−0.761540
$963$	1.79586	0.0578709
$964$	117.558	3.78630
$965$	−6.82172	−0.219599
$966$	7.81604	0.251477
$967$	−32.2235	−1.03624	−0.518119	−	0.855309i	$-0.673367\pi$
$967$	−32.2235	−1.03624	−0.518119	+	0.855309i	$0.673367\pi$
$968$	−10.1702	−0.326883
$969$	0.535149	0.0171915
$970$	8.44673	0.271208
$971$	−50.8484	−1.63180	−0.815902	−	0.578190i	$-0.803759\pi$
$971$	−50.8484	−1.63180	−0.815902	+	0.578190i	$0.803759\pi$
$972$	−9.09004	−0.291563
$973$	−4.90178	−0.157144
$974$	−40.0771	−1.28415
$975$	−12.3782	−0.396420
$976$	−208.216	−6.66483
$977$	−45.3566	−1.45109	−0.725543	−	0.688177i	$-0.758411\pi$
$977$	−45.3566	−1.45109	−0.725543	+	0.688177i	$0.758411\pi$
$978$	5.92956	0.189606
$979$	−9.62676	−0.307673
$980$	−34.5412	−1.10338
$981$	2.75634	0.0880030
$982$	−112.004	−3.57419
$983$	27.9429	0.891240	0.445620	−	0.895222i	$-0.352983\pi$
$983$	27.9429	0.891240	0.445620	+	0.895222i	$0.352983\pi$
$984$	−166.750	−5.31581
$985$	−18.2202	−0.580543
$986$	37.4614	1.19301
$987$	−5.59092	−0.177961
$988$	6.54807	0.208322
$989$	12.5574	0.399301
$990$	0.427568	0.0135890
$991$	23.7814	0.755442	0.377721	−	0.925919i	$-0.376708\pi$
$991$	23.7814	0.755442	0.377721	+	0.925919i	$0.376708\pi$
$992$	−141.548	−4.49414
$993$	−43.4630	−1.37926
$994$	−5.81557	−0.184459
$995$	7.03098	0.222897
$996$	118.202	3.74536
$997$	−18.8957	−0.598432	−0.299216	−	0.954185i	$-0.596725\pi$
$997$	−18.8957	−0.598432	−0.299216	+	0.954185i	$0.596725\pi$
$998$	6.52838	0.206652
$999$	6.18393	0.195651

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.i.1.1	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.i.1.1	✓	38	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:	\( N \)	\(=\)	\( 4015 = 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4015.a (trivial)

\(f(q)\)	\(=\)	\(q-2.76981 q^{2} +1.77605 q^{3} +5.67182 q^{4} +1.00000 q^{5} -4.91932 q^{6} -0.953960 q^{7} -10.1702 q^{8} +0.154367 q^{9} +O(q^{10})\)	\(q-2.76981 q^{2} +1.77605 q^{3} +5.67182 q^{4} +1.00000 q^{5} -4.91932 q^{6} -0.953960 q^{7} -10.1702 q^{8} +0.154367 q^{9} -2.76981 q^{10} -1.00000 q^{11} +10.0735 q^{12} -6.96950 q^{13} +2.64228 q^{14} +1.77605 q^{15} +16.8259 q^{16} -1.81899 q^{17} -0.427568 q^{18} -0.165649 q^{19} +5.67182 q^{20} -1.69428 q^{21} +2.76981 q^{22} +1.66553 q^{23} -18.0629 q^{24} +1.00000 q^{25} +19.3042 q^{26} -5.05400 q^{27} -5.41069 q^{28} +7.43543 q^{29} -4.91932 q^{30} +5.38941 q^{31} -26.2640 q^{32} -1.77605 q^{33} +5.03824 q^{34} -0.953960 q^{35} +0.875544 q^{36} -1.22357 q^{37} +0.458816 q^{38} -12.3782 q^{39} -10.1702 q^{40} +9.23166 q^{41} +4.69284 q^{42} +7.53958 q^{43} -5.67182 q^{44} +0.154367 q^{45} -4.61318 q^{46} +3.29987 q^{47} +29.8837 q^{48} -6.08996 q^{49} -2.76981 q^{50} -3.23062 q^{51} -39.5297 q^{52} -2.58351 q^{53} +13.9986 q^{54} -1.00000 q^{55} +9.70198 q^{56} -0.294202 q^{57} -20.5947 q^{58} +2.86589 q^{59} +10.0735 q^{60} -12.3747 q^{61} -14.9276 q^{62} -0.147260 q^{63} +39.0944 q^{64} -6.96950 q^{65} +4.91932 q^{66} +5.28055 q^{67} -10.3170 q^{68} +2.95807 q^{69} +2.64228 q^{70} -2.20096 q^{71} -1.56995 q^{72} -1.00000 q^{73} +3.38905 q^{74} +1.77605 q^{75} -0.939532 q^{76} +0.953960 q^{77} +34.2852 q^{78} -7.82311 q^{79} +16.8259 q^{80} -9.43927 q^{81} -25.5699 q^{82} +11.7340 q^{83} -9.60967 q^{84} -1.81899 q^{85} -20.8832 q^{86} +13.2057 q^{87} +10.1702 q^{88} +9.62676 q^{89} -0.427568 q^{90} +6.64862 q^{91} +9.44657 q^{92} +9.57188 q^{93} -9.14000 q^{94} -0.165649 q^{95} -46.6463 q^{96} -3.04958 q^{97} +16.8680 q^{98} -0.154367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9}+O(q^{10}) \) 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9	\( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9} + 4 q^{10} - 38 q^{11} + 12 q^{12} - q^{13} + 23 q^{14} + 5 q^{15} + 74 q^{16} + 26 q^{17} + 16 q^{18} - 10 q^{19} + 50 q^{20} + 21 q^{21} - 4 q^{22} + 10 q^{23} + 41 q^{24} + 38 q^{25} + 25 q^{26} + 5 q^{27} + 2 q^{28} + 28 q^{29} + 11 q^{30} + 24 q^{31} + 39 q^{32} - 5 q^{33} + 38 q^{34} + 111 q^{36} + 12 q^{37} + 19 q^{38} - 18 q^{39} + 15 q^{40} + 62 q^{41} - 17 q^{42} - 32 q^{43} - 50 q^{44} + 63 q^{45} - 9 q^{46} + 31 q^{47} + 53 q^{48} + 88 q^{49} + 4 q^{50} - 3 q^{51} - 21 q^{52} + 30 q^{53} + 49 q^{54} - 38 q^{55} + 32 q^{56} + 49 q^{57} + 12 q^{58} + 31 q^{59} + 12 q^{60} + 25 q^{61} + 12 q^{62} + 15 q^{63} + 137 q^{64} - q^{65} - 11 q^{66} + 20 q^{67} + 75 q^{68} + 92 q^{69} + 23 q^{70} + 32 q^{71} + 6 q^{72} - 38 q^{73} + 55 q^{74} + 5 q^{75} - 57 q^{76} - 17 q^{78} - 2 q^{79} + 74 q^{80} + 118 q^{81} + 14 q^{82} + 4 q^{83} + 22 q^{84} + 26 q^{85} + 5 q^{86} + 24 q^{87} - 15 q^{88} + 143 q^{89} + 16 q^{90} + 66 q^{91} + 29 q^{92} - 8 q^{93} - 7 q^{94} - 10 q^{95} + 59 q^{96} + 41 q^{97} - 10 q^{98} - 63 q^{99}+O(q^{100}) \) 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9 + 4 * q^10 - 38 * q^11 + 12 * q^12 - q^13 + 23 * q^14 + 5 * q^15 + 74 * q^16 + 26 * q^17 + 16 * q^18 - 10 * q^19 + 50 * q^20 + 21 * q^21 - 4 * q^22 + 10 * q^23 + 41 * q^24 + 38 * q^25 + 25 * q^26 + 5 * q^27 + 2 * q^28 + 28 * q^29 + 11 * q^30 + 24 * q^31 + 39 * q^32 - 5 * q^33 + 38 * q^34 + 111 * q^36 + 12 * q^37 + 19 * q^38 - 18 * q^39 + 15 * q^40 + 62 * q^41 - 17 * q^42 - 32 * q^43 - 50 * q^44 + 63 * q^45 - 9 * q^46 + 31 * q^47 + 53 * q^48 + 88 * q^49 + 4 * q^50 - 3 * q^51 - 21 * q^52 + 30 * q^53 + 49 * q^54 - 38 * q^55 + 32 * q^56 + 49 * q^57 + 12 * q^58 + 31 * q^59 + 12 * q^60 + 25 * q^61 + 12 * q^62 + 15 * q^63 + 137 * q^64 - q^65 - 11 * q^66 + 20 * q^67 + 75 * q^68 + 92 * q^69 + 23 * q^70 + 32 * q^71 + 6 * q^72 - 38 * q^73 + 55 * q^74 + 5 * q^75 - 57 * q^76 - 17 * q^78 - 2 * q^79 + 74 * q^80 + 118 * q^81 + 14 * q^82 + 4 * q^83 + 22 * q^84 + 26 * q^85 + 5 * q^86 + 24 * q^87 - 15 * q^88 + 143 * q^89 + 16 * q^90 + 66 * q^91 + 29 * q^92 - 8 * q^93 - 7 * q^94 - 10 * q^95 + 59 * q^96 + 41 * q^97 - 10 * q^98 - 63 * q^99

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