LMFDB - Embedded newform 8043.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$52$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.71076 q^{2} -1.00000 q^{3} +5.34821 q^{4} +0.130231 q^{5} +2.71076 q^{6} +1.00000 q^{7} -9.07620 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.71076 q^{2} -1.00000 q^{3} +5.34821 q^{4} +0.130231 q^{5} +2.71076 q^{6} +1.00000 q^{7} -9.07620 q^{8} +1.00000 q^{9} -0.353025 q^{10} +3.25358 q^{11} -5.34821 q^{12} +1.11985 q^{13} -2.71076 q^{14} -0.130231 q^{15} +13.9070 q^{16} +0.0623671 q^{17} -2.71076 q^{18} +2.72595 q^{19} +0.696503 q^{20} -1.00000 q^{21} -8.81967 q^{22} +6.67141 q^{23} +9.07620 q^{24} -4.98304 q^{25} -3.03565 q^{26} -1.00000 q^{27} +5.34821 q^{28} -7.51601 q^{29} +0.353025 q^{30} +8.48682 q^{31} -19.5460 q^{32} -3.25358 q^{33} -0.169062 q^{34} +0.130231 q^{35} +5.34821 q^{36} +10.9966 q^{37} -7.38938 q^{38} -1.11985 q^{39} -1.18200 q^{40} -6.34625 q^{41} +2.71076 q^{42} +2.41223 q^{43} +17.4008 q^{44} +0.130231 q^{45} -18.0846 q^{46} +7.87767 q^{47} -13.9070 q^{48} +1.00000 q^{49} +13.5078 q^{50} -0.0623671 q^{51} +5.98922 q^{52} -6.90598 q^{53} +2.71076 q^{54} +0.423717 q^{55} -9.07620 q^{56} -2.72595 q^{57} +20.3741 q^{58} +9.66024 q^{59} -0.696503 q^{60} -3.59541 q^{61} -23.0057 q^{62} +1.00000 q^{63} +25.1706 q^{64} +0.145840 q^{65} +8.81967 q^{66} +12.8812 q^{67} +0.333553 q^{68} -6.67141 q^{69} -0.353025 q^{70} -3.03164 q^{71} -9.07620 q^{72} -0.955600 q^{73} -29.8091 q^{74} +4.98304 q^{75} +14.5789 q^{76} +3.25358 q^{77} +3.03565 q^{78} -9.46286 q^{79} +1.81112 q^{80} +1.00000 q^{81} +17.2032 q^{82} +0.768405 q^{83} -5.34821 q^{84} +0.00812213 q^{85} -6.53898 q^{86} +7.51601 q^{87} -29.5301 q^{88} +2.54138 q^{89} -0.353025 q^{90} +1.11985 q^{91} +35.6801 q^{92} -8.48682 q^{93} -21.3545 q^{94} +0.355003 q^{95} +19.5460 q^{96} +2.85023 q^{97} -2.71076 q^{98} +3.25358 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) $ 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * 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 $

$2$	−2.71076	−1.91680	−0.958398	−	0.285435i	$-0.907862\pi$
$2$	−2.71076	−1.91680	−0.958398	+	0.285435i	$0.907862\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.34821	2.67411
$5$	0.130231	0.0582411	0.0291205	−	0.999576i	$-0.490729\pi$
$5$	0.130231	0.0582411	0.0291205	+	0.999576i	$0.490729\pi$
$6$	2.71076	1.10666
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−9.07620	−3.20892
$9$	1.00000	0.333333
$10$	−0.353025	−0.111636
$11$	3.25358	0.980991	0.490495	−	0.871444i	$-0.336816\pi$
$11$	3.25358	0.980991	0.490495	+	0.871444i	$0.336816\pi$
$12$	−5.34821	−1.54390
$13$	1.11985	0.310591	0.155296	−	0.987868i	$-0.450367\pi$
$13$	1.11985	0.310591	0.155296	+	0.987868i	$0.450367\pi$
$14$	−2.71076	−0.724481
$15$	−0.130231	−0.0336255
$16$	13.9070	3.47674
$17$	0.0623671	0.0151262	0.00756312	−	0.999971i	$-0.497593\pi$
$17$	0.0623671	0.0151262	0.00756312	+	0.999971i	$0.497593\pi$
$18$	−2.71076	−0.638932
$19$	2.72595	0.625375	0.312688	−	0.949856i	$-0.398771\pi$
$19$	2.72595	0.625375	0.312688	+	0.949856i	$0.398771\pi$
$20$	0.696503	0.155743
$21$	−1.00000	−0.218218
$22$	−8.81967	−1.88036
$23$	6.67141	1.39108	0.695542	−	0.718485i	$-0.255164\pi$
$23$	6.67141	1.39108	0.695542	+	0.718485i	$0.255164\pi$
$24$	9.07620	1.85267
$25$	−4.98304	−0.996608
$26$	−3.03565	−0.595340
$27$	−1.00000	−0.192450
$28$	5.34821	1.01072
$29$	−7.51601	−1.39569	−0.697844	−	0.716250i	$-0.745857\pi$
$29$	−7.51601	−1.39569	−0.697844	+	0.716250i	$0.745857\pi$
$30$	0.353025	0.0644532
$31$	8.48682	1.52428	0.762139	−	0.647414i	$-0.224149\pi$
$31$	8.48682	1.52428	0.762139	+	0.647414i	$0.224149\pi$
$32$	−19.5460	−3.45528
$33$	−3.25358	−0.566375
$34$	−0.169062	−0.0289939
$35$	0.130231	0.0220131
$36$	5.34821	0.891369
$37$	10.9966	1.80783	0.903915	−	0.427712i	$-0.140680\pi$
$37$	10.9966	1.80783	0.903915	+	0.427712i	$0.140680\pi$
$38$	−7.38938	−1.19872
$39$	−1.11985	−0.179320
$40$	−1.18200	−0.186891
$41$	−6.34625	−0.991118	−0.495559	−	0.868574i	$-0.665037\pi$
$41$	−6.34625	−0.991118	−0.495559	+	0.868574i	$0.665037\pi$
$42$	2.71076	0.418279
$43$	2.41223	0.367862	0.183931	−	0.982939i	$-0.441118\pi$
$43$	2.41223	0.367862	0.183931	+	0.982939i	$0.441118\pi$
$44$	17.4008	2.62327
$45$	0.130231	0.0194137
$46$	−18.0846	−2.66642
$47$	7.87767	1.14908	0.574538	−	0.818478i	$-0.305182\pi$
$47$	7.87767	1.14908	0.574538	+	0.818478i	$0.305182\pi$
$48$	−13.9070	−2.00730
$49$	1.00000	0.142857
$50$	13.5078	1.91029
$51$	−0.0623671	−0.00873314
$52$	5.98922	0.830555
$53$	−6.90598	−0.948609	−0.474304	−	0.880361i	$-0.657300\pi$
$53$	−6.90598	−0.948609	−0.474304	+	0.880361i	$0.657300\pi$
$54$	2.71076	0.368888
$55$	0.423717	0.0571339
$56$	−9.07620	−1.21286
$57$	−2.72595	−0.361060
$58$	20.3741	2.67525
$59$	9.66024	1.25766	0.628828	−	0.777545i	$-0.283535\pi$
$59$	9.66024	1.25766	0.628828	+	0.777545i	$0.283535\pi$
$60$	−0.696503	−0.0899182
$61$	−3.59541	−0.460345	−0.230172	−	0.973150i	$-0.573929\pi$
$61$	−3.59541	−0.460345	−0.230172	+	0.973150i	$0.573929\pi$
$62$	−23.0057	−2.92173
$63$	1.00000	0.125988
$64$	25.1706	3.14633
$65$	0.145840	0.0180892
$66$	8.81967	1.08563
$67$	12.8812	1.57369	0.786845	−	0.617151i	$-0.211713\pi$
$67$	12.8812	1.57369	0.786845	+	0.617151i	$0.211713\pi$
$68$	0.333553	0.0404492
$69$	−6.67141	−0.803143
$70$	−0.353025	−0.0421945
$71$	−3.03164	−0.359790	−0.179895	−	0.983686i	$-0.557576\pi$
$71$	−3.03164	−0.359790	−0.179895	+	0.983686i	$0.557576\pi$
$72$	−9.07620	−1.06964
$73$	−0.955600	−0.111844	−0.0559222	−	0.998435i	$-0.517810\pi$
$73$	−0.955600	−0.111844	−0.0559222	+	0.998435i	$0.517810\pi$
$74$	−29.8091	−3.46524
$75$	4.98304	0.575392
$76$	14.5789	1.67232
$77$	3.25358	0.370780
$78$	3.03565	0.343720
$79$	−9.46286	−1.06465	−0.532327	−	0.846539i	$-0.678682\pi$
$79$	−9.46286	−1.06465	−0.532327	+	0.846539i	$0.678682\pi$
$80$	1.81112	0.202489
$81$	1.00000	0.111111
$82$	17.2032	1.89977
$83$	0.768405	0.0843434	0.0421717	−	0.999110i	$-0.486572\pi$
$83$	0.768405	0.0843434	0.0421717	+	0.999110i	$0.486572\pi$
$84$	−5.34821	−0.583538
$85$	0.00812213	0.000880968 0
$86$	−6.53898	−0.705116
$87$	7.51601	0.805801
$88$	−29.5301	−3.14792
$89$	2.54138	0.269386	0.134693	−	0.990887i	$-0.456995\pi$
$89$	2.54138	0.269386	0.134693	+	0.990887i	$0.456995\pi$
$90$	−0.353025	−0.0372121
$91$	1.11985	0.117393
$92$	35.6801	3.71991
$93$	−8.48682	−0.880042
$94$	−21.3545	−2.20254
$95$	0.355003	0.0364225
$96$	19.5460	1.99491
$97$	2.85023	0.289397	0.144699	−	0.989476i	$-0.453779\pi$
$97$	2.85023	0.289397	0.144699	+	0.989476i	$0.453779\pi$
$98$	−2.71076	−0.273828
$99$	3.25358	0.326997
$100$	−26.6504	−2.66504
$101$	−10.0545	−1.00046	−0.500231	−	0.865892i	$-0.666752\pi$
$101$	−10.0545	−1.00046	−0.500231	+	0.865892i	$0.666752\pi$
$102$	0.169062	0.0167396
$103$	9.36566	0.922826	0.461413	−	0.887185i	$-0.347343\pi$
$103$	9.36566	0.922826	0.461413	+	0.887185i	$0.347343\pi$
$104$	−10.1640	−0.996664
$105$	−0.130231	−0.0127092
$106$	18.7204	1.81829
$107$	11.7033	1.13140	0.565700	−	0.824611i	$-0.308606\pi$
$107$	11.7033	1.13140	0.565700	+	0.824611i	$0.308606\pi$
$108$	−5.34821	−0.514632
$109$	5.17573	0.495745	0.247872	−	0.968793i	$-0.420269\pi$
$109$	5.17573	0.495745	0.247872	+	0.968793i	$0.420269\pi$
$110$	−1.14859	−0.109514
$111$	−10.9966	−1.04375
$112$	13.9070	1.31408
$113$	8.66917	0.815527	0.407764	−	0.913088i	$-0.366309\pi$
$113$	8.66917	0.815527	0.407764	+	0.913088i	$0.366309\pi$
$114$	7.38938	0.692079
$115$	0.868824	0.0810182
$116$	−40.1972	−3.73222
$117$	1.11985	0.103530
$118$	−26.1866	−2.41067
$119$	0.0623671	0.00571718
$120$	1.18200	0.107902
$121$	−0.414226	−0.0376569
$122$	9.74628	0.882387
$123$	6.34625	0.572222
$124$	45.3893	4.07608
$125$	−1.30010	−0.116285
$126$	−2.71076	−0.241494
$127$	−0.493993	−0.0438348	−0.0219174	−	0.999760i	$-0.506977\pi$
$127$	−0.493993	−0.0438348	−0.0219174	+	0.999760i	$0.506977\pi$
$128$	−29.1395	−2.57559
$129$	−2.41223	−0.212385
$130$	−0.395336	−0.0346733
$131$	−18.9040	−1.65165	−0.825824	−	0.563927i	$-0.809290\pi$
$131$	−18.9040	−1.65165	−0.825824	+	0.563927i	$0.809290\pi$
$132$	−17.4008	−1.51455
$133$	2.72595	0.236370
$134$	−34.9179	−3.01644
$135$	−0.130231	−0.0112085
$136$	−0.566056	−0.0485389
$137$	13.6960	1.17013	0.585063	−	0.810988i	$-0.301070\pi$
$137$	13.6960	1.17013	0.585063	+	0.810988i	$0.301070\pi$
$138$	18.0846	1.53946
$139$	2.92684	0.248252	0.124126	−	0.992266i	$-0.460387\pi$
$139$	2.92684	0.248252	0.124126	+	0.992266i	$0.460387\pi$
$140$	0.696503	0.0588653
$141$	−7.87767	−0.663419
$142$	8.21806	0.689644
$143$	3.64353	0.304687
$144$	13.9070	1.15891
$145$	−0.978817	−0.0812863
$146$	2.59040	0.214383
$147$	−1.00000	−0.0824786
$148$	58.8122	4.83433
$149$	−21.5069	−1.76191	−0.880957	−	0.473197i	$-0.843100\pi$
$149$	−21.5069	−1.76191	−0.880957	+	0.473197i	$0.843100\pi$
$150$	−13.5078	−1.10291
$151$	2.82899	0.230220	0.115110	−	0.993353i	$-0.463278\pi$
$151$	2.82899	0.230220	0.115110	+	0.993353i	$0.463278\pi$
$152$	−24.7412	−2.00678
$153$	0.0623671	0.00504208
$154$	−8.81967	−0.710709
$155$	1.10525	0.0887755
$156$	−5.98922	−0.479521
$157$	7.59643	0.606261	0.303130	−	0.952949i	$-0.401968\pi$
$157$	7.59643	0.606261	0.303130	+	0.952949i	$0.401968\pi$
$158$	25.6515	2.04073
$159$	6.90598	0.547680
$160$	−2.54550	−0.201239
$161$	6.67141	0.525780
$162$	−2.71076	−0.212977
$163$	5.41853	0.424412	0.212206	−	0.977225i	$-0.431935\pi$
$163$	5.41853	0.424412	0.212206	+	0.977225i	$0.431935\pi$
$164$	−33.9411	−2.65035
$165$	−0.423717	−0.0329863
$166$	−2.08296	−0.161669
$167$	12.8244	0.992382	0.496191	−	0.868214i	$-0.334732\pi$
$167$	12.8244	0.992382	0.496191	+	0.868214i	$0.334732\pi$
$168$	9.07620	0.700244
$169$	−11.7459	−0.903533
$170$	−0.0220171	−0.00168864
$171$	2.72595	0.208458
$172$	12.9011	0.983702
$173$	−3.42175	−0.260151	−0.130076	−	0.991504i	$-0.541522\pi$
$173$	−3.42175	−0.260151	−0.130076	+	0.991504i	$0.541522\pi$
$174$	−20.3741	−1.54456
$175$	−4.98304	−0.376682
$176$	45.2474	3.41065
$177$	−9.66024	−0.726108
$178$	−6.88907	−0.516357
$179$	11.4708	0.857368	0.428684	−	0.903454i	$-0.358977\pi$
$179$	11.4708	0.857368	0.428684	+	0.903454i	$0.358977\pi$
$180$	0.696503	0.0519143
$181$	−2.92554	−0.217454	−0.108727	−	0.994072i	$-0.534677\pi$
$181$	−2.92554	−0.217454	−0.108727	+	0.994072i	$0.534677\pi$
$182$	−3.03565	−0.225018
$183$	3.59541	0.265780
$184$	−60.5510	−4.46388
$185$	1.43210	0.105290
$186$	23.0057	1.68686
$187$	0.202916	0.0148387
$188$	42.1314	3.07275
$189$	−1.00000	−0.0727393
$190$	−0.962327	−0.0698145
$191$	−11.3119	−0.818501	−0.409250	−	0.912422i	$-0.634210\pi$
$191$	−11.3119	−0.818501	−0.409250	+	0.912422i	$0.634210\pi$
$192$	−25.1706	−1.81654
$193$	3.48149	0.250603	0.125302	−	0.992119i	$-0.460010\pi$
$193$	3.48149	0.250603	0.125302	+	0.992119i	$0.460010\pi$
$194$	−7.72629	−0.554716
$195$	−0.145840	−0.0104438
$196$	5.34821	0.382015
$197$	10.9479	0.780006	0.390003	−	0.920814i	$-0.372474\pi$
$197$	10.9479	0.780006	0.390003	+	0.920814i	$0.372474\pi$
$198$	−8.81967	−0.626786
$199$	23.1722	1.64264	0.821318	−	0.570471i	$-0.193239\pi$
$199$	23.1722	1.64264	0.821318	+	0.570471i	$0.193239\pi$
$200$	45.2271	3.19804
$201$	−12.8812	−0.908570
$202$	27.2554	1.91768
$203$	−7.51601	−0.527520
$204$	−0.333553	−0.0233534
$205$	−0.826478	−0.0577237
$206$	−25.3880	−1.76887
$207$	6.67141	0.463695
$208$	15.5738	1.07985
$209$	8.86908	0.613487
$210$	0.353025	0.0243610
$211$	20.3087	1.39811	0.699053	−	0.715070i	$-0.253605\pi$
$211$	20.3087	1.39811	0.699053	+	0.715070i	$0.253605\pi$
$212$	−36.9346	−2.53668
$213$	3.03164	0.207725
$214$	−31.7248	−2.16866
$215$	0.314147	0.0214247
$216$	9.07620	0.617557
$217$	8.48682	0.576123
$218$	−14.0302	−0.950242
$219$	0.955600	0.0645734
$220$	2.26613	0.152782
$221$	0.0698420	0.00469808
$222$	29.8091	2.00066
$223$	27.4714	1.83962	0.919812	−	0.392360i	$-0.128341\pi$
$223$	27.4714	1.83962	0.919812	+	0.392360i	$0.128341\pi$
$224$	−19.5460	−1.30597
$225$	−4.98304	−0.332203
$226$	−23.5000	−1.56320
$227$	−20.5618	−1.36474	−0.682369	−	0.731008i	$-0.739050\pi$
$227$	−20.5618	−1.36474	−0.682369	+	0.731008i	$0.739050\pi$
$228$	−14.5789	−0.965514
$229$	5.61860	0.371288	0.185644	−	0.982617i	$-0.440563\pi$
$229$	5.61860	0.371288	0.185644	+	0.982617i	$0.440563\pi$
$230$	−2.35517	−0.155295
$231$	−3.25358	−0.214070
$232$	68.2168	4.47865
$233$	7.80020	0.511008	0.255504	−	0.966808i	$-0.417759\pi$
$233$	7.80020	0.511008	0.255504	+	0.966808i	$0.417759\pi$
$234$	−3.03565	−0.198447
$235$	1.02592	0.0669234
$236$	51.6650	3.36311
$237$	9.46286	0.614679
$238$	−0.169062	−0.0109587
$239$	7.89339	0.510581	0.255290	−	0.966864i	$-0.417829\pi$
$239$	7.89339	0.510581	0.255290	+	0.966864i	$0.417829\pi$
$240$	−1.81112	−0.116907
$241$	−12.0448	−0.775872	−0.387936	−	0.921686i	$-0.626812\pi$
$241$	−12.0448	−0.775872	−0.387936	+	0.921686i	$0.626812\pi$
$242$	1.12287	0.0721806
$243$	−1.00000	−0.0641500
$244$	−19.2290	−1.23101
$245$	0.130231	0.00832015
$246$	−17.2032	−1.09683
$247$	3.05266	0.194236
$248$	−77.0281	−4.89129
$249$	−0.768405	−0.0486957
$250$	3.52426	0.222894
$251$	−6.39187	−0.403451	−0.201726	−	0.979442i	$-0.564655\pi$
$251$	−6.39187	−0.403451	−0.201726	+	0.979442i	$0.564655\pi$
$252$	5.34821	0.336906
$253$	21.7059	1.36464
$254$	1.33910	0.0840224
$255$	−0.00812213	−0.000508627 0
$256$	28.6489	1.79055
$257$	−26.8685	−1.67601	−0.838005	−	0.545662i	$-0.816278\pi$
$257$	−26.8685	−1.67601	−0.838005	+	0.545662i	$0.816278\pi$
$258$	6.53898	0.407099
$259$	10.9966	0.683296
$260$	0.779981	0.0483724
$261$	−7.51601	−0.465229
$262$	51.2441	3.16587
$263$	−18.2727	−1.12674	−0.563372	−	0.826204i	$-0.690496\pi$
$263$	−18.2727	−1.12674	−0.563372	+	0.826204i	$0.690496\pi$
$264$	29.5301	1.81745
$265$	−0.899372	−0.0552480
$266$	−7.38938	−0.453072
$267$	−2.54138	−0.155530
$268$	68.8915	4.20822
$269$	20.5732	1.25437	0.627186	−	0.778869i	$-0.284207\pi$
$269$	20.5732	1.25437	0.627186	+	0.778869i	$0.284207\pi$
$270$	0.353025	0.0214844
$271$	1.99413	0.121135	0.0605674	−	0.998164i	$-0.480709\pi$
$271$	1.99413	0.121135	0.0605674	+	0.998164i	$0.480709\pi$
$272$	0.867337	0.0525900
$273$	−1.11985	−0.0677766
$274$	−37.1265	−2.24289
$275$	−16.2127	−0.977663
$276$	−35.6801	−2.14769
$277$	−22.5582	−1.35539	−0.677697	−	0.735342i	$-0.737022\pi$
$277$	−22.5582	−1.35539	−0.677697	+	0.735342i	$0.737022\pi$
$278$	−7.93397	−0.475848
$279$	8.48682	0.508092
$280$	−1.18200	−0.0706382
$281$	1.42401	0.0849492	0.0424746	−	0.999098i	$-0.486476\pi$
$281$	1.42401	0.0849492	0.0424746	+	0.999098i	$0.486476\pi$
$282$	21.3545	1.27164
$283$	−5.41435	−0.321849	−0.160925	−	0.986967i	$-0.551448\pi$
$283$	−5.41435	−0.321849	−0.160925	+	0.986967i	$0.551448\pi$
$284$	−16.2139	−0.962117
$285$	−0.355003	−0.0210285
$286$	−9.87673	−0.584023
$287$	−6.34625	−0.374607
$288$	−19.5460	−1.15176
$289$	−16.9961	−0.999771
$290$	2.65334	0.155809
$291$	−2.85023	−0.167084
$292$	−5.11075	−0.299084
$293$	12.0874	0.706152	0.353076	−	0.935595i	$-0.385136\pi$
$293$	12.0874	0.706152	0.353076	+	0.935595i	$0.385136\pi$
$294$	2.71076	0.158095
$295$	1.25806	0.0732472
$296$	−99.8074	−5.80119
$297$	−3.25358	−0.188792
$298$	58.3000	3.37723
$299$	7.47099	0.432059
$300$	26.6504	1.53866
$301$	2.41223	0.139039
$302$	−7.66870	−0.441284
$303$	10.0545	0.577617
$304$	37.9097	2.17427
$305$	−0.468233	−0.0268110
$306$	−0.169062	−0.00966464
$307$	3.96360	0.226215	0.113107	−	0.993583i	$-0.463920\pi$
$307$	3.96360	0.226215	0.113107	+	0.993583i	$0.463920\pi$
$308$	17.4008	0.991505
$309$	−9.36566	−0.532794
$310$	−2.99606	−0.170165
$311$	−8.25019	−0.467825	−0.233913	−	0.972258i	$-0.575153\pi$
$311$	−8.25019	−0.467825	−0.233913	+	0.972258i	$0.575153\pi$
$312$	10.1640	0.575424
$313$	−3.41258	−0.192890	−0.0964452	−	0.995338i	$-0.530747\pi$
$313$	−3.41258	−0.192890	−0.0964452	+	0.995338i	$0.530747\pi$
$314$	−20.5921	−1.16208
$315$	0.130231	0.00733768
$316$	−50.6094	−2.84700
$317$	−1.19667	−0.0672116	−0.0336058	−	0.999435i	$-0.510699\pi$
$317$	−1.19667	−0.0672116	−0.0336058	+	0.999435i	$0.510699\pi$
$318$	−18.7204	−1.04979
$319$	−24.4539	−1.36916
$320$	3.27800	0.183246
$321$	−11.7033	−0.653215
$322$	−18.0846	−1.00781
$323$	0.170009	0.00945957
$324$	5.34821	0.297123
$325$	−5.58027	−0.309538
$326$	−14.6883	−0.813511
$327$	−5.17573	−0.286218
$328$	57.5998	3.18042
$329$	7.87767	0.434310
$330$	1.14859	0.0632280
$331$	0.820069	0.0450751	0.0225375	−	0.999746i	$-0.492825\pi$
$331$	0.820069	0.0450751	0.0225375	+	0.999746i	$0.492825\pi$
$332$	4.10960	0.225543
$333$	10.9966	0.602610
$334$	−34.7638	−1.90219
$335$	1.67753	0.0916534
$336$	−13.9070	−0.758687
$337$	−17.0703	−0.929878	−0.464939	−	0.885343i	$-0.653924\pi$
$337$	−17.0703	−0.929878	−0.464939	+	0.885343i	$0.653924\pi$
$338$	31.8404	1.73189
$339$	−8.66917	−0.470845
$340$	0.0434389	0.00235580
$341$	27.6125	1.49530
$342$	−7.38938	−0.399572
$343$	1.00000	0.0539949
$344$	−21.8939	−1.18044
$345$	−0.868824	−0.0467759
$346$	9.27555	0.498657
$347$	−10.3864	−0.557570	−0.278785	−	0.960354i	$-0.589932\pi$
$347$	−10.3864	−0.557570	−0.278785	+	0.960354i	$0.589932\pi$
$348$	40.1972	2.15480
$349$	−15.2538	−0.816516	−0.408258	−	0.912867i	$-0.633864\pi$
$349$	−15.2538	−0.816516	−0.408258	+	0.912867i	$0.633864\pi$
$350$	13.5078	0.722023
$351$	−1.11985	−0.0597733
$352$	−63.5946	−3.38960
$353$	−23.6146	−1.25688	−0.628438	−	0.777859i	$-0.716306\pi$
$353$	−23.6146	−1.25688	−0.628438	+	0.777859i	$0.716306\pi$
$354$	26.1866	1.39180
$355$	−0.394814	−0.0209546
$356$	13.5918	0.720366
$357$	−0.0623671	−0.00330082
$358$	−31.0946	−1.64340
$359$	0.529839	0.0279638	0.0139819	−	0.999902i	$-0.495549\pi$
$359$	0.529839	0.0279638	0.0139819	+	0.999902i	$0.495549\pi$
$360$	−1.18200	−0.0622970
$361$	−11.5692	−0.608906
$362$	7.93044	0.416814
$363$	0.414226	0.0217412
$364$	5.98922	0.313920
$365$	−0.124449	−0.00651394
$366$	−9.74628	−0.509446
$367$	−34.0933	−1.77966	−0.889829	−	0.456294i	$-0.849177\pi$
$367$	−34.0933	−1.77966	−0.889829	+	0.456294i	$0.849177\pi$
$368$	92.7790	4.83644
$369$	−6.34625	−0.330373
$370$	−3.88207	−0.201819
$371$	−6.90598	−0.358540
$372$	−45.3893	−2.35333
$373$	−16.9473	−0.877497	−0.438749	−	0.898610i	$-0.644578\pi$
$373$	−16.9473	−0.877497	−0.438749	+	0.898610i	$0.644578\pi$
$374$	−0.550057	−0.0284428
$375$	1.30010	0.0671369
$376$	−71.4993	−3.68729
$377$	−8.41683	−0.433489
$378$	2.71076	0.139426
$379$	25.2513	1.29707	0.648537	−	0.761183i	$-0.275381\pi$
$379$	25.2513	1.29707	0.648537	+	0.761183i	$0.275381\pi$
$380$	1.89863	0.0973977
$381$	0.493993	0.0253080
$382$	30.6638	1.56890
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	29.1395	1.48702
$385$	0.423717	0.0215946
$386$	−9.43749	−0.480355
$387$	2.41223	0.122621
$388$	15.2437	0.773879
$389$	9.41733	0.477477	0.238739	−	0.971084i	$-0.423266\pi$
$389$	9.41733	0.477477	0.238739	+	0.971084i	$0.423266\pi$
$390$	0.395336	0.0200186
$391$	0.416076	0.0210419
$392$	−9.07620	−0.458417
$393$	18.9040	0.953580
$394$	−29.6771	−1.49511
$395$	−1.23236	−0.0620066
$396$	17.4008	0.874425
$397$	20.0566	1.00661	0.503306	−	0.864108i	$-0.332117\pi$
$397$	20.0566	1.00661	0.503306	+	0.864108i	$0.332117\pi$
$398$	−62.8143	−3.14860
$399$	−2.72595	−0.136468
$400$	−69.2990	−3.46495
$401$	−1.89294	−0.0945290	−0.0472645	−	0.998882i	$-0.515050\pi$
$401$	−1.89294	−0.0945290	−0.0472645	+	0.998882i	$0.515050\pi$
$402$	34.9179	1.74154
$403$	9.50399	0.473427
$404$	−53.7737	−2.67534
$405$	0.130231	0.00647123
$406$	20.3741	1.01115
$407$	35.7783	1.77346
$408$	0.566056	0.0280240
$409$	−8.73941	−0.432136	−0.216068	−	0.976378i	$-0.569323\pi$
$409$	−8.73941	−0.432136	−0.216068	+	0.976378i	$0.569323\pi$
$410$	2.24038	0.110645
$411$	−13.6960	−0.675572
$412$	50.0896	2.46774
$413$	9.66024	0.475349
$414$	−18.0846	−0.888808
$415$	0.100070	0.00491225
$416$	−21.8887	−1.07318
$417$	−2.92684	−0.143328
$418$	−24.0419	−1.17593
$419$	−15.8166	−0.772692	−0.386346	−	0.922354i	$-0.626263\pi$
$419$	−15.8166	−0.772692	−0.386346	+	0.922354i	$0.626263\pi$
$420$	−0.696503	−0.0339859
$421$	0.369195	0.0179935	0.00899673	−	0.999960i	$-0.497136\pi$
$421$	0.369195	0.0179935	0.00899673	+	0.999960i	$0.497136\pi$
$422$	−55.0519	−2.67989
$423$	7.87767	0.383025
$424$	62.6800	3.04401
$425$	−0.310778	−0.0150749
$426$	−8.21806	−0.398166
$427$	−3.59541	−0.173994
$428$	62.5918	3.02549
$429$	−3.64353	−0.175911
$430$	−0.851577	−0.0410667
$431$	35.1026	1.69083	0.845416	−	0.534109i	$-0.179353\pi$
$431$	35.1026	1.69083	0.845416	+	0.534109i	$0.179353\pi$
$432$	−13.9070	−0.669099
$433$	−13.6961	−0.658192	−0.329096	−	0.944297i	$-0.606744\pi$
$433$	−13.6961	−0.658192	−0.329096	+	0.944297i	$0.606744\pi$
$434$	−23.0057	−1.10431
$435$	0.978817	0.0469307
$436$	27.6809	1.32568
$437$	18.1859	0.869949
$438$	−2.59040	−0.123774
$439$	−20.7546	−0.990563	−0.495281	−	0.868733i	$-0.664935\pi$
$439$	−20.7546	−0.990563	−0.495281	+	0.868733i	$0.664935\pi$
$440$	−3.84574	−0.183338
$441$	1.00000	0.0476190
$442$	−0.189325	−0.00900526
$443$	−20.9446	−0.995106	−0.497553	−	0.867434i	$-0.665768\pi$
$443$	−20.9446	−0.995106	−0.497553	+	0.867434i	$0.665768\pi$
$444$	−58.8122	−2.79110
$445$	0.330966	0.0156893
$446$	−74.4685	−3.52618
$447$	21.5069	1.01724
$448$	25.1706	1.18920
$449$	30.2255	1.42643	0.713214	−	0.700946i	$-0.247239\pi$
$449$	30.2255	1.42643	0.713214	+	0.700946i	$0.247239\pi$
$450$	13.5078	0.636765
$451$	−20.6480	−0.972277
$452$	46.3646	2.18081
$453$	−2.82899	−0.132917
$454$	55.7382	2.61592
$455$	0.145840	0.00683706
$456$	24.7412	1.15861
$457$	−17.0866	−0.799277	−0.399638	−	0.916673i	$-0.630864\pi$
$457$	−17.0866	−0.799277	−0.399638	+	0.916673i	$0.630864\pi$
$458$	−15.2307	−0.711683
$459$	−0.0623671	−0.00291105
$460$	4.64665	0.216651
$461$	−29.9374	−1.39432	−0.697161	−	0.716914i	$-0.745554\pi$
$461$	−29.9374	−1.39432	−0.697161	+	0.716914i	$0.745554\pi$
$462$	8.81967	0.410328
$463$	16.4784	0.765815	0.382907	−	0.923787i	$-0.374923\pi$
$463$	16.4784	0.765815	0.382907	+	0.923787i	$0.374923\pi$
$464$	−104.525	−4.85245
$465$	−1.10525	−0.0512546
$466$	−21.1445	−0.979499
$467$	−25.0695	−1.16008	−0.580039	−	0.814589i	$-0.696963\pi$
$467$	−25.0695	−1.16008	−0.580039	+	0.814589i	$0.696963\pi$
$468$	5.98922	0.276852
$469$	12.8812	0.594799
$470$	−2.78101	−0.128278
$471$	−7.59643	−0.350025
$472$	−87.6782	−4.03572
$473$	7.84838	0.360869
$474$	−25.6515	−1.17821
$475$	−13.5835	−0.623254
$476$	0.333553	0.0152884
$477$	−6.90598	−0.316203
$478$	−21.3971	−0.978679
$479$	−37.6782	−1.72156	−0.860781	−	0.508976i	$-0.830024\pi$
$479$	−37.6782	−1.72156	−0.860781	+	0.508976i	$0.830024\pi$
$480$	2.54550	0.116186
$481$	12.3146	0.561497
$482$	32.6505	1.48719
$483$	−6.67141	−0.303559
$484$	−2.21537	−0.100699
$485$	0.371189	0.0168548
$486$	2.71076	0.122963
$487$	−31.7521	−1.43882	−0.719412	−	0.694584i	$-0.755588\pi$
$487$	−31.7521	−1.43882	−0.719412	+	0.694584i	$0.755588\pi$
$488$	32.6326	1.47721
$489$	−5.41853	−0.245034
$490$	−0.353025	−0.0159480
$491$	−22.7319	−1.02588	−0.512938	−	0.858425i	$-0.671443\pi$
$491$	−22.7319	−1.02588	−0.512938	+	0.858425i	$0.671443\pi$
$492$	33.9411	1.53018
$493$	−0.468752	−0.0211115
$494$	−8.27503	−0.372311
$495$	0.423717	0.0190446
$496$	118.026	5.29952
$497$	−3.03164	−0.135988
$498$	2.08296	0.0933397
$499$	1.39611	0.0624983	0.0312492	−	0.999512i	$-0.490051\pi$
$499$	1.39611	0.0624983	0.0312492	+	0.999512i	$0.490051\pi$
$500$	−6.95322	−0.310957
$501$	−12.8244	−0.572952
$502$	17.3268	0.773333
$503$	26.4249	1.17823	0.589114	−	0.808050i	$-0.299477\pi$
$503$	26.4249	1.17823	0.589114	+	0.808050i	$0.299477\pi$
$504$	−9.07620	−0.404286
$505$	−1.30941	−0.0582680
$506$	−58.8396	−2.61574
$507$	11.7459	0.521655
$508$	−2.64198	−0.117219
$509$	34.0757	1.51038	0.755189	−	0.655507i	$-0.227545\pi$
$509$	34.0757	1.51038	0.755189	+	0.655507i	$0.227545\pi$
$510$	0.0220171	0.000974935 0
$511$	−0.955600	−0.0422732
$512$	−19.3811	−0.856534
$513$	−2.72595	−0.120353
$514$	72.8340	3.21257
$515$	1.21970	0.0537464
$516$	−12.9011	−0.567941
$517$	25.6306	1.12723
$518$	−29.8091	−1.30974
$519$	3.42175	0.150198
$520$	−1.32367	−0.0580467
$521$	−17.2741	−0.756792	−0.378396	−	0.925644i	$-0.623524\pi$
$521$	−17.2741	−0.756792	−0.378396	+	0.925644i	$0.623524\pi$
$522$	20.3741	0.891750
$523$	17.7980	0.778251	0.389126	−	0.921185i	$-0.372777\pi$
$523$	17.7980	0.778251	0.389126	+	0.921185i	$0.372777\pi$
$524$	−101.103	−4.41669
$525$	4.98304	0.217478
$526$	49.5329	2.15974
$527$	0.529298	0.0230566
$528$	−45.2474	−1.96914
$529$	21.5076	0.935115
$530$	2.43798	0.105899
$531$	9.66024	0.419218
$532$	14.5789	0.632078
$533$	−7.10687	−0.307833
$534$	6.88907	0.298119
$535$	1.52413	0.0658940
$536$	−116.912	−5.04985
$537$	−11.4708	−0.495002
$538$	−55.7691	−2.40438
$539$	3.25358	0.140142
$540$	−0.696503	−0.0299727
$541$	−6.78690	−0.291791	−0.145896	−	0.989300i	$-0.546606\pi$
$541$	−6.78690	−0.291791	−0.145896	+	0.989300i	$0.546606\pi$
$542$	−5.40561	−0.232191
$543$	2.92554	0.125547
$544$	−1.21903	−0.0522654
$545$	0.674040	0.0288727
$546$	3.03565	0.129914
$547$	19.6562	0.840439	0.420220	−	0.907422i	$-0.361953\pi$
$547$	19.6562	0.840439	0.420220	+	0.907422i	$0.361953\pi$
$548$	73.2490	3.12904
$549$	−3.59541	−0.153448
$550$	43.9488	1.87398
$551$	−20.4882	−0.872828
$552$	60.5510	2.57722
$553$	−9.46286	−0.402402
$554$	61.1500	2.59801
$555$	−1.43210	−0.0607892
$556$	15.6534	0.663852
$557$	24.2238	1.02640	0.513199	−	0.858270i	$-0.328460\pi$
$557$	24.2238	1.02640	0.513199	+	0.858270i	$0.328460\pi$
$558$	−23.0057	−0.973909
$559$	2.70134	0.114255
$560$	1.81112	0.0765337
$561$	−0.202916	−0.00856713
$562$	−3.86014	−0.162830
$563$	16.6008	0.699641	0.349821	−	0.936817i	$-0.386243\pi$
$563$	16.6008	0.699641	0.349821	+	0.936817i	$0.386243\pi$
$564$	−42.1314	−1.77405
$565$	1.12899	0.0474972
$566$	14.6770	0.616920
$567$	1.00000	0.0419961
$568$	27.5158	1.15454
$569$	27.7424	1.16302	0.581511	−	0.813539i	$-0.302462\pi$
$569$	27.7424	1.16302	0.581511	+	0.813539i	$0.302462\pi$
$570$	0.962327	0.0403074
$571$	24.3865	1.02054	0.510272	−	0.860013i	$-0.329545\pi$
$571$	24.3865	1.02054	0.510272	+	0.860013i	$0.329545\pi$
$572$	19.4864	0.814767
$573$	11.3119	0.472562
$574$	17.2032	0.718046
$575$	−33.2439	−1.38637
$576$	25.1706	1.04878
$577$	−23.9450	−0.996845	−0.498423	−	0.866934i	$-0.666087\pi$
$577$	−23.9450	−0.996845	−0.498423	+	0.866934i	$0.666087\pi$
$578$	46.0724	1.91636
$579$	−3.48149	−0.144686
$580$	−5.23492	−0.217368
$581$	0.768405	0.0318788
$582$	7.72629	0.320265
$583$	−22.4691	−0.930577
$584$	8.67322	0.358900
$585$	0.145840	0.00602972
$586$	−32.7659	−1.35355
$587$	−33.8816	−1.39844	−0.699222	−	0.714905i	$-0.746470\pi$
$587$	−33.8816	−1.39844	−0.699222	+	0.714905i	$0.746470\pi$
$588$	−5.34821	−0.220557
$589$	23.1346	0.953245
$590$	−3.41030	−0.140400
$591$	−10.9479	−0.450337
$592$	152.929	6.28536
$593$	46.5077	1.90984	0.954921	−	0.296860i	$-0.0959395\pi$
$593$	46.5077	1.90984	0.954921	+	0.296860i	$0.0959395\pi$
$594$	8.81967	0.361875
$595$	0.00812213	0.000332975 0
$596$	−115.024	−4.71155
$597$	−23.1722	−0.948376
$598$	−20.2521	−0.828169
$599$	25.3698	1.03658	0.518291	−	0.855204i	$-0.326568\pi$
$599$	25.3698	1.03658	0.518291	+	0.855204i	$0.326568\pi$
$600$	−45.2271	−1.84639
$601$	16.8004	0.685303	0.342651	−	0.939463i	$-0.388675\pi$
$601$	16.8004	0.685303	0.342651	+	0.939463i	$0.388675\pi$
$602$	−6.53898	−0.266509
$603$	12.8812	0.524563
$604$	15.1300	0.615632
$605$	−0.0539450	−0.00219318
$606$	−27.2554	−1.10717
$607$	−31.3882	−1.27401	−0.637004	−	0.770861i	$-0.719827\pi$
$607$	−31.3882	−1.27401	−0.637004	+	0.770861i	$0.719827\pi$
$608$	−53.2814	−2.16085
$609$	7.51601	0.304564
$610$	1.26927	0.0513911
$611$	8.82183	0.356893
$612$	0.333553	0.0134831
$613$	−44.4574	−1.79562	−0.897808	−	0.440387i	$-0.854841\pi$
$613$	−44.4574	−1.79562	−0.897808	+	0.440387i	$0.854841\pi$
$614$	−10.7444	−0.433608
$615$	0.826478	0.0333268
$616$	−29.5301	−1.18980
$617$	9.45031	0.380455	0.190228	−	0.981740i	$-0.439077\pi$
$617$	9.45031	0.380455	0.190228	+	0.981740i	$0.439077\pi$
$618$	25.3880	1.02126
$619$	18.3975	0.739457	0.369729	−	0.929140i	$-0.379451\pi$
$619$	18.3975	0.739457	0.369729	+	0.929140i	$0.379451\pi$
$620$	5.91109	0.237395
$621$	−6.67141	−0.267714
$622$	22.3643	0.896726
$623$	2.54138	0.101818
$624$	−15.5738	−0.623449
$625$	24.7459	0.989835
$626$	9.25068	0.369732
$627$	−8.86908	−0.354197
$628$	40.6273	1.62121
$629$	0.685826	0.0273457
$630$	−0.353025	−0.0140648
$631$	−17.5690	−0.699409	−0.349705	−	0.936860i	$-0.613718\pi$
$631$	−17.5690	−0.699409	−0.349705	+	0.936860i	$0.613718\pi$
$632$	85.8868	3.41639
$633$	−20.3087	−0.807197
$634$	3.24388	0.128831
$635$	−0.0643332	−0.00255299
$636$	36.9346	1.46455
$637$	1.11985	0.0443702
$638$	66.2887	2.62439
$639$	−3.03164	−0.119930
$640$	−3.79486	−0.150005
$641$	22.8566	0.902782	0.451391	−	0.892326i	$-0.350928\pi$
$641$	22.8566	0.902782	0.451391	+	0.892326i	$0.350928\pi$
$642$	31.7248	1.25208
$643$	−2.16724	−0.0854676	−0.0427338	−	0.999086i	$-0.513607\pi$
$643$	−2.16724	−0.0854676	−0.0427338	+	0.999086i	$0.513607\pi$
$644$	35.6801	1.40599
$645$	−0.314147	−0.0123695
$646$	−0.460854	−0.0181321
$647$	35.0366	1.37743	0.688715	−	0.725033i	$-0.258175\pi$
$647$	35.0366	1.37743	0.688715	+	0.725033i	$0.258175\pi$
$648$	−9.07620	−0.356547
$649$	31.4303	1.23375
$650$	15.1268	0.593321
$651$	−8.48682	−0.332625
$652$	28.9795	1.13492
$653$	−26.0011	−1.01750	−0.508752	−	0.860913i	$-0.669893\pi$
$653$	−26.0011	−1.01750	−0.508752	+	0.860913i	$0.669893\pi$
$654$	14.0302	0.548622
$655$	−2.46188	−0.0961938
$656$	−88.2571	−3.44586
$657$	−0.955600	−0.0372815
$658$	−21.3545	−0.832483
$659$	37.3860	1.45635	0.728175	−	0.685391i	$-0.240369\pi$
$659$	37.3860	1.45635	0.728175	+	0.685391i	$0.240369\pi$
$660$	−2.26613	−0.0882089
$661$	−34.3439	−1.33582	−0.667912	−	0.744240i	$-0.732812\pi$
$661$	−34.3439	−1.33582	−0.667912	+	0.744240i	$0.732812\pi$
$662$	−2.22301	−0.0863998
$663$	−0.0698420	−0.00271244
$664$	−6.97420	−0.270651
$665$	0.355003	0.0137664
$666$	−29.8091	−1.15508
$667$	−50.1423	−1.94152
$668$	68.5876	2.65373
$669$	−27.4714	−1.06211
$670$	−4.54739	−0.175681
$671$	−11.6979	−0.451594
$672$	19.5460	0.754005
$673$	35.1605	1.35534	0.677669	−	0.735367i	$-0.262990\pi$
$673$	35.1605	1.35534	0.677669	+	0.735367i	$0.262990\pi$
$674$	46.2735	1.78239
$675$	4.98304	0.191797
$676$	−62.8197	−2.41614
$677$	13.2935	0.510911	0.255456	−	0.966821i	$-0.417775\pi$
$677$	13.2935	0.510911	0.255456	+	0.966821i	$0.417775\pi$
$678$	23.5000	0.902513
$679$	2.85023	0.109382
$680$	−0.0737181	−0.00282696
$681$	20.5618	0.787932
$682$	−74.8509	−2.86619
$683$	15.3320	0.586661	0.293331	−	0.956011i	$-0.405236\pi$
$683$	15.3320	0.586661	0.293331	+	0.956011i	$0.405236\pi$
$684$	14.5789	0.557440
$685$	1.78364	0.0681493
$686$	−2.71076	−0.103497
$687$	−5.61860	−0.214363
$688$	33.5468	1.27896
$689$	−7.73368	−0.294630
$690$	2.35517	0.0896598
$691$	−36.1856	−1.37657	−0.688283	−	0.725443i	$-0.741635\pi$
$691$	−36.1856	−1.37657	−0.688283	+	0.725443i	$0.741635\pi$
$692$	−18.3003	−0.695672
$693$	3.25358	0.123593
$694$	28.1550	1.06875
$695$	0.381166	0.0144584
$696$	−68.2168	−2.58575
$697$	−0.395797	−0.0149919
$698$	41.3493	1.56509
$699$	−7.80020	−0.295031
$700$	−26.6504	−1.00729
$701$	37.9834	1.43461	0.717307	−	0.696757i	$-0.245374\pi$
$701$	37.9834	1.43461	0.717307	+	0.696757i	$0.245374\pi$
$702$	3.03565	0.114573
$703$	29.9761	1.13057
$704$	81.8947	3.08652
$705$	−1.02592	−0.0386382
$706$	64.0134	2.40918
$707$	−10.0545	−0.378139
$708$	−51.6650	−1.94169
$709$	−6.57688	−0.247000	−0.123500	−	0.992345i	$-0.539412\pi$
$709$	−6.57688	−0.247000	−0.123500	+	0.992345i	$0.539412\pi$
$710$	1.07025	0.0401656
$711$	−9.46286	−0.354885
$712$	−23.0661	−0.864438
$713$	56.6190	2.12040
$714$	0.169062	0.00632699
$715$	0.474501	0.0177453
$716$	61.3483	2.29269
$717$	−7.89339	−0.294784
$718$	−1.43626	−0.0536009
$719$	6.11475	0.228042	0.114021	−	0.993478i	$-0.463627\pi$
$719$	6.11475	0.228042	0.114021	+	0.993478i	$0.463627\pi$
$720$	1.81112	0.0674964
$721$	9.36566	0.348795
$722$	31.3614	1.16715
$723$	12.0448	0.447950
$724$	−15.6464	−0.581495
$725$	37.4526	1.39095
$726$	−1.12287	−0.0416735
$727$	2.46466	0.0914092	0.0457046	−	0.998955i	$-0.485447\pi$
$727$	2.46466	0.0914092	0.0457046	+	0.998955i	$0.485447\pi$
$728$	−10.1640	−0.376703
$729$	1.00000	0.0370370
$730$	0.337350	0.0124859
$731$	0.150444	0.00556437
$732$	19.2290	0.710724
$733$	3.34622	0.123595	0.0617977	−	0.998089i	$-0.480317\pi$
$733$	3.34622	0.123595	0.0617977	+	0.998089i	$0.480317\pi$
$734$	92.4188	3.41124
$735$	−0.130231	−0.00480364
$736$	−130.400	−4.80659
$737$	41.9100	1.54378
$738$	17.2032	0.633257
$739$	−24.8920	−0.915666	−0.457833	−	0.889038i	$-0.651374\pi$
$739$	−24.8920	−0.915666	−0.457833	+	0.889038i	$0.651374\pi$
$740$	7.65917	0.281557
$741$	−3.05266	−0.112142
$742$	18.7204	0.687249
$743$	−5.19266	−0.190500	−0.0952501	−	0.995453i	$-0.530365\pi$
$743$	−5.19266	−0.190500	−0.0952501	+	0.995453i	$0.530365\pi$
$744$	77.0281	2.82399
$745$	−2.80086	−0.102616
$746$	45.9400	1.68198
$747$	0.768405	0.0281145
$748$	1.08524	0.0396803
$749$	11.7033	0.427629
$750$	−3.52426	−0.128688
$751$	24.9740	0.911315	0.455658	−	0.890155i	$-0.349404\pi$
$751$	24.9740	0.911315	0.455658	+	0.890155i	$0.349404\pi$
$752$	109.554	3.99504
$753$	6.39187	0.232933
$754$	22.8160	0.830909
$755$	0.368422	0.0134082
$756$	−5.34821	−0.194513
$757$	−26.0322	−0.946158	−0.473079	−	0.881020i	$-0.656857\pi$
$757$	−26.0322	−0.946158	−0.473079	+	0.881020i	$0.656857\pi$
$758$	−68.4503	−2.48623
$759$	−21.7059	−0.787876
$760$	−3.22208	−0.116877
$761$	9.77322	0.354279	0.177139	−	0.984186i	$-0.443316\pi$
$761$	9.77322	0.354279	0.177139	+	0.984186i	$0.443316\pi$
$762$	−1.33910	−0.0485104
$763$	5.17573	0.187374
$764$	−60.4985	−2.18876
$765$	0.00812213	0.000293656 0
$766$	2.71076	0.0979437
$767$	10.8180	0.390617
$768$	−28.6489	−1.03378
$769$	44.8402	1.61698	0.808490	−	0.588510i	$-0.200285\pi$
$769$	44.8402	1.61698	0.808490	+	0.588510i	$0.200285\pi$
$770$	−1.14859	−0.0413924
$771$	26.8685	0.967645
$772$	18.6198	0.670140
$773$	−15.1776	−0.545899	−0.272950	−	0.962028i	$-0.587999\pi$
$773$	−15.1776	−0.545899	−0.272950	+	0.962028i	$0.587999\pi$
$774$	−6.53898	−0.235039
$775$	−42.2901	−1.51911
$776$	−25.8693	−0.928653
$777$	−10.9966	−0.394501
$778$	−25.5281	−0.915227
$779$	−17.2995	−0.619820
$780$	−0.779981	−0.0279278
$781$	−9.86370	−0.352951
$782$	−1.12788	−0.0403330
$783$	7.51601	0.268600
$784$	13.9070	0.496677
$785$	0.989290	0.0353093
$786$	−51.2441	−1.82782
$787$	−6.14414	−0.219015	−0.109507	−	0.993986i	$-0.534927\pi$
$787$	−6.14414	−0.219015	−0.109507	+	0.993986i	$0.534927\pi$
$788$	58.5518	2.08582
$789$	18.2727	0.650525
$790$	3.34062	0.118854
$791$	8.66917	0.308240
$792$	−29.5301	−1.04931
$793$	−4.02633	−0.142979
$794$	−54.3686	−1.92947
$795$	0.899372	0.0318974
$796$	123.930	4.39258
$797$	−30.1003	−1.06621	−0.533104	−	0.846050i	$-0.678975\pi$
$797$	−30.1003	−1.06621	−0.533104	+	0.846050i	$0.678975\pi$
$798$	7.38938	0.261581
$799$	0.491307	0.0173812
$800$	97.3987	3.44356
$801$	2.54138	0.0897952
$802$	5.13131	0.181193
$803$	−3.10912	−0.109718
$804$	−68.8915	−2.42961
$805$	0.868824	0.0306220
$806$	−25.7630	−0.907464
$807$	−20.5732	−0.724212
$808$	91.2569	3.21041
$809$	5.38338	0.189270	0.0946348	−	0.995512i	$-0.469832\pi$
$809$	5.38338	0.189270	0.0946348	+	0.995512i	$0.469832\pi$
$810$	−0.353025	−0.0124040
$811$	0.721790	0.0253455	0.0126727	−	0.999920i	$-0.495966\pi$
$811$	0.721790	0.0253455	0.0126727	+	0.999920i	$0.495966\pi$
$812$	−40.1972	−1.41065
$813$	−1.99413	−0.0699373
$814$	−96.9864	−3.39937
$815$	0.705660	0.0247182
$816$	−0.867337	−0.0303629
$817$	6.57561	0.230052
$818$	23.6904	0.828317
$819$	1.11985	0.0391308
$820$	−4.42018	−0.154359
$821$	28.8889	1.00823	0.504115	−	0.863637i	$-0.331819\pi$
$821$	28.8889	1.00823	0.504115	+	0.863637i	$0.331819\pi$
$822$	37.1265	1.29493
$823$	−6.54311	−0.228078	−0.114039	−	0.993476i	$-0.536379\pi$
$823$	−6.54311	−0.228078	−0.114039	+	0.993476i	$0.536379\pi$
$824$	−85.0046	−2.96128
$825$	16.2127	0.564454
$826$	−26.1866	−0.911147
$827$	39.3333	1.36775	0.683876	−	0.729599i	$-0.260293\pi$
$827$	39.3333	1.36775	0.683876	+	0.729599i	$0.260293\pi$
$828$	35.6801	1.23997
$829$	1.81794	0.0631396	0.0315698	−	0.999502i	$-0.489949\pi$
$829$	1.81794	0.0631396	0.0315698	+	0.999502i	$0.489949\pi$
$830$	−0.271266	−0.00941578
$831$	22.5582	0.782537
$832$	28.1874	0.977223
$833$	0.0623671	0.00216089
$834$	7.93397	0.274731
$835$	1.67013	0.0577974
$836$	47.4338	1.64053
$837$	−8.48682	−0.293347
$838$	42.8750	1.48109
$839$	38.7823	1.33891	0.669457	−	0.742851i	$-0.266527\pi$
$839$	38.7823	1.33891	0.669457	+	0.742851i	$0.266527\pi$
$840$	1.18200	0.0407830
$841$	27.4904	0.947944
$842$	−1.00080	−0.0344898
$843$	−1.42401	−0.0490454
$844$	108.615	3.73869
$845$	−1.52968	−0.0526227
$846$	−21.3545	−0.734181
$847$	−0.414226	−0.0142330
$848$	−96.0412	−3.29807
$849$	5.41435	0.185820
$850$	0.842443	0.0288956
$851$	73.3628	2.51484
$852$	16.2139	0.555479
$853$	33.2662	1.13901	0.569506	−	0.821987i	$-0.307135\pi$
$853$	33.2662	1.13901	0.569506	+	0.821987i	$0.307135\pi$
$854$	9.74628	0.333511
$855$	0.355003	0.0121408
$856$	−106.222	−3.63058
$857$	2.22266	0.0759245	0.0379622	−	0.999279i	$-0.487913\pi$
$857$	2.22266	0.0759245	0.0379622	+	0.999279i	$0.487913\pi$
$858$	9.87673	0.337186
$859$	47.6152	1.62461	0.812305	−	0.583233i	$-0.198212\pi$
$859$	47.6152	1.62461	0.812305	+	0.583233i	$0.198212\pi$
$860$	1.68013	0.0572918
$861$	6.34625	0.216280
$862$	−95.1546	−3.24098
$863$	−2.91403	−0.0991948	−0.0495974	−	0.998769i	$-0.515794\pi$
$863$	−2.91403	−0.0991948	−0.0495974	+	0.998769i	$0.515794\pi$
$864$	19.5460	0.664970
$865$	−0.445618	−0.0151515
$866$	37.1268	1.26162
$867$	16.9961	0.577218
$868$	45.3893	1.54061
$869$	−30.7882	−1.04442
$870$	−2.65334	−0.0899566
$871$	14.4251	0.488775
$872$	−46.9760	−1.59081
$873$	2.85023	0.0964658
$874$	−49.2976	−1.66752
$875$	−1.30010	−0.0439514
$876$	5.11075	0.172676
$877$	7.83945	0.264720	0.132360	−	0.991202i	$-0.457745\pi$
$877$	7.83945	0.264720	0.132360	+	0.991202i	$0.457745\pi$
$878$	56.2607	1.89871
$879$	−12.0874	−0.407697
$880$	5.89261	0.198640
$881$	−38.3717	−1.29278	−0.646389	−	0.763008i	$-0.723721\pi$
$881$	−38.3717	−1.29278	−0.646389	+	0.763008i	$0.723721\pi$
$882$	−2.71076	−0.0912760
$883$	17.7826	0.598431	0.299216	−	0.954186i	$-0.403275\pi$
$883$	17.7826	0.598431	0.299216	+	0.954186i	$0.403275\pi$
$884$	0.373530	0.0125632
$885$	−1.25806	−0.0422893
$886$	56.7756	1.90741
$887$	1.22071	0.0409875	0.0204938	−	0.999790i	$-0.493476\pi$
$887$	1.22071	0.0409875	0.0204938	+	0.999790i	$0.493476\pi$
$888$	99.8074	3.34932
$889$	−0.493993	−0.0165680
$890$	−0.897170	−0.0300732
$891$	3.25358	0.108999
$892$	146.923	4.91935
$893$	21.4741	0.718603
$894$	−58.3000	−1.94984
$895$	1.49385	0.0499340
$896$	−29.1395	−0.973482
$897$	−7.47099	−0.249449
$898$	−81.9340	−2.73417
$899$	−63.7870	−2.12741
$900$	−26.6504	−0.888345
$901$	−0.430706	−0.0143489
$902$	55.9718	1.86366
$903$	−2.41223	−0.0802740
$904$	−78.6831	−2.61696
$905$	−0.380996	−0.0126647
$906$	7.66870	0.254776
$907$	−22.1739	−0.736274	−0.368137	−	0.929772i	$-0.620004\pi$
$907$	−22.1739	−0.736274	−0.368137	+	0.929772i	$0.620004\pi$
$908$	−109.969	−3.64945
$909$	−10.0545	−0.333487
$910$	−0.395336	−0.0131053
$911$	−13.8113	−0.457588	−0.228794	−	0.973475i	$-0.573478\pi$
$911$	−13.8113	−0.457588	−0.228794	+	0.973475i	$0.573478\pi$
$912$	−37.9097	−1.25531
$913$	2.50007	0.0827401
$914$	46.3176	1.53205
$915$	0.468233	0.0154793
$916$	30.0495	0.992863
$917$	−18.9040	−0.624265
$918$	0.169062	0.00557988
$919$	−54.5506	−1.79946	−0.899730	−	0.436448i	$-0.856236\pi$
$919$	−54.5506	−1.79946	−0.899730	+	0.436448i	$0.856236\pi$
$920$	−7.88562	−0.259981
$921$	−3.96360	−0.130605
$922$	81.1530	2.67263
$923$	−3.39500	−0.111748
$924$	−17.4008	−0.572445
$925$	−54.7965	−1.80170
$926$	−44.6689	−1.46791
$927$	9.36566	0.307609
$928$	146.908	4.82250
$929$	22.3940	0.734724	0.367362	−	0.930078i	$-0.380261\pi$
$929$	22.3940	0.734724	0.367362	+	0.930078i	$0.380261\pi$
$930$	2.99606	0.0982446
$931$	2.72595	0.0893393
$932$	41.7172	1.36649
$933$	8.25019	0.270099
$934$	67.9574	2.22363
$935$	0.0264260	0.000864222 0
$936$	−10.1640	−0.332221
$937$	−16.2670	−0.531419	−0.265710	−	0.964053i	$-0.585606\pi$
$937$	−16.2670	−0.531419	−0.265710	+	0.964053i	$0.585606\pi$
$938$	−34.9179	−1.14011
$939$	3.41258	0.111365
$940$	5.48682	0.178960
$941$	42.6848	1.39149	0.695743	−	0.718291i	$-0.255075\pi$
$941$	42.6848	1.39149	0.695743	+	0.718291i	$0.255075\pi$
$942$	20.5921	0.670926
$943$	−42.3384	−1.37873
$944$	134.345	4.37254
$945$	−0.130231	−0.00423641
$946$	−21.2751	−0.691712
$947$	−2.02208	−0.0657088	−0.0328544	−	0.999460i	$-0.510460\pi$
$947$	−2.02208	−0.0657088	−0.0328544	+	0.999460i	$0.510460\pi$
$948$	50.6094	1.64372
$949$	−1.07013	−0.0347379
$950$	36.8216	1.19465
$951$	1.19667	0.0388047
$952$	−0.566056	−0.0183460
$953$	44.7699	1.45024	0.725119	−	0.688624i	$-0.241785\pi$
$953$	44.7699	1.45024	0.725119	+	0.688624i	$0.241785\pi$
$954$	18.7204	0.606097
$955$	−1.47316	−0.0476703
$956$	42.2155	1.36535
$957$	24.4539	0.790483
$958$	102.137	3.29988
$959$	13.6960	0.442266
$960$	−3.27800	−0.105797
$961$	41.0260	1.32342
$962$	−33.3819	−1.07627
$963$	11.7033	0.377134
$964$	−64.4180	−2.07476
$965$	0.453398	0.0145954
$966$	18.0846	0.581862
$967$	22.2040	0.714032	0.357016	−	0.934098i	$-0.383794\pi$
$967$	22.2040	0.714032	0.357016	+	0.934098i	$0.383794\pi$
$968$	3.75960	0.120838
$969$	−0.170009	−0.00546149
$970$	−1.00620	−0.0323072
$971$	16.5395	0.530778	0.265389	−	0.964141i	$-0.414500\pi$
$971$	16.5395	0.530778	0.265389	+	0.964141i	$0.414500\pi$
$972$	−5.34821	−0.171544
$973$	2.92684	0.0938303
$974$	86.0722	2.75793
$975$	5.58027	0.178712
$976$	−50.0012	−1.60050
$977$	39.3154	1.25781	0.628906	−	0.777482i	$-0.283503\pi$
$977$	39.3154	1.25781	0.628906	+	0.777482i	$0.283503\pi$
$978$	14.6883	0.469681
$979$	8.26858	0.264265
$980$	0.696503	0.0222490
$981$	5.17573	0.165248
$982$	61.6207	1.96640
$983$	3.20276	0.102152	0.0510761	−	0.998695i	$-0.483735\pi$
$983$	3.20276	0.102152	0.0510761	+	0.998695i	$0.483735\pi$
$984$	−57.5998	−1.83622
$985$	1.42576	0.0454284
$986$	1.27067	0.0404665
$987$	−7.87767	−0.250749
$988$	16.3263	0.519408
$989$	16.0930	0.511727
$990$	−1.14859	−0.0365047
$991$	19.0357	0.604689	0.302344	−	0.953199i	$-0.402231\pi$
$991$	19.0357	0.604689	0.302344	+	0.953199i	$0.402231\pi$
$992$	−165.884	−5.26681
$993$	−0.820069	−0.0260241
$994$	8.21806	0.260661
$995$	3.01774	0.0956689
$996$	−4.10960	−0.130218
$997$	55.0507	1.74347	0.871736	−	0.489976i	$-0.162994\pi$
$997$	55.0507	1.74347	0.871736	+	0.489976i	$0.162994\pi$
$998$	−3.78451	−0.119797
$999$	−10.9966	−0.347917

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.t.1.2	✓	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.t.1.2	✓	52	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-2.71076 q^{2} -1.00000 q^{3} +5.34821 q^{4} +0.130231 q^{5} +2.71076 q^{6} +1.00000 q^{7} -9.07620 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.71076 q^{2} -1.00000 q^{3} +5.34821 q^{4} +0.130231 q^{5} +2.71076 q^{6} +1.00000 q^{7} -9.07620 q^{8} +1.00000 q^{9} -0.353025 q^{10} +3.25358 q^{11} -5.34821 q^{12} +1.11985 q^{13} -2.71076 q^{14} -0.130231 q^{15} +13.9070 q^{16} +0.0623671 q^{17} -2.71076 q^{18} +2.72595 q^{19} +0.696503 q^{20} -1.00000 q^{21} -8.81967 q^{22} +6.67141 q^{23} +9.07620 q^{24} -4.98304 q^{25} -3.03565 q^{26} -1.00000 q^{27} +5.34821 q^{28} -7.51601 q^{29} +0.353025 q^{30} +8.48682 q^{31} -19.5460 q^{32} -3.25358 q^{33} -0.169062 q^{34} +0.130231 q^{35} +5.34821 q^{36} +10.9966 q^{37} -7.38938 q^{38} -1.11985 q^{39} -1.18200 q^{40} -6.34625 q^{41} +2.71076 q^{42} +2.41223 q^{43} +17.4008 q^{44} +0.130231 q^{45} -18.0846 q^{46} +7.87767 q^{47} -13.9070 q^{48} +1.00000 q^{49} +13.5078 q^{50} -0.0623671 q^{51} +5.98922 q^{52} -6.90598 q^{53} +2.71076 q^{54} +0.423717 q^{55} -9.07620 q^{56} -2.72595 q^{57} +20.3741 q^{58} +9.66024 q^{59} -0.696503 q^{60} -3.59541 q^{61} -23.0057 q^{62} +1.00000 q^{63} +25.1706 q^{64} +0.145840 q^{65} +8.81967 q^{66} +12.8812 q^{67} +0.333553 q^{68} -6.67141 q^{69} -0.353025 q^{70} -3.03164 q^{71} -9.07620 q^{72} -0.955600 q^{73} -29.8091 q^{74} +4.98304 q^{75} +14.5789 q^{76} +3.25358 q^{77} +3.03565 q^{78} -9.46286 q^{79} +1.81112 q^{80} +1.00000 q^{81} +17.2032 q^{82} +0.768405 q^{83} -5.34821 q^{84} +0.00812213 q^{85} -6.53898 q^{86} +7.51601 q^{87} -29.5301 q^{88} +2.54138 q^{89} -0.353025 q^{90} +1.11985 q^{91} +35.6801 q^{92} -8.48682 q^{93} -21.3545 q^{94} +0.355003 q^{95} +19.5460 q^{96} +2.85023 q^{97} -2.71076 q^{98} +3.25358 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * q^99

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