LMFDB - Embedded newform 8038.2.a.a.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.25539 q^{3} +1.00000 q^{4} -2.85398 q^{5} -3.25539 q^{6} +2.81527 q^{7} +1.00000 q^{8} +7.59759 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.25539 q^{3} +1.00000 q^{4} -2.85398 q^{5} -3.25539 q^{6} +2.81527 q^{7} +1.00000 q^{8} +7.59759 q^{9} -2.85398 q^{10} -4.01480 q^{11} -3.25539 q^{12} +2.56777 q^{13} +2.81527 q^{14} +9.29083 q^{15} +1.00000 q^{16} -5.40140 q^{17} +7.59759 q^{18} +6.04980 q^{19} -2.85398 q^{20} -9.16482 q^{21} -4.01480 q^{22} -4.00444 q^{23} -3.25539 q^{24} +3.14520 q^{25} +2.56777 q^{26} -14.9670 q^{27} +2.81527 q^{28} +3.52662 q^{29} +9.29083 q^{30} -0.948510 q^{31} +1.00000 q^{32} +13.0698 q^{33} -5.40140 q^{34} -8.03472 q^{35} +7.59759 q^{36} +1.32820 q^{37} +6.04980 q^{38} -8.35910 q^{39} -2.85398 q^{40} -2.76617 q^{41} -9.16482 q^{42} -8.60816 q^{43} -4.01480 q^{44} -21.6834 q^{45} -4.00444 q^{46} -4.54535 q^{47} -3.25539 q^{48} +0.925751 q^{49} +3.14520 q^{50} +17.5837 q^{51} +2.56777 q^{52} +10.8674 q^{53} -14.9670 q^{54} +11.4582 q^{55} +2.81527 q^{56} -19.6945 q^{57} +3.52662 q^{58} +5.49354 q^{59} +9.29083 q^{60} +14.3046 q^{61} -0.948510 q^{62} +21.3893 q^{63} +1.00000 q^{64} -7.32836 q^{65} +13.0698 q^{66} +6.93307 q^{67} -5.40140 q^{68} +13.0360 q^{69} -8.03472 q^{70} -10.7906 q^{71} +7.59759 q^{72} +5.80848 q^{73} +1.32820 q^{74} -10.2389 q^{75} +6.04980 q^{76} -11.3028 q^{77} -8.35910 q^{78} +6.90231 q^{79} -2.85398 q^{80} +25.9306 q^{81} -2.76617 q^{82} -17.1131 q^{83} -9.16482 q^{84} +15.4155 q^{85} -8.60816 q^{86} -11.4805 q^{87} -4.01480 q^{88} -15.0321 q^{89} -21.6834 q^{90} +7.22896 q^{91} -4.00444 q^{92} +3.08778 q^{93} -4.54535 q^{94} -17.2660 q^{95} -3.25539 q^{96} +5.03150 q^{97} +0.925751 q^{98} -30.5028 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) $ 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.25539	−1.87950	−0.939751	−	0.341859i	$-0.888944\pi$
$3$	−3.25539	−1.87950	−0.939751	+	0.341859i	$0.888944\pi$
$4$	1.00000	0.500000
$5$	−2.85398	−1.27634	−0.638169	−	0.769896i	$-0.720308\pi$
$5$	−2.85398	−1.27634	−0.638169	+	0.769896i	$0.720308\pi$
$6$	−3.25539	−1.32901
$7$	2.81527	1.06407	0.532036	−	0.846722i	$-0.321427\pi$
$7$	2.81527	1.06407	0.532036	+	0.846722i	$0.321427\pi$
$8$	1.00000	0.353553
$9$	7.59759	2.53253
$10$	−2.85398	−0.902507
$11$	−4.01480	−1.21051	−0.605254	−	0.796032i	$-0.706929\pi$
$11$	−4.01480	−1.21051	−0.605254	+	0.796032i	$0.706929\pi$
$12$	−3.25539	−0.939751
$13$	2.56777	0.712171	0.356085	−	0.934453i	$-0.384111\pi$
$13$	2.56777	0.712171	0.356085	+	0.934453i	$0.384111\pi$
$14$	2.81527	0.752413
$15$	9.29083	2.39888
$16$	1.00000	0.250000
$17$	−5.40140	−1.31003	−0.655016	−	0.755615i	$-0.727338\pi$
$17$	−5.40140	−1.31003	−0.655016	+	0.755615i	$0.727338\pi$
$18$	7.59759	1.79077
$19$	6.04980	1.38792	0.693959	−	0.720014i	$-0.255865\pi$
$19$	6.04980	1.38792	0.693959	+	0.720014i	$0.255865\pi$
$20$	−2.85398	−0.638169
$21$	−9.16482	−1.99993
$22$	−4.01480	−0.855959
$23$	−4.00444	−0.834984	−0.417492	−	0.908681i	$-0.637091\pi$
$23$	−4.00444	−0.834984	−0.417492	+	0.908681i	$0.637091\pi$
$24$	−3.25539	−0.664505
$25$	3.14520	0.629039
$26$	2.56777	0.503581
$27$	−14.9670	−2.88039
$28$	2.81527	0.532036
$29$	3.52662	0.654876	0.327438	−	0.944873i	$-0.393815\pi$
$29$	3.52662	0.654876	0.327438	+	0.944873i	$0.393815\pi$
$30$	9.29083	1.69626
$31$	−0.948510	−0.170358	−0.0851788	−	0.996366i	$-0.527146\pi$
$31$	−0.948510	−0.170358	−0.0851788	+	0.996366i	$0.527146\pi$
$32$	1.00000	0.176777
$33$	13.0698	2.27515
$34$	−5.40140	−0.926333
$35$	−8.03472	−1.35812
$36$	7.59759	1.26627
$37$	1.32820	0.218355	0.109177	−	0.994022i	$-0.465178\pi$
$37$	1.32820	0.218355	0.109177	+	0.994022i	$0.465178\pi$
$38$	6.04980	0.981407
$39$	−8.35910	−1.33853
$40$	−2.85398	−0.451254
$41$	−2.76617	−0.432003	−0.216001	−	0.976393i	$-0.569302\pi$
$41$	−2.76617	−0.432003	−0.216001	+	0.976393i	$0.569302\pi$
$42$	−9.16482	−1.41416
$43$	−8.60816	−1.31273	−0.656366	−	0.754442i	$-0.727907\pi$
$43$	−8.60816	−1.31273	−0.656366	+	0.754442i	$0.727907\pi$
$44$	−4.01480	−0.605254
$45$	−21.6834	−3.23236
$46$	−4.00444	−0.590423
$47$	−4.54535	−0.663007	−0.331504	−	0.943454i	$-0.607556\pi$
$47$	−4.54535	−0.663007	−0.331504	+	0.943454i	$0.607556\pi$
$48$	−3.25539	−0.469876
$49$	0.925751	0.132250
$50$	3.14520	0.444798
$51$	17.5837	2.46221
$52$	2.56777	0.356085
$53$	10.8674	1.49275	0.746375	−	0.665526i	$-0.231793\pi$
$53$	10.8674	1.49275	0.746375	+	0.665526i	$0.231793\pi$
$54$	−14.9670	−2.03675
$55$	11.4582	1.54502
$56$	2.81527	0.376206
$57$	−19.6945	−2.60860
$58$	3.52662	0.463068
$59$	5.49354	0.715198	0.357599	−	0.933875i	$-0.383595\pi$
$59$	5.49354	0.715198	0.357599	+	0.933875i	$0.383595\pi$
$60$	9.29083	1.19944
$61$	14.3046	1.83151	0.915755	−	0.401737i	$-0.131593\pi$
$61$	14.3046	1.83151	0.915755	+	0.401737i	$0.131593\pi$
$62$	−0.948510	−0.120461
$63$	21.3893	2.69480
$64$	1.00000	0.125000
$65$	−7.32836	−0.908971
$66$	13.0698	1.60878
$67$	6.93307	0.847010	0.423505	−	0.905894i	$-0.360800\pi$
$67$	6.93307	0.847010	0.423505	+	0.905894i	$0.360800\pi$
$68$	−5.40140	−0.655016
$69$	13.0360	1.56935
$70$	−8.03472	−0.960333
$71$	−10.7906	−1.28060	−0.640302	−	0.768123i	$-0.721191\pi$
$71$	−10.7906	−1.28060	−0.640302	+	0.768123i	$0.721191\pi$
$72$	7.59759	0.895385
$73$	5.80848	0.679831	0.339916	−	0.940456i	$-0.389601\pi$
$73$	5.80848	0.679831	0.339916	+	0.940456i	$0.389601\pi$
$74$	1.32820	0.154400
$75$	−10.2389	−1.18228
$76$	6.04980	0.693959
$77$	−11.3028	−1.28807
$78$	−8.35910	−0.946481
$79$	6.90231	0.776571	0.388285	−	0.921539i	$-0.373067\pi$
$79$	6.90231	0.776571	0.388285	+	0.921539i	$0.373067\pi$
$80$	−2.85398	−0.319085
$81$	25.9306	2.88118
$82$	−2.76617	−0.305472
$83$	−17.1131	−1.87841	−0.939204	−	0.343360i	$-0.888435\pi$
$83$	−17.1131	−1.87841	−0.939204	+	0.343360i	$0.888435\pi$
$84$	−9.16482	−0.999963
$85$	15.4155	1.67204
$86$	−8.60816	−0.928242
$87$	−11.4805	−1.23084
$88$	−4.01480	−0.427979
$89$	−15.0321	−1.59340	−0.796700	−	0.604375i	$-0.793423\pi$
$89$	−15.0321	−1.59340	−0.796700	+	0.604375i	$0.793423\pi$
$90$	−21.6834	−2.28563
$91$	7.22896	0.757801
$92$	−4.00444	−0.417492
$93$	3.08778	0.320187
$94$	−4.54535	−0.468817
$95$	−17.2660	−1.77145
$96$	−3.25539	−0.332252
$97$	5.03150	0.510871	0.255436	−	0.966826i	$-0.417781\pi$
$97$	5.03150	0.510871	0.255436	+	0.966826i	$0.417781\pi$
$98$	0.925751	0.0935149
$99$	−30.5028	−3.06565
$100$	3.14520	0.314520
$101$	11.0171	1.09624	0.548119	−	0.836400i	$-0.315344\pi$
$101$	11.0171	1.09624	0.548119	+	0.836400i	$0.315344\pi$
$102$	17.5837	1.74104
$103$	−2.21020	−0.217778	−0.108889	−	0.994054i	$-0.534729\pi$
$103$	−2.21020	−0.217778	−0.108889	+	0.994054i	$0.534729\pi$
$104$	2.56777	0.251790
$105$	26.1562	2.55258
$106$	10.8674	1.05553
$107$	18.5674	1.79498	0.897489	−	0.441036i	$-0.145389\pi$
$107$	18.5674	1.79498	0.897489	+	0.441036i	$0.145389\pi$
$108$	−14.9670	−1.44020
$109$	−10.1999	−0.976976	−0.488488	−	0.872570i	$-0.662451\pi$
$109$	−10.1999	−0.976976	−0.488488	+	0.872570i	$0.662451\pi$
$110$	11.4582	1.09249
$111$	−4.32381	−0.410398
$112$	2.81527	0.266018
$113$	−14.3639	−1.35124	−0.675622	−	0.737249i	$-0.736125\pi$
$113$	−14.3639	−1.35124	−0.675622	+	0.737249i	$0.736125\pi$
$114$	−19.6945	−1.84456
$115$	11.4286	1.06572
$116$	3.52662	0.327438
$117$	19.5089	1.80359
$118$	5.49354	0.505722
$119$	−15.2064	−1.39397
$120$	9.29083	0.848132
$121$	5.11863	0.465330
$122$	14.3046	1.29507
$123$	9.00497	0.811951
$124$	−0.948510	−0.0851788
$125$	5.29357	0.473472
$126$	21.3893	1.90551
$127$	10.2520	0.909719	0.454859	−	0.890563i	$-0.349690\pi$
$127$	10.2520	0.909719	0.454859	+	0.890563i	$0.349690\pi$
$128$	1.00000	0.0883883
$129$	28.0230	2.46728
$130$	−7.32836	−0.642739
$131$	6.45229	0.563739	0.281870	−	0.959453i	$-0.409045\pi$
$131$	6.45229	0.563739	0.281870	+	0.959453i	$0.409045\pi$
$132$	13.0698	1.13758
$133$	17.0318	1.47685
$134$	6.93307	0.598926
$135$	42.7154	3.67636
$136$	−5.40140	−0.463166
$137$	8.07482	0.689878	0.344939	−	0.938625i	$-0.387900\pi$
$137$	8.07482	0.689878	0.344939	+	0.938625i	$0.387900\pi$
$138$	13.0360	1.10970
$139$	7.51806	0.637673	0.318837	−	0.947810i	$-0.396708\pi$
$139$	7.51806	0.637673	0.318837	+	0.947810i	$0.396708\pi$
$140$	−8.03472	−0.679058
$141$	14.7969	1.24612
$142$	−10.7906	−0.905524
$143$	−10.3091	−0.862089
$144$	7.59759	0.633133
$145$	−10.0649	−0.835844
$146$	5.80848	0.480713
$147$	−3.01368	−0.248564
$148$	1.32820	0.109177
$149$	22.2479	1.82262	0.911309	−	0.411723i	$-0.135073\pi$
$149$	22.2479	1.82262	0.911309	+	0.411723i	$0.135073\pi$
$150$	−10.2389	−0.835999
$151$	17.3068	1.40841	0.704204	−	0.709998i	$-0.251304\pi$
$151$	17.3068	1.40841	0.704204	+	0.709998i	$0.251304\pi$
$152$	6.04980	0.490703
$153$	−41.0376	−3.31770
$154$	−11.3028	−0.910802
$155$	2.70703	0.217434
$156$	−8.35910	−0.669263
$157$	−19.7147	−1.57340	−0.786700	−	0.617335i	$-0.788212\pi$
$157$	−19.7147	−1.57340	−0.786700	+	0.617335i	$0.788212\pi$
$158$	6.90231	0.549118
$159$	−35.3776	−2.80563
$160$	−2.85398	−0.225627
$161$	−11.2736	−0.888484
$162$	25.9306	2.03730
$163$	−6.97741	−0.546513	−0.273257	−	0.961941i	$-0.588101\pi$
$163$	−6.97741	−0.546513	−0.273257	+	0.961941i	$0.588101\pi$
$164$	−2.76617	−0.216001
$165$	−37.3008	−2.90387
$166$	−17.1131	−1.32823
$167$	6.56893	0.508319	0.254159	−	0.967162i	$-0.418201\pi$
$167$	6.56893	0.508319	0.254159	+	0.967162i	$0.418201\pi$
$168$	−9.16482	−0.707081
$169$	−6.40657	−0.492813
$170$	15.4155	1.18231
$171$	45.9639	3.51495
$172$	−8.60816	−0.656366
$173$	−21.9296	−1.66728	−0.833640	−	0.552309i	$-0.813747\pi$
$173$	−21.9296	−1.66728	−0.833640	+	0.552309i	$0.813747\pi$
$174$	−11.4805	−0.870337
$175$	8.85458	0.669343
$176$	−4.01480	−0.302627
$177$	−17.8836	−1.34422
$178$	−15.0321	−1.12670
$179$	11.0306	0.824463	0.412232	−	0.911079i	$-0.364749\pi$
$179$	11.0306	0.824463	0.412232	+	0.911079i	$0.364749\pi$
$180$	−21.6834	−1.61618
$181$	−12.6759	−0.942196	−0.471098	−	0.882081i	$-0.656142\pi$
$181$	−12.6759	−0.942196	−0.471098	+	0.882081i	$0.656142\pi$
$182$	7.22896	0.535846
$183$	−46.5670	−3.44233
$184$	−4.00444	−0.295211
$185$	−3.79065	−0.278694
$186$	3.08778	0.226407
$187$	21.6856	1.58581
$188$	−4.54535	−0.331504
$189$	−42.1361	−3.06495
$190$	−17.2660	−1.25261
$191$	15.9570	1.15461	0.577303	−	0.816530i	$-0.304105\pi$
$191$	15.9570	1.15461	0.577303	+	0.816530i	$0.304105\pi$
$192$	−3.25539	−0.234938
$193$	−4.62618	−0.333000	−0.166500	−	0.986041i	$-0.553247\pi$
$193$	−4.62618	−0.333000	−0.166500	+	0.986041i	$0.553247\pi$
$194$	5.03150	0.361240
$195$	23.8567	1.70841
$196$	0.925751	0.0661250
$197$	−8.72576	−0.621684	−0.310842	−	0.950462i	$-0.600611\pi$
$197$	−8.72576	−0.621684	−0.310842	+	0.950462i	$0.600611\pi$
$198$	−30.5028	−2.16774
$199$	−13.1987	−0.935633	−0.467817	−	0.883826i	$-0.654959\pi$
$199$	−13.1987	−0.935633	−0.467817	+	0.883826i	$0.654959\pi$
$200$	3.14520	0.222399
$201$	−22.5699	−1.59196
$202$	11.0171	0.775158
$203$	9.92838	0.696836
$204$	17.5837	1.23110
$205$	7.89459	0.551382
$206$	−2.21020	−0.153992
$207$	−30.4241	−2.11462
$208$	2.56777	0.178043
$209$	−24.2887	−1.68009
$210$	26.1562	1.80495
$211$	−8.17724	−0.562945	−0.281472	−	0.959569i	$-0.590823\pi$
$211$	−8.17724	−0.562945	−0.281472	+	0.959569i	$0.590823\pi$
$212$	10.8674	0.746375
$213$	35.1276	2.40690
$214$	18.5674	1.26924
$215$	24.5675	1.67549
$216$	−14.9670	−1.01837
$217$	−2.67031	−0.181273
$218$	−10.1999	−0.690827
$219$	−18.9089	−1.27774
$220$	11.4582	0.772509
$221$	−13.8695	−0.932967
$222$	−4.32381	−0.290195
$223$	−24.9605	−1.67148	−0.835739	−	0.549126i	$-0.814961\pi$
$223$	−24.9605	−1.67148	−0.835739	+	0.549126i	$0.814961\pi$
$224$	2.81527	0.188103
$225$	23.8959	1.59306
$226$	−14.3639	−0.955473
$227$	−6.93787	−0.460483	−0.230241	−	0.973134i	$-0.573952\pi$
$227$	−6.93787	−0.460483	−0.230241	+	0.973134i	$0.573952\pi$
$228$	−19.6945	−1.30430
$229$	−12.2860	−0.811885	−0.405942	−	0.913899i	$-0.633057\pi$
$229$	−12.2860	−0.811885	−0.405942	+	0.913899i	$0.633057\pi$
$230$	11.4286	0.753579
$231$	36.7949	2.42093
$232$	3.52662	0.231534
$233$	3.44387	0.225615	0.112808	−	0.993617i	$-0.464016\pi$
$233$	3.44387	0.225615	0.112808	+	0.993617i	$0.464016\pi$
$234$	19.5089	1.27533
$235$	12.9723	0.846221
$236$	5.49354	0.357599
$237$	−22.4697	−1.45957
$238$	−15.2064	−0.985685
$239$	−16.6622	−1.07779	−0.538893	−	0.842374i	$-0.681157\pi$
$239$	−16.6622	−1.07779	−0.538893	+	0.842374i	$0.681157\pi$
$240$	9.29083	0.599720
$241$	0.662582	0.0426806	0.0213403	−	0.999772i	$-0.493207\pi$
$241$	0.662582	0.0426806	0.0213403	+	0.999772i	$0.493207\pi$
$242$	5.11863	0.329038
$243$	−39.5134	−2.53479
$244$	14.3046	0.915755
$245$	−2.64207	−0.168796
$246$	9.00497	0.574136
$247$	15.5345	0.988435
$248$	−0.948510	−0.0602305
$249$	55.7099	3.53047
$250$	5.29357	0.334795
$251$	−2.35998	−0.148960	−0.0744802	−	0.997222i	$-0.523730\pi$
$251$	−2.35998	−0.148960	−0.0744802	+	0.997222i	$0.523730\pi$
$252$	21.3893	1.34740
$253$	16.0770	1.01076
$254$	10.2520	0.643268
$255$	−50.1835	−3.14261
$256$	1.00000	0.0625000
$257$	19.3162	1.20491	0.602456	−	0.798152i	$-0.294189\pi$
$257$	19.3162	1.20491	0.602456	+	0.798152i	$0.294189\pi$
$258$	28.0230	1.74463
$259$	3.73924	0.232345
$260$	−7.32836	−0.454485
$261$	26.7938	1.65849
$262$	6.45229	0.398624
$263$	−28.1430	−1.73537	−0.867687	−	0.497110i	$-0.834394\pi$
$263$	−28.1430	−1.73537	−0.867687	+	0.497110i	$0.834394\pi$
$264$	13.0698	0.804388
$265$	−31.0153	−1.90525
$266$	17.0318	1.04429
$267$	48.9354	2.99480
$268$	6.93307	0.423505
$269$	−17.3551	−1.05816	−0.529079	−	0.848573i	$-0.677462\pi$
$269$	−17.3551	−1.05816	−0.529079	+	0.848573i	$0.677462\pi$
$270$	42.7154	2.59958
$271$	−23.8023	−1.44589	−0.722944	−	0.690907i	$-0.757211\pi$
$271$	−23.8023	−1.44589	−0.722944	+	0.690907i	$0.757211\pi$
$272$	−5.40140	−0.327508
$273$	−23.5331	−1.42429
$274$	8.07482	0.487818
$275$	−12.6273	−0.761457
$276$	13.0360	0.784677
$277$	−12.6385	−0.759373	−0.379686	−	0.925115i	$-0.623968\pi$
$277$	−12.6385	−0.759373	−0.379686	+	0.925115i	$0.623968\pi$
$278$	7.51806	0.450903
$279$	−7.20639	−0.431436
$280$	−8.03472	−0.480167
$281$	−2.84184	−0.169530	−0.0847648	−	0.996401i	$-0.527014\pi$
$281$	−2.84184	−0.169530	−0.0847648	+	0.996401i	$0.527014\pi$
$282$	14.7969	0.881143
$283$	−6.53062	−0.388205	−0.194102	−	0.980981i	$-0.562179\pi$
$283$	−6.53062	−0.388205	−0.194102	+	0.980981i	$0.562179\pi$
$284$	−10.7906	−0.640302
$285$	56.2076	3.32945
$286$	−10.3091	−0.609589
$287$	−7.78751	−0.459682
$288$	7.59759	0.447692
$289$	12.1751	0.716185
$290$	−10.0649	−0.591031
$291$	−16.3795	−0.960183
$292$	5.80848	0.339916
$293$	−4.40255	−0.257200	−0.128600	−	0.991697i	$-0.541048\pi$
$293$	−4.40255	−0.257200	−0.128600	+	0.991697i	$0.541048\pi$
$294$	−3.01368	−0.175762
$295$	−15.6785	−0.912835
$296$	1.32820	0.0772000
$297$	60.0894	3.48674
$298$	22.2479	1.28879
$299$	−10.2825	−0.594651
$300$	−10.2389	−0.591140
$301$	−24.2343	−1.39684
$302$	17.3068	0.995894
$303$	−35.8649	−2.06038
$304$	6.04980	0.346980
$305$	−40.8249	−2.33763
$306$	−41.0376	−2.34597
$307$	−20.2650	−1.15658	−0.578291	−	0.815830i	$-0.696280\pi$
$307$	−20.2650	−1.15658	−0.578291	+	0.815830i	$0.696280\pi$
$308$	−11.3028	−0.644034
$309$	7.19509	0.409314
$310$	2.70703	0.153749
$311$	−9.11159	−0.516671	−0.258335	−	0.966055i	$-0.583174\pi$
$311$	−9.11159	−0.516671	−0.258335	+	0.966055i	$0.583174\pi$
$312$	−8.35910	−0.473241
$313$	−34.3290	−1.94039	−0.970194	−	0.242330i	$-0.922088\pi$
$313$	−34.3290	−1.94039	−0.970194	+	0.242330i	$0.922088\pi$
$314$	−19.7147	−1.11256
$315$	−61.0445	−3.43947
$316$	6.90231	0.388285
$317$	12.2056	0.685535	0.342768	−	0.939420i	$-0.388636\pi$
$317$	12.2056	0.685535	0.342768	+	0.939420i	$0.388636\pi$
$318$	−35.3776	−1.98388
$319$	−14.1587	−0.792733
$320$	−2.85398	−0.159542
$321$	−60.4442	−3.37367
$322$	−11.2736	−0.628253
$323$	−32.6774	−1.81822
$324$	25.9306	1.44059
$325$	8.07613	0.447983
$326$	−6.97741	−0.386443
$327$	33.2048	1.83623
$328$	−2.76617	−0.152736
$329$	−12.7964	−0.705488
$330$	−37.3008	−2.05334
$331$	−3.43464	−0.188785	−0.0943926	−	0.995535i	$-0.530091\pi$
$331$	−3.43464	−0.188785	−0.0943926	+	0.995535i	$0.530091\pi$
$332$	−17.1131	−0.939204
$333$	10.0911	0.552989
$334$	6.56893	0.359436
$335$	−19.7868	−1.08107
$336$	−9.16482	−0.499982
$337$	35.6609	1.94258	0.971288	−	0.237908i	$-0.0764615\pi$
$337$	35.6609	1.94258	0.971288	+	0.237908i	$0.0764615\pi$
$338$	−6.40657	−0.348471
$339$	46.7602	2.53966
$340$	15.4155	0.836022
$341$	3.80808	0.206219
$342$	45.9639	2.48544
$343$	−17.1007	−0.923349
$344$	−8.60816	−0.464121
$345$	−37.2046	−2.00303
$346$	−21.9296	−1.17894
$347$	−9.65263	−0.518181	−0.259090	−	0.965853i	$-0.583423\pi$
$347$	−9.65263	−0.518181	−0.259090	+	0.965853i	$0.583423\pi$
$348$	−11.4805	−0.615421
$349$	31.1950	1.66983	0.834916	−	0.550377i	$-0.185516\pi$
$349$	31.1950	1.66983	0.834916	+	0.550377i	$0.185516\pi$
$350$	8.85458	0.473297
$351$	−38.4317	−2.05133
$352$	−4.01480	−0.213990
$353$	28.1430	1.49790	0.748951	−	0.662625i	$-0.230558\pi$
$353$	28.1430	1.49790	0.748951	+	0.662625i	$0.230558\pi$
$354$	−17.8836	−0.950505
$355$	30.7961	1.63448
$356$	−15.0321	−0.796700
$357$	49.5029	2.61997
$358$	11.0306	0.582983
$359$	6.59379	0.348007	0.174003	−	0.984745i	$-0.444330\pi$
$359$	6.59379	0.348007	0.174003	+	0.984745i	$0.444330\pi$
$360$	−21.6834	−1.14281
$361$	17.6000	0.926318
$362$	−12.6759	−0.666233
$363$	−16.6632	−0.874590
$364$	7.22896	0.378901
$365$	−16.5773	−0.867695
$366$	−46.5670	−2.43409
$367$	−7.41052	−0.386826	−0.193413	−	0.981117i	$-0.561956\pi$
$367$	−7.41052	−0.386826	−0.193413	+	0.981117i	$0.561956\pi$
$368$	−4.00444	−0.208746
$369$	−21.0162	−1.09406
$370$	−3.79065	−0.197067
$371$	30.5946	1.58839
$372$	3.08778	0.160094
$373$	17.7799	0.920610	0.460305	−	0.887761i	$-0.347740\pi$
$373$	17.7799	0.920610	0.460305	+	0.887761i	$0.347740\pi$
$374$	21.6856	1.12133
$375$	−17.2327	−0.889891
$376$	−4.54535	−0.234408
$377$	9.05554	0.466384
$378$	−42.1361	−2.16725
$379$	7.83716	0.402568	0.201284	−	0.979533i	$-0.435489\pi$
$379$	7.83716	0.402568	0.201284	+	0.979533i	$0.435489\pi$
$380$	−17.2660	−0.885727
$381$	−33.3743	−1.70982
$382$	15.9570	0.816429
$383$	−31.2455	−1.59657	−0.798286	−	0.602278i	$-0.794260\pi$
$383$	−31.2455	−1.59657	−0.798286	+	0.602278i	$0.794260\pi$
$384$	−3.25539	−0.166126
$385$	32.2578	1.64401
$386$	−4.62618	−0.235467
$387$	−65.4013	−3.32454
$388$	5.03150	0.255436
$389$	0.352701	0.0178827	0.00894133	−	0.999960i	$-0.497154\pi$
$389$	0.352701	0.0178827	0.00894133	+	0.999960i	$0.497154\pi$
$390$	23.8567	1.20803
$391$	21.6296	1.09386
$392$	0.925751	0.0467575
$393$	−21.0048	−1.05955
$394$	−8.72576	−0.439597
$395$	−19.6991	−0.991167
$396$	−30.5028	−1.53282
$397$	−24.3547	−1.22233	−0.611163	−	0.791505i	$-0.709298\pi$
$397$	−24.3547	−1.22233	−0.611163	+	0.791505i	$0.709298\pi$
$398$	−13.1987	−0.661593
$399$	−55.4453	−2.77574
$400$	3.14520	0.157260
$401$	−5.25334	−0.262339	−0.131170	−	0.991360i	$-0.541873\pi$
$401$	−5.25334	−0.262339	−0.131170	+	0.991360i	$0.541873\pi$
$402$	−22.5699	−1.12568
$403$	−2.43556	−0.121324
$404$	11.0171	0.548119
$405$	−74.0054	−3.67736
$406$	9.92838	0.492737
$407$	−5.33245	−0.264320
$408$	17.5837	0.870522
$409$	10.2445	0.506558	0.253279	−	0.967393i	$-0.418491\pi$
$409$	10.2445	0.506558	0.253279	+	0.967393i	$0.418491\pi$
$410$	7.89459	0.389886
$411$	−26.2867	−1.29663
$412$	−2.21020	−0.108889
$413$	15.4658	0.761023
$414$	−30.4241	−1.49526
$415$	48.8405	2.39748
$416$	2.56777	0.125895
$417$	−24.4742	−1.19851
$418$	−24.2887	−1.18800
$419$	−24.8859	−1.21576	−0.607878	−	0.794030i	$-0.707979\pi$
$419$	−24.8859	−1.21576	−0.607878	+	0.794030i	$0.707979\pi$
$420$	26.1562	1.27629
$421$	9.68577	0.472055	0.236028	−	0.971746i	$-0.424154\pi$
$421$	9.68577	0.472055	0.236028	+	0.971746i	$0.424154\pi$
$422$	−8.17724	−0.398062
$423$	−34.5337	−1.67909
$424$	10.8674	0.527767
$425$	−16.9885	−0.824062
$426$	35.1276	1.70194
$427$	40.2712	1.94886
$428$	18.5674	0.897489
$429$	33.5601	1.62030
$430$	24.5675	1.18475
$431$	−0.436752	−0.0210376	−0.0105188	−	0.999945i	$-0.503348\pi$
$431$	−0.436752	−0.0210376	−0.0105188	+	0.999945i	$0.503348\pi$
$432$	−14.9670	−0.720099
$433$	−33.3017	−1.60038	−0.800188	−	0.599749i	$-0.795267\pi$
$433$	−33.3017	−1.60038	−0.800188	+	0.599749i	$0.795267\pi$
$434$	−2.67031	−0.128179
$435$	32.7652	1.57097
$436$	−10.1999	−0.488488
$437$	−24.2261	−1.15889
$438$	−18.9089	−0.903502
$439$	−22.0371	−1.05178	−0.525888	−	0.850554i	$-0.676267\pi$
$439$	−22.0371	−1.05178	−0.525888	+	0.850554i	$0.676267\pi$
$440$	11.4582	0.546246
$441$	7.03347	0.334927
$442$	−13.8695	−0.659707
$443$	19.4623	0.924684	0.462342	−	0.886702i	$-0.347009\pi$
$443$	19.4623	0.924684	0.462342	+	0.886702i	$0.347009\pi$
$444$	−4.32381	−0.205199
$445$	42.9013	2.03372
$446$	−24.9605	−1.18191
$447$	−72.4257	−3.42562
$448$	2.81527	0.133009
$449$	6.69766	0.316082	0.158041	−	0.987433i	$-0.449482\pi$
$449$	6.69766	0.316082	0.158041	+	0.987433i	$0.449482\pi$
$450$	23.8959	1.12646
$451$	11.1056	0.522943
$452$	−14.3639	−0.675622
$453$	−56.3405	−2.64711
$454$	−6.93787	−0.325611
$455$	−20.6313	−0.967211
$456$	−19.6945	−0.922278
$457$	−2.18915	−0.102404	−0.0512020	−	0.998688i	$-0.516305\pi$
$457$	−2.18915	−0.102404	−0.0512020	+	0.998688i	$0.516305\pi$
$458$	−12.2860	−0.574089
$459$	80.8426	3.77341
$460$	11.4286	0.532861
$461$	−0.321222	−0.0149608	−0.00748040	−	0.999972i	$-0.502381\pi$
$461$	−0.321222	−0.0149608	−0.00748040	+	0.999972i	$0.502381\pi$
$462$	36.7949	1.71185
$463$	−0.688974	−0.0320193	−0.0160097	−	0.999872i	$-0.505096\pi$
$463$	−0.688974	−0.0320193	−0.0160097	+	0.999872i	$0.505096\pi$
$464$	3.52662	0.163719
$465$	−8.81245	−0.408667
$466$	3.44387	0.159534
$467$	−22.4578	−1.03922	−0.519611	−	0.854403i	$-0.673923\pi$
$467$	−22.4578	−1.03922	−0.519611	+	0.854403i	$0.673923\pi$
$468$	19.5089	0.901797
$469$	19.5185	0.901280
$470$	12.9723	0.598369
$471$	64.1790	2.95721
$472$	5.49354	0.252861
$473$	34.5601	1.58907
$474$	−22.4697	−1.03207
$475$	19.0278	0.873055
$476$	−15.2064	−0.696985
$477$	82.5659	3.78043
$478$	−16.6622	−0.762110
$479$	−7.09779	−0.324306	−0.162153	−	0.986766i	$-0.551844\pi$
$479$	−7.09779	−0.324306	−0.162153	+	0.986766i	$0.551844\pi$
$480$	9.29083	0.424066
$481$	3.41051	0.155506
$482$	0.662582	0.0301798
$483$	36.7000	1.66991
$484$	5.11863	0.232665
$485$	−14.3598	−0.652044
$486$	−39.5134	−1.79237
$487$	−23.2643	−1.05421	−0.527104	−	0.849801i	$-0.676722\pi$
$487$	−23.2643	−1.05421	−0.527104	+	0.849801i	$0.676722\pi$
$488$	14.3046	0.647537
$489$	22.7142	1.02717
$490$	−2.64207	−0.119357
$491$	−40.2610	−1.81696	−0.908478	−	0.417934i	$-0.862754\pi$
$491$	−40.2610	−1.81696	−0.908478	+	0.417934i	$0.862754\pi$
$492$	9.00497	0.405975
$493$	−19.0487	−0.857909
$494$	15.5345	0.698929
$495$	87.0544	3.91280
$496$	−0.948510	−0.0425894
$497$	−30.3784	−1.36266
$498$	55.7099	2.49642
$499$	−41.0580	−1.83801	−0.919004	−	0.394249i	$-0.871005\pi$
$499$	−41.0580	−1.83801	−0.919004	+	0.394249i	$0.871005\pi$
$500$	5.29357	0.236736
$501$	−21.3845	−0.955387
$502$	−2.35998	−0.105331
$503$	−37.2762	−1.66206	−0.831031	−	0.556226i	$-0.812249\pi$
$503$	−37.2762	−1.66206	−0.831031	+	0.556226i	$0.812249\pi$
$504$	21.3893	0.952754
$505$	−31.4425	−1.39917
$506$	16.0770	0.714712
$507$	20.8559	0.926243
$508$	10.2520	0.454859
$509$	−16.1031	−0.713757	−0.356878	−	0.934151i	$-0.616159\pi$
$509$	−16.1031	−0.713757	−0.356878	+	0.934151i	$0.616159\pi$
$510$	−50.1835	−2.22216
$511$	16.3524	0.723390
$512$	1.00000	0.0441942
$513$	−90.5471	−3.99775
$514$	19.3162	0.852001
$515$	6.30788	0.277958
$516$	28.0230	1.23364
$517$	18.2487	0.802576
$518$	3.73924	0.164293
$519$	71.3896	3.13366
$520$	−7.32836	−0.321370
$521$	−13.3112	−0.583173	−0.291587	−	0.956544i	$-0.594183\pi$
$521$	−13.3112	−0.583173	−0.291587	+	0.956544i	$0.594183\pi$
$522$	26.7938	1.17273
$523$	38.1539	1.66835	0.834176	−	0.551499i	$-0.185944\pi$
$523$	38.1539	1.66835	0.834176	+	0.551499i	$0.185944\pi$
$524$	6.45229	0.281870
$525$	−28.8251	−1.25803
$526$	−28.1430	−1.22710
$527$	5.12329	0.223174
$528$	13.0698	0.568788
$529$	−6.96443	−0.302801
$530$	−31.0153	−1.34722
$531$	41.7377	1.81126
$532$	17.0318	0.738423
$533$	−7.10288	−0.307660
$534$	48.9354	2.11764
$535$	−52.9910	−2.29100
$536$	6.93307	0.299463
$537$	−35.9088	−1.54958
$538$	−17.3551	−0.748230
$539$	−3.71671	−0.160090
$540$	42.7154	1.83818
$541$	11.5053	0.494653	0.247326	−	0.968932i	$-0.420448\pi$
$541$	11.5053	0.494653	0.247326	+	0.968932i	$0.420448\pi$
$542$	−23.8023	−1.02240
$543$	41.2652	1.77086
$544$	−5.40140	−0.231583
$545$	29.1104	1.24695
$546$	−23.5331	−1.00712
$547$	26.9199	1.15101	0.575506	−	0.817798i	$-0.304805\pi$
$547$	26.9199	1.15101	0.575506	+	0.817798i	$0.304805\pi$
$548$	8.07482	0.344939
$549$	108.680	4.63836
$550$	−12.6273	−0.538431
$551$	21.3353	0.908915
$552$	13.0360	0.554851
$553$	19.4319	0.826327
$554$	−12.6385	−0.536958
$555$	12.3401	0.523807
$556$	7.51806	0.318837
$557$	12.2239	0.517944	0.258972	−	0.965885i	$-0.416616\pi$
$557$	12.2239	0.517944	0.258972	+	0.965885i	$0.416616\pi$
$558$	−7.20639	−0.305071
$559$	−22.1038	−0.934890
$560$	−8.03472	−0.339529
$561$	−70.5950	−2.98052
$562$	−2.84184	−0.119876
$563$	−18.8342	−0.793766	−0.396883	−	0.917869i	$-0.629908\pi$
$563$	−18.8342	−0.793766	−0.396883	+	0.917869i	$0.629908\pi$
$564$	14.7969	0.623062
$565$	40.9943	1.72464
$566$	−6.53062	−0.274502
$567$	73.0017	3.06578
$568$	−10.7906	−0.452762
$569$	−42.4808	−1.78089	−0.890443	−	0.455094i	$-0.849606\pi$
$569$	−42.4808	−1.78089	−0.890443	+	0.455094i	$0.849606\pi$
$570$	56.2076	2.35428
$571$	−7.33674	−0.307033	−0.153517	−	0.988146i	$-0.549060\pi$
$571$	−7.33674	−0.307033	−0.153517	+	0.988146i	$0.549060\pi$
$572$	−10.3091	−0.431044
$573$	−51.9462	−2.17008
$574$	−7.78751	−0.325045
$575$	−12.5948	−0.525238
$576$	7.59759	0.316566
$577$	40.2044	1.67373	0.836866	−	0.547407i	$-0.184385\pi$
$577$	40.2044	1.67373	0.836866	+	0.547407i	$0.184385\pi$
$578$	12.1751	0.506419
$579$	15.0601	0.625874
$580$	−10.0649	−0.417922
$581$	−48.1780	−1.99876
$582$	−16.3795	−0.678952
$583$	−43.6304	−1.80699
$584$	5.80848	0.240357
$585$	−55.6778	−2.30200
$586$	−4.40255	−0.181868
$587$	−7.98252	−0.329474	−0.164737	−	0.986338i	$-0.552678\pi$
$587$	−7.98252	−0.329474	−0.164737	+	0.986338i	$0.552678\pi$
$588$	−3.01368	−0.124282
$589$	−5.73829	−0.236442
$590$	−15.6785	−0.645472
$591$	28.4058	1.16846
$592$	1.32820	0.0545886
$593$	26.8600	1.10301	0.551505	−	0.834172i	$-0.314054\pi$
$593$	26.8600	1.10301	0.551505	+	0.834172i	$0.314054\pi$
$594$	60.0894	2.46550
$595$	43.3988	1.77918
$596$	22.2479	0.911309
$597$	42.9671	1.75853
$598$	−10.2825	−0.420482
$599$	24.7014	1.00927	0.504635	−	0.863333i	$-0.331627\pi$
$599$	24.7014	1.00927	0.504635	+	0.863333i	$0.331627\pi$
$600$	−10.2389	−0.417999
$601$	32.9442	1.34382	0.671910	−	0.740633i	$-0.265474\pi$
$601$	32.9442	1.34382	0.671910	+	0.740633i	$0.265474\pi$
$602$	−24.2343	−0.987717
$603$	52.6747	2.14508
$604$	17.3068	0.704204
$605$	−14.6085	−0.593919
$606$	−35.8649	−1.45691
$607$	12.8587	0.521918	0.260959	−	0.965350i	$-0.415961\pi$
$607$	12.8587	0.521918	0.260959	+	0.965350i	$0.415961\pi$
$608$	6.04980	0.245352
$609$	−32.3208	−1.30970
$610$	−40.8249	−1.65295
$611$	−11.6714	−0.472174
$612$	−41.0376	−1.65885
$613$	−38.4582	−1.55331	−0.776656	−	0.629925i	$-0.783086\pi$
$613$	−38.4582	−1.55331	−0.776656	+	0.629925i	$0.783086\pi$
$614$	−20.2650	−0.817828
$615$	−25.7000	−1.03632
$616$	−11.3028	−0.455401
$617$	8.44663	0.340049	0.170024	−	0.985440i	$-0.445615\pi$
$617$	8.44663	0.340049	0.170024	+	0.985440i	$0.445615\pi$
$618$	7.19509	0.289429
$619$	−38.7677	−1.55820	−0.779102	−	0.626897i	$-0.784325\pi$
$619$	−38.7677	−1.55820	−0.779102	+	0.626897i	$0.784325\pi$
$620$	2.70703	0.108717
$621$	59.9344	2.40508
$622$	−9.11159	−0.365341
$623$	−42.3195	−1.69549
$624$	−8.35910	−0.334632
$625$	−30.8337	−1.23335
$626$	−34.3290	−1.37206
$627$	79.0694	3.15773
$628$	−19.7147	−0.786700
$629$	−7.17414	−0.286052
$630$	−61.0445	−2.43207
$631$	7.35049	0.292618	0.146309	−	0.989239i	$-0.453261\pi$
$631$	7.35049	0.292618	0.146309	+	0.989239i	$0.453261\pi$
$632$	6.90231	0.274559
$633$	26.6202	1.05806
$634$	12.2056	0.484746
$635$	−29.2590	−1.16111
$636$	−35.3776	−1.40281
$637$	2.37711	0.0941847
$638$	−14.1587	−0.560547
$639$	−81.9823	−3.24317
$640$	−2.85398	−0.112813
$641$	41.6331	1.64441	0.822205	−	0.569192i	$-0.192744\pi$
$641$	41.6331	1.64441	0.822205	+	0.569192i	$0.192744\pi$
$642$	−60.4442	−2.38554
$643$	27.0286	1.06590	0.532951	−	0.846146i	$-0.321083\pi$
$643$	27.0286	1.06590	0.532951	+	0.846146i	$0.321083\pi$
$644$	−11.2736	−0.444242
$645$	−79.9770	−3.14909
$646$	−32.6774	−1.28567
$647$	−4.68449	−0.184166	−0.0920831	−	0.995751i	$-0.529353\pi$
$647$	−4.68449	−0.184166	−0.0920831	+	0.995751i	$0.529353\pi$
$648$	25.9306	1.01865
$649$	−22.0555	−0.865754
$650$	8.07613	0.316772
$651$	8.69292	0.340703
$652$	−6.97741	−0.273257
$653$	23.0964	0.903832	0.451916	−	0.892060i	$-0.350741\pi$
$653$	23.0964	0.903832	0.451916	+	0.892060i	$0.350741\pi$
$654$	33.2048	1.29841
$655$	−18.4147	−0.719522
$656$	−2.76617	−0.108001
$657$	44.1305	1.72169
$658$	−12.7964	−0.498855
$659$	38.6799	1.50675	0.753377	−	0.657589i	$-0.228424\pi$
$659$	38.6799	1.50675	0.753377	+	0.657589i	$0.228424\pi$
$660$	−37.3008	−1.45193
$661$	20.2087	0.786025	0.393013	−	0.919533i	$-0.371433\pi$
$661$	20.2087	0.786025	0.393013	+	0.919533i	$0.371433\pi$
$662$	−3.43464	−0.133491
$663$	45.1508	1.75351
$664$	−17.1131	−0.664117
$665$	−48.6084	−1.88495
$666$	10.0911	0.391023
$667$	−14.1221	−0.546811
$668$	6.56893	0.254159
$669$	81.2563	3.14155
$670$	−19.7868	−0.764433
$671$	−57.4299	−2.21706
$672$	−9.16482	−0.353540
$673$	−49.4858	−1.90754	−0.953769	−	0.300540i	$-0.902833\pi$
$673$	−49.4858	−1.90754	−0.953769	+	0.300540i	$0.902833\pi$
$674$	35.6609	1.37361
$675$	−47.0740	−1.81188
$676$	−6.40657	−0.246406
$677$	−13.1696	−0.506149	−0.253074	−	0.967447i	$-0.581442\pi$
$677$	−13.1696	−0.506149	−0.253074	+	0.967447i	$0.581442\pi$
$678$	46.7602	1.79581
$679$	14.1650	0.543604
$680$	15.4155	0.591157
$681$	22.5855	0.865479
$682$	3.80808	0.145819
$683$	−13.4775	−0.515702	−0.257851	−	0.966185i	$-0.583014\pi$
$683$	−13.4775	−0.515702	−0.257851	+	0.966185i	$0.583014\pi$
$684$	45.9639	1.75747
$685$	−23.0454	−0.880518
$686$	−17.1007	−0.652906
$687$	39.9959	1.52594
$688$	−8.60816	−0.328183
$689$	27.9049	1.06309
$690$	−37.2046	−1.41635
$691$	−20.0156	−0.761429	−0.380715	−	0.924693i	$-0.624322\pi$
$691$	−20.0156	−0.761429	−0.380715	+	0.924693i	$0.624322\pi$
$692$	−21.9296	−0.833640
$693$	−85.8737	−3.26207
$694$	−9.65263	−0.366409
$695$	−21.4564	−0.813887
$696$	−11.4805	−0.435168
$697$	14.9412	0.565938
$698$	31.1950	1.18075
$699$	−11.2112	−0.424045
$700$	8.85458	0.334672
$701$	0.431964	0.0163151	0.00815753	−	0.999967i	$-0.497403\pi$
$701$	0.431964	0.0163151	0.00815753	+	0.999967i	$0.497403\pi$
$702$	−38.4317	−1.45051
$703$	8.03533	0.303058
$704$	−4.01480	−0.151314
$705$	−42.2300	−1.59048
$706$	28.1430	1.05918
$707$	31.0160	1.16648
$708$	−17.8836	−0.672109
$709$	−46.3515	−1.74077	−0.870383	−	0.492374i	$-0.836129\pi$
$709$	−46.3515	−1.74077	−0.870383	+	0.492374i	$0.836129\pi$
$710$	30.7961	1.15576
$711$	52.4409	1.96669
$712$	−15.0321	−0.563352
$713$	3.79826	0.142246
$714$	49.5029	1.85260
$715$	29.4219	1.10032
$716$	11.0306	0.412232
$717$	54.2419	2.02570
$718$	6.59379	0.246078
$719$	−50.0299	−1.86580	−0.932899	−	0.360137i	$-0.882730\pi$
$719$	−50.0299	−1.86580	−0.932899	+	0.360137i	$0.882730\pi$
$720$	−21.6834	−0.808091
$721$	−6.22232	−0.231731
$722$	17.6000	0.655005
$723$	−2.15696	−0.0802184
$724$	−12.6759	−0.471098
$725$	11.0919	0.411943
$726$	−16.6632	−0.618428
$727$	−33.7941	−1.25335	−0.626677	−	0.779279i	$-0.715585\pi$
$727$	−33.7941	−1.25335	−0.626677	+	0.779279i	$0.715585\pi$
$728$	7.22896	0.267923
$729$	50.8400	1.88296
$730$	−16.5773	−0.613553
$731$	46.4962	1.71972
$732$	−46.5670	−1.72116
$733$	−9.91470	−0.366208	−0.183104	−	0.983094i	$-0.558615\pi$
$733$	−9.91470	−0.366208	−0.183104	+	0.983094i	$0.558615\pi$
$734$	−7.41052	−0.273527
$735$	8.60099	0.317252
$736$	−4.00444	−0.147606
$737$	−27.8349	−1.02531
$738$	−21.0162	−0.773617
$739$	−18.0236	−0.663007	−0.331504	−	0.943454i	$-0.607556\pi$
$739$	−18.0236	−0.663007	−0.331504	+	0.943454i	$0.607556\pi$
$740$	−3.79065	−0.139347
$741$	−50.5708	−1.85777
$742$	30.5946	1.12316
$743$	16.9618	0.622267	0.311133	−	0.950366i	$-0.399291\pi$
$743$	16.9618	0.622267	0.311133	+	0.950366i	$0.399291\pi$
$744$	3.08778	0.113203
$745$	−63.4950	−2.32628
$746$	17.7799	0.650970
$747$	−130.018	−4.75712
$748$	21.6856	0.792903
$749$	52.2723	1.90999
$750$	−17.2327	−0.629248
$751$	22.0743	0.805501	0.402751	−	0.915310i	$-0.368054\pi$
$751$	22.0743	0.805501	0.402751	+	0.915310i	$0.368054\pi$
$752$	−4.54535	−0.165752
$753$	7.68265	0.279971
$754$	9.05554	0.329783
$755$	−49.3932	−1.79760
$756$	−42.1361	−1.53247
$757$	51.3789	1.86740	0.933698	−	0.358061i	$-0.116562\pi$
$757$	51.3789	1.86740	0.933698	+	0.358061i	$0.116562\pi$
$758$	7.83716	0.284658
$759$	−52.3371	−1.89972
$760$	−17.2660	−0.626303
$761$	7.01948	0.254456	0.127228	−	0.991873i	$-0.459392\pi$
$761$	7.01948	0.254456	0.127228	+	0.991873i	$0.459392\pi$
$762$	−33.3743	−1.20902
$763$	−28.7156	−1.03957
$764$	15.9570	0.577303
$765$	117.121	4.23450
$766$	−31.2455	−1.12895
$767$	14.1061	0.509343
$768$	−3.25539	−0.117469
$769$	4.77663	0.172250	0.0861248	−	0.996284i	$-0.472552\pi$
$769$	4.77663	0.172250	0.0861248	+	0.996284i	$0.472552\pi$
$770$	32.2578	1.16249
$771$	−62.8818	−2.26463
$772$	−4.62618	−0.166500
$773$	8.80334	0.316634	0.158317	−	0.987388i	$-0.449393\pi$
$773$	8.80334	0.316634	0.158317	+	0.987388i	$0.449393\pi$
$774$	−65.4013	−2.35080
$775$	−2.98325	−0.107162
$776$	5.03150	0.180620
$777$	−12.1727	−0.436693
$778$	0.352701	0.0126449
$779$	−16.7348	−0.599585
$780$	23.8567	0.854206
$781$	43.3220	1.55018
$782$	21.6296	0.773473
$783$	−52.7828	−1.88630
$784$	0.925751	0.0330625
$785$	56.2652	2.00819
$786$	−21.0048	−0.749215
$787$	−47.0457	−1.67700	−0.838499	−	0.544903i	$-0.816567\pi$
$787$	−47.0457	−1.67700	−0.838499	+	0.544903i	$0.816567\pi$
$788$	−8.72576	−0.310842
$789$	91.6167	3.26164
$790$	−19.6991	−0.700861
$791$	−40.4383	−1.43782
$792$	−30.5028	−1.08387
$793$	36.7308	1.30435
$794$	−24.3547	−0.864315
$795$	100.967	3.58093
$796$	−13.1987	−0.467817
$797$	−2.67037	−0.0945895	−0.0472947	−	0.998881i	$-0.515060\pi$
$797$	−2.67037	−0.0945895	−0.0472947	+	0.998881i	$0.515060\pi$
$798$	−55.4453	−1.96274
$799$	24.5513	0.868561
$800$	3.14520	0.111199
$801$	−114.208	−4.03534
$802$	−5.25334	−0.185502
$803$	−23.3199	−0.822941
$804$	−22.5699	−0.795979
$805$	32.1746	1.13401
$806$	−2.43556	−0.0857888
$807$	56.4976	1.98881
$808$	11.0171	0.387579
$809$	12.6602	0.445110	0.222555	−	0.974920i	$-0.428560\pi$
$809$	12.6602	0.445110	0.222555	+	0.974920i	$0.428560\pi$
$810$	−74.0054	−2.60028
$811$	−20.6382	−0.724705	−0.362353	−	0.932041i	$-0.618026\pi$
$811$	−20.6382	−0.724705	−0.362353	+	0.932041i	$0.618026\pi$
$812$	9.92838	0.348418
$813$	77.4859	2.71755
$814$	−5.33245	−0.186902
$815$	19.9134	0.697536
$816$	17.5837	0.615552
$817$	−52.0776	−1.82197
$818$	10.2445	0.358191
$819$	54.9227	1.91915
$820$	7.89459	0.275691
$821$	−2.17409	−0.0758762	−0.0379381	−	0.999280i	$-0.512079\pi$
$821$	−2.17409	−0.0758762	−0.0379381	+	0.999280i	$0.512079\pi$
$822$	−26.2867	−0.916855
$823$	−12.3208	−0.429476	−0.214738	−	0.976672i	$-0.568890\pi$
$823$	−12.3208	−0.429476	−0.214738	+	0.976672i	$0.568890\pi$
$824$	−2.21020	−0.0769961
$825$	41.1070	1.43116
$826$	15.4658	0.538124
$827$	−19.7050	−0.685210	−0.342605	−	0.939480i	$-0.611309\pi$
$827$	−19.7050	−0.685210	−0.342605	+	0.939480i	$0.611309\pi$
$828$	−30.4241	−1.05731
$829$	16.3125	0.566558	0.283279	−	0.959038i	$-0.408578\pi$
$829$	16.3125	0.566558	0.283279	+	0.959038i	$0.408578\pi$
$830$	48.8405	1.69528
$831$	41.1432	1.42724
$832$	2.56777	0.0890213
$833$	−5.00035	−0.173252
$834$	−24.4742	−0.847474
$835$	−18.7476	−0.648787
$836$	−24.2887	−0.840043
$837$	14.1963	0.490697
$838$	−24.8859	−0.859670
$839$	27.8921	0.962944	0.481472	−	0.876462i	$-0.340102\pi$
$839$	27.8921	0.962944	0.481472	+	0.876462i	$0.340102\pi$
$840$	26.1562	0.902474
$841$	−16.5630	−0.571137
$842$	9.68577	0.333794
$843$	9.25129	0.318631
$844$	−8.17724	−0.281472
$845$	18.2842	0.628996
$846$	−34.5337	−1.18729
$847$	14.4103	0.495145
$848$	10.8674	0.373187
$849$	21.2597	0.729632
$850$	−16.9885	−0.582700
$851$	−5.31870	−0.182323
$852$	35.1276	1.20345
$853$	31.5112	1.07892	0.539462	−	0.842010i	$-0.318628\pi$
$853$	31.5112	1.07892	0.539462	+	0.842010i	$0.318628\pi$
$854$	40.2712	1.37805
$855$	−131.180	−4.48626
$856$	18.5674	0.634621
$857$	−31.4660	−1.07486	−0.537428	−	0.843309i	$-0.680604\pi$
$857$	−31.4660	−1.07486	−0.537428	+	0.843309i	$0.680604\pi$
$858$	33.5601	1.14572
$859$	−3.79461	−0.129471	−0.0647353	−	0.997902i	$-0.520620\pi$
$859$	−3.79461	−0.129471	−0.0647353	+	0.997902i	$0.520620\pi$
$860$	24.5675	0.837746
$861$	25.3514	0.863974
$862$	−0.436752	−0.0148758
$863$	−0.396896	−0.0135105	−0.00675524	−	0.999977i	$-0.502150\pi$
$863$	−0.396896	−0.0135105	−0.00675524	+	0.999977i	$0.502150\pi$
$864$	−14.9670	−0.509187
$865$	62.5867	2.12801
$866$	−33.3017	−1.13164
$867$	−39.6349	−1.34607
$868$	−2.67031	−0.0906364
$869$	−27.7114	−0.940045
$870$	32.7652	1.11084
$871$	17.8025	0.603216
$872$	−10.1999	−0.345413
$873$	38.2272	1.29380
$874$	−24.2261	−0.819459
$875$	14.9028	0.503808
$876$	−18.9089	−0.638872
$877$	−24.2093	−0.817488	−0.408744	−	0.912649i	$-0.634033\pi$
$877$	−24.2093	−0.817488	−0.408744	+	0.912649i	$0.634033\pi$
$878$	−22.0371	−0.743717
$879$	14.3320	0.483407
$880$	11.4582	0.386254
$881$	27.4689	0.925450	0.462725	−	0.886502i	$-0.346872\pi$
$881$	27.4689	0.925450	0.462725	+	0.886502i	$0.346872\pi$
$882$	7.03347	0.236829
$883$	1.87234	0.0630093	0.0315046	−	0.999504i	$-0.489970\pi$
$883$	1.87234	0.0630093	0.0315046	+	0.999504i	$0.489970\pi$
$884$	−13.8695	−0.466483
$885$	51.0396	1.71568
$886$	19.4623	0.653850
$887$	49.5074	1.66229	0.831147	−	0.556052i	$-0.187685\pi$
$887$	49.5074	1.66229	0.831147	+	0.556052i	$0.187685\pi$
$888$	−4.32381	−0.145098
$889$	28.8622	0.968007
$890$	42.9013	1.43806
$891$	−104.106	−3.48769
$892$	−24.9605	−0.835739
$893$	−27.4984	−0.920200
$894$	−72.4257	−2.42228
$895$	−31.4810	−1.05229
$896$	2.81527	0.0940516
$897$	33.4735	1.11765
$898$	6.69766	0.223504
$899$	−3.34503	−0.111563
$900$	23.8959	0.796530
$901$	−58.6991	−1.95555
$902$	11.1056	0.369777
$903$	78.8923	2.62537
$904$	−14.3639	−0.477737
$905$	36.1769	1.20256
$906$	−56.3405	−1.87179
$907$	41.8596	1.38993	0.694963	−	0.719046i	$-0.255421\pi$
$907$	41.8596	1.38993	0.694963	+	0.719046i	$0.255421\pi$
$908$	−6.93787	−0.230241
$909$	83.7031	2.77626
$910$	−20.6313	−0.683921
$911$	13.9963	0.463719	0.231859	−	0.972749i	$-0.425519\pi$
$911$	13.9963	0.463719	0.231859	+	0.972749i	$0.425519\pi$
$912$	−19.6945	−0.652149
$913$	68.7057	2.27383
$914$	−2.18915	−0.0724105
$915$	132.901	4.39358
$916$	−12.2860	−0.405942
$917$	18.1649	0.599859
$918$	80.8426	2.66820
$919$	−46.7728	−1.54289	−0.771447	−	0.636294i	$-0.780466\pi$
$919$	−46.7728	−1.54289	−0.771447	+	0.636294i	$0.780466\pi$
$920$	11.4286	0.376790
$921$	65.9705	2.17380
$922$	−0.321222	−0.0105789
$923$	−27.7077	−0.912009
$924$	36.7949	1.21046
$925$	4.17744	0.137354
$926$	−0.688974	−0.0226411
$927$	−16.7922	−0.551529
$928$	3.52662	0.115767
$929$	−46.7735	−1.53459	−0.767294	−	0.641295i	$-0.778397\pi$
$929$	−46.7735	−1.53459	−0.767294	+	0.641295i	$0.778397\pi$
$930$	−8.81245	−0.288971
$931$	5.60060	0.183552
$932$	3.44387	0.112808
$933$	29.6618	0.971084
$934$	−22.4578	−0.734841
$935$	−61.8901	−2.02402
$936$	19.5089	0.637667
$937$	19.3083	0.630775	0.315387	−	0.948963i	$-0.397866\pi$
$937$	19.3083	0.630775	0.315387	+	0.948963i	$0.397866\pi$
$938$	19.5185	0.637301
$939$	111.754	3.64696
$940$	12.9723	0.423111
$941$	9.60795	0.313210	0.156605	−	0.987661i	$-0.449945\pi$
$941$	9.60795	0.313210	0.156605	+	0.987661i	$0.449945\pi$
$942$	64.1790	2.09106
$943$	11.0770	0.360716
$944$	5.49354	0.178800
$945$	120.255	3.91191
$946$	34.5601	1.12364
$947$	12.9410	0.420527	0.210263	−	0.977645i	$-0.432568\pi$
$947$	12.9410	0.420527	0.210263	+	0.977645i	$0.432568\pi$
$948$	−22.4697	−0.729783
$949$	14.9148	0.484156
$950$	19.0278	0.617343
$951$	−39.7341	−1.28846
$952$	−15.2064	−0.492843
$953$	40.6683	1.31738	0.658688	−	0.752416i	$-0.271112\pi$
$953$	40.6683	1.31738	0.658688	+	0.752416i	$0.271112\pi$
$954$	82.5659	2.67317
$955$	−45.5408	−1.47367
$956$	−16.6622	−0.538893
$957$	46.0920	1.48994
$958$	−7.09779	−0.229319
$959$	22.7328	0.734081
$960$	9.29083	0.299860
$961$	−30.1003	−0.970978
$962$	3.41051	0.109959
$963$	141.068	4.54584
$964$	0.662582	0.0213403
$965$	13.2030	0.425021
$966$	36.7000	1.18080
$967$	33.9137	1.09059	0.545296	−	0.838244i	$-0.316417\pi$
$967$	33.9137	1.09059	0.545296	+	0.838244i	$0.316417\pi$
$968$	5.11863	0.164519
$969$	106.378	3.41735
$970$	−14.3598	−0.461065
$971$	20.7273	0.665170	0.332585	−	0.943073i	$-0.392079\pi$
$971$	20.7273	0.665170	0.332585	+	0.943073i	$0.392079\pi$
$972$	−39.5134	−1.26739
$973$	21.1654	0.678531
$974$	−23.2643	−0.745438
$975$	−26.2910	−0.841986
$976$	14.3046	0.457878
$977$	−15.5393	−0.497146	−0.248573	−	0.968613i	$-0.579962\pi$
$977$	−15.5393	−0.497146	−0.248573	+	0.968613i	$0.579962\pi$
$978$	22.7142	0.726321
$979$	60.3509	1.92882
$980$	−2.64207	−0.0843979
$981$	−77.4949	−2.47422
$982$	−40.2610	−1.28478
$983$	−4.51159	−0.143897	−0.0719487	−	0.997408i	$-0.522922\pi$
$983$	−4.51159	−0.143897	−0.0719487	+	0.997408i	$0.522922\pi$
$984$	9.00497	0.287068
$985$	24.9031	0.793479
$986$	−19.0487	−0.606633
$987$	41.6573	1.32597
$988$	15.5345	0.494217
$989$	34.4709	1.09611
$990$	87.0544	2.76677
$991$	−15.5382	−0.493587	−0.246793	−	0.969068i	$-0.579377\pi$
$991$	−15.5382	−0.493587	−0.246793	+	0.969068i	$0.579377\pi$
$992$	−0.948510	−0.0301152
$993$	11.1811	0.354822
$994$	−30.3784	−0.963544
$995$	37.6689	1.19418
$996$	55.7099	1.76524
$997$	33.7532	1.06897	0.534487	−	0.845176i	$-0.320505\pi$
$997$	33.7532	1.06897	0.534487	+	0.845176i	$0.320505\pi$
$998$	−41.0580	−1.29967
$999$	−19.8791	−0.628947

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.a.1.5	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.a.1.5	✓	75	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.25539 q^{3} +1.00000 q^{4} -2.85398 q^{5} -3.25539 q^{6} +2.81527 q^{7} +1.00000 q^{8} +7.59759 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.25539 q^{3} +1.00000 q^{4} -2.85398 q^{5} -3.25539 q^{6} +2.81527 q^{7} +1.00000 q^{8} +7.59759 q^{9} -2.85398 q^{10} -4.01480 q^{11} -3.25539 q^{12} +2.56777 q^{13} +2.81527 q^{14} +9.29083 q^{15} +1.00000 q^{16} -5.40140 q^{17} +7.59759 q^{18} +6.04980 q^{19} -2.85398 q^{20} -9.16482 q^{21} -4.01480 q^{22} -4.00444 q^{23} -3.25539 q^{24} +3.14520 q^{25} +2.56777 q^{26} -14.9670 q^{27} +2.81527 q^{28} +3.52662 q^{29} +9.29083 q^{30} -0.948510 q^{31} +1.00000 q^{32} +13.0698 q^{33} -5.40140 q^{34} -8.03472 q^{35} +7.59759 q^{36} +1.32820 q^{37} +6.04980 q^{38} -8.35910 q^{39} -2.85398 q^{40} -2.76617 q^{41} -9.16482 q^{42} -8.60816 q^{43} -4.01480 q^{44} -21.6834 q^{45} -4.00444 q^{46} -4.54535 q^{47} -3.25539 q^{48} +0.925751 q^{49} +3.14520 q^{50} +17.5837 q^{51} +2.56777 q^{52} +10.8674 q^{53} -14.9670 q^{54} +11.4582 q^{55} +2.81527 q^{56} -19.6945 q^{57} +3.52662 q^{58} +5.49354 q^{59} +9.29083 q^{60} +14.3046 q^{61} -0.948510 q^{62} +21.3893 q^{63} +1.00000 q^{64} -7.32836 q^{65} +13.0698 q^{66} +6.93307 q^{67} -5.40140 q^{68} +13.0360 q^{69} -8.03472 q^{70} -10.7906 q^{71} +7.59759 q^{72} +5.80848 q^{73} +1.32820 q^{74} -10.2389 q^{75} +6.04980 q^{76} -11.3028 q^{77} -8.35910 q^{78} +6.90231 q^{79} -2.85398 q^{80} +25.9306 q^{81} -2.76617 q^{82} -17.1131 q^{83} -9.16482 q^{84} +15.4155 q^{85} -8.60816 q^{86} -11.4805 q^{87} -4.01480 q^{88} -15.0321 q^{89} -21.6834 q^{90} +7.22896 q^{91} -4.00444 q^{92} +3.08778 q^{93} -4.54535 q^{94} -17.2660 q^{95} -3.25539 q^{96} +5.03150 q^{97} +0.925751 q^{98} -30.5028 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * q^99

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