LMFDB - Embedded newform 1155.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1155,2,Mod(1,1155)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1155, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1155.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1155 = 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1155.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:	$9.22272143346$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 4x + 4 $ x^4 - 2x^3 - 5x^2 + 4*x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.63640$ of defining polynomial
Character	$\chi$	$=$	1155.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.222191 q^{2} -1.00000 q^{3} -1.95063 q^{4} +1.00000 q^{5} +0.222191 q^{6} +1.00000 q^{7} +0.877796 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.222191 q^{2} -1.00000 q^{3} -1.95063 q^{4} +1.00000 q^{5} +0.222191 q^{6} +1.00000 q^{7} +0.877796 q^{8} +1.00000 q^{9} -0.222191 q^{10} +1.00000 q^{11} +1.95063 q^{12} +0.900012 q^{13} -0.222191 q^{14} -1.00000 q^{15} +3.70622 q^{16} -0.506248 q^{17} -0.222191 q^{18} -4.95063 q^{19} -1.95063 q^{20} -1.00000 q^{21} -0.222191 q^{22} -6.60749 q^{23} -0.877796 q^{24} +1.00000 q^{25} -0.199975 q^{26} -1.00000 q^{27} -1.95063 q^{28} +3.40626 q^{29} +0.222191 q^{30} -3.54437 q^{31} -2.57908 q^{32} -1.00000 q^{33} +0.112484 q^{34} +1.00000 q^{35} -1.95063 q^{36} +11.4456 q^{37} +1.09999 q^{38} -0.900012 q^{39} +0.877796 q^{40} +6.80127 q^{41} +0.222191 q^{42} +9.26309 q^{43} -1.95063 q^{44} +1.00000 q^{45} +1.46813 q^{46} +12.6581 q^{47} -3.70622 q^{48} +1.00000 q^{49} -0.222191 q^{50} +0.506248 q^{51} -1.75559 q^{52} -1.39501 q^{53} +0.222191 q^{54} +1.00000 q^{55} +0.877796 q^{56} +4.95063 q^{57} -0.756842 q^{58} +6.93938 q^{59} +1.95063 q^{60} +5.39501 q^{61} +0.787529 q^{62} +1.00000 q^{63} -6.83940 q^{64} +0.900012 q^{65} +0.222191 q^{66} -10.3012 q^{67} +0.987504 q^{68} +6.60749 q^{69} -0.222191 q^{70} +6.35689 q^{71} +0.877796 q^{72} +3.55562 q^{73} -2.54312 q^{74} -1.00000 q^{75} +9.65685 q^{76} +1.00000 q^{77} +0.199975 q^{78} +1.98875 q^{79} +3.70622 q^{80} +1.00000 q^{81} -1.51118 q^{82} -4.16310 q^{83} +1.95063 q^{84} -0.506248 q^{85} -2.05818 q^{86} -3.40626 q^{87} +0.877796 q^{88} +2.25059 q^{89} -0.222191 q^{90} +0.900012 q^{91} +12.8888 q^{92} +3.54437 q^{93} -2.81252 q^{94} -4.95063 q^{95} +2.57908 q^{96} +0.0506185 q^{97} -0.222191 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9	\( 4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9} + 2 q^{10} + 4 q^{11} - 6 q^{12} + 4 q^{13} + 2 q^{14} - 4 q^{15} + 6 q^{16} + 6 q^{17} + 2 q^{18} - 6 q^{19} + 6 q^{20} - 4 q^{21} + 2 q^{22} + 10 q^{23} - 6 q^{24} + 4 q^{25} - 4 q^{27} + 6 q^{28} + 6 q^{29} - 2 q^{30} - 8 q^{31} + 14 q^{32} - 4 q^{33} - 16 q^{34} + 4 q^{35} + 6 q^{36} + 12 q^{37} + 4 q^{38} - 4 q^{39} + 6 q^{40} - 2 q^{42} + 6 q^{43} + 6 q^{44} + 4 q^{45} - 4 q^{46} - 6 q^{48} + 4 q^{49} + 2 q^{50} - 6 q^{51} - 12 q^{52} + 14 q^{53} - 2 q^{54} + 4 q^{55} + 6 q^{56} + 6 q^{57} + 20 q^{58} + 2 q^{59} - 6 q^{60} + 2 q^{61} + 20 q^{62} + 4 q^{63} - 2 q^{64} + 4 q^{65} - 2 q^{66} - 12 q^{67} + 20 q^{68} - 10 q^{69} + 2 q^{70} + 4 q^{71} + 6 q^{72} + 20 q^{73} - 32 q^{74} - 4 q^{75} + 16 q^{76} + 4 q^{77} - 4 q^{79} + 6 q^{80} + 4 q^{81} - 16 q^{82} + 14 q^{83} - 6 q^{84} + 6 q^{85} + 40 q^{86} - 6 q^{87} + 6 q^{88} - 6 q^{89} + 2 q^{90} + 4 q^{91} + 40 q^{92} + 8 q^{93} + 4 q^{94} - 6 q^{95} - 14 q^{96} - 14 q^{97} + 2 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9 + 2 * q^10 + 4 * q^11 - 6 * q^12 + 4 * q^13 + 2 * q^14 - 4 * q^15 + 6 * q^16 + 6 * q^17 + 2 * q^18 - 6 * q^19 + 6 * q^20 - 4 * q^21 + 2 * q^22 + 10 * q^23 - 6 * q^24 + 4 * q^25 - 4 * q^27 + 6 * q^28 + 6 * q^29 - 2 * q^30 - 8 * q^31 + 14 * q^32 - 4 * q^33 - 16 * q^34 + 4 * q^35 + 6 * q^36 + 12 * q^37 + 4 * q^38 - 4 * q^39 + 6 * q^40 - 2 * q^42 + 6 * q^43 + 6 * q^44 + 4 * q^45 - 4 * q^46 - 6 * q^48 + 4 * q^49 + 2 * q^50 - 6 * q^51 - 12 * q^52 + 14 * q^53 - 2 * q^54 + 4 * q^55 + 6 * q^56 + 6 * q^57 + 20 * q^58 + 2 * q^59 - 6 * q^60 + 2 * q^61 + 20 * q^62 + 4 * q^63 - 2 * q^64 + 4 * q^65 - 2 * q^66 - 12 * q^67 + 20 * q^68 - 10 * q^69 + 2 * q^70 + 4 * q^71 + 6 * q^72 + 20 * q^73 - 32 * q^74 - 4 * q^75 + 16 * q^76 + 4 * q^77 - 4 * q^79 + 6 * q^80 + 4 * q^81 - 16 * q^82 + 14 * q^83 - 6 * q^84 + 6 * q^85 + 40 * q^86 - 6 * q^87 + 6 * q^88 - 6 * q^89 + 2 * q^90 + 4 * q^91 + 40 * q^92 + 8 * q^93 + 4 * q^94 - 6 * q^95 - 14 * q^96 - 14 * q^97 + 2 * q^98 + 4 * 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 $

$2$	−0.222191	−0.157113	−0.0785565	−	0.996910i	$-0.525031\pi$
$2$	−0.222191	−0.157113	−0.0785565	+	0.996910i	$0.525031\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.95063	−0.975315
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.222191	0.0907092
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.877796	0.310348
$9$	1.00000	0.333333
$10$	−0.222191	−0.0702631
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.95063	0.563099
$13$	0.900012	0.249619	0.124809	−	0.992181i	$-0.460168\pi$
$13$	0.900012	0.249619	0.124809	+	0.992181i	$0.460168\pi$
$14$	−0.222191	−0.0593831
$15$	−1.00000	−0.258199
$16$	3.70622	0.926556
$17$	−0.506248	−0.122783	−0.0613916	−	0.998114i	$-0.519554\pi$
$17$	−0.506248	−0.122783	−0.0613916	+	0.998114i	$0.519554\pi$
$18$	−0.222191	−0.0523710
$19$	−4.95063	−1.13575	−0.567876	−	0.823114i	$-0.692235\pi$
$19$	−4.95063	−1.13575	−0.567876	+	0.823114i	$0.692235\pi$
$20$	−1.95063	−0.436174
$21$	−1.00000	−0.218218
$22$	−0.222191	−0.0473714
$23$	−6.60749	−1.37776	−0.688878	−	0.724877i	$-0.741896\pi$
$23$	−6.60749	−1.37776	−0.688878	+	0.724877i	$0.741896\pi$
$24$	−0.877796	−0.179179
$25$	1.00000	0.200000
$26$	−0.199975	−0.0392183
$27$	−1.00000	−0.192450
$28$	−1.95063	−0.368635
$29$	3.40626	0.632527	0.316263	−	0.948671i	$-0.397572\pi$
$29$	3.40626	0.632527	0.316263	+	0.948671i	$0.397572\pi$
$30$	0.222191	0.0405664
$31$	−3.54437	−0.636588	−0.318294	−	0.947992i	$-0.603110\pi$
$31$	−3.54437	−0.636588	−0.318294	+	0.947992i	$0.603110\pi$
$32$	−2.57908	−0.455922
$33$	−1.00000	−0.174078
$34$	0.112484	0.0192908
$35$	1.00000	0.169031
$36$	−1.95063	−0.325105
$37$	11.4456	1.88165	0.940825	−	0.338892i	$-0.110052\pi$
$37$	11.4456	1.88165	0.940825	+	0.338892i	$0.110052\pi$
$38$	1.09999	0.178442
$39$	−0.900012	−0.144117
$40$	0.877796	0.138792
$41$	6.80127	1.06218	0.531090	−	0.847315i	$-0.321782\pi$
$41$	6.80127	1.06218	0.531090	+	0.847315i	$0.321782\pi$
$42$	0.222191	0.0342849
$43$	9.26309	1.41261	0.706304	−	0.707909i	$-0.250361\pi$
$43$	9.26309	1.41261	0.706304	+	0.707909i	$0.250361\pi$
$44$	−1.95063	−0.294069
$45$	1.00000	0.149071
$46$	1.46813	0.216463
$47$	12.6581	1.84637	0.923187	−	0.384351i	$-0.125575\pi$
$47$	12.6581	1.84637	0.923187	+	0.384351i	$0.125575\pi$
$48$	−3.70622	−0.534947
$49$	1.00000	0.142857
$50$	−0.222191	−0.0314226
$51$	0.506248	0.0708889
$52$	−1.75559	−0.243457
$53$	−1.39501	−0.191620	−0.0958099	−	0.995400i	$-0.530544\pi$
$53$	−1.39501	−0.191620	−0.0958099	+	0.995400i	$0.530544\pi$
$54$	0.222191	0.0302364
$55$	1.00000	0.134840
$56$	0.877796	0.117300
$57$	4.95063	0.655727
$58$	−0.756842	−0.0993782
$59$	6.93938	0.903431	0.451715	−	0.892162i	$-0.350812\pi$
$59$	6.93938	0.903431	0.451715	+	0.892162i	$0.350812\pi$
$60$	1.95063	0.251825
$61$	5.39501	0.690761	0.345380	−	0.938463i	$-0.387750\pi$
$61$	5.39501	0.690761	0.345380	+	0.938463i	$0.387750\pi$
$62$	0.787529	0.100016
$63$	1.00000	0.125988
$64$	−6.83940	−0.854925
$65$	0.900012	0.111633
$66$	0.222191	0.0273499
$67$	−10.3012	−1.25849	−0.629247	−	0.777206i	$-0.716636\pi$
$67$	−10.3012	−1.25849	−0.629247	+	0.777206i	$0.716636\pi$
$68$	0.987504	0.119752
$69$	6.60749	0.795448
$70$	−0.222191	−0.0265569
$71$	6.35689	0.754424	0.377212	−	0.926127i	$-0.376883\pi$
$71$	6.35689	0.754424	0.377212	+	0.926127i	$0.376883\pi$
$72$	0.877796	0.103449
$73$	3.55562	0.416154	0.208077	−	0.978112i	$-0.433280\pi$
$73$	3.55562	0.416154	0.208077	+	0.978112i	$0.433280\pi$
$74$	−2.54312	−0.295632
$75$	−1.00000	−0.115470
$76$	9.65685	1.10772
$77$	1.00000	0.113961
$78$	0.199975	0.0226427
$79$	1.98875	0.223752	0.111876	−	0.993722i	$-0.464314\pi$
$79$	1.98875	0.223752	0.111876	+	0.993722i	$0.464314\pi$
$80$	3.70622	0.414368
$81$	1.00000	0.111111
$82$	−1.51118	−0.166882
$83$	−4.16310	−0.456960	−0.228480	−	0.973549i	$-0.573376\pi$
$83$	−4.16310	−0.456960	−0.228480	+	0.973549i	$0.573376\pi$
$84$	1.95063	0.212831
$85$	−0.506248	−0.0549103
$86$	−2.05818	−0.221939
$87$	−3.40626	−0.365189
$88$	0.877796	0.0935734
$89$	2.25059	0.238562	0.119281	−	0.992861i	$-0.461941\pi$
$89$	2.25059	0.238562	0.119281	+	0.992861i	$0.461941\pi$
$90$	−0.222191	−0.0234210
$91$	0.900012	0.0943469
$92$	12.8888	1.34375
$93$	3.54437	0.367534
$94$	−2.81252	−0.290089
$95$	−4.95063	−0.507924
$96$	2.57908	0.263227
$97$	0.0506185	0.00513953	0.00256976	−	0.999997i	$-0.499182\pi$
$97$	0.0506185	0.00513953	0.00256976	+	0.999997i	$0.499182\pi$
$98$	−0.222191	−0.0224447
$99$	1.00000	0.100504
$100$	−1.95063	−0.195063
$101$	−3.25690	−0.324074	−0.162037	−	0.986785i	$-0.551806\pi$
$101$	−3.25690	−0.324074	−0.162037	+	0.986785i	$0.551806\pi$
$102$	−0.112484	−0.0111376
$103$	−0.0481195	−0.00474136	−0.00237068	−	0.999997i	$-0.500755\pi$
$103$	−0.0481195	−0.00474136	−0.00237068	+	0.999997i	$0.500755\pi$
$104$	0.790027	0.0774686
$105$	−1.00000	−0.0975900
$106$	0.309960	0.0301060
$107$	15.8900	1.53615	0.768073	−	0.640362i	$-0.221216\pi$
$107$	15.8900	1.53615	0.768073	+	0.640362i	$0.221216\pi$
$108$	1.95063	0.187700
$109$	−17.4706	−1.67338	−0.836691	−	0.547675i	$-0.815513\pi$
$109$	−17.4706	−1.67338	−0.836691	+	0.547675i	$0.815513\pi$
$110$	−0.222191	−0.0211851
$111$	−11.4456	−1.08637
$112$	3.70622	0.350205
$113$	19.4200	1.82688	0.913440	−	0.406973i	$-0.133416\pi$
$113$	19.4200	1.82688	0.913440	+	0.406973i	$0.133416\pi$
$114$	−1.09999	−0.103023
$115$	−6.60749	−0.616151
$116$	−6.64436	−0.616913
$117$	0.900012	0.0832062
$118$	−1.54187	−0.141941
$119$	−0.506248	−0.0464077
$120$	−0.877796	−0.0801315
$121$	1.00000	0.0909091
$122$	−1.19873	−0.108528
$123$	−6.80127	−0.613250
$124$	6.91376	0.620874
$125$	1.00000	0.0894427
$126$	−0.222191	−0.0197944
$127$	10.9394	0.970713	0.485357	−	0.874316i	$-0.338690\pi$
$127$	10.9394	0.970713	0.485357	+	0.874316i	$0.338690\pi$
$128$	6.67782	0.590242
$129$	−9.26309	−0.815570
$130$	−0.199975	−0.0175390
$131$	11.3469	0.991383	0.495691	−	0.868499i	$-0.334915\pi$
$131$	11.3469	0.991383	0.495691	+	0.868499i	$0.334915\pi$
$132$	1.95063	0.169781
$133$	−4.95063	−0.429274
$134$	2.28884	0.197726
$135$	−1.00000	−0.0860663
$136$	−0.444383	−0.0381055
$137$	13.1900	1.12690	0.563448	−	0.826152i	$-0.309475\pi$
$137$	13.1900	1.12690	0.563448	+	0.826152i	$0.309475\pi$
$138$	−1.46813	−0.124975
$139$	−19.5581	−1.65890	−0.829449	−	0.558583i	$-0.811345\pi$
$139$	−19.5581	−1.65890	−0.829449	+	0.558583i	$0.811345\pi$
$140$	−1.95063	−0.164858
$141$	−12.6581	−1.06600
$142$	−1.41245	−0.118530
$143$	0.900012	0.0752628
$144$	3.70622	0.308852
$145$	3.40626	0.282875
$146$	−0.790027	−0.0653831
$147$	−1.00000	−0.0824786
$148$	−22.3262	−1.83520
$149$	−0.568114	−0.0465417	−0.0232708	−	0.999729i	$-0.507408\pi$
$149$	−0.568114	−0.0465417	−0.0232708	+	0.999729i	$0.507408\pi$
$150$	0.222191	0.0181418
$151$	4.44688	0.361882	0.180941	−	0.983494i	$-0.442086\pi$
$151$	4.44688	0.361882	0.180941	+	0.983494i	$0.442086\pi$
$152$	−4.34564	−0.352478
$153$	−0.506248	−0.0409277
$154$	−0.222191	−0.0179047
$155$	−3.54437	−0.284691
$156$	1.75559	0.140560
$157$	8.15186	0.650589	0.325294	−	0.945613i	$-0.394537\pi$
$157$	8.15186	0.650589	0.325294	+	0.945613i	$0.394537\pi$
$158$	−0.441884	−0.0351544
$159$	1.39501	0.110632
$160$	−2.57908	−0.203894
$161$	−6.60749	−0.520743
$162$	−0.222191	−0.0174570
$163$	6.63311	0.519545	0.259773	−	0.965670i	$-0.416352\pi$
$163$	6.63311	0.519545	0.259773	+	0.965670i	$0.416352\pi$
$164$	−13.2668	−1.03596
$165$	−1.00000	−0.0778499
$166$	0.925005	0.0717943
$167$	3.75559	0.290616	0.145308	−	0.989386i	$-0.453583\pi$
$167$	3.75559	0.290616	0.145308	+	0.989386i	$0.453583\pi$
$168$	−0.877796	−0.0677234
$169$	−12.1900	−0.937691
$170$	0.112484	0.00862713
$171$	−4.95063	−0.378584
$172$	−18.0689	−1.37774
$173$	−15.7038	−1.19394	−0.596968	−	0.802265i	$-0.703628\pi$
$173$	−15.7038	−1.19394	−0.596968	+	0.802265i	$0.703628\pi$
$174$	0.756842	0.0573760
$175$	1.00000	0.0755929
$176$	3.70622	0.279367
$177$	−6.93938	−0.521596
$178$	−0.500062	−0.0374813
$179$	−8.95682	−0.669464	−0.334732	−	0.942313i	$-0.608646\pi$
$179$	−8.95682	−0.669464	−0.334732	+	0.942313i	$0.608646\pi$
$180$	−1.95063	−0.145391
$181$	−13.8025	−1.02593	−0.512967	−	0.858408i	$-0.671454\pi$
$181$	−13.8025	−1.02593	−0.512967	+	0.858408i	$0.671454\pi$
$182$	−0.199975	−0.0148231
$183$	−5.39501	−0.398811
$184$	−5.80002	−0.427584
$185$	11.4456	0.841500
$186$	−0.787529	−0.0577444
$187$	−0.506248	−0.0370205
$188$	−24.6913	−1.80080
$189$	−1.00000	−0.0727393
$190$	1.09999	0.0798015
$191$	6.91376	0.500262	0.250131	−	0.968212i	$-0.419526\pi$
$191$	6.91376	0.500262	0.250131	+	0.968212i	$0.419526\pi$
$192$	6.83940	0.493591
$193$	2.14317	0.154269	0.0771344	−	0.997021i	$-0.475423\pi$
$193$	2.14317	0.154269	0.0771344	+	0.997021i	$0.475423\pi$
$194$	−0.0112470	−0.000807487 0
$195$	−0.900012	−0.0644512
$196$	−1.95063	−0.139331
$197$	18.5901	1.32449	0.662243	−	0.749289i	$-0.269605\pi$
$197$	18.5901	1.32449	0.662243	+	0.749289i	$0.269605\pi$
$198$	−0.222191	−0.0157905
$199$	−3.60005	−0.255201	−0.127600	−	0.991826i	$-0.540727\pi$
$199$	−3.60005	−0.255201	−0.127600	+	0.991826i	$0.540727\pi$
$200$	0.877796	0.0620696
$201$	10.3012	0.726591
$202$	0.723656	0.0509163
$203$	3.40626	0.239073
$204$	−0.987504	−0.0691391
$205$	6.80127	0.475022
$206$	0.0106917	0.000744929 0
$207$	−6.60749	−0.459252
$208$	3.33565	0.231286
$209$	−4.95063	−0.342442
$210$	0.222191	0.0153327
$211$	−4.22191	−0.290649	−0.145324	−	0.989384i	$-0.546423\pi$
$211$	−4.22191	−0.290649	−0.145324	+	0.989384i	$0.546423\pi$
$212$	2.72116	0.186890
$213$	−6.35689	−0.435567
$214$	−3.53062	−0.241349
$215$	9.26309	0.631737
$216$	−0.877796	−0.0597265
$217$	−3.54437	−0.240608
$218$	3.88182	0.262910
$219$	−3.55562	−0.240266
$220$	−1.95063	−0.131512
$221$	−0.455630	−0.0306490
$222$	2.54312	0.170683
$223$	−9.94938	−0.666260	−0.333130	−	0.942881i	$-0.608105\pi$
$223$	−9.94938	−0.666260	−0.333130	+	0.942881i	$0.608105\pi$
$224$	−2.57908	−0.172322
$225$	1.00000	0.0666667
$226$	−4.31496	−0.287027
$227$	14.8519	0.985755	0.492877	−	0.870099i	$-0.335945\pi$
$227$	14.8519	0.985755	0.492877	+	0.870099i	$0.335945\pi$
$228$	−9.65685	−0.639541
$229$	3.24316	0.214314	0.107157	−	0.994242i	$-0.465825\pi$
$229$	3.24316	0.214314	0.107157	+	0.994242i	$0.465825\pi$
$230$	1.46813	0.0968054
$231$	−1.00000	−0.0657952
$232$	2.99000	0.196303
$233$	11.1694	0.731733	0.365866	−	0.930667i	$-0.380773\pi$
$233$	11.1694	0.731733	0.365866	+	0.930667i	$0.380773\pi$
$234$	−0.199975	−0.0130728
$235$	12.6581	0.825724
$236$	−13.5362	−0.881130
$237$	−1.98875	−0.129183
$238$	0.112484	0.00729125
$239$	2.45057	0.158514	0.0792571	−	0.996854i	$-0.474745\pi$
$239$	2.45057	0.158514	0.0792571	+	0.996854i	$0.474745\pi$
$240$	−3.70622	−0.239236
$241$	−26.1012	−1.68133	−0.840664	−	0.541557i	$-0.817835\pi$
$241$	−26.1012	−1.68133	−0.840664	+	0.541557i	$0.817835\pi$
$242$	−0.222191	−0.0142830
$243$	−1.00000	−0.0641500
$244$	−10.5237	−0.673710
$245$	1.00000	0.0638877
$246$	1.51118	0.0963496
$247$	−4.45563	−0.283505
$248$	−3.11123	−0.197564
$249$	4.16310	0.263826
$250$	−0.222191	−0.0140526
$251$	−9.37758	−0.591908	−0.295954	−	0.955202i	$-0.595637\pi$
$251$	−9.37758	−0.591908	−0.295954	+	0.955202i	$0.595637\pi$
$252$	−1.95063	−0.122878
$253$	−6.60749	−0.415409
$254$	−2.43064	−0.152512
$255$	0.506248	0.0317025
$256$	12.1950	0.762190
$257$	−13.9718	−0.871538	−0.435769	−	0.900059i	$-0.643523\pi$
$257$	−13.9718	−0.871538	−0.435769	+	0.900059i	$0.643523\pi$
$258$	2.05818	0.128137
$259$	11.4456	0.711197
$260$	−1.75559	−0.108877
$261$	3.40626	0.210842
$262$	−2.52118	−0.155759
$263$	−18.1232	−1.11752	−0.558761	−	0.829328i	$-0.688723\pi$
$263$	−18.1232	−1.11752	−0.558761	+	0.829328i	$0.688723\pi$
$264$	−0.877796	−0.0540246
$265$	−1.39501	−0.0856950
$266$	1.09999	0.0674446
$267$	−2.25059	−0.137734
$268$	20.0939	1.22743
$269$	32.1793	1.96201	0.981005	−	0.193984i	$-0.0621409\pi$
$269$	32.1793	1.96201	0.981005	+	0.193984i	$0.0621409\pi$
$270$	0.222191	0.0135221
$271$	−24.0669	−1.46196	−0.730979	−	0.682400i	$-0.760936\pi$
$271$	−24.0669	−1.46196	−0.730979	+	0.682400i	$0.760936\pi$
$272$	−1.87627	−0.113766
$273$	−0.900012	−0.0544712
$274$	−2.93070	−0.177050
$275$	1.00000	0.0603023
$276$	−12.8888	−0.775813
$277$	16.8443	1.01208	0.506039	−	0.862511i	$-0.331109\pi$
$277$	16.8443	1.01208	0.506039	+	0.862511i	$0.331109\pi$
$278$	4.34564	0.260634
$279$	−3.54437	−0.212196
$280$	0.877796	0.0524584
$281$	−0.0444327	−0.00265063	−0.00132532	−	0.999999i	$-0.500422\pi$
$281$	−0.0444327	−0.00265063	−0.00132532	+	0.999999i	$0.500422\pi$
$282$	2.81252	0.167483
$283$	−0.455630	−0.0270844	−0.0135422	−	0.999908i	$-0.504311\pi$
$283$	−0.455630	−0.0270844	−0.0135422	+	0.999908i	$0.504311\pi$
$284$	−12.4000	−0.735802
$285$	4.95063	0.293250
$286$	−0.199975	−0.0118248
$287$	6.80127	0.401467
$288$	−2.57908	−0.151974
$289$	−16.7437	−0.984924
$290$	−0.756842	−0.0444433
$291$	−0.0506185	−0.00296731
$292$	−6.93570	−0.405881
$293$	31.1337	1.81885	0.909424	−	0.415870i	$-0.136523\pi$
$293$	31.1337	1.81885	0.909424	+	0.415870i	$0.136523\pi$
$294$	0.222191	0.0129585
$295$	6.93938	0.404027
$296$	10.0469	0.583966
$297$	−1.00000	−0.0580259
$298$	0.126230	0.00731231
$299$	−5.94682	−0.343913
$300$	1.95063	0.112620
$301$	9.26309	0.533916
$302$	−0.988059	−0.0568564
$303$	3.25690	0.187104
$304$	−18.3481	−1.05234
$305$	5.39501	0.308918
$306$	0.112484	0.00643028
$307$	−27.9038	−1.59255	−0.796276	−	0.604934i	$-0.793200\pi$
$307$	−27.9038	−1.59255	−0.796276	+	0.604934i	$0.793200\pi$
$308$	−1.95063	−0.111148
$309$	0.0481195	0.00273742
$310$	0.787529	0.0447286
$311$	−20.4274	−1.15833	−0.579167	−	0.815209i	$-0.696622\pi$
$311$	−20.4274	−1.15833	−0.579167	+	0.815209i	$0.696622\pi$
$312$	−0.790027	−0.0447265
$313$	22.0531	1.24652	0.623258	−	0.782016i	$-0.285809\pi$
$313$	22.0531	1.24652	0.623258	+	0.782016i	$0.285809\pi$
$314$	−1.81127	−0.102216
$315$	1.00000	0.0563436
$316$	−3.87932	−0.218229
$317$	−17.9038	−1.00558	−0.502788	−	0.864410i	$-0.667692\pi$
$317$	−17.9038	−1.00558	−0.502788	+	0.864410i	$0.667692\pi$
$318$	−0.309960	−0.0173817
$319$	3.40626	0.190714
$320$	−6.83940	−0.382334
$321$	−15.8900	−0.886894
$322$	1.46813	0.0818155
$323$	2.50625	0.139451
$324$	−1.95063	−0.108368
$325$	0.900012	0.0499237
$326$	−1.47382	−0.0816273
$327$	17.4706	0.966128
$328$	5.97013	0.329645
$329$	12.6581	0.697864
$330$	0.222191	0.0122312
$331$	18.5112	1.01747	0.508735	−	0.860923i	$-0.330113\pi$
$331$	18.5112	1.01747	0.508735	+	0.860923i	$0.330113\pi$
$332$	8.12068	0.445680
$333$	11.4456	0.627217
$334$	−0.834460	−0.0456596
$335$	−10.3012	−0.562815
$336$	−3.70622	−0.202191
$337$	24.8187	1.35196	0.675981	−	0.736919i	$-0.263720\pi$
$337$	24.8187	1.35196	0.675981	+	0.736919i	$0.263720\pi$
$338$	2.70851	0.147323
$339$	−19.4200	−1.05475
$340$	0.987504	0.0535549
$341$	−3.54437	−0.191938
$342$	1.09999	0.0594805
$343$	1.00000	0.0539949
$344$	8.13110	0.438400
$345$	6.60749	0.355735
$346$	3.48925	0.187583
$347$	1.03194	0.0553972	0.0276986	−	0.999616i	$-0.491182\pi$
$347$	1.03194	0.0553972	0.0276986	+	0.999616i	$0.491182\pi$
$348$	6.64436	0.356175
$349$	18.6100	0.996170	0.498085	−	0.867128i	$-0.334037\pi$
$349$	18.6100	0.996170	0.498085	+	0.867128i	$0.334037\pi$
$350$	−0.222191	−0.0118766
$351$	−0.900012	−0.0480391
$352$	−2.57908	−0.137466
$353$	−8.89126	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$353$	−8.89126	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$354$	1.54187	0.0819495
$355$	6.35689	0.337389
$356$	−4.39008	−0.232674
$357$	0.506248	0.0267935
$358$	1.99013	0.105182
$359$	−10.1395	−0.535141	−0.267571	−	0.963538i	$-0.586221\pi$
$359$	−10.1395	−0.535141	−0.267571	+	0.963538i	$0.586221\pi$
$360$	0.877796	0.0462639
$361$	5.50875	0.289934
$362$	3.06680	0.161188
$363$	−1.00000	−0.0524864
$364$	−1.75559	−0.0920180
$365$	3.55562	0.186110
$366$	1.19873	0.0626584
$367$	−15.4631	−0.807165	−0.403583	−	0.914943i	$-0.632235\pi$
$367$	−15.4631	−0.807165	−0.403583	+	0.914943i	$0.632235\pi$
$368$	−24.4888	−1.27657
$369$	6.80127	0.354060
$370$	−2.54312	−0.132211
$371$	−1.39501	−0.0724255
$372$	−6.91376	−0.358462
$373$	0.470008	0.0243361	0.0121681	−	0.999926i	$-0.496127\pi$
$373$	0.470008	0.0243361	0.0121681	+	0.999926i	$0.496127\pi$
$374$	0.112484	0.00581641
$375$	−1.00000	−0.0516398
$376$	11.1112	0.573018
$377$	3.06568	0.157890
$378$	0.222191	0.0114283
$379$	−25.2225	−1.29559	−0.647797	−	0.761813i	$-0.724310\pi$
$379$	−25.2225	−1.29559	−0.647797	+	0.761813i	$0.724310\pi$
$380$	9.65685	0.495386
$381$	−10.9394	−0.560442
$382$	−1.53618	−0.0785977
$383$	−21.2257	−1.08458	−0.542290	−	0.840191i	$-0.682443\pi$
$383$	−21.2257	−1.08458	−0.542290	+	0.840191i	$0.682443\pi$
$384$	−6.67782	−0.340776
$385$	1.00000	0.0509647
$386$	−0.476194	−0.0242376
$387$	9.26309	0.470869
$388$	−0.0987380	−0.00501266
$389$	8.93182	0.452861	0.226431	−	0.974027i	$-0.427294\pi$
$389$	8.93182	0.452861	0.226431	+	0.974027i	$0.427294\pi$
$390$	0.199975	0.0101261
$391$	3.34503	0.169165
$392$	0.877796	0.0443354
$393$	−11.3469	−0.572375
$394$	−4.13055	−0.208094
$395$	1.98875	0.100065
$396$	−1.95063	−0.0980229
$397$	−17.6563	−0.886144	−0.443072	−	0.896486i	$-0.646111\pi$
$397$	−17.6563	−0.886144	−0.443072	+	0.896486i	$0.646111\pi$
$398$	0.799900	0.0400954
$399$	4.95063	0.247842
$400$	3.70622	0.185311
$401$	−25.1494	−1.25590	−0.627951	−	0.778253i	$-0.716106\pi$
$401$	−25.1494	−1.25590	−0.627951	+	0.778253i	$0.716106\pi$
$402$	−2.28884	−0.114157
$403$	−3.18998	−0.158904
$404$	6.35302	0.316074
$405$	1.00000	0.0496904
$406$	−0.756842	−0.0375614
$407$	11.4456	0.567339
$408$	0.444383	0.0220002
$409$	29.0150	1.43470	0.717350	−	0.696713i	$-0.245355\pi$
$409$	29.0150	1.43470	0.717350	+	0.696713i	$0.245355\pi$
$410$	−1.51118	−0.0746321
$411$	−13.1900	−0.650614
$412$	0.0938634	0.00462432
$413$	6.93938	0.341465
$414$	1.46813	0.0721545
$415$	−4.16310	−0.204359
$416$	−2.32121	−0.113807
$417$	19.5581	0.957765
$418$	1.09999	0.0538021
$419$	−26.1294	−1.27650	−0.638251	−	0.769828i	$-0.720342\pi$
$419$	−26.1294	−1.27650	−0.638251	+	0.769828i	$0.720342\pi$
$420$	1.95063	0.0951810
$421$	37.0963	1.80796	0.903982	−	0.427571i	$-0.140631\pi$
$421$	37.0963	1.80796	0.903982	+	0.427571i	$0.140631\pi$
$422$	0.938073	0.0456647
$423$	12.6581	0.615458
$424$	−1.22454	−0.0594688
$425$	−0.506248	−0.0245566
$426$	1.41245	0.0684333
$427$	5.39501	0.261083
$428$	−30.9956	−1.49823
$429$	−0.900012	−0.0434530
$430$	−2.05818	−0.0992542
$431$	9.60255	0.462539	0.231269	−	0.972890i	$-0.425712\pi$
$431$	9.60255	0.462539	0.231269	+	0.972890i	$0.425712\pi$
$432$	−3.70622	−0.178316
$433$	−31.1137	−1.49523	−0.747615	−	0.664132i	$-0.768801\pi$
$433$	−31.1137	−1.49523	−0.747615	+	0.664132i	$0.768801\pi$
$434$	0.787529	0.0378026
$435$	−3.40626	−0.163318
$436$	34.0787	1.63208
$437$	32.7112	1.56479
$438$	0.790027	0.0377490
$439$	−34.0644	−1.62580	−0.812902	−	0.582401i	$-0.802113\pi$
$439$	−34.0644	−1.62580	−0.812902	+	0.582401i	$0.802113\pi$
$440$	0.877796	0.0418473
$441$	1.00000	0.0476190
$442$	0.101237	0.00481535
$443$	19.0375	0.904498	0.452249	−	0.891892i	$-0.350622\pi$
$443$	19.0375	0.904498	0.452249	+	0.891892i	$0.350622\pi$
$444$	22.3262	1.05955
$445$	2.25059	0.106688
$446$	2.21067	0.104678
$447$	0.568114	0.0268709
$448$	−6.83940	−0.323131
$449$	18.6806	0.881592	0.440796	−	0.897607i	$-0.354696\pi$
$449$	18.6806	0.881592	0.440796	+	0.897607i	$0.354696\pi$
$450$	−0.222191	−0.0104742
$451$	6.80127	0.320260
$452$	−37.8813	−1.78178
$453$	−4.44688	−0.208933
$454$	−3.29996	−0.154875
$455$	0.900012	0.0421932
$456$	4.34564	0.203503
$457$	27.2875	1.27646	0.638228	−	0.769847i	$-0.279668\pi$
$457$	27.2875	1.27646	0.638228	+	0.769847i	$0.279668\pi$
$458$	−0.720602	−0.0336715
$459$	0.506248	0.0236296
$460$	12.8888	0.600942
$461$	−37.2644	−1.73558	−0.867788	−	0.496934i	$-0.834459\pi$
$461$	−37.2644	−1.73558	−0.867788	+	0.496934i	$0.834459\pi$
$462$	0.222191	0.0103373
$463$	−8.50119	−0.395084	−0.197542	−	0.980294i	$-0.563296\pi$
$463$	−8.50119	−0.395084	−0.197542	+	0.980294i	$0.563296\pi$
$464$	12.6244	0.586071
$465$	3.54437	0.164366
$466$	−2.48175	−0.114965
$467$	−20.5163	−0.949381	−0.474691	−	0.880153i	$-0.657440\pi$
$467$	−20.5163	−0.949381	−0.474691	+	0.880153i	$0.657440\pi$
$468$	−1.75559	−0.0811523
$469$	−10.3012	−0.475666
$470$	−2.81252	−0.129732
$471$	−8.15186	−0.375618
$472$	6.09136	0.280378
$473$	9.26309	0.425917
$474$	0.441884	0.0202964
$475$	−4.95063	−0.227151
$476$	0.987504	0.0452621
$477$	−1.39501	−0.0638733
$478$	−0.544495	−0.0249046
$479$	12.2806	0.561117	0.280559	−	0.959837i	$-0.409480\pi$
$479$	12.2806	0.561117	0.280559	+	0.959837i	$0.409480\pi$
$480$	2.57908	0.117718
$481$	10.3012	0.469695
$482$	5.79947	0.264159
$483$	6.60749	0.300651
$484$	−1.95063	−0.0886650
$485$	0.0506185	0.00229847
$486$	0.222191	0.0100788
$487$	−20.5901	−0.933024	−0.466512	−	0.884515i	$-0.654490\pi$
$487$	−20.5901	−0.933024	−0.466512	+	0.884515i	$0.654490\pi$
$488$	4.73572	0.214376
$489$	−6.63311	−0.299960
$490$	−0.222191	−0.0100376
$491$	6.32684	0.285526	0.142763	−	0.989757i	$-0.454401\pi$
$491$	6.32684	0.285526	0.142763	+	0.989757i	$0.454401\pi$
$492$	13.2668	0.598112
$493$	−1.72441	−0.0776637
$494$	0.990002	0.0445423
$495$	1.00000	0.0449467
$496$	−13.1362	−0.589834
$497$	6.35689	0.285146
$498$	−0.925005	−0.0414505
$499$	−0.627480	−0.0280899	−0.0140449	−	0.999901i	$-0.504471\pi$
$499$	−0.627480	−0.0280899	−0.0140449	+	0.999901i	$0.504471\pi$
$500$	−1.95063	−0.0872349
$501$	−3.75559	−0.167787
$502$	2.08362	0.0929964
$503$	−7.07686	−0.315542	−0.157771	−	0.987476i	$-0.550431\pi$
$503$	−7.07686	−0.315542	−0.157771	+	0.987476i	$0.550431\pi$
$504$	0.877796	0.0391001
$505$	−3.25690	−0.144930
$506$	1.46813	0.0652662
$507$	12.1900	0.541376
$508$	−21.3387	−0.946752
$509$	3.06062	0.135659	0.0678297	−	0.997697i	$-0.478393\pi$
$509$	3.06062	0.135659	0.0678297	+	0.997697i	$0.478393\pi$
$510$	−0.112484	−0.00498087
$511$	3.55562	0.157291
$512$	−16.0653	−0.709992
$513$	4.95063	0.218576
$514$	3.10442	0.136930
$515$	−0.0481195	−0.00212040
$516$	18.0689	0.795438
$517$	12.6581	0.556703
$518$	−2.54312	−0.111738
$519$	15.7038	0.689320
$520$	0.790027	0.0346450
$521$	−36.2018	−1.58603	−0.793016	−	0.609201i	$-0.791490\pi$
$521$	−36.2018	−1.58603	−0.793016	+	0.609201i	$0.791490\pi$
$522$	−0.756842	−0.0331261
$523$	35.7412	1.56285	0.781426	−	0.623998i	$-0.214493\pi$
$523$	35.7412	1.56285	0.781426	+	0.623998i	$0.214493\pi$
$524$	−22.1336	−0.966911
$525$	−1.00000	−0.0436436
$526$	4.02681	0.175577
$527$	1.79433	0.0781623
$528$	−3.70622	−0.161293
$529$	20.6589	0.898211
$530$	0.309960	0.0134638
$531$	6.93938	0.301144
$532$	9.65685	0.418678
$533$	6.12123	0.265140
$534$	0.500062	0.0216398
$535$	15.8900	0.686985
$536$	−9.04236	−0.390571
$537$	8.95682	0.386515
$538$	−7.14997	−0.308257
$539$	1.00000	0.0430730
$540$	1.95063	0.0839418
$541$	44.1644	1.89878	0.949388	−	0.314105i	$-0.101704\pi$
$541$	44.1644	1.89878	0.949388	+	0.314105i	$0.101704\pi$
$542$	5.34745	0.229693
$543$	13.8025	0.592323
$544$	1.30566	0.0559796
$545$	−17.4706	−0.748359
$546$	0.199975	0.00855814
$547$	−11.1644	−0.477353	−0.238677	−	0.971099i	$-0.576714\pi$
$547$	−11.1644	−0.477353	−0.238677	+	0.971099i	$0.576714\pi$
$548$	−25.7288	−1.09908
$549$	5.39501	0.230254
$550$	−0.222191	−0.00947427
$551$	−16.8631	−0.718394
$552$	5.80002	0.246865
$553$	1.98875	0.0845704
$554$	−3.74267	−0.159011
$555$	−11.4456	−0.485840
$556$	38.1507	1.61795
$557$	−15.5137	−0.657336	−0.328668	−	0.944446i	$-0.606600\pi$
$557$	−15.5137	−0.657336	−0.328668	+	0.944446i	$0.606600\pi$
$558$	0.787529	0.0333387
$559$	8.33690	0.352613
$560$	3.70622	0.156617
$561$	0.506248	0.0213738
$562$	0.00987257	0.000416449 0
$563$	29.7581	1.25415	0.627077	−	0.778957i	$-0.284251\pi$
$563$	29.7581	1.25415	0.627077	+	0.778957i	$0.284251\pi$
$564$	24.6913	1.03969
$565$	19.4200	0.817006
$566$	0.101237	0.00425531
$567$	1.00000	0.0419961
$568$	5.58005	0.234134
$569$	−11.2949	−0.473507	−0.236753	−	0.971570i	$-0.576083\pi$
$569$	−11.2949	−0.473507	−0.236753	+	0.971570i	$0.576083\pi$
$570$	−1.09999	−0.0460734
$571$	33.3300	1.39482	0.697408	−	0.716675i	$-0.254337\pi$
$571$	33.3300	1.39482	0.697408	+	0.716675i	$0.254337\pi$
$572$	−1.75559	−0.0734050
$573$	−6.91376	−0.288826
$574$	−1.51118	−0.0630756
$575$	−6.60749	−0.275551
$576$	−6.83940	−0.284975
$577$	−47.6300	−1.98286	−0.991432	−	0.130622i	$-0.958303\pi$
$577$	−47.6300	−1.98286	−0.991432	+	0.130622i	$0.958303\pi$
$578$	3.72031	0.154744
$579$	−2.14317	−0.0890671
$580$	−6.64436	−0.275892
$581$	−4.16310	−0.172715
$582$	0.0112470	0.000466203 0
$583$	−1.39501	−0.0577756
$584$	3.12111	0.129152
$585$	0.900012	0.0372109
$586$	−6.91763	−0.285765
$587$	20.5844	0.849607	0.424804	−	0.905285i	$-0.360343\pi$
$587$	20.5844	0.849607	0.424804	+	0.905285i	$0.360343\pi$
$588$	1.95063	0.0804427
$589$	17.5469	0.723006
$590$	−1.54187	−0.0634778
$591$	−18.5901	−0.764693
$592$	42.4201	1.74345
$593$	−7.47882	−0.307118	−0.153559	−	0.988139i	$-0.549074\pi$
$593$	−7.47882	−0.307118	−0.153559	+	0.988139i	$0.549074\pi$
$594$	0.222191	0.00911662
$595$	−0.506248	−0.0207542
$596$	1.10818	0.0453928
$597$	3.60005	0.147340
$598$	1.32133	0.0540333
$599$	−28.5512	−1.16657	−0.583285	−	0.812268i	$-0.698233\pi$
$599$	−28.5512	−1.16657	−0.583285	+	0.812268i	$0.698233\pi$
$600$	−0.877796	−0.0358359
$601$	−23.5950	−0.962460	−0.481230	−	0.876594i	$-0.659810\pi$
$601$	−23.5950	−0.962460	−0.481230	+	0.876594i	$0.659810\pi$
$602$	−2.05818	−0.0838851
$603$	−10.3012	−0.419498
$604$	−8.67423	−0.352949
$605$	1.00000	0.0406558
$606$	−0.723656	−0.0293965
$607$	20.2401	0.821520	0.410760	−	0.911744i	$-0.365263\pi$
$607$	20.2401	0.821520	0.410760	+	0.911744i	$0.365263\pi$
$608$	12.7681	0.517814
$609$	−3.40626	−0.138029
$610$	−1.19873	−0.0485350
$611$	11.3925	0.460889
$612$	0.987504	0.0399175
$613$	−31.4644	−1.27083	−0.635417	−	0.772169i	$-0.719172\pi$
$613$	−31.4644	−1.27083	−0.635417	+	0.772169i	$0.719172\pi$
$614$	6.19998	0.250211
$615$	−6.80127	−0.274254
$616$	0.877796	0.0353674
$617$	35.6813	1.43647	0.718237	−	0.695798i	$-0.244949\pi$
$617$	35.6813	1.43647	0.718237	+	0.695798i	$0.244949\pi$
$618$	−0.0106917	−0.000430085 0
$619$	5.41245	0.217545	0.108772	−	0.994067i	$-0.465308\pi$
$619$	5.41245	0.217545	0.108772	+	0.994067i	$0.465308\pi$
$620$	6.91376	0.277663
$621$	6.60749	0.265149
$622$	4.53880	0.181989
$623$	2.25059	0.0901681
$624$	−3.33565	−0.133533
$625$	1.00000	0.0400000
$626$	−4.90001	−0.195844
$627$	4.95063	0.197709
$628$	−15.9013	−0.634529
$629$	−5.79433	−0.231035
$630$	−0.222191	−0.00885232
$631$	−32.0863	−1.27734	−0.638668	−	0.769483i	$-0.720514\pi$
$631$	−32.0863	−1.27734	−0.638668	+	0.769483i	$0.720514\pi$
$632$	1.74572	0.0694410
$633$	4.22191	0.167806
$634$	3.97806	0.157989
$635$	10.9394	0.434116
$636$	−2.72116	−0.107901
$637$	0.900012	0.0356598
$638$	−0.756842	−0.0299637
$639$	6.35689	0.251475
$640$	6.67782	0.263964
$641$	33.8075	1.33532	0.667658	−	0.744468i	$-0.267297\pi$
$641$	33.8075	1.33532	0.667658	+	0.744468i	$0.267297\pi$
$642$	3.53062	0.139343
$643$	−1.26309	−0.0498114	−0.0249057	−	0.999690i	$-0.507929\pi$
$643$	−1.26309	−0.0498114	−0.0249057	+	0.999690i	$0.507929\pi$
$644$	12.8888	0.507889
$645$	−9.26309	−0.364734
$646$	−0.556867	−0.0219096
$647$	−21.8000	−0.857047	−0.428524	−	0.903531i	$-0.640966\pi$
$647$	−21.8000	−0.857047	−0.428524	+	0.903531i	$0.640966\pi$
$648$	0.877796	0.0344831
$649$	6.93938	0.272395
$650$	−0.199975	−0.00784367
$651$	3.54437	0.138915
$652$	−12.9388	−0.506721
$653$	−0.827455	−0.0323808	−0.0161904	−	0.999869i	$-0.505154\pi$
$653$	−0.827455	−0.0323808	−0.0161904	+	0.999869i	$0.505154\pi$
$654$	−3.88182	−0.151791
$655$	11.3469	0.443360
$656$	25.2070	0.984170
$657$	3.55562	0.138718
$658$	−2.81252	−0.109644
$659$	19.7769	0.770399	0.385199	−	0.922833i	$-0.374133\pi$
$659$	19.7769	0.770399	0.385199	+	0.922833i	$0.374133\pi$
$660$	1.95063	0.0759282
$661$	−2.73185	−0.106257	−0.0531283	−	0.998588i	$-0.516919\pi$
$661$	−2.73185	−0.106257	−0.0531283	+	0.998588i	$0.516919\pi$
$662$	−4.11304	−0.159858
$663$	0.455630	0.0176952
$664$	−3.65436	−0.141816
$665$	−4.95063	−0.191977
$666$	−2.54312	−0.0985439
$667$	−22.5068	−0.871467
$668$	−7.32577	−0.283443
$669$	9.94938	0.384665
$670$	2.28884	0.0884256
$671$	5.39501	0.208272
$672$	2.57908	0.0994903
$673$	43.2212	1.66605	0.833027	−	0.553233i	$-0.186606\pi$
$673$	43.2212	1.66605	0.833027	+	0.553233i	$0.186606\pi$
$674$	−5.51450	−0.212411
$675$	−1.00000	−0.0384900
$676$	23.7781	0.914544
$677$	2.20504	0.0847464	0.0423732	−	0.999102i	$-0.486508\pi$
$677$	2.20504	0.0847464	0.0423732	+	0.999102i	$0.486508\pi$
$678$	4.31496	0.165715
$679$	0.0506185	0.00194256
$680$	−0.444383	−0.0170413
$681$	−14.8519	−0.569126
$682$	0.787529	0.0301560
$683$	−5.42757	−0.207680	−0.103840	−	0.994594i	$-0.533113\pi$
$683$	−5.42757	−0.207680	−0.103840	+	0.994594i	$0.533113\pi$
$684$	9.65685	0.369239
$685$	13.1900	0.503963
$686$	−0.222191	−0.00848331
$687$	−3.24316	−0.123734
$688$	34.3311	1.30886
$689$	−1.25553	−0.0478319
$690$	−1.46813	−0.0558906
$691$	7.72378	0.293826	0.146913	−	0.989149i	$-0.453066\pi$
$691$	7.72378	0.293826	0.146913	+	0.989149i	$0.453066\pi$
$692$	30.6323	1.16447
$693$	1.00000	0.0379869
$694$	−0.229287	−0.00870363
$695$	−19.5581	−0.741882
$696$	−2.99000	−0.113336
$697$	−3.44313	−0.130418
$698$	−4.13498	−0.156511
$699$	−11.1694	−0.422466
$700$	−1.95063	−0.0737269
$701$	32.1988	1.21613	0.608066	−	0.793887i	$-0.291946\pi$
$701$	32.1988	1.21613	0.608066	+	0.793887i	$0.291946\pi$
$702$	0.199975	0.00754757
$703$	−56.6631	−2.13709
$704$	−6.83940	−0.257769
$705$	−12.6581	−0.476732
$706$	1.97556	0.0743513
$707$	−3.25690	−0.122488
$708$	13.5362	0.508721
$709$	−25.0963	−0.942511	−0.471256	−	0.881997i	$-0.656199\pi$
$709$	−25.0963	−0.942511	−0.471256	+	0.881997i	$0.656199\pi$
$710$	−1.41245	−0.0530082
$711$	1.98875	0.0745841
$712$	1.97556	0.0740373
$713$	23.4194	0.877062
$714$	−0.112484	−0.00420961
$715$	0.900012	0.0336586
$716$	17.4714	0.652939
$717$	−2.45057	−0.0915182
$718$	2.25291	0.0840777
$719$	9.36433	0.349230	0.174615	−	0.984637i	$-0.444132\pi$
$719$	9.36433	0.349230	0.174615	+	0.984637i	$0.444132\pi$
$720$	3.70622	0.138123
$721$	−0.0481195	−0.00179206
$722$	−1.22400	−0.0455524
$723$	26.1012	0.970715
$724$	26.9236	1.00061
$725$	3.40626	0.126505
$726$	0.222191	0.00824630
$727$	−13.9145	−0.516061	−0.258030	−	0.966137i	$-0.583073\pi$
$727$	−13.9145	−0.516061	−0.258030	+	0.966137i	$0.583073\pi$
$728$	0.790027	0.0292804
$729$	1.00000	0.0370370
$730$	−0.790027	−0.0292402
$731$	−4.68942	−0.173445
$732$	10.5237	0.388966
$733$	−1.16554	−0.0430502	−0.0215251	−	0.999768i	$-0.506852\pi$
$733$	−1.16554	−0.0430502	−0.0215251	+	0.999768i	$0.506852\pi$
$734$	3.43576	0.126816
$735$	−1.00000	−0.0368856
$736$	17.0413	0.628149
$737$	−10.3012	−0.379450
$738$	−1.51118	−0.0556275
$739$	43.5856	1.60332	0.801661	−	0.597779i	$-0.203950\pi$
$739$	43.5856	1.60332	0.801661	+	0.597779i	$0.203950\pi$
$740$	−22.3262	−0.820728
$741$	4.45563	0.163682
$742$	0.309960	0.0113790
$743$	0.508255	0.0186461	0.00932304	−	0.999957i	$-0.497032\pi$
$743$	0.508255	0.0186461	0.00932304	+	0.999957i	$0.497032\pi$
$744$	3.11123	0.114063
$745$	−0.568114	−0.0208141
$746$	−0.104432	−0.00382352
$747$	−4.16310	−0.152320
$748$	0.987504	0.0361067
$749$	15.8900	0.580609
$750$	0.222191	0.00811328
$751$	−11.1188	−0.405731	−0.202865	−	0.979207i	$-0.565025\pi$
$751$	−11.1188	−0.405731	−0.202865	+	0.979207i	$0.565025\pi$
$752$	46.9138	1.71077
$753$	9.37758	0.341738
$754$	−0.681167	−0.0248066
$755$	4.44688	0.161839
$756$	1.95063	0.0709438
$757$	−3.60255	−0.130937	−0.0654684	−	0.997855i	$-0.520854\pi$
$757$	−3.60255	−0.130937	−0.0654684	+	0.997855i	$0.520854\pi$
$758$	5.60423	0.203555
$759$	6.60749	0.239837
$760$	−4.34564	−0.157633
$761$	41.5980	1.50793	0.753963	−	0.656917i	$-0.228140\pi$
$761$	41.5980	1.50793	0.753963	+	0.656917i	$0.228140\pi$
$762$	2.43064	0.0880527
$763$	−17.4706	−0.632479
$764$	−13.4862	−0.487913
$765$	−0.506248	−0.0183034
$766$	4.71616	0.170402
$767$	6.24553	0.225513
$768$	−12.1950	−0.440051
$769$	−27.7113	−0.999297	−0.499648	−	0.866228i	$-0.666537\pi$
$769$	−27.7113	−0.999297	−0.499648	+	0.866228i	$0.666537\pi$
$770$	−0.222191	−0.00800722
$771$	13.9718	0.503183
$772$	−4.18054	−0.150461
$773$	−10.8663	−0.390833	−0.195416	−	0.980720i	$-0.562606\pi$
$773$	−10.8663	−0.390833	−0.195416	+	0.980720i	$0.562606\pi$
$774$	−2.05818	−0.0739797
$775$	−3.54437	−0.127318
$776$	0.0444327	0.00159504
$777$	−11.4456	−0.410610
$778$	−1.98457	−0.0711504
$779$	−33.6706	−1.20637
$780$	1.75559	0.0628603
$781$	6.35689	0.227467
$782$	−0.743236	−0.0265781
$783$	−3.40626	−0.121730
$784$	3.70622	0.132365
$785$	8.15186	0.290952
$786$	2.52118	0.0899276
$787$	−52.8938	−1.88546	−0.942730	−	0.333558i	$-0.891751\pi$
$787$	−52.8938	−1.88546	−0.942730	+	0.333558i	$0.891751\pi$
$788$	−36.2623	−1.29179
$789$	18.1232	0.645202
$790$	−0.441884	−0.0157215
$791$	19.4200	0.690496
$792$	0.877796	0.0311911
$793$	4.85558	0.172427
$794$	3.92308	0.139225
$795$	1.39501	0.0494760
$796$	7.02237	0.248901
$797$	7.31621	0.259153	0.129577	−	0.991569i	$-0.458638\pi$
$797$	7.31621	0.259153	0.129577	+	0.991569i	$0.458638\pi$
$798$	−1.09999	−0.0389391
$799$	−6.40814	−0.226704
$800$	−2.57908	−0.0911844
$801$	2.25059	0.0795208
$802$	5.58798	0.197319
$803$	3.55562	0.125475
$804$	−20.0939	−0.708656
$805$	−6.60749	−0.232883
$806$	0.708785	0.0249659
$807$	−32.1793	−1.13277
$808$	−2.85890	−0.100576
$809$	−44.1282	−1.55146	−0.775732	−	0.631063i	$-0.782619\pi$
$809$	−44.1282	−1.55146	−0.775732	+	0.631063i	$0.782619\pi$
$810$	−0.222191	−0.00780701
$811$	43.0394	1.51132	0.755659	−	0.654965i	$-0.227317\pi$
$811$	43.0394	1.51132	0.755659	+	0.654965i	$0.227317\pi$
$812$	−6.64436	−0.233171
$813$	24.0669	0.844062
$814$	−2.54312	−0.0891363
$815$	6.63311	0.232348
$816$	1.87627	0.0656826
$817$	−45.8581	−1.60437
$818$	−6.44688	−0.225410
$819$	0.900012	0.0314490
$820$	−13.2668	−0.463296
$821$	−45.9099	−1.60227	−0.801134	−	0.598485i	$-0.795770\pi$
$821$	−45.9099	−1.60227	−0.801134	+	0.598485i	$0.795770\pi$
$822$	2.93070	0.102220
$823$	1.96931	0.0686459	0.0343230	−	0.999411i	$-0.489073\pi$
$823$	1.96931	0.0686459	0.0343230	+	0.999411i	$0.489073\pi$
$824$	−0.0422391	−0.00147147
$825$	−1.00000	−0.0348155
$826$	−1.54187	−0.0536486
$827$	3.60937	0.125510	0.0627550	−	0.998029i	$-0.480011\pi$
$827$	3.60937	0.125510	0.0627550	+	0.998029i	$0.480011\pi$
$828$	12.8888	0.447916
$829$	−5.28622	−0.183598	−0.0917989	−	0.995778i	$-0.529262\pi$
$829$	−5.28622	−0.183598	−0.0917989	+	0.995778i	$0.529262\pi$
$830$	0.925005	0.0321074
$831$	−16.8443	−0.584323
$832$	−6.15554	−0.213405
$833$	−0.506248	−0.0175405
$834$	−4.34564	−0.150477
$835$	3.75559	0.129968
$836$	9.65685	0.333989
$837$	3.54437	0.122511
$838$	5.80572	0.200555
$839$	18.3268	0.632713	0.316356	−	0.948640i	$-0.397540\pi$
$839$	18.3268	0.632713	0.316356	+	0.948640i	$0.397540\pi$
$840$	−0.877796	−0.0302868
$841$	−17.3974	−0.599910
$842$	−8.24248	−0.284055
$843$	0.0444327	0.00153034
$844$	8.23540	0.283474
$845$	−12.1900	−0.419348
$846$	−2.81252	−0.0966965
$847$	1.00000	0.0343604
$848$	−5.17023	−0.177547
$849$	0.455630	0.0156372
$850$	0.112484	0.00385817
$851$	−75.6268	−2.59245
$852$	12.4000	0.424815
$853$	38.1150	1.30503	0.652516	−	0.757775i	$-0.273713\pi$
$853$	38.1150	1.30503	0.652516	+	0.757775i	$0.273713\pi$
$854$	−1.19873	−0.0410195
$855$	−4.95063	−0.169308
$856$	13.9482	0.476739
$857$	−19.0912	−0.652144	−0.326072	−	0.945345i	$-0.605725\pi$
$857$	−19.0912	−0.652144	−0.326072	+	0.945345i	$0.605725\pi$
$858$	0.199975	0.00682703
$859$	16.2357	0.553954	0.276977	−	0.960877i	$-0.410667\pi$
$859$	16.2357	0.553954	0.276977	+	0.960877i	$0.410667\pi$
$860$	−18.0689	−0.616143
$861$	−6.80127	−0.231787
$862$	−2.13360	−0.0726708
$863$	32.0076	1.08955	0.544775	−	0.838582i	$-0.316615\pi$
$863$	32.0076	1.08955	0.544775	+	0.838582i	$0.316615\pi$
$864$	2.57908	0.0877422
$865$	−15.7038	−0.533945
$866$	6.91320	0.234920
$867$	16.7437	0.568646
$868$	6.91376	0.234668
$869$	1.98875	0.0674638
$870$	0.756842	0.0256593
$871$	−9.27122	−0.314143
$872$	−15.3356	−0.519331
$873$	0.0506185	0.00171318
$874$	−7.26815	−0.245849
$875$	1.00000	0.0338062
$876$	6.93570	0.234335
$877$	−3.88495	−0.131186	−0.0655928	−	0.997846i	$-0.520894\pi$
$877$	−3.88495	−0.131186	−0.0655928	+	0.997846i	$0.520894\pi$
$878$	7.56881	0.255435
$879$	−31.1337	−1.05011
$880$	3.70622	0.124937
$881$	26.2331	0.883816	0.441908	−	0.897060i	$-0.354302\pi$
$881$	26.2331	0.883816	0.441908	+	0.897060i	$0.354302\pi$
$882$	−0.222191	−0.00748157
$883$	−9.26622	−0.311833	−0.155917	−	0.987770i	$-0.549833\pi$
$883$	−9.26622	−0.311833	−0.155917	+	0.987770i	$0.549833\pi$
$884$	0.888765	0.0298924
$885$	−6.93938	−0.233265
$886$	−4.22997	−0.142108
$887$	−16.2593	−0.545935	−0.272968	−	0.962023i	$-0.588005\pi$
$887$	−16.2593	−0.545935	−0.272968	+	0.962023i	$0.588005\pi$
$888$	−10.0469	−0.337153
$889$	10.9394	0.366895
$890$	−0.500062	−0.0167621
$891$	1.00000	0.0335013
$892$	19.4076	0.649814
$893$	−62.6656	−2.09702
$894$	−0.126230	−0.00422176
$895$	−8.95682	−0.299393
$896$	6.67782	0.223090
$897$	5.94682	0.198559
$898$	−4.15067	−0.138510
$899$	−12.0730	−0.402659
$900$	−1.95063	−0.0650210
$901$	0.706223	0.0235277
$902$	−1.51118	−0.0503169
$903$	−9.26309	−0.308256
$904$	17.0468	0.566968
$905$	−13.8025	−0.458811
$906$	0.988059	0.0328261
$907$	−44.6806	−1.48359	−0.741797	−	0.670624i	$-0.766026\pi$
$907$	−44.6806	−1.48359	−0.741797	+	0.670624i	$0.766026\pi$
$908$	−28.9706	−0.961422
$909$	−3.25690	−0.108025
$910$	−0.199975	−0.00662911
$911$	−53.3601	−1.76790	−0.883949	−	0.467583i	$-0.845125\pi$
$911$	−53.3601	−1.76790	−0.883949	+	0.467583i	$0.845125\pi$
$912$	18.3481	0.607568
$913$	−4.16310	−0.137779
$914$	−6.06305	−0.200548
$915$	−5.39501	−0.178354
$916$	−6.32620	−0.209024
$917$	11.3469	0.374707
$918$	−0.112484	−0.00371252
$919$	42.4769	1.40118	0.700591	−	0.713563i	$-0.252920\pi$
$919$	42.4769	1.40118	0.700591	+	0.713563i	$0.252920\pi$
$920$	−5.80002	−0.191221
$921$	27.9038	0.919460
$922$	8.27983	0.272682
$923$	5.72128	0.188318
$924$	1.95063	0.0641710
$925$	11.4456	0.376330
$926$	1.88889	0.0620728
$927$	−0.0481195	−0.00158045
$928$	−8.78503	−0.288383
$929$	36.4812	1.19691	0.598455	−	0.801157i	$-0.295782\pi$
$929$	36.4812	1.19691	0.598455	+	0.801157i	$0.295782\pi$
$930$	−0.787529	−0.0258241
$931$	−4.95063	−0.162250
$932$	−21.7874	−0.713670
$933$	20.4274	0.668764
$934$	4.55855	0.149160
$935$	−0.506248	−0.0165561
$936$	0.790027	0.0258229
$937$	−35.4087	−1.15675	−0.578376	−	0.815770i	$-0.696313\pi$
$937$	−35.4087	−1.15675	−0.578376	+	0.815770i	$0.696313\pi$
$938$	2.28884	0.0747333
$939$	−22.0531	−0.719676
$940$	−24.6913	−0.805341
$941$	16.8161	0.548191	0.274095	−	0.961703i	$-0.411622\pi$
$941$	16.8161	0.548191	0.274095	+	0.961703i	$0.411622\pi$
$942$	1.81127	0.0590144
$943$	−44.9393	−1.46343
$944$	25.7189	0.837079
$945$	−1.00000	−0.0325300
$946$	−2.05818	−0.0669172
$947$	−58.8375	−1.91196	−0.955980	−	0.293431i	$-0.905203\pi$
$947$	−58.8375	−1.91196	−0.955980	+	0.293431i	$0.905203\pi$
$948$	3.87932	0.125995
$949$	3.20010	0.103880
$950$	1.09999	0.0356883
$951$	17.9038	0.580569
$952$	−0.444383	−0.0144025
$953$	13.7575	0.445650	0.222825	−	0.974858i	$-0.428472\pi$
$953$	13.7575	0.445650	0.222825	+	0.974858i	$0.428472\pi$
$954$	0.309960	0.0100353
$955$	6.91376	0.223724
$956$	−4.78015	−0.154601
$957$	−3.40626	−0.110109
$958$	−2.72865	−0.0881588
$959$	13.1900	0.425927
$960$	6.83940	0.220741
$961$	−18.4374	−0.594756
$962$	−2.28884	−0.0737952
$963$	15.8900	0.512049
$964$	50.9139	1.63983
$965$	2.14317	0.0689911
$966$	−1.46813	−0.0472362
$967$	28.8681	0.928337	0.464168	−	0.885747i	$-0.346353\pi$
$967$	28.8681	0.928337	0.464168	+	0.885747i	$0.346353\pi$
$968$	0.877796	0.0282134
$969$	−2.50625	−0.0805123
$970$	−0.0112470	−0.000361119 0
$971$	−39.4282	−1.26531	−0.632656	−	0.774433i	$-0.718035\pi$
$971$	−39.4282	−1.26531	−0.632656	+	0.774433i	$0.718035\pi$
$972$	1.95063	0.0625665
$973$	−19.5581	−0.627004
$974$	4.57493	0.146590
$975$	−0.900012	−0.0288235
$976$	19.9951	0.640028
$977$	−2.81758	−0.0901425	−0.0450712	−	0.998984i	$-0.514351\pi$
$977$	−2.81758	−0.0901425	−0.0450712	+	0.998984i	$0.514351\pi$
$978$	1.47382	0.0471276
$979$	2.25059	0.0719293
$980$	−1.95063	−0.0623106
$981$	−17.4706	−0.557794
$982$	−1.40577	−0.0448599
$983$	49.0000	1.56286	0.781429	−	0.623995i	$-0.214491\pi$
$983$	49.0000	1.56286	0.781429	+	0.623995i	$0.214491\pi$
$984$	−5.97013	−0.190321
$985$	18.5901	0.592328
$986$	0.383150	0.0122020
$987$	−12.6581	−0.402912
$988$	8.69129	0.276507
$989$	−61.2057	−1.94623
$990$	−0.222191	−0.00706171
$991$	7.64874	0.242970	0.121485	−	0.992593i	$-0.461234\pi$
$991$	7.64874	0.242970	0.121485	+	0.992593i	$0.461234\pi$
$992$	9.14123	0.290234
$993$	−18.5112	−0.587437
$994$	−1.41245	−0.0448001
$995$	−3.60005	−0.114129
$996$	−8.12068	−0.257313
$997$	32.7092	1.03591	0.517956	−	0.855408i	$-0.326693\pi$
$997$	32.7092	1.03591	0.517956	+	0.855408i	$0.326693\pi$
$998$	0.139421	0.00441329
$999$	−11.4456	−0.362124

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1155.2.a.v.1.2	✓	4
3.2	odd	2		3465.2.a.bj.1.3		4
5.4	even	2		5775.2.a.by.1.3		4
7.6	odd	2		8085.2.a.bq.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.v.1.2	✓	4	1.1	even	1	trivial
3465.2.a.bj.1.3		4	3.2	odd	2
5775.2.a.by.1.3		4	5.4	even	2
8085.2.a.bq.1.2		4	7.6	odd	2

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:	\( N \)	\(=\)	\( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1155.a (trivial)

\(f(q)\)	\(=\)	\(q-0.222191 q^{2} -1.00000 q^{3} -1.95063 q^{4} +1.00000 q^{5} +0.222191 q^{6} +1.00000 q^{7} +0.877796 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.222191 q^{2} -1.00000 q^{3} -1.95063 q^{4} +1.00000 q^{5} +0.222191 q^{6} +1.00000 q^{7} +0.877796 q^{8} +1.00000 q^{9} -0.222191 q^{10} +1.00000 q^{11} +1.95063 q^{12} +0.900012 q^{13} -0.222191 q^{14} -1.00000 q^{15} +3.70622 q^{16} -0.506248 q^{17} -0.222191 q^{18} -4.95063 q^{19} -1.95063 q^{20} -1.00000 q^{21} -0.222191 q^{22} -6.60749 q^{23} -0.877796 q^{24} +1.00000 q^{25} -0.199975 q^{26} -1.00000 q^{27} -1.95063 q^{28} +3.40626 q^{29} +0.222191 q^{30} -3.54437 q^{31} -2.57908 q^{32} -1.00000 q^{33} +0.112484 q^{34} +1.00000 q^{35} -1.95063 q^{36} +11.4456 q^{37} +1.09999 q^{38} -0.900012 q^{39} +0.877796 q^{40} +6.80127 q^{41} +0.222191 q^{42} +9.26309 q^{43} -1.95063 q^{44} +1.00000 q^{45} +1.46813 q^{46} +12.6581 q^{47} -3.70622 q^{48} +1.00000 q^{49} -0.222191 q^{50} +0.506248 q^{51} -1.75559 q^{52} -1.39501 q^{53} +0.222191 q^{54} +1.00000 q^{55} +0.877796 q^{56} +4.95063 q^{57} -0.756842 q^{58} +6.93938 q^{59} +1.95063 q^{60} +5.39501 q^{61} +0.787529 q^{62} +1.00000 q^{63} -6.83940 q^{64} +0.900012 q^{65} +0.222191 q^{66} -10.3012 q^{67} +0.987504 q^{68} +6.60749 q^{69} -0.222191 q^{70} +6.35689 q^{71} +0.877796 q^{72} +3.55562 q^{73} -2.54312 q^{74} -1.00000 q^{75} +9.65685 q^{76} +1.00000 q^{77} +0.199975 q^{78} +1.98875 q^{79} +3.70622 q^{80} +1.00000 q^{81} -1.51118 q^{82} -4.16310 q^{83} +1.95063 q^{84} -0.506248 q^{85} -2.05818 q^{86} -3.40626 q^{87} +0.877796 q^{88} +2.25059 q^{89} -0.222191 q^{90} +0.900012 q^{91} +12.8888 q^{92} +3.54437 q^{93} -2.81252 q^{94} -4.95063 q^{95} +2.57908 q^{96} +0.0506185 q^{97} -0.222191 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9	\( 4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9} + 2 q^{10} + 4 q^{11} - 6 q^{12} + 4 q^{13} + 2 q^{14} - 4 q^{15} + 6 q^{16} + 6 q^{17} + 2 q^{18} - 6 q^{19} + 6 q^{20} - 4 q^{21} + 2 q^{22} + 10 q^{23} - 6 q^{24} + 4 q^{25} - 4 q^{27} + 6 q^{28} + 6 q^{29} - 2 q^{30} - 8 q^{31} + 14 q^{32} - 4 q^{33} - 16 q^{34} + 4 q^{35} + 6 q^{36} + 12 q^{37} + 4 q^{38} - 4 q^{39} + 6 q^{40} - 2 q^{42} + 6 q^{43} + 6 q^{44} + 4 q^{45} - 4 q^{46} - 6 q^{48} + 4 q^{49} + 2 q^{50} - 6 q^{51} - 12 q^{52} + 14 q^{53} - 2 q^{54} + 4 q^{55} + 6 q^{56} + 6 q^{57} + 20 q^{58} + 2 q^{59} - 6 q^{60} + 2 q^{61} + 20 q^{62} + 4 q^{63} - 2 q^{64} + 4 q^{65} - 2 q^{66} - 12 q^{67} + 20 q^{68} - 10 q^{69} + 2 q^{70} + 4 q^{71} + 6 q^{72} + 20 q^{73} - 32 q^{74} - 4 q^{75} + 16 q^{76} + 4 q^{77} - 4 q^{79} + 6 q^{80} + 4 q^{81} - 16 q^{82} + 14 q^{83} - 6 q^{84} + 6 q^{85} + 40 q^{86} - 6 q^{87} + 6 q^{88} - 6 q^{89} + 2 q^{90} + 4 q^{91} + 40 q^{92} + 8 q^{93} + 4 q^{94} - 6 q^{95} - 14 q^{96} - 14 q^{97} + 2 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9 + 2 * q^10 + 4 * q^11 - 6 * q^12 + 4 * q^13 + 2 * q^14 - 4 * q^15 + 6 * q^16 + 6 * q^17 + 2 * q^18 - 6 * q^19 + 6 * q^20 - 4 * q^21 + 2 * q^22 + 10 * q^23 - 6 * q^24 + 4 * q^25 - 4 * q^27 + 6 * q^28 + 6 * q^29 - 2 * q^30 - 8 * q^31 + 14 * q^32 - 4 * q^33 - 16 * q^34 + 4 * q^35 + 6 * q^36 + 12 * q^37 + 4 * q^38 - 4 * q^39 + 6 * q^40 - 2 * q^42 + 6 * q^43 + 6 * q^44 + 4 * q^45 - 4 * q^46 - 6 * q^48 + 4 * q^49 + 2 * q^50 - 6 * q^51 - 12 * q^52 + 14 * q^53 - 2 * q^54 + 4 * q^55 + 6 * q^56 + 6 * q^57 + 20 * q^58 + 2 * q^59 - 6 * q^60 + 2 * q^61 + 20 * q^62 + 4 * q^63 - 2 * q^64 + 4 * q^65 - 2 * q^66 - 12 * q^67 + 20 * q^68 - 10 * q^69 + 2 * q^70 + 4 * q^71 + 6 * q^72 + 20 * q^73 - 32 * q^74 - 4 * q^75 + 16 * q^76 + 4 * q^77 - 4 * q^79 + 6 * q^80 + 4 * q^81 - 16 * q^82 + 14 * q^83 - 6 * q^84 + 6 * q^85 + 40 * q^86 - 6 * q^87 + 6 * q^88 - 6 * q^89 + 2 * q^90 + 4 * q^91 + 40 * q^92 + 8 * q^93 + 4 * q^94 - 6 * q^95 - 14 * q^96 - 14 * q^97 + 2 * q^98 + 4 * q^99

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