LMFDB - Embedded newform 6050.2.a.cv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6050,2,Mod(1,6050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6050 = 2 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6050.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.3094932229$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 242)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	6050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} +4.73205 q^{7} +1.00000 q^{8} -2.46410 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} +4.73205 q^{7} +1.00000 q^{8} -2.46410 q^{9} -0.732051 q^{12} +3.00000 q^{13} +4.73205 q^{14} +1.00000 q^{16} -5.19615 q^{17} -2.46410 q^{18} -1.26795 q^{19} -3.46410 q^{21} +1.26795 q^{23} -0.732051 q^{24} +3.00000 q^{26} +4.00000 q^{27} +4.73205 q^{28} +3.00000 q^{29} +0.196152 q^{31} +1.00000 q^{32} -5.19615 q^{34} -2.46410 q^{36} +7.19615 q^{37} -1.26795 q^{38} -2.19615 q^{39} +0.803848 q^{41} -3.46410 q^{42} +1.26795 q^{46} +8.19615 q^{47} -0.732051 q^{48} +15.3923 q^{49} +3.80385 q^{51} +3.00000 q^{52} -11.1962 q^{53} +4.00000 q^{54} +4.73205 q^{56} +0.928203 q^{57} +3.00000 q^{58} +9.46410 q^{59} -2.53590 q^{61} +0.196152 q^{62} -11.6603 q^{63} +1.00000 q^{64} +10.1962 q^{67} -5.19615 q^{68} -0.928203 q^{69} -9.46410 q^{71} -2.46410 q^{72} -0.928203 q^{73} +7.19615 q^{74} -1.26795 q^{76} -2.19615 q^{78} +4.73205 q^{79} +4.46410 q^{81} +0.803848 q^{82} +8.19615 q^{83} -3.46410 q^{84} -2.19615 q^{87} +6.46410 q^{89} +14.1962 q^{91} +1.26795 q^{92} -0.143594 q^{93} +8.19615 q^{94} -0.732051 q^{96} +1.00000 q^{97} +15.3923 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{12} + 6 q^{13} + 6 q^{14} + 2 q^{16} + 2 q^{18} - 6 q^{19} + 6 q^{23} + 2 q^{24} + 6 q^{26} + 8 q^{27} + 6 q^{28} + 6 q^{29} - 10 q^{31} + 2 q^{32} + 2 q^{36} + 4 q^{37} - 6 q^{38} + 6 q^{39} + 12 q^{41} + 6 q^{46} + 6 q^{47} + 2 q^{48} + 10 q^{49} + 18 q^{51} + 6 q^{52} - 12 q^{53} + 8 q^{54} + 6 q^{56} - 12 q^{57} + 6 q^{58} + 12 q^{59} - 12 q^{61} - 10 q^{62} - 6 q^{63} + 2 q^{64} + 10 q^{67} + 12 q^{69} - 12 q^{71} + 2 q^{72} + 12 q^{73} + 4 q^{74} - 6 q^{76} + 6 q^{78} + 6 q^{79} + 2 q^{81} + 12 q^{82} + 6 q^{83} + 6 q^{87} + 6 q^{89} + 18 q^{91} + 6 q^{92} - 28 q^{93} + 6 q^{94} + 2 q^{96} + 2 q^{97} + 10 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^12 + 6 * q^13 + 6 * q^14 + 2 * q^16 + 2 * q^18 - 6 * q^19 + 6 * q^23 + 2 * q^24 + 6 * q^26 + 8 * q^27 + 6 * q^28 + 6 * q^29 - 10 * q^31 + 2 * q^32 + 2 * q^36 + 4 * q^37 - 6 * q^38 + 6 * q^39 + 12 * q^41 + 6 * q^46 + 6 * q^47 + 2 * q^48 + 10 * q^49 + 18 * q^51 + 6 * q^52 - 12 * q^53 + 8 * q^54 + 6 * q^56 - 12 * q^57 + 6 * q^58 + 12 * q^59 - 12 * q^61 - 10 * q^62 - 6 * q^63 + 2 * q^64 + 10 * q^67 + 12 * q^69 - 12 * q^71 + 2 * q^72 + 12 * q^73 + 4 * q^74 - 6 * q^76 + 6 * q^78 + 6 * q^79 + 2 * q^81 + 12 * q^82 + 6 * q^83 + 6 * q^87 + 6 * q^89 + 18 * q^91 + 6 * q^92 - 28 * q^93 + 6 * q^94 + 2 * q^96 + 2 * q^97 + 10 * q^98

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
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−0.732051	−0.298858
$7$	4.73205	1.78855	0.894274	−	0.447521i	$-0.147693\pi$
$7$	4.73205	1.78855	0.894274	+	0.447521i	$0.147693\pi$
$8$	1.00000	0.353553
$9$	−2.46410	−0.821367
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−0.732051	−0.211325
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	4.73205	1.26469
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.19615	−1.26025	−0.630126	−	0.776493i	$-0.716997\pi$
$17$	−5.19615	−1.26025	−0.630126	+	0.776493i	$0.716997\pi$
$18$	−2.46410	−0.580794
$19$	−1.26795	−0.290887	−0.145444	−	0.989367i	$-0.546461\pi$
$19$	−1.26795	−0.290887	−0.145444	+	0.989367i	$0.546461\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	0	0
$23$	1.26795	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$23$	1.26795	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$24$	−0.732051	−0.149429
$25$	0	0
$26$	3.00000	0.588348
$27$	4.00000	0.769800
$28$	4.73205	0.894274
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	0.196152	0.0352300	0.0176150	−	0.999845i	$-0.494393\pi$
$31$	0.196152	0.0352300	0.0176150	+	0.999845i	$0.494393\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.19615	−0.891133
$35$	0	0
$36$	−2.46410	−0.410684
$37$	7.19615	1.18304	0.591520	−	0.806290i	$-0.298528\pi$
$37$	7.19615	1.18304	0.591520	+	0.806290i	$0.298528\pi$
$38$	−1.26795	−0.205689
$39$	−2.19615	−0.351666
$40$	0	0
$41$	0.803848	0.125540	0.0627700	−	0.998028i	$-0.480007\pi$
$41$	0.803848	0.125540	0.0627700	+	0.998028i	$0.480007\pi$
$42$	−3.46410	−0.534522
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	1.26795	0.186949
$47$	8.19615	1.19553	0.597766	−	0.801671i	$-0.296055\pi$
$47$	8.19615	1.19553	0.597766	+	0.801671i	$0.296055\pi$
$48$	−0.732051	−0.105662
$49$	15.3923	2.19890
$50$	0	0
$51$	3.80385	0.532645
$52$	3.00000	0.416025
$53$	−11.1962	−1.53791	−0.768955	−	0.639303i	$-0.779223\pi$
$53$	−11.1962	−1.53791	−0.768955	+	0.639303i	$0.779223\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	4.73205	0.632347
$57$	0.928203	0.122944
$58$	3.00000	0.393919
$59$	9.46410	1.23212	0.616061	−	0.787699i	$-0.288728\pi$
$59$	9.46410	1.23212	0.616061	+	0.787699i	$0.288728\pi$
$60$	0	0
$61$	−2.53590	−0.324689	−0.162344	−	0.986734i	$-0.551906\pi$
$61$	−2.53590	−0.324689	−0.162344	+	0.986734i	$0.551906\pi$
$62$	0.196152	0.0249114
$63$	−11.6603	−1.46905
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	10.1962	1.24566	0.622829	−	0.782358i	$-0.285983\pi$
$67$	10.1962	1.24566	0.622829	+	0.782358i	$0.285983\pi$
$68$	−5.19615	−0.630126
$69$	−0.928203	−0.111743
$70$	0	0
$71$	−9.46410	−1.12318	−0.561591	−	0.827415i	$-0.689811\pi$
$71$	−9.46410	−1.12318	−0.561591	+	0.827415i	$0.689811\pi$
$72$	−2.46410	−0.290397
$73$	−0.928203	−0.108638	−0.0543190	−	0.998524i	$-0.517299\pi$
$73$	−0.928203	−0.108638	−0.0543190	+	0.998524i	$0.517299\pi$
$74$	7.19615	0.836536
$75$	0	0
$76$	−1.26795	−0.145444
$77$	0	0
$78$	−2.19615	−0.248665
$79$	4.73205	0.532397	0.266199	−	0.963918i	$-0.414232\pi$
$79$	4.73205	0.532397	0.266199	+	0.963918i	$0.414232\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0.803848	0.0887701
$83$	8.19615	0.899645	0.449822	−	0.893118i	$-0.351487\pi$
$83$	8.19615	0.899645	0.449822	+	0.893118i	$0.351487\pi$
$84$	−3.46410	−0.377964
$85$	0	0
$86$	0	0
$87$	−2.19615	−0.235452
$88$	0	0
$89$	6.46410	0.685193	0.342597	−	0.939483i	$-0.388694\pi$
$89$	6.46410	0.685193	0.342597	+	0.939483i	$0.388694\pi$
$90$	0	0
$91$	14.1962	1.48816
$92$	1.26795	0.132193
$93$	−0.143594	−0.0148900
$94$	8.19615	0.845369
$95$	0	0
$96$	−0.732051	−0.0747146
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	15.3923	1.55486
$99$	0	0
$100$	0	0
$101$	−16.3923	−1.63110	−0.815548	−	0.578690i	$-0.803564\pi$
$101$	−16.3923	−1.63110	−0.815548	+	0.578690i	$0.803564\pi$
$102$	3.80385	0.376637
$103$	8.39230	0.826918	0.413459	−	0.910523i	$-0.364320\pi$
$103$	8.39230	0.826918	0.413459	+	0.910523i	$0.364320\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−11.1962	−1.08747
$107$	12.5885	1.21697	0.608486	−	0.793565i	$-0.291777\pi$
$107$	12.5885	1.21697	0.608486	+	0.793565i	$0.291777\pi$
$108$	4.00000	0.384900
$109$	2.07180	0.198442	0.0992211	−	0.995065i	$-0.468365\pi$
$109$	2.07180	0.198442	0.0992211	+	0.995065i	$0.468365\pi$
$110$	0	0
$111$	−5.26795	−0.500012
$112$	4.73205	0.447137
$113$	−10.8564	−1.02128	−0.510642	−	0.859793i	$-0.670592\pi$
$113$	−10.8564	−1.02128	−0.510642	+	0.859793i	$0.670592\pi$
$114$	0.928203	0.0869342
$115$	0	0
$116$	3.00000	0.278543
$117$	−7.39230	−0.683419
$118$	9.46410	0.871241
$119$	−24.5885	−2.25402
$120$	0	0
$121$	0	0
$122$	−2.53590	−0.229589
$123$	−0.588457	−0.0530594
$124$	0.196152	0.0176150
$125$	0	0
$126$	−11.6603	−1.03878
$127$	13.8564	1.22956	0.614779	−	0.788700i	$-0.289245\pi$
$127$	13.8564	1.22956	0.614779	+	0.788700i	$0.289245\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	10.1962	0.880813
$135$	0	0
$136$	−5.19615	−0.445566
$137$	9.46410	0.808573	0.404286	−	0.914632i	$-0.367520\pi$
$137$	9.46410	0.808573	0.404286	+	0.914632i	$0.367520\pi$
$138$	−0.928203	−0.0790139
$139$	−20.1962	−1.71302	−0.856508	−	0.516134i	$-0.827371\pi$
$139$	−20.1962	−1.71302	−0.856508	+	0.516134i	$0.827371\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	−9.46410	−0.794210
$143$	0	0
$144$	−2.46410	−0.205342
$145$	0	0
$146$	−0.928203	−0.0768186
$147$	−11.2679	−0.929365
$148$	7.19615	0.591520
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	4.39230	0.357441	0.178720	−	0.983900i	$-0.442804\pi$
$151$	4.39230	0.357441	0.178720	+	0.983900i	$0.442804\pi$
$152$	−1.26795	−0.102844
$153$	12.8038	1.03513
$154$	0	0
$155$	0	0
$156$	−2.19615	−0.175833
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	4.73205	0.376462
$159$	8.19615	0.649997
$160$	0	0
$161$	6.00000	0.472866
$162$	4.46410	0.350733
$163$	−22.1962	−1.73854	−0.869268	−	0.494340i	$-0.835410\pi$
$163$	−22.1962	−1.73854	−0.869268	+	0.494340i	$0.835410\pi$
$164$	0.803848	0.0627700
$165$	0	0
$166$	8.19615	0.636145
$167$	16.3923	1.26847	0.634237	−	0.773138i	$-0.281314\pi$
$167$	16.3923	1.26847	0.634237	+	0.773138i	$0.281314\pi$
$168$	−3.46410	−0.267261
$169$	−4.00000	−0.307692
$170$	0	0
$171$	3.12436	0.238925
$172$	0	0
$173$	−4.39230	−0.333941	−0.166970	−	0.985962i	$-0.553398\pi$
$173$	−4.39230	−0.333941	−0.166970	+	0.985962i	$0.553398\pi$
$174$	−2.19615	−0.166490
$175$	0	0
$176$	0	0
$177$	−6.92820	−0.520756
$178$	6.46410	0.484505
$179$	−13.8564	−1.03568	−0.517838	−	0.855479i	$-0.673263\pi$
$179$	−13.8564	−1.03568	−0.517838	+	0.855479i	$0.673263\pi$
$180$	0	0
$181$	7.19615	0.534886	0.267443	−	0.963574i	$-0.413821\pi$
$181$	7.19615	0.534886	0.267443	+	0.963574i	$0.413821\pi$
$182$	14.1962	1.05229
$183$	1.85641	0.137230
$184$	1.26795	0.0934745
$185$	0	0
$186$	−0.143594	−0.0105288
$187$	0	0
$188$	8.19615	0.597766
$189$	18.9282	1.37682
$190$	0	0
$191$	−21.4641	−1.55309	−0.776544	−	0.630063i	$-0.783029\pi$
$191$	−21.4641	−1.55309	−0.776544	+	0.630063i	$0.783029\pi$
$192$	−0.732051	−0.0528312
$193$	10.2679	0.739103	0.369552	−	0.929210i	$-0.379511\pi$
$193$	10.2679	0.739103	0.369552	+	0.929210i	$0.379511\pi$
$194$	1.00000	0.0717958
$195$	0	0
$196$	15.3923	1.09945
$197$	−13.3923	−0.954162	−0.477081	−	0.878859i	$-0.658305\pi$
$197$	−13.3923	−0.954162	−0.477081	+	0.878859i	$0.658305\pi$
$198$	0	0
$199$	−0.392305	−0.0278098	−0.0139049	−	0.999903i	$-0.504426\pi$
$199$	−0.392305	−0.0278098	−0.0139049	+	0.999903i	$0.504426\pi$
$200$	0	0
$201$	−7.46410	−0.526477
$202$	−16.3923	−1.15336
$203$	14.1962	0.996375
$204$	3.80385	0.266323
$205$	0	0
$206$	8.39230	0.584720
$207$	−3.12436	−0.217158
$208$	3.00000	0.208013
$209$	0	0
$210$	0	0
$211$	25.8564	1.78003	0.890014	−	0.455933i	$-0.150694\pi$
$211$	25.8564	1.78003	0.890014	+	0.455933i	$0.150694\pi$
$212$	−11.1962	−0.768955
$213$	6.92820	0.474713
$214$	12.5885	0.860529
$215$	0	0
$216$	4.00000	0.272166
$217$	0.928203	0.0630105
$218$	2.07180	0.140320
$219$	0.679492	0.0459158
$220$	0	0
$221$	−15.5885	−1.04859
$222$	−5.26795	−0.353562
$223$	20.3923	1.36557	0.682785	−	0.730619i	$-0.260769\pi$
$223$	20.3923	1.36557	0.682785	+	0.730619i	$0.260769\pi$
$224$	4.73205	0.316173
$225$	0	0
$226$	−10.8564	−0.722157
$227$	−16.3923	−1.08800	−0.543998	−	0.839087i	$-0.683090\pi$
$227$	−16.3923	−1.08800	−0.543998	+	0.839087i	$0.683090\pi$
$228$	0.928203	0.0614718
$229$	−13.1962	−0.872026	−0.436013	−	0.899940i	$-0.643610\pi$
$229$	−13.1962	−0.872026	−0.436013	+	0.899940i	$0.643610\pi$
$230$	0	0
$231$	0	0
$232$	3.00000	0.196960
$233$	−6.80385	−0.445735	−0.222867	−	0.974849i	$-0.571542\pi$
$233$	−6.80385	−0.445735	−0.222867	+	0.974849i	$0.571542\pi$
$234$	−7.39230	−0.483250
$235$	0	0
$236$	9.46410	0.616061
$237$	−3.46410	−0.225018
$238$	−24.5885	−1.59383
$239$	14.1962	0.918273	0.459136	−	0.888366i	$-0.348159\pi$
$239$	14.1962	0.918273	0.459136	+	0.888366i	$0.348159\pi$
$240$	0	0
$241$	−26.7846	−1.72535	−0.862674	−	0.505760i	$-0.831212\pi$
$241$	−26.7846	−1.72535	−0.862674	+	0.505760i	$0.831212\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	−2.53590	−0.162344
$245$	0	0
$246$	−0.588457	−0.0375187
$247$	−3.80385	−0.242033
$248$	0.196152	0.0124557
$249$	−6.00000	−0.380235
$250$	0	0
$251$	9.12436	0.575924	0.287962	−	0.957642i	$-0.407022\pi$
$251$	9.12436	0.575924	0.287962	+	0.957642i	$0.407022\pi$
$252$	−11.6603	−0.734527
$253$	0	0
$254$	13.8564	0.869428
$255$	0	0
$256$	1.00000	0.0625000
$257$	19.3923	1.20966	0.604829	−	0.796355i	$-0.293241\pi$
$257$	19.3923	1.20966	0.604829	+	0.796355i	$0.293241\pi$
$258$	0	0
$259$	34.0526	2.11592
$260$	0	0
$261$	−7.39230	−0.457572
$262$	0	0
$263$	9.80385	0.604531	0.302266	−	0.953224i	$-0.402257\pi$
$263$	9.80385	0.604531	0.302266	+	0.953224i	$0.402257\pi$
$264$	0	0
$265$	0	0
$266$	−6.00000	−0.367884
$267$	−4.73205	−0.289597
$268$	10.1962	0.622829
$269$	9.58846	0.584619	0.292309	−	0.956324i	$-0.405576\pi$
$269$	9.58846	0.584619	0.292309	+	0.956324i	$0.405576\pi$
$270$	0	0
$271$	16.3923	0.995762	0.497881	−	0.867245i	$-0.334112\pi$
$271$	16.3923	0.995762	0.497881	+	0.867245i	$0.334112\pi$
$272$	−5.19615	−0.315063
$273$	−10.3923	−0.628971
$274$	9.46410	0.571747
$275$	0	0
$276$	−0.928203	−0.0558713
$277$	−8.32051	−0.499931	−0.249965	−	0.968255i	$-0.580419\pi$
$277$	−8.32051	−0.499931	−0.249965	+	0.968255i	$0.580419\pi$
$278$	−20.1962	−1.21128
$279$	−0.483340	−0.0289368
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−6.00000	−0.357295
$283$	6.92820	0.411839	0.205919	−	0.978569i	$-0.433982\pi$
$283$	6.92820	0.411839	0.205919	+	0.978569i	$0.433982\pi$
$284$	−9.46410	−0.561591
$285$	0	0
$286$	0	0
$287$	3.80385	0.224534
$288$	−2.46410	−0.145199
$289$	10.0000	0.588235
$290$	0	0
$291$	−0.732051	−0.0429136
$292$	−0.928203	−0.0543190
$293$	23.7846	1.38951	0.694756	−	0.719246i	$-0.255512\pi$
$293$	23.7846	1.38951	0.694756	+	0.719246i	$0.255512\pi$
$294$	−11.2679	−0.657160
$295$	0	0
$296$	7.19615	0.418268
$297$	0	0
$298$	15.0000	0.868927
$299$	3.80385	0.219982
$300$	0	0
$301$	0	0
$302$	4.39230	0.252749
$303$	12.0000	0.689382
$304$	−1.26795	−0.0727219
$305$	0	0
$306$	12.8038	0.731947
$307$	25.2679	1.44212	0.721059	−	0.692874i	$-0.243656\pi$
$307$	25.2679	1.44212	0.721059	+	0.692874i	$0.243656\pi$
$308$	0	0
$309$	−6.14359	−0.349497
$310$	0	0
$311$	8.78461	0.498130	0.249065	−	0.968487i	$-0.419877\pi$
$311$	8.78461	0.498130	0.249065	+	0.968487i	$0.419877\pi$
$312$	−2.19615	−0.124333
$313$	31.7846	1.79657	0.898286	−	0.439411i	$-0.144813\pi$
$313$	31.7846	1.79657	0.898286	+	0.439411i	$0.144813\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	4.73205	0.266199
$317$	7.85641	0.441260	0.220630	−	0.975358i	$-0.429189\pi$
$317$	7.85641	0.441260	0.220630	+	0.975358i	$0.429189\pi$
$318$	8.19615	0.459617
$319$	0	0
$320$	0	0
$321$	−9.21539	−0.514353
$322$	6.00000	0.334367
$323$	6.58846	0.366592
$324$	4.46410	0.248006
$325$	0	0
$326$	−22.1962	−1.22933
$327$	−1.51666	−0.0838715
$328$	0.803848	0.0443851
$329$	38.7846	2.13826
$330$	0	0
$331$	28.7846	1.58215	0.791073	−	0.611722i	$-0.209523\pi$
$331$	28.7846	1.58215	0.791073	+	0.611722i	$0.209523\pi$
$332$	8.19615	0.449822
$333$	−17.7321	−0.971710
$334$	16.3923	0.896947
$335$	0	0
$336$	−3.46410	−0.188982
$337$	2.66025	0.144913	0.0724566	−	0.997372i	$-0.476916\pi$
$337$	2.66025	0.144913	0.0724566	+	0.997372i	$0.476916\pi$
$338$	−4.00000	−0.217571
$339$	7.94744	0.431646
$340$	0	0
$341$	0	0
$342$	3.12436	0.168946
$343$	39.7128	2.14429
$344$	0	0
$345$	0	0
$346$	−4.39230	−0.236132
$347$	−28.3923	−1.52418	−0.762089	−	0.647472i	$-0.775826\pi$
$347$	−28.3923	−1.52418	−0.762089	+	0.647472i	$0.775826\pi$
$348$	−2.19615	−0.117726
$349$	21.9282	1.17379	0.586895	−	0.809663i	$-0.300350\pi$
$349$	21.9282	1.17379	0.586895	+	0.809663i	$0.300350\pi$
$350$	0	0
$351$	12.0000	0.640513
$352$	0	0
$353$	−21.9282	−1.16712	−0.583560	−	0.812070i	$-0.698341\pi$
$353$	−21.9282	−1.16712	−0.583560	+	0.812070i	$0.698341\pi$
$354$	−6.92820	−0.368230
$355$	0	0
$356$	6.46410	0.342597
$357$	18.0000	0.952661
$358$	−13.8564	−0.732334
$359$	6.58846	0.347725	0.173863	−	0.984770i	$-0.444375\pi$
$359$	6.58846	0.347725	0.173863	+	0.984770i	$0.444375\pi$
$360$	0	0
$361$	−17.3923	−0.915384
$362$	7.19615	0.378221
$363$	0	0
$364$	14.1962	0.744081
$365$	0	0
$366$	1.85641	0.0970359
$367$	−8.58846	−0.448314	−0.224157	−	0.974553i	$-0.571963\pi$
$367$	−8.58846	−0.448314	−0.224157	+	0.974553i	$0.571963\pi$
$368$	1.26795	0.0660964
$369$	−1.98076	−0.103114
$370$	0	0
$371$	−52.9808	−2.75062
$372$	−0.143594	−0.00744498
$373$	−11.3205	−0.586154	−0.293077	−	0.956089i	$-0.594679\pi$
$373$	−11.3205	−0.586154	−0.293077	+	0.956089i	$0.594679\pi$
$374$	0	0
$375$	0	0
$376$	8.19615	0.422684
$377$	9.00000	0.463524
$378$	18.9282	0.973562
$379$	3.60770	0.185315	0.0926574	−	0.995698i	$-0.470464\pi$
$379$	3.60770	0.185315	0.0926574	+	0.995698i	$0.470464\pi$
$380$	0	0
$381$	−10.1436	−0.519672
$382$	−21.4641	−1.09820
$383$	0.679492	0.0347204	0.0173602	−	0.999849i	$-0.494474\pi$
$383$	0.679492	0.0347204	0.0173602	+	0.999849i	$0.494474\pi$
$384$	−0.732051	−0.0373573
$385$	0	0
$386$	10.2679	0.522625
$387$	0	0
$388$	1.00000	0.0507673
$389$	32.6603	1.65594	0.827970	−	0.560772i	$-0.189496\pi$
$389$	32.6603	1.65594	0.827970	+	0.560772i	$0.189496\pi$
$390$	0	0
$391$	−6.58846	−0.333193
$392$	15.3923	0.777429
$393$	0	0
$394$	−13.3923	−0.674695
$395$	0	0
$396$	0	0
$397$	−5.58846	−0.280477	−0.140238	−	0.990118i	$-0.544787\pi$
$397$	−5.58846	−0.280477	−0.140238	+	0.990118i	$0.544787\pi$
$398$	−0.392305	−0.0196645
$399$	4.39230	0.219890
$400$	0	0
$401$	−7.39230	−0.369154	−0.184577	−	0.982818i	$-0.559092\pi$
$401$	−7.39230	−0.369154	−0.184577	+	0.982818i	$0.559092\pi$
$402$	−7.46410	−0.372276
$403$	0.588457	0.0293131
$404$	−16.3923	−0.815548
$405$	0	0
$406$	14.1962	0.704543
$407$	0	0
$408$	3.80385	0.188319
$409$	2.66025	0.131541	0.0657705	−	0.997835i	$-0.479049\pi$
$409$	2.66025	0.131541	0.0657705	+	0.997835i	$0.479049\pi$
$410$	0	0
$411$	−6.92820	−0.341743
$412$	8.39230	0.413459
$413$	44.7846	2.20371
$414$	−3.12436	−0.153554
$415$	0	0
$416$	3.00000	0.147087
$417$	14.7846	0.724005
$418$	0	0
$419$	−39.3731	−1.92350	−0.961750	−	0.273928i	$-0.911677\pi$
$419$	−39.3731	−1.92350	−0.961750	+	0.273928i	$0.911677\pi$
$420$	0	0
$421$	7.58846	0.369839	0.184919	−	0.982754i	$-0.440798\pi$
$421$	7.58846	0.369839	0.184919	+	0.982754i	$0.440798\pi$
$422$	25.8564	1.25867
$423$	−20.1962	−0.981971
$424$	−11.1962	−0.543733
$425$	0	0
$426$	6.92820	0.335673
$427$	−12.0000	−0.580721
$428$	12.5885	0.608486
$429$	0	0
$430$	0	0
$431$	−16.3923	−0.789590	−0.394795	−	0.918769i	$-0.629184\pi$
$431$	−16.3923	−0.789590	−0.394795	+	0.918769i	$0.629184\pi$
$432$	4.00000	0.192450
$433$	−31.7846	−1.52747	−0.763735	−	0.645529i	$-0.776637\pi$
$433$	−31.7846	−1.52747	−0.763735	+	0.645529i	$0.776637\pi$
$434$	0.928203	0.0445552
$435$	0	0
$436$	2.07180	0.0992211
$437$	−1.60770	−0.0769065
$438$	0.679492	0.0324674
$439$	21.1244	1.00821	0.504105	−	0.863642i	$-0.331822\pi$
$439$	21.1244	1.00821	0.504105	+	0.863642i	$0.331822\pi$
$440$	0	0
$441$	−37.9282	−1.80610
$442$	−15.5885	−0.741467
$443$	−17.0718	−0.811106	−0.405553	−	0.914072i	$-0.632921\pi$
$443$	−17.0718	−0.811106	−0.405553	+	0.914072i	$0.632921\pi$
$444$	−5.26795	−0.250006
$445$	0	0
$446$	20.3923	0.965604
$447$	−10.9808	−0.519372
$448$	4.73205	0.223568
$449$	−9.92820	−0.468541	−0.234270	−	0.972171i	$-0.575270\pi$
$449$	−9.92820	−0.468541	−0.234270	+	0.972171i	$0.575270\pi$
$450$	0	0
$451$	0	0
$452$	−10.8564	−0.510642
$453$	−3.21539	−0.151072
$454$	−16.3923	−0.769329
$455$	0	0
$456$	0.928203	0.0434671
$457$	−5.19615	−0.243066	−0.121533	−	0.992587i	$-0.538781\pi$
$457$	−5.19615	−0.243066	−0.121533	+	0.992587i	$0.538781\pi$
$458$	−13.1962	−0.616616
$459$	−20.7846	−0.970143
$460$	0	0
$461$	−33.0000	−1.53696	−0.768482	−	0.639872i	$-0.778987\pi$
$461$	−33.0000	−1.53696	−0.768482	+	0.639872i	$0.778987\pi$
$462$	0	0
$463$	−28.7846	−1.33773	−0.668867	−	0.743382i	$-0.733220\pi$
$463$	−28.7846	−1.33773	−0.668867	+	0.743382i	$0.733220\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−6.80385	−0.315182
$467$	−21.4641	−0.993240	−0.496620	−	0.867968i	$-0.665426\pi$
$467$	−21.4641	−0.993240	−0.496620	+	0.867968i	$0.665426\pi$
$468$	−7.39230	−0.341709
$469$	48.2487	2.22792
$470$	0	0
$471$	2.92820	0.134924
$472$	9.46410	0.435621
$473$	0	0
$474$	−3.46410	−0.159111
$475$	0	0
$476$	−24.5885	−1.12701
$477$	27.5885	1.26319
$478$	14.1962	0.649317
$479$	8.78461	0.401379	0.200690	−	0.979655i	$-0.435682\pi$
$479$	8.78461	0.401379	0.200690	+	0.979655i	$0.435682\pi$
$480$	0	0
$481$	21.5885	0.984349
$482$	−26.7846	−1.22001
$483$	−4.39230	−0.199857
$484$	0	0
$485$	0	0
$486$	−15.2679	−0.692568
$487$	12.9808	0.588214	0.294107	−	0.955772i	$-0.404978\pi$
$487$	12.9808	0.588214	0.294107	+	0.955772i	$0.404978\pi$
$488$	−2.53590	−0.114795
$489$	16.2487	0.734792
$490$	0	0
$491$	33.3731	1.50611	0.753053	−	0.657960i	$-0.228581\pi$
$491$	33.3731	1.50611	0.753053	+	0.657960i	$0.228581\pi$
$492$	−0.588457	−0.0265297
$493$	−15.5885	−0.702069
$494$	−3.80385	−0.171143
$495$	0	0
$496$	0.196152	0.00880750
$497$	−44.7846	−2.00886
$498$	−6.00000	−0.268866
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	9.12436	0.407240
$503$	−14.1962	−0.632975	−0.316488	−	0.948597i	$-0.602504\pi$
$503$	−14.1962	−0.632975	−0.316488	+	0.948597i	$0.602504\pi$
$504$	−11.6603	−0.519389
$505$	0	0
$506$	0	0
$507$	2.92820	0.130046
$508$	13.8564	0.614779
$509$	12.9282	0.573033	0.286516	−	0.958075i	$-0.407503\pi$
$509$	12.9282	0.573033	0.286516	+	0.958075i	$0.407503\pi$
$510$	0	0
$511$	−4.39230	−0.194304
$512$	1.00000	0.0441942
$513$	−5.07180	−0.223925
$514$	19.3923	0.855358
$515$	0	0
$516$	0	0
$517$	0	0
$518$	34.0526	1.49618
$519$	3.21539	0.141140
$520$	0	0
$521$	9.46410	0.414630	0.207315	−	0.978274i	$-0.433528\pi$
$521$	9.46410	0.414630	0.207315	+	0.978274i	$0.433528\pi$
$522$	−7.39230	−0.323552
$523$	2.53590	0.110887	0.0554435	−	0.998462i	$-0.482343\pi$
$523$	2.53590	0.110887	0.0554435	+	0.998462i	$0.482343\pi$
$524$	0	0
$525$	0	0
$526$	9.80385	0.427468
$527$	−1.01924	−0.0443987
$528$	0	0
$529$	−21.3923	−0.930100
$530$	0	0
$531$	−23.3205	−1.01202
$532$	−6.00000	−0.260133
$533$	2.41154	0.104456
$534$	−4.73205	−0.204776
$535$	0	0
$536$	10.1962	0.440407
$537$	10.1436	0.437728
$538$	9.58846	0.413388
$539$	0	0
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	16.3923	0.704110
$543$	−5.26795	−0.226069
$544$	−5.19615	−0.222783
$545$	0	0
$546$	−10.3923	−0.444750
$547$	−24.0000	−1.02617	−0.513083	−	0.858339i	$-0.671497\pi$
$547$	−24.0000	−1.02617	−0.513083	+	0.858339i	$0.671497\pi$
$548$	9.46410	0.404286
$549$	6.24871	0.266688
$550$	0	0
$551$	−3.80385	−0.162049
$552$	−0.928203	−0.0395070
$553$	22.3923	0.952218
$554$	−8.32051	−0.353505
$555$	0	0
$556$	−20.1962	−0.856508
$557$	37.1769	1.57524	0.787618	−	0.616164i	$-0.211314\pi$
$557$	37.1769	1.57524	0.787618	+	0.616164i	$0.211314\pi$
$558$	−0.483340	−0.0204614
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−18.0000	−0.759284
$563$	−0.588457	−0.0248005	−0.0124003	−	0.999923i	$-0.503947\pi$
$563$	−0.588457	−0.0248005	−0.0124003	+	0.999923i	$0.503947\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	6.92820	0.291214
$567$	21.1244	0.887140
$568$	−9.46410	−0.397105
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−5.66025	−0.236874	−0.118437	−	0.992962i	$-0.537788\pi$
$571$	−5.66025	−0.236874	−0.118437	+	0.992962i	$0.537788\pi$
$572$	0	0
$573$	15.7128	0.656412
$574$	3.80385	0.158770
$575$	0	0
$576$	−2.46410	−0.102671
$577$	−14.6077	−0.608126	−0.304063	−	0.952652i	$-0.598343\pi$
$577$	−14.6077	−0.608126	−0.304063	+	0.952652i	$0.598343\pi$
$578$	10.0000	0.415945
$579$	−7.51666	−0.312382
$580$	0	0
$581$	38.7846	1.60906
$582$	−0.732051	−0.0303445
$583$	0	0
$584$	−0.928203	−0.0384093
$585$	0	0
$586$	23.7846	0.982533
$587$	−30.5885	−1.26252	−0.631260	−	0.775571i	$-0.717462\pi$
$587$	−30.5885	−1.26252	−0.631260	+	0.775571i	$0.717462\pi$
$588$	−11.2679	−0.464682
$589$	−0.248711	−0.0102480
$590$	0	0
$591$	9.80385	0.403276
$592$	7.19615	0.295760
$593$	−23.1962	−0.952552	−0.476276	−	0.879296i	$-0.658014\pi$
$593$	−23.1962	−0.952552	−0.476276	+	0.879296i	$0.658014\pi$
$594$	0	0
$595$	0	0
$596$	15.0000	0.614424
$597$	0.287187	0.0117538
$598$	3.80385	0.155551
$599$	−3.80385	−0.155421	−0.0777105	−	0.996976i	$-0.524761\pi$
$599$	−3.80385	−0.155421	−0.0777105	+	0.996976i	$0.524761\pi$
$600$	0	0
$601$	8.66025	0.353259	0.176630	−	0.984277i	$-0.443481\pi$
$601$	8.66025	0.353259	0.176630	+	0.984277i	$0.443481\pi$
$602$	0	0
$603$	−25.1244	−1.02314
$604$	4.39230	0.178720
$605$	0	0
$606$	12.0000	0.487467
$607$	26.1962	1.06327	0.531635	−	0.846974i	$-0.321578\pi$
$607$	26.1962	1.06327	0.531635	+	0.846974i	$0.321578\pi$
$608$	−1.26795	−0.0514221
$609$	−10.3923	−0.421117
$610$	0	0
$611$	24.5885	0.994743
$612$	12.8038	0.517565
$613$	20.3205	0.820738	0.410369	−	0.911920i	$-0.365400\pi$
$613$	20.3205	0.820738	0.410369	+	0.911920i	$0.365400\pi$
$614$	25.2679	1.01973
$615$	0	0
$616$	0	0
$617$	−10.6077	−0.427050	−0.213525	−	0.976938i	$-0.568494\pi$
$617$	−10.6077	−0.427050	−0.213525	+	0.976938i	$0.568494\pi$
$618$	−6.14359	−0.247132
$619$	−30.1962	−1.21369	−0.606843	−	0.794822i	$-0.707564\pi$
$619$	−30.1962	−1.21369	−0.606843	+	0.794822i	$0.707564\pi$
$620$	0	0
$621$	5.07180	0.203524
$622$	8.78461	0.352231
$623$	30.5885	1.22550
$624$	−2.19615	−0.0879165
$625$	0	0
$626$	31.7846	1.27037
$627$	0	0
$628$	−4.00000	−0.159617
$629$	−37.3923	−1.49093
$630$	0	0
$631$	−20.5885	−0.819614	−0.409807	−	0.912172i	$-0.634404\pi$
$631$	−20.5885	−0.819614	−0.409807	+	0.912172i	$0.634404\pi$
$632$	4.73205	0.188231
$633$	−18.9282	−0.752329
$634$	7.85641	0.312018
$635$	0	0
$636$	8.19615	0.324999
$637$	46.1769	1.82960
$638$	0	0
$639$	23.3205	0.922545
$640$	0	0
$641$	3.67949	0.145331	0.0726656	−	0.997356i	$-0.476849\pi$
$641$	3.67949	0.145331	0.0726656	+	0.997356i	$0.476849\pi$
$642$	−9.21539	−0.363702
$643$	25.8038	1.01760	0.508802	−	0.860883i	$-0.330088\pi$
$643$	25.8038	1.01760	0.508802	+	0.860883i	$0.330088\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	6.58846	0.259219
$647$	23.3205	0.916824	0.458412	−	0.888740i	$-0.348418\pi$
$647$	23.3205	0.916824	0.458412	+	0.888740i	$0.348418\pi$
$648$	4.46410	0.175366
$649$	0	0
$650$	0	0
$651$	−0.679492	−0.0266314
$652$	−22.1962	−0.869268
$653$	1.85641	0.0726468	0.0363234	−	0.999340i	$-0.488435\pi$
$653$	1.85641	0.0726468	0.0363234	+	0.999340i	$0.488435\pi$
$654$	−1.51666	−0.0593061
$655$	0	0
$656$	0.803848	0.0313850
$657$	2.28719	0.0892317
$658$	38.7846	1.51198
$659$	−16.9808	−0.661477	−0.330738	−	0.943723i	$-0.607298\pi$
$659$	−16.9808	−0.661477	−0.330738	+	0.943723i	$0.607298\pi$
$660$	0	0
$661$	−16.4115	−0.638335	−0.319168	−	0.947698i	$-0.603403\pi$
$661$	−16.4115	−0.638335	−0.319168	+	0.947698i	$0.603403\pi$
$662$	28.7846	1.11875
$663$	11.4115	0.443188
$664$	8.19615	0.318072
$665$	0	0
$666$	−17.7321	−0.687103
$667$	3.80385	0.147286
$668$	16.3923	0.634237
$669$	−14.9282	−0.577158
$670$	0	0
$671$	0	0
$672$	−3.46410	−0.133631
$673$	0.928203	0.0357796	0.0178898	−	0.999840i	$-0.494305\pi$
$673$	0.928203	0.0357796	0.0178898	+	0.999840i	$0.494305\pi$
$674$	2.66025	0.102469
$675$	0	0
$676$	−4.00000	−0.153846
$677$	4.60770	0.177088	0.0885441	−	0.996072i	$-0.471779\pi$
$677$	4.60770	0.177088	0.0885441	+	0.996072i	$0.471779\pi$
$678$	7.94744	0.305220
$679$	4.73205	0.181599
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	5.41154	0.207067	0.103533	−	0.994626i	$-0.466985\pi$
$683$	5.41154	0.207067	0.103533	+	0.994626i	$0.466985\pi$
$684$	3.12436	0.119463
$685$	0	0
$686$	39.7128	1.51624
$687$	9.66025	0.368562
$688$	0	0
$689$	−33.5885	−1.27962
$690$	0	0
$691$	−41.1769	−1.56644	−0.783222	−	0.621742i	$-0.786425\pi$
$691$	−41.1769	−1.56644	−0.783222	+	0.621742i	$0.786425\pi$
$692$	−4.39230	−0.166970
$693$	0	0
$694$	−28.3923	−1.07776
$695$	0	0
$696$	−2.19615	−0.0832449
$697$	−4.17691	−0.158212
$698$	21.9282	0.829995
$699$	4.98076	0.188390
$700$	0	0
$701$	17.7846	0.671715	0.335858	−	0.941913i	$-0.390974\pi$
$701$	17.7846	0.671715	0.335858	+	0.941913i	$0.390974\pi$
$702$	12.0000	0.452911
$703$	−9.12436	−0.344132
$704$	0	0
$705$	0	0
$706$	−21.9282	−0.825279
$707$	−77.5692	−2.91729
$708$	−6.92820	−0.260378
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	−11.6603	−0.437294
$712$	6.46410	0.242252
$713$	0.248711	0.00931431
$714$	18.0000	0.673633
$715$	0	0
$716$	−13.8564	−0.517838
$717$	−10.3923	−0.388108
$718$	6.58846	0.245879
$719$	−46.7321	−1.74281	−0.871406	−	0.490563i	$-0.836791\pi$
$719$	−46.7321	−1.74281	−0.871406	+	0.490563i	$0.836791\pi$
$720$	0	0
$721$	39.7128	1.47898
$722$	−17.3923	−0.647275
$723$	19.6077	0.729218
$724$	7.19615	0.267443
$725$	0	0
$726$	0	0
$727$	36.9808	1.37154	0.685770	−	0.727818i	$-0.259465\pi$
$727$	36.9808	1.37154	0.685770	+	0.727818i	$0.259465\pi$
$728$	14.1962	0.526144
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	0	0
$732$	1.85641	0.0686148
$733$	16.6077	0.613419	0.306710	−	0.951803i	$-0.400772\pi$
$733$	16.6077	0.613419	0.306710	+	0.951803i	$0.400772\pi$
$734$	−8.58846	−0.317006
$735$	0	0
$736$	1.26795	0.0467372
$737$	0	0
$738$	−1.98076	−0.0729129
$739$	−10.7321	−0.394785	−0.197392	−	0.980325i	$-0.563247\pi$
$739$	−10.7321	−0.394785	−0.197392	+	0.980325i	$0.563247\pi$
$740$	0	0
$741$	2.78461	0.102295
$742$	−52.9808	−1.94498
$743$	−38.1962	−1.40128	−0.700640	−	0.713514i	$-0.747102\pi$
$743$	−38.1962	−1.40128	−0.700640	+	0.713514i	$0.747102\pi$
$744$	−0.143594	−0.00526439
$745$	0	0
$746$	−11.3205	−0.414473
$747$	−20.1962	−0.738939
$748$	0	0
$749$	59.5692	2.17661
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	8.19615	0.298883
$753$	−6.67949	−0.243414
$754$	9.00000	0.327761
$755$	0	0
$756$	18.9282	0.688412
$757$	13.1962	0.479622	0.239811	−	0.970820i	$-0.422914\pi$
$757$	13.1962	0.479622	0.239811	+	0.970820i	$0.422914\pi$
$758$	3.60770	0.131037
$759$	0	0
$760$	0	0
$761$	−19.9808	−0.724302	−0.362151	−	0.932119i	$-0.617958\pi$
$761$	−19.9808	−0.724302	−0.362151	+	0.932119i	$0.617958\pi$
$762$	−10.1436	−0.367464
$763$	9.80385	0.354923
$764$	−21.4641	−0.776544
$765$	0	0
$766$	0.679492	0.0245510
$767$	28.3923	1.02519
$768$	−0.732051	−0.0264156
$769$	−34.5167	−1.24470	−0.622351	−	0.782738i	$-0.713822\pi$
$769$	−34.5167	−1.24470	−0.622351	+	0.782738i	$0.713822\pi$
$770$	0	0
$771$	−14.1962	−0.511262
$772$	10.2679	0.369552
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	0	0
$776$	1.00000	0.0358979
$777$	−24.9282	−0.894294
$778$	32.6603	1.17093
$779$	−1.01924	−0.0365180
$780$	0	0
$781$	0	0
$782$	−6.58846	−0.235603
$783$	12.0000	0.428845
$784$	15.3923	0.549725
$785$	0	0
$786$	0	0
$787$	44.7846	1.59640	0.798199	−	0.602393i	$-0.205786\pi$
$787$	44.7846	1.59640	0.798199	+	0.602393i	$0.205786\pi$
$788$	−13.3923	−0.477081
$789$	−7.17691	−0.255505
$790$	0	0
$791$	−51.3731	−1.82662
$792$	0	0
$793$	−7.60770	−0.270157
$794$	−5.58846	−0.198327
$795$	0	0
$796$	−0.392305	−0.0139049
$797$	−36.9282	−1.30806	−0.654032	−	0.756467i	$-0.726924\pi$
$797$	−36.9282	−1.30806	−0.654032	+	0.756467i	$0.726924\pi$
$798$	4.39230	0.155486
$799$	−42.5885	−1.50667
$800$	0	0
$801$	−15.9282	−0.562795
$802$	−7.39230	−0.261031
$803$	0	0
$804$	−7.46410	−0.263239
$805$	0	0
$806$	0.588457	0.0207275
$807$	−7.01924	−0.247089
$808$	−16.3923	−0.576679
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	−6.92820	−0.243282	−0.121641	−	0.992574i	$-0.538816\pi$
$811$	−6.92820	−0.243282	−0.121641	+	0.992574i	$0.538816\pi$
$812$	14.1962	0.498187
$813$	−12.0000	−0.420858
$814$	0	0
$815$	0	0
$816$	3.80385	0.133161
$817$	0	0
$818$	2.66025	0.0930136
$819$	−34.9808	−1.22233
$820$	0	0
$821$	16.3923	0.572095	0.286048	−	0.958215i	$-0.407658\pi$
$821$	16.3923	0.572095	0.286048	+	0.958215i	$0.407658\pi$
$822$	−6.92820	−0.241649
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	8.39230	0.292360
$825$	0	0
$826$	44.7846	1.55826
$827$	−20.1962	−0.702289	−0.351145	−	0.936321i	$-0.614207\pi$
$827$	−20.1962	−0.702289	−0.351145	+	0.936321i	$0.614207\pi$
$828$	−3.12436	−0.108579
$829$	14.8038	0.514159	0.257079	−	0.966390i	$-0.417240\pi$
$829$	14.8038	0.514159	0.257079	+	0.966390i	$0.417240\pi$
$830$	0	0
$831$	6.09103	0.211296
$832$	3.00000	0.104006
$833$	−79.9808	−2.77117
$834$	14.7846	0.511949
$835$	0	0
$836$	0	0
$837$	0.784610	0.0271201
$838$	−39.3731	−1.36012
$839$	−50.4449	−1.74155	−0.870775	−	0.491682i	$-0.836382\pi$
$839$	−50.4449	−1.74155	−0.870775	+	0.491682i	$0.836382\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	7.58846	0.261516
$843$	13.1769	0.453837
$844$	25.8564	0.890014
$845$	0	0
$846$	−20.1962	−0.694358
$847$	0	0
$848$	−11.1962	−0.384477
$849$	−5.07180	−0.174064
$850$	0	0
$851$	9.12436	0.312779
$852$	6.92820	0.237356
$853$	−10.8564	−0.371716	−0.185858	−	0.982577i	$-0.559506\pi$
$853$	−10.8564	−0.371716	−0.185858	+	0.982577i	$0.559506\pi$
$854$	−12.0000	−0.410632
$855$	0	0
$856$	12.5885	0.430265
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−24.7846	−0.845640	−0.422820	−	0.906214i	$-0.638960\pi$
$859$	−24.7846	−0.845640	−0.422820	+	0.906214i	$0.638960\pi$
$860$	0	0
$861$	−2.78461	−0.0948992
$862$	−16.3923	−0.558324
$863$	3.80385	0.129484	0.0647422	−	0.997902i	$-0.479377\pi$
$863$	3.80385	0.129484	0.0647422	+	0.997902i	$0.479377\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	−31.7846	−1.08009
$867$	−7.32051	−0.248617
$868$	0.928203	0.0315053
$869$	0	0
$870$	0	0
$871$	30.5885	1.03645
$872$	2.07180	0.0701599
$873$	−2.46410	−0.0833972
$874$	−1.60770	−0.0543811
$875$	0	0
$876$	0.679492	0.0229579
$877$	−6.46410	−0.218277	−0.109139	−	0.994027i	$-0.534809\pi$
$877$	−6.46410	−0.218277	−0.109139	+	0.994027i	$0.534809\pi$
$878$	21.1244	0.712912
$879$	−17.4115	−0.587277
$880$	0	0
$881$	41.5359	1.39938	0.699690	−	0.714447i	$-0.253321\pi$
$881$	41.5359	1.39938	0.699690	+	0.714447i	$0.253321\pi$
$882$	−37.9282	−1.27711
$883$	15.6077	0.525241	0.262620	−	0.964899i	$-0.415413\pi$
$883$	15.6077	0.525241	0.262620	+	0.964899i	$0.415413\pi$
$884$	−15.5885	−0.524297
$885$	0	0
$886$	−17.0718	−0.573538
$887$	−45.8038	−1.53794	−0.768971	−	0.639283i	$-0.779231\pi$
$887$	−45.8038	−1.53794	−0.768971	+	0.639283i	$0.779231\pi$
$888$	−5.26795	−0.176781
$889$	65.5692	2.19912
$890$	0	0
$891$	0	0
$892$	20.3923	0.682785
$893$	−10.3923	−0.347765
$894$	−10.9808	−0.367252
$895$	0	0
$896$	4.73205	0.158087
$897$	−2.78461	−0.0929754
$898$	−9.92820	−0.331308
$899$	0.588457	0.0196261
$900$	0	0
$901$	58.1769	1.93815
$902$	0	0
$903$	0	0
$904$	−10.8564	−0.361079
$905$	0	0
$906$	−3.21539	−0.106824
$907$	−37.5692	−1.24747	−0.623733	−	0.781638i	$-0.714385\pi$
$907$	−37.5692	−1.24747	−0.623733	+	0.781638i	$0.714385\pi$
$908$	−16.3923	−0.543998
$909$	40.3923	1.33973
$910$	0	0
$911$	−31.6077	−1.04721	−0.523605	−	0.851961i	$-0.675413\pi$
$911$	−31.6077	−1.04721	−0.523605	+	0.851961i	$0.675413\pi$
$912$	0.928203	0.0307359
$913$	0	0
$914$	−5.19615	−0.171873
$915$	0	0
$916$	−13.1962	−0.436013
$917$	0	0
$918$	−20.7846	−0.685994
$919$	−22.1436	−0.730450	−0.365225	−	0.930919i	$-0.619008\pi$
$919$	−22.1436	−0.730450	−0.365225	+	0.930919i	$0.619008\pi$
$920$	0	0
$921$	−18.4974	−0.609511
$922$	−33.0000	−1.08680
$923$	−28.3923	−0.934544
$924$	0	0
$925$	0	0
$926$	−28.7846	−0.945921
$927$	−20.6795	−0.679204
$928$	3.00000	0.0984798
$929$	39.0000	1.27955	0.639774	−	0.768563i	$-0.279028\pi$
$929$	39.0000	1.27955	0.639774	+	0.768563i	$0.279028\pi$
$930$	0	0
$931$	−19.5167	−0.639633
$932$	−6.80385	−0.222867
$933$	−6.43078	−0.210534
$934$	−21.4641	−0.702327
$935$	0	0
$936$	−7.39230	−0.241625
$937$	35.4449	1.15793	0.578967	−	0.815351i	$-0.303456\pi$
$937$	35.4449	1.15793	0.578967	+	0.815351i	$0.303456\pi$
$938$	48.2487	1.57538
$939$	−23.2679	−0.759321
$940$	0	0
$941$	34.6077	1.12818	0.564089	−	0.825714i	$-0.309227\pi$
$941$	34.6077	1.12818	0.564089	+	0.825714i	$0.309227\pi$
$942$	2.92820	0.0954060
$943$	1.01924	0.0331910
$944$	9.46410	0.308030
$945$	0	0
$946$	0	0
$947$	−5.90897	−0.192016	−0.0960078	−	0.995381i	$-0.530607\pi$
$947$	−5.90897	−0.192016	−0.0960078	+	0.995381i	$0.530607\pi$
$948$	−3.46410	−0.112509
$949$	−2.78461	−0.0903923
$950$	0	0
$951$	−5.75129	−0.186498
$952$	−24.5885	−0.796916
$953$	−39.5885	−1.28240	−0.641198	−	0.767376i	$-0.721562\pi$
$953$	−39.5885	−1.28240	−0.641198	+	0.767376i	$0.721562\pi$
$954$	27.5885	0.893209
$955$	0	0
$956$	14.1962	0.459136
$957$	0	0
$958$	8.78461	0.283818
$959$	44.7846	1.44617
$960$	0	0
$961$	−30.9615	−0.998759
$962$	21.5885	0.696040
$963$	−31.0192	−0.999581
$964$	−26.7846	−0.862674
$965$	0	0
$966$	−4.39230	−0.141320
$967$	−32.4449	−1.04336	−0.521678	−	0.853142i	$-0.674694\pi$
$967$	−32.4449	−1.04336	−0.521678	+	0.853142i	$0.674694\pi$
$968$	0	0
$969$	−4.82309	−0.154940
$970$	0	0
$971$	−4.05256	−0.130053	−0.0650264	−	0.997884i	$-0.520713\pi$
$971$	−4.05256	−0.130053	−0.0650264	+	0.997884i	$0.520713\pi$
$972$	−15.2679	−0.489720
$973$	−95.5692	−3.06381
$974$	12.9808	0.415930
$975$	0	0
$976$	−2.53590	−0.0811721
$977$	−32.3205	−1.03402	−0.517012	−	0.855978i	$-0.672956\pi$
$977$	−32.3205	−1.03402	−0.517012	+	0.855978i	$0.672956\pi$
$978$	16.2487	0.519576
$979$	0	0
$980$	0	0
$981$	−5.10512	−0.162994
$982$	33.3731	1.06498
$983$	−32.1051	−1.02399	−0.511997	−	0.858987i	$-0.671094\pi$
$983$	−32.1051	−1.02399	−0.511997	+	0.858987i	$0.671094\pi$
$984$	−0.588457	−0.0187593
$985$	0	0
$986$	−15.5885	−0.496438
$987$	−28.3923	−0.903737
$988$	−3.80385	−0.121017
$989$	0	0
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0.196152	0.00622785
$993$	−21.0718	−0.668693
$994$	−44.7846	−1.42048
$995$	0	0
$996$	−6.00000	−0.190117
$997$	12.4641	0.394742	0.197371	−	0.980329i	$-0.436760\pi$
$997$	12.4641	0.394742	0.197371	+	0.980329i	$0.436760\pi$
$998$	16.0000	0.506471
$999$	28.7846	0.910705

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6050.2.a.cv.1.1		2
5.4	even	2		242.2.a.c.1.2	✓	2
11.10	odd	2		6050.2.a.cc.1.1		2
15.14	odd	2		2178.2.a.y.1.2		2
20.19	odd	2		1936.2.a.y.1.1		2
40.19	odd	2		7744.2.a.bt.1.2		2
40.29	even	2		7744.2.a.cs.1.1		2
55.4	even	10		242.2.c.g.27.1		8
55.9	even	10		242.2.c.g.81.2		8
55.14	even	10		242.2.c.g.9.1		8
55.19	odd	10		242.2.c.f.9.1		8
55.24	odd	10		242.2.c.f.81.2		8
55.29	odd	10		242.2.c.f.27.1		8
55.39	odd	10		242.2.c.f.3.2		8
55.49	even	10		242.2.c.g.3.2		8
55.54	odd	2		242.2.a.e.1.2	yes	2
165.164	even	2		2178.2.a.s.1.2		2
220.219	even	2		1936.2.a.v.1.1		2
440.109	odd	2		7744.2.a.cv.1.1		2
440.219	even	2		7744.2.a.bq.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.a.c.1.2	✓	2	5.4	even	2
242.2.a.e.1.2	yes	2	55.54	odd	2
242.2.c.f.3.2		8	55.39	odd	10
242.2.c.f.9.1		8	55.19	odd	10
242.2.c.f.27.1		8	55.29	odd	10
242.2.c.f.81.2		8	55.24	odd	10
242.2.c.g.3.2		8	55.49	even	10
242.2.c.g.9.1		8	55.14	even	10
242.2.c.g.27.1		8	55.4	even	10
242.2.c.g.81.2		8	55.9	even	10
1936.2.a.v.1.1		2	220.219	even	2
1936.2.a.y.1.1		2	20.19	odd	2
2178.2.a.s.1.2		2	165.164	even	2
2178.2.a.y.1.2		2	15.14	odd	2
6050.2.a.cc.1.1		2	11.10	odd	2
6050.2.a.cv.1.1		2	1.1	even	1	trivial
7744.2.a.bq.1.2		2	440.219	even	2
7744.2.a.bt.1.2		2	40.19	odd	2
7744.2.a.cs.1.1		2	40.29	even	2
7744.2.a.cv.1.1		2	440.109	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:	\( N \)	\(=\)	\( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6050.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} +4.73205 q^{7} +1.00000 q^{8} -2.46410 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} +4.73205 q^{7} +1.00000 q^{8} -2.46410 q^{9} -0.732051 q^{12} +3.00000 q^{13} +4.73205 q^{14} +1.00000 q^{16} -5.19615 q^{17} -2.46410 q^{18} -1.26795 q^{19} -3.46410 q^{21} +1.26795 q^{23} -0.732051 q^{24} +3.00000 q^{26} +4.00000 q^{27} +4.73205 q^{28} +3.00000 q^{29} +0.196152 q^{31} +1.00000 q^{32} -5.19615 q^{34} -2.46410 q^{36} +7.19615 q^{37} -1.26795 q^{38} -2.19615 q^{39} +0.803848 q^{41} -3.46410 q^{42} +1.26795 q^{46} +8.19615 q^{47} -0.732051 q^{48} +15.3923 q^{49} +3.80385 q^{51} +3.00000 q^{52} -11.1962 q^{53} +4.00000 q^{54} +4.73205 q^{56} +0.928203 q^{57} +3.00000 q^{58} +9.46410 q^{59} -2.53590 q^{61} +0.196152 q^{62} -11.6603 q^{63} +1.00000 q^{64} +10.1962 q^{67} -5.19615 q^{68} -0.928203 q^{69} -9.46410 q^{71} -2.46410 q^{72} -0.928203 q^{73} +7.19615 q^{74} -1.26795 q^{76} -2.19615 q^{78} +4.73205 q^{79} +4.46410 q^{81} +0.803848 q^{82} +8.19615 q^{83} -3.46410 q^{84} -2.19615 q^{87} +6.46410 q^{89} +14.1962 q^{91} +1.26795 q^{92} -0.143594 q^{93} +8.19615 q^{94} -0.732051 q^{96} +1.00000 q^{97} +15.3923 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{12} + 6 q^{13} + 6 q^{14} + 2 q^{16} + 2 q^{18} - 6 q^{19} + 6 q^{23} + 2 q^{24} + 6 q^{26} + 8 q^{27} + 6 q^{28} + 6 q^{29} - 10 q^{31} + 2 q^{32} + 2 q^{36} + 4 q^{37} - 6 q^{38} + 6 q^{39} + 12 q^{41} + 6 q^{46} + 6 q^{47} + 2 q^{48} + 10 q^{49} + 18 q^{51} + 6 q^{52} - 12 q^{53} + 8 q^{54} + 6 q^{56} - 12 q^{57} + 6 q^{58} + 12 q^{59} - 12 q^{61} - 10 q^{62} - 6 q^{63} + 2 q^{64} + 10 q^{67} + 12 q^{69} - 12 q^{71} + 2 q^{72} + 12 q^{73} + 4 q^{74} - 6 q^{76} + 6 q^{78} + 6 q^{79} + 2 q^{81} + 12 q^{82} + 6 q^{83} + 6 q^{87} + 6 q^{89} + 18 q^{91} + 6 q^{92} - 28 q^{93} + 6 q^{94} + 2 q^{96} + 2 q^{97} + 10 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^12 + 6 * q^13 + 6 * q^14 + 2 * q^16 + 2 * q^18 - 6 * q^19 + 6 * q^23 + 2 * q^24 + 6 * q^26 + 8 * q^27 + 6 * q^28 + 6 * q^29 - 10 * q^31 + 2 * q^32 + 2 * q^36 + 4 * q^37 - 6 * q^38 + 6 * q^39 + 12 * q^41 + 6 * q^46 + 6 * q^47 + 2 * q^48 + 10 * q^49 + 18 * q^51 + 6 * q^52 - 12 * q^53 + 8 * q^54 + 6 * q^56 - 12 * q^57 + 6 * q^58 + 12 * q^59 - 12 * q^61 - 10 * q^62 - 6 * q^63 + 2 * q^64 + 10 * q^67 + 12 * q^69 - 12 * q^71 + 2 * q^72 + 12 * q^73 + 4 * q^74 - 6 * q^76 + 6 * q^78 + 6 * q^79 + 2 * q^81 + 12 * q^82 + 6 * q^83 + 6 * q^87 + 6 * q^89 + 18 * q^91 + 6 * q^92 - 28 * q^93 + 6 * q^94 + 2 * q^96 + 2 * q^97 + 10 * q^98

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