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Properties

Label 560.2.k.a.111.3
Level $560$
Weight $2$
Character 560.111
Analytic conductor $4.472$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [560,2,Mod(111,560)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("560.111"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(i, \sqrt{3}, \sqrt{11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 111.3
Root \(-0.396143 - 1.68614i\) of defining polynomial
Character \(\chi\) \(=\) 560.111
Dual form 560.2.k.a.111.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.792287 q^{3} -1.00000i q^{5} +(-0.792287 + 2.52434i) q^{7} -2.37228 q^{9} -0.792287i q^{11} -5.37228i q^{13} +0.792287i q^{15} -3.37228i q^{17} -3.46410 q^{19} +(0.627719 - 2.00000i) q^{21} -1.87953i q^{23} -1.00000 q^{25} +4.25639 q^{27} -5.37228 q^{29} -8.51278 q^{31} +0.627719i q^{33} +(2.52434 + 0.792287i) q^{35} -0.744563 q^{37} +4.25639i q^{39} -2.74456i q^{41} -3.46410i q^{43} +2.37228i q^{45} -11.1846 q^{47} +(-5.74456 - 4.00000i) q^{49} +2.67181i q^{51} +11.4891 q^{53} -0.792287 q^{55} +2.74456 q^{57} +6.63325 q^{59} -0.744563i q^{61} +(1.87953 - 5.98844i) q^{63} -5.37228 q^{65} -6.63325i q^{67} +1.48913i q^{69} -6.63325i q^{71} -2.74456i q^{73} +0.792287 q^{75} +(2.00000 + 0.627719i) q^{77} +14.0588i q^{79} +3.74456 q^{81} +10.3923 q^{83} -3.37228 q^{85} +4.25639 q^{87} +17.4891i q^{89} +(13.5615 + 4.25639i) q^{91} +6.74456 q^{93} +3.46410i q^{95} +2.11684i q^{97} +1.87953i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9} + 28 q^{21} - 8 q^{25} - 20 q^{29} + 40 q^{37} - 24 q^{57} - 20 q^{65} + 16 q^{77} - 16 q^{81} - 4 q^{85} + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.792287 −0.457427 −0.228714 − 0.973494i \(-0.573452\pi\)
−0.228714 + 0.973494i \(0.573452\pi\)
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) −0.792287 + 2.52434i −0.299456 + 0.954110i
\(8\) 0 0
\(9\) −2.37228 −0.790760
\(10\) 0 0
\(11\) − 0.792287i − 0.238884i −0.992841 − 0.119442i \(-0.961890\pi\)
0.992841 − 0.119442i \(-0.0381105\pi\)
\(12\) 0 0
\(13\) − 5.37228i − 1.49000i −0.667063 − 0.745001i \(-0.732449\pi\)
0.667063 − 0.745001i \(-0.267551\pi\)
\(14\) 0 0
\(15\) 0.792287i 0.204568i
\(16\) 0 0
\(17\) − 3.37228i − 0.817898i −0.912557 − 0.408949i \(-0.865895\pi\)
0.912557 − 0.408949i \(-0.134105\pi\)
\(18\) 0 0
\(19\) −3.46410 −0.794719 −0.397360 − 0.917663i \(-0.630073\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) 0.627719 − 2.00000i 0.136979 − 0.436436i
\(22\) 0 0
\(23\) − 1.87953i − 0.391909i −0.980613 − 0.195954i \(-0.937220\pi\)
0.980613 − 0.195954i \(-0.0627804\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 4.25639 0.819142
\(28\) 0 0
\(29\) −5.37228 −0.997608 −0.498804 − 0.866715i \(-0.666227\pi\)
−0.498804 + 0.866715i \(0.666227\pi\)
\(30\) 0 0
\(31\) −8.51278 −1.52894 −0.764470 − 0.644659i \(-0.776999\pi\)
−0.764470 + 0.644659i \(0.776999\pi\)
\(32\) 0 0
\(33\) 0.627719i 0.109272i
\(34\) 0 0
\(35\) 2.52434 + 0.792287i 0.426691 + 0.133921i
\(36\) 0 0
\(37\) −0.744563 −0.122405 −0.0612027 − 0.998125i \(-0.519494\pi\)
−0.0612027 + 0.998125i \(0.519494\pi\)
\(38\) 0 0
\(39\) 4.25639i 0.681568i
\(40\) 0 0
\(41\) − 2.74456i − 0.428629i −0.976765 − 0.214314i \(-0.931248\pi\)
0.976765 − 0.214314i \(-0.0687517\pi\)
\(42\) 0 0
\(43\) − 3.46410i − 0.528271i −0.964486 − 0.264135i \(-0.914913\pi\)
0.964486 − 0.264135i \(-0.0850865\pi\)
\(44\) 0 0
\(45\) 2.37228i 0.353639i
\(46\) 0 0
\(47\) −11.1846 −1.63144 −0.815720 − 0.578447i \(-0.803659\pi\)
−0.815720 + 0.578447i \(0.803659\pi\)
\(48\) 0 0
\(49\) −5.74456 − 4.00000i −0.820652 − 0.571429i
\(50\) 0 0
\(51\) 2.67181i 0.374129i
\(52\) 0 0
\(53\) 11.4891 1.57815 0.789076 − 0.614295i \(-0.210560\pi\)
0.789076 + 0.614295i \(0.210560\pi\)
\(54\) 0 0
\(55\) −0.792287 −0.106832
\(56\) 0 0
\(57\) 2.74456 0.363526
\(58\) 0 0
\(59\) 6.63325 0.863576 0.431788 − 0.901975i \(-0.357883\pi\)
0.431788 + 0.901975i \(0.357883\pi\)
\(60\) 0 0
\(61\) − 0.744563i − 0.0953315i −0.998863 − 0.0476657i \(-0.984822\pi\)
0.998863 − 0.0476657i \(-0.0151782\pi\)
\(62\) 0 0
\(63\) 1.87953 − 5.98844i 0.236798 − 0.754472i
\(64\) 0 0
\(65\) −5.37228 −0.666349
\(66\) 0 0
\(67\) − 6.63325i − 0.810380i −0.914232 − 0.405190i \(-0.867205\pi\)
0.914232 − 0.405190i \(-0.132795\pi\)
\(68\) 0 0
\(69\) 1.48913i 0.179270i
\(70\) 0 0
\(71\) − 6.63325i − 0.787222i −0.919277 − 0.393611i \(-0.871226\pi\)
0.919277 − 0.393611i \(-0.128774\pi\)
\(72\) 0 0
\(73\) − 2.74456i − 0.321227i −0.987017 − 0.160613i \(-0.948653\pi\)
0.987017 − 0.160613i \(-0.0513472\pi\)
\(74\) 0 0
\(75\) 0.792287 0.0914854
\(76\) 0 0
\(77\) 2.00000 + 0.627719i 0.227921 + 0.0715352i
\(78\) 0 0
\(79\) 14.0588i 1.58174i 0.611986 + 0.790869i \(0.290371\pi\)
−0.611986 + 0.790869i \(0.709629\pi\)
\(80\) 0 0
\(81\) 3.74456 0.416063
\(82\) 0 0
\(83\) 10.3923 1.14070 0.570352 − 0.821401i \(-0.306807\pi\)
0.570352 + 0.821401i \(0.306807\pi\)
\(84\) 0 0
\(85\) −3.37228 −0.365775
\(86\) 0 0
\(87\) 4.25639 0.456333
\(88\) 0 0
\(89\) 17.4891i 1.85384i 0.375255 + 0.926922i \(0.377555\pi\)
−0.375255 + 0.926922i \(0.622445\pi\)
\(90\) 0 0
\(91\) 13.5615 + 4.25639i 1.42163 + 0.446191i
\(92\) 0 0
\(93\) 6.74456 0.699379
\(94\) 0 0
\(95\) 3.46410i 0.355409i
\(96\) 0 0
\(97\) 2.11684i 0.214933i 0.994209 + 0.107466i \(0.0342738\pi\)
−0.994209 + 0.107466i \(0.965726\pi\)
\(98\) 0 0
\(99\) 1.87953i 0.188900i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 560.2.k.a.111.3 ✓ 8
3.2 odd 2 5040.2.d.e.4591.6 8
4.3 odd 2 inner 560.2.k.a.111.5 yes 8
5.2 odd 4 2800.2.e.i.2799.5 8
5.3 odd 4 2800.2.e.j.2799.4 8
5.4 even 2 2800.2.k.l.2351.5 8
7.6 odd 2 inner 560.2.k.a.111.6 yes 8
8.3 odd 2 2240.2.k.c.1791.4 8
8.5 even 2 2240.2.k.c.1791.6 8
12.11 even 2 5040.2.d.e.4591.7 8
20.3 even 4 2800.2.e.j.2799.5 8
20.7 even 4 2800.2.e.i.2799.4 8
20.19 odd 2 2800.2.k.l.2351.4 8
21.20 even 2 5040.2.d.e.4591.3 8
28.27 even 2 inner 560.2.k.a.111.4 yes 8
35.13 even 4 2800.2.e.i.2799.6 8
35.27 even 4 2800.2.e.j.2799.3 8
35.34 odd 2 2800.2.k.l.2351.3 8
56.13 odd 2 2240.2.k.c.1791.3 8
56.27 even 2 2240.2.k.c.1791.5 8
84.83 odd 2 5040.2.d.e.4591.2 8
140.27 odd 4 2800.2.e.j.2799.6 8
140.83 odd 4 2800.2.e.i.2799.3 8
140.139 even 2 2800.2.k.l.2351.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
560.2.k.a.111.3 ✓ 8 1.1 even 1 trivial
560.2.k.a.111.4 yes 8 28.27 even 2 inner
560.2.k.a.111.5 yes 8 4.3 odd 2 inner
560.2.k.a.111.6 yes 8 7.6 odd 2 inner
2240.2.k.c.1791.3 8 56.13 odd 2
2240.2.k.c.1791.4 8 8.3 odd 2
2240.2.k.c.1791.5 8 56.27 even 2
2240.2.k.c.1791.6 8 8.5 even 2
2800.2.e.i.2799.3 8 140.83 odd 4
2800.2.e.i.2799.4 8 20.7 even 4
2800.2.e.i.2799.5 8 5.2 odd 4
2800.2.e.i.2799.6 8 35.13 even 4
2800.2.e.j.2799.3 8 35.27 even 4
2800.2.e.j.2799.4 8 5.3 odd 4
2800.2.e.j.2799.5 8 20.3 even 4
2800.2.e.j.2799.6 8 140.27 odd 4
2800.2.k.l.2351.3 8 35.34 odd 2
2800.2.k.l.2351.4 8 20.19 odd 2
2800.2.k.l.2351.5 8 5.4 even 2
2800.2.k.l.2351.6 8 140.139 even 2
5040.2.d.e.4591.2 8 84.83 odd 2
5040.2.d.e.4591.3 8 21.20 even 2
5040.2.d.e.4591.6 8 3.2 odd 2
5040.2.d.e.4591.7 8 12.11 even 2