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Properties

Label 560.2.k.a.111.1
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 111.1
Root \(-1.26217 + 1.18614i\) of defining polynomial
Character \(\chi\) \(=\) 560.111
Dual form 560.2.k.a.111.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.52434 q^{3} -1.00000i q^{5} +(-2.52434 + 0.792287i) q^{7} +3.37228 q^{9} -2.52434i q^{11} +0.372281i q^{13} +2.52434i q^{15} +2.37228i q^{17} +3.46410 q^{19} +(6.37228 - 2.00000i) q^{21} +8.51278i q^{23} -1.00000 q^{25} -0.939764 q^{27} +0.372281 q^{29} +1.87953 q^{31} +6.37228i q^{33} +(0.792287 + 2.52434i) q^{35} +10.7446 q^{37} -0.939764i q^{39} +8.74456i q^{41} +3.46410i q^{43} -3.37228i q^{45} +7.86797 q^{47} +(5.74456 - 4.00000i) q^{49} -5.98844i q^{51} -11.4891 q^{53} -2.52434 q^{55} -8.74456 q^{57} +6.63325 q^{59} +10.7446i q^{61} +(-8.51278 + 2.67181i) q^{63} +0.372281 q^{65} -6.63325i q^{67} -21.4891i q^{69} -6.63325i q^{71} +8.74456i q^{73} +2.52434 q^{75} +(2.00000 + 6.37228i) q^{77} +15.7908i q^{79} -7.74456 q^{81} -10.3923 q^{83} +2.37228 q^{85} -0.939764 q^{87} -5.48913i q^{89} +(-0.294954 - 0.939764i) q^{91} -4.74456 q^{93} -3.46410i q^{95} -15.1168i q^{97} -8.51278i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9} + 28 q^{21} - 8 q^{25} - 20 q^{29} + 40 q^{37} - 24 q^{57} - 20 q^{65} + 16 q^{77} - 16 q^{81} - 4 q^{85} + 8 q^{93}+O(q^{100}) \) 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
\(3\) −2.52434 −1.45743 −0.728714 − 0.684819i \(-0.759881\pi\)
−0.728714 + 0.684819i \(0.759881\pi\)
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) −2.52434 + 0.792287i −0.954110 + 0.299456i
\(8\) 0 0
\(9\) 3.37228 1.12409
\(10\) 0 0
\(11\) − 2.52434i − 0.761116i −0.924757 − 0.380558i \(-0.875732\pi\)
0.924757 − 0.380558i \(-0.124268\pi\)
\(12\) 0 0
\(13\) 0.372281i 0.103252i 0.998666 + 0.0516261i \(0.0164404\pi\)
−0.998666 + 0.0516261i \(0.983560\pi\)
\(14\) 0 0
\(15\) 2.52434i 0.651781i
\(16\) 0 0
\(17\) 2.37228i 0.575363i 0.957726 + 0.287681i \(0.0928844\pi\)
−0.957726 + 0.287681i \(0.907116\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 − 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) 6.37228 − 2.00000i 1.39055 − 0.436436i
\(22\) 0 0
\(23\) 8.51278i 1.77504i 0.460772 + 0.887518i \(0.347572\pi\)
−0.460772 + 0.887518i \(0.652428\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −0.939764 −0.180858
\(28\) 0 0
\(29\) 0.372281 0.0691309 0.0345655 − 0.999402i \(-0.488995\pi\)
0.0345655 + 0.999402i \(0.488995\pi\)
\(30\) 0 0
\(31\) 1.87953 0.337573 0.168787 − 0.985653i \(-0.446015\pi\)
0.168787 + 0.985653i \(0.446015\pi\)
\(32\) 0 0
\(33\) 6.37228i 1.10927i
\(34\) 0 0
\(35\) 0.792287 + 2.52434i 0.133921 + 0.426691i
\(36\) 0 0
\(37\) 10.7446 1.76640 0.883198 − 0.469001i \(-0.155386\pi\)
0.883198 + 0.469001i \(0.155386\pi\)
\(38\) 0 0
\(39\) − 0.939764i − 0.150483i
\(40\) 0 0
\(41\) 8.74456i 1.36567i 0.730572 + 0.682836i \(0.239253\pi\)
−0.730572 + 0.682836i \(0.760747\pi\)
\(42\) 0 0
\(43\) 3.46410i 0.528271i 0.964486 + 0.264135i \(0.0850865\pi\)
−0.964486 + 0.264135i \(0.914913\pi\)
\(44\) 0 0
\(45\) − 3.37228i − 0.502710i
\(46\) 0 0
\(47\) 7.86797 1.14766 0.573830 − 0.818974i \(-0.305457\pi\)
0.573830 + 0.818974i \(0.305457\pi\)
\(48\) 0 0
\(49\) 5.74456 − 4.00000i 0.820652 − 0.571429i
\(50\) 0 0
\(51\) − 5.98844i − 0.838549i
\(52\) 0 0
\(53\) −11.4891 −1.57815 −0.789076 − 0.614295i \(-0.789440\pi\)
−0.789076 + 0.614295i \(0.789440\pi\)
\(54\) 0 0
\(55\) −2.52434 −0.340382
\(56\) 0 0
\(57\) −8.74456 −1.15825
\(58\) 0 0
\(59\) 6.63325 0.863576 0.431788 − 0.901975i \(-0.357883\pi\)
0.431788 + 0.901975i \(0.357883\pi\)
\(60\) 0 0
\(61\) 10.7446i 1.37570i 0.725853 + 0.687850i \(0.241445\pi\)
−0.725853 + 0.687850i \(0.758555\pi\)
\(62\) 0 0
\(63\) −8.51278 + 2.67181i −1.07251 + 0.336617i
\(64\) 0 0
\(65\) 0.372281 0.0461758
\(66\) 0 0
\(67\) − 6.63325i − 0.810380i −0.914232 − 0.405190i \(-0.867205\pi\)
0.914232 − 0.405190i \(-0.132795\pi\)
\(68\) 0 0
\(69\) − 21.4891i − 2.58699i
\(70\) 0 0
\(71\) − 6.63325i − 0.787222i −0.919277 − 0.393611i \(-0.871226\pi\)
0.919277 − 0.393611i \(-0.128774\pi\)
\(72\) 0 0
\(73\) 8.74456i 1.02347i 0.859142 + 0.511737i \(0.170998\pi\)
−0.859142 + 0.511737i \(0.829002\pi\)
\(74\) 0 0
\(75\) 2.52434 0.291485
\(76\) 0 0
\(77\) 2.00000 + 6.37228i 0.227921 + 0.726189i
\(78\) 0 0
\(79\) 15.7908i 1.77661i 0.459256 + 0.888304i \(0.348116\pi\)
−0.459256 + 0.888304i \(0.651884\pi\)
\(80\) 0 0
\(81\) −7.74456 −0.860507
\(82\) 0 0
\(83\) −10.3923 −1.14070 −0.570352 − 0.821401i \(-0.693193\pi\)
−0.570352 + 0.821401i \(0.693193\pi\)
\(84\) 0 0
\(85\) 2.37228 0.257310
\(86\) 0 0
\(87\) −0.939764 −0.100753
\(88\) 0 0
\(89\) − 5.48913i − 0.581846i −0.956746 − 0.290923i \(-0.906038\pi\)
0.956746 − 0.290923i \(-0.0939624\pi\)
\(90\) 0 0
\(91\) −0.294954 − 0.939764i −0.0309195 − 0.0985140i
\(92\) 0 0
\(93\) −4.74456 −0.491988
\(94\) 0 0
\(95\) − 3.46410i − 0.355409i
\(96\) 0 0
\(97\) − 15.1168i − 1.53488i −0.641119 − 0.767441i \(-0.721530\pi\)
0.641119 − 0.767441i \(-0.278470\pi\)
\(98\) 0 0
\(99\) − 8.51278i − 0.855566i
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.111.1 ✓ 8
3.2 odd 2 5040.2.d.e.4591.5 8
4.3 odd 2 inner 560.2.k.a.111.7 yes 8
5.2 odd 4 2800.2.e.i.2799.7 8
5.3 odd 4 2800.2.e.j.2799.2 8
5.4 even 2 2800.2.k.l.2351.7 8
7.6 odd 2 inner 560.2.k.a.111.8 yes 8
8.3 odd 2 2240.2.k.c.1791.2 8
8.5 even 2 2240.2.k.c.1791.8 8
12.11 even 2 5040.2.d.e.4591.8 8
20.3 even 4 2800.2.e.j.2799.7 8
20.7 even 4 2800.2.e.i.2799.2 8
20.19 odd 2 2800.2.k.l.2351.2 8
21.20 even 2 5040.2.d.e.4591.4 8
28.27 even 2 inner 560.2.k.a.111.2 yes 8
35.13 even 4 2800.2.e.i.2799.8 8
35.27 even 4 2800.2.e.j.2799.1 8
35.34 odd 2 2800.2.k.l.2351.1 8
56.13 odd 2 2240.2.k.c.1791.1 8
56.27 even 2 2240.2.k.c.1791.7 8
84.83 odd 2 5040.2.d.e.4591.1 8
140.27 odd 4 2800.2.e.j.2799.8 8
140.83 odd 4 2800.2.e.i.2799.1 8
140.139 even 2 2800.2.k.l.2351.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
560.2.k.a.111.1 ✓ 8 1.1 even 1 trivial
560.2.k.a.111.2 yes 8 28.27 even 2 inner
560.2.k.a.111.7 yes 8 4.3 odd 2 inner
560.2.k.a.111.8 yes 8 7.6 odd 2 inner
2240.2.k.c.1791.1 8 56.13 odd 2
2240.2.k.c.1791.2 8 8.3 odd 2
2240.2.k.c.1791.7 8 56.27 even 2
2240.2.k.c.1791.8 8 8.5 even 2
2800.2.e.i.2799.1 8 140.83 odd 4
2800.2.e.i.2799.2 8 20.7 even 4
2800.2.e.i.2799.7 8 5.2 odd 4
2800.2.e.i.2799.8 8 35.13 even 4
2800.2.e.j.2799.1 8 35.27 even 4
2800.2.e.j.2799.2 8 5.3 odd 4
2800.2.e.j.2799.7 8 20.3 even 4
2800.2.e.j.2799.8 8 140.27 odd 4
2800.2.k.l.2351.1 8 35.34 odd 2
2800.2.k.l.2351.2 8 20.19 odd 2
2800.2.k.l.2351.7 8 5.4 even 2
2800.2.k.l.2351.8 8 140.139 even 2
5040.2.d.e.4591.1 8 84.83 odd 2
5040.2.d.e.4591.4 8 21.20 even 2
5040.2.d.e.4591.5 8 3.2 odd 2
5040.2.d.e.4591.8 8 12.11 even 2