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Properties

Label 2116.1.g.e.1559.1
Level $2116$
Weight $1$
Character 2116.1559
Analytic conductor $1.056$
Analytic rank $0$
Dimension $20$
Projective image $D_{6}$
CM discriminant -4
Inner twists $40$

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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] = [2116,1,Mod(255,2116)] 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("2116.255"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2116, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2116 = 2^{2} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2116.g (of order \(22\), degree \(10\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,2,0,-2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.05602156673\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: 20.0.344255425354822086003595935744.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 3 x^{18} + 9 x^{16} + 27 x^{14} + 81 x^{12} + 243 x^{10} + 729 x^{8} + 2187 x^{6} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.102981488.1

Embedding invariants

Embedding label 1559.1
Root \(1.13425 - 1.30900i\) of defining polynomial
Character \(\chi\) \(=\) 2116.1559
Dual form 2116.1.g.e.699.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.959493 + 0.281733i) q^{2} +(0.841254 + 0.540641i) q^{4} +(-0.246497 + 1.71442i) q^{5} +(0.654861 + 0.755750i) q^{8} +(-0.142315 - 0.989821i) q^{9} +(-0.719520 + 1.57553i) q^{10} +(-0.415415 + 0.909632i) q^{13} +(0.415415 + 0.909632i) q^{16} +(0.142315 - 0.989821i) q^{18} +(-1.13425 + 1.30900i) q^{20} +(-1.91899 - 0.563465i) q^{25} +(-0.654861 + 0.755750i) q^{26} +(0.841254 - 0.540641i) q^{29} +(0.142315 + 0.989821i) q^{32} +(0.415415 - 0.909632i) q^{36} +(-1.45709 + 0.936417i) q^{40} +(0.142315 - 0.989821i) q^{41} +1.73205 q^{45} +(-0.654861 + 0.755750i) q^{49} +(-1.68251 - 1.08128i) q^{50} +(-0.841254 + 0.540641i) q^{52} +(-0.719520 - 1.57553i) q^{53} +(0.959493 - 0.281733i) q^{58} +(1.13425 + 1.30900i) q^{61} +(-0.142315 + 0.989821i) q^{64} +(-1.45709 - 0.936417i) q^{65} +(0.654861 - 0.755750i) q^{72} +(0.841254 + 0.540641i) q^{73} +(-1.66189 + 0.487975i) q^{80} +(-0.959493 + 0.281733i) q^{81} +(0.415415 - 0.909632i) q^{82} +(1.13425 - 1.30900i) q^{89} +(1.66189 + 0.487975i) q^{90} +(0.246497 - 1.71442i) q^{97} +(-0.841254 + 0.540641i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} - 2 q^{4} + 2 q^{8} - 2 q^{9} + 2 q^{13} - 2 q^{16} + 2 q^{18} - 4 q^{25} - 2 q^{26} - 2 q^{29} + 2 q^{32} - 2 q^{36} + 2 q^{41} - 2 q^{49} + 4 q^{50} + 2 q^{52} + 2 q^{58} - 2 q^{64} + 2 q^{72}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2116\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1059\)
\(\chi(n)\) \(e\left(\frac{6}{11}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(3\) 0 0 0.654861 − 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(4\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(5\) −0.246497 + 1.71442i −0.246497 + 1.71442i 0.371662 + 0.928368i \(0.378788\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(6\) 0 0
\(7\) 0 0 −0.415415 − 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(8\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(9\) −0.142315 − 0.989821i −0.142315 − 0.989821i
\(10\) −0.719520 + 1.57553i −0.719520 + 1.57553i
\(11\) 0 0 0.959493 − 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(12\) 0 0
\(13\) −0.415415 + 0.909632i −0.415415 + 0.909632i 0.580057 + 0.814576i \(0.303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(17\) 0 0 −0.540641 − 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(18\) 0.142315 − 0.989821i 0.142315 − 0.989821i
\(19\) 0 0 −0.841254 − 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(20\) −1.13425 + 1.30900i −1.13425 + 1.30900i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −1.91899 − 0.563465i −1.91899 − 0.563465i
\(26\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(27\) 0 0
\(28\) 0 0
\(29\) 0.841254 − 0.540641i 0.841254 − 0.540641i −0.0475819 − 0.998867i \(-0.515152\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(30\) 0 0
\(31\) 0 0 −0.654861 − 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(32\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.415415 − 0.909632i 0.415415 − 0.909632i
\(37\) 0 0 0.989821 − 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.45709 + 0.936417i −1.45709 + 0.936417i
\(41\) 0.142315 − 0.989821i 0.142315 − 0.989821i −0.786053 − 0.618159i \(-0.787879\pi\)
0.928368 − 0.371662i \(-0.121212\pi\)
\(42\) 0 0
\(43\) 0 0 0.654861 − 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(44\) 0 0
\(45\) 1.73205 1.73205
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(50\) −1.68251 − 1.08128i −1.68251 − 1.08128i
\(51\) 0 0
\(52\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(53\) −0.719520 − 1.57553i −0.719520 − 1.57553i −0.814576 − 0.580057i \(-0.803030\pi\)
0.0950560 − 0.995472i \(-0.469697\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0.959493 − 0.281733i 0.959493 − 0.281733i
\(59\) 0 0 0.415415 − 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(60\) 0 0
\(61\) 1.13425 + 1.30900i 1.13425 + 1.30900i 0.945001 + 0.327068i \(0.106061\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(65\) −1.45709 − 0.936417i −1.45709 − 0.936417i
\(66\) 0 0
\(67\) 0 0 −0.959493 − 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.959493 − 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(72\) 0.654861 − 0.755750i 0.654861 − 0.755750i
\(73\) 0.841254 + 0.540641i 0.841254 + 0.540641i 0.888835 − 0.458227i \(-0.151515\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.415415 − 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(80\) −1.66189 + 0.487975i −1.66189 + 0.487975i
\(81\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(82\) 0.415415 − 0.909632i 0.415415 − 0.909632i
\(83\) 0 0 −0.142315 − 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.13425 − 1.30900i 1.13425 − 1.30900i 0.189251 − 0.981929i \(-0.439394\pi\)
0.945001 − 0.327068i \(-0.106061\pi\)
\(90\) 1.66189 + 0.487975i 1.66189 + 0.487975i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.246497 − 1.71442i 0.246497 − 1.71442i −0.371662 − 0.928368i \(-0.621212\pi\)
0.618159 − 0.786053i \(-0.287879\pi\)
\(98\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(99\) 0 0
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 2116.1.g.e.1559.1 20
4.3 odd 2 CM 2116.1.g.e.1559.1 20
23.2 even 11 inner 2116.1.g.e.647.1 20
23.3 even 11 2116.1.c.d.1059.2 yes 2
23.4 even 11 inner 2116.1.g.e.863.1 20
23.5 odd 22 inner 2116.1.g.e.795.1 20
23.6 even 11 inner 2116.1.g.e.487.2 20
23.7 odd 22 inner 2116.1.g.e.399.2 20
23.8 even 11 inner 2116.1.g.e.255.1 20
23.9 even 11 inner 2116.1.g.e.699.1 20
23.10 odd 22 inner 2116.1.g.e.995.1 20
23.11 odd 22 inner 2116.1.g.e.1235.1 20
23.12 even 11 inner 2116.1.g.e.1235.2 20
23.13 even 11 inner 2116.1.g.e.995.2 20
23.14 odd 22 inner 2116.1.g.e.699.2 20
23.15 odd 22 inner 2116.1.g.e.255.2 20
23.16 even 11 inner 2116.1.g.e.399.1 20
23.17 odd 22 inner 2116.1.g.e.487.1 20
23.18 even 11 inner 2116.1.g.e.795.2 20
23.19 odd 22 inner 2116.1.g.e.863.2 20
23.20 odd 22 2116.1.c.d.1059.1 ✓ 2
23.21 odd 22 inner 2116.1.g.e.647.2 20
23.22 odd 2 inner 2116.1.g.e.1559.2 20
92.3 odd 22 2116.1.c.d.1059.2 yes 2
92.7 even 22 inner 2116.1.g.e.399.2 20
92.11 even 22 inner 2116.1.g.e.1235.1 20
92.15 even 22 inner 2116.1.g.e.255.2 20
92.19 even 22 inner 2116.1.g.e.863.2 20
92.27 odd 22 inner 2116.1.g.e.863.1 20
92.31 odd 22 inner 2116.1.g.e.255.1 20
92.35 odd 22 inner 2116.1.g.e.1235.2 20
92.39 odd 22 inner 2116.1.g.e.399.1 20
92.43 even 22 2116.1.c.d.1059.1 ✓ 2
92.51 even 22 inner 2116.1.g.e.795.1 20
92.55 odd 22 inner 2116.1.g.e.699.1 20
92.59 odd 22 inner 2116.1.g.e.995.2 20
92.63 even 22 inner 2116.1.g.e.487.1 20
92.67 even 22 inner 2116.1.g.e.647.2 20
92.71 odd 22 inner 2116.1.g.e.647.1 20
92.75 odd 22 inner 2116.1.g.e.487.2 20
92.79 even 22 inner 2116.1.g.e.995.1 20
92.83 even 22 inner 2116.1.g.e.699.2 20
92.87 odd 22 inner 2116.1.g.e.795.2 20
92.91 even 2 inner 2116.1.g.e.1559.2 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2116.1.c.d.1059.1 ✓ 2 23.20 odd 22
2116.1.c.d.1059.1 ✓ 2 92.43 even 22
2116.1.c.d.1059.2 yes 2 23.3 even 11
2116.1.c.d.1059.2 yes 2 92.3 odd 22
2116.1.g.e.255.1 20 23.8 even 11 inner
2116.1.g.e.255.1 20 92.31 odd 22 inner
2116.1.g.e.255.2 20 23.15 odd 22 inner
2116.1.g.e.255.2 20 92.15 even 22 inner
2116.1.g.e.399.1 20 23.16 even 11 inner
2116.1.g.e.399.1 20 92.39 odd 22 inner
2116.1.g.e.399.2 20 23.7 odd 22 inner
2116.1.g.e.399.2 20 92.7 even 22 inner
2116.1.g.e.487.1 20 23.17 odd 22 inner
2116.1.g.e.487.1 20 92.63 even 22 inner
2116.1.g.e.487.2 20 23.6 even 11 inner
2116.1.g.e.487.2 20 92.75 odd 22 inner
2116.1.g.e.647.1 20 23.2 even 11 inner
2116.1.g.e.647.1 20 92.71 odd 22 inner
2116.1.g.e.647.2 20 23.21 odd 22 inner
2116.1.g.e.647.2 20 92.67 even 22 inner
2116.1.g.e.699.1 20 23.9 even 11 inner
2116.1.g.e.699.1 20 92.55 odd 22 inner
2116.1.g.e.699.2 20 23.14 odd 22 inner
2116.1.g.e.699.2 20 92.83 even 22 inner
2116.1.g.e.795.1 20 23.5 odd 22 inner
2116.1.g.e.795.1 20 92.51 even 22 inner
2116.1.g.e.795.2 20 23.18 even 11 inner
2116.1.g.e.795.2 20 92.87 odd 22 inner
2116.1.g.e.863.1 20 23.4 even 11 inner
2116.1.g.e.863.1 20 92.27 odd 22 inner
2116.1.g.e.863.2 20 23.19 odd 22 inner
2116.1.g.e.863.2 20 92.19 even 22 inner
2116.1.g.e.995.1 20 23.10 odd 22 inner
2116.1.g.e.995.1 20 92.79 even 22 inner
2116.1.g.e.995.2 20 23.13 even 11 inner
2116.1.g.e.995.2 20 92.59 odd 22 inner
2116.1.g.e.1235.1 20 23.11 odd 22 inner
2116.1.g.e.1235.1 20 92.11 even 22 inner
2116.1.g.e.1235.2 20 23.12 even 11 inner
2116.1.g.e.1235.2 20 92.35 odd 22 inner
2116.1.g.e.1559.1 20 1.1 even 1 trivial
2116.1.g.e.1559.1 20 4.3 odd 2 CM
2116.1.g.e.1559.2 20 23.22 odd 2 inner
2116.1.g.e.1559.2 20 92.91 even 2 inner