↑

Properties

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

Related objects

Downloads

Learn more

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

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{18} q^{2} + \beta_{14} q^{4} - \beta_{9} q^{5} - \beta_{10} q^{8} + \beta_{2} q^{9} + \beta_{5} q^{10} - \beta_{16} q^{13} + \beta_{6} q^{16} + (\beta_{18} + \beta_{16} + \beta_{14} + \cdots + 1) q^{18}+ \cdots - \beta_{8} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} - 2 q^{4} + 2 q^{8} - 2 q^{9} + 2 q^{13} - 2 q^{16} + 2 q^{18} - 4 q^{25} - 2 q^{26} - 2 q^{29} + 2 q^{32} - 2 q^{36} + 2 q^{41} - 2 q^{49} + 4 q^{50} + 2 q^{52} + 2 q^{58} - 2 q^{64} + 2 q^{72}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 3 x^{18} + 9 x^{16} + 27 x^{14} + 81 x^{12} + 243 x^{10} + 729 x^{8} + 2187 x^{6} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 27 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} ) / 81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} ) / 81 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} ) / 243 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} ) / 243 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{12} ) / 729 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{13} ) / 729 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{14} ) / 2187 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{15} ) / 2187 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( \nu^{16} ) / 6561 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( \nu^{17} ) / 6561 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( \nu^{18} ) / 19683 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( \nu^{19} ) / 19683 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27\beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 81\beta_{8} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 81\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 243\beta_{10} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 243\beta_{11} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 729\beta_{12} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 729\beta_{13} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2187\beta_{14} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2187\beta_{15} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 6561\beta_{16} \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 6561\beta_{17} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 19683\beta_{18} \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 19683\beta_{19} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(5\) \(1059\)
\(\chi(n)\) \(\beta_{18}\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
0.246497 + 1.71442i
−0.246497 − 1.71442i
−0.719520 − 1.57553i
0.719520 + 1.57553i
−1.66189 + 0.487975i
1.66189 − 0.487975i
−0.719520 + 1.57553i
0.719520 − 1.57553i
1.13425 + 1.30900i
−1.13425 − 1.30900i
1.45709 + 0.936417i
−1.45709 − 0.936417i
0.246497 − 1.71442i
−0.246497 + 1.71442i
−1.66189 − 0.487975i
1.66189 + 0.487975i
1.45709 − 0.936417i
−1.45709 + 0.936417i
1.13425 − 1.30900i
−1.13425 + 1.30900i
−0.841254 − 0.540641i 0 0.415415 + 0.909632i −1.66189 − 0.487975i 0 0 0.142315 − 0.989821i −0.959493 + 0.281733i 1.13425 + 1.30900i
255.2 −0.841254 − 0.540641i 0 0.415415 + 0.909632i 1.66189 + 0.487975i 0 0 0.142315 − 0.989821i −0.959493 + 0.281733i −1.13425 − 1.30900i
399.1 0.142315 − 0.989821i 0 −0.959493 − 0.281733i −1.13425 − 1.30900i 0 0 −0.415415 + 0.909632i −0.654861 + 0.755750i −1.45709 + 0.936417i
399.2 0.142315 − 0.989821i 0 −0.959493 − 0.281733i 1.13425 + 1.30900i 0 0 −0.415415 + 0.909632i −0.654861 + 0.755750i 1.45709 − 0.936417i
487.1 −0.415415 − 0.909632i 0 −0.654861 + 0.755750i −1.45709 − 0.936417i 0 0 0.959493 + 0.281733i 0.841254 − 0.540641i −0.246497 + 1.71442i
487.2 −0.415415 − 0.909632i 0 −0.654861 + 0.755750i 1.45709 + 0.936417i 0 0 0.959493 + 0.281733i 0.841254 − 0.540641i 0.246497 − 1.71442i
647.1 0.142315 + 0.989821i 0 −0.959493 + 0.281733i −1.13425 + 1.30900i 0 0 −0.415415 − 0.909632i −0.654861 − 0.755750i −1.45709 − 0.936417i
647.2 0.142315 + 0.989821i 0 −0.959493 + 0.281733i 1.13425 − 1.30900i 0 0 −0.415415 − 0.909632i −0.654861 − 0.755750i 1.45709 + 0.936417i
699.1 0.959493 − 0.281733i 0 0.841254 − 0.540641i −0.246497 − 1.71442i 0 0 0.654861 − 0.755750i −0.142315 + 0.989821i −0.719520 − 1.57553i
699.2 0.959493 − 0.281733i 0 0.841254 − 0.540641i 0.246497 + 1.71442i 0 0 0.654861 − 0.755750i −0.142315 + 0.989821i 0.719520 + 1.57553i
795.1 0.654861 + 0.755750i 0 −0.142315 + 0.989821i −0.719520 + 1.57553i 0 0 −0.841254 + 0.540641i 0.415415 + 0.909632i −1.66189 + 0.487975i
795.2 0.654861 + 0.755750i 0 −0.142315 + 0.989821i 0.719520 − 1.57553i 0 0 −0.841254 + 0.540641i 0.415415 + 0.909632i 1.66189 − 0.487975i
863.1 −0.841254 + 0.540641i 0 0.415415 − 0.909632i −1.66189 + 0.487975i 0 0 0.142315 + 0.989821i −0.959493 − 0.281733i 1.13425 − 1.30900i
863.2 −0.841254 + 0.540641i 0 0.415415 − 0.909632i 1.66189 − 0.487975i 0 0 0.142315 + 0.989821i −0.959493 − 0.281733i −1.13425 + 1.30900i
995.1 −0.415415 + 0.909632i 0 −0.654861 − 0.755750i −1.45709 + 0.936417i 0 0 0.959493 − 0.281733i 0.841254 + 0.540641i −0.246497 − 1.71442i
995.2 −0.415415 + 0.909632i 0 −0.654861 − 0.755750i 1.45709 − 0.936417i 0 0 0.959493 − 0.281733i 0.841254 + 0.540641i 0.246497 + 1.71442i
1235.1 0.654861 − 0.755750i 0 −0.142315 − 0.989821i −0.719520 − 1.57553i 0 0 −0.841254 − 0.540641i 0.415415 − 0.909632i −1.66189 − 0.487975i
1235.2 0.654861 − 0.755750i 0 −0.142315 − 0.989821i 0.719520 + 1.57553i 0 0 −0.841254 − 0.540641i 0.415415 − 0.909632i 1.66189 + 0.487975i
1559.1 0.959493 + 0.281733i 0 0.841254 + 0.540641i −0.246497 + 1.71442i 0 0 0.654861 + 0.755750i −0.142315 − 0.989821i −0.719520 + 1.57553i
1559.2 0.959493 + 0.281733i 0 0.841254 + 0.540641i 0.246497 − 1.71442i 0 0 0.654861 + 0.755750i −0.142315 − 0.989821i 0.719520 − 1.57553i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 255.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
23.b odd 2 1 inner
23.c even 11 9 inner
23.d odd 22 9 inner
92.b even 2 1 inner
92.g odd 22 9 inner
92.h even 22 9 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2116.1.g.e 20
4.b odd 2 1 CM 2116.1.g.e 20
23.b odd 2 1 inner 2116.1.g.e 20
23.c even 11 1 2116.1.c.d ✓ 2
23.c even 11 9 inner 2116.1.g.e 20
23.d odd 22 1 2116.1.c.d ✓ 2
23.d odd 22 9 inner 2116.1.g.e 20
92.b even 2 1 inner 2116.1.g.e 20
92.g odd 22 1 2116.1.c.d ✓ 2
92.g odd 22 9 inner 2116.1.g.e 20
92.h even 22 1 2116.1.c.d ✓ 2
92.h even 22 9 inner 2116.1.g.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2116.1.c.d ✓ 2 23.c even 11 1
2116.1.c.d ✓ 2 23.d odd 22 1
2116.1.c.d ✓ 2 92.g odd 22 1
2116.1.c.d ✓ 2 92.h even 22 1
2116.1.g.e 20 1.a even 1 1 trivial
2116.1.g.e 20 4.b odd 2 1 CM
2116.1.g.e 20 23.b odd 2 1 inner
2116.1.g.e 20 23.c even 11 9 inner
2116.1.g.e 20 23.d odd 22 9 inner
2116.1.g.e 20 92.b even 2 1 inner
2116.1.g.e 20 92.g odd 22 9 inner
2116.1.g.e 20 92.h even 22 9 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2116, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{20} + 3 T_{5}^{18} + 9 T_{5}^{16} + 27 T_{5}^{14} + 81 T_{5}^{12} + 243 T_{5}^{10} + 729 T_{5}^{8} + \cdots + 59049 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + 3 T^{18} + \cdots + 59049 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} \) Copy content Toggle raw display
$23$ \( T^{20} \) Copy content Toggle raw display
$29$ \( (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} \) Copy content Toggle raw display
$37$ \( T^{20} \) Copy content Toggle raw display
$41$ \( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} \) Copy content Toggle raw display
$47$ \( T^{20} \) Copy content Toggle raw display
$53$ \( T^{20} + 3 T^{18} + \cdots + 59049 \) Copy content Toggle raw display
$59$ \( T^{20} \) Copy content Toggle raw display
$61$ \( T^{20} + 3 T^{18} + \cdots + 59049 \) Copy content Toggle raw display
$67$ \( T^{20} \) Copy content Toggle raw display
$71$ \( T^{20} \) Copy content Toggle raw display
$73$ \( (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} \) Copy content Toggle raw display
$83$ \( T^{20} \) Copy content Toggle raw display
$89$ \( T^{20} + 3 T^{18} + \cdots + 59049 \) Copy content Toggle raw display
$97$ \( T^{20} + 3 T^{18} + \cdots + 59049 \) Copy content Toggle raw display
show more
show less