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Properties

Label 2116.1.g.d
Level $2116$
Weight $1$
Character orbit 2116.g
Analytic conductor $1.056$
Analytic rank $0$
Dimension $20$
Projective image $D_{6}$
CM discriminant -23
Inner twists $40$

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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,-1,0,1,0] 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: \(\Q(\zeta_{33})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.33856.2

$q$-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)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{66}^{19} q^{2} + (\zeta_{66}^{31} + \zeta_{66}^{20}) q^{3} - \zeta_{66}^{5} q^{4} + ( - \zeta_{66}^{17} - \zeta_{66}^{6}) q^{6} - \zeta_{66}^{24} q^{8} + ( - \zeta_{66}^{29} + \cdots - \zeta_{66}^{7}) q^{9} + \cdots - \zeta_{66}^{28} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} + q^{4} + 3 q^{6} + 2 q^{8} + 4 q^{9} + 3 q^{12} + 2 q^{13} + q^{16} + 2 q^{18} + 2 q^{25} + q^{26} - 2 q^{29} - q^{32} - 2 q^{36} + 2 q^{41} - 3 q^{48} - 2 q^{49} + q^{50} - q^{52}+ \cdots - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(5\) \(1059\)
\(\chi(n)\) \(\zeta_{66}^{30}\) \(-1\)

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
−0.327068 + 0.945001i
0.981929 − 0.189251i
−0.888835 − 0.458227i
0.0475819 + 0.998867i
0.928368 − 0.371662i
−0.786053 − 0.618159i
−0.888835 + 0.458227i
0.0475819 − 0.998867i
0.580057 − 0.814576i
−0.995472 − 0.0950560i
0.235759 − 0.971812i
0.723734 + 0.690079i
−0.327068 − 0.945001i
0.981929 + 0.189251i
0.928368 + 0.371662i
−0.786053 + 0.618159i
0.235759 + 0.971812i
0.723734 − 0.690079i
0.580057 + 0.814576i
−0.995472 + 0.0950560i
−0.0475819 + 0.998867i 1.71442 − 0.246497i −0.995472 − 0.0950560i 0 0.164642 + 1.72421i 0 0.142315 − 0.989821i 1.91899 − 0.563465i 0
255.2 0.888835 − 0.458227i −1.71442 + 0.246497i 0.580057 − 0.814576i 0 −1.41089 + 1.00469i 0 0.142315 − 0.989821i 1.91899 − 0.563465i 0
399.1 −0.928368 + 0.371662i −1.57553 + 0.719520i 0.723734 − 0.690079i 0 1.19525 − 1.25354i 0 −0.415415 + 0.909632i 1.30972 − 1.51150i 0
399.2 0.786053 + 0.618159i 1.57553 − 0.719520i 0.235759 + 0.971812i 0 1.68323 + 0.408346i 0 −0.415415 + 0.909632i 1.30972 − 1.51150i 0
487.1 −0.580057 + 0.814576i −0.487975 − 1.66189i −0.327068 − 0.945001i 0 1.63679 + 0.566498i 0 0.959493 + 0.281733i −1.68251 + 1.08128i 0
487.2 0.995472 + 0.0950560i 0.487975 + 1.66189i 0.981929 + 0.189251i 0 0.327793 + 1.70075i 0 0.959493 + 0.281733i −1.68251 + 1.08128i 0
647.1 −0.928368 − 0.371662i −1.57553 − 0.719520i 0.723734 + 0.690079i 0 1.19525 + 1.25354i 0 −0.415415 − 0.909632i 1.30972 + 1.51150i 0
647.2 0.786053 − 0.618159i 1.57553 + 0.719520i 0.235759 − 0.971812i 0 1.68323 − 0.408346i 0 −0.415415 − 0.909632i 1.30972 + 1.51150i 0
699.1 −0.723734 − 0.690079i 1.30900 − 1.13425i 0.0475819 + 0.998867i 0 −1.73009 − 0.0824143i 0 0.654861 − 0.755750i 0.284630 − 1.97964i 0
699.2 −0.235759 + 0.971812i −1.30900 + 1.13425i −0.888835 − 0.458227i 0 −0.793672 − 1.53951i 0 0.654861 − 0.755750i 0.284630 − 1.97964i 0
795.1 −0.981929 + 0.189251i 0.936417 − 1.45709i 0.928368 − 0.371662i 0 −0.643738 + 1.60798i 0 −0.841254 + 0.540641i −0.830830 − 1.81926i 0
795.2 0.327068 − 0.945001i −0.936417 + 1.45709i −0.786053 − 0.618159i 0 1.07068 + 1.36148i 0 −0.841254 + 0.540641i −0.830830 − 1.81926i 0
863.1 −0.0475819 − 0.998867i 1.71442 + 0.246497i −0.995472 + 0.0950560i 0 0.164642 − 1.72421i 0 0.142315 + 0.989821i 1.91899 + 0.563465i 0
863.2 0.888835 + 0.458227i −1.71442 − 0.246497i 0.580057 + 0.814576i 0 −1.41089 − 1.00469i 0 0.142315 + 0.989821i 1.91899 + 0.563465i 0
995.1 −0.580057 − 0.814576i −0.487975 + 1.66189i −0.327068 + 0.945001i 0 1.63679 − 0.566498i 0 0.959493 − 0.281733i −1.68251 − 1.08128i 0
995.2 0.995472 − 0.0950560i 0.487975 − 1.66189i 0.981929 − 0.189251i 0 0.327793 − 1.70075i 0 0.959493 − 0.281733i −1.68251 − 1.08128i 0
1235.1 −0.981929 − 0.189251i 0.936417 + 1.45709i 0.928368 + 0.371662i 0 −0.643738 − 1.60798i 0 −0.841254 − 0.540641i −0.830830 + 1.81926i 0
1235.2 0.327068 + 0.945001i −0.936417 − 1.45709i −0.786053 + 0.618159i 0 1.07068 − 1.36148i 0 −0.841254 − 0.540641i −0.830830 + 1.81926i 0
1559.1 −0.723734 + 0.690079i 1.30900 + 1.13425i 0.0475819 − 0.998867i 0 −1.73009 + 0.0824143i 0 0.654861 + 0.755750i 0.284630 + 1.97964i 0
1559.2 −0.235759 − 0.971812i −1.30900 − 1.13425i −0.888835 + 0.458227i 0 −0.793672 + 1.53951i 0 0.654861 + 0.755750i 0.284630 + 1.97964i 0
\(n\): 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
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
4.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.d 20
4.b odd 2 1 inner 2116.1.g.d 20
23.b odd 2 1 CM 2116.1.g.d 20
23.c even 11 1 2116.1.c.e ✓ 2
23.c even 11 9 inner 2116.1.g.d 20
23.d odd 22 1 2116.1.c.e ✓ 2
23.d odd 22 9 inner 2116.1.g.d 20
92.b even 2 1 inner 2116.1.g.d 20
92.g odd 22 1 2116.1.c.e ✓ 2
92.g odd 22 9 inner 2116.1.g.d 20
92.h even 22 1 2116.1.c.e ✓ 2
92.h even 22 9 inner 2116.1.g.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2116.1.c.e ✓ 2 23.c even 11 1
2116.1.c.e ✓ 2 23.d odd 22 1
2116.1.c.e ✓ 2 92.g odd 22 1
2116.1.c.e ✓ 2 92.h even 22 1
2116.1.g.d 20 1.a even 1 1 trivial
2116.1.g.d 20 4.b odd 2 1 inner
2116.1.g.d 20 23.b odd 2 1 CM
2116.1.g.d 20 23.c even 11 9 inner
2116.1.g.d 20 23.d odd 22 9 inner
2116.1.g.d 20 92.b even 2 1 inner
2116.1.g.d 20 92.g odd 22 9 inner
2116.1.g.d 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 \(S_{1}^{\mathrm{new}}(2116, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + T^{19} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{20} - 3 T^{18} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( T^{20} \) 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
$31$ \( T^{20} - 3 T^{18} + \cdots + 59049 \) 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
$47$ \( (T^{2} + 3)^{10} \) Copy content Toggle raw display
$53$ \( T^{20} \) Copy content Toggle raw display
$59$ \( T^{20} \) Copy content Toggle raw display
$61$ \( T^{20} \) Copy content Toggle raw display
$67$ \( T^{20} \) Copy content Toggle raw display
$71$ \( T^{20} - 3 T^{18} + \cdots + 59049 \) 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
$89$ \( T^{20} \) Copy content Toggle raw display
$97$ \( T^{20} \) Copy content Toggle raw display
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