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Properties

Label 3381.1.o.c
Level $3381$
Weight $1$
Character orbit 3381.o
Analytic conductor $1.687$
Analytic rank $0$
Dimension $16$
Projective image $D_{8}$
CM discriminant -23
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3381,1,Mod(68,3381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3381.68");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3381.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.68733880771\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.18667348829703.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q + ( - \zeta_{48}^{22} + \zeta_{48}^{10}) q^{2} + \zeta_{48}^{5} q^{3} + \zeta_{48}^{8} q^{4} + (\zeta_{48}^{15} + \zeta_{48}^{3}) q^{6} - \zeta_{48}^{6} q^{8} + \zeta_{48}^{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{48}^{22} + \zeta_{48}^{10}) q^{2} + \zeta_{48}^{5} q^{3} + \zeta_{48}^{8} q^{4} + (\zeta_{48}^{15} + \zeta_{48}^{3}) q^{6} - \zeta_{48}^{6} q^{8} + \zeta_{48}^{10} q^{9} + \zeta_{48}^{13} q^{12} + (\zeta_{48}^{21} + \zeta_{48}^{3}) q^{13} - \zeta_{48}^{16} q^{16} + (\zeta_{48}^{20} + \zeta_{48}^{8}) q^{18} + \zeta_{48}^{4} q^{23} - \zeta_{48}^{8} q^{25} + (\zeta_{48}^{19} + \zeta_{48}^{13} + \cdots + \zeta_{48}) q^{26} + \cdots + ( - \zeta_{48}^{19} + \zeta_{48}^{7}) q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 8 q^{16} + 8 q^{18} - 8 q^{25} + 8 q^{39} + 16 q^{64} - 16 q^{78}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3381\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(2255\)
\(\chi(n)\) \(-\zeta_{48}^{16}\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
−0.130526 + 0.991445i
0.608761 + 0.793353i
−0.608761 − 0.793353i
0.130526 − 0.991445i
0.793353 − 0.608761i
−0.991445 − 0.130526i
0.991445 + 0.130526i
−0.793353 + 0.608761i
−0.130526 − 0.991445i
0.608761 − 0.793353i
−0.608761 + 0.793353i
0.130526 + 0.991445i
0.793353 + 0.608761i
−0.991445 + 0.130526i
0.991445 − 0.130526i
−0.793353 − 0.608761i
−1.22474 − 0.707107i −0.608761 + 0.793353i 0.500000 + 0.866025i 0 1.30656 − 0.541196i 0 0 −0.258819 − 0.965926i 0
68.2 −1.22474 − 0.707107i −0.130526 − 0.991445i 0.500000 + 0.866025i 0 −0.541196 + 1.30656i 0 0 −0.965926 + 0.258819i 0
68.3 −1.22474 − 0.707107i 0.130526 + 0.991445i 0.500000 + 0.866025i 0 0.541196 − 1.30656i 0 0 −0.965926 + 0.258819i 0
68.4 −1.22474 − 0.707107i 0.608761 − 0.793353i 0.500000 + 0.866025i 0 −1.30656 + 0.541196i 0 0 −0.258819 − 0.965926i 0
68.5 1.22474 + 0.707107i −0.991445 + 0.130526i 0.500000 + 0.866025i 0 −1.30656 − 0.541196i 0 0 0.965926 − 0.258819i 0
68.6 1.22474 + 0.707107i −0.793353 − 0.608761i 0.500000 + 0.866025i 0 −0.541196 − 1.30656i 0 0 0.258819 + 0.965926i 0
68.7 1.22474 + 0.707107i 0.793353 + 0.608761i 0.500000 + 0.866025i 0 0.541196 + 1.30656i 0 0 0.258819 + 0.965926i 0
68.8 1.22474 + 0.707107i 0.991445 − 0.130526i 0.500000 + 0.866025i 0 1.30656 + 0.541196i 0 0 0.965926 − 0.258819i 0
2138.1 −1.22474 + 0.707107i −0.608761 − 0.793353i 0.500000 − 0.866025i 0 1.30656 + 0.541196i 0 0 −0.258819 + 0.965926i 0
2138.2 −1.22474 + 0.707107i −0.130526 + 0.991445i 0.500000 − 0.866025i 0 −0.541196 − 1.30656i 0 0 −0.965926 − 0.258819i 0
2138.3 −1.22474 + 0.707107i 0.130526 − 0.991445i 0.500000 − 0.866025i 0 0.541196 + 1.30656i 0 0 −0.965926 − 0.258819i 0
2138.4 −1.22474 + 0.707107i 0.608761 + 0.793353i 0.500000 − 0.866025i 0 −1.30656 − 0.541196i 0 0 −0.258819 + 0.965926i 0
2138.5 1.22474 − 0.707107i −0.991445 − 0.130526i 0.500000 − 0.866025i 0 −1.30656 + 0.541196i 0 0 0.965926 + 0.258819i 0
2138.6 1.22474 − 0.707107i −0.793353 + 0.608761i 0.500000 − 0.866025i 0 −0.541196 + 1.30656i 0 0 0.258819 − 0.965926i 0
2138.7 1.22474 − 0.707107i 0.793353 − 0.608761i 0.500000 − 0.866025i 0 0.541196 − 1.30656i 0 0 0.258819 − 0.965926i 0
2138.8 1.22474 − 0.707107i 0.991445 + 0.130526i 0.500000 − 0.866025i 0 1.30656 − 0.541196i 0 0 0.965926 + 0.258819i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner
69.c even 2 1 inner
161.c even 2 1 inner
161.f odd 6 1 inner
161.g even 6 1 inner
483.c odd 2 1 inner
483.m even 6 1 inner
483.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3381.1.o.c 16
3.b odd 2 1 inner 3381.1.o.c 16
7.b odd 2 1 inner 3381.1.o.c 16
7.c even 3 1 3381.1.c.a ✓ 8
7.c even 3 1 inner 3381.1.o.c 16
7.d odd 6 1 3381.1.c.a ✓ 8
7.d odd 6 1 inner 3381.1.o.c 16
21.c even 2 1 inner 3381.1.o.c 16
21.g even 6 1 3381.1.c.a ✓ 8
21.g even 6 1 inner 3381.1.o.c 16
21.h odd 6 1 3381.1.c.a ✓ 8
21.h odd 6 1 inner 3381.1.o.c 16
23.b odd 2 1 CM 3381.1.o.c 16
69.c even 2 1 inner 3381.1.o.c 16
161.c even 2 1 inner 3381.1.o.c 16
161.f odd 6 1 3381.1.c.a ✓ 8
161.f odd 6 1 inner 3381.1.o.c 16
161.g even 6 1 3381.1.c.a ✓ 8
161.g even 6 1 inner 3381.1.o.c 16
483.c odd 2 1 inner 3381.1.o.c 16
483.m even 6 1 3381.1.c.a ✓ 8
483.m even 6 1 inner 3381.1.o.c 16
483.o odd 6 1 3381.1.c.a ✓ 8
483.o odd 6 1 inner 3381.1.o.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3381.1.c.a ✓ 8 7.c even 3 1
3381.1.c.a ✓ 8 7.d odd 6 1
3381.1.c.a ✓ 8 21.g even 6 1
3381.1.c.a ✓ 8 21.h odd 6 1
3381.1.c.a ✓ 8 161.f odd 6 1
3381.1.c.a ✓ 8 161.g even 6 1
3381.1.c.a ✓ 8 483.m even 6 1
3381.1.c.a ✓ 8 483.o odd 6 1
3381.1.o.c 16 1.a even 1 1 trivial
3381.1.o.c 16 3.b odd 2 1 inner
3381.1.o.c 16 7.b odd 2 1 inner
3381.1.o.c 16 7.c even 3 1 inner
3381.1.o.c 16 7.d odd 6 1 inner
3381.1.o.c 16 21.c even 2 1 inner
3381.1.o.c 16 21.g even 6 1 inner
3381.1.o.c 16 21.h odd 6 1 inner
3381.1.o.c 16 23.b odd 2 1 CM
3381.1.o.c 16 69.c even 2 1 inner
3381.1.o.c 16 161.c even 2 1 inner
3381.1.o.c 16 161.f odd 6 1 inner
3381.1.o.c 16 161.g even 6 1 inner
3381.1.o.c 16 483.c odd 2 1 inner
3381.1.o.c 16 483.m even 6 1 inner
3381.1.o.c 16 483.o odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3381, [\chi])\):

\( T_{2}^{4} - 2T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{443} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{8} + 1 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4)^{8} \) Copy content Toggle raw display
$73$ \( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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