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Properties

Label 3751.1.t.d
Level $3751$
Weight $1$
Character orbit 3751.t
Analytic conductor $1.872$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -31
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3751,1,Mod(2138,3751)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3751, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3751.2138");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3751 = 11^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3751.t (of order \(10\), degree \(4\), 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: \(1.87199286239\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.324000000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.1279091.1

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + 2 \beta_{4} q^{4} + (\beta_{6} + \beta_{4} + \beta_{2} + 1) q^{5} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{7} - \beta_1 q^{8} + \beta_{2} q^{9} + \beta_{5} q^{10} + 3 \beta_{6} q^{14}+ \cdots - 2 \beta_{5} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4} + 2 q^{5} - 2 q^{9} - 6 q^{14} - 2 q^{16} + 4 q^{20} + 2 q^{31} - 4 q^{36} + 6 q^{38} - 8 q^{45} + 4 q^{47} - 4 q^{49} + 24 q^{56} + 2 q^{59} + 2 q^{64} + 16 q^{67} + 6 q^{70} - 2 q^{71} + 2 q^{80}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) : 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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3751\mathbb{Z}\right)^\times\).

\(n\) \(2421\) \(2543\)
\(\chi(n)\) \(-1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2138.1
−0.535233 + 1.64728i
0.535233 − 1.64728i
−0.535233 − 1.64728i
0.535233 + 1.64728i
−1.40126 + 1.01807i
1.40126 − 1.01807i
−1.40126 − 1.01807i
1.40126 + 1.01807i
−1.40126 − 1.01807i 0 0.618034 + 1.90211i 0.809017 − 0.587785i 0 0.535233 + 1.64728i 0.535233 − 1.64728i −0.809017 − 0.587785i −1.73205
2138.2 1.40126 + 1.01807i 0 0.618034 + 1.90211i 0.809017 − 0.587785i 0 −0.535233 − 1.64728i −0.535233 + 1.64728i −0.809017 − 0.587785i 1.73205
2665.1 −1.40126 + 1.01807i 0 0.618034 − 1.90211i 0.809017 + 0.587785i 0 0.535233 − 1.64728i 0.535233 + 1.64728i −0.809017 + 0.587785i −1.73205
2665.2 1.40126 − 1.01807i 0 0.618034 − 1.90211i 0.809017 + 0.587785i 0 −0.535233 + 1.64728i −0.535233 − 1.64728i −0.809017 + 0.587785i 1.73205
2913.1 −0.535233 + 1.64728i 0 −1.61803 − 1.17557i −0.309017 − 0.951057i 0 1.40126 + 1.01807i 1.40126 − 1.01807i 0.309017 − 0.951057i 1.73205
2913.2 0.535233 − 1.64728i 0 −1.61803 − 1.17557i −0.309017 − 0.951057i 0 −1.40126 − 1.01807i −1.40126 + 1.01807i 0.309017 − 0.951057i −1.73205
3657.1 −0.535233 − 1.64728i 0 −1.61803 + 1.17557i −0.309017 + 0.951057i 0 1.40126 − 1.01807i 1.40126 + 1.01807i 0.309017 + 0.951057i 1.73205
3657.2 0.535233 + 1.64728i 0 −1.61803 + 1.17557i −0.309017 + 0.951057i 0 −1.40126 + 1.01807i −1.40126 − 1.01807i 0.309017 + 0.951057i −1.73205
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2138.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
341.b even 2 1 inner
341.t odd 10 3 inner
341.ba even 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3751.1.t.d 8
11.b odd 2 1 inner 3751.1.t.d 8
11.c even 5 1 3751.1.d.c ✓ 2
11.c even 5 3 inner 3751.1.t.d 8
11.d odd 10 1 3751.1.d.c ✓ 2
11.d odd 10 3 inner 3751.1.t.d 8
31.b odd 2 1 CM 3751.1.t.d 8
341.b even 2 1 inner 3751.1.t.d 8
341.t odd 10 1 3751.1.d.c ✓ 2
341.t odd 10 3 inner 3751.1.t.d 8
341.ba even 10 1 3751.1.d.c ✓ 2
341.ba even 10 3 inner 3751.1.t.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3751.1.d.c ✓ 2 11.c even 5 1
3751.1.d.c ✓ 2 11.d odd 10 1
3751.1.d.c ✓ 2 341.t odd 10 1
3751.1.d.c ✓ 2 341.ba even 10 1
3751.1.t.d 8 1.a even 1 1 trivial
3751.1.t.d 8 11.b odd 2 1 inner
3751.1.t.d 8 11.c even 5 3 inner
3751.1.t.d 8 11.d odd 10 3 inner
3751.1.t.d 8 31.b odd 2 1 CM
3751.1.t.d 8 341.b even 2 1 inner
3751.1.t.d 8 341.t odd 10 3 inner
3751.1.t.d 8 341.ba even 10 3 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 3T_{2}^{6} + 9T_{2}^{4} + 27T_{2}^{2} + 81 \) acting on \(S_{1}^{\mathrm{new}}(3751, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T - 2)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
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