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Properties

Label 3751.1.t.b
Level $3751$
Weight $1$
Character orbit 3751.t
Analytic conductor $1.872$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -11, -31, 341
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: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-11}, \sqrt{-31})\)
Artin image: $C_5\times D_4$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{10} q^{4} + \zeta_{10}^{2} q^{5} - \zeta_{10}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{10} q^{4} + \zeta_{10}^{2} q^{5} - \zeta_{10}^{3} q^{9} + \zeta_{10}^{2} q^{16} + 2 \zeta_{10}^{3} q^{20} + 3 \zeta_{10}^{4} q^{25} + \zeta_{10}^{3} q^{31} - \zeta_{10}^{4} q^{36} + 2 q^{45} - \zeta_{10}^{4} q^{47} - \zeta_{10}^{2} q^{49} - \zeta_{10} q^{59} + \zeta_{10}^{3} q^{64} + q^{67} - \zeta_{10}^{2} q^{71} + 2 \zeta_{10}^{4} q^{80} - \zeta_{10} q^{81} + \zeta_{10}^{3} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{4} - 2 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{4} - 2 q^{5} - q^{9} - q^{16} + 2 q^{20} - 3 q^{25} + q^{31} + q^{36} + 8 q^{45} + 2 q^{47} + q^{49} - 2 q^{59} + q^{64} + 8 q^{67} + 2 q^{71} - 2 q^{80} - q^{81} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(2421\) \(2543\)
\(\chi(n)\) \(-1\) \(-\zeta_{10}\)

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
−0.309017 − 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 − 0.587785i
0 0 −0.309017 − 0.951057i −1.61803 + 1.17557i 0 0 0 −0.809017 − 0.587785i 0
2665.1 0 0 −0.309017 + 0.951057i −1.61803 − 1.17557i 0 0 0 −0.809017 + 0.587785i 0
2913.1 0 0 0.809017 + 0.587785i 0.618034 + 1.90211i 0 0 0 0.309017 − 0.951057i 0
3657.1 0 0 0.809017 − 0.587785i 0.618034 − 1.90211i 0 0 0 0.309017 + 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
341.b even 2 1 RM by \(\Q(\sqrt{341}) \)
11.c even 5 3 inner
11.d odd 10 3 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.b 4
11.b odd 2 1 CM 3751.1.t.b 4
11.c even 5 1 3751.1.d.a ✓ 1
11.c even 5 3 inner 3751.1.t.b 4
11.d odd 10 1 3751.1.d.a ✓ 1
11.d odd 10 3 inner 3751.1.t.b 4
31.b odd 2 1 CM 3751.1.t.b 4
341.b even 2 1 RM 3751.1.t.b 4
341.t odd 10 1 3751.1.d.a ✓ 1
341.t odd 10 3 inner 3751.1.t.b 4
341.ba even 10 1 3751.1.d.a ✓ 1
341.ba even 10 3 inner 3751.1.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3751.1.d.a ✓ 1 11.c even 5 1
3751.1.d.a ✓ 1 11.d odd 10 1
3751.1.d.a ✓ 1 341.t odd 10 1
3751.1.d.a ✓ 1 341.ba even 10 1
3751.1.t.b 4 1.a even 1 1 trivial
3751.1.t.b 4 11.b odd 2 1 CM
3751.1.t.b 4 11.c even 5 3 inner
3751.1.t.b 4 11.d odd 10 3 inner
3751.1.t.b 4 31.b odd 2 1 CM
3751.1.t.b 4 341.b even 2 1 RM
3751.1.t.b 4 341.t odd 10 3 inner
3751.1.t.b 4 341.ba even 10 3 inner

Hecke kernels

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

Hecke characteristic polynomials

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