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Properties

Label 3751.1.o.a
Level $3751$
Weight $1$
Character orbit 3751.o
Analytic conductor $1.872$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3751,1,Mod(717,3751)]
 
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([1, 6]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3751.717");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3751 = 11^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3751.o (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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 341)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.111746041.1

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{3} + q^{4} + (\zeta_{10}^{2} + 1) q^{5} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{3} + q^{4} + (\zeta_{10}^{2} + 1) q^{5} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{9} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{12} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \cdots + 1) q^{15}+ \cdots + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{4} + 3 q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 4 q^{4} + 3 q^{5} - 3 q^{9} - 2 q^{12} + q^{15} + 4 q^{16} + 3 q^{20} + 3 q^{23} + 2 q^{25} + q^{27} - q^{31} - 3 q^{36} - 2 q^{37} - q^{45} + 3 q^{47} - 2 q^{48} + 4 q^{49} - 2 q^{53} + 3 q^{59} + q^{60} + 4 q^{64} - 2 q^{67} + q^{69} - 2 q^{71} - q^{75} + 3 q^{80} - 2 q^{89} + 3 q^{92} - 2 q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(2421\) \(2543\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
717.1
−0.309017 − 0.951057i
0.809017 − 0.587785i
−0.309017 + 0.951057i
0.809017 + 0.587785i
0 −0.500000 − 1.53884i 1.00000 0.190983 + 0.587785i 0 0 0 −1.30902 + 0.951057i 0
1008.1 0 −0.500000 + 0.363271i 1.00000 1.30902 − 0.951057i 0 0 0 −0.190983 + 0.587785i 0
1062.1 0 −0.500000 + 1.53884i 1.00000 0.190983 − 0.587785i 0 0 0 −1.30902 − 0.951057i 0
2296.1 0 −0.500000 − 0.363271i 1.00000 1.30902 + 0.951057i 0 0 0 −0.190983 − 0.587785i 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 \(\Q(\sqrt{-11}) \)
341.k even 5 1 inner
341.o odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3751.1.o.a 4
11.b odd 2 1 CM 3751.1.o.a 4
11.c even 5 1 341.1.z.a ✓ 4
11.c even 5 1 3751.1.x.a 4
11.c even 5 1 3751.1.y.a 4
11.c even 5 1 3751.1.be.a 4
11.d odd 10 1 341.1.z.a ✓ 4
11.d odd 10 1 3751.1.x.a 4
11.d odd 10 1 3751.1.y.a 4
11.d odd 10 1 3751.1.be.a 4
31.d even 5 1 3751.1.x.a 4
33.f even 10 1 3069.1.dg.a 4
33.h odd 10 1 3069.1.dg.a 4
341.f even 5 1 3751.1.be.a 4
341.g even 5 1 3751.1.y.a 4
341.j even 5 1 341.1.z.a ✓ 4
341.k even 5 1 inner 3751.1.o.a 4
341.o odd 10 1 inner 3751.1.o.a 4
341.x odd 10 1 3751.1.y.a 4
341.y odd 10 1 3751.1.be.a 4
341.z odd 10 1 3751.1.x.a 4
341.be odd 10 1 341.1.z.a ✓ 4
1023.bl even 10 1 3069.1.dg.a 4
1023.cb odd 10 1 3069.1.dg.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
341.1.z.a ✓ 4 11.c even 5 1
341.1.z.a ✓ 4 11.d odd 10 1
341.1.z.a ✓ 4 341.j even 5 1
341.1.z.a ✓ 4 341.be odd 10 1
3069.1.dg.a 4 33.f even 10 1
3069.1.dg.a 4 33.h odd 10 1
3069.1.dg.a 4 1023.bl even 10 1
3069.1.dg.a 4 1023.cb odd 10 1
3751.1.o.a 4 1.a even 1 1 trivial
3751.1.o.a 4 11.b odd 2 1 CM
3751.1.o.a 4 341.k even 5 1 inner
3751.1.o.a 4 341.o odd 10 1 inner
3751.1.x.a 4 11.c even 5 1
3751.1.x.a 4 11.d odd 10 1
3751.1.x.a 4 31.d even 5 1
3751.1.x.a 4 341.z odd 10 1
3751.1.y.a 4 11.c even 5 1
3751.1.y.a 4 11.d odd 10 1
3751.1.y.a 4 341.g even 5 1
3751.1.y.a 4 341.x odd 10 1
3751.1.be.a 4 11.c even 5 1
3751.1.be.a 4 11.d odd 10 1
3751.1.be.a 4 341.f even 5 1
3751.1.be.a 4 341.y odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3751, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + \cdots + 1 \) 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
$23$ \( T^{4} - 3 T^{3} + \cdots + 1 \) 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
$37$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + T - 1)^{2} \) 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
$89$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
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