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Properties

Label 3479.1.g.c
Level $3479$
Weight $1$
Character orbit 3479.g
Analytic conductor $1.736$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -71
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3479,1,Mod(851,3479)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3479, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3479.851");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3479 = 7^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3479.g (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: \(1.73624717895\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.1729063.1
Artin image: $C_3\times D_8$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{2} + 2) q^{2} + ( - \beta_{3} - \beta_1) q^{3} + 3 \beta_{2} q^{4} + \beta_1 q^{5} - 2 \beta_{3} q^{6} - 4 q^{8} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{2} + 2) q^{2} + ( - \beta_{3} - \beta_1) q^{3} + 3 \beta_{2} q^{4} + \beta_1 q^{5} - 2 \beta_{3} q^{6} - 4 q^{8} + ( - \beta_{2} - 1) q^{9} + (2 \beta_{3} + 2 \beta_1) q^{10} + 3 \beta_1 q^{12} + 2 q^{15} + ( - 5 \beta_{2} - 5) q^{16} - 2 \beta_{2} q^{18} + \beta_1 q^{19} + 3 \beta_{3} q^{20} + (4 \beta_{3} + 4 \beta_1) q^{24} + \beta_{2} q^{25} + (4 \beta_{2} + 4) q^{30} - 6 \beta_{2} q^{32} + 3 q^{36} + (2 \beta_{3} + 2 \beta_1) q^{38} - 4 \beta_1 q^{40} - 2 q^{43} + ( - \beta_{3} - \beta_1) q^{45} + 5 \beta_{3} q^{48} - 2 q^{50} + 2 q^{57} + 6 \beta_{2} q^{60} + 7 q^{64} - q^{71} + (4 \beta_{2} + 4) q^{72} + ( - \beta_{3} - \beta_1) q^{73} + \beta_1 q^{75} + 3 \beta_{3} q^{76} + ( - 5 \beta_{3} - 5 \beta_1) q^{80} - \beta_{2} q^{81} - \beta_{3} q^{83} + ( - 4 \beta_{2} - 4) q^{86} - \beta_1 q^{89} - 2 \beta_{3} q^{90} + 2 \beta_{2} q^{95} - 6 \beta_1 q^{96}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 6 q^{4} - 16 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 6 q^{4} - 16 q^{8} - 2 q^{9} + 8 q^{15} - 10 q^{16} + 4 q^{18} - 2 q^{25} + 8 q^{30} + 12 q^{32} + 12 q^{36} - 8 q^{43} - 8 q^{50} + 8 q^{57} - 12 q^{60} + 28 q^{64} - 4 q^{71} + 8 q^{72} + 2 q^{81} - 8 q^{86} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3479\mathbb{Z}\right)^\times\).

\(n\) \(640\) \(1569\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
851.1
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 − 1.22474i
1.00000 + 1.73205i −0.707107 + 1.22474i −1.50000 + 2.59808i −0.707107 − 1.22474i −2.82843 0 −4.00000 −0.500000 − 0.866025i 1.41421 − 2.44949i
851.2 1.00000 + 1.73205i 0.707107 − 1.22474i −1.50000 + 2.59808i 0.707107 + 1.22474i 2.82843 0 −4.00000 −0.500000 − 0.866025i −1.41421 + 2.44949i
1206.1 1.00000 − 1.73205i −0.707107 − 1.22474i −1.50000 − 2.59808i −0.707107 + 1.22474i −2.82843 0 −4.00000 −0.500000 + 0.866025i 1.41421 + 2.44949i
1206.2 1.00000 − 1.73205i 0.707107 + 1.22474i −1.50000 − 2.59808i 0.707107 − 1.22474i 2.82843 0 −4.00000 −0.500000 + 0.866025i −1.41421 − 2.44949i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.b odd 2 1 CM by \(\Q(\sqrt{-71}) \)
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
497.b even 2 1 inner
497.g odd 6 1 inner
497.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3479.1.g.c 4
7.b odd 2 1 inner 3479.1.g.c 4
7.c even 3 1 3479.1.d.b ✓ 2
7.c even 3 1 inner 3479.1.g.c 4
7.d odd 6 1 3479.1.d.b ✓ 2
7.d odd 6 1 inner 3479.1.g.c 4
71.b odd 2 1 CM 3479.1.g.c 4
497.b even 2 1 inner 3479.1.g.c 4
497.g odd 6 1 3479.1.d.b ✓ 2
497.g odd 6 1 inner 3479.1.g.c 4
497.i even 6 1 3479.1.d.b ✓ 2
497.i even 6 1 inner 3479.1.g.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3479.1.d.b ✓ 2 7.c even 3 1
3479.1.d.b ✓ 2 7.d odd 6 1
3479.1.d.b ✓ 2 497.g odd 6 1
3479.1.d.b ✓ 2 497.i even 6 1
3479.1.g.c 4 1.a even 1 1 trivial
3479.1.g.c 4 7.b odd 2 1 inner
3479.1.g.c 4 7.c even 3 1 inner
3479.1.g.c 4 7.d odd 6 1 inner
3479.1.g.c 4 71.b odd 2 1 CM
3479.1.g.c 4 497.b even 2 1 inner
3479.1.g.c 4 497.g odd 6 1 inner
3479.1.g.c 4 497.i even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3479, [\chi])\):

\( T_{2}^{2} - 2T_{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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