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Properties

Label 3744.2.g.d
Level $3744$
Weight $2$
Character orbit 3744.g
Analytic conductor $29.896$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3744,2,Mod(1873,3744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3744, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3744.1873");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.g (of order \(2\), degree \(1\), 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: \(29.8959905168\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{5} + ( - \beta_{5} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{5} + ( - \beta_{5} - 1) q^{7} + ( - \beta_{6} - \beta_{2}) q^{11} + \beta_1 q^{13} + (2 \beta_{5} + \beta_{4} - \beta_{3} + 2) q^{17} + (\beta_{7} + \beta_{6} + \cdots + 2 \beta_1) q^{19}+ \cdots + ( - 2 \beta_{5} - 6 \beta_{4} + \cdots - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{7} + 16 q^{17} - 8 q^{23} + 8 q^{25} + 4 q^{31} - 36 q^{41} + 24 q^{47} - 24 q^{49} + 40 q^{55} + 4 q^{65} + 16 q^{71} + 32 q^{73} - 60 q^{89} - 24 q^{95} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{20}^{5} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{20}^{7} + \zeta_{20}^{6} + \zeta_{20}^{4} + \zeta_{20}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{20}^{7} - \zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20}^{4} - \zeta_{20}^{3} + 2\zeta_{20} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{20}^{7} + \zeta_{20}^{6} + \zeta_{20}^{5} - \zeta_{20}^{4} - \zeta_{20}^{3} + 2\zeta_{20} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{20}^{7} - \zeta_{20}^{6} + \zeta_{20}^{4} + \zeta_{20}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{20}^{7} + \zeta_{20}^{6} - \zeta_{20}^{5} - \zeta_{20}^{4} + \zeta_{20}^{3} + 2\zeta_{20}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{20}^{7} - \zeta_{20}^{6} - \zeta_{20}^{5} - \zeta_{20}^{4} + \zeta_{20}^{3} \) Copy content Toggle raw display

Display \(\zeta_{20}^j\) in terms of \(\beta_i\)

\(\zeta_{20}\)\(=\) \( ( \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{2}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{3}\)\(=\) \( ( \beta_{7} + 2\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{4}\)\(=\) \( ( -\beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{5}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{20}^{6}\)\(=\) \( ( -\beta_{7} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{7}\)\(=\) \( ( \beta_{7} - 2\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{20}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1873.1
−0.587785 − 0.809017i
0.951057 − 0.309017i
−0.951057 + 0.309017i
0.587785 − 0.809017i
0.587785 + 0.809017i
−0.951057 − 0.309017i
0.951057 + 0.309017i
−0.587785 + 0.809017i
0 0 0 − 2.79360i 0 −1.28408 0 0 0
1873.2 0 0 0 − 2.52015i 0 −2.79360 0 0 0
1873.3 0 0 0 − 1.28408i 0 −0.442463 0 0 0
1873.4 0 0 0 − 0.442463i 0 2.52015 0 0 0
1873.5 0 0 0 0.442463i 0 2.52015 0 0 0
1873.6 0 0 0 1.28408i 0 −0.442463 0 0 0
1873.7 0 0 0 2.52015i 0 −2.79360 0 0 0
1873.8 0 0 0 2.79360i 0 −1.28408 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1873.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.2.g.d 8
3.b odd 2 1 1248.2.g.a 8
4.b odd 2 1 936.2.g.d 8
8.b even 2 1 inner 3744.2.g.d 8
8.d odd 2 1 936.2.g.d 8
12.b even 2 1 312.2.g.a ✓ 8
24.f even 2 1 312.2.g.a ✓ 8
24.h odd 2 1 1248.2.g.a 8
48.i odd 4 1 9984.2.a.s 4
48.i odd 4 1 9984.2.a.bh 4
48.k even 4 1 9984.2.a.y 4
48.k even 4 1 9984.2.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.g.a ✓ 8 12.b even 2 1
312.2.g.a ✓ 8 24.f even 2 1
936.2.g.d 8 4.b odd 2 1
936.2.g.d 8 8.d odd 2 1
1248.2.g.a 8 3.b odd 2 1
1248.2.g.a 8 24.h odd 2 1
3744.2.g.d 8 1.a even 1 1 trivial
3744.2.g.d 8 8.b even 2 1 inner
9984.2.a.s 4 48.i odd 4 1
9984.2.a.y 4 48.k even 4 1
9984.2.a.bb 4 48.k even 4 1
9984.2.a.bh 4 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 16T_{5}^{6} + 76T_{5}^{4} + 96T_{5}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(3744, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 16 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} - 6 T^{2} + \cdots - 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 40 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 8 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 64 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} - 24 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 104 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} + \cdots + 316)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 176 T^{6} + \cdots + 952576 \) Copy content Toggle raw display
$41$ \( (T^{4} + 18 T^{3} + \cdots - 2644)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 344 T^{6} + \cdots + 49336576 \) Copy content Toggle raw display
$47$ \( (T^{4} - 12 T^{3} + \cdots - 964)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 224 T^{6} + \cdots + 1478656 \) Copy content Toggle raw display
$59$ \( T^{8} + 104 T^{6} + \cdots + 5776 \) Copy content Toggle raw display
$61$ \( (T^{4} + 60 T^{2} + 400)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 176 T^{6} + \cdots + 5776 \) Copy content Toggle raw display
$71$ \( (T^{4} - 8 T^{3} + \cdots + 956)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 16 T^{3} + \cdots - 304)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 120 T^{2} + \cdots + 80)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 184 T^{6} + \cdots + 2572816 \) Copy content Toggle raw display
$89$ \( (T^{4} + 30 T^{3} + \cdots - 2420)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 28 T^{3} + \cdots - 18544)^{2} \) Copy content Toggle raw display
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