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Properties

Label 1248.2.c.d
Level $1248$
Weight $2$
Character orbit 1248.c
Analytic conductor $9.965$
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] = [1248,2,Mod(961,1248)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1248, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1248.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.c (of order \(2\), degree \(1\), 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: \(9.96533017226\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.134560000.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{6} + 13x^{4} + 7x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} + \beta_{6} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_1 q^{5} + \beta_{6} q^{7} + q^{9} + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{11} + \beta_{3} q^{13} + \beta_1 q^{15} + \beta_{5} q^{17} + (\beta_{7} + \beta_{6} + \cdots + \beta_{3}) q^{19}+ \cdots + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 8 q^{9} + 8 q^{23} - 16 q^{25} + 8 q^{27} - 8 q^{29} + 24 q^{35} - 32 q^{49} + 24 q^{53} + 16 q^{55} + 8 q^{61} + 24 q^{65} + 8 q^{69} - 16 q^{75} - 32 q^{77} - 16 q^{79} + 8 q^{81} - 8 q^{87} + 16 q^{91} + 56 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{6} + 13x^{4} + 7x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{6} + 24\nu^{4} + 32\nu^{2} + 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{7} + 2\nu^{6} + 14\nu^{5} + 12\nu^{4} + 26\nu^{3} + 14\nu^{2} + 12\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{7} - 2\nu^{6} + 14\nu^{5} - 12\nu^{4} + 26\nu^{3} - 14\nu^{2} + 12\nu - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 4\nu^{6} + 28\nu^{4} + 48\nu^{2} + 14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -4\nu^{7} - 28\nu^{5} - 50\nu^{3} - 20\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -8\nu^{7} - 52\nu^{5} - 80\nu^{3} - 24\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} - 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} + \beta_{4} + \beta_{3} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{5} - 4\beta_{4} + 4\beta_{3} - 5\beta_{2} + 23 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 12\beta_{6} - 10\beta_{4} - 10\beta_{3} + 12\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -6\beta_{5} + 16\beta_{4} - 16\beta_{3} + 23\beta_{2} - 91 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} + 58\beta_{6} + 45\beta_{4} + 45\beta_{3} - 44\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
− 2.09529i
− 1.35567i
− 0.737640i
− 0.477260i
0.477260i
0.737640i
1.35567i
2.09529i
0 1.00000 0 − 4.19059i 0 3.54445i 0 1.00000 0
961.2 0 1.00000 0 − 2.71135i 0 − 2.91177i 0 1.00000 0
961.3 0 1.00000 0 − 1.47528i 0 0.324294i 0 1.00000 0
961.4 0 1.00000 0 − 0.954520i 0 4.78051i 0 1.00000 0
961.5 0 1.00000 0 0.954520i 0 − 4.78051i 0 1.00000 0
961.6 0 1.00000 0 1.47528i 0 − 0.324294i 0 1.00000 0
961.7 0 1.00000 0 2.71135i 0 2.91177i 0 1.00000 0
961.8 0 1.00000 0 4.19059i 0 − 3.54445i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1248.2.c.d yes 8
3.b odd 2 1 3744.2.c.o 8
4.b odd 2 1 1248.2.c.c ✓ 8
8.b even 2 1 2496.2.c.q 8
8.d odd 2 1 2496.2.c.r 8
12.b even 2 1 3744.2.c.p 8
13.b even 2 1 inner 1248.2.c.d yes 8
39.d odd 2 1 3744.2.c.o 8
52.b odd 2 1 1248.2.c.c ✓ 8
104.e even 2 1 2496.2.c.q 8
104.h odd 2 1 2496.2.c.r 8
156.h even 2 1 3744.2.c.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.c.c ✓ 8 4.b odd 2 1
1248.2.c.c ✓ 8 52.b odd 2 1
1248.2.c.d yes 8 1.a even 1 1 trivial
1248.2.c.d yes 8 13.b even 2 1 inner
2496.2.c.q 8 8.b even 2 1
2496.2.c.q 8 104.e even 2 1
2496.2.c.r 8 8.d odd 2 1
2496.2.c.r 8 104.h odd 2 1
3744.2.c.o 8 3.b odd 2 1
3744.2.c.o 8 39.d odd 2 1
3744.2.c.p 8 12.b even 2 1
3744.2.c.p 8 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1248, [\chi])\):

\( T_{5}^{8} + 28T_{5}^{6} + 208T_{5}^{4} + 448T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{23}^{4} - 4T_{23}^{3} - 48T_{23}^{2} + 64T_{23} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 28 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{8} + 44 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{8} + 64 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{2} - 20)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} + 92 T^{6} + \cdots + 92416 \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + \cdots + 176)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 108 T^{6} + \cdots + 92416 \) Copy content Toggle raw display
$37$ \( T^{8} + 112 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( T^{8} + 76 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( (T^{4} - 96 T^{2} + \cdots - 256)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 224 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
$53$ \( (T^{4} - 12 T^{3} + \cdots - 16)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 144 T^{6} + \cdots + 430336 \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} + \cdots - 1936)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 92 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( T^{8} + 336 T^{6} + \cdots + 92416 \) Copy content Toggle raw display
$73$ \( T^{8} + 496 T^{6} + \cdots + 62980096 \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} + \cdots + 2816)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 208 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( T^{8} + 332 T^{6} + \cdots + 215296 \) Copy content Toggle raw display
$97$ \( T^{8} + 432 T^{6} + \cdots + 7929856 \) Copy content Toggle raw display
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