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Properties

Label 1248.2.c.b
Level $1248$
Weight $2$
Character orbit 1248.c
Analytic conductor $9.965$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} + ( - \beta_{2} + \beta_1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_1 q^{5} + ( - \beta_{2} + \beta_1) q^{7} + q^{9} + \beta_{2} q^{11} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{13} + \beta_1 q^{15} + 2 q^{17} + (\beta_{2} - \beta_1) q^{19} + ( - \beta_{2} + \beta_1) q^{21} + (\beta_{5} + \beta_{4} - 1) q^{23} + (\beta_{5} - \beta_{3} - 2) q^{25} + q^{27} + (\beta_{5} + \beta_{4} + 1) q^{29} + ( - \beta_{2} - \beta_1) q^{31} + \beta_{2} q^{33} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 5) q^{35} + (\beta_{4} - \beta_{3} + 2 \beta_{2}) q^{37} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{39} + 3 \beta_1 q^{41} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + 4) q^{43} + \beta_1 q^{45} + (\beta_{4} - \beta_{3} + \beta_{2}) q^{47} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 2) q^{49} + 2 q^{51} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 3) q^{53} + (\beta_{4} + \beta_{3} - 2) q^{55} + (\beta_{2} - \beta_1) q^{57} + (\beta_{2} - 2 \beta_1) q^{59} + (\beta_{5} + 2 \beta_{4} + \beta_{3} + 3) q^{61} + ( - \beta_{2} + \beta_1) q^{63} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + \cdots - 3) q^{65}+ \cdots + \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 6 q^{9} + 2 q^{13} + 12 q^{17} - 8 q^{23} - 10 q^{25} + 6 q^{27} + 4 q^{29} - 24 q^{35} + 2 q^{39} + 24 q^{43} - 6 q^{49} + 12 q^{51} - 12 q^{53} - 16 q^{55} + 12 q^{61} - 16 q^{65} - 8 q^{69} - 10 q^{75} + 24 q^{77} + 8 q^{79} + 6 q^{81} + 4 q^{87} - 32 q^{91} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 5x^{4} + 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{5} + 2\nu^{4} + 8\nu^{3} + 6\nu^{2} + 6\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{5} + 2\nu^{4} - 8\nu^{3} + 6\nu^{2} - 6\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{5} + 2\nu^{4} + 8\nu^{3} + 10\nu^{2} + 6\nu + 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{3} - 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} - 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{5} + \beta_{4} + 4\beta_{3} + 19 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{4} + \beta_{3} - 8\beta_{2} + 18\beta_1 ) / 4 \) 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
− 1.80194i
− 1.24698i
− 0.445042i
0.445042i
1.24698i
1.80194i
0 1.00000 0 − 3.60388i 0 − 4.49396i 0 1.00000 0
961.2 0 1.00000 0 − 2.49396i 0 1.10992i 0 1.00000 0
961.3 0 1.00000 0 − 0.890084i 0 1.60388i 0 1.00000 0
961.4 0 1.00000 0 0.890084i 0 − 1.60388i 0 1.00000 0
961.5 0 1.00000 0 2.49396i 0 − 1.10992i 0 1.00000 0
961.6 0 1.00000 0 3.60388i 0 4.49396i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.6
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.b yes 6
3.b odd 2 1 3744.2.c.m 6
4.b odd 2 1 1248.2.c.a ✓ 6
8.b even 2 1 2496.2.c.o 6
8.d odd 2 1 2496.2.c.p 6
12.b even 2 1 3744.2.c.l 6
13.b even 2 1 inner 1248.2.c.b yes 6
39.d odd 2 1 3744.2.c.m 6
52.b odd 2 1 1248.2.c.a ✓ 6
104.e even 2 1 2496.2.c.o 6
104.h odd 2 1 2496.2.c.p 6
156.h even 2 1 3744.2.c.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.c.a ✓ 6 4.b odd 2 1
1248.2.c.a ✓ 6 52.b odd 2 1
1248.2.c.b yes 6 1.a even 1 1 trivial
1248.2.c.b yes 6 13.b even 2 1 inner
2496.2.c.o 6 8.b even 2 1
2496.2.c.o 6 104.e even 2 1
2496.2.c.p 6 8.d odd 2 1
2496.2.c.p 6 104.h odd 2 1
3744.2.c.l 6 12.b even 2 1
3744.2.c.l 6 156.h even 2 1
3744.2.c.m 6 3.b odd 2 1
3744.2.c.m 6 39.d odd 2 1

Hecke kernels

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

\( T_{5}^{6} + 20T_{5}^{4} + 96T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{23}^{3} + 4T_{23}^{2} - 32T_{23} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 20 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{6} + 24 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{6} + 20 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T - 2)^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 24 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( (T^{3} + 4 T^{2} - 32 T - 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} - 36 T + 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 56 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$37$ \( T^{6} + 160 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( T^{6} + 180 T^{4} + \cdots + 46656 \) Copy content Toggle raw display
$43$ \( (T^{3} - 12 T^{2} + \cdots + 832)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 84 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$53$ \( (T^{3} + 6 T^{2} + \cdots + 232)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 68 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( (T^{3} - 6 T^{2} + \cdots + 664)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 216 T^{4} + \cdots + 118336 \) Copy content Toggle raw display
$71$ \( T^{6} + 132 T^{4} + \cdots + 10816 \) Copy content Toggle raw display
$73$ \( T^{6} + 80 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( (T^{3} - 4 T^{2} - 144 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 356 T^{4} + \cdots + 10816 \) Copy content Toggle raw display
$89$ \( T^{6} + 276 T^{4} + \cdots + 10816 \) Copy content Toggle raw display
$97$ \( T^{6} + 336 T^{4} + \cdots + 200704 \) Copy content Toggle raw display
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