Properties

Label 1248.2.ca.a
Level $1248$
Weight $2$
Character orbit 1248.ca
Analytic conductor $9.965$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1248,2,Mod(49,1248)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1248, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1248.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.ca (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: \(9.96533017226\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{7} + \beta_{4} - \beta_{2} - 1) q^{7} + (\beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{7} + \beta_{4} - \beta_{2} - 1) q^{7} + (\beta_{4} + 1) q^{9} + ( - \beta_{6} + 2 \beta_{5} + \beta_1) q^{11} + ( - 4 \beta_{5} - 3 \beta_1) q^{13} - \beta_{2} q^{15} + ( - \beta_{7} - 2 \beta_{4} - \beta_{2} - 2) q^{17} + ( - \beta_{6} + \beta_{5} + \beta_{3} - \beta_1) q^{19} + (\beta_{5} + \beta_{3} + 2 \beta_1) q^{21} + ( - 2 \beta_{7} - 5 \beta_{4} + \beta_{2}) q^{23} + \beta_{5} q^{27} + (\beta_{6} - 2 \beta_{3} - 6 \beta_1) q^{29} - 2 \beta_{7} q^{31} + ( - \beta_{7} + \beta_{4} + \beta_{2} - 1) q^{33} + ( - \beta_{6} - 5 \beta_{5} - \beta_{3} - 5 \beta_1) q^{35} + (4 \beta_{6} - 2 \beta_{5} - \beta_1) q^{37} + ( - \beta_{4} + 3) q^{39} + (2 \beta_{4} - \beta_{2} + 4) q^{41} + ( - \beta_{6} - 5 \beta_{5} - \beta_{3} - 5 \beta_1) q^{43} + ( - \beta_{6} + \beta_{3}) q^{45} + ( - \beta_{7} - 2 \beta_{4} - 1) q^{47} + ( - 4 \beta_{7} - \beta_{4} + 2 \beta_{2}) q^{49} + ( - 2 \beta_{6} - 2 \beta_{5} + \beta_{3}) q^{51} + ( - 2 \beta_{6} - 4 \beta_{5} + \beta_{3}) q^{53} + ( - 2 \beta_{7} + 5 \beta_{4} + \beta_{2}) q^{55} + ( - \beta_{7} + 2 \beta_{4} + 1) q^{57} + (2 \beta_{6} - 4 \beta_{5} - 2 \beta_{3} + 4 \beta_1) q^{59} + (\beta_{5} + \beta_1) q^{61} + ( - \beta_{4} - \beta_{2} - 2) q^{63} + (4 \beta_{7} - 3 \beta_{2}) q^{65} + (3 \beta_{6} - 10 \beta_{5} - 5 \beta_1) q^{67} + ( - \beta_{6} - 5 \beta_{5} - \beta_{3} - 5 \beta_1) q^{69} + ( - 3 \beta_{7} + \beta_{4} + 3 \beta_{2} - 1) q^{71} + ( - 2 \beta_{7} - 10 \beta_{4} - 5) q^{73} + 2 \beta_{5} q^{77} - 14 q^{79} + \beta_{4} q^{81} + ( - 3 \beta_{5} + \beta_{3} - 6 \beta_1) q^{83} + (2 \beta_{6} + 5 \beta_{5} - 2 \beta_{3} - 5 \beta_1) q^{85} + (\beta_{7} + 6 \beta_{4} + \beta_{2} + 6) q^{87} + (4 \beta_{4} - 4 \beta_{2} + 8) q^{89} + (4 \beta_{6} + 7 \beta_{5} - \beta_{3} + 2 \beta_1) q^{91} - 2 \beta_{6} q^{93} + ( - \beta_{7} + 5 \beta_{4} - \beta_{2} + 5) q^{95} + (10 \beta_{4} - 10) q^{97} + (\beta_{5} - \beta_{3} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{7} + 4 q^{9} - 8 q^{17} + 20 q^{23} - 12 q^{33} + 28 q^{39} + 24 q^{41} + 4 q^{49} - 20 q^{55} - 12 q^{63} - 12 q^{71} - 112 q^{79} - 4 q^{81} + 24 q^{87} + 48 q^{89} + 20 q^{95} - 120 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 13\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 29\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 9 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} - 8\nu^{4} + 24\nu^{2} - 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{6} + 24\nu^{4} - 56\nu^{2} + 21 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 8\nu^{5} + 22\nu^{3} - \nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 3\beta_{4} - \beta_{3} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{6} + 7\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} + 11\beta_{5} - 5\beta_{2} + 11\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{2} + 29\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(1\) \(1 + \beta_{4}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−1.40126 + 0.809017i
0.535233 0.309017i
1.40126 0.809017i
−0.535233 + 0.309017i
−1.40126 0.809017i
0.535233 + 0.309017i
1.40126 + 0.809017i
−0.535233 0.309017i
0 −0.866025 + 0.500000i 0 −2.23607 0 0.436492 + 0.252009i 0 0.500000 0.866025i 0
49.2 0 −0.866025 + 0.500000i 0 2.23607 0 −3.43649 1.98406i 0 0.500000 0.866025i 0
49.3 0 0.866025 0.500000i 0 −2.23607 0 −3.43649 1.98406i 0 0.500000 0.866025i 0
49.4 0 0.866025 0.500000i 0 2.23607 0 0.436492 + 0.252009i 0 0.500000 0.866025i 0
433.1 0 −0.866025 0.500000i 0 −2.23607 0 0.436492 0.252009i 0 0.500000 + 0.866025i 0
433.2 0 −0.866025 0.500000i 0 2.23607 0 −3.43649 + 1.98406i 0 0.500000 + 0.866025i 0
433.3 0 0.866025 + 0.500000i 0 −2.23607 0 −3.43649 + 1.98406i 0 0.500000 + 0.866025i 0
433.4 0 0.866025 + 0.500000i 0 2.23607 0 0.436492 0.252009i 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
13.e even 6 1 inner
104.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1248.2.ca.a 8
4.b odd 2 1 312.2.bk.a 8
8.b even 2 1 inner 1248.2.ca.a 8
8.d odd 2 1 312.2.bk.a 8
12.b even 2 1 936.2.dg.c 8
13.e even 6 1 inner 1248.2.ca.a 8
24.f even 2 1 936.2.dg.c 8
52.i odd 6 1 312.2.bk.a 8
104.p odd 6 1 312.2.bk.a 8
104.s even 6 1 inner 1248.2.ca.a 8
156.r even 6 1 936.2.dg.c 8
312.ba even 6 1 936.2.dg.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bk.a 8 4.b odd 2 1
312.2.bk.a 8 8.d odd 2 1
312.2.bk.a 8 52.i odd 6 1
312.2.bk.a 8 104.p odd 6 1
936.2.dg.c 8 12.b even 2 1
936.2.dg.c 8 24.f even 2 1
936.2.dg.c 8 156.r even 6 1
936.2.dg.c 8 312.ba even 6 1
1248.2.ca.a 8 1.a even 1 1 trivial
1248.2.ca.a 8 8.b even 2 1 inner
1248.2.ca.a 8 13.e even 6 1 inner
1248.2.ca.a 8 104.s even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 5 \) acting on \(S_{2}^{\mathrm{new}}(1248, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 6 T^{3} + 10 T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 16 T^{6} + 252 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 4 T^{3} + 27 T^{2} - 44 T + 121)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 16 T^{6} + 252 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( (T^{4} - 10 T^{3} + 90 T^{2} - 100 T + 100)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 102 T^{6} + 9963 T^{4} + \cdots + 194481 \) Copy content Toggle raw display
$31$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + 166 T^{6} + \cdots + 35153041 \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} + 55 T^{2} - 84 T + 49)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 80 T^{6} + 6300 T^{4} + \cdots + 10000 \) Copy content Toggle raw display
$47$ \( (T^{4} + 16 T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 62 T^{2} + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 136 T^{6} + 17712 T^{4} + \cdots + 614656 \) Copy content Toggle raw display
$61$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 240 T^{6} + 56700 T^{4} + \cdots + 810000 \) Copy content Toggle raw display
$71$ \( (T^{4} + 6 T^{3} - 30 T^{2} - 252 T + 1764)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 190 T^{2} + 3025)^{2} \) Copy content Toggle raw display
$79$ \( (T + 14)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 64 T^{2} + 484)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 24 T^{3} + 160 T^{2} + 768 T + 1024)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 30 T + 300)^{4} \) Copy content Toggle raw display
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