Properties

Label 1008.1.dc.a
Level $1008$
Weight $1$
Character orbit 1008.dc
Analytic conductor $0.503$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1008.dc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.21168.2

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + ( -1 + \beta_{2} ) q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + ( -1 + \beta_{2} ) q^{7} + ( \beta_{1} - \beta_{3} ) q^{11} - q^{13} + ( -1 + \beta_{2} ) q^{19} + \beta_{2} q^{25} -\beta_{2} q^{31} + ( -\beta_{1} + \beta_{3} ) q^{35} + ( -1 + \beta_{2} ) q^{37} -\beta_{3} q^{41} + q^{43} + \beta_{1} q^{47} -\beta_{2} q^{49} + 2 q^{55} -\beta_{1} q^{65} -\beta_{2} q^{67} -\beta_{3} q^{71} -\beta_{2} q^{73} + \beta_{3} q^{77} + ( 1 - \beta_{2} ) q^{79} + \beta_{3} q^{83} + ( 1 - \beta_{2} ) q^{91} + ( -\beta_{1} + \beta_{3} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{7} + O(q^{10}) \) \( 4q - 2q^{7} - 4q^{13} - 2q^{19} + 2q^{25} - 2q^{31} - 2q^{37} + 4q^{43} - 2q^{49} + 8q^{55} - 2q^{67} - 2q^{73} + 2q^{79} + 2q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
0 0 0 −1.22474 + 0.707107i 0 −0.500000 0.866025i 0 0 0
305.2 0 0 0 1.22474 0.707107i 0 −0.500000 0.866025i 0 0 0
737.1 0 0 0 −1.22474 0.707107i 0 −0.500000 + 0.866025i 0 0 0
737.2 0 0 0 1.22474 + 0.707107i 0 −0.500000 + 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.1.dc.a 4
3.b odd 2 1 inner 1008.1.dc.a 4
4.b odd 2 1 504.1.cu.a 4
7.c even 3 1 inner 1008.1.dc.a 4
12.b even 2 1 504.1.cu.a 4
21.h odd 6 1 inner 1008.1.dc.a 4
28.d even 2 1 3528.1.cu.a 4
28.f even 6 1 3528.1.d.b 2
28.f even 6 1 3528.1.cu.a 4
28.g odd 6 1 504.1.cu.a 4
28.g odd 6 1 3528.1.d.a 2
84.h odd 2 1 3528.1.cu.a 4
84.j odd 6 1 3528.1.d.b 2
84.j odd 6 1 3528.1.cu.a 4
84.n even 6 1 504.1.cu.a 4
84.n even 6 1 3528.1.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.cu.a 4 4.b odd 2 1
504.1.cu.a 4 12.b even 2 1
504.1.cu.a 4 28.g odd 6 1
504.1.cu.a 4 84.n even 6 1
1008.1.dc.a 4 1.a even 1 1 trivial
1008.1.dc.a 4 3.b odd 2 1 inner
1008.1.dc.a 4 7.c even 3 1 inner
1008.1.dc.a 4 21.h odd 6 1 inner
3528.1.d.a 2 28.g odd 6 1
3528.1.d.a 2 84.n even 6 1
3528.1.d.b 2 28.f even 6 1
3528.1.d.b 2 84.j odd 6 1
3528.1.cu.a 4 28.d even 2 1
3528.1.cu.a 4 28.f even 6 1
3528.1.cu.a 4 84.h odd 2 1
3528.1.cu.a 4 84.j odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1008, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 4 - 2 T^{2} + T^{4} \)
$7$ \( ( 1 + T + T^{2} )^{2} \)
$11$ \( 4 - 2 T^{2} + T^{4} \)
$13$ \( ( 1 + T )^{4} \)
$17$ \( T^{4} \)
$19$ \( ( 1 + T + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( 1 + T + T^{2} )^{2} \)
$37$ \( ( 1 + T + T^{2} )^{2} \)
$41$ \( ( 2 + T^{2} )^{2} \)
$43$ \( ( -1 + T )^{4} \)
$47$ \( 4 - 2 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( 1 + T + T^{2} )^{2} \)
$71$ \( ( 2 + T^{2} )^{2} \)
$73$ \( ( 1 + T + T^{2} )^{2} \)
$79$ \( ( 1 - T + T^{2} )^{2} \)
$83$ \( ( 2 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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