Properties

Label 1008.2.a.n
Level $1008$
Weight $2$
Character orbit 1008.a
Self dual yes
Analytic conductor $8.049$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - q^{7} +O(q^{10})\) \( q + \beta q^{5} - q^{7} + \beta q^{11} + 2 q^{13} -\beta q^{17} + 4 q^{19} -\beta q^{23} + 7 q^{25} + 4 q^{31} -\beta q^{35} + 2 q^{37} -3 \beta q^{41} + 4 q^{43} + 2 \beta q^{47} + q^{49} + 2 \beta q^{53} + 12 q^{55} -2 \beta q^{59} -10 q^{61} + 2 \beta q^{65} + 4 q^{67} -3 \beta q^{71} + 14 q^{73} -\beta q^{77} -8 q^{79} -12 q^{85} + \beta q^{89} -2 q^{91} + 4 \beta q^{95} + 14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} + O(q^{10}) \) \( 2q - 2q^{7} + 4q^{13} + 8q^{19} + 14q^{25} + 8q^{31} + 4q^{37} + 8q^{43} + 2q^{49} + 24q^{55} - 20q^{61} + 8q^{67} + 28q^{73} - 16q^{79} - 24q^{85} - 4q^{91} + 28q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
0 0 0 −3.46410 0 −1.00000 0 0 0
1.2 0 0 0 3.46410 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.a.n 2
3.b odd 2 1 inner 1008.2.a.n 2
4.b odd 2 1 63.2.a.b 2
7.b odd 2 1 7056.2.a.cm 2
8.b even 2 1 4032.2.a.bq 2
8.d odd 2 1 4032.2.a.bt 2
12.b even 2 1 63.2.a.b 2
20.d odd 2 1 1575.2.a.q 2
20.e even 4 2 1575.2.d.i 4
21.c even 2 1 7056.2.a.cm 2
24.f even 2 1 4032.2.a.bt 2
24.h odd 2 1 4032.2.a.bq 2
28.d even 2 1 441.2.a.g 2
28.f even 6 2 441.2.e.i 4
28.g odd 6 2 441.2.e.j 4
36.f odd 6 2 567.2.f.j 4
36.h even 6 2 567.2.f.j 4
44.c even 2 1 7623.2.a.bi 2
60.h even 2 1 1575.2.a.q 2
60.l odd 4 2 1575.2.d.i 4
84.h odd 2 1 441.2.a.g 2
84.j odd 6 2 441.2.e.i 4
84.n even 6 2 441.2.e.j 4
132.d odd 2 1 7623.2.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.a.b 2 4.b odd 2 1
63.2.a.b 2 12.b even 2 1
441.2.a.g 2 28.d even 2 1
441.2.a.g 2 84.h odd 2 1
441.2.e.i 4 28.f even 6 2
441.2.e.i 4 84.j odd 6 2
441.2.e.j 4 28.g odd 6 2
441.2.e.j 4 84.n even 6 2
567.2.f.j 4 36.f odd 6 2
567.2.f.j 4 36.h even 6 2
1008.2.a.n 2 1.a even 1 1 trivial
1008.2.a.n 2 3.b odd 2 1 inner
1575.2.a.q 2 20.d odd 2 1
1575.2.a.q 2 60.h even 2 1
1575.2.d.i 4 20.e even 4 2
1575.2.d.i 4 60.l odd 4 2
4032.2.a.bq 2 8.b even 2 1
4032.2.a.bq 2 24.h odd 2 1
4032.2.a.bt 2 8.d odd 2 1
4032.2.a.bt 2 24.f even 2 1
7056.2.a.cm 2 7.b odd 2 1
7056.2.a.cm 2 21.c even 2 1
7623.2.a.bi 2 44.c even 2 1
7623.2.a.bi 2 132.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5}^{2} - 12 \)
\( T_{11}^{2} - 12 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -12 + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -12 + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -12 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( -12 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( -108 + T^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -48 + T^{2} \)
$53$ \( -48 + T^{2} \)
$59$ \( -48 + T^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( -108 + T^{2} \)
$73$ \( ( -14 + T )^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( -12 + T^{2} \)
$97$ \( ( -14 + T )^{2} \)
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