Properties

Label 1008.3.d.a
Level $1008$
Weight $3$
Character orbit 1008.d
Analytic conductor $27.466$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 \beta_{1} - \beta_{3} ) q^{5} + \beta_{2} q^{7} +O(q^{10})\) \( q + ( 2 \beta_{1} - \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{11} + ( -8 - 4 \beta_{2} ) q^{13} + ( -2 \beta_{1} + 5 \beta_{3} ) q^{17} -20 q^{19} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{23} + ( -33 + 8 \beta_{2} ) q^{25} + ( -4 \beta_{1} - 5 \beta_{3} ) q^{29} + ( -4 - 8 \beta_{2} ) q^{31} + ( -\beta_{1} + 5 \beta_{3} ) q^{35} + 38 q^{37} + ( 6 \beta_{1} + 7 \beta_{3} ) q^{41} + ( -20 - 24 \beta_{2} ) q^{43} -4 \beta_{3} q^{47} + 7 q^{49} + ( -24 \beta_{1} + 7 \beta_{3} ) q^{53} + ( 116 - 16 \beta_{2} ) q^{55} + ( -8 \beta_{1} - 4 \beta_{3} ) q^{59} + ( 58 - 16 \beta_{2} ) q^{61} + ( -12 \beta_{1} - 12 \beta_{3} ) q^{65} + ( 48 + 32 \beta_{2} ) q^{67} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{71} + ( -24 - 20 \beta_{2} ) q^{73} + ( 2 \beta_{1} - 10 \beta_{3} ) q^{77} + ( -76 + 16 \beta_{2} ) q^{79} + ( 8 \beta_{1} - 24 \beta_{3} ) q^{83} + ( 82 - 56 \beta_{2} ) q^{85} + ( -18 \beta_{1} + 23 \beta_{3} ) q^{89} + ( -28 - 8 \beta_{2} ) q^{91} + ( -40 \beta_{1} + 20 \beta_{3} ) q^{95} + ( -72 - 44 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 32q^{13} - 80q^{19} - 132q^{25} - 16q^{31} + 152q^{37} - 80q^{43} + 28q^{49} + 464q^{55} + 232q^{61} + 192q^{67} - 96q^{73} - 304q^{79} + 328q^{85} - 112q^{91} - 288q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} - 16 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\( \nu^{2} + 4 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} - 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{3} - 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 4\)
\(\nu^{3}\)\(=\)\((\)\(-16 \beta_{3} + 15 \beta_{1}\)\()/6\)

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
2.57794i
1.16372i
1.16372i
2.57794i
0 0 0 8.89753i 0 −2.64575 0 0 0
449.2 0 0 0 6.06910i 0 2.64575 0 0 0
449.3 0 0 0 6.06910i 0 2.64575 0 0 0
449.4 0 0 0 8.89753i 0 −2.64575 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.d.a 4
3.b odd 2 1 inner 1008.3.d.a 4
4.b odd 2 1 126.3.b.a 4
8.b even 2 1 4032.3.d.j 4
8.d odd 2 1 4032.3.d.i 4
12.b even 2 1 126.3.b.a 4
20.d odd 2 1 3150.3.e.e 4
20.e even 4 2 3150.3.c.b 8
24.f even 2 1 4032.3.d.i 4
24.h odd 2 1 4032.3.d.j 4
28.d even 2 1 882.3.b.f 4
28.f even 6 2 882.3.s.i 8
28.g odd 6 2 882.3.s.e 8
36.f odd 6 2 1134.3.q.c 8
36.h even 6 2 1134.3.q.c 8
60.h even 2 1 3150.3.e.e 4
60.l odd 4 2 3150.3.c.b 8
84.h odd 2 1 882.3.b.f 4
84.j odd 6 2 882.3.s.i 8
84.n even 6 2 882.3.s.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.b.a 4 4.b odd 2 1
126.3.b.a 4 12.b even 2 1
882.3.b.f 4 28.d even 2 1
882.3.b.f 4 84.h odd 2 1
882.3.s.e 8 28.g odd 6 2
882.3.s.e 8 84.n even 6 2
882.3.s.i 8 28.f even 6 2
882.3.s.i 8 84.j odd 6 2
1008.3.d.a 4 1.a even 1 1 trivial
1008.3.d.a 4 3.b odd 2 1 inner
1134.3.q.c 8 36.f odd 6 2
1134.3.q.c 8 36.h even 6 2
3150.3.c.b 8 20.e even 4 2
3150.3.c.b 8 60.l odd 4 2
3150.3.e.e 4 20.d odd 2 1
3150.3.e.e 4 60.h even 2 1
4032.3.d.i 4 8.d odd 2 1
4032.3.d.i 4 24.f even 2 1
4032.3.d.j 4 8.b even 2 1
4032.3.d.j 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{4} + 116 T_{5}^{2} + 2916 \)
\( T_{11}^{4} + 464 T_{11}^{2} + 46656 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 2916 + 116 T^{2} + T^{4} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( 46656 + 464 T^{2} + T^{4} \)
$13$ \( ( -48 + 16 T + T^{2} )^{2} \)
$17$ \( 79524 + 788 T^{2} + T^{4} \)
$19$ \( ( 20 + T )^{4} \)
$23$ \( 46656 + 464 T^{2} + T^{4} \)
$29$ \( 248004 + 1892 T^{2} + T^{4} \)
$31$ \( ( -432 + 8 T + T^{2} )^{2} \)
$37$ \( ( -38 + T )^{4} \)
$41$ \( 910116 + 3924 T^{2} + T^{4} \)
$43$ \( ( -3632 + 40 T + T^{2} )^{2} \)
$47$ \( ( 288 + T^{2} )^{2} \)
$53$ \( 64738116 + 16164 T^{2} + T^{4} \)
$59$ \( 9216 + 3392 T^{2} + T^{4} \)
$61$ \( ( 1572 - 116 T + T^{2} )^{2} \)
$67$ \( ( -4864 - 96 T + T^{2} )^{2} \)
$71$ \( 46656 + 464 T^{2} + T^{4} \)
$73$ \( ( -2224 + 48 T + T^{2} )^{2} \)
$79$ \( ( 3984 + 152 T + T^{2} )^{2} \)
$83$ \( 53231616 + 18176 T^{2} + T^{4} \)
$89$ \( 443556 + 19476 T^{2} + T^{4} \)
$97$ \( ( -8368 + 144 T + T^{2} )^{2} \)
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