Properties

Label 1008.2.s.r
Level $1008$
Weight $2$
Character orbit 1008.s
Analytic conductor $8.049$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( 1 + \beta_{1} - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( 1 + \beta_{1} - \beta_{3} ) q^{7} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{11} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -4 + 4 \beta_{2} ) q^{17} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{19} -4 \beta_{2} q^{23} + ( -9 - \beta_{1} + 10 \beta_{2} + 2 \beta_{3} ) q^{25} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{29} + ( -1 + \beta_{2} ) q^{31} + ( -11 + 3 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{35} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{37} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} + 6 \beta_{2} q^{47} + ( -4 + 3 \beta_{1} - 3 \beta_{3} ) q^{49} + ( 6 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -15 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{55} + ( -2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{59} -10 \beta_{2} q^{61} + ( -2 + 4 \beta_{1} + 14 \beta_{2} - 2 \beta_{3} ) q^{65} + ( 3 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{67} + 2 q^{71} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{73} + ( -5 + 2 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{77} + ( -2 + 4 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{79} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{83} + ( 4 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{85} + ( 2 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{89} + ( -3 + 4 \beta_{1} + 11 \beta_{2} - \beta_{3} ) q^{91} + ( 14 - 2 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} ) q^{95} + ( 13 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{5} + 6q^{7} + O(q^{10}) \) \( 4q - q^{5} + 6q^{7} + q^{11} + 10q^{13} - 8q^{17} - 5q^{19} - 8q^{23} - 19q^{25} - 6q^{29} - 2q^{31} - 30q^{35} - 3q^{37} - 12q^{41} + 14q^{43} + 12q^{47} - 10q^{49} + 11q^{53} - 58q^{55} - 5q^{59} - 20q^{61} + 26q^{65} + 7q^{67} + 8q^{71} + q^{73} - 27q^{77} + 8q^{79} + 14q^{83} + 8q^{85} - 6q^{89} + 15q^{91} + 26q^{95} + 50q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 4 \nu - 5 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu + 5 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-4 \beta_{3} + 4 \beta_{1} + 5\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−1.63746 + 1.52274i
2.13746 0.656712i
−1.63746 1.52274i
2.13746 + 0.656712i
0 0 0 −2.13746 + 3.70219i 0 1.50000 + 2.17945i 0 0 0
289.2 0 0 0 1.63746 2.83616i 0 1.50000 2.17945i 0 0 0
865.1 0 0 0 −2.13746 3.70219i 0 1.50000 2.17945i 0 0 0
865.2 0 0 0 1.63746 + 2.83616i 0 1.50000 + 2.17945i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.s.r 4
3.b odd 2 1 336.2.q.g 4
4.b odd 2 1 504.2.s.i 4
7.c even 3 1 inner 1008.2.s.r 4
7.c even 3 1 7056.2.a.cu 2
7.d odd 6 1 7056.2.a.ch 2
12.b even 2 1 168.2.q.c 4
21.c even 2 1 2352.2.q.bf 4
21.g even 6 1 2352.2.a.ba 2
21.g even 6 1 2352.2.q.bf 4
21.h odd 6 1 336.2.q.g 4
21.h odd 6 1 2352.2.a.bf 2
24.f even 2 1 1344.2.q.w 4
24.h odd 2 1 1344.2.q.x 4
28.d even 2 1 3528.2.s.bk 4
28.f even 6 1 3528.2.a.bd 2
28.f even 6 1 3528.2.s.bk 4
28.g odd 6 1 504.2.s.i 4
28.g odd 6 1 3528.2.a.bk 2
84.h odd 2 1 1176.2.q.l 4
84.j odd 6 1 1176.2.a.n 2
84.j odd 6 1 1176.2.q.l 4
84.n even 6 1 168.2.q.c 4
84.n even 6 1 1176.2.a.k 2
168.s odd 6 1 1344.2.q.x 4
168.s odd 6 1 9408.2.a.dp 2
168.v even 6 1 1344.2.q.w 4
168.v even 6 1 9408.2.a.ec 2
168.ba even 6 1 9408.2.a.dw 2
168.be odd 6 1 9408.2.a.dj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 12.b even 2 1
168.2.q.c 4 84.n even 6 1
336.2.q.g 4 3.b odd 2 1
336.2.q.g 4 21.h odd 6 1
504.2.s.i 4 4.b odd 2 1
504.2.s.i 4 28.g odd 6 1
1008.2.s.r 4 1.a even 1 1 trivial
1008.2.s.r 4 7.c even 3 1 inner
1176.2.a.k 2 84.n even 6 1
1176.2.a.n 2 84.j odd 6 1
1176.2.q.l 4 84.h odd 2 1
1176.2.q.l 4 84.j odd 6 1
1344.2.q.w 4 24.f even 2 1
1344.2.q.w 4 168.v even 6 1
1344.2.q.x 4 24.h odd 2 1
1344.2.q.x 4 168.s odd 6 1
2352.2.a.ba 2 21.g even 6 1
2352.2.a.bf 2 21.h odd 6 1
2352.2.q.bf 4 21.c even 2 1
2352.2.q.bf 4 21.g even 6 1
3528.2.a.bd 2 28.f even 6 1
3528.2.a.bk 2 28.g odd 6 1
3528.2.s.bk 4 28.d even 2 1
3528.2.s.bk 4 28.f even 6 1
7056.2.a.ch 2 7.d odd 6 1
7056.2.a.cu 2 7.c even 3 1
9408.2.a.dj 2 168.be odd 6 1
9408.2.a.dp 2 168.s odd 6 1
9408.2.a.dw 2 168.ba even 6 1
9408.2.a.ec 2 168.v even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{4} + T_{5}^{3} + 15 T_{5}^{2} - 14 T_{5} + 196 \)
\( T_{11}^{4} - T_{11}^{3} + 15 T_{11}^{2} + 14 T_{11} + 196 \)
\( T_{13}^{2} - 5 T_{13} - 8 \)
\( T_{17}^{2} + 4 T_{17} + 16 \)