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: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 2 \beta_1 - 1) q^{5} + ( - \beta_{3} + \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 2 \beta_1 - 1) q^{5} + ( - \beta_{3} + \beta_1 + 1) q^{7} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{11} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{13} + (4 \beta_{2} - 4) q^{17} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{19} - 4 \beta_{2} q^{23} + (2 \beta_{3} + 10 \beta_{2} - \beta_1 - 9) q^{25} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{29} + (\beta_{2} - 1) q^{31} + ( - \beta_{3} + 5 \beta_{2} + 3 \beta_1 - 11) q^{35} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{37} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{41} + ( - \beta_{3} - \beta_{2} - \beta_1 + 4) q^{43} + 6 \beta_{2} q^{47} + ( - 3 \beta_{3} + 3 \beta_1 - 4) q^{49} + (2 \beta_{3} - 5 \beta_{2} - \beta_1 + 6) q^{53} + (\beta_{3} + \beta_{2} + \beta_1 - 15) q^{55} + (2 \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{59} - 10 \beta_{2} q^{61} + ( - 2 \beta_{3} + 14 \beta_{2} + 4 \beta_1 - 2) q^{65} + ( - 2 \beta_{3} - 4 \beta_{2} + \beta_1 + 3) q^{67} + 2 q^{71} + (2 \beta_{3} - \beta_1 + 1) q^{73} + ( - 3 \beta_{3} - 6 \beta_{2} + 2 \beta_1 - 5) q^{77} + ( - 2 \beta_{3} + 5 \beta_{2} + 4 \beta_1 - 2) q^{79} + (\beta_{3} + \beta_{2} + \beta_1 + 3) q^{83} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 4) q^{85} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 2) q^{89} + ( - \beta_{3} + 11 \beta_{2} + 4 \beta_1 - 3) q^{91} + (4 \beta_{3} - 12 \beta_{2} - 2 \beta_1 + 14) q^{95} + ( - \beta_{3} - \beta_{2} - \beta_1 + 13) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{5} + 6 q^{7} + q^{11} + 10 q^{13} - 8 q^{17} - 5 q^{19} - 8 q^{23} - 19 q^{25} - 6 q^{29} - 2 q^{31} - 30 q^{35} - 3 q^{37} - 12 q^{41} + 14 q^{43} + 12 q^{47} - 10 q^{49} + 11 q^{53} - 58 q^{55} - 5 q^{59} - 20 q^{61} + 26 q^{65} + 7 q^{67} + 8 q^{71} + q^{73} - 27 q^{77} + 8 q^{79} + 14 q^{83} + 8 q^{85} - 6 q^{89} + 15 q^{91} + 26 q^{95} + 50 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 5 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu + 5 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{3} + 4\beta _1 + 5 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + 15 T^{2} - 14 T + 196 \) Copy content Toggle raw display
$7$ \( (T^{2} - 3 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} + 15 T^{2} + 14 T + 196 \) Copy content Toggle raw display
$13$ \( (T^{2} - 5 T - 8)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 5 T^{3} + 33 T^{2} - 40 T + 64 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T - 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 3 T^{3} + 21 T^{2} - 36 T + 144 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 7 T - 2)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 11 T^{3} + 105 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( T^{4} + 5 T^{3} + 33 T^{2} - 40 T + 64 \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 7 T^{3} + 51 T^{2} + 14 T + 4 \) Copy content Toggle raw display
$71$ \( (T - 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - T^{3} + 15 T^{2} + 14 T + 196 \) Copy content Toggle raw display
$79$ \( T^{4} - 8 T^{3} + 105 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$83$ \( (T^{2} - 7 T - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} + 84 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$97$ \( (T^{2} - 25 T + 142)^{2} \) Copy content Toggle raw display
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