Properties

Label 1008.2.s.m
Level $1008$
Weight $2$
Character orbit 1008.s
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.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 2 \zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} + ( 6 - 6 \zeta_{6} ) q^{11} -3 q^{13} + ( 4 - 4 \zeta_{6} ) q^{17} -5 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + 4 q^{29} + ( 7 - 7 \zeta_{6} ) q^{31} + ( 2 - 6 \zeta_{6} ) q^{35} + 9 \zeta_{6} q^{37} + 2 q^{41} + q^{43} -2 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + ( 8 - 8 \zeta_{6} ) q^{53} + 12 q^{55} -10 \zeta_{6} q^{61} -6 \zeta_{6} q^{65} + ( -15 + 15 \zeta_{6} ) q^{67} -6 q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} + ( -18 + 12 \zeta_{6} ) q^{77} + \zeta_{6} q^{79} + 6 q^{83} + 8 q^{85} -8 \zeta_{6} q^{89} + ( 6 + 3 \zeta_{6} ) q^{91} + ( 10 - 10 \zeta_{6} ) q^{95} -14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 5q^{7} + O(q^{10}) \) \( 2q + 2q^{5} - 5q^{7} + 6q^{11} - 6q^{13} + 4q^{17} - 5q^{19} + 4q^{23} + q^{25} + 8q^{29} + 7q^{31} - 2q^{35} + 9q^{37} + 4q^{41} + 2q^{43} - 2q^{47} + 11q^{49} + 8q^{53} + 24q^{55} - 10q^{61} - 6q^{65} - 15q^{67} - 12q^{71} + 11q^{73} - 24q^{77} + q^{79} + 12q^{83} + 16q^{85} - 8q^{89} + 15q^{91} + 10q^{95} - 28q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\) \(1\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 1.00000 1.73205i 0 −2.50000 + 0.866025i 0 0 0
865.1 0 0 0 1.00000 + 1.73205i 0 −2.50000 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
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.m 2
3.b odd 2 1 336.2.q.a 2
4.b odd 2 1 504.2.s.g 2
7.c even 3 1 inner 1008.2.s.m 2
7.c even 3 1 7056.2.a.i 1
7.d odd 6 1 7056.2.a.bn 1
12.b even 2 1 168.2.q.b 2
21.c even 2 1 2352.2.q.v 2
21.g even 6 1 2352.2.a.e 1
21.g even 6 1 2352.2.q.v 2
21.h odd 6 1 336.2.q.a 2
21.h odd 6 1 2352.2.a.x 1
24.f even 2 1 1344.2.q.i 2
24.h odd 2 1 1344.2.q.t 2
28.d even 2 1 3528.2.s.d 2
28.f even 6 1 3528.2.a.y 1
28.f even 6 1 3528.2.s.d 2
28.g odd 6 1 504.2.s.g 2
28.g odd 6 1 3528.2.a.f 1
84.h odd 2 1 1176.2.q.e 2
84.j odd 6 1 1176.2.a.e 1
84.j odd 6 1 1176.2.q.e 2
84.n even 6 1 168.2.q.b 2
84.n even 6 1 1176.2.a.d 1
168.s odd 6 1 1344.2.q.t 2
168.s odd 6 1 9408.2.a.f 1
168.v even 6 1 1344.2.q.i 2
168.v even 6 1 9408.2.a.cd 1
168.ba even 6 1 9408.2.a.cs 1
168.be odd 6 1 9408.2.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.b 2 12.b even 2 1
168.2.q.b 2 84.n even 6 1
336.2.q.a 2 3.b odd 2 1
336.2.q.a 2 21.h odd 6 1
504.2.s.g 2 4.b odd 2 1
504.2.s.g 2 28.g odd 6 1
1008.2.s.m 2 1.a even 1 1 trivial
1008.2.s.m 2 7.c even 3 1 inner
1176.2.a.d 1 84.n even 6 1
1176.2.a.e 1 84.j odd 6 1
1176.2.q.e 2 84.h odd 2 1
1176.2.q.e 2 84.j odd 6 1
1344.2.q.i 2 24.f even 2 1
1344.2.q.i 2 168.v even 6 1
1344.2.q.t 2 24.h odd 2 1
1344.2.q.t 2 168.s odd 6 1
2352.2.a.e 1 21.g even 6 1
2352.2.a.x 1 21.h odd 6 1
2352.2.q.v 2 21.c even 2 1
2352.2.q.v 2 21.g even 6 1
3528.2.a.f 1 28.g odd 6 1
3528.2.a.y 1 28.f even 6 1
3528.2.s.d 2 28.d even 2 1
3528.2.s.d 2 28.f even 6 1
7056.2.a.i 1 7.c even 3 1
7056.2.a.bn 1 7.d odd 6 1
9408.2.a.f 1 168.s odd 6 1
9408.2.a.bk 1 168.be odd 6 1
9408.2.a.cd 1 168.v even 6 1
9408.2.a.cs 1 168.ba even 6 1

Hecke kernels

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

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{11}^{2} - 6 T_{11} + 36 \)
\( T_{13} + 3 \)
\( T_{17}^{2} - 4 T_{17} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 - 2 T + T^{2} \)
$7$ \( 7 + 5 T + T^{2} \)
$11$ \( 36 - 6 T + T^{2} \)
$13$ \( ( 3 + T )^{2} \)
$17$ \( 16 - 4 T + T^{2} \)
$19$ \( 25 + 5 T + T^{2} \)
$23$ \( 16 - 4 T + T^{2} \)
$29$ \( ( -4 + T )^{2} \)
$31$ \( 49 - 7 T + T^{2} \)
$37$ \( 81 - 9 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( ( -1 + T )^{2} \)
$47$ \( 4 + 2 T + T^{2} \)
$53$ \( 64 - 8 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 100 + 10 T + T^{2} \)
$67$ \( 225 + 15 T + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( 121 - 11 T + T^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 64 + 8 T + T^{2} \)
$97$ \( ( 14 + T )^{2} \)
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