Properties

Label 1008.2.s.i
Level $1008$
Weight $2$
Character orbit 1008.s
Analytic conductor $8.049$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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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 252)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 - \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -2 - \zeta_{6} ) q^{7} + 5 q^{13} -\zeta_{6} q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + ( 11 - 11 \zeta_{6} ) q^{31} -11 \zeta_{6} q^{37} + 13 q^{43} + ( 3 + 5 \zeta_{6} ) q^{49} -14 \zeta_{6} q^{61} + ( 5 - 5 \zeta_{6} ) q^{67} + ( -17 + 17 \zeta_{6} ) q^{73} + 17 \zeta_{6} q^{79} + ( -10 - 5 \zeta_{6} ) q^{91} + 14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 5q^{7} + O(q^{10}) \) \( 2q - 5q^{7} + 10q^{13} - q^{19} + 5q^{25} + 11q^{31} - 11q^{37} + 26q^{43} + 11q^{49} - 14q^{61} + 5q^{67} - 17q^{73} + 17q^{79} - 25q^{91} + 28q^{97} + O(q^{100}) \)

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 0 0 −2.50000 + 0.866025i 0 0 0
865.1 0 0 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.s.i 2
3.b odd 2 1 CM 1008.2.s.i 2
4.b odd 2 1 252.2.k.b 2
7.c even 3 1 inner 1008.2.s.i 2
7.c even 3 1 7056.2.a.be 1
7.d odd 6 1 7056.2.a.z 1
12.b even 2 1 252.2.k.b 2
21.g even 6 1 7056.2.a.z 1
21.h odd 6 1 inner 1008.2.s.i 2
21.h odd 6 1 7056.2.a.be 1
28.d even 2 1 1764.2.k.f 2
28.f even 6 1 1764.2.a.d 1
28.f even 6 1 1764.2.k.f 2
28.g odd 6 1 252.2.k.b 2
28.g odd 6 1 1764.2.a.f 1
36.f odd 6 1 2268.2.i.c 2
36.f odd 6 1 2268.2.l.e 2
36.h even 6 1 2268.2.i.c 2
36.h even 6 1 2268.2.l.e 2
84.h odd 2 1 1764.2.k.f 2
84.j odd 6 1 1764.2.a.d 1
84.j odd 6 1 1764.2.k.f 2
84.n even 6 1 252.2.k.b 2
84.n even 6 1 1764.2.a.f 1
252.o even 6 1 2268.2.i.c 2
252.u odd 6 1 2268.2.l.e 2
252.bb even 6 1 2268.2.l.e 2
252.bl odd 6 1 2268.2.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.k.b 2 4.b odd 2 1
252.2.k.b 2 12.b even 2 1
252.2.k.b 2 28.g odd 6 1
252.2.k.b 2 84.n even 6 1
1008.2.s.i 2 1.a even 1 1 trivial
1008.2.s.i 2 3.b odd 2 1 CM
1008.2.s.i 2 7.c even 3 1 inner
1008.2.s.i 2 21.h odd 6 1 inner
1764.2.a.d 1 28.f even 6 1
1764.2.a.d 1 84.j odd 6 1
1764.2.a.f 1 28.g odd 6 1
1764.2.a.f 1 84.n even 6 1
1764.2.k.f 2 28.d even 2 1
1764.2.k.f 2 28.f even 6 1
1764.2.k.f 2 84.h odd 2 1
1764.2.k.f 2 84.j odd 6 1
2268.2.i.c 2 36.f odd 6 1
2268.2.i.c 2 36.h even 6 1
2268.2.i.c 2 252.o even 6 1
2268.2.i.c 2 252.bl odd 6 1
2268.2.l.e 2 36.f odd 6 1
2268.2.l.e 2 36.h even 6 1
2268.2.l.e 2 252.u odd 6 1
2268.2.l.e 2 252.bb even 6 1
7056.2.a.z 1 7.d odd 6 1
7056.2.a.z 1 21.g even 6 1
7056.2.a.be 1 7.c even 3 1
7056.2.a.be 1 21.h odd 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{11} \)
\( T_{13} - 5 \)
\( T_{17} \)