Properties

Label 1008.2.s.b
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q -3 \zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -3 \zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + ( -3 + 3 \zeta_{6} ) q^{11} + 2 q^{13} + ( -6 + 6 \zeta_{6} ) q^{17} + 2 \zeta_{6} q^{19} -6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + 9 q^{29} + ( -7 + 7 \zeta_{6} ) q^{31} + ( 9 - 6 \zeta_{6} ) q^{35} + 10 \zeta_{6} q^{37} + 4 q^{43} + 12 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 3 - 3 \zeta_{6} ) q^{53} + 9 q^{55} + ( -3 + 3 \zeta_{6} ) q^{59} + 4 \zeta_{6} q^{61} -6 \zeta_{6} q^{65} + ( 2 - 2 \zeta_{6} ) q^{67} + ( -2 + 2 \zeta_{6} ) q^{73} + ( -6 - 3 \zeta_{6} ) q^{77} + 5 \zeta_{6} q^{79} -9 q^{83} + 18 q^{85} + 6 \zeta_{6} q^{89} + ( -2 + 6 \zeta_{6} ) q^{91} + ( 6 - 6 \zeta_{6} ) q^{95} -13 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} + q^{7} + O(q^{10}) \) \( 2q - 3q^{5} + q^{7} - 3q^{11} + 4q^{13} - 6q^{17} + 2q^{19} - 6q^{23} - 4q^{25} + 18q^{29} - 7q^{31} + 12q^{35} + 10q^{37} + 8q^{43} + 12q^{47} - 13q^{49} + 3q^{53} + 18q^{55} - 3q^{59} + 4q^{61} - 6q^{65} + 2q^{67} - 2q^{73} - 15q^{77} + 5q^{79} - 18q^{83} + 36q^{85} + 6q^{89} + 2q^{91} + 6q^{95} - 26q^{97} + O(q^{100}) \)

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 −1.50000 + 2.59808i 0 0.500000 2.59808i 0 0 0
865.1 0 0 0 −1.50000 2.59808i 0 0.500000 + 2.59808i 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.b 2
3.b odd 2 1 1008.2.s.o 2
4.b odd 2 1 126.2.g.a 2
7.c even 3 1 inner 1008.2.s.b 2
7.c even 3 1 7056.2.a.by 1
7.d odd 6 1 7056.2.a.h 1
12.b even 2 1 126.2.g.d yes 2
21.g even 6 1 7056.2.a.bx 1
21.h odd 6 1 1008.2.s.o 2
21.h odd 6 1 7056.2.a.e 1
28.d even 2 1 882.2.g.e 2
28.f even 6 1 882.2.a.h 1
28.f even 6 1 882.2.g.e 2
28.g odd 6 1 126.2.g.a 2
28.g odd 6 1 882.2.a.j 1
36.f odd 6 1 1134.2.e.k 2
36.f odd 6 1 1134.2.h.f 2
36.h even 6 1 1134.2.e.g 2
36.h even 6 1 1134.2.h.j 2
84.h odd 2 1 882.2.g.g 2
84.j odd 6 1 882.2.a.e 1
84.j odd 6 1 882.2.g.g 2
84.n even 6 1 126.2.g.d yes 2
84.n even 6 1 882.2.a.a 1
252.o even 6 1 1134.2.e.g 2
252.u odd 6 1 1134.2.h.f 2
252.bb even 6 1 1134.2.h.j 2
252.bl odd 6 1 1134.2.e.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.g.a 2 4.b odd 2 1
126.2.g.a 2 28.g odd 6 1
126.2.g.d yes 2 12.b even 2 1
126.2.g.d yes 2 84.n even 6 1
882.2.a.a 1 84.n even 6 1
882.2.a.e 1 84.j odd 6 1
882.2.a.h 1 28.f even 6 1
882.2.a.j 1 28.g odd 6 1
882.2.g.e 2 28.d even 2 1
882.2.g.e 2 28.f even 6 1
882.2.g.g 2 84.h odd 2 1
882.2.g.g 2 84.j odd 6 1
1008.2.s.b 2 1.a even 1 1 trivial
1008.2.s.b 2 7.c even 3 1 inner
1008.2.s.o 2 3.b odd 2 1
1008.2.s.o 2 21.h odd 6 1
1134.2.e.g 2 36.h even 6 1
1134.2.e.g 2 252.o even 6 1
1134.2.e.k 2 36.f odd 6 1
1134.2.e.k 2 252.bl odd 6 1
1134.2.h.f 2 36.f odd 6 1
1134.2.h.f 2 252.u odd 6 1
1134.2.h.j 2 36.h even 6 1
1134.2.h.j 2 252.bb even 6 1
7056.2.a.e 1 21.h odd 6 1
7056.2.a.h 1 7.d odd 6 1
7056.2.a.bx 1 21.g even 6 1
7056.2.a.by 1 7.c even 3 1

Hecke kernels

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

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{13} - 2 \)
\( T_{17}^{2} + 6 T_{17} + 36 \)