Properties

Label 1008.1.f.a
Level $1008$
Weight $1$
Character orbit 1008.f
Self dual yes
Analytic conductor $0.503$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -3, -7, 21
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1008.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\)
Artin image $D_4$
Artin field Galois closure of 4.0.3024.2

$q$-expansion

\(f(q)\) \(=\) \( q + q^{7} + O(q^{10}) \) \( q + q^{7} + q^{25} - 2q^{37} + 2q^{43} + q^{49} - 2q^{67} - 2q^{79} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
0
0 0 0 0 0 1.00000 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.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
21.c even 2 1 RM by \(\Q(\sqrt{21}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.1.f.a 1
3.b odd 2 1 CM 1008.1.f.a 1
4.b odd 2 1 63.1.d.a 1
7.b odd 2 1 CM 1008.1.f.a 1
12.b even 2 1 63.1.d.a 1
20.d odd 2 1 1575.1.h.b 1
20.e even 4 2 1575.1.e.b 2
21.c even 2 1 RM 1008.1.f.a 1
28.d even 2 1 63.1.d.a 1
28.f even 6 2 441.1.m.a 2
28.g odd 6 2 441.1.m.a 2
36.f odd 6 2 567.1.l.b 2
36.h even 6 2 567.1.l.b 2
60.h even 2 1 1575.1.h.b 1
60.l odd 4 2 1575.1.e.b 2
84.h odd 2 1 63.1.d.a 1
84.j odd 6 2 441.1.m.a 2
84.n even 6 2 441.1.m.a 2
140.c even 2 1 1575.1.h.b 1
140.j odd 4 2 1575.1.e.b 2
252.n even 6 2 3969.1.k.b 2
252.o even 6 2 3969.1.k.b 2
252.r odd 6 2 3969.1.t.c 2
252.s odd 6 2 567.1.l.b 2
252.u odd 6 2 3969.1.t.c 2
252.bb even 6 2 3969.1.t.c 2
252.bi even 6 2 567.1.l.b 2
252.bj even 6 2 3969.1.t.c 2
252.bl odd 6 2 3969.1.k.b 2
252.bn odd 6 2 3969.1.k.b 2
420.o odd 2 1 1575.1.h.b 1
420.w even 4 2 1575.1.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.1.d.a 1 4.b odd 2 1
63.1.d.a 1 12.b even 2 1
63.1.d.a 1 28.d even 2 1
63.1.d.a 1 84.h odd 2 1
441.1.m.a 2 28.f even 6 2
441.1.m.a 2 28.g odd 6 2
441.1.m.a 2 84.j odd 6 2
441.1.m.a 2 84.n even 6 2
567.1.l.b 2 36.f odd 6 2
567.1.l.b 2 36.h even 6 2
567.1.l.b 2 252.s odd 6 2
567.1.l.b 2 252.bi even 6 2
1008.1.f.a 1 1.a even 1 1 trivial
1008.1.f.a 1 3.b odd 2 1 CM
1008.1.f.a 1 7.b odd 2 1 CM
1008.1.f.a 1 21.c even 2 1 RM
1575.1.e.b 2 20.e even 4 2
1575.1.e.b 2 60.l odd 4 2
1575.1.e.b 2 140.j odd 4 2
1575.1.e.b 2 420.w even 4 2
1575.1.h.b 1 20.d odd 2 1
1575.1.h.b 1 60.h even 2 1
1575.1.h.b 1 140.c even 2 1
1575.1.h.b 1 420.o odd 2 1
3969.1.k.b 2 252.n even 6 2
3969.1.k.b 2 252.o even 6 2
3969.1.k.b 2 252.bl odd 6 2
3969.1.k.b 2 252.bn odd 6 2
3969.1.t.c 2 252.r odd 6 2
3969.1.t.c 2 252.u odd 6 2
3969.1.t.c 2 252.bb even 6 2
3969.1.t.c 2 252.bj even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1008, [\chi])\).