Properties

Label 504.1.l.a
Level 504
Weight 1
Character orbit 504.l
Self dual yes
Analytic conductor 0.252
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -7, -56, 8
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 504.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.251528766367\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{2}, \sqrt{-7})\)
Artin image $D_4$
Artin field Galois closure of 4.0.3528.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} - q^{7} + q^{8} - q^{14} + q^{16} - 2q^{23} - q^{25} - q^{28} + q^{32} - 2q^{46} + q^{49} - q^{50} - q^{56} + q^{64} + 2q^{71} - 2q^{79} - 2q^{92} + q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
0
1.00000 0 1.00000 0 0 −1.00000 1.00000 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.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
8.b even 2 1 RM by \(\Q(\sqrt{2}) \)
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.1.l.a 1
3.b odd 2 1 56.1.h.a 1
4.b odd 2 1 2016.1.l.a 1
7.b odd 2 1 CM 504.1.l.a 1
7.c even 3 2 3528.1.bw.a 2
7.d odd 6 2 3528.1.bw.a 2
8.b even 2 1 RM 504.1.l.a 1
8.d odd 2 1 2016.1.l.a 1
12.b even 2 1 224.1.h.a 1
15.d odd 2 1 1400.1.m.a 1
15.e even 4 2 1400.1.c.a 2
21.c even 2 1 56.1.h.a 1
21.g even 6 2 392.1.j.a 2
21.h odd 6 2 392.1.j.a 2
24.f even 2 1 224.1.h.a 1
24.h odd 2 1 56.1.h.a 1
28.d even 2 1 2016.1.l.a 1
48.i odd 4 2 1792.1.c.b 1
48.k even 4 2 1792.1.c.a 1
56.e even 2 1 2016.1.l.a 1
56.h odd 2 1 CM 504.1.l.a 1
56.j odd 6 2 3528.1.bw.a 2
56.p even 6 2 3528.1.bw.a 2
84.h odd 2 1 224.1.h.a 1
84.j odd 6 2 1568.1.n.a 2
84.n even 6 2 1568.1.n.a 2
105.g even 2 1 1400.1.m.a 1
105.k odd 4 2 1400.1.c.a 2
120.i odd 2 1 1400.1.m.a 1
120.w even 4 2 1400.1.c.a 2
168.e odd 2 1 224.1.h.a 1
168.i even 2 1 56.1.h.a 1
168.s odd 6 2 392.1.j.a 2
168.v even 6 2 1568.1.n.a 2
168.ba even 6 2 392.1.j.a 2
168.be odd 6 2 1568.1.n.a 2
336.v odd 4 2 1792.1.c.a 1
336.y even 4 2 1792.1.c.b 1
840.u even 2 1 1400.1.m.a 1
840.bp odd 4 2 1400.1.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.1.h.a 1 3.b odd 2 1
56.1.h.a 1 21.c even 2 1
56.1.h.a 1 24.h odd 2 1
56.1.h.a 1 168.i even 2 1
224.1.h.a 1 12.b even 2 1
224.1.h.a 1 24.f even 2 1
224.1.h.a 1 84.h odd 2 1
224.1.h.a 1 168.e odd 2 1
392.1.j.a 2 21.g even 6 2
392.1.j.a 2 21.h odd 6 2
392.1.j.a 2 168.s odd 6 2
392.1.j.a 2 168.ba even 6 2
504.1.l.a 1 1.a even 1 1 trivial
504.1.l.a 1 7.b odd 2 1 CM
504.1.l.a 1 8.b even 2 1 RM
504.1.l.a 1 56.h odd 2 1 CM
1400.1.c.a 2 15.e even 4 2
1400.1.c.a 2 105.k odd 4 2
1400.1.c.a 2 120.w even 4 2
1400.1.c.a 2 840.bp odd 4 2
1400.1.m.a 1 15.d odd 2 1
1400.1.m.a 1 105.g even 2 1
1400.1.m.a 1 120.i odd 2 1
1400.1.m.a 1 840.u even 2 1
1568.1.n.a 2 84.j odd 6 2
1568.1.n.a 2 84.n even 6 2
1568.1.n.a 2 168.v even 6 2
1568.1.n.a 2 168.be odd 6 2
1792.1.c.a 1 48.k even 4 2
1792.1.c.a 1 336.v odd 4 2
1792.1.c.b 1 48.i odd 4 2
1792.1.c.b 1 336.y even 4 2
2016.1.l.a 1 4.b odd 2 1
2016.1.l.a 1 8.d odd 2 1
2016.1.l.a 1 28.d even 2 1
2016.1.l.a 1 56.e even 2 1
3528.1.bw.a 2 7.c even 3 2
3528.1.bw.a 2 7.d odd 6 2
3528.1.bw.a 2 56.j odd 6 2
3528.1.bw.a 2 56.p even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} \) acting on \(S_{1}^{\mathrm{new}}(504, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ 1
$5$ \( 1 + T^{2} \)
$7$ \( 1 + T \)
$11$ \( ( 1 - T )( 1 + T ) \)
$13$ \( 1 + T^{2} \)
$17$ \( ( 1 - T )( 1 + T ) \)
$19$ \( 1 + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( 1 - T )( 1 + T ) \)
$31$ \( ( 1 - T )( 1 + T ) \)
$37$ \( ( 1 - T )( 1 + T ) \)
$41$ \( ( 1 - T )( 1 + T ) \)
$43$ \( ( 1 - T )( 1 + T ) \)
$47$ \( ( 1 - T )( 1 + T ) \)
$53$ \( ( 1 - T )( 1 + T ) \)
$59$ \( 1 + T^{2} \)
$61$ \( 1 + T^{2} \)
$67$ \( ( 1 - T )( 1 + T ) \)
$71$ \( ( 1 - T )^{2} \)
$73$ \( ( 1 - T )( 1 + T ) \)
$79$ \( ( 1 + T )^{2} \)
$83$ \( 1 + T^{2} \)
$89$ \( ( 1 - T )( 1 + T ) \)
$97$ \( ( 1 - T )( 1 + T ) \)
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