Properties

Label 507.1.c.a
Level $507$
Weight $1$
Character orbit 507.c
Self dual yes
Analytic conductor $0.253$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -3, -39, 13
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 507.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.253025961405\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{13})\)
Artin image: $D_4$
Artin field: Galois closure of 4.2.6591.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + q^{4} + q^{9} + O(q^{10}) \) \( q - q^{3} + q^{4} + q^{9} - q^{12} + q^{16} + q^{25} - q^{27} + q^{36} - 2q^{43} - q^{48} - q^{49} - 2q^{61} + q^{64} - q^{75} - 2q^{79} + q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
170.1
0
0 −1.00000 1.00000 0 0 0 0 1.00000 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 \(\Q(\sqrt{-3}) \)
13.b even 2 1 RM by \(\Q(\sqrt{13}) \)
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.1.c.a 1
3.b odd 2 1 CM 507.1.c.a 1
13.b even 2 1 RM 507.1.c.a 1
13.c even 3 2 507.1.i.a 2
13.d odd 4 2 39.1.d.a 1
13.e even 6 2 507.1.i.a 2
13.f odd 12 4 507.1.h.a 2
39.d odd 2 1 CM 507.1.c.a 1
39.f even 4 2 39.1.d.a 1
39.h odd 6 2 507.1.i.a 2
39.i odd 6 2 507.1.i.a 2
39.k even 12 4 507.1.h.a 2
52.f even 4 2 624.1.l.a 1
65.f even 4 2 975.1.e.a 2
65.g odd 4 2 975.1.g.a 1
65.k even 4 2 975.1.e.a 2
91.i even 4 2 1911.1.h.a 1
91.z odd 12 4 1911.1.w.b 2
91.bb even 12 4 1911.1.w.a 2
104.j odd 4 2 2496.1.l.b 1
104.m even 4 2 2496.1.l.a 1
117.y odd 12 4 1053.1.n.b 2
117.z even 12 4 1053.1.n.b 2
156.l odd 4 2 624.1.l.a 1
195.j odd 4 2 975.1.e.a 2
195.n even 4 2 975.1.g.a 1
195.u odd 4 2 975.1.e.a 2
273.o odd 4 2 1911.1.h.a 1
273.cb odd 12 4 1911.1.w.a 2
273.cd even 12 4 1911.1.w.b 2
312.w odd 4 2 2496.1.l.a 1
312.y even 4 2 2496.1.l.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.1.d.a 1 13.d odd 4 2
39.1.d.a 1 39.f even 4 2
507.1.c.a 1 1.a even 1 1 trivial
507.1.c.a 1 3.b odd 2 1 CM
507.1.c.a 1 13.b even 2 1 RM
507.1.c.a 1 39.d odd 2 1 CM
507.1.h.a 2 13.f odd 12 4
507.1.h.a 2 39.k even 12 4
507.1.i.a 2 13.c even 3 2
507.1.i.a 2 13.e even 6 2
507.1.i.a 2 39.h odd 6 2
507.1.i.a 2 39.i odd 6 2
624.1.l.a 1 52.f even 4 2
624.1.l.a 1 156.l odd 4 2
975.1.e.a 2 65.f even 4 2
975.1.e.a 2 65.k even 4 2
975.1.e.a 2 195.j odd 4 2
975.1.e.a 2 195.u odd 4 2
975.1.g.a 1 65.g odd 4 2
975.1.g.a 1 195.n even 4 2
1053.1.n.b 2 117.y odd 12 4
1053.1.n.b 2 117.z even 12 4
1911.1.h.a 1 91.i even 4 2
1911.1.h.a 1 273.o odd 4 2
1911.1.w.a 2 91.bb even 12 4
1911.1.w.a 2 273.cb odd 12 4
1911.1.w.b 2 91.z odd 12 4
1911.1.w.b 2 273.cd even 12 4
2496.1.l.a 1 104.m even 4 2
2496.1.l.a 1 312.w odd 4 2
2496.1.l.b 1 104.j odd 4 2
2496.1.l.b 1 312.y even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(507, [\chi])\).

Hecke characteristic polynomials

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