Properties

Label 507.1.h.a
Level $507$
Weight $1$
Character orbit 507.h
Analytic conductor $0.253$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -3, -39, 13
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 507.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.253025961405\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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: $C_3\times D_4$
Artin field: Galois closure of 12.6.3722179279923.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{3} -\zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{6} q^{3} -\zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{9} + q^{12} -\zeta_{6} q^{16} - q^{25} - q^{27} + \zeta_{6} q^{36} + 2 \zeta_{6}^{2} q^{43} -\zeta_{6}^{2} q^{48} -\zeta_{6} q^{49} -2 \zeta_{6}^{2} q^{61} - q^{64} -\zeta_{6} q^{75} -2 q^{79} -\zeta_{6} q^{81} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + q^{4} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + q^{4} - q^{9} + 2q^{12} - q^{16} - 2q^{25} - 2q^{27} + q^{36} - 2q^{43} + q^{48} - q^{49} + 2q^{61} - 2q^{64} - q^{75} - 4q^{79} - q^{81} + O(q^{100}) \)

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 + 0.866025i 0.500000 0.866025i 0 0 0 0 −0.500000 + 0.866025i 0
485.1 0 0.500000 0.866025i 0.500000 + 0.866025i 0 0 0 0 −0.500000 0.866025i 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 \(\Q(\sqrt{-39}) \)
13.c even 3 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner
39.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.1.h.a 2
3.b odd 2 1 CM 507.1.h.a 2
13.b even 2 1 RM 507.1.h.a 2
13.c even 3 1 39.1.d.a 1
13.c even 3 1 inner 507.1.h.a 2
13.d odd 4 2 507.1.i.a 2
13.e even 6 1 39.1.d.a 1
13.e even 6 1 inner 507.1.h.a 2
13.f odd 12 2 507.1.c.a 1
13.f odd 12 2 507.1.i.a 2
39.d odd 2 1 CM 507.1.h.a 2
39.f even 4 2 507.1.i.a 2
39.h odd 6 1 39.1.d.a 1
39.h odd 6 1 inner 507.1.h.a 2
39.i odd 6 1 39.1.d.a 1
39.i odd 6 1 inner 507.1.h.a 2
39.k even 12 2 507.1.c.a 1
39.k even 12 2 507.1.i.a 2
52.i odd 6 1 624.1.l.a 1
52.j odd 6 1 624.1.l.a 1
65.l even 6 1 975.1.g.a 1
65.n even 6 1 975.1.g.a 1
65.q odd 12 2 975.1.e.a 2
65.r odd 12 2 975.1.e.a 2
91.g even 3 1 1911.1.w.b 2
91.h even 3 1 1911.1.w.b 2
91.k even 6 1 1911.1.w.b 2
91.l odd 6 1 1911.1.w.a 2
91.m odd 6 1 1911.1.w.a 2
91.n odd 6 1 1911.1.h.a 1
91.p odd 6 1 1911.1.w.a 2
91.t odd 6 1 1911.1.h.a 1
91.u even 6 1 1911.1.w.b 2
91.v odd 6 1 1911.1.w.a 2
104.n odd 6 1 2496.1.l.a 1
104.p odd 6 1 2496.1.l.a 1
104.r even 6 1 2496.1.l.b 1
104.s even 6 1 2496.1.l.b 1
117.f even 3 1 1053.1.n.b 2
117.h even 3 1 1053.1.n.b 2
117.k odd 6 1 1053.1.n.b 2
117.l even 6 1 1053.1.n.b 2
117.m odd 6 1 1053.1.n.b 2
117.r even 6 1 1053.1.n.b 2
117.u odd 6 1 1053.1.n.b 2
117.v odd 6 1 1053.1.n.b 2
156.p even 6 1 624.1.l.a 1
156.r even 6 1 624.1.l.a 1
195.x odd 6 1 975.1.g.a 1
195.y odd 6 1 975.1.g.a 1
195.bf even 12 2 975.1.e.a 2
195.bl even 12 2 975.1.e.a 2
273.r even 6 1 1911.1.w.a 2
273.s odd 6 1 1911.1.w.b 2
273.u even 6 1 1911.1.h.a 1
273.x odd 6 1 1911.1.w.b 2
273.y even 6 1 1911.1.w.a 2
273.bf even 6 1 1911.1.w.a 2
273.bm odd 6 1 1911.1.w.b 2
273.bn even 6 1 1911.1.h.a 1
273.bp odd 6 1 1911.1.w.b 2
273.br even 6 1 1911.1.w.a 2
312.ba even 6 1 2496.1.l.a 1
312.bg odd 6 1 2496.1.l.b 1
312.bh odd 6 1 2496.1.l.b 1
312.bn even 6 1 2496.1.l.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.1.d.a 1 13.c even 3 1
39.1.d.a 1 13.e even 6 1
39.1.d.a 1 39.h odd 6 1
39.1.d.a 1 39.i odd 6 1
507.1.c.a 1 13.f odd 12 2
507.1.c.a 1 39.k even 12 2
507.1.h.a 2 1.a even 1 1 trivial
507.1.h.a 2 3.b odd 2 1 CM
507.1.h.a 2 13.b even 2 1 RM
507.1.h.a 2 13.c even 3 1 inner
507.1.h.a 2 13.e even 6 1 inner
507.1.h.a 2 39.d odd 2 1 CM
507.1.h.a 2 39.h odd 6 1 inner
507.1.h.a 2 39.i odd 6 1 inner
507.1.i.a 2 13.d odd 4 2
507.1.i.a 2 13.f odd 12 2
507.1.i.a 2 39.f even 4 2
507.1.i.a 2 39.k even 12 2
624.1.l.a 1 52.i odd 6 1
624.1.l.a 1 52.j odd 6 1
624.1.l.a 1 156.p even 6 1
624.1.l.a 1 156.r even 6 1
975.1.e.a 2 65.q odd 12 2
975.1.e.a 2 65.r odd 12 2
975.1.e.a 2 195.bf even 12 2
975.1.e.a 2 195.bl even 12 2
975.1.g.a 1 65.l even 6 1
975.1.g.a 1 65.n even 6 1
975.1.g.a 1 195.x odd 6 1
975.1.g.a 1 195.y odd 6 1
1053.1.n.b 2 117.f even 3 1
1053.1.n.b 2 117.h even 3 1
1053.1.n.b 2 117.k odd 6 1
1053.1.n.b 2 117.l even 6 1
1053.1.n.b 2 117.m odd 6 1
1053.1.n.b 2 117.r even 6 1
1053.1.n.b 2 117.u odd 6 1
1053.1.n.b 2 117.v odd 6 1
1911.1.h.a 1 91.n odd 6 1
1911.1.h.a 1 91.t odd 6 1
1911.1.h.a 1 273.u even 6 1
1911.1.h.a 1 273.bn even 6 1
1911.1.w.a 2 91.l odd 6 1
1911.1.w.a 2 91.m odd 6 1
1911.1.w.a 2 91.p odd 6 1
1911.1.w.a 2 91.v odd 6 1
1911.1.w.a 2 273.r even 6 1
1911.1.w.a 2 273.y even 6 1
1911.1.w.a 2 273.bf even 6 1
1911.1.w.a 2 273.br even 6 1
1911.1.w.b 2 91.g even 3 1
1911.1.w.b 2 91.h even 3 1
1911.1.w.b 2 91.k even 6 1
1911.1.w.b 2 91.u even 6 1
1911.1.w.b 2 273.s odd 6 1
1911.1.w.b 2 273.x odd 6 1
1911.1.w.b 2 273.bm odd 6 1
1911.1.w.b 2 273.bp odd 6 1
2496.1.l.a 1 104.n odd 6 1
2496.1.l.a 1 104.p odd 6 1
2496.1.l.a 1 312.ba even 6 1
2496.1.l.a 1 312.bn even 6 1
2496.1.l.b 1 104.r even 6 1
2496.1.l.b 1 104.s even 6 1
2496.1.l.b 1 312.bg odd 6 1
2496.1.l.b 1 312.bh odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( 4 + 2 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 4 - 2 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 2 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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