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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,1,Mod(23,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.23");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 507.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.253025961405\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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

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