Properties

Label 507.2.k.b
Level $507$
Weight $2$
Character orbit 507.k
Analytic conductor $4.048$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(80,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.80");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.k (of order \(12\), degree \(4\), 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: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \cdots + 1) q^{7}+ \cdots + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \cdots + 1) q^{7}+ \cdots + (11 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + \cdots + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{7} - 6 q^{9} + 8 q^{16} + 2 q^{19} + 18 q^{21} + 16 q^{28} - 14 q^{31} + 22 q^{37} + 6 q^{43} - 18 q^{49} + 12 q^{57} - 30 q^{63} - 10 q^{67} - 34 q^{73} + 30 q^{75} - 4 q^{76} - 18 q^{81} - 24 q^{84} - 36 q^{93} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(\zeta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 0.866025 + 1.50000i 1.73205 + 1.00000i 0 0 2.86603 0.767949i 0 −1.50000 + 2.59808i 0
89.1 0 −0.866025 1.50000i −1.73205 1.00000i 0 0 1.13397 + 4.23205i 0 −1.50000 + 2.59808i 0
188.1 0 −0.866025 + 1.50000i −1.73205 + 1.00000i 0 0 1.13397 4.23205i 0 −1.50000 2.59808i 0
488.1 0 0.866025 1.50000i 1.73205 1.00000i 0 0 2.86603 + 0.767949i 0 −1.50000 2.59808i 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.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.k.b 4
3.b odd 2 1 CM 507.2.k.b 4
13.b even 2 1 507.2.k.a 4
13.c even 3 1 39.2.k.a 4
13.c even 3 1 507.2.f.c 4
13.d odd 4 1 39.2.k.a 4
13.d odd 4 1 507.2.k.c 4
13.e even 6 1 507.2.f.b 4
13.e even 6 1 507.2.k.c 4
13.f odd 12 1 507.2.f.b 4
13.f odd 12 1 507.2.f.c 4
13.f odd 12 1 507.2.k.a 4
13.f odd 12 1 inner 507.2.k.b 4
39.d odd 2 1 507.2.k.a 4
39.f even 4 1 39.2.k.a 4
39.f even 4 1 507.2.k.c 4
39.h odd 6 1 507.2.f.b 4
39.h odd 6 1 507.2.k.c 4
39.i odd 6 1 39.2.k.a 4
39.i odd 6 1 507.2.f.c 4
39.k even 12 1 507.2.f.b 4
39.k even 12 1 507.2.f.c 4
39.k even 12 1 507.2.k.a 4
39.k even 12 1 inner 507.2.k.b 4
52.f even 4 1 624.2.cn.b 4
52.j odd 6 1 624.2.cn.b 4
65.f even 4 1 975.2.bp.d 4
65.g odd 4 1 975.2.bo.c 4
65.k even 4 1 975.2.bp.a 4
65.n even 6 1 975.2.bo.c 4
65.q odd 12 1 975.2.bp.a 4
65.q odd 12 1 975.2.bp.d 4
156.l odd 4 1 624.2.cn.b 4
156.p even 6 1 624.2.cn.b 4
195.j odd 4 1 975.2.bp.a 4
195.n even 4 1 975.2.bo.c 4
195.u odd 4 1 975.2.bp.d 4
195.x odd 6 1 975.2.bo.c 4
195.bl even 12 1 975.2.bp.a 4
195.bl even 12 1 975.2.bp.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.a 4 13.c even 3 1
39.2.k.a 4 13.d odd 4 1
39.2.k.a 4 39.f even 4 1
39.2.k.a 4 39.i odd 6 1
507.2.f.b 4 13.e even 6 1
507.2.f.b 4 13.f odd 12 1
507.2.f.b 4 39.h odd 6 1
507.2.f.b 4 39.k even 12 1
507.2.f.c 4 13.c even 3 1
507.2.f.c 4 13.f odd 12 1
507.2.f.c 4 39.i odd 6 1
507.2.f.c 4 39.k even 12 1
507.2.k.a 4 13.b even 2 1
507.2.k.a 4 13.f odd 12 1
507.2.k.a 4 39.d odd 2 1
507.2.k.a 4 39.k even 12 1
507.2.k.b 4 1.a even 1 1 trivial
507.2.k.b 4 3.b odd 2 1 CM
507.2.k.b 4 13.f odd 12 1 inner
507.2.k.b 4 39.k even 12 1 inner
507.2.k.c 4 13.d odd 4 1
507.2.k.c 4 13.e even 6 1
507.2.k.c 4 39.f even 4 1
507.2.k.c 4 39.h odd 6 1
624.2.cn.b 4 52.f even 4 1
624.2.cn.b 4 52.j odd 6 1
624.2.cn.b 4 156.l odd 4 1
624.2.cn.b 4 156.p even 6 1
975.2.bo.c 4 65.g odd 4 1
975.2.bo.c 4 65.n even 6 1
975.2.bo.c 4 195.n even 4 1
975.2.bo.c 4 195.x odd 6 1
975.2.bp.a 4 65.k even 4 1
975.2.bp.a 4 65.q odd 12 1
975.2.bp.a 4 195.j odd 4 1
975.2.bp.a 4 195.bl even 12 1
975.2.bp.d 4 65.f even 4 1
975.2.bp.d 4 65.q odd 12 1
975.2.bp.d 4 195.u odd 4 1
975.2.bp.d 4 195.bl even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{4} - 8T_{7}^{3} + 41T_{7}^{2} - 130T_{7} + 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 8 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 14 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$37$ \( T^{4} - 22 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$67$ \( T^{4} + 10 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 34 T^{3} + \cdots + 20449 \) Copy content Toggle raw display
$79$ \( (T^{2} - 147)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 28 T^{3} + \cdots + 28561 \) Copy content Toggle raw display
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