Properties

Label 507.2.b.a
Level $507$
Weight $2$
Character orbit 507.b
Analytic conductor $4.048$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(337,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} - q^{3} + q^{4} + 2 i q^{5} - i q^{6} + 4 i q^{7} + 3 i q^{8} + q^{9} - 2 q^{10} - 4 i q^{11} - q^{12} - 4 q^{14} - 2 i q^{15} - q^{16} - 2 q^{17} + i q^{18} + 2 i q^{20} - 4 i q^{21} + \cdots - 4 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{9} - 4 q^{10} - 2 q^{12} - 8 q^{14} - 2 q^{16} - 4 q^{17} + 8 q^{22} + 2 q^{25} - 2 q^{27} - 20 q^{29} + 4 q^{30} - 16 q^{35} + 2 q^{36} - 12 q^{40} + 8 q^{42} + 24 q^{43}+ \cdots - 4 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
1.00000i −1.00000 1.00000 2.00000i 1.00000i 4.00000i 3.00000i 1.00000 −2.00000
337.2 1.00000i −1.00000 1.00000 2.00000i 1.00000i 4.00000i 3.00000i 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.b.a 2
3.b odd 2 1 1521.2.b.b 2
13.b even 2 1 inner 507.2.b.a 2
13.c even 3 2 507.2.j.e 4
13.d odd 4 1 39.2.a.a 1
13.d odd 4 1 507.2.a.a 1
13.e even 6 2 507.2.j.e 4
13.f odd 12 2 507.2.e.a 2
13.f odd 12 2 507.2.e.b 2
39.d odd 2 1 1521.2.b.b 2
39.f even 4 1 117.2.a.a 1
39.f even 4 1 1521.2.a.e 1
52.f even 4 1 624.2.a.i 1
52.f even 4 1 8112.2.a.s 1
65.f even 4 1 975.2.c.f 2
65.g odd 4 1 975.2.a.f 1
65.k even 4 1 975.2.c.f 2
91.i even 4 1 1911.2.a.f 1
104.j odd 4 1 2496.2.a.q 1
104.m even 4 1 2496.2.a.e 1
117.y odd 12 2 1053.2.e.b 2
117.z even 12 2 1053.2.e.d 2
143.g even 4 1 4719.2.a.c 1
156.l odd 4 1 1872.2.a.h 1
195.j odd 4 1 2925.2.c.e 2
195.n even 4 1 2925.2.a.p 1
195.u odd 4 1 2925.2.c.e 2
273.o odd 4 1 5733.2.a.e 1
312.w odd 4 1 7488.2.a.by 1
312.y even 4 1 7488.2.a.bl 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 13.d odd 4 1
117.2.a.a 1 39.f even 4 1
507.2.a.a 1 13.d odd 4 1
507.2.b.a 2 1.a even 1 1 trivial
507.2.b.a 2 13.b even 2 1 inner
507.2.e.a 2 13.f odd 12 2
507.2.e.b 2 13.f odd 12 2
507.2.j.e 4 13.c even 3 2
507.2.j.e 4 13.e even 6 2
624.2.a.i 1 52.f even 4 1
975.2.a.f 1 65.g odd 4 1
975.2.c.f 2 65.f even 4 1
975.2.c.f 2 65.k even 4 1
1053.2.e.b 2 117.y odd 12 2
1053.2.e.d 2 117.z even 12 2
1521.2.a.e 1 39.f even 4 1
1521.2.b.b 2 3.b odd 2 1
1521.2.b.b 2 39.d odd 2 1
1872.2.a.h 1 156.l odd 4 1
1911.2.a.f 1 91.i even 4 1
2496.2.a.e 1 104.m even 4 1
2496.2.a.q 1 104.j odd 4 1
2925.2.a.p 1 195.n even 4 1
2925.2.c.e 2 195.j odd 4 1
2925.2.c.e 2 195.u odd 4 1
4719.2.a.c 1 143.g even 4 1
5733.2.a.e 1 273.o odd 4 1
7488.2.a.bl 1 312.y even 4 1
7488.2.a.by 1 312.w odd 4 1
8112.2.a.s 1 52.f even 4 1

Hecke kernels

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

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 36 \) Copy content Toggle raw display
$43$ \( (T - 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 144 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( T^{2} + 4 \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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