Properties

Label 507.2.f.b
Level $507$
Weight $2$
Character orbit 507.f
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(239,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.f (of order \(4\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2} + \beta_1) q^{3} + 2 \beta_{2} q^{4} + ( - 3 \beta_{3} + 2 \beta_{2} - 2) q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2} + \beta_1) q^{3} + 2 \beta_{2} q^{4} + ( - 3 \beta_{3} + 2 \beta_{2} - 2) q^{7} + 3 q^{9} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{12} - 4 q^{16} + (5 \beta_{2} - 2 \beta_1 + 5) q^{19} + ( - \beta_{3} + 5 \beta_{2} - 5) q^{21} + 5 \beta_{2} q^{25} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{27} + (2 \beta_{2} - 6 \beta_1 + 2) q^{28} + (6 \beta_{2} - 5 \beta_1 + 6) q^{31} + 6 \beta_{2} q^{36} + (4 \beta_{3} - 7 \beta_{2} + 7) q^{37} + (\beta_{3} - \beta_1 + 1) q^{43} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{48} + (3 \beta_{3} - 7 \beta_{2} - 3 \beta_1 + 3) q^{49} + ( - 7 \beta_{2} + 8 \beta_1 - 7) q^{57} + (5 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{61} + ( - 9 \beta_{3} + 6 \beta_{2} - 6) q^{63} - 8 \beta_{2} q^{64} + (2 \beta_{2} + 7 \beta_1 + 2) q^{67} + ( - \beta_{3} - 8 \beta_{2} + 8) q^{73} + ( - 5 \beta_{3} + 5 \beta_1 - 5) q^{75} + (4 \beta_{3} + 6 \beta_{2} - 6) q^{76} + ( - 7 \beta_{3} + 7 \beta_{2} - 7 \beta_1) q^{79} + 9 q^{81} + ( - 8 \beta_{2} - 2 \beta_1 - 8) q^{84} + ( - 11 \beta_{2} + 7 \beta_1 - 11) q^{93} + ( - 8 \beta_{2} + 11 \beta_1 - 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{7} + 12 q^{9} - 16 q^{16} + 16 q^{19} - 18 q^{21} - 4 q^{28} + 14 q^{31} + 20 q^{37} - 12 q^{57} - 6 q^{63} + 22 q^{67} + 34 q^{73} - 32 q^{76} + 36 q^{81} - 36 q^{84} - 30 q^{93} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0 −1.73205 2.00000i 0 0 2.09808 + 2.09808i 0 3.00000 0
239.2 0 1.73205 2.00000i 0 0 −3.09808 3.09808i 0 3.00000 0
437.1 0 −1.73205 2.00000i 0 0 2.09808 2.09808i 0 3.00000 0
437.2 0 1.73205 2.00000i 0 0 −3.09808 + 3.09808i 0 3.00000 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.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.f.b 4
3.b odd 2 1 CM 507.2.f.b 4
13.b even 2 1 507.2.f.c 4
13.c even 3 1 507.2.k.a 4
13.c even 3 1 507.2.k.c 4
13.d odd 4 1 inner 507.2.f.b 4
13.d odd 4 1 507.2.f.c 4
13.e even 6 1 39.2.k.a 4
13.e even 6 1 507.2.k.b 4
13.f odd 12 1 39.2.k.a 4
13.f odd 12 1 507.2.k.a 4
13.f odd 12 1 507.2.k.b 4
13.f odd 12 1 507.2.k.c 4
39.d odd 2 1 507.2.f.c 4
39.f even 4 1 inner 507.2.f.b 4
39.f even 4 1 507.2.f.c 4
39.h odd 6 1 39.2.k.a 4
39.h odd 6 1 507.2.k.b 4
39.i odd 6 1 507.2.k.a 4
39.i odd 6 1 507.2.k.c 4
39.k even 12 1 39.2.k.a 4
39.k even 12 1 507.2.k.a 4
39.k even 12 1 507.2.k.b 4
39.k even 12 1 507.2.k.c 4
52.i odd 6 1 624.2.cn.b 4
52.l even 12 1 624.2.cn.b 4
65.l even 6 1 975.2.bo.c 4
65.o even 12 1 975.2.bp.d 4
65.r odd 12 1 975.2.bp.a 4
65.r odd 12 1 975.2.bp.d 4
65.s odd 12 1 975.2.bo.c 4
65.t even 12 1 975.2.bp.a 4
156.r even 6 1 624.2.cn.b 4
156.v odd 12 1 624.2.cn.b 4
195.y odd 6 1 975.2.bo.c 4
195.bc odd 12 1 975.2.bp.a 4
195.bf even 12 1 975.2.bp.a 4
195.bf even 12 1 975.2.bp.d 4
195.bh even 12 1 975.2.bo.c 4
195.bn odd 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.e even 6 1
39.2.k.a 4 13.f odd 12 1
39.2.k.a 4 39.h odd 6 1
39.2.k.a 4 39.k even 12 1
507.2.f.b 4 1.a even 1 1 trivial
507.2.f.b 4 3.b odd 2 1 CM
507.2.f.b 4 13.d odd 4 1 inner
507.2.f.b 4 39.f even 4 1 inner
507.2.f.c 4 13.b even 2 1
507.2.f.c 4 13.d odd 4 1
507.2.f.c 4 39.d odd 2 1
507.2.f.c 4 39.f even 4 1
507.2.k.a 4 13.c even 3 1
507.2.k.a 4 13.f odd 12 1
507.2.k.a 4 39.i odd 6 1
507.2.k.a 4 39.k even 12 1
507.2.k.b 4 13.e even 6 1
507.2.k.b 4 13.f odd 12 1
507.2.k.b 4 39.h odd 6 1
507.2.k.b 4 39.k even 12 1
507.2.k.c 4 13.c even 3 1
507.2.k.c 4 13.f odd 12 1
507.2.k.c 4 39.i odd 6 1
507.2.k.c 4 39.k even 12 1
624.2.cn.b 4 52.i odd 6 1
624.2.cn.b 4 52.l even 12 1
624.2.cn.b 4 156.r even 6 1
624.2.cn.b 4 156.v odd 12 1
975.2.bo.c 4 65.l even 6 1
975.2.bo.c 4 65.s odd 12 1
975.2.bo.c 4 195.y odd 6 1
975.2.bo.c 4 195.bh even 12 1
975.2.bp.a 4 65.r odd 12 1
975.2.bp.a 4 65.t even 12 1
975.2.bp.a 4 195.bc odd 12 1
975.2.bp.a 4 195.bf even 12 1
975.2.bp.d 4 65.o even 12 1
975.2.bp.d 4 65.r odd 12 1
975.2.bp.d 4 195.bf even 12 1
975.2.bp.d 4 195.bn odd 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
\( T_{7}^{4} + 2T_{7}^{3} + 2T_{7}^{2} - 26T_{7} + 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 169 \) 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
$19$ \( T^{4} - 16 T^{3} + \cdots + 676 \) 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
$37$ \( T^{4} - 20 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 3)^{2} \) 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
$61$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 22 T^{3} + \cdots + 169 \) 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
$97$ \( T^{4} + 10 T^{3} + \cdots + 28561 \) Copy content Toggle raw display
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