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Properties

Label 504.2.p.b
Level $504$
Weight $2$
Character orbit 504.p
Analytic conductor $4.024$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,2,Mod(307,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("504.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.p (of order \(2\), degree \(1\), 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: \(4.02446026187\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} - 2 \beta_1 q^{4} + 2 \beta_{3} q^{5} + (\beta_{3} - 2 \beta_1) q^{7} + (2 \beta_1 + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} - 2 \beta_1 q^{4} + 2 \beta_{3} q^{5} + (\beta_{3} - 2 \beta_1) q^{7} + (2 \beta_1 + 2) q^{8} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{10} + 2 q^{11} - 2 \beta_{3} q^{13} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{14} - 4 q^{16} - 2 \beta_{2} q^{17} - 4 \beta_{2} q^{19} - 4 \beta_{2} q^{20} + (2 \beta_1 - 2) q^{22} - 2 \beta_1 q^{23} + 7 q^{25} + (2 \beta_{3} - 2 \beta_{2}) q^{26} + ( - 2 \beta_{2} - 4) q^{28} + 8 \beta_1 q^{29} - 2 \beta_{3} q^{31} + ( - 4 \beta_1 + 4) q^{32} + (2 \beta_{3} + 2 \beta_{2}) q^{34} + ( - 4 \beta_{2} + 6) q^{35} + 4 \beta_1 q^{37} + (4 \beta_{3} + 4 \beta_{2}) q^{38} + (4 \beta_{3} + 4 \beta_{2}) q^{40} + 6 \beta_{2} q^{41} + 6 q^{43} - 4 \beta_1 q^{44} + (2 \beta_1 + 2) q^{46} - 4 \beta_{3} q^{47} + ( - 4 \beta_{2} - 1) q^{49} + (7 \beta_1 - 7) q^{50} + 4 \beta_{2} q^{52} + 4 \beta_1 q^{53} + 4 \beta_{3} q^{55} + (2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{56} + ( - 8 \beta_1 - 8) q^{58} - 4 \beta_{2} q^{59} + 6 \beta_{3} q^{61} + (2 \beta_{3} - 2 \beta_{2}) q^{62} + 8 \beta_1 q^{64} - 12 q^{65} + 10 q^{67} - 4 \beta_{3} q^{68} + (4 \beta_{3} + 4 \beta_{2} + 6 \beta_1 - 6) q^{70} + 2 \beta_1 q^{71} + ( - 4 \beta_1 - 4) q^{74} - 8 \beta_{3} q^{76} + (2 \beta_{3} - 4 \beta_1) q^{77} + 12 \beta_1 q^{79} - 8 \beta_{3} q^{80} + ( - 6 \beta_{3} - 6 \beta_{2}) q^{82} + 4 \beta_{2} q^{83} - 12 \beta_1 q^{85} + (6 \beta_1 - 6) q^{86} + (4 \beta_1 + 4) q^{88} + 6 \beta_{2} q^{89} + (4 \beta_{2} - 6) q^{91} - 4 q^{92} + (4 \beta_{3} - 4 \beta_{2}) q^{94} - 24 \beta_1 q^{95} - 8 \beta_{2} q^{97} + (4 \beta_{3} + 4 \beta_{2} - \beta_1 + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 8 q^{8} + 8 q^{11} + 8 q^{14} - 16 q^{16} - 8 q^{22} + 28 q^{25} - 16 q^{28} + 16 q^{32} + 24 q^{35} + 24 q^{43} + 8 q^{46} - 4 q^{49} - 28 q^{50} + 16 q^{56} - 32 q^{58} - 48 q^{65} + 40 q^{67} - 24 q^{70} - 16 q^{74} - 24 q^{86} + 16 q^{88} - 24 q^{91} - 16 q^{92} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

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

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−1.00000 − 1.00000i 0 2.00000i −3.46410 0 −1.73205 + 2.00000i 2.00000 − 2.00000i 0 3.46410 + 3.46410i
307.2 −1.00000 − 1.00000i 0 2.00000i 3.46410 0 1.73205 + 2.00000i 2.00000 − 2.00000i 0 −3.46410 − 3.46410i
307.3 −1.00000 + 1.00000i 0 − 2.00000i −3.46410 0 −1.73205 − 2.00000i 2.00000 + 2.00000i 0 3.46410 − 3.46410i
307.4 −1.00000 + 1.00000i 0 − 2.00000i 3.46410 0 1.73205 − 2.00000i 2.00000 + 2.00000i 0 −3.46410 + 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.p.b ✓ 4
3.b odd 2 1 504.2.p.e yes 4
4.b odd 2 1 2016.2.p.b 4
7.b odd 2 1 inner 504.2.p.b ✓ 4
8.b even 2 1 2016.2.p.b 4
8.d odd 2 1 inner 504.2.p.b ✓ 4
12.b even 2 1 2016.2.p.f 4
21.c even 2 1 504.2.p.e yes 4
24.f even 2 1 504.2.p.e yes 4
24.h odd 2 1 2016.2.p.f 4
28.d even 2 1 2016.2.p.b 4
56.e even 2 1 inner 504.2.p.b ✓ 4
56.h odd 2 1 2016.2.p.b 4
84.h odd 2 1 2016.2.p.f 4
168.e odd 2 1 504.2.p.e yes 4
168.i even 2 1 2016.2.p.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.p.b ✓ 4 1.a even 1 1 trivial
504.2.p.b ✓ 4 7.b odd 2 1 inner
504.2.p.b ✓ 4 8.d odd 2 1 inner
504.2.p.b ✓ 4 56.e even 2 1 inner
504.2.p.e yes 4 3.b odd 2 1
504.2.p.e yes 4 21.c even 2 1
504.2.p.e yes 4 24.f even 2 1
504.2.p.e yes 4 168.e odd 2 1
2016.2.p.b 4 4.b odd 2 1
2016.2.p.b 4 8.b even 2 1
2016.2.p.b 4 28.d even 2 1
2016.2.p.b 4 56.h odd 2 1
2016.2.p.f 4 12.b even 2 1
2016.2.p.f 4 24.h odd 2 1
2016.2.p.f 4 84.h odd 2 1
2016.2.p.f 4 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\):

\( T_{5}^{2} - 12 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T - 2)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$43$ \( (T - 6)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$67$ \( (T - 10)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
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