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Properties

Label 504.2.t.b
Level $504$
Weight $2$
Character orbit 504.t
Analytic conductor $4.024$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,2,Mod(193,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("504.193"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.t (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content 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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} + 1) q^{3} - q^{5} + (3 \zeta_{6} - 2) q^{7} + 3 \zeta_{6} q^{9} + 3 q^{11} - 3 \zeta_{6} q^{13} + ( - \zeta_{6} - 1) q^{15} + 5 \zeta_{6} q^{17} + (7 \zeta_{6} - 7) q^{19} + (4 \zeta_{6} - 5) q^{21} + \cdots + 9 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 2 q^{5} - q^{7} + 3 q^{9} + 6 q^{11} - 3 q^{13} - 3 q^{15} + 5 q^{17} - 7 q^{19} - 6 q^{21} + 10 q^{23} - 8 q^{25} + q^{29} + 8 q^{31} + 9 q^{33} + q^{35} - 3 q^{37} + 5 q^{41} + 7 q^{43}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(1\) \(1\) \(-\zeta_{6}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 1.50000 − 0.866025i 0 −1.00000 0 −0.500000 − 2.59808i 0 1.50000 − 2.59808i 0
457.1 0 1.50000 + 0.866025i 0 −1.00000 0 −0.500000 + 2.59808i 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
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.t.b yes 2
3.b odd 2 1 1512.2.t.b 2
4.b odd 2 1 1008.2.t.a 2
7.c even 3 1 504.2.q.b ✓ 2
9.c even 3 1 504.2.q.b ✓ 2
9.d odd 6 1 1512.2.q.a 2
12.b even 2 1 3024.2.t.e 2
21.h odd 6 1 1512.2.q.a 2
28.g odd 6 1 1008.2.q.d 2
36.f odd 6 1 1008.2.q.d 2
36.h even 6 1 3024.2.q.c 2
63.g even 3 1 inner 504.2.t.b yes 2
63.n odd 6 1 1512.2.t.b 2
84.n even 6 1 3024.2.q.c 2
252.o even 6 1 3024.2.t.e 2
252.bl odd 6 1 1008.2.t.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.b ✓ 2 7.c even 3 1
504.2.q.b ✓ 2 9.c even 3 1
504.2.t.b yes 2 1.a even 1 1 trivial
504.2.t.b yes 2 63.g even 3 1 inner
1008.2.q.d 2 28.g odd 6 1
1008.2.q.d 2 36.f odd 6 1
1008.2.t.a 2 4.b odd 2 1
1008.2.t.a 2 252.bl odd 6 1
1512.2.q.a 2 9.d odd 6 1
1512.2.q.a 2 21.h odd 6 1
1512.2.t.b 2 3.b odd 2 1
1512.2.t.b 2 63.n odd 6 1
3024.2.q.c 2 36.h even 6 1
3024.2.q.c 2 84.n even 6 1
3024.2.t.e 2 12.b even 2 1
3024.2.t.e 2 252.o even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$17$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$23$ \( (T - 5)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$41$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$47$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$89$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$97$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
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