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Properties

Label 624.2.bv.b
Level $624$
Weight $2$
Character orbit 624.bv
Analytic conductor $4.983$
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] = [624,2,Mod(49,624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 5])) 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("624.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.bv (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.98266508613\)
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: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} + (4 \zeta_{6} - 2) q^{5} + ( - \zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} + (2 \zeta_{6} + 2) q^{11} + ( - \zeta_{6} + 4) q^{13} + ( - 2 \zeta_{6} - 2) q^{15} + (2 \zeta_{6} - 4) q^{19} + \cdots + ( - 4 \zeta_{6} + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 3 q^{7} - q^{9} + 6 q^{11} + 7 q^{13} - 6 q^{15} - 6 q^{19} - 6 q^{23} - 14 q^{25} + 2 q^{27} - 6 q^{29} - 6 q^{33} + 6 q^{35} - 2 q^{39} - 12 q^{41} - q^{43} + 6 q^{45} - 4 q^{49} + 24 q^{53}+ \cdots - 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(\zeta_{6}\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 −0.500000 − 0.866025i 0 − 3.46410i 0 1.50000 + 0.866025i 0 −0.500000 + 0.866025i 0
433.1 0 −0.500000 + 0.866025i 0 3.46410i 0 1.50000 − 0.866025i 0 −0.500000 − 0.866025i 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
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.bv.b 2
3.b odd 2 1 1872.2.by.f 2
4.b odd 2 1 39.2.j.a ✓ 2
12.b even 2 1 117.2.q.a 2
13.e even 6 1 inner 624.2.bv.b 2
13.f odd 12 2 8112.2.a.bu 2
20.d odd 2 1 975.2.bc.c 2
20.e even 4 2 975.2.w.d 4
39.h odd 6 1 1872.2.by.f 2
52.b odd 2 1 507.2.j.b 2
52.f even 4 2 507.2.e.f 4
52.i odd 6 1 39.2.j.a ✓ 2
52.i odd 6 1 507.2.b.c 2
52.j odd 6 1 507.2.b.c 2
52.j odd 6 1 507.2.j.b 2
52.l even 12 2 507.2.a.e 2
52.l even 12 2 507.2.e.f 4
156.p even 6 1 1521.2.b.f 2
156.r even 6 1 117.2.q.a 2
156.r even 6 1 1521.2.b.f 2
156.v odd 12 2 1521.2.a.h 2
260.w odd 6 1 975.2.bc.c 2
260.bg even 12 2 975.2.w.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a ✓ 2 4.b odd 2 1
39.2.j.a ✓ 2 52.i odd 6 1
117.2.q.a 2 12.b even 2 1
117.2.q.a 2 156.r even 6 1
507.2.a.e 2 52.l even 12 2
507.2.b.c 2 52.i odd 6 1
507.2.b.c 2 52.j odd 6 1
507.2.e.f 4 52.f even 4 2
507.2.e.f 4 52.l even 12 2
507.2.j.b 2 52.b odd 2 1
507.2.j.b 2 52.j odd 6 1
624.2.bv.b 2 1.a even 1 1 trivial
624.2.bv.b 2 13.e even 6 1 inner
975.2.w.d 4 20.e even 4 2
975.2.w.d 4 260.bg even 12 2
975.2.bc.c 2 20.d odd 2 1
975.2.bc.c 2 260.w odd 6 1
1521.2.a.h 2 156.v odd 12 2
1521.2.b.f 2 156.p even 6 1
1521.2.b.f 2 156.r even 6 1
1872.2.by.f 2 3.b odd 2 1
1872.2.by.f 2 39.h odd 6 1
8112.2.a.bu 2 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{2} + 12 \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 12 \) Copy content Toggle raw display
$7$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$13$ \( T^{2} - 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 3 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$71$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$73$ \( T^{2} + 3 \) Copy content Toggle raw display
$79$ \( (T - 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 192 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
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