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Properties

Label 624.2.q.j
Level $624$
Weight $2$
Character orbit 624.q
Analytic conductor $4.983$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,2,Mod(289,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.q (of order \(3\), degree \(2\), 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: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.2101707.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 6x^{4} - 4x^{3} - 12x^{2} - 18x + 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{2} q^{5} + (\beta_{5} - 2 \beta_{4} - 1) q^{7} + ( - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_{2} q^{5} + (\beta_{5} - 2 \beta_{4} - 1) q^{7} + ( - \beta_{3} - 1) q^{9} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 2) q^{11}+ \cdots + (\beta_{5} + \beta_{4} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 3 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 3 q^{7} - 3 q^{9} - 3 q^{13} + 6 q^{19} - 6 q^{21} + 30 q^{25} + 6 q^{27} + 12 q^{29} - 6 q^{31} + 12 q^{35} - 6 q^{37} - 6 q^{39} - 21 q^{43} + 24 q^{47} - 24 q^{49} - 12 q^{53} - 18 q^{55} - 12 q^{57} + 6 q^{59} - 3 q^{61} + 3 q^{63} + 18 q^{65} + 27 q^{67} + 12 q^{71} - 18 q^{73} - 15 q^{75} + 60 q^{77} - 18 q^{79} - 3 q^{81} + 36 q^{83} - 12 q^{85} + 12 q^{87} - 24 q^{89} - 21 q^{91} + 3 q^{93} - 42 q^{95} - 21 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 6x^{4} - 4x^{3} - 12x^{2} - 18x + 31 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} - 3\nu^{4} + 5\nu^{3} - 19\nu^{2} + 5\nu + 17 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{5} + 4\nu^{4} - 15\nu^{3} + 7\nu^{2} + 60\nu - 51 ) / 50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\nu^{5} + 19\nu^{4} + 135\nu^{3} + 77\nu^{2} - 65\nu - 461 ) / 100 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9\nu^{5} + 13\nu^{4} + 70\nu^{3} + 79\nu^{2} - 5\nu - 222 ) / 50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{5} - 16\nu^{4} - 90\nu^{3} - 53\nu^{2} + 110\nu + 354 ) / 50 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{4} - \beta_{3} - 2\beta_{2} - \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{5} - 3\beta_{4} + 8\beta_{3} - \beta_{2} - 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -10\beta_{5} - 5\beta_{4} - 5\beta_{3} + 15\beta_{2} + \beta _1 + 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -41\beta_{5} + 48\beta_{4} - 109\beta_{3} + 12\beta_{2} + 39\beta _1 - 20 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(-1 - \beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−1.18073 − 1.01456i
1.26446 − 0.0756139i
−0.0837246 + 2.82222i
−1.18073 + 1.01456i
1.26446 + 0.0756139i
−0.0837246 − 2.82222i
0 −0.500000 − 0.866025i 0 −3.93800 0 0.892467 − 1.54580i 0 −0.500000 + 0.866025i 0
289.2 0 −0.500000 − 0.866025i 0 0.133492 0 −1.96217 + 3.39858i 0 −0.500000 + 0.866025i 0
289.3 0 −0.500000 − 0.866025i 0 3.80451 0 2.56970 − 4.45086i 0 −0.500000 + 0.866025i 0
529.1 0 −0.500000 + 0.866025i 0 −3.93800 0 0.892467 + 1.54580i 0 −0.500000 − 0.866025i 0
529.2 0 −0.500000 + 0.866025i 0 0.133492 0 −1.96217 − 3.39858i 0 −0.500000 − 0.866025i 0
529.3 0 −0.500000 + 0.866025i 0 3.80451 0 2.56970 + 4.45086i 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.q.j 6
3.b odd 2 1 1872.2.t.u 6
4.b odd 2 1 312.2.q.e ✓ 6
12.b even 2 1 936.2.t.h 6
13.c even 3 1 inner 624.2.q.j 6
13.c even 3 1 8112.2.a.ck 3
13.e even 6 1 8112.2.a.cl 3
39.i odd 6 1 1872.2.t.u 6
52.i odd 6 1 4056.2.a.y 3
52.j odd 6 1 312.2.q.e ✓ 6
52.j odd 6 1 4056.2.a.z 3
52.l even 12 2 4056.2.c.m 6
156.p even 6 1 936.2.t.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.q.e ✓ 6 4.b odd 2 1
312.2.q.e ✓ 6 52.j odd 6 1
624.2.q.j 6 1.a even 1 1 trivial
624.2.q.j 6 13.c even 3 1 inner
936.2.t.h 6 12.b even 2 1
936.2.t.h 6 156.p even 6 1
1872.2.t.u 6 3.b odd 2 1
1872.2.t.u 6 39.i odd 6 1
4056.2.a.y 3 52.i odd 6 1
4056.2.a.z 3 52.j odd 6 1
4056.2.c.m 6 52.l even 12 2
8112.2.a.ck 3 13.c even 3 1
8112.2.a.cl 3 13.e even 6 1

Hecke kernels

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

\( T_{5}^{3} - 15T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{6} - 3T_{7}^{5} + 27T_{7}^{4} - 18T_{7}^{3} + 432T_{7}^{2} - 648T_{7} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{3} - 15 T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + \cdots + 1296 \) Copy content Toggle raw display
$11$ \( T^{6} + 24 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} + 21 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{6} - 6 T^{5} + \cdots + 43264 \) Copy content Toggle raw display
$23$ \( T^{6} + 24 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{6} - 12 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$31$ \( (T^{3} + 3 T^{2} - 12 T - 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 6 T^{5} + \cdots + 324 \) Copy content Toggle raw display
$41$ \( T^{6} + 57 T^{4} + \cdots + 24336 \) Copy content Toggle raw display
$43$ \( T^{6} + 21 T^{5} + \cdots + 44944 \) Copy content Toggle raw display
$47$ \( (T^{3} - 12 T^{2} + \cdots + 24)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 6 T^{2} + \cdots - 208)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 6 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$61$ \( T^{6} + 3 T^{5} + \cdots + 27889 \) Copy content Toggle raw display
$67$ \( T^{6} - 27 T^{5} + \cdots + 274576 \) Copy content Toggle raw display
$71$ \( T^{6} - 12 T^{5} + \cdots + 1600 \) Copy content Toggle raw display
$73$ \( (T^{3} + 9 T^{2} + \cdots - 169)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 9 T^{2} - 120 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 18 T^{2} + \cdots + 992)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 24 T^{5} + \cdots + 2304 \) Copy content Toggle raw display
$97$ \( T^{6} + 21 T^{5} + \cdots + 55696 \) Copy content Toggle raw display
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