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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,2,Mod(49,624)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//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

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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
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

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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} + (4 \zeta_{6} - 2) q^{5} + ( - \zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} + (4 \zeta_{6} - 2) q^{5} + ( - \zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} + (2 \zeta_{6} + 2) q^{11} + ( - \zeta_{6} + 4) q^{13} + ( - 2 \zeta_{6} - 2) q^{15} + (2 \zeta_{6} - 4) q^{19} + (2 \zeta_{6} - 1) q^{21} + (6 \zeta_{6} - 6) q^{23} - 7 q^{25} + q^{27} + (6 \zeta_{6} - 6) q^{29} + ( - 2 \zeta_{6} + 1) q^{31} + (2 \zeta_{6} - 4) q^{33} + 6 \zeta_{6} q^{35} + (4 \zeta_{6} - 3) q^{39} + ( - 4 \zeta_{6} - 4) q^{41} - \zeta_{6} q^{43} + ( - 2 \zeta_{6} + 4) q^{45} + ( - 4 \zeta_{6} + 2) q^{47} + (4 \zeta_{6} - 4) q^{49} + 12 q^{53} + (12 \zeta_{6} - 12) q^{55} + ( - 4 \zeta_{6} + 2) q^{57} + ( - 2 \zeta_{6} + 4) q^{59} - \zeta_{6} q^{61} + ( - \zeta_{6} - 1) q^{63} + (14 \zeta_{6} - 4) q^{65} + ( - 5 \zeta_{6} - 5) q^{67} - 6 \zeta_{6} q^{69} + ( - 6 \zeta_{6} + 12) q^{71} + (2 \zeta_{6} - 1) q^{73} + ( - 7 \zeta_{6} + 7) q^{75} + 6 q^{77} + 11 q^{79} + (\zeta_{6} - 1) q^{81} + (16 \zeta_{6} - 8) q^{83} - 6 \zeta_{6} q^{87} + (4 \zeta_{6} + 4) q^{89} + ( - 5 \zeta_{6} + 7) q^{91} + (\zeta_{6} + 1) q^{93} - 12 \zeta_{6} q^{95} + (3 \zeta_{6} - 6) q^{97} + ( - 4 \zeta_{6} + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 3 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 3 q^{7} - q^{9} + 6 q^{11} + 7 q^{13} - 6 q^{15} - 6 q^{19} - 6 q^{23} - 14 q^{25} + 2 q^{27} - 6 q^{29} - 6 q^{33} + 6 q^{35} - 2 q^{39} - 12 q^{41} - q^{43} + 6 q^{45} - 4 q^{49} + 24 q^{53} - 12 q^{55} + 6 q^{59} - q^{61} - 3 q^{63} + 6 q^{65} - 15 q^{67} - 6 q^{69} + 18 q^{71} + 7 q^{75} + 12 q^{77} + 22 q^{79} - q^{81} - 6 q^{87} + 12 q^{89} + 9 q^{91} + 3 q^{93} - 12 q^{95} - 9 q^{97}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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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