Properties

Label 1008.6.a.br
Level $1008$
Weight $6$
Character orbit 1008.a
Self dual yes
Analytic conductor $161.667$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,6,Mod(1,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1008.a (trivial)

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: yes
Analytic conductor: \(161.666890371\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{114}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 114 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 504)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 6\sqrt{114}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 14) q^{5} + 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 14) q^{5} + 49 q^{7} + (7 \beta + 298) q^{11} + ( - 4 \beta + 266) q^{13} + (11 \beta + 1050) q^{17} + ( - 12 \beta + 1372) q^{19} + (21 \beta + 1830) q^{23} + (28 \beta + 1175) q^{25} + (56 \beta - 360) q^{29} + ( - 92 \beta - 1988) q^{31} + (49 \beta + 686) q^{35} + (140 \beta + 3394) q^{37} + (89 \beta - 2002) q^{41} + ( - 112 \beta - 9740) q^{43} + (158 \beta - 10780) q^{47} + 2401 q^{49} + ( - 14 \beta + 28788) q^{53} + (396 \beta + 32900) q^{55} + (58 \beta - 11172) q^{59} + ( - 380 \beta - 19362) q^{61} + (210 \beta - 12692) q^{65} + (504 \beta - 3644) q^{67} + (175 \beta - 1966) q^{71} + ( - 616 \beta + 9646) q^{73} + (343 \beta + 14602) q^{77} + ( - 392 \beta - 7816) q^{79} + ( - 456 \beta - 11312) q^{83} + (1204 \beta + 59844) q^{85} + ( - 235 \beta + 56406) q^{89} + ( - 196 \beta + 13034) q^{91} + (1204 \beta - 30040) q^{95} + ( - 1448 \beta - 18018) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 28 q^{5} + 98 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 28 q^{5} + 98 q^{7} + 596 q^{11} + 532 q^{13} + 2100 q^{17} + 2744 q^{19} + 3660 q^{23} + 2350 q^{25} - 720 q^{29} - 3976 q^{31} + 1372 q^{35} + 6788 q^{37} - 4004 q^{41} - 19480 q^{43} - 21560 q^{47} + 4802 q^{49} + 57576 q^{53} + 65800 q^{55} - 22344 q^{59} - 38724 q^{61} - 25384 q^{65} - 7288 q^{67} - 3932 q^{71} + 19292 q^{73} + 29204 q^{77} - 15632 q^{79} - 22624 q^{83} + 119688 q^{85} + 112812 q^{89} + 26068 q^{91} - 60080 q^{95} - 36036 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−10.6771
10.6771
0 0 0 −50.0625 0 49.0000 0 0 0
1.2 0 0 0 78.0625 0 49.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.6.a.br 2
3.b odd 2 1 1008.6.a.bj 2
4.b odd 2 1 504.6.a.q yes 2
12.b even 2 1 504.6.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.6.a.n 2 12.b even 2 1
504.6.a.q yes 2 4.b odd 2 1
1008.6.a.bj 2 3.b odd 2 1
1008.6.a.br 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5}^{2} - 28T_{5} - 3908 \) Copy content Toggle raw display
\( T_{11}^{2} - 596T_{11} - 112292 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 28T - 3908 \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 596T - 112292 \) Copy content Toggle raw display
$13$ \( T^{2} - 532T + 5092 \) Copy content Toggle raw display
$17$ \( T^{2} - 2100 T + 605916 \) Copy content Toggle raw display
$19$ \( T^{2} - 2744 T + 1291408 \) Copy content Toggle raw display
$23$ \( T^{2} - 3660 T + 1539036 \) Copy content Toggle raw display
$29$ \( T^{2} + 720 T - 12740544 \) Copy content Toggle raw display
$31$ \( T^{2} + 3976 T - 30784112 \) Copy content Toggle raw display
$37$ \( T^{2} - 6788 T - 68919164 \) Copy content Toggle raw display
$41$ \( T^{2} + 4004 T - 28499780 \) Copy content Toggle raw display
$43$ \( T^{2} + 19480 T + 43387024 \) Copy content Toggle raw display
$47$ \( T^{2} + 21560 T + 13756144 \) Copy content Toggle raw display
$53$ \( T^{2} - 57576 T + 827944560 \) Copy content Toggle raw display
$59$ \( T^{2} + 22344 T + 111007728 \) Copy content Toggle raw display
$61$ \( T^{2} + 38724 T - 217730556 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 1029202928 \) Copy content Toggle raw display
$71$ \( T^{2} + 3932 T - 121819844 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 1464242108 \) Copy content Toggle raw display
$79$ \( T^{2} + 15632 T - 569547200 \) Copy content Toggle raw display
$83$ \( T^{2} + 22624 T - 725408000 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 2954993436 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 8280224892 \) Copy content Toggle raw display
show more
show less