Properties

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

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{106}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 106 \) 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 \(\beta = 6\sqrt{106}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 38) q^{5} - 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 38) q^{5} - 49 q^{7} + (5 \beta + 282) q^{11} + (4 \beta + 258) q^{13} + ( - 9 \beta + 38) q^{17} + (4 \beta - 1180) q^{19} + ( - 41 \beta - 1018) q^{23} + ( - 76 \beta + 2135) q^{25} + ( - 20 \beta - 4160) q^{29} + (84 \beta + 3140) q^{31} + ( - 49 \beta + 1862) q^{35} + ( - 60 \beta - 1230) q^{37} + ( - 211 \beta + 7154) q^{41} + ( - 32 \beta - 11564) q^{43} + ( - 222 \beta + 6356) q^{47} + 2401 q^{49} + ( - 26 \beta + 4948) q^{53} + (92 \beta + 8364) q^{55} + (198 \beta + 30444) q^{59} + (700 \beta + 8086) q^{61} + (106 \beta + 5460) q^{65} + (40 \beta + 31284) q^{67} + ( - 283 \beta - 366) q^{71} + ( - 344 \beta + 622) q^{73} + ( - 245 \beta - 13818) q^{77} + ( - 1512 \beta - 5800) q^{79} + ( - 80 \beta + 66144) q^{83} + (380 \beta - 35788) q^{85} + ( - 703 \beta - 46726) q^{89} + ( - 196 \beta - 12642) q^{91} + ( - 1332 \beta + 60104) q^{95} + ( - 312 \beta - 136466) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 76 q^{5} - 98 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 76 q^{5} - 98 q^{7} + 564 q^{11} + 516 q^{13} + 76 q^{17} - 2360 q^{19} - 2036 q^{23} + 4270 q^{25} - 8320 q^{29} + 6280 q^{31} + 3724 q^{35} - 2460 q^{37} + 14308 q^{41} - 23128 q^{43} + 12712 q^{47} + 4802 q^{49} + 9896 q^{53} + 16728 q^{55} + 60888 q^{59} + 16172 q^{61} + 10920 q^{65} + 62568 q^{67} - 732 q^{71} + 1244 q^{73} - 27636 q^{77} - 11600 q^{79} + 132288 q^{83} - 71576 q^{85} - 93452 q^{89} - 25284 q^{91} + 120208 q^{95} - 272932 q^{97}+O(q^{100}) \) 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
−10.2956
10.2956
0 0 0 −99.7738 0 −49.0000 0 0 0
1.2 0 0 0 23.7738 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.bg 2
3.b odd 2 1 1008.6.a.bv 2
4.b odd 2 1 504.6.a.k 2
12.b even 2 1 504.6.a.u yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.6.a.k 2 4.b odd 2 1
504.6.a.u yes 2 12.b even 2 1
1008.6.a.bg 2 1.a even 1 1 trivial
1008.6.a.bv 2 3.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 76T_{5} - 2372 \) Copy content Toggle raw display
\( T_{11}^{2} - 564T_{11} - 15876 \) 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
$5$ \( T^{2} + 76T - 2372 \) Copy content Toggle raw display
$7$ \( (T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 564T - 15876 \) Copy content Toggle raw display
$13$ \( T^{2} - 516T + 5508 \) Copy content Toggle raw display
$17$ \( T^{2} - 76T - 307652 \) Copy content Toggle raw display
$19$ \( T^{2} + 2360 T + 1331344 \) Copy content Toggle raw display
$23$ \( T^{2} + 2036 T - 5378372 \) Copy content Toggle raw display
$29$ \( T^{2} + 8320 T + 15779200 \) Copy content Toggle raw display
$31$ \( T^{2} - 6280 T - 17066096 \) Copy content Toggle raw display
$37$ \( T^{2} + 2460 T - 12224700 \) Copy content Toggle raw display
$41$ \( T^{2} - 14308 T - 118712420 \) Copy content Toggle raw display
$43$ \( T^{2} + 23128 T + 129818512 \) Copy content Toggle raw display
$47$ \( T^{2} - 12712 T - 147669008 \) Copy content Toggle raw display
$53$ \( T^{2} - 9896 T + 21903088 \) Copy content Toggle raw display
$59$ \( T^{2} - 60888 T + 777234672 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 1804456604 \) Copy content Toggle raw display
$67$ \( T^{2} - 62568 T + 972583056 \) Copy content Toggle raw display
$71$ \( T^{2} + 732 T - 305485668 \) Copy content Toggle raw display
$73$ \( T^{2} - 1244 T - 451183292 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 8690285504 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 4350606336 \) Copy content Toggle raw display
$89$ \( T^{2} + 93452 T + 297417532 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 18251504452 \) Copy content Toggle raw display
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