Properties

Label 1008.6.a.bn
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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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: \(\Q(\sqrt{1129}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 282 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 168)
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 \(\beta = 2\sqrt{1129}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{5} + 49 q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{5} + 49 q^{7} + ( - \beta + 50) q^{11} + (8 \beta + 270) q^{13} + ( - 3 \beta - 576) q^{17} + (18 \beta - 1472) q^{19} + ( - 49 \beta + 1342) q^{23} + 1391 q^{25} + (30 \beta - 498) q^{29} + ( - 62 \beta - 1308) q^{31} - 49 \beta q^{35} + (124 \beta - 1746) q^{37} + (83 \beta - 8508) q^{41} + (116 \beta - 6820) q^{43} + (254 \beta + 5916) q^{47} + 2401 q^{49} + (240 \beta - 18774) q^{53} + ( - 50 \beta + 4516) q^{55} + (382 \beta - 3160) q^{59} + ( - 154 \beta + 19882) q^{61} + ( - 270 \beta - 36128) q^{65} + (806 \beta + 13688) q^{67} + ( - 415 \beta + 17730) q^{71} + (418 \beta + 32094) q^{73} + ( - 49 \beta + 2450) q^{77} + ( - 406 \beta + 55044) q^{79} + ( - 156 \beta - 2948) q^{83} + (576 \beta + 13548) q^{85} + ( - 549 \beta - 83580) q^{89} + (392 \beta + 13230) q^{91} + (1472 \beta - 81288) q^{95} + ( - 338 \beta - 5938) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 98 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 98 q^{7} + 100 q^{11} + 540 q^{13} - 1152 q^{17} - 2944 q^{19} + 2684 q^{23} + 2782 q^{25} - 996 q^{29} - 2616 q^{31} - 3492 q^{37} - 17016 q^{41} - 13640 q^{43} + 11832 q^{47} + 4802 q^{49} - 37548 q^{53} + 9032 q^{55} - 6320 q^{59} + 39764 q^{61} - 72256 q^{65} + 27376 q^{67} + 35460 q^{71} + 64188 q^{73} + 4900 q^{77} + 110088 q^{79} - 5896 q^{83} + 27096 q^{85} - 167160 q^{89} + 26460 q^{91} - 162576 q^{95} - 11876 q^{97}+O(q^{100}) \) 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
17.3003
−16.3003
0 0 0 −67.2012 0 49.0000 0 0 0
1.2 0 0 0 67.2012 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.bn 2
3.b odd 2 1 336.6.a.t 2
4.b odd 2 1 504.6.a.o 2
12.b even 2 1 168.6.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.a.j 2 12.b even 2 1
336.6.a.t 2 3.b odd 2 1
504.6.a.o 2 4.b odd 2 1
1008.6.a.bn 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5}^{2} - 4516 \) Copy content Toggle raw display
\( T_{11}^{2} - 100T_{11} - 2016 \) 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
$5$ \( T^{2} - 4516 \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 100T - 2016 \) Copy content Toggle raw display
$13$ \( T^{2} - 540T - 216124 \) Copy content Toggle raw display
$17$ \( T^{2} + 1152 T + 291132 \) Copy content Toggle raw display
$19$ \( T^{2} + 2944 T + 703600 \) Copy content Toggle raw display
$23$ \( T^{2} - 2684 T - 9041952 \) Copy content Toggle raw display
$29$ \( T^{2} + 996 T - 3816396 \) Copy content Toggle raw display
$31$ \( T^{2} + 2616 T - 15648640 \) Copy content Toggle raw display
$37$ \( T^{2} + 3492 T - 66389500 \) Copy content Toggle raw display
$41$ \( T^{2} + 17016 T + 41275340 \) Copy content Toggle raw display
$43$ \( T^{2} + 13640 T - 14254896 \) Copy content Toggle raw display
$47$ \( T^{2} - 11832 T - 256355200 \) Copy content Toggle raw display
$53$ \( T^{2} + 37548 T + 92341476 \) Copy content Toggle raw display
$59$ \( T^{2} + 6320 T - 649007184 \) Copy content Toggle raw display
$61$ \( T^{2} - 39764 T + 288192468 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 2746394832 \) Copy content Toggle raw display
$71$ \( T^{2} - 35460 T - 463415200 \) Copy content Toggle raw display
$73$ \( T^{2} - 64188 T + 240971252 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 2285442560 \) Copy content Toggle raw display
$83$ \( T^{2} + 5896 T - 101210672 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 5624489484 \) Copy content Toggle raw display
$97$ \( T^{2} + 11876 T - 480666060 \) Copy content Toggle raw display
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