Properties

Label 1008.6.a.bx
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: \(\Q(\sqrt{429}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 107 \) 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 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 \(\beta = 2\sqrt{429}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 40) q^{5} - 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 40) q^{5} - 49 q^{7} + (3 \beta - 216) q^{11} + (16 \beta - 438) q^{13} + ( - 15 \beta - 424) q^{17} + ( - 16 \beta + 1676) q^{19} + (\beta + 2648) q^{23} + ( - 80 \beta + 191) q^{25} + (188 \beta - 128) q^{29} + ( - 208 \beta + 428) q^{31} + (49 \beta - 1960) q^{35} + (176 \beta + 1818) q^{37} + ( - 405 \beta - 1528) q^{41} + (224 \beta + 13108) q^{43} + ( - 82 \beta - 1456) q^{47} + 2401 q^{49} + (306 \beta - 13616) q^{53} + (336 \beta - 13788) q^{55} + (394 \beta - 29520) q^{59} + (560 \beta - 20210) q^{61} + (1078 \beta - 44976) q^{65} + ( - 256 \beta - 26460) q^{67} + (1219 \beta + 26376) q^{71} + ( - 1440 \beta + 9406) q^{73} + ( - 147 \beta + 10584) q^{77} + (864 \beta - 25144) q^{79} + ( - 464 \beta + 33024) q^{83} + ( - 176 \beta + 8780) q^{85} + (839 \beta - 65752) q^{89} + ( - 784 \beta + 21462) q^{91} + ( - 2316 \beta + 94496) q^{95} + (416 \beta + 82174) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 80 q^{5} - 98 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 80 q^{5} - 98 q^{7} - 432 q^{11} - 876 q^{13} - 848 q^{17} + 3352 q^{19} + 5296 q^{23} + 382 q^{25} - 256 q^{29} + 856 q^{31} - 3920 q^{35} + 3636 q^{37} - 3056 q^{41} + 26216 q^{43} - 2912 q^{47} + 4802 q^{49} - 27232 q^{53} - 27576 q^{55} - 59040 q^{59} - 40420 q^{61} - 89952 q^{65} - 52920 q^{67} + 52752 q^{71} + 18812 q^{73} + 21168 q^{77} - 50288 q^{79} + 66048 q^{83} + 17560 q^{85} - 131504 q^{89} + 42924 q^{91} + 188992 q^{95} + 164348 q^{97}+O(q^{100}) \) 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
10.8562
−9.85616
0 0 0 −1.42463 0 −49.0000 0 0 0
1.2 0 0 0 81.4246 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.bx 2
3.b odd 2 1 1008.6.a.be 2
4.b odd 2 1 504.6.a.w yes 2
12.b even 2 1 504.6.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.6.a.j 2 12.b even 2 1
504.6.a.w yes 2 4.b odd 2 1
1008.6.a.be 2 3.b odd 2 1
1008.6.a.bx 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5}^{2} - 80T_{5} - 116 \) Copy content Toggle raw display
\( T_{11}^{2} + 432T_{11} + 31212 \) 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
$5$ \( T^{2} - 80T - 116 \) Copy content Toggle raw display
$7$ \( (T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 432T + 31212 \) Copy content Toggle raw display
$13$ \( T^{2} + 876T - 247452 \) Copy content Toggle raw display
$17$ \( T^{2} + 848T - 206324 \) Copy content Toggle raw display
$19$ \( T^{2} - 3352 T + 2369680 \) Copy content Toggle raw display
$23$ \( T^{2} - 5296 T + 7010188 \) Copy content Toggle raw display
$29$ \( T^{2} + 256 T - 60633920 \) Copy content Toggle raw display
$31$ \( T^{2} - 856 T - 74057840 \) Copy content Toggle raw display
$37$ \( T^{2} - 3636 T - 49849692 \) Copy content Toggle raw display
$41$ \( T^{2} + 3056 T - 279132116 \) Copy content Toggle raw display
$43$ \( T^{2} - 26216 T + 85717648 \) Copy content Toggle raw display
$47$ \( T^{2} + 2912 T - 9418448 \) Copy content Toggle raw display
$53$ \( T^{2} + 27232 T + 24716080 \) Copy content Toggle raw display
$59$ \( T^{2} + 59040 T + 605045424 \) Copy content Toggle raw display
$61$ \( T^{2} + 40420 T - 129693500 \) Copy content Toggle raw display
$67$ \( T^{2} + 52920 T + 587671824 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 1854215700 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 3469824764 \) Copy content Toggle raw display
$79$ \( T^{2} + 50288 T - 648766400 \) Copy content Toggle raw display
$83$ \( T^{2} - 66048 T + 721136640 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 3115397068 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 6455602180 \) Copy content Toggle raw display
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