Properties

Label 8008.2.a.h
Level $8008$
Weight $2$
Character orbit 8008.a
Self dual yes
Analytic conductor $63.944$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8008,2,Mod(1,8008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8008.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: \(63.9442019386\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + (\beta + 2) q^{5} + q^{7} + (3 \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + (\beta + 2) q^{5} + q^{7} + (3 \beta + 1) q^{9} - q^{11} - q^{13} + (4 \beta + 5) q^{15} + (\beta - 3) q^{17} + ( - 3 \beta + 2) q^{19} + (\beta + 1) q^{21} - 8 q^{23} + (5 \beta + 2) q^{25} + (4 \beta + 7) q^{27} + 4 q^{29} + (4 \beta - 4) q^{31} + ( - \beta - 1) q^{33} + (\beta + 2) q^{35} + (2 \beta + 4) q^{37} + ( - \beta - 1) q^{39} - 2 \beta q^{41} + ( - 3 \beta + 1) q^{43} + (10 \beta + 11) q^{45} + ( - 4 \beta + 4) q^{47} + q^{49} - \beta q^{51} + (3 \beta + 5) q^{53} + ( - \beta - 2) q^{55} + ( - 4 \beta - 7) q^{57} + (4 \beta - 6) q^{59} + ( - 7 \beta + 3) q^{61} + (3 \beta + 1) q^{63} + ( - \beta - 2) q^{65} + (\beta + 7) q^{67} + ( - 8 \beta - 8) q^{69} - 7 \beta q^{71} + (2 \beta - 6) q^{73} + (12 \beta + 17) q^{75} - q^{77} + ( - 3 \beta - 2) q^{79} + (6 \beta + 16) q^{81} + ( - 3 \beta + 14) q^{83} - 3 q^{85} + (4 \beta + 4) q^{87} + (\beta + 3) q^{89} - q^{91} + (4 \beta + 8) q^{93} + ( - 7 \beta - 5) q^{95} - 10 q^{97} + ( - 3 \beta - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 5 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + 5 q^{5} + 2 q^{7} + 5 q^{9} - 2 q^{11} - 2 q^{13} + 14 q^{15} - 5 q^{17} + q^{19} + 3 q^{21} - 16 q^{23} + 9 q^{25} + 18 q^{27} + 8 q^{29} - 4 q^{31} - 3 q^{33} + 5 q^{35} + 10 q^{37} - 3 q^{39} - 2 q^{41} - q^{43} + 32 q^{45} + 4 q^{47} + 2 q^{49} - q^{51} + 13 q^{53} - 5 q^{55} - 18 q^{57} - 8 q^{59} - q^{61} + 5 q^{63} - 5 q^{65} + 15 q^{67} - 24 q^{69} - 7 q^{71} - 10 q^{73} + 46 q^{75} - 2 q^{77} - 7 q^{79} + 38 q^{81} + 25 q^{83} - 6 q^{85} + 12 q^{87} + 7 q^{89} - 2 q^{91} + 20 q^{93} - 17 q^{95} - 20 q^{97} - 5 q^{99}+O(q^{100}) \) 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
−1.30278
2.30278
0 −0.302776 0 0.697224 0 1.00000 0 −2.90833 0
1.2 0 3.30278 0 4.30278 0 1.00000 0 7.90833 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)
\(13\) \(1\)

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 8008.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.h 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{2} - 3T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 5T_{5} + 3 \) Copy content Toggle raw display
\( T_{17}^{2} + 5T_{17} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 5T + 3 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 5T + 3 \) Copy content Toggle raw display
$19$ \( T^{2} - T - 29 \) Copy content Toggle raw display
$23$ \( (T + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T - 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 12 \) Copy content Toggle raw display
$41$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 29 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 48 \) Copy content Toggle raw display
$53$ \( T^{2} - 13T + 13 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T - 36 \) Copy content Toggle raw display
$61$ \( T^{2} + T - 159 \) Copy content Toggle raw display
$67$ \( T^{2} - 15T + 53 \) Copy content Toggle raw display
$71$ \( T^{2} + 7T - 147 \) Copy content Toggle raw display
$73$ \( T^{2} + 10T + 12 \) Copy content Toggle raw display
$79$ \( T^{2} + 7T - 17 \) Copy content Toggle raw display
$83$ \( T^{2} - 25T + 127 \) Copy content Toggle raw display
$89$ \( T^{2} - 7T + 9 \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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