Properties

Label 8512.2.a.r
Level $8512$
Weight $2$
Character orbit 8512.a
Self dual yes
Analytic conductor $67.969$
Analytic rank $0$
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] = [8512,2,Mod(1,8512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8512.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8512.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: \(67.9686622005\)
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: no (minimal twist has level 1064)
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.

\(f(q)\) \(=\) \( q - \beta q^{3} + q^{5} + q^{7} + \beta q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + q^{5} + q^{7} + \beta q^{9} + (\beta - 1) q^{11} + ( - 2 \beta + 1) q^{13} - \beta q^{15} + (\beta + 3) q^{17} + q^{19} - \beta q^{21} + (2 \beta + 3) q^{23} - 4 q^{25} + (2 \beta - 3) q^{27} + ( - \beta + 3) q^{29} + ( - \beta + 7) q^{31} - 3 q^{33} + q^{35} + (2 \beta - 1) q^{37} + (\beta + 6) q^{39} + ( - \beta - 1) q^{41} + ( - 4 \beta + 2) q^{43} + \beta q^{45} + ( - 2 \beta + 5) q^{47} + q^{49} + ( - 4 \beta - 3) q^{51} - 3 \beta q^{53} + (\beta - 1) q^{55} - \beta q^{57} + ( - 2 \beta + 3) q^{59} + (4 \beta - 3) q^{61} + \beta q^{63} + ( - 2 \beta + 1) q^{65} + (3 \beta - 7) q^{67} + ( - 5 \beta - 6) q^{69} + (6 \beta - 1) q^{71} + (3 \beta - 2) q^{73} + 4 \beta q^{75} + (\beta - 1) q^{77} + 14 q^{79} + ( - 2 \beta - 6) q^{81} + (5 \beta - 4) q^{83} + (\beta + 3) q^{85} + ( - 2 \beta + 3) q^{87} + ( - 6 \beta + 6) q^{89} + ( - 2 \beta + 1) q^{91} + ( - 6 \beta + 3) q^{93} + q^{95} + ( - 6 \beta + 9) q^{97} + 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 2 q^{5} + 2 q^{7} + q^{9} - q^{11} - q^{15} + 7 q^{17} + 2 q^{19} - q^{21} + 8 q^{23} - 8 q^{25} - 4 q^{27} + 5 q^{29} + 13 q^{31} - 6 q^{33} + 2 q^{35} + 13 q^{39} - 3 q^{41} + q^{45} + 8 q^{47} + 2 q^{49} - 10 q^{51} - 3 q^{53} - q^{55} - q^{57} + 4 q^{59} - 2 q^{61} + q^{63} - 11 q^{67} - 17 q^{69} + 4 q^{71} - q^{73} + 4 q^{75} - q^{77} + 28 q^{79} - 14 q^{81} - 3 q^{83} + 7 q^{85} + 4 q^{87} + 6 q^{89} + 2 q^{95} + 12 q^{97} + 6 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
0 −2.30278 0 1.00000 0 1.00000 0 2.30278 0
1.2 0 1.30278 0 1.00000 0 1.00000 0 −1.30278 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(19\) \(-1\)

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 8512.2.a.r 2
4.b odd 2 1 8512.2.a.x 2
8.b even 2 1 2128.2.a.i 2
8.d odd 2 1 1064.2.a.b 2
24.f even 2 1 9576.2.a.bs 2
56.e even 2 1 7448.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.a.b 2 8.d odd 2 1
2128.2.a.i 2 8.b even 2 1
7448.2.a.bb 2 56.e even 2 1
8512.2.a.r 2 1.a even 1 1 trivial
8512.2.a.x 2 4.b odd 2 1
9576.2.a.bs 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8512))\):

\( T_{3}^{2} + T_{3} - 3 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} + T_{11} - 3 \) Copy content Toggle raw display
\( T_{23}^{2} - 8T_{23} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$13$ \( T^{2} - 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 7T + 9 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 3 \) Copy content Toggle raw display
$29$ \( T^{2} - 5T + 3 \) Copy content Toggle raw display
$31$ \( T^{2} - 13T + 39 \) Copy content Toggle raw display
$37$ \( T^{2} - 13 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T - 1 \) Copy content Toggle raw display
$43$ \( T^{2} - 52 \) Copy content Toggle raw display
$47$ \( T^{2} - 8T + 3 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T - 27 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T - 51 \) Copy content Toggle raw display
$67$ \( T^{2} + 11T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 113 \) Copy content Toggle raw display
$73$ \( T^{2} + T - 29 \) Copy content Toggle raw display
$79$ \( (T - 14)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3T - 79 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 108 \) Copy content Toggle raw display
$97$ \( T^{2} - 12T - 81 \) Copy content Toggle raw display
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