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Properties

Label 8512.2.a.ca
Level $8512$
Weight $2$
Character orbit 8512.a
Self dual yes
Analytic conductor $67.969$
Analytic rank $1$
Dimension $6$
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.41027408.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 6x^{4} + 15x^{3} + 12x^{2} - 17x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4256)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{3} q^{5} + q^{7} + (\beta_{3} + \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{3} q^{5} + q^{7} + (\beta_{3} + \beta_{2} + \beta_1) q^{9} + (\beta_{5} + \beta_{2} - 1) q^{11} + (\beta_{5} + \beta_{3} - \beta_1 - 1) q^{13} + (\beta_{4} + \beta_1) q^{15} + ( - \beta_{5} - \beta_{3} - \beta_{2}) q^{17} + q^{19} - \beta_1 q^{21} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3}) q^{23} + (\beta_{4} - \beta_{3} - \beta_{2}) q^{25} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} + \cdots - 4) q^{27}+ \cdots + \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 7 q^{11} - 4 q^{13} + 4 q^{15} - 2 q^{17} + 6 q^{19} - 3 q^{21} - 2 q^{23} + q^{25} - 18 q^{27} - 5 q^{29} + 14 q^{31} + 3 q^{33} - 3 q^{35} + q^{37} + 22 q^{39} - 5 q^{41} - 15 q^{43} - 22 q^{45} - 7 q^{47} + 6 q^{49} + 4 q^{51} - 15 q^{53} + 20 q^{55} - 3 q^{57} - 23 q^{59} - 9 q^{61} + 3 q^{63} - 20 q^{65} - 2 q^{67} + 2 q^{69} + 23 q^{71} + 2 q^{73} - 4 q^{75} - 7 q^{77} + 17 q^{79} + 6 q^{81} - 22 q^{83} + 14 q^{85} - 10 q^{87} + 3 q^{89} - 4 q^{91} - 10 q^{93} - 3 q^{95} - 21 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 6x^{4} + 15x^{3} + 12x^{2} - 17x - 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 4\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} + 7\nu^{2} - 6\nu - 9 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} + 7\nu^{3} - 6\nu^{2} - 11\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 4\nu^{4} - \nu^{3} + 12\nu^{2} - 3\nu - 2 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 2\beta_{3} + 4\beta_{2} + 5\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 2\beta_{4} + 9\beta_{3} + 15\beta_{2} + 11\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{5} + 9\beta_{4} + 26\beta_{3} + 52\beta_{2} + 40\beta _1 + 50 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
3.24685
1.96359
1.26066
−0.365550
−1.38372
−1.72183
0 −3.24685 0 −1.31770 0 1.00000 0 7.54204 0
1.2 0 −1.96359 0 −3.24194 0 1.00000 0 0.855673 0
1.3 0 −1.26066 0 1.97889 0 1.00000 0 −1.41073 0
1.4 0 0.365550 0 2.99343 0 1.00000 0 −2.86637 0
1.5 0 1.38372 0 −1.87017 0 1.00000 0 −1.08531 0
1.6 0 1.72183 0 −1.54252 0 1.00000 0 −0.0353081 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.ca 6
4.b odd 2 1 8512.2.a.cf 6
8.b even 2 1 4256.2.a.n yes 6
8.d odd 2 1 4256.2.a.i ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4256.2.a.i ✓ 6 8.d odd 2 1
4256.2.a.n yes 6 8.b even 2 1
8512.2.a.ca 6 1.a even 1 1 trivial
8512.2.a.cf 6 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8512))\):

\( T_{3}^{6} + 3T_{3}^{5} - 6T_{3}^{4} - 15T_{3}^{3} + 12T_{3}^{2} + 17T_{3} - 7 \) Copy content Toggle raw display
\( T_{5}^{6} + 3T_{5}^{5} - 11T_{5}^{4} - 38T_{5}^{3} + 9T_{5}^{2} + 103T_{5} + 73 \) Copy content Toggle raw display
\( T_{11}^{6} + 7T_{11}^{5} + 2T_{11}^{4} - 35T_{11}^{3} + 22T_{11}^{2} - T_{11} - 1 \) Copy content Toggle raw display
\( T_{23}^{6} + 2T_{23}^{5} - 74T_{23}^{4} - 308T_{23}^{3} + 465T_{23}^{2} + 3338T_{23} + 3436 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 3 T^{5} + \cdots - 7 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + \cdots + 73 \) Copy content Toggle raw display
$7$ \( (T - 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 7 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} + \cdots + 100 \) Copy content Toggle raw display
$17$ \( T^{6} + 2 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 2 T^{5} + \cdots + 3436 \) Copy content Toggle raw display
$29$ \( T^{6} + 5 T^{5} + \cdots + 3115 \) Copy content Toggle raw display
$31$ \( T^{6} - 14 T^{5} + \cdots - 4 \) Copy content Toggle raw display
$37$ \( T^{6} - T^{5} + \cdots - 88861 \) Copy content Toggle raw display
$41$ \( T^{6} + 5 T^{5} + \cdots - 1631 \) Copy content Toggle raw display
$43$ \( T^{6} + 15 T^{5} + \cdots - 39280 \) Copy content Toggle raw display
$47$ \( T^{6} + 7 T^{5} + \cdots - 32935 \) Copy content Toggle raw display
$53$ \( T^{6} + 15 T^{5} + \cdots - 28649 \) Copy content Toggle raw display
$59$ \( T^{6} + 23 T^{5} + \cdots - 29369 \) Copy content Toggle raw display
$61$ \( T^{6} + 9 T^{5} + \cdots + 11465 \) Copy content Toggle raw display
$67$ \( T^{6} + 2 T^{5} + \cdots - 140636 \) Copy content Toggle raw display
$71$ \( T^{6} - 23 T^{5} + \cdots - 49993 \) Copy content Toggle raw display
$73$ \( T^{6} - 2 T^{5} + \cdots - 602300 \) Copy content Toggle raw display
$79$ \( T^{6} - 17 T^{5} + \cdots + 6224 \) Copy content Toggle raw display
$83$ \( T^{6} + 22 T^{5} + \cdots - 124880 \) Copy content Toggle raw display
$89$ \( T^{6} - 3 T^{5} + \cdots + 22336 \) Copy content Toggle raw display
$97$ \( T^{6} + 21 T^{5} + \cdots - 25595 \) Copy content Toggle raw display
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