Properties

Label 3864.2.a.x
Level $3864$
Weight $2$
Character orbit 3864.a
Self dual yes
Analytic conductor $30.854$
Analytic rank $0$
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] = [3864,2,Mod(1,3864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3864, 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("3864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3864.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: \(30.8541953410\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 13x^{4} + 7x^{3} + 31x^{2} - 17x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{5} q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_{5} q^{5} - q^{7} + q^{9} + (\beta_{4} + 1) q^{11} + (\beta_{5} + \beta_{2}) q^{13} + \beta_{5} q^{15} + ( - \beta_{2} + 2) q^{17} + (\beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{19} - q^{21} + q^{23} + (\beta_{4} + \beta_{3} - \beta_1 + 3) q^{25} + q^{27} + ( - \beta_{4} + \beta_1) q^{29} - \beta_{2} q^{31} + (\beta_{4} + 1) q^{33} - \beta_{5} q^{35} + ( - \beta_{4} + \beta_1) q^{37} + (\beta_{5} + \beta_{2}) q^{39} + (\beta_{5} - \beta_{4} - \beta_{3} + 1) q^{41} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{43} + \beta_{5} q^{45} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{47} + q^{49} + ( - \beta_{2} + 2) q^{51} + (\beta_{5} + \beta_1 - 1) q^{53} + (2 \beta_{5} - \beta_{3} - 2) q^{55} + (\beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{57} + ( - \beta_{5} - \beta_1 + 3) q^{59} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + 3) q^{61} - q^{63} + (\beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{65} + ( - 2 \beta_{5} - \beta_{3} + 2) q^{67} + q^{69} + ( - \beta_{5} - \beta_{4} - \beta_1 + 2) q^{71} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 4) q^{73} + (\beta_{4} + \beta_{3} - \beta_1 + 3) q^{75} + ( - \beta_{4} - 1) q^{77} + ( - 3 \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 + 2) q^{79} + q^{81} + (2 \beta_{5} + \beta_{2}) q^{83} + (2 \beta_{5} - \beta_{3} + 2 \beta_{2} + 2) q^{85} + ( - \beta_{4} + \beta_1) q^{87} + (2 \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{89} + ( - \beta_{5} - \beta_{2}) q^{91} - \beta_{2} q^{93} + (2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{95} + ( - \beta_{5} + \beta_{3} - \beta_{2} + 6) q^{97} + (\beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} - 6 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} - 6 q^{7} + 6 q^{9} + 5 q^{11} + 2 q^{13} + 10 q^{17} + 3 q^{19} - 6 q^{21} + 6 q^{23} + 18 q^{25} + 6 q^{27} + 3 q^{29} - 2 q^{31} + 5 q^{33} + 3 q^{37} + 2 q^{39} + 4 q^{41} + 6 q^{43} - 2 q^{47} + 6 q^{49} + 10 q^{51} - 4 q^{53} - 15 q^{55} + 3 q^{57} + 16 q^{59} + 22 q^{61} - 6 q^{63} + 35 q^{65} + 9 q^{67} + 6 q^{69} + 11 q^{71} + 24 q^{73} + 18 q^{75} - 5 q^{77} + 18 q^{79} + 6 q^{81} + 2 q^{83} + 13 q^{85} + 3 q^{87} + 7 q^{89} - 2 q^{91} - 2 q^{93} + 5 q^{95} + 37 q^{97} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 13x^{4} + 7x^{3} + 31x^{2} - 17x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} - 12\nu^{3} + 5\nu^{2} + 21\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 14\nu^{3} - 5\nu^{2} + 32\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu^{4} + 14\nu^{3} - 7\nu^{2} - 39\nu + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{5} - 2\nu^{4} - 40\nu^{3} + 9\nu^{2} + 98\nu - 26 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 3\beta _1 + 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{5} + 7\beta_{4} - 5\beta_{3} - 3\beta_{2} - 3\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 19\beta_{5} + 27\beta_{4} - 11\beta_{3} - 19\beta_{2} + 29\beta _1 + 166 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 113\beta_{5} + 169\beta_{4} - 105\beta_{3} - 57\beta_{2} - 37\beta _1 + 346 ) / 4 \) 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
0.174924
−1.90697
1.67441
3.52361
−2.82561
0.359645
0 1.00000 0 −4.39875 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −2.43390 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 −0.344522 0 −1.00000 0 1.00000 0
1.4 0 1.00000 0 1.16465 0 −1.00000 0 1.00000 0
1.5 0 1.00000 0 1.74592 0 −1.00000 0 1.00000 0
1.6 0 1.00000 0 4.26660 0 −1.00000 0 1.00000 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
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

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 3864.2.a.x 6
4.b odd 2 1 7728.2.a.cg 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.x 6 1.a even 1 1 trivial
7728.2.a.cg 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3864))\):

\( T_{5}^{6} - 24T_{5}^{4} + 5T_{5}^{3} + 100T_{5}^{2} - 60T_{5} - 32 \) Copy content Toggle raw display
\( T_{11}^{6} - 5T_{11}^{5} - 33T_{11}^{4} + 145T_{11}^{3} + 100T_{11}^{2} - 464T_{11} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 24 T^{4} + 5 T^{3} + 100 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$7$ \( (T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 5 T^{5} - 33 T^{4} + 145 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} - 58 T^{4} + \cdots - 3160 \) Copy content Toggle raw display
$17$ \( T^{6} - 10 T^{5} - 7 T^{4} + 372 T^{3} + \cdots - 896 \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} - 81 T^{4} + \cdots + 6272 \) Copy content Toggle raw display
$23$ \( (T - 1)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} - 3 T^{5} - 75 T^{4} + 31 T^{3} + \cdots - 184 \) Copy content Toggle raw display
$31$ \( T^{6} + 2 T^{5} - 47 T^{4} + 76 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$37$ \( T^{6} - 3 T^{5} - 75 T^{4} + 31 T^{3} + \cdots - 184 \) Copy content Toggle raw display
$41$ \( T^{6} - 4 T^{5} - 122 T^{4} + \cdots - 268 \) Copy content Toggle raw display
$43$ \( T^{6} - 6 T^{5} - 134 T^{4} + \cdots - 5312 \) Copy content Toggle raw display
$47$ \( T^{6} + 2 T^{5} - 249 T^{4} + \cdots - 184000 \) Copy content Toggle raw display
$53$ \( T^{6} + 4 T^{5} - 90 T^{4} + \cdots + 2800 \) Copy content Toggle raw display
$59$ \( T^{6} - 16 T^{5} + 10 T^{4} + \cdots + 14048 \) Copy content Toggle raw display
$61$ \( T^{6} - 22 T^{5} + 14 T^{4} + \cdots - 89480 \) Copy content Toggle raw display
$67$ \( T^{6} - 9 T^{5} - 177 T^{4} + \cdots + 220928 \) Copy content Toggle raw display
$71$ \( T^{6} - 11 T^{5} - 97 T^{4} + \cdots + 17920 \) Copy content Toggle raw display
$73$ \( T^{6} - 24 T^{5} + 47 T^{4} + \cdots + 219520 \) Copy content Toggle raw display
$79$ \( T^{6} - 18 T^{5} - 89 T^{4} + \cdots + 145408 \) Copy content Toggle raw display
$83$ \( T^{6} - 2 T^{5} - 117 T^{4} + \cdots - 3968 \) Copy content Toggle raw display
$89$ \( T^{6} - 7 T^{5} - 177 T^{4} + \cdots - 22832 \) Copy content Toggle raw display
$97$ \( T^{6} - 37 T^{5} + 469 T^{4} + \cdots - 4840 \) Copy content Toggle raw display
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