Properties

Label 2888.2.a.t
Level $2888$
Weight $2$
Character orbit 2888.a
Self dual yes
Analytic conductor $23.061$
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] = [2888,2,Mod(1,2888)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2888, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2888.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2888 = 2^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2888.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: \(23.0607961037\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.3022625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 7x^{4} + 7x^{3} + 17x^{2} + 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{3} + ( - \beta_{5} - \beta_1) q^{5} + (\beta_{5} + \beta_{2}) q^{7} + (\beta_{4} + \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{3} + ( - \beta_{5} - \beta_1) q^{5} + (\beta_{5} + \beta_{2}) q^{7} + (\beta_{4} + \beta_{2} - \beta_1 + 2) q^{9} + (\beta_{5} - \beta_{4} + \beta_1 - 1) q^{11} + (\beta_{3} - \beta_{2} - 2) q^{13} + (\beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{15} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2} + 2) q^{17} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_1 - 4) q^{21} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 - 2) q^{23} + ( - \beta_{5} - \beta_{3} + 2 \beta_1 + 3) q^{25} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{27} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + 5 \beta_1 + 3) q^{29} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{31} + ( - \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{33} + (\beta_{5} + 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 7) q^{35} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{37} + (\beta_{5} + \beta_{4} + 2 \beta_{2} + 3 \beta_1 + 9) q^{39} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_1 - 1) q^{41} + (\beta_{5} - \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 4) q^{43} + ( - \beta_{5} + 2 \beta_{3} - 7 \beta_1 - 2) q^{45} + (2 \beta_{5} - \beta_{3} + \beta_1 + 3) q^{47} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{49} + (2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 7) q^{51} + (\beta_{5} - \beta_{4} + \beta_{2} - 4 \beta_1 - 1) q^{53} + (\beta_{5} + \beta_{3} + \beta_{2} + 3 \beta_1 - 5) q^{55} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 1) q^{59} + (2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{61} + (\beta_{5} + \beta_{4} + 4 \beta_{2} + 3 \beta_1 + 7) q^{63} + (3 \beta_{5} + \beta_{4} - 3 \beta_{3} + \beta_{2} + 2 \beta_1) q^{65} + ( - \beta_{4} + 2 \beta_{3} + 3 \beta_1 - 2) q^{67} + (\beta_{5} + \beta_{4} + \beta_{3} + 5 \beta_{2} - 6 \beta_1 - 1) q^{69} + (2 \beta_{5} - \beta_{3} - 3 \beta_1 - 1) q^{71} + ( - \beta_{5} + \beta_{3} - 2 \beta_1 - 1) q^{73} + (\beta_{4} + \beta_{3} - \beta_{2} - 6 \beta_1 - 6) q^{75} + ( - \beta_{5} - 3 \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{77} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_1 - 3) q^{79} + (\beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_{2} - 4 \beta_1 - 2) q^{81} + ( - \beta_{5} + \beta_{4} + 3 \beta_{2} - 5 \beta_1 - 4) q^{83} + ( - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_1 + 7) q^{85} + ( - 2 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} + \beta_{2} - 11 \beta_1 + 2) q^{87} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - 1) q^{89} + ( - 4 \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 7) q^{91} + (2 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 11 \beta_1 + 3) q^{93} + (\beta_{5} + 3 \beta_{4} - 3 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{97} + ( - \beta_{3} - \beta_{2} + 9 \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 2 q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 2 q^{5} - 2 q^{7} + 9 q^{9} - 5 q^{11} - 10 q^{13} + 4 q^{17} - 23 q^{21} - 15 q^{23} + 12 q^{25} - 18 q^{27} + 7 q^{29} - 5 q^{31} + q^{33} - 38 q^{35} - 17 q^{37} + 37 q^{39} - 4 q^{41} - 11 q^{43} + 6 q^{45} + 18 q^{47} + 20 q^{49} - 48 q^{51} + 7 q^{53} - 42 q^{55} - q^{59} - 4 q^{61} + 19 q^{63} - 6 q^{65} - 20 q^{67} - 6 q^{69} + 6 q^{71} - 2 q^{73} - 19 q^{75} + 41 q^{77} - 8 q^{79} - 6 q^{81} - 22 q^{83} + 32 q^{85} + 29 q^{87} - 14 q^{89} - 26 q^{91} - 5 q^{93} - 41 q^{97} - 47 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 7x^{4} + 7x^{3} + 17x^{2} + 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 10\nu^{2} + 7\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 10\nu^{2} + 9\nu - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 8\nu^{2} + 10\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - 2\beta_{3} - \beta_{2} + \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + 5\beta_{4} - 6\beta_{3} - \beta_{2} + 3\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{5} + 9\beta_{4} - 15\beta_{3} - 4\beta_{2} + 8\beta _1 + 37 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{5} + 15\beta_{4} - 20\beta_{3} - 3\beta_{2} + 13\beta _1 + 46 \) 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
0.149742
2.70201
−0.496915
−1.65046
−1.20509
2.50071
0 −3.03477 0 1.96231 0 1.69050 0 6.20985 0
1.2 0 −2.77311 0 −3.10181 0 4.25689 0 4.69013 0
1.3 0 −1.45214 0 3.79176 0 −3.95766 0 −0.891299 0
1.4 0 0.354424 0 −1.55742 0 1.82103 0 −2.87438 0
1.5 0 1.60721 0 −1.92601 0 −1.29923 0 −0.416870 0
1.6 0 2.29838 0 2.83118 0 −4.51153 0 2.28257 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
\(2\) \(-1\)
\(19\) \(-1\)

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 2888.2.a.t 6
4.b odd 2 1 5776.2.a.bz 6
19.b odd 2 1 2888.2.a.u yes 6
76.d even 2 1 5776.2.a.bx 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2888.2.a.t 6 1.a even 1 1 trivial
2888.2.a.u yes 6 19.b odd 2 1
5776.2.a.bx 6 76.d even 2 1
5776.2.a.bz 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2888))\):

\( T_{3}^{6} + 3T_{3}^{5} - 9T_{3}^{4} - 24T_{3}^{3} + 24T_{3}^{2} + 40T_{3} - 16 \) Copy content Toggle raw display
\( T_{5}^{6} - 2T_{5}^{5} - 19T_{5}^{4} + 26T_{5}^{3} + 109T_{5}^{2} - 70T_{5} - 196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 3 T^{5} - 9 T^{4} - 24 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} - 19 T^{4} + 26 T^{3} + \cdots - 196 \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} - 29 T^{4} - 38 T^{3} + \cdots - 304 \) Copy content Toggle raw display
$11$ \( T^{6} + 5 T^{5} - 29 T^{4} - 166 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$13$ \( T^{6} + 10 T^{5} + 6 T^{4} - 194 T^{3} + \cdots - 311 \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} - 69 T^{4} + 140 T^{3} + \cdots - 76 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 15 T^{5} + 27 T^{4} + \cdots - 6224 \) Copy content Toggle raw display
$29$ \( T^{6} - 7 T^{5} - 122 T^{4} + \cdots + 63541 \) Copy content Toggle raw display
$31$ \( T^{6} + 5 T^{5} - 55 T^{4} + \cdots + 2000 \) Copy content Toggle raw display
$37$ \( T^{6} + 17 T^{5} + 40 T^{4} + \cdots + 379 \) Copy content Toggle raw display
$41$ \( T^{6} + 4 T^{5} - 116 T^{4} + \cdots - 12211 \) Copy content Toggle raw display
$43$ \( T^{6} + 11 T^{5} - 153 T^{4} + \cdots - 64976 \) Copy content Toggle raw display
$47$ \( T^{6} - 18 T^{5} + 47 T^{4} + \cdots + 496 \) Copy content Toggle raw display
$53$ \( T^{6} - 7 T^{5} - 102 T^{4} + \cdots + 1621 \) Copy content Toggle raw display
$59$ \( T^{6} + T^{5} - 169 T^{4} - 302 T^{3} + \cdots - 1264 \) Copy content Toggle raw display
$61$ \( T^{6} + 4 T^{5} - 92 T^{4} + \cdots + 1319 \) Copy content Toggle raw display
$67$ \( T^{6} + 20 T^{5} + 43 T^{4} + \cdots + 13264 \) Copy content Toggle raw display
$71$ \( T^{6} - 6 T^{5} - 133 T^{4} + \cdots + 17744 \) Copy content Toggle raw display
$73$ \( T^{6} + 2 T^{5} - 38 T^{4} + 48 T^{3} + \cdots + 71 \) Copy content Toggle raw display
$79$ \( T^{6} + 8 T^{5} - 116 T^{4} + \cdots - 61504 \) Copy content Toggle raw display
$83$ \( T^{6} + 22 T^{5} + 12 T^{4} + \cdots - 41024 \) Copy content Toggle raw display
$89$ \( T^{6} + 14 T^{5} - 38 T^{4} + \cdots - 1769 \) Copy content Toggle raw display
$97$ \( T^{6} + 41 T^{5} + 344 T^{4} + \cdots + 324539 \) Copy content Toggle raw display
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