Properties

Label 2128.2.a.r
Level $2128$
Weight $2$
Character orbit 2128.a
Self dual yes
Analytic conductor $16.992$
Analytic rank $0$
Dimension $3$
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] = [2128,2,Mod(1,2128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2128, 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("2128.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2128 = 2^{4} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2128.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: \(16.9921655501\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.733.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 8 \) 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 532)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{2} - \beta_1 + 1) q^{5} + q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{2} - \beta_1 + 1) q^{5} + q^{7} + (\beta_{2} + 2) q^{9} + (\beta_{2} + 1) q^{11} + ( - \beta_{2} - \beta_1 + 3) q^{13} + ( - 2 \beta_{2} - \beta_1 - 2) q^{15} + (\beta_{2} + 3) q^{17} + q^{19} + \beta_1 q^{21} + ( - \beta_{2} - \beta_1 - 1) q^{23} + ( - \beta_{2} + \beta_1 + 2) q^{25} + (\beta_{2} + \beta_1 - 3) q^{27} + (\beta_{2} + 2 \beta_1 + 3) q^{29} + (\beta_{2} + 2 \beta_1 + 1) q^{31} + (\beta_{2} + 3 \beta_1 - 3) q^{33} + ( - \beta_{2} - \beta_1 + 1) q^{35} + ( - \beta_{2} + \beta_1 + 1) q^{37} + ( - 2 \beta_{2} + \beta_1 - 2) q^{39} + ( - \beta_{2} + 2 \beta_1 + 3) q^{41} + (2 \beta_{2} - 2 \beta_1 + 2) q^{43} + ( - 3 \beta_1 - 2) q^{45} + ( - 3 \beta_{2} - \beta_1 + 1) q^{47} + q^{49} + (\beta_{2} + 5 \beta_1 - 3) q^{51} + (3 \beta_1 + 2) q^{53} + (\beta_{2} - 2 \beta_1 - 3) q^{55} + \beta_1 q^{57} + (3 \beta_{2} + 3 \beta_1 + 3) q^{59} + ( - \beta_{2} - 5 \beta_1 + 1) q^{61} + (\beta_{2} + 2) q^{63} + ( - 3 \beta_{2} - \beta_1 + 9) q^{65} + (\beta_{2} + 3) q^{67} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{69} + ( - \beta_{2} - 3 \beta_1 + 1) q^{71} + (2 \beta_{2} + 3 \beta_1 - 8) q^{73} + 8 q^{75} + (\beta_{2} + 1) q^{77} + ( - 4 \beta_{2} - 2) q^{79} + ( - \beta_{2} - \beta_1 - 4) q^{81} + ( - 2 \beta_{2} + \beta_1 + 2) q^{83} + ( - \beta_{2} - 4 \beta_1 - 1) q^{85} + (3 \beta_{2} + 5 \beta_1 + 7) q^{87} + (2 \beta_{2} - 6) q^{89} + ( - \beta_{2} - \beta_1 + 3) q^{91} + (3 \beta_{2} + 3 \beta_1 + 7) q^{93} + ( - \beta_{2} - \beta_1 + 1) q^{95} + (5 \beta_{2} + \beta_1 + 9) q^{97} + (\beta_{2} - \beta_1 + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + 2 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} + 2 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 8 q^{13} - 7 q^{15} + 9 q^{17} + 3 q^{19} + q^{21} - 4 q^{23} + 7 q^{25} - 8 q^{27} + 11 q^{29} + 5 q^{31} - 6 q^{33} + 2 q^{35} + 4 q^{37} - 5 q^{39} + 11 q^{41} + 4 q^{43} - 9 q^{45} + 2 q^{47} + 3 q^{49} - 4 q^{51} + 9 q^{53} - 11 q^{55} + q^{57} + 12 q^{59} - 2 q^{61} + 6 q^{63} + 26 q^{65} + 9 q^{67} - 9 q^{69} - 21 q^{73} + 24 q^{75} + 3 q^{77} - 6 q^{79} - 13 q^{81} + 7 q^{83} - 7 q^{85} + 26 q^{87} - 18 q^{89} + 8 q^{91} + 24 q^{93} + 2 q^{95} + 28 q^{97} + 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 7x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) 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
−2.69639
1.17819
2.51820
0 −2.69639 0 1.42586 0 1.00000 0 4.27053 0
1.2 0 1.17819 0 3.43366 0 1.00000 0 −1.61186 0
1.3 0 2.51820 0 −2.85952 0 1.00000 0 3.34132 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 2128.2.a.r 3
4.b odd 2 1 532.2.a.e 3
8.b even 2 1 8512.2.a.bl 3
8.d odd 2 1 8512.2.a.bn 3
12.b even 2 1 4788.2.a.o 3
28.d even 2 1 3724.2.a.i 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.a.e 3 4.b odd 2 1
2128.2.a.r 3 1.a even 1 1 trivial
3724.2.a.i 3 28.d even 2 1
4788.2.a.o 3 12.b even 2 1
8512.2.a.bl 3 8.b even 2 1
8512.2.a.bn 3 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2128))\):

\( T_{3}^{3} - T_{3}^{2} - 7T_{3} + 8 \) Copy content Toggle raw display
\( T_{5}^{3} - 2T_{5}^{2} - 9T_{5} + 14 \) Copy content Toggle raw display
\( T_{11}^{3} - 3T_{11}^{2} - 7T_{11} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 7T + 8 \) Copy content Toggle raw display
$5$ \( T^{3} - 2 T^{2} + \cdots + 14 \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 3 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$13$ \( T^{3} - 8 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{3} - 9 T^{2} + \cdots + 14 \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 4 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$29$ \( T^{3} - 11 T^{2} + \cdots + 2 \) Copy content Toggle raw display
$31$ \( T^{3} - 5 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$37$ \( T^{3} - 4 T^{2} + \cdots + 50 \) Copy content Toggle raw display
$41$ \( T^{3} - 11 T^{2} + \cdots + 280 \) Copy content Toggle raw display
$43$ \( T^{3} - 4 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$47$ \( T^{3} - 2 T^{2} + \cdots - 184 \) Copy content Toggle raw display
$53$ \( T^{3} - 9 T^{2} + \cdots + 322 \) Copy content Toggle raw display
$59$ \( T^{3} - 12 T^{2} + \cdots + 108 \) Copy content Toggle raw display
$61$ \( T^{3} + 2 T^{2} + \cdots - 202 \) Copy content Toggle raw display
$67$ \( T^{3} - 9 T^{2} + \cdots + 14 \) Copy content Toggle raw display
$71$ \( T^{3} - 55T + 58 \) Copy content Toggle raw display
$73$ \( T^{3} + 21 T^{2} + \cdots - 302 \) Copy content Toggle raw display
$79$ \( T^{3} + 6 T^{2} + \cdots - 1016 \) Copy content Toggle raw display
$83$ \( T^{3} - 7 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$89$ \( T^{3} + 18 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$97$ \( T^{3} - 28 T^{2} + \cdots + 2536 \) Copy content Toggle raw display
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