Properties

Label 4400.2.a.by
Level $4400$
Weight $2$
Character orbit 4400.a
Self dual yes
Analytic conductor $35.134$
Analytic rank $1$
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] = [4400,2,Mod(1,4400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4400, 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("4400.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.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: \(35.1341768894\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 6 \) 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 2200)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{2} - 1) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{2} - 1) q^{7} + (\beta_{2} + 2) q^{9} - q^{11} + q^{13} + (\beta_1 - 2) q^{17} + (2 \beta_1 - 3) q^{19} + (\beta_{2} + 3 \beta_1 - 1) q^{21} + ( - \beta_1 - 3) q^{23} + ( - \beta_{2} - \beta_1 + 1) q^{27} + ( - \beta_{2} - 2 \beta_1 + 4) q^{29} + (2 \beta_{2} + 1) q^{31} + \beta_1 q^{33} + ( - \beta_{2} + \beta_1 - 3) q^{37} - \beta_1 q^{39} + ( - \beta_{2} + \beta_1 + 1) q^{41} + (\beta_{2} - 3 \beta_1 - 2) q^{43} + (\beta_{2} + 3 \beta_1 - 1) q^{47} + (\beta_1 + 3) q^{49} + ( - \beta_{2} + 2 \beta_1 - 5) q^{51} + (3 \beta_{2} + 2 \beta_1 + 1) q^{53} + ( - 2 \beta_{2} + 3 \beta_1 - 10) q^{57} + (\beta_{2} + 3 \beta_1 - 1) q^{59} + (\beta_{2} + 2 \beta_1 - 1) q^{61} + ( - \beta_{2} - \beta_1 - 11) q^{63} + (2 \beta_{2} + 2 \beta_1 + 4) q^{67} + (\beta_{2} + 3 \beta_1 + 5) q^{69} + ( - \beta_{2} + \beta_1 - 6) q^{71} + ( - 3 \beta_{2} - 7) q^{73} + (\beta_{2} + 1) q^{77} + (2 \beta_{2} + \beta_1 - 4) q^{79} + ( - \beta_{2} + \beta_1 - 2) q^{81} + (\beta_{2} + 4 \beta_1 - 8) q^{83} + (3 \beta_{2} - 2 \beta_1 + 9) q^{87} + ( - \beta_{2} + 10) q^{89} + ( - \beta_{2} - 1) q^{91} + ( - 2 \beta_{2} - 5 \beta_1 + 2) q^{93} + (2 \beta_{2} - \beta_1 - 3) q^{97} + ( - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - 3 q^{7} + 6 q^{9} - 3 q^{11} + 3 q^{13} - 5 q^{17} - 7 q^{19} - 10 q^{23} + 2 q^{27} + 10 q^{29} + 3 q^{31} + q^{33} - 8 q^{37} - q^{39} + 4 q^{41} - 9 q^{43} + 10 q^{49} - 13 q^{51} + 5 q^{53} - 27 q^{57} - q^{61} - 34 q^{63} + 14 q^{67} + 18 q^{69} - 17 q^{71} - 21 q^{73} + 3 q^{77} - 11 q^{79} - 5 q^{81} - 20 q^{83} + 25 q^{87} + 30 q^{89} - 3 q^{91} + q^{93} - 10 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 7x + 6 \) : 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
2.75153
0.841083
−2.59261
0 −2.75153 0 0 0 −3.57093 0 4.57093 0
1.2 0 −0.841083 0 0 0 3.29258 0 −2.29258 0
1.3 0 2.59261 0 0 0 −2.72165 0 3.72165 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(11\) \( +1 \)

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 4400.2.a.by 3
4.b odd 2 1 2200.2.a.v yes 3
5.b even 2 1 4400.2.a.bz 3
5.c odd 4 2 4400.2.b.bb 6
20.d odd 2 1 2200.2.a.u 3
20.e even 4 2 2200.2.b.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.u 3 20.d odd 2 1
2200.2.a.v yes 3 4.b odd 2 1
2200.2.b.m 6 20.e even 4 2
4400.2.a.by 3 1.a even 1 1 trivial
4400.2.a.bz 3 5.b even 2 1
4400.2.b.bb 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4400))\):

\( T_{3}^{3} + T_{3}^{2} - 7T_{3} - 6 \) Copy content Toggle raw display
\( T_{7}^{3} + 3T_{7}^{2} - 11T_{7} - 32 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 7T - 6 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 3 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 5T^{2} + T - 4 \) Copy content Toggle raw display
$19$ \( T^{3} + 7 T^{2} + \cdots - 27 \) Copy content Toggle raw display
$23$ \( T^{3} + 10 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{3} - 10 T^{2} + \cdots + 201 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} + \cdots + 207 \) Copy content Toggle raw display
$37$ \( T^{3} + 8T^{2} - T - 44 \) Copy content Toggle raw display
$41$ \( T^{3} - 4 T^{2} + \cdots + 24 \) Copy content Toggle raw display
$43$ \( T^{3} + 9 T^{2} + \cdots - 508 \) Copy content Toggle raw display
$47$ \( T^{3} - 77T - 192 \) Copy content Toggle raw display
$53$ \( T^{3} - 5 T^{2} + \cdots + 142 \) Copy content Toggle raw display
$59$ \( T^{3} - 77T - 192 \) Copy content Toggle raw display
$61$ \( T^{3} + T^{2} + \cdots - 114 \) Copy content Toggle raw display
$67$ \( T^{3} - 14 T^{2} + \cdots + 96 \) Copy content Toggle raw display
$71$ \( T^{3} + 17 T^{2} + \cdots + 52 \) Copy content Toggle raw display
$73$ \( T^{3} + 21 T^{2} + \cdots - 1052 \) Copy content Toggle raw display
$79$ \( T^{3} + 11 T^{2} + \cdots - 144 \) Copy content Toggle raw display
$83$ \( T^{3} + 20 T^{2} + \cdots - 829 \) Copy content Toggle raw display
$89$ \( T^{3} - 30 T^{2} + \cdots - 879 \) Copy content Toggle raw display
$97$ \( T^{3} + 10 T^{2} + \cdots - 23 \) Copy content Toggle raw display
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