Properties

Label 2200.2.a.w
Level $2200$
Weight $2$
Character orbit 2200.a
Self dual yes
Analytic conductor $17.567$
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] = [2200,2,Mod(1,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, 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("2200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2200.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: \(17.5670884447\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
2.52892
−0.167449
−2.36147
0 −1.52892 0 0 0 −1.39543 0 −0.662410 0
1.2 0 1.16745 0 0 0 4.97196 0 −1.63706 0
1.3 0 3.36147 0 0 0 −0.576535 0 8.29947 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 2200.2.a.w yes 3
4.b odd 2 1 4400.2.a.bx 3
5.b even 2 1 2200.2.a.t 3
5.c odd 4 2 2200.2.b.l 6
20.d odd 2 1 4400.2.a.ca 3
20.e even 4 2 4400.2.b.bc 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.t 3 5.b even 2 1
2200.2.a.w yes 3 1.a even 1 1 trivial
2200.2.b.l 6 5.c odd 4 2
4400.2.a.bx 3 4.b odd 2 1
4400.2.a.ca 3 20.d odd 2 1
4400.2.b.bc 6 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2200))\):

\( T_{3}^{3} - 3T_{3}^{2} - 3T_{3} + 6 \) Copy content Toggle raw display
\( T_{7}^{3} - 3T_{7}^{2} - 9T_{7} - 4 \) Copy content Toggle raw display
\( T_{13}^{3} + 3T_{13}^{2} - 21T_{13} - 15 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 3 T^{2} + \cdots + 6 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 3 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 3 T^{2} + \cdots - 15 \) Copy content Toggle raw display
$17$ \( T^{3} + 3 T^{2} + \cdots - 144 \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 6 T^{2} + \cdots + 5 \) Copy content Toggle raw display
$29$ \( T^{3} - 12 T^{2} + \cdots - 31 \) Copy content Toggle raw display
$31$ \( T^{3} + 3 T^{2} + \cdots + 73 \) Copy content Toggle raw display
$37$ \( T^{3} - 12 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$41$ \( T^{3} - 6 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$43$ \( T^{3} + 3 T^{2} + \cdots - 12 \) Copy content Toggle raw display
$47$ \( T^{3} - 12 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$53$ \( T^{3} + 9 T^{2} + \cdots - 542 \) Copy content Toggle raw display
$59$ \( T^{3} - 24 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$61$ \( T^{3} + 3 T^{2} + \cdots - 346 \) Copy content Toggle raw display
$67$ \( T^{3} - 6 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$71$ \( T^{3} + 15 T^{2} + \cdots - 1500 \) Copy content Toggle raw display
$73$ \( T^{3} + 33 T^{2} + \cdots + 1184 \) Copy content Toggle raw display
$79$ \( T^{3} - 15 T^{2} + \cdots + 172 \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} + \cdots - 13 \) Copy content Toggle raw display
$89$ \( T^{3} - 6 T^{2} + \cdots + 453 \) Copy content Toggle raw display
$97$ \( T^{3} + 18 T^{2} + \cdots - 2477 \) Copy content Toggle raw display
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