Properties

Label 2200.2.a.y
Level $2200$
Weight $2$
Character orbit 2200.a
Self dual yes
Analytic conductor $17.567$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.54764.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{9} + q^{11} + 4 q^{13} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{17} + (\beta_{3} - 2 \beta_{2} - 2) q^{19} + ( - \beta_{3} + 2 \beta_1 - 2) q^{21} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{23} + (2 \beta_{2} + 3 \beta_1 + 2) q^{27} + (\beta_{3} + 2 \beta_1) q^{29} + ( - \beta_{3} + \beta_{2} + \beta_1 - 4) q^{31} + \beta_1 q^{33} + ( - 2 \beta_{2} - \beta_1 + 2) q^{37} + 4 \beta_1 q^{39} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{41} + ( - 2 \beta_{3} + 2 \beta_{2} + 2) q^{43} + ( - 2 \beta_{2} - 2) q^{47} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 3) q^{49} + ( - \beta_{3} + 2 \beta_1 - 6) q^{51} + ( - \beta_{3} - 2 \beta_{2} + 2) q^{53} + ( - \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{57} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{59} + ( - \beta_{3} - 2 \beta_1 + 8) q^{61} + (2 \beta_{2} - 2 \beta_1 + 10) q^{63} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{67} + ( - 2 \beta_{3} - 3 \beta_{2} - \beta_1 - 6) q^{69} + (\beta_{3} + \beta_{2} - 3 \beta_1) q^{71} + (2 \beta_{3} + 4) q^{73} + \beta_{3} q^{77} + ( - 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{79} + (2 \beta_{2} + 6 \beta_1 + 9) q^{81} + ( - 2 \beta_{2} + 4 \beta_1 + 6) q^{83} + (\beta_{3} + 2 \beta_{2} + 4 \beta_1 + 6) q^{87} + ( - \beta_{3} - \beta_{2} - \beta_1 + 6) q^{89} + 4 \beta_{3} q^{91} + (2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 6) q^{93} + ( - 3 \beta_{3} - \beta_1) q^{97} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} + q^{7} + 7 q^{9} + 4 q^{11} + 16 q^{13} + 7 q^{17} - 9 q^{19} - 7 q^{21} + 6 q^{23} + 13 q^{27} + 3 q^{29} - 15 q^{31} + q^{33} + 5 q^{37} + 4 q^{39} + 10 q^{41} + 8 q^{43} - 10 q^{47} + 9 q^{49} - 23 q^{51} + 5 q^{53} - 15 q^{57} + 6 q^{59} + 29 q^{61} + 40 q^{63} + 6 q^{67} - 30 q^{69} - q^{71} + 18 q^{73} + q^{77} - 20 q^{79} + 44 q^{81} + 26 q^{83} + 31 q^{87} + 21 q^{89} + 4 q^{91} + 25 q^{93} - 4 q^{97} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 9\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} + 7\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 9\beta _1 + 2 \) 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
−2.67673
−0.339102
0.655762
3.36007
0 −2.67673 0 0 0 4.38559 0 4.16490 0
1.2 0 −0.339102 0 0 0 −4.05237 0 −2.88501 0
1.3 0 0.655762 0 0 0 −0.415806 0 −2.56998 0
1.4 0 3.36007 0 0 0 1.08258 0 8.29009 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.y 4
4.b odd 2 1 4400.2.a.cb 4
5.b even 2 1 2200.2.a.x 4
5.c odd 4 2 440.2.b.d 8
15.e even 4 2 3960.2.d.f 8
20.d odd 2 1 4400.2.a.ce 4
20.e even 4 2 880.2.b.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.b.d 8 5.c odd 4 2
880.2.b.j 8 20.e even 4 2
2200.2.a.x 4 5.b even 2 1
2200.2.a.y 4 1.a even 1 1 trivial
3960.2.d.f 8 15.e even 4 2
4400.2.a.cb 4 4.b odd 2 1
4400.2.a.ce 4 20.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2200))\):

\( T_{3}^{4} - T_{3}^{3} - 9T_{3}^{2} + 3T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{4} - T_{7}^{3} - 18T_{7}^{2} + 12T_{7} + 8 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} - 9 T^{2} + 3 T + 2 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} - 18 T^{2} + 12 T + 8 \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( (T - 4)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 7 T^{3} - 44 T^{2} + 292 T + 32 \) Copy content Toggle raw display
$19$ \( T^{4} + 9 T^{3} - 20 T^{2} - 256 T - 128 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} - 59 T^{2} + 218 T + 856 \) Copy content Toggle raw display
$29$ \( T^{4} - 3 T^{3} - 38 T^{2} + 32 \) Copy content Toggle raw display
$31$ \( T^{4} + 15 T^{3} + 43 T^{2} - 83 T + 16 \) Copy content Toggle raw display
$37$ \( T^{4} - 5 T^{3} - 55 T^{2} + 191 T - 148 \) Copy content Toggle raw display
$41$ \( T^{4} - 10 T^{3} - 48 T^{2} + \cdots - 1024 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} - 64 T^{2} + \cdots + 1136 \) Copy content Toggle raw display
$47$ \( T^{4} + 10 T^{3} - 12 T^{2} + \cdots - 640 \) Copy content Toggle raw display
$53$ \( T^{4} - 5 T^{3} - 76 T^{2} + \cdots + 1280 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} - 47 T^{2} - 76 T - 32 \) Copy content Toggle raw display
$61$ \( T^{4} - 29 T^{3} + 274 T^{2} + \cdots + 160 \) Copy content Toggle raw display
$67$ \( T^{4} - 6 T^{3} - 83 T^{2} + 298 T + 568 \) Copy content Toggle raw display
$71$ \( T^{4} + T^{3} - 137 T^{2} - 865 T - 1336 \) Copy content Toggle raw display
$73$ \( T^{4} - 18 T^{3} + 48 T^{2} + \cdots - 1024 \) Copy content Toggle raw display
$79$ \( T^{4} + 20 T^{3} - 92 T^{2} + \cdots - 19456 \) Copy content Toggle raw display
$83$ \( T^{4} - 26 T^{3} + 76 T^{2} + \cdots - 6176 \) Copy content Toggle raw display
$89$ \( T^{4} - 21 T^{3} + 121 T^{2} + \cdots - 346 \) Copy content Toggle raw display
$97$ \( T^{4} + 4 T^{3} - 147 T^{2} + \cdots + 512 \) Copy content Toggle raw display
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