Properties

Label 5550.2.a.cd
Level $5550$
Weight $2$
Character orbit 5550.a
Self dual yes
Analytic conductor $44.317$
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] = [5550,2,Mod(1,5550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5550, 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("5550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.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: \(44.3169731218\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.993.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} + ( - \beta_{2} + \beta_1) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + q^{6} + ( - \beta_{2} + \beta_1) q^{7} - q^{8} + q^{9} + (\beta_{2} - 2 \beta_1 + 1) q^{11} - q^{12} + ( - \beta_1 + 3) q^{13} + (\beta_{2} - \beta_1) q^{14} + q^{16} + (\beta_{2} - 2 \beta_1 + 1) q^{17} - q^{18} + (\beta_1 + 3) q^{19} + (\beta_{2} - \beta_1) q^{21} + ( - \beta_{2} + 2 \beta_1 - 1) q^{22} + ( - \beta_{2} - \beta_1 - 2) q^{23} + q^{24} + (\beta_1 - 3) q^{26} - q^{27} + ( - \beta_{2} + \beta_1) q^{28} + ( - 2 \beta_1 + 6) q^{31} - q^{32} + ( - \beta_{2} + 2 \beta_1 - 1) q^{33} + ( - \beta_{2} + 2 \beta_1 - 1) q^{34} + q^{36} - q^{37} + ( - \beta_1 - 3) q^{38} + (\beta_1 - 3) q^{39} + ( - 2 \beta_1 + 6) q^{41} + ( - \beta_{2} + \beta_1) q^{42} + (2 \beta_{2} + 1) q^{43} + (\beta_{2} - 2 \beta_1 + 1) q^{44} + (\beta_{2} + \beta_1 + 2) q^{46} + (2 \beta_{2} - 4 \beta_1) q^{47} - q^{48} + ( - 2 \beta_{2} - \beta_1 + 4) q^{49} + ( - \beta_{2} + 2 \beta_1 - 1) q^{51} + ( - \beta_1 + 3) q^{52} + (\beta_{2} + 5 \beta_1 - 4) q^{53} + q^{54} + (\beta_{2} - \beta_1) q^{56} + ( - \beta_1 - 3) q^{57} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{59} + (2 \beta_1 + 3) q^{61} + (2 \beta_1 - 6) q^{62} + ( - \beta_{2} + \beta_1) q^{63} + q^{64} + (\beta_{2} - 2 \beta_1 + 1) q^{66} + (4 \beta_1 - 6) q^{67} + (\beta_{2} - 2 \beta_1 + 1) q^{68} + (\beta_{2} + \beta_1 + 2) q^{69} + (2 \beta_1 - 2) q^{71} - q^{72} + ( - 2 \beta_{2} - 3 \beta_1 + 7) q^{73} + q^{74} + (\beta_1 + 3) q^{76} + (\beta_{2} + 4 \beta_1 - 14) q^{77} + ( - \beta_1 + 3) q^{78} - 4 \beta_{2} q^{79} + q^{81} + (2 \beta_1 - 6) q^{82} + ( - 3 \beta_{2} - 2 \beta_1 + 5) q^{83} + (\beta_{2} - \beta_1) q^{84} + ( - 2 \beta_{2} - 1) q^{86} + ( - \beta_{2} + 2 \beta_1 - 1) q^{88} + (\beta_{2} + 2 \beta_1 + 1) q^{89} + ( - 3 \beta_{2} + 5 \beta_1 - 3) q^{91} + ( - \beta_{2} - \beta_1 - 2) q^{92} + (2 \beta_1 - 6) q^{93} + ( - 2 \beta_{2} + 4 \beta_1) q^{94} + q^{96} + ( - 2 \beta_{2} - 4 \beta_1) q^{97} + (2 \beta_{2} + \beta_1 - 4) q^{98} + (\beta_{2} - 2 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{8} + 3 q^{9} + 2 q^{11} - 3 q^{12} + 8 q^{13} + 3 q^{16} + 2 q^{17} - 3 q^{18} + 10 q^{19} - 2 q^{22} - 8 q^{23} + 3 q^{24} - 8 q^{26} - 3 q^{27} + 16 q^{31} - 3 q^{32} - 2 q^{33} - 2 q^{34} + 3 q^{36} - 3 q^{37} - 10 q^{38} - 8 q^{39} + 16 q^{41} + 5 q^{43} + 2 q^{44} + 8 q^{46} - 2 q^{47} - 3 q^{48} + 9 q^{49} - 2 q^{51} + 8 q^{52} - 6 q^{53} + 3 q^{54} - 10 q^{57} + 8 q^{59} + 11 q^{61} - 16 q^{62} + 3 q^{64} + 2 q^{66} - 14 q^{67} + 2 q^{68} + 8 q^{69} - 4 q^{71} - 3 q^{72} + 16 q^{73} + 3 q^{74} + 10 q^{76} - 37 q^{77} + 8 q^{78} - 4 q^{79} + 3 q^{81} - 16 q^{82} + 10 q^{83} - 5 q^{86} - 2 q^{88} + 6 q^{89} - 7 q^{91} - 8 q^{92} - 16 q^{93} + 2 q^{94} + 3 q^{96} - 6 q^{97} - 9 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x + 3 \) : 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
−2.25342
2.77339
0.480031
−1.00000 −1.00000 1.00000 0 1.00000 −3.33131 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 −0.918290 −1.00000 1.00000 0
1.3 −1.00000 −1.00000 1.00000 0 1.00000 4.24960 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(37\) \(1\)

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 5550.2.a.cd 3
5.b even 2 1 5550.2.a.ci yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5550.2.a.cd 3 1.a even 1 1 trivial
5550.2.a.ci yes 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5550))\):

\( T_{7}^{3} - 15T_{7} - 13 \) Copy content Toggle raw display
\( T_{11}^{3} - 2T_{11}^{2} - 27T_{11} - 21 \) Copy content Toggle raw display
\( T_{13}^{3} - 8T_{13}^{2} + 15T_{13} - 3 \) Copy content Toggle raw display
\( T_{17}^{3} - 2T_{17}^{2} - 27T_{17} - 21 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 15T - 13 \) Copy content Toggle raw display
$11$ \( T^{3} - 2 T^{2} - 27 T - 21 \) Copy content Toggle raw display
$13$ \( T^{3} - 8 T^{2} + 15 T - 3 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 27 T - 21 \) Copy content Toggle raw display
$19$ \( T^{3} - 10 T^{2} + 27 T - 15 \) Copy content Toggle raw display
$23$ \( T^{3} + 8 T^{2} - 5 T - 9 \) Copy content Toggle raw display
$29$ \( T^{3} \) Copy content Toggle raw display
$31$ \( T^{3} - 16 T^{2} + 60 T - 24 \) Copy content Toggle raw display
$37$ \( (T + 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 16 T^{2} + 60 T - 24 \) Copy content Toggle raw display
$43$ \( T^{3} - 5 T^{2} - 49 T + 173 \) Copy content Toggle raw display
$47$ \( T^{3} + 2 T^{2} - 112 T - 392 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 189 T - 1033 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 84 T + 600 \) Copy content Toggle raw display
$61$ \( T^{3} - 11 T^{2} + 15 T + 51 \) Copy content Toggle raw display
$67$ \( T^{3} + 14 T^{2} - 36 T - 312 \) Copy content Toggle raw display
$71$ \( T^{3} + 4 T^{2} - 20 T - 24 \) Copy content Toggle raw display
$73$ \( T^{3} - 16 T^{2} - 63 T + 1323 \) Copy content Toggle raw display
$79$ \( T^{3} + 4 T^{2} - 224 T - 960 \) Copy content Toggle raw display
$83$ \( T^{3} - 10 T^{2} - 155 T + 1119 \) Copy content Toggle raw display
$89$ \( T^{3} - 6 T^{2} - 39 T - 45 \) Copy content Toggle raw display
$97$ \( T^{3} + 6 T^{2} - 192 T + 712 \) Copy content Toggle raw display
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