Properties

Label 9054.2.a.ba
Level $9054$
Weight $2$
Character orbit 9054.a
Self dual yes
Analytic conductor $72.297$
Analytic rank $0$
Dimension $5$
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] = [9054,2,Mod(1,9054)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9054, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("9054.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9054 = 2 \cdot 3^{2} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9054.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: \(72.2965539901\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.205225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1006)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{5} + (\beta_{4} + \beta_{3} - \beta_1 - 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{5} + (\beta_{4} + \beta_{3} - \beta_1 - 1) q^{7} - q^{8} + ( - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{10} + (\beta_{3} + 2 \beta_{2} + 4) q^{11} + (2 \beta_{4} + \beta_{2} + \beta_1) q^{13} + ( - \beta_{4} - \beta_{3} + \beta_1 + 1) q^{14} + q^{16} + ( - 2 \beta_{3} - \beta_1 + 1) q^{17} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{19} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{20} + ( - \beta_{3} - 2 \beta_{2} - 4) q^{22} + (2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{23} + ( - \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 2) q^{25} + ( - 2 \beta_{4} - \beta_{2} - \beta_1) q^{26} + (\beta_{4} + \beta_{3} - \beta_1 - 1) q^{28} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 2) q^{29} + ( - \beta_{3} - \beta_{2} - \beta_1 - 7) q^{31} - q^{32} + (2 \beta_{3} + \beta_1 - 1) q^{34} + ( - \beta_{4} - 6 \beta_1 + 3) q^{35} + ( - \beta_{4} - 4 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{37} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{38} + ( - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{40} + ( - \beta_{3} + 2 \beta_{2} + 3) q^{41} + (\beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{43} + (\beta_{3} + 2 \beta_{2} + 4) q^{44} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{46} + (5 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 4) q^{47} + ( - \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 8) q^{49} + (\beta_{4} - 2 \beta_{2} + 2 \beta_1 - 2) q^{50} + (2 \beta_{4} + \beta_{2} + \beta_1) q^{52} + (2 \beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{53} + (5 \beta_{4} + 7 \beta_{3} + \beta_{2} + 6 \beta_1 + 3) q^{55} + ( - \beta_{4} - \beta_{3} + \beta_1 + 1) q^{56} + (\beta_{4} + \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 2) q^{58} + ( - \beta_{4} + \beta_{2} + 2 \beta_1 - 4) q^{59} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 1) q^{61} + (\beta_{3} + \beta_{2} + \beta_1 + 7) q^{62} + q^{64} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{65} + (5 \beta_{4} + 2 \beta_{3} + \beta_{2} + 4) q^{67} + ( - 2 \beta_{3} - \beta_1 + 1) q^{68} + (\beta_{4} + 6 \beta_1 - 3) q^{70} + ( - 3 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 6) q^{71} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{73} + (\beta_{4} + 4 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{74} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{76} + (3 \beta_{4} + 3 \beta_{3} - 7 \beta_{2} - 6 \beta_1 - 3) q^{77} + (3 \beta_{2} - 3) q^{79} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{80} + (\beta_{3} - 2 \beta_{2} - 3) q^{82} + (\beta_{4} + 2 \beta_{3} - 5 \beta_{2} - \beta_1 - 1) q^{83} + (4 \beta_{4} - 3 \beta_{2} + 4 \beta_1 - 6) q^{85} + ( - \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{86} + ( - \beta_{3} - 2 \beta_{2} - 4) q^{88} + ( - \beta_{4} + \beta_{3} + 5 \beta_{2} + 4 \beta_1 - 2) q^{89} + ( - 6 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 1) q^{91} + (2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{92} + ( - 5 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 4) q^{94} + (2 \beta_{4} + 6 \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 6) q^{95} + ( - 5 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 3) q^{97} + (\beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} + 5 q^{4} + 3 q^{5} - 9 q^{7} - 5 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{2} + 5 q^{4} + 3 q^{5} - 9 q^{7} - 5 q^{8} - 3 q^{10} + 15 q^{11} - 5 q^{13} + 9 q^{14} + 5 q^{16} + 6 q^{17} + 10 q^{19} + 3 q^{20} - 15 q^{22} + 12 q^{23} + 6 q^{25} + 5 q^{26} - 9 q^{28} + 16 q^{29} - 33 q^{31} - 5 q^{32} - 6 q^{34} + 11 q^{35} + 2 q^{37} - 10 q^{38} - 3 q^{40} + 12 q^{41} - 17 q^{43} + 15 q^{44} - 12 q^{46} + 7 q^{47} + 36 q^{49} - 6 q^{50} - 5 q^{52} + 2 q^{55} + 9 q^{56} - 16 q^{58} - 18 q^{59} + q^{61} + 33 q^{62} + 5 q^{64} + 14 q^{65} + 6 q^{67} + 6 q^{68} - 11 q^{70} + 26 q^{71} + 9 q^{73} - 2 q^{74} + 10 q^{76} - 16 q^{77} - 21 q^{79} + 3 q^{80} - 12 q^{82} - 28 q^{85} + 17 q^{86} - 15 q^{88} - 15 q^{89} + 26 q^{91} + 12 q^{92} - 7 q^{94} + 18 q^{95} + 26 q^{97} - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 2\nu^{2} + 3\nu - 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 6\beta_{3} + 7\beta_{2} + 9\beta _1 + 14 \) 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
−1.35347
1.17073
0.418933
2.54180
−1.77799
−1.00000 0 1.00000 −3.94839 0 −3.24145 −1.00000 0 3.94839
1.2 −1.00000 0 1.00000 0.411173 0 −3.93028 −1.00000 0 −0.411173
1.3 −1.00000 0 1.00000 1.45190 0 −1.38596 −1.00000 0 −1.45190
1.4 −1.00000 0 1.00000 2.22418 0 −4.85942 −1.00000 0 −2.22418
1.5 −1.00000 0 1.00000 2.86114 0 4.41712 −1.00000 0 −2.86114
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(503\) \(1\)

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 9054.2.a.ba 5
3.b odd 2 1 1006.2.a.h 5
12.b even 2 1 8048.2.a.o 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1006.2.a.h 5 3.b odd 2 1
8048.2.a.o 5 12.b even 2 1
9054.2.a.ba 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9054))\):

\( T_{5}^{5} - 3T_{5}^{4} - 11T_{5}^{3} + 50T_{5}^{2} - 55T_{5} + 15 \) Copy content Toggle raw display
\( T_{7}^{5} + 9T_{7}^{4} + 5T_{7}^{3} - 156T_{7}^{2} - 479T_{7} - 379 \) Copy content Toggle raw display
\( T_{11}^{5} - 15T_{11}^{4} + 60T_{11}^{3} + 35T_{11}^{2} - 425T_{11} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 3 T^{4} - 11 T^{3} + 50 T^{2} + \cdots + 15 \) Copy content Toggle raw display
$7$ \( T^{5} + 9 T^{4} + 5 T^{3} - 156 T^{2} + \cdots - 379 \) Copy content Toggle raw display
$11$ \( T^{5} - 15 T^{4} + 60 T^{3} + 35 T^{2} + \cdots - 9 \) Copy content Toggle raw display
$13$ \( T^{5} + 5 T^{4} - 27 T^{3} - 85 T^{2} + \cdots + 15 \) Copy content Toggle raw display
$17$ \( T^{5} - 6 T^{4} - 22 T^{3} + 77 T^{2} + \cdots + 219 \) Copy content Toggle raw display
$19$ \( T^{5} - 10 T^{4} + 2 T^{3} + 143 T^{2} + \cdots - 333 \) Copy content Toggle raw display
$23$ \( T^{5} - 12 T^{4} + 9 T^{3} + \cdots + 1497 \) Copy content Toggle raw display
$29$ \( T^{5} - 16 T^{4} + 23 T^{3} + \cdots + 4119 \) Copy content Toggle raw display
$31$ \( T^{5} + 33 T^{4} + 424 T^{3} + \cdots + 9797 \) Copy content Toggle raw display
$37$ \( T^{5} - 2 T^{4} - 106 T^{3} + \cdots - 1895 \) Copy content Toggle raw display
$41$ \( T^{5} - 12 T^{4} + 2 T^{3} + 281 T^{2} + \cdots - 687 \) Copy content Toggle raw display
$43$ \( T^{5} + 17 T^{4} + 55 T^{3} + \cdots - 7067 \) Copy content Toggle raw display
$47$ \( T^{5} - 7 T^{4} - 193 T^{3} + \cdots - 66081 \) Copy content Toggle raw display
$53$ \( T^{5} - 88 T^{3} + 51 T^{2} + \cdots - 3347 \) Copy content Toggle raw display
$59$ \( T^{5} + 18 T^{4} + 85 T^{3} + \cdots - 1563 \) Copy content Toggle raw display
$61$ \( T^{5} - T^{4} - 111 T^{3} + 248 T^{2} + \cdots - 8863 \) Copy content Toggle raw display
$67$ \( T^{5} - 6 T^{4} - 207 T^{3} + \cdots + 19989 \) Copy content Toggle raw display
$71$ \( T^{5} - 26 T^{4} + 47 T^{3} + \cdots + 51891 \) Copy content Toggle raw display
$73$ \( T^{5} - 9 T^{4} + 7 T^{3} + 8 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$79$ \( T^{5} + 21 T^{4} + 99 T^{3} + \cdots - 729 \) Copy content Toggle raw display
$83$ \( T^{5} - 312 T^{3} + 223 T^{2} + \cdots - 729 \) Copy content Toggle raw display
$89$ \( T^{5} + 15 T^{4} - 153 T^{3} + \cdots + 45027 \) Copy content Toggle raw display
$97$ \( T^{5} - 26 T^{4} + 72 T^{3} + \cdots - 30449 \) Copy content Toggle raw display
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