Properties

Label 5390.2.a.bv
Level $5390$
Weight $2$
Character orbit 5390.a
Self dual yes
Analytic conductor $43.039$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5390,2,Mod(1,5390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5390, 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("5390.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.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: \(43.0393666895\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) 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 770)
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.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} - q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} - q^{8} + (\beta_{2} - 1) q^{9} + q^{10} - q^{11} - \beta_1 q^{12} + (2 \beta_{2} + 2 \beta_1 - 2) q^{13} + \beta_1 q^{15} + q^{16} + ( - \beta_{2} - 4) q^{17} + ( - \beta_{2} + 1) q^{18} + ( - 3 \beta_{2} + 5 \beta_1 + 2) q^{19} - q^{20} + q^{22} + (6 \beta_{2} - 2 \beta_1 + 2) q^{23} + \beta_1 q^{24} + q^{25} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{26} + (3 \beta_1 - 1) q^{27} + (5 \beta_{2} - 5 \beta_1 - 2) q^{29} - \beta_1 q^{30} + (5 \beta_{2} - 3 \beta_1) q^{31} - q^{32} + \beta_1 q^{33} + (\beta_{2} + 4) q^{34} + (\beta_{2} - 1) q^{36} + ( - 3 \beta_{2} + 2 \beta_1 + 6) q^{37} + (3 \beta_{2} - 5 \beta_1 - 2) q^{38} + ( - 2 \beta_{2} - 6) q^{39} + q^{40} + ( - 2 \beta_{2} - 8) q^{41} + ( - 6 \beta_{2} + 6 \beta_1 + 2) q^{43} - q^{44} + ( - \beta_{2} + 1) q^{45} + ( - 6 \beta_{2} + 2 \beta_1 - 2) q^{46} + ( - 2 \beta_{2} - 6 \beta_1 + 4) q^{47} - \beta_1 q^{48} - q^{50} + (5 \beta_1 + 1) q^{51} + (2 \beta_{2} + 2 \beta_1 - 2) q^{52} + (2 \beta_{2} - \beta_1) q^{53} + ( - 3 \beta_1 + 1) q^{54} + q^{55} + ( - 5 \beta_{2} + \beta_1 - 7) q^{57} + ( - 5 \beta_{2} + 5 \beta_1 + 2) q^{58} + ( - 4 \beta_{2} - 4) q^{59} + \beta_1 q^{60} + (5 \beta_{2} - 2 \beta_1 + 2) q^{61} + ( - 5 \beta_{2} + 3 \beta_1) q^{62} + q^{64} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{65} - \beta_1 q^{66} + (4 \beta_{2} - 2 \beta_1 + 5) q^{67} + ( - \beta_{2} - 4) q^{68} + (2 \beta_{2} - 8 \beta_1 - 2) q^{69} + (\beta_{2} + \beta_1) q^{71} + ( - \beta_{2} + 1) q^{72} + ( - 4 \beta_1 - 1) q^{73} + (3 \beta_{2} - 2 \beta_1 - 6) q^{74} - \beta_1 q^{75} + ( - 3 \beta_{2} + 5 \beta_1 + 2) q^{76} + (2 \beta_{2} + 6) q^{78} + ( - 2 \beta_1 + 4) q^{79} - q^{80} + ( - 6 \beta_{2} + \beta_1 - 3) q^{81} + (2 \beta_{2} + 8) q^{82} + (4 \beta_{2} - 4 \beta_1 + 8) q^{83} + (\beta_{2} + 4) q^{85} + (6 \beta_{2} - 6 \beta_1 - 2) q^{86} + (5 \beta_{2} - 3 \beta_1 + 5) q^{87} + q^{88} + (8 \beta_{2} - 5 \beta_1 + 4) q^{89} + (\beta_{2} - 1) q^{90} + (6 \beta_{2} - 2 \beta_1 + 2) q^{92} + (3 \beta_{2} - 5 \beta_1 + 1) q^{93} + (2 \beta_{2} + 6 \beta_1 - 4) q^{94} + (3 \beta_{2} - 5 \beta_1 - 2) q^{95} + \beta_1 q^{96} + ( - 2 \beta_1 - 6) q^{97} + ( - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{8} - 3 q^{9} + 3 q^{10} - 3 q^{11} - 6 q^{13} + 3 q^{16} - 12 q^{17} + 3 q^{18} + 6 q^{19} - 3 q^{20} + 3 q^{22} + 6 q^{23} + 3 q^{25} + 6 q^{26} - 3 q^{27} - 6 q^{29} - 3 q^{32} + 12 q^{34} - 3 q^{36} + 18 q^{37} - 6 q^{38} - 18 q^{39} + 3 q^{40} - 24 q^{41} + 6 q^{43} - 3 q^{44} + 3 q^{45} - 6 q^{46} + 12 q^{47} - 3 q^{50} + 3 q^{51} - 6 q^{52} + 3 q^{54} + 3 q^{55} - 21 q^{57} + 6 q^{58} - 12 q^{59} + 6 q^{61} + 3 q^{64} + 6 q^{65} + 15 q^{67} - 12 q^{68} - 6 q^{69} + 3 q^{72} - 3 q^{73} - 18 q^{74} + 6 q^{76} + 18 q^{78} + 12 q^{79} - 3 q^{80} - 9 q^{81} + 24 q^{82} + 24 q^{83} + 12 q^{85} - 6 q^{86} + 15 q^{87} + 3 q^{88} + 12 q^{89} - 3 q^{90} + 6 q^{92} + 3 q^{93} - 12 q^{94} - 6 q^{95} - 18 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.87939
−0.347296
−1.53209
−1.00000 −1.87939 1.00000 −1.00000 1.87939 0 −1.00000 0.532089 1.00000
1.2 −1.00000 0.347296 1.00000 −1.00000 −0.347296 0 −1.00000 −2.87939 1.00000
1.3 −1.00000 1.53209 1.00000 −1.00000 −1.53209 0 −1.00000 −0.652704 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)

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 5390.2.a.bv 3
7.b odd 2 1 5390.2.a.bx 3
7.d odd 6 2 770.2.i.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.i.j 6 7.d odd 6 2
5390.2.a.bv 3 1.a even 1 1 trivial
5390.2.a.bx 3 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5390))\):

\( T_{3}^{3} - 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{13}^{3} + 6T_{13}^{2} - 24T_{13} - 136 \) Copy content Toggle raw display
\( T_{17}^{3} + 12T_{17}^{2} + 45T_{17} + 51 \) Copy content Toggle raw display
\( T_{19}^{3} - 6T_{19}^{2} - 45T_{19} + 269 \) Copy content Toggle raw display
\( T_{31}^{3} - 57T_{31} + 107 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 6 T^{2} - 24 T - 136 \) Copy content Toggle raw display
$17$ \( T^{3} + 12 T^{2} + 45 T + 51 \) Copy content Toggle raw display
$19$ \( T^{3} - 6 T^{2} - 45 T + 269 \) Copy content Toggle raw display
$23$ \( T^{3} - 6 T^{2} - 72 T + 456 \) Copy content Toggle raw display
$29$ \( T^{3} + 6 T^{2} - 63 T - 267 \) Copy content Toggle raw display
$31$ \( T^{3} - 57T + 107 \) Copy content Toggle raw display
$37$ \( T^{3} - 18 T^{2} + 87 T - 107 \) Copy content Toggle raw display
$41$ \( T^{3} + 24 T^{2} + 180 T + 408 \) Copy content Toggle raw display
$43$ \( T^{3} - 6 T^{2} - 96 T + 424 \) Copy content Toggle raw display
$47$ \( T^{3} - 12 T^{2} - 108 T + 1272 \) Copy content Toggle raw display
$53$ \( T^{3} - 9T + 9 \) Copy content Toggle raw display
$59$ \( T^{3} + 12T^{2} - 192 \) Copy content Toggle raw display
$61$ \( T^{3} - 6 T^{2} - 45 T + 269 \) Copy content Toggle raw display
$67$ \( T^{3} - 15 T^{2} + 39 T + 127 \) Copy content Toggle raw display
$71$ \( T^{3} - 9T - 9 \) Copy content Toggle raw display
$73$ \( T^{3} + 3 T^{2} - 45 T + 17 \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} + 36 T - 8 \) Copy content Toggle raw display
$83$ \( T^{3} - 24 T^{2} + 144 T - 192 \) Copy content Toggle raw display
$89$ \( T^{3} - 12 T^{2} - 99 T + 921 \) Copy content Toggle raw display
$97$ \( T^{3} + 18 T^{2} + 96 T + 152 \) Copy content Toggle raw display
show more
show less