Properties

Label 5290.2.a.bb
Level $5290$
Weight $2$
Character orbit 5290.a
Self dual yes
Analytic conductor $42.241$
Analytic rank $0$
Dimension $4$
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] = [5290,2,Mod(1,5290)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5290, 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("5290.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5290 = 2 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5290.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: \(42.2408626693\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_{2} + 1) q^{3} + q^{4} + q^{5} + (\beta_{2} + 1) q^{6} + (\beta_{2} + 2) q^{7} + q^{8} + (\beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_{2} + 1) q^{3} + q^{4} + q^{5} + (\beta_{2} + 1) q^{6} + (\beta_{2} + 2) q^{7} + q^{8} + (\beta_1 + 1) q^{9} + q^{10} + (\beta_1 + 1) q^{11} + (\beta_{2} + 1) q^{12} - 2 \beta_{2} q^{13} + (\beta_{2} + 2) q^{14} + (\beta_{2} + 1) q^{15} + q^{16} + (\beta_{3} - \beta_{2}) q^{17} + (\beta_1 + 1) q^{18} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{19} + q^{20} + (\beta_{2} + \beta_1 + 5) q^{21} + (\beta_1 + 1) q^{22} + (\beta_{2} + 1) q^{24} + q^{25} - 2 \beta_{2} q^{26} + (2 \beta_{3} - \beta_{2} + \beta_1) q^{27} + (\beta_{2} + 2) q^{28} + (\beta_{2} - 2 \beta_1) q^{29} + (\beta_{2} + 1) q^{30} - \beta_{2} q^{31} + q^{32} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{33} + (\beta_{3} - \beta_{2}) q^{34} + (\beta_{2} + 2) q^{35} + (\beta_1 + 1) q^{36} + (2 \beta_{2} + 6) q^{37} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{38} + (2 \beta_{2} - 2 \beta_1 - 6) q^{39} + q^{40} + ( - 4 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{41} + (\beta_{2} + \beta_1 + 5) q^{42} + ( - 3 \beta_{3} + 5 \beta_{2} + 2) q^{43} + (\beta_1 + 1) q^{44} + (\beta_1 + 1) q^{45} + (\beta_{2} - \beta_1 - 3) q^{47} + (\beta_{2} + 1) q^{48} + (2 \beta_{2} + \beta_1) q^{49} + q^{50} + (\beta_{3} - 2) q^{51} - 2 \beta_{2} q^{52} + (\beta_{2} + 8) q^{53} + (2 \beta_{3} - \beta_{2} + \beta_1) q^{54} + (\beta_1 + 1) q^{55} + (\beta_{2} + 2) q^{56} + ( - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1 - 8) q^{57} + (\beta_{2} - 2 \beta_1) q^{58} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{59} + (\beta_{2} + 1) q^{60} + ( - 2 \beta_{3} - 2 \beta_{2} + 2) q^{61} - \beta_{2} q^{62} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{63} + q^{64} - 2 \beta_{2} q^{65} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{66} + ( - \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 2) q^{67} + (\beta_{3} - \beta_{2}) q^{68} + (\beta_{2} + 2) q^{70} + ( - 4 \beta_{3} + \beta_{2}) q^{71} + (\beta_1 + 1) q^{72} + (6 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 1) q^{73} + (2 \beta_{2} + 6) q^{74} + (\beta_{2} + 1) q^{75} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{76} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{77} + (2 \beta_{2} - 2 \beta_1 - 6) q^{78} + (\beta_{2} - \beta_1 + 1) q^{79} + q^{80} + (4 \beta_{3} - \beta_1 - 2) q^{81} + ( - 4 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{82} + (3 \beta_{3} + \beta_{2} - \beta_1 + 7) q^{83} + (\beta_{2} + \beta_1 + 5) q^{84} + (\beta_{3} - \beta_{2}) q^{85} + ( - 3 \beta_{3} + 5 \beta_{2} + 2) q^{86} + ( - 4 \beta_{3} - 3 \beta_{2} - \beta_1 - 1) q^{87} + (\beta_1 + 1) q^{88} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 1) q^{89} + (\beta_1 + 1) q^{90} + ( - 2 \beta_1 - 6) q^{91} + (\beta_{2} - \beta_1 - 3) q^{93} + (\beta_{2} - \beta_1 - 3) q^{94} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{95} + (\beta_{2} + 1) q^{96} - 4 \beta_{3} q^{97} + (2 \beta_{2} + \beta_1) q^{98} + (4 \beta_{3} + 2 \beta_1 + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} + 4 q^{5} + 2 q^{6} + 6 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} + 4 q^{5} + 2 q^{6} + 6 q^{7} + 4 q^{8} + 4 q^{9} + 4 q^{10} + 4 q^{11} + 2 q^{12} + 4 q^{13} + 6 q^{14} + 2 q^{15} + 4 q^{16} + 2 q^{17} + 4 q^{18} + 8 q^{19} + 4 q^{20} + 18 q^{21} + 4 q^{22} + 2 q^{24} + 4 q^{25} + 4 q^{26} + 2 q^{27} + 6 q^{28} - 2 q^{29} + 2 q^{30} + 2 q^{31} + 4 q^{32} + 8 q^{33} + 2 q^{34} + 6 q^{35} + 4 q^{36} + 20 q^{37} + 8 q^{38} - 28 q^{39} + 4 q^{40} - 8 q^{41} + 18 q^{42} - 2 q^{43} + 4 q^{44} + 4 q^{45} - 14 q^{47} + 2 q^{48} - 4 q^{49} + 4 q^{50} - 8 q^{51} + 4 q^{52} + 30 q^{53} + 2 q^{54} + 4 q^{55} + 6 q^{56} - 38 q^{57} - 2 q^{58} + 4 q^{59} + 2 q^{60} + 12 q^{61} + 2 q^{62} + 12 q^{63} + 4 q^{64} + 4 q^{65} + 8 q^{66} + 2 q^{67} + 2 q^{68} + 6 q^{70} - 2 q^{71} + 4 q^{72} + 2 q^{73} + 20 q^{74} + 2 q^{75} + 8 q^{76} + 12 q^{77} - 28 q^{78} + 2 q^{79} + 4 q^{80} - 8 q^{81} - 8 q^{82} + 26 q^{83} + 18 q^{84} + 2 q^{85} - 2 q^{86} + 2 q^{87} + 4 q^{88} + 4 q^{90} - 24 q^{91} - 14 q^{93} - 14 q^{94} + 8 q^{95} + 2 q^{96} - 4 q^{98} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 2\beta _1 + 4 \) 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
1.21969
−1.49551
−0.219687
2.49551
1.00000 −2.33225 1.00000 1.00000 −2.33225 −1.33225 1.00000 2.43937 1.00000
1.2 1.00000 −0.0947876 1.00000 1.00000 −0.0947876 0.905212 1.00000 −2.99102 1.00000
1.3 1.00000 1.60020 1.00000 1.00000 1.60020 2.60020 1.00000 −0.439374 1.00000
1.4 1.00000 2.82684 1.00000 1.00000 2.82684 3.82684 1.00000 4.99102 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 5290.2.a.bb yes 4
23.b odd 2 1 5290.2.a.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5290.2.a.ba 4 23.b odd 2 1
5290.2.a.bb yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5290))\):

\( T_{3}^{4} - 2T_{3}^{3} - 6T_{3}^{2} + 10T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 6T_{7}^{3} + 6T_{7}^{2} + 12T_{7} - 12 \) Copy content Toggle raw display
\( T_{11}^{4} - 4T_{11}^{3} - 12T_{11}^{2} + 32T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} - 6 T^{2} + 10 T + 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} + 6 T^{2} + 12 T - 12 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} - 12 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} - 24 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} - 6 T^{2} + 10 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} - 54 T^{2} + \cdots - 1079 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} - 66 T^{2} - 100 T + 916 \) Copy content Toggle raw display
$31$ \( T^{4} - 2 T^{3} - 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$37$ \( T^{4} - 20 T^{3} + 120 T^{2} + \cdots - 176 \) Copy content Toggle raw display
$41$ \( T^{4} + 8 T^{3} - 60 T^{2} - 592 T - 944 \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{3} - 150 T^{2} + \cdots + 4969 \) Copy content Toggle raw display
$47$ \( T^{4} + 14 T^{3} + 54 T^{2} + 44 T + 4 \) Copy content Toggle raw display
$53$ \( T^{4} - 30 T^{3} + 330 T^{2} + \cdots + 2724 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} - 78 T^{2} - 124 T + 157 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} - 24 T^{2} + \cdots - 192 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} - 270 T^{2} + \cdots + 17953 \) Copy content Toggle raw display
$71$ \( T^{4} + 2 T^{3} - 78 T^{2} + \cdots + 1108 \) Copy content Toggle raw display
$73$ \( T^{4} - 2 T^{3} - 210 T^{2} + \cdots - 1331 \) Copy content Toggle raw display
$79$ \( T^{4} - 2 T^{3} - 18 T^{2} + 28 T + 52 \) Copy content Toggle raw display
$83$ \( T^{4} - 26 T^{3} + 162 T^{2} + \cdots - 1727 \) Copy content Toggle raw display
$89$ \( T^{4} - 156 T^{2} + 48 T + 5136 \) Copy content Toggle raw display
$97$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
show more
show less