Properties

Label 5290.2.a.x
Level $5290$
Weight $2$
Character orbit 5290.a
Self dual yes
Analytic conductor $42.241$
Analytic rank $1$
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] = [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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.13888.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 6x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 7x^{2} + 6x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 4\nu + 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 3\beta_{2} + 6\beta _1 + 5 \) 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
1.53728
3.31829
−1.95150
−0.904079
1.00000 −2.95150 1.00000 1.00000 −2.95150 3.36571 1.00000 5.71133 1.00000
1.2 1.00000 −1.90408 1.00000 1.00000 −1.90408 −0.510134 1.00000 0.625518 1.00000
1.3 1.00000 0.537282 1.00000 1.00000 0.537282 −0.123069 1.00000 −2.71133 1.00000
1.4 1.00000 2.31829 1.00000 1.00000 2.31829 −4.73251 1.00000 2.37448 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.x yes 4
23.b odd 2 1 5290.2.a.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5290.2.a.w 4 23.b odd 2 1
5290.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5290))\):

\( T_{3}^{4} + 2T_{3}^{3} - 7T_{3}^{2} - 10T_{3} + 7 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} - 15T_{7}^{2} - 10T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 10T_{11}^{3} + 29T_{11}^{2} + 18T_{11} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} - 7 T^{2} - 10 T + 7 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} - 15 T^{2} - 10 T - 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 10 T^{3} + 29 T^{2} + 18 T - 9 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} - 7 T^{2} - 6 T + 9 \) Copy content Toggle raw display
$17$ \( T^{4} + 14 T^{3} + 55 T^{2} + 42 T - 63 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} - 69 T^{2} - 70 T + 647 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} - 10 T^{2} - 24 T + 36 \) Copy content Toggle raw display
$31$ \( T^{4} + 10 T^{3} - 47 T^{2} + \cdots - 729 \) Copy content Toggle raw display
$37$ \( T^{4} + 8 T^{3} - 36 T^{2} - 208 T + 644 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} - 109 T^{2} + \cdots - 279 \) Copy content Toggle raw display
$43$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} - 38 T^{2} + 432 T - 828 \) Copy content Toggle raw display
$53$ \( T^{4} + 24 T^{3} + 178 T^{2} + \cdots - 324 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} - 118 T^{2} + \cdots + 1764 \) Copy content Toggle raw display
$61$ \( T^{4} + 38 T^{3} + 525 T^{2} + \cdots + 6713 \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} - 106 T^{2} + \cdots + 2044 \) Copy content Toggle raw display
$71$ \( T^{4} + 10 T^{3} - 17 T^{2} + \cdots - 657 \) Copy content Toggle raw display
$73$ \( T^{4} - 62 T^{2} + 240 T - 252 \) Copy content Toggle raw display
$79$ \( T^{4} - 20 T^{3} - 24 T^{2} + \cdots - 9508 \) Copy content Toggle raw display
$83$ \( T^{4} - 4 T^{3} - 10 T^{2} + 24 T + 36 \) Copy content Toggle raw display
$89$ \( T^{4} - 4 T^{3} - 130 T^{2} + \cdots + 2628 \) Copy content Toggle raw display
$97$ \( T^{4} + 10 T^{3} - 45 T^{2} + \cdots + 449 \) Copy content Toggle raw display
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