Properties

Label 5082.2.a.ca
Level $5082$
Weight $2$
Character orbit 5082.a
Self dual yes
Analytic conductor $40.580$
Analytic rank $1$
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] = [5082,2,Mod(1,5082)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5082, 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("5082.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5082.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: \(40.5799743072\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + (\beta_{3} + \beta_{2} + 1) q^{5} - q^{6} - q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + (\beta_{3} + \beta_{2} + 1) q^{5} - q^{6} - q^{7} - q^{8} + q^{9} + ( - \beta_{3} - \beta_{2} - 1) q^{10} + q^{12} + ( - \beta_{2} + \beta_1 - 1) q^{13} + q^{14} + (\beta_{3} + \beta_{2} + 1) q^{15} + q^{16} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{17} - q^{18} + ( - 2 \beta_{2} - \beta_1 - 2) q^{19} + (\beta_{3} + \beta_{2} + 1) q^{20} - q^{21} + ( - \beta_{3} + \beta_{2}) q^{23} - q^{24} + ( - \beta_{2} + 2 \beta_1) q^{25} + (\beta_{2} - \beta_1 + 1) q^{26} + q^{27} - q^{28} + (\beta_{3} + \beta_{2} + \beta_1 - 4) q^{29} + ( - \beta_{3} - \beta_{2} - 1) q^{30} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{31} - q^{32} + (2 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{34} + ( - \beta_{3} - \beta_{2} - 1) q^{35} + q^{36} + (\beta_{3} - \beta_{2} - 3) q^{37} + (2 \beta_{2} + \beta_1 + 2) q^{38} + ( - \beta_{2} + \beta_1 - 1) q^{39} + ( - \beta_{3} - \beta_{2} - 1) q^{40} + (2 \beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{41} + q^{42} + ( - 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{43} + (\beta_{3} + \beta_{2} + 1) q^{45} + (\beta_{3} - \beta_{2}) q^{46} + ( - 3 \beta_{3} - 5 \beta_{2} - \beta_1 - 4) q^{47} + q^{48} + q^{49} + (\beta_{2} - 2 \beta_1) q^{50} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{51} + ( - \beta_{2} + \beta_1 - 1) q^{52} + (\beta_{3} - 2 \beta_1 - 1) q^{53} - q^{54} + q^{56} + ( - 2 \beta_{2} - \beta_1 - 2) q^{57} + ( - \beta_{3} - \beta_{2} - \beta_1 + 4) q^{58} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{59} + (\beta_{3} + \beta_{2} + 1) q^{60} + ( - \beta_{3} + 6 \beta_{2} + 2 \beta_1 + 1) q^{61} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{62} - q^{63} + q^{64} + (\beta_{3} + 2 \beta_{2} - 1) q^{65} + ( - \beta_{3} - 4 \beta_1 - 3) q^{67} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{68} + ( - \beta_{3} + \beta_{2}) q^{69} + (\beta_{3} + \beta_{2} + 1) q^{70} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 6) q^{71} - q^{72} + (\beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{73} + ( - \beta_{3} + \beta_{2} + 3) q^{74} + ( - \beta_{2} + 2 \beta_1) q^{75} + ( - 2 \beta_{2} - \beta_1 - 2) q^{76} + (\beta_{2} - \beta_1 + 1) q^{78} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 2) q^{79} + (\beta_{3} + \beta_{2} + 1) q^{80} + q^{81} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{82} + (5 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 3) q^{83} - q^{84} + ( - 4 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 8) q^{85} + (2 \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{86} + (\beta_{3} + \beta_{2} + \beta_1 - 4) q^{87} + ( - 2 \beta_{3} - \beta_{2} + 6 \beta_1 - 1) q^{89} + ( - \beta_{3} - \beta_{2} - 1) q^{90} + (\beta_{2} - \beta_1 + 1) q^{91} + ( - \beta_{3} + \beta_{2}) q^{92} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{93} + (3 \beta_{3} + 5 \beta_{2} + \beta_1 + 4) q^{94} + ( - \beta_{3} - 5 \beta_{2} - 3 \beta_1 - 5) q^{95} - q^{96} + (\beta_{3} - 2 \beta_{2} - 1) q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} + 4 q^{12} - 2 q^{13} + 4 q^{14} + 2 q^{15} + 4 q^{16} - 6 q^{17} - 4 q^{18} - 4 q^{19} + 2 q^{20} - 4 q^{21} - 2 q^{23} - 4 q^{24} + 2 q^{25} + 2 q^{26} + 4 q^{27} - 4 q^{28} - 18 q^{29} - 2 q^{30} + 8 q^{31} - 4 q^{32} + 6 q^{34} - 2 q^{35} + 4 q^{36} - 10 q^{37} + 4 q^{38} - 2 q^{39} - 2 q^{40} - 14 q^{41} + 4 q^{42} - 10 q^{43} + 2 q^{45} + 2 q^{46} - 6 q^{47} + 4 q^{48} + 4 q^{49} - 2 q^{50} - 6 q^{51} - 2 q^{52} - 4 q^{53} - 4 q^{54} + 4 q^{56} - 4 q^{57} + 18 q^{58} - 10 q^{59} + 2 q^{60} - 8 q^{61} - 8 q^{62} - 4 q^{63} + 4 q^{64} - 8 q^{65} - 12 q^{67} - 6 q^{68} - 2 q^{69} + 2 q^{70} + 20 q^{71} - 4 q^{72} - 10 q^{73} + 10 q^{74} + 2 q^{75} - 4 q^{76} + 2 q^{78} + 14 q^{79} + 2 q^{80} + 4 q^{81} + 14 q^{82} - 20 q^{83} - 4 q^{84} - 26 q^{85} + 10 q^{86} - 18 q^{87} - 2 q^{89} - 2 q^{90} + 2 q^{91} - 2 q^{92} + 8 q^{93} + 6 q^{94} - 10 q^{95} - 4 q^{96} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 7x^{2} + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) 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
1.54336
−2.14896
−1.54336
2.14896
−1.00000 1.00000 1.00000 −3.11525 −1.00000 −1.00000 −1.00000 1.00000 3.11525
1.2 −1.00000 1.00000 1.00000 0.289903 −1.00000 −1.00000 −1.00000 1.00000 −0.289903
1.3 −1.00000 1.00000 1.00000 1.87918 −1.00000 −1.00000 −1.00000 1.00000 −1.87918
1.4 −1.00000 1.00000 1.00000 2.94617 −1.00000 −1.00000 −1.00000 1.00000 −2.94617
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)

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 5082.2.a.ca 4
11.b odd 2 1 5082.2.a.cf 4
11.d odd 10 2 462.2.j.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.j.e 8 11.d odd 10 2
5082.2.a.ca 4 1.a even 1 1 trivial
5082.2.a.cf 4 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5082))\):

\( T_{5}^{4} - 2T_{5}^{3} - 9T_{5}^{2} + 20T_{5} - 5 \) Copy content Toggle raw display
\( T_{13}^{4} + 2T_{13}^{3} - 8T_{13}^{2} - 4T_{13} + 4 \) Copy content Toggle raw display
\( T_{17}^{4} + 6T_{17}^{3} - 41T_{17}^{2} - 250T_{17} - 155 \) Copy content Toggle raw display
\( T_{19}^{4} + 4T_{19}^{3} - 11T_{19}^{2} - 20T_{19} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} - 9 T^{2} + 20 T - 5 \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} - 8 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} - 41 T^{2} - 250 T - 155 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} - 11 T^{2} - 20 T - 5 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} - 9 T^{2} + 5 \) Copy content Toggle raw display
$29$ \( T^{4} + 18 T^{3} + 106 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} - 21 T^{2} + 198 T - 139 \) Copy content Toggle raw display
$37$ \( T^{4} + 10 T^{3} + 27 T^{2} - 49 \) Copy content Toggle raw display
$41$ \( T^{4} + 14 T^{3} + 19 T^{2} - 110 T - 55 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} - 72 T^{2} + \cdots + 116 \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} - 122 T^{2} + \cdots - 236 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} - 34 T^{2} - 76 T + 236 \) Copy content Toggle raw display
$59$ \( T^{4} + 10 T^{3} - 30 T^{2} + \cdots - 500 \) Copy content Toggle raw display
$61$ \( T^{4} + 8 T^{3} - 106 T^{2} + \cdots + 596 \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} - 58 T^{2} + \cdots + 404 \) Copy content Toggle raw display
$71$ \( T^{4} - 20 T^{3} + 72 T^{2} + \cdots - 2864 \) Copy content Toggle raw display
$73$ \( T^{4} + 10 T^{3} - 42 T^{2} + \cdots + 916 \) Copy content Toggle raw display
$79$ \( T^{4} - 14 T^{3} - 66 T^{2} + \cdots - 4220 \) Copy content Toggle raw display
$83$ \( T^{4} + 20 T^{3} - 162 T^{2} + \cdots - 11564 \) Copy content Toggle raw display
$89$ \( T^{4} + 2 T^{3} - 309 T^{2} + \cdots + 20725 \) Copy content Toggle raw display
$97$ \( T^{4} - 18 T^{2} - 20 T - 4 \) Copy content Toggle raw display
show more
show less