Properties

Label 6160.2.a.br
Level $6160$
Weight $2$
Character orbit 6160.a
Self dual yes
Analytic conductor $49.188$
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] = [6160,2,Mod(1,6160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("6160.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.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: \(49.1878476451\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.11348.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 385)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{3} + q^{5} + q^{7} + (\beta_{3} - \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{3} + q^{5} + q^{7} + (\beta_{3} - \beta_{2} + 2) q^{9} + q^{11} + (\beta_{3} - \beta_1 - 1) q^{13} + ( - \beta_{3} - 1) q^{15} + (\beta_{3} - 1) q^{17} + (2 \beta_{2} - \beta_1) q^{19} + ( - \beta_{3} - 1) q^{21} + (\beta_{3} + \beta_{2} + \beta_1 - 3) q^{23} + q^{25} + (\beta_1 - 4) q^{27} + (2 \beta_1 - 2) q^{29} + ( - 3 \beta_{2} - 4) q^{31} + ( - \beta_{3} - 1) q^{33} + q^{35} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{37} + (\beta_{3} + 3 \beta_{2} + 2 \beta_1 - 3) q^{39} + ( - 2 \beta_{3} - \beta_{2} - 4) q^{41} + (\beta_{3} + \beta_{2} - \beta_1 + 5) q^{43} + (\beta_{3} - \beta_{2} + 2) q^{45} + ( - 3 \beta_{3} + 4 \beta_{2} - \beta_1 - 3) q^{47} + q^{49} + (\beta_{3} + \beta_{2} - 3) q^{51} + ( - \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{53} + q^{55} + (2 \beta_{3} + 4 \beta_{2} + 2) q^{57} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 - 6) q^{59} + (\beta_{2} - \beta_1 - 2) q^{61} + (\beta_{3} - \beta_{2} + 2) q^{63} + (\beta_{3} - \beta_1 - 1) q^{65} + (\beta_{3} - 5 \beta_{2} - \beta_1 + 1) q^{67} + (4 \beta_{3} - 3 \beta_1) q^{69} - 8 q^{71} + ( - \beta_{3} - 2 \beta_1 - 3) q^{73} + ( - \beta_{3} - 1) q^{75} + q^{77} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{79} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{81} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{83} + (\beta_{3} - 1) q^{85} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 2) q^{87} + ( - \beta_1 - 2) q^{89} + (\beta_{3} - \beta_1 - 1) q^{91} + (\beta_{3} - 3 \beta_{2} + 3 \beta_1 + 1) q^{93} + (2 \beta_{2} - \beta_1) q^{95} + (3 \beta_1 - 2) q^{97} + (\beta_{3} - \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 8 q^{9} + 4 q^{11} - 8 q^{13} - 2 q^{15} - 6 q^{17} - 6 q^{19} - 2 q^{21} - 14 q^{23} + 4 q^{25} - 14 q^{27} - 4 q^{29} - 10 q^{31} - 2 q^{33} + 4 q^{35} - 2 q^{37} - 16 q^{39} - 10 q^{41} + 14 q^{43} + 8 q^{45} - 16 q^{47} + 4 q^{49} - 16 q^{51} + 2 q^{53} + 4 q^{55} - 4 q^{57} - 26 q^{59} - 12 q^{61} + 8 q^{63} - 8 q^{65} + 10 q^{67} - 14 q^{69} - 32 q^{71} - 14 q^{73} - 2 q^{75} + 4 q^{77} - 20 q^{79} - 16 q^{81} + 10 q^{83} - 6 q^{85} + 4 q^{87} - 10 q^{89} - 8 q^{91} + 14 q^{93} - 6 q^{95} - 2 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} + 2\beta_{2} + 5\beta _1 + 6 ) / 2 \) 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
−1.77571
2.64119
−0.589216
0.723742
0 −2.92887 0 1.00000 0 1.00000 0 5.57827 0
1.2 0 −2.33468 0 1.00000 0 1.00000 0 2.45073 0
1.3 0 1.06361 0 1.00000 0 1.00000 0 −1.86874 0
1.4 0 2.19994 0 1.00000 0 1.00000 0 1.83973 0
\(n\): 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 6160.2.a.br 4
4.b odd 2 1 385.2.a.h 4
12.b even 2 1 3465.2.a.bk 4
20.d odd 2 1 1925.2.a.x 4
20.e even 4 2 1925.2.b.p 8
28.d even 2 1 2695.2.a.l 4
44.c even 2 1 4235.2.a.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.h 4 4.b odd 2 1
1925.2.a.x 4 20.d odd 2 1
1925.2.b.p 8 20.e even 4 2
2695.2.a.l 4 28.d even 2 1
3465.2.a.bk 4 12.b even 2 1
4235.2.a.r 4 44.c even 2 1
6160.2.a.br 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6160))\):

\( T_{3}^{4} + 2T_{3}^{3} - 8T_{3}^{2} - 10T_{3} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} + 8T_{13}^{3} - 8T_{13}^{2} - 162T_{13} - 236 \) Copy content Toggle raw display
\( T_{17}^{4} + 6T_{17}^{3} + 4T_{17}^{2} - 14T_{17} + 4 \) Copy content Toggle raw display
\( T_{19}^{4} + 6T_{19}^{3} - 28T_{19}^{2} - 120T_{19} + 32 \) Copy content Toggle raw display
\( T_{23}^{4} + 14T_{23}^{3} + 28T_{23}^{2} - 284T_{23} - 976 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} - 8 T^{2} - 10 T + 16 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{3} - 8 T^{2} - 162 T - 236 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + 4 T^{2} - 14 T + 4 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} - 28 T^{2} - 120 T + 32 \) Copy content Toggle raw display
$23$ \( T^{4} + 14 T^{3} + 28 T^{2} + \cdots - 976 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} - 80 T^{2} - 272 T + 304 \) Copy content Toggle raw display
$31$ \( T^{4} + 10 T^{3} - 30 T^{2} + \cdots + 304 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} - 28 T^{2} - 20 T + 8 \) Copy content Toggle raw display
$41$ \( T^{4} + 10 T^{3} - 14 T^{2} + \cdots - 428 \) Copy content Toggle raw display
$43$ \( T^{4} - 14 T^{3} + 36 T^{2} + \cdots - 272 \) Copy content Toggle raw display
$47$ \( T^{4} + 16 T^{3} - 72 T^{2} + \cdots - 8408 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} - 236 T^{2} + \cdots + 12728 \) Copy content Toggle raw display
$59$ \( T^{4} + 26 T^{3} + 110 T^{2} + \cdots - 11848 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} + 30 T^{2} - 38 T - 4 \) Copy content Toggle raw display
$67$ \( T^{4} - 10 T^{3} - 192 T^{2} + \cdots + 11168 \) Copy content Toggle raw display
$71$ \( (T + 8)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} - 20 T^{2} + \cdots + 124 \) Copy content Toggle raw display
$79$ \( T^{4} + 20 T^{3} + 52 T^{2} + \cdots - 544 \) Copy content Toggle raw display
$83$ \( T^{4} - 10 T^{3} - 28 T^{2} + \cdots - 992 \) Copy content Toggle raw display
$89$ \( T^{4} + 10 T^{3} + 16 T^{2} - 32 T - 32 \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} - 192 T^{2} + \cdots + 2272 \) Copy content Toggle raw display
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