Properties

Label 6160.2.a.bu
Level $6160$
Weight $2$
Character orbit 6160.a
Self dual yes
Analytic conductor $49.188$
Analytic rank $0$
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: \(0\)
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 3080)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + q^{5} + q^{7} + ( - \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + q^{5} + q^{7} + ( - \beta_{3} - \beta_{2}) q^{9} + q^{11} + ( - \beta_{2} + \beta_1 - 2) q^{13} - \beta_{2} q^{15} + (2 \beta_{3} - \beta_{2}) q^{17} + \beta_1 q^{19} - \beta_{2} q^{21} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{23} + q^{25} + ( - 2 \beta_{3} + \beta_1 + 2) q^{27} + (2 \beta_{3} - 2 \beta_{2}) q^{29} + (\beta_{3} + 1) q^{31} - \beta_{2} q^{33} + q^{35} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{37} + ( - 3 \beta_{3} + \beta_{2} + 1) q^{39} + ( - 3 \beta_{3} - 1) q^{41} + ( - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{43} + ( - \beta_{3} - \beta_{2}) q^{45} + (2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{47} + q^{49} + (\beta_{3} + 3 \beta_{2} - 2 \beta_1 + 5) q^{51} + ( - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{53} + q^{55} + ( - 2 \beta_{3} - 2) q^{57} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 + 5) q^{59} + ( - \beta_{3} - \beta_1 + 1) q^{61} + ( - \beta_{3} - \beta_{2}) q^{63} + ( - \beta_{2} + \beta_1 - 2) q^{65} + (\beta_{3} + \beta_{2} - \beta_1 + 5) q^{67} + ( - 2 \beta_{3} - \beta_1 + 2) q^{69} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{71} + (2 \beta_{3} - 3 \beta_{2}) q^{73} - \beta_{2} q^{75} + q^{77} + (\beta_{3} + 3 \beta_{2} + 5) q^{79} + ( - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 4) q^{81} + ( - 2 \beta_{3} - \beta_1 + 6) q^{83} + (2 \beta_{3} - \beta_{2}) q^{85} + (2 \beta_{2} - 2 \beta_1 + 8) q^{87} + ( - 2 \beta_{3} - 3 \beta_1 + 4) q^{89} + ( - \beta_{2} + \beta_1 - 2) q^{91} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{93} + \beta_1 q^{95} + (2 \beta_{2} - \beta_1 - 2) q^{97} + ( - \beta_{3} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} + 2 q^{15} - 2 q^{17} + 2 q^{19} + 2 q^{21} + 6 q^{23} + 4 q^{25} + 14 q^{27} + 2 q^{31} + 2 q^{33} + 4 q^{35} - 6 q^{37} + 8 q^{39} + 2 q^{41} + 14 q^{43} + 4 q^{45} + 4 q^{49} + 8 q^{51} - 10 q^{53} + 4 q^{55} - 4 q^{57} + 10 q^{59} + 4 q^{61} + 4 q^{63} - 4 q^{65} + 14 q^{67} + 10 q^{69} + 4 q^{71} + 2 q^{73} + 2 q^{75} + 4 q^{77} + 12 q^{79} - 4 q^{81} + 26 q^{83} - 2 q^{85} + 24 q^{87} + 14 q^{89} - 4 q^{91} - 2 q^{93} + 2 q^{95} - 14 q^{97} + 4 q^{99}+O(q^{100}) \) 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
−0.589216
2.64119
−1.77571
0.723742
0 −1.80513 0 1.00000 0 1.00000 0 0.258482 0
1.2 0 −0.883951 0 1.00000 0 1.00000 0 −2.21863 0
1.3 0 1.64940 0 1.00000 0 1.00000 0 −0.279464 0
1.4 0 3.03967 0 1.00000 0 1.00000 0 6.23961 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.bu 4
4.b odd 2 1 3080.2.a.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3080.2.a.o 4 4.b odd 2 1
6160.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6160))\):

\( T_{3}^{4} - 2T_{3}^{3} - 6T_{3}^{2} + 6T_{3} + 8 \) Copy content Toggle raw display
\( T_{13}^{4} + 4T_{13}^{3} - 18T_{13}^{2} - 42T_{13} + 116 \) Copy content Toggle raw display
\( T_{17}^{4} + 2T_{17}^{3} - 38T_{17}^{2} - 54T_{17} + 196 \) Copy content Toggle raw display
\( T_{19}^{4} - 2T_{19}^{3} - 20T_{19}^{2} + 8T_{19} + 32 \) Copy content Toggle raw display
\( T_{23}^{4} - 6T_{23}^{3} - 16T_{23}^{2} + 84T_{23} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} - 6 T^{2} + 6 T + 8 \) 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
$13$ \( T^{4} + 4 T^{3} - 18 T^{2} - 42 T + 116 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} - 38 T^{2} - 54 T + 196 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} - 20 T^{2} + 8 T + 32 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} - 16 T^{2} + 84 T - 64 \) Copy content Toggle raw display
$29$ \( T^{4} - 56 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$31$ \( T^{4} - 2 T^{3} - 8 T^{2} + 10 T + 16 \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} - 28 T^{2} - 228 T - 344 \) Copy content Toggle raw display
$41$ \( T^{4} - 2 T^{3} - 84 T^{2} + \cdots + 1516 \) Copy content Toggle raw display
$43$ \( T^{4} - 14 T^{3} + 28 T^{2} + \cdots - 976 \) Copy content Toggle raw display
$47$ \( T^{4} - 70 T^{2} + 78 T - 16 \) Copy content Toggle raw display
$53$ \( T^{4} + 10 T^{3} - 4 T^{2} - 28 T - 8 \) Copy content Toggle raw display
$59$ \( T^{4} - 10 T^{3} - 76 T^{2} + \cdots + 1688 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} - 24 T^{2} + 138 T - 172 \) Copy content Toggle raw display
$67$ \( T^{4} - 14 T^{3} + 36 T^{2} + \cdots - 272 \) Copy content Toggle raw display
$71$ \( T^{4} - 4 T^{3} - 144 T^{2} + \cdots + 2048 \) Copy content Toggle raw display
$73$ \( T^{4} - 2 T^{3} - 86 T^{2} + 230 T - 4 \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} - 32 T^{2} + \cdots - 1088 \) Copy content Toggle raw display
$83$ \( T^{4} - 26 T^{3} + 196 T^{2} + \cdots - 1376 \) Copy content Toggle raw display
$89$ \( T^{4} - 14 T^{3} - 152 T^{2} + \cdots - 11072 \) Copy content Toggle raw display
$97$ \( T^{4} + 14 T^{3} + 32 T^{2} + \cdots - 256 \) Copy content Toggle raw display
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