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Properties

Label 2960.2.a.ba
Level $2960$
Weight $2$
Character orbit 2960.a
Self dual yes
Analytic conductor $23.636$
Analytic rank $1$
Dimension $5$
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] = [2960,2,Mod(1,2960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.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: \(23.6357189983\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.368464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 185)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{4} + \beta_{3} + \beta_{2} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{4} + \beta_{3} + \beta_{2} + \cdots - 1) q^{9}+ \cdots + ( - 6 \beta_{4} - 2 \beta_{3} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} + 5 q^{5} - 7 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{3} + 5 q^{5} - 7 q^{7} + 2 q^{9} - 7 q^{11} + 2 q^{13} + q^{15} - 8 q^{17} - 14 q^{19} - 9 q^{21} - 2 q^{23} + 5 q^{25} + 7 q^{27} + 2 q^{29} - 8 q^{31} - 21 q^{33} - 7 q^{35} - 5 q^{37} - 12 q^{39} - 9 q^{41} - 14 q^{43} + 2 q^{45} + 5 q^{47} + 2 q^{49} - 10 q^{51} - 15 q^{53} - 7 q^{55} + 4 q^{57} - 12 q^{59} + 12 q^{61} - 24 q^{63} + 2 q^{65} + 2 q^{67} - 30 q^{69} - 13 q^{71} - 5 q^{73} + q^{75} - 7 q^{77} - 36 q^{79} + 21 q^{81} + 9 q^{83} - 8 q^{85} + 6 q^{87} - 16 q^{89} + 4 q^{91} - 16 q^{93} - 14 q^{95} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 6\nu^{2} + 4\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 10\nu^{2} - 4\nu + 12 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 3\beta_{3} + 2\beta_{2} + 6\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} - 10\beta_{3} + 10\beta_{2} + 14\beta _1 + 18 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
3.14884
−1.09027
1.17837
−1.78948
0.552543
0 −2.51369 0 1.00000 0 −0.281955 0 3.31863 0
1.2 0 −0.744131 0 1.00000 0 −3.94357 0 −2.44627 0
1.3 0 0.518894 0 1.00000 0 1.33546 0 −2.73075 0
1.4 0 0.671838 0 1.00000 0 0.305070 0 −2.54863 0
1.5 0 3.06709 0 1.00000 0 −4.41500 0 6.40702 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(37\) \(1\)

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 2960.2.a.ba 5
4.b odd 2 1 185.2.a.d ✓ 5
12.b even 2 1 1665.2.a.q 5
20.d odd 2 1 925.2.a.h 5
20.e even 4 2 925.2.b.g 10
28.d even 2 1 9065.2.a.j 5
60.h even 2 1 8325.2.a.cc 5
148.b odd 2 1 6845.2.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.a.d ✓ 5 4.b odd 2 1
925.2.a.h 5 20.d odd 2 1
925.2.b.g 10 20.e even 4 2
1665.2.a.q 5 12.b even 2 1
2960.2.a.ba 5 1.a even 1 1 trivial
6845.2.a.g 5 148.b odd 2 1
8325.2.a.cc 5 60.h even 2 1
9065.2.a.j 5 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

\( T_{3}^{5} - T_{3}^{4} - 8T_{3}^{3} + 4T_{3}^{2} + 4T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{5} + 7T_{7}^{4} + 6T_{7}^{3} - 24T_{7}^{2} + 2 \) Copy content Toggle raw display
\( T_{13}^{5} - 2T_{13}^{4} - 20T_{13}^{3} + 20T_{13}^{2} + 76T_{13} - 88 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - T^{4} - 8 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 7 T^{4} + \cdots + 2 \) Copy content Toggle raw display
$11$ \( T^{5} + 7 T^{4} + \cdots + 32 \) Copy content Toggle raw display
$13$ \( T^{5} - 2 T^{4} + \cdots - 88 \) Copy content Toggle raw display
$17$ \( T^{5} + 8 T^{4} + \cdots - 32 \) Copy content Toggle raw display
$19$ \( T^{5} + 14 T^{4} + \cdots - 2224 \) Copy content Toggle raw display
$23$ \( T^{5} + 2 T^{4} + \cdots - 1504 \) Copy content Toggle raw display
$29$ \( T^{5} - 2 T^{4} + \cdots + 32 \) Copy content Toggle raw display
$31$ \( T^{5} + 8 T^{4} + \cdots + 3016 \) Copy content Toggle raw display
$37$ \( (T + 1)^{5} \) Copy content Toggle raw display
$41$ \( T^{5} + 9 T^{4} + \cdots - 928 \) Copy content Toggle raw display
$43$ \( T^{5} + 14 T^{4} + \cdots + 1312 \) Copy content Toggle raw display
$47$ \( T^{5} - 5 T^{4} + \cdots + 3994 \) Copy content Toggle raw display
$53$ \( T^{5} + 15 T^{4} + \cdots - 16 \) Copy content Toggle raw display
$59$ \( T^{5} + 12 T^{4} + \cdots - 5456 \) Copy content Toggle raw display
$61$ \( T^{5} - 12 T^{4} + \cdots - 512 \) Copy content Toggle raw display
$67$ \( T^{5} - 2 T^{4} + \cdots + 7568 \) Copy content Toggle raw display
$71$ \( T^{5} + 13 T^{4} + \cdots + 291136 \) Copy content Toggle raw display
$73$ \( T^{5} + 5 T^{4} + \cdots + 176 \) Copy content Toggle raw display
$79$ \( T^{5} + 36 T^{4} + \cdots - 5912 \) Copy content Toggle raw display
$83$ \( T^{5} - 9 T^{4} + \cdots - 314 \) Copy content Toggle raw display
$89$ \( T^{5} + 16 T^{4} + \cdots + 1856 \) Copy content Toggle raw display
$97$ \( T^{5} - 464 T^{3} + \cdots - 193408 \) Copy content Toggle raw display
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