Properties

Label 8960.2.a.cc
Level $8960$
Weight $2$
Character orbit 8960.a
Self dual yes
Analytic conductor $71.546$
Analytic rank $0$
Dimension $6$
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] = [8960,2,Mod(1,8960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8960, 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("8960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8960 = 2^{8} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8960.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: \(71.5459602111\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.33267456.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 8x^{4} + 8x^{3} + 13x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 2240)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - q^{5} + q^{7} + (\beta_{5} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - q^{5} + q^{7} + (\beta_{5} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{9} + ( - \beta_{3} - \beta_{2} - 1) q^{11} + (\beta_{4} + \beta_{3} - \beta_1 - 1) q^{13} + \beta_{3} q^{15} + ( - \beta_{3} - \beta_1 + 1) q^{17} + ( - \beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{19} - \beta_{3} q^{21} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{23} + q^{25} + ( - 2 \beta_{4} - 3 \beta_{3} - 2) q^{27} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 3) q^{29} + (\beta_{5} + \beta_{4} + \beta_{2} + 1) q^{31} + (\beta_{5} + \beta_{3} - 2 \beta_{2} - \beta_1 + 5) q^{33} - q^{35} + (\beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{37} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{39} + ( - \beta_{4} + \beta_{2} - \beta_1) q^{41} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{43} + ( - \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{45} + (\beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 2) q^{47} + q^{49} + (2 \beta_{5} - \beta_{3} - \beta_{2} - 2 \beta_1 + 5) q^{51} + (\beta_{5} - 2 \beta_{3} + 2 \beta_{2} - 2) q^{53} + (\beta_{3} + \beta_{2} + 1) q^{55} + (\beta_{5} + 3 \beta_{4} + 2 \beta_{3} - \beta_{2} - 1) q^{57} + (\beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 - 2) q^{59} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 3) q^{61} + (\beta_{5} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{63} + ( - \beta_{4} - \beta_{3} + \beta_1 + 1) q^{65} + ( - 2 \beta_{5} + 2 \beta_{4}) q^{67} + (3 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{69} + ( - \beta_{5} - 4 \beta_{3} + 4) q^{71} + ( - \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{73} - \beta_{3} q^{75} + ( - \beta_{3} - \beta_{2} - 1) q^{77} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{79} + (4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 7) q^{81} + (2 \beta_{5} + 2 \beta_{4} + 6) q^{83} + (\beta_{3} + \beta_1 - 1) q^{85} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_1 - 2) q^{87} + ( - \beta_{4} - \beta_{2} - \beta_1 - 2) q^{89} + (\beta_{4} + \beta_{3} - \beta_1 - 1) q^{91} + ( - 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1) q^{93} + (\beta_{5} + \beta_{4} + \beta_{2} - \beta_1) q^{95} + ( - 2 \beta_{5} + \beta_{3} + \beta_1 + 3) q^{97} + ( - 2 \beta_{4} - 8 \beta_{3} + 2 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{5} + 6 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{5} + 6 q^{7} + 10 q^{9} - 4 q^{11} - 4 q^{13} + 8 q^{17} + 4 q^{23} + 6 q^{25} - 12 q^{27} - 16 q^{29} + 4 q^{31} + 36 q^{33} - 6 q^{35} + 4 q^{37} - 16 q^{39} - 12 q^{43} - 10 q^{45} + 16 q^{47} + 6 q^{49} + 36 q^{51} - 16 q^{53} + 4 q^{55} - 4 q^{57} - 4 q^{59} + 16 q^{61} + 10 q^{63} + 4 q^{65} + 12 q^{69} + 24 q^{71} - 4 q^{73} - 4 q^{77} + 8 q^{79} + 46 q^{81} + 36 q^{83} - 8 q^{85} - 16 q^{87} - 8 q^{89} - 4 q^{91} - 4 q^{93} + 16 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 8x^{4} + 8x^{3} + 13x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 6\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 10\nu^{2} + 18\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 6\nu^{2} + 8\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - 2\beta_{4} - 4\beta_{3} + 4\beta_{2} + 9\beta _1 + 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - 2\beta_{4} - 9\beta_{3} + 11\beta_{2} + 13\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 13\beta_{5} - 22\beta_{4} - 52\beta_{3} + 60\beta_{2} + 95\beta _1 + 189 ) / 2 \) 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
0.631475
−0.234875
3.38043
−2.03144
1.53427
−1.27985
0 −3.32646 0 −1.00000 0 1.00000 0 8.06534 0
1.2 0 −2.11677 0 −1.00000 0 1.00000 0 1.48073 0
1.3 0 0.191804 0 −1.00000 0 1.00000 0 −2.96321 0
1.4 0 0.639640 0 −1.00000 0 1.00000 0 −2.59086 0
1.5 0 1.47713 0 −1.00000 0 1.00000 0 −0.818075 0
1.6 0 3.13466 0 −1.00000 0 1.00000 0 6.82607 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

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 8960.2.a.cc 6
4.b odd 2 1 8960.2.a.cb 6
8.b even 2 1 8960.2.a.ch 6
8.d odd 2 1 8960.2.a.ce 6
16.e even 4 2 2240.2.b.g 12
16.f odd 4 2 2240.2.b.h yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.b.g 12 16.e even 4 2
2240.2.b.h yes 12 16.f odd 4 2
8960.2.a.cb 6 4.b odd 2 1
8960.2.a.cc 6 1.a even 1 1 trivial
8960.2.a.ce 6 8.d odd 2 1
8960.2.a.ch 6 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8960))\):

\( T_{3}^{6} - 14T_{3}^{4} + 4T_{3}^{3} + 37T_{3}^{2} - 28T_{3} + 4 \) Copy content Toggle raw display
\( T_{11}^{6} + 4T_{11}^{5} - 42T_{11}^{4} - 204T_{11}^{3} + 257T_{11}^{2} + 2288T_{11} + 2704 \) Copy content Toggle raw display
\( T_{13}^{6} + 4T_{13}^{5} - 50T_{13}^{4} - 160T_{13}^{3} + 517T_{13}^{2} + 1812T_{13} + 772 \) Copy content Toggle raw display
\( T_{23}^{6} - 4T_{23}^{5} - 80T_{23}^{4} + 160T_{23}^{3} + 1504T_{23}^{2} + 2304T_{23} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 14 T^{4} + 4 T^{3} + 37 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T - 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 4 T^{5} - 42 T^{4} + \cdots + 2704 \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} - 50 T^{4} - 160 T^{3} + \cdots + 772 \) Copy content Toggle raw display
$17$ \( T^{6} - 8 T^{5} - 34 T^{4} + 228 T^{3} + \cdots + 88 \) Copy content Toggle raw display
$19$ \( T^{6} - 104 T^{4} - 64 T^{3} + \cdots - 5888 \) Copy content Toggle raw display
$23$ \( T^{6} - 4 T^{5} - 80 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( T^{6} + 16 T^{5} + 2 T^{4} - 984 T^{3} + \cdots - 944 \) Copy content Toggle raw display
$31$ \( T^{6} - 4 T^{5} - 80 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$37$ \( T^{6} - 4 T^{5} - 124 T^{4} + \cdots - 58496 \) Copy content Toggle raw display
$41$ \( T^{6} - 140 T^{4} + 224 T^{3} + \cdots - 32192 \) Copy content Toggle raw display
$43$ \( T^{6} + 12 T^{5} - 144 T^{4} + \cdots + 5632 \) Copy content Toggle raw display
$47$ \( T^{6} - 16 T^{5} - 78 T^{4} + \cdots + 50272 \) Copy content Toggle raw display
$53$ \( T^{6} + 16 T^{5} - 60 T^{4} + \cdots - 76544 \) Copy content Toggle raw display
$59$ \( T^{6} + 4 T^{5} - 264 T^{4} + \cdots - 80384 \) Copy content Toggle raw display
$61$ \( T^{6} - 16 T^{5} - 44 T^{4} + \cdots - 5312 \) Copy content Toggle raw display
$67$ \( T^{6} - 272 T^{4} - 832 T^{3} + \cdots + 123904 \) Copy content Toggle raw display
$71$ \( T^{6} - 24 T^{5} - 20 T^{4} + \cdots + 29248 \) Copy content Toggle raw display
$73$ \( T^{6} + 4 T^{5} - 200 T^{4} + \cdots + 22528 \) Copy content Toggle raw display
$79$ \( T^{6} - 8 T^{5} - 138 T^{4} + \cdots - 13628 \) Copy content Toggle raw display
$83$ \( T^{6} - 36 T^{5} + 268 T^{4} + \cdots - 430016 \) Copy content Toggle raw display
$89$ \( T^{6} + 8 T^{5} - 60 T^{4} + \cdots + 11584 \) Copy content Toggle raw display
$97$ \( T^{6} - 16 T^{5} - 50 T^{4} + \cdots + 2776 \) Copy content Toggle raw display
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