Properties

Label 960.2.k.e
Level $960$
Weight $2$
Character orbit 960.k
Analytic conductor $7.666$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,2,Mod(481,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("960.481");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.k (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_1 q^{5} - 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_1 q^{5} - 2 q^{7} - q^{9} + ( - \beta_{2} + 2 \beta_1) q^{11} + (\beta_{2} - 2 \beta_1) q^{13} + q^{15} - \beta_{3} q^{17} - 2 \beta_{2} q^{19} + 2 \beta_1 q^{21} - 4 q^{23} - q^{25} + \beta_1 q^{27} + ( - 2 \beta_{2} + 2 \beta_1) q^{29} + ( - \beta_{3} - 4) q^{31} + ( - \beta_{3} + 2) q^{33} - 2 \beta_1 q^{35} + ( - \beta_{2} + 6 \beta_1) q^{37} + (\beta_{3} - 2) q^{39} + ( - 2 \beta_{3} - 2) q^{41} - 2 \beta_{2} q^{43} - \beta_1 q^{45} + 4 q^{47} - 3 q^{49} + \beta_{2} q^{51} + (2 \beta_{2} + 6 \beta_1) q^{53} + (\beta_{3} - 2) q^{55} - 2 \beta_{3} q^{57} + (\beta_{2} - 6 \beta_1) q^{59} - 4 \beta_1 q^{61} + 2 q^{63} + ( - \beta_{3} + 2) q^{65} - 2 \beta_{2} q^{67} + 4 \beta_1 q^{69} - 2 \beta_{3} q^{71} + 10 q^{73} + \beta_1 q^{75} + (2 \beta_{2} - 4 \beta_1) q^{77} + (3 \beta_{3} - 4) q^{79} + q^{81} + 8 \beta_1 q^{83} - \beta_{2} q^{85} + ( - 2 \beta_{3} + 2) q^{87} + ( - 2 \beta_{3} - 6) q^{89} + ( - 2 \beta_{2} + 4 \beta_1) q^{91} + (\beta_{2} + 4 \beta_1) q^{93} + 2 \beta_{3} q^{95} + (2 \beta_{3} - 6) q^{97} + (\beta_{2} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{7} - 4 q^{9} + 4 q^{15} - 16 q^{23} - 4 q^{25} - 16 q^{31} + 8 q^{33} - 8 q^{39} - 8 q^{41} + 16 q^{47} - 12 q^{49} - 8 q^{55} + 8 q^{63} + 8 q^{65} + 40 q^{73} - 16 q^{79} + 4 q^{81} + 8 q^{87} - 24 q^{89} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\zeta_{12}^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{12}^{3} + 4\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
0 1.00000i 0 1.00000i 0 −2.00000 0 −1.00000 0
481.2 0 1.00000i 0 1.00000i 0 −2.00000 0 −1.00000 0
481.3 0 1.00000i 0 1.00000i 0 −2.00000 0 −1.00000 0
481.4 0 1.00000i 0 1.00000i 0 −2.00000 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.k.e 4
3.b odd 2 1 2880.2.k.f 4
4.b odd 2 1 960.2.k.f yes 4
5.b even 2 1 4800.2.k.o 4
5.c odd 4 1 4800.2.d.i 4
5.c odd 4 1 4800.2.d.r 4
8.b even 2 1 inner 960.2.k.e 4
8.d odd 2 1 960.2.k.f yes 4
12.b even 2 1 2880.2.k.k 4
16.e even 4 1 3840.2.a.be 2
16.e even 4 1 3840.2.a.bn 2
16.f odd 4 1 3840.2.a.bf 2
16.f odd 4 1 3840.2.a.bi 2
20.d odd 2 1 4800.2.k.i 4
20.e even 4 1 4800.2.d.m 4
20.e even 4 1 4800.2.d.n 4
24.f even 2 1 2880.2.k.k 4
24.h odd 2 1 2880.2.k.f 4
40.e odd 2 1 4800.2.k.i 4
40.f even 2 1 4800.2.k.o 4
40.i odd 4 1 4800.2.d.i 4
40.i odd 4 1 4800.2.d.r 4
40.k even 4 1 4800.2.d.m 4
40.k even 4 1 4800.2.d.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.k.e 4 1.a even 1 1 trivial
960.2.k.e 4 8.b even 2 1 inner
960.2.k.f yes 4 4.b odd 2 1
960.2.k.f yes 4 8.d odd 2 1
2880.2.k.f 4 3.b odd 2 1
2880.2.k.f 4 24.h odd 2 1
2880.2.k.k 4 12.b even 2 1
2880.2.k.k 4 24.f even 2 1
3840.2.a.be 2 16.e even 4 1
3840.2.a.bf 2 16.f odd 4 1
3840.2.a.bi 2 16.f odd 4 1
3840.2.a.bn 2 16.e even 4 1
4800.2.d.i 4 5.c odd 4 1
4800.2.d.i 4 40.i odd 4 1
4800.2.d.m 4 20.e even 4 1
4800.2.d.m 4 40.k even 4 1
4800.2.d.n 4 20.e even 4 1
4800.2.d.n 4 40.k even 4 1
4800.2.d.r 4 5.c odd 4 1
4800.2.d.r 4 40.i odd 4 1
4800.2.k.i 4 20.d odd 2 1
4800.2.k.i 4 40.e odd 2 1
4800.2.k.o 4 5.b even 2 1
4800.2.k.o 4 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\):

\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{23} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$13$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$47$ \( (T - 4)^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$59$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$61$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$73$ \( (T - 10)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T - 92)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T - 12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 12 T - 12)^{2} \) Copy content Toggle raw display
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