Properties

Label 960.3.j.b
Level $960$
Weight $3$
Character orbit 960.j
Analytic conductor $26.158$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,3,Mod(319,960)]
 
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([1, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.j (of order \(2\), degree \(1\), not 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: \(26.1581053786\)
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - 5 \beta_1 q^{5} - 6 \beta_{3} q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - 5 \beta_1 q^{5} - 6 \beta_{3} q^{7} + 3 q^{9} + 6 \beta_{2} q^{11} - 18 \beta_1 q^{13} + 5 \beta_{2} q^{15} - 10 \beta_1 q^{17} - 8 \beta_{2} q^{19} + 18 q^{21} + 4 \beta_{3} q^{23} - 25 q^{25} - 3 \beta_{3} q^{27} - 36 q^{29} + 4 \beta_{2} q^{31} - 18 \beta_1 q^{33} + 30 \beta_{2} q^{35} + 54 \beta_1 q^{37} + 18 \beta_{2} q^{39} + 18 q^{41} - 12 \beta_{3} q^{43} - 15 \beta_1 q^{45} + 59 q^{49} + 10 \beta_{2} q^{51} + 26 \beta_1 q^{53} + 30 \beta_{3} q^{55} + 24 \beta_1 q^{57} + 18 \beta_{2} q^{59} + 74 q^{61} - 18 \beta_{3} q^{63} - 90 q^{65} + 24 \beta_{3} q^{67} - 12 q^{69} + 60 \beta_{2} q^{71} + 36 \beta_1 q^{73} + 25 \beta_{3} q^{75} - 108 \beta_1 q^{77} + 52 \beta_{2} q^{79} + 9 q^{81} + 52 \beta_{3} q^{83} - 50 q^{85} + 36 \beta_{3} q^{87} + 18 q^{89} + 108 \beta_{2} q^{91} - 12 \beta_1 q^{93} - 40 \beta_{3} q^{95} + 72 \beta_1 q^{97} + 18 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} + 72 q^{21} - 100 q^{25} - 144 q^{29} + 72 q^{41} + 236 q^{49} + 296 q^{61} - 360 q^{65} - 48 q^{69} + 36 q^{81} - 200 q^{85} + 72 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0 −1.73205 0 5.00000i 0 −10.3923 0 3.00000 0
319.2 0 −1.73205 0 5.00000i 0 −10.3923 0 3.00000 0
319.3 0 1.73205 0 5.00000i 0 10.3923 0 3.00000 0
319.4 0 1.73205 0 5.00000i 0 10.3923 0 3.00000 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.3.j.b 4
4.b odd 2 1 inner 960.3.j.b 4
5.b even 2 1 inner 960.3.j.b 4
8.b even 2 1 60.3.f.a 4
8.d odd 2 1 60.3.f.a 4
20.d odd 2 1 inner 960.3.j.b 4
24.f even 2 1 180.3.f.e 4
24.h odd 2 1 180.3.f.e 4
40.e odd 2 1 60.3.f.a 4
40.f even 2 1 60.3.f.a 4
40.i odd 4 1 300.3.c.a 2
40.i odd 4 1 300.3.c.c 2
40.k even 4 1 300.3.c.a 2
40.k even 4 1 300.3.c.c 2
120.i odd 2 1 180.3.f.e 4
120.m even 2 1 180.3.f.e 4
120.q odd 4 1 900.3.c.f 2
120.q odd 4 1 900.3.c.j 2
120.w even 4 1 900.3.c.f 2
120.w even 4 1 900.3.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.a 4 8.b even 2 1
60.3.f.a 4 8.d odd 2 1
60.3.f.a 4 40.e odd 2 1
60.3.f.a 4 40.f even 2 1
180.3.f.e 4 24.f even 2 1
180.3.f.e 4 24.h odd 2 1
180.3.f.e 4 120.i odd 2 1
180.3.f.e 4 120.m even 2 1
300.3.c.a 2 40.i odd 4 1
300.3.c.a 2 40.k even 4 1
300.3.c.c 2 40.i odd 4 1
300.3.c.c 2 40.k even 4 1
900.3.c.f 2 120.q odd 4 1
900.3.c.f 2 120.w even 4 1
900.3.c.j 2 120.q odd 4 1
900.3.c.j 2 120.w even 4 1
960.3.j.b 4 1.a even 1 1 trivial
960.3.j.b 4 4.b odd 2 1 inner
960.3.j.b 4 5.b even 2 1 inner
960.3.j.b 4 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{2} - 108 \) Copy content Toggle raw display
\( T_{11}^{2} + 108 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 324)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$29$ \( (T + 36)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2916)^{2} \) Copy content Toggle raw display
$41$ \( (T - 18)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 432)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 676)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 972)^{2} \) Copy content Toggle raw display
$61$ \( (T - 74)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 1728)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 10800)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 1296)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8112)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 8112)^{2} \) Copy content Toggle raw display
$89$ \( (T - 18)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 5184)^{2} \) Copy content Toggle raw display
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