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Properties

Label 960.3.j.e
Level $960$
Weight $3$
Character orbit 960.j
Analytic conductor $26.158$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.389136420864.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 24x^{4} + 80x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} - 1) q^{5} + ( - \beta_{4} - 3 \beta_1) q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} - 1) q^{5} + ( - \beta_{4} - 3 \beta_1) q^{7} + 3 q^{9} + \beta_{6} q^{11} + ( - \beta_{5} - \beta_{3}) q^{13} + (\beta_{4} + \beta_{2}) q^{15} + \beta_{7} q^{17} + ( - 2 \beta_{4} + 4 \beta_{2} + 2 \beta_1) q^{19} + (\beta_{5} - \beta_{3} - 8) q^{21} + ( - 2 \beta_{4} + 6 \beta_1) q^{23} + ( - \beta_{7} + 2 \beta_{5} + 5) q^{25} + 3 \beta_1 q^{27} + (3 \beta_{5} - 3 \beta_{3} + 26) q^{29} + (2 \beta_{6} - 2 \beta_{4} + \cdots + 2 \beta_1) q^{31}+ \cdots + 3 \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{5} + 24 q^{9} - 72 q^{21} + 32 q^{25} + 184 q^{29} - 256 q^{41} - 12 q^{45} - 24 q^{49} - 304 q^{61} + 168 q^{65} + 144 q^{69} + 72 q^{81} - 24 q^{85} + 560 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 24x^{4} + 80x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + \nu^{3} + 4\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 5\nu^{5} - 12\nu^{4} - 24\nu^{3} - 16\nu - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 4\nu^{6} + 5\nu^{5} + 20\nu^{4} + 24\nu^{3} + 32\nu^{2} + 144\nu + 192 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 3\nu^{5} - 22\nu^{3} - 8\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 4\nu^{6} + 5\nu^{5} - 20\nu^{4} + 24\nu^{3} - 32\nu^{2} + 144\nu - 192 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 5\nu^{4} + 40\nu^{2} + 80 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} - 9\nu^{5} + 24\nu^{3} - 16\nu ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_{3} - 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 5\beta_{4} - 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{6} - 5\beta_{5} + 4\beta_{4} + 5\beta_{3} - 8\beta_{2} - 4\beta _1 - 28 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - 4\beta_{5} + \beta_{4} - 4\beta_{3} + 71\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3\beta_{6} - 15\beta_{5} - 20\beta_{4} + 15\beta_{3} + 40\beta_{2} + 20\beta _1 - 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -19\beta_{7} + 4\beta_{5} - 29\beta_{4} + 4\beta_{3} - 139\beta_1 ) / 4 \) Copy content Toggle raw display

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

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
1.52274 − 1.29664i
1.52274 + 1.29664i
−0.656712 − 1.88911i
−0.656712 + 1.88911i
−1.52274 − 1.29664i
−1.52274 + 1.29664i
0.656712 − 1.88911i
0.656712 + 1.88911i
0 −1.73205 0 −4.27492 − 2.59328i 0 0.837253 0 3.00000 0
319.2 0 −1.73205 0 −4.27492 + 2.59328i 0 0.837253 0 3.00000 0
319.3 0 −1.73205 0 3.27492 − 3.77822i 0 9.55505 0 3.00000 0
319.4 0 −1.73205 0 3.27492 + 3.77822i 0 9.55505 0 3.00000 0
319.5 0 1.73205 0 −4.27492 − 2.59328i 0 −0.837253 0 3.00000 0
319.6 0 1.73205 0 −4.27492 + 2.59328i 0 −0.837253 0 3.00000 0
319.7 0 1.73205 0 3.27492 − 3.77822i 0 −9.55505 0 3.00000 0
319.8 0 1.73205 0 3.27492 + 3.77822i 0 −9.55505 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.8
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.e 8
4.b odd 2 1 inner 960.3.j.e 8
5.b even 2 1 inner 960.3.j.e 8
8.b even 2 1 60.3.f.b ✓ 8
8.d odd 2 1 60.3.f.b ✓ 8
20.d odd 2 1 inner 960.3.j.e 8
24.f even 2 1 180.3.f.h 8
24.h odd 2 1 180.3.f.h 8
40.e odd 2 1 60.3.f.b ✓ 8
40.f even 2 1 60.3.f.b ✓ 8
40.i odd 4 2 300.3.c.f 8
40.k even 4 2 300.3.c.f 8
120.i odd 2 1 180.3.f.h 8
120.m even 2 1 180.3.f.h 8
120.q odd 4 2 900.3.c.r 8
120.w even 4 2 900.3.c.r 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.b ✓ 8 8.b even 2 1
60.3.f.b ✓ 8 8.d odd 2 1
60.3.f.b ✓ 8 40.e odd 2 1
60.3.f.b ✓ 8 40.f even 2 1
180.3.f.h 8 24.f even 2 1
180.3.f.h 8 24.h odd 2 1
180.3.f.h 8 120.i odd 2 1
180.3.f.h 8 120.m even 2 1
300.3.c.f 8 40.i odd 4 2
300.3.c.f 8 40.k even 4 2
900.3.c.r 8 120.q odd 4 2
900.3.c.r 8 120.w even 4 2
960.3.j.e 8 1.a even 1 1 trivial
960.3.j.e 8 4.b odd 2 1 inner
960.3.j.e 8 5.b even 2 1 inner
960.3.j.e 8 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 \(S_{3}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{4} - 92T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{4} + 348T_{11}^{2} + 24576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{3} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 92 T^{2} + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 348 T^{2} + 24576)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 84 T^{2} + 1536)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1044 T^{2} + 221184)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1008 T^{2} + 221184)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 368 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 46 T + 16)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2304 T^{2} + 393216)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 756 T^{2} + 124416)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 64 T - 1028)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2528 T^{2} + 1364224)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 3296 T^{2} + 614656)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 756 T^{2} + 124416)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 16668 T^{2} + 69033984)^{2} \) Copy content Toggle raw display
$61$ \( (T + 38)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 9392 T^{2} + 7573504)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 17136 T^{2} + 884736)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 4176 T^{2} + 3538944)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2304 T^{2} + 393216)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 4064 T^{2} + 2027776)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 140 T + 3988)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 45888 T^{2} + 495550464)^{2} \) Copy content Toggle raw display
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