↑

Properties

Label 960.4.f.t
Level $960$
Weight $4$
Character orbit 960.f
Analytic conductor $56.642$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,4,Mod(769,960)]
 
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, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.769");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.f (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: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.989122628102400.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 87x^{6} + 1985x^{4} + 7248x^{2} + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 480)
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.

\(f(q)\) \(=\) \( q - 3 \beta_1 q^{3} + (\beta_{4} - 2) q^{5} + ( - \beta_{6} - \beta_{5} + 4 \beta_1) q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta_1 q^{3} + (\beta_{4} - 2) q^{5} + ( - \beta_{6} - \beta_{5} + 4 \beta_1) q^{7} - 9 q^{9} + ( - \beta_{6} + \beta_{5} + \beta_{2}) q^{11} + ( - 2 \beta_{7} - \beta_{4} + \beta_{3} + 1) q^{13} + (3 \beta_{6} + 3 \beta_1) q^{15} + (3 \beta_{7} + \beta_{4} - \beta_{3} - 1) q^{17} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{19} + (3 \beta_{4} + 3 \beta_{3} + 9) q^{21} + ( - 6 \beta_{6} - 6 \beta_{5} - 20 \beta_1) q^{23} + ( - 5 \beta_{7} - \beta_{4} + \cdots - 18) q^{25}+ \cdots + (9 \beta_{6} - 9 \beta_{5} - 9 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{5} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{5} - 72 q^{9} + 72 q^{21} - 128 q^{25} - 888 q^{29} - 192 q^{41} + 108 q^{45} + 1000 q^{49} + 1008 q^{61} + 1704 q^{65} - 624 q^{69} + 648 q^{81} - 1992 q^{85} - 1872 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 87x^{6} + 1985x^{4} + 7248x^{2} + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -57\nu^{7} - 4927\nu^{5} - 113561\nu^{3} - 489392\nu ) / 181376 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -62\nu^{6} - 4862\nu^{4} - 95978\nu^{2} - 226448 ) / 7085 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 121 \nu^{7} - 128 \nu^{6} - 11071 \nu^{5} - 12288 \nu^{4} - 274969 \nu^{3} - 294912 \nu^{2} + \cdots - 582272 ) / 69760 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 121 \nu^{7} - 128 \nu^{6} + 11071 \nu^{5} - 12288 \nu^{4} + 274969 \nu^{3} - 294912 \nu^{2} + \cdots - 512512 ) / 69760 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3249 \nu^{7} - 1072 \nu^{6} - 280839 \nu^{5} - 76752 \nu^{4} - 6291601 \nu^{3} - 891568 \nu^{2} + \cdots + 6206592 ) / 906880 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3249 \nu^{7} + 1072 \nu^{6} - 280839 \nu^{5} + 76752 \nu^{4} - 6291601 \nu^{3} + 891568 \nu^{2} + \cdots - 6206592 ) / 906880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 57\nu^{7} + 4927\nu^{5} + 110073\nu^{3} + 356848\nu ) / 4360 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{6} + 2\beta_{5} - 4\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -8\beta_{6} + 8\beta_{5} + 2\beta_{4} + 2\beta_{3} - 3\beta_{2} - 174 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -12\beta_{7} - 19\beta_{6} - 19\beta_{5} - 66\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 344\beta_{6} - 344\beta_{5} - 210\beta_{4} - 210\beta_{3} + 181\beta_{2} + 7198 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2215\beta_{7} + 3322\beta_{6} + 3322\beta_{5} + 456\beta_{4} - 456\beta_{3} + 21436\beta _1 - 456 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3648\beta_{6} + 3648\beta_{5} + 3343\beta_{4} + 3343\beta_{3} - 2616\beta_{2} - 81081 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 104417 \beta_{7} - 152906 \beta_{6} - 152906 \beta_{5} - 39416 \beta_{4} + 39416 \beta_{3} + \cdots + 39416 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
− 0.383633i
7.11205i
− 2.07432i
− 5.65409i
− 7.11205i
0.383633i
5.65409i
2.07432i
0 − 3.00000i 0 −8.72842 − 6.98676i 0 − 11.4568i 0 −9.00000 0
769.2 0 − 3.00000i 0 −8.72842 + 6.98676i 0 − 11.4568i 0 −9.00000 0
769.3 0 − 3.00000i 0 5.72842 − 9.60131i 0 17.4568i 0 −9.00000 0
769.4 0 − 3.00000i 0 5.72842 + 9.60131i 0 17.4568i 0 −9.00000 0
769.5 0 3.00000i 0 −8.72842 − 6.98676i 0 11.4568i 0 −9.00000 0
769.6 0 3.00000i 0 −8.72842 + 6.98676i 0 11.4568i 0 −9.00000 0
769.7 0 3.00000i 0 5.72842 − 9.60131i 0 − 17.4568i 0 −9.00000 0
769.8 0 3.00000i 0 5.72842 + 9.60131i 0 − 17.4568i 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.8
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 960.4.f.t 8
4.b odd 2 1 inner 960.4.f.t 8
5.b even 2 1 inner 960.4.f.t 8
8.b even 2 1 480.4.f.g ✓ 8
8.d odd 2 1 480.4.f.g ✓ 8
20.d odd 2 1 inner 960.4.f.t 8
24.f even 2 1 1440.4.f.l 8
24.h odd 2 1 1440.4.f.l 8
40.e odd 2 1 480.4.f.g ✓ 8
40.f even 2 1 480.4.f.g ✓ 8
40.i odd 4 1 2400.4.a.by 4
40.i odd 4 1 2400.4.a.cb 4
40.k even 4 1 2400.4.a.by 4
40.k even 4 1 2400.4.a.cb 4
120.i odd 2 1 1440.4.f.l 8
120.m even 2 1 1440.4.f.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.4.f.g ✓ 8 8.b even 2 1
480.4.f.g ✓ 8 8.d odd 2 1
480.4.f.g ✓ 8 40.e odd 2 1
480.4.f.g ✓ 8 40.f even 2 1
960.4.f.t 8 1.a even 1 1 trivial
960.4.f.t 8 4.b odd 2 1 inner
960.4.f.t 8 5.b even 2 1 inner
960.4.f.t 8 20.d odd 2 1 inner
1440.4.f.l 8 24.f even 2 1
1440.4.f.l 8 24.h odd 2 1
1440.4.f.l 8 120.i odd 2 1
1440.4.f.l 8 120.m even 2 1
2400.4.a.by 4 40.i odd 4 1
2400.4.a.by 4 40.k even 4 1
2400.4.a.cb 4 40.i odd 4 1
2400.4.a.cb 4 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{4} + 436T_{7}^{2} + 40000 \) Copy content Toggle raw display
\( T_{11}^{4} - 1380T_{11}^{2} + 288000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 6 T^{3} + \cdots + 15625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 436 T^{2} + 40000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 1380 T^{2} + 288000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 5556 T^{2} + 486720)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 11364 T^{2} + 6094080)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2256 T^{2} + 1152000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 16400 T^{2} + 46895104)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 222 T - 4608)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 154176 T^{2} + 5709496320)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 157620 T^{2} + 3976200000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 48 T - 260)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 114208 T^{2} + 2490409216)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 140192 T^{2} + 98089216)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 533076 T^{2} + 1096384320)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 571044 T^{2} + 33019188480)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 252 T - 465660)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 162832 T^{2} + 1569427456)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 549264 T^{2} + 36172615680)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 253584 T^{2} + 12556062720)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 1863744 T^{2} + 806669844480)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 621728 T^{2} + 53298186496)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 468 T - 697644)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 218496 T^{2} + 8206110720)^{2} \) Copy content Toggle raw display
show more
show less