↑

Properties

Label 768.4.f.b
Level $768$
Weight $4$
Character orbit 768.f
Analytic conductor $45.313$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,4,Mod(383,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.383");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.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: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 48)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + \beta_{6} q^{5} + 5 \beta_{3} q^{7} + (\beta_{5} - 21) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + \beta_{6} q^{5} + 5 \beta_{3} q^{7} + (\beta_{5} - 21) q^{9} + ( - 6 \beta_{4} + 3 \beta_{2}) q^{11} + 13 \beta_1 q^{13} + ( - 3 \beta_{7} + 24 \beta_{3}) q^{15} + 4 \beta_{5} q^{17} - 31 \beta_{2} q^{19} + ( - 5 \beta_{6} + 15 \beta_1) q^{21} + 18 \beta_{7} q^{23} + 163 q^{25} + ( - 15 \beta_{4} - 27 \beta_{2}) q^{27} + \beta_{6} q^{29} - 9 \beta_{3} q^{31} + ( - 3 \beta_{5} + 144) q^{33} + (60 \beta_{4} - 30 \beta_{2}) q^{35} + 103 \beta_1 q^{37} + (13 \beta_{7} + 13 \beta_{3}) q^{39} + 18 \beta_{5} q^{41} + 27 \beta_{2} q^{43} + ( - 21 \beta_{6} + 144 \beta_1) q^{45} + 12 \beta_{7} q^{47} + 43 q^{49} + (24 \beta_{4} - 108 \beta_{2}) q^{51} - 3 \beta_{6} q^{53} - 144 \beta_{3} q^{55} + ( - 31 \beta_{5} - 186) q^{57} + (114 \beta_{4} - 57 \beta_{2}) q^{59} - 139 \beta_1 q^{61} + (30 \beta_{7} - 105 \beta_{3}) q^{63} + 26 \beta_{5} q^{65} + 257 \beta_{2} q^{67} + ( - 18 \beta_{6} - 432 \beta_1) q^{69} + 6 \beta_{7} q^{71} + 422 q^{73} + 163 \beta_{4} q^{75} + 30 \beta_{6} q^{77} - 193 \beta_{3} q^{79} + ( - 42 \beta_{5} + 153) q^{81} + ( - 6 \beta_{4} + 3 \beta_{2}) q^{83} + 576 \beta_1 q^{85} + ( - 3 \beta_{7} + 24 \beta_{3}) q^{87} + 22 \beta_{5} q^{89} - 130 \beta_{2} q^{91} + (9 \beta_{6} - 27 \beta_1) q^{93} + 186 \beta_{7} q^{95} - 1070 q^{97} + (126 \beta_{4} + 81 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 168 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 168 q^{9} + 1304 q^{25} + 1152 q^{33} + 344 q^{49} - 1488 q^{57} + 3376 q^{73} + 1224 q^{81} - 8560 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\zeta_{24}^{6} + 4\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\zeta_{24}^{4} - 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 4\zeta_{24}^{7} - \zeta_{24}^{6} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24}^{2} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 12\zeta_{24}^{5} + 12\zeta_{24}^{3} - 12\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -12\zeta_{24}^{5} + 12\zeta_{24}^{3} + 12\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 8\zeta_{24}^{7} - 4\zeta_{24}^{5} - 4\zeta_{24}^{3} - 4\zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( -3\beta_{7} + \beta_{6} - \beta_{5} + 6\beta_{4} - 3\beta_{2} ) / 48 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{6} + \beta_{5} ) / 24 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{3} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -3\beta_{7} - \beta_{6} + \beta_{5} + 6\beta_{4} - 3\beta_{2} ) / 48 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( 3\beta_{7} + \beta_{6} + \beta_{5} + 6\beta_{4} - 3\beta_{2} ) / 48 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1
0.258819 − 0.965926i
−0.258819 − 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
−0.965926 − 0.258819i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.965926 + 0.258819i
0 −1.73205 − 4.89898i 0 −16.9706 0 17.3205i 0 −21.0000 + 16.9706i 0
383.2 0 −1.73205 − 4.89898i 0 16.9706 0 − 17.3205i 0 −21.0000 + 16.9706i 0
383.3 0 −1.73205 + 4.89898i 0 −16.9706 0 − 17.3205i 0 −21.0000 − 16.9706i 0
383.4 0 −1.73205 + 4.89898i 0 16.9706 0 17.3205i 0 −21.0000 − 16.9706i 0
383.5 0 1.73205 − 4.89898i 0 −16.9706 0 17.3205i 0 −21.0000 − 16.9706i 0
383.6 0 1.73205 − 4.89898i 0 16.9706 0 − 17.3205i 0 −21.0000 − 16.9706i 0
383.7 0 1.73205 + 4.89898i 0 −16.9706 0 − 17.3205i 0 −21.0000 + 16.9706i 0
383.8 0 1.73205 + 4.89898i 0 16.9706 0 17.3205i 0 −21.0000 + 16.9706i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 383.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.f.b 8
3.b odd 2 1 inner 768.4.f.b 8
4.b odd 2 1 inner 768.4.f.b 8
8.b even 2 1 inner 768.4.f.b 8
8.d odd 2 1 inner 768.4.f.b 8
12.b even 2 1 inner 768.4.f.b 8
16.e even 4 1 48.4.c.b ✓ 4
16.e even 4 1 192.4.c.c 4
16.f odd 4 1 48.4.c.b ✓ 4
16.f odd 4 1 192.4.c.c 4
24.f even 2 1 inner 768.4.f.b 8
24.h odd 2 1 inner 768.4.f.b 8
48.i odd 4 1 48.4.c.b ✓ 4
48.i odd 4 1 192.4.c.c 4
48.k even 4 1 48.4.c.b ✓ 4
48.k even 4 1 192.4.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.4.c.b ✓ 4 16.e even 4 1
48.4.c.b ✓ 4 16.f odd 4 1
48.4.c.b ✓ 4 48.i odd 4 1
48.4.c.b ✓ 4 48.k even 4 1
192.4.c.c 4 16.e even 4 1
192.4.c.c 4 16.f odd 4 1
192.4.c.c 4 48.i odd 4 1
192.4.c.c 4 48.k even 4 1
768.4.f.b 8 1.a even 1 1 trivial
768.4.f.b 8 3.b odd 2 1 inner
768.4.f.b 8 4.b odd 2 1 inner
768.4.f.b 8 8.b even 2 1 inner
768.4.f.b 8 8.d odd 2 1 inner
768.4.f.b 8 12.b even 2 1 inner
768.4.f.b 8 24.f even 2 1 inner
768.4.f.b 8 24.h 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 \(S_{4}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} - 288 \) Copy content Toggle raw display
\( T_{19}^{2} - 11532 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 42 T^{2} + 729)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 288)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 300)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 864)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 676)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4608)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 11532)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 31104)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 288)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 972)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 42436)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 93312)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 8748)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 13824)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2592)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 311904)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 77284)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 792588)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 3456)^{4} \) Copy content Toggle raw display
$73$ \( (T - 422)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 446988)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 864)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 139392)^{4} \) Copy content Toggle raw display
$97$ \( (T + 1070)^{8} \) Copy content Toggle raw display
show more
show less