Properties

Label 768.4.c.r
Level $768$
Weight $4$
Character orbit 768.c
Analytic conductor $45.313$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,4,Mod(767,768)]
 
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, 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("768.767");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.c (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: \(4\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 192)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_{3} q^{5} + 17 \beta_1 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_{3} q^{5} + 17 \beta_1 q^{7} + 27 q^{9} + 14 \beta_{2} q^{11} + 27 \beta_1 q^{15} - 17 \beta_{3} q^{21} + 17 q^{25} + 27 \beta_{2} q^{27} + 21 \beta_{3} q^{29} + 35 \beta_1 q^{31} + 378 q^{33} - 68 \beta_{2} q^{35} - 27 \beta_{3} q^{45} - 813 q^{49} - 49 \beta_{3} q^{53} + 378 \beta_1 q^{55} - 138 \beta_{2} q^{59} + 459 \beta_1 q^{63} - 322 q^{73} + 17 \beta_{2} q^{75} - 238 \beta_{3} q^{77} - 685 \beta_1 q^{79} + 729 q^{81} + 170 \beta_{2} q^{83} - 567 \beta_1 q^{87} - 35 \beta_{3} q^{93} - 574 q^{97} + 378 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 108 q^{9} + 68 q^{25} + 1512 q^{33} - 3252 q^{49} - 1288 q^{73} + 2916 q^{81} - 2296 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -3\zeta_{12}^{3} + 6\zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 12\zeta_{12}^{2} - 6 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( 2\beta_{2} - 3\beta_1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{3} + 6 ) / 12 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_1 ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
767.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0 −5.19615 0 10.3923i 0 34.0000i 0 27.0000 0
767.2 0 −5.19615 0 10.3923i 0 34.0000i 0 27.0000 0
767.3 0 5.19615 0 10.3923i 0 34.0000i 0 27.0000 0
767.4 0 5.19615 0 10.3923i 0 34.0000i 0 27.0000 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.c.r 4
3.b odd 2 1 inner 768.4.c.r 4
4.b odd 2 1 inner 768.4.c.r 4
8.b even 2 1 inner 768.4.c.r 4
8.d odd 2 1 inner 768.4.c.r 4
12.b even 2 1 inner 768.4.c.r 4
16.e even 4 2 192.4.f.a 4
16.f odd 4 2 192.4.f.a 4
24.f even 2 1 inner 768.4.c.r 4
24.h odd 2 1 CM 768.4.c.r 4
48.i odd 4 2 192.4.f.a 4
48.k even 4 2 192.4.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.4.f.a 4 16.e even 4 2
192.4.f.a 4 16.f odd 4 2
192.4.f.a 4 48.i odd 4 2
192.4.f.a 4 48.k even 4 2
768.4.c.r 4 1.a even 1 1 trivial
768.4.c.r 4 3.b odd 2 1 inner
768.4.c.r 4 4.b odd 2 1 inner
768.4.c.r 4 8.b even 2 1 inner
768.4.c.r 4 8.d odd 2 1 inner
768.4.c.r 4 12.b even 2 1 inner
768.4.c.r 4 24.f even 2 1 inner
768.4.c.r 4 24.h odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} + 108 \) Copy content Toggle raw display
\( T_{7}^{2} + 1156 \) Copy content Toggle raw display
\( T_{11}^{2} - 5292 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1156)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 5292)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 47628)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4900)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 259308)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 514188)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T + 322)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 1876900)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 780300)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T + 574)^{4} \) Copy content Toggle raw display
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