Properties

Label 768.2.f.g
Level $768$
Weight $2$
Character orbit 768.f
Analytic conductor $6.133$
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] = [768,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.383");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 96)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} - \beta_{3} q^{5} - \beta_1 q^{7} + (2 \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} - \beta_{3} q^{5} - \beta_1 q^{7} + (2 \beta_{2} - 1) q^{9} + 2 \beta_{2} q^{11} + \beta_1 q^{13} + ( - \beta_{3} - 2 \beta_1) q^{15} - 6 q^{19} + (\beta_{3} - \beta_1) q^{21} - 2 \beta_{3} q^{23} + 3 q^{25} + (\beta_{2} - 5) q^{27} - \beta_{3} q^{29} + \beta_1 q^{31} + (2 \beta_{2} - 4) q^{33} + 4 \beta_{2} q^{35} + 3 \beta_1 q^{37} + ( - \beta_{3} + \beta_1) q^{39} - 4 \beta_{2} q^{41} - 2 q^{43} + (\beta_{3} - 4 \beta_1) q^{45} - 4 \beta_{3} q^{47} + 3 q^{49} + 3 \beta_{3} q^{53} - 4 \beta_1 q^{55} + ( - 6 \beta_{2} - 6) q^{57} + 2 \beta_{2} q^{59} + \beta_1 q^{61} + (2 \beta_{3} + \beta_1) q^{63} - 4 \beta_{2} q^{65} + 2 q^{67} + ( - 2 \beta_{3} - 4 \beta_1) q^{69} + 2 \beta_{3} q^{71} + 6 q^{73} + (3 \beta_{2} + 3) q^{75} + 2 \beta_{3} q^{77} - 7 \beta_1 q^{79} + ( - 4 \beta_{2} - 7) q^{81} + 2 \beta_{2} q^{83} + ( - \beta_{3} - 2 \beta_1) q^{87} + 12 \beta_{2} q^{89} + 4 q^{91} + ( - \beta_{3} + \beta_1) q^{93} + 6 \beta_{3} q^{95} + 10 q^{97} + ( - 2 \beta_{2} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{9} - 24 q^{19} + 12 q^{25} - 20 q^{27} - 16 q^{33} - 8 q^{43} + 12 q^{49} - 24 q^{57} + 8 q^{67} + 24 q^{73} + 12 q^{75} - 28 q^{81} + 16 q^{91} + 40 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0 1.00000 1.41421i 0 −2.82843 0 2.00000i 0 −1.00000 2.82843i 0
383.2 0 1.00000 1.41421i 0 2.82843 0 2.00000i 0 −1.00000 2.82843i 0
383.3 0 1.00000 + 1.41421i 0 −2.82843 0 2.00000i 0 −1.00000 + 2.82843i 0
383.4 0 1.00000 + 1.41421i 0 2.82843 0 2.00000i 0 −1.00000 + 2.82843i 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
3.b odd 2 1 inner
8.d odd 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 768.2.f.g 4
3.b odd 2 1 inner 768.2.f.g 4
4.b odd 2 1 768.2.f.a 4
8.b even 2 1 768.2.f.a 4
8.d odd 2 1 inner 768.2.f.g 4
12.b even 2 1 768.2.f.a 4
16.e even 4 1 96.2.c.a 4
16.e even 4 1 192.2.c.b 4
16.f odd 4 1 96.2.c.a 4
16.f odd 4 1 192.2.c.b 4
24.f even 2 1 inner 768.2.f.g 4
24.h odd 2 1 768.2.f.a 4
48.i odd 4 1 96.2.c.a 4
48.i odd 4 1 192.2.c.b 4
48.k even 4 1 96.2.c.a 4
48.k even 4 1 192.2.c.b 4
80.i odd 4 1 2400.2.o.a 4
80.j even 4 1 2400.2.o.a 4
80.k odd 4 1 2400.2.h.c 4
80.q even 4 1 2400.2.h.c 4
80.s even 4 1 2400.2.o.h 4
80.t odd 4 1 2400.2.o.h 4
144.u even 12 2 2592.2.s.e 8
144.v odd 12 2 2592.2.s.e 8
144.w odd 12 2 2592.2.s.e 8
144.x even 12 2 2592.2.s.e 8
240.t even 4 1 2400.2.h.c 4
240.z odd 4 1 2400.2.o.h 4
240.bb even 4 1 2400.2.o.a 4
240.bd odd 4 1 2400.2.o.a 4
240.bf even 4 1 2400.2.o.h 4
240.bm odd 4 1 2400.2.h.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.c.a 4 16.e even 4 1
96.2.c.a 4 16.f odd 4 1
96.2.c.a 4 48.i odd 4 1
96.2.c.a 4 48.k even 4 1
192.2.c.b 4 16.e even 4 1
192.2.c.b 4 16.f odd 4 1
192.2.c.b 4 48.i odd 4 1
192.2.c.b 4 48.k even 4 1
768.2.f.a 4 4.b odd 2 1
768.2.f.a 4 8.b even 2 1
768.2.f.a 4 12.b even 2 1
768.2.f.a 4 24.h odd 2 1
768.2.f.g 4 1.a even 1 1 trivial
768.2.f.g 4 3.b odd 2 1 inner
768.2.f.g 4 8.d odd 2 1 inner
768.2.f.g 4 24.f even 2 1 inner
2400.2.h.c 4 80.k odd 4 1
2400.2.h.c 4 80.q even 4 1
2400.2.h.c 4 240.t even 4 1
2400.2.h.c 4 240.bm odd 4 1
2400.2.o.a 4 80.i odd 4 1
2400.2.o.a 4 80.j even 4 1
2400.2.o.a 4 240.bb even 4 1
2400.2.o.a 4 240.bd odd 4 1
2400.2.o.h 4 80.s even 4 1
2400.2.o.h 4 80.t odd 4 1
2400.2.o.h 4 240.z odd 4 1
2400.2.o.h 4 240.bf even 4 1
2592.2.s.e 8 144.u even 12 2
2592.2.s.e 8 144.v odd 12 2
2592.2.s.e 8 144.w odd 12 2
2592.2.s.e 8 144.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} - 8 \) Copy content Toggle raw display
\( T_{19} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T + 6)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T + 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$67$ \( (T - 2)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$73$ \( (T - 6)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 288)^{2} \) Copy content Toggle raw display
$97$ \( (T - 10)^{4} \) Copy content Toggle raw display
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