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Properties

Label 768.2.j.f
Level $768$
Weight $2$
Character orbit 768.j
Analytic conductor $6.133$
Analytic rank $0$
Dimension $8$
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(193,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.j (of order \(4\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{6} - \beta_{2} + 1) q^{5} + \beta_{7} q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{6} - \beta_{2} + 1) q^{5} + \beta_{7} q^{7} + \beta_{2} q^{9} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3}) q^{11} + (3 \beta_{2} + 3) q^{13} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{15} + (\beta_{6} + \beta_{4}) q^{17} + (\beta_{7} + \beta_{5} + 4 \beta_1) q^{19} + \beta_{6} q^{21} + (2 \beta_{3} - 2 \beta_1) q^{23} + (2 \beta_{6} - 2 \beta_{4} - 3 \beta_{2}) q^{25} + \beta_{3} q^{27} + ( - 3 \beta_{4} - \beta_{2} - 1) q^{29} - 3 \beta_{5} q^{31} + ( - \beta_{6} - \beta_{4} + 2) q^{33} + (\beta_{7} + \beta_{5} + 6 \beta_1) q^{35} + (2 \beta_{6} + 3 \beta_{2} - 3) q^{37} + (3 \beta_{3} - 3 \beta_1) q^{39} + (\beta_{6} - \beta_{4} + 8 \beta_{2}) q^{41} + ( - \beta_{7} + \beta_{5}) q^{43} + (\beta_{4} + \beta_{2} + 1) q^{45} + ( - 2 \beta_{3} - 2 \beta_1) q^{47} + q^{49} + ( - \beta_{7} - \beta_{5}) q^{51} + ( - \beta_{6} - 5 \beta_{2} + 5) q^{53} + (4 \beta_{3} - 4 \beta_1) q^{55} + (\beta_{6} - \beta_{4} - 4 \beta_{2}) q^{57} + ( - 4 \beta_{7} + 4 \beta_{5}) q^{59} + (2 \beta_{4} - 3 \beta_{2} - 3) q^{61} - \beta_{5} q^{63} + (3 \beta_{6} + 3 \beta_{4} + 6) q^{65} + ( - 2 \beta_{7} - 2 \beta_{5} - 8 \beta_1) q^{67} + (2 \beta_{2} - 2) q^{69} + ( - 2 \beta_{7} - 8 \beta_{3} + 8 \beta_1) q^{71} + 4 \beta_{2} q^{73} + (2 \beta_{7} - 2 \beta_{5} - 3 \beta_{3}) q^{75} + ( - 2 \beta_{4} + 6 \beta_{2} + 6) q^{77} - \beta_{5} q^{79} - q^{81} + (\beta_{7} + \beta_{5} + 2 \beta_1) q^{83} + (2 \beta_{6} - 6 \beta_{2} + 6) q^{85} + (3 \beta_{7} - \beta_{3} + \beta_1) q^{87} + ( - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{2}) q^{89} + (3 \beta_{7} - 3 \beta_{5}) q^{91} + 3 \beta_{4} q^{93} + (6 \beta_{5} + 10 \beta_{3} + 10 \beta_1) q^{95} + ( - 2 \beta_{6} - 2 \beta_{4} + 8) q^{97} + (\beta_{7} + \beta_{5} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} + 24 q^{13} - 8 q^{29} + 16 q^{33} - 24 q^{37} + 8 q^{45} + 8 q^{49} + 40 q^{53} - 24 q^{61} + 48 q^{65} - 16 q^{69} + 48 q^{77} - 8 q^{81} + 48 q^{85} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

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

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{5} + 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + \beta_{4} + 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( -\beta_{6} + \beta_{4} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} + \beta_{5} - 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} - \beta_{5} + 2\beta_1 ) / 4 \) 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 \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
−0.258819 + 0.965926i
0.965926 − 0.258819i
0.258819 − 0.965926i
−0.965926 + 0.258819i
−0.258819 − 0.965926i
0.965926 + 0.258819i
0.258819 + 0.965926i
−0.965926 − 0.258819i
0 −0.707107 + 0.707107i 0 −0.732051 − 0.732051i 0 2.44949i 0 − 1.00000i 0
193.2 0 −0.707107 + 0.707107i 0 2.73205 + 2.73205i 0 − 2.44949i 0 − 1.00000i 0
193.3 0 0.707107 − 0.707107i 0 −0.732051 − 0.732051i 0 − 2.44949i 0 − 1.00000i 0
193.4 0 0.707107 − 0.707107i 0 2.73205 + 2.73205i 0 2.44949i 0 − 1.00000i 0
577.1 0 −0.707107 − 0.707107i 0 −0.732051 + 0.732051i 0 − 2.44949i 0 1.00000i 0
577.2 0 −0.707107 − 0.707107i 0 2.73205 − 2.73205i 0 2.44949i 0 1.00000i 0
577.3 0 0.707107 + 0.707107i 0 −0.732051 + 0.732051i 0 2.44949i 0 1.00000i 0
577.4 0 0.707107 + 0.707107i 0 2.73205 − 2.73205i 0 − 2.44949i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.j.f yes 8
3.b odd 2 1 2304.2.k.e 8
4.b odd 2 1 inner 768.2.j.f yes 8
8.b even 2 1 768.2.j.e ✓ 8
8.d odd 2 1 768.2.j.e ✓ 8
12.b even 2 1 2304.2.k.e 8
16.e even 4 1 768.2.j.e ✓ 8
16.e even 4 1 inner 768.2.j.f yes 8
16.f odd 4 1 768.2.j.e ✓ 8
16.f odd 4 1 inner 768.2.j.f yes 8
24.f even 2 1 2304.2.k.l 8
24.h odd 2 1 2304.2.k.l 8
32.g even 8 1 3072.2.a.l 4
32.g even 8 1 3072.2.a.r 4
32.g even 8 2 3072.2.d.h 8
32.h odd 8 1 3072.2.a.l 4
32.h odd 8 1 3072.2.a.r 4
32.h odd 8 2 3072.2.d.h 8
48.i odd 4 1 2304.2.k.e 8
48.i odd 4 1 2304.2.k.l 8
48.k even 4 1 2304.2.k.e 8
48.k even 4 1 2304.2.k.l 8
96.o even 8 1 9216.2.a.bc 4
96.o even 8 1 9216.2.a.bi 4
96.p odd 8 1 9216.2.a.bc 4
96.p odd 8 1 9216.2.a.bi 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.e ✓ 8 8.b even 2 1
768.2.j.e ✓ 8 8.d odd 2 1
768.2.j.e ✓ 8 16.e even 4 1
768.2.j.e ✓ 8 16.f odd 4 1
768.2.j.f yes 8 1.a even 1 1 trivial
768.2.j.f yes 8 4.b odd 2 1 inner
768.2.j.f yes 8 16.e even 4 1 inner
768.2.j.f yes 8 16.f odd 4 1 inner
2304.2.k.e 8 3.b odd 2 1
2304.2.k.e 8 12.b even 2 1
2304.2.k.e 8 48.i odd 4 1
2304.2.k.e 8 48.k even 4 1
2304.2.k.l 8 24.f even 2 1
2304.2.k.l 8 24.h odd 2 1
2304.2.k.l 8 48.i odd 4 1
2304.2.k.l 8 48.k even 4 1
3072.2.a.l 4 32.g even 8 1
3072.2.a.l 4 32.h odd 8 1
3072.2.a.r 4 32.g even 8 1
3072.2.a.r 4 32.h odd 8 1
3072.2.d.h 8 32.g even 8 2
3072.2.d.h 8 32.h odd 8 2
9216.2.a.bc 4 96.o even 8 1
9216.2.a.bc 4 96.p odd 8 1
9216.2.a.bi 4 96.o even 8 1
9216.2.a.bi 4 96.p odd 8 1

Hecke kernels

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

\( T_{5}^{4} - 4T_{5}^{3} + 8T_{5}^{2} + 16T_{5} + 16 \) Copy content Toggle raw display
\( T_{13}^{2} - 6T_{13} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 896T^{4} + 4096 \) Copy content Toggle raw display
$13$ \( (T^{2} - 6 T + 18)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} + 3104T^{4} + 256 \) Copy content Toggle raw display
$23$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 54)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 12 T^{3} + \cdots + 36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 152 T^{2} + 2704)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 20 T^{3} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 36864)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 12 T^{3} + \cdots + 36)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 49664 T^{4} + 65536 \) Copy content Toggle raw display
$71$ \( (T^{4} + 304 T^{2} + 10816)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 6)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 896T^{4} + 4096 \) Copy content Toggle raw display
$89$ \( (T^{4} + 104 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T + 16)^{4} \) Copy content Toggle raw display
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