Properties

Label 768.3.h.f
Level $768$
Weight $3$
Character orbit 768.h
Analytic conductor $20.926$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,3,Mod(641,768)]
 
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([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.641");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.h (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: \(20.9264843029\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{4} q^{5} + (\beta_{6} + \beta_{5}) q^{7} + (3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{4} q^{5} + (\beta_{6} + \beta_{5}) q^{7} + (3 \beta_{2} + 3) q^{9} + ( - \beta_{7} - 4 \beta_{3}) q^{11} - 3 \beta_1 q^{13} - \beta_{5} q^{15} + 8 \beta_{2} q^{17} + (\beta_{7} - 2 \beta_{3}) q^{19} + (9 \beta_{4} - 9 \beta_1) q^{21} + ( - 4 \beta_{6} + 2 \beta_{5}) q^{23} - 17 q^{25} + ( - 3 \beta_{7} + 3 \beta_{3}) q^{27} + 11 \beta_{4} q^{29} + (3 \beta_{6} + 3 \beta_{5}) q^{31} + ( - 9 \beta_{2} - 36) q^{33} + ( - 2 \beta_{7} - 8 \beta_{3}) q^{35} + 19 \beta_1 q^{37} - 3 \beta_{6} q^{39} + 2 \beta_{2} q^{41} + ( - \beta_{7} + 2 \beta_{3}) q^{43} + ( - 3 \beta_{4} + 12 \beta_1) q^{45} + ( - 8 \beta_{6} + 4 \beta_{5}) q^{47} + 59 q^{49} - 8 \beta_{7} q^{51} - 5 \beta_{4} q^{53} + (4 \beta_{6} + 4 \beta_{5}) q^{55} + ( - 9 \beta_{2} + 18) q^{57} + ( - \beta_{7} - 4 \beta_{3}) q^{59} - 11 \beta_1 q^{61} + ( - 9 \beta_{6} + 9 \beta_{5}) q^{63} - 6 \beta_{2} q^{65} + ( - 11 \beta_{7} + 22 \beta_{3}) q^{67} + ( - 18 \beta_{4} - 36 \beta_1) q^{69} + (4 \beta_{6} - 2 \beta_{5}) q^{71} + 30 q^{73} - 17 \beta_{3} q^{75} - 54 \beta_{4} q^{77} + (3 \beta_{6} + 3 \beta_{5}) q^{79} + (18 \beta_{2} - 63) q^{81} + ( - 5 \beta_{7} - 20 \beta_{3}) q^{83} + 32 \beta_1 q^{85} + 11 \beta_{5} q^{87} + 2 \beta_{2} q^{89} + ( - 6 \beta_{7} + 12 \beta_{3}) q^{91} + (27 \beta_{4} - 27 \beta_1) q^{93} + ( - 4 \beta_{6} + 2 \beta_{5}) q^{95} + 90 q^{97} + (9 \beta_{7} - 36 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} - 136 q^{25} - 288 q^{33} + 472 q^{49} + 144 q^{57} + 240 q^{73} - 504 q^{81} + 720 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} - 2\zeta_{24}^{4} + \zeta_{24}^{3} + \zeta_{24} + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -4\zeta_{24}^{7} - 4\zeta_{24}^{6} - 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 8\zeta_{24}^{2} - 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 4\zeta_{24}^{7} - 2\zeta_{24}^{6} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 4\zeta_{24}^{2} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -4\zeta_{24}^{7} + 2\zeta_{24}^{5} + 8\zeta_{24}^{4} + 2\zeta_{24}^{3} + 2\zeta_{24} - 4 \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - \beta_{5} + 3\beta_{4} + 4\beta_{3} + 3\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} + 3\beta_1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{4} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{7} - 2\beta_{3} + 6 ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - \beta_{5} - 3\beta_{4} + 4\beta_{3} - 3\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} - \beta_{5} + 3\beta_{4} - 4\beta_{3} - 3\beta_{2} ) / 24 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
−0.258819 + 0.965926i
−0.965926 0.258819i
−0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
0 −2.44949 1.73205i 0 −2.82843 0 −10.3923 0 3.00000 + 8.48528i 0
641.2 0 −2.44949 1.73205i 0 2.82843 0 10.3923 0 3.00000 + 8.48528i 0
641.3 0 −2.44949 + 1.73205i 0 −2.82843 0 −10.3923 0 3.00000 8.48528i 0
641.4 0 −2.44949 + 1.73205i 0 2.82843 0 10.3923 0 3.00000 8.48528i 0
641.5 0 2.44949 1.73205i 0 −2.82843 0 10.3923 0 3.00000 8.48528i 0
641.6 0 2.44949 1.73205i 0 2.82843 0 −10.3923 0 3.00000 8.48528i 0
641.7 0 2.44949 + 1.73205i 0 −2.82843 0 10.3923 0 3.00000 + 8.48528i 0
641.8 0 2.44949 + 1.73205i 0 2.82843 0 −10.3923 0 3.00000 + 8.48528i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 641.8
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 768.3.h.f 8
3.b odd 2 1 inner 768.3.h.f 8
4.b odd 2 1 inner 768.3.h.f 8
8.b even 2 1 inner 768.3.h.f 8
8.d odd 2 1 inner 768.3.h.f 8
12.b even 2 1 inner 768.3.h.f 8
16.e even 4 1 96.3.e.a 4
16.e even 4 1 192.3.e.e 4
16.f odd 4 1 96.3.e.a 4
16.f odd 4 1 192.3.e.e 4
24.f even 2 1 inner 768.3.h.f 8
24.h odd 2 1 inner 768.3.h.f 8
48.i odd 4 1 96.3.e.a 4
48.i odd 4 1 192.3.e.e 4
48.k even 4 1 96.3.e.a 4
48.k even 4 1 192.3.e.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.e.a 4 16.e even 4 1
96.3.e.a 4 16.f odd 4 1
96.3.e.a 4 48.i odd 4 1
96.3.e.a 4 48.k even 4 1
192.3.e.e 4 16.e even 4 1
192.3.e.e 4 16.f odd 4 1
192.3.e.e 4 48.i odd 4 1
192.3.e.e 4 48.k even 4 1
768.3.h.f 8 1.a even 1 1 trivial
768.3.h.f 8 3.b odd 2 1 inner
768.3.h.f 8 4.b odd 2 1 inner
768.3.h.f 8 8.b even 2 1 inner
768.3.h.f 8 8.d odd 2 1 inner
768.3.h.f 8 12.b even 2 1 inner
768.3.h.f 8 24.f even 2 1 inner
768.3.h.f 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 \(S_{3}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} - 8 \) Copy content Toggle raw display
\( T_{7}^{2} - 108 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 6 T^{2} + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 216)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 512)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 864)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 968)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 972)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1444)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 3456)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 200)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 216)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 484)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 13068)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 864)^{4} \) Copy content Toggle raw display
$73$ \( (T - 30)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 972)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 5400)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$97$ \( (T - 90)^{8} \) Copy content Toggle raw display
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