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Properties

Label 768.3.h.e
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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Show commands: Magma / PariGP / SageMath

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: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{2}) q^{3} + \beta_{7} q^{5} - \beta_{3} q^{7} + (\beta_{6} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_{2}) q^{3} + \beta_{7} q^{5} - \beta_{3} q^{7} + (\beta_{6} - 5) q^{9} + 5 \beta_{4} q^{11} - 5 \beta_1 q^{13} + ( - 7 \beta_{5} + 2 \beta_{3}) q^{15} + (5 \beta_{4} - 10 \beta_{2}) q^{19} + ( - \beta_{7} - 7 \beta_1) q^{21} - 6 \beta_{5} q^{23} + 31 q^{25} + ( - 10 \beta_{4} + \beta_{2}) q^{27} + 5 \beta_{7} q^{29} + 5 \beta_{3} q^{31} + (5 \beta_{6} + 20) q^{33} - 14 \beta_{4} q^{35} + 5 \beta_1 q^{37} + (5 \beta_{5} + 5 \beta_{3}) q^{39} - 2 \beta_{6} q^{41} + (11 \beta_{4} - 22 \beta_{2}) q^{43} + ( - 5 \beta_{7} + 28 \beta_1) q^{45} + 4 \beta_{5} q^{47} - 21 q^{49} + 5 \beta_{7} q^{53} + 20 \beta_{3} q^{55} + (5 \beta_{6} - 70) q^{57} - 35 \beta_{4} q^{59} - 45 \beta_1 q^{61} + (14 \beta_{5} + 5 \beta_{3}) q^{63} - 10 \beta_{6} q^{65} + (\beta_{4} - 2 \beta_{2}) q^{67} + ( - 6 \beta_{7} + 12 \beta_1) q^{69} - 10 \beta_{5} q^{71} + 30 q^{73} + (31 \beta_{4} - 31 \beta_{2}) q^{75} - 10 \beta_{7} q^{77} + 5 \beta_{3} q^{79} + ( - 10 \beta_{6} - 31) q^{81} + 9 \beta_{4} q^{83} + ( - 35 \beta_{5} + 10 \beta_{3}) q^{87} - 10 \beta_{6} q^{89} + (10 \beta_{4} - 20 \beta_{2}) q^{91} + (5 \beta_{7} + 35 \beta_1) q^{93} - 70 \beta_{5} q^{95} + 10 q^{97} + ( - 5 \beta_{4} - 40 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{9} + 248 q^{25} + 160 q^{33} - 168 q^{49} - 560 q^{57} + 240 q^{73} - 248 q^{81} + 80 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 5\nu^{2} ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 4\nu^{5} - 16\nu^{4} + 7\nu^{3} + 4\nu - 8 ) / 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 3\nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 4\nu^{5} + 7\nu^{3} + 4\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 4\nu^{5} - 7\nu^{3} + 4\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} + 4\nu^{5} + 13\nu^{3} + 44\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 4\nu^{5} - 13\nu^{3} + 44\nu ) / 12 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + \beta_{6} - 5\beta_{5} + 5\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{4} - 6\beta_{2} - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} + \beta_{6} - 11\beta_{5} - 11\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{3} + 9\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} + 7\beta_{6} + 13\beta_{5} - 13\beta_{4} ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
−1.28897 + 0.581861i
0.581861 + 1.28897i
−1.28897 − 0.581861i
0.581861 − 1.28897i
−0.581861 − 1.28897i
1.28897 − 0.581861i
−0.581861 + 1.28897i
1.28897 + 0.581861i
0 −1.41421 − 2.64575i 0 −7.48331 0 −5.29150 0 −5.00000 + 7.48331i 0
641.2 0 −1.41421 − 2.64575i 0 7.48331 0 5.29150 0 −5.00000 + 7.48331i 0
641.3 0 −1.41421 + 2.64575i 0 −7.48331 0 −5.29150 0 −5.00000 − 7.48331i 0
641.4 0 −1.41421 + 2.64575i 0 7.48331 0 5.29150 0 −5.00000 − 7.48331i 0
641.5 0 1.41421 − 2.64575i 0 −7.48331 0 5.29150 0 −5.00000 − 7.48331i 0
641.6 0 1.41421 − 2.64575i 0 7.48331 0 −5.29150 0 −5.00000 − 7.48331i 0
641.7 0 1.41421 + 2.64575i 0 −7.48331 0 5.29150 0 −5.00000 + 7.48331i 0
641.8 0 1.41421 + 2.64575i 0 7.48331 0 −5.29150 0 −5.00000 + 7.48331i 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.e 8
3.b odd 2 1 inner 768.3.h.e 8
4.b odd 2 1 inner 768.3.h.e 8
8.b even 2 1 inner 768.3.h.e 8
8.d odd 2 1 inner 768.3.h.e 8
12.b even 2 1 inner 768.3.h.e 8
16.e even 4 1 96.3.e.b ✓ 4
16.e even 4 1 192.3.e.f 4
16.f odd 4 1 96.3.e.b ✓ 4
16.f odd 4 1 192.3.e.f 4
24.f even 2 1 inner 768.3.h.e 8
24.h odd 2 1 inner 768.3.h.e 8
48.i odd 4 1 96.3.e.b ✓ 4
48.i odd 4 1 192.3.e.f 4
48.k even 4 1 96.3.e.b ✓ 4
48.k even 4 1 192.3.e.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.e.b ✓ 4 16.e even 4 1
96.3.e.b ✓ 4 16.f odd 4 1
96.3.e.b ✓ 4 48.i odd 4 1
96.3.e.b ✓ 4 48.k even 4 1
192.3.e.f 4 16.e even 4 1
192.3.e.f 4 16.f odd 4 1
192.3.e.f 4 48.i odd 4 1
192.3.e.f 4 48.k even 4 1
768.3.h.e 8 1.a even 1 1 trivial
768.3.h.e 8 3.b odd 2 1 inner
768.3.h.e 8 4.b odd 2 1 inner
768.3.h.e 8 8.b even 2 1 inner
768.3.h.e 8 8.d odd 2 1 inner
768.3.h.e 8 12.b even 2 1 inner
768.3.h.e 8 24.f even 2 1 inner
768.3.h.e 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 \(S_{3}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} - 56 \) Copy content Toggle raw display
\( T_{7}^{2} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 10 T^{2} + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 56)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 200)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{2} + 700)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 288)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 1400)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 700)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 224)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 3388)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 128)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 1400)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 9800)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8100)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 800)^{4} \) Copy content Toggle raw display
$73$ \( (T - 30)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 700)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 648)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 5600)^{4} \) Copy content Toggle raw display
$97$ \( (T - 10)^{8} \) Copy content Toggle raw display
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