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Properties

Label 768.3.g.h
Level $768$
Weight $3$
Character orbit 768.g
Analytic conductor $20.926$
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] = [768,3,Mod(511,768)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.511");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.g (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: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_1 q^{5} - \beta_{7} q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_1 q^{5} - \beta_{7} q^{7} - 3 q^{9} - \beta_{6} q^{11} + (\beta_{4} + \beta_1) q^{13} - \beta_{3} q^{15} + ( - \beta_{5} - 2) q^{17} + ( - \beta_{6} + 4 \beta_{2}) q^{19} + \beta_{4} q^{21} + 2 \beta_{7} q^{23} + ( - 2 \beta_{5} + 11) q^{25} + 3 \beta_{2} q^{27} + ( - 2 \beta_{4} + \beta_1) q^{29} + ( - \beta_{7} + 4 \beta_{3}) q^{31} - \beta_{5} q^{33} + ( - 3 \beta_{6} - 4 \beta_{2}) q^{35} + ( - \beta_{4} - 3 \beta_1) q^{37} + (3 \beta_{7} - \beta_{3}) q^{39} + ( - 3 \beta_{5} - 10) q^{41} + (\beta_{6} - 12 \beta_{2}) q^{43} - 3 \beta_1 q^{45} + (2 \beta_{7} - 4 \beta_{3}) q^{47} + ( - 4 \beta_{5} - 11) q^{49} + (3 \beta_{6} + 2 \beta_{2}) q^{51} + ( - 2 \beta_{4} - 3 \beta_1) q^{53} + ( - 4 \beta_{7} + 4 \beta_{3}) q^{55} + ( - \beta_{5} + 12) q^{57} + ( - 4 \beta_{6} + 12 \beta_{2}) q^{59} + ( - \beta_{4} - 7 \beta_1) q^{61} + 3 \beta_{7} q^{63} + ( - 5 \beta_{5} + 24) q^{65} + (8 \beta_{6} - 12 \beta_{2}) q^{67} - 2 \beta_{4} q^{69} + ( - 6 \beta_{7} - 4 \beta_{3}) q^{71} + ( - 4 \beta_{5} - 50) q^{73} + (6 \beta_{6} - 11 \beta_{2}) q^{75} + (4 \beta_{4} - 4 \beta_1) q^{77} + ( - 5 \beta_{7} - 4 \beta_{3}) q^{79} + 9 q^{81} + ( - 5 \beta_{6} + 16 \beta_{2}) q^{83} + (4 \beta_{4} + 10 \beta_1) q^{85} + ( - 6 \beta_{7} - \beta_{3}) q^{87} + ( - 6 \beta_{5} + 50) q^{89} + (9 \beta_{6} + 56 \beta_{2}) q^{91} + (\beta_{4} + 12 \beta_1) q^{93} + ( - 4 \beta_{7} + 8 \beta_{3}) q^{95} + ( - 6 \beta_{5} + 14) q^{97} + 3 \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{9} - 16 q^{17} + 88 q^{25} - 80 q^{41} - 88 q^{49} + 96 q^{57} + 192 q^{65} - 400 q^{73} + 72 q^{81} + 400 q^{89} + 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{7} + 2\nu^{6} + \nu^{5} - 4\nu^{4} + \nu^{3} + 6\nu^{2} - 10\nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 30 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 5\nu^{7} - 11\nu^{6} + 3\nu^{5} + 19\nu^{4} - 19\nu^{3} - 27\nu^{2} + 58\nu - 34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -5\nu^{7} + 15\nu^{6} - 9\nu^{5} - 19\nu^{4} + 33\nu^{3} + 15\nu^{2} - 76\nu + 66 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -6\nu^{7} + 18\nu^{6} - 10\nu^{5} - 26\nu^{4} + 42\nu^{3} + 26\nu^{2} - 108\nu + 80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -8\nu^{7} + 22\nu^{6} - 12\nu^{5} - 34\nu^{4} + 48\nu^{3} + 30\nu^{2} - 124\nu + 96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 10\nu^{7} - 26\nu^{6} + 14\nu^{5} + 38\nu^{4} - 54\nu^{3} - 38\nu^{2} + 148\nu - 114 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} - 2\beta_{3} - 2\beta _1 + 16 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} - 2\beta_{3} - 4\beta_{2} - 2\beta _1 + 12 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{7} - \beta_{6} + 3\beta_{5} + 2\beta_{4} - 2\beta_{3} + 24\beta_{2} - 6\beta _1 - 8 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} - 3\beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{2} - 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{7} - 2\beta_{6} + 3\beta_{5} + 2\beta_{4} + 2\beta_{3} + 20\beta_{2} + 8\beta _1 + 28 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{6} + 4\beta_{4} - 12\beta_{2} + 6\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10\beta_{7} + 13\beta_{6} + 15\beta_{5} + 2\beta_{4} - 2\beta_{3} - 112\beta_{2} + 22\beta _1 + 128 ) / 32 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
511.1
1.20036 + 0.747754i
1.40994 − 0.109843i
−1.27597 + 0.609843i
0.665665 − 1.24775i
1.20036 − 0.747754i
1.40994 + 0.109843i
−1.27597 − 0.609843i
0.665665 + 1.24775i
0 − 1.73205i 0 −7.98203 0 − 2.13878i 0 −3.00000 0
511.2 0 − 1.73205i 0 −2.87875 0 10.7436i 0 −3.00000 0
511.3 0 − 1.73205i 0 2.87875 0 − 10.7436i 0 −3.00000 0
511.4 0 − 1.73205i 0 7.98203 0 2.13878i 0 −3.00000 0
511.5 0 1.73205i 0 −7.98203 0 2.13878i 0 −3.00000 0
511.6 0 1.73205i 0 −2.87875 0 − 10.7436i 0 −3.00000 0
511.7 0 1.73205i 0 2.87875 0 10.7436i 0 −3.00000 0
511.8 0 1.73205i 0 7.98203 0 − 2.13878i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 511.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d 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.g.h 8
3.b odd 2 1 2304.3.g.z 8
4.b odd 2 1 inner 768.3.g.h 8
8.b even 2 1 inner 768.3.g.h 8
8.d odd 2 1 inner 768.3.g.h 8
12.b even 2 1 2304.3.g.z 8
16.e even 4 1 24.3.b.a ✓ 4
16.e even 4 1 96.3.b.a 4
16.f odd 4 1 24.3.b.a ✓ 4
16.f odd 4 1 96.3.b.a 4
24.f even 2 1 2304.3.g.z 8
24.h odd 2 1 2304.3.g.z 8
48.i odd 4 1 72.3.b.b 4
48.i odd 4 1 288.3.b.b 4
48.k even 4 1 72.3.b.b 4
48.k even 4 1 288.3.b.b 4
80.i odd 4 1 600.3.p.a 8
80.i odd 4 1 2400.3.p.a 8
80.j even 4 1 600.3.p.a 8
80.j even 4 1 2400.3.p.a 8
80.k odd 4 1 600.3.g.a 4
80.k odd 4 1 2400.3.g.a 4
80.q even 4 1 600.3.g.a 4
80.q even 4 1 2400.3.g.a 4
80.s even 4 1 600.3.p.a 8
80.s even 4 1 2400.3.p.a 8
80.t odd 4 1 600.3.p.a 8
80.t odd 4 1 2400.3.p.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.b.a ✓ 4 16.e even 4 1
24.3.b.a ✓ 4 16.f odd 4 1
72.3.b.b 4 48.i odd 4 1
72.3.b.b 4 48.k even 4 1
96.3.b.a 4 16.e even 4 1
96.3.b.a 4 16.f odd 4 1
288.3.b.b 4 48.i odd 4 1
288.3.b.b 4 48.k even 4 1
600.3.g.a 4 80.k odd 4 1
600.3.g.a 4 80.q even 4 1
600.3.p.a 8 80.i odd 4 1
600.3.p.a 8 80.j even 4 1
600.3.p.a 8 80.s even 4 1
600.3.p.a 8 80.t odd 4 1
768.3.g.h 8 1.a even 1 1 trivial
768.3.g.h 8 4.b odd 2 1 inner
768.3.g.h 8 8.b even 2 1 inner
768.3.g.h 8 8.d odd 2 1 inner
2304.3.g.z 8 3.b odd 2 1
2304.3.g.z 8 12.b even 2 1
2304.3.g.z 8 24.f even 2 1
2304.3.g.z 8 24.h odd 2 1
2400.3.g.a 4 80.k odd 4 1
2400.3.g.a 4 80.q even 4 1
2400.3.p.a 8 80.i odd 4 1
2400.3.p.a 8 80.j even 4 1
2400.3.p.a 8 80.s even 4 1
2400.3.p.a 8 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{4} - 72T_{5}^{2} + 528 \) Copy content Toggle raw display
\( T_{7}^{4} + 120T_{7}^{2} + 528 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 72 T^{2} + 528)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 120 T^{2} + 528)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 384 T^{2} + 33792)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 188)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 224 T^{2} + 256)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 480 T^{2} + 8448)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 1608 T^{2} + 528)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 3384 T^{2} + 279312)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 864 T^{2} + 76032)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 20 T - 1628)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 992 T^{2} + 135424)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 3552 T^{2} + 8448)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 1800 T^{2} + 803088)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 2912 T^{2} + 350464)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 3552 T^{2} + 8448)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 9056 T^{2} + 13424896)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 8928 T^{2} + 12849408)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 100 T - 572)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 7416 T^{2} + 10797072)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 4736 T^{2} + 692224)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 100 T - 4412)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 28 T - 6716)^{4} \) Copy content Toggle raw display
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