Properties

Label 864.3.h.e
Level $864$
Weight $3$
Character orbit 864.h
Analytic conductor $23.542$
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] = [864,3,Mod(593,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, 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("864.593");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 864.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: \(23.5422948407\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.242095489024.11
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 2x^{4} - 32x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 216)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + (\beta_{7} - 6) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + (\beta_{7} - 6) q^{7} + \beta_{3} q^{11} + ( - \beta_{6} + \beta_1) q^{13} + ( - 3 \beta_{5} - \beta_{4}) q^{17} + (2 \beta_{6} + 5 \beta_1) q^{19} + (4 \beta_{5} - \beta_{4}) q^{23} + ( - 2 \beta_{7} - 7) q^{25} + ( - 10 \beta_{3} + 6 \beta_{2}) q^{29} + (6 \beta_{7} + 10) q^{31} + ( - 5 \beta_{3} + 10 \beta_{2}) q^{35} + ( - 7 \beta_{6} - 7 \beta_1) q^{37} + (8 \beta_{5} - 6 \beta_{4}) q^{41} + (10 \beta_{6} - 6 \beta_1) q^{43} + (2 \beta_{5} - 7 \beta_{4}) q^{47} + ( - 12 \beta_{7} + 18) q^{49} + (2 \beta_{3} + 12 \beta_{2}) q^{53} + ( - 2 \beta_{7} - 2) q^{55} + ( - 19 \beta_{3} + 8 \beta_{2}) q^{59} + ( - 3 \beta_{6} - 5 \beta_1) q^{61} + (7 \beta_{5} - \beta_{4}) q^{65} + ( - 12 \beta_{6} + 7 \beta_1) q^{67} + ( - 12 \beta_{5} - 6 \beta_{4}) q^{71} + (4 \beta_{7} + 31) q^{73} + ( - 2 \beta_{3} + 3 \beta_{2}) q^{77} + ( - 5 \beta_{7} - 24) q^{79} + ( - 10 \beta_{3} - 10 \beta_{2}) q^{83} + (14 \beta_{6} - 4 \beta_1) q^{85} + (3 \beta_{5} - 11 \beta_{4}) q^{89} + (10 \beta_{6} - 15 \beta_1) q^{91} - 5 \beta_{4} q^{95} + (14 \beta_{7} - 9) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 48 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 48 q^{7} - 56 q^{25} + 80 q^{31} + 144 q^{49} - 16 q^{55} + 248 q^{73} - 192 q^{79} - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 2x^{4} - 32x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 14\nu^{4} - 14\nu^{2} - 16 ) / 80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 6\nu^{5} - 26\nu^{3} + 136\nu ) / 80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 2\nu^{5} + 42\nu^{3} + 208\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 6\nu^{5} + 18\nu^{3} - 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 2\nu^{5} + 2\nu^{3} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 2\nu^{4} + 18\nu^{2} - 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + 2\nu^{4} + 14\nu^{2} + 16 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{4} + 3\beta_{3} - 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{5} + 6\beta_{4} - 3\beta_{3} + 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{7} + 7\beta_{6} + 10\beta _1 + 23 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 19\beta_{5} + 5\beta_{4} - 14\beta_{3} + \beta_{2} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
0.926315 + 1.77255i
0.926315 1.77255i
1.90839 + 0.598380i
1.90839 0.598380i
−1.90839 0.598380i
−1.90839 + 0.598380i
−0.926315 1.77255i
−0.926315 + 1.77255i
0 0 0 −5.39773 0 −11.5678 0 0 0
593.2 0 0 0 −5.39773 0 −11.5678 0 0 0
593.3 0 0 0 −2.62001 0 −0.432236 0 0 0
593.4 0 0 0 −2.62001 0 −0.432236 0 0 0
593.5 0 0 0 2.62001 0 −0.432236 0 0 0
593.6 0 0 0 2.62001 0 −0.432236 0 0 0
593.7 0 0 0 5.39773 0 −11.5678 0 0 0
593.8 0 0 0 5.39773 0 −11.5678 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b 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 864.3.h.e 8
3.b odd 2 1 inner 864.3.h.e 8
4.b odd 2 1 216.3.h.f 8
8.b even 2 1 inner 864.3.h.e 8
8.d odd 2 1 216.3.h.f 8
12.b even 2 1 216.3.h.f 8
24.f even 2 1 216.3.h.f 8
24.h odd 2 1 inner 864.3.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.h.f 8 4.b odd 2 1
216.3.h.f 8 8.d odd 2 1
216.3.h.f 8 12.b even 2 1
216.3.h.f 8 24.f even 2 1
864.3.h.e 8 1.a even 1 1 trivial
864.3.h.e 8 3.b odd 2 1 inner
864.3.h.e 8 8.b even 2 1 inner
864.3.h.e 8 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 36T_{5}^{2} + 200 \) acting on \(S_{3}^{\mathrm{new}}(864, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 36 T^{2} + 200)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 12 T + 5)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 28 T^{2} + 72)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 94 T^{2} + 225)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 764 T^{2} + 145800)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 586 T^{2} + 2025)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 932 T^{2} + 8712)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 3616 T^{2} + 2880000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 20 T - 1016)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3430 T^{2} + 1750329)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 7952 T^{2} + 15235200)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 7768 T^{2} + 5363856)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 7812 T^{2} + 5604552)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 5488 T^{2} + 6480000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 11196 T^{2} + 31331528)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 846 T^{2} + 18225)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 11106 T^{2} + 11390625)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 16272 T^{2} + 58320000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 62 T + 465)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 48 T - 199)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 7200 T^{2} + 8000000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 19292 T^{2} + 32805000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 18 T - 5995)^{4} \) Copy content Toggle raw display
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