Properties

Label 48.27.e.b
Level $48$
Weight $27$
Character orbit 48.e
Analytic conductor $205.581$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,27,Mod(17,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 27, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.17");
 
S:= CuspForms(chi, 27);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 27 \)
Character orbit: \([\chi]\) \(=\) 48.e (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: \(205.580601950\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 11609466 x^{6} - 3416571600 x^{5} + 38618090622117 x^{4} + \cdots + 60\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{65}\cdot 3^{36}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 6)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 274365) q^{3} + (\beta_{7} + 717 \beta_{2} - 23 \beta_1) q^{5} + ( - 13 \beta_{7} + 11 \beta_{6} - 20 \beta_{5} + \cdots - 22901407970) q^{7}+ \cdots + ( - 1722 \beta_{7} + 291 \beta_{6} - 264 \beta_{5} + \cdots - 606741412887) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 274365) q^{3} + (\beta_{7} + 717 \beta_{2} - 23 \beta_1) q^{5} + ( - 13 \beta_{7} + 11 \beta_{6} - 20 \beta_{5} + \cdots - 22901407970) q^{7}+ \cdots + (22\!\cdots\!41 \beta_{7} + \cdots - 61\!\cdots\!92) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2194920 q^{3} - 183211263760 q^{7} - 4853931303096 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2194920 q^{3} - 183211263760 q^{7} - 4853931303096 q^{9} + 512429022548560 q^{13} + 10\!\cdots\!40 q^{15}+ \cdots - 49\!\cdots\!36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 11609466 x^{6} - 3416571600 x^{5} + 38618090622117 x^{4} + \cdots + 60\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 64\!\cdots\!10 \nu^{7} + \cdots + 19\!\cdots\!18 ) / 45\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 50\!\cdots\!76 \nu^{7} + \cdots - 37\!\cdots\!00 ) / 23\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 77\!\cdots\!60 \nu^{7} + \cdots + 20\!\cdots\!36 ) / 55\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 90\!\cdots\!38 \nu^{7} + \cdots + 45\!\cdots\!38 ) / 11\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22\!\cdots\!56 \nu^{7} + \cdots - 10\!\cdots\!64 ) / 18\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 81\!\cdots\!44 \nu^{7} + \cdots - 22\!\cdots\!12 ) / 61\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 38\!\cdots\!70 \nu^{7} + \cdots + 25\!\cdots\!22 ) / 11\!\cdots\!10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 744 \beta_{7} - 584 \beta_{6} + 1168 \beta_{5} + 720 \beta_{4} - 160 \beta_{3} + 429121 \beta_{2} + 1212840 \beta_1 ) / 2579890176 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3452 \beta_{7} + 1140 \beta_{6} + 11492 \beta_{5} - 10236 \beta_{4} + 16168 \beta_{3} + 1518627 \beta_{2} + 4357856 \beta _1 + 77997779380224 ) / 26873856 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2169744696 \beta_{7} - 1936273048 \beta_{6} + 3250800080 \beta_{5} + 1005364368 \beta_{4} - 847613600 \beta_{3} + 1330443105797 \beta_{2} + \cdots + 16\!\cdots\!00 ) / 1289945088 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 13139782376 \beta_{7} - 8217275736 \beta_{6} + 174895396352 \beta_{5} - 93210043200 \beta_{4} + 182095992448 \beta_{3} + \cdots + 77\!\cdots\!16 ) / 53747712 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 714812320650516 \beta_{7} - 652120843327168 \beta_{6} + \cdots + 99\!\cdots\!00 ) / 80621568 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 65\!\cdots\!88 \beta_{7} + \cdots + 28\!\cdots\!04 ) / 35831808 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 30\!\cdots\!72 \beta_{7} + \cdots + 60\!\cdots\!00 ) / 644972544 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−2107.01 1.41421i
−2107.01 + 1.41421i
530.624 + 1.41421i
530.624 1.41421i
−891.970 1.41421i
−891.970 + 1.41421i
2468.36 + 1.41421i
2468.36 1.41421i
0 −1.54011e6 412209.i 0 1.23273e9i 0 6.52628e10 0 2.20203e12 + 1.26970e12i 0
17.2 0 −1.54011e6 + 412209.i 0 1.23273e9i 0 6.52628e10 0 2.20203e12 1.26970e12i 0
17.3 0 −793293. 1.38295e6i 0 1.58663e9i 0 −2.87923e10 0 −1.28324e12 + 2.19417e12i 0
17.4 0 −793293. + 1.38295e6i 0 1.58663e9i 0 −2.87923e10 0 −1.28324e12 2.19417e12i 0
17.5 0 388622. 1.54623e6i 0 1.99247e8i 0 4.03334e9 0 −2.23981e12 1.20180e12i 0
17.6 0 388622. + 1.54623e6i 0 1.99247e8i 0 4.03334e9 0 −2.23981e12 + 1.20180e12i 0
17.7 0 847324. 1.35052e6i 0 2.16695e9i 0 −1.32110e11 0 −1.10595e12 2.28866e12i 0
17.8 0 847324. + 1.35052e6i 0 2.16695e9i 0 −1.32110e11 0 −1.10595e12 + 2.28866e12i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.27.e.b 8
3.b odd 2 1 inner 48.27.e.b 8
4.b odd 2 1 6.27.b.a 8
12.b even 2 1 6.27.b.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.27.b.a 8 4.b odd 2 1
6.27.b.a 8 12.b even 2 1
48.27.e.b 8 1.a even 1 1 trivial
48.27.e.b 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + \cdots + 71\!\cdots\!00 \) acting on \(S_{27}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2194920 T^{7} + \cdots + 41\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + 91605631880 T^{3} + \cdots + 10\!\cdots\!56)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 74\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{4} - 256214511274280 T^{3} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 37\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 42\!\cdots\!76)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 58\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 37\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 21\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 10\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 80\!\cdots\!56)^{2} \) Copy content Toggle raw display
show more
show less