Properties

Label 48.19.e.b
Level $48$
Weight $19$
Character orbit 48.e
Analytic conductor $98.585$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,19,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, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.17");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 19 \)
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: \(98.5853461007\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.601940665.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 123x^{2} - 1744x + 16016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{11} \)
Twist minimal: no (minimal twist has level 3)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 - 3969) q^{3} + ( - 9 \beta_{3} - 61 \beta_{2} - 45 \beta_1) q^{5} + (287 \beta_{3} - 287 \beta_1 + 23936038) q^{7} + (567 \beta_{3} + 14499 \beta_{2} + \cdots - 221335335) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 - 3969) q^{3} + ( - 9 \beta_{3} - 61 \beta_{2} - 45 \beta_1) q^{5} + (287 \beta_{3} - 287 \beta_1 + 23936038) q^{7} + (567 \beta_{3} + 14499 \beta_{2} + \cdots - 221335335) q^{9}+ \cdots + ( - 8675986498263 \beta_{3} + \cdots - 75\!\cdots\!40) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 15876 q^{3} + 95744152 q^{7} - 885341340 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 15876 q^{3} + 95744152 q^{7} - 885341340 q^{9} - 5426221528 q^{13} + 68287821120 q^{15} - 191416649480 q^{19} - 843499414296 q^{21} + 11407599454180 q^{25} + 4632207691356 q^{27} - 35728415085608 q^{31} + 12242871023040 q^{33} - 475299833502232 q^{37} - 416909545005096 q^{39} - 15\!\cdots\!92 q^{43}+ \cdots - 30\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 123x^{2} - 1744x + 16016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -1062\nu^{3} - 21006\nu^{2} - 259830\nu + 69732 ) / 169 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 576\nu^{3} - 3456\nu^{2} + 185472\nu - 958464 ) / 169 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4950\nu^{3} + 2322\nu^{2} - 197622\nu + 6210828 ) / 169 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 4\beta_{3} + 27\beta_{2} - 4\beta _1 + 7776 ) / 31104 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{3} - 91\beta_{2} - 68\beta _1 - 635040 ) / 10368 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -608\beta_{3} - 603\beta_{2} + 32\beta _1 + 18911232 ) / 15552 \) 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
6.57949 + 5.90892i
6.57949 5.90892i
−6.07949 12.9551i
−6.07949 + 12.9551i
0 −12172.0 15468.1i 0 787628.i 0 3.80616e7 0 −9.11041e7 + 3.76556e8i 0
17.2 0 −12172.0 + 15468.1i 0 787628.i 0 3.80616e7 0 −9.11041e7 3.76556e8i 0
17.3 0 4234.02 19222.2i 0 1.14247e6i 0 9.81043e6 0 −3.51567e8 1.62775e8i 0
17.4 0 4234.02 + 19222.2i 0 1.14247e6i 0 9.81043e6 0 −3.51567e8 + 1.62775e8i 0
\(n\): e.g. 2-40 or 990-1000
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 48.19.e.b 4
3.b odd 2 1 inner 48.19.e.b 4
4.b odd 2 1 3.19.b.b 4
12.b even 2 1 3.19.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.19.b.b 4 4.b odd 2 1
3.19.b.b 4 12.b even 2 1
48.19.e.b 4 1.a even 1 1 trivial
48.19.e.b 4 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 1925594804160T_{5}^{2} + 809714226235114137600000 \) acting on \(S_{19}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} + \cdots + 373401093446500)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots - 17\!\cdots\!40)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 38\!\cdots\!40 \) Copy content Toggle raw display
$19$ \( (T^{2} + \cdots - 79\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 12\!\cdots\!60 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots - 34\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots + 14\!\cdots\!40)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots - 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 66\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 14\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 84\!\cdots\!80)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 64\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 24\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 74\!\cdots\!20)^{2} \) Copy content Toggle raw display
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