Properties

Label 48.15.e.b
Level $48$
Weight $15$
Character orbit 48.e
Analytic conductor $59.678$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,15,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, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.17");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 15 \)
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: \(59.6779047129\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.1929141760.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 364x^{2} + 3640 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{7} \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 549) q^{3} + (3 \beta_{3} - 8 \beta_{2} - 22 \beta_1) q^{5} + (11 \beta_{3} + 11 \beta_{2} + \cdots - 206402) q^{7}+ \cdots + ( - 189 \beta_{3} + 540 \beta_{2} + \cdots - 406215) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 549) q^{3} + (3 \beta_{3} - 8 \beta_{2} - 22 \beta_1) q^{5} + (11 \beta_{3} + 11 \beta_{2} + \cdots - 206402) q^{7}+ \cdots + ( - 1364449779 \beta_{3} + \cdots + 21526343879040) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2196 q^{3} - 825608 q^{7} - 1624860 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2196 q^{3} - 825608 q^{7} - 1624860 q^{9} + 201696872 q^{13} + 449729280 q^{15} - 1314947240 q^{19} + 1947743784 q^{21} - 6882482780 q^{25} - 20862898164 q^{27} + 34970710072 q^{31} - 59537237760 q^{33} + 55576789928 q^{37} - 18888686856 q^{39} + 323678929048 q^{43} + 1007022481920 q^{45} - 2246577120564 q^{49} - 2656416881664 q^{51} + 832322050560 q^{55} + 1073653456968 q^{57} - 4171641626392 q^{61} - 2946514688712 q^{63} + 10964239937752 q^{67} + 15227331485184 q^{69} - 44644130922808 q^{73} - 36265563003060 q^{75} - 41215442578760 q^{79} - 17388818777916 q^{81} + 45292279818240 q^{85} + 41651639527680 q^{87} - 23445744391888 q^{91} - 145717264072728 q^{93} + 70529980615688 q^{97} + 86105375516160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 364x^{2} + 3640 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{3} + 8\nu^{2} + 652\nu + 1456 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 8\nu^{2} + 1036\nu + 1456 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -16\nu^{3} + 80\nu^{2} - 5600\nu + 14560 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 384 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 7\beta _1 - 26208 ) / 144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -16\beta_{3} - 505\beta_{2} + 665\beta_1 ) / 576 \) 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
18.8072i
18.8072i
3.20795i
3.20795i
0 −1922.67 1042.26i 0 23159.5i 0 −478389. 0 2.61037e6 + 4.00784e6i 0
17.2 0 −1922.67 + 1042.26i 0 23159.5i 0 −478389. 0 2.61037e6 4.00784e6i 0
17.3 0 824.672 2025.56i 0 122931.i 0 65585.1 0 −3.42280e6 3.34084e6i 0
17.4 0 824.672 + 2025.56i 0 122931.i 0 65585.1 0 −3.42280e6 + 3.34084e6i 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 48.15.e.b 4
3.b odd 2 1 inner 48.15.e.b 4
4.b odd 2 1 3.15.b.a 4
12.b even 2 1 3.15.b.a 4
20.d odd 2 1 75.15.c.d 4
20.e even 4 2 75.15.d.b 8
60.h even 2 1 75.15.c.d 4
60.l odd 4 2 75.15.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.15.b.a 4 4.b odd 2 1
3.15.b.a 4 12.b even 2 1
48.15.e.b 4 1.a even 1 1 trivial
48.15.e.b 4 3.b odd 2 1 inner
75.15.c.d 4 20.d odd 2 1
75.15.c.d 4 60.h even 2 1
75.15.d.b 8 20.e even 4 2
75.15.d.b 8 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 15648272640T_{5}^{2} + 8105489557401600000 \) acting on \(S_{15}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + \cdots + 22876792454961 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} + 412804 T - 31375221500)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 43\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( (T^{2} + \cdots + 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 14\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots - 45\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 16\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 17\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 90\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots - 37\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 44\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
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