Properties

Label 48.3.e.a
Level $48$
Weight $3$
Character orbit 48.e
Self dual yes
Analytic conductor $1.308$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 48.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.30790526893\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} - 2q^{7} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} - 2q^{7} + 9q^{9} - 22q^{13} - 26q^{19} - 6q^{21} + 25q^{25} + 27q^{27} + 46q^{31} + 26q^{37} - 66q^{39} + 22q^{43} - 45q^{49} - 78q^{57} + 74q^{61} - 18q^{63} - 122q^{67} - 46q^{73} + 75q^{75} + 142q^{79} + 81q^{81} + 44q^{91} + 138q^{93} + 2q^{97} + O(q^{100}) \)

Expression as an eta quotient

\(f(z) = \dfrac{\eta(4z)^{9}\eta(12z)^{9}}{\eta(2z)^{3}\eta(6z)^{3}\eta(8z)^{3}\eta(24z)^{3}}=q\prod_{n=1}^\infty(1 - q^{2n})^{-3}(1 - q^{4n})^{9}(1 - q^{6n})^{-3}(1 - q^{8n})^{-3}(1 - q^{12n})^{9}(1 - q^{24n})^{-3}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0
0 3.00000 0 0 0 −2.00000 0 9.00000 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 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.3.e.a 1
3.b odd 2 1 CM 48.3.e.a 1
4.b odd 2 1 12.3.c.a 1
5.b even 2 1 1200.3.l.b 1
5.c odd 4 2 1200.3.c.c 2
8.b even 2 1 192.3.e.a 1
8.d odd 2 1 192.3.e.b 1
9.c even 3 2 1296.3.q.b 2
9.d odd 6 2 1296.3.q.b 2
12.b even 2 1 12.3.c.a 1
15.d odd 2 1 1200.3.l.b 1
15.e even 4 2 1200.3.c.c 2
16.e even 4 2 768.3.h.b 2
16.f odd 4 2 768.3.h.a 2
20.d odd 2 1 300.3.g.b 1
20.e even 4 2 300.3.b.a 2
24.f even 2 1 192.3.e.b 1
24.h odd 2 1 192.3.e.a 1
28.d even 2 1 588.3.c.c 1
28.f even 6 2 588.3.p.b 2
28.g odd 6 2 588.3.p.c 2
36.f odd 6 2 324.3.g.b 2
36.h even 6 2 324.3.g.b 2
44.c even 2 1 1452.3.e.b 1
48.i odd 4 2 768.3.h.b 2
48.k even 4 2 768.3.h.a 2
60.h even 2 1 300.3.g.b 1
60.l odd 4 2 300.3.b.a 2
84.h odd 2 1 588.3.c.c 1
84.j odd 6 2 588.3.p.b 2
84.n even 6 2 588.3.p.c 2
132.d odd 2 1 1452.3.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.c.a 1 4.b odd 2 1
12.3.c.a 1 12.b even 2 1
48.3.e.a 1 1.a even 1 1 trivial
48.3.e.a 1 3.b odd 2 1 CM
192.3.e.a 1 8.b even 2 1
192.3.e.a 1 24.h odd 2 1
192.3.e.b 1 8.d odd 2 1
192.3.e.b 1 24.f even 2 1
300.3.b.a 2 20.e even 4 2
300.3.b.a 2 60.l odd 4 2
300.3.g.b 1 20.d odd 2 1
300.3.g.b 1 60.h even 2 1
324.3.g.b 2 36.f odd 6 2
324.3.g.b 2 36.h even 6 2
588.3.c.c 1 28.d even 2 1
588.3.c.c 1 84.h odd 2 1
588.3.p.b 2 28.f even 6 2
588.3.p.b 2 84.j odd 6 2
588.3.p.c 2 28.g odd 6 2
588.3.p.c 2 84.n even 6 2
768.3.h.a 2 16.f odd 4 2
768.3.h.a 2 48.k even 4 2
768.3.h.b 2 16.e even 4 2
768.3.h.b 2 48.i odd 4 2
1200.3.c.c 2 5.c odd 4 2
1200.3.c.c 2 15.e even 4 2
1200.3.l.b 1 5.b even 2 1
1200.3.l.b 1 15.d odd 2 1
1296.3.q.b 2 9.c even 3 2
1296.3.q.b 2 9.d odd 6 2
1452.3.e.b 1 44.c even 2 1
1452.3.e.b 1 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{3}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( T \)
$7$ \( 2 + T \)
$11$ \( T \)
$13$ \( 22 + T \)
$17$ \( T \)
$19$ \( 26 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( -46 + T \)
$37$ \( -26 + T \)
$41$ \( T \)
$43$ \( -22 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( -74 + T \)
$67$ \( 122 + T \)
$71$ \( T \)
$73$ \( 46 + T \)
$79$ \( -142 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( -2 + T \)
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