Properties

Label 48.4.a.b
Level 48
Weight 4
Character orbit 48.a
Self dual yes
Analytic conductor 2.832
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.83209168028\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} + 14q^{5} + 24q^{7} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} + 14q^{5} + 24q^{7} + 9q^{9} + 28q^{11} - 74q^{13} - 42q^{15} + 82q^{17} - 92q^{19} - 72q^{21} - 8q^{23} + 71q^{25} - 27q^{27} - 138q^{29} - 80q^{31} - 84q^{33} + 336q^{35} + 30q^{37} + 222q^{39} + 282q^{41} - 4q^{43} + 126q^{45} - 240q^{47} + 233q^{49} - 246q^{51} - 130q^{53} + 392q^{55} + 276q^{57} - 596q^{59} - 218q^{61} + 216q^{63} - 1036q^{65} + 436q^{67} + 24q^{69} - 856q^{71} - 998q^{73} - 213q^{75} + 672q^{77} + 32q^{79} + 81q^{81} + 1508q^{83} + 1148q^{85} + 414q^{87} - 246q^{89} - 1776q^{91} + 240q^{93} - 1288q^{95} + 866q^{97} + 252q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −3.00000 0 14.0000 0 24.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.4.a.b 1
3.b odd 2 1 144.4.a.b 1
4.b odd 2 1 24.4.a.a 1
5.b even 2 1 1200.4.a.u 1
5.c odd 4 2 1200.4.f.p 2
7.b odd 2 1 2352.4.a.w 1
8.b even 2 1 192.4.a.g 1
8.d odd 2 1 192.4.a.a 1
12.b even 2 1 72.4.a.b 1
16.e even 4 2 768.4.d.b 2
16.f odd 4 2 768.4.d.o 2
20.d odd 2 1 600.4.a.h 1
20.e even 4 2 600.4.f.b 2
24.f even 2 1 576.4.a.u 1
24.h odd 2 1 576.4.a.v 1
28.d even 2 1 1176.4.a.a 1
36.f odd 6 2 648.4.i.b 2
36.h even 6 2 648.4.i.k 2
60.h even 2 1 1800.4.a.bg 1
60.l odd 4 2 1800.4.f.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.a.a 1 4.b odd 2 1
48.4.a.b 1 1.a even 1 1 trivial
72.4.a.b 1 12.b even 2 1
144.4.a.b 1 3.b odd 2 1
192.4.a.a 1 8.d odd 2 1
192.4.a.g 1 8.b even 2 1
576.4.a.u 1 24.f even 2 1
576.4.a.v 1 24.h odd 2 1
600.4.a.h 1 20.d odd 2 1
600.4.f.b 2 20.e even 4 2
648.4.i.b 2 36.f odd 6 2
648.4.i.k 2 36.h even 6 2
768.4.d.b 2 16.e even 4 2
768.4.d.o 2 16.f odd 4 2
1176.4.a.a 1 28.d even 2 1
1200.4.a.u 1 5.b even 2 1
1200.4.f.p 2 5.c odd 4 2
1800.4.a.bg 1 60.h even 2 1
1800.4.f.q 2 60.l odd 4 2
2352.4.a.w 1 7.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 14 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(48))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T \)
$5$ \( 1 - 14 T + 125 T^{2} \)
$7$ \( 1 - 24 T + 343 T^{2} \)
$11$ \( 1 - 28 T + 1331 T^{2} \)
$13$ \( 1 + 74 T + 2197 T^{2} \)
$17$ \( 1 - 82 T + 4913 T^{2} \)
$19$ \( 1 + 92 T + 6859 T^{2} \)
$23$ \( 1 + 8 T + 12167 T^{2} \)
$29$ \( 1 + 138 T + 24389 T^{2} \)
$31$ \( 1 + 80 T + 29791 T^{2} \)
$37$ \( 1 - 30 T + 50653 T^{2} \)
$41$ \( 1 - 282 T + 68921 T^{2} \)
$43$ \( 1 + 4 T + 79507 T^{2} \)
$47$ \( 1 + 240 T + 103823 T^{2} \)
$53$ \( 1 + 130 T + 148877 T^{2} \)
$59$ \( 1 + 596 T + 205379 T^{2} \)
$61$ \( 1 + 218 T + 226981 T^{2} \)
$67$ \( 1 - 436 T + 300763 T^{2} \)
$71$ \( 1 + 856 T + 357911 T^{2} \)
$73$ \( 1 + 998 T + 389017 T^{2} \)
$79$ \( 1 - 32 T + 493039 T^{2} \)
$83$ \( 1 - 1508 T + 571787 T^{2} \)
$89$ \( 1 + 246 T + 704969 T^{2} \)
$97$ \( 1 - 866 T + 912673 T^{2} \)
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