Properties

Label 48.4.a.a
Level 48
Weight 4
Character orbit 48.a
Self dual yes
Analytic conductor 2.832
Analytic rank 1
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} - 18q^{5} - 8q^{7} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} - 18q^{5} - 8q^{7} + 9q^{9} - 36q^{11} - 10q^{13} + 54q^{15} + 18q^{17} + 100q^{19} + 24q^{21} - 72q^{23} + 199q^{25} - 27q^{27} - 234q^{29} + 16q^{31} + 108q^{33} + 144q^{35} - 226q^{37} + 30q^{39} + 90q^{41} - 452q^{43} - 162q^{45} - 432q^{47} - 279q^{49} - 54q^{51} + 414q^{53} + 648q^{55} - 300q^{57} + 684q^{59} + 422q^{61} - 72q^{63} + 180q^{65} - 332q^{67} + 216q^{69} + 360q^{71} + 26q^{73} - 597q^{75} + 288q^{77} - 512q^{79} + 81q^{81} + 1188q^{83} - 324q^{85} + 702q^{87} - 630q^{89} + 80q^{91} - 48q^{93} - 1800q^{95} - 1054q^{97} - 324q^{99} + O(q^{100}) \)

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 −18.0000 0 −8.00000 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.a 1
3.b odd 2 1 144.4.a.g 1
4.b odd 2 1 12.4.a.a 1
5.b even 2 1 1200.4.a.be 1
5.c odd 4 2 1200.4.f.d 2
7.b odd 2 1 2352.4.a.bk 1
8.b even 2 1 192.4.a.l 1
8.d odd 2 1 192.4.a.f 1
12.b even 2 1 36.4.a.a 1
16.e even 4 2 768.4.d.j 2
16.f odd 4 2 768.4.d.g 2
20.d odd 2 1 300.4.a.b 1
20.e even 4 2 300.4.d.e 2
24.f even 2 1 576.4.a.b 1
24.h odd 2 1 576.4.a.a 1
28.d even 2 1 588.4.a.c 1
28.f even 6 2 588.4.i.e 2
28.g odd 6 2 588.4.i.d 2
36.f odd 6 2 324.4.e.h 2
36.h even 6 2 324.4.e.a 2
44.c even 2 1 1452.4.a.d 1
52.b odd 2 1 2028.4.a.c 1
52.f even 4 2 2028.4.b.c 2
60.h even 2 1 900.4.a.g 1
60.l odd 4 2 900.4.d.c 2
84.h odd 2 1 1764.4.a.b 1
84.j odd 6 2 1764.4.k.o 2
84.n even 6 2 1764.4.k.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 4.b odd 2 1
36.4.a.a 1 12.b even 2 1
48.4.a.a 1 1.a even 1 1 trivial
144.4.a.g 1 3.b odd 2 1
192.4.a.f 1 8.d odd 2 1
192.4.a.l 1 8.b even 2 1
300.4.a.b 1 20.d odd 2 1
300.4.d.e 2 20.e even 4 2
324.4.e.a 2 36.h even 6 2
324.4.e.h 2 36.f odd 6 2
576.4.a.a 1 24.h odd 2 1
576.4.a.b 1 24.f even 2 1
588.4.a.c 1 28.d even 2 1
588.4.i.d 2 28.g odd 6 2
588.4.i.e 2 28.f even 6 2
768.4.d.g 2 16.f odd 4 2
768.4.d.j 2 16.e even 4 2
900.4.a.g 1 60.h even 2 1
900.4.d.c 2 60.l odd 4 2
1200.4.a.be 1 5.b even 2 1
1200.4.f.d 2 5.c odd 4 2
1452.4.a.d 1 44.c even 2 1
1764.4.a.b 1 84.h odd 2 1
1764.4.k.b 2 84.n even 6 2
1764.4.k.o 2 84.j odd 6 2
2028.4.a.c 1 52.b odd 2 1
2028.4.b.c 2 52.f even 4 2
2352.4.a.bk 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 \( T_{5} + 18 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(48))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T \)
$5$ \( 1 + 18 T + 125 T^{2} \)
$7$ \( 1 + 8 T + 343 T^{2} \)
$11$ \( 1 + 36 T + 1331 T^{2} \)
$13$ \( 1 + 10 T + 2197 T^{2} \)
$17$ \( 1 - 18 T + 4913 T^{2} \)
$19$ \( 1 - 100 T + 6859 T^{2} \)
$23$ \( 1 + 72 T + 12167 T^{2} \)
$29$ \( 1 + 234 T + 24389 T^{2} \)
$31$ \( 1 - 16 T + 29791 T^{2} \)
$37$ \( 1 + 226 T + 50653 T^{2} \)
$41$ \( 1 - 90 T + 68921 T^{2} \)
$43$ \( 1 + 452 T + 79507 T^{2} \)
$47$ \( 1 + 432 T + 103823 T^{2} \)
$53$ \( 1 - 414 T + 148877 T^{2} \)
$59$ \( 1 - 684 T + 205379 T^{2} \)
$61$ \( 1 - 422 T + 226981 T^{2} \)
$67$ \( 1 + 332 T + 300763 T^{2} \)
$71$ \( 1 - 360 T + 357911 T^{2} \)
$73$ \( 1 - 26 T + 389017 T^{2} \)
$79$ \( 1 + 512 T + 493039 T^{2} \)
$83$ \( 1 - 1188 T + 571787 T^{2} \)
$89$ \( 1 + 630 T + 704969 T^{2} \)
$97$ \( 1 + 1054 T + 912673 T^{2} \)
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