Properties

Label 48.6.a.d
Level 48
Weight 6
Character orbit 48.a
Self dual yes
Analytic conductor 7.698
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.69842335102\)
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

\(f(q)\) \(=\) \( q + 9q^{3} - 34q^{5} + 240q^{7} + 81q^{9} + O(q^{10}) \) \( q + 9q^{3} - 34q^{5} + 240q^{7} + 81q^{9} + 124q^{11} + 46q^{13} - 306q^{15} + 1954q^{17} + 1924q^{19} + 2160q^{21} - 2840q^{23} - 1969q^{25} + 729q^{27} - 8922q^{29} + 4648q^{31} + 1116q^{33} - 8160q^{35} - 4362q^{37} + 414q^{39} - 2886q^{41} - 11332q^{43} - 2754q^{45} - 7008q^{47} + 40793q^{49} + 17586q^{51} - 22594q^{53} - 4216q^{55} + 17316q^{57} + 28q^{59} - 6386q^{61} + 19440q^{63} - 1564q^{65} + 39076q^{67} - 25560q^{69} + 54872q^{71} + 21034q^{73} - 17721q^{75} + 29760q^{77} - 26632q^{79} + 6561q^{81} - 56188q^{83} - 66436q^{85} - 80298q^{87} + 64410q^{89} + 11040q^{91} + 41832q^{93} - 65416q^{95} - 116158q^{97} + 10044q^{99} + O(q^{100}) \)

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 9.00000 0 −34.0000 0 240.000 0 81.0000 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 48.6.a.d 1
3.b odd 2 1 144.6.a.i 1
4.b odd 2 1 24.6.a.a 1
8.b even 2 1 192.6.a.f 1
8.d odd 2 1 192.6.a.n 1
12.b even 2 1 72.6.a.e 1
16.e even 4 2 768.6.d.a 2
16.f odd 4 2 768.6.d.r 2
20.d odd 2 1 600.6.a.i 1
20.e even 4 2 600.6.f.f 2
24.f even 2 1 576.6.a.k 1
24.h odd 2 1 576.6.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.6.a.a 1 4.b odd 2 1
48.6.a.d 1 1.a even 1 1 trivial
72.6.a.e 1 12.b even 2 1
144.6.a.i 1 3.b odd 2 1
192.6.a.f 1 8.b even 2 1
192.6.a.n 1 8.d odd 2 1
576.6.a.k 1 24.f even 2 1
576.6.a.l 1 24.h odd 2 1
600.6.a.i 1 20.d odd 2 1
600.6.f.f 2 20.e even 4 2
768.6.d.a 2 16.e even 4 2
768.6.d.r 2 16.f odd 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 34 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(48))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 9 T \)
$5$ \( 1 + 34 T + 3125 T^{2} \)
$7$ \( 1 - 240 T + 16807 T^{2} \)
$11$ \( 1 - 124 T + 161051 T^{2} \)
$13$ \( 1 - 46 T + 371293 T^{2} \)
$17$ \( 1 - 1954 T + 1419857 T^{2} \)
$19$ \( 1 - 1924 T + 2476099 T^{2} \)
$23$ \( 1 + 2840 T + 6436343 T^{2} \)
$29$ \( 1 + 8922 T + 20511149 T^{2} \)
$31$ \( 1 - 4648 T + 28629151 T^{2} \)
$37$ \( 1 + 4362 T + 69343957 T^{2} \)
$41$ \( 1 + 2886 T + 115856201 T^{2} \)
$43$ \( 1 + 11332 T + 147008443 T^{2} \)
$47$ \( 1 + 7008 T + 229345007 T^{2} \)
$53$ \( 1 + 22594 T + 418195493 T^{2} \)
$59$ \( 1 - 28 T + 714924299 T^{2} \)
$61$ \( 1 + 6386 T + 844596301 T^{2} \)
$67$ \( 1 - 39076 T + 1350125107 T^{2} \)
$71$ \( 1 - 54872 T + 1804229351 T^{2} \)
$73$ \( 1 - 21034 T + 2073071593 T^{2} \)
$79$ \( 1 + 26632 T + 3077056399 T^{2} \)
$83$ \( 1 + 56188 T + 3939040643 T^{2} \)
$89$ \( 1 - 64410 T + 5584059449 T^{2} \)
$97$ \( 1 + 116158 T + 8587340257 T^{2} \)
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