Properties

Label 48.6.a.a
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 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 9q^{3} + 6q^{5} + 40q^{7} + 81q^{9} + O(q^{10}) \) \( q - 9q^{3} + 6q^{5} + 40q^{7} + 81q^{9} + 564q^{11} + 638q^{13} - 54q^{15} + 882q^{17} + 556q^{19} - 360q^{21} + 840q^{23} - 3089q^{25} - 729q^{27} + 4638q^{29} - 4400q^{31} - 5076q^{33} + 240q^{35} - 2410q^{37} - 5742q^{39} - 6870q^{41} - 9644q^{43} + 486q^{45} + 18672q^{47} - 15207q^{49} - 7938q^{51} + 33750q^{53} + 3384q^{55} - 5004q^{57} + 18084q^{59} + 39758q^{61} + 3240q^{63} + 3828q^{65} + 23068q^{67} - 7560q^{69} + 4248q^{71} - 41110q^{73} + 27801q^{75} + 22560q^{77} - 21920q^{79} + 6561q^{81} - 82452q^{83} + 5292q^{85} - 41742q^{87} - 94086q^{89} + 25520q^{91} + 39600q^{93} + 3336q^{95} + 49442q^{97} + 45684q^{99} + O(q^{100}) \)

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 6.00000 0 40.0000 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.a 1
3.b odd 2 1 144.6.a.f 1
4.b odd 2 1 3.6.a.a 1
8.b even 2 1 192.6.a.l 1
8.d odd 2 1 192.6.a.d 1
12.b even 2 1 9.6.a.a 1
16.e even 4 2 768.6.d.h 2
16.f odd 4 2 768.6.d.k 2
20.d odd 2 1 75.6.a.e 1
20.e even 4 2 75.6.b.b 2
24.f even 2 1 576.6.a.s 1
24.h odd 2 1 576.6.a.t 1
28.d even 2 1 147.6.a.a 1
28.f even 6 2 147.6.e.k 2
28.g odd 6 2 147.6.e.h 2
36.f odd 6 2 81.6.c.c 2
36.h even 6 2 81.6.c.a 2
44.c even 2 1 363.6.a.d 1
52.b odd 2 1 507.6.a.b 1
60.h even 2 1 225.6.a.a 1
60.l odd 4 2 225.6.b.b 2
68.d odd 2 1 867.6.a.a 1
76.d even 2 1 1083.6.a.c 1
84.h odd 2 1 441.6.a.i 1
132.d odd 2 1 1089.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.6.a.a 1 4.b odd 2 1
9.6.a.a 1 12.b even 2 1
48.6.a.a 1 1.a even 1 1 trivial
75.6.a.e 1 20.d odd 2 1
75.6.b.b 2 20.e even 4 2
81.6.c.a 2 36.h even 6 2
81.6.c.c 2 36.f odd 6 2
144.6.a.f 1 3.b odd 2 1
147.6.a.a 1 28.d even 2 1
147.6.e.h 2 28.g odd 6 2
147.6.e.k 2 28.f even 6 2
192.6.a.d 1 8.d odd 2 1
192.6.a.l 1 8.b even 2 1
225.6.a.a 1 60.h even 2 1
225.6.b.b 2 60.l odd 4 2
363.6.a.d 1 44.c even 2 1
441.6.a.i 1 84.h odd 2 1
507.6.a.b 1 52.b odd 2 1
576.6.a.s 1 24.f even 2 1
576.6.a.t 1 24.h odd 2 1
768.6.d.h 2 16.e even 4 2
768.6.d.k 2 16.f odd 4 2
867.6.a.a 1 68.d odd 2 1
1083.6.a.c 1 76.d even 2 1
1089.6.a.b 1 132.d 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 \( T_{5} - 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(48))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 9 T \)
$5$ \( 1 - 6 T + 3125 T^{2} \)
$7$ \( 1 - 40 T + 16807 T^{2} \)
$11$ \( 1 - 564 T + 161051 T^{2} \)
$13$ \( 1 - 638 T + 371293 T^{2} \)
$17$ \( 1 - 882 T + 1419857 T^{2} \)
$19$ \( 1 - 556 T + 2476099 T^{2} \)
$23$ \( 1 - 840 T + 6436343 T^{2} \)
$29$ \( 1 - 4638 T + 20511149 T^{2} \)
$31$ \( 1 + 4400 T + 28629151 T^{2} \)
$37$ \( 1 + 2410 T + 69343957 T^{2} \)
$41$ \( 1 + 6870 T + 115856201 T^{2} \)
$43$ \( 1 + 9644 T + 147008443 T^{2} \)
$47$ \( 1 - 18672 T + 229345007 T^{2} \)
$53$ \( 1 - 33750 T + 418195493 T^{2} \)
$59$ \( 1 - 18084 T + 714924299 T^{2} \)
$61$ \( 1 - 39758 T + 844596301 T^{2} \)
$67$ \( 1 - 23068 T + 1350125107 T^{2} \)
$71$ \( 1 - 4248 T + 1804229351 T^{2} \)
$73$ \( 1 + 41110 T + 2073071593 T^{2} \)
$79$ \( 1 + 21920 T + 3077056399 T^{2} \)
$83$ \( 1 + 82452 T + 3939040643 T^{2} \)
$89$ \( 1 + 94086 T + 5584059449 T^{2} \)
$97$ \( 1 - 49442 T + 8587340257 T^{2} \)
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