Properties

Label 48.8.a.f
Level 48
Weight 8
Character orbit 48.a
Self dual yes
Analytic conductor 14.994
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.9944812232\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + 27q^{3} - 26q^{5} - 1056q^{7} + 729q^{9} + O(q^{10}) \) \( q + 27q^{3} - 26q^{5} - 1056q^{7} + 729q^{9} - 6412q^{11} + 5206q^{13} - 702q^{15} - 6238q^{17} - 41492q^{19} - 28512q^{21} + 29432q^{23} - 77449q^{25} + 19683q^{27} - 210498q^{29} - 185240q^{31} - 173124q^{33} + 27456q^{35} + 507630q^{37} + 140562q^{39} + 360042q^{41} - 620044q^{43} - 18954q^{45} + 847680q^{47} + 291593q^{49} - 168426q^{51} + 1423750q^{53} + 166712q^{55} - 1120284q^{57} + 2548724q^{59} - 706058q^{61} - 769824q^{63} - 135356q^{65} + 2418796q^{67} + 794664q^{69} - 265976q^{71} - 5791238q^{73} - 2091123q^{75} + 6771072q^{77} - 2955688q^{79} + 531441q^{81} - 3462932q^{83} + 162188q^{85} - 5683446q^{87} - 2211126q^{89} - 5497536q^{91} - 5001480q^{93} + 1078792q^{95} - 15594814q^{97} - 4674348q^{99} + O(q^{100}) \)

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 27.0000 0 −26.0000 0 −1056.00 0 729.000 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 48.8.a.f 1
3.b odd 2 1 144.8.a.f 1
4.b odd 2 1 24.8.a.a 1
8.b even 2 1 192.8.a.d 1
8.d odd 2 1 192.8.a.l 1
12.b even 2 1 72.8.a.c 1
20.d odd 2 1 600.8.a.e 1
20.e even 4 2 600.8.f.e 2
24.f even 2 1 576.8.a.p 1
24.h odd 2 1 576.8.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.a.a 1 4.b odd 2 1
48.8.a.f 1 1.a even 1 1 trivial
72.8.a.c 1 12.b even 2 1
144.8.a.f 1 3.b odd 2 1
192.8.a.d 1 8.b even 2 1
192.8.a.l 1 8.d odd 2 1
576.8.a.o 1 24.h odd 2 1
576.8.a.p 1 24.f even 2 1
600.8.a.e 1 20.d odd 2 1
600.8.f.e 2 20.e even 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 \( T_{5} + 26 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(48))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 27 T \)
$5$ \( 1 + 26 T + 78125 T^{2} \)
$7$ \( 1 + 1056 T + 823543 T^{2} \)
$11$ \( 1 + 6412 T + 19487171 T^{2} \)
$13$ \( 1 - 5206 T + 62748517 T^{2} \)
$17$ \( 1 + 6238 T + 410338673 T^{2} \)
$19$ \( 1 + 41492 T + 893871739 T^{2} \)
$23$ \( 1 - 29432 T + 3404825447 T^{2} \)
$29$ \( 1 + 210498 T + 17249876309 T^{2} \)
$31$ \( 1 + 185240 T + 27512614111 T^{2} \)
$37$ \( 1 - 507630 T + 94931877133 T^{2} \)
$41$ \( 1 - 360042 T + 194754273881 T^{2} \)
$43$ \( 1 + 620044 T + 271818611107 T^{2} \)
$47$ \( 1 - 847680 T + 506623120463 T^{2} \)
$53$ \( 1 - 1423750 T + 1174711139837 T^{2} \)
$59$ \( 1 - 2548724 T + 2488651484819 T^{2} \)
$61$ \( 1 + 706058 T + 3142742836021 T^{2} \)
$67$ \( 1 - 2418796 T + 6060711605323 T^{2} \)
$71$ \( 1 + 265976 T + 9095120158391 T^{2} \)
$73$ \( 1 + 5791238 T + 11047398519097 T^{2} \)
$79$ \( 1 + 2955688 T + 19203908986159 T^{2} \)
$83$ \( 1 + 3462932 T + 27136050989627 T^{2} \)
$89$ \( 1 + 2211126 T + 44231334895529 T^{2} \)
$97$ \( 1 + 15594814 T + 80798284478113 T^{2} \)
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