Properties

Label 48.8.a.c
Level 48
Weight 8
Character orbit 48.a
Self dual yes
Analytic conductor 14.994
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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: \(0\)
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

\(f(q)\) \(=\) \( q - 27q^{3} + 110q^{5} - 504q^{7} + 729q^{9} + O(q^{10}) \) \( q - 27q^{3} + 110q^{5} - 504q^{7} + 729q^{9} - 3812q^{11} + 9574q^{13} - 2970q^{15} + 26098q^{17} + 38308q^{19} + 13608q^{21} + 71128q^{23} - 66025q^{25} - 19683q^{27} + 74262q^{29} + 275680q^{31} + 102924q^{33} - 55440q^{35} - 266610q^{37} - 258498q^{39} + 684762q^{41} - 245956q^{43} + 80190q^{45} - 478800q^{47} - 569527q^{49} - 704646q^{51} - 569410q^{53} - 419320q^{55} - 1034316q^{57} + 1525324q^{59} - 2640458q^{61} - 367416q^{63} + 1053140q^{65} - 1416236q^{67} - 1920456q^{69} + 3511304q^{71} + 4738618q^{73} + 1782675q^{75} + 1921248q^{77} - 4661488q^{79} + 531441q^{81} + 5729252q^{83} + 2870780q^{85} - 2005074q^{87} + 11993514q^{89} - 4825296q^{91} - 7443360q^{93} + 4213880q^{95} + 7150754q^{97} - 2778948q^{99} + O(q^{100}) \)

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 0 110.000 0 −504.000 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.c 1
3.b odd 2 1 144.8.a.d 1
4.b odd 2 1 24.8.a.c 1
8.b even 2 1 192.8.a.k 1
8.d odd 2 1 192.8.a.c 1
12.b even 2 1 72.8.a.b 1
20.d odd 2 1 600.8.a.a 1
20.e even 4 2 600.8.f.d 2
24.f even 2 1 576.8.a.t 1
24.h odd 2 1 576.8.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.a.c 1 4.b odd 2 1
48.8.a.c 1 1.a even 1 1 trivial
72.8.a.b 1 12.b even 2 1
144.8.a.d 1 3.b odd 2 1
192.8.a.c 1 8.d odd 2 1
192.8.a.k 1 8.b even 2 1
576.8.a.s 1 24.h odd 2 1
576.8.a.t 1 24.f even 2 1
600.8.a.a 1 20.d odd 2 1
600.8.f.d 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 \( T_{5} - 110 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(48))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 27 T \)
$5$ \( 1 - 110 T + 78125 T^{2} \)
$7$ \( 1 + 504 T + 823543 T^{2} \)
$11$ \( 1 + 3812 T + 19487171 T^{2} \)
$13$ \( 1 - 9574 T + 62748517 T^{2} \)
$17$ \( 1 - 26098 T + 410338673 T^{2} \)
$19$ \( 1 - 38308 T + 893871739 T^{2} \)
$23$ \( 1 - 71128 T + 3404825447 T^{2} \)
$29$ \( 1 - 74262 T + 17249876309 T^{2} \)
$31$ \( 1 - 275680 T + 27512614111 T^{2} \)
$37$ \( 1 + 266610 T + 94931877133 T^{2} \)
$41$ \( 1 - 684762 T + 194754273881 T^{2} \)
$43$ \( 1 + 245956 T + 271818611107 T^{2} \)
$47$ \( 1 + 478800 T + 506623120463 T^{2} \)
$53$ \( 1 + 569410 T + 1174711139837 T^{2} \)
$59$ \( 1 - 1525324 T + 2488651484819 T^{2} \)
$61$ \( 1 + 2640458 T + 3142742836021 T^{2} \)
$67$ \( 1 + 1416236 T + 6060711605323 T^{2} \)
$71$ \( 1 - 3511304 T + 9095120158391 T^{2} \)
$73$ \( 1 - 4738618 T + 11047398519097 T^{2} \)
$79$ \( 1 + 4661488 T + 19203908986159 T^{2} \)
$83$ \( 1 - 5729252 T + 27136050989627 T^{2} \)
$89$ \( 1 - 11993514 T + 44231334895529 T^{2} \)
$97$ \( 1 - 7150754 T + 80798284478113 T^{2} \)
show more
show less