Properties

Label 48.8.a.b
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 6)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 27q^{3} - 114q^{5} + 1576q^{7} + 729q^{9} + O(q^{10}) \) \( q - 27q^{3} - 114q^{5} + 1576q^{7} + 729q^{9} - 7332q^{11} - 3802q^{13} + 3078q^{15} - 6606q^{17} - 24860q^{19} - 42552q^{21} - 41448q^{23} - 65129q^{25} - 19683q^{27} - 41610q^{29} - 33152q^{31} + 197964q^{33} - 179664q^{35} - 36466q^{37} + 102654q^{39} - 639078q^{41} + 156412q^{43} - 83106q^{45} + 433776q^{47} + 1660233q^{49} + 178362q^{51} + 786078q^{53} + 835848q^{55} + 671220q^{57} - 745140q^{59} - 1660618q^{61} + 1148904q^{63} + 433428q^{65} + 3290836q^{67} + 1119096q^{69} - 5716152q^{71} + 2659898q^{73} + 1758483q^{75} - 11555232q^{77} - 3807440q^{79} + 531441q^{81} - 2229468q^{83} + 753084q^{85} + 1123470q^{87} + 5991210q^{89} - 5991952q^{91} + 895104q^{93} + 2834040q^{95} - 4060126q^{97} - 5345028q^{99} + O(q^{100}) \)

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 −114.000 0 1576.00 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.b 1
3.b odd 2 1 144.8.a.h 1
4.b odd 2 1 6.8.a.a 1
8.b even 2 1 192.8.a.n 1
8.d odd 2 1 192.8.a.f 1
12.b even 2 1 18.8.a.a 1
20.d odd 2 1 150.8.a.e 1
20.e even 4 2 150.8.c.k 2
24.f even 2 1 576.8.a.h 1
24.h odd 2 1 576.8.a.i 1
28.d even 2 1 294.8.a.l 1
28.f even 6 2 294.8.e.d 2
28.g odd 6 2 294.8.e.c 2
36.f odd 6 2 162.8.c.d 2
36.h even 6 2 162.8.c.i 2
60.h even 2 1 450.8.a.ba 1
60.l odd 4 2 450.8.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.8.a.a 1 4.b odd 2 1
18.8.a.a 1 12.b even 2 1
48.8.a.b 1 1.a even 1 1 trivial
144.8.a.h 1 3.b odd 2 1
150.8.a.e 1 20.d odd 2 1
150.8.c.k 2 20.e even 4 2
162.8.c.d 2 36.f odd 6 2
162.8.c.i 2 36.h even 6 2
192.8.a.f 1 8.d odd 2 1
192.8.a.n 1 8.b even 2 1
294.8.a.l 1 28.d even 2 1
294.8.e.c 2 28.g odd 6 2
294.8.e.d 2 28.f even 6 2
450.8.a.ba 1 60.h even 2 1
450.8.c.a 2 60.l odd 4 2
576.8.a.h 1 24.f even 2 1
576.8.a.i 1 24.h 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 \( T_{5} + 114 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(48))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 27 T \)
$5$ \( 1 + 114 T + 78125 T^{2} \)
$7$ \( 1 - 1576 T + 823543 T^{2} \)
$11$ \( 1 + 7332 T + 19487171 T^{2} \)
$13$ \( 1 + 3802 T + 62748517 T^{2} \)
$17$ \( 1 + 6606 T + 410338673 T^{2} \)
$19$ \( 1 + 24860 T + 893871739 T^{2} \)
$23$ \( 1 + 41448 T + 3404825447 T^{2} \)
$29$ \( 1 + 41610 T + 17249876309 T^{2} \)
$31$ \( 1 + 33152 T + 27512614111 T^{2} \)
$37$ \( 1 + 36466 T + 94931877133 T^{2} \)
$41$ \( 1 + 639078 T + 194754273881 T^{2} \)
$43$ \( 1 - 156412 T + 271818611107 T^{2} \)
$47$ \( 1 - 433776 T + 506623120463 T^{2} \)
$53$ \( 1 - 786078 T + 1174711139837 T^{2} \)
$59$ \( 1 + 745140 T + 2488651484819 T^{2} \)
$61$ \( 1 + 1660618 T + 3142742836021 T^{2} \)
$67$ \( 1 - 3290836 T + 6060711605323 T^{2} \)
$71$ \( 1 + 5716152 T + 9095120158391 T^{2} \)
$73$ \( 1 - 2659898 T + 11047398519097 T^{2} \)
$79$ \( 1 + 3807440 T + 19203908986159 T^{2} \)
$83$ \( 1 + 2229468 T + 27136050989627 T^{2} \)
$89$ \( 1 - 5991210 T + 44231334895529 T^{2} \)
$97$ \( 1 + 4060126 T + 80798284478113 T^{2} \)
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