Properties

Label 48.8.a.d
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 12)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 27q^{3} + 270q^{5} - 1112q^{7} + 729q^{9} + O(q^{10}) \) \( q - 27q^{3} + 270q^{5} - 1112q^{7} + 729q^{9} + 5724q^{11} - 4570q^{13} - 7290q^{15} - 36558q^{17} - 51740q^{19} + 30024q^{21} - 22248q^{23} - 5225q^{25} - 19683q^{27} - 157194q^{29} + 103936q^{31} - 154548q^{33} - 300240q^{35} - 94834q^{37} + 123390q^{39} + 659610q^{41} + 75772q^{43} + 196830q^{45} - 405648q^{47} + 413001q^{49} + 987066q^{51} - 1346274q^{53} + 1545480q^{55} + 1396980q^{57} + 1303884q^{59} + 1833782q^{61} - 810648q^{63} - 1233900q^{65} - 1369388q^{67} + 600696q^{69} - 2714040q^{71} + 2868794q^{73} + 141075q^{75} - 6365088q^{77} + 1129648q^{79} + 531441q^{81} - 5912028q^{83} - 9870660q^{85} + 4244238q^{87} - 897750q^{89} + 5081840q^{91} - 2806272q^{93} - 13969800q^{95} + 13719074q^{97} + 4172796q^{99} + O(q^{100}) \)

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 0 270.000 0 −1112.00 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.d 1
3.b odd 2 1 144.8.a.c 1
4.b odd 2 1 12.8.a.b 1
8.b even 2 1 192.8.a.j 1
8.d odd 2 1 192.8.a.b 1
12.b even 2 1 36.8.a.a 1
20.d odd 2 1 300.8.a.a 1
20.e even 4 2 300.8.d.a 2
24.f even 2 1 576.8.a.v 1
24.h odd 2 1 576.8.a.u 1
28.d even 2 1 588.8.a.a 1
28.f even 6 2 588.8.i.g 2
28.g odd 6 2 588.8.i.b 2
36.f odd 6 2 324.8.e.b 2
36.h even 6 2 324.8.e.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.8.a.b 1 4.b odd 2 1
36.8.a.a 1 12.b even 2 1
48.8.a.d 1 1.a even 1 1 trivial
144.8.a.c 1 3.b odd 2 1
192.8.a.b 1 8.d odd 2 1
192.8.a.j 1 8.b even 2 1
300.8.a.a 1 20.d odd 2 1
300.8.d.a 2 20.e even 4 2
324.8.e.b 2 36.f odd 6 2
324.8.e.e 2 36.h even 6 2
576.8.a.u 1 24.h odd 2 1
576.8.a.v 1 24.f even 2 1
588.8.a.a 1 28.d even 2 1
588.8.i.b 2 28.g odd 6 2
588.8.i.g 2 28.f even 6 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 \( T_{5} - 270 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(48))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 27 T \)
$5$ \( 1 - 270 T + 78125 T^{2} \)
$7$ \( 1 + 1112 T + 823543 T^{2} \)
$11$ \( 1 - 5724 T + 19487171 T^{2} \)
$13$ \( 1 + 4570 T + 62748517 T^{2} \)
$17$ \( 1 + 36558 T + 410338673 T^{2} \)
$19$ \( 1 + 51740 T + 893871739 T^{2} \)
$23$ \( 1 + 22248 T + 3404825447 T^{2} \)
$29$ \( 1 + 157194 T + 17249876309 T^{2} \)
$31$ \( 1 - 103936 T + 27512614111 T^{2} \)
$37$ \( 1 + 94834 T + 94931877133 T^{2} \)
$41$ \( 1 - 659610 T + 194754273881 T^{2} \)
$43$ \( 1 - 75772 T + 271818611107 T^{2} \)
$47$ \( 1 + 405648 T + 506623120463 T^{2} \)
$53$ \( 1 + 1346274 T + 1174711139837 T^{2} \)
$59$ \( 1 - 1303884 T + 2488651484819 T^{2} \)
$61$ \( 1 - 1833782 T + 3142742836021 T^{2} \)
$67$ \( 1 + 1369388 T + 6060711605323 T^{2} \)
$71$ \( 1 + 2714040 T + 9095120158391 T^{2} \)
$73$ \( 1 - 2868794 T + 11047398519097 T^{2} \)
$79$ \( 1 - 1129648 T + 19203908986159 T^{2} \)
$83$ \( 1 + 5912028 T + 27136050989627 T^{2} \)
$89$ \( 1 + 897750 T + 44231334895529 T^{2} \)
$97$ \( 1 - 13719074 T + 80798284478113 T^{2} \)
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