Properties

Label 48.8.a.e
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

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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: \(0\)
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

\(f(q)\) \(=\) \( q + 27q^{3} - 378q^{5} + 832q^{7} + 729q^{9} + O(q^{10}) \) \( q + 27q^{3} - 378q^{5} + 832q^{7} + 729q^{9} + 2484q^{11} + 14870q^{13} - 10206q^{15} - 22302q^{17} + 16300q^{19} + 22464q^{21} + 115128q^{23} + 64759q^{25} + 19683q^{27} + 157086q^{29} + 16456q^{31} + 67068q^{33} - 314496q^{35} - 149266q^{37} + 401490q^{39} - 241110q^{41} + 443188q^{43} - 275562q^{45} - 922752q^{47} - 131319q^{49} - 602154q^{51} - 697626q^{53} - 938952q^{55} + 440100q^{57} - 870156q^{59} + 2067062q^{61} + 606528q^{63} - 5620860q^{65} + 1680748q^{67} + 3108456q^{69} + 1070280q^{71} - 2403334q^{73} + 1748493q^{75} + 2066688q^{77} - 2301512q^{79} + 531441q^{81} - 4708692q^{83} + 8430156q^{85} + 4241322q^{87} + 4143690q^{89} + 12371840q^{91} + 444312q^{93} - 6161400q^{95} - 1622974q^{97} + 1810836q^{99} + O(q^{100}) \)

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 −378.000 0 832.000 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.e 1
3.b odd 2 1 144.8.a.j 1
4.b odd 2 1 12.8.a.a 1
8.b even 2 1 192.8.a.g 1
8.d odd 2 1 192.8.a.o 1
12.b even 2 1 36.8.a.c 1
20.d odd 2 1 300.8.a.g 1
20.e even 4 2 300.8.d.c 2
24.f even 2 1 576.8.a.d 1
24.h odd 2 1 576.8.a.e 1
28.d even 2 1 588.8.a.d 1
28.f even 6 2 588.8.i.a 2
28.g odd 6 2 588.8.i.h 2
36.f odd 6 2 324.8.e.f 2
36.h even 6 2 324.8.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.8.a.a 1 4.b odd 2 1
36.8.a.c 1 12.b even 2 1
48.8.a.e 1 1.a even 1 1 trivial
144.8.a.j 1 3.b odd 2 1
192.8.a.g 1 8.b even 2 1
192.8.a.o 1 8.d odd 2 1
300.8.a.g 1 20.d odd 2 1
300.8.d.c 2 20.e even 4 2
324.8.e.a 2 36.h even 6 2
324.8.e.f 2 36.f odd 6 2
576.8.a.d 1 24.f even 2 1
576.8.a.e 1 24.h odd 2 1
588.8.a.d 1 28.d even 2 1
588.8.i.a 2 28.f even 6 2
588.8.i.h 2 28.g odd 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 \( T_{5} + 378 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(48))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 27 T \)
$5$ \( 1 + 378 T + 78125 T^{2} \)
$7$ \( 1 - 832 T + 823543 T^{2} \)
$11$ \( 1 - 2484 T + 19487171 T^{2} \)
$13$ \( 1 - 14870 T + 62748517 T^{2} \)
$17$ \( 1 + 22302 T + 410338673 T^{2} \)
$19$ \( 1 - 16300 T + 893871739 T^{2} \)
$23$ \( 1 - 115128 T + 3404825447 T^{2} \)
$29$ \( 1 - 157086 T + 17249876309 T^{2} \)
$31$ \( 1 - 16456 T + 27512614111 T^{2} \)
$37$ \( 1 + 149266 T + 94931877133 T^{2} \)
$41$ \( 1 + 241110 T + 194754273881 T^{2} \)
$43$ \( 1 - 443188 T + 271818611107 T^{2} \)
$47$ \( 1 + 922752 T + 506623120463 T^{2} \)
$53$ \( 1 + 697626 T + 1174711139837 T^{2} \)
$59$ \( 1 + 870156 T + 2488651484819 T^{2} \)
$61$ \( 1 - 2067062 T + 3142742836021 T^{2} \)
$67$ \( 1 - 1680748 T + 6060711605323 T^{2} \)
$71$ \( 1 - 1070280 T + 9095120158391 T^{2} \)
$73$ \( 1 + 2403334 T + 11047398519097 T^{2} \)
$79$ \( 1 + 2301512 T + 19203908986159 T^{2} \)
$83$ \( 1 + 4708692 T + 27136050989627 T^{2} \)
$89$ \( 1 - 4143690 T + 44231334895529 T^{2} \)
$97$ \( 1 + 1622974 T + 80798284478113 T^{2} \)
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