Properties

Label 48.5.g.a
Level 48
Weight 5
Character orbit 48.g
Analytic conductor 4.962
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 48.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.96175822802\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - 6 \zeta_{6} ) q^{3} -42 q^{5} + ( -44 + 88 \zeta_{6} ) q^{7} -27 q^{9} +O(q^{10})\) \( q + ( 3 - 6 \zeta_{6} ) q^{3} -42 q^{5} + ( -44 + 88 \zeta_{6} ) q^{7} -27 q^{9} + ( 12 - 24 \zeta_{6} ) q^{11} -182 q^{13} + ( -126 + 252 \zeta_{6} ) q^{15} -246 q^{17} + ( 68 - 136 \zeta_{6} ) q^{19} + 396 q^{21} + ( 432 - 864 \zeta_{6} ) q^{23} + 1139 q^{25} + ( -81 + 162 \zeta_{6} ) q^{27} + 78 q^{29} + ( -852 + 1704 \zeta_{6} ) q^{31} -108 q^{33} + ( 1848 - 3696 \zeta_{6} ) q^{35} + 530 q^{37} + ( -546 + 1092 \zeta_{6} ) q^{39} -918 q^{41} + ( 492 - 984 \zeta_{6} ) q^{43} + 1134 q^{45} + ( -2184 + 4368 \zeta_{6} ) q^{47} -3407 q^{49} + ( -738 + 1476 \zeta_{6} ) q^{51} -4626 q^{53} + ( -504 + 1008 \zeta_{6} ) q^{55} -612 q^{57} + ( 132 - 264 \zeta_{6} ) q^{59} + 1346 q^{61} + ( 1188 - 2376 \zeta_{6} ) q^{63} + 7644 q^{65} + ( -628 + 1256 \zeta_{6} ) q^{67} -3888 q^{69} + ( 1056 - 2112 \zeta_{6} ) q^{71} -926 q^{73} + ( 3417 - 6834 \zeta_{6} ) q^{75} + 1584 q^{77} + ( 2540 - 5080 \zeta_{6} ) q^{79} + 729 q^{81} + ( -6924 + 13848 \zeta_{6} ) q^{83} + 10332 q^{85} + ( 234 - 468 \zeta_{6} ) q^{87} + 11586 q^{89} + ( 8008 - 16016 \zeta_{6} ) q^{91} + 7668 q^{93} + ( -2856 + 5712 \zeta_{6} ) q^{95} -13118 q^{97} + ( -324 + 648 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 84q^{5} - 54q^{9} + O(q^{10}) \) \( 2q - 84q^{5} - 54q^{9} - 364q^{13} - 492q^{17} + 792q^{21} + 2278q^{25} + 156q^{29} - 216q^{33} + 1060q^{37} - 1836q^{41} + 2268q^{45} - 6814q^{49} - 9252q^{53} - 1224q^{57} + 2692q^{61} + 15288q^{65} - 7776q^{69} - 1852q^{73} + 3168q^{77} + 1458q^{81} + 20664q^{85} + 23172q^{89} + 15336q^{93} - 26236q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 0.866025i
0.500000 0.866025i
0 5.19615i 0 −42.0000 0 76.2102i 0 −27.0000 0
31.2 0 5.19615i 0 −42.0000 0 76.2102i 0 −27.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.5.g.a 2
3.b odd 2 1 144.5.g.f 2
4.b odd 2 1 inner 48.5.g.a 2
5.b even 2 1 1200.5.e.b 2
5.c odd 4 2 1200.5.j.b 4
8.b even 2 1 192.5.g.b 2
8.d odd 2 1 192.5.g.b 2
12.b even 2 1 144.5.g.f 2
16.e even 4 2 768.5.b.c 4
16.f odd 4 2 768.5.b.c 4
20.d odd 2 1 1200.5.e.b 2
20.e even 4 2 1200.5.j.b 4
24.f even 2 1 576.5.g.d 2
24.h odd 2 1 576.5.g.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.5.g.a 2 1.a even 1 1 trivial
48.5.g.a 2 4.b odd 2 1 inner
144.5.g.f 2 3.b odd 2 1
144.5.g.f 2 12.b even 2 1
192.5.g.b 2 8.b even 2 1
192.5.g.b 2 8.d odd 2 1
576.5.g.d 2 24.f even 2 1
576.5.g.d 2 24.h odd 2 1
768.5.b.c 4 16.e even 4 2
768.5.b.c 4 16.f odd 4 2
1200.5.e.b 2 5.b even 2 1
1200.5.e.b 2 20.d odd 2 1
1200.5.j.b 4 5.c odd 4 2
1200.5.j.b 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 42 \) acting on \(S_{5}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 27 T^{2} \)
$5$ \( ( 1 + 42 T + 625 T^{2} )^{2} \)
$7$ \( 1 + 1006 T^{2} + 5764801 T^{4} \)
$11$ \( 1 - 28850 T^{2} + 214358881 T^{4} \)
$13$ \( ( 1 + 182 T + 28561 T^{2} )^{2} \)
$17$ \( ( 1 + 246 T + 83521 T^{2} )^{2} \)
$19$ \( 1 - 246770 T^{2} + 16983563041 T^{4} \)
$23$ \( 1 + 190 T^{2} + 78310985281 T^{4} \)
$29$ \( ( 1 - 78 T + 707281 T^{2} )^{2} \)
$31$ \( 1 + 330670 T^{2} + 852891037441 T^{4} \)
$37$ \( ( 1 - 530 T + 1874161 T^{2} )^{2} \)
$41$ \( ( 1 + 918 T + 2825761 T^{2} )^{2} \)
$43$ \( 1 - 6111410 T^{2} + 11688200277601 T^{4} \)
$47$ \( 1 + 4550206 T^{2} + 23811286661761 T^{4} \)
$53$ \( ( 1 + 4626 T + 7890481 T^{2} )^{2} \)
$59$ \( 1 - 24182450 T^{2} + 146830437604321 T^{4} \)
$61$ \( ( 1 - 1346 T + 13845841 T^{2} )^{2} \)
$67$ \( 1 - 39119090 T^{2} + 406067677556641 T^{4} \)
$71$ \( 1 - 47477954 T^{2} + 645753531245761 T^{4} \)
$73$ \( ( 1 + 926 T + 28398241 T^{2} )^{2} \)
$79$ \( 1 - 58545362 T^{2} + 1517108809906561 T^{4} \)
$83$ \( 1 + 48908686 T^{2} + 2252292232139041 T^{4} \)
$89$ \( ( 1 - 11586 T + 62742241 T^{2} )^{2} \)
$97$ \( ( 1 + 13118 T + 88529281 T^{2} )^{2} \)
show more
show less