Properties

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

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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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + ( 3 - 6 \zeta_{6} ) q^{3} + 6 q^{5} + ( 36 - 72 \zeta_{6} ) q^{7} -27 q^{9} +O(q^{10})\) \( q + ( 3 - 6 \zeta_{6} ) q^{3} + 6 q^{5} + ( 36 - 72 \zeta_{6} ) q^{7} -27 q^{9} + ( 108 - 216 \zeta_{6} ) q^{11} -86 q^{13} + ( 18 - 36 \zeta_{6} ) q^{15} + 426 q^{17} + ( 36 - 72 \zeta_{6} ) q^{19} -324 q^{21} + ( -432 + 864 \zeta_{6} ) q^{23} -589 q^{25} + ( -81 + 162 \zeta_{6} ) q^{27} + 1182 q^{29} + ( -900 + 1800 \zeta_{6} ) q^{31} -972 q^{33} + ( 216 - 432 \zeta_{6} ) q^{35} -430 q^{37} + ( -258 + 516 \zeta_{6} ) q^{39} + 2250 q^{41} + ( 1548 - 3096 \zeta_{6} ) q^{43} -162 q^{45} + ( 216 - 432 \zeta_{6} ) q^{47} -1487 q^{49} + ( 1278 - 2556 \zeta_{6} ) q^{51} -1602 q^{53} + ( 648 - 1296 \zeta_{6} ) q^{55} -324 q^{57} + ( 2052 - 4104 \zeta_{6} ) q^{59} + 2114 q^{61} + ( -972 + 1944 \zeta_{6} ) q^{63} -516 q^{65} + ( -756 + 1512 \zeta_{6} ) q^{67} + 3888 q^{69} + ( -1728 + 3456 \zeta_{6} ) q^{71} + 4066 q^{73} + ( -1767 + 3534 \zeta_{6} ) q^{75} -11664 q^{77} + ( -3204 + 6408 \zeta_{6} ) q^{79} + 729 q^{81} + ( -5292 + 10584 \zeta_{6} ) q^{83} + 2556 q^{85} + ( 3546 - 7092 \zeta_{6} ) q^{87} -2046 q^{89} + ( -3096 + 6192 \zeta_{6} ) q^{91} + 8100 q^{93} + ( 216 - 432 \zeta_{6} ) q^{95} -2942 q^{97} + ( -2916 + 5832 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 12q^{5} - 54q^{9} + O(q^{10}) \) \( 2q + 12q^{5} - 54q^{9} - 172q^{13} + 852q^{17} - 648q^{21} - 1178q^{25} + 2364q^{29} - 1944q^{33} - 860q^{37} + 4500q^{41} - 324q^{45} - 2974q^{49} - 3204q^{53} - 648q^{57} + 4228q^{61} - 1032q^{65} + 7776q^{69} + 8132q^{73} - 23328q^{77} + 1458q^{81} + 5112q^{85} - 4092q^{89} + 16200q^{93} - 5884q^{97} + O(q^{100}) \)

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 6.00000 0 62.3538i 0 −27.0000 0
31.2 0 5.19615i 0 6.00000 0 62.3538i 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.b 2
3.b odd 2 1 144.5.g.d 2
4.b odd 2 1 inner 48.5.g.b 2
5.b even 2 1 1200.5.e.a 2
5.c odd 4 2 1200.5.j.a 4
8.b even 2 1 192.5.g.a 2
8.d odd 2 1 192.5.g.a 2
12.b even 2 1 144.5.g.d 2
16.e even 4 2 768.5.b.d 4
16.f odd 4 2 768.5.b.d 4
20.d odd 2 1 1200.5.e.a 2
20.e even 4 2 1200.5.j.a 4
24.f even 2 1 576.5.g.g 2
24.h odd 2 1 576.5.g.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.5.g.b 2 1.a even 1 1 trivial
48.5.g.b 2 4.b odd 2 1 inner
144.5.g.d 2 3.b odd 2 1
144.5.g.d 2 12.b even 2 1
192.5.g.a 2 8.b even 2 1
192.5.g.a 2 8.d odd 2 1
576.5.g.g 2 24.f even 2 1
576.5.g.g 2 24.h odd 2 1
768.5.b.d 4 16.e even 4 2
768.5.b.d 4 16.f odd 4 2
1200.5.e.a 2 5.b even 2 1
1200.5.e.a 2 20.d odd 2 1
1200.5.j.a 4 5.c odd 4 2
1200.5.j.a 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 6 \) acting on \(S_{5}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 27 T^{2} \)
$5$ \( ( 1 - 6 T + 625 T^{2} )^{2} \)
$7$ \( 1 - 914 T^{2} + 5764801 T^{4} \)
$11$ \( 1 + 5710 T^{2} + 214358881 T^{4} \)
$13$ \( ( 1 + 86 T + 28561 T^{2} )^{2} \)
$17$ \( ( 1 - 426 T + 83521 T^{2} )^{2} \)
$19$ \( 1 - 256754 T^{2} + 16983563041 T^{4} \)
$23$ \( 1 + 190 T^{2} + 78310985281 T^{4} \)
$29$ \( ( 1 - 1182 T + 707281 T^{2} )^{2} \)
$31$ \( 1 + 582958 T^{2} + 852891037441 T^{4} \)
$37$ \( ( 1 + 430 T + 1874161 T^{2} )^{2} \)
$41$ \( ( 1 - 2250 T + 2825761 T^{2} )^{2} \)
$43$ \( 1 + 351310 T^{2} + 11688200277601 T^{4} \)
$47$ \( 1 - 9619394 T^{2} + 23811286661761 T^{4} \)
$53$ \( ( 1 + 1602 T + 7890481 T^{2} )^{2} \)
$59$ \( 1 - 11602610 T^{2} + 146830437604321 T^{4} \)
$61$ \( ( 1 - 2114 T + 13845841 T^{2} )^{2} \)
$67$ \( 1 - 38587634 T^{2} + 406067677556641 T^{4} \)
$71$ \( 1 - 41865410 T^{2} + 645753531245761 T^{4} \)
$73$ \( ( 1 - 4066 T + 28398241 T^{2} )^{2} \)
$79$ \( 1 - 47103314 T^{2} + 1517108809906561 T^{4} \)
$83$ \( 1 - 10900850 T^{2} + 2252292232139041 T^{4} \)
$89$ \( ( 1 + 2046 T + 62742241 T^{2} )^{2} \)
$97$ \( ( 1 + 2942 T + 88529281 T^{2} )^{2} \)
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