Properties

Label 48.3.g.a
Level 48
Weight 3
Character orbit 48.g
Analytic conductor 1.308
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.30790526893\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + 6 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + 6 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} -3 q^{9} + ( -12 + 24 \zeta_{6} ) q^{11} -14 q^{13} + ( 6 - 12 \zeta_{6} ) q^{15} -6 q^{17} + ( -4 + 8 \zeta_{6} ) q^{19} -12 q^{21} + 11 q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + 30 q^{29} + ( 12 - 24 \zeta_{6} ) q^{31} + 36 q^{33} + ( 24 - 48 \zeta_{6} ) q^{35} + 26 q^{37} + ( -14 + 28 \zeta_{6} ) q^{39} -54 q^{41} + ( -12 + 24 \zeta_{6} ) q^{43} -18 q^{45} + ( 24 - 48 \zeta_{6} ) q^{47} + q^{49} + ( -6 + 12 \zeta_{6} ) q^{51} -18 q^{53} + ( -72 + 144 \zeta_{6} ) q^{55} + 12 q^{57} + ( 12 - 24 \zeta_{6} ) q^{59} -70 q^{61} + ( -12 + 24 \zeta_{6} ) q^{63} -84 q^{65} + ( 68 - 136 \zeta_{6} ) q^{67} + ( -48 + 96 \zeta_{6} ) q^{71} + 82 q^{73} + ( 11 - 22 \zeta_{6} ) q^{75} + 144 q^{77} + ( 44 - 88 \zeta_{6} ) q^{79} + 9 q^{81} + ( 12 - 24 \zeta_{6} ) q^{83} -36 q^{85} + ( 30 - 60 \zeta_{6} ) q^{87} + 114 q^{89} + ( -56 + 112 \zeta_{6} ) q^{91} -36 q^{93} + ( -24 + 48 \zeta_{6} ) q^{95} + 34 q^{97} + ( 36 - 72 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 12q^{5} - 6q^{9} + O(q^{10}) \) \( 2q + 12q^{5} - 6q^{9} - 28q^{13} - 12q^{17} - 24q^{21} + 22q^{25} + 60q^{29} + 72q^{33} + 52q^{37} - 108q^{41} - 36q^{45} + 2q^{49} - 36q^{53} + 24q^{57} - 140q^{61} - 168q^{65} + 164q^{73} + 288q^{77} + 18q^{81} - 72q^{85} + 228q^{89} - 72q^{93} + 68q^{97} + O(q^{100}) \)

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 1.73205i 0 6.00000 0 6.92820i 0 −3.00000 0
31.2 0 1.73205i 0 6.00000 0 6.92820i 0 −3.00000 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 48.3.g.a 2
3.b odd 2 1 144.3.g.b 2
4.b odd 2 1 inner 48.3.g.a 2
5.b even 2 1 1200.3.e.h 2
5.c odd 4 2 1200.3.j.a 4
7.b odd 2 1 2352.3.m.a 2
8.b even 2 1 192.3.g.a 2
8.d odd 2 1 192.3.g.a 2
9.c even 3 1 1296.3.o.a 2
9.c even 3 1 1296.3.o.c 2
9.d odd 6 1 1296.3.o.n 2
9.d odd 6 1 1296.3.o.p 2
12.b even 2 1 144.3.g.b 2
15.d odd 2 1 3600.3.e.t 2
15.e even 4 2 3600.3.j.i 4
16.e even 4 2 768.3.b.b 4
16.f odd 4 2 768.3.b.b 4
20.d odd 2 1 1200.3.e.h 2
20.e even 4 2 1200.3.j.a 4
24.f even 2 1 576.3.g.i 2
24.h odd 2 1 576.3.g.i 2
28.d even 2 1 2352.3.m.a 2
36.f odd 6 1 1296.3.o.a 2
36.f odd 6 1 1296.3.o.c 2
36.h even 6 1 1296.3.o.n 2
36.h even 6 1 1296.3.o.p 2
48.i odd 4 2 2304.3.b.n 4
48.k even 4 2 2304.3.b.n 4
60.h even 2 1 3600.3.e.t 2
60.l odd 4 2 3600.3.j.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.g.a 2 1.a even 1 1 trivial
48.3.g.a 2 4.b odd 2 1 inner
144.3.g.b 2 3.b odd 2 1
144.3.g.b 2 12.b even 2 1
192.3.g.a 2 8.b even 2 1
192.3.g.a 2 8.d odd 2 1
576.3.g.i 2 24.f even 2 1
576.3.g.i 2 24.h odd 2 1
768.3.b.b 4 16.e even 4 2
768.3.b.b 4 16.f odd 4 2
1200.3.e.h 2 5.b even 2 1
1200.3.e.h 2 20.d odd 2 1
1200.3.j.a 4 5.c odd 4 2
1200.3.j.a 4 20.e even 4 2
1296.3.o.a 2 9.c even 3 1
1296.3.o.a 2 36.f odd 6 1
1296.3.o.c 2 9.c even 3 1
1296.3.o.c 2 36.f odd 6 1
1296.3.o.n 2 9.d odd 6 1
1296.3.o.n 2 36.h even 6 1
1296.3.o.p 2 9.d odd 6 1
1296.3.o.p 2 36.h even 6 1
2304.3.b.n 4 48.i odd 4 2
2304.3.b.n 4 48.k even 4 2
2352.3.m.a 2 7.b odd 2 1
2352.3.m.a 2 28.d even 2 1
3600.3.e.t 2 15.d odd 2 1
3600.3.e.t 2 60.h even 2 1
3600.3.j.i 4 15.e even 4 2
3600.3.j.i 4 60.l odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} \)
$5$ \( ( 1 - 6 T + 25 T^{2} )^{2} \)
$7$ \( 1 - 50 T^{2} + 2401 T^{4} \)
$11$ \( 1 + 190 T^{2} + 14641 T^{4} \)
$13$ \( ( 1 + 14 T + 169 T^{2} )^{2} \)
$17$ \( ( 1 + 6 T + 289 T^{2} )^{2} \)
$19$ \( 1 - 674 T^{2} + 130321 T^{4} \)
$23$ \( ( 1 - 23 T )^{2}( 1 + 23 T )^{2} \)
$29$ \( ( 1 - 30 T + 841 T^{2} )^{2} \)
$31$ \( 1 - 1490 T^{2} + 923521 T^{4} \)
$37$ \( ( 1 - 26 T + 1369 T^{2} )^{2} \)
$41$ \( ( 1 + 54 T + 1681 T^{2} )^{2} \)
$43$ \( 1 - 3266 T^{2} + 3418801 T^{4} \)
$47$ \( 1 - 2690 T^{2} + 4879681 T^{4} \)
$53$ \( ( 1 + 18 T + 2809 T^{2} )^{2} \)
$59$ \( 1 - 6530 T^{2} + 12117361 T^{4} \)
$61$ \( ( 1 + 70 T + 3721 T^{2} )^{2} \)
$67$ \( 1 + 4894 T^{2} + 20151121 T^{4} \)
$71$ \( 1 - 3170 T^{2} + 25411681 T^{4} \)
$73$ \( ( 1 - 82 T + 5329 T^{2} )^{2} \)
$79$ \( 1 - 6674 T^{2} + 38950081 T^{4} \)
$83$ \( 1 - 13346 T^{2} + 47458321 T^{4} \)
$89$ \( ( 1 - 114 T + 7921 T^{2} )^{2} \)
$97$ \( ( 1 - 34 T + 9409 T^{2} )^{2} \)
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