Properties

Label 48.4.c.a
Level 48
Weight 4
Character orbit 48.c
Analytic conductor 2.832
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.83209168028\)
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: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 3 - 6 \zeta_{6} ) q^{3} + ( 18 - 36 \zeta_{6} ) q^{7} -27 q^{9} +O(q^{10})\) \( q + ( 3 - 6 \zeta_{6} ) q^{3} + ( 18 - 36 \zeta_{6} ) q^{7} -27 q^{9} + 70 q^{13} + ( -90 + 180 \zeta_{6} ) q^{19} -162 q^{21} + 125 q^{25} + ( -81 + 162 \zeta_{6} ) q^{27} + ( 90 - 180 \zeta_{6} ) q^{31} + 110 q^{37} + ( 210 - 420 \zeta_{6} ) q^{39} + ( 126 - 252 \zeta_{6} ) q^{43} -629 q^{49} + 810 q^{57} + 182 q^{61} + ( -486 + 972 \zeta_{6} ) q^{63} + ( -378 + 756 \zeta_{6} ) q^{67} -1190 q^{73} + ( 375 - 750 \zeta_{6} ) q^{75} + ( -630 + 1260 \zeta_{6} ) q^{79} + 729 q^{81} + ( 1260 - 2520 \zeta_{6} ) q^{91} -810 q^{93} + 1330 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 54q^{9} + O(q^{10}) \) \( 2q - 54q^{9} + 140q^{13} - 324q^{21} + 250q^{25} + 220q^{37} - 1258q^{49} + 1620q^{57} + 364q^{61} - 2380q^{73} + 1458q^{81} - 1620q^{93} + 2660q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
0.500000 + 0.866025i
0.500000 0.866025i
0 5.19615i 0 0 0 31.1769i 0 −27.0000 0
47.2 0 5.19615i 0 0 0 31.1769i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.4.c.a 2
3.b odd 2 1 CM 48.4.c.a 2
4.b odd 2 1 inner 48.4.c.a 2
8.b even 2 1 192.4.c.a 2
8.d odd 2 1 192.4.c.a 2
12.b even 2 1 inner 48.4.c.a 2
16.e even 4 2 768.4.f.a 4
16.f odd 4 2 768.4.f.a 4
24.f even 2 1 192.4.c.a 2
24.h odd 2 1 192.4.c.a 2
48.i odd 4 2 768.4.f.a 4
48.k even 4 2 768.4.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.4.c.a 2 1.a even 1 1 trivial
48.4.c.a 2 3.b odd 2 1 CM
48.4.c.a 2 4.b odd 2 1 inner
48.4.c.a 2 12.b even 2 1 inner
192.4.c.a 2 8.b even 2 1
192.4.c.a 2 8.d odd 2 1
192.4.c.a 2 24.f even 2 1
192.4.c.a 2 24.h odd 2 1
768.4.f.a 4 16.e even 4 2
768.4.f.a 4 16.f odd 4 2
768.4.f.a 4 48.i odd 4 2
768.4.f.a 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{4}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 27 T^{2} \)
$5$ \( ( 1 - 125 T^{2} )^{2} \)
$7$ \( ( 1 - 20 T + 343 T^{2} )( 1 + 20 T + 343 T^{2} ) \)
$11$ \( ( 1 + 1331 T^{2} )^{2} \)
$13$ \( ( 1 - 70 T + 2197 T^{2} )^{2} \)
$17$ \( ( 1 - 4913 T^{2} )^{2} \)
$19$ \( ( 1 - 56 T + 6859 T^{2} )( 1 + 56 T + 6859 T^{2} ) \)
$23$ \( ( 1 + 12167 T^{2} )^{2} \)
$29$ \( ( 1 - 24389 T^{2} )^{2} \)
$31$ \( ( 1 - 308 T + 29791 T^{2} )( 1 + 308 T + 29791 T^{2} ) \)
$37$ \( ( 1 - 110 T + 50653 T^{2} )^{2} \)
$41$ \( ( 1 - 68921 T^{2} )^{2} \)
$43$ \( ( 1 - 520 T + 79507 T^{2} )( 1 + 520 T + 79507 T^{2} ) \)
$47$ \( ( 1 + 103823 T^{2} )^{2} \)
$53$ \( ( 1 - 148877 T^{2} )^{2} \)
$59$ \( ( 1 + 205379 T^{2} )^{2} \)
$61$ \( ( 1 - 182 T + 226981 T^{2} )^{2} \)
$67$ \( ( 1 - 880 T + 300763 T^{2} )( 1 + 880 T + 300763 T^{2} ) \)
$71$ \( ( 1 + 357911 T^{2} )^{2} \)
$73$ \( ( 1 + 1190 T + 389017 T^{2} )^{2} \)
$79$ \( ( 1 - 884 T + 493039 T^{2} )( 1 + 884 T + 493039 T^{2} ) \)
$83$ \( ( 1 + 571787 T^{2} )^{2} \)
$89$ \( ( 1 - 704969 T^{2} )^{2} \)
$97$ \( ( 1 - 1330 T + 912673 T^{2} )^{2} \)
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