Properties

Label 48.4.c.b
Level 48
Weight 4
Character orbit 48.c
Analytic conductor 2.832
Analytic rank 0
Dimension 4
CM no
Inner twists 4

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} -\beta_{3} q^{5} -5 \beta_{1} q^{7} + ( 21 - \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} -\beta_{3} q^{5} -5 \beta_{1} q^{7} + ( 21 - \beta_{3} ) q^{9} + ( -3 \beta_{1} - 6 \beta_{2} ) q^{11} -26 q^{13} + ( 27 \beta_{1} + 6 \beta_{2} ) q^{15} + 4 \beta_{3} q^{17} -31 \beta_{1} q^{19} + ( 30 + 5 \beta_{3} ) q^{21} + ( 18 \beta_{1} + 36 \beta_{2} ) q^{23} -163 q^{25} + ( 27 \beta_{1} - 15 \beta_{2} ) q^{27} + \beta_{3} q^{29} -9 \beta_{1} q^{31} + ( 144 - 3 \beta_{3} ) q^{33} + ( -30 \beta_{1} - 60 \beta_{2} ) q^{35} + 206 q^{37} + 26 \beta_{2} q^{39} -18 \beta_{3} q^{41} -27 \beta_{1} q^{43} + ( -288 - 21 \beta_{3} ) q^{45} + ( -12 \beta_{1} - 24 \beta_{2} ) q^{47} + 43 q^{49} + ( -108 \beta_{1} - 24 \beta_{2} ) q^{51} + 3 \beta_{3} q^{53} + 144 \beta_{1} q^{55} + ( 186 + 31 \beta_{3} ) q^{57} + ( 57 \beta_{1} + 114 \beta_{2} ) q^{59} + 278 q^{61} + ( -135 \beta_{1} - 60 \beta_{2} ) q^{63} + 26 \beta_{3} q^{65} + 257 \beta_{1} q^{67} + ( -864 + 18 \beta_{3} ) q^{69} + ( 6 \beta_{1} + 12 \beta_{2} ) q^{71} -422 q^{73} + 163 \beta_{2} q^{75} + 30 \beta_{3} q^{77} -193 \beta_{1} q^{79} + ( 153 - 42 \beta_{3} ) q^{81} + ( 3 \beta_{1} + 6 \beta_{2} ) q^{83} + 1152 q^{85} + ( -27 \beta_{1} - 6 \beta_{2} ) q^{87} -22 \beta_{3} q^{89} + 130 \beta_{1} q^{91} + ( 54 + 9 \beta_{3} ) q^{93} + ( -186 \beta_{1} - 372 \beta_{2} ) q^{95} -1070 q^{97} + ( 81 \beta_{1} - 126 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 84q^{9} + O(q^{10}) \) \( 4q + 84q^{9} - 104q^{13} + 120q^{21} - 652q^{25} + 576q^{33} + 824q^{37} - 1152q^{45} + 172q^{49} + 744q^{57} + 1112q^{61} - 3456q^{69} - 1688q^{73} + 612q^{81} + 4608q^{85} + 216q^{93} - 4280q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{2} - 2 \)
\(\beta_{2}\)\(=\)\( -\nu^{3} - \nu^{2} + 4 \nu + 1 \)
\(\beta_{3}\)\(=\)\( 6 \nu^{3} \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 6 \beta_{2} + 3 \beta_{1}\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{1} + 2\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)\(/6\)

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
1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
−1.22474 0.707107i
0 −4.89898 1.73205i 0 16.9706i 0 17.3205i 0 21.0000 + 16.9706i 0
47.2 0 −4.89898 + 1.73205i 0 16.9706i 0 17.3205i 0 21.0000 16.9706i 0
47.3 0 4.89898 1.73205i 0 16.9706i 0 17.3205i 0 21.0000 16.9706i 0
47.4 0 4.89898 + 1.73205i 0 16.9706i 0 17.3205i 0 21.0000 + 16.9706i 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 inner
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.b 4
3.b odd 2 1 inner 48.4.c.b 4
4.b odd 2 1 inner 48.4.c.b 4
8.b even 2 1 192.4.c.c 4
8.d odd 2 1 192.4.c.c 4
12.b even 2 1 inner 48.4.c.b 4
16.e even 4 2 768.4.f.b 8
16.f odd 4 2 768.4.f.b 8
24.f even 2 1 192.4.c.c 4
24.h odd 2 1 192.4.c.c 4
48.i odd 4 2 768.4.f.b 8
48.k even 4 2 768.4.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.4.c.b 4 1.a even 1 1 trivial
48.4.c.b 4 3.b odd 2 1 inner
48.4.c.b 4 4.b odd 2 1 inner
48.4.c.b 4 12.b even 2 1 inner
192.4.c.c 4 8.b even 2 1
192.4.c.c 4 8.d odd 2 1
192.4.c.c 4 24.f even 2 1
192.4.c.c 4 24.h odd 2 1
768.4.f.b 8 16.e even 4 2
768.4.f.b 8 16.f odd 4 2
768.4.f.b 8 48.i odd 4 2
768.4.f.b 8 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 288 \) acting on \(S_{4}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 42 T^{2} + 729 T^{4} \)
$5$ \( ( 1 + 38 T^{2} + 15625 T^{4} )^{2} \)
$7$ \( ( 1 - 386 T^{2} + 117649 T^{4} )^{2} \)
$11$ \( ( 1 + 1798 T^{2} + 1771561 T^{4} )^{2} \)
$13$ \( ( 1 + 26 T + 2197 T^{2} )^{4} \)
$17$ \( ( 1 - 5218 T^{2} + 24137569 T^{4} )^{2} \)
$19$ \( ( 1 - 2186 T^{2} + 47045881 T^{4} )^{2} \)
$23$ \( ( 1 - 6770 T^{2} + 148035889 T^{4} )^{2} \)
$29$ \( ( 1 - 48490 T^{2} + 594823321 T^{4} )^{2} \)
$31$ \( ( 1 - 58610 T^{2} + 887503681 T^{4} )^{2} \)
$37$ \( ( 1 - 206 T + 50653 T^{2} )^{4} \)
$41$ \( ( 1 - 44530 T^{2} + 4750104241 T^{4} )^{2} \)
$43$ \( ( 1 - 150266 T^{2} + 6321363049 T^{4} )^{2} \)
$47$ \( ( 1 + 193822 T^{2} + 10779215329 T^{4} )^{2} \)
$53$ \( ( 1 - 295162 T^{2} + 22164361129 T^{4} )^{2} \)
$59$ \( ( 1 + 98854 T^{2} + 42180533641 T^{4} )^{2} \)
$61$ \( ( 1 - 278 T + 226981 T^{2} )^{4} \)
$67$ \( ( 1 + 191062 T^{2} + 90458382169 T^{4} )^{2} \)
$71$ \( ( 1 + 712366 T^{2} + 128100283921 T^{4} )^{2} \)
$73$ \( ( 1 + 422 T + 389017 T^{2} )^{4} \)
$79$ \( ( 1 - 539090 T^{2} + 243087455521 T^{4} )^{2} \)
$83$ \( ( 1 + 1142710 T^{2} + 326940373369 T^{4} )^{2} \)
$89$ \( ( 1 - 1270546 T^{2} + 496981290961 T^{4} )^{2} \)
$97$ \( ( 1 + 1070 T + 912673 T^{2} )^{4} \)
show more
show less