Properties

Label 3072.2.a.n
Level $3072$
Weight $2$
Character orbit 3072.a
Self dual yes
Analytic conductor $24.530$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 48)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 + \beta_{3} ) q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 + \beta_{3} ) q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} + ( 2 - \beta_{2} - \beta_{3} ) q^{13} + ( -1 - \beta_{3} ) q^{15} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + ( 1 + \beta_{2} ) q^{21} + 2 \beta_{1} q^{23} + ( 1 + 2 \beta_{1} + 2 \beta_{3} ) q^{25} - q^{27} + ( 3 - 2 \beta_{1} - \beta_{3} ) q^{29} + ( -3 + \beta_{2} ) q^{31} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{33} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{35} + ( 4 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{37} + ( -2 + \beta_{2} + \beta_{3} ) q^{39} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{41} + ( \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{43} + ( 1 + \beta_{3} ) q^{45} -2 \beta_{1} q^{47} + ( 1 - 4 \beta_{1} + 2 \beta_{2} ) q^{49} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( 5 + 2 \beta_{1} + \beta_{3} ) q^{53} + ( -4 - 4 \beta_{1} ) q^{55} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{57} + 4 \beta_{1} q^{59} + ( 4 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{61} + ( -1 - \beta_{2} ) q^{63} + ( -2 - 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{65} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{67} -2 \beta_{1} q^{69} + ( 2 + 2 \beta_{2} ) q^{71} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -1 - 2 \beta_{1} - 2 \beta_{3} ) q^{75} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( -3 + 4 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{79} + q^{81} + ( -5 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{83} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{85} + ( -3 + 2 \beta_{1} + \beta_{3} ) q^{87} + ( -2 + 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{89} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{91} + ( 3 - \beta_{2} ) q^{93} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{95} + ( -6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{97} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 4q^{5} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 4q^{5} - 4q^{7} + 4q^{9} + 8q^{13} - 4q^{15} + 4q^{21} + 4q^{25} - 4q^{27} + 12q^{29} - 12q^{31} + 16q^{37} - 8q^{39} + 4q^{45} + 4q^{49} + 20q^{53} - 16q^{55} + 16q^{61} - 4q^{63} - 8q^{65} + 16q^{67} + 8q^{71} - 8q^{73} - 4q^{75} + 24q^{77} - 12q^{79} + 4q^{81} + 24q^{85} - 12q^{87} - 8q^{89} + 16q^{91} + 12q^{93} + 24q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 6 x^{2} - 4 x + 2\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.27133
0.334904
−1.74912
2.68554
0 −1.00000 0 −1.79793 0 0.158942 0 1.00000 0
1.2 0 −1.00000 0 −0.473626 0 −4.55765 0 1.00000 0
1.3 0 −1.00000 0 2.47363 0 2.55765 0 1.00000 0
1.4 0 −1.00000 0 3.79793 0 −2.15894 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.2.a.n 4
3.b odd 2 1 9216.2.a.x 4
4.b odd 2 1 3072.2.a.t 4
8.b even 2 1 3072.2.a.o 4
8.d odd 2 1 3072.2.a.i 4
12.b even 2 1 9216.2.a.y 4
16.e even 4 2 3072.2.d.i 8
16.f odd 4 2 3072.2.d.f 8
24.f even 2 1 9216.2.a.bo 4
24.h odd 2 1 9216.2.a.bn 4
32.g even 8 2 192.2.j.a 8
32.g even 8 2 384.2.j.a 8
32.h odd 8 2 48.2.j.a 8
32.h odd 8 2 384.2.j.b 8
96.o even 8 2 144.2.k.b 8
96.o even 8 2 1152.2.k.c 8
96.p odd 8 2 576.2.k.b 8
96.p odd 8 2 1152.2.k.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 32.h odd 8 2
144.2.k.b 8 96.o even 8 2
192.2.j.a 8 32.g even 8 2
384.2.j.a 8 32.g even 8 2
384.2.j.b 8 32.h odd 8 2
576.2.k.b 8 96.p odd 8 2
1152.2.k.c 8 96.o even 8 2
1152.2.k.f 8 96.p odd 8 2
3072.2.a.i 4 8.d odd 2 1
3072.2.a.n 4 1.a even 1 1 trivial
3072.2.a.o 4 8.b even 2 1
3072.2.a.t 4 4.b odd 2 1
3072.2.d.f 8 16.f odd 4 2
3072.2.d.i 8 16.e even 4 2
9216.2.a.x 4 3.b odd 2 1
9216.2.a.y 4 12.b even 2 1
9216.2.a.bn 4 24.h odd 2 1
9216.2.a.bo 4 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3072))\):

\( T_{5}^{4} - 4 T_{5}^{3} - 4 T_{5}^{2} + 16 T_{5} + 8 \)
\( T_{7}^{4} + 4 T_{7}^{3} - 8 T_{7}^{2} - 24 T_{7} + 4 \)
\( T_{11}^{4} - 24 T_{11}^{2} - 32 T_{11} + 32 \)
\( T_{19}^{4} - 32 T_{19}^{2} - 64 T_{19} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( 8 + 16 T - 4 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 4 - 24 T - 8 T^{2} + 4 T^{3} + T^{4} \)
$11$ \( 32 - 32 T - 24 T^{2} + T^{4} \)
$13$ \( 4 + 48 T + 4 T^{2} - 8 T^{3} + T^{4} \)
$17$ \( 16 - 64 T - 32 T^{2} + T^{4} \)
$19$ \( 16 - 64 T - 32 T^{2} + T^{4} \)
$23$ \( ( -8 + T^{2} )^{2} \)
$29$ \( -248 + 80 T + 28 T^{2} - 12 T^{3} + T^{4} \)
$31$ \( -28 + 24 T + 40 T^{2} + 12 T^{3} + T^{4} \)
$37$ \( -1052 + 224 T + 52 T^{2} - 16 T^{3} + T^{4} \)
$41$ \( -112 + 192 T - 64 T^{2} + T^{4} \)
$43$ \( -112 - 256 T - 96 T^{2} + T^{4} \)
$47$ \( ( -8 + T^{2} )^{2} \)
$53$ \( 136 - 272 T + 124 T^{2} - 20 T^{3} + T^{4} \)
$59$ \( ( -32 + T^{2} )^{2} \)
$61$ \( -1052 + 224 T + 52 T^{2} - 16 T^{3} + T^{4} \)
$67$ \( 256 + 256 T - 16 T^{3} + T^{4} \)
$71$ \( 64 + 192 T - 32 T^{2} - 8 T^{3} + T^{4} \)
$73$ \( 64 + 64 T - 96 T^{2} + 8 T^{3} + T^{4} \)
$79$ \( -10108 - 2888 T - 168 T^{2} + 12 T^{3} + T^{4} \)
$83$ \( 32 - 160 T - 216 T^{2} + T^{4} \)
$89$ \( -1904 - 1632 T - 200 T^{2} + 8 T^{3} + T^{4} \)
$97$ \( 512 + 768 T - 224 T^{2} + T^{4} \)
show more
show less