Properties

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

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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: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \(x^{4} - 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1536)
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{2} q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{2} q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + q^{9} + ( 2 + \beta_{3} ) q^{11} -\beta_{1} q^{13} + \beta_{2} q^{15} + ( -2 + \beta_{3} ) q^{17} + ( -2 - \beta_{3} ) q^{19} + ( -\beta_{1} + \beta_{2} ) q^{21} + 4 \beta_{1} q^{23} + 5 q^{25} + q^{27} + ( 2 \beta_{1} + \beta_{2} ) q^{29} + ( -5 \beta_{1} + \beta_{2} ) q^{31} + ( 2 + \beta_{3} ) q^{33} + ( 10 - \beta_{3} ) q^{35} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{37} -\beta_{1} q^{39} + ( 2 - \beta_{3} ) q^{41} + ( 6 - \beta_{3} ) q^{43} + \beta_{2} q^{45} + ( 5 - 2 \beta_{3} ) q^{49} + ( -2 + \beta_{3} ) q^{51} + ( -6 \beta_{1} - \beta_{2} ) q^{53} + ( 10 \beta_{1} + 2 \beta_{2} ) q^{55} + ( -2 - \beta_{3} ) q^{57} + 2 \beta_{3} q^{59} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -\beta_{1} + \beta_{2} ) q^{63} -\beta_{3} q^{65} + 12 q^{67} + 4 \beta_{1} q^{69} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -6 + 2 \beta_{3} ) q^{73} + 5 q^{75} + 8 \beta_{1} q^{77} + ( -3 \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( -2 - \beta_{3} ) q^{83} + ( 10 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 2 \beta_{1} + \beta_{2} ) q^{87} -10 q^{89} + ( 2 - \beta_{3} ) q^{91} + ( -5 \beta_{1} + \beta_{2} ) q^{93} + ( -10 \beta_{1} - 2 \beta_{2} ) q^{95} + ( -4 - 2 \beta_{3} ) q^{97} + ( 2 + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{9} + 8q^{11} - 8q^{17} - 8q^{19} + 20q^{25} + 4q^{27} + 8q^{33} + 40q^{35} + 8q^{41} + 24q^{43} + 20q^{49} - 8q^{51} - 8q^{57} + 48q^{67} - 24q^{73} + 20q^{75} + 4q^{81} - 8q^{83} - 40q^{89} + 8q^{91} - 16q^{97} + 8q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 4 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 8 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 6\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} + 4 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.874032
−2.28825
2.28825
0.874032
0 1.00000 0 −3.16228 0 −4.57649 0 1.00000 0
1.2 0 1.00000 0 −3.16228 0 −1.74806 0 1.00000 0
1.3 0 1.00000 0 3.16228 0 1.74806 0 1.00000 0
1.4 0 1.00000 0 3.16228 0 4.57649 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.2.a.q 4
3.b odd 2 1 9216.2.a.bd 4
4.b odd 2 1 3072.2.a.k 4
8.b even 2 1 3072.2.a.k 4
8.d odd 2 1 inner 3072.2.a.q 4
12.b even 2 1 9216.2.a.bj 4
16.e even 4 2 3072.2.d.g 8
16.f odd 4 2 3072.2.d.g 8
24.f even 2 1 9216.2.a.bd 4
24.h odd 2 1 9216.2.a.bj 4
32.g even 8 2 1536.2.j.g 8
32.g even 8 2 1536.2.j.h yes 8
32.h odd 8 2 1536.2.j.g 8
32.h odd 8 2 1536.2.j.h yes 8
96.o even 8 2 4608.2.k.bf 8
96.o even 8 2 4608.2.k.bg 8
96.p odd 8 2 4608.2.k.bf 8
96.p odd 8 2 4608.2.k.bg 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.g 8 32.g even 8 2
1536.2.j.g 8 32.h odd 8 2
1536.2.j.h yes 8 32.g even 8 2
1536.2.j.h yes 8 32.h odd 8 2
3072.2.a.k 4 4.b odd 2 1
3072.2.a.k 4 8.b even 2 1
3072.2.a.q 4 1.a even 1 1 trivial
3072.2.a.q 4 8.d odd 2 1 inner
3072.2.d.g 8 16.e even 4 2
3072.2.d.g 8 16.f odd 4 2
4608.2.k.bf 8 96.o even 8 2
4608.2.k.bf 8 96.p odd 8 2
4608.2.k.bg 8 96.o even 8 2
4608.2.k.bg 8 96.p odd 8 2
9216.2.a.bd 4 3.b odd 2 1
9216.2.a.bd 4 24.f even 2 1
9216.2.a.bj 4 12.b even 2 1
9216.2.a.bj 4 24.h odd 2 1

Hecke kernels

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

\( T_{5}^{2} - 10 \)
\( T_{7}^{4} - 24 T_{7}^{2} + 64 \)
\( T_{11}^{2} - 4 T_{11} - 16 \)
\( T_{19}^{2} + 4 T_{19} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( ( -10 + T^{2} )^{2} \)
$7$ \( 64 - 24 T^{2} + T^{4} \)
$11$ \( ( -16 - 4 T + T^{2} )^{2} \)
$13$ \( ( -2 + T^{2} )^{2} \)
$17$ \( ( -16 + 4 T + T^{2} )^{2} \)
$19$ \( ( -16 + 4 T + T^{2} )^{2} \)
$23$ \( ( -32 + T^{2} )^{2} \)
$29$ \( 4 - 36 T^{2} + T^{4} \)
$31$ \( 1600 - 120 T^{2} + T^{4} \)
$37$ \( 484 - 116 T^{2} + T^{4} \)
$41$ \( ( -16 - 4 T + T^{2} )^{2} \)
$43$ \( ( 16 - 12 T + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( 3844 - 164 T^{2} + T^{4} \)
$59$ \( ( -80 + T^{2} )^{2} \)
$61$ \( 484 - 116 T^{2} + T^{4} \)
$67$ \( ( -12 + T )^{4} \)
$71$ \( 1024 - 96 T^{2} + T^{4} \)
$73$ \( ( -44 + 12 T + T^{2} )^{2} \)
$79$ \( 64 - 56 T^{2} + T^{4} \)
$83$ \( ( -16 + 4 T + T^{2} )^{2} \)
$89$ \( ( 10 + T )^{4} \)
$97$ \( ( -64 + 8 T + T^{2} )^{2} \)
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