Properties

Label 3072.2.a.l
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: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 768)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} + \beta_{2} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} + \beta_{2} q^{7} + q^{9} + ( -2 + \beta_{3} ) q^{11} + 3 \beta_{1} q^{13} + ( -\beta_{1} - \beta_{2} ) q^{15} -\beta_{3} q^{17} + ( -4 - \beta_{3} ) q^{19} -\beta_{2} q^{21} + 2 \beta_{1} q^{23} + ( 3 + 2 \beta_{3} ) q^{25} - q^{27} + ( \beta_{1} + 3 \beta_{2} ) q^{29} + 3 \beta_{2} q^{31} + ( 2 - \beta_{3} ) q^{33} + ( 6 + \beta_{3} ) q^{35} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{37} -3 \beta_{1} q^{39} + ( 8 - \beta_{3} ) q^{41} + \beta_{3} q^{43} + ( \beta_{1} + \beta_{2} ) q^{45} -2 \beta_{1} q^{47} - q^{49} + \beta_{3} q^{51} + ( -5 \beta_{1} + \beta_{2} ) q^{53} + 4 \beta_{1} q^{55} + ( 4 + \beta_{3} ) q^{57} -4 \beta_{3} q^{59} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{61} + \beta_{2} q^{63} + ( 6 + 3 \beta_{3} ) q^{65} + ( -8 - 2 \beta_{3} ) q^{67} -2 \beta_{1} q^{69} + ( 8 \beta_{1} - 2 \beta_{2} ) q^{71} + 4 q^{73} + ( -3 - 2 \beta_{3} ) q^{75} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{77} -\beta_{2} q^{79} + q^{81} + ( -2 - \beta_{3} ) q^{83} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{85} + ( -\beta_{1} - 3 \beta_{2} ) q^{87} + ( 2 - 2 \beta_{3} ) q^{89} + 3 \beta_{3} q^{91} -3 \beta_{2} q^{93} + ( -10 \beta_{1} - 6 \beta_{2} ) q^{95} + ( 8 - 2 \beta_{3} ) q^{97} + ( -2 + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 4q^{9} - 8q^{11} - 16q^{19} + 12q^{25} - 4q^{27} + 8q^{33} + 24q^{35} + 32q^{41} - 4q^{49} + 16q^{57} + 24q^{65} - 32q^{67} + 16q^{73} - 12q^{75} + 4q^{81} - 8q^{83} + 8q^{89} + 32q^{97} - 8q^{99} + O(q^{100}) \)

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
−1.93185
−0.517638
0.517638
1.93185
0 −1.00000 0 −3.86370 0 −2.44949 0 1.00000 0
1.2 0 −1.00000 0 −1.03528 0 −2.44949 0 1.00000 0
1.3 0 −1.00000 0 1.03528 0 2.44949 0 1.00000 0
1.4 0 −1.00000 0 3.86370 0 2.44949 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.l 4
3.b odd 2 1 9216.2.a.bi 4
4.b odd 2 1 3072.2.a.r 4
8.b even 2 1 3072.2.a.r 4
8.d odd 2 1 inner 3072.2.a.l 4
12.b even 2 1 9216.2.a.bc 4
16.e even 4 2 3072.2.d.h 8
16.f odd 4 2 3072.2.d.h 8
24.f even 2 1 9216.2.a.bi 4
24.h odd 2 1 9216.2.a.bc 4
32.g even 8 2 768.2.j.e 8
32.g even 8 2 768.2.j.f yes 8
32.h odd 8 2 768.2.j.e 8
32.h odd 8 2 768.2.j.f yes 8
96.o even 8 2 2304.2.k.e 8
96.o even 8 2 2304.2.k.l 8
96.p odd 8 2 2304.2.k.e 8
96.p odd 8 2 2304.2.k.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.e 8 32.g even 8 2
768.2.j.e 8 32.h odd 8 2
768.2.j.f yes 8 32.g even 8 2
768.2.j.f yes 8 32.h odd 8 2
2304.2.k.e 8 96.o even 8 2
2304.2.k.e 8 96.p odd 8 2
2304.2.k.l 8 96.o even 8 2
2304.2.k.l 8 96.p odd 8 2
3072.2.a.l 4 1.a even 1 1 trivial
3072.2.a.l 4 8.d odd 2 1 inner
3072.2.a.r 4 4.b odd 2 1
3072.2.a.r 4 8.b even 2 1
3072.2.d.h 8 16.e even 4 2
3072.2.d.h 8 16.f odd 4 2
9216.2.a.bc 4 12.b even 2 1
9216.2.a.bc 4 24.h odd 2 1
9216.2.a.bi 4 3.b odd 2 1
9216.2.a.bi 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3072))\):

\( T_{5}^{4} - 16 T_{5}^{2} + 16 \)
\( T_{7}^{2} - 6 \)
\( T_{11}^{2} + 4 T_{11} - 8 \)
\( T_{19}^{2} + 8 T_{19} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( 16 - 16 T^{2} + T^{4} \)
$7$ \( ( -6 + T^{2} )^{2} \)
$11$ \( ( -8 + 4 T + T^{2} )^{2} \)
$13$ \( ( -18 + T^{2} )^{2} \)
$17$ \( ( -12 + T^{2} )^{2} \)
$19$ \( ( 4 + 8 T + T^{2} )^{2} \)
$23$ \( ( -8 + T^{2} )^{2} \)
$29$ \( 2704 - 112 T^{2} + T^{4} \)
$31$ \( ( -54 + T^{2} )^{2} \)
$37$ \( 36 - 84 T^{2} + T^{4} \)
$41$ \( ( 52 - 16 T + T^{2} )^{2} \)
$43$ \( ( -12 + T^{2} )^{2} \)
$47$ \( ( -8 + T^{2} )^{2} \)
$53$ \( 1936 - 112 T^{2} + T^{4} \)
$59$ \( ( -192 + T^{2} )^{2} \)
$61$ \( 36 - 84 T^{2} + T^{4} \)
$67$ \( ( 16 + 16 T + T^{2} )^{2} \)
$71$ \( 10816 - 304 T^{2} + T^{4} \)
$73$ \( ( -4 + T )^{4} \)
$79$ \( ( -6 + T^{2} )^{2} \)
$83$ \( ( -8 + 4 T + T^{2} )^{2} \)
$89$ \( ( -44 - 4 T + T^{2} )^{2} \)
$97$ \( ( 16 - 16 T + T^{2} )^{2} \)
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