Properties

Label 3072.2.a.g
Level $3072$
Weight $2$
Character orbit 3072.a
Self dual yes
Analytic conductor $24.530$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} -\beta q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} -\beta q^{7} + q^{9} + \beta q^{13} -6 q^{17} -6 q^{19} -\beta q^{21} + 6 \beta q^{23} -5 q^{25} + q^{27} -6 \beta q^{29} + \beta q^{31} + 5 \beta q^{37} + \beta q^{39} -6 q^{41} -6 q^{43} -6 \beta q^{47} -5 q^{49} -6 q^{51} + 6 \beta q^{53} -6 q^{57} -5 \beta q^{61} -\beta q^{63} + 4 q^{67} + 6 \beta q^{69} -6 \beta q^{71} -5 q^{75} + \beta q^{79} + q^{81} + 12 q^{83} -6 \beta q^{87} + 6 q^{89} -2 q^{91} + \beta q^{93} -12 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{9} - 12q^{17} - 12q^{19} - 10q^{25} + 2q^{27} - 12q^{41} - 12q^{43} - 10q^{49} - 12q^{51} - 12q^{57} + 8q^{67} - 10q^{75} + 2q^{81} + 24q^{83} + 12q^{89} - 4q^{91} - 24q^{97} + O(q^{100}) \)

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
1.41421
−1.41421
0 1.00000 0 0 0 −1.41421 0 1.00000 0
1.2 0 1.00000 0 0 0 1.41421 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.g 2
3.b odd 2 1 9216.2.a.n 2
4.b odd 2 1 3072.2.a.a 2
8.b even 2 1 3072.2.a.a 2
8.d odd 2 1 inner 3072.2.a.g 2
12.b even 2 1 9216.2.a.o 2
16.e even 4 2 3072.2.d.a 4
16.f odd 4 2 3072.2.d.a 4
24.f even 2 1 9216.2.a.n 2
24.h odd 2 1 9216.2.a.o 2
32.g even 8 2 768.2.j.b 4
32.g even 8 2 768.2.j.c yes 4
32.h odd 8 2 768.2.j.b 4
32.h odd 8 2 768.2.j.c yes 4
96.o even 8 2 2304.2.k.b 4
96.o even 8 2 2304.2.k.c 4
96.p odd 8 2 2304.2.k.b 4
96.p odd 8 2 2304.2.k.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.b 4 32.g even 8 2
768.2.j.b 4 32.h odd 8 2
768.2.j.c yes 4 32.g even 8 2
768.2.j.c yes 4 32.h odd 8 2
2304.2.k.b 4 96.o even 8 2
2304.2.k.b 4 96.p odd 8 2
2304.2.k.c 4 96.o even 8 2
2304.2.k.c 4 96.p odd 8 2
3072.2.a.a 2 4.b odd 2 1
3072.2.a.a 2 8.b even 2 1
3072.2.a.g 2 1.a even 1 1 trivial
3072.2.a.g 2 8.d odd 2 1 inner
3072.2.d.a 4 16.e even 4 2
3072.2.d.a 4 16.f odd 4 2
9216.2.a.n 2 3.b odd 2 1
9216.2.a.n 2 24.f even 2 1
9216.2.a.o 2 12.b even 2 1
9216.2.a.o 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3072))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -2 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -2 + T^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( ( 6 + T )^{2} \)
$23$ \( -72 + T^{2} \)
$29$ \( -72 + T^{2} \)
$31$ \( -2 + T^{2} \)
$37$ \( -50 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( 6 + T )^{2} \)
$47$ \( -72 + T^{2} \)
$53$ \( -72 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( -50 + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( -72 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( -2 + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( ( 12 + T )^{2} \)
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