Properties

Label 3072.2.d.g
Level $3072$
Weight $2$
Character orbit 3072.d
Analytic conductor $24.530$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 1536)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{6} q^{5} + ( \beta_{2} - \beta_{7} ) q^{7} - q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{6} q^{5} + ( \beta_{2} - \beta_{7} ) q^{7} - q^{9} + ( -2 \beta_{1} - \beta_{5} ) q^{11} -\beta_{3} q^{13} + \beta_{7} q^{15} + ( -2 - \beta_{4} ) q^{17} + ( -2 \beta_{1} - \beta_{5} ) q^{19} + ( \beta_{3} - \beta_{6} ) q^{21} -4 \beta_{2} q^{23} -5 q^{25} -\beta_{1} q^{27} + ( 2 \beta_{3} + \beta_{6} ) q^{29} + ( -5 \beta_{2} + \beta_{7} ) q^{31} + ( 2 - \beta_{4} ) q^{33} + ( 10 \beta_{1} - \beta_{5} ) q^{35} + ( 3 \beta_{3} + 2 \beta_{6} ) q^{37} + \beta_{2} q^{39} + ( -2 - \beta_{4} ) q^{41} + ( -6 \beta_{1} + \beta_{5} ) q^{43} + \beta_{6} q^{45} + ( 5 + 2 \beta_{4} ) q^{49} + ( -2 \beta_{1} + \beta_{5} ) q^{51} + ( 6 \beta_{3} + \beta_{6} ) q^{53} + ( -10 \beta_{2} - 2 \beta_{7} ) q^{55} + ( 2 - \beta_{4} ) q^{57} -2 \beta_{5} q^{59} + ( 3 \beta_{3} - 2 \beta_{6} ) q^{61} + ( -\beta_{2} + \beta_{7} ) q^{63} + \beta_{4} q^{65} + 12 \beta_{1} q^{67} -4 \beta_{3} q^{69} + ( 2 \beta_{2} + 2 \beta_{7} ) q^{71} + ( 6 + 2 \beta_{4} ) q^{73} -5 \beta_{1} q^{75} + 8 \beta_{3} q^{77} + ( -3 \beta_{2} - \beta_{7} ) q^{79} + q^{81} + ( -2 \beta_{1} - \beta_{5} ) q^{83} + ( -10 \beta_{3} + 2 \beta_{6} ) q^{85} + ( -2 \beta_{2} - \beta_{7} ) q^{87} + 10 q^{89} + ( -2 \beta_{1} + \beta_{5} ) q^{91} + ( -5 \beta_{3} + \beta_{6} ) q^{93} + ( -10 \beta_{2} - 2 \beta_{7} ) q^{95} + ( -4 + 2 \beta_{4} ) q^{97} + ( 2 \beta_{1} + \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{9} + O(q^{10}) \) \( 8q - 8q^{9} - 16q^{17} - 40q^{25} + 16q^{33} - 16q^{41} + 40q^{49} + 16q^{57} + 48q^{73} + 8q^{81} + 80q^{89} - 32q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 8 \nu^{2} \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{7} - \nu^{5} - 13 \nu^{3} - 5 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( 4 \nu^{4} + 14 \)\()/3\)
\(\beta_{5}\)\(=\)\( -2 \nu^{6} - 12 \nu^{2} \)
\(\beta_{6}\)\(=\)\((\)\( 4 \nu^{7} + \nu^{5} + 29 \nu^{3} + 11 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( -4 \nu^{7} + \nu^{5} - 29 \nu^{3} + 11 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{3} + \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 6 \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} + 2 \beta_{3} - 2 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{4} - 14\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{7} - 5 \beta_{6} - 11 \beta_{3} - 11 \beta_{2}\)\()/4\)
\(\nu^{6}\)\(=\)\(-2 \beta_{5} - 9 \beta_{1}\)
\(\nu^{7}\)\(=\)\((\)\(13 \beta_{7} - 13 \beta_{6} - 29 \beta_{3} + 29 \beta_{2}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1537.1
−1.14412 + 1.14412i
−0.437016 + 0.437016i
0.437016 0.437016i
1.14412 1.14412i
0.437016 + 0.437016i
1.14412 + 1.14412i
−1.14412 1.14412i
−0.437016 0.437016i
0 1.00000i 0 3.16228i 0 1.74806 0 −1.00000 0
1537.2 0 1.00000i 0 3.16228i 0 4.57649 0 −1.00000 0
1537.3 0 1.00000i 0 3.16228i 0 −4.57649 0 −1.00000 0
1537.4 0 1.00000i 0 3.16228i 0 −1.74806 0 −1.00000 0
1537.5 0 1.00000i 0 3.16228i 0 −4.57649 0 −1.00000 0
1537.6 0 1.00000i 0 3.16228i 0 −1.74806 0 −1.00000 0
1537.7 0 1.00000i 0 3.16228i 0 1.74806 0 −1.00000 0
1537.8 0 1.00000i 0 3.16228i 0 4.57649 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1537.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
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.d.g 8
4.b odd 2 1 inner 3072.2.d.g 8
8.b even 2 1 inner 3072.2.d.g 8
8.d odd 2 1 inner 3072.2.d.g 8
16.e even 4 1 3072.2.a.k 4
16.e even 4 1 3072.2.a.q 4
16.f odd 4 1 3072.2.a.k 4
16.f odd 4 1 3072.2.a.q 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
48.i odd 4 1 9216.2.a.bd 4
48.i odd 4 1 9216.2.a.bj 4
48.k even 4 1 9216.2.a.bd 4
48.k even 4 1 9216.2.a.bj 4
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 16.e even 4 1
3072.2.a.k 4 16.f odd 4 1
3072.2.a.q 4 16.e even 4 1
3072.2.a.q 4 16.f odd 4 1
3072.2.d.g 8 1.a even 1 1 trivial
3072.2.d.g 8 4.b odd 2 1 inner
3072.2.d.g 8 8.b even 2 1 inner
3072.2.d.g 8 8.d odd 2 1 inner
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 48.i odd 4 1
9216.2.a.bd 4 48.k even 4 1
9216.2.a.bj 4 48.i odd 4 1
9216.2.a.bj 4 48.k even 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 + T^{2} )^{4} \)
$5$ \( ( 10 + T^{2} )^{4} \)
$7$ \( ( 64 - 24 T^{2} + T^{4} )^{2} \)
$11$ \( ( 256 + 48 T^{2} + T^{4} )^{2} \)
$13$ \( ( 2 + T^{2} )^{4} \)
$17$ \( ( -16 + 4 T + T^{2} )^{4} \)
$19$ \( ( 256 + 48 T^{2} + T^{4} )^{2} \)
$23$ \( ( -32 + T^{2} )^{4} \)
$29$ \( ( 4 + 36 T^{2} + T^{4} )^{2} \)
$31$ \( ( 1600 - 120 T^{2} + T^{4} )^{2} \)
$37$ \( ( 484 + 116 T^{2} + T^{4} )^{2} \)
$41$ \( ( -16 + 4 T + T^{2} )^{4} \)
$43$ \( ( 256 + 112 T^{2} + T^{4} )^{2} \)
$47$ \( T^{8} \)
$53$ \( ( 3844 + 164 T^{2} + T^{4} )^{2} \)
$59$ \( ( 80 + T^{2} )^{4} \)
$61$ \( ( 484 + 116 T^{2} + T^{4} )^{2} \)
$67$ \( ( 144 + T^{2} )^{4} \)
$71$ \( ( 1024 - 96 T^{2} + T^{4} )^{2} \)
$73$ \( ( -44 - 12 T + T^{2} )^{4} \)
$79$ \( ( 64 - 56 T^{2} + T^{4} )^{2} \)
$83$ \( ( 256 + 48 T^{2} + T^{4} )^{2} \)
$89$ \( ( -10 + T )^{8} \)
$97$ \( ( -64 + 8 T + T^{2} )^{4} \)
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