Properties

Label 1152.3.e.d
Level $1152$
Weight $3$
Character orbit 1152.e
Analytic conductor $31.390$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{2} ) q^{5} + ( -2 + 2 \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{2} ) q^{5} + ( -2 + 2 \beta_{3} ) q^{7} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{11} + ( 8 - 3 \beta_{3} ) q^{13} + ( \beta_{1} - 6 \beta_{2} ) q^{17} + ( -8 + 2 \beta_{3} ) q^{19} -10 \beta_{1} q^{23} + ( 11 + 2 \beta_{3} ) q^{25} + ( -3 \beta_{1} - 7 \beta_{2} ) q^{29} + ( 30 - 2 \beta_{3} ) q^{31} + ( 26 \beta_{1} - 6 \beta_{2} ) q^{35} + ( 22 + 4 \beta_{3} ) q^{37} + ( -21 \beta_{1} + 10 \beta_{2} ) q^{41} + ( -36 - 8 \beta_{3} ) q^{43} + ( -10 \beta_{1} + 8 \beta_{2} ) q^{47} + ( 51 - 8 \beta_{3} ) q^{49} + ( -27 \beta_{1} + 5 \beta_{2} ) q^{53} + ( 28 - 4 \beta_{3} ) q^{55} + ( -20 \beta_{1} + 24 \beta_{2} ) q^{59} + ( 50 + 8 \beta_{3} ) q^{61} + ( -44 \beta_{1} + 14 \beta_{2} ) q^{65} + ( -28 - 2 \beta_{3} ) q^{67} + ( -54 \beta_{1} - 12 \beta_{2} ) q^{71} + ( 68 - 4 \beta_{3} ) q^{73} + ( -52 \beta_{1} + 12 \beta_{2} ) q^{77} + ( 122 - 6 \beta_{3} ) q^{79} + ( 78 \beta_{1} + 14 \beta_{2} ) q^{83} + ( 74 - 7 \beta_{3} ) q^{85} + ( -17 \beta_{1} - 28 \beta_{2} ) q^{89} + ( -160 + 22 \beta_{3} ) q^{91} + ( 32 \beta_{1} - 12 \beta_{2} ) q^{95} + ( 40 + 26 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{7} + O(q^{10}) \) \( 4q - 8q^{7} + 32q^{13} - 32q^{19} + 44q^{25} + 120q^{31} + 88q^{37} - 144q^{43} + 204q^{49} + 112q^{55} + 200q^{61} - 112q^{67} + 272q^{73} + 488q^{79} + 296q^{85} - 640q^{91} + 160q^{97} + O(q^{100}) \)

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
−1.22474 + 0.707107i
1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
0 0 0 4.87832i 0 −11.7980 0 0 0
1025.2 0 0 0 2.04989i 0 7.79796 0 0 0
1025.3 0 0 0 2.04989i 0 7.79796 0 0 0
1025.4 0 0 0 4.87832i 0 −11.7980 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.3.e.d yes 4
3.b odd 2 1 inner 1152.3.e.d yes 4
4.b odd 2 1 1152.3.e.h yes 4
8.b even 2 1 1152.3.e.b 4
8.d odd 2 1 1152.3.e.f yes 4
12.b even 2 1 1152.3.e.h yes 4
16.e even 4 2 2304.3.h.k 8
16.f odd 4 2 2304.3.h.i 8
24.f even 2 1 1152.3.e.f yes 4
24.h odd 2 1 1152.3.e.b 4
48.i odd 4 2 2304.3.h.k 8
48.k even 4 2 2304.3.h.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.e.b 4 8.b even 2 1
1152.3.e.b 4 24.h odd 2 1
1152.3.e.d yes 4 1.a even 1 1 trivial
1152.3.e.d yes 4 3.b odd 2 1 inner
1152.3.e.f yes 4 8.d odd 2 1
1152.3.e.f yes 4 24.f even 2 1
1152.3.e.h yes 4 4.b odd 2 1
1152.3.e.h yes 4 12.b even 2 1
2304.3.h.i 8 16.f odd 4 2
2304.3.h.i 8 48.k even 4 2
2304.3.h.k 8 16.e even 4 2
2304.3.h.k 8 48.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{4} + 28 T_{5}^{2} + 100 \)
\( T_{7}^{2} + 4 T_{7} - 92 \)
\( T_{13}^{2} - 16 T_{13} - 152 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 100 + 28 T^{2} + T^{4} \)
$7$ \( ( -92 + 4 T + T^{2} )^{2} \)
$11$ \( 1600 + 112 T^{2} + T^{4} \)
$13$ \( ( -152 - 16 T + T^{2} )^{2} \)
$17$ \( 184900 + 868 T^{2} + T^{4} \)
$19$ \( ( -32 + 16 T + T^{2} )^{2} \)
$23$ \( ( 200 + T^{2} )^{2} \)
$29$ \( 324900 + 1212 T^{2} + T^{4} \)
$31$ \( ( 804 - 60 T + T^{2} )^{2} \)
$37$ \( ( 100 - 44 T + T^{2} )^{2} \)
$41$ \( 101124 + 4164 T^{2} + T^{4} \)
$43$ \( ( -240 + 72 T + T^{2} )^{2} \)
$47$ \( 322624 + 1936 T^{2} + T^{4} \)
$53$ \( 1340964 + 3516 T^{2} + T^{4} \)
$59$ \( 37356544 + 15424 T^{2} + T^{4} \)
$61$ \( ( 964 - 100 T + T^{2} )^{2} \)
$67$ \( ( 688 + 56 T + T^{2} )^{2} \)
$71$ \( 16842816 + 15120 T^{2} + T^{4} \)
$73$ \( ( 4240 - 136 T + T^{2} )^{2} \)
$79$ \( ( 14020 - 244 T + T^{2} )^{2} \)
$83$ \( 96353856 + 29040 T^{2} + T^{4} \)
$89$ \( 77968900 + 19972 T^{2} + T^{4} \)
$97$ \( ( -14624 - 80 T + T^{2} )^{2} \)
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