Properties

Label 1152.3.e.a
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: \(\Q(\sqrt{-2}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{3} ) q^{5} -2 q^{7} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{3} ) q^{5} -2 q^{7} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{11} + ( -8 + \beta_{2} ) q^{13} + ( -\beta_{1} - 2 \beta_{3} ) q^{17} + ( 8 + 2 \beta_{2} ) q^{19} + ( -6 \beta_{1} + 4 \beta_{3} ) q^{23} + ( -21 + 2 \beta_{2} ) q^{25} + ( -13 \beta_{1} - \beta_{3} ) q^{29} + ( -2 - 4 \beta_{2} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{35} -26 q^{37} + ( -11 \beta_{1} - 2 \beta_{3} ) q^{41} + ( 28 - 4 \beta_{2} ) q^{43} + ( -38 \beta_{1} - 4 \beta_{3} ) q^{47} -45 q^{49} + ( -21 \beta_{1} + 3 \beta_{3} ) q^{53} + ( -100 + 8 \beta_{2} ) q^{55} + 52 \beta_{1} q^{59} + ( -30 - 4 \beta_{2} ) q^{61} + ( -52 \beta_{1} + 10 \beta_{3} ) q^{65} + ( 84 + 2 \beta_{2} ) q^{67} + ( 22 \beta_{1} - 8 \beta_{3} ) q^{71} + ( -28 - 12 \beta_{2} ) q^{73} + ( -12 \beta_{1} + 4 \beta_{3} ) q^{77} + ( -38 + 4 \beta_{2} ) q^{79} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{83} + ( -86 + \beta_{2} ) q^{85} + ( -47 \beta_{1} + 12 \beta_{3} ) q^{89} + ( 16 - 2 \beta_{2} ) q^{91} + ( -80 \beta_{1} - 4 \beta_{3} ) q^{95} + ( -88 + 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{7} + O(q^{10}) \) \( 4q - 8q^{7} - 32q^{13} + 32q^{19} - 84q^{25} - 8q^{31} - 104q^{37} + 112q^{43} - 180q^{49} - 400q^{55} - 120q^{61} + 336q^{67} - 112q^{73} - 152q^{79} - 344q^{85} + 64q^{91} - 352q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{3} - 3 \nu^{2} + 19 \nu - 9 \)\()/3\)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \nu + 10 \)
\(\beta_{3}\)\(=\)\((\)\( 8 \nu^{3} - 12 \nu^{2} + 88 \nu - 42 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - 4 \beta_{1} + 2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2} - 4 \beta_{1} - 18\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-8 \beta_{3} + 3 \beta_{2} + 38 \beta_{1} - 28\)\()/4\)

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
0.500000 + 3.07253i
0.500000 + 0.244099i
0.500000 0.244099i
0.500000 3.07253i
0 0 0 8.04746i 0 −2.00000 0 0 0
1025.2 0 0 0 5.21904i 0 −2.00000 0 0 0
1025.3 0 0 0 5.21904i 0 −2.00000 0 0 0
1025.4 0 0 0 8.04746i 0 −2.00000 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.a 4
3.b odd 2 1 inner 1152.3.e.a 4
4.b odd 2 1 1152.3.e.e yes 4
8.b even 2 1 1152.3.e.c yes 4
8.d odd 2 1 1152.3.e.g yes 4
12.b even 2 1 1152.3.e.e yes 4
16.e even 4 2 2304.3.h.l 8
16.f odd 4 2 2304.3.h.j 8
24.f even 2 1 1152.3.e.g yes 4
24.h odd 2 1 1152.3.e.c yes 4
48.i odd 4 2 2304.3.h.l 8
48.k even 4 2 2304.3.h.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.e.a 4 1.a even 1 1 trivial
1152.3.e.a 4 3.b odd 2 1 inner
1152.3.e.c yes 4 8.b even 2 1
1152.3.e.c yes 4 24.h odd 2 1
1152.3.e.e yes 4 4.b odd 2 1
1152.3.e.e yes 4 12.b even 2 1
1152.3.e.g yes 4 8.d odd 2 1
1152.3.e.g yes 4 24.f even 2 1
2304.3.h.j 8 16.f odd 4 2
2304.3.h.j 8 48.k even 4 2
2304.3.h.l 8 16.e even 4 2
2304.3.h.l 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 \(S_{3}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{4} + 92 T_{5}^{2} + 1764 \)
\( T_{7} + 2 \)
\( T_{13}^{2} + 16 T_{13} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 1764 + 92 T^{2} + T^{4} \)
$7$ \( ( 2 + T )^{4} \)
$11$ \( 10816 + 496 T^{2} + T^{4} \)
$13$ \( ( -24 + 16 T + T^{2} )^{2} \)
$17$ \( 30276 + 356 T^{2} + T^{4} \)
$19$ \( ( -288 - 16 T + T^{2} )^{2} \)
$23$ \( 399424 + 1552 T^{2} + T^{4} \)
$29$ \( 86436 + 764 T^{2} + T^{4} \)
$31$ \( ( -1404 + 4 T + T^{2} )^{2} \)
$37$ \( ( 26 + T )^{4} \)
$41$ \( 4356 + 836 T^{2} + T^{4} \)
$43$ \( ( -624 - 56 T + T^{2} )^{2} \)
$47$ \( 4769856 + 7184 T^{2} + T^{4} \)
$53$ \( 236196 + 2556 T^{2} + T^{4} \)
$59$ \( ( 5408 + T^{2} )^{2} \)
$61$ \( ( -508 + 60 T + T^{2} )^{2} \)
$67$ \( ( 6704 - 168 T + T^{2} )^{2} \)
$71$ \( 3415104 + 7568 T^{2} + T^{4} \)
$73$ \( ( -11888 + 56 T + T^{2} )^{2} \)
$79$ \( ( 36 + 76 T + T^{2} )^{2} \)
$83$ \( 2286144 + 3312 T^{2} + T^{4} \)
$89$ \( 3678724 + 21508 T^{2} + T^{4} \)
$97$ \( ( 7392 + 176 T + T^{2} )^{2} \)
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