Properties

Label 1152.3.g.a
Level $1152$
Weight $3$
Character orbit 1152.g
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.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{5} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{7} +O(q^{10})\) \( q + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{5} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{7} + ( 6 \zeta_{8} + 10 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{11} + ( -6 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{13} + ( 2 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{17} + ( -2 \zeta_{8} + 18 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{19} + ( -4 \zeta_{8} + 28 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{23} + ( 11 - 16 \zeta_{8} + 16 \zeta_{8}^{3} ) q^{25} + ( -34 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{29} + ( 32 \zeta_{8} + 32 \zeta_{8}^{3} ) q^{31} + ( -8 \zeta_{8} - 24 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{35} + ( 10 + 28 \zeta_{8} - 28 \zeta_{8}^{3} ) q^{37} + ( -2 + 16 \zeta_{8} - 16 \zeta_{8}^{3} ) q^{41} + ( 34 \zeta_{8} - 2 \zeta_{8}^{2} + 34 \zeta_{8}^{3} ) q^{43} + ( 8 \zeta_{8} - 24 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{47} + ( 1 - 32 \zeta_{8} + 32 \zeta_{8}^{3} ) q^{49} + ( 22 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{53} + ( 28 \zeta_{8} + 28 \zeta_{8}^{2} + 28 \zeta_{8}^{3} ) q^{55} + ( 22 \zeta_{8} - 22 \zeta_{8}^{2} + 22 \zeta_{8}^{3} ) q^{59} + ( 74 + 28 \zeta_{8} - 28 \zeta_{8}^{3} ) q^{61} + ( -20 - 16 \zeta_{8} + 16 \zeta_{8}^{3} ) q^{65} + ( 6 \zeta_{8} - 54 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{67} + ( 20 \zeta_{8} - 12 \zeta_{8}^{2} + 20 \zeta_{8}^{3} ) q^{71} + ( 22 - 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{73} + ( 88 + 64 \zeta_{8} - 64 \zeta_{8}^{3} ) q^{77} + ( 24 \zeta_{8} - 104 \zeta_{8}^{2} + 24 \zeta_{8}^{3} ) q^{79} + ( 74 \zeta_{8} - 10 \zeta_{8}^{2} + 74 \zeta_{8}^{3} ) q^{83} + ( -68 + 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{85} + ( -54 - 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{89} + ( 40 \zeta_{8} + 56 \zeta_{8}^{2} + 40 \zeta_{8}^{3} ) q^{91} + ( 76 \zeta_{8} - 52 \zeta_{8}^{2} + 76 \zeta_{8}^{3} ) q^{95} + ( -82 + 40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{5} + O(q^{10}) \) \( 4q - 8q^{5} - 24q^{13} + 8q^{17} + 44q^{25} - 136q^{29} + 40q^{37} - 8q^{41} + 4q^{49} + 88q^{53} + 296q^{61} - 80q^{65} + 88q^{73} + 352q^{77} - 272q^{85} - 216q^{89} - 328q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
0 0 0 −7.65685 0 1.65685i 0 0 0
127.2 0 0 0 −7.65685 0 1.65685i 0 0 0
127.3 0 0 0 3.65685 0 9.65685i 0 0 0
127.4 0 0 0 3.65685 0 9.65685i 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
4.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.g.a 4
3.b odd 2 1 128.3.c.b yes 4
4.b odd 2 1 inner 1152.3.g.a 4
8.b even 2 1 1152.3.g.b 4
8.d odd 2 1 1152.3.g.b 4
12.b even 2 1 128.3.c.b yes 4
16.e even 4 1 2304.3.b.j 4
16.e even 4 1 2304.3.b.p 4
16.f odd 4 1 2304.3.b.j 4
16.f odd 4 1 2304.3.b.p 4
24.f even 2 1 128.3.c.a 4
24.h odd 2 1 128.3.c.a 4
48.i odd 4 1 256.3.d.d 4
48.i odd 4 1 256.3.d.e 4
48.k even 4 1 256.3.d.d 4
48.k even 4 1 256.3.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.3.c.a 4 24.f even 2 1
128.3.c.a 4 24.h odd 2 1
128.3.c.b yes 4 3.b odd 2 1
128.3.c.b yes 4 12.b even 2 1
256.3.d.d 4 48.i odd 4 1
256.3.d.d 4 48.k even 4 1
256.3.d.e 4 48.i odd 4 1
256.3.d.e 4 48.k even 4 1
1152.3.g.a 4 1.a even 1 1 trivial
1152.3.g.a 4 4.b odd 2 1 inner
1152.3.g.b 4 8.b even 2 1
1152.3.g.b 4 8.d odd 2 1
2304.3.b.j 4 16.e even 4 1
2304.3.b.j 4 16.f odd 4 1
2304.3.b.p 4 16.e even 4 1
2304.3.b.p 4 16.f odd 4 1

Hecke kernels

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

\( T_{5}^{2} + 4 T_{5} - 28 \)
\( T_{13}^{2} + 12 T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -28 + 4 T + T^{2} )^{2} \)
$7$ \( 256 + 96 T^{2} + T^{4} \)
$11$ \( 784 + 344 T^{2} + T^{4} \)
$13$ \( ( 4 + 12 T + T^{2} )^{2} \)
$17$ \( ( -124 - 4 T + T^{2} )^{2} \)
$19$ \( 99856 + 664 T^{2} + T^{4} \)
$23$ \( 565504 + 1632 T^{2} + T^{4} \)
$29$ \( ( 1124 + 68 T + T^{2} )^{2} \)
$31$ \( ( 2048 + T^{2} )^{2} \)
$37$ \( ( -1468 - 20 T + T^{2} )^{2} \)
$41$ \( ( -508 + 4 T + T^{2} )^{2} \)
$43$ \( 5326864 + 4632 T^{2} + T^{4} \)
$47$ \( 200704 + 1408 T^{2} + T^{4} \)
$53$ \( ( 452 - 44 T + T^{2} )^{2} \)
$59$ \( 234256 + 2904 T^{2} + T^{4} \)
$61$ \( ( 3908 - 148 T + T^{2} )^{2} \)
$67$ \( 8088336 + 5976 T^{2} + T^{4} \)
$71$ \( 430336 + 1888 T^{2} + T^{4} \)
$73$ \( ( -668 - 44 T + T^{2} )^{2} \)
$79$ \( 93392896 + 23936 T^{2} + T^{4} \)
$83$ \( 117765904 + 22104 T^{2} + T^{4} \)
$89$ \( ( -284 + 108 T + T^{2} )^{2} \)
$97$ \( ( 3524 + 164 T + T^{2} )^{2} \)
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