Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.b (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.

\(f(q)\) \(=\) \( q + 4 \zeta_{8}^{2} q^{5} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{7} +O(q^{10})\) \( q + 4 \zeta_{8}^{2} q^{5} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{7} + ( -10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{11} + 20 \zeta_{8}^{2} q^{13} + 10 q^{17} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{19} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{23} + 9 q^{25} -20 \zeta_{8}^{2} q^{29} + ( 32 \zeta_{8} - 32 \zeta_{8}^{3} ) q^{35} -20 \zeta_{8}^{2} q^{37} + 30 q^{41} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{43} + ( -48 \zeta_{8} - 48 \zeta_{8}^{3} ) q^{47} -79 q^{49} -60 \zeta_{8}^{2} q^{53} + ( -40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{55} + ( 30 \zeta_{8} - 30 \zeta_{8}^{3} ) q^{59} -28 \zeta_{8}^{2} q^{61} -80 q^{65} + ( 58 \zeta_{8} - 58 \zeta_{8}^{3} ) q^{67} + ( 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{71} -10 q^{73} + 160 \zeta_{8}^{2} q^{77} + ( -80 \zeta_{8} - 80 \zeta_{8}^{3} ) q^{79} + ( -18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{83} + 40 \zeta_{8}^{2} q^{85} -22 q^{89} + ( 160 \zeta_{8} - 160 \zeta_{8}^{3} ) q^{91} + ( 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{95} + 150 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 40q^{17} + 36q^{25} + 120q^{41} - 316q^{49} - 320q^{65} - 40q^{73} - 88q^{89} + 600q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 0 0 4.00000i 0 11.3137i 0 0 0
703.2 0 0 0 4.00000i 0 11.3137i 0 0 0
703.3 0 0 0 4.00000i 0 11.3137i 0 0 0
703.4 0 0 0 4.00000i 0 11.3137i 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
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 1152.3.b.g 4
3.b odd 2 1 128.3.d.c 4
4.b odd 2 1 inner 1152.3.b.g 4
8.b even 2 1 inner 1152.3.b.g 4
8.d odd 2 1 inner 1152.3.b.g 4
12.b even 2 1 128.3.d.c 4
16.e even 4 1 2304.3.g.h 2
16.e even 4 1 2304.3.g.m 2
16.f odd 4 1 2304.3.g.h 2
16.f odd 4 1 2304.3.g.m 2
24.f even 2 1 128.3.d.c 4
24.h odd 2 1 128.3.d.c 4
48.i odd 4 1 256.3.c.c 2
48.i odd 4 1 256.3.c.f 2
48.k even 4 1 256.3.c.c 2
48.k even 4 1 256.3.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.3.d.c 4 3.b odd 2 1
128.3.d.c 4 12.b even 2 1
128.3.d.c 4 24.f even 2 1
128.3.d.c 4 24.h odd 2 1
256.3.c.c 2 48.i odd 4 1
256.3.c.c 2 48.k even 4 1
256.3.c.f 2 48.i odd 4 1
256.3.c.f 2 48.k even 4 1
1152.3.b.g 4 1.a even 1 1 trivial
1152.3.b.g 4 4.b odd 2 1 inner
1152.3.b.g 4 8.b even 2 1 inner
1152.3.b.g 4 8.d odd 2 1 inner
2304.3.g.h 2 16.e even 4 1
2304.3.g.h 2 16.f odd 4 1
2304.3.g.m 2 16.e even 4 1
2304.3.g.m 2 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 \(S_{3}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} + 16 \)
\( T_{7}^{2} + 128 \)
\( T_{17} - 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 16 + T^{2} )^{2} \)
$7$ \( ( 128 + T^{2} )^{2} \)
$11$ \( ( -200 + T^{2} )^{2} \)
$13$ \( ( 400 + T^{2} )^{2} \)
$17$ \( ( -10 + T )^{4} \)
$19$ \( ( -200 + T^{2} )^{2} \)
$23$ \( ( 128 + T^{2} )^{2} \)
$29$ \( ( 400 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( 400 + T^{2} )^{2} \)
$41$ \( ( -30 + T )^{4} \)
$43$ \( ( -8 + T^{2} )^{2} \)
$47$ \( ( 4608 + T^{2} )^{2} \)
$53$ \( ( 3600 + T^{2} )^{2} \)
$59$ \( ( -1800 + T^{2} )^{2} \)
$61$ \( ( 784 + T^{2} )^{2} \)
$67$ \( ( -6728 + T^{2} )^{2} \)
$71$ \( ( 3200 + T^{2} )^{2} \)
$73$ \( ( 10 + T )^{4} \)
$79$ \( ( 12800 + T^{2} )^{2} \)
$83$ \( ( -648 + T^{2} )^{2} \)
$89$ \( ( 22 + T )^{4} \)
$97$ \( ( -150 + T )^{4} \)
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