Properties

Label 1152.3.h.d
Level $1152$
Weight $3$
Character orbit 1152.h
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.h (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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{5} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{7} +O(q^{10})\) \( q + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{5} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{7} + 4 q^{11} + 18 \zeta_{8}^{2} q^{13} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{17} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{19} + 36 \zeta_{8}^{2} q^{23} -7 q^{25} + ( 9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{29} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{31} + 36 q^{35} -36 \zeta_{8}^{2} q^{37} + ( -21 \zeta_{8} - 21 \zeta_{8}^{3} ) q^{41} + ( 48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{43} -36 \zeta_{8}^{2} q^{47} + 23 q^{49} + ( -57 \zeta_{8} + 57 \zeta_{8}^{3} ) q^{53} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{55} + 80 q^{59} -36 \zeta_{8}^{2} q^{61} + ( 54 \zeta_{8} + 54 \zeta_{8}^{3} ) q^{65} + ( 84 \zeta_{8} + 84 \zeta_{8}^{3} ) q^{67} -108 \zeta_{8}^{2} q^{71} -56 q^{73} + ( 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{77} + ( -18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{79} + 76 q^{83} + 18 \zeta_{8}^{2} q^{85} + ( -63 \zeta_{8} - 63 \zeta_{8}^{3} ) q^{89} + ( 108 \zeta_{8} + 108 \zeta_{8}^{3} ) q^{91} + 72 \zeta_{8}^{2} q^{95} + 104 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 16q^{11} - 28q^{25} + 144q^{35} + 92q^{49} + 320q^{59} - 224q^{73} + 304q^{83} + 416q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 −4.24264 0 −8.48528 0 0 0
449.2 0 0 0 −4.24264 0 −8.48528 0 0 0
449.3 0 0 0 4.24264 0 8.48528 0 0 0
449.4 0 0 0 4.24264 0 8.48528 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
8.d odd 2 1 inner
12.b even 2 1 inner
24.h 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.h.d yes 4
3.b odd 2 1 1152.3.h.a 4
4.b odd 2 1 1152.3.h.a 4
8.b even 2 1 1152.3.h.a 4
8.d odd 2 1 inner 1152.3.h.d yes 4
12.b even 2 1 inner 1152.3.h.d yes 4
16.e even 4 1 2304.3.e.e 4
16.e even 4 1 2304.3.e.l 4
16.f odd 4 1 2304.3.e.e 4
16.f odd 4 1 2304.3.e.l 4
24.f even 2 1 1152.3.h.a 4
24.h odd 2 1 inner 1152.3.h.d yes 4
48.i odd 4 1 2304.3.e.e 4
48.i odd 4 1 2304.3.e.l 4
48.k even 4 1 2304.3.e.e 4
48.k even 4 1 2304.3.e.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.h.a 4 3.b odd 2 1
1152.3.h.a 4 4.b odd 2 1
1152.3.h.a 4 8.b even 2 1
1152.3.h.a 4 24.f even 2 1
1152.3.h.d yes 4 1.a even 1 1 trivial
1152.3.h.d yes 4 8.d odd 2 1 inner
1152.3.h.d yes 4 12.b even 2 1 inner
1152.3.h.d yes 4 24.h odd 2 1 inner
2304.3.e.e 4 16.e even 4 1
2304.3.e.e 4 16.f odd 4 1
2304.3.e.e 4 48.i odd 4 1
2304.3.e.e 4 48.k even 4 1
2304.3.e.l 4 16.e even 4 1
2304.3.e.l 4 16.f odd 4 1
2304.3.e.l 4 48.i odd 4 1
2304.3.e.l 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 \(S_{3}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} - 18 \)
\( T_{7}^{2} - 72 \)
\( T_{11} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -18 + T^{2} )^{2} \)
$7$ \( ( -72 + T^{2} )^{2} \)
$11$ \( ( -4 + T )^{4} \)
$13$ \( ( 324 + T^{2} )^{2} \)
$17$ \( ( 18 + T^{2} )^{2} \)
$19$ \( ( 288 + T^{2} )^{2} \)
$23$ \( ( 1296 + T^{2} )^{2} \)
$29$ \( ( -162 + T^{2} )^{2} \)
$31$ \( ( -72 + T^{2} )^{2} \)
$37$ \( ( 1296 + T^{2} )^{2} \)
$41$ \( ( 882 + T^{2} )^{2} \)
$43$ \( ( 4608 + T^{2} )^{2} \)
$47$ \( ( 1296 + T^{2} )^{2} \)
$53$ \( ( -6498 + T^{2} )^{2} \)
$59$ \( ( -80 + T )^{4} \)
$61$ \( ( 1296 + T^{2} )^{2} \)
$67$ \( ( 14112 + T^{2} )^{2} \)
$71$ \( ( 11664 + T^{2} )^{2} \)
$73$ \( ( 56 + T )^{4} \)
$79$ \( ( -648 + T^{2} )^{2} \)
$83$ \( ( -76 + T )^{4} \)
$89$ \( ( 7938 + T^{2} )^{2} \)
$97$ \( ( -104 + T )^{4} \)
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