Properties

Label 2304.3.h.b
Level $2304$
Weight $3$
Character orbit 2304.h
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 288)
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.

\(f(q)\) \(=\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} -8 q^{7} +O(q^{10})\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} -8 q^{7} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{11} -8 \zeta_{8}^{2} q^{13} + ( -9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{17} + 32 \zeta_{8}^{2} q^{19} + ( -24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{23} -23 q^{25} + ( -31 \zeta_{8} + 31 \zeta_{8}^{3} ) q^{29} + 40 q^{31} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{35} + 26 \zeta_{8}^{2} q^{37} + ( 47 \zeta_{8} + 47 \zeta_{8}^{3} ) q^{41} -16 \zeta_{8}^{2} q^{43} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{47} + 15 q^{49} + ( 23 \zeta_{8} - 23 \zeta_{8}^{3} ) q^{53} + 16 q^{55} + ( 16 \zeta_{8} - 16 \zeta_{8}^{3} ) q^{59} -54 \zeta_{8}^{2} q^{61} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{65} -80 \zeta_{8}^{2} q^{67} + ( 56 \zeta_{8} + 56 \zeta_{8}^{3} ) q^{71} -96 q^{73} + ( -64 \zeta_{8} + 64 \zeta_{8}^{3} ) q^{77} -104 q^{79} + ( 72 \zeta_{8} - 72 \zeta_{8}^{3} ) q^{83} -18 \zeta_{8}^{2} q^{85} + ( 55 \zeta_{8} + 55 \zeta_{8}^{3} ) q^{89} + 64 \zeta_{8}^{2} q^{91} + ( 32 \zeta_{8} + 32 \zeta_{8}^{3} ) q^{95} -80 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 32q^{7} + O(q^{10}) \) \( 4q - 32q^{7} - 92q^{25} + 160q^{31} + 60q^{49} + 64q^{55} - 384q^{73} - 416q^{79} - 320q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2177.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
0 0 0 −1.41421 0 −8.00000 0 0 0
2177.2 0 0 0 −1.41421 0 −8.00000 0 0 0
2177.3 0 0 0 1.41421 0 −8.00000 0 0 0
2177.4 0 0 0 1.41421 0 −8.00000 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
8.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 2304.3.h.b 4
3.b odd 2 1 inner 2304.3.h.b 4
4.b odd 2 1 2304.3.h.g 4
8.b even 2 1 inner 2304.3.h.b 4
8.d odd 2 1 2304.3.h.g 4
12.b even 2 1 2304.3.h.g 4
16.e even 4 1 288.3.e.d yes 2
16.e even 4 1 576.3.e.g 2
16.f odd 4 1 288.3.e.a 2
16.f odd 4 1 576.3.e.b 2
24.f even 2 1 2304.3.h.g 4
24.h odd 2 1 inner 2304.3.h.b 4
48.i odd 4 1 288.3.e.d yes 2
48.i odd 4 1 576.3.e.g 2
48.k even 4 1 288.3.e.a 2
48.k even 4 1 576.3.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.e.a 2 16.f odd 4 1
288.3.e.a 2 48.k even 4 1
288.3.e.d yes 2 16.e even 4 1
288.3.e.d yes 2 48.i odd 4 1
576.3.e.b 2 16.f odd 4 1
576.3.e.b 2 48.k even 4 1
576.3.e.g 2 16.e even 4 1
576.3.e.g 2 48.i odd 4 1
2304.3.h.b 4 1.a even 1 1 trivial
2304.3.h.b 4 3.b odd 2 1 inner
2304.3.h.b 4 8.b even 2 1 inner
2304.3.h.b 4 24.h odd 2 1 inner
2304.3.h.g 4 4.b odd 2 1
2304.3.h.g 4 8.d odd 2 1
2304.3.h.g 4 12.b even 2 1
2304.3.h.g 4 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} - 2 \)
\( T_{7} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -2 + T^{2} )^{2} \)
$7$ \( ( 8 + T )^{4} \)
$11$ \( ( -128 + T^{2} )^{2} \)
$13$ \( ( 64 + T^{2} )^{2} \)
$17$ \( ( 162 + T^{2} )^{2} \)
$19$ \( ( 1024 + T^{2} )^{2} \)
$23$ \( ( 1152 + T^{2} )^{2} \)
$29$ \( ( -1922 + T^{2} )^{2} \)
$31$ \( ( -40 + T )^{4} \)
$37$ \( ( 676 + T^{2} )^{2} \)
$41$ \( ( 4418 + T^{2} )^{2} \)
$43$ \( ( 256 + T^{2} )^{2} \)
$47$ \( ( 128 + T^{2} )^{2} \)
$53$ \( ( -1058 + T^{2} )^{2} \)
$59$ \( ( -512 + T^{2} )^{2} \)
$61$ \( ( 2916 + T^{2} )^{2} \)
$67$ \( ( 6400 + T^{2} )^{2} \)
$71$ \( ( 6272 + T^{2} )^{2} \)
$73$ \( ( 96 + T )^{4} \)
$79$ \( ( 104 + T )^{4} \)
$83$ \( ( -10368 + T^{2} )^{2} \)
$89$ \( ( 6050 + T^{2} )^{2} \)
$97$ \( ( 80 + T )^{4} \)
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