Properties

Label 2304.3.h.d
Level $2304$
Weight $3$
Character orbit 2304.h
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

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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: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} +O(q^{10})\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} + 24 \zeta_{8}^{2} q^{13} + ( 23 \zeta_{8} + 23 \zeta_{8}^{3} ) q^{17} -23 q^{25} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{29} -70 \zeta_{8}^{2} q^{37} + ( -49 \zeta_{8} - 49 \zeta_{8}^{3} ) q^{41} -49 q^{49} + ( -73 \zeta_{8} + 73 \zeta_{8}^{3} ) q^{53} -22 \zeta_{8}^{2} q^{61} + ( 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{65} + 96 q^{73} + 46 \zeta_{8}^{2} q^{85} + ( 119 \zeta_{8} + 119 \zeta_{8}^{3} ) q^{89} -144 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 92q^{25} - 196q^{49} + 384q^{73} - 576q^{97} + O(q^{100}) \)

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 0 0 0 0
2177.2 0 0 0 −1.41421 0 0 0 0 0
2177.3 0 0 0 1.41421 0 0 0 0 0
2177.4 0 0 0 1.41421 0 0 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 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f 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.d 4
3.b odd 2 1 inner 2304.3.h.d 4
4.b odd 2 1 CM 2304.3.h.d 4
8.b even 2 1 inner 2304.3.h.d 4
8.d odd 2 1 inner 2304.3.h.d 4
12.b even 2 1 inner 2304.3.h.d 4
16.e even 4 1 288.3.e.c 2
16.e even 4 1 576.3.e.d 2
16.f odd 4 1 288.3.e.c 2
16.f odd 4 1 576.3.e.d 2
24.f even 2 1 inner 2304.3.h.d 4
24.h odd 2 1 inner 2304.3.h.d 4
48.i odd 4 1 288.3.e.c 2
48.i odd 4 1 576.3.e.d 2
48.k even 4 1 288.3.e.c 2
48.k even 4 1 576.3.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.e.c 2 16.e even 4 1
288.3.e.c 2 16.f odd 4 1
288.3.e.c 2 48.i odd 4 1
288.3.e.c 2 48.k even 4 1
576.3.e.d 2 16.e even 4 1
576.3.e.d 2 16.f odd 4 1
576.3.e.d 2 48.i odd 4 1
576.3.e.d 2 48.k even 4 1
2304.3.h.d 4 1.a even 1 1 trivial
2304.3.h.d 4 3.b odd 2 1 inner
2304.3.h.d 4 4.b odd 2 1 CM
2304.3.h.d 4 8.b even 2 1 inner
2304.3.h.d 4 8.d odd 2 1 inner
2304.3.h.d 4 12.b even 2 1 inner
2304.3.h.d 4 24.f even 2 1 inner
2304.3.h.d 4 24.h odd 2 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -2 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 576 + T^{2} )^{2} \)
$17$ \( ( 1058 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( -2 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( 4900 + T^{2} )^{2} \)
$41$ \( ( 4802 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( -10658 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( ( 484 + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -96 + T )^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 28322 + T^{2} )^{2} \)
$97$ \( ( 144 + T )^{4} \)
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