Properties

Label 2304.3.e.h
Level $2304$
Weight $3$
Character orbit 2304.e
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.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + \beta_{3} q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + \beta_{3} q^{7} + 2 \beta_{2} q^{11} + 3 \beta_{3} q^{13} -3 \beta_{2} q^{17} -4 q^{19} -14 \beta_{1} q^{23} + 19 q^{25} -19 \beta_{1} q^{29} -11 \beta_{3} q^{31} -2 \beta_{2} q^{35} -8 \beta_{3} q^{37} + 7 \beta_{2} q^{41} + 56 q^{43} + 26 \beta_{1} q^{47} -37 q^{49} + 29 \beta_{1} q^{53} + 6 \beta_{3} q^{55} + 24 \beta_{2} q^{59} + 20 \beta_{3} q^{61} -6 \beta_{2} q^{65} + 28 q^{67} -22 \beta_{1} q^{71} -20 q^{73} -12 \beta_{1} q^{77} -21 \beta_{3} q^{79} + 18 \beta_{2} q^{83} -9 \beta_{3} q^{85} + 13 \beta_{2} q^{89} + 36 q^{91} -4 \beta_{1} q^{95} -8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 16q^{19} + 76q^{25} + 224q^{43} - 148q^{49} + 112q^{67} - 80q^{73} + 144q^{91} - 32q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 5 \nu \)
\(\beta_{2}\)\(=\)\( -3 \nu^{3} - 9 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 4\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{2} - 9 \beta_{1}\)\()/6\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
1.93185i
0.517638i
1.93185i
0.517638i
0 0 0 2.44949i 0 −3.46410 0 0 0
1025.2 0 0 0 2.44949i 0 3.46410 0 0 0
1025.3 0 0 0 2.44949i 0 −3.46410 0 0 0
1025.4 0 0 0 2.44949i 0 3.46410 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.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.e.h 4
3.b odd 2 1 inner 2304.3.e.h 4
4.b odd 2 1 2304.3.e.i 4
8.b even 2 1 2304.3.e.i 4
8.d odd 2 1 inner 2304.3.e.h 4
12.b even 2 1 2304.3.e.i 4
16.e even 4 2 576.3.h.a 8
16.f odd 4 2 576.3.h.a 8
24.f even 2 1 inner 2304.3.e.h 4
24.h odd 2 1 2304.3.e.i 4
48.i odd 4 2 576.3.h.a 8
48.k even 4 2 576.3.h.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.3.h.a 8 16.e even 4 2
576.3.h.a 8 16.f odd 4 2
576.3.h.a 8 48.i odd 4 2
576.3.h.a 8 48.k even 4 2
2304.3.e.h 4 1.a even 1 1 trivial
2304.3.e.h 4 3.b odd 2 1 inner
2304.3.e.h 4 8.d odd 2 1 inner
2304.3.e.h 4 24.f even 2 1 inner
2304.3.e.i 4 4.b odd 2 1
2304.3.e.i 4 8.b even 2 1
2304.3.e.i 4 12.b even 2 1
2304.3.e.i 4 24.h odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 6 \)
\( T_{7}^{2} - 12 \)
\( T_{13}^{2} - 108 \)
\( T_{19} + 4 \)
\( T_{31}^{2} - 1452 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 6 + T^{2} )^{2} \)
$7$ \( ( -12 + T^{2} )^{2} \)
$11$ \( ( 72 + T^{2} )^{2} \)
$13$ \( ( -108 + T^{2} )^{2} \)
$17$ \( ( 162 + T^{2} )^{2} \)
$19$ \( ( 4 + T )^{4} \)
$23$ \( ( 1176 + T^{2} )^{2} \)
$29$ \( ( 2166 + T^{2} )^{2} \)
$31$ \( ( -1452 + T^{2} )^{2} \)
$37$ \( ( -768 + T^{2} )^{2} \)
$41$ \( ( 882 + T^{2} )^{2} \)
$43$ \( ( -56 + T )^{4} \)
$47$ \( ( 4056 + T^{2} )^{2} \)
$53$ \( ( 5046 + T^{2} )^{2} \)
$59$ \( ( 10368 + T^{2} )^{2} \)
$61$ \( ( -4800 + T^{2} )^{2} \)
$67$ \( ( -28 + T )^{4} \)
$71$ \( ( 2904 + T^{2} )^{2} \)
$73$ \( ( 20 + T )^{4} \)
$79$ \( ( -5292 + T^{2} )^{2} \)
$83$ \( ( 5832 + T^{2} )^{2} \)
$89$ \( ( 3042 + T^{2} )^{2} \)
$97$ \( ( 8 + T )^{4} \)
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