Properties

Label 2304.3.e.f
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: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 1152)
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.

\(f(q)\) \(=\) \( q + 3 \beta_{1} q^{5} + \beta_{2} q^{7} +O(q^{10})\) \( q + 3 \beta_{1} q^{5} + \beta_{2} q^{7} + \beta_{3} q^{11} -10 q^{13} -17 \beta_{1} q^{17} + 2 \beta_{2} q^{19} -\beta_{3} q^{23} + 7 q^{25} + 11 \beta_{1} q^{29} + \beta_{2} q^{31} + 3 \beta_{3} q^{35} + 64 q^{37} + 9 \beta_{1} q^{41} + 4 \beta_{2} q^{43} -\beta_{3} q^{47} + 111 q^{49} -13 \beta_{1} q^{53} -6 \beta_{2} q^{55} -30 \beta_{1} q^{65} -6 \beta_{2} q^{67} + 7 \beta_{3} q^{71} -96 q^{73} + 160 \beta_{1} q^{77} -5 \beta_{2} q^{79} -7 \beta_{3} q^{83} + 102 q^{85} + 39 \beta_{1} q^{89} -10 \beta_{2} q^{91} + 6 \beta_{3} q^{95} + 64 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 40q^{13} + 28q^{25} + 256q^{37} + 444q^{49} - 384q^{73} + 408q^{85} + 256q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 4 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{3} + 28 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( 8 \nu^{2} - 16 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 4 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 16\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} + 28 \beta_{1}\)\()/8\)

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
−1.58114 0.707107i
1.58114 0.707107i
−1.58114 + 0.707107i
1.58114 + 0.707107i
0 0 0 4.24264i 0 −12.6491 0 0 0
1025.2 0 0 0 4.24264i 0 12.6491 0 0 0
1025.3 0 0 0 4.24264i 0 −12.6491 0 0 0
1025.4 0 0 0 4.24264i 0 12.6491 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
4.b odd 2 1 inner
12.b 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.f 4
3.b odd 2 1 inner 2304.3.e.f 4
4.b odd 2 1 inner 2304.3.e.f 4
8.b even 2 1 2304.3.e.k 4
8.d odd 2 1 2304.3.e.k 4
12.b even 2 1 inner 2304.3.e.f 4
16.e even 4 2 1152.3.h.e 8
16.f odd 4 2 1152.3.h.e 8
24.f even 2 1 2304.3.e.k 4
24.h odd 2 1 2304.3.e.k 4
48.i odd 4 2 1152.3.h.e 8
48.k even 4 2 1152.3.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.h.e 8 16.e even 4 2
1152.3.h.e 8 16.f odd 4 2
1152.3.h.e 8 48.i odd 4 2
1152.3.h.e 8 48.k even 4 2
2304.3.e.f 4 1.a even 1 1 trivial
2304.3.e.f 4 3.b odd 2 1 inner
2304.3.e.f 4 4.b odd 2 1 inner
2304.3.e.f 4 12.b even 2 1 inner
2304.3.e.k 4 8.b even 2 1
2304.3.e.k 4 8.d odd 2 1
2304.3.e.k 4 24.f even 2 1
2304.3.e.k 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} + 18 \)
\( T_{7}^{2} - 160 \)
\( T_{13} + 10 \)
\( T_{19}^{2} - 640 \)
\( T_{31}^{2} - 160 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 18 + T^{2} )^{2} \)
$7$ \( ( -160 + T^{2} )^{2} \)
$11$ \( ( 320 + T^{2} )^{2} \)
$13$ \( ( 10 + T )^{4} \)
$17$ \( ( 578 + T^{2} )^{2} \)
$19$ \( ( -640 + T^{2} )^{2} \)
$23$ \( ( 320 + T^{2} )^{2} \)
$29$ \( ( 242 + T^{2} )^{2} \)
$31$ \( ( -160 + T^{2} )^{2} \)
$37$ \( ( -64 + T )^{4} \)
$41$ \( ( 162 + T^{2} )^{2} \)
$43$ \( ( -2560 + T^{2} )^{2} \)
$47$ \( ( 320 + T^{2} )^{2} \)
$53$ \( ( 338 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( -5760 + T^{2} )^{2} \)
$71$ \( ( 15680 + T^{2} )^{2} \)
$73$ \( ( 96 + T )^{4} \)
$79$ \( ( -4000 + T^{2} )^{2} \)
$83$ \( ( 15680 + T^{2} )^{2} \)
$89$ \( ( 3042 + T^{2} )^{2} \)
$97$ \( ( -64 + T )^{4} \)
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