Properties

Label 2304.3.e.g
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: \( 2^{2}\cdot 3^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} -\beta_{3} q^{7} +O(q^{10})\) \( q + \beta_{2} q^{5} -\beta_{3} q^{7} + 2 \beta_{1} q^{11} + \beta_{3} q^{13} + 5 \beta_{1} q^{17} -20 q^{19} + 2 \beta_{2} q^{23} -29 q^{25} + 5 \beta_{2} q^{29} -5 \beta_{3} q^{31} -18 \beta_{1} q^{35} + 4 \beta_{3} q^{37} -17 \beta_{1} q^{41} + 40 q^{43} + 10 \beta_{2} q^{47} + 59 q^{49} + 5 \beta_{2} q^{53} -6 \beta_{3} q^{55} + 8 \beta_{1} q^{59} + 18 \beta_{1} q^{65} -100 q^{67} + 10 \beta_{2} q^{71} -20 q^{73} -12 \beta_{2} q^{77} + 5 \beta_{3} q^{79} -30 \beta_{1} q^{83} -15 \beta_{3} q^{85} -3 \beta_{1} q^{89} -108 q^{91} -20 \beta_{2} q^{95} + 40 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 80q^{19} - 116q^{25} + 160q^{43} + 236q^{49} - 400q^{67} - 80q^{73} - 432q^{91} + 160q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 3 \nu^{3} + 9 \nu \)
\(\beta_{2}\)\(=\)\( 3 \nu^{3} + 15 \nu \)
\(\beta_{3}\)\(=\)\( 6 \nu^{2} + 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 12\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{2} + 5 \beta_{1}\)\()/6\)

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
0.517638i
1.93185i
0.517638i
1.93185i
0 0 0 7.34847i 0 −10.3923 0 0 0
1025.2 0 0 0 7.34847i 0 10.3923 0 0 0
1025.3 0 0 0 7.34847i 0 −10.3923 0 0 0
1025.4 0 0 0 7.34847i 0 10.3923 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.g 4
3.b odd 2 1 inner 2304.3.e.g 4
4.b odd 2 1 2304.3.e.j 4
8.b even 2 1 2304.3.e.j 4
8.d odd 2 1 inner 2304.3.e.g 4
12.b even 2 1 2304.3.e.j 4
16.e even 4 2 576.3.h.b 8
16.f odd 4 2 576.3.h.b 8
24.f even 2 1 inner 2304.3.e.g 4
24.h odd 2 1 2304.3.e.j 4
48.i odd 4 2 576.3.h.b 8
48.k even 4 2 576.3.h.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.3.h.b 8 16.e even 4 2
576.3.h.b 8 16.f odd 4 2
576.3.h.b 8 48.i odd 4 2
576.3.h.b 8 48.k even 4 2
2304.3.e.g 4 1.a even 1 1 trivial
2304.3.e.g 4 3.b odd 2 1 inner
2304.3.e.g 4 8.d odd 2 1 inner
2304.3.e.g 4 24.f even 2 1 inner
2304.3.e.j 4 4.b odd 2 1
2304.3.e.j 4 8.b even 2 1
2304.3.e.j 4 12.b even 2 1
2304.3.e.j 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} + 54 \)
\( T_{7}^{2} - 108 \)
\( T_{13}^{2} - 108 \)
\( T_{19} + 20 \)
\( T_{31}^{2} - 2700 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 54 + T^{2} )^{2} \)
$7$ \( ( -108 + T^{2} )^{2} \)
$11$ \( ( 72 + T^{2} )^{2} \)
$13$ \( ( -108 + T^{2} )^{2} \)
$17$ \( ( 450 + T^{2} )^{2} \)
$19$ \( ( 20 + T )^{4} \)
$23$ \( ( 216 + T^{2} )^{2} \)
$29$ \( ( 1350 + T^{2} )^{2} \)
$31$ \( ( -2700 + T^{2} )^{2} \)
$37$ \( ( -1728 + T^{2} )^{2} \)
$41$ \( ( 5202 + T^{2} )^{2} \)
$43$ \( ( -40 + T )^{4} \)
$47$ \( ( 5400 + T^{2} )^{2} \)
$53$ \( ( 1350 + T^{2} )^{2} \)
$59$ \( ( 1152 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( 100 + T )^{4} \)
$71$ \( ( 5400 + T^{2} )^{2} \)
$73$ \( ( 20 + T )^{4} \)
$79$ \( ( -2700 + T^{2} )^{2} \)
$83$ \( ( 16200 + T^{2} )^{2} \)
$89$ \( ( 162 + T^{2} )^{2} \)
$97$ \( ( -40 + T )^{4} \)
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