Properties

Label 2304.3.e.l
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: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{5} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{7} +O(q^{10})\) \( q + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{5} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{7} + 4 \zeta_{8}^{2} q^{11} + 18 q^{13} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{17} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{19} -36 \zeta_{8}^{2} q^{23} + 7 q^{25} + ( -9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{29} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{31} -36 \zeta_{8}^{2} q^{35} + 36 q^{37} + ( 21 \zeta_{8} + 21 \zeta_{8}^{3} ) q^{41} + ( -48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{43} -36 \zeta_{8}^{2} q^{47} + 23 q^{49} + ( -57 \zeta_{8} - 57 \zeta_{8}^{3} ) q^{53} + ( -12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{55} + 80 \zeta_{8}^{2} q^{59} -36 q^{61} + ( 54 \zeta_{8} + 54 \zeta_{8}^{3} ) q^{65} + ( 84 \zeta_{8} - 84 \zeta_{8}^{3} ) q^{67} + 108 \zeta_{8}^{2} q^{71} + 56 q^{73} + ( -24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{77} + ( -18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{79} -76 \zeta_{8}^{2} q^{83} -18 q^{85} + ( 63 \zeta_{8} + 63 \zeta_{8}^{3} ) q^{89} + ( -108 \zeta_{8} + 108 \zeta_{8}^{3} ) q^{91} + 72 \zeta_{8}^{2} q^{95} + 104 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 72q^{13} + 28q^{25} + 144q^{37} + 92q^{49} - 144q^{61} + 224q^{73} - 72q^{85} + 416q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0 0 0 4.24264i 0 −8.48528 0 0 0
1025.2 0 0 0 4.24264i 0 8.48528 0 0 0
1025.3 0 0 0 4.24264i 0 −8.48528 0 0 0
1025.4 0 0 0 4.24264i 0 8.48528 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.l 4
3.b odd 2 1 inner 2304.3.e.l 4
4.b odd 2 1 inner 2304.3.e.l 4
8.b even 2 1 2304.3.e.e 4
8.d odd 2 1 2304.3.e.e 4
12.b even 2 1 inner 2304.3.e.l 4
16.e even 4 1 1152.3.h.a 4
16.e even 4 1 1152.3.h.d yes 4
16.f odd 4 1 1152.3.h.a 4
16.f odd 4 1 1152.3.h.d yes 4
24.f even 2 1 2304.3.e.e 4
24.h odd 2 1 2304.3.e.e 4
48.i odd 4 1 1152.3.h.a 4
48.i odd 4 1 1152.3.h.d yes 4
48.k even 4 1 1152.3.h.a 4
48.k even 4 1 1152.3.h.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.h.a 4 16.e even 4 1
1152.3.h.a 4 16.f odd 4 1
1152.3.h.a 4 48.i odd 4 1
1152.3.h.a 4 48.k even 4 1
1152.3.h.d yes 4 16.e even 4 1
1152.3.h.d yes 4 16.f odd 4 1
1152.3.h.d yes 4 48.i odd 4 1
1152.3.h.d yes 4 48.k even 4 1
2304.3.e.e 4 8.b even 2 1
2304.3.e.e 4 8.d odd 2 1
2304.3.e.e 4 24.f even 2 1
2304.3.e.e 4 24.h odd 2 1
2304.3.e.l 4 1.a even 1 1 trivial
2304.3.e.l 4 3.b odd 2 1 inner
2304.3.e.l 4 4.b odd 2 1 inner
2304.3.e.l 4 12.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{2} + 18 \)
\( T_{7}^{2} - 72 \)
\( T_{13} - 18 \)
\( T_{19}^{2} - 288 \)
\( T_{31}^{2} - 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 18 + T^{2} )^{2} \)
$7$ \( ( -72 + T^{2} )^{2} \)
$11$ \( ( 16 + T^{2} )^{2} \)
$13$ \( ( -18 + T )^{4} \)
$17$ \( ( 18 + T^{2} )^{2} \)
$19$ \( ( -288 + T^{2} )^{2} \)
$23$ \( ( 1296 + T^{2} )^{2} \)
$29$ \( ( 162 + T^{2} )^{2} \)
$31$ \( ( -72 + T^{2} )^{2} \)
$37$ \( ( -36 + T )^{4} \)
$41$ \( ( 882 + T^{2} )^{2} \)
$43$ \( ( -4608 + T^{2} )^{2} \)
$47$ \( ( 1296 + T^{2} )^{2} \)
$53$ \( ( 6498 + T^{2} )^{2} \)
$59$ \( ( 6400 + T^{2} )^{2} \)
$61$ \( ( 36 + T )^{4} \)
$67$ \( ( -14112 + T^{2} )^{2} \)
$71$ \( ( 11664 + T^{2} )^{2} \)
$73$ \( ( -56 + T )^{4} \)
$79$ \( ( -648 + T^{2} )^{2} \)
$83$ \( ( 5776 + T^{2} )^{2} \)
$89$ \( ( 7938 + T^{2} )^{2} \)
$97$ \( ( -104 + T )^{4} \)
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