Properties

Label 2304.3.h.a
Level $2304$
Weight $3$
Character orbit 2304.h
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.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 72)
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.

\(f(q)\) \(=\) \( q + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{5} -12 q^{7} +O(q^{10})\) \( q + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{5} -12 q^{7} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{11} -8 \zeta_{8}^{2} q^{13} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{17} -16 \zeta_{8}^{2} q^{19} + ( -28 \zeta_{8} - 28 \zeta_{8}^{3} ) q^{23} + 25 q^{25} + ( -21 \zeta_{8} + 21 \zeta_{8}^{3} ) q^{29} -4 q^{31} + ( 60 \zeta_{8} - 60 \zeta_{8}^{3} ) q^{35} -30 \zeta_{8}^{2} q^{37} + ( -15 \zeta_{8} - 15 \zeta_{8}^{3} ) q^{41} + 8 \zeta_{8}^{2} q^{43} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{47} + 95 q^{49} + ( -35 \zeta_{8} + 35 \zeta_{8}^{3} ) q^{53} -40 q^{55} + ( -56 \zeta_{8} + 56 \zeta_{8}^{3} ) q^{59} -14 \zeta_{8}^{2} q^{61} + ( 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{65} -88 \zeta_{8}^{2} q^{67} + ( -20 \zeta_{8} - 20 \zeta_{8}^{3} ) q^{71} + 80 q^{73} + ( -48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{77} + 100 q^{79} + ( -92 \zeta_{8} + 92 \zeta_{8}^{3} ) q^{83} + 70 \zeta_{8}^{2} q^{85} + ( 105 \zeta_{8} + 105 \zeta_{8}^{3} ) q^{89} + 96 \zeta_{8}^{2} q^{91} + ( 80 \zeta_{8} + 80 \zeta_{8}^{3} ) q^{95} -112 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 48q^{7} + O(q^{10}) \) \( 4q - 48q^{7} + 100q^{25} - 16q^{31} + 380q^{49} - 160q^{55} + 320q^{73} + 400q^{79} - 448q^{97} + O(q^{100}) \)

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 −7.07107 0 −12.0000 0 0 0
2177.2 0 0 0 −7.07107 0 −12.0000 0 0 0
2177.3 0 0 0 7.07107 0 −12.0000 0 0 0
2177.4 0 0 0 7.07107 0 −12.0000 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.b 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.a 4
3.b odd 2 1 inner 2304.3.h.a 4
4.b odd 2 1 2304.3.h.h 4
8.b even 2 1 inner 2304.3.h.a 4
8.d odd 2 1 2304.3.h.h 4
12.b even 2 1 2304.3.h.h 4
16.e even 4 1 72.3.e.a 2
16.e even 4 1 576.3.e.h 2
16.f odd 4 1 144.3.e.a 2
16.f odd 4 1 576.3.e.a 2
24.f even 2 1 2304.3.h.h 4
24.h odd 2 1 inner 2304.3.h.a 4
48.i odd 4 1 72.3.e.a 2
48.i odd 4 1 576.3.e.h 2
48.k even 4 1 144.3.e.a 2
48.k even 4 1 576.3.e.a 2
80.i odd 4 1 1800.3.c.a 4
80.j even 4 1 3600.3.c.c 4
80.k odd 4 1 3600.3.l.l 2
80.q even 4 1 1800.3.l.a 2
80.s even 4 1 3600.3.c.c 4
80.t odd 4 1 1800.3.c.a 4
112.l odd 4 1 3528.3.d.a 2
144.u even 12 2 1296.3.q.k 4
144.v odd 12 2 1296.3.q.k 4
144.w odd 12 2 648.3.m.a 4
144.x even 12 2 648.3.m.a 4
240.t even 4 1 3600.3.l.l 2
240.z odd 4 1 3600.3.c.c 4
240.bb even 4 1 1800.3.c.a 4
240.bd odd 4 1 3600.3.c.c 4
240.bf even 4 1 1800.3.c.a 4
240.bm odd 4 1 1800.3.l.a 2
336.y even 4 1 3528.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.e.a 2 16.e even 4 1
72.3.e.a 2 48.i odd 4 1
144.3.e.a 2 16.f odd 4 1
144.3.e.a 2 48.k even 4 1
576.3.e.a 2 16.f odd 4 1
576.3.e.a 2 48.k even 4 1
576.3.e.h 2 16.e even 4 1
576.3.e.h 2 48.i odd 4 1
648.3.m.a 4 144.w odd 12 2
648.3.m.a 4 144.x even 12 2
1296.3.q.k 4 144.u even 12 2
1296.3.q.k 4 144.v odd 12 2
1800.3.c.a 4 80.i odd 4 1
1800.3.c.a 4 80.t odd 4 1
1800.3.c.a 4 240.bb even 4 1
1800.3.c.a 4 240.bf even 4 1
1800.3.l.a 2 80.q even 4 1
1800.3.l.a 2 240.bm odd 4 1
2304.3.h.a 4 1.a even 1 1 trivial
2304.3.h.a 4 3.b odd 2 1 inner
2304.3.h.a 4 8.b even 2 1 inner
2304.3.h.a 4 24.h odd 2 1 inner
2304.3.h.h 4 4.b odd 2 1
2304.3.h.h 4 8.d odd 2 1
2304.3.h.h 4 12.b even 2 1
2304.3.h.h 4 24.f even 2 1
3528.3.d.a 2 112.l odd 4 1
3528.3.d.a 2 336.y even 4 1
3600.3.c.c 4 80.j even 4 1
3600.3.c.c 4 80.s even 4 1
3600.3.c.c 4 240.z odd 4 1
3600.3.c.c 4 240.bd odd 4 1
3600.3.l.l 2 80.k odd 4 1
3600.3.l.l 2 240.t even 4 1

Hecke kernels

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

\( T_{5}^{2} - 50 \)
\( T_{7} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -50 + T^{2} )^{2} \)
$7$ \( ( 12 + T )^{4} \)
$11$ \( ( -32 + T^{2} )^{2} \)
$13$ \( ( 64 + T^{2} )^{2} \)
$17$ \( ( 98 + T^{2} )^{2} \)
$19$ \( ( 256 + T^{2} )^{2} \)
$23$ \( ( 1568 + T^{2} )^{2} \)
$29$ \( ( -882 + T^{2} )^{2} \)
$31$ \( ( 4 + T )^{4} \)
$37$ \( ( 900 + T^{2} )^{2} \)
$41$ \( ( 450 + T^{2} )^{2} \)
$43$ \( ( 64 + T^{2} )^{2} \)
$47$ \( ( 288 + T^{2} )^{2} \)
$53$ \( ( -2450 + T^{2} )^{2} \)
$59$ \( ( -6272 + T^{2} )^{2} \)
$61$ \( ( 196 + T^{2} )^{2} \)
$67$ \( ( 7744 + T^{2} )^{2} \)
$71$ \( ( 800 + T^{2} )^{2} \)
$73$ \( ( -80 + T )^{4} \)
$79$ \( ( -100 + T )^{4} \)
$83$ \( ( -16928 + T^{2} )^{2} \)
$89$ \( ( 22050 + T^{2} )^{2} \)
$97$ \( ( 112 + T )^{4} \)
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