Properties

Label 2304.2.f.e
Level $2304$
Weight $2$
Character orbit 2304.f
Analytic conductor $18.398$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
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 288)
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.

\(f(q)\) \(=\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} + 4 \zeta_{8}^{2} q^{7} +O(q^{10})\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} + 4 \zeta_{8}^{2} q^{7} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{11} + 4 \zeta_{8}^{2} q^{13} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{17} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{23} -3 q^{25} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{29} -4 \zeta_{8}^{2} q^{31} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{35} + 6 \zeta_{8}^{2} q^{37} + ( 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{41} + 8 q^{43} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{47} -9 q^{49} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{53} -8 \zeta_{8}^{2} q^{55} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{59} -2 \zeta_{8}^{2} q^{61} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{65} -8 q^{67} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} + ( 16 \zeta_{8} - 16 \zeta_{8}^{3} ) q^{77} + 4 \zeta_{8}^{2} q^{79} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{83} + 6 \zeta_{8}^{2} q^{85} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{89} -16 q^{91} -8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 12q^{25} + 32q^{43} - 36q^{49} - 32q^{67} - 64q^{91} - 32q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 −1.41421 0 4.00000i 0 0 0
1151.2 0 0 0 −1.41421 0 4.00000i 0 0 0
1151.3 0 0 0 1.41421 0 4.00000i 0 0 0
1151.4 0 0 0 1.41421 0 4.00000i 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 2304.2.f.e 4
3.b odd 2 1 inner 2304.2.f.e 4
4.b odd 2 1 2304.2.f.c 4
8.b even 2 1 2304.2.f.c 4
8.d odd 2 1 inner 2304.2.f.e 4
12.b even 2 1 2304.2.f.c 4
16.e even 4 1 288.2.c.a 4
16.e even 4 1 576.2.c.c 4
16.f odd 4 1 288.2.c.a 4
16.f odd 4 1 576.2.c.c 4
24.f even 2 1 inner 2304.2.f.e 4
24.h odd 2 1 2304.2.f.c 4
48.i odd 4 1 288.2.c.a 4
48.i odd 4 1 576.2.c.c 4
48.k even 4 1 288.2.c.a 4
48.k even 4 1 576.2.c.c 4
80.i odd 4 1 7200.2.o.n 4
80.j even 4 1 7200.2.o.n 4
80.k odd 4 1 7200.2.h.d 4
80.q even 4 1 7200.2.h.d 4
80.s even 4 1 7200.2.o.a 4
80.t odd 4 1 7200.2.o.a 4
144.u even 12 2 2592.2.s.d 8
144.v odd 12 2 2592.2.s.d 8
144.w odd 12 2 2592.2.s.d 8
144.x even 12 2 2592.2.s.d 8
240.t even 4 1 7200.2.h.d 4
240.z odd 4 1 7200.2.o.a 4
240.bb even 4 1 7200.2.o.n 4
240.bd odd 4 1 7200.2.o.n 4
240.bf even 4 1 7200.2.o.a 4
240.bm odd 4 1 7200.2.h.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.c.a 4 16.e even 4 1
288.2.c.a 4 16.f odd 4 1
288.2.c.a 4 48.i odd 4 1
288.2.c.a 4 48.k even 4 1
576.2.c.c 4 16.e even 4 1
576.2.c.c 4 16.f odd 4 1
576.2.c.c 4 48.i odd 4 1
576.2.c.c 4 48.k even 4 1
2304.2.f.c 4 4.b odd 2 1
2304.2.f.c 4 8.b even 2 1
2304.2.f.c 4 12.b even 2 1
2304.2.f.c 4 24.h odd 2 1
2304.2.f.e 4 1.a even 1 1 trivial
2304.2.f.e 4 3.b odd 2 1 inner
2304.2.f.e 4 8.d odd 2 1 inner
2304.2.f.e 4 24.f even 2 1 inner
2592.2.s.d 8 144.u even 12 2
2592.2.s.d 8 144.v odd 12 2
2592.2.s.d 8 144.w odd 12 2
2592.2.s.d 8 144.x even 12 2
7200.2.h.d 4 80.k odd 4 1
7200.2.h.d 4 80.q even 4 1
7200.2.h.d 4 240.t even 4 1
7200.2.h.d 4 240.bm odd 4 1
7200.2.o.a 4 80.s even 4 1
7200.2.o.a 4 80.t odd 4 1
7200.2.o.a 4 240.z odd 4 1
7200.2.o.a 4 240.bf even 4 1
7200.2.o.n 4 80.i odd 4 1
7200.2.o.n 4 80.j even 4 1
7200.2.o.n 4 240.bb even 4 1
7200.2.o.n 4 240.bd odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} - 2 \)
\( T_{7}^{2} + 16 \)
\( T_{23}^{2} - 32 \)
\( T_{43} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 8 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + 2 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 + 10 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 16 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 19 T^{2} )^{4} \)
$23$ \( ( 1 + 14 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 56 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 46 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 38 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 16 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 62 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 88 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 10 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 118 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 8 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 + 110 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 73 T^{2} )^{4} \)
$79$ \( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 134 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 160 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 8 T + 97 T^{2} )^{4} \)
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