Properties

Label 2304.2.f.d
Level $2304$
Weight $2$
Character orbit 2304.f
Analytic conductor $18.398$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

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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 36)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} +O(q^{10})\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} + 4 \zeta_{8}^{2} q^{13} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{17} -3 q^{25} + ( -7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{29} + 2 \zeta_{8}^{2} q^{37} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{41} + 7 q^{49} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{53} + 10 \zeta_{8}^{2} q^{61} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{65} + 16 q^{73} + 10 \zeta_{8}^{2} q^{85} + ( 13 \zeta_{8} + 13 \zeta_{8}^{3} ) q^{89} + 8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 12q^{25} + 28q^{49} + 64q^{73} + 32q^{97} + O(q^{100}) \)

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 0 0 0 0
1151.2 0 0 0 −1.41421 0 0 0 0 0
1151.3 0 0 0 1.41421 0 0 0 0 0
1151.4 0 0 0 1.41421 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f 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 2304.2.f.d 4
3.b odd 2 1 inner 2304.2.f.d 4
4.b odd 2 1 CM 2304.2.f.d 4
8.b even 2 1 inner 2304.2.f.d 4
8.d odd 2 1 inner 2304.2.f.d 4
12.b even 2 1 inner 2304.2.f.d 4
16.e even 4 1 36.2.b.a 2
16.e even 4 1 576.2.c.b 2
16.f odd 4 1 36.2.b.a 2
16.f odd 4 1 576.2.c.b 2
24.f even 2 1 inner 2304.2.f.d 4
24.h odd 2 1 inner 2304.2.f.d 4
48.i odd 4 1 36.2.b.a 2
48.i odd 4 1 576.2.c.b 2
48.k even 4 1 36.2.b.a 2
48.k even 4 1 576.2.c.b 2
80.i odd 4 1 900.2.h.a 4
80.j even 4 1 900.2.h.a 4
80.k odd 4 1 900.2.e.b 2
80.q even 4 1 900.2.e.b 2
80.s even 4 1 900.2.h.a 4
80.t odd 4 1 900.2.h.a 4
112.j even 4 1 1764.2.e.b 2
112.l odd 4 1 1764.2.e.b 2
144.u even 12 2 324.2.h.c 4
144.v odd 12 2 324.2.h.c 4
144.w odd 12 2 324.2.h.c 4
144.x even 12 2 324.2.h.c 4
240.t even 4 1 900.2.e.b 2
240.z odd 4 1 900.2.h.a 4
240.bb even 4 1 900.2.h.a 4
240.bd odd 4 1 900.2.h.a 4
240.bf even 4 1 900.2.h.a 4
240.bm odd 4 1 900.2.e.b 2
336.v odd 4 1 1764.2.e.b 2
336.y even 4 1 1764.2.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.b.a 2 16.e even 4 1
36.2.b.a 2 16.f odd 4 1
36.2.b.a 2 48.i odd 4 1
36.2.b.a 2 48.k even 4 1
324.2.h.c 4 144.u even 12 2
324.2.h.c 4 144.v odd 12 2
324.2.h.c 4 144.w odd 12 2
324.2.h.c 4 144.x even 12 2
576.2.c.b 2 16.e even 4 1
576.2.c.b 2 16.f odd 4 1
576.2.c.b 2 48.i odd 4 1
576.2.c.b 2 48.k even 4 1
900.2.e.b 2 80.k odd 4 1
900.2.e.b 2 80.q even 4 1
900.2.e.b 2 240.t even 4 1
900.2.e.b 2 240.bm odd 4 1
900.2.h.a 4 80.i odd 4 1
900.2.h.a 4 80.j even 4 1
900.2.h.a 4 80.s even 4 1
900.2.h.a 4 80.t odd 4 1
900.2.h.a 4 240.z odd 4 1
900.2.h.a 4 240.bb even 4 1
900.2.h.a 4 240.bd odd 4 1
900.2.h.a 4 240.bf even 4 1
1764.2.e.b 2 112.j even 4 1
1764.2.e.b 2 112.l odd 4 1
1764.2.e.b 2 336.v odd 4 1
1764.2.e.b 2 336.y even 4 1
2304.2.f.d 4 1.a even 1 1 trivial
2304.2.f.d 4 3.b odd 2 1 inner
2304.2.f.d 4 4.b odd 2 1 CM
2304.2.f.d 4 8.b even 2 1 inner
2304.2.f.d 4 8.d odd 2 1 inner
2304.2.f.d 4 12.b even 2 1 inner
2304.2.f.d 4 24.f even 2 1 inner
2304.2.f.d 4 24.h odd 2 1 inner

Hecke kernels

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

\( T_{5}^{2} - 2 \)
\( T_{7} \)
\( T_{23} \)
\( T_{43} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 8 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 7 T^{2} )^{4} \)
$11$ \( ( 1 - 11 T^{2} )^{4} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 + 16 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 19 T^{2} )^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{4} \)
$29$ \( ( 1 - 40 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 80 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 + 56 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 59 T^{2} )^{4} \)
$61$ \( ( 1 - 12 T + 61 T^{2} )^{2}( 1 + 12 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 + 67 T^{2} )^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{4} \)
$73$ \( ( 1 - 16 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 - 79 T^{2} )^{4} \)
$83$ \( ( 1 - 83 T^{2} )^{4} \)
$89$ \( ( 1 + 160 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 8 T + 97 T^{2} )^{4} \)
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