Properties

Label 2304.2.d.r
Level $2304$
Weight $2$
Character orbit 2304.d
Analytic conductor $18.398$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{5} + 4 q^{7} +O(q^{10})\) \( q + 2 i q^{5} + 4 q^{7} -2 i q^{11} + 2 i q^{13} + 2 q^{17} + 2 i q^{19} + 4 q^{23} + q^{25} + 6 i q^{29} + 8 i q^{35} -10 i q^{37} -6 q^{41} -6 i q^{43} + 8 q^{47} + 9 q^{49} -6 i q^{53} + 4 q^{55} + 14 i q^{59} + 2 i q^{61} -4 q^{65} + 10 i q^{67} + 12 q^{71} -14 q^{73} -8 i q^{77} -8 q^{79} + 6 i q^{83} + 4 i q^{85} -2 q^{89} + 8 i q^{91} -4 q^{95} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{7} + O(q^{10}) \) \( 2q + 8q^{7} + 4q^{17} + 8q^{23} + 2q^{25} - 12q^{41} + 16q^{47} + 18q^{49} + 8q^{55} - 8q^{65} + 24q^{71} - 28q^{73} - 16q^{79} - 4q^{89} - 8q^{95} - 4q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1153.1
1.00000i
1.00000i
0 0 0 2.00000i 0 4.00000 0 0 0
1153.2 0 0 0 2.00000i 0 4.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.d.r 2
3.b odd 2 1 256.2.b.c 2
4.b odd 2 1 2304.2.d.b 2
8.b even 2 1 inner 2304.2.d.r 2
8.d odd 2 1 2304.2.d.b 2
12.b even 2 1 256.2.b.a 2
16.e even 4 1 1152.2.a.c 1
16.e even 4 1 1152.2.a.m 1
16.f odd 4 1 1152.2.a.h 1
16.f odd 4 1 1152.2.a.r 1
24.f even 2 1 256.2.b.a 2
24.h odd 2 1 256.2.b.c 2
48.i odd 4 1 128.2.a.a 1
48.i odd 4 1 128.2.a.d yes 1
48.k even 4 1 128.2.a.b yes 1
48.k even 4 1 128.2.a.c yes 1
96.o even 8 4 1024.2.e.m 4
96.p odd 8 4 1024.2.e.i 4
240.t even 4 1 3200.2.a.e 1
240.t even 4 1 3200.2.a.u 1
240.z odd 4 1 3200.2.c.f 2
240.z odd 4 1 3200.2.c.k 2
240.bb even 4 1 3200.2.c.e 2
240.bb even 4 1 3200.2.c.l 2
240.bd odd 4 1 3200.2.c.f 2
240.bd odd 4 1 3200.2.c.k 2
240.bf even 4 1 3200.2.c.e 2
240.bf even 4 1 3200.2.c.l 2
240.bm odd 4 1 3200.2.a.h 1
240.bm odd 4 1 3200.2.a.x 1
336.v odd 4 1 6272.2.a.b 1
336.v odd 4 1 6272.2.a.g 1
336.y even 4 1 6272.2.a.a 1
336.y even 4 1 6272.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 48.i odd 4 1
128.2.a.b yes 1 48.k even 4 1
128.2.a.c yes 1 48.k even 4 1
128.2.a.d yes 1 48.i odd 4 1
256.2.b.a 2 12.b even 2 1
256.2.b.a 2 24.f even 2 1
256.2.b.c 2 3.b odd 2 1
256.2.b.c 2 24.h odd 2 1
1024.2.e.i 4 96.p odd 8 4
1024.2.e.m 4 96.o even 8 4
1152.2.a.c 1 16.e even 4 1
1152.2.a.h 1 16.f odd 4 1
1152.2.a.m 1 16.e even 4 1
1152.2.a.r 1 16.f odd 4 1
2304.2.d.b 2 4.b odd 2 1
2304.2.d.b 2 8.d odd 2 1
2304.2.d.r 2 1.a even 1 1 trivial
2304.2.d.r 2 8.b even 2 1 inner
3200.2.a.e 1 240.t even 4 1
3200.2.a.h 1 240.bm odd 4 1
3200.2.a.u 1 240.t even 4 1
3200.2.a.x 1 240.bm odd 4 1
3200.2.c.e 2 240.bb even 4 1
3200.2.c.e 2 240.bf even 4 1
3200.2.c.f 2 240.z odd 4 1
3200.2.c.f 2 240.bd odd 4 1
3200.2.c.k 2 240.z odd 4 1
3200.2.c.k 2 240.bd odd 4 1
3200.2.c.l 2 240.bb even 4 1
3200.2.c.l 2 240.bf even 4 1
6272.2.a.a 1 336.y even 4 1
6272.2.a.b 1 336.v odd 4 1
6272.2.a.g 1 336.v odd 4 1
6272.2.a.h 1 336.y even 4 1

Hecke kernels

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

\( T_{5}^{2} + 4 \)
\( T_{7} - 4 \)
\( T_{11}^{2} + 4 \)
\( T_{17} - 2 \)
\( T_{23} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 + T^{2} \)
$7$ \( ( -4 + T )^{2} \)
$11$ \( 4 + T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 4 + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( 36 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 36 + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( 196 + T^{2} \)
$61$ \( 4 + T^{2} \)
$67$ \( 100 + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( ( 14 + T )^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( ( 2 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
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