Properties

Label 2304.2.d.c
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 96)
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.

\(f(q)\) \(=\) \( q + 2 i q^{5} -4 q^{7} +O(q^{10})\) \( q + 2 i q^{5} -4 q^{7} -4 i q^{11} -2 i q^{13} + 6 q^{17} + 4 i q^{19} + q^{25} -2 i q^{29} -4 q^{31} -8 i q^{35} + 2 i q^{37} + 2 q^{41} + 4 i q^{43} + 8 q^{47} + 9 q^{49} + 10 i q^{53} + 8 q^{55} + 4 i q^{59} + 6 i q^{61} + 4 q^{65} -4 i q^{67} + 16 q^{71} + 6 q^{73} + 16 i q^{77} -4 q^{79} + 12 i q^{83} + 12 i q^{85} + 10 q^{89} + 8 i q^{91} -8 q^{95} -14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{7} + O(q^{10}) \) \( 2q - 8q^{7} + 12q^{17} + 2q^{25} - 8q^{31} + 4q^{41} + 16q^{47} + 18q^{49} + 16q^{55} + 8q^{65} + 32q^{71} + 12q^{73} - 8q^{79} + 20q^{89} - 16q^{95} - 28q^{97} + O(q^{100}) \)

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.c 2
3.b odd 2 1 768.2.d.a 2
4.b odd 2 1 2304.2.d.s 2
8.b even 2 1 inner 2304.2.d.c 2
8.d odd 2 1 2304.2.d.s 2
12.b even 2 1 768.2.d.h 2
16.e even 4 1 288.2.a.c 1
16.e even 4 1 576.2.a.h 1
16.f odd 4 1 288.2.a.b 1
16.f odd 4 1 576.2.a.g 1
24.f even 2 1 768.2.d.h 2
24.h odd 2 1 768.2.d.a 2
48.i odd 4 1 96.2.a.a 1
48.i odd 4 1 192.2.a.c 1
48.k even 4 1 96.2.a.b yes 1
48.k even 4 1 192.2.a.a 1
80.i odd 4 1 7200.2.f.x 2
80.j even 4 1 7200.2.f.f 2
80.k odd 4 1 7200.2.a.bx 1
80.q even 4 1 7200.2.a.e 1
80.s even 4 1 7200.2.f.f 2
80.t odd 4 1 7200.2.f.x 2
144.u even 12 2 2592.2.i.h 2
144.v odd 12 2 2592.2.i.w 2
144.w odd 12 2 2592.2.i.b 2
144.x even 12 2 2592.2.i.q 2
240.t even 4 1 2400.2.a.q 1
240.t even 4 1 4800.2.a.co 1
240.z odd 4 1 2400.2.f.r 2
240.z odd 4 1 4800.2.f.e 2
240.bb even 4 1 2400.2.f.a 2
240.bb even 4 1 4800.2.f.bh 2
240.bd odd 4 1 2400.2.f.r 2
240.bd odd 4 1 4800.2.f.e 2
240.bf even 4 1 2400.2.f.a 2
240.bf even 4 1 4800.2.f.bh 2
240.bm odd 4 1 2400.2.a.r 1
240.bm odd 4 1 4800.2.a.f 1
336.v odd 4 1 4704.2.a.e 1
336.v odd 4 1 9408.2.a.ct 1
336.y even 4 1 4704.2.a.t 1
336.y even 4 1 9408.2.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.a.a 1 48.i odd 4 1
96.2.a.b yes 1 48.k even 4 1
192.2.a.a 1 48.k even 4 1
192.2.a.c 1 48.i odd 4 1
288.2.a.b 1 16.f odd 4 1
288.2.a.c 1 16.e even 4 1
576.2.a.g 1 16.f odd 4 1
576.2.a.h 1 16.e even 4 1
768.2.d.a 2 3.b odd 2 1
768.2.d.a 2 24.h odd 2 1
768.2.d.h 2 12.b even 2 1
768.2.d.h 2 24.f even 2 1
2304.2.d.c 2 1.a even 1 1 trivial
2304.2.d.c 2 8.b even 2 1 inner
2304.2.d.s 2 4.b odd 2 1
2304.2.d.s 2 8.d odd 2 1
2400.2.a.q 1 240.t even 4 1
2400.2.a.r 1 240.bm odd 4 1
2400.2.f.a 2 240.bb even 4 1
2400.2.f.a 2 240.bf even 4 1
2400.2.f.r 2 240.z odd 4 1
2400.2.f.r 2 240.bd odd 4 1
2592.2.i.b 2 144.w odd 12 2
2592.2.i.h 2 144.u even 12 2
2592.2.i.q 2 144.x even 12 2
2592.2.i.w 2 144.v odd 12 2
4704.2.a.e 1 336.v odd 4 1
4704.2.a.t 1 336.y even 4 1
4800.2.a.f 1 240.bm odd 4 1
4800.2.a.co 1 240.t even 4 1
4800.2.f.e 2 240.z odd 4 1
4800.2.f.e 2 240.bd odd 4 1
4800.2.f.bh 2 240.bb even 4 1
4800.2.f.bh 2 240.bf even 4 1
7200.2.a.e 1 80.q even 4 1
7200.2.a.bx 1 80.k odd 4 1
7200.2.f.f 2 80.j even 4 1
7200.2.f.f 2 80.s even 4 1
7200.2.f.x 2 80.i odd 4 1
7200.2.f.x 2 80.t odd 4 1
9408.2.a.bj 1 336.y even 4 1
9408.2.a.ct 1 336.v odd 4 1

Hecke kernels

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

\( T_{5}^{2} + 4 \)
\( T_{7} + 4 \)
\( T_{11}^{2} + 16 \)
\( T_{17} - 6 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \)
$7$ \( ( 1 + 4 T + 7 T^{2} )^{2} \)
$11$ \( 1 - 6 T^{2} + 121 T^{4} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 6 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 22 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( 1 - 54 T^{2} + 841 T^{4} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 8 T + 47 T^{2} )^{2} \)
$53$ \( 1 - 6 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 102 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 86 T^{2} + 3721 T^{4} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 16 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 6 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 10 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 14 T + 97 T^{2} )^{2} \)
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