Properties

Label 2304.3.b.c
Level $2304$
Weight $3$
Character orbit 2304.b
Analytic conductor $62.779$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
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 32)
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.

\(f(q)\) \(=\) \( q + 2 i q^{5} -8 i q^{7} +O(q^{10})\) \( q + 2 i q^{5} -8 i q^{7} -4 q^{11} -14 i q^{13} -18 q^{17} -12 q^{19} -40 i q^{23} + 21 q^{25} + 14 i q^{29} + 32 i q^{31} + 16 q^{35} + 30 i q^{37} -14 q^{41} -28 q^{43} + 16 i q^{47} -15 q^{49} + 66 i q^{53} -8 i q^{55} -52 q^{59} + 82 i q^{61} + 28 q^{65} + 4 q^{67} -56 i q^{71} -66 q^{73} + 32 i q^{77} + 16 i q^{79} + 140 q^{83} -36 i q^{85} -30 q^{89} -112 q^{91} -24 i q^{95} -14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 8q^{11} - 36q^{17} - 24q^{19} + 42q^{25} + 32q^{35} - 28q^{41} - 56q^{43} - 30q^{49} - 104q^{59} + 56q^{65} + 8q^{67} - 132q^{73} + 280q^{83} - 60q^{89} - 224q^{91} - 28q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.00000i
1.00000i
0 0 0 2.00000i 0 8.00000i 0 0 0
127.2 0 0 0 2.00000i 0 8.00000i 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
8.d 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.b.c 2
3.b odd 2 1 256.3.d.c 2
4.b odd 2 1 2304.3.b.g 2
8.b even 2 1 2304.3.b.g 2
8.d odd 2 1 inner 2304.3.b.c 2
12.b even 2 1 256.3.d.a 2
16.e even 4 1 288.3.g.b 2
16.e even 4 1 576.3.g.g 2
16.f odd 4 1 288.3.g.b 2
16.f odd 4 1 576.3.g.g 2
24.f even 2 1 256.3.d.c 2
24.h odd 2 1 256.3.d.a 2
48.i odd 4 1 32.3.c.a 2
48.i odd 4 1 64.3.c.b 2
48.k even 4 1 32.3.c.a 2
48.k even 4 1 64.3.c.b 2
240.t even 4 1 800.3.b.a 2
240.t even 4 1 1600.3.b.e 2
240.z odd 4 1 800.3.h.b 2
240.z odd 4 1 1600.3.h.c 2
240.bb even 4 1 800.3.h.a 2
240.bb even 4 1 1600.3.h.a 2
240.bd odd 4 1 800.3.h.a 2
240.bd odd 4 1 1600.3.h.a 2
240.bf even 4 1 800.3.h.b 2
240.bf even 4 1 1600.3.h.c 2
240.bm odd 4 1 800.3.b.a 2
240.bm odd 4 1 1600.3.b.e 2
336.v odd 4 1 1568.3.d.b 2
336.y even 4 1 1568.3.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.3.c.a 2 48.i odd 4 1
32.3.c.a 2 48.k even 4 1
64.3.c.b 2 48.i odd 4 1
64.3.c.b 2 48.k even 4 1
256.3.d.a 2 12.b even 2 1
256.3.d.a 2 24.h odd 2 1
256.3.d.c 2 3.b odd 2 1
256.3.d.c 2 24.f even 2 1
288.3.g.b 2 16.e even 4 1
288.3.g.b 2 16.f odd 4 1
576.3.g.g 2 16.e even 4 1
576.3.g.g 2 16.f odd 4 1
800.3.b.a 2 240.t even 4 1
800.3.b.a 2 240.bm odd 4 1
800.3.h.a 2 240.bb even 4 1
800.3.h.a 2 240.bd odd 4 1
800.3.h.b 2 240.z odd 4 1
800.3.h.b 2 240.bf even 4 1
1568.3.d.b 2 336.v odd 4 1
1568.3.d.b 2 336.y even 4 1
1600.3.b.e 2 240.t even 4 1
1600.3.b.e 2 240.bm odd 4 1
1600.3.h.a 2 240.bb even 4 1
1600.3.h.a 2 240.bd odd 4 1
1600.3.h.c 2 240.z odd 4 1
1600.3.h.c 2 240.bf even 4 1
2304.3.b.c 2 1.a even 1 1 trivial
2304.3.b.c 2 8.d odd 2 1 inner
2304.3.b.g 2 4.b odd 2 1
2304.3.b.g 2 8.b even 2 1

Hecke kernels

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

\( T_{5}^{2} + 4 \)
\( T_{7}^{2} + 64 \)
\( T_{11} + 4 \)
\( T_{17} + 18 \)
\( T_{19} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 + T^{2} \)
$7$ \( 64 + T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 196 + T^{2} \)
$17$ \( ( 18 + T )^{2} \)
$19$ \( ( 12 + T )^{2} \)
$23$ \( 1600 + T^{2} \)
$29$ \( 196 + T^{2} \)
$31$ \( 1024 + T^{2} \)
$37$ \( 900 + T^{2} \)
$41$ \( ( 14 + T )^{2} \)
$43$ \( ( 28 + T )^{2} \)
$47$ \( 256 + T^{2} \)
$53$ \( 4356 + T^{2} \)
$59$ \( ( 52 + T )^{2} \)
$61$ \( 6724 + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( 3136 + T^{2} \)
$73$ \( ( 66 + T )^{2} \)
$79$ \( 256 + T^{2} \)
$83$ \( ( -140 + T )^{2} \)
$89$ \( ( 30 + T )^{2} \)
$97$ \( ( 14 + T )^{2} \)
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