Properties

Label 2304.1.b.a
Level 2304
Weight 1
Character orbit 2304.b
Analytic conductor 1.150
Analytic rank 0
Dimension 2
Projective image \(D_{2}\)
CM/RM discs -3, -4, 12
Inner twists 8

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 144)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\zeta_{12})\)
Artin image $D_4:C_2$
Artin field Galois closure of 8.0.3057647616.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q -2 i q^{13} + q^{25} -2 i q^{37} + q^{49} + 2 i q^{61} + 2 q^{73} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 2q^{25} + 2q^{49} + 4q^{73} - 4q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.00000i
1.00000i
0 0 0 0 0 0 0 0 0
127.2 0 0 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
12.b even 2 1 RM by \(\Q(\sqrt{3}) \)
8.b even 2 1 inner
8.d odd 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 2304.1.b.a 2
3.b odd 2 1 CM 2304.1.b.a 2
4.b odd 2 1 CM 2304.1.b.a 2
8.b even 2 1 inner 2304.1.b.a 2
8.d odd 2 1 inner 2304.1.b.a 2
12.b even 2 1 RM 2304.1.b.a 2
16.e even 4 1 144.1.g.a 1
16.e even 4 1 576.1.g.a 1
16.f odd 4 1 144.1.g.a 1
16.f odd 4 1 576.1.g.a 1
24.f even 2 1 inner 2304.1.b.a 2
24.h odd 2 1 inner 2304.1.b.a 2
48.i odd 4 1 144.1.g.a 1
48.i odd 4 1 576.1.g.a 1
48.k even 4 1 144.1.g.a 1
48.k even 4 1 576.1.g.a 1
80.i odd 4 1 3600.1.j.a 2
80.j even 4 1 3600.1.j.a 2
80.k odd 4 1 3600.1.e.b 1
80.q even 4 1 3600.1.e.b 1
80.s even 4 1 3600.1.j.a 2
80.t odd 4 1 3600.1.j.a 2
144.u even 12 2 1296.1.o.b 2
144.v odd 12 2 1296.1.o.b 2
144.w odd 12 2 1296.1.o.b 2
144.x even 12 2 1296.1.o.b 2
240.t even 4 1 3600.1.e.b 1
240.z odd 4 1 3600.1.j.a 2
240.bb even 4 1 3600.1.j.a 2
240.bd odd 4 1 3600.1.j.a 2
240.bf even 4 1 3600.1.j.a 2
240.bm odd 4 1 3600.1.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.1.g.a 1 16.e even 4 1
144.1.g.a 1 16.f odd 4 1
144.1.g.a 1 48.i odd 4 1
144.1.g.a 1 48.k even 4 1
576.1.g.a 1 16.e even 4 1
576.1.g.a 1 16.f odd 4 1
576.1.g.a 1 48.i odd 4 1
576.1.g.a 1 48.k even 4 1
1296.1.o.b 2 144.u even 12 2
1296.1.o.b 2 144.v odd 12 2
1296.1.o.b 2 144.w odd 12 2
1296.1.o.b 2 144.x even 12 2
2304.1.b.a 2 1.a even 1 1 trivial
2304.1.b.a 2 3.b odd 2 1 CM
2304.1.b.a 2 4.b odd 2 1 CM
2304.1.b.a 2 8.b even 2 1 inner
2304.1.b.a 2 8.d odd 2 1 inner
2304.1.b.a 2 12.b even 2 1 RM
2304.1.b.a 2 24.f even 2 1 inner
2304.1.b.a 2 24.h odd 2 1 inner
3600.1.e.b 1 80.k odd 4 1
3600.1.e.b 1 80.q even 4 1
3600.1.e.b 1 240.t even 4 1
3600.1.e.b 1 240.bm odd 4 1
3600.1.j.a 2 80.i odd 4 1
3600.1.j.a 2 80.j even 4 1
3600.1.j.a 2 80.s even 4 1
3600.1.j.a 2 80.t odd 4 1
3600.1.j.a 2 240.z odd 4 1
3600.1.j.a 2 240.bb even 4 1
3600.1.j.a 2 240.bd odd 4 1
3600.1.j.a 2 240.bf even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2304, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$7$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$11$ \( ( 1 + T^{2} )^{2} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( ( 1 + T^{2} )^{2} \)
$19$ \( ( 1 + T^{2} )^{2} \)
$23$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$29$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$31$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$37$ \( ( 1 + T^{2} )^{2} \)
$41$ \( ( 1 + T^{2} )^{2} \)
$43$ \( ( 1 + T^{2} )^{2} \)
$47$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$53$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$59$ \( ( 1 + T^{2} )^{2} \)
$61$ \( ( 1 + T^{2} )^{2} \)
$67$ \( ( 1 + T^{2} )^{2} \)
$71$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$73$ \( ( 1 - T )^{4} \)
$79$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$83$ \( ( 1 + T^{2} )^{2} \)
$89$ \( ( 1 + T^{2} )^{2} \)
$97$ \( ( 1 + T )^{4} \)
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