Properties

Label 2304.3.b.m
Level $2304$
Weight $3$
Character orbit 2304.b
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 144)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -8 + 16 \zeta_{12}^{2} ) q^{7} +O(q^{10})\) \( q + ( -8 + 16 \zeta_{12}^{2} ) q^{7} -22 \zeta_{12}^{3} q^{13} + ( -32 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{19} + 25 q^{25} + ( -24 + 48 \zeta_{12}^{2} ) q^{31} + 26 \zeta_{12}^{3} q^{37} + ( -96 \zeta_{12} + 48 \zeta_{12}^{3} ) q^{43} -143 q^{49} -74 \zeta_{12}^{3} q^{61} + ( 64 \zeta_{12} - 32 \zeta_{12}^{3} ) q^{67} -46 q^{73} + ( 40 - 80 \zeta_{12}^{2} ) q^{79} + ( 352 \zeta_{12} - 176 \zeta_{12}^{3} ) q^{91} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 100q^{25} - 572q^{49} - 184q^{73} - 8q^{97} + O(q^{100}) \)

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
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 0 0 0 0 13.8564i 0 0 0
127.2 0 0 0 0 0 13.8564i 0 0 0
127.3 0 0 0 0 0 13.8564i 0 0 0
127.4 0 0 0 0 0 13.8564i 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 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 2304.3.b.m 4
3.b odd 2 1 CM 2304.3.b.m 4
4.b odd 2 1 inner 2304.3.b.m 4
8.b even 2 1 inner 2304.3.b.m 4
8.d odd 2 1 inner 2304.3.b.m 4
12.b even 2 1 inner 2304.3.b.m 4
16.e even 4 1 144.3.g.c 2
16.e even 4 1 576.3.g.f 2
16.f odd 4 1 144.3.g.c 2
16.f odd 4 1 576.3.g.f 2
24.f even 2 1 inner 2304.3.b.m 4
24.h odd 2 1 inner 2304.3.b.m 4
48.i odd 4 1 144.3.g.c 2
48.i odd 4 1 576.3.g.f 2
48.k even 4 1 144.3.g.c 2
48.k even 4 1 576.3.g.f 2
80.i odd 4 1 3600.3.j.e 4
80.j even 4 1 3600.3.j.e 4
80.k odd 4 1 3600.3.e.h 2
80.q even 4 1 3600.3.e.h 2
80.s even 4 1 3600.3.j.e 4
80.t odd 4 1 3600.3.j.e 4
144.u even 12 1 1296.3.o.f 2
144.u even 12 1 1296.3.o.k 2
144.v odd 12 1 1296.3.o.f 2
144.v odd 12 1 1296.3.o.k 2
144.w odd 12 1 1296.3.o.f 2
144.w odd 12 1 1296.3.o.k 2
144.x even 12 1 1296.3.o.f 2
144.x even 12 1 1296.3.o.k 2
240.t even 4 1 3600.3.e.h 2
240.z odd 4 1 3600.3.j.e 4
240.bb even 4 1 3600.3.j.e 4
240.bd odd 4 1 3600.3.j.e 4
240.bf even 4 1 3600.3.j.e 4
240.bm odd 4 1 3600.3.e.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.g.c 2 16.e even 4 1
144.3.g.c 2 16.f odd 4 1
144.3.g.c 2 48.i odd 4 1
144.3.g.c 2 48.k even 4 1
576.3.g.f 2 16.e even 4 1
576.3.g.f 2 16.f odd 4 1
576.3.g.f 2 48.i odd 4 1
576.3.g.f 2 48.k even 4 1
1296.3.o.f 2 144.u even 12 1
1296.3.o.f 2 144.v odd 12 1
1296.3.o.f 2 144.w odd 12 1
1296.3.o.f 2 144.x even 12 1
1296.3.o.k 2 144.u even 12 1
1296.3.o.k 2 144.v odd 12 1
1296.3.o.k 2 144.w odd 12 1
1296.3.o.k 2 144.x even 12 1
2304.3.b.m 4 1.a even 1 1 trivial
2304.3.b.m 4 3.b odd 2 1 CM
2304.3.b.m 4 4.b odd 2 1 inner
2304.3.b.m 4 8.b even 2 1 inner
2304.3.b.m 4 8.d odd 2 1 inner
2304.3.b.m 4 12.b even 2 1 inner
2304.3.b.m 4 24.f even 2 1 inner
2304.3.b.m 4 24.h odd 2 1 inner
3600.3.e.h 2 80.k odd 4 1
3600.3.e.h 2 80.q even 4 1
3600.3.e.h 2 240.t even 4 1
3600.3.e.h 2 240.bm odd 4 1
3600.3.j.e 4 80.i odd 4 1
3600.3.j.e 4 80.j even 4 1
3600.3.j.e 4 80.s even 4 1
3600.3.j.e 4 80.t odd 4 1
3600.3.j.e 4 240.z odd 4 1
3600.3.j.e 4 240.bb even 4 1
3600.3.j.e 4 240.bd odd 4 1
3600.3.j.e 4 240.bf even 4 1

Hecke kernels

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

\( T_{5} \)
\( T_{7}^{2} + 192 \)
\( T_{11} \)
\( T_{17} \)
\( T_{19}^{2} - 768 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 192 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( 484 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( -768 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( 1728 + T^{2} )^{2} \)
$37$ \( ( 676 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -6912 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 5476 + T^{2} )^{2} \)
$67$ \( ( -3072 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( ( 46 + T )^{4} \)
$79$ \( ( 4800 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( 2 + T )^{4} \)
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