Properties

Label 2304.3.b.n
Level $2304$
Weight $3$
Character orbit 2304.b
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 6 \zeta_{12}^{3} q^{5} + ( -4 + 8 \zeta_{12}^{2} ) q^{7} +O(q^{10})\) \( q + 6 \zeta_{12}^{3} q^{5} + ( -4 + 8 \zeta_{12}^{2} ) q^{7} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{11} -14 \zeta_{12}^{3} q^{13} + 6 q^{17} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{19} -11 q^{25} -30 \zeta_{12}^{3} q^{29} + ( 12 - 24 \zeta_{12}^{2} ) q^{31} + ( -48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{35} -26 \zeta_{12}^{3} q^{37} -54 q^{41} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{43} + ( -24 + 48 \zeta_{12}^{2} ) q^{47} + q^{49} -18 \zeta_{12}^{3} q^{53} + ( 72 - 144 \zeta_{12}^{2} ) q^{55} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{59} -70 \zeta_{12}^{3} q^{61} + 84 q^{65} + ( 136 \zeta_{12} - 68 \zeta_{12}^{3} ) q^{67} + ( -48 + 96 \zeta_{12}^{2} ) q^{71} -82 q^{73} -144 \zeta_{12}^{3} q^{77} + ( 44 - 88 \zeta_{12}^{2} ) q^{79} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{83} + 36 \zeta_{12}^{3} q^{85} + 114 q^{89} + ( 112 \zeta_{12} - 56 \zeta_{12}^{3} ) q^{91} + ( 24 - 48 \zeta_{12}^{2} ) q^{95} + 34 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 24q^{17} - 44q^{25} - 216q^{41} + 4q^{49} + 336q^{65} - 328q^{73} + 456q^{89} + 136q^{97} + O(q^{100}) \)

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 6.00000i 0 6.92820i 0 0 0
127.2 0 0 0 6.00000i 0 6.92820i 0 0 0
127.3 0 0 0 6.00000i 0 6.92820i 0 0 0
127.4 0 0 0 6.00000i 0 6.92820i 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
4.b odd 2 1 inner
8.b even 2 1 inner
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.n 4
3.b odd 2 1 768.3.b.b 4
4.b odd 2 1 inner 2304.3.b.n 4
8.b even 2 1 inner 2304.3.b.n 4
8.d odd 2 1 inner 2304.3.b.n 4
12.b even 2 1 768.3.b.b 4
16.e even 4 1 144.3.g.b 2
16.e even 4 1 576.3.g.i 2
16.f odd 4 1 144.3.g.b 2
16.f odd 4 1 576.3.g.i 2
24.f even 2 1 768.3.b.b 4
24.h odd 2 1 768.3.b.b 4
48.i odd 4 1 48.3.g.a 2
48.i odd 4 1 192.3.g.a 2
48.k even 4 1 48.3.g.a 2
48.k even 4 1 192.3.g.a 2
80.i odd 4 1 3600.3.j.i 4
80.j even 4 1 3600.3.j.i 4
80.k odd 4 1 3600.3.e.t 2
80.q even 4 1 3600.3.e.t 2
80.s even 4 1 3600.3.j.i 4
80.t odd 4 1 3600.3.j.i 4
144.u even 12 1 1296.3.o.a 2
144.u even 12 1 1296.3.o.c 2
144.v odd 12 1 1296.3.o.n 2
144.v odd 12 1 1296.3.o.p 2
144.w odd 12 1 1296.3.o.a 2
144.w odd 12 1 1296.3.o.c 2
144.x even 12 1 1296.3.o.n 2
144.x even 12 1 1296.3.o.p 2
240.t even 4 1 1200.3.e.h 2
240.z odd 4 1 1200.3.j.a 4
240.bb even 4 1 1200.3.j.a 4
240.bd odd 4 1 1200.3.j.a 4
240.bf even 4 1 1200.3.j.a 4
240.bm odd 4 1 1200.3.e.h 2
336.v odd 4 1 2352.3.m.a 2
336.y even 4 1 2352.3.m.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.g.a 2 48.i odd 4 1
48.3.g.a 2 48.k even 4 1
144.3.g.b 2 16.e even 4 1
144.3.g.b 2 16.f odd 4 1
192.3.g.a 2 48.i odd 4 1
192.3.g.a 2 48.k even 4 1
576.3.g.i 2 16.e even 4 1
576.3.g.i 2 16.f odd 4 1
768.3.b.b 4 3.b odd 2 1
768.3.b.b 4 12.b even 2 1
768.3.b.b 4 24.f even 2 1
768.3.b.b 4 24.h odd 2 1
1200.3.e.h 2 240.t even 4 1
1200.3.e.h 2 240.bm odd 4 1
1200.3.j.a 4 240.z odd 4 1
1200.3.j.a 4 240.bb even 4 1
1200.3.j.a 4 240.bd odd 4 1
1200.3.j.a 4 240.bf even 4 1
1296.3.o.a 2 144.u even 12 1
1296.3.o.a 2 144.w odd 12 1
1296.3.o.c 2 144.u even 12 1
1296.3.o.c 2 144.w odd 12 1
1296.3.o.n 2 144.v odd 12 1
1296.3.o.n 2 144.x even 12 1
1296.3.o.p 2 144.v odd 12 1
1296.3.o.p 2 144.x even 12 1
2304.3.b.n 4 1.a even 1 1 trivial
2304.3.b.n 4 4.b odd 2 1 inner
2304.3.b.n 4 8.b even 2 1 inner
2304.3.b.n 4 8.d odd 2 1 inner
2352.3.m.a 2 336.v odd 4 1
2352.3.m.a 2 336.y even 4 1
3600.3.e.t 2 80.k odd 4 1
3600.3.e.t 2 80.q even 4 1
3600.3.j.i 4 80.i odd 4 1
3600.3.j.i 4 80.j even 4 1
3600.3.j.i 4 80.s even 4 1
3600.3.j.i 4 80.t odd 4 1

Hecke kernels

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

\( T_{5}^{2} + 36 \)
\( T_{7}^{2} + 48 \)
\( T_{11}^{2} - 432 \)
\( T_{17} - 6 \)
\( T_{19}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 36 + T^{2} )^{2} \)
$7$ \( ( 48 + T^{2} )^{2} \)
$11$ \( ( -432 + T^{2} )^{2} \)
$13$ \( ( 196 + T^{2} )^{2} \)
$17$ \( ( -6 + T )^{4} \)
$19$ \( ( -48 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( ( 900 + T^{2} )^{2} \)
$31$ \( ( 432 + T^{2} )^{2} \)
$37$ \( ( 676 + T^{2} )^{2} \)
$41$ \( ( 54 + T )^{4} \)
$43$ \( ( -432 + T^{2} )^{2} \)
$47$ \( ( 1728 + T^{2} )^{2} \)
$53$ \( ( 324 + T^{2} )^{2} \)
$59$ \( ( -432 + T^{2} )^{2} \)
$61$ \( ( 4900 + T^{2} )^{2} \)
$67$ \( ( -13872 + T^{2} )^{2} \)
$71$ \( ( 6912 + T^{2} )^{2} \)
$73$ \( ( 82 + T )^{4} \)
$79$ \( ( 5808 + T^{2} )^{2} \)
$83$ \( ( -432 + T^{2} )^{2} \)
$89$ \( ( -114 + T )^{4} \)
$97$ \( ( -34 + T )^{4} \)
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