# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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