# Properties

 Label 2304.3.b.k Level $2304$ Weight $3$ Character orbit 2304.b Analytic conductor $62.779$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 96) 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.

 $$f(q)$$ $$=$$ $$q + ( 4 - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + ( -4 + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{7} +O(q^{10})$$ $$q + ( 4 - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + ( -4 + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{7} + ( -8 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{11} + ( 8 - 16 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{13} + ( 6 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{17} + ( 24 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{19} + 8 \zeta_{12}^{3} q^{23} + ( -27 + 32 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{25} + ( 4 - 8 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{29} + ( -20 + 40 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{31} + ( 56 - 48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{35} + ( 16 - 32 \zeta_{12}^{2} + 18 \zeta_{12}^{3} ) q^{37} + ( -22 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{41} + ( -40 + 40 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{43} + ( 8 - 16 \zeta_{12}^{2} - 56 \zeta_{12}^{3} ) q^{47} + ( -15 + 64 \zeta_{12} - 32 \zeta_{12}^{3} ) q^{49} + ( -4 + 8 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{53} + ( -24 + 48 \zeta_{12}^{2} + 32 \zeta_{12}^{3} ) q^{55} + ( -32 - 88 \zeta_{12} + 44 \zeta_{12}^{3} ) q^{59} + 14 \zeta_{12}^{3} q^{61} + ( -76 - 48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{65} + ( -32 + 56 \zeta_{12} - 28 \zeta_{12}^{3} ) q^{67} + ( 48 - 96 \zeta_{12}^{2} + 40 \zeta_{12}^{3} ) q^{71} + ( 30 - 64 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{73} + ( 16 - 32 \zeta_{12}^{2} - 16 \zeta_{12}^{3} ) q^{77} + ( 12 - 24 \zeta_{12}^{2} - 76 \zeta_{12}^{3} ) q^{79} + ( -56 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{83} + ( 8 - 16 \zeta_{12}^{2} - 84 \zeta_{12}^{3} ) q^{85} + ( -78 + 32 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{89} + ( 56 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{91} + ( 88 - 176 \zeta_{12}^{2} ) q^{95} + ( -62 - 96 \zeta_{12} + 48 \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 32q^{11} + 24q^{17} + 96q^{19} - 108q^{25} + 224q^{35} - 88q^{41} - 160q^{43} - 60q^{49} - 128q^{59} - 304q^{65} - 128q^{67} + 120q^{73} - 224q^{83} - 312q^{89} + 224q^{91} - 248q^{97} + O(q^{100})$$

## 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 8.92820i 0 10.9282i 0 0 0
127.2 0 0 0 4.92820i 0 2.92820i 0 0 0
127.3 0 0 0 4.92820i 0 2.92820i 0 0 0
127.4 0 0 0 8.92820i 0 10.9282i 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
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.k 4
3.b odd 2 1 768.3.b.d 4
4.b odd 2 1 2304.3.b.o 4
8.b even 2 1 2304.3.b.o 4
8.d odd 2 1 inner 2304.3.b.k 4
12.b even 2 1 768.3.b.a 4
16.e even 4 1 288.3.g.d 4
16.e even 4 1 576.3.g.j 4
16.f odd 4 1 288.3.g.d 4
16.f odd 4 1 576.3.g.j 4
24.f even 2 1 768.3.b.d 4
24.h odd 2 1 768.3.b.a 4
48.i odd 4 1 96.3.g.a 4
48.i odd 4 1 192.3.g.c 4
48.k even 4 1 96.3.g.a 4
48.k even 4 1 192.3.g.c 4
240.t even 4 1 2400.3.e.a 4
240.z odd 4 1 2400.3.j.a 4
240.bb even 4 1 2400.3.j.b 4
240.bd odd 4 1 2400.3.j.b 4
240.bf even 4 1 2400.3.j.a 4
240.bm odd 4 1 2400.3.e.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.g.a 4 48.i odd 4 1
96.3.g.a 4 48.k even 4 1
192.3.g.c 4 48.i odd 4 1
192.3.g.c 4 48.k even 4 1
288.3.g.d 4 16.e even 4 1
288.3.g.d 4 16.f odd 4 1
576.3.g.j 4 16.e even 4 1
576.3.g.j 4 16.f odd 4 1
768.3.b.a 4 12.b even 2 1
768.3.b.a 4 24.h odd 2 1
768.3.b.d 4 3.b odd 2 1
768.3.b.d 4 24.f even 2 1
2304.3.b.k 4 1.a even 1 1 trivial
2304.3.b.k 4 8.d odd 2 1 inner
2304.3.b.o 4 4.b odd 2 1
2304.3.b.o 4 8.b even 2 1
2400.3.e.a 4 240.t even 4 1
2400.3.e.a 4 240.bm odd 4 1
2400.3.j.a 4 240.z odd 4 1
2400.3.j.a 4 240.bf even 4 1
2400.3.j.b 4 240.bb even 4 1
2400.3.j.b 4 240.bd odd 4 1

## Hecke kernels

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

 $$T_{5}^{4} + 104 T_{5}^{2} + 1936$$ $$T_{7}^{4} + 128 T_{7}^{2} + 1024$$ $$T_{11}^{2} + 16 T_{11} + 16$$ $$T_{17}^{2} - 12 T_{17} - 156$$ $$T_{19}^{2} - 48 T_{19} + 528$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$1936 + 104 T^{2} + T^{4}$$
$7$ $$1024 + 128 T^{2} + T^{4}$$
$11$ $$( 16 + 16 T + T^{2} )^{2}$$
$13$ $$8464 + 584 T^{2} + T^{4}$$
$17$ $$( -156 - 12 T + T^{2} )^{2}$$
$19$ $$( 528 - 48 T + T^{2} )^{2}$$
$23$ $$( 64 + T^{2} )^{2}$$
$29$ $$2704 + 296 T^{2} + T^{4}$$
$31$ $$1401856 + 2432 T^{2} + T^{4}$$
$37$ $$197136 + 2184 T^{2} + T^{4}$$
$41$ $$( 292 + 44 T + T^{2} )^{2}$$
$43$ $$( 400 + 80 T + T^{2} )^{2}$$
$47$ $$8667136 + 6656 T^{2} + T^{4}$$
$53$ $$144 + 168 T^{2} + T^{4}$$
$59$ $$( -4784 + 64 T + T^{2} )^{2}$$
$61$ $$( 196 + T^{2} )^{2}$$
$67$ $$( -1328 + 64 T + T^{2} )^{2}$$
$71$ $$28217344 + 17024 T^{2} + T^{4}$$
$73$ $$( -2172 - 60 T + T^{2} )^{2}$$
$79$ $$28558336 + 12416 T^{2} + T^{4}$$
$83$ $$( 3088 + 112 T + T^{2} )^{2}$$
$89$ $$( 5316 + 156 T + T^{2} )^{2}$$
$97$ $$( -3068 + 124 T + T^{2} )^{2}$$