# Properties

 Label 2304.3.g.y Level $2304$ Weight $3$ Character orbit 2304.g Analytic conductor $62.779$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.3317760000.5 Defining polynomial: $$x^{8} - 7 x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{22}$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + \beta_{5} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + \beta_{5} q^{7} + \beta_{4} q^{11} + \beta_{1} q^{13} + \beta_{7} q^{17} -\beta_{3} q^{19} + \beta_{6} q^{23} - q^{25} + 5 \beta_{2} q^{29} + \beta_{5} q^{31} -3 \beta_{4} q^{35} + 3 \beta_{1} q^{37} -\beta_{7} q^{41} -5 \beta_{3} q^{43} + \beta_{6} q^{47} -11 q^{49} -3 \beta_{2} q^{53} -8 \beta_{5} q^{55} -2 \beta_{4} q^{59} -\beta_{1} q^{61} -3 \beta_{7} q^{65} -10 \beta_{3} q^{67} + 10 q^{73} -20 \beta_{2} q^{77} -7 \beta_{5} q^{79} + 11 \beta_{4} q^{83} -8 \beta_{1} q^{85} + 2 \beta_{7} q^{89} -15 \beta_{3} q^{91} + \beta_{6} q^{95} + 50 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{25} - 88q^{49} + 80q^{73} + 400q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 7 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{6} + 11 \nu^{2}$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{5} + \nu^{3} + 20 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$-2 \nu^{6} + 6 \nu^{2}$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{7} + 4 \nu^{5} - 13 \nu^{3} - 4 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$4 \nu^{4} - 14$$ $$\beta_{6}$$ $$=$$ $$-2 \nu^{7} - 8 \nu^{5} - 2 \nu^{3} + 40 \nu$$ $$\beta_{7}$$ $$=$$ $$-3 \nu^{7} + 4 \nu^{5} + 13 \nu^{3} - 4 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{7} + \beta_{6} + 4 \beta_{4} + 8 \beta_{2}$$$$)/64$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{1}$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} - 3 \beta_{6} - 4 \beta_{4} + 24 \beta_{2}$$$$)/64$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{5} + 14$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$10 \beta_{7} + \beta_{6} + 20 \beta_{4} + 8 \beta_{2}$$$$)/64$$ $$\nu^{6}$$ $$=$$ $$($$$$-11 \beta_{3} + 6 \beta_{1}$$$$)/16$$ $$\nu^{7}$$ $$=$$ $$($$$$-2 \beta_{7} - 13 \beta_{6} + 4 \beta_{4} + 104 \beta_{2}$$$$)/64$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1279.1
 −0.178197 − 1.40294i 1.40294 − 0.178197i −0.178197 + 1.40294i 1.40294 + 0.178197i 0.178197 + 1.40294i −1.40294 + 0.178197i 0.178197 − 1.40294i −1.40294 − 0.178197i
0 0 0 −4.89898 0 7.74597i 0 0 0
1279.2 0 0 0 −4.89898 0 7.74597i 0 0 0
1279.3 0 0 0 −4.89898 0 7.74597i 0 0 0
1279.4 0 0 0 −4.89898 0 7.74597i 0 0 0
1279.5 0 0 0 4.89898 0 7.74597i 0 0 0
1279.6 0 0 0 4.89898 0 7.74597i 0 0 0
1279.7 0 0 0 4.89898 0 7.74597i 0 0 0
1279.8 0 0 0 4.89898 0 7.74597i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1279.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
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.g.y 8
3.b odd 2 1 inner 2304.3.g.y 8
4.b odd 2 1 inner 2304.3.g.y 8
8.b even 2 1 inner 2304.3.g.y 8
8.d odd 2 1 inner 2304.3.g.y 8
12.b even 2 1 inner 2304.3.g.y 8
16.e even 4 1 72.3.b.c 4
16.e even 4 1 288.3.b.c 4
16.f odd 4 1 72.3.b.c 4
16.f odd 4 1 288.3.b.c 4
24.f even 2 1 inner 2304.3.g.y 8
24.h odd 2 1 inner 2304.3.g.y 8
48.i odd 4 1 72.3.b.c 4
48.i odd 4 1 288.3.b.c 4
48.k even 4 1 72.3.b.c 4
48.k even 4 1 288.3.b.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.b.c 4 16.e even 4 1
72.3.b.c 4 16.f odd 4 1
72.3.b.c 4 48.i odd 4 1
72.3.b.c 4 48.k even 4 1
288.3.b.c 4 16.e even 4 1
288.3.b.c 4 16.f odd 4 1
288.3.b.c 4 48.i odd 4 1
288.3.b.c 4 48.k even 4 1
2304.3.g.y 8 1.a even 1 1 trivial
2304.3.g.y 8 3.b odd 2 1 inner
2304.3.g.y 8 4.b odd 2 1 inner
2304.3.g.y 8 8.b even 2 1 inner
2304.3.g.y 8 8.d odd 2 1 inner
2304.3.g.y 8 12.b even 2 1 inner
2304.3.g.y 8 24.f even 2 1 inner
2304.3.g.y 8 24.h odd 2 1 inner

## Hecke kernels

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

 $$T_{5}^{2} - 24$$ $$T_{7}^{2} + 60$$ $$T_{11}^{2} + 160$$ $$T_{13}^{2} - 240$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( -24 + T^{2} )^{4}$$
$7$ $$( 60 + T^{2} )^{4}$$
$11$ $$( 160 + T^{2} )^{4}$$
$13$ $$( -240 + T^{2} )^{4}$$
$17$ $$( -640 + T^{2} )^{4}$$
$19$ $$( 64 + T^{2} )^{4}$$
$23$ $$( 1536 + T^{2} )^{4}$$
$29$ $$( -600 + T^{2} )^{4}$$
$31$ $$( 60 + T^{2} )^{4}$$
$37$ $$( -2160 + T^{2} )^{4}$$
$41$ $$( -640 + T^{2} )^{4}$$
$43$ $$( 1600 + T^{2} )^{4}$$
$47$ $$( 1536 + T^{2} )^{4}$$
$53$ $$( -216 + T^{2} )^{4}$$
$59$ $$( 640 + T^{2} )^{4}$$
$61$ $$( -240 + T^{2} )^{4}$$
$67$ $$( 6400 + T^{2} )^{4}$$
$71$ $$T^{8}$$
$73$ $$( -10 + T )^{8}$$
$79$ $$( 2940 + T^{2} )^{4}$$
$83$ $$( 19360 + T^{2} )^{4}$$
$89$ $$( -2560 + T^{2} )^{4}$$
$97$ $$( -50 + T )^{8}$$