# Properties

 Label 2304.3.e.m Level $2304$ Weight $3$ Character orbit 2304.e 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.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.5473632256.1 Defining polynomial: $$x^{8} + 49 x^{4} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: no (minimal twist has level 1152) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{5} + \beta_{5} q^{7} +O(q^{10})$$ $$q + \beta_{6} q^{5} + \beta_{5} q^{7} -\beta_{3} q^{11} -\beta_{4} q^{13} -5 \beta_{2} q^{17} + \beta_{7} q^{19} + 5 \beta_{1} q^{23} -9 q^{25} + 5 \beta_{6} q^{29} -15 \beta_{5} q^{31} + \beta_{3} q^{35} -6 \beta_{4} q^{37} -19 \beta_{2} q^{41} -11 \beta_{1} q^{47} -41 q^{49} + 5 \beta_{6} q^{53} + 34 \beta_{5} q^{55} + 4 \beta_{3} q^{59} -10 \beta_{4} q^{61} -34 \beta_{2} q^{65} -5 \beta_{7} q^{67} + 25 \beta_{1} q^{71} -40 q^{73} -8 \beta_{6} q^{77} -45 \beta_{5} q^{79} -5 \beta_{3} q^{83} + 5 \beta_{4} q^{85} -81 \beta_{2} q^{89} -\beta_{7} q^{91} -34 \beta_{1} q^{95} -40 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 72q^{25} - 328q^{49} - 320q^{73} - 320q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 49 x^{4} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 65 \nu^{2}$$$$)/36$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{7} - 16 \nu^{5} - 181 \nu^{3} - 464 \nu$$$$)/576$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 33 \nu^{2}$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$4 \nu^{4} + 98$$$$)/9$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{7} + 16 \nu^{5} - 181 \nu^{3} + 464 \nu$$$$)/288$$ $$\beta_{6}$$ $$=$$ $$($$$$-13 \nu^{7} - 16 \nu^{5} - 701 \nu^{3} - 1616 \nu$$$$)/576$$ $$\beta_{7}$$ $$=$$ $$($$$$-13 \nu^{7} + 16 \nu^{5} - 701 \nu^{3} + 1616 \nu$$$$)/144$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - 4 \beta_{6} - 2 \beta_{5} + 4 \beta_{2}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 9 \beta_{1}$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{7} - 20 \beta_{6} + 26 \beta_{5} + 52 \beta_{2}$$$$)/16$$ $$\nu^{4}$$ $$=$$ $$($$$$9 \beta_{4} - 98$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-29 \beta_{7} + 116 \beta_{6} + 202 \beta_{5} - 404 \beta_{2}$$$$)/16$$ $$\nu^{6}$$ $$=$$ $$($$$$-65 \beta_{3} - 297 \beta_{1}$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$181 \beta_{7} + 724 \beta_{6} - 1402 \beta_{5} - 2804 \beta_{2}$$$$)/16$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 −1.10418 + 1.10418i 1.81129 + 1.81129i 1.10418 + 1.10418i −1.81129 + 1.81129i −1.10418 − 1.10418i 1.81129 − 1.81129i 1.10418 − 1.10418i −1.81129 − 1.81129i
0 0 0 5.83095i 0 −2.82843 0 0 0
1025.2 0 0 0 5.83095i 0 −2.82843 0 0 0
1025.3 0 0 0 5.83095i 0 2.82843 0 0 0
1025.4 0 0 0 5.83095i 0 2.82843 0 0 0
1025.5 0 0 0 5.83095i 0 −2.82843 0 0 0
1025.6 0 0 0 5.83095i 0 −2.82843 0 0 0
1025.7 0 0 0 5.83095i 0 2.82843 0 0 0
1025.8 0 0 0 5.83095i 0 2.82843 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1025.8 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.e.m 8
3.b odd 2 1 inner 2304.3.e.m 8
4.b odd 2 1 inner 2304.3.e.m 8
8.b even 2 1 inner 2304.3.e.m 8
8.d odd 2 1 inner 2304.3.e.m 8
12.b even 2 1 inner 2304.3.e.m 8
16.e even 4 2 1152.3.h.f 8
16.f odd 4 2 1152.3.h.f 8
24.f even 2 1 inner 2304.3.e.m 8
24.h odd 2 1 inner 2304.3.e.m 8
48.i odd 4 2 1152.3.h.f 8
48.k even 4 2 1152.3.h.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.h.f 8 16.e even 4 2
1152.3.h.f 8 16.f odd 4 2
1152.3.h.f 8 48.i odd 4 2
1152.3.h.f 8 48.k even 4 2
2304.3.e.m 8 1.a even 1 1 trivial
2304.3.e.m 8 3.b odd 2 1 inner
2304.3.e.m 8 4.b odd 2 1 inner
2304.3.e.m 8 8.b even 2 1 inner
2304.3.e.m 8 8.d odd 2 1 inner
2304.3.e.m 8 12.b even 2 1 inner
2304.3.e.m 8 24.f even 2 1 inner
2304.3.e.m 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 $$S_{3}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}^{2} + 34$$ $$T_{7}^{2} - 8$$ $$T_{13}^{2} - 68$$ $$T_{19}^{2} - 544$$ $$T_{31}^{2} - 1800$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 34 + T^{2} )^{4}$$
$7$ $$( -8 + T^{2} )^{4}$$
$11$ $$( 272 + T^{2} )^{4}$$
$13$ $$( -68 + T^{2} )^{4}$$
$17$ $$( 50 + T^{2} )^{4}$$
$19$ $$( -544 + T^{2} )^{4}$$
$23$ $$( 400 + T^{2} )^{4}$$
$29$ $$( 850 + T^{2} )^{4}$$
$31$ $$( -1800 + T^{2} )^{4}$$
$37$ $$( -2448 + T^{2} )^{4}$$
$41$ $$( 722 + T^{2} )^{4}$$
$43$ $$T^{8}$$
$47$ $$( 1936 + T^{2} )^{4}$$
$53$ $$( 850 + T^{2} )^{4}$$
$59$ $$( 4352 + T^{2} )^{4}$$
$61$ $$( -6800 + T^{2} )^{4}$$
$67$ $$( -13600 + T^{2} )^{4}$$
$71$ $$( 10000 + T^{2} )^{4}$$
$73$ $$( 40 + T )^{8}$$
$79$ $$( -16200 + T^{2} )^{4}$$
$83$ $$( 6800 + T^{2} )^{4}$$
$89$ $$( 13122 + T^{2} )^{4}$$
$97$ $$( 40 + T )^{8}$$