# Properties

 Label 2304.3.b.s Level $2304$ Weight $3$ Character orbit 2304.b Analytic conductor $62.779$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{18}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + ( -\beta_{1} - \beta_{6} ) q^{7} +O(q^{10})$$ $$q + \beta_{2} q^{5} + ( -\beta_{1} - \beta_{6} ) q^{7} + \beta_{4} q^{11} + ( 3 \beta_{1} + 2 \beta_{6} ) q^{13} + ( -\beta_{3} + \beta_{4} ) q^{17} + ( 12 + \beta_{7} ) q^{19} + ( -5 \beta_{2} - \beta_{5} ) q^{23} + ( -11 - 2 \beta_{7} ) q^{25} + ( -\beta_{2} - 2 \beta_{5} ) q^{29} + ( 9 \beta_{1} + \beta_{6} ) q^{31} + ( 8 \beta_{3} + 3 \beta_{4} ) q^{35} + ( -\beta_{1} + 2 \beta_{6} ) q^{37} + ( 7 \beta_{3} + \beta_{4} ) q^{41} + ( 20 - \beta_{7} ) q^{43} + ( -7 \beta_{2} + 5 \beta_{5} ) q^{47} + ( -11 - 2 \beta_{7} ) q^{49} + ( \beta_{2} + 6 \beta_{5} ) q^{53} + ( 28 \beta_{1} + 4 \beta_{6} ) q^{55} + ( -8 \beta_{3} + 2 \beta_{4} ) q^{59} + ( -7 \beta_{1} - 2 \beta_{6} ) q^{61} + ( -17 \beta_{3} - 7 \beta_{4} ) q^{65} + ( 24 - 6 \beta_{7} ) q^{67} + ( -8 \beta_{2} - 8 \beta_{5} ) q^{71} + ( 38 + 6 \beta_{7} ) q^{73} + ( -10 \beta_{2} - 6 \beta_{5} ) q^{77} + ( 29 \beta_{1} - 3 \beta_{6} ) q^{79} + ( 16 \beta_{3} + \beta_{4} ) q^{83} + 20 \beta_{1} q^{85} + ( 14 \beta_{3} - 6 \beta_{4} ) q^{89} + ( 124 + 5 \beta_{7} ) q^{91} + ( 25 \beta_{2} + 5 \beta_{5} ) q^{95} + ( 14 + 4 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 96q^{19} - 88q^{25} + 160q^{43} - 88q^{49} + 192q^{67} + 304q^{73} + 992q^{91} + 112q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{2}$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} + 16 \nu^{4} + 7 \nu^{3} - 4 \nu + 8$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 4 \nu^{5} + 7 \nu^{3} + 4 \nu$$$$)/6$$ $$\beta_{4}$$ $$=$$ $$-\nu^{6} + 3 \nu^{2}$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 12 \nu^{5} - 16 \nu^{4} + 21 \nu^{3} - 12 \nu - 8$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{7} + 4 \nu^{5} + 13 \nu^{3} + 44 \nu$$$$)/12$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 4 \nu^{5} - 13 \nu^{3} + 44 \nu$$$$)/6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{7} + 4 \beta_{6} - \beta_{5} + 2 \beta_{3} - \beta_{2}$$$$)/32$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + 6 \beta_{1}$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} + 4 \beta_{6} + 5 \beta_{5} + 10 \beta_{3} + 5 \beta_{2}$$$$)/32$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{5} + 9 \beta_{2} - 8$$$$)/16$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{7} + 4 \beta_{6} + 11 \beta_{5} - 22 \beta_{3} + 11 \beta_{2}$$$$)/32$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{4} + 18 \beta_{1}$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$-14 \beta_{7} + 28 \beta_{6} - 13 \beta_{5} - 26 \beta_{3} - 13 \beta_{2}$$$$)/32$$

## 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
 0.581861 + 1.28897i 1.28897 − 0.581861i −1.28897 + 0.581861i −0.581861 − 1.28897i −0.581861 + 1.28897i −1.28897 − 0.581861i 1.28897 + 0.581861i 0.581861 − 1.28897i
0 0 0 8.11993i 0 9.48331i 0 0 0
127.2 0 0 0 8.11993i 0 9.48331i 0 0 0
127.3 0 0 0 2.46308i 0 5.48331i 0 0 0
127.4 0 0 0 2.46308i 0 5.48331i 0 0 0
127.5 0 0 0 2.46308i 0 5.48331i 0 0 0
127.6 0 0 0 2.46308i 0 5.48331i 0 0 0
127.7 0 0 0 8.11993i 0 9.48331i 0 0 0
127.8 0 0 0 8.11993i 0 9.48331i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.8 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
8.d odd 2 1 inner
24.f even 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.s 8
3.b odd 2 1 inner 2304.3.b.s 8
4.b odd 2 1 2304.3.b.r 8
8.b even 2 1 2304.3.b.r 8
8.d odd 2 1 inner 2304.3.b.s 8
12.b even 2 1 2304.3.b.r 8
16.e even 4 1 1152.3.g.d 8
16.e even 4 1 1152.3.g.e yes 8
16.f odd 4 1 1152.3.g.d 8
16.f odd 4 1 1152.3.g.e yes 8
24.f even 2 1 inner 2304.3.b.s 8
24.h odd 2 1 2304.3.b.r 8
48.i odd 4 1 1152.3.g.d 8
48.i odd 4 1 1152.3.g.e yes 8
48.k even 4 1 1152.3.g.d 8
48.k even 4 1 1152.3.g.e yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.g.d 8 16.e even 4 1
1152.3.g.d 8 16.f odd 4 1
1152.3.g.d 8 48.i odd 4 1
1152.3.g.d 8 48.k even 4 1
1152.3.g.e yes 8 16.e even 4 1
1152.3.g.e yes 8 16.f odd 4 1
1152.3.g.e yes 8 48.i odd 4 1
1152.3.g.e yes 8 48.k even 4 1
2304.3.b.r 8 4.b odd 2 1
2304.3.b.r 8 8.b even 2 1
2304.3.b.r 8 12.b even 2 1
2304.3.b.r 8 24.h odd 2 1
2304.3.b.s 8 1.a even 1 1 trivial
2304.3.b.s 8 3.b odd 2 1 inner
2304.3.b.s 8 8.d odd 2 1 inner
2304.3.b.s 8 24.f even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{4} + 72 T_{5}^{2} + 400$$ $$T_{7}^{4} + 120 T_{7}^{2} + 2704$$ $$T_{11}^{2} - 112$$ $$T_{17}^{4} - 288 T_{17}^{2} + 6400$$ $$T_{19}^{2} - 24 T_{19} - 80$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 400 + 72 T^{2} + T^{4} )^{2}$$
$7$ $$( 2704 + 120 T^{2} + T^{4} )^{2}$$
$11$ $$( -112 + T^{2} )^{4}$$
$13$ $$( 35344 + 520 T^{2} + T^{4} )^{2}$$
$17$ $$( 6400 - 288 T^{2} + T^{4} )^{2}$$
$19$ $$( -80 - 24 T + T^{2} )^{4}$$
$23$ $$( 4096 + 1920 T^{2} + T^{4} )^{2}$$
$29$ $$( 132496 + 840 T^{2} + T^{4} )^{2}$$
$31$ $$( 71824 + 760 T^{2} + T^{4} )^{2}$$
$37$ $$( 48400 + 456 T^{2} + T^{4} )^{2}$$
$41$ $$( 2119936 - 3360 T^{2} + T^{4} )^{2}$$
$43$ $$( 176 - 40 T + T^{2} )^{4}$$
$47$ $$( 12390400 + 9088 T^{2} + T^{4} )^{2}$$
$53$ $$( 4787344 + 7176 T^{2} + T^{4} )^{2}$$
$59$ $$( 2560000 - 4992 T^{2} + T^{4} )^{2}$$
$61$ $$( 784 + 840 T^{2} + T^{4} )^{2}$$
$67$ $$( -7488 - 48 T + T^{2} )^{4}$$
$71$ $$( 8192 + T^{2} )^{4}$$
$73$ $$( -6620 - 76 T + T^{2} )^{4}$$
$79$ $$( 8179600 + 7736 T^{2} + T^{4} )^{2}$$
$83$ $$( 65286400 - 16608 T^{2} + T^{4} )^{2}$$
$89$ $$( 5017600 - 20608 T^{2} + T^{4} )^{2}$$
$97$ $$( -3388 - 28 T + T^{2} )^{4}$$