# Properties

 Label 2304.3.h.j Level $2304$ Weight $3$ Character orbit 2304.h Analytic conductor $62.779$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.959512576.1 Defining polynomial: $$x^{8} + 7 x^{4} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} - \beta_{5} ) q^{5} -2 q^{7} +O(q^{10})$$ $$q + ( \beta_{3} - \beta_{5} ) q^{5} -2 q^{7} + ( -4 \beta_{3} + 2 \beta_{5} ) q^{11} + ( 4 \beta_{1} - \beta_{6} ) q^{13} + ( -\beta_{2} + \beta_{4} ) q^{17} + ( 4 \beta_{1} + 2 \beta_{6} ) q^{19} + ( -6 \beta_{2} - 2 \beta_{4} ) q^{23} + ( 21 - \beta_{7} ) q^{25} + ( 6 \beta_{3} + \beta_{5} ) q^{29} + ( 2 + 2 \beta_{7} ) q^{31} + ( -2 \beta_{3} + 2 \beta_{5} ) q^{35} -13 \beta_{1} q^{37} + ( 11 \beta_{2} - \beta_{4} ) q^{41} + ( -14 \beta_{1} + 4 \beta_{6} ) q^{43} + ( 38 \beta_{2} - 2 \beta_{4} ) q^{47} -45 q^{49} + ( -12 \beta_{3} + 3 \beta_{5} ) q^{53} + ( -100 + 4 \beta_{7} ) q^{55} -26 \beta_{3} q^{59} + ( 15 \beta_{1} + 4 \beta_{6} ) q^{61} + ( -52 \beta_{2} - 5 \beta_{4} ) q^{65} + ( 42 \beta_{1} + 2 \beta_{6} ) q^{67} + ( 22 \beta_{2} + 4 \beta_{4} ) q^{71} + ( 28 + 6 \beta_{7} ) q^{73} + ( 8 \beta_{3} - 4 \beta_{5} ) q^{77} + ( 38 - 2 \beta_{7} ) q^{79} + ( -6 \beta_{3} + 6 \beta_{5} ) q^{83} + ( -43 \beta_{1} + \beta_{6} ) q^{85} + ( 47 \beta_{2} + 6 \beta_{4} ) q^{89} + ( -8 \beta_{1} + 2 \beta_{6} ) q^{91} + ( 80 \beta_{2} - 2 \beta_{4} ) q^{95} + ( -88 + \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 16q^{7} + O(q^{10})$$ $$8q - 16q^{7} + 168q^{25} + 16q^{31} - 360q^{49} - 800q^{55} + 224q^{73} + 304q^{79} - 704q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7 x^{4} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{6} + 32 \nu^{2}$$$$)/45$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{7} - 9 \nu^{5} - 13 \nu^{3} - 9 \nu$$$$)/135$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{7} - 18 \nu^{5} + 26 \nu^{3} - 18 \nu$$$$)/135$$ $$\beta_{4}$$ $$=$$ $$($$$$8 \nu^{4} + 28$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{7} - 30 \nu^{6} - 9 \nu^{5} + 13 \nu^{3} + 60 \nu^{2} - 9 \nu$$$$)/135$$ $$\beta_{6}$$ $$=$$ $$($$$$16 \nu^{7} + 18 \nu^{5} + 166 \nu^{3} + 558 \nu$$$$)/135$$ $$\beta_{7}$$ $$=$$ $$($$$$-32 \nu^{7} + 36 \nu^{5} - 332 \nu^{3} + 1116 \nu$$$$)/135$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + 2 \beta_{6} + 2 \beta_{3} + 4 \beta_{2}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{5} - \beta_{3} + 10 \beta_{1}$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{6} + 8 \beta_{3} - 16 \beta_{2}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{4} - 28$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{7} - 2 \beta_{6} - 62 \beta_{3} - 124 \beta_{2}$$$$)/16$$ $$\nu^{6}$$ $$=$$ $$($$$$-8 \beta_{5} + 4 \beta_{3} + 5 \beta_{1}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{7} + 26 \beta_{6} - 166 \beta_{3} + 332 \beta_{2}$$$$)/16$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2177.1
 −1.52616 + 0.819051i −1.52616 − 0.819051i 1.52616 + 0.819051i 1.52616 − 0.819051i 0.819051 + 1.52616i 0.819051 − 1.52616i −0.819051 + 1.52616i −0.819051 − 1.52616i
0 0 0 −8.04746 0 −2.00000 0 0 0
2177.2 0 0 0 −8.04746 0 −2.00000 0 0 0
2177.3 0 0 0 −5.21904 0 −2.00000 0 0 0
2177.4 0 0 0 −5.21904 0 −2.00000 0 0 0
2177.5 0 0 0 5.21904 0 −2.00000 0 0 0
2177.6 0 0 0 5.21904 0 −2.00000 0 0 0
2177.7 0 0 0 8.04746 0 −2.00000 0 0 0
2177.8 0 0 0 8.04746 0 −2.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2177.8 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.b 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.h.j 8
3.b odd 2 1 inner 2304.3.h.j 8
4.b odd 2 1 2304.3.h.l 8
8.b even 2 1 inner 2304.3.h.j 8
8.d odd 2 1 2304.3.h.l 8
12.b even 2 1 2304.3.h.l 8
16.e even 4 1 1152.3.e.e yes 4
16.e even 4 1 1152.3.e.g yes 4
16.f odd 4 1 1152.3.e.a 4
16.f odd 4 1 1152.3.e.c yes 4
24.f even 2 1 2304.3.h.l 8
24.h odd 2 1 inner 2304.3.h.j 8
48.i odd 4 1 1152.3.e.e yes 4
48.i odd 4 1 1152.3.e.g yes 4
48.k even 4 1 1152.3.e.a 4
48.k even 4 1 1152.3.e.c yes 4

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

## Hecke kernels

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

 $$T_{5}^{4} - 92 T_{5}^{2} + 1764$$ $$T_{7} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 1764 - 92 T^{2} + T^{4} )^{2}$$
$7$ $$( 2 + T )^{8}$$
$11$ $$( 10816 - 496 T^{2} + T^{4} )^{2}$$
$13$ $$( 576 + 304 T^{2} + T^{4} )^{2}$$
$17$ $$( 30276 + 356 T^{2} + T^{4} )^{2}$$
$19$ $$( 82944 + 832 T^{2} + T^{4} )^{2}$$
$23$ $$( 399424 + 1552 T^{2} + T^{4} )^{2}$$
$29$ $$( 86436 - 764 T^{2} + T^{4} )^{2}$$
$31$ $$( -1404 - 4 T + T^{2} )^{4}$$
$37$ $$( 676 + T^{2} )^{4}$$
$41$ $$( 4356 + 836 T^{2} + T^{4} )^{2}$$
$43$ $$( 389376 + 4384 T^{2} + T^{4} )^{2}$$
$47$ $$( 4769856 + 7184 T^{2} + T^{4} )^{2}$$
$53$ $$( 236196 - 2556 T^{2} + T^{4} )^{2}$$
$59$ $$( -5408 + T^{2} )^{4}$$
$61$ $$( 258064 + 4616 T^{2} + T^{4} )^{2}$$
$67$ $$( 44943616 + 14816 T^{2} + T^{4} )^{2}$$
$71$ $$( 3415104 + 7568 T^{2} + T^{4} )^{2}$$
$73$ $$( -11888 - 56 T + T^{2} )^{4}$$
$79$ $$( 36 - 76 T + T^{2} )^{4}$$
$83$ $$( 2286144 - 3312 T^{2} + T^{4} )^{2}$$
$89$ $$( 3678724 + 21508 T^{2} + T^{4} )^{2}$$
$97$ $$( 7392 + 176 T + T^{2} )^{4}$$
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