# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.540942598144.16 Defining polynomial: $$x^{8} + 16 x^{6} + 71 x^{4} + 56 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{12}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + \beta_{3} q^{7} +O(q^{10})$$ $$q + \beta_{2} q^{5} + \beta_{3} q^{7} + ( \beta_{2} + \beta_{4} ) q^{11} + \beta_{5} q^{13} + ( -\beta_{1} - 2 \beta_{6} ) q^{17} + ( -\beta_{5} - \beta_{7} ) q^{19} + ( -3 \beta_{1} - \beta_{6} ) q^{23} + ( -5 - 4 \beta_{3} ) q^{25} + ( -\beta_{2} + 2 \beta_{4} ) q^{29} + ( -16 - 3 \beta_{3} ) q^{31} + ( 7 \beta_{2} - \beta_{4} ) q^{35} + ( -\beta_{5} - \beta_{7} ) q^{37} + ( \beta_{1} - 2 \beta_{6} ) q^{41} + ( 2 \beta_{5} - 2 \beta_{7} ) q^{43} + ( -7 \beta_{1} + 3 \beta_{6} ) q^{47} + 3 q^{49} + ( -5 \beta_{2} - 2 \beta_{4} ) q^{53} + ( -32 - 2 \beta_{3} ) q^{55} + ( 4 \beta_{2} - 4 \beta_{4} ) q^{59} + ( -3 \beta_{5} + \beta_{7} ) q^{61} + 2 \beta_{6} q^{65} + ( 3 \beta_{5} - \beta_{7} ) q^{67} + ( -15 \beta_{1} + 3 \beta_{6} ) q^{71} + ( 20 + 8 \beta_{3} ) q^{73} + ( 4 \beta_{2} - 8 \beta_{4} ) q^{77} + ( -48 + 11 \beta_{3} ) q^{79} + ( -7 \beta_{2} + \beta_{4} ) q^{83} + ( 2 \beta_{5} + 5 \beta_{7} ) q^{85} + ( -7 \beta_{1} + 4 \beta_{6} ) q^{89} + ( -7 \beta_{5} + \beta_{7} ) q^{91} + ( -34 \beta_{1} - 14 \beta_{6} ) q^{95} + ( -24 - 4 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 40q^{25} - 128q^{31} + 24q^{49} - 256q^{55} + 160q^{73} - 384q^{79} - 192q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 16 x^{6} + 71 x^{4} + 56 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{5} + 9 \nu^{3} + 11 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{7} + 29 \nu^{5} + 115 \nu^{3} + 73 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{4} + 32 \nu^{2} + 14$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{7} - 35 \nu^{5} - 157 \nu^{3} - 43 \nu$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$2 \nu^{6} + 24 \nu^{4} + 68 \nu^{2} + 16$$ $$\beta_{6}$$ $$=$$ $$($$$$4 \nu^{7} + 61 \nu^{5} + 245 \nu^{3} + 107 \nu$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$2 \nu^{6} + 24 \nu^{4} + 76 \nu^{2} + 48$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + \beta_{4} - \beta_{2} + \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{5} - 32$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{6} - 3 \beta_{4} + 5 \beta_{2} - 2 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{7} + 4 \beta_{5} + \beta_{3} + 114$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$61 \beta_{6} + 43 \beta_{4} - 79 \beta_{2} + 33 \beta_{1}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$31 \beta_{7} - 29 \beta_{5} - 12 \beta_{3} - 856$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-461 \beta_{6} - 315 \beta_{4} + 619 \beta_{2} - 285 \beta_{1}$$$$)/8$$

## 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
 2.78961i − 0.467011i − 0.854013i − 2.69642i 0.854013i 2.69642i − 2.78961i 0.467011i
0 0 0 7.67101i 0 7.21110 0 0 0
1025.2 0 0 0 7.67101i 0 7.21110 0 0 0
1025.3 0 0 0 1.07498i 0 −7.21110 0 0 0
1025.4 0 0 0 1.07498i 0 −7.21110 0 0 0
1025.5 0 0 0 1.07498i 0 −7.21110 0 0 0
1025.6 0 0 0 1.07498i 0 −7.21110 0 0 0
1025.7 0 0 0 7.67101i 0 7.21110 0 0 0
1025.8 0 0 0 7.67101i 0 7.21110 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
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.e.n 8
3.b odd 2 1 inner 2304.3.e.n 8
4.b odd 2 1 2304.3.e.o 8
8.b even 2 1 inner 2304.3.e.n 8
8.d odd 2 1 2304.3.e.o 8
12.b even 2 1 2304.3.e.o 8
16.e even 4 2 72.3.h.a 8
16.f odd 4 2 288.3.h.a 8
24.f even 2 1 2304.3.e.o 8
24.h odd 2 1 inner 2304.3.e.n 8
48.i odd 4 2 72.3.h.a 8
48.k even 4 2 288.3.h.a 8

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

 $$T_{5}^{4} + 60 T_{5}^{2} + 68$$ $$T_{7}^{2} - 52$$ $$T_{13}^{4} - 472 T_{13}^{2} + 2448$$ $$T_{19}^{4} - 1504 T_{19}^{2} + 352512$$ $$T_{31}^{2} + 32 T_{31} - 212$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 68 + 60 T^{2} + T^{4} )^{2}$$
$7$ $$( -52 + T^{2} )^{4}$$
$11$ $$( 9792 + 304 T^{2} + T^{4} )^{2}$$
$13$ $$( 2448 - 472 T^{2} + T^{4} )^{2}$$
$17$ $$( 171396 + 836 T^{2} + T^{4} )^{2}$$
$19$ $$( 352512 - 1504 T^{2} + T^{4} )^{2}$$
$23$ $$( 576 + 464 T^{2} + T^{4} )^{2}$$
$29$ $$( 103428 + 988 T^{2} + T^{4} )^{2}$$
$31$ $$( -212 + 32 T + T^{2} )^{4}$$
$37$ $$( 352512 - 1504 T^{2} + T^{4} )^{2}$$
$41$ $$( 133956 + 932 T^{2} + T^{4} )^{2}$$
$43$ $$( 10027008 - 6400 T^{2} + T^{4} )^{2}$$
$47$ $$( 46656 + 4176 T^{2} + T^{4} )^{2}$$
$53$ $$( 1591812 + 2524 T^{2} + T^{4} )^{2}$$
$59$ $$( 4456448 + 4608 T^{2} + T^{4} )^{2}$$
$61$ $$( 3172608 - 5472 T^{2} + T^{4} )^{2}$$
$67$ $$( 3172608 - 5472 T^{2} + T^{4} )^{2}$$
$71$ $$( 13483584 + 11088 T^{2} + T^{4} )^{2}$$
$73$ $$( -2928 - 40 T + T^{2} )^{4}$$
$79$ $$( -3988 + 96 T + T^{2} )^{4}$$
$83$ $$( 183872 + 3120 T^{2} + T^{4} )^{2}$$
$89$ $$( 171396 + 5828 T^{2} + T^{4} )^{2}$$
$97$ $$( -256 + 48 T + T^{2} )^{4}$$