# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.22581504.2 Defining polynomial: $$x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{24}$$ Twist minimal: no (minimal twist has level 24) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{5} -\beta_{7} q^{7} +O(q^{10})$$ $$q -\beta_{1} q^{5} -\beta_{7} q^{7} + \beta_{6} q^{11} + ( \beta_{1} - \beta_{4} ) q^{13} + ( 2 + \beta_{5} ) q^{17} + ( -\beta_{3} - \beta_{6} ) q^{19} -2 \beta_{7} q^{23} + ( 11 - 2 \beta_{5} ) q^{25} + ( -\beta_{1} - 2 \beta_{4} ) q^{29} -\beta_{2} q^{31} + ( -\beta_{3} + 3 \beta_{6} ) q^{35} + ( -3 \beta_{1} + \beta_{4} ) q^{37} + ( 10 + 3 \beta_{5} ) q^{41} + ( 3 \beta_{3} + \beta_{6} ) q^{43} + ( -\beta_{2} - \beta_{7} ) q^{47} + ( -11 - 4 \beta_{5} ) q^{49} + ( 3 \beta_{1} - 2 \beta_{4} ) q^{53} + ( -\beta_{2} - 3 \beta_{7} ) q^{55} + ( 3 \beta_{3} + 4 \beta_{6} ) q^{59} + ( -7 \beta_{1} + \beta_{4} ) q^{61} + ( -24 + 5 \beta_{5} ) q^{65} + ( 3 \beta_{3} + 8 \beta_{6} ) q^{67} + ( -\beta_{2} + 7 \beta_{7} ) q^{71} + ( -50 - 4 \beta_{5} ) q^{73} + ( 4 \beta_{1} + 4 \beta_{4} ) q^{77} + ( \beta_{2} - 6 \beta_{7} ) q^{79} + ( 4 \beta_{3} + 5 \beta_{6} ) q^{83} + ( 10 \beta_{1} - 4 \beta_{4} ) q^{85} + ( -50 + 6 \beta_{5} ) q^{89} + ( -14 \beta_{3} + 9 \beta_{6} ) q^{91} + ( 2 \beta_{2} + 2 \beta_{7} ) q^{95} + ( 14 - 6 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 16q^{17} + 88q^{25} + 80q^{41} - 88q^{49} - 192q^{65} - 400q^{73} - 400q^{89} + 112q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{7} - 2 \nu^{6} - \nu^{5} + 4 \nu^{4} - \nu^{3} - 6 \nu^{2} + 10 \nu - 2$$ $$\beta_{2}$$ $$=$$ $$-10 \nu^{7} + 18 \nu^{6} + 2 \nu^{5} - 38 \nu^{4} + 22 \nu^{3} + 70 \nu^{2} - 84 \nu + 22$$ $$\beta_{3}$$ $$=$$ $$-6 \nu^{7} + 14 \nu^{6} - 6 \nu^{5} - 22 \nu^{4} + 30 \nu^{3} + 22 \nu^{2} - 80 \nu + 60$$ $$\beta_{4}$$ $$=$$ $$-5 \nu^{7} + 15 \nu^{6} - 9 \nu^{5} - 19 \nu^{4} + 33 \nu^{3} + 15 \nu^{2} - 76 \nu + 66$$ $$\beta_{5}$$ $$=$$ $$-6 \nu^{7} + 18 \nu^{6} - 10 \nu^{5} - 26 \nu^{4} + 42 \nu^{3} + 26 \nu^{2} - 108 \nu + 80$$ $$\beta_{6}$$ $$=$$ $$8 \nu^{7} - 22 \nu^{6} + 12 \nu^{5} + 34 \nu^{4} - 48 \nu^{3} - 30 \nu^{2} + 124 \nu - 96$$ $$\beta_{7}$$ $$=$$ $$10 \nu^{7} - 26 \nu^{6} + 14 \nu^{5} + 38 \nu^{4} - 54 \nu^{3} - 38 \nu^{2} + 148 \nu - 114$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + \beta_{2} + 4 \beta_{1} + 32$$$$)/64$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + \beta_{2} + 4 \beta_{1} + 24$$$$)/32$$ $$\nu^{3}$$ $$=$$ $$($$$$11 \beta_{7} + 2 \beta_{6} + 6 \beta_{5} + 4 \beta_{4} + 12 \beta_{3} + \beta_{2} + 12 \beta_{1} - 16$$$$)/64$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{7} + 6 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - 4$$$$)/16$$ $$\nu^{5}$$ $$=$$ $$($$$$9 \beta_{7} + 4 \beta_{6} + 6 \beta_{5} + 4 \beta_{4} + 10 \beta_{3} - \beta_{2} - 16 \beta_{1} + 56$$$$)/32$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{6} + 4 \beta_{4} - 3 \beta_{3} - 6 \beta_{1}$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$19 \beta_{7} - 26 \beta_{6} + 30 \beta_{5} + 4 \beta_{4} - 56 \beta_{3} + \beta_{2} - 44 \beta_{1} + 256$$$$)/64$$

## 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
 1.20036 + 0.747754i 1.20036 − 0.747754i 1.40994 + 0.109843i 1.40994 − 0.109843i −1.27597 + 0.609843i −1.27597 − 0.609843i 0.665665 + 1.24775i 0.665665 − 1.24775i
0 0 0 −7.98203 0 2.13878i 0 0 0
1279.2 0 0 0 −7.98203 0 2.13878i 0 0 0
1279.3 0 0 0 −2.87875 0 10.7436i 0 0 0
1279.4 0 0 0 −2.87875 0 10.7436i 0 0 0
1279.5 0 0 0 2.87875 0 10.7436i 0 0 0
1279.6 0 0 0 2.87875 0 10.7436i 0 0 0
1279.7 0 0 0 7.98203 0 2.13878i 0 0 0
1279.8 0 0 0 7.98203 0 2.13878i 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d 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.z 8
3.b odd 2 1 768.3.g.h 8
4.b odd 2 1 inner 2304.3.g.z 8
8.b even 2 1 inner 2304.3.g.z 8
8.d odd 2 1 inner 2304.3.g.z 8
12.b even 2 1 768.3.g.h 8
16.e even 4 1 72.3.b.b 4
16.e even 4 1 288.3.b.b 4
16.f odd 4 1 72.3.b.b 4
16.f odd 4 1 288.3.b.b 4
24.f even 2 1 768.3.g.h 8
24.h odd 2 1 768.3.g.h 8
48.i odd 4 1 24.3.b.a 4
48.i odd 4 1 96.3.b.a 4
48.k even 4 1 24.3.b.a 4
48.k even 4 1 96.3.b.a 4
240.t even 4 1 600.3.g.a 4
240.t even 4 1 2400.3.g.a 4
240.z odd 4 1 600.3.p.a 8
240.z odd 4 1 2400.3.p.a 8
240.bb even 4 1 600.3.p.a 8
240.bb even 4 1 2400.3.p.a 8
240.bd odd 4 1 600.3.p.a 8
240.bd odd 4 1 2400.3.p.a 8
240.bf even 4 1 600.3.p.a 8
240.bf even 4 1 2400.3.p.a 8
240.bm odd 4 1 600.3.g.a 4
240.bm odd 4 1 2400.3.g.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.b.a 4 48.i odd 4 1
24.3.b.a 4 48.k even 4 1
72.3.b.b 4 16.e even 4 1
72.3.b.b 4 16.f odd 4 1
96.3.b.a 4 48.i odd 4 1
96.3.b.a 4 48.k even 4 1
288.3.b.b 4 16.e even 4 1
288.3.b.b 4 16.f odd 4 1
600.3.g.a 4 240.t even 4 1
600.3.g.a 4 240.bm odd 4 1
600.3.p.a 8 240.z odd 4 1
600.3.p.a 8 240.bb even 4 1
600.3.p.a 8 240.bd odd 4 1
600.3.p.a 8 240.bf even 4 1
768.3.g.h 8 3.b odd 2 1
768.3.g.h 8 12.b even 2 1
768.3.g.h 8 24.f even 2 1
768.3.g.h 8 24.h odd 2 1
2304.3.g.z 8 1.a even 1 1 trivial
2304.3.g.z 8 4.b odd 2 1 inner
2304.3.g.z 8 8.b even 2 1 inner
2304.3.g.z 8 8.d odd 2 1 inner
2400.3.g.a 4 240.t even 4 1
2400.3.g.a 4 240.bm odd 4 1
2400.3.p.a 8 240.z odd 4 1
2400.3.p.a 8 240.bb even 4 1
2400.3.p.a 8 240.bd odd 4 1
2400.3.p.a 8 240.bf even 4 1

## Hecke kernels

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

 $$T_{5}^{4} - 72 T_{5}^{2} + 528$$ $$T_{7}^{4} + 120 T_{7}^{2} + 528$$ $$T_{11}^{2} + 64$$ $$T_{13}^{4} - 384 T_{13}^{2} + 33792$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 528 - 72 T^{2} + T^{4} )^{2}$$
$7$ $$( 528 + 120 T^{2} + T^{4} )^{2}$$
$11$ $$( 64 + T^{2} )^{4}$$
$13$ $$( 33792 - 384 T^{2} + T^{4} )^{2}$$
$17$ $$( -188 - 4 T + T^{2} )^{4}$$
$19$ $$( 256 + 224 T^{2} + T^{4} )^{2}$$
$23$ $$( 8448 + 480 T^{2} + T^{4} )^{2}$$
$29$ $$( 528 - 1608 T^{2} + T^{4} )^{2}$$
$31$ $$( 279312 + 3384 T^{2} + T^{4} )^{2}$$
$37$ $$( 76032 - 864 T^{2} + T^{4} )^{2}$$
$41$ $$( -1628 - 20 T + T^{2} )^{4}$$
$43$ $$( 135424 + 992 T^{2} + T^{4} )^{2}$$
$47$ $$( 8448 + 3552 T^{2} + T^{4} )^{2}$$
$53$ $$( 803088 - 1800 T^{2} + T^{4} )^{2}$$
$59$ $$( 350464 + 2912 T^{2} + T^{4} )^{2}$$
$61$ $$( 8448 - 3552 T^{2} + T^{4} )^{2}$$
$67$ $$( 13424896 + 9056 T^{2} + T^{4} )^{2}$$
$71$ $$( 12849408 + 8928 T^{2} + T^{4} )^{2}$$
$73$ $$( -572 + 100 T + T^{2} )^{4}$$
$79$ $$( 10797072 + 7416 T^{2} + T^{4} )^{2}$$
$83$ $$( 692224 + 4736 T^{2} + T^{4} )^{2}$$
$89$ $$( -4412 + 100 T + T^{2} )^{4}$$
$97$ $$( -6716 - 28 T + T^{2} )^{4}$$