# Properties

 Label 2304.2.l.d Level $2304$ Weight $2$ Character orbit 2304.l Analytic conductor $18.398$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.3975326257$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{7} ) q^{7} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{7} ) q^{7} + ( -1 + \beta_{2} - \beta_{6} ) q^{11} + ( 2 + 2 \beta_{2} + \beta_{5} ) q^{13} + ( \beta_{1} + \beta_{3} - 3 \beta_{4} + \beta_{7} ) q^{17} + ( -\beta_{3} + \beta_{4} ) q^{19} -6 \beta_{2} q^{23} + ( 3 \beta_{2} + \beta_{5} - \beta_{6} ) q^{25} + ( 3 \beta_{3} - 3 \beta_{4} + \beta_{7} ) q^{29} + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{31} + ( 7 + 7 \beta_{2} + \beta_{5} ) q^{35} + ( 1 - \beta_{2} ) q^{37} + ( -\beta_{1} + 2 \beta_{3} + \beta_{7} ) q^{41} + ( 2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{43} -6 q^{47} + 7 q^{49} + ( 3 \beta_{1} - 3 \beta_{4} ) q^{53} + ( 2 \beta_{1} + 8 \beta_{3} - 2 \beta_{7} ) q^{55} + ( -4 + 4 \beta_{2} + 2 \beta_{6} ) q^{59} + ( 5 + 5 \beta_{2} - 2 \beta_{5} ) q^{61} + ( -3 \beta_{1} - 3 \beta_{3} - 6 \beta_{4} - 3 \beta_{7} ) q^{65} + ( -6 \beta_{3} + 6 \beta_{4} - 2 \beta_{7} ) q^{67} + ( -8 \beta_{2} - \beta_{5} + \beta_{6} ) q^{71} + ( 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{73} + ( 6 \beta_{3} - 6 \beta_{4} - 2 \beta_{7} ) q^{77} + ( 3 \beta_{1} + 3 \beta_{3} - 5 \beta_{4} + 3 \beta_{7} ) q^{79} + ( 3 + 3 \beta_{2} - 3 \beta_{5} ) q^{83} + ( 5 - 5 \beta_{2} - \beta_{6} ) q^{85} + ( 2 \beta_{1} + 5 \beta_{3} - 2 \beta_{7} ) q^{89} + ( -4 \beta_{1} - 9 \beta_{3} - 5 \beta_{4} ) q^{91} + ( 2 + \beta_{5} + \beta_{6} ) q^{95} + ( 8 + \beta_{5} + \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{11} + 16q^{13} + 56q^{35} + 8q^{37} - 48q^{47} + 56q^{49} - 32q^{59} + 40q^{61} + 24q^{83} + 40q^{85} + 16q^{95} + 64q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 5 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{2}$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 4 \nu^{5} + 7 \nu^{3} + 4 \nu$$$$)/24$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} + 7 \nu^{3} - 4 \nu$$$$)/24$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{6} + 8 \nu^{4} + 9 \nu^{2} + 4$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{6} - 8 \nu^{4} + 9 \nu^{2} - 4$$$$)/12$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 4 \nu^{5} + 13 \nu^{3} - 4 \nu$$$$)/24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} + 6 \beta_{2}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + 2 \beta_{4} + 3 \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{6} + 3 \beta_{5} - 2$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{4} - 5 \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{6} - 5 \beta_{5} + 18 \beta_{2}$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$7 \beta_{7} - 10 \beta_{4} - 3 \beta_{3}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
575.1
 1.28897 − 0.581861i 0.581861 − 1.28897i −0.581861 + 1.28897i −1.28897 + 0.581861i 1.28897 + 0.581861i 0.581861 + 1.28897i −0.581861 − 1.28897i −1.28897 − 0.581861i
0 0 0 −2.57794 + 2.57794i 0 −3.74166 0 0 0
575.2 0 0 0 −1.16372 + 1.16372i 0 −3.74166 0 0 0
575.3 0 0 0 1.16372 1.16372i 0 3.74166 0 0 0
575.4 0 0 0 2.57794 2.57794i 0 3.74166 0 0 0
1727.1 0 0 0 −2.57794 2.57794i 0 −3.74166 0 0 0
1727.2 0 0 0 −1.16372 1.16372i 0 −3.74166 0 0 0
1727.3 0 0 0 1.16372 + 1.16372i 0 3.74166 0 0 0
1727.4 0 0 0 2.57794 + 2.57794i 0 3.74166 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1727.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner
16.e even 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.l.d yes 8
3.b odd 2 1 2304.2.l.h yes 8
4.b odd 2 1 2304.2.l.h yes 8
8.b even 2 1 2304.2.l.e yes 8
8.d odd 2 1 2304.2.l.a 8
12.b even 2 1 inner 2304.2.l.d yes 8
16.e even 4 1 inner 2304.2.l.d yes 8
16.e even 4 1 2304.2.l.e yes 8
16.f odd 4 1 2304.2.l.a 8
16.f odd 4 1 2304.2.l.h yes 8
24.f even 2 1 2304.2.l.e yes 8
24.h odd 2 1 2304.2.l.a 8
48.i odd 4 1 2304.2.l.a 8
48.i odd 4 1 2304.2.l.h yes 8
48.k even 4 1 inner 2304.2.l.d yes 8
48.k even 4 1 2304.2.l.e yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2304.2.l.a 8 8.d odd 2 1
2304.2.l.a 8 16.f odd 4 1
2304.2.l.a 8 24.h odd 2 1
2304.2.l.a 8 48.i odd 4 1
2304.2.l.d yes 8 1.a even 1 1 trivial
2304.2.l.d yes 8 12.b even 2 1 inner
2304.2.l.d yes 8 16.e even 4 1 inner
2304.2.l.d yes 8 48.k even 4 1 inner
2304.2.l.e yes 8 8.b even 2 1
2304.2.l.e yes 8 16.e even 4 1
2304.2.l.e yes 8 24.f even 2 1
2304.2.l.e yes 8 48.k even 4 1
2304.2.l.h yes 8 3.b odd 2 1
2304.2.l.h yes 8 4.b odd 2 1
2304.2.l.h yes 8 16.f odd 4 1
2304.2.l.h yes 8 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}^{8} + 184 T_{5}^{4} + 1296$$ $$T_{11}^{4} + 4 T_{11}^{3} + 8 T_{11}^{2} - 48 T_{11} + 144$$ $$T_{13}^{4} - 8 T_{13}^{3} + 32 T_{13}^{2} + 48 T_{13} + 36$$ $$T_{37}^{2} - 2 T_{37} + 2$$