# Properties

 Label 2304.2.f.d Level $2304$ Weight $2$ Character orbit 2304.f Analytic conductor $18.398$ Analytic rank $0$ Dimension $4$ CM discriminant -4 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$18.3975326257$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} +O(q^{10})$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} + 4 \zeta_{8}^{2} q^{13} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{17} -3 q^{25} + ( -7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{29} + 2 \zeta_{8}^{2} q^{37} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{41} + 7 q^{49} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{53} + 10 \zeta_{8}^{2} q^{61} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{65} + 16 q^{73} + 10 \zeta_{8}^{2} q^{85} + ( 13 \zeta_{8} + 13 \zeta_{8}^{3} ) q^{89} + 8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 12q^{25} + 28q^{49} + 64q^{73} + 32q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
0 0 0 −1.41421 0 0 0 0 0
1151.2 0 0 0 −1.41421 0 0 0 0 0
1151.3 0 0 0 1.41421 0 0 0 0 0
1151.4 0 0 0 1.41421 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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 CM by $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f 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 2304.2.f.d 4
3.b odd 2 1 inner 2304.2.f.d 4
4.b odd 2 1 CM 2304.2.f.d 4
8.b even 2 1 inner 2304.2.f.d 4
8.d odd 2 1 inner 2304.2.f.d 4
12.b even 2 1 inner 2304.2.f.d 4
16.e even 4 1 36.2.b.a 2
16.e even 4 1 576.2.c.b 2
16.f odd 4 1 36.2.b.a 2
16.f odd 4 1 576.2.c.b 2
24.f even 2 1 inner 2304.2.f.d 4
24.h odd 2 1 inner 2304.2.f.d 4
48.i odd 4 1 36.2.b.a 2
48.i odd 4 1 576.2.c.b 2
48.k even 4 1 36.2.b.a 2
48.k even 4 1 576.2.c.b 2
80.i odd 4 1 900.2.h.a 4
80.j even 4 1 900.2.h.a 4
80.k odd 4 1 900.2.e.b 2
80.q even 4 1 900.2.e.b 2
80.s even 4 1 900.2.h.a 4
80.t odd 4 1 900.2.h.a 4
112.j even 4 1 1764.2.e.b 2
112.l odd 4 1 1764.2.e.b 2
144.u even 12 2 324.2.h.c 4
144.v odd 12 2 324.2.h.c 4
144.w odd 12 2 324.2.h.c 4
144.x even 12 2 324.2.h.c 4
240.t even 4 1 900.2.e.b 2
240.z odd 4 1 900.2.h.a 4
240.bb even 4 1 900.2.h.a 4
240.bd odd 4 1 900.2.h.a 4
240.bf even 4 1 900.2.h.a 4
240.bm odd 4 1 900.2.e.b 2
336.v odd 4 1 1764.2.e.b 2
336.y even 4 1 1764.2.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.b.a 2 16.e even 4 1
36.2.b.a 2 16.f odd 4 1
36.2.b.a 2 48.i odd 4 1
36.2.b.a 2 48.k even 4 1
324.2.h.c 4 144.u even 12 2
324.2.h.c 4 144.v odd 12 2
324.2.h.c 4 144.w odd 12 2
324.2.h.c 4 144.x even 12 2
576.2.c.b 2 16.e even 4 1
576.2.c.b 2 16.f odd 4 1
576.2.c.b 2 48.i odd 4 1
576.2.c.b 2 48.k even 4 1
900.2.e.b 2 80.k odd 4 1
900.2.e.b 2 80.q even 4 1
900.2.e.b 2 240.t even 4 1
900.2.e.b 2 240.bm odd 4 1
900.2.h.a 4 80.i odd 4 1
900.2.h.a 4 80.j even 4 1
900.2.h.a 4 80.s even 4 1
900.2.h.a 4 80.t odd 4 1
900.2.h.a 4 240.z odd 4 1
900.2.h.a 4 240.bb even 4 1
900.2.h.a 4 240.bd odd 4 1
900.2.h.a 4 240.bf even 4 1
1764.2.e.b 2 112.j even 4 1
1764.2.e.b 2 112.l odd 4 1
1764.2.e.b 2 336.v odd 4 1
1764.2.e.b 2 336.y even 4 1
2304.2.f.d 4 1.a even 1 1 trivial
2304.2.f.d 4 3.b odd 2 1 inner
2304.2.f.d 4 4.b odd 2 1 CM
2304.2.f.d 4 8.b even 2 1 inner
2304.2.f.d 4 8.d odd 2 1 inner
2304.2.f.d 4 12.b even 2 1 inner
2304.2.f.d 4 24.f even 2 1 inner
2304.2.f.d 4 24.h odd 2 1 inner

## Hecke kernels

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

 $$T_{5}^{2} - 2$$ $$T_{7}$$ $$T_{23}$$ $$T_{43}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 8 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 - 7 T^{2} )^{4}$$
$11$ $$( 1 - 11 T^{2} )^{4}$$
$13$ $$( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2}$$
$17$ $$( 1 + 16 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 19 T^{2} )^{4}$$
$23$ $$( 1 + 23 T^{2} )^{4}$$
$29$ $$( 1 - 40 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 31 T^{2} )^{4}$$
$37$ $$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 80 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 43 T^{2} )^{4}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$( 1 + 56 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 59 T^{2} )^{4}$$
$61$ $$( 1 - 12 T + 61 T^{2} )^{2}( 1 + 12 T + 61 T^{2} )^{2}$$
$67$ $$( 1 + 67 T^{2} )^{4}$$
$71$ $$( 1 + 71 T^{2} )^{4}$$
$73$ $$( 1 - 16 T + 73 T^{2} )^{4}$$
$79$ $$( 1 - 79 T^{2} )^{4}$$
$83$ $$( 1 - 83 T^{2} )^{4}$$
$89$ $$( 1 + 160 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 8 T + 97 T^{2} )^{4}$$