# Properties

 Label 2304.3.b.d Level $2304$ Weight $3$ Character orbit 2304.b Analytic conductor $62.779$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 8 i q^{5} +O(q^{10})$$ $$q + 8 i q^{5} + 10 i q^{13} -16 q^{17} -39 q^{25} + 40 i q^{29} -70 i q^{37} -80 q^{41} + 49 q^{49} + 56 i q^{53} + 22 i q^{61} -80 q^{65} -110 q^{73} -128 i q^{85} + 160 q^{89} -130 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 32q^{17} - 78q^{25} - 160q^{41} + 98q^{49} - 160q^{65} - 220q^{73} + 320q^{89} - 260q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 − 1.00000i 1.00000i
0 0 0 8.00000i 0 0 0 0 0
127.2 0 0 0 8.00000i 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 $$\Q(\sqrt{-1})$$
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.b.d 2
3.b odd 2 1 2304.3.b.e 2
4.b odd 2 1 CM 2304.3.b.d 2
8.b even 2 1 inner 2304.3.b.d 2
8.d odd 2 1 inner 2304.3.b.d 2
12.b even 2 1 2304.3.b.e 2
16.e even 4 1 36.3.d.a 1
16.e even 4 1 576.3.g.a 1
16.f odd 4 1 36.3.d.a 1
16.f odd 4 1 576.3.g.a 1
24.f even 2 1 2304.3.b.e 2
24.h odd 2 1 2304.3.b.e 2
48.i odd 4 1 36.3.d.b yes 1
48.i odd 4 1 576.3.g.c 1
48.k even 4 1 36.3.d.b yes 1
48.k even 4 1 576.3.g.c 1
80.i odd 4 1 900.3.f.b 2
80.j even 4 1 900.3.f.b 2
80.k odd 4 1 900.3.c.c 1
80.q even 4 1 900.3.c.c 1
80.s even 4 1 900.3.f.b 2
80.t odd 4 1 900.3.f.b 2
144.u even 12 2 324.3.f.e 2
144.v odd 12 2 324.3.f.f 2
144.w odd 12 2 324.3.f.e 2
144.x even 12 2 324.3.f.f 2
240.t even 4 1 900.3.c.b 1
240.z odd 4 1 900.3.f.a 2
240.bb even 4 1 900.3.f.a 2
240.bd odd 4 1 900.3.f.a 2
240.bf even 4 1 900.3.f.a 2
240.bm odd 4 1 900.3.c.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.d.a 1 16.e even 4 1
36.3.d.a 1 16.f odd 4 1
36.3.d.b yes 1 48.i odd 4 1
36.3.d.b yes 1 48.k even 4 1
324.3.f.e 2 144.u even 12 2
324.3.f.e 2 144.w odd 12 2
324.3.f.f 2 144.v odd 12 2
324.3.f.f 2 144.x even 12 2
576.3.g.a 1 16.e even 4 1
576.3.g.a 1 16.f odd 4 1
576.3.g.c 1 48.i odd 4 1
576.3.g.c 1 48.k even 4 1
900.3.c.b 1 240.t even 4 1
900.3.c.b 1 240.bm odd 4 1
900.3.c.c 1 80.k odd 4 1
900.3.c.c 1 80.q even 4 1
900.3.f.a 2 240.z odd 4 1
900.3.f.a 2 240.bb even 4 1
900.3.f.a 2 240.bd odd 4 1
900.3.f.a 2 240.bf even 4 1
900.3.f.b 2 80.i odd 4 1
900.3.f.b 2 80.j even 4 1
900.3.f.b 2 80.s even 4 1
900.3.f.b 2 80.t odd 4 1
2304.3.b.d 2 1.a even 1 1 trivial
2304.3.b.d 2 4.b odd 2 1 CM
2304.3.b.d 2 8.b even 2 1 inner
2304.3.b.d 2 8.d odd 2 1 inner
2304.3.b.e 2 3.b odd 2 1
2304.3.b.e 2 12.b even 2 1
2304.3.b.e 2 24.f even 2 1
2304.3.b.e 2 24.h odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 64$$ $$T_{7}$$ $$T_{11}$$ $$T_{17} + 16$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$64 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$100 + T^{2}$$
$17$ $$( 16 + T )^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$1600 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$4900 + T^{2}$$
$41$ $$( 80 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$3136 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$484 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 110 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$( -160 + T )^{2}$$
$97$ $$( 130 + T )^{2}$$
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