# Properties

 Label 2304.3.b.g Level $2304$ Weight $3$ Character orbit 2304.b Analytic conductor $62.779$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{5} + 8 i q^{7} +O(q^{10})$$ $$q + 2 i q^{5} + 8 i q^{7} + 4 q^{11} -14 i q^{13} -18 q^{17} + 12 q^{19} + 40 i q^{23} + 21 q^{25} + 14 i q^{29} -32 i q^{31} -16 q^{35} + 30 i q^{37} -14 q^{41} + 28 q^{43} -16 i q^{47} -15 q^{49} + 66 i q^{53} + 8 i q^{55} + 52 q^{59} + 82 i q^{61} + 28 q^{65} -4 q^{67} + 56 i q^{71} -66 q^{73} + 32 i q^{77} -16 i q^{79} -140 q^{83} -36 i q^{85} -30 q^{89} + 112 q^{91} + 24 i q^{95} -14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 8q^{11} - 36q^{17} + 24q^{19} + 42q^{25} - 32q^{35} - 28q^{41} + 56q^{43} - 30q^{49} + 104q^{59} + 56q^{65} - 8q^{67} - 132q^{73} - 280q^{83} - 60q^{89} + 224q^{91} - 28q^{97} + O(q^{100})$$

## 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 2.00000i 0 8.00000i 0 0 0
127.2 0 0 0 2.00000i 0 8.00000i 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
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.g 2
3.b odd 2 1 256.3.d.a 2
4.b odd 2 1 2304.3.b.c 2
8.b even 2 1 2304.3.b.c 2
8.d odd 2 1 inner 2304.3.b.g 2
12.b even 2 1 256.3.d.c 2
16.e even 4 1 288.3.g.b 2
16.e even 4 1 576.3.g.g 2
16.f odd 4 1 288.3.g.b 2
16.f odd 4 1 576.3.g.g 2
24.f even 2 1 256.3.d.a 2
24.h odd 2 1 256.3.d.c 2
48.i odd 4 1 32.3.c.a 2
48.i odd 4 1 64.3.c.b 2
48.k even 4 1 32.3.c.a 2
48.k even 4 1 64.3.c.b 2
240.t even 4 1 800.3.b.a 2
240.t even 4 1 1600.3.b.e 2
240.z odd 4 1 800.3.h.a 2
240.z odd 4 1 1600.3.h.a 2
240.bb even 4 1 800.3.h.b 2
240.bb even 4 1 1600.3.h.c 2
240.bd odd 4 1 800.3.h.b 2
240.bd odd 4 1 1600.3.h.c 2
240.bf even 4 1 800.3.h.a 2
240.bf even 4 1 1600.3.h.a 2
240.bm odd 4 1 800.3.b.a 2
240.bm odd 4 1 1600.3.b.e 2
336.v odd 4 1 1568.3.d.b 2
336.y even 4 1 1568.3.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.3.c.a 2 48.i odd 4 1
32.3.c.a 2 48.k even 4 1
64.3.c.b 2 48.i odd 4 1
64.3.c.b 2 48.k even 4 1
256.3.d.a 2 3.b odd 2 1
256.3.d.a 2 24.f even 2 1
256.3.d.c 2 12.b even 2 1
256.3.d.c 2 24.h odd 2 1
288.3.g.b 2 16.e even 4 1
288.3.g.b 2 16.f odd 4 1
576.3.g.g 2 16.e even 4 1
576.3.g.g 2 16.f odd 4 1
800.3.b.a 2 240.t even 4 1
800.3.b.a 2 240.bm odd 4 1
800.3.h.a 2 240.z odd 4 1
800.3.h.a 2 240.bf even 4 1
800.3.h.b 2 240.bb even 4 1
800.3.h.b 2 240.bd odd 4 1
1568.3.d.b 2 336.v odd 4 1
1568.3.d.b 2 336.y even 4 1
1600.3.b.e 2 240.t even 4 1
1600.3.b.e 2 240.bm odd 4 1
1600.3.h.a 2 240.z odd 4 1
1600.3.h.a 2 240.bf even 4 1
1600.3.h.c 2 240.bb even 4 1
1600.3.h.c 2 240.bd odd 4 1
2304.3.b.c 2 4.b odd 2 1
2304.3.b.c 2 8.b even 2 1
2304.3.b.g 2 1.a even 1 1 trivial
2304.3.b.g 2 8.d 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 $$S_{3}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}^{2} + 4$$ $$T_{7}^{2} + 64$$ $$T_{11} - 4$$ $$T_{17} + 18$$ $$T_{19} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 + T^{2}$$
$7$ $$64 + T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$196 + T^{2}$$
$17$ $$( 18 + T )^{2}$$
$19$ $$( -12 + T )^{2}$$
$23$ $$1600 + T^{2}$$
$29$ $$196 + T^{2}$$
$31$ $$1024 + T^{2}$$
$37$ $$900 + T^{2}$$
$41$ $$( 14 + T )^{2}$$
$43$ $$( -28 + T )^{2}$$
$47$ $$256 + T^{2}$$
$53$ $$4356 + T^{2}$$
$59$ $$( -52 + T )^{2}$$
$61$ $$6724 + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$3136 + T^{2}$$
$73$ $$( 66 + T )^{2}$$
$79$ $$256 + T^{2}$$
$83$ $$( 140 + T )^{2}$$
$89$ $$( 30 + T )^{2}$$
$97$ $$( 14 + T )^{2}$$