# Properties

 Label 2304.2.a.t Level 2304 Weight 2 Character orbit 2304.a Self dual yes Analytic conductor 18.398 Analytic rank 1 Dimension 2 CM discriminant -8 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.3975326257$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 128) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q +O(q^{10})$$ $$q + 2 \beta q^{11} -6 q^{17} -6 \beta q^{19} -5 q^{25} -6 q^{41} + 6 \beta q^{43} -7 q^{49} + 10 \beta q^{59} -6 \beta q^{67} + 2 q^{73} -2 \beta q^{83} -18 q^{89} -10 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 12q^{17} - 10q^{25} - 12q^{41} - 14q^{49} + 4q^{73} - 36q^{89} - 20q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 0 0 0 0 0 0 0 0
1.2 0 0 0 0 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.a.t 2
3.b odd 2 1 256.2.a.e 2
4.b odd 2 1 inner 2304.2.a.t 2
8.b even 2 1 inner 2304.2.a.t 2
8.d odd 2 1 CM 2304.2.a.t 2
12.b even 2 1 256.2.a.e 2
15.d odd 2 1 6400.2.a.by 2
16.e even 4 2 1152.2.d.c 2
16.f odd 4 2 1152.2.d.c 2
24.f even 2 1 256.2.a.e 2
24.h odd 2 1 256.2.a.e 2
48.i odd 4 2 128.2.b.a 2
48.k even 4 2 128.2.b.a 2
60.h even 2 1 6400.2.a.by 2
96.o even 8 2 1024.2.e.a 2
96.o even 8 2 1024.2.e.f 2
96.p odd 8 2 1024.2.e.a 2
96.p odd 8 2 1024.2.e.f 2
120.i odd 2 1 6400.2.a.by 2
120.m even 2 1 6400.2.a.by 2
240.t even 4 2 3200.2.d.c 2
240.z odd 4 2 3200.2.f.o 4
240.bb even 4 2 3200.2.f.o 4
240.bd odd 4 2 3200.2.f.o 4
240.bf even 4 2 3200.2.f.o 4
240.bm odd 4 2 3200.2.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.a 2 48.i odd 4 2
128.2.b.a 2 48.k even 4 2
256.2.a.e 2 3.b odd 2 1
256.2.a.e 2 12.b even 2 1
256.2.a.e 2 24.f even 2 1
256.2.a.e 2 24.h odd 2 1
1024.2.e.a 2 96.o even 8 2
1024.2.e.a 2 96.p odd 8 2
1024.2.e.f 2 96.o even 8 2
1024.2.e.f 2 96.p odd 8 2
1152.2.d.c 2 16.e even 4 2
1152.2.d.c 2 16.f odd 4 2
2304.2.a.t 2 1.a even 1 1 trivial
2304.2.a.t 2 4.b odd 2 1 inner
2304.2.a.t 2 8.b even 2 1 inner
2304.2.a.t 2 8.d odd 2 1 CM
3200.2.d.c 2 240.t even 4 2
3200.2.d.c 2 240.bm odd 4 2
3200.2.f.o 4 240.z odd 4 2
3200.2.f.o 4 240.bb even 4 2
3200.2.f.o 4 240.bd odd 4 2
3200.2.f.o 4 240.bf even 4 2
6400.2.a.by 2 15.d odd 2 1
6400.2.a.by 2 60.h even 2 1
6400.2.a.by 2 120.i odd 2 1
6400.2.a.by 2 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2304))$$:

 $$T_{5}$$ $$T_{7}$$ $$T_{11}^{2} - 8$$ $$T_{13}$$ $$T_{19}^{2} - 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 5 T^{2} )^{2}$$
$7$ $$( 1 + 7 T^{2} )^{2}$$
$11$ $$1 + 14 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$( 1 + 6 T + 17 T^{2} )^{2}$$
$19$ $$1 - 34 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 23 T^{2} )^{2}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$( 1 + 37 T^{2} )^{2}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$1 + 14 T^{2} + 1849 T^{4}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 + 53 T^{2} )^{2}$$
$59$ $$1 - 82 T^{2} + 3481 T^{4}$$
$61$ $$( 1 + 61 T^{2} )^{2}$$
$67$ $$1 + 62 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 - 2 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 79 T^{2} )^{2}$$
$83$ $$1 + 158 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 18 T + 89 T^{2} )^{2}$$
$97$ $$( 1 + 10 T + 97 T^{2} )^{2}$$