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: $$x^{2} - 2$$ 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 $$q + \beta q^{11} - 6 q^{17} - 3 \beta q^{19} - 5 q^{25} - 6 q^{41} + 3 \beta q^{43} - 7 q^{49} + 5 \beta q^{59} - 3 \beta q^{67} + 2 q^{73} - \beta q^{83} - 18 q^{89} - 10 q^{97} +O(q^{100})$$ q + b * q^11 - 6 * q^17 - 3*b * q^19 - 5 * q^25 - 6 * q^41 + 3*b * q^43 - 7 * q^49 + 5*b * q^59 - 3*b * q^67 + 2 * q^73 - b * q^83 - 18 * q^89 - 10 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 12 q^{17} - 10 q^{25} - 12 q^{41} - 14 q^{49} + 4 q^{73} - 36 q^{89} - 20 q^{97}+O(q^{100})$$ 2 * q - 12 * q^17 - 10 * q^25 - 12 * q^41 - 14 * q^49 + 4 * q^73 - 36 * q^89 - 20 * q^97

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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2304))$$:

 $$T_{5}$$ T5 $$T_{7}$$ T7 $$T_{11}^{2} - 8$$ T11^2 - 8 $$T_{13}$$ T13 $$T_{19}^{2} - 72$$ T19^2 - 72

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 8$$
$13$ $$T^{2}$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} - 72$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} - 72$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} - 200$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 72$$
$71$ $$T^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 8$$
$89$ $$(T + 18)^{2}$$
$97$ $$(T + 10)^{2}$$