# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$18.3975326257$$ 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: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{5} +O(q^{10})$$ $$q + 2 i q^{5} -6 i q^{13} -2 q^{17} + q^{25} -10 i q^{29} -2 i q^{37} + 10 q^{41} -7 q^{49} -14 i q^{53} + 10 i q^{61} + 12 q^{65} + 6 q^{73} -4 i q^{85} + 10 q^{89} + 18 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 4q^{17} + 2q^{25} + 20q^{41} - 14q^{49} + 24q^{65} + 12q^{73} + 20q^{89} + 36q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1153.1
 − 1.00000i 1.00000i
0 0 0 2.00000i 0 0 0 0 0
1153.2 0 0 0 2.00000i 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 2304.2.d.j 2
3.b odd 2 1 256.2.b.b 2
4.b odd 2 1 CM 2304.2.d.j 2
8.b even 2 1 inner 2304.2.d.j 2
8.d odd 2 1 inner 2304.2.d.j 2
12.b even 2 1 256.2.b.b 2
16.e even 4 1 288.2.a.d 1
16.e even 4 1 576.2.a.c 1
16.f odd 4 1 288.2.a.d 1
16.f odd 4 1 576.2.a.c 1
24.f even 2 1 256.2.b.b 2
24.h odd 2 1 256.2.b.b 2
48.i odd 4 1 32.2.a.a 1
48.i odd 4 1 64.2.a.a 1
48.k even 4 1 32.2.a.a 1
48.k even 4 1 64.2.a.a 1
80.i odd 4 1 7200.2.f.m 2
80.j even 4 1 7200.2.f.m 2
80.k odd 4 1 7200.2.a.v 1
80.q even 4 1 7200.2.a.v 1
80.s even 4 1 7200.2.f.m 2
80.t odd 4 1 7200.2.f.m 2
96.o even 8 4 1024.2.e.j 4
96.p odd 8 4 1024.2.e.j 4
144.u even 12 2 2592.2.i.t 2
144.v odd 12 2 2592.2.i.e 2
144.w odd 12 2 2592.2.i.t 2
144.x even 12 2 2592.2.i.e 2
240.t even 4 1 800.2.a.d 1
240.t even 4 1 1600.2.a.n 1
240.z odd 4 1 800.2.c.e 2
240.z odd 4 1 1600.2.c.l 2
240.bb even 4 1 800.2.c.e 2
240.bb even 4 1 1600.2.c.l 2
240.bd odd 4 1 800.2.c.e 2
240.bd odd 4 1 1600.2.c.l 2
240.bf even 4 1 800.2.c.e 2
240.bf even 4 1 1600.2.c.l 2
240.bm odd 4 1 800.2.a.d 1
240.bm odd 4 1 1600.2.a.n 1
336.v odd 4 1 1568.2.a.e 1
336.v odd 4 1 3136.2.a.m 1
336.y even 4 1 1568.2.a.e 1
336.y even 4 1 3136.2.a.m 1
336.bo even 12 2 1568.2.i.f 2
336.br odd 12 2 1568.2.i.f 2
336.bt odd 12 2 1568.2.i.g 2
336.bu even 12 2 1568.2.i.g 2
528.s odd 4 1 3872.2.a.f 1
528.s odd 4 1 7744.2.a.v 1
528.x even 4 1 3872.2.a.f 1
528.x even 4 1 7744.2.a.v 1
624.v even 4 1 5408.2.a.g 1
624.bi odd 4 1 5408.2.a.g 1
816.w even 4 1 9248.2.a.f 1
816.bg odd 4 1 9248.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 48.i odd 4 1
32.2.a.a 1 48.k even 4 1
64.2.a.a 1 48.i odd 4 1
64.2.a.a 1 48.k even 4 1
256.2.b.b 2 3.b odd 2 1
256.2.b.b 2 12.b even 2 1
256.2.b.b 2 24.f even 2 1
256.2.b.b 2 24.h odd 2 1
288.2.a.d 1 16.e even 4 1
288.2.a.d 1 16.f odd 4 1
576.2.a.c 1 16.e even 4 1
576.2.a.c 1 16.f odd 4 1
800.2.a.d 1 240.t even 4 1
800.2.a.d 1 240.bm odd 4 1
800.2.c.e 2 240.z odd 4 1
800.2.c.e 2 240.bb even 4 1
800.2.c.e 2 240.bd odd 4 1
800.2.c.e 2 240.bf even 4 1
1024.2.e.j 4 96.o even 8 4
1024.2.e.j 4 96.p odd 8 4
1568.2.a.e 1 336.v odd 4 1
1568.2.a.e 1 336.y even 4 1
1568.2.i.f 2 336.bo even 12 2
1568.2.i.f 2 336.br odd 12 2
1568.2.i.g 2 336.bt odd 12 2
1568.2.i.g 2 336.bu even 12 2
1600.2.a.n 1 240.t even 4 1
1600.2.a.n 1 240.bm odd 4 1
1600.2.c.l 2 240.z odd 4 1
1600.2.c.l 2 240.bb even 4 1
1600.2.c.l 2 240.bd odd 4 1
1600.2.c.l 2 240.bf even 4 1
2304.2.d.j 2 1.a even 1 1 trivial
2304.2.d.j 2 4.b odd 2 1 CM
2304.2.d.j 2 8.b even 2 1 inner
2304.2.d.j 2 8.d odd 2 1 inner
2592.2.i.e 2 144.v odd 12 2
2592.2.i.e 2 144.x even 12 2
2592.2.i.t 2 144.u even 12 2
2592.2.i.t 2 144.w odd 12 2
3136.2.a.m 1 336.v odd 4 1
3136.2.a.m 1 336.y even 4 1
3872.2.a.f 1 528.s odd 4 1
3872.2.a.f 1 528.x even 4 1
5408.2.a.g 1 624.v even 4 1
5408.2.a.g 1 624.bi odd 4 1
7200.2.a.v 1 80.k odd 4 1
7200.2.a.v 1 80.q even 4 1
7200.2.f.m 2 80.i odd 4 1
7200.2.f.m 2 80.j even 4 1
7200.2.f.m 2 80.s even 4 1
7200.2.f.m 2 80.t odd 4 1
7744.2.a.v 1 528.s odd 4 1
7744.2.a.v 1 528.x even 4 1
9248.2.a.f 1 816.w even 4 1
9248.2.a.f 1 816.bg odd 4 1

## Hecke kernels

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

 $$T_{5}^{2} + 4$$ $$T_{7}$$ $$T_{11}$$ $$T_{17} + 2$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$100 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$196 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$100 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$( -18 + T )^{2}$$