# Properties

 Label 2304.2.d.d Level $2304$ Weight $2$ Character orbit 2304.d Analytic conductor $18.398$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 384) 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.

 $$f(q)$$ $$=$$ $$q -2 q^{7} +O(q^{10})$$ $$q -2 q^{7} + 4 i q^{11} -6 i q^{13} -6 q^{17} + 4 q^{23} + 5 q^{25} + 4 i q^{29} + 10 q^{31} + 2 i q^{37} -2 q^{41} + 8 i q^{43} + 12 q^{47} -3 q^{49} + 12 i q^{53} + 4 i q^{59} -2 i q^{61} -4 i q^{67} -4 q^{71} + 10 q^{73} -8 i q^{77} -6 q^{79} + 12 i q^{83} + 2 q^{89} + 12 i q^{91} -6 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{7} + O(q^{10})$$ $$2q - 4q^{7} - 12q^{17} + 8q^{23} + 10q^{25} + 20q^{31} - 4q^{41} + 24q^{47} - 6q^{49} - 8q^{71} + 20q^{73} - 12q^{79} + 4q^{89} - 12q^{97} + O(q^{100})$$

## 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 0 0 −2.00000 0 0 0
1153.2 0 0 0 0 0 −2.00000 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.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.d.d 2
3.b odd 2 1 768.2.d.b 2
4.b odd 2 1 2304.2.d.m 2
8.b even 2 1 inner 2304.2.d.d 2
8.d odd 2 1 2304.2.d.m 2
12.b even 2 1 768.2.d.g 2
16.e even 4 1 1152.2.a.k 1
16.e even 4 1 1152.2.a.l 1
16.f odd 4 1 1152.2.a.i 1
16.f odd 4 1 1152.2.a.j 1
24.f even 2 1 768.2.d.g 2
24.h odd 2 1 768.2.d.b 2
48.i odd 4 1 384.2.a.c yes 1
48.i odd 4 1 384.2.a.g yes 1
48.k even 4 1 384.2.a.b 1
48.k even 4 1 384.2.a.f yes 1
240.t even 4 1 9600.2.a.w 1
240.t even 4 1 9600.2.a.bw 1
240.bm odd 4 1 9600.2.a.h 1
240.bm odd 4 1 9600.2.a.bh 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.b 1 48.k even 4 1
384.2.a.c yes 1 48.i odd 4 1
384.2.a.f yes 1 48.k even 4 1
384.2.a.g yes 1 48.i odd 4 1
768.2.d.b 2 3.b odd 2 1
768.2.d.b 2 24.h odd 2 1
768.2.d.g 2 12.b even 2 1
768.2.d.g 2 24.f even 2 1
1152.2.a.i 1 16.f odd 4 1
1152.2.a.j 1 16.f odd 4 1
1152.2.a.k 1 16.e even 4 1
1152.2.a.l 1 16.e even 4 1
2304.2.d.d 2 1.a even 1 1 trivial
2304.2.d.d 2 8.b even 2 1 inner
2304.2.d.m 2 4.b odd 2 1
2304.2.d.m 2 8.d odd 2 1
9600.2.a.h 1 240.bm odd 4 1
9600.2.a.w 1 240.t even 4 1
9600.2.a.bh 1 240.bm odd 4 1
9600.2.a.bw 1 240.t even 4 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{7} + 2$$ $$T_{11}^{2} + 16$$ $$T_{17} + 6$$ $$T_{23} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 5 T^{2} )^{2}$$
$7$ $$( 1 + 2 T + 7 T^{2} )^{2}$$
$11$ $$1 - 6 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$17$ $$( 1 + 6 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 19 T^{2} )^{2}$$
$23$ $$( 1 - 4 T + 23 T^{2} )^{2}$$
$29$ $$( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )$$
$31$ $$( 1 - 10 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{2}$$
$43$ $$1 - 22 T^{2} + 1849 T^{4}$$
$47$ $$( 1 - 12 T + 47 T^{2} )^{2}$$
$53$ $$1 + 38 T^{2} + 2809 T^{4}$$
$59$ $$1 - 102 T^{2} + 3481 T^{4}$$
$61$ $$1 - 118 T^{2} + 3721 T^{4}$$
$67$ $$1 - 118 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 4 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 10 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 6 T + 79 T^{2} )^{2}$$
$83$ $$1 - 22 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 2 T + 89 T^{2} )^{2}$$
$97$ $$( 1 + 6 T + 97 T^{2} )^{2}$$