# Properties

 Label 4608.2.k.n Level $4608$ Weight $2$ Character orbit 4608.k Analytic conductor $36.795$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4608 = 2^{9} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4608.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7950652514$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (i + 1) q^{5} + 4 i q^{7}+O(q^{10})$$ q + (i + 1) * q^5 + 4*i * q^7 $$q + (i + 1) q^{5} + 4 i q^{7} + ( - 4 i - 4) q^{11} + ( - 3 i + 3) q^{13} + 6 q^{17} + (4 i - 4) q^{19} - 8 i q^{23} - 3 i q^{25} + ( - 3 i + 3) q^{29} + 4 q^{31} + (4 i - 4) q^{35} + (i + 1) q^{37} + 2 i q^{41} + ( - 4 i - 4) q^{43} + 8 q^{47} - 9 q^{49} + ( - 7 i - 7) q^{53} - 8 i q^{55} + (3 i - 3) q^{61} + 6 q^{65} + ( - 8 i + 8) q^{67} - 10 i q^{73} + ( - 16 i + 16) q^{77} - 12 q^{79} + ( - 4 i + 4) q^{83} + (6 i + 6) q^{85} + 16 i q^{89} + (12 i + 12) q^{91} - 8 q^{95} + 8 q^{97} +O(q^{100})$$ q + (i + 1) * q^5 + 4*i * q^7 + (-4*i - 4) * q^11 + (-3*i + 3) * q^13 + 6 * q^17 + (4*i - 4) * q^19 - 8*i * q^23 - 3*i * q^25 + (-3*i + 3) * q^29 + 4 * q^31 + (4*i - 4) * q^35 + (i + 1) * q^37 + 2*i * q^41 + (-4*i - 4) * q^43 + 8 * q^47 - 9 * q^49 + (-7*i - 7) * q^53 - 8*i * q^55 + (3*i - 3) * q^61 + 6 * q^65 + (-8*i + 8) * q^67 - 10*i * q^73 + (-16*i + 16) * q^77 - 12 * q^79 + (-4*i + 4) * q^83 + (6*i + 6) * q^85 + 16*i * q^89 + (12*i + 12) * q^91 - 8 * q^95 + 8 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5}+O(q^{10})$$ 2 * q + 2 * q^5 $$2 q + 2 q^{5} - 8 q^{11} + 6 q^{13} + 12 q^{17} - 8 q^{19} + 6 q^{29} + 8 q^{31} - 8 q^{35} + 2 q^{37} - 8 q^{43} + 16 q^{47} - 18 q^{49} - 14 q^{53} - 6 q^{61} + 12 q^{65} + 16 q^{67} + 32 q^{77} - 24 q^{79} + 8 q^{83} + 12 q^{85} + 24 q^{91} - 16 q^{95} + 16 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 - 8 * q^11 + 6 * q^13 + 12 * q^17 - 8 * q^19 + 6 * q^29 + 8 * q^31 - 8 * q^35 + 2 * q^37 - 8 * q^43 + 16 * q^47 - 18 * q^49 - 14 * q^53 - 6 * q^61 + 12 * q^65 + 16 * q^67 + 32 * q^77 - 24 * q^79 + 8 * q^83 + 12 * q^85 + 24 * q^91 - 16 * q^95 + 16 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$$.

 $$n$$ $$2053$$ $$3583$$ $$4097$$ $$\chi(n)$$ $$i$$ $$1$$ $$1$$

## 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 1.00000 1.00000i 0 4.00000i 0 0 0
3457.1 0 0 0 1.00000 + 1.00000i 0 4.00000i 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
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4608.2.k.n yes 2
3.b odd 2 1 4608.2.k.l yes 2
4.b odd 2 1 4608.2.k.u yes 2
8.b even 2 1 4608.2.k.k yes 2
8.d odd 2 1 4608.2.k.d 2
12.b even 2 1 4608.2.k.e yes 2
16.e even 4 1 4608.2.k.k yes 2
16.e even 4 1 inner 4608.2.k.n yes 2
16.f odd 4 1 4608.2.k.d 2
16.f odd 4 1 4608.2.k.u yes 2
24.f even 2 1 4608.2.k.t yes 2
24.h odd 2 1 4608.2.k.m yes 2
32.g even 8 2 9216.2.a.t 2
32.h odd 8 2 9216.2.a.a 2
48.i odd 4 1 4608.2.k.l yes 2
48.i odd 4 1 4608.2.k.m yes 2
48.k even 4 1 4608.2.k.e yes 2
48.k even 4 1 4608.2.k.t yes 2
96.o even 8 2 9216.2.a.c 2
96.p odd 8 2 9216.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.k.d 2 8.d odd 2 1
4608.2.k.d 2 16.f odd 4 1
4608.2.k.e yes 2 12.b even 2 1
4608.2.k.e yes 2 48.k even 4 1
4608.2.k.k yes 2 8.b even 2 1
4608.2.k.k yes 2 16.e even 4 1
4608.2.k.l yes 2 3.b odd 2 1
4608.2.k.l yes 2 48.i odd 4 1
4608.2.k.m yes 2 24.h odd 2 1
4608.2.k.m yes 2 48.i odd 4 1
4608.2.k.n yes 2 1.a even 1 1 trivial
4608.2.k.n yes 2 16.e even 4 1 inner
4608.2.k.t yes 2 24.f even 2 1
4608.2.k.t yes 2 48.k even 4 1
4608.2.k.u yes 2 4.b odd 2 1
4608.2.k.u yes 2 16.f odd 4 1
9216.2.a.a 2 32.h odd 8 2
9216.2.a.c 2 96.o even 8 2
9216.2.a.t 2 32.g even 8 2
9216.2.a.v 2 96.p odd 8 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4608, [\chi])$$:

 $$T_{5}^{2} - 2T_{5} + 2$$ T5^2 - 2*T5 + 2 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11}^{2} + 8T_{11} + 32$$ T11^2 + 8*T11 + 32 $$T_{13}^{2} - 6T_{13} + 18$$ T13^2 - 6*T13 + 18 $$T_{19}^{2} + 8T_{19} + 32$$ T19^2 + 8*T19 + 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 2$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2} + 8T + 32$$
$13$ $$T^{2} - 6T + 18$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} + 8T + 32$$
$23$ $$T^{2} + 64$$
$29$ $$T^{2} - 6T + 18$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} - 2T + 2$$
$41$ $$T^{2} + 4$$
$43$ $$T^{2} + 8T + 32$$
$47$ $$(T - 8)^{2}$$
$53$ $$T^{2} + 14T + 98$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 6T + 18$$
$67$ $$T^{2} - 16T + 128$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T + 12)^{2}$$
$83$ $$T^{2} - 8T + 32$$
$89$ $$T^{2} + 256$$
$97$ $$(T - 8)^{2}$$