Properties

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

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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: $$x^{2} + 1$$ 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.

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

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.u yes 2
3.b odd 2 1 4608.2.k.e yes 2
4.b odd 2 1 4608.2.k.n yes 2
8.b even 2 1 4608.2.k.d 2
8.d odd 2 1 4608.2.k.k yes 2
12.b even 2 1 4608.2.k.l yes 2
16.e even 4 1 4608.2.k.d 2
16.e even 4 1 inner 4608.2.k.u yes 2
16.f odd 4 1 4608.2.k.k yes 2
16.f odd 4 1 4608.2.k.n yes 2
24.f even 2 1 4608.2.k.m yes 2
24.h odd 2 1 4608.2.k.t yes 2
32.g even 8 2 9216.2.a.a 2
32.h odd 8 2 9216.2.a.t 2
48.i odd 4 1 4608.2.k.e yes 2
48.i odd 4 1 4608.2.k.t yes 2
48.k even 4 1 4608.2.k.l yes 2
48.k even 4 1 4608.2.k.m yes 2
96.o even 8 2 9216.2.a.v 2
96.p odd 8 2 9216.2.a.c 2

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

Hecke kernels

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

 $$T_{5}^{2} - 2 T_{5} + 2$$ $$T_{7}^{2} + 16$$ $$T_{11}^{2} - 8 T_{11} + 32$$ $$T_{13}^{2} - 6 T_{13} + 18$$ $$T_{19}^{2} - 8 T_{19} + 32$$

Hecke characteristic polynomials

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