# Properties

 Label 4608.2.k.bd Level $4608$ Weight $2$ Character orbit 4608.k Analytic conductor $36.795$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 512) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{16}^{2} - \zeta_{16}^{4} ) q^{5} + ( -2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} ) q^{7} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{16}^{2} - \zeta_{16}^{4} ) q^{5} + ( -2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} ) q^{7} + ( \zeta_{16} + \zeta_{16}^{3} + 3 \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{11} + ( -1 + \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{13} + ( 2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} ) q^{17} + ( -\zeta_{16} + \zeta_{16}^{3} + 3 \zeta_{16}^{5} + 3 \zeta_{16}^{7} ) q^{19} + ( 4 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{23} + ( -4 \zeta_{16}^{2} + \zeta_{16}^{4} - 4 \zeta_{16}^{6} ) q^{25} + ( -1 + \zeta_{16}^{4} + 6 \zeta_{16}^{6} ) q^{29} + ( -4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} ) q^{31} + ( -2 \zeta_{16} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{35} + ( -5 - 2 \zeta_{16}^{2} - 5 \zeta_{16}^{4} ) q^{37} -4 \zeta_{16}^{4} q^{41} + ( \zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{43} + ( -4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} ) q^{47} + ( -1 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{49} + ( 1 - 6 \zeta_{16}^{2} + \zeta_{16}^{4} ) q^{53} + ( 8 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 8 \zeta_{16}^{7} ) q^{55} + ( -\zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{59} + ( -5 + 5 \zeta_{16}^{4} - 6 \zeta_{16}^{6} ) q^{61} -2 q^{65} + ( -5 \zeta_{16} + 5 \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{67} + ( 4 \zeta_{16} - 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{71} + ( 6 \zeta_{16}^{2} - 2 \zeta_{16}^{4} + 6 \zeta_{16}^{6} ) q^{73} + ( 8 - 8 \zeta_{16}^{4} - 4 \zeta_{16}^{6} ) q^{77} + ( -8 \zeta_{16} + 8 \zeta_{16}^{7} ) q^{79} + ( -\zeta_{16} + \zeta_{16}^{3} + 7 \zeta_{16}^{5} + 7 \zeta_{16}^{7} ) q^{83} + ( 4 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{4} ) q^{85} + ( -2 \zeta_{16}^{2} - 2 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{89} + ( 6 \zeta_{16} + 6 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{91} + ( -2 \zeta_{16} + 2 \zeta_{16}^{7} ) q^{95} + ( 8 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{5} + O(q^{10})$$ $$8q - 8q^{5} - 8q^{13} - 8q^{29} - 40q^{37} - 8q^{49} + 8q^{53} - 40q^{61} - 16q^{65} + 64q^{77} + 32q^{85} + 64q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$2053$$ $$3583$$ $$4097$$ $$\chi(n)$$ $$\zeta_{16}^{2}$$ $$1$$ $$1$$

## 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
 −0.382683 − 0.923880i 0.382683 + 0.923880i −0.923880 + 0.382683i 0.923880 − 0.382683i 0.382683 − 0.923880i −0.382683 + 0.923880i 0.923880 + 0.382683i −0.923880 − 0.382683i
0 0 0 −2.41421 + 2.41421i 0 1.53073i 0 0 0
1153.2 0 0 0 −2.41421 + 2.41421i 0 1.53073i 0 0 0
1153.3 0 0 0 0.414214 0.414214i 0 3.69552i 0 0 0
1153.4 0 0 0 0.414214 0.414214i 0 3.69552i 0 0 0
3457.1 0 0 0 −2.41421 2.41421i 0 1.53073i 0 0 0
3457.2 0 0 0 −2.41421 2.41421i 0 1.53073i 0 0 0
3457.3 0 0 0 0.414214 + 0.414214i 0 3.69552i 0 0 0
3457.4 0 0 0 0.414214 + 0.414214i 0 3.69552i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3457.4 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 inner
16.e even 4 1 inner
16.f odd 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.bd 8
3.b odd 2 1 512.2.e.j yes 8
4.b odd 2 1 inner 4608.2.k.bd 8
8.b even 2 1 4608.2.k.bi 8
8.d odd 2 1 4608.2.k.bi 8
12.b even 2 1 512.2.e.j yes 8
16.e even 4 1 inner 4608.2.k.bd 8
16.e even 4 1 4608.2.k.bi 8
16.f odd 4 1 inner 4608.2.k.bd 8
16.f odd 4 1 4608.2.k.bi 8
24.f even 2 1 512.2.e.i 8
24.h odd 2 1 512.2.e.i 8
32.g even 8 1 9216.2.a.w 4
32.g even 8 1 9216.2.a.bp 4
32.h odd 8 1 9216.2.a.w 4
32.h odd 8 1 9216.2.a.bp 4
48.i odd 4 1 512.2.e.i 8
48.i odd 4 1 512.2.e.j yes 8
48.k even 4 1 512.2.e.i 8
48.k even 4 1 512.2.e.j yes 8
96.o even 8 1 1024.2.a.h 4
96.o even 8 1 1024.2.a.i 4
96.o even 8 2 1024.2.b.g 8
96.p odd 8 1 1024.2.a.h 4
96.p odd 8 1 1024.2.a.i 4
96.p odd 8 2 1024.2.b.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.i 8 24.f even 2 1
512.2.e.i 8 24.h odd 2 1
512.2.e.i 8 48.i odd 4 1
512.2.e.i 8 48.k even 4 1
512.2.e.j yes 8 3.b odd 2 1
512.2.e.j yes 8 12.b even 2 1
512.2.e.j yes 8 48.i odd 4 1
512.2.e.j yes 8 48.k even 4 1
1024.2.a.h 4 96.o even 8 1
1024.2.a.h 4 96.p odd 8 1
1024.2.a.i 4 96.o even 8 1
1024.2.a.i 4 96.p odd 8 1
1024.2.b.g 8 96.o even 8 2
1024.2.b.g 8 96.p odd 8 2
4608.2.k.bd 8 1.a even 1 1 trivial
4608.2.k.bd 8 4.b odd 2 1 inner
4608.2.k.bd 8 16.e even 4 1 inner
4608.2.k.bd 8 16.f odd 4 1 inner
4608.2.k.bi 8 8.b even 2 1
4608.2.k.bi 8 8.d odd 2 1
4608.2.k.bi 8 16.e even 4 1
4608.2.k.bi 8 16.f odd 4 1
9216.2.a.w 4 32.g even 8 1
9216.2.a.w 4 32.h odd 8 1
9216.2.a.bp 4 32.g even 8 1
9216.2.a.bp 4 32.h odd 8 1

## Hecke kernels

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

 $$T_{5}^{4} + 4 T_{5}^{3} + 8 T_{5}^{2} - 8 T_{5} + 4$$ $$T_{7}^{4} + 16 T_{7}^{2} + 32$$ $$T_{11}^{8} + 816 T_{11}^{4} + 153664$$ $$T_{13}^{4} + 4 T_{13}^{3} + 8 T_{13}^{2} - 8 T_{13} + 4$$ $$T_{19}^{8} + 1584 T_{19}^{4} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 4 - 8 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$7$ $$( 32 + 16 T^{2} + T^{4} )^{2}$$
$11$ $$153664 + 816 T^{4} + T^{8}$$
$13$ $$( 4 - 8 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$17$ $$( -8 + T^{2} )^{4}$$
$19$ $$64 + 1584 T^{4} + T^{8}$$
$23$ $$( 1568 + 80 T^{2} + T^{4} )^{2}$$
$29$ $$( 1156 - 136 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$31$ $$( 512 - 64 T^{2} + T^{4} )^{2}$$
$37$ $$( 2116 + 920 T + 200 T^{2} + 20 T^{3} + T^{4} )^{2}$$
$41$ $$( 16 + T^{2} )^{4}$$
$43$ $$64 + 48 T^{4} + T^{8}$$
$47$ $$( 512 - 64 T^{2} + T^{4} )^{2}$$
$53$ $$( 1156 + 136 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$59$ $$64 + 48 T^{4} + T^{8}$$
$61$ $$( 196 + 280 T + 200 T^{2} + 20 T^{3} + T^{4} )^{2}$$
$67$ $$153664 + 10032 T^{4} + T^{8}$$
$71$ $$( 9248 + 208 T^{2} + T^{4} )^{2}$$
$73$ $$( 4624 + 152 T^{2} + T^{4} )^{2}$$
$79$ $$( 8192 - 256 T^{2} + T^{4} )^{2}$$
$83$ $$5345344 + 35376 T^{4} + T^{8}$$
$89$ $$( 16 + 24 T^{2} + T^{4} )^{2}$$
$97$ $$( 56 - 16 T + T^{2} )^{4}$$