# Properties

 Label 1536.2.d.c Level $1536$ Weight $2$ Character orbit 1536.d Analytic conductor $12.265$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1536 = 2^{9} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1536.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.2650217505$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + \beta_{2} q^{5} - 3 \beta_{3} q^{7} - q^{9}+O(q^{10})$$ q + b1 * q^3 + b2 * q^5 - 3*b3 * q^7 - q^9 $$q + \beta_1 q^{3} + \beta_{2} q^{5} - 3 \beta_{3} q^{7} - q^{9} + 6 \beta_1 q^{11} - 4 \beta_{2} q^{13} - \beta_{3} q^{15} - 6 q^{17} - 4 \beta_1 q^{19} - 3 \beta_{2} q^{21} + 2 \beta_{3} q^{23} + 3 q^{25} - \beta_1 q^{27} - \beta_{2} q^{29} + \beta_{3} q^{31} - 6 q^{33} - 6 \beta_1 q^{35} - 6 \beta_{2} q^{37} + 4 \beta_{3} q^{39} + 2 q^{41} - \beta_{2} q^{45} - 2 \beta_{3} q^{47} + 11 q^{49} - 6 \beta_1 q^{51} - 7 \beta_{2} q^{53} - 6 \beta_{3} q^{55} + 4 q^{57} + 4 \beta_1 q^{59} + 6 \beta_{2} q^{61} + 3 \beta_{3} q^{63} + 8 q^{65} - 8 \beta_1 q^{67} + 2 \beta_{2} q^{69} - 2 \beta_{3} q^{71} - 8 q^{73} + 3 \beta_1 q^{75} - 18 \beta_{2} q^{77} - 9 \beta_{3} q^{79} + q^{81} - 2 \beta_1 q^{83} - 6 \beta_{2} q^{85} + \beta_{3} q^{87} - 2 q^{89} + 24 \beta_1 q^{91} + \beta_{2} q^{93} + 4 \beta_{3} q^{95} + 2 q^{97} - 6 \beta_1 q^{99}+O(q^{100})$$ q + b1 * q^3 + b2 * q^5 - 3*b3 * q^7 - q^9 + 6*b1 * q^11 - 4*b2 * q^13 - b3 * q^15 - 6 * q^17 - 4*b1 * q^19 - 3*b2 * q^21 + 2*b3 * q^23 + 3 * q^25 - b1 * q^27 - b2 * q^29 + b3 * q^31 - 6 * q^33 - 6*b1 * q^35 - 6*b2 * q^37 + 4*b3 * q^39 + 2 * q^41 - b2 * q^45 - 2*b3 * q^47 + 11 * q^49 - 6*b1 * q^51 - 7*b2 * q^53 - 6*b3 * q^55 + 4 * q^57 + 4*b1 * q^59 + 6*b2 * q^61 + 3*b3 * q^63 + 8 * q^65 - 8*b1 * q^67 + 2*b2 * q^69 - 2*b3 * q^71 - 8 * q^73 + 3*b1 * q^75 - 18*b2 * q^77 - 9*b3 * q^79 + q^81 - 2*b1 * q^83 - 6*b2 * q^85 + b3 * q^87 - 2 * q^89 + 24*b1 * q^91 + b2 * q^93 + 4*b3 * q^95 + 2 * q^97 - 6*b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 24 q^{17} + 12 q^{25} - 24 q^{33} + 8 q^{41} + 44 q^{49} + 16 q^{57} + 32 q^{65} - 32 q^{73} + 4 q^{81} - 8 q^{89} + 8 q^{97}+O(q^{100})$$ 4 * q - 4 * q^9 - 24 * q^17 + 12 * q^25 - 24 * q^33 + 8 * q^41 + 44 * q^49 + 16 * q^57 + 32 * q^65 - 32 * q^73 + 4 * q^81 - 8 * q^89 + 8 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$517$$ $$1025$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
769.1
 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
0 1.00000i 0 1.41421i 0 −4.24264 0 −1.00000 0
769.2 0 1.00000i 0 1.41421i 0 4.24264 0 −1.00000 0
769.3 0 1.00000i 0 1.41421i 0 4.24264 0 −1.00000 0
769.4 0 1.00000i 0 1.41421i 0 −4.24264 0 −1.00000 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 inner
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 1536.2.d.c 4
3.b odd 2 1 4608.2.d.n 4
4.b odd 2 1 inner 1536.2.d.c 4
8.b even 2 1 inner 1536.2.d.c 4
8.d odd 2 1 inner 1536.2.d.c 4
12.b even 2 1 4608.2.d.n 4
16.e even 4 1 1536.2.a.d 2
16.e even 4 1 1536.2.a.i yes 2
16.f odd 4 1 1536.2.a.d 2
16.f odd 4 1 1536.2.a.i yes 2
24.f even 2 1 4608.2.d.n 4
24.h odd 2 1 4608.2.d.n 4
48.i odd 4 1 4608.2.a.g 2
48.i odd 4 1 4608.2.a.l 2
48.k even 4 1 4608.2.a.g 2
48.k even 4 1 4608.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.d 2 16.e even 4 1
1536.2.a.d 2 16.f odd 4 1
1536.2.a.i yes 2 16.e even 4 1
1536.2.a.i yes 2 16.f odd 4 1
1536.2.d.c 4 1.a even 1 1 trivial
1536.2.d.c 4 4.b odd 2 1 inner
1536.2.d.c 4 8.b even 2 1 inner
1536.2.d.c 4 8.d odd 2 1 inner
4608.2.a.g 2 48.i odd 4 1
4608.2.a.g 2 48.k even 4 1
4608.2.a.l 2 48.i odd 4 1
4608.2.a.l 2 48.k even 4 1
4608.2.d.n 4 3.b odd 2 1
4608.2.d.n 4 12.b even 2 1
4608.2.d.n 4 24.f even 2 1
4608.2.d.n 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1536, [\chi])$$:

 $$T_{5}^{2} + 2$$ T5^2 + 2 $$T_{7}^{2} - 18$$ T7^2 - 18 $$T_{23}^{2} - 8$$ T23^2 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$(T^{2} + 2)^{2}$$
$7$ $$(T^{2} - 18)^{2}$$
$11$ $$(T^{2} + 36)^{2}$$
$13$ $$(T^{2} + 32)^{2}$$
$17$ $$(T + 6)^{4}$$
$19$ $$(T^{2} + 16)^{2}$$
$23$ $$(T^{2} - 8)^{2}$$
$29$ $$(T^{2} + 2)^{2}$$
$31$ $$(T^{2} - 2)^{2}$$
$37$ $$(T^{2} + 72)^{2}$$
$41$ $$(T - 2)^{4}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} - 8)^{2}$$
$53$ $$(T^{2} + 98)^{2}$$
$59$ $$(T^{2} + 16)^{2}$$
$61$ $$(T^{2} + 72)^{2}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} - 8)^{2}$$
$73$ $$(T + 8)^{4}$$
$79$ $$(T^{2} - 162)^{2}$$
$83$ $$(T^{2} + 4)^{2}$$
$89$ $$(T + 2)^{4}$$
$97$ $$(T - 2)^{4}$$