# Properties

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

# 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + ( - \beta_{2} + 2 \beta_1) q^{5} + ( - \beta_{3} - 2) q^{7} - q^{9}+O(q^{10})$$ q - b1 * q^3 + (-b2 + 2*b1) * q^5 + (-b3 - 2) * q^7 - q^9 $$q - \beta_1 q^{3} + ( - \beta_{2} + 2 \beta_1) q^{5} + ( - \beta_{3} - 2) q^{7} - q^{9} + 2 \beta_1 q^{11} - 2 \beta_{2} q^{13} + ( - \beta_{3} + 2) q^{15} + (4 \beta_{3} - 2) q^{17} + 4 \beta_{2} q^{19} + (\beta_{2} + 2 \beta_1) q^{21} + (2 \beta_{3} - 4) q^{23} + (4 \beta_{3} - 1) q^{25} + \beta_1 q^{27} + (\beta_{2} - 2 \beta_1) q^{29} + ( - \beta_{3} + 6) q^{31} + 2 q^{33} - 2 \beta_1 q^{35} + ( - 4 \beta_{2} - 4 \beta_1) q^{37} - 2 \beta_{3} q^{39} + (4 \beta_{3} + 6) q^{41} + (4 \beta_{2} - 4 \beta_1) q^{43} + (\beta_{2} - 2 \beta_1) q^{45} + (6 \beta_{3} + 4) q^{47} + (4 \beta_{3} - 1) q^{49} + ( - 4 \beta_{2} + 2 \beta_1) q^{51} + (7 \beta_{2} + 2 \beta_1) q^{53} + (2 \beta_{3} - 4) q^{55} + 4 \beta_{3} q^{57} - 4 \beta_1 q^{59} + (4 \beta_{2} + 4 \beta_1) q^{61} + (\beta_{3} + 2) q^{63} + (4 \beta_{3} - 4) q^{65} + 8 \beta_1 q^{67} + ( - 2 \beta_{2} + 4 \beta_1) q^{69} + (2 \beta_{3} - 12) q^{71} + (4 \beta_{3} - 4) q^{73} + ( - 4 \beta_{2} + \beta_1) q^{75} + ( - 2 \beta_{2} - 4 \beta_1) q^{77} + ( - 3 \beta_{3} + 10) q^{79} + q^{81} + ( - 8 \beta_{2} + 2 \beta_1) q^{83} + (10 \beta_{2} - 12 \beta_1) q^{85} + (\beta_{3} - 2) q^{87} - 2 q^{89} + (4 \beta_{2} + 4 \beta_1) q^{91} + (\beta_{2} - 6 \beta_1) q^{93} + ( - 8 \beta_{3} + 8) q^{95} + (8 \beta_{3} + 2) q^{97} - 2 \beta_1 q^{99}+O(q^{100})$$ q - b1 * q^3 + (-b2 + 2*b1) * q^5 + (-b3 - 2) * q^7 - q^9 + 2*b1 * q^11 - 2*b2 * q^13 + (-b3 + 2) * q^15 + (4*b3 - 2) * q^17 + 4*b2 * q^19 + (b2 + 2*b1) * q^21 + (2*b3 - 4) * q^23 + (4*b3 - 1) * q^25 + b1 * q^27 + (b2 - 2*b1) * q^29 + (-b3 + 6) * q^31 + 2 * q^33 - 2*b1 * q^35 + (-4*b2 - 4*b1) * q^37 - 2*b3 * q^39 + (4*b3 + 6) * q^41 + (4*b2 - 4*b1) * q^43 + (b2 - 2*b1) * q^45 + (6*b3 + 4) * q^47 + (4*b3 - 1) * q^49 + (-4*b2 + 2*b1) * q^51 + (7*b2 + 2*b1) * q^53 + (2*b3 - 4) * q^55 + 4*b3 * q^57 - 4*b1 * q^59 + (4*b2 + 4*b1) * q^61 + (b3 + 2) * q^63 + (4*b3 - 4) * q^65 + 8*b1 * q^67 + (-2*b2 + 4*b1) * q^69 + (2*b3 - 12) * q^71 + (4*b3 - 4) * q^73 + (-4*b2 + b1) * q^75 + (-2*b2 - 4*b1) * q^77 + (-3*b3 + 10) * q^79 + q^81 + (-8*b2 + 2*b1) * q^83 + (10*b2 - 12*b1) * q^85 + (b3 - 2) * q^87 - 2 * q^89 + (4*b2 + 4*b1) * q^91 + (b2 - 6*b1) * q^93 + (-8*b3 + 8) * q^95 + (8*b3 + 2) * q^97 - 2*b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q - 8 * q^7 - 4 * q^9 $$4 q - 8 q^{7} - 4 q^{9} + 8 q^{15} - 8 q^{17} - 16 q^{23} - 4 q^{25} + 24 q^{31} + 8 q^{33} + 24 q^{41} + 16 q^{47} - 4 q^{49} - 16 q^{55} + 8 q^{63} - 16 q^{65} - 48 q^{71} - 16 q^{73} + 40 q^{79} + 4 q^{81} - 8 q^{87} - 8 q^{89} + 32 q^{95} + 8 q^{97}+O(q^{100})$$ 4 * q - 8 * q^7 - 4 * q^9 + 8 * q^15 - 8 * q^17 - 16 * q^23 - 4 * q^25 + 24 * q^31 + 8 * q^33 + 24 * q^41 + 16 * q^47 - 4 * q^49 - 16 * q^55 + 8 * q^63 - 16 * q^65 - 48 * q^71 - 16 * q^73 + 40 * q^79 + 4 * q^81 - 8 * q^87 - 8 * q^89 + 32 * q^95 + 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 0.585786i 0 −3.41421 0 −1.00000 0
769.2 0 1.00000i 0 3.41421i 0 −0.585786 0 −1.00000 0
769.3 0 1.00000i 0 3.41421i 0 −0.585786 0 −1.00000 0
769.4 0 1.00000i 0 0.585786i 0 −3.41421 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
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 1536.2.d.a 4
3.b odd 2 1 4608.2.d.c 4
4.b odd 2 1 1536.2.d.f 4
8.b even 2 1 inner 1536.2.d.a 4
8.d odd 2 1 1536.2.d.f 4
12.b even 2 1 4608.2.d.o 4
16.e even 4 1 1536.2.a.b 2
16.e even 4 1 1536.2.a.l yes 2
16.f odd 4 1 1536.2.a.e yes 2
16.f odd 4 1 1536.2.a.g yes 2
24.f even 2 1 4608.2.d.o 4
24.h odd 2 1 4608.2.d.c 4
48.i odd 4 1 4608.2.a.e 2
48.i odd 4 1 4608.2.a.r 2
48.k even 4 1 4608.2.a.a 2
48.k even 4 1 4608.2.a.n 2

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

## Hecke kernels

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

 $$T_{5}^{4} + 12T_{5}^{2} + 4$$ T5^4 + 12*T5^2 + 4 $$T_{7}^{2} + 4T_{7} + 2$$ T7^2 + 4*T7 + 2 $$T_{23}^{2} + 8T_{23} + 8$$ T23^2 + 8*T23 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4} + 12T^{2} + 4$$
$7$ $$(T^{2} + 4 T + 2)^{2}$$
$11$ $$(T^{2} + 4)^{2}$$
$13$ $$(T^{2} + 8)^{2}$$
$17$ $$(T^{2} + 4 T - 28)^{2}$$
$19$ $$(T^{2} + 32)^{2}$$
$23$ $$(T^{2} + 8 T + 8)^{2}$$
$29$ $$T^{4} + 12T^{2} + 4$$
$31$ $$(T^{2} - 12 T + 34)^{2}$$
$37$ $$T^{4} + 96T^{2} + 256$$
$41$ $$(T^{2} - 12 T + 4)^{2}$$
$43$ $$T^{4} + 96T^{2} + 256$$
$47$ $$(T^{2} - 8 T - 56)^{2}$$
$53$ $$T^{4} + 204T^{2} + 8836$$
$59$ $$(T^{2} + 16)^{2}$$
$61$ $$T^{4} + 96T^{2} + 256$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} + 24 T + 136)^{2}$$
$73$ $$(T^{2} + 8 T - 16)^{2}$$
$79$ $$(T^{2} - 20 T + 82)^{2}$$
$83$ $$T^{4} + 264 T^{2} + 15376$$
$89$ $$(T + 2)^{4}$$
$97$ $$(T^{2} - 4 T - 124)^{2}$$