# Properties

 Label 1536.2.d.g Level $1536$ Weight $2$ Character orbit 1536.d Analytic conductor $12.265$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.18939904.2 Defining polynomial: $$x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2$$ x^8 - 4*x^7 + 14*x^6 - 28*x^5 + 43*x^4 - 44*x^3 + 30*x^2 - 12*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{3} + \beta_{5} q^{5} - \beta_{2} q^{7} - q^{9}+O(q^{10})$$ q - b4 * q^3 + b5 * q^5 - b2 * q^7 - q^9 $$q - \beta_{4} q^{3} + \beta_{5} q^{5} - \beta_{2} q^{7} - q^{9} + ( - \beta_{6} + 2 \beta_{4}) q^{11} + (\beta_{7} - \beta_{5}) q^{13} + \beta_1 q^{15} + (\beta_{3} + 2) q^{17} - \beta_{6} q^{19} + \beta_{7} q^{21} + 2 \beta_1 q^{23} + (2 \beta_{3} - 5) q^{25} + \beta_{4} q^{27} + ( - 2 \beta_{7} + \beta_{5}) q^{29} - \beta_{2} q^{31} + ( - \beta_{3} + 2) q^{33} + ( - 3 \beta_{6} - 2 \beta_{4}) q^{35} + ( - \beta_{7} - \beta_{5}) q^{37} + (\beta_{2} - \beta_1) q^{39} + ( - \beta_{3} - 6) q^{41} + ( - 3 \beta_{6} - 4 \beta_{4}) q^{43} - \beta_{5} q^{45} - 2 \beta_1 q^{47} + (4 \beta_{3} + 7) q^{49} + ( - \beta_{6} - 2 \beta_{4}) q^{51} + ( - 2 \beta_{7} - \beta_{5}) q^{53} + (2 \beta_{2} - 4 \beta_1) q^{55} - \beta_{3} q^{57} + ( - 2 \beta_{6} + 4 \beta_{4}) q^{59} + (\beta_{7} + \beta_{5}) q^{61} + \beta_{2} q^{63} + ( - 5 \beta_{3} + 8) q^{65} + ( - 2 \beta_{6} + 8 \beta_{4}) q^{67} - 2 \beta_{5} q^{69} + 2 \beta_{2} q^{71} - 4 q^{73} + ( - 2 \beta_{6} + 5 \beta_{4}) q^{75} + 2 \beta_{5} q^{77} + \beta_{2} q^{79} + q^{81} + (\beta_{6} - 6 \beta_{4}) q^{83} + 2 \beta_{7} q^{85} + ( - 2 \beta_{2} + \beta_1) q^{87} - 10 q^{89} + ( - \beta_{6} - 12 \beta_{4}) q^{91} + \beta_{7} q^{93} + (2 \beta_{2} - 2 \beta_1) q^{95} + ( - 2 \beta_{3} + 6) q^{97} + (\beta_{6} - 2 \beta_{4}) q^{99}+O(q^{100})$$ q - b4 * q^3 + b5 * q^5 - b2 * q^7 - q^9 + (-b6 + 2*b4) * q^11 + (b7 - b5) * q^13 + b1 * q^15 + (b3 + 2) * q^17 - b6 * q^19 + b7 * q^21 + 2*b1 * q^23 + (2*b3 - 5) * q^25 + b4 * q^27 + (-2*b7 + b5) * q^29 - b2 * q^31 + (-b3 + 2) * q^33 + (-3*b6 - 2*b4) * q^35 + (-b7 - b5) * q^37 + (b2 - b1) * q^39 + (-b3 - 6) * q^41 + (-3*b6 - 4*b4) * q^43 - b5 * q^45 - 2*b1 * q^47 + (4*b3 + 7) * q^49 + (-b6 - 2*b4) * q^51 + (-2*b7 - b5) * q^53 + (2*b2 - 4*b1) * q^55 - b3 * q^57 + (-2*b6 + 4*b4) * q^59 + (b7 + b5) * q^61 + b2 * q^63 + (-5*b3 + 8) * q^65 + (-2*b6 + 8*b4) * q^67 - 2*b5 * q^69 + 2*b2 * q^71 - 4 * q^73 + (-2*b6 + 5*b4) * q^75 + 2*b5 * q^77 + b2 * q^79 + q^81 + (b6 - 6*b4) * q^83 + 2*b7 * q^85 + (-2*b2 + b1) * q^87 - 10 * q^89 + (-b6 - 12*b4) * q^91 + b7 * q^93 + (2*b2 - 2*b1) * q^95 + (-2*b3 + 6) * q^97 + (b6 - 2*b4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{9}+O(q^{10})$$ 8 * q - 8 * q^9 $$8 q - 8 q^{9} + 16 q^{17} - 40 q^{25} + 16 q^{33} - 48 q^{41} + 56 q^{49} + 64 q^{65} - 32 q^{73} + 8 q^{81} - 80 q^{89} + 48 q^{97}+O(q^{100})$$ 8 * q - 8 * q^9 + 16 * q^17 - 40 * q^25 + 16 * q^33 - 48 * q^41 + 56 * q^49 + 64 * q^65 - 32 * q^73 + 8 * q^81 - 80 * q^89 + 48 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{6} - 3\nu^{5} + 10\nu^{4} - 15\nu^{3} + 17\nu^{2} - 10\nu$$ v^6 - 3*v^5 + 10*v^4 - 15*v^3 + 17*v^2 - 10*v $$\beta_{2}$$ $$=$$ $$-\nu^{6} + 3\nu^{5} - 12\nu^{4} + 19\nu^{3} - 31\nu^{2} + 22\nu - 10$$ -v^6 + 3*v^5 - 12*v^4 + 19*v^3 - 31*v^2 + 22*v - 10 $$\beta_{3}$$ $$=$$ $$2\nu^{6} - 6\nu^{5} + 20\nu^{4} - 30\nu^{3} + 38\nu^{2} - 24\nu + 8$$ 2*v^6 - 6*v^5 + 20*v^4 - 30*v^3 + 38*v^2 - 24*v + 8 $$\beta_{4}$$ $$=$$ $$-8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31$$ -8*v^7 + 28*v^6 - 98*v^5 + 175*v^4 - 256*v^3 + 223*v^2 - 126*v + 31 $$\beta_{5}$$ $$=$$ $$18\nu^{7} - 63\nu^{6} + 219\nu^{5} - 390\nu^{4} + 565\nu^{3} - 489\nu^{2} + 272\nu - 66$$ 18*v^7 - 63*v^6 + 219*v^5 - 390*v^4 + 565*v^3 - 489*v^2 + 272*v - 66 $$\beta_{6}$$ $$=$$ $$-20\nu^{7} + 70\nu^{6} - 246\nu^{5} + 440\nu^{4} - 650\nu^{3} + 570\nu^{2} - 332\nu + 84$$ -20*v^7 + 70*v^6 - 246*v^5 + 440*v^4 - 650*v^3 + 570*v^2 - 332*v + 84 $$\beta_{7}$$ $$=$$ $$38\nu^{7} - 133\nu^{6} + 465\nu^{5} - 830\nu^{4} + 1215\nu^{3} - 1059\nu^{2} + 608\nu - 152$$ 38*v^7 - 133*v^6 + 465*v^5 - 830*v^4 + 1215*v^3 - 1059*v^2 + 608*v - 152
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} + 2 ) / 4$$ (b7 + b6 - b5 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - 2\beta _1 - 6 ) / 4$$ (b7 + b6 - b5 + b3 - 2*b1 - 6) / 4 $$\nu^{3}$$ $$=$$ $$( -4\beta_{7} - 7\beta_{6} + 6\beta_{5} + 12\beta_{4} + 3\beta_{3} - 6\beta _1 - 20 ) / 8$$ (-4*b7 - 7*b6 + 6*b5 + 12*b4 + 3*b3 - 6*b1 - 20) / 8 $$\nu^{4}$$ $$=$$ $$( -5\beta_{7} - 8\beta_{6} + 7\beta_{5} + 12\beta_{4} - 4\beta_{3} - 2\beta_{2} + 6\beta _1 + 14 ) / 4$$ (-5*b7 - 8*b6 + 7*b5 + 12*b4 - 4*b3 - 2*b2 + 6*b1 + 14) / 4 $$\nu^{5}$$ $$=$$ $$( 6\beta_{7} + 13\beta_{6} - 16\beta_{5} - 40\beta_{4} - 25\beta_{3} - 10\beta_{2} + 40\beta _1 + 104 ) / 8$$ (6*b7 + 13*b6 - 16*b5 - 40*b4 - 25*b3 - 10*b2 + 40*b1 + 104) / 8 $$\nu^{6}$$ $$=$$ $$( 22\beta_{7} + 40\beta_{6} - 42\beta_{5} - 90\beta_{4} + 8\beta_{3} + 5\beta_{2} - 7\beta _1 - 12 ) / 4$$ (22*b7 + 40*b6 - 42*b5 - 90*b4 + 8*b3 + 5*b2 - 7*b1 - 12) / 4 $$\nu^{7}$$ $$=$$ $$( 14\beta_{7} + 19\beta_{6} - 8\beta_{5} + 147\beta_{3} + 70\beta_{2} - 196\beta _1 - 472 ) / 8$$ (14*b7 + 19*b6 - 8*b5 + 147*b3 + 70*b2 - 196*b1 - 472) / 8

## 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
 0.5 + 0.691860i 0.5 − 0.0297061i 0.5 + 1.44392i 0.5 − 2.10607i 0.5 + 2.10607i 0.5 − 1.44392i 0.5 + 0.0297061i 0.5 − 0.691860i
0 1.00000i 0 3.95687i 0 −1.63899 0 −1.00000 0
769.2 0 1.00000i 0 2.08402i 0 5.03127 0 −1.00000 0
769.3 0 1.00000i 0 2.08402i 0 −5.03127 0 −1.00000 0
769.4 0 1.00000i 0 3.95687i 0 1.63899 0 −1.00000 0
769.5 0 1.00000i 0 3.95687i 0 1.63899 0 −1.00000 0
769.6 0 1.00000i 0 2.08402i 0 −5.03127 0 −1.00000 0
769.7 0 1.00000i 0 2.08402i 0 5.03127 0 −1.00000 0
769.8 0 1.00000i 0 3.95687i 0 −1.63899 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 769.8 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.g 8
3.b odd 2 1 4608.2.d.p 8
4.b odd 2 1 inner 1536.2.d.g 8
8.b even 2 1 inner 1536.2.d.g 8
8.d odd 2 1 inner 1536.2.d.g 8
12.b even 2 1 4608.2.d.p 8
16.e even 4 1 1536.2.a.m 4
16.e even 4 1 1536.2.a.n yes 4
16.f odd 4 1 1536.2.a.m 4
16.f odd 4 1 1536.2.a.n yes 4
24.f even 2 1 4608.2.d.p 8
24.h odd 2 1 4608.2.d.p 8
48.i odd 4 1 4608.2.a.t 4
48.i odd 4 1 4608.2.a.ba 4
48.k even 4 1 4608.2.a.t 4
48.k even 4 1 4608.2.a.ba 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.m 4 16.e even 4 1
1536.2.a.m 4 16.f odd 4 1
1536.2.a.n yes 4 16.e even 4 1
1536.2.a.n yes 4 16.f odd 4 1
1536.2.d.g 8 1.a even 1 1 trivial
1536.2.d.g 8 4.b odd 2 1 inner
1536.2.d.g 8 8.b even 2 1 inner
1536.2.d.g 8 8.d odd 2 1 inner
4608.2.a.t 4 48.i odd 4 1
4608.2.a.t 4 48.k even 4 1
4608.2.a.ba 4 48.i odd 4 1
4608.2.a.ba 4 48.k even 4 1
4608.2.d.p 8 3.b odd 2 1
4608.2.d.p 8 12.b even 2 1
4608.2.d.p 8 24.f even 2 1
4608.2.d.p 8 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 $$S_{2}^{\mathrm{new}}(1536, [\chi])$$:

 $$T_{5}^{4} + 20T_{5}^{2} + 68$$ T5^4 + 20*T5^2 + 68 $$T_{7}^{4} - 28T_{7}^{2} + 68$$ T7^4 - 28*T7^2 + 68 $$T_{23}^{4} - 80T_{23}^{2} + 1088$$ T23^4 - 80*T23^2 + 1088

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} + 1)^{4}$$
$5$ $$(T^{4} + 20 T^{2} + 68)^{2}$$
$7$ $$(T^{4} - 28 T^{2} + 68)^{2}$$
$11$ $$(T^{4} + 24 T^{2} + 16)^{2}$$
$13$ $$(T^{4} + 40 T^{2} + 272)^{2}$$
$17$ $$(T^{2} - 4 T - 4)^{4}$$
$19$ $$(T^{2} + 8)^{4}$$
$23$ $$(T^{4} - 80 T^{2} + 1088)^{2}$$
$29$ $$(T^{4} + 116 T^{2} + 3332)^{2}$$
$31$ $$(T^{4} - 28 T^{2} + 68)^{2}$$
$37$ $$(T^{4} + 56 T^{2} + 272)^{2}$$
$41$ $$(T^{2} + 12 T + 28)^{4}$$
$43$ $$(T^{4} + 176 T^{2} + 3136)^{2}$$
$47$ $$(T^{4} - 80 T^{2} + 1088)^{2}$$
$53$ $$(T^{4} + 148 T^{2} + 68)^{2}$$
$59$ $$(T^{4} + 96 T^{2} + 256)^{2}$$
$61$ $$(T^{4} + 56 T^{2} + 272)^{2}$$
$67$ $$(T^{4} + 192 T^{2} + 1024)^{2}$$
$71$ $$(T^{4} - 112 T^{2} + 1088)^{2}$$
$73$ $$(T + 4)^{8}$$
$79$ $$(T^{4} - 28 T^{2} + 68)^{2}$$
$83$ $$(T^{4} + 88 T^{2} + 784)^{2}$$
$89$ $$(T + 10)^{8}$$
$97$ $$(T^{2} - 12 T + 4)^{4}$$