# Properties

 Label 1512.2.s.a Level $1512$ Weight $2$ Character orbit 1512.s Analytic conductor $12.073$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1512 = 2^{3} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1512.s (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0733807856$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -4 \zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -4 \zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} + ( 2 - 2 \zeta_{6} ) q^{11} + 5 q^{13} + ( 6 - 6 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} -6 q^{29} + ( 7 - 7 \zeta_{6} ) q^{31} + ( -8 - 4 \zeta_{6} ) q^{35} -7 \zeta_{6} q^{37} -2 q^{41} -7 q^{43} -2 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 6 - 6 \zeta_{6} ) q^{53} -8 q^{55} + ( 6 - 6 \zeta_{6} ) q^{59} + 9 \zeta_{6} q^{61} -20 \zeta_{6} q^{65} + ( 7 - 7 \zeta_{6} ) q^{67} + 8 q^{71} + ( -10 + 10 \zeta_{6} ) q^{73} + ( 2 - 6 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} -14 q^{83} -24 q^{85} + 12 \zeta_{6} q^{89} + ( 15 - 10 \zeta_{6} ) q^{91} + ( 16 - 16 \zeta_{6} ) q^{95} -15 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} + 4q^{7} + O(q^{10})$$ $$2q - 4q^{5} + 4q^{7} + 2q^{11} + 10q^{13} + 6q^{17} + 4q^{19} + 6q^{23} - 11q^{25} - 12q^{29} + 7q^{31} - 20q^{35} - 7q^{37} - 4q^{41} - 14q^{43} - 2q^{47} + 2q^{49} + 6q^{53} - 16q^{55} + 6q^{59} + 9q^{61} - 20q^{65} + 7q^{67} + 16q^{71} - 10q^{73} - 2q^{77} - q^{79} - 28q^{83} - 48q^{85} + 12q^{89} + 20q^{91} + 16q^{95} - 30q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1081$$ $$1135$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −2.00000 3.46410i 0 2.00000 1.73205i 0 0 0
1297.1 0 0 0 −2.00000 + 3.46410i 0 2.00000 + 1.73205i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.s.a 2
3.b odd 2 1 1512.2.s.j yes 2
7.c even 3 1 inner 1512.2.s.a 2
21.h odd 6 1 1512.2.s.j yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.s.a 2 1.a even 1 1 trivial
1512.2.s.a 2 7.c even 3 1 inner
1512.2.s.j yes 2 3.b odd 2 1
1512.2.s.j yes 2 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1512, [\chi])$$:

 $$T_{5}^{2} + 4 T_{5} + 16$$ $$T_{11}^{2} - 2 T_{11} + 4$$ $$T_{13} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4}$$
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 5 T + 13 T^{2} )^{2}$$
$17$ $$1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$
$37$ $$1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 7 T + 43 T^{2} )^{2}$$
$47$ $$1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4}$$
$53$ $$1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4}$$
$59$ $$1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4}$$
$61$ $$1 - 9 T + 20 T^{2} - 549 T^{3} + 3721 T^{4}$$
$67$ $$1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 8 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$
$79$ $$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 14 T + 83 T^{2} )^{2}$$
$89$ $$1 - 12 T + 55 T^{2} - 1068 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 15 T + 97 T^{2} )^{2}$$