# Properties

 Label 3024.2.b.q Level 3024 Weight 2 Character orbit 3024.b Analytic conductor 24.147 Analytic rank 0 Dimension 4 CM discriminant -84 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-6}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 24 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} -\beta_{2} q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} -\beta_{2} q^{7} -\beta_{1} q^{11} + ( \beta_{1} - \beta_{3} ) q^{17} + ( -6 + \beta_{2} ) q^{19} + \beta_{3} q^{23} + ( -7 + 3 \beta_{2} ) q^{25} + ( -3 + 2 \beta_{2} ) q^{31} + ( 2 \beta_{1} - \beta_{3} ) q^{35} + ( 4 + 3 \beta_{2} ) q^{37} -\beta_{3} q^{41} + 7 q^{49} + ( 12 - 3 \beta_{2} ) q^{55} + ( \beta_{1} - 2 \beta_{3} ) q^{71} + ( -2 \beta_{1} + \beta_{3} ) q^{77} + ( -9 + 9 \beta_{2} ) q^{85} + ( 4 \beta_{1} + \beta_{3} ) q^{89} + ( -8 \beta_{1} + \beta_{3} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 24q^{19} - 28q^{25} - 12q^{31} + 16q^{37} + 28q^{49} + 48q^{55} - 36q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 24 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} + 12$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 18 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2} - 12$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3} - 18 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 − 4.46512i − 2.01563i 2.01563i 4.46512i
0 0 0 4.46512i 0 2.64575 0 0 0
1567.2 0 0 0 2.01563i 0 −2.64575 0 0 0
1567.3 0 0 0 2.01563i 0 −2.64575 0 0 0
1567.4 0 0 0 4.46512i 0 2.64575 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
84.h odd 2 1 CM by $$\Q(\sqrt{-21})$$
3.b odd 2 1 inner
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.b.q 4
3.b odd 2 1 inner 3024.2.b.q 4
4.b odd 2 1 3024.2.b.t yes 4
7.b odd 2 1 3024.2.b.t yes 4
12.b even 2 1 3024.2.b.t yes 4
21.c even 2 1 3024.2.b.t yes 4
28.d even 2 1 inner 3024.2.b.q 4
84.h odd 2 1 CM 3024.2.b.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.2.b.q 4 1.a even 1 1 trivial
3024.2.b.q 4 3.b odd 2 1 inner
3024.2.b.q 4 28.d even 2 1 inner
3024.2.b.q 4 84.h odd 2 1 CM
3024.2.b.t yes 4 4.b odd 2 1
3024.2.b.t yes 4 7.b odd 2 1
3024.2.b.t yes 4 12.b even 2 1
3024.2.b.t yes 4 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3024, [\chi])$$:

 $$T_{5}^{4} + 24 T_{5}^{2} + 81$$ $$T_{11}^{4} + 24 T_{11}^{2} + 81$$ $$T_{13}$$ $$T_{17}^{2} + 54$$ $$T_{19}^{2} + 12 T_{19} + 29$$ $$T_{29}$$ $$T_{47}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 4 T^{2} - 9 T^{4} + 100 T^{6} + 625 T^{8}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$1 - 20 T^{2} + 279 T^{4} - 2420 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 13 T^{2} )^{4}$$
$17$ $$( 1 + 20 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 + 4 T^{2} - 513 T^{4} + 2116 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 29 T^{2} )^{4}$$
$31$ $$( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$1 - 68 T^{2} + 2943 T^{4} - 114308 T^{6} + 2825761 T^{8}$$
$43$ $$( 1 - 43 T^{2} )^{4}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$( 1 + 53 T^{2} )^{4}$$
$59$ $$( 1 + 59 T^{2} )^{4}$$
$61$ $$( 1 - 61 T^{2} )^{4}$$
$67$ $$( 1 - 67 T^{2} )^{4}$$
$71$ $$1 + 100 T^{2} + 4959 T^{4} + 504100 T^{6} + 25411681 T^{8}$$
$73$ $$( 1 - 73 T^{2} )^{4}$$
$79$ $$( 1 - 79 T^{2} )^{4}$$
$83$ $$( 1 + 83 T^{2} )^{4}$$
$89$ $$1 + 172 T^{2} + 21663 T^{4} + 1362412 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 - 97 T^{2} )^{4}$$