# Properties

 Label 3024.2.k.i Level 3024 Weight 2 Character orbit 3024.k Analytic conductor 24.147 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 189) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + ( 2 - \beta_{2} ) q^{7} +O(q^{10})$$ $$q + \beta_{3} q^{5} + ( 2 - \beta_{2} ) q^{7} -\beta_{1} q^{11} -2 \beta_{2} q^{13} + 3 \beta_{2} q^{19} -\beta_{1} q^{23} + 10 q^{25} -2 \beta_{1} q^{29} -\beta_{2} q^{31} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{35} - q^{37} + \beta_{3} q^{41} -2 q^{43} -2 \beta_{3} q^{47} + ( 1 - 4 \beta_{2} ) q^{49} + 4 \beta_{1} q^{53} -5 \beta_{2} q^{55} + 2 \beta_{3} q^{59} -4 \beta_{2} q^{61} -6 \beta_{1} q^{65} + 10 q^{67} + 5 \beta_{1} q^{71} -6 \beta_{2} q^{73} + ( -2 \beta_{1} - \beta_{3} ) q^{77} -2 q^{79} -2 \beta_{3} q^{83} -3 \beta_{3} q^{89} + ( -6 - 4 \beta_{2} ) q^{91} + 9 \beta_{1} q^{95} + 8 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{7} + O(q^{10})$$ $$4q + 8q^{7} + 40q^{25} - 4q^{37} - 8q^{43} + 4q^{49} + 40q^{67} - 8q^{79} - 24q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/5$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{2} - 5$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 10 \nu$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$5 \beta_{2} + 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1889.1
 −1.93649 − 1.11803i −1.93649 + 1.11803i 1.93649 + 1.11803i 1.93649 − 1.11803i
0 0 0 −3.87298 0 2.00000 1.73205i 0 0 0
1889.2 0 0 0 −3.87298 0 2.00000 + 1.73205i 0 0 0
1889.3 0 0 0 3.87298 0 2.00000 1.73205i 0 0 0
1889.4 0 0 0 3.87298 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c 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.k.i 4
3.b odd 2 1 inner 3024.2.k.i 4
4.b odd 2 1 189.2.c.b 4
7.b odd 2 1 inner 3024.2.k.i 4
12.b even 2 1 189.2.c.b 4
21.c even 2 1 inner 3024.2.k.i 4
28.d even 2 1 189.2.c.b 4
36.f odd 6 1 567.2.o.c 4
36.f odd 6 1 567.2.o.d 4
36.h even 6 1 567.2.o.c 4
36.h even 6 1 567.2.o.d 4
84.h odd 2 1 189.2.c.b 4
252.s odd 6 1 567.2.o.c 4
252.s odd 6 1 567.2.o.d 4
252.bi even 6 1 567.2.o.c 4
252.bi even 6 1 567.2.o.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.c.b 4 4.b odd 2 1
189.2.c.b 4 12.b even 2 1
189.2.c.b 4 28.d even 2 1
189.2.c.b 4 84.h odd 2 1
567.2.o.c 4 36.f odd 6 1
567.2.o.c 4 36.h even 6 1
567.2.o.c 4 252.s odd 6 1
567.2.o.c 4 252.bi even 6 1
567.2.o.d 4 36.f odd 6 1
567.2.o.d 4 36.h even 6 1
567.2.o.d 4 252.s odd 6 1
567.2.o.d 4 252.bi even 6 1
3024.2.k.i 4 1.a even 1 1 trivial
3024.2.k.i 4 3.b odd 2 1 inner
3024.2.k.i 4 7.b odd 2 1 inner
3024.2.k.i 4 21.c even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{2} - 15$$ $$T_{11}^{2} + 5$$ $$T_{13}^{2} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 5 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 - 4 T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 17 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 17 T^{2} )^{4}$$
$19$ $$( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 41 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 38 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + T + 37 T^{2} )^{4}$$
$41$ $$( 1 + 67 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 2 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 34 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 26 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 58 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 10 T + 67 T^{2} )^{4}$$
$71$ $$( 1 - 17 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 38 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 2 T + 79 T^{2} )^{4}$$
$83$ $$( 1 + 106 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 43 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$