# Properties

 Label 3024.2.t.b Level $3024$ Weight $2$ Character orbit 3024.t Analytic conductor $24.147$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 252) 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.

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

## Character values

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

 $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −2.00000 0 −2.00000 1.73205i 0 0 0
1873.1 0 0 0 −2.00000 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
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.t.b 2
3.b odd 2 1 1008.2.t.b 2
4.b odd 2 1 756.2.l.a 2
7.c even 3 1 3024.2.q.e 2
9.c even 3 1 3024.2.q.e 2
9.d odd 6 1 1008.2.q.f 2
12.b even 2 1 252.2.l.a yes 2
21.h odd 6 1 1008.2.q.f 2
28.d even 2 1 5292.2.l.b 2
28.f even 6 1 5292.2.i.b 2
28.f even 6 1 5292.2.j.b 2
28.g odd 6 1 756.2.i.a 2
28.g odd 6 1 5292.2.j.c 2
36.f odd 6 1 756.2.i.a 2
36.f odd 6 1 2268.2.k.b 2
36.h even 6 1 252.2.i.a 2
36.h even 6 1 2268.2.k.a 2
63.g even 3 1 inner 3024.2.t.b 2
63.n odd 6 1 1008.2.t.b 2
84.h odd 2 1 1764.2.l.b 2
84.j odd 6 1 1764.2.i.b 2
84.j odd 6 1 1764.2.j.a 2
84.n even 6 1 252.2.i.a 2
84.n even 6 1 1764.2.j.c 2
252.n even 6 1 5292.2.l.b 2
252.o even 6 1 252.2.l.a yes 2
252.r odd 6 1 1764.2.j.a 2
252.s odd 6 1 1764.2.i.b 2
252.u odd 6 1 2268.2.k.b 2
252.u odd 6 1 5292.2.j.c 2
252.bb even 6 1 1764.2.j.c 2
252.bb even 6 1 2268.2.k.a 2
252.bi even 6 1 5292.2.i.b 2
252.bj even 6 1 5292.2.j.b 2
252.bl odd 6 1 756.2.l.a 2
252.bn odd 6 1 1764.2.l.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.a 2 36.h even 6 1
252.2.i.a 2 84.n even 6 1
252.2.l.a yes 2 12.b even 2 1
252.2.l.a yes 2 252.o even 6 1
756.2.i.a 2 28.g odd 6 1
756.2.i.a 2 36.f odd 6 1
756.2.l.a 2 4.b odd 2 1
756.2.l.a 2 252.bl odd 6 1
1008.2.q.f 2 9.d odd 6 1
1008.2.q.f 2 21.h odd 6 1
1008.2.t.b 2 3.b odd 2 1
1008.2.t.b 2 63.n odd 6 1
1764.2.i.b 2 84.j odd 6 1
1764.2.i.b 2 252.s odd 6 1
1764.2.j.a 2 84.j odd 6 1
1764.2.j.a 2 252.r odd 6 1
1764.2.j.c 2 84.n even 6 1
1764.2.j.c 2 252.bb even 6 1
1764.2.l.b 2 84.h odd 2 1
1764.2.l.b 2 252.bn odd 6 1
2268.2.k.a 2 36.h even 6 1
2268.2.k.a 2 252.bb even 6 1
2268.2.k.b 2 36.f odd 6 1
2268.2.k.b 2 252.u odd 6 1
3024.2.q.e 2 7.c even 3 1
3024.2.q.e 2 9.c even 3 1
3024.2.t.b 2 1.a even 1 1 trivial
3024.2.t.b 2 63.g even 3 1 inner
5292.2.i.b 2 28.f even 6 1
5292.2.i.b 2 252.bi even 6 1
5292.2.j.b 2 28.f even 6 1
5292.2.j.b 2 252.bj even 6 1
5292.2.j.c 2 28.g odd 6 1
5292.2.j.c 2 252.u odd 6 1
5292.2.l.b 2 28.d even 2 1
5292.2.l.b 2 252.n even 6 1

## Hecke kernels

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

 $$T_{5} + 2$$ $$T_{11} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 2 T + 5 T^{2} )^{2}$$
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$( 1 - 4 T + 11 T^{2} )^{2}$$
$13$ $$1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4}$$
$17$ $$1 - 7 T + 32 T^{2} - 119 T^{3} + 289 T^{4}$$
$19$ $$1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4}$$
$23$ $$( 1 - 4 T + 23 T^{2} )^{2}$$
$29$ $$1 + T - 28 T^{2} + 29 T^{3} + 841 T^{4}$$
$31$ $$1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4}$$
$37$ $$( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$
$41$ $$1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4}$$
$43$ $$( 1 - 13 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} )$$
$47$ $$1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4}$$
$53$ $$1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4}$$
$59$ $$1 - 7 T - 10 T^{2} - 413 T^{3} + 3481 T^{4}$$
$61$ $$1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4}$$
$67$ $$1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 8 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$
$79$ $$1 + 9 T + 2 T^{2} + 711 T^{3} + 6241 T^{4}$$
$83$ $$1 + T - 82 T^{2} + 83 T^{3} + 6889 T^{4}$$
$89$ $$1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4}$$
$97$ $$1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4}$$