# Properties

 Label 3024.2.t.f 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 126) 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.

 $$f(q)$$ $$=$$ $$q + 3 q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 3 q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} -3 q^{11} + ( -5 + 5 \zeta_{6} ) q^{13} + ( 3 - 3 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} -3 q^{23} + 4 q^{25} -3 \zeta_{6} q^{29} -4 \zeta_{6} q^{31} + ( -3 + 9 \zeta_{6} ) q^{35} + 7 \zeta_{6} q^{37} + ( -9 + 9 \zeta_{6} ) q^{41} + 11 \zeta_{6} q^{43} + ( -8 + 3 \zeta_{6} ) q^{49} + ( -3 + 3 \zeta_{6} ) q^{53} -9 q^{55} -12 \zeta_{6} q^{59} + ( -2 + 2 \zeta_{6} ) q^{61} + ( -15 + 15 \zeta_{6} ) q^{65} -4 \zeta_{6} q^{67} + ( -11 + 11 \zeta_{6} ) q^{73} + ( 3 - 9 \zeta_{6} ) q^{77} + ( 8 - 8 \zeta_{6} ) q^{79} -3 \zeta_{6} q^{83} + ( 9 - 9 \zeta_{6} ) q^{85} + 15 \zeta_{6} q^{89} + ( -10 - 5 \zeta_{6} ) q^{91} + 15 \zeta_{6} q^{95} + \zeta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{5} + q^{7} + O(q^{10})$$ $$2q + 6q^{5} + q^{7} - 6q^{11} - 5q^{13} + 3q^{17} + 5q^{19} - 6q^{23} + 8q^{25} - 3q^{29} - 4q^{31} + 3q^{35} + 7q^{37} - 9q^{41} + 11q^{43} - 13q^{49} - 3q^{53} - 18q^{55} - 12q^{59} - 2q^{61} - 15q^{65} - 4q^{67} - 11q^{73} - 3q^{77} + 8q^{79} - 3q^{83} + 9q^{85} + 15q^{89} - 25q^{91} + 15q^{95} + q^{97} + O(q^{100})$$

## 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 3.00000 0 0.500000 2.59808i 0 0 0
1873.1 0 0 0 3.00000 0 0.500000 + 2.59808i 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.f 2
3.b odd 2 1 1008.2.t.c 2
4.b odd 2 1 378.2.h.b 2
7.c even 3 1 3024.2.q.a 2
9.c even 3 1 3024.2.q.a 2
9.d odd 6 1 1008.2.q.e 2
12.b even 2 1 126.2.h.a yes 2
21.h odd 6 1 1008.2.q.e 2
28.d even 2 1 2646.2.h.f 2
28.f even 6 1 2646.2.e.e 2
28.f even 6 1 2646.2.f.i 2
28.g odd 6 1 378.2.e.a 2
28.g odd 6 1 2646.2.f.e 2
36.f odd 6 1 378.2.e.a 2
36.f odd 6 1 1134.2.g.f 2
36.h even 6 1 126.2.e.b 2
36.h even 6 1 1134.2.g.d 2
63.g even 3 1 inner 3024.2.t.f 2
63.n odd 6 1 1008.2.t.c 2
84.h odd 2 1 882.2.h.e 2
84.j odd 6 1 882.2.e.h 2
84.j odd 6 1 882.2.f.a 2
84.n even 6 1 126.2.e.b 2
84.n even 6 1 882.2.f.e 2
252.n even 6 1 2646.2.h.f 2
252.n even 6 1 7938.2.a.c 1
252.o even 6 1 126.2.h.a yes 2
252.o even 6 1 7938.2.a.r 1
252.r odd 6 1 882.2.f.a 2
252.s odd 6 1 882.2.e.h 2
252.u odd 6 1 1134.2.g.f 2
252.u odd 6 1 2646.2.f.e 2
252.bb even 6 1 882.2.f.e 2
252.bb even 6 1 1134.2.g.d 2
252.bi even 6 1 2646.2.e.e 2
252.bj even 6 1 2646.2.f.i 2
252.bl odd 6 1 378.2.h.b 2
252.bl odd 6 1 7938.2.a.o 1
252.bn odd 6 1 882.2.h.e 2
252.bn odd 6 1 7938.2.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.b 2 36.h even 6 1
126.2.e.b 2 84.n even 6 1
126.2.h.a yes 2 12.b even 2 1
126.2.h.a yes 2 252.o even 6 1
378.2.e.a 2 28.g odd 6 1
378.2.e.a 2 36.f odd 6 1
378.2.h.b 2 4.b odd 2 1
378.2.h.b 2 252.bl odd 6 1
882.2.e.h 2 84.j odd 6 1
882.2.e.h 2 252.s odd 6 1
882.2.f.a 2 84.j odd 6 1
882.2.f.a 2 252.r odd 6 1
882.2.f.e 2 84.n even 6 1
882.2.f.e 2 252.bb even 6 1
882.2.h.e 2 84.h odd 2 1
882.2.h.e 2 252.bn odd 6 1
1008.2.q.e 2 9.d odd 6 1
1008.2.q.e 2 21.h odd 6 1
1008.2.t.c 2 3.b odd 2 1
1008.2.t.c 2 63.n odd 6 1
1134.2.g.d 2 36.h even 6 1
1134.2.g.d 2 252.bb even 6 1
1134.2.g.f 2 36.f odd 6 1
1134.2.g.f 2 252.u odd 6 1
2646.2.e.e 2 28.f even 6 1
2646.2.e.e 2 252.bi even 6 1
2646.2.f.e 2 28.g odd 6 1
2646.2.f.e 2 252.u odd 6 1
2646.2.f.i 2 28.f even 6 1
2646.2.f.i 2 252.bj even 6 1
2646.2.h.f 2 28.d even 2 1
2646.2.h.f 2 252.n even 6 1
3024.2.q.a 2 7.c even 3 1
3024.2.q.a 2 9.c even 3 1
3024.2.t.f 2 1.a even 1 1 trivial
3024.2.t.f 2 63.g even 3 1 inner
7938.2.a.c 1 252.n even 6 1
7938.2.a.o 1 252.bl odd 6 1
7938.2.a.r 1 252.o even 6 1
7938.2.a.bd 1 252.bn odd 6 1

## Hecke kernels

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

 $$T_{5} - 3$$ $$T_{11} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 3 T + 5 T^{2} )^{2}$$
$7$ $$1 - T + 7 T^{2}$$
$11$ $$( 1 + 3 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4}$$
$19$ $$1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4}$$
$23$ $$( 1 + 3 T + 23 T^{2} )^{2}$$
$29$ $$1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$
$37$ $$1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4}$$
$41$ $$1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4}$$
$43$ $$1 - 11 T + 78 T^{2} - 473 T^{3} + 1849 T^{4}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4}$$
$59$ $$1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4}$$
$61$ $$1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4}$$
$67$ $$1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4}$$
$79$ $$1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$1 + 3 T - 74 T^{2} + 249 T^{3} + 6889 T^{4}$$
$89$ $$1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4}$$
$97$ $$1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4}$$