# Properties

 Label 3024.2.cc.d Level 3024 Weight 2 Character orbit 3024.cc Analytic conductor 24.147 Analytic rank 0 Dimension 48 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.cc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 504) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 12q^{23} - 24q^{25} + 36q^{29} - 12q^{43} + 6q^{49} - 36q^{65} + 60q^{77} + 12q^{79} + 12q^{91} - 108q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1 0 0 0 −1.91834 + 3.32266i 0 −2.41978 1.06989i 0 0 0
881.2 0 0 0 −1.79302 + 3.10561i 0 −2.25543 + 1.38312i 0 0 0
881.3 0 0 0 −1.79123 + 3.10250i 0 2.56863 0.634134i 0 0 0
881.4 0 0 0 −1.60290 + 2.77631i 0 0.555071 + 2.58687i 0 0 0
881.5 0 0 0 −1.26858 + 2.19724i 0 −0.521158 2.59391i 0 0 0
881.6 0 0 0 −1.16173 + 2.01217i 0 1.39373 + 2.24889i 0 0 0
881.7 0 0 0 −1.11364 + 1.92889i 0 2.58429 0.566975i 0 0 0
881.8 0 0 0 −0.965651 + 1.67256i 0 −2.53170 + 0.768427i 0 0 0
881.9 0 0 0 −0.422480 + 0.731757i 0 −0.327684 2.62538i 0 0 0
881.10 0 0 0 −0.0977451 + 0.169300i 0 2.48762 + 0.900962i 0 0 0
881.11 0 0 0 −0.0868503 + 0.150429i 0 −1.72196 + 2.00869i 0 0 0
881.12 0 0 0 −0.00869840 + 0.0150661i 0 0.514871 2.59517i 0 0 0
881.13 0 0 0 0.00869840 0.0150661i 0 −2.50492 0.851694i 0 0 0
881.14 0 0 0 0.0868503 0.150429i 0 2.60056 0.486915i 0 0 0
881.15 0 0 0 0.0977451 0.169300i 0 −0.463555 + 2.60483i 0 0 0
881.16 0 0 0 0.422480 0.731757i 0 −2.10980 1.59647i 0 0 0
881.17 0 0 0 0.965651 1.67256i 0 1.93133 1.80831i 0 0 0
881.18 0 0 0 1.11364 1.92889i 0 −1.78316 + 1.95457i 0 0 0
881.19 0 0 0 1.16173 2.01217i 0 1.25073 + 2.33145i 0 0 0
881.20 0 0 0 1.26858 2.19724i 0 −1.98582 1.74829i 0 0 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2897.24 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.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.cc.d 48
3.b odd 2 1 1008.2.cc.d 48
4.b odd 2 1 1512.2.bu.a 48
7.b odd 2 1 inner 3024.2.cc.d 48
9.c even 3 1 1008.2.cc.d 48
9.d odd 6 1 inner 3024.2.cc.d 48
12.b even 2 1 504.2.bu.a 48
21.c even 2 1 1008.2.cc.d 48
28.d even 2 1 1512.2.bu.a 48
36.f odd 6 1 504.2.bu.a 48
36.f odd 6 1 4536.2.k.a 48
36.h even 6 1 1512.2.bu.a 48
36.h even 6 1 4536.2.k.a 48
63.l odd 6 1 1008.2.cc.d 48
63.o even 6 1 inner 3024.2.cc.d 48
84.h odd 2 1 504.2.bu.a 48
252.s odd 6 1 1512.2.bu.a 48
252.s odd 6 1 4536.2.k.a 48
252.bi even 6 1 504.2.bu.a 48
252.bi even 6 1 4536.2.k.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bu.a 48 12.b even 2 1
504.2.bu.a 48 36.f odd 6 1
504.2.bu.a 48 84.h odd 2 1
504.2.bu.a 48 252.bi even 6 1
1008.2.cc.d 48 3.b odd 2 1
1008.2.cc.d 48 9.c even 3 1
1008.2.cc.d 48 21.c even 2 1
1008.2.cc.d 48 63.l odd 6 1
1512.2.bu.a 48 4.b odd 2 1
1512.2.bu.a 48 28.d even 2 1
1512.2.bu.a 48 36.h even 6 1
1512.2.bu.a 48 252.s odd 6 1
3024.2.cc.d 48 1.a even 1 1 trivial
3024.2.cc.d 48 7.b odd 2 1 inner
3024.2.cc.d 48 9.d odd 6 1 inner
3024.2.cc.d 48 63.o even 6 1 inner
4536.2.k.a 48 36.f odd 6 1
4536.2.k.a 48 36.h even 6 1
4536.2.k.a 48 252.s odd 6 1
4536.2.k.a 48 252.bi even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{48} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(3024, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database