# Properties

 Label 6030.2.d.k Level 6030 Weight 2 Character orbit 6030.d Analytic conductor 48.150 Analytic rank 0 Dimension 24 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6030.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.1497924188$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{2} + 24q^{4} - 24q^{5} - 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{2} + 24q^{4} - 24q^{5} - 24q^{8} + 24q^{10} - 12q^{11} + 24q^{16} + 4q^{19} - 24q^{20} + 12q^{22} + 24q^{25} - 24q^{32} - 16q^{37} - 4q^{38} + 24q^{40} - 8q^{41} - 12q^{44} - 20q^{49} - 24q^{50} - 24q^{53} + 12q^{55} + 24q^{64} - 32q^{67} - 4q^{73} + 16q^{74} + 4q^{76} - 24q^{80} + 8q^{82} + 12q^{88} - 4q^{95} + 20q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2411.1 −1.00000 0 1.00000 −1.00000 0 4.73932i −1.00000 0 1.00000
2411.2 −1.00000 0 1.00000 −1.00000 0 3.69493i −1.00000 0 1.00000
2411.3 −1.00000 0 1.00000 −1.00000 0 3.48375i −1.00000 0 1.00000
2411.4 −1.00000 0 1.00000 −1.00000 0 3.30563i −1.00000 0 1.00000
2411.5 −1.00000 0 1.00000 −1.00000 0 2.95623i −1.00000 0 1.00000
2411.6 −1.00000 0 1.00000 −1.00000 0 2.79591i −1.00000 0 1.00000
2411.7 −1.00000 0 1.00000 −1.00000 0 2.65115i −1.00000 0 1.00000
2411.8 −1.00000 0 1.00000 −1.00000 0 2.64016i −1.00000 0 1.00000
2411.9 −1.00000 0 1.00000 −1.00000 0 1.58772i −1.00000 0 1.00000
2411.10 −1.00000 0 1.00000 −1.00000 0 1.26508i −1.00000 0 1.00000
2411.11 −1.00000 0 1.00000 −1.00000 0 0.376360i −1.00000 0 1.00000
2411.12 −1.00000 0 1.00000 −1.00000 0 0.0653098i −1.00000 0 1.00000
2411.13 −1.00000 0 1.00000 −1.00000 0 0.0653098i −1.00000 0 1.00000
2411.14 −1.00000 0 1.00000 −1.00000 0 0.376360i −1.00000 0 1.00000
2411.15 −1.00000 0 1.00000 −1.00000 0 1.26508i −1.00000 0 1.00000
2411.16 −1.00000 0 1.00000 −1.00000 0 1.58772i −1.00000 0 1.00000
2411.17 −1.00000 0 1.00000 −1.00000 0 2.64016i −1.00000 0 1.00000
2411.18 −1.00000 0 1.00000 −1.00000 0 2.65115i −1.00000 0 1.00000
2411.19 −1.00000 0 1.00000 −1.00000 0 2.79591i −1.00000 0 1.00000
2411.20 −1.00000 0 1.00000 −1.00000 0 2.95623i −1.00000 0 1.00000
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2411.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
201.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6030.2.d.k 24
3.b odd 2 1 6030.2.d.l yes 24
67.b odd 2 1 6030.2.d.l yes 24
201.d even 2 1 inner 6030.2.d.k 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6030.2.d.k 24 1.a even 1 1 trivial
6030.2.d.k 24 201.d even 2 1 inner
6030.2.d.l yes 24 3.b odd 2 1
6030.2.d.l yes 24 67.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(6030, [\chi])$$:

 $$T_{7}^{24} + \cdots$$ $$T_{11}^{12} + \cdots$$ $$T_{41}^{12} + \cdots$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database