# Properties

 Label 2100.2.ce.d Level 2100 Weight 2 Character orbit 2100.ce Analytic conductor 16.769 Analytic rank 0 Dimension 24 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.ce (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 8q^{11} + 8q^{21} - 12q^{31} - 8q^{51} - 48q^{61} + 64q^{71} + 12q^{81} + 116q^{91} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
157.1 0 −0.965926 + 0.258819i 0 0 0 2.25190 + 1.38886i 0 0.866025 0.500000i 0
157.2 0 −0.965926 + 0.258819i 0 0 0 −0.728357 2.54352i 0 0.866025 0.500000i 0
157.3 0 −0.965926 + 0.258819i 0 0 0 −2.48947 + 0.895840i 0 0.866025 0.500000i 0
157.4 0 0.965926 0.258819i 0 0 0 0.728357 + 2.54352i 0 0.866025 0.500000i 0
157.5 0 0.965926 0.258819i 0 0 0 2.48947 0.895840i 0 0.866025 0.500000i 0
157.6 0 0.965926 0.258819i 0 0 0 −2.25190 1.38886i 0 0.866025 0.500000i 0
493.1 0 −0.258819 0.965926i 0 0 0 1.38886 2.25190i 0 −0.866025 + 0.500000i 0
493.2 0 −0.258819 0.965926i 0 0 0 −2.54352 + 0.728357i 0 −0.866025 + 0.500000i 0
493.3 0 −0.258819 0.965926i 0 0 0 0.895840 + 2.48947i 0 −0.866025 + 0.500000i 0
493.4 0 0.258819 + 0.965926i 0 0 0 2.54352 0.728357i 0 −0.866025 + 0.500000i 0
493.5 0 0.258819 + 0.965926i 0 0 0 −1.38886 + 2.25190i 0 −0.866025 + 0.500000i 0
493.6 0 0.258819 + 0.965926i 0 0 0 −0.895840 2.48947i 0 −0.866025 + 0.500000i 0
1657.1 0 −0.258819 + 0.965926i 0 0 0 1.38886 + 2.25190i 0 −0.866025 0.500000i 0
1657.2 0 −0.258819 + 0.965926i 0 0 0 −2.54352 0.728357i 0 −0.866025 0.500000i 0
1657.3 0 −0.258819 + 0.965926i 0 0 0 0.895840 2.48947i 0 −0.866025 0.500000i 0
1657.4 0 0.258819 0.965926i 0 0 0 2.54352 + 0.728357i 0 −0.866025 0.500000i 0
1657.5 0 0.258819 0.965926i 0 0 0 −1.38886 2.25190i 0 −0.866025 0.500000i 0
1657.6 0 0.258819 0.965926i 0 0 0 −0.895840 + 2.48947i 0 −0.866025 0.500000i 0
1993.1 0 −0.965926 0.258819i 0 0 0 2.25190 1.38886i 0 0.866025 + 0.500000i 0
1993.2 0 −0.965926 0.258819i 0 0 0 −0.728357 + 2.54352i 0 0.866025 + 0.500000i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1993.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.d odd 6 1 inner
35.i odd 6 1 inner
35.k even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.ce.d 24
5.b even 2 1 inner 2100.2.ce.d 24
5.c odd 4 2 inner 2100.2.ce.d 24
7.d odd 6 1 inner 2100.2.ce.d 24
35.i odd 6 1 inner 2100.2.ce.d 24
35.k even 12 2 inner 2100.2.ce.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.ce.d 24 1.a even 1 1 trivial
2100.2.ce.d 24 5.b even 2 1 inner
2100.2.ce.d 24 5.c odd 4 2 inner
2100.2.ce.d 24 7.d odd 6 1 inner
2100.2.ce.d 24 35.i odd 6 1 inner
2100.2.ce.d 24 35.k even 12 2 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2100, [\chi])$$:

 $$T_{11}^{6} + 2 T_{11}^{5} + 24 T_{11}^{4} + 20 T_{11}^{3} + 460 T_{11}^{2} + 600 T_{11} + 900$$ $$T_{17}^{24} - 392 T_{17}^{20} + 111408 T_{17}^{16} - 14944352 T_{17}^{12} + 1468049536 T_{17}^{8} - 34227360000 T_{17}^{4} + 656100000000$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database