# Properties

 Label 2100.2.ce.e Level 2100 Weight 2 Character orbit 2100.ce Analytic conductor 16.769 Analytic rank 0 Dimension 32 CM no Inner twists 4

# 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: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{12})$$ Twist minimal: no (minimal twist has level 420) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 8q^{11} - 8q^{21} - 16q^{23} + 24q^{31} - 12q^{33} - 20q^{37} + 24q^{43} - 12q^{47} - 8q^{51} - 40q^{53} + 16q^{57} - 24q^{61} + 12q^{63} - 16q^{71} + 60q^{73} + 84q^{77} + 16q^{81} + 48q^{87} + 40q^{91} - 8q^{93} + O(q^{100})$$

## 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 −0.106918 + 2.64359i 0 0.866025 0.500000i 0
157.2 0 −0.965926 + 0.258819i 0 0 0 1.79435 1.94431i 0 0.866025 0.500000i 0
157.3 0 −0.965926 + 0.258819i 0 0 0 −1.45817 2.20765i 0 0.866025 0.500000i 0
157.4 0 −0.965926 + 0.258819i 0 0 0 2.02043 + 1.70817i 0 0.866025 0.500000i 0
157.5 0 0.965926 0.258819i 0 0 0 −2.08616 + 1.62725i 0 0.866025 0.500000i 0
157.6 0 0.965926 0.258819i 0 0 0 2.42120 1.06666i 0 0.866025 0.500000i 0
157.7 0 0.965926 0.258819i 0 0 0 −1.05955 2.42432i 0 0.866025 0.500000i 0
157.8 0 0.965926 0.258819i 0 0 0 1.93892 1.80016i 0 0.866025 0.500000i 0
493.1 0 −0.258819 0.965926i 0 0 0 −1.94431 1.79435i 0 −0.866025 + 0.500000i 0
493.2 0 −0.258819 0.965926i 0 0 0 1.70817 2.02043i 0 −0.866025 + 0.500000i 0
493.3 0 −0.258819 0.965926i 0 0 0 −2.20765 + 1.45817i 0 −0.866025 + 0.500000i 0
493.4 0 −0.258819 0.965926i 0 0 0 2.64359 + 0.106918i 0 −0.866025 + 0.500000i 0
493.5 0 0.258819 + 0.965926i 0 0 0 −1.80016 1.93892i 0 −0.866025 + 0.500000i 0
493.6 0 0.258819 + 0.965926i 0 0 0 1.62725 + 2.08616i 0 −0.866025 + 0.500000i 0
493.7 0 0.258819 + 0.965926i 0 0 0 −2.42432 + 1.05955i 0 −0.866025 + 0.500000i 0
493.8 0 0.258819 + 0.965926i 0 0 0 −1.06666 2.42120i 0 −0.866025 + 0.500000i 0
1657.1 0 −0.258819 + 0.965926i 0 0 0 −1.94431 + 1.79435i 0 −0.866025 0.500000i 0
1657.2 0 −0.258819 + 0.965926i 0 0 0 1.70817 + 2.02043i 0 −0.866025 0.500000i 0
1657.3 0 −0.258819 + 0.965926i 0 0 0 −2.20765 1.45817i 0 −0.866025 0.500000i 0
1657.4 0 −0.258819 + 0.965926i 0 0 0 2.64359 0.106918i 0 −0.866025 0.500000i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1993.8 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.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.ce.e 32
5.b even 2 1 420.2.bo.a 32
5.c odd 4 1 420.2.bo.a 32
5.c odd 4 1 inner 2100.2.ce.e 32
7.d odd 6 1 inner 2100.2.ce.e 32
15.d odd 2 1 1260.2.dq.c 32
15.e even 4 1 1260.2.dq.c 32
35.i odd 6 1 420.2.bo.a 32
35.i odd 6 1 2940.2.x.c 32
35.j even 6 1 2940.2.x.c 32
35.k even 12 1 420.2.bo.a 32
35.k even 12 1 inner 2100.2.ce.e 32
35.k even 12 1 2940.2.x.c 32
35.l odd 12 1 2940.2.x.c 32
105.p even 6 1 1260.2.dq.c 32
105.w odd 12 1 1260.2.dq.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bo.a 32 5.b even 2 1
420.2.bo.a 32 5.c odd 4 1
420.2.bo.a 32 35.i odd 6 1
420.2.bo.a 32 35.k even 12 1
1260.2.dq.c 32 15.d odd 2 1
1260.2.dq.c 32 15.e even 4 1
1260.2.dq.c 32 105.p even 6 1
1260.2.dq.c 32 105.w odd 12 1
2100.2.ce.e 32 1.a even 1 1 trivial
2100.2.ce.e 32 5.c odd 4 1 inner
2100.2.ce.e 32 7.d odd 6 1 inner
2100.2.ce.e 32 35.k even 12 1 inner
2940.2.x.c 32 35.i odd 6 1
2940.2.x.c 32 35.j even 6 1
2940.2.x.c 32 35.k even 12 1
2940.2.x.c 32 35.l odd 12 1

## Hecke kernels

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

 $$T_{11}^{16} - \cdots$$ $$T_{17}^{32} - \cdots$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database