# Properties

 Label 2100.2.s.b Level 2100 Weight 2 Character orbit 2100.s Analytic conductor 16.769 Analytic rank 0 Dimension 24 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.s (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 420) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{3} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{3} - 24q^{13} + 4q^{21} - 8q^{27} - 16q^{31} + 20q^{33} - 32q^{37} + 8q^{43} + 52q^{51} + 28q^{57} - 8q^{63} + 24q^{67} - 12q^{81} + 20q^{87} - 24q^{91} - 20q^{93} + 104q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1457.1 0 −1.71916 0.210923i 0 0 0 0.707107 + 0.707107i 0 2.91102 + 0.725222i 0
1457.2 0 −1.61532 + 0.625101i 0 0 0 −0.707107 0.707107i 0 2.21850 2.01947i 0
1457.3 0 −1.16966 1.27746i 0 0 0 −0.707107 0.707107i 0 −0.263792 + 2.98838i 0
1457.4 0 −0.625101 + 1.61532i 0 0 0 −0.707107 0.707107i 0 −2.21850 2.01947i 0
1457.5 0 −0.587996 1.62919i 0 0 0 0.707107 + 0.707107i 0 −2.30852 + 1.91591i 0
1457.6 0 0.210923 + 1.71916i 0 0 0 0.707107 + 0.707107i 0 −2.91102 + 0.725222i 0
1457.7 0 0.522921 1.65123i 0 0 0 0.707107 + 0.707107i 0 −2.45311 1.72692i 0
1457.8 0 1.04182 1.38370i 0 0 0 −0.707107 0.707107i 0 −0.829228 2.88312i 0
1457.9 0 1.27746 + 1.16966i 0 0 0 −0.707107 0.707107i 0 0.263792 + 2.98838i 0
1457.10 0 1.38370 1.04182i 0 0 0 −0.707107 0.707107i 0 0.829228 2.88312i 0
1457.11 0 1.62919 + 0.587996i 0 0 0 0.707107 + 0.707107i 0 2.30852 + 1.91591i 0
1457.12 0 1.65123 0.522921i 0 0 0 0.707107 + 0.707107i 0 2.45311 1.72692i 0
1793.1 0 −1.71916 + 0.210923i 0 0 0 0.707107 0.707107i 0 2.91102 0.725222i 0
1793.2 0 −1.61532 0.625101i 0 0 0 −0.707107 + 0.707107i 0 2.21850 + 2.01947i 0
1793.3 0 −1.16966 + 1.27746i 0 0 0 −0.707107 + 0.707107i 0 −0.263792 2.98838i 0
1793.4 0 −0.625101 1.61532i 0 0 0 −0.707107 + 0.707107i 0 −2.21850 + 2.01947i 0
1793.5 0 −0.587996 + 1.62919i 0 0 0 0.707107 0.707107i 0 −2.30852 1.91591i 0
1793.6 0 0.210923 1.71916i 0 0 0 0.707107 0.707107i 0 −2.91102 0.725222i 0
1793.7 0 0.522921 + 1.65123i 0 0 0 0.707107 0.707107i 0 −2.45311 + 1.72692i 0
1793.8 0 1.04182 + 1.38370i 0 0 0 −0.707107 + 0.707107i 0 −0.829228 + 2.88312i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1793.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.s.b 24
3.b odd 2 1 inner 2100.2.s.b 24
5.b even 2 1 420.2.s.a 24
5.c odd 4 1 420.2.s.a 24
5.c odd 4 1 inner 2100.2.s.b 24
15.d odd 2 1 420.2.s.a 24
15.e even 4 1 420.2.s.a 24
15.e even 4 1 inner 2100.2.s.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.s.a 24 5.b even 2 1
420.2.s.a 24 5.c odd 4 1
420.2.s.a 24 15.d odd 2 1
420.2.s.a 24 15.e even 4 1
2100.2.s.b 24 1.a even 1 1 trivial
2100.2.s.b 24 3.b odd 2 1 inner
2100.2.s.b 24 5.c odd 4 1 inner
2100.2.s.b 24 15.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{12} + 70 T_{11}^{10} + 1489 T_{11}^{8} + 14112 T_{11}^{6} + 65032 T_{11}^{4} + 137504 T_{11}^{2} + 99856$$ acting on $$S_{2}^{\mathrm{new}}(2100, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database