Properties

 Label 2100.2.bi.n Level 2100 Weight 2 Character orbit 2100.bi Analytic conductor 16.769 Analytic rank 0 Dimension 32 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.bi (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 420) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 12q^{19} - 8q^{21} - 12q^{31} - 24q^{39} + 44q^{49} - 10q^{51} - 24q^{61} + 28q^{79} - 20q^{81} + 16q^{91} + 28q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1 0 −1.73192 0.0216419i 0 0 0 2.64396 0.0973684i 0 2.99906 + 0.0749640i 0
101.2 0 −1.64956 0.528155i 0 0 0 −1.63251 2.08204i 0 2.44211 + 1.74245i 0
101.3 0 −1.52899 + 0.813749i 0 0 0 −0.567546 + 2.58416i 0 1.67563 2.48843i 0
101.4 0 −1.28218 1.16448i 0 0 0 −1.63251 2.08204i 0 0.287951 + 2.98615i 0
101.5 0 −0.956690 + 1.44386i 0 0 0 −2.60236 + 0.477214i 0 −1.16949 2.76266i 0
101.6 0 −0.884700 1.48906i 0 0 0 2.64396 0.0973684i 0 −1.43461 + 2.63475i 0
101.7 0 −0.772078 + 1.55045i 0 0 0 2.60236 0.477214i 0 −1.80779 2.39414i 0
101.8 0 −0.0597684 1.73102i 0 0 0 −0.567546 + 2.58416i 0 −2.99286 + 0.206921i 0
101.9 0 0.0597684 + 1.73102i 0 0 0 0.567546 2.58416i 0 −2.99286 + 0.206921i 0
101.10 0 0.772078 1.55045i 0 0 0 −2.60236 + 0.477214i 0 −1.80779 2.39414i 0
101.11 0 0.884700 + 1.48906i 0 0 0 −2.64396 + 0.0973684i 0 −1.43461 + 2.63475i 0
101.12 0 0.956690 1.44386i 0 0 0 2.60236 0.477214i 0 −1.16949 2.76266i 0
101.13 0 1.28218 + 1.16448i 0 0 0 1.63251 + 2.08204i 0 0.287951 + 2.98615i 0
101.14 0 1.52899 0.813749i 0 0 0 0.567546 2.58416i 0 1.67563 2.48843i 0
101.15 0 1.64956 + 0.528155i 0 0 0 1.63251 + 2.08204i 0 2.44211 + 1.74245i 0
101.16 0 1.73192 + 0.0216419i 0 0 0 −2.64396 + 0.0973684i 0 2.99906 + 0.0749640i 0
1601.1 0 −1.73192 + 0.0216419i 0 0 0 2.64396 + 0.0973684i 0 2.99906 0.0749640i 0
1601.2 0 −1.64956 + 0.528155i 0 0 0 −1.63251 + 2.08204i 0 2.44211 1.74245i 0
1601.3 0 −1.52899 0.813749i 0 0 0 −0.567546 2.58416i 0 1.67563 + 2.48843i 0
1601.4 0 −1.28218 + 1.16448i 0 0 0 −1.63251 + 2.08204i 0 0.287951 2.98615i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1601.16 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.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.bi.n 32
3.b odd 2 1 inner 2100.2.bi.n 32
5.b even 2 1 inner 2100.2.bi.n 32
5.c odd 4 2 420.2.bn.a 32
7.d odd 6 1 inner 2100.2.bi.n 32
15.d odd 2 1 inner 2100.2.bi.n 32
15.e even 4 2 420.2.bn.a 32
21.g even 6 1 inner 2100.2.bi.n 32
35.i odd 6 1 inner 2100.2.bi.n 32
35.k even 12 2 420.2.bn.a 32
35.k even 12 2 2940.2.f.a 32
35.l odd 12 2 2940.2.f.a 32
105.p even 6 1 inner 2100.2.bi.n 32
105.w odd 12 2 420.2.bn.a 32
105.w odd 12 2 2940.2.f.a 32
105.x even 12 2 2940.2.f.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bn.a 32 5.c odd 4 2
420.2.bn.a 32 15.e even 4 2
420.2.bn.a 32 35.k even 12 2
420.2.bn.a 32 105.w odd 12 2
2100.2.bi.n 32 1.a even 1 1 trivial
2100.2.bi.n 32 3.b odd 2 1 inner
2100.2.bi.n 32 5.b even 2 1 inner
2100.2.bi.n 32 7.d odd 6 1 inner
2100.2.bi.n 32 15.d odd 2 1 inner
2100.2.bi.n 32 21.g even 6 1 inner
2100.2.bi.n 32 35.i odd 6 1 inner
2100.2.bi.n 32 105.p even 6 1 inner
2940.2.f.a 32 35.k even 12 2
2940.2.f.a 32 35.l odd 12 2
2940.2.f.a 32 105.w odd 12 2
2940.2.f.a 32 105.x even 12 2

Hecke kernels

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

 $$T_{11}^{16} - \cdots$$ $$T_{13}^{8} + 54 T_{13}^{6} + 1039 T_{13}^{4} + 8256 T_{13}^{2} + 21904$$ $$T_{19}^{8} + \cdots$$ $$T_{37}^{16} + \cdots$$

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database